diff --git a/data_all_eng_slimpj/shuffled/split2/finalzpgm b/data_all_eng_slimpj/shuffled/split2/finalzpgm new file mode 100644 index 0000000000000000000000000000000000000000..b77583224b8474a9a1d96f88e0c7e556da7195a8 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzpgm @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nLet $\\mathcal{B}(\\mathcal{H})$ denote the $C^*-$algebra of all bounded linear operators acting on a Hilbert space $\\mathcal{H}$. An operator $T\\in\\mathcal{B}(\\mathcal{H})$ is said to be positive if, for every $x\\in\\mathcal{H}$, one has $\\left\\geq 0$. In this case, we simply write $A\\geq O.$ Positive operators play an important role in understanding the geometry of a Hilbert space, and these operators constitute a special class of the wider class of self-adjoint operators; that is $A^*=A$, where $A^*$ denotes the conjugate of $A$. Among the most basic properties of self-adjoint operators is the fact that \n$$\\|T\\|=\\omega(T)=r(T),\\;T\\;{\\text{is\\;normal}},$$ where $\\|\\cdot\\|, \\omega(\\cdot)$, and $r(\\cdot)$ denote the operator norm, the numerical radius, and the spectral radius respectively. Actually, for a general $T\\in\\mathcal{B}(\\mathcal{H})$ one has \n $$\\|T\\|\\geq \\omega(T)\\geq r(T).$$\\\\\nWhile both $\\|\\cdot\\|$ and $\\omega(\\cdot)$ are norms on $\\mathcal{B}(\\mathcal{H})$, $r(\\cdot)$ is not. In fact, we have the equivalence relation \\cite[Theorem 1.3-1]{gust}\n\\begin{equation}\\label{eq_equiv_norms}\n\\frac{1}{2}\\|T\\|\\le\\omega(T)\\le\\|T\\|,\\;T\\in\\mathcal{B}(\\mathcal{H}).\n\\end{equation}\n\nNumerous researchers' core interests have been sharpening the above inequality and obtaining new possible relations between $\\|\\cdot\\|$ and $\\omega(\\cdot)$. This is because $\\|\\cdot\\|$ is much easier to compute than $\\omega(\\cdot)$, not to forget the math appetite for obtaining such new relations.\n\nThe Cartesian decomposition of $T\\in\\mathcal{B}(\\mathcal{H})$ is $T=\\mathfrak{R}T+\\textup i\\mathfrak{I}T$, where $\\mathfrak{R}T=\\frac{T+T^*}{2}$ and $\\mathfrak{I}T=\\frac{T-T^*}{2\\textup i}$ are the real and imaginary parts of $T$, respectively. Although $\\|T\\|\\geq \\omega(T)$ is always valid, the following reverses hold for the Cartesian components of $T$, see \\cite[Theorem 2.1]{1}\n\\begin{equation}\\label{eq_norm_reim}\n\\|\\mathfrak{R}T\\|\\le\\omega(T),\\;\\|\\mathfrak{I}T\\|\\le \\omega(T).\n\\end{equation}\n\nWhile the original definition of $\\omega(\\cdot)$ is based on a supremum over inner product values (i.e., $\\omega(T)=\\sup_{\\|x\\|=1}|\\left|$), the following identity is extremely useful \\cite{3}\n\\begin{equation}\\label{eq_w_re}\n\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left\\| {{\\operatorname{\\mathfrak Re}}^{\\textup i\\theta }}T \\right\\|=\\omega \\left( T \\right).\n\\end{equation}\n\nExploring further relations between $\\|\\cdot\\|$ and $\\omega(\\cdot),$ \nit has been shown in \\cite[Theorem 2.3]{1} that\n\\begin{equation}\\label{eq_fuad_mos}\n\\|A+B\\|\\le 2\\omega\\left(\\left[\\begin{array}{cc}O&A\\\\B^*&O\\end{array}\\right]\\right)\\le \\|A\\|+\\|B\\|;\n\\end{equation}\nas an interesting refinement of the triangle inequality of norms, using the numerical radius of a matrix operator.\n\n\nHaving the matrix operator term in \\eqref{eq_fuad_mos} is not a coincidence. In fact, numerous results have included such terms while studying numerical radius inequalities. For example, it has been shown in \\cite[Theorem 2.4]{4} that\n\\begin{equation}\\label{eq_max_w}\n\\frac{\\max \\left\\{ \\omega \\left( S+T \\right),\\omega \\left( S-T \\right) \\right\\}}{2}\\le \\omega \\left( \\left[ \\begin{matrix}\n O & S \\\\\n T & O \\\\\n\\end{matrix} \\right] \\right),\\text{ for any }S,T\\in \\mathcal B\\left( \\mathcal H \\right);\n\\end{equation}\n\nan inequality which has been reversed in a way or another by the form \\cite[Theorem 2.4]{4}\n\\begin{equation}\\label{eq_w_average_w}\n\\omega \\left( \\left[ \\begin{matrix}\n O & S \\\\\n T & O \\\\\n\\end{matrix} \\right] \\right)\\le \\frac{\\omega \\left( S+T \\right)+\\omega \\left( S-T \\right)}{2},\\text{ for any }S,T\\in \\mathcal B\\left( \\mathcal H \\right).\n\\end{equation}\n\n\n\nThe above matrix operator is not only comparable with numerical radius terms, as we also have \\cite[Theorem 2.1]{5} \n\\begin{equation}\\label{eq_need_prf}\n2\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)\\le \\max \\left\\{ \\left\\| A \\right\\|,\\left\\| B \\right\\| \\right\\}+\\frac{1}{2}\\left( \\left\\| {{\\left| A \\right|}^{\\frac{1}{2}}}{{\\left| B \\right|}^{\\frac{1}{2}}} \\right\\|+\\left\\| {{\\left| {{B}^{*}} \\right|}^{\\frac{1}{2}}}{{\\left| {{A}^{*}} \\right|}^{\\frac{1}{2}}} \\right\\| \\right),\n\\end{equation}\nfor any $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$.\n\nThe right-hand side of this latter inequality is related to the Davidson-Power inequality, which has been generalized in \\cite[Theorem 5]{6} to the form\n\\begin{equation}\\label{eq_abuamer}\n\\|A+B^*\\|\\le \\max\\{\\|A\\|,\\|B\\|\\}+\\max\\{\\|\\;|A|^{1\/2}|B^*|^{1\/2}\\|,\\|\\;|A^*|^{1\/2}|B|^{1\/2}\\|\\}.\n\\end{equation}\n\nAn important tool in obtaining matrix inequalities is convexity; whether it is scalar or operator convexity. Recall that a function $f:J\\to\\mathbb{R}$ is said to be convex on the interval $J$ if it satisfies $f((1-\\lambda)a+\\lambda b)\\le (1-\\lambda)f(a)+\\lambda f(b)$ for all $a,b\\in J$ and $0\\le \\lambda\\le 1$. In convex analysis, the Hermite-Hadamard inequality which states that for a convex function $f$ on $[0,1]$ one has\n\\begin{equation}\\label{eq_hh}\nf\\left( \\frac{1}{2} \\right)\\le \\int\\limits_{0}^{1}{f\\left( t \\right)dt}\\le \\frac{f\\left( 0 \\right)+f\\left( 1 \\right)}{2},\n\\end{equation}\nis a non-avoidable tool. Notice that this inequality provides a refinement of the mid-convexity condition of $f$.\n\nOur target in this paper is to further explore numerical radius and operator norm inequalities, via matrix operators and convex functions. For this, we begin by noting that \nsince $\\omega(\\cdot)$ and $||\\cdot||$ are norms, one can easily verify that the functions\n\\[f\\left( t \\right)=\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right),\\;{\\text{and}}\\;g(t)=\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|\\]\nare convex on $\\left[ 0,1 \\right]$.\n\n\n\n\n\n\n\n\n\n\n\n\nWith a considerable amount of research devoted to inequalities of convex functions, the following inequalities which have been shown in \\cite{2} for a convex function $f:\\left[ 0,1 \\right]\\to \\mathbb{R}$ have played a useful role in the literature\n\t\\[f\\left( t \\right)\\le \\left( 1-t \\right)f\\left( 0 \\right)+tf\\left( 1 \\right)-2r\\left( \\frac{f\\left( 0 \\right)+f\\left( 1 \\right)}{2}-f\\left( \\frac{1}{2} \\right) \\right),\\]\nand\n\\[\\left( 1-t \\right)f\\left( 0 \\right)+tf\\left( 1 \\right)\\le f\\left( t \\right)+2R\\left( \\frac{f\\left( 0 \\right)+f\\left( 1 \\right)}{2}-f\\left( \\frac{1}{2} \\right) \\right),\\]\nwhere $r=\\min \\left\\{ t,1-t \\right\\}$, $R=\\max \\left\\{ t,1-t \\right\\}$, and $0\\le t\\le 1$. \nWe refer the reader to \\cite{sab_mia,sab_mjom} for some applications and further discussion of these inequalities.\n\n\n\nApplying these later inequalities to the convex functions $f$ and $g$ above implies the following refinements and reverses of \\eqref{eq_norm_reim}.\n\\begin{equation}\\label{6}\n\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{2R}\\le \\omega \\left( T \\right)-\\left\\| \\mathfrak RT \\right\\|\\le \\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{2r}.\n\\end{equation}\nFurthermore,\n\\begin{equation}\\label{ned_quo}\n\\frac{\\left\\| T \\right\\|-\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|}{2R}\\le \\left\\| T \\right\\|-\\left\\| \\mathfrak RT \\right\\|\\le \\frac{\\left\\| T \\right\\|-\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|}{2r}.\n\\end{equation}\n\n\nUsing this approach, we will be able to present refined versions and generalizations of most of the above inequalities, with the conclusion of some product inequalities that entail some interesting relations. We refer to inequalities that govern $\\omega(AB)$ as product inequalities. It is well-known that $\\omega(\\cdot)$ is not sub-multiplicative. We refer the reader to \\cite{Li} for further discussion of this property. Interestingly, our approach will entail a relation between $\\omega(AB)$ and $\\|A+B\\|$, with an application to the matrix arithmetic-geometric mean inequality that states \\cite[Theorem IX.4.5]{bhatia}\n\\begin{equation*}\n\\|A^{1\/2}B^{1\/2}\\|\\le\\frac{1}{2}\\|A+B\\|,\\;A,B\\in\\mathcal{B}(\\mathcal{H}), A,B\\geq O. \n\\end{equation*}\nNamely, we obtain a new refinement of this inequality using the numerical radius; as a new approach to this direction, see Remark \\ref{remark_amgm} below.\n\nTo achieve our goal, some auxiliary results are needed as follows. \n\\begin{lemma}\nLet $A,B\\in\\mathcal{B}(\\mathcal{H})$. \n\\begin{enumerate}\n\\item If $n\\in\\mathbb{N}$, then \\cite[Theorem 2.1-1]{gust}\n\\begin{equation}\\label{eq_power_ineq}\n\\omega(A^n)\\le \\omega^n(A).\n\\end{equation}\n\\item The operator norm satisfies the identity\n\\begin{equation}\\label{eq_norm_blocks}\n\\left\\|\\left[\\begin{array}{cc}O&A\\\\A^*&O\\end{array}\\right]\\right\\|=\\|A\\|.\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\n\n\\section{Main Result}\nIn this section we present our results, starting with the following simple consequence that follows by applying \\eqref{eq_hh} on\n\\[f\\left( t \\right)=\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)\\;{\\text{and}}\\;g(t)=\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|\\] yielding refinements of \\eqref{eq_norm_reim}.\n\\begin{proposition}\nLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then\n\\begin{equation}\\label{2}\n\\left\\| \\mathfrak RT \\right\\|\\le \\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}dt\\le \\omega \\left( T \\right),\n\\end{equation}\nand\n\\begin{equation}\\label{3}\n\\left\\| \\mathfrak IT \\right\\|\\le \\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right)dt}\\le \\omega \\left( T \\right).\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{15}\n\\left\\| \\mathfrak RT \\right\\|\\le \\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|dt}\\le \\left\\| T \\right\\|,\n\\end{equation}\nand\n\\begin{equation}\\label{16}\n\\left\\| \\mathfrak IT \\right\\|\\le \\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right){{T}^{*}}-tT \\right\\|dt}\\le \\left\\| T \\right\\|.\n\\end{equation}\n\\end{proposition}\n\nThe identity \\eqref{eq_w_re} provides an alternative formula to evaluate the numerical radius without appealing to the inner product. Interestingly, the inequalities \\eqref{2} and \\eqref{3} provide the following alternative identities, which help better understand how the numerical radius behaves.\n\\begin{corollary}\nLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then\n\\[\\omega \\left( T \\right)=\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left(\\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right){{e}^{\\textup i\\theta }}T+t{{e}^{-\\textup i\\theta }}{{T}^{*}} \\right)dt}\\right)=\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left(\\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right){{e}^{-\\textup i\\theta }}{{T}^{*}}-t{{e}^{\\textup i\\theta }}T \\right)dt}\\right).\\]\n\\end{corollary}\n\\begin{proof}\nReplacing $T$ by ${{e}^{\\textup i\\theta }}T$ in \\eqref{2}, we get\n\\[\\left\\| {{\\operatorname{\\mathfrak Re}}^{\\textup i\\theta }}T \\right\\|\\le \\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right){{e}^{\\textup i\\theta }}T+t{{e}^{-\\textup i\\theta }}{{T}^{*}} \\right)dt}\\le \\omega \\left( T \\right).\\]\nTaking the supremum over $\\theta \\in \\mathbb{R}$, \\eqref{eq_w_re} implies the first identity. The second identity follows from \\eqref{3} and noting that\n\\[\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left\\| \\mathfrak I{{e}^{\\textup i\\theta }}T \\right\\|=\\omega \\left( T \\right).\\]\n\\end{proof}\n\n\n\nThe following result involves an integral refinement of the second inequality in \\eqref{eq_equiv_norms}.\n\\begin{proposition}\nLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then\n\\[\\omega \\left( T \\right)\\le \\min \\left\\{ {{\\lambda }_{1}},{{\\lambda }_{2}} \\right\\}\\le \\left\\| T \\right\\|,\\]\nwhere \n\\[{{\\lambda }_{1}}=\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left( \\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right){{e}^{\\textup i\\theta }}T+t{{e}^{-\\textup i\\theta }}{{T}^{*}} \\right\\|dt} \\right)\\text{ and }{{\\lambda }_{2}}=\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left( \\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right){{e}^{-\\textup i\\theta }}{{T}^{*}}-t{{e}^{\\textup i\\theta }}T \\right\\|dt} \\right).\\]\n\\end{proposition}\n\\begin{proof}\nBy the inequality \\eqref{15}, we have \n\\begin{equation*}\n\\sup_{\\theta \\in \\mathbb{R}}\\|{{\\operatorname{\\mathfrak Re}}^{\\textup i\\theta }}T\\|\\le \\sup_{\\theta \\in \\mathbb{R}}\\left( \\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right)e^{i\\theta}T+t{e^{-i\\theta}{T}^{*}} \\right\\|dt}\\right)\\le \\|T\\|.\n\\end{equation*}\nFinally, by \\eqref{eq_w_re} we get\n\\begin{equation*}\n \\omega(T)\\le\\sup_{\\theta \\in \\mathbb{R}}\\left( \\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right)e^{i\\theta}T+t{e^{-i\\theta}{T}^{*}} \\right\\|dt}\\right)\\le \\left\\| T \\right\\|.\n\\end{equation*}\nBy a similar proof and with the help of \\eqref{16}, we also have\n\\begin{equation*}\n\\omega(T)\\le \\sup_{\\theta\\in \\mathbb{R}}\\left( \\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right){e^{-i\\theta}{T}^{*}}-te^{i\\theta}T \\right\\|dt}\\right)\\le \\left\\| T \\right\\|.\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\n\n\nThe second inequality in the inequalities \\eqref{2} and \\eqref{3} can be reversed as follows. \n\\begin{proposition}\nLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then \n\\[\\frac{1}{2}\\omega \\left( T \\right)\\le \\left\\{ \\begin{aligned}\n & \\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)dt}, \\\\ \n & \\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right)dt}. \\\\ \n\\end{aligned} \\right.\\]\n\\end{proposition}\n\\begin{proof}\nFor any $0\\le t\\le 1$, it can be easily shown that\n\\[\\left| 1-2t \\right|\\omega \\left( T \\right)\\le \\min \\left\\{ \\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right),\\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right) \\right\\}.\\]\nIntegrating this over the interval $[0,1]$ implies the desired result.\n\\end{proof}\n\n\nThe following result holds as well.\n\\begin{theorem}\n\tLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then\n\\begin{equation*}\n\\left\\| T \\right\\|\\le 2\\int\\limits_{0}^{1}{\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|dt}\\le2 \\left\\| T \\right\\|,\n\\end{equation*}\nand\n\\[\\omega \\left( T \\right)\\le 2\\int\\limits_{0}^{1}{\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)dt}\\le2\\omega \\left( T \\right).\\]\n\\end{theorem}\n\\begin{proof}\nLet $h(T)=\\|T\\|$ for any $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then, $h$ is a convex function on $\\mathcal B\\left( \\mathcal H \\right)$. For each $t\\in [0, 1],$ we have \n\\begin{equation*}\nh((1-2t)T)+h((2t-1)T^*)=h((1-t)A+tB)+h((1-t)B+tA),\n\\end{equation*}\nwhere $A=(1-t)T+tT^*$ and $B=-(1-t)T^*-tT$. Then, \n\\[ \\begin{aligned}\nh((1-2t)T)+h((2t-1)T^*)&=h((1-t)A+tB)+h((1-t)B+tA)\\\\\n&\\le(1-t)h(A)+th(B)+(1-t)h(B)+th(A)\\\\\n&=h(A)+h(B)\\nonumber \\\\\n&=h((1-t)T+tT^*)+h(-(1-t)T^*-tT)\\\\\n&=h((1-t)T+tT^*)+h((1-t)T^*+tT).\n\\end{aligned} \\]\nIntegrating, the previous inequality, from $t=0$ to $t=1$, we obtain\n\\begin{equation*}\n\\int_0^1|1-2t|(\\|T\\|+\\|T^*\\|)\\:dt\\le 2\\int_0^1\\|(1-t)T+tT^*\\|\\:dt.\n\\end{equation*}\nThus, \n\\begin{equation*}\n\\|T\\|=\\frac12 (\\|T\\|+\\|T^*\\|)\\le 2\\int_0^1\\|(1-t)T+tT^*\\|\\:dt.\n\\end{equation*}\nOn the other hand,\n\\begin{align*}\n\\|(1-t)T+tT^*\\|&\\le (1-t)\\|T\\|+t\\|T^*\\|=\\|T\\|; 0\\le t\\le 1.\n\\end{align*}\nIntegrating this last inequality and then multiplying by 2 complete the proof of the first inequality. The second inequality is proved similarly.\n\\end{proof}\n\n\n\n\n\n\n\n\nContinuing with the convexity of the norms, the inequality \\eqref{6} may be used to get the following refinement of the first inequality in \\eqref{eq_equiv_norms}.\n\\begin{theorem}\\label{9}\nLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then for any $0\\le t\\le 1$,\n\\[\\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{4R}\\left( 2\\omega \\left( T \\right)-\\left( \\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right)+\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right) \\right) \\right)\\le \\omega \\left( T \\right),\\]\nwhere $R=\\max \\left\\{ t,1-t \\right\\}$.\n\\end{theorem}\n\\begin{proof}\nThe first inequality in \\eqref{6} can be written as\n\\begin{equation}\\label{4}\n\\left\\| \\mathfrak RT \\right\\|+\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{2R}\\le \\omega \\left( T \\right).\n\\end{equation}\nReplacing $\\textup i{{T}^{*}}$ by $T$, we infer that\n\\begin{equation}\\label{5}\n\\left\\| \\mathfrak IT \\right\\|+\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right)}{2R}\\le \\omega \\left( T \\right).\n\\end{equation}\nBy \\eqref{4} and \\eqref{5}, we get\n\\[\\begin{aligned}\n & \\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{4R}\\left( 2\\omega \\left( T \\right)-\\left( \\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right)+\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right) \\right) \\right) \\\\ \n & =\\frac{1}{2}\\left\\| \\mathfrak RT+\\textup i\\mathfrak IT \\right\\|+\\frac{1}{4R}\\left( 2\\omega \\left( T \\right)-\\left( \\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right)+\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right) \\right) \\right) \\\\ \n & \\le \\frac{1}{2}\\left( \\left\\| \\mathfrak RT \\right\\|+\\left\\| \\mathfrak IT \\right\\| \\right)+\\frac{1}{4R}\\left( 2\\omega \\left( T \\right)-\\left( \\omega \\left( \\left( 1-t \\right){{T}^{*}}-tT \\right)+\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right) \\right) \\right) \\\\ \n &\\qquad\\text{(by the triangle inequality for the usual operator norm)}\\\\\n & \\le \\omega \\left( T \\right).\n\\end{aligned}\\]\nThis completes the proof.\n\\end{proof}\n\n\nAs a consequence of Theorem \\ref{9}, we get the following corollaries. Our results considerably refines \\cite[(4.3)]{4} and \\cite[(4.2)]{4}, respectively.\n\\begin{corollary}\\label{refwn}\n\tLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then,\n\\begin{equation*}\n\\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{2}\\Big| \\|\\mathfrak IT \\|-\\|\\mathfrak RT \\|\\Big| \\le\n\\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{2}\\left( 2\\omega \\left( T \\right)-\\left( \\|\\mathfrak IT \\|+\\|\\mathfrak RT \\| \\right)\\right)\\le \\omega \\left( T \\right).\t\n\\end{equation*}\n\\end{corollary}\n\\begin{proof}\nThe second inequality can be deduced from Theorem \\ref{9} with $t=\\frac12.$ On the other hand, we have \n\\[\\begin{aligned}\n\\frac{1}{2}\\Big| \\|\\mathfrak IT \\|-\\|\\mathfrak RT \\|\\Big|=& \\frac{1}{2}\\Big| \\|\\mathfrak IT \\|-\\omega(T)+\\omega(T)-|\\mathfrak RT \\|\\Big| \\\\ \n&\\le\\frac{1}{2}\\left(\\Big| \\|\\mathfrak IT \\|-\\omega(T)\\Big|+\\Big|\\omega(T)-|\\mathfrak RT \\|\\Big|\\right) \\\\\n&= \\frac{1}{2}\\left( \\omega(T)-\\|\\mathfrak IT \\|+\\omega(T)-\\|\\mathfrak RT \\|\\right),\\quad({\\text{by\\;the\\;inequality\\;\\eqref{eq_w_re}}}). \\nonumber\\ \n\\end{aligned}\\]\t\nThis completes the proof.\n\\end{proof}\n\n\nAs a consequence of Corollary \\ref{refwn}, we characterize when the numerical radius to be equal to half the operator norm. The following result is related to Theorem 3.1 previously obtained by Yamazaki in \\cite{3}.\n\\begin{proposition}\n\tLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then, $\\frac{\\|T\\|}{2}=\\omega(T)$ if and only if $\\|\\mathfrak Ie^{\\textup i\\theta}T\\|=\\|\\mathfrak Re^{\\textup i\\theta}T\\|=\\frac{\\|T\\|}{2}$ for any $\\theta \\in \\mathbb{R}.$\n\\end{proposition}\n\\begin{proof}\n\tIf $\\|\\mathfrak Ie^{\\textup i\\theta}T\\|=\\|\\mathfrak Re^{\\textup i\\theta}T\\|=\\frac{\\|T\\|}{2}$ for any $\\theta \\in \\mathbb{R}$, then by \\eqref{eq_w_re} we conclude that $\\omega(T)=\\frac{\\|T\\|}{2}$. Conversely, we suppose that $\\omega(T)=\\frac{\\|T\\|}{2}$, thus from Corollary \\ref{refwn} we conclude that \n\\[\t\\frac{1}{2}\\left\\| T \\right\\|=\\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{2}\\Big| \\|\\mathfrak IT \\|-\\|\\mathfrak RT \\|\\Big|=\n\t\\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{2}\\left( 2\\omega \\left( T \\right)-\\left( \\|\\mathfrak IT \\|+\\|\\mathfrak RT \\| \\right)\\right)= \\omega \\left( T \\right).\\]\n\tIf we replace $T$ for $e^{\\textup i\\theta}T$ with $\\theta \\in \\mathbb{R}$, we have \n\t\\[\\frac{1}{2}\\left\\| T \\right\\|=\t\\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{2}\\Big| \\|\\mathfrak Ie^{\\textup i\\theta}T \\|-\\|\\mathfrak Re^{\\textup i\\theta}T \\|\\Big|=\n\t\\frac{1}{2}\\left\\| T \\right\\|+\\frac{1}{2}\\left( 2\\omega \\left( T \\right)-\\left( \\|\\mathfrak Ie^{\\textup i\\theta}T \\|+\\|\\mathfrak Re^{\\textup i\\theta}T \\| \\right)\\right)= \\omega \\left( T \\right).\\]\n\tThis implies that $\\|\\mathfrak Ie^{\\textup i\\theta}T \\|=\\|\\mathfrak Re^{\\textup i\\theta}T \\|$ and $2\\omega(T)=\\|\\mathfrak Ie^{\\textup i\\theta}T \\|+\\|\\mathfrak Re^{\\textup i\\theta}T \\|,$ i.e. for any $\\theta \\in \\mathbb{R}$ we get\n\t$$\\|\\mathfrak Ie^{\\textup i\\theta}T \\|=\\|\\mathfrak Re^{\\textup i\\theta}T \\|=\\frac{\\|T\\|}{2}.$$\n\t\\end{proof}\n\n\n\\begin{corollary}\nLet $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$. Then, \n\\begin{eqnarray}\n\\omega \\left( \\left[ \\begin{matrix}\nO & A \\\\\n{{B}} & O \\\\\n\\end{matrix} \\right]\\right)&\\geq&\n\\frac{1}{2}\\left\\| \\left[ \\begin{matrix}\nO & A \\\\\n{{B}} & O \\\\\n\\end{matrix} \\right] \\right\\|+\\frac{1}{2}\\left( 2\\omega \\left( \\left[ \\begin{matrix}\nO & A \\\\\n{{B}} & O \\\\\n\\end{matrix} \\right] \\right)-\\left( \\|A-B^* \\|+\\| A+B^* \\| \\right)\\right)\\nonumber\\\\\n&\\geq&\\frac{1}{2}\\left\\| \\left[ \\begin{matrix}\nO & A \\\\\n{{B}} & O \\\\\n\\end{matrix} \\right] \\right\\|+\\frac{1}{2}\\Big| \\|A-B^*\\|-\\|A+B^*\\|\\Big|.\\nonumber\\\\ \n\\nonumber\\ \\end{eqnarray}\n\\end{corollary}\n\\begin{proof}\n\tThis follows clearly from Corollary \\ref{refwn} by considering $T=\\left[ \\begin{matrix}\n\tO & A \\\\\n\t{{B}} & O \\\\\n\t\\end{matrix} \\right] $ and equality \\eqref{eq_norm_blocks}.\n\\end{proof}\n\n\n\nOn the other hand, the reverse for the second inequality in \\eqref{eq_w_re} may be obtained as follows. \n\\begin{theorem}\nLet $T\\in \\mathcal B\\left( \\mathcal H \\right)$. Then for any $0\\le t\\le 1$,\n\\[\\left\\| T \\right\\|\\le \\omega \\left( T \\right)+\\frac{\\left\\| T \\right\\|-\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|}{2r}-\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{2R},\\]\nwhere $r=\\min \\left\\{ t,1-t \\right\\}$ and $R=\\max \\left\\{ t,1-t \\right\\}$. In particular, \n\\[\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{2R}\\le \\frac{\\left\\| T \\right\\|-\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|}{2r}.\\]\n\n\\end{theorem}\n\\begin{proof}\nThe inequalities \\eqref{6} and \\eqref{ned_quo} imply\n\t\\[\\begin{aligned}\n \\left\\| T \\right\\|&\\le \\left\\| \\mathfrak RT \\right\\|+\\frac{\\left\\| T \\right\\|-\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|}{2r} \\\\ \n & \\le \\omega \\left( T \\right)+\\frac{\\left\\| T \\right\\|-\\left\\| \\left( 1-t \\right)T+t{{T}^{*}} \\right\\|}{2r}-\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{2R}. \n\\end{aligned}\\]\t\nThis proves the first assertion. The second assertion follows from the first, noting that $\\omega(T)\\le\\|T\\|.$\n\\end{proof}\n\n\n\n\nContinuing with the theme of this paper, in the following result, the numerical radius of convex combinations of operator matrices is used to refine the triangle inequality, thanks to\n\\[\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)\\le \\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right); 0\\le t\\le 1.\\]\n\\begin{theorem}\\label{10}\nLet $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$. Then for any $0\\le t\\le 1$,\n\\[\\left\\| A+B \\right\\|\\le \\left\\| A \\right\\|+\\left\\| B \\right\\|-\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{R},\\]\nwhere $R=\\max \\left\\{ t,1-t \\right\\}$. \n\\end{theorem}\n\\begin{proof}\nLet $T=\\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right]$ on $\\mathcal H\\oplus \\mathcal H$. Then by \\eqref{6}, we can write \n{\\footnotesize\n\\[\\begin{aligned}\n & \\left\\| A+B \\right\\| \\\\ \n & =\\left\\| T+{{T}^{*}} \\right\\| \\\\ \n & =2\\left\\| \\mathfrak RT \\right\\| \\\\ \n & \\le 2\\omega \\left( T \\right)-\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{R} \\\\ \n & =2\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left\\| \\mathfrak R{{e}^{\\textup i\\theta }}T \\right\\|-\\frac{\\omega \\left( T \\right)-\\omega \\left( \\left( 1-t \\right)T+t{{T}^{*}} \\right)}{R} \\\\ \n & =\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left\\| \\left[ \\begin{matrix}\n O & {{e}^{\\textup i\\theta }}A+{{e}^{-\\textup i\\theta }}B \\\\\n {{e}^{\\textup i\\theta }}{{B}^{*}}+{{e}^{-\\textup i\\theta }}{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right\\|-\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{R} \\\\ \n & =\\underset{\\theta \\in \\mathbb{R}}{\\mathop{\\sup }}\\,\\left\\| {{e}^{\\textup i\\theta }}A+{{e}^{-\\textup i\\theta }}B \\right\\|-\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{R} \\\\ \n & \\le \\left\\| A \\right\\|+\\left\\| B \\right\\|-\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{R}, \n\\end{aligned}\\]}\nwhere the triangle inequality for the operator norm has been used to obtain the last inequality. This completes the proof.\n\\end{proof}\n\n\n\\begin{remark}\nLetting $T=\\left[\\begin{array}{cc}O&A\\\\B^*&O\\end{array}\\right]$, we have\n\\[\\begin{aligned}\n & \\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)+\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)B-tA \\\\\n \\left( 1-t \\right){{A}^{*}}-t{{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n&=\\omega((1-t)T+tT^*)+\\omega((1-t)T^*-tT)\\\\\n&\\le 2\\omega(T)\\quad({\\text{by\\;the\\;triangle\\;inequality}})\\\\\n & = 2\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right),\n\\end{aligned}\\]\nfor any $0\\le t\\le 1$. \nThus, noting \\eqref{eq_max_w} we have\n\\[\\begin{aligned}\n & \\max \\left\\{ \\omega \\left( \\left( 1-t \\right)\\left( B+{{A}^{*}} \\right)-t\\left( A+{{B}^{*}} \\right) \\right),\\omega \\left( \\left( 1-t \\right)\\left( B-{{A}^{*}} \\right)+t\\left( {{B}^{*}}-A \\right) \\right) \\right\\} \\\\ \n &\\quad +\\max \\left\\{ \\omega \\left( \\left( 1-t \\right)\\left( A+{{B}^{*}} \\right)+t\\left( B+{{A}^{*}} \\right) \\right),\\omega \\left( \\left( 1-t \\right)\\left( A-{{B}^{*}} \\right)+t\\left( B-{{A}^{*}} \\right) \\right) \\right\\} \\\\ \n & \\le 2\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)+2\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)B-tA \\\\\n \\left( 1-t \\right){{A}^{*}}-t{{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & \\le 4\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right).\n\\end{aligned}\\]\n In particular,\n \\begin{equation}\\label{8}\n\\begin{aligned}\n & \\max \\left\\{ \\left\\| \\mathfrak IA-\\mathfrak IB \\right\\|,\\left\\| \\mathfrak RA-\\mathfrak RB \\right\\| \\right\\}+\\max \\left\\{ \\left\\| \\mathfrak RA+\\mathfrak RB \\right\\|,\\left\\| \\mathfrak IA+\\mathfrak IB \\right\\| \\right\\} \\\\ \n & \\le \\left\\| A+B \\right\\|+\\left\\| A-B \\right\\| \\\\ \n & \\le 4\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n \\end{matrix} \\right] \\right).\n \\end{aligned}\n \\end{equation}\n\nAlso noting \\eqref{eq_w_average_w}, by the second inequality in \\eqref{8}, we get the following interesting inequalities\n\\[\\begin{aligned}\n \\frac{\\left\\| A+B \\right\\|+\\left\\| A-B \\right\\|}{2}&\\le 2\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & \\le \\omega \\left( A+{{B}^{*}} \\right)+\\omega \\left( A-{{B}^{*}} \\right). \n\\end{aligned}\\]\n\\end{remark}\n\n\nThe following result provides an integral version of \\eqref{eq_fuad_mos}; where the numerical radius of convex combinations of operator matrices is used to refine the triangle inequality. Since its proof is similar to Theorem \\ref{10}, we state it without details.\n\\begin{theorem}\nLet $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$. Then\n\\[\\left\\| A+B \\right\\|\\le 2\\int\\limits_{0}^{1}{\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}dt\\le \\left\\| A \\right\\|+\\left\\| B \\right\\|.\\]\n\\end{theorem}\n\n\n\n\nThe matrix operator $\\left[\\begin{array}{cc}O&A\\\\B^*&O\\end{array}\\right]$ is further used to obtain the following improvement of \\eqref{eq_abuamer}.\n\\begin{theorem}\\label{11}\nLet $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$. Then for any $0 \\le t \\le 1$,\n\\[\\begin{aligned}\n & \\left\\| A+B \\right\\|+\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{R} \\\\ \n & \\le \\max \\left\\{ \\left\\| A \\right\\|,\\left\\| B \\right\\| \\right\\}+\\frac{1}{2}\\left( \\left\\| {{\\left| A \\right|}^{\\frac{1}{2}}}{{\\left| B \\right|}^{\\frac{1}{2}}} \\right\\|+\\left\\| {{\\left| {{B}^{*}} \\right|}^{\\frac{1}{2}}}{{\\left| {{A}^{*}} \\right|}^{\\frac{1}{2}}} \\right\\| \\right), \n\\end{aligned}\\]\nwhere $R=\\max \\left\\{ t,1-t \\right\\}$. In particular, if $A$ and $B$ are self-adjoint, we get\n{\\small\n\\[\\left\\| A+B \\right\\|+\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n B & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right)B+tA & O \\\\\n\\end{matrix} \\right] \\right)}{R}\\le \\max \\left\\{ \\left\\| A \\right\\|,\\left\\| B \\right\\| \\right\\}+\\left\\| {{\\left| A \\right|}^{\\frac{1}{2}}}{{\\left| B \\right|}^{\\frac{1}{2}}} \\right\\|.\\]\n}\n\\end{theorem}\n\\begin{proof}\n\nCombining \\eqref{eq_need_prf} with the inequality \\eqref{4}, we infer the desired result.\n\\end{proof}\n\n\n\\begin{remark}\nIt is worthwhile to mention here that if $A$ and $B$ are positive operators, then Theorem \\ref{11} reduces to \\cite{7}\n\\[\\left\\| A+B \\right\\|\\le \\max \\left\\{ \\left\\| A \\right\\|,\\left\\| B \\right\\| \\right\\}+\\left\\| {{A}^{\\frac{1}{2}}}{{B}^{\\frac{1}{2}}} \\right\\|.\\]\nThis follows from the following point for positive operators \\cite{8}\n\\begin{equation}\\label{eq_ned_pf_remark}\n\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right)B+tA & O \\\\\n\\end{matrix} \\right] \\right)=\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n B & O \\\\\n\\end{matrix} \\right] \\right)=\\frac{1}{2}\\left\\| A+B \\right\\|.\n\\end{equation}\n\n\\end{remark}\n\nNow we move to study inequalities for $\\omega(AB)$, where $A,B\\in\\mathcal{B}(\\mathcal{H})$. Interestingly, the following numerical radius inequality leads to a new proof of the arithmetic-geometric mean inequality for positive operators, as we shall see in Remark \\ref{remark_amgm} below.\n\\begin{theorem}\\label{12}\nLet $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$. Then for any $0 \\le t \\le 1$,\n\\[{{\\omega }^{\\frac{1}{2}}}\\left( AB \\right)\\le \\frac{1}{2}\\left\\| A+B^{*} \\right\\|+\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB^{*} \\\\\n \\left( 1-t \\right){{B}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{2r},\\]\nwhere $r=\\min \\left\\{ t,1-t \\right\\}$.\n\\end{theorem}\n\\begin{proof}\nBy the second inequality in \\eqref{6}, we have\n\\[2\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)\\le \\left\\| A+B \\right\\|+\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB \\\\\n \\left( 1-t \\right){{B}^{*}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{r}.\\]\nThus,\n\\[\\begin{aligned}\n & 2{{\\omega }^{\\frac{1}{2}}}\\left( AB \\right) \\\\ \n & \\le 2\\max \\left\\{ {{\\omega }^{\\frac{1}{2}}}\\left( AB \\right),{{\\omega }^{\\frac{1}{2}}}\\left( BA \\right) \\right\\} \\\\ \n & =2{{\\omega }^{\\frac{1}{2}}}\\left( \\left[ \\begin{matrix}\n AB & O \\\\\n O & BA \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & =2{{\\omega }^{\\frac{1}{2}}}\\left( {{\\left[ \\begin{matrix}\n O & A \\\\\n B & O \\\\\n\\end{matrix} \\right]}^{2}} \\right) \\\\ \n & \\le 2\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n B & O \\\\\n\\end{matrix} \\right] \\right)\\quad({\\text{by}}\\;\\eqref{eq_power_ineq}) \\\\ \n & \\le \\left\\| A+B^{*} \\right\\|+\\frac{\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}} & O \\\\\n\\end{matrix} \\right] \\right)-\\omega \\left( \\left[ \\begin{matrix}\n O & \\left( 1-t \\right)A+tB^{*} \\\\\n \\left( 1-t \\right){{B}}+t{{A}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)}{r}, \n\\end{aligned}\\]\nwhich completes the proof.\n\\end{proof}\n\nNow we use Theorem \\ref{12} to prove the following arithmetic-geometric mean inequality for positive operators.\n\\begin{remark}\\label{remark_amgm}\nLet $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$ be two positive operators. It follows from Theorem \\ref{12},\n\\[\\begin{aligned}\n\\left\\| {{A}^{\\frac{1}{2}}}{{B}^{\\frac{1}{2}}} \\right\\|&={{r}^{\\frac{1}{2}}}\\left( AB \\right)\\quad \\text{(by \\cite[(2.1)]{5})}\\\\\n&\\le {{\\omega }^{\\frac{1}{2}}}\\left( AB \\right)\\\\\n&\\le \\frac{1}{2}\\left\\| A+B \\right\\|,\n\\end{aligned}\\]\nwhere \\eqref{eq_ned_pf_remark} has been used together with the fact that $r(T)\\le \\omega(T)$ for any $T\\in\\mathcal{B}(\\mathcal{H})$.\n\\end{remark}\n\nWhile Theorem \\ref{12} provides an upper bound of $\\omega(AB)$ in terms of $ \\left[ \\begin{matrix}\n O & A \\\\\n {{B}} & O \\\\\n\\end{matrix} \\right]$, we have the following lower bound in terms of the same matrix operator.\n\\begin{theorem}\\label{1}\nLet $A,B\\in \\mathcal B\\left( \\mathcal H \\right)$. Then\n\\[\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n B & O \\\\\n\\end{matrix} \\right] \\right)\\le \\sqrt{\\max \\left\\{ \\omega \\left( AB \\right),\\omega \\left( BA \\right) \\right\\}+\\underset{\\lambda \\in \\mathbb{C}}{\\mathop{\\inf }}\\,{{\\left\\| \\left[ \\begin{matrix}\n -\\lambda I & A \\\\\n B & -\\lambda I \\\\\n\\end{matrix} \\right] \\right\\|}^{2}}},\\]\nwhere $I$ is the identity operator in $\\mathcal{B}(\\mathcal{H}).$\n\\end{theorem}\n\\begin{proof}\nBy the main result of \\cite{9}, we can write\n\\[\\begin{aligned}\n \\max \\left\\{ \\omega \\left( A{{B}^{*}} \\right),\\omega \\left( {{B}^{*}}A \\right) \\right\\}&=\\omega \\left( \\left[ \\begin{matrix}\n A{{B}^{*}} & O \\\\\n O & {{B}^{*}}A \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & =\\omega \\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right]\\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & =\\omega \\left( {{T}^{2}} \\right) \\\\ \n & \\ge {{\\omega }^{2}}\\left( T \\right)-\\underset{\\lambda \\in \\mathbb{C}}{\\mathop{\\inf }}\\,{{\\left\\| T-\\lambda I \\right\\|}^{2}} \\\\ \n & ={{\\omega }^{2}}\\left( \\left[ \\begin{matrix}\n O & A \\\\\n {{B}^{*}} & O \\\\\n\\end{matrix} \\right] \\right)-\\underset{\\lambda \\in \\mathbb{C}}{\\mathop{\\inf }}\\,{{\\left\\| \\left[ \\begin{matrix}\n -\\lambda I & A \\\\\n {{B}^{*}} & -\\lambda I \\\\\n\\end{matrix} \\right] \\right\\|}^{2}},\n\\end{aligned}\\]\nwhich completes the proof.\n\\end{proof}\n\n\\begin{remark}\nIt follows from Theorem \\ref{1} that for $X_i\\in\\mathcal{B}(\\mathcal{H})$ $\\left( i=1,2,3,4 \\right)$,\n\\[\\begin{aligned}\n & \\omega \\left( \\left[ \\begin{matrix}\n {{X}_{1}} & {{X}_{2}} \\\\\n {{X}_{3}} & {{X}_{4}} \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & =\\omega \\left( \\left[ \\begin{matrix}\n {{X}_{1}} & O \\\\\n O & {{X}_{4}} \\\\\n\\end{matrix} \\right]+\\left[ \\begin{matrix}\n O & {{X}_{2}} \\\\\n {{X}_{3}} & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & \\le \\omega \\left( \\left[ \\begin{matrix}\n {{X}_{1}} & O \\\\\n O & {{X}_{4}} \\\\\n\\end{matrix} \\right] \\right)+\\omega \\left( \\left[ \\begin{matrix}\n O & {{X}_{2}} \\\\\n {{X}_{3}} & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & \\le \\max \\left\\{ \\omega \\left( {{X}_{1}} \\right),\\omega \\left( {{X}_{4}} \\right) \\right\\}+\\sqrt{\\max \\left\\{ \\omega \\left( {{X}_{2}}{{X}_{3}} \\right),\\omega \\left( {{X}_{3}}{{X}_{2}} \\right) \\right\\}+\\underset{\\lambda \\in \\mathbb{C}}{\\mathop{\\inf }}\\,{{\\left\\| \\left[ \\begin{matrix}\n -\\lambda I & {{X}_{2}} \\\\\n {{X}_{3}} & -\\lambda I \\\\\n\\end{matrix} \\right] \\right\\|}^{2}}}. \n\\end{aligned}\\]\n\\end{remark}\n\\begin{remark}\nNotice that\n\\[\\begin{aligned}\n r\\left( {{X}_{1}}{{X}_{2}}+{{X}_{3}}{{X}_{4}} \\right) & =r\\left( \\left[ \\begin{matrix}\n {{X}_{1}}{{X}_{2}}+{{X}_{3}}{{X}_{4}} & O \\\\\n O & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & =r\\left( \\left[ \\begin{matrix}\n {{X}_{1}} & {{X}_{3}} \\\\\n O & O \\\\\n\\end{matrix} \\right]\\left[ \\begin{matrix}\n {{X}_{2}} & O \\\\\n {{X}_{4}} & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & =r\\left( \\left[ \\begin{matrix}\n {{X}_{2}} & O \\\\\n {{X}_{4}} & O \\\\\n\\end{matrix} \\right]\\left[ \\begin{matrix}\n {{X}_{1}} & {{X}_{3}} \\\\\n O & O \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & =r\\left( \\left[ \\begin{matrix}\n {{X}_{2}}{{X}_{1}} & {{X}_{2}}{{X}_{3}} \\\\\n {{X}_{4}}{{X}_{1}} & {{X}_{4}}{{X}_{3}} \\\\\n\\end{matrix} \\right] \\right) \\\\ \n & \\le \\omega \\left( \\left[ \\begin{matrix}\n {{X}_{2}}{{X}_{1}} & {{X}_{2}}{{X}_{3}} \\\\\n {{X}_{4}}{{X}_{1}} & {{X}_{4}}{{X}_{3}} \\\\\n\\end{matrix} \\right] \\right). \n\\end{aligned}\\]\nIf in the above inequality we put ${{X}_{1}}={{e}^{\\textup i\\theta }}A$, ${{X}_{2}}=B$, ${{X}_{3}}={{e}^{-\\textup i\\theta }}{{B}^{*}}$, and ${{X}_{4}}={{A}^{*}}$, we reach\n\\begin{equation}\\label{13}\n\\left\\| {{\\operatorname{\\mathfrak Re}}^{\\textup i\\theta }}AB \\right\\|\\le \\frac{1}{2}\\omega \\left( \\left[ \\begin{matrix}\n {{e}^{\\textup i\\theta }}BA & {{e}^{-\\textup i\\theta }}B{{B}^{*}} \\\\\n {{e}^{\\textup i\\theta }}{{A}^{*}}A & {{\\left( {{e}^{\\textup i\\theta }}BA \\right)}^{*}} \\\\\n\\end{matrix} \\right] \\right).\n\\end{equation}\nThis indicates the relation between the numerical radius of the product of two operators and the numerical radius of $2\\times 2$ operator matrices.\n\nThe case $A=U{{\\left| T \\right|}^{1-t}}$ and $B={{\\left| T \\right|}^{t}}$, in \\eqref{13}, implies\n\\[\\left\\| \\mathfrak R{{e}^{\\textup i\\theta }}T \\right\\|\\le \\frac{1}{2}\\omega \\left( \\left[ \\begin{matrix}\n {{e}^{\\textup i\\theta }}\\widetilde{{{T}_{t}}} & {{e}^{-\\textup i\\theta }}{{\\left| T \\right|}^{2t}} \\\\\n {{e}^{\\textup i\\theta }}{{\\left| T \\right|}^{2\\left( 1-t \\right)}} & {{\\left( {{e}^{\\textup i\\theta }}\\widetilde{{{T}_{t}}} \\right)}^{*}} \\\\\n\\end{matrix} \\right] \\right),\\quad 0\\le t \\le 1,\\]\nwhere $\\widetilde{{{T}_{t}}}$ is the weighted Aluthge transform of $T$ defined by $\\widetilde{{{T}_{t}}}=|T|^tU|T|^{1-t},$ where $U$ is the partial isometry appearing in the polar decomposition in $T=U|T|.$ \n\nNotice that, if we replace $A=\\sqrt{\\frac{\\left\\| B \\right\\|}{\\left\\| A \\right\\|}}A$ and $B=\\sqrt{\\frac{\\left\\| A \\right\\|}{\\left\\| B \\right\\|}}B$, in \\eqref{13}, we also have\n\\begin{equation}\\label{14}\n\\left\\| {{\\operatorname{\\mathfrak Re}}^{\\textup i\\theta }}AB \\right\\|\\le \\frac{1}{2}\\omega \\left( \\left[ \\begin{matrix}\n {{e}^{\\textup i\\theta }}BA & {{e}^{-\\textup i\\theta }}\\frac{\\left\\| A \\right\\|}{\\left\\| B \\right\\|}B{{B}^{*}} \\\\\n {{e}^{\\textup i\\theta }}\\frac{\\left\\| B \\right\\|}{\\left\\| A \\right\\|}{{A}^{*}}A & {{\\left( {{e}^{\\textup i\\theta }}BA \\right)}^{*}} \\\\\n\\end{matrix} \\right] \\right),\n\\end{equation}\nand\n\\[\\left\\| {{\\operatorname{\\mathfrak Re}}^{\\textup i\\theta }}T \\right\\|\\le \\frac{1}{2}\\omega \\left( \\left[ \\begin{matrix}\n {{e}^{\\textup i\\theta }}\\widetilde{{{T}_{t}}} & {{e}^{-\\textup i\\theta }}{{\\left\\| T \\right\\|}^{1-2t}}{{\\left| T \\right|}^{2t}} \\\\\n {{e}^{\\textup i\\theta }}{{\\left\\| T \\right\\|}^{2t-1}}{{\\left| T \\right|}^{2\\left( 1-t \\right)}} & {{\\left( {{e}^{\\textup i\\theta }}\\widetilde{{{T}_{t}}} \\right)}^{*}} \\\\\n\\end{matrix} \\right] \\right).\\]\n\\end{remark}\nTo better understand how the above relations help obtain the numerical radius of the product of two operators, we give an example. Recall that in \\cite[Corollary 2]{8}, Abu-Omar and Kittaneh proved that if $\\mathcal H_1$ and $\\mathcal H_2$ are Hilbert spaces and $\\mathbb X=\\left[ \\begin{matrix}\n {{X}_{1}} & {{X}_{2}} \\\\\n {{X}_{3}} & {{X}_{4}} \\\\\n\\end{matrix} \\right]$ is an operator matrix with\n$X_1\\in \\mathcal B(\\mathcal H_1)$, $X_2\\in \\mathcal B(\\mathcal H_2,\\mathcal H_1)$, $X_3\\in \\mathcal B(\\mathcal H_1,\\mathcal H_2)$, and $X_4\\in \\mathcal B(\\mathcal H_2)$, then\n\\begin{equation*}\n\\omega \\left( \\mathbb X \\right)\\le \\frac{1}{2}\\left( \\omega \\left( {{X}_{1}} \\right)+\\omega \\left( {{X}_{4}} \\right)+\\sqrt{{{\\left( \\omega \\left( {{X}_{1}} \\right)-\\omega \\left( {{X}_{4}} \\right) \\right)}^{2}}+4{{\\omega }^{2}}\\left( \\mathbb E \\right)} \\right),\n\\end{equation*}\nwhere $\\mathbb E=\\left[ \\begin{matrix}\n O & {{X}_{2}} \\\\\n {{X}_{3}} & O \\\\\n\\end{matrix} \\right]$.\nIn the same paper (see \\cite[Remark 6]{8}), it has been shown that\n\\[{{\\omega }}\\left( \\mathbb E \\right)\\le \\min \\left\\{ {{\\alpha }_{1}},{{\\alpha }_{2}} \\right\\}\\]\nwhere\n\\[{{\\alpha }_{1}}=\\frac{1}{4}\\sqrt{\\left\\| {{\\left| {{X}_{2}} \\right|}^{2}}+{{\\left| X_{3}^{*} \\right|}^{2}} \\right\\|+2\\omega \\left( {{X}_{3}}{{X}_{2}} \\right)}\\;\\text{ and }\\;{{\\alpha }_{2}}=\\frac{1}{4}\\sqrt{\\left\\| {{\\left| X_{2}^{*} \\right|}^{2}}+{{\\left| {{X}_{3}} \\right|}^{2}} \\right\\|+2\\omega \\left( {{X}_{2}}{{X}_{3}} \\right)}.\\]\nCombining these two inequalities we get \n\\[\\omega \\left( \\left[ \\begin{matrix}\n {{X}_{1}} & {{X}_{2}} \\\\\n {{X}_{3}} & {{X}_{4}} \\\\\n\\end{matrix} \\right] \\right)\\le \\frac{1}{2}\\left( \\omega \\left( {{X}_{1}} \\right)+\\omega \\left( {{X}_{4}} \\right)+\\sqrt{{{\\left( \\omega \\left( {{X}_{1}} \\right)-\\omega \\left( {{X}_{4}} \\right) \\right)}^{2}}+4\\min \\left\\{ \\alpha _{1}^{2},\\alpha _{2}^{2} \\right\\}} \\right).\\]\nNow, using this and \\eqref{14}, we have\n\\[\\left\\| {{\\operatorname{\\mathfrak Re}}^{\\textup i\\theta }}AB \\right\\|\\le \\frac{1}{2}\\left( \\omega \\left( BA \\right)+\\min \\left\\{ {{\\beta }_{1}},{{\\beta }_{2}} \\right\\} \\right),\\]\nwhere\n\\[{{\\beta }_{1}}=\\frac{1}{2}\\sqrt{\\left\\| \\frac{{{\\left\\| A \\right\\|}^{2}}}{{{\\left\\| B \\right\\|}^{2}}}{{\\left| {{B}^{*}} \\right|}^{4}}+\\frac{{{\\left\\| B \\right\\|}^{2}}}{{{\\left\\| A \\right\\|}^{2}}}{{\\left| A \\right|}^{4}} \\right\\|+2\\omega \\left( {{\\left| A \\right|}^{2}}{{\\left| {{B}^{*}} \\right|}^{2}} \\right)},\\]\nand\n\\[{{\\beta }_{2}}=\\frac{1}{2}\\sqrt{\\left\\| \\frac{{{\\left\\| A \\right\\|}^{2}}}{{{\\left\\| B \\right\\|}^{2}}}{{\\left| {{B}^{*}} \\right|}^{4}}+\\frac{{{\\left\\| B \\right\\|}^{2}}}{{{\\left\\| A \\right\\|}^{2}}}{{\\left| A \\right|}^{4}} \\right\\|+2\\omega \\left( {{\\left| {{B}^{*}} \\right|}^{2}}{{\\left| A \\right|}^{2}} \\right)}.\\]\nThis implies,\n\\[\\omega \\left( AB \\right)\\le \\frac{1}{2}\\omega \\left( BA \\right)+\\frac{1}{4}\\sqrt{\\left\\| \\frac{{{\\left\\| A \\right\\|}^{2}}}{{{\\left\\| B \\right\\|}^{2}}}{{\\left| {{B}^{*}} \\right|}^{4}}+\\frac{{{\\left\\| B \\right\\|}^{2}}}{{{\\left\\| A \\right\\|}^{2}}}{{\\left| A \\right|}^{4}} \\right\\|+2\\min \\left\\{ \\omega \\left( {{\\left| A \\right|}^{2}}{{\\left| {{B}^{*}} \\right|}^{2}} \\right),\\omega \\left( {{\\left| {{B}^{*}} \\right|}^{2}}{{\\left| A \\right|}^{2}} \\right) \\right\\}}.\\]\nWe also have by \\eqref{13},\n\\[\\omega \\left( AB \\right)\\le \\frac{1}{2}\\omega \\left( BA \\right)+\\frac{1}{4}\\sqrt{\\left\\| {{\\left| {{B}^{*}} \\right|}^{4}}+{{\\left| A \\right|}^{4}} \\right\\|+2\\min \\left\\{ \\omega \\left( {{\\left| A \\right|}^{2}}{{\\left| {{B}^{*}} \\right|}^{2}} \\right),\\omega \\left( {{\\left| {{B}^{*}} \\right|}^{2}}{{\\left| A \\right|}^{2}} \\right) \\right\\}}.\\]\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nIt is well known result that realistic interpretations of quantum\ntheory are nonlocal \\cite{bell}. This was first shown by means of\nBell's inequality. Afterwards, the proof of the same for three\nspin-1\/2 particles as well as for two spin-1 particles, without\nusing inequality caused much interest among physicists\n\\cite{green}. Surprisingly Hardy gave a proof of nonlocality\nwithout using inequality, for two spin-1\/2 particles which\nrequires two measurement settings on both the sides as happens in\ncase of Bell's argument \\cite{hardy92}. Later Hardy showed this\nkind of nonlocality argument can be made for almost all entangled\nstate of two spin-1\/2 particles except for maximally entangled\none.\\cite{hardy93}. He considered the cases where the measurement\nchoices were same for both the parties. Jordan showed that for\nany given entangled state of two spin-1\/2 particles except\nmaximally entangled state there are many set of observables on\neach side which satisfy Hardy's nonlocality conditions\n\\cite{jordan}. Jordan also showed that the set of observables\nwhich gives maximum probability of success in showing the\ncontradiction with local-realism, is the same as\nchosen by Hardy.\\\\\nRecently Cabello has introduced a logical structure to prove\nBell's theorem without inequality for three particles GHZ and W\nstate \\cite{cabello02}. Logical structure presented by Cabello is\nas follows : Consider four events D, E, F and G where D and F may\nhappen in one system and E and G happen in another system which is\nfar apart from the first. The probability of joint occurrence of D\nand E is non-zero, E always implies F, D always implies G, but F\nand G happen with lower probability than D and E. These four\nstatements are not compatible with local realism. The difference\nbetween these two probabilities is the measure of violation of\nlocal realism. Though Cabello's logical structure was originally\nproposed for showing nonlocality for three particle states but\nLiang and Li \\cite{liang05} exploited it in establishing\nnonlocality without inequality for a class of two qubit mixed\nentangled state. In this sense, Hardy's logical structure is an\nspecial case of Cabello's structure as the logical structure of\nHardy for establishing nonlocality is as follows: D and E\nsometimes happen, E always implies F, D always implies G, but F\nand G never happen. Recently based on Cabello's logical structure\nKunkri and Choudhary \\cite{kunkri} have shown that there may be\nmany classes of two qubit mixed states which exhibit nonlocality\nwithout inequality. It is noteworthy here that in contrast there\nis no two qubit mixed state which shows Hardy type nonlocality\n\\cite{karpra}. So it seems interesting to study that whether\nmaximally entangled states follow this more general (than\nHardy's), Cabello's nonlocality argument or not , because Hardy's\nnonlocality argument is not followed by a maximally entangled\nstate. In this paper we have studied it and found that maximally\nentangled states do not respond even to this argument. However,\nfor all other pure entangled states , Cabello's argument runs. We\nfurther have enquired about the highest value of difference\nbetween the two probabilities which appear in Cabello's argument.\nSurprisingly this value differs from the highest value of\nprobability which\nappears in Hardy's argument.\\\\\n\\section*{Cabello's argument for two qubits}\nLet us consider two spin-1\/2 particles A and B. Let F, D, G and E\nrepresent the spin observables along $n_F (\n\\sin{\\theta_F}\\cos{\\phi_F}, \\sin{\\theta_F}\\sin{\\phi_F},\n\\cos{\\theta_F})$, $n_D ( \\sin{\\theta_D}\\cos{\\phi_D},\n\\sin{\\theta_D}\\sin{\\phi_D}, \\cos{\\theta_D})$, $n_G (\n\\sin{\\theta_G}\\cos{\\phi_G}, \\sin{\\theta_G}\\sin{\\phi_G},\n\\cos{\\theta_G})$ and $n_E ( \\sin{\\theta_E}\\cos{\\phi_E},\n\\sin{\\theta_E}\\sin{\\phi_E}, \\cos{\\theta_E})$ respectively. Every\nobservable has the eigen value $\\pm 1$. Let F and D are measured\non particle A and G and E are measured on particle B. Now we\nconsider the following equations\n\\begin{equation}\nP(F = +1, G = +1) = q_1\n\\end{equation}\n\\begin{equation}\nP(D = +1, G = -1) = 0\n\\end{equation}\n\\begin{equation}\nP(F = -1, E = +1) = 0\n\\end{equation}\n\\begin{equation}\nP(D = +1, E = +1) = q_4\n\\end{equation}\n\nEquation (1) tells that if F is measured on particle A and G is\nmeasured on particle B, then the probability that both will get +1\neigen value is $q_1$. Other equations can be analyzed in a similar\nfashion. These equations form the basis of Cabello's nonlocality\nargument. It can easily be seen that these equations contradict\nlocal-realism if $q_1 < q_4$. To show this, let us consider those\nhidden variable states $\\lambda$ for which $D = +1$ and $E = +1$.\nNow for these states equations $(2)$ and $(3)$ tell that the\nvalues of $G$ and $F$ must be equal to $+1$. Thus according to\nlocal realism $P(F = +1, G = +1)$ should be at least equal to\n$q_4$, which contradicts equation $(1)$ as $q_1 < q_4$. It should\nbe noted here that $q_1=0$ reduces this argument to Hardy's one.\nSo by Cabello's argument we specifically mean that the above\nargument runs even with nonzero $q_1$.\\\\\nNow we will show that for almost all two qubit pure entangled\nstate other than maximally entangled one this kind of nonlocality\n argument runs. Following Schmidt\n decomposition procedure any\n entangled state of two particles A and B can be written as\n \\begin{equation}\n |\\psi\\rangle = (\\cos{\\beta}) |0\\rangle_A |0\\rangle_B + (\\sin\n {\\beta})e^{i\\gamma} |1\\rangle_A |1\\rangle_B\n \\end{equation}\nIf either $\\cos{\\beta}$ or $\\sin{\\beta}$ is zero, we have a\nproduct state not an entangled state. Then it is not possible to\nsatisfy equation $(1)-(4)$. Hence we assume that neither\n$\\cos{\\beta}$ nor $\\sin{\\beta}$ is zero; both are positive.\\\\\nThe density matrix for the above state is\n\\begin{equation}\n\\begin{array}{lcl}\n \\rho = \\frac{1}{4}[I^A \\otimes I^B + (\\cos^2{\\beta} - \\sin^2{\\beta})I^A\\otimes \\sigma_z^B +\n(\\cos^2{\\beta} - \\sin^2{\\beta})\\sigma_z^A \\otimes I^B \\\\ +\n(2\\cos{\\beta}\\sin{\\beta}\\cos{\\gamma})\\sigma_x^A \\otimes \\sigma_x^B\n+ (2\\cos{\\beta}\\sin{\\beta}\\sin{\\gamma})\\sigma_x^A \\otimes\n\\sigma_y^B \\\\ + (2\\cos{\\beta}\\sin{\\beta}\\sin{\\gamma})\\sigma_y^A\n\\otimes \\sigma_x^B -\n(2\\cos{\\beta}\\sin{\\beta}\\cos{\\gamma})\\sigma_y^A \\otimes \\sigma_y^B\n+ \\sigma_z^A \\otimes \\sigma_z^B]\n\\end{array}\n\\end{equation}\n\nWhere $\\sigma_x$, $\\sigma_y$ and $\\sigma_z$ are Pauli operators.\nNow for this state if F is measured on particle A and G is\nmeasured on particle B, then the probability that both will get +1\neigen value is given by\n\n\\begin{equation}\n\\begin{array}{lcl}\n P(F = +1, G = +1) = (\\frac{1}{4})[1 + (\\cos^2{\\beta} -\n \\sin^2{\\beta})(\\cos{\\theta_F} + \\cos{\\theta_G})\\\\\n + \\cos{\\theta_F}\\cos{\\theta_G} + 2\\cos{\\beta}\\sin{\\beta}\\sin{\\theta_F}\\sin{\\theta_G}\n \\times \\cos{(\\phi_F + \\phi_G - \\gamma)}]\n\\end{array}\n\\end{equation}\nRearranging the above expression we get\n\\begin{equation}\n\\begin{array}{lcl}\n P(F = +1, G = +1) =\n \\cos^2{\\beta}\\cos^2{\\frac{\\theta_F}{2}}\\cos^2{\\frac{\\theta_G}{2}} +\n \\sin^2{\\beta}\\sin^2{\\frac{\\theta_F}{2}}\\sin^2{\\frac{\\theta_G}{2}} +\n \\\\\n + 2\\cos{\\beta}\\sin{\\beta}\\cos{\\frac{\\theta_F}{2}}\\sin{\\frac{\\theta_F}{2}}\n \\cos{\\frac{\\theta_G}{2}}\\sin{\\frac{\\theta_G}{2}}\n \\times \\cos{(\\phi_F + \\phi_G - \\gamma)}]= q_1(say)\n\\end{array}\n\\end{equation}\nSimilar calculations for other probabilities give us:\n\\begin{equation}\n\\begin{array}{lcl}\n P(D = +1, G = -1) =\n \\cos^2{\\beta}\\cos^2{\\frac{\\theta_D}{2}}\\sin^2{\\frac{\\theta_G}{2}} +\n \\sin^2{\\beta}\\sin^2{\\frac{\\theta_D}{2}}\\cos^2{\\frac{\\theta_G}{2}} +\n \\\\\n + 2\\cos{\\beta}\\sin{\\beta}\\cos{\\frac{\\theta_D}{2}}\\sin{\\frac{\\theta_D}{2}}\n \\cos{\\frac{\\theta_G}{2}}\\sin{\\frac{\\theta_G}{2}}\n \\times \\cos{(\\phi_D + \\phi_G + \\pi - \\gamma)}]= q_2(say)\n\\end{array}\n\\end{equation}\n\n\\begin{equation}\n\\begin{array}{lcl}\n P(F = -1, E = +1) =\n \\cos^2{\\beta}\\cos^2{\\frac{\\theta_E}{2}}\\sin^2{\\frac{\\theta_F}{2}} +\n \\sin^2{\\beta}\\sin^2{\\frac{\\theta_E}{2}}\\cos^2{\\frac{\\theta_F}{2}} +\n \\\\\n + 2\\cos{\\beta}\\sin{\\beta}\\cos{\\frac{\\theta_F}{2}}\\sin{\\frac{\\theta_F}{2}}\n \\cos{\\frac{\\theta_E}{2}}\\sin{\\frac{\\theta_E}{2}}\n \\times \\cos{(\\phi_F + \\phi_E + \\pi - \\gamma)}]=q_3(say)\n\\end{array}\n\\end{equation}\n\n\\begin{equation}\n\\begin{array}{lcl}\n P(D = +1, E = +1) =\n \\cos^2{\\beta}\\cos^2{\\frac{\\theta_D}{2}}\\cos^2{\\frac{\\theta_E}{2}} +\n \\sin^2{\\beta}\\sin^2{\\frac{\\theta_D}{2}}\\sin^2{\\frac{\\theta_E}{2}} +\n \\\\\n + 2\\cos{\\beta}\\sin{\\beta}\\cos{\\frac{\\theta_D}{2}}\\sin{\\frac{\\theta_D}{2}}\n \\cos{\\frac{\\theta_E}{2}}\\sin{\\frac{\\theta_E}{2}}\n \\times \\cos{(\\phi_D + \\phi_E - \\gamma)}]=q_4(say)\n\\end{array}\n\\end{equation}\nFor running Cabello's nonlocality argument, following conditions\nshould be satisfied:\n\\begin{equation}\nq_2=0,~~ q_3=0,~~ (q_4 - q_1) > 0, ~~ q_1 > 0\n\\end{equation}\n\n\n\n\n\n\n\nSince $q_2$ represents probability, it can not be negative. If it\nis zero, it is at its minimum value. Then its derivative must be\nzero. From it's derivative with respect to $\\phi_D$ we see that\n$\\sin{(\\phi_D + \\phi_G + \\pi - \\gamma)}$ must be zero. Evidently\n\n\\begin{equation}\n\\cos{(\\phi_D + \\phi_G + \\pi - \\gamma)} = -1\n\\end{equation}\nWe conclude that if $q_2$ is zero, then\n\\begin{equation}\n\\cos{\\beta}\\cos{\\frac{\\theta_D}{2}}\\sin{\\frac{\\theta_G}{2}} =\n\\sin{\\beta}\\sin{\\frac{\\theta_D}{2}}\\cos{\\frac{\\theta_G}{2}}\n\\end{equation}\nSimilar sort of argument for $q_3$ to be zero will give:\n\\begin{equation}\n\\cos{(\\phi_F + \\phi_E + \\pi - \\gamma)} = -1\n\\end{equation}\nand\n\\begin{equation}\n\\cos{\\beta}\\cos{\\frac{\\theta_E}{2}}\\sin{\\frac{\\theta_F}{2}} =\n\\sin{\\beta}\\sin{\\frac{\\theta_E}{2}}\\cos{\\frac{\\theta_F}{2}}\n\\end{equation}\n\\section*{Maximally entangled states of two spin-1\/2 particles do not\nexhibit Cabello type nonlocality-}\n For maximally entangled state\n$\\tan{\\beta} = 1$, then from equations $(14)$ and $(16)$ we get\n\\begin{equation}\n\\frac{\\theta_G}{2} = \\frac{\\theta_D}{2} + n\\pi\n\\end{equation}\n\\begin{equation}\n\\frac{\\theta_F}{2} = \\frac{\\theta_E}{2} + m\\pi\n\\end{equation}\nUsing equations $(17)$ and $(18)$ first in equation $(8)$ and then\nin equation (11) we get $q_1$ and $q_4$ for maximally entangled\nstate as:\n\\begin{equation}\n\\begin{array}{lcl}\nq_1 =\n \\frac{1}{2}\\cos^2{\\frac{\\theta_D}{2}}\\cos^2{\\frac{\\theta_E}{2}} +\n \\frac{1}{2}\\sin^2{\\frac{\\theta_D}{2}}\\sin^2{\\frac{\\theta_E}{2}}\n \\\\\n + \\cos{\\frac{\\theta_D}{2}}\\sin{\\frac{\\theta_D}{2}}\n \\cos{\\frac{\\theta_E}{2}}\\sin{\\frac{\\theta_E}{2}}\n \\times \\cos{(\\phi_F + \\phi_G - \\gamma)}]\n\\end{array}\n\\end{equation}\n\n\\begin{equation}\n\\begin{array}{lcl}\nq_4 =\n \\frac{1}{2}\\cos^2{\\frac{\\theta_D}{2}}\\cos^2{\\frac{\\theta_E}{2}} +\n \\frac{1}{2}\\sin^2{\\frac{\\theta_D}{2}}\\sin^2{\\frac{\\theta_E}{2}}\n \\\\\n + \\cos{\\frac{\\theta_D}{2}}\\sin{\\frac{\\theta_D}{2}}\n \\cos{\\frac{\\theta_E}{2}}\\sin{\\frac{\\theta_E}{2}}\n \\times \\cos{(\\phi_D + \\phi_E - \\gamma)}]\n\\end{array}\n\\end{equation}\nFrom equations $(19)$ and $(20)$ it is clear that $q_4$ will be\ngrater than $ q_1$ for a maximally entangled state only when\n$\\cos{(\\phi_D + \\phi_E - \\gamma)}\n> \\cos{(\\phi_F + \\phi_G - \\gamma)}$. But equation $(13)$ together\nwith equation $(15)$ says that $\\cos{(\\phi_D + \\phi_E - \\gamma)} =\n\\cos{(\\phi_F + \\phi_G - \\gamma)}$ {\\it i.e} $ q_4 = q_1$. So one\ncan conclude that there is no choice of observable which can make\nmaximally entangled state to show Cabello type of\nnonlocality .\\\\\n\\section*{Cabello's argument runs for other two particle pure\nentangled states-}\n To show that for every pure entangled state other than maximally\n entangled state of two spin-1\/2 particles, Cabello like argument runs\n it will be sufficient to show that one can always choose a set of observables for which\n set of conditions given\nby equation (12) is satisfied. This is equivalent of saying that\nfor $ 0<\\beta<\\frac{\\pi}{2}$ except when $\\beta=\\frac{\\pi}{4}$\nthere is at least one value for each of\n$\\theta_D$,$\\theta_E$,$\\theta_G$,$\\theta_F,\\phi_D$,$\\phi_E$,$\\phi_G$,$\\phi_F$\nfor which conditions mentioned in(12) are satisfied.\\\\\nLet us choose our $\\phi's$ in such a manner that\\\\\n$$cos{(\\phi_F + \\phi_G - \\gamma)}= cos{(\\phi_D + \\phi_E -\n\\gamma)}=-1$$\nFor these $\\phi's$ equations (8) and (11) respectively will read\nas:\\\\\n\\begin{equation}\nq_1 = (\\cos{\\beta}\\cos{\\frac{\\theta_F}{2}}\\cos{\\frac{\\theta_G}{2}}\n- \\sin{\\beta}\\sin{\\frac{\\theta_F}{2}}\\sin{\\frac{\\theta_G}{2}})^2\n\\end{equation}\n\\begin{equation}\nq_4 = (\\cos{\\beta}\\cos{\\frac{\\theta_D}{2}}\\cos{\\frac{\\theta_E}{2}}\n- \\sin{\\beta}\\sin{\\frac{\\theta_D}{2}}\\sin{\\frac{\\theta_E}{2}})^2\n\\end{equation}\nSo\n\\begin{equation}\n\\begin{array}{lcl}\n (q_4 - q_1) =\n \\cos^2{\\beta}(\\cos^2{\\frac{\\theta_D}{2}}\\cos^2{\\frac{\\theta_E}{2}}\n - \\cos^2{\\frac{\\theta_F}{2}}\\cos^2{\\frac{\\theta_G}{2}}) +\n \\sin^2{\\beta}(\\sin^2{\\frac{\\theta_D}{2}}\\sin^2{\\frac{\\theta_E}{2}}\n - \\sin^2{\\frac{\\theta_F}{2}}\\sin^2{\\frac{\\theta_G}{2}})\\\\\n + 2\\sin{\\beta}\\cos{\\beta}(\\cos{\\frac{\\theta_F}{2}}\\cos{\\frac{\\theta_G}{2}}\\sin{\\frac{\\theta_F}{2}}\n \\sin{\\frac{\\theta_G}{2}} - \\cos{\\frac{\\theta_D}{2}}\\cos{\\frac{\\theta_E}{2}}\\sin{\\frac{\\theta_D}{2}}\n \\sin{\\frac{\\theta_E}{2}})\n\\end{array}\n\\end{equation}\nNow we will have to choose at least one set of values of\n$\\theta's$ in such a way that $(q_4 - q_1)$ and $q_1$ are nonzero\nand positive. Moreover, these values of $\\theta's$ should also not\nviolate conditions given in equations $(14)$ and $(16)$.\\\\\nlet us try with $ \\frac{\\theta_D}{2} = 0$ {\\it i.e}\n$$ \\sin{\\frac{\\theta_D}{2}} = 0,~~~\\cos{\\frac{\\theta_D}{2}} = 1$$\nThis makes equation $(14)$ to read as\n $$\\sin{\\frac{\\theta_G}{2}} = 0,\\Rightarrow {\\frac{\\theta_G}{2}} =\n 0$$\n Then from equation $(23)$ we get $$ (q_4 - q_1) =\n \\cos^2{\\beta}(\\cos^2{\\frac{\\theta_E}{2}}-\n \\cos^2{\\frac{\\theta_F}{2}})$$\nThus $(q_4 - q_1) > 0$ if\n\\begin{equation}\n\\cos{\\frac{\\theta_E}{2}}> \\cos{\\frac{\\theta_F}{2}}\n\\end{equation}\n Rewriting equation $(16)$ as\n \\begin{equation}\n \\tan{\\frac{\\theta_F}{2}} = \\tan{\\beta}\\tan{\\frac{\\theta_E}{2}}\n\\end{equation}\n Values of $\\theta's$ satisfying inequality (24) will not violate\n equation (25) provided $\\tan{\\beta} > 1$.\nNow for these values of $\\theta's$, from equation (21), we get:\n$q_1 = (\\cos{\\beta}\\cos{\\frac{\\theta_F}{2}})^2$\nwhich is greater than zero.\\\\\n So for the above values of $\\theta's$ {\\it i.e} for\n$\\frac{\\theta_D}{2} = \\frac{\\theta_G}{2} = 0$ and\n$\\cos{\\frac{\\theta_E}{2}}> \\cos{\\frac{\\theta_F}{2}}$, all the\nstates for which $\\tan{\\beta} > 1$ ; Cabello's nonlocality\nargument runs.\\\\\nFor other states {\\it i.e} for the states for which $\\tan{\\beta}\n< 1$, let us choose $\\frac{\\theta_D}{2} = \\frac{\\theta_G}{2} =\n\\frac {\\pi}{2}$. Then from equation $(23)$ we get $$ (q_4 - q_1) =\n \\sin^2{\\beta}(\\sin^2{\\frac{\\theta_E}{2}}-\n \\sin^2{\\frac{\\theta_F}{2}})$$\n Thus $(q_4 - q_1) > 0$ if\n\\begin{equation}\n\\sin{\\frac{\\theta_E}{2}}> \\sin{\\frac{\\theta_F}{2}}\n\\end{equation}\nOne can easily check that for abovementioned values of $\\theta's$\n; $q_1$ is also positive and equation (25) is satisfied too.\n\nThus if we choose $\\frac{\\theta_D}{2} = \\frac{\\theta_G}{2} =\n\\frac{\\pi}{2}$ and $\\sin{\\frac{\\theta_E}{2}}>\n\\sin{\\frac{\\theta_F}{2}}$, then all the states for which,\n$\\tan{\\beta} < 1$ satisfy Cabello's nonlocality argument. So for\nevery $\\beta$ (except for $\\beta=\\frac{\\pi}{4}$); we can choose\n$\\theta's$ and $\\phi's$ and hence the observables in such a way\nthat Cabello's argument\nruns.\\\\\n\\section*{Maximum probability of success}\nFor getting maximum probability of success of Cabello's argument\nin contradicting local-realism we will have to maximize the\nquantity $(q_4 - q_1)$ for a given $\\beta$ over all observable\nparameters $\\theta's$ and $\\phi's$ under the restrictions given by\nequation's $(13)-(16)$. Using the equations $(13)-(16)$, we have\n\\begin{equation}\n\\begin{array}{lcl}\n (q_4 - q_1) =\n \\cos^2{\\beta}[(k_2 - k_1) + \\tan^2{\\beta}\\tan^2{\\frac{\\theta_D}{2}}\n \\tan^2{\\frac{\\theta_E}{2}} (k_2 - k_1 \\tan^4{\\beta}) + \\\\ 2 \\tan{\\beta}\\tan{\\frac{\\theta_D}{2}}\n \\tan{\\frac{\\theta_E}{2}} (k_2 - k_1 \\tan^2{\\beta})\\cos{(\\phi_D + \\phi_E -\n \\gamma)}]\n\\end{array}\n\\end{equation}\nwhere $$ k_1 = \\frac{1}\n {(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_D}{2}} + 1)(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_E}{2}} +\n 1)},~~~~~ k_2 = \\frac{1}\n {(\\tan^2{\\frac{\\theta_D}{2}} + 1)(\\tan^2{\\frac{\\theta_E}{2}} +\n 1)} $$\n It is clear from the equation $(27)$ that one can obtain maximum value of $(q_4 - q_1)$,\n when $\\cos{(\\phi_D + \\phi_E - \\gamma)}= \\pm 1$.\nLet us first consider $\\cos{(\\phi_D + \\phi_E - \\gamma)}= -1$, then\nfrom equation $(27)$ we have\n\\begin{equation}\n\\begin{array}{lcl}\n (q_4 - q_1) =\n \\cos^2{\\beta}[\\frac{(1- \\tan{\\beta}\\tan{\\frac{\\theta_D}{2}}\\tan{\\frac{\\theta_E}{2}})^2}\n {(\\tan^2{\\frac{\\theta_D}{2}} + 1)(\\tan^2{\\frac{\\theta_E}{2}} +\n 1)} -\n\\frac{(1-\n\\tan^3{\\beta}\\tan{\\frac{\\theta_D}{2}}\\tan{\\frac{\\theta_E}{2}})^2}\n {(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_D}{2}} + 1)(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_E}{2}} +\n 1)} ]\n\\end{array}\n\\end{equation}\nFrom the above equation one can show that $(q_4 - q_1)$ will be\nmaximum when $\\theta_D = \\theta_E$ (see Appendix) which in turn\nimplies $\\theta_G = \\theta_F$ {\\it i.e } $(q_4 - q_1)$ becomes\nmaximum when measurement settings in both the sides is same as\nwas in Hardy's case. Now for the optimal case {\\it i.e } for\n$\\theta_G = \\theta_F$ and $\\theta_D = \\theta_E$, $(q_4 - q_1)$\nbecomes\n\\begin{equation}\n\\begin{array}{lcl}\n (q_4 - q_1) =\n \\cos^2{\\beta}[\\frac{(1- \\tan{\\beta}\\tan^2{\\frac{\\theta_D}{2}})^2}\n {(\\tan^2{\\frac{\\theta_D}{2}} + 1)^2} -\n\\frac{(1- \\tan^3{\\beta}\\tan^2{\\frac{\\theta_D}{2}})^2}\n {(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_D}{2}} + 1)^2} ]\n\\end{array}\n\\end{equation}\nNumerically we have checked that $(q_4 - q_1)$ has a maximum value\nof $.1078$ when $\\cos{\\beta} = .485$ with $\\theta_D = \\theta_E =\n.59987$. This is interesting as maximum probability of success of\n Hardy's argument is only $9\\%$, whereas in case of Cabello's\nargument it is approximately $11\\%$.\\\\ Here we are comparing the\nmaximum probability of success of Hardy's argument with that of\nCabello's argument for all states.\\\\\n\\begin{figure}[hp]\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics{samir1.ps}}\n\\end{center}\n\\caption{Comparison of the maximum probability of success between\nHardy's and Cabello's case}\n\\end{figure}\n\nGraph shows that for $\\cos{\\beta} \\approx.7$ {\\it i.e } for\n$\\beta=\\frac{\\pi}{4}$ and for $\\cos{\\beta}=1$ {\\it i.e } for\n$\\beta=0$; maximum of $(q_4 - q_1)$ vanishes. This is as expected\nbecause these values of $\\beta$ represent respectively the\nmaximally entangled and product states for which Cabello's\nargument does not run. For most of the other values of $\\beta$\n{\\it i.e } for most of the other entangled states , maximum\n probability of success of Cabello's argument in establishing their\n nonlocal feature is more than the maximum probability of success\n of hardy's argument in doing the same.\\\\\nAs we have mentioned earlier (just before equation 28) that\n$\\cos{(\\phi_D + \\phi_E - \\gamma)}= 1$ also optimizes $(q_4 -\nq_1)$. This also gives the same maximum value for $(q_4 - q_1)$as\ngiven by $\\cos{(\\phi_D + \\phi_E - \\gamma)}= -1$\nbut for $\\theta_D = -\\theta_E$.\\\\\n\\section*{Conclusion}\n\nIn conclusion, here we have shown that maximally entangled states\ndo not respond even to Cabello's argument which is a relaxed one\nand is more general than Hardy's argument. All other pure\nentangled states response to Cabello's argument. These states also\nexhibit Hardy type nonlocality. But, interestingly for most of\nthese nonmaximally entangled states, fraction of runs in which\nCabello's argument succeeds in demonstrating their nonlocal\nfeature can be made more than the fraction of runs in which\nHardy's argument succeeds in doing the same. So it seems that in\nsome sense, for demonstrating the nonlocal features of most of\nthe entangled\nstates, Cabello's argument is a better candidate.\\\\\n{\\bf Appendix-}\\\\ We want to optimize $(q_4 - q_1)$ given in\nequation $(28)$ with respect to $\\theta_D$ and $\\theta_E$ for a\ngiven $\\beta$. Differentiating equation $(28)$ with respect to\n$\\theta_D$ and equating it to zero, we have the following two\nequations\n\\begin{equation}\n(\\tan{\\beta}\\tan{\\frac{\\theta_E}{2}} + \\tan {\\frac{\\theta_D}{2}})=\n0\n\\end{equation}\nand\n\\begin{equation}\n(\\tan{\\beta}\\tan{\\frac{\\theta_E}{2}}\\tan{\\frac{\\theta_D}{2}} -\n1)(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_E}{2}} +\n1)(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_D}{2}} + 1)^2 = $$\n$$ (\\tan^2{\\beta}\\sec^2{\\frac{\\theta_D}{2}})(\\tan^3{\\beta}\\tan{\\frac{\\theta_E}{2}}\\tan{\\frac{\\theta_D}{2}}\n- 1)(\\sec^2{\\frac{\\theta_E}{2}}\\sec^2{\\frac{\\theta_D}{2}})\n\\end{equation}\n Similarly differentiating equation $(28)$ with\nrespect to $\\theta_E$ and equating it to zero, we have\n\\begin{equation}\n(\\tan{\\beta}\\tan{\\frac{\\theta_D}{2}} + \\tan {\\frac{\\theta_E}{2}})=\n0\n\\end{equation}\nand\n\\begin{equation}\n(\\tan{\\beta}\\tan{\\frac{\\theta_D}{2}}\\tan{\\frac{\\theta_E}{2}} -\n1)(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_D}{2}} +\n1)(\\tan^2{\\beta}\\tan^2{\\frac{\\theta_E}{2}} + 1)^2 = $$\n$$ (\\tan^2{\\beta}\\sec^2{\\frac{\\theta_E}{2}})(\\tan^3{\\beta}\\tan{\\frac{\\theta_D}{2}}\\tan{\\frac{\\theta_E}{2}}\n- 1)(\\sec^2{\\frac{\\theta_D}{2}}\\sec^2{\\frac{\\theta_E}{2}})\n\\end{equation}\nAnalyzing above four conditions we have\n$$\\theta_D = \\theta_E$$ will give the optimal solution.\nSimilarly for $cos{(\\phi_D + \\phi_E - \\gamma)}= +1$, we will get\nsame kind of results.\n\\section*{Acknowledgement}\nAuthors would like to thank Guruprasad Kar, Debasis Sarkar for\nuseful discussions. We also thank Swarup Poria to help us in\nnumerical calculation. S.K acknowledges the support by the Council\nof Scientific and Industrial Research, Government of India, New\nDelhi.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{$\\protect\\bigskip $Introduction}\n\nLet $G$ be a Lie group which acts transitively on some space $M.$ In this\nframework, Felix Klein defined geometry (later known as \\textit{Erlangen\nProgramm}) as the study of properties which are invariant under the action\nof $G.$ Realizing $M$ as the coset space $G\/H$ where $H\\subset G$ is the\nstabilizer of some point $p\\in M$, we may thus speak of the Klein pair (or\nKlein geometry) $(G,G\/H)$. In Euclidean geometry, the best known example, $G$\nis the isometry group of $\\mathbb{R}^{n}$ and is the semi-direct product of\nthe orthogonal group $O(n)$ and $\\mathbb{R}^{n},$ $H=O(n)$ and $G\/H=\\mathbb{R%\n}^{n}.$ Non-Euclidean geometries, which were known at Klein's time, are\nother examples. Riemannian geometry can not be realized in this way unless\nthe metric has constant curvature in which case the isometry group acts\ntransitively and we have again a Klein geometry. We will refer the reader to\n[14], pg. 133 so that he\/she can feel the philosophical disturbance created\nby this situation at that time. As an attempt to unify (among others)\nRiemannian geometry and \\textit{Erlangen Programm}, Elie Cartan introduced\nin 1922 his generalized spaces (principal bundles) which are curved analogs\nof the principal $H$-bundle $G\\rightarrow G\/H.$ We will mention here the two\noutstanding books: [29] for Klein geometries, their generalizations called\nCartan geometries in [29] and also for the notions of Cartan and Ehresmann\nconnections, and [14] for more history of this subject (see in particular\npg. 34-42 for \\textit{Erlangen Programm)}. Cartan's approach, which is later\ndevoloped mainly by topologists from the point of view of fiber bundles,\nturned out to be extremely powerful. The spectacular achievements of the\ntheory of principle bundles and connections in geometry, topology and\nphysics are well known and it is therefore redundant to elobarate on them\nhere. However, it is also a fact that this theory leaves us with the\nunpleasent question: What happened to \\textit{Erlangen Programm}? The main\nreason for this question is that the total space $P$ of the principal bundle \n$P\\rightarrow M$ does not have any group-like structure and therefore does\nnot act on the base manifold $M.$ Thus $P$ emerges in this framework as a\nnew entity whose relation to the geometry of $M$ may not be immediate and we\nmust deal now with $P$ as a seperate problem. Consequently, it seems that\nthe most essential feature of Klein's realization of geometry is given up by\nthis approach. Some mathematicians already expressed their dissatisfaction\nof this state of affairs in literature with varying tones, among which we\nwill mention [29], [35], [26] and other contraversial works of the author of\n[26] (see the references in [26]).\n\nThe purpose of this work is to present another such unification which we by\nno means claim to be the ultimate and correct one but believe that it is\nfaithful to Klein's original conception of geometry. This unification is\nbased on the ideas which S. Lie and F. Klein communicated to each other\nbefore 1872 (see [14] for the extremely interesting history of this subject)\nand the works of D.C. Spencer and his co-workers around 1970 on the formal\nintegrability of PDEs. The main idea is simple and seems to be the only\npossible one: We concentrate on the action of $G$ on $M=G\/H$ and generalize\nthis action rather than generalizing the action of $H$ on $G$ in the\nprincipal $H$-bundle $G\\rightarrow G\/H.$ Now $G\\subset Diff(M)$ and $G$ may\nbe far from being a Lie group. As the natural generalization of \\textit{%\nErlangen Programm, }we may now deal directly with the group $G$ as in [2]\n(see also [23]), but this approach again does not incorporate Riemannian\ngeometry unless the metric is homogeneous. We consider here the Lie\npseudogroup $\\widetilde{G}$ determined by $G$ and filter the action of $%\n\\widetilde{G}$ on $M$ via its jets, thus realizing $\\widetilde{G}$ as a\nprojective limit $Lim_{\\leftarrow k}$ $\\mathcal{S}_{k}(M)$ of Lie equations $%\n\\mathcal{S}_{k}(M).$ Lie equations (in finite form) are very special\ngroupoids and are very concrete objects which are extensively studied by\nSpencer and his co-workers culminating in the difficult work [11]. We will\nrefer to [11], [25], [26] for Lie equations and [19], [20] for\ndifferentiable groupoids and algebroids. On the infinitesimal level, we\nobtain the approximation $Lim_{\\leftarrow k}$ $\\frak{s}_{k}(M)$ where $\\frak{%\ns}_{k}(M)$ is the infinitesimal Lie equation (or the algebroid) of $\\mathcal{%\nS}_{k}(M).$ The idea is now to start with the expression $Lim_{\\leftarrow k}$\n$\\mathcal{S}_{k}(M)$ as our definition of homogeneous geometry $\\mathcal{S}%\n_{\\infty }(M)$ (Section 3, Definition 2). Any transitive pseudogroup (in\nparticular a complex, symplectic or contact structure) determines a\nhomogeneous geometry and Klein geometries are special cases (Section 4).\nSome $\\mathcal{S}_{k}(M)$ may not prolong to a homogeneous geometry due to\nthe lack of formal integrability and Riemannian geometry (almost complex,\nalmost symplectic...structures) emerge as truncated geometries (Section 5).\nWe associate various spectral sequences to a homogeneous geometry $\\mathcal{S%\n}_{\\infty }(M)$ (in particular to a truncated geometry $\\mathcal{S}_{k}(M))$\n(Sections 2, 3). For a complex structure, we believe that one these spectral\nsequences is directly related to the Fr\\\"{o}licher spectral sequence which\nconverges to de Rham cohomology with $E_{1}$ term equal to Dolbeault\ncohomology. This unification is also a natural specialization of the\nstandard principal bundle approach initiated by E. Cartan (Sections 6, 7, 8).\n\nThe idea of filtering an object via jets (for instance, the solution space\nof some PDE called a diffiety in [32], [33]) is not new and is used by\nA.M.Vinogradov in 1978 in the construction of his $\\mathcal{C}$-spectral\nsequence and in the variational bicomplex approach to Euler-Lagrange\nequations (see [32], [33] and the references therein). In fact, this paper\ncan also be considered as the first step towards the realization of a\nprogram stated in [33] for quasi-homogeneous geometries ([33], Section 6.4).\nFurther, there is definitely a relation between the higher order de Rham\ncomplexes constructed here and those in [31]. We also believe that the main\nidea of this paper, though it may not have been stated as explicitly as in\nthis paper, is contained in [26] and traces back to [11] and [25]. In\nparticular, we would like to emphasize that all the ingredients of this\nunification are known and exist in the literature.\n\nThis paper consists of nine sections. Section 2 contains the technical core\nin terms of which the geometric concepts substantiate. This section may be\nsomewhat demanding for the reader who is not much familiar with jets and the\nformalism of Spencer operator. However, as we proceed, technical points\nslowly evaporate and the main geometric concepts which we are all familiar\nwith, start to take the center stage.\n\n\\section{Universal homogeneous envelope}\n\nLet $M$ be a differentiable manifold and $Diff(M)$ be the group of\ndiffeomorphisms of $M.$ Consider the map $Diff(M)\\times M\\rightarrow M$\ndefined by $(g,x)\\rightarrow g(x)=y$ and let $j_{k}(g)_{y}^{x}$ denote the $%\nk $-jet of $g$ with source at $x$ and target at $y.$ By choosing coordinates \n$(U,x^{i})$ and $(V,y^{i})$ around the points $x,y,$ we may think $%\nj_{k}(g)_{y}^{x}$ as the coefficients of the Taylor expansion of $g(x)=y$ up\nto order $k.$ We will call $j_{k}(g)_{y}^{x}$ the $k$-arrow induced by $g$\nwith source at $x$ and target at $y$ and imagine $j_{k}(g)_{y}^{x}$ as an\narrow starting at $x$ and ending at $y.$ Let $(f_{k})_{y}^{x}$ denote \n\\textit{any }$k$-arrow, i.e., $(f_{k})_{y}^{x}$ is the $k$-jet induced by\nsome arbitrary local diffeomorphism which maps $x$ to $y$. With some\nassumptions on orientability which involve only 1-jets (see [24] for\ndetails), there exists some $g\\in Diff(M)$ with $%\nj_{k}(g)_{y}^{x}=(f_{k})_{y}^{x}.$ Therefore, with the assumption imposed by\n[24], the pseudogroup $\\widetilde{Diff(M)}$ of local diffeomorphisms on $M$\ninduces the same $k$-arrows as $Diff(M),$ but this fact will not be used in\nthis paper. We can compose successive $k$-arrows and invert all $k$-arrows.\n\nNow let $(\\mathcal{G}_{k})_{y}^{x}$ denote the set of all $k$-arrows with\nsource at $x$ and target at $y.$ We define $\\mathcal{G}_{k}(M)\\doteq \\cup\n_{x,y\\in M}(\\mathcal{G}_{k})_{y}^{x}$ and obtain the projections $\\pi _{k,m}:%\n\\mathcal{G}_{k}(M)\\rightarrow \\mathcal{G}_{m}(M),$ $1\\leq m\\leq k-1,$ and $%\n\\pi _{k,m}$ is compatible with composition and inversion of arrows. We will\ndenote all projections arising from the projection of jets by the same\nnotation $\\pi _{k,m}.$ Now $\\mathcal{G}_{k}(M)$ is a transitive Lie equation\nin finite form $(TLEFF)$ on $M$ which is a very special groupoid (see [11],\n[25], [26] for Lie equations and [19], [20] for groupoids). We also have the\nlocally trivial map $\\mathcal{G}_{k}(M)\\rightarrow M\\times M$ which maps $(%\n\\mathcal{G}_{k})_{y}^{x}$ to $(x,y).$ Note that $(\\mathcal{G}_{k})_{x}^{x}$\nis a Lie group and can be identified (not in a canonical way) with $k^{th}$%\n-order jet group. Thus we obtain the sequence of \\ homomorphisms\n\n\\begin{equation}\n......\\longrightarrow \\mathcal{G}_{k+1}(M)\\longrightarrow \\mathcal{G}%\n_{k}(M)\\longrightarrow .....\\longrightarrow \\mathcal{G}_{1}(M)%\n\\longrightarrow M\\times M\\longrightarrow 1\n\\end{equation}\n\nwhere the last arrow is used with no algebraic meaning but to express\nsurjectivity. (1) gives the vague formula $Diff(M)\\times M=Lim_{k\\rightarrow\n\\infty }\\mathcal{G}_{k}(M)$ or more precisely $\\widetilde{Diff(M)}%\n=Lim_{k\\rightarrow \\infty }\\mathcal{G}_{k}(M).$ The ambiguity in this last\nformula is that a formal Taylor expansion may fail to recapture a local\ndiffeomorphism. However, this ambiguity is immaterial for our purpose for\nthe following reason: Let $(j_{\\infty }g)_{x}^{x}$ denote the $\\infty $-jet\nof some local diffeomorphism $g$ where $g(x)=x.$ Now $(j_{\\infty }g)_{x}^{x}$\ndetermines $g$ modulo the $\\infty $-jet of the identity diffeomorphism. This\nis a consequence of the following elementary but remarkable fact: For \n\\textit{any }sequence of real numbers $a_{0},a_{1},....$, there exists a\nreal valued differentiable function $f$ defined, say, near the origin $o\\in \n\\mathbb{R}$, satisfying $f^{(n)}(o)=a_{n}.$ In particular, the same\ninterpretation is valid for the $\\infty $-jets of all objects to be defined\nbelow.\n\nSince $\\mathcal{G}_{k}(M)$ is a differentiable groupoid (we will call the\nobject called a Lie groupoid in [19], [20] a differentiable groupoid,\nreserving the term ``Lie'' for Lie equations), it has an algebroid $\\frak{g}%\n_{k}(M)$ which can be constructed using jets only. To do this, we start by\nletting $J_{k}(T(M))_{p}$ denote the vector space of $k$-jets of vector\nfields at $p\\in M$ where $T(M)\\rightarrow M$ is the tangent bundle of $M.$\nAn element of $J_{k}(T(M))_{p}$ is of the form $(p,\\xi ^{i}(p),\\xi\n_{j_{1}}^{i}(p),\\xi _{j_{2}j_{1}}^{i}(p),....,\\xi\n_{j_{k}j_{k-1}....j_{1}}^{i}(p))$ in some coordinates $(U,x^{i})$ around $p.$\nIf $X=(\\xi ^{i}(x))$, $Y=(\\eta ^{i}(x))$ are two vector fields on $U,$\ndifferentiating the usual bracket formula $[X,Y](x)=\\xi ^{a}(x)\\partial\n_{a}\\eta ^{i}(x)-\\eta ^{a}(x)\\partial _{a}\\xi ^{i}(x)$ successively $k$%\n-times and evaluating at $p$, we obtain the \\textit{algebraic bracket }$\\{$ $%\n,$ $\\}_{k,p}:$ $J_{k}(T(M))_{p}$ $\\times J_{k}(T(M))_{p}\\rightarrow\nJ_{k-1}(T(M))_{p}.$ Note that this bracket does \\textit{not} endow $%\nJ_{k}(T(M))_{p}$ with a Lie algebra structure. However, for $k=\\infty ,$ $%\nJ_{\\infty }(T(M))_{p}$ is a graded Lie algebra with the bracket $\\{$ $,$ $%\n\\}_{\\infty ,p},$ and is the well known Lie algebra of formal vector fields\nwhich is extensively studied in literature ([10]). However, let $%\nJ_{k,0}(T(M))_{p}$ be the kernel of $\\ J_{k}(T(M))_{p}\\rightarrow\nJ_{0}(T(M))_{p}=T(M)_{p}$. Now $J_{k,0}(T(M))_{p}$ \\textit{is} a Lie algebra\nwith the bracket $\\{$ $,$ $\\}_{k,p}$ which is in fact the Lie algebra of $%\n\\mathcal{G}_{k}(M)_{p}^{p}.$\n\nWe now define the vector bundle $J_{k}(T(M))\\doteq \\cup _{x\\in\nM}J_{k}(T(M))_{x}\\rightarrow M.$ We will denote a section of $%\nJ_{k}(T(M))\\rightarrow M$ by $\\overset{(k)}{X}.$ To simplify our notation,\nwe will use the same notation $E$ for both the total space $E$ of a vector\nbundle $E\\rightarrow M$ and also for the space $\\Gamma E$ of global sections\nof $E\\rightarrow M.$ In a coordinate system $(U,x^{i}),$ $\\overset{(k)}{X}$\nis of the form $\\overset{(k)}{X}(x)=(x,\\xi ^{i}(x),\\xi _{j_{1}}^{i}(x),\\xi\n_{j_{2}j_{1}}^{i}(x),....,\\xi _{j_{k}j_{k-1}....j_{1}}^{i}(x)),$ but we may\nnot have $\\xi _{j_{m}j_{m-1}....j_{1}}^{i}(x)=\\frac{\\partial \\xi\n_{j_{m-1}....j_{1}}^{i}}{\\partial x^{j_{m}}}(x)$ $,$ $1\\leq m\\leq k.$ We can\nthink $\\overset{(k)}{X}$ also as the function $\\overset{(k)}{X}(x,y)=\\frac{1%\n}{\\alpha !}\\xi _{\\alpha }^{i}(x)(y-x)^{\\alpha }$ where $\\alpha $ is a\nmulti-index with $\\left| \\alpha \\right| \\leq k$ and we used summation\nconvention. For some $\\overline{x}\\in U,$ $\\overset{(k)}{X}(\\overline{x},y)$\nis some Taylor polynomial which is \\textit{not }necessarily the Taylor\npolynomial of $\\ \\xi ^{i}(x)$ at $x=\\overline{x}$ since we may \\textit{not }%\nhave $\\xi _{\\alpha +j}^{i}(\\overline{x})=\\frac{\\partial \\xi _{\\alpha }^{i}}{%\n\\partial x^{j}}(\\overline{x}),$ $\\left| \\alpha \\right| \\leq k.$ Note that we\nhave the bundle projections $\\pi _{k,m}:J_{k}(T(M))\\rightarrow J_{m}(T(M))$\nfor $0\\leq m\\leq k-1,$ where $J_{0}(T(M))\\doteq T(M).$ We will denote $%\nJ_{k}(T(M))$ by $\\frak{g}_{k}(M)$ for the reason which will be clear below.\n\nWe now have the Spencer bracket $[$ $,$ $]$ defined on $\\frak{g}_{k}(M)$ by\nthe formula\n\n\\begin{equation}\n\\lbrack \\overset{(k)}{X},\\overset{(k)}{Y}]=\\{\\overset{(k+1)}{X},\\overset{%\n(k+1)}{Y}\\}+i(\\overset{(0)}{X})D\\overset{(k+1)}{Y}-i(\\overset{(0)}{Y})D%\n\\overset{(k+1)}{X}\\qquad k\\geq 0\n\\end{equation}\n\nIn (2), $\\overset{(k+1)}{X}$ and $\\overset{(k+1)}{Y}$are arbitrary lifts of $%\n\\overset{(k)}{X}$ and $\\overset{(k)}{Y}$, $\\{$ $,$ $\\}:\\frak{g}%\n_{k+1}(M)\\times \\frak{g}_{k+1}(M)\\rightarrow \\frak{g}_{k}(M)$ is the\nalgebraic bracket defined pointwise by $\\{\\overset{(k+1)}{X},\\overset{(k+1)}{%\nY}\\}(p)\\doteq \\{\\overset{(k+1)}{X}(p),$ $\\overset{(k+1)}{Y}(p)\\}_{k+1,p}$\nand $D$ $:\\frak{g}_{k+1}(M)\\rightarrow T^{\\ast }\\otimes \\frak{g}_{k}(M)$ is\nthe Spencer operator given locally by the formula $(x,\\xi ^{i}(x),\\xi\n_{j_{1}}^{i}(x),\\xi _{j_{2}j_{1}}^{i}(x),....,\\xi\n_{j_{k+1}j_{k}....j_{1}}^{i}(x))\\rightarrow (x,\\frac{\\partial \\xi ^{i}}{%\n\\partial x^{j_{1}}}-\\xi _{j_{1}}^{i}(x),$ $.....,\\frac{\\partial \\xi\n_{j_{k}....j_{1}}^{i}(x)}{\\partial x^{j_{k+1}}}-$ $\\xi\n_{j_{k+1}j_{k}....j_{1}}^{i}(x)).$ Finally, the vector fields $\\overset{(0)}{%\nX}$ and $\\overset{(0)}{Y}$ are the projections of $\\overset{(k)}{X}$ and $%\n\\overset{(k)}{Y}$ and $i(\\overset{(0)}{X})$ denotes the interior product\nwith respect to the vector field $\\overset{(0)}{X}.$ It turns out that RHS\nof (2) does not depend on the lifts $\\overset{(k+1)}{X},$ $\\overset{(k+1)}{Y}%\n.$ The bracket $\\ [$ $,$ $]$ satisfies Jacobi identity. We will refer to\n[25], [26] for further details. In view of the local formulas for $\\{$ $,$ $%\n\\}_{k,p}$ and $D$, it is elementary to make local computations using (2)\nwhich however become formidable starting already with $k=3$. It is easy to\ncheck that (2) gives the usual bracket formula for vector fields for $k=0.$\nIn fact, letting $\\mathcal{X}(M)$ denote the Lie algebra of vector fields on \n$M$, we have the prolongation map $j_{k}:\\mathcal{X}(M)\\rightarrow \\frak{g}%\n_{k}(M)$ defined by $(x,\\xi ^{i}(x))\\rightarrow (x,\\xi ^{i}(x),\\partial\n_{j_{1}}\\xi ^{i}(x),\\partial _{j_{2}j_{1}}\\xi ^{i}(x),....,\\partial\n_{j_{k+1}j_{k}....j_{1}}\\xi ^{i}(x))$ which satisfies $j_{k}\\overset{(0)}{[X}%\n,$ $\\overset{(0)}{Y}]=[j_{k}\\overset{(0)}{X},$ $j_{k}\\overset{(0)}{Y}].$\nThus (2) gives the usual bracket and its derivatives when restricted to $%\nj_{k}(\\mathcal{X}(M)).$\n\nNow $\\frak{g}_{k}(M)$ is the transitive Lie equation in infinitesimal form $%\n(TLEIF)$ determined by $\\mathcal{G}_{k}(M).$ If we regard $\\mathcal{G}%\n_{k}(M) $ as a differentiable groupoid and construct its algebroid as in\n[19], [20], we end up with $\\frak{g}_{k}(M),$ justifying our notation $\\frak{%\ng}_{k}(M)$ for $J_{k}(T(M)).$ The projection $\\pi _{k,m}:\\frak{g}%\n_{k}(M)\\rightarrow \\frak{g}_{m}(M)$ respects the bracket, i.e., it is a\nhomomorphism of $TLEIF^{\\prime }$s.\n\nIn this way we arrive at the infinitesimal analog of (1):\n\n\\begin{equation}\n......\\longrightarrow \\frak{g}_{k+1}(M)\\longrightarrow \\frak{g}%\n_{k}(M)\\longrightarrow ......\\longrightarrow \\frak{g}_{1}(M)\\longrightarrow \n\\frak{g}_{0}(M)\\longrightarrow 0\n\\end{equation}\n\nProceeding formally, the formula $\\widetilde{Diff(M)}=Lim_{k\\rightarrow\n\\infty }\\mathcal{G}_{k}(M)$ now gives $\\mathcal{A}\\widetilde{Diff(M)}%\n=Lim_{k\\rightarrow \\infty }\\frak{g}_{k}(M)$ where $\\mathcal{A}$ stands for\nthe functor which assigns to a groupoid its algebroid. However note that $%\n\\widetilde{Diff(M)}$ is not a groupoid but rather a pseudogroup. Since a\nvector field integrates to some 1-parameter group of local diffeomorphisms\n(no condition on vector fields and diffeomorphisms since we have not imposed\na geometric structure yet), we naturally expect $\\mathcal{A}\\widetilde{%\nDiff(M)}=J_{\\infty }(T(M))$. As above, the vagueness in this formula is that\na vector field need not be locally determined by the Taylor expansion of its\ncoefficients at some point.\n\nWe now define the vector space $J_{k}(M)_{x}\\doteq \\{j_{k}(f)_{x}\\mid f\\in\nC^{\\infty }(M)\\}$ where $C^{\\infty }(M)$ denotes the set of smooth functions\non $M$ and $j_{k}(f)_{x}$ denotes the $k$-jet of $f$ at $x\\in M.$ Now $%\nJ_{k}(M)_{x}$ is a commutative algebra over $\\mathbb{R}$ with the\nmultiplication $\\bullet $ defined by $j_{k}(f)_{x}\\bullet j_{k}(g)_{x}\\doteq\nj_{k}(fg)_{x}.$ We define the vector bundle $J_{k}(M)\\doteq $ $\\cup _{x\\in\nM}J_{k}(M)_{x}\\rightarrow M$ with the obvious differentiable structure and\nprojection map. The vector space of global sections of $J_{k}(M)\\rightarrow\nM $ is a commutative algebra with the fiberwise defined operations. We have\nthe projection homomorphism $\\pi _{k,m}:J_{k}(M)\\rightarrow J_{m}(M).$ We\nwill denote an element of $J_{k}(M)$ by $\\overset{(k)}{f}$ which is locally\nof the form $%\n(x,f(x),f_{i_{1}}(x),f_{i_{2}i_{1}}(x),....,f_{i_{k}....i_{1}}(x))=(f_{%\n\\alpha }(x)),$ $\\left| \\alpha \\right| \\leq k.$\n\nNow let $\\overset{(k)}{X}\\in \\frak{g}_{k}(M)$ and $\\overset{(k)}{f}\\in\nJ_{k}(M).$ We define $\\overset{(k)}{X}\\overset{(k)}{f}\\in J_{k}(M)$ by\n\n\\begin{equation}\n\\overset{(k)}{X}\\overset{(k)}{f}\\doteq \\overset{(k)}{X}\\ast \\overset{(k+1)}{f%\n}+i(\\overset{(0)}{X})D\\overset{(k+1)}{f}\n\\end{equation}\n\nIn (4), $\\ast :$ $\\frak{g}_{k}(M)\\times J_{k+1}(M)\\rightarrow J_{k}(M)$ is\nthe algebraic action of $\\frak{g}_{k}(M)$ on $J_{k+1}(M)$ whose local\nformula is obtained by differentiating the standard formula $\\overset{(0)}{X}%\nf=\\xi ^{a}\\partial _{a}f$ successively $k$-times and substituting jets$,$ $%\nD: $ $J_{k+1}(M)\\rightarrow T^{\\ast }\\otimes J_{k}(M)$ is the Spencer\noperator defined by $%\n(x,f(x),f_{i_{1}}(x),f_{i_{2}i_{1}}(x),....,f_{i_{k}....i_{1}}(x))$\n\n$\\rightarrow (x,\\partial _{i_{1}}f(x)-f_{i_{1}}(x),\\partial\n_{i_{2}}f_{i_{1}}(x)-f_{i_{2}i_{1}}(x),....,\\partial\n_{i_{k}}f_{i_{k-1}....i_{1}}(x)-f_{i_{k}....i_{1}}(x))$ and $\\overset{(k+1}{f%\n}$ is some lift of $\\overset{(k)}{f}.$ The RHS of (4) does not depend on the\nlift $\\overset{(k+1}{f}$. It is easy to check that $\\overset{(0)}{X}\\overset{%\n(0)}{f}=\\xi ^{a}\\partial _{a}f.$ Like (2), (4) is compatible with\nprojection, i.e., we have $\\pi _{k,m}(\\overset{(k)}{X}\\overset{(k)}{f})=$ $%\n(\\pi _{k,m}\\overset{(k)}{X})($ $\\pi _{k,m}\\overset{(k)}{f}),$ $0\\leq m\\leq\nk. $ Since $J_{k}(M)_{x}$ is a vector space over $\\mathbb{R}$, $J_{k}(M)$ is\na module over $C^{\\infty }(M).$ We will call some $\\overset{(k)}{f}%\n=(f_{\\alpha })\\in J_{k}(M)$ a smooth function if $f_{\\alpha }(x)=0$ for $%\n1\\leq \\left| \\alpha \\right| \\leq k.$ This definition does not depend on\ncoordinates. Thus we have an injection $C^{\\infty }(M)\\rightarrow J_{k}(M)$\nof algebras. If $\\overset{(k)}{f}$ $\\in C^{\\infty }(M),$ then $\\overset{(k)}{%\nf}\\bullet \\overset{(k)}{g}=(\\pi _{k,0}\\overset{(k)}{f})\\overset{(k)}{g}%\n\\doteq \\overset{(0)}{f}\\overset{(k)}{g}.$ Similar considerations apply to $%\nJ_{k}(T(M)).$\n\nWe thus obtain the $k^{th}$-order analogs of the well known formulas:\n\n\\begin{equation}\n\\lbrack \\overset{(k)}{X},\\overset{(k)}{Y}]\\overset{(k)}{f}=\\overset{(k)}{X}(%\n\\overset{(k)}{Y}\\overset{(k)}{f})-\\overset{(k)}{Y}(\\overset{(k)}{X}\\overset{%\n(k)}{f})\n\\end{equation}\n\n\\begin{equation}\n\\overset{(k)}{X}(\\overset{(k)}{f}\\bullet \\overset{(k)}{g})=(\\overset{(k)}{X}%\n\\overset{(k)}{f})\\bullet \\overset{(k)}{g}+\\overset{(k)}{f}\\bullet (\\overset{%\n(k)}{X}\\overset{(k)}{g})\n\\end{equation}\n\nIn particular, (6) gives\n\n\\begin{equation}\n\\overset{(k)}{X}(\\overset{(0)}{f}\\overset{(k)}{g})=(\\overset{(0)}{X}\\overset{%\n(0)}{f})\\overset{(k)}{g}+\\overset{(0)}{f}(\\overset{(k)}{X}\\overset{(k)}{g})\n\\end{equation}\n\nwhere $\\overset{(0)}{X}=\\pi _{k,0}\\overset{(k)}{X}$. In the language of\n[19], [20], (5) and (7) define a representation of the algebroid $\\frak{g}%\n_{k}(M)$ on the vector bundle $J_{k}(M)\\rightarrow M$ (see also [25], pg.362\nand [11], III, pg. 419). All constructions of this paper work also for other\nsuch ``intrinsic'' representations of $\\frak{g}_{k}(M).$ The passage from\nsuch intrinsic representations to general representations, we believe, is\nquite crucial for \\textit{Erlangen Programm }and will be touched at the end\nof Section 4 and in $5)$ of Section 9.\n\nNow let $\\overset{[k,r]}{\\wedge }(M)_{x}$ denote the vector space of $r$%\n-linear and alternating maps $\\frak{g}_{k}(M)_{x}\\times $ $.....\\times \\frak{%\ng}_{k}(M)_{x}\\rightarrow J_{k}(M)_{x}$ where we assume $r\\geq 1,$ $k\\geq 0.$\nWe define the vector bundle $\\overset{[k,r]}{\\wedge }(M)\\doteq \\cup _{x\\in M}%\n\\overset{[k,r]}{\\wedge }(M)_{x}\\rightarrow M.$ If $\\overset{[k,r]}{\\omega }$ \n$\\in \\overset{\\lbrack k,r]}{\\wedge }(M)$ $(=\\Gamma \\overset{\\lbrack k,r]}{%\n\\wedge }(M))$ and $\\overset{(k)}{X}_{1},....,\\overset{(k)}{X}_{r}\\in \\frak{g}%\n_{k}(M),$ then $\\overset{[k,r]}{\\omega }(\\overset{(k)}{X}_{1},....,\\overset{%\n(k)}{X}_{r})\\in J_{k}(M)$ is defined by $(\\overset{[k,r]}{\\omega }(\\overset{%\n(k)}{X}_{1},....,\\overset{(k)}{X}_{r}))(x)\\doteq \\overset{\\lbrack k,r]}{%\n\\omega }(x)(\\overset{(k)}{X}_{1}(x),....,$ $\\overset{(k)}{X}_{r}(x))\\in\nJ_{k}(M)_{x}.$ We define $d\\overset{[k,r]}{\\omega }$ by the standard\nformula: $(d\\overset{[k,r]}{\\omega })(\\overset{(k)}{X}_{1},....,\\overset{(k)%\n}{X}_{r+1})=$\n\n\\begin{eqnarray}\n&&\\frac{1}{r+1}\\sum_{1\\leq i\\leq n+1}(-1)^{i+1}\\overset{(k)}{X}_{i}\\overset{%\n[k,r]}{\\omega }(\\overset{(k)}{X}_{1},....,\\overset{(i)}{\\parallel },....,%\n\\overset{(k)}{X}_{r}) \\\\\n&&+\\frac{1}{r+1}\\sum_{i\\leq j+1}(-1)^{i+j}\\overset{[k,r]}{\\omega }([\\overset{%\n(k)}{X}_{i},\\overset{(k)}{X}_{j}],...,\\overset{(i)}{\\parallel },...,\\overset{%\n(j)}{\\parallel },...,\\overset{(k)}{X}_{r+1}) \\notag\n\\end{eqnarray}\n\nWe also define $\\overset{[k,0]}{\\wedge }(M)\\doteq J_{k}(M)$ and $d:\\overset{%\n[k,0]}{\\wedge }(M)\\rightarrow \\overset{\\lbrack k,1]}{\\wedge }(M)$ by $(d%\n\\overset{(k)}{f})\\overset{(k)}{X}\\doteq \\overset{(k)}{X}\\overset{(k)}{f}.$\nWe have $d:\\overset{[k,r]}{\\wedge }(M)\\rightarrow \\overset{\\lbrack k,r+1]}{%\n\\wedge }(M)$: this follows from (6) or can be checked directly as in [9],\npg. 489, using $\\overset{(k)}{f}\\bullet \\overset{(k)}{g}=\\overset{(0)}{f}%\n\\overset{(k)}{g}$ if $\\overset{(k)}{f}\\in C^{\\infty }(M).$ In view of the\nJacobi identity and the alternating character of $\\overset{[k,r]}{\\omega }$,\nthe standard computation shows $d^{2}=0.$ Thus we obtain the complex\n\n\\begin{equation}\n\\overset{\\lbrack k,0]}{\\wedge }(M)\\longrightarrow \\overset{\\lbrack k,1]}{%\n\\wedge }(M)\\longrightarrow \\overset{\\lbrack k,2]}{\\wedge }(M)\\longrightarrow\n.....\\longrightarrow \\overset{\\lbrack k,n]}{\\wedge }(M)\\qquad k\\geq 0\n\\end{equation}\n\nFor $k=0,$ (9) gives de Rham complex.\n\nWe now assume $r\\geq 1$ and define the subspace $\\overset{(k,r)}{\\wedge }%\n(M)_{x}\\subset \\overset{\\lbrack k,r]}{\\wedge }(M)_{x}$ by the following\ncondition: $\\overset{[k,r]}{\\omega }_{x}\\in \\overset{\\lbrack k,r]}{\\wedge }%\n(M)_{x}$ belongs to $\\overset{(k,r)}{\\wedge }(M)_{x}$ iff $\\pi _{k,m}%\n\\overset{[k,r]}{\\omega }(\\overset{(k)}{X}_{1},$\n\n$....,$ $\\overset{(k)}{X}_{r})(x)\\in J_{m}(M)_{x}$ depends on $\\overset{(m)}{%\nX}_{1}(x),....,$ $\\overset{(m)}{X}_{r}(x),$ $m\\leq k.$ This condition holds\nvacuously for $k=0.$ Thus we obtain the projection $\\pi _{k,m}:\\overset{(k,r)%\n}{\\wedge }(M)_{x}\\rightarrow \\overset{(m,r)}{\\wedge }(M)_{x}.$ We define the\nvector bundle $\\overset{(k,r)}{\\wedge }(M)\\doteq \\cup _{x\\in M}\\overset{(k,r)%\n}{\\wedge }(M)_{x}\\rightarrow M$ and set $\\overset{(k,0)}{\\wedge }=\\overset{%\n[k,0]}{\\wedge }=J_{k}(M).$\n\n\\begin{definition}\nAn exterior $(k,r)$-form $\\overset{(k,r)}{\\omega }$ on $M$ is a smooth\nsection of the vector bundle $\\overset{(k,r)}{\\wedge }(M)\\rightarrow M.$\n\\end{definition}\n\nAn explicit description of $\\overset{(k,r)}{\\omega }$ in local coordinates\nis not without interest but will not be done here. Applying $\\pi _{k,m}$ to\n(8), we deduce $d:\\overset{(k,r)}{\\wedge }(M)\\rightarrow \\overset{(k,r+1)}{%\n\\wedge }(M)$ and the commutative diagram\n\n\\begin{equation}\n\\begin{array}{ccc}\n\\overset{(k+1,r)}{\\wedge } & \\overset{d}{\\longrightarrow } & \\overset{%\n(k+1,r+1)}{\\wedge } \\\\ \n\\downarrow _{\\pi } & & \\downarrow _{\\pi } \\\\ \n\\overset{(k,r)}{\\wedge } & \\overset{d}{\\longrightarrow } & \\overset{(k,r+1)}{%\n\\wedge }\n\\end{array}\n\\end{equation}\n\nThus we obtain the array\n\n\\begin{equation}\n\\begin{array}{ccccccccc}\n\\overset{(\\infty ,0)}{\\wedge } & \\longrightarrow & \\overset{(\\infty ,1)}{%\n\\wedge } & \\longrightarrow & \\overset{(\\infty ,2)}{\\wedge } & \\longrightarrow\n& ..... & \\longrightarrow & \\overset{(\\infty ,n)}{\\wedge } \\\\ \n..... & & ..... & & ..... & & ..... & & ..... \\\\ \n\\downarrow & & \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow\n\\\\ \n\\overset{(2,0)}{\\wedge } & \\longrightarrow & \\overset{(2,1)}{\\wedge } & \n\\longrightarrow & \\overset{(2,2)}{\\wedge } & \\longrightarrow & ..... & \n\\longrightarrow & \\overset{(2,n)}{\\wedge } \\\\ \n\\downarrow & & \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow\n\\\\ \n\\overset{(1,0)}{\\wedge } & \\longrightarrow & \\overset{(1,1)}{\\wedge } & \n\\longrightarrow & \\overset{(1,2)}{\\wedge } & \\longrightarrow & ..... & \n\\longrightarrow & \\overset{(1,n)}{\\wedge } \\\\ \n\\downarrow & & \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow\n\\\\ \n\\overset{(0,0)}{\\wedge } & \\longrightarrow & \\overset{(0,1)}{\\wedge } & \n\\longrightarrow & \\overset{(0,2)}{\\wedge } & \\longrightarrow & ..... & \n\\longrightarrow & \\overset{(0,n)}{\\wedge }\n\\end{array}\n\\end{equation}\n\nwhere all horizontal maps are given by $d$ and all vertical maps are\nprojections. The top sequence is defined algebraically by taking projective\nlimits in each coloumn.\n\nUsing $\\bullet $, we also define the wedge product of $(k,r)$-forms in each\nrow by the standard formula which turns $\\overset{(k,\\ast )}{\\wedge }%\n(M)\\doteq \\oplus _{0\\leq r\\leq n}\\overset{(k,r)}{\\wedge }(M)$ into an\nalgebra. This algebra structure descends to the cohomology. In particular,\nwe obtain an algebra structure on the cohomology of the top row of (11),\nwhich we will denote by $H^{\\ast }(\\frak{g}_{\\infty }(M),J_{\\infty }(M))$.\n\nNow let $C^{k+1,r}(M)\\doteq $ $Kernel$ $(\\pi _{\\infty ,k}:\\overset{(\\infty\n,r)}{\\wedge }(M)\\rightarrow \\overset{(k,r)}{\\wedge }(M))$ and $\\mathcal{C}%\n^{k+1,\\ast }(M)\\doteq \\oplus _{0\\leq r\\leq n}C^{k+1,r}(M),$ $k\\geq 0.$ This\ngives the array\n\n\\begin{equation}\n\\begin{array}{ccccccccc}\n\\overset{(\\infty ,0)}{\\wedge }(M) & \\longrightarrow & \\overset{(\\infty ,1)}{%\n\\wedge }(M) & \\longrightarrow & \\overset{(\\infty ,2)}{\\wedge }(M) & \n\\longrightarrow & ..... & \\longrightarrow & \\overset{(\\infty ,n)}{\\wedge }(M)\n\\\\ \n\\uparrow & & \\uparrow & & \\uparrow & & \\uparrow & & \\uparrow \\\\ \nC^{1,0}(M) & \\longrightarrow & C^{1,1}(M) & \\longrightarrow & C^{1,2}(M) & \n\\longrightarrow & ..... & \\longrightarrow & C^{1,n}(M) \\\\ \n\\uparrow & & \\uparrow & & \\uparrow & & \\uparrow & & \\uparrow \\\\ \nC^{2,0}(M) & \\longrightarrow & C^{2,1}(M) & \\longrightarrow & C^{2,2}(M) & \n\\longrightarrow & .... & \\longrightarrow & C^{2,n}(M) \\\\ \n\\uparrow & & \\uparrow & & \\uparrow & & \\uparrow & & \\uparrow \\\\ \n..... & & ..... & & ..... & & ..... & & ..... \\\\ \nC^{\\infty ,0}(M) & \\longrightarrow & C^{\\infty ,1}(M) & \\longrightarrow & \nC^{\\infty ,2}(M) & \\longrightarrow & ..... & \\longrightarrow & C^{\\infty\n,n}(M)\n\\end{array}\n\\end{equation}\n\nwhere all horizontal maps in (12) are restrictions of $d$ and all vertical\nmaps are inclusions. The filtration in (12) preserves wedge product. In\nfact, we have $C^{k,r}(M)\\wedge $ $C^{s,t}(M)\\subset C^{k+s,r+t}(M)$ which\nfollows easily from the definition of $\\bullet .$ We will denote the\nspectral sequence of algebras determined by the filtration in (12) by $%\n\\mathcal{U}_{M}$ and call $\\mathcal{U}_{M}$ the universal spectral sequence\nof $M.$\n\nThe above construction will be relevant in the next section. However, we\nremark here that (11) contains no information other than $H_{deR}^{\\ast }(M,%\n\\mathbb{R)}.$ To see this, we observe that if $\\overset{(k)}{X}\\overset{(k)}{%\nf}=0$ for all $\\overset{(k)}{X}\\in \\frak{g}_{k}(M))$, then $\\overset{(k)}{f}%\n\\in \\mathbb{R}\\subset C^{\\infty }(M).$ Thus the kernel of the first\noperators in the horizontal rows of (11) define the constant sheaf $\\mathbb{R%\n}$ on $M.$ Since $\\overset{(k,r)}{\\wedge }(M)$ is a module over $C^{\\infty\n}(M)$, each row of (10) (which is easily shown to be locally exact) is a\nsoft resolution of the constant sheaf $\\mathbb{R}$ and thus computes $%\nH_{deR}^{\\ast }(M,\\mathbb{R)}.$\n\n\\section{Homogeneous geometries}\n\n\\begin{definition}\nA homogeneous geometry on a differentiable manifold $M$ is a diagram\n\n\\begin{equation}\n\\begin{array}{ccccccccc}\n..... & \\longrightarrow & \\mathcal{G}_{2}(M) & \\longrightarrow & \\mathcal{G%\n}_{1}(M) & \\longrightarrow & M\\times M & \\longrightarrow & 1 \\\\ \n\\uparrow & & \\uparrow & & \\uparrow & & \\parallel & & \\\\ \n..... & \\longrightarrow & \\mathcal{S}_{2}(M) & \\longrightarrow & \\mathcal{S%\n}_{1}(M) & \\longrightarrow & M\\times M & \\longrightarrow & 1\n\\end{array}\n\\end{equation}\n\nwhere $i)$ $\\mathcal{S}_{k}(M)$ is a $TLEFF$ for all $k\\geq 1$ and therefore \n$\\mathcal{S}_{k}(M)$ $\\subset \\mathcal{G}_{k}(M)$ and the vertical maps are\ninclusions $ii)$ The horizontal maps in the bottom sequence of (13) are\nrestrictions of the projection maps in the top sequence and are surjective\nmorphisms.\n\\end{definition}\n\nWith an abuse of notation, we will denote (13) by $\\mathcal{S}_{\\infty }(M)$\nand call $\\mathcal{S}_{\\infty }(M)$ a homogeneous geometry on $M.$ We thus\nimagine that the lower sequence of (13) ``converges'' to some (pseudo)group $%\nG\\subset \\widetilde{Diff(M)}$ which acts transitively on $M.$ However, $G$\nmay be far from being a Lie group and it may be intractible to deal with $G$\ndirectly. The idea of Definition 2 is to work with the arrows of $G$ rather\nthan to work with $G$ itself.\n\n\\begin{definition}\nLet $\\mathcal{S}_{\\infty }(M)$ be a homogeneous geometry on $M.$ $\\mathcal{S}%\n_{\\infty }(M)$ \\ is called a Klein geometry if there exists an integer $%\nm\\geqslant 1$ with the following property: If $(f_{m})_{y}^{x}\\in \\mathcal{S}%\n_{m}(M)_{y}^{x}$, then there exists a unique local diffeomorphism $g$ with $%\ng(x)=y$ satisfying $i)$ $j_{k}(g)_{y}^{x}=(f_{m})_{y}^{x}$ \\ $ii)$ $%\nj_{k}(g)_{g(z)}^{z}\\in \\mathcal{S}_{m}(M)_{g(z)}^{z}$ for all $z$ near $x.$\nThe smallest such integer (uniquely determined if $M$ is connected) is\ncalled the order of the Klein geometry.\n\\end{definition}\n\nIn short, a Klein geometry is a transitive pseudogroup whose local\ndiffeomorphisms are uniquely determined by any of their $m$-arrows and we\nrequire $m$ to be the smallest such integer. Once we have a Klein geometry $%\n\\mathcal{S}_{\\infty }(M)$ of order $m,$ then all $\\mathcal{S}_{k}(M),$ $%\nk\\geqslant m+1$ are uniquely determined by $\\mathcal{S}_{m}(M)$ and $%\n\\mathcal{S}_{k+1}(M)\\rightarrow \\mathcal{S}_{k}(M)$ is an isomorphism for\nall $k\\geqslant m,$ i.e., the Klein geometry $\\mathcal{S}_{\\infty }(M)$\nprolongs in a unique way to a homogeneous geometry and all the information\nis contained in terms up to order $m.$ We will thus denote a Klein geometry\nby $\\mathcal{S}_{(m)}(M).$ We will take a closer look at these geometries in\nthe next section. For instance, let $M^{2n}$ be a complex manifold. We\ndefine $\\mathcal{S}_{k}(M)_{y}^{x}$ by the following condition: $%\n(f_{k})_{y}^{x}\\in \\mathcal{G}_{k}(M)_{y}^{x}$ belongs to $\\mathcal{S}%\n_{k}(M)_{y}^{x}$ if there exists a local holomorphic diffeomorphism $g$ with \n$g(x)=y$ and $j^{k}(g)_{y}^{x}=(f_{k})_{y}^{x}.$ We see that $TLEFF$ $%\n\\mathcal{S}_{k}(M)$ defined by $\\mathcal{S}_{k}(M)\\doteq \\cup _{x,y\\in M}%\n\\mathcal{S}_{k}(M)_{y}^{x}$ satisfies conditions of Definition 3 and\ntherefore a complex structure determines a homogeneous geometry which is not\nnecessarily a Klein geometry. Similarly, a symplectic or contact structure\ndetermines a homogeneous geometry (since these structures have no local\ninvariants) which need not be Klein. More generally, any transitive\npseudogroup determines a homogeneous geometry via its arrows.\n\nNow given a homogeneous geometry $\\mathcal{S}_{\\infty }(M)$, we will sketch\nthe construction of its infinitesimal geometry $\\frak{s}_{\\infty }(M)$\nreferring to [25], [26] for further details. Let $x\\in M$ and $X$ be a\nvector field defined near $x.$ Let $f_{t}$ be the 1-parameter group of local\ndiffeomorphisms generated by $X.$ Suppose $X$ has the property that $%\nj_{k}(f_{t})_{y_{t}}^{x}$ belongs to $\\mathcal{S}_{k}(M)_{y_{t}}^{x}$ for\nall small $t$ with $t\\geqslant 0$ where $y_{t}=f_{t}(x)$. This is actually a\ncondition only on the $k$-jet of $X$ at $x.$ In this way we define the\nsubspace $\\frak{s}_{k}(M)_{x}\\subset \\frak{g}_{k}(M)_{x}$ which consists of\nthose $\\overset{(k)}{X}(x)$ satisfying this condition. We define the vector\nsubbundle $\\frak{s}_{k}(M)\\doteq \\cup _{x\\in M}\\frak{s}_{k}(M)_{x}%\n\\rightarrow M$ of $\\frak{g}_{k}(M)\\rightarrow M$ and the bracket (2) on $%\n\\frak{g}_{k}(M)$ restricts to a bracket on $\\frak{s}_{k}(M)$ and $\\frak{s}%\n_{k}(M)$ is the $TLEIF$ determined by $\\mathcal{S}_{k}(M).$\n\nIn this way we arrive at the diagram\n\n\\begin{equation}\n\\begin{array}{ccccccccccc}\n.... & \\longrightarrow & \\frak{g}_{k}(M) & \\longrightarrow & .... & \n\\longrightarrow & \\frak{g}_{1}(M) & \\longrightarrow & \\frak{g}_{0}(M) & \n\\longrightarrow & 0 \\\\ \n& & \\uparrow & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\ \n.... & \\longrightarrow & \\frak{s}_{k}(M) & \\longrightarrow & .... & \n\\longrightarrow & \\frak{s}_{1}(M) & \\longrightarrow & \\frak{s}_{0}(M) & \n\\longrightarrow & 0\n\\end{array}\n\\end{equation}\n\nwhere the bottom horizontal maps are restrictions of the upper horizontal\nmaps and are surjective morphisms. The vertical maps are injective morphisms\ninduced by inclusion. Thus (14) is the infinitesimal analog of (13). We will\ndenote (14) by $\\frak{s}_{\\infty }(M)$ and call $\\frak{s}_{\\infty }(M)$ the\ninfinitesimal geometry of $\\mathcal{S}_{\\infty }(M).$\n\nNow we define $L_{\\overset{(k)}{Y}}:\\overset{(k,r)}{\\wedge }\\rightarrow \n\\overset{(k,r)}{\\wedge }$ and $i_{\\overset{(k)}{Y}}:\\overset{(k,r+1)}{\\wedge \n}\\rightarrow \\overset{(k,r)}{\\wedge }$ by the standard formulas: $(L_{%\n\\overset{(k)}{Y}}\\overset{(k,r)}{\\omega })(\\overset{(k)}{X}_{1},....,%\n\\overset{(k)}{X}_{r})\\doteq \\overset{(k)}{Y}(\\overset{(k,r)}{\\omega }(%\n\\overset{(k)}{X}_{1},....,\\overset{(k)}{X}_{r}))-\\overset{(k,r)}{\\omega }([%\n\\overset{(k)}{Y},\\overset{(k)}{X}_{1}],\\overset{(k)}{X}_{2}....,$\n\n$\\overset{(k)}{X}_{r})-....-\\overset{(k,r)}{\\omega }([\\overset{(k)}{X}_{1},%\n\\overset{(k)}{X}_{2}....,[\\overset{(k)}{Y},\\overset{(k)}{X}_{r}])$ and $(i_{%\n\\overset{(k)}{Y}}\\overset{(k+1,r)}{\\omega })(\\overset{(k)}{X}_{1},....,%\n\\overset{(k)}{X}_{r})\\doteq (r+1)$ $\\ \\overset{(k+1,r)}{\\omega }(\\overset{(k)%\n}{Y},\\overset{(k)}{X}_{1},....,\\overset{(k)}{X}_{r}).$ We also define $(\\pi\n_{k,m}L_{\\overset{(k)}{Y}}):\\overset{(m,r)}{\\wedge }\\rightarrow \\overset{%\n(m,r)}{\\wedge }$ by $\\pi _{k,m}L_{\\overset{(k)}{Y}}\\doteq L_{\\pi _{k,m}%\n\\overset{(k)}{Y}}$ and similarly $\\pi _{k,m}i_{\\overset{(k)}{Y}}=i_{\\pi\n_{k,m}\\overset{(k)}{Y}}.$ With these definitions we obtain the well known\nformulas\n\n\\begin{equation}\nL_{\\overset{(k)}{X}}=d\\circ i_{\\overset{(k)}{X}}+i_{\\overset{(k)}{X}}\\circ d\n\\end{equation}\n\n\\begin{equation}\nL_{\\overset{(k)}{X}}\\circ d=d\\circ L_{\\overset{(k)}{X}}\n\\end{equation}\n\nWe will now indicate briefly how a homogeneous geometry $\\mathcal{S}_{\\infty\n}(M)$ gives rise to various spectral sequences.\n\n$1)$ We define $\\overset{(k,r)}{\\frak{s}}(M)\\subset \\overset{(k,r)}{\\wedge }%\n(M)$ by the following condition: Some $\\overset{(k,r)}{\\omega }$ belongs to $%\n\\overset{(k,r)}{\\frak{s}}(M)$ if $\\quad i)$ $L_{\\overset{(k)}{X}}\\overset{%\n(k,r)}{\\omega }=0,$ $\\overset{(k)}{X}\\in \\frak{s}_{k}(M)$ $\\quad ii)$ $i_{%\n\\overset{(k)}{X}}\\overset{(k,r)}{\\omega }=0,$ $\\overset{(k)}{X}\\in \\frak{s}%\n_{k}(M).$ (15) and (16) show that $d:\\overset{(k,r)}{\\frak{s}}(M)\\rightarrow \n\\overset{(k,r+1)}{\\frak{s}}(M)$ and (10) holds. Thus we arrive at (11) and\n(12). Recall that if $\\frak{g}$ is any Lie algebra with a representation on $%\nV$ and $\\frak{s}\\subset \\frak{g}$ is a subalgebra, then we can define the\nrelative Lie algebra cohomology groups $H^{\\ast }(\\frak{g},\\frak{s},V)$ ([6])%\n$.$ Since our construction is modelled on the definition of $H^{\\ast }(\\frak{%\ng},\\frak{s},V),$ we will denote the cohomology of the top row of (11) by $%\nH^{\\ast }(\\frak{g}_{\\infty }(M),\\frak{s}_{\\infty }(M),J_{\\infty }(M))$ in\nthis case.\n\n$2)$ We will first make\n\n\\begin{definition}\n$\\Theta (\\mathcal{S}_{k}(M))\\doteq \\{\\overset{(k)}{f}\\in J_{k}(M)\\mid \n\\overset{(k)}{X}\\overset{(k)}{f}=0$ for all $\\overset{(k)}{X}\\in \\frak{s}%\n_{k}(M)\\}$\n\\end{definition}\n\n(6) shows that $\\Theta (\\mathcal{S}_{k}(M))$ is a subalgebra of $J_{k}(M).$\n\\ We will call $\\Theta (\\mathcal{S}_{k}(M))$ the $k^{th}$ order structure\nalgebra of the homogeneous geometry $\\mathcal{S}_{\\infty }(M).$ Note that $%\n\\Theta (\\mathcal{G}_{k}(M))=\\mathbb{R}.$ For $k\\geq 1,$ we define $\\overset{%\n[k,r]}{\\frak{s}}(M)_{x}$ as the space of alternating maps $\\frak{s}%\n_{k}(M)_{x}\\times ....\\times \\frak{s}_{k}(M)_{x}\\rightarrow J_{k}(M)_{x}$\nand define $\\overset{[k,r]}{\\frak{s}}(M)$ as in Section 2. We have the\nrestriction map $\\theta _{(k,r)}:\\overset{[k,r]}{\\wedge }(M)\\rightarrow \n\\overset{\\lbrack k,r]}{\\frak{s}}(M)$ whose kernel will be denoted by $%\n\\overset{[k,r]}{\\wedge }_{\\frak{s}}(M).$ Since $\\Theta (\\mathcal{S}%\n_{k}(M))=Ker(\\theta _{(k,1)}\\circ d),$ we obtain the commutative diagram\n\n\\begin{equation}\n\\begin{array}{ccccccccccc}\n& & 0 & & 0 & & 0 & & .... & & 0 \\\\ \n& & \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow & & \n\\downarrow \\\\ \n& & \\Theta (\\mathcal{S}_{k}(M)) & \\longrightarrow & \\overset{[k,1]}{\\wedge }%\n_{\\frak{s}}(M) & \\longrightarrow & \\overset{[k,2]}{\\wedge }_{\\frak{s}}(M) & \n\\longrightarrow & .... & \\longrightarrow & \\overset{[k,n]}{\\wedge }_{\\frak{s}%\n}(M) \\\\ \n& & \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow & & \n\\downarrow \\\\ \n\\mathbb{R} & \\longrightarrow & \\overset{[k,0]}{\\wedge }(M) & \\longrightarrow\n& \\overset{[k,1]}{\\wedge }(M) & \\longrightarrow & \\overset{[k,2]}{\\wedge }(M)\n& \\longrightarrow & .... & \\longrightarrow & \\overset{[k,2]}{\\wedge }(M) \\\\ \n& & \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow & & \n\\downarrow \\\\ \n0 & \\longrightarrow & \\frac{\\overset{[k,0]}{\\wedge }(M)}{\\Theta (\\mathcal{S}%\n_{k}(M))} & \\longrightarrow & \\overset{[k,1]}{\\frak{s}}(M) & \\longrightarrow\n& \\overset{[k,2]}{\\frak{s}}(M) & \\longrightarrow & .... & \\longrightarrow & \n\\overset{[k,n]}{\\frak{s}}(M) \\\\ \n& & \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow & & \n\\downarrow \\\\ \n& & 0 & & 0 & & 0 & & .... & & 0\n\\end{array}\n\\end{equation}\n\nWe will call (17) the horizontal crossection of the representation triple $(%\n\\mathcal{G}_{\\infty }(M),$\n\n$\\mathcal{S}_{\\infty }(M),J_{\\infty }(M))$ at order $k$. Passing to the long\nexact sequence in (17), we see that local cohomology of the top and bottom\nsequences coincide (with a shift in order) in view of the local exactness of\nthe middle row. Now defining $\\overset{(k,r)}{\\frak{s}}(M)$ as in Section 2,\n(17) defines an exact sequence of three spectral sequences where the middle\nspectral sequence is (12). Note that local exactness of the top and bottom\nsequences would imply that their cohomologies coincide with the sheaf\ncohomology groups $H^{\\ast }(M,\\Theta (\\mathcal{S}_{k}(M))$ and $H^{\\ast }(M,%\n\\frac{\\overset{[k,0]}{\\wedge }(M)}{\\Theta (\\mathcal{S}_{k}(M))})$\nrespectively since partition of unity applies to the spaces in these\nsequences. We will denote the limiting cohomology of the top and bottom\nsequences in (17) respectively by $H^{\\ast }(\\frak{s}_{\\infty }(M),0)$ and $%\nH^{\\ast }(\\frak{s}_{\\infty }(M),J_{\\infty }(M)).$ The reader may compare\n(17) to the diagram on page 183 in [25] which relates Spencer sequence to\nJanet sequence, called $\\mathcal{P}$-sequence in [25].\n\nBefore we end this section, note that $(f_{k+1})_{y}^{x}\\in \\mathcal{S}%\n_{k}(M)_{y}^{x}$ defines an isomorphism $(f_{k+1})_{y}^{x}:J_{k}(M)_{x}%\n\\rightarrow J_{k}(M)_{y}\\ $(in fact, $(f_{k})_{y}^{x}$ does it) and also an\nisomorphism $(f_{k+1})_{y}^{x}:\\frak{s}_{k}(M)_{x}\\rightarrow \\frak{s}%\n_{k}(M)_{y}.$ Let us assume that $\\mathcal{S}_{\\infty }(M)$ is defined by\nsome $G\\subset Diff(M)$ as in the case of a symplectic structure,\nhomogeneous complex structure or a Klein geometry (see Section 4). Now $G$\nacts on $\\overset{(k,r)}{\\frak{s}}(M)$ (defined as in 1) or 2) above) by $(g%\n\\overset{(k,r)}{\\omega })(\\overset{(k)}{X}_{1},....,\\overset{(k)}{X}%\n_{r})(p)\\doteq j_{k+1}(g)_{p}^{q}(\\overset{(k,r)}{\\omega }%\n(q)(j_{k+1}(g^{-1})_{q}^{p}\\overset{(k)}{X}%\n_{1}(p),....,j_{k+1}(g^{-1})_{q}^{p}\\overset{(k)}{X}_{r}(p))$ where $g(q)=p.$\nThe cochains which are invariant under this action form a subcomplex whose\ncohomology can be localized, i.e., can be computed at any point of $M.$ If $%\n\\mathcal{S}_{\\infty }(M)=\\mathcal{G}_{\\infty }(M)$ and $G=Diff(M)$, then an\ninvariant form must vanish but this need not be the case for a homogeneous\ngeometry. We will not go into the precise description of this cohomology\nhere though it is quite relevant for Klein geometries in Section 4 and can\nbe expressed in terms of some relative Lie algebra cohomology groups in this\ncase.\n\n\\section{Klein geometries}\n\nLet $G$ be a Lie group and $H$ a closed subgroup. $G$ acts on the left coset\nspace $G\/H$ on the left. Let $o$ denote the coset of $H.$ Now $H$ fixes $o$\nand therefore acts on the tangent space $T(G\/H)_{o}$ at $o$. However some\nelements of $H$ may act as identity on $T(G\/H)_{o}.$ The action of $h\\in H$\n(we regard $h$ as a transformation and use the same notation) on $T(G\/H)_{o}$\ndepends only on $1$-arrow of $h$ with source and target at $o.$ Let $%\nH_{1}\\subset H$ be the normal subgroup of $H$ consisting of elements which\nact as identity on $T(G\/H)_{o}.$ To recover $H_{1},$ we consider $1$-jets of\nvector fields at $o$ which we will denote by $J_{1}T(G(H)_{o}.$ The action\nof $h$ on $J_{1}T(G(H)_{o}$ depends only on $2$-arrow of $h.$ Now some\nelements $h\\in H_{1}$ may still act as identity at $J_{1}T(G(H)_{o}$ and we\ndefine the normal subgroup $H_{2}\\subset $ $H_{1}$ consisting of those\nelements. Iterating this procedure, we obtain a decreasing sequence of\nnormal subgroups $\\{1\\}\\subset ...\\subset H_{k}\\subset H_{k-1}\\subset\n.....\\subset H_{2}\\subset H_{1}\\subset H_{0}=H$ which stabilizes at some\ngroup $N$ which is the largest normal subgroup of $G$ contained in $H$ $($%\nsee [29], pg. 161). We will call the smallest integer $m$ satisfying $%\nN=H_{m} $ the order of the Klein pair $(G,H)$. In this case, it is easy to\nshow that $g\\in G$ is uniquely determined modulo $N$ by any of its $m$%\n-arrows. $G$ acts effectively on $G\/H$ iff $N=\\{1\\}.$ If $(G,H)$ is a Klein\npair of order $m,$ then so is $(G\/N,H\/N)$ which is further effective. We\nwill call $N$ the ghost of the Klein pair $(G,H)$ since it can not be\ndetected from the action and therefore \\textit{may} have implications that\nfall outside the scope of \\textit{Erlangen Programm. }We will touch this\nissue again in $5)$ of Section 9. Thus we see that an \\textit{effective }%\nKlein pair $(G,H)$ of order $m$ determines a Klein geometry $\\mathcal{S}%\n_{(m)}(G\/H)$ according to Definition 3 where the local diffeomorphisms\nrequired by Definition 3 are restrictions of global diffeomorphisms of $G\/H$\nwhich are induced by the elements of $G.$ Conversely, let $\\mathcal{S}%\n_{(m)}(M)$ be a Klein geometry according to Definition 3 and let $\\widetilde{%\nM}$ be the universal covering space of $M.$ We pull back the pseudogroup on $%\nM$ to a pseudogroup on $\\widetilde{M}$ using the local diffeomorphism $\\pi :%\n\\widetilde{M}\\rightarrow M.$ Using simple connectedness of $\\widetilde{M}$\nand a mild technical assumption which guarantees that the domains of the\nlocal diffeomorphisms do not become arbitrarily small, the standard\nmonodromy argument shows that a local diffeomorphism defined on $\\widetilde{M%\n}$ in this way uniquely extends to some global diffeomorphism on $\\widetilde{%\nM}.$ This construction is essentially the same as the one given in [30] on\npage 139-146. The global diffeomorphisms on $\\widetilde{M}$ obtained in this\nway form a Lie group $G.$ If $H\\subset G$ is the stabilizer of some $p\\in \n\\widetilde{M},$ then $H$ is isomorphic to $\\mathcal{S}_{(m)}(M)_{q}^{q}$\nwhere $\\pi (p)=q.$ To summarize, a Klein geometry $\\mathcal{S}_{(m)}(M)$\naccording to Definition 3 becomes an effective Klein pair $(G,H)$ of order $%\nm $ when pulled back to $\\widetilde{M}$. Conversely, an effective Klein pair \n$(G,H)$ of order $m$ defines a Klein geometry $\\mathcal{S}_{(m)}(M)$ if we\nmode out by the action of a discrete subgroup. Keeping this relation in\nmind, we will consider an effective Klein pair $(G,H)$ as our main object\nbelow.\n\nNow the above filtration of normal subgroups gives the diagram\n\n\\begin{equation}\n\\begin{array}{ccccccccccc}\n\\mathcal{G}_{m}(G\/H)_{o}^{o} & \\longrightarrow & \\mathcal{G}%\n_{m-1}(G\/H)_{o}^{o} & \\longrightarrow & .... & \\longrightarrow & \\mathcal{G%\n}_{1}(G\/H)_{o}^{o} & \\longrightarrow & 1 & & \\\\ \n\\uparrow & & \\uparrow & & \\uparrow & & \\uparrow & & & & \\\\ \nH & \\longrightarrow & H\/H_{m} & \\longrightarrow & ... & \\longrightarrow & \nH\/H_{1} & \\longrightarrow & 1 & & \n\\end{array}\n\\end{equation}\n\nwhere the spaces in the top sequence are jet groups in our universal\nsituation in Section 2 and the vertical maps are injections. Since the\nkernels in the upper sequence are vector groups, we see that $H_{1}$ is\nsolvable. As before, we now define $\\mathcal{S}_{m}(M)_{y}^{x}$ which\nconsists of $m$-arrows of elements of $G$ and define $\\mathcal{S}%\n_{m}(M)\\doteq \\cup _{x,y\\in M}\\mathcal{S}_{m}(M)_{y}^{x}.$ As in the case $%\nDiff(M)\\times M\\rightarrow M$ in Section 2, we obtain the map $G\\times\nG\/H\\rightarrow \\mathcal{S}_{m}(M)$ defined by $(g,x)\\rightarrow $ $m$-arrow\nof $g$ starting at $x$ and ending at $g(x).$ This map is surjective by\ndefinition and this time also injective by the definition of $m.$ Thus we\nobtain a concrete realization of $\\mathcal{S}_{m}(M)$ as $G\\times G\/H.$ Note\nthat $\\mathcal{S}_{m}(M)_{o}^{o}=H.$ Going downwards in the filtration, we\nobtain the commutative diagram\n\n\\begin{equation}\n\\begin{array}{ccc}\nG\\times G\/H & \\longrightarrow & \\mathcal{S}_{m}(M) \\\\ \n\\downarrow & & \\downarrow \\\\ \nG\/H_{m-1}\\times G\/H & \\longrightarrow & \\mathcal{S}_{m-1}(M) \\\\ \n\\downarrow & & \\downarrow \\\\ \n.... & \\longrightarrow & ... \\\\ \n\\downarrow & & \\downarrow \\\\ \nG\/H_{2}\\times G\/H & \\longrightarrow & \\mathcal{S}_{2}(M) \\\\ \n\\downarrow & & \\downarrow \\\\ \nG\/H_{1}\\times G\/H & \\longrightarrow & \\mathcal{S}_{1}(M)\n\\end{array}\n\\end{equation}\n\nFor instance, the bottom map in (19) is defined by $\\{xH_{1}\\}\\times\n\\{yH\\}\\rightarrow 1$-arrow of the diffeomorphism $\\{xH_{1}\\}$ starting at\nthe coset $\\{yH\\}$ and ending at the coset $\\{xyH\\}.$ Note that this is not\na group action since $G\/H_{1}$ is not a group but the composition and\ninversion of $1$-arrows are well defined. This map is a bijection by the\ndefinition of $H_{1}.$ Fixing one such $1$-arrow$,$ all other $1$-arrows\nstarting at $\\{yH\\}$ are generated by composing this arrow with elements of $%\n\\mathcal{S}_{1}(M)_{y}^{y}=I_{xy^{-1}}H\/I_{xy^{-1}}H_{1}$ where $I_{xy^{-1}}$\nis the inner automorphism of $G$ determined by $xy^{-1}\\in G.$ The vertical\nprojections on the right coloumn of (19) are induced by projection of jets\nas in Sections 2, 3 and the projections on the left coloumn are induced by\nprojections on the first factor and identity map on the second factor.\n\nA Lie group $G$ is clearly an effective Klein pair $(G,\\{1\\})$ with order $%\nm=0.$ For many Klein geometries we have $m=1$. This is the case, for\ninstance, if $H$ is compact, in which case we have an invariant metric, $H$\nis discrete or $(G,H)$ is a reductive pair which is extensively studied in\nliterature from the point of view of principal bundles ([12]). If $G$ is\nsemisimple, it is not difficult to show that the order of $(G,H)$ is at most\ntwo (see [18], pg. 131). For instance, let $M$ be a homogeneous complex\nmanifold, i.e., $Aut(M)$ acts transitively on $M.$ If $M$ is compact, then $%\nM=G\/H$ for some complex Lie group $G$ and a closed complex subgroup $H$\n([34]). If futher $\\pi _{1}(M)$ is finite, then $G\/H=\\overline{G}\/\\overline{H%\n}$ as complex manifolds for some semisimple Lie group $\\overline{G}$ ([34]).\nThus it follows that jets of order greater than two do not play any role in\nthe complex structure of $M$ in this case. If $G$ is reductive, it is stated\nin [35] that the order of $(G,H)$ is at most three. On the other hand, for\nany positive integer $m$, an effective Klein pair $(G;H)$ of order $m$ is\nconstructed in [1] such that $G\/H$ is open and dense in some weighted\nprojective space. Other examples of Klein pairs of arbitrary order are\ncommunicated to us by the author of [35]. However, we do not know the answer\nto the following question\n\n$\\mathbf{Q1:}$ For some positive integer $m,$ does there exist a Klein pair $%\n(G,H)$ of order $m$ such that $G\/H$ compact?\n\nIt is crucial to observe $\\ $that $(G_{1},H_{1})$ and $(G_{2},H_{2})$ may be\ntwo Klein pairs with different orders with $G_{1}\/H_{1}$ homeomorphic to $%\nG_{2}\/H_{2}.$ For instance, let $H\\subset G$ be complex Lie groups with $\\pi\n_{0}(G)=\\pi _{1}(G\/H)=0$ and $G\/H$ compact. If $M$ $\\subset G$ is a maximal\ncompact subgroup, then $(M,M\\cap H)$ is a Klein pair of order one and $%\nG\/H=M\/M\\cap H$ as topological manifolds (in fact, $G\/H$ is Kaehler iff its\nEuler characteristic is nonzero, see [34], [4]). The crucial fact here is\nthat an abundance of Lie groups may act transitively on a homogeneous space $%\nM$ with different orders and the topology (but not the analytic structure)\nof $M$ is determined by actions of order one only (the knowledge of a\nparticular such action suffices, see Theorem VI in [34] where a detailed\ndescription of complex homogeneous spaces is given. It turns out that ``they\nare many more than we expect'' as stated there).\n\nNow let $(G,H)$ be an effective Klein pair of order $m$ and let $\\mathcal{L}%\n(G)$ be the Lie algebra of $G.$ We have the map $\\sigma :\\mathcal{L}%\n(G)\\rightarrow J_{m}(T(G\/H))_{p}$ defined by $X\\rightarrow j_{m}(X^{\\ast\n})_{p}$ where $X^{\\ast }$ is the vector field on $G\/H$ induced by $X$ and $%\np\\in G\/H.$ $\\sigma $ is a homomorphism of Lie algebras where the bracket on $%\nJ_{m}(T(G\/H))_{o}$ is the algebraic bracket defined in Section 2. Note that\nthe map $X\\rightarrow X^{\\ast }$ is injective due to effectiveness and also\nsurjective due to transitivity. It is now easy to give a description of the\ninfinitesimal analog of (19). We can thus express everything defined in\nSections 2, 3 in concrete terms which will enable us to use the highly\ndeveloped structure theory of (semisimple) Lie groups ([17]). A detailed\ndescription of $\\frak{s}_{m}(M)$ is given in [26] (Theorem 15 on pg. 199).\nThe formula on pg. 200 in [26] is the same as the formula (15), Example\n3.3.7, pg. 104 in the recent book [20], but higher order jets and Spencer\noperator remain hidden in (15) and also in (4), Example 3.2.9, pg. 98 in\n[20].\n\nThe following situation deserves special mention: Let $G$ be a complex Lie\ngroup, $H$ a closed complex subgroup with a holomorphic representation on\nthe vector space $V$ and let $E\\rightarrow G\/H$ be the associated\nhomogeneous vector bundle of $G\\rightarrow G\/H.$ We now have the sheaf\ncohomology groups $H^{\\ast }(G\/H,\\mathbf{E})$ where $\\mathbf{E}$ denotes the\nsheaf of holomorphic sections of $E\\rightarrow G\/H.$ Borel-Weil theorem is\nderived in [4] from $H^{0}(G\/H,\\mathbf{E)}$. If the Klein pair $(G,H)$ has\norder $m$ and is effective, the principal bundle $G\\rightarrow G\/H$ can be\nidentified with the principal bundle $\\mathcal{S}_{m}(M)^{(o)}\\rightarrow\nG\/H $ where $\\mathcal{S}_{m}(M)^{(o)}$ consists of $m$-arrows in $G\/H$ with\nsource at the coset $o$ of $H$ (see Section 7). If $gx=y$, $g\\in G,$ $x,y\\in\nG\/H$, then the $m$-arrow of $g$ gives an isomorphism $E_{x}\\rightarrow E_{y}$\nbetween the fibers. Consequently, the action of $G$ on sections of $%\nE\\rightarrow G\/H$ is equivalent to the representation of the $TLEFF$ $%\n\\mathcal{S}_{m}(M)=G\\times G\/H$ on $E\\rightarrow G\/H.$ On the infinitesimal\nlevel, this gives a representation of $\\frak{s}_{m}(M)$ on $E\\rightarrow G\/H$\nand we can also define the cohomology groups $H^{\\ast }(\\frak{s}_{m}(M),E)$\nas in [19], [20]. Letting $\\Theta $ denote the sections of $E$ killed by $%\n\\frak{s}_{m}(M),$ we see that these two cohomology groups are related by a\ndiagram similar to (17). It is crucial to observe here how the order of jets\nremains hidden in $H^{\\ast }(G\/H,\\mathbf{E})$.\n\n\\section{Truncated geometries}\n\n\\begin{definition}\nSome $TLEFF$ $\\mathcal{S}_{k}(M)$ is called a truncated geometry on $M$ of\norder $k.$\n\\end{definition}\n\nWe will view a truncated geometry $\\mathcal{S}_{k}(M)$ as a diagram (13)\nwhere all $\\mathcal{S}_{m}(M)$ for $m\\leq k$ are defined as projections of $%\n\\mathcal{S}_{k}(M)$ and $\\mathcal{S}_{m}(M)$ for $m\\geqslant k+1$ do not\nexist. A homogeneous geometry defines a truncated geometry of any order. The\nquestion arises whether some truncated geometry always prolongs uniquely to\nsome homogeneous geometry. The answer turns out to be negative. For\ninstance, let $(M,g)$ be a Riemannian manifold and consider all $1$-arrows\non $M$ which preserve the metric $g.$ Such $1$-arrows define a $TLEFF$ $%\n\\mathcal{S}_{1}(M).$ We may fix some point $p\\in M$ and fix some coordinates\naround $p$ once and for all so that $g_{ij}(p)=\\delta _{ij}$, thus\nidentifying $\\mathcal{S}_{1}(M)_{p}^{p}$ with the orthogonal group $O(n).$\nNow any $1$-arrow with source at $p$ defines an orthogonal frame at its\ntarget $q$ by mapping the fixed orthogonal coordinate frame at $p$ to $q.$\nThe group $O(n)$ acts on all such $1$-arrows by composing on the source. Now\nforgetting $1$-arrows but keeping the orthogonal frames defined by them, we\nrecover the orthogonal frame bundle of the metric $g$. However we will not\nadapt this point of view. In view of the existence of geodesic coordinates,\nwe can now construct $2$-arrows on $M$ which preserve $1$-jet of $g$ , i.e., \n$1$-jet of $g$ at all $x\\in M$ can be identified (in various ways). Thus we\nobtain $\\mathcal{S}_{2}(M)$ and the projection $\\pi _{2,1}:\\mathcal{S}%\n_{2}(M)\\rightarrow \\mathcal{S}_{1}(M).$ As a remarkable fact, $\\pi _{2,1}$\nturns out to be an isomorphism. This fact is equivalent to the well known\nGauss trick of shifting the indices and showing the uniqueness of a metric\nconnection which is symmetric (Levi-Civita connection). The Christoffel\nsymbols are obtained now by twisting the $2$-arrows of $\\mathcal{S}_{2}(M)$\nby the $1$-arrows of $\\mathcal{S}_{1}(M)$. Now we may not be able to\nidentify $2$-jet of $g$ over $M$ due to curvature of $g$ and thus we may\nfail to construct the surjection $\\pi _{3,2}:\\mathcal{S}_{3}(M)\\rightarrow \n\\mathcal{S}_{2}(M).$ If we achive this (and $\\pi _{3,2}$ will be again an\nisomorphism), the next obstruction comes from the $3$-jet of $g$ which is\nessentially the covariant derivative of the curvature. However, if $g$ has\nconstant curvature, then we can prolong $\\mathcal{S}_{2}(M)$ uniquely to a\nhomogeneous geometry $\\mathcal{S}_{\\infty }(M)$ which, as a remarkable fact,\nturns out to be a Klein geometry of order one since in this case $%\nLim_{k\\rightarrow \\infty }S_{k}(M)$ recaptures the isometry group of $(M,g)$\nwhich acts transitively on $M$ and any isometry is uniquely determined by\nany of its $1$-arrows$.$ Thus we may view a truncated geometry $\\mathcal{S}%\n_{k}(M)$ as a candidate for some homogeneous geometry $\\mathcal{S}_{\\infty\n}(M)$ but $\\mathcal{S}_{k}(M)$ must overcome the obstructions, if any, put\nforward by $M.$ Almost all geometric structures (Riemannian, almost complex,\nalmost symplectic, almost quaternionic, ..) may be viewed as truncated\ngeometries of order at least one, each being a potential canditate for a\nhomogeneous geometry.\n\n\\begin{definition}\nA truncated geometry $\\mathcal{S}_{k}(M)$ is called formally integrable if\nit prolongs to a homogeneous geometry.\n\\end{definition}\n\nHowever, we require the prolongation required by Definition 6 to be\nintrinsically determined by $\\mathcal{S}_{k}(M)$ in some sense and not be\ncompletely arbitrary. Given some $\\mathcal{S}_{k}(M),$ note that Definition\n6 requires the surjectivity of $\\mathcal{S}_{j+1}(M)\\rightarrow \\mathcal{S}%\n_{j}(M),$ $j\\geqslant k.$ For instance, we may construct some $\\mathcal{S}%\n_{k+1}(M)$ in an intrinsic way without $\\mathcal{S}_{k+1}(M)\\rightarrow \n\\mathcal{S}_{k}(M)$ being surjective. We may now redifine all lower terms by \n$\\widetilde{\\mathcal{S}_{j}(M)}=\\pi _{k+1,j}\\mathcal{S}_{k+1}(M)$ and start\nanew at order $k+1.$ This is not allowed by Definition 6. For instance, let $%\n\\mathcal{S}_{1}(M)$ be defined by some almost symplectic form $\\omega $ (or\nalmost complex structure $J).$ Then $\\pi _{2,1}:\\mathcal{S}%\n_{2}(M)\\rightarrow \\mathcal{S}_{1}(M)$ will be surjective if $d\\omega =0$ $($%\nor $N(J)=0$ where $N(J)$ is the Nijenhuis tensor of $J$).\n\nThis prolongation process which we tried to sketch above is centered around\nthe concept of formal integrability which can be defined in full generality\nturning the ambigious Definition 6 into a precise one. However, this\nfundamental concept turns out to be highly technical and is fully developed\nby D.C. Spencer and his co-workers from the point of view of PDEs,\nculminating in [11]. More geometric aspects of this concept are emphasized\nin [25], [26] and other books by this author .\n\n\\section{Bundle maps}\n\nIn this section we will briefly indicate the allowable bundle maps in the\npresent framework. Consider the universal $TLEFF$ $\\mathcal{G}_{k}(M)$ of\norder $k.$ We define the group bundle $\\mathcal{AG}_{k}(M)\\doteq \\cup _{x\\in\nM}\\mathcal{G}_{k}(M)_{x}^{x}$. The sections of this bundle form a group with\nthe operation defined fiberwise. We will call such a section a universal\nbundle map (or a universal gauge transformation) of order $k.$ We will\ndenote the group of universal bundle maps by $\\Gamma \\mathcal{AG}_{k}(M).$\nWe obtain the projection $\\pi _{k+1,k}:\\Gamma \\mathcal{AG}%\n_{k+1}(M)\\rightarrow \\Gamma \\mathcal{AG}_{k}(M)$ which is a homomorphism. If \n$\\mathcal{S}_{k}(M)\\subset \\mathcal{G}_{k}(M),$ we similarly define $%\n\\mathcal{AS}_{k}(M)\\subset \\mathcal{AG}_{k}(M)$ and call elements of $\\Gamma \n\\mathcal{AS}_{k}(M)$ automorphisms (or gauge transformations) of $\\mathcal{S}%\n_{k}(M)$. Now let $\\mathcal{S}_{k}(M)\\subset \\mathcal{G}_{k}(M$ and $%\ng_{k}\\in \\Gamma \\mathcal{AG}_{k}(M).$ We will denote $g_{k}(x)$ by $%\n(g_{k})_{x}^{x},$ $x\\in M.$ We define the $TLEFF$ $(Adg)\\mathcal{S}_{k}(M)$\nby defining its $k$-arrows as $(Adg)\\mathcal{S}_{k}(M)_{y}^{x}\\doteq\n\\{(g_{k})_{y}^{y}(f_{k})_{y}^{x}(g_{k}^{-1})_{x}^{x}\\mid (f_{k})_{y}^{x}\\in \n\\mathcal{S}_{k}(M)_{y}^{x}\\}.$\n\n\\begin{definition}\n$\\mathcal{S}_{k}(M)$ is called equivalent to $\\mathcal{H}_{k}(M)$ if there\nexists some $g\\in \\Gamma \\mathcal{AG}_{k}(M)$ satisfying $(Adg)\\mathcal{S}%\n_{k}(M)=\\mathcal{H}_{k}(M)$\n\\end{definition}\n\nA necessary condition for the equivalence of $\\mathcal{S}_{k}(M)$ and $%\n\\mathcal{H}_{k}(M)$ is that $\\mathcal{S}_{k}(M)_{x}^{x}$ and $\\mathcal{H}%\n_{k}(M)_{x}^{x}$ be conjugate in $\\mathcal{G}_{k}(M)_{x}^{x}$, i.e., they\nmust be compatible structures (like both Riemannian,...). Let $\\frak{s}%\n_{k}(M)$ and $\\frak{h}_{k}(M)$ be the corresponding $TLEIF^{\\prime }$s. The\nabove action induces an action of $g$ on $\\frak{s}_{k}(M)$ which we will\ndenote also by $(Adg)\\frak{s}_{k}(M).$ This latter action uses $1$-jet of $g$\nsince the geometric order of $\\frak{s}_{k}(M)$ is $k+1,$ i.e., the\ntransformation rule of the elements (sections) of $\\frak{s}_{k}(M)$ uses\nderivatives up to order $k+1.$ This construction is functorial. In\nparticular, $(Adg)\\mathcal{S}_{k}(M)=\\mathcal{H}_{k}(M)$ implies $(Adg)\\frak{%\ns}_{k}(M)=\\frak{h}_{k}(M)$. Clearly these actions commute with projections.\nIn this way we define the moduli spaces of geometries.\n\nSome $g_{k}\\in \\Gamma \\mathcal{AG}_{k}(M)$ acts on any $k^{th}$ order object\ndefined in Section 2. However, the action of $g_{k}$ does not commute with $%\nd $ and therefore $g_{k}$ does not act on the horizontal complexes in (10).\nTo do this, we define $\\mathcal{AG}_{\\infty }(M)\\doteq \\cup _{x\\in M}%\n\\mathcal{G}_{\\infty }(M)_{x}^{x}$ where an element $(g_{\\infty })_{x}^{x}$\nof $\\mathcal{G}_{\\infty }(M)_{x}^{x}$ is the $\\infty $-jet of some local\ndiffeomorphism with source and target at $x.$ As we noted above, $(g_{\\infty\n})_{x}^{x}$ is far from being a formal object: it determines this\ndiffeomorphism modulo the $\\infty $-jet of identity diffeomorphism. Now some \n$g_{\\infty }\\in \\Gamma \\mathcal{AG}_{\\infty }(M)$ does act on the horizontal\ncomplexes in (10).\n\n\\section{Principal bundles}\n\nLet $\\mathcal{S}_{k}(M)\\subset \\mathcal{G}_{k}(M).$ We fix some $p\\in M$ and\ndefine $\\mathcal{S}_{k}(M)^{(p)}\\doteq \\cup _{x\\in M}\\mathcal{S}%\n_{k}(M)_{x}^{p}.$ The group $\\mathcal{S}_{k}(M)_{p}^{p}$ acts on $\\mathcal{S}%\n_{k}(M)_{x}^{p}$ by composing with $k$-arrows of $\\mathcal{S}_{k}(M)_{x}^{p}$\nat the source as $(f_{k})_{x}^{p}\\rightarrow (f_{k})_{x}^{p}(h_{k})_{p}^{p}$\nand the projection $\\mathcal{S}_{k}(M)^{(p)}\\rightarrow M$ with fiber $%\n\\mathcal{S}_{k}(M)_{x}^{p}$ over $x$ is a principal bundle with group $%\n\\mathcal{S}_{k}(M)_{p}^{p}.$ Considering the adjoint action of $\\mathcal{S}%\n_{k}(M)_{p}^{p}$ on itself, we construct the associated bundle whose\nsections are automorphisms (or gauge transformations) which we use in gauge\ntheory. This associated bundle can be identified with $\\mathcal{AS}_{k}(M)$\nin Section 6 and therefore the two concepts of automorphisms coincide. In\ngauge theory, we let $h_{k}\\in \\Gamma \\mathcal{AS}_{k}(M)$ act on $\\mathcal{S%\n}_{k}(M)^{(p)}$ on the target as $(f_{k})_{x}^{p}\\rightarrow\n(h_{k})_{x}^{x}(f_{k})_{x}^{p}$ reserving the source for the group $\\mathcal{%\nS}_{k}(M)_{p}^{p}.$ We will denote this transformation by $f\\rightarrow\nh\\odot f.$ We can regard the object $h_{k}\\odot \\mathcal{S}_{1}(g)^{(p)}$ as\nanother principal $\\mathcal{S}_{k}(M)_{p}^{p}$-bundle: we imagine two copies\nof $\\mathcal{S}_{k}(g)_{p}^{p}$, one belonging to principal bundle and one\noutside which is the group of the principal bundle and $h_{k}$ acts only on\nthe principal bundle without changing the group. To be consistent with $%\n\\odot ,$ we now regard $h_{k}\\in \\Gamma \\mathcal{AG}_{k}(M)$ as a general\nbundle map and define the transform of $\\mathcal{S}_{k}(M)^{(p)}$ by $h_{k}$\nusing $\\odot .$ Now $\\odot $ has a drawback from geometric point of view. To\nsee this, let $(M,g)$ be a Riemannian manifold. We will denote the $TLEFF$\ndetermined by $g$ by $\\mathcal{S}_{1}(g),$ identifying the principal $%\n\\mathcal{S}_{1}(g)^{(p)}\\rightarrow M$ with the orthogonal frame bundle of $%\ng $ and the group $\\mathcal{S}_{1}(g)_{p}^{p}$ with $O(n)$ as in Section 5.\nNow the transformed object $h\\odot \\mathcal{S}_{1}(g)^{(p)}$, which is\nanother $O(n)$-principal bundle, is not related to any metric in sight\nunless $h$ = identity !! Thus we see that $\\odot $ dispenses with the\nconcept of a metric but keeps the concept of an $O(n)$-principal bundle,\ncarrying us from our geometric envelope outside into the topological world\nof general principal bundles. On the other hand, the action of $(\\mathcal{G}%\n_{1})_{x}^{x}$ on metrics at $x$ gives an action of $h$ on metrics on $M$\nwhich we will denote by $g\\rightarrow h\\boxdot g.$ Changing our notation $%\n(Adh)\\mathcal{S}_{1}(g)$ defined in Section 6 to $h\\boxdot \\mathcal{S}%\n_{1}(g) $ (using the same notation $\\boxdot $), we see that $h\\boxdot \n\\mathcal{S}_{1}(g)=\\mathcal{S}_{1}(h\\boxdot g).$ Thus $\\boxdot $ preserves\nboth metrics and also $1$-arrows determined by them.\n\nConsider the naive inclusion\n\n\\begin{equation}\n\\text{differential geometry}\\subset \\text{topology}\n\\end{equation}\n\nIf we drop the word differential in (20), we may adapt the point of view\nthat the opposite inclusion holds now in (20). This is the point of view of\nA. Grothendieck who views geometry as the study of \\textit{form}, which\ncontains topology as a special branch (see his Promenade \\#12, translated by\nRoy Lisker). This broad perspective is clearly a far reaching generalization\nof \\textit{Erlangen Programm }presented here.\n\nIn view of (20), we believe that any theorem in the framework whose main\ningredients we attempted to outline here, however deep and far reaching, can\nbe formulated and proved also using principal bundles. To summarize, we may\nsay that differential geometry is the study \\textit{smooth forms }and the\nconcepts of a \\textit{form }and continuous deformation of \\textit{forms }%\ncome afterwards as higher level of abstractions. We feel that it may be\nfruitful to start with differential geometry rather than starting at a\nhigher level and then specializing to it.\n\n\\section{Connection and curvature}\n\nRecall that a right principal $G$-bundle $P\\rightarrow M$ determines the\ngroupoid $\\frac{P\\times P}{G}\\rightarrow M\\times M$ where the action of $G$\non $P\\times P$ is given by $(x,y)g\\doteq (xg,yg).$ Let $\\mathcal{A}%\n(P)\\rightarrow M$ be the automorphism bundle obtained as the associated\nbundle of $P\\rightarrow M$ using the adjoint action of $G$ on itself as in\nSection 7, whose sections are gauge transformations. We obtain in this way\nthe groupoid extension\n\n\\begin{equation}\n1\\longrightarrow \\mathcal{A}(P)\\longrightarrow \\frac{P\\times P}{G}%\n\\longrightarrow M\\times M\\longrightarrow 1\n\\end{equation}\n\nwhere we again use the first and last arrows to indicate injectivity and\nsurjectivity without any algebraic meaning. On the infinitesimal level, (21)\ngives the Atiyah sequence of $P\\rightarrow M$\n\n\\begin{equation}\n0\\longrightarrow \\mathcal{LA}(P)\\longrightarrow \\frac{TP}{G}\\overset{\\pi }{%\n\\longrightarrow }TM\\longrightarrow 0\n\\end{equation}\n\n(see [19], [20] for the details of the Atiyah sequence) where $\\mathcal{LA}%\n(P)\\longrightarrow M$ is the Lie algebra bundle obtained as the associated\nbundle using the adjoint action of $G$ on its Lie algebra $\\mathcal{L}(G).$\nConnection forms $\\omega $ on $P\\rightarrow M$ are in 1-1 correspondence\nwith transversals in (22), i.e., vector bundle maps $\\omega :TM\\rightarrow \n\\frac{TP}{G}$ with $\\pi \\circ \\omega =id$ and curvature $\\kappa $ of $\\omega \n$ is defined by $\\kappa (X,Y)=\\omega \\lbrack X,Y]-[\\omega X,\\omega Y].$ The\nextension (22) splits iff (22) admits a flat transversal. Thus Atiyah\nsequence completely recovers the formalism of connection and curvature on $%\nP\\rightarrow M$ in the framework of algebroid extensions as long as we work\nover a fixed base manifold $M$.\n\nNow let $\\mathcal{S}_{\\infty }(M)$ be a homogeneous geometry with\ninfinitesimal geometry $\\frak{s}_{\\infty }(M).$ The groupoid $\\frac{\\mathcal{%\nS}_{k}(M)^{(p)}\\times \\mathcal{S}_{k}(M)^{(p)}}{\\mathcal{S}_{k}(M)_{p}^{p}}$\nin (21) determined by the principal bundle $\\mathcal{S}_{k}(M)^{(p)}%\n\\rightarrow M$ as defined in Section 7 can be identified with $\\mathcal{S}%\n_{k}(M)$ and the algebroid $\\frac{T\\mathcal{S}_{k}(M)^{(p)}}{\\mathcal{S}%\n_{k}(M)_{p}^{p}}$ in (22) can be identified with $\\frak{s}_{k}(M).$ We have $%\n\\mathcal{A}(\\mathcal{S}_{k}(M)^{(p)})=\\cup _{x\\in M}\\mathcal{S}%\n_{k}(M)_{x}^{x}$ as already indicated in Section 7 and $\\mathcal{LA}(%\n\\mathcal{S}_{k}(M)^{(p)})$ $\\doteq \\cup _{x\\in M}\\mathcal{L}(\\mathcal{S}%\n_{k}(M)_{x}^{x})$ where the bracket of sections is defined fiberwise. Thus\n(21) becomes\n\n\\begin{equation}\n1\\longrightarrow \\mathcal{AS}_{k}(M)\\longrightarrow \\mathcal{S}%\n_{k}(M)\\longrightarrow M\\times M\\longrightarrow 1\n\\end{equation}\n\nand the Atiyah sequence (22) is now\n\n\\begin{equation}\n0\\longrightarrow \\mathcal{LAS}_{k}(M)\\longrightarrow \\frak{s}%\n_{k}(M)\\longrightarrow TM\\longrightarrow 0\n\\end{equation}\n\nIt is easy to check exactness of (24) in local coordinates using (2).\n\nOur purpose is now to indicate how the present framework captures\ninformation peculiar to jets by changing the base manifold, which Atiyah\nsequence does not detect. To see this, let $m\\leq k+1$ and consider the Lie\ngroup extension at $x\\in M:$\n\n\\begin{equation}\n1\\longrightarrow \\mathcal{S}_{k,m}(M)_{x}^{x}\\longrightarrow \\mathcal{S}%\n_{k}(M)_{x}^{x}\\longrightarrow \\mathcal{S}_{m}(M)_{x}^{x}\\longrightarrow 1\n\\end{equation}\n\nwhere the kernel $\\mathcal{S}_{k,m}(M)_{x}^{x}$ is nilpotent if $m\\geq 1$\nand is abelian if $k=m+1\\geq 2.$ Consider the $\\mathcal{S}_{k,m}(M)_{p}^{p}$%\n-principal bundle $\\mathcal{S}_{k}(M)^{(p)}\\rightarrow \\mathcal{S}%\n_{m}(M)^{(p)}.$ If $\\mathcal{S}_{k,m}(M)_{p}^{p}$ is contractible (this is\nthe case in many examples for $m\\geq 1)$, this principal bundle is trivial\nand its Atiyah sequence admits flat transversals. Thus nothing is gained by\nconsidering higher order jets and all information is contained in the Atiyah\nsequence of $\\mathcal{S}_{1}(M)^{(p)}\\rightarrow M.$ On the other hand, we\nhave the following extension of $TLEIF^{\\prime }s:$%\n\\begin{equation}\n0\\longrightarrow \\mathcal{LS}_{k,m}(M)\\longrightarrow \\frak{s}%\n_{k}(M)\\longrightarrow \\frak{s}_{m}(M)\\longrightarrow 0\n\\end{equation}\n\nwhere $\\mathcal{LS}_{k,m}(M)\\doteq \\cup _{x\\in M}\\mathcal{L}(\\mathcal{S}%\n_{k,m}(M)_{x}^{x}).$ Using (2), it is easy to check that the existence of a\nflat transversal in (26) a priori forces the splitting of the Lie algebra\nextension\n\n\\begin{equation}\n0\\longrightarrow \\mathcal{L}(\\mathcal{S}_{k,m}(M)_{x}^{x})\\longrightarrow \n\\mathcal{L}(\\mathcal{S}_{k}(M)_{x}^{x})\\longrightarrow \\mathcal{L}(\\mathcal{S%\n}_{m}(M)_{x}^{x})\\longrightarrow 0\n\\end{equation}\n\nfor all $x\\in M$ where (27) is (25) in infinitesimal form. However (25) and\n(27) do not split in general. For instance, (27) does not split even in the\nuniversal situation $\\mathcal{S}_{\\infty }(M)=\\mathcal{G}_{\\infty }(M)$ when \n$m=2$ and $k=3.$ In fact, the dimensions of the extension groups $H^{2}(%\n\\mathcal{L}(\\mathcal{S}_{m}(M)_{x}^{x}),\\mathcal{S}_{m+1,m}(M)_{x}^{x})$ are\ncomputed in [28] for all $m$ when $\\dim M=1.$\n\nThus we see that the theory of principal bundles, which is essentially\ntopological, concentrates on the maximal compact subgroup of the structure\ngroup of the principal bundle as it is this group which produces nontrivial\ncharacteristic classes as invariants of equivalence classes of principal\nbundles modulo bundle maps. Consequently, this theory concentrates on the\ncontractibility of the kernel in (25) whereas it is the types of the\nextensions in (25), (27) which emerge as the new ingredient in the present\nframework.\n\nThe connections considered above are transversals and involve only $%\nTLEIF^{\\prime }s$ (algebroids). There is another notion of connection based\non Maurer-Cartan form, which is incorporated by the nonlinear Spencer\nsequence (see Theorem 31 on page 224 in [25]), where the passage from $TLEFF$\n(groupoid) to its $TLEIF$ (algebroid) is used in a crucial way. The passage\nfrom extensions of $TLEFF^{\\prime }s$ (torsionfree connections in finite\nform) to extensions of their $TLEIF^{\\prime }s$ (torsionfree connections in\ninfinitesimal form) relates these two notions by means of a single diagram\nwhich we hope to discuss elsewhere. The approach to parabolic geometries\nadapted in [5] is a complicated mixture of these two notions whose\nintricacies, we believe, will be clearly depicted by this diagram.\n\nHowever we view a connection, the main point here seems to be that it\nbelongs to the group rather than to the space on which the group acts. Since\nthere is an abundance of groups acting transitively on some given space, it\nseems meaningles to speak of the curvature of some space unless we specify\nthe group. However, it turns out that the knowledge of the $k$-arrows of\nsome ideal group is sufficient to define a connection but this connection\nwill not be unique except in some special cases.\n\n\\section{Some remarks}\n\nIn this section (unfortunately somewhat long) we would like to make some\nremarks on the relations between the present framework and some other\nframeworks.\n\n$1)$ Let $\\mathcal{E}\\rightarrow M$ be a differentiable fibered space. It\nturns out that we have an exterior calculus on $J^{\\infty }(\\mathcal{E}%\n)\\rightarrow M.$ Decomposing exterior derivative and forms into their\nhorizontal and vertical components, we obtain a spectral sequence, called\nVinogradov $\\mathcal{C}$-spectral sequence, which is fundamental in the\nstudy of calculus of variations (see [32], [33] and the references therein).\nThe limit term of $\\mathcal{C}$-spectral sequence is $H_{deR}^{\\ast\n}(J^{\\infty }(\\mathcal{E})).$ In particular, if $\\mathcal{E}=T(M),$ we\nobtain $H_{deR}^{\\ast }(\\frak{g}_{\\infty }(M)).$ The $\\mathcal{C}$-spectral\nsequence can be defined also with coefficients ([21]). On the other hand, we\ndefined $H^{\\ast }(\\frak{g}_{\\infty }(M),J_{\\infty }(M))$ and $H^{\\ast }(%\n\\frak{g}_{\\infty }(M),\\frak{s}_{\\infty }(M),J_{\\infty }(M))$ in Sections 2,\n3. As we indicated in Section 2, we can consider representations of $\\frak{g}%\n_{\\infty }(M)$ other than $J_{\\infty }(M)$ (for instance, see $2)$ below).\nThese facts hint, we feel, at the existence of a very general Van Est type\ntheorem which relates these cohomology groups.\n\n$2)$ Recall that (5) and (7) define a representation of the algebroid $\\frak{%\ng}_{k}(M)$ on the vector bundle $J_{k}(M)\\rightarrow M$. Thus we can\nconsider the cohomology groups $H^{\\ast }(\\frak{g}_{k}(M),J_{k}(M)$ as\ndefined in [19], [20] (see also the references there for original sources)\nand $H^{\\ast }(\\frak{g}_{k}(M),J_{k}(M))$ coincides with the cohomology of\nthe bottom sequence of (17) by definition. Now $\\frak{g}_{k}(M)$ has other\n``intrinsic'' representations, for instance $\\frak{g}_{k-1}(M).$ Lemma 8.32\nand the formula on page 383 in [25] (which looks very similar to (4)) define\nthis representation. In particular, the cohomology groups $H^{\\ast }(\\frak{g}%\n_{k}(M),$ $\\frak{g}_{k-1}(M))$ are defined and are given by the bottom\nsequence of (17) for this case. Using this representation of $\\frak{g}%\n_{k}(M) $ on $\\frak{g}_{k-1}(M)$, deformations of $TLEIF^{\\prime }$s are\nstudied in [25] in detail using Janet sequence (Chapter 7, Section 8 of\n[25]). Recently, some deformation cohomology groups are introduced in [8] in\nthe general framework of algebroids. However, if the algebroid is a $TLEIF,$\nit seems to us that these cohomology groups coincide with those in [25] (and\ntherefore also with the bottom sequence of (17)) and are not new (see also\n[20], pg. 309 for a similar claim). In view of the last paragraph of Section\n4, the fact that deformation cohomology arises as sheaf cohomology in\nKodaira-Spencer theory and as algebroid cohomology in [25], [8] is no\ncoincidence.\n\n$3)$ Let $\\mathcal{S}_{2}(M)$ be a truncated geometry on $M.$ We fix some $%\nx\\in M$ and consider the following diagram of Lie group extensions\n\n\\begin{equation}\n\\begin{array}{ccccccccc}\n0 & \\longrightarrow & \\mathcal{G}_{2,1}(M)_{x}^{x} & \\longrightarrow & \n\\mathcal{G}_{2}(M)_{x}^{x} & \\longrightarrow & \\mathcal{G}_{1}(M)_{x}^{x} & \n\\longrightarrow & 1 \\\\ \n& & \\uparrow & & \\uparrow & & \\uparrow & & \\\\ \n0 & \\longrightarrow & \\mathcal{S}_{2,1}(M)_{x}^{x} & \\longrightarrow & \n\\mathcal{S}_{2}(M)_{x}^{x} & \\longrightarrow & \\mathcal{S}_{1}(M)_{x}^{x} & \n\\longrightarrow & 1\n\\end{array}\n\\end{equation}\n\nwhere the vertical maps are inclusions. The top row of (28) splits and the\ncomponents of these splittings are naturally interpreted as the Christoffel\nsymbols of symmetric ``point connections''. Some $k_{x}^{x}\\in \\mathcal{G}%\n_{2,1}(M)_{x}^{x}$, which is a particular bundle map as defined in Section 6\nwhen $M=\\{x\\},$ transforms $\\mathcal{S}_{2}(M)_{x}^{x}$ by conjugation but\nacts as identity on $\\mathcal{S}_{2,1}(M)_{x}^{x}$ and $\\mathcal{S}%\n_{1}(M)_{x}^{x}.$ Using $\\mathcal{G}_{2,1}(M)_{x}^{x}$ as allowable\nisomorphisms, we defined in [13] the group of \\textit{restricted }extensions \n$H_{res}^{2}(\\mathcal{S}_{1}(M)_{x}^{x},\\mathcal{S}_{2,1}(M)_{x}^{x})$. This\ngroup vanishes iff the restriction of some splitting of the top row of (28)\nto the bottom row splits also the bottom row or equivalently, iff the bottom\nrow admits a symmetric point connection. The main point is that this group\nis sensitive to phenomena happening only inside the top row of (28) which is\nour universal envelope as in Section 2. Using the Lie algebra anolog of\n(28), we defined also the group $H_{res}^{2}(\\mathcal{L}(\\mathcal{S}%\n_{1}(M)_{x}^{x}),\\mathcal{S}_{2,1}(M)_{x}^{x})$ (note that $\\mathcal{S}%\n_{2,1}(M)_{x}^{x}\\subset \\mathcal{G}_{2,1}(M)_{x}^{x}$ are vector groups)\nobtaining the homomorphism $H_{res}^{2}(\\mathcal{S}_{1}(M)_{x}^{x},\\mathcal{S%\n}_{2,1}(M)_{x}^{x})\\rightarrow H_{res}^{2}(\\mathcal{L}(\\mathcal{S}%\n_{1}(M)_{x}^{x}),\\mathcal{S}_{2,1}(M)_{x}^{x})$ ([13]). On the other hand,\nregarding the bottom row of (28) as an \\textit{arbitrary }Lie group\nextension as in [15] without any reference to our universal envelope, we can\ndefine $H^{2}(\\mathcal{S}_{1}(M)_{x}^{x},\\mathcal{S}_{2,1}(M)_{x}^{x})\\ $\\\nand the homomorphism $H^{2}(\\mathcal{S}_{1}(M)_{x}^{x},\\mathcal{S}%\n_{2,1}(M)_{x}^{x})\\rightarrow H^{2}(\\mathcal{L}(\\mathcal{S}_{1}(M)_{x}^{x}),%\n\\mathcal{S}_{2,1}(M)_{x}^{x})$. Thus we obtain the following commutative\ndiagram\n\n\\begin{equation}\n\\begin{array}{ccc}\nH_{res}^{2}(\\mathcal{S}_{1}(M)_{x}^{x},\\mathcal{S}_{2,1}(M)_{x}^{x}) & \n\\longrightarrow & H_{res}^{2}(\\mathcal{L}(\\mathcal{S}_{1}(M)_{x}^{x}),%\n\\mathcal{S}_{2,1}(M)_{x}^{x}) \\\\ \n\\downarrow & & \\downarrow \\\\ \nH^{2}(\\mathcal{S}_{1}(M)_{x}^{x},\\mathcal{S}_{2,1}(M)_{x}^{x} & \n\\longrightarrow & H^{2}(\\mathcal{L}(\\mathcal{S}_{1}(M)_{x}^{x}),\\mathcal{S}%\n_{2,1}(M)_{x}^{x})\n\\end{array}\n\\end{equation}\n\nwhere the vertical homomorphisms are induced by inclusion. In [13], we gave\nexamples where $H_{res}^{2}(\\mathcal{L}(\\mathcal{S}_{1}(M)_{x}^{x}),\\mathcal{%\nS}_{2,1}(M)_{x}^{x})$ is nontrivial whereas $H^{2}(\\mathcal{L}(\\mathcal{S}%\n_{1}(M)_{x}^{x}),$\n\n$\\mathcal{S}_{2,1}(M)_{x}^{x})$ is trivial. This fact shows that we may\nloose information when we pass from the top row to the bottom row in (29).\n\nNow the action of $\\mathcal{S}_{1}(M)_{x}^{x}$ on $\\mathcal{S}%\n_{2,1}(M)_{x}^{x}$ gives a representation of the $TLEFF$ $\\mathcal{S}_{1}(M)$\non the vector bundle $\\mathcal{S}_{2,1}(M)\\doteq \\cup _{x\\in M}\\mathcal{S}%\n_{2,1}(M)_{x}^{x}$ and thus we can define the cohomology groups $%\nH_{res}^{\\ast }(\\mathcal{S}_{1}(M),\\mathcal{S}_{2,1}(M))$ in such a way that\nthey will respect our universal envelope. In this way we arrive at the\nglobal analog of (29). We believe that we will loose information also in\nthis global case. To summarize, even though the constructions in this paper\ncan be formulated in the general framework of groupoids and algebroids as in\n[7], [8],[19], [20], we believe that this general framework will not be\nsensitive in general to certain phenomena peculiar to jets unless it takes\nthe universal homogeneous envelope into account. In particular, we would\nlike to express here our belief that Lie equations form the geometric core\nof groupoids which are the ultimate generalizations of (pseudo)group actions\nin which we dispense with the action but retain the symmetry that the action\ninduces on the space (compare to the introduction of [36]).\n\n$4)$ Let $\\mathcal{X}$ $(M)$ be the Lie algebra of smooth vector fields on $%\nM.$ Recalling that $j_{k}(\\mathcal{X}(M))\\subset \\frak{g}_{k}(M)$, (5) gives\na representation of $\\mathcal{X}(M)$ on $J_{k}(M)$. Denoting the cochains\ncomputing this cohomology by $\\overset{(k,r)}{\\wedge }_{GF},$ we obtain the\nchain map $\\overset{(k,\\ast )}{\\wedge }\\rightarrow $ $\\overset{(k,\\ast )}{%\n\\wedge }_{GF}$ which indicates that Gelfand-Fuks cohomology is involved in\nthe present framework and plays a central role.\n\n$5)$ This remark can be considered as the continuation of Section 7 and the\nlast paragraph of Section 4.\n\nLet $Q$ be the subgroup of $\\mathcal{G}_{1}(n+1)=GL(n+1,\\mathbb{R)}$\nconsisting of matrices of the form\n\n\\begin{equation}\n\\left[ \n\\begin{array}{cc}\nA & 0 \\\\ \n\\xi & \\lambda\n\\end{array}\n\\right]\n\\end{equation}\nwhere $A$ is an invertible $n\\times n$ matrix, $\\xi =(\\xi _{1},...,\\xi _{n})$\nand $\\lambda \\neq 0.$ We will denote (30) by $(A,\\xi ,\\lambda ).$ We have\nthe homomorphism $Q\/\\lambda I\\rightarrow \\mathcal{G}_{1}(n)$ defined by $%\n(A,\\xi ,1)\\rightarrow A,$ where $\\lambda I$ denotes the subgroup $\\{\\lambda\nI\\mid \\lambda \\in \\mathbb{R\\}},$ with the abelian kernel $K$ consisting of\nelements of the form $(I,\\xi ,1).$ We also have the injective homomorphism $%\nQ\/\\lambda I\\rightarrow \\mathcal{G}_{2}(n)$ defined by $(A_{j}^{i},\\xi\n_{j},1)\\rightarrow (A_{j}^{i},\\xi _{j}A_{k}^{i}+\\xi _{k}A_{j}^{i})$ which\ngives the diagram\n\n\\begin{equation}\n\\begin{array}{ccccccccc}\n0 & \\longrightarrow & \\mathcal{G}_{2,1}(n) & \\longrightarrow & \\mathcal{G}%\n_{2}(n) & \\longrightarrow & \\mathcal{G}_{1}(n) & \\longrightarrow & 1 \\\\ \n& & \\uparrow & & \\uparrow & & \\parallel & & \\\\ \n0 & \\longrightarrow & K & \\longrightarrow & Q\/\\lambda I & \\longrightarrow & \n\\mathcal{G}_{1}(n) & \\longrightarrow & 1\n\\end{array}\n\\end{equation}\n\nNow $(\\mathcal{G}_{1}(n+1),Q)$ is a Klein pair of order two with ghost $N=$ $%\n\\lambda I$. We also have the effective Klein pair $(\\mathcal{G}%\n_{1}(n+1)\/\\lambda I,Q\/\\lambda I)$ which is of order two. Note that the\nstandard action of $\\mathcal{G}_{1}(n+1)$ on $\\mathbb{R}^{n+1}\\backslash 0$\ninduces a transitive action of $\\mathcal{G}_{1}(n+1)$ on the the real\nprojective space $\\mathbb{R}P(n)$ and $Q$ is the stabilizer of the coloumn\nvector $p=$ $(0,0,...,1)^{T}$ so that both Klein pairs $(\\mathcal{G}%\n_{1}(n+1),Q)$ and $(\\mathcal{G}_{1}(n+1)\/\\lambda I,Q\/\\lambda I)$ define the\nsame base $\\mathbb{R}P(n).$ Clearly, $\\mathcal{G}_{1}(n+1)$ and $\\mathcal{G}%\n_{1}(n+1)\/\\lambda I$ induce the same $2$-arrows on $\\mathbb{R}P(n).$\n\nAt this stage, we have two relevant principle bundles.\n\n$i)$ The principle bundle $\\mathcal{G}_{1}(n+1)^{(p)}\\rightarrow \\mathbb{R}%\nP(n)$ which consists of all $2$-arrows eminating from $p$ and has the\nstructure group $Q\/\\lambda I.$ This is the same principle bundle as $(%\n\\mathcal{G}_{1}(n+1)\/\\lambda I)^{(p)}\\rightarrow \\mathbb{R}P(n)$. In\nparticular, (18) reduces to (31) in this case (the reader may refer to\nExample 4.1 on pg. 132 of \\ [18] and also to the diagram on pg.142).\n\n$ii)$ The principle bundle $\\mathcal{G}_{1}(n+1)\\rightarrow \\mathbb{R}P(n)$\nwith structure group $Q.$ Note that we have the central extension\n\n\\begin{equation}\n1\\longrightarrow \\lambda I\\longrightarrow Q\\longrightarrow Q\/\\lambda\nI\\longrightarrow 1\n\\end{equation}\n\nwhich splits: some $q=(A,\\xi ,\\lambda )\\in Q$ factors as $q=ab$ where $%\na=(\\lambda ^{-1}A,\\lambda ^{-1}\\xi ,1)$\n\n$\\in \\mathcal{G}_{2}(n)$ and $b=\\lambda I.$ Note that $ii)$ is obtained from \n$i)$ by lifting the structure group $Q\/\\lambda I$ to $Q$ in (31).\n\nNow we have a representation of $Q$ on $\\mathbb{R}$ defined by the\nhomomorphism $(A,\\xi ,\\lambda )\\rightarrow \\lambda ^{-N}$ for some integer $%\nN\\geq 0.$ Replacing $\\mathbb{R}$ by $\\mathbb{C}$ and working with complex\ngroups and holomorphic actions, it is known that the holomorphic sections of\nthe associated line bundle of $\\mathcal{G}_{1}(n+1,\\mathbb{C})\\rightarrow \n\\mathbb{C}P(n)$ realize all irreducable representations of the unitary group \n$U(n)$ as $N$ varies (see [16], pg. 138-152 and [17]). As a very crucial\nfact, we can repeat this construction by replacing the Klein pair $(\\mathcal{%\nG}_{1}(n+1,\\mathbb{C}),Q)$ of order two by the effective Klein pair $%\n(U(n),U(n-1)\\times U(1))$ of order one and this latter construction recovers\nthe same line bundle (see [16] for details). The following question\ntherefore arises naturally: Let $(G,H)$ be a Klein pair with ghost $N.$ Let $%\n\\rho :H\\rightarrow GL(V)$ be a representation and $E\\rightarrow M=G\/H$ be\nthe associated homogeneous vector bundle of $G\\rightarrow G\/H.$ Can we\nalways find some \\textit{effective} Klein pair $(\\overline{G},\\overline{H})$\n(not necessarily of the same order) with $\\overline{G}\/\\overline{H}=M,$ a\nrepresentation $\\overline{\\rho }:\\overline{H}\\rightarrow GL(V)$ such that $%\nE\\rightarrow M$ is associated with $(\\overline{G})^{(p)}\\rightarrow M$, or\nshortly\n\n$\\mathbf{Q2:}$ Can we always avoid ghosts in Klein geometry?\n\nReplacing $\\lambda I,$ $Q,$ $Q\/\\lambda I$ in (31) respectively by $U(1),$ $%\nSpin^{c}(4),$ $SO(4)$ and recalling the construction of $Spin^{c}$-bundle on\na 4-manifold ([22]), we see that $\\mathbf{Q2}$ is quite relevant as it asks\nessentially the scope and limitations of \\textit{Erlangen Programm.}\n\n$6)$ The assumption of transitivity, i.e., the surjectivity of the right\narrows in (13) , (14), is imposed upon us by \\textit{Erlangen Programm.}\nHowever, many of the constructions in this paper can be carried out without\nthe assumption of transitivity. For instance, foliations give rise to\nintransitive Lie equations but they are studied in the literature mostly\nfrom the point of view of general groupoids and algebroids (see the\nreferences in [7], [8]).\n\nOur main object of study in this paper has been a differentiable manifold $%\nM. $ The sole reason for this is that this author has been obsessed years\nago by the question ``What are Christoffel symbols? '' and he could not\nlearn algebraic geometry from books and he did not have the chance to learn\nit from experts (this last remark applies also to differential geometry) as\nhe has always been at the wrong place at the right time. We feel (and\nsometimes almost see, for instance [27], [3]) that the present framework has\nalso an algebraic counterpart valid for algebraic varieties.\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\textbf{References}\n\n\\bigskip\n\n[1] E.Abado\\u{g}lu, preprint\n\n[2] A.Banyaga, The structure of Classical Diffeomorphism Groups, Kluwer\nAcademic Publishers, Volume 400, 1997\n\n[3] A.Beilinson, V.Ginzburg: Infinitesimal structure of moduli spaces of\nG-bundles, Internat. Math. Res. Notices, no 4, 93-106, 1993\n\n[4] R.Bott: Homogeneous vector bundles, Ann. of Math., Vol. 66, No. 2,\n203-248, 1957\n\n[5] A.Cap, J.Slovak, V.Soucek: Bernstein-Gelfand-Gelfand sequences, Ann. of\nMath. 154, 97-113, 2001\n\n[6] C.Chevalley, S.Eilenberg: Cohomology theory of Lie groups and Lie\nalgebras. Trans. Amer. Math. Soc. 63, (1948). 85--124\n\n[7] M.Crainic: Differentiable and algebroid cohomology, Van Est isomorphism,\nand characteristic classes, Comment. Math. Helv. 78, 2003, 681-721\n\n[8] M.Crainic, I.Moerdijk: Deformations of Lie brackets: cohomological\naspects, arXiv: 0403434\n\n[9] W.T.Van Est: Group cohomology and Lie Algebra cohomology in Lie groups,\nIndagationes Math., 15, 484-492, 1953\n\n[10] D.B.Fuks: Cohomology of infinite-dimensional Lie algebras. Translated\nfrom the Russian by A.B. Sosinski, Contemporary Soviet Mathematics,\nConsultants Bureau, New York, 1986\n\n[11] H.Goldschmidt, D.C.Spencer: On the nonlinear cohomology of Lie\nequations, I, II, III, IV, Acta Math. 136, 103-170, 171-239, 1976, J.\nDifferential Geometry, 13, 409-453, 455-526, 1979\n\n[12] W.Greub, S.Halperin, R.Vanstone: Connections, Curvature and Cohomology,\nVol.III, Academic Press, New York San Fransisco London, 1976\n\n[13] B.G\\\"{u}rel, E.Orta\\c{c}gil, F.\\\"{O}zt\\\"{u}rk: Group extensions in\nsecond order jet group, preprint\n\n[14] T.Hawkins: Emergence of the Theory of Lie Groups, An Essay in the\nHistory of Mathematics 1869-1926, Sources and Studies in the History of\nMathematics and Physical Sciences, 2000 Springer-Verlag New York, Inc.\n\n[15] G.Hochschild: Group extensions of Lie groups. Ann. of Math. (2) 54,\n(1951). 96--109\n\n[16] A.W.Knapp: Lie Groups, Lie Algebras and Cohomology, Mathematical Notes\n34, Princeton University Press, NJ, 1988\n\n[17] A.W.Knapp: Representation Theory of Semisimple Groups: An Overview\nbased on Examples, Mathematical Notes, Princeton University Press,\nPrinceton, NJ, 1986\n\n[18] S.Kobayashi: Transformation Groups in Differential Geometry, Ergebnisse\nder Mathematic und ihrer Grenzgebiete. Band 70, Springer-Verlag, 1972\n\n[19] K.Mackenzie: Lie Groupoids and Lie Algebroids in Differential Geometry,\nLondon Mathematical Society Lecture Note Series, 124, Cambridge University\nPress, Cambridge, 1987\n\n[20] K.Mackenzie: General Theory of Lie Groupoids and Lie Algebroids, London\nMathematical Society Lecture Note Series, 213, Cambridge University Press,\nCambridge, 2005\n\n[21] M. Marvan: On the $\\mathcal{C}$-spectral sequence with ''general''\ncoefficients. Differential geometry and its applications (Brno, 1989),\n361--371, World Sci. Publishing, Teaneck, NJ, 1990\n\n[22] J.W.Morgan: The Seiberg-Witten Equations and Applications to the\ntopology of smooth Four-Manifolds, Mathematical Notes 44, Princeton\nUnivertsity Press, Princeton, NJ, 1996\n\n[23] H.Omori: Infinite dimensional Lie transformation groups. Lecture Notes\nin Mathematics, Vol. 427. Springer-Verlag, Berlin-New York, 1974\n\n[24] R.S.Palais: Extending diffeomorphisms, Proc. Amer. Math. Soc. 11 1960\n274--277\n\n[25] J.F. Pommaret: Systems of Partial Differential Equations and Lie\nPseudogroups, Gordon and Breach Science Publishers, New York, London, Paris,\n1978\n\n[26] J.F.Pommaret: Partial Differential Equations and Group Theory. New\nperspectives for applications. Mathematics and its Applications, 293. Kluwer\nAcademic Publishers Group, Dordrecht, 1994\n\n[27] Z.Ran: Derivatives of Moduli, Internat. Math. Res. Notices, no 4,\n63-74, 1992\n\n[28] B.K.Reinhart: Some remarks on the structure of the Lie algebra of\nformal vector fields. Transversal structure of foliations (Toulouse, 1982).\nAst\\'{e}risque No. 116 (1984), 190--194\n\n[29] R.W.Sharpe: Differential Geometry, Cartan's Generalization of Klein's\nErlangen Program, Graduate Texts in Mathematics, Springer-Verlag, New York\nBerlin Heidelberg, 1997\n\n[30] W.Thurston: Three-dimensional Geometry and Topology Vol 1, Edited by\nSilvio Levy, Princeton Mathematical Series, 35, Princeton University Press,\nNJ, 1997\n\n[31] G.Vezzosi, A.M.Vinogradov: On higher order analogues of de Rham\ncohomology, Differential Geom. Appl. 19 (2003), no. 1, 29--59\n\n[32] A.M.Vinogradov: Cohomological Analysis of Partial Differential\nEquations and Secondary Calculus, Translations of Mathematical Monographs,\nVolume 204, AMS, 2000\n\n[33] A.M.Vinogradov: Scalar differential invariants, diffieties and\ncharacteristic classes, Mechanics, Analysis and Geometry, 200 years after\nLagrange, M.Francaviglia (editor), Elsevier Science Publishers B.V., 1991,\n379-414\n\n[34] H.C.Wang: Closed manifolds with homogeneous structures, Amer.J.Math.,\n76, 1954, 1-32\n\n[35] G.Weingart: Holonomic and semi-holonomic geometries, Seminaires \\&\nCongres, 4, 2000, 307-328\n\n\\bigskip \\lbrack 36] A.Weinstein: Groupoids: unifying internal and external\nsymmetry. A tour through some examples. Groupoids in analysis, geometry, and\nphysics (Boulder, CO, 1999), 1--19, Contemp. Math., 282, Amer. Math. Soc.,\nProvidence, RI, 2001\n\n\\bigskip\n\n\\bigskip\n\nErc\\\"{u}ment Orta\\c{c}gil, Bo\\u{g}azi\\c{c}i University, Bebek, 34342,\nIstanbul, Turkey\n\ne-mail: ortacgil@boun.edu.tr\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWith the end of Dennard Scaling, High Energy and Nuclear Physics (HENP) libraries and applications are now required to be multi-thread safe.\nThis is to ensure the scaling of performances with new CPU architectures.\nThe new \\texttt{C++} standards introduced new constructs and library components: \\texttt{std::thread}, \\texttt{std::mutex}, \\texttt{std::lock}, \\ldots\nThese components are however quite low-level and hard to use and compose, or easy to misuse.\nMoreover, \\texttt{C++} is still plagued with scalability issues.\nDevelopment speed is hindered by the interplay between the compilation model (based on \\texttt{\\#include}) and \\texttt{C++} templates.\n\\texttt{C++} is a very large language, hard and subtle to teach and understand: this has an impact on code maintenance and the ability to bring newcomers up-to-speed on any given project.\nOn the user side, \\texttt{C++} is showing deficiencies as well: installing a project's many dependencies can be quite difficult.\n\\texttt{Python} users will be surprised to learn there is still no \\texttt{pip}-like mechanism to install, automatically and recursively, dependencies for a given project.\nFinally, most HEP software stacks rely on applications dynamically linked to hundreds shared libraries: packaging and deployment can be time consuming tasks.\n\nFixing the whole \\texttt{C++} ecosystem sounds like a rather daunting task.\nIt might be easier instead to start from a blank page, with a new language that addresses all the current deficiencies of \\texttt{C++}, and improves the day-to-day life of a typical software programmer in a multi-core environment.\n\n\\texttt{Go}~\\cite{ref-golang} was created in 2009 to address these issues.\nWe will first describe briefly the \\texttt{Go} programming language, its concurrency primitives and the features of its tooling that make \\texttt{Go} a great fit for HENP applications.\nWe will then introduce \\texttt{Go-HEP}, a set of libraries and applications that aims to provide physicists with the basic tools to perform HENP analyses and integrate them in already existing \\texttt{C++\/Python} analysis pipelines.\n\n\\section{Introduction to \\texttt{Go}}\n\n\\texttt{Go} is an open source language, released under the BSD-3 license in 2009.\nIt is a compiled language with a garbage collector and builtin support for reflection.\nThe language is reminiscent of \\texttt{C\/C++}: it shows a syntax similar to its older peers but adds first-class functions, closures and object-oriented programming via the concept of interfaces.\n\\texttt{Go} is already available, via the \\texttt{gc} toolchain, on all major platforms (Linux, Windows, macOS, Android, iOS, \\ldots) and for many architectures (\\texttt{amd64}, \\texttt{arm64}, \\texttt{i386}, \\texttt{s390x}, \\texttt{mips64}, \\ldots)\nThere is also a \\texttt{Go} frontend to \\texttt{GCC}, aptly named \\texttt{gccgo}, that can can target all the platforms and architectures that the \\texttt{GCC} toolchain supports.\nGriesemer, Pike and Thompson created \\texttt{Go} to replace the multi-threaded \\texttt{C++} web servers that were hard to develop and slow to compile, even when using Google's massive infrastructure.\nAs such, \\texttt{Go} exposes two builtin concurrency primitives: the \\emph{goroutines} -- very lightweight green threads -- and the \\emph{channels} -- typed conduits that connect goroutines together.\n\n\n\nLaunching goroutines is done by prepending a function (or method) call with the keyword \\texttt{go}.\nGoroutines can be described as green threads: very lightweight execution contexts that are multiplexed on native threads.\nEach goroutine starts with a small stack (around $4KB$) that can grow and shrink as needed.\nThis allows to write applications with thousands of goroutines without the need to buy high-end servers.\nWriting real world concurrent programs needs synchronization and communication: the ability to exchange data between goroutines.\nThis is achieved \\emph{via} channels.\nNot only do channels allow to exchange data in a type safe manner but they are also a synchronization point: a goroutine trying to send a token of data through a channel will block until there is another goroutine on the other end of the channel trying to extract data from the channel, and \\emph{vice versa}.\nThe last piece that makes building concurrent programs in \\texttt{Go} efficient is the keyword \\texttt{select}.\nThis keyword is like a \\texttt{switch} statement for controling the data flow between multiple channels and thus between multiple goroutines.\nThe concurrency builtin tools of \\texttt{Go} are easier to reason about and more composable than mutexes and locks.\n\nMoreover, \\texttt{Go} comes with a package system and a builtin tool to build \\texttt{Go} code.\nThere are no header files like in \\texttt{C++} as they require to recursively process all dependent headers and thus slow the build.\nOnce compiled, \\texttt{Go} packages are completely self-contained and do not require a client of package \\texttt{p1} -- which itself depends on \\texttt{p2} and \\texttt{p3} -- to know anything about these packages.\nThe client only needs \\texttt{p1}.\nThis greatly improves the scalability of the build process as well as its speed.\nThanks to the way third-party packages are imported in \\texttt{Go}, \\emph{e.g.} \\mintinline{go}{import \"github.com\/pkg\/errors\"}, a build tool only needs to parse and understand \\texttt{Go} code to discover (recursively) dependencies of a given package.\nCoupled to the fact that package import strings contain the URL to the repository (GitHub, BitBucket, \\ldots) holding the actual code, this allows a simple command -- \\texttt{go get} -- to fetch code from the internet, (recursively) discover and fetch its dependencies and, finally build the whole artefact.\nThe instructions to install code are valid on all platforms that are supported by a given \\texttt{Go} toolchain: one just needs to point \\texttt{go get} at a repository.\nFinally, \\texttt{Go} code being very quick to compile makes static compilation manageable and enables simple deployment scenarii that boil down to \\texttt{scp}-ing the resulting binary.\n\\texttt{Go} exposes a productive development environment that is concurrency friendly.\nIt is being used by many companies~\\footnote{A non-exhaustive list is maintained here: \\texttt{https:\/\/github.com\/golang\/go\/wiki\/GoUsers}.} beside Google, and in a variety of scenarii: from rocket telemetry to cloud systems to container orchestration.\nBut for \\texttt{Go} to be useful in HENP, physicists need libraries and tools to carry analyses.\nMoreover, these tools need to be interoperable with existing analyses pipelines.\n\n\\section{\\texttt{Go-HEP}}\n\n\\texttt{Go-HEP} is a set of libraries and applications released under the BSD-3 license.\nThey allow High Energy and Nuclear Physicists to write efficient analysis code in the \\texttt{Go} programming language.\nThe \\texttt{go-hep.org\/x\/hep} organization provides a set of pure-\\texttt{Go} packages and building blocks to:\n\\begin{itemize}\n\t\\item write analyses in \\texttt{Go},\n\t\\item write data acquisition and monitoring systems in \\texttt{Go},\n\t\\item write control frameworks in \\texttt{Go}.\n\\end{itemize}\n\nAs \\texttt{Go-HEP} is written in pure-\\texttt{Go}, the whole software suite and its dependencies are installable on all \\texttt{Go} supported platforms (Linux, macOS, Windows, RPi3, \\ldots) with:\n\\begin{verbatim}\n $> go get go-hep.org\/x\/hep\/...\n\\end{verbatim}\nThe ellipsis (\\ldots) at the end instructs the \\texttt{go} tool to compile and install all the libraries and applications that are part of the \\texttt{go-hep.org\/x\/hep} repository.\n\nThe libraries provided by \\texttt{Go-HEP} can be broadly organized around two categories:\n\\begin{itemize}\n\t\\item libraries that provide physics and statistical tools (Lorentz vectors, histograms and n-tuples, fits, jet clustering, fast detector simulation, plots, \\ldots)\n\t\\item libraries that provide low level interoperability with \\texttt{C++} libraries, to allow \\texttt{Go-HEP} users to integrate with existing analyses pipelines.\n\\end{itemize}\n\nIndeed, analyses in HENP are mainly written in \\texttt{C++} and \\texttt{Python}.\nEven though \\texttt{Go} has a native foreign function interface called \\texttt{cgo} that allows \\texttt{Go} code to use \\texttt{C} code and vice versa, the libraries under \\texttt{go-hep.org\/x\/hep} do not call any \\texttt{C++} library (via a \\texttt{C} shim library.)\nThis decision allows to retain the quick edit-compile-run development cycle of \\texttt{Go} and the easy deployment and cross-compilation of \\texttt{Go} applications.\nInstead, the strategy for interoperating with \\texttt{C++} is to integrate with \\emph{e.g.} ROOT~\\cite{ref-ROOT} at the data file level.\n\\texttt{Go-HEP} provides read\/write access for LCIO, LHEF, HepMC, SLHA and YODA files.\n\\texttt{Go-HEP} provides -- at the moment -- only read access to ROOT files via its \\texttt{go-hep.org\/x\/hep\/rootio} package.\n\nIn the following, we will describe two components of \\texttt{Go-HEP}, \\texttt{fads} and \\texttt{hep\/rootio}, that enable physics analyses.\n\n\\section{\\texttt{fads}}\n\n\\subsection{\\texttt{fads-app}}\n\n\\texttt{fads} is a \"FAst Detector Simulation toolkit\".\nThis library is built on top of \\texttt{Go-HEP}'s control framework, \\texttt{fwk}.\nThe control framework exposes a traditional API, with a \\texttt{Task} type that can be started, process some event data and then stopped.\nEach event is processed by a goroutine.\nThe framework runs these tasks in their own goroutine and lets the \\texttt{Go} runtime deal with work stealing.\nData between tasks are exchanged via an event store service which itself utilizes \\texttt{channel}s to ensure there are no data races.\nBesides an event store service, \\texttt{fwk} also provides services for message logging, histogramming and n-tupling. \nN-tuple and histograms (1D and 2D) are provided by the \\texttt{hep\/hbook} and \\texttt{hep\/hbook\/ntup} packages and are persistified using \\texttt{hep\/rio}, a binary file format that heavily draws inspiration from \\texttt{SIO}, the binary file format the LCIO community uses.\n\nData dependencies between tasks are described at the job configuration level: each task declares what are its inputs (the type of input and a name that identifies it) and its outputs.\n\\texttt{fwk} ensures there are no cycles between the tasks and connects the goroutines together via channels.\n\\texttt{fwk} enables task-level parallelism and event-level parallelism via the concurrency building blocks of \\texttt{Go}.\nFor sub-task parallelism, users are required -- by construction -- to use the same building block, so everything is consistent and the \\texttt{Go} runtime has the complete picture.\n\n\\texttt{fads} is itself a transliteration of Delphes~\\cite{ref-delphes} (v3.0.12) but with \\texttt{fwk} underneath instead of \\texttt{ROOT}.\nTo assess the performances of \\texttt{Go} and \\texttt{fads} in a realistic settings, the whole ATLAS data card provided with Delphes has been implemented and packaged with \\texttt{fads} under \\texttt{hep\/fads\/cmd\/fads-app}.\nThis application is composed of:\n\\begin{itemize}\n\\item a HepMC file reader task,\n\\item a particle propagator, a calorimeter simulator,\n\\item energy rescalers, momentum smearers,\n\\item electron, photon and muon isolation tasks,\n\\item b-tagging and tau-tagging tasks, and a jet-finder task.\n\\end{itemize}\n\nThe jet finder task is based on a naive re-implementation of the \\texttt{C++} FastJet~\\cite{ref-fastjet} library: only the $N^3$ \"dumb\" clustering has been implemented so far.\nA small part of the directed acyclic graph of the resulting \\texttt{fads-app} application can be found in figure~\\ref{fig-dflow}.\n\n\\begin{figure}[h]\n\t\\begin{center}\n \\includegraphics[width=0.75\\textwidth]{figs\/fads-dflow-detail.png}\n\t\\end{center}\n\t\\caption{\\label{fig-dflow}Part of the directed acyclic graph of data dependencies between tasks composing the \\texttt{fads-app}, a \\texttt{Go} transliteration of the Delphes' ATLAS data card example. Rectangles are tasks, ellipses are data collections.}\n\\end{figure}\n\nDelphes was compiled with \\texttt{gcc-4.8} and the $N^3$ clustering strategy hard-coded, \\texttt{fads} was compiled with \\texttt{Go-1.9}.\nTimings were obtained on a Linux Xeon CPU E5-4620@2.20GHz server with 64 cores and 128Gb RAM with a 10000 HepMC events input file.\n\\begin{figure}[h]\n\n \\includegraphics[width=0.5\\textwidth]{figs\/linux-64-cores-rss.png}\n \\includegraphics[width=0.5\\textwidth]{figs\/linux-64-cores-hz.png}\n\n\t\\caption{\\label{fig-fads-perfs}Memory usage (left) and event processing rate (right) for \\texttt{fads} (solid red curve) and \\texttt{Delphes} (green dash line).}\n\\end{figure}\n\nThe results for various numbers of threads are shown in figure~\\ref{fig-fads-perfs}.\nAs Delphes is not thread-safe, we only show the data for one Delphes process and then draw a flat line for visual aid.\nThe graph on the left shows a smaller memory footprint for \\texttt{fads} that only matches that of Delphes' when 30 threads are used.\nThe graph on the right shows that \\texttt{fads} achieves a better event processing rate even when using only one thread, or when using only one thread but with the sequential event loop instead of the concurrent event loop (data point for $n_{thread}=0$.)\n\n\\texttt{fads} achieves better performances than Delphes-3.0.12 on this data set.\nThe output simulated data for \\texttt{fads} was matched bit-by-bit with that of Delphes up to the calorimetry stage~\\footnote{We could not manage to match data further down the sequence because of Delphes' usage of PRNG to seed other PRNGs in the calorimeter. Control histograms agreed at the statistical level.}.\n\\texttt{fads} also achieves these performances without the need to merge output files.\n\n\n\\subsection{\\texttt{fads-rivet-mc-generic}}\n\n\\texttt{fads} exports another application, \\texttt{fads-rivet-mc-generic}, a reimplementation of the \\texttt{MC\\_GENERIC} analysis from the Rivet~\\cite{ref-rivet} toolkit.\n\n\\begin{table}[h]\n\\begin{center}\n \\begin{tabular}{ l | c | c | c | c }\n\t & \\texttt{MaxRSS} ($Mb$) & Real ($s$) & CPU ($s$) \\\\\n \\hline\n\t \\texttt{Rivet} & 27 & 13.3 & 13.3 \\\\\n\t \\texttt{fads} & 23 & 5.7 & 5.7 \\\\\n \\hline\n \\end{tabular}\n\t\\caption{\\label{fig-rivet-perf-tab}Runtime performances of \\texttt{Rivet} and \\texttt{fads} to read a file with $Z$ events.}\n\\end{center}\n\\end{table}\n\n\\texttt{fads} shows better performances than the \\texttt{C++} application.\nAs Rivet does not use ROOT for this application, the memory usage of Rivet is closer to that of \\texttt{fads}.\nHowever, the event processing rate of \\texttt{fads} with only one thread is twice as good as the one of Rivet.\nThis is because the job steering and the event loop of Rivet are written in Python.\n\n\\section{\\texttt{hep\/rootio}}\nThe \\texttt{hep\/rootio} package is a pure-\\texttt{Go} package that:\n\\begin{itemize}\n\\item decodes and understands the structure of \\texttt{TFile}s, \\texttt{TKey}s, \\texttt{TDirectory} and \\texttt{TStreamerInfo}s,\n\\item decodes and deserializes \\texttt{TH1x}, \\texttt{TH2x}, \\texttt{TLeaf}, \\texttt{TBranch} and \\texttt{TTree}s. \n\\end{itemize}\nAt the moment, \\texttt{hep\/rootio} only concerns itself with reading ROOT files, although writing ROOT files is on the roadmap.\nNonetheless, \\texttt{hep\/rootio} can already be useful and provides the following commands:\n\\begin{itemize}\n\t\\item \\texttt{cmd\/root-ls} lists the content of ROOT files,\n\t\\item \\texttt{cmd\/root-print} extracts histograms from ROOT files and saves them as PDF or PNG,\n\t\\item \\texttt{cmd\/root-dump} prints the contents of \\texttt{TTree}s, event by event,\n\t\\item \\texttt{cmd\/root-diff} prints the differences between the contents of two \\texttt{TTree}s,\n\t\\item \\texttt{root-cnv-npy} converts simple \\texttt{TTree}s to \\texttt{NumPy} array data files,\n\t\\item \\texttt{cmd\/root-srv} allows to interactively plot histograms from a file and branches from a tree. \\texttt{root-srv} being pure-\\texttt{Go}, it can be hosted on Google AppEngine.\n\\end{itemize}\n\n\\texttt{rootio} can read flat \\texttt{TTree}s with \\texttt{C++} builtins, static and dynamic arrays of \\texttt{C++} builtins.\n\\texttt{rootio} can also read \\texttt{TTree}s with user defined classes containing \\texttt{std::vector}~\\footnote{where \\texttt{T} is a \\texttt{C++} builtin or a \\texttt{std::string} or a \\texttt{TString}}, another user defined class, \\texttt{std::string} or \\texttt{TString}, arrays of \\texttt{C++} builtins and \\texttt{C++ builtins}.\n\nTo assess the performances of \\texttt{hep\/rootio} with regard to the original \\texttt{C++} implementation, we created two input files of $10^6$ entries and 100 branches of \\texttt{float64}.\nOne file was compressed with the default settings of ROOT-6.10 ($686 Mb$) and the other had no compression ($764 Mb$).\n\n\\begin{table}[h]\n\\begin{center}\n \\begin{tabular}{ l | c | c | c | c }\n\t \\texttt{fCompression=0} & \\texttt{VMem} ($Mb$) & \\texttt{MaxRSS} ($Mb$) & Real ($s$) & CPU ($s$) \\\\\n \\hline\n\t \\texttt{ROOT} & 517 & 258 & 6.7 & 6.6 \\\\\n\t \\texttt{hep\/rootio} & 43 & 42 & 12.9 & 12.9 \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\n\\begin{center}\n \\begin{tabular}{ l | c | c | c | c }\n\t \\texttt{fCompression=1} & \\texttt{VMem} ($Mb$) & \\texttt{MaxRSS} ($Mb$) & Real ($s$) & CPU ($s$) \\\\\n \\hline\n\t \\texttt{ROOT} & 529 & 292 & 12.6 & 12.0 \\\\\n\t \\texttt{hep\/rootio} & 83 & 82 & 35.8 & 35.8 \\\\\n \\hline\n \\end{tabular}\n\t\\caption{\\label{fig-rootio-perf-tab}Runtime performances of \\texttt{C++\/ROOT} and \\texttt{hep\/rootio} to read a file with $10^6$ events, using no compression (up) and with default compression (down).}\n\\end{center}\n\\end{table}\n\nTable~\\ref{fig-rootio-perf-tab} shows the results obtained running the two programs with the compressed file and the uncompressed file as inputs.\nWhile the memory footprint of the \\texttt{Go} program is almost an order of magnitude lower than its \\texttt{C++} counterpart, it is twice as slow in the non-compressed case and almost thrice in the compressed case.\nThe degraded performances in the compressed case have been tracked back to the decompression package from the standard library which could probably be further optimized.\nOn the otherhand, the degraded performances in the non-compressed case come from the still young implementation of \\texttt{hep\/rootio} which could be also further optimized, \\emph{e.g.} preemptively loading multiple consecutive entries.\n\n\\section{Conclusions}\n\n\\texttt{Go} improves on \\texttt{C++\/Python} and addresses their deficiences with regard to code distribution, code installation, compilation, development and runtime speeds.\n\\texttt{Go} also provides builtin facilities to tackle concurrency efficiently and easily.\n\\texttt{Go-HEP} provides some building blocks that are already competitive with battle-tested \\texttt{C++} programs, both in terms of CPU, memory usage and cores' utilization.\nFurther improvements are still necessary in the ROOT I\/O area, both in terms of performances and features, so that \\texttt{Go-HEP} can be part of physics analyses pipelines.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nVisual recognition systems (\\emph{e.g.}, image classification) play key roles in a wide range of applications such as autonomous driving, robot autonomy, smart medical diagnosis and video surveillance. Recently, remarkable progress has been made through big data and powerful GPUs driven deep neural networks (DNNs) under supervised learning framework~\\cite{LeCunCNN,AlexNet}. DNNs significantly increase prediction accuracy in visual recognition tasks and even outperform humans in some image classification tasks~\\cite{ResidualNet,InceptionNet}.\nDespite the dramatic improvement, it has been shown that DNNs trained for visual recognition tasks can be easily fooled by so-called \\textbf{adversarial attacks} which utilize visually imperceptible, carefully-crafted perturbations to cause networks to misclassify inputs in arbitrarily chosen ways in the close set of labels used in training~\\cite{FoolDeepNet,LBFGS, AdversarialExampl, CWAttack}, even with one-pixel attack~\\cite{OnePixelAttack}. Assuming full access to DNNs pretrained with clean images, white-box targeted attacks are powerful ways of investigating the brittleness of DNNs and their sensitivity to non-robust yet well-generalizing features in the data, and of exploiting adversarial examples as useful features~\\cite{AttackNotBugs}. \n\n\\begin{figure} [t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{top5-1.pdf}\n \\caption{\\small Examples of ordered Top-$5$ attacks for a pretrained ResNet-50 model~\\cite{ResidualNet} in the ImageNet-1000 dataset. We compare a modified C\\&W method~\\cite{CWAttack} (see details in Sec.~\\ref{sec:modifiedCW}) and our proposed adversarial distillation method in terms of $\\ell_2$ distance between a clean image and the learned adversarial example. \n \n Here, ${9\\times30}$ and ${9\\times1000}$ refer to the settings of hyperparameter search: we perform $9$ binary searches for the trade-off parameter of perturbation energy penalty in the objective function and each search step takes $30$ and $1000$ iterations of optimization respectively. Our method consistently outperforms the modified C\\&W method. Similarly, our method obtains better results for the bottom image of \\textit{Volleyball}, where the modified C\\&W$_{9\\times30}$ method fails to attack. (Best viewed in color and magnification)}\\label{fig:ex-top5-attack}\n\\end{figure}\n\nIn this paper, we focus on learning visually-imperceptible targeted attacks under the white-box setting. One scheme of learning these attacks is to design a proper adversarial objective function that leads to the imperceptible perturbation for any test image, \\emph{e.g.}, the widely used Carlini-Wagner (C\\&W) method~\\cite{CWAttack}. However, most methods address targeted attacks in the Top-$1$ manner, which limits the flexibility of attacks, and may lead to less rich perturbations. We propose to generalize this setting to account for \\textbf{ordered Top-$k$ targeted attacks}, that is to enforce the Top-$k$ predicted labels of an adversarial example to be the $k$ (randomly) selected and ordered labels ($k\\geq 1$, the ground-truth (GT) label is exclusive). Figure~\\ref{fig:ex-top5-attack} shows two examples. \n\nTo see why Top-$k$ targeted attacks are entailed, let's take a close look at the ``robustness\" of an attack method itself under the traditional Top-$1$ protocol. One crucial question is, \n\n\\textit{How far is the attack method able to push the underlying ground-truth label in the prediction of the learned adversarial examples?} \n\nConsider a white-box targeted attack method such as the C\\&W method~\\cite{CWAttack}. Although it can achieve $100$\\% attack success rate (ASR) under the given Top-$1$ protocol, if the ground-truth labels of adversarial examples still largely appear in the Top-$5$ of the prediction, we may be over-confident about the $100$\\% ASR, especially when some downstream modules may rely on Top-$5$ predictions in their decision making. Table~\\ref{tab:gt-rank} shows the results. The C\\&W method does not push the GT labels very far, especially when smaller perturbation energy is aimed using larger search range (\\emph{e.g.}, the average rank of the GT label is $2.6$ for C\\&W$_{9\\times1000}$). On the contrary, the three untargeted attack approaches work much better in terms of pushing the GT labels, although their perturbation energy are usually much larger.\nWhat is interesting to us is the difference of the objective functions used by the C\\&W method and the three untargeted attack methods respectively. The former maximizes the margin of the logits between the target and the runner-up (either GT or not), while the latter maximizes the cross-entropy between the prediction probabilities (softmax of logits) and the one-hot distribution of the ground-truth. Furthermore, the label smoothing methods~\\cite{LabelSmoothing, ConfidencePenalty} is often used to improve the performance of DNNs, which address the over-confidence in the one-hot vector encoding of annotations. And, the network distillation method~\\cite{distillation,distillation1} views the knowledge of a DNN as the conditional distribution it produces over outputs given an input. One question naturally arises,\n\n\\textit{Can we design a proper adversarial distribution similar in spirit to label smoothing to guide the ordered Top-$k$ attack by leveraging the view of point of network distillation?}\n\nOur proposed method aims to harness the best of the above strategies in designing proper target distributions and objective functions to achieve both high ASR and low perturbation energy. Our proposed ordered Top-$k$ attacks explicitly push the GT labels to a ``safe\" zone of retaining the ASR. \n \n \n\n \n\\begin{table}\n\\caption{\\small Results of showing where the ground-truth (GT) labels are in the prediction of learned adversarial examples for different attack methods. The test is done in ImageNet-1000 {\\tt validation} dataset using a pretrained ResNet-${50}$ model~\\cite{ResidualNet}. Please see Sec.~\\ref{sec:exp-setup} for details of experimental settings. }\n\\label{tab:gt-rank}\n\\centering\n\\resizebox{0.8\\textwidth}{!}{\n\\begin{tabular}{llllllll} \n\\toprule\n\\multirow{2}{*}{Method} &\\multirow{2}{*}{ASR} & \\multicolumn{5}{c}{Proportion of GT Labels in Top-$k$ {\\small (smaller is better)}}&\\multirow{2}{3cm}{{\\footnotesize Average Rank of GT Labels (larger is better)}}\\\\ \\cmidrule(r){3-7}\n&&Top-$3$ &Top-${5}$ &Top-${10}$ &Top-${50}$ &Top-${100}$\\\\ \\midrule\nC\\&W$_{9\\times30}$~\\cite{CWAttack} &99.9 &36.9 &50.5 &66.3 &90.0 &95.1 &20.4\\\\\nC\\&W$_{9\\times1000}$~\\cite{CWAttack} &100&71.9&87.0&96.1&99.9&100&2.6\\\\ \n\\hline \nFGSM~\\cite{FGSM} &80.7 &25.5&37.8 &52.8 &81.2 &89.2 &44.2\\\\\nPGD$_{10}$~\\cite{IFGSM, PGD} &100 &3.3 &6.7 &12 &34.7 &43.9 &306.5\\\\\nMIFGSM$_{10}$~\\cite{MIFGSM} &99.9 &0.7 &1.9 &6.0 &22.5 &32.3 &404.4\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\nTowards learning the generalized ordered Top-$k$ attacks, we present an \\textbf{adversarial distillation} framework: First, we compute an adversarial probability distribution for any given ordered Top-$k$ targeted labels with respect to the ground-truth of a test image. Then, we learn adversarial examples by minimizing the Kullback-Leibler (KL) divergence together with the perturbation energy penalty, similar in spirit to the network distillation method~\\cite{distillation}. More specifically, we explore how to leverage label semantic similarities in computing the targeted distributions, leading to \\textbf{knowledge-oriented attacks}. We measure label semantic similarities using the cosine distance between some off-the-shelf word2vec embedding of labels such as the pretrained Glove embedding~\\cite{Glove}.\nAlong this direction, a few questions of interest that naturally arise are studied: Are all Top-$k$ targets equally challenging for an attack approach? How can we leverage the semantic knowledge between labels to guide an attack approach to learn better adversarial examples and to find the weak spots of different attack approaches? We found that KL is a stronger alternative than the C\\&W loss function, and label semantic knowledge is useful in designing effective adversarial distributions. \n\nIn experiments, we develop a modified C\\&W approach for ordered Top-$k$ attacks as a strong baseline. We thoroughly test Top-$1$ and Top-$5$ attacks in the ImageNet-1000~\\cite{ImageNet} {\\tt validation} dataset using two popular DNNs trained with clean ImageNet-1000 {\\tt train} dataset, ResNet-50~\\cite{ResidualNet} and DenseNet-121~\\cite{DenseNet}. For both models, our proposed adversarial distillation approach outperforms the vanilla C\\&W method in the Top-$1$ setting, as well as other baseline methods such as the PGD method~\\cite{PGD,IFGSM}. Our approach shows significant improvement in the Top-$5$ setting against the modified C\\&W method. We observe that Top-$k$ targets that are distant from the GT label in terms of either label semantic distance or prediction scores of clean images are actually more difficulty to attack. \n\n\\textbf{Our Contributions.} This paper makes three main contributions to the field of learning adversarial attacks: \n (i) To our knowledge, this is the first work of learning ordered Top-$k$ attacks. This generalized setting is a straightforward extension of the widely used Top-$1$ attack and able to improve the robustness of adversarial attacks themselves. \n (ii) A conceptually simple yet effective framework, adversarial distillation is proposed to learn ordered Top-$k$ attacks under the white-box settings. It outperforms a strong baseline, the C\\&W method~\\cite{CWAttack} under both the traditional Top-$1$ and the proposed ordered Top-$5$ in the ImageNet-1000 dataset using two popular DNNs, ResNet-50~\\cite{ResidualNet} and DenseNet-121~\\cite{DenseNet}. \n (iii) Knowledge-oriented design of adversarial target distributions are studied whose effectiveness is supported by the experimental results.\n\n\\textbf{Paper organization.} The remainder of this paper is organized as follows. In Section~\\ref{sec:formulation}, we overview the white-box targeted attacks and the C\\&W method, and then present details of our proposed adversarial distillation framework for the ordered Top-$k$ targeted attack. In Section~\\ref{sec:exp-setup}, we present thorough comparisons in ImageNet-1000. In Section~\\ref{sec:related}, we briefly review the related work. Finally, we conclude this paper in Section~\\ref{sec:conclusion}. \n\n\\section{Problem formulation}\\label{sec:formulation}\nIn this section, we first briefly introduce, to be self-contained, the white-box attack setting and the widely used C\\&W method~\\cite{CWAttack} under the Top-$1$ protocol. We then define the ordered Top-$k$ attack setting and develop a modified C\\&W method for it ($k> 1$) as a strong baseline. Finally, we present our proposed adversarial distillation framework. \n\n\\subsection{Background on white-box targeted attack under the Top-$1$ setting}\nWe focus on classification tasks using DNNs. Denote by $(x, y)$ a pair of a clean input $x\\in \\mathcal{X}$ and its ground-truth label $y\\in \\mathcal{Y}$. For example, in the ImageNet-1000 classification task, $x$ represents a RGB image defined in the lattice of $224\\times 224$ and we have $\\mathcal{X}\\triangleq R^{3\\times 224\\times 224}$. $y$ is the category label and we have $\\mathcal{Y}\\triangleq \\{1, \\cdots, 1000\\}$. Let $f(\\cdot;\\Theta)$ be a DNN pretrained on clean training data where $\\Theta$ collects all estimated parameters and is fixed in learning adversarial examples. For notation simplicity, we denote by $f(\\cdot)$ a pretrained DNN. \nThe prediction for an input $x$ from $f(\\cdot)$ is usually defined using softmax function by,\n\\begin{equation}\n P =f(x)=softmax(z(x)),\n\\end{equation}\nwhere $P\\in R^{|\\mathcal{Y}|}$ represents the estimated confidence\/probability vector ($P_c\\geq 0$ and $\\sum_c P_c=1$) and $z(x)$ is the logit vector. The predicted label is then inferred by $\\hat{y}=\\arg\\max_{c\\in [1,|\\mathcal{Y}|]} P_c$. \n\nIn learning targeted attacks under the Top-$1$ protocol, for an input $(x, y)$, given a target label $t\\neq y$, we seek to compute some visually-imperceptible perturbation $\\delta(x, t, f)$ using the pretrained and fixed DNN $f(\\cdot)$ under the white-box setting. \\textit{White-box attacks} assume the complete knowledge of the pretrained DNN $f$, including its parameter values, architecture, training method, etc. The perturbed example $x'=x+\\delta(x, t, f)$ is called \\textbf{an adversarial example} of $x$ if $t=\\hat{y}'=\\arg\\max_c f(x')_c$ and the perturbation $\\delta(x, t, f)$ is sufficiently small according to some energy metric. We usually focus on the subset of inputs $(x,y)$'s that are correctly classified by the model, \\emph{i.e.}, $y=\\hat{y}=\\arg\\max_c f(x)_c$. Learning $\\delta(x, t, f)$ under the Top-$1$ protocol is posed as a constrained optimization problem~\\cite{AdversarialExampl, CWAttack}, \n\\begin{align}\n \\text{minimize}\\quad &\\mathcal{E}(\\delta)=||\\delta||_p, \\label{eq:formulation}\\\\\n \\nonumber \\text{subject to}\\quad & t=\\arg\\max_c f(x+\\delta)_c,\\\\\n \\nonumber & x+\\delta \\in [0, 1]^n,\n\\end{align}\nwhere $\\mathcal{E}(\\cdot)$ is defined by a $\\ell_p$ norm (\\emph{e.g.}, the $\\ell_2$ norm) and $n$ the size of the input domain (e.g., the number of pixels). \nTo overcome the difficulty (non-linear and non-convex constraints) of directly solving Eqn.~\\ref{eq:formulation}, the C\\&W method expresses it in a different form by designing some loss functions $L(x')=L(x+\\delta)$ such that the first constraint $t=\\arg\\max_c f(x')_c$ is satisfied if and only if $L(x')\\leq 0$. The best loss function proposed by the C\\&W method is defined by the hinge loss of logits between the target label and the runner-up, \n\\begin{equation}\n L_{CW}(x') = \\max(0, \\max_{c\\neq t}z(x')_c - z(x')_t). \\label{eq:cwloss}\n\\end{equation}\nThen, the learning problem becomes, \n\\begin{align}\n \\text{minimize}\\quad & ||\\delta||_p + \\lambda\\cdot L(x+\\delta), \\label{eq:formulation1}\\\\\n \\nonumber \\text{subject to}\\quad & x+\\delta \\in [0, 1]^n,\n\\end{align}\nwhich can be solved via back-propagation with the constraint satisfied via introducing a {\\tt tanh} layer. For the trade-off parameter $\\lambda$, a binary search will be performed during the learning (\\emph{e.g.}, $9\\times 1000$). \n\n\\subsection{The proposed ordered Top-$k$ attack setting}\nIt is straightforward to extend Eqn.~\\ref{eq:formulation} for learning ordered Top-$k$ attacks ($k\\geq 1$). Denote by $(t_1, \\cdots, t_k)$ the ordered Top-$k$ targets ($t_i\\neq y$). We have, \n\\begin{align}\n \\text{minimize}\\quad &\\mathcal{E}(\\delta)=||\\delta||_p, \\label{eq:formulation-topk}\\\\\n \\nonumber \\text{subject to}\\quad & t_i=\\arg\\max_{c\\in [1, |\\mathcal{Y}|], c\\notin \\{t_1, t_{i-1}\\} } f(x+\\delta)_c, \\quad i\\in \\{1,\\cdots, k\\}, \\\\\n \\nonumber & x+\\delta \\in [0, 1]^n .\n\\end{align}\n\n\\subsubsection{A modified C\\&W method}\\label{sec:modifiedCW}\nWe can modify the loss function (Eqn.~\\ref{eq:cwloss}) of the C\\&W method accordingly to solve Eqn.~\\ref{eq:formulation-topk}. We have, \n\\begin{equation}\n L^{(k)}_{CW}(x') = \\sum_{i=1}^k \\max(0, \\max_{j\\notin \\{t_1,\\cdots, t_{i}\\}}z(x')_j - z(x')_{t_i}). \\label{eq:cwloss-topk}\n\\end{equation}\nSo, the vanilla C\\&W loss (Eqn.~\\ref{eq:cwloss}) is the special case of Eqn.~\\ref{eq:cwloss-topk} (\\emph{i.e.}, when $k=1$).\n\n\\subsubsection{Our proposed knowledge-oriented adversarial distillation framework}\nIn the C\\&W loss functions, only the margin of logits between the targeted labels and the runner-ups is taken into account. In our adversarial distillation framework, we adopt the view of point proposed in the network distillation method~\\cite{distillation} that the full confidence\/probability distribution summarizes the knowledge of a trained DNN. We hypothesize that we can leverage the network distillation framework to learn the ordered Top-$k$ attacks by designing a proper adversarial probability distribution across the entire set of labels that satisfies the specification of the given ordered Top-$k$ targets. \n\nConsider the Top-$k$ targets, $(t_1, \\cdots, t_k)$, we want to define the adversarial probability distribution, denoted by $P^{adv}$ in which $P^{adv}_{t_i}> P^{adv}_{t_j}$ ($\\forall iP^{adv}_j$ ($\\forall j\\notin (t_1, \\cdots, t_k)$). The space of candidate distributions are huge. We present a simple knowledge-oriented approach to define the adversarial distribution. We first specify the logit distribution and then compute the probability distribution using softmax. Denote by $Z$ the maximum logit (\\emph{e.g.}, $Z=10$ in our experiments). We define the adversarial logits for the ordered Top-$k$ targets by,\n\\begin{equation}\n z^{adv}_{t_i}=Z - (i-1)\\times \\gamma, \\quad i\\in [1, \\cdots, k],\n\\end{equation}\nwhere $\\gamma$ is an empirically chosen decreasing factor (\\emph{e.g.}, $\\gamma=0.3$ in our experiments). For the remaining categories $j\\notin (t_1, \\cdots, t_k)$, we define the adversarial logits by,\n\\begin{equation}\n z^{adv}_j = \\alpha \\times \\frac{1}{k}\\sum_{i=1}^k s(t_i, j) + \\epsilon, \\label{eq:logit-others}\n\\end{equation}\nwhere $0\\leq \\alpha < z^{adv}_{t_k}$ is the maximum logit that can be assigned to any $j$, $s(a, b)$ is the semantic similarity between the label $a$ and label $b$, and $\\epsilon$ is a small position for numerical consideration (\\emph{e.g.}, $\\epsilon=1e$-$5$). We compute $s(a, b)$ using the cosine distance between the Glove~\\cite{Glove} embedding vectors of category names and $-1\\leq s(a, b) \\leq 1$. Here, when $\\alpha=0$, we discard the semantic knowledge and treat all the remaining categories equally. Note that our design of $P^{adv}$ is similar in spirit to the label smoothing technique and its variants~\\cite{LabelSmoothing,ConfidencePenalty} except that we target attack labels and exploit label semantic knowledge. The design choice is still preliminary, although we observe its effectiveness in experiments. We hope this can encourage more sophisticated work to be explored. \n\nWith the adversarial probability distribution $P^{adv}$ defined above as the target, we use the KL divergence as the loss function in our adversarial distillation framework as done in network distillation~\\cite{distillation} and we have, \n\\begin{equation}\n L^{(k)}_{adv}(x') = KL(f(x')||P^{adv}),\n\\end{equation}\nand then we follow the same optimization scheme as done in the C\\&W method (Eqn.~\\ref{eq:formulation1}). \n\n\\section{Experiments}\\label{sec:exp-setup}\nIn this section, we present results of our proposed method tested in ImageNet-1000~\\cite{ImageNet} using two pretrained DNNs, ResNet-50~\\cite{ResidualNet} and DenseNet-121~\\cite{DenseNet} from the PyTorch model zoo~\\footnote{https:\/\/github.com\/pytorch\/vision\/tree\/master\/torchvision\/models}. We implement our method using the AdverTorch toolkit~\\footnote{https:\/\/github.com\/BorealisAI\/advertorch}. Our source code will be released. \n\n\\textbf{Data.} In ImageNet-1000~\\cite{ImageNet}, there are $50,000$ images for validation. We obtain the subset of images for which the predictions of both the ResNet-50 and DenseNet-121 are correct. To reduce the computational demand, we further test our method in a randomly sampled subset, as commonly done in the literature. To enlarge the coverage of categories, we first randomly select 500 categories and then randomly chose 2 images per selected categories, resulting in 1000 test images in total. \n\n\\textbf{Settings.} We follow the protocol used in the C\\&W method. We only test $\\ell_2$ norm as the energy penalty for perturbations in learning. But we evaluate learned adversarial examples in terms of three norms ($\\ell_1$, $\\ell_2$ and $\\ell_{\\infty}$). We test two search schema for the trade-off parameter $\\lambda$ in optimization: both use $9$ steps of binary search, and $30$ and $1000$ iterations of optimization are performed for each trial of $\\lambda$. Only $\\alpha=1$ is used in Eqn.~\\ref{eq:logit-others} in experiments for simplicity due to computational demand.\nWe compare the results under three scenarios proposed in the C\\&W method~\\cite{CWAttack}: \\textit{The Best Case} settings test the attack against all incorrect classes, and report the target class(es) that was least difficult to attack.\n\\textit{The Worst Case} settings test the attack against all incorrect classes, and report the target class(es) that was most difficult to attack.\n\\textit{The Average Case} settings select the target class(es) uniformly at random among the labels that are not the GT.\n\n\\begin{figure} [h]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{top1-1.pdf}\n \\caption{\\small Adversarial examples learned with the Top-$1$ attack setting using ResNet-50~\\cite{ResidualNet}. The perturbation is shown by $\\ell_{2}$ distance between the clean image and adversarial examples. For better visualization, we use different scales in showing the heat maps for different methods. (Best viewed in color and magnification)}\\label{fig:ex-top1}\n\\end{figure}\n\n\\begin{table*} [h]\n\\caption{\\small Results and comparisons under the Top-$1$ targeted attack setting. We also test against three state-of-the-art untargeted attack methods, FGSM and PGD and MIFGSM, and the last two use 10 steps in optimization.}\n\\label{tab:top1}\n\\centering\n\\resizebox{1\\textwidth}{!}{\n\\begin{tabular}{llllllllllllll} \n\\toprule\n\\multirow{2}{*}{Model}&\\multirow{2}{*}{Attack Method} & \\multicolumn{4}{c}{Best Case}&\\multicolumn{4}{c}{Average Case} &\\multicolumn{4}{c}{Worst Case} \\\\ \n\\cmidrule(r){3-14}\n&&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$\\\\\n\\midrule\n\\multirow{7}{*}{ResNet-50~\\cite{ResidualNet}}&FGSM~\\cite{FGSM} &2.3 &9299 &24.1 &0.063 &0.46 &9299 &24.1 &0.063 &0 &N.A. &N.A. &N.A.\\\\\n&PGD$_{10}$~\\cite{IFGSM, PGD} &99.6 &4691 &14.1 &0.063 &88.1 &4714 &14.2 &0.063 &57.1 &4748 &14.3 &0.063\\\\\n&MIFGSM$_{10}$~\\cite{MIFGSM} &100 &5961 &17.4 &0.063 &99.98 &6082 &17.6 &0.063 &99.9 &6211 &17.9 &0.063\\\\\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &100 &209.7 &0.777 &0.022 &99.92 &354.1 &1.273 &0.031 &99.9 &560.9 &1.987 &0.042\\\\\n&Ours$_{9\\times30}$ &100 &140.9 &0.542 &0.018 &99.9 &184.6 &0.696 &0.025 &99.9 &238.6 &0.880 &0.032\\\\\n&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &95.6 &0.408 &0.017 &100 &127.2 &0.516 &\\textbf{0.023} &100 &164.1 &0.635 &0.030\\\\\n&Ours$_{9\\times1000}$ &100 &\\textbf{81.3} &\\textbf{0.380} &\\textbf{0.016} &100 &\\textbf{109.6} &\\textbf{0.472} &\\textbf{0.023} &100 &\\textbf{143.9} &\\textbf{0.579} &\\textbf{0.029}\\\\\n\\midrule\n\\multirow{7}{*}{DenseNet-121~\\cite{DenseNet}}&FGSM~\\cite{FGSM} &6.4 &9263 &24.0 &0.063 &1.44 &9270 &24.0 &0.063 &0 &N.A. &N.A. &N.A.\\\\\n&PGD$_{10}$~\\cite{IFGSM, PGD} &100 &4617 &14.2 &0.063 &97.2 &4716 &14.2 &0.063 &87.6 &4716 &14.2 &0.063\\\\\n&MIFGSM$_{10}$~\\cite{MIFGSM} &100 &5979 &17.6 &0.063 &100 &6095 &17.6 &0.063 &100 &6218 &17.9 &0.063\\\\\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &99.9 &188.6 &0.694 &0.019 &99.9 &279.4 &1.008 &0.028 &99.9 &396.5 &1.404 &0.037\\\\\n&Ours$_{9\\times30}$ &99.9 &136.4&0.523 &0.017 &99.9 &181.8 &0.678 &0.024 &99.9 &240.0 &0.870 &0.031\\\\\n&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &98.5 &0.415 &\\textbf{0.016} &100 &132.3 &0.528 &\\textbf{0.023} &100 &174.8 &0.657 &\\textbf{0.030}\\\\\n&Ours$_{9\\times1000}$ &100 &\\textbf{83.8} &\\textbf{0.384} &\\textbf{0.016} &100 &\\textbf{115.9} &\\textbf{0.485} &\\textbf{0.023} &100 &\\textbf{158.69} &\\textbf{0.610} &\\textbf{0.030}\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\caption{\\small Results of Top-$1$ targeted attacks using 5 most-like labels and 5 least-like labels as targets respectively, based on the label semantic similarities.}\n\\label{tab:top1-sim}\n\\centering\n\\resizebox{1\\textwidth}{!}{\n\\begin{tabular}{lllllllllllllllll} \n\\toprule\n\\multirow{2}{*}{Model}&\\multirow{2}{*}{Similarity} &\\multirow{2}{*}{Attack Method}& \\multicolumn{4}{c}{Best Case}&\\multicolumn{4}{c}{Average Case} &\\multicolumn{4}{c}{Worst Case} \\\\\n\\cmidrule(r){4-15}\n&&&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$ \\ &\\\\\n\\midrule\n\\multirow{14}{*}{ResNet-50~\\cite{ResidualNet}}&\\multirow{7}{*}{Most like}\n&FGSM~\\cite{FGSM} &32.4 &9137 &23.8 &0.063 &0.0862 &9137 &23.8 &0.063 &0 &N.A. &N.A. &N.A.\\\\\n&&PGD$_{10}$~\\cite{IFGSM, PGD} &99.9 &4687 &14.1 &0.063 &94.6 &4708 &14.2 &0.063 &78.4 &4737 &14.3 &0.063\\\\\n&&MIFGSM$_{10}$~\\cite{MIFGSM}&100 &5993 &17.4 &0.063 &99.98 &6110 &17.7 &0.063 &99.9 &6228 &17.9 &0.063 \\\\\n&&C\\&W$_{9\\times30}$~\\cite{CWAttack} &100 &138 &0.51 &0.012 &99.94 &249 &0.89 &0.023 &99.9 &401 &1.40 &0.035\\\\\n&&Ours$_{9\\times30}$ &100 &114 &0.43 &0.011 &99.96 &171 &0.62 &0.020 &99.9 &251 &0.87 &0.028\\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &82 &0.34 &\\textbf{0.010} &100 &126 &0.48 &\\textbf{0.018} &100 &194 &0.68 &0.026\\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{75} &\\textbf{0.32} &\\textbf{0.010} &100 &\\textbf{117} &\\textbf{0.46} &\\textbf{0.018} &100 &\\textbf{187} &\\textbf{0.65} &\\textbf{0.025}\\\\\n\\cmidrule(r){2-15}\n&\\multirow{7}{*}{Least like}\n&FGSM~\\cite{FGSM} &0.4 &8860 &23.4 &0.063 &0.08 &8860 &23.4 &0.063 &0 &N.A. &N.A. &N.A.\\\\\n&&PGD$_{10}$~\\cite{IFGSM, PGD} &99.4 &4696 &14.1 &0.063 &84.82 &4721 &14.2 &0.063 &50 &4762 &14.3 &0.063\\\\\n&&MIFGSM$_{10}$~\\cite{MIFGSM}&100 &5960 &17.4 &0.0625 &99.96 &6069 &17.6 &0.063 &99.8 &6194 &17.9 &0.063\\\\\n&&C\\&W$_{9\\times30}$~\\cite{CWAttack} &99.9 &259 &0.95 &0.025 &99.9 &421 & 1.51&0.035 &99.9 &639 &2.25 &0.046\\\\\n&&Ours$_{9\\times30}$ &99.9 &154 &0.59 &0.020 &99.9 &194 & 0.73 &0.026 &99.9 &240 &0.89 &0.033\\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &102 &0.44 &\\textbf{0.019} &100 &132 &0.54 &0.025 &100 &165 &0.65 &0.032\\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{85} &\\textbf{0.40} &\\textbf{0.019} &100 &\\textbf{111} &\\textbf{0.49} &\\textbf{0.024} &100 &\\textbf{142} &\\textbf{0.59} &\\textbf{0.030}\\\\\n\\midrule\n\\multirow{14}{*}{DenseNet-121~\\cite{DenseNet}}&\\multirow{7}{*}{Most like}\n&FGSM~\\cite{FGSM} &46.1 &9132 &23.8 &0.063 &14.62 &9143 &23.8 &0.063 &0.3 &9263 &24.0. &0.063\\\\\n&&PGD$_{10}$~\\cite{IFGSM, PGD} &100 &4692 &14.1 &0.063 &98.92 &4712 &14.2 &0.063 &94.7 &4733 &14.3 &0.063\\\\\n&&MIFGSM$_{10}$~\\cite{MIFGSM}&100 &6010 &17.5 &0.063 &100 &6128 &17.7 &0.063 &100 &6245 &18.0 &0.063 \\\\\n&&C\\&W$_{9\\times30}$~\\cite{CWAttack} &100 &130 &0.48 &0.010 &100 &218 &0.77 &0.021 &100 &332 &1.14 &0.031\\\\\n&&Ours$_{9\\times30}$ &100 &114 &0.42 &0.010 &100 &170 &0.61 &0.019 &100 &250 &0.85 &0.028\\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack}&100 &85 &0.34 &\\textbf{0.010} &100 &134 &0.50 &\\textbf{0.018} &100 &210 &0.71 &\\textbf{0.026}\\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{77} &\\textbf{0.33} &\\textbf{0.010} &100 &\\textbf{124} &\\textbf{0.48} &\\textbf{0.018} &100 &\\textbf{202} &\\textbf{0.69} &\\textbf{0.026}\\\\\n\\cmidrule(r){2-15}\n&\\multirow{7}{*}{Least like}\n&FGSM~\\cite{FGSM} &2.1 &9101 &23.7 &0.063 &0.42 &9101 &23.7 &0.063 &0 &N.A. &N.A. &N.A.\\\\\n&&PGD$_{10}$~\\cite{IFGSM, PGD} &100 &4698 &14.2 &0.063 &96.32 &4718 &14.2 &0.063 &83.6 &4745 &14.3 &0.063\\\\\n&&MIFGSM$_{10}$~\\cite{MIFGSM}&100 &5973 &17.4 &0.0625 &99.98 &6082 &17.9 &0.063 &99.9 &6203 &17.9 &0.063\\\\\n&&C\\&W$_{9\\times30}$~\\cite{CWAttack} &99.9 &215 &0.79 &0.023 &99.9 &310 & 1.12&0.030 &99.9 &428 &1.52 &0.039\\\\\n&&Ours$_{9\\times30}$ &99.9 &145 &0.56 &0.019 &99.9 &188 & 0.70 &0.025 &99.9 &240 &0.88 &0.032\\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &102 &0.43 &\\textbf{0.018} &100 &134 &0.54 &\\textbf{0.024} &100 &170 &0.65 &\\textbf{0.031}\\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{85} &\\textbf{0.40} &\\textbf{0.018} &100 &\\textbf{114} &\\textbf{0.49} &\\textbf{0.024} &100 &\\textbf{149} &\\textbf{0.59} &\\textbf{0.031}\\\\\n\n\\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\\subsection{Results for the Top-$1$ attack setting}\nWe first evaluate whether the proposed adversarial distillation framework is effective for the traditional Top-$1$ attack setting. The results show that the proposed method can consistently outperform the C\\&W method, as well as some other state-of-the-art untargeted attack methods including the PGD method~\\cite{PGD,IFGSM}. \n\n\nFigure~\\ref{fig:ex-top1} shows two qualitative results. We can see the C\\&W method and our proposed method ``attend\" to different regions in images to achieve the attacks. Table~\\ref{tab:top1} shows the quantitative results. Our proposed method obtains smaller $\\ell_1$ and $\\ell_2$ norm, while the $\\ell_{\\infty}$ norm are almost the same. Note that we only use the $\\ell_2$ norm in the objective function in learning. We will evaluate the results of explicitly using $\\ell_1$ and $\\ell_{\\infty}$ norm as penalty respectively in future work. \n\n\n\nAs shown in Table~\\ref{tab:top1-sim}, we also test whether the label semantic knowledge can help identify the weak spots of different attack methods, and whether the proposed method can gain more in those weak spots. We observe that attacks are more challenging if the Top-$1$ target is selected from the least-like set in terms of the label semantic similarity (see Eqn.~\\ref{eq:logit-others}). \n\n\\subsection{Results for the ordered Top-$5$ attack setting}\nWe test ordered Top-$5$ attacks and compare with the modified C\\&W method. Our proposed method significantly outperforms the modified C\\&W method, especially for the $9\\times 30$ optimization scheme, as shown in Table~\\ref{tab:top5}. We also observe improvement on the $\\ell_{\\infty}$ norm for the ordered Top-$5$ attacks (please see Figure~\\ref{fig:ex-top5-attack} for two visual examples). \n\nWe also test the effectiveness of knowledge-oriented specifications of selecting the ordered Top-$5$ targets with similar observation obtained as in the Top-$1$ experiments (see Table~\\ref{tab:top5-sim}). \n\nTo further evaluate the proposed method, we also test the ordered Top-$5$ attacks using labels with 5 highest and 5 lowest clean prediction scores as targets respectively, as shown in Table~\\ref{tab:top5-clean-logit}. We observe similar patterns that the 5 labels with the lowest clean prediction scores are more challenging to attack. This shed lights on learning data-driven knowledge: Instead of using the label semantic knowledge which may have some discrepancy in guiding the design of adversarial loss functions, we can leverage the similarities measured based on the confusion matrix in the training data if available. We leave this for future work. \n\n\\begin{table*}\n\\caption{\\small Results and comparisons under the ordered Top-$5$ targeted attack protocol using randomly selected and ordered 5 targets (GT exclusive).}\n\\label{tab:top5}\n\\centering\n\\resizebox{1\\textwidth}{!}{\n\\begin{tabular}{llllllllllllll} \n\\toprule\n\\multirow{2}{*}{Model}&\\multirow{2}{*}{Attack Method} & \\multicolumn{4}{c}{Best Case}&\\multicolumn{4}{c}{Average Case} &\\multicolumn{4}{c}{Worst Case} \\\\ \n\\cmidrule(r){3-14}\n&&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$\\\\\n\\midrule\n\\multirow{4}{*}{ResNet-50~\\cite{ResidualNet}}&C\\&W$_{9\\times30}$~\\cite{CWAttack} &75.8 &2370 &7.76 &0.083 &29.34 &2425 &7.94 &0.086 &0.7 &2553 &8.37 &0.094\\\\\n&Ours$_{9\\times30}$ &96.1 &1060 &3.58 &0.056 &80.68 &1568 &5.13 &0.070 &49.8 &2215 &7.07 &0.087\\\\\n&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &437 &1.59 &0.044 &100 &600 &2.16 &0.058 &100 &779 &2.77 &0.074\\\\\n&Ours$_{9\\times1000}$ &100 &\\textbf{285} &\\textbf{1.09} &\\textbf{0.034} &100 &\\textbf{359} &\\textbf{1.35} &\\textbf{0.043} &100 &\\textbf{456} &\\textbf{1.68} &\\textbf{0.055}\\\\\n\\midrule\n\\multirow{4}{*}{DenseNet-121~\\cite{DenseNet}}&C\\&W$_{9\\times30}$~\\cite{CWAttack} &96.6 &2161 &7.09 &0.071 &73.68 &2329 &7.65 &0.080 &35.6 &2530 &8.28 &0.088\\\\\n&Ours$_{9\\times30}$ &97.7 &6413 &2.14 &0.043 &92.66 &1063 &3.57 &0.057 &83.3 &1636 &5.35 &0.072\\\\\n&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &392 &1.42 &0.040 &100 &527 &1.89 &0.052&100 &669 &2.37 &0.065\\\\\n&Ours$_{9\\times1000}$ &100 &\\textbf{273} &\\textbf{1.05} &\\textbf{0.033} &100 &\\textbf{344} &\\textbf{1.29} &\\textbf{0.042} &100 &\\textbf{425} &\\textbf{1.57} &\\textbf{0.052}\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\\begin{table}\n\\caption{\\small Results of ordered Top-$5$ targeted attacks using 5 most-like labels and 5 least-like labels as targets respectively, based on the label semantic similarities.}\n\\label{tab:top5-sim}\n\\centering\n\\resizebox{0.8\\textwidth}{!}{\n\\begin{tabular}{lllllll} \n\\toprule\nModel&Similarity &Attack Method&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$\\\\\n\\midrule\n\\multirow{8}{*}{ResNet-50~\\cite{ResidualNet}}&\\multirow{4}{*}{Most like}\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &80 &1922 &6.30 &0.066\\\\\n&&Ours$_{9\\times30}$ &96.5 &1286 &4.20 &0.054 \\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack}&100 &392 &1.43 &0.042 \\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{277} &\\textbf{1.05} &\\textbf{0.035} \\\\\n\\cmidrule(r){2-7}\n&\\multirow{4}{*}{Least like}\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &27.1 &2418 &7.90 &0.085 \\\\\n&&Ours$_{9\\times30}$ &77.1 &1635 &5.35 &0.072\\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &596 &2.15 &0.060 \\\\\n&&Ours$_{9\\times1000}$ &100 &\\textbf{370} &\\textbf{1.39} &\\textbf{0.045} \\\\\n\\midrule\n\\multirow{8}{*}{DenseNet-121~\\cite{DenseNet}}&\\multirow{4}{*}{Most like}\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &92.1 &1798 &5.88 &0.059\\\\\n&&Ours$_{9\\times30}$ &98.4 &1228 &4.00 &0.050 \\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack}&100 &361 &1.31 &0.039 \\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{265} &\\textbf{1.00} &\\textbf{0.034} \\\\\n\\cmidrule(r){2-7}\n&\\multirow{4}{*}{Least like}\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &75.7 &2325 &7.64 &0.080 \\\\\n&&Ours$_{9\\times30}$ &92.8 &1076 &3.63 &0.057\\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &529 &1.90 &0.052 \\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{343} &\\textbf{1.29} &\\textbf{0.042} \\\\\n\\bottomrule\n\\end{tabular}\n\n\\end{table}\n\n\n\n\n\n\n\n\\begin{table}\n\\caption{\\small Results of ordered Top-$5$ targeted attacks using labels with 5 highest and 5 lowest prediction scores of clean images as targets respectively. }\n\\label{tab:top5-clean-logit}\n\\centering\n\\resizebox{0.8\\textwidth}{!}{\n\\begin{tabular}{lllllll} \n\\toprule\nModel&Clean prediction &Attack Method&ASR&$\\ell_{1}$&$\\ell_{2}$&$\\ell_{\\infty}$\\\\\n\\midrule\n\\multirow{8}{*}{ResNet-50~\\cite{ResidualNet}}&\\multirow{4}{*}{Highest}\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &93 &1546 &4.98 &0.042\\\\\n&&Ours$_{9\\times30}$ &99.9 &1182 &3.78 &0.039 \\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack}&100 &205 &0.75 &0.025 \\\\\n&&Ours$_{9\\times1000}$&100 &\\textbf{170} &\\textbf{0.65} &\\textbf{0.023} \\\\\n\\cmidrule(r){2-7}\n&\\multirow{4}{*}{Lowest}\n&C\\&W$_{9\\times30}$~\\cite{CWAttack} &13.4 &2231 &7.30 &0.082 \\\\\n&&Ours$_{9\\times30}$ &68.6 &1791 &5.86&0.077\\\\\n&&C\\&W$_{9\\times1000}$~\\cite{CWAttack} &100 &621 &2.25 &0.064 \\\\\n&&Ours$_{9\\times1000}$ &100 &\\textbf{392} &\\textbf{1.47} &\\textbf{0.047} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\\section{Related work}\\label{sec:related}\nThe growing ubiquity of DNNs in advanced machine learning and AI systems dramatically increases their capabilities, but also increases the potential for new vulnerabilities to attacks. This situation has become critical as many powerful approaches have been developed where imperceptible perturbations to DNN inputs could deceive a well-trained DNN, significantly altering its prediction.\nPlease refer to~\\cite{AttackSurvey} for a comprehensive survey of attack methods in computer vision. We review some related work that motivate our work and show the difference. \n\n\\textbf{Distillation.}\nThe central idea of our proposed work is built on distillation. Network distillation~\\cite{distillation1,distillation} is a powerful training scheme proposed to train a new, usually lightweight model (a.k.a., the student) to mimic another already trained model (a.k.a. the teacher). It takes a functional viewpoint of the knowledge learned by the teacher as the conditional distribution it produces over outputs given an input. It teaches the student to keep up or emulate by adding some regularization terms to the loss in order to encourage the two models to be similar directly based on the distilled knowledge, replacing the training labels. Label smoothing~\\cite{LabelSmoothing} can be treated as a simple hand-crafted knowledge to help improve model performance. \nDistillation has been exploited to develop defense models~\\cite{distillation_defense} to improve model robustness. Our proposed adversarial distillation method utilizes the distillation idea in an opposite direction, leveraging label semantic driven knowledge for learning ordered Top-$k$ attacks and improving attack robustness. \n\n\\textbf{Adversarial Attack.} For image classification tasks using DNNs, the discovery of the existence of visually-imperceptible adversarial attacks~\\cite{LBFGS} was a big shock in developing DNNs.\nWhite-box attacks provide a powerful way of evaluating model brittleness. In a plain and loose explanation, DNNs are universal function approximator~\\cite{UniversalApproximator} and capable of even fitting random labels~\\cite{DNNGeneralization} in large scale classification tasks as ImageNet-1000~\\cite{ImageNet}. Thus, adversarial attacks are always learnable provided proper objective functions are given, especially when DNNs are trained with fully differentible back-propagation. Many white-box attack methods focus on norm-ball constrained objective functions~\\cite{LBFGS,IFGSM,CWAttack,MIFGSM}. The C\\&W method investigates 7 different loss functions. The best performing loss function found by the C\\&W method has been appliedin many attack methods and achieved strong results~\\cite{ZOOAttack, PGD, EADAttack}. By introducing momentum in the MIFGSM method~\\cite{MIFGSM} and the $\\ell_{p}$ gradient projection in the PGD method~\\cite{PGD}, they usually achieve better performance in generating adversarial examples. In the meanwhile, some other attack methods such as the StrAttack~\\cite{StrAttack} also investigate different loss functions for better interpretability of attacks. Our proposed method leverage label semantic knowledge in the loss function design for the first time.\n\n\n\n\n\\section{Conclusions}\\label{sec:conclusion}\nThis paper proposes to extend the traditional Top-$1$ targeted attack setting to the ordered Top-$k$ setting ($k\\geq 1$) under the white-box attack protocol. The ordered Top-$k$ targeted attacks can improve the robustness of attacks themselves. To our knowledge, it is the first work studying this ordered Top-$k$ attacks. To learn the ordered Top-$k$ attacks, we present a conceptually simple yet effective adversarial distillation framework motivated by network distillation. We also develop a modified C\\&W method as the strong baseline for the ordered Top-$k$ targeted attacks. In experiments, the proposed method is tested in ImageNet-1000 using two popular DNNs, ResNet-50 and DenseNet-121, with consistently better results obtained. We investigate the effectiveness of label semantic knowledge in designing the adversarial distribution for distilling the ordered Top-$k$ targeted attacks. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments} \nThis work is supported by ARO grant W911NF1810295 and ARO DURIP grant W911NF1810209, and NSF IIS 1822477. \n\n\n\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbkno b/data_all_eng_slimpj/shuffled/split2/finalzzbkno new file mode 100644 index 0000000000000000000000000000000000000000..70ca4d4c7374f1689cdeebcb51da7388fd9c26e0 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzbkno @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\n\\subsection{Hankel operators}\nLet $\\alpha = \\{\\alpha(j)\\}_{j\\ge 0}$\nbe a sequence of complex numbers. The \\emph{Hankel matrix} $H(\\alpha)$\nis the ``infinite matrix'' $\\{\\alpha(j+k)\\}_{j,k\\ge 0}$, considered as a linear operator on $\\ell^2({\\mathbb Z}_+)$, \n${\\mathbb Z}_+=\\{0,1,2,\\dots\\}$, so that\n$$\n(H(\\alpha)x)(k) = \\sum_{j\\ge 0} \\alpha(j+k)x(j), \\quad k\\geq0,\\quad\nx=\\{x(j)\\}_{j\\geq0}\\in \\ell^2({\\mathbb Z}_+).\n$$\nSimilarly, for a \\emph{kernel function} ${\\mathbf a}\\in L^1_\\mathrm{loc}(0,\\infty)$, the \\emph{integral Hankel operator}\non $L^2(0,\\infty)$ is defined by the formula\n$$\n({\\mathbf H}({\\mathbf a}){\\mathbf f})(t) = \\int_0^\\infty {\\mathbf a}(t+s){\\mathbf f}(s) \\,\\,ds, \\quad t>0,\\quad {\\mathbf f}\\in L^2(0,\\infty). \n$$\nIn order to distinguish between these two classes of operators, we \nuse boldface font for objects associated with integral Hankel operators. \n\nFor general background on Hankel operators, see \\cite{Nikolski,Peller}. \nIn what follows, we will only consider\nbounded Hankel matrices and bounded integral Hankel operators. \n\n\n\n\\subsection{Restrictions}\n\nThe purpose of this paper is to examine the linear map, which we call \n\\emph{the restriction map}, \nbetween the set of integral Hankel operators and the set of Hankel matrices. \nTo set the scene, let us consider the \\emph{pointwise restriction} of integral kernels to integers. \nFor a given kernel function ${\\mathbf a}$, define the sequence\n\\[\n\\alpha(j):={\\mathbf a}(j+1), \\quad j\\geq0.\n\\label{a1}\n\\]\nOf course, for this operation to make sense, the kernel function ${\\mathbf a}$ has to be \ncontinuous. Here is our first result; we denote by $\\mathbf{S}_p$, $00$, so the restriction \\eqref{a1} is well defined.\nThe operator $H(\\alpha)$ is in $\\mathbf{S}_p$ and we have the estimate\n\\[\n\\norm{H(\\alpha)}_{\\mathbf{S}_p}\\leq C_p\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}. \n\\label{a2}\n\\]\n\\end{theorem}\nThe continuity of the kernel function ${\\mathbf a}$ for trace class integral Hankel operators\nis well known (see e.g. \\cite[Corollary 7.10]{Partington}); the main point here\nis the estimate \\eqref{a2}. \nIn Section~\\ref{sec.c} we give a slightly more precise version of Theorem~\\ref{thm.a1} and \nshow that it does not extend to $p>1$. Further, we show that if \nwe restrict the map ${\\mathbf H}({\\mathbf a})\\mapsto H(\\alpha)$ to \\emph{non-negative} integral \nHankel operators, then it is bounded in $\\mathbf{S}_p$ norm for all $00,\n$$\nand one can consider the map\n$$\nH(\\alpha)\\mapsto {\\mathbf H}(\\mathcal{E}_{{\\bm{\\varphi}}}\\alpha). \n$$\nAlthough some Schatten norm boundedness results for this map are not\ndifficult to prove, we have not succeeded in finding a coherent set of estimates\nfor it and therefore we do not discuss extensions here. \n\n\n\n\\subsection{Symbols}\nFor a bounded Hankel matrix $H(\\alpha)$, its analytic symbol is the function \n$$\n\\widecheck \\alpha(z)=\\sum_{m\\geq0} \\alpha_m z^m, \n\\quad \\abs{z}<1.\n$$\nSimilarly, for a bounded integral Hankel operator ${\\mathbf H}({\\mathbf a})$, its analytic symbol\nis the function \n$$\n\\widecheck{\\mathbf a}(\\xi)=\\int_0^\\infty {\\mathbf a}(t)e^{2\\pi i t\\xi}dt, \\quad \\Im\\xi>0.\n$$\nIt is instructive to view restriction maps on Hankel operators\nin terms of the symbols. \nIf $\\alpha=\\mathcal{R}_{\\bm{\\varphi}}{\\mathbf a}$, then for the symbols we have\n\\begin{equation}\n\\widecheck\\alpha(z)=\\int_{\\mathbb R} \\frac{\\widecheck{\\mathbf a}(\\xi+i0)\\widecheck{\\bm{\\varphi}}(-\\xi+i0)}{1-ze^{-2\\pi i\\xi}}d\\xi, \\quad \\abs{z}<1.\n\\label{r}\n\\end{equation}\nIn particular, for the pointwise restriction \\eqref{a1} we have\n\\[\n\\widecheck\\alpha(e^{2\\pi i\\xi})=e^{-2\\pi i\\xi}\\sum_{j\\in{\\mathbb Z}}\\widecheck {\\mathbf a}(\\xi-j), \\quad \\Im \\xi>0.\n\\label{r1}\n\\]\nSince Schatten norms of Hankel operators correspond to Besov norms of the symbols\n(see Section~\\ref{sec.b}), \none can view the topic of this paper as the study of the map induced by \\eqref{r} between Besov classes. \nWe prefer to use an operator theoretic viewpoint whenever possible, although sometimes we \nhave to resort to proofs in terms of Besov classes. \n\n\\section{Preliminaries}\\label{sec.b}\n\nThroughout this paper, the symbol `$C$' with a (possibly empty) set of subscripts\nwill denote a positive constant, depending only on the subscripts, whose precise\nvalue may change with each occurrence. Moreover, we write $X\\asymp Y$ for two\nexpressions $X$ and $Y$ if $X\\le C Y$ and $Y\\le C X$.\n\n\\subsection{Operator theory, Schatten classes}\nFor a bounded linear operator $A$ in a Hilbert space, we denote by $\\norm{A}_{\\mathcal{B}}$ the operator norm of $A$. \n\nFor a compact operator $A$ in a Hilbert space, let $\\{s_n(A)\\}_{n=1}^\\infty$ \nbe the sequence of singular values of $A$, enumerated with multiplicities \ntaken into account. For $00.\n$$\nWe set ${\\mathbf w}_m(t)={\\mathbf w}(t\/2^m)$. For $m\\geq0$, we denote by $w_m$ the restriction of the function ${\\mathbf w}_m$ onto \n${\\mathbb Z}_+$, i.e. $w_m(j)={\\mathbf w}_m(j)$, $j\\geq0$. \n\\begin{proposition}\\cite[Theorem 6.7.4]{Peller}\\label{thm.peller}\nLet $00$, let $\\delta_\\lambda(t)=\\delta(t-\\lambda)$, where $\\delta(t)$ is the\nDirac delta function, so that if ${\\mathbf a}\\in C(0,\\infty)$, then\n$$\n(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})(j) = {\\mathbf a}(j+\\lambda), \\quad j\\ge 0. \n$$\nIf ${\\mathbf a}$ is the kernel function of an integral Hankel operator\nof class $\\mathbf{S}_1$, then ${\\mathbf a}$ is almost everywhere equal to a continuous\nfunction on $(0,\\infty)$ \\cite[Corollary 7.10]{Partington}, and the estimate\n\\begin{equation}\n\\abs{{\\mathbf a}(t)}\\leq C\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_1}\/t, \\quad t>0,\n\\label{cc1}\n\\end{equation}\nholds true with some absolute constant $C$. \nThus, the\ndefinition of $\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a}$ makes sense without any further\nrestriction on ${\\mathbf a}$. \n\nThe aim of this section is to prove the following.\n\\begin{theorem}\\label{pointwise}\nLet $00$. If ${\\mathbf H}({\\mathbf a})\\in\\mathbf{S}_p$ then $H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})\\in\\mathbf{S}_p$\nand \n\\[\n\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})}_{\\mathbf{S}_p}\n\\leq \nC_p(1+1\/\\lambda)\n\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}.\n\\label{pointwise1}\n\\]\n\\end{theorem}\n\nThe main component in the proof of Theorem \\ref{pointwise} is the estimate\n\\eqref{b3.3}.\n\n\\begin{proof}\nDenote ${\\mathbf b}(t)={\\mathbf a}(t+\\lambda)$. \nBy Proposition~\\ref{thm.peller}(i), we have\n\\[\n\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})}_{\\mathbf{S}_p}^p \n\\leq \nC_p \\abs{{\\mathbf b}(0)}^p+C_p\\sum_{m\\geq0} 2^m \\Norm{\\sum_{j\\geq0}{\\mathbf b}(j)w_m(j)z^j}_{L^p({\\mathbb T})}^p.\n\\label{c0}\n\\]\nLet us first estimate the series in the right hand side of \\eqref{c0}. \nApplying \\eqref{b3.3} to ${\\mathbf f}=\\boldsymbol\\mathcal{F}^{-1}({\\mathbf b} {\\mathbf w}_m)$, we obtain \n$$\n\\Norm{\\sum_{j\\geq0}{\\mathbf b}(j)w_m(j)z^j}_{L^p({\\mathbb T})}^p\n\\leq \n\\norm{\\boldsymbol\\mathcal{F}^{-1}({\\mathbf b} {\\mathbf w}_m)}_{L^p({\\mathbb R})}^p\n$$\nfor every $m\\geq0$. \nBy Proposition~\\ref{thm.peller}(ii), this yields\n$$\n\\sum_{m\\geq0} 2^m \\Norm{\\sum_{j\\geq0}{\\mathbf b}(j)w_m(j)z^j}_{L^p({\\mathbb T})}^p\n\\leq\n\\sum_{m\\in{\\mathbb Z}} 2^m \\norm{\\boldsymbol\\mathcal{F}^{-1}({\\mathbf b} {\\mathbf w}_m)}_{L^p({\\mathbb R})}^p\n\\leq\nC_p \\norm{{\\mathbf H}({\\mathbf b})}_{\\mathbf{S}_p}^p. \n$$\nLet us relate the norm of ${\\mathbf H}({\\mathbf b})$ to the norm of ${\\mathbf H}({\\mathbf a})$. \nWriting \n$$\n\\int_0^\\infty \\int_0^\\infty {\\mathbf b}(t+s){\\mathbf f}(t)\\overline{{\\mathbf{g}}(s)}dt\\, ds\n=\n\\int_{\\lambda\/2}^\\infty \\int_{\\lambda\/2}^\\infty \n{\\mathbf a}(t+s){\\mathbf f}(t-\\lambda\/2)\\overline{{\\mathbf{g}}(s-\\lambda\/2)}dt\\, ds, \n$$\nwe see that ${\\mathbf H}({\\mathbf b})$ is unitarily equivalent to the restriction of ${\\mathbf H}({\\mathbf a})$ onto \nthe subspace $L^2(\\lambda\/2,\\infty)\\subset L^2(0,\\infty)$.\nIt follows that \n\\[\n\\norm{{\\mathbf H}({\\mathbf b})}_{\\mathbf{S}_p}\\leq \\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}\n\\label{c6}\n\\]\nfor all $p>0$. \nFinally, consider the first term in the right hand side of \\eqref{c0}. \nBy \\eqref{cc1} we have\n$$\n\\abs{{\\mathbf b}(0)}=\n\\abs{{\\mathbf a}(\\lambda)}\\leq C\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_1}\/\\lambda\\leq C\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}\/\\lambda.\n$$\nCombining the above estimates, we arrive at the required statement. \n\\end{proof}\n\n\\begin{remark*}\nOne can also consider restrictions of ${\\mathbf a}$ to the scaled lattice $\\{\\gamma j+\\lambda\\}_{j\\geq0}$\nfor some $\\gamma>0$. \nFor $\\gamma>0$, let ${\\mathbf a}_\\gamma(t) = {\\mathbf a}(\\gamma t)$ and let $V_\\gamma:L^2(0,\\infty)\n\\to L^2(0,\\infty)$ be the unitary operator\n$$\nV_\\gamma {\\mathbf f}(t) = \\sqrt{\\gamma} {\\mathbf f}(\\gamma t), \\quad t>0.\n$$\nThen $\\gamma{\\mathbf H}({\\mathbf a}_\\gamma) = V_\\gamma {\\mathbf H}({\\mathbf a}) V_\\gamma^*$ and so\n$\\gamma\\norm{{\\mathbf H}({\\mathbf a}_\\gamma)}_{\\mathbf{S}_p} = \\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}$ for all $00}\\gamma\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a}_\\gamma)}_{\\mathbf{S}_p}\n\\le C_{p} \\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}.\n\\label{c11}\n\\]\n\\end{remark*}\n\n\n\\subsection{Counterexample for $p>1$}\n\n\nFor $p>1$, is it no longer the case that the kernel of an integral\nHankel operator of class $\\mathbf{S}_p$ is necessarily continuous. However,\neven if we restrict to operators with continuous kernels, the conclusions\nof Theorem~\\ref{pointwise} still fail and thus the condition $01$. \nIndeed, by the assumption on the support of ${\\mathbf a}$ we have\n$$\n{\\mathbf a}^{(N)}={\\mathbf a}^{(N)} {\\mathbf w}_{-1}+{\\mathbf a}^{(N)} {\\mathbf w}_{0}+{\\mathbf a}^{(N)} {\\mathbf w}_{1}\n$$\nfor all $N$, where ${\\mathbf w}_m$ are defined in Section \\ref{sec.b2}. \nIt is easy to conclude that\n$$\n\\sum_{m=-1,0,1}2^m\\norm{\\boldsymbol\\mathcal{F}^{-1}({\\mathbf a}^{(N)}{\\mathbf w}_m)}_{L^p({\\mathbb R})}^p\n\\leq\nC_p\\norm{\\boldsymbol\\mathcal{F}^{-1}({\\mathbf a}^{(N)})}_{L^p({\\mathbb R})}^p=CN^{1-p}.\n$$\n\\subsection{Partial converse of Theorem~\\ref{pointwise}}\\label{sec.c.2}\nIt is clear that one cannot bound ${\\mathbf H}({\\mathbf a})$ by $H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})$ in any norm. \nHowever, one can achieve a partial converse if we vary our restriction operators\nin an appropriate sense and take a supremum over all restrictions in the right side. \nHere we briefly sketch a sample argument of this nature. \nFix $00$,\nlet ${\\mathbf a}_\\gamma(t) = {\\mathbf a}(\\gamma t)$. Observe that, by a change of variable, \n\\begin{align*}\n\\norm{\\mathcal{F}^{-1}(w^\\tau\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a}_\\gamma)}_{L^p({\\mathbb T})}^p\n&=\n\\int_{-1\/2}^{1\/2}\\left|\n\\sum_{j\\ge 0} {\\mathbf w}(\\tau j){\\mathbf a}(\\gamma (j+\\lambda))\ne^{2\\pi ij s} \\right|^p\\,ds\n\\\\\n&=\n\\gamma^{1-p}\\int_{-1\/2\\gamma}^{1\/2\\gamma}\\left|\n\\gamma\\sum_{j\\ge 0} {\\mathbf w}(\\tau j){\\mathbf a}(\\gamma (j+\\lambda))\ne^{2\\pi ij\\gamma s} \\right|^p\\,ds.\n\\end{align*}\nBy another change of variable, \nit then follows from \\eqref{d.1} that\n\\begin{align}\n\\gamma^p\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a}_\\gamma)}_{\\mathbf{S}_p}^p\n&\\ge C_p\n\\gamma\\int_0^2\\int_{-1\/2\\gamma}^{1\/2\\gamma}\\left|\n\\gamma\\sum_{j\\ge 0} {\\mathbf w}(\\tau j){\\mathbf a}(\\gamma (j+\\lambda))\ne^{2\\pi ij\\gamma s} \\right|^p\\,ds\\frac{d\\tau}{\\tau^2} \\nonumber\\\\\n&= C_p\n\\int_0^{2\/\\gamma}\\int_{-1\/2\\gamma}^{1\/2\\gamma}\\left|\n\\gamma\\sum_{j\\ge 0} {\\mathbf w}(\\tau \\gamma j){\\mathbf a}(\\gamma (j+\\lambda))\ne^{2\\pi ij\\gamma s} \\right|^p\\,ds\\frac{d\\tau}{\\tau^2}.\n\\label{Riemann sum}\n\\end{align}\nSince ${\\mathbf a}$ is continuous, for each $s\\in{\\mathbb R}$ and $\\tau>0$ the integrand\nin \\eqref{Riemann sum} converges to $|\\boldsymbol\\mathcal{F}^{-1}({\\mathbf a} {\\mathbf w}^\\tau)(s)|^p$\nas $\\gamma\\to 0$. Then by Fatou's Lemma we see that\n\\begin{align}\n\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}^p\n&\\asymp\n\\int_0^\\infty \\norm{\\boldsymbol\\mathcal{F}^{-1}({\\mathbf a} {\\mathbf w}^\\tau)}_{L^p({\\mathbb R})}^p\n\\frac{d\\tau}{\\tau^2} \\nonumber\\\\\n&\\le \\lim_{\\gamma\\to 0}\n\\int_0^{2\/\\gamma}\\int_{-1\/2\\gamma}^{1\/2\\gamma}\\left|\n\\gamma\\sum_{j\\ge 0} {\\mathbf w}(\\tau \\gamma j){\\mathbf a}(\\gamma (j+\\lambda))\ne^{2\\pi ij\\gamma s} \\right|^p\\,ds\\frac{d\\tau}{\\tau^2} \\nonumber\\\\\n&\\le C_p \\lim_{\\gamma\\to 0}\n\\gamma^p\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a}_\\gamma)}_{\\mathbf{S}_p}^p.\n\\label{d.3}\n\\end{align}\nThis gives an analogue of Igari's theorem for Fourier multipliers \\cite{Igari}.\nCombining \\eqref{d.3} with \\eqref{c11} gives the estimate\n$$\n\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}\n\\asymp\n\\sup_{\\gamma>0}\\gamma\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a}_\\gamma)}_{\\mathbf{S}_p},\n\\quad \n01$, the estimate \\eqref{pointwise1}\nremains valid for all $00.\n$$\nIn particular, it follows that the kernel function ${\\mathbf a}(t)$ is continuous in $t>0$, \nand therefore the restriction $\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a}$ is well defined for all $\\lambda>0$. \n\n\n\n\nWe have the following theorem.\n\n\n\\begin{theorem}\\label{positive}\nLet ${\\mathbf H}({\\mathbf a})\\ge 0$ be a bounded integral Hankel operator and $\\lambda>0$. \nThen the following hold:\n\\begin{enumerate}[\\rm (i)]\n\\item \nIf ${\\mathbf H}({\\mathbf a})\\in\\mathcal{B}$ then $H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})\\in\\mathcal{B}$ and\n\\[\n\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})}_\\mathcal{B}\n\\leq\nC(1+1\/\\lambda)\n\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathcal{B}}. \n\\label{c7}\n\\]\n\n\\item\nIf ${\\mathbf H}({\\mathbf a})\\in\\mathbf{S}_p$ for some $01$, as for $00$. \nThen \n$$\n\\lambda{\\mathbf a}(\\lambda)\\leq 2\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}. \n$$\n\\end{lemma}\n\\begin{proof}\nTake ${\\mathbf f}(t)=e^{-t\/\\lambda}$; then $\\norm{{\\mathbf f}}^2_{L^2(0,\\infty}=\\lambda\/2$ \nand using the monotonicity of ${\\mathbf a}(t)$, \n\\begin{multline*}\n({\\mathbf H}({\\mathbf a}){\\mathbf f},{\\mathbf f})\n=\n\\int_0^\\infty \\int_0^\\infty {\\mathbf a}(t+s)e^{-(t+s)\/\\lambda}dt\\, ds\n=\n\\int_0^\\infty {\\mathbf a}(t)e^{-t\/\\lambda}tdt\n\\\\\n\\geq\n\\int_0^\\lambda {\\mathbf a}(t)e^{-t\/\\lambda}tdt\n\\geq\n{\\mathbf a}(\\lambda)\\int_0^\\lambda e^{-t\/\\lambda}tdt\n=(1-2e^{-1})\\lambda^2{\\mathbf a}(\\lambda). \n\\end{multline*}\nOn the other hand, \n$$\n({\\mathbf H}({\\mathbf a}){\\mathbf f},{\\mathbf f})\\leq \\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}\\norm{{\\mathbf f}}^2_{L^2(0,\\infty)}\n=(\\lambda\/2) \\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}. \n$$\nCombining these two estimates, we obtain \n$$\n\\lambda{\\mathbf a}(\\lambda)\\leq\n\\frac{1}{2(1-2e^{-1})}\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}\\leq 2\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B},\n$$\nas required. \n\\end{proof}\n\n\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{positive} for $p\\in2{\\mathbb N}\\cup\\{\\infty\\}$]\n\nFirst let us \\textbf{assume that $\\lambda\\geq2$.}\nLet $K$ be the integral operator in $L^2(0,\\infty)$ with the integral kernel\n$$\nK(t,s)={\\mathbf a}(\\lambda+\\floor{t}+\\floor{s}),\n$$\nwhere $\\floor{t}$ is the largest integer less than or equal to $t$. \nSince \n$$\n\\lambda+\\floor{t}+\\floor{s}\\geq \\lambda+(t-1)+(s-1)\\geq t+s,\n$$ \nby monotonicity of ${\\mathbf a}$ we have\n$$\nK(t,s)\\leq {\\mathbf a}(t+s).\n$$\nIn the terminology of \\cite[Chapter 2]{Simon}, this means that $K$ is \n\\emph{pointwise dominated} by ${\\mathbf H}({\\mathbf a})$. \nBy \\cite[Theorem 2.13]{Simon}, it follows that \n$$\n\\norm{K}\\leq \\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B} \\quad \\text{ and} \\quad \n\\norm{K}_{\\mathbf{S}_p}\\leq \\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}\n$$\nfor all $p\\in 2{\\mathbb N}$. \n(This implication does not extend to $p\\not\\in 2{\\mathbb N}$; see e.g. \\cite{Peller2,Simon2}.) \nIt is also true (see \\cite{Pitt,DoddsFremlin}) that the compactness of ${\\mathbf H}({\\mathbf a})$ \nimplies the compactness of $K$. \n\nNext, let us relate $K$ to $H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})$. \nFor ${\\mathbf f}\\in L^2(0,\\infty)$ let us write the quadratic form of $K$ as\n$$\n(K{\\mathbf f},{\\mathbf f})=\\sum_{j,k\\geq0}{\\mathbf a}(\\lambda+j+k)f_j \\overline{f_k}, \n\\qquad\nf_j=\\int_j^{j+1}{\\mathbf f}(t)dt.\n$$\nThis means that, writing \n$L^2(0,\\infty)=\\ell^2({\\mathbb Z}_+)\\otimes L^2(0,1)$, the operator $K$ can be represented\nas \n$$\nK=H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})\\otimes (\\cdot,\\mathbbm{1})\\mathbbm{1},\n$$\nwhere $(\\cdot,\\mathbbm{1})\\mathbbm{1}$ is the rank one operator in $L^2(0,1)$ acting as \n$$\n{\\mathbf f}\\mapsto \\int_0^1{\\mathbf f}(t)dt. \n$$\nIt follows that \n$$\n\\norm{K}_\\mathcal{B}=\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})}_\\mathcal{B}\n\\quad\\text{ and }\\quad\n\\norm{K}_{\\mathbf{S}_p}=\\norm{H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})}_{\\mathbf{S}_p}\n$$\nfor all $p>0$. This completes the proof for $\\lambda\\geq2$ and $p\\in 2{\\mathbb N}\\cup\\{\\infty\\}$. \n\n\nLet us consider the case $0<\\lambda<2$. \nLet $P_2$ be the projection onto $\\ell^2(\\{2,3,\\dots\\})$ in $\\ell^2({\\mathbb Z}_+)$. \nWrite\n$$\nH(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})=P_2H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})P_2+\\widetilde H.\n$$\nThe operator $\\widetilde H$ is of rank $\\leq4$. Inspecting the matrix elements of $\\widetilde H$ and using Lemma~\\ref{lma.c1}, it is easy to see\nthat \n$$\n\\norm{\\widetilde H}_{\\mathbf{S}_1}\\leq C\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathcal{B}}\/\\lambda, \\quad \\lambda>0.\n$$\nOn the other hand, \nthe operator $P_2 H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})P_2$ is unitarily equivalent to \n$H(\\mathcal{R}_{\\delta_{\\lambda+2}}{\\mathbf a})$. Thus, applying the previous step of the proof, we\nobtain \n$$\n\\norm{P_2 H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})P_2}_\\mathcal{B}\n\\leq\n\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}\n\\quad \\text{ and }\\quad\n\\norm{P_2 H(\\mathcal{R}_{\\delta_\\lambda}{\\mathbf a})P_2}_{\\mathbf{S}_p}\n\\leq\n\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}\n$$\nfor $p\\in2{\\mathbb N}$. Combining these estimates, we arrive at \n\\eqref{c7} and \\eqref{c8} for $p\\in 2{\\mathbb N}$. \n\\end{proof}\n\nAs already mentioned, this proof does not extend to $p\\not\\in 2{\\mathbb N}$; see e.g. \\cite{Peller2,Simon2}. \nBelow we give a different proof which works for all $1\\leq p<\\infty$, but does not give\nprecise information about the constants in the estimates. \n\n\n\n\n\\subsection{Proof of Theorem~\\ref{positive} for $1\\leq p<\\infty$}\n\nIn order to simplify our notation, we set ${\\mathbf b}(t)={\\mathbf a}(t+\\lambda)$, $b(k)={\\mathbf a}(k+\\lambda)$, and \n\\begin{align*}\n\\widecheck b_m(z)&=\\sum_{k\\geq0}b(k)w_m(k)z^k, \\quad m\\in{\\mathbb Z}_+, \\quad z\\in {\\mathbb T},\n\\\\\n\\widecheck {\\mathbf b}_m(\\xi)&=\\int_0^\\infty {\\mathbf b}(t) {\\mathbf w}_m(t) e^{2\\pi i \\xi t} dt, \\quad m\\in {\\mathbb Z}, \\quad \\xi\\in{\\mathbb R}.\n\\end{align*}\nThe core of the proof is the bound\n\\[\n\\sum_{m\\geq1} 2^m \\norm{\\widecheck b_m}_{L^p({\\mathbb T})}^p\n\\leq\nC_p\n\\sum_{m\\in{\\mathbb Z}}2^m \\norm{\\widecheck {\\mathbf b}_m}_{L^p({\\mathbb R})}^p,\n\\label{c5}\n\\]\nwhich we prove below. \nThroughout the proof, we use the property that ${\\mathbf b}$ and $b$ are positive and monotone decreasing. \n\n\n\\textbf{First step: upper bound for $\\norm{\\widecheck b_m}_{L^p({\\mathbb T})}$.}\nFix $m\\geq1$. \nFirst we prepare two pointwise bounds for $\\widecheck b_m(z)$. The first one is trivial:\n\\[\n\\abs{\\widecheck b_m(z)}\n\\leq \n\\sum_k b(k)w_m(k)\n\\leq \n2^{m+1}b(2^{m-1}).\n\\label{c9}\n\\]\nThe second one is obtained through a discrete version of integration by parts (Abel summation). \nWe have\n\\begin{multline*}\n\\widecheck b_m(z)\n=\n\\frac{1}{z-1}\\sum_k b(k)w_m(k)(z^{k+1}-z^k)\n\\\\\n=\n\\frac{1}{z-1}\\sum_k \\bigl(b(k)w_m(k)-b(k+1)w_m(k+1)\\bigr)z^{k+1}\n\\\\\n=\n\\frac{1}{z-1}\\sum_k \\bigl((b(k)-b(k+1))w_m(k)+b(k+1)(w_m(k)-w_m(k+1))\\bigr)z^{k+1},\n\\end{multline*}\nand therefore \n\\begin{multline*}\n\\abs{\\widecheck b_m(z)}\n\\leq\n\\frac{1}{\\abs{z-1}}\\sum_{k=2^{m-1}}^{2^{m+1}} (b(k)-b(k+1)) \\\\\n+\n\\frac{1}{\\abs{z-1}}\\sum_{k=2^{m-1}}^{1+2^{m+1}} b(k)\\abs{w_m(k-1)-w_m(k)}.\n\\end{multline*}\nClearly, the first sum here is telescoping. For the second sum, we use the estimate\n$$\n\\abs{w_m(k-1)-w_m(k)}\\leq C2^{-m}. \n$$\nPutting this together, we obtain\n\\begin{multline}\n\\abs{\\widecheck b_m(z)}\n\\leq\n\\frac{1}{\\abs{z-1}}\n(b(2^{m-1})-b(1+2^{m+1})) \\\\\n+\n\\frac{C}{\\abs{z-1}}2^{-m}\\sum_{k=2^{m-1}}^{1+2^{m+1}} b(k)\n\\leq\n\\frac{C}{\\abs{z-1}}b(2^{m-1}),\n\\label{c10}\n\\end{multline}\nwhich is our second bound for $\\widecheck b_m(z)$. \n\nNow we can estimate the norm $\\norm{\\widecheck b_m}_{L^p({\\mathbb T})}$. \nWe split the integral over the unit circle into two parts and estimate them separately. \nUsing \\eqref{c9}, we obtain\n$$\n2^m\\int_{\\abs{t}<2^{-m}}\\abs{\\widecheck b_m(e^{2\\pi it})}^pdt\n\\leq \nC2^{pm}b(2^{m-1})^p.\n$$\nUsing \\eqref{c10}, we get\n\\begin{multline*}\n2^m\\int_{\\abs{t}>2^{-m}}\\abs{\\widecheck b_m(e^{2\\pi it})}^p dt\n\\leq \nC2^m\\int_{\\abs{t}>2^{-m}}\\frac{dt}{\\abs{e^{2\\pi it}-1}^p}b(2^{m-1})^p\n\\\\\n\\leq\nC2^m\\int_{2^{-m}}^1\\frac{dt}{t^p}b(2^{m-1})^p\n\\leq\nC2^{pm}b(2^{m-1})^p. \n\\end{multline*}\nCombining the estimates for two integrals above, we obtain\n$$\n2^m\\norm{\\widecheck b_m}^p_{L^p({\\mathbb T})}\\leq C2^{pm}b(2^{m-1})^p.\n$$\n\n\n\n\\textbf{Second step: lower bound for $\\norm{\\widecheck {\\mathbf b}_m}_{L^p({\\mathbb R})}$.}\nFor the derivative of ${\\mathbf b}_m$ we have\n$$\n\\widecheck {\\mathbf b}_m'(\\xi)=2\\pi i\\int_0^\\infty {\\mathbf b}(t){\\mathbf w}_m(t)te^{2\\pi it\\xi}dt,\n$$\nand therefore\n$$\n\\abs{\\widecheck {\\mathbf b}_m'(\\xi)}\n\\leq\n2\\pi \n\\int_0^\\infty {\\mathbf b}(t){\\mathbf w}_m(t)tdt\n\\leq \n2^{m+2}\\pi \\int_0^\\infty {\\mathbf b}(t){\\mathbf w}_m(t)dt\n=\n2^{m+2}\\pi \\widecheck {\\mathbf b}_m(0).\n$$\nIt follows that \n$$\n\\abs{\\widecheck {\\mathbf b}_m(\\xi)-\\widecheck {\\mathbf b}_m(0)}\\leq \\abs{\\xi}2^{m+2}\\pi \\widecheck{\\mathbf b}_m(0),\n$$\nand therefore for $\\abs{\\xi}<2^{-m-5}$ we have \n$$\n\\abs{\\widecheck {\\mathbf b}_m(\\xi)}\\geq \\widecheck{\\mathbf b}_m(0)\/2.\n$$\nWe use this to obtain a lower bound for the integral of $\\abs{\\widecheck {\\mathbf b}_m}^p$: \n\\begin{multline*}\n2^m\\int_{\\mathbb R}\\abs{\\widecheck{\\mathbf b}_m(\\xi)}^pd\\xi\n\\geq \n2^m \\int_{\\abs{\\xi}<2^{-m-5}}\\abs{\\widecheck{\\mathbf b}_m(\\xi)}^pd\\xi\n\\geq\n2^{-5}( \\widecheck{\\mathbf b}_m(0)\/2)^p=C\\widecheck{\\mathbf b}_m(0)^p.\n\\end{multline*}\nFinally,\n$$\n\\widecheck{\\mathbf b}_m(0)\n=\n\\int_0^\\infty {\\mathbf b}(t){\\mathbf w}_m(t)dt\n\\geq\n{\\mathbf b}(2^{m+1})\\int_0^\\infty {\\mathbf w}_m(t)dt\n=C2^{m}{\\mathbf b}(2^{m+1}),\n$$\nand so we obtain \n$$\n2^m\\norm{\\widecheck {\\mathbf b}_m}_{L^p({\\mathbb R})}^p\\geq C 2^{mp}b(2^{m+1})^p.\n$$\n\n\n\\textbf{Combining the two steps and completing the proof.}\nCombining the upper bound for $\\norm{\\widecheck b_m}_{L^p({\\mathbb T})}$ and the lower bound\nfor $\\norm{\\widecheck{\\mathbf b}_m}_{L^p({\\mathbb R})}$, we obtain\n$$\n2^m\\norm{\\widecheck b_m}_{L^p({\\mathbb T})}^p\n\\leq \nC2^{p(m-1)}b(2^{m-1})^p\n\\leq\nC2^{m-2}\\norm{\\widecheck {\\mathbf b}_{m-2}}_{L^p({\\mathbb R})}^p, \\quad m\\geq1.\n$$\nSumming over $m$, we obtain the bound \\eqref{c5}. \n\nBy Proposition~\\ref{thm.peller}(i), we have\n\\[\n\\norm{H(b)}_{\\mathbf{S}_p}^p\n\\leq\nC_p\\abs{b(0)}^p+\nC_p\\sum_{m\\geq0} 2^m \\norm{\\widecheck b_m}_{L^p({\\mathbb T})}^p.\n\\label{dd}\n\\]\nBy Lemma~\\ref{lma.c1}, we have\n$$\n\\abs{b(0)}^p=\\abs{{\\mathbf a}(\\lambda)}^p\\leq 2^p \\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}^p\/\\lambda^p. \n$$\nSimilarly, the $m=0$ term in the series in \\eqref{dd} can be estimated as follows:\n$$\n\\norm{\\widecheck b_0}_{L^p({\\mathbb T})}^p=\\abs{b(1)}^p=\\abs{{\\mathbf a}(\\lambda+1)}^p\n\\leq 2^p\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}^p\/(1+\\lambda)^p\n\\leq 2^p\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}^p\/\\lambda^p.\n$$\nCombining this with \\eqref{c5} and using \n Proposition~\\ref{thm.peller}(ii), we obtain \n$$\n\\norm{H(b)}_{\\mathbf{S}_p}^p\n\\leq\nC_p\n\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}^p\/\\lambda^p\n+\nC_p\\sum_{m\\in{\\mathbb Z}}2^m \\norm{\\widecheck {\\mathbf b}_m}_{L^p({\\mathbb R})}^p\n\\leq\nC_p\n\\norm{{\\mathbf H}({\\mathbf a})}_\\mathcal{B}^p\/\\lambda^p\n+\nC_p\\norm{{\\mathbf H}({\\mathbf b})}_{\\mathbf{S}_p}^p.\n$$\nFinally, as in \\eqref{c6}, we have $\\norm{{\\mathbf H}({\\mathbf b})}_{\\mathbf{S}_p}\\leq \\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}$, \nand we arrive at the required estimate \\eqref{c8}. \n\n\n\n\n\n\\section{Restriction by averaging}\\label{sec.d}\n\n\\subsection{Boundedness of restrictions by averaging}\n\nThe main result of this section says that if the function ${\\bm{\\varphi}}$ is sufficiently regular,\nthen the map ${\\mathbf H}({\\mathbf a})\\mapsto H(\\mathcal{R}_{\\bm{\\varphi}}{\\mathbf a})$ is bounded with respect to all\nSchatten norms. We will make use of the periodisation operator $\\mathcal{P}$ from Section\n\\ref{sec.b3}.\n\n\n\\begin{theorem}\\label{average}\n\tLet ${\\bm{\\varphi}}\\in C({\\mathbb R})$ be such that $\\supp{\\bm{\\varphi}}\\subset[0,\\infty)$ and \n\t$\\mathcal{P}(\\abs{\\widecheck{\\bm{\\varphi}}})\\in L^\\infty({\\mathbb T})$. \n\tThen there exist bounded operators $\\Phi_1$ and $\\Phi_2$ acting from\n\t$\\ell^2({\\mathbb Z}_+)$ to $L^2(0,\\infty)$ such that\n\t\\[\n\t\\Phi_2^*{\\mathbf H}({\\mathbf a})\\Phi_1=H(\\mathcal{R}_{\\bm{\\varphi}}{\\mathbf a})\n\t\\label{average2}\n\t\\]\n\tand $\\norm{\\Phi_1}_\\mathcal{B}=\\norm{\\Phi_2}_\\mathcal{B} =\\sqrt{A}$, where $A=\\norm{\\mathcal{P}(\\abs{\\widecheck{\\bm{\\varphi}}})}_{L^\\infty({\\mathbb T})}$. \n\tConsequently, we have\n$$\n\\norm{H(\\mathcal{R}_{{\\bm{\\varphi}}}{\\mathbf a})}_{\\mathcal{B}}\\leq A\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathcal{B}}\n\\quad \\text{ and }\\quad\n\\norm{H(\\mathcal{R}_{{\\bm{\\varphi}}}{\\mathbf a})}_{\\mathbf{S}_p}\\leq A\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}\n$$\nfor every $00} {\\mathbf a}(s+t){\\bm{\\varphi}}_1(t-j)\\overline{{\\bm{\\varphi}}_2(s-k)}\\,dt\\, ds.\n\t\\end{multline*}\n\tSince both ${\\bm{\\varphi}}_1$ and ${\\bm{\\varphi}}_2$ are supported on $(0,\\infty)$, we\n\tcan rewrite this as\n\t$$\n\t\\mathcal{R}_{\\bm{\\varphi}}{\\mathbf a}(j+k)=\n\t\\int_0^\\infty \\int_0^\\infty {\\mathbf a}(s+t){\\bm{\\varphi}}_1(t-j)\\overline{{\\bm{\\varphi}}_2(s-k)}\\,ds\\, dt.\n\t$$\n\t\n\t\n\tNow for $x=\\{x(j)\\}_{j\\ge 0}\\in \\ell^2({\\mathbb Z}_+)$, let us compute\n\tthe quadratic form\n\t\\begin{multline*}\n\t({\\mathbf H}({\\mathbf a})\\Phi_1 x,\\Phi_2 x)\n\t=\n\t\\sum_{j,k\\ge 0}\\int_0^\\infty \\int_0^\\infty {\\mathbf a}(t+s) x(j) \\overline{x(k)}\n\t{\\bm{\\varphi}}_1(t-j)\\overline{{\\bm{\\varphi}}_2(s-k)}\\,dt\\,ds\n\t\\\\\n\t=\n\t\\sum_{j,k\\ge 0}\\mathcal{R}_{\\bm{\\varphi}}{\\mathbf a}(j+k)x(j) \\overline{x(k)},\n\t\\end{multline*}\n\twhich yields \\eqref{average2}.\n\\end{proof}\n\n\n\\subsection{Unitary equivalence and restrictions associated to general convolutions}\n\nLet $L_n = L_n^{(0)}$ be the $n$-th Laguerre polynomial (see \\cite[Ch. V]{Szego}\nfor the definition) and let\n\\[\n{\\mathbf u}_n (t) = -2i\\sqrt{\\pi}L_n(4\\pi t)e^{-2\\pi t}, \\quad t>0.\n\\label{laguerre}\n\\]\nThen $\\{{\\mathbf u}_n\\}_{n\\ge 0}$ is an orthonormal basis of\n$L^2(0,\\infty)$. It is well known that the matrix of an integral Hankel operator\nis a Hankel matrix in the basis $\\{{\\mathbf u}_n\\}_{n\\ge 0}$ and hence the classes\nof Hankel matrices and integral Hankel operators are unitarily equivalent\n\\cite[Ch. 1, Thm 8.9]{Peller}. \n\nIn this subsection we discuss how this unitary equivalence fits into our ``restriction by averaging\" framework. \nThis requires looking at restrictions by averaging of a more\ngeneral type than considered above. \nTo a given integral Hankel operator ${\\mathbf H}({\\mathbf a})$ we\nassociate the Hankel matrix $H(\\alpha)$ with\n$$\n\\alpha_j=\\int_0^\\infty {\\mathbf a}(t){\\bm{\\varphi}}_j(t)dt,\\quad j\\geq0,\n$$\nwhere ${\\bm{\\varphi}}_j$ is a certain sequence of smooth functions, \na more general one than just translations of a single function. \nOur sequence ${\\bm{\\varphi}}_j$ will be given by the multiple convolution of the form\n$$\n{\\bm{\\varphi}}_j={\\bm{\\varphi}}*\\underbrace{\\nu*\\nu*\\cdots*\\nu}_{\\text{$j$ terms}},\\quad j\\geq0,\n$$\nwhere ${\\bm{\\varphi}}$ is a sufficiently regular function supported on $[0,\\infty)$, \nand $\\nu$ is a positive finite measure supported on $[0,\\infty)$. \nObserve that if $d\\nu(t)=\\delta(t-1)dt$, then ${\\bm{\\varphi}}_j(t)={\\bm{\\varphi}}(t-j)$, \nso we recover the definition of $\\mathcal{R}_\\varphi$. \n\n\nTo make the multiple convolution notation more readable, we introduce the \n(formal) convolution with $\\nu$ operator\n$$\nT_\\nu{\\mathbf f}={\\mathbf f}*\\nu;\n$$\nthen ${\\bm{\\varphi}}_j=T_\\nu^j {\\bm{\\varphi}}$. \n\n\n\n\\begin{theorem}\\label{convolution}\nLet $\\nu$ be a positive measure on $[0,\\infty)$ with $\\nu([0,\\infty))\\leq1$, \nand let ${\\bm{\\varphi}}\\in C({\\mathbb R})$ satisfy $\\supp{\\bm{\\varphi}}\\subset[0,\\infty)$ and \n\\[\n\\abs{\\widecheck {\\bm{\\varphi}}(\\xi)} \\le \\frac{C}{1+\\xi^2}, \\quad \\xi\\in{\\mathbb R}.\n\\label{convolution1}\n\\]\nFor $j\\geq0$, set ${\\bm{\\varphi}}_j=T_\\nu^j {\\bm{\\varphi}}$ and consider the map\n$$\n{\\mathbf a}(t)\\mapsto \\alpha=\\{\\alpha(j)\\}_{j=0}^\\infty, \n\\quad\n\\alpha(j)=\\int_0^\\infty {\\mathbf a}(t){\\bm{\\varphi}}_j(t) dt. \n$$\nThen there exist bounded operators $\\Phi_1$ and $\\Phi_2$ acting\nfrom $\\ell^2({\\mathbb Z}_+)$ to $L^2(0,\\infty)$ such that\n\\[\n\\Phi_2^*{\\mathbf H}({\\mathbf a})\\Phi_1=H(\\alpha). \n\\label{d2}\n\\]\nConsequently, \n$$\n\\norm{H(\\alpha)}\\leq A\\norm{{\\mathbf H}({\\mathbf a})}\n\\quad\\text{ and }\\quad\n\\norm{H(\\alpha)}_{\\mathbf{S}_p}\\leq A\\norm{{\\mathbf H}({\\mathbf a})}_{\\mathbf{S}_p}\n$$\nfor all $00$, which we assume. Provided the Lorentz invariant expression $(\\vec B)^2-(\\vec E)^2$ is positive, $(\\vec B)^2$ will be positive in any Lorentz frame. In discussing undamped wave propagation we will not use the inequality $(\\vec B)^2-(\\vec E)>0$, but in treating damped longitudinal mode propagation in Appendix B, we\nwill assume that $(\\vec E)^2\/(\\vec B)^2$ is small, as motivated by the fact that when $(\\vec E)^2$ is of order\n $(\\vec B)^2$ the vacuum is highly unstable against pair creation. \\big(Strictly speaking, the vacuum is stable against\npair production only when $\\vec E \\cdot \\vec B=0$ and $(\\vec B)^2 -(\\vec E)^2 >0$, that is, when there is\na Lorentz frame in which the Abelian field has vanishing $\\vec E$ \\cite{schwinger}.\\big)\n\nGiven that $(\\vec B)^2>0$, we can solve the constraint $\\omega=0$ of Eq. \\eqref{eq:omegatheta2} for\n$\\Psi_0$, giving\n\\begin{equation}\\label{eq:solve1}\n\\Psi_0=\\frac{\\vec Q \\cdot \\vec \\Psi}{(\\vec B)^2}~~~,\n\\end{equation}\nwhere we have defined\n\\begin{equation}\\label{eq:qdef}\n\\vec Q\\equiv \\vec \\sigma \\cdot \\vec B (\\vec B + \\vec \\sigma \\times \\vec E)=\n\\vec B\\times \\vec E + \\vec B \\vec \\sigma \\cdot (\\vec B +i \\vec E)-i \\vec B \\cdot \\vec E \\vec \\sigma~~~.\n\\end{equation}\nSubstituting the solution for $\\Psi_0$ into Eq. \\eqref{eq:d0psi2}, we get an equation of motion for $\\vec \\Psi$\nby itself,\n\\begin{equation}\\label{eq:d0psi3}\nD_0 \\vec \\Psi=\\vec D \\frac{\\vec Q \\cdot \\vec \\Psi}{(\\vec B)^2}+ i \\vec D \\times \\vec \\Psi~~~.\n\\end{equation}\n\n\nTo determine the wave propagation velocity in the neighborhood of a spacetime point $x_*=(t_*,\\vec x_*)$, we need to calculate the equation for\nthe wavefronts, or characteristics, at that point. Writing the first order Eq. \\eqref{eq:d0psi3} in the form\n\\begin{equation}\\label{eq:d0psi4}\n\\partial_0 \\vec \\Psi=\\vec \\nabla \\frac{\\vec Q_* \\cdot \\vec \\Psi}{(\\vec B_*)^2}+ i \\vec \\nabla\\times \\vec \\Psi+\n\\vec \\Delta[\\vec \\Psi, x_*, x]~~~,\n\\end{equation}\nwith $\\vec B_*$ and $\\vec Q_*$ the values of the respective quantities at $ x_*$,\nwe see that $\\vec \\Delta[\\vec \\Psi, x_*, x]$ involves no first derivatives of $\\vec \\Psi$ at $ x_*$, and so is not needed \\cite{courant},\n \\cite{madore} for determining the wavefronts of Eq. \\eqref{eq:d0psi2}. The reason is that when taking an infinitesimal line integral\nof Eq. \\eqref{eq:d0psi4}, according to\n\\begin{equation}\\label{eq:lineint}\n\\lim_{\\delta \\to 0} \\int_{-\\delta}^{\\delta} d\\ell [\\partial_0 \\vec \\Psi = ...]~~~,\n\\end{equation}\ndiscontinuities across wavefronts contribute through the first derivative terms, but when the external fields are smooth\nthe term $\\vec \\Delta[\\vec \\Psi, x_*, x]$ makes a vanishing contribution as $\\delta \\to 0$.\nDropping $\\vec \\Delta$, and multiplying through by $(\\vec B_*)^2$, we get the equation determining the wavefronts in the form\n\\begin{equation}\\label{eq:d0psifinal}\n(\\vec B_*)^2 \\partial_0 \\vec \\Psi=\\vec \\nabla \\vec Q_* \\cdot \\vec \\Psi+ i (\\vec B_*)^2 \\vec \\nabla\\times \\vec \\Psi~~~.\n\\end{equation}\nBy similar reasoning, the constraint $\\chi$ can be simplified, for purposes of determining the wavefronts, by replacing\n$\\vec D$ by $\\vec \\nabla$, giving\n\\begin{equation}\\label{eq:newchi}\n0=\\vec \\sigma \\cdot \\vec \\nabla \\times \\vec{\\Psi}~~~.\n\\end{equation}\n\n\nSince these are now linear equations with constant coefficients, the solutions are plane waves, and without loss of generality we can take the negative $z=x_3$ axis as the direction of wave propagation. So making the Ansatz\n\\begin{equation}\\label{eq:ansatz}\n\\vec \\Psi= \\vec C \\exp(i \\Omega t + i K z)~~~,\n\\end{equation}\n Eq. \\eqref{eq:d0psifinal} for the wavefronts or characteristics takes the form\n\\begin{equation}\\label{eq:ceq}\n0=\\vec F\\equiv (\\vec B_*)^2 \\Omega \\vec C- K \\hat z\\vec Q_* \\cdot \\vec C- i (\\vec B_*)^2 K \\hat z \\times \\vec C~~~,\n\\end{equation}\nwith $\\hat z$ a unit vector along the $z$ axis, and the constraint Eq. \\eqref{eq:newchi} becomes an admissability\ncondition on $\\vec C$,\n\\begin{equation}\\label{eq:newchi1}\n0=\\vec \\sigma \\cdot \\hat z \\times \\vec C~~~.\n\\end{equation}\n\n\nWriting $F_m$ as a matrix times $C_n$ (and dropping the subscripts $*$, which are implicit from here on) we have\n\\begin{align}\\label{eq:matrix}\nF_m=&N_{mn}C_n~~~,\\cr\nN_{mn}=&(\\vec B)^2 \\Omega \\delta_{mn}- K \\delta_{m3} Q_n - i (\\vec B)^2 K \\epsilon_{m3n}~~~.\\cr\n\\end{align}\nThe equation for the characteristics is now\n\\begin{equation}\\label{eq:char}\n{\\rm det} (N)=0~~~,\n\\end{equation}\nsince this is the condition for Eq. \\eqref{eq:ceq} to have a solution with nonzero $\\vec C$. However, since evaluation of\nthe determinant shows that it factorizes into blocks that determine $C_{1,2}$ and a block that determines $C_3$, a simpler\nway to proceed is to work directly from the equations $F_m=0$, which decouple in a corresponding way. Calculating from\nEq. \\eqref{eq:ceq}, we find\n\\begin{align}\\label{eq:fcomp}\n0=&F_1^{\\uparrow,\\downarrow}=(\\vec B)^2 \\big(\\Omega C_1^{\\uparrow,\\,\\downarrow}+iK C_2^{\\uparrow,\\,\\downarrow}\\big)~~~,\\cr\n0=&F_2^{\\uparrow,\\,\\downarrow}=(\\vec B)^2 \\big(\\Omega C_2^{\\uparrow,\\,\\downarrow}-iK C_1^{\\uparrow,\\,\\downarrow}\\big)~~~,\\cr\n0=&F_3^{\\uparrow,\\,\\downarrow}=(\\vec B)^2 \\Omega C_3^{\\uparrow,\\,\\downarrow}- K (\\vec Q \\cdot \\vec C)^{\\uparrow,\\,\\downarrow}~~~,\\cr\n\\end{align}\nwhere $\\uparrow,\\, \\downarrow$ indicate the up and down spinor components, labeled in Eq. \\eqref{eq:Psidef} by $\\alpha=1,\\,2$. Similarly,\nthe constraint Eq. \\eqref{eq:newchi1} becomes $0= -\\sigma_1 C_2 +\\sigma_2 C_1$, that is\n\\begin{align}\\label{eq:newchi2}\nC_2^{\\uparrow}=&iC_1^{\\uparrow}~~~,\\cr\nC_2^{\\downarrow}=&-i C_1^{\\downarrow}~~~,\\cr\n\\end{align}\nwith no corresponding condition on $C_3^{\\uparrow,\\,\\downarrow}$.\nThe first two lines of Eq. \\eqref{eq:fcomp} together with Eq. \\eqref{eq:newchi2} have the solution\n\\begin{align}\\label{eq:c12soln}\nC_1^{\\uparrow}=&C~,~~C_2^{\\uparrow}=iC~,~~\\Omega=K~~~,\\cr\nC_1^{\\downarrow}=&C~,~~C_2^{\\downarrow}=-iC~,~~\\Omega=-K~~~,\\cr\n\\end{align}\nwith C arbitrary, corresponding to waves with velocity of magnitude $|\\Omega\/K|=1$. Thus the modes with $C_{1,2}\\neq 0$ are\nexactly luminal. Because general background gauge fields are a non-isotropic medium, these modes have nonzero longitudinal\ncomponents given by solving the third line of Eq. \\eqref{eq:fcomp},\n\\begin{equation}\\label{eq:solveforc3}\nC_3=K \\big((\\vec B)^2 \\Omega-K Q_3\\big)^{-1} (Q_1 C_1+ Q_2 C_2)~~~.\n\\end{equation}\n\n\nThe effect on the characteristics of a gauge change $\\vec \\Psi \\to \\vec \\Psi + \\vec D \\epsilon$, $\\epsilon= E \\exp(i \\Omega t + i K z)f(t,z)$, where $f$ has a unit slope\ndiscontinuity along the $z$ axis at $x_*$, is to shift $C_3^{\\uparrow,\\downarrow} \\to C_3^{\\uparrow,\\downarrow} + E^{\\uparrow,\\downarrow} $,\nand thus $ C_3^{\\uparrow,\\downarrow}$ are gauge degrees of freedom.\nIn Appendix B, we continue this discussion and show that the longitudinal gauge mode with $C_1=C_2=0, C_3 \\neq 0$ also does not propagate superluminally, although in general it is subluminal.\n\n\n\n\\section{Failure of adiabatic decoupling and inapplicability of the $S$-matrix ``no-go'' theorems}\n\nWe show in this section that various ``no-go'' theorems that claim to rule out gauging of higher spin theories\ndo not apply to the gauged Rarita-Schwinger field. The reason is that there is a failure of adiabatic decoupling, arising\nfrom the fact that the $\\omega$ secondary constraint is homogeneous in the gauge fields. For a recent paper on\n``no-go'' theorems see \\cite{mcgady}, which has extensive references to the earlier literature. In our analysis here\nwe shall refer specifically to the paper of Porrati \\cite{porrati}, which uses so called ``on-shell'' methods to give limits on\nmassless high-spin particles.\n\nThe analysis of Porrati assumes that ``the general helicity-conserving matrix element of a $U(1)$ current\nbetween on-shell spin $s$ states is $\\langle v, p+q|J_{\\mu}|u,p\\rangle$...'', where $u$ and $v$ are free-space spinors that obey the massless\nDirac equation. Porrati assumes that the matrix element is bilinear in $u$ and $v$, and ``otherwise depends only on\nthe momenta''. We shall see in the following subsections that this assumed form is not realized in the gauged Rarita-Schwinger\ntheory, where because of the failure of adiabatic decoupling the matrix element in question also depends on\nthe $U(1)$ gauge field polarization through the dual field-strength $\\hat F_{\\mu \\nu}=\\frac{1}{2}\\epsilon_{\\mu\\nu\\lambda\\sigma} F^{\\lambda\\sigma}$. In fact, the initial and final Rarita-Schwinger\nspinors both must have a $\\hat F_{\\mu \\nu}$ dependence in order to obey the secondary constraint of Eq. \\eqref{eq:constraint2}, and\nso the matrix element has the more complicated form $\\langle v, p+q, \\hat F_{\\mu \\nu} |J_{\\mu}|u,p, \\hat F_{\\mu \\nu}\\rangle$.\n\nWe show in Sec. 7A that the initial and final Rarita-Schwinger spinors in the limit of zero gauge field amplitude are equal to free-space spinors $u,v$ of the form assumed by\nPorrati, plus a fermionic gauge transformation that depends explicitly on the photon field strength $\\hat F_{\\mu \\nu}$. This structure arises from the homogeneous form of the secondary constraint, and corresponds to\nan intrinsically non-perturbative aspect of the gauged Rarita-Schwinger\nequation. As another reflection of this, we show in Sec. 7B that one cannot set up a covariant Lippmann-Schwinger equation \\cite{lippmann} for the\nRarita-Scwhinger wave function, and so the matrix element that enters into the ``no-go'' theorems does not admit a Born approximation.\nIn Sec. 7C, we show that a matrix element that has all the required invariances can be formulated using an analog of the\ndistorted wave Born approximation, in which the initial and final Rarita-Schwinger states have an explicit dependence on the photon\npolarizations.\n\n\n\\subsection{The zero amplitude limit of the $\\vec\\Psi$ equation: retained memory of the gauge field}\n\nAs in Sec. 6, let us consider a Rarita-Schwinger field propagating in an external Abelian gauge field. For convenience, we\nassume that the ratio $|\\vec E(\\vec x)|\/|\\vec B(\\vec x)|\\equiv r(\\vec x)$ is bounded from above. In the limit as the\nvector potential amplitude $\\vec A$ is scaled to zero, Eqs. \\eqref{eq:solve1} and \\eqref{eq:qdef} become\n\\begin{align}\\label{eq:psi0lim}\n\\Psi_0(\\vec x)=&\\vec R(\\vec x) \\cdot \\vec \\Psi(\\vec x)~~~,\\cr\n\\vec R(\\vec x)=&\\vec \\sigma \\cdot \\hat B(\\vec x) \\big(\\hat B(\\vec x) + r(\\vec x)\\vec \\sigma \\times \\hat E(\\vec x)\\big)~~~,\\cr\n\\end{align}\nwith $\\hat B = \\vec B \/|\\vec B|$ and $\\hat E=\\vec E\/|\\vec E|$ unit vectors along the $\\vec E$ and $\\vec B$ fields. When the\nexternal field is a propagating plane wave with wave vector direction $\\hat q$, the unit vectors $\\hat q$, $\\hat B$ and $\\hat E$\nform an orthonormal set of constant unit vectors, and $|\\vec r(\\vec x)|=1$. We see that because the secondary constraint of Eq. \\eqref{eq:constraint2} is homogeneous in the field strengths, the relation between $\\Psi_0$ and $\\vec \\Psi$ retains a memory of the gauge field orientations, and thus of the photon polarization, even in the limit as the field amplitude approaches zero.\n\nIn the zero amplitude limit, $D_0=\\partial_0$ and $\\vec D =\\vec \\nabla$, so substituting Eq. \\eqref{eq:psi0lim} into Eq. \\eqref{eq:d0psi3}, the zero amplitude limit for the equation of motion for $\\vec \\Psi$ becomes\n\\begin{equation}\\label{eq:limd0psi}\n\\partial_0 \\vec \\Psi=\\vec \\nabla \\vec R \\cdot\\vec \\Psi+ i \\vec \\nabla \\times \\vec \\Psi~~~.\n\\end{equation}\nwith the primary constraint now $\\vec \\sigma \\cdot \\vec \\nabla \\times \\vec \\Psi=0$.\nHence through $\\vec R$ the $\\vec \\Psi$ equation of motion retains a memory of the external fields in the limit of zero amplitude, that is, adiabatic\ndecoupling has failed. Let us now consider the situation in which the Rarita-Schwinger field and the external gauge fields are plane waves, so that $\\vec R$ is a constant and $\\vec \\Psi$ has the form\n\\begin{equation}\\label{eq:psiwave}\n\\vec \\Psi=\\vec C e^{i(\\Omega t+\\vec k \\cdot \\vec x)} ~~~.\n\\end{equation}\nMaking the fermionic gauge transformation\n\\begin{align}\\label{eq:gaugechange}\n\\vec \\Psi &\\to \\vec \\Psi^{\\prime}=\\vec \\Psi + \\vec \\nabla \\epsilon~~~,\\cr\n\\epsilon =& E e^{i(\\Omega t + \\vec k \\cdot \\vec x)}~~~,\\cr\n\\end{align}\n$\\vec \\Psi^{\\prime}$ still obeys the zero amplitude primary constraint since $\\vec \\sigma \\cdot \\vec \\nabla \\times \\vec \\nabla \\epsilon=0$.\nThen the gauge choice\n\\begin{equation}\\label{eq:echoice}\nE=i\\frac{\\vec R \\cdot \\vec C}{\\vec R \\cdot \\vec k}\n\\end{equation}\nreduces Eq. \\eqref{eq:limd0psi} to the free-space form\n\\begin{equation}\\label{eq:limd0psi1}\n\\partial_0 \\vec \\Psi^{\\prime}= i \\vec \\nabla \\times \\vec \\Psi^{\\prime}~~~.\n\\end{equation}\nThus a Rarita-Schwinger plane wave in a zero amplitude gauge field plane wave background is equal to a free-space solution plus\na gauge term that has a memory of the photon polarizations.\n\n\\subsection{Breakdown of the Lippmann-Schwinger equation: no Born approximation to scattering}\n\nLet us now examine what happens if one tries to set up a covariant Lippmann-Schwinger equation, so as to generate\na Born perturbation series for the Rarita-Schwinger wave function in an external gauge field. Let us start from\nthe Rarita-Schwinger equation in the form (see Eq. \\eqref{eq:a6})\n\\begin{equation}\\label{eq:gammars}\n\\gamma^{\\eta\\nu\\rho}D_{\\nu}\\psi_{\\rho}=0~~~.\n\\end{equation}\nSplitting $D_{\\nu}$ into $\\partial_{\\nu}$ and $gA_{\\nu}$,\nthis equation takes the form\n\\begin{equation}\\label{eq:gammars1}\n\\gamma^{\\eta\\nu\\rho}\\partial_{\\nu}\\psi_{\\rho}=-\\gamma^{\\eta\\nu\\rho}gA_{\\nu}\\psi_{\\rho}~~~.\n\\end{equation}\nLet us now try to solve this equation as a perturbation series around a free-space\nsolution by writing\n\\begin{equation}\\label{eq:lippmann1}\n\\psi_{\\rho}(x)=\\psi_{\\rho}^{\\rm free}(x)+ \\int d^4y S_{\\rho \\alpha}(x-y) \\gamma^{\\alpha\\beta\\kappa}g A_{\\beta}(y) \\psi_{\\kappa}(y)~~~,\n\\end{equation}\nwhere $\\psi_{\\rho}^{\\rm free}$ obeys the free-space Rarita-Schwinger equation\n\\begin{equation}\\label{eq:freespace1}\n\\gamma^{\\eta\\nu\\rho}\\partial_{\\nu} \\psi_{\\rho}^{\\rm free}=0.\n\\end{equation}\n If the free-space Green's Rarita-Schwinger Green's function $S_{\\rho \\alpha}(x-y)$ obeyed\n\\begin{equation}\\label{eq:freegreen1}\n\\gamma^{\\eta\\nu\\rho}\\partial_{x \\nu} S_{\\rho \\alpha}(x-y)=-\\delta^{\\eta}_{\\alpha}\\delta^4(x-y)~~~,\n\\end{equation}\nthen Eq. \\eqref{eq:lippmann1} would reproduce Eq. \\eqref{eq:gammars1}. But in fact the free-space\nGreen's function cannot obey Eq. \\eqref{eq:freegreen1}, because $\\partial_{x\\eta}\\gamma^{\\eta\\nu\\rho}\\partial_{x\\nu} S_{\\rho \\alpha}(x-y)=0$;\ninstead it obeys \\cite{freed}\n\\begin{equation}\\label{eq:freegreen2}\n\\gamma^{\\eta\\nu\\rho}\\partial_{x \\nu} S_{\\rho \\alpha}(x-y)=-\\delta^{\\eta}_{\\alpha}\\delta^4(x-y)+\\partial_{y\\alpha}\\Omega^{\\eta}(x-y)~~~,\n\\end{equation}\nwith $\\Omega$ necessarily nonvanishing.\nIntegrating $\\partial_{y\\alpha}$ by parts onto the factor $\\gamma^{\\alpha\\beta\\kappa}g A_{\\beta}(y) \\psi_{\\kappa}(y)$, one\ngets\n\\begin{equation}\\label{eq:intbyparts}\n\\gamma^{\\alpha\\beta\\kappa}gF_{\\alpha \\beta}(y) \\psi_{\\kappa}(y)\n+\\gamma^{\\alpha\\beta\\kappa}g A_{\\beta}(y)\\partial_{y\\alpha} \\psi_{\\kappa}(y)~~~.\n\\end{equation}\nThe first term of this expression vanishes by virtue of the secondary constraint, but the second term is\nnon-vanishing because the Rarita-Schwinger equation for the exact wave function $\\psi_{\\kappa}(y)$ is\n\\begin{equation}\\label{eq:intby parts1}\n\\gamma^{\\alpha\\beta\\kappa}D_{y\\alpha} \\psi_{\\kappa}(y)=0~~~,\n\\end{equation}\nthat is, it requires the full covariant derivative $D_{y\\alpha}$ in place of its free-space restriction $\\partial_{y\\alpha}$.\nThe conclusion from this analysis is that one cannot set up a covariant Lippmann-Schwinger equation for the gauged Rarita-Schwinger\nwave function, and thus one cannot develop this wave function into a Born approximation series expansion in powers of the\ncoupling $g$ to the external gauge field.\n\n\n\\subsection{Lorentz covariance and mode counting in on-shell Rarita-Schwinger field-photon scattering: a distorted\nwave Born approximation analog}\nWe address finally the question \\cite{witten} of whether one can write down an amplitude for leading order on-shell scattering of Rarita-Schwinger fields from an external electromagnetic field, which has the requisite relativistic covariance while preserving the correct counting of massless spin $\\frac{3}{2}$ propagation modes. Looking ahead to quantization, an operator effective action for this scattering process can be inferred from the interaction term in Eq. \\eqref{eq:action},\n\\begin{align}\\label{eq:effaction}\nS_{\\rm eff}(\\psi_{\\mu},A_{\\nu}) = &\\int d^4x {\\cal L}_{\\rm eff}(\\psi_{\\mu},A_{\\nu})~~~,\\cr\n{\\cal L}_{\\rm eff}(\\psi_{\\mu},A_{\\nu})=& \\frac{1}{2}g\\,\\overline{\\psi}_{\\mu} (x)\ni\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}A_{\\nu}(x) \\psi_{\\rho}(x)~~~,\\cr\n\\end{align}\nwhere we have suppressed spinor indices as in the text from Eq. \\eqref{eq:eqmo} onwards.\nFor Abelian external fields $A_{\\nu}$, the covariant derivatives in the\nequations of motion and constraints are given by\n\\begin{equation}\\label{eq:covdeviv}\nD_{\\nu}=\\partial_{\\nu}+gA_{\\nu}~~,~~\\overleftarrow{D}_{\\nu}=\\overleftarrow{\\partial}_{\\nu}-gA_{\\nu}~~~.\n\\end{equation}\nAt the outset we shall assume that $A_{\\nu}(x)$ is of short range, and vanishes for $|\\vec x|>R$ for\nsome radius $R$.\nThis effective action, the equations of motion of Eqs. \\eqref{eq:eqmo} and \\eqref{eq:eqmo1a}, and the primary and secondary\nconstraints following from them, given in Eqs. \\eqref{eq:constraint1} and \\eqref{eq:constraint2}, are all relativistically covariant, and so\nprovide a starting point for calculating a covariant scattering amplitude.\nTaking the matrix element of Eq. \\eqref{eq:effaction} between an incoming Rarita-Schwinger state of four-momentum $p$, and an outgoing Rarita-Schwinger\nstate of four momentum $p'$, we get the corresponding scattering amplitude\n\\begin{equation}\\label{eq:amplitude}\n{\\cal A}_S = \\frac{1}{2}ig\\,\\int d^4x\n \\overline{\\psi}_{\\mu }(p^{\\prime},x)\n\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}A_{\\nu}(x) \\psi_{\\rho}(p,x)~~~,\n\\end{equation}\nwhere $\\psi_{\\rho}$ and $\\overline{\\psi}_{\\mu}$ are now wave functions, rather than operators, that\nobey the Rarita-Schwinger equations of motion in the presence of the external field $A_{\\nu}$.\n\n\nWe now introduce source currents for the gauge potential $A_{\\nu}$ and the Rarita-Schwinger\nwave functions $\\psi_{\\rho}$ and $\\overline{\\psi}_{\\mu}$, and study their conservation properties. The source current\nto which the gauge potential $A_{\\nu}$ couples is defined by writing the scattering amplitude as\n\\begin{align}\\label{eq:source1}\n{\\cal A}_S=&\\frac{1}{2}ig\\,\\int d^4x A_{\\nu}(x) J^{\\nu}(x)~~~,\\cr\nJ^{\\nu}(x)=& \\overline{\\psi}_{\\mu }(p^{\\prime},x)\n\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta} \\psi_{\\rho}(p,x)~~~.\\cr\n\\end{align}\nThe source current for the Rarita-Schwinger field $\\overline{\\psi}_{\\mu }(p^{\\prime},x)$ is\ndefined by writing the scattering amplitude as\n\\begin{align}\\label{eq:source2}\n{\\cal A}_S=&\\frac{1}{2}ig\\,\\int d^4x\\overline{\\psi}_{\\mu }(p^{\\prime},x){\\cal J}^{\\mu}(p,x)~~~,\\cr\n{\\cal J}^{\\mu}(p,x)=&\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}A_{\\nu}(x) \\psi_{\\rho}(p,x)~~~.\\cr\n\\end{align}\nFinally, the source current for the Rarita-Schwinger field $ \\psi_{\\rho}(p,x)$ is\ndefined by writing the scattering amplitude as\n\\begin{align}\\label{eq:source3}\n{\\cal A}_S=&\\frac{1}{2}ig\\,\\int d^4x \\overline{\\cal J}^{\\rho}(p^{\\prime},x) \\psi_{\\rho}(p,x)~~~,\\cr\n\\overline{\\cal J}^{\\rho}(p^{\\prime},x)=&\\overline{\\psi}_{\\mu }(p^{\\prime},x)\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}A_{\\nu}(x) ~~~.\\cr\n\\end{align}\n\nWe now show that the three currents that we have just defined are conserved. For the source current $J^{\\nu}$ for the gauge potential,\nwe have\n\\begin{align}\\label{eq:cons1}\n \\partial_{\\nu}J^{\\nu}=&\n \\overline{\\psi}_{\\mu }(p^{\\prime},x)\\overleftarrow{D}_{\\nu}\n\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta} \\psi_{\\rho}(p,x)\\cr\n +& \\overline{\\psi}_{\\mu }(p^{\\prime},x)\n\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta} D_{\\nu}\\psi_{\\rho}(p,x)\\cr\n=&0~~~,\\cr\n\\end{align}\nwhere the first and second terms on the right vanish by the Rarita-Schwinger equations\nfor $ \\overline{\\psi}_{\\mu }(p^{\\prime},x)$ and $\\psi_{\\rho}(p,x)$ respectively.\nFor the source current ${\\cal J}^{\\mu}(p,x)$ for the spinor $\\overline{\\psi}_{\\mu }(p^{\\prime},x)$ , we\nhave\n\\begin{align}\\label{eq:cons2}\nD_{\\mu}{\\cal J}^{\\mu}(p,x)=&\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}\\big(\\partial_{\\mu}A_{\\nu}(x)\\big) \\psi_{\\rho}(p,x)\\cr\n+&\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}A_{\\nu}(x) D_{\\mu}\\psi_{\\rho}(p,x)\\cr\n=&0~~~.\\cr\n\\end{align}\nThe second term on the right vanishes by the Rarita-Schwinger equation\nfor $\\psi_{\\rho}(p,x)$, while the first term on the right can be rewritten as\n\\begin{equation}\\label{eq:cons2a}\n\\frac{1}{2}\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}F_{\\mu\\nu}(x)\\psi_{\\rho}(p,x)\n\\end{equation}\nand vanishes by the secondary constraint of Eq. \\eqref{eq:constraint2}.\nFinally, for the source current $\\overline{\\cal J}^{\\rho}(p^{\\prime},x)$ for the spinor $ \\psi_{\\rho}(p,x)$, we have\n\\begin{align}\\label{eq:cons3}\n\\overline{\\cal J}^{\\rho}(p^{\\prime},x)\\overleftarrow{D}_{\\rho}=&\\overline{\\psi}_{\\mu }(p^{\\prime},x)\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}\\big(\\partial_{\\rho}A_{\\nu}(x)\\big)\\cr\n+&\\overline{\\psi}_{\\mu }(p^{\\prime},x)\\overleftarrow{D}_{\\rho}\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}A_{\\nu}(x)\\cr\n=&0~~~.\\cr\n\\end{align}\nAgain, the second term on the right vanishes by the Rarita-Schwinger equation, while the first term on the right\nvanishes by the secondary constraint of Eq. \\eqref{eq:constraint2}.\n\n\nConsider now the following three gauge transformations,\n\\begin{align}\\label{eq:gauge1}\nA_{\\nu}(x) \\to & A_{\\nu}(x) + \\partial_{\\nu}\\Lambda~~~,\\cr\n\\psi_{\\rho}(p,x) &\\to \\psi_{\\rho}(p,x)+ D_{\\rho} \\alpha~~~,\\cr\n\\overline{\\psi}_{\\mu }(p^{\\prime},x) \\to & \\overline{\\psi}_{\\mu }(p^{\\prime},x) + \\overline{\\beta}\\overleftarrow{D}_{\\mu}~~~,\\cr\n\\end{align}\nwith $\\alpha$ and $\\beta$ independent spinorial gauge parameters. From Eqs. \\eqref{eq:source1}-\\eqref{eq:source3},\ntogether with Eqs. \\eqref{eq:cons1}-\\eqref{eq:cons3},\nwe find that these transformations each leave the amplitude ${\\cal A}_S$ invariant,\n\\begin{align}\n\\delta_{\\Lambda} {\\cal A}_S= & \\frac{1}{2}ig\\,\\int d^4x \\big(\\partial_{\\nu}\\Lambda\\big) J^{\\nu}(x)\n=-\\frac{1}{2}ig\\,\\int d^4x \\Lambda \\partial_{\\nu} J^{\\nu}(x)=0~~~,\\cr\n\\delta_{\\alpha} {\\cal A_S}=&\\frac{1}{2}ig\\,\\int d^4x \\overline{\\cal J}^{\\rho}(p^{\\prime},x) D_{\\rho} \\alpha\n=-\\frac{1}{2}ig\\,\\int d^4x \\overline{\\cal J}^{\\rho}(p^{\\prime},x)\\overleftarrow{D}_{\\rho} \\alpha=0~~~,\\cr\n\\delta_{\\beta} {\\cal A}_S=& \\frac{1}{2}ig\\,\\int d^4x \\overline{\\beta}\\overleftarrow{D}_{\\mu}{\\cal J}^{\\mu}(p,x)\n= -\\frac{1}{2}ig\\,\\int d^4x \\overline{\\beta}D_{\\mu}{\\cal J}^{\\mu}(p,x)=0~~~.\\cr\n\\end{align}\nThis, together with the primary and secondary constraints, implies the correct mode-counting for the Rarita-Schwinger\nwave functions, since the gauge degrees of freedom do not change the amplitude and so are redundant.\n\n We next must\nspecify more precisely the structure of the spinor wave functions entering the formula for ${\\cal A}_S$. Since the\ngauge field $A_{\\nu}$ is assumed to vanish in the external region $|\\vec x|>R$, the Rarita-Schwinger wave functions obey\nfree field equations in this region. So for $|\\vec x|>>R$ they can be taken asymptotically as plane waves at $t \\to \\pm \\infty$,\n\\begin{align}\\label{eq:extsoln}\n\\psi_{\\mu}(p^{\\prime},x)\\sim &u_{\\mu}(p^{\\prime})e^{i p^{\\prime} \\cdot x}~~,~~t \\to +\\infty~~~,\\cr\n\\psi_{\\rho}(p,x)\\sim & u_{\\rho}(p) e^{i p \\cdot x}~~,~~t \\to -\\infty ~~~.\\cr\n\\end{align}\nWith these boundary conditions, the formula for the amplitude takes the final form\n\\begin{equation}\\label{eq:amplitudefinal}\n{\\cal A}_S = \\frac{1}{2}ig\\,\\int d^4x\n \\overline{\\psi}^{(-)}_{\\mu }(p^{\\prime},x)\n\\epsilon^{\\mu\\eta\\nu\\rho}\\gamma_5\\gamma_{\\eta}A_{\\nu}(x) \\psi^{(+)}_{\\rho}(p,x)~~~.\n\\end{equation}\nThe out state (-) and in state (+) boundary conditions used here are analogs of the\nboundary conditions used in\nthe distorted wave Born approximation \\cite{dwba}, which the construction of Eq. \\eqref{eq:amplitudefinal} resembles.\nEquation \\eqref{eq:amplitudefinal} then gives an approximation to the matrix element for\nRarita-Schwinger scattering by the gauge potential.\n\nRather than invoking the presence of redundant degrees of freedom to count physical Rarita-Schwinger states, we can\nfollow the usual procedure of imposing a gauge-fixing constraint. To preserve relativistic and gauge covariance, this\ncan be taken as the gauge covariant Lorentz gauge condition\n\\begin{equation}\\label{eq:gaugecovlor1}\n \\overline{\\psi}_{\\mu}(p^{\\prime},x)\\overleftarrow{D}^{\\mu}= D^{\\rho}\\psi_{\\rho}(p,x)=0~~~.\n \\end{equation}\nwhich is attainable from a generic gauge by the gauge transformation of Eq. \\eqref{eq:gaugetrans}, provided that $D^{\\mu}D_{\\mu}$ is invertible.\nIn the external region where the gauge field vanishes, one can instead use the condition $\\gamma^{\\rho}\\psi_{\\rho}=0$ in place\nof the secondary constraint together with the gauge condition $\\partial^{\\rho}\\psi_{\\rho}=0$, giving the usual covariant\ndegree of freedom counting for the incoming and outgoing Rarita-Schwinger wave functions \\cite{alvarez}. Alternatively,\nif we are not concerned to maintain manifest Lorentz covariance, we can make a gauge transformation in the external region to\nthe gauge $\\psi_0= \\vec \\nabla \\cdot \\vec \\psi=0$ used in \\cite{dasb}, \\cite{freed} to enumerate Rarita-Schwinger degrees of freedom. When\na non-Lorentz covariant radiation gauge condition is used, scattering matrix elements depend on a unit timelike vector in addition to the\nparticle momenta, and so the conditions assumed in \\cite{porrati} are not obeyed.\n\n\nNote that if one were to attempt to construct a Born approximation amplitude, in which the Rarita-Schwinger wave functions\nin the presence of the gauge field are replaced by plane waves in the interior region where the potential is nonzero, the\narguments given above for compatibility of Lorentz covariance with degree of freedom counting would fail.\nThe reason for this is that the spinor source\ncurrents would then no longer be conserved, even to zeroth order in the gauge coupling $g$, because the free particle plane wave solutions\ndo not obey the secondary constraint of Eq. \\eqref{eq:constraint2}. The non-existence of a satisfactory Born approximation for\nRarita-Schwinger photon scattering agrees with the result obtained in Sec. 7B, that one cannot construct a Lippmann-Schwinger\nequation for this process.\nTo establish compatibility, we have had to use an analog\nof the distorted wave Born approximation \\cite{dwba}, in which the leading approximation to the amplitude is constructed using interacting rather than free fermion wave functions and does not have a perturbation expansion for small coupling, $g$ .\n\nWhen the external Abelian potential is a plane wave field which extends to infinity, there is no large $|\\vec x|$ region where the\nRarita-Schwinger solutions reduce to free-space ones. Rather, as shown in Sec. 7A, in the adiabatic decoupling limit of a zero\namplitude gauge field, the Rarita-Schwinger solutions become free-space solutions plus gauge terms that remember the photon polarization,\nand which are necessary to enforce the secondary constraint. Thus one cannot attain the kinematic form assumed in the on-shell\n``no-go'' theorems. But as shown here, using distorted Born approximation waves one can write down a consistent\ncovariant scattering amplitude.\n\n\\section{Summary and Remarks}\nTo conclude, we see that unlike the massive case, the massless gauged Rarita-Schwinger equation leads to a consistent classical theory. The theory has the correct counting of propagating non-gauge degrees of freedom with no superluminal wave propagation. The theory admits a generalized fermionic gauge transformation, and infinitesimal gauge transformations are an invariance of the constrained\nflat and curved spacetime actions and of the fermion number. The\ngauged Rarita-Schwinger equation has a non-perturbative aspect when the secondary constraint $\\omega$ is eliminated, resulting in a breakdown of\nadiabatic decoupling, leading to the inapplicability of various $S$-matrix ``no-go''theorems that claim to forbid gauged massless Rarita-Schwinger\nfields. The extension of these results to the quantized\nRarita-Schwinger theory is given in the following paper, where we show that a consistent quantization by the Dirac bracket and path integral\nmethods is possible, with a manifestly positive semi-definite canonical anticommutator in covariant radiation gauge.\nThus, in the massless case our analysis eliminates the various objections that have been raised to gauging Rarita-Schwinger fields, showing that\nnon-Abelian gauging of Rarita-Schwinger fields can be contemplated as part of the anomaly cancelation mechanism in constructing grand unified\nmodels.\n\nWe conclude with several remarks:\n\n\\begin{enumerate}\n\n\\item We have introduced gauge fixing to make time evolution of the Rarita-Schwinger fields unique, but the analysis of\nthis paper does not {\\it require} gauge fixing. Specifically, if gauge fixing is not imposed, the correct helicity counting\nis still obtained because fermionic gauge degrees of freedom are redundant degrees of freedom, and are not physical. Gauge\nfixing makes this redundancy manifest by providing a condition that excludes the gauge degrees of freedom, but in analogy to\nthe case of Maxwell electrodynamics, gauge fixing is not needed to get the correct physical state counting. On the other hand,\nin the following paper, where we turn to quantization, gauge fixing is needed. This can already be anticipated from the form\nof the constraint matrix $N$ of Eq. \\eqref{eq:detp}, which when gauge fixing is omitted reduces to the single element\n${\\cal A}= -2ig \\vec \\sigma \\cdot \\vec B(\\vec x) \\delta^3(\\vec x-\\vec y)$ which is not invertible in the small $\\vec B$ limit.\nInversion of the constraint matrix does not enter into the calculations of this paper, but is needed in the following paper\nboth for Dirac bracket and path integral quantization.\n\n\n\\item A possible exception to the non-perturbative behavior detailed in Sec. 7 is when the $\\vec E$ and $\\vec B$\ngauge fields are random, since if Eq. \\eqref{eq:solve1} is replaced by an average, denoted by AV,\n\\begin{equation}\\label{eq:average1}\n\\langle \\Psi_0\\rangle_{\\rm AV} \\simeq \\Big\\langle \\frac{\\vec Q}{(\\vec B)^2} \\Big\\rangle_{\\rm AV} \\cdot \\langle \\vec \\Psi \\rangle_{\\rm AV}~~~,\n\\end{equation}\nit becomes\n\\begin{equation}\\label{eq:average2}\n\\langle \\Psi_0\\rangle_{\\rm AV} \\simeq \\frac{1}{3}\\vec \\sigma \\cdot \\langle \\vec \\Psi \\rangle_{\\rm AV}~~~,\n\\end{equation}\nwhich is compatible with $\\langle \\Psi_0\\rangle_{\\rm AV} =\\vec \\sigma \\cdot \\langle \\vec \\Psi \\rangle_{\\rm AV}=0$, the customary\nfree Rarita-Schwinger constraints employed in \\cite{dasb}, \\cite{freed}. This heuristic observation suggests that Rarita-Schwinger fields coupled to\nquantized gauge fields with zero background gauge field may have a perturbative $g \\to 0$ limit.\n\n\\item In showing in the Abelian case that there is no superluminal propagation, the inversion of $\\vec \\sigma \\cdot \\vec B$ to get\n$\\Psi_0$ only required $(\\vec B)^2 \\neq 0$. In the non-Abelian case, where $\\vec B$ is itself a matrix, the conditions for\ninvertibility are nontrivial and have yet to be analyzed. We will see in the following paper that this issue is side-stepped when\nthe constraints are dealt with by the Dirac bracket or path integral procedures, since these do not require inversion of $\\vec \\sigma \\cdot \\vec B$ when a gauge constraint is included.\n\n\n\\end{enumerate}\n\n\\section{Acknowledgements}\n\nI wish to thank Edward Witten for conversations about gauging Rarita-Schwinger fields and Rarita-Schwinger scattering from photons,\namong other topics. I also wish to acknowledge the various people who\nasked about the status of gauged Rarita-Schwinger fields when I gave seminars on \\cite{adler}. Following on the initial draft\nof this paper, I had a fruitful correspondence with Stanley Deser and Andrew Waldron about gauge invariance and counting degrees\nof freedom when invariance of the action is conditional on a constraint. I wish to thank Thomas Spencer for a very helpful conversation which emphasized the significance of the gauge invariants, and Laurentiu Rodina for an explication of the paper \\cite{mcgady} that uses\n ``on-shell'' methods. This work was supported in part by the National Science Foundation under Grant\nNo. PHYS-1066293 through the hospitality of the Aspen Center for Physics.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\nEspecially during the last three years one notices a significant further boost of interest in octonionic analysis both from mathematicians and from theoretical physicists, see for instance \\cite{JRS,KO2018,KO2019,Kra2019-1,Nolder2018}. \nIn fact, many physicists currently believe that the octonions provide the adequate setting to describe the symmetries arising in a possible unified world theory combining the standard model of particle physics and aspects of supergravity. See also \\cite{Kra2019-2} for the references therein. \n\n\\par\\medskip\\par\n\nAlready during the 1970s, but particularly in the first decade of this century, a lot of effort has been made to carry over fundamental tools from Clifford analysis to the non-associative octonionic setting. \n\nMany analogues of important theorems from Clifford analysis could also be established in the non-associative setting, such as for instance a Cauchy integral formula or Taylor and Laurent series representations involving direct analogues of the Fueter polynomials, see for example \\cite{Imaeda,Nono,XL2000,XL2001,XL2002,XZL}. Of course, one carefully has to put parenthesis in order to take care of the non-associative nature. \n\nAlthough some of these fundamental theorems formally look very similar to those in the associative Clifford algebra setting, Clifford analysis and octononic analysis are two different function theories. \n\nIn \\cite{KO2018,KO2019} the authors describe a number of substantial and structural differences between the set of Clifford monogenic functions from $\\mathbb{R}^8 \\to Cl_8 \\cong \\mathbb{R}^{128}$ and the set of octonionic monogenic functions from $\\mathbb{O} \\to \\mathbb{O}$. This is not only reflected in the different mapping property, but also in the fact that unlike in the Clifford case, left octonionic monogenic functions do not form an octonionic right module anymore. \n\nThe fact that one cannot interchange the parenthesis arbitrarily in a product of octonionic expressions does not permit to carry over a number of standard arguments from the Clifford analysis setting to the octonionic setting. \n\nIn this paper we depart from the octonionic Cauchy integral formula for left or right octonionic monogenic functions, taking special care of the non-associativity by bracketing the terms together in a particular way. First we derive a topological generalized version of this Cauchy integral formula involving the winding number of $7$-dimensional hypersurfaces in the sense of the Kronecker index. From the physical point of view this winding number represents the fourth Chern number of the $G_2$-principal bundles that arise in the application of a generalization of 't Hoofd ansatz to construct special solutions of generalized $G_2$-Yang-Mills gauge fields, see \\cite{Burdik,GTBook}.\n\nThis homological version of Cauchy's integral formula is the starting point to introduce first the notion of the order of an isolated zero, or more generally, of an isolated $a$-point of a left (right) octonionic monogenic function. This notion of the order represents the topological mapping degree counting how often the image of a small sphere around zero (or around an arbitrary point $a$) wraps around zero (or $a$, respectively). An application of the transformation formula then leads to an explicit argument principle for isolated zeroes and $a$-points of octonionic monogenic functions. On the one-hand this argument principle naturally relates the fundamental solution of the octonionic Cauchy-Riemann equation with the fourth Chern number of the $G_2$-principal bundles that are related to special solutions of the $G_2$-Yang-Mills equation from 't Hoofd' ansatz. However, this topic will be investigated in detail in one of our follow-up papers. \n\nOn the other hand this argument principle allows us to establish a generalization of Rouch\\'e's theorem using a classical homotopy argument. \n\nIn turn, this version of Rouch\\'e's theorem enables us to prove that the limit function of a normally convergent sequence of octonionic monogenic functions that have no isolated $a$-points inside an octonionic domain either vanishes identically over the whole domain or it satisfies $\\sum_{c \\in C}{\\rm ord}(f;c)=0$. Note that this statement is slightly weaker than the classical Hurwitz theorem, because in the higher dimensional cases the condition ${\\rm ord}(f;c)=0$ does not immediately mean that $f(c)\\neq 0$. It is a sufficient but not necessary condition for being zero-free. Anyway, this statement is also new for the associative Clifford analysis setting, of course one has to restrict oneself to paravector-valued functions when addressing this case. \n\nA big goal and novelty of this paper consists in addressing also the context of non-isolated zeroes and $a$-points which lie on special simply-connected compact manifolds of dimension $k \\in \\{1,\\ldots,6\\}$. Instead of taking small spheres, the adequate geometric tool is the use of tubular domains that surround these zero or $a$-point varieties. This geometrical setting allows us to introduce the winding number of a surface wrapping around such a compact zero or $a$-point variety and gives a meaningful definition for the order of a zero variety of an octonionic monogenic function. We also manage to establish an argument principle for these classes of non-isolated zero varieties. These results are even new for the associative Clifford analysis setting and can also be applied to left and right monogenic paravector valued functions in $\\mathbb{R}^{n+1}$ for general dimensions $n \\in \\mathbb{N}$. \n \n\nTo finish we would like to mention that octonions also offer an alternative function theory of octonionic slice-regular functions, see for example \\cite{GPzeroes,GP,JRS}.There are of course also connections between octonionic slice-regular functions and octonionic solutions of the generalized octonionic Cauchy-Riemann equations. In the slice-regular context one even gets explicit relations between poles and zeroes as well as a simpler classification of zeroes in a very general situation. In the slice-regular setting only isolated and spherical zeroes can appear and their multiplicity can simply be described in terms of a power exponent appearing in a factorization that makes use of the so-called slice-product. This is a very prosperous direction for developing further powerful function theoretical tools to address problems in the octonionic setting. Note that slice-regular functions also are connected with concrete physical applications, see for instance \\cite{Burdik}, in particular also in the construction of special solutions of 't Hoofd ansatz of $G_2$-Yang-Mills solutions. \n\nHowever, in this paper we entirely restrict ourselves to solutions of the octonionic Cauchy-Riemann equation, but it is an interesting challenge to pay more attention to these topics in the framework of other octonionic generalized function theories. \n\n\n \n \\section{Basic notions of octonions}\n \n The octonions form an eight-dimensional real non-associative normed division algebra over the real numbers. They serve as a confortable number system to describe the symmetries in recent unifying physical models connecting the standard model of particle physics and supergravity, see \\cite{Burdik,G}. \n\nFollowing \\cite{Baez,WarrenDSmith} and others, the octonions can be constructed by the usual Cayley-Dickson doubling process. The latter is initiated by taking two pairs of complex numbers $(a,b)$ and $(c,d)$ and forming an addition and multiplication operation by $$\n (a,b)+(c,d) :=(a+c,b+d),\\quad\\quad (a,b)\\cdot (c,d) := (ac-d\\overline{b},\\overline{a}d+cb) \n $$ \n where $\\overline{\\cdot}$ denotes the conjugation (anti-)automorphism which will be extended by $\\overline{(a,b)}:=(\\overline{a},-b)$ to the set of pairs $(a,b)$. \n \n In the first step of this doubling procedure we get the real quaternions $\\mathbb{H}$. Each quaternion can be written in the form $z=x_0 + x_ 1e_1 + x_2 e_2 + x_3 e_3$ where $e_i^2=-1$ for $i=1,2,3$ and $e_1 e_2 = e_3$, $e_2 e_3 = e_1$, $e_3 e_1 = e_2$ and $e_i e_j = - e_j e_i$ for all mutually distinct $i,j$ from $\\{1,2,3\\}$. Already the commutativity has been lost in this first step of the doubling process. However, $\\mathbb{H}$ is still associative.\n \n The next duplification in which one considers pairs of quaternions already leads to the octonions $\\mathbb{O}$ which are not even associative anymore. However, in contrast to Clifford algebras, the octonions still form a division algebra. In real coordinates octonions can be expressed in the form \n $$\n z = x_0 + x_1 e_1 + x_2 e_2 + x_3 e_3 + x_4 e_4 + x_5 e_5 + x_6 e_6 + x_7 e_7\n $$\n where $e_4=e_1 e_2$, $e_5=e_1 e_3$, $e_6= e_2 e_3$ and $e_7 = e_4 e_3 = (e_1 e_2) e_3$. \n Like in the quaternionic case, we have $e_i^2=-1$ for all $i =1,\\ldots,7$ and $e_i e_j = -e_j e_i$ for all mutual distinct $i,j \\in \\{1,\\ldots,7\\}$. Their mutual multiplication is illustrated as follows, \n\\begin{center}\n \\begin{tabular}{|l|rrrrrrr|}\n $\\cdot$ & $e_1$& $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\\\ \\hline\n $e_1$ & $-1$ & $e_4$ & $e_5$ & $-e_2$ &$-e_3$ & $-e_7$ & $e_6$ \\\\\n $e_2$ & $-e_4$& $-1$ & $e_6$ & $e_1$ & $e_7$ & $-e_3$ & $-e_5$ \\\\\n $e_3$ & $-e_5$& $-e_6$ & $-1$ & $-e_7$&$e_1$ & $e_2$ & $e_4$ \\\\\n $e_4$ & $e_2$ & $-e_1$ & $e_7$ & $-1$ &$-e_6$ & $e_5$ & $-e_3$\\\\\n $e_5$ & $e_3$ & $-e_7$ & $-e_1$& $e_6$& $-1$ & $-e_4$ & $e_2$ \\\\\n $e_6$ & $e_7$ & $e_3$ & $-e_2$& $-e_5$& $e_4$ & $-1$ & $-e_1$ \\\\\n $e_7$ & $-e_6$ & $e_5$ & $-e_4$& $e_3$ & $-e_2$& $e_1$ & $-1$ \\\\ \\hline \t\n \\end{tabular}\n\\end{center}\nFortunately, the octonions still form an alternative and composition algebra. \n\nIn particular, the Moufang rule $(ab)(ca) = a((bc)a)$ holds for all $a,b,c \\in \\mathbb{O}$. In the special case $c=1$, one obtains the flexibility condition $(ab)a= a(ba)$. \n\nLet $a = a_0 + \\sum\\limits_{i=1}^7 a_i e_i$ be an octonion represented with the seven imaginary units as mentioned above. We call $a_0$ the real part of $a$ and write $\\Re{a} = a_0$. The conjugation leaves the real part invariant, but $\\overline{e_j}=-e_j$ for all $j =1,\\ldots,7$. On two general octonions $a,b \\in \\mathbb{O}$ one has $\\overline{a\\cdot b} = \\overline{b}\\cdot \\overline{a}$. \n\nThe Euclidean norm and the Euclidean scalar product from $\\mathbb{R}^8$ naturally extends to the octonionic case by $\\langle a,b \\rangle := \\sum\\limits_{i=0}^7 a_i b_i = \\Re\\{a \\overline{b}\\}$ and $|a|:= \\sqrt{\\langle a,a\\rangle} = \\sqrt{\\sum\\limits_{i=0}^7 a_i^2}$. We have the important norm composition property $|a\\cdot b|=|a|\\cdot|b|$ for all $a,b \\in \\mathbb{O}$. Every non-zero element $a \\in \\mathbb{O}$ is invertible with $a^{-1} =\\overline{a}\/|a|^2$. \n\nThe famous theorems of Frobenius and Hurwitz theorem tell us that \n$\\mathbb{R},\\mathbb{C},\\mathbb{H}$ and $\\mathbb{O}$ are the only real normed division algebras.\n \n\nFurther important rules are \n$$\n(a\\overline{b})b = \\overline{b}(ba) =a(\\overline{b}b)=a(b \\overline{b})\n$$ \nfor all $a,b \\in \\mathbb{O}$ and, \n$\\Re\\{b(\\overline{a}a)c\\} =\\Re\\{(b \\overline{a})(ac)\\}$ for all $a,b,c \\in \\mathbb{O}$, as stated for instance in \\cite{CDieckmann} Proposition 1.6. \n\nWe also use the notation $B_8(z,r) :=\\{z \\in \\mathbb{O} \\mid |z| < r\\}$ and $\\overline{B_8(z,r)} :=\\{z \\in \\mathbb{O} \\mid |z| \\le r\\}$ for the eight-dimensional solid open and closed ball of radius $r$ in the octonions and $S_7(z,r)$ for the seven-dimensional sphere $S_7(z,r) :=\\{z \\in \\mathbb{O} \\mid |z| = r\\}$. If $z=0$ and $r=1$ then we denote the unit ball and unit sphere by $B_8$ and $S_7$, respectively. The notation $\\partial B_8(z,r)$ means the same as $S_7(z,r)$. \n \n\n\n \n\n\\section{Argument principle for isolated zeroes of octonionic monogenic functions}\nWe start this section by recalling the definition of octonionic regularity or octonionic monogenicity in the sense of the Riemann approach. From \\cite{Imaeda,XL2000} and elsewhere we quote \n\\begin{definition}\n\tLet $U \\subseteq \\mathbb{O}$ be an open set. A real differentiable function $f:U \\to \\mathbb{O}$ is called left (right) octonionic monogenic or equivalently left (right) ${\\mathbb{O}}$-regular for short if it satisfies ${\\cal{D}} f = 0$ or $f {\\cal{D}} = 0$ where $\n\t{\\cal{D}}:= \\frac{\\partial }{\\partial x_0} + \\sum\\limits_{i=1}^7 e_i \\frac{\\partial }{\\partial x_i}$ stands for the octonionic Cauchy-Riemann operator, \n\twhere $e_i$ are the octonionic units like defined in the preliminary section before. \n\\end{definition}\n In contrast to the associative Clifford analysis setting, the set of left (right) ${\\mathbb{O}}$-regular functions do not form an ${\\mathbb{O}}$-right (left) module. The following example given in \\cite{KO2019} provides a counter-example. Take the function $f(z):= x_1 - x_2 e_4$. A direct computation gives ${\\cal{D}}[f(z)] = e_1 - e_2 e_4 = e_1 - e_1 = 0$. But the function $g(z):=(f(z))\\cdot e_3 = (x_1 - x_2 e_4) e_3 = x_1 e_3 - x_2 e_7$ satisfies ${\\cal{D}}[g(z)] = e_1 e_3 - e_2 e_7 = e_5 -(-e_5) = 2 e_5 \\neq 0$. It is clearly the lack of associativity that destroys the modular structure of ${\\mathbb{O}}$-regular functions. \n This is one significant structural difference to Clifford analysis. However, note that the composition with an arbitrary translation of the form $z \\mapsto z + \\omega$ where $\\omega \\in \\mathbb{O}$ still preserves monogenicity also in the octonionic case, i.e. ${\\cal{D}}f(z+\\omega) = 0$ if and only if ${\\cal{D}}f (z) = 0$. This is a simple consequence of the chain rule, because the differential remains invariant under an arbitrary octonionic translation.\n \n An important property of left or right ${\\mathbb{O}}$-regular functions is that they satisfy the following Cauchy integral theorem, cf. for instance \\cite{XL2002}: \n \\begin{proposition} (Cauchy's integral theorem)\\\\\nLet $G \\subseteq \\mathbb{O}$ be a bounded $8$-dimensional connected star-like domain with an orientable strongly Lipschitz boundary $\\partial G$. Let $f \\in C^1(\\overline{G},\\mathbb{O})$. If $f$ is left (resp.) right $\\mathbb{O}$-regular inside of $G$, then \n$$\n\\int\\limits_{\\partial G} d\\sigma(z) f(z) = 0,\\quad {\\rm resp.}\\;\\;\\int\\limits_{\\partial G} f(z) d\\sigma(z) = 0\n$$ \t\nwhere $d\\sigma(z) = \\sum\\limits_{i=0}^7 (-1)^j e_i \\stackrel{\\wedge}{d x_i} = n(z) dS(z)$, where $\\stackrel{\\wedge}{dx_i} = dx_0 \\wedge dx_1 \\wedge \\cdots dx_{i-1} \\wedge dx_{i+1} \\cdots \\wedge dx_7$ and where $n(z)$ is the outward directed unit normal field at $z \\in \\partial G$ and $dS =|d \\sigma(z)|$ the ordinary scalar surface Lebesgue measure of the $7$-dimensional boundary surface. \n \\end{proposition} \n \n An important left and right ${\\mathbb{O}}$-regular function is the function $q_{\\bf 0}: \\mathbb{O} \\backslash\\{0\\} \\to \\mathbb{O},\\;q_{\\bf 0}(z) := \\frac{x_0 - x_1 e_1 - \\cdots - x_7 e_7}{(x_0^2+x_1^2+\\cdots + x_7^2)^4} = \\frac{\\overline{z}}{|z|^8}$ whose only singular point is an isolated point singularity of order $7$ at the origin. This function serves as Cauchy kernel in the following Cauchy integral formula for ${\\mathbb{O}}$-regular functions. Before we recall this formula, we point out another essential difference to the associative setting: \n \\begin{remark}\n \tAs already mentioned in {\\rm \\cite{GTBook}}, in contrast to quaternionic and Clifford analysis, octonionic analysis does {\\em not} offer an analogy of a general Borel-Pompeiu formula of the form \n \t$$\n \t\\int\\limits_{\\partial G} g(z) d\\sigma(z) f(z) = 0,\n \t$$\n \tnot even if $g$ is right $\\mathbb{O}$-regular and $f$ left $\\mathbb{O}$-regular, independently how we bracket these terms together. The lack of such an identity is again a consequence of the lack of associativity. However, if one of these functions is the Cauchy kernel, then one obtains a generalization. \n \\end{remark}\n For convenience we recall from \\cite{Imaeda,Nono,XL2002}:\n \\begin{proposition}\nLet $U \\subseteq \\mathbb{O}$ be a non-empty open set and $G \\subseteq U$ be an $8$-dimensional compact oriented manifold with a strongly Lipschitz boundary $\\partial G$. If $f: U \\to \\mathbb{O}$ is left (resp. right) $\\mathbb{O}$-regular, then for all $z \\not\\in \\partial G$\n$$\n\\chi(z)f(z)= \\frac{3}{\\pi^4} \\int\\limits_{\\partial G} q_{\\bf 0}(w-z) \\Big(d\\sigma(w) f(w)\\Big),\\quad\\quad \\chi(z) f(z)= \\frac{3}{\\pi^4} \\int\\limits_{\\partial G} \\Big(f(w)d\\sigma(w)\\Big) q_{\\bf 0}(w-z),\n$$\nwhere $\\chi(z) = 1$ if $z$ is in the interior of $G$ and $\\chi(z)=0$ if $z$ in the exterior of $G$. \n \\end{proposition}\nNote that the way how the parenthesis are put is very important. Putting the parenthesis in the other way around, would lead in the left $\\mathbb{O}$-regular case to a different formula of the form\n$$\n\\frac{3}{\\pi^4} \\int\\limits_{\\partial G} \\Big( q_{\\bf 0}(w-z) d\\sigma(w)\\Big) f(w) = \\chi(z) f(z) + \\int\\limits_G \\sum\\limits_{i=0}^7 \\Big[q_{\\bf 0}(w-z),{\\cal{D}}f_i(w),e_i \\Big] dw_0 \\wedge \\cdots \\wedge dw_7, \n$$\nwhere $[a,b,c] := (ab)c - a(bc)$ stands for the associator of three octonionic elements. The volume integral which appears additionally always vanishes in algebras where one has the associativity, such as in Clifford algebras. \nSee \\cite{XL2002}. \n \nAn important subcase is obtained when we take for $f$ the constant function $f(z) = 1$ for all $z \\in U$ which is trivially left and right $\\mathbb{O}$-regular. Then the Cauchy integral simplifies to the constant value \n$$\n\\chi(z) = \\frac{3}{\\pi^4} \\int\\limits_{\\partial G} q_{\\bf 0}(w-z) d\\sigma(w),\\quad {\\rm resp.}\\; \\chi(z)= \\frac{3}{\\pi^4} \\int\\limits_{\\partial G} d\\sigma(w) q_{\\bf 0}(w-z),\n$$\nsimply indicating if $z$ belongs to the interior or to the exterior component of $\\partial G$. \nThis is the starting point to introduce a following generalized topological version of the above stated Cauchy integral formula. Following for instance \\cite{Hempfling} \none can consider more generally for $G$ a bounded Lipschitz domain whose boundary $\\partial G$ could be a $7$-chain, homologous to a differentiable $7$-chain with image $\\partial B(z,r)$, parametrized as \n\\begin{equation}\\label{param}\n\\partial G = \\{x_0(\\lambda_1,\\ldots,\\lambda_7) + \\sum\\limits_{i=1}^7 x_i(\\lambda_1,\\ldots,\\lambda_7) e_i\\}. \n\\end{equation}\nIn this more general case one has \n$$\nw_{\\partial G}(z) = \\frac{3}{\\pi^4} \\int\\limits_{\\partial G} q_{\\bf 0}(w-z) d\\sigma(w),\\quad {\\rm resp.}\\; w_{\\partial G}(z)= \\frac{3}{\\pi^4} \\int\\limits_{\\partial G} d\\sigma(w) q_{\\bf 0}(w-z),\n$$\nwhere $w_{\\partial G}(z)$ represents the topological winding number, sometimes called the Kronecker-index (cf. \\cite{ghs}), counting how often $\\partial G$ wraps around $z$. Note that this is a purely topological entity induced by\n$$\nH_8(\\partial G,\\partial G - z) \\cong H_8(B_8,S_7) \\cong \\tilde{H}_7(S_7) \\cong \\mathbb{Z},\n$$\nwhere $H_8$ is the related homology group and $\\tilde{H}_7$ the related reduced homology group. Due to this property, the winding number $w_{\\partial G}(z)$ is always an integer. This is the basis for the more general topogical version of Cauchy's integral formula: \n\\begin{theorem}\\label{topcauchy}(Topological generalized octonionic Cauchy integral formula)\\\\\t\nLet $U \\subseteq \\mathbb{O}$ be an open set and $G$ be a closed manifold whose boundary $\\Gamma$ is a strongly Lipschitz $7$-chain. If $f:U \\to \\mathbb{O}$ is left $\\mathbb{O}$-regular, then we have the identity\n$$\nw_{\\Gamma}(z)f(z)= \\frac{3}{\\pi^4} \\int\\limits_{\\Gamma} q_{\\bf 0}(w-z) \\Big(d\\sigma(w) f(w)\\Big),\\quad z \\not\\in \\Gamma\n$$\nwhere $w_{\\Gamma}(z)$ is the topological winding number counting how often $\\Gamma$ wraps around $z$. The latter equals zero if $z$ in a point from the exterior of $G$.\n\\end{theorem}\n\\begin{remark}\nNote that if we put the parenthesis the other way around, then we get the identity \n$$\n\\frac{3}{\\pi^4} \\int\\limits_{\\partial G} \\Big( q_{\\bf 0}(w-z) d\\sigma(w)\\Big) f(w) = w_{\\partial G}(z) f(z) + \\int\\limits_G \\sum\\limits_{i=0}^7 \\Big[q_{\\bf 0}(w-z),{\\cal{D}}f_i(w),e_i \\Big] dw_0 \\wedge \\cdots \\wedge dw_7. \n$$\nThe volume integral is not affected in the topological version, because we simply integrate over the volume and orientation does not play any role, because the scalar differential $dV = dw_0 \\wedge \\cdots \\wedge dw_7$ has no orientation. \n\\end{remark}\n\\begin{remark}\nComparing with {\\rm \\cite{Burdik,GTBook}}, we can relate the octonionic winding number with the fourth Chern number of the $G_2$-principal bundles associated to special solutions of $G_2$ Yang-Mills gauge fields arising in generalizing 't Hoofd ansatz, see {\\rm \\cite{Burdik,GTBook}}. This allows us to explicitly relate the fundamental solution of the octonionic Cauchy-Riemann equation with Chern numbers of the related $G_2$-principal bundles. We will shed some more light on this interesting connection in a follow-up paper. \n\\end{remark}\nThe topological winding number is also the key tool to define a generalized notion of multiplicity of zeroes and $a$-points of $\\mathbb{O}$-regular functions. To proceed to the definition and classification of $a$-points we first need the octonionic identity theorem:\n\\begin{proposition}\\label{identity} \nLet $G \\subseteq \\mathbb{O}$ be an $8$-dimensional domain. Suppose that $f,g: G \\to \\mathbb{O}$ are two left (right) $\\mathbb{O}$-regular functions. If there exists a $7$-dimensional smooth sub-manifold $V$ where $f(z)=g(z)$ for all $z \\in V$, then we have $f(z) = g(z)$ for all $z \\in G$. \n\\end{proposition}\t\nIn particular, a left (right) $\\mathbb{O}$-regular function satisfying $f(z)=0$ on a $7$-dimensional sub-manifold vanishes identically. Similarly, if there is an octonion $a \\in \\mathbb{O}$ such that $f(z)=a$ for all $z \\in V$, then $f(z) = a$ for all $z \\in G$. Although the proof only uses basic tools of octonionic analysis, we prefer to present it in detail, as we are not aware of a direct reference in the literature addressing the particular octonionic setting. For the proof of the statement in the associative Clifford analysis setting we refer to \\cite{ghs}, p. 187. \t\n\\begin{proof}\nThe proof can be done by extending R. Fueter's argumentation from the quaternionic case presented in \\cite{Fueter1948-49} on pp.185-189. Without loss of generality we consider the situation where $g(z)$ is the zero function. Suppose now that $V$ is a seven dimensional smooth manifold where $f|_V = 0$. Consider an arbitrary point $c \\in V$ with $f(c)=0$. Since $V$ is $7$-dimensional and smooth one can find seven $\\mathbb{R}$-linearly independent unit octonions, say ${\\bf n}_1, \\ldots, {\\bf n}_7$ with $|{\\bf n}_h|=1$ $(h=1,\\ldots,7)$ that lie in the $7$-dimensional tangent space $T_V(c)$. Next define $\\xi^{(h)}_0 := \\langle {\\bf n}_h,1\\rangle$ and $\\xi^{(h)}_j :=\\langle {\\bf n}_h,e_j\\rangle$ for $j=1,\\ldots,7$ where $\\langle\\cdot,\\cdot\\rangle$ is the scalar product on $\\mathbb{O}$ defined in Section~2. \nNotice that all the values $\\xi^{(h)}_j$ are real for all $j=0,1\\ldots,7$ and all $h=1,\\ldots,7$. Next consider for each point $c \\in V$ the real $7\\times8$-matrix composed by the seven rows constituted by the eight real coordinates of the seven octonions ${\\bf n}_1,\\ldots,{\\bf n}_7$, respectively, i.e.\n$$\nA:=\\left(\\begin{array}{cccc} \\xi^{(1)}_0 & \\xi^{(1)}_1 & \\cdots & \\xi^{(1)}_7\\\\ \n\\xi^{(2)}_0 & \\xi^{(2)}_1 & \\cdots & \\xi^{(2)}_7\\\\\n\\vdots & \\vdots & \\cdots & \\vdots \\\\\n\\xi^{(7)}_0 & \\xi^{(7)}_1 & \\cdots & \\xi^{(7)}_7\\\\\n\\end{array} \\right)\n$$ \nRe-interpreting the seven octonions ${\\bf n}_j$ as column vectors from $\\mathbb{R}^8$, we have $rank({\\bf n}_1,\\ldots,{\\bf n}_7)=7$ in view of the $\\mathbb{R}$-linear independency. Consequently, also the rank of the largest non-vanishing sub-determinant must equal $7$. Without loss of generality we may suppose that \n\\begin{equation}\\label{domega}\n\\det\\left(\\begin{array}{cccc} \\xi^{(1)}_1 & \\xi^{(1)}_2 & \\cdots & \\xi^{(1)}_7\\\\ \n\\xi^{(2)}_1 & \\xi^{(2)}_2 & \\cdots & \\xi^{(2)}_7\\\\\n\\vdots & \\vdots & \\cdots & \\vdots \\\\\n\\xi^{(7)}_1 & \\xi^{(7)}_2 & \\cdots & \\xi^{(7)}_7\\\\\n\\end{array} \\right) \\neq 0.\n\\end{equation}\nOtherwise, we change the labels of the components. \n\nNext we use that $f(z) = f_0(z) + \\sum\\limits_{k=0}^7 f_k(z) e_k \\equiv 0$ on $V$. Therefore, the directional derivatives also vanish all, i.e. $\\frac{\\partial f}{\\partial {\\bf n}_h} = 0$ for each $h=1,2,\\ldots,7$. Using the ordinary chain rule gives seven equations:\n$$\n\\frac{\\partial f}{\\partial {\\bf n}_h} = \\sum\\limits_{k=0}^7 \\frac{\\partial f}{\\partial x_k} \\frac{\\partial x_k}{\\partial {\\bf n}_h} = \\sum\\limits_{k=0}^7 \\frac{\\partial f}{\\partial x_k} \\xi^{(h)}_k=0,\\quad h=1,\\ldots,7.\n$$\nAdditionally, as eighth condition, $f$ has to satisfy the octonionic left Cauchy-Riemann equation $\\sum\\limits_{k=0}^7 e_k \\frac{\\partial f}{\\partial x_k} = 0$. \n\nConsider the formal octonionc determinant \n$$\n\\det(\\Omega) := \\det\\left(\\begin{array}{cccc} 1 & e_1 & \\cdots & e_7\\\\ \n\\xi^{(1)}_0 & \\xi^{(1)}_1 & \\cdots & \\xi^{(1)}_7\\\\\n\\vdots & \\vdots & \\cdots & \\vdots \\\\\n\\xi^{(7)}_0 & \\xi^{(7)}_1 & \\cdots & \\xi^{(7)}_7\\\\\n\\end{array} \\right),\n$$\ndefined formally in the usual way. Note that the non-associativity does not lead to ambiguous interpretations, because only the entities $e_1,\\ldots,e_7$ are octonions, while the other entries $\\xi^{(h)}_k$ are all real-valued expressions. So, this formal determinant is a well-defined octonion. \nThe eight equations mentioned above could be satisfied under two particular circumstances only. Firstly, they could be satisfied if $\\det(\\Omega)$ vanished. However, this is impossible. Notice that $\\det(\\Omega)$ represents an octonion. An octonion only vanishes if {\\em all} its real components vanish. However, we obviously have $\\Re\\{\\det(\\Omega))\\}\\neq 0$ in view of (\\ref{domega}). The only remaining second option is that \n$$\n\\frac{\\partial f}{\\partial x_k} = 0,\\quad k=0,1,\\ldots,7\n$$ \nat each $z \\in V$. Note that also the octonionic Cauchy integral formula implies that the left $\\mathbb{O}$-regularity of $f$ is also inherited by all partial derivatives of $f$. Consequently, the same argumentation is also true for all partial derivatives $\\frac{\\partial^{n_1+\\cdots+n_7}}{\\partial x_1^{n_1} \\cdots \\partial x_7^{n_7}} f(z) = 0$. Following \\cite{Imaeda,XL2001} we can expand $f$ into a Taylor series around each left $\\mathbb{O}$-regular point $z=c \\in V$ \nof the form $f(z) = \\sum\\limits_{n=0}^{\\infty} \\sum\\limits_{n=n_1+\\cdots+n_7} V_{\\bf n}(z-c) c_{n_1,\\ldots,n_7} \\equiv 0$ where \n$$\nV_{\\bf n}(z) = \\frac{1}{|{\\bf n}|!} \\sum\\limits_{\\pi \\in perm({\\bf n})} (Z_{\\pi(n_1)}(Z_{\\pi(n_2)}( \\cdots (Z_{\\pi(n_{6})} Z_{\\pi(n_{7})})\\cdots))).\n$$\nOne has to apply the parenthesis in this particular way. Due to the lack of associativity, the parenthesis cannot be neglected. \nHere, $perm({\\bf n})$ denotes the set of all distinguishable permutations of the sequence $(n_1,n_2,\\ldots,n_7)$ and $Z_i := V_{\\tau(i)}(z) := x_i - x_0 e_i$ for all $i=1,\\ldots,7$, cf. \\cite{XL2001} Theorem C p.208. Here $\\tau(i)$ is the multi-index $(n_1,\\ldots,n_7)$ where $n_j = 0$ for all $j \\neq i$ and $n_i=1$. \n\n\nHowever, following also from \\cite{XL2001}, $c_{n_1,\\ldots,n_7} :=\\Bigg( \\frac{\\partial^{n_1+\\cdots+n_7}}{\\partial x_1^{n_1} \\cdots \\partial x_7^{n_7}} f(z)\\Bigg)_{z=c} = 0$. The uniqueness of the Taylor series representation implies that $f$ must be identically zero over the whole domain $G$. \n\\end{proof}\n\n\\begin{remark}\nIf one considers instead of $\\mathbb{O}$-regular functions, the set of slice-regular functions from {\\rm \\cite{GP,JRS}}, then one even gets a much stronger version of the identity theorem, namely stating that two slice-regular functions already coincide with each other, when they coincide with each other on a one-dimensional set with an accumulation point. This has a strong consequence on the structure of the zeroes. \n\\end{remark}\n\nSince also the octonions form a normed algebra, we can introduce the notion of an isolated $a$-point of an $\\mathbb{O}$-regular function as follows, compare with \\cite{Hempfling,Kra2004}:\n\\begin{definition}\nLet $U \\subseteq \\mathbb{O}$ be an open set and $f:U \\to \\mathbb{O}$ be a function. Then we say that $f$ has an isolated $a$-point at $c \\in U$, if $f(c)=a$ and if there exists a positive real $\\varepsilon > 0$, such that $f(z) \\neq a$ for all $z \\in B(c,\\varepsilon) \\backslash\\{c\\}$. If $a=0$, then we call $c$ an isolated zero. \n\\end{definition}\nLet $U \\subseteq \\mathbb{O}$ be an open set, $c \\in U$ and $f:U \\to \\mathbb{O}$ be a real differentiable function, i.e. we suppose that each real component function $$f_i:U \\to \\mathbb{R}\\;\\;(i=0,1,\\ldots,7)\\;\\; {\\rm of}\\;\\; f(z)= f_0(z) + f_1(z) e_1 + \\cdots + f_7(z)e_7$$ is partial differentiable. \nAccording to the implicit function theorem in $\\mathbb{R}^8$ a sufficient criterion for an $a$-point of a real-differentiable function $f:U \\to \\mathbb{O}$ of being an isolated $a$-point with $f(c)=a$ is that the Jacobian determinant does not vanish $\\det(Jf)(c) := \\det\\Big(\\frac{\\partial f_j}{\\partial x_j} \\Big)_{0 \\le i,j \\le 7} \\neq 0$. However, this clearly is just a sufficient criterion, as the following example illustrates. \nTake for instance the function $:\\mathbb{O} \\to \\mathbb{O}$ defined by \n\\begin{eqnarray*}\nf(z)&:=& V_{2,0,0,\\ldots,0}(z) + V_{0,2,0,\\ldots,0}(z)+\\cdots+V_{0,\\ldots,0,2}(z)\\\\ &=& Z_1^2+Z_2^2+\\cdots+Z_7^2 = (x_1^2+\\cdots+x_7^2-7x_0^2) - 2 \\sum\\limits_{i=1}^7 x_0x_i e_i\n\\end{eqnarray*}\nwhich is clearly left and right $\\mathbb{O}$-regular in the whole algebra $\\mathbb{O}$. \nObviously, one has $f(0)=0$. In general, $f(z)=0$ implies that first \n$$x_1^2+x_2^2+\\cdots+x_7^2=7x_0^2$$\nand one has $x_0 x_i = 0$ for each $i=1,\\ldots,7$. The first relation implies that $x_0 = \\pm \\frac{1}{\\sqrt{7}} \\sqrt{x_1^2+\\cdots+x_7^2}$. Inserting this expression into the other relations yields $\\sqrt{x_1^2+\\cdots+x_7^2} x_i = 0$ for all $i=1,\\ldots,7$. Since $x_1^2+\\cdots+x_7^2 > 0$ whenever $(x_1,x_2,\\ldots,x_7) \\neq (0,0,\\ldots,0)$ we must have $x_i = 0$ for all $i=1,\\ldots,7$. Therefore, also $x_0=0$. Summarizing $z=0$ is the only zero of $f$ and therefore it must be an isolated zero. The Jacobian matrix however is:\n$$\n(Jf)(z):= \\left(\\begin{array}{cccccc} -14x_0 & 2x_1 & 2x_2 & \\cdots & 2x_6 & 2x_7\\\\\n\t\t\t\t\t\t\t\t\t -2x_1 & -2x_0 & 0 & \\cdots & 0 & 0 \\\\\n\t\t\t\t\t\t\t\t\t -2x_2 & 0 &-2x_0 & \\cdots &0 & 0\\\\\n\t\t\t\t\t\t\t\t\t \\vdots&\\vdots &\\vdots&\\vdots &\\vdots& \\vdots\\\\\n\t\t\t\t\t\t\t\t\t -2x_6 & 0 & 0 & \\cdots & -2x_0 & 0 \\\\\n\t\t\t\t\t\t\t\t\t -2x_7 & 0 & 0 & \\cdots & 0 & -2x_0 \n\t\t\t\t\t\t\t\t\t \\end{array} \\right).\n$$\nInserting $z=0$ yields $\\det(Jf)(z)=0$. \n\nA typical example of a non-linear left $\\mathbb{O}$-regular function with one single octonionic isolated zero $z^*$ satisfying $Jf(z^*) \\neq 0$ can be constructed by applying T. Hempfling's construction from \\cite{Hempfling} p.111. \nAdapting from \\cite{Hempfling}, the octonionic version of the function \n$$\nf(z)= (x_1 x_2 \\cdots x_7-1)-(x_0 x_2 \\cdots x_7-1)e_1 - \\cdots -(x_0 x_1 \\cdots x_6-1)e_7 \n$$ \nactually is left $\\mathbb{O}$-regular. We have \n$$\n\\frac{\\partial f}{\\partial x_0} = -\\sum\\limits_{j=1}^7\\Big(\\prod\\limits_{i\\neq 0, i \\neq j} x_i \\Big) e_j \n$$ \nand for $k \\in \\{1,\\ldots,7\\}$ \n$$\ne_k \\frac{\\partial f}{\\partial x_k} = \\Big(\\prod\\limits_{i\\neq 0, i \\neq k} x_i \\Big) e_k - \\Big(\\prod\\limits_{i\\neq 0, i \\neq k} x_i \\Big) e_k e_1 - \\cdots - \\Big(\\prod\\limits_{i\\neq 0, i \\neq k} x_i \\Big) e_k e_7. \n$$\nSo, $f$ actually satisfies $\\frac{\\partial f}{\\partial x_0} + \\sum\\limits_{k=1}^7 e_k \\frac{\\partial f}{\\partial x_k} = 0$. \n\nAs one readily observes, one has $f(z^*)=0$ when inserting $z^{*}=1+e_1+\\cdots+e_7$. Furthermore, $\\frac{\\partial f_i}{\\partial x_j} = \\delta_{ij} \\prod\\limits_{k=0,k\\neq i,j}^7 x_k$, where $\\delta_{ij}$ denotes the ordinary Kronecker symbol. Thus,\n$$\nJf((1+e_1+\\cdots+e_7)) = \\left( \\begin{array}{ccccc} 0 & 1 & 1 & \\cdots & 1\\\\\n\t\t\t\t\t\t\t\t\t\t\t 1 & 0 & 1 & \\cdots & 1 \\\\\n\t\t\t\t\t\t\t\t\t\t\t 1 & 1 & 0 & \\cdots & 1\\\\\n\t\t\t\t\t\t\t\t\t\t\t \\vdots & \\vdots & \\vdots & \\vdots & \\vdots\\\\\n\t\t\t\t\t\t\t\t\t\t\t 1 & 1 & 1 & \\cdots & 0 \\end{array}\n\\right),\n$$\nand therefore $\\det(Jf((1+e_1+\\cdots+e_7))) = -7 \\neq 0$. $z^*$ clearly is an isolated zero of $f$. \n\nNote that in general a left or right $\\mathbb{O}$-regular function can possess also zeroes that lie on $k$-dimensional manifolds with $k \\le 6$. The case $k=7$ cannot appear as direct a consequence of Proposition~\\ref{identity}, because if a left $\\mathbb{O}$-regular function vanishes on a $7$-dimensional manifold, then it must be identically zero over the whole $8$-dimensional space. Furthermore, note that the zero sets of left or right $\\mathbb{O}$-regular functions must be real analytic manifolds. Already very simple octonionic functions can have connected sets of zeroes. Adapting from \\cite{Zoell} and \\cite{Hempfling}, in the octonionic case the simplest examples (for each dimension) are \n\\begin{eqnarray*}\nf(z)=Z_1^2+\\cdots+Z_7^2 & & {\\rm isolated\\;zero\\;at}\\;z^*=0\\\\\nf(z)=Z_1^2+\\cdots+Z_6^2 & & {\\rm 1-dimensional\\;zero\\;set \\;at}\\;z \\in e_7 \\mathbb{R}\\\\\nf(z)=Z_1^2+\\cdots+Z_5^2 & & {\\rm 2-dimensional\\;zero\\;set \\;at}\\;z \\in e_6 \\mathbb{R} \\oplus e_7 \\mathbb{R}\\\\\nf(z)=Z_1^2+\\cdots+Z_4^2 & & {\\rm 3-dimensional\\;zero\\;set \\;at}\\;z \\in e_5 \\mathbb{R} \\oplus e_6 \\mathbb{R} \\oplus e_7 \\mathbb{R}\\\\\nf(z)=Z_1^2+Z_2^2+Z_3^2 & & {\\rm 4-dimensional\\;zero\\;set \\;at}\\;z \\in e_4 \\mathbb{R} \\oplus \\cdots \\oplus e_7 \\mathbb{R}\\\\\nf(z)=Z_1^2+Z_2^2 & & {\\rm 5-dimensional\\;zero\\;set \\;at}\\;z \\in e_3 \\mathbb{R} \\oplus \\cdots \\oplus e_7 \\mathbb{R}\\\\\nf(z)=Z_1^2 & & {\\rm 6-dimensional\\;zero\\;set \\;at}\\;z \\in e_2 \\mathbb{R} \\oplus \\cdots \\oplus e_7 \\mathbb{R}\\\\\n\\end{eqnarray*}\n where $Z_i$ are again the octonionic Fueter polynomials $Z_i = x_i-x_0e_i$ for $i=1,\\ldots,7$. \n \n Generalizing the construction from \\cite{Hempfling} a further class of interesting examples can be gained from the following construction. Let $k \\in \\{2,\\ldots,6\\}$ be an integer and consider the function $f:\\mathbb{O} \\to \\mathbb{O}$, $f(z):= Z_1^2+\\cdots+Z_k^2-\\sum\\limits_{j=k+1}^7 Z_je_j$ composed by the octonionic Fueter polynomials. Again, this function is both left and right $\\mathbb{O}$-regular and can be written in the form \n $$\n f(z)=\\Big(\\sum\\limits_{i=1}^k x_i^2\\Big) - k x_0^2+(7-k)x_0 - 2 x_0\\sum\\limits_{i=1}^k x_i e_i + \\sum\\limits_{i=k+1}^7 x_i e_i.\n $$\n when switching to the ordinary variables $x_i$.\n Now consider the function $g(z):=f(z)-R^2$, where $R>0$ is a real number. Then $g(z)=0$ if and only if the following system of equations is satisfied\n \\begin{eqnarray*}\n \\sum\\limits_{i=1}^k x_i^2-k x_0^2-R^2+(7-k)x_0\n &=& 0 \\\\\n x_0 x_i &=& 0,\\;\\;i=1,\\ldots,k\\\\\n x_i & = & 0,\\;\\;i=k+1,\\ldots,7.\n \\end{eqnarray*}\nFirst case: $x_0=0$. Then $g(z)=0$ if and only if $\\sum\\limits_{i=1}^k x_i^2-R^2 = 0$. Now the zero variety of $g$ is the compact $k-1$-dimensional sphere of radius $R$ centered around the origin in the subspace generated by $e_1,e_2,\\ldots,e_k$. \\\\\nSecond case $x_0 \\neq 0$. Then $x_i = 0$ for all $i=1,\\ldots,7$. In this case $g(z)=0$ if and only if $-kx_0^2-R^2+(7-k)x_0=0$. This condition can only be satisfied if $x_0 = - \\frac{1-\\frac{7}{k}}{2} \\pm \\sqrt{\\frac{(1-\\frac{7}{k})^2}{4} - \\frac{R}{k}}$, provided the value in the square root expression is not negative. In this case the zero set consists at most of two isolated points $(x_0,0,\\ldots,0)$ on the real axis. \n\\par\\medskip\\par\nIn the spirit of \\cite{Hempfling,HeKra,Kra2004} we now proceed to introduce the order of an isolated zero or isolated $a$-point of an $\\mathbb{O}$-regular function. This can be done like in the quaternionic and Clifford analysis case in terms of the topological Cauchy integral mentioned above and then represents the order of an isolated $a$-point in the sense of the topological mapping degree. \n\\begin{definition}\\label{isolatedorder}\n\tLet $U \\subseteq \\mathbb{O}$ be an open set, $U \\neq \\emptyset$. Let $f:U \\to \\mathbb{O}$ be left $\\mathbb{O}$-regular (resp. right $\\mathbb{O}$-regular) and suppose that $c \\in U$ is an isolated $a$-point of $f$, i.e. $f(c)=a$ with $a \\in \\mathbb{O}$. Choose an $\\varepsilon > 0$ such that $\\overline{B}(c,\\varepsilon) \\subseteq U$ and suppose that $f(z) \\neq 0$ for all $z \\in \\overline{B}(c,\\varepsilon) \\backslash\\{c\\}$. Then, \n\t$$\n\t{\\rm ord}(f-a;c) := \\frac{3}{\\pi^4} \\int\\limits_{(f-a)(\\partial B(c,\\varepsilon))} q_{\\bf 0}(w) d\\sigma(w),\\quad {\\rm resp.}\\;{\\rm ord}(f-a;c) := \\frac{3}{\\pi^4} \\int\\limits_{(f-a)(\\partial B(c,\\varepsilon))} d\\sigma(w) q_{\\bf 0}(w) \n\t$$\n\tis called the order of the isolated $a$-point of the octonionic left (resp. right) $\\mathbb{O}$-regular function $f$ at $c$.\n\\end{definition}\n In the case where $a=0$, we address the order of isolated zeroes of $f$, which in the left $\\mathbb{O}$-regular case equals the Cauchy integral\n $$\n {\\rm ord}(f;c) := \\frac{3}{\\pi^4} \\int\\limits_{f(\\partial B(c,\\varepsilon))} q_{\\bf 0}(w) d\\sigma(w).\n $$\n \\begin{proposition}\n The numbers ord$(f-a;c)$ are integers and count how often the image of the sphere around the octonionic $a$-point wraps around $a$ and therefore represents the notion of the order of an $a$-point in the sense of the topological mapping degree. \t\n \\end{proposition}\n\\begin{proof}\nThe topological generalized version of the octonionic Cauchy integral formula (Theorem~\\ref{topcauchy}) tells us that every octonionic function $h:U \\to \\mathbb{O}$ that is left $\\mathbb{O}$ regular over an open set $U$ which contains a closed manifold $G$ whose boundary $\\Gamma$ is a strongly Lipschitz $7$-chain satisfies \n$$\nw_{\\Gamma}(y)h(y) = \\frac{3}{\\pi^4} \\int\\limits_{\\Gamma} q_{\\bf 0}(w-y) \\Big(d\\sigma(w) h(w)\\Big).\n$$\nSo, in the case where $h(z) = 1$ for all $z \\in U$, one has \n$$\nw_{\\Gamma}(y) = \\frac{3}{\\pi^4} \\int\\limits_{\\Gamma} q_{\\bf 0}(w-y) d\\sigma(w).\n$$ \nIn view of the mentioned property $H_8(\\Gamma,\\Gamma - c) \\cong \\tilde{H}_7(S_7)$ one can replace in the latter equation $\\Gamma$ by the homeomorphic equivalent small sphere $\\partial B(c,\\varepsilon)$, so we have \n$$\nw_{\\Gamma}(y) = \\frac{3}{\\pi^4} \\int\\limits_{\\partial B(c,\\varepsilon)} q_{\\bf 0}(w-y) d\\sigma(w).\n$$\nNext we replace the octonion $y$ by $f(c)-a$ and $\\partial B(c,\\varepsilon)$ by $(f-a)(\\partial B(c,\\varepsilon))$ and one obtains \n\\begin{eqnarray*}\nw_{\\Gamma}(y) &=& \\frac{3}{\\pi^4} \\int\\limits_{(f-a)\\partial B(c,\\varepsilon)} q_{\\bf 0}(w-(f(c)-a)) d\\sigma(w)\\\\\n&= & \\frac{3}{\\pi^4} \\int\\limits_{(f-a)\\partial B(c,\\varepsilon)} q_{\\bf 0}(w) d\\sigma(w) \\\\\n&=& w_{(f-a)(\\partial B(c,\\varepsilon))}(0).\n\\end{eqnarray*}\nWe recall that also $f(\\partial B(c,\\varepsilon))$ and hence also the translated expression $(f-a)(\\partial B(c,\\varepsilon))$ represents a $7$-dimensional cycle, cf. \\cite{AH} p. 470. \n\\end{proof}\n\\begin{remark}\\label{zero-order}\nIn contrast to complex analysis it can happen that one has ord$(f;c) = 0$ even if $f(c)=0$. This can occur for instance when the outward normal field of the surface of the image of the boundary cycle $\\Gamma$ turns into an inward directed one after one loop of the parametrization of $f(\\Gamma)$, so that in total all the contributions of the integration over the complete cycle $f(\\Gamma)$ can cancel out each other symmetrically when this happens. \nThis phenomenon already occurs in the quaternionic setting, as pointed out in {\\rm \\cite{Fueter1948-49}, p. 199}. \n\\end{remark}\n\\begin{remark}\nAs explained in {\\rm \\cite{Zoell}}, already in the quaternionic case there is no direct correspondence anymore between the order of an $a$-point and the number of vanishing coefficients in the octonionic Taylor series expansion. Note that in complex analysis one has the relation \n$$\n{\\rm ord}(f-a;c) = n, \\quad \\Longleftrightarrow \\quad (f-a)^{(k)}(c) = 0,\\; \\forall k < n,\\;(f-a)^{(n)}(c) \\neq 0. \n$$\nSince the situation is already so complicated in the quaternions, it cannot be expected that one gets a simpler relation for the octonionic case. Actually, analogues of the counter-example presented in {\\rm \\cite{Zoell}} on p.131-132 can easily be constructed. \n\n \n In the octonionic slice-regular setting described for instance in {\\rm \\cite{GPzeroes}}, the situation is much simpler. As mentioned previously, in the slice function theoretical setting an octonionic slice-regular function either has isolated zeroes or spherical zeroes, similarly to the slice-monogenic setting in $\\mathbb{R}^{n+1}$ cf. {\\rm \\cite{CSS,GPzeroes}}. In terms of the symmetric slice product the multiplicity of such a zero then can be described by the exponent of the (slice) power, namely in the usual way like in classical real and complex analysis: A slice-regular function $f$ can be decomposed uniquely in the way $f(z)=(z-a)^{*k}*g(z)$ where $g(z)$ is a uniquely defined and zero-free slice-regular function around $a$, see {\\rm \\cite{CSS,GPzeroes}} and elsewhere. Note that ordinary powers of $z$ are intrinsic slice regular functions, also in the octonions. The slice-product gives some kind of symmetric structure. In the setting of $\\mathbb{O}$-regular functions in the sense of the Cauchy-Riemann operator, such a decomposition is not possible, because of the lack of commutativity (and also of non-associativity). \n\n \n\n\\end{remark}\n\nThe definition of the order of an isolated $a$-point of an octonionic left or right $\\mathbb{O}$-regular function in the sense of Definition~\\ref{isolatedorder} is very natural from the topological point of view and so far the only meaningful tool to introduce a notion of ``multiplicity'' of an $a$-point. However, using this definition to calculate the value of the order of a concrete practical example is very difficult in general. Note that one has to perform the integration over the {\\em image} of the sphere. Now, a significant advantage of the octonionic setting in comparison to the Clifford analysis setting is that octonionic functions represent maps from $\\mathbb{O} \\to \\mathbb{O}$ which can be uniquely identified with a map from $\\mathbb{R}^8 \\to \\mathbb{R}^8$, by identifying the map $$x_0 + x_1 e_1 + \\ldots + x_7 e_7 \\mapsto f_0(z) + f_1(z) e_1 + \\cdots + f_7 (z)e_7$$ with the corresponding map $\\left(\\begin{array}{c} x_0 \\\\ x_1\\\\ \\vdots \\\\ x_7 \\end{array}\\right) \\mapsto \\left(\\begin{array}{c} f_0(x_0,\\ldots,x_7) \\\\ f_1(x_0,x_1,\\ldots,x_7)\\\\ \\vdots \\\\ f_7(x_0,x_1,\\ldots,x_7) \\end{array}\\right)$. However, in Clifford analysis one deals with maps from $\\mathbb{R}^8$ to $Cl_8 \\cong \\mathbb{R}^{128}$. Now, if the $7$-dimensional surface $\\partial G$ is parametrized as in (\\ref{param}), the image of that surface $f(\\partial G)$ can be parametrized as\n$$\nf(\\partial G) = \\{f_0(x(\\lambda_1,\\ldots,\\lambda_7)) + \\sum\\limits_{i=1}^7 f_i(x(\\lambda_1,\\ldots,\\lambda_7)e_i)\\}\n$$\nand one can simply apply the chain rule for ordinary real differentiable functions from $\\mathbb{R}^8 \\to \\mathbb{R}^8$, as indicated in \\cite{Hempfling} for purely paravector-valued functions. Applying the chain rule and exploiting the special mapping property that the image of octonionic functions are again octonions leads to the following octonionic generalization of the transformation formula from \\cite{Kra2004} p. 32. In the Clifford analysis case one had to restrict onself to particular paravector-valued functions. This restriction is not necessary in the octonionic setting:\n\\begin{lemma}\n\tLet $G \\subseteq \\mathbb{O}$ be a domain and suppose that each real component function of an octonionic function $f:G \\to \\mathbb{O}$ is real differentiable in the ordinary sense. Then we have \n\t$$\n\td\\sigma(f(z)) = [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)],\n\t$$\n\twhere $[(Jf)^{adj}(z)]$ stands for the adjunct real component $8 \\times 8$ matrix of the Jacobian $(Jf) = (\\frac{\\partial f_i}{\\partial x_j})_{ij}$.Furthermore, $[d\\sigma(z)]$ represents the $\\mathbb{R}^8$-vector composed by $\\stackrel{\\wedge}{dx_i}$ for $i=0,\\ldots,7$ and $\\circledcirc$ means the usual matrix-vector product, multiplying the real $8\\times8$-matrix in the usual way with the $8$-dimensional real vector. The resulting $\\mathbb{R}^8$-vector on the right-hand side then is re-interpreted as on octonion on the left-hand side identifying the unit vectors with the corresponding octonionic units. \n\\end{lemma} \nIt should be pointed out very clearly that $\\circledcirc$ does not mean the usual octonionic product. To be more explicit $[d \\sigma(z)]$ is interpreted as the vector \n$$\n[d\\sigma(z)] :=\n\\left(\\begin{array}{c} (-1)^0 \\stackrel{\\wedge}{dx_0} \\\\ (-1)^1\\stackrel{\\wedge}{dx_1}\\\\ \\vdots \\\\ (-1)^7 \\stackrel{\\wedge}{dx_7} \\end{array}\\right).$$ \nThe adjunct matrix $[(Jf)^{adj}(z)]$ has the form \n$$\n(Jf)^{adj}(z) =\\Bigg((-1)^{i+j} \\det \\Big( \\frac{\\partial f_i}{\\partial x_j}(z) \\Big)^{adj} \\Bigg)_{i,j}.\n$$\nThis also provides a correction to \\cite{Kra2004} p. 32 where the index $i$ of the function $f$ has been forgotten as well as the star after $(Jf)$ (indicating the adjunct) in the second line of the proof. The proof for the octonionic case can be done along the same lines as presented for the paravector-valued Clifford case in \\cite{Kra2004} p. 32. The chain rule leads to \n$$\nd\\sigma(f(z)) = \\sum\\limits_{i=0}^7 \\sum\\limits_{j=0}^7 (-1)^{i+j} e_i \\det \\Big(\\frac{\\partial f_i}{\\partial x_j}(z)\\Big)^{adj}(-1)^j \\stackrel{\\wedge}{dx_j}\n$$\nand the stated formula follows, because no associativity property is required. \n\\par\\medskip\\par\nThis lemma allows us to reformulate the definition of the order given in Definition~\\ref{isolatedorder} in the way that the integration is performed over the simple sphere $S_7(c,\\varepsilon)$. In contrast to the Clifford analysis case presented in \\cite{Kra2004} p. 33 we do not need to worry about a possible restriction of the range of values. All octonion-valued functions satisfying the left or right octonionic Cauchy-Riemann system are admitted here. \nHowever, the way how we put the brackets in the following theorem is crucially important. In the left $\\mathbb{O}$-regular case we have\n\\begin{theorem}\\label{order-reformulated}\nLet $G \\subseteq \\mathbb{O}$ be a domain. Let $f:G \\to \\mathbb{O}$ be a left $\\mathbb{O}$-regular function and suppose that $c \\in G$ is an isolated $a$-point of $f$ with $f(c)=a$. Choose $\\varepsilon > 0$ such that $\\overline{B}(c,\\varepsilon) \\subseteq G$ and $f(z) \\neq 0$ for all $z \\in \\overline{B}(c,\\varepsilon) \\backslash\\{c\\}$. Then the order of the $a$-point can be re-expressed by\n\\begin{eqnarray*}\n{\\rm ord}(f-a;c) &=& \\frac{3}{\\pi^4} \\int\\limits_{S_7(c,\\varepsilon)} q_{\\bf 0}(f(z)-a) \\cdot \\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg)\\\\ &=& \\frac{3}{\\pi^4} \\int\\limits_{S_7(c,\\varepsilon)} \\frac{\\overline{f(z)-a}}{|f(z)-a|^8}\\cdot \\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg).\n\\end{eqnarray*}\t\n\\end{theorem}\nHere, $\\cdot$ stands for the octonionic product, where the term inside the large parenthesis on the right is re-interpreted as octonion.\n\nNote that the Jacobian determinant is invariant under translations. Therefore $J(f-a)(z) = Jf(z)$. In the complex case the Jacobian simplifies to $(f-a)'(z)= f'(z)$ and one re-obtains the usual integrand $\\frac{f'(z)}{f(z)-a}$ because the Cauchy kernel then coincides with the simple inverse. \n \nFor the sake of completeness, in the right $\\mathbb{O}$-regular case one obtains \n\\begin{eqnarray*}\n\t{\\rm ord}(f-a;c) &=& \\frac{3}{\\pi^4} \\int\\limits_{S_7(c,\\varepsilon)} \\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg) \\cdot q_{\\bf 0}(f(z)-a) \\\\ &=& \\frac{3}{\\pi^4} \\int\\limits_{S_7(c,\\varepsilon)} \\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg) \\cdot \\frac{\\overline{f(z)-a}}{|f(z)-a|^8}.\n\\end{eqnarray*}\t\n \n\\par\\medskip\\par\nNote that we always have ord$(f-a;c)=0$ in all points $c$ where $f(c) \\neq a$. \nAs a direct application this property and the statement of Theorem~\\ref{order-reformulated} we can deduce the following argument principle for isolated $a$-points of $\\mathbb{O}$-regular functions which provides an extension of Theorem 1.34 from \\cite{Kra2004} where the paravector-valued Clifford holomorphic case has been treated. But also in the octonionic case we have \n\\begin{theorem} (Octonionic argument principle)\\\\\nLet $G \\subseteq \\mathbb{O}$ be a domain and suppose that $f:G \\to \\mathbb{O}$ is left $\\mathbb{O}$-regular over $G$. Now, consider a nullhomologous $7$-dimensional cycle $\\Gamma$ that parametrizes the surface of an $8$-dimensional oriented compact manifold $C \\subset G$. Under the assumption that $f$ has only isolated $a$-points in the interior of $C$ and no further $a$-points on the boundary $\\Gamma$, we have the order relation\n$$\n\\sum\\limits_{c \\in C} {\\rm ord}(f-a;c) = \\frac{3}{\\pi^4} \\int\\limits_{\\Gamma} \\frac{\\overline{f(z)-a}}{|f(z)-a|^8}\\cdot \\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg). \n$$\n\\end{theorem} \n\\begin{proof}\n\tThe proof follows along the same lines as in the Clifford analysis case given in \\cite{Kra2004} pp.33. This is a consequence of its predominant topological nature. The crucial point is that any oriented compact manifold can have atmost finitely many isolated $a$-points in its interior, let us call them $c_1,\\ldots,c_n$. Thus, one can find a sufficiently small real number $\\varepsilon > 0$ such that there are no $a$-points in the union of the sets $\\bigcup_{i=1}^n B(c_i,\\varepsilon) \\backslash\\{c_i\\}$. Since $f$ has neither further $a$-points nor singular points in the remaining part $C \\backslash \\bigcup_{i=1}^n B_i$ one obtains in view of Theorem~\\ref{order-reformulated} that \n\t$$\n\t\\int\\limits_{\\Gamma} \\frac{\\overline{f(z)-a}}{|f(z)-a|^8}\\cdot \\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg) = \\sum\\limits_{i=1}^n \\int\\limits_{S(c_i,\\varepsilon)} \\frac{\\overline{f(z)-a}}{|f(z)-a|^8}\\cdot \\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg).\n\t$$\n\tThe assertion now follows directly, when we take into account the mentioned property that ord$(f-a;c)=0$ at all $c \\in C$ with $f(c) \\neq a$. \n\\end{proof}\nThe big goal of the argument principle is that it provides us with a toplogical tool to control the isolated $a$-points or zeroes of an octonionic regular function under special circumstances. Its classical application is Rouch\\'e's theorem that presents a sufficient criterion to describe by which function an octonionic regular function may be distorted in the way that it has no influence on the numbers of isolated zeroes inside a domain, when particular requirements are met. Alternatively, it gives a criterion to decide whether two octonionic monogenic functions have the same number of isolated zeroes inside such a domain. In close analogy to the associative Clifford analysis case, cf. \\cite{Kra2004} Theorem 1.35, we may establish \n\\begin{theorem}\\label{rouche} (Generalized classical Rouch\\'e's theorem)\\\\\nSuppose that $G \\subseteq \\mathbb{O}$ is a domain and that $\\Gamma$ is a nullhomologous $7$-dimensional cycle parametrizing the boundary of an oriented compact $8$-dimensional manifold $C \\subset G$. Let $f,g:G \\to \\mathbb{O}$ be two $\\mathbb{O}$-regular functions that have only a finite number of zeroes inside of int $C$ and no zeroes on $\\Gamma$. Provided that $|f(z)-g(z)| < |f(z)|$ for all $z \\in \\Gamma$, then \n$$\n\\sum\\limits_{c \\in C} {\\rm ord}(f;c) = \\sum\\limits_{c \\in C} {\\rm ord}(g,c).\n$$\t\n\\end{theorem} \nAlso the nature of this theorem is predominantly topological. The topological aspects play a much more profound role than the function theoretical aspects, which nevertheless are also needed because the proof uses the argument principle involving the particular Cauchy-kernel of the octonionic Cauchy-Riemann system. Let us define a family of left $\\mathbb{O}$-regular functions depending on a {\\em continuous} real parameter $t \\in [0,1]$ by \n$$\nh_{t}(z) := f(z)+ t(g(z)-f(z)), \\quad z \\in G.\n$$\nFor each $t \\in [0,1]$ each function $h_z$ is left $\\mathbb{O}$-regular over $G$, since $t$ is only a {\\em real} parameter. Note that otherwise, the left $\\mathbb{O}$-regularity would be destroyed in general. Let $z \\in \\Gamma$. Then we have $|t(g(z)-f(z)|=|t||f(z)-g(z)| \\le |f(z)-g(z)| < |f(z)|$, where the latter inequality follows from the assumption. Therefore $h_t(z) \\neq 0$ for all $z \\in \\Gamma$. \n\n Furthermore, for each $t \\in [0,1]$ the entity ord$(h_t;c)$ is an integer. Since the number of zeroes is supposed to be finite in $G$, for each $t$ the sum $\\sum\\limits_{c \\in C} {\\rm ord}(h_t;c)$ is finite and represents a finite integer $N(t) \\in \\mathbb{Z}$. Per definition we have \n\\begin{eqnarray*}\n\tN(t) &=& \\sum\\limits_{c \\in C} {\\rm ord}(h_t;c)\\\\\n\t & =& \\frac{3}{\\pi^4} \\int\\limits_{\\Gamma} q_{\\bf 0}(h_t(z)) \\cdot \\Bigg(\n\t [(Jh_t)^{adj}(z)] \\circledcirc [d\\sigma(z)] \n\t \\Bigg)\\\\\n\t &=& \\frac{3}{\\pi^4} \\int\\limits_{\\Gamma} q_{\\bf 0}(f(z)+t g(z)-t f(z)) \\cdot \\Bigg(\n\t [(J (f+tg-tf))^{adj}(z)] \\circledcirc [d\\sigma(z)] \n\t \\Bigg).\n\\end{eqnarray*}\nSince all terms under the latter integral are continuous functions in the variable $t$, also the expression $N(t)$ on the left-hand side must be continuous in the variable $t$. However, $N(t)$ is an integer-valued expression for any $t \\in [0,1]$.Therefore, $N(t)$ must be a constant expression, hence $N(0)= \\sum\\limits_{c \\in C} {\\rm ord}(h_0;c) = \\sum\\limits_{c \\in C} {\\rm ord}(f;c) $ and $N(1) = \\sum\\limits_{c \\in C} {\\rm ord}(h_1;c) = \\sum\\limits_{c \\in C} {\\rm ord}(g;c)$ must be equal. \n\\par\\medskip\\par\nAs a nice application of Theorem~\\ref{rouche} we can establish the following weakened version of Hurwitz' theorem. The following statement can also be carried over to the quaternionic monogenic setting and to the context of paravector-valued monogenic functions in Clifford algebras, for which this statement has not been established so far, at least as far as we know. We prove \n\\begin{theorem}(Generalized Hurwitz theorem)\\\\\n\tLet $G \\subset \\mathbb{O}$ be a domain. Suppose that $f_n: G \\to \\mathbb{O}$ is a normally convergent sequence of $\\mathbb{O}$-regular functions with $f_n(z) \\neq 0$ at all $z \\in G$ and for each $n \\in \\mathbb{N}$. Then the limit function $f(z):= \\lim\\limits_{n \\to \\infty} f_n(z)$ has the property that either $\\sum\\limits_{c \\in G} {\\rm ord}(f;c)=0$ for all $c \\in G$ or $f$ vanishes identically over $G$. \n \\end{theorem}\n \n\\begin{proof}\nAccording to \\cite{XL2002} Theorem 11, left (or right) $\\mathbb{O}$-regular functions satisfy Weierstra{\\ss}' convergence theorem. \nTherefore, the limit function $f$ is a well-defined $\\mathbb{O}$-regular function over the whole domain $G$. Let us assume now that $f \\not\\equiv 0$ over $G$. Take an arbitrary point $z^* \\in G$. In view of the identity theorem of left $\\mathbb{O}$-regular functions (Proposition~\\ref{identity}) there must exist a positive real $R > 0$ such that the closed ball $\\overline{B(z^*,R)}$ is entirely contained inside $G$ and $M:= \\min_{z \\in S_7(z^*,R)} |f(z)| > 0$. Moreover, since $S_7(z^*,R)$ is compact there must exist an index $n_0 \\in \\mathbb{N}$ such that \n$$\n\\max_{z \\in S_7(z^*,R)} |f(z)-f_n(z)| < M,\\quad \\forall n \\ge n_0.\n$$ \nSummarizing, for all indices $n \\ge n_0$, we have the inequality\n$$\n|f(z)-f_n(z)| < M \\le |f(z)| \\quad\\quad \\forall z \\in S_7(z^*,R) \n$$\t\nwhich is the required condition of Rouch\\'e's theorem in Theorem~\\ref{rouche}. \n\nNow Rouch\\'e's theorem tells us that \n$$\n\\sum\\limits_{c \\in S_7(z^*,R)} {\\rm ord}(f;c) = \\sum\\limits_{c \\in S_7(z^*,R)} \\underbrace{{\\rm ord}(f_n;c)}_{=0}.\n$$\nNote that since $f_n(z) \\neq 0$ for all $z \\in G$ we have ${\\rm ord}(f_n;c)=0$ for all $n \\in \\mathbb{N}$. Since the points $z^*$ can be chosen arbitrarily inside of $G$, we can conclude that \n$$\n\\sum\\limits_{c \\in G} {\\rm ord}(f;c) = 0\n$$\nand the statement is proven.\n\\end{proof}\n\\begin{remark}\nNote that in contrast to the complex analytic case, ord$(f;c) = 0$ does not guarantee that $f(c)\\neq 0$, as pointed out in Remark~\\ref{zero-order}. Therefore, we can only establish this weaker statement.\n\\end{remark}\n\\begin{remark}\nIn the context of other regularity concepts, such as for slice-regular octonionic functions and generalized octonionic holomorphic functions in the sense of S.V. Ludkovski, generalized statements of Rouch\\'e and Hurwitz type could be established, see {\\rm \\cite{GPzeroes,L2007}}. \n\\end{remark} \n \n \n \\section{Rudiments for the treatment of non-isolated zeroes}\nThe following section presents results which are even new for quaternionic functions and paravector-valued functions in associative Clifford algebras. \n\nThe aim is to present a meaningful definition of the order of zeroes or $a$-points of an $\\mathbb{O}$-regular function that are not-isolated but lying on a $k$-dimensional simply connected compact manifold of dimension $1 \\le k \\le 6$, including in the simplest case compact algebraic varieties in eight variables. \n\nThe case $k=0$ is the isolated case which has been treated in the previous section. As mentioned in the previous section, the case $k=7$ does not appear in the $\\mathbb{O}$-regular setting, because of the identity theorem for $\\mathbb{O}$-regular functions (Proposition~\\ref{identity}), which excludes this situation. Without loss of generality we focus on the treatment of compact varieties of zeroes, because varieties of $a$-points can be studied in the same way by looking at the function $f(z)-a$. \n\nLet us recall that in the isolated case one can always consider a small sphere around that zero with the property that no zeroes lie inside or on the boundary of that sphere. \n\nLet us now suppose that we have a $k$-dimensional simply-connected compact variety of zeroes ($k \\le 6$), that we call $M$. To leave it simple we restrict ourselves in all that follows to those varieties that do not have auto-intersections. \n\nIn the case of dealing with a variety of non-isolated zeroes with these properties, \nthe proper analogue of a sphere surrounding an isolated point is a tubular domain of thickness $\\varepsilon >0$ of the form\n$$\nT_M^{\\varepsilon} :=\\{z \\in \\mathbb{O} \\backslash M \\mid \\min_{c \\in M}\\{|z-c|\\} =\\varepsilon\\}.\n$$ \nIn the case where $k = {\\rm dim}\\;M=1$ and where $M$ is a finite closed line segment, parametrized in the form $[\\gamma] = \\gamma(t),\\quad t \\in [0,1]$, the domain \n$$\nT_{[\\gamma]}^{\\varepsilon} :=\\{z \\in \\mathbb{O} \\backslash M \\mid \\min_{t \\in [0,1]}\\{|z-\\gamma(t)|\\} =\\varepsilon\\}\n$$ \nis nothing else than an ordinary symmetric circular tube of thickness $\\varepsilon$ around that line segment. In the case where $M$ is a closed circle, the associated tubular domain $T_{[\\gamma]}^{\\varepsilon}$ is a generalized torus, more precisely it is homeomorphically equivalent to the real Hopf manifold $S_1 \\times S_6 \\cong \\mathbb{R}^8 \\backslash\\{0\\}\/\\mathbb{Z}$. A concrete example of a left and right $\\mathbb{O}$-regular function where the zero set is up to atmost two isolated points the unit circle lying in the subspace generated by $e_1$ and $e_2$ is the function $f(z)=Z_1^2+Z_2^2-1+\\sum\\limits_{j=3}^7 Z_j e_j$, where again $Z_i = x_i - x_0 e_i$ for all $i=1,\\ldots,7$. \n\nIn the particular case where $M$ is just an isolated point, say $M =\\{z_0\\}$, the tube then reduces to the set $T_{z_0} = \\{z \\in \\mathbb{O} \\mid |z-z_0| = \\varepsilon\\}$ which is the ordinary sphere $\\partial B_8(z_0;\\varepsilon)$ of the eight-dimensional ball. Thus, tubular domains provide us with a natural analogue for circular symmetric neighborhoods around closed simply connected manifolds with no auto-intersections.\n \nIn this framework, this is an adequate geometry to meaningfully introduce the notion of the order of a compact simply connected zero manifold of a left $\\mathbb{O}$-regular function, generalizing the definitions given above for the isolated case. \nWe introduce \n\\begin{definition}\nSuppose that $U \\subseteq \\mathbb{O}$ is a non-empty open set. Let $f:U \\to \\mathbb{O}$ be left $\\mathbb{O}$-regular and suppose that $M$ is a compact simply connected manifold of dimension $k \\in \\{0,1\\ldots,6\\}$ with the above mentioned properties and with $M \\subset U$ and $f(z)=0$ for all $z \\in M$. Further assume that there is a real positive $\\varepsilon > 0$ such that $T_M^{\\varepsilon} \\subset U$ and that $f(z) \\neq 0$ for all $z \\in T_M^{\\varepsilon}$ and for all $z \\in int T_M^{\\varepsilon} \\backslash M$.Then we can define the order of the non-isolated zero variety $M$ of $f$ by\n\\begin{eqnarray*}\n\t{\\rm ord}(f;M) &:=& \\frac{3}{\\pi^4} \\int\\limits_{f(T_M^{\\varepsilon})} q_{\\bf 0}(w) d \\sigma(w)\\\\\n\t&=& \\frac{3}{\\pi^4} \\int\\limits_{T_M^{\\varepsilon}} q_{\\bf 0}(f(z)) \\cdot \n\t\\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg).\n\\end{eqnarray*}\t \n\\end{definition} \n\\begin{remark}\n\tThe integral counts how often the image of the tubular surface $T_M^{\\varepsilon}$ under $f$ wraps around zero. \n\tAll zeroes belonging to the same zero variety $M$ have the same order, because the winding number is in view of its homotopic property a continuous and hence constant expression. The zero variety $M$ is simply-connected. Therefore ${\\rm ord}(f;c_i) = {\\rm ord}(f;c_j) = {\\rm ord}(f;M)$ for all $c_i,c_j \\in M$. \n\\end{remark}\nNotice further that the integral expressions are really well-defined because we do not integrate over any zeroes of $f$; $f(z) \\neq 0$ for all $z \\in T_M^{\\varepsilon}$. \n\nThis generalized notion allows us to set up a generalized version of the octonionic argument principle where we now may admit left $\\mathbb{O}$-regular functions having a finite number of compact simply-connected zero varieties $M_1,\\ldots,M_p$ with no auto-intersections of dimension $k_1,\\ldots,k_p$, respectively, lying inside a domain $G \\subset \\mathbb{O}$. We can prove\n\\begin{theorem} (Generalized octonionic argument principle for non-isolated zeroes)\\\\\n\tLet $G \\subset \\mathbb{O}$ be a domain. Suppose that $f:G \\to \\mathbb{O}$ is a left $\\mathbb{O}$-regular function over $G$. Assume that $C$ is an $8$-dimensional oriented compact manifold $C \\subset G$ whose boundary is parametrized by a $7$-dimensional null-homologous cycle $\\Gamma$. Furthermore, suppose that $f$ has a finite number of simply-connected closed zero varieties $M_1,\\ldots,M_p$ with no auto-intersections of dimension $k_1,\\ldots,k_p$, respectively, and that $f$ has no further zeroes inside of $C$ nor on its boundary $\\Gamma$. Then we have \n\t$$\n\t\\frac{3}{\\pi^4} \\int\\limits_{f(\\Gamma)} q_{\\bf 0}(w) d \\sigma(w) = \\sum\\limits_{i=1}^p {\\rm ord}(f;M_i).\n\t$$ \n\\end{theorem}\n\\begin{proof}\nSince $f$ has no zeroes on $\\Gamma$ and since $C$ is compact the following integral and integral transformation is well-defined:\n\n\\begin{equation}\\label{preveq}\n\\frac{3}{\\pi^4} \\int\\limits_{f(\\Gamma)} q_{\\bf 0}(w)d\\sigma(w) = \\frac{3}{\\pi^4} \\int\\limits_{\\Gamma} q_{\\bf 0}(f(z)) \\cdot \n\\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg).\n\\end{equation}\nSince $f$ has no zeroes in $C \\backslash \\bigcup_{i=1}^p M_i$, we have that $\\sum\\limits_{c \\in C \\backslash \\bigcup_{i=1}^p M_i} {\\rm ord}(f;c) = 0$, so that \nthe latter integral from (\\ref{preveq}) can be expressed in the form \n$$\n\\frac{3}{\\pi^4}\\sum\\limits_{i=1}^p\\int\\limits_{T_{M_i}^{\\varepsilon_i}} q_{\\bf 0}(f(z)) \\cdot \n\\Bigg( [(Jf)^{adj}(z)] \\circledcirc [d\\sigma(z)] \\Bigg) = \\sum\\limits_{i=1}^p {\\rm ord}(f;M_i),\n$$\nbecause the contribution of this integral over the boundary of a domain that contains no zeroes inside is zero. \n\\end{proof}\n\\begin{remark}\n\tThe statement remains valid in the Clifford analysis setting, addressing paravector-valued functions with zero varieties that have the above mentioned properties. \n\\end{remark}\n\\section{Perspectives}\nThe previous section suggests an approach how to address orders of non-isolated zeroes of octonionic regular or Clifford monogenic functions in the sense of the Riemann approach. A further step would consist in applying this argument principle to establish generalizations of Rouch\\'e's theorem and Hurwitz' theorem to the non-isolated context. Obviously, the geometric conditions claimed in the previous section are very strong. As mentioned in Section~3 it is very easy to also construct $\\mathbb{O}$-regular functions that have zero varieties of infinite extension. If we want to address varieties with auto-intersections, then we have to adapt the use of tubular domains. An important question is to investigate which genus do the arising zero manifolds have in the most general case. To get some insight in these kinds of questions a profound study of algebraic geometrical methods, in particular a deep study of understanding the nature of the appearing zero varities of $\\mathbb{O}$-regular functions is required. Working on the intersection of algebraic geometry and hypercomplex function theories represents a promising branch for future investigation. \n\n\nFurthermore, this paper shows that the argument principle is more a topological theorem than an analytic one, although the Cauchy kernel is explicitly needed in its definition. However, the predominant topological character gives the hope that these kinds of theorems can be carried over to many more hypercomplex function theories, in particular to the context of null-solutions to other differential equations. However, a really substantial question is to ask whether these tools can be carried over to functions that are defined in other algebras beyond octonions and paravector-valued subspaces of Clifford algebras. Both paravector spaces and octonions are normed and are free of zero-divisors. Following K. Imaeda \\cite{Imaeda}, already in the context of sedenions it is not anymore possible to set up a direct analogue of Cauchy's integral formula. Cauchy's integral formula however is the basic tool for establishing all these results. The appearance of zero divisors will also have an impact on the topological properties. There remain a lot of open questions and challenges for future research. \n\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIt is well-known that improving pass rates in mathematics courses is of paramount importance for academic institutions all over the world. This objective becomes even more critical for public universities as they subside partially or totally the education of its enrolled undergraduates; subject to the country's legislation and the student's economic stratification.\n\nThe general consensus is that creating better conditions for the students will improve students' success. Hence, most of the work done in order to address this challenge has two principal directions: \n\\begin{enumerate}[(i)]\n\\item The traditional pedagogical approach which, essentially aims to improve the presentation of the course contents on two fronts: presentation of mathematical discourse i.e., curricula reform, development of course materials and improvement of the lecturer's instructional practice. Part of the latter are the teaching evaluations' open questions, giving feedback to the instructor about how people felt during his\/her classes. \n\n\\item The uses of technology in the learning of mathematics. One stream goes in the recollection of data and the measurement of the resource impact in the cognitive process: development of LMS platforms and real-time feedback interfaces \\cite{Theroux}. The other stream explores the use of the aforementioned harvested information to improve the learning process \\cite{CastroEtAl}: targeted problem sets and training tests \\cite{BeckMostow}, identification of favorable pedagogical approaches and learning patterns\/styles \\cite{LevyWilenski}, identification of fortitudes and weaknesses, assessment of study exercise vs skills building \\cite{LevyWilenski}, problem solving approaches, platforms for interaction between students through the learning process \\cite{HerrenkohlTasker}. There is also the use of big data to asses learning rather than improve instructional techniques, such as early detection of students at risk \\cite{MaacfaydenDawson}. \n\\end{enumerate}\nThe present work fits in the second category, in this case, the use of big data to define policies enhancing the higher education production \\cite{Lazear} and without raising the costs. More specifically, the method will suggest an optimal design of student body\/composition to maximize the pass rate chances. The design is driven by favorable teacher-student partnership, rather than peer diversity or a peer interaction criterion (see \\cite{DeGiorgiPellizaariWoolstonGui, DeGiorgiPellizaariRedaelli}). A second aspect of the method is that, it is based on the computation of expectations (conditioned to the students' segmentation) and not on statistical regression (linear or not), as it pursues the construction of a probabilistic model, rather than the construction of a production function (see \\cite{DeGiorgiPellizaariWoolstonGui, Kruegger}). In addition, the input database (which is the method's input), is updated from one academic term to the next, therefore, it seems more adequate to recompute the conditional expectations term-wisely instead of pursuing rigid regression models. Moreover, given the current computational tools and possibilities, once the method is coded as an algorithm, the proposed updating approach will come at zero cost increase. This is the main reason while our method will be presented and explained, mostly in the format of an algorithm.\n\nOur approach starts from data bases of a public university for 15 semesters, between February 2010 and July 2017, containing the information of academic performance and demographics of its population. A first stage of descriptive statistics allows to identify the success factors by correlation. A second stage revisits the historical performance (15 semesters in total), it defines segmentation profiles and then computes the historical efficiency for each of the involved instructors, conditioned to the quantiles of segmentation. Next, it uses integer programming to find the optimal matching of students-instructors in two different ways. A third stage of the method randomizes the involved factors, namely the profile of the group taking the class, the number of Tenured Track instructors, the number of sections (group) in the course and such. This is aimed to produce Monte Carlo simulations and find the expected values of improvement in the long run. The factors are regarded as random variables with probabilistic distributions computed from the empirical knowledge, recorded in the database and each Monte Carlo simulation arises from one random realization of the algorithm. \n\\subsection{Economic Justification}\\label{economic_justification}\nWhether or not public higher education constitutes a public good, is a subject that has been extensively debated in economics (see, e.g, \\cite{grace1989education} and \\cite{tilak2008higher}). On one side, it benefits the whole society by playing a redistributive role, where low income class students can access higher education to improve their future labor perspectives. On the other side, public higher education can be of limited access.\\footnote{In Colombia, by 2016 only 49,42\\% of the students had access to higher education (official information at \\url{https:\/\/www.mineducacion.gov.co\/portal\/}.) }. This last feature shows why the classification of public higher education as a public good, is a matter of debate among economists, because a public good must be accessible to all individuals. Such a debate is not the matter of this paper, but it highlights two important properties of public higher education (its redistribution role, and its limited access), which are relevant to understand the contribution of this work. \n\nThe Colombian government has implemented different strategies to address the limited access aspect of education, implementing programs like ``Familias en Acci\\'on'', a welfare program designed to improve school attendance rates among the young; and ``Ser Pilo Paga'', a merit-based financial aid program designed to increase the higher education attendance rates among the poorest. Although these strategies have increased the attendance rates in education (as showed by \\cite{londono2017intended} in the case of higher education), a report from the World Bank (\\cite{marta2017crossroads}) showed that 37 percent of students starting a bachelors' degree program withdraw from the higher education system (with percentages going as high as 53 percent, when including short-cycle programs). These numbers show that the limited access aspect of public higher education, can not be addressed only by means of increasing the raw coverage. Higher education institutions need strategies to decrease drop out rates, and require mechanisms to help students to improve their pass rates, grades and others. \n\nThis paper provides a mechanism--an algorithm- to potentially help higher education institutions to improve such indicators of student welfare (pass rates and grades). \nOur approach is to understand the University, not only as an agent that provides education, but also as a \\textbf{rational regulator agent}, capable to optimally allocate some of its resources for enhancement of social welfare of its students body.\n\nThroughout the paper we will make remarks emphasizing the interpretation and\/or importance of the problem from the economic point of view. To that end we have introduced the environment ``Analysis\" where these annotations will be included.\n\\subsection{Organization and notation}\nThe paper is organized as follows. In \\textsc{Section} \\ref{Sec The study case and its databases}, we a brief description of the study case setting and its databases. In \\textsc{Section} \\ref{Sec Quantification: Variables, Segmentation and Profiling} the available historical information is quantified by a process of statistical segmentation and profiling. In \\textsc{Section} \\ref{Sec Optimization and Historical Assessment} the two optimization mechanisms are presented, formulated a problems of integer programming and an assessment of the historical behavior is performed, i.e., compute the theoretical outcome should the method would have been applied in the past. In \\textsc{Section} \\ref{Sec Randomization and Asymptotic Assesment} we randomize the study case, using the historical records in order to generate random and plausible instances of the study case and apply the optimization method in order to perform Monte Carlo simulations, as well as observing its asymptotic behavior. Finally, \\textsc{Section} \\ref{Sec Conclusions} delivers the conclusions.\n\nWe close the introduction describing the mathematical notation. For any natural number $ N\\in \\N $, the symbol $ [N] \\defining \\{ 1, 2, \\ldots , N \\} $ indicates the set\/window of the first $ N $ natural numbers. For any set $ E $ we denote by $ \\vert E \\vert $ its cardinal and $ \\wp(E) $ its power set. We understand $ \\Omega $ as a generic finite probability space $ \\big( \\Omega, \\wp(\\Omega), \\prob \\big) $ in which all outcomes are equally likely, i.e. the event probability function satisfies $ \\prob(\\{\\omega\\}) = \\vert \\Omega \\vert^{-1} $ for all $ \\omega \\in \\Omega $. In particular for any event $ E \\subseteq \\Omega $ it holds that \n\\begin{equation}\\label{Eq Probabiity Measure}\n\\prob( E ) \\defining \\dfrac{\\vert E \\vert}{\\vert \\Omega\\vert } = \\dfrac{\\text{number of favourable outcomes}}{\\text{total number of possible outcomes} } .\n\\end{equation}\nA particularly important probability space is $ \\itS_{N} $, where $ \\itS_{N} $ denotes the set of all permutations in $ [N] $, its elements will be usually denoted by $ \\pi, \\sigma, \\tau $, etc. Random variables will be represented with upright capital letters, namely $ \\X, \\Y, \\Z, ... $, expectation and variance of such variables with $ \\Exp(\\X) $ and $ \\Var(\\X) $ respectively.\nVectors (deterministic or random) are indicated with bold letters, namely $ \\p, \\g,\\mathbf{X} ... $ etc. Deterministic matrices are represented with capital letters i.e., $ A, G, T $.\n\\section{The study case and its databases}\\label{Sec The study case and its databases}\nIn this work our study case is the performance of lower division mathematics courses at Universidad Nacional de Colombia, Sede Medell\\'in (National University of Colombia at Medell\\'in). The Institution is a branch of the National University of Colombia, the best ranked higher education Institution in Colombia; it is divided in five colleges: Architecture, Science, Humanities \\& Economical Sciences, Agriculture and Engineering (Facultad de Minas). The colleges are divided in schools and\/or departments. The University offers 27 undergraduate programs and 85 graduate programs divided in Specializations, MSc and PhD levels, depending on the school\/department. Each semester, the University has an average enrollment of 10000 undergraduates and 2000 graduates with graduation rates of 1240 and 900 respectively. The College of Engineering is the most numerous, consequently, the mathematics lower division courses are highly demanded and have a profound impact in the campus' life; its teaching and evaluation is in charge of the School of Mathematics.\n\nThe School of Mathematics is part of the College of Science, it teaches two types of courses: specialization (advanced undergraduate and graduate courses in mathematics) and service (lower division). The latter are: \\textit{Basic Mathematics} (BM, college algebra), \\textit{Differential Calculus} (DC), \\textit{Integral Calculus} (IC), \\textit{Vector Calculus} (VC), \\textit{Differential Equations} (ODE), \\textit{Vector \\& Analytic Geometry} (VAG), \\textit{Linear Algebra} (LA), \\textit{Numerical Methods} (NM), \\textit{Discrete Mathematics} and \\textit{Applied Mathematics}. The total demand of these courses amounts to an average of 7200 enrollment registrations per semester. The last two courses, \\textit{Discrete Mathematics} and \\textit{Applied Mathematics} have very low an unstable enrollment, therefore, their data are not suitable for statistical analysis and they will be omitted in the following. On the other hand, the remaining courses are ideally suited for big data analysis, due to its massive nature; see Table \\ref{Tb Historical Enrollment Table} below.\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{ c c c c c c c c c c c}\n \\hline\n \\rowcolor{gray!80}\nYear \n& Semester\n& DC\n& IC\n& VC\n& VAG\n& LA\n& ODE\n& BM\n& NM \n& Total\\\\\n \\toprule\n2010 &\t1 &\t1631 &\t782\t & 381\t& 1089\t & 983\t& 668 &\t848\t& 142 & 6524 \\\\\n2013 &\t2 &\t1446 &\t1212 &\t549\t& 1187\t& 1103 & 786 &\t846\t& 326 & 7455 \\\\\n2016 &\t2 &\t1569 &\t1296 &\t594\t& 1355\t& 1009 & 1019 &\t1111 &\t284\t& 8237 \\\\\n\\rowcolor{gray!80}\n\\hline\nMean & Does not apply & 1445.9\t& 1122.9 & 549.8 & 1146.5 & 988.7 & 801.8 & 905.4\t& 267.5 & 7228.5 \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Historical Enrollment Sample. The table shows the mean enrollment for each course for the period from 2010-1 to 2017-1, together with a sample of three semesters in that time window.}\\label{Tb Historical Enrollment Table}\n\\end{table}\nOn a typical semester these courses are divided in sections (between 8 to 22, depending on the enrollment) of sizes ranging from 80 to 140 (because of classroom physical capacities). There is no graded homework but students have problem sheets as well as optional recitation classes. As for the \\textbf{grading scale} 5.0 is the maximum, 3.0 is the \\textbf{minimum pass grade} and grades contain only \\textbf{one decimal}. The evaluation consists in three exams which the students take simultaneously; the activity is executed with the aid of the software packages SiDEx-$ \\Omega $ and RaPID-$ \\Omega$ which manage the logistics of the evaluation and proctoring activities, including the organization of the grading stage. More specifically, for fairness and consistency of the grading process a particular problem is graded by one single grader for all the students, i.e., it is a \\textbf{centralized} grading process. As a consequence of the institutional policies described above, the data are statistically comparable. Moreover, the paper-based tests administrator SiDEx-$ \\Omega $ introduces high levels of fraud control, because of its \\textit{students seating assignment algorithm}; this increases even more the \\textbf{reliability} of the data.\n\\subsection{The Databases}\nThe University allowed limited access to its data bases for the production of this work. The information was delivered in five separate tables which were merged in one single database using Pandas: the file \\textit{Assembled\\_Data.csv} which contains 108940 rows, each of them with the following fields\n\\begin{enumerate}[(i)]\n\\item Student's Personal Information: \n$ \\bullet $ Year of Birth\n$ \\bullet $ e-mail\n$ \\bullet $ ID Number\n$ \\bullet $ Last Name and Names\n$ \\bullet $ Gender\n\n\\item Student's General Academic Information:\n$ \\bullet $ University Entrance Year (AA, Academic Age)\n$ \\bullet $ Career\n$ \\bullet $ Academic Average (GPA)\n\n\n\\item Student's Academic Information Relative to the Course:\n$ \\bullet $ Course\n$ \\bullet $ Course Code\n$ \\bullet $ Academic Year \n$ \\bullet $ Academic Semester\n$ \\bullet $ Grade\n$ \\bullet $ Completed vs Canceled \n$ \\bullet $ Number of Attempts\n$ \\bullet $ Number of Cancellations \n\n\n\\item Student's Administrative Information Relative to the Course:\n$ \\bullet $ Section Number\n$ \\bullet $ Schedule\n$ \\bullet $ Section Capacity\n$ \\bullet $ Number of Enrolled Students\n$ \\bullet $ Instructor's ID Number\n$ \\bullet $ Instructor's Name\n$ \\bullet $ Tenured vs. Adjoint Instructor\n\n\\end{enumerate}\n\\begin{remark}[Meaning of a row]\\label{Rem Row Meaning}\nIt is understood that one registration corresponds to one row, for instance if a particular student registers for DC and LA in the same term, one row is created for each registration, repeating all the information listed in items (i) and (ii) above. The same holds when an individual needs to repeat a course because of previous failure or cancellation. \n\\end{remark}\n\\section{Quantification: Variables, Segmentation and Profiling}\\label{Sec Quantification: Variables, Segmentation and Profiling}\nIn the present work we postulate the Lecturer as one of the most important factors of success, more specifically the aim is to attain the optimal Instructor-Student partnership; in this we differ from \\cite{DeGiorgiPellizaariWoolstonGui, DeGiorgiPellizaariRedaelli} where its proposed that the class composition should be driven by the peers interaction. To that end, it becomes necessary to profile the students' population according to its relevant features. \n\\subsection{Determination of the Segmentation Factor}\nComputing the correlation matrix of the quantitative factors considered in the \\textit{Assembled\\_Data.csv}; in the table \\ref{Tb Correlations DC} we display the correlation matrix for the Differential Calculus course. From the \\textit{Grade} row it is clear that the most significant factor in the \\textit{Grade} variable is the \\textit{Academic Average} (GPA) followed by the \\textit{Academic Age} (AA) and the \\textit{Age}. However, the impact of the GPA is about four times the impact of AA and the same holds for the \\textit{Age} factor. Moreover, from the GPA row, it is clear that the most significant factor after the \\textit{Grade} are precisely the \\textit{Academic Age} and the \\textit{Age} (the younger the student, the higher the GPA). In addition, for the remaining courses, similar correlation matrices are observed. Hence, we keep the GPA as the only significant quantitative factor in the \\textit{Grade} variable. \n\\begin{remark}\\label{Rem sections size}\nIt is important to mention that the impact of the class size in the students' performance has been subject of extensive discussion without consensus. While \n\\cite{AngrisLavy, Kruegger} report a significant advantage in reducing class sizes, \\cite{Hoxby} finds no effect. In our particular case \\textsc{Table} \\ref{Tb Correlations DC} shows that the \\textit{Section Capacity} is uncorrelated, not only to the \\textit{Grade} variable, but also to the \\textit{GPA} variable. Moreover the \\textit{Section Capacity} is uncorrelated with the \\textit{Cancellations} (drop out) variable. \n\\end{remark}\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\scriptsize{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{ c | c c c c c c c c }\n \\hline\n \\rowcolor{gray!80}\nFACTOR\n& Section \n& Age\n& AA\n& \\# Enrolled \n& Grade\n& \\# Cancellations\n& \\# Attempts\n& GPA\n\\\\\n\\rowcolor{gray!80}\n& Capacity\n&\n& \n& Students\n& \n& \n& \n& \n\\\\\n\n\\hline \nSection Capacity\n& 1.0000 \t & -0.0108\t& 0.0570\t& 0.8334 & 0.0074\t& 0.0003 &\t0.0393 & 0.0180 \\\\\nAge\n& -0.0108\t& 1.0000\t& 0.3069\t& -0.0168\t& -0.2031 & 0.0775\t & 0.1384 &\t-0.2082 \\\\\nAA\n& 0.0570\t & 0.3069\t & 1.0000\t & 0.0416\t & -0.2164\t& 0.1668 &\t0.4294 &\t-0.1667 \\\\\n\\# Enrolled Students\n& 0.8334\t& -0.0168 & 0.0416\t& 1.0000\t& 0.0252 & 0.0131 &\t-0.0041\t& 0.0325 \\\\\n\\ding{217} Grade\n& 0.0074\t& -0.2031\t& -0.2164 & 0.0252\t & 1.0000\t & -0.1247 &\t-0.0241\t& $ \\star $ 0.8207 \\\\\n\\# Cancellations\n& 0.0003\t& 0.0775\t& 0.1668\t& 0.0131 \t& -0.1247\t& 1.0000 &\t0.3101 & -0.0686 \\\\\n\\# Attempts\n& 0.0393\t& 0.1384\t& 0.4294\t& -0.0041 & -0.0241 &\t0.3101 &\t1.0000 &\t-0.0401 \\\\\n\\ding{217} GPA \n& 0.0180\t& -0.2082\t& -0.1667 &\t0.0325\t & $ \\star $ 0.8207\t & -0.0686 &\t-0.0401\t& 1.0000 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Quantitative Factors Correlations Table, Course: \\textbf{Differential Calculus}. The table displays the correlation matrix for the variables of interest (Section Capacity, Age, AA, \\# Enrolled Students, Grade, \\# Cancellations, \\# Attemps, GPA).\n}\\label{Tb Correlations DC}\n\\end{table}\n\nTwo binary variables remain to be analyzed namely \\textit{Pass\/Fail} (PF) and \\textit{Gender}. If we generically denote by $ X $ the binary variables and by $ Y $ a variable of interest, the point-biserial correlation coefficient is given by\n\\begin{equation}\\label{Eq point-biserial correlation}\nr_{pb} \\defining \\frac{M_{1} - M_{0} }{\\sigma} \\sqrt{\\frac{N_{1}\\, N_{0}}{N^{2}}}.\n\\end{equation}\nHere, the indexes $ 0,1 $ are the values of the binary variable $ X $. For $ i = 0, 1 $, $ M_{i} $ is the mean value of the variable $ Y $ for the data points in the group\/event $ \\{ X = i \\} $, $ N_{i} $ denotes the population of each group $ \\{ X = i \\} $, $ N = N_{1} + N_{0} $ stands for the total population and $ \\sigma $ indicates the standard deviation of the variable $ Y $. \n\nThe correlation analysis between the binary \\textit{Pass\/Fail} (PF) variable vs the quantitative factors is displayed in Table \\ref{Tb PF Biserial Correllations}. As in the \\textit{Grade} variable analysis, the most significant factor in the \\textit{Pass\/Fail} variable, is the \\textit{Academic Average} (GPA) followed by the \\textit{Academic Age} (AA) and the \\textit{Age}. In this case however, the impact of the GPA is only three times the impact of AA as well as the \\textit{Age} factor. Again, we keep the GPA as the only significant quantitative factor in the \\textit{Pass\/Fail} variable. \n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{ c | c c c c c c c c }\n \\hline\n \\rowcolor{gray!80}\n\\diagbox{FACTOR}{COURSE}\n& DC\n& IC\n& VC\n& VAG\n& LA\n& ODE\n& BM\n& NM\n \\\\\n\n\\hline\nSection Capacity \n& 0.0058 &\t0.0139 & -0.0296 & -0.0986 & -0.0147 &\t0.0448 & -0.0451 & 0.0840 \\\\\nAge\n& -0.1757 &\t-0.2554 & -0.3048 &\t-0.1879\t& -0.2444 &\t-0.2555 & -0.0892 & -0.3333 \\\\\nAA\n& -0.1783 &\t-0.2855 & -0.3012 &\t-0.1550 & -0.2230 &\t-0.3841\t& -0.0209 &\t-0.2731 \\\\\n\\# Enrolled Students\n& 0.0070 &\t0.0276 & 0.0172\t& -0.0410 & 0.0602 & 0.1258\t& -0.0374 &\t0.1916 \\\\\nGrade\n& 0.8072 & 0.7987 &\t0.7864 & 0.8145\t& 0.7959 & 0.7999 &\t0.7988 & 0.7922 \\\\\n\\# Cancellations\n& -0.1129 &\t-0.1397 & -0.1420 &\t-0.1299\t& -0.1178 &\t-0.1647\t& -0.0096 &\t-0.1489 \\\\\n\\# Attempts\n& -0.0988 &\t-0.1635\t& -0.1683 &\t-0.1374\t& -0.1399 &\t-0.1577 & -0.0054 &\t-0.2065 \\\\\n\\ding{217} GPA\n& 0.6062 & 0.5892 &\t0.5884 & 0.6445 & 0.6063 & 0.5341 &\t0.6125 & 0.5828\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Pass\/Fail vs Quantitative Factors, Biserial Correlations Table, \n\tCourse: \\textbf{All}. We display the biserial correlations for each of the variables (Section Capacity, Age, AA, \\# Enrolled Students, Grade, \\# Cancellations, \\# Attemps, GPA) compared to the Pass\/Fail variable, per course.}\\label{Tb PF Biserial Correllations}\n\\end{table}\n\nNext, the correlation analysis \\textit{Gender} variable vs the Academic Performance Variables, i.e., \\textit{Grade}, GPA and \\textit{Pass\/Fail} (PF) is summarized in the table \\ref{Tb Gender Correllations} below. Clearly, the \\textit{Gen}der variable has negligible incidence in the Academic Performance Variables, with the exception of the GPA for the Basic Mathematics (BM) course case, where females do slightly better. Since this unique correlation phenomenon is not present in the remaining courses, the \\textit{Gender} variable will be neglected from now on. Finally, it is important to stress that, given the binary nature of the \\textit{Gender} and the \\textit{Pass\/Fail} (PF) variables, all the correlation coefficients agree i.e., point-biserial, Pearson and Spearman and Kendall. \n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{ c | c c c c c c c c }\n \\hline\n \\rowcolor{gray!80}\n\\diagbox{FACTOR}{COURSE}\n& DC\n& IC\n& VC\n& VAG\n& LA\n& ODE\n& BM\n& NM\n \\\\\n\n\\hline\nGrade\n& 0.0545 &\t0.0584 & 0.0240\t& -0.0485 &\t0.0154 & 0.0164\t& -0.0740 &\t-0.0405 \\\\\nGPA\n& -0.0425 &\t-0.0441\t& -0.0419 &\t-0.0766\t& -0.0343 & -0.0824\t& -0.1159 & -0.0663 \\\\\nPass\/Fail (PF)\n& 0.0509 &\t0.0382\t& 0.0166 &\t-0.0373\t& 0.0103 &\t0.0085 & -0.0535 & -0.0226 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Gender vs Academic Performance Variables,\n\tCourse: \\textbf{All}. The biserial correlations of each of the variables in (Grade, GPA, Pass\/Fail) compared to the Gender variable, is displaye for each course.}\\label{Tb Gender Correllations}\n\\end{table}\n\nFrom the previous discussion, it is clear that out of the analyzed variables, the GPA is the only one with significant incidence on the academic performance variables \\textit{Grade} and \\textit{Pass\/Fail}. Consequently, this will be used as the unique criterion for the segmentation of students' population. \nFrom now on, our analysis will be focused on the \\textit{Grade Average} and the \\textit{Pass\/Fail} variables as measures of success, while the \\textit{GPA} will be used for segmentation purposes discussed in Section \\ref{Sec Segmentation Process}. In \\textsc{Table} \\ref{Tb Academic Performance Variables Average}, the global averages (from 2010-1 to 2017-1) of these variables for all the service courses are displayed. \n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\rowcolors{2}{gray!25}{white}\n\\begin{center}\n\\begin{tabular}{ c | c c c c c c c c }\n \\hline\n \\rowcolor{gray!80}\n\\diagbox{VARIABLE}{COURSE}\n& DC\n& IC\n& VC\n& VAG\n& LA\n& ODE\n& BM\n& NM\n \\\\\n\n\\hline\nGrade &\n2.6849\t& 2.7829 &\t3.2198\t& 2.8616 &\t3.0233 & 3.1170 & 2.8308 &\t3.1893\\\\\nGPA &\n3.2213 & 3.3527\t& 3.4969 & 3.2386 & 3.3201\t& 3.4548 &\t3.2696 & 3.5330 \\\\\nPass\/Fail &\n0.5010 & 0.5339 & 0.7151 & 0.5901 &\t0.6398\t& 0.6846 &\t0.5441\t& 0.6924 \\\\\nNumber of Tries &\n1.7382 & 1.9140\t& 1.4996 & 1.5205 &\t1.5495\t& 1.7710 &\t1.0549\t& 1.4115 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Academic Performance Variables Average Values,\n\tCourse: \\textbf{All}. The average performance (as captured by the variables Grade, GPA, and Pass\/Fail) of students from 2010-1 to 2017-1, is displayed for each course.}\\label{Tb Academic Performance Variables Average}\n\\end{table}\n\\subsection{Segmentation Process}\\label{Sec Segmentation Process}\nThe profiling of students' population is to be made course-wise. For each group taking a course, the algorithm computes a partition of the interval $ [0,5] $ of ten numerical GPA intervals $\\big(I_{\\ell}: \\ell \\in [10] \\big)$, so that approximately ten percent of the population is contained in $ I_\\ell $ for all $ \\ell \\in [10] $. Equivalently, if a histogram of relative frequencies is drawn, as in \\textsc{Figure} \\ref{Fig DC GPA Histograms}, the area between the curve and any interval should be around 0.1. Hence, if $ f_{\\gpa} $ is the relative frequency of the $ \\gpa $ variable then $ \\int_{I_{\\ell} } f_{\\gpa} \\, dx \\sim 0.1 $ for all $ \\ell \\in [10] $. The process described above is summarized in the pseudocode \\ref{Alg Segmentation of Students} below.\n\n\n\\begin{figure}[h!]\n \\centering\n \\begin{subfigure}[Example DC. GPA Histogram, Semesters from 2010-1 to 2017-1. ]\n \n{\\includegraphics[scale = 0.38]{DC_Global_GPA.eps} } \n \\end{subfigure}\n \n ~\n \n \\begin{subfigure}[Example DC. GPA Histogram, Semesters 2010-1 and 2015-1.]\n \n{\\includegraphics[scale = 0.38]{DC_Sample_GPA.eps} } \n \\end{subfigure}\n \n\\caption{Differential Calculus GPA Normalized Histogram. Figure (a) Shows the normalized frequencies histograms for all the semesters available in the database \\textit{Assembled\\_Data.csv}. Figure (b) displays the normalized histogram of only two semesters for optical purposes. Observe that in both cases the area beneath the normalized histogram is exactly equal to one. \n\\label{Fig DC GPA Histograms} }\n\\end{figure}\n\n\\begin{algorithm}[H]\n\\KwData{\nDatabase: Assembled\\_Data.csv. \\\\\nYear ($ y $) and Semester ($ s $). \\\\\n\\textit{Analyzed Course}: DC, IC, ..., NM. \n}\n\\KwResult{Extremes of the GPA Segmentation Intervals $ \\big(I_{\\ell}: \\ell \\in [L]\\big) $; extremes = [0, n\\_1, n\\_2,\\ldots, 5 ]}\n\\textbf{Initialization}\\;\nList\\_GPA $ \\leftarrow $ \\textbf{sorted} (\\textbf{hash} $ \\gpa $ variable from \\\\\nAssembled\\_Data.csv[(Course = \\textit{Analyzed Course}) \\& (Year = $ y $) \\& (Semester = $ s $ )] ) \\;\nextremes\\_0 $ \\leftarrow $ 0\\; \n\\For{ $ i \\in [10] $ }{\nextremes\\_i $ \\leftarrow \\big\\lfloor i\\times \\frac{\\text{length of List\\_GPA} }{10} \\big\\rfloor$\\;\n}\n\\If{list {extremes} contains repetitions}{\nextremes $ \\leftarrow $ remove repetitions from extremes}\n\\caption{Segmentation of Students}\\label{Alg Segmentation of Students}\n\\end{algorithm}\n\\begin{remark}\\label{Rem Segmentation Algorithm}\nObserve that Algorithm \\ref{Alg Segmentation of Students} is aimed to produce ten segmentation intervals, however the last instruction considers removing some points out of the eleven extremes previously defined, in case of repetition. Such situation could arise when a particular GPA value is too frequent as it can be seen in \\textsc{Figure} \\ref{Fig DC GPA Histograms} (b), for the case of Semester 2011-1, which has a peak at GPA = 3. Similar peaks can be observed in other semesters as \\textsc{Figure} \\ref{Fig DC GPA Histograms} (a) shows.\n\\end{remark}\n\\subsection{Computation of the Lecturer's Performance}\\label{Sec Computation Lecturer Performance}\nThe treatment of the lecturer as a success factor is completely tailored to the case of study and it can not be considered as a general method, the expected (average) performance will be computed for the \\textit{Grade} and the \\textit{Pass\/Fail} variables. For the computation of instructors' performance, first a segmentation process $ \\big(I_{\\ell}: \\ell \\in [L]\\big) $ (as described in Subsection \\ref{Sec Segmentation Process}) has to be done. Next, the computation is subject to the following two principles \n\\begin{enumerate}[(i)]\n\\item Adjunct and Tenured (Track or not) lecturers are separated in different groups. \n\n\\item If the experience of a particular instructor (the full personal teaching log inside the database \\textit{Assembled\\_Data.csv}) within a segment $ I_{\\ell} $ of analysis, accumulates less than 30 individuals, his\/her performance within such $ I_{\\ell} $ is replaced by the average performance of the group he\/she belongs to (Adjunct or Tenured) within such group, i.e., the conditional expectation of the Academic Performance Variable ($ \\apv $) subject to the segment of analysis: $ \\Exp\\big( \\apv \\big\\vert \\text{Instructor} = x\\big) $, see \\cite{MendenhallBeaver, BillingsleyProb}.\n\\end{enumerate}\n\\begin{remark}\\label{Rem Adjunct lecturers treatment}\nThe separation of Adjunct and Tenured instructors is done because the working conditions, expectations, as well as the results, are significantly different from one group to the other inside the Institution of analysis. In particular, the adjunct instructors are not stable nor full-time personnel. Consequently, these two groups are hardly comparable. On the other hand, there is an internal policy of rotating teaching faculty through the lower division courses, according to the needs of the School of Mathematics. Hence, due to the hiring and teaching-rotation policies, an Adjunct instructor rarely accumulates 30 or more students of experience within a profile segment $ I_{\\ell} $. \n\\end{remark}\n\\begin{analysis}\\label{measurament_of_instructor}\nMeasuring instructor performance through the students \\textit{Grade} and \\textit{Pass\/Fail} variables, treating instructors as a transformation function in which output (student results) is measured with respect to the input (students background), was the norm in the past (see, e.g., \\cite{higgins1989performance}). Nonetheless this approach has several problems, as pointed out in \\cite{higgins1989performance} and \\cite{rockoff2010subjective}. Some of these problems are: the difficulty in accurately measuring students' background, the existing bias charged on instructors tasked with students who are more difficult to teach, and the non-comparability of students' grades across different instructors. Nonetheless, time and again, new developments on how to measure instructors' performance appear. In \\cite{berk2005survey}, R. A. Berk presents 12 strategies to measure teaching effectiveness, some of these measures are: peer ratings, self-evaluation, alumni ratings, teaching awards and others. Finally, given the algorithmic nature of our work we only need one variable of instructor performance in order to present the optimization method, but the algorithm itself applies to any quantitative measure, as the ones just mentioned above. \n\\end{analysis}\nThe performance computation is described in the following pseucode\n\n\\begin{algorithm}[H]\n\\KwData{\nDatabase: Assembled\\_Data.csv. \\\\\n\\textit{Analyzed Course}: DC, IC, ..., NM.\\\\\nAcademic Performance Variable ($ \\apv $): Grade, Pass\/Fail \\\\\nGroup Segmentation $ \\big( I_{\\ell} : \\ell \\in [L]\\big) $. }\n\\KwResult{Hash tables of performance for the analyzed course, conditioned to each segmentation\\\\ interval $ \\big( I_{\\ell}: \\ell \\in [L]\\big) $ and for the chosen Academic Performance Variable ($ \\apv $): \nAPV\\_Performance\\_Tenured$ [\\ell] $,\nAPV\\_Performance\\_Adjoint$ [\\ell] $,\nAPV\\_Performance\\_Instructor$ [\\ell] ,\\, \\ell \\in [L]$}\n\\textbf{Initialization}\\;\nInstructors\\_List $ \\leftarrow $ \\textbf{hash} lecturer list from Assembled\\_Data.csv[Course = \\textit{Analyzed Course}] \\; \nTenured\\_List $ \\leftarrow $ \\textbf{choose} from Instructors\\_List the Tenured lecturers\\; \nAdjoint\\_List $ \\leftarrow $ Instructors\\_List $ - $ Tenured\\_List\\; \n\n\\For{ $ \\ell \\in [L] $ }{\nX = \\textbf{hash} $ \\apv $ field from table:\\newline\nAssembled\\_Data.csv[(Course = \\textit{Analyzed Course}) \\& (GPA $ \\in I_{\\ell} $) \\& (\\textit{Instructor} $ \\in $ Tenured\\_List) ]\\;\nAPV\\_Performance\\_Tenured$ [\\ell]\\leftarrow \\Exp(\\X) $ \\;\nX = \\textbf{hash} $ \\apv $ field from table: \\newline\nAssembled\\_Data.csv[(Course = \\textit{Analyzed Course}) \\& (GPA $ \\in I_{\\ell} $) \\& (\\textit{Instructor} $ \\in $ Adjoint\\_List) ]\\;\nAPV\\_Performance\\_Adjoint$ [\\ell] \\leftarrow \\Exp(\\X) $.\n}\n\\For{ $ \\ell \\in [L] $ }{\n\t\\For{instructor $ \\in $ Instructors\\_List}{\n\tX = \\textbf{hash} $ \\apv $ field from table:\\newline\n\tAssembled\\_Data.csv[(Course = \\textit{Analyzed Course}) \\& (GPA $ \\in I_{\\ell} $) \\& (\\textit{Instructor} = instructor) ]\\;\n\t\t\\eIf{length of $ \\X >= 30 $ }{\n\t\tAPV\\_Performance\\_Instructor$ [\\ell]\\leftarrow \\Exp(\\X) $ .}\n\t\t{\n\t\t\t\\eIf{instructor $ \\in $ Tenured\\_List}{\n\t\t\tAPV\\_Performance\\_Instructor$ [\\ell]\\leftarrow $APV\\_Performance\\_Tenured$ [\\ell] $ }\n\t\t\t{APV\\_Performance\\_Instructor$ [\\ell]\\leftarrow $APV\\_Performance\\_Adjoint$ [\\ell] $ }\n\t\t}\n}\n}\n\\caption{Computation of Instructors' Performance}\\label{Alg Instructors' performance}\n\\end{algorithm}\n\\section{Core Optimization Algorithm and Historical Assessment}\\label{Sec Optimization and Historical Assessment}\nIn this section we describe the core optimization algorithm. Essentially, it is the integration of the previous algorithms with an integer programming module whose objective function is to maximize the \\textit{Expectation} of the academic performance variables (\\textit{Grade} and \\textit{Pass\/Fail}), according to the big data analysis described in \\textsc{Section} \\ref{Sec Computation Lecturer Performance}. Two methods are implemented for each course and semester recorded in the database.\n\\begin{enumerate}[I.]\n\\item \\textbf{Instructors Assignment (IA).} Assuming that the groups of students are already decided, assign the instructors pursuing the optimal expected performance partnership: Instructor-Conformed Group. This is known in integer programming as the Job Assignment Problem.\n\n\\item \\textbf{Students Assignment (SA).} Assuming that the sections (with a given capacity) and their corresponding lecturers are fixed, assign the students to the available sections in order to optimize the expected performance of the Student-Instructor partnership. This is the integer programming version of the Production Problem in linear optimization. \n\\end{enumerate} \nIn order to properly model the integer programs we first introduce some notation\n\\begin{definition}\\label{Def Characteristic Numbers}\nLet $ N, L, J \\in \\N $ be respectively the total number of students, the total number of segmentation profiles and the total number of sections. Let $ \\p = \\big(p_{\\ell}: \\ell \\in [L] \\big) \\in \\N^{L} $, $ \\g = \\big(g_{j}: j \\in [J] \\big) \\in \\N^{J} $ be respectively the population of students in each profiling segment and the capacities of each section, in particular the following sum condition holds.\n\\begin{equation}\\label{Eq Sum Condition}\n\\sum_{\\ell \\, = \\, 1}^{L} p_{\\ell} = \\sum_{j \\, = \\, 1}^{J} g_{j} = N .\n\\end{equation}\n\\end{definition}\n\\begin{remark}\\label{Rem Salck Variables Absence}\nObserve that the condition $ \\sum_{j \\, = \\, 1}^{J} g_{j} = N $ implies there are no slack variables for the capacity of the sections. This is due to the study case, in contrast with other Universities where substantial slack capacities can be afforded.\n\\end{remark}\n\\begin{definition}\\label{Def Performance and Group Assignment Matrices}\nLet $ N, L, J \\in \\N $ , $ \\p \\in \\N^{L}, \\g \\in \\N^{J} $ be as in Definition \\ref{Def Characteristic Numbers}. \n\\begin{enumerate}[(i)]\n\\item \nWe say a matrix $ G \\in \\R^{L\\times J} $ is a \\textbf{group assignment matrix} if all its entries are non-negative integers and \n\\begin{align}\\label{Eq Row Sum Column Sum Condition}\n& \\sum\\limits_{j \\,\\in\\,[J] } G(\\ell,j) = p_{\\ell}, \\; \\forall \\, \\ell \\in [L] , &\n& \\sum\\limits_{\\ell \\,\\in\\,[L] } G(\\ell,j) = g_{j}, \\; \\forall \\, j \\in [J] .\n\\end{align}\nFurthermore, define the \\textbf{group assignment space} by $ \\itG \\defining \\big\\{ G: G \\text{ is a group assignment matrix} \\big\\} $. \n\n\\item Let $ \\big(t_{j}: j\\in [J]\\big) $ be the intructors assigned to the course. For a fixed $ \\apv \\in \\big\\{\\text{Grade, Pass\/Fail} \\big\\} $ define \\textbf{expected performance matrix} $ T_{\\apv} \\in \\R^{J \\times L} $ as the matrix whose entries $ T_{\\apv}(j, \\ell) $ are the $ \\apv $ variable performance, corresponding to the instructor $ t_{j} $ within the segmentation interval $ I_{\\ell} $.\n\n\\item Given a group assignment matrix $ G $ and a faculty team $ \\big(t_{j}: j\\in {J}\\big) $, define the \\textbf{choice performance matrix} $ C_{\\apv} $ by\n\\begin{equation}\\label{Eq Costs Table APV}\nC_{\\apv} \\defining T_{\\apv} G .\n\\end{equation}\n\n\\end{enumerate}\n\\end{definition}\n\\begin{remark}\\label{Rem Group Assignment Matrix and Performance Matrix}\n\tObserve the following\n\t%\n\t\\begin{enumerate}[(i)]\n\t\t\\item $ C_{\\apv}(j,i) $ measures the average performance of the instructor $ t_{ j } $ over the partition $ \\{G(k,i)\\}_{k=1}^{L} $ of the section $ g_{ i } $. \n\t\t\n\t\t\\item Recall from combinatorics that a weak composition of $ n $ in $ m $ pars is a sequence of inon-negative ntegers $ (a_{ 1 }, \\ldots, a_{ m } ) $ satisfying $ \\sum_{ i \\, = \\, 1 }^{ m } a_{ i } = n $ (see \\cite{BonaWalk}). Notice that $ \\big(G(\\ell,j): j\\in [J]\\big) $ is a weak composition of $ p_{\\ell} $ for every $ \\ell\\in [L] $ and that $ \\big(G(\\ell,j): \\ell\\in [L]\\big) $ is a weak composition of $ g_{j} $ for every $ j\\in [J] $.\n\n\\item Recall that the expected performance matrix $ T_{\\apv} $ for $ \\apv \\in \\big\\{\\text{Grade, Pass\/Fail} \\big\\} $, is recovered from the file APV\\_Performance\\_Instructor hash table constructed in Algorithm \\ref{Alg Instructors' performance}, Section \\ref{Sec Computation Lecturer Performance}. \n\t\\end{enumerate}\n\\end{remark}\nNext we introduce the integer problems\n\\begin{problem}[Instructors Assignment Method]\\label{Pblm Instructors Assignment}\nLet $ N, L, J \\in \\N $ be as in \\textsc{Definition} \\ref{Def Characteristic Numbers}, let $ \\xi = \\big(\\xi(i, j): i \\in[I], j\\in [J] \\big) \\in \\big\\{0,1\\big\\}^{L\\times J} $ and let $ C_{\\apv} $ be as in \\textsc{Definition} \\ref{Def Performance and Group Assignment Matrices} for a fixed group assignment matrix $ G $ and faculty team $ \\big(t_{j}: j\\in {J}\\big) $. Then, the instructors assignment problem is given by\n\\begin{subequations}\\label{Eq Instructors Assignment}\n\\begin{equation}\\label{Eq Instructors Assignment Objective Function}\nv_{\\ia} \\defining \\max\\limits_{ \\xi \\, \\in \\, \\{0,1\\}^{L\\times J} }\n\\sum_{i\\, = \\, 1}^{J} \\sum_{j\\, = \\, 1}^{J} C_{\\apv}(i, j) \\, \\xi(i, j) \n,\n\\end{equation}\nsubject to:\n\\begin{align}\\label{Eq Instructors Assignment Constraints}\n& \\sum_{i\\, = \\, 1}^{J} \\xi(i, j) = 1, \\; \\forall\\, j \\in [J] , & \n& \\sum_{j\\, = \\, 1}^{J} \\xi(i, j) = 1 , \\; \\forall\\, i \\in [J] .\n\\end{align}\n\\end{subequations}\n\\end{problem}\n\\begin{problem}[Students Assignment Problem]\\label{Pblm Students Assignment Problem}\nWith the notation introduced in \\textsc{Definition} \\ref{Def Characteristic Numbers} and a chosen faculty team $ \\big(t_{j}: j\\in {J}\\big) $, let $ \\pi $ be a permutation in $ \\itS_{J} $ such that $ t_{\\pi(j)} $ is the instructor of section $ j $ for all $ j \\in [J] $ i.e., a chosen assignment of lecturers to the sections. Then, the students assignment problem is given by\n\\begin{equation}\\label{Eq Students Assignment Problem}\nv_{\\sa} = \\max\\limits_{G\\, \\in\\, \\itG }\n\\sum_{j\\, = \\, 1}^{J} \\big(T_{\\apv} G \\big)\\big(j, \\pi(j) \\big) \n= \\max\\limits_{G\\, \\in\\, \\itG }\n\\sum_{j\\, = \\, 1}^{J} C_{\\apv}\\big(j, \\pi(j) \\big) \n.\n\\end{equation}\n\\end{problem}\n\\begin{remark}\\label{Rem Integer Programming Problems}\n\\begin{enumerate}[(i)]\n\\item Observe that the constraints of the problem \\ref{Pblm Students Assignment Problem} are only those of \\textsc{Equation} \\eqref{Eq Row Sum Column Sum Condition}; these are fully contained in the condition $ G \\in \\itG $.\n\n\\item Notice that although the search space of \\textsc{Problem} \\ref{Pblm Students Assignment Problem} is significantly bigger than the search space of \\textsc{Problem} \\ref{Pblm Instructors Assignment}, the optimum of the former need not be bigger or equal than the optimum of the latter. However, in practice, the numerical results below show that this is the case, not because of search spaces inclusion, but due to the overwhelming difference of sesarch space sizes.\n\\end{enumerate}\n\\end{remark}\nIn order to asses the enhancement introduced by the method, it is necessary to compute rates of optimal performance over the historical one i.e., if $ G_{h} \\in \\itG $, $ \\pi_{h} \\in \\itS_{J} $ are respectively the historical group composition and instructors assignation for a given semester $ h $ then, the relative enhancement $ \\rho_{\\mt} $, due to a method $ \\mt $ is given by \n\\begin{align}\\label{Eq Performance Rates}\n& \\rho_{\\mt} \\defining \n100\\frac{v_{\\mt} - \\sum\\limits_{j\\, = \\, 1}^{J} \\big(T_{\\apv} G_{h}\\big) \\big(j, \\pi_{h}(j) \\big) }\n{\\sum\\limits_{j\\, = \\, 1}^{J} \\big(T_{\\apv} G_{h}\\big) \\big(j, \\pi_{h}(j) \\big) }\n\\, , &\n& \\mt \\in \\{ \\ia, \\sa\\} .\n\\end{align} \nFinally, we describe in \\textsc{Algorithm} \\ref{Alg Optimization Algoritm} below the optimization algorithm\n\\begin{analysis}[$\\ia $ and $ \\sa $ solutions as Pareto equilibria]\\label{An Pareto Balance}\n\\begin{enumerate}[(i)]\n\\item The $ \\ia $ and $ \\sa $ problems are two scheduling formulations driven by social welfare. The University as a central regulator agent aims to solve such scheduloing problems in order to improve the social welfare of its community (i.e, students and professors). Given a group assignment matrix $G$, the $ \\ia $ method seeks to find the matching pairs $(\\mathit{instructor}, \\mathit{section})$ in order to maximize the total average performance of the instructors, subject to the constraint that each instructor must teach only one section. On the other hand, the $ \\sa $ method seeks to find a group assignment matrix $G$, given a complete matching pairs $(\\mathit{instructor},\\mathit{section})$. More specifically, a distribution of the students population that maximizes the total average performance.\\footnote{ Notice that the $ \\ia $ method is easire to implement than the $ \\sa $ method, the former only requires to allocate the instructors, while the later requires a redistribution of the whole students population. } \n\n\\item \nThe solutions $ v_{\\ia} $ and $ v_{\\sa} $ are configurations corresponding to Pareto equilibria, i.e. situations where no individual can improve his\/her welfare (success chances in this particular case) without decreasing the well-being of another individual of the system. In this same spirit, the parameters $ \\rho_{\\mt} $ are measures of deviation from the Pareto equilibrium. \n\\end{enumerate}\n\\end{analysis}\n\\begin{algorithm}[t]\n\\KwData{\nDatabase: Assembled\\_Data.csv. \\\\\nYear ($ y $) and Semester ($ s $). \\\\\nAnalyzed Course: DC, IC, ..., NM.\\\\\nAcademic Performance Variable ($ \\apv $): Grade, Pass\/Fail. \\\\\nGroup Segmentation $ \\big( I_{\\ell} : \\ell \\in [L]\\big) $.\\\\\nAPV\\_Performance\\_Instructor$ [\\ell] ,\\, \\ell \\in [L]$. \\\\\nOptimization Method: $ \\mt \\in \\{\\ia, \\sa\\} $. }\n\\KwResult{Relative enhancement value $ \\rho_{\\mt} $ for method $ \\mt $, for chosen course, year and semester. \n}\n\\textbf{Initialization}\\;\nCourse\\_Table $ \\leftarrow $ \\textbf{hash} \\\\\nAssembled\\_Data.csv[(Course = \\textit{Analyzed Course}) \\& (Year = $ y $) \\& (Semester = $ s $) ]\\;\nInstructors\\_List $ \\leftarrow $ \\textbf{hash} lecturer list from Course\\_Table \\;\nInstructors\\_Performance $ \\leftarrow $ \\textbf{hash} APV\\_Performance\\_Instructor[Instructor $ \\in \\text{Instructors\\_List}$ ]\\;\nSection\\_List $ \\leftarrow $ \\textbf{hash} section list from Course\\_Table \n\n\\For{ $ \\ell \\in [L] $ }{\n\t\\For{ $ j \\in J $ }{\n\t$ T_{\\apv} (\\ell, j) \\leftarrow $ \\textbf{hash} \\\\\n\tAPV\\_Instructors\\_Performance[(Instructor = Instructors\\_List$ (j) $ )\n\t\\& (Segmentation = $ I_{\\ell} $)]\\;\n\t$ G_{h}(\\ell, j) \\leftarrow $ lenght(\\textbf{hash} Course\\_Table[(Section = $ j $) \\& (Segmentation = $ I_{\\ell} $)])\n\t}\n}\n\\eIf{$ \\mt = \\ia $}{\n\t$ C_{\\apv} \\leftarrow T_{\\apv} G_{h} $,\n\t$v_{\\ia} \\leftarrow $ \\textbf{solve} Problem \\ref{Pblm Instructors Assignment}, \\textbf{input}: $ C_{\\apv} $.\n\t}\n\t{\n\t$ \\p \\leftarrow \\Big(\\sum_{j\\, = \\, 1}^{J}G(\\ell, j) : \\ell \\in [L]\\Big) $,\n\t$ \\g \\leftarrow \\Big(\\sum_{\\ell\\, = \\, 1}^{L}G(\\ell, i) : i \\in [J]\\Big) $,\n\t$ \\pi \\leftarrow $ Section\\_List\\;\n\t$v_{\\sa} \\leftarrow $ \\textbf{solve} Problem \\ref{Pblm Students Assignment Problem}, \\textbf{input}: $ ( T_{\\apv}, \\p, \\g, \\pi) $.\n} \n$ \\pi_{h} \\leftarrow $ Section\\_List\\;\n$ \\rho_{\\mt} \\leftarrow $ \\textbf{compute} Equation \\eqref{Eq Performance Rates}, \\textbf{input}: \n$ \\big( T_{\\apv} , G_{h}, \\pi_{h}, v_{\\mt}\\big) $.\n\\caption{Optimization Algorithm}\\label{Alg Optimization Algoritm}\n\\end{algorithm}\n\\subsection{Historical Assessment}\\label{Sec Historical Assessment}\nIn the current section, we are to assess the enhancement of the proposed method with respect to the average of the historical results. To that end, we merely integrate \\textsc{Algorithms} \\ref{Alg Segmentation of Students}, \\ref{Alg Instructors' performance} and \\ref{Alg Optimization Algoritm} in a master algorithm going through a time loop to evaluate the performance of each semester and then store the results in a table, this is done in \\textsc{Algorithm} \\ref{Alg Analytica Omega}. It is important to observe that excepting for the database, all the remaining input data must be defined by the user.\n\nThe numerical results for the Differential Calculus course are summarized in \\textsc{Table} \\ref{Tb Historical Enhnacements} and illustrated in \\textsc{Figure} \\ref{Fig Historical Enhancement}. The results clearly show that the Students Assignment method (SA) yields better results than the Instructor Assignment method (IA), which holds for both Academic Performance Variables: \\textit{Pass\/Fail} and \\textit{Average}. Such difference happens not only for the mean value, but on every observed instance (semester), this is due to the difference of size between search spaces for the problems \\ref{Pblm Instructors Assignment} and \\ref{Pblm Students Assignment Problem} as discussed in Remark \\ref{Rem Integer Programming Problems}. On the other hand, it can be observed that the \\textit{Pass\/Fail} variable is considerably more sensitive to the optimization process than the \\textit{Average} variable. Again, the phenomenon takes place not only for the enhancement's mean value, the former around three times the latter, but the domination occurs for every semester analyzed by the algorithm. The latter holds because, for an improvement on the \\textit{Average} variable to occur, a general improvement in the students' grades should take place, while the improvement of the pass rate is not as demanding.\n\nThe results of the optimization methods yield similar behavior for all the remaining lower division courses. Consequently, in the following we will only be concerned with the analysis of the \\textit{Pass\/Fail} variable, which gives the title to the present paper. The two optimization methods will be kept for further analysis, not because of efficiencies (clearly SA yields better results), but because of the administrative limitations a Higher Education Institution could face when implementing the solution. Clearly, from the administrative point of view, it is way easier for an Institution implementing IA instead of SA, \n\n\nIt is also important to mention, that although enhancements of 1.4 or 7 percent may not appear significant at first sight, the benefit is substantial considering the typical enrollments displayed in \\textsc{Table} \\ref{Tb Historical Enrollment Table}, as well as the average \\textit{Number of Tries} a student needs to pass de course displayed in \\textsc{Table} \\ref{Tb Academic Performance Variables Average}. In addition, the fact that Latin American public universities heavily subside its students despite having serious budgeting limitations (as in our study case), gives more relevance to the method's results. \n\n\\begin{algorithm}[H]\n\\KwData{\nDatabase: Assembled\\_Data.csv. \\\\\nAnalyzed Course: DC, IC, ..., NM.\\\\\nAcademic Performance Variable ($ \\apv$ ): Grade, Pass\/Fail. \\\\\nOptimization Method: $ \\mt \\in \\{\\ia, \\sa\\} $. }\n\\KwResult{Table of Relative Enhancement Values $ \\rho_{\\mt} $ for chosen method, course and academic performance variable.\n}\n\\textbf{Initialization}\\;\n\\For{ Year $ \\in [2010, 2017]$ }{\n\t\\For{ Semester $ \\in [2] $ }{\n\t\\textbf{call} Algorithm \\ref{Alg Segmentation of Students}, \\textbf{input}: (\\textit{Assembled\\_Data.csv}, \\textit{Year}, \\textit{Semester}, \\textit{Analyzed Course})\\;\n\t\\textbf{call} Algorithm \\ref{Alg Instructors' performance}, \\textbf{input}: (\\textit{Assembled\\_Data.csv}, \\textit{Analyzed Course}, $ \\apv $, Group Segmentation $ \\big( I_{\\ell}: \\ell \\in [L] \\big) $)\\;\n\t\\textbf{call} Algorithm \\ref{Alg Optimization Algoritm}, \\textbf{input}: (\\textit{Assembled\\_Data.csv},\n\t\\textit{Year}, \\textit{Semester}, \\textit{Analyzed Course}, $ \\apv $, Group Segmentation $ \\big( I_{\\ell}: \\ell \\in [L] \\big) $, $ \\mt $ )\\;\n\t\n\tAPV\\_mt\\_Assessment[Year, Semester]$ \\leftarrow \\rho_{\\mt} $. \n\t\n\t}\n}\n\\caption{Historical Assessment Algorithm}\\label{Alg Analytica Omega}\n\\end{algorithm}\n\\begin{figure}[h!]\n \\centering\n \\begin{subfigure}[Example DC. Enhancement Results $ \\apv = $ \\textit{Pass Rate}. ]\n \n{\\includegraphics[scale = 0.38]{CD_Historical_Enhancement_Pass.eps} } \n \\end{subfigure}\n \n ~\n \n \\begin{subfigure}[Example DC. Enhancement Results $ \\apv = $ \\textit{Average}.]\n \n{\\includegraphics[scale = 0.38]{CD_Historical_Enhancement_Mean.eps} } \n \\end{subfigure}\n \n\\caption{Example: \\textbf{Differential Calculus} course. Both figures show the enhancement results $ \\rho_{\\mt} $ for $ \\mt \\in \\{ \\ia, \\sa\\} $ optimization methods. The Instructor Assignment method (IA) is depicted in blue, while the Students Assignment method (SA) is represented in red. \nFigure (a) shows results for the \\textit{Pass\/Fail} variable. \nFigure (b) shows the results for \\textit{Average} variable. \n\\label{Fig Historical Enhancement} }\n\\end{figure}\n\\begin{analysis}[Figure \\ref{Fig Historical Enhancement}]\n As it was already mentioned in the beginning of subsection \\ref{Sec Historical Assessment}, the Students Assignment method ($ \\sa $) yields better results than the Instructor Assignment method ($ \\ia $). This is aligned with the following idea from the economic empirical perceptions: when individuals have more instruments to participate, their well-being in terms of social welfare increases.\n\\end{analysis} \n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{p{1.5cm} | p{1.5cm} p{1.5cm} | p{1.5cm} p{1.5cm} }\n \\hline\n \\rowcolor{gray!80}\nAcademic\n&\n\\multicolumn{2}{c |}{\\textit{APV} = Pass\/Fail} &\n\\multicolumn{2}{c}{\\textit{APV} = Average} \\\\\n\\rowcolor{gray!80}\nSemester\n& $ \\mt = \\ia $\n& $ \\mt = \\sa $\n& $ \\mt = \\ia $\n& $ \\mt = \\sa $\n \\\\\n\n\\hline\n2010-1\t& 2.0482 &\t5.8891 & 0.7868 &\t1.8393 \\\\\n2013-1\t& 0.9090 &\t5.8924\t& 0.4230 & 2.0870 \\\\\n2016-1\t& 2.0939 &\t8.2952\t& 0.4391 &\t2.1050 \\\\\n\\hline\n\\rowcolor{gray!80}\nMean & 1.3811 &\t7.0432\t& 0.5501 &\t2.1584 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Relative Enhancements Sample, $ \\rho_{\\mt} $,\n\tCourse: \\textbf{Differential Calculus}. We display the mean relative enhancement for the Differential Calculus course during the period from 2010-1 to 2017-1; together with a sample of three semesters in the same time window.}\\label{Tb Historical Enhnacements}\n\\end{table}\n\\section{Randomization and Predictive Assessment}\\label{Sec Randomization and Asymptotic Assesment}\nSo far, the method has been assessed with respect to the historical log i.e., comparing its optimization outputs with those of 15 recorded semesters. The aim of the present section is to perform Monte Carlo simulations on the efficiency of the method and apply the Law of Large Numbers to estimate the expected enhancement of the algorithm. We present below for the sake of completeness, its proof and details can be found in \\cite{BillingsleyProb}.\n\\begin{theorem}[Law of Large Numbers]\\label{Th the Law of Large Numbers}\nLet $ \\big(\\Z^{(n)}:n\\in \\N\\big) $ be a sequence of independent, identically distributed random variables with expectation $ \\mu = \\Exp(\\Z^{(1)}) $, then\n\\begin{equation}\\label{Eq the Law of Large Numbers}\n\\prob\\bigg[ \\Big\\vert \\frac{\\Z^{(1)} + \\Z^{(2)} + \\ldots \\Z^{(n)}}{n} - \\mu \\Big\\vert > 0 \\bigg]\n\\xrightarrow[n\\, \\rightarrow \\,\\infty]{} 0 ,\n\\end{equation}\ni.e. , the sequence $ \\big(\\Z^{(n)}:n\\in \\N\\big) $ converges to $ \\mu $ in the Ces\\`aro sense.\n\\end{theorem}\nIn order to achieve Monte Carlo simulations, we first randomize several factors\/variables which define the setting of a semester for each course, in Section \\ref{Sec Randomization of variables}. Next we discuss normalization criteria in Section \\ref{Sec Normalization of the method}, to make the enhancement simulations comparable. Finally, we present in Section \\ref{Sec Numerical Simulations}, the Monte Carlo simulations results for both, the random variable simulating the benefits of the method ($ \\Z^{(n)} $ in Theorem \\ref{Th the Law of Large Numbers}) as well as the evolution of its Ces\\'aro means ($ \\frac{\\Z^{(1)} + \\Z^{(2)} + \\ldots + \\Z^{(n)}}{n} $ in Theorem \\ref{Th the Law of Large Numbers}) to determine the asymptotic performance of the proposed algorithm. \n\nThroughout this section we adopt a notational convention, the label \\textbf{RandInputAlgorithm} will refer to the random versions of the respective algorithm developed in the previous sections. For instance\nRandInputAlgorithm \\ref{Alg Optimization Algoritm}, \\textbf{input}: \n(\\underline{Group Assignment Matrix \\textit{G} , List of Lecturers \\textit{L\\_list} }, \\textit{Analyzed Course}, $ \\apv $, Group Segmentation $ \\big( I_{\\ell}: \\ell \\in [L] \\big) $, $ \\mt $ ), refers to Algorithm \\ref{Alg Optimization Algoritm} above, but with a different set of input data; for clarity the randomly generated input data are underlined. This notation is introduced for exposition brevity: avoiding to write an algorithm whose logic is basically identical to its deterministic version. \n\\subsection{Randomization of Variables}\\label{Sec Randomization of variables}\nFour factors will be randomized in the same fashion: Number of Tenured Lecturers, Number of Enrolled Students, List of Students' GPA and Number of Groups. First, we randomize the integer-valued statistical variables by merely computing 95 percent confidence intervals from the empirical data and then assuming that, the impact of the factor can be modeled by a random variable uniformly distributed on such confidence interval, see \\cite{MendenhallBeaver}.\n\\begin{definition}\\label{Def Confidence Interval}\nLet $ x $ be a scalar statistical variable with mean $ \\bar{x} $, standard deviation $ \\sigma $ and $ n $ its sample size.\n\\begin{enumerate}[(i)]\n\\item If $ x $ is real-valued, its \\textbf{95 percent confidence interval} is given by \n\\begin{equation}\\label{Eq Confidence Interval Real-Valued}\n I_{x} \\defining \\Big[\\bar{x} - 1.96 \\, \\frac{\\sigma}{\\sqrt{n}}, \\bar{x} + 1.96 \\, \\frac{\\sigma}{\\sqrt{n}} \\Big] .\n\\end{equation}\n\\item \nIf $ x $ is integer-valued, its \\textbf{95 percent confidence interval} is given by \n\\begin{equation}\\label{Eq Confidence Interval Integer-Valued}\n I_{x} \\defining \\Big[\\big\\lfloor\\bar{x} - 1.96 \\, \\frac{\\sigma}{\\sqrt{n}} \\big\\rfloor,\n \\big\\lceil \\bar{x} + 1.96 \\, \\frac{\\sigma}{\\sqrt{n}} \\big\\rceil\\Big] \\cap \\setZ,\n\\end{equation}\nwhere $ x $, $ \\lfloor \\cdot \\rfloor ,\\lceil \\cdot \\rceil: \\R \\rightarrow \\R $ denote the floor and ceiling functions respectively. \n\\end{enumerate}\n\\end{definition}\nThe randomization of the statistical variables listed above, heavily relies on the empirical distributions mined from the database.\n\\begin{hypothesis}\\label{Hyp Variables Randomization}\n\\begin{enumerate}[(i)]\n\\item \nLet $ x $ be a scalar statistical variable, then its associated random variable $ \\X $ is uniformly distributed on its confidence interval $ I_{x} $, i.e. $ \\X \\sim \\unif (I_{x}) $, where the confidence interval is defined by \\eqref{Eq Confidence Interval Integer-Valued} or \\eqref{Eq Confidence Interval Real-Valued} depending on whether the variable $ x $ is integer or real valued.\n\n\\item Let $ \\mathbf{x} = \\big(x_{i}, \\ldots, x_{d} \\big)\\in \\R^{d} $ be a vector statistical variable, then its associated random variable is given by $ \\textbf{X} = \\big(\\X_{1} , \\ldots, \\X_{d} \\big) $, where $ \\X_{i} $ is the random variable associated to $ x_{i} $ for all $ i \\in [d] $ as defined above.\n\\end{enumerate}\n\\end{hypothesis} \nFrom here, it is not hard to compute the confidence intervals (or ranges) of the random variables as it is shown in \\textsc{Tables} \\ref{Tb Tenured Random Variable} and \\ref{Tb Enrollement Random Variable}. In contrast, the \\textit{Sections} and the \\textit{GPA} variables will need further considerations in its treatment. \n\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{ c | c c c c c c c c }\n \\hline\n \\rowcolor{gray!80}\n\\diagbox\n{PARAMETERS}{COURSE}\n& DC\n& IC\n& VC\n& VAG\n& LA\n& ODE\n& BM\n& NM\n \\\\\n\n\\hline\nUpper Extreme &\n8 &\t5 &\t3 & 7 &\t5 &\t4 &\t4 &\t2 \\\\\nLower Extreme &\n6 &\t3 &\t2 &\t4 &\t3 &\t3 &\t2 &\t1 \\\\\nAverage &\n7.2667 & 4.0667\t& 2.6000 &\t5.1333 & 3.7333\t& 3.5333 & 3.3333 &\t1.6667 \\\\\nStandard Deviation &\n0.7037 &\t1.1629\t& 0.7368 & 1.9952 &\t1.2228 & 0.9155\t& 1.0465 & 0.6172 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Random Variable: Number of Tenured Instructors $ \\upnt $,\n\tCourse: \\textbf{All}. The upper \\& lower extremes, average and standard deviation for the random variable ``Number of Tenure Instructors NT'' across all courses are displayed.}\\label{Tb Tenured Random Variable}\n\\end{table}\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\scriptsize{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{ c | c c c c c c c c }\n \\hline\n \\rowcolor{gray!80}\n\\diagbox\n{PARAMETERS}{COURSE}\n& DC\n& IC\n& VC\n& VAG\n& LA\n& ODE\n& BM\n& NM\n \\\\\n\n\\hline\nUpper Extreme &\n1554 &\t1203 &\t586\t& 1243\t& 1045\t& 882 & 974\t& 301 \\\\\nLower Extreme &\n1337 &\t1043 & 513 & 1050 &\t932\t& 721 &\t837 & 234 \\\\\nAverage &\n1445.9333 &\t1122.8667 &\t549.8000 & 1146.5333 &\t988.6667 &\t801.8000 &\t905.4000 &\t267.5333 \\\\\nStandard Deviation &\n213.4446 &\t156.8642 &\t70.9969\t& 188.8304 & 110.7229 &\t158.4433 &\t134.2971 &\t65.4847 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Random Variable: Number of Enrolled Students $ \\upne $,\n\tCourse: \\textbf{All}. \nThe upper \\& lower extremes, average and standard deviation for the random variable ``Number of Tenure Students NE'' across all courses are displayed.\n}\\label{Tb Enrollement Random Variable}\n\\end{table}\n\nThe \\textit{Sections} variable is a list of several sections with different capacities. A statistical scan of the data shows that this list is a most unpredictable variable, because section capacities range from 15 to 150 with very low relative frequencies in each of its values. Consequently, it was decided to group the section capacities in integer intervals\n\\begin{definition}\\label{Def Gathering of Section Capacities}\nGiven the list of integer intervals \n\\begin{equation}\\label{Eq Intervals Agrupation} \n\\mathcal{I} \\defining \\big\\{ [15, 30], [31, 45], [46, 60], [61, 75], [76, 90], \n[91, 105], [ 106, 120], [121, 135], [136, 150]\\big\\}, \n\\end{equation}\nfor each semester and for each course, the \\textbf{sections frequency variable} is given by \n\\begin{equation}\\label{Eq Sections Frequency Variable}\n\\boldsf \\defining \n\\Big(\\frac{\\itns_{I}}{\\sum\\limits_{K\\, \\in \\,\\mathcal{I} } \\itns_{K} } : I \\in \\mathcal{I} \\Big) ,\n\\end{equation}\nwhere $ \\itns_{I} $ is the number of sections whose capacity belongs to the interval $ I \\in \\mathcal{I} $.\n\\end{definition}\n\n\\begin{hypothesis}\\label{Hyp Group Sizes Variable}\nThe Sections variable $ \\textbf{S} $ is completely defined by the number of groups variable $ \\upns $ in the following way\n\\begin{equation}\\label{Eq Group Sizes Variable}\n\\textbf{S} \\defining \\Big\\lceil \\upns \\, \\overline{\\boldsf} \\Big\\rceil .\n\\end{equation}\nHere $ \\overline{\\boldsf} $ is the average vector of $ \\boldsf $ introduced in \\textsc{Equation} \\eqref{Eq Sections Frequency Variable} and it is understood that the ceiling function $ \\lceil \\cdot \\rceil $ applies to each component of the vector. \n\\end{hypothesis}\nFinally, the \\textit{GPA} variable is treated as follows\n\\begin{hypothesis}\\label{Hyp GPA random variable}\nFor each semester define $\\mathbf{x} \\defining \\big(x_{i}: i\\in [50] \\big)$, where $ x_{i} $ is the relative frequency of registering students whose GPA is equal to $ \\frac{i}{10} $; in particular $ \\sum_{i \\, \\in\\, [50] } x_{i} = 1 $. Let $ \\textbf{X}_{\\gpa} $ be the associated random variable to the list of relative frequencies $ \\mathbf{x} $, as introduced in \\textsc{Hypothesis} \\ref{Hyp Variables Randomization}. Then, the random variable $ \\textbf{GPA} $ is given by\n\\begin{equation}\\label{Eq GPA random variable}\n\\textbf{GPA}\\defining \\big\\lceil \\upne \\, \\textbf{X}_{\\gpa} \\big\\rceil ,\n\\end{equation}\nwhere $ \\upne $ is the number of enrolled students random variable and it is understood that the ceiling function $ \\lceil \\cdot \\rceil $ applies to each component of the vector. \n\\end{hypothesis} \n\\begin{remark}\\label{Rem Sections and GPA rand var}\n\nNotice that both random variables $ \\textbf{S} $ and $ \\textbf{GPA} $ are the product of a scalar and a vector. However, for $ \\textbf{S} $ the scalar is a random variable $ \\upns $ and the vector $ \\overline{\\boldsf} $ is deterministic, while in the case of $ \\textbf{GPA} $ the scalar $ \\upne $ and the vector $ \\textbf{X}_{\\text{GPA}} $ are random variables. There lies the difference in the randomization of the vector variables.\n\n\\end{remark}\nSo far, $ \\textbf{S} = \\big( s_{K}^{(1)}: K\\in \\mathcal{I}\\big) $ is producing a list of sections whose capacity lies within the ranges declared in $ \\mathcal{I} $ (\\textsc{Equation} \\eqref{Eq Intervals Agrupation}), this introduces a set of slacks which will be used later on, to match the number of enrolled students $ \\upne $ with the total sections capacity. The match will be done in several steps, once the equality $ \\sum_{K\\, \\in \\, \\mathcal{I} }s_{K} = \\upne $ is attained, a group assignment matrix $ G $ (as in Definition \\ref{Def Performance and Group Assignment Matrices} (i)) will be generated randomly.\n\\begin{enumerate}[{step} 1.]\n\\item\\label{Step 1 Initial Approximation Solution} Solve the following Data Fitting Problem (see \\cite{Bertsimas} for its solution)\n\\begin{problem}\\label{Pblm Optimization Students Sections Capacity}\nGiven two realizations of $ \\textbf{S} = \\big( s_{K}^{(1)}: K\\in \\mathcal{I}\\big) $ and $ \\upne $, consider the integer problem\n\\begin{equation}\\label{Eq Optimization Students Sections Capacity}\ndf^{(1)} = \\min \\bigg\\{\\Big\\vert \\sum_{K\\, \\in \\, \\mathcal{I}}\\sum_{i\\, = \\, 1}^{s_{K}^{(1)}} x_{K, i}\n- \\upne \\Big\\vert : \nx_{K, i}\\, \\in K , \\text{for all } i \\in \\big[ s_{K}^{(1)} \\big] \\text{ and } K \\in \\mathcal{I} \\bigg\\}.\n\\end{equation}\nDenote by $ \\big(x_{K, i}^{(1)}: i \\in [s_{K}^{(1)}], K \\in \\mathcal{I}\\big) $ the optimal solution to problem \\eqref{Eq Optimization Students Sections Capacity}. If $ df^{(1)} \\equiv 0 $ then jump to \\textsc{step} \\ref{Step 4 Initial Generate randomly a group assignment matrix} below.\n\\end{problem}\n\n\n\\item\\label{Step 2 Increase\/Decrease Sections} Decide whether or not is more convenient increase or decrease the number of sections $ s_{K}^{(1)} \\mapsto s_{K}^{(2)} $ according the case, using the \\textbf{Greedy Algorithm} \\ref{Alg Greedy Algorithm}, to modify the sections' capacity, starting from the large sections to the small ones and get\n\\begin{equation}\\label{Eq Opening or closineg sections}\ndf^{(2)} \\defining \\Big\\vert \\sum_{K\\, \\in \\, \\mathcal{I}}\\sum_{i\\, = \\, 1}^{s_{K}^{(2)}} \nx_{K, i}^{(1)}\n- \\upne \\Big\\vert < \ndf^{(1)} \n.\n\\end{equation}\n\n\n\n\\begin{algorithm}[H]\n\\KwData{\n$ \\big(s_{K}^{(1)}: K \\in \\mathcal{I} \\big) $ ,\n$ \\Sigma = \\sum_{K\\, \\in \\, \\mathcal{I}}\\sum_{i\\, = \\, 1}^{s_{K}^{(1)}} x_{K, i}^{(1)} $,\n$ \\upne $. }\n\\KwResult{New set of sections quantities $ \\big(s_{K}^{(2)}: K \\in \\mathcal{I} \\big) $.\n}\n\\textbf{Initialization}\\;\n\\textbf{sort} the list of intervals $ \\mathcal{I} $ from large values to small ones\\;\n\t\\eIf{ $ \\Sigma > \\upne $ }{\n\t\\textbf{define} $ df^{(2)} \\defining \\Sigma - \\upne $, $ K^{(r)} \\defining \\text{ right extreme of } K $, for all $ K \\in \\mathcal{I} $\\;\n\t\\While{ $ df^{(2)} > \\min\\{K^{(r)}: K \\in \\mathcal{I} \\} $ }{\n\t\t\\If{ $ \\Sigma - \\upne > K^{(r)} $ }{\n\t\t$ s_{K}^{(2)} = s_{K}^{(1)} - 1 $, $ df^{(2)} = df^{(2)} - K^{(r)} $\n\t\t}\n\t\t}\n\t\t}\n\t{\n\t\\textbf{define} $ df^{(2)} \\defining \\upne - \\Sigma $, $ K^{(\\ell)} \\defining \\text{ left extreme of } K $, for all $ K \\in \\mathcal{I} $\\;\n\t\\While{ $ df^{(2)} > \\min\\{K^{(\\ell)}: K \\in \\mathcal{I} \\} $ }{\n\t\\If{ $ \\Sigma - \\upne > K^{(\\ell)} $ }{\n\t$ s_{K}^{(2)} = s_{K}^{(1)} + 1 $, $ df^{(2)} = df^{(1)} - K^{(\\ell)} $\n\t\t}\n\t\t}\n\t}\n\\caption{Greedy Algorithm Increase\/Decrease Number of Sections}\\label{Alg Greedy Algorithm}\n\\end{algorithm}\nIf $ df^{(2)} \\equiv 0 $ then jump to \\textsc{step} \\ref{Step 4 Initial Generate randomly a group assignment matrix} below.\n\n\\item\\label{Step 3 Modify Sections Capacity} If $ 0 < df^{(2)} $ (once the Greedy Algorithm \\ref{Alg Greedy Algorithm} has been applied), apply \\textbf{increase\/decrease capacities Algorithm} \\ref{Alg Increase Decrease Capacities} (breaking the constraints $ x_{K, i}^{(2)} \\in K $ of \\textsc{Equation} \\eqref{Eq Optimization Students Sections Capacity}). Firstly changing $ x_{K, i}^{(1)} \\mapsto x_{K, i}^{(2)} $ as evenly as possible. Secondly, distributing the reminder in randomly chosen sections $ x_{K, i}^{(2)} \\mapsto x_{K, i}^{(3)} $ and get\n\\begin{equation}\\label{Eq Evenly Distribute Excess or Absence}\ndf^{(3)} \\defining \\Big\\vert \\sum_{K\\, \\in \\, \\mathcal{I}}\\sum_{i\\, = \\, 1}^{s_{K}^{(2)}} \nx_{K, i}^{(3)}\n- \\upne \\Big\\vert \\equiv 0 < df^{(2)}.\n\\end{equation}\n\\begin{algorithm}[H]\n\\KwData{\n$ df^{(2)} $, \n$ \\big(s_{K}^{(2)}: K \\in \\mathcal{I} \\big) $ ,\n$ \\Sigma = \\sum_{K\\, \\in \\, \\mathcal{I}}\\sum_{i\\, = \\, 1}^{s_{K}^{(2)}} x_{K, i}^{(1)} $,\n$ \\upne $. }\n\\KwResult{New set of sections capacities $ \\big(x_{K, i}^{(2)}: i \\in [ s_{K}^{(2)} ], K \\in \\mathcal{I} \\big) $.\n}\n\\textbf{initialization}\\;\n\\textbf{sort} the list of sections capacities $ \\big( x_{K, i}^{(1)}: i \\in [s_{K}^{(2)}], K \\in \\mathcal{I} \\big) $ from large values to small ones\\;\n\\textbf{define} total number of sections $ s \\defining \\sum_{K \\in \\mathcal{I} } s_{K}^{(2)} $\\;\n\t\\eIf{ $ \\Sigma > \\upne $ }{\n\t\\textbf{define} $ u = \\big\\lfloor \\dfrac{\\Sigma - \\upne }{s} \\big\\rfloor $ \\;\n\t\\textbf{define} $ x_{K, i}^{(2)} \\defining x_{K, i}^{(1)} - u $ for all $ i \\in [ s_{K}^{(2)} ], K \\in \\mathcal{I} $\\;\n\t\\textbf{choose} $ \\Sigma - \\upne - u \\times s $ sections $ K \\in \\mathcal{I}, i \\in [s_{K}^{2} ] $ denote this set by $ S $\\;\n\t\\eIf{ $ (K, i) \\in S $ }{\n\t$ x_{K, i}^{(3)} \\defining x_{K, i}^{(2)} - 1 $\n\t}\n\t{\n\t$ x_{K, i}^{(3)} \\defining x_{K, i}^{(2)} $\n\t}\n\t}\n\t{\n\t\\textbf{define} $ u = \\big\\lfloor \\dfrac{\\upne - \\Sigma }{s} \\big\\rfloor $ \\;\n\t\\textbf{define} $ x_{K, i}^{(2)} \\defining x_{K, i}^{(1)} + u $ for all $ i \\in [ s_{K}^{(2)} ], K \\in \\mathcal{I} $\\;\n\t\\textbf{choose} $ \\upne - \\Sigma - u \\times s $ sections $ K \\in \\mathcal{I}, i \\in [s_{K}^{2} ] $ denote this set by $ S $\\;\n\t\\eIf{ $ (K, i) \\in S $ }{\n\t$ x_{K, i}^{(3)} \\defining x_{K, i}^{(2)} + 1 $\n\t}\n\t{\n\t$ x_{K, i}^{(3)} \\defining x_{K, i}^{(2)} $\n\t}\n\t}\n\\caption{Greedy Algorithm Increase\/Decrease Sections' Capacity}\n\\label{Alg Greedy Algorithm Sections Capacity}\n\\end{algorithm}\n\n\\item\\label{Step 4 Initial Generate randomly a group assignment matrix} Generate randomly a group assignment matrix $ G $.\n\\end{enumerate}\nThe random setting described above is summarized in the pseudocode \\ref{Alg Random Setting MC Analytica Omega}\n\n\\begin{algorithm}[t]\n\\KwData{ \n$ \\upne $ random variable distribution,\n$ \\textbf{X}_{\\gpa} $ random variable distribution \\\\\n$ \\upns $ random variable distribution, average sections frequency $ \\overline{\\boldsf} $ variable \\\\\nAnalyzed Course: DC, IC, ..., NM. \\\\\nList of Tenured Lecturers}\n\\KwResult{Random Group Assignment Matrix $ G $.\\\\\n}\n\\textbf{Initialization}\\;\n\\textbf{compute} a realization of $ \\upne $ and a realization for $ \\textbf{X}_{\\gpa} $\\;\n\\textbf{compute} $ \\textbf{GPA}, $ \\textbf{input}: ($ \\upne, \\textbf{X}_{\\gpa} $)\\;\n\\textbf{call} RandInputAlgorithm \\ref{Alg Segmentation of Students}, \\textbf{input}: \n(\\underline{\\textbf{GPA} list})\\;\n\\textbf{compute} a realization of $ \\upns $ \\;\n\\textbf{compute} $ \\textbf{S} $, \\textbf{input}: ($ \\upns, { \\overline{\\boldsf} }$, \\textit{Analyzed Course})\\;\n\\textbf{solve} Problem \\ref{Eq Optimization Students Sections Capacity} \\textbf{input}: ($ \\textbf{S}, \\upne $)\\;\n\\eIf{ $ df^{(1)} > 0 $ }{\n\t\\textbf{run} the increase\/decrease number of sections Greedy Algorithm \\ref{Alg Greedy Algorithm}\\;\n\t\\eIf{ $ df^{(2)} > 0 $ }{\n\t\t\\textbf{run} the increase\/decrease capacities algorithm \n\t\t\\ref{Alg Greedy Algorithm Sections Capacity} \\; \n\t\t$ \\textbf{S} \\leftarrow \\big(x_{K, i}^{(3)}: i\\in [s_{K}^{(2)}], K \\in \\mathcal{I} \\big) $\\;\n\t\t\\textbf{return} $ \\textbf{S} $\n\t\t}\n\t\t{$ \\textbf{S} \\leftarrow \\big(x_{K, i}^{(3)}: i\\in [s_{K}^{(2)}], K \\in \\mathcal{I} \\big) $\\;\n\t\t\\textbf{return} $ \\textbf{S} $}\n\t}\n\t{$ \\textbf{S} \\leftarrow \\big(x_{K, i}^{(3)}: i\\in [s_{K}^{(1)}], K \\in \\mathcal{I} \\big) $\\;\n\t\\textbf{return} $ \\textbf{S} $ }\n\n\\textbf{compute} a random group matrix assignment $ G $, \\textbf{input}: $ (\\textbf{S},\\textbf{GPA} ) $\\;\n\\caption{Random Setting Algorithm}\n\\label{Alg Random Setting MC Analytica Omega}\n\\end{algorithm}\nWe close this section displaying tree tables. \\textsc{Table} \\ref{Tb Sections Random Variable} contains the confidence intervals for the number of sections random variable $ \\upns $. The capacities distribution vector $ \\overline{\\boldsf} $ , as well as the confidence intervals of $ \\textbf{X}_{\\gpa} $ are shown in \\textsc{Table} \\ref{Tb GPA Random Variable} for the course of Differential Calculus only; due to the $ \\gpa $ range length, the table has been split in five rows to fit the page format. Finally, \\textsc{Table} \\ref{Tb Sections Realization Example} presents an example of a group assignment matrix $ G $ produced by Algorithm \\ref{Alg Random Setting MC Analytica Omega}\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{ c | c c c c c c c c }\n \\hline\n \\rowcolor{gray!80}\n\\diagbox\n{PARAMETERS}{COURSE}\n& DC\n& IC\n& VC\n& VAG\n& LA\n& ODE\n& BM\n& NM\n \\\\\n\n\\hline\nUpper Extreme &\n20\t& 11 &\t5 &\t17 & 10\t& 7\t& 17 &\t3 \\\\\nLower Extreme &\n16\t& 9\t& 3\t& 15 &\t8 &\t5 &\t13 & 1 \\\\\n$ [15, 30] $ &\n0.0202 & 0.0000 &\t0.0133 & 0.0000\t& 0.0000\t& 0.0000\t& 0.0064\t& 0.0000 \\\\\n$ [31, 45] $ &\n0.0030 & 0.0000 &\t0.0000\t& 0.0000\t& 0.0000\t& 0.0000\t& 0.0712\t& 0.0000 \\\\\n$ [46, 60] $ &\n0.0403\t& 0.0310 &\t0.0000\t& 0.0044\t& 0.0000\t& 0.0000\t& 0.0609\t& 0.0000 \\\\\n$ [61, 75] $ &\n0.3802\t& 0.0330 &\t0.0000\t& 0.7112\t& 0.0572\t& 0.0222\t& 0.6125\t& 0.0667 \\\\\n$ [76, 90] $ &\n0.1155\t& 0.0048 &\t0.0000\t& 0.1134\t& 0.0083\t& 0.0000\t& 0.2056\t& 0.0000 \\\\\n$ [91, 105] $ &\n0.1034\t& 0.0588 &\t0.0133\t& 0.0675\t& 0.0763\t& 0.0095\t& 0.0300\t& 0.0333 \\\\\n$ [106, 120] $ &\n0.1640\t& 0.2115 &\t0.0300\t& 0.1035\t& 0.2939\t& 0.0429\t& 0.0133\t& 0.0333 \\\\\n$ [121, 135] $ &\n0.0629\t& 0.3417 &\t0.2600\t& 0.0000\t& 0.1791\t& 0.3937\t& 0.0000\t& 0.2889 \\\\\n$ [135, 150] $ &\n0.1104\t& 0.3193 &\t0.6833\t& 0.0000\t& 0.3852\t& 0.5317\t& 0.0000\t& 0.5778 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Random Variable Number of Sections $ \\upns $ and Capacities Distribution Vector \n\t$ \\overline{\\boldsf} $,\n\tCourse: \\textbf{All}. \nThe upper \\& lower extreme and confidence intervals for the Variable number of sections $ \\upns $ are displayed for all courses.}\\label{Tb Sections Random Variable}\n\\end{table}\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\scriptsize{\n\\begin{center}\n\\begin{tabular}{ c | c c c c c c c c c c}\n \\hline\n \\rowcolor{gray!80}\n\\diagbox\n{INTERVALS}{GPA}\n& 0.1\n& 0.2\n& 0.3\n& 0.4\n& 0.5\n& 0.6\n& 0.7\n& 0.8\n& 0.9\n& 1.0\n \\\\\n\\hline\nUpper Extreme &\n0.0021 &\t0.0027 & 0.0031 & 0.0031 &\t0.0035\t& 0.0031\t& 0.0044\t& 0.0040 &\t0.0047 & 0.0046\\\\\n\\rowcolor{gray!25}\nLower Extreme &\n0.0012& 0.0013 &\t0.0017\t& 0.0017 &\t0.0024\t& 0.0016\t& 0.0031\t& 0.0021 & 0.0020 & 0.0028 \\\\ \n\\hline\n\\rowcolor{gray!80}\n\\diagbox\n{INTERVALS}{GPA}\n& 1.1\n& 1.2\n& 1.3\n& 1.4\n& 1.5\n& 1.6\n& 1.7\n& 1.8\n& 1.9\n& 2.0\n \\\\\n\\hline\nUpper Extreme & \n0.0047\t& 0.0045\t& 0.0055\t& 0.0053\t& 0.0055\t& 0.0056\t& 0.0051\t& 0.0073\t& 0.0082 & 0.0084\\\\\n\\rowcolor{gray!25}\nLower Extreme &\n0.0026\t& 0.0025\t& 0.0032\t& 0.0028\t& 0.0031\t& 0.0038\t& 0.0033\t& 0.0049\t& 0.0058 & 0.0056\n\\\\\n\\hline\n\\rowcolor{gray!80}\n\\diagbo\n{INTERVALS}{GPA}\n& 2.1\n& 2.2\n& 2.3\n& 2.4\n& 2.5\n& 2.6\n& 2.7\n& 2.8\n& 2.9\n& 3.0\n \\\\\n\\hline\nUpper Extreme &\n0.0114\t& 0.0119\t& 0.0147\t& 0.0191\t& 0.0208\t& 0.0277\t& 0.0328\t& 0.0348\t& 0.0420 & 0.0858\\\\\n\\rowcolor{gray!25}\nLower Extreme & \n0.0077\t& 0.0076\t& 0.0105\t& 0.0142\t& 0.0165\t& 0.0210\t& 0.0250\t& 0.0287\t& 0.0332 & 0.0672\n\\\\\n\\hline\n\\rowcolor{gray!80}\n\\diagbox\n{INTERVALS}{GPA}\n& 3.1\n& 3.2\n& 3.3\n& 3.4\n& 3.5\n& 3.6\n& 3.7\n& 3.8\n& 3.9\n& 4.0\n \\\\\n\\hline\nUpper Extreme &\n 0.0571\t& 0.0566\t& 0.0671\t& 0.0668\t& 0.0630\t& 0.0668\t& 0.0625\t& 0.0527 & 0.0496 & 0.0421\n\\\\\n\\rowcolor{gray!25}\nLower Extreme &\n0.0506\t& 0.0489\t& 0.0565\t& 0.0594\t& 0.0558\t& 0.0549 &\t0.0526\t& 0.0440\t& 0.0404 & 0.0357\n\\\\\n\\hline\n\\rowcolor{gray!80}\n\\diagbo\n{INTERVALS}{GPA}\n& 4.1\n& 4.2\n& 4.3\n& 4.4\n& 4.5\n& 4.6\n& 4.7\n& 4.8\n& 4.9\n& 5.0\n \\\\\n\\hline\nUpper Extreme &\n0.0373\t& 0.0255\t& 0.0209\t& 0.0161\t& 0.0136\t& 0.0081\t& 0.0055\t& 0.0029\t& 0.0011 & 0.0002\n\\\\\n\\rowcolor{gray!25}\nLower Extreme &\n0.0282\t& 0.0201\t& 0.0160\t& 0.0115\t& 0.0088\t& 0.0054\t& 0.0032\t& 0.0014\t& 0.0002 & 0.0000\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Random Variable: $ \\textbf{X}_{\\gpa} $,\n\tCourse: \\textbf{Differential Calculus}. \n\tThe distribution capacities vector $ \\overline{\\boldsf} $ and the confidence intervals of $ \\textbf{X}_{\\gpa} $ for the course of Differential Calculus are displayed. }\\label{Tb GPA Random Variable}\n\\end{table}\n\\subsection{Normalization of the method and probabilistic spaces}\\label{Sec Normalization of the method}\nIn order to measure the proposed method's enhancement, now there is need to normalize the results as in the case of the historical assessment of the algorithm, \\textsc{Section} \\ref{Sec Optimization and Historical Assessment}, \\textsc{Equation} \\eqref{Eq Performance Rates} where the improvement in the academic performance variable was divided over the historical performance of the semester at hand. In the case of Monte Carlo simulations, the concept of ``historical performance\" simply does not apply, as the assignation of students and\/or lecturers actually did not happen. We approach this fact in two different ways\n\\begin{definition}[Normalization Methods]\\label{Def Normalization Methods}\nWe introduce the following normalization methods.\n\\begin{enumerate}[(a)]\n\\item\\label{Mthd Random Normalization} \\textbf{Random Normalization}. Normalize with respect to a random assignation of instructors or students, depending on the method \\textbf{IA} or \\textbf{SA} respectively.\n\n\\item\\label{Mthd Expected Normalization} \\textbf{Expected Normalization}. Normalize with respect to the \\textit{expected assignation} of instructors or students, depending on the method \\textbf{IA} or \\textbf{SA} respectively. \n\\end{enumerate}\n\\end{definition}\nIn the first case, it is straightforward to compute the normalization, in the second case, the concept of \\textit{expected assignation} needs to be stated in neater terms. To that end, we need to present some intermediate mathematical results and definitions\n\\begin{theorem}\\label{Th expected performance molecule computation}\nLet $ K \\in \\N $ be fixed and let $\\big( T(i, k): i, k\\in [K]\\big) $ be a matrix. Define the Random Variable \n\\begin{align*}\n& \\X_{ \\ia }: \\itS_{K} \\rightarrow \\R, & \n& \\X_{ \\ia }(\\sigma) \\defining \\sum\\limits_{k \\, \\in \\, [K]} T\\big(k, \\sigma(k)\\big) .\n\\end{align*}\nThen,\n\\begin{equation}\\label{Eq expected performance molecule computation}\n\\Exp\\big(\\X_{ \\ia }\\big) = \n\\frac{1}{K}\\sum\\limits_{i, k \\, \\in\\, [K]} T(k, i) = \\frac{1}{K}\\, \\textup{sum}(T),\n\\end{equation}\nwhere $ \\textup{sum}(T) \\defining \\sum\\limits_{(k,i) \\in [K] \\times [K]}T(k, i) = \\sum\\limits_{k \\, \\in \\, [K] }\\sum\\limits_{i \\, \\in [K]} T(k, i) $.\n\\end{theorem}\n\\begin{proof}\nConsider the following calculation\n\\begin{equation*}\n\\begin{split}\n\\Exp\\big(\\X_{ \\ia }\\big) \n& = \\frac{1}{K!}\\sum\\limits_{\\sigma\\,\\in \\, \\itS_{K}} \\X_{ \\ia }(\\sigma) \n= \n\\frac{1}{K!}\\sum\\limits_{\\sigma\\,\\in \\, \\itS_{K} } \\sum\\limits_{k \\, \\in\\, [K]} T\\big(k,\\sigma(k)\\big) \n= \\frac{1}{K!} \\sum\\limits_{k \\, \\in\\, [K]} \\sum\\limits_{\\sigma\\,\\in \\, \\itS_{K}} T\\big(k,\\sigma(k)\\big) \n\\\\\n& = \\frac{1}{K!} \\sum\\limits_{k \\, \\in\\, [K]} \\sum\\limits_{i \\, \\in\\, [K]}\n\\sum\\limits_{\\substack{\\sigma\\,\\in \\, \\itS_{K} \\\\ \\sigma(k) \\, = \\, i } } T\\big(k,\\sigma(k)\\big) \n= \\frac{1}{K!} \\sum\\limits_{k \\, \\in\\, [K]} \\sum\\limits_{i \\, \\in\\, [K]}\nT\\big(k, i\\big)\\sum\\limits_{\\substack{\\sigma\\,\\in \\, \\itS_{K} \\\\ \\sigma(k) \\, = \\, i } } 1\n= \\frac{(K - 1)!}{K!} \\sum\\limits_{k \\, \\in\\, [K]} \\sum\\limits_{i \\, \\in\\, [K]}\nT\\big(k, i\\big) .\n\\end{split}\n\\end{equation*}\nFrom here, Equation \\eqref{Eq expected performance molecule computation} follows trivially.\n\\end{proof}\n\\begin{remark}\\label{Rem expected performance molecule computation}\nNotice that in Proposition \\ref{Th expected performance molecule computation} the following holds\n\\begin{enumerate}[(i)]\n\\item It is understood that the probabilistic space is $ \\Omega \\equiv \\itS_{K} $ where all the outcomes are equally likely.\n\n\\item Assume that the setting of a semester is given, namely: number of sections with capacities, conformation of sections and a set of instructors teaching the course. Then, $ T = C_{\\apv} $, $ J $ is the number of sections and $ \\Exp(X_{ \\ia }) $ is the expected performance, when assigning instructors randomly to the available sections with defined students i.e., the $ \\ia $ method.\n\n\\end{enumerate}\n\\end{remark}\nOur next step is to be able to compute the expected performance of a group when assigning students randomly to available sections with defined instructors. This task is far more complicated due to the richness of the search space. We begin introducing some notation.\n\\begin{definition}\\label{Def labeling and asignation functions}\nLet $ N, L, J \\in \\N $ , $ \\p = (p_{1}, \\ldots, p_{L}) \\in \\N^{L}, \\g = (g_{1}, \\ldots, g_{J}) \\in \\N^{J} $ be as in Definition \\ref{Def Characteristic Numbers}. \n\\begin{enumerate}[(i)]\n\\item Let $ c: [N] \\rightarrow [L] $ be the \\textbf{classification function} of each student i.e., for each student $ n \\in [N] $ it assigns the label $ c(n) \\in [L] $ describing the profile to which he\/she belongs to. \n\n\\item Define the \\textbf{student assignation} probabilistic space by\n\\begin{equation}\\label{Eq student assignation probabilitic space}\n\\Omega\\defining \\Big\\{\\omega: [N] \\rightarrow [J]: \n\\big\\vert \\omega^{-1}(j) \\big\\vert = g_{j} \\, , \\, \\text{for all } j \\in [J]\\Big\\} .\n\\end{equation}\n\n\\item For a fixed element $ \\omega \\in \\Omega $, define the matrix $ G^{\\omega} \\in \\R^{L\\times J} $ whose entries are given by \n\\begin{align*}\n& G^{\\omega}(\\ell, j) = \n\\big\\vert \\big\\{n\\in [N]: c(n) = \\ell, \\omega(n) = j \\big\\}\\big\\vert = \n\\big\\vert c^{-1}(\\ell) \\cap \\omega^{-1}(j)\\big\\vert\\, ,\n&\n& \\forall\\, \\ell \\in [L], j\\in [J] .\n\\end{align*}\n\\end{enumerate}\n\\end{definition}\n\\begin{remark}\\label{Rem Student Assignment Setting}\nIn Definition \\ref{Def labeling and asignation functions} notice the following\n\\begin{enumerate}[(i)]\n\\item The student classification function satisfies that $ p_{\\ell} \\defining \\big \\vert c^{-1}(\\ell) \\big\\vert $ for all $ \\ell \\in [L] $.\n\n\\item An element $ \\omega $ of the student assignation space is such that every individual is assigned to a section and every section is full (recall the sum condition \\eqref{Eq Sum Condition}). \n\n\\item In our study, a list of $ N $ enrolled students is completely characterized a classification function $ c: [N] \\rightarrow [L] $ and a section assignment function $ \\omega \\in \\Omega $\n\\begin{equation*}\n\\begin{array}{cccc}\n1, & 2, & \\ldots, & N , \\\\\nc(1), & c(2), & \\ldots, & c(N) ,\\\\\n\\omega(1), & \\omega(2), & \\ldots , & \\omega(N) .\n\\end{array}\n\\end{equation*}\nThe first row represents identity and the second indicates profile classification. Therefore, only the third row is subject to decision or randomization as it is done in this model.\n\n\\item For every $ \\omega \\in \\Omega $ the matrix $ G^{\\omega} $ is clearly a group assignment matrix as introduced in Definition \\ref{Def Performance and Group Assignment Matrices}.\n\\end{enumerate}\n\\end{remark}\n\\begin{proposition}\\label{Th measuring global performance}\nLet $ N, L, J \\in \\N $ , $ \\p = (p_{1}, \\ldots, p_{L}) \\in \\N^{L}, \\g = (g_{1}, \\ldots, g_{J}) \\in \\N^{J} $ be as in Definition \\ref{Def Characteristic Numbers}. \nThen\n\\begin{align}\\label{Eq measuring global performance}\n& \\trace\\big(T_{\\apv} G^{\\omega} \\big) = \n\\sum\\limits_{j\\, = \\, 1}^{J} \\big(T_{\\apv} G^{\\omega} \\big)\\big(j, j \\big) =\n\\sum_{n\\, \\in\\,[N]} T_{\\apv}\\big( \\omega(n), c(n) \\big), \n& \n& \\text{for all } \\omega \\in \\Omega .\n\\end{align} \n\\end{proposition}\n\\begin{proof}\nConsider the following identities\n\\begin{equation*}\n\\begin{split}\n\\sum_{n\\, \\in\\,[N]} T_{APV}\\big( \\omega(n), c(n) \\big) & =\n\\sum_{(\\ell, j)\\, \\in \\,[L]\\times[J]}\n\\sum_{\\substack{n\\, =\\, 1\\\\ c(n) \\, = \\, \\ell, \\, \\omega(n) \\, = \\, j}}^{N} T_{APV} \n\\big(\\omega(n), c(n)\\big) \\\\\n& = \\sum_{j\\, \\in [J]} \\sum_{\\ell \\, \\in \\,[L]} \nT_{APV} \\big( j, \\ell \\big)\n\\sum_{\\substack{n\\, =\\, 1\\\\ c(n) \\, = \\, \\ell, \\, \\omega(n) \\, = \\, j}}^{N} 1 \\\\\n& = \\sum_{j\\, \\in [J]} \\sum_{\\ell \\, \\in \\,[L]} \nT_{APV} \\big( j, \\ell \\big)\nG^{\\omega}(\\ell, j) \n\\\\\n& = \\sum_{j\\, \\in [J]} \\big(T_{APV} G^{\\omega}\\big) \\big( j, j \\big) ,\n\\end{split}\n\\end{equation*}\ni.e., the result holds.\n\\end{proof}\n\\begin{remark}\\label{Rem Probabilistic Modeling Student Assignation}\nObserve that if it is assumed that that the instructor $t_{j}$ is assigned to the section $ j $ for all $ j \\in {J}$, i.e., the instructor assignment function $ \\pi \\in \\itS_{J} $ of Problem \\ref{Pblm Students Assignment Problem}\nis the identity then, the previous result states that \n\\begin{equation}\\label{Eq measuring global performance comment}\n\\trace\\big(T_{\\apv} G^{\\omega} \\big) = \n\\sum\\limits_{j\\, = \\, 1}^{J} \\big(T_{\\apv} G^{\\omega} \\big)\\big(j, \\pi(j) \\big) =\n\\sum_{n\\, \\in\\,[N]} T_{\\apv}\\big( \\omega(n), c(n) \\big), \n\\end{equation} \nfor each $ \\omega \\in \\Omega $. Since the expression of the middle measures the global performance of the group, so does the right hand side. \nHence, it makes sense to declare the left hand side in the expression above as a random variable.\n\n\n\\end{remark}\n\\begin{definition}\\label{Def Random assignment matrix}\n\tLet $ N, L, J \\in \\N $ , $ \\p = (p_{1}, \\ldots, p_{L}) \\in \\N^{L}, \\g = (g_{1}, \\ldots, g_{J}) \\in \\N^{J} $ be as in Definition \\ref{Def Characteristic Numbers} and let $ T \\in \\R^{ J\\times L } = \\big( T(j, \\ell) : k \\in [J] , \\ell \\in [L] \\big) $, be a fixed matrix.\n\tDefine the \\textbf{student assignment performance} random variable\n\t%\n\t\\begin{align}\\label{Eq Student Assignment Performance}\n\t& \\X_{ \\sa }: \\Omega \\rightarrow \\R\\, , & \n\t& \\X_{ \\sa }(\\omega) \\defining \\sum\\limits_{n\\,\\in\\, [N] } T\\big(\\omega(n), c(n)\\big) \\, .\n\t\\end{align}\n\t%\n\\end{definition}\nBefore computing the expectation of the random variable $ \\X_{\\sa} $ some previous results from combinatorics are needed.\n\\begin{lemma}\\label{Th cardinal of student assignment space}\n\t%\n\t\\begin{enumerate}[(i)]\n\t\t\\item The cardinal of the student assignment space is given by\n\t\t%\n\t\t\\begin{equation}\\label{Eq cardinal of student assignment space}\n\t\t\\vert \\Omega \\vert = N!\\prod\\limits_{j \\, = \\, 1}^{J}\\dfrac{1}{g_{j}!} .\n\t\t\\end{equation} \n\t\t%\n\t\t\\item Let $ n\\in [N] $, $ j\\in [J] $ be fixed, and define the set\n\t\t%\n\t\t\\begin{equation*}\n\t\t\\Omega_{n, j} \\defining \\big\\{ \\omega\\in \\Omega: \\omega (n) = j\\big\\}.\n\t\t\\end{equation*}\n\t\t%\n\t\tThen $ \\vert \\Omega_{n, j} \\vert = \\dfrac{(N - 1)! }{(g_{j} - 1)!} \n\t\t\\prod\\limits_{\\substack { i \\, \\in \\, [K]\\\\\n\t\t\t\ti \\, \\neq \\, j} }\\dfrac{1}{g_{i}!} $\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\t\\begin{enumerate}[(i)]\n\t\t\\item Let $ \\omega $ be an element of $ \\Omega $ and write it in the extended way i,e,\n\t\t%\n\t\t\\begin{equation*}\n\t\t\\begin{array}{cccc}\n\t\t\t\t\\omega(1), & \\omega(2), & \\ldots , & \\omega(N), \\\\\n\t\t\t1 , & 2 , & \\ldots, & N .\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\t%\n\t\tClearly, $ \\omega $ is a permutation of of the multiset\n\t\t%\n\t\t\\begin{equation}\\label{Eq assignment section multiset}\n\t\t\\big\\{\\underbrace{1,1,\\ldots, 1}_{g_{1} \\text{-times}},\n\t\t\\underbrace{2,2,\\ldots, 2}_{g_{2} \\text{-times}},\n\t\t\\ldots\n\t\t,\\underbrace{J,J,\\ldots, J}_{g_{J} \\text{-times}}\\big\\} =\n\t\t\\big\\{1\\cdot g_{1}, 2\\cdot g_{2}, \\ldots J\\cdot g_{J} \\big\\}.\n\t\t\\end{equation}\n\t\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tFrom elementary combinatorics, it is known that the number of permutations of the multiset \\eqref{Eq assignment section multiset} is given given by the expression \\eqref{Eq cardinal of student assignment space}, see Theorem 3.5 in \\cite{BonaWalk}.\n\t\n\t\\item First we analyze the case of the set $ \\Omega_{N, j} $. Recalling the expression \\eqref{Eq student assignation probabilitic space}, we can write $ \\Omega_{N, j} = \\big\\{\\omega: [N] \\rightarrow [J]: \\, \\omega(N) = j , \\, \n\t\\vert \\omega^{-1}(i) \\vert = g_{i}, \\text{ for all } i \\in [J]\\big\\} $. It is direct to see that there is a bijection with the set $ \\widetilde{\\Omega} \\defining \\big\\{\\omega: [N - 1] \\rightarrow [J]: \n\t\\big\\vert \\omega^{-1}(i) \\big\\vert = \\widetilde{g}_{i} \\, , \\, \\text{for all } i \\in [J]\\big\\} $ where $ \\widetilde{g}_{i} $ is defined as follows\n\t%\n\t\\begin{equation*}\n\t\\widetilde{g}_{i} \\defining \n\t\\begin{cases}\n\tg_{i}, & i \\neq j , \\\\\n\tg_{i} - 1, & j = i .\n\t\\end{cases}\n\t\\end{equation*}\n\t%\n\tApplying the previous part on the set $ \\widetilde{\\Omega} $, it follows that $ \\Omega_{N, j} $ satisfies the result. For the general case $ \\Omega_{n,j} $, take the permutation $ \\sigma \\in \\itS_{N} $ defined by \n\t%\n\t\\begin{equation*}\n\t\\sigma(k) \\defining\n\t\\begin{cases}\n\tN, & k = n , \\\\\n\tn, & k = N , \\\\\n\tk & \\text{otherwise}. \n\t\\end{cases} \n\t\\end{equation*}\n\n\tObserve that the map $ \\varphi: \\Omega_{n,j} \\rightarrow \\Omega_{N,j} $ defined by $ \\varphi(\\omega) \\defining \\omega\\circ\\sigma $ is clearly a bijection. Consequently, $ \\vert \\Omega_{n,j} \\vert = \\vert \\Omega_{N,j} \\vert $ and the proof is complete.\n\t\\end{enumerate}\t\n\\end{proof}\n\\begin{theorem}\\label{Def Expectation of random student assignment}\n\tThe expectation of the random variable $ \\X_{ \\sa } $ is given by \n\t%\n\t\\begin{equation}\\label{Eq Expectation of random student assignment}\n\t\\Exp\\big(\\X_{ \\sa } \\big) = \\frac{1}{N} \\, \\g^{t} T \\, \\p .\n\t\\end{equation}\n\t%\n\\end{theorem}\n\\begin{proof}\n\tBy definition\n\t%\n\t\\begin{equation}\n\t\\begin{split}\n\t\\vert \\Omega \\vert\\, \\Exp\\big(\\X_{ \\sa }\\big) \n\t& = \n\t\\sum\\limits_{\\omega\\,\\in\\,\\Omega} \\sum\\limits_{n\\,= \\,1 }^{N}\n\tT\\big(\\omega(n), c(n)\\big)\n\t= \\sum\\limits_{n\\,= \\,1 }^{N}\n\t\\sum\\limits_{\\omega\\,\\in\\,\\Omega}\n\tT\\big(\\omega(n), c(n)\\big)\\\\\n\t& = \\sum\\limits_{n\\,= \\,1 }^{N}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\t\\sum\\limits_{\\substack{\\omega\\,\\in\\,\\Omega\\\\\\omega(n) \\, = \\, j}}\n\tT\\big(\\omega(n), c(n)\\big) \n\t= \\sum\\limits_{n\\,= \\,1 }^{N}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\tT\\big( j, c(n) \\big)\\sum\\limits_{\\substack{\\omega\\,\\in\\,\\Omega\\\\\\omega(n) \\, = \\, j}} 1 \n\t\\end{split}\n\t\\end{equation}\n\t%\n\tRecalling Lemma \\ref{Th cardinal of student assignment space} (ii), it follows that\n\t%\n\t\\begin{equation}\n\t\\begin{split}\n\t\\Exp\\big(\\X_{ \\sa }\\big) \n\t& = \\frac{1}{\\vert \\Omega \\vert} \n\t\\sum\\limits_{n\\,= \\,1 }^{N}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\tT\\big( j, c(n) \\big)\n\t\\dfrac{(N - 1)! }{(g_{j} - 1)!} \n\t\\prod\\limits_{\\substack { i \\, \\in \\, [J]\\\\\n\t\t\ti \\, \\neq \\, j} }\\dfrac{1}{g_{i}!} \\\\\n\t& = \\frac{1}{\\vert \\Omega \\vert}\\sum\\limits_{n\\,= \\,1 }^{N}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\tg_{j} \\, T\\big( j, c(n) \\big) (N - 1)! \n\t\\prod\\limits_{ i \\, \\in \\, [J] } \\dfrac{1}{g_{i}!} = \n\t\\frac{1}{N}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\t\\sum\\limits_{n\\,= \\,1 }^{N}\n\tg_{j} \\, T\\big( j, c(n) \\big) \\\\\n\t& = \\frac{1}{N}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\tg_{j}\n\t\\sum_{\\ell\\, = \\, 1}^{L}\n\t\\sum\\limits_{\\substack { n \\, \\in \\, [N]\\\\\n\t\t\tc(n) \\, = \\, \\ell} }\n\t\\, T\\big( j, c(n) \\big)\n\t= \\frac{1}{N}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\tg_{j}\n\t\\sum_{\\ell\\, = \\, 1}^{L}\n\tT\\big(j, \\ell\\big) p_{\\ell} \\\\\n\t& = \\frac{1}{N}\n\t\\sum_{\\ell\\, = \\, 1}^{L}\n\tp_{\\ell}\n\t\\sum\\limits_{j\\,= \\,1 }^{J}\n\tT\\big(j, \\ell \\big) g_{j} .\n\t\\end{split}\n\t\\end{equation}\n\t%\n\tHere, the second equality uses the identity $ \\frac{1}{(g_{j} - 1)!} = \\frac{ g_{j}}{g_{j}!} $ and the third uses the expression \\eqref{Eq cardinal of student assignment space}, together with an obvious exchange of indexes. The fourth equality is a convenient association of summands, while the fifth merely uses the fact $ \\vert c^{-1}(\\ell) \\vert = p_{\\ell} $. From here, the result follows trivially.\n\\end{proof}\n\\begin{remark}\\label{Rem random student assignment tool}\nLet $ \\pi \\in \\itS_{J} $ be a permutation and let $ A^{\\pi} $ its associated permutation matrix\n\\begin{equation*}\nA^{\\pi} = \\big[\\mathbf{\\widehat{e}}_{\\pi(1)}, \\mathbf{\\widehat{e}}_{\\pi(2)}, \\ldots, \\mathbf{\\widehat{e}}_{\\pi(J)} \\big] ,\n\\end{equation*}\nwhere $ \\big(\\mathbf{\\widehat{e}}_{j}: j\\in [J]\\big) $ is the canonical basis of $ \\R^{J} $. Then, if the instructors $ \\big\\{ t_{j}: j\\in [J] \\big\\}$ are assigned to their corresponding sections by a permutation $ \\pi \\in \\itS_{J} $, other than the identity, by taking\n\\begin{equation*}\nT \\defining T_{\\apv} \\, A^{\\pi},\n\\end{equation*}\nthe random variable $ \\X_{\\sa}(\\omega) $ (as defined in \\eqref{Eq Student Assignment Performance}), computes the global performance of the group for each $ \\omega \\in \\Omega $ (as discussed in Remark \\ref{Rem Probabilistic Modeling Student Assignation}). Therefore, without loss of generality, it can be assumed that $ \\pi \\in \\itS_{J} $ is the identity.\n\\end{remark}\nFinally we define \n\\begin{definition}\\label{Def Normalization Methods Optimization Random Version}\n\tThe random version of Algorithm \\ref{Alg Optimization Algoritm} will have two methods. \n\t%\n\t\\begin{enumerate}[(i)]\n\t\t\\item The Random Normalization method introduced in \\textsc{Definition} \\ref{Def Normalization Methods} \\ref{Mthd Random Normalization} defined in Equation \\ref{Eq Performance Rates}. However, it is important to observe that this time $ v_{\\mt} $, $ \\rho_{\\mt} $ and $ \\X_{\\mt} \\defining \\sum_{j\\, = \\, 1}^{J} \\big(T_{APV} G_{h}\\big) \\big(j, \\pi_{h}(j) \\big) $ are all random variables. \n\t\t\n\t\t\\item Second, the Expected Normalization method introduced in \\textsc{Definition} \\ref{Def Normalization Methods} \\ref{Mthd Expected Normalization}, which is computed using \n\t\t%\n\t\t%\n\t\t\\begin{align}\\label{Eq Performance Rates Normalized}\n\t\t& \\gamma_{\\mt} \\defining \n\t\t100\\frac{v_{\\mt} - \\Exp\\big(\\X_{\\mt}\\big) }{ \\Exp\\big(\\X_{\\mt}\\big) }\n\t\t\\, , &\n\t\t& \\mt \\in \\{ \\ia, \\sa\\} .\n\t\t\\end{align} \n\t\t%\n\t\t%\n\t\tHere, $ \\Exp\\big(\\X_{\\mt}\\big) $ is given by Theorem \\ref{Th expected performance molecule computation} if $ \\mt = \\ia $ and by Theorem \\ref{Def Expectation of random student assignment} if $ \\mt = \\sa $. Again, $ v_{\\mt} $ and $ \\gamma_{\\mt} $ are both random variables. \n\t\\end{enumerate} \n\t%\n\\end{definition}\n\\begin{remark}\\label{Rem Harmonic Mean Comments}\n\\begin{enumerate}[(i)]\n\\item It is understood that, for the application of the Law of Large Numbers \\ref{Th the Law of Large Numbers} in the numerical experiments, the random variables above will be considered as sequences of independent, identically distributed, variables i.e., \n $ \\big(v_{\\mt}^{(n)}: n\\in \\N \\big) $, $ \\big(\\X_{\\mt}^{(n)}: n\\in \\N\\big) $, $ \\big(\\rho_{\\mt}^{(n)}: n\\in \\N \\big) $ and $ \\big(\\gamma_{\\mt}^{(n)}: n\\in \\N \\big) $; where the index $ n $ indicates an iteration of the Monte Carlo simulation. \n \n\\item It is direct to see that $ \\big(\\gamma_{\\mt}^{(n)}: n\\in \\N\\big) $ converges in the Ces\\`aro sense to $ \\Exp\\big(v_{\\mt}^{(1)}\\big) \\big(\\Exp\\big(\\X_{\\mt}^{(n)}\\big) \\big) ^{-1} - 1 $. \n\n\\item Define $ \\Z_{\\mt}^{(n)} \\defining \\dfrac{1}{\\X_{\\mt}^{(n)} }$, since $ \\big(v_{\\mt}^{(n)}: n\\in \\N \\big) $ and $ \\big(\\X_{\\mt}^{(n)}: n\\in \\N\\big) $ are independent, it holds that \n\\begin{equation}\\label{Eq Harmonic Mean Comments}\n\\rho_{\\mt}^{(n)} = \\frac{v_{\\mt}^{(n)} - \\X_{\\mt}^{(n)} }{ \\X_{\\mt}^{(n)} } \n= \\frac{v_{\\mt}^{(n)} }{ \\X_{\\mt}^{(n)} } - 1\n= v_{\\mt}^{(n)}\\Z_{\\mt}^{(n)} - 1 \n\\xrightarrow[n\\,\\rightarrow \\,\\infty]{\\text{Ces\\`aro}} \n\\Exp\\big(v_{\\mt}^{(1)}\\big) \\Exp\\big(\\Z_{\\mt}^{(n)} \\big) - 1\n= \\Exp\\big(v_{\\mt}^{(1)}\\big) \\Exp\\Big(\\frac{1}{\\X_{\\mt}^{(n)} } \\Big) - 1.\n\\end{equation}\nThe right hand side of the expression above involves the reciprocal of the harmonic mean of the variable $ \\big(\\X_{\\mt}^{(n)}: n\\in \\N\\big) $. Clearly, $ \\big(\\gamma_{\\mt}^{(n)}: n\\in \\N\\big) $ and $ \\big(\\rho_{\\mt}^{(n)}: n\\in \\N\\big) $ converge (in the C\\`esaro sense) to different limits. Unfortunately, the harmonic mean has no simple expression equivalent to that of Equation \\eqref{Eq Expectation of random student assignment} for the arithmetic mean. Consequently, it can be handled only numerically; this will be done in the next section. \n\\end{enumerate}\n\\end{remark}\n\\subsection{The Monte Carlo Simulation Algorithm and Numerical Results}\\label{Sec Numerical Simulations}\nThe randomization of the variables as well as its normalization discussed in the sections \\ref{Sec Randomization of variables} and \\ref{Sec Normalization of the method} respectively are summarized in the pseudocode \\ref{Alg Monte Carlo Analytica Omega} below. A particular example of the Monte Carlo simulation results is depicted in \\textsc{Figure} \\ref{Fig Asymptotic Enhancement}, while the corresponding body\/composition of enrolled students displayed presented in \\textsc{Table} \\ref{Tb Sections Realization Example}. \n\nThe results of several simulations for the Differential Calculus course are summarized in \\textsc{Table} \\ref{Tb Monte Carlo Experiments Summary}. Out several experiments, it is observed that a reasonable level of convergence of the Ces\\`aro means is attained above 800 iterations. Given that we are simulating the behavior of a highly complex random process, it is clear that no convergence rate can actually be concluded, the threshold for which the Ces\\`aro mean stabilizes shifts significantly from one experiment to the other. This is because every experiment defines a number of sections $ \\upns $, an enrollment body\/composition of students as in \\textsc{Table} \\ref{Tb Sections Realization Example}, a group matrix assignment $ G $ and a number of tenured lecturers $ \\upnt $, from here, the iteration process begins as it is shown in \\textsc{Algorithm} \\ref{Alg Monte Carlo Analytica Omega}. Therefore, the starting triple $ ( \\upns, G, \\upnt ) $ changes substantially between simulations as it can be seen in \\textsc{Table} \\ref{Tb Monte Carlo Experiments Summary}. These changes become even more dramatical when shifting from one course to another, as it is the case of \\textsc{Table} \\ref{Tb Monte Carlo Experiments All Courses Summary}, reporting the algorithm's performance for all the remaining seven service courses.\n\nIt is also important to observe the difference between Random (\\textsc{Definition} \\ref{Def Normalization Methods} \\ref{Mthd Random Normalization}) vs. Expected (\\textsc{Definition} \\ref{Def Normalization Methods} \\ref{Mthd Expected Normalization}) normalization methods. It is not significant in the simulation of the method's performance (see Figure \\ref{Fig Asymptotic Enhancement} (a) and (b)) and it is negligible in the behavior of their corresponding Ces\\`aro means, i.e., regardless of the chosen normalization method (see figure Figure \\ref{Fig Asymptotic Enhancement} (c) and (d)), the asymptotic behavior difference is negligible at least, from the numerical point of view. The latter can be also observed on the Tables \\ref{Tb Monte Carlo Experiments Summary} and \\ref{Tb Monte Carlo Experiments All Courses Summary}.\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{p{1.5cm}|p{2.0cm}p{2.0cm}p{2.0cm}p{2.0cm}p{1.2cm}p{1.2cm}p{1.2cm} }\n \\hline\n \\rowcolor{gray!80}\nExperiment\n&\n\\multicolumn{2}{c}{Random Normalization, $ \\rho_{\\mt} $ } &\n\\multicolumn{2}{c}{Expected Normalization, $ \\gamma_{\\mt} $ } \n& Enrollment \n& Sections\n& Lecturers\\\\\n\\rowcolor{gray!80}\nNumber\n& $ 100 \\times \\dfrac{v_{\\ia} }{ \\rho_{\\ia} } $\n& $ 100 \\times \\dfrac{v_{\\sa} }{ \\rho_{\\sa} } $\n& $ 100 \\times \\dfrac{v_{\\ia} }{ \\gamma_{\\ia} } $\n& $ 100 \\times \\dfrac{v_{\\sa} }{ \\gamma_{\\sa} } $\n& $ \\upns $ \n& $ \\sum\\limits_{K\\,\\in\\, \\mathcal{I}} s_{K} $\n& $ \\upnt $\n \\\\[5pt]\n\n\\hline\n1 &\n0.3097 &\t2.9059 &\t0.3056 &\t2.9044 &\t1355\t &15\t & 6 \\\\\n2 &\n0.4595 &\t3.2880 & 0.4588 & 3.2854\t & 1445 &\t14 & 8 \\\\\n3 &\n0.4373 & 3.2655 & 0.4414 & 3.2653 & 1225 & 14 & 7 \\\\\n4 &\n0.4158 &\t3.2689 &\t0.4130 &\t3.2663 &\t1456\t &15\t& 7 \\\\\n5 &\n0.4357 &\t2.9680 & 0.4315 &\t2.9651 &\t1296\t & 14 & 6 \\\\\n6 &\n0.4943 &\t3.1690 &\t0.5053 &\t3.1697 &\t1547\t & 15 & 8 \\\\\n7 &\n0.5099 &\t3.4486 &\t0.5008 &\t3.4439 &\t1556 & 16\t & 8 \\\\\n8 &\n0.4937 & 3.3720 & 0.4841 & 3.3666 & 1532\t& 16 & 8 \\\\\n9 & \n0.4080 & 3.1254 & 0.4009 & 3.1301 & 1444 & 15\t& 7 \\\\\n10 &\n0.4843 & 3.4498 & 0.4807 & 3.4454 & 1546 & 16 & 8 \\\\\n\\rowcolor{gray!80}\nMean &\n0.4448 &\t3.2261 &\t0.4422 &\t3.2242 & 1440.2 & 15.0 & 7.3\n \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{Monte Carlo Simulations Summary. The table shows a summary of the Monte Carlo Simulations with 10 experiments and 800 iterations each, for the \\textbf{Differential Calculus} course.\n}\n\\label{Tb Monte Carlo Experiments Summary}\n\\end{table}\n\n\\begin{algorithm}[H]\n\t\\KwData{\n\t\tDatabase: \\textit{Assembled\\_Data.csv}\n\t\tAnalyzed Course: DC, IC, ..., NM.\\\\\n\t\tOptimization Method: $ \\mt \\in \\{\\ia, \\sa\\} $. \\\\\n\t\t$ \\upnt $ random variable distribution \\\\\n\t\tNumer of Iterations: $ NI $}\n\t\\KwResult{Table of Relative Enhancement Values $ \\rho_{\\mt} $, $ \\gamma_{\\mt} $ for chosen method, course and academic performance variable.\n\t}\n\t\\textbf{Initialization}\\;\n\t\\textbf{call} Algorithm \\ref{Alg Random Setting MC Analytica Omega}\\;\n\t$ \\mathit{nt} \\leftarrow $\\textbf{compute} a realization of $ \\upnt $\\;\n\t\\textbf{call} Algorithm \\ref{Alg Instructors' performance}, \\textbf{input}: (\\textit{Assembled\\_Data.csv},\t\\textit{Analyzed Course}, $ \\apv $, Group Segmentation $ \\big( I_{\\ell}: \\ell \\in [L] \\big) $)\\;\n\t\\For{ \\text{iteration} $ \\in [NI] $ }{\n\t\t $ \\mathit{list} \\leftarrow $ \\textbf{compute} a random list of $ nt $-lecturers\\;\n\t\t\\textbf{call} RandInputAlgorithm \\ref{Alg Optimization Algoritm}, \\textbf{input}: \n\t\t(\\underline{Group Assignment Matrix \\textit{G} , List of Lecturers \\textit{L\\_list} }, \\textit{Analyzed Course}, $ \\apv $, Group Segmentation $ \\big( I_{\\ell}: \\ell \\in [L] \\big) $, $\\mt$ )\\;\n\t\t\n\t\tAPV\\_mt\\_Assessment[\\textit{iteration}]$ \\leftarrow \\big[ \\rho_{\\mt}, \\gamma_{\\mt} \\big] $. \t\t\n\t}\n\t\\caption{Monte Carlo Simulation Algorithm}\n\t\\label{Alg Monte Carlo Analytica Omega}\n\\end{algorithm}\n\\begin{figure}[h!]\n \\centering\n \\begin{subfigure}[Example DC. Enhancement Results Pass Rate Monte Carlo Simulation. Instructor Assignment Method ($ \\ia $). ]\n \n{\\includegraphics[scale = 0.38]{CD_Simulation_IA.eps} } \n \\end{subfigure}\n \n ~\n \n \\begin{subfigure}[Example DC. Enhancement Results Pass Rate Monte Carlo Simulation. Student Assignment Method ($ \\sa $). ]\n \n{\\includegraphics[scale = 0.38]{CD_Simulation_SA.eps} } \n \\end{subfigure}\n \n \\begin{subfigure}[Example DC. Ces\\`aro Means Enhancement Results Pass Rate Monte Carlo Simulation. Instructor Assignment Method ($ \\ia $). ]\n \n{\\includegraphics[scale = 0.38]{CD_Cesaro_Simulation_IA.eps} } \n \\end{subfigure}\n \n ~\n \n \\begin{subfigure}[Example DC. Ces\\`aro Means Enhancement Results Pass Rate Monte Carlo Simulation. Student Assignment Method ($ \\sa $). ]\n \n{\\includegraphics[scale = 0.38]{CD_Cesaro_Simulation_SA.eps} } \n \\end{subfigure}\n \n \n\\caption{Example: \\textbf{Differential Calculus} course.\nEnrollment of 1441 Students, 15 Sections, 8 Tenured Lecturers, 100 Iterations. All figures display the normalization $ \\rho_{\\mt} $ vs $ \\gamma_{\\mt} $ for $ \\mt \\in \\{ \\ia, \\sa \\} $. \n\\label{Fig Asymptotic Enhancement} }\n\\end{figure}\n\\begin{table}[h!]\n\n\t\\def1.4{1.4}\n\t\\scriptsize{\n\t\t\\begin{center}\n\t\t\\rowcolors{2}{gray!25}{white}\n\t\t\t\\begin{tabular}{ c | c c c c c c c c c c | c }\n\t\t\t\t\\hline\n\t\t\t\t\\rowcolor{gray!80}\n\t\t\t\t\\tiny{\n\t\t\t\t\\diagbox\n\t\t\t\t{SECTION}{SEGMENT} }\n\t\t\t\t& \\tiny{[0, 2.2]}\n\t\t\t\t& \\tiny{(2.2, 2.7]}\n\t\t\t\t& \\tiny{(2.7, 3.0]}\n\t\t\t\t& \\tiny{(3.0, 3.1]}\n\t\t\t\t& \\tiny{(3.1, 3.3]}\n\t\t\t\t& \\tiny{(3.3, 3.5]}\n\t\t\t\t& \\tiny{(3.5, 3.7]}\n\t\t\t\t& \\tiny{(3.7, 3.8]}\n\t\t\t\t& \\tiny{(3.8, 4.1]}\n\t\t\t\t& \\tiny{(4.1, 5.0]} \n\t\t\t\t& Total \\\\\n\t\t\t\t\\hline\n\t\t\t\t1&\n\t\t\t\t6 &\t1 &\t15 &\t5 &\t13 &\t8 &\t8 &\t4 &\t8 &\t6 &\t74 \\\\\n\t\t\t\t2 &\n\t\t\t\t10 & 6 & 14 &\t3 &\t5 &\t7\t& 11 &\t5 & 8 &\t5 & 74 \\\\\n\t\t\t\t3 &\n\t\t\t\t9 & 9\t& 9\t & 8 &\t6 &\t3 &\t9 &\t1 &\t15 &\t5 &\t74 \\\\\n\t\t\t\t4 &\n\t\t\t\t11 & 12\t& 13 &\t1 &\t5 & 3 &\t8 &\t7 &\t8 &\t6 &\t74 \\\\\n\t\t\t\t5 &\n\t\t\t\t5 &\t9 &\t12\t& 7\t& 11 &\t6 &\t9 &\t2 &\t5 &\t9 & 75 \\\\\n\t\t\t\t6 &\n\t\t\t\t9 & 7 &\t6 &\t3 &\t13 & 11 & 11 &\t1 &\t7 &\t6 &\t74 \\\\\n\t\t\t\t7 &\n\t\t\t\t12 & 5 & 7 & 2 & 7 & 10 & 12 &\t6 &\t21 & 7 & 89 \\\\\n\t\t\t\t8 &\n\t\t\t\t7 &\t11 & 11 & 2 & 6 & 12 & 14 &\t8 &\t9 &\t9 &\t89 \\\\\n\t\t\t\t9 &\n\t\t\t\t11 & 18 & 14 &\t7 & 6 &\t12 & 10 & 7 & 10 &\t9 &\t104 \\\\\n\t\t\t\t10 &\n\t\t\t\t7 &\t9 &\t15 & 7 & 14 & 14 & 16 &\t8 &\t8 &\t6 &\t104 \\\\\n\t\t\t\t11 &\n\t\t\t\t14\t& 8\t& 20 &\t4 &\t15 & 17 & 14 &\t0 & 13 & 14 & 119 \\\\\n\t\t\t\t12 &\n\t\t\t\t7 &\t11 & 14 & 6\t& 13 &\t21 & 14 & 4\t& 18 & 11 &\t119 \\\\\n\t\t\t\t13 &\n\t\t\t\t15\t& 15 &\t15 & 9\t& 15 &\t16 & 11 & 3 & 11 &\t9 &\t119 \\\\\n\t\t\t\t14 &\n\t\t\t\t14\t& 14 &\t20 & 5\t& 17 & 15 &\t12 & 4 & 9 & 9 & 119 \\\\\n\t\t\t\t16 &\n\t\t\t\t15 & 13 & 25 &\t6 &\t15 & 16\t& 10 &\t9\t& 17 &\t8 & 134 \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\rowcolor{gray!80}\n\t\t\t\tTotal &\n\t\t\t\t152\t& 148 & 210\t& 75 & 161 & 171 & 169 & 69\t& 167 &\t119 & 1441\n\t\t\t\t\\\\\n\t\t\t\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t}\n\t%\n\\caption{An example of a random realization of Algorithm \\ref{Alg Random Setting MC Analytica Omega}, i.e. a group matrix assignment $ G $ and a number of tenured lecturers $ \\upnt $ for the \\textbf{Differential Calculus} course.\n}\\label{Tb Sections Realization Example}\n\\end{table}\n\\begin{table}[h!]\n\\def1.4{1.4}\n\\small{\n\\begin{center}\n\\rowcolors{2}{gray!25}{white}\n\\begin{tabular}{p{1.5cm}|p{2.0cm}p{2.0cm}p{2.0cm}p{2.0cm}p{1.2cm}p{1.2cm}p{1.2cm} }\n \\hline\n \\rowcolor{gray!80}\nCourse\n&\n\\multicolumn{2}{c}{Random Normalization, $ \\rho_{\\mt} $ } &\n\\multicolumn{2}{c}{Expected Normalization, $ \\gamma_{\\mt} $ } \n& Enrollment \n& Sections\n& Lecturers\\\\\n\\rowcolor{gray!80}\n\n& $ 100 \\times \\dfrac{v_{\\ia} }{ \\rho_{\\ia} } $\n& $ 100 \\times \\dfrac{v_{\\sa} }{ \\rho_{\\sa} } $\n& $ 100 \\times \\dfrac{v_{\\ia} }{ \\gamma_{\\ia} } $\n& $ 100 \\times \\dfrac{v_{\\sa} }{ \\gamma_{\\sa} } $\n& $ \\upns $ \n& $ \\sum\\limits_{K\\,\\in\\, \\mathcal{I}} s_{K} $\n& $ \\upnt $\n \\\\[5pt]\n\n\\hline\nDC &\n0.4448 &\t3.2261 &\t0.4422 &\t3.2242 & 1440.2 & 15.0 & 7.3 \\\\\nIC\t& 0.3267\t& 2.8094\t& 0.3196\t& 2.8104\t& 1068.0\t& 8.2\t& 5.1 \\\\\nVC\t& 0.1684\t& 1.9797\t& 0.1800\t& 1.9923\t& 586.6\t& 4.1\t& 2.4 \\\\\nVAG\t& 0.4070\t& 3.1366\t& 0.4079\t& 3.1474\t& 1080.8\t& 14.5\t& 6.2 \\\\\nLA\t& 0.2131\t& 3.2906\t& 0.2009\t& 3.2788\t& 1078.2\t& 8.0\t& 4.2 \\\\\nODE\t& 0.4323\t& 5.6270\t& 0.4269\t& 5.6332\t& 798.2\t& 6.4\t& 3.1 \\\\\nBM\t& 0.5706\t& 3.0775\t& 0.5909\t& 3.0791\t& 910.9\t& 11.1\t& 3.0 \\\\\nNM\t& 0.3750\t& 3.3825\t& 0.3405\t& 3.3920\t& 263.1\t& 2.3\t& 1.7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n}\n\\caption{A summary of the Monte Carlo Simulations with 800 iterations for each course.\n}\\label{Tb Monte Carlo Experiments All Courses Summary}\n\\end{table}\n\\section{Conclusions and Future Work}\\label{Sec Conclusions}\nThe present work delivers several conclusions. \n\\begin{enumerate}[(i)]\n\\item A method has been implemented aimed to increase the academic performance for massive university lower division courses in mathematics. It is based on integer programming and big data analysis to compute the associated cost functions, while the constraints (such as the number of sections and corresponding capacities) are defined by administrative sources. The integer programs come from two mechanisms: assign instructors optimally ($ \\ia$ method) or assign students optimally ($ \\sa $ method). \n\n\\item The academic performance was explored using two measures; Pass Rate and Grade. After correlation analysis of the data, it is determined that the one relevant factor, known at the time when the semester begins and incident on these statistical variables is the $ \\gpa $. Consequently the profiling of students as well as the student body composition is defined in terms of the $ \\gpa $ (see \\textsc{Table} \\ref{Tb Historical Enhnacements})\n\n\\item The historical assessment of the method yields poor enhancement levels for the Grade variable, due to its typical statistical robustness. However, the Pass Rate yields more satisfactory results; good enough to pursue a deeper analysis such as the method's randomization and its asymptotic assessment, presented in \\textsc{Section} \\ref{Sec Randomization and Asymptotic Assesment}.\n\n\\item The asymptotic analysis of the algorithm is done by randomizing the enrollment population and the administrative factors, statistically based on the empirical observations reported in the database \\textit{Assembled\\_Data.csv}. The Monte Carlo experiments establish that the method does not deliver a fixed value of relative enhancement, it depends on the starting parameters $ \\big(\\upns, G, \\upnt\\big) $ whose remarkable randomness inherit uncertainty to the algorithm's output values. \n\n\\item Computing a weighted average across the courses by crossing the tables \\ref{Tb Enrollement Random Variable} and \\ref{Tb Monte Carlo Experiments All Courses Summary}, gives a rough estimate of 3.3 percent full scale benefit, if the students assignment method ($ \\sa $) is implemented. This is approximately 240 extra students per semester passing their respective courses which, in the long run represent a significant gain for the Institution. \n\n\\item A 3.3 \\% enhancement for the method's benefit may seem low at first sight. However, it is important to stress that this enhancement corresponds to a detailed treatment of the tenured lecturers only, while the adjunct lecturers are treated in general terms because of insufficient data as they are unstable personnel. Tenured lecturers represent a fraction of less than 50 percent from the involved faculty team as \\textsc{Table} \\ref{Tb Monte Carlo Experiments All Courses Summary} shows. Consequently, should the stable personnel fraction increase, the method would deliver more accurate and perhaps more optimistic results. \n\n\\item The algorithms \\ref{Alg Analytica Omega} and \\ref{Alg Monte Carlo Analytica Omega}, could have been adjusted to keep only the sections with tenured lecturers. However, the Authors chose to discard this artificial setting because it is biased with respect to the study case.\n\n\\item It is the perception of the Authors that no general conclusions can be derived for the method's enhancement level. On one hand it is sufficiently general and flexible to be implemented at any Institution with massive courses and therefore big databases available. On the other hand, the experiments performed in the present work, suggest that its effectiveness needs to be evaluated on a case-wise basis.\n\n\\item Considering age as a factor is also possible by merely applying the segmentation process described in Section \\ref{Sec Segmentation Process} (\\textsc{Algorithm} \\ref{Alg Segmentation of Students} with an adequate number of segmentation intervals $ \\big( \\widetilde{I}_{\\ell}: \\ell \\in [\\widetilde{L}]\\big) $). First, computing the lecturers performance conditioned to the \\textit{Age} variable as in \\textsc{Section} \\ref{Sec Computation Lecturer Performance} (\\textsc{Algorithm} \\ref{Alg Instructors' performance}, output $ T_{\\textit{Age}} $). Second, weighting its impact according to the correlation values, namely the costs table in \\textsc{Equation} \\eqref{Eq Costs Table APV} can be modified as\n\\begin{equation}\\label{Eq Costs Table Age and APV}\nC = \\frac{4}{5} \\, T_{\\textit{APV} } G + \\frac{1}{5}T_{\\textit{Age}} \\widetilde{G}.\n\\end{equation}\nHere it is understood that the group matrix $ \\widetilde{G} $ is constructed according to the \\textit{Age} variable segmentation $ \\big( \\widetilde{I}_{\\ell}: \\ell \\in [\\widetilde{L}]\\big) $. The weighting coefficients were proposed, according to the correlation with the \\textit{Grade} variable reported in \\textsc{Table} \\ref{Tb Correlations DC}: \\textit{Age}: 0.2, $ \\gpa $: 0.8, i.e., the second is 4 times the first one (see \\cite{DeGiorgiPellizaariWoolstonGui} for further discussion on these type of models). Yet again, the flexibility of the method, allows to introduce in the same fashion any number of variables fitting to the case at hand.\n\n\\item This paper has worked two methods, assign instructors while keeping the students fixed ($ \\ia $) and assign students while keeping the instructors fixed ($ \\sa $), both of them come down to a linear optimization problem, \\ref{Pblm Instructors Assignment} and \\ref{Pblm Students Assignment Problem} respectively. However, moving both instructors and students simultaneously is no longer a linear, but a bilinear optimization question (see \\cite{Orlov, CapraraMonaci}). This view will be further explored in future work. \n\n\\item So far, the present work assumed that allocating students and\/or instructors is a decision centralized by the administrative departments of the analyzed Institution. However, in our study case, student location is decided differently, using a $ \\gpa $ competition-based mechanism to assign priority starting from the highest to the lowest scorers. This competitive scenario is better modeled using game theory which will be explored in future work. \n\n\\item The algorithm presented in this work offers a mechanism for higher education institutions to help their students improving their pass rates and grades. This is done by solving two different social welfare schedules ($ \\ia $ and $ \\sa $ methods). The work also provides a way to measure (and monitor) how far from the Pareto equilibrium is an Institution at a given time. As mentioned in the economic justification (Subsection \\ref{economic_justification}), this is particularly relevant in countries like Colombia where the drop out rates from college are high.\n\\end{enumerate}\n\\section{Acknowledgements}\nThe Authors wish to thank Universidad Nacional de Colombia, Sede Medell\\'in for its support in the production of this work, in particular, to the Academic Director of the University for allowing access to their databases for this study. The first author was supported by grant Hermes 45713 from Universidad Nacional de Colombia, Sede Medell\\'in. \n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nNumerical approximations for infinite-dimensional SEEs with superlinearly growing nonlinearities have been intensively studied in the literature \n(cf., e.g., \\cite{j08b,Doersek2012,KamraniBloemker2017,Kamrani2015,HairerMatetski2016,CoxWelti2016,YangZhang2017,CuiHongLiu2017,CuiHongLiuZhou2017} and the references mentioned therein).\nIn applications one is often interested in statistical quantities of the solution process of the considered SEE and, in view of this, one is especially interested in strong and weak numerical approximations for the considered SEE (cf., e.g., Heinrich~\\cite{Heinrich1998,Heinrich2001}, Giles~\\cite{Giles2008}, and Creutzig et al.~\\cite{Creutzig2009}).\nIt has been established in the literature that the linear-implicit Euler scheme, the explicit Euler scheme,\nand the exponential Euler scheme\nconverge, in general, neither numerically weakly nor strongly \nin the case of such SEEs; cf., e.g., Hutzenthaler, Jentzen, \\& Kloeden~\\cite[Theorem~2.1]{hjk11} and Hutzenthaler, Jentzen, \\& Kloeden~\\cite[Theorem~2.1]{HutzenthalerJentzenKloeden2013}. Fully drift-implicit Euler schemes, in contrast, do converge strongly in the case of several SEEs with superlinearly growing nonlinearities;\ncf., e.g.,\nHu~\\cite[Theorem~2.4]{Hu1996}\nand\nHigham, Mao, \\& Stuart~\\cite[Theorem~3.3]{Higham2002}\nfor finite-dimensional SEEs\nand cf., e.g.,\nGy{\\\"o}ngy \\& Millet~\\cite[Theorem~2.10]{gm05}, Brze{\\'z}niak, Carelli, \\& Prohl~\\cite[Theorem~7.1]{Brzezniak2013}, Kov{\\'a}cs, Larsson, \\& Lindgren~\\cite[Theorem~1.1]{kll2015},\nand\nFurihata et al.~\\cite[Theorem~5.4]{FurihataKovacsLarssonLindgren2016}\nfor infinite-dimensional SEEs.\nIn order to implement these schemes, a nonlinear equation has to be solved approximatively in each time step and this results in additional computational effort (see, e.g., Hutzenthaler, Jentzen, \\& Kloeden~\\cite[Figure~4]{HutzenthalerJentzenKloeden2012}). Moreover, it has not yet been shown in the literature that these approximate implementations of fully drift-implicit Euler methods converge strongly. Lately, a series of appropriately modified versions of the explicit Euler method has been introduced and proven to converge strongly for some SEEs with superlinearly growing nonlinearities; cf., e.g., Hutzenthaler, Jentzen, \\& Kloeden~\\cite{HutzenthalerJentzenKloeden2012}, Wang \\& Gan~\\cite{WangGan2013},\nHutzenthaler \\& Jentzen~\\cite{Hutzenthaler2015}, Tretyakov \\& Zhang~\\cite{TretyakovZhang2013},\nand Sabanis~\\cite{Sabanis2013,Sabanis2013E}\nfor finite-dimensional SEEs\nand cf., e.g.,\nGy{\\\"o}ngy, Sabanis, \\& {\\v{S}}i{\\v{s}}ka~\\cite{GoengySabanisS2015},\nJentzen \\& Pu{\\v s}nik~\\cite{jp2015}, Becker \\& Jentzen~\\cite{BeckerJentzen2016}, and Hutzenthaler et al.~\\cite{Salimova2016}\nfor infinite-dimensional SEEs.\nThese modified versions are easily realizable, explicit, and truncate\/tame superlinearly growing nonlinearities in order to prevent strong divergence. \nHowever, except for Becker \\& Jentzen~\\cite{BeckerJentzen2016} and Hutzenthaler et al.~\\cite{Salimova2016}, each of the above mentioned \n strong convergence results applies\nonly to trace class noise driven SEEs and excludes SEEs driven by the more irregular space-time white noise.\nIn \\cite{BeckerJentzen2016} a coercivity\/Lyapunov-type condition\nhas been employed to obtain\nstrong convergence for stochastic Allen-Cahn equations\nwith additive space-time white noise; \ncf.\\ \\cite[(85), Lemma~6.2, and Corollaries~6.16--6.17]{BeckerJentzen2016}.\nHowever, the machinery in \\cite{BeckerJentzen2016}\nassumes the coercivity\/Lyapunov-type coefficient in the coercivity\/Lyapunov-type condition to be a constant\n(cf.\\ \\cite[(85)]{BeckerJentzen2016} with \\eqref{eq:coer:burgers} below)\nand,\ntherefore, applies only to temporal semi-discrete approximation methods for stochastic Allen-Cahn equations \nbut excludes a series of important additive space-time white noise driven \nSEEs \nsuch as stochastic Burgers equations with space-time white noise. The approach in Hutzenthaler et al.~\\cite{Salimova2016}, in turn, does not require the coercivity\/Lyapunov-type coefficient to be a constant and allows it to be a function of the noise process. Nevertheless, the article~\\cite{Salimova2016} imposes some serious restrictions on the coercivity\/Lyapunov-type coefficient, which are satisfied in the case of stochastic Kuramoto-Sivashinsky equations (see, e.g., \\cite[Lemma~5.2 and Theorem~4.6]{Salimova2016}) but not in the case of stochastic Burgers equations (see, e.g., Lemma~\\ref{coer:burgers} below). More precisely, the composition of the coercivity\/Lyapunov-type coefficient and the driving Ornstein-Uhlenbeck process needs to admit suitable exponential integrability properties (cf., e.g., \\cite[Theorem~4.6 and Corollary~5.8]{Salimova2016}) because the coercivity\/Lyapunov-type coefficient is employed in a Gronwall-type argument (cf., e.g., \\cite[Corollary~3.2 and Corollary~2.6]{Salimova2016}), which, in turn, requires a suitable exponential term to be integrable.\nTo the best of our knowledge, there exists neither a strong nor a weak temporal numerical approximation result for stochastic Burgers equations with space-time white noise in the \nscientific literature. It is a key contribution of this paper to relax the restrictions on the coercivity\/Lyapunov-type coefficient in \\cite{BeckerJentzen2016} and \\cite{Salimova2016} so that strong convergence for numerical approximations for stochastic Burgers equations with space-time white noise can be achieved.\nIn order to obtain a strong convergence result for stochastic Burgers equations, we prove that suitable exponential integrability properties of the composition of the coercivity\/Lyapunov-type coefficient and a transformed driving Ornstein-Uhlenbeck process also yield strong convergence (see Theorem~\\ref{thm:strong} and Proposition~\\ref{prop:exists} below). Additional important ingredients in the proof of the convergence result are transformations of semigroups for solutions of SPDEs (see Proposition~\\ref{prop:transform_SG} below) and Fernique's theorem (see, e.g., Proposition~\\ref{thm:fernique} below). \n\nTo illustrate the main result of this paper (see Theorem~\\ref{thm:strong} below) we specialize it to the case of stochastic Burgers equations. This is the subject of the following theorem. \n\n\n\n\n\n\\begin{theorem}\n\t\\label{thm:intro}\nLet $ T \\in (0,\\infty)$, \n$\\varrho \\in (\\nicefrac{1}{8}, \\nicefrac{1}{4})$,\n$ \\chi \\in (0, \\nicefrac{\\varrho }{2 } - \\nicefrac{1}{16}] $, $ H = L^2((0,1); \\mathbb{R})$, \nlet $ A \\colon D(A) \\subseteq H \\to H $ be the Laplace operator with Dirichlet boundary conditions on $H$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $, be a family of interpolation spaces associated to $ -A $,\nlet\n$\\xi \\in H_{\\nicefrac{1}{2}} $,\nlet\n$F \\colon H_{\\nicefrac{1}{8}} \\to H_{-\\nicefrac{1}{2}} $,\n$(e_n)_{n \\in \\mathbb{N} = \\{1,2,3, \\ldots\\}} \\colon \\mathbb{N} \\to H $,\n$(P_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H) $,\nand\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $\nbe functions which satisfy\nfor all $v \\in H_{\\nicefrac{1}{8}}$, $n \\in \\mathbb{N}$ that\n$F(v)= -\\frac{1}{2}(v^2)'$,\n$ e_n = (\\sqrt{2} \\sin(n \\pi x) )_{x \\in (0,1)}$,\n$ P_n(v) = \\sum_{k = 1 }^n \\langle e_k, v \\rangle_H e_k $,\nand\n$ \\limsup_{ m \\to \\infty} h_m =0$,\nlet $ ( \\Omega, \\F, \\P ) $ be a probability space,\nlet $(W_t)_{t \\in [0, T]}$ be an $\\mathrm{Id}_H$-cylindrical $( \\Omega, \\F, \\P )$-Wiener process,\nlet $ \\mathcal{X}^n, \\mathcal{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $ n \\in \\mathbb{N}$, be stochastic processes, and assume that for all $ n \\in \\mathbb{N} $, $t \\in [0, T]$ it holds $\\P$-a.s.~that $\\mathcal{O}_t^n = \\int_0^t P_n \\, e^{(t-s)A} \\, dW_s$\nand\n\\begin{equation}\n\\label{eq:intro}\n \\mathcal{X}_t^n = P_n \\, e^{ t A } \\xi + \\int_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds + \\mathcal{O}_t^n. \n\\end{equation} Then\n\\begin{enumerate}[(i)]\n\t\\item \\label{item:intro:exists} there exists an up-to-indistinguishability unique stochastic process $ X\\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ with continuous sample paths which satisfies that for all $t \\in [0,T]$ it holds $\\P$-a.s.\\ that \n\t\\begin{equation}\n\tX_t = e^{ t A } \\xi + \\int_0^t e^{ ( t - s ) A} \\, F ( X_s ) \\, ds + \\int_0^t e^{(t-s)A} \\, dW_s\n\t\\end{equation}\nand \n\t\\item \\label{item:intro:conv} it holds for all $p \\in (0, \\infty)$ that\n\t\\begin{align}\n\t\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[ \\| X_t -\\mathcal{X}_t^n \\|_H^p \\big] = 0.\n\t\\end{align} \n\\end{enumerate}\n\n\n\n \n\\end{theorem}\n\n\nIn the framework of Theorem~\\ref{thm:intro} we note that the stochastic process $ X $ in \\eqref{item:intro:exists} in Theorem~\\ref{thm:intro} is a mild solution process of the stochastic Burgers equation\n\\begin{align}\n\\tfrac{\\partial}{\\partial t} X_t(x) = \\tfrac{\\partial^2}{\\partial x^2} X_t(x) - X_t(x) \\cdot \\tfrac{\\partial}{\\partial x} X_t(x) + \\tfrac{\\partial}{\\partial t} W_t(x)\n\\end{align}\nwith $X_0(x) = \\xi(x)$ and $X_t(0)= X_t(1)=0$\nfor $t \\in [0,T]$, $x \\in (0,1)$.\nObserve that~\\eqref{item:intro:exists} in Theorem~\\ref{thm:intro} follows, e.g., from Bl\\\"omker \\& Jentzen~\\cite[Theorem~3.1 and Subsection~4.3]{BloemkerJentzen2013}, while~\\eqref{item:intro:conv} in Theorem~\\ref{thm:intro} is an immediate consequence of Corollary~\\ref{cor:burgers:short} below. \nMoreover, we note that the scheme proposed in~\\eqref{eq:intro} is a modified version of the scheme proposed in \\cite{Salimova2016} for stochastic Kuramoto-Sivashinsky equations (cf. \\cite[(90)]{Salimova2016} with \\eqref{eq:intro} above).\n\n\n\nThe remainder of this paper is organized as follows. In Section~\\ref{sec:a_priori} the required a priori moment bounds for the proposed scheme are established. In Section~\\ref{sec:main} the error analysis is first performed in the pathwise sense to obtain pathwise convergence in Proposition~\\ref{prop:main_det} and pathwise a priori bounds in Proposition~\\ref{prop:main_det:2}. Combining these pathwise results allows\nus to accomplish strong convergence in Theorem~\\ref{thm:strong} for a large class of SEEs on general separable $\\mathbb{R}$-Hilbert spaces. In Section~\\ref{sec:abstract} we verify the assumptions of Theorem~\\ref{thm:strong} in the case of more concrete SPDEs on the Hilbert space $L^2((0,1);\\mathbb{R})$ and we prove strong convergence in Proposition~\\ref{abs:prop:last}. To derive Proposition~\\ref{abs:prop:last} we also employ the properties of stochastic convolution processes in Proposition~\\ref{prop:exists}, which are obtained using Fernique's theorem (see Section~\\ref{sec:fernique}). In Section~\\ref{sec:examples} we apply Proposition~\\ref{abs:prop:last} in the case of stochastic Burgers and stochastic Allen-Cahn equations and establish strong convergence in Corollary~\\ref{cor:burgers:short} and in Corollary~\\ref{cor:cahn:short}, respectively.\n\\subsection{Notation}\n\\label{sec:notation}\n\nThroughout this article the following notation is frequently used.\nFor every set $A$ we denote by $\\mathcal{P}(A)$ the power set of $A$,\nwe denote by $\\#_{A} \\in \\mathbb{N}_0 \\cup \\{\\infty\\}$ the number of elements of $A$,\nand we denote by $\\mathcal{P}_0(A)$ the set given by $\\mathcal{P}_0(A) = \\{ B \\in \\mathcal{P}(A) \\colon \\#_B < \\infty \\}$.\nFor all measurable spaces $( A, \\mathcal{A})$ and $( B, \\mathcal{B})$ we denote by $ \\mathcal{M}(\\mathcal{A}, \\mathcal{B})$ the set of all $\\mathcal{A} \\slash \\mathcal{B}$-measurable functions.\nFor every set $A \\in \\mathcal{B}(\\mathbb{R})$ we denote by $\\lambda_A \\colon \\mathcal{B}(A) \\to [0, \\infty]$ the Lebesgue-Borel measure on $ ( A, \\mathcal{B}(A) ) $.\nFor every measure space $(\\Omega, \\F, \\mu)$, every measurable space $(S, \\mathcal{S})$, every set $R$, and every function $f \\colon \\Omega \\to R$ we denote by $\\left[ f \\right]_{\\mu, \\mathcal{S}} $ the set given by $\\left[f \\right]_{\\mu, \\mathcal{S}} = \\left\\{ g \\in \\mathcal{M}(\\F, \\mathcal{S}) \\colon ( \\exists \\, A \\in \\F \\colon \\mu(A)=0 \\,\\, \\text{and} \\,\\, \\{ \\omega \\in \\Omega \\colon f(\\omega) \\neq g(\\omega)\\} \\subseteq A ) \\right\\}$.\nFor every $ h \\in (0, \\infty)$ we denote by $ \\lfloor \\cdot \\rfloor_h \\colon \\mathbb{R} \\to \\mathbb{R}$ the function which satisfies for all $t \\in \\mathbb{R}$ that $\\lfloor t \\rfloor_h = \\max( (-\\infty, t] \\cap \\{0, h, -h, 2h, -2h, \\ldots\\} )$.\nWe denote by $\\underline{(\\cdot)} \\colon \\bigl\\{ [v]_{\\lambda_{(0,1)}, \\mathcal{B}(\\mathbb{R})} \\in \\mathcal{P} \\bigl( \\mathcal{M}(\\mathcal{B}((0,1)), \\mathcal{B}(\\mathbb{R})) \\bigr)\\colon v \\in \\mathcal{C}( (0,1), \\mathbb{R} ) \\bigr\\} \\to \\mathcal{C}((0,1), \\mathbb{R})$ the function which satisfies for all $v \\in \\mathcal{C}( (0,1), \\mathbb{R} ) $ that $\\underline{[v]_{\\lambda_{(0,1)}, \\mathcal{B}(\\mathbb{R})}}=v$.\nFor all real numbers $\\theta \\in (0,1)$ and $p \\in [1, \\infty)$ we denote by $ \\left\\| \\cdot \\right\\|_{\\mathcal{W}^{\\theta, p}((0,1), \\mathbb{R})} \\colon \\mathcal{M}(\\mathcal{B}((0,1)) ,\\mathcal{B}(\\mathbb{R})) \\to [0,\\infty]$ the function which satisfies for all $v \\in \\mathcal{M}(\\mathcal{B}((0,1)) ,\\mathcal{B}(\\mathbb{R}))$ that\n\\begin{align}\n\t\\|v\\|_{\\mathcal{W}^{\\theta, p}((0,1), \\mathbb{R})} = \\left[ \\int_0^1 |v(x)|^p \\, dx + \\int_0^1 \\int_0^1 \\frac{|v(x)-v(y)|^p}{|x-y|^{1+ \\theta p}} \\, dx \\, dy\\right]^{\\nicefrac{1}{p}}.\n\\end{align}\n\n\n\n\n\\section{A priori bounds}\n\\label{sec:a_priori}\n\nIn Proposition~\\ref{prop:priori_bound} and Corollary~\\ref{cor:a_priori} below we establish general\na priori bounds which will be used in Section~\\ref{sec:main} to obtain a~priori bounds for the proposed approximation method.\nBefore we state Proposition~\\ref{prop:priori_bound} and Corollary~\\ref{cor:a_priori},\nwe present in five auxiliary lemmas, Lemmas~\\ref{lem:fund:calc}--\\ref{lemma:lyapunov3},\nelementary results which are used in the proofs of Proposition~\\ref{prop:priori_bound} and Corollary~\\ref{cor:a_priori}.\nIn particular,\nwe prove a~priori bounds for general Lyapunov-type functions\nin Lemmas~\\ref{lemma:lyapunov2}--\\ref{lemma:lyapunov3},\nwhereas in Proposition~\\ref{prop:priori_bound} and Corollary~\\ref{cor:a_priori}\nwe establish a priori bounds for a particular choice for the Lyapunov-type function.\n\n\n\n\n\n\\subsection{On strong and mild solutions of semilinear evolution equations}\n\\label{subsec:strong_mild}\n\n\n\n\n\n\n\n\n\n\nThe next elementary and well-known result, Lemma~\\ref{lem:fund:calc}, presents a version of the fundamental theorem of calculus and the chain rule.\nIt is employed in the proof of Lemma~\\ref{lemma:fund:gen} below.\n\n\n\\begin{lemma}\n\\label{lem:fund:calc}\nLet $(V, \\left\\| \\cdot \\right\\|_V)$ be a nontrivial $\\mathbb{R}$-Banach space, let $(W, \\left\\|\\cdot\\right\\|_W)$ be an $\\mathbb{R}$-Banach space, let $U \\subseteq V$ be an open set, let $f \\in \\mathcal{C}^1(U,W)$, $a \\in \\mathbb{R}$, $b \\in (a,\\infty)$,\nlet $ x \\colon [ a, b ] \\to U $ be a function,\nand let $y \\colon [a,b] \\to V$ be a strongly $\\mathcal{B}([a,b]) \\slash (V, \\left\\| \\cdot \\right\\|_V)$-measurable function which satisfies for all $t \\in [a,b]$ that\n $\\int_a^b \\|y_s\\|_V \\, ds < \\infty$ and \n\\begin{align}\nx_t = x_a + \\int_a^t y_s \\, ds.\n\\end{align}\nThen \n\\begin{enumerate}[(i)]\n\t\\item it holds that the function $[a,b] \\ni t \\mapsto f'(x_t) \\, y_t \\in W$ is strongly $\\mathcal{B}([a,b]) \\slash (W, \\left\\| \\cdot \\right\\|_W)$-measurable,\n\t\\item it holds that $\\int_a^b \\| f'(x_s) \\, y_s \\|_W \\, ds < \\infty$, and \n\t\\item it holds for all $t_0 \\in [a,b]$, $t \\in [t_0,b]$ that\n\t\\begin{align}\n\tf(x_t) = f(x_{t_0}) + \\int_{t_0}^t f'(x_s) \\, y_s \\, ds.\n\t\\end{align}\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\n\n\\begin{lemma}\n\\label{lemma:fund:gen}\nLet $(V, \\left\\|\\cdot \\right\\|_V)$ be a separable $\\mathbb{R}$-Banach space,\nlet $A \\in L(V)$, $T \\in (0, \\infty)$,\nlet $Y \\colon [0,T] \\to V $ be a function,\nand let $Z \\colon [0,T] \\to V $ be a $\\mathcal{B}([0,T]) $\/$ \\mathcal{B}(V) $-measurable function which satisfies for all $t \\in [0, T]$ that $\\sup_{s \\in [0,T]} \\|Z_s\\|_V <\\infty$ and $Y_t = \\int_0^t e^{(t-s)A} \\, Z_s \\, ds$. Then \n\\begin{enumerate}[(i)]\n\\item it holds that $Y$ is continuous and\n\\item it holds for all $t \\in [0,T]$ that $Y_t = \\int_0^t A Y_s + Z_s \\, ds$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:fund:gen}]\nFirst, note that for all $t \\in [0,T]$ it holds that $Y_t =e^{tA} \\int_0^t e^{-sA} \\, Z_s \\, ds$. The assumption that $ A \\in L(V)$ and the assumption that $\\sup_{s \\in [0,T]} \\|Z_s\\|_V <\\infty$ hence prove that $Y$ is continuous. Moreover, Lemma~\\ref{lem:fund:calc} (with $V= \\mathbb{R} \\times V$, $W = V$, $U= \\mathbb{R} \\times V$, $f= ( \\mathbb{R} \\times V \\ni (t,v) \\mapsto e^{tA} \\,v \\in V)$, $a=0$, $b=T$, $x = ([0,T] \\ni t \\mapsto (t, \\int_0^t e^{-sA} \\, Z_s \\, ds) \\in \\mathbb{R} \\times V)$, $ y = ([0,T] \\ni t \\mapsto (1, e^{-tA} \\, Z_t) \\in \\mathbb{R} \\times V)$, $t_0 =0$ in the notation of Lemma~\\ref{lem:fund:calc}) ensures for all $t \\in [0,T]$ that\n\\begin{align}\n\\begin{split}\nY_t = e^{tA} \\int_0^t e^{-sA} \\, Z_s \\, ds = \\int_0^t A e^{sA} \\int_0^s e^{-uA} \\, Z_u \\, du \\, ds+ \\int_0^t e^{sA} \\, e^{-sA} \\, Z_s \\, ds = \\int_0^t AY_s +Z_s \\, ds.\n\\end{split}\n\\end{align}\nThe proof of Lemma~\\ref{lemma:fund:gen} is thus completed. \n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:lyapunov1}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( V, \\left\\| \\cdot \\right\\|_V ) $ be a separable $\\mathbb{R}$-Banach space,\nlet $T \\in (0, \\infty)$, $\\eta \\in [0, \\infty)$, $ h \\in (0, T] $, $ A \\in L(V)$,\nand let $ Z \\colon [0, T] \\to \\mathbb{R} $,\n$Y, O, \\mathbb{O} \\colon [0, T] \\to V $,\nand $F \\colon V \\to V $\nbe functions which satisfy\nfor all $t \\in [0,T]$ that $\\eta O \\in \\mathcal{C}([0, T], V)$, $ \\mathbb{O}_t = O_t - \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta O_s \\, ds$, and\n\\begin{equation}\nY_t = e^{ t A}\\, ( Y_0 - O_0 ) + \\int_0^t e^{ ( t - s ) A } \\, Z_{ \\fl{s} } F \\big( Y_{ \\fl{s} } \\big) \\, ds + O_t.\n\\end{equation}\nThen \n\\begin{enumerate}[(i)]\n\\item it holds that the functions $ [0,T] \\ni t \\mapsto Y_{t} -\\mathbb{O}_t \\in V $ and $ [0,T] \\ni t \\mapsto \\eta Y_{t} \\in V $ are continuous and\n\\item it holds for all $ t \\in [0,T] $ that $ Y_t- \\mathbb{O}_t= Y_0 - \\mathbb{O}_0 + \\int_0^t (A-\\eta) (Y_s - \\mathbb{O}_s)+ Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) + \\eta Y_s \\, ds $. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:lyapunov1}]\nThroughout this proof let $ \\mathbb{A} \\in L(V) $\nbe the linear operator given by $\\mathbb{A}= A-\\eta$\nand let $\\bar{Y}, E \\colon [0,T] \\to V $\nbe the functions which satisfy for all $ t \\in [0,T] $ that\n$\\bar{Y}_t = Y_t - \\mathbb{O}_t$ and $E_t = \\int_0^t e^{(t-s)\\mathbb{A}} \\, \\eta O_s \\, ds$.\nNote that the assumption that $\\eta O \\in \\mathcal{C}([0, T], V)$ and Lemma~\\ref{lemma:fund:gen} (with $V=V$, $A=\\mathbb{A}$, $T=T$, $Y=E$, $Z=\\eta O$ in the notation of Lemma~\\ref{lemma:fund:gen}) prove that $E$ is continuous and that for all $t \\in [0, T]$ it holds that \n\\begin{align}\n\\label{eq:diff:E}\nE_t = \\int_0^t \\mathbb{A} E_s + \\eta O_s \\, ds.\n\\end{align} \nMoreover, observe that for all $t \\in [0,T]$ it holds that \n\\begin{align}\nY_t - O_t = e^{ t A } \\, (Y_0 -O_0) \n+ \\int_0^t e^{ ( t - s ) A} \\, Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) \\, ds .\n\\end{align}\nThis and Lemma~\\ref{lemma:fund:gen} (with $V=V$, $A=A$, $T=T$, $Y=([0,T] \\ni t \\mapsto (Y_t-O_t-e^{tA}\\, (Y_0-O_0)) \\in V)$,\n$Z=([0,T] \\ni t \\mapsto Z_{ \\fl{t} } F(Y_{ \\lfloor t \\rfloor_{h } }) \\in V) $\nin the notation of Lemma~\\ref{lemma:fund:gen}) prove that for all $ t \\in [0,T] $ it holds that\n$(Y- O) \\in \\mathcal{C}([0, T], V)$ and\n\\begin{align}\n\\label{eq:diff:Y-O}\n\\begin{split}\nY_t - O_t = Y_0- O_0 + \\int_0^t A (Y_s -O_s)+ Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) \\, ds.\n\\end{split}\n\\end{align}\nThis and the fact that $E, \\eta O \\in \\mathcal{C}([0, T], V)$ ensure that the functions $ [0,T] \\ni t \\mapsto \\bar{Y}_t \\in V $ and $ [0,T] \\ni t \\mapsto \\eta Y_{t} \\in V $ are continuous. In the next step we combine \\eqref{eq:diff:E} and \\eqref{eq:diff:Y-O} to obtain that for all $t \\in [0,T]$ it holds that\n\\begin{align}\n\\begin{split}\n\\bar{Y}_t & = \\bar{Y}_0 + \\int_0^t A (Y_s -O_s)+ Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) \\, ds + E_t \\\\\n& = \\bar{Y}_0 + \\int_0^t A (Y_s -O_s)+ Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) + \\mathbb{A} E_s + \\eta O_s \\, ds \\\\\n& = \\bar{Y}_0 + \\int_0^t \\mathbb{A} (Y_s -O_s)+ Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) + \\mathbb{A} E_s + \\eta O_s + \\eta(Y_s - O_s) \\, ds \\\\\n & = \\bar{Y}_0 + \\int_0^t \\mathbb{A} (Y_s -O_s+ E_s)+ Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) + \\eta Y_s \\, ds \\\\\n & = \\bar{Y}_0 + \\int_0^t \\mathbb{A} \\bar{Y}_s + Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) + \\eta Y_s \\, ds.\n\\end{split}\n\\end{align}\nThe proof of Lemma~\\ref{lemma:lyapunov1} is thus completed.\n\\end{proof}\n\n\n\n\n\\subsection{General a priori bounds}\n\\label{subsec:gen_a_priori}\n\n\\begin{lemma}\n\\label{lemma:lyapunov2}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( V, \\left\\| \\cdot \\right\\|_V ) $ be a nontrivial separable $\\mathbb{R}$-Banach space,\nlet $T \\in (0, \\infty)$, $\\eta \\in [0, \\infty)$, $ h \\in (0, T] $, $ A \\in L(V)$, $\\mathbb{V} \\in \\mathcal{C}^1(V, \\mathbb{R})$, $F \\in \\mathcal{C}(V, V)$,\nand let\n$ Z \\colon [0, T] \\to \\mathbb{R} $, $Y, O, \\mathbb{O} \\colon [0, T] \\to V $, and $\\phi, f \\colon V \\to \\mathbb{R} $\nbe functions which \nsatisfy for all $t \\in [0,T]$ that $f \\circ \\mathbb{O} \\in \\mathcal{C}([0,T], \\mathbb{R})$, $\\eta O \\in \\mathcal{C}([0, T], V)$, $ \\mathbb{O}_t = O_t - \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta O_s \\, ds$, and\n\\begin{equation}\nY_t = e^{ t A} \\, ( Y_0 - O_0 ) + \\int_0^t e^{ ( t - s ) A } \\, Z_{ \\fl{s} } F \\big( Y_{ \\fl{s} } \\big) \\, ds + O_t.\n\\end{equation}\nThen \n\\begin{enumerate}[(i)]\n\\item it holds that the functions $ [0,T] \\ni t \\mapsto Y_{t} -\\mathbb{O}_t \\in V $ and $ [0,T] \\ni t \\mapsto \\eta Y_{t} \\in V $ are continuous,\n\\item it holds that $\\sup_{s\\in [0,T]} \\big\\|F\\big( Y_s - \\mathbb{O}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\big\\|_V < \\infty$, and \n\\item it holds for all $ t \\in [0, T]$ that\n\\begin{align}\n&e^{- \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds} \\, \\mathbb{V}( Y_t - \\mathbb{O}_t ) = \\mathbb{V}( Y_0 - \\mathbb{O}_0 ) \\nonumber \\\\\n& + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, \\mathbb{V}' ( Y_s - \\mathbb{O}_s) \\! \\left[ (A-\\eta) ( Y_s - \\mathbb{O}_s) + Z_{ \\fl{s} } F\\big( Y_s - \\mathbb{O}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) + \\eta Y_s \\right] ds \\nonumber \\\\\n& + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' ( Y_s - \\mathbb{O}_s)\\! \\left[\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( Y_s - \\mathbb{O}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds \\\\\n&- \\int_0^t \\big[ \\phi\\big( \\mathbb{O}_{\\fl{s} } \\big) + f(\\mathbb{O}_s)\\big] e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u)\\, du} \\, \\mathbb{V}( Y_s - \\mathbb{O}_s)\\, ds \\nonumber.\n\\end{align}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:lyapunov2}]\nThroughout this proof let $\\bar{Y} \\colon [0,T] \\to V $ be the function which satisfies for all $ t \\in [0,T] $ that $\\bar{Y}_t = Y_t - \\mathbb{O}_t$.\nNote that Lemma~\\ref{lemma:lyapunov1} (with $V=V$, $T = T$, $\\eta = \\eta$, $h=h$, $ A = A$, $ Z =Z$, $Y=Y$, $O=O$, $\\mathbb{O}=\\mathbb{O}$, $F =F$ in the notation of Lemma~\\ref{lemma:lyapunov1}) and the assumption that $F \\in \\mathcal{C}(V,V)$ establish \nthat for all $t \\in [0,T]$ it holds that $\\bar{Y} \\in \\mathcal{C}([0,T],V)$, $\\eta Y \\in \\mathcal{C}([0,T],V)$, $\\sup_{s\\in [0,T]} \\big\\|F\\big( Y_s - \\mathbb{O}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\big\\|_V < \\infty$, and \n\\begin{align}\n\\begin{split}\n\\bar{Y}_t = \\bar{Y}_0 + \\int_0^t (A-\\eta) \\bar{Y}_s + Z_{ \\fl{s} } F\\big(Y_{ \\fl{s} }\\big) + \\eta Y_s \\, ds.\n\\end{split}\n\\end{align}\nThis and Lemma~\\ref{lem:fund:calc} (with $V= \\mathbb{R} \\times V$, $W=\\mathbb{R}$, $U= \\mathbb{R} \\times V$, $f= (\\mathbb{R} \\times V \\ni (t,v) \\mapsto e^{-t} \\,\\mathbb{V}(v) \\in \\mathbb{R})$, $a=0$, $b=T$, $x = ([0,T] \\ni t \\mapsto (\\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds, \\bar{Y}_t) \\in \\mathbb{R} \\times V)$, $y = ([0,T] \\ni t \\mapsto ( \\phi( \\mathbb{O}_{\\fl{t} } ) + f(\\mathbb{O}_t), (A-\\eta) \\bar{Y}_t + Z_{ \\fl{t} } F(Y_{ \\fl{t} }) + \\eta Y_t ) \\in \\mathbb{R} \\times V)$, $t_0=0$ in the notation of Lemma~\\ref{lem:fund:calc}) ensure for all $ t \\in [0,T] $ that\n\\begin{align}\n\\begin{split}\n& e^{- \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds} \\, \\mathbb{V}( \\bar{Y}_t ) \\\\\n&=\\mathbb{V}(\\bar{Y}_0 ) + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, \\mathbb{V}' (\\bar{Y}_s) \\! \\left[ (A-\\eta)\\bar{Y}_s + Z_{ \\fl{s} }\nF\\big( Y_{ \\fl{s} }\\big) + \\eta Y_s \\right] ds \\\\\n& \\quad - \\int_0^t \\big[ \\phi\\big( \\mathbb{O}_{\\fl{s} } \\big) + f(\\mathbb{O}_s) \\big] e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, \\mathbb{V}(\\bar{Y}_s)\\, ds \\\\\n& = \\mathbb{V}(\\bar{Y}_0 ) + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, \\mathbb{V}' (\\bar{Y}_s)\\! \\left[ (A-\\eta) \\bar{Y}_s + Z_{ \\fl{s} } F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) + \\eta Y_s \\right] ds \\\\\n& \\quad + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' (\\bar{Y}_s)\\! \\left[\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds \\\\\n&\\quad - \\int_0^t \\big[ \\phi\\big( \\mathbb{O}_{\\fl{s} } \\big) + f(\\mathbb{O}_s)\\big] e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u)\\, du} \\, \\mathbb{V}(\\bar{Y}_s)\\, ds .\n\\end{split}\n\\end{align}\nThe proof of Lemma~\\ref{lemma:lyapunov2} is thus completed.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lemma:lyapunov3}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( V, \\left\\| \\cdot \\right\\|_V ) $ be a nontrivial separable $\\mathbb{R}$-Banach space,\nlet $T \\in (0, \\infty)$, $\\eta \\in [0, \\infty)$, $ h \\in (0, T] $, $ A \\in L(V)$,\n$\\mathbb{V} \\in \\mathcal{C}^1(V, [0, \\infty))$,\n$F \\in \\mathcal{C}(V, V)$,\n$\\varphi \\in \\mathcal{C}(V, [0,\\infty))$, \nand let\n$ Z \\colon [0, T] \\to [0,1] $, $Y, O, \\mathbb{O} \\colon [0, T] \\to V $, and $\\phi, \\Phi, f, g \\colon V \\to [0, \\infty)$\nbe functions which satisfy for all $v, w \\in V $, $t \\in [0,T]$ that $f \\circ \\mathbb{O}$, $ g \\circ \\mathbb{O} \\in \\mathcal{C}([0,T], [0, \\infty))$, $\\eta O \\in \\mathcal{C}([0, T], V)$, $ \\mathbb{V}'(v)F(v+w) \\leq \\phi(w) \\mathbb{V}(v) + \\varphi(v)+ \\Phi(w)$, $\\eta \\mathbb{V}'(v) (v+w) \\leq f(w) \\mathbb{V}(v) + g(w)$, $ \\mathbb{O}_t = O_t - \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta O_s \\, ds$, and\n\\begin{equation}\nY_t = e^{ t A}\\, ( Y_0 - O_0 ) + \\int_0^t e^{ ( t - s ) A } \\, Z_{ \\fl{s} } F \\big( Y_{ \\fl{s} } \\big) \\, ds + O_t.\n\\end{equation}\nThen\n\\begin{enumerate}[(i)]\n\\item it holds that the function $ [0,T] \\ni t \\mapsto Y_{t} -\\mathbb{O}_t \\in V $ is continuous,\n\\item it holds that $\\sup_{s\\in [0,T]} \\big\\|F\\big( Y_s - \\mathbb{O}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\big\\|_V < \\infty$, and \n\\item it holds for all $ t \\in [0, T]$ that\n\\begin{align}\n&\\mathbb{V}( Y_t - \\mathbb{O}_t ) \\leq e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds} \\, \\mathbb{V}(Y_0 - \\mathbb{O}_0) \\nonumber \\\\\n& + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\left[ \\mathbb{V}' (Y_s - \\mathbb{O}_s) (A-\\eta)(Y_s - \\mathbb{O}_s)+ \\varphi( Y_s - \\mathbb{O}_s)+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + g(\\mathbb{O}_s) \\right] ds \\nonumber \\\\\n& + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' (Y_s - \\mathbb{O}_s)\\! \\left[\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( Y_s - \\mathbb{O}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds.\n\\end{align}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:lyapunov3}]\nThroughout this proof let $\\bar{Y} \\colon [0,T] \\to V $ be the function which satisfies for all $ t \\in [0,T] $ that $\\bar{Y}_t = Y_t - \\mathbb{O}_t$\nand let $ \\mathbb{A} \\in L(V)$ be the linear operator given by $\\mathbb{A}= A-\\eta$.\nNote that Lemma~\\ref{lemma:lyapunov2} (with $ V=V$, $T=T$, $\\eta=\\eta$, $ h=h$, $ A =A$, $\\mathbb{V} = (V \\ni v \\mapsto \\mathbb{V}(v) \\in \\mathbb{R})$, $F=F$, $ Z=([0,T] \\ni t \\mapsto Z_t \\in \\mathbb{R})$, $Y=Y$, $O=O$, $\\mathbb{O}= \\mathbb{O}$, $\\phi=(V \\ni v \\mapsto \\phi(v) \\in \\mathbb{R})$, $f=(V \\ni v \\mapsto f(v) \\in \\mathbb{R})$ in the notation of Lemma~\\ref{lemma:lyapunov2}) ensures that for all $ t \\in [0,T] $ it holds that $\\bar{Y} \\in \\mathcal{C}([0,T], V)$, $\\sup_{s\\in [0,T]} \\big\\|F\\big( Y_s - \\mathbb{O}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\big\\|_V < \\infty$, and\n\\begin{align}\n\\label{eq:lyapunov:fund}\n\\begin{split}\n& e^{- \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds} \\, \\mathbb{V}( \\bar{Y}_t ) \\\\\n& = \\mathbb{V}(\\bar{Y}_0 ) + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, \\mathbb{V}' (\\bar{Y}_s) \\!\\left[ \\mathbb{A} \\bar{Y}_s + Z_{ \\fl{s} } F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) + \\eta Y_s \\right] ds \\\\\n& \\quad + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' (\\bar{Y}_s) \\! \\left[\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds \\\\\n& \\quad - \\int_0^t \\big[ \\phi\\big( \\mathbb{O}_{\\fl{s} } \\big) + f(\\mathbb{O}_s)\\big] e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u)\\, du} \\, \\mathbb{V}(\\bar{Y}_s)\\, ds .\n\\end{split}\n\\end{align}\nFurthermore, the assumption that $\\forall \\, v, w \\in V \\colon \\mathbb{V}'(v)F(v+w) \\leq \\phi(w) \\mathbb{V}(v) + \\varphi(v)+ \\Phi(w)$ implies for all $s \\in [0, T]$ that\n\\begin{align}\n\\begin{split}\n& \\mathbb{V}' (\\bar{Y}_s) Z_{ \\fl{s} } F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) = Z_{ \\fl{s} } \\mathbb{V}' (\\bar{Y}_s) F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) \\\\\n&\\leq Z_{ \\fl{s} } \\! \\left[ \\phi\\big( \n\\mathbb{O}_{ \\fl{s} } \\big) \\mathbb{V}( \\bar{Y}_s ) + \\varphi( \\bar{Y}_s)+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big)\\right] \\leq \\phi\\big( \n\\mathbb{O}_{ \\fl{s} } \\big) \\mathbb{V}( \\bar{Y}_s ) + \\varphi( \\bar{Y}_s)+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big).\n\\end{split}\n\\end{align}\nThis together with \\eqref{eq:lyapunov:fund} proves for all $t \\in [0, T]$ that\n\\begin{align}\n\\begin{split}\n& e^{- \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds} \\,\\mathbb{V}( \\bar{Y}_t ) \\leq \\mathbb{V}(\\bar{Y}_0 ) + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, \\mathbb{V}' (\\bar{Y}_s) \\mathbb{A} \\bar{Y}_s \\, ds \\\\\n& \\quad + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\left[ \\phi\\big( \n\\mathbb{O}_{ \\fl{s} } \\big) \\mathbb{V}( \\bar{Y}_s ) + \\varphi( \\bar{Y}_s)+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\eta \\mathbb{V}' (\\bar{Y}_s) \\big( \\bar{Y}_s + \\mathbb{O}_s\\big) \\right] ds \\\\\n&\\quad + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' (\\bar{Y}_s)\\! \\left[\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds \\\\\n& \\quad- \\int_0^t \\big[ \\phi\\big( \\mathbb{O}_{\\fl{s} } \\big) + f(\\mathbb{O}_s) \\big] e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u)\\, du} \\, \\mathbb{V}(\\bar{Y}_s)\\, ds.\n\\end{split}\n\\end{align}\nThe assumption that $ \\forall \\, v, w \\in V \\colon \\eta \\mathbb{V}'(v) (v+w) \\leq f(w) \\mathbb{V}(v) + g(w)$ hence establishes for all $t \\in [0, T]$ that\n\\begin{align}\n\\begin{split}\n&e^{- \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds} \\, \\mathbb{V}( \\bar{Y}_t ) \\leq \\mathbb{V}(\\bar{Y}_0 ) + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, \\mathbb{V}' (\\bar{Y}_s) \\mathbb{A} \\bar{Y}_s \\, ds \\\\\n& \\quad + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\left[ \\phi\\big( \n\\mathbb{O}_{ \\fl{s} } \\big) \\mathbb{V}( \\bar{Y}_s ) + \\varphi( \\bar{Y}_s )+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + f(\\mathbb{O}_s) \\mathbb{V} (\\bar{Y}_s) + g (\\mathbb{O}_s) \\right] ds \\\\\n& \\quad + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' (\\bar{Y}_s) \\!\\left[F\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds \\\\\n& \\quad - \\int_0^t \\big[ \\phi\\big( \\mathbb{O}_{\\fl{s} } \\big) + f(\\mathbb{O}_s) \\big] e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u)\\, du} \\, \\mathbb{V}(\\bar{Y}_s)\\, ds\\\\\n& = \\mathbb{V}(\\bar{Y}_0 ) + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\left[ \\mathbb{V}' (\\bar{Y}_s) \\mathbb{A} \\bar{Y}_s + \\varphi( \\bar{Y}_s )+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + g(\\mathbb{O}_s) \\right] ds \\\\\n& \\quad + \\int_0^t e^{- \\int_0^s \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' (\\bar{Y}_s) \\!\\left[F\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds.\n\\end{split}\n\\end{align}\nThis assures for all $t \\in [0, T]$ that\n\\begin{align}\n\\begin{split}\n\\mathbb{V}( \\bar{Y}_t ) & \\leq e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + f(\\mathbb{O}_s) \\, ds} \\, \\mathbb{V}(\\bar{Y}_0 ) \\\\\n& \\quad + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\left[ \\mathbb{V}' (\\bar{Y}_s) \\mathbb{A} \\bar{Y}_s + \\varphi( \\bar{Y}_s )+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + g(\\mathbb{O}_s) \\right] ds\\\\\n& \\quad + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + f(\\mathbb{O}_u) \\, du} \\, Z_{ \\fl{s} } \\mathbb{V}' (\\bar{Y}_s) \\!\\left[\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big)\\right] ds.\n\\end{split}\n\\end{align}\nThe proof of Lemma~\\ref{lemma:lyapunov3} is thus completed.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{A priori bounds based on a coercivity-type assumption}\n\n\n\\begin{prop}[A priori bounds]\n\\label{prop:priori_bound}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) $ be a separable $\\mathbb{R}$-Hilbert space,\nlet $ \\mathbb{H} \\subseteq H$ be a nonempty orthonormal basis of $ H $,\nlet $\\beta, T \\in (0, \\infty)$, $\\eta, \\theta , \\vartheta, \\kappa, \\chi, \\varphi \\in [0, \\infty)$, $\\alpha \\in \\mathbb{R}$, $\\rho \\in (-\\infty,1-\\alpha)$, $ \\varrho \\in [\\rho, \\rho +1]$, $\\psi \\in (-\\infty, 2-2\\varphi)$, $ h \\in (0, T] $, $F \\in \\mathcal{C}(H, H)$, $ A \\in L(H) $,\nlet $\\lambda \\colon \\mathbb{H} \\to \\mathbb{R} $,\n$ Y, O, \\mathbb{O} \\colon [0, T] \\to H $,\nand\n$\\phi, \\Phi \\colon H \\to [0,\\infty) $\nbe functions which satisfy\n$\\eta O \\in \\mathcal{C}([0, T], H)$,\n$\\sup_{b \\in \\mathbb{H}} \\lambda_b < \\min\\{\\eta, \\kappa\\}$,\nand\n$ \\forall \\, b \\in \\mathbb{H} \\colon A b = \\lambda_b \\, b $,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $, be a family of interpolation spaces associated to $ \\kappa- A $ (cf., e.g., \\cite[Section~3.7]{sy02}),\nand assume for all $v, w \\in H $, $t \\in [0,T]$ that\n$\\left< v, F( v + w ) \\right>_H \\leq \\frac{1}{2} \\phi(w) \\| v \\|^2_H+ \\varphi \\|(\\eta-A)^{\\nicefrac{1}{2}} v \\|^2_{ H }+ \\frac{1}{2}\\Phi( w )$,\n$\\| F(v)\\|_{H_{-\\alpha}}^2 \\leq \\theta \\max\\{ 1, \\| v\\|_{ H_{\\varrho} }^{2 + \\vartheta} \\} $,\n$ \\|(\\eta-A)^{\\nicefrac{-1}{2}} [F(v) - F(w) ]\\|_{H}^2\\leq \\theta \\max\\{ 1, \\|v\\|_{ H_{\\varrho} }^{\\vartheta} \\} \\|v-w\\|_{ H_{\\rho} }^2 + \\theta \\, \\|v-w\\|^{2 + \\vartheta}_{ H_{\\rho}}$,\n$ \\mathbb{O}_t = O_t - \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta O_s \\, ds$,\nand\n\\begin{equation}\n\\label{eq:scheme_continuous}\nY_t = e^{ t A} ( Y_0 - O_0 ) + \\int_0^t e^{ ( t - s ) A } \\, \\one_{[0, h^{ - \\chi }]} \\big( \\big\\| Y_{ \\fl{s} } \\big\\|_{H_{\\varrho}} + \\big\\| O_{ \\fl{s} } \\big\\|_{H_{\\varrho}} \\big) F \\big( Y_{ \\fl{s} } \\big) \\, ds + O_t.\n\\end{equation}\nThen\n\\begin{enumerate}[(i)]\n\\item it holds that the functions $ [0,T] \\ni t \\mapsto Y_{t} -\\mathbb{O}_t \\in H $ and $ [0,T] \\ni t \\mapsto \\eta \\mathbb{O}_{t} \\in H $ are continuous and\n\\item it holds for all $ t \\in [0, T]$ that\n\\begin{align}\n&\\| Y_t - \\mathbb{O}_t \\|_H^2 + \\psi \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\, \\|(\\eta - A)^{\\nicefrac{1}{2}}( Y_s - \\mathbb{O}_s) \\|_{H}^2 \\, ds \\nonumber \\\\\n& \\leq e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + 2\\eta (1+\\beta) \\, ds} \\, \\|Y_0 - O_0\\|_H^2 + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\, \\Big[ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\nonumber \\\\\n& + \\tfrac{ \\theta e^{h \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ h\\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)}+ \\sqrt{\\eta} ]^{2+\\vartheta}\\left|\\max \\{1 , \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} O_u \\|_{H_{\\varrho}} \\, du \\}\\right|^{2+\\vartheta} }{(1- \\varphi - \\nicefrac{\\psi}{2})( 1-\\max\\{\\alpha+\\rho,0\\})^{2+\\vartheta}} \\nonumber \\\\\n& \\cdot \\max \\!\\big\\{h^2, h^{2(\\varrho - \\rho -\\chi)},h^{ 2(1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n& \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\Big] \\, ds. \\nonumber\n\\end{align}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}[Proof of Proposition~\\ref{prop:priori_bound}]\nThroughout this proof let $ Z \\colon [0, T] \\to [0, 1] $ be the function which satisfies for all $ s \\in [0, T]$ that $ Z_s = \\one_{[0, h^{ - \\chi }]} \\! \\left( \\| Y_{s } \\|_{H_{\\varrho}} + \\| O_{s } \\|_{H_{\\varrho}} \\right)$, let $ \\mathbb{A} \\in L(H) $ be the linear operator given by $\\mathbb{A}= A-\\eta$, and let $\\bar{Y} \\colon [0,T] \\to H $ be the function which satisfies for all $ t \\in [0,T] $ that $\\bar{Y}_t = Y_t - \\mathbb{O}_t$. Observe that the Cauchy-Schwartz inequality and the fact that $\\forall \\, a , b \\in \\mathbb{R} , \\, \\varepsilon \\in (0,\\infty) \\colon a b \\leq \\varepsilon a^2 + \\frac{ b^2 }{ 4 \\varepsilon }$ prove for all $v, w \\in H$ that\n\\begin{align}\n\\label{eq:eta:prod}\n\\begin{split}\n2\\eta \\langle v, v+w \\rangle_H & = 2\\eta \\|v\\|_H^2 +2\\eta \\langle v, w \\rangle_H \\leq 2 \\eta \\|v\\|_H^2 +2 \\eta \\|v\\|_H \\|w\\|_H \\\\\n&\\leq 2 \\eta \\|v\\|_H^2 +2 \\eta \\beta \\|v\\|_H^2 + \\tfrac{\\eta}{2\\beta} \\|w\\|_H^2 = 2\\eta(1+\\beta)\\|v\\|_H^2 + \\tfrac{\\eta}{2\\beta} \\|w\\|_H^2 .\n\\end{split}\n\\end{align}\nIn addition, the assumption that $\\eta O \\in \\mathcal{C}([0, T], H)$ and Lemma~\\ref{lemma:fund:gen} (with $V=H$, $A=\\mathbb{A}$, $T=T$, $Y= ([0,T] \\ni t \\mapsto \\int_0^t e^{(t-s)\\mathbb{A}} \\, \\eta O_s \\, ds \\in H)$, $Z= \\eta O$ in the notation of Lemma~\\ref{lemma:fund:gen}) ensure that $\\eta \\mathbb{O} \\in \\mathcal{C}([0,T],H)$. This, \\eqref{eq:eta:prod}, and Lemma~\\ref{lemma:lyapunov3} (with \n$ V=H$, $T=T$, $\\eta=\\eta$, $ h=h $, $ A=A$, $\\mathbb{V}= (H \\ni v \\mapsto \\|v\\|_H^2 \\in [0, \\infty)) \\in \\mathcal{C}^1(H, [0, \\infty))$, $F =F$, $\\varphi=(H \\ni v \\mapsto 2\\varphi \\|(\\eta-A)^{\\nicefrac{1}{2}} v \\|^2_{ H } \\in [0, \\infty)) $, $ Z=Z$, $Y=Y$, $O=O$, $\\mathbb{O}=\\mathbb{O}$, $\\phi=\\phi$, $\\Phi=\\Phi$, $f= (H \\ni v \\mapsto 2\\eta(1+\\beta) \\in [0, \\infty))$, $g= (H \\ni v \\mapsto\\nicefrac{\\eta}{(2 \\beta)} \\|v\\|_H^2 \\in [0, \\infty))$ in the notation of Lemma~\\ref{lemma:lyapunov3}) establish that for all $ t \\in [0, T]$ it holds that $\\bar{Y} \\in \\mathcal{C}([0,T],H)$ and\n\\begin{align}\n\\label{eq:prop:lemma}\n&\\| \\bar{Y}_t \\|_H^2 \\leq e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + 2\\eta (1+\\beta) \\, ds} \\, \\|\\bar{Y}_0 \\|_H^2 \\nonumber \\\\\n&+ \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + 2\\eta (1+\\beta) \\, du} \\left[ 2 \\langle \\bar{Y}_s , \\mathbb{A} \\bar{Y}_s \\rangle_H + 2 \\varphi \\|(\\eta-A)^{\\nicefrac{1}{2}} \\bar{Y}_s \\|_{H}^2+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\right] ds \\nonumber\\\\\n& + 2 \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + 2\\eta (1+\\beta) \\, du} \\, Z_{ \\fl{s} } \\big<\\bar{Y}_s, \nF\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) \\big>_H \\, ds \\\\\n& = e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + 2\\eta (1+\\beta) \\, ds} \\, \\|\\bar{Y}_0\\|_H^2 + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\left[ - \\psi \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s \\|_{H}^2+ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\right] ds \\nonumber \\\\\n& + \\int_0^t \\! e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + 2\\eta (1+\\beta) \\, du} \\bigl[ -(2- 2\\varphi -\\psi) \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s\\|_{H}^2 + 2 Z_{ \\fl{s} } \\big\\langle \\bar{Y}_s,\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) \\big\\rangle_H \\bigr]\\, ds \\nonumber.\n\\end{align}\nNext note that the fact that $2- 2\\varphi -\\psi > 0$ and the Cauchy-Schwartz inequality show for all $ s \\in [0,T] $ that\n\\begin{align}\n& -(2- 2\\varphi -\\psi) \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s\\|_{H}^2 + 2 Z_{ \\fl{s} } \\big\\langle \\bar{Y}_s,\n F\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n \\mathbb{O}_{ \\fl{s} } \\big) \\big\\rangle_H \\nonumber \\\\\n & \\leq Z_{ \\fl{s} } \\!\\left[-(2- 2\\varphi -\\psi) \\|(-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s\\|_{H}^2 + 2 \\big< (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s, ( - \\mathbb{A} )^{\\nicefrac{-1}{2}}\\big[ F\\big(Y_{ \\fl{s} }\\big)- F\\big( \\bar{Y}_s +\n \\mathbb{O}_{\\fl{s} } \\big) \\big] \\big>_H \\right] \\\\\n & \\leq Z_{ \\fl{s} }\\!\\left[ -(2- 2\\varphi -\\psi) \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s\\|_{H}^2 + 2 \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s\\|_{H} \\big\\|(-\\mathbb{A})^{\\nicefrac{-1}{2}} \\big[F\\big(Y_{ \\fl{s} }\\big)- F\\big( \\bar{Y}_s +\n \\mathbb{O}_{\\fl{s} } \\big) \\big]\\big\\|_{H} \\right]. \\nonumber\n\\end{align}\nThe fact that $\\forall \\, a , b \\in \\mathbb{R} , \\, \\varepsilon \\in (0,\\infty) \\colon 2 a b \\leq \\varepsilon a^2 + \\frac{ b^2 }{ \\varepsilon }$ hence proves for all $ s \\in [0, T]$ that\n\\begin{equation}\n\\begin{split}\n& -(2- 2\\varphi -\\psi) \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s\\|_{H}^2 + 2 Z_{ \\fl{s} } \\big\\langle \\bar{Y}_s,\nF\\big( Y_{ \\fl{s} }\\big) - F\\big( \\bar{Y}_s +\n\\mathbb{O}_{ \\fl{s} } \\big) \\big\\rangle_H \\\\\n&\\leq \\tfrac{1}{(2- 2\\varphi -\\psi)} \\, Z_{ \\fl{s} } \\big\\|(-\\mathbb{A})^{\\nicefrac{-1}{2}} \\big[ F\\big(Y_{ \\fl{s} }\\big)- F\\big( \\bar{Y}_s +\n\\mathbb{O}_{\\fl{s} } \\big) \\big]\\big\\|_{H}^2.\n\\end{split}\n\\end{equation}\nThis together with \\eqref{eq:prop:lemma} ensures for all $t \\in [0, T]$ that\n\\begin{align}\n\\label{eq:estimate0}\n\\begin{split}\n&\\| \\bar{Y}_t \\|_H^2 + \\psi \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\, \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s \\|_{H}^2 \\, ds \\\\\n& \\leq e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + 2\\eta (1+\\beta) \\, ds} \\, \\|\\bar{Y}_0\\|_H^2+ \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\,\\big[ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\big] \\, ds \\\\\n& \\quad + \\tfrac{1}{(2- 2\\varphi -\\psi)} \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + 2\\eta (1+\\beta) \\, du} \\, Z_{ \\fl{s} } \\big\\|(-\\mathbb{A})^{\\nicefrac{-1}{2}} \\big[ F\\big(Y_{ \\fl{s} }\\big)- F\\big( \\bar{Y}_s +\n\\mathbb{O}_{\\fl{s} } \\big) \\big]\\big\\|_{H}^2 \\, ds.\n\\end{split}\n\\end{align}\nFurthermore, the assumption that $ \\forall \\, v, w \\in H \\colon \\| (-\\mathbb{A})^{\\nicefrac{-1}{2}} [F(v) - F(w)] \\|_{H}^2\\leq \\theta \\max\\{ 1, \\|v\\|_{ H_{\\varrho} }^{\\vartheta} \\}\\| v-w\\|_{ H_{\\rho} }^2 + \\theta \\, \\| v-w\\|^{2 + \\vartheta}_{ H_{\\rho}}$ shows for all $ s \\in [0, T]$ that\n\\begin{equation}\n\\label{eq:estimate1}\n\\begin{split}\n& Z_{ \\fl{s} } \\big\\|(-\\mathbb{A})^{\\nicefrac{-1}{2}} \\big[ F\\big(Y_{ \\fl{s} }\\big)- F\\big( \\bar{Y}_s +\n\\mathbb{O}_{\\fl{s} } \\big) \\big]\\big\\|_{H}^2 \\\\\n& \\leq Z_{ \\fl{s} } \\theta \\left[ \\max\\Big\\{ 1, \\big\\| Y_{ \\fl{s} }\\big\\|_{ H_{\\varrho} }^{\\vartheta} \\Big\\} \\, \\big\\| \\bar{Y}_{ \\fl{s} } - \\bar{Y}_s \\big\\|_{ H_{\\rho}}^2 + \\big\\| \\bar{Y}_{ \\fl{s} } - \\bar{Y}_s \\big\\|_{ H_{\\rho} }^{2 + \\vartheta} \\right] \\\\\n& \\leq Z_{ \\fl{s} } \\theta \\,\\big\\| \\bar{Y}_{ \\fl{s} } - \\bar{Y}_s \\big\\|_{ H_{\\rho}}^2 \\left[ \\max \\{1, h^{-\\vartheta \\chi} \\} + \\big\\| \\bar{Y}_{ \\fl{s} } - \\bar{Y}_s \\big\\|_{ H_{\\rho}}^{\\vartheta} \\right].\n\\end{split}\n\\end{equation}\n Moreover, observe that for all $ s \\in [0, T]$ it holds that \n\\begin{equation}\n\\label{eq:priori:Z}\n\\begin{split}\n&Z_{ \\fl{s} } \\| Y_{ \\fl{s} }- O_{ \\fl{s} } - Y_s + O_s \\|_{ H_{\\rho} } \\\\\n&= Z_{ \\fl{s} } \\Big\\| \\left( e^{ ( s - \\fl{s} ) A }- \\operatorname{Id}_H \\right) \\big(Y_{ \\fl{s} }- O_{ \\fl{s} }\\big) + \\smallint_{\\fl{s}}^s e^{ ( s - u ) A } \\, F\\big( Y_{ \\fl{s} } \\big) \\, du \\Big\\|_{ H_{\\rho}} \\\\ \n&\\leq Z_{ \\fl{s} } \\Big[ \\big\\|\\left( e^{ ( s - \\fl{s} ) A }- \\operatorname{Id}_H \\right) \\big(Y_{ \\fl{s} }- O_{ \\fl{s} }\\big) \\big\\|_{ H_{\\rho}} + \\smallint_{\\fl{s}}^s \\big\\| e^{ ( s - u ) A } \\, F\\big( Y_{ \\fl{s} } \\big) \\big\\|_{ H_{\\rho}} \\, du \\Big] \\\\ \n&\\leq Z_{ \\fl{s} } \\big\\| (\\kappa-A)^{\\rho -\\varrho} \\left( e^{ ( s - \\fl{s} ) A }- \\mathrm{Id}_H \\right) \\! \\big\\|_{L(H)} \\big\\| Y_{ \\fl{s} }- O_{ \\fl{s} }\\big\\|_{ H_{\\varrho}} \\\\\n& \\quad + Z_{ \\fl{s} } \\smallint_{\\fl{s}}^s \\big\\|(\\kappa-A)^{\\rho+\\alpha} e^{ ( s - u ) A } \\big\\|_{L(H)} \\big\\| F\\big( Y_{ \\fl{s} } \\big) \\big\\|_{ H_{-\\alpha}} \\, du.\n\\end{split}\n\\end{equation}\nNote that\nthe fact that\n$ \\forall \\, q \\in [ 0, 1 ], \\, t \\in ( 0, \\infty ) \\colon\n\\bigl( \n\\| (\\kappa-A)^{-q} ( e^{ t (A-\\kappa) } - \\operatorname{Id}_H ) \\|_{L(H)} \\leq t^q $\nand\n$ \\| (\\kappa-A)^{q} \\, e^{ t (A-\\kappa) } \\|_{L(H)} \\leq t^{-q}\n\\bigr) $\n(cf., e.g., Lemma~11.36 in Renardy \\& Rogers~\\cite{RenardyRogers1993})\nand the fact that $\\forall \\, x \\in \\mathbb{R} \\colon |e^x -1| \\leq |x| e^{|x|}$ imply that for all $ s \\in [0, T] \\backslash \\{0, h, 2h, 3h, \\ldots\\}$, $u \\in [\\lfloor s \\rfloor_h, s)$, $r \\in \\mathbb{R}$ it holds that \n\\begin{align}\n\\label{eq:priori:smooth}\n\\begin{split}\n&\\big\\| (\\kappa-A)^{\\rho -\\varrho} \\left( e^{ ( s - \\fl{s} ) (A-r) }- \\operatorname{Id}_H \\right) \\! \\big\\|_{L(H)} \\\\\n&\\leq \\big\\| (\\kappa-A)^{\\rho -\\varrho} \\left( e^{ ( s - \\fl{s} ) (A-\\kappa) }- \\operatorname{Id}_H \\right) \\! \\big\\|_{L(H)} + \\big\\| (\\kappa-A)^{\\rho -\\varrho} \\left( e^{ ( s - \\fl{s} ) (A-r) }- e^{ ( s - \\fl{s} ) (A-\\kappa) } \\right) \\! \\big\\|_{L(H)} \\\\\n& \\leq \\left| s - \\fl{s} \\right|^{ \\varrho - \\rho } + \\big\\| (\\kappa-A)^{\\rho -\\varrho} \\, e^{ ( s - \\fl{s} ) (A-\\kappa) } \\big\\|_{L(H)} |e^{ ( s - \\fl{s} )(\\kappa -r)}-1| \\\\\n& \\leq h^{ \\varrho - \\rho } + ( s - \\fl{s} )|\\kappa -r| e^{ ( s - \\fl{s} )|\\kappa -r|} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\\\\n& \\leq h^{ \\varrho - \\rho } + h |\\kappa -r| e^{ h|\\kappa -r|} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)}\n\\end{split}\n\\end{align}\nand\n\\begin{align}\n\\begin{split}\n \\big\\|(\\kappa-A)^{\\rho+\\alpha} e^{ ( s - u ) A } \\big\\|_{L(H)} & \\leq \\big\\|(\\kappa-A)^{\\rho+\\alpha} e^{ ( s - u ) (A-\\kappa) } \\big\\|_{L(H)} e^{(s-u)\\kappa}\\\\\n & \\leq ( s - u )^{ - \\max\\{\\alpha+\\rho,0\\}} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} e^{h \\kappa}.\n\\end{split}\n\\end{align}\nThis, \\eqref{eq:priori:Z}, \\eqref{eq:priori:smooth}, and the assumption that $\\forall \\, v \\in H \\colon \\| F(v)\\|_{H_{-\\alpha}}^2 \\leq \\theta \\max\\{ 1, \\| v\\|_{ H_{\\varrho} }^{2 + \\vartheta} \\}$ yield that for all $ s \\in [0, T] \\backslash \\{0, h, 2h, 3h, \\ldots\\}$ it holds that \n\\begin{align*}\n&Z_{ \\fl{s} } \\| Y_{ \\fl{s} }- O_{ \\fl{s} } - Y_s + O_s \\|_{ H_{\\rho} } \\leq Z_{ \\fl{s} } h^{-\\chi} \\left[ h^{ \\varrho - \\rho } + h \\kappa e^{h \\kappa} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)}\\right] \\\\\n& \\quad + Z_{ \\fl{s} } e^{h \\kappa} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} \\smallint_{ \\fl{s}}^s \\left( s - u \\right)^{ - \\max\\{\\alpha+\\rho,0\\}} \\big\\| F\\big( Y_{ \\fl{s} } \\big) \\big\\|_{ H_{-\\alpha}} \\, du \\\\ \n& \\leq h^{ \\varrho - \\rho -\\chi } + h^{1-\\chi} \\kappa e^{h \\kappa} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\\\\n& \\quad + Z_{ \\fl{s} } e^{h\\kappa} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} \\cdot \\tfrac{\\left| s - \\fl{s} \\right|^{ 1-\\max\\{\\alpha+\\rho,0\\}}}{1-\\max\\{\\alpha+\\rho,0\\}} \\sqrt{\\theta} \\max\\big\\{ 1, \\big\\| Y_{ \\fl{s} }\\big\\|_{ H_{\\varrho} }^{1 + \\nicefrac{\\vartheta}{2}} \\big\\} \\\\\n& \\leq h^{ \\varrho - \\rho -\\chi } + h^{1-\\chi} \\kappa e^{h \\kappa} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\\\\n&\\quad + e^{h\\kappa} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} \\cdot \\tfrac{h^{ 1-\\max\\{\\alpha+\\rho,0\\}}}{1-\\max\\{\\alpha+\\rho,0\\}} \\sqrt{\\theta} \\max\\big\\{ 1,h^{-(1+\\nicefrac{\\vartheta}{2})\\chi} \\big\\} \\numberthis \\label{eq:estimate2} \\\\\n& \\leq \\tfrac{e^{h \\kappa}}{ 1-\\max\\{\\alpha+\\rho,0\\}} \\Big[ h^{\\varrho - \\rho -\\chi}+ h^{1-\\chi} \\kappa \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\\\\n&\\quad + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)}\\, h^{ 1-\\max\\{\\alpha+\\rho,0\\} } \\max\\{ 1, h^{-(1+\\nicefrac{\\vartheta}{2})\\chi} \\}\\Big] \\\\\n& \\leq \\frac{ e^{h \\kappa}(1+ \\kappa \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} ) }{( 1-\\max\\{\\alpha+\\rho,0\\})}\\\\\n& \\quad \\cdot \\max\\! \\left\\{ h^{\\varrho - \\rho -\\chi}, h^{1-\\chi}, h^{ 1-\\max\\{\\alpha+\\rho,0\\} } , h^{ 1-\\max\\{\\alpha+\\rho,0\\} -(1+\\nicefrac{\\vartheta}{2})\\chi} \\right\\} \\\\\n& \\leq \\frac{ e^{h \\kappa}(1+ \\kappa \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} ) \\max\\! \\left\\{ h, h^{\\varrho - \\rho -\\chi}, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -(1+\\nicefrac{\\vartheta}{2}) \\chi} \\right\\} }{( 1-\\max\\{\\alpha+\\rho,0\\})} .\n\\end{align*}\nMoreover,\nthe fact that\n$ \\forall \\, t \\in [ 0, \\infty ) \\colon\n\\| e^{ t \\mathbb{A} } \\|_{L(H)} \\leq 1 $\nand \\eqref{eq:priori:smooth} prove for all $ s \\in [0, T]\\backslash \\{0, h, 2h, 3h, \\ldots\\}$ that \n\\begin{align*}\n& \\Big\\| \\smallint_0^s e^{(s-u)\\mathbb{A}} \\, \\eta O_u \\, du - \\smallint_0^{\\lfloor s \\rfloor_h} e^{(\\lfloor s \\rfloor_h-u)\\mathbb{A}} \\, \\eta O_u \\, du \\Big\\|_{H_{\\rho}}\\\\\n&= \\Big\\| \\left(e^{(s-\\fl{s})\\mathbb{A}}- \\mathrm{Id}_H\\right) \\smallint_0^{\\lfloor s \\rfloor_h} e^{(\\lfloor s \\rfloor_h-u)\\mathbb{A}} \\, \\eta O_u \\, du + \\smallint_{\\lfloor s \\rfloor_h}^s e^{(s-u)\\mathbb{A}} \\, \\eta O_u \\, du \\Big\\|_{H_{\\rho}} \\\\\n& \\leq \\|(\\kappa-A)^{\\rho- \\varrho} \\left(e^{(s-\\fl{s})\\mathbb{A}}- \\mathrm{Id}_H\\right) \\big\\|_{L(H)} \\smallint_0^{\\lfloor s \\rfloor_h} \\big\\| e^{(\\lfloor s \\rfloor_h-u)\\mathbb{A}} \\big\\|_{L(H)} \\, \\| \\eta O_u \\|_{H_{\\varrho}} \\, du \\numberthis \\\\\n& \\quad + \\smallint_{\\lfloor s \\rfloor_h}^s \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)} \\|e^{(s-u)\\mathbb{A}} \\|_{L(H)} \\| \\eta O_u \\|_{H_{\\varrho}} \\, du \\\\\n& \\leq \\big[ h^{ \\varrho - \\rho } + h |\\kappa -\\eta| e^{ h|\\kappa -\\eta|} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\big] \\smallint_0^{\\lfloor s \\rfloor_h} \\| \\eta O_u \\|_{H_{\\varrho}} \\, du + \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)} \\smallint_{\\lfloor s \\rfloor_h}^s \\| \\eta O_u \\|_{H_{\\varrho}} \\, du \\\\\n&\\leq \\big[ h^{ \\varrho - \\rho } + h |\\kappa -\\eta| e^{ h|\\kappa -\\eta|} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\big] \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du + \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)} \\smallint_{\\lfloor s \\rfloor_h}^s \\| \\eta O_u \\|_{H_{\\varrho}} \\, du.\n\\end{align*}\nThis and \\eqref{eq:estimate2} ensure for all $s \\in [0, T]$ that\n\\begin{align*}\n& Z_{ \\fl{s} } \\| \\bar{Y}_{ \\fl{s} } - \\bar{Y}_s \\|_{ H_{\\rho} } = Z_{ \\fl{s} } \\Big\\| Y_{ \\fl{s} }- O_{ \\fl{s} } + \\smallint_0^{\\lfloor s \\rfloor_h} e^{(\\lfloor s \\rfloor_h-u)\\mathbb{A}} \\, \\eta O_u \\, du - Y_s + O_s - \\smallint_0^s e^{(s-u)\\mathbb{A}} \\, \\eta O_u \\, du \\Big\\|_{ H_{\\rho} } \\\\\n& \\leq Z_{ \\fl{s} } \\| Y_{ \\fl{s} }- O_{ \\fl{s} } - Y_s + O_s \\|_{ H_{\\rho} } \\\\\n& \\quad + Z_{ \\fl{s} } \\Big\\|\\smallint_0^s e^{(s-u)\\mathbb{A}} \\, \\eta O_u \\, du - \\smallint_0^{\\lfloor s \\rfloor_h} e^{(\\lfloor s \\rfloor_h-u)\\mathbb{A}} \\, \\eta O_u \\, du \\Big\\|_{H_{\\rho}} \\\\\n& \\leq \\frac{ e^{h \\kappa}(1+ \\kappa \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} ) \\max\\! \\left\\{ h, h^{\\varrho - \\rho -\\chi}, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -(1+\\nicefrac{\\vartheta}{2}) \\chi} \\right\\} }{( 1-\\max\\{\\alpha+\\rho,0\\})}\\\\\n& \\quad + \\big[ h^{ \\varrho - \\rho } + h |\\kappa -\\eta| e^{ h|\\kappa -\\eta|} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\big] \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du + \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)} \\smallint_{\\lfloor s \\rfloor_h}^s \\| \\eta O_u \\|_{H_{\\varrho}} \\, du\\\\\n& \\leq \\frac{ e^{h \\kappa}(1+ \\kappa \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} ) \\max\\! \\left\\{ h, h^{\\varrho - \\rho -\\chi}, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -(1+\\nicefrac{\\vartheta}{2}) \\chi} \\right\\} }{( 1-\\max\\{\\alpha+\\rho,0\\})}\\\\\n& \\quad + e^{h \\kappa} \\big[ h^{ \\varrho - \\rho } + h |\\kappa -\\eta| e^{ h\\eta} \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} \\big] \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du \\numberthis \\label{eq:bar:Y} \\\\\n& \\quad + e^{h \\kappa}\\sqrt{\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)} \\smallint_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{\\eta} O_u \\|_{H_{\\varrho}} \\, du \\\\\n& \\leq \\tfrac{ e^{h \\kappa} [1+ \\kappa \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} + (1+ |\\kappa-\\eta|e^{ h\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)}) \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du + \\sqrt{\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)} ] }{( 1-\\max\\{\\alpha+\\rho,0\\})} \\\\\n&\\quad \\cdot \\max \\!\\big\\{h, h^{\\varrho - \\rho -\\chi},h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\} .\n\\end{align*}\nNext note that for all $a \\in [1, \\infty)$, $b,c \\in [0, \\infty)$ it holds that\n\\begin{align}\n\\begin{split}\n|ab|^2 (c+ |ab|^{\\vartheta}) \\leq a^{2+\\vartheta}b^2 (c+ b^{\\vartheta}) \\leq 2 a^{2+\\vartheta} b^2 \\max\\{ c, b^{\\vartheta}\\} .\n\\end{split}\n\\end{align}\nThis, \\eqref{eq:estimate1}, and \\eqref{eq:bar:Y} show for all $s \\in [0, T]$ that\n\\begin{align}\n\\label{eq:F:term}\n\\begin{split}\n& Z_{ \\fl{s} } \\big\\| (-\\mathbb{A})^{\\nicefrac{-1}{2}} \\big[F\\big(Y_{ \\fl{s} }\\big)- F\\big(\\bar{Y}_s + \\mathbb{O}_{ \\fl{s} }\\big) \\big] \\big\\|_{ H }^2 \\\\ \n& \\leq \\tfrac{2 \\theta e^{h \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta}) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} + (1+ |\\kappa-\\eta|e^{ h\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)}) \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du ]^{2+\\vartheta} }{( 1-\\max\\{\\alpha+\\rho,0\\})^{2+\\vartheta}}\\\\\n& \\quad \\cdot \\left|\\max \\!\\big\\{h, h^{\\varrho - \\rho -\\chi},h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\} \\right|^2 \\\\\n& \\quad \\cdot \\left|\\max \\!\\big\\{1, h^{-\\chi}, h, h^{\\varrho - \\rho -\\chi},h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\\\\n&= \\tfrac{2 \\theta e^{h \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta}) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} + (1+ |\\kappa-\\eta|e^{ h\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)}) \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du ]^{2+\\vartheta} }{( 1-\\max\\{\\alpha+\\rho,0\\})^{2+\\vartheta}}\\\\\n& \\quad \\cdot \\left|\\max \\!\\big\\{h, h^{\\varrho - \\rho -\\chi},h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\} \\right|^2 \\\\\n& \\quad \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}.\n\\end{split}\n\\end{align}\nObserve that H\\\"olders inequality implies for all $s \\in [0,T]$ that\n\\begin{align}\n\\begin{split}\n&\\left|\\max \\!\\big\\{h, h^{\\varrho - \\rho -\\chi},h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\} \\right|^2 \\\\\n& \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\\\\n&\\leq \\left| \\max \\!\\Big\\{h, h^{\\varrho - \\rho -\\chi},h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\sqrt{h} \\sqrt{ \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du} \\Big\\} \\right|^2 \\\\\n&\\quad \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{\\lfloor s \\rfloor_h}^s \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\\\\n&\\leq \\max \\!\\big\\{h^2, h^{2(\\varrho - \\rho -\\chi)},h^{ 2(1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n&\\quad \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}.\n\\end{split}\n\\end{align}\nThis and \\eqref{eq:F:term} establish for all $s \\in [0, T]$ that\n\\begin{align}\n\\label{eq:estimate3}\n\\begin{split}\n& Z_{ \\fl{s} } \\big\\| (-\\mathbb{A})^{\\nicefrac{-1}{2}} \\big[F\\big(Y_{ \\fl{s} }\\big)- F\\big(\\bar{Y}_s + \\mathbb{O}_{ \\fl{s} }\\big) \\big] \\big\\|_{ H }^2 \\\\\n& \\leq \\tfrac{2 \\theta e^{h \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta}) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} + (1+ |\\kappa-\\eta|e^{ h\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)}) \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du ]^{2+\\vartheta} }{( 1-\\max\\{\\alpha+\\rho,0\\})^{2+\\vartheta}}\\\\\n& \\quad \\cdot \\max \\!\\big\\{h^2, h^{2(\\varrho - \\rho -\\chi)},h^{ 2(1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n&\\quad \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\}-( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}.\n\\end{split}\n\\end{align}\nCombining \\eqref{eq:estimate3} with \\eqref{eq:estimate0} yields that for all $t \\in [0, T]$ it holds that\n\\begin{align}\n\\begin{split}\n&\\| \\bar{Y}_t \\|_H^2 + \\psi \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du}\\, \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s \\|_{H}^2 \\, ds \\\\\n& \\leq e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + 2\\eta (1+\\beta) \\, ds} \\, \\|\\bar{Y}_0\\|_H^2+ \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\left[ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\right] ds \\\\\n& + \\tfrac{2 \\theta e^{h \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta}) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} \\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} + (1+ |\\kappa-\\eta|e^{ h\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)}) \\smallint_0^T \\| \\eta O_u \\|_{H_{\\varrho}} \\, du ]^{2+\\vartheta} }{(2- 2\\varphi -\\psi)( 1-\\max\\{\\alpha+\\rho,0\\})^{2+\\vartheta}} \\\\\n& \\cdot \\max \\!\\big\\{h^2, h^{2(\\varrho - \\rho -\\chi)},h^{ 2(1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n& \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta} \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) + 2\\eta (1+\\beta) \\, du} \\, ds.\n\\end{split}\n\\end{align}\nThis assures for all $ t \\in [0, T]$ that\n\\begin{align*}\n&\\| \\bar{Y}_t \\|_H^2 + \\psi \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\, \\| (-\\mathbb{A})^{\\nicefrac{1}{2}} \\bar{Y}_s \\|_{H}^2 \\, ds \\\\\n&\\leq e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + 2\\eta (1+\\beta) \\, ds} \\, \\|\\bar{Y}_0\\|_H^2 + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\, \\Big[ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\\\\n& + \\tfrac{ \\theta e^{h \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta}) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta}\\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} + (1+ |\\kappa-\\eta|e^{ h\\eta} \\|(\\kappa-A)^{\\rho-\\varrho} \\|_{L(H)}) \\sqrt{\\eta} ]^{2+\\vartheta} }{(1- \\varphi -\\nicefrac{\\psi}{2})( 1-\\max\\{\\alpha+\\rho,0\\})^{2+\\vartheta}} \\\\\n& \\cdot \\left|\\max \\{1 , \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} O_u \\|_{H_{\\varrho}} \\, du \\}\\right|^{2+\\vartheta} \\max \\!\\big\\{h^2, h^{2(\\varrho - \\rho -\\chi)},h^{ 2(1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n& \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\Big] \\, ds \\numberthis \\\\\n&= e^{ \\int_0^t \\phi( \\mathbb{O}_{\\fl{s} } ) + 2\\eta (1+\\beta) \\, ds} \\, \\|\\bar{Y}_0\\|_H^2 + \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\, \\Big[ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\\\\n& + \\tfrac{ \\theta e^{h \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ h\\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta}\\|(\\kappa-A)^{ \\min\\{\\alpha+\\rho,0\\}} \\|_{L(H)} + \\sqrt{\\eta} ]^{2+\\vartheta}\\left|\\max \\{1 , \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} O_u \\|_{H_{\\varrho}} \\, du \\}\\right|^{2+\\vartheta} }{(1- \\varphi -\\nicefrac{\\psi}{2})( 1-\\max\\{\\alpha+\\rho,0\\})^{2+\\vartheta}} \\\\\n& \\cdot \\max \\!\\big\\{h^2, h^{2(\\varrho - \\rho -\\chi)},h^{ 2(1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n& \\cdot \\left|\\max \\!\\big\\{ h^{-\\chi}, h, h^{ 1-\\max\\{\\alpha+\\rho,0\\} -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi} , \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\Big] \\, ds.\n\\end{align*}\nThe proof of Proposition~\\ref{prop:priori_bound} is thus completed.\n\\end{proof}\n\nThe next result, Corollary~\\ref{cor:a_priori}, follows immediately from Proposition~\\ref{prop:priori_bound} above.\n\n\\begin{cor}\n\\label{cor:a_priori}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) $ be a separable $\\mathbb{R}$-Hilbert space,\nlet $ \\mathbb{H} \\subseteq H$ be a nonempty orthonormal basis of $ H $,\nlet $\\beta, T \\in (0, \\infty)$, $\\eta, \\theta , \\vartheta, \\kappa \\in [0, \\infty)$, $ \\varphi \\in [0,1)$, $\\alpha \\in \\mathbb{R}$, $\\rho \\in [-\\alpha,1-\\alpha)$, $ \\varrho \\in [\\rho, \\rho +1]$, $\\chi \\in [0, \\nicefrac{(2-2\\alpha-2\\rho)}{(1+\\vartheta)} ]$, \n$ h \\in (0, \\min\\{1,T\\}] $,\n$F \\in \\mathcal{C}(H, H)$,\n$ A \\in L(H) $,\nlet $\\lambda \\colon \\mathbb{H} \\to \\mathbb{R} $,\n$ Y, O, \\mathbb{O} \\colon [0, T] \\to H $,\nand\n$\\phi, \\Phi \\colon H \\to [0,\\infty) $\nbe functions which satisfy\n$\\eta O \\in \\mathcal{C}([0, T], H)$,\n$\\sup_{b \\in \\mathbb{H}} \\lambda_b < \\min\\{\\eta, \\kappa\\}$,\nand\n$ \\forall \\, b \\in \\mathbb{H} \\colon A b = \\lambda_b \\, b $,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $, be a family of interpolation spaces associated to $ \\kappa- A $,\nand assume for all $v, w \\in H $, $t \\in [0,T]$ that\n$\\left< v, F( v + w ) \\right>_H \\leq \\frac{1}{2} \\phi(w) \\| v \\|^2_H+ \\varphi \\|(\\eta-A)^{\\nicefrac{1}{2}} v \\|^2_{ H }+ \\frac{1}{2}\\Phi( w )$,\n$\\| F(v)\\|_{H_{-\\alpha}}^2 \\leq \\theta \\max\\{ 1, \\| v\\|_{ H_{\\varrho} }^{2 + \\vartheta} \\} $,\n$ \\|(\\eta-A)^{\\nicefrac{-1}{2}} [F(v) - F(w) ]\\|_{H}^2\\leq \\theta \\max\\{ 1, \\|v\\|_{ H_{\\varrho} }^{\\vartheta} \\} \\|v-w\\|_{ H_{\\rho} }^2 + \\theta \\, \\|v-w\\|^{2 + \\vartheta}_{ H_{\\rho}}$,\n$ \\mathbb{O}_t = O_t - \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta O_s \\, ds$,\nand\n\\begin{equation}\nY_t = \\int_0^t e^{ ( t - s ) A } \\, \\one_{[0, h^{ - \\chi }]} \\big( \\big\\| Y_{ \\fl{s} } \\big\\|_{H_{\\varrho}} + \\big\\| O_{ \\fl{s} } \\big\\|_{H_{\\varrho}} \\big) F \\big( Y_{ \\fl{s} } \\big) \\, ds + O_t.\n\\end{equation}\nThen it holds for all $ t \\in [0, T]$ that $\\eta \\mathbb{O} \\in \\mathcal{C}([0,T],H)$ and\n\\begin{align}\n\\begin{split}\n&\\| Y_t - \\mathbb{O}_t \\|_H^2 \\leq \\int_0^t e^{ \\int_s^t \\phi( \\mathbb{O}_{\\fl{u} } ) +2\\eta (1+\\beta) \\, du} \\, \\Big[ \\Phi\\big(\\mathbb{O}_{ \\fl{s} } \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s\\|_H^2 \\\\\n&\\quad + \\tfrac{ \\theta e^{ \\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta} ]^{2+\\vartheta} \\left|\\max \\{1 , \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} O_u \\|_{H_{\\varrho}} \\, du \\}\\right|^{2+\\vartheta}}{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}} \\\\\n& \\quad \\cdot \\max \\!\\big\\{ h^{2(\\varrho - \\rho -\\chi)},h^{ 2(1-\\alpha - \\rho -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\left|\\max \\!\\big\\{ h^{-\\chi}, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} O_u \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\Big] ds.\n\\end{split}\n\\end{align}\n\\end{cor}\n\n\n\n\\section{Main result}\n\\label{sec:main}\n\nIn the main result of this article, Theorem~\\ref{thm:strong} below,\nwe establish strong convergence for an explicit space-time discrete numerical approximation scheme\nfor a large class of SEEs.\nBefore presenting Theorem~\\ref{thm:strong},\nwe provide a few elementary and well-known results in Lemmas~\\ref{lem:fast_convergence}--\\ref{lem:subsequence} below.\nThese auxiliary lemmas as well as\na pathwise convergence result (see Proposition~\\ref{prop:main_det} below)\nand pathwise a~priori bounds (see Proposition~\\ref{prop:main_det:2} below)\nare required in the proof of Theorem~\\ref{thm:strong}.\n\n\\subsection{Fast convergence in probability}\n\n\\begin{lemma}\n\\label{lem:fast_convergence}\nLet $ ( \\Omega, \\F, \\P ) $ be a probability space, let $ ( E, d ) $ be a metric space, and let $ X_n \\colon \\Omega \\to E $, $ n \\in \\mathbb{N}_0 $, be strongly $ \\F $\/$ ( E, d ) $-measurable functions which satisfy $ \\sum_{ n = 1 }^{ \\infty } \\mathbb{E}\\big[\\! \\min\\!\\big\\{ 1 , d( X_n, X_0 ) \\big\\} \\big] < \\infty $. Then it holds that $\\{ \\limsup_{n \\to \\infty} d(X_n, X_0)=0\\} \\in \\F$ and $\\P \\bigl( \\limsup_{n \\to \\infty} d(X_n, X_0)=0 \\bigr)=1$. \n\\end{lemma} \n\\begin{proof}[Proof of Lemma~\\ref{lem:fast_convergence}] \nNote that the assumption that $ \\sum_{ n = 1 }^{ \\infty } \\mathbb{E}\\big[ \\!\\min\\!\\big\\{ 1 , d( X_n, X_0 ) \\big\\} \\big] < \\infty $ and Markov's inequality ensure for all $ \\varepsilon \\in ( 0, 1 ] $ that \n\\begin{equation} \n\\begin{split} \n\\sum_{ n = 1 }^{ \\infty } \\P\\bigl( d( X_n, X_0 ) \\geq \\varepsilon \\bigr) & = \\sum_{ n = 1 }^{ \\infty } \\P\\bigl( \\min\\{ 1, d( X_n, X_0 ) \\} \\geq \\varepsilon \\bigr) \\\\ \n& \\leq \\frac{1}{\\varepsilon} \\sum_{ n = 1 }^{ \\infty } \\mathbb{E}\\big[ \\!\\min\\{ 1, d( X_n, X_0 ) \\} \\big] < \\infty . \n\\end{split} \n\\end{equation} \nThe Borel-Cantelli lemma hence implies for all $ \\varepsilon \\in ( 0, 1 ] $ that \n\\begin{equation} \n\\P\\bigg( \\bigcap_{n=1}^{\\infty} \\bigcup_{m=n}^{\\infty} \\left\\{ d( X_m , X_0 ) \\geq \\varepsilon \\right\\} \\bigg) = 0 . \n\\end{equation} \nThis proves for all $ \\varepsilon \\in ( 0, 1 ] $ that \n\\begin{equation} \n\\label{eq:fast_conv}\n\\P\\bigg( \\bigcup_{n=1}^{\\infty} \\bigcap_{m=n}^{\\infty} \\left\\{ d( X_m , X_0 ) < \\varepsilon \\right\\} \\bigg) = 1 . \n\\end{equation} \nMoreover, note that\n\\begin{align}\n\\begin{split}\n&\\bigg\\{\\! \\limsup_{n \\to \\infty} d(X_n, X_0)=0\\bigg\\} = \\bigcap_{\\varepsilon \\in (0,\\infty) \\cap \\mathbb{Q}} \\bigcup_{n=1}^{\\infty} \\bigcap_{m=n}^{\\infty} \\left\\{ d( X_m , X_0 ) < \\varepsilon \\right\\} \\in \\mathcal{F}.\n\\end{split}\n\\end{align}\nEquation~\\eqref{eq:fast_conv} hence shows that \n\\begin{equation} \n\\begin{split} \n\\P\\biggl( \\limsup_{n \\to \\infty} d(X_n, X_0)=0 \\biggr) & = \\P\\bigg( \\bigcap_{\\varepsilon \\in (0,\\infty) \\cap \\mathbb{Q}} \\bigcup_{n=1}^{\\infty} \\bigcap_{m=n}^{\\infty} \\left\\{ d( X_m , X_0 ) < \\varepsilon \\right\\} \\bigg) \\\\ \n& = \\lim_{ \\varepsilon \\searrow 0 } \\P\\bigg( \\bigcup_{n=1}^{\\infty} \\bigcap_{m=n}^{\\infty} \\left\\{ d( X_m , X_0 ) < \\varepsilon \\right\\} \\bigg) = 1 . \n\\end{split} \n\\end{equation} \nThe proof of Lemma~\\ref{lem:fast_convergence} is thus completed. \n\\end{proof}\n\n\\subsection{A characterization of convergent sequences in topological spaces}\n\\begin{lemma}\n\\label{lem:subsequence}\nLet $ ( E, \\mathcal{E} ) $ be a topological space and let $ e \\colon \\mathbb{N}_0 \\to E $ be a function. Then the following seven statements are equivalent: \\begin{enumerate}[(i)] \n\\item \\label{item:con} It holds that $ e_n \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $. \n\\item \\label{item:infty:inc} For every function $ k \\colon \\mathbb{N} \\to \\mathbb{N} $ with $ \\liminf_{ n \\to \\infty } k(n) = \\infty $ there exists a strictly increasing function $ l \\colon \\mathbb{N} \\to \\mathbb{N} $ such that $ e_{ k( l(n) ) } \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $.\n\\item \\label{item:infty:infty} For every function $ k \\colon \\mathbb{N} \\to \\mathbb{N} $ with $ \\liminf_{ n \\to \\infty } k(n) = \\infty $ there exists a function $ l \\colon \\mathbb{N} \\to \\mathbb{N} $ with $ \\liminf_{ n \\to \\infty } l(n) = \\infty $ such that $ e_{ k( l(n) ) } \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $.\n\\item \\label{item:infty:exist} For every function $ k \\colon \\mathbb{N} \\to \\mathbb{N} $ with $ \\liminf_{ n \\to \\infty } k(n) = \\infty $ there exists a function $ l \\colon \\mathbb{N} \\to \\mathbb{N} $ such that $ e_{ k( l(n) ) } \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $. \n\\item \\label{item:inc:inc} For every strictly increasing function $ k \\colon \\mathbb{N} \\to \\mathbb{N} $ there exists a strictly increasing function $ l \\colon \\mathbb{N} \\to \\mathbb{N} $ such that $ e_{ k( l(n) ) } \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $. \n\\item \\label{item:inc:infty} For every strictly increasing function $ k \\colon \\mathbb{N} \\to \\mathbb{N} $ there exists a function $ l \\colon \\mathbb{N} \\to \\mathbb{N} $ with $ \\liminf_{ n \\to \\infty }$ $l(n) = \\infty $ such that $ e_{ k( l(n) ) } \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $. \n\\item \\label{item:inc:exist} For every strictly increasing function $ k \\colon \\mathbb{N} \\to \\mathbb{N} $ there exists a function $ l \\colon \\mathbb{N} \\to \\mathbb{N} $ such that $ e_{ k( l(n) ) } \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $. \n\\end{enumerate} \n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lem:subsequence}]\nConsider the notation in Subsection~\\ref{sec:notation}.\nIt is clear that $ (\\eqref{item:con} \\Rightarrow \\eqref{item:infty:inc}) $, $ (\\eqref{item:infty:inc} \\Rightarrow \\eqref{item:infty:infty}) $, $ (\\eqref{item:infty:infty} \\Rightarrow \\eqref{item:infty:exist}) $, $ (\\eqref{item:inc:inc} \\Rightarrow \\eqref{item:inc:infty}) $, and $ (\\eqref{item:inc:infty} \\Rightarrow \\eqref{item:inc:exist}) $. The fact that $ (\\eqref{item:infty:exist} \\Rightarrow \\eqref{item:inc:exist})$ and the fact that $(\\eqref{item:infty:inc} \\Rightarrow \\eqref{item:inc:inc}) $ hence ensure that it is sufficient to prove that $ ( \\eqref{item:inc:exist} \\Rightarrow \\eqref{item:con} ) $ in order to complete the proof of Lemma~\\ref{lem:subsequence}. We show $ ( \\eqref{item:inc:exist} \\Rightarrow \\eqref{item:con} ) $ by a contradiction and for this we assume $ \\big( \\eqref{item:inc:exist} \\wedge (\\neg \\eqref{item:con} ) \\big) $ in the following.\nObserve that $ (\\neg \\eqref{item:con}) $ assures that there exists a set $ A \\in \\mathcal{E} $ with $ e_0 \\in A $ and $ \\#_{ \\{ n \\in \\mathbb{N} \\colon e_n \\notin A \\} } = \\infty $.\nThis implies that there exists a strictly increasing function $ k \\colon \\mathbb{N} \\to \\mathbb{N} $ such that for all $ n \\in \\mathbb{N} $ it holds that \n\\begin{equation} \\label{eq:subsequence_contra} \ne_{ k(n) } \\notin A . \n\\end{equation} \nNext note that $ \\eqref{item:inc:exist} $ ensures that there exists a function $ l \\colon \\mathbb{N} \\to \\mathbb{N} $ such that $ e_{ k( l(n) ) } \\in E $, $ n \\in \\mathbb{N} $, converges in $ ( E, \\mathcal{E} ) $ to $ e_0 $. This proves that there exists a natural number $ N \\in \\mathbb{N} $ such that for all $ n \\in \\{ N , N + 1, \\dots \\} $ it holds that $ e_{ k( l(n) ) } \\in A $. In particular, we obtain that $ e_{ k( l(N) ) } \\in A $. This contradicts to \\eqref{eq:subsequence_contra}. The proof of Lemma~\\ref{lem:subsequence} is thus completed.\n\\end{proof}\n\n\\subsection{Pathwise convergence}\n\n\\begin{prop}\\label{prop:main_det}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) $ be a separable $\\mathbb{R}$-Hilbert space,\nlet $ \\mathbb{H} \\subseteq H$ \nbe a nonempty orthonormal basis of $ H $,\nlet $ \\kappa \\in [0, \\infty)$,\nlet\n$\\lambda \\colon \\mathbb{H} \\to \\mathbb{R} $\nbe a function which satisfies\n$\\inf_{b \\in \\mathbb{H}} \\lambda_b > -\\kappa$,\nlet $ A \\colon D(A) \\subseteq H \\to H $ be the linear operator which satisfies\n$ D(A) = \\{ v \\in H \\colon \\sum_{b \\in \\mathbb{H}} | \\lambda_b \\langle b , v \\rangle_H |^2 < \\infty \\} $\nand\n$ \\forall \\, v \\in D(A) \\colon A v = \\sum_{b \\in \\mathbb{H}} - \\lambda_b \\langle b , v \\rangle_H b$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $,\nbe a family of interpolation spaces associated to $ \\kappa- A $,\nlet $ \\alpha \\in [ 0, 1)$, $\\varrho \\in [0, 1-\\alpha)$, $ T , \\chi \\in (0,\\infty)$, $F \\in \\mathcal{C}(H_{\\varrho}, H_{-\\alpha})$, $ X, O \\in \\mathcal{C}( [0, T], H_{ \\varrho}) $,\nlet\n$ ( \\mathbb{H}_n )_{ n \\in \\mathbb{N} } \\colon \\mathbb{N} \\to \\mathcal{P}_0( \\mathbb{H} ) $,\n$ ( P_n )_{ n \\in \\mathbb{N} } \\colon \\mathbb{N} \\to L( H ) $,\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $,\nand\n$ \\mathcal{X}^n, \\mathcal{O}^n \\colon [0,T] \\to H_{\\varrho} $, $ n \\in \\mathbb{N} $,\nbe functions,\nand assume for all $ v \\in H $, $ n \\in \\mathbb{N} $, $ t \\in [0,T] $, $r \\in (0,\\infty)$ that\n$P_n(v) = \\sum_{ b \\in \\mathbb{H}_n } \\langle b, v \\rangle_H b$,\n$\\sup (\\{\\|F(x)-F(y)\\|_{H_{-\\alpha}}\/\\|x-y\\|_{H_{\\varrho}} \\colon x, y \\in H_{\\varrho}, x\\neq y, \\max\\{\\|x\\|_{H_{\\varrho}},\\|y\\|_{H_{\\varrho}}\\} \\leq r \\}) < \\infty$,\n$ \\liminf_{ m \\to \\infty } \\inf( \\{\\lambda_b \\colon b \\in \\mathbb{H} \\backslash \\mathbb{H}_m \\} \\cup \\{\\infty\\} ) = \\infty $,\n$ \\limsup_{m\\to \\infty} ( h_m + \\sup_{s \\in [0, T]} \\| O_s - \\mathcal{O}_s^m\\|_{H_{\\varrho}} ) = 0$,\n$ X_t = \\int_0^t e^{ ( t - s ) A } \\, F( X_s ) \\, ds + O_t$,\nand \n\\begin{align}\n\\mathcal{X}_t^n = \\int_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{[0,|h_n|^{-\\chi}]} \\big( \\big\\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\big\\|_{ H_{\\varrho} } + \\big\\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big\\|_{ H_{\\varrho} } \\big) \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds +\\mathcal{O}_t^n.\n\\end{align}\nThen it holds that $ \\limsup_{n \\to \\infty} \\sup_{t \\in [0, T]} \\| X_t- \\mathcal{X}_t^n \\|_{H_\\varrho}=0 $.\n\\end{prop}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:main_det}]\nNote that\nthe fact that\n$ \\forall \\, r \\in [ 0, 1 ], \\, t \\in ( 0, \\infty ) \\colon\n\\| (\\kappa-A)^{r} \\, e^{ t (A-\\kappa) } \\|_{L(H)} \\leq t^{-r} $\n(cf., e.g., Lemma~11.36 in Renardy \\& Rogers~\\cite{RenardyRogers1993})\nproves for all $ n \\in \\mathbb{N} $, $ t \\in [0, T] $, $ \\varepsilon \\in [0, 1 - \\varrho - \\alpha) $ that \n\\begin{align} \n\\label{eq:sup_s_alpha_finite} \n\\begin{split} \n\\sup_{ s \\in (0, T] } \\big( s^{ ( \\varrho + \\alpha ) } \\| e^{ s A } \\|_{ L( H_{ - \\alpha }, H_{ \\varrho } ) } \\big) & = \\sup_{ s \\in (0, T] } \\big( s^{ ( \\varrho + \\alpha ) } \\| (\\kappa-A)^{ ( \\varrho + \\alpha ) } \\, e^{ s A } \\|_{ L(H) } \\big) \\\\ \n& \\leq e^{T \\kappa} \\cdot \\sup_{ s \\in (0, T] } \\big( s^{ ( \\varrho + \\alpha ) } \\| (\\kappa-A)^{ ( \\varrho + \\alpha ) } \\, e^{ s (A-\\kappa) } \\|_{ L(H) } \\big) \\leq e^{T \\kappa} < \\infty \n\\end{split} \n\\end{align} \nand \n\\begin{align} \n\\begin{split} \n& \\int_0^t \\| ( \\mathrm{Id}_{H_{\\varrho}} - P_n ) \\, e^{ s A } \\|_{ L( H_{ - \\alpha } , H_{ \\varrho } ) } \\, ds \\leq \\int_0^t \\| \\mathrm{Id}_{H_{\\varrho+\\varepsilon}} - P_n|_{H_{\\varrho+\\varepsilon}} \\|_{ L( H_{ \\varrho + \\varepsilon } , H_{ \\varrho } ) } \\, \\| e^{ s A } \\|_{ L( H_{ - \\alpha }, H_{ \\varrho + \\varepsilon } ) } \\, ds \\\\ \n& = \\| ( \\kappa- A )^{ - \\varepsilon } ( \\mathrm{Id}_H - P_n ) \\|_{ L( H ) } \\int_0^t \\| (\\kappa - A )^{ ( \\varrho + \\varepsilon + \\alpha ) } \\, e^{ s A } \\|_{ L(H) } \\, ds \\\\ \n& = \\| ( \\kappa- A )^{ -1 } ( \\mathrm{Id}_H - P_n ) \\|_{ L( H ) }^{\\varepsilon} \\int_0^t e^{s \\kappa} \\, \\| (\\kappa - A )^{( \\varrho + \\varepsilon + \\alpha ) }\\, e^{ s (A-\\kappa) } \\|_{ L(H) } \\, ds \\\\ \n& \\leq e^{T \\kappa} \\, \\| (\\kappa- A )^{ - 1 } ( \\mathrm{Id}_H - P_n ) \\|_{ L( H ) }^{ \\varepsilon } \\int_0^t s^{ - ( \\varrho + \\varepsilon + \\alpha ) } \\, ds \\\\ \n& \\leq \\frac{ e^{T \\kappa} \\, \\| ( \\kappa - A )^{ - 1 } ( \\mathrm{Id}_H - P_n ) \\|_{ L( H ) }^{ \\varepsilon } \\, T^{ ( 1 - \\varrho - \\varepsilon - \\alpha ) } }{ ( 1 - \\varrho - \\varepsilon -\\alpha) } . \n\\end{split} \n\\end{align} \nThis and the assumption that $ \\liminf_{ n \\to \\infty} \\inf( \\{\\lambda_b \\colon b \\in \\mathbb{H} \\backslash \\mathbb{H}_n \\} \\cup \\{\\infty\\} ) = \\infty $ imply that\n\\begin{align}\n\\label{eq:sup_Id_H_0} \n\\limsup_{ n \\to \\infty } \\left( \\int_0^T \\| (\\mathrm{Id}_{H_{\\varrho}} - P_n ) e^{ s A } \\|_{ L( H_{ - \\alpha } , H_{ \\varrho } ) } \\, ds \\right) = 0 . \n\\end{align} \nCombining the fact that $\\limsup_{ n \\to \\infty} h_n =0 $, the fact that $ \\limsup_{n\\to \\infty} \\sup_{t \\in [0, T]} \\| O_t - \\mathcal{O}_t^n\\|_{H_{\\varrho}} = 0$, \\eqref{eq:sup_s_alpha_finite}, \\eqref{eq:sup_Id_H_0}, and the fact that $ \\limsup_{ n \\to \\infty } \\| P_n |_{H_{\\varrho}} \\|_{ L( H_{ \\varrho } ) } = 1 < \\infty $ with, e.g., Proposition~3.3 in Hutzenthaler et al.~\\cite{Salimova2016} (with $ V = H_{ \\varrho } $, $ W = H_{ - \\alpha } $, $ T = T $, $ \\chi = \\chi $, $\\Upsilon = \\sup_{t \\in (0,T]}(t^{(\\varrho+\\alpha)}\\|e^{tA}\\|_{L(H_{-\\alpha},H_{\\varrho})})$, $ \\alpha = \\varrho + \\alpha $, $ (P_n)_{n \\in \\mathbb{N}} = ( H_{ \\varrho } \\ni v \\mapsto P_n( v ) \\in H_{ \\varrho } )_{n \\in \\mathbb{N}} $, $ (h_n)_{n \\in \\mathbb{N}} = (h_n)_{n \\in \\mathbb{N}} $, $ F = F $, $\\Psi= ([0,\\infty] \\ni r \\mapsto \\sup (\\{ 0 \\} \\cup \\{\\|F(x)-F(y)\\|_{H_{-\\alpha}}\/\\|x-y\\|_{H_{\\varrho}} \\colon x, y \\in H_{\\varrho}, x\\neq y, \\max\\{\\|x\\|_{H_{\\varrho}},\\|y\\|_{H_{\\varrho}}\\} \\leq r \\}) \\in [0,\\infty])$, $X=X$, $O=O$, $(\\mathcal{X}_n)_{n \\in \\mathbb{N}} = (\\mathcal{X}_n)_{n \\in \\mathbb{N}}$, $(\\mathcal{O}_n)_{n \\in \\mathbb{N}} = (\\mathcal{O}_n)_{n \\in \\mathbb{N}}$, $ S = ((0,T] \\ni t \\mapsto ( H_{ - \\alpha } \\ni v \\mapsto e^{ t A } v \\in H_{ \\varrho } ) \\in L(H_{-\\alpha},H_{\\varrho}) ) $ in the notation of Proposition~3.3 in Hutzenthaler et al.~\\cite{Salimova2016}) shows that $ \\limsup_{n \\to \\infty} \\sup_{t \\in [0, T]} \\| X_t- \\mathcal{X}_t^n \\|_{H_\\varrho}=0 $. The proof of Proposition~\\ref{prop:main_det} is thus completed. \n\\end{proof}\n\n\n\\subsection{Pathwise a priori bounds}\n\n\\begin{prop}\\label{prop:main_det:2}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) $ be a separable $\\mathbb{R}$-Hilbert space,\nlet $ \\mathbb{H} \\subseteq H$ be a nonempty orthonormal basis of $ H $,\nlet $\\eta , \\theta , \\vartheta, \\kappa \\in [0, \\infty)$,\nlet $\\lambda \\colon \\mathbb{H} \\to \\mathbb{R} $\nbe a function which satisfies\n$\\inf_{b \\in \\mathbb{H}} \\lambda_b > -\\min\\{\\eta, \\kappa\\}$,\nlet $ A \\colon D(A) \\subseteq H \\to H $ be the linear operator which satisfies\n$ D(A) = \\{ v \\in H \\colon \\sum_{b \\in \\mathbb{H}} | \\lambda_b \\langle b , v \\rangle_H |^2 < \\infty \\} $\nand\n$ \\forall \\, v \\in D(A) \\colon A v = \\sum_{b \\in \\mathbb{H}} - \\lambda_b \\langle b , v \\rangle_H b$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $,\nbe a family of interpolation spaces associated to $ \\kappa- A $,\nlet $\\varphi \\in [0,1)$, $ \\alpha \\in [ 0, \\nicefrac{1}{2}]$, $\\rho \\in [0, 1- \\alpha)$, $\\varrho \\in (\\rho, 1-\\alpha)$, $ \\beta, T \\in (0,\\infty)$, $ \\chi \\in ( 0, \\min\\{ \\nicefrac{ ( 1 - \\alpha - \\rho ) }{( 1 + \\vartheta ) }, \\nicefrac{( \\varrho - \\rho ) }{( 1 + \\nicefrac{\\vartheta}{2} ) } \\}] $, $p \\in [2, \\infty)$, $F \\in \\mathcal{C}(H_{\\varrho}, H_{-\\alpha})$,\nlet\n$ \\phi, \\Phi \\colon H_1 \\to [0,\\infty) $,\n$ ( \\mathbb{H}_n )_{ n \\in \\mathbb{N} } \\colon \\mathbb{N} \\to \\mathcal{P}_0( \\mathbb{H} ) $,\n$ ( P_n )_{ n \\in \\mathbb{N} } \\colon \\mathbb{N} \\to L( H_{ -1 } ) $,\nand\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $\nbe functions,\nlet\n$ \\mathcal{X}^n, \\mathbb{O}^n \\colon [0,T] \\to H_{\\varrho} $, $ n \\in \\mathbb{N} $,\nand\n$ \\mathcal{O}^n \\colon [0,T] \\to P_n(H) $, $ n \\in \\mathbb{N} $,\nbe functions,\nand assume\nfor all $u \\in H$, $ n \\in \\mathbb{N} $, $ v, w \\in P_n(H) $, $ t \\in [0,T] $ that\n$P_n(u) = \\sum_{ b \\in \\mathbb{H}_n } \\langle b, u \\rangle_H b$,\n$ \\left< v, P_n F( v + w ) \\right>_H \\leq \\phi( w ) \\| v \\|^2_H + \\varphi \\| (\\eta-A)^{\\nicefrac{1}{2}} v \\|^2_{ H } + \\Phi( w ) $,\n$\\| F(v)\\|_{H_{-\\alpha}}^2 \\leq \\theta \\max\\{ 1, \\| v\\|_{ H_{\\varrho} }^{2 + \\vartheta} \\} $,\n$ \\|(\\eta-A)^{\\nicefrac{-1}{2}} [F(v) - F(w) ]\\|_{H}^2\\leq \\theta \\max\\{ 1, \\|v\\|_{ H_{\\varrho} }^{\\vartheta} \\} \\|v-w\\|_{ H_{\\rho} }^2 + \\theta \\, \\|v-w\\|^{2 + \\vartheta}_{ H_{\\rho}}$,\n$\\eta \\mathcal{O}^n \\in \\mathcal{C}([0, T], P_n( H ) )$,\n$\\mathbb{O}_t^n = \\mathcal{O}_t^n - \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta \\mathcal{O}_s^n \\, ds$,\nand \n\\begin{align}\n\\mathcal{X}_t^n = \\int_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{[0,|h_n|^{-\\chi}]} \\big( \\big\\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\big\\|_{ H_{\\varrho} } + \\big\\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big\\|_{ H_{\\varrho} } \\big) \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds +\\mathcal{O}_t^n.\n\\end{align}\nThen\n\\begin{enumerate}[(i)]\n\\item \\label{item:Pn} it holds for all $n \\in \\mathbb{N}$ that $ \\mathcal{X}^n( [0,T] ) \\cup \\mathbb{O}^n( [0,T] ) \\subseteq P_n( H )$ and\n\\item \\label{item:bound} it holds for all $t \\in [0,T]$, $n \\in \\mathbb{N}$ with $ h_n \\leq 1 $ that $\\eta \\mathbb{O}^n \\in \\mathcal{C}([0, T], H_{\\varrho})$ and \n\\begin{align*}\n\\| \\mathcal{X}^n_t \\|_H^p \n&\\leq 2^{p-1}\\| \\mathbb{O}^n_t \\|_H^p + 2^{p-1} t^{\\nicefrac{p}{2}-1} \\bigg[1 + \\tfrac{ \\theta e^{ \\kappa(2+ \\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta} ]^{2+\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}}\\bigg]^{\\nicefrac{p}{2}} \\\\\n& \\quad \\cdot \\int_0^t e^{ \\int_s^t p \\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +p\\eta (1+\\beta) \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\numberthis \\\\\n& \\quad + \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 2\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\Big]^{\\nicefrac{p}{2}} ds.\n\\end{align*}\n\\end{enumerate}\n\n\\end{prop}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:main_det:2}]\nWithout loss of generality we assume for all $n \\in \\mathbb{N}$ that $\\mathbb{H}_n \\neq \\emptyset$.\nThroughout this proof let $ \\tilde{\\phi}_n \\colon P_n( H ) \\to [0,\\infty) $, $ n \\in \\mathbb{N} $, and $\\tilde{\\Phi}_n \\colon P_n( H ) \\to [0,\\infty) $, $ n \\in \\mathbb{N} $, be the functions which satisfy for all $ n \\in \\mathbb{N} $, $ v \\in P_n( H ) $ that $\\tilde{\\phi}_n( v ) = 2 \\cdot \\phi( v )$ and $\\tilde{\\Phi}_n( v ) = 2 \\cdot \\Phi( v )$.\nNote that for all $n \\in \\mathbb{N}$ it holds that $P_n(H)$ is a finite-dimensional $\\mathbb{R}$-vector space and $ \\mathcal{X}^n( [0,T] ) \\cup \\mathbb{O}^n( [0,T] ) \\subseteq P_n( H )$. Corollary~\\ref{cor:a_priori} (with \n$ H = P_n( H ) $,\n$ \\mathbb{H} = \\mathbb{H}_n $,\n$\\beta=\\beta$, $T =T$, $\\eta=\\eta$, $ \\theta= \\theta$, $\\vartheta = \\vartheta$, $\\kappa= \\kappa$,\n$ \\varphi= \\varphi$, $\\alpha = \\alpha$, $\\rho= \\rho$, $ \\varrho = \\varrho$,\n$\\chi = \\chi $, \n$ h = h_n$,\n$ F = ( P_n( H ) \\ni v \\mapsto P_n F( v ) \\in P_n(H) ) \\in \\mathcal{C}( P_n( H ), P_n( H ) ) $,\n$ A = ( P_n( H ) \\ni v \\mapsto A v \\in P_n( H ) ) \\in L( P_n( H ) ) $, \n$ Y = ( [0,T] \\ni t\\mapsto \\mathcal{X}^n_t \\in P_n( H ) ) $, \n$ O = \\mathcal{O}^n $, \n$ \\mathbb{O} = ( [0,T] \\ni t \\mapsto \\mathbb{O}^n_t \\in P_n( H ) ) $,\n$ \\phi = \\tilde{\\phi}_n $, $ \\Phi = \\tilde{\\Phi}_n $\nfor $ n \\in \\{ m \\in \\mathbb{N} \\colon h_m \\leq 1 \\}$ in the notation of Corollary~\\ref{cor:a_priori}) hence proves that for all $t \\in [0, T]$, $ n \\in \\mathbb{N} $ with $h_n \\leq 1$ it holds that $\\eta \\mathbb{O}^n \\in \\mathcal{C}([0, T], H_{\\varrho})$ and\n\\begin{align*}\n\\label{eq:thm_application}\n& \\| \\mathcal{X}^n_t - \\mathbb{O}^n_t \\|_H^2 \\leq \\int_0^t e^{ \\int_s^t \\tilde{\\phi}_n( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2\\eta (1+\\beta) \\, du} \\Big[ \\tilde{\\Phi}_n\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\\\\n& + \\tfrac{ \\theta e^{\\kappa(2+\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta} ]^{2+\\vartheta} \\left|\\max \\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\}\\right|^{2+ \\vartheta}}{(1- \\varphi )( 1-\\alpha - \\rho)^{2+\\vartheta}} \\\\\n& \\cdot \\max \\!\\big\\{ |h_n|^{2(\\varrho - \\rho -\\chi)},|h_n|^{ 2(1-\\alpha - \\rho -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h_n \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\left|\\max \\!\\big\\{ |h_n|^{-\\chi}, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{\\vartheta}\\Big] ds \\\\\n& = \\int_0^t e^{ \\int_s^t 2\\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2\\eta (1+\\beta) \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\numberthis \\\\\n& + \\tfrac{\\theta e^{ \\kappa(2+ \\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta} ]^{2+\\vartheta} \\left|\\max \\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\}\\right|^{2+ \\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}} \\\\\n& \\cdot \\max \\!\\big\\{ |h_n|^{2(\\varrho - \\rho -\\chi)},|h_n|^{ 2(1-\\alpha - \\rho -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h_n \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\left|\\max \\!\\big\\{ |h_n|^{-\\chi}, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{ \\vartheta}\\Big] ds.\n\\end{align*}\nMoreover, the assumption that $ \\chi \\in ( 0, \\min\\{ \\nicefrac{ ( 1 - \\alpha - \\rho ) }{( 1 + \\vartheta ) }, \\nicefrac{( \\varrho - \\rho ) }{( 1 + \\nicefrac{\\vartheta}{2} ) } \\}] $ implies that\n\\begin{equation}\n1 - \\alpha - \\rho -( 1 + \\vartheta ) \\chi\\geq 0, \\qquad 1-\\vartheta \\chi \\geq 0, \\qquad \\text{and} \\qquad \\varrho - \\rho -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi\\geq 0.\n\\end{equation}\nThis ensures for all $ n \\in \\mathbb{N} $ with $ h_n \\leq 1 $ that\n\\begin{equation}\n\\begin{split}\n& \\max \\!\\big\\{ |h_n|^{2(\\varrho - \\rho -\\chi)},|h_n|^{ 2(1-\\alpha - \\rho -( 1 + \\nicefrac{\\vartheta}{2} ) \\chi)} , h_n \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\left|\\max \\!\\big\\{ |h_n|^{-\\chi}, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{ \\vartheta}\\\\\n& = \\max \\!\\big\\{ |h_n|^{2(\\varrho - \\rho -(1+\\nicefrac{\\vartheta}{2})\\chi)},|h_n|^{ 2(1-\\alpha - \\rho -( 1 + \\vartheta ) \\chi)} , |h_n|^{1-\\vartheta \\chi} \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n& \\quad \\cdot \\left|\\max \\!\\big\\{ 1, |h_n|^{\\chi} \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{ \\vartheta} \\\\\n& \\leq \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{ \\vartheta}.\n\\end{split}\n\\end{equation}\nThe fact that $\\forall \\, x,y \\in H \\colon \\|x+y\\|_H^2 \\leq 2\\|x\\|_H^2 + 2\\|y\\|_H^2$ and \\eqref{eq:thm_application} hence show that for all $t \\in [0,T]$, $n \\in \\mathbb{N}$ with $ h_n \\leq 1 $ it holds that\n\\begin{align}\n\\begin{split}\n\\| \\mathcal{X}^n_t \\|_H^2 \n&\\leq 2 \\, \\| \\mathbb{O}^n_t \\|_H^2 + 2\\int_0^t e^{ \\int_s^t 2\\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2\\eta (1+\\beta) \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\\\\n&\\quad + \\tfrac{ \\theta e^{ \\kappa(2+ \\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta}]^{2+\\vartheta} \\left|\\max \\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\}\\right|^{2+ \\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}} \\\\\n& \\quad \\cdot \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{ \\vartheta}\\Big] ds\\\\\n& = 2 \\, \\| \\mathbb{O}^n_t \\|_H^2 + 2\\int_0^t e^{ \\int_s^t 2\\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2\\eta (1+\\beta) \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\\\\n& \\quad + \\tfrac{ \\theta e^{ \\kappa(2+ \\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta} ]^{2+\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}} \\\\\n& \\quad \\cdot \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 2\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\Big] ds \\\\\n& \\leq 2 \\, \\| \\mathbb{O}^n_t \\|_H^2 + 2 \\bigg[1 + \\tfrac{ \\theta e^{ \\kappa(2+ \\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta}]^{2+\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}}\\bigg] \\\\\n& \\quad \\cdot \\int_0^t e^{ \\int_s^t 2\\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2\\eta (1+\\beta) \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\\\\n& \\quad + \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 2\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\Big] ds.\n\\end{split}\n\\end{align}\nThis, the assumption that $p \\in [2, \\infty)$, the fact that $\\forall \\, a, b \\in \\mathbb{R} \\colon |a+b|^{\\nicefrac{p}{2}} \\leq 2^{\\nicefrac{p}{2}-1} |a|^{\\nicefrac{p}{2}} + 2^{\\nicefrac{p}{2}-1} |b|^{\\nicefrac{p}{2}} $, and H\\\"older's inequality prove for all $t \\in [0,T]$, $n \\in \\mathbb{N}$ with $ h_n \\leq 1 $ that\n\\begin{align}\n\\label{eq:thm:bound}\n\\begin{split}\n&\\| \\mathcal{X}^n_t \\|_H^p \n\\leq 2^{p-1}\\| \\mathbb{O}^n_t \\|_H^p + 2^{p-1} \\bigg[1 + \\tfrac{ \\theta e^{ \\kappa(2+ \\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta} ]^{2+\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}}\\bigg]^{\\nicefrac{p}{2}} \\\\\n& \\quad \\cdot \\bigg[\\int_0^t e^{ \\int_s^t 2\\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2\\eta (1+\\beta) \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\\\\n& \\quad + \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 2\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\Big] ds \\bigg]^{\\nicefrac{p}{2}}\\\\\n&\\leq 2^{p-1}\\| \\mathbb{O}^n_t \\|_H^p + 2^{p-1} t^{\\nicefrac{p}{2}-1} \\bigg[1 + \\tfrac{ \\theta e^{ \\kappa(2+ \\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + \\sqrt{\\theta} + \\sqrt{\\eta} ]^{2+\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\vartheta}}\\bigg]^{\\nicefrac{p}{2}} \\\\\n& \\quad \\cdot \\int_0^t e^{ \\int_s^t p \\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +p\\eta (1+\\beta) \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 \\beta} \\|\\mathbb{O}_s^n\\|_H^2 \\\\\n& \\quad + \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 2\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\Big]^{\\nicefrac{p}{2}} ds .\n\\end{split}\n\\end{align}\nThe proof of Proposition~\\ref{prop:main_det:2} is thus completed.\n\\end{proof}\n\n\n\n\\subsection{Strong convergence}\n\\label{subsec:thm:strong}\n\n\n\n\n\\begin{theorem}\n\\label{thm:strong}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ ( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) $ be a separable $\\mathbb{R}$-Hilbert space,\nlet $ ( \\Omega, \\F, \\P ) $ be a probability space,\nlet $ \\mathbb{H} \\subseteq H$ be a nonempty orthonormal basis of $ H $,\nlet $\\eta, \\theta, \\kappa \\in [0, \\infty)$,\nlet $\\lambda \\colon \\mathbb{H} \\to \\mathbb{R} $\nbe a function which satisfies $\\inf_{b \\in \\mathbb{H}} \\lambda_b > -\\min\\{\\eta, \\kappa\\}$,\nlet $ A \\colon D(A) \\subseteq H \\to H $\nbe the linear operator which satisfies\n$ D(A) = \\{ v \\in H \\colon \\sum_{b \\in \\mathbb{H}} | \\lambda_b \\langle b , v \\rangle_H |^2 < \\infty \\} $\nand\n$ \\forall \\, v \\in D(A) \\colon A v = \\sum_{b \\in \\mathbb{H}} - \\lambda_b \\langle b , v \\rangle_H b$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $,\nbe a family of interpolation spaces associated to $ \\kappa- A $,\nlet $\\varphi \\in [0,1)$, $ \\alpha \\in [ 0, \\nicefrac{1}{2}]$, $\\rho \\in [0, 1- \\alpha)$, $\\varrho \\in (\\rho, 1-\\alpha)$, $ \\vartheta,T \\in (0,\\infty)$, $ \\chi \\in ( 0, \\min\\{ \\nicefrac{ ( 1 - \\alpha - \\rho ) }{( 1 + 2 \\vartheta ) }, \\nicefrac{( \\varrho - \\rho ) }{( 1 + \\vartheta) } \\}] $, $p \\in [2, \\infty)$, $F \\in \\mathcal{C}(H_{\\varrho}, H_{-\\alpha} )$,\nlet\n$ \\phi, \\Phi \\colon H_1 \\to [0,\\infty) $,\n$ ( \\mathbb{H}_n )_{ n \\in \\mathbb{N} } \\colon \\mathbb{N} \\to \\mathcal{P}_0( \\mathbb{H} ) $,\n$ ( P_n )_{ n \\in \\mathbb{N} } \\colon \\mathbb{N} \\to L( H_{ -1 } ) $,\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $,\nand\n$ \\mathbb{X}^n, \\mathbb{O}^n \\colon [0,T]\\times \\Omega \\to H_{\\varrho} $, $ n \\in \\mathbb{N} $,\nbe functions,\nlet $ \\mathcal{X}^n \\colon [0,T] \\times \\Omega \\to H_{ \\varrho } $, $ n \\in \\mathbb{N} $, be stochastic processes,\nlet $ \\mathcal{O}^n \\colon [0,T] \\times \\Omega \\to P_n( H ) $, $ n \\in \\mathbb{N} $,\nand\n$ X, O \\colon [0, T] \\times \\Omega \\to H_{ \\varrho} $\nbe stochastic processes with continuous sample paths,\nand assume\nfor all $u \\in H$, $n \\in \\mathbb{N}$, $v, w \\in P_n(H)$, $ t \\in [0,T] $ that\n$ P_n(u) = \\sum_{ b \\in \\mathbb{H}_n } \\langle b, u \\rangle_H b $,\n$ \\left< v, P_n F( v + w ) \\right>_H \\leq \\phi( w ) \\| v \\|^2_H + \\varphi \\| (\\eta-A)^{\\nicefrac{1}{2}} v \\|^2_{ H} + \\Phi( w ) $,\n$ \\left\\| F(v) - F(w) \\right\\|_{ H_{ - \\alpha } } \\leq \\theta \\, ( 1 + \\| v \\|_{ H_{ \\rho } }^{ \\vartheta } + \\|w\\|_{H_{\\rho}}^{\\vartheta}) \\, \\|v-w\\|_{H_{\\rho}} $,\n$ \\liminf_{ m \\to \\infty } \\inf( \\{\\lambda_b \\colon b \\in \\mathbb{H} \\backslash \\mathbb{H}_m \\} \\cup \\{\\infty\\} ) = \\infty $,\n$\\limsup_{m \\to \\infty} \\mathbb{E} \\big[\\!\\min\\!\\big\\{ 1, \\sup_{s \\in [0,T]}\\| O_s- \\mathcal{O}_s^{m} \\|_{ H_{ \\varrho } } \\big\\} + h_m \\big] = 0$,\n$\\mathbb{O}_t^n = \\mathcal{O}_t^n - \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta \\mathcal{O}_s^n \\, ds$,\n$ \\mathbb{X}_t^n = \\smallint\\nolimits_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{ \\varrho } } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{ \\varrho } } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds +\\mathcal{O}_t^n$,\n$ \\limsup_{ m \\to \\infty} \\sup_{ s \\in [0,T]} \\mathbb{E} \\bigl[ \\| \\mathbb{O}_s^m \\|_H^p \\bigr] < \\infty$,\n$ \\P\\big( X_t = \\int_0^t e^{ ( t - s ) A } \\, F( X_s ) \\, ds + O_t \\big)=\\P (\\mathbb{X}_t^n = \\mathcal{X}_t^n )=1$,\nand\n\\begin{equation}\n\\label{eq:integrability}\n \\limsup_{ m \\to \\infty} \\mathbb{E}\\biggl[ \\int_0^T e^{ \\int_s^T p\\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_m} }^m ) \\, du} \\max\\bigl\\{ 1, |\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_m} }^m )|^{\\nicefrac{p}{2}}, \\|\\mathbb{O}_s^m\\|_H^p, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^m \\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\bigr\\} \\, ds\\biggr] < \\infty.\n\\end{equation}\nThen\n\\begin{enumerate}[(i)]\n\\item \\label{item:exp} it holds that $\\limsup_{n \\to \\infty} \\mathbb{E} \\big[\\!\\min\\!\\big\\{ 1, \\sup_{t \\in [0,T]}\\| X_t- \\mathbb{X}_t^{n} \\|_{ H_{ \\varrho } } \\big\\} \\big] = 0$, \n\\item \\label{item:pnorm} it holds that $ \\limsup_{ n \\to \\infty } \\sup_{ t \\in [0,T] } \\mathbb{E}\\big[ \\| X_t \\|^p_H + \\| \\mathcal{X}^n_t \\|_H^p \\big] < \\infty $, and\n\\item \\label{item:strong} it holds for all $ q \\in (0, p) $ that $ \\limsup_{ n \\to \\infty } \\sup_{ t \\in [0,T] } \\mathbb{E}\\big[ \\| X_t - \\mathcal{X}_t^n \\|_H^q \\big] = 0 $.\n\\end{enumerate}\n\\end{theorem}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:strong}]\nThroughout this proof let $ \\tilde{\\Omega} $ be the set given by\n\\begin{equation}\n\\begin{split}\n\\tilde{\\Omega} & =\n\\Bigl\\{\\omega \\in \\Omega \\colon \\Bigl(\\forall \\, m \\in \\mathbb{N}, s \\in [0, T] \\colon \\mathbb{X}_{\\lfloor s \\rfloor_{h_m}}^m(\\omega)= \\mathcal{X}_{\\lfloor s \\rfloor_{h_m}}^m(\\omega) \\Bigr)\\Bigr\\}\n\\\\ & \\quad\n\\cap\n\\biggl\\{ \\omega \\in \\Omega \\colon \\biggl( \\forall \\, s \\in [0,T] \\colon X_s(\\omega) = \\int_0^s e^{ ( s - u ) A } \\, F( X_u(\\omega) ) \\, du + O_s(\\omega)\\biggr)\\biggr\\},\n\\end{split}\n\\end{equation}\nlet $ \\tilde{\\mathcal{X}}^n \\colon [0,T] \\times \\Omega \\to H_{\\varrho} $, $ n \\in \\mathbb{N} $,\nbe the sequence which satisfies for all $n \\in \\mathbb{N}$, $t \\in [0, T]$ that\n\\begin{align}\n\\label{eq:tilde_X}\n\\tilde{\\mathcal{X}}_t^n = \\int_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\|\\tilde{\\mathcal{X}}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{ \\varrho } } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{ \\varrho } } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\tilde{\\mathcal{X}}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds +\\mathcal{O}_t^n,\n\\end{align} \nand let $ \\tilde{\\theta} \\in [0, \\infty)$, $\\tilde{\\vartheta} \\in (0,\\infty)$ \nbe the real numbers given by $ \\tilde{\\vartheta} = 2 \\vartheta $ \nand \n\\begin{multline}\n\\tilde{\\theta} = \n\\max\\{1, \\|(\\eta-A)^{-1} (\\kappa-A)\\|_{L(H)}\\} \\max\\!\\bigg\\{ \\big(8 \\theta^2 + 2 \\, \\| F(0) \\|_{ H_{ - \\alpha } }^2 \\big) \\max\\!\\bigg\\{ 1, \\sup_{\n\tu \\in H_{ \\varrho } \\backslash \\{ 0 \\} }\\tfrac{ \\| u \\|_{ H_{ \\rho } }^{ 2 + 2 \\vartheta } \n}{\\| u \\|_{ H_{ \\varrho } }^{ 2 + 2 \\vartheta } } \\bigg\\} ,\\\\\n3 \\, \\theta^2 \\bigg[\\sup_{ u \\in H_{ - \\alpha } \\backslash\\{ 0 \\} }\\tfrac{ \\| u \\|_{ H_{ \\nicefrac{-1}{2} } }^2 }{\\| u \\|_{ H_{ - \\alpha } }^2 } \\bigg] \\bigg[1 + \\sup_{ u \\in H_{ \\varrho } \\backslash \\{ 0 \\} }\\tfrac{\n\t\\| u \\|_{ H_{ \\rho } }^{ 2 \\vartheta } }{\\| u \\|_{H_{\\varrho}}^{2 \\vartheta}} \\bigg]\n\\big(1+2^{\\max\\{2\\vartheta-1, 0\\}}\\big) \\! \\bigg\\}.\n\\end{multline}\nObserve that for all $n \\in \\mathbb{N}$ it holds that $X$, $O$, $\\mathbb{X}^n$, $\\mathcal{O}^n$, $\\mathbb{O}^n$, $\\tilde{\\mathcal{X}}^n$ are stochastic processes with continuous sample paths.\nIn addition, note that the assumption that $X, O \\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ are stochastic processes with continuous sample paths\nand\nthe assumption that\n$\\forall \\, t \\in [0, T] \\colon \\P\\big(X_t = \\int_0^t e^{ ( t - s ) A } \\, F( X_s ) \\, ds + O_t\\big)=1$\nshow that\n\\begin{equation}\n\\biggl\\{ \\omega \\in \\Omega \\colon \\biggl(\\forall \\, t \\in [0,T] \\colon X_t(\\omega) = \\int_0^t e^{ ( t - s ) A } \\, F( X_s(\\omega) ) \\, ds + O_t(\\omega) \\biggr)\\biggr\\} \\in \\F\n\\end{equation}\nand\n\\begin{equation}\n\\P\\biggl(\\forall \\, t \\in [0,T] \\colon X_t = \\int_0^t e^{ ( t - s ) A } \\, F( X_s ) \\, ds + O_t\\biggr)=1.\n\\end{equation}\nThis and the assumption that\n$\\forall \\, n\\in \\mathbb{N}, \\, t \\in [0, T] \\colon \\P(\\mathbb{X}_t^n = \\mathcal{X}_t^n)=1$\nyield that $\\tilde{\\Omega} \\in \\F$ and $\\P(\\tilde{\\Omega})=1$.\nIn the next step let $k \\colon \\mathbb{N} \\to \\mathbb{N} $ be a strictly increasing function.\nThe fact that\n$\\limsup_{n \\to \\infty} \\mathbb{E} \\bigl[\\min\\bigl\\{ 1, \\sup_{t \\in [0,T]}\\| O_t- \\mathcal{O}_t^{n} \\|_{ H_{ \\varrho } } \\bigr\\} \\bigr] = 0$\nassures that\n\\begin{equation}\n\\limsup_{n \\to \\infty} \\mathbb{E} \\biggl[ \\min \\biggl\\{ 1, \\sup_{t \\in [0,T]}\\| O_t- \\mathcal{O}_t^{k(n)} \\|_{ H_{ \\varrho } } \\biggr\\} \\biggr] = 0.\n\\end{equation}\nThis implies that there exists a strictly increasing function $l \\colon \\mathbb{N} \\to \\mathbb{N} $ such that\n\\begin{equation}\n\\sum_{n=1}^{\\infty} \\mathbb{E} \\biggl[ \\min \\biggl\\{ 1, \\sup_{t \\in [0,T]}\\| O_t- \\mathcal{O}_t^{k(l(n))} \\|_{ H_{ \\varrho } } \\biggr\\} \\biggr] < \\infty.\n\\end{equation}\nLemma~\\ref{lem:fast_convergence}\n(with\n$(\\Omega, \\mathcal{F}, \\P)= (\\Omega, \\mathcal{F}, \\P)$,\n$E=\\mathbb{R}$,\n$d= ( \\mathbb{R} \\times \\mathbb{R} \\ni (x,y) \\mapsto | x - y | \\in [0,\\infty) )$,\n$(X_n)_{n \\in \\mathbb{N}} = ( \\Omega \\ni \\omega \\mapsto\n\\sup_{t \\in [0,T]}\\| O_t( \\omega ) - \\mathcal{O}_t^{k(l(n))}( \\omega ) \\|_{ H_{ \\varrho } } \\in \\mathbb{R} )_{n \\in \\mathbb{N}}$,\n$X_0 = ( \\Omega \\ni \\omega \\mapsto 0 \\in \\mathbb{R} ) $\nin the notation of Lemma~\\ref{lem:fast_convergence})\nhence proves that\n\\begin{equation}\n\\P\\biggl(\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]}\\| O_t- \\mathcal{O}_t^{k(l(n))} \\|_{ H_{ \\varrho } } = 0\\biggr)=1.\n\\end{equation}\nCombining this,\n\\eqref{eq:tilde_X},\nthe fact that $\\forall \\, \\omega \\in \\tilde{\\Omega}, \\, t \\in [0,T] \\colon X_t(\\omega) = \\int_0^t e^{ ( t - s ) A } \\, F( X_s(\\omega) ) \\, ds + O_t(\\omega)$,\nand\nthe fact that $\\P(\\tilde{ \\Omega })=1$\nwith Proposition~\\ref{prop:main_det}\n(with $ H = H $,\n$ \\mathbb{H}= \\mathbb{H}$,\n$\\kappa=\\kappa$,\n$A=A$,\n$\\alpha = \\alpha$,\n$ \\varrho = \\varrho$,\n$T =T$,\n$\\chi = \\chi $,\n$ F = F $,\n$X=([0,T] \\ni t \\mapsto X_t(\\omega) \\in H_{\\varrho})$,\n$O= ([0,T] \\ni t \\mapsto O_t(\\omega) \\in H_{\\varrho})$,\n$ ( \\mathbb{H}_n )_{ n \\in \\mathbb{N} } = ( \\mathbb{H}_{k(l(n))} )_{ n \\in \\mathbb{N} }$,\n$ ( P_n )_{ n \\in \\mathbb{N} } = ( H \\ni v \\mapsto P_{k(l(n))}(v) \\in H )_{ n \\in \\mathbb{N} }$,\n$ (h_n)_{n \\in \\mathbb{N}} = (h_{k(l(n))})_{n \\in \\mathbb{N}}$,\n$ (\\mathcal{X}^n)_{n \\in \\mathbb{N}} = ( [0,T] \\ni t \\mapsto \\tilde{\\mathcal{X}}^{k(l(n))}_t( \\omega ) \\in H_{\\varrho} )_{n \\in \\mathbb{N}} $,\n$(\\mathcal{O}^n)_{n \\in \\mathbb{N}} = ( [0,T] \\ni t \\mapsto \\mathcal{O}^{k(l(n))}_t( \\omega ) \\in H_{\\varrho} )_{n \\in \\mathbb{N}} $\nfor $\\omega \\in \\{ \\varpi \\in \\Omega \\colon \\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]}\\| O_t(\\varpi)- \\mathcal{O}_t^{k(l(n))}(\\varpi) \\|_{ H_{ \\varrho } } = 0\\} \\cap \\tilde{\\Omega}$ in the notation of Proposition~\\ref{prop:main_det})\nestablishes that\n\\begin{equation}\n\\P\\biggl(\\limsup_{n \\to \\infty} \\sup_{t \\in [0, T]} \\| X_t- \\tilde{\\mathcal{X}}_t^{k(l(n))} \\|_{H_\\varrho}=0\\biggr)=1.\n\\end{equation}\nThe fact that\n$\\forall \\, \\omega \\in \\tilde{\\Omega}, \\, t\\in [0,T], \\, n \\in \\mathbb{N} \\colon \\tilde{\\mathcal{X}}^n_t(\\omega) = \\mathbb{X}_t^n(\\omega)$\nand\nthe fact that\n$\\P(\\tilde{\\Omega})=1$\nhence show that\n\\begin{equation}\n\\P\\biggl(\\limsup_{n \\to \\infty} \\sup_{t \\in [0, T]} \\| X_t- \\mathbb{X}_t^{k(l(n))} \\|_{H_\\varrho}=0\\biggr)=1.\n\\end{equation}\nThis and Lebesgue's theorem of dominated convergence imply that\n\\begin{equation}\n\\limsup_{n \\to \\infty} \\mathbb{E} \\biggl[\\min\\biggl\\{ 1, \\sup_{t \\in [0,T]}\\| X_t- \\mathbb{X}_t^{k(l(n))} \\|_{ H_{ \\varrho } } \\biggr\\} \\biggr] = 0.\n\\end{equation}\nAs $k \\colon \\mathbb{N} \\to \\mathbb{N} $ was an arbitrary strictly increasing function,\nLemma~\\ref{lem:subsequence} proves that\n\\begin{equation}\n\\limsup_{n \\to \\infty}\n\\mathbb{E} \\biggl[\\min\\biggl\\{ 1, \\sup_{t \\in [0,T]}\\| X_t- \\mathbb{X}_t^{n} \\|_{ H_{ \\varrho } } \\biggr\\} \\biggr] = 0.\n\\end{equation}\nThis concludes \\eqref{item:exp}.\nNext\nnote that, e.g., Lemma~2.4 in Hutzenthaler et al.~\\cite{Salimova2016} (with $V=H_{\\varrho}$, $\\mathcal{V}=H_{\\rho}$, $W=H_{-\\alpha}$, $\\mathcal{W}= H_{\\nicefrac{-1}{2}}$, $\\epsilon=\\theta$, $\\theta=(\\max\\{1, \\|(\\eta-A)^{-1} (\\kappa-A)\\|_{L(H)}\\})^{-1} \\tilde{\\theta}$, $\\varepsilon= \\vartheta$, $\\vartheta = \\tilde{\\vartheta}$, $F=F$ in the notation of Lemma~2.4 in Hutzenthaler et al.~\\cite{Salimova2016}) ensures that for all $v, w \\in H_{\\varrho}$ it holds that \n\\begin{align}\n\\label{eq:F_condition_proof_1}\n\\begin{split}\n\\| (\\eta -A)^{\\nicefrac{-1}{2}} [F(v) - F(w)] \\|_{H}^2 & \\leq \\|(\\eta-A)^{\\nicefrac{-1}{2}} (\\kappa-A)^{\\nicefrac{1}{2}} \\|_{L(H)}^2 \\|F(v)- F(w)\\|_{H_{\\nicefrac{-1}{2}}}^2 \\\\\n&\\leq \\max\\{1, \\|(\\eta-A)^{-1} (\\kappa-A)\\|_{L(H)}\\} \\|F(v)- F(w)\\|_{H_{\\nicefrac{-1}{2}}}^2 \\\\\n&\\leq \\tilde{\\theta} \\max\\{1, \\|v\\|_{H_{\\varrho}}^{\\tilde{\\vartheta}}\\}\\|v-w\\|_{H_{\\rho}}^2 + \\tilde{\\theta} \\, \\|v-w\\|_{H_{\\rho}}^{2+\\tilde{\\vartheta}}\n\\end{split}\n\\end{align}\nand \n\\begin{equation}\n\\label{eq:F_condition_proof_2}\n\\|F(v)\\|_{H_{-\\alpha}}^2 \\leq \\tilde{\\theta} \\max\\{ 1, \\|v\\|_{H_{\\varrho}}^{2 + \\tilde{\\vartheta}} \\}.\n\\end{equation}\nFurthermore, observe that the assumption that $\\chi \\in ( 0, \\nicefrac{ ( 1 - \\alpha - \\rho ) }{( 1 + 2 \\vartheta) }]\\cap (0, \\nicefrac{( \\varrho - \\rho ) }{( 1 + \\vartheta) }]$ assures that $\\chi \\in ( 0, \\nicefrac{ ( 1 - \\alpha - \\rho ) }{( 1 + \\tilde{ \\vartheta}) }]\\cap (0, \\nicefrac{( \\varrho - \\rho ) }{( 1 + \\nicefrac{\\tilde{\\vartheta}}{2} ) }]$. Combining this, \\eqref{eq:F_condition_proof_1}, and \\eqref{eq:F_condition_proof_2} with Proposition~\\ref{prop:main_det:2} (with\n $ H = H $,\n $ \\mathbb{H}= \\mathbb{H}$,\n$\\eta=\\eta$,\n$ \\theta= \\tilde{\\theta}$, $\\vartheta = \\tilde{\\vartheta}$, $\\kappa=\\kappa$, $A=A$,\n$ \\varphi= \\varphi$, $\\alpha = \\alpha$,\n$\\rho= \\rho$, $ \\varrho = \\varrho$,\n$\\beta=1$,\n$T =T$,\n$\\chi = \\chi $, $p=p$, \n$ F = F $, $ \\phi = \\phi$, $ \\Phi = \\Phi $,\n$ ( \\mathbb{H}_n )_{ n \\in \\mathbb{N} } = ( \\mathbb{H}_n )_{ n \\in \\mathbb{N} }$,\n$ ( P_n )_{ n \\in \\mathbb{N} } = ( P_n )_{ n \\in \\mathbb{N} }$,\n$ (h_n)_{n \\in \\mathbb{N}} = (h_n)_{n \\in \\mathbb{N}}$,\n$ (\\mathcal{X}^n)_{n \\in \\mathbb{N}} = ( [0,T] \\ni t \\mapsto \\tilde{\\mathcal{X}}^{n}_t( \\omega ) \\in H_{\\varrho} )_{n \\in \\mathbb{N}} $, \n$ (\\mathbb{O}^n)_{n \\in \\mathbb{N}} = ( [0,T] \\ni t \\mapsto \\mathbb{O}^{n}_t( \\omega ) \\in H_{\\varrho} )_{n \\in \\mathbb{N}} $, \n$(\\mathcal{O}^n)_{n \\in \\mathbb{N}} = ( [0,T] \\ni t \\mapsto \\mathcal{O}^{n}_t( \\omega ) \\in P_n(H) )_{n \\in \\mathbb{N}} $ for $\\omega \\in \\Omega$ in the notation of Proposition~\\ref{prop:main_det:2}) shows that for all $t \\in [0, T]$, $ n \\in \\mathbb{N} $ with $ h_n \\leq 1 $ it holds that\n\\begin{align}\n\\label{eq:cor_1bound}\n\\begin{split}\n\\| \\tilde{\\mathcal{X}}^n_t \\|_H^p \n&\\leq 2^{p-1}\\| \\mathbb{O}^n_t \\|_H^p + 2^{p-1} \\, t^{\\nicefrac{p}{2}-1} \\bigg[ 1+ \\tfrac{ \\tilde{\\theta} e^{ \\kappa(2+ \\tilde{\\vartheta})} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + |\\tilde{\\theta}|^{\\nicefrac{1}{2}} + \\sqrt{\\eta} ]^{2+\\tilde{\\vartheta}} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+\\tilde{\\vartheta}}} \\bigg]^{\\nicefrac{p}{2}} \\\\\n& \\quad \\cdot\\int_0^t e^{ \\int_s^t p \\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2p\\eta \\, du} \\Big[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2} \\|\\mathbb{O}_s^n\\|_H^2 \\\\\n& \\quad + \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 2\\tilde{\\vartheta}} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\Big]^{\\nicefrac{p}{2}} ds.\n\\end{split}\n\\end{align}\nNext observe that H\\\"older's inequality implies for all $n \\in \\mathbb{N}$ that \n\\begin{align}\n\\begin{split}\n& \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 2\\tilde{\\vartheta}} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n& = \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{\\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 4\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\sqrt{ \\eta} \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n& \\leq |\\max\\{1, \\eta\\}|^{2+2\\vartheta} \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}} \\, du \\big\\}\\right|^{2+ 4\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n&\\leq |\\max\\{1, \\eta\\}|^{2+2\\vartheta} \\left|\\max \\!\\big\\{ 1, T \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\}\\right|^{1+ 2\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\} \\\\\n&\\leq |\\max\\{1, \\eta\\}|^{2+2\\vartheta} \\, |\\max\\{1, T\\}|^{1+2\\vartheta} \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^2 \\, du \\big\\}\\right|^{2+ 2\\vartheta} \\\\ \n&\\leq |\\max\\{1, \\eta\\}|^{2+2\\vartheta} \\, |\\max\\{1, T\\}|^{1+2\\vartheta} \\left|\\max \\!\\big\\{ 1, T^{p+p\\vartheta-1} \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^{2p+2p\\vartheta} \\, du \\big\\}\\right|^{\\nicefrac{2}{p}} \\\\ \n&\\leq |\\max\\{1, \\eta, T\\}|^{5+6\\vartheta} \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^{2p+2p\\vartheta} \\, du \\big\\}\\right|^{\\nicefrac{2}{p}}.\n\\end{split}\n\\end{align}\nThe fact that $\\forall \\, a, b, c \\in \\mathbb{R} \\colon |a+b+c|^{\\nicefrac{p}{2}} \\leq 3^{\\nicefrac{p}{2}-1} (|a|^{\\nicefrac{p}{2}} + |b|^{\\nicefrac{p}{2}} + |c|^{\\nicefrac{p}{2}})$ together with \\eqref{eq:cor_1bound} hence establishes for all $t \\in [0,T]$, $ n \\in \\mathbb{N} $ with $ h_n \\leq 1 $ that \n\\begin{align}\n\\begin{split}\n&\\| \\tilde{\\mathcal{X}}^n_t \\|_H^p \n\\leq 2^{p-1}\\| \\mathbb{O}^n_t \\|_H^p + 2^{p-1} \\, t^{\\nicefrac{p}{2}-1} \\bigg[ 1+ \\tfrac{ \\tilde{\\theta} e^{ \\kappa(2+ 2\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + |\\tilde{\\theta}|^{\\nicefrac{1}{2}} + \\sqrt{\\eta} ]^{2+2\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+2\\vartheta}} \\bigg]^{\\nicefrac{p}{2}} \\\\\n & \\quad \\cdot\\int_0^t e^{ \\int_s^t p \\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2 p\\eta \\, du} \\bigg[ 2 \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) + \\tfrac{\\eta}{2 } \\|\\mathbb{O}_s^n\\|_H^2\\\\\n& \\quad + |\\max\\{1, \\eta, T\\}|^{5+6\\vartheta} \\left|\\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^{2p+2p\\vartheta} \\, du \\big\\}\\right|^{\\nicefrac{2}{p}} \\bigg]^{\\nicefrac{p}{2}} ds\\\\\n&\\leq 2^{p-1}\\|\\mathbb{O}^n_t \\|_H^p + 2^{p-1} \\, |3t|^{\\nicefrac{p}{2}-1} \\bigg[ 1+ \\tfrac{ \\tilde{\\theta} e^{ \\kappa(2+ 2\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + |\\tilde{\\theta}|^{\\nicefrac{1}{2}} + \\sqrt{\\eta} ]^{2+2\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+2\\vartheta}} \\bigg]^{\\nicefrac{p}{2}} \\\\\n& \\quad \\cdot\\int_0^t e^{ \\int_s^t p\\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2 p\\eta \\, du} \\Big[ 2^{\\nicefrac{p}{2}} \\big| \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) \\big|^{\\nicefrac{p}{2}} + \\big|\\tfrac{\\eta}{2 } \\big|^{\\nicefrac{p}{2}} \\|\\mathbb{O}_s^n\\|_H^p \\\\\n& \\quad + |\\max\\{1, \\eta, T\\}|^{3p+3p\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\big\\} \\Big] ds.\n\\end{split}\n\\end{align}\nThis, the fact that $\\forall \\, \\omega \\in \\tilde{\\Omega}, \\, t\\in [0,T], \\, n \\in \\mathbb{N} \\colon \\tilde{\\mathcal{X}}^n_t(\\omega) = \\mathbb{X}_t^n(\\omega)$, and the fact that $\\P(\\tilde{\\Omega})=1$ yield that for all $t \\in [0, T]$, $n \\in \\mathbb{N} $\nwith $ h_n \\leq 1 $ it holds that \n\\begin{align}\n\\begin{split}\n&\\mathbb{E}\\big[ \\| \\mathbb{X}^n_t \\|_H^p\\big] = \\mathbb{E}\\big[ \\| \\tilde{\\mathcal{X}}^n_t \\|_H^p\\big] \\\\\n& \\leq 2^{p-1} \\, \\mathbb{E}\\big[\\| \\mathbb{O}^n_t \\|_H^p \\big] + 2^{p-1} \\, |3t|^{\\nicefrac{p}{2}-1} \\bigg[ 1+ \\tfrac{ \\tilde{\\theta} e^{ \\kappa(2+ 2\\vartheta)} [ 1+ (\\kappa + \\sqrt{\\eta} +\\sqrt{\\eta}|\\kappa-\\eta|e^{ \\eta} ) \\|(\\kappa-A)^{ \\rho - \\varrho } \\|_{L(H)} + |\\tilde{\\theta}|^{\\nicefrac{1}{2}} + \\sqrt{\\eta} ]^{2+2\\vartheta} }{(1- \\varphi)( 1-\\alpha - \\rho)^{2+2\\vartheta}} \\bigg]^{\\nicefrac{p}{2}} \\\\\n& \\quad \\cdot \\mathbb{E} \\bigg[\\int_0^T e^{ \\int_s^T p \\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) +2p\\eta \\, du} \\Big[ 2^{\\nicefrac{p}{2}} \\big| \\Phi\\big(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n \\big) \\big|^{\\nicefrac{p}{2}} + \\big|\\tfrac{\\eta}{2 } \\big|^{\\nicefrac{p}{2}} \\|\\mathbb{O}_s^n\\|_H^p \\\\\n& \\quad + |\\max\\{1, \\eta, T\\}|^{3p+3p\\vartheta} \\max \\!\\big\\{ 1, \\smallint\\nolimits_{0}^T \\| \\mathcal{O}_u^n \\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\big\\} \\Big] ds \\bigg].\n\\end{split}\n\\end{align}\nThe assumption that $\\forall \\, t \\in [0,T], \\, n \\in \\mathbb{N} \\colon \\P (\\mathbb{X}_t^n= \\mathcal{X}_t^n)=1$, the fact that $ \\limsup_{ n \\to \\infty} h_n = 0$, the assumption that $ \\limsup_{ n \\to \\infty} \\sup_{ t \\in [0,T]} \\mathbb{E}[ \\| \\mathbb{O}_t^n \\|_H^p] < \\infty$, and \\eqref{eq:integrability} hence ensure that\n\\begin{equation}\n\\label{eq:a_priori_X} \n\\limsup_{ n \\to \\infty } \\sup_{ t \\in [0,T] } \\mathbb{E}\\big[ \\| \\mathcal{X}^n_t \\|_H^p \\big]= \\limsup_{ n \\to \\infty } \\sup_{ t \\in [0,T] } \\mathbb{E}\\big[ \\| \\mathbb{X}^n_t \\|_H^p \\big] < \\infty.\n\\end{equation}\nNext note that \\eqref{item:exp} and the fact $H_{\\varrho} \\subseteq H$ continuously\nimply that\n\\begin{equation}\n\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[\\!\\min\\!\\big\\{ 1,\\| X_t- \\mathbb{X}_t^{n} \\|_{ H } \\big\\} \\big] = 0.\n\\end{equation}\nThe assumption that\n$\\forall \\, t \\in [0,T], \\, n \\in \\mathbb{N} \\colon \\P (\\mathbb{X}_t^n= \\mathcal{X}_t^n)=1$\nhence assures that\n\\begin{equation}\n\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[\\!\\min\\!\\big\\{ 1,\\| X_t- \\mathcal{X}_t^{n} \\|_{ H } \\big\\} \\big] = 0.\n\\end{equation}\nCombining this with, e.g., Lemma~4.2 in Hutzenthaler et al.~\\cite{Salimova2016}\nproves for all $\\varepsilon \\in (0, \\infty)$ that\n\\begin{equation}\n\\limsup_{n \\to \\infty} \\sup_{t \\in [0, T]} \\P \\bigl(\\|X_t - \\mathcal{X}_t^n\\|_H \\geq \\varepsilon \\bigr)=0.\n\\end{equation}\nE.g., Proposition~4.5 in Hutzenthaler et al.~\\cite{Salimova2016} together with \\eqref{eq:a_priori_X}\nhence shows for all $q \\in (0, p)$ that\n$ \\sup_{ t \\in [0,T] } \\mathbb{E}\\big[ \\| X_t \\|^p_H \\big] < \\infty $\nand\n\\begin{equation}\n\\limsup_{ n \\to \\infty } \\sup_{ t \\in [0,T] } \\mathbb{E}\\big[ \\| X_t - \\mathcal{X}_t^n \\|_H^q \\big] = 0.\n\\end{equation}\nCombining this with \\eqref{eq:a_priori_X} establishes \\eqref{item:pnorm} and \\eqref{item:strong}.\nThe proof of Theorem~\\ref{thm:strong} is thus completed.\n\\end{proof}\n\n\n\\section{Fernique's theorem}\n\\label{sec:fernique}\n\nIn this section we present a number of elementary and well-known results\nand, in particular, Fernique's theorem, which is crucial for the derivations in Section~\\ref{sec:abstract} below. \n\n\\subsection{Uniqueness theorem for measures}\n\nProposition~\\ref{thm:measure_uniqueness} can, e.g., be found \nas Lemma~1.42 in Klenke~\\cite{Klenke2008}.\n\n\n\\begin{prop}[Uniqueness theorem for measures]\n\t\\label{thm:measure_uniqueness}\n\tConsider the notation in Subsection~\\ref{sec:notation},\n\tlet \n\t$ \\Omega $ be a set,\n\tlet\n\t$\n\t\\mathcal{E} \\subseteq \\mathcal{P}( \\Omega )\n\t$\n\tbe an $ \\cap $-stable subset of $ \\mathcal{P}( \\Omega ) $ (see, e.g., \\cite[Definition~2.1]{JentzenPusnik2016}), and let\n\t$ \n\t\\mu_1, \\mu_2 \\colon \\sigma_{ \\Omega }( \\mathcal{E} ) \\to [0,\\infty]\n\t$\n\tbe measures which satisfy that there exists a sequence\n\t$ \\Omega_n \\in \\{ A \\in \\mathcal{E} \\colon \\mu_1( A ) < \\infty \\} $, $ n \\in \\mathbb{N} $,\n\tsuch that\n\t$\n\t\\cup_{ n \\in \\mathbb{N} } \\Omega_n = \\Omega\n\t$\n\tand \n\t$ \\mu_1|_{ \\mathcal{E} } = \\mu_2|_{ \\mathcal{E} } $.\n\tThen it holds that $ \\mu_1 = \\mu_2 $.\n\\end{prop}\n\\begin{proof}[Proof of Proposition~\\ref{thm:measure_uniqueness}]\n\t%\n\tThroughout this proof let \n\t$ \\mathcal{S} \\subseteq \\mathcal{E} $\n\tbe the set given by\n\t$ \\mathcal{S} = \\{ A \\in \\mathcal{E} \\colon \\mu_1(A) < \\infty \\} $\n\tand let\n\t$E_n \\in \\sigma_{\\Omega}(\\mathcal{E}) $, $ n \\in \\mathbb{N}_0 $,\n\tand\n\t$ \\mathcal{D}_E \\in \\mathcal{P}( \\sigma_{ \\Omega }( \\mathcal{E} ) ) $, \n\t$ E \\in \\mathcal{S} $,\n\tbe the sets which satisfy\n\tfor all $n \\in \\mathbb{N} $, $ E \\in \\mathcal{S} $ that\n\t$E_0= \\emptyset$, $E_n = \\bigcup_{i=1}^n \\Omega_n$, \n\tand\n\t$ \\mathcal{D}_E = \\{ A \\in \\sigma_{ \\Omega }( \\mathcal{E} ) \\colon\n\t\\mu_1( A \\cap E ) = \\mu_2 ( A \\cap E ) \\} $.\n\t%\n\tFirst, note that for all $ E \\in \\mathcal{S} $ \n\tit holds that\n\t$ \\Omega \\in \\mathcal{D}_E $.\n\tNext observe that for all \n\t$ E \\in \\mathcal{S} $,\n\t$ A, B \\in \\mathcal{D}_E $ with $ B \\subseteq A $ it holds that\n\t\\begin{equation}\n\t\\begin{split}\n\t\\mu_1 ( ( A \\backslash B ) \\cap E ) \n\t&\n\t= \\mu_1 ( A \\cap E ) - \\mu_1 ( B \\cap E )\n\t\\\\\n\t&\n\t= \\mu_2 ( A \\cap E ) - \\mu_2 ( B \\cap E )\n\t= \\mu_2 ( ( A \\backslash B ) \\cap E ) .\n\t\\end{split}\n\t\\end{equation}\n\t%\n\tThis shows for all \n\t$ E \\in \\mathcal{S} $,\n\t$ A, B \\in \\mathcal{D}_E $ \n\twith \n\t$ B \\subseteq A $ \n\tthat \n\t$ A \\backslash B \\in \\mathcal{D}_E $.\n\t%\n\tMoreover, note that for all \n\tsets $ E \\in \\mathcal{S} $\n\tand all sequences\n\t$ A_n \\in \\mathcal{D}_E $, $ n \\in \\mathbb{N} $,\n\twith $ \\forall \\, i \\in \\mathbb{N}, \\, j \\in \\mathbb{N} \\backslash \\{ i \\} \\colon A_i \\cap A_j = \\emptyset $\n\tit holds that\n\t\\begin{equation}\n\t\\begin{split}\n\t\\mu_1 \\big( \\big( \\cup_{n\\in \\mathbb{N}} A_n \\big) \\cap E \\big)\n\t=\n\t\\sum_{n=1}^\\infty \\mu_1 ( A_n \\cap E )\n\t=\n\t\\sum_{n=1}^\\infty \\mu_2 ( A_n \\cap E ) \n\t=\n\t\\mu_2 \\big ( \\big( \\cup_{n\\in \\mathbb{N}} A_n \\big) \\cap E \\big ).\n\t\\end{split}\n\t\\end{equation}\n\t%\n\tThis proves for all\n\tsets $ E \\in \\mathcal{S} $\n\tand all sequences\n\t$ A_n \\in \\mathcal{D}_E $, $ n \\in \\mathbb{N} $,\n\twith $ \\forall \\, i \\in \\mathbb{N}, \\, j \\in \\mathbb{N} \\backslash \\{ i \\} \\colon A_i \\cap A_j = \\emptyset $ \n\tthat $ \\cup_{n\\in \\mathbb{N}} A_n \\in \\mathcal{D}_E $.\n\t%\n\tTherefore, we have established that for all\n\t$ E \\in \\mathcal{S} $ it holds that\n\t$ \\mathcal{D}_E $ is a Dynkin system on $\\Omega$ (see, e.g., Definition~2.2 in Jentzen \\& Pu\\v{s}nik~\\cite{JentzenPusnik2016}).\n\tThe fact that\n\t $ \\forall \\, E \\in \\mathcal{S} \\colon \\mathcal{E} \\subseteq \\mathcal{D}_E $\n\thence shows for all\n\t$ E \\in \\mathcal{S} $ \n\tthat\n\t$ \\delta_{ \\Omega } ( \\mathcal{E} ) \\subseteq \\mathcal{D}_E $ (see, e.g., Definition~2.3 in~\\cite{JentzenPusnik2016}).\n\tCombining this, the assumption that $ \\mathcal{E} $ is $ \\cap $-stable,\n\tand, e.g., Theorem~2.5 in~\\cite{JentzenPusnik2016} ensures \n\tfor all $ E \\in \\mathcal{S} $ that\n\t$ \\sigma_{ \\Omega } ( \\mathcal{E} ) = \\delta_{ \\Omega } ( \\mathcal{E} )\n\t\\subseteq \\mathcal{D}_E \\subseteq \\sigma_{ \\Omega } ( \\mathcal{E} ) .$\n\tThis assures for all $ E \\in \\mathcal{S} $ that\n\t\\begin{equation}\n\t\\label{eq:valid_for_finiteM_sets}\n\t\\sigma_{ \\Omega } ( \\mathcal{E} ) = \\mathcal{D}_E \n\t.\n\t\\end{equation}\n\t%\n\tNext note that for all $n \\in \\mathbb{N}$, $m \\in \\mathbb{N} \\backslash \\{n\\}$ it holds that $E_n = \\bigcup_{i=1}^n ((\\Omega\\backslash E_{i-1}) \\cap \\Omega_i)$ and $((\\Omega\\backslash E_{n-1}) \\cap \\Omega_n)\\cap ((\\Omega\\backslash E_{m-1}) \\cap \\Omega_m) = \\emptyset $.\n\tEquation \\eqref{eq:valid_for_finiteM_sets} hence proves for all $n \\in \\mathbb{N}$, $ A \\in \\sigma_{ \\Omega } ( \\mathcal{E} ) $ that\n\t\\begin{equation}\n\t\\mu_1( A \\cap E_n )\n\t=\n\t\\sum_{i=1}^n \\mu_1\\big((A \\cap(\\Omega\\backslash E_{i-1})) \\cap \\Omega_i\\big)\n\t=\n\t\\sum_{i=1}^n \\mu_2\\big((A \\cap(\\Omega\\backslash E_{i-1})) \\cap \\Omega_i\\big)\n\t=\n\t\\mu_2 ( A \\cap E_n ).\n\t\\end{equation}\n\tThis implies for all\n\t$ A \\in \\sigma_{ \\Omega } ( \\mathcal{E} ) $ that\n\t\\begin{equation}\n\t\\mu_1( A )\n\t=\n\t\\lim\\nolimits_{n\\to \\infty} \n\t\\mu_1 ( A \\cap E_n )\n\t=\n\t\\lim\\nolimits_{n\\to \\infty}\n\t\\mu_2 (A \\cap E_n )\n\t=\n\t\\mu_2 ( A ).\n\t\\end{equation}\n\t%\n\tThe proof of Proposition~\\ref{thm:measure_uniqueness} is thus completed.\n\\end{proof}\n\n\\subsection{Borel sigma-algebras on normed vector spaces}\n\n\nIn this subsection we first recall the Hahn-Banach theorem (see, e.g., Werner~\\cite[Theorem~III.1.5]{w05a}).\n\n\n\\begin{prop}[Hahn-Banach theorem; Extension of continuous linear functionals]\n\t\\label{thm:Hahn_Banach}\n\tLet \n\t$ \\mathbb{K} \\in \\{ \\mathbb{R}, \\mathbb{C} \\} $,\n\tlet\n\t$ ( V , \\left\\| \\cdot \\right\\|_V ) $\n\tbe a normed $ \\mathbb{K} $-vector space,\n\tlet $ U \\subseteq V $ be a $ \\mathbb{K} $-subspace of $ V $,\n\tand let $ \\phi \\in U' $.\n\tThen there exists a function $ \\varphi \\in V' $\n\tsuch that\n\t\\begin{equation}\n\t\\varphi|_U = \\phi \n\t\\qquad \n\t\\text{and}\n\t\\qquad\n\t\\left\\| \\varphi \\right\\|_{ V' }\n\t=\n\t\\left\\| \\phi \\right\\|_{ U' }\n\t.\n\t\\end{equation}\n\\end{prop}\n\n\n\n\nThe proof of Proposition~\\ref{thm:Hahn_Banach}\nemploys the axiom of choice.\nThe next result, Corollary~\\ref{cor:Hahn_Banach}, is a direct consequence\nof the Hahn-Banach theorem.\n\n\n\n\n\n\n\\begin{cor}[Projections into $ 1 $-dimensional subspaces]\n\t\\label{cor:Hahn_Banach}\n\tLet \n\t$ \\mathbb{K} \\in \\{ \\mathbb{R}, \\mathbb{C} \\} $,\n\tlet\n\t$ ( V , \\left\\| \\cdot \\right\\|_V ) $\n\tbe a nontrivial normed $ \\mathbb{K} $-vector space,\n\tand let $ v \\in V $.\n\tThen there exists a function $ \\varphi \\in V' $\n\tsuch that \n\t\\begin{equation}\n\t\\label{eq:cor_Hahn_Banach}\n\t\\varphi( v )\n\t=\n\t\\left\\| v \\right\\|_V\n\t\\qquad \n\t\\text{and}\n\t\\qquad\n\t\\left\\| \\varphi \\right\\|_{ V' }\n\t=\n\t1\n\t.\n\t\\end{equation}\n\\end{cor}\n\n\n\n\n\\begin{proof}[Proof\tof Corollary~\\ref{cor:Hahn_Banach}]\n\tWe show Corollary~\\ref{cor:Hahn_Banach} in two steps.\n\tIn the first step we assume that $ v \\neq 0 $.\n\tLet \n\t$ U \\subseteq V$\n\tbe the $ \\mathbb{K} $-subspace of $ V $\n\tgiven by\n\t$ \n\tU \n\t= \\left\\{ \\lambda v \\in V \\colon \\lambda \\in \\mathbb{K} \\right\\} \n\t= \\operatorname{span}_V(\\{ v \\})\n\t$\n\tand let\n\t$ \\phi \\colon U \\to \\mathbb{K} $\n\tbe the function which satisfies \n\tfor all $ \\lambda \\in \\mathbb{K} $ that\n\t\\begin{equation}\n\t\\phi( \\lambda v ) = \n\t\\lambda \\left\\| v \\right\\|_V\n\t.\n\t\\end{equation}\n\tProposition~\\ref{thm:Hahn_Banach} implies that there exists a function $ \\varphi \\in V' $ such that\n\t\\begin{equation}\n\t\\varphi|_U = \\phi\n\t\\qquad\n\t\\text{and}\n\t\\qquad\n\t\\left\\| \\varphi \\right\\|_{ V' }\n\t=\n\t\\left\\| \\phi \\right\\|_{ U' }\n\t= \n\t1\n\t.\n\t\\end{equation}\n\tThis proves \\eqref{eq:cor_Hahn_Banach}\n\tin the case $ v \\neq 0 $.\n\n\n\n\tIn the second step we assume that $ v = 0 $.\n\tNote that the assumption that $V$ is nontrivial ensures that\n\tthere exists a vector $ u \\in V $ such that $ u \\neq 0 $.\n\tThe first step hence shows that there exists a function $ \\varphi \\in V' $\n\tsuch that\n\t\\begin{equation}\n\t\\varphi( u ) = \\left\\| u \\right\\|_V\n\t\\qquad\n\t\\text{and}\n\t\\qquad \n\t\\left\\| \\varphi \\right\\|_{ V' } = 1\n\t.\n\t\\end{equation}\n\tIn addition, observe that $ \\varphi( v ) = \\varphi( 0 ) = 0 = \\left\\| v \\right\\|_V $.\n\tThe proof of Corollary~\\ref{cor:Hahn_Banach}\n\tis thus completed.\n\\end{proof}\n\n\n\n\n\n\nThe next result, Corollary~\\ref{cor:norm},\nis an immediate consequence of Corollary~\\ref{cor:Hahn_Banach}\nabove.\n\n\n\n\\begin{cor}[Norm via the dual space]\n\t\\label{cor:norm}\n\tLet \n\t$ \\mathbb{K} \\in \\{ \\mathbb{R}, \\mathbb{C} \\} $,\n\tlet\n\t$ \n\t( V, \\left\\| \\cdot \\right\\|_V ) \n\t$\n\tbe a nontrivial normed $ \\mathbb{K} $-vector space,\n\tand let $ v \\in V $.\n\tThen \n\t\\begin{equation}\n\t\\label{eq:cor_norm}\n\t\\left\\| \n\tv \n\t\\right\\|_V\n\t=\n\t\\sup_{ \n\t\t\\varphi \\in V' \\backslash \\{ 0 \\}\n\t}\n\t\\frac{\n\t\t\\Re(\\varphi( v )) \n\t}{\n\t\\left\\| \\varphi \\right\\|_{ V' }\n}\n=\n\\sup_{ \n\t\\varphi \\in V' \\backslash \\{ 0 \\}\n}\n\\frac{\n\t\\left| \\varphi( v ) \\right|\n}{\n\\left\\| \\varphi \\right\\|_{ V' }\n}\n.\n\\end{equation}\n\\end{cor}\n\n\n\n\nIf the normed vector space in\nCorollary~\\ref{cor:norm} is separable,\nthen the following result, Corollary~\\ref{cor:norm2}, can be obtained.\nCorollary~\\ref{cor:norm2}\nis also an immediate consequence of Corollary~\\ref{cor:Hahn_Banach}\nabove.\n\n\n\n\\begin{cor}[Norm of a separable normed vector space via the dual space]\n\t\\label{cor:norm2}\n\tLet \n\t$ \\mathbb{K} \\in \\{ \\mathbb{R}, \\mathbb{C} \\} $\n\tand let\n\t$ ( V, \\left\\| \\cdot \\right\\|_V ) $\n\tbe a separable normed $ \\mathbb{K} $-vector space.\n\tThen there exists a sequence\n\t$ \\varphi_n \\in V' $, $n \\in \\mathbb{N}$,\n\twhich satisfies for all $ v \\in V $ that\n\t\\begin{equation}\n\t\\left\\| \n\tv \n\t\\right\\|_V\n\t=\n\t\\sup_{ \n\t\tn \\in \\mathbb{N}\n\t}\n\t\\Re(\\varphi_n( v ))\n\t=\n\t\\sup_{ \n\t\tn \\in \\mathbb{N}\n\t}\n\t\\left| \\varphi_n( v ) \\right|\n\t.\n\t\\end{equation}\n\\end{cor}\n\n\n\n\n\\begin{proof}[Proof\tof Corollary~\\ref{cor:norm2}]\n\tWithout loss of generality we assume that $V$ is nontrivial.\n\tThe assumption that $ ( V, \\left\\| \\cdot \\right\\|_V ) $\n\tis separable implies that there exists\n\ta sequence \n\t$ v_n \\in V $, $ n \\in \\mathbb{N} $,\n\tsuch that the set\n\t$\n\t\\left\\{ v_n \\colon n \\in \\mathbb{N} \\right\\}\n\t$\n\tis dense in $ V $.\n\tCorollary~\\ref{cor:Hahn_Banach}\n\thence shows that there exists\n\ta sequence\n\t$ \\varphi_n \\in V' $, $ n \\in \\mathbb{N} $,\n\twhich satisfies\n\tfor all $ n \\in \\mathbb{N} $ that\n\t\\begin{equation}\n\t\\varphi_n( v_n )\n\t=\n\t\\left\\| v_n \\right\\|_V\n\t\\qquad \n\t\\text{and}\n\t\\qquad\n\t\\left\\| \\varphi_n \\right\\|_{ V' }\n\t= 1\n\t.\n\t\\end{equation}\n\tThis ensures for all $ k \\in \\mathbb{N} $ that\n\t\\begin{equation}\n\t\\left\\| v_k \\right\\|_V\n\t=\n\t\\sup_{ \n\t\tn \\in \\mathbb{N}\n\t}\n\t|\\varphi_n( v_k )|\n\t.\n\n\n\n\n\n\n\n\t\\end{equation}\n\tNext let \n\t$ v \\in V $ and $ \\varepsilon \\in ( 0,\\infty) $.\n\tNote that\n\t\\begin{equation}\n\t\\sup_{ n \\in \\mathbb{N} } \n\t|\\varphi_n( v )|\n\t\\leq\n\t\\sup_{ n \\in \\mathbb{N} }\n\t\\left[\n\t\\left\\| \\varphi_n \\right\\|_{ V' }\n\t\\left\\| v \\right\\|_V\n\t\\right]\n\t=\n\t\\left\\| v \\right\\|_V\n\t.\n\t\\end{equation}\n\tIt thus remains to prove that\n\t\\begin{equation}\n\t\\left\\| v \\right\\|_V\n\t\\leq\n\t\\varepsilon\n\t+\n\t\\sup_{ n \\in \\mathbb{N} } \n\t\\Re(\\varphi_n( v ))\n\t.\n\t\\end{equation}\n\tFor this observe that\n\tthe fact that\n\t$\n\t\\{ v_n \\in V \\colon n \\in \\mathbb{N} \\}\n\t$\n\tis dense in $ V $ ensures\n\tthat there exists a natural number $ k \\in \\mathbb{N} $\n\tsuch that $ \\left\\| v - v_k \\right\\|_V \\leq \\frac{ \\varepsilon }{ 2 } $.\n\tThis implies that\n\t\\begin{equation}\n\t\\begin{split}\n\t\\left\\| v \\right\\|_V\n\t& \\leq\n\t\\left\\| v_k \\right\\|_V\n\t+\n\t\\left\\| v - v_k \\right\\|_V\n\t=\n\t\\Re(\\varphi_k( v_k ))\n\t+\n\t\\left\\| v - v_k \\right\\|_V\n\t\\\\ &\n\t=\n\t\\Re(\\varphi_k( v ))\n\t+\n\t\\left\\| v - v_k \\right\\|_V\n\t+\n\t\\Re(\\varphi_k( v_k - v ))\n\t\\\\ &\n\t\\leq\n\t\\Re(\\varphi_k( v ))\n\t+\n\t\\left\\| v - v_k \\right\\|_V\n\t+\n\t\\left\\| \\varphi_k \\right\\|_{ V' }\n\t\\left\\| v - v_k \\right\\|_V\n\t\\\\ &\n\t=\n\t\\Re(\\varphi_k( v ))\n\t+\n\t2 \\left\\| v - v_k \\right\\|_V\n\t\\leq\n\t\\sup_{ n \\in \\mathbb{N} }\n\t\\Re(\\varphi_n( v ))\n\t+\n\t2 \\left\\| v - v_k \\right\\|_V\n\t\\leq\n\t\\sup_{ n \\in \\mathbb{N} }\n\t\\Re(\\varphi_n( v ))\n\t+\n\t\\varepsilon\n\t.\n\t\\end{split}\n\t\\end{equation}\n\tThe proof of Corollary~\\ref{cor:norm2}\n\tis thus completed.\n\\end{proof}\n\n\n\nThe last result of this subsection, Proposition~\\ref{prop:charac_BE_lin} below, follows from Corollary~\\ref{cor:norm2} above. We refer to the statement of Proposition~\\ref{prop:charac_BE_lin} as \\emph{linear\ncharacterization of the Borel sigma-algebra}.\n\n\n\n\n\\begin{prop}[Linear characterization of the Borel sigma-algebra]\n\t\\label{prop:charac_BE_lin}\n\tLet $ \\mathbb{K} \\in \\{ \\mathbb{R} , \\mathbb{C} \\} $ and\n\tlet $ ( V, \\left\\| \\cdot \\right\\|_V ) $ be a separable normed $ \\mathbb{K} $-vector space.\n\tThen there exists a sequence $ \\varphi_n \\in V' $, $ n \\in \\mathbb{N} $,\n\tsuch that\n\t\\begin{equation}\n\t\\begin{split}\n\t\\mathcal{B}( V ) \n\t& =\n\t\\sigma_V\\!\\left( \n\t\\varphi \\colon \n\t\\varphi \\in V'\n\t\\right)\n\t=\n\t\\sigma_V\\!\\left( \n\t\\varphi_n \\colon \n\tn \\in \\mathbb{N}\n\t\\right)\n\n\n\n\n\n\n\n\n\n\n\n\t.\n\t\\end{split}\n\t\\end{equation}\n\\end{prop}\n\n\n\n\\begin{proof}[Proof\tof Proposition~\\ref{prop:charac_BE_lin}]\nThroughout this proof\n\tlet $ f_v \\colon V \\to [0,\\infty) $,\n\t$ v \\in V $,\n\tbe the functions which satisfy\n\tfor all $ x, v \\in V $ that\n\t\\begin{equation}\n\tf_v( x )\n\t=\n\t\\left\\| x - v \\right\\|_V\n\t.\n\t\\end{equation}\n\tNote that\n\t\\begin{equation}\n\t\\label{eq:Borel-sigma-algebra}\n\t\\mathcal{B}( V ) \n\t=\n\t\\sigma_V\\!\\left(\n\tf_v \\colon v \\in V\n\t\\right)\n\t.\n\t\\end{equation}\n\tNext observe that \n\tCorollary~\\ref{cor:norm2}\n\tshows that there\n\texists a sequence \n\t$ \\varphi_n \\in V' $,\n\t$ n \\in \\mathbb{N} $,\n\twhich satisfies for all $ v \\in V $ that\n\t\\begin{equation}\n\t\\left\\| v \\right\\|_V\n\t=\n\t\\sup_{ n \\in \\mathbb{N} }\n\t\\Re(\\varphi_n( v ))\n\t.\n\t\\end{equation}\n\tThis implies that\n\t\\begin{equation}\n\t\\begin{split}\n\t\\mathcal{B}( V ) & \\supseteq\n\t\\sigma_V\\!\\left(\n\t\\varphi_n \n\t\\colon\n\tn \\in \\mathbb{N} \n\t\\right) \\supseteq \t\\sigma_V\\big(\n\t( V \\ni u \\mapsto \\Re(\\varphi_n(u)) \\in \\mathbb{R} ) \n\t\\colon\n\tn \\in \\mathbb{N} \n\t\\big)\n\t\\\\ & = \n\t\\sigma_V\\big(\n\t( V \\ni u \\mapsto \n\t\\Re(\\varphi_n( u + v ))\n\t\\in \\mathbb{R} )\n\t\\colon\n\tn \\in \\mathbb{N} , v \\in V \n\t\\big)\n\t\\supseteq\n\t\\sigma_V\\!\\left(\n\tf_v\n\t\\colon\n\tv \\in V\n\t\\right)\n\t.\n\t\\end{split}\n\t\\end{equation}\n\tCombining this with \\eqref{eq:Borel-sigma-algebra} completes the proof of Proposition~\\ref{prop:charac_BE_lin}.\n\\end{proof}\n\n\n\\subsection{Fourier transform of a measure}\n\nIn Lemma~\\ref{lem:charac_fct2} further below we present a well-known result which states that the Fourier transform of a finite measure on a separable normed $ \\mathbb{R} $-vector space determines the measure uniquely. The proof of Lemma~\\ref{lem:charac_fct2} employs Proposition~\\ref{prop:charac_BE_lin} and Proposition~\\ref{thm:measure_uniqueness} above.\n\n\\begin{definition}[Image measure\/Pushforward measure]\n\t\\label{def:image_measure}\n\tLet\n\t$ \\left( \\Omega, \\mathcal{A}, \\mu \\right) $\n\tbe a measure space, \n\tlet\n\t$\n\t( \\tilde{\\Omega}, \\tilde{\\mathcal{A}} \n\t)\n\t$\n\tbe a measurable space,\n\tand \n\tlet\n\t$ f \\colon \\Omega \\to \\tilde{ \\Omega } $\n\tbe an $ \\mathcal{A} $\/$ \\tilde{ \\mathcal{A} } $-measurable function.\n\tThen we denote by\n\t$ f( \\mu )_{ \\tilde{\\mathcal{A}} } \\colon \\tilde{ \\mathcal{A} } \\to [0,\\infty] $\n\tthe function which satisfies\n\tfor all $ A \\in \\tilde{ \\mathcal{A} } $ that\n\t\\begin{equation}\n\t\\big(\n\tf( \\mu )_{ \\tilde{\\mathcal{A}} }\n\t\\big)( A )\n\t=\n\t\\mu\\!\\left(\n\tf^{ - 1 }( A )\n\t\\right)\n\t\\end{equation}\n\tand we call $ f( \\mu )_{ \\tilde{\\mathcal{A}} } $\n\tthe image measure of $ \\mu $ under $ f $ associated to $ \\tilde{\\mathcal{A}} $.\n\\end{definition} \n\n\\begin{prop}[Characteristic function] \n\t\\label{prop:charac_fct} \n\tLet $ d \\in \\mathbb{N} $ and let $ \\mu_k \\colon \\mathcal{B}( \\mathbb{R}^d ) \\to [0,\\infty] $, $ k \\in \\{ 1, 2 \\} $, be finite measures which satisfy for all $ \\xi \\in \\mathbb{R}^d $ that \n\t\\begin{equation} \n\t\\int_{ \\mathbb{R}^d } e^{ i \\left< \\xi, x \\right>_{ \\mathbb{R}^d } } \\, \\mu_1( dx ) = \\int_{ \\mathbb{R}^d } e^{ i \\left< \\xi, x \\right>_{ \\mathbb{R}^d } } \\, \\mu_2( dx ) . \n\t\\end{equation} \n\tThen it holds that $ \\mu_1 = \\mu_2 $. \n\\end{prop} \n\nProposition~\\ref{prop:charac_fct} is, e.g., proved as Theorem~15.8 in Klenke~\\cite{Klenke2008}. \n\n\\begin{definition}[Characteristic functional]\nLet $ ( V, \\left\\| \\cdot \\right\\|_V ) $ be a normed $ \\mathbb{R} $-vector space,\nlet $ \\mathscr{M} $ be the set of all finite measures $ \\mu \\colon \\mathcal{B}( V ) \\to [0,\\infty] $ on $ ( V, \\mathcal{B}( V ) ) $,\nand let\n$ \\mathbb{M} $ be the set of all functions from $ V' $ to $ \\mathbb{C} $.\nThen we denote by\n$ \\mathbb{F}_V \\colon \\mathscr{M} \\to \\mathbb{M} $\nthe function which satisfies for all\n$ \\mu \\in \\mathscr{M} $, $ \\varphi \\in V' $ that \n\\begin{equation} \n( \\mathbb{F}_V \\mu )( \\varphi ) = \\big( \\mathbb{F}_V( \\mu ) \\big)( \\varphi ) = \\int_{ V } e^{ i \\, \\varphi( x ) } \\, \\mu( d x ) \n\\end{equation} \nand for every\n$ \\mu \\in \\mathscr{M} $ we call $ \\mathbb{F}_V( \\mu ) $ the characteristic functional of $ \\mu $. \n\\end{definition} \n\n\n\n\n\\begin{lemma}[Characteristic functional determines measure uniquely] \\label{lem:charac_fct2} Let $ ( V, \\left\\| \\cdot \\right\\|_V ) $ be a separable normed $ \\mathbb{R} $-vector space. Then $ \\mathbb{F}_V $ is injective. \n\\end{lemma} \n\\begin{proof}[Proof of Lemma~\\ref{lem:charac_fct2}]\nConsider the notation in Subsection~\\ref{sec:notation} and\nlet $ \\mu_1, \\mu_2 \\colon \\mathcal{B}( V ) \\to [0,\\infty] $\nbe finite measures on $ ( V, \\mathcal{B}( V ) ) $ which satisfy\n$ \\mathbb{F}_V( \\mu_1 ) = \\mathbb{F}_V( \\mu_2 ) $. Note that for all $ n \\in \\mathbb{N} $, $ \\phi = ( \\phi_1, \\dots, \\phi_n ) \\in L( V, \\mathbb{R}^n ) $, $ \\xi \\in \\mathbb{R}^n $ it holds that\n\\begin{equation} \n\t\\begin{split} \n\t\\int_{ \\mathbb{R}^n } e^{ i \\left< \\xi, x \\right>_{ \\mathbb{R}^n } } \\, \\bigl( \\phi( \\mu_1 )_{\\mathcal{B}(\\mathbb{R}^n)} \\bigr)( dx ) & = \\int_{ V } e^{ i \\left< \\xi, \\phi( v ) \\right>_{ \\mathbb{R}^n } } \\, ( \\mu_1 )( dv ) = \\bigl( \\mathbb{F}_V \\mu_1 \\bigr)\n\t\\bigl( V \\ni v \\mapsto \\langle \\xi , \\phi( v ) \\rangle_{ \\mathbb{R}^n } \\in \\mathbb{R} \\bigr) \\\\ \n\t& = \\bigl( \\mathbb{F}_V \\mu_2 \\bigr)\\bigl( V \\ni v \\mapsto \\langle \\xi , \\phi( v ) \\rangle_{ \\mathbb{R}^n } \\in \\mathbb{R} \\bigr)\n\t= \\int_{ V } e^{ i \\left< \\xi, \\phi( v ) \\right>_{ \\mathbb{R}^n } } \\, ( \\mu_2 )( dv ) \\\\\n\t& = \\int_{ \\mathbb{R}^n } e^{ i \\left< \\xi, x \\right>_{ \\mathbb{R}^n } } \\, \\bigl( \\phi( \\mu_2 )_{\\mathcal{B}(\\mathbb{R}^n)} \\bigr)( dx ) . \n\t\\end{split} \n\t\\end{equation} \n\tProposition~\\ref{prop:charac_fct} hence implies for all $ n \\in \\mathbb{N} $, $ \\phi \\in L( V, \\mathbb{R}^n ) $ that \n\t\\begin{equation} \\label{eq:coincide_phi} \n\t\\phi( \\mu_1 )_{\\mathcal{B}(\\mathbb{R}^n)} = \\phi( \\mu_2 )_{\\mathcal{B}(\\mathbb{R}^n)} .\n\t\\end{equation} \n\tIn the next step let $ \\mathcal{E} \\subseteq \\mathcal{P}( V ) $ be the set given by\n\t\\begin{equation}\n\t\\mathcal{E} = \\bigcup_{ n \\in \\mathbb{N} } \\left\\{ \\phi^{ - 1 }( B ) \\in \\mathcal{P}( V ) \\colon \\phi \\in L( V, \\mathbb{R}^n ) , B \\in \\mathcal{B}( \\mathbb{R}^n ) \\right\\} . \n\t\\end{equation} \n\tNote that $ \\mathcal{E} \\subseteq \\mathcal{B}( V ) $.\n\tIn addition, observe that \\eqref{eq:coincide_phi} shows that \n\t\\begin{equation} \\mu_1|_{ \\mathcal{E} } = \\mu_2|_{ \\mathcal{E} } . \n\t\\end{equation}\n\tThis, the fact that $ \\mathcal{E} $ is $ \\cap $-stable, the fact $ V \\in \\mathcal{E} $, and Proposition~\\ref{thm:measure_uniqueness} prove that \n\t\\begin{equation} \\label{eq:mu12_coincide} \\mu_1|_{ \\sigma_{ V }( \\mathcal{E} ) } = \\mu_2|_{ \\sigma_{ V }( \\mathcal{E} ) } . \n\t\\end{equation} \n\tMoreover, observe that Proposition~\\ref{prop:charac_BE_lin} establishes that \n\t\\begin{equation} \\sigma_V( \\mathcal{E} ) = \\mathcal{B}( V ) . \n\t\\end{equation} \n\tCombining this with \\eqref{eq:mu12_coincide} completes the proof of Lemma~\\ref{lem:charac_fct2}. \n\\end{proof}\n\n\n\n\n\\subsection{Fernique's theorem}\n\n\nThe proof of Fernique's theorem (see Proposition~\\ref{thm:fernique} below) requires the two following well-known auxiliary lemmas, Lemma~\\ref{lem:indep} and Lemma~\\ref{lem:iid} below.\n\n\n\n\\begin{lemma}[Independent projections of random variables]\n\\label{lem:indep}\nLet $(V_1, \\left\\| \\cdot \\right\\|_{V_1})$ and $(V_2, \\left\\| \\cdot \\right\\|_{V_2})$ be separable normed $ \\mathbb{R} $-vector spaces, \nlet $(\\Omega, \\mathcal{F}, \\P)$ be a probability space, and let $X_1 \\colon \\Omega \\to V_1$ and $X_2 \\colon \\Omega \\to V_2$ be \nfunctions which satisfy \nfor all $\\varphi_1 \\in (V_1)'$, $\\varphi_2 \\in (V_2)'$ \nthat \n$ \\varphi_1 \\circ X_1\n\\colon \\Omega \\to \\mathbb{R} $ and \n$ \\varphi_2 \\circ X_2\n\\colon \\Omega \\to \\mathbb{R} $ are independent random variables. Then it holds that $X_1$ and $X_2$ are independent random variables.\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lem:indep}]\nNote that\nthe assumption that\n$ \\forall \\, \\varphi_1 \\in (V_1)', \\, \\varphi_2 \\in (V_2)' \\colon\n(\n\\varphi_1 \\circ X_1 $\nand\n$ \\varphi_2 \\circ X_2 $\nare\n$\\mathcal{ F } $\/$ \\mathcal{ B }( \\mathbb{R} )$-measurable$)$\nand Proposition~\\ref{prop:charac_BE_lin}\nshow that\n$X_1$ is $ \\mathcal{ F } $\/$\\mathcal{ B }( V_1 ) $-measurable\nand that\n$X_2$ is $ \\mathcal{ F } $\/$\\mathcal{ B }( V_2 ) $-measurable.\nThroughout this proof let \n$\n ( \\tilde{\\Omega}, \\tilde{\\mathcal{F}}, \\tilde{\\P} )\n$\nbe the probability space given by\n\\begin{equation}\n \\tilde{\\Omega} = V_1 \\times V_2, \n \\quad\n \\tilde{\\mathcal{F}} = \\mathcal{B}(V_1) \\otimes \\mathcal{B}(V_2) , \n \\quad\n \\text{and}\n \\quad\n \\tilde{\\P} = X_1(\\P)_{\\mathcal{B}(V_1)} \\otimes X_2(\\P)_{\\mathcal{B}(V_2)} \n . \n\\end{equation}\nNext note that for all $\\varphi \\in L( \\tilde{\\Omega} ,\\mathbb{R})$ it holds that\n\t\\begin{align}\n\t\\label{eq:indep}\n\t\\begin{split}\n\t\\big( \\mathbb{F}_{\\tilde{\\Omega}} \\, \\tilde{\\P}\\big)(\\varphi)\n\t& = \n\t\\int_{\\tilde{\\Omega}} \\exp\\!\\big( i \\, \\varphi( x_1, x_2 )\\big) \\, \\tilde{\\P}(d(x_1,x_2)) \n\t\\\\\n\t&\n\t= \\int_{\\tilde{\\Omega}} \n\t\\exp\\!\\big( i \\, \\varphi( x_1, 0 )\\big) \n\t\\exp\\!\\big( i \\, \\varphi( 0, x_2 )\\big)\\, \\big(X_1(\\P)_{\\mathcal{B}(V_1)} \\otimes X_2(\\P)_{\\mathcal{B}(V_2)} \\big)(d(x_1,x_2)) \n\t\\\\\n\t&= \\int_{V_1} \n\t\\exp\\!\\big( i \\, \\varphi( x_1, 0 )\\big) \\, X_1(\\P)_{\\mathcal{B}(V_1)}(dx_1) \n\t\\int_{V_2} \\exp\\!\\big( i \\, \\varphi( 0, x_2 )\\big) \\, X_2(\\P)_{\\mathcal{B}(V_2)}(dx_2) \\\\\n\t& = \n\t\\mathbb{E}\\!\\left[e^{ i \\, \\varphi( X_1, 0 )}\\right] \\mathbb{E}\\!\\left[e^{ i \\, \\varphi( 0, X_2 )}\\right].\n\t\\end{split}\n\t\\end{align}\n\tMoreover, observe that for all $\\varphi \\in L( \\tilde{\\Omega} ,\\mathbb{R})$ it holds that $(V_1 \\ni v \\mapsto \\varphi( v, 0 ) \\in \\mathbb{R}) \\in (V_1)' $ \n\tand $(V_2 \\ni v \\mapsto \\varphi( 0,v ) \\in \\mathbb{R}) \\in (V_2)' $. \n\tThis and \\eqref{eq:indep} imply for all $\\varphi \\in L( \\tilde{\\Omega} ,\\mathbb{R})$ that\n\t\\begin{align}\n\t\\begin{split}\n\t \\big( \\mathbb{F}_{\\tilde{\\Omega}} \\, \\tilde{\\P} \\big)(\\varphi) \n\t & = \n \t \\mathbb{E}\\!\\left[\n \t e^{ i \\, \\varphi( X_1, 0 )}\n \t \\, \n \t e^{ i \\, \\varphi( 0, X_2 )}\n \t \\right]\n\t =\n\t \\mathbb{E}\\!\\left[\n\t e^{ i \\{ \\varphi( X_1, 0 ) + \\varphi( 0, X_2 ) \\} }\n\t \\right] \n\t\\\\ &\n\t =\n\t \\mathbb{E}\\!\\left[e^{ i \\, \\varphi( X_1, X_2 )}\\right] \n\t = \\bigl(\n\t \\mathbb{F}_{\\tilde{\\Omega}}\\bigl[ (X_1,X_2)( \\P )_{\\tilde{\\mathcal{F}}} \\bigr]\n\t \\bigr) (\\varphi).\n\t\\end{split}\n\t\\end{align}\n\tCombining this with Lemma~\\ref{lem:charac_fct2} yields that\n\t\\begin{align}\n \t X_1(\\P)_{\\mathcal{B}(V_1)} \\otimes X_2(\\P)_{\\mathcal{B}(V_2)} \n \t =\n\t \\tilde{\\P}\n\t =\n \t (X_1,X_2)(\\P)_{\\mathcal{B}(V_1) \\otimes \\mathcal{B}(V_2)} .\n\t\\end{align}\n\tThe proof of Lemma~\\ref{lem:indep} is thus completed.\n\\end{proof}\n\n\n\nLemma~\\ref{lem:iid} below demonstrates under suitable hypotheses that an appropriate orthogonal transformation \nof two appropriate independent random variables also results in independent random variables. \nObserve that the columns of the $ 2 \\times 2 $-matrix \n\\begin{equation}\n\\label{eq:orthogonal_transform}\n \\left(\n \\begin{array}{cc}\n \\nicefrac{ 1 }{ \\sqrt{2} }\n &\n \\nicefrac{ 1 }{ \\sqrt{2} }\n \\\\\n \\nicefrac{ 1 }{ \\sqrt{2} }\n &\n - \\nicefrac{ 1 }{ \\sqrt{2} }\n \\end{array}\n \\right)\n\\end{equation}\nconstitute an orthonormal basis of $ \\mathbb{R}^2 $.\nRoughly speaking, the orthogonal transformation \nassociated to \\eqref{eq:orthogonal_transform} is employed in the next lemma.\n\n\n\\begin{lemma}[Orthogonal transformations of independent random variables]\n\\label{lem:iid}\nLet $(V, \\left\\| \\cdot \\right\\|_V)$ be a separable normed $ \\mathbb{R} $-vector space, \nlet $(\\Omega, \\mathcal{F}, \\P)$ be a probability space, \nlet $X_1, X_2 \\colon \\Omega \\to V$ be independent random variables \nwhich satisfy for every $\\varphi \\in V'$ \nthat $ \\varphi \\circ X_1\n\\colon \\Omega \\to \\mathbb{R} $ and \n$ \\varphi \\circ X_2\n\\colon \\Omega \\to \\mathbb{R} $ \nare identically distributed centered Gaussian random variables, \nand let $Y_1, Y_2 \\colon \\Omega \\to V$ satisfy \n\\begin{equation}\n Y_1= 2^{-\\nicefrac{1}{2}}(X_1 + X_2)\n\\qquad \n \\text{and}\n\\qquad \n Y_2 = 2^{-\\nicefrac{1}{2}}(X_1 - X_2)\n .\n\\end{equation}\nThen \n\\begin{enumerate}[(i)]\n\\item \nit holds that $Y_1$ and $Y_2$ are independent random variables and\n\\item \\label{item:equal_dist}\nit holds that\n$Y_1(\\P)_{\\mathcal{B}(V)}=Y_2(\\P)_{\\mathcal{B}(V)}=X_1(\\P)_{\\mathcal{B}(V)}=X_2(\\P)_{\\mathcal{B}(V)}$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:iid}]\nObserve that the assumption that\n$ \\forall \\, \\varphi \\in V' \\colon\n( \\varphi \\circ X_1\n\\text{ and }\n\\varphi \\circ X_2 $\nare identically distributed random variables$ ) $\nproves\nfor all $\\varphi \\in V'$ that\n\\begin{equation}\n\t \\mathbb{E}\\!\\left[ \n\t e^{ \n\t i \\,\n\t \\varphi(X_1) \n\t } \n\t \\right] \n\t= \n\t \\mathbb{E}\\!\\left[ \n\t e^{ \n\t i \\,\n\t \\varphi(X_2) \n\t } \n\t \\right]\n\t .\n\\end{equation}\nLemma~\\ref{lem:charac_fct2}\nhence shows that\n\\begin{equation} \\label{eq:equal_dist_X}\nX_1(\\P)_{\\mathcal{B}(V)}=X_2(\\P)_{\\mathcal{B}(V)}.\n\\end{equation}\nIn addition, note that the hypothesis that $ X_1 $ and $ X_2 $ are independent random variables ensures \nfor all $\\varphi_1, \\varphi_2 \\in V'$, $\\xi = (\\xi_1, \\xi_2) \\in \\mathbb{R}^2 $ that \n\t\\begin{equation}\n\t\\begin{split}\n\t \\mathbb{E}\\!\\left[ \n\t e^{ i \\langle \\xi, (\\varphi_1(Y_1), \\varphi_2(Y_2)) \\rangle_{\\mathbb{R}^2} } \n\t \\right] \n\t & =\n\t \\mathbb{E}\\!\\left[ \n\t e^{ i \\, \\xi_1 \\varphi_1(Y_1) + i \\, \\xi_2 \\varphi_2(Y_2) } \n\t \\right] \n\t\\\\ &\n\t = \\mathbb{E}\\big[ e^{ i \\,2^{-\\nicefrac{1}{2}} \\xi_1 \\varphi_1 (X_1+X_2) + i \\, 2^{-\\nicefrac{1}{2}} \\xi_2 \\varphi_2 (X_1-X_2)}\\big] \n\t\\\\\n\t& \n\t = \\mathbb{E}\\big[ \n\t e^{ i \\, 2^{-\\nicefrac{1}{2}} (\\xi_1\\varphi_1 +\\xi_2\\varphi_2) ( X_1) + i \\, 2^{-\\nicefrac{1}{2}} (\\xi_1\\varphi_1 - \\xi_2\\varphi_2) (X_2)}\n\t \\big] \n\t\\\\\n\t& \n\t = \n\t \\mathbb{E}\\big[ e^{ i \\, 2^{-\\nicefrac{1}{2}} (\\xi_1\\varphi_1 +\\xi_2\\varphi_2) ( X_1)} \\big] \n\t \\,\n\t \\mathbb{E}\\big[ e^{ i \\, 2^{-\\nicefrac{1}{2}} (\\xi_1\\varphi_1 - \\xi_2\\varphi_2) (X_2)}\n\t \\big] .\n\\end{split}\n\\end{equation}\nHence, we obtain that for all $\\varphi_1, \\varphi_2 \\in V'$, $\\xi = (\\xi_1, \\xi_2) \\in \\mathbb{R}^2 $ it holds that \n\\begin{equation}\n\\begin{split}\n\t&\n\t \\mathbb{E}\\!\\left[ \n\t e^{ \n\t i \\langle \\xi, (\\varphi_1(Y_1), \\varphi_2(Y_2)) \\rangle_{\\mathbb{R}^2} \n\t } \n\t \\right] \n\t\\\\\n\t& = \n \t \\exp\\!\\left( - \\tfrac{1}{2} \\mathbb{E}\\!\\left[| 2^{ - \\nicefrac{1}{2} } (\\xi_1\\varphi_1 +\\xi_2\\varphi_2) ( X_1)|^2\\right] \\right) \n \t \\exp\\!\\left( - \\tfrac{1}{2} \\mathbb{E}\\!\\left[| 2^{ - \\nicefrac{1}{2} } (\\xi_1\\varphi_1 -\\xi_2\\varphi_2) ( X_2)|^2\\right]\\right) \n\t\\\\\n\t&\n\t= \n \t \\exp\\!\\left( - \\tfrac{1}{4} \\mathbb{E}\\!\\left[| (\\xi_1\\varphi_1 +\\xi_2\\varphi_2) ( X_1)|^2\\right] \\right) \n \t \\exp\\!\\left( \n \t - \\tfrac{1}{4} \n \t \\mathbb{E}\\!\\left[| (\\xi_1\\varphi_1 -\\xi_2\\varphi_2) ( X_2)|^2\\right]\n \t \\right) \n \t\\\\\n \t&\n \t=\n \t \\exp\\!\\left( \n \t - \n \t \\tfrac{ 1 }{ 4 } \n \t \\left\\{\n\t \\mathbb{E}\\!\\left[\n\t\t| ( \\xi_1 \\varphi_1 + \\xi_2 \\varphi_2 )( X_1 ) |^2 \n\t \\right]\n\t +\n\t \\mathbb{E}\\!\\left[\n\t | ( \\xi_1 \\varphi_1 - \\xi_2 \\varphi_2 )( X_1 ) |^2\n\t \\right]\n\t \\right\\}\n \t \\right) \n \t\\\\\n \t&\n \t=\n \t \\exp\\!\\left( \n \t - \n \t \\tfrac{ 1 }{ 4 } \n \t \\left\\{\n \t 2\n \t \\,\n\t \\mathbb{E}\\!\\left[\n\t\t| ( \\xi_1 \\varphi_1 )( X_1 ) |^2 \n\t \\right]\n\t +\n\t 2 \\,\n\t \\mathbb{E}\\!\\left[\n\t | ( \\xi_2 \\varphi_2 )( X_1 ) |^2\n\t \\right]\n\t \\right\\}\n \t \\right) \n \t .\n\\end{split}\n\\end{equation}\nThis shows for all $\\varphi_1, \\varphi_2 \\in V'$, $\\xi = (\\xi_1, \\xi_2) \\in \\mathbb{R}^2 $ that \n\\begin{equation}\n\\label{eq:distribution_Fernique_same}\n\\begin{split}\n\t \\mathbb{E}\\!\\left[ \n\t e^{ \n\t i \\langle \\xi, (\\varphi_1(Y_1), \\varphi_2(Y_2)) \\rangle_{\\mathbb{R}^2} \n\t } \n\t \\right]\n\t& = \\exp \\! \\left( - \\tfrac{1}{2}\\mathbb{E}\\!\\left[|\\xi_1 \\varphi_1(X_1)|^2\\right]\\right) \\exp \\! \\left( - \\tfrac{1}{2}\\mathbb{E}\\!\\left[|\\xi_2 \\varphi_2(X_2)|^2\\right]\\right) \\\\\n\t&= \n\t \\mathbb{E}\\big[ \n\t e^{ \n\t i \\, \\xi_1 \\varphi_1( X_1 ) \n\t }\n\t \\big] \n\t \\, \n\t \\mathbb{E}\\big[ \n\t e^{ \n\t i \\, \\xi_2 \\varphi_2( X_2 ) \n\t }\n\t \\big] \n\t=\n\t \\mathbb{E}\\big[ \n\t e^{ \n\t i \\, \\xi_1 \\varphi_1( X_1 ) \n\t +\n\t i \\, \\xi_2 \\varphi_2( X_2 ) \n\t }\n\t \\big] \n\t\\\\ &\n\t= \n\t \\mathbb{E} \\!\\left[ e^{ i \\langle \\xi,(\\varphi_1(X_1), \\varphi_2(X_2))\\rangle_{\\mathbb{R}^2}}\\right].\n\t\\end{split}\n\t\\end{equation}\nIn particular, this implies\nfor all $\\varphi \\in V'$ that\n\\begin{equation}\n\\mathbb{E}\\!\\left[ \n e^{ \n i \\,\n \\varphi(Y_1) \n } \n\\right] \n= \n\\mathbb{E}\\!\\left[ \n e^{ \n i \\,\n \\varphi(X_1) \n } \n\\right]\n\\qquad \\text{and} \\qquad\n\\mathbb{E}\\!\\left[ \n e^{ \n i \\,\n \\varphi(Y_2) \n } \n\\right] \n= \n\\mathbb{E}\\!\\left[ \n e^{ \n i \\,\n \\varphi(X_2) \n } \n\\right].\n\\end{equation}\nLemma~\\ref{lem:charac_fct2}\nhence establishes that\n\\begin{equation}\n Y_1(\\P)_{\\mathcal{B}(V)}=X_1(\\P)_{\\mathcal{B}(V)}\n \\qquad \n \\text{and}\n \\qquad\n Y_2(\\P)_{\\mathcal{B}(V)}=X_2(\\P)_{\\mathcal{B}(V)}\n .\n\\end{equation}\nThis and \\eqref{eq:equal_dist_X} prove~\\eqref{item:equal_dist}.\nNext note that\nLemma~\\ref{lem:charac_fct2} \nand \\eqref{eq:distribution_Fernique_same} show \nfor all $\\varphi_1, \\varphi_2 \\in V'$ \nthat\n\\begin{equation}\n( \\varphi_1 \\circ Y_1, \\varphi_2 \\circ Y_2 )( \\P )_{ \\mathcal{B}( \\mathbb{R}^2 ) }\n=\n( \\varphi_1 \\circ X_1, \\varphi_2 \\circ X_2 )( \\P )_{ \\mathcal{B}( \\mathbb{R}^2 ) }.\n\\end{equation}\nThis, the assumption that $ X_1 $ and $ X_2 $ are independent, and~\\eqref{item:equal_dist} ensure\nfor all $ \\varphi_1, \\varphi_2 \\in V' $ that\n\\begin{equation}\n\\begin{split}\n( \\varphi_1 \\circ Y_1 )( \\P )_{ \\mathcal{B}( \\mathbb{R} ) } \\otimes ( \\varphi_2 \\circ Y_2 )( \\P )_{ \\mathcal{B}( \\mathbb{R} ) }\n& =\n( \\varphi_1 \\circ X_1 )( \\P )_{ \\mathcal{B}( \\mathbb{R} ) } \\otimes ( \\varphi_2 \\circ X_2 )( \\P )_{ \\mathcal{B}( \\mathbb{R} ) }\n\\\\ & =\n( \\varphi_1 \\circ X_1, \\varphi_2 \\circ X_2 )( \\P )_{ \\mathcal{B}( \\mathbb{R}^2 ) }\n=\n( \\varphi_1 \\circ Y_1, \\varphi_2 \\circ Y_2 )( \\P )_{ \\mathcal{B}( \\mathbb{R}^2 ) }.\n\\end{split}\n\\end{equation}\nThis proves\nfor every $ \\varphi_1, \\varphi_2 \\in V' $\nthat\n$\n \\varphi_1 \\circ Y_1\n$\nand\n$\n \\varphi_2 \\circ Y_2\n$\nare independent random variables.\nLemma~\\ref{lem:indep} hence establishes that $Y_1$ and $Y_2$ are independent random variables. The proof of Lemma~\\ref{lem:iid} is thus completed.\n\\end{proof}\n\n\nIn the next result, Proposition~\\ref{thm:fernique} below, we present Fernique's theorem (see, e.g., Theorem~8.2.1 in Stroock~\\cite{Stroock2010}).\n\n\n\\begin{prop}[Fernique's theorem]\n\\label{thm:fernique}\nLet $ (V, \\left\\| \\cdot \\right\\|_V) $ be a separable normed $ \\mathbb{R} $-vector space, \nlet $ (\\Omega, \\mathcal{F}, \\P) $ be a probability space, \nlet $X \\colon \\Omega \\to V$ be a function which satisfies for all $\\varphi \\in V'$ that $\\varphi \\circ X \\colon \\Omega \\to \\mathbb{R} $ is a\ncentered Gaussian random variable, \nand let $ R \\in (0, \\infty) $ satisfy \n\\begin{equation}\n R \\geq \n \\inf\\!\\big( \n \\{ r \\in [0, \\infty) \\colon \\P(\\|X\\|_V \\leq r) \\geq \\nicefrac{9}{10}\n \\} \\big)\n .\n\\end{equation}\nThen\n\t\\begin{align}\\label{eq:th:fern0}\n\t\\begin{split}\n\t\\mathbb{E} \\! \\left[ \\exp\\! \\left( \\frac{\\|X\\|_V^2}{18 R^2}\\right)\\right] \\leq \\sqrt{e} + \\sum_{n=0}^{\\infty} \\bigg[\\frac{e}{3}\\bigg]^{(2^n)}< 13 < \\infty.\n\t\\end{split}\n\t\\end{align}\t\n\\end{prop}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{thm:fernique}]\n\tThroughout this proof let\n\t$\n\t ( \\tilde{\\Omega}, \\tilde{\\mathcal{F}}, \\tilde{\\P} )\n\t$\n\tbe the probability space given by\n\t$\\tilde{\\Omega} = \\Omega \\times \\Omega$, \n\t$\\tilde{\\mathcal{F}} = \\mathcal{F} \\otimes \\mathcal{F}$, \n\tand\n\t$\\tilde{\\P} = \\P \\otimes \\P$, \n\tlet $ Y_1, Y_2 \\colon \\tilde{\\Omega} \\to V $ \n\tbe the functions which satisfy for all \n\t$\\omega_1, \\omega_2 \\in \\Omega$ that \n\t\\begin{equation}\n\t Y_1(\\omega_1, \\omega_2) = X(\\omega_1) \n\t\\qquad \n\t \\text{and}\n\t\\qquad\n\t Y_2(\\omega_1, \\omega_2) = X(\\omega_2) , \n\t\\end{equation}\n\tlet $ Z_1, Z_2 \\colon \\tilde{\\Omega} \\to V $ \n\tbe the functions which satisfy \n\t\\begin{equation}\n\t Z_1 = 2^{-\\nicefrac{1}{2}}(Y_1 + Y_2) \n\t \\qquad\n\t \\text{and}\n\t \\qquad\n\t Z_2 = 2^{-\\nicefrac{1}{2}}(Y_1 - Y_2) , \n\t\\end{equation}\n\tand let $t_n \\in (0, \\infty)$, $n \\in \\mathbb{N}_0$, be the sequence \n\tof real numbers which satisfies for all $n \\in \\mathbb{N}$ that \n\t\\begin{equation}\n\t t_0 = R \n\t \\qquad \n\t \\text{and} \n\t \\qquad \n\t t_n = R + \\sqrt{2} \\, t_{ n - 1 } . \n\t\\end{equation}\n\tObserve that \n\t$ Y_1 $ and $ Y_2 $ are independent random variables.\n\tIn addition, note that\n\tfor every $\\varphi \\in V'$ it holds that \n\tthe random variables \n\t$ \\varphi \\circ Y_1 \\colon \\tilde{\\Omega }\\to \\mathbb{R} $ and \n\t$ \\varphi\\circ Y_2 \\colon \\tilde{\\Omega} \\to \\mathbb{R} $ \n\thave the same distribution on $ ( \\mathbb{R}, \\mathcal{B}(\\mathbb{R}) ) $ \n\tas the random variable $ \\varphi \\circ X \\colon \\Omega \\to \\mathbb{R} $. \n\tLemma~\\ref{lem:iid} \n\thence ensures that $Z_1$ and $Z_2$ are \n\tindependent random variables and \n\t\\begin{equation}\n\t Z_1(\\tilde{\\P})_{\\mathcal{B}(V)}=\n\t Z_2(\\tilde{\\P})_{\\mathcal{B}(V)}=\n\t Y_1(\\tilde{\\P})_{\\mathcal{B}(V)}=X(\\P)_{\\mathcal{B}(V)}\n\t .\n\t\\end{equation}\n\tThis proves for all $s, t \\in (0, \\infty)$ with $s \\leq t$ that\n\t\\begin{align}\n\t\\begin{split}\n\t&\\P(\\|X\\|_V \\leq s) \\, \\P(\\|X\\|_V > t)= \\tilde{\\P}(\\|Z_2\\|_V \\leq s) \\, \\tilde{\\P}(\\|Z_1\\|_V > t) \\\\\n\t&= \\tilde{\\P}\\big(\\{\\|Z_2\\|_V \\leq s\\} \\cap \\{\\|Z_1\\|_V > t\\}\\big) \\\\\n\t&= \\tilde{\\P}\\big(\\{\\|Y_1- Y_2\\|_V \\leq \\sqrt{2} \\,s \\} \\cap \\{ \\|Y_1 +Y_2\\|_V > \\sqrt{2} \\,t \\}\\big) \\\\\n\t& \\leq \\tilde{\\P}\\big(\\{| \\|Y_1\\|_V- \\|Y_2\\|_V | \\leq \\sqrt{2} \\, s \\} \\cap \\{ \\|Y_1\\|_V +\\|Y_2\\|_V > \\sqrt{2} \\, t \\}\\big) \\\\\n\t& \\leq \\tilde{\\P}\\big(\\!\\min\\{\\|Y_1\\|_V, \\|Y_2\\|_V\\} > 2^{-\\nicefrac{1}{2}}(t-s)\\big) = \\big| \\P\\big(\\|X\\|_V > 2^{-\\nicefrac{1}{2}}(t-s) \\big) \\big|^2.\n\t\\end{split}\n\t\\end{align}\n\tThis, in turn, implies for all $n \\in \\mathbb{N}$ that\n\t\\begin{align}\n\t\\P(\\|X\\|_V \\leq R) \\,\\P(\\|X\\|_V > t_n) \\leq | \\P(\\|X\\|_V > t_{n-1} ) |^2.\n\t\\end{align}\n\tThe fact that $\\P(\\|X\\|_V \\leq R) \\geq \\nicefrac{9}{10} >0$ hence shows for all $n \\in \\mathbb{N}$ that\n\t\\begin{align}\n\t\\frac{\\P(\\|X\\|_V > t_n)}{\\P(\\|X\\|_V \\leq R) } \\leq \\left( \\frac{\\P(\\|X\\|_V > t_{n-1})}{\\P(\\|X\\|_V \\leq R) } \\right)^2.\n\t\\end{align}\n\tThis and induction on $n \\in \\mathbb{N}_0$ establish for all $n \\in \\mathbb{N}_0$ that\n\t\\begin{align}\n\t\\label{eq:thm:fern1}\n\t\\frac{\\P(\\|X\\|_V > t_n)}{\\P(\\|X\\|_V \\leq R) } \\leq \\left( \\frac{\\P(\\|X\\|_V > R)}{\\P(\\|X\\|_V \\leq R) } \\right)^{(2^n)}.\n\t\\end{align}\n\tMoreover, induction on $n \\in \\mathbb{N}_0$ ensures for all $n \\in \\mathbb{N}_0$ that\n\t\\begin{align}\n\tt_n = R \\cdot \\frac{2^{\\frac{n+1}{2}}-1}{\\sqrt{2}-1} \\leq (\\sqrt{2}+1) \\, 2^{\\frac{n+1}{2}} R \\leq 3 \\cdot 2^{\\frac{n+1}{2}} R .\n\t\\end{align}\n\tCombining this with \\eqref{eq:thm:fern1} \n\tand the fact that \n\t$\\P(\\|X\\|_V \\leq R) \\geq \\nicefrac{9}{10} \\geq 9 \\, \\P(\\|X\\|_V > R)$ \n\tyields that for all $n \\in \\mathbb{N}_0$ it holds that\n\t\\begin{align}\n\t\\P(\\|X\\|_V > 3 \\cdot 2^{\\frac{n}{2}} R ) \\leq 3^{-(2^n)}.\n\t\\end{align} \n\tThe fact that $\\nicefrac{e}{3}<1$ hence shows that\n\t\\begin{align}\n\t\\begin{split}\n\t\\mathbb{E} \\! \\left[ \\exp\\! \\left( \\frac{\\|X\\|_V^2}{18 R^2}\\right)\\right]\n\t&\\leq \\sqrt{e} \\, \\P(\\|X\\|_V \\leq 3 R) + \\sum_{n=0}^{\\infty} e^{(2^n)} \\, \\P \\big( 3 \\cdot 2^{\\frac{n}{2}} R < \\|X\\|_V \\leq 3 \\cdot 2^{\\frac{n+1}{2}} R\\big) \\\\\n\t&\\leq \\sqrt{e}+ \\sum_{n=0}^{\\infty} e^{(2^n)} \\, \\P \\big( \\|X\\|_V > 3 \\cdot 2^{\\frac{n}{2}} R \\big) \\leq \\sqrt{e} + \\sum_{n=0}^{\\infty} \\bigg[\\frac{e}{3}\\bigg]^{(2^n)}\\\\\n\t& \\leq \\sqrt{e} + \\sum_{n=0}^{\\infty} \\bigg[\\frac{e}{3}\\bigg]^{n} = \\sqrt{e} + \\frac{3}{3-e} < 13 < \\infty.\n\t\\end{split}\n\t\\end{align}\n\tThe proof of Proposition~\\ref{thm:fernique} is thus completed.\n\\end{proof}\n\n\n\n\\section{Abstract examples}\n\\label{sec:abstract}\n\nIn this section we verify the assumptions of Theorem~\\ref{thm:strong} above\nin the case of more specific SPDEs (see the setting in Subsection~\\ref{setting:example} below)\nand establish strong convergence in this setting in Proposition~\\ref{abs:prop:last} below.\nFirst, we show a result on transformations of semigroups for solutions of SPDEs in Proposition~\\ref{prop:transform_SG} below.\nNext we combine this with Fernique's theorem (see Proposition~\\ref{thm:fernique} above)\nand the elementary results in Lemmas~\\ref{abs:phi(w)_finite}--\\ref{lem:conv:series},\nProposition~\\ref{prop:abstract}, and Lemma~\\ref{lemma:conv:rate}\nto derive certain properties of stochastic convolution processes (see Proposition~\\ref{prop:exists} below).\nFinally, the latter allow us to apply Theorem~\\ref{thm:strong} in order to prove Proposition~\\ref{abs:prop:last}.\n\n\n\\subsection{Transformations of semigroups for solutions of SPDEs}\n\nRoughly speaking, Proposition~\\ref{prop:transform_SG} below\nproves that a mild solution of an SPDE does not depend on a shift of the linear part of the drift coefficient function\nif the nonlinear part of the drift coefficient function is shifted accordingly.\nThis result is achieved under optimal hypotheses in the sense that\nthe hypotheses of Proposition~\\ref{prop:transform_SG}\nare required for the mathematical formulation of the statement to be meaningful\n(see, in particular, \\eqref{eq:assume1}--\\eqref{eq:assume2}).\nTo the best of our knowledge, Proposition~\\ref{prop:transform_SG} is the first result in the literature\nto establish this assertion under optimal hypotheses,\neven in the special case of partial differential equations (where the diffusion coefficient function is zero).\n\n\n\\begin{prop}\n\\label{prop:transform_SG}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet \n$ ( H, \\left< \\cdot, \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) $\nand\n$ ( U, \\left< \\cdot, \\cdot \\right>_U, \\left\\| \\cdot \\right\\|_U ) $\nbe separable $ \\mathbb{R} $-Hilbert spaces,\nlet $ \\mathbb{H} \\subseteq H$ be a nonempty orthonormal basis of $ H $,\nlet $T \\in (0, \\infty)$, $ \\alpha, \\beta, \\gamma, \\eta, \\kappa \\in \\mathbb{R} $,\nlet $\\lambda \\colon \\mathbb{H} \\to \\mathbb{R} $\nbe a function which satisfies\n$\\sup_{b \\in \\mathbb{H}} \\lambda_b < \\kappa $,\nlet $ A \\colon D(A) \\subseteq H \\to H $\nbe the linear operator which satisfies\n$ D(A) = \\{ v \\in H \\colon \\sum_{b \\in \\mathbb{H}} | \\lambda_b \\langle b , v \\rangle_H |^2 < \\infty \\} $\nand\n$ \\forall \\, v \\in D(A) \\colon A v = \\sum_{b \\in \\mathbb{H}} \\lambda_b \\langle b , v \\rangle_H b $,\nlet \n$ \n( \nH_r \n,\n\\left< \\cdot, \\cdot \\right>_{ H_r }\n,\n\\left\\| \\cdot \\right\\|_{ H_r } \n) \n$, $ r \\in \\mathbb{R} $,\nbe a family of interpolation spaces associated \nto $ \\kappa - A $,\nlet $ ( \\Omega, \\mathcal{F}, \\P)$ be a probability space with a normal filtration $( \\mathbb{F}_t )_{ t \\in [0,T] } $,\nlet $ ( W_t )_{ t \\in [0,T] } $ be an\n$ \\operatorname{Id}_U $-cylindrical \n$ ( \\Omega, \\mathcal{F}, \\P, ( \\mathbb{F}_t )_{ t \\in [0,T] } ) $-Wiener process,\nlet\n$ O \\in \\mathcal{B}( H_{ \\gamma } ) $,\nlet\n$ F \\colon O \\to H_{ \\alpha } $\nbe a $ \\mathcal{B}( O ) $\/$ \\mathcal{B}( H_{ \\alpha } ) $-measurable function,\nlet\n$ \\tilde{F} \\colon O \\to H_{ \\min\\{ \\alpha , \\gamma \\} } $\nbe the function which satisfies for all $ v \\in O $ that\n$\n\\tilde{F}( v ) = \\eta v + F( v )\n$,\nlet\n$ B \\colon O \\to \\mathrm{HS}( U, H_{ \\beta } ) $\nbe a $ \\mathcal{B}( O ) $\/$ \\mathcal{B}( \\mathrm{HS}( U, H_{ \\beta } ) ) $-measurable function,\nlet\n$ \\xi \\colon \\Omega \\to O $\nbe an $ \\mathbb{F}_0 $\/$ \\mathcal{B}( O ) $-measurable function,\nand let\n$ X \\colon [0,T] \\times \\Omega \\to O $ \nbe an $ ( \\mathbb{F}_t )_{ t \\in [0,T] } $\/$ \\mathcal{B}( O ) $-predictable stochastic process \nwhich satisfies for all $ t \\in [0,T] $ that\n\\begin{equation}\n\\label{eq:assume1}\n\\P\\!\\left(\n\\int_0^t \n\\| \ne^{ ( t - s )A } F( X_s ) \n\\|_{ H_{ \\gamma } } \n+ \n\\| \ne^{ ( t - s )A } \nB( X_s ) \n\\|^2_{ \\mathrm{HS}( U, H_{ \\gamma } ) } \\, ds < \\infty\n\\right) = 1\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:assume2}\n\\begin{split}\n\\left[ \nX_t \n\\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n& =\n\\left[ \ne^{ tA } \\xi \n+\n\\int_0^t\n\\mathbbm{1}_{\n\t\\{\n\t\\int_0^t \\| e^{ ( t - u )A } F( X_u ) \\|_{ H_{ \\gamma } } du < \\infty \n\t\\}\n}\n\\,\ne^{ ( t - s )A } F( X_s ) \\, ds\n\\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n\\\\ & \\quad +\n\\int_0^t\ne^{ ( t - s ) A} B( X_s ) \\, dW_s\n.\n\\end{split}\n\\end{equation}\nThen it holds for all $ t \\in [0,T] $ that\n\\begin{equation}\n\\P\\!\\left(\n\\int_0^t \n\\| \ne^{ ( t - s )(A- \\eta) } \\tilde{F}( X_s ) \n\\|_{ H_{ \\gamma } } \n+ \n\\| \ne^{ ( t - s )(A-\\eta) } \nB( X_s ) \n\\|^2_{ \\mathrm{HS}( U , H_{ \\gamma } ) } \\, ds < \\infty\n\\right) = 1\n\\end{equation}\nand\n\\begin{equation}\n\\begin{split}\n\\left[ \nX_t \n\\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n& =\n\\left[ \ne^{ t(A-\\eta) } \\xi \n+\n\\int_0^t\n\\mathbbm{1}_{\n\t\\{\n\t\\int_0^t \\| e^{ ( t - u )(A-\\eta) } \\tilde{F}( X_u ) \\|_{ H_{ \\gamma } } du < \\infty \n\t\\}\n}\n\\,\ne^{ ( t - s )(A- \\eta)} \\tilde{F}( X_s ) \\, ds\n\\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n\\\\ & \\quad +\n\\int_0^t\ne^{ ( t - s ) (A-\\eta)} B( X_s ) \\, dW_s\n.\n\\end{split}\n\\end{equation}\n\\end{prop}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:transform_SG}]\nThroughout this proof let $ \\psi, \\psi_1 \\colon [0,T] \\times H_{ \\gamma } \\to H_{ \\gamma } $\nand\n$ \\psi_2 \\colon [0,T] \\times H_{ \\gamma } \\to L( H_{ \\gamma } ) $\nbe the functions which satisfy\nfor all $ (t, x) \\in [0,T] \\times H_{ \\gamma } $,\n$ v \\in H_{ \\gamma } $\nthat\n\\begin{equation}\n\\psi( t, x ) \n= \ne^{ \\eta t } x\n,\n\\quad\n\\psi_1( t, x )\n=\n\\tfrac{ \\partial }{ \\partial t } \\psi( t, x )\n=\n\\eta \n\\,\n\\psi( t, x )\n,\n\\quad\n\\text{and}\n\\quad\n\\psi_2( t, x ) \\, v\n=\n\\tfrac{ \\partial }{ \\partial x } \\psi( t, x ) \\, v\n=\n\\psi( t, v )\n.\n\\end{equation}\nNext observe that\n\\eqref{eq:assume1}--\\eqref{eq:assume2}\nimply that for all $ t \\in [0,T] $ it holds that\n\\begin{equation}\n\\begin{split}\n&\n\\P\\!\\left(\t\\int_0^t \n\\| \ne^{ ( t - s )( A - \\eta ) } \ne^{ - \\eta s }\nF( X_s ) \n\\|_{ H_{ \\gamma } } \n+ \n\\| \ne^{ ( t - s ) ( A - \\eta ) } \ne^{ - \\eta s }\nB( X_s ) \n\\|^2_{ \\mathrm{HS}( U, H_{ \\gamma } ) } \\, ds < \\infty \\right)\n\\\\ & =\n\\P \\!\\left(\ne^{ - \\eta t }\n\\int_0^t \n\\| \ne^{ ( t - s ) A } \nF( X_s ) \n\\|_{ H_{ \\gamma } } \\, ds \n+ \ne^{ - 2 \\eta t }\n\\int_0^t \n\\| \ne^{ ( t - s ) A } \nB( X_s ) \n\\|^2_{ \\mathrm{HS}( U, H_{ \\gamma } ) } \\, ds\n< \\infty \\right)\n\\\\ & =\n\\P \\!\\left(\n\\int_0^t \n\\| \ne^{ ( t - s ) A } \nF( X_s ) \n\\|_{ H_{ \\gamma } }\n+\n\\| \ne^{ ( t - s ) A } \nB( X_s ) \n\\|^2_{ \\mathrm{HS}( U, H_{ \\gamma } ) } \\, ds\n< \\infty \\right) =1\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{split}\n\\left[ \ne^{ - \\eta t } X_t \n\\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n& =\n\\left[ \ne^{ t(A-\\eta) } \\xi \n+\n\\int_0^t\n\\mathbbm{1}_{\n\t\\{\n\t\\int_0^t \\| e^{ ( t - u )A }\n\tF( X_u ) \\|_{ H_{ \\gamma } } du < \\infty \n\t\\}\n}\n\\,\ne^{ ( t - s )(A- \\eta)}\ne^{ - \\eta s } F( X_s ) \\, ds\n\\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n\\\\ & \\quad +\n\\int_0^t\ne^{ ( t - s ) (A-\\eta)}\ne^{ - \\eta s } B( X_s ) \\, dW_s .\n\\end{split}\n\\end{equation}\nNote that this establishes that the stochastic process\n$\n\\big( [0,T] \\times \\Omega \\ni (t,\\omega) \\mapsto e^{ - \\eta t } X_t(\\omega) \\in H_{\\gamma} \\big)\n$\nis an $ ( \\Omega, \\mathcal{F}, \\P, ( \\mathbb{F}_t )_{ t \\in [0,T] } ) $-mild It\\^{o} process with evolution family\n$ \\big( \\{ (t_1, t_2) \\in [0,T]^2 \\colon t_1 < t_2 \\} \\ni ( s , t ) \\mapsto e^{ ( t - s )( A - \\eta ) } \\in L(H_{\\min\\{\\alpha,\\beta,\\gamma\\}}, H_{\\gamma}) \\big)\n$,\nmild drift\n$ \\big( \n[0,T] \\times \\Omega \\ni (t, \\omega) \\mapsto\te^{ - \\eta t } F( X_t(\\omega) ) \\in H_{\\min\\{\\alpha,\\beta,\\gamma\\}} \\big)\n$,\nand mild diffusion \n$ \\big( \n[0,T] \\times \\Omega \\ni (t, \\omega) \\mapsto\ne^{ - \\eta t } B( X_t(\\omega) ) \\in \\mathrm{HS}(U, H_{\\min\\{\\alpha,\\beta,\\gamma\\}}) \\big)\n$ (see Definition~1 in Da~Prato, Jentzen, \\& R\\\"ockner~\\cite{DaPratoJentzenRoeckner_Mild}).\nThe mild It\\^{o} formula in Theorem~1 in Da~Prato, Jentzen, \\& R\\\"ockner~\\cite{DaPratoJentzenRoeckner_Mild} \nhence proves that\nfor all $ t \\in [0,T] $ it holds that\n\\begin{gather}\n\\P\\! \\left( \\int_0^t\n\\big\\|\n\t\\psi_1 \\bigl( \n\ts, \n\te^{ ( t - s )( A - \\eta ) } \n\te^{ - \\eta s } \n\tX_s\n\t\\bigr)\n\\big\\|_{ H_{ \\gamma } }\n\\,\nds\n< \\infty \\right)=1\n,\\\\\n \\P \\! \\left(\\int_0^t\n\\big\\|\n\t\\psi_2\\!\\left( \n\ts, \n\te^{ ( t - s ) ( A - \\eta ) } \n\te^{ - \\eta s } \n\tX_s \n\t\\right)\n\te^{\n\t\t ( t - s ) ( A - \\eta )\n\t}\n\te^{ - \\eta s } \n\tF( X_s )\n\\big\\|_{ H_{ \\gamma } }\n\\,\nds\n< \\infty \\right)=1\n,\\\\\n\\P \\! \\left( \\int_0^t\n\\big\\|\n\t\\psi_2\\!\\left( \n\ts, \n\te^{ ( t - s ) ( A - \\eta ) } \n\te^{ - \\eta s } \n\tX_s\n\t\\right)\n\te^{\n\t\t ( t - s ) ( A - \\eta )\n\t}\n\te^{ - \\eta s } \n\tB( X_s )\n\\big\\|_{ \\mathrm{HS}( U, H_{ \\gamma } ) }^2\n\\,\nds\n< \\infty \\right)=1 ,\n\\end{gather}\nand\n\\begin{equation}\n\\begin{split}\n& \\left[ X_t \\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n=\n\\left[ \\psi\\!\\left( \nt, e^{ - \\eta t } X_t \n\\right) \\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n\\\\ & = \n\\bigg[ \\psi\\!\\left( 0, \ne^{ t ( A - \\eta )}\ne^{ - \\eta \\cdot 0 } X_0 \n\\right) \n+\n \\int_0^t \\mathbbm{1}_{\\{ \\int_0^t \\|\\psi_1( \n\tu, \n\te^{ ( t - u )( A - \\eta ) } \n\te^{ - \\eta u } \n\tX_u\n\t)\\|_{H_{\\gamma}} du < \\infty\\}}\n\t\\,\n\t\\psi_1\\!\\left( \ns, \ne^{ ( t - s )( A - \\eta ) } \ne^{ - \\eta s } \nX_s\n\\right)\nds \\\\\n& \\quad + \\int_0^t \\mathbbm{1}_{\\{ \\int_0^t \\|\t\\psi_2( \n\tu, \n\te^{ ( t - u ) ( A - \\eta ) } \n\te^{ - \\eta u } \n\tX_u ) \\,\n\te^{\n\t\t( t - u ) ( A - \\eta )\n\t}\n\te^{ - \\eta u } \n\tF( X_u )\\|_{H_{\\gamma}} du < \\infty\\}} \t\\\\\n& \\qquad \\quad \\cdot \\psi_2\\!\\left( \ns, \ne^{ ( t - s ) ( A - \\eta ) } \ne^{ - \\eta s } \nX_s \n\\right)\ne^{\n\t( t - s ) ( A - \\eta )\n}\ne^{ - \\eta s } \nF( X_s ) \\, ds \\bigg]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) } \\\\\n& \\quad +\n\\int_0^t\n\\psi_2\\!\\left( \ns, \ne^{ ( t - s ) ( A - \\eta ) } \ne^{ - \\eta s } \nX_s\n\\right)\ne^{\n\t( t - s ) ( A - \\eta )\n}\ne^{ - \\eta s } \nB( X_s ) \\,\ndW_s\n.\n\\end{split}\n\\end{equation}\nThis ensures that for all $t \\in [0,T]$ it holds that\n\\begin{equation}\n\\label{eq:trans:first}\n\\P\\! \\left( \\int_0^t\n\\big\\|\ne^{ ( t - s ) ( A - \\eta ) }\n\\eta\nX_s\n\\big\\|_{ H_{ \\gamma } }\n+\n\\big\\|\n\te^{\n\t ( t - s ) ( A - \\eta )\n}\nF( X_s )\n\\big\\|_{ H_{ \\gamma } }\n+\n\\big\\|\ne^{\n\t ( t - s ) ( A - \\eta )\n}\nB( X_s )\n\\big\\|_{ \\mathrm{HS}( U, H_{ \\gamma } ) }^2\n\\,\nds\n< \\infty \\right)=1\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:trans:last}\n\\begin{split}\n&\n\\left[ X_t \\right]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n=\n\\bigg[e^{ t( A - \\eta ) } \n\\xi + \\int_0^t \\mathbbm{1}_{\\{ \\int_0^t\n\t\\|\n\te^{ ( t - u ) ( A - \\eta ) }\n\t\\eta\n\tX_u\\|_{ H_{ \\gamma } }\n\tdu\n\t< \\infty \\}}\n\\,\ne^{ ( t - s ) ( A - \\eta ) }\n\\eta\nX_s\n\\,\nds\n\\\\ & \\quad\n+ \\int_0^t \\mathbbm{1}_{\\{ \\int_0^t\n\t\\|\n\te^{ ( t - u ) ( A - \\eta ) }\n\tF(X_u)\\|_{ H_{ \\gamma } }\n\tdu\n\t< \\infty \\}} \\,\ne^{ ( t - s ) ( A - \\eta ) }\nF(X_s)\n\\,\nds \n \\bigg]_{ \\P, \\mathcal{B}( H_{ \\gamma } ) }\n+\n\\int_0^t\ne^{ ( t - s )( A - \\eta ) }\nB( X_s ) \\,\ndW_s\n.\n\\end{split}\n\\end{equation}\nMoreover, \\eqref{eq:trans:first} shows for all $t \\in [0,T]$ that\n\\begin{equation}\n\\begin{split}\n&\n\\P \\biggl(\n\\int_0^t\n\\big\\|\ne^{ ( t - s ) ( A - \\eta ) }\n\\tilde{F}(X_s)\\big\\|_{ H_{ \\gamma } } \n\\,\nds\n< \\infty \\biggr)\n= \n\\P \\biggl(\n\\int_0^t\n\\bigl\\|\ne^{ ( t - s ) ( A - \\eta ) } \\big[\n\\eta X_s + F(X_s) \\bigr]\\big\\|_{ H_{ \\gamma } } \\, ds < \\infty \\biggr) \\\\\n& \\geq\n\\P \\biggl(\n\\int_0^t\n\\big\\|\ne^{ ( t - s ) ( A - \\eta ) }\n\\eta\nX_s\\big\\|_{ H_{ \\gamma } }\n+ \\big\\|\ne^{ ( t - s ) ( A - \\eta ) } F(X_s) \\big\\|_{ H_{ \\gamma } } \\, ds < \\infty \\biggr)\n= 1.\n\\end{split}\n\\end{equation}\nCombining this with \\eqref{eq:trans:first}--\\eqref{eq:trans:last} completes the proof of Proposition~\\ref{prop:transform_SG}.\n\\end{proof}\n\n\n\\subsection{Setting}\\label{setting:example}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet\n$ T, c_0, \\gamma, \\theta, \\vartheta \\in (0,\\infty)$,\n$ \\alpha \\in [ 0, \\nicefrac{1}{2}]$,\n$\\varphi \\in [0,1)$,\n$\\rho \\in [0, \\nicefrac{1}{4})$,\n$\\varrho \\in (\\rho, \\nicefrac{1}{4})$,\n$ \\chi \\in (0, \\nicefrac{( \\varrho - \\rho ) }{( 1 + \\vartheta) }] $,\n$( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) = (L^2(\\lambda_{(0,1)}; \\mathbb{R}), \\langle \\cdot , \\cdot \\rangle_{L^2(\\lambda_{(0,1)}; \\mathbb{R})}, \\left\\| \\cdot \\right\\|_{L^2(\\lambda_{(0,1)}; \\mathbb{R})} )$,\nlet\n$(e_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to H $\nand\n$(\\lambda_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, \\infty)$\nbe the functions which satisfy\nfor all $ n \\in \\mathbb{N} $ that\n$ e_n = [ (\\sqrt{2} \\sin(n \\pi x) )_{x \\in (0,1)}]_{\\lambda_{(0,1)} , \\mathcal{B}(\\mathbb{R})}$\nand\n$ \\lambda_n = c_0 \\pi^2 n^2 $,\nlet\n$ A \\colon D(A) \\subseteq H \\to H $\nbe the linear operator which satisfies\n$ D(A) = \\{ v \\in H \\colon \\sum_{k = 1}^\\infty | \\lambda_k \\langle e_k , v \\rangle_H |^2 < \\infty \\} $\nand\n$ \\forall \\, v \\in D(A) \\colon A v = \\sum_{k = 1}^\\infty - \\lambda_k \\langle e_k , v \\rangle_H e_k$,\nlet\n$ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $,\nbe a family of interpolation spaces associated to $ -A $,\nlet $\\xi \\in H_{\\nicefrac{1}{2}} $,\nlet\n$F \\in \\mathcal{C}(H_{\\varrho}, H_{-\\alpha} )$,\n$(P_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H_{-1}) $,\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $,\nand\n$ \\phi, \\Phi \\colon H_1 \\to [0,\\infty) $\nbe functions which\nsatisfy\nfor all $ u \\in H_1 $, $ n \\in \\mathbb{N} $, $ v, w \\in P_n( H ) $ that\n$\\phi(u)= \\gamma + \\gamma \\left[\\sup\\nolimits_{x \\in (0,1)} |\\underline{u}(x)|^2 \\right]$,\n$\\Phi(u)= \\gamma + \\gamma \\left[\\sup\\nolimits_{x \\in (0,1)} |\\underline{u}(x)|^{\\gamma} \\right]$,\n$ P_n(u) = \\sum_{k =1 }^n \\langle e_k, u \\rangle_H e_k $,\n$ \\limsup_{ m \\to \\infty} h_m =0$,\n$ \\left< v, P_n F( v + w ) \\right>_H \\leq \\phi( w ) \\| v \\|^2_H + \\varphi \\| v \\|^2_{ H_{ \\nicefrac{1}{2} }} + \\Phi( w ) $,\nand\n$ \\left\\| F(v) - F(w) \\right\\|_{ H_{ - \\alpha } } \\leq \\theta \\, ( 1 + \\| v \\|_{ H_{ \\rho } }^{ \\vartheta } + \\|w\\|_{H_{\\rho}}^{\\vartheta}) \\, \\|v-w\\|_{H_{\\rho}} $,\nlet $ ( \\Omega, \\F, \\P ) $ be a probability space,\nlet $(W_t)_{t \\in [0, T]}$ be an $\\mathrm{Id}_H$-cylindrical $( \\Omega, \\F, \\P )$-Wiener process,\nand let\n$ \\mathcal{X}^n, \\mathcal{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $ n \\in \\mathbb{N}$,\nbe stochastic processes which satisfy\nfor all $ n \\in \\mathbb{N} $, $ t \\in [ 0, T ] $ that\n$\\left[\\mathcal{O}_t^n \\right]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)A} \\, dW_s$\nand\n\\begin{equation}\\label{eq:set:abstract}\n\\P \\Big( \\mathcal{X}_t^n = P_n \\, e^{ t A } \\, \\xi + \\smallint_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds + \\mathcal{O}_t^n \\Big)=1. \n\\end{equation}\n\n\n\\subsection{Properties of the stochastic convolution process}\n\nThe proof of the next result, Lemma~\\ref{abs:phi(w)_finite} below, is a slight adaptation of the proof of Lemma~5.6 in Hutzenthaler et al.~\\cite{Salimova2016}.\n\n\\begin{lemma}\\label{abs:phi(w)_finite}\nAssume the setting in Subsection~\\ref{setting:example}, let $\\beta \\in (0, \\nicefrac{1}{2}]$, $p \\in (\\nicefrac{1}{\\beta}, \\infty)$, $t \\in [0,T]$, $n \\in \\mathbb{N}$, $\\eta \\in [0,\\infty)$,\nlet\n$ \\mathbb{O} \\colon \\Omega \\to P_n(H) $\nbe an $ \\F $\/$ \\mathcal{B}(P_n(H)) $-measurable function which satisfies \n$\\left[\\mathbb{O} \\right]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)(A-\\eta)} \\, dW_s$, and let $Y \\colon \\Omega \\to \\mathbb{R}$ be a standard normally distributed random variable. Then \n\\begin{align}\\label{abs:norm:eq}\n\\begin{split}\n& \\bigl(\\mathbb{E} \\! \\left[\\sup\\nolimits_{x \\in (0,1)} | \\underline{\\mathbb{O}}(x) |^2 \\right] \\bigr)^{\\nicefrac{1}{2}} \\leq \\pi^{2} \\big( \\mathbb{E} \\big[|Y|^p \\big] \\big)^{\\nicefrac{1}{p}} \\left[ \\sum_{k =1}^n \\frac{ k^{4 \\beta}}{\\lambda_k+\\eta}\\right]^{\\nicefrac{1}{2}} \\\\\n& \\quad \\cdot \\sup \\! \\Big(\\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} |v(x)| \\colon \\big[v \\in \\mathcal{C}((0,1), \\mathbb{R}) \\text{ and } \\|v\\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})} \\leq 1\\big] \\Big\\}\\Big) < \\infty.\n\\end{split}\n\\end{align}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{abs:phi(w)_finite}] \nFirst, observe that Jensen's inequality shows that\n\\begin{align*}\\label{abs:phi(w)_eq1}\n&\\mathbb{E} \\! \\left[\\sup\\nolimits_{x \\in (0,1)} | \\underline{\\mathbb{O}}(x) |^2 \\right] \\\\\n&\\leq \\Big[\\sup \\! \\Big(\\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} |v(x)| \\colon \\big[v \\in \\mathcal{C}((0,1), \\mathbb{R}) \\text{ and } \\|v\\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})} \\leq 1\\big] \\Big\\}\\Big)\\Big]^2 \\, \\mathbb{E} \\big[\\| \\underline{\\mathbb{O}} \\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})}^2 \\big] \\numberthis \\\\\n& \\leq \\Big[\\sup \\!\\Big(\\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} |v(x)| \\colon \\big[v \\in \\mathcal{C}((0,1), \\mathbb{R}) \\text{ and } \\|v\\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})} \\leq 1\\big] \\Big\\}\\Big)\\Big]^2 \\big( \\mathbb{E} \\big[\\| \\underline{\\mathbb{O}} \\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})}^p \\big]\\big)^{\\!\\nicefrac{2}{p}}.\n\\end{align*}\nIn addition, note that\n\\begin{align}\\label{abs:phi(w)_all}\n\\begin{split}\n&\\mathbb{E} \\big[\\| \\underline{\\mathbb{O}} \\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})}^p \\big] = \\mathbb{E} \\! \\left[ \\int_0^1 |\\underline{\\mathbb{O}} (x)|^p \\, dx + \\int_0^1 \\int_0^1 \\frac{|\\underline{\\mathbb{O}} (x)- \\underline{\\mathbb{O}}(y)|^p}{|x-y|^{1+ \\beta p}} \\, dx \\, dy\\right]\\\\\n& = \\mathbb{E} \\big[|Y|^p \\big] \\int_0^1 \\left( \\mathbb{E} \\! \\left[ |\\underline{\\mathbb{O}} (x)|^2 \\right] \\right)^{\\nicefrac{p}{2}} dx + \\mathbb{E} \\big[|Y|^p \\big] \\int_0^1 \\int_0^1 \\frac{\\left( \\mathbb{E} \\! \\left[ |\\underline{\\mathbb{O}} (x) - \\underline{\\mathbb{O}}(y) |^2 \\right] \\right)^{\\nicefrac{p}{2}}}{|x-y|^{1+ \\beta p}}\\, dx \\, dy.\n\\end{split}\n\\end{align}\nFurthermore, It\\^o's isometry yields for all $ x \\in (0,1)$ that\n\\begin{align}\n\\begin{split}\n\\mathbb{E} \\! \\left[ |\\underline{\\mathbb{O}}(x)|^2 \\right] & = \\mathbb{E} \\Bigg[ \\left| \\sum\\limits_{k =1}^n \\underline{e_k} (x) \\int_0^t e^{-(\\lambda_k+\\eta)(t-s)} \\, d \\! \\left< e_k, W_s \\right>_H \\right|^2 \\Bigg]= \\sum\\limits_{k =1}^n |\\underline{e_k} (x)|^2 \\int_0^t e^{-2(\\lambda_k+\\eta)(t-s)} \\, ds \\\\\n& \\leq \\sum_{k =1}^n \\frac{|\\underline{e_k}(x)|^2 }{ 2(\\lambda_k+\\eta)} \\leq \\sum_{k =1}^n \\frac{1}{ \\lambda_k+\\eta}.\n\\end{split}\n\\end{align}\nThis implies that \n\\begin{align}\\label{abs:phi(w)_1}\n\\int_0^1\\left( \\mathbb{E} \\! \\left[ |\\underline{\\mathbb{O}}(x)|^2 \\right] \\right)^{\\nicefrac{p}{2}} dx \\leq \\left[\\sum_{k =1}^n \\frac{1}{\\lambda_k+\\eta} \\right]^{\\nicefrac{p}{2}}.\n\\end{align}\nNext note that again It\\^o's isometry ensures for all $ x, y \\in (0,1)$ that\n\\begin{align}\\label{eq:o_diff}\n\\begin{split}\n\\mathbb{E} \\! \\left[ |\\underline{\\mathbb{O}} (x) - \\underline{\\mathbb{O}}(y) |^2 \\right] &= \\mathbb{E} \\Bigg[ \\left| \\sum\\limits_{k =1}^n \\left[\\underline{e_k}(x) - \\underline{e_k}(y)\\right] \\int_0^t e^{-(\\lambda_k+\\eta)(t-s)} \\ d \\! \\left< e_k, W_s \\right>_H \\right|^2 \\Bigg] \\\\\n&\\leq \\sum_{k =1}^n \\frac{|\\underline{e_k}(x) - \\underline{e_k}(y)|^2 }{ 2(\\lambda_k+\\eta)}.\n\\end{split}\n\\end{align}\nMoreover, the fact that $\\beta \\leq \\nicefrac{1}{2}$ and the fact that $\\forall \\, x, y \\in \\mathbb{R} \\colon |\\sin(x)-\\sin(y)|\\leq |x-y|$ prove that for all $x, y \\in (0,1)$, $k \\in \\mathbb{N}$ it holds that\n\\begin{align}\n\\begin{split}\n|\\underline{e_k}(x) - \\underline{e_k}(y)|^2 & =2 \\, |\\sin(k\\pi x) - \\sin(k \\pi y)|^2 \\\\\n&= 2 \\, |\\sin(k\\pi x) - \\sin(k \\pi y)|^{2-4\\beta} |\\sin(k\\pi x) - \\sin(k \\pi y)|^{4\\beta} \\leq 2^{3-4\\beta} |k\\pi|^{4\\beta} |x-y|^{4\\beta}.\n\\end{split}\n\\end{align}\nThis together with \\eqref{eq:o_diff} establishes for all $x, y \\in (0, 1)$ that\n\\begin{align}\n\\mathbb{E} \\! \\left[ |\\underline{\\mathbb{O}} (x) - \\underline{\\mathbb{O}}(y) |^2 \\right] \\leq 2^{2-4\\beta} \\,\\pi^{4\\beta} \\, |x-y|^{4 \\beta} \\sum_{k =1}^n \\frac{ k^{4 \\beta} \\, }{ \\lambda_k+\\eta}.\n\\end{align}\nThe fact that $\\beta p \\geq 1$ hence ensures that\n\\begin{align}\\label{abs:phi(w)_2}\n\\begin{split}\n&\\int_0^1\\int_0^1 \\frac{\\left( \\mathbb{E} \\! \\left[ |\\underline{\\mathbb{O}} (x) - \\underline{\\mathbb{O}}(y) |^2 \\right] \\right)^{\\nicefrac{p}{2}}}{|x-y|^{1+ \\beta p}} \\, dx \\, dy \\\\\n&\\leq 2^{p(1-2\\beta)} \\pi^{2p\\beta} \\! \\left[ \\sum_{k =1}^n \\frac{k^{4 \\beta} }{\\lambda_k+\\eta}\\right]^{\\nicefrac{p}{2}} \\int_0^1 \\int_0^1 |x-y|^{\\beta p -1} \\, dx \\, dy \\leq 2^{p(1-2\\beta)} \\pi^{2p\\beta} \\! \\left[ \\sum_{k =1}^n \\frac{k^{4 \\beta} }{\\lambda_k+\\eta}\\right]^{\\nicefrac{p}{2}}.\n\\end{split}\n\\end{align}\nCombining this, \\eqref{abs:phi(w)_all}, and \\eqref{abs:phi(w)_1} proves that \n\\begin{align}\\label{abs:phi(w)_last}\n\\begin{split}\n& \\big( \\mathbb{E} \\big[\\| \\underline{\\mathbb{O}} \\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})}^p \\big]\\big)^{\\!\\nicefrac{1}{p}}\n\\leq \\big( \\mathbb{E} \\big[|Y|^p \\big] \\big)^{\\nicefrac{1}{p}} \\left\\{ \\left[\\sum_{k =1}^n \\frac{1}{\\lambda_k+\\eta} \\right]^{\\nicefrac{p}{2}} + 2^{p(1-2\\beta)} \\pi^{2p\\beta}\\! \\left[ \\sum_{k =1}^n \\frac{k^{4 \\beta} }{\\lambda_k+\\eta}\\right]^{\\nicefrac{p}{2}} \\right\\}^{\\!\\nicefrac{1}{p}} \\\\\n& \\leq \\big( \\mathbb{E} \\big[|Y|^p \\big] \\big)^{\\nicefrac{1}{p}} \\left\\{ 2^{p(1-2\\beta)+1} \\pi^{2p\\beta}\\! \\left[ \\sum_{k =1}^n \\frac{k^{4 \\beta} }{\\lambda_k+\\eta}\\right]^{\\nicefrac{p}{2}} \\right\\}^{\\!\\nicefrac{1}{p}} \\\\\n& \\leq 2^{2-2\\beta} \\,\\pi^{2\\beta} \\big( \\mathbb{E} \\big[|Y|^p \\big] \\big)^{\\nicefrac{1}{p}} \\left[ \\sum_{k =1}^n \\frac{ k^{4 \\beta}}{\\lambda_k+\\eta}\\right]^{\\nicefrac{1}{2}} \\leq \\pi^{2} \\big( \\mathbb{E} \\big[|Y|^p \\big] \\big)^{\\nicefrac{1}{p}} \\left[ \\sum_{k =1}^n \\frac{ k^{4 \\beta}}{\\lambda_k+\\eta}\\right]^{\\nicefrac{1}{2}}.\n\\end{split}\n\\end{align}\nIn addition, note that the fact that $\\beta p > 1$ and the Sobolev embedding theorem yield that\n\\begin{align}\n\\sup \\! \\Big(\\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} |v(x)| \\colon \\big[v \\in \\mathcal{C}((0,1), \\mathbb{R}) \\text{ and } \\|v\\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})} \\leq 1\\big] \\Big\\}\\Big) < \\infty.\n\\end{align}\nThis, \\eqref{abs:phi(w)_eq1}, and \\eqref{abs:phi(w)_last} show \\eqref{abs:norm:eq}. The proof of Lemma~\\ref{abs:phi(w)_finite} is thus completed.\n\\end{proof}\n\n\n\\begin{lemma}\n\t\\label{lem:conv:series}\nLet $\\alpha \\in \\mathbb{R}$, $\\beta \\in (1+ \\alpha, \\infty)$. Then it holds that\n\\begin{align}\n\\limsup_{\\eta \\to \\infty} \\left( \\sum_{k = 1}^\\infty \\frac{k^{\\alpha}}{k^{\\beta} + \\eta} \\right) = 0.\n\\end{align}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lem:conv:series}]\nObserve that for all $\\eta \\in [0, \\infty)$, $k \\in \\mathbb{N}$ it holds that\n\\begin{align}\n\\frac{k^{\\alpha}}{k^{\\beta} + \\eta} \\leq \\frac{1}{k^{\\beta -\\alpha}} \\qquad \\text{and} \\qquad \\sum_{n = 1}^\\infty \\frac{1}{n^{\\beta - \\alpha} } < \\infty.\n\\end{align}\nLebesgue's theorem of dominated convergence hence ensures that\n\\begin{align}\n\\limsup_{\\eta \\to \\infty} \\left( \\sum_{k = 1}^\\infty \\frac{k^{\\alpha}}{k^{\\beta} + \\eta} \\right) = \\sum_{k = 1}^\\infty \\limsup_{\\eta \\to \\infty} \\left( \\frac{k^{\\alpha}}{k^{\\beta} + \\eta} \\right) = 0.\n\\end{align} \nThe proof of Lemma~\\ref{lem:conv:series} is thus completed.\n\\end{proof}\n\\pagebreak\n\n\\begin{prop}\\label{prop:abstract}\nAssume the setting in Subsection~\\ref{setting:example}, let $\\beta \\in (0, \\nicefrac{1}{4})$, $p \\in (\\nicefrac{1}{\\beta}, \\infty)$, $\\eta \\in [0,\\infty)$, let $\\tilde{ \\mathcal{O} }^n, \\mathbb{O}^n \\colon [0,T] \\times \\Omega \\to P_n(H)$, $n \\in \\mathbb{N}$, be stochastic processes with continuous sample paths which satisfy for all $n \\in \\mathbb{N}$, $t \\in [0,T]$ that $[\\tilde{\\mathcal{O}}_t^n ]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)A} \\, dW_s$ and $\\left[\\mathbb{O}_t^n \\right]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)(A-\\eta)} \\, dW_s$, and assume\n\\begin{align}\\label{abstract:gamma}\n\\begin{split}\n720 p^3 T \\gamma \\pi^{4} \\left[ \\sum_{k = 1}^\\infty \\frac{ k^{4 \\beta}}{\\lambda_k+\\eta}\\right] \\Big[\\sup \\! \\Big(\\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} |v(x)| \\colon \\big[v \\in \\mathcal{C}((0,1), \\mathbb{R}) \\text{ and } \\|v\\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})} \\leq 1\\big] \\Big\\}\\Big)\\Big]^2 \\leq 1.\n\\end{split}\n\\end{align}\nThen\n\\begin{enumerate}[(i)]\n\\item\n\\label{item:finite1}\nit holds that $ \\sup_{ n \\in \\mathbb{N} } \\sup_{ s \\in [0,T]} \\mathbb{E}[ \\| \\mathbb{O}_s^n + P_n \\, e^{s(A-\\eta)} \\xi \\|_H^p] < \\infty$ and\n\\item\nit holds that\n\\begin{align}\n& \\nonumber \\sup_{ n \\in \\mathbb{N} } \\mathbb{E}\\biggl[ \\int_0^T \\exp \\left( \\smallint_s^T p\\,\\phi\\big( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\max\\Big\\{ 1, \\big|\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi)\\big|^{\\nicefrac{p}{2}}, \\\\\n& \\quad \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^p, \\smallint\\nolimits_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\Big\\} \\, ds \\biggr]< \\infty.\n\\end{align}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}[Proof of Proposition~\\ref{prop:abstract}]\nFirst, note that the Burkholder-Davis-Gundy inequality proves that\nfor all standard normally distributed random variables $ Y \\colon \\Omega \\to \\mathbb{R} $ it holds that\n\\begin{equation}\n\\big( \\mathbb{E} \\big[|Y|^p \\big] \\big)^{\\nicefrac{1}{p}}\n\\leq\n\\sqrt{\\tfrac{p(p-1)}{2}}\n\\leq p.\n\\end{equation}\nMarkov's inequality, Lemma~\\ref{abs:phi(w)_finite}, and \\eqref{abstract:gamma} hence imply that for all $n \\in \\mathbb{N}$, $t \\in [0,T]$ it holds that\n\\begin{align}\n\\begin{split}\n& \\P \\! \\left( \\sup\\nolimits_{x \\in (0,1)} | \\underline{\\mathbb{O}_t^n}(x) |^2 \\geq \\frac{1}{72 p T \\gamma} \\right) \\leq 72 p T \\gamma \\, \\mathbb{E} \\! \\left[\\sup\\nolimits_{x \\in (0,1)} | \\underline{\\mathbb{O}_t^n}(x) |^2 \\right] \\leq 72 p^3 T \\gamma \\pi^{4} \\left[ \\sum_{k =1}^n \\frac{ k^{4 \\beta}}{\\lambda_k+\\eta}\\right]\\\\\n& \\quad \\cdot \\Big[\\sup \\! \\Big(\\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} |v(x)| \\colon \\big[v \\in \\mathcal{C}((0,1), \\mathbb{R}) \\text{ and } \\|v\\|_{\\mathcal{W}^{\\beta, p}((0,1), \\mathbb{R})} \\leq 1\\big] \\Big\\}\\Big)\\Big]^2 \\leq \\frac{1}{10}.\n\\end{split}\n\\end{align}\nThis and Proposition~\\ref{thm:fernique} (with $V = P_n(H)$, $ \\left\\|\\cdot \\right\\|_V = (P_n(H) \\ni v \\mapsto \\sup_{x \\in (0,1)} |\\underline{v}(x)| \\in [ 0, \\infty ) )$, $X= \\mathbb{O}_t^n$, $R= (72pT\\gamma)^{\\nicefrac{-1}{2}}$ for $t \\in [0,T]$, $n \\in \\mathbb{N}$ in the notation of Proposition~\\ref{thm:fernique}) show that for all $n \\in \\mathbb{N}$, $t \\in [0, T]$ it holds that\n\\begin{align}\\label{eq:abstract:13}\n\\mathbb{E} \\! \\left[ \\exp \\! \\left(4 p T \\gamma \\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} | \\underline{\\mathbb{O}_t^n}(x) |^2\\Big\\} \\right)\\right] \\leq 13.\n\\end{align}\nIn addition, H\\\"older's inequality yields for all $n \\in \\mathbb{N}$ that\n\\begin{align*}\\label{eq:abstract:holder}\n&\\mathbb{E}\\bigg[ \\int_0^T \\exp \\left( \\smallint_s^T p\\,\\phi\\big( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\max\\Big\\{ 1, \\big|\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi)\\big|^{\\nicefrac{p}{2}}, \\\\\n& \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^p, \\smallint\\nolimits_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\Big\\} \\, ds \\bigg]^2 \\\\\n& = \\bigg( \\int_0^T \\mathbb{E}\\bigg[\\! \\exp \\left( \\smallint_s^T p\\,\\phi\\big( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\max\\Big\\{ 1, \\big|\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi)\\big|^{\\nicefrac{p}{2}}, \\\\\n& \\quad \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^p, \\smallint\\nolimits_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\Big\\} \\bigg] \\, ds \\bigg)^2 \\\\\n& \\leq T \\int_0^T \\mathbb{E}\\bigg[\\! \\exp \\left( \\smallint_s^T p\\,\\phi\\big( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\max\\Big\\{ 1, \\big|\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi)\\big|^{\\nicefrac{p}{2}}, \\numberthis \\\\\n& \\quad \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^p, \\smallint\\nolimits_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\Big\\} \\bigg]^2 \\, ds \\\\\n& \\leq T \\int_0^T \\mathbb{E}\\bigg[\\! \\exp \\left( \\smallint_s^T 2p \\,\\phi\\big( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\bigg] \\mathbb{E} \\bigg[ \\! \\max\\Big\\{ 1, \\big|\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi)\\big|^{p}, \\\\\n& \\quad \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^{2p}, T \\smallint\\nolimits_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\, du \\Big\\} \\bigg] \\, ds \\\\\n& \\leq T \\, \\mathbb{E}\\bigg[ \\! \\exp \\left( \\smallint_0^T 2p \\,\\phi\\big( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\bigg] \\int_0^T \\mathbb{E} \\bigg[ 1 + \\big|\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi)\\big|^{p} \\\\\n& \\quad + \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^{2p} + T \\smallint\\nolimits_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\, du \\bigg] \\, ds.\n\\end{align*}\nNext note that the fact that $ \\forall \\, x, y \\in \\mathbb{R} \\colon |x+y|^2 \\leq 2 x^2 + 2y^2$ ensures that for all $n \\in \\mathbb{N}$ it holds that\n\\begin{align}\\label{eq:abstract:phi1}\n\\begin{split}\n& \\mathbb{E} \\!\\left[ \\exp \\! \\left( \\smallint_0^T 2p \\,\\phi\\big(\\mathbb{O}_{\\lfloor u \\rfloor_{h_n}}^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\right] \\\\\n&= \\mathbb{E} \\!\\left[ \\exp \\! \\left( \\smallint_0^T 2p\\gamma + 2p\\gamma \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor u \\rfloor_{h_n}}^n} \\!(x) + \\underline{ P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi}(x)\\big|^2 \\Big\\} \\, du \\right) \\right] \\\\\n& \\leq \\exp \\!\\left( 2p\\gamma T + 4p\\gamma \\smallint_0^T \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{ P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi}(x)|^2 \\Big\\} \\, du \\right) \\\\\n& \\quad \\cdot \\mathbb{E} \\!\\left[ \\exp \\! \\left( \\smallint_0^T 4p\\gamma \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor u \\rfloor_{h_n}}^n} \\!(x)\\big|^2 \\Big\\} \\, du \\right) \\right].\n\\end{split}\n\\end{align}\nFurthermore, e.g., Lemma~2.22 in Cox, Hutzenthaler, \\& Jentzen~\\cite{CoxHutzenthalerJentzen2014} and \\eqref{eq:abstract:13} prove for all $n \\in \\mathbb{N}$ that\n\\begin{equation}\\label{abstract:prop:jen}\n\\begin{split}\n& \\mathbb{E} \\! \\left[ \\exp \\! \\left( \\smallint_0^T 4 p \\gamma \\Big\\{\\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor u \\rfloor_{h_n}}^n} \\! (x)\\big|^2 \\Big\\} \\, du \\right)\\right] \\\\\\\n& \\leq \\frac{1}{T} \\int_0^T \\mathbb{E}\\! \\left[ \\exp \\! \\left( 4 p T \\gamma \\Big\\{\\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor u \\rfloor_{h_n}}^n}\\!(x)\\big|^2 \\Big\\} \\right)\\right] du \\leq 13 .\n\\end{split}\n\\end{equation} \nThis and \\eqref{eq:abstract:phi1} show for all $n \\in \\mathbb{N}$ that\n\\begin{align}\n\\label{eq:abs:phi0}\n\\begin{split}\n& \\mathbb{E} \\!\\left[ \\exp \\! \\left( \\smallint_0^T 2p \\,\\phi\\big(\\mathbb{O}_{\\lfloor u \\rfloor_{h_n}}^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\right] \\\\\n& \\leq 13 \\exp \\!\\left( 2p\\gamma T + 4p\\gamma \\smallint_0^T \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{ P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi}(x)|^2 \\Big\\} \\, du \\right).\n\\end{split}\n\\end{align}\nIn addition, the Sobolev embedding theorem ensures that \n\\begin{align}\n\\sup \\! \\Big(\\Big\\{\\! \\sup\\nolimits_{x \\in (0,1)} |\\underline{v}(x)| \\colon \\big[ v \\in H_{\\nicefrac{1}{2}} \\text{ and } \\|v\\|_{H_{\\nicefrac{1}{2}}} \\leq 1\\big] \\Big\\}\\Big) < \\infty.\n\\end{align}\nThis establishes for all $n \\in \\mathbb{N}$, $s \\in [0, T]$ that\n\\begin{align}\n\\label{eq:abs:xi}\n\\begin{split}\n&\\sup\\nolimits_{x \\in (0,1)} | \\underline{P_n \\, e^{s(A-\\eta)} \\xi} (x) | \\\\\n&\\leq \\Big[\\sup \\! \\Big(\\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{v}(x)| \\colon \\big[v \\in H_{\\nicefrac{1}{2}} \\text{ and } \\|v\\|_{H_{\\nicefrac{1}{2}}} \\leq 1\\big] \\Big\\}\\Big) \\Big] \\|P_n \\, e^{s(A-\\eta)} \\xi \\|_{H_{\\nicefrac{1}{2}}} \\\\\n& \\leq \\Big[\\sup \\! \\Big(\\Big\\{ \\sup\\nolimits_{x \\in (0,1)} |\\underline{v}(x)| \\colon \\big[v \\in H_{\\nicefrac{1}{2}} \\text{ and } \\|v\\|_{H_{\\nicefrac{1}{2}}} \\leq 1\\big] \\Big\\}\\Big) \\Big] \\| \\xi \\|_{H_{\\nicefrac{1}{2}}} < \\infty.\n\\end{split}\n\\end{align}\nCombing this with \\eqref{eq:abs:phi0} implies that \n\\begin{align}\\label{eq:abstract:phi}\n\\sup_{ n \\in \\mathbb{N} } \\mathbb{E} \\!\\left[ \\exp \\! \\left( \\smallint_0^T 2p \\,\\phi\\big(\\mathbb{O}_{\\lfloor u \\rfloor_{h_n}}^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\right] < \\infty.\n\\end{align}\nNext note that the fact that $ \\forall \\, x, y \\in \\mathbb{R}, \\, a \\in [0, \\infty) \\colon |x+y|^a \\leq 2^{\\max\\{a-1,0\\}} |x|^a + 2^{\\max\\{a-1,0\\}} |y|^a$\nand the triangle inequality show that for all $n \\in \\mathbb{N}$, $s \\in [0, T]$ it holds that\n\\begin{align}\n\\label{eq:abstract:Phi0}\n\\begin{split}\n& \\mathbb{E} \\! \\left[ \\big| \\Phi\\big( \\mathbb{O}_{ \\lfloor s \\rfloor_{h_n } }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\big|^p \\right]\\\\\n&= \\mathbb{E} \\bigg[ \\left|\\gamma + \\gamma \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor s \\rfloor_{h_n}}^n} \\! (x) + \\underline{ P_n \\,e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi}(x)\\big|^{\\gamma} \\Big\\} \\right|^p \\bigg] \\\\\n& \\leq \\mathbb{E} \\! \\left[ 2^{p-1} \\gamma^p + 2^{p-1} \\gamma^p \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor s \\rfloor_{h_n}}^n} \\!(x) + \\underline{ P_n \\,e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi}(x)\\big|^{p\\gamma} \\Big\\} \\right] \\\\\n& \\leq \\mathbb{E} \\! \\left[ 2^{p-1} \\gamma^p + 2^{p-1} \\gamma^p \\, 2^{\\max\\{p\\gamma-1,0\\}} \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor s \\rfloor_{h_n}}^n} \\!(x)\\big|^{p\\gamma}+ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{ P_n \\,e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi}(x)\\big|^{p\\gamma} \\Big\\} \\right] \\\\\n& \\leq 2^{p-1} \\gamma^p + 2^{p(\\gamma+1)-1} \\gamma^p \\, \\mathbb{E} \\! \\left[ \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{\\lfloor s \\rfloor_{h_n}}^n} \\!(x)\\big|^{p\\gamma} \\Big\\}+ \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{ P_n \\,e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi}(x)\\big|^{p\\gamma} \\Big\\} \\right] .\n\\end{split}\n\\end{align}\nFurthermore, observe that, e.g., Lemma~5.7 in Hutzenthaler et al.~\\cite{Salimova2016} (with $a=4pT\\gamma$, $x=\\sup\\nolimits_{x \\in (0,1)} |\\underline{\\mathbb{O}_{s}^n (\\omega)}(x)|^2$, $r=\\nicefrac{r}{2}$ for $\\omega \\in \\Omega$, $s \\in [0, T]$, $n \\in \\mathbb{N}$, $ r \\in [ 0, \\infty ) $ in the notation of Lemma~5.7 in Hutzenthaler et al.~\\cite{Salimova2016})\nand \\eqref{eq:abstract:13}\nensure that for all $ r \\in [ 0, \\infty ) $, $n \\in \\mathbb{N}$, $s \\in [0, T]$ it holds that\n\\begin{equation}\n\\label{eq:abstract:estimate-exponential}\n\\mathbb{E} \\! \\left[\\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{s}^n}(x)\\big|^{r} \\Big\\}\\right]\n\\leq\n\\tfrac{(\\lfloor \\nicefrac{r}{2} \\rfloor_1 +1)!}{ |4pT\\gamma|^{\\nicefrac{r}{2}}} \\,\n\\mathbb{E} \\! \\left[\\exp \\! \\left( 4pT\\gamma\\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{s}^n}(x)\\big|^{2} \\Big\\} \\right)\\right]\n\\leq\n\\tfrac{ 13 \\, (\\lfloor \\nicefrac{r}{2} \\rfloor_1 +1)!}{ |4pT\\gamma|^{\\nicefrac{r}{2}}}.\n\\end{equation}\nCombining this and \\eqref{eq:abstract:Phi0} proves for all $n \\in \\mathbb{N}$, $s \\in [0, T]$ that\n\\begin{align}\n\\begin{split}\n&\\mathbb{E} \\! \\left[ \\big| \\Phi\\big( \\mathbb{O}_{ \\lfloor s \\rfloor_{h_n } }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\big|^p \\right] \\\\\n&\\leq 2^{p-1} \\gamma^p+ 2^{p(\\gamma+1)-1} \\gamma^p \\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{ P_n \\,e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi}(x)\\big|^{p\\gamma} \\Big\\} +\\tfrac{13 \\cdot 2^{p(\\gamma+1)-1} \\gamma^p (\\lfloor \\nicefrac{p\\gamma}{2} \\rfloor_1 +1)!}{ |4pT\\gamma|^{\\nicefrac{p\\gamma}{2}}} .\n\\end{split}\n\\end{align}\nThis together with \\eqref{eq:abs:xi} yields that \n\\begin{align}\\label{eq:abstract:Phi}\n\\begin{split}\n\\sup_{ n \\in \\mathbb{N} } \\int_0^T \\mathbb{E} \\! \\left[ \\big| \\Phi\\big( \\mathbb{O}_{ \\lfloor s \\rfloor_{h_n } }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\big|^p \\right] ds < \\infty.\n\\end{split}\n\\end{align}\nMoreover, \\eqref{eq:abstract:estimate-exponential} establishes for all $ r \\in [ 1, \\infty ) $, $n \\in \\mathbb{N}$, $s \\in [0, T]$ that\n\\begin{align}\n\\label{eq:abstract:O0}\n\\begin{split}\n\\mathbb{E} \\!\\left[\\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^{r} \\right] & \\leq \\mathbb{E}\\! \\left[ 2^{r-1} \\|\\mathbb{O}_s^n\\|_H^{r} + 2^{r-1} \\| P_n \\, e^{s (A-\\eta)} \\xi\\|_H^{r} \\right]\\\\\n& \\leq 2^{r-1} \\, \\mathbb{E} \\! \\left[\\Big\\{ \\!\\sup\\nolimits_{x \\in (0,1)} \\big|\\underline{\\mathbb{O}_{s}^n}(x)\\big|^{r} \\Big\\}\\right] + 2^{r-1} \\| P_n \\, e^{s (A-\\eta)} \\xi\\|_H^{r}\\\\\n& \\leq \\tfrac{13 \\cdot 2^{r-1} (\\lfloor \\nicefrac{r}{2} \\rfloor_1 +1)!}{ |4pT\\gamma|^{\\nicefrac{r}{2}}} + 2^{r-1} \\|\\xi\\|_H^{r}.\n\\end{split}\n\\end{align}\nObserve that this implies that \n\\begin{equation}\n\\sup_{ n \\in \\mathbb{N} } \\sup_{s \\in [0,T]} \\mathbb{E} \\big[ \\big\\| \\mathbb{O}_s^n + P_n \\, e^{s(A-\\eta)} \\xi \\big\\|_H^p \\big]\n\\leq \\tfrac{13 \\cdot 2^{p-1} (\\lfloor \\nicefrac{p}{2} \\rfloor_1 +1)!}{ |4pT\\gamma|^{\\nicefrac{p}{2}}} + 2^{p-1} \\|\\xi\\|_H^{p}\n< \\infty.\n\\end{equation}\nThis proves \\eqref{item:finite1}.\nIn addition, \\eqref{eq:abstract:O0} shows that \n\\begin{align}\\label{eq:abstract:O}\n\\begin{split}\n\\sup_{ n \\in \\mathbb{N} } \\int_0^T \\mathbb{E} \\! \\left[ \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^{2p} \\right] ds < \\infty.\n\\end{split}\n\\end{align}\nIn the next step note that for all $n \\in \\mathbb{N}$ it holds that\n\\begin{align}\n\\label{eq:abstract:int}\n\\begin{split}\n\\mathbb{E} \\! \\left[\\smallint_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\, du \\right] &\\leq 2^{4p+ 4p\\vartheta-1} \\, \\mathbb{E} \\!\\left[\\smallint_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n \\big\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta}+ \\|P_n \\, e^{uA} \\xi \\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\, du \\right] \\\\\n&\\leq 2^{4p+ 4p\\vartheta-1} \\, \\mathbb{E} \\!\\left[\\smallint_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n \\big\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta}+ \\| \\xi \\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\, du \\right].\n\\end{split}\n\\end{align}\nFurthermore, observe that the Burkholder-Davis-Gundy-type inequality in Da~Prato \\& Zabczyk~\\cite[Lemma~7.7]{dz92} proves for all $n \\in \\mathbb{N}$, $u \\in [0, T]$ that\n\\begin{align}\n\\begin{split}\n\\mathbb{E} \\!\\left[ \\big\\| \\tilde{\\mathcal{O}}_u^n \\big\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\right] & = \\mathbb{E} \\!\\left[\\left\\| \\int_0^u P_n \\, e^{(u-s)A} \\, dW_s \\right\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\right] \\\\\n&\\leq \\left[\\tfrac{(4p+ 4p\\vartheta)(4p+ 4p\\vartheta-1)}{2}\\right]^{2p+ 2p\\vartheta} \\left[ \\int_0^u \\|P_n\\, e^{(u-s)A} \\|_{\\mathrm{HS}(H, H_{\\varrho})}^2 \\,ds\\right]^{2p+ 2p\\vartheta} \\\\\n& \\leq \\left[4p+ 4p\\vartheta \\right]^{4p+ 4p\\vartheta} \\left[ \\int_0^u \\| (-A)^{\\varrho} \\,e^{(u-s)A} \\|_{\\mathrm{HS}(H)}^2 \\, ds\\right]^{2p+ 2p\\vartheta}\\\\\n& = \\left[4p+ 4p\\vartheta \\right]^{4p+ 4p\\vartheta} \\left[ \\sum_{k = 1}^\\infty \\int_0^u (\\lambda_k)^{2\\varrho} \\,e^{-2\\lambda_k s} \\, ds\\right]^{2p+ 2p\\vartheta}\\\\\n& = \\left[4p+ 4p\\vartheta\\right]^{4p+ 4p\\vartheta} \\left[ \\sum_{k = 1}^\\infty \\frac{(\\lambda_k)^{2\\varrho} (1-e^{-2\\lambda_k u})}{2\\lambda_k} \\right]^{2p+ 2p\\vartheta}\\\\\n& \\leq \\left[4p+ 4p\\vartheta \\right]^{4p+ 4p\\vartheta} \\left[ \\sum_{k = 1}^\\infty (\\lambda_k)^{2\\varrho-1} \\right]^{2p+ 2p\\vartheta} < \\infty.\n\\end{split}\n\\end{align}\nCombining this with \\eqref{eq:abstract:int} yields that\n\\begin{align}\n\\begin{split}\n\\sup_{ n \\in \\mathbb{N} } \\mathbb{E} \\! \\left[ \\smallint_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{4p+ 4p\\vartheta} \\, du \\right] < \\infty.\n\\end{split}\n\\end{align}\nThis, \\eqref{eq:abstract:holder}, \\eqref{eq:abstract:phi}, \\eqref{eq:abstract:Phi}, and \\eqref{eq:abstract:O} ensure that\n\\begin{align}\\label{eq:abstract:last}\n\\begin{split}\n&\\sup_{ n \\in \\mathbb{N} } \\mathbb{E}\\biggl[ \\int_0^T \\exp \\left( \\smallint_s^T p\\,\\phi\\big( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor u \\rfloor_{h_n} (A-\\eta)} \\xi\\big) \\, du \\right) \\max\\Big\\{ 1, \\big|\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n + P_n \\, e^{\\lfloor s \\rfloor_{h_n} (A-\\eta)} \\xi)\\big|^{\\nicefrac{p}{2}}, \\\\\n& \\quad \\big\\|\\mathbb{O}_s^n+ P_n \\, e^{s (A-\\eta)} \\xi\\big\\|_H^p, \\smallint\\nolimits_{0}^T \\big\\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\big\\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\Big\\} \\, ds \\biggr]< \\infty.\n\\end{split}\n\\end{align}\nThe proof of Proposition~\\ref{prop:abstract} is thus completed.\n\\end{proof}\n\n\nThe proof of the next elementary result, Lemma~\\ref{lemma:conv:rate}, is a slight adaptation of the proof of Lemma~5.9 in Hutzenthaler et al.~\\cite{Salimova2016}.\n\n\\begin{lemma}\\label{lemma:conv:rate}\nAssume the setting in Subsection~\\ref{setting:example}, let $p \\in [2, \\infty)$, $n \\in \\mathbb{N}$, $\\varepsilon \\in [0, \\nicefrac{1}{4} -\\varrho)$, and let $O \\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ be a stochastic process which satisfies for all $t \\in [0, T]$ that $ [O_t ]_{\\P, \\mathcal{B}(H)} = \\int_0^t e^{(t-s)A} \\, dW_s$. Then \n\\begin{align}\n\\sup_{t \\in [0, T]} \\bigl( \\mathbb{E} \\bigl[ \\|O_t - \\mathcal{O}_t^n \\|_{ H_{\\varrho} }^p \\bigr] \\bigr)^{ \\nicefrac{1}{p} }\n\\leq\n\\left[\\tfrac{p(p-1)}{4 ( c_0 \\pi^2 )^{2\\varepsilon} } \\sum_{k = 1}^\\infty (\\lambda_k)^{2\\varrho+2\\varepsilon-1}\\right]^{\\nicefrac{1}{2}} n^{-2\\varepsilon} < \\infty.\n\\end{align}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:conv:rate}]\nFirst, note that the Burkholder-Davis-Gundy-type inequality in Da~Prato \\& Zabczyk~\\cite[Lemma~7.7]{dz92} shows for all $t \\in [0, T]$ that\n\\begin{align}\\label{eq:burk}\n\\begin{split}\n\\bigl( \\mathbb{E} \\bigl[ \\|O_t - \\mathcal{O}_t^n \\|_{ H_{\\varrho} }^p \\bigr] \\bigr)^{ \\nicefrac{1}{p} }\n& =\n\\biggl( \\mathbb{E} \\biggl[\n \\biggl\\| \\int_0^t (\\mathrm{Id}_{H_{\\varrho}}- P_n) \\, e^{(t-s)A} \\, d W_s \\biggr\\|_{ H_{\\varrho} }^p\n\\biggr] \\biggr)^{ \\nicefrac{1}{p} } \\\\\n& \\leq\n\\left[ \\tfrac{p(p-1)}{2} \\int_0^t \\big \\|(\\mathrm{Id}_{H_{\\varrho}}- P_n) \\, e^{(t-s)A} \\big\\|^2_{\\mathrm{HS}(H, H_{\\varrho})} \\, ds \\right]^{\\nicefrac{1}{2}}.\n\\end{split}\n\\end{align}\nNext observe that for all $ t \\in [0, T] $ it holds that\n\\begin{align}\n\\begin{split}\n\\int_0^t \\big \\|(\\mathrm{Id}_{H_{\\varrho}}- P_n) \\, e^{(t-s)A} \\big\\|^2_{\\mathrm{HS}(H, H_{\\varrho})} \\, ds\n& \\leq \\int_0^t \\|\\mathrm{Id}_{H_{\\varrho+\\varepsilon}}- P_n|_{H_{\\varrho+\\varepsilon}} \\|^2_{L(H_{\\varrho+\\varepsilon}, H_{\\varrho})} \\, \\| e^{(t-s)A} \\|^2_{\\mathrm{HS}(H, H_{\\varrho+\\varepsilon})} \\, ds\\\\ \n& = \\| ( - A )^{ - \\varepsilon } ( \\mathrm{Id}_H - P_n |_H ) \\|^2_{ L( H ) } \\int_0^t \\| e^{sA} \\|^2_{\\mathrm{HS}(H, H_{\\varrho+\\varepsilon})} \\, ds\\\\\n& = \\| ( - A )^{ - 1} ( \\mathrm{Id}_H - P_n |_H ) \\|^{2\\varepsilon}_{ L( H ) } \\sum_{k = 1}^\\infty \\int_0^t (\\lambda_k)^{2\\varrho+2\\varepsilon} e^{-2\\lambda_k s} \\, ds \\\\ \n& \\leq |\\lambda_{n+1}|^{-2\\varepsilon} \\sum_{k = 1}^\\infty \\frac{ (\\lambda_k)^{2\\varrho+2\\varepsilon}}{2\\lambda_k } \\leq \\tfrac{1}{2} ( c_0 \\pi^2 n^2 )^{-2\\varepsilon} \\sum_{k = 1}^\\infty (\\lambda_k)^{2\\varrho+2\\varepsilon-1}.\n\\end{split}\n\\end{align}\nThis and \\eqref{eq:burk} ensure that for all $t \\in [0, T]$ it holds that\n\\begin{align}\n\\bigl( \\mathbb{E} \\bigl[ \\|O_t - \\mathcal{O}_t^n \\|_{ H_{\\varrho} }^p \\bigr] \\bigr)^{ \\nicefrac{1}{p} }\n\\leq\n\\left[\\tfrac{p(p-1)}{4(c_0 \\pi^2)^{2\\varepsilon}} \\sum_{k = 1}^\\infty (\\lambda_k)^{2\\varrho+2\\varepsilon-1} \\right]^{\\nicefrac{1}{2}} n^{-2\\varepsilon} < \\infty.\n\\end{align}\nThe proof of Lemma~\\ref{lemma:conv:rate} is thus completed.\n\\end{proof}\n\nSome of the arguments in the proof of the next result, Proposition~\\ref{prop:exists} below, are similar to the arguments in the proof of Corollary~5.10 in Hutzenthaler et al.~\\cite{Salimova2016}.\n\n\\begin{prop}\n\\label{prop:exists}\nAssume the setting in Subsection~\\ref{setting:example} and let $p \\in (0, \\infty)$. Then there exist a real number $\\eta \\in [0,\\infty)$ and stochastic processes\n$ O \\colon [0, T] \\times \\Omega \\to H_{\\varrho}$\nand $ \\tilde{\\mathcal{O}}^n, \\mathbb{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $n \\in \\mathbb{N}$,\nwith continuous sample paths\nsuch that\n\\begin{enumerate}[(i)]\n\t\\item \\label{item:O}\n\tit holds for all $t \\in [0, T]$ that\n\t$[O_t]_{\\P, \\mathcal{B}(H)} = \\int_0^{t} e^{(t-s)A} \\, dW_s$,\n\t\\item \\label{item:mathcal:O}\n\tit holds for all $n \\in \\mathbb{N}$, $t \\in [0, T]$ that\n\t$[\\tilde{\\mathcal{O}}^n_t]_{\\P, \\mathcal{B}(H)} = \\int_0^{t} P_n \\, e^{(t-s)A} \\, dW_s$,\n\t\\item \\label{item:mathbb:O}\n\tit holds for all $n \\in \\mathbb{N}$, $t \\in [0, T]$ that\n\t$ \\mathbb{O}^n_t\n\t= \\tilde{\\mathcal{O}}_t^n + P_n\\, e^{tA} \\xi\n\t- \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta \\, ( \\tilde{\\mathcal{O}}_s^n + P_n \\, e^{sA} \\xi ) \\, ds $,\n\t\\item \\label{item:conv}\n\tit holds that\n\t$\\P \\bigl( \\limsup_{n \\to \\infty} \\sup_{s \\in [0, T]} \\| (O_s + e^{sA} \\xi) - (\\tilde{\\mathcal{O}}_s^n + P_n \\, e^{sA} \\xi ) \\|_{H_{\\varrho}} =0 \\bigr)=1$,\n\t\\item \\label{item:scheme}\n\tit holds for all $n \\in \\mathbb{N}$, $t \\in [0, T]$ that\n\t\\begin{equation*}\n \\P \\Bigl( \\mathcal{X}_t^n = P_n \\, e^{ t A } \\xi + \\smallint_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\tilde{\\mathcal{O}}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds + \\tilde{\\mathcal{O}}_t^n \\Bigr)=1,\n\t\\end{equation*}\n\tand\n\t\\item \\label{item:regularity}\n\tit holds that \n\t\\begin{align}\n\t\\nonumber\n\t& \\sup_{ n \\in \\mathbb{N} } \\mathbb{E}\\biggl[ \\int_0^T e^{ \\int_s^T p\\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) \\, du} \\max\\bigl\\{ 1, |\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n )|^{\\nicefrac{p}{2}}, \\|\\mathbb{O}_s^n\\|_H^p, \\smallint\\nolimits_{0}^T \\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\|_{H_{\\varrho}}^{2p+ 2p\\vartheta} \\, du \\bigr\\} \\, ds \\biggr]\n\t\\\\ & +\n\t\\sup_{ n \\in \\mathbb{N} } \\sup_{ s \\in [0,T]} \\mathbb{E}[ \\| \\mathbb{O}_s^n \\|_H^p]\n\t< \\infty. \n\t\\end{align}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}[Proof of Proposition~\\ref{prop:exists}]\nThroughout this proof let $\\varepsilon \\in (0, \\nicefrac{1}{4}- \\varrho)$, $\\beta \\in (0, \\nicefrac{1}{4})$, $q \\in (\\max\\{p, \\nicefrac{1}{\\beta},\\nicefrac{4}{\\varepsilon} \\}, $ $ \\infty)$.\nObserve that Lemma~\\ref{abs:phi(w)_finite} and Lemma~\\ref{lem:conv:series} ensure that there exists a real number $\\eta \\in [0, \\infty)$ such that\n\\begin{align}\n\\label{eq:eps:bound}\n720 q^3 T \\gamma \\pi^{4} \\left[ \\sum_{k = 1}^\\infty \\frac{ k^{4 \\beta}}{\\lambda_k+\\eta}\\right] \\Big[\\sup \\! \\Big(\\Big\\{ \\! \\sup\\nolimits_{x \\in (0,1)} |v(x)| \\colon \\big[v \\in \\mathcal{C}((0,1), \\mathbb{R}) \\text{ and } \\|v\\|_{\\mathcal{W}^{\\beta, q}((0,1), \\mathbb{R})} \\leq 1\\big] \\Big\\}\\Big)\\Big]^2 \\leq 1.\n\\end{align}\nNext note that the Burkholder-Davis-Gundy-type inequality in Da~Prato \\& Zabczyk~\\cite[Lemma~7.7]{dz92} yields for all $n \\in \\mathbb{N}$, $t_1, t_2 \\in [0, T]$ with $t_1 \\leq t_2$ that\n\\begin{align}\n\\begin{split}\n& \\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_1}P_n \\, e^{(t_1-s)A} \\, dW_s - \\int_0^{t_2} P_n \\, e^{(t_2-s)A} \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n& + \\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_1} e^{(t_1-s)A} \\, dW_s - \\int_0^{t_2} e^{(t_2-s)A} \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n& =\n\\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_2} \\bigl( \\one_{(-\\infty,t_1)}(s)\\, P_n \\, e^{\\max\\{ t_1-s, 0 \\}A} - P_n \\, e^{(t_2-s)A} \\bigr) \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n& \\quad +\n\\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_2} \\bigl( \\one_{(-\\infty,t_1)}(s)\\, e^{ \\max\\{ t_1-s, 0 \\}A} - e^{(t_2-s)A} \\bigr) \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n&\\leq \\tfrac{q(q-1)}{2} \\int_0^{t_2} \\left\\| \\one_{(-\\infty,t_1)}(s)\\, P_n \\, e^{\\max\\{ t_1-s, 0 \\}A} - P_n \\, e^{(t_2-s)A} \\right\\|^2_{\\mathrm{HS}(H, H_{\\varrho})} ds \\\\\n& \\quad + \\tfrac{q(q-1)}{2} \\int_0^{t_2} \\left\\| \\one_{(-\\infty,t_1)}(s)\\, e^{\\max\\{ t_1-s, 0 \\} A} - e^{(t_2-s)A} \\right\\|^2_{\\mathrm{HS}(H, H_{\\varrho})} ds \\\\\n& \\leq q (q-1) \\bigg[ \\int_{t_1}^{t_2} \\big \\| e^{(t_2-s)A} \\big\\|^2_{\\mathrm{HS}(H, H_{\\varrho})} \\, ds + \\int_{0}^{t_1} \\big \\| e^{(t_1-s)A}- e^{(t_2-s)A} \\big\\|^2_{\\mathrm{HS}(H, H_{\\varrho})} \\, ds \\bigg] \\\\\n& = q (q-1) \\bigg[ \\int_{t_1}^{t_2} \\big \\| e^{(t_2-s)A} \\big\\|^2_{\\mathrm{HS}(H, H_{\\varrho})} \\, ds+ \\int_0^{t_1} \\big\\| e^{(t_1-s)A} \\big(\\mathrm{Id}_H - e^{(t_2-t_1)A}\\big) \\big\\|_{\\mathrm{HS}(H, H_{\\varrho})}^2 \\, ds \\bigg].\n\\end{split}\n\\end{align}\nThis proves for all $n \\in \\mathbb{N}$, $t_1, t_2 \\in [0, T]$ with $t_1 \\leq t_2$ that\n\\begin{align}\n\\label{eq:O:Holder3}\n\\begin{split}\n& \\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_1}P_n \\, e^{(t_1-s)A} \\, dW_s - \\int_0^{t_2} P_n \\, e^{(t_2-s)A} \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n& + \\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_1} e^{(t_1-s)A} \\, dW_s - \\int_0^{t_2} e^{(t_2-s)A} \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n& \\leq q(q-1) \\int_{t_1}^{t_2} \\big \\| (-A)^{\\varrho} \\, e^{(t_2-s)A} \\big\\|^2_{\\mathrm{HS}(H)} \\, ds \\\\\n&\\quad + q (q-1) \\int_0^{t_1} \\big\\| (-A)^{\\varrho+\\varepsilon} \\, e^{(t_1-s)A} \\big\\|_{\\mathrm{HS}(H)}^2 \\, \\big\\|(-A)^{-\\varepsilon} \\big(\\mathrm{Id}_H - e^{(t_2-t_1)A}\\big) \\big\\|_{L(H)}^2 \\, ds \\\\\n& = q (q-1) \\sum_{k = 1}^\\infty \\int_{t_1}^{t_2} (\\lambda_k)^{2\\varrho} \\,e^{-2(t_2-s) \\lambda_k} \\, ds \\\\\n& \\quad + q (q-1) \\big\\|(-A)^{-\\varepsilon} \\big(\\mathrm{Id}_H - e^{(t_2-t_1)A}\\big) \\big\\|_{L(H)}^2 \\sum_{k = 1}^\\infty \\int_0^{t_1} (\\lambda_k)^{2\\varrho+2\\varepsilon} \\,e^{-2(t_1-s) \\lambda_k} \\, ds \\\\\n& = q(q-1) \\sum_{k = 1}^\\infty\\frac{( \\lambda_k)^{2\\varrho} (1-e^{-2\\lambda_k(t_2-t_1)})}{2\\lambda_k} \\\\\n&\\quad + q(q-1) \\big\\|(-A)^{-\\varepsilon} \\big(\\mathrm{Id}_H - e^{(t_2-t_1)A}\\big) \\big\\|_{L(H)}^2 \\sum_{k = 1}^\\infty\\frac{( \\lambda_k)^{2\\varrho+2\\varepsilon} (1-e^{-2\\lambda_k t_1})}{2\\lambda_k}.\n\\end{split}\n\\end{align}\nMoreover, note that the fact that\n$ \\forall \\, r \\in [ 0, 1 ], \\, t \\in [ 0, \\infty ) \\colon\n\\| (-A)^{-r} ( \\operatorname{Id}_H - e^{ t A } ) \\|_{L(H)} \\leq t^r $\n(cf., e.g., Lemma~11.36 in Renardy \\& Rogers~\\cite{RenardyRogers1993})\nimplies that \n\\begin{align}\n\\label{eq:O:Holder2}\n\\begin{split}\n& \\sup_{ t_1, t_2 \\in [ 0, T ], \\, t_1 < t_2 }\n\\frac{\\|(-A)^{-\\varepsilon} (\\mathrm{Id}_H - e^{(t_2-t_1)A}) \\|_{L(H)}^2 }{(t_2-t_1)^{2\\varepsilon}}\n= \\sup_{ t \\in ( 0, T ] } \\bigl( t^{ -\\varepsilon }\n\\| (-A)^{-\\varepsilon} (\\mathrm{Id}_H - e^{tA}) \\|_{L(H)} \\bigr)^2 \n\\leq 1.\n\\end{split}\n\\end{align}\nThe fact that $\\forall \\, x\\in \\mathbb{R} \\colon 1-e^{-x} \\leq x$ and \\eqref{eq:O:Holder3} hence establish for all $n \\in \\mathbb{N}$, $t_1, t_2 \\in [0, T]$ with $t_1 < t_2$ that\n\\begin{align}\\label{eq:O:Holder}\n\\begin{split}\n& \\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_1}P_n \\, e^{(t_1-s)A} \\, dW_s - \\int_0^{t_2} P_n \\, e^{(t_2-s)A} \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n& + \\biggl( \\mathbb{E} \\biggl[\n\\biggl\\| \\int_0^{t_1} e^{(t_1-s)A} \\, dW_s - \\int_0^{t_2} e^{(t_2-s)A} \\, dW_s \\biggr\\|_{ H_{\\varrho} }^q\n\\biggr] \\biggr)^{ \\nicefrac{2}{q} } \\\\\n& \\leq q(q-1) \\sum_{k = 1}^\\infty\\frac{(\\lambda_k)^{2\\varrho-1} (1-e^{-2\\lambda_k(t_2-t_1)})^{2\\varepsilon}}{2} \\\\\n&\\quad + q(q-1) \\big\\|(-A)^{-\\varepsilon} \\big(\\mathrm{Id}_H - e^{(t_2-t_1)A}\\big) \\big\\|_{L(H)}^2 \\sum_{k = 1}^\\infty ( \\lambda_k)^{2\\varrho+2\\varepsilon-1} \\\\\n& \\leq q(q-1) \\Bigg[ \\sum_{k = 1}^\\infty( \\lambda_k)^{2\\varrho+2\\varepsilon-1} \\Bigg] \\Bigg( 1+ \\frac{\\|(-A)^{-\\varepsilon} (\\mathrm{Id}_H - e^{(t_2-t_1)A}) \\|_{L(H)}^2 }{(t_2-t_1)^{2\\varepsilon}}\\Bigg) (t_2-t_1)^{2\\varepsilon} \\\\\n& \\leq 2 \\, q(q-1) \\Bigg[ \\sum_{k = 1}^\\infty ( \\lambda_k)^{2\\varrho+2\\varepsilon-1} \\Bigg] (t_2-t_1)^{2\\varepsilon} < \\infty.\n\\end{split}\n\\end{align}\nCombining this with the fact that $ q \\varepsilon > 1$ and the Kolmogorov-Chentsov continuity theorem shows that there exist stochastic processes\n$ O \\colon [0, T] \\times \\Omega \\to H_{\\varrho}$,\n$ \\tilde{\\mathcal{O}}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $n \\in \\mathbb{N}$,\nand $ \\mathbb{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $n \\in \\mathbb{N}$,\nwith continuous sample paths\nwhich satisfy\nfor all $n \\in \\mathbb{N}$, $t \\in [0, T]$ that\n$[O_t]_{\\P, \\mathcal{B}(H)} = \\int_0^{t} e^{(t-s)A} \\, dW_s$,\n$[\\tilde{\\mathcal{O}}^n_t]_{\\P, \\mathcal{B}(H)} = \\int_0^{t} P_n \\, e^{(t-s)A} \\, dW_s$,\nand\n\\begin{equation}\n\\label{eq:mathbbO_def}\n\\mathbb{O}^n_t\n= \\tilde{\\mathcal{O}}_t^n + P_n\\, e^{tA} \\xi\n- \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta \\, ( \\tilde{\\mathcal{O}}_s^n + P_n \\, e^{sA} \\xi ) \\, ds.\n\\end{equation}\nThis proves \\eqref{item:O}--\\eqref{item:mathbb:O}.\nNext observe that the fact that\n$ \\forall \\, n \\in \\mathbb{N}, \\, t \\in [0, T] \\colon \\P(\\mathcal{O}_t^n = \\tilde{\\mathcal{O}}^n_t )=1$\nand\nLemma~\\ref{lemma:conv:rate} demonstrate that \n\\begin{align}\\\n\\sup_{n \\in \\mathbb{N}} \\biggl\\{n^{\\varepsilon} \\sup_{t \\in [0, T]} \\bigl( \\mathbb{E} \\bigl[ \\|O_t - \\tilde{\\mathcal{O}}_t^n \\|_{ H_{\\varrho} }^q \\bigr] \\bigr)^{ \\nicefrac{1}{q} } \\biggr\\} < \\infty.\n\\end{align}\nThe fact that $ O \\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ and $\\tilde{\\mathcal{O}}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $n \\in \\mathbb{N}$, are stochastic processes with continuous sample paths, \\eqref{eq:O:Holder}, and Cox et al.~\\cite[Corollary~2.11]{CoxWelti2016} (with $T=T$, $p=q$, $\\beta= \\varepsilon$, $\\theta^N= \\{ \\frac{k T}{N} \\in [0, T] \\colon k \\in \\{ 0, 1, \\ldots, N \\} \\}$, $ E = H_{\\varrho} $, $Y^N= ([0,T] \\times \\Omega \\ni (t, \\omega) \\mapsto \\tilde{\\mathcal{O}}^N_t(\\omega) \\in H_{\\varrho})$, $Y^0= O$, $\\alpha=0$, $\\varepsilon= \\nicefrac{\\varepsilon}{2}$ for $N \\in \\mathbb{N}$ in the notation of Cox et al.~\\cite[Corollary~2.11]{CoxWelti2016}) hence prove that\n\\begin{align}\n\\sup_{n \\in \\mathbb{N}}\n\\biggl\\{ n^{(\\nicefrac{\\varepsilon}{2}- \\nicefrac{1}{q})}\n\\biggl(\n\\mathbb{E} \\biggl[\n \\sup_{t \\in [0, T]} \\|O_t - \\tilde{\\mathcal{O}}_t^n \\|_{H_{\\varrho}}^q\n\\biggr] \\biggr)^{ \\nicefrac{1}{q} }\n\\biggr\\} < \\infty.\n\\end{align}\nThis, the fact that $\\nicefrac{\\varepsilon}{2}- \\nicefrac{1}{q} > \\nicefrac{1}{q}$, and Hutzenthaler \\& Jentzen~\\cite[Lemma~3.21]{Hutzenthaler2015} (cf., e.g., Graham \\& Talay~\\cite[Theorem~7.12]{Graham2013} and Kloeden \\& Neuenkirch~\\cite[Lemma~2.1]{Kloeden2007}) ensure that \n\\begin{align}\\label{eq:O:conv}\n\\P \\bigg(\\limsup_{n \\to \\infty} \\sup_{s \\in [0, T]} \\| O_s - \\tilde{\\mathcal{O}}_s^n \\|_{H_{\\varrho}} =0 \\bigg)=1.\n\\end{align}\nNext note that for all $n \\in \\mathbb{N}$ it holds that \n\\begin{align}\n\\begin{split}\n\\sup_{ s \\in [ 0, T ] }\n\\|(\\mathrm{Id}_H- P_n) \\, e^{sA} \\xi \\|_{H_{\\varrho}} &\\leq \\|(-A)^{(\\varrho-\\nicefrac{1}{2})} (\\mathrm{Id}_H-P_n |_H ) \\|_{L(H)} \\|\\xi\\|_{H_{\\nicefrac{1}{2}}} \\\\\n&= \\|(-A)^{-1} (\\mathrm{Id}_H-P_n |_H ) \\|_{L(H)}^{(\\nicefrac{1}{2}-\\varrho)} \\|\\xi\\|_{H_{\\nicefrac{1}{2}}}\n\\leq ( c_0 \\pi^2 n^2 )^{(\\varrho - \\nicefrac{1}{2})} \\|\\xi\\|_{H_{\\nicefrac{1}{2}}} .\n\\end{split}\n\\end{align}\nCombining this with \\eqref{eq:O:conv} shows that \n\\begin{align}\n\\P \\bigg( \\limsup_{n \\to \\infty} \\sup_{s \\in [0, T]} \\| (O_s + e^{sA} \\xi) - (\\tilde{\\mathcal{O}}_s^n + P_n \\, e^{sA} \\xi ) \\|_{H_{\\varrho}} =0 \\bigg)=1.\n\\end{align}\nThis establishes~\\eqref{item:conv}. Furthermore, the fact that $ \\forall \\, n \\in \\mathbb{N}, \\, t \\in [0, T] \\colon \\P(\\mathcal{O}_t^n = \\tilde{\\mathcal{O}}^n_t )=1$ and \\eqref{eq:set:abstract}\nprove~\\eqref{item:scheme}.\nMoreover, Proposition~\\ref{prop:transform_SG}\n(with $H=H$,\n$U=H$,\n$ \\mathbb{H} = \\{ e_k \\in H \\colon k \\in \\mathbb{N} \\} $,\n$T=T$,\n$\\alpha= 0$,\n$\\beta=0$,\n$\\gamma=0$,\n$\\eta=\\eta$,\n$\\kappa=0$,\n$A=A$,\n$ ( W_t )_{ t \\in [ 0, T ] } = ( W_t )_{ t \\in [ 0, T ] } $,\n$O=P_n(H)$,\n$F= (P_n(H) \\ni v \\mapsto 0 \\in H)$,\n$ \\tilde{F} = ( P_n(H) \\ni v \\mapsto \\eta v \\in H) $,\n$B= ( P_n(H) \\ni v \\mapsto ( H \\ni u \\mapsto P_n(u) \\in H ) \\in \\mathrm{HS}(H) ) $,\n$\\xi= ( \\Omega \\ni \\omega \\mapsto P_n \\, \\xi \\in P_n(H) ) $,\n$X= ( [0,T] \\times \\Omega \\ni (t,\\omega) \\mapsto (\\tilde{\\mathcal{O}}^n_t( \\omega ) + P_n \\, e^{tA} \\xi) \\in P_n( H ) ) $\nfor $n \\in \\mathbb{N}$\nin the notation of Proposition~\\ref{prop:transform_SG})\nensures that for all $n \\in \\mathbb{N}$, $t \\in [0,T]$ it holds that \n\\begin{equation}\n\\begin{split}\n[ \\tilde{\\mathcal{O}}_t^n + P_n\\, e^{tA} \\xi ]_{\\P, \\mathcal{B}(H)}\n& =\n\\left[\nP_n \\, e^{t(A-\\eta)} \\xi\n+ \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta \\, ( \\tilde{\\mathcal{O}}_s^n + P_n \\, e^{sA} \\xi ) \\, ds\n\\right]_{\\P, \\mathcal{B}(H)}\n\\\\ & \\quad \n+ \\int_0^{t} P_n \\, e^{(t-s)(A-\\eta)} \\, dW_s.\n\\end{split}\n\\end{equation}\nThis and \\eqref{eq:mathbbO_def} imply for all $n \\in \\mathbb{N}$, $t \\in [0,T]$ that\n\\begin{equation}\n\\label{eq:trans:O}\n[ \\mathbb{O}^n_t - P_n \\, e^{t(A-\\eta)} \\xi ]_{\\P, \\mathcal{B}(H)}\n= \\int_0^{t} P_n \\, e^{(t-s)(A-\\eta)} \\, dW_s.\n\\end{equation}\nIn addition, note that for all $ n \\in \\mathbb{N} $ it holds that\n$ [0,T] \\times \\Omega \\ni (t,\\omega) \\mapsto ( \\mathbb{O}^n_t( \\omega ) - P_n \\, e^{t(A-\\eta)} \\xi ) \\in P_n( H ) $\nis a stochastic process with continuous sample paths.\nProposition~\\ref{prop:abstract}, \\eqref{eq:trans:O}, and \\eqref{eq:eps:bound} hence show that \n\\begin{align}\n\\begin{split}\n& \\sup_{ n \\in \\mathbb{N} } \\mathbb{E}\\biggl[ \\int_0^T e^{ \\int_s^T q\\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_n} }^n ) \\, du} \\max\\bigl\\{ 1, |\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_n} }^n )|^{\\nicefrac{q}{2}}, \\|\\mathbb{O}_s^n\\|_H^q, \\smallint\\nolimits_{0}^T \\| \\tilde{\\mathcal{O}}_u^n+ P_n \\, e^{uA} \\xi \\|_{H_{\\varrho}}^{2q+ 2q\\vartheta} \\, du \\bigr\\} \\, ds \\biggr]\n\\\\ & +\n\\sup_{ n \\in \\mathbb{N} } \\sup_{ s \\in [0,T]} \\mathbb{E}[ \\| \\mathbb{O}_s^n \\|_H^q]\n< \\infty.\n\\end{split}\n\\end{align}\nThis establishes~\\eqref{item:regularity}.\nThe proof of Proposition~\\ref{prop:exists} is thus completed. \n\\end{proof}\n\n\n\\subsection{Strong convergence}\n\n\n\\begin{prop}\n\\label{abs:prop:last}\nAssume the setting in Subsection~\\ref{setting:example} and let $ X\\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ be a stochastic process with continuous sample paths which satisfies for all $t \\in [0, T]$ that $ [X_t ]_{\\P, \\mathcal{B}(H)} = [ e^{ t A } \\xi + \\smallint_0^t e^{ ( t - s ) A} \\, F ( X_s ) \\, ds ]_{\\P, \\mathcal{B}(H)} + \\int_0^t e^{(t-s)A} \\, dW_s$. Then it holds for all $p \\in (0, \\infty)$ that\n\\begin{align}\n\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[ \\| X_t -\\mathcal{X}_t^n \\|_H^p \\big] = 0.\n\\end{align}\n\\end{prop}\n\\begin{proof}[Proof of Proposition~\\ref{abs:prop:last}]\nThroughout this proof let $p \\in (0,\\infty)$, $ q \\in ( \\max\\{ p, 2 \\}, \\infty ) $.\nNote that Proposition~\\ref{prop:exists}\nshows that there exist a real number $\\eta \\in [0,\\infty)$ and stochastic processes\n$ O \\colon [0, T] \\times \\Omega \\to H_{\\varrho}$,\n$ \\tilde{\\mathcal{O}}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $n \\in \\mathbb{N}$,\nand $ \\mathbb{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $n \\in \\mathbb{N}$,\nwith continuous sample paths\nwhich satisfy\nfor all $n \\in \\mathbb{N}$, $t \\in [0, T]$ that\n\\begin{gather}\n\\label{eq:O:as}\n[O_t]_{\\P, \\mathcal{B}(H)} = \\int_0^{t} e^{(t-s)A} \\, dW_s,\n\\qquad\\qquad\\!\\!\n[\\tilde{\\mathcal{O}}^n_t]_{\\P, \\mathcal{B}(H)} = \\int_0^{t} P_n \\, e^{(t-s)A} \\, dW_s,\n\\\\\n\\mathbb{O}^n_t\n= \\tilde{\\mathcal{O}}_t^n + P_n\\, e^{tA} \\xi\n- \\int_0^t e^{(t-s)(A-\\eta)} \\, \\eta \\, ( \\tilde{\\mathcal{O}}_s^n + P_n \\, e^{sA} \\xi ) \\, ds,\n\\\\\n\\P \\bigg( \\limsup_{m \\to \\infty} \\sup_{s \\in [0, T]} \\| (O_s + e^{sA} \\xi) - (\\tilde{\\mathcal{O}}_s^m + P_m \\, e^{sA} \\xi ) \\|_{H_{\\varrho}} =0 \\bigg)=1,\n\\\\\\label{eq:X^n}\n\\!\\P \\Big( \\mathcal{X}_t^n = P_n \\, e^{ t A } \\xi \\!+\\! \\smallint_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\tilde{\\mathcal{O}}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds \\!+\\! \\tilde{\\mathcal{O}}_t^n \\Big)=1,\n\\end{gather}\nand\n\\begin{align}\\label{eq:strong:limsup}\n\\nonumber\n& \\limsup_{ m \\to \\infty} \\mathbb{E}\\biggl[ \\int_0^T e^{ \\int_s^T q\\, \\phi( \\mathbb{O}_{\\lfloor u \\rfloor_{h_m} }^m ) \\, du} \\max\\bigl\\{ 1, |\\Phi(\\mathbb{O}_{ \\lfloor s \\rfloor_{h_m} }^m )|^{\\nicefrac{q}{2}}, \\|\\mathbb{O}_s^m\\|_H^q, \\smallint\\nolimits_{0}^T \\| \\tilde{\\mathcal{O}}_u^m+ P_m \\, e^{uA} \\xi \\|_{H_{\\varrho}}^{2q+ 2q\\vartheta} \\, du \\bigr\\} \\, ds \\biggr]\n\\\\ & +\n\\limsup_{ m \\to \\infty} \\sup_{ s \\in [0,T]} \\mathbb{E}[ \\| \\mathbb{O}_s^m \\|_H^q]\n< \\infty.\n\\end{align}\nIn addition, observe that the fact that\n$\\eta \\in [0, \\infty)$\nand the assumption that\n$\\forall \\, n \\in \\mathbb{N}, \\, v, w \\in P_n( H ) \\colon \\left< v, P_n F( v + w ) \\right>_H \\leq \\phi( w ) \\| v \\|^2_H + \\varphi \\| v \\|^2_{ H_{ \\nicefrac{1}{2} } } + \\Phi( w )$\nestablish\nthat for all $ n \\in \\mathbb{N} $, $ v, w \\in P_n( H ) $ it holds that\n\\begin{equation}\n\\left< v, P_n F( v + w ) \\right>_H \\leq \\phi( w ) \\| v \\|^2_H + \\varphi \\| (\\eta-A)^{\\nicefrac{1}{2}} v \\|^2_{ H} + \\Phi( w ).\n\\end{equation}\nCombining this,\nthe fact that\n$ \\nicefrac{ ( 1 - \\alpha - \\rho ) }{( 1 + 2 \\vartheta ) }\n\\geq\n\\nicefrac{ ( 2 \\varrho - \\rho ) }{( 1 + 2 \\vartheta ) }\n\\geq\n\\nicefrac{ ( 2 \\varrho - 2 \\rho ) }{( 2 + 2 \\vartheta ) }\n=\n\\nicefrac{( \\varrho - \\rho ) }{( 1 + \\vartheta) } $,\nthe fact that $ \\forall \\, t \\in [0, T] \\colon \\P(X_t = \\int_0^t e^{(t-s)A} \\, F(X_s) \\, ds +O_t + e^{tA} \\xi)=1$,\nand\n\\eqref{eq:O:as}--\\eqref{eq:strong:limsup}\nwith \\eqref{item:strong} in Theorem~\\ref{thm:strong} (with\n$ H = H $,\n$\\mathbb{H}= \\{ e_k \\in H \\colon k \\in \\mathbb{N} \\}$,\n$\\eta=\\eta$,\n$ \\theta= \\theta$,\n$\\kappa= 0$,\n$ A = A $,\n$\\varphi= \\varphi$,\n$\\alpha= \\alpha$,\n$\\rho= \\rho$,\n$\\varrho =\\varrho$,\n$\\vartheta=\\vartheta$,\n$ T = T $,\n$ \\chi = \\chi $,\n$p=q$,\n$F=F$,\n$\\phi=\\phi$,\n$\\Phi=\\Phi$,\n$ ( \\mathbb{H}_n )_{ n \\in \\mathbb{N} } = ( \\{ e_k \\in H \\colon k \\in \\{1, 2, \\ldots, n\\}\\} )_{ n \\in \\mathbb{N} } $,\n$ ( P_n )_{ n \\in \\mathbb{N} } = ( P_n )_{ n \\in \\mathbb{N} } $,\n$ ( h_n )_{ n \\in \\mathbb{N} } = ( h_n )_{ n \\in \\mathbb{N} } $,\n$ ( \\mathbb{O}^n )_{ n \\in \\mathbb{N} } = ( [0,T] \\times \\Omega \\ni (t,\\omega) \\mapsto \\mathbb{O}^n_t( \\omega ) \\in H_{\\varrho} )_{ n \\in \\mathbb{N} } $,\n$ ( \\mathcal{X}^n )_{ n \\in \\mathbb{N} } = ([0, T] \\times \\Omega \\ni (t,\\omega) \\mapsto \\mathcal{X}_t^n(\\omega) \\in H_{\\varrho})_{ n \\in \\mathbb{N} } $,\n$ ( \\mathcal{O}^n )_{ n \\in \\mathbb{N} } = ( [0,T] \\times \\Omega \\ni (t,\\omega) \\mapsto (\\tilde{\\mathcal{O}}^n_t( \\omega ) + P_n \\, e^{tA} \\xi) \\in P_n(H) )_{ n \\in \\mathbb{N} } $,\n$X=X$,\n$ O = ( [0,T] \\times \\Omega \\ni (t,\\omega) \\mapsto (O_t( \\omega ) + e^{tA} \\xi) \\in H_{\\varrho}) $,\n$q=p$\nin the notation of \\eqref{item:strong} in Theorem~\\ref{thm:strong})\ncompletes the proof of Proposition~\\ref{abs:prop:last}. \n\\end{proof}\n\n\n\\section{Examples}\n\\label{sec:examples}\n\nIn this section we demonstrate how Proposition~\\ref{abs:prop:last} can be applied to stochastic Burgers and stochastic Allen-Cahn equations (see Corollary~\\ref{cor:burgers:short} and Corollary~\\ref{cor:cahn:short} below).\n\n\n\\subsection{Stochastic Burgers equations}\n\\label{sec:Burgers}\n\n\n\\subsubsection{Setting}\\label{setting:burgers}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $c_1 \\in \\mathbb{R}$, $ T, c_0 \\in (0,\\infty)$, \n$\\varrho \\in (\\nicefrac{1}{8}, \\nicefrac{1}{4})$,\n$ \\chi \\in (0, \\nicefrac{\\varrho }{2 } - \\nicefrac{1}{16}] $,\n$( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) = (L^2(\\lambda_{(0,1)}; \\mathbb{R}), \\langle \\cdot , \\cdot \\rangle_{L^2(\\lambda_{(0,1)}; \\mathbb{R})}, \\left\\| \\cdot \\right\\|_{L^2(\\lambda_{(0,1)}; \\mathbb{R})} )$,\nlet $(e_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to H $\nbe the function which satisfies\nfor all $ n \\in \\mathbb{N} $ that\n$ e_n = [ (\\sqrt{2} \\sin(n \\pi x) )_{x \\in (0,1)}]_{\\lambda_{(0,1)} , \\mathcal{B}(\\mathbb{R})}$,\nlet $ A \\colon D(A) \\subseteq H \\to H $ be the linear operator\nwhich satisfies\n$ D(A) = \\{ v \\in H \\colon $ $ \\sum_{k = 1}^\\infty k^4 | \\langle e_k , v \\rangle_H |^2 < \\infty \\} $\nand $ \\forall \\, v \\in D(A) \\colon A v = - c_0 \\pi^2 \\sum_{k = 1}^\\infty k^2 \\langle e_k , v \\rangle_H e_k$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $, be a family of interpolation spaces associated to $ -A $,\nlet\n$\\xi \\in H_{\\nicefrac{1}{2}} $,\nlet\n$F \\colon H_{\\nicefrac{1}{8}} \\to H_{-\\nicefrac{1}{2}} $,\n$(P_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H) $,\nand\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $\nbe functions which satisfy\nfor all $v \\in H_{\\nicefrac{1}{8}}$, $n \\in \\mathbb{N}$ that\n$F(v)= c_1(v^2)'$,\n$ P_n(v) = \\sum_{k = 1 }^n \\langle e_k, v \\rangle_H e_k $,\nand\n$ \\limsup_{ m \\to \\infty} h_m =0$,\nlet $ ( \\Omega, \\F, \\P ) $ be a probability space,\nlet $(W_t)_{t \\in [0, T]}$ be an $\\mathrm{Id}_H$-cylindrical $( \\Omega, \\F, \\P )$-Wiener process,\nand let $ \\mathcal{X}^n, \\mathcal{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $ n \\in \\mathbb{N}$, be stochastic processes\nwhich satisfy\nfor all $ n \\in \\mathbb{N} $, $ t \\in [0,T] $ that\n$\\left[\\mathcal{O}_t^n \\right]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)A} \\, dW_s$\nand\n\\begin{equation}\\label{eq:set:burgers}\n\\P \\Big( \\mathcal{X}_t^n = P_n \\, e^{ t A } \\xi + \\smallint_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds + \\mathcal{O}_t^n \\Big)=1. \n\\end{equation}\n\n\n\\subsubsection{Properties of the nonlinearity}\n\nIn this subsection we establish a few elementary properties of the function $F$ in Subsection~\\ref{setting:burgers} above, see Lemmas~\\ref{coer:burgers}--\\ref{nonlin:burgers} below.\nFor the proof of these properties we present two elementary and well-known facts in Lemmas~\\ref{burgers_norms}--\\ref{lem:sup:v} below.\nSee, e.g., Section~4 in Jentzen, Kloeden, \\& Winkel~\\cite{jkw09} for the next result, Lemma~\\ref{burgers_norms}.\n\\begin{lemma}\\label{burgers_norms}\nAssume the setting in Subsection~\\ref{setting:burgers}. Then it holds for all $v \\in H_{\\nicefrac{1}{2}}$ that $\\|v\\|_{H_{\\nicefrac{1}{2}}} = \\sqrt{c_0} \\|v'\\|_H$.\n\\end{lemma}\t\n\\begin{proof}[Proof of Lemma~\\ref{burgers_norms}]\nNote that integration by parts proves for all $v \\in H_1$ that \n\\begin{align}\\label{burgers_norms_1}\n\\begin{split}\n\\|v\\|_{H_{\\nicefrac{1}{2}}}^2 & =\\| (-A)^{\\nicefrac{1}{2}} v \\|^2_{ H} = \\langle (-A)^{\\nicefrac{1}{2}} v, (-A)^{\\nicefrac{1}{2}} v \\rangle_H= -\\langle v, Av \\rangle_H \\\\ & = -c_0 \\langle v, v'' \\rangle_H\n= c_0 \\langle v', v' \\rangle_H = c_0 \\|v'\\|_H^2.\n\\end{split}\n\\end{align}\nThe fact that $H_1 \\subseteq H_{\\nicefrac{1}{2}}$ is dense in $H_{\\nicefrac{1}{2}}$ and the fact that $(H_{\\nicefrac{1}{2}} \\ni v \\mapsto v' \\in H) \\in L(H_{\\nicefrac{1}{2}}, H)$ thus complete the proof of Lemma~\\ref{burgers_norms}.\n\\end{proof}\nCf., e.g., Lemma~4.7 in Bl\\\"omker \\& Jentzen~\\cite{BloemkerJentzen2013} for the next lemma.\n\\begin{lemma}\n\t\\label{lem:sup:v}\nAssume the setting in Subsection~\\ref{setting:burgers}. Then it holds that\n\\begin{align}\n\\sup_{v \\in H \\backslash \\{0\\}} \\frac{\\|v'\\|_{H_{-\\nicefrac{1}{2}}}}{\\|v\\|_H} = (c_0)^{-\\nicefrac{1}{2}}.\n\\end{align}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lem:sup:v}]\nObserve that, e.g., (iii) in Lemma~3.10 in Jacobe de Naurois et al.~\\cite{Naurois2015} and Lemma~\\ref{burgers_norms} show that\n\\begin{equation}\n\\begin{split}\n\\sup_{v \\in H \\backslash \\{0\\}} \\frac{\\|v'\\|_{H_{-\\nicefrac{1}{2}}}}{\\|v\\|_H} & = \\sup_{v \\in H \\backslash \\{0\\}} \\sup_{w \\in H_{\\nicefrac{1}{2}} \\backslash \\{0\\}} \\frac{|\\langle v',w \\rangle_H |}{\\|v\\|_H \\|w \\|_{H_{\\nicefrac{1}{2}}}} \\\\\n&= \\sup_{w \\in H_{\\nicefrac{1}{2}} \\backslash \\{0\\}} \\sup_{v \\in H \\backslash \\{0\\}} \\frac{|\\langle v,w' \\rangle_H |}{\\|v\\|_H \\|w \\|_{H_{\\nicefrac{1}{2}}}}\\\\\n& = \\sup_{w \\in H_{\\nicefrac{1}{2}} \\backslash \\{0\\}} \\frac{\\|w'\\|_H}{\\|w\\|_{H_{\\nicefrac{1}{2}}}}= (c_0)^{-\\nicefrac{1}{2}}.\n\\end{split}\n\\end{equation}\nThe proof of Lemma~\\ref{lem:sup:v} is thus completed.\n\\end{proof}\n\n\nThe next simple lemma is a slight modification of Lemma~5.7 in Bl\\\"omker \\& Jentzen~\\cite{BloemkerJentzen2013}.\n\\begin{lemma}\n\\label{coer:burgers}\nAssume the setting in Subsection~\\ref{setting:burgers} and let $v, w \\in H_{ \\nicefrac{1}{2} }$. Then it holds that $F(v+w) \\in H$ and\n\\begin{align}\n\\label{eq:coer:burgers}\n\\begin{split}\n\\bigl| \\left< v, F( v + w ) \\right>_H \\bigr| &\\leq \\max\\big\\{\\tfrac{2|c_1|^2 }{c_0} , 4\\big\\} \\| v \\|_H^2 \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^2\\big]+ \\tfrac{3}{4}\\| v \\|^2_{ H_{ \\nicefrac{1}{2} } } \\\\\n&\\quad + \\max\\big\\{\\tfrac{2|c_1|^2 }{c_0} , 4\\big\\} \\big[1+\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^{ \\max\\{\\nicefrac{2|c_1|^2 }{c_0} , 4\\}}\\big].\n\\end{split}\n\\end{align}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{coer:burgers}]\nObserve that, e.g., Lemma~4.5 in Jentzen \\& Pu\\v{s}nik~\\cite{JentzenPusnik2016} ensures for all $ u \\in H_{ \\nicefrac{1}{2} } $ that $F(u) \\in H$. Next note that integration by parts and, e.g., again Lemma~4.5 in Jentzen \\& Pu\\v{s}nik~\\cite{JentzenPusnik2016} yield that\n\\begin{align}\n\\begin{split}\n3 \\, \\langle v', v^2 \\rangle_H &= 2 \\, \\langle v \\cdot v', v \\rangle_H + \\langle v', v^2 \\rangle_H = \\langle (v^2)', v \\rangle_H + \\langle v', v^2 \\rangle_H \\\\\n& = - \\langle v^2, v' \\rangle_H + \\langle v', v^2 \\rangle_H = 0.\n\\end{split}\n\\end{align}\nApplying integration by parts again hence shows that\n\\begin{align}\n\\begin{split}\n\\langle v, F(v+w) \\rangle_H &= c_1 \\langle v, [(v+w)^2]' \\rangle_H = - c_1 \\langle v', (v+w)^2 \\rangle_H\\\\\n& = - c_1 \\langle v', v^2 \\rangle_H - 2 \\, c_1 \\langle v', v \\cdot w \\rangle_H - c_1 \\langle v', w^2 \\rangle_H\\\\\n& = - 2 \\, c_1 \\langle v', v \\cdot w \\rangle_H - c_1 \\langle v', w^2 \\rangle_H.\n\\end{split}\n\\end{align}\nThis, the Cauchy-Schwartz inequality, the fact that $\\forall \\, a,b \\in \\mathbb{R}, \\, \\varepsilon \\in (0, \\infty) \\colon 2ab \\leq \\varepsilon a^2 + \\frac{b^2}{\\varepsilon}$, and Lemma~\\ref{burgers_norms} prove that\n\\begin{align}\n\\begin{split}\n\\bigl| \\left< v, F( v + w ) \\right>_H \\bigr| &\\leq 2 \\, |c_1|\\|v'\\|_H \\| v \\cdot w \\|_H + |c_1| \\|v'\\|_H \\|w^2\\|_H \\\\\n& \\leq \\tfrac{c_0}{2} \\|v'\\|_H^2 + \\tfrac{2|c_1|^2}{c_0} \\| v \\cdot w \\|_H^2 + \\tfrac{c_0}{4}\\|v'\\|_H^2 + \\tfrac{|c_1|^2 }{c_0}\\| w^2 \\|_H^2 \\\\\n& \\leq \\tfrac{2|c_1|^2}{c_0} \\| v \\|_H^2 \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^2\\big]+ \\tfrac{3 c_0}{4}\\|v'\\|_H^2 + \\tfrac{|c_1|^2 }{c_0} \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^4\\big] \\\\\n& = \\tfrac{2|c_1|^2}{c_0} \\| v \\|_H^2 \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^2\\big]+ \\tfrac{3}{4}\\| v \\|^2_{ H_{ \\nicefrac{1}{2} } } + \\tfrac{|c_1|^2 }{c_0} \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^4\\big]\\\\\n& \\leq \\max\\big\\{\\tfrac{2|c_1|^2 }{c_0} , 4\\big\\} \\| v \\|_H^2 \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^2\\big]+ \\tfrac{3}{4}\\| v \\|^2_{ H_{ \\nicefrac{1}{2} } } \\\\\n&\\quad + \\max\\big\\{\\tfrac{2|c_1|^2 }{c_0} , 4\\big\\} \\big[1+ \\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^{ \\max\\{\\nicefrac{2|c_1|^2 }{c_0} , 4\\}}\\big] . \n\\end{split}\n\\end{align}\nThe proof of Lemma~\\ref{coer:burgers} is thus completed.\n\\end{proof}\n\n\\begin{lemma}\\label{nonlin:burgers}\nAssume the setting in Subsection~\\ref{setting:burgers} and let $v, w \\in H_{\\nicefrac{1}{8}}$. Then it holds that $F \\in \\mathcal{C}(H_{\\nicefrac{1}{8}}, H_{-\\nicefrac{1}{2}})$ and\n\\begin{align}\n\\begin{split}\n&\\|F(v) - F(w) \\|_{H_{-\\nicefrac{1}{2}}} \\\\\n& \\leq |c_1| |c_0|^{-\\nicefrac{1}{2}} \\left[\\sup_{u \\in H_{\\nicefrac{1}{8}} \\setminus \\{0\\}} \\frac{\\|u\\|^2_{L^4(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^2_{H_{\\nicefrac{1}{8}}}} \\right] \\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{8}}} + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{8}}} \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{8}}} < \\infty.\n\\end{split}\n\\end{align}\t\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{nonlin:burgers}]\nObserve that the fact that $v^2, w^2 \\in H$ and Lemma~\\ref{lem:sup:v} establish that\n\\begin{align}\n\\begin{split}\n\\|(v^2)' - (w^2)' \\|_{H_{-\\nicefrac{1}{2}}} &\\leq |c_0|^{-\\nicefrac{1}{2}} \\| v^2 - w^2 \\|_H \\\\\n&\\leq |c_0|^{-\\nicefrac{1}{2}} \\|v+w \\|_{L^4(\\lambda_{(0,1)}; \\mathbb{R})} \\|v -w \\|_{L^4(\\lambda_{(0,1)}; \\mathbb{R})} \\\\\n& \\leq |c_0|^{-\\nicefrac{1}{2}} \\left[\\sup_{u \\in H_{ \\nicefrac{1}{8} } \\setminus \\{0\\}} \\frac{\\|u\\|_{L^4(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|_{H_{\\nicefrac{1}{8}}}} \\right]^2 \\! \\! \\left\\| v + w \\right\\|_{ H_{ \\nicefrac{1}{8} }} \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{8} }} \\\\\n& \\leq |c_0|^{-\\nicefrac{1}{2}} \\left[\\sup_{u \\in H_{ \\nicefrac{1}{8} } \\setminus \\{0\\}} \\frac{\\|u\\|^2_{L^4(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^2_{H_{\\nicefrac{1}{8}}}} \\right] \\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{8} }} + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{8}}} \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{8}}}.\n\\end{split}\n\\end{align}\nThis shows that\n\\begin{align}\n\\label{nonlin:bur:last}\n\\begin{split}\n& \\|F(v) - F(w) \\|_{H_{-\\nicefrac{1}{2}}} = \\left\\| c_1\\! \\left((v^2)' - (w^2)'\\right) \\right\\|_{H_{-\\nicefrac{1}{2}}} \\\\\n& \\leq |c_1| |c_0|^{-\\nicefrac{1}{2}} \\left[\\sup_{u \\in H_{ \\nicefrac{1}{8}} \\setminus \\{0\\}} \\frac{\\|u\\|^2_{L^4(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^2_{H_{\\nicefrac{1}{8}}}} \\right] \\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{8}}} + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{8}}} \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{8}}}.\n\\end{split}\n\\end{align}\nNext note that the Sobolev embedding theorem yields that\n\\begin{align}\n\\sup_{u \\in H_{ \\nicefrac{1}{8} }\\setminus \\{0\\}} \\frac{\\|u\\|^2_{L^4(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^2_{H_{\\nicefrac{1}{8}}}} < \\infty.\n\\end{align}\nInequality \\eqref{nonlin:bur:last} hence completes the proof of Lemma~\\ref{nonlin:burgers}.\n\\end{proof}\n\n\n\n\n\\subsubsection{Strong convergence}\n\n\\begin{cor}\n\\label{burgers:cor:last}\nAssume the setting in Subsection~\\ref{setting:burgers} and let $ X\\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ be a stochastic process with continuous sample paths which satisfies for all $t \\in [0, T]$ that $ [X_t ]_{\\P, \\mathcal{B}(H)} = [ e^{ t A } \\xi + \\smallint_0^t e^{ ( t - s ) A} \\, F ( X_s ) \\, ds ]_{\\P, \\mathcal{B}(H)} + \\int_0^t e^{(t-s)A} \\, dW_s$. Then it holds for all $p \\in (0, \\infty)$ that\n\\begin{align}\n\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[ \\| X_t -\\mathcal{X}_t^n \\|_H^p \\big] = 0.\n\\end{align}\n\\end{cor}\n\\begin{proof}[Proof of Corollary~\\ref{burgers:cor:last}]\nThroughout this proof\nlet $(\\tilde{P}_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H_{-1}) $\nbe the function which satisfies\nfor all $ n \\in \\mathbb{N} $, $ v \\in H $ that\n$ \\tilde{P}_n(v) = P_n(v) $\nand let $ \\phi , \\Phi \\colon H_1 \\to [0,\\infty) $ be the functions which satisfy\nfor all $ v \\in H_{1}$ that $\\phi(v) = \\max\\big\\{\\frac{2|c_1|^2 }{c_0} , 4\\big\\} \\big[1+\\sup\\nolimits_{x \\in (0,1)} |\\underline{v}(x)|^2\\big]$ and $\\Phi(v)= \\max\\big\\{\\frac{2|c_1|^2 }{c_0} , 4\\big\\} \\big[1+ \\sup\\nolimits_{x \\in (0,1)} |\\underline{v}(x)|^{ \\max\\{\\nicefrac{2|c_1|^2 }{c_0} , 4\\}}\\big]$.\nNext note that Lemma~\\ref{coer:burgers} proves for all $n \\in \\mathbb{N}$, $v, w \\in \\tilde{P}_n(H) = P_n(H)$ that\n\\begin{align}\\label{burgers:cor:coer}\n\\langle v, \\tilde{P}_n F( v + w ) \\rangle_H = \\left< v, F( v + w ) \\right>_H \\leq \\phi(w) \\| v \\|^2_H+ \\tfrac{3}{4} \\| v \\|^2_{ H_{ \\nicefrac{1}{2} } }+ \\Phi( w ).\n\\end{align}\nFurthermore, Lemma~\\ref{nonlin:burgers} ensures that for all $n \\in \\mathbb{N}$, $v, w \\in \\tilde{P}_n(H)$ it holds that $F \\in \\mathcal{C}(H_{\\nicefrac{1}{8}}, H_{-\\nicefrac{1}{2}})$ and\n\\begin{align}\\label{burgers:cor:con}\n\\begin{split}\n&\\|F(v) - F(w) \\|_{H_{-\\nicefrac{1}{2}}} \\\\\n& \\leq |c_1| |c_0|^{-\\nicefrac{1}{2}} \\left[\\sup_{u \\in H_{ \\nicefrac{1}{8} } \\setminus \\{0\\}} \\frac{\\|u\\|^2_{L^4(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^2_{H_{\\nicefrac{1}{8}}}} \\right] \\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{8} }} + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{8} }} \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{8} }} < \\infty.\n\\end{split}\n\\end{align}\nCombining \\eqref{burgers:cor:coer}--\\eqref{burgers:cor:con} and Proposition~\\ref{abs:prop:last} (with\n$T=T$,\n$c_0 = c_0$,\t\n$\\gamma=\\max\\{\\nicefrac{2|c_1|^2 }{c_0} , 4\\}$,\t\n$ \\theta= 1 + |c_1| |c_0|^{-\\nicefrac{1}{2}} [\\sup_{u \\in H_{ \\nicefrac{1}{8}}\\backslash \\{0\\}} \\nicefrac{\\|u\\|^2_{L^4(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^2_{H_{\\nicefrac{1}{8}}}} ]$,\n$\\vartheta=1$,\n$\\alpha= \\nicefrac{1}{2}$,\n$\\varphi= \\nicefrac{3}{4}$,\n$\\rho= \\nicefrac{1}{8}$,\n$\\varrho =\\varrho$, \n$\\chi = \\chi$,\n$\\xi = \\xi$,\n $F=F|_{H_{\\varrho}}$,\n $(P_n)_{n \\in \\mathbb{N}} = (\\tilde{P}_n)_{n \\in \\mathbb{N}}$,\n $(h_n)_{n \\in \\mathbb{N}} = (h_n)_{n \\in \\mathbb{N}}$,\n$\\phi=\\phi$, $\\Phi=\\Phi$, $(\\mathcal{X}^n)_{n \\in \\mathbb{N}} = (\\mathcal{X}^n)_{n \\in \\mathbb{N}}$, \n$ (\\mathcal{O}^n)_{n \\in \\mathbb{N}} = (\\mathcal{O}^n)_{n \\in \\mathbb{N}}$, $X=X$ in the notation of Proposition~\\ref{abs:prop:last}) completes the proof of Corollary~\\ref{burgers:cor:last}. \n\\end{proof}\n\n\nThe next result, Corollary~\\ref{cor:burgers:short} below, is a direct consequence of Corollary~\\ref{burgers:cor:last}.\nIt establishes strong convergence for the stochastic Burgers equation, also see Remark~\\ref{rem:burgers} below.\n\n\\begin{cor}\\label{cor:burgers:short}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ T \\in (0,\\infty)$, \n$\\varrho \\in (\\nicefrac{1}{8}, \\nicefrac{1}{4})$,\n$ \\chi \\in (0, \\nicefrac{\\varrho }{2 } - \\nicefrac{1}{16}] $,\n$( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) = (L^2(\\lambda_{(0,1)}; \\mathbb{R}), \\langle \\cdot , \\cdot \\rangle_{L^2(\\lambda_{(0,1)}; \\mathbb{R})}, \\left\\| \\cdot \\right\\|_{L^2(\\lambda_{(0,1)}; \\mathbb{R})} )$,\nlet $ A \\colon D(A) \\subseteq H \\to H $ be the Laplace operator with Dirichlet boundary conditions on $H$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $, be a family of interpolation spaces associated to $ -A $,\nlet\n$\\xi \\in H_{\\nicefrac{1}{2}} $,\nlet\n$F \\colon H_{\\nicefrac{1}{8}} \\to H_{-\\nicefrac{1}{2}} $,\n$(e_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to H $,\n$(P_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H) $,\nand\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $\nbe functions which satisfy\nfor all $v \\in H_{\\nicefrac{1}{8}}$, $n \\in \\mathbb{N}$ that\n$F(v)= -\\frac{1}{2}(v^2)'$,\n$ e_n = [ (\\sqrt{2} \\sin(n \\pi x) )_{x \\in (0,1)}]_{\\lambda_{(0,1)} , \\mathcal{B}(\\mathbb{R})}$,\n$ P_n(v) = \\sum_{k = 1 }^n \\langle e_k, v \\rangle_H e_k $,\nand\n$ \\limsup_{ m \\to \\infty} h_m =0$,\nlet $ ( \\Omega, \\F, \\P ) $ be a probability space,\nlet $(W_t)_{t \\in [0, T]}$ be an $\\mathrm{Id}_H$-cylindrical $( \\Omega, \\F, \\P )$-Wiener process,\nlet $ \\mathcal{X}^n, \\mathcal{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $ n \\in \\mathbb{N}$, be stochastic processes, let $ X\\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ be a stochastic process with continuous sample paths, and assume for all $ n \\in \\mathbb{N} $, $t \\in [0, T]$ that $ [X_t ]_{\\P, \\mathcal{B}(H)} = [ e^{ t A } \\xi + \\smallint_0^t e^{ ( t - s ) A} \\, F ( X_s ) \\, ds ]_{\\P, \\mathcal{B}(H)} + \\int_0^t e^{(t-s)A} \\, dW_s$, $\\left[\\mathcal{O}_t^n \\right]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)A} \\, dW_s$,\nand\n\\begin{equation}\\label{eq:cor:burgers}\n\\P \\Big( \\mathcal{X}_t^n = P_n \\, e^{ t A } \\xi + \\smallint_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds + \\mathcal{O}_t^n \\Big)=1. \n\\end{equation} Then it holds for all $p \\in (0, \\infty)$ that\n\\begin{align}\n\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[ \\| X_t -\\mathcal{X}_t^n \\|_H^p \\big] = 0.\n\\end{align}\n\\end{cor}\n\n\n\\begin{remark}\\label{rem:burgers}\nConsider the setting in Corollary~\\ref{cor:burgers:short}.\nRoughly speaking, Corollary~\\ref{cor:burgers:short} demonstrates that\nthe full-discrete explicit numerical approximation scheme described by~\\eqref{eq:cor:burgers}\nconverges strongly to a mild solution process of the stochastic Burgers equation \n\\begin{align}\n\\tfrac{\\partial}{\\partial t} X_t(x) = \\tfrac{\\partial^2}{\\partial x^2} X_t(x) - X_t(x) \\cdot \\tfrac{\\partial}{\\partial x} X_t(x) + \\tfrac{\\partial}{\\partial t} W_t(x)\n\\end{align}\nwith $X_0(x) = \\xi(x)$ and $X_t(0)= X_t(1)=0$\nfor $t \\in [0,T]$, $x \\in (0,1)$\n(cf., e.g.,\nDa~Prato, Debussche, \\& Temam~\\cite[Section~1]{DaPrato1994}).\n\\end{remark}\n\n\n\\subsection{Stochastic Allen-Cahn equations}\n\\label{sec:cahn}\n\n\\subsubsection{Setting}\n\\label{setting:cahn}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $c_1 \\in \\mathbb{R}$, $ c_0, c_2, T \\in (0,\\infty)$, \n$\\varrho \\in (\\nicefrac{1}{6}, \\nicefrac{1}{4})$,\n$ \\chi \\in (0, \\nicefrac{\\varrho}{3}-\\nicefrac{1}{18}] $,\n$( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H )\n= (L^2(\\lambda_{(0,1)}; \\mathbb{R}), \\langle \\cdot , \\cdot \\rangle_{L^2(\\lambda_{(0,1)}; \\mathbb{R})}, \\left\\| \\cdot \\right\\|_{L^2(\\lambda_{(0,1)}; \\mathbb{R})} )$,\nlet $(e_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to H $\nbe the function which satisfies\nfor all $ n \\in \\mathbb{N} $ that\n$e_n = [ (\\sqrt{2} \\sin(n \\pi x) )_{x \\in (0,1)}]_{\\lambda_{(0,1)} , \\mathcal{B}(\\mathbb{R})}$,\nlet $ A \\colon D(A) \\subseteq H \\to H $\nbe the linear operator which satisfies\n$ D(A) = \\{ v \\in H \\colon $ $ \\sum_{k = 1}^\\infty k^4 | \\langle e_k , v \\rangle_H |^2 < \\infty \\} $\nand\n$ \\forall \\, v \\in D(A) \\colon A v = -c_0 \\pi^2 \\sum_{k = 1}^\\infty k^2 \\langle e_k , v \\rangle_H e_k$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $,\nbe a family of interpolation spaces associated to $ - A $,\nlet $\\xi \\in H_{\\nicefrac{1}{2}} $,\nlet\n$F \\colon H_{\\nicefrac{1}{6}} \\to H $,\n$(P_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H) $,\nand\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $\nbe functions which satisfy\nfor all $v \\in H_{\\nicefrac{1}{6}}$, $n \\in \\mathbb{N}$ that\n$F(v)= c_1 v - c_2 v^3$,\n$ P_n(v) = \\sum_{k = 1 }^n \\langle e_k, v \\rangle_H e_k $,\nand $ \\limsup_{ m \\to \\infty} h_m =0$,\nlet $ ( \\Omega, \\F, \\P ) $ be a probability space,\nlet $(W_t)_{t \\in [0, T]}$ be an $\\mathrm{Id}_H$-cylindrical $( \\Omega, \\F, \\P )$-Wiener process,\nand let $ \\mathcal{X}^n, \\mathcal{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $ n \\in \\mathbb{N}$,\nbe stochastic processes which satisfy\nfor all $ n \\in \\mathbb{N} $, $ t \\in [0,T] $ that\n$\\left[\\mathcal{O}_t^n \\right]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)A} \\, dW_s$\nand \n\\begin{equation}\\label{eq:set:cahn}\n\\P \\Big( \\mathcal{X}_t^n = P_n \\, e^{ t A } \\xi + \\smallint_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds + \\mathcal{O}_t^n \\Big)=1. \n\\end{equation}\n\n\\subsubsection{Properties of the nonlinearity}\n\nIn the results in this subsection, Lemmas~\\ref{coer:cahn}--\\ref{nonlin:cahn} below,\nwe collect a few elementary properties of the function $F$ in Subsection~\\ref{setting:cahn} above.\n\n\\begin{lemma}\n\\label{coer:cahn}\nAssume the setting in Subsection~\\ref{setting:cahn} and let $r \\in (\\nicefrac{1}{4}, \\infty)$, $v, w \\in H_r$. Then\n\\begin{align}\n\\begin{split}\n\\left< v, F( v + w ) \\right>_H &\\leq \\max\\big\\{6, c_1+ |c_1|^2+|c_2|^2\\big\\}\\big[\\|v\\|_H^2 + 1+ \\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^{\\max\\{6, c_1+ |c_1|^2+|c_2|^2\\}}\\big].\n\\end{split}\n\\end{align}\n\\begin{proof}[Proof of Lemma~\\ref{coer:cahn}]\nNote that the fact that $\\forall \\, a,b \\in \\mathbb{R} \\colon a^2 + 3ab+3b^2 \\geq 0$ and the Cauchy-Schwartz inequality ensure that\n\\begin{align}\n\\begin{split}\n&\\left< v, F( v + w ) \\right>_H = \\left< v, c_1( v + w ) - c_2( v + w )^3 \\right>_H = c_1 \\|v\\|_H^2 + c_1 \\left_H -c_2 \\left< v, ( v + w )^3 \\right>_H\\\\\n&=c_1 \\|v\\|_H^2 + c_1 \\left_H -c_2 \\int_0^1 [\\underline{v}(x)]^2 \\big( [\\underline{v}(x)]^2 + 3 \\, \\underline{v}(x) \\underline{w}(x) + 3 \\, [\\underline{w}(x)]^2\\big) + \\underline{v}(x) [\\underline{w}(x)]^3 \\, dx\\\\\n& \\leq c_1 \\|v\\|_H^2 + c_1 \\left_H -c_2 \\int_0^1 \\underline{v}(x) [\\underline{w}(x)]^3 \\, dx\\\\\n& \\leq c_1 \\|v\\|_H^2 + |c_1| \\|v\\|_H \\|w\\|_H + c_2 \\|v\\|_H \\|w^3\\|_H.\n\\end{split}\n\\end{align}\nThe fact that $\\forall \\, a, b \\in \\mathbb{R} \\colon ab \\leq a^2 + b^2$ and the fact that $\\forall \\, a \\in \\mathbb{R}, \\, b \\in [1, \\infty) \\colon a \\leq 1+|a|^b$ hence prove that\n\\begin{align}\n\\begin{split}\n\\left< v, F( v + w ) \\right>_H & \\leq c_1 \\|v\\|_H^2 + |c_1|^2\\|v\\|_H^2 + \\|w\\|_H^2 + |c_2|^2\\|v\\|_H^2 + \\|w^3\\|_H^2 \\\\\n& \\leq (c_1+ |c_1|^2+|c_2|^2) \\|v\\|_H^2 + \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^2\\big]+ \\big[\\!\\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^6\\big]\\\\\n& \\leq (c_1+ |c_1|^2+|c_2|^2) \\|v\\|_H^2 + 2 \\, \\big( 1+ \\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^{\\max\\{6, c_1+ |c_1|^2+|c_2|^2\\}} \\big)\\\\\n& \\leq \\max\\big\\{6, c_1+ |c_1|^2+|c_2|^2\\big\\}\\big[\\|v\\|_H^2 + 1+ \\sup\\nolimits_{x \\in (0,1)} |\\underline{w}(x)|^{\\max\\{6, c_1+ |c_1|^2+|c_2|^2\\}}\\big].\n\\end{split}\n\\end{align}\nThe proof of Lemma~\\ref{coer:cahn} is thus completed.\n\\end{proof}\n\\end{lemma}\n\n\\begin{lemma}\\label{nonlin:cahn}\nAssume the setting in Subsection~\\ref{setting:cahn} and let $v, w \\in H_{\\nicefrac{1}{6}}$. Then it holds that $F \\in \\mathcal{C}(H_{\\nicefrac{1}{6}}, H)$ and\n\\begin{align}\n\\label{eq:cahn-cont}\n\\begin{split}\n&\\|F(v) - F(w) \\|_{H} \\\\\n& \\leq \\left(\\frac{|c_1|}{(c_0 \\pi^2)^{\\nicefrac{1}{6}}}+ 2 \\, c_2 \\!\\left[\\sup_{u \\in H_{ \\nicefrac{1}{6}} \\setminus \\{0\\}} \\frac{\\|u\\|^3_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^3_{H_{\\nicefrac{1}{6}}}} \\right] \\right)\\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{6}}} < \\infty.\n\\end{split}\n\\end{align}\t\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{nonlin:cahn}]\nFirst, observe that\n\\begin{align}\\label{v-w:cahn}\n\\begin{split}\n\\|v-w \\|_{H} &\\leq \\|(-A)^{\\nicefrac{-1}{6}} \\|_{L(H)} \\|v-w \\|_{H_{\\nicefrac{1}{6}}} = (c_0 \\pi^2)^{\\nicefrac{-1}{6}} \\|v-w \\|_{H_{\\nicefrac{1}{6}}} \\\\\n& \\leq (c_0 \\pi^2)^{\\nicefrac{-1}{6}} \\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 \\big) \\|v-w \\|_{H_{\\nicefrac{1}{6}}}.\n\\end{split}\n\\end{align}\nIn addition, note that H\\\"older's inequality implies that\n\\begin{align}\n\\begin{split}\n&\\|v^3 -w^3\\|_H = \\|(v-w) \\cdot (v^2 + v \\cdot w + w^2)\\|_{L^2(\\lambda_{(0,1)}; \\mathbb{R})}\\\\\n&\\leq \\|v-w\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})} \\|v^2+v \\cdot w+w^2\\|_{L^3(\\lambda_{(0,1)}; \\mathbb{R})}\\\\\n& \\leq \\|v-w\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})} \\big(\\|v\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}^2 + \\|v\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})} \\|w\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})} + \\|w\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}^2\\big)\\\\\n& \\leq 2\\, \\|v-w\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})} \\big(\\|v\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}^2 + \\|w\\|_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}^2\\big)\\\\\n& \\leq 2 \\left[\\sup_{u \\in H_{\\nicefrac{1}{6} } \\setminus \\{0\\}} \\frac{\\|u\\|^3_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^3_{H_{\\nicefrac{1}{6}}}} \\right] \\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{6} }}.\n\\end{split}\n\\end{align}\nCombining this with \\eqref{v-w:cahn} shows that\n\\begin{align}\n\\label{nonlin:cahn:last}\n\\begin{split}\n& \\|F(v) - F(w) \\|_{H} = \\|c_1(v-w)-c_2(v^3-w^3) \\|_{H} \\leq |c_1| \\|v-w\\|_H+ c_2 \\|v^3-w^3\\|_H \\\\\n& \\leq \\left(\\frac{|c_1|}{(c_0 \\pi^2)^{\\nicefrac{1}{6}}}+ 2 \\, c_2 \\!\\left[\\sup_{u \\in H_{ \\nicefrac{1}{6}} \\setminus \\{0\\}} \\frac{\\|u\\|^3_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^3_{H_{\\nicefrac{1}{6}}}} \\right] \\right)\\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{6}}} .\n\\end{split}\n\\end{align}\nFurthermore, observe that the Sobolev embedding theorem proves that\n\\begin{align}\n\\sup_{u \\in H_{ \\nicefrac{1}{6} }\\setminus \\{0\\}} \\frac{\\|u\\|^3_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^3_{H_{\\nicefrac{1}{6}}}} < \\infty.\n\\end{align}\nThis and \\eqref{nonlin:cahn:last} establish \\eqref{eq:cahn-cont}.\nThe proof of Lemma~\\ref{nonlin:cahn} is thus completed.\n\\end{proof}\n\n\\subsubsection{Strong convergence}\n\n\n\\begin{cor}\n\\label{cahn:cor:last}\nAssume the setting in Subsection~\\ref{setting:cahn} and let $ X\\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ be a stochastic process with continuous sample paths which satisfies for all $t \\in [0, T]$ that $ [X_t ]_{\\P, \\mathcal{B}(H)} = [ e^{ t A } \\xi + \\smallint_0^t e^{ ( t - s ) A} \\, F ( X_s ) \\, ds ]_{\\P, \\mathcal{B}(H)} + \\int_0^t e^{(t-s)A} \\, dW_s$. Then it holds for all $p \\in (0, \\infty)$ that\n\\begin{align}\n\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[ \\| X_t -\\mathcal{X}_t^n \\|_H^p \\big] = 0.\n\\end{align}\n\\end{cor}\n\\begin{proof}[Proof of Corollary~\\ref{cahn:cor:last}]\nThroughout this proof\nlet $(\\tilde{P}_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H_{-1}) $\nbe the function which satisfies\nfor all $ n \\in \\mathbb{N} $, $ v \\in H $ that\n$ \\tilde{P}_n(v) = P_n(v) $\nand let $ \\phi , \\Phi \\colon H_1 \\to [0,\\infty) $ be the functions which satisfy for all $ v \\in H_{1}$ that $\\phi(v) = \\max\\{6, c_1+ |c_1|^2+|c_2|^2\\} \\big[1+\\sup\\nolimits_{x \\in (0,1)} |\\underline{v}(x)|^2\\big]$ and $\\Phi(v)= \\max\\{6, c_1+ |c_1|^2+|c_2|^2\\} \\big[1+ \\sup\\nolimits_{x \\in (0,1)} |\\underline{v}(x)|^{ \\max\\{6, c_1+ |c_1|^2+|c_2|^2\\}}\\big]$.\nObserve that Lemma~\\ref{coer:cahn} ensures for all $n \\in \\mathbb{N}$, $v, w \\in \\tilde{P}_n(H) = P_n(H)$ that\n\\begin{align}\\label{cahn:cor:coer}\n\\langle v, \\tilde{P}_n F( v + w ) \\rangle_H = \\left< v, F( v + w ) \\right>_H \\leq \\phi(w) \\| v \\|^2_H+\\Phi( w ).\n\\end{align}\nIn addition, Lemma~\\ref{nonlin:cahn} shows that for all $n \\in \\mathbb{N}$, $v, w \\in \\tilde{P}_n(H)$ it holds that $F \\in \\mathcal{C}(H_{\\nicefrac{1}{6}}, H)$ and\n\\begin{align}\\label{cahn:cor:con}\n\\begin{split}\n&\\|F(v) - F(w) \\|_H \\\\\n& \\leq \\left(\\frac{|c_1|}{(c_0 \\pi^2)^{\\nicefrac{1}{6}}}+ 2 \\, c_2 \\!\\left[\\sup_{u \\in H_{ \\nicefrac{1}{6}} \\setminus \\{0\\}} \\frac{\\|u\\|^3_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^3_{H_{\\nicefrac{1}{6}}}} \\right] \\right)\\big( 1+ \\left\\| v \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 + \\left\\| w \\right\\|_{ H_{\\nicefrac{1}{6} }}^2 \\big) \\left\\| v - w \\right\\|_{ H_{\\nicefrac{1}{6}}} < \\infty.\n\\end{split}\n\\end{align}\nCombining \\eqref{cahn:cor:coer}--\\eqref{cahn:cor:con} with Proposition~\\ref{abs:prop:last}\n(with\n$T=T$,\n$c_0 = c_0$,\t\n$\\gamma=\\max\\{6, c_1+ |c_1|^2+|c_2|^2\\}$,\t\n$ \\theta= \\nicefrac{|c_1|}{(c_0 \\pi^2)^{\\nicefrac{1}{6}}}+ 2 \\, c_2 [\\sup_{u \\in H_{ \\nicefrac{1}{6} }\\backslash \\{0\\}} \\nicefrac{\\|u\\|^3_{L^6(\\lambda_{(0,1)}; \\mathbb{R})}}{\\|u\\|^3_{H_{\\nicefrac{1}{6}}}} ]$,\n$\\vartheta=2$,\n$\\alpha= 0$,\n$\\varphi= 0$,\n$\\rho= \\nicefrac{1}{6}$,\n$\\varrho =\\varrho$, \n$\\chi = \\chi$,\n$\\xi = \\xi$,\n$F=F|_{H_{\\varrho}}$,\n$(P_n)_{n \\in \\mathbb{N}} = (\\tilde{P}_n)_{n \\in \\mathbb{N}}$,\n$(h_n)_{n \\in \\mathbb{N}} = (h_n)_{n \\in \\mathbb{N}}$,\n$\\phi=\\phi$, $\\Phi=\\Phi$, $(\\mathcal{X}^n)_{n \\in \\mathbb{N}} = (\\mathcal{X}^n)_{n \\in \\mathbb{N}}$, \n$ (\\mathcal{O}^n)_{n \\in \\mathbb{N}} = (\\mathcal{O}^n)_{n \\in \\mathbb{N}}$, $X=X$ in the notation of Proposition~\\ref{abs:prop:last})\ncompletes the proof of Corollary~\\ref{cahn:cor:last}. \n\\end{proof}\n\n\nThe next result, Corollary~\\ref{cor:cahn:short} below,\nproves strong convergence for the stochastic Allen-Cahn equation, also see Remark~\\ref{rem:allen-cahn} below.\nCorollary~\\ref{cor:cahn:short} follows immediately from Corollary~\\ref{cahn:cor:last} above.\n\n\\begin{cor}\\label{cor:cahn:short}\nConsider the notation in Subsection~\\ref{sec:notation},\nlet $ T \\in (0,\\infty)$, \n$\\varrho \\in (\\nicefrac{1}{6}, \\nicefrac{1}{4})$,\n$ \\chi \\in (0, \\nicefrac{\\varrho }{3 } - \\nicefrac{1}{18}] $,\n$( H, \\left< \\cdot , \\cdot \\right>_H, \\left\\| \\cdot \\right\\|_H ) = (L^2(\\lambda_{(0,1)}; \\mathbb{R}), \\langle \\cdot , \\cdot \\rangle_{L^2(\\lambda_{(0,1)}; \\mathbb{R})}, \\left\\| \\cdot \\right\\|_{L^2(\\lambda_{(0,1)}; \\mathbb{R})} )$,\nlet $ A \\colon D(A) \\subseteq H \\to H $ be the Laplace operator with Dirichlet boundary conditions on $H$,\nlet $ ( H_r, \\left< \\cdot , \\cdot \\right>_{ H_r }, \\left\\| \\cdot \\right\\|_{ H_r } ) $, $ r \\in \\mathbb{R} $, be a family of interpolation spaces associated to $ -A $,\nlet\n$\\xi \\in H_{\\nicefrac{1}{2}} $,\nlet\n$F \\colon H_{\\nicefrac{1}{6}} \\to H$,\n$(e_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to H $,\n$(P_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to L(H) $,\nand\n$(h_n)_{n \\in \\mathbb{N}} \\colon \\mathbb{N} \\to (0, T] $\nbe functions which satisfy\nfor all $v \\in H_{\\nicefrac{1}{6}}$, $n \\in \\mathbb{N}$ that\n$F(v)= v- v^3$,\n$ e_n = [ (\\sqrt{2} \\sin(n \\pi x) )_{x \\in (0,1)}]_{\\lambda_{(0,1)} , \\mathcal{B}(\\mathbb{R})}$,\n$ P_n(v) = \\sum_{k = 1 }^n \\langle e_k, v \\rangle_H e_k $,\nand\n$ \\limsup_{ m \\to \\infty} h_m =0$,\nlet $ ( \\Omega, \\F, \\P ) $ be a probability space,\nlet $(W_t)_{t \\in [0, T]}$ be an $\\mathrm{Id}_H$-cylindrical $( \\Omega, \\F, \\P )$-Wiener process,\nlet $ \\mathcal{X}^n, \\mathcal{O}^n \\colon [0, T] \\times \\Omega \\to P_n(H)$, $ n \\in \\mathbb{N}$, be stochastic processes, let $ X\\colon [0, T] \\times \\Omega \\to H_{\\varrho}$ be a stochastic process with continuous sample paths, and assume for all $ n \\in \\mathbb{N} $, $t \\in [0, T]$ that $ [X_t ]_{\\P, \\mathcal{B}(H)} = [ e^{ t A } \\xi + \\smallint_0^t e^{ ( t - s ) A} \\, F ( X_s ) \\, ds ]_{\\P, \\mathcal{B}(H)} + \\int_0^t e^{(t-s)A} \\, dW_s$, $\\left[\\mathcal{O}_t^n \\right]_{\\P, \\mathcal{B}(H)} = \\int_0^t P_n \\, e^{(t-s)A} \\, dW_s$,\nand\n\\begin{equation}\\label{eq:short:cahn}\n\t\\P \\Big( \\mathcal{X}_t^n = P_n \\, e^{ t A } \\xi + \\smallint_0^t P_n \\, e^{ ( t - s ) A } \\, \\one_{ \\{ \\| \\mathcal{X}_{ \\lfloor s \\rfloor_{h_n} }^n \\|_{ H_{\\varrho} } + \\| \\mathcal{O}_{ \\lfloor s \\rfloor_{h_n} }^n +P_n \\, e^{ \\lfloor s \\rfloor_{ h_n } A } \\xi \\|_{ H_{\\varrho} } \\leq | h_n|^{ - \\chi } \\}} \\, F \\big( \\mathcal{X}_{ \\lfloor s \\rfloor_{ h_n } }^n \\big) \\, ds + \\mathcal{O}_t^n \\Big)=1. \n\\end{equation} Then it holds for all $p \\in (0, \\infty)$ that\n\\begin{align}\n\t\\limsup_{n \\to \\infty} \\sup_{t \\in [0,T]} \\mathbb{E} \\big[ \\| X_t -\\mathcal{X}_t^n \\|_H^p \\big] = 0.\n\\end{align}\n\\end{cor}\n\n\n\\begin{remark}\\label{rem:allen-cahn}\nConsider the setting in Corollary~\\ref{cor:cahn:short}.\nRoughly speaking, Corollary~\\ref{cor:cahn:short}\nreveals that the full-discrete explicit numerical approximation scheme described by \\eqref{eq:short:cahn}\nconverges strongly to a mild solution process of the stochastic Allen-Cahn equation \n\\begin{align}\n\\tfrac{\\partial}{\\partial t} X_t(x) = \\tfrac{\\partial^2}{\\partial x^2} X_t(x) + X_t(x) - [X_t(x)]^3 + \\tfrac{\\partial}{\\partial t} W_t(x)\n\\end{align}\nwith $X_0(x) = \\xi(x)$ and $X_t(0)= X_t(1) = 0$ \nfor $x \\in (0,1)$, $t \\in [0,T]$\n(cf., e.g., Kov\\'acs, Larsson, \\& Lindgren~\\cite[Section~1]{Kovacs2015}).\n\\end{remark}\n\n\n\\section*{Acknowledgements}\n\nMario Hefter is gratefully acknowledged for bringing a few typos into our notice.\nThis project has been partially supported through the ETH Research Grant \\mbox{ETH-47 15-2}\n``Mild stochastic calculus and numerical approximations for nonlinear stochastic evolution equations with L\\'evy noise''.\n\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdkqh b/data_all_eng_slimpj/shuffled/split2/finalzzdkqh new file mode 100644 index 0000000000000000000000000000000000000000..6731882f0c38d58a2f8d3dfc47d28f956b8ce03e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdkqh @@ -0,0 +1,5 @@ +{"text":"\\section*{Introduction}\n\nTechnological innovation has lead to a world full of data of an increasingly\ngrowing dimension. These data in turn contain information, the extraction\nof which is a basic task of Statistics (c.f. \\citet{lindsay2004}).\nAn important type of information is the kind of interdependence among\nvariables being represented by data. This calls for statistical means\nof extracting, quantifying and, if possible, modeling such interdependence.\nAt the very least, coefficients that somehow summarize the type and\nintensity of multivariate interdependence are very desirable in applied\nscience.\n\nThe introduction of the correlation concept by Francis Galton (1822-1911)\nhad a tremendous impact on many sciences due to its straightforward\ninterpretation as a measure of ``partial causation'' or ``average\nassociation'', as summarized in a single parameter. However elementary\nthis concept now may seem, it was welcomed as an important scientific\ncontribution at the end of the XIX century. As \\citet{pearson_life_2011}\nwrites, it did ``open to quantitative analysis wide fields of medical,\npsychological and social research {[}...{]}, {[}it{]} was to replace\nnot only in the minds of many of us the old category of causation,\nbut deeply to influence our outlook on the universe''. \n\nKarl Pearson would develop the original correlation coefficient of\nGalton into the widely used product-moment correlation coefficient.\nGiven the first paradigmatic step, new implementations of the concept\nwould appear, in the form of other association coefficients more adequate\nfor specific applications in psychology and the social sciences: Spearman's\n$\\rho$, Kendall's $\\tau$, Ginni's $\\gamma$, Blonqvist's $q$, etc.\n(The reader is referred to \\citet{joe_relative_1989} for more coefficients).\nThese are coefficients intended to represent the degree of association\nbetween two random variables.\n\nLater on, \\citet{renyi_measures_1959} would attempt to give some\nmathematical rigor to the concept of dependence, providing ``seven\nrather natural postulates which should be fulfilled by a suitable\nmeasure of dependence''. R\\'enyi's work would be revised by \\citet{schweizer_nonparametric_1981},\nwho made some ``reasonable modifications'' to the postulates, since\nthey were found to be too restrictive. Additionally, \\citet{schweizer_nonparametric_1981}\nused the concept of copulas to introduce a number of measures of pair-wise\ndependence which fulfilled their new postulates. With these conceptual\ntools (i.e. a set of reasonable postulates and the unifying concept\nof copulas), \\citet{wolff1980n} extends the measures of dependence\nbetween two variables given by \\citet{schweizer_nonparametric_1981},\nand proposes an extension to more than two variables of Spearman's\n$\\text{\\ensuremath{\\rho}}$. This course of action has been further\nfollowed and developed by \\citet{jaworski_copula-based_2010}. Indeed,\n\\citet{jaworski_copula-based_2010} introduce a series of measures\nthat can be considered as extension to more that two variables of\nsome of the well-established, pair-wise measures of dependence mentioned\nabove.\n\nAnother course of action, traceable back to \\citet{linfoot1957informational},\nis to use entropy or mutual information as association coefficient.\n\\citet{joe_relative_1989} proposes a number of measures of this type\nthat apply to more than two variables; \\citet{pena2007dimensionless}\nintroduce a measure which adjusts itself to dimension, so as to compare\nthe intensity of association of two vectors of different dimensions.\n\\citet{Micheas2006765} deal with the general case of $\\varphi$-dependence,\nof which mutual information is one particular case. The intensity\nof association is measured by \\citet{Micheas2006765} in terms of\nthe deviance of the joint distribution from the distribution given\nby the product of the marginal distributions (the independence case).\nThe specific definition of deviance depends on the specific selection\nof function $\\varphi:\\left[0,+\\infty\\right)\\rightarrow\\mathbb{R}$,\nwhich is continuous and convex, satisfying some basic conditions. \n\nApart from their theoretical interest, measures of dependence for\nmore than two variables are required and sought in applied research.\nIn the area of \\emph{neuronal science}, an influential theory of behavior\nintroduced by \\citet{hebb_organisation_1949} suggests that ``fundamental\ninsight into the nature of neuronal computation requires the understanding\nof the cooperative dynamics of populations of neurons'' (\\citet{grun_analysis_2010},\nchapter 12), and further evidence in the course of the years has lead\nbrain theorists to build models that ``rely on groups of neurons,\nrather than single nerve cells, as the functional\\emph{ }building\nblocks for representation and processing of information'' (\\citet{grun_analysis_2010},\npreface); this has lead to the development of techniques to quantify\nbeyond pair-wise association in that research area. Concerning applied\n\\emph{atmospheric research}, \\citet{bardossy_copula_2009} in the\ncontext of daily precipitation modeling, and \\citet{bardossy_multiscale_2012}\nin the context of downscaling, have found evidence that explicit quantification\nof interactions among more than two variables, and their proper incorporation\ninto modeling and forecasting, may be of an importance hitherto unexplored:\npredictions based on statistical models can be otherwise severely\nbiased, particularly for very high (or extreme) values of the multivariate\nprocess modeled. More recently, \\citet{ellipticalSpatialRodriguezBardossy}\ninvestigate the consequences for inference of ignoring multivariate\ninterdependence in the context of Spatial Statistics, and propose\na model that can deal with this type of interdependence explicitly.\nThe present paper comprises the theoretical basis for the work of\n\\citet{ellipticalSpatialRodriguezBardossy}. In the area of \\emph{finance},\n``herding'' behavior, the degree to which several economic actors\nbehave as a herd (\\citet{Dhaene2012357}), doing basically the same\nthing, is important for estimation of loss risks: If a single underlying\nfactor or small number of factors are inducing a high degree of herd\nbehavior, financial assets practically independent or very loosely\ncorrelated with each other can interact \\emph{en bloc}, rendering\nportfolio diversification ineffective. As \\citet{Dhaene2012357} indicate,\npair-wise correlations, or a measure based on these, may be misleading\nin this case. \\citet{RePEc:ner:leuven:urn:hdl:123456789\/410070} present\na related interdependence measure for aggregating risks. \n\nIn a recent paper, \\citet{reimherr2013} note that most of the theory\non measures of association has left out the important issue of interpretability\nof the measures for the research at hand. These authors argue (correctly,\nin our opinion) that the lack of interpretability limits the their\nuse as summary tools. This problem is greatly exacerbated if what\nwe intend to quantify or represent is the association among several\nvariables. In a more general manner, the interpretability of parameters\nand coefficients of a statistical model has been considered an important\ncharacteristic of the model by \\citet{cox1995relation}. \n\nThis paper deals with some of the issues inherent in formulating interaction\ncoefficients that pertain to more than two variables. We propose an\napproach for dealing with these issues. Section \\ref{sec:Difficulties-of-defining}\nintroduces some issues that one encounters when dealing with measures\nof interaction for more than two variables. Section \\ref{sec:Interaction-parameters-versus-manifestations}\nstates the approach we suggest for dealing with these issues: to discriminate\nbetween interaction ``parameters'' and interaction ``manifestations''.\nWe illustrate what we mean by the names \\emph{interaction parameter}\nand \\emph{interaction manifestation}. Section \\ref{sec:The-Lancaster-Interaction}\nintroduces joint cumulants and Lancaster Interactions. The relation\nbetween the two is exhibited, and a justification of joint cumulants\nas legitimate extensions to covariance coefficients indicated. Section\n\\ref{sec:Interaction-manifestations-in-terms} exhibits the relation\nbetween joint cumulants and some illustrative interaction manifestations,\nas defined in this paper. Section \\ref{sec:Illustration:-Runoff-to}\nillustrates the ideas presented, in that a specific model is introduced,\nand the ideas of this paper applied to simulated data. In section\n\\ref{sec:Discussion}, a discussion of the results is provided. \n\n\n\\section{\\label{sec:Difficulties-of-defining}Difficulties of defining a measure\nof multivariate interaction}\n\n\n\\subsection{\\label{sub:Interpretability}Interpretability }\n\nFor a two dimensional dataset, interpretability of a dependence coefficient\nis aided by the possibility of plotting the data. One looks at several\ndatasets and computes the respective coefficient of dependence. After\nmany such data sets, one has an idea of what, say, a correlation coefficient\nwith a value of $-0.8$ stands for. This visual aid is still possible\nfor three dimensional datasets, but is not available for higher dimensions.\nAssuming we have a coefficient of interdependence, $\\lambda$, applicable\nto multivariate vectors; how is one supposed to interpret a value\nof $\\lambda\\left(X_{1},X_{2},X_{3},X_{4}\\right)=-0.8$? Can one visualize\na dataset producing such a coefficient, so as to relate it to the\nphenomenon one is investigating? \n\nIt has been claimed that major advances in the science of statistics\nusually occur as a result of the theory-practice interaction (\\citet{box1976}),\nand that the parameters of a model should have clear subject-matter\ninterpretations (\\citet{cox1995relation}). These statements suggest\nthat interaction parameters as mere abstract constructions will not\nfind much application in statistical modeling, unless one can ``paraphrase''\ntheir meaning and relate them to the problem at hand. \n\nOur approach to interaction quantification and modeling consists in\ndiscriminating between interaction \\emph{manifestations} and interaction\n\\emph{parameters}. So, we can focus on quantification and modeling\nof what \\emph{really interests us} about dependence in data (i.e.\nits subject-matter relevant manifestation), while trying to reproduce\nsuch manifestations with as few parameters as possible. \n\n\n\\subsubsection*{Relation to the ``probability inversion'' technique in Probabilistic\nRisk Assessment}\n\nAn analogous approach has found successful application in the area\nof probabilistic risk assessment (\\citet{bedford2001probabilistic}).\nWith the aid of mathematical models, it is often possible to predict\n(approximately) what the consequences of a given event may be. This\nmathematical model has parameters that ideally should be calibrated\non the basis of past data. However, absence of data for certain events\n(e.g. a nuclear accident in a given region) makes the use of expert\nknowledge necessary, whereby the model parameters are to be estimated\non the basis of the experience of a group of experts. Experts usually\ncannot give an adequate direct evaluation of the joint probability\ndistribution of the model parameters, or \\emph{target variables}.\nHence each expert is asked to express his uncertainty judgments in\nterms of \\emph{elicitation variables}, i.e. \\emph{observable quantities\n}within the area of his\/her expertise. A target variable-set for the\nmodel is then recovered, such that the elicitation variables produced\nby the mathematical model look as similar as possible like the elicited\nvariables provided by the experts. This is an inverse problem, labeled\n``probabilistic inversion''. The interested reader is referred for\nmore details to (\\citet{bedford2001probabilistic,Du20061164}) and\nthe references therein.\n\nWe suggest in section \\ref{sec:Interaction-parameters-versus-manifestations}\na course of action that is analogous to ``probabilistic inversion''\nfor the problem of interactions quantification and modeling. \n\n\n\\subsection{\\label{sub:High-parametric-dimensionality}High parametric dimensionality}\n\nA second issue when defining an interaction coefficient, is the issue\nof high dimensionality. As dimension of the random vector under analysis\nincreases, a naive use of interaction coefficients becomes prohibiting.\nFor example, the correlation matrix of a 10-dimensional random vector\nis an array having $45$ correlation coefficients. Assume symmetry\non the variables with respect to the association coefficient (i.e.\nthe order of the variables plays no role on the coefficient's value):\nIf, for the same 10-dimensional vector, one intends to consider 3-wise,\n4-wise and 5-wise \\textquotedbl{}correlation coefficients\\textquotedbl{},\nthe corresponding arrays would have 450, 4500, and 45000 coefficients,\nrespectively. \n\nHence, it is necessary to be able to select judiciously the interaction\nparameters with which to work, and impose reasonable constraints on\nthem.\n\nAnother aspect that can be considered a sort of ``curse'' of dimensionality,\nis the coefficient of interdependence to use: there are too many features\nthat multivariate datasets can exhibit. \n\nIn the one-dimensional case, parameters such as mean, standard deviation,\nskewness and kurtosis (basically, the first four cumulants) give a\nlot of information about the distribution of data, provided these\ndata come from an unimodal distribution. Those parameters (mean, skewness\ncoefficient, etc.) describe data to some extent, since they can be\nreadily connected to specific questions about data: the location of\ndata, how informative this location about data is, how symmetric the\ndistribution is, to what extend can one expect values very far away\nfrom the mean. As a reference one may have in mind these characteristics\nfor the normal distribution. \n\nBut as dimension grows, one must focus on that feature of data interaction\nwhich is most connected with the research questions at hand, rather\nthan on an abstract dependence coefficient.\n\n\n\\subsubsection{How high dimensionality is dealt with in the realm of Spatial Statistics}\n\nWe present now an example of how the issue of high dimensionality\nhas been addressed in the context of Spatial Statistics. This will\ngive us a basis method from which to generalize. \n\nIn the area of spatial statistics (see, for example \\citet{cressie_statistics_1991,cressie2011statistics,diggle2007model}),\nthe studied random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$ spans hundreds\nor thousands of components, each of which component represents the\nvalue of an environmental process at a given location $j=1,\\ldots,J$.\nThe way high dimensionality is addressed in spatial statistics is\nan apt introduction for the method we advocate in this paper. We give\nhere a very basic form of a spatial statistical model, since it suffices\nfor our introducing purposes.\n\nOne focuses on the correlation between every two components of $\\mathbf{X}$.\nThe covariance among every two components, $\\left(X_{i},X_{j}\\right)$,\nof $\\mathbf{X}$, is expressed as a function $Cov\\left(d\\right)$\nof the distance between the locations represented by these two components,\n$d\\geq0$. The covariance function $Cov\\left(d\\right)$ must be such\nthat the resulting covariance matrix is positive definite. To this\nend there are a number of covariance functions often used in practice,\nfor example, one popular covariance function is the powered exponential\none, \n\\begin{equation}\nCov\\left(d\\right)=\\sigma_{0}^{2}.I\\left(d=0\\right)+\\sigma_{1}^{2}\\exp\\left(-\\left(d\/\\theta_{1}\\right)^{\\theta_{2}}\\right)\\label{eq:Power-exp_cov}\n\\end{equation}\nwhere $\\theta_{1}>0$, $0<\\theta_{2}\\leq2$, $\\sigma_{0}^{2}\\geq0$,\n$\\sigma_{1}^{2}\\geq0$ are the covariance function parameters.\n\nNote that:\n\\begin{enumerate}\n\\item Function (\\ref{eq:Power-exp_cov}) allows to have the covariance between\nevery two components of $\\mathbf{X}$ as a function of the distance\nbetween the locations these components represent, and only 4 additional\nparameters. In this way, the whole dependence structure of $\\mathbf{X}\\in\\mathbb{R}^{J}$\n(with $J>>2$) is \\emph{low dimensionally} obtained, built on the\nbasis of 2-dimensional dependence coefficients. \n\\item The interesting \\emph{dependence manifestation} to recover is covariance\nbetween every two components of $\\mathbf{X}$, whereas the (interaction)\nparameters to estimate are the function parameters, $\\theta_{1},\\theta_{2},\\sigma_{0}^{2}$\nand $\\sigma_{1}^{2}$. This is entirely analogous to the probability\ninversion technique mentioned in section \\ref{sub:Interpretability}:\ncovariance takes the place of the elicitation variables, whereas the\ncovariance function parameters are the target variables.\n\\item There is a functional relation between $\\theta_{1},\\theta_{2},\\sigma_{0}^{2},\\sigma_{1}^{2}$\nand the dependence manifestation. Covariance can be written in terms\nof the (interaction) parameters, $\\theta_{1},\\theta_{2},\\sigma_{0}^{2}$\nand $\\sigma_{1}^{2}$.\n\\end{enumerate}\nItems 1 through 3 summarize a technique to tackle the problem of high\ndimensionality in an ingenious low-dimensional way. The issue of interpretability\ngoes relatively unnoticed, since in this case parameters have a relatively\nstraightforward interpretation: $\\sigma_{0}^{2}$ represents a micro\nscale variability of the environmental process; $\\sigma_{0}^{2}+\\sigma_{1}^{2}$\nrepresents the variance of the marginal distribution of each component\n$X_{j}$ of $\\mathbf{X}$; $\\theta_{1}$ (often called ``range'')\nrepresent the distance at which correlation between data from two\nlocations is relatively insignificant. The parameter $\\theta_{2}$\nmight even receive a suitable interpretation, depending on the context.\n\nIn the next section, this approach is extended to deal with the interdependence\namong more than two variables at a time, keeping basically the same\nideas.\n\n\n\\section{\\label{sec:Interaction-parameters-versus-manifestations}Interaction\nparameters versus interaction manifestations}\n\nThe approach we advocate in this paper can be summarized as follows:\nfirst select an interaction ``manifestation'' relevant for the research\nin question. Then fit (low-dimensional) interactions ``parameters''\nthat make the fitted distribution reproduce, as close as possible,\nthe observed interaction manifestation. In this way, we circumvent\nthe issues of interpretability and high dimensionality mentioned above. \n\nBy interaction manifestation, we mean any function of more than one\ncomponent of the random vector analyzed, $\\mathbf{X}\\in\\mathbb{R}^{J}$,\nwhich can be interpreted as relevant for the research objectives at\nhand. For the sake of illustration:\n\\begin{enumerate}\n\\item The distribution of the sum of subsets of components of a random vector.\nIn the context of financial analysis, this sum is readily interpreted\nas ``risk'' (see also section \\ref{sec:Illustration:-Runoff-to}\nbelow). \n\\item The joint distribution of subsets of components, or the probability\nof trespassing simultaneously a threshold defined for each component.\nThis is useful in many applications. For example, in the context of\nseries systems reliability, such trespassing probability is the probability\nof ``failure''. \n\\item Differential entropy, any information-based dependence measure, or\nany of the copula-based generalizations to correlation measures studied\nby \\citet{jaworski_copula-based_2010}, of subsets of components.\nDepending on the specific research carried out, these may have subject-matter\ninterpretations, or can readily provide the versed researcher of a\nspecific area with a summary picture of the dependence in the data. \n\\end{enumerate}\nInteraction manifestations are interesting for the problem at hand,\nwe would like our model to reproduce them properly. But they are not\nvery helpful for building a model that integrates them, let alone\na low-dimensional model. \\emph{If we had }interaction parameters or\ncoefficients which:\n\\begin{enumerate}\n\\item Provide us with an idea of the number of variables interacting within\nthe random vector analyzed, $\\mathbf{X}\\in\\mathbb{R}^{J}$.\n\\item Can be somehow (functionally) connected with the interaction manifestations\nthat are interesting for the research carried out. \n\\item Can be built into a parametric or semi-parametric model. This would\nimmediately open up the possibility of a low-dimensional model, via\na judicious selection of assumptions and\/or constraints on the interaction\nparameters.\n\\end{enumerate}\n\\emph{Then we could} proceed, in the manner of an inverse problem,\nas follows:\n\\begin{enumerate}\n\\item We find data-based estimates or approximations to the interesting\ninteraction manifestations\n\\item We fit the interactions parameters so as to match best the observed\ninteraction manifestations\n\\end{enumerate}\nIn the next section, we introduce a reasonable interaction measure,\nand through it, a reasonable type of interaction parameter with which\none can work along the lines above; namely the joint cumulant. We\nclaim that using joint cumulants as building blocks of a multivariate\nstatistical model allows for an adequate consideration of dependence,\nboth of pairs of variables, and of groups of more variables. \n\nIt might be argued that moments (and hence cumulants) of sufficiently\nhigh orders might not exist for the ``true'' probability distribution\nof the process under analysis. We would answer that such distributions\ncan always be sufficiently (i.e. for practical purposes) approximated\nby a distribution with existing moments of all orders. See, for example\n\\citet{1987}, where the authors introduce a semi-parametric model,\nsimilar to an Edgeworth expansion. This model possesses moments of\nall orders. Yet, under minimal conditions it can approximate \\emph{any}\ncontinuous distribution on $\\mathbb{R}^{J}$, provided sufficiently\nmany factors are added to the sum defining the model. Additionally,\n\\citet{del_brio_gramcharlier_2009,mauleon2000testing,perote_multivariate_2004}\npresent variants of the model of \\citet{1987}, and show how they\ncan be effectively applied to modeling heavy tailed data, both univariate\nand multivariate. \n\n\n\\section{\\label{sec:The-Lancaster-Interaction}The Lancaster Interaction Measure\nand Joint Cumulants}\n\nIn this section, the connection between the Lancaster Interaction\nmeasure of a random variable and its joint cumulants is established.\nTo our knowledge, this connection has not been pointed out before\nas a justification of joint cumulants as reasonable interdependence\nparameters.\n\n\n\\subsection{A review of Lancaster Interactions}\n\nWe review now the function called ``additive interaction measure''\nor ``Lancaster interaction measure'', introduced by \\citet{lancaster_chi-squared_1969}\nand later modified by \\citet{streitberg_lancaster_1990}. This function\ncan be built for every random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$,\nand has the property of being identically zero if any sub-vector of\n$\\mathbf{X}$ is independent of the other. \n\nAn additive interaction measure $\\Delta F$$\\left(\\mathbf{X}\\right)$\nis a signed measure determined by a given distribution $F\\left(\\mathbf{X}\\right)$\non $\\mathbb{R}^{J}$. Its defining characteristic is that it is equal\nto zero for all $\\mathbf{X}\\in\\mathbb{R}^{J}$, if $F\\left(\\mathbf{X}\\right)$\ncan be written as the non-trivial product of two or more of its (multivariate)\nmarginal distributions (\\citet{streitberg_lancaster_1990}). For example,\nif $J=4$ and $F$ can be written as $F_{124}F_{3}$, being $F_{124}$\nand $F_{3}$ the marginal distributions of $\\left(X_{1},X_{2},X_{4}\\right)$\nand $X_{3}$, respectively, then $\\Delta F\\left(\\mathbf{X}\\right)\\equiv0$,\nfor all $\\mathbf{X}\\in\\mathbb{R}^{J}$. \n\nAn alternative explanation is that $\\Delta F\\equiv0$, if one subset\nof $\\mathbf{X}$'s components is independent of another subset of\ncomponents. If $\\Delta F\\equiv0$, then $F$ is said to be \\textquotedbl{}decomposable\\textquotedbl{}. \n\nLancaster Interaction measure is defined by \n\\begin{equation}\n\\Delta F\\left(\\mathbf{X}\\right)=\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(|\\pi|-1\\right)!\\right)F_{\\pi}\\left(\\mathbf{X}\\right)\\right\\} \\label{eq:Lancaster_measure_definition}\n\\end{equation}\nwhere the sum is over all partitions, $\\pi$, of index set $C=\\left\\{ 1,\\ldots,J\\right\\} $. \n\nAn example will help clarify the notation: for index set $C=\\left\\{ 1,2,3,4\\right\\} $\nthere are 15 partitions, three of which are: $\\pi_{1}=\\left\\{ \\left\\{ 1\\right\\} ,\\left\\{ 2\\right\\} ,\\left\\{ 3,4\\right\\} \\right\\} $,\n$\\pi_{2}=\\left\\{ \\left\\{ 1,4\\right\\} ,\\left\\{ 2,3\\right\\} \\right\\} $,\n$\\pi_{3}=\\left\\{ \\left\\{ 1,2,3,4\\right\\} \\right\\} $. Their cardinalities\nare $\\left|\\pi_{1}\\right|=3$ , $\\left|\\pi_{2}\\right|=2$ and $\\left|\\pi_{3}\\right|=1$,\nrespectively. In general, a set of $J$ elements has a total of $B_{J}$\npossible partitions%\n\\footnote{The number $B_{J}$ is often called Bell's number.%\n}, where $B_{0}=B_{1}=1$ and any subsequent $B_{k>1}$ can be found\n(see e.g. \\citet{rota_number_1964}) by the recurrence relation $B_{k+1}=\\sum_{r=0}^{k}{k \\choose r}B_{r}$.\nThe reader is referred to the textbook of \\citet{AignerDiskrete}\nfor more on partitions and their enumeration.\n\nThe symbol $F_{\\pi_{1}}$ is further to be interpreted as \n\\begin{equation}\nF_{\\pi_{1}}\\left(\\mathbf{X}\\right)=F_{1}\\left(X_{1}\\right)F_{2}\\left(X_{2}\\right)F_{34}\\left(X_{3},X_{4}\\right)\\label{eq:FActorized_distribution_example}\n\\end{equation}\nthat is, the product of the (multivariate) marginal distributions\ndefined by partition $\\pi_{1}$. The same explanation holds at (\\ref{eq:Lancaster_measure_definition})\nfor any of the $B_{J}$ partitions, $\\pi$, of index set $C=\\left\\{ 1,\\ldots,J\\right\\} $. \n\nIt will be convenient to define partition operator $J_{\\pi}$, to\nbe applied to $F$ for a given partition $\\pi$, by \n\\begin{equation}\nJ_{\\pi}F\\rightarrow F_{\\pi}\\label{eq:partition_operator}\n\\end{equation}\nwhere $F_{\\pi}$ is as in the example at equation (\\ref{eq:FActorized_distribution_example}). \n\n\\citet{streitberg_lancaster_1990,streitberg_alternative_1999} shows\nan important result concerning $\\Delta F$: given a probability distribution\nfunction $F$, function $\\Delta F$ as in (\\ref{eq:Lancaster_measure_definition})\nis the \\emph{only} function built as a linear combination of products\nof (multivariate) marginal distributions of $F$, such that $\\Delta F\\left(\\mathbf{X}\\right):=0$,\nwhenever one subset of $\\mathbf{X}$'s components is independent of\nanother components subset. \n\nSince the interaction measure is defined in terms of a given distribution\n$F$, we can define the interaction operator:\n\\begin{equation}\n\\Delta=\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(\\left|\\pi\\right|-1\\right)!\\right)J_{\\pi}\\right\\} \\label{eq:interactions_operator}\n\\end{equation}\nwhich, upon application to the distribution in question, returns the\nadditive interaction measure. \n\n\n\\subsection{A review of Joint Cumulants}\n\nMoments and cumulants can be defined as constants summarizing important\ninformation about a probability distribution and sometimes, even determining\nit completely (cf. \\citet{kendall_advanced_1969}). In this section\nwe deal with random variables having a probability density function.\nThe development is also valid for discreet distributions, under simple\nmodifications. The reader is referred to \\citet{kendall_advanced_1969,muirhead_aspects_1982,billingsley_probability_1986,mccullagh_tensor_1987}\nfor more details on moments and cumulants. \n\nThe Cumulant Generating Function (c.g.f.), $K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)$,\nof a random vector, $\\mathbf{X}\\in\\mathbb{R}^{J}$, is defined as\nthe logarithm of the moment generating function (m.g.f.),\n\\begin{equation}\nK_{\\mathbf{X}}\\left(\\mathbf{t}\\right)=\\log\\left(M_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\right)=E\\left(\\exp\\left(\\sum_{j=1}^{J}t_{j}X_{j}\\right)\\right)\n\\end{equation}\nwhere $\\mathbf{t}\\in\\mathbb{R}^{J}$, assuming these functions exist.\n\nJoint cumulants are then defined to be the coefficients of the Taylor\nexpansion for $K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)$, \n\\begin{equation}\nK_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\sim\\sum_{r_{1=0}}^{\\infty}\\ldots\\sum_{r_{J}=0}^{\\infty}\\frac{\\kappa_{r_{1},\\ldots,r_{J}}.t_{1}^{r_{1}}\\ldots t_{J}^{r_{J}}}{r_{1}!\\ldots r_{J}!}\n\\end{equation}\nand hence can be found by differentiating $K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)$\nand evaluating at $\\mathbf{t}=\\mathbf{0}$,\n\\begin{equation}\n\\kappa_{r_{1},\\ldots,r_{J}}=\\frac{\\partial^{r_{1}+\\ldots+r_{J}}}{\\partial^{r_{J}}t_{J}\\ldots\\partial^{r_{1}}t_{1}}K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\mid_{\\mathbf{t}=\\mathbf{0}}\n\\end{equation}\nwhere $r_{j}\\geq0$ is a non-negative integer. An important particular\ncase is the covariance coefficient, or second order joint cumulant,\n\\[\n\\frac{\\partial^{2}}{\\partial t_{i}\\partial t_{j}}K_{\\mathbf{X}}\\left(t_{i},t_{j}\\right)\\mid_{\\left(t_{i},t_{j}\\right)=\\left(0,0\\right)}=cov\\left(X_{i},X_{j}\\right)\n\\]\n\n\nThe c.g.f. of a sub-vector $\\mathbf{Y}=\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$,\nwith indexes in an index set, $j_{i}\\in I$, can be readily found\nin terms of that of $\\mathbf{X}$, by setting the indexes not corresponding\nto $\\mathbf{Y}$ to zero: \n\\begin{multline*}\nK_{\\mathbf{Y}}\\left(\\mathbf{s}\\right)=\\left(E\\left(\\exp\\left(\\sum_{i=1}^{k}s_{i}X_{j_{i}}\\right)\\right)\\right)=\\log\\left(E\\left(\\exp\\left(\\sum_{j=1}^{J}g_{j}\\left(\\mathbf{s}\\right)X_{j}\\right)\\right)\\right)=K_{\\mathbf{X}}\\left(g\\left(\\mathbf{s}\\right)\\right)\n\\end{multline*}\nwhere $g:\\mathbb{R}^{k}\\rightarrow\\mathbb{R}^{J}$, and \n\\[\ng_{j}\\left(\\mathbf{s}\\right)=\\begin{cases}\n1, & j\\in I\\\\\n0, & j\\notin I\n\\end{cases}\n\\]\n\n\nAn alternative definition for joint cumulants uses product moments\nas departing point (see, for example, \\citet{brillinger_time_1974}).\nLet $\\mathbf{X}\\in\\mathbb{R}^{J}$ be a random vector. For a set $\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)$\nof $\\mathbf{X}$\\textasciiacute{}s components, where some sub-indexes\n$j_{r}$ may be repeated, consider joint moments \n\\[\nE\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)\n\\]\n\n\nConsider partition operator $J_{\\pi}^{*}$, analogous to (\\ref{eq:partition_operator}),\nrelated to each partition $\\pi$ of $\\left(j_{1},\\ldots,j_{d}\\right)$.\nThis operator converts $E\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)$\ninto the product of the factors determined by partition $\\pi$. \n\nFor example, for $d=4$ , $\\left(j_{1},j_{2},j_{3},j_{3}\\right)$\nand $\\pi=\\left\\{ \\left\\{ 1\\right\\} ,\\left\\{ 2,3\\right\\} ,\\left\\{ 4\\right\\} \\right\\} $,\none has partition components $v_{1}=\\left\\{ 1\\right\\} $, $v_{2}=\\left\\{ 2,3\\right\\} $\nand $v_{3}=\\left\\{ 4\\right\\} $. Upon application of $J_{\\pi}^{*}$,\nwe have, \n\\[\nJ_{\\pi}^{*}E\\left(X_{j_{1}}\\ldots X_{j_{4}}\\right)=E\\left(X_{j_{1}}\\right)E\\left(X_{j_{2}}X_{j_{3}}\\right)E\\left(X_{j_{3}}\\right)\n\\]\n\n\nIn the general case\n\n\\[\nJ_{\\pi}^{*}E\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)=\\prod_{v\\in\\pi}E\\left(\\prod_{j_{r}\\in v}X_{j_{r}}\\right)\n\\]\n\n\nThe alternative definition of joint cumulants can now be given.\n\nFor random variables $\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)$, their\njoint cumulant of order \\emph{d} is given by, \n\\begin{multline}\ncum\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right):=\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(\\left|\\pi\\right|-1\\right)!\\right)J_{\\pi}^{*}\\right\\} E\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)\\label{eq:Joint_cumulants_Brillinger}\n\\end{multline}\n\n\nTwo examples are:\n\n\\begin{eqnarray*}\ncum\\left(X_{1},X_{2}\\right) & = & E\\left(X_{1}X_{2}\\right)-E\\left(X_{1}\\right)E\\left(X_{2}\\right)\n\\end{eqnarray*}\nand\n\\begin{multline*}\ncum\\left(X_{1},X_{2},X_{3}\\right)=E\\left(X_{1}X_{2}X_{3}\\right)-E\\left(X_{1}X_{2}\\right)E\\left(X_{3}\\right)-E\\left(X_{1}X_{3}\\right)E\\left(X_{2}\\right)\\\\\n-E\\left(X_{2}X_{3}\\right)E\\left(X_{1}\\right)+2E\\left(X_{1}\\right)E\\left(X_{2}\\right)E\\left(X_{3}\\right)\n\\end{multline*}\n\n\nHence joint cumulants can be seen, from a merely formalistic point\nof view, to form a kind of higher order covariance coefficient. The\nsecond order joint cumulant is just the typical covariance coefficient. \n\n\n\\subsection{Relationship between Lancaster Interactions and Joint Cumulants}\n\nThe similarity between (\\ref{eq:Lancaster_measure_definition}) and\n(\\ref{eq:Joint_cumulants_Brillinger}) is evident. Indeed, if we concentrate\nfor now on the case $\\mathbf{X}\\in\\mathbb{R}^{2}$, then \\citet{lehmann_concepts_1966}\nreports that: \n\\begin{multline}\nCov\\left(X_{1},X_{2}\\right)=cum\\left(X_{1},X_{2}\\right)=\\\\\n\\intop_{-\\infty}^{+\\infty}\\intop_{-\\infty}^{+\\infty}\\left[F_{12}\\left(x_{1},x_{2}\\right)-F_{1}\\left(x_{1}\\right)F_{2}\\left(x_{2}\\right)\\right]dx_{1}dx_{2}\\label{eq:Hoeffding_Formula}\n\\end{multline}\nunder the condition that $E\\left(\\left|X_{1}^{k_{1}}X_{2}^{k_{2}}\\right|\\right)<+\\infty$,\nfor $k_{j}=0,1$.\n\nThis equation is often called \\textquotedbl{}Hoeffding's formula\\textquotedbl{}\nsince it was first discovered by \\citet{hoeffding_masstabinvariante_1940}.\nOf course, the above equation can be written in terms of the Lancaster\ninteraction measure (\\ref{eq:Lancaster_measure_definition}), as\n\\begin{equation}\ncum\\left(X_{1},X_{2}\\right)=\\intop_{-\\infty}^{+\\infty}\\intop_{-\\infty}^{+\\infty}\\Delta F\\left(x_{1},x_{2}\\right)dx_{1}dx_{2}\\label{eq:Hoeffding_Formula2}\n\\end{equation}\n\n\nIt turns out that this equation can be extended to higher dimensions.\nLet $\\mathbf{X}\\in\\mathbb{R}^{J}$ be a random vector. As shown by\n\\citet{block_multivariate_1988}, we have that (page 1808):\n\\begin{equation}\ncum\\left(\\mathbf{X}\\right)=\\left(-1\\right)^{J}\\intop_{-\\infty}^{+\\infty}\\ldots\\intop_{-\\infty}^{+\\infty}\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(\\left|\\pi\\right|-1\\right)!\\right)F_{\\pi}\\right\\} d\\mathbf{X}\\label{eq:cumul_lancaster}\n\\end{equation}\nunder the condition that $E\\left(\\left|X_{j}^{J}\\right|\\right)<+\\infty$,\nfor $j=1,\\ldots,J$. Again, this is the same as saying that\n\n\\begin{equation}\ncum\\left(\\mathbf{X}\\right)=\\left(-1\\right)^{J}\\intop_{-\\infty}^{+\\infty}\\ldots\\intop_{-\\infty}^{+\\infty}\\Delta F\\left(\\mathbf{X}\\right)d\\mathbf{X}\\label{eq:cumul_lancaster-1}\n\\end{equation}\n\n\nThus, joint cumulants are equal (up to a known constant) to the integral\nof Lancaster Interaction measure; they are ``summary'' or ``integral''\nmeasures of additive interaction. To our knowledge, this connection\nhad not been pointed out elsewhere.\n\nIt goes without much explanation that the joint cumulants of a random\nvector $\\mathbf{X}$ vanish whenever a subset of the vector is independent\nof another, since then the integrating function is identically zero.\nThis property is well-known and oftentimes the reason why joint cumulants\nare used in practice (e.g. in \\citet{brillinger_time_1974,mendel_tutorial_1991}).\nThe inverse is true only if the distribution of $\\mathbf{X}$ is determined\nby its moments, which may or may not be a reasonable assumption, depending\non the application. Again, based on the work of \\citet{1987,perote_multivariate_2004,mauleon2000testing,del_brio_gramcharlier_2009},\nwe argue that this is not an extreme limitation to our approach, since\nall we are seeking is a good approximation to the distribution under\nanalysis. \n\nIn particular, whenever we have $cum\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)\\neq0$,\nwhere no index $j_{k}$ is repeated, this means that one cannot decompose\nthe distribution of $\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)$: At\nleast $d$ variables within $\\mathbf{X}$ are interacting simultaneously\nwith each other. \n\nOur theoretical contribution here is that joint cumulants are seen\nas the integral of the Lancaster interaction measure. As shown by\n\\citet{streitberg_lancaster_1990}, $\\Delta F$ is the only additive\nmeasure, built very elementarily with the marginal distributions of\nthe random vector, which vanishes whenever one subset of $\\mathbf{X}$'s\ncomponents is independent of another subset of components.\n\nWe have provided a theoretical basis for declaring joint cumulants\n``interaction parameters'', and the cumulant generating function\na ``dependence structure''. The functional character of the c.g.f.\nopens up the possibility of parametric modeling, with its respective\nlow-dimensionality advantage. It is just another way of defining a\nmodel, alternative to the density specification. \n\nWe shall see below, how the parameters of a model expressed as a c.g.f.\ncan be connected with some interesting interaction manifestations.\n\n\n\\section{\\label{sec:Interaction-manifestations-in-terms}Interaction manifestations\nin terms of interaction parameters}\n\nThe connection between interaction parameters (i.e. joint cumulants)\nand interaction manifestations relies on the concepts of the Edgeworth\nexpansion and the saddlepoint approximation to the density of a random\nvector. A brief review of these topics is provided at the appendix. \n\n\n\\subsection{Connection of dependence structure with interaction manifestations}\n\nWe shall show explicitly the connection of joint cumulants and the\nc.g.f. with three of the interaction manifestations listed at section\n\\ref{sec:Interaction-parameters-versus-manifestations}, which manifestations\nrefer to subsets of components, $\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$,\n$1\\leq k\\leq J$, of the random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$.\nNamely: the distribution of the sum of components; parameters related\nto the joint probability of the components; and the differential entropy\nof the components.\n\nA relevant point here is that, except for the distribution of the\nsum of components, even with a lot of data at hand, estimation of\nthe interaction manifestations mentioned can be done only for (multivariate)\nmarginals of relatively low dimension, such as $k$ equal to 3, 4\nor 5. But armed with a sensible c.g.f., we can consistently integrate\nthese manifestations into the whole distribution (in much the same\nway as thousand of covariance coefficients are integrated into a Spatial\nStatistics model that spans thousands of variables). This we can attain\nwith the aid of the overarching dependence structure, that is, the\nc.g.f. \n\nAssume for the moment you have a reasonable type of c.g.f., that is,\none that seems reasonable for the problem at hand (for an illustration\nsee section \\ref{sec:Illustration:-Runoff-to}). \n\n\n\\subsubsection{\\label{sub:Connection-of-dependence-sumas}Connection of dependence\nstructure with Sums of components}\n\nGiven a random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$ representing\nthe variables under analysis, we are interested in the distribution\nof variable $S_{\\mathbf{X}}=\\sum_{i=1}^{k}X_{j_{i}}$, where $\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$,\n$1\\leq k\\leq J$, is a sub-vector of the random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$.\nThe distribution of $S_{\\mathbf{X}}$ is the interaction manifestation\nwe in which we are interested. We want to fit the distribution of\nthe whole vector, $\\mathbf{X}\\in\\mathbb{R}^{J}$, in such a way the\nwe fit this interaction manifestation properly.\n\n\\emph{One course of action} is to find the cumulants of $S_{\\mathbf{X}}$\nin terms of the joint cumulants of $\\mathbf{X}$, and then approximate\nthe density of $S_{\\mathbf{X}}$, by using the Edgeworth Expansion.\nSince $S_{\\mathbf{X}}$ is a one-dimensional random variable, one\ncan alternatively find research-relevant quantiles of its distribution\nby inverting the Edgeworth Expansion, i.e. by using the Cornish-Fisher\ninversion. \n\nTo find the cumulants of $S_{\\mathbf{X}}$, note that two of the properties\nof joint cumulants are \\citet{brillinger_time_1974}: symmetry and\nmulti-linearity. Symmetry means that $cum\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)=cum\\left(P\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)\\right)$\nfor any permutation $P\\left(j_{1},\\ldots,j_{k}\\right)$ of the indexes\n$\\left(j_{1},\\ldots,j_{k}\\right)$. Concerning multi-linearity, for\nany random variable $Z\\in\\mathbb{R}$, one has\n\\[\ncum\\left(Z+X_{j_{1}},\\ldots,X_{j_{k}}\\right)=cum\\left(Z,\\ldots,X_{j_{k}}\\right)+cum\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)\n\\]\n Combining these two properties, it can be shown that\n\\begin{equation}\n\\kappa_{r}\\left(S_{\\mathbf{X}}\\right)=cum\\left(\\underbrace{S_{\\mathbf{X}},\\ldots,S_{\\mathbf{X}}}_{r}\\right)=\\sum_{i_{1}=1}^{k}\\left[\\sum_{i_{2}=1}^{k}\\ldots\\left[\\sum_{i_{r}=1}^{k}cum\\left(X_{j_{i_{1}}},\\ldots,X_{j_{i_{r}}}\\right)\\right]\\right]\\label{eq:cumulants-of-sums}\n\\end{equation}\nwhere $\\kappa_{r}\\left(S_{\\mathbf{X}}\\right)$ denotes the \\emph{r}-th\ncumulant of random variable $S_{\\mathbf{X}}=\\sum_{i=1}^{k}X_{j_{i}}$.\nThen the interesting quantiles of $S_{\\mathbf{X}}$ can be (approximately)\nwritten in terms of the $\\kappa_{r}$ via the Cornish-Fisher inversion. \n\nAs the dimension $k$ of the sub-vector increases, this approach becomes\nimpractical, since the sum at (\\ref{eq:cumulants-of-sums}) comprises\ntoo many elements. Fortunately, knowing the c.g.f. of $\\mathbf{X}$\ntells much about the c.g.f. of sums of its components.\n\n\\emph{A second course of action} uses all the information provided\nby the c.g.f. and is now given.\n\nIn a somewhat more general context as before, consider a random vector\n$\\mathbf{X}=\\left(X_{1},\\ldots,X_{J}\\right)$. One wishes to study\nthe joint distribution of aggregated variables of the form:\n\\begin{eqnarray}\n\\xi_{1} & = & \\sum_{j_{1}\\in I_{1}}X_{j_{1}}\\nonumber \\\\\n\\xi_{2} & = & \\sum_{j_{2}\\in I_{2}}X_{j_{2}}\\\\\n\\vdots & \\vdots & \\vdots\\nonumber \\\\\n\\xi_{l} & = & \\sum_{j_{l}\\in I_{l}}X_{j_{l}}\n\\end{eqnarray}\n\n\nwhere $I_{k}$, for $k=1,\\ldots,l$ represent non-overlapping index\nsets such that\n\n\\[\nI_{1}\\cup\\ldots\\cup I_{l}=\\left\\{ 1,\\ldots,J\\right\\} \n\\]\n(Note that $S_{\\mathbf{X}}$ above is the specific case in which $I_{1}=\\left\\{ 1,\\ldots,J\\right\\} $).\n\nThe cumulant generating function of the $l$-dimensional vector so\nobtained is given by\n\\begin{multline}\nK_{\\mathbf{\\xi}}\\left(\\mathbf{t}\\right)=\\log\\left(E\\left(\\exp\\left(\\mathbf{t}.\\mathbf{\\xi}^{'}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(t_{1}\\xi_{1}+\\ldots+t_{l}\\xi_{l}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(t_{1}\\sum_{I_{1}}X_{j_{1}}+\\ldots+t_{l}\\sum_{I_{l}}X_{j_{l}}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(g_{1}\\left(\\mathbf{t}\\right)X_{1}+\\ldots+g_{J}\\left(\\mathbf{t}\\right)X_{J}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(g\\left(\\mathbf{t}\\right).\\mathbf{X}^{'}\\right)\\right)\\right)=K_{\\mathbf{X}}\\left(g\\left(\\mathbf{t}\\right)\\right)\\label{eq:cum_gen_fun_suma}\n\\end{multline}\n\n\nFunction $g:\\mathbb{R}^{l}\\rightarrow\\mathbb{R}^{J}$ is a vector\nfunction defined by\n\n\\begin{eqnarray}\ng\\left(\\mathbf{t}\\right) & = & \\left(g_{1}\\left(\\mathbf{t}\\right),\\ldots,g_{J}\\left(\\mathbf{t}\\right)\\right)\\nonumber \\\\\ng_{j}\\left(\\mathbf{t}\\right) & = & \\mathbf{t}.\\left(\\mathbf{1}\\left(j\\in I_{1}\\right),\\ldots,\\mathbf{1}\\left(j\\in I_{l}\\right)\\right)^{'}\\label{eq:transf_cums}\n\\end{eqnarray}\nwhere\n\\[\n\\mathbf{1}\\left(j\\in I_{k}\\right)=\\begin{cases}\n1, & j\\in I_{k}\\\\\n0, & j\\notin I_{k}\n\\end{cases}\n\\]\n\n\nIt is hence possible to find the cumulant generating function of random\nvector $\\mathbf{\\xi}\\in\\mathbb{R}^{l}$ in terms of that of the original\nvector $\\mathbf{X}\\in\\mathbb{R}^{J}$. If we know the c.g.f. of the\noriginal random vector $\\mathbf{X}$, then the cumulants, the cumulant\ngenerating function, and hence the approximate joint density of the\naggregated variables, via Saddlepoint approximation at (\\ref{eq:Saddlepoint})\nof $\\mathbf{\\xi}\\in\\mathbb{R}^{l}$ are also determined (see section\n\\ref{sec:Illustration:-Runoff-to}). We can use this fact in order\nto fit the modeol for $\\mathbf{X}$ in such a way that the interesting\ninteraction manifestation (the sums of components) are explicitly\nconsidered in the estimation. \n\n\n\\subsubsection{Joint probabilities of (multivariate) marginals}\n\nJoint marginal distributions are usually important interaction manifestations.\nGiven a sub-vector $\\mathbf{Y}:=\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$\nof $\\mathbf{X}$, in order to find probabilities of the form \n\\[\n\\Pr\\left(X_{j_{1}}\\geq x_{j_{1}},\\ldots,X_{j_{k}}\\geq x_{j_{k}}\\right)\n\\]\none should in principle integrate expression (\\ref{eq:Saddlepoint}),\nfor the c.f.g. of $\\mathbf{Y}$.\n\nIn the uni-variate case, it is a well-established practice \\citet{Huzurbazar_saddlepoint1999}\nto employ instead an accurate approximation to that integral, which\nis due to \\citet{lugannani_rice1980}. Namely, in the univariate case,\nwe have: \n\\begin{multline}\nF_{X}\\left(x_{0}\\right)\\approx\\intop_{-\\infty}^{x_{0}}\\frac{\\exp\\left(K_{X}\\left(\\hat{\\lambda}\\left(x\\right)\\right)-x\\hat{\\lambda}\\left(x\\right)\\right)}{\\left(2\\pi\\right)^{1\/2}\\left(\\frac{d^{2}K_{X}\\left(\\mathbf{\\lambda}\\right)}{d\\lambda^{2}}\\mid_{\\lambda=\\hat{\\lambda}\\left(x\\right)}\\right)^{1\/2}}dx\\\\\n\\approx\\Phi\\left(r\\right)+\\phi\\left(r\\right)\\left\\{ \\frac{1}{r}-\\frac{1}{q}\\right\\} \\label{eq:Lugganani}\n\\end{multline}\n\n\nWhere $\\hat{\\tau}$ is such that $K_{X}^{'}\\left(\\hat{\\tau}\\right)=x_{0}$,\nand: \n\\begin{eqnarray*}\nr & = & sign\\left(\\hat{\\tau}\\right)\\left\\{ 2\\left[\\hat{\\tau}x_{0}-K_{X}\\left(\\hat{\\tau}\\right)\\right]\\right\\} ^{\\frac{1}{2}}\\\\\nq & = & \\hat{\\tau}\\left\\{ \\frac{d^{2}K_{X}\\left(\\lambda\\right)}{d\\lambda^{2}}\\mid_{\\lambda=\\hat{\\tau}}\\right\\} ^{\\frac{1}{2}}\n\\end{eqnarray*}\n\n\nThus, one must not perform the numerical integration at all. \n\nFor the multivariate case, \\citet{kolassa2010multivariate} have provided\na generalization of the Lugannani-Rice formula, which produces an\napproximation to probability $\\Pr\\left(\\mathbf{Y}\\geq\\mathbf{y}\\right)$\nof order $O\\left(n^{-1}\\right)$, for $\\mathbf{X}\\in\\mathbb{R}^{J}$.\nThis formula is extremely complicated and writing it here will most\nlikely obscure rather than clarify anything. Only the probability\ndistribution function of a multivariate Normal distribution with covariance\nmatrix given by\n\n\\[\n\\Gamma_{ij}=\\frac{\\partial^{2}}{\\partial t_{i}\\partial t_{j}}K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\mid_{\\mathbf{t}=\\mathbf{0}}\n\\]\nmust be computed. For this task there are accurate methods available\nfor up to 20 dimensions \\citet{Genz93comparisonof}.\n\nIf one intends to deal with vectors of dimension at most 5, corresponding\nto multidimensional marginals of the random field modeled, we consider\nmore convenient to use numerical integration of (\\ref{eq:Saddlepoint}).\nFor higher dimensions it would be better to use the result of \\citet{kolassa2010multivariate}\nin order to avoid difficult and inaccurate integrations.\n\n\n\\subsubsection{Differential entropy}\n\nThis also an important interaction manifestation, often encountered\nin statistical research. Using the shorthand notation of \\ref{eq:shorthand_not},\ndefine $Z\\left(\\mathbf{x}\\right):=\\frac{1}{3!}\\kappa^{j_{1},j_{2},j_{3}}h_{j_{1}j_{2}j_{3}}\\left(\\mathbf{x};\\Gamma\\right)$.\n\\citet{barros_2005_2005} studies an approximation to the differential\nentropy of $\\mathbf{X}$, which utilizes only the first correction\nterm in \\ref{eq:edgeworth_series}:\n\n\\begin{multline}\n\\intop f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)\\log\\left(f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)\\right)d\\mathbf{x}=H\\left(\\phi_{\\Gamma}\\right)-\\intop f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)\\log\\left(\\frac{f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)}{\\phi_{\\Gamma}\\left(\\mathbf{x}\\right)}\\right)d\\mathbf{x}\\\\\n\\approx H\\left(\\phi_{\\Gamma}\\right)-\\int\\phi_{\\Gamma}\\left(\\mathbf{x}\\right)\\left(1+Z\\left(\\mathbf{x}\\right)\\right)\\log\\left(1+Z\\left(\\mathbf{x}\\right)\\right)d\\mathbf{x}\\\\\n\\approx H\\left(\\phi_{\\Gamma}\\right)-\\int\\phi_{\\Gamma}\\left(\\mathbf{x}\\right)\\left(Z\\left(\\mathbf{x}\\right)+\\frac{1}{2}Z\\left(\\mathbf{x}\\right)^{2}\\right)d\\mathbf{x}=H\\left(\\phi_{\\Gamma}\\right)-\\frac{1}{12}\\Big\\{\\sum_{j=1}^{J}\\left(k^{j,j,j}\\right)^{2}\\\\\n+3\\sum_{i,j=1,i\\neq j}^{J}\\left(\\kappa^{i,i,j}\\right)^{2}+\\frac{1}{6}\\sum_{i,j,k=1,i1}=0$. In order to avoid identifiability problems\nof the covariance matrix, we set $c_{1}=1$ and declare $\\Gamma$\nto be a true covariance matrix. This model is treated in detail at\n\\citet{ellipticalSpatialRodriguezBardossy}, in the context of spatial\nstatistics; it is shown at \\citet{ellipticalSpatialRodriguezBardossy}\nthat it covers a span of tail dependence going from zero (i.e. Normal)\nto that of the Student-t.\n\n\n\\subsection{Some data}\n\nIn figure \\ref{fig:dataset_Y} an 8-dimensional dataset is presented,\nwith a size of $n=10950$ realization. This dataset may represent\nthe daily (log) return of 8 stocks, or they could represent some daily\nmeasured environmental variable at 8 locations, possibly after transformation.\nIn either case this dataset would amount to a 30 year record. A plot\nof the data appears in figure \\ref{fig:dataset_Y}. We are interested\nin fitting a model that recovers properly the distribution of the\nsum of the components of the 8-dimensional random vector, $S_{\\mathbf{X}}=\\sum_{i=1}^{8}X_{i}$. \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_01}\n\\par\\end{centering}\n\n\\caption{\\label{fig:dataset_Y}8-dimensional test data set}\n\n\n\\end{figure}\n\n\nWe shall employ the model given by c.g.f. (\\ref{eq:archetypal_cgf-1}),\ndue to the shape of data, and to the flexibility of the mentioned\nmodel to represent tail dependence. Specifically, we are interested\nin fitting a model that captures the correlation among the 8 components\nproperly, but \\emph{additionally} provides a good estimation to the\ndistribution of interaction manifestation \n\\begin{equation}\nS_{\\mathbf{X}}=\\sum_{j=1}^{8}X_{j}\n\\end{equation}\n\n\nWe assume for simplicity a mean vector $\\mathbf{m}=\\left(0,\\ldots,0\\right)$\nof zeros (otherwise, data could be standardized to have zero means,\nfirst). As in section \\ref{sub:Connection-of-dependence-sumas}, we\nhave that the c.g.f. of $S_{\\mathbf{X}}$ is given by\n\\begin{equation}\nK_{S_{\\mathbf{X}}}\\left(t\\right)=K_{\\mathbf{X}}\\left(g\\left(t\\right)\\right)=\\frac{c_{1}}{1!}\\left(\\frac{1}{2}g\\left(t\\right)\\Gamma g\\left(t\\right)^{T}\\right)+\\frac{c_{2}}{2!}\\left(\\frac{1}{2}g\\left(t\\right)\\Gamma g\\left(t\\right)^{T}\\right)^{2}+\\frac{c_{3}}{3!}\\left(\\frac{1}{2}g\\left(t\\right)\\Gamma g\\left(t\\right)^{T}\\right)^{3}+\\ldots\\label{eq:CGF_Suma}\n\\end{equation}\nwhere \n\\begin{equation}\ng\\left(t\\right)=\\underbrace{\\left(t,\\ldots,t\\right)}_{8}\n\\end{equation}\n\n\n\n\\subsection{\\label{sub:Parameter-estimation}Parameter estimation}\n\nOur estimating strategy consists of:\n\n\n\\paragraph*{Step 1: Estimate Covariance matrix $\\Gamma$}\n\nIn this way capture much of the 8-dimensional dependence structure.\nSince our model is a member of the elliptical family, we can use the\nestimator for the correlation matrix which uses Kendall's $\\tau$\ncorrelation coefficient (see \\citet{muller_kendalls_2003}),\n\\begin{equation}\n\\hat{cor}\\left(X_{i},X_{j}\\right)=\\sin\\left(\\frac{\\pi}{2}\\tau\\left(X_{i},X_{j}\\right)\\right)\n\\end{equation}\nwhereby a complete correlation matrix, $\\hat{R}$, is obtained.\n\nThen the covariance matrix estimate can be found by \n\\begin{equation}\n\\hat{\\Gamma}=\\Sigma^{\\frac{1}{2}}\\hat{R}\\Sigma^{\\frac{1}{2}}\n\\end{equation}\nwith \n\\[\n\\Sigma=\\left(\\begin{array}{ccc}\nS^{2}\\left(X_{1}\\right) & \\ldots & 0\\\\\n\\vdots & \\ddots & \\vdots\\\\\n0 & \\ldots & S^{2}\\left(X_{8}\\right)\n\\end{array}\\right)\n\\]\nand $S^{2}\\left(X_{j}\\right)$ stands for the sample variance of $X_{j}$.\nThis procedure was followed, resulting in the covariance matrix given\nin table \\ref{tab:Estimated-covariance-for}, at the appendix.\n\nAlternatively, if data represents an environmental variable sampled\nat several locations, standard geostatistical tools can be used to\nestimate $\\Gamma$ (see \\citet{ellipticalSpatialRodriguezBardossy}).\nThe covariance matrix will be in the following considered as known.\n\n\n\\paragraph*{Step 2: Interaction manifestation fitting}\n\nWe do this in a ``method-of-moments'' fashion (method of cumulants,\nshould we say). The $r$-th order cumulant of $S_{\\mathbf{X}}$, $\\kappa_{r}\\left(S_{\\mathbf{X}}\\right)$,\ncan be found by differentiating (\\ref{eq:CGF_Suma}) $r$ times with\nrespect to $t$, and then setting $t=0$. Performing the necessary\ncomputations, one has for the mean and the variance: \n\\begin{eqnarray}\n\\kappa_{1}\\left(S_{\\mathbf{X}}\\right) & = & 0\\\\\n\\kappa_{2}\\left(S_{\\mathbf{X}}\\right) & = & \\frac{c_{1}}{1!}\\frac{2}{2}\\sum_{i,j=1}^{8}\\Gamma_{ij}\n\\end{eqnarray}\nand in general, odd-ordered cumulants will be zero, while even-ordered\ncumulants are given by\n\\begin{equation}\n\\kappa_{2r}\\left(S_{\\mathbf{X}}\\right)=\\frac{c_{r}}{r!}\\frac{\\left(2r\\right)!}{2^{r}}\\left(\\sum_{i_{1},\\ldots,i_{r}=1}^{8}\\sum_{j_{1},\\ldots,j_{r}=1}^{8}\\Gamma_{i_{1}j_{1}}\\ldots\\Gamma_{i_{r}j_{r}}\\right)\\label{eq:cumulantes_de_suma}\n\\end{equation}\n\n\nWe compute the sample cumulants, $\\hat{\\kappa}_{2r}$ (for $r=1,2,3$),\nof $S_{\\mathbf{X}}$. These are found to be 37.426, 463.509 and 105098.112,\nrespectively. Substituting these sample cumulants for the theoretical\ncumulants in (\\ref{eq:cumulantes_de_suma}), and using the already\navailable covariance matrix, $\\Gamma$, we can estimate $c_{1}$,\n$c_{2}$ and $c_{3}$. These estimates are given by $\\hat{c}_{1}=0.999$,\n$\\hat{c}_{2}=0.1101$ and $\\hat{c}_{3}=0.1332$. Note that by considering\ncumulants of $S_{\\mathbf{X}}$ of order $\\geq4$, we can capture important\ntail characteristics of its distribution.\n\n\n\\subsection{Evaluation of the fit}\n\nWe use the Monte Carlo approach to evaluate the fit carried out in\nthe previous sub-section. One can sample from a random vector, $\\mathbf{Y}\\in\\mathbb{R}^{8}$,\nhaving c.g.f. as in (\\ref{eq:archetypal_cgf-1}), by sampling two\nindependent random variables: 1. a non-negative random variable $V>0$,\nwith cumulants $c_{1},\\ldots,c_{r}$ (in our case, $r=3)$; 2. a normally\ndistributed random vector $\\mathbf{X}\\sim N\\left(\\mathbf{0},\\Gamma\\right)$.\nThen one sets: \n\\begin{equation}\n\\mathbf{Y}=\\mathbf{m}+\\sqrt{V}\\times\\mathbf{X}\\label{eq:construccion_Kano}\n\\end{equation}\n\n\nFor more details, the reader is referred to \\citet{ellipticalSpatialRodriguezBardossy}. \n\nWe fitted $V$ as a mixture of 5 gamma random variables, in such a\nway that the cumulants of this mixture are $\\hat{c}_{1}=0.999$, $\\hat{c}_{2}=0.1101$\nand $\\hat{c}_{3}=0.1332$, up to a small error. Then we were able\nto simulate 1000 samples of $\\mathbf{Y}$, each of size $n=10950$,\nusing the fitted parameters. One of the realizations is shown in figure\n\\ref{fig:One-sample-of-Y-new}. Note that the covariance structure\nis mostly recovered, though there are some outliers of a magnitude\nsomewhat larger than those displayed in figure \\ref{fig:dataset_Y}.\nThis is because, once we fitted covariance matrix $\\Gamma$, we focus\non recovering the distribution of the sum of the components of the\nvector $\\mathbf{X}$, i.e. $S_{\\mathbf{X}}$. The outliers there presented\nare part of the mechanism that helps recover the distribution of the\ncomponents sum.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_02}\n\\par\\end{centering}\n\n\\caption{\\label{fig:One-sample-of-Y-new}One sample of size n=10950, generated\nusing the parameters fitted in section \\ref{sub:Parameter-estimation}.}\n\n\n\\end{figure}\n\n\nTo see how well the fitted parameters reproduce $S_{\\mathbf{X}}$,\nwe present several sample quantiles of it, together with confidence\nbands built out of the 1000 Monte Carlo simulations. See table \\ref{tab:Representative-quantiles-of}.\nWe see an excellent cover of the given quantiles, particularly at\nthe tails of the distribution of $S_{\\mathbf{X}}$. \n\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{|c||c||c||c|}\n\\hline \nQuantile (\\%) & 2.75\\% & 97.5\\% & Observed\\tabularnewline\n\\hline \n\\hline \n0 (min) & -41.921 & -22.644 & -29.191\\tabularnewline\n\\hline \n\\hline \n0.1 & -21.549 & -18.876 & -20.72\\tabularnewline\n\\hline \n\\hline \n0.5 & -16.956 & -15.72 & -17.032\\tabularnewline\n\\hline \n\\hline \n1 & -15.033 & -14.124 & -14.771\\tabularnewline\n\\hline \n\\hline \n5 & -10.248 & -9.748 & -10.049\\tabularnewline\n\\hline \n\\hline \n10 & -7.897 & -7.518 & -7.774\\tabularnewline\n\\hline \n\\hline \n20 & -5.159 & -4.838 & -5.194\\tabularnewline\n\\hline \n\\hline \n25 & -4.138 & -3.838 & -4.212\\tabularnewline\n\\hline \n\\hline \n50 & -0.136 & 0.137 & -0.18\\tabularnewline\n\\hline \n\\hline \n75 & 3.836 & 4.145 & 4.013\\tabularnewline\n\\hline \n\\hline \n80 & 4.84 & 5.155 & 4.987\\tabularnewline\n\\hline \n\\hline \n90 & 7.507 & 7.903 & 7.573\\tabularnewline\n\\hline \n\\hline \n95 & 9.765 & 10.273 & 9.911\\tabularnewline\n\\hline \n\\hline \n99 & 14.114 & 15.058 & 14.293\\tabularnewline\n\\hline \n\\hline \n99.5 & 15.718 & 17.034 & 16.01\\tabularnewline\n\\hline \n\\hline \n99.9 & 18.93 & 21.625 & 20.159\\tabularnewline\n\\hline \n\\hline \n99.99 & 21.908 & 29.07 & 28.542\\tabularnewline\n\\hline \n\\hline \n100 (max) & 22.897 & 43.735 & 28.983\\tabularnewline\n\\hline \n\\end{tabular}\n\\par\\end{centering}\n\n\\caption{\\label{tab:Representative-quantiles-of}Representative quantiles of\n$S_{\\mathbf{X}}$ and confidence bands of 1000 Monte Carlo simulations\nof 10950 sized samples each. The parameters fitted in section \\ref{sub:Parameter-estimation}\nhave been used for the simulation. Simulations reproduce quantiles\nvery similar to those observed.}\n\n\n\\end{table}\n\n\nAdditionally, the distribution of the 365-block maxima of the components\nsums is also acceptably recovered. In figure \\ref{fig:Empirical-Cumulative-Distributio-maxima}\nwe show the empirical distribution function of the 30 sample 365-block\nmaxima (i.e. yearly maxima). The Monte Carlo based 95\\% confidence\nbands for the 365-block maxima of $S_{\\mathbf{X}}$ are also presented\nin figure \\ref{fig:Empirical-Cumulative-Distributio-maxima}.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_03}\n\\par\\end{centering}\n\n\\caption{\\label{fig:Empirical-Cumulative-Distributio-maxima}Empirical Cumulative\nDistribution Function of the 365-block maxima, made out of the 10950\nsized sample presented at figure \\ref{fig:dataset_Y}. Monte Carlo\nsimulation based 95\\% confidence bands have been added from data simulated\nusing the parameters fitted in this section.}\n\n\n\\end{figure}\n\n\n\n\\subsection{More complicated questions}\n\nThe techniques presented in this section can also be used to investigate\nmore complex situations. For example, one would like to model jointly\nthe random variables\n\\begin{eqnarray}\nZ_{1} & := & X_{1}+\\ldots+X_{4}\\\\\nZ_{2} & := & X_{5}+\\ldots+X_{8}\n\\end{eqnarray}\n\n\nThis may be the case if each group of components, $X_{1},\\ldots,X_{4}$\nand $X_{5},\\ldots,X_{8}$, refers each to a geographical area (in\nenvironmental modeling); or if there is some economical reason to\ngroup them (stock price modeling). We may then wish to model the distributions\nof $Z_{1}$ and $Z_{2}$, but also model properly at least the correlation\nbetween them.\n\nApplying a similar computation as before, we find that\n\\begin{equation}\ncov\\left(Z_{1},Z_{2}\\right)=\\frac{c_{1}}{2}\\sum_{i=1}^{4}\\sum_{j=5}^{8}\\Gamma_{ij}\\label{eq:cov_ZZ}\n\\end{equation}\nfor the covariance. Regarding each $Z_{j}$, all odd-ordered cumulants\nare zero, whereas all even-ordered cumulants are given by\n\\begin{equation}\n\\kappa_{2r}\\left(Z_{j}\\right)=\\left(2r-1\\right)!\\times c_{r}\\times\\left(\\frac{R_{j}}{2}\\right)^{r}\\label{eq:cums_ZZ}\n\\end{equation}\nfor $j=1,2$, where\n\\begin{eqnarray*}\nR_{1} & = & 2\\sum_{i=1}^{4}\\Gamma_{ii}+4\\sum_{11}=0$ as in the original model by \\citet{sanso_venezuelan_1999}.\nAs shown by \\citet{ellipticalSpatialRodriguezBardossy}, a random\nfield with $\\left(c_{1},\\ldots,c_{5}\\right)$ as above is practically\nindistinguishable in its one and two dimensional marginal distributions\nfrom a Gaussian field with the same covariance function and mean.\nHowever, implications for the interaction manifestation ``average\nof fields components'', where each component represents daily precipitation\nover an 500 mt $\\times$ 500 mt squared area on the Saalach river\ncatchment, are significant. \n\nThe authors obtained 3000 conditional simulations, given the rainfall\ndata available, of the rainfall field over the Saalach river catchment\nfor June 1st 2013, a day of intense rainfall during the 2013 central\nEuropean floods. In figure \\ref{fig:Two-conditionally-simulated-fields},\ntwo of the obtained conditional fields are presented, using the Gaussian\nand the almost-Gaussian latent structure. In figure \\ref{fig:Boxplots-of-the-conditional},\nwe show the distribution of the conditional values of mean precipitation\nover the catchment, for both latent structures. Note that the multivariate\ninteractions, hardly noticeable on the one and two dimensional marginal\ndistributions, increase dramatically the probability of a very high\nmean precipitation over the studied catchment. The consequence is\nthat substantial under-estimation of flood return periods may me incurred,\nif one does not account for interaction among more than tow components,\nin one's spatio-temporal precipitation models.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.5\\textwidth]{fig_04}\\includegraphics[width=0.5\\textwidth]{fig_05}\n\\par\\end{centering}\n\n\\caption{\\label{fig:Two-conditionally-simulated-fields}Two conditionally simulated\nfields for June 1st 2013, for part of the Saalach river catchment: Field with Gaussian latent structure (left),\nand field with non-Gaussian latent structure (right). Stations providing\nthe observed data are indicated in red. Stations indicated by blue\npoints have no available data for that day. Note the intense precipitation\nclusters predictable by the model with latent field having multivariate\ninteractions.}\n\n\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_06}\n\\par\\end{centering}\n\n\\caption{\\label{fig:Boxplots-of-the-conditional}Boxplots of the average of\nthe conditionally simulated random fields for June 1st 2013, in millimeters,\nfor the Saalach river catchment. The field with high oder interacting\nlatent structure shows much more variability. In particular, average\nprecipitation over the catchment above 120 mm are quite probable under\nthis model.}\n\n\n\\end{figure}\n\n\n\n\\section{\\label{sec:Discussion}Discussion}\n\nAn approach for considering interactions that go beyond correlations\nhas been presented. We have seen that the discrimination between interactions\n``parameters'' and interactions ``manifestations'' can help to\ncircumvent two major problems one is confronted with when attempting\nto quantify and model higher order interactions: the problem of interpretability,\nby working with subject-matter relevant manifestations of interdependence;\nand the problem of high dimensionality, by recoursing to joint cumulants\nas building blocks of a dependence model. By using the cumulant generating\nfunction, we are recoursing to a well-studied object: the characteristic\nfunction of a distribution. \n\nAs dimension of vector $\\mathbf{X}$ increases, interactions of high\norder may be more and more difficult to assess. For example, a random\nvector having c.f.g. (\\ref{eq:archetypal_cgf-1}), with $c_{1}=1$\n, $c_{r}\\approx0$ for $2\\leq r\\leq3$ but then $c_{r\\geq4}\\neq0$,\nwould have one and two dimensional marginals practically equal to\nthose of a Guassian distribution. But the interaction coefficients\nof groups of 14 components or more will be very different, producing\nvery different interaction manifestations. The difference in the overall\ndependence structures may grow tremendously as the dimension of the\nrandom vector $\\mathbf{X}$ grow (i.e. $J>>2$), even though these\nfact may go totally unnoticed in the one and two dimensional marginal\nanalysis of data. \n\nIn \\citet{ellipticalSpatialRodriguezBardossy}, these issues are dealt\nwith and illustrated in the context of Spatial Statistics, where the\nissue of low dimensionality is essential, and where interaction manifestations\ncan differ drastically between two models having very similar 1 and\n2 dimensional marginals, due to the big dimension of the field.\n\n\n\\subsection*{Acknowledgments}\n\nThis research forms part of the Ph.D thesis of the first author, which\nwas funded by a scholarship of the German Academic Exchange Service\n(DAAD). This Ph.D work was carried out within the framework of the\nENWAT program at the University of Stuttgart.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{Introduction}}\n\nStrongly correlated electron systems, such as heavy fermion compounds,\nhigh-temperature superconductors, have gained much attention from both\ntheoretical and experimental point of view. The competition between the\nkinetic energy and strong Coulomb interaction of fermions generates a lot\nof fascinating phenomena. Various theoretical approaches have been developed\nto treat the regime of intermediate coupling. The widely used perturbative\nmethods, such as random phase approximation (RPA), fluctuation exchange\n(FLEX)\\cite{Bickers-1989,Bickers-1991}, and the two-particle self-consistent\n(TPSC)\\cite{Tremblay-1995,Tremblay-1994} method are based on the expansion in\nthe Coulomb interaction which is only valid in weak-coupling. To go beyond the\nperturbative approximation and to gain insight of the correlation effects of\nthe fermion systems, new theoretical methods are needed. Dynamical mean field\ntheory (DMFT)\\cite{Metzner-1989,Hartmann-1984,George-1996} is a big step\nforward in the understanding Metal-Insulator transition. \n\nDynamical mean field theory maps a many-body interacting system on a lattice\nonto a single impurity embedded in a non-interacting bath. Such a mapping\nbecomes exact in the limit of infinite coordination number. All local temporal\nfluctuations are taken into account in this theory, however spatial\nfluctuations are treated on the mean field level. DMFT has been proven a\nsuccessful theory describing the basic physics of the Mott-Hubbard\ntransition. But the non-local correlation effect can't always be\nomitted. Although, straight forward extensions of\nDMFT\\cite{Hettler-1998,Kotliar-2001,Okamoto-2003,Potthoff-2003,Maier-2005} \nhave captured the influence of short-range correlation, these methods are\nstill not capable of describing the collective behavior, e.g. spin wave\nexcitations of\nmany-body system. At the same time, most of the numerically exact impurity\nsolvers require a substantial amount of time to achieve a desired accuracy even\non a small cluster, which makes the investigation of larger lattice to be\nimpossible. \n\nRecently, some efforts have been made to take the spatial fluctuations into\naccount in different ways\\cite{Toschi-2007, Rubtsov-2006, Kusunose-2006,\n Tokar-2007, Slezak-2006}. All these methods construct the non-local\ncontribution of DMFT from the local two-particle vertex. The electron\nself-energy is expressed as a function of the two-particle vertex and \nthe single-particle propagator. The cluster extention of DMFT considers\nthe correlation within the small cluster. Compared to these, the \ndiagrammatic re-summation technique involved in these new methods makes them\nonly approximately include the non-local corrections. While, long range\ncorrelations are also considered in these methods and the computational burden\nis not serious. \n\nIn this paper we will apply the method of Rubtsov\\cite{Rubtsov-2006} to\nconsider the vertex renormalization of the DF through the Bethe-Salpeter\nequation. Lattice susceptibility is calculated from the renormalized DF\nvertex. \n\nThe paper is organized as follows: In Sec. \\ref{Details} we summarize the\nbasic idea of the DF method and give details of the calculation. The\nDMFT two-particle Green's function and the corresponding vertex calculation\nare implemented in CT-QMC in Sec. \\ref{Vertex}. The frequency dependent vertex\nis modified through the Bethe-Salpeter Equation to obtain the momentum\ndependence in Sec. \\ref{Mom-Vex}. In Sec. \\ref{application} we present the\ncalculation of the lattice susceptibility and compare it with QMC results and\nalso the works from Toschi\\cite{Toschi-2007}. The conclusions are summarized\nin Sec. \\ref{conclusion}, where we also present possible application.\n\n\\section{The DF method\\label{Details}}\n\nWe study the general one-band Hubbard model at two dimensions\n\\begin{equation}\n H=\\sum_{k,\\sigma}\\epsilon_{k,\\sigma}c_{k\\sigma}^{\\dagger}\n c_{k\\sigma}^{\\phantom\\dagger}+U\\sum_{i}n_{i\\uparrow}n_{i\\downarrow} \n\\end{equation}\n$c_{k\\sigma}^{\\dagger}(c_{k\\sigma})$ creates (annihilates) an electron with\nspin-$\\sigma$ and momentum $k$. The dispersion relation is\n$\\epsilon_{k}=-2t\\sum_{i=1}^{N}\\cos k_{i}$, where $N$ is the number of lattice\nsites. The basic idea of the DF method\\cite{Rubtsov-2006} is to\ntransform the hopping between different sites into coupling to an auxiliary \nfield $f(f^{\\dagger})$. By doing so, each lattice site can be viewed as an\nisolated impurity. The interacting lattice problem is reduced to solving a\nmulti-impurity problem which couples to the auxiliary field. This can be done\nusing the standard DMFT calculation. After integrating out the lattice\nfermions $c(c^{\\dagger})$ one can obtain an effective theory of the auxiliary\nfield where DMFT serves as an starting point of the expansion over the\ncoupling between each impurity site with the auxiliary field. \n\nTo explicitly demonstrate the above idea we start from the action of DMFT\nwhich can be written as \n\\begin{equation}\\label{original_fermion}\n S[c^{+},c]=\\sum_{i}S_{imp}^{i}-\\sum_{\\nu,k,\\sigma}(\\Delta_{\\nu}\n -\\epsilon_{k\\nu})c_{\\nu k\\sigma}^{\\dagger}c_{\\nu k\\sigma}^{\\phantom\\dagger}\n\\end{equation} \nwhere $\\Delta_{\\nu}$ is the hybridization function of the impurity problem\ndefined by $S_{imp}^{i}$ which is the action of an isolated impurity at site\n$i$ with the local Green's function $g_{\\nu}$. Using the Gaussian identity, we\ndecouple the lattice sites into many impurities which couple only to the field\n$f$ \n\\begin{eqnarray}\\label{auxiliary_field}\n S[c^{\\dagger},c;f^{\\dagger},f]&=&\\sum_{i}S_{imp}^{i}+\\sum_{k,\\nu,\\sigma}\n [g_{\\nu}^{-1}(c_{k\\nu\\sigma}^{\\dagger}f_{k\\nu\\sigma}^{\\phantom\\dagger}+h.c.)\n \\nonumber\\\\\n &&\\hspace{1cm}+g_{\\nu}^{-2}(\\Delta_{\\nu}-\\epsilon_{k})^{-1}\n f_{k\\nu\\sigma}^{\\dagger}f_{k\\nu\\sigma}^{\\phantom\\dagger}] \n\\end{eqnarray}\nThe equivalence of Eqs. (\\ref{original_fermion}) and (\\ref{auxiliary_field})\nform an exact relation between the Green's funtion of the lattice electrons and\nthe DF. \n\\begin{equation}\\label{relation}\n G_{\\nu,k}=g_{\\nu}^{-2}(\\Delta_{\\nu}-\\epsilon_{k})^{-2}\n G_{\\nu,k}^{d}+(\\Delta_{\\nu}-\\epsilon_{k})^{-1}\n\\end{equation}\nThis relation is easily derived by considering the derivative over\n$\\epsilon_{k}$ in the two actions. Eq. (\\ref{relation}) allows now to solve\nthe many-body ``lattice'' problem based on DMFT which is different from the\nstraight forward cluster extension. The problem is now to solve the Green's\nfunction of the DF $G^{d}_{\\nu,k}$. It is determined by integrating \nEq. (\\ref{auxiliary_field}) over $c^{\\dagger}$ and $c$ yielding a Taylor\nexpansion series in powers of $f^{\\dagger}$ and $f$. The Grassmann integral\nensures that $\\bar{f}$ and $f$ appear only in pairs associated with the\nlattice fermion n-particle vertex obtained from the single-site DMFT\ncalculation. In this paper we restrict our considerations to the two-particle\nvertex $\\gamma^{(4)}$.\n \n\\begin{figure}[b]\n \\includegraphics[width=200pt]{Self-Energy}%\n \\caption{The first two self-energy diagrams. They are composed of the local\n vertices function and DF propagator. \\label{Self-Energy}}\n\\end{figure}\n\nExpanding the Luttinger-Ward functional in $\\gamma^{4}$, the first two\ncontributions to the self energy function are the diagrams shown in\nFig. \\ref{Self-Energy}. Diagram (a) vanishes for the bare DF since this\ndiagram exactly corresponds to the DMFT self consistency. Therefore the first\nnon-local contribution is given by diagram (b). The self-energy for these two\ndiagrams are \n\\begin{subequations}\n \\begin{align}\n \\Sigma^{(1)}_{\\sigma}(k_{1}) &= -\\frac{T}{N}\\sum_{\\sigma^{\\prime},\n k_{2}}G_{\\sigma^{\\prime}}^{d}\n (k_{2})\\gamma^{(4)}_{\\sigma\\sigma^{\\prime}}\n (\\nu,\\nu^{\\prime};\\nu^{\\prime},\\nu) \\label{self-energy1} \\\\\n \\Sigma^{(2)}_{\\sigma}(k_{1}) &= -\\frac{T^{2}}{2N^{2}}\\sum_{2,3,4}\n G^{d}_{\\sigma_{2}}(k_{2})G^{d}_{\\sigma_{3}}(k_{3})\n G^{d}_{\\sigma_{4}}(k_{4})\\nonumber\\\\\n & \\gamma^{(4)}_{\\sigma_{1234}}(\\nu_{1},\\nu_{2};\\nu_{3},\\nu_{4})\n \\gamma^{(4)}_{\\sigma_{4321}}(\\nu_{4},\\nu_{3};\\nu_{2},\\nu_{1})\\nonumber\\\\\n & \\delta_{k_{1}+k_{2}, k_{3}+k_{4}}\\delta_{\\sigma_{1}+\\sigma_{2},\n \\sigma_{3}+\\sigma_{4}} \\label{self-energy2}\n \\end{align}\n\\end{subequations}\nHere space-time notation is used, $k=(\\vec{k},\\nu)$,\n$q=(\\vec{q},\\omega)$. Fermionic Matsubara frequency is \n$\\nu_{n}=(2n+1)\\pi\/\\beta$, bosonic frequency is $\\omega_{m} = 2m\\pi\/\\beta$\nwhere $\\beta$ is the inverse temperature. Together with the bare DF\nGreen's function\n$G^{d}_{0}(k)=-g_{\\nu}^{2}\/[(\\Delta_{\\nu}-\\epsilon_{k})^{-1}+g_{\\nu}]$, the\nnew Green's function can be derived from the Dyson equation \n\\begin{equation}\\label{Dyson}\n [G^{d}(k)]^{-1}=[G_{0}^{d}(k)]^{-1}-\\Sigma^{d}(k)\n\\end{equation}\n\nThe algorithm of the whole calculation is: \n\\begin{enumerate}\n\\item Set initial value of $\\Delta_{\\nu}$ for the first DMFT loop.\n\\item Determine the single-site DMFT Green's function $g_{\\nu}$ from\n the hybridization function $\\Delta_{\\nu}$. The self-consistency condition\n ensures that the first diagram of the DF self-energy is very small. \n\\item Go through the DMFT loop once again to calculate the two-particle\n Green's function and corresponding $\\gamma$-function. The method for\n determining the $\\gamma$-function is implemented for both strong and\n weak-coupling CT-QMC in the next section of this paper. \n\\item Start an inner loop calculation to determine the DF Green's\n function and in the end the lattice Green's function.\n \\begin{enumerate}\n \\item From Eqs. (\\ref{self-energy1}), (\\ref{self-energy2}) and the Dyson\n equation (\\ref{Dyson}) to calculate the self-energy of the DF. \n \\item Repeatly use Eq. (\\ref{self-energy1}), (\\ref{self-energy2}) and\n Eq. (\\ref{Dyson}) until the convergence of the DF Green's function\n is achived. \n \\item The lattice Green's function is then given by Eq. (\\ref{relation})\n from that of the DF.\n \\end{enumerate}\n\\item Fourier transform the momentum lattice Green's function into real space.\n And from the on-site component $G_{ii}$ to determine a new hybridization\n function $\\Delta_{\\nu}$ which is given by Eq. (\\ref{Old-New}).\n\\item Go back to the Step 3. and iteratively perform the outer loop until the\n hybridization $\\Delta_{\\nu}$ doesn't change any more. \n\\end{enumerate} \n\nAlthough diagram (a) is exactly zero for the bare DF Green's\nfunction, it gives non-zero contribution from the second loop where the DF\nGreen's function is updated from Eq. (\\ref{Dyson}). As a result, the \nhybridization function should also be updated before the next DMFT loop is\nperformed . This is simply done by setting the local full DF Green's\nfunction to zero, together with the condition that the old hybrization\nfunction forces the bare local DF Green's function to be zero\n($\\sum_{k}G^{0,d}_{\\nu,k}=0$), we obtain a set of equations \n\\begin{subequations}\n \\begin{align}\n & \\frac{1}{N}\\sum_{k}[G_{\\nu,k} - (\\Delta^{New}_{\\nu}-\\epsilon_{k})^{-1}]\n g_{\\nu}^{2}(\\Delta^{New}_{\\nu}-\\epsilon_{k})^{2} = 0 \\\\\n & \\frac{1}{N}\\sum_{k}[G^{0}_{\\nu,k} -\n (\\Delta^{Old}_{\\nu}-\\epsilon_{k})^{-1}] \n g_{\\nu}^{2}(\\Delta^{Old}_{\\nu}-\\epsilon_{k})^{2} = 0 \n \\end{align}\n\\end{subequations}\nwhich yields \n\\begin{equation}\n \\Delta_{\\nu}^{New}-\\Delta_{\\nu}^{Old}\\approx\\frac{1}{N}\\sum_{k}(G_{\\nu,k}-\n G^{0}_{\\nu,k})(\\Delta^{Old}_{\\nu}-\\epsilon_{k})^{2}\n\\end{equation}\nThis equation finally gives us the relation between the new and old\nhybridization function. \n\\begin{equation}\\label{Old-New}\n \\Delta_{\\nu}^{New}=\\Delta_{\\nu}^{Old}+g_{\\nu}^{2}G_{loc}^{d}\n\\end{equation}\n\nIn the whole calculation, the DF perturbation calculation converges\nquickly. The most time consuming part of this method is the DMFT calculation\nof the two particle Green's function. There are some useful symmetries to\naccelerate the calculation. As already pointed out\\cite{Abrikosov:QFT,\n Nozieres:1964}, it is convenient to take the symmetric form of the\ninteraction term. The two particle Green's function is then a fully\nantisymmetric function. Such fully antisymmetric form is very useful to speed\nup the calculation of the two particle Green's function. One does not need to\ncalculate all the frequency points within the cutoff in Mastsubara space, a\nfew special points are calculated and the values for the other points are\ngiven by that of those special points through antisymmetric property. In the\nDF self energy calculation, we always have the convolution type of momentum\nsummation which is very easy to be calculated by fast fourier transform (FFT). \n\n\\section{CT-QMC and two-particle vertex}\\label{Vertex}\n\nFrom the above analysis, the key idea of the DF method is to\nconstruct the nonlocal contribution from the auxiliary field and the DMFT\ntwo-particle Green's function. Therefore it is quite important to accurately \ndetermine the two-particle vertex. Here we adapt the newly developed CT-QMC\nmethod\\cite{Rubtsov-2005, Werner-2006(1), Werner-2006(2)} to calculate the two\nparticle Green's function $\\chi$. \n\nFirst we briefly outline the CT-QMC technique. For more details, we refer the\nreaders to\\cite{Rubtsov-2005, Werner-2006(1), Werner-2006(2)}. Here we discuss\nthe two-particle Green's function and some numerical implemetations in more\ndetailed. Two variants of the CT-QMC methods have been proposed based on the\ndiagrammatic expansion. Unlike the Hirsch-Fye method, these methods don't have\na Trotter error and can approach the low temperature region easily. In the\nweak-coupling method\\cite{Rubtsov-2005} the non-interacting part of the\npartition function is kept and expanded the interaction term into Taylor\nseries. Wick's theorem ensures that the corresponding expansion can be written\ninto a determinant at each order \n\\begin{equation}\n {\\cal Z}=\\sum_{k}\\frac{(-U)^{k}}{k!}\\int d\\tau_{1}\\cdots d\\tau_{k}e^{-S_{0}}\n \\det[D_{\\uparrow}D_{\\downarrow}]\n\\end{equation} \nwith\n\\begin{equation}\n D_{\\uparrow}D_{\\downarrow}=\n \\left(\n \\begin{array}{cc}\n \\cdots & G_{\\uparrow}(\\tau_{1}-\\tau_{k}) \\cr\n \\cdots & \\cdots \n \\end{array}\n \\right)\n \\left(\n \\begin{array}{cc}\n \\cdots & \\cdots \\cr\n G_{\\downarrow}(\\tau_{k}-\\tau_{1}) & \\cdots\n \\end{array}\n \\right)\n\\end{equation}\nwhere $S_{0}$ is the non-interacting action and $G^{0}$ is the Weiss field, \nand the one-particle Green's function is measused as \n\\begin{equation}\n G(\\nu)=G^{0}(\\nu)-\\frac{1}{\\beta}G^{0}(\\nu)\\sum_{i,j}M_{i,j}\n e^{i\\nu(\\tau_{i}-\\tau_{j})}G^{0}(\\nu)\n\\end{equation}\n\nIn the strong coupling method the effective action is expanded in the\nhybridization function by integrating over the non-interacting bath degrees of\nfreedom. Such an expansion also yields a determinant. \n\\begin{eqnarray}\n &&{\\cal Z}=TrT_{\\tau}e^{-S_{loc}}\\prod_{\\sigma}\n \\sum_{k_{\\sigma}}\\frac{1}{k_{\\sigma}!}\\int d\\tau_{1}^{s}\\cdots\n d\\tau_{k_{\\sigma}}^{s}\\int d\\tau_{1}^{e}\\cdots d\\tau_{k_{\\sigma}}^{e}\n \\nonumber\\\\\n &&\\Psi_{\\sigma}(\\tau^{e})\\left(\n \\begin{array}{ccc}\n \\Delta(\\tau_{1}^{e}-\\tau_{1}^{s}) & \\cdots & \\Delta(\\tau_{1}^{e}\n -\\tau_{k_{\\sigma}}^{s})\\\\\n \\cdots & \\ddots & \\cdots \\\\\n \\Delta(\\tau_{k_{\\sigma}}^{e}-\\tau_{1}^{s}) & \\cdots & \n \\Delta(\\tau_{k_{\\sigma}}^{e}-\\tau_{k_{\\sigma}}^{s})\n \\end{array}\\right)\n \\Psi_{\\sigma}^{\\dagger}(\\tau^{s})\n\\end{eqnarray}\nHere $\\Psi(\\tau) = (c_{1}(\\tau), c_{2}(\\tau),\\cdots,\nc_{k_{\\sigma}(\\tau)})$. The action is evaluated by a Monte Carlo random walk\nin the space of expansion order $k$. Therefore the corresponding hybridization\nmatrix changes in every Monte Carlo step. One particle Green's function is\nmeasured from the expansion of hybridization function as\n$G(\\tau_{j}^{e}-\\tau_{i}^{s})=M_{i,j}$. $M$ is the inverse matrix of the\nhybridization function. Apparently one needs to calculate this inverse matrix\nin every update step which is time consuming, fortunately it can be obtained by\nthe fast-update algorithm\\cite{Rubtsov-2005}. \n\nAt the same time such a relation allows direct measurement of the Matsubara\nGreen's function \n\\begin{equation}\n G(i\\nu_{n})=\\frac{1}{\\beta}\\sum_{i,j}e^{-i\\nu_{n}\\tau_{i}^{s}}M_{i,j}\n e^{i\\nu_{n}\\tau_{j}^{e}}\n\\end{equation}\n\nCompared with the imaginary time measurement, it seems additional\ncomputational time is needed for the sum over every matrix elements\n$M_{i,j}$. K. Haule proposed to implement such measurement in every fast update\nprocedure which makes sure that only linear amount of time is\nneeded\\cite{Haule-2007}. \n\nIn our calculation the Green's function is measured in the weak-coupling\nCT-QMC at each accepted update which greatly reduces the computational\ntime. The weak-coupling CT-QMC normally yields a higher perturbation order $k$\nthan the strong-coupling CT-QMC. It seems that the performance of the\nstrong-coupling CT-QMC is better\\cite{Emanuel-2007}. Concerning the\nconvergence speed, the weak-coupling CT-QMC is almost same as the\nstrong-coupling one under the above implementation together with a proper\nchoice of $\\alpha$, since in strong-coupling CT-QMC more Monte Carlo steps are \nneeded usually in order to smooth the noise of Green's function at imaginary\ntime around $\\beta\/2$ or at large Matsubara frequency points. Furthermore, the\nweak-coupling CT-QMC is much easier implemented for large cluster DMFT\ncalculation, in which case the strong-coupling method needs to handle a big\neigenspace. In this paper we mainly use weak-coupling CT-QMC as impurity \nsolver, while all the results can be obtained in the strong-coupling CT-QMC\nwhich was used as an accuracy check. \n\nSimilarly, we adapt K. Haule's implementation to calculate the two-particle\nGreen's function in frequency space. In the weak coupling CT-QMC, the\nnon-interacting action has Gaussian form which ensures the applicability of\nWick's theorem for measuring the two particle Green's function \n\\begin{eqnarray}\\label{2PG}\n \\chi_{\\sigma\\sigma^{\\prime}}(\\nu_{1},\\nu_{2},\\nu_{3},\\nu_{4})&=&\n T[\\overline{G_{\\sigma}(\\nu_{1},\\nu_{2})G_{\\sigma^{\\prime}}\n (\\nu_{3},\\nu_{4})}\\nonumber\\\\\n &-&\\delta_{\\sigma\\sigma^{\\prime}}\\overline{\n G_{\\sigma}(\\nu_{1},\\nu_{4})G_{\\sigma}(\\nu_{3},\\nu_{2})}]\n\\end{eqnarray} \nThe over-line indicates the Monte Carlo average. In each Monte Carlo\nmeasurement, $G(\\nu,\\nu^{\\prime})$ depends on two different argument $\\nu$ \nand $\\nu^{\\prime}$, only in the average level,\n$\\overline{G(\\nu,\\nu^{\\prime})}=G(\\nu)\\delta_{\\nu,\\nu^{\\prime}}$ is a function\nof single frequency. In each fast-update procedure, the new and old\n$G(\\nu,\\nu^{\\prime})$ have a closed relation which ensures that one can\ndetermine the updated Green's function $G^{New}(\\nu,\\nu^{\\prime})$ from the\nold one $G^{Old}(\\nu,\\nu^{\\prime})$. For example, adding pair of kinks and \nsupposing before updating the perturbation order is $k$, then it is $k+1$\nfor the new M-matrix. The new inserted pair is at $k+1$ row and $k+1$\ncolumn. \n\\begin{eqnarray}\\label{strong-update}\n &&G^{New}(\\nu,\\nu^{\\prime})-G^{old}(\\nu,\\nu^{\\prime})\\nonumber\\\\\n &=&\\frac{M^{New}_{k+1,k+1}}{\\beta}G^{0}(\\nu)\\left\\{XL\\cdot XR-XR\\cdot\n e^{-i\\nu\\tau_{k+1}^{s}}\\right.\\nonumber\\\\\n &&\\left.\\hspace{0.5cm}-XL\\cdot e^{i\\nu^{\\prime}\\tau_{k+1}^{e}}\n +e^{-i\\nu\\tau_{k+1}^{s}+i\\nu^{\\prime}\\tau_{k+1}^{e}}\\right\\}\n G^{0}(\\nu^{\\prime}) \n\\end{eqnarray}\nHere, $XL=\\sum_{i=1}^{k}e^{-i\\nu\\tau_{i}^{s}}L_{i}$,\n$XR=\\sum_{j=1}^{k}e^{i\\nu^{\\prime}\\tau_{j}^{e}}R_{j}$ and $L_{i}, R_{j}$ have\nthe same definition as in Ref\\cite{Rubtsov-2005}. In every step, one only needs\nto calculates the Green's function when the update is accepted and only a few\ncalculations are needed. A similar procedure for removing pairs, shiftting\nend-point operation can be used. Such method is also applicable in the segment\npicture of strong-coupling CT-QMC. In the weak-coupling CT-QMC, such an\nimplementation greatly improves the calculating speed in low temperature and\nstrong interaction regime\\footnote{In fact, the improvement is more obvious\n for larger M-matrices. The strong coupling CT-QMC and the weak coupling\n CT-QMC require approximately the same amount of CPU time although in the\n weak coupling case the average perturbation order is higher than in the\n strong coupling case}. Once one obtains the two frequency dependent\nGreen's function in every monte carlo step, the two-particle Green's function\ncan be determined easily from Eq. (\\ref{2PG}). The two-particle vertex is then\ngiven from the following equation: \n\\begin{equation}\n \\gamma^{\\sigma\\sigma^{\\prime}}_{\\omega}(\\nu,\\nu^{\\prime})=\n \\frac{\\beta^{2}[\\chi^{\\sigma\\sigma^{\\prime}}_{\\omega}(\\nu,\\nu^{\\prime})\n -\\chi^{0}_{\\omega}(\\nu,\\nu^{\\prime})]}\n {g_{\\sigma}(\\nu)g_{\\sigma}\n (\\nu+\\omega)g_{\\sigma^{\\prime}}(\\nu^{\\prime}+\\omega) \n g_{\\sigma^{\\prime}}(\\nu^{\\prime})}\n\\end{equation}\nwhere \n\\begin{equation}\n \\chi^{0}_{\\omega}(\\nu,\\nu^{\\prime})=T[\\delta_{\\omega,0}g_{\\sigma}(\\nu)\ng_{\\sigma^{\\prime}}(\\nu^{\\prime})-\\delta_{\\sigma\\sigma^{\\prime}}\n\\delta_{\\nu,\\nu^{\\prime}}g_{\\sigma}(\\nu)g_{\\sigma}(\\nu+\\omega)]\n\\end{equation} \nis the bare susceptibility. For the multi-particle Green's function, it still\ncan be constructed from the two frequency dependent Green's function\n$G(\\nu,\\nu^{\\prime})$, but more terms appear from Wicks theorem. Simply, when\nset $\\nu=\\nu^{\\prime}$ one can calculate the one-particle Green's funtion\neasily. \n\n\\section{Momentum dependece of Vertex}\\label{Mom-Vex}\n\nAs mentioned earlier diagram (a) in Fig. \\ref{Self-Energy} only gives the\nlocal contribution. The first non-local correction in the DF method\nis from diagram (b). Momentum dependences comes into this theory through the\nbubble-like diagram between the two vertices which yields the momentum\ndependence of the DF vertex. The natural way to renormalize vertex is\nthrough the Bethe-Salpeter equation. Since the DMFT vertex is only a function\nof Matsubara frequency, the integral over internal momentum $k$ and\n$k^{\\prime}$ ensures that the full vertex only depends on the center of mass\nmomentum $Q$. The Bethe-Salpeter equation in the particle-hole\nchannel\\cite{Abrikosov:QFT, Nozieres:1964} are shown in\nFig. \\ref{BSE-channel}. \n\nFrom the construction of the DF method, we know the interaction of the\nDF is coming from the two particle vertex of lattice fermion which is\nobtained through DMFT calculation. In the Bethe-Salpeter equation, it plays\nthe role of the building-block. The corresponding Bethe-Salpeter equation for\nthese two channels are\n\\begin{subequations}\\label{BSE}\n \\begin{align}\n & \\Gamma^{ph0,\\sigma\\sigma^{\\prime}}_{Q}(\\nu,\\nu^{\\prime}) = \n \\gamma^{\\sigma\\sigma^{\\prime}}_{\\omega}(\\nu,\\nu^{\\prime})- \\nonumber\\\\\n &\\frac{T}{N}\\sum_{k^{\\prime\\prime}\\sigma^{\\prime\\prime}}\n \\gamma^{\\sigma\\sigma^{\\prime\\prime}}_{\\omega}(\\nu,\\nu^{\\prime\\prime})\n G^{d}(k^{\\prime\\prime})G^{d}(k^{\\prime\\prime}+Q)\n \\Gamma^{ph0,\\sigma^{\\prime\\prime}\\sigma^{\\prime}}_{Q}\n (\\nu^{\\prime\\prime},\\nu^{\\prime}) \\\\\n & \\Gamma^{ph1,\\sigma\\bar{\\sigma}}_{Q}(\\nu,\\nu^{\\prime}) = \n \\gamma^{\\sigma\\bar{\\sigma}}_{\\omega}(\\nu,\\nu^{\\prime})- \\nonumber\\\\\n &\\frac{T}{N}\\sum_{k^{\\prime\\prime}}\n \\gamma^{\\sigma\\bar{\\sigma}}_{\\omega}(\\nu,\\nu^{\\prime\\prime})\n G^{d}(k^{\\prime\\prime})G^{d}(k^{\\prime\\prime}+Q)\n \\Gamma^{ph1,\\sigma\\bar{\\sigma}}_{Q}(\\nu^{\\prime\\prime},\\nu^{\\prime})\n \\end{align}\n\\end{subequations} \nHere, the short hand notation of spin configuration is\nused. $\\gamma^{\\sigma\\sigma^{\\prime}}$ represents\n$\\gamma^{\\sigma\\sigma\\sigma^{\\prime}\\sigma^{\\prime}}$, while\n$\\gamma^{\\sigma\\bar{\\sigma}\\bar{\\sigma}\\sigma}$ is denoted by\n$\\gamma^{\\sigma\\bar{\\sigma}}$ where\n$\\bar{\\sigma}=-\\sigma$. $\\Gamma^{ph0(ph1)}$ are the full vertices in the\n$S_{z}=0$ and $S_{z}=\\pm1$ channel, respectively. $G^{d}$ is the full DF\nGreen's function obtained from section \\ref{Details} which is kept unchanged\nin the calculation of the Bethe-Salpeter Equation. Different from the work of\nS. Brener\\cite{Brener-2007}, we solve the above equations directly in momentum\nspace with the advantage that in this way we can calculate the susceptibility\nfor any specific center of mass momentum $Q$ and it's convenient to use FFT for\ninvestigating larger lattice. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=230pt]{BSE-channel}\n \\caption{$S_{z}=0$ (ph0) and $S_{z}=\\pm1$ (ph1) particle-hole channels of\n the DF vertex, between vertices there are two full DF\n Green's function. The $S_{z}=\\pm1$ component is the triplet channel,\n while that for $S_{z}=0$ can be either singlet or triplet.} \n \\label{BSE-channel}\n \\end{center}\n\\end{figure}\n\nIn Eq. (\\ref{BSE}) one has to sum over the internal spin indices in the\n$S_{z}=0$ channel which is not present in $S_{z}=\\pm1$ channel. One can\ndecouple the $S_{z}=0$ channel into the charge and spin channels\n$\\gamma_{c(s)}=\\gamma^{\\sigma\\sigma}\\pm\\gamma^{\\sigma\\bar{\\sigma}}$ which can\nbe solved seperately, and it turns out that the spin channel vertex function \nis exactly same as the that in $S_{z}=\\pm1$ channel, see e.g.\nP. Nozieres\\cite{Nozieres:1964}. Such relation is true for the DMFT vertex,\nand was also verified for the momentum dependent vertex in the DF\nmethod\\cite{Brener-2007}. In our calculation, we have solved the $S_{z}=0$\nchannel by decoupling it to the charge and spin channel, while the $ph1$\nchannel is not used. \n\nOnce the converged momentum dependent DF vertex is obtained, one can\ndetermine the corresponding DF susceptibility in the standard way by\nattaching four Green's functions to the DF vertex. \n\\begin{subequations}\n \\begin{align}\n & \\chi^{\\sigma\\sigma^{\\prime}}_{d}(Q) = \\chi^{0}_{d}(Q)+\\nonumber\\\\\n & \\frac{T^{2}}{N^{2}}\n \\sum_{k,k^{\\prime}}G^{d}_{\\sigma}(k)G^{d}_{\\sigma}(k+Q)\n \\Gamma^{\\sigma\\sigma^{\\prime}}(Q)\n G^{d}_{\\sigma^{\\prime}}(k^{\\prime})G^{d}_{\\sigma^{\\prime}}(k^{\\prime}+Q) \\\\\n & \\chi^{\\sigma\\bar{\\sigma}}_{d}(Q) = \\chi^{0}_{d}(Q)+\\nonumber\\\\\n & \\frac{T^{2}}{N^{2}}\n \\sum_{k,k^{\\prime}}G^{d}_{\\sigma}(k)G^{d}_{\\bar{\\sigma}}(k+Q)\n \\Gamma^{\\sigma\\bar{\\sigma}}(Q)G^{d}_{\\sigma}(k^{\\prime})\n G^{d}_{\\bar{\\sigma}}(k^{\\prime}+Q)\n \\end{align}\n\\end{subequations}\nThe momentum sum over $\\vec{k}$ and $\\vec{k}^{\\prime}$ can be performed\nindependently by FFT becasue the DF vertx $\\Gamma^{\\sigma\\sigma^{\\prime}}(Q)$\nonly depends on the center of mass momentum $Q$.\n\nNow the z-component DF spin susceptibility $\\langle S^{z}\\cdot\nS^{z}\\rangle=\\frac{1}{2}(\\chi^{\\uparrow\\uparrow}_{d} \n-\\chi^{\\uparrow\\downarrow}_{d})$ can be determined from the spin channel\ncomponent calculated above. In Fig. \\ref{momentum-distribution}, \n$\\tilde{\\chi}^{zz}=\\chi^{zz}-\\chi_{0}^{zz}$ is shown for $U\/t=4$ at\ntemperatures $\\beta t = 4.0$ (left panel) and $\\beta t=1.0$ (right\npanel). With the lowing down of temperature the DF susceptibility grows up, \nespecially at wave vector $(\\pi, \\pi)$. The momentum $\\vec{k}_{x}$ and\n$\\vec{k}_{y}$ run from $0$ to $2\\pi$. \n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=230pt]{Sz}\n \\caption{The nontrivial part of the DF spin susceptibilities as a function\n of momentum in 2D Hubbard Model at $U\/t=4.0$, $\\beta t = 1.0$ (right\n panel) and $\\beta t = 4.0$ (left panel). Here 32 $\\times$ 32 momentum\n points are used in the first Brillouin zone.} \n \\label{momentum-distribution}\n \\end{center}\n\\end{figure}\nThe susceptibility is strongly peaked at the wave vector $(\\pi,\\pi)$ at the low\ntemperature case and the peak value becomes higher and higher. The magnetic\ninstability of the DF system is indicated by the enhancement of the\nDF susceptiblity. The effect of momentum dependence of vertex is clearly\nvisible in this diagram. The bare vertex which is only a function of frequency\nbecomes momentum dependent through the Bethe-Salpeter equation. Later on we\nwill see that such momentum dependent vertex plays a very important role in the\ncalculation of the lattice fermion susceptibility. \n \n\\section{Lattice susceptibility}\\label{application} \n\nThe strong antiferromagnetic fluctuation in 2D system is indicated by\nthe enhancement of the DF susceptibility at the wave vector\n$(\\pi,\\pi)$ shown in Fig \\ref{momentum-distribution}. This is the consequence\nof the deep relation between the the Green's function of the lattice and the\nDF, see Eq. (\\ref{relation}). In order to observe the magnetic\ninstability of the lattice fermion directly, we have calculated the\nlattice susceptibility based on the DF method. By differentiating\nthe partition function in Eqns. (\\ref{original_fermion},\n\\ref{auxiliary_field}) twice over the kinetic energy, we obtain an exact\nrelation between the susceptibility of DF and lattice fermions. After some\nsimplifications\\cite{Brener-2007}, it is given by \n\\begin{eqnarray}\n && \\chi_{f}(Q) = \\chi^{0}_{f}(Q) + \\nonumber\\\\\n && \\frac{T^{2}}{N^{2}}\\sum_{k,k^{\\prime}}G^{\\prime}(k)G^{\\prime}(k+Q)\n \\Gamma^{d}_{Q}(\\nu,\\nu^{\\prime})G^{\\prime}(k^{\\prime})G^{\\prime}\n (k^{\\prime}+Q)\n\\end{eqnarray} \nHere $G^{\\prime}$ cannt be interpreted as a particle propagator, it is\ndefined as: \n\\begin{equation}\n G^{\\prime}(k) = \\frac{G^{d}(k)}{g_{\\nu}[\\Delta_{\\nu}-\\epsilon(k)]}\n\\end{equation}\nAgain, the sum is performed over internal momentum and frequency $k,\nk^{\\prime}$ which is performed by FFT and rough summing over a few Matsubara\npoints. Again as in Eq. (\\ref{relation}), this equation established a\nconnection between the lattice susceptibility and the DF\nsusceptibility. From this point of view, it is easy to understand that the\ninstability of DFs generates the instability of the lattice\nfermions. \n\nOne can also find relations for the higher order Green's function of the\nDF and the lattice fermions in the same way. This emphasizes the similar\nnature of the DF and lattice fermions except that DF possess only\nnon-local information, since the DMFT self-consistency ensures that the local\nDF Green's function is exactly zero.\n\nThe lattice magnetic susceptibility is calculated using the following\ndefinition \n\\begin{eqnarray}\n \\chi_{m}(q) &=& \\frac{1}{N}\\sum_{i}e^{iq \\cdot\n r_{i}}\\int_{0}^{\\beta}d\\tau e^{-i\\omega_{m}\\tau}\\chi_{f}(i,\n \\tau)\\nonumber\\\\ \n &=& 2(\\chi_{f}^{\\uparrow\\uparrow}-\\chi_{f}^{\\uparrow\\downarrow})\n\\end{eqnarray} \nwhere $\\chi_{f}(i,\n\\tau)=\\langle\n[n_{i,\\uparrow}(\\tau)-n_{i,\\downarrow}(\\tau)]\\times[n_{0,\\uparrow}(0) -\nn_{0,\\downarrow}(0)]\\rangle$. $\\chi_{f}$ represents the lattice susceptibility\nin order to distinguish with that of the DF. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_0}\n \\caption{The uniform spin suscetibility of the DF using the bare\n vertex (only frequency dependent) and the full vertex(vertex from the\n Bethe-Salpeter quqation) for half filled 2D Hubbard model at $U\/t = 4.0$\n and various temperatures. These results reproduce the similiar solution\n in comparison with the calculation of finite size of\n QMC.}\\label{uniform_susceptibility} \n \\end{center}\n\\end{figure}\n\nWe have used two different ways to calculate the lattice susceptibility. First\nwe have solved the above equation using the bare vertex\n$\\Gamma(\\nu,\\nu^{\\prime}; \\omega)$ which is obtained from the DMFT \ncalculation. In contrast, the second calculation was performed using the full\nDF vertex. In both of these calculations, the full one particle DF Green's\nfunction was used. The momentum dependent of the DF vertex is\nobtained through the calculation of the Bethe-Salpeter equation. The lattice\nsusceptibility is expected to be improved if we use the momentum\ndependence DF vertex. In this way, we can understand the effect of momentum\ndependence in the DF vertex.\n\nIn Fig. \\ref{uniform_susceptibility} we plotted the results for the uniform\nsusceptibility $\\chi_{m=0}(0,0)$ by using both the bare and full DF\nvertex. The lattice QMC result\\cite{Moreo-1993} is shown for comparison. The\ncalculation is done for $U\/t = 4.0$ and several values of temperature. The\nmomentum sum is approximated over 32 $\\times$ 32 points here. Both of these\ncalculations reproduce the well known Curie-Weiss law behavior. Surprisingly\nenough, the results for the bare vertex fit the QMC results better than that\nfor the momentum dependent vertex. We believe that this is the finite size\neffect of QMC\\cite{Moreo-1993}. A. Moreo showed that $\\chi$ becomes smaller\nwhen increasing the cluster size $N$. The 4 $\\times$ 4 cluster calculation \nresult at the same temperature located above of that from 8 $\\times$ 8 cluster\ncalculation. Therefore the results obtained from the full vertex is expected\nto be more reliable. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_pi}\n \\caption{Uniform spin susceptibility at the wave vector $(\\pi, \\pi)$. The\n QMC results are obtained from Ref.\\cite{Bickers-1991(2)}.}\n \\label{chi_pi}\n \\end{center}\n\\end{figure}\n\nThe importance of the momentum dependence of the DF vertex is more clearly\nobserved in the calculation of $\\chi_{m}(\\pi, \\pi)$, see\nFig. \\ref{chi_pi}. Again, in this diagram QMC results\\cite{Bickers-1991(2)}\nare shown for comparison. The same parameters are used as in \nFig. \\ref{uniform_susceptibility}. The result from the DF with bare vertex\ndoes not produce the same results compared with QMC solution. Evenmore\ninteresting, with decreasing temperature the deviation becomes larger. On the\nother hand, the momentum dependent vertex in the DF method gives a\nsatisfactory answer. This shows the importance of the momentum\ndependence in the DF vertex function. Fig.~\\ref{chi_q} shows the evolution of\n$\\chi$ against $q$ for fixed transfer frequency $\\omega_{m}=0$. The path in\nmomentum space is shown in the inset. From this diagram we can see that\n$\\chi(q,0)$ reaches its maximum value at wave vector $(\\pi,\\pi)$. \n\nThe comparison between the DF and QMC results shows the good\nperformance of DF method. The DF calculation started from a\nsingle site DMFT calculation and by introducing an auxiliary field, the\nnon-local information is introduced and nicely reproduces the QMC results. Our\ncalculation could be done within four hours for each value of the\ntemperature on average. In this sense, this method is cheap and reliable\ncompared with the more computationally intensive lattice QMC calculation. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_q}\n \\caption{$\\chi(q,0)$ vs $q$ at $\\beta t= 2.0$, $U\/t = 4.0$ for various $q$\n which is along the trajectory shown in the inset.} \\label{chi_q} \n \\end{center}\n\\end{figure}\n\nSimilar as the DF method, Dynamical Vertex Approximation\n(D$\\Gamma$A)\\cite{Toschi-2007} is also based on the two particle local\nvertex. It deals with the lattice fermion directly, without introducing any\nauxiliary field. The perturbative nature of this method ensures its validity\nat weak-coupling regime. Unlike in the DF method, D$\\Gamma$A takes the\nirreducible two particle local vertex as building blocks. \n\\begin{subequations}\\label{DGA-BSE}\n \\begin{align}\n & \\gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime};\\omega) = \n \\gamma^{-1}_{c(s),ir}(\\nu,\\nu^{\\prime};\\omega) - \n \\chi_{0}(\\nu;\\omega)\\delta_{\\nu,\\nu^{\\prime}} \\\\\n & \\Gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime};Q) = \n \\gamma^{-1}_{c(s),ir}(\\nu,\\nu^{\\prime};\\omega) -\n \\chi_{0}(\\nu;Q)\\delta_{\\nu,\\nu^{\\prime}}\n \\end{align}\n\\end{subequations}\nwith the spin and charge vertex defined as\n$\\gamma_{c(s)}=\\gamma^{\\uparrow\\uparrow}\\pm\\gamma^{\\uparrow\\downarrow}$. The\nbare susceptibility is defined as \n\\begin{subequations}\n \\begin{align}\n & \\chi_{0}(\\nu;\\omega) = -TG_{loc}(\\nu)G_{loc}(\\nu+\\omega) \\\\\n & \\chi_{0}(\\nu,Q) = -\\frac{T}{N}\\sum_{\\vec{k}}\n G^{0}(\\vec{k},\\nu)G^{0}(\\vec{k}+\\vec{q},\\nu+\\omega)\n \\end{align}\n\\end{subequations}\nAnd the self-energy is calculated through the standard Schwinger-Dyson\nequation \n\\begin{equation}\\label{DGA-Selfenergy}\n \\Sigma(k) = -U\\frac{T^{2}}{N^{2}}\\sum_{k^{\\prime},Q}\n \\Gamma_{f}(k,k^{\\prime};Q)G^{0}(k^{\\prime})G^{0}(k^{\\prime}+Q)G^{0}(k+Q)\n\\end{equation}\nHere, the full vertex $\\Gamma_{f}(k,k^{\\prime};Q)$ is obtained by summing all\nthe channel dependent vertices and subtracting the double counted diagrams.\n\\begin{eqnarray}\\label{DGA-FullVertex}\n \\Gamma_{f}(k,k^{\\prime};Q)&=&\\frac{1}{2}\\bigg\\{\n [3\\Gamma_{c}(\\nu,\\nu^{\\prime};Q)-\\Gamma_{s}(\\nu,\\nu^{\\prime};Q)]\\nonumber\\\\\n &&-[\\Gamma_{c}(\\nu,\\nu^{\\prime};\\omega)-\n \\Gamma_{s}(\\nu,\\nu^{\\prime};\\omega)]\\bigg\\}\n\\end{eqnarray}\nThe one particle propagator is given by the DMFT lattice Green's function\nwhere the self energy is purely local $G^{0}(k)=\n1\/[i\\nu-\\epsilon(k)-\\Sigma(\\nu)]$, the local Green's function is $G_{loc}(\\nu)\n= 1\/[i\\nu-\\Delta(\\nu)-\\Sigma(\\nu)]$. Then the Dyson equation gives the lattice\nGreen's function from the self-energy function $G^{-1} = G^{-1}_{0}-\\Sigma$. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_0_DGA}\n \\caption{Comparison with the D$\\Gamma$A susceptibilities $\\chi(0,0)$ which\n obtained from both the DMFT lattice Green's function (D$\\Gamma$A\n $(G^{0}$)) and the full Green's function (D$\\Gamma$A $(G$)), see context\n for more details.}\n \\label{DGA_0}\n \\end{center}\n\\end{figure}\n\nBefore presenting the comparison, we take a deeper look at the analysis of \nEq. (\\ref{DGA-BSE}), \n\\begin{eqnarray}\n \\Gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime};Q) &=& \n \\gamma^{-1}_{c(s)}(\\nu,\\nu^{\\prime};\\omega) - \\nonumber \\\\\n &&[\\chi_{0}(\\nu;Q)-\\chi_{0}(\\nu,\\omega)]\\delta_{\\nu,\\nu^{\\prime}}\n\\end{eqnarray} \nThe second term in the brackets on RHS removes the local term from the\nbare susceptibility. The whole term in the brackets then represents only the\nnon-local bare susceptibility. In order to compare with the DF\nmethod, we take the inverse form of Eq. (\\ref{BSE})\n\\begin{eqnarray} \n \\Gamma_{d,c{s}}^{-1}(\\nu,\\nu^{\\prime};Q) &=&\n \\gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime}, \\omega) - \\nonumber\\\\\n && \\frac{T}{N}\\sum_{\\vec{k}}G^{d}(k)G^{d}(k+q) \n\\end{eqnarray}\nThe above two equations are same except for the last term. Since the local\nDF Green's function $G^{d}_{loc}$ is zero, the bare DF\nsusceptibility is purely non-local which coincides with the analysis of\nD$\\Gamma$A Bethe-Salpeter equation. Therefore, it is not surprising that \nthese two methods generate similar results. It is not easy to perform a term\nto term comparison between the DF method and D$\\Gamma$A although the\nbare susceptibilities have no local term in both of these method. The\none particle Green's functions have different meaning in these two methods. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_pi_DGA}\n \\caption{D$\\Gamma$A susceptibilities $\\chi(\\pi,\\pi)$ at $U\/t=4.0$. The\n susceptibility are determined from both of the DMFT and full lattice\n Green's function together with the vertex obtained from\n Eq. (\\ref{DGA-FullVertex})}. \n \\label{DGA_pi}\n \\end{center}\n\\end{figure}\n\nThe lattice susceptibility within the D$\\Gamma$A method is obtained by\nattaching four Green's functions on the vertex obtained in\nEq. (\\ref{DGA-FullVertex}). There are two possible choices of the lattice\nGreen's function, one is the DMFT lattice Green's function $G^{0}$, the other\none is the Green's function $G$ constructed by the non-local self-energy from\nthe Dyson equation. In Fig. \\ref{DGA_0} and \\ref{DGA_pi}, we presented\nthe D$\\Gamma$A lattice susceptibility calculated from both the DMFT lattice\nGreen's function labeled as D$\\Gamma$A($G^{0}$) and the full Green's function\nlabeled as D$\\Gamma$A($G$). The DF result from the calculation with\nthe full DF vertex is re-plotted for comparison. In Fig. \\ref{DGA_0}, the\nD$\\Gamma$A susceptibility calculated from the DMFT Green's function\n(D$\\Gamma$A($G^{0}$)) is basically the same as the DF susceptibility\nonly with some small deviation. The results for $T\/t > 1.0$ which are not shown\nhere which nicely repeat the DF and QMC results, the deviation\nbetween the D$\\Gamma$A and the DF method becomes smaller with the\nincreasing of temperature. The D$\\Gamma$A susceptibility is calculated from\nthe full Green's function (D$\\Gamma$A($G$)) shows a different behavior at low\ntemperature regime which reached its maximum value at $T\/t\\approx0.36$. As we\nknow, the Hubbard Model at half filling with strong coupling maps to the\nHeisenberg model, $\\chi$ reasches a maximum at $T\\approx J$ where \n$J$ is the effective spin coupling constant given as $4t^{2}\/U$. The\ncalculation uses the parameter $U\/t=4.0$ which is in the intermediate coupling\nregime. Therefore we further calculated the lattice susceptibility at\n$U\/t=10.0$ which are shown in Fig. \\ref{chi_0_U10}. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=220pt]{chi_0_U10}\n \\caption{The comparison of the DF resulsts and that of QMC for\n the uniform susceptibility at $U\/t=10$. 4$\\times$4 QMC\n results\\cite{Moreo-1993} also shows the errorbars.} \n \\label{chi_0_U10}\n \\end{center}\n\\end{figure}\n\nWhen the temperature is greater than 0.4, the DF method and\nD$\\Gamma$A (D$\\Gamma$A($G^{0}$)) generate the similar results to the QMC\ncalculation. Reducing the temperature further, the QMC susceptibility\ngreatly drops and shows a peak around 0.4 which coincides with the\nbehavior of the Heisenberg model. The DF femion and D$\\Gamma$A susceptibility\ncontinuously grows up with the decreasing of temperature. Although the\nD$\\Gamma$A with the full Green's function (D$\\Gamma$A($G$)) shows a peak, it\nlocates at $T\/t=0.6667$ which is larger than the peak position of the QMC. And\nD$\\Gamma$A($G$) generated a large deviation from that of QMC. In \nthis diagram, we only show the results of the DF approach for\n$T\/t>0.3$ and the D$\\Gamma$A results for $T\/t>0.4$. The Bethe-salpeter\nequation of the D$\\Gamma$A have a eigenvalue approaching one when further\nlowering the temperature, which makes the access of lower temperature region\nimpossible. \n \n\\begin{figure}[b]\n \\begin{center}\n \\includegraphics[width=220pt]{eigenvalue_T}\n \\caption{The evolution of maximum eigenvalue in spin channel against\n temperature for DF method and D$\\Gamma$A.}\n \\label{eigenvalue-T}\n \\end{center}\n\\end{figure}\n\nFig. \\ref{DGA_pi} shows the results of D$\\Gamma$A susceptibilities at wave\nvector $(\\pi, \\pi)$. In contrast to the comparison for $\\chi(0,0)$ results, the\nD$\\Gamma$A susceptibility calculated from the full Green's function D$\\Gamma$A\n($G$) yields better results than that from the calculation with the DMFT\nGreen's function D$\\Gamma$A ($G^{0}$). D$\\Gamma$A ($G$) results are almost on\ntop of the DF results, the results with DMFT Green's function D$\\Gamma$A\n($G^{0}$) is large than the DF results. The deviation becomes\nlarger at lower temperature. Summarizing, the D$\\Gamma$A calculation\nusing the full Green's function generated the same result as the DF\nmethod for $\\chi(\\pi,\\pi)$ while failed to produce $\\chi(0,0)$ correctly. In\ncontrast, the calculation with the DMFT Green's function in D$\\Gamma$A nicely\nproduced the results calculated with the DF method for $\\chi(0,0)$\nwhile generated larger devivation for $\\chi(\\pi,\\pi)$ at lower temperature\nregime compared to that from the DF method. Together with\nFig. \\ref{uniform_susceptibility} and \\ref{chi_pi}, we can see that the DF\nfermion calculation with the full DF vertex generated basically the same\nresults for both $\\chi(0,0)$ and $\\chi(\\pi,\\pi)$ compared to the results of\nQMC. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=230pt]{chi_0_Away}\n \\caption{Uniform magentic susceptibility is plotted as a function of\n dopping at $\\beta t=2.5$ and $U\/t =4.0, 10.0$.}\\label{away-half}\n \\end{center}\n\\end{figure}\n\nIn both the DF method and the D$\\Gamma$A, the operation of inverting\nlarge matrices is required for solving the Bethe-Salpeter\nequation. Fig. \\ref{eigenvalue-T} shows the leading eigenvalue of \nEqns. (\\ref{BSE}) and (\\ref{DGA-BSE}). As expected, the leading eigenvalue\napproaches one with decreasing temperature which directly indicates the\nmagnetic instability of 2D system. The eigenvalues corresponding to the DF\nfermion method always lies below of that from D$\\Gamma$A indicating the\nbetter convergence of the DF method. When the leading eigenvalues are closed to\none, the matrix inversion in Eqns. (\\ref{BSE}) and (\\ref{DGA-BSE}) are ill\ndefined, which prevents the investigation at very low temperature. \n\nConcerning the performance of the DF method, we also calculated the \nuniform susceptibility at away half-filling. In the strong-coupling limit,\nthe Hubbard model is equivalent to the Heisenberg model with coupling constant\n$J=4t^{2}\/U$. The consequence of doping is to effectively decrease the\ncoupling $J$, which yields the increasing behavior of $\\chi$ with doping. The\nfinite size QMC calulation\\cite{Moreo-1993, Chen-1993} observed a \nslightly increasing $\\chi$ with very small doping at strong interaction\nor in the low temperature region. Here, we did a similar calculation at $\\beta\nt=2.5$ and $U\/t=4, 10$. Since the DF method and the D$\\Gamma$A do not\nsuffer from the finite size problem. We would expect to observe results similar\nto those of QMC\\cite{Moreo-1993,Chen-1993}. In D$\\Gamma$A the\nsuseceptibility is calculated from the DMFT Green's function $G^{0}$ and the\nvertex obtained from Eq. (\\ref{DGA-FullVertex}). As shown in\nFig. \\ref{away-half} at $U\/t=4.0$, the susceptibility $\\chi$ slightly\nincreases in the weak dopping region where $\\delta$ is around $0.05$, DF\nfermion results clearly showed such behavior, D$\\Gamma$A also gave a signal of\nit. Further doping the system, both the D$\\Gamma$A and the DF method\nreproduce the decrease with doping as already seen in the QMC. With the\nincreasing of interaction, we would expect to see the enhancement of this\neffect, however our calculation indicates that such increasing-decreasing\nbehaviro dissappear. Both the D$\\Gamma$A and the DF method give the\nsame decreasing curve which contradict to QMC result\\cite{Moreo-1993}. The\nresults will most likely be further improved by including the higher order\nvertex or calculating the cluster DMFT plus DF\/D$\\Gamma$A\\cite{Hafermann-2007}.\n \n\\section{Conclusion}\\label{conclusion}\n\nIn this paper, we extended both the DF method and D$\\Gamma$A to\ncalculate the lattice susceptibility. Both of these methods gave equally good\nresults compared with QMC calculation at $U\/t=4.0$. Although they are supposed\nto be weak-coupling methods, at $U\/t=10.0$ these two methods generated right\nresults at high temperature region. While both of them failed to reproduce\nthe Heisenberg physics at low temperature. The investigation of the lattice\nsusceptibility suffers from hard determined matrix inversion problem at low\ntemperature regime. The DF methods always generates smaller eigenvalues\ncompared to D$\\Gamma$A indicating the better convergence. The implementation\nof DF method in momentum space greatly improves the calculational\nspeed and makes it easier to deal with larger size lattice. \n\n\\begin{acknowledgments}\nWe would like to thank the condensed matter group of\nA. Lichtenstein at Hamburg University for their hospitality\nin particular for the discussions and open exchange\nof data with H. Hafermann. Gang Li and Hunpyo Lee would like\nto thank Philipp Werner for his help in implementing the\nstrong-coupling CT-QMC code.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nReactions on surfaces play an important role in many technological \napplications like the heterogenous catalysis, growth of semiconductor \ndevices, corrosion and lubrication of mechanical parts, or hydrogen\nstorage in metals. In spite of this significance the understanding\nof the microscopic details of these reactions is still rather incomplete.\nOf particular importance are processes in which chemical bonds of\nmolecules are breaking due to the presence of a substrate because\nthere processes represent the first elementary step in, e.g.,\nheterogeneous catalysis or corrosion. Often this is the rate-limiting\nstep, for example in the ammonia synthesis. Hydrogen dissociation\non metal surfaces has become {\\em the} model system for the\nbond-breaking process on surfaces in the last years\nbecause it can be studied in detail experimentally as well as\ntheoretically. In particular in the theoretical description\nthere has been much progress recently due to the improvement of \ncomputer power and the development of efficient algorithms.\nIt has become possible to map out detailed potential energy surfaces\nof the dissociation of hydrogen on metal surfaces \nby density functional theory calculations \n\\cite{Ham94,Whi94,Wil95,Wil96PRB,Whi96PRB,Wie96,Eich96,Dong96}.\nThe availability of high-dimensional\nreliable potential energy surfaces has challenged the dynamics community\nto improve their methods in order to perform high-dimensional dynamical \nstudies on these potentials. Now quantum studies of the dissociation of \nhydrogen on surfaces are possible in which all six degrees of \nfreedom of the molecule are treated dynamically\n\\cite{Gro95PRL,Gro98PRB,Kro97PRL,Kro97JCP,Dai97}.\nIn this brief review I will illustrate this progress by focusing\non the hydrogen dissociation on the clean and sulfur-covered\npalladium surface. \n\nHydrogen is the simplest molecule which\nmakes it accessible to a relatively complete theoretical treatment.\nAt the same time hydrogen is also well-suited for performing\nexperiments which allows a fruitful interaction between theory\nand experiment. I will show that general concepts relating to\nthe reactivity of surfaces as well as to dynamical reaction\nmechanisms can be deduced from the detailed comparison\nof theoretical and experimental results of the hydrogen dissociation\nof metal surfaces. These concepts are applicable to any reaction\nsystem making hydrogen the ideal candidate\nfor studying reactions on surfaces.\n\n\n\n\n\\section{General concepts in the adsorption dynamics at surfaces}\n\nThe sticking or adsorption probability is defined as the fraction\nof atoms or molecules impinging on a surface that are not\nscattered back, i.e. that remain on the surface. It should be noted\nhere that there is no unambiguous definition of the sticking probability\nbecause for surfaces with non-zero temperature every adsorbed particle\nwill sooner or later desorb again. Hence the sticking probability\ndepends on the time-scale of the required residence time on the surface.\n\n\n\n\nAtomic adsorption is often very efficient. \nHydrogen atoms, e.g, stick at metal surfaces \\cite{Eilm96} and \nsemiconductor surfaces \\cite{Schu83} with a probability of order unity. \nHowever, dissociative adsorption probabilities can differ by many orders \nof magnitude. Whereas the sticking probability of hydrogen molecules on \nmany transition metal surfaces is about 0.5 \\cite{Eilm96,Ren89},\nat room temperature the dissociation probability of H$_2$\/Si\nis only 10$^{-8}$ \\cite{Bra96PRB}, and for N$_2$\/Ru it is even as low as\n10$^{-13}$ \\cite{Hin97}. The investigation of processes that occur within\nsuch a wide range of probabilities represents of course a great challenge\nfor the theory as well as for the experiment.\n\n\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\put(-1.,0.){ \\rotate[l]{\\epsfysize=10.cm \n \\epsffile{mol_ads_bw.ps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{Atomic adsorption probability for Xe and Ar on Pt(111) and\n molecular adsorption probability of N$_2$\/W(100). These\n examples are taken from the textbook by Zangwill \n \\protect\\cite{Zan88}. The inset illustrates the adsorption\n process.\n } \n\n\\label{mol_ads}\n\\end{figure} \n\n\nThere is one fundamental difference between atomic and molecular adsorption \non the one side and dissociative adsorption on the other side that is very\nimportant for the theoretical description of these processes. Here with \nmolecular adsorption a sticking process is meant in which the molecule \nstays intact on the surface. This fundamental difference will be illustrated\nin the following. In atomic and molecular adsorption it is crucial that\nthe impinging particles transfer their kinetic energy to the surface,\notherwise they would be scattered back into the gas phase.\nIf $P_E (\\epsilon)$ is the probability that an incoming particle with\nkinetic energy $E$ will transfer the energy $\\epsilon$ to the surface,\nthen the atomic or molecular sticking probability can be expressed as\n\\begin{equation}\nS(E) \\ = \\ \\int_{E}^{\\infty} \\ P_E(\\epsilon) \\ d\\epsilon,\n\\end{equation}\ni.e., it corresponds to the fraction of particles that transfer more\nenergy to the surface than their initial kinetic energy. This excess\nenergy has to be transferred to substrate excitation, i.e., either\nphonons or electron-hole pairs. Hence any theoretical description\nof atomic or molecular adsorption has to consider dissipation to the continous\nexcitation spectrum of the substrate. In Fig.~\\ref{mol_ads}\nsticking probabilities for atomic and molecular adsorption as a function\nof the initial kinetic energy are shown that correspond indeed to textbook \nexamples \\cite{Zan88}. These curves show a typical behavior, namely the\ndecrease of the sticking probability with increasing kinetic energy.\nThis is due to the fact that the energy transfer to the surface becomes\nless efficient at higher kinetic energies. Of course, the higher the kinetic \nenergy is, the more energy is transfered to the surface. But the fraction\nof particles that loose more energy than their initial kinetic energy\nbecomes smaller at higher kinetic energy. There is still more\ninteresting physics in these sticking probabilities. For example, at low \nkinetic energies classically the sticking probability should become\nunity if there is no barrier before the adsorption well. Every\nimpinging particle transfers energy to the substrate so that\nin the limit of zero initial kinetic energy all particles will stick.\nQuantum mechanically, however, there is a non-zero probability for\nelastic scattering at the surface so that the sticking probabilities\nbecome less than unity in the zero-energy limit \\cite{Sch88}. In \nFig.~\\ref{mol_ads} these quantum effects at low energies are evident in \nthe sticking probability of the light noble gas argon on Pt(111) compared\nto the sticking probability of the heavier noble gas xenon on the same\nsurface.\n\n\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\put(-1.,0.){ \\rotate[l]{\\epsfysize=10.cm \n \\epsffile{diss_ads_bw.ps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{Dissociative adsorption probability versus kinetic energy \n of H$_2$\/Cu(111) for molecules initially in the vibrational ground state. \n Solid line: results of five-dimensional calculations in which the molecular\n axis was kept parallel to the surface \n (from ref.~\\protect\\cite{Gro94PRL}).\n Dashed line: Experimental curve (from ref.~\\protect\\cite{Ret95}).\n The inset illustrates the dissociation process. }\n\n\\label{diss_ads}\n\\end{figure} \n\n\nNow in the case of dissociative adsorption there is another channel\nfor energy transfer, which is the conversion of the kinetic and internal \nenergy of the molecule into translational energy of the atomic fragments \non the surface relative to each other. This represents the fundamental\ndifference to atomic or molecular adsorption. It is true that eventually\nthe atomic fragments will also dissipate their kinetic energy and come\nto rest at the surface. However, especially in the case of light molecules\nlike hydrogen dissociating on metal surfaces the energy transfer to the\nsubstrate is very small due to the large mass mismatch. Whether a molecule\nsticks on the surface or not is almost entirely determined by the \nbond-breaking process for which the energy transfer to the substrate\ncan be neglected. This makes it possible to describe the dissociative\nadsorption process within low-dimensional potential energy surfaces\nneglecting the surface degrees of freedom\nif furthermore no substantial surface rearrangement upon adsorption occurs,\nas it is usually case in the dissociative adsorption on close-packed\nmetal surfaces. Fig.~\\ref{diss_ads} shows the dissociative adsorption \nprobability of a system which also corresponds to a textbook example,\nnamely the dissociative adsorption of H$_2$ on Cu(111). In this system\nthe dissociation is hindered by a noticeable barrier so that the\ndependence of the sticking probability on the kinetic energy exhibits\na behavior typical for activated systems \\cite{Gro94PRL,Ret95}.\n\n\n\n\\section{Dissociative adsorption at a transition metal surface}\n\nTransition metal surfaces are usually very reactive as fas as\nhydrogen dissociation is concerned \\cite{Ren89,Aln89,Ber92,But94}.\nIn Fig.~\\ref{h2pdstick} the results of molecular beam experiments\nof Rendulic, Anger and Winkler \\cite{Ren89} and of Rettner and\nAuerbach \\cite{Ret96} for the dissociative adsorption of H$_2$ \non Pd(100) are shown. At low kinetic energies\nthese experiments yield a sticking probability above 0.5. But even \nmore interestingly, the sticking probability initially decreases\nwith increasing kinetic energy. This is reminiscent of the dependence\nof the sticking probability on the kinetic energy in atomic or\nmolecular adsorption illustrated in Fig.~\\ref{mol_ads}. Therefore\nfor a long time it was believed that such an initially decreasing\nsticking probability in dissociative adsorption is a signature of\nthe so-called precursor mechanism \\cite{Ren94}. In this mechanism the \nmolecule does not directly dissociate but is first trapped molecularly\nin a precursor state from which it then dissociates. This trapping\nprobability decreases with increasing kinetic energy and thus\ndetermines the sticking at low kinetic energies.\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\centerline{ \\rotate[r]{\\epsfysize=8.cm \n \\epsffile{h2pd100.eps}} }\n\n \\end{picture}\n\\end{center}\n \\caption{Sticking probability versus kinetic energy for\na hydrogen beam under normal incidence on a Pd(100) surface.\nTheory: six-dimensional results for H$_2$ molecules initially in the \nrotational and vibrational ground state (dashed line)\nand with an initial rotational and energy distribution \nadequate for molecular beam experiments (solid line) \\protect\\cite{Gro95PRL}.\nH$_2$ molecular beam adsorption experiment under normal incidence\n(Rendulic {\\it et al.}~\\protect\\onlinecite{Ren89}): circles;\nH$_2$ effusive beam scattering experiment with an incident angle of\nof $\\theta_i = 15^{\\circ}$\n(Rettner and Auerbach~\\protect\\onlinecite{Ret96}): long-dashed line. }\n\\label{h2pdstick}\n\\end{figure} \n\n \n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,11.0)\n\\put(-1.5,-0.5){ {\\epsfysize=12.cm \n \\epsffile{H2Pd_PES_inset_bw.eps}} }\n\\put(2.5,-0.5){ {\\epsfysize=12.cm \n \\epsffile{H2Pd_PES_hth_bw.eps}} }\n\n \\end{picture}\n\n\\end{center}\n \\caption{Contour plots of the PES along a two two-dimensional cuts \n through the six-dimensional coordinate space of \n H$_2$\/Pd\\,(100), so-called elbow plots, determined by GGA \n calculations \\protect\\cite{Wil95,Wil96PRB}. The coordinates \n in the figure are the H$_2$ center-of-mass distance \n from the surface $Z$ and the H-H interatomic distance $d$.\n The dissociation process in the $Zd$ plane is illustrated\n in the inset. The lateral H$_2$ center-of-mass coordinates \n in the surface unit cell and the orientation of the molecular \n axis, i.e. the coordinates $X$, $Y$, $\\theta$, and $\\phi$ \n are kept fixed for each 2D cut and depicted above the \n elbow plots. Energies are in eV per H$_2$ molecule.\n The contour spacing in a) is 0.1~eV, while it is 0.05~eV in b).}\n\n\\label{h2pdelbow}\n\\end{figure} \n\n\nWilke and Scheffler have performed density-functional theory (DFT) calculations\nof the interaction of H$_2$ with Pd(100) in order to elucidate the dissociation\nprocess. In Fig.~\\ref{h2pdelbow} two so-called elbow plots are shown.\nThey represent a two-dimensional cut through the potential energy surface (PES)\nof H$_2$\/Pd(100) in which the orientation of the molecule, its \ncenter-of-mass lateral coordinates and the substrate are kept fixed. \nThe molecule is oriented parallel to the surface, and the PES is plotted\nas a function of the center-of-mass distance of the molecule from the\nsurface $Z$ and the interatomic distance $d_{\\rm H-H}$. Fig.~\\ref{h2pdelbow}a \ndemonstrates that the dissociation of H$_2$ on Pd(100) is non-activated, i.e.,\nthere are reaction paths towards dissociative adsorption with no energy\nbarrier. The majority of reaction pathways, however, is hindered by\nbarriers \\cite{Gro96CPLa}. \nFurthermore, in these calculations no molecular adsorption\nstate has been found. It looks like there is such a well in \nFig.~\\ref{h2pdelbow}b. However, the detailed DFT study of the PES\nhas shown that this apparent well does not correspond to a local minimum\nof the PES, it is rather a saddle point in the multi-dimensional PES.\n\n\nAn analytical representation of this {\\it ab initio} PES has been used\nfor a quantum dynamical study in which all six hydrogen degrees of freedom \nwere taken into account explicitly while the substrate was kept fixed\n\\cite{Gro96CPLa}. The results are also plotted in Fig.~\\ref{h2pdstick}.\nThe sticking curve for a monoenergetic beam initially in the vibrational and\nrotational ground state shows a strong oscillatory structure which will be \ndiscussed below. An experimental molecular beam, however, does not correspond\nto a monoenergetic beam in one specific quantum state. \nIf one assumes an energy spread and a distribution of internal\nmolecular states typical for a beam experiment, the oscillations\nare almost entirely smoothed out in the 6D quantum results\n(solid line in fig.~\\ref{h2pdstick}). The results corresponding to the\nbeam simulation agree with the experimental results semi-quantitatively.\nMore importantly, they reproduce the general trend found in the \nexperiment, namely the initial decrease of the sticking probability\nas a function of the kinetic energy followed by an increase at higher\nenergies. Now in the {\\it ab initio} PES there is no molecular adsorption\nstate, furthermore in the 6D quantum dynamical calculation no energy\ntransfer to the substrate is considered. Hence the precursor mechanism\ncannot be operative in the simulation. So what is the reason for the\ninitial decrease of the sticking probability?\n\nSince energy transfer cannot play a crucial role in the adsorption process,\nthis decrease in the sticking probability has to be caused by a purely \ndynamical effect, namely the steering\neffect \\cite{Gro95PRL,Gro95JCP,King78,Kay95,Whi96}:\nAlthough the majority of pathways to dissociative adsorption\nhas non-vanishing barriers with a rather broad distribution of\nheights and positions, slow molecules can be very efficiently\nsteered to non-activated pathways towards dissociative adsorption \nby the attractive forces of the potential. This mechanism becomes \nless effective at higher kinetic energies where the molecules are \ntoo fast to be focused into favourable configurations towards \ndissociative adsorption. If the\nkinetic energy is further increased, the molecules will eventually\nhave enough energy to directly traverse the barrier region leading\nto the final rise in the sticking probability.\n\n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n \\special{psfile=h2pd_snap.ps \n vscale=85 hscale=85 angle=-90 voffset=370 hoffset=-240 }\n\n \\end{picture}\n\n\\end{center}\n \\caption{Snapshots of classical trajectories of hydrogen molecules \nimpinging on a Pd(100) surface. The initial conditions are chosen in \nsuch a way that the trajectories are restricted to the $xz$-plane.\nLeft trajectory: initial kinetic energy $E_i = 0.01$~eV. \nRight trajectory: same initial conditions as in the left trajectory\nexcept that the molecule has a higher kinetic energy of 0.12 eV. }\n\n\\label{traj2run}\n\\end{figure} \n\n\n\nIn order to illustrate the steering effects, we use the results\nof classical molecular dynamics calculations which have been\nperformed on exactly the same PES as the quantum dynamical calculations\n\\cite{Gro97Vac}. In these classical calculations significant\ndifferences in the sticking probability compared to the quantum\nresults have been found which are mainly due to zero-point effects.\nThe steering effect, however, is a general mechanism operative in\nquantum as well as in classical dynamics. I will therefore use\nsnapshots of two typical trajectories in order to illustrate\nthe dynamical mechanism responsible for the initial decrease\nof the sticking probaiblity. These trajectories are plotted\nin Fig.~\\ref{traj2run}. The initial conditions are chosen in \nsuch a way that the trajectories are restricted to the $xz$-plane.\nThe left trajectory demonstrates why the sticking probability\nis so large at low kinetic energies due to the steering effect.\nThe incident kinetic energy is $E_i = 0.01$~eV. In this particular \nexample the molecular axis is initially almost perpendicular to the\nsurface. In such a configuration the molecule cannot dissociate\nat the surface. But the molecule is so slow that the forces \nacting upon it can reorient the molecule. It is turned parallel \nto the surface and then follows a non-activated path towards \ndissociative adsorption. This shows how molecule with unfavorable\ninitial conditions can still dissociate due to very efficient\nsteering towards favorable configurations.\n\nThis process becomes less effective at higher kinetic energies,\nwhich is demonstrated with the right trajectory in Fig.~\\ref{traj2run}.\nThe initial conditions are the same as for the left trajectory, except \nfor the higher kinetic energy of 0.12~eV. Of course the same forces\nact upon the molecule, and due to the anisotropy of the PES the molecule also\nstarts to rotate to a configuration parallel to the surface. However,\nnow the molecule is too fast to finish this rotation. It hits the \nrepulsive wall of the PES at the surface with its molecular axis tilted by \nabout 45$^{\\circ}$ with respect to the surface normal. At the classical \nturning point there is a very rapid rotation corresponding\nto a flip-flop motion, and then the molecule is scattered back into \nthe gas-phase rotationally excited.\n\n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.1)\n \\special{psfile=h2pd_scat.ps \n vscale=35 hscale=35 angle=-90 voffset=200 hoffset=-15 }\n \\end{picture}\n\n\\end{center}\n \\caption{Angular distribution of the in-plane and out-of-plane\n of scattering of H$_2$\/Pd(100). \n The initial kinetic energy is $E_i = 76$~meV, the\n incident angle is $\\theta_i = 32^{\\circ}$ along the\n $\\langle0 \\bar 1 1\\rangle$ direction. The molecules are initially\n in the rotational ground state $j_i = 0$.\n Open circles correspond to rotationally elastic, filled\n circles to rotationally inelastic diffraction.\n The radii of the circles are proportional to the logarithm \n of the scattering intensity. \n $x$ denotes the $\\langle0 \\bar 1 1\\rangle$ direction, $y$ the\n $\\langle0 1 1\\rangle$ direction. The specular peak is the largest\n open circle (from Ref.~\\protect\\cite{Gro96CPLb}). }\n\n\\label{inout}\n\\end{figure} \n\nNow I like to come back to the strong oscillatory structure of the\nsticking curve for a monoenergetic beam initially in one particular\nquantum state in Fig.~\\ref{h2pdstick}. \nThese oscillations are a consequence of the quantum\nnature of the hydrogen particle, in classical calculations they\ndo not appear \\cite{Gro98PRB,Gro97Vac}. If a quantum particle is\ninteracting with a periodic surface, coherent scattering leads to\ndiffraction. Such a calculated diffraction pattern is shown in \nFig.~\\ref{inout} for hydrogen molecules in the ground state\nscattering at a Pd(100) surface with an kinetic energy of $E_i = 76$~meV under \nan angle of incidence of $\\theta_i = 32^{\\circ}$. There are rather\nmany diffraction peaks since also rotationally inelastic diffraction \ncan occur, i.e., scattering in which the rotational state of the\nmolecule is changed. Still the number of diffraction\npeaks is finite and increases discontinously with increasing energy.\nAn analysis of the oscillatory structure of the sticking\nprobability in Fig.~\\ref{h2pdstick} reveals that these oscillations\ncan be related to threshold effects associated with the opening of \nnew scattering channels \\cite{Gro96CPLb,Gro96PRL}.\nThese oscillations have not been observed experimentally yet,\nalthough they have been carefully searched for \\cite{Ret96,Ret96PRL}.\nThey are very sensitive to the symmetry of the initial conditions\n\\cite{Gro96CPLb,Gro96PRL}. Furthermore, since Pd(100) is a very\nreactive surface, a large fraction of the incoming hydrogen molecules\nis not scattered back coherently but adsorbs dissociatively. These adatoms\nthen disturb the periodicity of the surface and thus suppress,\nin addition to already existing surface imperfections like steps\nand vacancies, the coherence of the scattering events. Hence \nthe experimental observation of this oscillations actually represents\na challenging task. \n\n\nThe dependence of adsorption and desorption \non kinetic energy, molecular rotation and orientation\n\\cite{Gro95PRL,Gro96SSb}, molecular vibration \\cite{Gro96CPLa}, ro-vibrational\ncoupling \\cite{Gro96Prog}, angle of incidence \\cite{Gro98PRB}, \nand the rotationally elastic and inelastic \ndiffraction of H$_2$\/Pd(100) \\cite{Gro96CPLb} have been studied so far\nby six-dimensional {\\it ab initio} dynamics calculations\non the same PES. The results of these calculations have been compared \nto a number of independent experiments \\cite{Ren89,Sch92,Beu95,Wet96}, and \nthey are at least in semi-quantitative agreement with all of these experiments\nThis shows that {\\it ab initio} dynamics calculations are indeed\ncapable of adequately describing the hydrogen dissociation\non transition metal surfaces. \n\n\n\n\\section{Dissociative adsorption at a sulfur-covered transition metal surface}\n\n\n\nThe presence of an adsorbate on a surface can profoundly change the \nsurface reactivity. A well-known example is the reduction of the activity\nof the car-exhaust catalyst by lead. But also adsorbed sulfur ``poisons''\nthis catalyst. An understanding of the underlying mechanisms and their \nconsequences on the reaction rates is therefore of decisive importance\nfor, e.g., designing better catalysts. \nTraditionally an ``trial and error'' approach\nwas used to improve the activity of a catalyst by adding some\nsubstances. \nOn Pd(100) it is experimentally well-known that the presence of sulfur \nleads to a large reducting of hydrogen dissociation \nprobability \\cite{Ren89,Bur90}. While at the clean surface the dissociation\nprobability is about 60\\% for a kinetic energy of $E_{\\rm i} = 0.05$~eV,\nat the sulfur-covered surface it drops below 1\\% at the same\nenergy \\cite{Ren89}.\n\nTheoretically this problem had only been addressed by a small \nnumber of studies. These focused either on the \nadsorbate induced change of the density of states (DOS) at the Fermi level\n\\cite{Fei84,Fei85,MacL86} or on the adlayer induced electrostatic\nfield \\cite{Nor84,Nor93,Ham93}. Just recently, the poisoning of hydrogen \ndissociation on Pd(100) by sulfur adsorption has been the subject of\ndetailed DFT studies \\cite{Wil95,Wil96S,Wei97}.\n\n\n\n\\begin{figure*}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(18,8.5)\n\\put(-1.,-4.){ \\rotate[r]{\\epsfysize=20.cm \n \\epsffile{H2SPd_PES_applphys.ps}} }\n \\end{picture}\n\n\\end{center}\n\\caption{ Cuts through the six-dimensional potential \n energy surface (PES) of H$_2$ dissociation over\n (2$\\times$2)S\/Pd(100) at four different sites with\n the molecular axis parallel to the surface:\n a) at the fourfold hollow site;\n b) at the bridge site between two Pd atoms;\n c) on top of a Pd atom;\n d) on top of a S atom.\n The energy contours, given in eV per molecule, \n are displayed as a function of the H-H distance, \n $d_{H-H}$, and the height $Z$\n of the center-of-mass of H$_2$ above the topmost Pd layer. \n The geometry of each dissociation pathway is indicated \n in the panel above the contour plots. The large open circles\n are the sulfur atoms, the large filled circles are the\n palladium atoms.}\n\n\\label{H2SPd_PES}\n\\end{figure*} \n\n\n\n\n\nIn Fig.~\\ref{H2SPd_PES} I have collected four elbow plots of the hydrogen\ndissociation on the (2$\\times$2) sulfur covered Pd(100) surface \\cite{Wei97}.\nIn contrast to the clean Pd(100) surface, the hydrogen dissociation \non the sulfur covered surface is no longer non-activated. The minimum\nbarrier, which is shown in Fig.~\\ref{H2SPd_PES}a, has a height of\n0.1~eV and corresponds to a configuration in which the H$_2$ center of\nmass is located above the fourfold hollow site. This is the\nsite which is farthest away from the sulfur atoms in the surface unit cell.\nRecall that the most favorable reaction path on the {\\em clean} Pd(100)\nsurface corresponds to the H$_2$ molecule dissociating at the bridge position\nbetween two Pd atoms (see Fig.~\\ref{h2pdelbow}a). For this approach\ngeometry the dissociation at the sulfur covered surface is now\nhindered by a barrier of height 0.15~eV (Fig.~\\ref{H2SPd_PES}b). \nThere is a peculiar local minimum at the dissociation path for this\nconfiguration when the molecule is still 1\\mbox{\\AA} above the Pd atoms.\nThere are apparently subtle compensating effects between the attractive\ninteraction of H$_2$ with the Pd atoms and the repulsion originating\nfrom the S atoms. Figs.~\\ref{H2SPd_PES}a and b show \nthat the dissociation is hindered by the formation of energy barriers\nin the entrance channel of the potential energy surface, however, \nthe hydrogen dissociation is still exothermic, i.e., the poisoning\nis not due to site-blocking. This result is actually at variance\nwith measurements of the hydrogen saturation coverage as a function\nof the sulfur coverage \\cite{Bur90}.\nIn these experiments a linear decrease of the hydrogen saturation \ncoverage with increasing sulfur coverages was found. At a sulfur coverage\nof $\\Theta_{\\rm S} = 0.28$, which is close to the one used in the\ncalculations, hydrogen adsorption should be completely suppressed, i.e.,\nthere should be no attractive sites for hydrogen adsorption any more.\nA possible explanation for this apparent contradiction will be given\nbelow.\n\nCloser to the sulfur atoms the PES becomes strongly repulsive.\nThis is illustrated in Fig.~\\ref{H2SPd_PES}c and d. While the dissociation \npath over the Pd on-top position on the clean surface is hindered by a \nbarrier of height 0.15~eV \\cite{Wil96PRB} (Fig.~\\ref{h2pdelbow}b), \nthe adsorbed sulfur leads to an increase in this barrier height \nto 1.3~eV (Fig.~\\ref{H2SPd_PES}c). Directly above the sulfur atoms \nthe barrier towards dissociation even increases to values \nof about 2.5~eV for molecules oriented parallel to\nthe surface (Fig.~\\ref{H2SPd_PES}d). \n\n\nThe goal of any theoretical study should be to provide a qualitative\npicture that explains the calculated results. There are many different\nways of illustrating the electronic factors that determine the\nreactivity of a particular system \n(see, e.g., Refs.~\\cite{Wil96PRB,Eich96,Fei84,Ham95PRL,Wil96AP}). \nCurrent studies have emphasized that the reactivity of surfaces\ncannot be solely understood by the electronic density of states\nat the Fermi level \\cite{Ham95,WCoh96}. In order to understand the origins of \nthe formation of the small energy barriers at the hollow and bridge site and \nthe large energy barriers at the top sites, we will therefore compare the \nwhole relevant DOS for the H$_2$ molecule in these different geometries. \nFor a discussion of the reactivity of the clean Pd(100) surface I refer\nto Ref.~\\cite{Wil96PRB}. Here I focus on the changes of the density of states \ninduced by the presence of sulfur on the surface (Fig.~\\ref{DOS}).\n\n\n\n\\begin{figure*}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(18,6.5)\n\\put(-3.,-8.){ \\rotate[r]{\\epsfysize=24.cm \n \\epsffile{H2SPd_DOS.ps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{ Density of states (DOS) for a H$_2$ molecule situated at \n (a) ($ Z,d_{\\rm H-H}$)= (4.03\\AA, 0.75\\AA) and\n (b) ($Z,d_{\\rm H-H}$)= (1.61\\AA, 0.75\\AA) above the fourfold\n hollow site which corresponds to the configuration\n depicted in Fig.~\\protect\\ref{H2SPd_PES}a,\n and for a H$_2$ molecule situated at\n (c) (Z,d$_{H-H}$)= (3.38\\AA, 0.75\\AA) above the sulfur\n atom which corresponds to the configuration\n depicted in Fig.~\\protect\\ref{H2SPd_PES}d.\n $Z$ and $d_{\\rm H-H}$ denote the H$_2$ center-of-mass distance \n from the surface and the H-H interatomic distance, respectively.\n Given is the local DOS at the H atoms, the S adatoms, \n the surface Pd atoms, and the bulk Pd atoms. \n The energies are given in eV.}\n\n\\label{DOS}\n\\end{figure*} \n\n\nThe information provided by the density of states alone is often\nnot sufficient to assess the reactivity of a particular system.\nIt is also important to know the character of the occupied and\nunoccupied states. For the dissociation the occupation\nof the bonding $\\sigma_{\\rm g}$ and the anti-bonding $\\sigma_{\\rm u}^*$ \nH$_2$ molecular levels {\\em and} of the bonding and anti-bonding\nstates with respect to the surface-molecule interaction are of\nparticular importance.\n\n\n\nFigure~\\ref{DOS}a shows the DOS when the H$_2$ molecule is \nstill far away from the surface above the fourfold hollow site,\ni.e, in the configuration that corresponds to Fig.~\\ref{H2SPd_PES}a.\nThe H-H distance $d$ is 0.75 \\AA\\ and the center of mass of the H$_2$ \nmolecule is 4.03 \\AA\\ above the topmost Pd layer so that \nthere is no interaction between the hydrogen molecule and \nthe sulfur covered palladium surface.\nThe large peak in the sulfur DOS at -13~eV corresponds to the S 3$s$ state.\nThe sulfur {\\it p} orbitals strongly interact with the Pd {\\it d} states, \nwhich is evident from the peak in the sulfur DOS at the \nPd {\\it d} band edge (at E-E$_F$ = -4.8 eV) and from\nthe broad band at higher energies which has substantial \nweight close to the Fermi level. \nThe {\\it d} band at the surface Pd atoms is broadened and shifted\ndown somewhat with respect to the clean surface due to the interaction \nwith the S atoms \\cite{Wil96S}. There is one intense peak in the\nhydrogen DOS at -4.8 eV which corresponds to the $\\sigma_{\\rm g}$ state.\nThis peak is degenerate with the sulfur related bonding state at -4.8 eV, \nthis degeneracy, however, is accidental, as will become evident \nimmediately.\n\n\nThe density of states for the molecule at the minimum barrier position\nof Fig.~\\ref{H2SPd_PES}a is shown in Fig.~\\ref{DOS}b. Now the\n$\\sigma_{\\rm g}$ state has shifted down to -7.1 eV while the\nsulfur state at -4.8 eV remains almost unchanged. This indicates that\nthere is no direct interaction between hydrogen and sulfur. It also\nproofs that the degeneracy between these two states in Fig.~\\ref{DOS}a\nis accidental. Furthermore, we find a broad distribution of hydrogen \nstates with a small, but still significant weight below the Fermi level.\nThese are states of mainly H$_2$-surface antibonding character\n\\cite{Wil96S,Wei97}. A comparison with the hydrogen dissociation\nat the clean Pd surface yields that more H$_2$-surface antibonding\nstates are populated at the sulfur covered surface. This is caused\nby the sulfur induced downshift of the Pd $d$-band. These \nH$_2$-surface antibonding states lead to a repulsive interaction and thus \nto the building up of the barriers in the entrance channel of\nthe PES \\cite{Wil96S}. It is therefore an indirect interaction\nbetween sulfur and hydrogen that is responsible for the barriers\nat this site. A similar picture explains why for example noble\nmetals are so unreactive for hydrogen dissociation: The low-lying\n$d$-bands of the noble metals cause a downshift and a substantial \noccupation of the antibonding H$_2$-surface states resulting\nin high barriers for hydrogen dissociation \\cite{Ham95PRL}. \n\n\nThe situation is entirely different if the molecule approaches\nthe surface above the sulfur atom. This is demonstrated\nin Fig.~\\ref{DOS}c. The center of mass of the H$_2$ molecule is\nstill 3.38~\\AA\\ above the topmost Pd layer, but already at this\ndistance the hydrogen and the sulfur states strongly couple. The intense \npeak of the DOS at \\mbox{-4.8 eV} has split into a sharp bonding state \nat -6.6 eV and a narrow anti-bonding state at -4.0 eV.\nThus it is a direct interaction of the hydrogen with the sulfur related\nstates that causes the high barriers towards hydrogen dissociation\nclose to the sulfur atoms. In conclusion, the poisoning of hydrogen\ndissociation on Pd(100) by adsorbed sulfur is due to a combination\nof a indirect effect, namely the sulfur-related downshift of the Pd $d$-bands \nresulting in a larger occupation of H$_2$-surface antibonding states,\nwith a direct repulsive interaction between H$_2$ and S close to the\nsulfur atoms. \n\nIn order to assess the dynamical consequences of the sulfur adsorption\non the hydrogen dissociation, six-dimensional dynamical calculations \non the analytical representation of the {\\it ab initio} PES of H$_2$ at \nS(2$\\times$2)\/Pd(100) have been performed \\cite{Gro98PRL}.\nThe results of these quantum and classical calculations for the\nH$_2$ dissociative adsorption probability as a function of the incident\nenergy are compared with the experiment \\cite{Ren89} in Fig.~\\ref{stick}.\nIn addition, also the integrated barrier distribution $P_b(E)$,\n\\begin{eqnarray} \\label{barr}\nP_b (E) & = & \\frac{1}{2\\pi A} \\ \\int \n\\Theta (E - E_{\\rm b} (\\theta, \\phi, X, Y)) \\nonumber \\\\\n& & \\times \\ \\cos \\theta d\\theta \\ d\\phi \\ dX \\ dY\n\\end{eqnarray}\nis plotted. Here $\\theta$ and $\\phi$ are the polar and azimuthal\norientation of the molecule, $X$ and $Y$ are the lateral coordinates of \nthe hydrogen center-of-mass. $A$ is the area of the surface unit cell.\nEach quadruple defines a cut through the six-dimensional space (see\nFig.~\\ref{H2SPd_PES} for examples), \nand $E_{\\rm b}$ is the minimum energy barrier \nalong such a cut. The function $\\Theta$ is the Heavyside step function. \nThe quantity $P_b(E)$ is the fraction \nof the configuration space for which the barrier towards dissociation\nis less than $E$. If there were no steering effects, $P_b(E)$ would\ngive the classical sticking probability, i.e., it corresponds to the \nsticking probability in the classical sudden approximation or the so-called \n``hole model'' \\cite{Kar87}. \n\n\n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\centerline{ \\rotate[r]{\\epsfysize=8.cm \n \\epsffile{h2spd_stick.eps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{Sticking probability versus kinetic energy for\na H$_2$ beam under normal incidence on a S(2$\\times$2)\/Pd(100) surface.\nFull dots: experiment (from ref.~\\protect\\cite{Ren89});\nDashed-dotted line: Integrated barrier distribution,\nwhich corresponds to the sticking probability in the hole \nmodel \\protect\\cite{Kar87};\nSolid line: Quantum mechanical results for molecules initially in the\nrotational and vibrational ground-state;\nDashed line: Classical results for initially non-rotating and non-vibrating\nmolecules. The inset shows the quantum and classical results at low \nenergies.}\n\n\\label{stick}\n\\end{figure} \n\n\nFirst of all it is evident that the calculated sticking probabilities are \nsignificantly larger than the experimental results. \nOnly the onset of dissociative adsorption at $E_{\\rm i} \\approx 0.12$~eV \nis reproduced by the calculations. This onset is indeed also in agreement\nwith the experimentally measured mean kinetic energy of hydrogen \nmolecules desorbing from sulfur covered Pd(100) \\cite{Com80}.\nThe question arises where these large differences between theory and\nexperiment come from. It might be that uncertainties in the experimental \ndetermination are responsible for the difference. \nThe exact sulfur coverage in the \nexperiment was not very well characterized. The sulfur adlayer was obtained \nby simply heating up the sample which leads to segregation of bulk sulfur \nat the surface. The sulfur coverage was then monitored\nthrough the ratio of the Auger peaks S$_{132}$\/Pd$_{330}$ \\cite{Ren89}.\nSince the hydrogen sticking probability depends sensitively on the\nsulfur coverage \\cite{Ren89,Bur90}, a small uncertainty in the\nsulfur coverage can have a decisive influence. However, as noted above,\nwhile the DFT calculations yield that the poisoning is caused by\nthe building up of barriers hindering the dissociation, the vanishing\nhydrogen saturation coverage for roughly a quarter monolayer of adsorbed\nsulfur \\cite{Bur90} suggests that any attractive adsorption sites \nfor hydrogen have disappeared due to the presence of sulfur.\nThese seemingly contradicting\nresults and also the discrepancy between calculated and measured molecular\nbeam sticking probabilities could be reconciled if subsurface sulfur \nplays an important role for the hydrogen adsorption energies.\nSubsurface sulfur is not considered in the calculations but \nmight well be present in the experimental samples. The possible\ninfluence of subsurface species on reactions at surfaces certainly\nrepresents a very interesting and important research subject\nfor future investigations.\n\nExcept for this open question, there are further interesting\nresults obtained by the dynamical calculations. The calculated\nsticking probabilities are not only much larger than the\nexperimental ones, they are also much larger than what one would\nexpect from the hole model. This demonstrates that steering is\nnot only operative for potential energy surfaces with non-activated\nreaction paths like for H$_2$\/Pd(100), but also for activated\nsystems as H$_2$\/S(2$\\times$2)\/Pd(100). As Fig.~\\ref{H2SPd_PES} \ndemonstrates, the sulfur covered Pd surface represents a \nstrongly corrugated system with barrier heights varying by\nmore than 2~eV for molecules with their axis parallel to the\nsurface. And the large barriers above the sulfur atoms extend\nrather far into the gas phase (see Fig.~\\ref{H2SPd_PES}d).\nMolecules with unfavorable initial conditions are very effectively\nreoriented to low-barrier sites \\cite{Gro98PRL}. This leads to\nan enhancement of the sticking probability with respect to the\nhole model by a factor of three to four. \n\n\nFigure \\ref{stick} shows furthermore that the classical molecular\ndynamics calculations over-estimate the sticking probability \nof H$_2$ at S(2$\\times$2)\/Pd(100) compared to the quantum results. \nAt small energies below the minimum barrier height the quantum calculations \nstill show some dissociation due to tunneling, as the inset of \nFig.~\\ref{stick} reveals, whereas the classical results are of course zero. \nBut for higher energies the classical sticking probability is up to \nalmost 50\\% larger than the quantum sticking probabilities. \nThis suppression is also caused by the large corrugation and the\nanisotropy of the PES. The wave function describing the \nmolecule has to pass narrow valleys in the PES in the angular and lateral \ndegrees of freedom in order to dissociate. This leads to a localization\nof the wave function and thereby to the building up of zero-point energies\nwhich act as additional effective barriers. While\nthe vibrational H-H mode becomes softer upon dissociation so that\nthe zero-point energy in this particular mode decreases,\nfor the system H$_2$\/S(2$\\times$2)\/Pd(100) this\ndecrease is over-compensated by the increase in the zero-point\nenergies of the four other modes perpendicular to the reaction\npath, i.e., the sum of {\\em all} zero-point energies increases\nupon adsorption \\cite{Gro98PRL}. Therefore the quantum particles\nexperience an effectively higher barrier region causing the\nsuppressed sticking probability compared to the classical\nparticles. Interestingly enough, if the sum of all zero-point energies\nremains approximately constant along the reaction path as in the\nsystem H$_2$\/Pd(100), then these quantum effects almost cancel \nout \\cite{Gro98PRB,Gro97Vac}. \n\n\n\n\\section{Conclusions}\n\nIn this review {\\it ab initio} studies of reactions\non surfaces have been presented. In the last years the interaction between\nelectronic structure calculations on the one side and dynamical calculations\non the other side has been very fruitful. The availability of high-dimensional\nreliable potential energy surfaces has challenged the dynamics community\nto improve their methods in order to perform high-dimensional dynamical \nstudies on these potentials. Now quantum studies of the dissociation of \nhydrogen on surfaces are possible in which all six degrees of \nfreedom of the molecule are treated dynamically. In this review I have\ntried to show that this achievement represents an important step forward \nin our understanding of the interaction of molecules with surfaces.\nNot only the quantitative agreement with experiment is improved, but\nalso important qualitative concepts emerge from the electronic structure\ncalculations as well as from the high-dimensional dynamical simulations.\nThese concepts are applicable to any reaction system. This represents the \nimportance of hydrogen as a model system for studying reactions on surfaces.\nHowever, the {\\it ab initio} treatment of reactions on surfaces has now \nmatured enough so that in the future there will be also more studies\non other reaction systems like for example the important class of\noxidation reactions on surfaces. This will be the next step towards\na full microscopic description of catalytic reactions, one of the\nultimate goals of surface science. \n\n\n\\section*{Acknowledgements}\n\nI am very grateful to my collegues and coworkers who have made this work \npossible. I would like to mention in particular Thomas Brunner, \nBj{\\o}rk~Hammer, Ralf~Russ, Ching-Ming Wei, Steffen~Wilke, \nand last but not least my supervisors\nHelmar Teichler, Wilhelm Brenig and Matthias Scheffler.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Verification approaches}\nIn this section, we briefly discuss our experience in using hybrid system verification tools. \n{\\bf SpaceEx\\\/}~\\cite{DBLP:conf\/cav\/FrehseGDCRLRGDM11}, is a well-established reachability analysis tool for linear and affine hybrid systems. It implements the support function-based reachability algorithm, includes the PHAVer algorithm for rectangular dynamics~\\cite{Frehse:hscc05}, and also a simulator for nonlinear models. \nThe support function representation of sets is amenable to effective computation of convex hulls, linear transforms, Minkowski sums, etc.---operations that are necessary for safety verification. \n\n{\\bf C2E2\\\/}~\\cite{DuggiralaMVP15,FanQM0D16:CAV} is a simulation-driven bounded verification tool for nonlinear hybrid models. \nThe core algorithm of C2E2 relies on computing reachset over-approximations from validated numerical simulations and what are called {\\em discrepancy functions.\\\/} A discrepancy function for a model bounds the sensitivity of the trajectories of the hybrid system to changes in initial states and inputs. Candidate discrepancy functions can be obtained using a global Lipschitz or using a matrix norm for linear systems.\nHowever, typically these approaches give discrepancy functions that blow-up exponentially with time, and therefore, are not useful for verifying problems with long time horizons.\nThe automatic on-the-fly approach implemented in~\\cite{FanQM0D16:CAV} uses bounds on the Jaobian matrix of the system to get tighter local discrepancy functions and it has been used to verify several benchmark problems. Recently the tool has been extended to handle nonlinear models with dynamics with exponential and trigonometric functions. \n\nFor a (possibly nonlinear) mode with $\\dot x = f(x(t))$, the discrepancy computed by the algorithm of~\\cite{FanQM0D16:CAV} uses the Jacobian matrix $J(x)$ of $f(x)$ and the condition number of $J(x_0)$ evaluated at certain points $x_0$ in the state space. For ill-conditioned matrices, such as what we have in the passive mode, (the $A$-matrix representation of \\eqref{eq:lin1}), the over-approximation error may still blow-up.\nIll-conditioned systems may not only arise from passive dynamics but also from extremely large and small coefficients appearing together in $J(x_0)$.\n\nIn order to address this problem, we have created a MATLAB implementation of C2E2's verification algorithm ({\\bf{\\sf SDVTool\\\/}{}}) for linear models.\nUnlike C2E2, {\\sf SDVTool\\\/}{} \\ does not rely on discrepancy, but instead computes the reachable states under a given linear mode directly.\nThe particular algorithm implemented is the one presented in~\\cite{Duggirala2016}:\nFor an $n$-dimensional system, $n+1$ simulations are performed. From these simulations, special sets called \\emph{generalized star sets}, are generated to represent the exact reachsets. \nFor our purposes, a generalized star set is represented by a pair $\\langle x_0,V \\rangle$, where \n\t$x_0 \\in \\mathbb{R}^n$ is the center state and \n\t$V=\\{v_1,...,v_n \\} \\subseteq \\mathbb{R}^n$ is a standard basis (not necessarily unit vectors), and \n\nthe set defined by $\\langle x_0,V \\rangle$ is \n $$\\{x \\in \\mathbb{R}^n \\ | \\ \\exists \\alpha_1, \\ldots, \\alpha_n \\in [-1,1], x = x_0 + \\sum_{i=1}^n \\alpha_i v_i \\}.$$\nAs reachsets are calculated for time steps, $x_0$ and $V$ are transformed.\nWhen the reachtube from a given mode intersects the guards for a transition, the star sets are aggregated and over-approximated with hyperrectangles. \n If $R_i ^*= $ is the star set reachset obtained at time $t_i$, then the hyperrectangular reachset is: \n\\begin{equation*}\n\\begin{aligned}\nR_i = \\{x \\ | \\ x\\leq x(t_i)+\\sum_{j=1}^n max(-v_j,v_j) \\text{ }\\\\ \\text{and } x\\geq x(t_i)+\\sum_{j=1}^n min(-v_j,v_j) \\}.\n\\end{aligned}\n\\end{equation*}\n\nC2E2 and {\\sf SDVTool\\\/}{} currently accumulates all the reachable sets in ProxA and ProxB that \\emph{may} transition to Passive, and uses their convex hull to begin reachset computations under the Passive mode. It follows that if the time interval during which a transition may occur is large, then the initial set of states under the Passive mode is large, making it very difficult to prove safety. One solution is to allow partitioning and refinement of the initial passive mode set. Since this is not currently implemented in C2E2 or {\\sf SDVTool\\\/}{}, we restrict our experiments to transition interval lengths of 5 minutes or less. \nFor example, checking if the system is safe for a transition $clock\\in [50, 200\\text{ min}]$ could be achieved by running several experiments with small subintervals that cover the original interval.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe are grateful for the support of Richard S. Erwin in navigating and modeling the problem presented in this paper, and for Yu Meng's support with the C2E2 experiments. We acknowledge the support of the Air Force Research Laboratory through the Space Scholars Program. \n\n\\bibliographystyle{abbrv}\n\n\\section{Conclusions and future directions}\n\\label{sec:conclusion}\n\n\nIn this case study paper, we present a sequence of linear and nonlinear, nondeterministic benchmark models of autonomous rendezvous between spacecraft with several physical and geometric safety requirements. We designed an LQR controller and verified its safety across the different models, a variety of initial conditions, parameter ranges, and using three different hybrid system verification approaches. The models and requirements are made available online.\n\nThis case study, and in particular the requirement for passive safety, has shed light on the weakness of simulation-driven verification in handling ill-conditioned models.\n\nThe results provide a foundation for verifying more sophisticated maneuvers in future autonomous space operations.\nFor example, we proposed a continuous full-state feedback controller, but it is also possible to consider a situation where full state measurement is not possible and a simple bang-bang controller is required. Control theory tells us that this system maintains marginal stability which implies that errors will never recede, so for reasonably-sized initial sets, the reachable sets may not satisfy tight constraints such as LOS. \n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nA new age of deep space exploration is underway with several ongoing public-private partnerships. A groundwork for a possible mission to Mars in the 2030s is also underway~\\cite{obama_mars}.\nAutonomous operations where a spacecraft can operate independent of human control in a wide variety of conditions are essential for deployment, construction, and maintenance missions in space. \nDespite many spectacular successes like the Mars landing of the Curiosity rover, ensuring safety of autonomous spacecraft operations remains a daunting challenge. The cost of failures can be extreme. \nFor example, NASA's DART spacecraft was designed to rendezvous with the MUBLCOM satellite. In 2005, approximately 11 hours into a 24-hour mission, DART's propellant supply depleted due to excessive use of thrusters, and it began maneuvers for departure. In the process it collided with MUBLCOM; it met only 11 of 27 mission objectives, and the failure resulted in a loss exceeding \\$1 million. \nIn another incident, a navigation error caused the Mars Climate Orbiter to reach as low as 57 kilometers, where it was intended to enter an orbit at an altitude of 140-150 kilometers. The spacecraft was destroyed by the resulting atmospheric stresses and friction and the cost incurred was \\$85 million. \nThese incidents, and several others in~\\cite{wong:sw} highlight the consequences of failures in space applications and demonstrate the need for more rigorous testing before deployment. \n\n\nAlthough formal verification has played an important role in design and safety analysis of spacecraft hardware and software (see, for example~\\cite{Holzmann:2014:MC} and the references therein), they have not been used for model-based design and system-level verification and validation.\nIn this paper, we present and verify a realistic and challenging spacecraft maneuver problem called autonomous rendezvous, proximity operations, and docking (ARPOD). \nThe original hybrid control design problem and its variants are introduced by Jewison and Erwin in~\\cite{erwin-cdc16}. \nARPOD is a fundamental set of operations for a variety of space missions such as on-orbit transfer of personnel~\\cite{Woffinden:rendezvous}, resupply for on-orbit personnel~\\cite{Pinard2007}, assembly~\\cite{Zimpfer:rendezvous}, \nservicing, repair, and refueling~\\cite{Galabova2003}.\n\nThe basic setup for ARPOD consists of a passive module or a {\\em target\\\/} (launched separately into orbit) and a {\\em chaser\\\/} spacecraft that must transport the passive module to an on-orbit assembly location. The chaser maintains a relative bearing measurement to the target, but initially it may be too far away from the target to use its range sensors. Range measurements become available within a given range, giving the chaser accurate relative positioning data so that it can stage itself to dock the target. The target must be docked with a specific angle of approach and closing velocity, so as to avoid collision and ensure that the docking mechanisms on each body will mate. Furthermore, the docking procedure must be completed before the chaser goes into eclipse and loses vision-based sensor data. \nFinally, it is necessary for the system to ensure {\\em passive safety\\\/}. That is, the chaser spacecraft should maintain safe separation from the target even if it loses power and communication during its mission. \n\nIn this paper, we present a suite of hybrid models for the rendezvous portion of the ARPOD mission that can serve as benchmarks for verification tools and serve as building-blocks for more complex operations. \nWe present nonlinear models that consist of nonlinear orbital dynamics ({\\sf NLinProx\\\/}{} and {\\sf NLinProxTh\\\/}{}) and linearized models ({\\sf LinProx\\\/}{} and {\\sf LinProxTh\\\/}{}) using the Clohessy-Wiltshire-Hill (CWH) equations~\\cite{1960}.\nThe rendezvous operation is further subdivided into two phases: Proximity Operations A and B, such that phase A captures an interval of larger ranges than the interval of ranges in phase B. In other words, the chaser enters phase A first and as it moves closer to the target, it enters phase B. After this two-phase rendezvous, the chaser enters a docking phase\/maneuver (i.e. when the chaser is less than 10 meters from its target). We disregard the requirements for this phase for this paper. We develop a switched state-feedback controller using Linear Quadratic Regulation (LQR) for the rendezvous phases. \nThe position-triggered transitions brought about by the switching controller are urgent, resulting in a deterministic hybrid automaton. However, we extend the ARPOD problem in~\\cite{erwin-cdc16} to include the passive safety requirement by introducing a nondeterministic time-triggered transition to the passive mode.\n\nWe have successfully verified the requirements for most of the models using existing hybrid verification tools SpaceEx~\\cite{DBLP:conf\/cav\/FrehseGDCRLRGDM11} and C2E2~\\cite{DuggiralaMVP15,FanQM0D16:CAV},\nand also our new MatLab implementation of a simulation-driven verification algorithm ({\\sf SDVTool\\\/}{}) for linear hybrid models. {\\sf SDVTool\\\/}{} improves C2E2's reachability algorithm, with a new technique~\\cite{Duggirala2016} for obtaining reachsets for linear systems.\nWe obtain verification results for an array of varied initial state configurations and passive transitions times to show the robustness and limits of the switched LQR controller. \nThe experiments, in particular on the passive safety requirement, have demonstrated a weakness in simulation-driven verification approaches in handling ill-conditioned models which suggest a need for further research.\nOverall, we believe that our results and approaches establish feasibility of system-level verification of autonomous space operations, and they provide a foundation for the analysis of more sophisticated maneuvers in the future.\n\n\n\\section{Related work}\n\\label{sec:related}\n\nThere are few academic works on system-level verification of autonomous spacecraft. \nA survey of general verification approaches and how they may apply to small satellite systems is presented in~\\cite{nasa_survey}.\n Architecture and Analysis Design Language (AADL) and verification and validation (V\\&V) over AADL models for satellite systems have been reported in~\\cite{bozzano2010formal}\n\nAn feasibility study for applying formal verification of \nautonomous satellite maneuvers is presented in~\\cite{JGMDE:satellite2012}.\nThat approach relied on creating rectangular abstractions (dynamics of the form $\\dot x \\in [a,b]$) of the satellites dynamics through hybridization and verification using PHAVer~\\cite{Frehse:hscc05} and SpaceEx~\\cite{DBLP:conf\/cav\/FrehseGDCRLRGDM11}. \nThe generated abstract models have simple dynamics but hundreds of locations, and also, the analysis is necessarily conservative. In contrast, the approaches presented in this paper work directly with the linear (nonlinear) hybrid dynamics.\n\nThe ARPOD challenge~\\cite{erwin-cdc16} has been taken up by several researchers in proposing a variety of control strategies. \nA two-stage optimal control strategy is developed in~\\cite{Farahani-cdc16}, where the first part involves trajectory planning under a differentially-flat system and the second part implements Model Predictive Control on a linearized model. \nA supervisor is introduced to robustly coordinate a family of hybrid controllers in~\\cite{sanfelice2016cdc}. \nSafe reachsets (i.e. ReachAvoid sets) are computed for the ARPOD mission in~\\cite{oishi2016cdc} and used to solve for minimum fuel and minimum time trajectories. \n\n\\section{Spacecraft Rendezvous Model}\n\\label{sec:model}\n\nIn this section, we present the detailed development of the hybrid models. First we present the orbital dynamics of the spacecraft in Sections~\\ref{ssec:dynamics}-\\ref{ssec:lindynamics}. Then in Sections~\\ref{sec:hybrid-control}-\\ref{sec:lqr} we present a hybrid controller. Finally, we state the various mission constraints in Section~\\ref{sec:safeSets}. \n\n\\subsection{Nonlinear relative motion dynamics}\n\\label{ssec:dynamics}\n\nThe dynamics of the two spacecraft in orbit---the {\\em target\\\/} and the {\\em chaser\\\/}---are derived from Kepler's laws. We use the simplest case for relative motion in space, where the two spacecraft are restricted to the same orbital plane, resulting in two-dimensional, planar motion. The so called Hill's relative coordinate frame is used. As shown in Figure~\\ref{Hill}, Hill's frame is centered on the target spacecraft, with $+\\hat{\\*i}$-direction pointing radially outward from the Earth, $+\\hat{\\*k}$-direction normal from the orbital plane, and $+\\hat{\\*j}$-direction completing a right-handed system. We further assume that the target moves on a circular orbit, and thus, the $\\hat{\\*j}$-direction aligns with the tangential velocity of the target. \n\nThe restriction on the target's orbit implies that the target-centered frame rotates with constant angular velocity. We will assume the target is in geostationary equatorial orbit (GEO), so its angular velocity is $n = \\sqrt{\\frac{\\mu}{r^3}}$, where $\\mu=3.698\\times 10^{14} m^3\/s^2$ and $r=42164 km$. The chaser's position is represented by the vector $x \\*i + y\\*j $, and the chaser's thrusters provide acceleration in the form of $F_x\\*i + F_y\\*j$. The following equations are derived using Kepler's laws and constitute the nonlinear model of the spacecraft dynamics.\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= n^2 x + 2n\\dot{y} + \\frac{\\mu}{r^2} - \\frac{\\mu}{r_c^3}(r+x) + \\frac{F_x}{m_c}, \\\\ \n\\ddot{y} &= n^2 y - 2n\\dot{x} - \\frac{\\mu}{r_c^3}y + \\frac{F_y}{m_c}, \n\\end{split}\n\\label{eq:nonlin1}\n\\end{align}\nwhere $r_c = \\sqrt{(r+x)^2 + y^2}$ is the distance between the chaser and Earth and $m_c = 500$kg is the mass of the chaser.\n\n\\subsection{Linear dynamics}\n\\label{ssec:lindynamics}\nLinearization of these equations about the system's equilibrium\n point results in the Clohessy-Wiltshire-Hill (CWH) equations \\cite{1960}, which are commonly used to capture the relative motion dynamics of two satellites within a reasonably close range. These equations are:\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= 3n^2 x + 2n\\dot{y} + \\frac{F_x}{m_c}, \\\\\n\\ddot{y} &= - 2n\\dot{x} + \\frac{F_y}{m_c}. \n\\end{split}\n\\label{eq:lin1}\n\\end{align}\n\nLet the state vector be denoted by $\\vec{x} = [x, y, \\dot{x}, \\dot{y}]^{T}$. The state-space form of these linear time-invariant (LTI) equations is:\n$$\\dot{\\vec{x}} = A\\vec{x} + B\\vec{u}, \\ \\text{where},$$\n$$A = \\begin{bmatrix} 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 3n^2 & 0 & 0 & 2n \\\\ 0 & 0 & -2n & 0 \\end{bmatrix}, \nB = \\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\\\ \\frac{1}{m_c} & 0 \\\\ 0 & \\frac{1}{m_c} \\end{bmatrix},\n\\vec{u} = \\begin{bmatrix} F_x \\\\ F_y \\end{bmatrix}.$$\n\n\n\n\\subsection{Hybrid controller model}\n\\label{sec:hybrid-control}\n\nComplete hybrid automaton models for the system with additional documentation are available from~\\href{https:\/\/wiki.illinois.edu\/wiki\/display\/MitraResearch\/Autonomous+Satellite+System+Verification}{from this link}\\footnote{https:\/\/tinyurl.com\/verifysat}. \nVarying ranges of the relative distance between the spacecraft give rise to different constraints and requirements, and therefore, require separate controllers. We present a two-stage hybrid controller for achieving the rendezvous maneuver\\footnote{The rendezvous mission presented in this paper is a subset of the four-stage problem presented in~\\cite{erwin-cdc16}. Our two stages of rendezvous are almost identical to ``Phase 2'' and ``Phase 3'' in~\\cite{erwin-cdc16}.}. We refer to these discrete stages as \\emph{modes}. \nEach discrete mode has an \\emph{invariant} which specifies the conditions under which the system may operate in that mode, which we will first describe in words.\n \n \n\\begin{description}\n\n\\item{{\\em Mode 1\\\/}} or Proximity Operations A (ProxA): the chaser is attempting to rendezvous and its separation distance ($\\rho=\\sqrt{x^2+y^2}$) from the target is in the range 100-1000m.\n\n\\item{{\\em Mode 2}} or Proximity Operations B (ProxB): the chaser is attempting to rendezvous and its separation distance is less than 100m.\n\n\\item{{\\em Mode 3}} or Passive mode: the chaser is no longer attempting to rendezvous and is not using its thrusters, regardless of its separation distance. The system may transition to the Passive mode as a result of a failure or loss of power.\n\\end{description}\n\nThe state of the overall hybrid system is defined by the mode and the valuations of a set of continuous variables: relative position $x$, $y$,\n\tthrusts $F_x$, $F_y$, and a global timer $\\mathit{clock}$.\t\nThere are two timing parameters of the model $t_1$ and $t_2$ that specify the time interval over which the chaser spacecraft may enter the Passive mode.\nWhen the system is in a particular mode, the continuous variables $(x,y)$ evolve according to the (linear or nonlinear) differential equations of the previous section. The thrust inputs $F_x$ and $F_y$ are computed according the full-state feedback controller designed in Section~\\ref{sec:lqr}. \n\n\\begin{figure*}[t!]\n\\centering\n\\subfloat[]{\n\t\\label{hyMod}\n\t\\begin{tikzpicture}[scale=0.45]\n\t\\tikzstyle{every node}=[font=\\scriptsize, circle, font=\\scriptsize, minimum size = 1.6cm,text width=2cm]\n\t\\draw (-7,0) node[fill=blue!20] (p1) {\\center{\\vspace{-0.8cm}ProxA: $$\\dot{\\bar{x}} = (A-BK_1)\\bar{x}$$}};\n\t\\draw (0,0) node[fill=green!20] (p2) {\\center{\\vspace{-0.8cm}ProxB: $$\\dot{\\bar{x}} = (A-BK_2)\\bar{x}$$}};\n\t\\draw (7,0) node[fill=gray!20] (p3) {\\center{\\vspace{-0.8cm}Passive:$$\\dot{\\bar{x}} = A\\bar{x}$$}};\n\t\\tikzstyle{every node}=[font=\\scriptsize,fill=none]\n\t\\draw [ shorten >=1pt,->] (-7,4) to node[left] {$\\bar{x}_0$} (p1);\n\t\\draw [ shorten >=1pt,->] (p1) to [out=45,in=135] node[above] {$\\rho\\leq \\rho_t$} (p2);\n\t\\draw [ shorten >=1pt,->] (p2) to [out=-135,in=-45] node[above=4pt] {$\\rho \\geq \\rho_t$} (p1);\n\t\\draw [ shorten >=1pt,->] (p1) to [out=-50,in=-140] node[above=1pt] {$\\mathit{clock} \\in [t_1,t_2]$} (p3);\n\t\\draw [ shorten >=1pt,->] (p2) to [out=45,in=135] node[above] {$\\mathit{clock} \\in [t_1,t_2]$} (p3);\n\t\\end{tikzpicture}}\n\\subfloat[]{\n\t\\label{rhofig}\n\t\\hspace{5mm}\n\t\\begin{tikzpicture}[scale=0.7]\n\n\t\\fill [blue!20] (-2.4,-2.4) rectangle (2.4,2.4);\n\n\t\\fill [green!20] (0,0) circle (2cm);\n\n\t\\draw [ thick,->] (-2.5,0) to node[right=2cm] {$\\*i$} (2.5,0);\n\t\\draw [ thick,->] (0,-2.5) to node[above=2cm,right] {$\\*j$} (0,2.5);\n\n\t\\draw [shorten >=1pt,->] (0,0) to node[above,right]{$\\rho_t = 100$m} (1.41,1.41);\n\n\t\\draw[thick] (0.83,2) -- (-0.83,2);\n\t\\draw[thick] (0.83,2) -- (2,0.83);\n\t\\draw[thick] (2,0.83) -- (2,-0.83);\n\t\\draw[thick] (2,-0.83) -- (0.83,-2);\n\t\\draw[thick] (-0.83,-2) -- (0.83,-2);\n\t\\draw[thick] (-0.83,-2) -- (-2,-0.83);\n\t\\draw[thick] (-2,-0.83) -- (-2,0.83);\n\t\\draw[thick] (-2,0.83) -- (-0.83,2);\n\t\\end{tikzpicture}}\n\\caption{(a) Hybrid model for spacecraft rendezvous, with linear flow equations shown. The invariants in ProxA and ProxB are defined exclusively by the chaser's position, as shown by corresponding colors in the plane of motion in (b). The transition guards between ProxA and ProxB align exactly with their invariant sets, resulting in urgent transitions. The invariant for Passive mode is $clock > t_1$, irrespective of position. A transition to Passive occurs sometime within an interval of time, and hence is nondeterministic. In (b), the octagon represents how the invariants\/guards are approximated and modeled in the verification tools.}\n\\end{figure*}\n\nWe refer to the time elapsed in the mission with the variable $clock$ but do not consider it an explicit state variable. The invariants in each mode can be more precisely described as $\\rho \\geq 100$ and $\\mathit{clock} \\leq t_2$ for mode 1, $\\rho \\leq 100$ and $\\mathit{clock} \\leq t_2$ for mode 2, and $\\mathit{clock} \\geq t_1$ for mode 3. \nA transition from one mode to another is described by a \\emph{guard}. When the state satisfies the guard condition, the system \\emph{may} take the transition. \nIf a transition is required to occur as soon as possible, this is a called an \\emph{urgent transition}. In this system, the distance-based transitions between modes 1 and 2 are urgent. However, the transitions to mode 3 (Passive mode) are not urgent. There is an interval of time, $\\mathit{clock} \\in[t_1,t_2]$, within which the chaser could nondeterministically transition to the Passive mode. Roughly, a larger $[t_1, t_2]$ interval implies a bigger passive-safety envelope for the mission. These transitions to the Passive mode make the system nondeterministic. Indeed, for some choices of this interval, it is possible for the hybrid system to occupy any one of the three modes at a given time.\n\n\n\\begin{figure*}\n\\centering\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\includegraphics[width=.65\\textwidth]{Hillframe.png}\n\t\\caption{Hill's relative coordinate frame. The chaser's relative position vector is $x\\hat{\\*i}+y\\hat{\\*j}$.}\n\t\\label{Hill}\n\\end{minipage}\\qquad\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\begin{tikzpicture}[scale=0.8]\n\n\t\\fill [blue!20] (-2.5,-2.5) rectangle (2.5,2.5);\n\n\t\\fill [green!20] (0,0) circle (1.5cm);\n\n\t\\fill [red!60] (0,0) -- (-1.3,0.75) arc (150:210:1.5) -- cycle;\n\n\t\\draw [ thick,->] (-3,0) to node[right=2.5cm] {$\\*i$} (3,0);\n\t\\draw [ thick,->] (0,-3) to node[above=2.5cm,right] {$\\*j$} (0,3);\n\t\n\t\\draw [shorten >=1pt,->] (0,0) to node[above]{$\\rho$} (1,1.33);\n\t\\draw (0,0) circle (1.67cm);\n\t\\draw (-0.87,0.5) arc (150:210:1) node[right] {$60^{\\circ}$};\n\t\\end{tikzpicture}\n\t\\caption{The hybrid model (see Figure~\\ref{hyMod}) captures the chaser's motion in ProxA (blue) and ProxB (green), and we use verification tools to show that whenever $\\rho \\in$ ProxB, the chaser does not leave the LOS region (red).}\n\t\\label{LOSfig}\n\\end{minipage}\n\\end{figure*}\n\n\n\\subsection{Linear Quadratic Control}\n\\label{sec:lqr}\nWe have developed a full-state feedback controller, namely, a Linear Quadratic Regulator (LQR), to drive the chaser towards the target's position. \nClosed-loop feedback is desirable because the system can measure and adjust for errors, and ultimately guarantee liveness (i.e. eventually the target will be reached). LQR is specifically chosen because it is constructed by minimizing a quadratic cost function, which we can choose so as to roughly satisfy our safety constraints. \nLQR is only applicable to linear systems, so we design the control for the linearized model in~\\eqref{eq:lin1}, but we will use the same control with nonlinear dynamics~\\eqref{eq:nonlin1} when applying verification tools.\n\nThe form of an LQR solution is: $\\vec{u} = -K\\vec{x}$, where $K \\in \\mathbb{R}^{4\\times2}$ is a constant matrix and $\\vec{u} = [\\frac{F_x}{m_c}, \\frac{F_y}{m_c}]^T$. The $K$ matrix is found by minimizing the following cost function with respect to $\\vec{u}$: $\\int_{0}^{\\infty} (\\vec{x}^{T}Q\\vec{x}+\\vec{u}^{T}R\\vec{u}) dt $, where $Q$ and $R$ are positive definite matrices.\n\nGiven the form of the control law, we update the definition of the\nmodel given in~\\eqref{eq:lin1} to the following:\n\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= \\left(3n^2-\\frac{k_{11}}{m_c}\\right) x - \\frac{k_{12}}{m_c}y - \\frac{k_{13}}{m_c}\\dot{x} + \\left(2n-\\frac{k_{14}}{m_c}\\right)\\dot{y},\\\\\n\\ddot{y} &= - \\frac{k_{21}}{m_c}x - \\frac{k_{22}}{m_c}y - \\left(2n+ \\frac{k_{23}}{m_c}\\right)\\dot{x} - \\frac{k_{24}}{m_c}\\dot{y}.\n\\label{eq:cl}\n\\end{split}\n\\end{align}\n\nThese equations are the expanded version of the closed-loop form shown in Figure~\\ref{hyMod}. \nLater, we will discuss why we distinguish between the dependence of~\\eqref{eq:lin1} on $\\vec{x}$ and $\\vec{u}$ and~\\eqref{eq:cl} on only $\\vec{x}$.\n\nBryson's method \\cite{brysonrule} is used to help determine an appropriate cost function. Begin with $Q$ and $R$ as diagonal matrices, and choose their values so as to normalize each of the state and input variables. In other words, choose the diagonal elements so that $Q_{ii} = \\frac{1}{max(x_i^2)}$ and $R_{ii} = \\frac{1}{max(u_i^2)}$. Here, the denominators refer to the largest \\emph{desired} value of each variable, which will be determined by the safety constraints and mode invariants. While the LQR gains are obtained with our constraints in mind, the resulting controller does not guarantee these constraints are never violated. This is why further verification is still required.\nThis design process is repeated for modes 1 and 2, and the result is two distinct LQR controllers for each of these modes in our hybrid system. \n\n\n\\subsection{Constraints and safety requirements}\n\\label{sec:safeSets}\nIn this section, we enumerate the properties that define a safe and successful mission, and how they are modeled for verification tools. \n\n\t\\begin{description}\n\t\t\\item[ Thrust constraints]\nDuring the rendezvous stages (ProxA and ProxB), the thrusters cannot provide more than $10 N$ of force in any single direction, therefore, we have the constraints:\n \\[\n |F_x|, |F_y| \\leq 10.\n \\] \n\\item [LOS cone and proximity]\nDuring close-range rendezvous ProxB, the chaser must remain within a line-of-sight (LOS) cone (see Figure \\ref{LOSfig}), and its total velocity must remain under $5$cm\/s, so $\\sqrt{\\dot{x}^2+\\dot{y}^2} \\leq 5$cm\/s. The total velocity constraints cannot be exactly modeled using linear constraints, and a polytopic approximation over $\\dot{x},\\dot{y}$ is used. This is done in the same way as $\\sqrt{x^2+y^2}\\leq \\rho_t$ is approximated (see Figure~\\ref{rhofig}).\n\\item[Separation]\nDuring the Passive mode, the chaser must avoid collision with the target, which is theoretically a point mass at the origin. Even in a theoretical model, a small ball or box should be used to bound this point to account for limitations in numerical precision. In reality, the target satellite's dimensions may range from the order of $1$m\nto $100$m,\nso the size of this bounding box will vary depending on the situation. We use a box with a $0.1$m circumradius. \n\n\\end{description}\n\n\n\\section{Verification results}\n\\label{sec:results}\n\n\nIn this section, we discuss and compare verification results from SpaceEx, C2E2, and our implementation of {\\sf SDVTool\\\/}{}. Based on these results, we reach the broad conclusion that with some manual tweaks, the current hybrid system verification tools are indeed capable of analyzing realistic system-level properties of autonomous spacecraft maneuvers. \n\nIn the following presentation, we pick arbitrary model parameters, but to a large extent our results are robust with respect to parameter variations.\nThat is, the parameters can be tuned to the specific requirements of a real mission. \n\nFor subsequent discussion, we label our models as follows: {\\sf LinProx\\\/}{} denotes the equations in \\eqref{eq:cl}, {\\sf NLinProx\\\/}{} denotes the equations in \\eqref{eq:nonlin1} with the same controller as {\\sf LinProx\\\/}{} substituted into $F_x,F_y$, and {\\sf LinProxTh\\\/}{} denotes a model that will soon be introduced to account for explicit thrust values. \n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\subfloat[]{\\label{mat1:a}\\includegraphics[width=.37\\textwidth]{mat1_pos.png}}%\n\t\\subfloat[]{\\label{space1:a}\\includegraphics[width=.31\\textwidth]{space1_pos.png}}%\n\t\\subfloat[]{\\label{c2e2:a}\\includegraphics[width=.36\\textwidth]{c2e2_pos.png}}\\\\\n\t\\subfloat[]{\\label{mat1:b}\\includegraphics[width=.37\\textwidth]{mat1_vel.png}} %\n\t\\subfloat[]{\\label{space1:b}\\includegraphics[width=.31\\textwidth]{space1_vel.png}}%\n\t\\subfloat[]{\\label{mat3}\\includegraphics[width=.36\\textwidth]{mat3_thrust.png}}%\n\t\\caption{Examples of various generated reachsets. Reachable positions using {\\sf LinProx\\\/}{} in {\\sf SDVTool\\\/}{} shown in (a) and SpaceEx in (b). Reachable velocities using {\\sf LinProx\\\/}{} in {\\sf SDVTool\\\/}{} shown in (d) and SpaceEx in (e). Reachable positions of {\\sf NLinProx\\\/}{} without Passive mode in C2E2 shown in (c). Reachable thrusts using {\\sf LinProxTh\\\/}{} in {\\sf SDVTool\\\/}{} is shown in (f).}\n\t\\label{results}\n\\end{figure*}\n\n\\subsection{Hybrid safety proofs}\nFigure~\\ref{results} shows the typical reachset computations obtained from {\\sf SDVTool\\\/}{}, C2E2, and SpaceEx on the {\\sf LinProx\\\/}, {\\sf LinProxTh\\\/}, and {\\sf NLinProx\\\/} \\ models. These computations also establish the safety of the corresponding systems with respect to the requirements in Section~\\ref{sec:safeSets}. \nOverall, the plots show that the reachsets from the different tools are qualitatively similar. \nFrom the more detailed MatLab plots we can check that no part of the reachable sets intersect with unsafe regions. It is clear from the zoomed in portion of Figure~\\ref{mat1:a} that a reasonably larger collision region would violate safety. \n\nIn C2E2 and SpaceEx, each safety property is loaded and checked individually. In C2E2, the running time for a single property for the nonlinear model {\\sf NLinProx\\\/}{} is in the neighborhood of 5-10 minutes; in SpaceEx, the running time for a single property is on the order of a few seconds. {\\sf SDVTool\\\/}{} checks all (12) properties simultaneously and the running time varies from around 30 seconds to 10 minutes. We do not compare absolute running times in further detail in this paper as each of the tools have different semi-automatic workflows and require widely different execution environments.\n\n\n\n\n\\paragraph{Time horizon} \nTiming is obviously critical for space applications to ensure there is sufficient fuel, but with over-approximated reachability, we can only guarantee the mission is completed within a time upper-bound. This upper-bound obtained from reachability analysis may differ significantly from the actual mission time.\n Therefore, we do not impose any strict completion requirements. Instead, we choose a time horizon that is representative of what we might expect in practice, and focus on observing what behaviors are possible within these limits. Typically, Proximity A operations take 1-5 orbits (at under 4 hours an orbit) and proximity B operations take 45-90 minutes~\\cite{wertz:phases}. We choose a sum of approximately 4.5 hours to be our time horizon.\n\n\\paragraph{Initial states} We calculate a set of initial states assuming that the chaser spacecraft is performing the encompassing mission from~\\cite{erwin-cdc16}. \nWe choose an initial set radius of $[25m,25m,0,0]$\naround the point $\\vec{x}_0 = [-900m,-400m,0m\/s,0m\/s]^T$. This can be interpreted as uncertainty in the chaser's initial position, typically due to loss of precision from sensors and computations, or it can be used to explore multiple initial states of interest. We have successfully verified scenarios with uncertainty in the velocity dimensions as well.\n\n\\paragraph{Unsafe sets} For SpaceEx, C2E2, and {\\sf SDVTool\\\/}{}, we model the safety requirements as a collection of linear inequalities. \nThe LOS cone is approximated with a triangle, so we check three properties to prove the system remains within LOS constraints, and so on. \nMax thrust is effectively a one-dimensional constraint, a nonconvex interval, so two properties will capture the unsafe set for each thrust input (one along $x$-direction, one along $y$-direction). But in order to treat it this way, we must introduce extra variables $u_x,u_y$ to explicitly track the thruster values. These extra variables are the difference between {\\sf LinProx\\\/}{} and {\\sf LinProxTh\\\/}{}.\n\n\\paragraph{Passive transition time} The interval of time during which a transition to the passive mode may occur is trivially bounded by the mission time horizon. For this first example, we choose a small interval at $[120,125\n\\text{min}]$. This ensures that the chaser will operate in mode ProxB before transitioning to the passive mode. \n\n\n\n\\subsection{Adding thrust constraints}\n\\label{sec:6dim}\n In Section~\\ref{sec:safeSets}, we described a constraint on thrust that mimics the physical limitations of our spacecraft. We now set up the 6-dimensional model {\\sf LinProxTh\\\/} \\ so that we can verify this additional requirement. We introduce $u_x,u_y$ as explicit state variables, and solve for their differential equations to obtain:\n\\begin{align}\n\\begin{split}\n\\dot{u}_x &= k_{11}\\dot{x} + k_{12}\\dot{y} + k_{13}\\ddot{x} + k_{14}\\ddot{y},\\\\\n\\dot{u}_y &= k_{21}\\dot{x} + k_{22}\\dot{y} + k_{23}\\ddot{x} + k_{24}\\ddot{y}.\n\\end{split}\n\\label{eq:thrustode}\n\\end{align}\nThere are two equivalent numerical models that will produce different over-approximated reachsets. The first model consists of \\eqref{eq:thrustode} and the following:\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= 3n^2 x + 2n\\dot{y} - \\frac{u_x}{m_c},\\\\\n\\ddot{y} &= - 2n\\dot{x} - \\frac{u_y}{m_c}.\n\\end{split}\n\\label{eq:6dim1}\n\\end{align}\nHere $\\ddot{x}$ and $\\ddot{y}$ account for the effects of thrust inputs by explicitly adding $u_x,u_y$. Since each dimension is over-approximated in the reachset computation and $u_x,u_y$ are functions of position and velocity, the computation for subsequent reachable sets of position and velocity have even more uncertainty. \nRoughly, $u_x,u_y$ act as filters for $x,y,\\dot{x},\\dot{y}$, adding distortion and introducing more uncertainty. \nFigure~\\ref{mat2} shows the effects of these compounding errors. The overarching verification algorithm will partition the initial set to reduce errors stemming from the data structure, but it will have to do this numerous times and may time out in practice.\n\nThe second 6-dimensional model (a variant on {\\sf LinProxTh\\\/}) consists of~\\eqref{eq:cl} and \\eqref{eq:thrustode}. In this case, $\\ddot{x}$ and $\\ddot{y}$ implicitly calculate the thrust, and $u_x,u_y$ are independent ``tracking'' variables. The calculations for $\\ddot{x}$ and $\\ddot{y}$ are equivalent to those in \\eqref{eq:6dim1}, but they bypass the ``filter'' when constructing reachsets. The results are identical to those shown in Figures~\\ref{mat1:a}-\\ref{mat1:b}, with the addition of reachable sets of thrusts shown in Figure~\\ref{mat3}.\n\n Once again, our choice of data structure introduces some uncertainty to the explicit representation of the initial set of $u_x,u_y$ values. This is propagated throughout the analysis.\n\n\nWe use the \\eqref{eq:cl}-\\eqref{eq:thrustode} model to obtain safe thrusting results from {\\sf SDVTool\\\/}{}. The fine reachable sets in Figure~\\ref{mat3} show that the LQR controller operates well within the thrust constraints ($|u_x|,|u_y| \\leq 10N$).\n\n\n\\begin{figure*}\n\\centering\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{mat2_pos.png}\n\t\\caption{Coarse overapproximation of the reachable positions, when using the 6-dimensional {\\sf LinProxTh\\\/}{} model from Equation~\\ref{eq:6dim1}.}\n\t\\label{mat2}\n\\end{minipage}\\qquad\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{fig2.png}\n\t\\caption{Initial positions (with zero initial velocities) of {\\sf LinProx\\\/}{} that have been verified to be safe. They are safe for Passive transition times up to the time shown by the color map.}\n\t\\label{param1}\n\\end{minipage}\n\\end{figure*}\n\n\n\n\\subsection{Robustness of verification}\nTo demonstrate the robustness of the verification approaches (and the designed controller), we performed several experiments varying the initial set and passive transition times with the {\\sf LinProx\\\/} \\ model. \n\nThe scenarios that guarantee a safe mission are summarized in Figure~\\ref{param1}. \nRoughly, choosing an initial position or subset of positions within the shaded region will result in a safe mission for a transition time or interval within $[0,T]$, where $T$ is the time corresponding to the color at that initial position(s). \nFor these experiments, we consider initial separation between the chaser and the target to be near $1000m$, where this LQR control would start being used. We assume the initial chaser velocity to be zero. \nGenerally, we can conclude that, the closer to the $x$-axis the chaser starts, the later the chaser may safely abort to the passive mode. \nOn the other hand, the neighborhood of states along $\\sim 230^{\\circ}$ are not safe for a passive transition at any time. \n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Basic hypergeometric series and equations}\nThe theory of hypergeometric functions and equations dates back at\nleast as far as Gauss. It has long been and is still an integral\npart of the mathematical literature. In particular, the Galois\ntheory of (generalized) hypergeometric equations attracted the\nattention of many authors. For this issue, we refer the reader to\n\\cite{beukersheckman,beukersbrownheck,katzexpsum} and to the\nreferences therein. We also single out the papers\n\\cite{duvalmitschi,mitschihyperconf}, devoted to the calculation\nof some Galois groups by means of a density theorem (Ramis\ntheorem).\n\nIn this paper we focus our attention on the Galois theory of the\nbasic hypergeometric equations, the later being natural\n$q$-analogues of the hypergeometric equations.\n\nThe \\textit{basic hypergeometric series}\n$\\phi(z)=\\qhyper{a}{b}{c}{z}$ with parameters $(a,b,c) \\in\n(\\mathbb C^*) ^3$ defined by :\n\\begin{eqnarray*}\\qhyper{a}{b}{c}{z}&=&\\sum_{n=0}^{+\\infty}\n\\frac{\\pochamer{a,b}{q}{n} }{\\pochamer{c,q}{q}{n} } z^n\\\\\n&=& \\sum_{n=0}^{+\\infty} \\frac{(1-a)(1-aq)\\cdots\n(1-aq^{n-1})(1-b)(1-bq)\\cdots (1-bq^{n-1})}{(1-q)(1-q^2)\\cdots\n(1-q^{n})(1-c)(1-cq)\\cdots (1-cq^{n-1})} z^n\n\\end{eqnarray*}\nwas first introduced by Heine and was later generalized by\nRamanujan. As regards functional equations, the basic\nhypergeometric series provides us with a solution of the following\nsecond order $q$-difference equation, called the \\textit{basic\nhypergeometric equation} with parameters $(a,b,c)$ :\n\\begin{equation}\\label{equa hypergeo} \\phi(q^2z) - \\frac{(a+b)z-(1+c\/q)}{abz-c\/q} \\phi(qz) + \\frac{z-1}{abz-c\/q}\\phi(z)=0.\\end{equation}\nThis functional equation is equivalent to a functional system.\nIndeed, with the notations :\n$$\\lambda(a,b;c;z)= \\frac{(a+b)z-(1+c\/q)}{abz-c\/q}, \\ \\ \\ \\\n\\mu(a,b;c;z)=\\frac{z-1}{abz-c\/q},$$ a function $\\phi$ is solution\nof $(\\ref{equa hypergeo})$ if and only if the vector\n$\\Phi(z)=\\binom{\\phi(z)}{\\phi(qz)}$ satisfies the functional\nsystem : \\begin{equation}\\label{syst hypergeo} \\Phi(qz) =\n\\qhypermatrice{a}{b}{c}{z} \\Phi(z) \\end{equation} with :\n$$\\qhypermatrice{a}{b}{c}{z}=\\left( \\begin{array}{cc} 0&1\\\\ -\\mu(a,b;c;z) & \\lambda(a,b;c;z) \\end{array}\\right).$$\nThe present paper focuses on the calculation of the Galois group\nof the $q$-difference equation (\\ref{equa hypergeo}) or,\nequivalently, that of the $q$-difference system (\\ref{syst\nhypergeo}). A number of authors have developed $q$-difference\nGalois theories over the past years, among whom Franke\n\\cite{frankepvdifference}, Etingof \\cite{etingofgalois}, Van der\nPut and Singer \\cite{psgaloistheory}, Van der Put and Reversat\n\\cite{prev}, Chatzidakis and Hrushovski \\cite{chatzihru}, Sauloy\n\\cite{sauloyqgaloisfuchs}, Andr\\'e \\cite{andrenoncomm}, etc. The\nexact relations between the existing Galois theories for\n$q$-difference equations are partially understood. For this\nquestion, we refer the reader to \\cite{chatziharsing}, and also to\nour Remark section \\ref{constr}.\n\n\nIn this paper we follow the approach of Sauloy (initiated by\nEtingof in the regular case). Our method for computing the Galois\ngroups of the basic hypergeometric equations is based on a\n$q$-analogue of Schlesinger's density theorem stated and\nestablished in \\cite{sauloyqgaloisfuchs}. Note that some of these\ngroups were previously computed by Hendriks in\n\\cite{hendriksqhyper} using a radically different method\n(actually, the author dealt with the Galois groups defined by Van\nder Put and Singer, but these do coincide with those defined by\nSauloy : see our Remark section \\ref{constr}). On a related topic, \nwe also point out the appendix of \\cite{divizio} which contains the \n$q$-analogue of Schwarz's list. \n\nThe paper is organized as follows. In a first part, we give a\nbrief overview of some results from \\cite{sauloyqgaloisfuchs}. In\na second part, we compute the Galois groups of the basic\nhypergeometric equations in all non-resonant (but possibly\nlogarithmic) cases.\n\n\\section{Galois theory for regular singular $q$-difference equations}\n\nUsing analytic tools together with Tannakian duality, Sauloy\ndeveloped in \\cite{sauloyqgaloisfuchs} a Galois theory for regular\nsingular $q$-difference systems. In this section, we shall first\nrecall the principal notions used in \\cite{sauloyqgaloisfuchs},\nmainly the Birkhoff matrix and the twisted Birkhoff matrix. Then\nwe shall explain briefly that this lead to a Galois theory for\nregular singular $q$-difference systems. Last, we shall state a\ndensity theorem for these Galois groups, which will be of main\nimportance in our calculations.\n\n\\subsection{Basic notions}\\label{section the basic objects}\n\nLet us consider $A \\in \\text{Gl}_n(\\mathbb C(\\{z\\}))$. Following Sauloy\nin \\cite{sauloyqgaloisfuchs}, the $q$-difference system :\n\\begin{equation}\\label{general q diff system}Y(qz)=A(z)Y(z)\\end{equation}\nis said to be \\textit{Fuchsian} at $0$ if $A$ is holomorphic at\n$0$ and if $A(0)\\in \\text{Gl}_n(\\mathbb C)$. Such a system is\nnon-resonant at $0$ if, in addition, $Sp(A(0)) \\cap\nq^{\\mathbb Z^*} Sp(A(0))=\\emptyset$. Last we say that the above\n$q$-difference system is \\textit{regular singular} at $0$ if there\nexists $R^{(0)}\\in \\text{Gl}_n(\\mathbb C(\\{z\\}))$ such that the\n$q$-difference system defined by\n$(R^{(0)}(qz))^{-1}A(z)R^{(0)}(z)$ is Fuchsian at $0$. We have\nsimilar notions at $\\infty$ using the change of variable $z\n\\leftarrow 1\/z$.\n\nIn the case of a global system, that is $A\\in \\text{Gl}_n(\\mathbb C(z))$,\nwe will use the following terminology. If $A\\in\n\\text{Gl}_n(\\mathbb C(z))$, then the system (\\ref{general q diff system})\nis called \\textit{Fuchsian} (resp. \\textit{Fuchsian and non-resonnant}, \\textit{regular\nsingular}) if it is Fuchsian (resp. Fuchsian and non-resonnant,\nregular singular) both at $0$ and at $\\infty$.\n\nFor instance, the basic hypergeometric system (\\ref{syst hypergeo}) is Fuchsian.\\\\\n\n\n\\textit{Local fundamental system of solutions at $0$}. Suppose\nthat (\\ref{general q diff system}) is Fuchsian and non-resonant at\n$0$ and consider $J^{(0)}$ a Jordan normal form of $A(0)$.\nAccording to \\cite{sauloyqgaloisfuchs} there exists $F^{(0)} \\in\n\\text{Gl}_n(\\mathbb C\\{z\\})$ such that :\n\\begin{equation} \\label{transfo jauge}\nF^{(0)}(qz)J^{(0)}=A(z)F^{(0)}(z).\n\\end{equation}\nTherefore, if $e^{(0)}_{J^{(0)}}$ denotes a fundamental system of\nsolutions of the $q$-difference system with constant coefficients\n$X(qz)=J^{(0)}X(z)$, the matrix-valued function\n$Y^{(0)}=F^{(0)}e^{(0)}_{J^{(0)}}$ is a fundamental system of\nsolutions of (\\ref{general q diff system}). We are going to\ndescribe a possible choice for $e^{(0)}_{J^{(0)}}$. We denote by\n$\\theta_q$ the Jacobi theta function defined by\n$\\theta_q(z)=\\pochamer{q}{q}{\\infty}\\pochamer{z}{q}{\\infty}\\pochamer{q\/z}{q}{\\infty}$.\nThis is a meromorphic function over $\\mathbb C^*$ whose zeros are\nsimple and located on the discrete logarithmic spiral\n$q^\\mathbb Z$. Moreover, the functional equation\n$\\theta_q(qz)=-z^{-1}\\theta_q(z)$ holds. Now we introduce, for all\n$\\lambda \\in \\mathbb C^*$ such that $|q| \\leq |\\lambda| < 1$, the\n$q$-character\n$e^{(0)}_{\\lambda}=\\frac{\\theta_q}{\\theta_{q,\\lambda}}$ with\n$\\theta_{q,\\lambda}(z)=\\theta_q(\\lambda z)$ and we extend this\ndefinition to an arbitrary non-zero complex number $\\lambda \\in\n\\mathbb C^*$ requiring the equality\n$e^{(0)}_{q\\lambda}=ze^{(0)}_{\\lambda}$. If\n$D=P\\text{diag}(\\lambda_1,...,\\lambda_n)P^{-1}$ is a semisimple\nmatrix then we set\n$e^{(0)}_{D}:=P\\text{diag}(e^{(0)}_{\\lambda_1},...,e^{(0)}_{\\lambda_n})P^{-1}$.\nIt is easily seen that this does not depend on the chosen\ndiagonalization. Furthermore, consider\n$\\ell_q(z)=-z\\frac{\\theta_q'(z)}{\\theta_q(z)}$ and, if $U$ is a\nunipotent matrix, $e^{(0)}_{U}=\\Sigma_{k=0}^n \\ell_q^{(k)}\n(U-I_n)^k$ with $\\ell_q^{(k)}=\\binom{\\ell_q}{k}$. If\n$J^{(0)}=D^{(0)}U^{(0)}$ is the multiplicative Dunford\ndecomposition of $J^{(0)}$, with $D^{(0)}$ semi-simple and\n$U^{(0)}$ unipotent, we set\n$e^{(0)}_{J^{(0)}}=e^{(0)}_{D^{(0)}}e^{(0)}_{U^{(0)}}$.\\\\\n\n\n\n\\textit{Local fundamental system of solutions at $\\infty$}. Using\nthe variable change $z \\leftarrow 1\/z$, we have a similar\nconstruction at $\\infty$. The corresponding fundamental system of\nsolutions is denoted by\n$Y^{(\\infty)}=F^{(\\infty)}e^{(\\infty)}_{J^{(\\infty)}}$.\\\\\n\nThroughout this section we assume that the system (\\ref{general q\ndiff system}) is global and that it is Fuchsian\nand non-resonant.\\\\\n\n\\textit{Birkhoff matrix}. The linear relations between the two\nfundamental systems of solutions introduced above are given by the\nBirkhoff matrix (also called connection matrix)\n$P=(Y^{(\\infty)})^{-1}Y^{(0)}$. Its entries are elliptic functions\n\\textit{i.e.} meromorphic functions over the elliptic curve\n$\\mathbb{E}_q=\\mathbb C^* \/ q^\\mathbb Z$. \\\\\n\n\\textit{Twisted Birkhoff matrix}. In order to describe a\nZariki-dense set of generators of the Galois group associated to\nthe system (\\ref{general q diff system}), we introduce a\n``twisted\" connection matrix. According to\n\\cite{sauloyqgaloisfuchs}, we choose for all $z \\in \\mathbb C^*$ a\ngroup endomorphism $g_z$ of $\\mathbb C^*$ sending $q$ to $z$.\nBefore giving an explicit example, we have to introduce more\nnotations. Let us, for any fixed $\\tau \\in \\mathbb C$ such that\n$q=e^{-2\\pi i \\tau}$, write $q^y=e^{-2\\pi i \\tau y}$ for all $y\n\\in \\mathbb C$. We also define the (non continuous) function\n$\\log_q$ on the whole punctured complex plane $\\mathbb C^*$ by\n$\\log_q(q^y)=y$ if $y\\in \\mathbb C^* \\setminus \\mathbb R^+$ and we\nrequire that its discontinuity is located just before the cut\n(that is $\\mathbb R^+$) when turning counterclockwise around $0$. We\ncan now give an explicit example of endomorphism $g_z$ namely the\nfunction $g_z: \\mathbb C^*=\\mathbb{U} \\times q^\\mathbb R \\rightarrow\n\\mathbb C^*$ sending $uq^\\omega$ to\n$g_z(uq^\\omega)=z^{\\omega}=e^{-2\\pi i \\tau \\log_q(z) \\omega}$ for\n$(u,\\omega)\\in \\mathbb{U} \\times \\mathbb R$, where $\\mathbb{U}\\subset\n\\mathbb C$ is the unit circle.\n\nThen we set, for all $z$ in $\\mathbb C^*$,\n$\\psi_z^{(0)}(\\lambda)=\\frac{\\qcar{\\lambda}(z)}{g_z(\\lambda)}$ and\nwe define $\\psi_z^{(0)}\\left(D^{(0)}\\right)$, the \\textit{twisted\nfactor} at $0$, by\n$\\psi_z^{(0)}\\left(D^{(0)}\\right)=P\\text{diag}(\\psi_z^{(0)}(\\lambda_1),...,\\psi_z^{(0)}(\\lambda_n))P^{-1}$\nwith $D^{(0)}=P\\text{diag}(\\lambda_1,...,\\lambda_n)P^{-1}$. We\nhave a similar construction at $\\infty$ by using the variable\nchange $z \\leftarrow 1\/z$. The corresponding twisting factor is\ndenoted by $\\psi_z^{(\\infty)}(J^{(\\infty)})$.\n\nFinally, the twisted connection matrix $\\breve{P}(z)$ is :\n\\begin{eqnarray*}\n\\breve{P}(z)&=&\\psi_z^{(\\infty)}\\left(D^{(\\infty)}\\right)P(z)\\psi_z^{(0)}\\left(D^{(0)}\\right)^{-1}.\n\\end{eqnarray*}\n\n\n\n\\subsection{Definition of the Galois groups}\\label{constr}\n\nThe definition of the Galois groups of regular singular\n$q$-difference systems given by Sauloy in\n\\cite{sauloyqgaloisfuchs} is somewhat technical and long. Here we\ndo no more than describe the underlying idea.\\\\\n\n\\textit{(Global) Galois group.} Let us denote by $\\mathcal{E}$ the\ncategory of regular singular $q$-difference systems with\ncoefficients in $\\mathbb C(z)$ (so, the base field is\n$\\mathbb C(z)$; the difference field is $(\\mathbb C(z),f(z) \\mapsto\nf(qz))$). This category is naturally equipped with a tensor\nproduct $\\otimes$ such that $(\\mathcal{E},\\otimes)$ satisfies all\nthe axioms defining a Tannakian category over $\\mathbb C$ except\nthe existence of a \\textit{fiber functor} which is not obvious.\nThis problem can be overcome using an analogue of the\nRiemann-Hilbert correspondance.\n\nThe Riemann-Hilbert correspondance for regular singular\n$q$-difference systems entails that $\\mathcal{E}$ is equivalent to\nthe category $\\mathcal{C}$ of connection triples whose objects are\ntriples $(A^{(0)},P,A^{(\\infty)}) \\in \\text{Gl}_n(\\mathbb C) \\times\n\\text{Gl}_n(\\mathcal{M}(\\mathbb{E}_q)) \\times \\text{Gl}_n(\\mathbb C)$ (we refer to\n\\cite{sauloyqgaloisfuchs} for the complete definition of\n$\\mathcal{C}$). Furthermore $\\mathcal{C}$ can be endowed with a\ntensor product $\\und \\otimes$ making the above equivalence of\ncategories compatible with the tensor products. Let us emphasize\nthat $\\und \\otimes$ is not the usual tensor product for matrices.\nIndeed some twisting factors appear because of the bad\nmultiplicative properties of the $q$-characters $e_{q,c}$ : in\ngeneral $e_{q,c}e_{q,d}\\neq e_{q,cd}$.\n\nThe category $\\mathcal{C}$ allows us to define a Galois group :\n$\\mathcal{C}$ is a Tannakian category over $\\mathbb C$. The functor\n$\\omega_0$ from $\\mathcal{C}$ to $Vect_\\mathbb C$ sending an object\n$(A^{(0)},P,A^{(\\infty)})$ to the underlying vector space\n$\\mathbb C^n$ on which $A ^{(0)}$ acts is a fiber functor. Let us\nremark that there is a similar fiber functor $\\omega_\\infty$ at\n$\\infty$. Following the general formalism of the theory of\nTannakian categories (see \\cite{deligne}), the \\textit{absolute\nGalois group} of $\\mathcal{C}$ (or, using the above equivalence of\ncategories, of $\\mathcal E$) is defined as the pro-algebraic group\n$Aut^{\\und \\otimes}(\\omega_0)$ and the \\textit{global Galois group\nof an object $\\chi$ }of $\\mathcal{C}$ (or, using the above\nequivalence of categories, of an object of $\\mathcal{E}$) is the\ncomplex linear algebraic group $Aut^{\\und\n\\otimes}(\\omega_{0|\\langle \\chi \\rangle})$ where $\\langle \\chi\n\\rangle$ denotes the Tannakian subcategory of $\\mathcal{C}$\ngenerated by $\\chi$. For the sake of simplicity, we will often\ncall $Aut^{\\und \\otimes}(\\omega_{0|\\langle \\chi \\rangle})$ the\n\\textit{Galois group} of $\\chi$\n(or, using the above equivalence of categories, of the corresponding object of $\\mathcal{E}$). \\\\\n\n\\textit{Local Galois groups.} Let us point out that notions of\nlocal Galois groups at $0$ and at $\\infty$ are also available\n(here the difference fields are respectively\n$(\\mathbb C(\\{z\\}),f(z) \\mapsto f(qz) )$ and\n$(\\mathbb C(\\{z^{-1}\\}),f(z) \\mapsto f(qz) )$). As expected, they\nare subgroups of the (global) Galois group. Nevertheless, since\nthese groups are of second importance in what follows, we omit the\ndetails and\nwe refer the interesting reader to \\cite{sauloyqgaloisfuchs}.\\\\\n\n\\noindent \\textbf{Remark.} In \\cite{psgaloistheory}, Van der Put\nand Singer showed that the Galois groups defined using a\nPicard-Vessiot theory can be recovered by means of Tannakian\nduality : it is the group of tensor automorphisms of some suitable\ncomplex valued fiber functor over $\\mathcal E$. Since two complex\nvalued fiber functors on a same Tannakian category are necessarily\nisomorphic, we conclude that\nSauloy's and Van der Put and Singer's theories coincide.\\\\\n\n\nIn the rest of this section we exhibit some natural elements of\nthe Galois group of a given Fuchsian $q$-difference system and we\nstate the density theorem of Sauloy.\n\n\n\\subsection{The density theorem}\n\n\nFix a ``base point\" $y_0\\in \\Omega=\\mathbb C^* \\setminus\n\\{\\text{zeros of } \\det(P(z)) \\text{ or poles of } P(z)\\}$ .\nSauloy exhibits in \\cite{sauloyqgaloisfuchs} the following\nelements of the (global) Galois group associated to the\n$q$-difference system (\\ref{general q diff system}) :\n\\begin{itemize}\n\\item[1.a)] $\\gamma_1(D^{(0)})$ and $\\gamma_2(D^{(0)})$ where :\n$$\\gamma_1:\\mathbb C^*=\\mathbb{U} \\times q^\\mathbb R \\rightarrow \\mathbb{U}$$\nis the projection over the first factor and : $$\\gamma_2 :\n\\mathbb C^*=\\mathbb{U} \\times q^\\mathbb R \\rightarrow \\mathbb C^*$$ is defined by\n$\\gamma_2(uq^\\omega)=e^{2\\pi i \\omega}$.\n\\item[1.b)] $U^{(0)}$.\n\\item[2.a)] $\\breve{P}(y_0)^{-1}\\gamma_1(D^{(\\infty)})\\breve{P}(y_0)$ and $\\breve{P}(y_0)^{-1}\\gamma_2(D^{(\\infty)})\\breve{P}(y_0)$.\n\\item[2.b)] $\\breve{P}(y_0)^{-1} U^{(\\infty)} \\breve{P}(y_0)$.\n\\item[3)] $\\breve{P}(y_0)^{-1}\\breve{P}(z)$, $z \\in \\Omega$. \\\\\n\\end{itemize}\n\nThe following result is due to Sauloy \\cite{sauloyqgaloisfuchs}.\n\n\\begin{theo}\\label{density theo}\nThe algebraic group generated by the matrices 1.a. to 3. is the\n(global) Galois group $G$ of the $q$-difference system\n(\\ref{general q diff system}). The algebraic group generated by\nthe matrices 1.a) and 1.b) is the local Galois group at $0$ of the\n$q$-difference system (\\ref{general q diff system}). The algebraic\ngroup generated by the matrices 2.a) and 2.b) is the local Galois\ngroup at $\\infty$ of the $q$-difference system (\\ref{general q\ndiff system}).\n\\end{theo}\n\nThe algebraic group generated by the matrices 3) is called the\n\\textit{connection component} of the Galois group $G$. The\nfollowing result is easy but very useful. Its proof is left to the\nreader.\n\n\\begin{lem} The connection component of the Galois group $G$ of a regular singular $q$-difference system is a subgroup of the\nidentity component $G^I$ of $G$.\n\\end{lem}\n\n\\section{Galois groups of the basic hypergeometric equations : non-resonant and non-logarithmic cases}\\label{section generique}\n\nWe write $a=uq^\\alpha$, $b=vq^\\beta$ and $c=wq^\\gamma$ with\n$u,v,w\\in\\mathbb{U}$ and $\\alpha,\\beta,\\gamma \\in \\mathbb R$ (we\nchoose a logarithm of $q$).\\\\\n\nIn this section we are aiming to compute the Galois group of the\nbasic hypergeometric system (\\ref{syst hypergeo}) under the\nfollowing assumptions :\n$$a\/b \\not \\in q^\\mathbb Z \\text{ and }c \\not \\in q^\\mathbb Z.$$\n\nFirst, we give explicit formulas for the generators of the Galois group of (\\ref{syst hypergeo}) involved in Theorem \\ref{density theo}.\\\\\n\n\\textit{Local fundamental system of solutions at $0$.} We have :\n$$\\qhypermatrice{a}{b}{c}{0}=\\pmatrice{1}{1}{1}{q\/c}\\pmatrice{1}{0}{0}{q\/c} \\pmatrice{1}{1}{1}{q\/c}^{-1}.$$\nHence the system (\\ref{syst hypergeo}) is non-resonant, and\nnon-logarithmic at $0$ since $\\qhypermatrice{a}{b}{c}{0}$ is\nsemi-simple. A fundamental system of solutions at $0$ of\n(\\ref{syst hypergeo}) as described in section \\ref{section the\nbasic objects} is given by\n$\\yz{a}{b}{c}{z}=\\fz{a}{b}{c}{z}e^{(0)}_{\\jz{c}}(z)$ with $\\jz{c}=\n\\text{diag}(1,q\/c)$ and :\n$$\\fz{a}{b}{c}{z}=\\pmatrice{\\qhyper{a}{b}{c}{z}}{\\qhyper{aq\/c}{bq\/c}{q^2\/c}{z}}\n{\\qhyper{a}{b}{c}{qz}}{(q\/c) \\qhyper{aq\/c}{bq\/c}{q^2\/c}{qz}}.$$\\\\\n\n\\textit{Generators of the local Galois group at $0$.} We have two generators :\n$$\\pmatrice{1}{0}{0}{e^{2\\pi i \\gamma}} \\text{ and }\n\\pmatrice{1}{0}{0}{w}.$$\n\n\\textit{Local fundamental system of solutions at $\\infty$.} We\nhave :\n$$\\qhypermatrice{a}{b}{c}{\\infty}=\\pmatrice{1}{1}{1\/a}{1\/b} \\pmatrice{1\/a}{0}{0}{1\/b} \\pmatrice{1}{1}{1\/a}{1\/b}^{-1}.$$\nHence the system (\\ref{syst hypergeo}) is non-resonant and\nnon-logarithmic at $\\infty$ and a fundamental system of solutions\nat $\\infty$ of (\\ref{syst hypergeo}) as described in section\n\\ref{section the basic objects} is given by\n$\\yinf{a}{b}{c}{z}=\\finf{a}{b}{c}{z}e^{(\\infty)}_{\\jinf{a}{b}}(z)$\nwith $\\jinf{a}{b}= \\text{diag}(1\/a,1\/b)$ and :\n$$\\finf{a}{b}{c}{z}=\n\\pmatrice{\\qhyper{a}{aq\/c}{aq\/b}{\\frac{cq}{ab}z^{-1}}}{\\qhyper{b}{bq\/c}{bq\/a}{\\frac{cq}{ab}z^{-1}}}\n{\\frac{1}{a} \\qhyper{a}{aq\/c}{aq\/b}{\\frac{c}{ab}z^{-1}}}{\\frac{1}{b} \\qhyper{b}{bq\/c}{bq\/a}{\\frac{c}{ab}z^{-1}}}.$$\\\\\n\n\\textit{Generators of the local Galois group at $\\infty$.} We have two generators :\n$$\\breve{P}(y_0)^{-1}\\pmatrice{e^{2\\pi i \\alpha}}{0}{0}{e^{2\\pi i \\beta}}\\breve{P}(y_0) \\text{ and } \\breve{P}(y_0)^{-1}\\pmatrice{u}{0}{0}{v}\\breve{P}(y_0).$$\\\\\n\n\\textit{Birkhoff matrix.} The Barnes-Mellin-Watson formula (\\textit{cf.} \\cite{gasperrahman}) entails that :\n$$P(z)= (e^{(\\infty)}_{\\jinf{a}{b}}(z))^{-1}\n\\pmatrice{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}} \\frac{\\theta_q (a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}} \\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}} \\frac{\\theta_q (b z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}} \\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}}\ne^{(0)}_{\\jz{c}}(z).$$\\\\\n\n\\textit{Twisted Birkhoff matrix.} We have :\n$$\\breve{P}(z)= \\pmatrice{(1\/z)^{-\\alpha}}{0}{0}{(1\/z)^{-\\beta}}\n\\pmatrice{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}}\n\\pmatrice{1}{0}{0}{z^{1-\\gamma}}.$$\n\n$_{}$\\vskip 10 pt\n\nWe need to consider different cases.\n\n$_{}$\\vskip 10 pt\n\n\\condun{1} $\\underline{a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z\n\\text{ and } a\/b \\text{ or } c \\not \\in \\pm q^{\\mathbb Z\/2}}$.\n\n$_{}$\\vskip 5 pt\n\nUnder this assumption the four numbers $\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}$,\n$\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}$,\n$\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}$ and\n$\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}$ are non-zero.\n\n\\begin{prop}\nSuppose that \\condun{1} holds. Then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\n\\begin{proof}\nSuppose, at the contrary, that the action of $G^I$ is reducible\nand let $L \\subset \\mathbb C^2$ be an invariant line.\n\nRemark that $L$ is distinct from $\\mathbb C \\vect{1}{0}$ and\n$\\mathbb C \\vect{0}{1}$. Indeed, assume at the contrary that\n$L=\\mathbb C \\vect{1}{0}$ (the case $L=\\mathbb C \\vect{0}{1}$ is\nsimilar). The line $L=\\mathbb C \\vect{1}{0}$ being in particular\ninvariant by the connection component, we see that the line\ngenerated by $\\breve{P}(z) \\vect{1}{0}$ does not depend on $z \\in\n\\Omega$. This yields a contradiction because the ratio of the\ncomponents of $\\breve{P}(z) \\vect{1}{0}=\n\\vect{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}$ depends on $z$\n(remember that $a\/b \\not \\in q^\\mathbb Z$).\n\nOn the other hand, since, for all $n\\in\\mathbb N$, both matrices\n$\\pmatrice{1}{0} {0}{e^{2\\pi i \\gamma n}}$ and\n$\\pmatrice{1}{0}{0}{w^n}$ belong to $G$ and since $G^I$ is a\nnormal subgroup of $G$, both lines $L_n:=\\pmatrice{1}{0}\n{0}{e^{2\\pi i \\gamma n}}L$ and $L'_n:=\\pmatrice{1}{0}{0}{w^n} L$\nare also invariant by $G^I$.\n\n\nNote that because \\condun{1} holds, at least one of the complex\nnumbers $w,e^{2\\pi i \\gamma},u\/v,e^{2\\pi i (\\alpha-\\beta)}$ is\ndistinct from $\\pm 1$.\n\nFirst, suppose that $w \\neq \\pm 1$. We have seen that $L \\neq\n\\mathbb C \\vect{1}{0}, \\mathbb C \\vect{0}{1}$, hence $L_0,L_1,L_2$\nare three distinct lines invariant by the action of $G^I$. This\nimplies that $G^I$ consists of scalar matrices : this is a\ncontradiction (because, for instance, $\\mathbb C \\vect{1}{0}$ is\nnot invariant for the action of $G^I$). Hence, if $w \\neq \\pm 1$\nwe have proved that $G^I$ acts irreducibly.\n\nThe case $e^{2\\pi i \\gamma} \\neq \\pm 1$ is similar.\n\nLast, the proof is analogous if $u\/v \\neq \\pm 1$ or $e^{2\\pi i\n(\\alpha-\\beta)} \\neq \\pm 1$ (we then use the fact that, for all\n$z\\in \\Omega$, $G^I$ is normalized by $\\breve{P}(z)^{-1}\n\\pmatrice{u}{0}{0}{v} \\breve{P}(z)$ and $\\breve{P}(z)^{-1}\n\\pmatrice{e^{2\\pi i \\alpha}}{0}{0}{e^{2\\pi i \\beta}} \\breve{P}(z)$\nand that there exists $z \\in \\Omega$ such that $\\breve{P}(z)L$ is\ndistinct from $\\mathbb C \\vect{1}{0}$ and $\\mathbb C \\vect{0}{1}$).\n\\end{proof}\n\nWe have the following theorem.\n\n\\begin{theo}\\label{theo un}\nSuppose that \\condun{1} holds.\nThen we have the following dichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $abq\/c \\not \\in q^\\mathbb Z$ then $G=\\text{Gl}_2(\\mathbb C)$;\n\\item[$\\bullet$] if $abq\/c \\in q^\\mathbb Z$ then $G=\\overline{\\langle \\text{Sl}_2(\\mathbb C),\\sqrt{w}I,e^{\\pi i \\gamma}I \\rangle}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nSince $G^I$ acts irreducibly on $\\mathbb C^2$, the general theory\nof algebraic groups entails that $G^I$ is generated by its center\n$Z(G^I)$ together with its derived subgroup $G^{I,der}$ and that\n$Z(G^I)$ acts as scalars. Hence, $G^{I,der} \\subset\n\\text{Sl}_2(\\mathbb C)$ also acts irreducibly on $\\mathbb C^2$. Therefore\n$G^{I,der}=\\text{Sl}_2(\\mathbb C)$ (a connected algebraic group of\ndimension less than or equal to $2$ is solvable hence $\\text{dim}\n(G^{I,der}) =3$ and $G^{I,der}=\\text{Sl}_2(\\mathbb C)$). In order to\ncomplete the proof, it is sufficient to determine $\\text{det}(G)$.\nWe have :\n\\begin{eqnarray*}\n\\text{det}(\\breve{P}(z)\n&=&\\frac{(1\/z)^{-(\\alpha+\\beta)}z^{1-\\gamma}}{\\pochamer{q^2\/c,a\/b,c,b\/a}{q}{\\infty}}\n\\left(\\underbrace{\\theta_q(b)\\theta_q(c\/a) \\frac{\\theta_q (a\nz)}{\\theta_q(z)} \\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}\n-\\theta_q(c\/b)\\theta_q(a)\\frac{\\theta_q (\\frac{aq}{c}\nz)}{\\theta_q(z)} \\frac{\\theta_q (b\nz)}{\\theta_q(z)}}_{\\psi(z)}\\right).\n\\end{eqnarray*}\nA straightforward calculation shows that the function :\n$$\\theta_q(b)\\theta_q(c\/a) \\theta_q (a z) \\theta_q (\\frac{bq}{c}\nz) -\\theta_q(c\/b)\\theta_q(a)\\theta_q (\\frac{aq}{c} z)\\theta_q (b\nz)$$ vanishes for $z \\in q^\\mathbb Z$ and for $z \\in\n\\frac{c}{abq}q^\\mathbb Z$. On the other hand $\\psi$ is a solution\nof the first order $q$-difference equation $y(qz)=\\frac{c}{abq}\ny(z)$. Hence, if we suppose that $abq\/c \\not \\in q^\\mathbb Z$, we\ndeduce that the ratio $\\chi(z)=\\frac{\\psi(z)}{\\frac{\\theta_q\n(\\frac{abq}{c} z)}{\\theta_q(z)}}$ defines an holomorphic elliptic\nfunction over $\\mathbb C^*$. Therefore $\\chi$ is constant and,\nevaluating $\\chi$ at $z=1\/b$,\n we get :\n$$\\chi=-b \\theta_q(a\/b)\n \\theta_q (c).$$\nFinally, we obtain the identity :\n\\begin{eqnarray} \\label{det}\n\\text{det}(\\breve{P}(z))&=&\\frac{1-q\/c}{1\/a-1\/b}(1\/z)^{-(\\alpha+\\beta)}z^{1-\\gamma}\n\\frac{\\theta_q (\\frac{abq}{c} z)}{\\theta_q(z)}.\n\\end{eqnarray}\nBy analytic continuation (with respect to the parameters) we see\nthat this formula also holds if $abq\/c \\in q^\\mathbb Z$.\n\nConsequently, if $abq\/c \\not \\in q^\\mathbb Z$, for any fixed $y_0\n\\in \\Omega$, $\\text{det}(\\breve{P}(y_0)^{-1}\\breve{P}(z))$ is a non\nconstant holomorphic function (with respect to $z$). This implies\nthat $G=G^I=\\text{Gl}_2(\\mathbb C)$. On the other hand, if $abq\/c \\in\nq^\\mathbb Z$, then we have that\n$\\text{det}(\\breve{P}(y_0)^{-1}\\breve{P}(z))=1$, so that the\nconnection component of the Galois group is a subgroup of\n$\\text{Sl}_2(\\mathbb C)$ and the Galois group $G$ is the smallest\nalgebraic group which contains $\\text{Sl}_2(\\mathbb C)$ and $\\{\n\\sqrt{w}I,e^{\\pi i \\gamma}I\\}$.\n\\end{proof}\n\n\nWe are going to study the case $a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z$ and $a\/b,c \\in \\pm q^{\\mathbb Z+1\/2}$ in two steps.\\\\\n\n$_{}$\\vskip 10 pt\n\n\\condun{2} $\\underline{a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z\n\\text{ and } q^\\mathbb Z a \\cup q^\\mathbb Z b \\cup q^\\mathbb Z\naq\/c \\cup q^\\mathbb Z bq\/c=q^\\mathbb Z a \\cup -q^\\mathbb Z a \\cup\nq^{\\mathbb Z+1\/2} a \\cup -q^{\\mathbb Z+1\/2} a}$\n\n$_{}$\\vskip 5 pt\n\n We first establish a preliminary result.\n\n\\begin{lem}\\label{hg}\nSuppose that \\condun{2} holds. Then any functional equation of the\nform $Az^{n\/2}\\theta_q(q^Naz)+Bz^{m\/2}\\theta_q(-q^M\naz)+Cz^{l\/2}\\theta_q(q^Lq^{1\/2}az)+Dz^{k\/2}\n\\theta_q(-q^Kq^{1\/2}az)=0$ with $A,B,C,D \\in \\mathbb C$,\n$n,m,l,k,N,M,L,K\\in \\mathbb Z$ is trivial, that is $A=B=C=D=0$.\n\\end{lem}\n\n\\begin{proof}\nUsing the non-trivial monodromy of $z^{1\/2}$, we reduce the\nproblem to the case of $n,m,l,k$ being odd numbers. In this case,\nusing the functional equation $\\theta_q(qz)=-z^{-1}\\theta_q(z)$,\nwe can assume without loss of generality that $n=l=m=k=0$. The\nexpansion of $\\theta_q$ as an infinite Laurent series\n$\\theta_q(z)=\\sum_{j\\in\\mathbb Z}q^{\\frac{j(j-1)}{2}}(-z)^j$\nensures that, for all $j \\in \\mathbb Z$, the following equality\nholds :\n$$A(q^{N})^j+B(-q^{M})^j+C(q^{L+1\/2})^j+D(-q^{K+1\/2})^j=0.$$ Considering the associated generating series, this implies that :\n$$\\frac{A}{1-q^Nz}+\\frac{B}{1+q^Mz}+\\frac{C}{1-q^{L+1\/2}z}+\\frac{D}{1+q^{K+1\/2}z}=0.$$\nHence, considering the poles of this rational fraction, we obtain\n$A=B=C=D=0$.\n\\end{proof}\n\n\n\n\\begin{prop}\nSuppose that \\condun{2} holds. Then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\n\\begin{proof}\nSuppose, at the contrary, that the action of $G^I$ is reducible\nand consider an invariant line $L\\subset \\mathbb C^2$. In\nparticular, $L$ is invariant under the action of the connection\ncomponent. Consequently, the line $\\breve{P}(z)L$ does not depend\non $z\\in \\Omega$. This is impossible using Lemma \\ref{hg} (the cases\n$L=\\mathbb C \\vect{1}{0}$ or $\\mathbb C \\vect{0}{1}$ are excluded by\ndirect calculation; for the remaining cases consider the ratio of\nthe coordinates of a generator of $L$ and apply Lemma \\ref{hg}).\nWe get a contraction, hence prove that $G^I$ acts irreducibly.\n\\end{proof}\n\n\\begin{theo}\nIf \\condun{2} holds then we have the following dichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $abq\/c \\not \\in q^\\mathbb Z$ then $G=\\text{Gl}_2(\\mathbb C)$;\n\\item[$\\bullet$] if $abq\/c \\in q^\\mathbb Z$ then $G=\\overline{\\langle \\text{Sl}_2(\\mathbb C),\\sqrt{w}I,e^{\\pi i \\gamma}I \\rangle}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nThe proof follows the same lines as that of theorem \\ref{theo un}.\n\\end{proof}\n\n\nThe remaining subcases are $b\\in -aq^\\mathbb Z$ and $c\\in\n-q^\\mathbb Z$; $b\\in -aq^{\\mathbb Z+1\/2}$ and $c\\in\n-q^{\\mathbb Z+1\/2}$; $b\\in\naq^{\\mathbb Z+1\/2}$ and $c\\in q^{\\mathbb Z+1\/2}$.\\\\\n\n$_{}$\\vskip 10 pt\n\n\\condun{3} $\\underline{a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z\n\\text{ and } b\\in -aq^\\mathbb Z \\text{ and } c\\in -q^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\n We use the following notations : $b = -aq^\\delta\n\\text{ and } c = -q^\\gamma$ with\n$\\delta=\\beta-\\alpha,\\gamma\\in\\mathbb Z$.\n\nThe twisted connection matrix takes the following form :\n\\begin{eqnarray*}\n\\breve{P}(z) &=&(1\/z)^{-\\alpha} \\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}\n {\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n \\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}z^{1-\\gamma}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}z^{\\delta} }\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}z^{1+\\delta-\\gamma}} \\\\\n&=&(1\/z)^{-\\alpha} \\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}} {\n\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n q^{\\frac{\\gamma(1-\\gamma)}{2}}a^{\\gamma-1} \\frac{\\theta_q (-a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\nq^{\\frac{-\\delta(\\delta-1)}{2}}a^{-\\delta} \\frac{\\theta_q (-a\nz)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\nq^{-\\frac{(\\delta-\\gamma +1)(\\delta -\\gamma)}{2}}(-a)^{\\gamma -\\delta -1}\\frac{\\theta_q (a\nz)}{\\theta_q(z)}}\n\\end{eqnarray*}\n\n\\begin{theo}\nSuppose that \\condun{3} holds. We have $G=R\n\\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*} R^{-1} \\cup\n\\pmatrice{1}{0}{0}{-1} R \\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*}\nR^{-1}$ for some $R\\in \\text{Gl}_2(\\mathbb C)$ of the form\n$\\pmatrice{1}{1}{C}{-C}$, $C\\in\\mathbb C^*$.\n\\end{theo}\n\n\\begin{proof}\nRemark that there exist two nonzero constants $A,B$ such that, for\nall $z\\in \\Omega$ :\n\\begin{eqnarray*}\n\\breve{P}(-1\/a)^{-1}\\breve{P}(z)&=&(-a)^\\alpha \\frac{\\theta_q\n(-1\/a)}{\\theta_q(-1)} (1\/z)^{-\\alpha} \\pmatrice{\\frac{\\theta_q (a\nz)}{\\theta_q(z)}} {A \\frac{\\theta_q (-a z)}{\\theta_q(z)}} {B\n\\frac{\\theta_q (-a z)}{\\theta_q(z)} } {\\frac{\\theta_q (a\nz)}{\\theta_q(z)}}\\\\\n&=&(-a)^\\alpha \\frac{\\theta_q (-1\/a)}{\\theta_q(-1)} (1\/z)^{-\\alpha} R\n\\pmatrice{\\frac{\\theta_q (a z)}{\\theta_q(z)} + \\sqrt{BA}\n\\frac{\\theta_q (-a z)}{\\theta_q(z)}}{0} {0} {\\frac{\\theta_q (a\nz)}{\\theta_q(z)} - \\sqrt{BA} \\frac{\\theta_q (-a z)}{\\theta_q(z)}}\nR^{-1}\n\\end{eqnarray*}\nwith $R=\\pmatrice{1}{1}{\\sqrt{B\/A}}{-\\sqrt{B\/A}}$.\n\nFurthermore, we claim that the functions $X(z):=(1\/z)^{-\\alpha}\n(\\frac{\\theta_q (a z)}{\\theta_q(z)} + \\sqrt{BA} \\frac{\\theta_q (-a\nz)}{\\theta_q(z)})$ and $Y(z):=(1\/z)^{-\\alpha} (\\frac{\\theta_q (a\nz)}{\\theta_q(z)} - \\sqrt{BA} \\frac{\\theta_q (-a z)}{\\theta_q(z)})$\ndo not satisfy any non-trivial relation of the form $X^rY^s=1$\nwith $(r,s)\\in\\mathbb Z^2 \\setminus \\{(0,0\\}$. Indeed, suppose on\nthe contrary that such a relation holds. Then\n$\\frac{((1\/z)^{-\\alpha} (\\theta_q (a z) + \\sqrt{BA} \\theta_q (-a\nz)))^r}{((1\/z)^{-\\alpha} (\\theta_q (a z) - \\sqrt{BA} \\theta_q (-a\nz)))^s}=\\theta_q(z)^{s-r}$. Let us first exclude the case $r\\neq\ns$. If $s>r$ then we conclude that $\\theta_q (a z) + \\sqrt{BA}\n\\theta_q (-a z)$ must vanish on $q^\\mathbb Z$. In particular,\n$\\theta_q (a) + \\sqrt{BA} \\theta_q (-a)=0$ and $\\theta_q (aq) +\n\\sqrt{BA} \\theta_q (-aq) = -(az)^{-1}(\\theta_q (a z) - \\sqrt{BA}\n\\theta_q (-a z))=0$, so $\\theta_q (a)=0$ that is $a\\in\nq^\\mathbb Z$. This yields a contradiction. The case $r>s$ is\nsimilar by symmetry. Hence we have $r=s$ so that\n$\\left(\\frac{\\theta_q (a z) + \\sqrt{BA} \\theta_q (-a z)}{\\theta_q\n(a z) - \\sqrt{BA} \\theta_q (-a z)}\\right)^r=1$. Therefore the\nfunction $\\frac{\\theta_q (a z) + \\sqrt{BA} \\theta_q (-a\nz)}{\\theta_q (a z) - \\sqrt{BA} \\theta_q (-a z)}$ is constant. This\nis clearly impossible and our claim is proved.\n\n\n\n\nThis ensures that the connection component of $G^I$, generated by\nthe matrices $\\breve{P}(-1\/a)^{-1}\\breve{P}(z)$, $z \\in \\Omega$, is\nequal to $R\\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*}R^{-1}$.\nConsequently, $G$ is generated as an algebraic group by\n$R\\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*}R^{-1}$,\n$\\pmatrice{1}{0}{0}{-1}$,\n$\\breve{P}(-1\/a)^{-1}\\pmatrice{u}{0}{0}{-u}\\breve{P}(-1\/a)=\\pmatrice{u}{0}{0}{-u}$\nand $\\breve{P}(-1\/a)^{-1}\\pmatrice{e^{2\\pi i\n\\alpha}}{0}{0}{e^{2\\pi i \\alpha}}\\breve{P}(-1\/a)=\\pmatrice{e^{2\\pi\ni \\alpha}}{0}{0}{e^{2\\pi i \\alpha}}$. The theorem follows.\n\\end{proof}\n\n$\\bullet$ Both cases ($b\\in -aq^{\\mathbb Z+1\/2}$ and $c\\in -q^{\\mathbb Z+1\/2}$) and ($b\\in\naq^{\\mathbb Z+1\/2}$ and $c\\in q^{\\mathbb Z+1\/2}$) are similar.\\\\\n\n$_{}$\\vskip 10 pt\n\n\\condun{4} $\\underline{a \\in q^{\\mathbb N^*}}$.\n\n$_{}$\\vskip 5 pt\n\nIn this case, the twisted connection matrix is lower triangular :\n\n\\begin{eqnarray*}\n\\breve{P}(z) &=&\\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n(-1)^{\\alpha} q^{-\\frac{\\alpha (\\alpha -1)}{2}}} {0}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}\n\\end{eqnarray*}\n\n\\begin{theo}\\label{theo theo}\nSuppose that \\condun{4} holds.\nWe have the following trichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $b\/c \\not\\in q^\\mathbb Z$ then\n$G=\\pmatrice{1}{0} {\\mathbb C}{\\mathbb C^*}$;\n\\item[$\\bullet$] if $c\/b \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{0}{\\mathbb C}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$;\n\\item[$\\bullet$] if $bq\/c \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{0} {0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nRemark that in each case there exist two constants $A,B$ with $B\n\\neq 0$ such that, for all $z\\in \\Omega$,\n$\\breve{P}(1\/b)^{-1}\\breve{P}(z)= \\pmatrice{1}{0}{A\\frac{\\theta_q\n(b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}{B \\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}$. Hence, the\nconnection component is a subgroup of\n$\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$.\n\nAssume $b\/c \\not\\in q^\\mathbb Z$. Then $A\\neq 0$ and we claim that\nthe connection component is equal to\n$\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$. Indeed, for all\n$n\\in\\mathbb Z$, the following matrix :\n$$(\\breve{P}(1\/b)^{-1}\\breve{P}(z))^n=\\pmatrice{1}{0}{A\\frac{\\theta_q\n(b z)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1-\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}{1-B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}}{\\left(B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}$$ belongs to\nthe connection component. Consider a polynomial in two variables\n$K(X,Y) \\in \\mathbb C[X,Y]$ such that :\n$$K(A\\frac{\\theta_q (b\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1-\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}{1-B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}},\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n)=0.$$ If $K$\nwas non zero then we could assume that $K(X,0)\\neq 0$. But, for\nall $z\\in\\Omega$ in a neighborhood of $\\frac{c}{bq}$, we have\n$\\left|B \\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right|<1$, hence\nletting $n$ tend to $+\\infty$, we would get $K(A\\frac{\\theta_q (b\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1}{1-B \\frac{\\theta_q\n(\\frac{bq}{c} z)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}},0)=0$\nwhich would imply $K(X,0)=0$. This proves that $K=0$. In other\nwords the only algebraic subvariety of $\\mathbb C \\times\n\\mathbb C^*$ containing $(A\\frac{\\theta_q (b\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1-\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}{1-B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}},\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n)$ for all\n$n\\in\\mathbb Z$ is $\\mathbb C \\times \\mathbb C^*$ itself. In\nparticular, the algebraic group generated by the matrix\n$(\\breve{P}(1\/b)^{-1}\\breve{P}(z))^n$ for all $n\\in\\mathbb Z$ is\n$\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$, hence the connection\ncomponent is equal to $\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$.\n\nIt is now straightforward that\n$G=\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$.\n\n\nSuppose that $c\/b \\in q^{\\mathbb N^*}$. Then the function\n$\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}$ is constant. Hence the\nmatrix $\\breve{P}(1\/b)^{-1}\\breve{P}(z)$ simplifies as follows :\n$$\\breve{P}(1\/b)^{-1}\\breve{P}(z)= \\pmatrice{1}{0}{A\\frac{\\theta_q\n(b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}{1}$$ with $ A\\neq0$. The\nconnection component is equal to $\\pmatrice{1}{0}{\\mathbb C}{1}$\nand the whole Galois group $G$ is equal to\n$\\pmatrice{1}{0}{\\mathbb C}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\nLast, suppose that $bq\/c \\in q^{\\mathbb N^*}$. Then $A=0$ and the\nfunction $\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}$ is constant, hence\n$G=\\pmatrice{1}{0}{0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\\end{proof}\n\n$_{}$\\vskip 10 pt\n\n\\condun{5} $\\underline{a \\in q^{-\\mathbb N}}$.\n\n$_{}$\\vskip 5 pt\n\nIn this case, the twisted connection matrix is upper triangular :\n\n\\begin{eqnarray*}\n\\breve{P}(x) &=&\\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n(-1)^{\\alpha} q^{-\\frac{\\alpha (\\alpha -1)}{2}}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{aq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}z^{1-\\gamma}} {0}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}\n\\end{eqnarray*}\n\n\\begin{theo}\nSuppose that \\condun{5} holds.\nWe have the following trichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $b\/c \\not\\in q^\\mathbb Z$ then\n$G=\\pmatrice{1}{\\mathbb C}{0}{\\mathbb C^*}$;\n\\item[$\\bullet$] if $bq\/c \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{\\mathbb C}{0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$;\n\\item[$\\bullet$] if $c\/b \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{0} {0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nWe argue as for theorem \\ref{theo theo}.\n\\end{proof}\n\n$\\bullet$ The cases $\\underline{b \\in q^\\mathbb Z \\text{ or } a\/c \\in q^\\mathbb Z \\text{ or } b\/c \\in q^\\mathbb Z}$ is similar to the case $a \\in q^\\mathbb Z$. We leave the details to the reader.\\\\\n\n\n\\section{Galois groups of the basic hypergeometric equations : logarithmic cases}\n\nWe write $a=uq^\\alpha$, $b=vq^\\beta$ and $c=wq^\\gamma$ with\n$u,v,w\\in\\mathbb{U}$ and $\\alpha,\\beta,\\gamma \\in \\mathbb R$ (we\nchoose a logarithm of $q$).\n\n\n\n\n\\subsection{$c=q$ and $a\/b \\not \\in q^\\mathbb Z$}\n\n\nThe aim of this section is to compute the Galois group of the\nbasic hypergeometric system (\\ref{syst hypergeo}) under the\nassumption : $c=q$ and $a\/b \\not \\in q^\\mathbb Z$.\n\n$_{}$\n\n\\textit{Local fundamental system of solutions at $0$.} We have :\n$$\\qhypermatrice{a}{b}{q}{0}=\\pmatrice{1}{0}{1}{1} \\pmatrice{1}{1}{0}{1} \\pmatrice{1}{0}{1}{1}^{-1}.$$\nConsequently, we are in the non-resonant logarithmic case at $0$.\nWe consider this situation as a degenerate case as $c$ tends to $q$, $c\\neq q$.\n\nMore precisely, we consider the limit as $c$ tends to $q$, with\n$c\\not \\mathbb{E}_q q$, of the following matrix-valued function :\n\n\n\\begin{eqnarray*}\n&&\\fz{a}{b}{c}{z}\\pmatrice{1}{1}{1}{q\/c}^{-1}\\pmatrice{1}{0}{1}{1}\\\\\n&=&\\frac{-c}{c-q} \\pmatrice{(q\/c-1)\\qhyper{a}{b}{c}{z}}{\\qhyper{aq\/c}{bq\/c}{q^2\/c}{z}-\\qhyper{a}{b}{c}{z}}{(q\/c-1)\\qhyper{a}{b}{c}{qz}}{(q\/c) \\qhyper{aq\/c}{bq\/c}{q^2\/c}{qz}-\\qhyper{a}{b}{c}{qz}} \\\\\n\\end{eqnarray*}\n\nUsing the notations : $$\\qhyperc{a}{b}{z} =\n\\frac{d}{dc}_{|c=q}\\left[\\qhyper{a}{b}{c}{z}\\right] \\text{ and }\n\\qhyperabc{a}{b}{z} =\n\\frac{d}{dc}_{|c=q}\\left[\\qhyper{aq\/c}{bq\/c}{q^2\/c}{z}\\right]$$\nthe above limit is equal to :\n\\begin{eqnarray*}\n\\fz{a}{b}{q}{z}&:=&\\pmatrice{\\qhyper{a}{b}{q}{z}}{-q(\\qhyperabc{a}{b}{z} - \\qhyperc{a}{b}{z})}\n{\\qhyper{a}{b}{q}{qz}}{\\qhyper{a}{b}{q}{qz} -q(\\qhyperabc{a}{b}{qz} - \\qhyperc{a}{b}{qz})}. \\\\\n\\end{eqnarray*}\nFrom (\\ref{transfo jauge}) we deduce that $\\fz{a}{b}{q}{z}$\nsatisfies\n$\\fz{a}{b}{q}{qz}\\jz{q}=\\qhypermatrice{a}{b}{c}{z}\\fz{a}{b}{q}{z}$\nwith $\\jz{q}=\\pmatrice{1}{1}{0}{1}$. Hence, this matrix being\ninvertible as a matrix in the field of meromorphic functions, the\nmatrix-valued function\n$\\yz{a}{b}{q}{z}=\\fz{a}{b}{q}{z}e^{(0)}_{\\jz{q}}(z)$ is a\nfundamental system of solutions of the\nbasic hypergeometric equation with $c=q$. Let us recall that $e^{(0)}_{\\jz{q}}(z)=\\pmatrice{1}{\\ell_q(z)}{0}{1}$. \\\\\n\\\\\n\n\\textit{Generators of the local Galois group at $0$.} We have the following generator :\n$$\\pmatrice{1}{1}{0}{1}.$$\\\\\n\n\\textit{Local fundamental system of solutions at $\\infty$.} The\nsituation is as Section \\ref{section generique}. Hence we are in\nthe non-resonant and non-logarithmic case at $\\infty$ and a\nfundamental system of solutions at $\\infty$ of (\\ref{syst\nhypergeo}) as described in Section \\ref{section the basic objects}\nis given by\n$\\yinf{a}{b}{q}{z}=\\finf{a}{b}{q}{z}e^{(\\infty)}_{\\jinf{a}{b}}(z)$\nwith $\\jinf{a}{b} = \\text{diag}(1\/a,1\/b)$ and :\n$$\\finf{a}{b}{q}{z}=\n\\pmatrice{\\qhyper{a}{a}{aq\/b}{\\frac{q^2}{ab}z^{-1}}}{\\qhyper{b}{b}{bq\/a}{\\frac{q^2}{ab}z^{-1}}}\n{\\frac{1}{a} \\qhyper{a}{a}{aq\/b}{\\frac{q}{ab}z^{-1}}}{\\frac{1}{b} \\qhyper{b}{b}{bq\/a}{\\frac{q}{ab}z^{-1}}}.$$\\\\\n\n\n\\textit{Generators of the local Galois group at $\\infty$.} We have two generators :\n$$\\breve{P}(y_0)^{-1}\\pmatrice{e^{2\\pi i\\alpha}}{0}{0}{e^{2\\pi i \\beta}}\\breve{P}(y_0) \\text{ and } \\breve{P}(y_0)^{-1}\\pmatrice{u}{0}{0}{v}\\breve{P}(y_0).$$\\\\\n\n\n\\textit{Connection matrix}. The connection matrix is the limit as $c$ tends to $q$ of :\n\n\\begin{eqnarray*}\n(e^{(\\infty)}_{\\jinf{a}{b}}(z))^{-1}\\pmatrice{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}}\\pmatrice{1}{1}{1}{q\/c}^{-1}\\pmatrice{1}{0}{1}{1} e^{(0)}_{\\jz{q}}(z)\\\\\n\\end{eqnarray*}\nwhich is equal to :\n\n\\begin{eqnarray*}\nP(z)&:=&(e^{(\\infty)}_{\\jinf{a}{b}}(z))^{-1} \\pmatrice{ u(a,b;q)\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}{\n q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)} +az v(a,b;q) \\frac{\\theta'_q\n(a z)}{\\theta_q(z)}} {w(a,b;q) \\frac{\\theta_q (b\nz)}{\\theta_q(z)}} { q(w_c(a,b;q) -y_c(a,b;q)) \\frac{\\theta_q (b\nz)}{\\theta_q(z)} +bz y(a,b;q) \\frac{\\theta'_q (b\nz)}{\\theta_q(z)}}\ne^{(0)}_{\\jz{q}}(z)\\\\\n\\end{eqnarray*}\nwhere :\n\\begin{eqnarray*}\nu(a,b;c)= \\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}; \\ \\ v(a,b;c)=\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}};\\\\\nw(a,b;c)=\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}};\n\\ \\ y(a,b;c)=\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}.\\\\\n\\end{eqnarray*}\n and where the subscript $c$ means that we take\nthe derivative with respect to the third variable $c$.\\\\\n\n\n\\textit{Twisted connection matrix.}\n\\begin{small} \\begin{eqnarray*}\n\\breve{P}(z)&=&\\pmatrice{(1\/z)^{-\\alpha}}{0}{0}{(1\/z)^{-\\beta}}\n\\pmatrice{ u(a,b;q) \\frac{\\theta_q (a z)}{\\theta_q(z)}}{\n q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)} +az v(a,b;q)\n\\frac{\\theta'_q (a z)}{\\theta_q(z)}} {w(a,b;q) \\frac{\\theta_q\n(b z)}{\\theta_q(z)}} { q(w_c(a,b;q) -y_c(a,b;q))\n\\frac{\\theta_q (b z)}{\\theta_q(z)} +bz y(a,b;q)\n\\frac{\\theta'_q (b z)}{\\theta_q(z)}} \\pmatrice{1}{\\ell_q(z)}{0}{1}\n\\end{eqnarray*}\\end{small}\n\n$_{}$\\vskip 10 pt\n\nWe need to consider different cases.\n\n$_{}$\\vskip 10 pt\n\n\\conddeux{1} $\\underline{a \\not\\in q^\\mathbb Z \\text{ and } b\\not \\in\nq^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\nSubject to this condition, the complex numbers $u(a,b;q)$,\n$v(a,b;q)$, $w(a,b;q)$ and $y(a,b;q)$ are non-zero.\n\n\\begin{prop} \\label{blabla}\nIf \\conddeux{1} holds then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\n\\begin{proof} Assume, at the contrary, that the action of $G^I$ is\nreducible and let $L$ be an invariant line.\n\nLet us fist remark that $L \\neq \\mathbb C \\vect{1}{0}$ (in\nparticular, $G^I$ does not consist of scalar matrices). Indeed, if not,\n$\\mathbb C \\vect{1}{0}$ would be stabilized by the connection\ncomponent and the line spanned by $\\breve{P}(z) \\vect{1}{0}$\nwould be independent of $z\\in\\Omega$ : this is clearly false.\n\nThe group $G^I$ being normalized by $\\pmatrice{1}{1}{0}{1}$\n(since $G^I$ is a normal subgroup of $G$), the\nlines $\\pmatrice{1}{1}{0}{1}^n L$ are also invariant by the action\nof $G^I$. These lines being distinct (since $L\\neq \\mathbb C\n\\vect{1}{0}$) we conclude that $G^I$ consists of scalar matrices and\nwe get a contradiction. This proves that $G^I$ acts\nirreducibly.\n\\end{proof}\n\nAs a consequence we have the following theorem.\n\n\\begin{theo}\nSuppose that \\conddeux{1} holds. Then we have the following dichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $ab \\not \\in q^\\mathbb Z$ then $G=\\text{Gl}_2(\\mathbb C)$;\n\\item[$\\bullet$] if $ab \\in q^\\mathbb Z$ then $G= \\text{Sl}_2(\\mathbb C)$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nUsing the irreducibility of the natural action of $G^I$ and\narguing as for the proof of theorem \\ref{theo un}, we obtain the\nequality $G^{I,der}=\\text{Sl}_2(\\mathbb C)$. From the formula (\\ref{det})\nwe deduce that the determinant of the twisted connection matrices\nwhen $c=q$ is equal to the limit as $c$ tends to $q$ of\n$\\frac{-1}{1\/a-1\/b}(1\/z)^{-(\\alpha+\\beta)} z^{1-\\gamma}\n\\frac{\\theta_q (\\frac{abq}{c} z)}{\\theta_q(z)}$ \\textit{i.e.}\n$\\frac{-1}{1\/a-1\/b}(1\/z)^{-(\\alpha+\\beta)} z^{1-\\gamma}\n\\frac{\\theta_q (ab z)}{\\theta_q(z)}.$\n\nIf $ab \\not \\in q^\\mathbb Z$ then this determinant is a\nnon-constant holomorphic function and consequently\n$G=\\text{Gl}_2(\\mathbb C)$.\n\nIf $ab \\in q^\\mathbb Z$ then this determinant does not depend on\n$z$. This implies that the connection component of the Galois\ngroup is a sub-group of $\\text{Sl}_2(\\mathbb C)$. Furthermore, $ab \\in\nq^\\mathbb Z$ entails that $uv =1$ and $\\alpha+\\beta \\in\n\\mathbb Z$, that is, $e^{2\\pi i (\\alpha+\\beta)}=1$. Consequently,\nthe local Galois groups are subgroups of $\\text{Sl}_2(\\mathbb C)$ and the\nglobal Galois group $G$ is therefore a subgroup of\n$\\text{Sl}_2(\\mathbb C)$.\n\\end{proof}\n\n\n\n$_{}$\\vskip 10 pt\n\n\\conddeux{2} $\\underline{b\\in q^{\\mathbb N^*}}$.\n\n$_{}$\\vskip 5 pt\n\nThen the twisted\nconnection matrix simplifies as follows :\n\\begin{small}\\begin{eqnarray*}\n\\breve{P}(z)&=& \\pmatrice{ u(a,b;q) \\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}} {q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha} +az v(a,b;q)\n\\frac{\\theta'_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}} {0} {\nq(w_c(a,b;q) -y_c(a,b;q)) (-1)^\\beta\nq^{-\\frac{\\beta(\\beta-1)}{2}}}\n\\pmatrice{1}{\\ell_q(z)}{0}{1}\\\\\n\\end{eqnarray*}\\end{small}\n\n\\begin{theo}\\label{theou}\nSuppose that \\conddeux{2} holds.\nThen we have $G=\\pmatrice{\\mathbb C^*}{\\mathbb C}{0}{1}$.\n\\end{theo}\n\\begin{proof}\nFix a point $y_0 \\in \\Omega$ such that $\\breve{P}(y_0)$ is of the\nform $\\pmatrice{A}{B}{0}{C}$ with $A,C \\neq 0$. There exists a\nconstant $D\\in \\mathbb C^*$ such that :\n$$\\breve{P}(y_0)^{-1}\\breve{P}(z)= \\pmatrice{D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{*}{0}{1}.$$\nSince $G^I$ is normalized by $\\pmatrice{1}{1}{0}{1}$\n(remember that $G^I$ is a normal subgroup of $G$), it contains, for all $n\\in \\mathbb Z$, the matrix :\n$$\\pmatrice{D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{*+n(D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}-1)}{0}{1}.$$\nBecause $a\\not \\in q^\\mathbb Z$, the function $D\\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}-1$ is not identically equal to\nzero over $\\mathbb C^*$ and therefore $G^I$ contains, for all $z\n\\in \\Omega$ :\n$$\\pmatrice{D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{\\mathbb C}{0}{1}.$$\nIn particular, $\\pmatrice{D\\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{0}{0}{1}$ belongs to $G^I$, so that\n$\\pmatrice{\\mathbb C^*}{0}{0}{1}$ is a subgroup of $G^I$ and\n$\\pmatrice{\\mathbb C^*}{\\mathbb C}{0}{1} \\subset G$. The converse\ninclusion is clear.\n\\end{proof}\n\n$_{}$\\vskip 10 pt\n\n\\conddeux{3} $\\underline{b\\in q^{-\\mathbb N}}$.\n\n$_{}$\\vskip 5 pt\n\nUsing the identity :\n$$bz\\frac{\\theta_q'(bz)}{\\theta_q(z)}=(-\\beta-\\ell_q(z))(-1)^\\beta\nq^{-\\frac{\\beta(\\beta-1)}{2}}$$ we see that, in this case, the\ntwisted connection matrix takes the form :\n\n\\begin{eqnarray*}\n\\breve{P}(z\n&=&\\pmatrice{0} {q(u_c(a,b;q) -v_c(a,b;q)) \\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}} {w(a,b,q) (-1)^\\beta\nq^{-\\frac{\\beta(\\beta-1)}{2}}} { q(w_c(a,b;q)\n-y_c(a,b;q)-\\beta\/q) (-1)^\\beta q^{-\\frac{\\beta(\\beta-1)}{2}}}\\\\\n\\end{eqnarray*}\n\n\\begin{theo}\nSuppose that \\conddeux{3} holds. Then we have $G=\\pmatrice{1}{\\mathbb C}{0}{\\mathbb C^*}.$\n\\end{theo}\n\n\\begin{proof}\nFix a base point $y_0 \\in \\Omega$. There exist three constants\n$C,C',C''\\in \\mathbb C$ with $C\\neq 0$, such that the following\nidentity holds for all $z\\in \\Omega$ :\n$$\\breve{P}(y_0)^{-1}\\breve{P}(z)=\\pmatrice{1}{C'\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}+C''}\n{0}{C\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}.$$ The proof is similar to the proof of Theorem \\ref{theou}.\n\\end{proof}\n\n\nThe remaining case $a\\in q^\\mathbb Z$ is similar to\n\\conddeux{3}.\n\nThe case $a=b$ and $c\\not \\in q^\\mathbb Z$ is similar to the case treated in this section.\n\n\n\\subsection{$a=b$ and $c=q$}\n\nThe aim of this section is to compute the Galois group of the\nbasic hypergeometric system (\\ref{syst hypergeo}) under the\nassumption : $a=b$ and $c=q$.\n\n$_{}$\n\n\\textit{Local fundamental system of solutions at $0$.} The\nsituation is the same\nas in the case $c=q$ and $a\/b \\not\\in q^\\mathbb Z$.\\\\\n\n\\textit{Generator of the local Galois group at $0$}. We have the following generator :\n$$\\pmatrice{1}{1}{0}{1}.$$\\\\\n\n\\textit{Local fundamental system of solutions at $\\infty$.} We\nhave :\n$$\\qhypermatrice{a}{a}{q}{z}=\\pmatrice{1}{0}{1\/a}{1} \\pmatrice{1\/a}{1}{0}{1\/a} \\pmatrice{1}{0}{1\/a}{1}^{-1}.$$\nConsequently, we are in the non-resonant logarithmic case at\n$\\infty$. We consider the case $a=b$ and $c=q$ as a degenerate\ncase of the situation $c=q$ as $a$ tends to $b$, $a\/b \\neq 1$.\n\n\nWe consider the following matrix valued function :\n$$\\finf{a}{b}{q}{z}\\pmatrice{1}{1}{1\/a}{1\/b}^{-1}\\pmatrice{1}{0}{1\/a}{1}.$$\nA straightforward calculation, which we omit here because it is\nlong although easy, shows that this matrix-valued function does admit a limit as $a$ tends to $b$ that we denote $\\finf{a}{a}{q}{z}$. A fundamental\nsystem of solutions at $\\infty$ of (\\ref{syst hypergeo}) as\ndescribed in section \\ref{section the basic objects} is given by\n$\\yinf{a}{a}{q}{z}=\\finf{a}{a}{q}{z}e^{(\\infty)}_{\\jinf{a}{a}}(z)$\nwith $\\jinf{a}{a} = \\pmatrice{1\/a}{1}{0}{1\/a}$.\\\\\n\\\\\n\n\\textit{Generator of the local Galois group at $\\infty$}. We have the following generators :\n$$\\pmatrice{u}{0}{0}{u}, \\ \\ \\pmatrice{e^{2\\pi i \\alpha}}{0}{0}{e^{2\\pi i \\alpha}} \\text{ and } \\breve{P}(y_0)^{-1}\\pmatrice{1}{a}{0}{1}\\breve{P}(y_0).$$\\\\\n\n\\textit{Birkhoff matrix}. The Birkhoff matrix is equal to\n$(e^{(\\infty)}_{\\jinf{a}{a}}(z))^{-1}Qe^{(0)}_{\\jz{q}}(z)$ where\n$Q$ is the limit as $a$ tends to $b$ of :\n\n\\begin{eqnarray*}\n&&\\pmatrice{1}{0}{1\/a}{1}^{-1}\\pmatrice{1}{1}{1\/a}{1\/b}\\pmatrice{\nu(a,b;q) \\frac{\\theta_q (a z)}{\\theta_q(z)}}{\n q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)} +az v(a,b;q)\n\\frac{\\theta'_q (a z)}{\\theta_q(z)}} {w(a,b;q) \\frac{\\theta_q\n(b z)}{\\theta_q(z)}} { q(w_c(a,b;q) -y_c(a,b;q))\n\\frac{\\theta_q (b z)}{\\theta_q(z)} +bz y(a,b;q)\n\\frac{\\theta'_q (b z)}{\\theta_q(z)}}.\n\\end{eqnarray*}\nIt has the following form :\n\n\n\\begin{eqnarray*}\n&& \\pmatrice{ C \\frac{\\theta_q(az)}{\\theta_q(z)} +az\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2} \\frac{\\theta_q'(a\nz)}{\\theta_q(z)}} {*} {-(1\/a)\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\\frac{\\theta_q(az)}{\\theta_q(z)}}\n{C'\\frac{\\theta_q(az)}{\\theta_q(z)}-z\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\n\\frac{\\theta_q'(a z)}{\\theta_q(z)}}\n\\end{eqnarray*}\nwhere $*$ denotes some meromorphic function.\\\\\n\n\\textit{Twisted Birkhoff matrix}.\n\n$$\n (1\/z)^{-\\alpha} \\pmatrice{1}{-a\\ell_q(z)}{0}{1}\n\\pmatrice{C \\frac{\\theta_q(az)}{\\theta_q(z)}+az\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\n\\frac{\\theta_q'(a z)}{\\theta_q(z)}} {*} {-(1\/a)\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\\frac{\\theta_q(az)}{\\theta_q(z)}}\n{C'\\frac{\\theta_q(az)}{\\theta_q(z)}-z\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\n\\frac{\\theta_q'(a z)}{\\theta_q(z)}}\n \\pmatrice{1}{\\ell_q(z)}{0}{1}.\n$$\n\n$_{}$\\vskip 10 pt\n\nWe need to consider different cases.\n\n$_{}$\\vskip 10 pt\n\n\\condtrois{1} $\\underline{a \\not \\in q^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\n\\begin{prop}\nSuppose that \\condtrois{1} holds. Then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\\begin{proof}\nRemark that $\\mathbb C \\vect{1}{0}$ is not an invariant line.\nIndeed, if not, this line would be invariant by the action of the\nconnection component, hence the line spanned by $\\breve{P}(z)\n\\vect{1}{0}$ would be independent of $z\\in\\Omega$. Considering the\nratio of the coordinates of this line, this would imply the\nexistence of some constant $A \\in \\mathbb C$ such that the\nfollowing functional equation holds on $\\mathbb C^*$ :\n$$C \\frac{\\theta_q(az)}{\\theta_q(z)} +az\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2} \\frac{\\theta_q'(a\nz)}{\\theta_q(z)}+\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\\frac{\\theta_q(az)}{\\theta_q(z)}\n\\ell_q(z) = A \\frac{\\theta_q(az)}{\\theta_q(z)}.$$ The fact that\n$\\theta_q(az)$ vanishes exactly to the order one at $z=1\/a$,\nyields a contradiction.\n\nThe end of the proof is similar to the proof of Proposition\n\\ref{blabla}.\n\\end{proof}\n\n\\begin{theo}\nIf \\condtrois{1} holds then we have the following dichotomy :\n\\begin{itemize}\n \\item[$\\bullet$] if $a^2 \\not \\in q^\\mathbb Z$ then\n $G=\\text{Gl}_2(\\mathbb C)$;\n \\item[$\\bullet$] if $a^2 \\in q^\\mathbb Z$ then\n $G=\\text{Sl}_2(\\mathbb C)$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nThe proof follows the same line as that of theorem \\ref{theo un}.\n\\end{proof}\n\n$_{}$\\vskip 10 pt\n\n\\condtrois{2} $\\underline{a \\in q^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\nUnder this condition, the connection matrix simplifies as follows,\nfor some constants $C,C'\\in \\mathbb C$ :\n\\begin{eqnarray*}\n&&\n\\pmatrice{ C } {*}\n{0}{C'}.\\\\\n\\end{eqnarray*}\n\n\n\\begin{theo}\nSuppose that \\condtrois{2} holds. Then we have : $G = \\pmatrice{1}{\\mathbb C}{0}{1}$.\n\\end{theo}\n\\begin{proof}\nThe local Galois group at $0$ is generated by\n$\\pmatrice{1}{1}{0}{1}$, hence $G$ contains\n$\\pmatrice{1}{\\mathbb C}{0}{1}$.\n\nSince the twisted connection matrix is upper triangular with\nconstant diagonal entries, the connection component is a subgroup of\n$\\pmatrice{1}{\\mathbb C}{0}{1}$. The generators of the local Galois\ngroup at $0$ and at $\\infty$ also lie in\n$\\pmatrice{1}{\\mathbb C}{0}{1}$. Therefore, $G$ is a subgroup of\n$\\pmatrice{1}{\\mathbb C}{0}{1}$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdtrx b/data_all_eng_slimpj/shuffled/split2/finalzzdtrx new file mode 100644 index 0000000000000000000000000000000000000000..159b109dbf67d08659585b5a11fcdfc6eb004c2f --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdtrx @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nEver since the emergence of the field of nanophotonics, it is well-known that fruitful analogies can be drawn between the behavior of photons at the nanoscale on the one hand, and the physics of electrons, spins, and phonons in condensed matter on the other hand~\\cite{vanHaeringen1990Book, Lagendijk1996PR, vanRossum1999RMP,Soukoulis2001Book,Akkermans2007Book,Ghulinyan2015Book}.\nThe seminal phenomenon considered in this respect was the three-dimensional (3D) Anderson localization of light~\\cite{John1984PRL,Anderson1985PM} - in analogy to Anderson localization of spins~\\cite{Anderson1958PR} - that continues to receive attention to date~\\cite{vanderBeek2012PRB,Sperling2016NJP}.\nOther well-known examples are the analogy between weak localization or enhanced backscattering of light~\\cite{vanAlbada1985PRL,Wolf1985PRL} and of electrons~\\cite{Bergmann1984PR}, and the analogy between a complete 3D photonic band gap in a 3D photonic crystal~\\cite{Yablonovitch1987PRL,John1987PRL,Joannopoulos2008Book} and the electronic band gap in a semiconductor crystal such as silicon or germanium~\\cite{Ashcroft1976Book}. \n\nIn this work, we explore the propagation of light in a 3D superlattice of coupled resonant cavities inside a 3D photonic band gap.\nThe propagation of light in such a 3D cavity superlattice is analogous to electronic transport in an impurity band in a semiconductor~\\cite{Anderson1958PR,Feher1959PR,Ashcroft1976Book}.\nLight hops from cavity to cavity throughout the 3D superlattice, which differs fundamentally from the conventional spatially-extended Bloch wave propagation outside the gap.\nSince the light hops predominantly in a few high-symmetry directions including the Cartesian $(x, y, z)$ directions, we propose the name \"Cartesian light\" for the unusual propagation of light in the 3D superlattice of coupled cavities in a 3D photonic band gap. \n\nIn one dimension (1D), a chain of coupled resonant cavities is a well-known system that is known as a coupled resonator optical waveguide (CROW)~\\cite{Yariv1999OL}.\nThe weak coupling between cavities in a CROW has been demonstrated at optical frequencies \\cite{Notomi2008NP}.\nCROWs are widely studied for efficient nonlinear optical frequency conversion and for perfect transmission through bends, and for the one-dimensional (1D) localization of light~\\cite{Yariv1999OL,Mookherjea2008NatPhoton}. \nTwo dimensional (2D) arrays of coupled cavities have been studied, notably \nfor unusual discrete diffraction effects~\\cite{Christodoulides2003N}, for intricate coupled nanolasers~\\cite{Altug2005OE}, for topologically-protected propagation~\\cite{Hafezi2013NP}, and for transverse localization~\\cite{deRaedt1989PRL,Schwartz2007N}.\nThe coupling between the cavities at optical frequencies has been demonstrated to be significantly larger than the fabrication-induced disorder in the cavity frequencies~\\cite{Majumdar2012PRB}.\nIn 3D resonator arrays without band gap, topologically-protected propagation was studied~\\cite{Lin2016NatCommun,Zhang2017PhysRevA}, as well as the percolation of light through 3D lattices of coupled resonant microspheres~\\cite{Astratov2007OptExpress}, and dynamic localization of light~\\cite{Yuan2015PRL}. \nIn 3D photonic band gap crystals in the microwave regime, slow heavy-photon propagation was reported in a 1D array of weakly-coupled cavities~\\cite{Bayindir2000PRL,Bayindir2000PRB1,Bayindir2000PRB2}.\nNumerical calculations of a 2D array of cavities embedded in a 3D woodpile photonic crystal revealed ultraslow and negative group velocities~\\cite{Li2006OL}. \nTo the best of our knowledge, 3D superlattices of coupled cavities with resonances in a 3D photonic band gap have not yet been studied before. \n\n\\section{Methods}\n\\label{sec:structures}\n\nIn this paper, we study a 3D cavity superlattice that is embedded in a 3D photonic band gap crystal that has the inverse woodpile structure. \nThis structure has nearly the same symmetry as a diamond crystal of carbon atoms,\\cite{Ho1994SSC} yet thousandfold magnified, as illustrated in a youtube animation.\\cite{COPS2012youtube} \nThe inverse woodpile crystal structure consists of two perpendicular 2D arrays of nanopores with radius $r$ in a high-index medium such as silicon,\\cite{Ho1994SSC} as illustrated in Figure~\\ref{fig:crystalOfCavities}(a). \nEach 2D pore array corresponds to a diamond ${110}$ crystal face. \nIn view of the arrangement of the nanopores, it appears to be convenient to employ a tetragonal unit cell~\\cite{Hillebrand2003JAP,Woldering2009JAP} instead of the conventional cubic unit cell~\\cite{Ashcroft1976Book}. \nThe tetragonal unit cell has lattice parameters $c$ (in the $x$ and $z$ directions), and $a$ (in the $y$ direction) in a ratio $a\/c = \\sqrt{2}$ to ensure a cubic crystal structure. \nMore details, notably on the Brillouin zone, are presented in Appendix~\\ref{sec:BrillouinZoneTetragonalUnitCell}. \n\nInverse woodpile photonic crystals possess a broad 3D photonic band gap, whose width strongly depends on the radius $r$ of the pores~\\cite{Ho1994SSC,Hillebrand2003JAP,Woldering2009JAP}. \nFor a normalized pore radius $r\/a=0.24$ - as considered here - a maximum relative bandwidth $\\Delta \\omega_{pbg}\/\\omega_c = 25.3\\%$ occurs for $\\epsilon = 12.1$ typical of silicon,\\cite{Hillebrand2003JAP,Woldering2009JAP} with $\\Delta \\omega_{pbg}$ the frequency width of the band gap, and $\\omega_c$ the band gap's center frequency. \n3D inverse woodpile crystal nanostructures have been fabricated from a number of different high-refractive index backbones~\\cite{Schilling2005APL, Santamaria2007AdvMater, Hermatschweiler2007AdvFunctMater, Jia2007JApplPhys, vandenBroek2012AFM, Grishina2015NT}. \nIn nanophotonic experiments, the potential of silicon inverse woodpiles was demonstrated by the observation of a broad 3D photonic band gap for many angles~\\cite{Huisman2011PRB}, as well as a strong spontaneous emission inhibition of embedded quantum dots~\\cite{Leistikow2011PRL}.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=1.00\\columnwidth]{y18m9d20_cavitySuperlatticeCombinedNewLatticeParameters.eps}\n\\caption{\n(a) Design of a single cavity in an inverse woodpile photonic band gap crystal shown in a cut-out of a $M_x M_y M_z = 3\\times3\\times3$ supercell that is surrounded by boxed lines. \nThe high-index backbone is shown in gray. \nTwo proximal smaller defect pores are indicated in green, and the cavity region is highlighted as the bright region at the center. \nThe tetragonal lattice parameters $a$ and $c$ are shown, as well as the $x,y,z$ coordinate system. \n(b) $(x,z)$ and $(x,y)$ cross sections through a 3D superlattice of resonant cavities, with red circles indicating cavities and dashed rectangles representing unit cells of the underlying inverse woodpile crystal structure (see (a)). \nThe lattice parameters ($c_s^{x}$, $a_s$, $c_s^{z}$) of the superlattice are shown, as well as the $x,y,z$ coordinate system. \n}\n\\label{fig:crystalOfCavities}\n\\end{figure}\n\nTo create a resonant cavity in a inverse woodpile photonic crystal, Ref.~\\cite{Woldering2014PRB} proposed a design whereby two proximal perpendicular pores have a smaller radius $(r' < r)$ than all other pores, as shown in Figure \\ref{fig:crystalOfCavities}(a). \nNear the intersection region of the two smaller pores, the light is confined in all three directions to within a mode volume as small as $V_{\\text{mode}}=\\lambda^3$ where $\\lambda$ is the free-space wavelength.\\cite{Woldering2014PRB} \nSupercell band structures revealed up to five resonances within the band gap of the perfect crystal, depending on the defect pore radius $r'$\\cite{Woldering2014PRB}, that have quadrupolar symmetry.\\cite{Devashish2018Arxiv} \nThe best confinement occurs for a defect radius $r'\/r=0.5$ that is also considered here. \n\nFigure~\\ref{fig:crystalOfCavities}(b) shows a 3D superlattice of cavities as is studied here, where each sphere indicates one cavity, as shown in Fig.~\\ref{fig:crystalOfCavities}(a).\nThe cavity superlattice has lattice parameters ($c_s^{x}$, $a_s$, $c_s^{z}$) in the ($x, y, z$) directions that are integer multiples of the underlying inverse woodpile lattice parameters: $c_s^{x} = M_x c$, $a_s = M_y a$, $c_s^{z} = M_z c$. \nHere, we study the $M_x M_y M_z = 3\\times3\\times3$ superlattice such that the cavities are repeated every three unit cells with lattice parameters $c_s^{x} = 3c$, $a_s = 3a$, $c_s^{z} = 3c$. \nThus, the cavity superlattice is also cubic, similar to the underlying inverse woodpile structure (see section~\\ref{sec:discussion} for additional discussion). \n\nWe have calculated the band structure of the 3D cavity superlattice using the plane-wave expansion method.\\cite{Ashcroft1976Book,Joannopoulos2008Book,Johnson2001OptExpress} \nUsing the Richardson extrapolation method allows us to estimate the frequencies in the limit of infinite grid resolution~\\cite{Roache1998Book,Richardson1927PhilosTransRoyalSocA}. \nDetails on the calculations and the convergence are given in Appendix \\ref{sec:Calcs_Convrg}. \nAll calculations were performed on the ``Serendipity\" cluster in the MACS group at the MESA$^{+}$ Institute.\\cite{Serendipity} \nEven on this powerful computer cluster the calculations took 210 hours. \n\n\\section{Results}\n\\label{sec:results}\n\\subsection{Band structure of coupled cavity resonances}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=1.00\\columnwidth]{y18m5d1_bandStructureAndTightBindingDispersionNewEllipse.eps}\n\\caption{\n(a) Black curves are photonic bands of a 3D inverse woodpile photonic band gap crystal ($r\/a=0.24$, $\\epsilon_{\\text{Si}}=12.1$) with two proximate defect pores ($r'\/r=0.5$). \nThe abscissa indicates the wave vector between the high-symmetry points of the Brillouin zone (see Appendix~\\ref{sec:BrillouinZoneTetragonalUnitCell}). \nThe 3D photonic band gap of the perfect crystal is shown as a red bar, and the range of allowed modes outside the band gap is shown in grey. \nThe five bands of coupled-cavity modes are highlighted. \n(b) Zoom-in of the five coupled-cavity bands (blue circles), labeled $m = 1,\\ldots, 5$. \nThe $m = 3, 4, 5$ bands are accurately described by the tight-binding model (red curves). \n} \n\\label{fig:bandStructureCrystalOfCavities}\n\\end{figure}\n\nIn Fig.~\\ref{fig:bandStructureCrystalOfCavities}(a), the band structures are shown of a 3D cavity superlattice ($M_x M_y M_z = 3\\times3\\times3$) in an inverse woodpile photonic band gap crystal made of silicon. \nAs a result of the intentional defect pores many bands appear in the band gap of the perfect crystal between reduced frequencies $\\tilde{\\omega} = \\omega a\/ (2 \\pi c) = 0.492$ and $0.634$. \nThe lowest five bands between $\\tilde{\\omega} = 0.5$ and $0.55$ are dispersionless and correspond to the five cavity resonances that are from now on labeled as $m = 1,\\ldots,5$. \nThe $m=3,4,5$ bands are isolated in frequency, unlike the situation in solid state physics where the bands arising from $d$-orbitals are hybridized~\\cite{Ashcroft1976Book}.\nThe dispersions of the bands in Fig.~\\ref{fig:bandStructureCrystalOfCavities}(a) agree well with those of Woldering \\textit{et al.}~\\cite{Woldering2014PRB}.\nThe dispersive bands in the top half of the gap (between $\\tilde{\\omega} = 0.55$ and $0.634$) have unknown character, and may include waveguiding along the defect pores. \n\nA closer inspection of the five dispersionless cavity bands in Fig.~\\ref{fig:bandStructureCrystalOfCavities}(b) reveals that these bands have nonzero bandwidths, indicating that cavity resonances in the $M_x M_y M_z = 3\\times3\\times3$ superlattice are coupled, as is investigated in this paper. \nOur results agree well with a simultaneous investigation of a single cavity in an inverse woodpile crystal with finite support, studied by other numerical methods.\\cite{Devashish2018Arxiv} \nNotably, Ref.~\\cite{Devashish2018Arxiv} also reports that the first two bands are nearly degenerate. \nBased on the occurrence of five cavity bands, on degeneracies between bands, and on the field distribution (reported in Ref.~\\cite{Woldering2014PRB}), it has been concluded that the resonances of the inverse woodpile cavity have quadrupolar symmetry and are the optical analogues of d-orbitals in solid-state physics~\\cite{Devashish2018Arxiv}. \nTherefore, it is naively expected that neighboring cavities couple in diagonal directions. \n\n\\subsection{Dispersion bandwidths}\nFor a 1D coupled-resonator optical waveguide (CROW), it is well-known that the coupling coefficient along the waveguide is proportional to the dispersion bandwidth.\\cite{Yariv1999OL,Bayindir2000PRB2,Altug2004ApplPhysLett} \nA straightforward extension of this notion to a 3D cavity superlattice is to consider the dispersion bandwidth in various crystal directions, since this is straightforward to derive from photonic band structures as in Figure~\\ref{fig:bandStructureCrystalOfCavities}. \nFor a given crystal direction characterized by wave vector $\\mathbf{k}$, the dispersion bandwidth is defined as \n\\begin{equation}\n\\Delta \\tilde{\\omega} \\equiv |\\tilde{\\omega}_{max} - \\tilde{\\omega}_{min}|_{\\Gamma\\to\\mathbf{k}_{BZ}}\n,\n\\label{eq:dispersion-bandwidth}\n\\end{equation}\nin other words, the absolute value of the difference between the maximum and minimum frequencies on a trajectory in reciprocal space between the origin $\\Gamma$ and the edge of the Brillouin zone $\\mathbf{k}_{BZ}$ in the direction of $\\mathbf{k}$. \nAs an example, for the $m = 3$ coupled-cavity band in Figure~\\ref{fig:bandStructureCrystalOfCavities}(b), between $\\Gamma$ and $Z$ the minimum and maximum frequencies are nearly the same ($\\tilde{\\omega} = 0.521$) hence the bandwidth $\\Delta \\tilde{\\omega}$ is nearly zero. \nBetween $\\Gamma$ and $U$ the minimum and maximum frequencies differ much more ($\\tilde{\\omega} = 0.520$ to $0.521$) hence the dispersion bandwidth is much greater in the diagonal direction. \n\n\\begin{figure}[h!]\n\\includegraphics[width=1.00\\columnwidth]{y18m12d7_bandwidthThirdResonanceCombined.eps}\n\\caption{\n(a) Polar plot of the dispersion bandwidth for the $m = 3$ coupled-cavity band in the $(k_X,k_Z)$ plane. \nThe $X$, $U$, and $Z$ high-symmetry points are shown. \n(b) Polar plot of the dispersion bandwidth for the $m = 3$ coupled-cavity band in the $(k_Y,k_{U})$ plane. \nThe $Y$ and $U$ high-symmetry points are shown. \nThe black circles indicate the plane-wave results (cf. Fig.~\\ref{fig:bandStructureCrystalOfCavities}), and the red lines are guides to the eye. \n}\n\\label{fig:dispersionBandwidthThird}\n\\end{figure}\n\nA polar plot of the dispersion bandwidth $\\Delta \\omega$ versus wave vector $\\mathbf{k}$ in the $(k_X,k_Z)$ plane is shown in Figure~\\ref{fig:dispersionBandwidthThird}(a). \nFrom the band frequencies mentioned above, the dispersion bandwidth is very small in the real-space $x$ and $z$-directions (corresponding to $\\Gamma X$, $\\Gamma Z$, respectively, in reciprocal space). \nIn the diagonal directions that correspond to the $\\Gamma U$ high-symmetry trajectory the dispersion bandwidth is much greater. \nAs seen from a given central cavity in real space, the wave vector $\\mathbf{k}$ is then directed towards a second nearest neighboring cavity in the diagonal $1\/\\sqrt{2}.(1,0,1)$ direction (see Appendix~\\ref{sec:BrillouinZoneTetragonalUnitCell}). \nThe polar plot of the dispersion bandwidth for $m = 3$ therefore looks like a quadrupolar radiation pattern. \nBased on the 1D CROW-reasoning given above, one tentatively infers that light is transported through the 3D cavity superlattice preferentially in the $xz$-diagonal (corresponding to $\\Gamma U$) directions. \n\nFigure~\\ref{fig:dispersionBandwidthThird}(b) shows the dispersion bandwidth $\\Delta \\omega$ in the $(k_Y,k_{U})$ plane. \nThe largest bandwidth occurs at about $45^{o}$ off the $(XUZ)$ plane, which corresponds to the $\\Gamma R$ high-symmetry direction. \nThe bandwidth in the $\\Gamma Y$-direction is small, from which one tentatively infers that there is little light transport in the $y$-direction in real space. \n\n\\begin{figure}[h!]\n\\includegraphics[width=1.00\\columnwidth]{y18m12d7_fourthFifthResonanceCombined.eps}\n\\caption{\nPolar plots of the dispersion bandwidth versus wave vector for the \n(a,b) $m = 4$ and (c,d) $m = 5$ coupled-cavity bands. \nPanels (a,c) are in the $(k_X,k_Z)$ plane and (b,d) are in the $(k_Y,k_{U})$ plane.\n}\n\\label{fig:dispersionBandwidthFourthFifth}\n\\end{figure}\n\nThe dispersion bandwidth for the $m = 4,5$ bands is shown in Figures~\\ref{fig:dispersionBandwidthFourthFifth}(a,b). \nFor the $m = 4$ band, the dispersion bandwidth is large in the diagonal directions that correspond to the $\\Gamma U$ high-symmetry directions, and it is smaller in the $x$ and $z$-directions ($\\Gamma X$,$\\Gamma Z$, respectively). \nCompared to the $m = 3$ band, the dispersion bandwidth for the $m = 4$ band appears to be less strongly directional. \nTo quantify the directionality, we consider a directionality $D$ ratio between the maximum and the minimum bandwidths $D = \\Delta \\omega_{max}\/\\Delta \\omega_{min}$ in the $(k_X,k_Z)$ plane, which yields a directionality of about $D = 4$ that is much lower than $D = 15$ for the $m = 3$ band. \n\nFor the $m = 5$ band in Figure~\\ref{fig:dispersionBandwidthFourthFifth}(c), the polar plot of the dispersion bandwidth looks very much like a quadrupolar emission pattern. \nThe bandwidth is small in the diagonal directions that correspond to the $\\Gamma U$ high-symmetry directions, about $2.5 \\times$ smaller than for the $m = 4$ band. \nThe dispersion bandwidth is much smaller in the $x$ and $z$ directions, corresponding to a large directionality $D = 26$. \n\n\\subsection{Coupling coefficients}\nTo understand the coupling between the cavities in a 3D cavity superlattice more fundamentally, we derive the coupling coefficients of light from the dispersion relations using the tight-binding method, see Appendix \\ref{sec:photonictightbindingmethod} for details. \nFigure~\\ref{fig:bandStructureCrystalOfCavities}(b) shows that the $m = 3$ band is accurately described by the tight-binding model. \nIt appears that only $7$ independent coupling coefficients $\\kappa$ are needed in the tight-binding model, namely for the real-space directions $x$, $y$, $z$, $xz-diagonal$ (corresponding to $\\Gamma U$ in reciprocal space), $xy-diagonal$ (corresponding to $\\Gamma S$), $yz-diagonal$ (corresponding to $\\Gamma T$), and $xyz-diagonal$ (corresponding to $\\Gamma R$). \nThe reasons are as follows: \nsince the inverse woodpile cavity has mirror symmetry with respect to the $(y,z)$ and $(x,y)$ planes, the coupling coefficients in the $+x$ and $+z$ directions are symmetry related to those in the $-x$ and $-z$ directions, respectively, and the coefficients in the $xz-diagonal$ directions are symmetry related to each other. \nThe coupling coefficients in the $+y$ and $-y$ directions are equal by reciprocity, see Appendix \\ref{sec:hoppingRatesIn_yDirection}.\n\n\\begin{table*}[t]\n\\caption{(Nondimensional) coupling coefficients for the superlattice bands $m=3,4,5$, defined in Eq. \\eqref{eq:hoppingRatesDefined} in Appendix \\ref{sec:photonictightbindingmethod}. \nThe calculated coupling coefficients have in addition to the real part also an imaginary part, that is at least $100\\times$ smaller than the real part, that does not have physical significance, and that is not reported here. \nIn addition to the 7 coefficients $\\kappa$ per band (see text), we also provide the $\\beta$, defined in Eq. \\eqref{eq:betaCoefficient} in Appendix \\ref{sec:photonictightbindingmethod}.}\n\\label{table:hoppingRates}\n\\begin{tabular}{ |p{3.0cm}|p{3.0cm}|p{3.0cm}|p{3.0cm}| }\n\\hline\nCoupling coefficients& $m=3$ & $m=4$ & $m=5$ \\\\\n\\hline\n$\\beta$ & $-1.8\\cdot10^{-10}$ & $+3.3\\cdot10^{-10}$ & $-1.8\\cdot10^{-10}$ \\\\\n$\\kappa_{x}$ & $+5.2\\cdot10^{-4}$ & $-1.0\\cdot10^{-3}$ & $+5.4\\cdot10^{-4}$ \\\\\n$\\kappa_{y}$ & $+4.7\\cdot10^{-5}$ & $+4.3\\cdot10^{-5}$ & $+3.0\\cdot10^{-5}$ \\\\\n$\\kappa_{z}$ & $+5.1\\cdot10^{-4}$ & $-1.1\\cdot10^{-3}$ & $+5.3\\cdot10^{-4}$ \\\\\n$\\kappa_{xz}$ ($\/\/~\\Gamma U$) & $-4.2\\cdot10^{-4}$ & $+6.6\\cdot10^{-5}$ & $-1.5\\cdot10^{-4}$ \\\\\n$\\kappa_{xy}$ ($\/\/~\\Gamma S$) & $-1.4\\cdot10^{-6}$ & $+9.7\\cdot10^{-6}$ & $-4.6\\cdot10^{-5}$ \\\\\n$\\kappa_{yz}$ ($\/\/~\\Gamma T$) & $-1.6\\cdot10^{-6}$ & $+1.0\\cdot10^{-5}$ & $-4.6\\cdot10^{-5}$ \\\\\n$\\kappa_{xyz}$ ($\/\/~\\Gamma R$) & $+7.4\\cdot10^{-6}$ & $+2.4\\cdot10^{-6}$ & $-2.2\\cdot10^{-5}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=1.00\\columnwidth]{y18m5d1_3DhoppingRatesOfLightThirdResonanceNewFont.eps}\n\\caption{\nCoupling coefficients of light from a central cavity (black circle) to neighboring cavities (other circles) for the $m = 3$ coupled-cavity band, indicated with arrows. \nNonzero coefficients only occur in the $(x,y)$ and $(x,z)$ planes.\nBlue and red indicate negative and positive coupling coefficients, respectively, as shown by the color bar.\nThe $x, y, z$ coordinate system is shown. \nThis figure has been made using ParaView~\\cite{Ayachit2015Book}.\n}\n\\label{fig:hoppingRatesOfLightThirdBand}\n\\end{figure}\n\nThe coupling coefficients are given in Table \\ref{table:hoppingRates}. \nFor the $m = 3$ band the coupling coefficients of light are overlaid on the cavity superlattice structure in Figure~\\ref{fig:hoppingRatesOfLightThirdBand}. \nIn the $x$ and $z$ directions, the coupling coefficients are relatively large and positive, in the $xz-diagonal$ directions the coupling coefficients are large and negative, in the $y$ direction the coupling coefficient is about $10 \\times$ smaller, and in all other directions the coupling coefficients are vanishingly small (typically $100\\times$ less). \n\nRemarkably, the simultaneous occurrence of large coupling coefficients with near-vanishing dispersion bandwidths means that 1D CROW-like arguments \\textit{do not hold} for 3D cavity superlattices. \nIn other words, the bandwidth in a particular crystal direction for a 3D cavity superlattice is not necessarily proportional to the coupling coefficient in the same direction in real space. \nThe small difference between the $x$ and $z$ coefficients confirms that the $x$ and $z$ directions are not symmetry related for the inverse-woodpile cavity, as opposed to the perfect inverse-woodpile structure.\\cite{Devashish2017PRB} \n\nAccording to an 1D CROW-like argument, the large coupling coefficient in the $xz-diagonal$ directions agrees with the observation of a large dispersion bandwidth in the diagonal $U$ direction, see Fig.~\\ref{fig:dispersionBandwidthThird}. \nHowever, the negative sign disagrees with the fact that at the $U$ point the band frequency is lower than at the $\\Gamma$ point, since the reverse is true for an 1D CROW, see Ref.~\\cite{Yariv1999OL}. \n\nThe $\\kappa_y$ coupling coefficient in the $y$ direction is small, in agreement with the band frequencies that are almost the same at $\\Gamma$ and $Y$. \nIt is remarkable that the $\\kappa_y$ coupling coefficient is $10 \\times$ smaller than the coefficients for the $x$ or $z$ directions, while the nearest neighbor distance is only a $(\\sqrt2)$ greater than in the $x$ or $z$ directions, which would correspond to only a $(\\text{exp}(\\sqrt2) \\approx 4 \\times)$ smaller coefficient for cavities coupled by evanescent Bloch modes. \nWe surmise that the quadrupolar field pattern of each cavity in the $xz$ plane couples poorly to a neighboring cavity in a neighboring $xz$ plane. \nTherefore, light mostly hops in 2D $(x,z)$-layers, which is analogous to 2D electron transport in graphite or graphene layers.\\cite{Castroneto2009RMP,Jacqmin2014PRL} \nSince the light propagates very unusually by hopping only in a few discrete directions, we propose the name \"Cartesian light\" for the propagation of light in a 3D cavity superlattice. \n\n\n\nThe coupling coefficients of light for the $m = 4,5$ bands are shown in Fig.~\\ref{fig:hoppingRatesOfLightFourthFifthBand}. \nFor the $m = 4$ band, the nonzero coefficients are $\\kappa_x$, and $\\kappa_z$. \nThe coupling coefficients to all other neighboring cavities vanish, including the coefficient $\\kappa_{y}$ in the $y$ direction. \nIn the hopping of the $m = 4$ band we find the ultimate Cartesian light: light hops only in the $x$-$z$ directions.\nFor the $m = 5$ band, the nonzero coefficients are $\\kappa_x$, $\\kappa_y$, $\\kappa_z$, $\\kappa_{xz}$, $\\kappa_{xy}$, $\\kappa_{yz}$, and $\\kappa_{xyz}$, that is, nonzero coupling coefficients to all neighboring cavities. \nIn the $(x,z)$ plane, there is a mix of positive $x$ and $z$ coupling coefficients, and negative $xz-diagonal$ coupling coefficients, which is similar to the $m = 3$ band. \n\n\\begin{figure}[h]\n\\includegraphics[width=1.00\\columnwidth]{y18m3d29_3DHoppingRatesFourthFifthResonanceCombined.eps}\n\\caption{\n(a) Coupling coefficients of light from a central cavity (black circle) to neighboring cavities (other circles) for the $m = 4$ coupled-cavity band, indicated with arrows. \nNonzero coefficients only occur in the $(x,y)$ and $(x,z)$ planes. \nBlue and red indicate negative\/positive coupling coefficients, for which bonding\/antibonding resonances of the two coupled cavities are energetically favorable.\nThe $x, y, z$ coordinate system is shown.\n(b) Coupling coefficients of light from a central cavity (black circle) to neighboring cavities (other circles) for the $m = 5$ coupled-cavity band.\n}\n\\label{fig:hoppingRatesOfLightFourthFifthBand}\n\\end{figure}\n\nWe have not analyzed the $m = 1,2$ bands of coupled-cavity modes, since the band structures do not converge monotonically with increasing spatial resolution, as is elaborated in Appendix~\\ref{sec:Calcs_Convrg}. \n\n\\subsection{Propagation in 3D on the superlattice}\n\\label{sec:propagation-real-space}\n\nWe now discuss the propagation in real space, and why the dispersion bandwidth of the $m = 3$ band is much larger in the $U$ direction than in the other directions in the $(k_X, k_Z)$ plane in reciprocal space. \nWe first discuss the dispersion bandwidth in the $xz-diagonal$ directions that are symmetry related to each other, and that correspond to the $\\Gamma U$ direction in reciprocal space. \n\n\\begin{figure}[tbp]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{y18m5d7_checkerboardPatternResonatingCavitiesUpdated.eps}\n \\caption{Relative phase of the resonating cavities for a coupled-cavity mode with a wave vector at the $U$ high-symmetry point. \n As seen from a given cavity, the wave vector $\\mathbf{k}$ is directed towards an $xz$ diagonally neighboring cavity.\n The black dashed line indicates a wavefront of the Bloch wave.\n The blue and red cavities resonate out-of-phase with each other.\n The couplings are indicated with arrows, which are green if the corresponding cavities resonate with an energetically favorable relative phase, as is the case for all couplings. \n }\n\\label{fig:schematicUPointFavorable}\n\\end{figure}\n\nLet us consider the relative phase of the resonating cavities for a wave vector at the $U$-point ($\\mathbf{k} = \\mathbf{k}_{BZ} = U$), as shown in Figure~\\ref{fig:schematicUPointFavorable}. \nSince this coupled resonance eigenmode is a Bloch wave of the 3D superlattice (with a phase front as indicated in Fig.~\\ref{fig:schematicUPointFavorable}) it is clear that this collective oscillation differs fundamentally from waveguiding behavior in a 1D CROW; in other words, the superlattice Bloch modes differ from the ones in a CROW. \n\nIn Figure~\\ref{fig:schematicUPointFavorable}, neighboring cavities in the $xz-diagonal$ direction resonate in-phase with each other. \nNeighboring cavities in the $x$, or $z$-direction resonate out-of-phase with each other.\nHence, there is a checkerboard pattern of two sublattices of cavities that resonate out-of-phase with each other. \nThe in-phase resonance of neighboring cavities in the $xz$-diagonal direction is energetically favorable for the negative coupling coefficient in the $xz$-diagonal directions, as can be understood from Eq. \\eqref{eq:hoppingRatesDefined} in Appendix \\ref{sec:photonictightbindingmethod}. \nThe out-of-phase resonance of neighboring cavities in the $x$, or $z$-direction is energetically favorable for the positive coupling coefficient in the $x$, or $z$-direction.\nHence, the checkerboard pattern of out-of-phase resonating cavities is energetically favorable for all coupling coefficients in the ($x$, $z$)-plane, and the band frequency is relatively low at the $U$ high-symmetry point. \n\n\\begin{figure}[tbp]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{y18m11d6_checkerboardPatternResonatingCavitiesUpdated.eps}\n\\caption{ \nRelative phase of the resonating cavities for a coupled-cavity mode with a wave vector at the $X$ high-symmetry point. \nAs seen from a given cavity, the wave vector $\\mathbf{k}$ is directed towards a nearest-neighboring cavity. \nThe black dashed line indicates a wavefront of the Bloch wave.\nThe blue and red cavities resonate out-of-phase with each other.\nThe couplings are indicated with arrows, green arrows indicate an energetically favorable relative phase, and red arrows indicate an unfavorable phase. \n}\n\\label{fig:schematicXPointFavorable}\n\\end{figure}\n\nWe now consider the superlattice Bloch mode at the $\\mathbf{k} = \\Gamma$ point (the origin in reciprocal space). \nAll cavities resonate in-phase with each other, which is energetically favorable for the negative coupling coefficient in the $xz$-diagonal direction. \nHowever, it is not energetically favorable for the positive coupling coefficients in the $x$, or $z$-direction.\nHence, the band frequency is relatively high at the $\\Gamma$ point, and higher than at the $U$-point, in agreement with the observation of large dispersion bandwidth in Figure~\\ref{fig:bandStructureCrystalOfCavities}(b). \n\n\nWe now discuss the dispersion bandwidth in any direction in the $(k_X, k_Z)$ plane other than the $xz$-diagonal direction. \nFor example, the relative phase of the resonating cavities at the $X$ high-symmetry point is shown in Fig.~\\ref{fig:schematicXPointFavorable}.\nOnly the checkerboard pattern of out-of-phase resonating cavities at the $U$ high-symmetry point is energetically favorable for all coupling coefficients in the ($x$, $z$) plane.\nFor any other point in the Brillouin zone, the relative phase of the resonating cavities is not energetically favorable for all coupling coefficients in the ($x$, $z$) plane. \nHence, the dispersion bandwidth is small for directions other than the $xz$-diagonal direction, in agreement with the superlattice band structures in Figure~\\ref{fig:bandStructureCrystalOfCavities}(b). \n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection{Differences with other forms of light transport} \n\nCartesian light propagation on a 3D superlattice of cavities differs fundamentally from other known modes of propagation in periodic nanophotonic systems, notably from the conventional 3D spatially-extended Bloch wave propagation in crystals, from light tunneling through a band gap, from coupled-resonator optical waveguiding, and also from light diffusing at the edge of a gap. \n\n(1) A characteristic feature of the 3D superlattice Bloch modes is that they are constructed from modes where the light field is hopping from lattice site to lattice site. \nIn other words, the field pattern has its maxima on the lattice sites (i.e. the cavities) and decays exponentially in between lattice sites, since isolated cavities are tuned inside the photonic band gap where the wave vector is complex. \nIn contrast, the Bloch modes outside the photonic band gap are constructed from purely real modes of propagation; there is no reason for field maxima to be located on preferred positions in the crystal. \n\n(2) A second characteristic feature of the 3D superlattice Bloch modes is that they are genuine modes of propagation centred within the 3D photonic band gap. \nIn this sense, they are distinguished from the modes in photonic crystals with finite support that were recently described in Ref.~\\cite{Hasan2018PRL}. \nIn that study, it was found that the finite extent of a photonic band gap crystal leads to the filling of the density of states (DOS) in the band gap by states that are centred outside the band gap, while extending into the band gap due to their substantial band width. \n\n(3) The 3D Cartesian superlattice propagation differs fundamentally from the propagation in lower-dimensional 1D (a CROW) and 2D arrays of cavities. \nFirstly, in section~\\ref{sec:propagation-real-space} we have already discussed that the superlattice modes differ fundamentally from those of a CROW. \nIn other words, a 3D superlattice does not seem to be a \"3D CROW\". \nSecondly, if we perturb the frequency of one of the cavities in a superlattice, a bound state appears instantaneously in 1D and 2D, whereas a threshold frequency difference is required in 3D, see Ref.~\\cite{Economou2006Book}. \n\n(4) The propagation of light in a 3D cavity superlattice in a photonic band gap differs fundamentally from directional diffusion that was identified for 3D photonic band gap crystals with a certain degree of disorder~\\cite{Koenderink2003PhysRevLett}. \nIn the latter case, the modes of propagation are not waves but diffusive. \nMoreover, the typical frequencies are at the edge of the band gap, hence outside the gap, as opposed to the cavity superlattice modes that reside within the band gap, see Fig.~\\ref{fig:bandStructureCrystalOfCavities}. \n\n\n\\subsection{Crystal structures of the cavity superlattice } \nIf the magnification factors of the superlattice's lattice parameters, compared to the underlying crystal structure's lattice parameters, fulfill $M_i \\neq (M_j, M_k)$ ($i,j,k = x,y,z$), the cavity superlattice is not cubic anymore - in contrast to the underlying inverse woodpile structure - but has become tetragonal. \nIn the most general case with $M_x \\neq M_y \\neq M_z$ the superlattice has different cavity spacings in each direction $(x,y,z)$; the superlattice has then become orthorhombic. \nGiven that cavities in an inverse woodpile structure are necessarily located along the smaller-pore line defects, we currently doubt whether it is feasible to realize other 3D Bravais superlattices. \n\nWe have seen that for the $m = 3, 4, 5$ bands, the coupling coefficient in the $y$-direction is smaller than in the $(x,z)$ plane, by typically $10 \\times$. \nTo make the hopping of light more 3D, it is necessary to increase the coupling coefficient in the $y$-direction compared to the coefficients in the $(x,z)$-directions, which can be achieved by a closer cavity spacing in the $y$-direction, for example, in a $M_x M_y M_z = 3\\times2\\times3$ supercell in case of $M_x = M_z = 3$ (as studied here) or in general for supercells with $M_y < (M_x, M_z)$. \n\nConversely, if it is desired to realize a superlattice with effectively 2D transport of light in $(x, z)$ planes, the superlattice parameter in the $y$-direction should be made greater than the ones in the $(x, z)$-directions ($M_y >> (M_x, M_z)$). \nIn this situation, the 2D transport of light may hold analogies to that of charge carriers in graphite layers\\cite{Castroneto2009RMP} or in high-T$_c$ superconductors. \n\n\\subsection{Disorder} \nWe have studied the dispersion and hopping for superlattices without disorder. \nLet us briefly comment on the sensitivity of the results to a small degree of disorder, since we performed calculations for several grid resolutions in Appendix~\\ref{sec:Calcs_Convrg}, and a change of the grid resolution implies a slight shift in the geometry. \nOn the one hand, we observed that for all five bands of coupled-cavity modes, the center frequency is highly sensitive to the grid resolution and therefore to disorder. \nThis is likely the result of the lightning rod effect of the cavity mode field pattern identified in Ref.~\\cite{Woldering2014PRB}, wherein the inverse woodpile cavity resonances have regions of high intensity at sharp corners in the dielectric material. \nIf such sharp corners are slightly distorted, it is quite conceivable that the overlap with the field pattern changes, leading to a change in resonance frequency. \nOn the other hand, we observe for bands $m = 3, 4, 5$ that the features of the dispersion bands remain the same while the grid resolution is increased. \nTherefore, we expect the coupling coefficients for the $m = 3, 4, 5$ bands to be robust to small degrees of disorder. \n\nA 3D cavity superlattice is the photonic analogue of the Anderson model for spins and electrons~\\cite{Anderson1958PR}, albeit in the limit of zero disorder. \nThe 3D cavity superlattice also corresponds to the Hubbard model without interactions~\\cite{Hubbard1963ProcRSocA,Hubbard1964ProcRSocA}.\nWe anticipate that the present study may form the basis for further exploration of the physics of the 3D Anderson model for nanophotonic cavity superlattices that will proceed by introducing controlled degrees of disorder in the cavity resonance frequencies. \n\n\\subsection{Outlook}\n\nA possible application would be a scalable, coherently linked network of NV-based registers, see Ref.~\\cite{Childress2013MRSBull}.\n\n\n\n\n\n\n\n\\section{Summary}\nWe have studied for the first time ever the propagation of light in a 3D cavity superlattice within a 3D photonic band gap.\nSuch a 3D cavity superlattice is the photonic analogue of the Anderson model in the limit of zero disorder. \nThe light hops only in a few high-symmetry directions including the Cartesian $(x,y,z)$ directions, therefore we propose the name \"\\textit{Cartesian light}\".\n3D Cartesian hopping of light in a 3D band gap yields propagation as superlattice Bloch modes that differ fundamentally from the conventional 3D spatially-extended Bloch wave propagation in crystals, from light tunneling through a band gap, from coupled-resonator optical waveguiding, and also from light diffusing at the edge of a gap. \nThe large coupling coefficients in the Cartesian directions occur simultaneously with a near vanishing dispersion bandwidth in this direction. \nThis means that 1D CROW-like \\textit{do not} arguments hold for 3D cavity superlattices.\nThe unusually small dispersion bandwidth in the Cartesian directions is a result of interplay between positive and negative coupling coefficients in the Cartesian and diagonal directions.\n\n\\section{Acknowledgments}\nWe thank Bill Barnes (Exeter, Twente) for coining the term \"Cartesian light\" and for discussions, and we thank Geert Brocks, Ad Lagendijk, Jan Kl{\\\"a}rs and Bart van Tiggelen for useful discussions. \nThis research is supported by the 4TU federation, by the FOM\/NWO programme ``Stirring of light!,\" the STW\/NWO-Perspectief program ``Free-form scattering optics\", the ``Descartes-Huygens\" prize of the French Academy of Sciences, and the MESA$^{+}$ Institute for Nanotechnology section Applied Nanophotonics (ANP).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Intro} Type \\textrm{II} Magnetoelectric multiferroics are\nmaterials which exhibit coexistence between certain types of\nlong-range magnetic order and a ferroelectric order. These\nmaterials are usually characterized by a strong magnetoelectric\ncoupling between their electric and magnetic degrees of freedom.\nThe magnetoelectric effect enables the control of the electric\npolarization by a magnetic field, or the control of the\nmagnetization by an electric field. The study of magnetoelectric\nmultiferroics is thus of great interest in condensed matter\nphysics, both from basic research and technological applications\npoints of view.\\cite{KD09,CSW07,FM05,KT07} In recent years, the\ninterest in this field has grown after the discovery of new\nmaterials with a large magnetoelectric effect, such as\nTbMnO$^{}_{3}$,\\cite{KT03} TbMn$^{}_{2}$O$^{}_{5}$,\\cite{HN04}\nNi$^{}_{3}$V$^{}_{2}$O$^{}_{8}$,\\cite{LG05}\nCuFeO$^{}_{2}$,\\cite{KT06} and CoCr$^{}_{2}$O$^{}_{4}$.\\cite{YY06}\nIn those oxides, ferroelectricity appears in conjunction with a\nnoncollinear spiral magnetic phase, which breaks spatial inversion\nsymmetry, and therefore allows the appearance of an electric\npolarization.\n\nThere are two different approaches to the theoretical treatment of\nsuch noncollinear magnetoelectric multiferroics. One approach is\nbased on first principles calculations using the density\nfunctional theory (DFT).\\cite{PS09} The second approach constructs\na model Hamiltonian dictated by symmetry\nconsiderations.\\cite{LG05,MM06,HAB07} Different mechanisms for the\nmagnetoelectric coupling can then be\nsuggested.\\cite{HAB06,KD09,KH05,SI06} In this paper we develop a\nsemi-phenomenological model for describing the magnetic phase\ntransitions of Mn$^{}_{1-x}$M$^{}_{x}$WO$^{}_{4}$ (M=Fe, Zn, Mg)\nand the induced ferroelectric polarization. The model is\nsemi-phenomenological in the sense that some of the parameters can\nbe deduced from existing experimental data, while the others are\npurely phenomenological. The multiferroic MnWO$^{}_{4}$ is a\nnatural choice for such an approach, due to the vast experimental\ndata that exists in the literature.\n\nMnWO$^{}_{4}$ crystallizes in the wolframite structure, which\nbelongs to the monoclinic space group P2\/c with\n$\\beta\\approx91\\degree$. The unit cell includes two magnetic\nMn$^{2+}$ ions with spin $S=5\/2$ and orbital angular momentum\n$L=0$ at positions $\\tauv^{}_1=(0.5,y,0.25)$ and\n$\\tauv^{}_2=(0.5,1-y,0.75)$ (in units of the primitive lattice\nvectors) with $y=0.685$.\\cite{LG93} In zero magnetic field,\nMnWO$^{}_{4}$ undergoes three successive phase transitions at\ntemperatures $T^{}_{N3}\\approx13.5K$, $T^{}_{N2}\\approx12.3-12.7K$\nand $T^{}_{N1}\\approx7-8K$ to phases which are called AF3, AF2,\nand AF1, respectively.\\cite{LG93,AAH06,TK06} According to neutron\ndiffraction experiments,\\cite{LG93} AF3 is an incommensurate (IC)\nantiferromagnetic phase with a collinear sinusoidal structure, AF2\nis an incommensurate antiferromagnetic phase with an\nelliptical-spiral structure, and AF1 is a commensurate (C)\nantiferromagnetic phase with a collinear\n$\\uparrow\\uparrow\\downarrow\\downarrow$ structure. The propagation\nvectors are $\\qv^{}_{IC}=(-0.214,0.5,0.457)$ (in units of the\nprimitive reciprocal lattice vectors) for AF2 and AF3, and\n$\\qv^{}_{C1,2}=(\\pm0.25,0.5,0.5)$ for AF1. In AF3 and AF1, the\nmagnetic moments of the Mn$^{2+}$ ions align along the easy axis\nof magnetization, which lies in the $ac$-plane and forms an angle\nof $\\approx35\\degree-37\\degree$ with the $a$ axis. Different\nstudies\\cite{AAH06,TK06} reveal that a ferroelectric polarization,\nwhich is oriented along the $b$ axis, develops in the AF2 phase.\n\nAs opposed to MnWO$^{}_{4}$, other isomorphic wolframite\nstructures like FeWO$^{}_{4}$, CoWO$^{}_{4}$ and NiWO$^{}_{4}$\nshow only a single magnetic phase transition to a simple\ncommensurate antiferromagnetic phase with the propagation vector\n$\\qv^{}_{}=(0.5,0,0)$.\\cite{WH77} Those observations suggest that\nunlike the isomorphic structures, MnWO$^{}_{4}$ constitutes a\nhighly frustrated system with complex competing interactions. The\ncompetition between the different interactions manifests itself in\nthe sensitivity of the phase diagram to doping with different\ntransition metal ions at the Mn sites. It turns out that a small\nFe concentration suppresses the ferroelectric phase AF2 and\nexpands the stabilization range of AF3 and\nAF1.\\cite{CRP09,YF08,CRP08} In contrast to Fe doping, it has been\nreported\\cite{SYS09} that a small Co concentration stabilizes the\nferroelectric phase at the expense of the AF1 phase. A\nquantitative and microscopic understanding of the effect of Fe and\nCo doping on the multiferroic properties and the phase diagram of\nMnWO$^{}_{4}$ is quite complicated, since the exchange couplings\nof the M-M and M-Mn (M=Fe, Co) interactions, as well as the\nanisotropy parameters are not known. In order to overcome some of\nthese problems, a much simpler magnetic system has been achieved\nby the partial substitution of Mn ions by the non-magnetic ions\nZn$^{2+}$ and Mg$^{2+}$.\\cite{CRP11,ML09} Those studies reveal\nthat the AF1 phase is strongly suppressed as a result of magnetic\nions dilution by non-magnetic substituents.\n\nThe frustrated nature of MnWO$^{}_{4}$ was demonstrated by\nEhrenberg \\textit{et al.}.\\cite{EH99} Using inelastic neutron\nscattering they extracted 9 exchange couplings $J^{}_1-J^{}_9$ for\nthe superexchange interactions among the Mn ions. Later, Tian\n\\textit{et al.}\\cite{TC09} proposed different values for the 9\nexchange couplings based on DFT calculations. Those values depend\non an unknown on-site repulsion energy. Moreover, the authors have\nnoted that generally DFT calculations tend to overestimate the\nmagnitude of exchange interactions.\\cite{TC09} Recently, the\nexperimental data have been expanded.\\cite{YF11} In that study, Ye\n\\textit{et al.} suggested some corrections for the values of the\nexchange couplings, and included two additional ones, $J^{}_{10}$\nand $J^{}_{11}$. The two sets of experimental exchange couplings\nare summarized in Table ~\\ref{tab:Exchange couplings}. The model\nwe describe may help to compare these different sets of exchange\ncouplings, by examining their consistency with different\nexperimental observations.\n\\begin{table*} \\tiny\n\\centering \\caption{\\label{tab:Exchange couplings} Superexchange\ncouplings for the Mn$^{2+}$ ion at $\\tauv^{}_1=(0.5,y,0.25)$\naccording to different inelastic neutron scattering studies. We\ndenote $z=1-y$, $w=2-y$ and $u=1+y$. The values are presented in\nunits of $k^{}_{B}K$.\\cite{comment1}}\n \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c}\n \\hline\n \\hline\n & $J^{}_1$ & $J^{}_2$ & $J^{}_3$ & $J^{}_4$ & $J^{}_5$ & $J^{}_6$ & $J^{}_7$ & $J^{}_8$ & $J^{}_9$ & $J^{}_{10}$ & $J^{}_{11}$ & D \\\\ \\hline\n Neighbors & $\\left(\\frac{1}{2},z,\\frac{3}{4}\\right)$ & $\\left(\\frac{1}{2},w,\\frac{3}{4}\\right)$ & $\\left(\\frac{3}{2},y,\\frac{1}{4}\\right)$ & $\\left(\\frac{1}{2},y,\\frac{5}{4}\\right)$ & $\\left(\\frac{1}{2},u,\\frac{1}{4}\\right)$ & $\\left(\\frac{3}{2},z,\\frac{3}{4}\\right)$ & $\\left(-\\frac{1}{2},z,\\frac{3}{4}\\right)$ & $\\left(\\frac{3}{2},w,\\frac{3}{4}\\right)$ & $\\left(-\\frac{1}{2},w,\\frac{3}{4}\\right)$ & $\\left(\\frac{3}{2},y,\\frac{5}{4}\\right)$ & $\\left(\\frac{3}{2},y,-\\frac{3}{4}\\right)$ & \\\\\n & $\\left(\\frac{1}{2},z,-\\frac{1}{4}\\right)$ & $\\left(\\frac{1}{2},w,-\\frac{1}{4}\\right)$ & $\\left(-\\frac{1}{2},y,\\frac{1}{4}\\right)$ & $\\left(\\frac{1}{2},y,-\\frac{3}{4}\\right)$ & $\\left(\\frac{1}{2},-z,\\frac{1}{4}\\right)$ & $\\left(-\\frac{1}{2},z,-\\frac{1}{4}\\right)$ & $\\left(\\frac{3}{2},z,-\\frac{1}{4}\\right)$ & $\\left(\\frac{3}{2},w,-\\frac{1}{4}\\right)$ & $\\left(\\frac{3}{2},w,-\\frac{1}{4}\\right)$ & $\\left(-\\frac{1}{2},y,-\\frac{3}{4}\\right)$ & $\\left(-\\frac{1}{2},y,\\frac{5}{4}\\right)$ & \\\\\n Ref. \\onlinecite{EH99} & -0.195 & -0.135 & -0.423 &0.414 & 0.021 & -0.509 & 0.023 & 0.491 & -1.273 & - & - & 0.568 \\\\\n Ref. \\onlinecite{YF11} & -1.95(1) & -0.18(1) & -1.48(1) & -1.21(1) & 0.23(1) & -1.99(1) & -0.56(1) & 0.09(1) & -1.21(1) & -0.7(1) & 0.09(1) & 0.84(1) \\\\ \\hline \\hline\n \\end{tabular}\n\\end{table*}\n\nThe outline of the paper is as follows: in Sec. \\ref{Sec 1} we\ndefine the model. In Sec. \\ref{Sec 2} the results of the model are\nderived. In Sec. \\ref{Sec 3} the model parameters are fitted by\ncomparing its results with different experimental observations.\nHere we compare the two sets of experimental exchange couplings\nwith the fitted parameters. In Sec. \\ref{Sec 4} the Ginzburg\ncriterion is applied to the specific case of the multiferroic\nMnWO$^{}_{4}$, in order to examine whether the mean-field theory\napproach is valid. We conclude in Sec. \\ref{Sec 5} with a brief\nsummary.\n\\section{The model}\n\\label{Sec 1} In this section we develop the semi-phenomenological\nmodel. The spin Hamiltonian consists of a Heisenberg term with a\nsingle-ion anisotropy, which favors an easy axis in the\n$ac$-plane. According to experiments, the spin component along the\nhard axis in the $ac$-plane does not order in any of the phases.\nFurthermore, the transitions are almost not influenced by an\nexternal magnetic field along the hard axis. Hence we omit the\nhard axis component from the calculations and write the spin as\n$\\Sv(\\Rv+\\tauv)=S^{}_{x}(\\Rv+\\tauv)\\hat{\\bf\nx}+S^{}_{b}(\\Rv+\\tauv)\\hat{\\bf b}$, where $x$ denotes the easy\naxis in the $ac$-plane and $b$ denotes the axis perpendicular to\nthe $ac$-plane. Here $\\Sv(\\Rv+\\tauv)$ is the thermal average of\nthe dimensionless classical spin at position $\\Rv+\\tauv$, where\n$\\Rv$ is a lattice vector and $\\tauv$ is one of the two basis\nvectors $\\tauv^{}_{1}$, $\\tauv^{}_{2}$ in the unit cell,\nindicating the locations of the Mn$^{2+}$ ions. We study the\nfollowing Hamiltonian:\n\\begin{widetext}\n\\begin{align}\n\\label{eq:Hamiltonian}\n&H^{}_{mag}=-\\frac{1}{2}\\sum_{\\Rv,\\Rv'}\\sum_{\\tauv,\\tauv'=\\tauv^{}_{1},\\tauv^{}_{2}}J(\\Rv+\\tauv,\\Rv'+\\tauv')\n\\Sv(\\Rv+\\tauv)\\cdot\\Sv(\\Rv'+\\tauv')-\\frac{1}{2}D\\sum_{\\Rv}\\sum_{\\tauv=\\tauv^{}_{1},\\tauv^{}_{2}}{S^{}_{x}}^2(\\Rv+\\tauv).\n\\end{align}\n\\end{widetext}\nHere $J(\\Rv+\\tauv,\\Rv'+\\tauv')$ is the superexchange interaction\nenergy which couples the spins at $\\Rv+\\tauv$ and $\\Rv'+\\tauv'$,\nand $D$ is a positive single-ion anisotropy energy. To find an\nexpression for the magnetic free energy of the system, we expand\nthe entropy in the spin components up to the fourth order\n\\begin{align}\n\\label{eq:Entropy}\nT\\texttt{S}&=-\\frac{1}{2}aT\\sum_{\\Rv}\\sum_{\\tauv=\\tauv^{}_{1},\\tauv^{}_{2}}{\\Sv}^2(\\Rv+\\tauv)\\nonumber\\\\\n&-b\\sum_{\\Rv}\\sum_{\\tauv=\\tauv^{}_{1},\\tauv^{}_{2}}{\\Sv}^4(\\Rv+\\tauv),\n\\end{align}\nwhere $a$ and $b$ are positive parameters, and $T$ is the\ntemperature. Equation \\eqref{eq:Entropy} gives the entropy\nrelative to the high temperature paramagnetic phase (denoted by P)\nand thus the expression is negative. Combining Eqs.\n\\eqref{eq:Hamiltonian} and \\eqref{eq:Entropy} we obtain the\nmagnetic free energy\n\\begin{widetext}\n\\begin{align}\n&F^{}_{mag}=\\frac{1}{2}\\sum_{\\Rv,\\Rv'}\\sum_{\\tauv,\\tauv'=\\tauv^{}_{1},\\tauv^{}_{2}}\\sum_{\\alpha,\\beta=1}^2\\chi^{-1}_{\\alpha\\beta}\n(\\Rv+\\tauv,\\Rv'+\\tauv')S^{}_{\\alpha}(\\Rv+\\tauv)\nS^{}_{\\beta}(\\Rv'+\\tauv')+b\\sum_{\\Rv}\\sum_{\\tauv=\\tauv^{}_{1},\\tauv^{}_{2}}{\\Sv}^4(\\Rv+\\tauv),\n\\end{align}\n\\end{widetext}\nwhere the $4\\times4$ inverse susceptibility matrix is block\ndiagonal\n\\begin{align}\n\\chi^{-1}_{\\alpha\\beta}(\\Rv+\\tauv,\\Rv'+\\tauv')&=\\big[(aT-D^{}_{\\alpha})\\delta^{}_{\\Rv,\\Rv'}\\delta^{}_{\\tau,\\tau'}\\nonumber\\\\\n&-J(\\Rv+\\tauv,\\Rv'+\\tauv')\\big]\\delta^{}_{\\alpha,\\beta},\n\\end{align}\nwith $D^{}_{1}=D^{}_{x}=D$ and $D^{}_{2}=D^{}_{b}=0$. Below, we\nexploit the Fourier transforms of the spin components,\n\\label{eq:Fourier transform}\n\\begin{align}\n&S^{}_{\\alpha}(\\qv,\\tauv)=\\frac{1}{N}\\sum_{\\Rv}S^{}_{\\alpha}(\\Rv+\\tauv)e^{i\\qv\\cdot(\\Rv+\\tauv)},\\nonumber\\\\\n&S^{}_{\\alpha}(\\Rv+\\tauv)=\\sum_{\\qv}S^{}_{\\alpha}(\\qv,\\tauv)e^{-i\\qv\\cdot(\\Rv+\\tauv)}.\n\\end{align}\nHere $\\qv$ is in the first Brillouin zone and $N$ is the number of\nunit cells. In terms of the Fourier transform, the magnetic free\nenergy per unit cell, $f^{}_{mag}\\equiv{F^{}_{mag}\/N}$, is:\n\\begin{widetext}\n\\begin{align}\n\\label{eq:magnetic_free_energy}\nf^{}_{mag}&=\\frac{1}{2}\\sum_{\\tauv,\\tauv'=\\tauv^{}_{1},\\tauv^{}_{2}}\\sum_{\\alpha,\\beta=1}^2\\sum_{\\qv}\\chi^{-1}_{\\alpha\\beta}(\\qv;\\tauv,\\tauv')\n{S^{}_{\\alpha}}^\\ast(\\qv,\\tauv)S^{}_{\\beta}(\\qv,\\tauv')\\nonumber\\\\\n&+b\\sum_{\\Gv}\\sum_{\\tauv=\\tauv^{}_{1},\\tauv^{}_{2}}\\sum_{\\qv^{}_1,\\qv^{}_2,\\qv^{}_3,\\qv^{}_4}e^{-i\\Gv\\cdot\\tauv}\\big[S^{}_{x}(\\qv^{}_1,\\tauv)S^{}_{x}(\\qv^{}_2,\\tauv)S^{}_{x}(\\qv^{}_3,\\tauv)S^{}_{x}(\\qv^{}_4,\\tauv)\\nonumber\\\\\n&+S^{}_{b}(\\qv^{}_1,\\tauv)S^{}_{b}(\\qv^{}_2,\\tauv)S^{}_{b}(\\qv^{}_3,\\tauv)S^{}_{b}(\\qv^{}_4,\\tauv)\n+2S^{}_{x}(\\qv^{}_1,\\tauv)S^{}_{x}(\\qv^{}_2,\\tauv)S^{}_{b}(\\qv^{}_3,\\tauv)S^{}_{b}(\\qv^{}_4,\\tauv)\\big]\\delta(\\qv^{}_1+\\qv^{}_2+\\qv^{}_3+\\qv^{}_4-\\Gv),\n\\end{align}\n\\end{widetext}\nwhere $\\Gv$ is a reciprocal lattice vector and the Fourier\ntransform of the inverse susceptibility matrix is given by the\nblock diagonal hermitian matrix\n\\begin{align}\n\\label{eq:chi_matrix}\n\\chi^{-1}_{\\alpha\\beta}(\\qv;\\tauv,\\tauv')&=\\big[(aT-D^{}_{\\alpha})\\delta^{}_{\\tauv,\\tauv'}\\nonumber\\\\\n&-J(\\qv;\\tauv,\\tauv')\\big]\\delta^{}_{\\alpha,\\beta},\n\\end{align}\nwith $J(\\qv;\\tauv,\\tauv')$ being the Fourier transform of the\n$2\\times2$ matrix $J(\\Rv+\\tauv,\\Rv'+\\tauv')$\n\\begin{align}\n\\label{eq:J_matrix} J(\\qv;\\tauv,\\tauv')=\n\\sum_{\\Rv}J(\\tauv,\\Rv+\\tauv')e^{-i\\qv\\cdot(\\Rv+\\tauv'-\\tauv)}.\n\\end{align}\nIn the last expression the sum is over all lattice vectors $\\Rv$.\nThe four eigenvalues of the matrix \\eqref{eq:chi_matrix} are\n\\begin{align}\n\\label{eq:eigenvalues}\n&\\zeta^{}_{\\pm,x}(\\qv,T)=aT-D-\\lambda^{}_{\\pm}(\\qv), \\nonumber\\\\\n&\\zeta^{}_{\\pm,b}(\\qv,T)=aT-\\lambda^{}_{\\pm}(\\qv),\n\\end{align}\nand the corresponding eigenvectors are\n\\begin{align}\n\\label{subeq:eigenvectors}\n\\Sv^{}_{\\pm,x}(\\qv)=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1\\\\ \\pm{e}^{-i\\phi(\\qv)}\\\\ 0\\\\ 0 \\end{pmatrix}, \\nonumber\\\\\n\\Sv^{}_{\\pm,b}(\\qv)=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0\\\\ 0\\\\\n1\\\\ \\pm{e}^{-i\\phi(\\qv)} \\end{pmatrix}.\n\\end{align}\nHere, $\\lambda^{}_{\\pm}$ are the two eigenvalues of the matrix\n\\eqref{eq:J_matrix} and $\\phi(\\qv)$ is the phase of $J(\\qv;1,2)$.\nAssuming 11 exchange couplings as in Ref. \\onlinecite{YF11}, these\ntwo eigenvalues are given by\n\\begin{align}\n\\label{eq:lambda} \\lambda^{}_{\\pm}(\\qv)&=\\pm\n2\\sqrt{{\\Lambda^{}_{2}}^2(\\qv)+{\\Lambda^{}_{3}}^2(\\qv)+2\\cos(2\\pi\nq_{b})\\Lambda^{}_{2}(\\qv)\\Lambda^{}_{3}(\\qv)}\\nonumber\\\\\n&+2\\Lambda^{}_{1}(\\qv),\n\\end{align}\nwith the following definitions:\n\\begin{align}\n\\Lambda^{}_{1}(\\qv)&=J^{}_{3}\\cos(2\\pi q^{}_{a})+J^{}_{4}\\cos(2\\pi q^{}_{c})\\nonumber\\\\\n&+J^{}_{5}\\cos(2\\pi q^{}_{b})+J^{}_{10}\\cos\\big[2\\pi(q^{}_{a}+q^{}_{c})\\big]\\nonumber\\\\\n&+J^{}_{11}\\cos\\big[2\\pi(q^{}_{a}-q^{}_{c})\\big], \\nonumber\\\\\n\\Lambda^{}_{2}(\\qv)&=J^{}_{1}\\cos(\\pi\nq^{}_{c})+J^{}_{6}\\cos\\big[2\\pi(q^{}_{a}+\\frac{q^{}_{c}}{2})\\big]\\nonumber\\\\\n&+J^{}_{7}\\cos\\big[2\\pi(q^{}_{a}-\\frac{q^{}_{c}}{2})\\big], \\nonumber\\\\\n\\Lambda^{}_{3}(\\qv)&=J^{}_{2}\\cos(\\pi\nq^{}_{c})+J^{}_{8}\\cos\\big[2\\pi(q^{}_{a}+\\frac{q^{}_{c}}{2})\\big]\\nonumber\\\\\n&+J^{}_{9}\\cos\\big[2\\pi(q^{}_{a}-\\frac{q^{}_{c}}{2})\\big].\n\\end{align}\nNow let us transform to magnetic normal coordinates\n\\begin{align}\n\\begin{pmatrix} S^{}_x(\\qv,1) \\\\ S^{}_x(\\qv,2) \\\\\nS^{}_b(\\qv,1) \\\\ S^{}_b(\\qv,2)\n\\end{pmatrix}&=\\sigma^{}_{+,x}(\\qv)\\Sv^{}_{+,x}(\\qv)+\\sigma^{}_{-,x}(\\qv)\\Sv^{}_{-,x}(\\qv)\\nonumber\\\\\n&+\\sigma^{}_{+,b}(\\qv)\\Sv^{}_{+,b}(\\qv)+\\sigma^{}_{-,b}(\\qv)\\Sv^{}_{-,b}(\\qv).\n\\end{align}\nHere, $\\sigma^{}_{+,x}(\\qv)$, $\\sigma^{}_{-,x}(\\qv)$,\n$\\sigma^{}_{+,b}(\\qv)$ and $\\sigma^{}_{-,b}(\\qv)$ are the magnetic\norder parameters for a magnetic structure with wave vector $\\qv$.\nThe diagonal form of the magnetic free energy\n\\eqref{eq:magnetic_free_energy} is therefore\n\\begin{align}\nf^{}_{mag}&=\\frac{1}{2}\\sum_{\\qv}\\big[\\zeta^{}_{+,x}(\\qv,T)\\left|\\sigma^{}_{+,x}(\\qv)\\right|^2+\n\\zeta^{}_{-,x}(\\qv,T)\\left|\\sigma^{}_{-,x}(\\qv)\\right|^2 \\nonumber\\\\\n&+\\zeta^{}_{+,b}(\\qv,T)\\left|\\sigma^{}_{+,b}(\\qv)\\right|^2+\n\\zeta^{}_{-,b}(\\qv,T)\\left|\\sigma^{}_{-,b}(\\qv)\\right|^2\\big]\\nonumber\\\\\n&+O({\\sigma}^{4}).\n\\end{align}\nAt high enough temperatures, the eigenvalues\n\\eqref{eq:eigenvalues} are all positive and therefore the stable\nphase is the paramagnetic one. As we lower the temperature, we\nreach a critical temperature for which one of the eigenvalues\nvanishes. We denote the wave vector for which one of the\neigenvalues vanishes first as $\\qv^{}_{IC}$. Since\n$\\lambda^{}_{+}(\\qv)>\\lambda^{}_{-}(\\qv)$ and $D>0$, the first\neigenvalue which reaches zero is $\\zeta^{}_{+,x}$. At the\ntemperature $T^{(0)}_{N3}$ at which $\\zeta^{}_{+,x}=0$ there is a\nphase transition from the paramagnetic phase to the AF3 phase, in\nwhich $\\sigma^{}_{+,x}(\\qv^{}_{IC})\\neq0$ but all other order\nparameters remain zero. At the second transition\nAF3$\\rightarrow$AF2, the order parameter\n$\\sigma^{}_{+,b}(\\qv^{}_{IC})$ orders as well. This is true\nprovided that\n\\begin{equation}\n\\label{eq:condition}\n\\lambda^{}_{+}(\\qv^{}_{IC})-\\lambda^{}_{-}(\\qv^{}_{IC})>D.\n\\end{equation}\nThe last condition ensures that $\\zeta^{}_{+,b}(\\qv^{}_{IC},T)$\nvanishes before $\\zeta^{}_{-,x}(\\qv^{}_{IC},T)$ as the temperature\nis lowered. Henceforth, we will omit the plus sign in the order\nparameters subscript.\n\nTo describe the electric polarization, we need to add an electric\nfree energy and a magnetoelectric coupling term to the magnetic\nfree energy. Assuming a homogeneous polarization, the expression\nfor the electric free energy to lowest order is\n\\begin{equation}\nf^{}_{el}=V^{}_{cell}\\sum_{\\alpha=1}^3\\frac{{P^{}_{\\alpha}}^2}{2\\chi^{0}_{E,\\alpha}},\n\\end{equation}\nwhere $V^{}_{cell}$ is the volume of the unit cell, $\\Pv$ is the\nferroelectric order parameter and $\\chi^{0}_{E,\\alpha}$ is the\nhigh-temperature electric susceptibility along the $\\alpha$\ndirection. By symmetry considerations,\\cite{HAB07} the allowed\nmagnetoelectric coupling term of the lowest order in the\nincommensurate phases is\n\\begin{equation}\nf^{}_{int}=r\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|\\sin(\\varphi^{}_{x}-\\varphi^{}_{b})P^{}_{b},\n\\end{equation}\nwhere $\\varphi^{}_{x}$ and $\\varphi^{}_{b}$ are the phases of\n$\\sigma^{}_{x}(\\qv^{}_{IC})$ and $\\sigma^{}_{b}(\\qv^{}_{IC})$,\nrespectively, and $r$ is a small real magnetoelectric coupling\nparameter. Below we examine the results of the model.\n\n\\section{Phase boundaries and order parameters}\n\\label{Sec 2}\n\\subsection{\\textbf{MnWO}$^{}_{\\textbf{4}}$ without magnetic fields}\nThe wave vector $\\qv^{}_{IC}$ that characterizes the AF3 and AF2\nphases is determined by maximizing the eigenvalue\n$\\lambda^{}_{+}(\\qv)$ for a given set of coupling energies\n$\\{J^{}_i\\}$. After carrying out the maximization procedure, we\ncan find the first transition temperature by equating\n$\\zeta^{}_{+,x}$ to zero for $\\qv=\\qv^{}_{IC}$:\n\\begin{align}\n\\label{eq:T_N3}\n&T^{(0)}_{N3}=\\frac{\\lambda^{}_{+}(\\qv_{IC})+D}{a}.\n\\end{align}\nThe index 0 indicates that this is the transition temperature in\nthe absence of external magnetic fields. By transforming to normal\nmagnetic coordinates, the free energy of the incommensurate phases\nup to the fourth order in the magnetic order parameters is\n\\begin{widetext}\n\\begin{align}\n\\label{eq:free_energy_IC}\nf&=\\big(aT-D-\\lambda^{}_{+}(\\qv^{}_{IC})\\big)\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|^2+3b\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|^4\n+\\big(aT-\\lambda^{}_{+}(\\qv^{}_{IC})\\big)\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|^2+3b\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|^4\\nonumber\\\\\n&+2b\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|^2\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|^2\\big[2+\\cos(2\\varphi^{}_x-2\\varphi^{}_b)\\big]\n+V^{}_{cell}\\sum_{\\alpha=1}^3\\frac{{P^{}_{\\alpha}}^2}{2\\chi^{0}_{E,\\alpha}}+r\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|\\sin(\\varphi^{}_{x}-\\varphi^{}_{b})P^{}_{b}.\n\\end{align}\n\\end{widetext}\nThis expression is obtained by keeping the Fourier components\n$\\qv=\\pm\\qv^{}_{IC}$ in the total free energy\n$f=f^{}_{mag}+f^{}_{el}+f^{}_{int}$. Minimizing with respect to\nthe polarization components, we find the induced polarization\n\\begin{align}\n\\label{eq:polarization}\nP^{}_x&=P^{}_z=0, \\nonumber\\\\\nP^{}_b&=-\\frac{\\chi^{0}_{E,b}r}{V^{}_{cell}}\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|\\sin(\\varphi^{}_{x}-\\varphi^{}_{b}).\n\\end{align}\nInserting Eqs. \\eqref{eq:polarization} into Eq.\n\\eqref{eq:free_energy_IC}, we get\n\\begin{widetext}\n\\begin{align}\n\\label{eq:free_energy_IC2}\nf&=\\big(aT-D-\\lambda^{}_{+}(\\qv^{}_{IC})\\big)\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|^2+3b\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|^4+\\big(aT-\\lambda^{}_{+}(\\qv^{}_{IC})\\big)\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|^2+3b\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|^4\\nonumber\\\\\n&+2b\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|^2\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|^2\\big[2+\\cos(2\\varphi^{}_x-2\\varphi^{}_b)-2\\gamma\\sin^2(\\varphi^{}_x-\\varphi^{}_b)\\big],\n\\end{align}\nwhere $\\gamma$ is a dimensionless parameter given by\n\\begin{align}\n\\label{eq:gamma}\n\\gamma=\\frac{\\chi^{0}_{E,b}r^2}{8V^{}_{cell}b}.\n\\end{align}\nIn order to minimize the free energy \\eqref{eq:free_energy_IC2},\nthe phase difference $\\varphi^{}_x-\\varphi^{}_b$ should be\n$\\pm\\pi\/2$. In addition, we show below that $\\gamma$ is of order\n$10^{-5}$. Hence the last factor in the square brackets of Eq.\n\\eqref{eq:free_energy_IC2} will be neglected in the description of\nthe magnetic phase transitions. The minimization of the free\nenergy \\eqref{eq:free_energy_IC2} with respect to\n$\\left|\\sigma^{}_{x}(\\qv^{}_{IC})\\right|$ and\n$\\left|\\sigma^{}_{b}(\\qv^{}_{IC})\\right|$ yields\n\\label{eq:Incommensurate order parameters}\n\\begin{align}\n\\label{eq:Incommensurate order parameters}&\\left|\\sigma^{0}_{x}(\\qv^{}_{IC})\\right|=\\sqrt{\\frac{a\\big(T^{(0)}_{N3}-T\\big)}{6b}} \\quad,\\quad \\left|\\sigma^{0}_{b}(\\qv^{}_{IC})\\right|=0 \\quad,\\quad \\text{$T^{(0)}_{N2}T^{(0)}_{N3}\n\\\\\n\\chi^{0}_{E,b}\\big(1+\\frac{\\widetilde{T}^{(0)}_{N2}-T^{(0)}_{N2}}{T-\\tilde{T}^{(0)}_{N2}}\\big)\n& \\widetilde{T}^{(0)}_{N2}\\max\\bigg\\{2\\left(1-\\eta\\right),\\;\\frac{2}{3}\\left(1-\\sqrt{3\\eta^2-2}\\right)\\bigg\\},\n\\end{align}\nwhere $\\epsilon\\equiv\\frac{D}{\\lambda^{}_{+}(\\qv^{}_{IC})}$ and\n$\\eta\\equiv\\frac{\\lambda^{}_{+}(\\qv^{}_{C})}{\\lambda^{}_{+}(\\qv^{}_{IC})}$.\nIn this case, the transition temperature $T^{(0)}_{N1}$ is given\nby\n\\begin{align}\n\\label{eq:T_N1}\nT^{(0)}_{N1}=\\left[\\frac{4\\left(\\eta^2-1\\right)+4\\epsilon-3\\epsilon^2}{4\\left(2\\left(\\eta-1\\right)+\\epsilon\\right)}+\\epsilon\\right]\\frac{\\lambda^{}_{+}(\\qv^{}_{IC})}{a}.\n\\end{align}\nWe study below the effects of magnetic field on the transition\ntemperatures.\n\n\\subsection{The effect of an external magnetic field}\nThe formalism presented above can be generalized to take into\naccount the effect of a uniform external magnetic field $\\hv$.\nThis can be accomplished by adding to the free energy the Zeeman\nterm\n$F^{}_{Z}=g\\mu^{}_B\\sum_{\\Rv}\\sum_{\\tauv=\\tauv^{}_{1},\\tauv^{}_{2}}\\Sv(\\Rv+\\tauv)\\cdot\\hv$,\nor, equivalently\\cite{comment2}\n\\begin{align}\n\\label{eq:Zeeman}\nf^{}_{Z}\\equiv\\frac{F^{}_{Z}}{N}=g\\mu^{}_B\\sum_{\\tauv=\\tauv^{}_{1},\\tauv^{}_{2}}\\Sv(0,\\tauv)\\cdot\\hv.\n\\end{align}\nMinimizing the free energy with respect to\n$S^{}_{\\alpha}(0,\\tauv)$ at the paramagnetic phase, we find the\nresponse to the external magnetic field\n\\begin{align}\n\\label{Ferromagnetic Fourier component1}\nS^{}_{\\alpha}(0,\\tauv)=-\\frac{\\chi^{}_{\\alpha}(T)}{g\n\\mu^{}_B}h^{}_{\\alpha} \\qquad (\\alpha=x,b),\n\\end{align}\nwith the magnetic susceptibility following a Curie-Weiss law\n\\begin{align}\n\\label{eq:magnetic_susceptibility}\n\\chi^{}_{\\alpha}(T)=\\frac{(g\\mu^{}_B)^2}{aT-D^{}_{\\alpha}-2\\sum_{i=1}^{11}J^{}_i}.\n\\end{align}\nComparing Eq. \\eqref{eq:magnetic_susceptibility} with the general\nCurie-Weiss law\\cite{Ashcroft&Mermin}\n\\begin{align}\n\\label{eq:magnetic_susceptibility2}\n\\chi^{}_{\\alpha}(T)=\\frac{\\frac{(g\\mu^{}_B)^2J(J+1)}{3k^{}_{B}}}{T-\\theta^{}_{\\alpha}},\n\\end{align}\nwe identify the parameter $a$ introduced in the expansion of the\nentropy [see Eq. \\eqref{eq:Entropy}] as\n\\begin{align}\n\\label{eq:a parameter} a=\\frac{3k^{}_{B}}{J(J+1)}.\n\\end{align}\nFor Mn$^{2+}$ ions with $J=S=5\/2$ this parameter is\n$a^{}_{\\text{Mn}}=0.343k^{}_{B}$. The Curie-Weiss temperature is\nrelated to the exchange couplings and the anisotropy energy by:\n\\begin{align}\n\\label{eq:Curie-Weiss temperature}\n\\theta^{}_{\\alpha}=\\frac{J(J+1)}{3k^{}_{B}}\\left(D^{}_{\\alpha}+2\\sum_{i=1}^{11}J^{}_i\\right).\n\\end{align}\n\nIn the incommensurate phases AF3 and AF2, Eq. \\eqref{Ferromagnetic\nFourier component1} is replaced by\n\\begin{align}\n\\label{Ferromagnetic Fourier component2}\nS^{}_{\\alpha}(0,\\tauv)=\\frac{-\\chi^{}_{\\alpha}(T)h^{}_{\\alpha}}{g\n\\mu^{}_B\\left[1+\\frac{d^{}_{1\\alpha}\\left|\\sigma^{0}_{x}(\\qv^{}_{IC})\\right|^2+d^{}_{2\\alpha}\\left|\\sigma^{0}_{b}(\\qv^{}_{IC})\\right|^2}{a\\left(T-\\theta^{}_{\\alpha}\\right)}\\right]},\n\\end{align}\nwhere $d^{}_{1x}=d^{}_{2b}=12b$ and $d^{}_{2x}=d^{}_{1b}=4b$. The\ncorresponding form in the AF1 phase is\n\\begin{align}\n\\label{Ferromagnetic Fourier component3}\nS^{}_{\\alpha}(0,\\tauv)=\\frac{-\\chi^{}_{\\alpha}(T)h^{}_{\\alpha}}{g\n\\mu^{}_B\\left[1+\\frac{e^{}_{\\alpha}\\left|\\sigma^{0}_{x}(\\qv^{}_{C})\\right|^2}{a\\left(T-\\theta^{}_{\\alpha}\\right)}\\right]},\n\\end{align}\nwith $e^{}_{x}=12b$ and $e^{}_{b}=4b$. The ferromagnetic Fourier\ncomponent at $\\qv=0$ couples to the incommensurate and\ncommensurate wave vectors through the fourth order term in Eq.\n\\eqref{eq:magnetic_free_energy}. This coupling modifies the\ncoefficients of the free energy expansion and, consequently, the\ntransition temperatures. In the presence of an external magnetic\nfield, the first two transition temperatures are (to second order\nin the magnetic field)\n\\begin{align}\n\\label{eq:T_3 with magnetic field}\n\\begin{cases}\nT^{}_{N3}(h^{}_{x})=T^{(0)}_{N3}\\left[1-12\\frac{b\\chi^{2}_{x}(T^{(0)}_{N3})}{aT^{(0)}_{N3}(g\\mu^{}_B)^2}h^{2}_{x}\\right] \\quad & \\hv=h^{}_{x}\\hat{\\bf x}, \\\\\nT^{}_{N3}(h^{}_{b})=T^{(0)}_{N3}\\left[1-4\\frac{b\\chi^{2}_{b}(T^{(0)}_{N3})}{aT^{(0)}_{N3}(g\\mu^{}_B)^2}h^{2}_{b}\\right]\n\\quad & \\hv=h^{}_{b}\\hat{\\bf b},\n\\end{cases}\n\\end{align}\n\\begin{align}\n\\label{eq:T_2 with magnetic field}\n\\begin{cases}\nT^{}_{N2}(h^{}_{x})=T^{(0)}_{N2} \\quad & \\hv=h^{}_{x}\\hat{\\bf x},\\\\\nT^{}_{N2}(h^{}_{b})=T^{(0)}_{N2}\\left[1-16\\kappa\\frac{b\\chi^{2}_{b}(T^{(0)}_{N2})}{a(g\\mu^{}_B)^2}h^{2}_{b}\\right]\n\\quad & \\hv=h^{}_{b}\\hat{\\bf b},\n\\end{cases}\n\\end{align}\nwith\n$\\kappa=\\left(\\frac{1}{T^{(0)}_{N2}}+\\frac{8}{3\\left(T^{(0)}_{N2}-\\theta^{}_{b}\\right)}\\right)$.\nFor an external magnetic field along the easy axis direction, the\ninequality which determines the stability range of the AF1 phase\nis\n\\begin{widetext}\n\\begin{align}\n\\label{eq:T_1 with magnetic field_x}\nT&T^{(0)}_{N3}$ (in the\nparaelectric and paramagnetic phase), is experimentally found to\nbe $\\chi^{0}_{E,b}=11.3\\epsilon^{}_0$.\\cite{TK06} The\ndimensionless parameter $\\gamma$ [see Eq. \\eqref{eq:gamma}] is\nthen $\\gamma=5.9\\cdot10^{-5}$. This value supports the assumption\nthat the magnetic transitions are almost unaffected by the\nmagnetoelectric coupling. The dielectric constant\n$\\epsilon^{}_b=1+\\frac{\\chi^{0}_{E,b}}{\\epsilon^{}_0}$ is shown in\nFig. \\ref{fig:susceptibility}. This result is in good agreement\nwith the experimental measurements of Ref. \\onlinecite{TK06}. The\nnarrow width of the divergence region is a consequence of the\nsmall difference between $\\widetilde{T}^{(0)}_{N2}$ and\n${T}^{(0)}_{N2}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\subfigure[\\label{fig:polarization}]{ \\label{fig:polarization}\n\\includegraphics[width=0.45\\textwidth]{Polarization}\n} \\subfigure[\\label{fig:susceptibility}]{\n\\label{fig:susceptibility}\n\\includegraphics[width=0.45\\textwidth]{Dielectric_Constant}\n}\n\\end{center}\n\\caption{\\label{fig:electric}(Color online) (a) The ferroelectric\npolarization and (b) the dielectric constant $\\epsilon^{}_b$. The\nsolid lines are the calculated quantities and the dots are the\ndata points of Taniguchi \\textit{et al.}.\\cite{TK06} The\ncalculated polarization was obtained by setting\n$\\frac{\\chi^{0}_{E,b}\\left|r\\right|}{V^{}_{cell}}=21\\mu C\/m^2$.}\n\\end{figure}\nOnce again, the discrepancy between the linear behavior of the\ncalculated polarization and the observed one may be reconciled by\nassuming a critical exponent $\\beta\\approx\\frac{1}{3}$ for the\nmagnetic order parameters. The behavior of the calculated\npolarization in this case is given in Fig.\n\\ref{fig:polarization_critical}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Polarization_Critical}\n\\caption{\\label{fig:polarization_critical}(Color online) The\nferroelectric polarization calculated with the critical exponent\n$\\beta\\approx\\frac{1}{3}$. The solid lines are the calculated\nquantities and the dots are the experimental points of Taniguchi\n\\textit{et al.}.\\cite{TK06} The calculated polarization was\nobtained by setting\n$\\frac{\\chi^{0}_{E,b}\\left|r\\right|}{V^{}_{cell}}=27.5\\mu C\/m^2$.}\n\\end{figure}\nTo examine the effect of Fe doping, we use the relations\n\\eqref{eq:Fe a} and \\eqref{eq:Fe parameters} in the expressions\nfor the transition temperatures and fit the slope to the\nexperimental value according to the $x-T$ phase diagram of\nChaudhury \\textit{et al.}.\\cite{CRP09} This procedure yields the\nvalues $c^{}_1\\approx-3.26k^{}_{B}K$,\n$c^{}_2\\approx13.03k^{}_{B}K$ and $c^{}_3\\approx-1.3$. The\nanisotropy energy increases with increasing Fe concentration, as\nexpected, since as opposed to the Mn$^{2+}$ ion, the Fe$^{2+}$ ion\npossesses a non-vanishing angular momentum.\\cite{HN10}\n\nCalculating the different parameters for a small Fe concentration\n$x$ and repeating the calculations of the $T-H$ phase diagram, we\ncan check the consistency of the above results. The resulting\nphase diagram for $x=0.035$ is shown in Fig. \\ref{fig:Fe}. Except\nfor high fields or low temperatures, the result is in fine\nagreement with the measurement of Ye \\textit{et al.}.\\cite{YF08}\nThe reentrant ferroelectric phase observed at low\ntemperatures\\cite{CRP09,CRP09b} may be explained by higher order\nterms in the free energy expansion.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{T_H_Fe_Doping}\n\\caption{\\label{fig:Fe}(Color online) Magnetoelectric phase\ndiagram of Mn$^{}_{0.965}$Fe$^{}_{0.035}$WO$_4$ with the magnetic\nfield parallel to the easy axis. The solid lines are the\ncalculated transition temperatures and the dots are the\nexperimental points of Ye \\textit{et al.}.\\cite{YF08}}\n\\end{figure}\n\nThe effect of non-magnetic ions on the transition temperatures\n$T^{}_{N3}(x)$ and $T^{}_{N2}(x)$ is given by Eq.\n\\eqref{eq:non-magnetic}. These results are drawn in Fig.\n\\ref{fig:Zn} together with the experimental data of Chaudhury\n\\textit{et al.}\\cite{CRP11} of Mn$^{}_{1-x}$Zn$^{}_{x}$WO$^{}_4$.\nSimilar results have been observed in\nMn$^{}_{1-x}$Mg$^{}_{x}$WO$^{}_4$.\\cite{ML09} We stress that\nunlike the case of Fe doping, the results for the transition\ntemperatures $T^{}_{N3}(x)$ and $T^{}_{N2}(x)$ in the case of\nnon-magnetic ions doping do not require additional\nphenomenological parameters.\n\nAs opposed to $T^{}_{N3}(x)$ and $T^{}_{N2}(x)$, the calculated\ntransition temperature $T^{}_{N1}(x)$ does not coincide with the\nexperimentally measured one.\\cite{CRP11} The discrepancy may be\nexplained by allowing small changes in the exchange couplings\n$J^{\\text{Mn-Mn}}_{i}$ due to spin-lattice coupling (or exchange\nstriction). In other words, if we assume that\n$J^{\\text{Mn-Mn}}_{i}(x)=J^{\\text{Mn-Mn}}_{i}(1+\\xi^{}_{i}x)$ with\n$\\xi^{}_{i}x\\ll1$, then $T^{}_{N1}(x)$ changes dramatically while\n$T^{}_{N3}(x)$ and $T^{}_{N2}(x)$ are almost not influenced. The\nreason for this behavior is that the transition temperature\n$T^{}_{N1}$ [see Eq. \\eqref{eq:T_N1}] is much more sensitive to\nsmall changes in the exchange couplings than the transition\ntemperatures $T^{}_{N3}$ and $T^{}_{N2}$ [see Eqs. \\eqref{eq:T_N3}\nand \\eqref{eq:T_N2}]. A significant spin-lattice coupling in the\nmultiferroic MnWO$^{}_4$ has been demonstrated \\cite{TK08} by the\nappearance of an incommensurate lattice modulation in the AF3 and\nAF2 phases, with a lattice propagation vector equal to twice the\nmagnetic propagation vector. In addition, thermal expansion\nmeasurements reveal considerable discontinuities in the lattice\nparameters at the AF2$\\rightarrow$AF1 first order phase\ntransition.\\cite{CRP08b} Another indication for a dependence of\nthe Mn-Mn exchange couplings on the non-magnetic dopant\nconcentration is provided by the small change of the\nincommensurate propagation vector from\n$\\qv^{}_{IC}=(-0.214,0.5,0.457)$ in MnWO$^{}_4$ to\n$\\qv^{}_{IC}=(-0.209,0.5,0.453)$ in\nMn$^{}_{0.85}$Zn$^{}_{0.15}$WO$^{}_4$.\\cite{ML09}\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{X_T_Zn_Doping}\n\\caption{\\label{fig:Zn}(Color online) Transition temperatures\n$T^{}_{N3}(x)$ and $T^{}_{N2}(x)$ of\nMn$^{}_{1-x}$Zn$^{}_{x}$WO$^{}_4$. The solid lines are the\ncalculated transition temperatures and the dots are the data\npoints of Chaudhury \\textit{et al.}.\\cite{CRP11}}\n\\end{figure}\n\nThe next step is to compare the above fitted parameters with the\nparameters calculated directly from the experimental sets of\nexchange couplings of Ehrenberg \\textit{et al.} and Ye \\textit{et\nal.}. The calculated exchange couplings of Ref. \\onlinecite{TC09}\nyield much higher transition temperatures than the observed ones\nand thus will not be discussed here. Indeed, The problem of\noverestimation of exchange interactions by DFT calculations has\nbeen indicated by the authors.\\cite{TC09}\n\nThe first step is to maximize $\\lambda^{}_{+}(\\qv)$ [see Eq.\n\\eqref{eq:lambda}] in order to find the incommensurate wave vector\n$\\qv^{}_{IC}$ and the corresponding eigenvalue\n$\\lambda^{}_{+}(\\qv^{}_{IC})$. The maximization process yields\n$\\qv^{}_{IC}=(-0.28,0.5,0.44)$ and\n$\\lambda^{}_{+}(\\qv^{}_{IC})=3.82k^{}_{B}K$ for the\n$J^{}_1-J^{}_9$ values of Ehrenberg \\textit{et al.},\\cite{EH99}\nwhile for the $J^{}_1-J^{}_{11}$ values of Ye \\textit{et\nal.}\\cite{YF11} we find $\\qv^{}_{IC}=(-0.3,0.5,0.49)$ and\n$\\lambda^{}_{+}(\\qv^{}_{IC})=3.85k^{}_{B}K$. These results are in\nqualitative agreement with the incommensurate wave vector\n$\\qv^{}_{IC}=(-0.214,\\frac{1}{2},0.457)$ observed in experiments.\nHowever, the differences are not negligible, suggesting possible\nerrors in the experimental sets of exchange couplings. In\naddition, the transition temperatures $T^{(0)}_{N3}$ and\n$T^{(0)}_{N2}$ calculated from Eqs. \\eqref{eq:T_N3} and\n\\eqref{eq:T_N2} with $a^{}_{\\text{Mn}}=0.343k^{}_{B}$ [see Eq.\n\\eqref{eq:a parameter}] are found to be $T^{(0)}_{N3}=12.79K$,\n$T^{(0)}_{N2}=10.3K$ for the set of Ehrenberg \\textit{et al.} and\n$T^{(0)}_{N3}=13.67K$, $T^{(0)}_{N2}=10K$ for the set of Ye\n\\textit{et al.}. These values slightly differ from the observed\ntransition temperatures, especially the second one. The ratio\n$\\eta=\\frac{\\lambda^{}_{+}(\\qv^{}_{C})}{\\lambda^{}_{+}(\\qv^{}_{IC})}$\nis found to be $\\eta=0.974$ and $\\eta=0.97$ for the sets of\nEhrenberg \\textit{et al.} and Ye \\textit{et al.}, respectively.\nTable ~\\ref{tab:Parameters_ summary} summarizes the values of\n$\\lambda^{}_{+}(\\qv^{}_{IC})$, $D$ and $\\eta$ calculated from the\nexperimental sets of magnetic parameters and those fitted to the\nexperimental transition temperatures.\n\\begin{table}[ht]\n\\caption{\\label{tab:Parameters_ summary} Comparison between the\nmodel parameters calculated from the experimental sets of Ref.\n\\onlinecite{EH99} and Ref. \\onlinecite{YF11} and those fitted to\nthe experimental transition temperatures.}\n \\begin{tabular}{c|c|c|c}\n \\hline\n \\hline\n Parameter & Ref. \\onlinecite{EH99} & Ref. \\onlinecite{YF11} & This work \\\\ \\hline\n $\\lambda^{}_{+}(\\qv^{}_{IC})(k^{}_{B}K)$ & 3.82 & 3.85 & 4.36-4.45 \\\\\n & & \\\\ \\hline\n $D(k^{}_{B}K)$ & 0.568 & 0.83 & 0.27-0.18 \\\\\n & & \\\\ \\hline\n $\\eta$ & 0.974 & 0.97 & 0.97-0.98 \\\\\\hline \\hline\n \\end{tabular}\n\\end{table}\n\nThe calculation of the Curie-Weiss temperature reveals a much more\nserious discrepancy. According to Eq. \\eqref{eq:Curie-Weiss\ntemperature}, the Curie-Weiss temperature is $\\theta^{}_x=-7.6K$,\n$\\theta^{}_b=-9.25K$ for the set of Ehrenberg \\textit{et al.} and\n$\\theta^{}_x=-23.2K$, $\\theta^{}_b=-25.65K$ for the set of Ye\n\\textit{et al.}. These values do not fit the experimental\nCurie-Weiss temperature $\\theta\\approx-75K$.\\cite{AAH06,DH69} We\nsuspect that the origin of most of the discrepancies are errors in\nthe set of magnetic couplings. The results suggested by our model\nmay be used as additional constraints in the determination of\nthose couplings. As mentioned before, an additional possible cause\nfor the above discrepancies is related to fluctuations near the\ntransitions, as will be discussed in the next section.\n\\section{The Ginzburg criterion}\n\\label{Sec 4} The results of the preceding sections have been\nobtained within the mean-field approximation. Here we estimate the\nGinzburg range, in which fluctuations become important, near the\nfirst transition P$\\rightarrow$AF3, by two methods. First we\ncompare the mean square fluctuation of the order parameter\n$\\sigma^{}_{x}(\\qv^{}_{IC})$ with the mean-field value, and then\nwe compare the discontinuity in the heat capacity derived from the\nLandau theory with the divergent heat capacity, originating from\nthe fluctuations at quadratic order.\\cite{Landau&Lifshitz}\n\nLet us denote by\n$\\delta\\sigma^{}_{x}(\\qv)=\\sigma^{}_{x}(\\qv)-\\langle\\sigma^{}_{x}(\\qv)\\rangle$\nthe fluctuation of the order parameter in the AF3 phase. The\ncorrelation function of these deviations is\n\\begin{align}\n\\label{eq:Correlation}\n\\langle\\delta\\sigma^{}_{x}(\\qv)\\delta\\sigma^{}_{x}(\\qv')\\rangle=\\frac{k^{}_{B}T\\delta^{}_{\\qv',-\\qv}}{4N\\left(D+\\lambda^{}_{+}(\\qv)-aT\\right)},\n\\end{align}\nwhere $N$ is the number of unit cells in the correlation volume.\nWe can find the correlation lengths by expanding\n$\\lambda^{}_{+}(\\qv)$ to second order around $\\qv^{}_{IC}$:\n\\begin{align}\n\\label{eq:second order expansion} \\lambda^{}_{+}(\\qv)\\approx\n\\lambda^{}_{+}(\\qv^{}_{IC})+\\sum_{i,j}M^{}_{ij}\\left(q^{}_{i}-q^{}_{IC,i}\\right)\\left(q^{}_{j}-q^{}_{IC,j}\\right),\n\\end{align}\nwith $M^{}_{ij}\\equiv\\frac{1}{2}\\frac{\\partial^2\n\\lambda^{}_{+}(\\qv)}{\\partial q^{}_{i}\\partial\nq^{}_{j}}\\bigg|^{}_{\\qv=\\qv^{}_{IC}}$. Denoting by $\\mu^{}_1$,\n$\\mu^{}_2$ and $\\mu^{}_3$ the three eigenvalues of the positive\nmatrix $-M^{}_{ij}$, the three correlation lengths are\n\\begin{align}\n\\label{eq:correlation lengths}\n\\xi^{}_{i}=\\sqrt{\\frac{\\mu^{}_{i}}{a\\left(T^{(0)}_{N3}-T\\right)}}.\n\\end{align}\nSubstituting $\\qv=\\qv^{}_{IC}$ and\n$N=\\frac{\\xi^{}_{1}\\xi^{}_{2}\\xi^{}_{3}}{V^{}_{cell}}$ in Eq.\n\\eqref{eq:Correlation}, the condition\n$\\langle\\left|\\delta\\sigma^{}_{x}(\\qv^{}_{IC})\\right|^2\\rangle\\ll\\left|\\sigma^{0}_{x}(\\qv^{}_{IC})\\right|^2$\nfor the validity of the mean-field theory\nreads\\cite{Landau&Lifshitz}\n\\begin{align}\n\\label{eq:Ginzburg}\n\\frac{k^{}_{B}T^{(0)}_{N3}}{4a\\left(T^{(0)}_{N3}-T\\right)}\\frac{V^{}_{cell}}{\\xi^{}_{1}\\xi^{}_{2}\\xi^{}_{3}}\\ll\n\\frac{a\\left(T^{(0)}_{N3}-T\\right)}{6b}.\n\\end{align}\nInserting Eq. \\eqref{eq:correlation lengths} into Eq.\n\\eqref{eq:Ginzburg} at the Ginzburg temperature $T^{}_G$, we find\n\\begin{align}\n\\label{eq:Ginzburg temperature1}\n\\left|T^{}_{G}-T^{(0)}_{N3}\\right|\\approx\n\\frac{9k^{2}_{B}b^{2}V^{2}_{cell}\\left(T^{(0)}_{N3}\\right)^2}{4a\\mu^{}_{1}\\mu^{}_{2}\\mu^{}_{3}}.\n\\end{align}\nEquation \\eqref{eq:Ginzburg temperature1} estimates the\ntemperature range below $T^{(0)}_{N3}$, in which fluctuations are\nnot negligible.\n\nLet us now estimate the Ginzburg range according to the second\nmethod. On the one hand, according to Landau theory, the heat\ncapacity $c=-T\\frac{\\partial^2 f}{\\partial T^2}$ grows\ndiscontinuously at the transition P$\\rightarrow$AF3:\n\\begin{align}\n\\label{eq:heat capacity Landau} \\Delta c^{}_{L}\\equiv\nc^{}_{L}\\left(T^{(0)-}_{N3}\\right)-c^{}_{L}\\left(T^{(0)+}_{N3}\\right)=\\frac{a^2T^{(0)}_{N3}}{6b}.\n\\end{align}\nOn the other hand, assuming fluctuations at quadratic order, the\nsingular part of the heat capacity is given by\n\\begin{align}\n\\label{eq:heat capacity Fluctuations}\nc^{}_{G}=\\frac{V^{}_{cell}k^{}_{B}a^{2}T^{2}}{2\\left(2\\pi\\right)^3}\\int_{BZ}\\frac{d^3q}{\\left(aT-D-\\lambda^{}_{+}(\\qv)\\right)^2},\n\\end{align}\nwhere the integral is over the first Brillouin zone. In the\nneighborhood of $T^{(0)}_{N3}$, the main contribution to the\nintegral comes from the neighborhood of the incommensurate wave\nvector $\\qv^{}_{IC}$ in reciprocal space. Thus we can use the\nexpansion \\eqref{eq:second order expansion}. Replacing the first\nBrillouin zone by a sphere, and taking $T\\approx T^{(0)}_{N3}$, we\ncan estimate the integral in Eq. \\eqref{eq:heat capacity\nFluctuations}:\n\\begin{align}\n\\label{eq:heat capacity Fluctuations2} c^{}_{G}\\approx\n\\frac{k^{}_{B}a^{1.5}T^{2}\\left(T-T^{(0)}_{N3}\\right)^{-0.5}}{16\\pi\\sqrt{\\mu^{}_{1}\\mu^{}_{2}\\mu^{}_{3}}}.\n\\end{align}\nComparing Eqs. \\eqref{eq:heat capacity Landau} and \\eqref{eq:heat\ncapacity Fluctuations2} at the Ginzburg temperature $T^{}_G$, we\nfind\\cite{Landau&Lifshitz}\n\\begin{align}\n\\label{eq:Ginzburg temperature2}\n\\left|T^{}_{G}-T^{(0)}_{N3}\\right|\\approx\n\\left(\\frac{6}{16\\pi}\\right)^2\\frac{k^{2}_{B}b^{2}V^{2}_{cell}\\left(T^{(0)}_{N3}\\right)^2}{a\\mu^{}_{1}\\mu^{}_{2}\\mu^{}_{3}}.\n\\end{align}\nCalculating the eigenvalues $\\mu^{}_1$, $\\mu^{}_2$ and $\\mu^{}_3$\nfrom the experimental sets of exchange couplings, the Ginzburg\ntemperature is estimated to be\n$\\left|T^{}_{G}-T^{(0)}_{N3}\\right|\\approx 9.41K$ and\n$\\left|T^{}_{G}-T^{(0)}_{N3}\\right|\\approx 6.24K$ for the sets of\nEhrenberg \\textit{et al.} and Ye \\textit{et al.}, respectively, by\nthe first method [see Eq. \\eqref{eq:Ginzburg temperature1}] while\nit is $\\left|T^{}_{G}-T^{(0)}_{N3}\\right|\\approx 0.06K$ and\n$\\left|T^{}_{G}-T^{(0)}_{N3}\\right|\\approx 0.04K$ by the second\nmethod [see Eq. \\eqref{eq:Ginzburg temperature2}]. These values\nsuggest that fluctuations of the order parameters can also\ncontribute to the discrepancies between the experimental data and\nthe mean-field Landau theory results.\n\\section{Summary and Conclusions}\n\\label{Sec 5} We have studied the phase diagram of\nMn$^{}_{1-x}$M$^{}_{x}$WO$^{}_{4}$ (M=Fe, Zn, Mg) by a\nsemi-phenomenological Landau theory. The energy has been modelled\nby a Heisenberg Hamiltonian with a single-ion anisotropy, while\nthe entropy has been expanded in powers of the classical spins.\nThis approach is different from the previous theoretical\nstudies,\\cite{TP10,SVP10} which are purely phenomenological, since\nit enables to compare different sets of exchange couplings.\nAlthough a purely phenomenological approach may capture all the\nsymmetry aspects of the problem and may provide a full mapping of\nthe stable states allowed by the order parameter\nsymmetries,\\cite{TP10} it does not indicate a clear connection\nbetween the free energy coefficients and the microscopic\ninteractions. The advantage of our approach is the simple relation\nof the free energy coefficients with experimentally derived\nquantities such as the superexchange couplings and the anisotropy\ncoefficients. For instance, this simple relation allows us to\nconsider the effect of different dopants on the phase diagram, not\ndiscussed in Ref. \\onlinecite{TP10}. We emphasize that our\napproach does not contradict any symmetry requirement.\n\nWe used the superexchange interaction couplings from the inelastic\nneutron scattering studies of Ehrenberg \\textit{et al.}\\cite{EH99}\nand Ye \\textit{et al.}.\\cite{YF11} The results show that both sets\nyield transition temperatures $T^{(0)}_{N3}$ and $T^{(0)}_{N2}$\nthat slightly deviate from the experimental temperatures, and\nsignificantly underestimate the Curie-Weiss temperature\n$\\left|\\theta\\right|$. In addition, the calculated incommensurate\nwave vector $\\qv^{}_{IC}$ has non-negligible deviations from the\nexperimentally observed one. The results presented here can serve\nas additional constraints on a future determination of the\nmagnetic Hamiltonian parameters. Another possible cause for the\ndiscrepancies relates to fluctuations near the transitions. We\nhave demonstrated the possible important contribution of\nfluctuations in MnWO$^{}_{4}$. This issue should be further\nexamined in future experiments.\n\nBeyond that, the model clarifies the effect of different dopants\non the phase diagram. The sensitivity of the expression\n\\eqref{eq:T_N1} for the transition temperature $T^{}_{N1}(x)$ to\nsmall changes of the ratio\n$\\eta\\equiv\\frac{\\lambda^{}_{+}(\\qv^{}_{C})}{\\lambda^{}_{+}(\\qv^{}_{IC})}$\nreflects the frustrated nature of the multiferroic MnWO$^{}_4$.\nThe origin of the complex phase diagram lies in the competition\nbetween different superexchange interactions. Small changes in the\nlocal environment of the Mn$^{2+}$ ions due to a chemical doping\ncause a significant change in the phase diagram. The sensitivity\nfor the local environment manifests itself by the contrasting\nbehavior of doping with different ions.\n\nLooking to the future, two points should be further examined.\nFirstly, a new analysis of the inelastic scattering experiments,\ntogether with the additional constraints provided in this work,\nshould improve the exchange couplings for the multiferroic\nMnWO$^{}_4$. Secondly, the measurement of the critical exponents\nnear the transitions would shed light on the effect of\nfluctuations. This may contribute to the general understanding of\ncritical phenomena in multiferroics.\n\\begin{acknowledgments}\nWe thank H. Shaked for helpful discussions. We acknowledge support\nfrom the Israel Science Foundation (ISF).\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIt has become increasingly apparent that the presence of a nearby companion has a large impact on stellar evolution during various phases. A particularly interesting case is that of red giant evolution when the primary star becomes so large that the companion is (or at least is close to being) engulfed by the tenuous stellar hydrogen envelope of the giant star. This leads to a common-envelope (CE) evolution process which is difficult to understand in detail and very difficult to simulate numerically \\citep{ivanetal13}. There is good evidence of this phenomenon occurring. Companions around objects beyond the asymptotic giant branch (post-AGB) are ubiquitous in for example the Magellanic Clouds (MCs) \\citep{vanw03, kamaetal14, kamaetal15}, and they show an unexpected period distribution covering the range 100 to 2000 days, meaning that the companion lies closer than typical stellar radii on the AGB \\citep{vanwetal09}. In addition, they have dusty circumbinary disks. Recently, characteristics of the same kind have been found also for stars earlier in their evolution, for example, beyond the first red giant branch (post-RGB) \\citep{kamaetal16}. These are examples where the red giant evolution was terminated and the companion survived, but there are likely also cases that ended with a complete stellar merger.\n\nThe described scenario may be closely related to well-known phenomena, for example, planetary nebulae (PNe). That they are the descendants of AGB stars is well established \\citep{shkl78}, but it remains to be shown what the necessary conditions for this to actually happen are, since not all AGB stars become PNe. The increasingly energetic radiation from the star (essentially its core at this stage) as it evolves off the AGB is expected to produce a PN, with its highly excited and ionised circumstellar gas produced by an effective mass loss during the AGB, provided the conditions are the right ones \\citep{kwoketal78, vanw03}. In addition to this, there is, in many cases, an associated spectacular transformation of the circumstellar morphology and kinematics that poses an explanatory problem, where the effects of a nearby companion is often introduced. \n\nThere are other objects with spectacular circumstellar characteristics, whose explanation call for a companion and most likely a CE-evolution scenario. An example is the Boomerang nebula, potentially a stellar merger following RGB evolution of its primary, with its massive, high-velocity outflow of extremely cold gas \\citep{sahaetal17b}. Another type of possibly related objects are the red novae, explained to be the results of stellar mergers following CE evolution. Such objects also show circumstellar characteristics, for example, very-high-velocity molecular outflows, that are difficult to explain by single stellar evolution \\citep{kamietal18a, kamietal18b}. \n\nAGB stars and PNe are the most well-studied objects of the type discussed here. We will describe them in some more detail, and note that many of the discussed phenomena may have a more general application. AGB stars are largely spherical although inhomogeneities in their extended atmospheres appear to be a common phenomenon \\citep{ohnaetal16b, khouetal16a, vlemetal17}, most likely an effect of convection and pulsation \\citep{freyetal17, liljetal18}. Also their circumstellar envelopes (CSEs) of gas and microscopic dust grains have an overall spherical symmetry \\citep{cacaetal10}. Notable exceptions to this general behaviour exist though, for example, the carbon star V~Hya \\citep{sahaetal16}, the S-star $\\pi^1$~Gru \\citep{doanetal17} and the M-star L$_2$~Pup \\citep{kervetal15}, all three being semi-regular variables. The two former have bipolar high-velocity outflows and equatorial density enhancements, while the latter has a 10\\,au-sized circumbinary disk. Further, detections of spiral patterns or arcs in the circumstellar gas and dust distributions are becoming increasingly common \\citep{maurhugg06, maeretal12, kimetal17, ramsetal17, gueletal18}.\n\nThis is in stark contrast to the PNe where complex geometrical patterns and directed flows of high-velocity gas are the rule rather than the exception \\citep{sahaetal11a}. This applies in particular to the objects in transition from the AGB to the PN stage, the proto-PNe, where no examples of overall sphericity have been found \\citep{sahaetal07}. However, even though the PNe morphology is often complex, the presence of an equatorial density enhancement, and often an axial symmetry in the orthogonal direction in the form of a high-velocity bipolar outflow, are key features. Both these components are also commonly found already in the proto-PNe. Notable examples are AFGL618 \\citep{coxetal03, leeetal13}, OH231.8+4.2 \\citep{alcoetal01, bujaetal02, sacoetal18}, M1-6 \\citep{huggetal00}, M1-92 \\citep{bujaetal97, alcoetal07}, M2-9 \\citep{cacaetal17}, and M2-56 \\citep{cacaetal02}, and the young PN NGC6302 \\citep{sagaetal17}. Among the lower-mass post-AGB objects the presence of disks appear ubiquitous \\citep{hilletal14, hilletal17, gorletal15, bujaetal18}, and also jets are found \\citep{bolletal17}. Molecular bipolar outflows are more scarce, and if present of low velocity \\citep{bujaetal13}. The disks are often stable and in Keplerian rotation, for example, the Red Rectangle \\citep{bujaetal16} and IW~Car \\citep{bujaetal17}. Less well-characterised sources like IRAS\\,08005--2356 \\citep{sahapate15}, IRAS\\,16342--3814 \\citep{sahaetal17a}, and IRAS\\,22026+5306 \\citep{sahaetal06} have young bipolar outflows of very high velocities.\n\nThere is a general belief that the axial symmetry has its root in the AGB star not being alone \\citep{vanw03, doucetal15, joneboff17}. A companion in orbit provides the energy and the angular momentum required to produce equatorial density enhancements and high-velocity bipolar outflows \\citep{soke97, soke98a, dema09}, either through a CE evolution \\citep{sokelivi94, nordblac06, soke15} or through increasing the rotation and\/or magnetic field of the primary \\citep{blacetal01}. The outflows are thought to be driven by jets, originating from an accretion disk \\citep{reyelope99, chametal18}, that sculpt the CSE, and hence produce the apparently complex geometrical patterns seen in the circumstellar gas of PNe \\citep{sahatrau98, leesaha03}. Most likely, this phenomenon starts at the very end of the AGB evolution when the mass of the AGB star reaches its minimum, the radius its maximum, and the mass-loss rate its maximum. The alternative explanation where the AGB star itself produces the energy and the angular momentum of the outflow appears less likely \\citep{nordblac06}. Unfortunately, the identification of a (close) binary companion around an AGB star is a very difficult observational task. Indirect evidence in the form of UV and x-ray emission exists \\citep{sahaetal08, sahaetal15}, although there could be other causes for the emission \\citep{montetal17}.\n\nThe objects with spectacular circumstellar characteristics discussed here are not only physically complex, they are also chemically complex. It is well-known that AGB CSEs are rich in different molecular species at the end of the AGB evolution; more than 100 are now detected \\citep[][and references therein]{cernetal00a, velietal17, debeolof18}. This is the effect of a number of different processes, such as stellar atmosphere equilibrium chemistry, extended atmosphere non-equilibrium chemistry, and photo-induced circumstellar chemistry \\citep[for example,][]{mill16}. Additional processes become active during the post-AGB phase, for example, increased UV radiation and high-velocity outflows that lead to shocks. The result is that they often show molecular species that are not detected (or tend to be much weaker) in AGB CSEs \\citep{pardetal07, edwaziur13, sacoetal15, velietal15}. The red novae have a circumstellar medium that is most likely formed in the process leading up to the stellar merger. These envelopes are also relatively rich in different molecular species \\citep{kamietal18b}, and the first detection ever of a radioactive molecule, $^{26}$AlF, was done in such an environment \\citep{kamietal18a}.\n\nThe object of this paper, HD\\,101584, show many of the characteristics traditionally associated with a post-AGB object, but now also found in connection with post-RGB objects. The star is warm and there is a substantial far-IR excess due to a dusty circumstellar environment, and in addition there is good evidence of a companion. Its characteristics are summarised in Sect.~\\ref{s:hd_characteristics}. In this paper we present observations performed with the Atacama Large Millimeter\/submillimeter Array (ALMA) in observing bands centred on the $^{12}$CO and $^{13}$CO $J$\\,=\\,\\mbox{2--1} lines (from now on the more common isotope is meant unless rarer isotopes are specifically mentioned, that is, $^{12}$C$^{16}$O\\,=\\,CO). In total about 13\\,GHz, in high spectral resolution mode, have been covered using ALMA. Continuum emission at 1.3\\,mm is recovered from line-free regions. We further report complementary observations of line emission and continuum emission at 350\\,$\\mu$m using the Atacama Pathfinder Experiment telescope (APEX).\n\n\\section{HD101584}\n\n\\subsection{The characteristics}\n\\label{s:hd_characteristics}\n\n\\object{HD\\,101584} (V885~Cen, IRAS\\,11385--5517) is bright at optical wavelengths ($V$\\,$\\approx$\\,7 mag) and with an estimated effective temperature of $\\approx$\\,8500\\,K it has a spectral type A6Ia classification \\citep{sivaetal99,kipp05}. Its large far-IR excess led \\citet{partpott86} to infer an evolutionary status at, or shortly after, the end of the AGB, a conclusion corroborated by \\citet{bakketal96a} who presented optical and infrared data, and proposed that HD\\,101584 most likely has evolved from the asymptotic giant branch (AGB) at most a few hundred years ago. They estimated a (present-day) mass of $\\approx$\\,0.55\\,$M_\\odot$, and a luminosity of $\\approx$\\,5000\\,$L_\\odot$. This identification is consistent with its location well above the galactic plane (galactic latitude of 6$^\\circ$) and its high space velocity. \\citet{olofetal17} provided evidence that HD\\,101584 has gone through CNO-processing on the red giant branch and is of low initial mass ($\\approx$\\,1\\,$M_\\odot$), based on observations of circumstellar CO isotopologues. \\citet{kipp05} found the abundances of C, N, O, Na, and Mg to be close to solar, while the Si abundance is sub-solar by a factor of 20, possibly an effect of accretion of depleted gas. \n\nIn this paper we will argue that a post-RGB star is an alternative evolutionary status of HD\\,101584. The reason for this is that we envision a geometrically different circumstellar environment than that assumed by \\citet{bakketal96a}. The consequence is less circumstellar extinction, and the effect is that for a given luminosity the star must be located further away. This will impose some issues with a post-AGB interpretation as will be discussed below. \n\nPhotometric and radial-velocity variations show that HD\\,101584 has a companion. \\citet{bakketal96b} used the former and estimated a period of 218$^{\\rm d}$, while \\citet{diazetal07} found a period of 144$^{\\rm d}$ using the latter. The radial velocity estimate is presumably more accurate, but the data themselves have never been published and it is therefore not possible to estimate their uncertainty. Using these results, and assuming an 0.6\\,$M_\\odot$ primary star and an almost face-on orientation of the orbit plane (based on the circumstellar morphology, see below), \\citet{olofetal15} estimated a binary separation of $\\approx$\\,0.7\\,au and a companion mass of $\\approx$\\,0.6\\,$M_\\odot$, suggesting that it is a low-mass main-sequence star or possibly a low-luminosity white dwarf (WD), consistent with the absence of spectroscopic emission from the companion. \n\n\\citet{bakketal96b} inferred an essentially edge-on circumbinary disk to explain the presence of optical absorption lines, but this orientation of the orbit plane appears less likely for a number of reasons. Images from the Hubble Space Telescope (HST) show a diffuse circumstellar environment with evidence of an essentially circular ring of radius $\\approx$\\,1\\farcs5 roughly centred on the star \\citep{sahaetal07, siodetal08}, suggesting more of a face-on orientation. Further, the central star is bright despite significant amount of circumstellar material \\citep{olofetal15}, presumably because the polar axis of the system is oriented towards us and the region around it has been (at least) partially evacuated. In this paper we will provide additional arguments for the face-on orientation.\n\nThe molecular line emissions reveal much more morphological information than the visual images, and in addition provide kinematical information. \\citet{olofnyma99} obtained high-quality $^{12}$CO and $^{13}$CO $J$\\,=\\,\\mbox{1--0} and \\mbox{2--1} single-dish map data, and inferred the presence of relatively compact emission covering a velocity range of $\\approx$\\,100\\,km\\,s$^{-1}$, including a prominent central narrow-line feature, and a high-velocity ($\\approx$\\,150\\,km\\,s$^{-1}$) bipolar outflow having an east-west orientation and a Hubble-like velocity gradient. The most blue- and red-shifted emissions lie $\\approx$\\,5$\\arcsec$ to the W and E, respectively. The full complexity of the circumstellar material was revealed through ALMA observations in frequency regions centred on the $^{12}$CO and $^{13}$CO $J$\\,=\\,\\mbox{2--1} lines \\citep{olofetal15}. A double-peaked OH 1667\\,MHz maser line, with a total velocity coverage of $\\approx$\\,80\\,km\\,s$^{-1}$, was imaged by \\citet{telietal92}. The integrated OH emission is centred on the star (within 0\\farcs3), and the velocities of the maser spots increase systematically along a position angle (PA) $\\approx$\\,$-60^\\circ$ with the most blue- and red-shifted emission at $\\approx$\\,2$\\arcsec$ to the SE and the NW, respectively, that is, essentially in the opposite direction to the CO outflow. Therefore, \\citet{zijletal01} proposed that the OH maser emission comes from a second bipolar outflow.\n\nSo far, there are 12 circumstellar molecular species (not counting isotopologues) detected towards HD\\,101584, Table~\\ref{t:species} \\citep[][and this paper]{olofetal17}. This is based on data covering only a 13\\,GHz spectral range in ALMA band 6, complemented with targeted detections of HCN and HCO$^+$ using APEX. The different molecular line emissions sample different regions of the circumstellar medium depending on chemistry and excitation. \\citet{olofetal17} found the extreme-velocity spots of the high-velocity flow to be particularly rich in various species, and presented the first detection of methanol in an AGB-related object. In terms of detected species and their relative abundances they resemble the chemistry found in the so-called ``bullet-regions'' of bipolar outflows associated with young stellar objects (YSOs) \\citep{tafabach11}. The chemistry of the circumstellar environment of HD\\,101584 will be discussed in a forthcoming paper.\n\nIn summary, the circumstellar environment of HD\\,101584 consists of a central component and an orthogonal, bipolar, molecular outflow. OH emission, abundant oxygen-bearing molecules, and a 10$\\,\\mu$m feature indicate an O-rich (C\/O$<$1) circumstellar medium, that is, consistent with the chemistry of the primary star \\citep{sivaetal99}. \n\n\\begin{table}\n\\caption{Molecular species detected in the circumstellar environment of HD\\,101584 (not counting isotopologues), divided into the different components introduced in Sect.~\\ref{s:decomp}.}\n\\centering\n\\begin{tabular}{l l}\n\\hline\nCCS: CO, CS, SiO, SiS, SO, SO$_2$, OCS, H$_2$S \\\\\nEDE: CO, CS, SiO, SO, SO$_2$, H$_2$S \\\\\nHGS: CO \\\\\nHVO: CO, CS, SiO, SO, OCS, HCN, HCO$^+$, H$_2$CO, CH$_3$OH \\\\\n\\hline\n\\end{tabular}\n\\label{t:species}\n\\end{table}\n\n\\subsection{The scenario}\n\\label{s:hd_scenario}\n\nBased on the spectacular circumstellar characteristics of HD\\,101584, the following scenario for the evolution of the object has emerged \\citep{olofetal15}. The companion (of low mass and in a relatively close orbit) was eventually captured a few hundred years ago, for example, when the red giant star reached a critical size. It spiralled in towards the red giant, but stopped before it fell into the core of the primary. In this process, the outer parts of the red giant was ejected and most of the material formed an equatorial density enhancement in the plane of the binary system. The cease of the inward motion of the companion was likely connected to this. A smaller fraction of the circumstellar mass is now seen in the form of a high-velocity, bipolar outflow. During this CE evolution, the red giant evolution of HD\\,101584 was terminated and its core is becoming gradually revealed. HD\\,101584 may serve as an example where one version of the CE scenario can be studied observationally in some detail. \n\n\n\\subsection{The distance}\n\\label{s:hd_distance}\n\nBased on the identification of HD\\,101584 as a young post-AGB object and assuming a spherical dust envelope providing significant extinction, \\citet{bakketal96a} estimated the distance of HD\\,101584 to be about 0.7\\,kpc. However, our ALMA data rather suggest that the dust is located in a thick disk seen almost face-on, hence providing much less extinction along the line of sight, see Sect.~\\ref{s:sed_results}. As a consequence, for a given luminosity the star must be placed at a larger distance. This will lead to some problems with the post-AGB interpretation as discussed below, but opens up the possibility that HD\\,101584 is instead a post-RGB object of lower luminosity. We will therefore investigate two cases, a 500\\,$L_\\odot$ (the post-RGB case) and a 5000\\,$L_\\odot$ (the post-AGB case) star. The corresponding distances, taking into account the circumstellar extinction we estimate in Sect.~\\ref{s:sed}, are 0.56\\,kpc and 1.8\\,kpc, respectively.\n\nThe recent Gaia release-2 data suggest a distance of 2.0\\,(+0.19,--0.16)\\,kpc \\citep{gaiaetal18}. However, there are reasons why the Gaia result may not be correct in this particular case. First, the star is bright and estimated Gaia parallaxes for 7$^{\\rm m}$ stars are expected to be less reliable. Second, the estimated size of the orbit is of the same magnitude as the parallax. Third, even though the uncertainty of the result is formally small (parallax equals 0.48\\,$\\pm$\\,0.04), the goodness of fit (13.6) and the chi-square (741) values are very large. \n\nTherefore, we regard the Gaia estimate sufficiently uncertain to warrant giving all the distance-dependent results as their values at 1\\,kpc and the scaling of these values with distance in this paper. We will discuss the consequences of the uncertain distance for the evolutionary status of HD\\,101584 in Sect.~\\ref{s:evol_status}. Note that some quantities are constant, irrespective of the distance, for example, $L_\\ast\/D^2$ and $R_\\ast\/D$, where $D$ is the distance, and $L_\\ast$ and $R_\\ast$ the stellar luminosity and radius, respectively.\n\\section{Observations}\n\\label{s:obs_desc}\n\\subsection{ALMA}\n\nThe ALMA data were obtained during cycles~1 (May 2014, TA1) and 3 (October 2015, TA2; September 2016, TA3) with 35 to 39 antennas of the 12\\,m main array in two frequency settings in band 6, one for the $^{12}$CO($J$\\,=\\,\\mbox{2--1}) line (both cycles) and one for the $^{13}$CO($J$\\,=\\,\\mbox{2--1}) line (only cycle~1). In both settings, the data set contains four 1.875\\,GHz spectral windows with 3840 channels each. The baselines range from 13 to 16196\\,m. This means a highest angular resolution of 0\\farcs025, and a maximum recoverable scale of $\\approx$\\,8\\arcsec . Bandpass calibration was performed on J1107-4449, and gain calibration on J1131-5818 (TA1) and J1132-5606 (TA2 and TA3). Flux calibration was done using Ceres and Titan (TA1), J1131-5818 (TA2), and J1150-5416 (TA3). Based on the calibrator fluxes, we estimate the absolute flux calibration to be accurate to within 5\\%. However, the uncertainties in the reported flux densities are significantly larger than this. This is due to a combination of uncertainties introduced in the cleaning process and the difficulty in discriminating emission from the different components identified in the circumstellar medium of HD\\,101584. For this reason we do not report any formal error estimates since they would not reflect the real uncertainties that we estimate are at least of the order 20\\,\\%.\n\nThe data were reduced using various versions of CASA over the years, the last one being 4.7.3. After corrections for the time and frequency dependence of the system temperatures, and rapid atmospheric variations at each antenna using water vapour radiometer data, bandpass and gain calibration were done. For the $^{12}$CO($J$\\,=\\,\\mbox{2--1}) setting, data obtained in three different configurations were combined. Subsequently, for each individual tuning, self-calibration was performed on the strong continuum. Imaging was done using the CASA clean algorithm after a continuum subtraction was performed on the emission line data. The final line images were created using Briggs robust weighting. This resulted in close to circular beam sizes of about 0\\farcs65$\\times$0\\farcs55 (8$^\\circ$) and 0\\farcs09$\\times$0\\farcs08 (12$^\\circ$) for the cycle~1 and combined cycles~1 and 3 data, respectively. A beamsize of 0\\farcs15$\\times$0\\farcs14 (10$^\\circ$) is used for the presented continuum data. Typical channel rms noises are $\\approx$\\,2\\,mJy\\,beam$^{-1}$ and $\\approx$\\,0.7\\,mJy\\,beam$^{-1}$ for the cycle~1 and combined cycles~1 and 3 data at 1.5\\,km\\,s$^{-1}$ resolution, respectively.\n\\subsection{APEX}\n\nComplementary molecular line and continuum data on HD\\,101584 were obtained using APEX \\citep{gustetal06}. The Swedish heterodyne facility instruments SHeFI \\citep[A1,A2,A3;][]{vassetal08} and SEPIA \\citep[B5,B9;][]{belietal18} were used together with the facility FFT spectrometer covering about 4\\,GHz. The observations were made from August 2015 to August 2017 in dual-beamswitch mode with a beam throw of 2$\\arcmin$. In May 2018 observations with the PI230 receiver and a 16\\,GHz FFT spectrometer were used. Regular pointing checks were made on strong CO line emitters and continuum sources. Typically, the pointing was found to be consistent with the pointing model within 3$\\arcsec$. The antenna temperature, $T_{\\mathrm A}^{\\star}$, is corrected for atmospheric attenuation. The uncertainty in the absolute intensity scale is estimated to be about $\\pm 20$\\%. APEX telescope characteristics [beam width ($\\theta_{\\rm b}$), main beam efficiency ($\\eta_{\\rm mb}$), and Jy to K conversion] at representative observing frequencies are given in Table~\\ref{t:apex}. Low-order polynomial baselines were subtracted from the spectra.\n\nFinally, we have used the ArTeMiS bolometer camera to measure the 350\\,$\\mu$m flux of HD\\,101584. ArTeMiS is an ESO PI sub-mm camera arranged in 16$\\times$18 sub-arrays operating at 200, 350, and 450\\,$\\mu$m \\citep{reveetal14}. We observed for 3.5 hours on 26 Nov. 2016 at 350\\,$\\mu$m in spiral-raster-mapping mode under good weather conditions with precipitable water vapour in the range 0.4--0.6\\,mm. The resulting image has an angular resolution of 8\\arcsec , thus covering well the dust continuum emission region of HD\\,101584. The data were reduced using the ArTeMiS data reduction package provided by the ArTeMis team. G305.80--0.24 (aka B13134) was observed as a flux calibrator, and the uncertainty of the flux calibration is estimated to be 30\\%.\n\n\\begin{table}\n\\caption{APEX characteristics at representative observational frequencies.}\n\\centering\n\\begin{tabular}{l c l l}\n\\hline \\hline\nFrequency & $\\theta_{\\rm b}$ & $\\eta_{\\rm mb}$ & Jy\/K \\\\\n$[$GHz] & [\\arcsec ] \\\\\n\\hline \n170 & 37 & 0.75 & 38 \\\\\n185 & 34 & 0.75 & 38 \\\\\n200 & 31 & 0.75 & 38 \\\\\n220 & 28 & 0.75 & 39 \\\\\n260 & 24 & 0.75 & 39 \\\\\n300 & 21 & 0.74 & 40 \\\\\n345 & 18 & 0.73 & 41 \\\\\n460 & 14 & 0.60 & 48 \\\\\n690 & \\phantom{0}9 & 0.46 & 63 \\\\\n\\hline\n\\end{tabular}\n\\label{t:apex}\n\\end{table}\n\n\\begin{table*}\n\\caption{Molecular lines observed towards HD\\,101584.}\n\\centering\n\\begin{tabular}{l l c c l}\n\\hline \\hline\nMolecule & Line & Freq. & $E_{\\rm u}$\\,$^1$ & Telescope \\\\\n & & [GHz] & [K] & \\\\\n\\hline \nCO & $J\\,=\\,2-1$ & 230.538 & \\phantom{0}17 & ALMA\\\\\n & $J\\,=\\,3-2$ & 345.796 & \\phantom{0}33 & APEX\\\\\n & $J\\,=\\,4-3$ & 461.041 & \\phantom{0}55 & APEX \\\\\n & $J\\,=\\,6-5$ & 691.473 & 116 & APEX \\\\\n$^{13}$CO & $J\\,=\\,2-1$ & 220.399 & \\phantom{0}16 & ALMA, APEX \\\\\n & $J\\,=\\,3-2$ & 330.588 & \\phantom{0}32 & APEX \\\\\nC$^{17}$O & $J\\,=\\,2-1$ & 224.714 & \\phantom{0}16 & APEX \\\\\nC$^{18}$O & $J\\,=\\,2-1$ & 219.560 & \\phantom{0}16 & ALMA, APEX \\\\\n$^{13}$C$^{17}$O & $J\\,=\\,2-1$ & 214.574 & \\phantom{0}15 & ALMA \\\\\nSiO & $J\\,=\\,5-4$ & 217.105 & \\phantom{0}31 & ALMA \\\\\n$^{29}$SiO & $J\\,=\\,5-4$ & 214.386 & \\phantom{0}31 & ALMA \\\\\nSiS & $J\\,=\\,12-11$ & 217.818 & \\phantom{0}68 & ALMA \\\\\n & $J\\,=\\,13-12$ & 235.961 & \\phantom{0}79 & ALMA \\\\\nCS & $J\\,=\\,4-3$ & 195.954 & \\phantom{0}24 & APEX \\\\\n$^{13}$CS & $J\\,=\\,5-4$ & 231.221 & \\phantom{0}33 & ALMA \\\\\nSO & $N_J\\,=\\,5_5-4_4$ & 215.221 & \\phantom{0}44 & ALMA \\\\\n & $N_J\\,=\\,5_6-4_5$ & 219.949 & \\phantom{0}35 & ALMA \\\\\n & $N_J\\,=\\,6_5-5_4$ & 251.826 & \\phantom{0}51 & APEX \\\\\n & $N_J\\,=\\,8_7-7_7$ & 214.357 & \\phantom{0}81 & ALMA \\\\\n$^{33}$SO & $N_J\\,=\\,5_6-4_5$ & 217.831 & \\phantom{0}35 & ALMA \\\\ \n$^{34}$SO & $N_J\\,=\\,5_6-4_5$ & 215.840 & \\phantom{0}34 & ALMA \\\\\nSO$_2$ & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,4_{22}-3_{13}$ & 235.152 & \\phantom{00}9 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,16_{1,15}-15_{2,14}$ & 236.217 & 131 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,16_{3,13}-16_{2,14}$ & 214.689 & 148 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,22_{2,20}-22_{1,21}$ & 216.643 & 249 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,28_{3,25}-28_{2,26}$ & 234.187 & 403 & ALMA \\\\\nOCS & $J\\,=\\,18-17$ & 218.903 & 100 & ALMA \\\\ \n\t\t\t & $J\\,=\\,19-18$ & 231.061 & 111 & ALMA \\\\\nHCN & $J\\,=\\,3-2$ & 265.886 & \\phantom{0}26 & APEX \\\\ \nHCO$^+$ & $J\\,=\\,3-2$ & 267.558 & \\phantom{0}26 & APEX \\\\\n$p$-H$_2$O\\,$^2$ & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,3_{13}-2_{20}$ & 183.313 & 200 & APEX \\\\\n$p$-H$_2$S & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,2_{20}-2_{11}$ & 216.710 & \\phantom{0}84 & ALMA, APEX \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,2_{02}-1_{11}$ & 687.303 & \\phantom{0}55 & APEX \\\\\n$o$-H$_2$S & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,1_{10}-1_{01}$ & 168.763 & \\phantom{0}28 & APEX \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,3_{30}-3_{21}$ & 300.506 & 169 & APEX \\\\\n$p$-H$_2^{33}$S & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,2_{20}-2_{11}$ & 215.503 & \\phantom{0}84 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,2_{02}-1_{11}$ & 687.164 & \\phantom{0}55 & APEX \\\\\n$o$-H$_2^{33}$S & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,1_{10}-1_{01}$ & 168.319 & \\phantom{0}28 & APEX \\\\\n$p$-H$_2^{34}$S & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,2_{20}-2_{11}$ & 214.377 & \\phantom{0}84 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,2_{02}-1_{11}$ & 687.025 & \\phantom{0}55 & APEX \\\\\n$o$-H$_2^{34}$S & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,1_{10}-1_{01}$ & 167.911 & \\phantom{0}28 & APEX \\\\\n$p$-H$_2$CO & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,3_{03}-2_{02}$ & 218.222 & \\phantom{0}21 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,3_{22}-2_{21}$ & 218.476 & \\phantom{0}68 & ALMA \\\\\n & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,3_{21}-2_{20}$ & 218.760 & \\phantom{0}68 & ALMA \\\\\n$o$-H$_2^{13}$CO & $J_{K_{\\rm a},K_{\\rm c}}\\,=\\,3_{12}-2_{11}$ & 218.909 & \\phantom{0}33 & ALMA \\\\\n$E$-CH$_3$OH & $J_K\\,=\\,4_2-3_1$ & 218.440 & \\phantom{0}45 & ALMA \\\\\n & $J_K\\,=\\,8_{-1}-7_0$ & 229.759 & \\phantom{0}89 & ALMA \\\\\n\\hline\n\\end{tabular}\n\\label{t:obs_lines}\n\\tablefoot{(1) Energy of the upper energy level. (2) Only an upper limit is obtained.}\n\\end{table*}\n\\subsection{Observed lines}\n\nThe ALMA data cover the following frequency ranges: 214.02--215.90, 216.53--218.81, 219.13--221.01, 229.57--231.43, 231.63--236.70\\,GHz. In these ranges we have identified 30 lines. Only one line remains unidentified. The APEX data cover selected lines, 20 of them [only upper limit for the H$_2$O(\\mbox{$3_{13}-2_{20}$}) line], chosen to complement the ALMA data. Table~\\ref{t:obs_lines} summarises all the identified lines. They are typical for an oxygen-rich circumstellar chemistry with a significant contribution of sulphur species, but also weaker lines from carbon-species, other than CO, are present \\citep{mill16}. However, the presence of lines from H$_2$CO and CH$_3$OH point also to a non-standard circumstellar chemistry.\n\\subsection{Other data}\n\nIn our analysis, we will also make use of the CO and $^{13}$CO $J$\\,=\\,\\mbox{1--0} and the CO $J$\\,=\\,\\mbox{2--1} data obtained with the Swedish-ESO Submillimetre Telescope (SEST) and published by \\citet{olofnyma99}. In addition, we have constructed a spectral energy distribution (SED) using archive data and our ALMA and APEX data.\n\\subsection{Missing flux in ALMA data}\n\\label{s:alma_flux}\n\nThe amount of missing flux in the ALMA data varies over the line profile as illustrated in Fig.~\\ref{f:co_apex_global}, where we show the $^{13}$CO(\\mbox{2--1}) lines obtained with APEX and ALMA (the latter is integrated over the source). We have chosen this line because it is not as optically thick as that of the main isotopologue, and both the ALMA and APEX observations have a high signal-to-noise ratio. \n\nThe narrow central feature that stands on top of a broader plateau of emission has the same integrated intensity in the velocity range $\\mid$\\,$\\upsilon-\\upsilon_{\\rm sys}$\\,$\\mid$\\,$\\le$\\,10\\,km\\,s$^{-1}$ ($\\upsilon_{\\rm sys}$\\,$\\approx$\\,41.7\\,$\\pm$\\,0.2\\,km\\,s$^{-1}$; all velocities in this paper are with respect to the local standard of rest) in the APEX and ALMA data (with the plateau emission subtracted, and within the combined uncertainties). The ALMA to APEX line intensity ratio is 1.0$\\pm$0.15. This indicates that the relative calibration between the data sets is as good as can be expected. At the extreme velocities, 130\\,$\\le$\\,$\\mid$\\,$\\upsilon-\\upsilon_{\\rm sys}$\\,$\\mid$\\,$\\le$\\,150\\,km\\,s$^{-1}$ on either side of the systemic velocity, the amount of lost flux in the ALMA data is $\\approx$\\,30\\,\\%. Most of the flux in the ALMA data is lost at intermediate velocities. About 45\\,\\% is lost in the velocity ranges 20\\,$\\le$\\,$\\mid$\\,$\\upsilon-\\upsilon_{\\rm sys}$\\,$\\mid$\\,$\\le$\\,130\\,km\\,s$^{-1}$ on either side of the systemic velocity. In terms of total flux in the $^{13}$CO(\\mbox{2--1}) line, integrated over the full velocity range, the amount of flux lost in the ALMA data is $\\approx$\\,40\\,\\%. [Note, the beams of the APEX and ALMA antennae are the same so there is no need for a primary beam correction in this comparison.]\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=8cm]{34897_fig_1}\n \\caption{$^{13}$CO(\\mbox{2--1}) spectra of HD\\,101584 at 1.5\\,km\\,s$^{-1}$ resolution obtained with APEX (black histogram) and ALMA (red histogram). The ALMA spectrum is integrated over the whole source. \n }\n \\label{f:co_apex_global}\n\\end{figure} \n\n\n \\begin{figure*}\n \\includegraphics[width=\\textwidth]{34897_fig_2}\n \\caption{Single-dish CO isotopologue spectra of HD\\,101584 at 2\\,km\\,s$^{-1}$ resolution. {\\bf Left:} CO \\mbox{1--0}, \\mbox{2--1}, \\mbox{3--2}, \\mbox{4--3}, and \\mbox{6-5} spectra from top to bottom. {\\bf Middle left:} $^{13}$CO \\mbox{1--0}, \\mbox{2--1}, and \\mbox{3--2} spectra from top to bottom. {\\bf Middle right:} C$^{17}$O \\mbox{2--1} spectrum. {\\bf Right:} C$^{18}$O \\mbox{2--1} spectrum. We note that the emission at the velocity extremes in the CO \\mbox{4--3} and \\mbox{6--5} lines my be suppressed by up to $\\approx$\\,20\\,\\% and 50\\,\\%, respectively, due to the relative sizes of the source and the beam (the beam is always pointed towards the centre of the source).}\n \\label{f:co_spec}\n \\end{figure*} \n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=9cm]{34897_fig_3_left} \\hspace{5mm} \\includegraphics[width=7.9cm]{34897_fig_3_right}\n \\caption{{\\bf Left:} CO(\\mbox{2--1}) maximum-intensity image at 0\\farcs085 resolution. {\\bf Right:} 1.3\\,mm continuum image (red contours starting at 0.2\\,mJy\\,beam$^{-1}$ and with a spacing of 0.3\\,mJy\\,beam$^{-1}$) at 0\\farcs15 resolution overlaid an F606W HST image from \\citet{sahaetal07}. The latter has been shifted so that the diffraction cross coincides with the continuum peak. The CO line intensity peak at the centre coincides with the continuum peak within an uncertainty of about 0\\farcs01.}\n \\label{f:morphology}\n \\end{figure*} \n\\section{The circumstellar morphology and kinematics}\n\nThe complexity of the circumstellar environment of HD\\,101584 is evident already in the single-dish rotational-line data of the CO isotopologues. Figure~\\ref{f:co_spec} summarises data obtained with SEST and APEX. The differences in line shape can be attributed to the differences in optical depth of the lines and to some extent to the excitation (for example, the CO(\\mbox{6--5}) line clearly samples warmer gas), and to the existence of a number of components with different kinematics and physical conditions. The main CO isotopologue line emission is dominated by an $\\approx$\\,100\\,km\\,s$^{-1}$ broad component centred at $\\approx$\\,40\\,km\\,s$^{-1}$. This component tapers gradually into two distinct features at the extreme velocities, $\\approx$\\,140\\,km\\,s$^{-1}$ offset on either side of the centre (particularly prominent in the CO and $^{13}$CO \\mbox{2--1} line data). These features resemble the ``bullet'' emissions seen towards many high-velocity outflows of YSOs \\citep{bachetal91a, bachetal91b}. The total emission covers $\\approx$\\,310\\,km\\,s$^{-1}$. On the contrary, the line emissions from the rarer CO isotopologues are dominated by a narrow central feature, for example, the C$^{18}$O(\\mbox{2--1}) line is well fitted by a Gaussian having a full width at half maximum (FWHM) of 6.5\\,km\\,s$^{-1}$ centred at 41.5\\,km\\,s$^{-1}$. As judged by the CO isotopologue line intensity ratios, this feature has the highest optical depth in the CO lines. \n\\subsection{The overall morphology}\n\nThe ALMA data adds considerable morphological information. An overview is presented in Fig.~\\ref{f:morphology} that shows the circumstellar molecular gas, as traced by the CO(\\mbox{2--1}) maximum line intensity, and dust, as traced by the 1.3\\,mm continuum intensity, distributions around HD\\,101584. Although, these are not simply interpreted in the forms of gas and dust density distributions they provide substantially more information than the visual image in scattered light presented in Fig.~\\ref{f:morphology}, and serve as a base for our description of the circumstellar medium around HD\\,101584. The difference in morphology between the line and continuum maps is primarily due to the much lower optical depths in the extended emission. Adding kinematical information allows a decomposition into separate components as described below.\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=17cm]{34897_fig_4}\n \\caption{{\\bf Left panel:} Sketch of the morphology of the circumstellar medium of HD\\,101584 (not drawn to scale). The full extent and the identified morphological components: central compact source (CCS), equatorial density enhancement (EDE), hourglass structure (HGS) which forms the inner part of two diametrically orientated bubbles, and bipolar, high-velocity outflow (HVO) with extreme-velocity spots (EVSs), as well as a blow-up of the central region highlighting the CCS, EDE, and inner part of the HGS, are indicated. An hourglass structure suggesting a tentative second bipolar outflow, with a velocity gradient opposite to that of the HVO, is also shown. {\\bf Right panel:} CO(\\mbox{2--1}) PV-diagram along PA\\,=\\,90$^\\circ$ at resolutions of 0\\farcs085 and 1.5\\,km\\,s$^{-1}$ with the different components indicated. The flux scale is in mJy\\,beam$^{-1}$.}\n \\label{f:sketch_co21}\n \\end{figure*} \n\n\\subsection{Decomposition into different circumstellar components}\n\\label{s:decomp}\n\nThe combined morphological and kinematical information in the molecular line images make it possible to identify a number of distinct components in the circumstellar medium. The different components are exemplified below through selected molecular-line channel maps and position-velocity (PV) diagrams, specifically chosen because they highlight the different components that we will discuss and outline the reasons for their interpretations. We have identified the following components: \n\\begin{itemize}\n\\item\nCCS: A central compact source within a radius of $\\approx$\\,0\\farcs1 of the centre, Sect.~\\ref{s:obs_ccs}.\n\\item \nEDE: An equatorial density enhancement\\footnote{The term `equatorial' is used here to reflect that we believe that this is a flattened density distribution, for example, a disk or a torus, that lies at the waist of the bipolar outflow and in the plane of the binary orbit.} of diameter $\\approx$\\,3\\arcsec\\ and centred on the CCS, Sect.~\\ref{s:obs_ede}.\n\\item \nHVO: A bipolar, high-velocity outflow at PA\\,$\\approx$\\,90$^\\circ$, that is terminated in two extreme-velocity spots (EVSs) at $\\approx$\\,4\\arcsec\\ on each side of the CCS, Sects~\\ref{s:obs_hvo} and \\ref{s:obs_evs}.\n\\item \nHGS: An hourglass structure surrounding the initial $\\approx$\\,2\\arcsec\\ of the HVO, that develops into bubbles that close at the EVSs, Sect.~\\ref{s:obs_hgs}.\n\\end{itemize}\nA sketch of the proposed source structure and the nomenclature used in the remaining part of the paper is shown in Fig.~\\ref{f:sketch_co21} (left panel), and the different components are also indicated in the CO(\\mbox{2--1}) PV-diagram obtained along the major axis of the outflow (PA\\,=\\,90$^{\\circ}$), Fig.~\\ref{f:sketch_co21} (right panel).\n\nThe source structure, as well as how different lines probe different components, is further illustrated through six PV-diagrams of the CO(\\mbox{2--1}), SiO(\\mbox{5--4}), and $p$-H$_2$S(\\mbox{$2_{20}-2_{11}$}) line emissions that cut through the circumstellar medium of HD\\,101584. Figure~\\ref{f:pv_ra_de} (upper panel) shows the morphology in the (R.A.,$\\upsilon_{\\rm z}$)\\,-\\,plane at three different declinations (2\\farcs5\\,S, mid plane, and 2\\farcs5\\,N). There are several noteworthy features here. In the mid plane the HVO stretches along the line of sight in the CO(\\mbox{2--1}) and SiO(\\mbox{5--4}) lines, and it is clearly seen that the bright spots in the SiO line emission coincide with those of the CO line emission on either side of the source centre. The HGS is evident in the CO data, as well as the fact that it is the inner part of a bipolar bubble-like structure that closes at the EVSs. The $p$-H$_2$S line emission is confined to the region where the HGS closes towards the centre, interpreted by us as the EDE component. The CCS is seen in all three lines (it is particularly prominent in higher-excitation SO$_2$ lines as illustrated in Fig.~\\ref{f:so2_channels}). The PV-diagrams N and S of the mid plane shows a bipolar bubble-like structure that is inclined with respect to the mid plane and has a velocity gradient opposite to that of the HVO. It will be further discussed in Sect.~\\ref{s:multi_polar} in terms of a second bipolar outflow. In the lower panel of Fig.~\\ref{f:pv_ra_de} we see the morphology as seen in the (Decl.,$\\upsilon_{\\rm z}$)\\,-\\,plane at three different right ascensions [2\\farcs5\\,E, mid plane (vertical to the mid plane of the upper panel), and 2\\farcs5\\,W]. This shows once again the HGS component developing into bubbles that close at the EVSs, and the presence of a second bipolar bubble structure with a reversed velocity gradient, in the CO line emission. The CO(\\mbox{2--1})and SiO(\\mbox{5--4}) channel maps are presented in Figs~\\ref{f:co_channels} and \\ref{f:sio_channels}.\n\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=15cm]{34897_fig_5}\n \\caption{{\\bf Upper panel:} PV-diagrams in the right ascension direction as seen in the CO(\\mbox{2--1}) (colour), SiO(\\mbox{5--4}) (black contours), and $p$-H$_2$S(\\mbox{$2_{20}-2_{11}$}) (red contours) lines at declination offsets of 2\\farcs5\\,S (left), mid plane (middle), and 2\\farcs5\\,N (right). {\\bf Lower panel:} The same in the declination direction at right ascension offsets of 2\\farcs5\\,E (top), mid plane (middle), and 2\\farcs5\\,W (bottom). The flux scale is in mJy\\,beam$^{-1}$, and the contours start at 2\\,mJy\\,beam$^{-1}$ with a spacing of 2\\,mJy\\,beam$^{-1}$.}\n \\label{f:pv_ra_de}\n \\end{figure*} \n\n\\subsection{Continuum emission}\n\\label{s:obs_continuum}\n\nThe CCS and EDE components are particularly prominent in the 1.3\\,mm continuum, but there is also weak emission from parts of the HGS component, and extended diffuse emission that contributes significantly to the total flux, Fig.~\\ref{f:morphology}. The HVO component is not present in the continuum image. On the other hand, there is a low-surface-brightness region about 2\\farcs5\\,N of the centre that has no counterpart in the molecular gas. \n\nThe central peak of the 1.3\\,mm continuum has the coordinates $\\alpha$(2000) = 11$^{\\rm h}$40$^{\\rm m}$58\\fs7908 and $\\delta$(2000) = --55$^\\circ$34'25\\farcs802 (with an error of 0\\farcs004 in both directions). Within the uncertainties of both measurements this agrees with the Gaia position for HD101584 $\\alpha$(2000) = 11$^{\\rm h}$40$^{\\rm m}$58\\fs8052 and $\\delta$(2000) = --55$^\\circ$34'25\\farcs813. Thus, we draw the reasonable conclusion that the continuum peaks at the position of HD\\,101584. All position offsets in this paper refers to the continuum peak position.\n\nWe estimate that the 1.3\\,mm continuum fluxes are 12\\,mJy in the CCS (0\\farcs3 aperture), 120\\,mJy in the EDE (3\\arcsec\\ aperture, but excluding the CCS), and 70\\,mJy outside the EDE (but only covering the region where structures in the emission are clearly seen). The total flux density is therefore 202\\,mJy. However, there is substantial extended low-brightness emission for which the reliability is uncertain, for example, within a 10\\arcsec\\ aperture the total flux density is 245\\,mJy. In addition, it is possible that flux is missing in our ALMA 1.3\\,mm data due to extended emission. The ArTeMiS observations do not resolve the emission, and we can only report a total flux density of 9\\,Jy (350\\,$\\mu$m, 8\\arcsec\\ beam). The flux estimates are summarised in Table~\\ref{t:obs_cont}.\n\nThe 1.3\\,mm continuum image is overlayed the F606W HST image in Fig.~\\ref{f:morphology} assuming that the diffraction cross in the latter is the position of HD\\,101584, which coincides with the position of the continuum peak (within the errors of the position estimates). Some similarities are noticeable, and the predominance of scattered light to the west is naturally explained by the fact that this is the side of the HVO facing towards us as shown by the molecular line data. The eastern side is exposed to higher circumstellar extinction.\n\\begin{table}\n\\caption{Continuum measurements}\n\\centering\n\\begin{tabular}{c l c c}\n\\hline \\hline\nWavelength & Instrument & Aperture & $S$ \\\\\n$[$$\\mu$m] & & [\\arcsec ] & [Jy] \\\\\n\\hline \n\\phantom{1}350 & ArTeMiS & \\phantom{0}8\\phantom{.00} & \\phantom{10}9\\phantom{.000}\\\\\n1300 & ALMA & total\\,$^1$ & \\phantom{10}0.20\\phantom{0}\\\\\n1300 & ALMA & \\phantom{0}3\\phantom{.00} & \\phantom{10}0.12\\phantom{0} \\\\\n1300 & ALMA & \\phantom{0}0.3\\phantom{0} & \\phantom{10}0.012\\\\\n1300 & ALMA & \\phantom{0}0.05 & \\phantom{10}0.007\\\\\n\\hline\n\\end{tabular}\n\\label{t:obs_cont}\n\\tablefoot{(1) Inside the region where structure is seen in the continuum, that is, excluding extended very-low-brightness emission.}\n\\end{table}\n\\subsection{The central compact source, CCS}\n\\label{s:obs_ccs}\n\nThe CCS component is very compact in the 1.3\\,mm continuum. About 60\\,\\% of the flux density within an aperture of 0\\farcs3 comes from a region which is not resolved even when putting larger weight on the longest baselines: 7.0\\,mJy resides in a 0\\farcs027$\\times$0\\farcs026 (PA\\,=\\,100$^\\circ$) source (FWHM of a Gaussian 2D fit) when observed with a 0\\farcs025 beam. This means a Gaussian source size of $\\la$\\,0\\farcs01 ($\\la$\\,10\\,[$D$\/1\\,kpc]\\,au), meaning that this component is circumbinary in nature (the binary separation being of the order 1\\,mas). This is close in size to the mid-IR source studied by \\citet{hilletal17}. At 10.7\\,$\\mu$m they measure a source size of 0\\farcs028 (assuming a disk of uniform brightness) with a brightness temperature of $\\approx$\\,650\\,K using VLTI\/MIDI.\n\nThe CCS is seen in most of the molecular line emissions with emission peaks that coincide with the continuum peak. The results are summarised in Table~\\ref{t:obs_ccs}. It is particularly dominating in the emission of the higher-energy SO$_2$ lines that originate only from this component. As an example the channel maps of the SO$_2$(\\mbox{16$_{3,13}$--16$_{2,14}$}) line, that has an upper energy level at 148\\,K, is shown in Fig.~\\ref{f:so2_channels}. All the detected line emissions are resolved with deconvolved source sizes (FWHMs of 2D Gaussian fits) of $\\approx$0\\farcs15 ($\\approx$\\,150\\,[$D$\/1\\,kpc]\\,au). This is substantially larger than the compact continuum emission which probably arises in the inner, warmer region of the CCS. Notable exceptions are the SiO and $^{29}$SiO \\mbox{5--4} line emissions that are significantly smaller (still larger than the continuum source), perhaps indicating that they probe preferentially a warmer region closer to the centre. \n\nExcept for the CO, $^{13}$CO, SiO, and $^{29}$SiO lines, all lines are narrow, the average FWHM\\,=\\,3.0\\,km\\,s$^{-1}$, corresponds to a deconvolved FWHM\\,=\\,2.6\\,km\\,s$^{-1}$ at a spectral resolution of 1.5\\,km\\,s$^{-1}$. The SO$_2$(\\mbox{$16_{3,13}-16_{2,14}$}) line is shown as an example in Fig.~\\ref{f:so2_spec_ccs}. It is not clear why the CO and SiO lines are significantly broader, but confusion with emission along the line of sight could be part of an explanation (certainly in the case of CO).\n\nThe $^{13}$C$^{17}$O(\\mbox{2--1}), $^{13}$CS(\\mbox{5--4}), and SiS(\\mbox{12--11}, \\mbox{13--12}) lines all have the velocity characteristics of the CCS, but they are not peaked at the centre. Instead they show a patchy and extended structure over $\\approx$\\,0\\farcs5. This may be an effect of these lines being among the weaker ones, S\/N\\,$\\approx$\\,5 integrated over the area. \n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{34897_fig_6}\n \\caption{SO$_2$(\\mbox{16$_{3,13}$--16$_{2,14}$}) channel maps with a width and spacing of 1.5\\,km\\,s$^{-1}$ at a resolution of 0\\farcs085 (the beam is shown in the lower left corner of each panel). The flux scale is in mJy\\,beam$^{-1}$. Narrow line emission from the CCS dominates here.}\n \\label{f:so2_channels}\n \\end{figure} \n \n \\begin{figure}\n \\centering\n \\includegraphics[width=7cm]{34897_fig_7}\n \\caption{SO$_2$(\\mbox{16$_{3,13}$--16$_{2,14}$}) line, obtained with ALMA within an aperture of 0\\farcs3 centred on the continuum peak, at a resolution of 1.5\\,km\\,s$^{-1}$. This line is characteristic of the narrow-line-width emission from the CCS component.}\n \\label{f:so2_spec_ccs}\n \\end{figure} \n\n\\begin{table*}\n\\caption{Molecular line emission from the CCS component\\,$^1$.}\n\\centering\n\\begin{tabular}{l c c c c c }\n\\hline \\hline\nLine & Aperture\\,$^2$ & $S$ & \\phantom{0}$\\Delta \\upsilon$\\,$^3$ & $I$ & Size\\,$^4$ \\\\\n & [\\arcsec ] & [Jy] & [km\\,s$^{-1}$] & [Jy\\,km\\,s$^{-1}$] & \\\\\n\\hline \nCO(2--1) & 0.2 & 0.15\\phantom{0} & 18.7 & 3.0\\phantom{00} & 0\\farcs15 \\\\\n$^{13}$CO(2--1) & 0.6 & 0.35\\phantom{0} & \\phantom{1}6.2 & 2.3\\phantom{00} & \\ldots \\\\\nC$^{18}$O(2--1) & 0.6 & 0.20\\phantom{0} & \\phantom{1}2.9 & 0.61\\phantom{0} & \\ldots \\\\\n$^{13}$C$^{17}$O(2--1)\\,$^5$ & 0.6 & 0.015 & \\phantom{1}3.7 & 0.059 \\\\\nSiO(5--4) & 0.2 & 0.046 & 11.9 & 0.58\\phantom{0} & 0\\farcs10 \\\\\n$^{29}$SiO(5--4) & 0.2 & 0.013 & 10.0 & 0.14\\phantom{0} & 0\\farcs11\\\\\nSiS(12--11)\\,$^5$ & 0.6 & 0.011 & \\phantom{1}3.3 & 0.038 & \\ldots \\\\\nSiS(13--12)\\,$^5$ & 0.6 & 0.010 & \\phantom{1}4.2 & 0.045 & \\ldots \\\\\n$^{13}$CS(5--4)\\,$^5$ & 0.6 & 0.027 & \\phantom{1}2.5 & 0.072 & \\ldots \\\\\nSO($5_5-4_4$) & 0.2 & 0.073 & \\phantom{1}4.4 & 0.34\\phantom{0} & 0\\farcs15\\\\\nSO($5_6-4_5$) & 0.6 & 0.10\\phantom{0} & \\phantom{1}3.7 & 0.41\\phantom{0} & \\ldots \\\\\nSO($6_5-5_4$)\\,$^6$ & 25\\phantom{000} & 0.22\\phantom{0} & \\phantom{1}5.1 & 1.2\\phantom{00} & \\ldots \\\\\nSO($8_7-7_7$) & 0.2 & 0.011 & \\phantom{1}2.5 & 0.029 & \\ldots \\\\\n$^{33}$SO(5$_6$--4$_5$)\\,$^7$ & 0.6 & \\ldots & \\ldots & 0.019 & \\ldots \\\\\n$^{34}$SO(5$_6$--4$_5$) & 0.2 & 0.025 & \\phantom{1}2.7 & 0.072 & 0\\farcs19\\\\\nSO$_2$(4$_{22}$--3$_{13}$) & 0.6 & 0.043 & \\phantom{1}2.3 & 0.11\\phantom{0} & \\ldots \\\\\nSO$_2$(16$_{1,15}$--15$_{2,14}$) & 0.6 & 0.043 & \\phantom{1}2.2 & 0.10\\phantom{0} & \\ldots \\\\\nSO$_2$(16$_{3,13}$--16$_{2,14}$) & 0.2 & 0.025 & \\phantom{1}2.6 & 0.069 & 0\\farcs17 \\\\\nSO$_2$(22$_{2,20}$--22$_{1,21}$) & 0.2 & 0.015 & \\phantom{1}3.2 & 0.051 & 0\\farcs15\\\\\nSO$_2$(28$_{3,25}$--28$_{2,26}$) & 0.6 & 0.019 & \\phantom{1}1.8 & 0.036 & \\ldots \\\\\nOCS(19--18) & 0.2 & 0.023 & \\phantom{1}3.2 & 0.078 & 0\\farcs14\\\\\n$p$-H$_2$S(2$_{20}$--2$_{11}$) & 0.2 & 0.084 & \\phantom{1}3.6 & 0.32\\phantom{0} & 0\\farcs15 \\\\\n$p$-H$_2^{33}$S(2$_{20}$--2$_{11}$)\\,$^7$ & 0.2 & \\ldots & \\ldots & 0.15\\phantom{0} & \\ldots \\\\\n$p$-H$_2^{34}$S(2$_{20}$--2$_{11}$) & 0.2 & 0.060 & \\phantom{1}2.9 & 0.18\\phantom{0} & 0\\farcs26\\\\\n\\hline\n\\end{tabular}\n\\label{t:obs_ccs}\n\\tablefoot{(1) See Sect.~\\ref{s:obs_desc} for a discussion of the flux uncertainties. (2) The choice of aperture reflects the angular resolution of the ALMA data: the resolutions are $\\approx$\\,0\\farcs085 and $\\approx$\\,0\\farcs55 at 0\\farcs2 and 0\\farcs6 aperture, respectively. The apertures are centred on the continuum peak. (3) FWHM of Gaussian fit to the line within the given aperture at 1.5\\,km\\,s$^{-1}$ resolution. (4) Mean of the deconvolved FWHMs of a 2D Gaussian fit to the high-angular-resolution ALMA data. (5) These emissions are patchy and extended over a region of $\\approx$\\,0\\farcs5. (6) This is based on APEX data and the split into emission from the CCS and EDE components obtained using Gaussian decomposition is uncertain. (7) The integrated intensity of the sum of the hyperfine components.}\n\\end{table*}\n\n\\subsection{The equatorial density enhancement, EDE}\n\\label{s:obs_ede}\n\nThe EDE lies at the waist of the HGS and the HVO. It is the only component of the circumstellar medium around HD\\,101584 that is not particularly prominent in the CO(\\mbox{2--1}) line, nor in any of the other CO isotopologue lines. This is most likely due to contamination by emission from the HGS, a component only seen in the CO lines (see below). On the contrary, the EDE component is particularly prominent in the $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) line emission, and we will infer most of its characteristics using the emission of this line. In fact, the global $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) line is third in peak strength, only the global CO and $^{13}$CO \\mbox{2--1} lines are stronger in our ALMA data. The EDE component is not easily seen in the map data of any of the other line emissions. We have therefore identified the line emission from this component using line profiles obtained within a central 3\\arcsec\\ aperture. Three different types of line profiles can be identified, as exemplified in Fig.~\\ref{f:ede_spectra}. The $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}), $^{13}$CS(\\mbox{5--4}) and SiO(\\mbox{5--4}) lines are close to triangular with well-defined full widths at zero power (FWZP) of 15--20\\,km\\,s$^{-1}$. The SO(\\mbox{$5_6-5_4$}) and SO$_2$(\\mbox{$4_{22}-3_{13}$}) lines show a narrow feature (largely due to the CCS emission) centred on a plateau with well-defined FWZPs of $\\approx$\\,40\\,km\\,s$^{-1}$. Finally, the C$^{18}$O(\\mbox{2--1}) line is relatively narrow at the centre but have extended wings with no well-defined FWZP. The results for the different line emissions are summarised in Table~\\ref{t:obs_ede}. \n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=14cm]{34897_fig_8}\n \\caption{Three types of line profiles seen towards the EDE. All spectra are obtained with ALMA within an aperture of 3\\farcs0 and with 1.5\\,km\\,s$^{-1}$ resolution. {\\bf Left:} $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}), SiO(\\mbox{5--4}), and $^{13}$CS(\\mbox{5--4}) lines from top to bottom. {\\bf Middle:} SO(\\mbox{5$_6$--4$_5$}) and SO$_2$(\\mbox{$4_{22}-3_{13}$}) lines from top to middle. {\\bf Right:} C$^{18}$O(\\mbox{2--1}) line. }\n \\label{f:ede_spectra}\n \\end{figure*} \n\n\\begin{table*}\n\\caption{Molecular line emission from the EDE\\,$^1$}\n\\centering\n\\begin{tabular}{l c c c l}\n\\hline \\hline\nLine & Aperture\\,$^2$ & $S$ & FWZP\\,$^3$ & \\phantom{000}$I$ \\\\\n & [\\arcsec ] & [Jy] & [km\\,s$^{-1}$] & [Jy\\,km\\,s$^{-1}$] \\\\\n\\hline \nC$^{18}$O(2--1)\\,$^4$ & \\phantom{0}3 & 1.4\\phantom{11} & \\ldots & \\phantom{0}12 \\\\\nSiO(5--4) & \\phantom{0}3 & 0.38\\phantom{1} & 22.3 & \\phantom{00}4.5\\\\\nCS(4--3) & 32 & 0.35\\phantom{1} & 12.6 & \\phantom{00}5.0 \\\\\n$^{13}$CS(5--4) & \\phantom{0}3 & 0.11\\phantom{1} & 18.0 & \\phantom{00}1.9 \\\\ \nSO($5_5-4_4$) & \\phantom{0}3 & 0.30\\phantom{1} & 30.2 & \\phantom{00}5.4\\\\\nSO($5_6-4_5$) & \\phantom{0}3 & 0.18\\phantom{1} & 31.4 & \\phantom{00}4.0\\\\\nSO($6_5-5_4$)\\,$^5$ & 24 & 0.08\\phantom{1} & 20.0 & \\phantom{00}1.7\\\\\n$^{34}$SO($5_6-4_5$) & \\phantom{0}3 & 0.08:\\phantom{:} & 28:\\phantom{0} & \\phantom{00}1.5\\\\\nSO$_2$($4_{22}-3_{13}$) & \\phantom{0}3 & 0.035 & 35:\\phantom{0} & \\phantom{00}0.81\\\\\n$o$-H$_2$S($1_{10}-1_{01}$) & 37 & 1.9\\phantom{11} & 15.4 & \\phantom{0}19 \\\\\n$p$-H$_2$S($2_{20}-2_{11}$) & \\phantom{0}3 & 2.8\\phantom{11} & 17.9 & \\phantom{0}23 \\\\\n & 28 & 2.4\\phantom{11} & 15.3 & \\phantom{0}17 \\\\\n$o$-H$_2$S($3_{30}-3_{21}$) & 21 & 2.8\\phantom{11} & 12.7 & \\phantom{0}17 \\\\\n$p$-H$_2$S($2_{02}-1_{11}$) & \\phantom{0}9 & 5:\\phantom{111} & 6: & \\phantom{0}25: \\\\\n$o$-H$_2^{33}$S($1_{10}-1_{01}$)\\,$^6$ & 37 & \\ldots & \\ldots & \\phantom{0}12 \\\\\n$p$-H$_2^{33}$S($2_{20}-2_{11}$)\\,$^6$ & \\phantom{0}3 & \\ldots & \\ldots & \\phantom{00}3.4 \\\\\n$p$-H$_2^{33}$S($2_{02}-1_{11}$)\\,$^6$ & \\phantom{0}9 & \\ldots & \\ldots & \\phantom{0}24: \\\\\n$o$-H$_2^{34}$S($1_{10}-1_{01}$) & 37 & 1.6\\phantom{11} & 16.6 & \\phantom{0}12 \\\\\n$p$-H$_2^{34}$S($2_{20}-2_{11}$) & \\phantom{0}3 & 1.2\\phantom{11} & 15.0 & \\phantom{00}6.7 \\\\\n$p$-H$_2^{34}$S($2_{02}-1_{11}$) & \\phantom{0}9 & 5:\\phantom{111} & 6: & \\phantom{0}12 \\\\\n\\hline\n\\end{tabular}\n\\label{t:obs_ede}\n\\tablefoot{(1) See Sect.~\\ref{s:obs_desc} for a discussion of the flux uncertainties. (2) Centred on the continuum peak. (3) FWZP of the line within the given aperture at 1.5\\,km\\,s$^{-1}$ resolution. (4) Data obtained within a velocity range of 20\\,km\\,s$^{-1}$ centred on the systemic velocity. (5) This is based on APEX data and the split into emission from the CCS and EDE components obtained using Gaussian decomposition is uncertain. (6) The integrated intensity of the sum of the hyperfine components.}\n\\end{table*}\n\n\nThe $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) brightness distribution is dominated by emission from an essentially circular structure of size 2\\farcs3$\\times$2\\farcs1 (PA\\,$\\approx$\\,90$^\\circ$) centred on the CCS, Fig.~\\ref{f:h2s_channels}. The emission is very sharply truncated at the edge and limb-brightened. Notably, the size of the emitting region is independent of the line-of-sight velocity, and both blue- and redshifted emission is seen on either side of the centre channel map. As can be seen both in the channel maps and the PV-diagram, Fig.~\\ref{f:h2s_posvel}, the velocity gradient over the EDE is opposite to that of the HVO (compare right panel of Fig.~\\ref{f:sketch_co21}, and further discussed in Sect.~\\ref{s:obs_hvo}). The most straightforward interpretation is that the EDE has an expanding, flattened (possibly flared) density distribution that is oriented orthogonal to the HVO, that is, it is seen almost face-on. To estimate the expansion velocity and its dependence on the distance to the centre is difficult due to the essentially face-on orientation and unknown density distribution. It could be a disk or a torus. The dust modelling as described in Sect~\\ref{s:sed_results} suggests a disk, while the molecular line data appear more consistent with a torus.\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=14cm]{34897_fig_9}\n \\caption{$p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) channel maps with a width and spacing of 1.5\\,km\\,s$^{-1}$ at a resolution of 0\\farcs085 (he beam is shown in the lower left corner of each panel). The flux scale is in mJy\\,beam$^{-1}$. Emission from the EDE dominates for this line, although emission from the CCS is present at the centre.}\n \\label{f:h2s_channels}\n \\end{figure*} \n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{34897_fig_10}\n \\caption{$p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) PV-diagram along PA\\,=\\,90$^\\circ$ at resolutions of 0\\farcs085 and 1.5\\,km\\,s$^{-1}$. The flux scale is in mJy\\,beam$^{-1}$. }\n \\label{f:h2s_posvel}\n \\end{figure} \n \n \nWhen looked at in detail the EDE morphology exhibits some complications. There is an inner structure in the form of ``ears'' attached to the CCS (not necessarily physically though), that is particularly prominent in the rarer isotopologue $p$-H$_2^{34}$S(\\mbox{2$_{20}$--2$_{11}$}) line image at the systemic velocity, Fig.~\\ref{f:ede_h2(34)s_continuum}. This feature is also seen (weakly) in the CO(\\mbox{2--1}) and SiO(\\mbox{5--4}) data. The PA of the minor axis of this inner structure is $\\approx$\\,20$^\\circ$. Its velocity coverage is much lower than that of the outer circular structure. Both inner and outer structures can be partly traced in the form of ``arcs'' in the 1.3\\,mm continuum emission as seen in Fig.~\\ref{f:ede_h2(34)s_continuum}.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{34897_fig_11}\n \\caption{$p$-H$_2^{34}$S(\\mbox{2$_{20}$--2$_{11}$}) image in a 1.5\\,km\\,s$^{-1}$ channel at 41.5\\,km\\,s$^{-1}$ at a resolution of 0\\farcs085 (the beam is shown in the lower left corner). 1.3\\,mm continuum, at 0\\farcs15 resolution, is shown in red contours. The flux scale is in mJy\\,beam$^{-1}$, and the contours start at 0.6\\,mJy\\,beam$^{-1}$ with a spacing of 0.3\\,mJy\\,beam$^{-1}$ (the maximum value is 9.5\\,mJy\\,beam$^{-1}$).}\n \\label{f:ede_h2(34)s_continuum}\n \\end{figure} \n\n\nApart from H$_2$S, the EDE component is safely identified only in the ALMA SO and SO$_2$ images. Figure~\\ref{f:ede_so_so2} shows the channel maps of the SO(\\mbox{5$_6$--4$_5$}) and SO$_2$(\\mbox{$4_{22}-3_{13}$}) lines (note that the $4_{22}-3_{13}$ line is the one lowest in energy of our observed SO$_2$ lines, and hence less dominated by the CCS component). The EDE component is clearly visible. However, when looked at in more detail, a significant difference compared to the H$_2$S line emission from the EDE can be seen. In particular, the SO(\\mbox{5$_6$--4$_5$}) line extends over a velocity range larger than that of the H$_2$S line emission. It covers the range $\\approx$\\,$\\pm$20\\,km\\,s$^{-1}$ around the systemic velocity, Fig.~\\ref{f:ede_spectra}, and its emission in the velocity range outside that of the $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) line comes from a ring-like region that lies just outside that of the latter emission, Fig.~\\ref{f:ede_h2s_so}. The symmetry axis of this emission has a PA\\,$\\approx$\\,20$^\\circ$, that is, the same as the PA of the inner structure seen in the $p$-H$_2^{34}$S(\\mbox{2$_{20}$--2$_{11}$}) line data and discussed above, and the diameter of the ring-like region is $\\approx$\\,3\\arcsec . In addition, there is also an arc-like structure to the N at slightly blueshifted velocities, and a feature that stretches to the NE at slightly larger velocity offsets, in both the SO(\\mbox{5$_6$--4$_5$}) and SO$_2$(\\mbox{$4_{22}-3_{13}$}) lines. These have no apparent counterparts in the H$_2$S data. At present, we have no interpretation of these features. \n\nIt is difficult to reconcile the $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) and SO(\\mbox{5$_6$--4$_5$}) line brightness distributions. The sharp truncation of the H$_2$S line emission is most reasonably explained by a sharp density drop, and this is supported by the dust emission that is also truncated at roughly the same radius, see Fig.~\\ref{f:ede_h2(34)s_continuum}. Less likely explanations are excitation and\/or chemistry. However, the apparent continuation in space and velocity of the SO line emission with respect to that of H$_2$S rather suggests a smooth density distribution, and a chemistry where H$_2$S is destroyed at the expense of forming SO (for example, from S\\,+\\,OH). It is also possible that gas further away from the centre has been more accelerated through interaction with the HVO, and that this favours formation of SO. In fact, one may speculate that the SO line emission is coming from the HGS rather than the EDE. Against this interpretation, it may be argued that the SO line brightness distribution is circular and centred on the CCS emission. Further, it is not clear why, as opposed to the behaviour of the H$_2$S line emission that shows both blue- and redshifted emission over the area of the EDE, blue- and redshifted SO line emission is only seen towards the E and the W, respectively.\n\nIn summary, it is not clear whether the EDE component is defined by the spatial and kinematical characteristics of the H$_2$S line emission or whether it extends further both in space (reaching a diameter of $\\approx$\\,3\\arcsec) and velocity (reaching a maximum velocity of $\\approx$\\,20\\,km\\,s$^{-1}$). Higher angular resolution data for also the SO line emission may shed light on this issue. Furthermore, it may be that the CCS component gradually tapers into the EDE component, and that they are part of the same phenomenon.\n \n \\begin{figure*}\n \\centering\n \\includegraphics[width=14cm]{34897_fig_12_upper} \\includegraphics[width=14cm]{34897_fig_12_lower}\n \\caption{{\\bf Upper:} SO(\\mbox{5$_6$--4$_5$}) line channel maps. {\\bf Lower:} SO$_2$(\\mbox{$4_{22}-3_{13}$}) line channel maps. The channel width and spacing is 1.5\\,km\\,s$^{-1}$ at a resolution of 0\\farcs6 (he beam is shown in the lower left corner of each panel). The flux scale is in mJy\\,beam$^{-1}$. Emission from the EDE is clearly visible for these lines, but also emission from the CCS is present at the centre. }\n \\label{f:ede_so_so2}\n \\end{figure*} \n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{34897_fig_13}\n \\caption{EDE component as seen in the $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) (grey scale: velocity range \\mbox{32--52}\\,km\\,s$^{-1}$; 0\\farcs085 resolution) and SO(\\mbox{5$_6$--4$_5$}) lines (blue contours: velocity range \\mbox{22--32}\\,km\\,s$^{-1}$, red contours: velocity range \\mbox{52--62}\\,km\\,s$^{-1}$; 0\\farcs6 resolution). The contours start at 2\\,mJy\\,beam$^{-1}$ with a spacing of 3\\,mJy\\,beam$^{-1}$.}\n \\label{f:ede_h2s_so}\n \\end{figure} \n\n\nWe have complemented the ALMA H$_2$S data with APEX observations of the \\mbox{$1_{10}-1_{01}$}, \\mbox{$2_{02}-1_{11}$}, and \\mbox{$3_{30}-3_{21}$} lines (including isotopologues for the first two) and they emphasise the abundance of H$_2$S in this component, Fig.~\\ref{f:h2s_ede_spectra}. The relative contributions by emission from the CCS and the EDE in these lines are unknown, but the fact that the ALMA $p$-H$_2$S(\\mbox{2$_{20}$--2$_{11}$}) line is $\\approx$\\,75 times stronger in the latter is a strong argument in favour of also the APEX lines coming predominantly from the EDE component. The line widths of the \\mbox{$1_{10}-1_{01}$} and \\mbox{$3_{30}-3_{21}$} lines are consistent with this. On the other hand, the narrow widths of the \\mbox{2$_{02}$--1$_{11}$} lines (these are the lowest-energy lines of the observed H$_2$S lines) are more characteristic of the CCS component. Some guidance to the interpretation of these lines is obtained from the relative isotopologue line strengths. The three isotopologue \\mbox{2$_{02}$--1$_{11}$} lines are about equally strong indicating very high optical depths in the main isotopologue (the solar sulphur isotope ratios are $^{32}$S:$^{33}$S:$^{34}$S\\,=\\,127:1:23, and there is no reason to expect a low-mass star to alter this in any significant way during its evolution). Further, the \\mbox{2$_{02}$--1$_{11}$} line is $\\approx$\\,80 times stronger than the CCS emission in the \\mbox{2$_{20}$--2$_{11}$} line, while the expected ratio is about ten for optically thick emission at the same temperature, that is, following the black-body law of radiation. Thus, we conclude that also the \\mbox{2$_{02}$--1$_{11}$} lines originate mainly in the EDE, but we have no explanation for why the lines are so narrow.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=7cm]{34897_fig_14}\n \\caption{H$_2$S line spectra observed with APEX (the velocity resolution is 2\\,km\\,s$^{-1}$). {\\bf Top:} \\mbox{1$_{10}$--1$_{01}$} lines of $o$-H$_2$S (right), $o$-H$_2^{33}$S (middle), and $o$-H$_2^{34}$S (left). \n {\\bf Middle:} \\mbox{2$_{02}$--1$_{11}$} lines of $p$-H$_2$S (right), $p$-H$_2^{33}$S (middle), and $p$-H$_2^{34}$S (left). \n {\\bf Bottom:} \\mbox{3$_{30}$--3$_{21}$} line of $o$-H$_2$S.\n }\n \\label{f:h2s_ede_spectra}\n \\end{figure} \n\n\\subsection{The bipolar high-velocity outflow, HVO}\n\\label{s:obs_hvo}\n\nThe HVO component is clearly seen in the CO(\\mbox{2--1}) and SiO(\\mbox{5--4}) line emissions, but it is also present in the line emissions of SO, CS, and OCS, and it completely dominates the line emissions from HCN, HCO$^+$, H$_2$CO, and CH$_3$OH, where the EVSs are particularly prominent at $\\pm$\\,140\\,km\\,s$^{-1}$ and offset by $\\approx$\\,4\\arcsec\\ on either side of the centre, as discussed in Sect.~\\ref{s:obs_evs}. On the contrary, H$_2$S line emission, which is very strong in the EDE component, is markedly absent here. We exemplify the characteristics of the HVO through the CO(\\mbox{2--1}) and SiO(\\mbox{5--4}) channel maps (Figs \\ref{f:co_channels} and \\ref{f:sio_channels}) and an SiO(\\mbox{5--4}) PV diagram along PA\\,=\\,90$^\\circ$ (Fig.~\\ref{f:sio_posvel}).\n\nThe characteristics of the HVO as shown in the SiO(\\mbox{5--4}) data suggest that the driving outflow is highly collimated. The HVO is very symmetric with respect to the centre with a PA close to 90$^\\circ$ initially, turning gradually beyond an offset of $\\approx$\\,3\\arcsec\\ to reach $\\approx$\\,100$^\\circ$ at the EVSs. The HVO gas reaches a maximum velocity (not corrected for inclination angle) of $\\approx$\\,150\\,km\\,s$^{-1}$ at $\\pm$\\,4\\farcs2 from the centre. Particularly noticeable in the PV-diagram is the close to Hubble-like velocity dependence on distance to the centre, a phenomenon common in proto-PNe \\citep[for example,][]{alcoetal01}. \n\nThere are spots of enhanced line emission, symmetrically placed at $\\approx$\\,$\\pm$\\,0\\farcs8, $\\pm$\\,3\\farcs0, and $\\pm$\\,4\\farcs2 with respect to the centre. They outline a slightly S-shaped figure in the SiO(\\mbox{5--4}) PV-diagram, presumably an effect of a precessing driving jet. The EVSs are particularly prominent, both in the maps and as distinct features at the extreme velocities in the CO and $^{13}$CO single-dish spectra, Fig.~\\ref{f:co_spec}. The most reasonable explanation is that at these spots there are also piled-up material from an interaction between the HVO and a remnant wind [\\citet{olofetal17} provided evidence that the $^{12}$C\/$^{13}$C ratio is the same, $\\approx$\\,13, in the EVSs as in the CCS]. Another indication of this is that the bubbles of the HGS close at the EVSs.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{34897_fig_15}\n \\caption{SiO(\\mbox{5--4}) PV-diagram along PA\\,=\\,90$^\\circ$ at resolutions of 0\\farcs085 and 1.5\\,km\\,s$^{-1}$. The flux scale is in mJy\\,beam$^{-1}$. }\n \\label{f:sio_posvel}\n \\end{figure} \n\\subsection{The extreme-velocity spots, EVSs}\n\\label{s:obs_evs}\n\n\\citet{olofetal17} showed that the EVSs at the terminations of the HVO are particularly chemically rich (in a relative sense). They detected line emission from CO, $^{13}$CO, C$^{18}$O, $^{13}$CS, SO, SiO, $^{29}$SiO, H$_2$CO, H$_2^{13}$CO, and CH$_3$OH at these spots. Here we report the detections of also OCS using ALMA, as well as detections of CS, HCN, and HCO$^+$ using APEX where emissions from the two EVSs are clearly seen, Fig.~\\ref{f:cs_hcn_hco+_spec}. We searched unsuccessfully for the $p$-H$_2$O(\\mbox{3$_{13}$--2$_{20}$}) line with APEX and report an upper limit (which is high compared to the detection levels of the ALMA data).\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=7cm]{34897_fig_16}\n \\caption{Spectra obtained with APEX, and for comparison the SiO(\\mbox{5--4}) line extracted from the ALMA data integrated over the source (velocity resolution 2\\,km\\,s$^{-1}$). {\\bf From top to bottom:} Global ALMA SiO(\\mbox{5--4}) line, CS(\\mbox{4--3}) emission from the EDE and EVS components, HCN(\\mbox{3--2}) emission from the EVS components (and other features possibly related to the HVO component), and HCO$^+$(\\mbox{3--2}) emission from the EVS components.}\n \\label{f:cs_hcn_hco+_spec}\n \\end{figure} \n\n\\begin{table*}\n\\caption{Molecular line emission from the e-EVS\\,$^1$}\n\\centering\n\\begin{tabular}{l l c c c}\n\\hline \\hline\nLine & Instr. & $S$ & $\\Delta \\upsilon$\\,$^2$ & $I$ \\\\\n & & [Jy] & [km s$^{-1}$] & [Jy km s$^{-1}$] \\\\\n\\hline \nCO(1--0) & SEST & 0.38\\phantom{0} & \\ldots & \\phantom{0}6.2\\phantom{0} \\\\\nCO(2--1) & ALMA & 1.38\\phantom{0} & 10.8 & 16\\phantom{.00} \\\\\n & SEST & 1.48\\phantom{0} & \\ldots & 21\\phantom{.00} \\\\\nCO(3--2) & APEX & 1.64\\phantom{0} & \\ldots & 35\\phantom{.00} \\\\\nCO(4--3) & APEX & 0.86\\phantom{0} & \\ldots & 54\\phantom{.00} \\\\\nCO(6--5) & APEX & 0.86\\phantom{0} & \\ldots & 68\\phantom{.00} \\\\\n$^{13}$CO(2--1) & ALMA & 0.54\\phantom{0} & \\phantom{1}7.5 & \\phantom{0}4.3\\phantom{0} \\\\\n & SEST & 0.49\\phantom{0} & \\ldots & \\phantom{0}8.6\\phantom{0} \\\\\n$^{13}$CO(3--2) & APEX & 0.37\\phantom{0} & \\ldots & \\phantom{0}7.0\\phantom{0} \\\\\nC$^{18}$O(2--1) & ALMA & 0.06\\phantom{0} & \\phantom{1}4.8 & \\phantom{0}0.31 \\\\\nSiO(5--4) & ALMA & 0.45\\phantom{0} & \\phantom{1}9.5 & \\phantom{0}4.5\\phantom{0} \\\\\n$^{29}$SiO(5--4) & ALMA & 0.17\\phantom{0} & \\phantom{1}7.2 & \\phantom{0}1.3\\phantom{0} \\\\\nCS(4--3) & APEX & 0.2\\phantom{00} & \\ldots & \\phantom{0}1.9\\phantom{0} \\\\\n$^{13}$CS(5--4) & ALMA & 0.065 & \\phantom{1}9.6 & \\phantom{0}0.66 \\\\\nSO($5_5-4_4$) & ALMA & 0.078 & \\phantom{1}9.2 & \\phantom{0}0.76 \\\\\nSO($5_6-4_5$) & ALMA & 0.10\\phantom{0} & \\phantom{1}5.8 & \\phantom{0}0.61 \\\\\nOCS(18--17) & ALMA & 0.016 & \\phantom{1}8.6 & \\phantom{0}0.15 \\\\\nOCS(19--18) & ALMA & 0.016 & \\phantom{1}7.9 & \\phantom{0}0.13 \\\\\nHCN(3--2) & APEX & 0.08\\phantom{0} & \\ldots & \\phantom{0}1.4\\phantom{0} \\\\ \nHCO$^+$(3--2) & APEX & 0.06\\phantom{0} & \\ldots & \\phantom{0}1.0\\phantom{0} \\\\\n$p$-H$_2$O($3_{13}-2_{20}$) & APEX & \\ldots & \\ldots & $<$\\,3\\\\\n$p$-H$_2$CO($3_{03}-2_{02}$) & ALMA & 0.10\\phantom{0} & \\phantom{1}9.3 & \\phantom{0}1.0\\phantom{0} \\\\\n$p$-H$_2$CO($3_{22}-2_{21}$) & ALMA & 0.034 & \\phantom{1}9.4 & \\phantom{0}0.34 \\\\\n$p$-H$_2$CO($3_{21}-2_{20}$) & ALMA & 0.039 & \\phantom{1}7.4 & \\phantom{0}0.31 \\\\\n$o$-H$_2^{13}$CO($3_{12}-2_{11}$) & ALMA & 0.023 & \\phantom{1}8.7 & \\phantom{0}0.21 \\\\\n$E$-CH$_3$OH($4_2-3_1$) & ALMA & 0.051 & \\phantom{1}5.2 & \\phantom{0}0.28 \\\\\n$E$-CH$_3$OH($8_{-1}-7_0$) & ALMA & 0.065 & \\phantom{1}6.2 & \\phantom{0}0.43 \\\\\n \\hline\n\\end{tabular}\n\\label{t:obs_evs}\n\\tablefoot{(1) The line intensities apply to the velocity interval and region specified in the text for the ALMA data, and within the velocity interval specified in the text for the APEX and SEST data. The single-dish data are observed with the beam positioned at the centre of the source, and the reported intensities are not corrected for the beam response. Flux uncertainties are discussed in Sect.~\\ref{s:obs_evs}. (2) FWHM of a Gaussian fit to the line within the given aperture at 1.5\\,km\\,s$^{-1}$ resolution.}\n\\end{table*}\n\n\nIn Table~\\ref{t:obs_evs} we summarise the observational results for the eastern EVS (e-EVS; the western EVS shows very much the same line brightness pattern, but the emission is somewhat weaker). The data extraction is based on the appearance of the SiO(\\mbox{5--4}) line brightness distribution. Its emission at the most extreme redshifted velocities produces a line profile extending from 175 to 193\\,km\\,s$^{-1}$ at zero power (this is also the velocity range that contains for example all of the redshifted CH$_3$OH and essentially all of the redshifted H$_2$CO line emissions), and a brightness distribution that is essentially circular with a diameter of about 1\\arcsec\\ (determined from a 2D Gaussian fit) and centred 4\\farcs14\\,E and 0\\farcs37\\,S of the continuum peak. We have determined the data for all molecules observed with ALMA at this position, within an aperture of 1\\arcsec\\ and within the given velocity range (within this velocity range, all the brightness distributions are close to circular and have deconvolved sizes, FWHMs of 2D Gaussian fits, of 1\\arcsec\\ to within 0\\farcs3). The APEX and SEST data are estimated within the observing beams and in the above velocity range. The uncertainties in the observational results are dominated by the complexity of the brightness distributions for the ALMA data and by the uncertainty in which velocity range to use for the APEX and SEST data. They are difficult to estimate in a formal way, but may reach 50\\,\\% for some lines (especially for the single-dish data). \n\n\\subsection{The hourglass structure, HGS}\n\\label{s:obs_hgs}\n\nThe CO(\\mbox{2--1}) brightness distribution (and also those of its observed isotopologues, for example, Fig.~5 in \\citet{olofetal15}), Fig.~\\ref{f:co_channels}, shows an ellipse-like distribution whose size increases with velocity offset from the systemic velocity in the range $\\approx$\\,$\\pm$\\,30\\,km\\,s$^{-1}$. Assuming that the distance from the centre scales with the velocity offset (see below for a discussion on this), this gives an hourglass-like structure whose cross section increases with the distance from the centre. Also, the centre of the ``ellipse'' as a function of velocity is shifted consistently with the same sign of the velocity gradient as that of the HVO. Thus, an interpretation in the form of an hourglass structure (HGS), having the HVO along its symmetry axis, produced by pressure towards the sides from the outflow appears the most likely explanation for this component. The HGS is not seen in any of the other molecular line emissions, for example, it is absent in the SiO data, suggesting that the conditions in the walls are less extreme than in the HVO. \n\nThe velocity range in which we lose flux in the ALMA data is also the velocity range of the HGS, Sect.~\\ref{s:alma_flux}. This, most likely, means that we are not detecting material that has been accelerated to the same extent as the gas of the HGS component, because it is distributed in a more diffuse way.\n\nA slight distortion of the ellipse form starts at velocity offsets of $\\approx$\\,$\\pm$\\,30\\,km\\,s$^{-1}$ from the systemic velocity. This turns into a major complex structure in the velocity-offset ranges $\\approx$\\,40--80\\,km\\,s$^{-1}$ on either side of the systemic velocity, whose major components are two bright spots at velocity offsets of $\\pm$\\,60\\,km\\,s$^{-1}$. The morphology of the distortion, despite its complexity, is very symmetric with respect to the centre. We will come back to a possible explanation of this phenomenon in Sect.~\\ref{s:multi_polar}. Beyond these velocities the HGS becomes much fainter, but it can be traced as a bubble, on either side of the centre, that closes at the EVS. This provides further evidence for a connection between the HGS and the HVO. We will come back to this in Sects~\\ref{s:inc_angle} and \\ref{s:3D}.\n\n\\subsection{Inclination angle of the HVO}\n\\label{s:inc_angle}\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=16cm]{34897_fig_17} \n \\caption{CO(\\mbox{2--1}) PV-diagram along PA\\,=\\,90$^\\circ$ with an ellipse fitted to the observational HGS\/bubble structure shown in white. Results of the best-fit model (see text for details) for $i$\\,=\\,5$^\\circ$, 10$^\\circ$, and 20$^\\circ$ are shown as a red solid line.}\n \\label{f:inclination_angle}\n \\end{figure*} \n\nThe inclination angle, $i$, of the HVO cannot be estimated from the HVO emission itself since the maximum outflow velocity is not known. However, emission from the HGS\/bubble structure, as seen in the PV diagram (Fig.~\\ref{f:sketch_co21}, right panel) and channel maps (Fig.~\\ref{f:co_channels}) of the CO(\\mbox{2--1}) line, can be used to constrain it under certain assumptions. The latter are a source symmetric with respect to its centre, and an HVO axis direction constant with time (its projection on the sky is at PA\\,=\\,90$^\\circ$ in this case). We further assume that the expansion velocity of the HGS\/bubble structure is at each point linearly proportional to its distance from the centre. The latter is suggested by the close to Hubble-like appearance of the relation between line-of-sight velocity and apparent offset from the centre as shown by the HVO line emission, for example, Fig.~\\ref{f:sio_posvel}, that is, $\\upsilon_{\\rm z}$\\,$\\propto$\\,$p$ suggests $\\upsilon_{\\rm r}$\\,$\\propto$\\,$r$ for at least the emission involved in the HVO, that is, including the HGS.\n\nWe have used the publicly available code \\texttt{SHAPE} \\citep{stefwetal11} to determine the geometrical properties of the HGS\/bubble structure. The model is described by two, diametrically oriented, expanding, and thin-walled ellipsoids of homogeneous density (cigar-like and of the same geometry) that reach the maximum velocity at their tips. The results of this are compared to the observed PV-diagram and channel maps. The best fit to the data is found for $i$\\,$\\approx$\\,10$^\\circ$. This is examplified in Fig.~\\ref{f:inclination_angle} where the results of this model for three inclination angles, 5$^\\circ$, 10$^\\circ$, and 20$^\\circ$, are shown. A change of the cross section of the ellipsoids in order to make the results for the inclination angles of 5 and 20 degrees resemble better the observational data, leads to model channel maps that are inconsistent with the observed channel maps (primarily the cross section width of the ellipsoids is determined by the vertical sizes of the ellipses in the channel maps, and these are independent of the inclination angle for the chosen geometry). The estimated inclination angle of the HVO is therefore 10$^\\circ$\\,($-5^\\circ, +10^\\circ$). We conservatively estimate, using inspection by the eye, that the inclination angle lies in the range 5 to 20 degrees, that is, the quoted errors can be seen as 2$\\sigma$ limits. As argued above, the most reasonable conclusion is that the CCS and EDE components have flattened density distributions that are orthogonal to this direction.\n\\subsection{A 3D-reconstruction}\n\\label{s:3D}\n\nThe assumption that the line-of-sight velocity can be used as a measure of the spatial coordinate along the line of sight can be used to make a 3D-reconstruction of the source structure. The estimated inclination angle allows a determination of the correct scaling between the $z$ and $\\upsilon_{\\rm z}$ axes. Since the emission lines become gradually narrower towards the centre, the same relation can in principle be used also here to properly locate the emission, while the emergent morphology is more doubtful in this case. \n\nThe final 3D-reconstruction of the circumstellar medium of HD\\,101584 is shown in Fig.~\\ref{f:3d} in the form of an image of the CO(\\mbox{2--1}) line average intensity in the R.A. direction as seen from the side. There are a number of limitations to this 3D reconstruction. Among them, no corrections for radiative transfer effects, and no correction for the fact that within the same observed velocity channel, that is, gas moving with the same line-of-sight velocity, there will be gas moving at different absolute velocities and hence different distances to the centre (the small inclination angle and the high collimation of the outflow strongly limit the problem in our case). The movie in Fig.~\\ref{f:movie} shows the relation between the channel maps of the CO(\\mbox{2--1}) line emission and the 3D structure. \n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=17cm]{34897_fig_18} \n \\caption{Circumstellar environment of HD\\,101584 as seen from the side (the righthand side is facing towards us). The image is obtained by assuming radial expansion with a velocity that scales linearly with the distance from the centre, and using the estimated inclination angle. It gives the average intensity of the CO(\\mbox{2--1}) line in the R.A. direction at each pixel.}\n \\label{f:3d}\n \\end{figure*} \n\n\\subsection{A second bipolar outflow?}\n\\label{s:multi_polar}\n\nThere is evidence of a second bipolar, bubble-like structure in the CO(\\mbox{2--1}) line data, see Sect.~\\ref{s:decomp}. It covers $\\approx$\\,80\\,km\\,s$^{-1}$ in velocity, and has a velocity gradient opposite to that of the HVO in the direction of PA\\,$\\approx$\\,$-50^\\circ$. The most reasonable explanation to this structure is a second bipolar outflow, but there are no indications of bright spots in this case, for example, SiO line emission is not present. The inclination angle is unknown, but a difference in direction between this and the HVO may not be particularly large if we see both of them almost pole-on. Likewise, the opposite directions of the velocity gradients may become a natural consequence of only a smaller change in the direction of the outflow axis. The velocity of the second outflow is difficult to estimate since in the CO data we see only the bubbles, presumably of the same character as the HGS surrounding the HVO, and the inclination angle is unknown. The full velocity width of the CO line data indicates a maximum line-of-sight velocity of $\\approx$\\,40\\,km\\,s$^{-1}$, but the outflow velocity can be substantially higher.\n\nThe PA, the velocity gradient, and the velocity coverage of this bipolar outflow are interestingly close to the results for the OH\\,1667\\,MHz maser line data presented by \\citet{telietal92}. It is tempting to believe that this is more than a coincidence. The OH 1667\\,MHz maser peaks are distributed in two clusters, one with blueshifted spots and one with redshifted spots, separated by $\\approx$\\,3\\farcs5 along PA\\,$\\approx$\\,$-55^\\circ$. Based on these data \\citet{zijletal01} proposed the existence of a second bipolar outflow. In fact, the innermost OH maser spots border exactly outside the edge of the EDE emission as traced in the $p$-H$_2$S(\\mbox{$2_{20}-2_{11}$}) line, and coincides well with the blue- and red-shifted SO(\\mbox{5$_6$--4$_5$}) line emissions shown in Fig.~\\ref{f:ede_h2s_so}, except that the OH maser emission avoids the regions of peak emission in the SO line. The full velocity extent of the OH emission is $\\approx$\\,80\\,km\\,s$^{-1}$, and the dominant maser peaks are separated by $\\approx$\\,50\\,km\\,s$^{-1}$. Furthermore, the two clusters of red- and blue-shifted OH maser spots, and hence the SO emission peaks, are nearly coincident, in space but less so in velocity, with the areas where the ellipse-like pattern of the HGS in the CO(\\mbox{2--1}) channel maps is disturbed (on either side of the centre) and from where it develops eventually into the innermost bright spots as discussed in Sect.~\\ref{s:obs_hgs}. \n\nWe speculate here that the disturbance of the HGS structure and the existence of the OH 1667\\,MHz masers are connected to an interaction between the two outflows. It is noteworthy in this context that the HD\\,101584 OH maser has some rather peculiar characteristics: there is no detectable emission from the three other 18\\,cm lines (at 1612, 1665, and 1720\\,MHz) at very low levels (in a relative sense), the 1667\\,MHz emission lacks any time variability over a time scale of 25 years (time variability is a well-known characteristic of both circumstellar and interstellar cosmic masers), and there is no detectable polarisation (hence no indication of a magnetic field) (Vlemmings et al., in prep.). \n\\section{Quantitative estimates: gas}\n\\label{s:quant_est_gas}\n\nWe will here derive some quantitative results for the circumstellar molecular medium. However, our data base consists of only 13\\,GHz of ALMA data and some complementary APEX and SEST data. This means that the observational constraints on densities and temperatures are limited. Furthermore, the object is seen almost pole-on which restricts the information on its structure along the line of sight and its kinematics orthogonal to it. We will therefore perform rather simple analyses at this stage.\n\\subsection{Kinematical ages}\n\\label{s:age}\n\nA simple estimate of the age of an outflow is obtained by using its maximum apparent outflow velocity, its apparent length, and its inclination angle, that is, its kinematical age. This results in an age of $\\approx$\\,770\\,[$D$\/1\\,kpc]\\,yr for the HVO. An estimate like this suffers from a lack of knowledge of some key parameters required for a proper estimate, notably the velocity of the driving agent, presumably a collimated jet, and its evolution through interactions with the surrounding medium. Therefore, it should be regarded as a reasonable upper limit estimate of the time scale of the phenomenon \\citep{bujaetal01}. \n\nA comparison can be made with the kinematical age of the EDE using a line-of-sight expansion velocity of 10\\,km\\,s$^{-1}$ (from the H$_2$S line width, but the possible flaring of the EDE makes this an uncertain estimate), the estimated inclination angle, and the measured size. The result is an estimated age of $\\approx$\\,110\\,[$D$\/1\\,kpc]\\,yr, that is, considerably shorter than that of the HVO (though the HVO estimate is an upper limit). This goes against the conclusion by \\citet{hugg07} that in proto-PNe jets and tori develop nearly simultaneously, with the torus appearing first and the jet typically a few hundred years later. A possible explanation to our finding is that the EDE is more of a ``pattern'' structure, that is, a region where matter flows through, becomes excited, and hence observable. This would, on the other hand, mean that there must be an inner reservoir of gas and for this we find no evidence. Currently, we have no explanation for the relative ages of the EDE and HVO components.\n\nThe age of the second bipolar outflow is more difficult to estimate since there are no bright spots at the end of this outflow in the CO(\\mbox{2--1}) data, and the inclination angle is unknown. Using the separation of $\\approx$2\\arcsec\\ of the strongest OH masers, separated by $\\approx$\\,50\\,km\\,s$^{-1}$ in velocity, and the same inclination angle as for the HVO, we derive an age of $\\approx$\\,2100\\,[$D$\/1\\,kpc]\\,yr, that is, considerably older than the HVO. An inclination angle of $\\approx$\\,70$^\\circ$ for the second bipolar outflow is required to make the outflows of similar age. Alternatively, the outflow velocity is much higher than estimated from the bubble structure. Irrespective of this, the most important conclusion is that there is a recurrence aspect in the phenomenon responsible for the circumstellar structure of HD\\,101584.\n\n\\subsection{Mass, density, and temperature estimates}\n\nAs argued already above, we restrict ourselves here to simple calculations, which we think, nevertheless, provide us with good order-of-magnitude estimates. Only for the e-EVS do we attempt a radiative transfer analysis, since here we can use single-dish data on a number of CO isotopologue lines from different transitions as constraints. \n\\subsubsection{Molecular gas estimates}\n\nWe will here extensively use the CO(\\mbox{2--1}) line brightness temperatures ($T_{\\rm b}$) estimated in the various components. If the CO(\\mbox{2--1}) line emission is optically thick, which appears to be the case in most parts of the circumstellar medium of HD\\,101584, this will give us the excitation temperature of the CO \\mbox{2--1} transition ($T_{\\rm ex}$\\,=\\,$T_{\\rm b}$). If the population distribution of the rotational levels is thermalised by collisions ($T_{\\rm rot}$\\,=\\,$T_{\\rm ex}$\\,=\\,$T_{\\rm b}$), which considering the high densities estimated below also appears to be the case throughout the circumstellar medium of HD\\,101584, this will also give a good (but averaged over the beam) estimate of the gas kinetic temperature ($T_{\\rm k}$\\,=\\,$T_{\\rm b}$). \n\nWe will use the C$^{18}$O(\\mbox{2--1}) line data to estimate column densities assuming that this line is optically thin (which appears to be the case in most regions). These estimates are converted to H$_2$ column densities assuming that the CO\/C$^{18}$O ratio reflects the solar O\/$^{18}$O ratio of 480 \\citep{scotetal06}, and that the fractional CO abundance with respect to H$_2$, $f_{\\rm CO}$, is the one expected for O-rich AGB CSEs, 4$\\times$10$^{-4}$ (close to full association of carbon into CO assuming solar abundances, and close to the value that fits the e-EVS data, see Sect.~\\ref{s:physics_evs}), that is, $f_{\\rm C^{18}O}$\\,=\\,8$\\times$10$^{-7}$. The justification for using a solar value for the O\/$^{18}$O ratio is presented in Sect.~\\ref{s:evol_status}.\n\nIn the same way, a simple estimate of the gas mass is obtained using the equation,\n\\begin{equation}\n\\label{e:gasmass}\nM_{\\rm g} = \\frac{16\\pi m_{\\rm H}}{hc g_{\\rm u} A_{\\rm ul} f_{\\rm C^{18}O}}\\,\n I_{\\rm C^{18}O}\\, D^2\\, Q(T_{\\rm rot})\\, e^{E_{\\rm u}\/kT_{\\rm rot}}\\, ,\n\\end{equation}\nwhere the usual symbols are used for the constants, and $I_{\\rm C^{18}O}$ is the C$^{18}$O(\\mbox{2--1}) flux density integrated over a velocity range and an area, $Q$ the partition function, and $E_{\\rm u}$ the energy of the upper level. \n\\subsubsection{The CCS}\n\\label{s:physics_ccs}\n\nThe molecular line emissions come from a region $\\approx$\\,0\\farcs15 in diameter (corresponding to $\\approx$\\,150\\,[$D$\/1\\,kpc]\\,au), Table~\\ref{t:obs_ccs}. An estimate of the gas temperature can be obtained from the brightness temperature in the CO(\\mbox{2--1}) line, $\\approx$\\,160\\,K. As argued above, this is likely close to the kinetic temperature. This gas temperature estimate is very comparable to the equilibrium temperature of low-albedo dust at a distance of 75\\,[$D$\/1\\,kpc]\\,au from a star with the adopted HD\\,101584 characteristics, $\\approx$\\,200\\,K.\n\nAn estimate of the H$_2$ column density is obtained using the C$^{18}$O(\\mbox{2--1}) data. The strength of the ALMA line and assuming a source size of 0\\farcs15 and an excitation temperature of 160\\,K lead to a source-averaged C$^{18}$O column density of $\\approx$\\,10$^{18}$\\,cm$^{-2}$ (corresponds to a, source-averaged, optical depth in this line of $\\approx$\\,1, that is, there is some uncertainty in this estimate due to opacity). Assuming $f_{\\rm C^{18}O}$\\,=\\,8$\\times$10$^{-7}$ we find a source-averaged H$_2$ column density of $\\approx$\\,10$^{24}$\\,cm$^{-2}$. We use also the C$^{18}$O(\\mbox{2--1}) data and Eq.~(\\ref{e:gasmass}) to estimate the mass. The observed C$^{18}$O(\\mbox{2--1}) line intensity, its fractional abundance, and the estimated gas temperature result in a gas mass of $\\approx$\\,0.029\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$. Both the H$_2$ column density and the mass are likely lower limits considering the opacity of the C$^{18}$O(\\mbox{2--1}) line. A crude lower limit to the density can be obtained by making the reasonable assumption that the CCS has an extent along the line of sight that is not larger than its extent in the plane of the sky. With the estimated H$_2$ column density this points to an H$_2$ density in excess of 10$^9$\\,[1\\,kpc\/$D$]\\,cm$^{-3}$. This is a very high density meaning that for all molecules observed the excitation is collisionally dominated and the lines are thermalized.\n\nThe most likely interpretation of the CCS component is that of a circumbinary disk, presumably in slow rotation. Unfortunately, the spatial and velocity resolutions of our data are not high enough to allow a determination of the detailed kinematics of this component. \n\\subsubsection{The EDE}\n\\label{s:physics_ede}\n\nAn estimate of the gas column density and mass can be obtained in the same way as for the CCS using the C$^{18}$O(\\mbox{2--1}) data. The obtained CO(\\mbox{2--1}) line brightness temperature in the EDE area is $\\approx$\\,50\\,K. The strength of the ALMA C$^{18}$O(\\mbox{2--1}) line and assuming a source size of 3\\arcsec\\ combined with an excitation temperature of 50\\,K leads to a source-averaged C$^{18}$O column density of $\\approx$\\,2$\\times$10$^{17}$\\,cm$^{-2}$ (corresponds to a, source-averaged, optical depth in this line of $\\approx$\\,0.3, that is, there is some uncertainty in this estimate due to opacity). With the same assumptions on C$^{18}$O fractional abundance ratio as for the CCS, we estimate a source-averaged H$_2$ column density of $\\approx$\\,2$\\times$10$^{23}$\\,cm$^{-2}$. The gas mass is estimated using Eq.~(\\ref{e:gasmass}), the C$^{18}$O(\\mbox{2--1}) line intensity, its fractional abundance, and the estimated gas temperature. The result is 0.24\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$, that is, a significant fraction of the mass of the circumstellar medium around HD\\,101584 lies in this component, since the masses of the CCS (above) and HGS and HVO (below) components are lower. Making the same assumption on the geometry of the EDE as for the CCS, we get a lower limit to the H$_2$ density of $\\approx$\\,10$^7$\\,[1\\,kpc\/$D$]\\,cm$^{-3}$, that is, much lower than the lower limit for the CCS, but still a high-density region. Also here, all the observed molecular species are expected to be effectively excited by collisions.\n\nThe EDE component, probably a disk or a torus, contains most of the circumstellar mass, and it is in expansion. The connection between the CCS and EDE components, if any, is not clear. \n\\subsubsection{The HGS and HVO}\n\nThe HGS and HVO components are only clearly seen in the CO line data (except for SiO in the bright spots of the HVO). As for the CCS and EDE, the gas temperature is estimated using the CO brightness temperatures in these regions. The results are brightness temperatures of $\\approx$\\,25\\,K and $\\approx$\\,50\\,K in the HGS and HVO (for the latter this is estimated in the e-EVS). The observed CO\/$^{13}$CO \\mbox{2--1} line intensity ratios are about 1.5 and 3 in the HGS and HVO (as estimated for the e-EVS). Taking the estimated CO\/$^{13}$CO abundance ratio of $\\approx$13 \\citep{olofetal17} into account, we conclude that the CO optical depths are high in both regions, and higher in the HGS than in the HVO, and that the brightness temperatures are good estimates of the CO excitation temperature, and presumably the kinetic temperature. The C$^{18}$O(\\mbox{2--1}) integrated intensities are 8.3\\,Jy\\,km\\,s$^{-1}$ for the HGS (integrated over the velocity range 10\\,$\\le$\\,$\\mid$\\,$\\upsilon-\\upsilon_{\\rm sys}$\\,$\\mid$\\,$\\le$\\,45\\,km\\,s$^{-1}$) and 1.5\\,Jy\\,km\\,s$^{-1}$ for the HVO ($\\mid$\\,$\\upsilon-\\upsilon_{\\rm sys}$\\,$\\mid$\\,$\\ge$\\,45\\,km\\,s$^{-1}$). Using this, the adopted C$^{18}$O fractional abundance, and the estimated gas temperatures, the resulting masses are 0.12\\,[$D$\/1\\,kpc]$^2$ and 0.030\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$ for the HGS and HVO, respectively. A separate mass estimate for the e-EVS is given below. \n\\subsubsection{The EVSs}\n\\label{s:physics_evs}\n\nIn \\citet{olofetal17} we introduced a simple physical model for an EVS in order to derive molecular abundances through a radiative transfer analysis, noting that this is the only component for which we can identify emission from many CO transitions, hence providing observational constraints on the excitation. The observational data give limited information on the geometry except that the emission is largely confined to a region of diameter $\\la$\\,1\\arcsec . We will use the same method, although simplified to only one component, and assumptions here.\n\nHere we present the results of the CO line modelling, the result for the other molecules will be presented in a paper focussing on the circumstellar chemistry of HD\\,101584. The radiative transfer is solved using an Accelerated-Lambda-Iteration code, taking into account excitation through collisions with H$_2$. Collisional coefficients for CO is taken from \\citet{yangetal10}. Radiative excitation due to central star light (too distant) and dust emission inside the clump (too low optical depth) can be ignored. The CO line intensities for the e-EVS were used since they are slightly stronger than those of the western EVS, Table~\\ref{t:obs_evs}. We assume a spherical, homogenous, iso-thermal clump of radius 0\\farcs5. \n\nThe density and kinetic temperature are reasonably well-constrained by the observed CO line intensities. The best fit is obtained for $n_{\\rm H_2}$\\,=\\,(5$\\pm$2)$\\times$10$^5$\\,cm$^{-3}$ and $T_{\\rm k}$\\,=\\,60$\\pm$10\\,K, that is, consistent with thermal excitation, high optical depths, and the CO(\\mbox{2--1}) brightness temperature estimate above. Hence, the gas mass of the e-EVS becomes $\\approx$\\,10$^{-3}$\\,[$D$\/1\\,kpc]$^2$\\,M$_\\odot$. \n\nThe analysis also gives a best-fit CO abundance of (7$\\pm$2)$\\times$10$^{-4}$, that is, close to the level expected in an O-rich circumstellar gas (full association of CO and solar values for O and C results in a fractional CO abundance of 5$\\times$10$^{-4}$). The resulting CO\/$^{13}$CO ratio is 14$\\pm$6, that is, in very good agreement with the value 13$\\pm$6 estimated from C$^{17}$O and $^{13}$C$^{17}$O line emission from the CCS by \\citet{olofetal17}. An important conclusion from this is that the e-EVS material is dominated by circumstellar gas (possibly swept-up from a previous wind), and not by swept-up interstellar material. The C$^{16}$O\/C$^{18}$O ratio of 225$\\pm$75 is somewhat low compared to the solar value of 480 \\citep{scotetal06}. Low- and intermediate-mass stars are expected to destroy rather than produce $^{18}$O, but considering our simple model, the uncertainty in identifying emission from the EVS, and the use of a single C$^{18}$O line, we draw no conclusions based on this result.\n\\subsubsection{Summary of gas properties}\n\\label{s:mass_summary}\n\nWe summarize our findings on the densities, temperatures, and masses of the identified gas components in Table~\\ref{t:phys_char}. We conclude that the estimated total gas mass of the circumstellar material around HD\\,101584 is $\\approx$\\,0.42\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$. Finally, we note that, due to missing flux in the velocity range of the HGS component, there exists gas whose contribution to the total gas mass is not known, and the estimate of the latter should therefore be seen as a lower limit. If we assume that the characteristics of this gas is similar to that of the HGS component, we estimate the mass of this gas to be $\\approx$\\,0.12\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$, that is, a total circumstellar gas mass of $\\approx$\\,0.5\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$.\n\n\\begin{table}\n\\caption{Physical characteristics of identified components}\n\\centering\n\\begin{tabular}{l c c c c c}\n\\hline \\hline\nComponent & Size & $n_{\\rm H_2}$ & $M_{\\rm g}$ & $T_{\\rm k}$ & $\\Delta \\upsilon$\\,$^1$ \\\\\n & [au] & [cm$^{-3}$] & [$M_\\odot$] & [K] & [km\\,s$^{-1}$] \\\\\n\\hline \\hline\n\\multicolumn{6}{c}{Low-$L_\\ast$ case (500\\,$L_\\odot$, 0.56\\,kpc):}\\\\\n\\hline\nCCS & \\phantom{000}80 & $>$10$^9$ & 0.01\\phantom{00} & 160 & \\phantom{00}3 \\\\\nEDE & \\phantom{0}1700 & $>$10$^7$ & 0.08\\phantom{00} & \\phantom{0}50 & \\phantom{0}20 \\\\\nHGS & \\ldots & \\ldots & 0.04\\phantom{00} & \\phantom{0}25 & \\phantom{0}80\\\\\nHVO & 27000 & \\ldots & 0.01\\phantom{00} & \\phantom{0}50 & 150 \\\\\nEVS\\,$^2$ & \\phantom{00}560 & \\phantom{0}10$^6$ & 0.0002 & \\phantom{0}60 & \\phantom{00}8 \\\\\n\\hline \\hline\n\\multicolumn{6}{c}{High-$L_\\ast$ case (5000\\,$L_\\odot$, 1.8\\,kpc):}\\\\\n\\hline\nCCS & \\phantom{00}270 & $>$10$^{9}$ & 0.1\\phantom{00} & 160 & \\phantom{00}3 \\\\\nEDE & \\phantom{0}5400 & $>$10$^7$ & 0.8\\phantom{00} & \\phantom{0}50 & \\phantom{0}20 \\\\\nHGS & \\ldots & \\ldots & 0.4\\phantom{00} & \\phantom{0}25 & \\phantom{0}80\\\\\nHVO & 87000 & \\ldots & 0.1\\phantom{00} & \\phantom{0}50 & 150 \\\\\nEVS\\,$^2$ & \\phantom{0}1800 & \\phantom{0}10$^6$ & 0.005 & \\phantom{0}60 & \\phantom{00}8 \\\\\n\\hline\n\\end{tabular}\n\\label{t:phys_char}\n\\tablefoot{(1) FWHMs of Gaussian fits for CCS and EVS, FWZPs for EDE and HGS, and maximum expansion velocity for the HVO. (2) Using estimates for the e-EVS, see Sect.~\\ref{s:physics_evs}.}\n\\end{table}\n\\subsection{The energetics of the outflowing material}\n\\label{s:energy}\n\nThe energetics, that is, the energy, scalar momentum, and scalar momentum rate, of the outflowing material is of great interest in connection with discussing the driving mechanism and the energy source. We repeat here essentially the analysis in \\citep{olofetal15} since it is based on the same C$^{18}$O(\\mbox{2--1}) data. The differences are that we now apply this analysis to the EDE and HGS components in addition to the HVO component, since it is likely that the accelerations of these gas components have the same origin, and using the excitation temperatures estimated for these regions separately. We still use the line-of-sight velocities, so in this sense the estimates should be regarded as lower limits. The correction for the inclination angle will be largest for the EDE component, but its contributions to the energy and momentum are limited.\n\nThe results are that the kinetic energy of the accelerated gas is 7.2$\\times$10$^{45}$\\,[$D$\/1\\,kpc]$^2$\\,erg (360\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$\\,km$^2$\\,s$^{-2}$) and its scalar momentum is 1.8$\\times$10$^{39}$\\,[$D$\/1\\,kpc]$^2$\\,g\\,cm\\,s$^{-1}$ (9.0\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$\\,km\\,s$^{-1}$). Should this momentum be supplied by radiation, the corresponding time scale [momentum\/($L_\\ast\/c$)] is close to 3$\\times$10$^5$\\,yr which is of course unreasonably long in the case of HD\\,101584. Another acceleration mechanism must be at work. The corresponding values for the HVO component only is 6.4$\\times$10$^{45}$\\,[$D$\/1\\,kpc]$^2$\\,erg for the energy and 1.0$\\times$10$^{39}$\\,[$D$\/1\\,kpc]$^2$\\,g\\,cm\\,s$^{-1}$ for the scalar momentum.\n\\section{Quantitative estimates: dust}\n\\label{s:quant_est_dust}\n\nFor the analysis of the dust emission the situation is better in the sense that we have the full spectral energy distribution (SED) at our disposal. However, the analysis is still plagued by the uncertainty imposed by the geometry and the additional lack of knowledge of the characteristics and distribution of different dust types. We start by looking at the dust distribution, as traced by the 1.3\\,mm continuum, and its relation to the different components identified in the molecular line data.\n\\subsection{The CCS}\n\nThe CCS is estimated to have a 1.3\\,mm continuum flux density of 7\\,mJy coming from a source of size $\\la$\\,0\\farcs01 (corresponding to $\\la$\\,10\\,[$D$\/1\\,kpc]\\,au). Since this component is not (or just barely) resolved, it is, in principle, possible that this flux is coming from the central star. However, using the adopted values for HD\\,101584 (8500\\,K and $L_\\ast\/D^2$\\,=\\,1600\\,$L_\\odot$\/kpc$^2$) its flux density at 1.3\\,mm is estimated to be only about 0.01\\,mJy (the flux scales as $L_\\ast\/D^2$, hence the same result for the low- and high-$L_\\ast$ cases). It is in principle possible that also free-free emission from a region surrounding the warm star contributes to the flux, but it remains for the future to determine the properties of the immediate circumstellar surroundings of HD\\,101584. \n\nAnother possible explanation is that the emission is coming from heated dust in a disk, presumably the innermost warmest part of the region responsible for the narrow-line-width molecular line emission. The equilibrium temperature of low-albedo dust at a distance of 5\\,[$D$\/1\\,kpc]\\,au from a star with the adopted HD\\,101584 characteristics is $\\approx$\\,800\\,K. The blackbody emission at 1.3\\,mm from an optically thick dust disk of these characteristics is $\\approx$\\,3\\,mJy. Thus, this provides a possible explanation, but it requires a dust optical depth close to one (or higher) at 1.3\\,mm. Adopting a dust opacity of 0.5\\,cm$^2$\\,g$^{-1}$ [\\citet{liseetal15}; the average of the five values listed in their Table~6] such a disk become optically thick at a dust mass of about 2$\\times$10$^{-5}$\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$ (corresponding to a gas mass of about 0.004\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$, assuming a ``canonical'' gas-to-dust-mass ratio of 200). This value is not unreasonable. It can be further noted that the 0\\farcs028 disk with a brightness temperature of 650\\,K at 10.7\\,$\\mu$m observed by \\citet{hilletal17} will have a 1.3\\,mm continuum flux of 7\\,mJy (if optically thick at 1.3\\,mm and assuming that the brightness temperature equals the dust temperature). That is, it is very likely that this disk and the central 1.3\\,mm continuum source are two aspects of the same object. As noted in Sect.~\\ref{s:obs_ccs} increasing the aperture to 0\\farcs3 increases the flux by only $\\approx$\\,70\\,\\%, so the continuum flux from the central region is dominated by the very central part.\n\\subsection{The EDE}\n\nThe 1.3\\,mm continuum emission from HD\\,101584 is dominated by emission from the EDE, $\\approx$\\,120\\,mJy within an aperture of 3\\arcsec , Fig.~\\ref{f:morphology} and Table~\\ref{t:obs_ede}. This is likely also the emission that dominates at the far-IR wavelengths of the SED. An order of magnitude estimate of the dust mass can be obtained assuming optically thin dust emission,\n\\begin{equation}\n\\label{e:dustmass}\nM_{\\rm d} = \\frac{S_\\nu D^2}{\\kappa_\\nu B_\\nu}\\,\\, ,\n\\end{equation}\nwhere $S_{\\nu}$ is the flux density at the frequency $\\nu$ and within a given aperture, and $B_{\\nu}$ and $\\kappa_{\\nu}$ the blackbody brightness and the dust opacity at this frequency, respectively. Assuming a dust temperature of 50\\,K (the estimated gas temperature of the EDE component), and an opacity of 0.5\\,cm$^2$\\,g$^{-1}$ result in $M_{\\rm d}$\\,$\\approx$\\,0.014\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$ (the dust optical depth at 1.3\\,mm of a uniform disk of this size is well below one). This is not unreasonable, and consequently the EDE will be an important component of the SED modelling below. \n\n\\subsection{The SED}\n\\label{s:sed}\n\nThe SED of HD\\,101584 was constructed using photometric and spectroscopic measurements available from astronomical catalogues, Fig.~\\ref{f:sed}. The ISO short-wavelength spectrometer \\citep{degretal96} data were obtained from the NASA\/IPAC Infrared science archive, and the Herschel\/SPIRE \\citep{grifetal10} data from the level 2 product in the Herschel Space Observatory archive. Photometric measurements were obtained from the point-source catalogue of the IRAS satellite \\citep{neugetal84}, the WISE All-Sky Data Release \\citep{cutretal12}, the 2MASS All-Sky Catalog of Point Sources \\citep{cutretal03}, the MESS program \\citep{groeetal11} for Herschel\/PACS \\citep{pogletal10}, the AKARI Infrared Camera Mid-IR All-sky Survey \\citep{ishietal10}, the AKARI Far-infrared Surveyor \\citep[from catalogue;][]{kawadetal07}, and the Gaia second data release \\citep{gaiaetal18}. Finally, we have the results of our ArTeMiS and ALMA observations. There are several notable features: the very strong far-IR excess, the presence of a 10\\,$\\mu$m feature, strong features around 45\\,$\\mu$m [identified as due to crystalline water, \\citet{hoogetal02}], and the presence of high-$J$ CO lines in the SPIRE data.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=9cm]{34897_fig_19}\n \\caption{Observed SED of HD\\,101584, and a comparison with the model results. ISO SWS and Herschel\/SPIRE spectra are shown by the full violet and brown lines, respectively (note the presence of high-$J$ CO lines in the SPIRE spectrum). Photometric measurements are from Gaia (blue x:s), 2MASS (green pentagons), AKARI (orange circles), IRAS (black squares), WISE (red crosses), ALMA (purple circle), PACS (blue triangles), and ArTeMiS (orange inverted triangle). The red solid line represents a black-body at a temperature of 8500\\,K, and for a distance of 1\\,kpc the luminosity is 1600\\,$L_\\odot$. We show models with $a_{\\rm max}$\\,=\\,1\\,mm ($M_{\\rm d}$\\,=\\,10$^{-2}$\\,$M_\\odot$, green dashed line), $a_{\\rm max}$\\,=\\,0.1\\,mm ($M_{\\rm d}$\\,=\\,4$\\times$10$^{-3}$\\,$M_\\odot$, blue dash-dotted line), and $a_{\\rm max}$\\,=\\,0.01\\,mm ($M_{\\rm d}$\\,=\\,10$^{-3}$\\,$M_\\odot$, orange dotted line).}\n \\label{f:sed}\n\\end{figure}\n\\subsection{SED modelling}\n\\label{s:sed_fit}\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=7cm]{34897_fig_20}\n \\caption{Radial profile of the 1.3\\,mm continuum emission observed using ALMA (blue triangles) and the results of the best-fit model to the SED (orange inverted triangles).}\n \\label{f:cont_radial}\n\\end{figure}\n\n\nThe ALMA 1.3\\,mm continuum emission is dominated by the EDE component, and there is also a contribution from extended, diffuse emission. It is reasonable to assume that the optically thin emission at 1.3\\,mm is a good measure of the dust distribution, that is, we will model the SED assuming that the dust is located in a disk embedded in a spherically symmetric envelope. The dust density distribution is given by a simplified version of the analytical expression provided by \\cite{meixetal02}, with their $C$, $D$, and $E$ parameters set to 0, 0, and 1, respectively,\n\\begin{equation}\n\\rho(r,\\theta) = \\rho_{\\rm in} \\left( \\frac{r}{R_{\\rm in}}\\right)^{-B} \\times \\left\\{ 1+A(1-cos\\theta)^F\\left[\\frac{e^{(-r\/R_{\\rm disk})}}{e^{(-R_{\\rm in}\/R_{\\rm disk})}}\\right]\\right\\}.\n\\end{equation}\nThe dust density, both in the equatorial and the polar directions (defined by the radius, $r$, and the polar angle, $\\theta$), decreases radially following a power law. At the disks outer edge, the density in the equator decreases exponentially until it reaches the lower values of the density in the polar direction. The free parameters that define the density distribution are the inner radius of the envelope, $R_{\\rm in}$, the exponent of the radial profile, $B$, the equatorial density enhancement in the disk, $1+A$, the parameter that controls the shape (mainly opening angle) of the disk, $F$, and the disk radius, $R_{\\rm disk}$. The total dust mass, $M_{\\rm d}$, is used to calculate the overall density and define the density at the inner radius in the equatorial direction, $\\rho_{\\rm in}$. The outer radius of the envelope, $R_{\\rm out}$, is an input parameter that we set to 3000\\,[$D$\/1\\,kpc]\\,au. We consider an inclination angle of 10$^\\circ$ for the system, as determined from the analysis of the molecular lines, and a blackbody stellar spectrum with an effective temperature of 8500\\,K. For the continuum radiative transfer we used the state-of-the-art code \\texttt{MCMax} \\citep{minetal09} in its axisymmetric mode.\n\nThe dust opacity was calculated using the Mie approximation and optical constants for astronomical silicate dust \\citep{osseetal92} and water ice \\citep{warr84}. Astronomical silicates were used to simplify the fitting procedure, since a detailed study of the dust composition using optical constants for different species measured in the laboratory is out of the scope of this study. The water ice abundance by mass relative to the total amount of dust is $f_{\\rm ice}$. We considered a size distribution of the dust grains between a minimum size, $a_{\\rm min}$ (0.01\\,$\\mu$m), and a maximum size, $a_{\\rm max}$, which we vary to fit the observations. A value of $-3.5$ was used for the exponent of the size distribution, $p$, but also the effects of making the distribution steeper or shallower were explored as discussed below. The different grain sizes and the two species are all in thermal contact, and, hence, have a single temperature at a given radius and polar angle.\n\nWe apply no correction for interstellar extinction for two reasons. First, the total Galactic reddening, $E(B-V)$, in the direction of HD\\,101584 (Galactic coordinates: $\\ell$\\,=\\,293$^\\circ$ and $b$\\,=\\,6$^\\circ$) is 0.25$^{\\rm m}$ (using the estimator in the NASA\/IPAC Infrared Science Archive), while the distance to the source is only of the order 1\\,kpc. Second, most of the radiated energy emerges at near- to far-IR wavelengths where the extinction is negligible.\n\\subsection{SED model results}\n\\label{s:sed_results}\n\nThe comparison between the model results and the data was carried out by eye. We obtain good fits using models with disks that are optically thick at visual and near-infrared wavelengths in the equatorial direction. This causes the source to appear much more luminous when viewed pole-on than edge-on. For HD\\,101584, with an inferred low inclination, this implies a lower luminosity for a given distance than previously presented in the literature. For instance, for a distance of 0.7\\,kpc, we find a luminosity of $\\approx$\\,750\\,$L_\\odot$, which is significantly smaller than the value of 5000\\,$L_\\odot$ considered by \\citet{bakketal96a}. The value of $A$ affects the circumstellar extinction along the line of sight and the relative flux density between visible and far-IR wavelengths. By decreasing $A$, and increasing the opacity along the light of sight, the stellar luminosity can be made larger, but we were unable to obtain values significantly larger than $\\approx$\\,1000\\,$L_\\odot$ for a distance of 0.7\\,kpc in the context of our models. The circumstellar dust optical depth in the visual and in the direction of HD\\,101584 is 0.3 for the best-fit model.\n\nWe are able to reproduce the compact, strong continuum emission at 1.3\\,mm observed using ALMA, and also the Herschel\/SPIRE data, only if we consider a grain-size distribution including relatively large grains, $\\sim$\\,1\\,mm in size, Fig.~\\ref{f:sed}). Interestingly, we find that the water ice grains must be kept smaller in order to reproduce the strength of the 45\\,$\\mu$m feature. Therefore, we have used a water ice opacity calculated for small grains, sizes $\\lesssim$\\,10\\,$\\mu$m. The value of $B$ is well constrained by fitting the radial profile of the ALMA 1.3\\,mm continuum image, Fig.~\\ref{f:cont_radial}, while the other parameters do not have a strong effect on it. The values of $A$ and $F$ are constrained from fitting the SED and we estimate that they are uncertain by a factor of a few. The estimated dust mass is $\\approx$\\,0.01\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$. A summary of the best-fit model is given in Table~\\ref{t:sed_fit}. A comparison between the radial brightness distribution of the $p$-H$_2$S(\\mbox{$2_{20}-2_{11}$}) line and the dust density distribution suggests that the inner region, except for the CCS component, is dust-dominated.\n\n\\begin{table}\n\\caption[]{Derived parameters for the dust envelope of HD\\,101584\\,$^1$.}\n\\begin{tabular}{lcl}\n\\hline \\hline\nParameter & Preferred value & Constraint\\\\\n\\hline\n$R_\\star$ [$R_\\odot$] & 18.6 & SED ($L_\\star$\\,=\\,1600\\,$L_\\odot$) \\\\\n$R_{\\rm in}$ [au] & 2.15 & Near-IR interferometry\\\\ \n$A$ & 750 & SED\\\\ \n$B$ & 1.2 & ALMA B6 image\\\\ \n$F$ & 5.0 & SED \\\\ \n$R_{\\rm disk}$ [au] & 1500 & ALMA B6 image \\\\ \n$R_{\\rm out}$ [au] & 3000 & ALMA B6 image\\\\ \n$M_{\\rm d}$ [$M_\\odot$] & 0.010 & SED \\\\ \n$a_{\\rm min}$ [$\\mu$m] & $\\sim$\\,10$^{-2}$ & Not well constrained \\\\ \n$a_{\\rm max}$ [$\\mu$m] & $\\sim$\\,10$^3$ & SED + ALMA B6 image \\\\ \n$p$ & -3.5 & SED \\\\\n$f_{\\rm ice}$ & 7\\,\\% & Water ice feature in the far-IR \\\\\n\\hline\n\\end{tabular}\n\\label{t:sed_fit}\n\\tablefoot{(1) Values for a distance of 1\\,kpc.}\n\\end{table}\n\nThe dust mass we derive is relatively large compared to the gas mass obtained from the observation of molecular lines, $M_{\\rm d}$\\,$\\approx$\\,0.01\\,$M_\\odot$ and $M_{\\rm g}$\\,$\\approx$\\,0.5\\,$M_\\odot$, respectively, at a distance of 1\\,kpc, that is, a gas-to-dust-mass ratio of 50 (distance-independent estimate). This is mainly caused by the large amount of mass in large grains. We attempted to decrease the dust mass required by our model by changing the exponent of the size distribution of the dust grains and by adding metallic iron grains \\citep[using optical constants from][]{ordaetal88} to the dust mix but without success. Hollow or fluffy grains may decrease the dust mass somewhat, but not substantially. We also managed to get a good fit to the SED using small, pure iron grains in combination with water ice. The total dust mass remains the same, but it requires an excessive amount of iron. Nevertheless, the uncertainties in the gas and dust mass estimates, due to many simplifying assumptions, are such that a more physically reasonable gas-to-dust mass ratio lies within their limits.\n\nA simple, but independent of any assumptions on the dust density distribution, estimate of the dust mass can be obtained using Eq.~(\\ref{e:dustmass}). A flux density of 202\\,mJy at 1.3\\,mm, a dust temperature of 50\\,K (the estimated gas temperature of the EDE component), and an opacity of 0.5\\,cm$^2$\\,g$^{-1}$ result in a dust mass of 0.02\\,$M_\\odot$ at a distance of 1\\,kpc, only a factor of two different from the more sophisticated estimate above.\n\n\\section{Discussion}\n\nAs discussed already in Sect.~\\ref{s:hd_distance} the evolutionary status of HD\\,101584 remains unclear. We will here provide arguments that favour an interpretation in the form of a post-RGB object. We will also compare the characteristics of HD\\,101584 with those of seemingly similar objects, that is, objects with recent, energetic, bipolar outflows and early-spectral-type central stars, as well as discuss to what extent premature termination of the red giant evolution is a common phenomenon or not. \n\\subsection{Evolutionary status}\n\\label{s:evol_status}\n\nThe estimated circumstellar CO isotopologue ratios \\citep{olofetal17}, and the reasonable assumption that in this case they directly reflect the C and O isotope ratios, give crucial information on the characteristics and evolutionary status of HD\\,101584. The low $^{12}$C\/$^{13}$C ratio, $\\approx$\\,13, provides strong evidence of CNO-processing and hence a location on or beyond the RGB \\citep{tsuj07}. A classification as a RSG or a YSO is highly unlikely, as it is only during red giant evolution that such nucleo-processed material is brought to the surface of the star. The high effective temperature means that it must be beyond the RGB or the AGB. Furthermore, the essentially solar $^{17}$O\/$^{18}$O ratio, $\\approx$\\,0.2, is a strong indication of a low-mass star, $\\la$\\,1\\,$M_\\odot$ \\citep{denuetal17}. For more massive stars the ratio will go up, and for stars going though hot-bottom-burning on the AGB the ratio is expected to be very high [for example, $>$\\,10 for a sample of OH\/IR stars, \\citet{justetal15}]. The fact that HD\\,101584 shows a (close to) solar abundance of N is not at odds with this, since the abundance of a lower-abundance element (like $^{13}$C) changes (in a relative sense) more easily than that of a higher-abundance element (like N). Likewise, the abundances of $^{17}$O and $^{18}$O are not affected by CNO-processing for a low-mass star \\citep{denuetal17}. This is also the justification for adopting a solar value for the O\/$^{18}$O ratio, and hence a solar CO\/C$^{18}$O ratio, in the circumstellar mass estimates.\n\nBeyond this, it is, however, difficult to make a firm conclusion. At the time that \\citet{partpott86} and \\citet{bakketal96a} studied this object, a post-AGB identification seemed the most reasonable. However, the detections of post-RGB objects in the Magellanic Clouds with many characteristics similar to those of post-AGB objects \\citep{kamaetal16} have opened up the possibility of an alternative identification. We will discuss these two possibilities in the light of our findings.\n\nIn the absence of a reliable distance estimate, in Sect.~\\ref{s:hd_distance} we introduced two characteristics of the primary star that are examples of a post-RGB ($L_\\ast$\\,=\\,500\\,$L_\\odot$, $D$\\,=\\,0.56\\,kpc) and a post-AGB ($L_\\ast$\\,=\\,5000\\,$L_\\odot$, $D$\\,=\\,1.8\\,kpc) object, both having an effective temperature of 8500\\,K. The distance dependence of the circumstellar mass estimate means that the circumstellar mass, and hence the ejected mass from the primary in our scenario, will be very different in the two cases. This will make one identification more likely than the other, although it is too early to make a firm statement on this.\n\nIn the low-$L_\\ast$ case we use a $M_{\\rm c}$--$L_\\ast$ ($M_{\\rm c}$ is the core mass) relation for RGB stars \\citep{bootsack88} to estimate a present stellar mass of 0.36\\,$M_\\odot$ (the high surface temperature makes it likely that only the core of the star remains). The circumstellar mass is estimated to be 0.18\\,$M_\\odot$ (where we have made a correction also for the missing flux as discussed in Sect.~\\ref{s:mass_summary}), that is, an initial stellar mass of $\\approx$\\,0.54\\,$M_\\odot$. The estimated dust mass and a canonical gas-to-dust mass ratio of 200 would raise the circumstellar mass to $\\approx$\\,0.6\\,$M_\\odot$ and the stellar mass to $\\approx$\\,1\\,$M_\\odot$. Thus, in the low-$L_\\ast$ case we have a low-mass star in the mass range \\mbox{0.5--1}\\,$M_\\odot$. This is fully compatible with the estimated low $^{17}$O\/$^{18}$O ratio. \n\nIn the high-$L_\\ast$ case we obtain a present stellar mass of 0.55\\,$M_\\odot$ using instead an $M_{\\rm c}$--$L_\\ast$ relation for AGB stars \\citep{bootsack88}. The circumstellar mass is estimated to be 1.8\\,$M_\\odot$, hence an initial stellar mass of $\\approx$\\,2.4\\,$M_\\odot$. The estimated dust mass and a canonical gas-to-dust mass ratio of 200 would raise the circumstellar mass to $\\approx$\\,6.5\\,$M_\\odot$ and the stellar mass to $\\approx$\\,7\\,$M_\\odot$, that is, close to, or maybe even above, the upper mass limit for an AGB star. Thus, in the high-$L_\\ast$ case we have a intermediate-mass star in the mass range \\mbox{2--7}\\,$M_\\odot$. This is not compatible with the estimated low $^{17}$O\/$^{18}$O ratio, which suggests a low-mass star.\n\nTaken together, these results favour a post-RGB scenario over a post-AGB scenario. As far as we understand, it is observationally very difficult, if not impossible, to distinguish between a post-RGB and a post-AGB star if the distance is not known. Circumstellar-wise there may be a difference. In the former case we expect no, or very little, remnant circumstellar material, while this is not necessarily the case for a post-AGB star where the AGB mass loss can be substantial. Unfortunately, it is difficult to estimate from our data whether or not a remnant CSE exists in the case of HD\\,101584. It may be that complementary observations with the Atacama Compact Array (ACA) of ALMA can shed light on this.\n\\subsection{The energetics and common-envelope evolution}\n\nThe kinetic energy of the outflowing gas is estimated to be 2$\\times$10$^{45}$\\,erg and 2$\\times$10$^{46}$\\,erg in the low-$L_\\ast$ and high-$L_\\ast$ cases. Both values are very high, and it is remarked already in Sect.~\\ref{s:energy} that the scalar momentum rate cannot, by a large margin, be supplied by the stellar radiation. It is difficult to reconcile such high kinetic energies with anything but a CE evolution scenario where gravitational binding energy is released when the companion is captured by, and falls towards, the primary. Even more energy may be released if material falls towards the companion, possibly forming an accretion disk.\n\nFollowing the same procedure as in \\citet{olofetal15} we estimate the orbital characteristics from the mass function (assuming circular orbit; ps = primary star, cs = companion star),\n\\begin{equation}\n\\frac{(M_{\\rm cs} \\sin i)^3}{(M_{\\rm ps} + M_{\\rm cs})^2} = \\frac{4\\pi^2 (a_{\\rm ps} \\sin i)^3}{G P^2} \\,,\n\\end{equation}\nand the expression for the semi-major axis of the primary star's orbit\n\\begin{equation}\na_{\\rm ps} \\sin i = \\frac{K_{\\rm ps}P}{2\\pi} \\,.\n\\end{equation}\nwhere $K_{\\rm ps}$ is the semi-amplitude of the velocity curve of the primary, $P$ the orbital period, and $G$ the gravitational constant. The semi-major axis of the companion's orbit is obtained from $M_{\\rm ps}a_{\\rm ps}$\\,=\\,$M_{\\rm cs}a_{\\rm cs}$. The released gravitational energy is obtained using\n\\begin{equation}\nE_{\\rm rel} = -\\frac{G\\,M_{\\rm ps,i}\\,M_{\\rm cs}}{2a_{\\rm i}} + \\frac{G\\,M_{\\rm ps}\\,M_{\\rm cs}}{2a}\\,, \n\\end{equation}\nwhere $a$\\,=\\,$a_{\\rm ps}$\\,+\\,$a_{\\rm cs}$, and $a_{\\rm i}$ is the initial separation, and $M_{\\rm ps,i}$ the initial mass of the primary star. \n\nWe adopt here the orbit period estimated from the radial velocity data \\citep{diazetal07}, $P$\\,=\\,144$^{\\rm d}$, $K_{\\rm ps}$\\,=\\,3\\,km\\,s$^{-1}$ \\citep{bakketal96a}, and the inclination angle of 10$^\\circ$ estimated in this paper. For the two estimated $M_{\\rm ps}$ of 0.36 and 0.55\\,$M_\\odot$ we find companion masses of 0.27 and 0.41\\,$M_\\odot$, respectively, and a separations of 0.53\\,au in both cases. The released gravitational energy depends on the initial separation, $a_{\\rm i}$. We assume 1 and 4\\,au in the post-RGB and post-AGB cases, respectively [the latter is larger because an AGB star is larger, especially during a thermal pulse; for a detailed study of the capture process, see \\citet{madaetal16}]. The resulting released gravitational energies are 3$\\times$10$^{44}$ and 2$\\times$10$^{45}$\\,{\\rm erg} in the post-RGB and post-AGB cases, respectively. Consequently, the ratio between kinetic and released gravitational energy is about 10 in both cases. There is also energy released due to hydrogen recombination, but according to \\citet{sokeetal18} it contributes little to removing the stellar envelope.\n\nIt must be emphasized that there are considerable uncertainties in both the estimated energy released when the companion spirals inwards and the estimated kinetic energy of the outflowing gas. Furthermore, the CE evolution scenario is complex with an uncertain energy transfer efficiency \\citep{ivanetal13}. Nevertheless, the discrepancy between the released energy and kinetic energy estimates is so large that it must be concluded that energy released by the inward motion of the companion alone is not enough to explain the characteristics of HD\\,101584. A further possibility is that additional gravitational energy is released as circumstellar material falls towards the companion. The effect of such an infall may be the formation of a circum-companion accretion disk.\n\nIn turn, this may provide an efficient mechanism for driving an outflow via a jet \\citep{gorletal12, gorletal15}. \\citet{blaclucc14} have estimated the required accretion rate to drive an outflow, of given characteristics, under such circumstances. Assuming that the accretion disk surrounds an 0.27\\,$M_\\odot$ MS-star in the RGB case, we derive a required accretion rate of 2\\,$\\times$\\,10$^{-5}$\\,($Q$\/2)\\,$M_\\odot$\\,yr$^{-1}$ for the outflowing gas of HD\\,101584 ($Q$ is a numerical factor typically in the range 1 to 5 in jet models). The corresponding values for an 0.55\\,$M_\\odot$ MS-star or WD in the post-AGB case are 7\\,$\\times$\\,10$^{-5}$\\,($Q$\/2) and 10$^{-5}$\\,($Q$\/2)\\,$M_\\odot$\\,yr$^{-1}$, respectively. This is below the Eddington accretion rates in all cases, but well above what can be obtained from Bondi-Hoyle-Lyttleton accretion and wind-Roche-lobe overflow. Only accretion in connection with CE evolution, or possibly what is termed ``Red Rectangle Roche-lobe overflow'' in the MS-star case, will fit the requirements.\n\\subsection{Termination of red giant evolution}\n\nThere is good evidence that the red-giant evolution of HD\\,101584 was prematurely ended by a CE evolution. Whether it happened on the RGB or the AGB remains open. If sufficiently common, premature termination of red giant evolution may have a significant effect on for example the elemental synthesis of AGB stars and their contribution to the integrated light of galaxies, since such estimates are based on results from single-star evolution models. \n\nBased on the probability of having a companion and the period distribution of main sequence stars \\citep{raghetal10}, one can estimate how common binary systems, with the required characteristics to achieve CE evolution, are (it should be noted that the frequency of post-RGB binaries of this type will strongly affect the frequency of post-AGB binaries of this type). The estimate has been done for AGB stars. By considering only binaries with initial separations between that of a couple of maximum radii of an RGB star and the same for an AGB star, \\citet{madaetal16} estimated that only a few per cent of the AGB stars can interact strongly, that is, go through CE evolution, with a companion. \n\nThe observational value is substantially higher, but far from 100\\,\\%. Observationally-based constraints can be obtained from studies of PNe. Looking at purely statistical information, that is, disregarding whether or not close binaries are required to shape PNe, \\citet{miszetal09} used PNe in the Galactic bulge to estimate a close binary fraction of 12--21\\,\\%. \\citet{nieetal12} used a Monte Carlo simulation and the observed frequency of sequence E binaries in period-luminosity diagrams in the Large Magellanic Cloud as observational constraint. They found that the fraction of PNe with close binary central stars is 7--9\\,\\%, and the fraction having separations capable of influencing the nebula morphology (set as orbital periods less than $<$\\,500\\,yr) is 23--27\\,\\%. In summary, only about 1 in 5 PNe seem to have their origin in CE evolution. However, this may be a (significant) underestimate as argued by for example \\citet{demaetal17}. Therefore the question whether or not a close companion is a pre-requisite for the formation of PNe appears to remain open. A related issue is that post-CE-evolution objects may for some reason dominate in observed samples, hence giving a distorted view of the characteristics of AGB stars and PNe.\n\nLooking at the problem from the side of the PN morphology and kinematics, the evidence is strong that in those cases where the central star of a PNe has gone through CE evolution, there is also a link to the spatio-kinematic evolution of the nebula \\citep{hillwetal16}, and some understanding how this works \\citep{garcetal18}. This adds strength to the question whether the estimated low percentage of close-binary systems among PNe is wrong. Currently, there exist no viable way in which also single stars produce PNe with complex morphologies and kinematics.\n\nIf low-luminosity PNe have their origin in RGB stars, the CE-evolution process must be crucial for a successful result, since normal stellar mass loss on the RGB occurs at substantially lower rates than on the AGB and the evolution must be terminated before reaching the tip of the RGB and the start of He-burning. It remains to determine the fraction of this type of PNe that have their origin in RGB stars.\n\\subsection{Comparison with objects of similar characteristics}\n\n\\begin{table*}\n\\caption{Characteristics of selected evolved objects with extreme high-velocity outflows.}\n\\centering\n\\begin{tabular}{l c c c c c c c}\n\\hline \\hline\nName & Spectral type & $^{12}$C\/$^{13}$C & $\\upsilon_{\\rm max}$ & Kinematical age & $M_{\\rm HVO}$ & $E_{\\rm HVO}$ & $P_{\\rm HVO}$ \\\\\n & & & [km\\,s$^{-1}$] & [yr] &[$M_\\odot$] & [erg] & [g\\,cm\\,s$^{-1}$] \\\\\n\\hline \nHD\\,101584\\,$^1$ & A6Ie & 13\\,\\phantom{$^0$} & 150\\,\\phantom{$^0$} & 770 & 0.04\\phantom{0} & \\phantom{$>$}\\,6$\\times$10$^{45}$ & \\phantom{$>$}\\,1$\\times$10$^{39}$ \\\\\nIRAS\\,08005--2356 & F5Ie & 10\\,\\phantom{$^0$} & 200\\,$^2$ & 190 & 0.08\\phantom{0} & \\phantom{$>$}\\,3$\\times$10$^{45}$ & \\phantom{$>$}\\,3$\\times$10$^{39}$\\\\\nIRAS\\,16342--3814\\,$^3$ & obscured & \\phantom{0}\\,3\\,$^4$ & 250\\,\\phantom{$^0$} & 110 & 0.006 & $>$\\,3$\\times$10$^{45}$ & $>$\\,2$\\times$10$^{38}$\\\\\nIRAS\\,22036+5306 & F5\\,$^5$ &\\phantom{0}\\,6\\,\\phantom{$^0$} & 250\\,$^2$ & \\phantom{0}25 & 0.03\\phantom{0} & \\ldots & \\ldots \\\\\n\\hline\n\\end{tabular}\n\\label{t:hvos}\n\\tablefoot{(1) Values for a distance of 1\\,kpc. (2) Inclination angles of 30$^\\circ$ assumed. (3) This source also has an extreme-high-velocity outflow with a maximum velocity of $\\approx$\\,430\\,km\\,s$^{-1}$. (4) The ratio is estimated to be very low, but the value of 3 is assumed. (5) Uncertain, but not later than F5.}\n\\end{table*}\n\n\nThere exists in the literature a number of sources with characteristics similar to that of HD\\,101584, in particular the presence of a bipolar outflow with a very high maximum velocity, $>$\\,100\\,km\\,s$^{-1}$. These are particularly interesting since the estimated ages of the outflows are of the order only a few\\,$\\times$\\,10$^2$ years, i.e, they are objects for which the observed phenomenon is very energetic and recent. A number of them have been imaged in CO line emission, for example, IRAS\\,08005--2356 \\citep{sahapate15}, IRAS\\,16342--3814 \\citep{sahaetal17a}, and IRAS\\,22026+5306 \\citep{sahaetal06}. Some of their relevant characteristics (and those of HD\\,101584) are summarised in Table~\\ref{t:hvos}, and we will here give a brief comparison between HD\\,101584 and these sources. The central stars, when identified, are all of relatively early spectral type, F5 or earlier, and they all have low $^{12}$C\/$^{13}$C ratios, $\\la$\\,10. The maximum expansion velocities of the high-velocity outflows lie in the range 150 to 250\\,km\\,s$^{-1}$, and their masses are of the order 10$^{-(1-2)}$\\,$M_\\odot$. The energies and scalar momenta lie within a factor of a few, and within the ranges found by \\citet{bujaetal01} for a larger sample of proto-PNe. The momentum rates of their outflows are much higher than can be supplied by radiation. \n\nIRAS\\,16342--3814 has been studied in most detail using ALMA \\citep{sahaetal17a}, and some further comparison can therefore be made with this object. It has an additional extremely-high-velocity bipolar outflow with a maximum expansion velocity of $\\approx$\\,430\\,km\\,s$^{-1}$ (adopting an inclination angle of 43$^\\circ$) and a PA slightly different than that of the high-velocity outflow. In terms of mass, energy, and scalar momentum, this outflow has 7\\,\\%, 51\\,\\%, and 21\\,\\% of that of its high-velocity outflow, and it is estimated to be older suggesting a recurrent phenomenon. IRAS\\,16342--3814 further has an expanding equatorial density enhancement in the form of a torus \\citep[that obscures the central star even at 12\\,$\\mu$m;][]{verhetal09}. Its estimated size, expansion velocity, and particle density are 1300\\,au, 20\\,km\\,s$^{-1}$, and 10$^{(6-8)}$\\,cm$^{-3}$, respectively (the expansion age is 160\\,years). Thus, its characteristics are comparable to those of the EDE component of HD\\,101584. The molecular species detected towards this source is still limited, but include CO, SiO, SO, SO$_2$, and HCN (in some cases through their rare isotopologues). \n\nAnother type of possibly related objects are the red novae, most likely the stellar-merger end products of a CE evolution. These are objects that also show spectacular circumstellar characteristics in molecular line emission \\citep{kamietal18b}. An interesting example is CK~Vul, an object where the radioactive molecule $^{26}$AlF was recently detected \\citep{kamietal18a}. The Boomerang nebula could be of this type \\citep{sahaetal17b}. Its circumstellar characteristics are remarkable, in particular the spherical outflow of high velocity where the gas has cooled down to temperatures below that of the cosmic microwave background. In addition, there is an equatorial density enhancement and a bipolar high-velocity outflow. The circumstellar mass is high, $\\ga$\\,3\\,$M_\\odot$, and consequently the mass of the star, $\\ga$\\,4\\,$M_\\odot$. \\citet{sahaetal17b} argued that this object is in a post-CE-evolution phase after the companion merged into the core of an RGB star.\n\nIn terms of chemistry, OH231.8+4.2, an object with a morphology similar to that of HD\\,101584 \\citep{alcoetal01, bujaetal02}, is a suitable object to compare with \\citep{sacoetal15, velietal15}, but the chemical aspects will be discussed in a forthcoming paper. Also the red novae are interesting comparison sources in terms of circumstellar chemistry \\citep{kamietal18b}.\n\n\\section{Conclusions}\n\nWe have used ALMA and single-dish data to determine the physical and chemical characteristics of the circumstellar environment of the star HD\\,101584, a binary system with a period in the range 150--200 days consisting of a luminous, evolved star and a low-mass companion. The circumstellar medium is rich in molecules of different types, 12 (not counting isotopologues) have been detected, and a significant fraction of the line emissions have been mapped with angular resolutions in the range 0\\farcs1 to 0\\farcs6. The different chemistry and excitation conditions required by different molecules have been utilised in the analysis presented in this paper, where we focus on the physical characteristics of the circumstellar medium, and its consequences for the interpretation of the evolutionary status of HD\\,101584. An SED has been constructed, using also our ArTeMiS and ALMA data, to provide complementary information on the dusty circumstellar medium.\n\nThe circumstellar chemistry will be discussed in a forthcoming paper, but we note that the detected lines are typical for an oxygen-rich chemistry with a significant presence of sulphur species, but also weak lines from carbon-species, other than CO, are present. In addition, the existence of more complex species (in the circumstellar context) like H$_2$CO and CH$_3$OH points towards a chemistry that is most likely affected by also shocks and\/or dust grains. \n\nWe have identified four distinct components of the circumstellar medium, and the most likely interpretations are: {\\it i)} a slowly rotating circumbinary disk of diameter $\\approx$\\,150\\,[$D$\/1\\,kpc]\\,au, {\\it ii)} an expanding disk or torus of diameter $\\approx$\\,3000\\,[$D$\/1\\,kpc]\\,au, {\\it iii)} a bipolar high-velocity outflow reaching 24\\,000\\,[$D$\/1\\,kpc]\\,au from the centre, surrounded by {\\it iv)} an hourglass structure forming the inner part of two bubbles that enclose the outflow. The outflow is oriented essentially along the line of sight, an inclination angle of only 10$^\\circ$\\,(-5$^\\circ$,+10$^\\circ$, {\\bf 2$\\sigma$ errors}), and the central star is seen through a region that has been (at least partly) cleared from material. The circumbinary disk and the expanding equatorial disk\/torus are seen close to face-on. There is strong evidence of a second bipolar outflow in a direction different from that of the major outflow. In addition, there is structure in the hourglass component that is not understood, but it can possibly be related to an interaction between the two outflows. A 3D-reconstruction of the source has been attempted based on the assumption that the radial outflow velocity scales linearly with the distance to the centre. The mass of the circumstellar gas is $\\approx$\\,0.5\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$, about half of it lies in the expanding equatorial density enhancement. \n\nThe 1.3\\,mm continuum is dominated by emission from the equatorial disk\/torus, but 30--50\\,\\% of the flux comes from extended low-brightness emission whose morphology is difficult to determine. The innermost region of the circumbinary disk is particularly prominent, and the estimated size is $\\approx$\\,10\\,[$D$\/1\\,kpc]\\,au. The position of the continuum peak coincides with the Gaia position of HD\\,101584 within the uncertainties. Modelling the full SED in terms of a flared disk seen close to face-on (the equatorial density enhancement), surrounded by a much thinner spherical envelope, leads to a dust mass estimate of $\\approx$\\,0.01\\,[$D$\/1\\,kpc]$^2$\\,$M_\\odot$, and a substantial fraction of this mass must be in the form of large-sized, up to 1\\,mm, grains. About 7\\,\\% of the grains are in the form of crystalline water.\n\nThe absence of a reliable distance estimate makes the identification of the evolutionary status of HD\\,101584 difficult. The low estimated $^{12}$C\/$^{13}$C ratio and the high effective temperature are strong arguments for a phase beyond the RGB, that is, either post-RGB or post-AGB. We have therefore looked at two separate cases, one of lower luminosity (a post-RGB star) and one of higher luminosity (a post-AGB star). Relations between core mass and luminosity provide estimates of the present day masses of the stars. Combined with the circumstellar masses, this lead to estimated initial masses that lie in the ranges 0.5--1\\,$M_\\odot$ and 2--7\\,$M_\\odot$ in the post-RGB and post-AGB cases, respectively. The low estimated $^{17}$O\/$^{18}$O is consistent with a low-mass post-RGB star, while it is inconsistent with an intermediate-mass post-AGB star. Thus, based on these data we advocate a post-RGB identification.\n\nIrrespective of the evolutionary status, the results presented in this paper favour an interpretation of HD\\,101584 as an object where the red-giant evolution was terminated prematurely due to a CE evolution that ended avoiding a stellar merger. The remaining hydrogen envelope of HD\\,101584 was ejected during the interaction and it now forms a circumstellar medium of considerable complexity. The size and kinematics of the bipolar high-velocity outflow provide a time scale of $\\approx$\\,770\\,[$D$\/1\\,kpc]\\,yr for the circumstellar evolution. This is a time scale in line with those estimated for apparently similar objects. The considerably shorter kinematical age of the expanding equatorial disk\/torus is at odds with the conclusion by \\citet{hugg07} that the torus develops first and the jet follows shortly thereafter. However, the almost face-on orientation of the equatorial disk\/torus around HD\\,101584 makes it difficult to estimate reliably its kinematics and hence its age. Most likely, the ejection of the stellar envelope was powered by released gravitational energy. We estimate that the kinetic energy of the accelerated gas is 7$\\times$10$^{45}$\\,[$D$\/1\\,kpc]$^2$\\,erg, and that the kinetic to released gravitational energy is about 10, irrespective of the evolutionary status. This indicates that substantial additional energy must have been released, for example, due to material falling towards the companion. As a consequence a circum-companion accretion disk may have formed that now drives a highly collimated jet that, in turn, drives the expanding high-velocity molecular gas. The existence of the second bipolar outflow points to a recurrent phenomenon rather than a single explosive event, although its age is uncertain. Its interpretation in the light of a CE-evolution scenario is not clear.\n\n\n\\begin{acknowledgements}\nHO and WV acknowledge support from the Swedish Research Council. WV acknowledges support from the ERC through consolidator grant 614264.\nThis paper makes use of the following ALMA data: ADS\/JAO.ALMA\\#2012.1.00248.S and \\#2015.1.00078.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ.\nThis paper makes use of the following APEX data: O-093.F-9307, O-096.F-9303, O-098.F-9314, O-099.F-9310, O-0100.F-9302, O-0101.F-9302, and O-0102.F-9402A (SEPIA Band 9 Science Verification data). The Atacama Pathfinder EXperiment (APEX) is a collaboration between the Max-Planck-Institut f{\\\"u}r Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. This paper has also made use of the NASA\/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration, and data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory\/California Institute of Technology, funded by the National Aeronautics and Space Administration. Frederic Sch{\\\"u}ller is gratefully acknowledged for help with the reduction of the ArTeMiS data. Finally, we are grateful to the anonymous referee who provided a very constructive report.\n\\end{acknowledgements}\n\n\n\\bibpunct{(}{)}{;}{a}{}{,}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Background and motivation}\nThere are several lines of evidence that OH17.7$-$2.0 is an\noxygen-rich post-AGB object. Its central star has a spectral \ntype earlier than $\\sim$K5 with an effective temperature \n$\\sim$4000 $-$ 10000\\,K (\\cite{lebertre89}). This is a dusty\nobject with far-infrared colours consistent with a detached shell \nand mass-loss rate of $\\sim4\\times 10^{-5}$M$_{\\odot}$yr$^{-1}$\n(\\cite{vanderveen95}). At 2.2\\,$\\mu$m it appears as a prolate \nspheroid with the polar axis at a position angle of 20$^{\\circ}$\n(\\cite{bains03}). Non-variability or only slight variations at \nnear-infrared wavelengths (\\cite{lebertre89}), small amplitude\nOH flux variations at 1612\\,MHz (\\cite{herman85}), disappearance\nof the H$_2$O maser (\\cite{engels02}) and non-detection of SiO masers\n(\\cite{nyman98}) are consistent with a post-AGB status of \nOH17.7$-$2.0. \n\nIn 2003, the OH 1612-MHz maser line profile underwent a very \nunusual change; the intensity of the red-shifted emission at \nvelocity $\\sim$73\\,km\\,s$^{-1}$ increased nearly threefold over\na period of $\\sim$430\\,days (\\cite{szymczak05}). The flaring \nfeature exhibits strong (up to 80\\%) circular polarization\nand considerable ($\\sim$15\\%) linear polarization. \nIt is remarkable that adjacent red-shifted features do not show\nvariations within an absolute flux density uncertainty less than\n10\\%. Furthermore, the entire blue-shifted part of the 1612-MHz \nspectrum does not show any variations either. \nFinally, no changes were detected in the integrated flux \ndensities of the 1665 and 1667-MHz maser lines (\\cite{szymczak05}).\n \nThe outburst event in OH17.7$-$2.0 is spectacular and unique \nwhen compared to flare phenomena in circumstellar OH masers reported \nin the literature. Almost all eruptive changes in OH maser emission\nobserved so far took place in Mira-type variables and are\ncharacterized by global changes with time scales of months and years\nin the OH profiles of the 1612, 1665 and 1667-MHz transitions when \npresent (\\cite{jewell81}, \\cite{etoka97}), usually weakly associated \nwith changes in the optical and\/or infrared.\n\nIn this paper, we report the preliminary results of follow-up VLBI \nobservations taken in order to localize the burst in the shell and \nto determine the properties of the active regions. This should allow us \nto constrain possible causes of the outburst (\\cite{szymczak05}) \nand to understand the evolutionary status of the central star.\n\n\n\\section{Observations and reduction}\nThe OH 1612-MHz observations were carried out on 2005 March 3 \nusing eight EVN telescopes: Cambridge, Effelsberg, Hartebeesthoek,\nJodrell Bank, Medicina, Onsala and Toru\\'n. \nThe observations consisted of six 38-min. scans on the target \ninterleaved with 6-min. scans on the phase-calibrator source \nJ1733$-$1304. One-hour scan of the continuum source J2253+1608\nwas also made for the purpose of bandpass and amplitude\ncalibration. Dual circular polarization was recorded using the\nMarkV system with a spectral bandwidth of 0.5\\,MHz. \nFor OH17.7$-$2.0, this translates to LSR velocities from 15 to \n108\\,km\\,s$^{-1}$. The data were correlated with the JIVE correlator \nusing 1024 channels, yielding a spectral resolution of 0.09\\,km\\,s$^{-1}$. \nCorrelated data were calibrated and reduced using AIPS package, \nfollowing the standard procedures for spectral-line \nVLBI observations. The resulting synthesized beam size was \n64$\\times$20\\,mas at a position angle 9$^{\\circ}$.\nAn area of $2\\times2$~arcsec$^2$ was searched for maser emission\nabove 5$\\sigma$ limit over the entire band.\nFor single circular polarization, the rms noise in the final \nemission-free channel maps was $\\sim$9\\,mJy.\nThe relative positions of the maser components were determined\nwith 0.5\\,mas accuracy.\n\n\\section{Results}\nOH 1612-MHz maser emission was detected at velocities between\n47.9 and 75.0\\,km\\,s$^{-1}$; this is a slightly narrower range than\nthat reported in MERLIN observations (\\cite{bains03}). \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.6\\textwidth]{zmom_oh.ps}\n\\caption{Integrated intensity map of the OH 1612-MHz maser emission \n of OH17.7$-$2.0 for left-hand circular (LHC) polarization. \n The contours plotted at intervals of\n (-1, 1, 2, 4, 8, 16)$\\times$20\\,mJy\\,beam$^{-1}$\\,m\\,s$^{-1}$ \n show the blue-shifted emission ($V<$61.6\\,km\\,s$^{-1}$). \n The red-shifted emission ($V>$61.6\\,km\\,s$^{-1}$) is shown in grey \n that scales from 0 to 500\\,mJy\\,beam$^{-1}$\\,m\\,s$^{-1}$. \n Note that, for better presentation, the grey scale is cut at \n 500\\,mJy\\,beam$^{-1}$\\,m\\,s$^{-1}$ level and the peak intensity of \n the southern unresolved component is\n 9931\\,mJy\\,beam$^{-1}$\\,m\\,s$^{-1}$. \n The map origin is at RA(J2000)=18$^{\\rm h}$30$^{\\rm m}$30\\fsec695,\n Dec(J2000)=$-$14\\degr28\\arcmin56\\farcs82.\n The ellipse in the bottom left corner of the map represents \n the restoring beam.}\n\\label{fig1}\n\\end{figure}\n\nFig.\\,1 shows the LHC map of line emission integrated over all channels. \nMost of the blue-shifted emission originates from a ring-like\nstructure which resembles a symmetric shell of about 510~mas\ndiameter. No emission is seen from the S\nand the SE parts of the shell. Compact and unresolved \nemission in the velocity range of 48.6$-$49.6\\,km\\,s$^{-1}$ is found at \na position which coincides, within the synthesized beam, with the\nposition of the central star inferred from the model of expanding\nspherical shell. Thus, this compact emission is very likely the amplified\nstellar image. Weak and diffuse maser components at blue-shifted\nvelocities higher than 48.5\\,km\\,s$^{-1}$ are seen $\\sim$380\\,mas \nto the SE and $\\sim$600\\,mas to the NW of that compact emission \n(the amplified stellar image), at a position angle of \n$\\sim$160$^{\\circ}$. The red-shifted diffuse emission is also detected at\nroughly the same position angle but is weakly visible in Fig.\\,1 due \nto dynamic range limitations. Weak and scattered emission likely comes\nfrom an equatorial region. The red-shifted emission is dominated by\ntwo very strong (>150\\,Jy\\,beam$^{-1}$) unresolved components at\nvelocities $\\sim$73\\,km\\,s$^{-1}$ and at a projected distance of\n$\\sim$400\\,mas south of the central star. The brightness temperature\nof these bursting components is up to 10$^{11}$\\,K. The rest of the\nred-shifted emission follows a ring-like distribution with the SW part\nclearly seen. Clearly, there is a lack of red-shifted emission from\nthe northern side of the shell. \n\n\\section{Interpretation}\nIn general, the masers are located within a remnant spherical shell\n(Fig. 2). However, there is evidence for several major departures from \nthe simple shell model. (1) There is an offset of $\\sim$300\\,mas\nbetween the two emission peaks at extreme velocities along an axis \nat position angle of 8.4$^{\\circ}$. (2) Along the same position angle, \na biconal region can be seen; the emission appears to come from the sides \nof the cones as projected on the sky and likely represents\nthe interaction of a faster post-AGB wind or jet with the outer AGB wind.\nIn this case, the boundary appears to be thin and tangential emission\nmay dominate. The opening angle of the cone is less than 15$^{\\circ}$\nwhile the inclination angle between the equatorial plane and the line\nof sight is $\\sim30^{\\circ}$. The red-shifted bubble is at \nprojected distance of $\\sim400$\\,mas. (3) Low-brightness emission at\nmiddle radial velocities scattered along a plane roughly orthogonal to\nthe axis of the cones is likely due to the emission from the denser\nequatorial region or torus-like structure. Such a region can be\nproduced during a period of enhanced mass loss at the end of the AGB \nphase. The OH maser emission from this region, as predicted by a spherical \nshell model, should be weak due to short amplification path length. \n\nIt is remarkable that the position angle of the axis of biconal region \nseen in the OH 1612-MHz maser distribution ($\\sim10^{\\circ}$) is \nquite consistent with the position angle of the major axis of the\ninfrared nebulosity of size 2.5\\,arcsec ($20^{\\circ}$, \\cite{bains03}).\nThis confirms that OH17.7$-$2.0 recently started a bipolar outflow.\n\nThe position-velocity diagram (Fig. 3) implies that a spherical \nvelocity field for the bulk of the 1612-MHz emission, in the first \ninstance, satisfactorily fits all the data. The position of the central star \nis $\\Delta\\alpha$=74\\,mas, $\\Delta\\delta$=396\\,mas relative to the map\norigin (Fig. 1) and coincides within the beam with the position of \nthe compact blue-shifted emission which is interpreted as the\namplified stellar image. \nThe best-fitted systemic and expansion velocities are \n61.6\\,km\\,s$^{-1}$ and 13.8\\,km\\,s$^{-1}$, respectively. \nThe inner radius is 220\\,mas while the outer radius is up to 850\\,mas.\nAlthough the envelope is sparsely filled with the maser components,\nFig. 3 shows evidence for multiple shells which may result from\nperiodic enhancements of mass loss rate within a range of 300-800\nyears, for the assumed distance of 3.4\\,kpc. It appears that the maser\noutburst occurred in a shell of radius of 740\\,mas, i.e., 2500\\,AU from\nthe central star. For the first time, the present data provide evidence \nthat OH17.7$-$2.0 has a bipolar morphology of the 1612-MHz maser\nemission and the active region is confined to the surface of bubble.\nWe argue that one of most plausible causes of the burst is an\ninteraction of a jet-like outflow with the remnant outer AGB shell.\nIf the disappearance of the H$_2$O maser in 1990 (\\cite{engels02}) is \ncasually related to the 1612-MHz OH outburst in 2003 and assuming\nthat the H$_2$O maser arose in an inner AGB shell ($\\sim$50\\,AU), \nthe speed of the jet-like outflow should be $\\sim$800\\,km\\,s$^{-1}$. \nThe presence of such outflows is documented in many proto-planetary \nnebulae, but their effect on the onset of the OH maser emission is \nnot well understood (\\cite{szymczak05}). Our data suggest that \nthe outburst emerges from a narrow interface between the surface \nof the bubble and dense clumps of the detached AGB shell where the \nmagnetic field must play a role in the maser amplification since only \none sense of circular polarization is seen.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.6\\textwidth]{oh_spot.eps}\n\\caption{The overall distribution of all maser components\n brighter than 10$\\sigma$ ($>$100\\,mJy\\,beam$^{-1}$). The origin of\n diagram is the position of the brightest component. The size of the\n symbols is proportional to the logarithm of brightness. \n The colour corresponds to the velocity scale shown by a vertical bar.\n The dashed lines outline the polar cavities where the maser emission\n is quenched while it is present at the biconal surface.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.6\\textwidth]{oh17_r-v.eps}\n\\caption{Radial offset of the maser components from the best fitted\n stellar position as a function of the radial velocity. The circle\n size is proportional to the logarithm of the component brightness.\n Colours distinguish the components from the shells (red), the equatorial \n region (cyan) and amplification of stellar photons (blue). \n The dashed curves correspond to the models of spherical shells of\n radii 220, 300, 420 and 740\\,mas assuming a constant outflow velocity\n of 13.8\\,km\\,s$^{-1}$.}\n\\label{fig3}\n\\end{figure}\n\nMERLIN observations of OH17.7$-$2.0 revealed the southern outlying\ncomponents (\\cite{bains03}) which are interpreted as the result of the\ninteraction of a fast post-AGB wind with the remnant outer AGB wind.\nIn this scenario, the OH masers also trace the radial outflow away from\nthe central star along the surface of the cones; the velocities of\nthese maser components increase linearly with the distance from the star \n(\\cite{zijlstra01}).\nThe presence of such OH components was confirmed in some post-AGB\nobjects with broad ($>$50\\,km\\,s$^{-1}$) OH profiles\n(\\cite{zijlstra01}). We failed to find a similar phenomenon in our\ntarget. This suggests that OH17.7$-$2.0 is in an early post-AGB\nevolution phase.\n\n\n\n\n\\section{Concluding remarks}\nThe most striking result from the first-epoch EVN observations \nis the identification of an active region where the 1612-MHz outburst \noccurred. This finding strongly constrains a suite of possible causes\nof the outburst (\\cite{szymczak05}). It appears that one of the most\nprobable causes is the interaction of a jet-like outflow from the\npost-AGB star with dense clumps of the remnant outer AGB shell. \nThis confirms that OH17.7$-$2.0 is in a transitional, short-lasting\nphase from AGB to PN. Multi-epoch VLBI observations would allow us \nto trace the changes in the structure of OH maser emission and \nto explain a possible role of the magnetic field in the shaping\nof the bipolar outflow. It seems that this target is ideal\nfor optical and infrared observations in order to obtain a\ncomplete picture of the changes occurring in the whole envelope.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLet $\\mathcal{P}$ be a set of $v \\geq1$ elements and $\\mathcal{B}$ be a set of $k$-subsets of $\\mathcal{P},$ where $k$ is a positive integer with $1\\leq k \\leq v$. Let $t$ be a positive integer with $t\\leq k.$ If every $t$-subset of $\\mathcal{P}$ is contained in exactly $\\lambda$ elements of $\\mathcal{B}$, then we call the pair $\\mathbb{D}=(\\mathcal{P}, \\mathcal{B})$ a $t$-$(v,k,\\lambda)$ {\\em design}, or simply a {\\em $t$-design}. The elements of $\\mathcal{P}$ are called\n{\\em points}, and those of $\\mathcal{B}$ are referred to as {\\em blocks}. We often denote the number of blocks by $b$ and a $t$-design is simple when there is no repeated blocks in $\\mathcal{B}$. A $t$-design is called {\\em symmetric} if $v=b$ and {\\em trivial} if $k=t$ or $k=v$. Throughout this paper we study only simple $t$-designs with $t < k < v$. When $t\\geq 2$ and $\\lambda=1$, a $t$-design is called a {\\em Steiner system}. Clearly, the parameters of a $t$-$(v,k,\\lambda)$ design are restricted by the following identity. \n\\begin{equation}\\label{condition}\n b {k \\choose t} = \\lambda {v \\choose t}.\n\\end{equation}\n\nThe interplay between codes and $t$-designs has been ongoing for decades. On one hand, the incidence matrix of a $t$-design over any finite field can serve as a generator matrix of a linear code and much progress has been made (see~\\cite{AK92,Ding15b,KP95,KP03,Ton98,Ton07}). On the other hand, linear and nonlinear codes may hold $t$-designs. As a classical example, $4$-designs and $5$-designs with certain parameters were derived from binary and ternary Golay codes. Recently, Ding and Li~\\cite{DL17} obtained infinite families of $2$-designs from $p$-ary Hamming codes, ternary projective cyclic codes, binary codes with two zeros and their duals. They also obtained $3$-designs from the extended codes of these codes and RM codes. More recently, infinite families of $2$-designs and $3$-designs from some classes of binary linear codes with five weights were given by Ding ~\\cite{Ding182}. For other constructions of $t$-designs, for example, see~\\cite{BJL99,CM06,MS771,RR10}.\n\nThe objective of this paper is to construct $2$-designs from two classes of cyclic codes obtained from the triple-error correcting narrow-sense primitive BCH codes and the cyclic codes related to the generalized Kasami case, respectively. In the following, we will first present the weight distributions of these two classes of cyclic codes, and then explicitly determine the parameters of the derived $2$-designs.\n\n\n\\section{The classical construction of $t$-designs from affine-invariant codes }\n\nThroughout this paper, let $p=2$, $m=2s$, $\\gcd(s, l)=d$ and $\\gcd(s+l, 2l)=d'$, where both $s\\geq 2$ and $1\\leq l\\leq m-1$ are positive integers with $l\\neq s$. Let\n$\\mathbb{F}_q$ denote the finite field with $q=2^m$ elements and\n$\\mathbb{F}^*_q=\\mathbb{F}_q\\backslash\\{0\\}$. An $[n,k,d]$ {\\em linear code }\n $\\mathcal{C}$ over $\\mathbb{F}_2$ is a $k$-dimensional subspace of $\\mathbb{F}_2^n$\nwith minimum Hamming distance $d$, and is {\\em cyclic} if any cyclic shift of a codeword is another codeword of $\\mathcal{C}$.\n Any\n cyclic code $\\mathcal{C}$ can be expressed as $\\mathcal{C} = \\langle g(x) \\rangle,$\n where $g(x)$ is monic and has the least degree. The polynomial $g(x)$ is called the {\\em generator polynomial} and $h(x)=(x^n-1)\/g(x)$ is referred to\n as the\n{\\em parity-check polynomial} of $\\mathcal{C}$. If the generator polynomial\n$g(x)$ (resp. the parity-check polynomial $h(x)$) can be factored\ninto a product of $r$ irreducible polynomials over $\\mathbb{F}_p$, then\n$\\mathcal{C}$ is called a cyclic code with {\\em $r$ zeros} (resp. {\\em $r$ nonzeros}).\nThe code with the generator polynomial $x^kh(x^{-1})$ is called the {\\em dual} of $\\mathcal{C}$ and denoted by\n$\\mathcal{C}^{\\bot}$.\n\nFurthermore, we define the {\\em extended} code of a code $\\mathcal{C}$ to be\nthe code\n$$\n\\overline{\\mathcal{C}}=\\{(c_0, c_1, \\ldots, c_n) \\in \\mathbb{F}_2^{n+1}:(c_0, c_1, \\ldots, c_{n-1}) \\in \\mathcal{C} ~with ~\\sum^n_{i=0}c_i=0\\}.\n$$ \nThe\n {\\em support} of a codeword $\\mathbf{c}$ is defined by\n$$Suppt(\\mathbf{c})=\\{0\\leq i \\leq n-1: c_i\\neq 0\\}.$$\nLet $A_i$ be the number of codewords with Hamming weight $i$ in a code $\\mathcal{C}$. The {\\em weight enumerator} of $\\mathcal{C}$ is defined by\n$$1+A_1z+A_2z^2+\\ldots+A_nz^n.$$\nThe sequence $(1, A_1, \\ldots, A_n)$ is called the {\\em weight distribution} of the code $\\mathcal{C}.$ If $|\\{1\\leq i\\leq n: A_i\\neq 0\\}|=w,$ then we\ncall\n$\\mathcal{C}$ a {\\em $w$-weight code}.\n\nLet $n=q-1$, and $\\alpha$ be a generator of $\\mathbb{F}^*_q$. For any $i$ with $0\\leq i\n\\leq n-1$, let $M_i(x)$ denote the {\\em minimal polynomial} of $\\alpha^i$ over $\\mathbb{F}_2$. For any\n$2\\leq\\delta\\leq n$, the code $\\mathcal{C}_{(p, n, \\delta)}= \\langle g_{(p, n, \\delta, 1)} \\rangle$ with\n$$g_{(p, n, \\delta,\n1)}(x) = \\lcm(M_1(x), M_{2}(x), \\ldots, M_{1+\\delta-2}(x)),$$\nwhere $\\lcm$ denotes the least common multiple of the polynomials, is called a {\\em narrow-sense primitive BCH code} with designed distance $\\delta$.\n\nFor each $i$ with $A_i\\neq 0$, let $\\mathcal{B}_i$ denote the set of the supports of all codewords with weight $i$ in $\\mathcal{C}$, where the coordinates\nof a codeword are indexed by $(0, 1, 2, \\ldots, n-1).$\nLet $\\mathcal{P}=\\{0, 1, \\ldots, n-1\\}.$ The pair $(\\mathcal{P},\\mathcal{B}_i)$ could be a $t$-$(v,i,\\lambda)$ design for a certain positive $\\lambda$~\\cite{Ton98}. There exist two classical approaches to obtain $t$-designs from linear codes. The first one is to employ the Assmus-Mattson Theorem given in~\\cite{AM69}, and the second one is to study the automorphism group of a linear code $\\mathcal{C}$. If the permutation part of the automorphism group acts $t$-transitively on a code $\\mathcal{C}$, then the code $\\mathcal{C}$ holds $t$-designs~\\cite{AK92,MS771}. In the following, we will use the latter method to construct $2$-designs.\n\nWe conclude this section by summarizing some known results on affine-invariant codes related to $2$-designs.\n\nThe $2$-adic expansion of each $e\\in\\mathcal{P}$ is given by\n$$\ne=\\sum^{m-1}_{i=0}e_i2^i,~ ~0\\leq e_i\\leq 1 ,~0\\leq i \\leq m-1.\n$$\nFor any $r=\\sum^{m-1}_{i=0}r_i2^i\\in\\mathcal{P},$\nwe say that $r\\preceq e$ if $r_i \\leq e_i$ for all $0\\leq i\\leq m-1.$ By definition, we have $r\\leq e$ if $r\\preceq e$.\n\nThe set of coordinate permutations that map a code $\\mathcal{C}$ to itself forms a group, which is referred to as the {\\em permutation automorphism\ngroup} of $\\mathcal{C}$ and denoted by $PAut(\\mathcal{C})$. We define the {\\em affine group} $GA_1(\\mathbb{F}_q)$ by the set of all permutations\n$$\\sigma_{a,b}: x \\mapsto ax+b$$\nof $\\mathbb{F}_q$, where $a\\in \\mathbb{F}_q^*$ and $b \\in \\mathbb{F}_q$. An affine-invariant code is an extended cyclic code $\\overline{\\mathcal{C}}$ over\n$\\mathbb{F}_2$ such that $GA_1(\\mathbb{F}_q)\\subseteq PAut(\\overline{\\mathcal{C}})$~\\cite{HP03}.\n\n\n\nFor any integer $0\\leq j< n$, the $2$-cyclotomic coset of $j$ modulo $2^m-1$ is defined by\n$$C_j=\\{jp^i \\pmod {2^m-1} : 0 \\le i\\leq \\ell_j-1\\},$$\nwhere $\\ell_j$ is the smallest positive integer such that $j\\equiv jp^{\\ell_j}\\pmod {2^m-1}.$\nLet $g(x)=\\prod_j\\prod_{i\\in C_j}(x-\\alpha^i)$, where $j$ runs through some subset of representatives of the $2$-cyclotomic cosets\n$C_j$ modulo $2^m-1.$ The set $T=\\bigcup_jC_j$ is called the {\\em defining set} of $\\mathcal{C}$, which is the union of these $2$-cyclotomic cosets.\n\nAffine-invariance is an important property of an extended primitive cyclic code, for which the following lemma presented by Kasami, Lin and Peterson~\\cite{KLP67} provides a sufficient and necessary condition by examining the defining set of the code.\n\n\\begin{lemma}\\label{Kasami-Lin-Peterson}~\\cite{KLP67}\n Let $\\overline{\\mathcal{C}}$ be an extended cyclic code of\n length $2^m$ over $\\mathbb{F}_2$ with defining set $\\overline{T}$. The code $\\overline{\\mathcal{C}}$ is affine-invariant if and\nonly if whenever $e \\in \\overline{T}$ then $r \\in \\overline{T}$ for all $r \\in \\mathcal{P}$ with $r \\preceq e$.\n\\end{lemma}\n\n\\begin{lemma}\\label{The dual of an affine-invariant code}~\\cite{Ding18b} \n The dual of an affine-invariant code $\\overline{\\mathcal{C}}$ over $\\mathbb{F}_2$ of length $n+1$\nis also affine-invariant.\n\\end{lemma}\n\nThe importance of affine-invariant codes is partly due to Theorem \\ref{2-design} which will be used together with Lemmas~\\ref{Kasami-Lin-Peterson} and\n\\ref{The dual of an affine-invariant code} to derive the existence of $2$-designs.\n\n\\begin{theorem}\\label{2-design}~\\cite{Ding18b}\n For each $i$ with $\\overline{A_i} \\neq 0$ in an affine-invariant code $\\overline{\\mathcal{C}}$, the supports of the codewords of weight $i$\nform a $2$-design.\n\\end{theorem}\n\n\n\\section{Two classes of cyclic codes and their $t$-designs}\\label{theorem}\n\nIn this section, we introduce the main results on the weight distributions of two classes of cyclic codes and the corresponding $2$-designs. Their proofs will be presented in the subsequent section. In the following, let $\\tr_1^m$ denote the trace function from $\\mathbb{F}_{2^m}$ onto $\\mathbb{F}_2$. \n\n\n\n\\subsection{Results on the linear code derived from triple-error correcting BCH code}\n\nWe define\n\\begin{eqnarray}\\label{code-1}\n {\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}:=\\{ (\\tr_1^m (ax^5+bx^3+cx)+h )_{x \\in \\mathbb{F}_q}: a,b,c \\in \\mathbb{F}_q, h\\in \\mathbb{F}_2 \\},\n\\end{eqnarray}\nwhere $\\mathcal{C}_1$ is the cyclic code of length $n$ with the parity-check polynomial $M_1(x)M_3(x)M_5(x)$. It is easily seen that $\\mathcal{C}_1^{\\bot}$ is a BCH code with minimum distance $d\\geq\\delta=7$. Note that for $\\mathcal{C}_1^{\\bot}$, we only discuss the case of $m$ even since the complement case for $m$ odd has been studied in~\\cite{Ding182}.\n\nThe following two theorems constitute the first part of our main results in the present paper.\n\n\\begin{theorem}\\label{weight1}\nLet $s\\geq 3.$ The weight distributions of the code ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ over $\\mathbb{F}_2$ with length $n+1$ and\n$dim({\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot})=3m+1$ are given in Table \\ref{1}.\n\\begin{table}\n\\begin{center}\n\\caption{The weight distribution of ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$}\\label{1}\n\\begin{tabular}{ll}\n\\hline\\noalign{\\smallskip}\nWeight & Multiplicity \\\\\n\\noalign{\\smallskip}\n\\hline\\noalign{\\smallskip}\n$0$ & 1 \\\\\n$2^{2s-1}$ & $ 29\\times2^{6s-5}-33\\times2^{4s-5}+17\\times2^{2s-3}-2 $ \\\\\n$2^{2s-1}-2^{s-1} $ & $ \\frac{2}{15}\\times2^{2s}(3\\times2^{4s}+5\\times2^{2s}-8)$\\\\\n$ 2^{2s-1}+2^{s-1}$ & $ \\frac{2}{15}\\times2^{2s}(3\\times2^{4s}+5\\times2^{2s}-8)$\\\\\n$2^{2s-1}-2^s$ & $ \\frac{7}{3}\\times2^{4s-4}(2^{2s}-1)$ \\\\\n$ 2^{2s-1}+2^s$ & $ \\frac{7}{3}\\times2^{4s-4}(2^{2s}-1)$ \\\\\n$ 2^{2s-1}-2^{s+1} $ & $\\frac{1}{15}\\times2^{2s-4}(2^{4s-2}-5\\times2^{2s-2}+1)$ \\\\\n$ 2^{2s-1}+2^{s+1}$ & $\\frac{1}{15}\\times2^{2s-4}(2^{4s-2}-5\\times2^{2s-2}+1)$ \\\\\n$ 2^{2s}$ & $ 1$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\end{theorem}\n\nNote that the codes defined in (\\ref{code-1}) are eight-weight.\n\n\n\\begin{theorem}\\label{$2-$design-1}\nLet $s\\geq 3$ be a positive integer. Then the supports of the codewords of weight $i$ with ${\\overline{{A_i}^{\\bot}}}^{\\bot}\\neq0$ in\n${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ form a $2$-design.\nMoreover, let $\\mathcal{P}=\\{0, 1, \\ldots, 2^m-1\\}$ and $\\mathcal{B}$ be the set of the supports of the codewords of\n${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ with weight $i,$ where\n${\\overline{{A_i}^{\\bot}}}^{\\bot}\\neq 0.$ Then ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ holds $2$-$(2^m, i, \\lambda)$ designs for the following\npairs:\n\\begin{itemize}\n\\item $(i, \\lambda)=(2^{2s-1}, (29\\times2^{6s-5}-33\\times2^{4s-5}+17\\times2^{2s-3}-2)(2^{2s-1}-1)\/(2^{2s}-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}-2^{s-1}, \\frac{2}{15}\\times2^{s-1}(3\\times2^{4s}+5\\times2^{2s}-8)(2^{2s-1}-2^{s-1}-1)\/(2^s+1)).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^{s-1}, \\frac{2}{15}\\times2^{s-1}(3\\times2^{4s}+5\\times2^{2s}-8)(2^{2s-1}+2^{s-1}-1)\/(2^s-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}-2^s, \\frac{7}{3}\\times2^{3s-4}(2^{2s-1}-2^s-1)(2^{s-1}-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^s, \\frac{7}{3}\\times2^{3s-4}(2^{2s-1}+2^s-1)(2^{s-1}+1)).$\n\\item $(i, \\lambda)=(2^{2s-1}-2^{s+1}, \\frac{1}{15}\\times2^{s-3}(2^{4s-2}-5\\times2^{2s-2}+1)(2^{2s-1}-2^{s+1}-1)(2^{s-2}-1)\/(2^{2s}-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^{s+1}, \\frac{1}{15}\\times2^{s-3}(2^{4s-2}-5\\times2^{2s-2}+1)(2^{2s-1}+2^{s+1}-1)(2^{s-2}+1)\/(2^{2s}-1)).$\n\\end{itemize}\n\\end{theorem}\n\nThe following Examples \\ref{example1} and \\ref{example2} from Magma program confirm the main results in Theorems \\ref{weight1} and \\ref{$2-$design-1}.\n\n\n\\begin{example}\\label{example1}\nIf $s=3,$ then the code ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ has parameters $[64, 19, 16]$ and weight enumerator\n$1+252z^{16}+37632z^{24}+107520z^{28}+233478z^{32}+107520z^{36}+37632z^{40}+252z^{48}+z^{64}.$ It gives $2$-$(64, i, \\lambda)$ designs with the\nfollowing pairs $(i, \\lambda):$\n$$(16, 15), (24, 5152), (28, 20160), (32, 57443), (36, 33600), (40, 14560), (48, 141).$$\n\\end{example}\n\n\\begin{example}\\label{example2}\nIf $s=4,$ then the code ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ has parameters $[256, 25, 96]$ and weight enumerator\n$1+17136z^{96}+2437120z^{112}+6754304z^{120}+15137310z^{128}+6754304z^{136}+2437120z^{144}+17136z^{160}+z^{256}.$\n\\end{example}\n\n\nIt is worth noting that, for $m=4$,\nthe code ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ has parameters $[16, 11, 4]$ and weight enumerator\n $1+140z^4+448z^6+870z^8+448z^{10}+140z^{12}+z^{16}$. It forms $3$-$(16, i, \\lambda)$ designs with the following pairs $(i, \\lambda):$\n$$(4, 1), (6, 16), (8, 87), (10, 96), (12, 55).$$\n\n\n\\subsection{Results on the code related to the generalized Kasami case}\n\nWe define\n\\begin{eqnarray}\\label{code-2}\n \\lefteqn{{\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}:=\\{ (\\tr_1^s(ax^{2^s+1})+ \\tr_1^m(bx^{2^l+1}+cx)+h )_{x \\in \\mathbb{F}_q}:} \\\\\n& &\\qquad \\qquad \\qquad \\qquad \\qquad \\quad a \\in \\mathbb{F}_{2^s}, b, c \\in\n\\mathbb{F}_q, h\\in\n\\mathbb{F}_2 \\},\\nonumber\n\\end{eqnarray}\nwhere $\\mathcal{C}_2$ is the cyclic code of length $n$ with the parity-check polynomial $M_1(x)M_{2^l+1}(x)M_{2^s+1}(x)$. Note that $\\mathcal{C}_2^{\\bot}$ is the dual of the extended cyclic code of the parameters satisfying the generalized Kasami case.\n\nFor $\\overline{{\\mathcal{C}_2}^{\\bot}}^{\\bot}$, we present the main results in the following two theorems.\n\n\\begin{theorem}\\label{weight2}\nLet $1\\leq l\\leq m-1.$ The weight distributions of the code ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ over $\\mathbb{F}_2$ with length $n+1$ and\n$dim({\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot})=\\frac{5m}{2}+1$ are given in Tables \\ref{2} and \\ref{3}.\n\\begin{table}\n\\begin{center}\n\\caption{The weight distribution of ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ when $d'=d$}\\label{2}\n\\begin{tabular}{ll}\n\\hline\\noalign{\\smallskip}\nWeight & Multiplicity \\\\\n\\noalign{\\smallskip}\n\\hline\\noalign{\\smallskip}\n$0$ & 1 \\\\\n$2^{2s-1}-2^{s-1}$ & $ 2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})\/(2^{2d}-1)$ \\\\\n$2^{2s-1}+2^{s-1}$ & $ 2^{2s}(2^s-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})\/(2^{2d}-1)$ \\\\\n$ 2^{2s-1}-2^{s+d-1}$ & $ 2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)\/(2^{2d}-1)$\\\\\n$ 2^{2s-1}+2^{s+d-1}$ & $ 2^{2(s-d)}(2^{s+d}-1)(2^{2s}-1)\/(2^{2d}-1)$\\\\\n$ 2^{2s-1}$ & $ 2(2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$ \\\\\n$ 2^{2s}$ & $ 1$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{The weight distribution of ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ when $d'=2d$}\\label{3}\n\\begin{tabular}{ll}\n\\hline\\noalign{\\smallskip}\nWeight & Multiplicity \\\\\n\\noalign{\\smallskip}\n\\hline\\noalign{\\smallskip}\n$0$ & 1 \\\\\n$2^{2s-1}-2^{s-1}$ & $ 2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)\/(2^{2d}-1)(2^d+1)$ \\\\\n$2^{2s-1}+2^{s-1}$ & $ 2^{2s+3d}(2^s-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)\/(2^{2d}-1)(2^d+1)$ \\\\\n$ 2^{2s-1}-2^{s+d-1}$ & $ 2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)\/(2^d+1)^2$\\\\\n$ 2^{2s-1}+2^{s+d-1}$ & $ 2^{2s-d}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)\/(2^d+1)^2$\\\\\n$ 2^{2s-1}$ & $ 2(2^{2s}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d}+2^{2s-d}-2^{2s-2d+1}$\\\\\n&$+2^{2s-3d}-2^{2s-4d}+1)$ \\\\\n$ 2^{2s-1}-2^{s+2d-1}$ & $ 2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)\/(2^d+1)(2^{2d}-1)$\\\\\n$ 2^{2s-1}+2^{s+2d-1}$ & $ 2^{2s-4d}(2^{s-d}-1)(2^{2s}-1)\/(2^d+1)(2^{2d}-1)$\\\\\n$ 2^{2s}$ & $ 1$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\end{theorem}\nNote that the code are six-weight when $d'=d$ and eight-weight when $d'=2d$.\n\n\\begin{theorem}\\label{$2-$design-2}\nLet $s\\geq 2$ be a positive integer. Then the supports of the codewords of weight $i$ with ${\\overline{{A_i}^{\\bot}}}^{\\bot}\\neq0$ in\n${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ give a $2$-design.\nMoreover, let $\\mathcal{P}=\\{0, 1, \\ldots, 2^m-1\\}$ and $\\mathcal{B}$ be the set of the supports of the codewords of\n${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ with weight $i,$ where\n${\\overline{{A_i}^{\\bot}}}^{\\bot}\\neq 0.$ Then ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ holds $2$-$(2^m, i, \\lambda)$ designs for the following\npairs:\n\n(1) if $d'=d,$\n\\begin{itemize}\n\\item $(i, \\lambda)=(2^{2s-1}-2^{s-1}, 2^{s-1}(2^s-1)(2^{2s-1}-2^{s-1}-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})\/(2^{2d}-1)(2^s+1)).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^{s-1}, 2^{s-1}(2^{2s-1}+2^{s-1}-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})\/(2^{2d}-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}-2^{s+d-1}, 2^{s-d-1}(2^{s-d}-1)(2^{s+d}-1)(2^{2s-1}-2^{s+d-1}-1)\/(2^{2d}-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^{s+d-1}, 2^{s-d-1}(2^{s-d}+1)(2^{s+d}-1)(2^{2s-1}+2^{s+d-1}-1)\/(2^{2d}-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}, (2^{2s-1}-1)(2^{3s-d}-2^{2s-2d}+1)).$\n\\end{itemize}\n\\end{theorem}\n\n(2) if $d'=2d,$\n\\begin{itemize}\n\\item $(i, \\lambda)=(2^{2s-1}-2^{s-1},\n 2^{3d}(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)(2^{2s-1}-2^{s-1}-1)(2^{2s-1}-2^{s-1})\/(2^{2d}-1)(2^d+1)(2^s+1)).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^{s-1},\n 2^{3d}(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)(2^{2s-1}+2^{s-1}-1)(2^{2s-1}+2^{s-1})\/(2^{2d}-1)(2^d+1)(2^s+1)).$\n\\item $(i, \\lambda)=(2^{2s-1}-2^{s+d-1}, (2^s+2^{s-d}+2^{s-2d}+1)(2^{2s-1}-2^{s+d-1})(2^{2s-1}-2^{s+d-1}-1)\/2^d(2^d+1)^2).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^{s+d-1}, (2^s+2^{s-d}+2^{s-2d}+1)(2^{2s-1}+2^{s+d-1})(2^{2s-1}+2^{s+d-1}-1)\/2^d(2^d+1)^2).$\n\\item $(i, \\lambda)=(2^{2s-1}, 2(2^{2s-1}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d}+2^{2s-d}-2^{2s-2d+1}+2^{2s-3d}-2^{2s-4d}+1)\/2^d).$\n\\item $(i, \\lambda)=(2^{2s-1}-2^{s+2d-1}, (2^{s-d}-1)(2^{2s-1}-2^{s+2d-1})(2^{2s-1}-2^{s+2d-1}-1)\/2^{4d}(2^d+1)(2^{2d}-1)).$\n\\item $(i, \\lambda)=(2^{2s-1}+2^{s+2d-1}, (2^{s-d}-1)(2^{2s-1}+2^{s+2d-1})(2^{2s-1}+2^{s+2d-1}-1)\/2^{4d}(2^d+1)(2^{2d}-1)).$\n\\end{itemize}\n\nThe following two examples from Magma program confirm the results in Theorems~\\ref{weight2} and~\\ref{$2-$design-2}.\n\n\\begin{example}\\label{example4}\nIf $(s, l)=(2, 1),$ then the code ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ has parameters $[16, 11, 4]$ and weight enumerator\n$1+140z^4+448z^6+870z^8+448z^{10}+140z^{12}+z^{16}.$ It gives $2$-$(16, i, \\lambda)$ designs with the\nfollowing pairs $(i, \\lambda):$\n$$(4, 7), (6, 56), (8, 203), (10, 168), (12, 77).$$\n\\end{example}\n\n\\begin{example}\\label{example5}\nIf $(s, l)=(3, 2),$ then the code ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ has parameters $[64, 16, 24]$ and weight enumerator\n$1+5040z^{24}+12544z^{28}+30366z^{32}+12544z^{36}+5040z^{40}+z^{64}.$ It holds $2$-$(64, i, \\lambda)$ designs with the\nfollowing pairs $(i, \\lambda):$\n$$(24, 690), (28, 2352), (32, 7471), (36, 3920), (40, 1950).$$\n\\end{example}\n\n\\begin{example}\\label{example6}\nIf $(s, l)=(3, 1),$ then the code ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$ has parameters $[64, 16, 16]$ and weight enumerator\n$1+84z^{16}+3360z^{24}+17920z^{28}+22806z^{32}+17920z^{36}+3360z^{40}+84z^{48}+z^{64}.$ It forms $2$-$(64, i, \\lambda)$ designs with the\nfollowing pairs $(i, \\lambda):$\n$$(16, 5), (24, 460), (28, 3360), (32, 5611), (36, 5600), (40, 1300), (48, 47).$$\n\\end{example}\n\n\n\\section{Proofs of the main results}\\label{Proofs-main-results}\n\n\\subsection{Three lemmas related to the weights of codes}\n\nIn order to determine the weight distributions of the two classes of cyclic codes ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ and ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$, we need the following lemmas.\n\n\\begin{lemma}\\label{relation}~\\cite{Ding182}\nLet $\\mathcal{C}$ be an $[n, k, d]$ binary linear code, then ${\\overline{{\\mathcal{C}}^{\\bot}}}^{\\bot}$ has parameters $[n+1, k+1,\n{\\overline{d^{\\bot}}}^{\\bot}]$. Furthermore, ${\\overline{{\\mathcal{C}}^{\\bot}}}^{\\bot}$ has only even-weight codewords, and all the nonzero weights in\n${\\overline{{\\mathcal{C}}^{\\bot}}}^{\\bot}$ are the following:\n$$w_1, w_2, \\ldots, w_t; n+1-w_1, n+2-w_2, \\ldots, n+1-w_t; n+1,$$\nwhere $w_1, w_2, \\ldots, w_t$ denote all the nonzero weights of $\\mathcal{C}.$\n\\end{lemma}\n\nThe following Pless power moments given in~\\cite{HP03} are notable variations of the MacWilliams identities, which is a fundamental result about weight distributions and is a set of linear relations between the weight distributions of $\\mathcal{C}$ and $\\mathcal{C}^{\\bot}$.\n\n\\begin{lemma}\\label{power moment identities}\nLet $A_i$ and $A_i^{\\bot}$ denote the number of code vectors of weight $i$ in a code $\\mathcal{C}$ and $\\mathcal{C}^{\\bot}$, respectively. If\n$A_i^{\\bot}=0$ for $0 \\le i \\le 6,$ then the first seven Pless power moment identities hold:\n\\begin{eqnarray*}\n&&\\sum A_i=2^k,\\\\\n&&\\sum iA_i=2^{k-1}n,\\\\\n&&\\sum i^2A_i=2^{k-2}n(n+1),\\\\\n&&\\sum i^3A_i=2^{k-3}(n^3+3n^2),\\\\\n&&\\sum i^4A_i=2^{k-4}(n^4+6n^3+3n^2-2n),\\\\\n&&\\sum i^5A_i=2^{k-5}(n^5+10n^4+15n^3-10n^2),\\\\\n&&\\sum i^6A_i=2^{k-6}(n^6+15n^5+45n^4-15n^3-30n^2+16n),\n\\end{eqnarray*}\nwhere $k$ denotes the number of information digits.\n\\end{lemma}\n\nThe following lemma given by Luo, Tang and Wang~\\cite{LTW10}, gives the weight distributions of the cyclic codes related to the generalized Kasami case.\n\\begin{lemma}\\label{Weight distribution}\nThe weight distributions of $\\mathcal{C}_2$ are given in Tables \\ref{4} and \\ref{5}.\n\n\\begin{table}\n\\begin{center}\n\\caption{The weight distribution of ${\\mathcal{C}_2}$ when $d'=d$}\\label{4}\n\\begin{tabular}{ll}\n\\hline\\noalign{\\smallskip}\nWeight & Multiplicity \\\\\n\\noalign{\\smallskip}\n\\hline\\noalign{\\smallskip}\n$0$ & 1 \\\\\n$2^{2s-1}-2^{s-1}$ & $ 2^{s-1}(2^{2s}-1)(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})\/(2^{2d}-1)$ \\\\\n$2^{2s-1}+2^{s-1}$ & $ 2^{s-1}(2^s-1)^2(2^{2(s+d)}-2^{2s+d}-2^{2s}+2^{s+2d}-2^{s+d}+2^{2d})\/(2^{2d}-1)$ \\\\\n$ 2^{2s-1}-2^{s+d-1}$ & $ 2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}+1)\/(2^{2d}-1)$\\\\\n$ 2^{2s-1}+2^{s+d-1}$ & $ 2^{s-d-1}(2^{s+d}-1)(2^{2s}-1)(2^{s-d}-1)\/(2^{2d}-1)$\\\\\n$ 2^{2s-1}$ & $ (2^{3s-d}-2^{2(s-d)}+1)(2^{2s}-1)$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\end{lemma}\n\n\\begin{table}\n\\begin{center}\n\\caption{The weight distribution of ${\\mathcal{C}_2}$ when $d'=2d$}\\label{5}\n\\begin{tabular}{ll}\n\\hline\\noalign{\\smallskip}\nWeight & Multiplicity \\\\\n\\noalign{\\smallskip}\n\\hline\\noalign{\\smallskip}\n$0$ & 1 \\\\\n$2^{2s-1}-2^{s-1}$ & $ \\frac{2^{s+3d-1}(2^{2s}-1)(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)}{(2^{2d}-1)(2^d+1)}$ \\\\\n$2^{2s-1}+2^{s-1}$ & $ \\frac{2^{2s+3d-1}(2^s-1)^2(2^{2s}-2^{2(s-d)}-2^{2s-3d}+2^s-2^{s-d}+1)}{(2^{2d}-1)(2^d+1)}$ \\\\\n$ 2^{2s-1}-2^{s+d-1}$ & $ 2^{s-1}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)(2^{s-d}+1)\/(2^d+1)^2$\\\\\n$ 2^{2s-1}+2^{s+d-1}$ & $ 2^{s-1}(2^{2s}-1)(2^s+2^{s-d}+2^{s-2d}+1)(2^{s-d}-1)\/(2^d+1)^2$\\\\\n$ 2^{2s-1}$ & $ (2^{2s}-1)(2^{3s-d}-2^{3s-2d}+2^{3s-3d}-2^{3s-4d}+2^{3s-5d}$\\\\\n&$+2^{2s-d}-2^{2s-2d+1}+2^{2s-3d}-2^{2s-4d}+1)$ \\\\\n$ 2^{2s-1}-2^{s+2d-1}$ & $ 2^{s-2d-1}(2^{s-d}-1)(2^{2s}-1)(2^{s-2d}+1)\/(2^d+1)(2^{2d}-1)$\\\\\n$ 2^{2s-1}+2^{s+2d-1}$ & $ 2^{s-2d-1}(2^{s-d}-1)(2^{2s}-1)(2^{s-2d}-1)\/(2^d+1)(2^{2d}-1)$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Quadratic forms}\n\nTo determine the parameters of codes ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ defined in Eq.(\\ref{code-1}), we introduce the following function.\n\\begin{equation}\\label{S(a.b.c)}\nS(a,b,c)=\\sum\\limits_{x \\in \\mathbb{F}_q}(-1)^{\\tr_1^m(ax^5+bx^3+cx)},\\quad a,b,c \\in \\mathbb{F}_q.\\\\\n\\end{equation}\nThe first tool to determine the values of exponential sums $S(a,b,c)$ is quadratic forms over $\\mathbb{F}_2$.\nLet $H$ be an $m\\times m$\n matrix over $\\mathbb{F}_2$. For the quadratic form\n\\begin{equation}\\label{F(x)}\nF: \\mathbb{F}^m_2\\rightarrow \\mathbb{F}_2,\\quad F(X)=XHX^T \\quad (X=(x_1, x_2, \\ldots, x_m)\\in \\mathbb{F}^m_2),\n\\end{equation}\nwe define $r_F$ of $F$ to be the rank of $H+H^T$ over $\\mathbb{F}_2$.\n\nThe field $\\mathbb{F}_q$ is a vector space over $\\mathbb{F}_2$ with dimension $m$. We fix a basis $v_1, v_2, \\ldots, v_m$ of $\\mathbb{F}_q$ over\n$\\mathbb{F}_2$. Thus each $x\\in \\mathbb{F}_q$ can be uniquely expressed as\n$$x=x_1v_1+x_2v_2+\\ldots+x_mv_m \\quad (x_i\\in \\mathbb{F}_2).$$\nThen we have the following $\\mathbb{F}_2$-linear isomorphism $\\mathbb{F}_q \\rightarrow\\mathbb{F}^m_2:$\n$$ \\quad x=x_1v_1+x_2v_2+\\ldots+x_mv_m\\mapsto X=(x_1, \\ldots, x_m).$$\nWith the isomorphism, a function $f: \\mathbb{F}_q\\rightarrow\\mathbb{F}_2$ induces a function $F:\\mathbb{F}^m_2\\rightarrow\\mathbb{F}_2$ where for all\n$X=(x_1, \\ldots, x_m)\\in \\mathbb{F}^m_2, F(X)=f(x)$ where $x=x_1v_1+x_2v_2+\\ldots+x_mv_m.$\nIn this way, the function $f(x) = \\tr_1^m(wx)$ for $w\\in \\mathbb{F}_q$ induces a linear form\n $$F(X)=\\sum^m_{i=1} \\tr_1^m( w v_i)x_i=A_w X^T,$$\nwhere $A_w=(\\tr_1^m(w v_1), \\ldots, \\tr_1^m(w v_m)).$\n\nFor $(a, b, c)\\in \\mathbb{F}^3_q$, to determine the value of\n$$S(a,b,c)=\\sum\\limits_{x \\in \\mathbb{F}_q}(-1)^{\\tr_1^m(ax^5+bx^3+cx)}=\\sum\\limits_{X \\in \\mathbb{F}^m_2}(-1)^{XH_{a,b}X^T+A_cX^T},$$\nwhere $XH_{a,b}X^T$ is the quadratic form derived from\n$f_{a,b}(x) = \\tr_1^m (ax^5+bx^3)$ for $a, b\\in\\mathbb{F}_q$. We need to determine the rank of $H_{a,b}$ over $\\mathbb{F}_2.$ To this end, we have the following\nresult.\n\n\\begin{lemma}\\label{rank}\n For $(a, b)\\in \\mathbb{F}^2_q\/{\\{(0,0)\\}},$ let $r_{a,b}$ be the rank of $H_{a,b}$. Then $r_{a,b}=m$, $m-2$, or $m-4$.\n\\end{lemma}\n\n\\begin{proof}\n It is well known that the rank of the quadratic form $F(X)$ is defined as the codimension of the $ F_2$ -vector space\n$$V = \\{y\\in \\mathbb{F}_{q}:~ f(x+y)-f(x)-f(y) = 0 \\mathrm{~for~ all~} x\\in \\mathbb{F}_{q} \\}.$$\nThe cardinality of $V$ is $|V| = 2^{m-r_F}$, where $r_F$ is the rank of $f(x).$\n\nThe definition of the function $f_{a,b}(x)$ leads to\n$$\nf(x+y)-f_{a,b}(x)-f(y) = \\tr_1^m ((ax^4+bx^2+a^{2^{m-2}}x^{2^{m-2}}+b^{2^{m-1}}x^{2^{m-1}})y).\n$$\nLet $$\\Phi_{(a,b)}(x)=ax^4+bx^2+a^{2^{m-2}}x^{2^{m-2}}+b^{2^{m-1}}x^{2^{m-1}}.$$\n Then $r_{a,b}=r$ if and only if $\\Phi_{(a,b)}(x)=0$ has $2^{m-r_{a,b}}$ solutions in $\\mathbb{F}_q.$\nOn the other hand, since $\\Phi_{(a,b)}(x)$ is a $2$-linearized polynomial, then the set of the zeros to $\\Phi_{(a,b)}(x)=0$ is equivalent to that of\n$$a^4x^{16}+b^4x^8+b^2x^2+ax=0$$ in\n$\\mathbb{F}_q$ and forms an $\\mathbb{F}_2$-vector space. Since $r_{a,b}$ is even, $r_{a,b}=m, m-2, m-4.$\nWe then complete the proof. \n\\end{proof}\n\nThe following result, which was proved in~\\cite{LN97}, will be used in Section~\\ref{section-4}.\n\\begin{lemma}\\label{value}~\\cite{LN97}\nFor the fixed quadratic form defined in (\\ref{F(x)}), the value distribution of \\\\\n $\\sum\\limits_{X \\in \\mathbb{F}^m_2}(-1)^{F(X)+A_cX^T}$ when $A_c$ runs\nthrough $\\mathbb{F}^m_2$, is $0,$ $2^{m-\\frac{r_F}{2}},$ or $-2^{m-\\frac{r_F}{2}}$.\n\\end{lemma}\n\n\\subsection{Proofs of the main results}\\label{section-4}\n\nNow we are ready to give the proofs of our main results. We begin this subsection by proving the weight distribution of the code ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ given in Theorem \\ref{weight1}.\n\n\n \\begin{proof}[Proof of Theorem~\\ref{weight1}]\n\nFor each nonzero codeword $\\mathbf{c}(a, b, c)=(c_0, \\ldots, c_n)$ in $\\mathcal{C}_1,$ the Hamming weight of $\\mathbf{c}(a, b, c)$ is\n\\begin{eqnarray}\\label{weight formula-1}\nw_H(\\mathbf{c}(a,b,c))&=&|\\{i:0\\leq i\\leq n-1, c_i\\neq 0\\}| \\nonumber\\\\\n&=&n-|\\{i:0\\leq i\\leq n-1, c_i= 0\\}| \\nonumber\\\\\n&=&n-\\frac{1}{2}\\sum^{n-1}_{i=0}\\sum^1_{y=0}(-1)^{y\\cdot \\tr_1^m (a\\alpha^{5i}+b\\alpha^{3i}+c\\alpha^i)}\\nonumber\\\\\n&=&n-\\frac{n}{2}-\\frac{1}{2}\\sum_{x\\in \\mathbb{F}^*_q}(-1)^{\\tr_1^m (ax^5+bx^3+cx )} \\nonumber\\\\\n&=&\\frac{n}{2}+\\frac{1}{2}-\\frac{1}{2}S(a,b,c) \\nonumber\\\\\n&=&2^{2s-1}-\\frac{1}{2}S(a,b,c).\n\\end{eqnarray}\n By Lemmas \\ref{rank}-\\ref{value} and (\\ref{weight formula-1}), we have that the Hamming weight of $\\mathbf{c}(a,\nb, c)$ is\n$$2^{2s-1}, 2^{2s-1}-2^{s-1}, 2^{2s-1}+2^{s-1}, 2^{2s-1}-2^s, 2^{2s-1}+2^s, 2^{2s-1}-2^{s+1}, 2^{2s-1}+2^{s+1}.$$\nPlugging these values to the Pless power moments given by Lemma~\\ref{power moment\nidentities} and after tedious calculations, we obtain\n\\begin{eqnarray*}\n&&A_{2^{2s-1}}=29\\times2^{6s-6}-33\\times2^{4s-6}+17\\times2^{2s-4}-1,\\\\\n&&A_{2^{2s-1}-2^{s-1}}=\\frac{1}{15}(3\\times2^{6s}+3\\times2^{5s}+5\\times2^{4s}+5\\times2^{3s}-2^{2s+3}-2^{s+3}),\\\\\n&&A_{2^{2s-1}+2^{s-1}}=\\frac{1}{15}(3\\times2^{6s}-3\\times2^{5s}+5\\times2^{4s}-5\\times2^{3s}-2^{2s+3}+2^{s+3}),\\\\\n&&A_{2^{2s-1}-2^s}=\\frac{7}{3}\\times2^{3s-4}(2^{3s-1}+2^{2s}-2^{s-1}-1),\\\\\n&&A_{2^{2s-1}+2^s}=\\frac{7}{3}\\times2^{3s-4}(2^{3s-1}-2^{2s}-2^{s-1}+1),\\\\\n&&A_{2^{2s-1}-2^{s+1}}=\\frac{1}{15}\\times2^{s-3}(2^{5s-4}+2^{4s-2}-5\\times2^{3s-4}-5\\times2^{2s-2}+2^{s-2}+1),\\\\\n&&A_{2^{2s-1}+2^{s+1}}=\\frac{1}{15}\\times2^{s-3}(2^{5s-4}-2^{4s-2}-5\\times2^{3s-4}+5\\times2^{2s-2}+2^{s-2}-1).\n\\end{eqnarray*}\nThe desired conclusion then follows from Lemma \\ref{relation}. Thus the proof is completed.\n\\end{proof}\n\n\nThen we prove the affine-invariance of the code ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$.\n\n\n\\begin{lemma}\\label{affine invariant}\n The extended codes $\\overline{{\\mathcal{C}_1}^{\\bot}}$ and $\\overline{{\\mathcal{C}_2}^{\\bot}}$ are affine-invariant.\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the conclusion with Lemma \\ref{Kasami-Lin-Peterson}.\nThe defining set $T$ of the cyclic code ${\\mathcal{C}_1}^{\\bot}$ is $T =C_1 \\cup C_3 \\cup C_5$. Since $0 \\not \\in T$, the defining set $\\overline{T}$ of\n$\\overline{{\\mathcal{C}_1}^{\\bot}}$ is given by $\\overline{T} = C_1 \\cup C_3 \\cup C_5 \\cup \\{0\\}$.\nLet $e \\in \\overline{T} $ and $r \\in \\mathcal{P}$ . Assume that $e \\preceq s$. We need to prove that $r \\in \\overline{T}$ by Lemma~\\ref{Kasami-Lin-Peterson}.\n\nIf $r=0,$ then obviously $r\\in \\overline{T}$. Consider now the case $r>0$. If $e \\in C_1$, then the Hamming weight $wt(e) = 1.$ Since $r \\preceq e$,\n$wt(r)\n= 1.$ Consequently, $r \\in C_1\\subset \\overline{T}.$ If $e \\in C_3\\cup C_5$, then the Hamming weight $wt(e) = 2.$ Since $r \\preceq e$, either $wt(r) =\n1$\nor $r = e.$ In both cases, $r \\in \\overline{T}.$ The desired conclusion then follows\nfrom Lemma \\ref{Kasami-Lin-Peterson}.\n\nSimilarly, we can prove that $\\overline{{\\mathcal{C}_2}^{\\bot}}$ is affine-invariant.\n\nThus we complete the proof. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{$2-$design-1}]\nFrom the relation of ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ and $\\overline{{\\mathcal{C}_1}^{\\bot}}$, by Lemmas \\ref{The dual of an affine-invariant\ncode} and \\ref{affine invariant} , we have ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ is affine-invariant.\nThen ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ holds $2$-designs by Theorem \\ref{2-design}.\n\nMoreover, the number of supports of all codewords with weight $i\\neq 0$ in the code ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ is equal to\n${\\overline{{A_i}^{\\bot}}}^{\\bot}$ for each $i,$ where ${\\overline{{A_i}^{\\bot}}}^{\\bot}$ is given in Table \\ref{1}. Then the desired conclusions follow\nfrom Eq.(\\ref{condition}).\nThus, we finish the proof of Theorems \\ref{$2-$design-1}. \n\\end{proof}\n\nFrom all the above, we have finished the proof of the results related to ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$. Now we prove Theorems\n\\ref{weight2}\nand \\ref{$2-$design-2} related to ${\\overline{{\\mathcal{C}_2}^{\\bot}}}^{\\bot}$.\n\n\\begin{proof}[Proof of Theorem \\ref{weight2}]\nThe desired conclusion follows directly from Lemmas \\ref{relation} and \\ref{Weight distribution}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{$2-$design-2}]\nThe proof is similar to that of Theorem \\ref{$2-$design-1}, thus is omitted here.\n\\end{proof}\n\n\n\\section{Conclusion}\\label{section-5}\n\n\n\n\n\n\n\n\nIn this paper, we determined the weight distributions of two classes of binary cyclic codes. One is derived from the triple-error correcting BCH code\nand the other is from cyclic codes related to the generalized Kasami case. We proved that both classes of linear codes hold $2$-designs\nand explicitly computed their parameters. In particular, we get five $3$-designs in ${\\overline{{\\mathcal{C}_1}^{\\bot}}}^{\\bot}$ when $m=4.$\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdzas b/data_all_eng_slimpj/shuffled/split2/finalzzdzas new file mode 100644 index 0000000000000000000000000000000000000000..a4ce7dacf59e8bdbbd2949e40791f363f8151804 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdzas @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe recent strong evidence for the top quark in CDF~\\cite{cdf} completed the\ninformation we have on the masses and mixing angles of the quarks.\nIt emphasizes, however, at the same time our ignorance of their origin.\nAs the fermionic masses are free\nparameters in the standard model (SM), an embedding into a grand-\nunified-theory~(GUT) can help. This is also\nsuggested by the unification of the gauge coupling constants of the\nSM~\\cite{sun}, at $10^{16}$ GeV, provided the spectrum is extended\ninto that of the minimal SUSY-SM (MSSM)~\\cite{mssm}.\\footnote{ Another\npossibility is to\nintroduce an intermediate breaking scale at $\\approx 10^{12}$GeV.~\\cite{int} }\nGUTs give relations between the\nYukawa coupling constants of different flavours, like the successful\\quad\n$Y_{\\tau}(GUT)\\simeq Y_b(GUT)$\\quad\nlepton-quark one. Yet, the complete understanding of the mass-mixing\npattern requires\nrelations between the families. This can come only from outside\nthe GUT, by using a family-symmetry (or superstrings?).\nThe only phenomenological\nindication in this direction is that the mixing angles and masses\nof the quarks are consistent with the appearance of texture zeros in the\nYukawa matrices~\\cite{tex}.\n\nA recent study by Roberts, Ramond and Ross~(RRR)~\\cite{rrr} found five\ndifferent sets of symmetric quark mass matrices with texture zeros,\nwhich account for the quark masses and mixing. Special examples,\nlike the Fritzsch~\\cite{fr} texture, where known before. Also Dimopoulos,\nHall and Rabi~(DHR)~\\cite{dhr} discussed in detail quark and charged leptons\nmass matrices\nsuggested by Harvey, Ramond and Reiss~\\cite{hrr}, in terms of a SUSY-\nSO(10) broken directly into the MSSM.\n\nAll this is true for the quarks and charged leptons.\nThe neutrino-masses and mixing\nare completely unknown. Except for possible experimental indications coming\nfrom the solar-neutrino-puzzle~(SNP)~\\cite{snp},\nthe depletion of the atmospheric $\\nu_{\\mu}$~\\cite{atm} and some\ncosmological dark matter arguments~\\cite{cos}. All of which are consistent\nwith possible neutrino\nmasses in the range of $10^{-5}\\mbox{ eV} - 3\\mbox{ eV}$ .\\\\\nSuch small neutrino masses are obtained in in L-R symmetric GUTs, like\nSO(10), using the see-saw mechanism. This means that the SU(5) singlet\nRH-neutrinos acquire large Majorana masses. The diagonalization of the\ncomplete neutrino mass matrix leads then to three small eigenvalues.\n\nIn a previous paper~\\cite{ag}, we were able to predict the neutrino\nproperties, by requiring that all matrices, including the\nRH-neutrino Majorana mass one, have the same Fritzsch-texture.\nThe model was based on SUSY-SO(10) with the scale of the RH-neutrino\nmass matrix taken at the unification energy -- as is natural in\nSUSY theories. It gives\nneutrino masses and mixing angles consistent with a possible\nsolution of the solar neutrino puzzle, without the need for a free parameter.\nUnfortunately, if top was\nobserved at CDF, its mass is too high to be consistent with such a model.\n\nIn order to be able to predict the neutrino properties in terms of more\ncomplicate textures, we must use stronger assumptions.\nThe RH-neutrino scale becomes then a free parameter and to solve the SNP, it\nmust be lower then the GUT scale by several orders of magnitude.\nAssuming, as in almost all recent fermionic mass models,\nthat the SUSY-GUT is broken directly into the MSSM, it is not\nclear where this intermediate mass scale is coming from.\n\nIn this paper we use SUSY-SO(10) with\nthe three families of the quarks and the leptons,\nincluding $\\nu_R$, in the ${\\bf 16}$ representation $\\Psi_i$,\n$i=1,2,3$. In the view of the content of,\n$$\n{\\bf 16}\\times{\\bf 16}=({\\bf 10}+{\\bf 126})_{symmetric}+{\\bf 120}_{antisymm.}\n$$\nonly the \\ $\\phi_{\\bf 10}$\\ and \\ $\\phi_{\\bf \\overline{126}}$ \\ \nHiggs representations can contribute to the symmetric Yukawa terms.\\\\\nThe most general Yukawa Lagrangian at the GUT scale is then:\n\n\\begin{equation}\n{\\cal L}_Y = \\sum \\overline{\\Psi_i}^c \\Psi_j (Y_{ij}^{\\bf 10}\n\\phi_{\\bf 10}^{ij} +\nY^{\\bf \\overline{126}}_{ij}\\phi_{\\bf \\overline{126}}^{ij}) \\ .\n\\end{equation}\n\nNote, that one can absorb the difference between \\quad $Y_{ij}^{\\bf 10}$\n\\quad and \\quad $Y^{\\bf \\overline{126}}_{ij}$ in the VEVs, and use only one\neffective Yukawa matrix, if all the Higgs representations are different.\n(This is was used in the previous paper \\cite{ag} but is not true here,\nas we shall see later).\n\nOur aim is to predict the neutrino properties in terms of the\nmass matrices of the quarks and the charged leptons. In order to do this\nwe shall use the\nrequirements suggested by DHR~\\cite{dhr}\\footnote{They used those requirements\nfor ``their'' texture, which is very probably excluded experimentally as it\nrequires $|V_{cb}|>0.5$.} and apply them to the five\ntexture sets of RRR.\n\nThe requirements combine actually {\\em predictibility} and {\\em minimality}\nas follows:\n\\begin{enumerate}\n\\item{The textures of the mass matrices are dictated by discrete\nsymmetries and the directions of the VEVs in such a way that the\nminimal number of higgs multiplets is used.}\n\\item {Each fermion mass matrix element is generated by a VEV of only one\nof the \\ $\\phi_{\\bf 10}$\\ or \\ $\\phi_{\\bf \\overline{126}}$ \\ multiplets.}\n\\item {All entries of the RH-neutrino Majorana mass matrix, \\ $M_{\\nu_R}$ \\ ,\nmust be induced by one \\ $\\phi_{\\bf \\overline{126}}$ \\ \nmultiplet and in such a way the matrix is not singular.}\n\\end{enumerate}\n\nThe textures of RRR do not tell us if the non-vanishing entries are due to\nthe \\ $\\phi_{\\bf 10}$\\ or the \\ $\\phi_{\\bf \\overline{126}}$ \\ Higgs representation.\n More information about $M_d$, the down quarks mass matrix, can\nbe obtained using the ``connection'' between this matrix and that of the\ncharged leptons, $M_\\ell$.\nIn view of the fact that the \\ $\\phi_{\\bf \\overline{126}}$ \\ contributions come with a relative\nClebsch-Gordan coefficient of (-3), the fit of the $M_\\ell$ elements to\nthe lepton masses can tell us where \\ $\\phi_{\\bf \\overline{126}}$ \\ contributes.\nIt was already pointed out by Georgi and Jarlskog~\\cite{gj} that the Ansatz\n$$\nm_{\\tau} = m_b \\qquad m_{\\mu} = 3m_s \\qquad m_e = 1\/3 m_d\n$$\nat the GUT scale, works very well. This Ansatz can be generated by a factor\n(-3) in the \\ $(M_\\ell)_{22}$ \\ matrix element~\\cite{hrr}~\\cite{dhr}. RRR\nchecked it for their textures in terms of the SUSY-GUT broken directly\ninto the MSSM~\\cite{rrr}. This is obviously consistent with our\nrequirement (1). Note, also that the leading (3,3) elements are generated\nin such a case\nby \\ $\\phi_{\\bf 10}$\\ . Thus, they obey \\ $(M_d)_{33} = (M_{\\ell})_{33}$, and one has\nin addition, the successful approximate (Yukawa) \\quad\n$Y_b(GUT) \\simeq Y_\\tau(GUT)$ \\quad unification.\n\nThe structure of the mass matrix of the ``up\"-quarks, $M_u$, cannot be fixed\nusing similar arguments, as the\nrelated neutrino Dirac mass matrix, $M_{\\nu_D}$, is\nphenomenologically unknown. However, we will\nshow that our minimality and predictibility requirements limit\nconsiderably the possibilities.\n\nWe found using our method eight sets of symmetric textures which\npredict the neutrino properties -- up to the overall mass scale of the RH-\nneutrino masses. \\\\\nWe evolved then the Yukawa matrices, from the GUT scale to low energies, using\nthe renormalization group equations (RGEs). The resulting matrices are then\nfitted\nto the low-energy experimental data, and this fixes the quark parameters at\nthe GUT scale.\nThose parameters dictate the entries of the light-neutrino parameters, for each\none of the eight sets of textures. At the same time, we have also\npredictions for certain quantities in the quark-sector which we can\nused as a test. The light (see-saw) neutrino mass matrix is then evolved\nto low energies.\n\nThe resulting neutrino properties are given in terms of their mass ratios\nand mixing angles. The absolute neutrino masses can be obtained only\nwhen the intermediate scale, relevant for the overall scale of the RH-\nneutrino mass matrix, is given.\n\nThe mixing angles of all the eight texture sets are such that\n\\quad ${\\sin^2}2\\theta < 0.2 $,\\quad\nand hence, we cannot have vacuum oscillation as a solution to the solar\nneutrino puzzle. Also, the possible depletion of the atmospheric\n$\\nu_{\\mu}$\ncannot be accounted for. The values of the $\\nu_e-\\nu_{\\mu}$ mixing angle\n(i.e. \\quad ${\\sin^2}2\\theta_{e\\mu}$ \\quad)\nare generally in\nthe range of the small angle (i.e adiabatic) MSW~\\cite{msw} solution\nto the SNP. Requiring that \\quad ${\\Delta m}_{e\\mu}^2$ \\quad\nhas the right value for this solution, we obtain for the RH-neutrino scale:\n$$\nM_R \\sim 10^{13} - 10^{14} GeV.\n$$\nThe corresponding masses of $\\nu_\\tau$ are of few eV -- in\nthe range interesting for cosmology~\\cite{cos}. At the same time the\nvalues of \\quad ${\\sin^2}2\\theta_{\\mu\\tau}$ \\quad are such that $\\nu_\\tau$\noscillations\nwill be observed in experiments like CHORUS~\\cite{chorus}, NOMAD~\\cite\n{nomad} and P803~\\cite{p803}.\n\nThe plan of the paper is as follows. In sect.~2 the models (and their\ndiscrete symmetries) will be discussed in detail.\nSect.~3 will explain the details of our numerical analysis.\nIn sect.~4 we will give and discuss the results in the neutrino sector.\nConclusions and remarks can be found in sect.~5.\n\n\\section{The models}\n\nThe general Yukawa Lagrangian is given in eq.(1). To have the\nactual form of the mass matrices, one must give the Yukava coupling\nconstants $Y_{ij}^{\\bf 10}$ and $Y_{ij}^{\\bf \\overline {126}}$ and the\nVEVs of the Higgs representations \\ $\\phi_{\\bf 10}^{ij}$\\ and \\ $\\phi_{\\bf \\overline{126}}^{ij}$ \\ . The discrete symmetries,\nto be discussed later, will play here an important role. Those symmetries\nfix the non-zero entries of the Yukawa matrices and ensure the stability\nof our predictions.\n\nLet us, however, discuss first the possible entries to the matrices on\na pure phenomenological level.\nBoth \\ $\\phi_{\\bf 10}$\\ and \\ $\\phi_{\\bf \\overline{126}}$ \\ can develop VEVs in the directions of the down and\/or\nthe up quarks. However, only the \\ $\\phi_{\\bf \\overline{126}}$ \\ multiplets allow for a B--L\nviolating VEV, which generates the Majorana mass matrix of the RH-\nneutrino. We assume, as usual, that below the GUT scale one has\neffectively the MSSM with two doublets of Higgs \\ $H_u$ \\ and \\ $H_d$ \\ . These\nare mixtures of the SM doublet components of all scalar SO(10)\nrepresentations,\nalso those needed for the local symmetry breaking. We can therefore\nseparate the Yukawa terms into five groups, even at the SO(10) GUT\nscale as follows:\n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{equation}\n\\begin{array}{ccc}\n{\\cal L}_Y & = &\\sum Y_{ij} \\left \\{ a^{ij} \\left[ ( \\overline{d_{Ri}} d_{Lj} +\n\\overline{\\ell_{Ri}} \\ell_{Lj}) H_{{\\bf 10},d}^{ij}\n + (\\overline{u_{Ri}}{u_{Lj}} + \\overline{\\nu_{Ri}}\\nu_{Lj}) H_{{\\bf\n10},u}^{ij}\n\\right] \\right .\\\\\n & + & b^{ij} \\left [ (\\overline{d_{Ri}} d_{Lj} - 3\n\\overline{\\ell_{Ri}}{\\ell_{Lj}})\n H_{{\\bf 126},d}^{ij} + (\\overline{u_{Ri}} u_{Lj} - 3\n\\overline{\\nu_{Ri}}{\\nu_{Lj}})\n H_{{\\bf 126},u}^{ij} \\right . \\\\\n & + & \\left .\\left .\\overline{\\nu_{Ri}} \\nu_{Rj} (-3)\n\\phi_{{\\bf {\\overline 126},1}_{SU(5)}}^{ij} \\right ]\n \\right \\}\\ .\n\\end{array}\n\\end{equation}\n\nIn view of the requirement (2) that only one of the \\ $\\phi_{\\bf 10}^{ij}$\\ or \\ $\\phi_{\\bf \\overline{126}}^{ij}$ \\ can contribute\nto the mass matrices, we have for the non-vanishing Yukawa matrix elements\nonly one of the two possibilities:\n$$\n(a^{ij} , b^{ij})= (1,0) {\\rm\\quad or \\quad} (0,1).\n$$\nThe quark mass matrices develop below the GUT scale, in terms of the MSSM,\nthe following contributions, which define the effective Yukawa matrices\nused in the RGEs:\n\n\\begin{equation}\n(M_d)_{ij} = Y_{ij} (a_{ij}{\\gamma_d}^{ij} + b_{ij}{\\theta_d}^{ij})\n\\cos\\beta\\ \\upsilon\n\\end{equation}\n\n\\begin{equation}\n(M_u)_{ij} = Y_{ij} (a_{ij}{\\gamma_u}^{ij} + b_{ij}{\\theta_u}^{ij})\n\\sin\\beta\\ \\upsilon\n\\end{equation}\n\nwhere, $\\gamma^{ij}$ and $\\theta^{ij}$ account for the amount of\nmixing of the VEVs of the MSSM doublets $$ and $$ . Also,\nas usual in the MSSM:\n\n$$\n\\tan \\beta = \\frac{}{} {\\rm\\qquad and\\qquad}\n \\upsilon = \\sqrt{{}^2 + {}^2} = 174\\mbox{ GeV}\\ .\n $$\n\nNow, for the phenomenological ``good'' textures, there are additional\nrestrictions:\\\\\nThe Yukawa couplings \\quad $Y_{ij}$ \\quad vanish when the corresponding\ntexture\nzeros are common to both $M_d$ and $M_u$.\nE.g\\quad $Y_{ij}=0$\\quad in all texture sets.\\\\\nFor zero entries in only one matrix we have:\n$$\na_{ij}{\\gamma}_u^{ij} + b_{ij}{\\theta}_u^{ij} =0 \\qquad or \\qquad\na_{ij}{\\gamma}_d^{ij} + b_{ij}{\\theta}_d^{ij} =0 .\n$$\nAs for the non-vanishing $(i,j)$ matrix elements -- it is impossible\nto say which Higgs representation \\ $\\phi_{\\bf 10}^{ij}$\\ or \\ $\\phi_{\\bf \\overline{126}}^{ij}$ \\ contributes, as long as only\nthe quark masses and mixing angels are used.\n\nOur phenomenological discussion is based on the five sets of texture zeros\nfor the quarks, given in table \\ref{SQM}.\n\nNow, to predict the neutrino matrices we must know which of the Higgs\nrepresentations, \\ $\\phi_{\\bf 10}$\\ or \\ $\\phi_{\\bf \\overline{126}}$ \\ , contributes to the different matrix elements.\nAs was already discussed in the introduction, the structure of $M_d$\nand $M_{\\ell}$ is fixed by the need to have the approximate Yukawa\nunification and the Georgi-Jarlskog mass relations. The result is that\nall the non-vanishing matrix elements will be generated by \\ $\\phi_{\\bf 10}$\\ ~,\\linebreak[3]\nexcept for the (2,2) one which is due to \\ $\\phi_{\\bf \\overline{126}}$ \\ . (I.e. it obtains a relative\nfactor (-3) in \\ $(M_{\\ell})_{22}$ \\ relative to \\ $(M_d)_{22}$ ).\\\\\nThe explicit structure of those matrices , for the different textures,\ncan be found in table \\ref{STRUCT}.\n\nIn order to fix the structure of $M_u$ we must go in a different direction,\nas $M_{\\nu_D}$ is unknown phenomenology. We shall use our\npredictability and minimality requirements to restrict considerably\nthe number of possibilities. The resulting \\ $M_u$ \\ textures will dictate\nthe neutrino matrices.\\\\\nThe arguments go as follows:\\\\\nAll non-vanishing entries to \\quad $M_{\\nu_R}$ \\quad must be generated by\none \\ $\\phi_{\\bf \\overline{126}}$ \\ and\nthis should be induced via our discrete symmetries. Those, however,\nallow for one Higgs multiplet to couple to at most two $(i,j)$ entries.\nHence, only the following non-singular possibilities are open:\\\\\n\\renewcommand\\arraystretch{1.0}\n$$\n M_{\\nu_R}^I = \\left (\\begin{array}{ccc}\n0 & y & 0 \\\\\ny & 0 & 0 \\\\\n0 & 0 & x\n\\end{array} \\right )\\quad , \\quad\n M_{\\nu_R}^{II} = \\left (\\begin{array}{ccc}\n0 & 0 & y \\\\\n0 & x & 0 \\\\\ny & 0 & 0\n\\end{array} \\right )\n$$\nand\n$$\n M_{\\nu_R}^{III} = \\left (\\begin{array}{ccc}\nx & 0 & 0 \\\\\n0 & 0 & y \\\\\n0 & y & 0\n\\end{array} \\right ) .\n$$\nThe two last possibilities, however, cannot be realized in our textures.\nFor $M_{\\nu_R}^{III}$ it is clear because in all the five quark texture \\quad\n$Y_{11} = 0$. For $M_{\\nu_R}^{II}$ it is more complicated. If one Higgs\nrepresentation induces contributions to two entries (i,j) and (k,l) in\none of the matrices, it means that $\\overline {\\Psi_i^c} \\Psi_j$ and\n$\\overline {\\Psi_k^c}\\Psi_l$ have the same quantum numbers. Thus, {\\em all}\nmass matrices acquire a contribution in {\\em both} entries or {\\em no one at\nall}.\n(In the last case, all Higgs representations with the above quantum namber\ndo not develop a VEV in the relevant direction).\nIn our case \\ $(M_d)_{22} \\not=0$ \\ while \\ $(M_d)_{13} =\n(M_d)_{31} = 0 $ \\ in all textures and \\ $M_{\\nu_R}^{II}$ \\\nis inconsistent with the above requirement.\n\nNow, \\ $\\phi_{\\bf \\overline{126}}$ \\ which generates \\ $ M_{\\nu_R}^I$ \\ can contribute to \\ $M_u$ \\\nand\n \\ $M_{\\nu_R}$ \\ only, as \\quad $(M_d)_{33}$ \\quad must come from a \\ $\\phi_{\\bf 10}$\\ .\nHence, the \\ $\\phi_{\\bf \\overline{126}}$ \\ representation which induces the Majorana $M_{\\nu_R}$\ncan contribute to $M_u$ and $M_{\\nu_D}$ only. Minimality then requires\nthat this must be the case.\n\nOne finds, by explicit observation, that \\ $M_{\\nu_R}^I$ \\ is relevant\nfor the texture sets 1,2 and 4.\nThis fixes three matrix elements in \\ $M_u$.\nThe other entries required by the five quark textures can get\ncontributions from both \\ $\\phi_{\\bf 10}$\\ or \\ $\\phi_{\\bf \\overline{126}}$ \\ .\\ $M_{\\nu_D}$ \\ is then obtained from\n$M_u$ using suitable Clebsch-Gordan factors. Note, that the magnitude\nof those contributions is given by the phenomenology i.e by fitting\nthe evolved $M_u$ matrix to the observed masses and mixing angles. Only the\nfactors accompanying these contributions in \\ $M_{\\nu_D}$ \\ are\ndictated by\nthe choice of \\ $\\phi_{\\bf 10}$\\ or \\ $\\phi_{\\bf \\overline{126}}$ \\ .\n\nWe have, therefore, several possible combinations for each texture set and\nin total eight different ones. Those are presented in table \\ref{STRUCT}.\n\nOnce the neutrino matrices \\ $M_{\\nu_D}$ \\ and \\ $M_{\\nu_R}$ \\ are known, we can construct\nthe see-saw matrix for each model, in the form:\n\n\\begin{equation}\nM_{\\nu}^{light} \\simeq - M_{\\nu_D} M_{\\nu_R}^{-1} M_{\\nu_D} .\n\\end{equation}\n\nAfter this matrix is evolved to low energies (see next section) it gives us\nthe neutrino masses and mixing angles. To calculate the mixing angles one must\nobviously consider the charged lepton mass matrix as well. The angles\nare, however, independent on the overall mass scale of the RH-neutrinos.\nThe latter is a free parameter in our models and hence, we\npredict the neutrino mass ratios only. The RH-neutrino scale will be fixed\nlatter for models with mixing angles which allow for a solution to the\nSNP, such that \\ $\\Delta m_{e\\mu}^2$ \\ have the right value.\n\nWe know now phenomenologically what the different textures are and it remains\nonly to show how those textures can be induced using discrete symmetries.\nThis is actually strait forward and very similar in the\ndifferent models.\n\nLet the fermions and Higgs representation have the following\ntransformation properties under our symmetry:\n$$\n\\begin{array}{ccc}\n\\psi_j \\rightarrow e^{i\\alpha_j}\\psi_j & \\hbox{ and } &\n\\phi_j^{\\bf 10} \\rightarrow e^{i\\beta_j} \\phi_j^{\\bf 10} \\\\[10pt]\n\n & & \\phi_j^{\\bf\\overline{126}} \\rightarrow e^{i\\gamma_j}\n\\phi_j^{\\bf\\overline{126}} .\n\\end{array}\n$$\nWe must require that \\ $(M_u)_{12}$ \\ and \\ $(M_d)_{33}$ \\ are generated by\none \\ $\\phi_{\\bf \\overline{126}}$ \\ . Hence,\\\\\n$$\n{\\alpha_1} + {\\alpha_2} = 2{\\alpha_3} = -{\\gamma_1} .\n$$\nHowever, \\ $M_d$ \\ gets also contributions at the (1,2) and (3,3) entries,\nvia \\ $\\phi_{\\bf 10}$\\ . As our symmetry is on the SO(10) level, those\nmatrix elements also, must be due to the same Higgs representation i.e \\ $\\phi_{\\bf 10}$\\ \nin this case and \\\\\n$$\n\\beta_1 = \\gamma_1 .\n$$\n\nThis means that \\ $\\phi_{\\bf \\overline {126}}^1$ \\ generates a\nlight VEV in the u-direction while \\ $\\phi_{\\bf 10}^1$ \\ generates\none in the d-direction.\nThe other entries acquire contributions according to the corresponding\nquantum numbers.\\\\\n\nAs an explicit realization we can take:\\\\\n$$\n\\alpha_1 = 1 , \\qquad \\alpha_2 = 3 \\qquad \\hbox{ and } \\qquad \\alpha_3 = 2 .\n$$\nin this case:\\\\\n$$\n\\beta_1 = \\gamma_1 = -4 .\n$$\nE.g for the texture $1_I$ we have , in addition:\\\\\n$$\n\\gamma_2(\\phi_{\\bf 10}^{22}) = \\beta_2(\\phi_{\\bf \\overline{126}}^{22}) =\n -2\\alpha_2 = -6\n$$\nand\n$$\n\\beta_3(\\phi_{\\bf 10}^{23}) = -(\\alpha_2 + \\alpha_3) = -5 .\n$$\n\nSo finally for this texture we need:\\\\\n$$\n3 \\ \\times \\ \\phi_{\\bf 10} \\qquad and \\qquad 2 \\ \\times \\\n\\phi_{\\bf \\overline{126}} \\quad ,\n$$\n\nof which only \\ $\\phi_{\\bf \\overline{126}}^ 1$ \\ generates a heavy VEV.\n\nIn all other models very similar discrete symmetries are needed.\n\n\\section{Renormalization Group Equations and Fits}\n\nAll the matrix elements of our matrices are in principle complex numbers.\nOne can, however, use the freedom to redefine the nine phases of the three\nLH-doublets and six RH-singlets of the SM, to reduce considerably the\nnumber of the ``physical\" phases. In any case, symmetric quark matrices can be\nalways transformed into hermitian ones in this way~\\cite{rrr}.\nAs we are interested only in the neutrino\nsector and the leptonic phases cannot be observed - we use for simplicity\nonly one physical phase. Let us put it at the (1,2) matrix\nelement and in an hermitian way - as DHR~\\cite{dhr} do.\n\nAs an example, we give in the following the explicit matrix elements\nof the model\ndiscussed in the previous section.\n\nModel $1_I$ :\n\\begin{equation}\n{\\bf Y}_U = \\left(\\begin{array}{ccc} 0 & C_u & 0 \\\\ C_u & B_u & 0\\\\ 0 & 0 & A_u\n\t \\end{array} \\right)\n\\quad\n{\\bf Y}_D = \\left(\\begin{array}{ccc} 0 & D_d e^{i\\phi} & 0 \\\\\n\t\t\t\tD_d e^{-i\\phi} & C_d & B_d\\\\ 0 & B_d & A_d\n\t \\end{array} \\right)\n\\end{equation}\n\\begin{equation}\n{\\bf Y}_{\\nu_D} = \\left(\\begin{array}{ccc} 0 & -3 C_u & 0 \\\\ -3 C_u & B_u & 0\\\\\n\t\t\t\t\t0 & 0 & -3 A_u\n\t \\end{array} \\right)\n\\quad\n{\\bf Y}_L = \\left(\\begin{array}{ccc} 0 & D_d e^{i\\phi} & 0 \\\\\n\t\t\t\tD_d e^{-i\\phi} & -3 C_d & B_d\\\\ 0 & B_d & A_d\n\t \\end{array} \\right)\n\\end{equation}\n\\begin{equation}\n{\\bf Y}_{\\nu_M} = \\left(\\begin{array}{ccc} 0 & C_u & 0 \\\\ C_u & 0 & 0\\\\\n\t\t\t\t\t0 & 0 & A_u\n\\end{array} \\right)\n\\end{equation}\n\nIn order to extract the neutrino properties from these matrices, the matrix\nelements $A_u$, $B_u$, \\ldots, $D_d$ and $\\phi$ have to be determined. We do\nthis, as usual, by fitting masses and mixing angles as predicted by the\ntextures to their experimental values. Since these specific textures for the\nYukawa matrices are given at the unification scale $M_X$, we must evolve the\nmatrices from the GUT scale, $M_X$ to low energies using the renormalization\ngroup equations (RGEs) (see Appendix \\ref{App1}).\n\nIn our model, the SUSY-SO(10) is broken at $M_X$ directly into the MSSM.\nThe MSSM is broken at the effective \\ $M_{SUSY} \\approx 100 GeV$ into\nthe SM which in broken in its turn effectively at $M_Z$ into \\ $SU_C(3) \\times\nU_{EM}(1)$ .\\\nWe take, as it is done in many papers, $M_{SUSY} = M_Z$. A different choice\nwill have only a minor effect on the neutrino properties.\n\nThe renormalization group equations for the gauge and Yukawa coupling\nconstants~\\cite{rge} are coupled, non-linear first order differential\nequations, which do not have a complete analytical solution.\nWe therefore use a numerical procedure both to solve the RGEs and to fit the\nYukawa matrix parameters.\n\nFor the Yukawa RGEs we use the effective Yukawa matrices $\\Lambda^{u,d...}$\ndefined by\n\\begin{equation}\nM_u = \\Lambda^u\\ \\upsilon \\sin\\beta \\quad , \\quad M_d = \\Lambda^d\\ \\upsilon\n\\cos\\beta \\quad e.t.c.\n\\end{equation}\nusing equations (3) and (4). Thus, for a given value of $\\tan\\beta$ ,\n$\\Lambda^i$ are obtained in terms of the mass matrices.\nThe explicit calculations\nwere done using the semi-analytic form due to Barger, Berger and\nOhmann~\\cite{bbo} (see Appendix \\ref{App1}). This form reduces the number of\nvariables significantly.\nTo fix the parameters of a given texture, we obtained first the masses\nand mixing angles as functions of the GUT scale parameters and then evolved\nthem to low energies. Those are then fitted to the experimental values using\nthe 'shooting' method~\\cite{gauge1} (see Appendix \\ref{App2}).\n\nThe run of the Yukawa and gauge coupling constants from $M_X$\ndown to $M_Z$ is done in terms of the\ntwo loop RGEs of the MSSM~\\cite{rge}.\nThe appropriate boundary conditions for the Yukawa and gauge couplings are\napplied at $M_Z$. Below $M_Z$,\nthree loop QCD and one loop QED renormalization group equations, are used.\nWe compare then our parameters with the standard masses of the light quarks\nand charged leptons given at $\\mu = 1 GeV$, and the heavy quarks at\ntheir physical masses (see Table \\ref{DATA}).\n\nSince all of our textures have eight parameters, we have to fit to eight\nexperimental quantities. We use, out of the experimental data displayed\nin Table \\ref{DATA}\n$m_b$, $m_c$, $m_u$, $m_e$, $m_\\mu$, $m_\\tau$,\n$|V_{us}|$ and $|V_{cb}|$ to fix the texture parameters. In addition, we take\n$\\alpha_1(M_Z)$ and $\\alpha_2(M_Z)$ to fix the GUT scale $M_X$ and the gauge\ncoupling $\\alpha(M_X)$ at the GUT scale.\n\nNow, the texture parameters found with the shooting procedure define the\nsee-saw matrix (5). We evolve this matrix from $M_X$ to $M_Z$ using\none-loop renormalization group equations \\cite{SSRGE} (see Appendix\n\\ref{App3}).\n\nAs a result we obtain, for each texture, a set of solutions which give a\ngood fit to the data, i.e $\\chi^2 < 1$ .\nThose solutions are parametrized according to the value of $\\tan\\beta$.\nOne of the predicted parameters is $m_t$. The dependence of $m_t$ on\n$\\tan\\beta$ is given in Fig.~1, for the texture $1_I$. For all other textures\nit is practically the same, as we have in all our models the approximate\n$\\tau - b$ Yukawa unification. One sees , as it is already well known in this\ncase, that small and large values of $\\tan\\beta$ are preferred. As those\ncorrespond to two different physical situations~\\cite{tan}, we will\npresent our results for $\\tan\\beta = 1.5$ and $\\tan\\beta = 55$.\n\n\n\\section{Discussion of the results. }\n\n\n\nOur results for the different textures are displayed in table \\ref{RESULTS}.\\\\\nLooking at this table one sees clearly that we do not have large mixing\nangles. Practically speaking, all our solutions obey\n$$\n\\sin^2 2\\theta < 0.2 \\ .\n$$\nThis means that our models cannot allow for the\ndepletion of the atmospheric muon neutrinos~\\cite{atm}. Also, vacuum\noscillations will not be able to serve as a solution to the SNP~\\cite{snp}\nand only the small angle (i.e adiabatic) MSW mechanism can work.\n\nUsing the recent estimate for the small angle MSW region~\\cite{ks}:\n$$\n\\sin^2 2\\theta_{e\\mu} = 6 \\times 10^{-4} - 2 \\times 10^{-2} ,\n$$\none sees that the models\n$ 1_{II}, \\ 4_{III}, \\ 4_{IV}, \\ $\ncan explain the SNP.\nThis is obviously provided~\\footnote{Note, that this the mass difference\nrelevant for our mixing angles in the range $\\sin^2 2\\theta_{e\\mu} \\simeq\n(1-2) \\times 10^{-2}$ .}:\n$$\n\\Delta m_{e\\mu}^2 \\simeq 4 \\times 10^{-5} .\n$$\nThis requirement fixes the RH-neutrino scale to be:\n$$\nM_R = 10^{13} - 10^{14} GeV .\n$$\nKnowing the neutrino mass ratios we can compute the corresponding masses of\nthe $\\tau$-neutrinos. Those are found to be all in the few eV region i.e.\ninteresting for cosmology~\\cite{cos}. Also, the corresponding $\\nu_{\\tau} -\n\\nu_{\\mu}$ mixing angles are large enough for $1_{II}$ and $4_{IV}$\nto be observed in the already runing\nCERN CHORUS~\\cite{chorus} and NOMAD~\\cite{nomad} experiments, as well as in\nthe approved FERMILAB P803~\\cite{p803} one.\nAlso, some of the models\nwhich cannot solve the SNP have relatively\nlarge $\\sin^2 2\\theta_{\\mu\\tau}$\nwhich can be observed in the above experiments.\n\n\n\\section{Conclusions and remarks}\n\nWe looked in this paper for SUSY-SO(10) models which can predict the\nneutrino-spectrum in terms of the ``known\" parameters of the charged\nfermions.\nThe main idea is to dictate the mass matrix of the RH-neutrinos rather than\nsimply conjecture its form, as it is done in many models.\nTo do this we used the requirements of DHR and suitable discrete symmetries.\nStarting then from the general classification of ``good\"\" symmetric\ntextures for the quark mass matrices by RRR, we predicted correspondingly\neight neutrino mass matrices at the GUT scale.\nIn evolving these mass matrices to low energies we made some approximations:\na)we neglected threshold effects at the GUT scale as well as at\n$M_{SUSY}$ which we took to be $M_Z$.\nb)we started the renormalization of the see-saw matrix also from the\nGUT scale and not from $M_R$.\nc)we made a simplifying conjecture for the unobservable leptonic phases.\nThose approximations, however, cannot change the qualitative predictions\nof our models. They can at most change somewhat the neutrino mixing angles.\nPractically speaking, one must allow for up to 10\\% deviations from our\npredictions of the neutrino properties.\n\nWe also required that our SUSY-SO(10) is ``the whole story\". I.e. we did not\nuse possible non-renormalizable effective contributions due to physics\nat the Planck scale (like gravity or superstrings). Such contributions\nare frequently used in recent papers. There are very many possible\ncontributions of which one picks up those suitable for his arguments and\nneglects arbitrarily all others. Such a procedure destroys the predictibility\nwhich is the main ingredient of our models. Also, one can imagine scenarios\nwhere the non-renormalizable effects are negligible and that we actually\nassume.\n\nThere is, however, one indirect evidence that physics at the Planck-mass\nmay be relevant to our models. This is related to the RH-neutrino mass-scale\nwhich is a free parameter . Yet, to explain\nthe solar neutrino puzzle and get $\\tau$-neutrino masses relevant for\ncosmology, we need $M_R = 10^{13} - 10^{14} GeV$ which is equal to\n$\\frac{M_{GUT}^2}{M_{Planck}}$~. It is also intersting to\nembed such an intermediate scale into the local symmetry\nbreaking of SUSY-SO(10), in order to make it natural\\cite{ag2}.\n\n\n\\begin{appendix}\n\\section{Appendix}\n\\subsection{Semi-analytic approach\\label{App1}}\nIn the renormalization group equations for the Yukawa couplings the largest\nYukawa contributions come from the Yukawa couplings of the third generation\n$y_t$, $y_b$ and $y_\\tau$. In view of this fact, Barger et.~al.\\ \\cite{bbo}\nfind the following RGEs for the Yukawa couplings:\n\\begin{eqnarray}\n{{d\\lambda _i}\\over {dt}}&=&{{\\lambda _i}\\over {16\\pi ^2}}\n\\Bigg [x_1+x_2\\lambda _i^2+\na_u\\sum_ {\\alpha}\\lambda _{\\alpha}^2|V_{i\\alpha }|^2\\nonumber \\\\\n&&+{1\\over {16\\pi ^2}}\\Bigg (x_3+x_4\\lambda _i^2+x_5\\lambda _i^4\n+\\sum _{\\alpha}\\Big (b_u\\lambda _{\\alpha}^2+c_u\\lambda _{\\alpha}^4\n+(d_u+e_u)\\lambda _i^2\\lambda _{\\alpha}^2\\Big )\n|V_{i\\alpha }|^2\\Bigg )\\Bigg ]\\;, \\nonumber\\\\\n& &\\\\\n{{d\\lambda _{\\alpha}}\\over {dt}}&=&{{\\lambda _{\\alpha}}\\over {16\\pi ^2}}\n\\Bigg [x_6\n+x_7\\lambda _{\\alpha }^2+\na_d\\sum_ i\\lambda _i^2|V_{i\\alpha }|^2\\nonumber \\\\\n&&+{1\\over {16\\pi ^2}}\\Bigg (x_8+x_9\\lambda _{\\alpha }^2+x_{10}\n\\lambda _{\\alpha }^4\n+\\sum _i\\Big (b_d\\lambda _i^2+c_d\\lambda _i^4\n+(d_d+e_d)\\lambda _{\\alpha }^2\\lambda _i^2\\Big )\n|V_{i\\alpha }|^2\\Bigg )\\Bigg ]\\;, \\nonumber\\\\\n& &\\\\\n{{d\\lambda _a}\\over {dt}}&=&{{\\lambda _a}\\over {16\\pi ^2}}\n\\Bigg [x_{11}+x_{12}\\lambda _a^2\n+{1\\over {16\\pi ^2}}\\Bigg (x_{13}+x_{14}\\lambda _a^2+x_{15}\\lambda _a^4\n\\Bigg )\\Bigg ]\n\\;,\n\\end{eqnarray}\nwhere $i=u,c,t$, $\\alpha=d,s,b$ and $a=e,\\mu,\\tau$. The CKM matrix elements\n$W_1=|V_{cb}|^2$, $|V_{ub}|^2$, $|V_{ts}|^2$, $|V_{td}|^2$, $J$ evolve\naccording to\n\\begin{equation}\n{{dW_1}\\over {dt}}=-{{W_1}\\over {8\\pi ^2}}\n\\Bigg [\\left (a_d\\hat{\\lambda} _t^2\n+a_u\\hat{\\lambda} _b^2\\right )+{{1}\\over {(16\\pi ^2)}}(e_d+e_u)\n\\lambda _t^2\\lambda _b^2\\Bigg ] \\;, \\label{dW1dt}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\hat{\\lambda}_b^2&=&\\lambda _b^2\\left (\n1+{{b_u+c_u\\lambda _b^2}\n\\over {16\\pi ^2a_u}}\\right )\\;, \\\\\n\\hat{\\lambda} _t^2&=&\\lambda _t^2\\left (\n1+{{b_d+c_d\\lambda _t^2}\n\\over {16\\pi ^2a_d}}\\right )\\;.\n\\end{eqnarray}\nFor $W_2=|V_{us}|^2, |V_{cd}|^2, |V_{tb}|^2, |V_{cs}|^2, |V_{ud}|^2$ we have\n\\begin{equation}\n{{dW_2}\\over {dt}}=0\\;. \\label{dW2dt}\n\\end{equation}\nThe various coefficients for the RGEs in the MSSM are given in table\n\\ref{MSSMCoeff}.\n\\subsection{Determining the texture parameters\\label{App2}}\nTo determine the texture parameters, we fit the low energy predictions for\nmasses and mixing angles to the experimental values. In our case, all textures\nhave eight parameters, so we fit to eight experimentally known quantities,\nnamely $m_b$, $m_c$, $m_u$, $m_e$, $m_\\mu$, $m_\\tau$, $|V_{us}|$ and\n$|V_{cb}|$. The procedure employed here is called {\\em `shooting'}.\n\nLet $p_j$ be the texture parameters and $R_i(p_j)$ the low energy predictions\nobtained by the above mentioned running procedure. Further, let $r_i$ denote\ntheir experimental values. We then have to solve the system of equations\n\\begin{equation}\n\tR_i(p_j) - r_i = 0.\n\\end{equation}\nThis is done by an iterative numerical procedure.\n\\subsection{Renormalization of the see-saw matrix\\label{App3}}\nFollowing the authors of \\cite{SSRGE}, we define the neutrino see-saw mass\ncoefficient\n\\begin{equation}\n{1 \\over 2} c_1^{ab} =\n c_{21}^{ab} =\n c_{22}^{ab} =\n{1 \\over 2} c_3^{ab} =\nY_l^{ca} (M_R^{-1})^{cd} Y_l^{db}.\n\\end{equation}\n\nThe renormalization group equations for the see-saw matrix in the strict SUSY\nlimit are:\n\\begin{eqnarray}\n {d \\over dt} c^{ab}_1 &=& \\frac{1}{16\\pi^2}\\Big(\n \\left[{1\\over2} 2g_1^2 + 2g_2^2\n + 6~tr\\left(Y_uY^{\\dagger}_u\\right)\\right] c^{ab}_1\n + \\left(Y_l Y_l^\\dagger\\right)^{bc} c^{ca}_1\n + \\left(Y_l Y_l^\\dagger\\right)^{ac} c^{cb}_1\\nonumber \\\\\n & &-\\left(2g_1^2 + 6g_2^2\\right)\n \\left(c^{ab}_{21} + c^{ba}_{21}\\right)\n - \\left(2g_1^2 + 6g_2^2\\right)\n \\left(c^{ab}_{22} + c^{ba}_{22}\\right)\\Big)\n\\end{eqnarray}\n\\begin{eqnarray}\n {d \\over dt} c^{ab}_3 &=& \\frac{1}{16\\pi^2}\\Big(\\left[2 g_1^2+2 g^2_2\n +6~tr\\left(Y_u Y^{\\dagger}_u\\right)\\right]c^{ab}_3\n +\\left(Y_lY^{\\dagger}_l\\right)^{bc} c^{ca}_3\n +\\left(Y_l Y^{\\dagger}_l\\right)^{ac} c^{cb}_3\\nonumber\\\\\n & & -\\left(2~g_1^2 +6~g_2^2 \\right)\n \\left(c^{ab}_{21}+c^{ba}_{21}\\right)\n -\\left(2~g_1^2 +2~g_2^2 \\right)\n \\left(c^{ab}_{22}+c^{ba}_{22}\\right)\\Big)\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n {d \\over dt} c^{ab}_{21} &=&\\frac{1}{16\\pi^2}\\Big(\\left[4~g^2_2 - 2~g_1^2\n +6~tr\\left(Y_u Y^{\\dagger}_u\\right)\\right]c^{ab}_{21}\n + 2~g^2_2 ~c^{ba}_{21}\n +\\left(Y_l Y^{\\dagger}_l\\right)^{bc} c^{ca}_{21}\n +\\left(Y_l Y^{\\dagger}_l\\right)^{ac} c^{cb}_{21}\\nonumber \\\\\n & &+\\left(g_1^2-g_2^2\\right)\\left(c_{22}^{ab}+c_{22}^{ba}\\right)\n -{1\\over 2}\\left(g_1^2+5~g_2^2 \\right)\n \\left(c_1^{ab}+ c_3^{ab}\\right)\\Big)\n\\end{eqnarray}\n\n\\begin{eqnarray}\n {d \\over dt} c^{ab}_{22}&=&\\frac{1}{16\\pi^2}\\Big(\\left[-4~g_1^2-2~g^2_1\n +6~tr\\left(Y_u Y^{\\dagger}_u\\right)\\right]c^{ab}_{22}\n +2~g^2_2~c^{ba}_{22}\n +\\left(Y_l Y^{\\dagger}_l\\right)^{bc} c^{ca}_{22}\n +\\left(Y_l Y^{\\dagger}_l\\right)^{ac}c^{cb}_{22}\\nonumber \\\\\n & &+\\left(g_1^2-g_2^2\\right)\\left(c_{21}^{ab}+c_{21}^{ba}\\right)\n -4~g_2^2~c^{ab}_{21}\n -{1\\over2}\\left(g_1^2-g_2^2 \\right)\n \\left(c_1^{ab}+ c_3^{ab}\\right)\\Big)\n\\end{eqnarray}\n\\end{appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{figure\/model-architechture-v3.pdf}\n \\caption{Overview of the components of the proposed system.}\n \\label{fig:model_framework}\n\\end{figure*}\n\n\n\n\nVoice conversion (VC) aims to modify speech from a source speaker to sound like that of a target speaker without changing the linguistic content. Voice conversion based on neural networks, e.g. GAN~\\cite{GANHsu2017VoiceCF}, VAE~\\cite{VAEHsu2016Voicevae}, recognition-synthesis\\cite{PPGSun2016PhoneticPF} frameworks, have significantly improved the speech quality and speaker similarity. Despite recent progress, most current works focus on the transformation of timbre while ignoring the transformation of style contained in source speech. Recently, explicit prosodic features~\\cite{explicitMing2016ExemplarbasedSR,explicitRaitio2020prosodyControl} and implicit prosody extractor~\\cite{impiicitLiu2020TransferringSS,EnrichingSourceStyleTransfer} are used to model prosody and perform source style transfer, while requiring a large number of recordings. Data collection is expensive and time-consuming. Therefore, it remains a challenge to build a high-quality voice conversion framework for style transfer with limited data.\n\nRecently, one-shot voice conversion approaches are proposed, which refers to the conversion of the speaker identity given only one utterance of the target speaker, which is more practical for application as compared with previous methods in which sizeable data for the target speaker is needed. Some studies~\\cite{autovcqian2019autovc,AgainVC,OSSEDu2021ImprovingRO} propose to extract content and speaker representation from speech during training stage, and concatenate content representation from source speech and speaker representation from target speech at run-time to produce converted speech. However, the content representation may still contain speaker-related information, which results in unstable performance in terms of speaker similarity. Other works~\\cite{GSE,VQMIVC} decompose speech into three factors: content, speaker, and pitch with the premise that pitch is highly related to speaker characteristics. These methods further improve the disentanglement ability, while there is still a gap between the converted speech and target speech in terms of speaker similarity. To improve speaker similarity, many works~\\cite{Adaspeech,TTSADaptationArik2018NeuralVC,GCtts} in text to speech (TTS), a related task, focus on few-shot speaker adaptation. Unlike few-shot ($\\geq$10 utterances) TTS, in one-shot voice conversion, training on one utterance (3-4s) can easily lead to over-fitting and reduce the speaker similarity and speech quality. Besides delivering correct content and speaker information, transferring style or prosody from source speaker to the target is also desired, but more challenging~\\cite{explicitMing2016ExemplarbasedSR,explicitRaitio2020prosodyControl,impiicitLiu2020TransferringSS,EnrichingSourceStyleTransfer}, as prosodic aspects are entangled with content and speaker as well and the over-fit problem is rather complicated. \n\n\n\n\nIn our paper, we address the \\textit{one-shot style transfer} problem for voice conversion, aiming to transferring the source content and style to the target speaker while maintaining the good target speaker identity; more challengingly, we achieve the above goal with only one utterance with several seconds from the target speaker. Specifically, to solve the over-fit problem caused by training with only one utterance, we propose a novel one-shot voice conversion framework for style transfer based on recognition-synthesis framework~\\cite{PPGSun2016PhoneticPF}, integrating speaker normalization~\\cite{speakernormalzationGlarner2021}, weight regularization~\\cite{weightRegLi2020}, and prosody modeling. Speaker normalization is used to remove speaker-related information in the bottleneck feature (BN) extracted from ASR, which is helpful to improve speaker similarity. Weight regularization is applied to the adaptation with one utterance to prevent performance degradation. A prosody module~\\cite{EnrichingSourceStyleTransfer} is adopted to explicitly extract prosody information for transfer. We believe such a comprehensive characterization of speech factors can achieve style transfer and make the model focus on learning timbre in the adaptation process. The experimental results on VCTK~\\cite{VCTK} and CMU-ARCTIC~\\cite{CMU-Arctic} show that over-fitting is greatly alleviated after using these methods, and our framework is better than several state-of-the-art (SOTA) methods including AGAINVC~\\cite{AgainVC}, GSE~\\cite{GSE} and VQMIVC~\\cite{VQMIVC} in terms of style and speaker similarity.\n\nThe rest of the paper is organized as follows. Section 2 introduces our proposed methods. Section 3 presents the experiments and compares our proposed system with SOTA systems with subjective and objective evaluation. Section 5 concludes this paper.\n\n\n\n\n\n\\section{Proposed approach}\n\n\\subsection{System Overview}\n\n\nOur proposed one-shot voice conversion framework is based on a recognition-synthesis architecture, which is shown in Fig.~\\ref{fig:model_framework}. This framework consists of four main modules (light blue boxes in Fig.~\\ref{fig:model_framework}): content module, speaker module, prosody module and conversion module. The content module takes bottleneck features (BN) extracted from ASR as input, and the output is speaker-independent content representation. Specifically, we adopt speaker normalization technique to remove speaker-related information contained in BN. The speaker module takes mel spectrum from target speaker to extract the speaker representation. To ensure the speaker module is able to extract desired speaker representation without speaker confusion, we add an auxiliary speaker classifier after the reference encoder. The prosody module learns to extract speaker-independent prosody representations, which integrates explicit and implicit hybrid modeling methods~\\cite{EnrichingSourceStyleTransfer}. The explicit modeling includes raw logarithmic domain fundamental frequency (lf0), normalized short-term average amplitude (energy), and the voice\/unvoice flag (VUV). For implicit modeling, we adopt reference encoder to extract prosody representation. The conversion module takes prosody, content, and speaker representations as input, and the output is mel spectrum. Finally, modified LPCnet~\\cite{LPCnet} is adopted to reconstruct waveform from mel spectrum.\n\n\n\n\\subsection{Speaker Normalization}\n\nAlthough an ASR system is trained using multi-speaker data, targeting to speaker-independent acoustic representation, studies~\\cite{ppgdvectoradv,AccentVCWang2020,EnrichingSourceStyleTransfer} show that the bottleneck feature (BN) still inevitably contains speaker-related information, such as timbre and style. Thus for voice conversion, the speaker similarity of the converted speech may be degraded. As shown in Fig.~\\ref{fig:model_framework}, we specifically introduce a speaker normalization method to remove the speaker-related information. Previously, adversarial training ~\\cite{ppgdvectoradv,AccentVCWang2020} was usually used to achieve this goal. But taking into account the training cost and instability of adversarial training, we utilize a speaker normalization method~\\cite{speakernormalzationGlarner2021} instead in our framework. Specifically, we utilize an any-to-one VC network here to serve as a speaker `normalization' trick to `normalize' the source audio's BN to mel spectrum of a specific speaker to achieve the purpose of normalizing content information. By doing so, we actually relieve the burden of the rest of our framework.\n\n\n\n\\subsection{Weight Regularization}\n\\label{sec:sn}\n\nAs shown in Fig.~\\ref{fig:model_framework}, in order to alleviate the over-fitting problem caused by training with only one sentence and improve the stability of the training process, we introduce weight regularization~\\cite{weightRegLi2020}. The weight regularization is a variant of l2 regularization, defined as\n\\begin{equation}\n L_{wReg}=||\\theta-\\theta_{f}||^2,\n\\end{equation}\nwhere $\\theta$ represents the parameters of HighNet, Bi-GRU and postnet updated over time in adaptation process, and $\\theta_{f}$ represents these three layers' parameters before adaptation which is used as fixed value during the adaptation process. The core idea in $L_{wReg}$ is to prevent the parameters of adapted model to drift far away from those of the base model learnt in the large dataset. The regularization prevents model from changing too significantly, which is helpful in improving the stability of the training process.\n\n\\subsection{Source Style Transfer}\n\nMany one-shot voice conversion methods~\\cite{autovcqian2019autovc,INchou2019oneshot,AgainVC,GSE,VQMIVC} focus on disentangling content and speaker, while ignoring the prosody modeling, which results in that the prosody information may leak to the content and speaker representation. This reduces the stability of disentanglement and causes poor speaker similarity, although the prosody of source is much transferred to the target. Inspired by our previous work~\\cite{EnrichingSourceStyleTransfer}, we add a prosody module to the one-shot voice conversion framework. Recall that we explicitly decompose speech into three parts: content, speaker as well as style, and this prosody module is specifically in charge of the style representation for the final conversion module. In detail, the prosody module includes explicit and implicit modeling schemes. For explicit modeling, we extract energy, VUV, and raw lf0 from source audio. For implicit modeling, the BN is used as the input of a reference encoder, and the output is prosody representation. By adding the prosody module, the voice conversion framework is able to perform style transfer.\n\n\n\\begin{table*}[h]\n\\caption{MOS results with 95\\% confidence interval.}\n\\vspace{10pt}\n\\label{tab:mos}\n\\centering\n\\begin{tabular}{c|l|l|ccc}\n\\toprule\n\\multicolumn{3}{c|}{Model} & Speech Quality & Style Similarity & Speaker Similarity \\\\ \\midrule\nComparison & \\multicolumn{2}{l|}{AGAINVC~\\cite{AgainVC}} & 2.81$\\pm$0.12 & 3.03$\\pm$0.34 & 2.96$\\pm$0.34 \\\\ \\cline{2-6} \n & \\multicolumn{2}{l|}{VQMIVC~\\cite{VQMIVC}} & 2.94$\\pm$0.27 & 3.19$\\pm$0.30 & 3.02$\\pm$0.15 \\\\ \\cline{2-6} \n & \\multicolumn{2}{l|}{GSE~\\cite{GSE}} & \\textbf{3.46$\\pm$0.16} & 3.19$\\pm$0.12 & 3.03$\\pm$0.18 \\\\ \\cline{2-6} \n & \\multicolumn{2}{l|}{GSE-finetune} & 3.09$\\pm$0.14 & 2.88$\\pm$0.11 & 3.31$\\pm$0.12 \\\\ \\hline\nAblation & BL & Baseline & 2.93$\\pm$0.13 & 2.84$\\pm$0.16 & 3.25$\\pm$0.13 \\\\ \\cline{2-6} \n & P1 & \\ +Speaker Normalization & 3.16$\\pm$0.12 & 3.15$\\pm$0.12 & 3.37$\\pm$0.10 \\\\ \\cline{2-6} \n & P2 & \\ \\ +Weight Regularization & 3.27$\\pm$0.12 & 3.21$\\pm$0.13 & 3.43$\\pm$0.07 \\\\ \\hline\n\\multicolumn{2}{c|}{P3 (Proposed)} & \\ \\ \\ +Prosody Module & 3.43$\\pm$0.12 & \\textbf{3.76$\\pm$0.17} & \\textbf{3.56$\\pm$0.15} \\\\ \\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Training and Conversion Procedure}\n\n\n\n\nAs shown in Fig.~\\ref{fig:model_framework}, our approach is composed of three training phases and a conversion phase. Different phases are marked with numbers in the figure.\n\n\\textbf{Training phase 1.} As we mentioned in Section~\\ref{sec:sn}, we utilize an any-to-one VC method to implement our content module. The content module is trained on a large amount of speech data and finetunes on a specific speaker to ensure the performance of speaker normalization. In our experiment, the content module is optimized with mel reconstruction loss.\n\n\\textbf{Training phase 2.} The whole model is trained with a large amount of data. Note that the speaker module is trained with a randomly selected mel spectrum of the same speaker to make sure that the extracted speaker embedding is only related to the speaker information. The prosody and conversion modules are optimized with reconstruction loss $L_{recons}$. The speaker module is optimized with cross-entropy loss $L_{ce}$. The loss function of this stage can be described as follows:\n \\begin{equation}\n Loss_{stage2}=L_{recons}+ L_{ce}\n \\end{equation}\n\n\\textbf{Training phase 3.} This stage is for adaptation. In this stage, we only use one utterance from target speaker to fine-tune the model. Previous works~\\cite{Adaspeech,GCtts} confirm that a large number of parameters make model easy to over-fit. As shown in Fig.~\\ref{fig:model_framework}, we only optimize a part of the whole model. To further prevent performance degradation caused by over-fitting, we adopt the weight regularization~\\cite{weightRegLi2020} in this phase. The loss used in this phase is described as follows:\n \\begin{equation}\n Loss_{stage3}=L_{recons}+\\gamma L_{wReg}\n \\end{equation}\n\n\\textbf{Conversion phase.} The content and prosody representation are extracted from source speech. Note that we use linear transformation to obtain converted f0. The speaker representation is extracted from target speech. The conversion module takes content, prosody, and speaker representation to reconstruct converted speech. Through the proposed disentanglement and training procedure, the converted speech is expected to have the same prosody as source speech, while maintaining the content and target speaker's identity.\n\n\\section{EXPERIMENTS AND RESULTS}\n\\subsection{Dataset and Experimental Setup}\n\nIn our experiments, 102 speakers from VCTK~\\cite{VCTK} are used to train the conversion model. For one-shot testing, \\textit{p340}, \\textit{p363} from VCTK, as well as \\textit{slt}, \\textit{bdl} from CMU-ARCTIC~\\cite{CMU-Arctic} are used as the target speakers. The duration of target speech ranges from 3 to 4 seconds. For the source audio, we use CMU-ARCTIC (rms, clb) and ESD dataset~\\cite{ESD}. All speech utterances are downsampled to 16kHz. We use 80-dim mel spectrum computed with 50ms frame length and 12.5ms frame shift. The ASR system is a TDNN-F model trained with 1k hours standard English corpus. We use the 256-dim bottleneck features as the linguistic representation, which is extracted from the last fully-connected layer before softmax. Modified LPCnet~\\cite{LPCnet} is adopted to reconstruct waveform from mel spectrum. The vocoder is trained with speech data of 102 speakers from VCTK.\n\n\nTo validate our proposed method, we implement comparison and ablation systems. For comparison systems, we selected three SOTA one-shot VC methods, including AGAINVC~\\cite{AgainVC}, GSE~\\cite{GSE} and VQMIVC~\\cite{VQMIVC}. For a fair comparison, we finetune GSE with target speaker's utterance, which is referred as GSE-finetune. For ablation analysis, we implement four systems BL, P1, P2, and P3 (proposed). Among them, BL is composed of speaker and conversion module to convert BN to mel spectrum. P1 uses speaker normalization based on BL, and P2 adopts weight regularization based on P1. P3 is our final proposed system combining all the contributions.\n\n\n\nFor our proposed method, the content adopts the same model configuration as Tian \\textit{et al.}~\\cite{VCC2020_Tian2020}. The content module follows a typical encoder-decoder architecture using the CBHG as the encoder and an auto-regressive module consisting of prenet, decoder RNN and postnet as the decoder. For conversion module, the configuration of CBHG is the same as that of content module. Prenet consists of 2 fully connected layers with 80 and 256 hidden units respectively. Postnet contains 4 1D-convolution layers with 3*3 kernel size and 256 filters and a fully connected layers with 80 hidden units. The architecture and hyper-parameters of the reference encoder follow the origin configuration~\\cite{Rerferceencoder}. The speaker classifier consists of 3 fully connected layers. In training stage 2, the conversion model is trained for 120 epochs using batch size of 16. We use Adam optimizer to optimize our model with learning rate decay, which starts from 0.001 and decays every 20 epochs with decay rate 0.7. In training stage 3, we train the conversion model for 2000 steps using one utterance, and $\\gamma$ is set to 1. The learning rate starts from 0.001 and decays every 200 steps in decay rate 0.5.\n\n\n\\begin{figure*}[h]\n\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figure\/overfit_v3.pdf}\n\\caption{Spectrograms for a source audio utterance (left) and its corresponding converted utterances by different systems (right). The formants from the proposed system (esp. P3) are more similar to the source, indicating good style transfer. By contrast, the formants from BL are too flat.}\n\\label{fig:mel}\n\\vspace{-6pt}\n\\end{figure*}\n\n\n\n\\subsection{Subjective evaluation}\nWe conduct the following listening tests: three mean opinion score (MOS) tests to assess speech quality, style similarity, and speaker similarity respectively. Style similarity measures styles between converted audio and source audio. We select three sentences from rms, clb, and ESD respectively as source audio. Nine sentences are converted to four target speakers (p340, p363, slt, bdl), a total of 36 sentences are used for listening test. We highly recommend readers to listen to our samples\\footnote{Samples can be found in \\href{https:\/\/kerwinchao.github.io\/Oneshotvc.github.io\/}{\\url{https:\/\/kerwinchao.github.io\/Oneshotvc.github.io\/}}}.\n\n\n\\textbf{Comparison analysis.} We compare the proposed method with SOTA one-shot VC methods. The results of MOS tests are shown in Table~\\ref{tab:mos} comparison part. It is observed that GSE achieves the best result in terms of speech quality, and our proposed system P3 gets significantly higher MOS scores in terms of style and speaker similarity than the comparison systems. Previous one-shot methods lack prosody modeling ability, and target speaker's timber is unknown to model, which leads to low speaker similarity and unstable performance. Comparing GSE with GSE-finetune, we can see that making GSE learn from the target speaker utterance can significantly improve the speaker similarity.\n\n\\textbf{Ablation analysis.} As shown in Table~\\ref{tab:mos} ablation part, we evaluated the ablation systems. The BL system obtains poor results in three MOS tests, which indicates that BL system is easy to over-fit. By using our proposed method, the phenomenon of over-fitting is alleviated, and all MOS scores are improved. Adding the prosody module makes the speaker module focus on extracting timber from only one utterance, therefore that style and speaker similarity are able to be improved.\n\n\\textbf{Varying duration.} We further evaluate the performance of the proposed system under different duration of utterances (1, 3, 6, 9, 15 seconds) from CMU-ARCTIC speakers. Note that here only CMU-ARCTIC speakers are used for subjective test as listeners need to assess a large set of audio samples. As shown in Fig.~\\ref{fig:duration}, the result is affected by the duration of the target speech in general. For the extreme cases, e.g. 1-3s, the model still shows quality degradation caused by over-fitting even if our proposed method is used. But from 3 to 6s, benefiting from the proposed approach, the three MOS scores have clear increased while speech quality and style similarity undergo a quick boost. After 6 seconds, all three curves do not improve significantly and begin to stabilize from 9 seconds.\n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figure\/duration-v1.pdf}\n\t\\caption{The MOS performance of proposed system vs. training utterance with different duration}\n\t\\label{fig:duration}\n\\end{figure}\n\n\n\\subsection{Objective evaluation}\n\\textbf{Over-fitting on spectrograms}. We visualize the spectrogram of the converted speech for further investigation. The spectrogram of a testing sample is shown in Figure~\\ref{fig:mel}. We can see that BL has a very flat formant on the spectrogram, and noise appears in the silence part. Comparing these spectrograms, we observe that over-fitting phenomenon reduces the frequency of the formant and the fundamental frequency, which affects speech quality and speaker similarity. These figures suggest that the details of the frequency and the fluctuation of the formant become more abundant when the proposed method is used. This experiment shows that our method is effective in overcoming over-fitting.\n\n\n \n\n\n\n \n \n\n\n\\textbf{Prosody correlation}. To further verify the statistical significance of the expressiveness of each system, we extracted features related to the prosody: frame-level energy and lf0. We use 36 utterances to calculate the Pearson correlation coefficients between source audio and converted audio. The higher the Pearson correlation coefficient of the model, the higher the accuracy of the predicted prosodic attributes. As shown in Table~\\ref{tab:pearson}, P3 gets the highest scores from the perspective of energy and lf0. This indicates that the prosody module can improve the performance of style transfer. The conclusion from objective measurement is line with the that from the subjective listening.\n\n\n\\vspace{-15pt}\n\\begin{table}[h] \n\\centering\n\\caption{Pearson correlation in energy and lf0.}\n\\begin{tabular}{m{1cm}<{\\centering}|m{1cm}<{\\centering}m{1cm}<{\\centering}m{1cm}<{\\centering}m{1cm}}\n\\hline\n\\label{tab:pearson}\n\\vspace{5pt}\n\\textbf{} & BL & P1 & P2 & P3 \\\\ \\hline\nEnergy & 0.765 & 0.727 & 0.729 & \\textbf{0.755} \\\\ \\hline\nLf0 & 0.544 & 0.518 & 0.531 & \\textbf{0.733} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\vspace{-15pt}\n\n\n\\section{CONCLUSION}\n\n\nIn this study, we propose a novel one-shot voice conversion framework for style transfer. This task is challenging as training on one utterance is easy to over-fit, which results in serious degradation of speaker similarity and style. To mitigate this challenge, we build on the recognition-synthesis framework and introduce a disentangled structure to explicitly model content, speaker identity and prosody which are originally entangled in speech. Specifically, we first adopt speaker normalization in content module to normalize speaker-related information. Furthermore, we add weight regularization during adaptation process to prevent over-fitting. Finally, to improve the expressiveness of converted speech, prosody module is added to one-shot voice conversion framework, which can extract rich prosody representation from source audio. Experimental results show that our proposed system outperforms several SOTA one-shot systems in terms of speaker similarity and style.\n\n\n\n\n\n\n\\ninept\n\\bibliographystyle{IEEE}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\section{Acknowledgment}\n\\label{sec:acknowledgment}\n\nThe authors would like to acknowledge NVIDIA for providing GPU resources.\n\\section*{APPENDIX}\n\\label{sec:appendix}\n\n\\subsection{Memory Traffic Estimates Sensitivity to Convolution Configurations}\n\\label{subsec:evaluation_sensitivity}\n\nWe analyze the sensitivity of our memory traffic model to artificial \\abbrev{conv} layers. We use 256 input channels, a 13$\\times$13 IFmap, 128 output channels, a 3$\\times$3 filter, and stride 1, which is a commonly-used \\abbrev{conv} layer configuration~\\cite{he2016deep,szegedy2015going}. Then, we sweep each of these parameters to observe the model's sensitivity. \\fig{fig:sensitivity} shows the predicted data traffic at L1, L2, and DRAM channels, which are normalized to the measured data traffic on a TITAN Xp GPU.\n\n\\textit{\\textbf{Sensitivity to output channel count.}}\nFor different output channel counts, \\abbrev{DeLTA} shows geometric mean absolute error (GMAE) of 3.7\\%, 8.4\\%, and 3.7\\% for L1, L2, and DRAM respectively~(\\fig{fig:sensitivity_K}). The number of output channels affects the CTA tile width and the error is correlated with this CTA tile width (black line). Based on our profiled data, narrow CTA tiles (32 and 64) use \\(blk_K\\) size of 4 rather than 8 and this makes one warp access multiple distant filter data columns~(\\fig{fig:l1_traffic}), which increases L1 load inefficiency. In case of DRAM traffic, using a narrow CTA tile divides the GEMM matrix into more CTA tiles, increasing the number of CTA columns. As \\abbrev{DeLTA} calculates DRAM traffic by multiplying the IFmap matrix size and the number of columns of CTA tiles, data reuse between the columns can increase the error. The L2 estimates show the opposite error trend, which is because the ratio between \\(blk_K\\) and the filter size is used to calculate \\(DIST_H\\).\n\n\\begin{figure}[h]\n \\centering\n \\subfloat[Sensitivity to output channel count \\((=C_o)\\)]{\n \\includegraphics[width=0.47\\textwidth]{graph\/sweep_K.pdf}\n \\label{fig:sensitivity_K}\n }\\\\\n \\centering \n \\subfloat[Sensitivity to input channel count \\((=C_i)\\)]{\n \\includegraphics[width=0.47\\textwidth]{graph\/sweep_C.pdf} \n \\label{fig:sensitivity_C}\n }\\\\\n \\vspace*{-1mm}\n \\centering\n \\subfloat[Sensitivity to IFmap size \\((=H_i, W_i)\\)]{\n \\includegraphics[width=0.47\\textwidth]{graph\/sweep_HW.pdf}\n \\label{fig:sensitivity_HW}\n \n }\\\\\n \\centering \n \\subfloat[Sensitivity to mini-batch size \\((=B)\\)]{\n \\includegraphics[width=0.47\\textwidth]{graph\/sweep_B.pdf} \n \\label{fig:sensitivity_B}\n }\\\\\n \\caption{\\small{Predicted L1, L2, and DRAM traffic by \\abbrev{DeLTA} using artificial \\abbrev{conv} layer configurations (normalized to the measurement).}}\n \\label{fig:sensitivity}\n \\vspace*{-4mm}\n\\end{figure}\n\n\\textit{\\textbf{Sensitivity to input channel count.}}\nIn general, the estimated traffic is not sensitive to the number of input channels~(\\fig{fig:sensitivity_C}). Across different input channel counts ranging from 16 to 512, \\abbrev{DeLTA} shows only 4.1\\%, 5.7\\%, and 1.2\\% GMAE compared to measurements for L1, L2, and DRAM load traffic, respectively. DRAM traffic error has larger variation showing localized drops at some points, which are correlated with local peaks in the measured DRAM traffic (black line).\nThese are potentially caused by working set alignment to L2 cache associativity that may lead to cache thrashing, which is not considered in our model. Although \\abbrev{DeLTA} cannot predict such points, its estimated traffic follows the trend across different input channel counts with very small error.\n\n\\textit{\\textbf{Sensitivity to feature size.}}\n\\abbrev{DeLTA} over-predicts all data traffic of layers with small IFmap sizes (e.g., \\(H_i \\times W_i <20\\))~(\\fig{fig:sensitivity_HW}). For L1, decreasing IFmap size increases the range of data that a warp loads and therefore the load inefficiency. Our L1 model slightly overestimates the L1 traffic when the IFmap size is small. For large IFmaps, the predicted L1, L2, and DRAM traffic exhibit small error.\n\n\\textit{\\textbf{Sensitivity to mini-batch size.}}\nThe mini-batch size does not affect the im2col GEMM data layout. Therefore, both L1 and L2 traffic prediction results have very small variation for different mini-batch sizes ranging 16 to 512~\\fig{fig:sensitivity_B}. When the memory footprint size of the workload is close to the L2 cache capacity, DRAM traffic is slightly overestimated and deviates by up to 10\\% compared to measurements. We again observe localized drops in accuracy at some mini-batch sizes (maximum 20\\%). We believe this is caused by L2 cache thrashing, as explained earlier.\n\n\n\n\n\n\\renewcommand{\\thesubsection}{\\Alph{subsection}}\n\n\\begin{figure}[h]\n \\centering\n \\subfloat[Pascal TITAN Xp]{\n \\includegraphics[width=0.46\\textwidth]{graph\/appendix_MB_titanXp.pdf}\n }\\\\\n \\vspace*{-3mm}\n \\centering\n \\subfloat[Pascal TESLA P100]{\n \\includegraphics[width=0.46\\textwidth]{graph\/appendix_MB_gp100.pdf}\n }\\\\\n \\vspace*{-3mm}\n \\centering\n \\subfloat[Volta TESLA V100]{\n \\includegraphics[width=0.46\\textwidth]{graph\/appendix_MB_gv100.pdf}\n }\\\\\n \\caption{\\small{DRAM latency and effective DRAM bandwidth of different GPU devices measured using a micro benchmark.}}\n \\label{fig:micro_bench}\n \\vspace*{-3mm}\n\\end{figure}\n\n\\subsection{DRAM Latency and Effective Bandwidth Measurement}\n\\label{subsec:micro_bench}\n\\fig{fig:micro_bench} shows the turnaround latency to DRAM and the effective bandwidth at DRAM channels, which we measure on three GPU devices (TITAN Xp, P100, and V100) used for our experiment. We use a micro benchmark which generates a stream of DRAM traffic with increasing volume per time unit. When the DRAM traffic is small, the measured DRAM latency does not change with increasing memory access intensity. This is because the DRAM data channel is not busy and there is no or minor DRAM bank conflicts. The DRAM latency measured in this range is what we defined as physical latency in \\sect{sec:perf_model}. As the memory traffic intensity increases further, the measured DRAM turnaround latency exponentially increases. At this point, the large amount of traffic bottlenecks the DRAM data bus thus the transactions are stuck in the queue. We use the DRAM bandwidth measured for the peak DRAM channel bandwidth. Due to DRAM bank interleaving conflicts, the measured effective DRAM bandwidth is smaller than the theoretical maximum bandwidth of the GPU devices.\n\n\n\n\\subsection{Execution Cycle Comparison}\n\\label{subsec:execution_cycles}\n\\fig{fig:cycle_prediction} compares the estimated execution cycles of the conv layers used in four modern CNNs to the measured cycles on Pascal TITAN Xp. The execution cycles are different by an other of magnitude depending on the convolution configuration, and \\abbrev{DeLTA} shows high estimation accuracy regardless of the total execution cycles.\n\n\\begin{figure}[h]\n \\centering\n \\subfloat[AlexNet]{\n \\includegraphics[width=0.48\\textwidth]{graph\/appedix_perf_alexnet.pdf}\n }\\\\\n \\vspace*{-3mm}\n \\centering \n \\subfloat[VGG16]{\n \\includegraphics[width=0.48\\textwidth]{graph\/appedix_perf_vgg.pdf}\n }\\\\\n \\vspace*{-3mm}\n \\centering\n \\subfloat[GoogLeNet]{\n \\includegraphics[width=0.48\\textwidth]{graph\/appedix_perf_google.pdf}\n }\\\\\n \\vspace*{-3mm}\n \\centering \n \\subfloat[ResNet]{\n \\includegraphics[width=0.48\\textwidth]{graph\/appedix_perf_resnet.pdf}\n }\\\\\n \\vspace*{-2mm}\n \\caption{\\small{The estimated execution cycles by \\abbrev{DeLTA} and the measurement on TITAN Xp.}}\n \\label{fig:cycle_prediction}\n\\end{figure}\n\n\n\n\\subsection{Memory Traffic Comparison}\n\\label{subsec:memory_traffic}\n\\fig{fig:memory_traffic} compares the memory traffic at L1, L2, and DRAM channels that are estimated using our model and measured on NVIDIA TITAN Xp GPU. \\abbrev{DeLTA} shows highly accurate memory traffic prediction regardless of convolution kernel's total memory traffic volume.\n\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[width=1.\\textwidth]{graph\/appendix_TITANXp_Memory_Traffic.pdf}\n \\caption{\\small{Comparison of the model estimated L1, L2, and DRAM traffic to the measured data on NVIDIA TITAN Xp GPU.}}\n \\label{fig:memory_traffic}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\\section{Background}\n\\label{sec:background}\n\n\\subsection{GPU Architecture}\n\\label{sub_sec:gpu}\nGPUs are designed to accelerate compute-intensive highly-parallel workloads and therefore: (1) require applications to express parallelism with many threads, and (2) contain a large number of very wide SIMD cores, called streaming multiprocessors (SMs). Each SM processes a warp of 32 threads in lockstep, such that ideally all 32 threads execute the same instructions. In addition to the processing elements, each SM contains load store units (LSU), register files (RF), a shared memory RAM (SMEM), and an L1 cache. The SMs share access to the L2 cache and DRAM through a crossbar interconnection network. \n\nGPU workloads are tiled into thread groups called cooperative thread arrays (CTAs). The CTA scheduling mechanism is assumed to assign SMs to CTAs in a round-robin manner~\\cite{lee2014improving}. Each CTA typically consists of multiple thread warps that execute concurrently to hide memory access latencies. Given sufficient resources (RF and SMEM), multiple CTAs can be simultaneously executed within one SM (active CTAs). Interleaving multiple CTAs improves the ability of the SM to perform computation while some warps wait for memory.\n\n\\subsection{Convolution Layer Workload}\n\\label{sub_sec:conv_layer}\n\nCNNs consist of multiple layer types: convolution (\\abbrev{conv}) layers extract features from input images, pooling layers reduce feature dimensions, and fully-connected (FC) layers generate the score for each prediction class. Especially, \\abbrev{conv} layers have been the major focus of architecture research due to their high computation demands and data reuse~\\cite{chen2017eyeriss,kim2016neurocube}. \n\n\\fig{fig:conv_layer} shows the computation pattern within a \\abbrev{conv} layer. To compute a single data element of an output feature map (OFmap), a vector of input feature maps (IFmaps) are convolved by filters (matrices of weights) and accumulated then activated using a non-linear function such as ReLU~\\cite{lecun2015deep}. This \\abbrev{conv} layer computation is formulated as:\n\\begin{equation}\n\\footnotesize\n\\label{eq:conv}\nOFmap[k][x][y] \\!=\\!\\!\\! \\sum_{c=0}^{C_i-1} \\sum_{r=0}^{R-1} \\sum_{s=0}^{S-1}Fil[k][c][r][s]{\\times}IFmap[c][x+r][y+s]\n\\end{equation}\n\\(C,\\ H,\\ W\\) indicate the number of channels, the height and width of each feature and filter, and \\(i\\), \\(o\\), and \\(f\\) denotes input feature, output feature, and filter respectively. As each filter is mapped to a combination of IFmaps and OFmaps so there are \\(C_i \\times C_o\\) filters. \\((x,y)\\) indicates the position of filter in IFmap and it increments by the filter stride. Also, CNN model is generally trained using a \\emph{mini-batch} of samples (\\(B\\) in \\fig{fig:conv_layer}) in parallel which typically ranges 32--512~\\cite{keskar2016large}, and the samples in a mini-batch share filters. \\eq{eq:conv} also shows the high degree of data-reuse within a \\abbrev{conv} layer; all IFmaps, filters, and OFmaps are reused during convolution. Given the high math complexity and abundant data reuse, processing a \\abbrev{conv} layer is generally bounded by the compute throughput of a processor.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{fig\/conv_layer.pdf}\n \\vspace*{-2mm}\n \\caption{\\small{The dimensions and computation patterns of a \\abbrev{conv} layer.}}\n \\label{fig:conv_layer}\n \\vspace*{-2mm}\n\\end{figure}\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.38\\textwidth]{fig\/im2col.pdf}\n \\vspace*{-1mm}\n \\caption{\\small{Im2col GEMM converted from the convolution in \\fig{fig:conv_layer}. The red boxed data show duplicated accesses.}}\n \\label{fig:im2col}\n \\vspace*{-4mm}\n\\end{figure}\n\n\n\\subsection{Convolution Algorithm \\& GEMM Tiling}\n\\label{sub_sec:conv_lowering}\n\n\\textit{\\textbf{Parallel Convolution Algorithm.}}\nImage-to-column (\\emph{im2col}) is one of the most commonly used algorithms for GPU-accelerated convolution kernels~\\cite{chetlur2014cudnn,park2018deep} because it works well for a range of \\abbrev{conv} layers with different configurations (e.g., different \\(B, C, H, W\\))~\\cite{li2016performance}. To increase data parallelism and GPU resource utilization, im2col transforms the direct convolution described in \\fig{fig:conv_layer} into a single general matrix-matrix multiplication (GEMM) with three-dimensions by merging the IFmap and small filter matrices which is illustrated in \\fig{fig:im2col}. In this example, first, 2$\\times$2 filters for each OFmap channel are converted into columns and stacked \\ding{182}. Next, the data elements in the IFmaps are laid out in a way such that the elements to be multiplied by one filter (F0) are placed as a column \\ding{183}. The IFmap matrix data layout changes depending on the filter size and the convolution stride. The transformed im2col GEMM has three-dimensions (\\(M \\times N \\times K\\)), which are a function of the convolution configuration. Im2col duplicates many data elements which leads to IFmap data reuse in the \\abbrev{conv} layer (data in red dotted boxes). Therefore, compared to the typical GEMM, im2col GEMM has greater data reuse and benefits more from caches.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.48\\textwidth]{fig\/im2col_blk.pdf}\n \\vspace*{-1mm}\n \\caption{\\small{Im2col GEMM blocking and the data movement for processing a CTA per main loop.}}\n \\label{fig:im2col_blk}\n \\vspace*{-4mm}\n\\end{figure}\n\n\n\\textit{\\textbf{GEMM Blocking \\& Execution Flow.}}\nThe im2col GEMM is blocked for efficient execution on a GPU (\\fig{fig:im2col_blk}). The GEMM blocking divides OFmap matrix with \\(M \\times N\\) dimensions into \\(blk_M \\times blk_N\\) blocks of CTAs (\\(blk_M\\) and \\(blk_N\\) are the blocking factors for CTA height and width). Also, GEMM \\(K\\) dimension is also divided by a blocking factor of \\(blk_K\\).\n\nEach blocked GEMM execution flow (accumulation in K dimension) consists of three phases: \\emph{prologue}, \\emph{main loop}, and \\emph{epilogue}. \nDuring the prologue, each CTA loads blocked IFmap \\((blk_{M} \\!\\times\\! blk_{K})\\) and filter data \\((blk_{N} \\!\\times\\! blk_{K})\\) from the global memory (DRAM) to registers. GEMM kernels use input double buffering~\\cite{kurzak2012autotuning} to overlap these memory loads and the computation routine phases.\nThus, the data loaded in prologue are first fetched to registers and transferred to the shared memory (SMEM) to be used in the first main loop iteration.\n\nThe main loops account for the majority of GEMM execution time. The prefetched data in the SMEM in the prior main loop iteration (or prologue) is read and used as the input in the current loop. Specifically, each CTA tile is sub-blocked into warp tiles with a blocking factor of \\(blk_{WM} \\times blk_{WN}\\), and each warp reads both input data from the SMEM. Then, the computation pipeline multiplies the features and weights and accumulates their results in the registers for accumulation. The data loads, computation, and accumulation operations in the main loop routine are software pipelined for efficient resource utilization.\n\nAfter completing the main loops, at the epilogue stage, the accumulated results are written to the global memory. As all data are written to DRAM, the epilogue can significantly increase the GEMM execution time, particularly when main loops have a small number of iterations.\n\n\n\\section{Conclusion}\n\\label{sec:general}\n\nIn this paper, we introduce \\abbrev{DeLTA} which accurately models the memory traffic of a convolution layer at all memory hierarchy levels of a GPU. This is novel and unique in both accounting for the complex memory reuse and execution patterns of im2col which is the most-commonly used convolution algorithms for accelerating CNNs on GPUs. We use the traffic estimates to model the expected performance of a convolution layer on a GPU, where all important GPU characteristics are parameterized to enable the rapid identification of bottlenecks and the evaluation of design tradeoffs. We show how this can be used to explore the design space and better tune resource provisioning for CNNs when compared to equally scaling all resources or ignoring the need for higher memory bandwidth as arithmetic throughput increases.\n\n\n\\subsection{DRAM Traffic}\n\\label{sub_sec:dram_model}\n\n\\begin{figure}[b!]\n \\centering\n \\includegraphics[width=0.44\\textwidth]{fig\/cta_schedule.pdf}\n \\caption{\\small{IFmap and filter data reference at sequences of processing CTA batches (\\ding{182}, \\ding{183}, and \\ding{184})}}\n \\label{fig:cta_schedule}\n \\vspace*{-3mm}\n\\end{figure}\n\n\nCTAs in a GEMM kernel share the data in both input matrices. As L2 cache is shared by all SMs, the CTAs executed in parallel by all SMs (a CTA batch) can reuse such shared data. However, the specific inter-CTA data reuse depends on CTA scheduling mechanism. Our DRAM modeling assumes GPU uses column-wise CTA scheduling considering im2col GEMM's skinny shape. Then, we estimate the DRAM traffic by identifying the unique data elements within a CTA batch.\n\nCTA tile array of im2col GEMM is tall as its height is set by the product of the height and width of a feature, and the size of the mini-batch. In contrast, its width is relatively narrow as it equals to the number of output channels. Thus, im2col GEMM CTA tile array has a high aspect ratio making it has many orders of magnitude more CTAs in the column direction than in the row direction. This increases the chance of the CTAs in the same column to be executed in parallel. \\ding{182} to \\ding{184} in \\fig{fig:cta_schedule} illustrate the sequences of processing CTA batches. In this example, the GPU has 20 CTAs and there are 8 SMs so the CTA batch size is 8. At each step, the SMs fetch the red boxed IFmap and filter data.\n\nAt the sequences of \\ding{182}, \\ding{183}, and \\ding{184}, many CTAs refer to the same filter data, which makes the filter data have a short reuse distance thus increasing their chance to be resident in L2 cache. Also, each conv layer's total filter size is generally only a few mega-bytes maximum for recent CNNs~\\cite{szegedy2015going,he2016deep}, so we effectively consider filter data as loaded from DRAM just once. On the other hand, the IFmaps have long rereference distances between columns of CTA tiles in the CTA tile array. Therefore, the overlapping IFmap data across \\ding{182}, \\ding{183}, and \\ding{184} are fetched twice. This makes the effective IFmap data load count the same as the columns of CTA tiles in the GEMM.\nThus, we calculate the DRAM traffic of IFmap, Filter, and their sum as: \n\\begin{equation}\n\\scriptsize\n\\label{eq:dram_traffic_boundary_penalty}\n\\begin{split}\nT_{DRAM\\_IFmap} &= IFmap\\_size \\times \\frac{Columns\\_of\\_CTA\\_tiles}{CTA\\_tiles\\_array}\\\\\n \n \n &= (B \\!\\times\\! H_i \\!\\times\\! W_i \\!\\times\\! C_i) \\times \\frac{Columns\\_of\\_CTA\\_tiles}{CTA\\_tile\\_array}\\\\\nT_{DRAM\\_Filter} &= Filter\\_size \\\\\n \n &= (C_i \\!\\times\\! H_f \\!\\times\\! W_f) \\times C_o \\\\\nT_{DRAM\\_total} &= T_{DRAM\\_IFmap} + T_{DRAM\\_Filter}\n\\end{split}\n\\end{equation}\nBoth IFmap height (\\(H_i\\)) and width (\\(W_i\\)) are zero padded.\nSome IFMap data is no used by a \\(1 \\!\\times\\! 1\\) \\abbrev{conv} layer with a stride larger than 1 and is not loaded. These data elements are excluded from DRAM traffic.\n\n\\section{Experiment methodology}\n\\label{sub_sec:exp_env}\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=1\\textwidth]{graph\/cnn_traffic.pdf}\n \\vspace*{-6mm}\n \\caption{\\small{L1, L2, and DRAM traffic estimates for the \\abbrev{conv} layers by \\abbrev{DeLTA} normalized to the measured values on three different GPUs}}\n \\label{fig:cnn_traffic}\n \\vspace*{-3mm}\n\\end{figure*}\n\n\n\\noindent\\textit{\\textbf{Device Specification.}}\nWe compare the data traffic and performance estimates to the measured data on two Pascal GPUs (TITAN Xp and P100)~\\cite{pascal2016whitepaper} and one Volta GPU (V100)~\\cite{volta2017whitepaper} (\\tab{tab:device_spec}). The profiled memory traffic and execution cycles are the mean of 10 measurements. Also, since the memory access latencies and bandwidths are not specified by NVIDIA, we measure the access latency and bandwidth to L1, L2, and DRAM using microbenchmarks (both new and from prior work~\\cite{mei2017dissecting}).\n\n\\begin{table}[h] \n \\caption{GPU device specifications}\n \\vspace*{-1mm}\n \\centering\n \\noindent\\resizebox{0.97\\linewidth}{!}{\n \\tabulinesep=0.5mm\n \\renewcommand{\\arraystretch}{0.7}\n \\begin{tabu}{|[1.2pt]l|l|l|l|[1.2pt]}\n \\thickhline\n Specifications & Pascal TITAN Xp & Pascal P100 & Volta V100 \\tabularnewline\n \\midhline\n \\(Num_{SM}\\) & 30 & 56 & 84 \\tabularnewline\n \\hline\n \\(Core\\;clock\\) & 1.58 GHz & 1.2GHz & 1.38GHz\\tabularnewline\n \\hline\n \\(BW_{MAC}\\ (FP32)\\) & 12134 GFLOPS & 8602 GFLOPS & 14837 GFLOPS \\tabularnewline \n \\hline\n \\(Size_{REG}\\) & 256 KB\/SM & 256 KB\/SM & 256 KB\/SM \\tabularnewline\n \\hline\n \\(Size_{SMEM}\\) & 96 KB\/SM & 64 KB\/SM & $\\leq$94 KB\/SM \\tabularnewline\n \\hline\n \\(BW_{L1}\\) & 92 GB\/s\/SM & 38.1 GB\/s\/SM & 94.1 GB\/s\/SM \\tabularnewline\n \\hline\n \\(BW_{L2}\\) & 1051 GB\/s & 1382 GB\/s & 2167 GB\/s \\tabularnewline\n \\hline\n \\(BW_{DRAM}\\) & 450 GB\/s & 550 GB\/s & 850 GB\/s \\tabularnewline\n \\hline\n \\(Size_{L2}\\) & 3MB & 4MB & 6MB \\tabularnewline\n \\thickhline \n \\end{tabu}\n } \n \\label{tab:device_spec}\n \\vspace*{-1mm}\n\\end{table} \n\n\\noindent\\textit{\\textbf{Benchmarks.}}\nWe evaluate \\abbrev{DeLTA} on the \\abbrev{conv} layers of four popular CNNs (AlexNet~\\cite{krizhevsky2012imagenet}, VGGNet~\\cite{simonyan2014very}, GoogLeNet~\\cite{szegedy2015going}, and ResNet~\\cite{he2016deep}) used for ImageNet dataset training and prediction~\\cite{ILSVRC15}. Because many \\abbrev{conv} layers in these CNNs share configurations, we show the results on the unique subset. Unless specified, a mini-batch size of 256 is used for all evaluated layers. We use cuDNN ConvolutionForward API with {\\small IMPLICIT\\ PRECOMP\\ GEMM} algorithm to run \\abbrev{conv} layers compiled with CUDA v8.0 and cuDNN v7.0.\n\n\n\n\n\\section{Evaluation}\n\\label{sec:evaluation}\n\n\\subsection{Memory Traffic Model}\n\\label{sec:evaluation_cnn}\n\n\\fig{fig:cnn_traffic} shows \\abbrev{DeLTA} estimates of data traffic for L1, L2, and DRAM normalized to the measurement of three different GPUs for all unique layer configurations of the 4 evaluated CNNs. Both TITAN Xp and P100 use the same kernels and have an L1 request size of 128B so the measured and predicted L1 traffic is the same for both. \\abbrev{DeLTA} shows high accuracy with a (7.9\\% standard deviation). We are unsure of the L1 request size for Volta and experimented with 32B, 64B, and 128B granularity settings for \\abbrev{DeLTA}. We observed the best match to measurements with 32B L1 requests and \\abbrev{DeLTA} matches measured GV100 results with a shows geometric mean absolute error (GMAE) of 6.9\\% (13.3\\% stdev). \n\n\\textbf{The modeled L2 traffic} has a larger variation in error than L1 traffic. L2 traffic is the result of misses in L1 and our model makes the simplifying assumption that there is no overlap in data access to L1 between different CTAs. We hypothesize that the greater error is indeed the result of multiple concurrent CTAs reusing some data in L1. Two observations from the result strongly support our hypothesis. First, \\abbrev{DeLTA} over-estimates traffic primarily for layers that have larger features, which lead to greater opportunity for reuse between CTAs. Second, we see larger modeling errors for V100 than P100 and Titan XP. The V100 architecture has larger L1 caches (unified with SMEM)~\\cite{volta2017whitepaper} that can also contribute to more reuse across CTAs. Overall, \\abbrev{DeLTA} is still quite accurate with the GMAE and standard deviation (in parentheses) being 4.2\\% (7.2\\%) for Titan XP, 6.2\\% (14.4\\%) for P100, and 12.4\\% (25.0\\%) for V100.\n\n\n\\begin{figure}[b!]\n \\centering\n \\includegraphics[width=0.47\\textwidth]{graph\/cnn_traffic_all.pdf}\n \\caption{\\small{L2 and DRAM traffic estimates by \\abbrev{DeLTA} and prior methodology normalized to TITAN Xp measurement}}\n \\label{fig:traffic_comparison}\n \\vspace*{-3mm}\n\\end{figure}\n\n\\begin{figure*}[b!]\n \\centering\n \\includegraphics[width=0.99\\textwidth]{graph\/cnn_perf_titanxp.pdf}\n \\vspace*{-1mm}\n \\caption{\\small{Conv layer execution time estimates by \\abbrev{DeLTA} normalized to TITAN Xp's and their performance bottlenecks}}\n \\label{fig:performance_titan_xp}\n \\vspace*{-3mm}\n\\end{figure*}\n\n\\begin{figure*}[b!]\n \\centering\n \\includegraphics[width=0.99\\textwidth]{graph\/cnn_perf_v100.pdf}\n \\vspace*{-1mm}\n \\caption{\\small{Conv layer execution time estimates by \\abbrev{DeLTA} normalized to TESLA V100's and their performance bottlenecks}}\n \\label{fig:performance_titan_v100}\n \\vspace*{-3mm}\n\\end{figure*}\n\n\n\\textit{\\textbf{The modeled DRAM}} traffic is very accurate overall with a few notable outliers. Some of the layers in GoogLeNet and and ResNet have very small memory footprints that can completely fit within the L2 cache. The profiler we use to measure results reports anomalous numbers for these layers that suggest the impossible scenario where less data is read from DRAM than the actual footprint, leaving \\abbrev{DeLTA} with a large over-estimation. We attribute this to measurement errors and ignore these $>2\\times$ errors from the average numbers reported below. The other source of large modeling error relates to L2 cache behavior and CTA scheduling. \\abbrev{DeLTA} underestimates DRAM traffic for VGG16-conv1, GoogLeNet-4e\\_5{$\\times$}5, and a few ResNet152 layers. Our analysis indicates that these errors result from \\abbrev{DeLTA} identifying potential data reuse in L2 between CTAs that is not exploited by the hardware. For example, VGG16-conv1 uses large IFmaps \\(224 \\!\\times\\! 224\\) with a small filter stride of 1. This leads to a large overlap between layers given \\abbrev{DeLTA}'s assumptions on scheduling, which do not manifest in practice. Overall, the DRAM traffic model shows small GMAE (with standard deviation) of 2.8\\% (10.3\\%) for Titan Xp, 6.2\\% (14.4\\%) for P100, and 10.2\\% (9.2\\%) for V100 (without the anomalous measurements).\n\n\n\n\\textit{\\textbf{Memory Traffic Comparison with Prior Models.}}\n\\fig{fig:traffic_comparison} compares the normalized data traffic for all unique \\abbrev{conv} layers under evaluation for \\abbrev{DeLTA} and the prior models~\\cite{zhou2017performance,hong2009analytical}. As the prior models assume 100\\% cache miss rates for both levels of caches, we apply the L1 load traffic to both L2 and DRAM. Both L2 and DRAM traffic assumed by the prior models are far from the measurements because these prior models ignore the high data reuse in the \\abbrev{conv} layers. The deviation is relatively small for layers with \\(1 \\!\\times\\! 1\\) filters due to the low data reuse but the large filters have very large errors. These errors are multiple factors (up to nearly 100$\\times$) larger than those of \\abbrev{DeLTA}, and lead to wrong conclusions about performance bottlenecks of modern GPUs with their large arithmetic to memory throughput ratios.\n\n\n\\subsection{Performance Model}\n\\fig{fig:performance_titan_xp} and \\fig{fig:performance_titan_v100} show the performance estimations vs.~ the measured data on TITAN Xp and TESLA V100 respectively, along with their bottlenecks. The GMAE is 6.0\\% and 6.5\\% for TITAN Xp and V100 respectively. Although highly accurate, \\abbrev{DeLTA} underestimates the execution time for some layers regardless of their bottlenecks. One major reason is that the estimated memory traffic is uniform across each CTA's GEMM main loop but the actual data traffic is not. For example, if the data fetched by one iteration is reused in the next, the first iteration execution time is bound by the data loads but the second iteration is constrained by arithmetic throughput. Such non-uniform execution time bottlenecks make \\abbrev{DeLTA} provide conservative estimates that represent worst-case performance.\n\nOur evaluation shows that arithmetic throughput is the major performance bottleneck (90\\% of evaluated layers), which is expected due to the high data reuse of im2col GEMM. We also observe that some layers are bottlenecked by other resources. L1 BW restricts the first \\abbrev{conv} layer of AlexNet on TITAN Xp due to its poor L1 transaction efficiency. Many layers in GoogLeNet are bottlenecked by DRAM BW or latency. The layers bottlenecked by DRAM latency do not have enough CTAs to hide the load latency. The layers restricted by DRAM BW have more CTAs to interleave thus saturate the DRAM channels.\n\n\\textit{\\textbf{Performance Estimation with Different GPUs.}}\n\\fig{fig:performance_comp_all}{a} compares the performance estimation distribution for the three GPUs. \\abbrev{DeLTA} estimates performance best for Titan XP, but the overall accuracy is quite good For P100 and V100 as well with a robust low-variance estimation (10\\% standard deviation). The outliers correspond to those layers for which data traffic is poorly estimated because of dynamic behavior, as explained above. \n\n\\textit{\\textbf{Performance Model Comparison.}}\n\\fig{fig:performance_comp_all}{b} compares the normalized performance estimation results of \\abbrev{DeLTA} and the models; while prior work advocated using a 1.0 miss rate, we sweep a range of miss rates in the figure. Compared to \\abbrev{DeLTA}, the other models show wider estimation errors and a larger number of outliers. With the 1.0 miss rate advocated by the prior models, the layer execution time is over-predicted by 1.8$\\times$ on average and up to $7\\times$. The prediction error for the models using fixed miss rates becomes significantly larger when compute throughput scales as many layers become memory system resource bottleneck.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.48\\textwidth]{graph\/cnn_perf_unit.pdf}\n \\caption{\\small{Execution time estimates normalized to TITAN Xp's}}\n \\label{fig:performance_comp_all}\n \\vspace*{-3mm}\n\\end{figure}\n\n\\subsection{GPU Scaling Study}\nWe use \\abbrev{DeLTA} to explore GPU designs for efficient CNN performance scaling with less HW resources. We use the entire 152 \\abbrev{conv} layers in ResNet152 to evaluate the potential speedup over the Titan Xp baseline. As almost all compute kernels of ResNet are \\abbrev{conv} layers, its performance is dominated by their execution time. \\tab{tab:design options} shows the design options used in our experiment. Option 1 and 2 represent the conventional way to improve GPU performance, which keeps SM resources constant and scales the number of SMs and L2 and DRAM BW. These options are expensive as each extra SM involves the entire SM resources such as registers, SMEM capacity and BW, and L1. For the other design options, we use the resource bottleneck information from \\abbrev{DeLTA} to minimally scale independent resources for efficient performance gain.\n\n\\begin{figure}[t!]\n \\centering\n \\setlength\\tabcolsep{3pt}\n \\subfloat[GPU design options. Each column indicates a GPU design choice and xX indicates the magnitude of increase.]{\n \\label{tab:design options}\n \\noindent\\resizebox{0.95\\linewidth}{!}{\n \\tabulinesep=0.5mm\n \\centering\n \\renewcommand{\\arraystretch}{0.7}\n \\begin{tabu}{|[1.2pt]c|[1.2pt]c|c|c|c|c|c|c|c|c|c|[1.2pt]}\n \\thickhline \n Resources & TitanXp & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\tabularnewline\n \\thickhline \n \\(N_{SM}\\) & 1X & 2X & 4X & 1X & 1X & 1X & 1X & 1X & 2X & 1X \\tabularnewline\n \\hline\n \\(MAC_{BW}\\) \/ \\(SM\\) & 1X & 1X & 1X & 2X & 4X & 4X & 6X & 8X & 4X & 8X \\tabularnewline\n \\hline\n \\(REGS_{SIZE}\\) \/ \\(SM\\) & 1X & 1X & 1X & 1X & 1X & 2X & 2X & 3X & 2X & 3X \\tabularnewline\n \\hline\n \\(SMEM_{SIZE}\\) \/ \\(SM\\) & 1X & 1X & 1X & 1X & 1X & 2X & 2X & 3X & 2X & 3X \\tabularnewline\n \\hline\n \\(SMEM_{BW}\\) \/ \\(SM\\) & 1X & 1X & 1X & 1X & 1X & 2X & 2X & 3X & 2X & 3X \\tabularnewline\n \\hline\n \\(L1_{BW}\\) \/ \\(SM\\) & 1X & 1X & 1X & 1X & 1X & 1.5X & 2X & 2X & 2X & 2X \\tabularnewline\n \\hline\n \\(L2_{BW}\\) & 1X & 1.5X & 2X & 1X & 1X & 1.5X & 1.5X & 2X & 2X & 2X \\tabularnewline\n \\hline\n \\(DRAM_{BW}\\) & 1X & 1.5X & 2X & 1X & 1X & 1.5X & 2X & 2X & 2X & 3X \\tabularnewline\n \\hline\n CTA tile H,W & 128 & 128 & 128 & 128 & 128 & 128 & 128 & 256 & 256 & 256 \\tabularnewline\n \\thickhline \n \\end{tabu}\n } \n }\\\\\n \\vspace*{2mm}\n {\n \\includegraphics[width=0.48\\textwidth]{graph\/resnet.pdf}\n }\\\\\n \\caption{\\small{GPU resource scaling and speedup of \\abbrev{conv} layers in ResNet}}\n \\label{fig:resnet}\n \\vspace*{-3mm}\n\\end{figure}\n\n\n\n\\fig{fig:resnet}{b} shows the normalized performance gain for the different design options with their performance bottlenecks shown in \\fig{fig:resnet}{c}. \\abbrev{DeLTA} predicts that increasing the number of SMs by 2$\\times$ and 4$\\times$ along with L2 and DRAM BW improves ResNet forward propagation performance by 1.9$\\times$ and 3.4$\\times$, respectively. The limiting factor is memory BW increase, which restricts pipelined GEMM epilogue and some layers with larger memory BW requirements. Given \\abbrev{conv} layers are compute throughput hungry, options 3 and 4 increase only the per core arithmetic throughput by adding more MAC units. However, their performance headroom is only 2$\\times$ with most layers bottlenecked by DRAM BW and SM resources.\n\nBased on these observations, option 5 minimally increases resources to avoid bottlenecks, showing similar performance gains to option 2 with far lower hardware resource increases. Option 6 further increases compute throughput but \\abbrev{DeLTA} shows that now L2 BW becomes the limiter. From option 7 to 9, we increase the size of the GEMM tiles given the high compute throughput to memory bandwidth ratio. Such GEMM parameter choices are only beneficial for GPU designs with high arithmetic throughput, otherwise the performance is restricted by the memory system BW. Design option 8 increases the number of SMs by 2$\\times$, but \\abbrev{DeLTA} shows that increasing DRAM BW is more beneficial than doubling the SM resources (option 9). \n\nUsing \\abbrev{DeLTA} and a model of hardware resource costs, optimizing a future GPU for CNNs becomes a convex optimization problem. However, modeling precise hardware resource costs is outside the scope of the paper.\n\n\n\n\\section{Conclusion}\n\\label{sec:general}\n\n[Maybe this should be part of conclusion?]\nAlthough we model the memory traffic with im2col algorithm based matrix convolution workloads on Pascal GPU architecture, the proposed mechanism can be generalized to other GPU designs and convolution algorithms. We can predict the data load traffic at L1 channel by calculating the effective range of data elements touched by each warp which we defined as EEPW. The size of EEPW can vary depending on the kernel design but the mechanism remains the same. We estimated L2 traffic using intra-CTA data reuse and this mechanism can be applied to other algorithms such as FFT and Winograd. These algorithms have less data reuse within each matrix multiplication block can reuse the data reduce the reuse thus increase the data load traffic per request. On the other hand, bigger convolution GEMM blocking factor increase reuse thus reduce L2 traffic. Finally, understanding how CTAs are scheduled and how data elements are reused among CTAs can be used to predict the data traffic at DRAM channels. The size of L2 cache compared to the footprint of the entire workload or blocked working set impacts the prediction accuracy.\n\\section{introduction}\n\\label{sec:intro}\n\nConvolutional neural networks (CNNs) are the state of the art for various vision applications~\\cite{krizhevsky2012imagenet,ren2015faster,redmon2016you,wang2017fast}. The computation required for CNNs is a good match for GPUs~\\cite{krizhevsky2012imagenet,simonyan2014very}. The increasing demand for CNN computation is driving GPU arithmetic performance, which has been increasing at higher than its historical rate~\\cite{volta2017whitepaper}. However, based on our analysis, compared to the rapid GPU compute throughput increase of 32X for the past 9 years, its memory system bandwidth has improved by only 13X. This memory wall problem can bottleneck the performance of even the arithmetically-intensive CNNs, making performance scaling difficult.\n\nIt is therefore imperative to balance both arithmetic and memory performance in architecting a future GPU for efficient CNN performance scaling. This optimization benefits from analytical modeling, which can quickly provide insight and narrow the design space before slower and more resource-consuming modeling is used (e.g., simulators~\\cite{aamodt2012gpgpu}). Analytical models also aide in the optimization of software for efficient HW resource utilization~\\cite{zhou2017performance,lai2013performance}. Prior work on modeling CNN performance on GPUs focuses on modeling arithmetic performance with only simplistic and naive modeling of memory~\\cite{hong2009analytical,lai2013throughput,lai2013performance,zhou2017performance}. While these models are accurate when performance is bound by arithmetic throughput, the growing memory wall has shifted performance bottleneck to the memory system in many cases. \n\nWe present \\abbrev{DeLTA}---the first analytical GPU model for CNNs that accurately models both arithmetic performance and traffic in the memory hierarchy, and that can therefore be used to identify performance bottlenecks and explore the optimization space for future designs.\n\nOur GPU CNN performance model is both the first to model cache traffic in the GPU and the first to handle the most-commonly used algorithm for convolution (\\abbrev{conv}) and fully-connected (FC) layers in GPUs---\\emph{im2col} (image-to-column)~\\cite{chetlur2014cudnn}. While a \\abbrev{conv} layer can be computed by directly implementing the convolution operator, casting the convolution problem as a general matrix-matrix multiplication (GEMM) is both simpler and more efficient on a GPU. Our model is unique in that it accounts for the complex access locality that exists in im2col as the algorithm replicates and reorders input matrices to expose the parallelism needed for effective GPU acceleration. We demonstrate that only \\abbrev{DeLTA} is accurate enough to be use for architectural exploration, while prior models suggest orders of magnitude more traffic~\\cite{hong2009analytical,lai2013throughput,lai2013performance,zhou2017performance}. \n\nOur novel approach separately models the traffic at each level of the memory hierarchy using the access patterns exhibited by the cuDNN library~\\cite{cudnn2016userguide}.\nOur model accounts for how the computation is blocked for locality and parallelism and how the hardware handles memory accesses in the caches and the software-managed \\emph{shared memories}.\n\nWe then use the modeled traffic at each memory level and the bandwidths of memories and arithmetic to model overall performance and identify which resource bounds performance under different system configurations. We validate \\abbrev{DeLTA} with four popular CNNs (AlexNet~\\cite{krizhevsky2012imagenet}, VGG~\\cite{simonyan2014very}, GoogLeNet~\\cite{szegedy2015going}, and ResNet~\\cite{he2016deep}) executed on two NVIDIA Pascal GPUs (TITAN Xp and P100) and a Volta GPU (V100), and show that \\abbrev{DeLTA} is both accurate and robust across the diverse layers and architectures. We also demonstrate how \\abbrev{DeLTA} can be used for the design-space exploration of future GPUs and identify interesting tradeoffs for efficient CNN execution by independently scaling different GPU resources.\n\nTo summarize, our main contributions are:\n\\begin{itemize}[noitemsep,topsep=5pt,leftmargin=0.15in]\n \\item \n We introduce \\abbrev{DeLTA}, a GPU performance model for CNNs. Unlike prior work, \\abbrev{DeLTA} accurately models traffic across all memory hierarchy levels, capturing the data reuse at the different levels; accurately modeling memory traffic is critical for future GPU designs where compute throughput and memory bandwidth must be balanced.\n \\item \n We are first to analyze and model the memory access pattern of the im2col convolution algorithm, which is the most-commonly used algorithm for GPU-accelerated CNNs.\n \\item\n We validate the high accuracy and robustness of \\abbrev{DeLTA} across four popular CNNs and three different GPUs.\n \\item \n We demonstrate how \\abbrev{DeLTA} can be used to efficiently explore future GPU designs and pinpoint specific resource-scaling opportunities for future GPUs that effectively remove bottlenecks that \\abbrev{DeLTA} identifies.\n\\end{itemize}\n\n\n\\subsection{L1 Cache Traffic}\n\\label{sub_sec:l1_model}\n\nAs im2col rearranges the memory access layout, the memory addresses of adjacent IFmap data elements are not continuous. This lowers the spatial locality within each L1 request coalescence granularity (a warp of 32 threads) causing more memory requests than needed. Also, if the memory references are not aligned with the address of the L1 transactions, extra L1 transactions are made. We estimate traffic by calculating this L1 request inefficiency affected by each warp's non-continuous memory references and its address alignment.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.46\\textwidth]{fig\/l1.pdf}\n \\vspace*{-1mm}\n \n \\caption{\\small{IFmap and filter data requested by a single warp during a main loop (elements in the blue box). (a) The fraction of IFmap in the red box are the data visited by a single filter data element and it is stacked as a column in the IFmap matrix for GEMM. (b,c) The data layout of the filter matrix for GEMM.}}\n \\label{fig:l1_traffic}\n \\vspace*{-4mm}\n\\end{figure}\n\n\\fig{fig:l1_traffic}{a} shows the data layout of a single IFmap traversed by one element of a \\(3\\!\\times\\!3\\) filter with stride 1 and its im2col converted form. Convolution typically adds zero padding (\\emph{Pad}) around an IFmap boundary for better feature extraction. In our example, the original IFmap size is \\(4 \\!\\times\\! 4\\) and it includes pad of 1 which eventually increases the size of IFmap to \\(6 \\!\\times\\! 6\\). The number on each data element represents the relative location in the physical memory. As described in \\sect{sub_sec:conv_lowering}, IFmap data elements visited by a filter data are arranged as a column (\\textcolor{red}{\\ding{182}}). Then, one warp requests L1 loads for 32 threads which are a fraction of an IFmap column (blue boxed elements in \\fig{fig:l1_traffic}{a}). To avoid unnecessary data fetch and save L1 request bandwidth, the L1 loads are coalesced within a warp and this coalescing granularity becomes the fundamental access granularity to L1 cache. This corresponds to the L1 cache line size of 128B (4B $\\times$ 32) for Pascal GPU which we experimentally validated with assumption. However, even a warp reads 128B, it cannot be coalesced to a single L1 transaction as data are not continuous: {\\small\\(W_f\\!-\\!1\\)} elements are skipped every {\\small\\(W_i\\!+2Pad-W_f+1\\)} elements (\\fig{fig:l1_traffic}{a}). Also, if the stride is larger than 1, data between every two elements are skipped. Reflecting such non-continuous memory access pattern of each IFmap matrix column, its general memory access inefficiency can be calculated as:\n\\begin{equation}\n\\scriptsize\n\\label{eq:access_sparsity}\n\\begin{split}\n\\frac{Elements\\ requested}{Elements\\ used} \\!= \\! \\frac{(W_i+2Pad) \\times Strd}{W_i+ 2Pad- W_f+ 1}\n\\end{split}\n\\vspace*{-2mm}\n\\end{equation}\nAlso, if the coalesced L1 requests made by a warp are not aligned with the L1 transaction addresses, extra transactions are requested. Thus, the overall L1 load inefficiency per warp {\\small\\((MLI_{IFmap})\\)} is calculated by dividing the average L1 requests made per warp by the number of L1 requests per warp with perfect address alignment (\\eq{eq:mli_ifmap}).\n\\begin{equation}\n\\scriptsize\n\\label{eq:mli_ifmap}\n\\begin{split}\nMLI_{IFmap} \\!&= \\!\\frac{L1\\ requests\\ made}{Warp} \\times \\frac{Warp}{L1\\ requests}\\\\\n \\!&= \\!\\ceil{ \\frac{Elmts\\ requested}{Elmts\\ used} \\!\\times\\! \\frac{Bytes}{Warp} \\!\\!\\times\\!\\! \\frac{L1\\ requests}{Bytes} } \\!\\times\\! \\frac{Warp}{L1\\ requests}\n\\end{split}\n\\end{equation}\n\nUnlike IFmap, the data elements in the filter matrix are fully continuous in each column. However, based on our profiling result, Pascal GPU uses \\(blk_K\\) size of 4 or 8 depending on the CTA tile (\\fig{fig:l1_traffic}{b}, \\fig{fig:l1_traffic}{c}) which results in accessing multiple filter matrix columns by a warp of 32 threads. The data elements in different columns have distant memory addresses as the address is continuous in the column direction. As each 128B L1 request spans beyond the range of filter data at each column, loading an IFmap matrix tile has higher inefficiency \\(MLI_{Filter}\\) than IFmap's. For Pascal GPUs, considering all possible cache line address alignments, \\(MLI_{Filter}\\) is calculated as 2.0 and 2.75 when \\(blk_K\\) is 8 and 4 respectively. Eventually, we calculate the total L1 traffic by multiplying the size of both GEMM inputs (IFmaps and filters) and their MLIs as:\n\\begin{equation}\n\\footnotesize\n\\label{eq:l1_traffic}\n\\begin{split}\nT_{L1\\_total} &= GEMM\\_inputs \\times MLI\\\\\n &= (M \\! \\times \\! K) \\! \\times \\! MLI_{IFmap} + (N \\! \\times \\! K) \\! \\times \\! MLI_{Filter}\n\\end{split}\n\\end{equation}\n\n\\subsection{L2 Cache Traffic}\n\\label{sub_sec:l2_model}\n\nIn im2col GEMM, the IFmap matrix with many duplicated data accesses involves high spatial and temporal locality. As L1 cache is dedicated to a single SM, L1 can capture the reuse within one CTA's IFmap tile but not across active CTAs per SM given its small size ($\\sim$32KB). With this assumption, our L2 model estimates the traffic by identifying the unique data accesses made in a CTA input tile.\n\n\\textit{\\textbf{CTA Tile Selection.}}\nCTA tile size affects the spatial locality of the data elements accessed by a CTA; a bigger CTA tile has more data reuse. It also affects the number of CTAs per workload and the ratio between math operations and memory loads per main loop iteration. As it is a critical factor in modeling memory traffic, we analyze the CTA tile size \\(((blk_M \\times blk_N) \\times blk_K)\\) of the potential convolution GEMM kernels. Based on our analysis, cuDNN uses three types of CTA tilings: \\((128 \\times 128)\\times 8\\), \\((128 \\times 64)\\times 4\\), and \\((128 \\times 32)\\times 4\\). \\(Blk_M\\) is fixed to 128 for all tiles and \\(blk_N\\) and \\(blk_K\\) are chosen based on the width of the GEMM which equals to \\(C_o\\). We measure the CTA tile size by different \\(C_o\\) (\\fig{fig:cta_tile_sizes}), and use this as a look-up table to our L2 traffic model.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.48\\textwidth]{fig\/cta_tile_size.pdf}\n \\caption{\\small{Profiled CTA tile width by different output channel counts.}}\n \\label{fig:cta_tile_sizes}\n\\end{figure}\n\n\\textit{\\textbf{Intra-CTA Spatial Locality.}}\nWe extend the example used in \\sect{sub_sec:l1_model} to explain our L2 traffic model (\\fig{fig:l2_dist}). Again, IFmap elements visited by one filter data are remapped as a column in the IFmap matrix (\\textcolor{red}{\\ding{182}}) and the next column contains the IFmap data traversed by another filter data (\\textcolor{blue}{\\ding{183}}). Therefore, a \\(blk_M \\!\\times\\! blk_K\\) IFmap tile for one main loop iteration consists of multiple copies of data that are in continuous address range. This access pattern entails large data locality shown by the duplicated data elements in the white dotted boxes. As L1 cache captures the locality within each tile, only the unique elements are requested to L2. Based on our observation, if we know the address range of data within one IFmap tile (difference between the smallest and the largest address), we can estimate the size of memory requests to L2 in the tile. To be specific, within each IFmap tile, the accessed address increases from the top to the bottom and left to the right. Therefore, using the sum of the address distances between A and B (both edges of \\(blk_M\\)), and B and C (both edges of \\(blk_K\\)), we can estimate the total accessed data within an IFmap tile.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.48\\textwidth]{fig\/l2.pdf}\n \\caption{\\small{IFmap matrix layout of im2col GEMM. The elements in \\(blk_M \\!\\times\\! blk_K \\) tile are the input for one main loop. Other details on the figure are described in \\sect{sub_sec:l2_model}.}}\n \\label{fig:l2_dist}\n \\vspace*{-4mm}\n\\end{figure}\n\nWe first define the address distance between A and B as {\\small\\(DIST_V\\)} (vertical distance). As the data elements in a column are the IFmap data traversed by a filter data, we formulate {\\small\\((DIST_V)\\)} as \\eq{eq:dist_v} using the method used in L1 traffic modeling. Also, when IFmap tile width (\\(blk_K\\)) spans over more than one channel, multiple unique {\\small\\(DIST_V\\)} can be located within an IFmap tile. In \\fig{fig:l2_dist}, \\(blk_K\\) spans over red and green colored elements each belong to different channel thus an IFmap tile entails two unique {\\small\\(DIST_V\\)} (addition of the blue vertical dotted box). Therefore, using the ratio between \\(blk_K\\) and a channel width (\\(H_f \\!\\times\\! W_f\\)), the average vertical distance {\\small\\((A\\_DIST_V)\\)} per IFmap can be calculated using \\eq{eq:eff_dist_v}.\n\\begin{equation}\n\\scriptsize\n\\label{eq:dist_v}\nDIST_{V} = blk_{M} \\times \\frac{(W_i+2Pad) \\times Strd}{W_i+2Pad-Wf+1}\n\\end{equation}\n\\begin{equation}\n\\scriptsize\n\\label{eq:eff_dist_v}\nA\\_DIST_{V} = DIST_V \\times \\frac{blk_K}{H_f \\times W_f}\n\\end{equation}\n\nNext, we define the address distance between B and C of an IFmap tile as {\\small\\((DIST_H)\\)} (horizontal distance). Based on our observation, the distance between two adjacent columns within the same \\(W_f\\) range is 1 (\\ding{184}) as they are the traversals of the two adjacent elements of a filter. However, the distance between the two columns in different \\(W_f\\) ranges is {\\small\\(W_i\\!+2Pad-W_f+1\\)} (\\ding{185}) which is the width of the filter element traversal on a IFmap. Also, the alignment of \\(blk_K\\) to the filter width (\\(W_f\\)) affects the number of \\(W_f\\) edges within an IFmap tile; more \\(W_f\\) edges increases \\(DIST_H\\). Therefore, considering both the inter-column distances and the alignment of \\(blk_K\\) to the IFmap layout, we calculate {\\small\\(DIST_H\\)} using \\eq{eq:dist_h}. Also, if the elements of more than one data sample are located within a IFmap tile, multiple unique {\\small\\({DIST_H}\\)} can be found within an IFmap tile. In \\fig{fig:l2_dist}, the light colored elements belong to a different data sample which adds another unique {\\small\\(DIST_H\\)} (addition of the blue horizontal dotted box). The number of samples per IFmap tile is affected by the ratio between \\(blk_M\\) and the size of one IFmap which is the product of the height and width of an IFmap or \\((\\frac{W_i+2Pad-W_f+1}{Strd})^2\\) (assuming height and width are the same). Reflecting this, we calculate the average horizontal distance of an IFmap tile {\\small\\((A\\_DIST_H)\\)} (\\eq{eq:eff_dist_h}).\n\\begin{equation}\n\\scriptsize\n\\label{eq:dist_h}\n\\begin{split}\n DIST_H &= \\Big( \\frac{blk_{K} \\!-1}{W_f} \\Big) \\!\\times\\! \\big( (W_i \\!-\\! W_f+1) + Strd \\!\\times\\! (W_f \\!-\\! blk_{K} \\!+\\! 1) \\big)\\\\\n &+ \\Big( \\frac{W_f-blk_K+1}{W_f} \\Big) \\times \\big( Strd \\!\\times\\! (blk_K-1) \\big)\n\\end{split}\n\\vspace*{-3mm}\n\\end{equation}\n\\begin{equation}\n\\scriptsize\n\\label{eq:eff_dist_h}\nA\\_DIST_{H} = DIST_H \\!\\times\\! \\Bigg( 1+ \\frac{blk_M}{\\big(\\frac{H_i+2Pad-H_f+1}{Strd}\\big)^2} \\Bigg)\n\\end{equation}\n\nIn case of \\(1 \\!\\times\\! 1\\) convolution and FC layers, all IFmap data elements are unique and there is no data reuse within an IFmap tile. Therefore, we can simply calculate {\\small\\(DIST_V\\)} as the height of an IFmap tile and {\\small\\(DIST_H\\)} as its width. The same method is applied to filter data loads because they are all unique as well. Finally, using the average distance of IFmap and filter tiles, the total L2 load traffic within a main loop is calculated by adding {\\small\\(A\\_DIST_V\\)} and {\\small\\(A\\_DIST_H\\)}. Then, the total L2 traffic for a \\abbrev{conv} layer is calculated by multiplying the number of CTAs (\\(Num_{CTA}\\)) and the number of main loop iterations (\\eq{eq:l2_traffic}).\n\\begin{equation}\n\\scriptsize\n\\label{eq:l2_traffic}\n\\begin{split}\nT_{L2\\_total} &= A\\_DIST_{GEMM\\_tile} \\times \\frac{GEMM\\ tiles}{conv\\ layer}\\\\\n &= \\big(A\\_DIST_{IFmap} + DIST_{Filter} \\big) \\!\\times\\! \\frac{K}{blk_K} \\!\\times\\! \\frac{Num_{CTA}}{conv\\ layer}\n\\end{split}\n\\end{equation}\n\n\n\\subsection{L2 Cache Traffic}\n\\label{sub_sec:l2_model}\n\n\nIn im2col GEMM, the IFmap matrix with many duplicated data accesses involves high spatial and temporal locality.\nAs L1 cache is dedicated to a single SM, L1 can capture the reuse within one CTA's IFmap tile but not across active CTAs per SM given its small size ($\\sim$32KB).\nWith this assumption, our L2 model estimates the traffic by identifying the unique data accesses made in CTA input tile.\n\n\n\\textit{\\textbf{CTA Tile Selection.}}\nCTA tile size affects the spatial locality of the data elements accessed by a CTA; a bigger CTA tile has more data reuse.\nIt also affects the number of CTAs per workload and the ratio between math operations and memory loads per main loop iteration. \nAs it is a critical factor in modeling memory traffic, we analyze the CTA tile size (\\((blk_M + blk_N) \\times blk_K\\)) of the potential convolution GEMM kernels. \nBased on our analysis, cuDNN uses three types of CTA tilings: \\((128 \\times 128)\\times 8\\), \\((128 \\times 64)\\times 4\\), and \\((128 \\times 32)\\times 4\\).\n\\(blk_M\\) is fixed to 128 for all tiles and \\(blk_N\\) is chosen based on the width of the convolution GEMM \\((C_o)\\). \nWe measure the CTA tile size by different \\(C_o\\) and use this as a look-up table to our L2 traffic model.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.445\\textwidth]{fig\/l2.pdf}\n \n \\caption{\\small{IFmap data layout of im2col GEMM. The elements in \\(blk_M \\!\\times\\! blk_K \\) IFmap tile are the input for one main loop. Other details on the figure are described in \\sect{sub_sec:l2_model}.}}\n \\label{fig:l2_dist}\n \\vspace*{-4mm}\n\\end{figure}\n\\textit{\\textbf{Intra-CTA Spatial Locality.}}\nWe extend the example used in \\sect{sub_sec:l1_model} to explain our L2 traffic model (\\fig{fig:l2_dist}).\nAgain, IFmap elements visited by one filter data are remapped as a column in the IFmap matrix (\\textcolor{red}{\\ding{182}}) and the next column contains the IFmap data traversed by another filter data (\\textcolor{blue}{\\ding{183}}). \nTherefore, a \\(blk_M \\!\\times\\! blk_K\\) IFmap tile for one main loop iteration consists of multiple copies of data that are in continuous address range.\nThis access pattern entails large data locality shown by the duplicated data element in the white doted boxes.\nAs L1 cache captures the locality within each tile, only the unique elements are requested to L2.\nThus, based on our observation, if we know the address range of data within one IFmap tile (difference between the smallest and the largest address), we can estimate the size of memory requests to L2 in the tile.\nTo be specific, within each IFmap tile, the accessed address increases from the top to the bottom and left to the right.\nTherefore, the sum of the address distances between A and B (both edges of \\(blk_M\\)), and B and C (both edges of \\(blk_K\\)) in \\fig{fig:l2_dist} estimates the total accessed data within an IFmap tile.\n\n\nWe first define the address distance between A and B as {\\small\\(DIST_V\\)} (vertical distance).\nAs the data elements in a column are the IFmap data traversed by one filter data, we formulate {\\small\\((DIST_V)\\)} as \\eq{eq:dist_v} using the mechanism used in L1 traffic modeling.\nAlso, when IFmap tile spans over more than one channel, multiple unique {\\small\\(DIST_V\\)} can be located within an IFmap tile.\nIn this example, the red and green colored elements belong to different channels.\nIf an IFmap tile spans these two channels, it needs to fetch the data elements of both channels.\nTherefore, the ratio between the channel width and IFmap width affects the average size of vertical distances {\\small\\((A\\_DIST_V)\\)}.\nWe calculate {\\small\\((A\\_DIST_V)\\)} using \\eq{eq:eff_dist_v} where \\(H_f \\!\\times\\! W_f\\) indicates the width of a channel.\n\\eq{eq:eff_dist_v}.\n\\begin{equation}\n\\scriptsize\n\\label{eq:dist_v}\nDIST_{V} = blk_{M} \\times \\frac{(W_i+2Pad) \\times Strd}{W_i+2Pad-Wf+1}\n\\end{equation}\n\\begin{equation}\n\\scriptsize\n\\label{eq:eff_dist_v}\nA\\_DIST_{V} = DIST_V \\times \\frac{blk_K}{H_f \\times W_f}\n\\end{equation}\n\nNext, we define the address distance between B and C of an IFmap tile as {\\small\\((DIST_H)\\)} (horizontal distance).\nBased on our observation, the distance between two adjacent columns within the same \\(W_f\\) range is 1 (\\ding{184}) as they are traversed by the two adjacent element of a filter.\nHowever, the distance between two columns in different \\(W_f\\) ranges is {\\small\\(W_i\\!+2Pad-W_f+1\\)} (\\ding{185}) which is determined by the size of an IFmap.\nAlso, the alignment of the filter width (\\(W_f\\)) to \\(blk_K\\) affects the number of \\(H_f\\) edges within an IFmap tile.\nTherefore, considering both the inter-column distances and the alignment to \\(blk_K\\), we calculate {\\small\\(DIST_H\\)} using \\eq{eq:dist_h}.\nLike the vertical distances, if one IFmap spans more than one data sample, multiple unique {\\small\\({DIST_H}\\)} can be found within an IFmap tile.\nFor example, in \\fig{fig:l2_dist}, the light colored red and green elements belong to a different sample.\nTherefore, the number of unique horizontal distances per IFmap is affected by the ratio between the IFmap height (\\(blk_M\\)) and the entire elements traversed by one filter element \\((\\frac{W_i+2Pad-W_f+1}{Strd})^2\\) (assuming height and width are the same).\n\nIn \\fig{fig:l2_dist}, the data elements of two different samples present within a \\(blk_M\\) range which adds an additional {\\small\\(DIST_H\\)} (horizontal blue dotted box).\nThe number of samples per IFmap tile is affected by the ratio between \\(blk_M\\) and the size of one IFmap which is the product of the height and width of an IFmap or \\((\\frac{W_i+2Pad-W_f+1}{Strd})^2\\) (assuming height and width are the same).\nReflecting this, we calculate the average horizontal distance of an IFmap tile {\\small\\((A\\_DIST_H)\\)} (\\eq{eq:eff_dist_h}).\n\\begin{equation}\n\\scriptsize\n\\label{eq:dist_h}\n\\begin{split}\n DIST_H &= \\Big( \\frac{blk_{K} \\!-1}{W_f} \\Big) \\!\\times\\! \\big( (W_i \\!-\\! W_f+1) + Strd \\!\\times\\! (W_f \\!-\\! blk_{K} \\!+\\! 1) \\big)\\\\\n &+ \\Big( \\frac{W_f-blk_K+1}{W_f} \\Big) \\times \\big( Strd \\!\\times\\! (blk_K-1) \\big)\n\\end{split}\n\\vspace*{-3mm}\n\\end{equation}\n\\begin{equation}\n\\scriptsize\n\\label{eq:eff_dist_h}\nA\\_DIST_{H} = DIST_H \\!\\times\\! \\Big( 1+ \\frac{blk_M}{\\big(\\frac{H_i+2Pad-H_f+1}{Strd}\\big)^2} \\Big)\n\\end{equation}\n\nIn case of \\(1 \\!\\times\\! 1\\) convolution and FC layers, all IFmap data elements are unique and there is no data reuse within an IFmap tile.\nTherefore, we can simply calculate {\\small\\(DIST_V\\)} as the height of an IFmap tile and {\\small\\(DIST_H\\)} as its width.\nThe same method is applied to filter data elements because they are all unique as well.\nFinally, using the expected distance of IFmap and filter tiles, the total L2 load traffic within a main loop is calculated by adding {\\small\\(A\\_DIST_V\\)} and {\\small\\(A\\_DIST_H\\)}.\nThen, the total L2 traffic for a Conv layer is calculated by considering the number of CTAs and the number of main loop iterations (\\eq{eq:l2_traffic}).\n\\begin{equation}\n\\scriptsize\n\\label{eq:l2_traffic}\n\\begin{split}\nT_{L2\\_total} &= A\\_DIST_{GEMM\\_tile} \\times \\frac{GEMM\\ tiles}{Conv\\ layer}\\\\\n &= \\big(A\\_DIST_{IFmap} + DIST_{Filter} \\big) \\!\\times\\! \\frac{K}{blk_K} \\!\\times\\! \\frac{N_{CTA}}{Conv\\ layer}\n\\end{split}\n\\end{equation}\n\n\n\n\n\\subsection{Model parameters}\n\n\n\n\n\\section{Motivation and Related Work}\n\\label{sec:prior_work}\n\nTo analyze the performance bottlenecks of GPU-accelerated data parallel workloads, many GPU performance models have been proposed~\\cite{hong2009analytical,lai2013throughput,lai2013performance,zhou2017performance,wang2017gpgpu,sim2012performance,wu2015gpgpu}. In particular, prior models do in-depth analysis of the parallel GEMM and show high accuracy~\\cite{lai2013performance,zhou2017performance}. These prior models predict application performance based on potential execution time bottlenecks such as computation throughput, instruction fetch\/issue slots, and global memory bandwidth. However, they do not model the data traffic at each level of GPU memory system hierarchy which depends on each application's memory access and data reuse patterns. The models proposed by Zhou et al.~\\cite{zhou2017performance} and Sunpyo et al.~\\cite{hong2009analytical} include cache miss rate as a parameter but it is naively set to 1.\nSuch assumptions lead to poor performance estimation when the parallel workload has high spatial locality.\nCuMAPz{~\\cite{kim2011cumapz}} does consider an application's shared memory utilization by analyzing the device code, but does not consider cache and off-chip memory channel traffic.\n\n\n\\fig{fig:miss_rate} shows the cache miss rates of \\abbrev{conv} layers used in GoogLeNet~\\cite{szegedy2015going} measured on NVIDIA TITAN Xp GPU. Depending on the layer configuration, the L1 miss rate varies from 13\\% to 50\\% and L2 miss rate ranges from 8\\% to 90\\%. Due to such high traffic variation at different levels of memory hierarchy, the prior performance models fail to accurately predict CNN performance. Particularly, given faster GPU compute throughput scaling than its memory bandwidth, the architecture research for future GPU designs needs accurate data traffic modeling.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.46\\textwidth]{fig\/miss_rate.pdf}\n \\vspace*{-1mm}\n \\caption{\\small{L1 and L2 cache miss rates of the \\abbrev{conv} layers in GoogLeNet}}\n \\label{fig:miss_rate}\n \\vspace*{-2mm}\n\\end{figure}\n\n\n\\section{overview}\n\n\\noindent\\textbf{\\emph{Motivation.}}\n\\begin{itemize}[noitemsep,topsep=5pt,leftmargin=0.15in]\n \\item Deep learning kernels are designed to maximize GPU computing unit utilization as they are computation intensive in general\n \\item As GPU's compute throughput increases with larger number of cores or more SIMD units per core (or tensor cores), deep learning workload execution time is expected to be also restricted by the memory system\n \\item Existing GPU performance models assume the deep learning kernels are bottlenecked by compute throughput thus do not model memory system in detail\n \\item It is important to improve GPU HW resource efficiently so it leads to the optimal deep learning application performance with minimum needed cost\n\\end{itemize}\n\n\\noindent\\textbf{\\emph{Contribution.}}\n\\begin{itemize}[noitemsep,topsep=5pt,leftmargin=0.15in]\n \\item We analyze and model the memory traffic and the cache hit rate of deep learning applications running on GPU\n \\item We analyze deep learning application performance where the kernel execution is bottlenecked not only by computation throughput but by the memory system bandwidth using our model\n \\item We analyze the performance trade-offs in kernel configuration and HW resource cost\n \\item We discuss and suggest the optimal options of deep learning application kernel configuration and resource requirement for the performance of deep learning applications\n\\end{itemize}\n\n\\noindent\\textbf{\\emph{Background.}}\n\\begin{itemize}[noitemsep,topsep=5pt,leftmargin=0.15in]\n \\item Convolution lowering for deep learning application execution parallelism for GPU\n \\item GPU CTA tiling mechanism and resource allocation\n \\item Convolution kernel design and work process\n \\item Prior work on GEMM performance modeling\n\\end{itemize}\n\n\\noindent\\textbf{\\emph{Main Idea.}}\n\\begin{itemize}[noitemsep,topsep=5pt,leftmargin=0.15in]\n \\item Data locality in convolution matrix\n \\item L1 traffic: Sector based memory queries, Memory access inefficiency associated with kernel stride\n \\item L2 traffic: Input tensor size and its ratio to cache capacity\n \\item DRAM traffic: Data locality within CTA and inter-CTA (sliding window)\n \\item Multi-warp \/ Multi-CTA scheduling and its impact to GPU execution time\n\\end{itemize}\n\n\\noindent\\textbf{\\emph{Experiment.}}\n\\begin{itemize}[noitemsep,topsep=5pt,leftmargin=0.15in]\n \\item GPU performance estimation with compute throughput scaling\n \\item GPU performance bottleneck points with varying GPU HW resource configurations\n \\item GPU performance bottleneck points with varying kernel designs\n\\end{itemize}\n\n\\section{Performance Modeling}\n\\label{sec:perf_model}\n\n\\abbrev{DeLTA} predicts \\abbrev{conv} layer's execution time and its performance bottleneck using the memory traffic estimates extracted from our memory traffic model. To identify the execution bottleneck, our model analyzes the compute and memory access streams in the highly software-pipelined GEMM kernel each of which uses different GPU resources.\n\n\\fig{fig:loop_time} shows the execution time breakdown of the software pipelined GEMM main loop~\\cite{cutlass} (arrows indicate dependencies between execution blocks). Three execution streams proceed in parallel to maximize resource utilization: the global load stream \\emph{(GLS)}, shared memory access stream \\emph{(SAS)}, and compute stream \\emph{(CS)}, that each exercise a different resource.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.47\\textwidth]{fig\/loop_time.pdf}\n \\caption{\\small{Execution time breakdown of a software pipelined GEMM main loop~\\cite{cutlass}. Three execution streams: global load stream (GLS), shared load stream (SAS), and compute stream (CS)}}\n \\label{fig:loop_time}\n \\vspace*{-3mm}\n\\end{figure}\n\nFirst, the global load stream \\emph{(GLS)} loads inputs from the global memory to the registers for prefetch and then to SMEM. GLS execution time is determined by both the latency to load data from the global memory and to store it to SMEM. The total load time consists of the (empty) pipeline latency and the transfer latency. The pipeline latency is the data flight time and it includes cache tagging, buffer pipelining, and all circuit data paths. Pipeline latency is fixed regardless of memory traffic, however, data transfer time increases with the data volume because of the limited bus bandwidth. \\abbrev{DeLTA} calculates GLS execution time by comparing the load latency from L1, L2, and DRAM using the traffic estimated by our memory traffic model (\\eq{eq:gls_time}). In \\eq{eq:gls_time}, the load latency from each level is the sum of its pipeline latency (\\(LAT_{level}\\)) and the per-main loop volume (\\(TpL_{level}\\)) divided by bandwidth (\\(BW_{level}\\)). \\(Num_{SM}\\) is the number of SMs per GPU.\n\\begin{equation}\n\\scriptsize\n\\begin{aligned}\n\\label{eq:gls_time}\nt_{GLS} = & \\max\\Big( LAT_{L1} + \\frac{TpL_{L1}}{BW_{L1}},\\; LAT_{L2} + \\frac{TpL_{L2}}{BW_{L2}\/Num_{SM}}, \\\\\n & LAT_{DRAM} + \\frac{TpL_{DRAM}}{BW_{DRAM}\/Num_{SM}}\\Big)\n\\end{aligned}\n\\end{equation}\n\n\n\nSecond, the shared memory access stream \\emph{(SAS)} loads the prefetched data from the SMEM to registers. The execution time of SAS is bound by the SMEM bandwidth and the data volume of both the SMEM loads and the SMEM stores (from GLS). This is because the loads from SMEM share the same data path with the stores to SMEM. The data volume of the SMEM stores is set by the CTA blocking factors \\((blk_N+blk_M) \\times blk_K\\). Then, the SMEM loads transfer the stored data to registers for computation of each warp (Fig 3), thus their data volume is set by the warp blocking factors \\((blk_{WN}+blk_{WM}) \\times blk_K\\) multiplied by the number of warps per CTA (\\(Num_{warps}\\)). We calculate the SAS execution time per main loop by dividing the data volume of the SMEM stores and SMEM loads by their respective bandwidth \\eq{eq:sas_time}.\n\n\n\\begin{equation}\n\\scriptsize\n\\label{eq:sas_time}\nt_{SAS} = \\frac{(blk_M \\!+\\! blk_N) \\!\\times\\! blk_K}{BW_{SMEM\\_ST}} + \n \\frac{(blk_{WM} \\!+\\! blk_{WN}) \\!\\times\\! blk_K \\!\\times\\! Num_{warps}}{BW_{SMEM\\_LD}}\n\\end{equation}\n\n\\begin{figure}[t!]\n \\centering \n \n \\subfloat[Simplified CTA GEMM main loop execution timing model]{\n \\includegraphics[width=0.48\\textwidth]{fig\/cta_int_base.pdf} \n \\label{fig:loop_base}\n \n }\\\\\n \\vspace*{-2mm}\n \\centering \n \n \\subfloat[Case 1. \\(\\max(t_{CS},t_{SAS}) \\geq t_{GLS}\\)]{\n \\includegraphics[width=0.48\\textwidth]{fig\/cta_int1.pdf} \n \\label{fig:loop_case1}\n }\\\\\n \\vspace*{-2mm}\n \\centering \n \n \\subfloat[Case 2. \\(t_{GLS} \\geq t_{CS} \\times Num_{ACT\\_CTA}\\)]{\n \\includegraphics[width=0.48\\textwidth]{fig\/cta_int2.pdf} \n \\label{fig:loop_case2}\n }\\\\\n \\vspace*{-2mm}\n \\centering\n \n \\subfloat[Case 3. \\(\\max(t_{CS},t_{SAS}) \\times Num_{ACT\\_CTA} \\geq t_{GLS} \\)]{\n \\includegraphics[width=0.48\\textwidth]{fig\/cta_int3.pdf} \n \\label{fig:loop_case3}\n }\\\\\n \\vspace*{-2mm}\n \\centering\n \n \\subfloat[Case 4. \\(\\max(t_{L1\\;BW}, t_{L2\\;BW}, t_{DRAM\\;BW})\\) \\(\\geq t_{CS}\\)]{\n \\includegraphics[width=0.48\\textwidth]{fig\/cta_int4.pdf}\n \\label{fig:loop_case3}\n }\\\\\n\\caption{\\small{A GEMM loop execution model of active CTAs with different GPU resource bottlenecks.}}\n\\label{fig:cta_batch_exec_time}\n\\vspace*{-5mm}\n\\end{figure}\n\n\\begin{table*}[b!]\n\\renewcommand{\\arraystretch}{0.4}\n\\begin{tabular}{@{}*{2}{p{\\dimexpr1\\textwidth-\\tabcolsep\\relax}}@{}}\n\\footnotesize\n\\(\nt_{Prologue} = \\big( LAT_{DRAM} + \\frac{blk_M \\times blk_N}{BW_{DRAM}\/Num_{SM}} \\big) + \n \\big( LAT_{SMEM} +\\frac{blk_M \\times blk_N}{BW_{SMEM\\_ST}} \\big) +\n \\big( \\frac{(blk_{WM} + blk_{WN}) \\times blk_K \\times Num_{warps}}{BW_{SMEM\\_LD}} \\big)\\) \n \\small\\(\\;\\;\\;\\;\\;(14)\\) \n\\\\ \\\\\n\\footnotesize\n\\(\nt_{Epilogue} = \\frac{blk_N \\times blk_M}{BW_{DRAM}} ,\\;\\;\n t_{Epilogue\\_bottleneck} = \\frac{blk_N \\times blk_M}{BW_{Bottleneck(L1,\\; L2,\\; DRAM)}}\\) \n \\small\\(\\;\\;\\;\\;\\;(15)\\) \n\\\\ \\\\\n\\footnotesize\n\\(\nt_{MAC_{(sm)}(SMEM_{(sm)})} = \n t_{Prologue} + \n \\big( t_{CS}(t_{SMEM}) \\times \\frac{K}{blk_K} + t_{Epilogue} \\big) \\times \\frac{Num_{CTA}}{Num_{SM}}\\)\n \\small\\(\\;\\;\\;\\;\\;(16)\\)\n\\\\ \\\\\n\\footnotesize\n\\(\nt_{DRAM\\_LAT_{(sm)}} = \n t_{Prologue} + \n \\big(t_{GLS} + \\max\\big(\\frac{t_{CS}}{blk_K},\\frac{t_{SAS}}{blk_K}\\big)\\big) \\times \\frac{K}{blk_K} + t_{Epilogue} \\big)\\times \\frac{Num_{CTA}\/Num_{SM}}{Num_{ACT\\_CTA}}\\)\n \\small\\(\\;\\;\\;\\;\\;(17)\\)\n\\\\ \\\\\n\\footnotesize\n\\(\nt_{MEM\\_BW_{(sm)}} = t_{Prologue} + \n \\big( \\max(t_{L1\\_BW},t_{L2\\_BW},t_{DRAM\\_BW}) \\times \\frac{K}{blk_K} + t_{Epilogue\\_bottleneck} \\big)\n \\times \\frac{Num_{CTA}}{Num_{SM}}\\) \n \\small\\(\\;\\;\\;\\;\\;(18)\\)\n\\vspace*{-20mm}\n\\end{tabular}\n\\end{table*}\n\n\n\nFinally, the compute stream \\emph{(CS)} performs matrix multiplication and accumulation (MAC) operations and its execution time is determined by the compute throughput per SM. The GEMM kernel interleaves pieces of SAS over CS to hide the SMEM access latency. Therefore, if CS is the execution time bottleneck, the loop execution time is the number operations divided by MAC bandwidth (\\(BW_{MAC}\\)):\n\\begin{equation}\n\\small\n\\begin{split}\n\\label{eq:cs_time}\nt_{CS} = \\frac{blk_M \\times blk_N \\times blk_K}{BW_{MAC}}\n\\end{split}\n\\end{equation}\n\n\\textit{\\textbf{Multi-CTA Interleaving.}}\nWhen a SM has multiple active CTAs, \\(t_{GLS}\\) can be further hidden by other CTAs' \\(t_{SAS}\\) or \\(t_{CS}\\). Especially, the execution time prediction in the context of CTA interleaving is important for GPU with high compute throughput. This is because \\(t_{CS}\\) per main loop can be shorter than \\(t_{GLS}\\) thus needs multiple CTAs' \\(t_{CS}\\) to hide the load latency. The number of active CTAs per SM is set by the ratio between CTA's resource requirement and the size of registers and SMEM per SM. Since the highly-optimized GEMM kernel in cuDNN aggressively reuses registers to increase the number of active CTAs per SM, we use the hardware profiled information for accurate modeling.\n\nGiven multiple CTAs to interleave, we model the execution time of the active CTAs with four potential resource bottleneck cases (\\fig{fig:cta_batch_exec_time}). The examples (case 1--4) have multiple CTAs to interleave and each row is the time slot for one CTA. We use the simple CTA loop execution time model depicted in \\fig{fig:loop_base} as the base. The computes and SMEM accesses are spread over a loop iteration (gray background color).\n\nIn case 1, the loop execution time per CTA is bottlenecked by \\(t_{CS}\\) or \\(t_{SAS}\\) so we calculate the loop execution time of a CTA batch by adding \\(\\max(t_{CS},t_{SAS})\\) of all active CTAs per SM. Second, if \\(t_{CS}\\) of all CTAs is shorter than \\(t_{GLS}\\) but longer than the memory transfer latency, the loop execution time per CTA batch equals to a single CTA's \\(t_{GLS}\\) (case 2). In this case, each SM has insufficient number of CTAs to hide the loads from the global memory so SM resources are wasted until the entire data arrives for the next loop iteration. Third, if a SM has many CTAs to interleave, \\(t_{GLS}\\) can be completely hidden by the time for computation or SMEM access (case 3). Finally, if the memory bandwidth becomes the bottleneck due to the high data transfer time (red portion), CTA batch loop execution time is mostly dependent on the data transfer time of the active CTAs (case 4). \\abbrev{DeLTA} estimates the loop time by comparing the four possible performance bottlenecks.\n\n\n\\textit{\\textbf{GEMM Prologue \\& Epilogue.}} \nUnlike the GEMM main loop routine, GEMM prologue and epilogue are only constrained by the latency to the global memory or memory bandwidth. For GEMM prologue, considering its small memory access count, we model its impact only for the first CTA assuming the others are hidden by CTA interleaving (Eq. 14). However, GEMM epilogue is not negligible as it has many memory accesses of \\(blk_{M} \\times blk_{N}\\) and all its output are written to the global memory. Therefore, we add the DRAM transfer time of each CTA epilogue to the SM execution time. However, when performance is bottlenecked by the bandwidth of a specific memory level, we use its transfer time (Eq. 15).\n\nGiven the execution time of the main loops, prologue, and epilogue, we drive the total execution time of all CTAs allocated per SM by different execution constraints (Eq. 16, 17, and 18). Eq. 16 is used to compute the execution time of case 1 and 3. Eq. 17 and Eq. 18 are used for the case 2 and 4 respectively (Eq. 18 drives three sub-results each for L1, L2, and DRAM BW). Then, the largest value of Eq .16, 17, and 18 becomes the per-SM execution time and its performance bottleneck. Also, as CTAs are not often distributed to all SM, the execution time of the SM with the largest CTAs becomes the eventual \\abbrev{conv} layer execution time.\n\n\n\n\n\\section{Memory Traffic Modeling}\n\\label{sec:traffic_modeling}\n\nWe model the traffic at each level of GPU memory hierarchy using the granularities of data reuse based on the GEMM kernel blocking factors. Our traffic model understands the memory access patterns in the im2col GEMM formed with a specific \\abbrev{conv} layer configuration to identify the complex locality within a access granularity at each memory level. We use the implicit convolution kernels supported in cuDNN library, and NVIDIA Pascal GPU as the base architecture. We use 32b floating point data precision widely used for neural network training~\\cite{han2015deep}. The minimum memory transaction granularity is 32B, which corresponds to a single sector of one 128B cache line. We use the performance-efficient BCHW as the baseline tensor ordering{~\\cite{li2016optimizing,coppens2017gunreal}}.\n\n\n\n\n\n\n\n\\input{tex\/l1_model.tex}\n\\input{tex\/l2_model.tex}\n\\input{tex\/dram_model.tex}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nFor three points $x,y,z \\in \\mathbb{R}^{{N}}$, we denote by $c(x,y,z)$ the inverse of \nthe radius of the circumcircle determined by these three points.\nThis expression is called \\textit{Menger curvature} of $x,y,z$. \nFor a Borel set $E \\subset \\mathbb{R}^{{N}}$, we define by\n\\[\\mathcal{M}_{2}(E):=\\int_E \\int_E \\int_E c^2(x,y,z) \\ \\mathrm{d} \\mathcal{H}^{1}(x) \\mathrm{d} \\mathcal{H}^{1}(y) \\mathrm{d} \\mathcal{H}^{1} (z)\\]\nthe \\textit{total Menger curvature} of $E$, where $\\mathcal{H}^{1}$ denotes the one-dimensional Hausdorff measure.\nIn 1999, J.C. L\\'{e}ger proved the following theorem. \n\n\\begin{thm*}[\\cite{Leger}]\n\tIf $E\\subset \\mathbb{R}^{{N}}$ is some Borel set with $0< \\mathcal{H}^{1}(E)< \\infty$ and $\\mathcal{M}_{2}(E)< \\infty$, then\n\t$E$ is $1$-rectifiable, i.e., there exists a countable \n\tfamily of Lipschitz functions $f_i:\\mathbb{R} \\to \\mathbb{R}^{{N}}$ such that $\\mathcal{H}^{1}(E \\setminus \\bigcup_i f_{i}(\\mathbb{R})) =0$.\n\\end{thm*}\n\nThis result is an important step in the proof of Vitushkin's conjecture (for more details see \n\\cite{MR3154530,MR2676222}), which states that a compact \nset with finite one-dimensional Hausdorff measure is removable for bounded analytic functions \nif and only if it is purely $1$-unrectifiable, which means that every $1$-rectifiable subset of this set has\nHausdorff measure zero. A higher dimensional analogue of Vitushkin's conjecture is proven in \\cite{MR3264510} but\nwithout using a higher dimensional version of L\\'{e}ger's theorem since in the higher dimensional setting there\nseems to be no connection between the $n$-dimensional Riesz transform and curvature (cf.~introduction \nof \\cite{MR3264510}).\n\nThere exist several generalisations of L{\\'e}ger's result. Hahlomaa proved in \\cite{MR2456269, MR2297880, MR2163108} \nthat if $X$ is a metric space and $\\mathcal{M}_{2}(X)< \\infty$, then $X$ is 1-rectifiable.\nAnother version of this theorem dealing with sets of fractional Hausdorff dimension equal or less than $\\frac{1}{2}$\nis given by Lin and Mattila in \\cite{MR1814107}. \n\nIn the present work, we generalise the result of L{\\'e}ger to arbitrary dimension and co-dimension, i.e.,\nfor $n$-dimensional subsets of $\\mathbb{R}^{{N}}$ \nwhere $n \\in \\mathbb{N}$ satisfies $n < {N}$. In the case $n={N}$ every $E\\subset \\mathbb{R}^{{N}}$ is $n$-rectifiable. \nOn the one hand, it is quite clear which conclusion we want to obtain, namely that the set $E$ is $n$-rectifiable,\nwhich means that there exists a countable family of Lipschitz functions $f_i:\\mathbb{R}^{n} \\to \\mathbb{R}^{{N}}$ such that \n$\\mathcal{H}^{n}(E \\setminus \\bigcup_i f_{i}(\\mathbb{R}^{n})) =0$. \nOn the other hand, it is by no means clear how to define integral Menger curvature for $n$-dimensional sets.\nL\\'{e}ger himself suggested an expression \nwhich turns out to be improper for our proof\\footnote{Hence, we agree with a remark made by \nLerman and Whitehouse at the end of the introduction of \\cite{MR2558685}.} (cf.~section \\ref{22.10.2014.2}).\nWe characterise possible integrands for our result in Definition \\ref{4.10.12.1},\nbut for now let us start with an explicit example:\n\\[\\mathcal{K}(x_{0},\\dots,x_{n+1})= \n\t\\frac{\\mathcal{H}^{n+1}(\\Delta(x_{0},\\dots,x_{n+1}))}{\\Pi_{0 \\le i < j \\le n+1}d(x_{i},x_{j})},\\]\nwhere the numerator denotes the $(n+1)$-dimensional volume of the simplex ($\\Delta(x_{0},\\dots,x_{n+1})$)\n spanned by the vertices\n$x_{0}, \\dots,x_{n+1}$, and\n$d(x_{i},x_{j})$ is the distance between $x_i$ and $x_j$.\nUsing the law of sines, we obtain for $n=1$\n\\[\\mathcal{K}(x_{0},x_{1},x_{2})= \\frac{\\mathcal{H}^{2}(\\Delta(x_{0},x_{1},x_{2}))}{d(x_{0},x_{1})d(x_{0},x_{2})d(x_{1},x_{2})}\n\t=\\frac{1}{4}c(x_{0},x_{1},x_{2}).\\]\nHence, $\\mathcal{K}$ can be regarded as a generalisation of the original Menger curvature for higher dimensions.\nWe set \n\\begin{align}\\label{27.08.2015.1}\n\t\\mathcal{M}_{\\mathcal{K}^{2}}(E)&:=\\int_E \\dots \\int_E \\mathcal{K}^{2}(x_{0},\\dots,x_{n+1}) \\ \\mathrm{d} \\mathcal{H}^{n}(x_{0}) \\dots \\mathrm{d} \\mathcal{H}^{n} (x_{n+1}).\n\\end{align}\nNow we can state our main theorem for this specific integrand (see Theorem \\ref{maintheorem} for the general version).\n\\begin{thm}\\label{13.11.2014.1}\n\tIf $E\\subset \\mathbb{R}^{{N}}$ is some Borel set with $\\mathcal{M}_{\\mathcal{K}^{2}}(E)< \\infty$, then\n\t$E$ is $n$-rectifiable.\n\\end{thm}\n\nLet us briefly overview a couple of results for \nthe higher dimensional case. There exist well-known equivalent characterisations of $n$-rectifiability,\nfor example, in terms of approximating tangent planes \\cite[Thm. 15.19]{MR1333890}, \northogonal projections \\cite[Thm. 18.1, Besicovitch-Federer projection theorem]{MR1333890},\nand in terms of densities \\cite[Thm. 17.6 and Thm. 17.8 (Preiss's theorem)]{MR1333890}.\nRecently Tolsa and Azzam proved in \\cite{2015arXiv150101569T} and \\cite{TolsaAzzam} a characterisation of $n$-rectifiability using the so called\n$\\beta$-numbers\\footnote{Introduced by P. W. Jones in \\cite{MR1069238} and \\cite{MR1103619}.} \ndefined for $k > 1$, \n$x \\in \\mathbb{R}^{{N}}$, $t>0$, $p\\ge 1$ by\n\\[ \\beta_{p;k;\\mu} (x,t) := \\inf_{P \\in \\mathcal{P}({N},n)} \\left( \\frac{1}{t^{n}} \\int_{B(x,kt)} \n\t\\left( \\frac{d(y,P)}{t} \\right)^{p} \\mathrm{d} \\mu (y) \\right)^{\\frac{1}{p}}, \\]\nwhere $ \\mathcal{P}({N},n) $ denotes the set of all $n$-dimensional planes in $\\mathbb{R}^{{N}}$,\n$d(y,P)$ is the distance of $y$ to the $n$-dimensional plane $P$\nand $\\mu$ is a Borel measure on $\\mathbb{R}^{{N}}$.\nThey showed in particular that an $\\mathcal{H}^{n}$-measurable set $E \\subset \\mathbb{R}^{{N}}$ with $\\mathcal{H}^{n}(E)<\\infty$ is\n$n$-rectifiable \\textit{if and only if}\n\\begin{align}\\label{28.08.2015.1}\n\t\\int_{0}^{1} \\beta_{2;1;\\mathcal{H}^{n}|_{E}}(x,r)^{2}\\frac{\\mathrm{d} r}{r} &< \\infty \\quad \\quad \\quad \\text{ for } \\mathcal{H}^{n}-a.e. x \\in E.\n\\end{align}\nThis result is remarkable in relation to our result since the $\\beta$-numbers and even \nan expression similar to \\eqref{28.08.2015.1}\nplay an important role in our proof. Nevertheless at the moment, we do not see how Tolsa's result could be used to\nshorten our proof of Theorem \\ref{13.11.2014.1}.\nThere are further characterisations of rectifiability by Tolsa and Toro in \\cite{1402.2799} and \\cite{1408.6979}.\n\nNow we present some of our own intermediate results \nthat finally lead to the proof of Theorem \\ref{13.11.2014.1}, but that might also be of independent interest itself.\nThere is a connection between those $\\beta$-numbers and integral Menger curvature \\eqref{27.08.2015.1}. In section \\ref{1.11.2014.1},\nwe prove the following theorem (see Theorem \\ref{lem2.5} for a more general version):\n\n\\begin{thm}\\label{13.11.2014.2}\n\tLet $\\mu$ be some arbitrary Borel measure on $\\mathbb{R}^{{N}}$ with compact support such that there is a\n\tconstant $C \\ge 1$ with $\\mu(B) \\le C (\\operatorname{diam} B)^{n}$ for all balls $B \\subset \\mathbb{R}^{{N}}$, \n\twhere $\\operatorname{diam} B$ denotes the diameter of the ball $B$.\n\tLet $B(x,t)$ be a fixed ball with $\\mu(B(x,t)) \\ge \\lambda t^{n}$ for some $\\lambda > 0$ and let $k >2$.\n\tThen there exist some constants $k_{1}> 1$ and $C \\ge 1$ such that \n\t\\[ \\beta_{2;k}(x,t)^{2} \\le \\frac{C}{t^{n}}\n\t\t\\int_{B(x,k_{1}t)} \\dots \\int_{B(x,k_{1}t)} \\chi_{D}(x_{0},\\dots,x_{n})\n\t\t\\mathcal{K}^{2}(x_{0},\\dots,x_{n+1}) \\ \\mathrm{d} \\mu(x_{0}) \\dots \\mathrm{d} \\mu (x_{n+1}),\\]\n\twhere \n\t$D=\\{(x_{0},\\dots,x_{n+1})\\in B(x,k_{1}t)^{n+2} | d(x_{i},x_{j})\\ge \\frac{t}{k_{1}}, i\\neq j\\}$.\n\\end{thm}\n\nA measure $\\mu$ is said to be $n$-dimensional Ahlfors regular if and only if there exists\nsome constant $C\\ge 1$ so that $\\frac{1}{C} (\\operatorname{diam} B)^{n} \\le \\mu(B) \\le C (\\operatorname{diam} B)^{n}$\nfor all balls $B$ with centre on the support of $\\mu$.\nWe mention that we do not have to assume for this theorem that the measure $\\mu$ is $n$-dimensional Ahlfors regular.\nWe only need the upper bound on $\\mu(B)$ for each ball $B$ and the condition $\\mu(B(x,t)) \\ge \\lambda t^{n}$ \nfor \\textit{one} specific ball $B(x,t)$.\n\nLerman and Whitehouse obtain a comparable result in \\cite[Thm.~1.1]{MR2558685}. The main differences are that,\non the one hand, they have to use an $n$-dimensional Ahlfors regular measure, but, on the other hand,\nthey work in a real separable Hilbert space of possibly infinite dimension instead of $\\mathbb{R}^{{N}}$.\nThe higher dimensional Menger curvatures they used (see \\cite[introduction and section 6]{MR2558685}) \nare examples of integrands that also \nfit in our more general setting\\footnote{A characterisation of all possible integrands for our result \ncan be found at the beginning \nof section \\ref{1.11.2014.2}. In section \\ref{22.10.2014.2}, we discuss one of the integrands of Lerman and Whitehouse.}.\nThis means that all of our results are valid if one uses their integrands instead of the initial $\\mathcal{K}$\npresented as an example above.\n\nIn addition to rectifiability, there is the notion of \\textit{uniform rectifiability}, which implies rectifiability. \nA set is uniformly rectifiable if it is\nAhlfors regular\\footnote{A set $E$ is $n$-dimensional Ahlfors regular if and only if the restricted Hausdorff \nmeasure $\\mathcal{H}^{n} \\textsf{L} E$ is $n$-dimensional Ahlfors regular.} and if it fulfils a second condition \nin terms of $\\beta$-numbers (cf. \\cite[Thm.~1.57, (1.59)]{MR1251061}). \nIn \\cite{MR2558685} and \\cite{MR2848529}, Lerman and Whitehouse give an alternative characterisation of \nuniform rectifiability \nby proving that for an Ahlfors regular set this $\\beta$-number term is comparable to a term expressed \nwith integral Menger curvature. One of the two inequalities needed is given in\nin \\cite[Thm. 1.3]{MR2558685}, and is similar to our following theorem, which is\na consequence of Theorem \\ref{13.11.2014.2} in connection with Fubini's theorem \n(see Theorem \\ref{13.11.2014.4} for a more general version).\nWe emphasise again that in our \ncase the measure $\\mu$ does not have to be Ahlfors regular.\n\n\\begin{thm}\\label{13.11.2014.3}\n\tLet $\\mu,\\lambda$ and $k$ be as in the previous theorem. There exists a constant $C \\ge 1$ such that\n\t\\[\\int \\int_{0}^{\\infty} \\beta_{2;k}(x,t)^{2}\\Eins_{\\left\\{ \\mu(B(x,t)) \\ge \\lambda t^{n} \\right\\}} \\frac{\\mathrm{d} t}{t} \n\t\t \\mathrm{d} \\mu(x) \\le C\\mathcal{M}_{\\mathcal{K}^{2}}(\\mu).\\]\n\\end{thm}\n\nIn the last years, there occurred several papers working with integral Menger curvatures. \nSome deal with (one-dimensional) space curves and get higher regularity ($C^{1,\\alpha}$) of the arc length \nparametrisation if the integral Menger curvature is finite, e.g \\cite{MR2489022,MR2668877}.\nOthers handle higher dimensional objects in \\cite{1205.4112,MR3061777,SvdMsurface} occasionally \nusing versions of integral Menger curvatures \nsimilar to ours\\footnote{Our main result does not work with their integrands, but most of the partial results \nare valid, cf.~section \\ref{22.10.2014.2}.}. \nRemarkable are the results of Blatt and Kolasinski \\cite{MR2921162,MR3021472}. They proved among other things that\nfor $p > n(n+1)$ and some compact $n$-dimensional $C^{1}$ manifold $\\Sigma$ \n\\[\\int_{\\Sigma} \\dots \\int_{\\Sigma}\n\t\\left(\\frac{\\mathcal{H}^{n+1}(\\Delta(x_{0},\\dots,x_{n+1}))}{\\operatorname{diam}(\\Delta(x_{0},\\dots,x_{n+1}))^{n+2}}\\right)^{p} \n\t\\mathrm{d} \\mathcal{H}^{n}(x_{0}) \\dots, \\mathrm{d} \\mathcal{H}^{n}(x_{n+1}) < \\infty\\]\nis \\textit{equivalent} to having a local representation of $\\sigma$ as the graph of a function belonging to the Sobolev Slobodeckij space\n$W^{2-\\frac{n(n+1)}{p},p}$.\nFinally, we mention that in \\cite{MR3091327,MR3105400} Menger curvature energies are recently used as knot energies \nin geometric knot theory to avoid some of the drawbacks of self-repulsive potentials like the M\\\"obius energy\n\\cite{MR1098918,MR1259363}.\n\n\n\\textbf{Organisation of this work.}\nIn section 3, we give the precise formulation of our main result and\ndiscuss some examples of integrands known from several papers working with integral Menger \ncurvatures.\nIn section \\ref{beta}, we present some results for a Borel measure including the general versions of\nTheorems \\ref{13.11.2014.2} and \\ref{13.11.2014.3}, namely Theorem \\ref{lem2.5} and \\ref{13.11.2014.4}.\nThe following sections \\ref{27.10.2014.2} to \\ref{notanullset} give the proof of our main result.\nWe remark that all statements in section \\ref{construction}, \\ref{gamma} and \\ref{notanullset},\nexcept section \\ref{27.10.2014.3}, depend on the construction given in chapter \\ref{construction}.\n\n\n\\section{Preliminaries}\\label{27.10.2014.1}\n\\subsection{Basic notation and linear algebra facts}\nLet $n,m,{N} \\in \\mathbb{N}$ with $1\\le n < {N}$ and $1 \\le m < {N}$.\nIf $E \\subset \\mathbb{R}^{{N}}$ is some subset of $\\mathbb{R}^{{N}}$, we write $\\overline{E}$ for its closure and $\\mathring{E}$\nfor its interior.\\index{$\\overline{E}$}\\index{$\\mathring{E}$}\nWe set $d(x,y):=|x-y|$ \\index{$d(x,y)$} where $x,y \\in \\mathbb{R}^{{N}}$ and $|\\cdot|$ is the usual Euclidean norm.\nFurthermore, for $x \\in \\mathbb{R}^{{N}}$ and $E_{1},E_{2} \\subset \\mathbb{R}^{{N}}$, we set $d(x,E_{2})=\\inf_{y \\in E_{2}}d(x,y)$,\n$d(E_{1},E_{2})=\\inf_{z \\in E_{1}}d(z,E_{2})$\nand $\\# E$ means the number of elements of $E$.\\index{$\\#E$}\nBy $B(x,r)$ we denote the closed ball in $\\mathbb{R}^{{N}}$ with centre $x$ and radius $r$,\nand we define by $\\omega_{\\N}$ \\index{$\\omega_{\\N}$} the $n$-dimensional volume of the $n$-dimensional unit ball.\nLet $G({N},m)$ be the Grassmannian, the space of all $m $-dimensional linear subspaces of $\\mathbb{R}^{N}$ and\n$\\mathcal{P}({N},m )$ the set of all $m $-dimensional affine subspaces of $\\mathbb{R}^{N}$.\nFor $P \\in \\mathcal{P}({N},m )$, we define $\\pi_P$ as the orthogonal projection on $P$.\nIf $P \\in \\mathcal{P}({N},m )$, we have that $P - \\pi_{P}(0) \\in G({N},m )$, hence $P - \\pi_{P}(0)$\nis the linear subspace parallel to $P$.\nFurthermore, we set $\\pi_P^{\\perp}:=\\pi_{P-\\pi_{P}(0)}^{\\perp} := \\pi_{(P-\\pi_{P}(0))^{\\perp}}$\nwhere $\\pi_{(P-\\pi_{P}(0))^{\\perp}}$ is the orthogonal projection on the orthogonal complement of $P-\\pi_{P}(0)$.\nThis implies that $\\pi_P^{\\perp}=\\pi_{\\tilde P}^{\\perp}$ and $\\pi_P \\neq \\pi_{\\tilde P}$\nwhenever $P$ is parallel but not equal to $\\tilde P$.\n\nFurthermore, for $A \\subset \\mathbb{R}^{{N}}$ and $x \\in \\mathbb{R}^{{N}}$, we set $A+x:=\\{y \\in \\mathbb{R}^{n}|y-x \\in A\\}$. \\index{$A+x$}\nBy $\\operatorname{span}(A)$, we denote the linear subspace of $\\mathbb{R}^{{N}}$ spanned by the elements of $A$. If \n$A=\\{o_{1},\\dots,o_{m}\\}$ or $A=A_{1} \\cup A_{2}$, we may \nwrite $\\operatorname{span}(o_{1},\\dots,o_{m})$ resp. $\\operatorname{span}(A_{1},A_{2})$ instead of $\\operatorname{span}(A)$.\n\n\\begin{rem}\\label{24.04.2013.1}\n\tLet $P \\in \\mathcal{P}({N},m)$ and $a,x \\in \\mathbb{R}^{{N}}$.\n\tWe have $ \\pi_{P}(a) = \\pi_{P-x}(a-x)+x$.\n\\end{rem}\n\n\n\\begin{rem}\\label{2.11.11.1}\\label{12.4.11}\n\tLet $b,a,a_i \\in \\mathbb{R}^{N}$, $\\alpha_i \\in \\mathbb{R}$ for $i=1,..l$, $l \\in \\mathbb{N}$ with \n\t$b = a + \\sum_{i =1}^l \\alpha_i(a_i-a)$\n\tand $P \\in \\mathcal{P}({N},m)$.\n\tThen we have\n\t$\\pi_{P}(b)= \\pi_{P}\\left(a\\right) +\\sum_{i =1}^l \\alpha_i \\big[\\pi_{P} (a_i) - \\pi_{P}(a)\\big]$ and\n\t$ d(b,P) \\le d(a,P) + \\sum_{i =1}^l |\\alpha_i|\\left(d(a_i,P) + d(a,P) \\right)$.\n\\end{rem}\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\\begin{tikzpicture}[scale=0.49]\n\t\t\\draw[-] (21,4) -- (2,4) node [below] {$P_{2}$} ;\n\t\t\\draw[-] (21 ,4) -- (27,8);\n\t\t\\draw[-] (8,8) -- (17.2,8);\n\t\t\\draw[dotted] (17.2,8) -- (20.5,8);\n\t\t\\draw[-] (20.5,8) -- (27,8);\n\t\t\\draw[-] (2,4) -- (8,8);\n\n\t\t\\draw[-] (11.2,4) -- (17.2,8);\n\n\t\t\\draw[-] (19,7) -- (6,2) node [below] {$P_{1}$};\n\t\t\\draw[-] (19,7) -- (25,11);\n\t\t\\draw[dotted] (12,6) -- (17.2,8);\n\t\t\\draw[-] (17.2,8) -- (25,11);\n\t\t\\draw[-] (6,2) -- (9,4);\n\t\t\\draw[dotted] (9,4) -- (12,6);\n\t\t\n\t\t\\draw[dashed] (12.7,5) -- (17.5,5) node [right] {$\\pi_{P_{2}}(a_{1})$};\n\t\t\\draw (17.5,5) circle (0.03);\n\t\t\\draw[dashed] (17.5,6.85) -- (12.7,5);\n\t\t\\draw (12.7,5) circle (0.04) ;\n\t\t\\draw[-] (12.6,5.04) -- (10.6,5.5) node [left] {$\\pi_{P_1\\cap P_2}(a_{1})$};\n\t\t\\draw[dashed] (17.5,5) -- (17.5,6.85);\n\t\t\\draw (17.5,6.85) circle (0.03);\n\t\t\\draw[-] (17.6,6.8) -- (18.5,6.4) node [right] {$a_{1}$};\n\n\t\t\\draw[dashed] (15.7,7) -- (22,7) node [right] {$\\pi_{P_{2}}(a_{2})$};\n\t\t\\draw (22,7) circle (0.03);\n\t\t\\draw[dashed] (22,9.43) -- (15.7,7);\n\t\t\\draw (15.7,7) circle (0.04) ;\n\t\t\\draw[-] (15.6,7.04) -- (13.6,7.5) node [left] {$\\pi_{P_1\\cap P_2}(a_{1})$};\n\t\t\\draw[dashed] (22,9.43) -- (22,7);\n\t\t\\draw (22,9.43) circle (0.03) ;\n\t\t\\draw[-] (21.88,9.5) -- (21,10) node [left] {$a_{2}$};\n\t\t\n\t\t\\draw[-] (11.8,4.4) -- (13,3) node [right] {$P_{1}\\cap P_{2}$};\n\t\\end{tikzpicture}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption[...tba]{Illustration of Lemma \\ref{20.2.2012.4}: $\\frac{|a_1- \\pi_{P_2}(a_1)|}{|a_1- \\pi_{P_1 \\cap P_2}(a_1)|}= \\frac{|a_2- \\pi_{P_2}(a_2)|}{|a_2- \\pi_{P_1 \\cap P_2}(a_2)|}$}\n\\end{figure}\n\n\n\\begin{lem} \\label{20.2.2012.4}\n\tLet $P_1, P_2 \\in \\mathcal{P}({N},m)$ with $ \\dim P_1 = \\dim P_2 = m < {N}$ and $\\dim (P_1 \\cap P_2)= m -1$.\n\tFor $a_1,a_2 \\in P_1 \\setminus P_2$, we have\n\t$\\frac{|a_1- \\pi_{P_2}(a_1)|}{|a_1- \\pi_{P_1 \\cap P_2}(a_1)|}\n\t\t= \\frac{|a_2- \\pi_{P_2}(a_2)|}{|a_2- \\pi_{P_1 \\cap P_2}(a_2)|}$.\n\\end{lem}\n\\begin{proof}\n\tTranslate the whole setting so that $P_{1},P_{2}$ are linear subspaces. Then express $a_{1}$ by an orthonormal\n\tbase of $P_{1}$ and compute that $\\frac{|a_1- \\pi_{P_2}(a_1)|}{|a_1- \\pi_{P_1 \\cap P_2}(a_1)|}$\n\tis independent of $a_{1}$.\n\\end{proof}\n\n\n\\begin{rem} \\label{26.08.12.1}\n\tLet $A,B$ be affine subspaces of $\\mathbb{R}^{{N}}$ with $A \\subset B$ and let $a \\in \\mathbb{R}^{{N}}$. We have\n\t$\\pi_{A}(\\pi_{B}(a))=\\pi_{A}(a)=\\pi_{B}(\\pi_{A}(a))$.\n\\end{rem}\n\n\\subsection{Simplices}\n\\begin{dfn}\\label{24.09.13.1}\n\tLet $x_i \\in \\mathbb{R}^{{N}}$ for $i=0,1,\\dots,m$. We define \n\t$\\Delta(x_0,\\dots,x_m)=\\Delta(\\{x_0,\\dots,x_m\\})$ \\index{$\\Delta(x_0,\\dots,x_m)$} as the convex hull of the \n\tset $\\{x_0,\\dots,x_m \\}$ and call it \\textit{simplex} or $m$\\textit{-simplex}\n\tif $m$ \n\tis the Hausdorff dimension of $\\Delta(x_0,\\dots,x_m)$.\n\tIf the vertices of $T=\\Delta(x_0, \\dots, x_{m})$ are in some set $G \\subset \\mathbb{R}^{{N}}$, i.e., \n\t$x_0, \\dots, x_{m} \\in G$,\n\twe write $T=\\Delta(x_0, \\dots, x_{m}) \\in G$.\n\\end{dfn}\n\nWith $\\textnormal{aff}(E)$ we denote the smallest affine subspace of $\\mathbb{R}^{{N}}$ that contains the set $E \\subset \\mathbb{R}^{{N}}$.\n\\index{\\textnormal{aff}}\nIf $E=\\{x_{0}\\}$, we set $\\textnormal{aff}(E)=\\{x_{0}\\}$.\n\\begin{dfn}\n\tLet $T=\\Delta(x_0,\\dots,x_m) \\in \\mathbb{R}^{{N}}$. For $i,j \\in \\{0,1, \\dots ,m \\}$ we set\n\t\\begin{align*}\n\t\t\\face{i}T = \\face{x_i}T& = \\Delta(\\{x_0,\\dots ,x_m\\} \\setminus \\{x_i\\}), \\index{$\\face{i}T$} \\\\\n\t\t\\face{i,j}T = \\face{x_i,x_j}T& = \\Delta(\\{x_0,\\dots ,x_m\\} \\setminus \\{x_i,x_j\\}), \\\\\n\t\t\\height{i}T = \\height{x_i}T& = d\\big(x_i,\\textnormal{aff}(\\{x_0,\\dots,x_m\\}\\setminus \\{x_i\\})\\big).\\index{$\\height{i}T$}\n\t\\end{align*}\n\\end{dfn}\n\n\\begin{dfn}\n\tLet $T=\\Delta(x_0,\\dots,x_m)$ be an $m$-simplex in $\\mathbb{R}^{{N}}$. \n\tIf $\\height{i}T \\ge \\sigma$ for all $i=0,1,\\dots,m$, we call\n\t$T$ an $(m,\\sigma)$\\textit{-simplex}.\n\\end{dfn}\n\n\\begin{rem} \\label{25.09.2013.10}\n\tLet $T=\\Delta(x_0,\\dots,x_m)$ an $(m,\\sigma)$-simplex. For all $i \\in \\{0,\\dots, m \\}$, we have \n\t$ d(x_i,\\textnormal{aff}(A_i)) \\ge \\height{i}T \\ge \\sigma$\n\tfor every $\\emptyset \\neq A_i \\subset \\{x_0,\\dots,x_{m} \\} \\setminus \\{x_i\\}.$\n\\end{rem}\n\n\\begin{dfn} \\label{30.09.2013.2}\n\tLet $T=\\Delta(x_0,\\dots,x_m)$ be an $m$-simplex in $\\mathbb{R}^{{N}}$. \n\tBy $\\mathcal{H}^{m}(T)$ we denote the volume of $T$ and \n\twe define the \\textit{normalized volume}\n\t$\\volume(T):= m!\\ \\mathcal{H}^{m}(T)$\n\twhich is the volume of the parallelotope spanned by the simplex $T$ (cf. \\cite{Stein}).\n\tWe also have a characterisation of $\\volume(T)$ by the Gram determinant\n\t$\\volume(T)=\\sqrt{\\text{Gram}(x_{1}-x_{0},\\dots,x_{m}-x_{0})}$,\n\twhere the Gram determinant of vectors $v_1, \\dots, v_{m} \\in \\mathbb{R}^{{N}}$ is defined by\n\t$\\text{Gram}(v_1,\\dots,v_{m}):=\\det\\left( (v_{1},\\dots,v_{m})^{T}(v_1,\\dots,v_{m}) \\right)$.\n\\end{dfn}\n\n\\begin{rem} \\label{23.08.12.2}\n\tLet $T=\\Delta(x_0,\\dots,x_m)$ be an $m$-simplex.\n\tThe volume of the parallelotope, spanned by $T$, fulfils\n\t$\\volume(T)= \\height{i}T \\ \\volume(\\face{i}T)$\n\twhich implies\n\t$\\mathcal{H}^{m}(T)= \\frac{1}{m} \\height{i}T \\ \\mathcal{H}^{m-1}(\\face{i}T)$\n\tfor the volume of a simplex.\n\\end{rem}\n\n\\begin{lem} \\label{25.01.2012.1}\n\tLet $T=\\Delta(x_0,\\dots,x_m)$ be an $m$-simplex. \n\tWe have\n\t$\\frac{\\height{i}T}{\\height{i}\\face{j}T} = \\frac{\\height{j}T}{\\height{j}\\face{i}T}$.\n\\end{lem}\n\\begin{proof}\n\tWe have\n\t$\\frac{\\height{i}(T)}{\\height{i}(\\face{j}T)} \n\t\t = \\frac{\\volume(T)}{\\height{i}(\\face{j}T) \\ \\volume(\\face{i}T)}\n\t\t = \\frac{\\height{j}(T) \\ \\volume(\\face{j}T)}{\\height{i}(\\face{j}T) \\ \\height{j}(\\face{i}T) \\ \\volume(\\face{i,j}T)}\n\t\t = \\frac{\\height{j}(T) \\ \\volume(\\face{j}T)}{\\height{j}(\\face{i}T) \\ \\volume(\\face{j}T)}\n\t\t = \\frac{\\height{j}(T)}{\\height{j}(\\face{i}T)}.$\n\\end{proof}\n\n\\begin{lem} \\label{17.11.11.2}\n\tLet $0 < h < H$, $1 \\le m \\le {N}+1$ and $y_0, x_i \\in \\mathbb{R}^{N}$, $i=0,1,\\dots, m$. \n\tIf $T_x=\\Delta(x_0,\\dots,x_m)$ is an $(m,H)$-simplex and $d(y_0,x_0) \\le h$, then $T_y=\\Delta(y_0,x_1,\\dots,x_m)$ \n\tis an $(m,H-h)$-simplex.\n\\end{lem}\n\\begin{proof}\n\tWe have\n\t$\\height{0}T_y \\ge \\height{0}T_x - d(x_0,y_0) \\ge H-h$.\n\tNow, we show that $\\height{1}T_y \\ge H-h$. \n\tIf $m=1$, we have $\\height{1}T_{y} = d(y_0,x_1)= \\height{0}T_{y}$. So we can assume that $m\\ge 2$ for the rest of this \n\tproof.\n\tWe set $ z_0 := \\pi_{\\textnormal{aff}(\\face{1}T_y)}(x_0),$ \n\t$T_z:=\\Delta(z_0,x_1,\\dots,x_m)$\n\tand start with some intermediate results:\\\\\n\tI.\\, \tDue to $\\height{0}T_y \\ge H-h>0$, $T_y$ is an $m$-simplex.\\\\\n\tII.\\,\tWe have $d(x_0,z_0) = d(x_0,\\textnormal{aff}(\\face{1}T_y))\\le d(x_0,y_0) \\le h$.\\\\\n\tIII.\\, \tWe have $z_0=x_2 +r_0(y_0 - x_2) + \\sum_{j=3}^{m} r_j(x_j-x_2)$ for some\n\t\t$r_i \\in \\mathbb{R}$, $i=0,3,\\dots,m$ because $z_0 \\in \\textnormal{aff}(\\face{1}T_y)$.\\\\\n\tIV.\\,\tWith III., Remark \\ref{2.11.11.1} and because of $\\pi_{\\textnormal{aff}(\\face{0}T_x)}(x_i)=x_i$ for $i=2,\\dots m$\n\t\twe get\n\t\t\\begin{align*}\n\t\t\t\\height{0}T_z &= |z_0 - \\pi_{\\textnormal{aff}(\\face{0}T_x)}(z_0) |\n\t\t\t= | r_0 y_0 - r_0 \\pi_{\\textnormal{aff}(\\face{0}T_x)}(y_0)|\n\t\t\t= r_0 \\height{0}(T_y)\n\t\t\\end{align*}\n\t\tand analogously $\\height{0}(\\face{1}T_z) = r_0 \\height{0}(\\face{1}T_y)$.\\\\\n\tV.\\,\tWith Remark \\ref{26.08.12.1}, we get $\\pi_{\\textnormal{aff}(\\face{0,1}T_x)}(z_0)=\\pi_{\\textnormal{aff}(\\face{0,1}T_x)}(x_0)$ and \n\t\thence we obtain\n\t\t\\begin{align*}\n\t\t\t\\height{0}(\\face{1}T_z)\n\t\t\t&= d(\\pi_{\\textnormal{aff}(\\face{1}T_y)}(x_0),\\pi_{\\textnormal{aff}(\\face{0,1}T_x)}(z_0))\n\t\t\t= d(\\pi_{\\textnormal{aff}(\\face{1}T_y)}(x_0),\\pi_{\\textnormal{aff}(\\face{1}T_y)}(\\pi_{\\textnormal{aff}(\\face{0,1}T_x)}(z_0)))\\\\\n\t\t\t& \\le d(x_0,\\pi_{\\textnormal{aff}(\\face{0,1}T_x)}(z_0))\n\t\t\t = \\height{0}(\\face{1}T_x).\n\t\t\\end{align*}\n\t\n\tNow, with Lemma \\ref{25.01.2012.1} ($i=1$, $j=0$, $T=T_{y}$), IV and V we deduce\n\t\\begin{align*}\n\t\t\\height{1}T_y \n\t\t& \\ge \\height{0}T_z \\frac{\\height{1}(\\face{0}T_x)}{\\height{0}(\\face{1}T_x)}\n\t\t \\ge \\left( \\height{0}T_x - d(x_0,z_0) \\right) \\frac{\\height{1}(\\face{0}T_x)}{\\height{0}(\\face{1}T_x)}.\n\t\\end{align*}\n\tIf $\\frac{\\height{1}(\\face{0}T_x)}{\\height{0}(\\face{1}T_x)} \\ge 1$ this gives us directly $\\height{1}T_y \\ge H - h$.\n\tIn the other case, use Lemma \\ref{25.01.2012.1} and II to obtain\n\t$\\height{1}T_y > \\height{1}T_x - d(x_0,z_0) \\ge H - h$.\n\tSince, for $i=2,\\dots, m$, the points $x_i$ fulfil the same requirements as $x_1$, we are able to prove\n\t$\\height{i}T_y\\ge H-h$ for all $i=1,\\dots,m$ in the same way.\n\tSo, $T_y$ is an $(m,H-h)$-simplex.\n\\end{proof}\n\n\n\\begin{lem} \\label{29.02.2012.1}\n\tLet $C>0$, $1 \\le m \\le {N}$ and let $G \\subset \\mathbb{R}^{{N}}$\n\tbe a finite set so that for all $(m+1)$-simplices $S=\\Delta(x_0, \\dots , x_{m+1}) \\in G$, \n\tthere exists some $i \\in \\{0, \\dots, m +1 \\}$ so that $\\face{i}(S)$\n\tis no $(m,C)$-simplex.\n\n\tThen there exists some $m$-simplex $T_z=\\Delta(z_0, \\dots, z_{m}) \\in G$ so that for all $a \\in G$,\n\tthere exists some $i\\in \\{0,\\dots, m \\}$ with $d(a, \\textnormal{aff}(\\face{i}(T_z)) < 2 C$.\n\\end{lem}\n\\begin{proof}\n\tSince $G$ is finite, we are able to choose $T_z=\\Delta(z_0, \\dots, z_{m}) \\in G$ so that\n\t\\begin{align} \\label{20.2.2012.2}\n\t\t\\volume(T_z) &= \\max_{w_0,\\dots, w_{m}\\in G}\\volume(\\Delta(w_0, \\dots, w_{m})).\n\t\\end{align}\n\tWe can assume that $T_z$ is an $(m,2C)$-simplex, otherwise there would exist some $i\\in \\{0,\\dots,m\\}$ with\n\t$\\height{i}(T_z) < 2C$ and so for all $a \\in G$ with \\eqref{20.2.2012.2} we would obtain\n\t$d(a,\\textnormal{aff}(\\face{i}(T_z))) <2C.$\n\n\tNow, choose an arbitrary $y_0\\in G$. Set $S:=\\Delta(y_0,z_0, \\dots , z_{m})$. The properties of $G$ imply\n\tthat one face of $S$ is no $(m,C)$-simplex. Without loss of generality we assume that \n\t$T_y:=\\face{z_{0}}(S)$ is not an $(m,C)$-simplex (but an $m$-simplex). So there exists \n\tsome $i \\in \\{0,\\dots, m \\}$ with $\\height{i}(T_{y})0$, $1 \\le m \\le {N}$ and $D\\subset \\mathbb{R}^{{N}}$ be a bounded set. \n\tAssume that every simplex $S=\\Delta(y_0,\\dots,y_{m}) \\in D$ is not an $(m,H)$-simplex.\n\tThen there exists some $l \\in \\mathbb{N} \\cup \\{0\\}$, $l \\le m -1$ and $x_0,\\dots,x_l \\in \\overline{D}$ so that\n\t$\\overline{D} \\subset U_{H}(\\textnormal{aff}(x_0,\\dots,x_l))=\\{x \\in \\mathbb{R}^{{N}} | d(x,\\textnormal{aff}(x_0,\\dots,x_l)\\le H\\}$.\n\t\\index{$U_{H}$}\n\\end{lem}\n\\begin{proof}\n\tWe assume $\\# D \\ge 2$, otherwise the statement is trivial.\n\tLet $l \\in \\{0,\\dots,m-1\\}$ be the largest value such that there exists an $(l, H)$-simplex in $D$.\n\tIf $l=0$, we have $\\overline{D} \\subset U_H(\\textnormal{aff}(x_0))=B(x_0,H)$ for an arbitrary $x_0 \\in D$.\n\n\tNow suppose $ l \\ge 1$. Since $D$ is bounded, there exists $x_{0},\\dots,x_{l} \\in \\bar D$ such that\n\tthe volume $K:=\\volume(\\triangle(x_0,\\dots,x_l))$ is maximal. For some arbitrary $x_{l+1} \\in \\bar D$\n\tthe definition of $l$ and Lemma \\ref{17.11.11.2} imply that $\\triangle(x_0,\\dots,x_l)$ is not an\n\t$l+1,H$-simplex.\n\tHence there exists some $\\tilde l \\in \\{0,\\dots,l+1 \\}$ so that $\\height{\\tilde l}(T) 0$ for all \n\t$i \\in \\{1, \\dots, m+1 \\}$.\n\\end{lem}\n\\begin{proof}\n\tWe set $\\mu := \\mathcal{H}^{m} \\textsf{ L } F$. Since $\\mu(B)=\\infty$\n\tthere exists some $x_0 \\in B$ with $\\mu(B(x_0,h)) = \\infty$ for all $h>0$. \n\n\tThere exists some $c_1 > 0$ with $\\mu(B\\setminus \\mathring{B}(x_0,c_1))>0$. With Lemma \\ref{30.08.11.1}, there\n\texists some $x_1 \\in B \\setminus \\mathring{B}(x_0,c_1) $ with $\\mu(B(x_1,h))>0$ for all $h>0$ and\n\tthe simplex $T_1$ fulfils $\\height{1}(T_1) = d(x_0,x_1) \\ge c_1$.\n\n\tNow we assume that we already have $c_{l} >0$ and a simplex \n\t$T_{l}=\\Delta(x_0, \\dots, x_l) \\in \\mathbb{R}^{{N}}$ with $\\height{l}(T_l) \\ge c_l$ and $\\mu(B(x_i,h))>0$ \n\tfor all $i \\in \\{0,\\dots, l \\}$ and $h>0$ where $ l \\le m$.\n\tSo there exists some $00$\n\tand, with Lemma \\ref{30.08.11.1}, there exists some $x_{l+1} \\in F \\subset B$ so that\n\t$T_{l+1}:=\\Delta(x_0, \\dots, x_{l+1})$ fulfils $\\height{l+1}(T_{l+1}) \\ge c_{l+1}$\n\tand $\\mu(B(x_{l+1},h))>0$ for all $h>0$.\n\n\tSince $\\height{i}(T_{i}) \\ge C_{i}> 0$ for all $i \\in \\{1,\\dots,m+1\\}$ we obtain $\\volume(T)>0$ and hence\n\tthere exists some constant $c>0$ so that $T:=T_{m+1}$ is an $(m+1,c)$-simplex. \n\n\tTo conclude the proof set $\\sigma := \\frac{c}{m+3}$.\n\\end{proof}\n\n\\subsection{Angles between affine subspaces}\\label{23.04.2013.1}\n\\begin{dfn}\n\tFor $G_1,G_2 \\in G({N},m )$, we define $\\varangle(G_1,G_2):=\\|\\pi_{G_1} - \\pi_{G_2} \\|$,\n\twhere the right hand side is the usual norm of the linear map $\\pi_{G_1} - \\pi_{G_2}$.\n\tFor $P_1,P_2 \\in \\mathcal{P}({N},m )$, we define\n\t$\\varangle(P_1,P_2):=\\varangle(P_1-\\pi_{P_1}(0),P_2-\\pi_{P_2}(0))$.\n\\end{dfn}\n\n\\begin{rem}\\label{11.09.12.2}\\label{27.04.12.1}\n\tFor $P_{1},P_{2}, P_{3} \\in \\mathcal{P}({N},m )$ and $w \\in \\mathbb{R}^{N}$, we have\n\t$\\varangle(P_1,P_2)= \\varangle(P_1,P_2+w)$ and $\\varangle(P_1,P_3) \\le \\varangle(P_1,P_2) + \\varangle(P_2,P_3)$.\n\tThe angle $\\varangle$ is a metric on the Grassmannian $G({N},m )$ but not on $\\mathcal{P}({N},m )$\n\tbecause for $P \\in \\mathcal{P}({N},m )$, there exists some $w \\in \\mathbb{R}^{{N}}$ so that \n\t$\\varangle(P,P-w)=0$, but $P \\neq P-w$.\n\\end{rem}\n\n\\begin{lem} \\label{11.09.12.1}\n\tLet $U \\in G({N},m )$ and $v \\in \\mathbb{R}^{{N}}$ with $|v|=|\\pi_U(v)|$. Then we have $v = \\pi_U(v)$.\n\\end{lem}\n\\begin{proof}\n\tWe have $|\\pi_U(v)|^2 = |v|^2 = |\\pi_U(v) + \\pi_U^{\\perp}(v)|^2 = |\\pi_U(v)|^2 + |\\pi_U^{\\perp}(v)|^2$\n\tand so $\\pi_U^{\\perp}(v)=0$ which implies $v=\\pi_U(v)+\\pi_U^{\\perp}(v)=\\pi_U(v)$.\n\\end{proof}\n\n\\begin{lem} \\label{12.7.11.1}\\label{6.9.11.1}\n\tLet $P_1, P_2 \\in \\mathcal{P}({N},m )$ with $\\varangle(P_1,P_2) <1$ and $x,y \\in P_1$. We have\n\t\\[ d(x,y) \\le {\\textstyle \\frac{d(\\pi_{P_{2}}(x),\\pi_{P_{2}}(y))}{1-\\varangle(P_{1},P_{2})}} \\ \\\n\t\t\\text{ and } \\ \\ d(\\pi_{P_{2}}^\\perp(x),\\pi_{P_{2}}^\\perp(y)) \n\t\t\\le {\\textstyle \\frac{\\varangle(P_{1},P_{2})}{1-\\varangle(P_{1},P_{2})}} d(\\pi_{P_{2}}(x),\\pi_{P_{2}}(y)). \\]\n\\end{lem}\n\\begin{proof}\n\tFirst assume that $P_{1},P_{2} \\in G({N},m )$.\n\tWith $z:= \\frac{x-y}{|x-y|} \\in P_{1}$ and $\\pi_{P_{2}}^{\\perp}(z) + \\pi_{P_{2}}(z)=z=\\pi_{P_{1}}(z)$\n\twe get\n\t$| \\pi_{P_{2}}^{\\perp}(x)-\\pi_{P_{2}}^{\\perp}(y)| \n\t\t = |x-y | |\\pi_{P_{2}}^{\\perp}(z) + \\pi_{P_{2}}(z) - \\pi_{P_{2}}(z)|\n\t\t \\le |x-y | \\varangle(P_{1},P_{2})$,\n\tThis implies $d(x,y) \\le d(\\pi_{P_{2}}(x),\\pi_{P_{2}}(y)) + d(x,y) \\varangle(P_{1},P_{2})$.\n\tThese two estimates give the assertion in the case $P_{1},P_{2} \\in G({N},m )$.\n\tNow choose $t_1 \\in P_1$, $t_2 \\in P_2$ such that $P_1-t_1, P_2-t_2 \\in G({N},m )$ and\n\tuse Lemma \\ref{12.7.11.1}, Remark \\ref{24.04.2013.1} and Remark \\ref{11.09.12.2}\n\tto get the whole result.\n\\end{proof}\n\n\\begin{kor} \\label{24.04.2012.1}\n\tLet $P \\in \\mathcal{P}({N},m)$, $G \\in G({N},m)$ and $\\varangle(P,G)<1$. There exists some affine map \n\t$a :G \\to G^{\\perp}$ with $G(a)=P$, where $G(a)$ is the graph of the map $a$,\n\tand $a$ is Lipschitz continuous with Lipschitz constant $\\frac{\\varangle(P,G)}{1-\\varangle(P,G)}$.\n\\end{kor}\n\\begin{proof}\n\tSet $a(y)= \\pi_{P_2}^{\\perp}(\\pi_{P_2}^{-1}{}\\big|_{P_1}(y))$ and use Lemma \\ref{6.9.11.1}.\n\\end{proof}\n\n\\begin{kor}\\label{24.10.11.1}\n\tLet $G_1,G_2 \\in G({N},m)$ and\n\t$o_1,\\dots, o_{m}$ be an orthonormal basis of $G_1$. If\n\t$d(o_i,G_2) \\le \\tilde \\sigma \\le \\tilde \\sigma_1:=10^{-1}(10^{m}+1)^{-1}$, then \n\t$\\varangle(G_1,G_2)\\le 4m (10^{m} + 1) \\tilde \\sigma$.\n\\end{kor}\n\\begin{proof}\n\tFor $i=1,\\dots, m$, set $h_i:=\\pi_{P_2}(o_i)$ and use Lemma 2.3 from \\cite{MR3078345}.\n\\end{proof}\n\nFor $x,y \\in \\mathbb{R}^{{N}}$, we set $\\langle x,y \\rangle$ to be the usual scalar product in $\\mathbb{R}^{{N}}$. \n\\index{$\\langle x,y \\rangle$}\n\\begin{lem}\\label{30.05.12.1}\n\tLet $C, \\hat C\\ge 1$, $t > 0$ and \n\t$S=\\Delta(y_0,\\dots,y_{m })$ an $(m ,\\frac{t}{C})$-simplex with\n\t$S \\subset B(x,\\hat C t)$, $x \\in \\mathbb{R}^{N}$. \n\tThere exists an orthonormal basis $(o_{1},\\dots,o_{m})$ of $\\operatorname{span}(y_1-y_0, \\dots, y_{m} - y_0) $ and $\\gamma_{l,r} \\in \\mathbb{R}$ so that\n\tfor all $1 \\le l \\le m $ and $1 \\le r \\le l$ we have\n\t\\[o_l := \\sum_{r=1}^{l} \\gamma_{l,r} (y_r-y_0) \\ \\ \\ \\ \\text{ and } \\ \\ \\ \\\n\t\t|\\gamma_{l,r}| \\le (2lC \\hat C)^{l} \\frac{C}{t} \\le (2m C \\hat C)^{m } \\frac{C}{t}.\\]\n\\end{lem}\n\\begin{proof}\n\tWe set $z_i:=y_i - y_{0}$ for all $i=0,\\dots, m$, and $R:=\\Delta(z_0,\\dots,z_{m})= S - y_{0}$. \n\tWe obtain for all $i\\in \\{1, \\dots,m \\}$ ($S$ is an $(m,\\frac{t}{C})$-simplex)\n\t\\begin{align}\\label{30.05.12.3}\n\t\td(z_i,\\textnormal{aff}(z_0,\\dots,z_{i-1})) \\ge \\height{i}(R) & = \\height{i}(S) \\ge {\\textstyle \\frac{t}{C} }.\n\t\\end{align}\n\tDue to $\\height{i}(R) \\ge \\frac{t}{C}>0$, we have that $(z_1,\\dots,z_m)$ are linearly independent. \n\tSo with the Gram-Schmidt process\n\twe are able to define some orthonormal basis of the $m$-dimensional linear subspace $\\operatorname{span}(z_1,\\dots,z_{m} )$\n\t\\begin{align*}\n\t\to_1 & :=\\gamma_{1,1} z_1,\n\t\t& o_{l+1} := \\gamma_{l+1,l+1} z_{l+1}- \\gamma_{l+1,l+1}\\displaystyle{\\sum_{i=1}^{l} \\langle z_{l+1},o_i \\rangle o_i},\n\t\\end{align*}\n\twhere $\\gamma_{1,1}:=\\frac{1}{|z_{1}|}$ and $\\gamma_{l+1,l+1}:=\\frac{1}{d(z_{l+1},\\textnormal{aff}(z_0,\\dots,z_{l}))}$.\n\tFurthermore we define recursively \n\t\\[\\gamma_{l+1,r} := -\\sum_{i=r}^{l} \n\t\t\\gamma_{l+1,l+1} \\langle z_{l+1},o_i \\rangle \\gamma_{i,r}\\]\n\tfor $r\\in \\{1,\\dots,l\\}$.\n\tNow we prove by induction that $\\gamma_{l,r}$ fulfil the desired properties.\n\tWe have $o_1=\\gamma_{1,1} (y_1-y_0)$ and \\eqref{30.05.12.3} implies $|\\gamma_{1,1}| \\le \\frac{C}{t}$.\n\tNow let $1 \\le l \\le m $. We assume that, for all $i \\in \\{1,\\dots,l \\}$, $j \\in \\{1,\\dots,i\\}$, we have\n\t$o_i = \\sum_{r=1}^{i} \\gamma_{i,r} z_r$ and $|\\gamma_{i,j}| \\le (2lC \\hat C)^{l} \\frac{C}{t}$.\n\tWe obtain\n\t\\begin{align*}\n\t\to_{l+1} & = \\gamma_{l+1,l+1} z_{l+1} - \\sum_{i=1}^{l} \\sum_{r=1}^{i} \n\t\t\t\\gamma_{l+1,l+1}\\langle z_{l+1},o_i \\rangle \\gamma_{i,r} z_r\n\t\t = \\sum_{r=1}^{l+1} \\gamma_{l+1,r} z_r.\n\t\\end{align*}\n\tIf $r=l+1$, \\eqref{30.05.12.3} implies $|\\gamma_{l+1,r}| \\le \\frac{C}{t}$ and if $1 \\le r \\le l$, we get\n\twith $|z_{l+1}| \\le 2 \\hat C t$ \n\t\\begin{align*}\n\t\t|\\gamma_{l+1,r}| & \\stackrel{\\eqref{30.05.12.3}}{\\le} \\sum_{i=r}^{l}\n\t\t\t\\frac{C}{t}|z_{l+1}| (2lC \\hat C)^l \\frac{C}{t}\n\t\t < (2(l+1)C \\hat C)^{l+1} \\frac{C}{t}.\n\t\\end{align*}\n\\end{proof}\n\n\n\\begin{lem}\\label{21.11.11.2}\n\tLet $C, \\hat C \\ge 1$, $t > 0$,\n\t$0<\\sigma \\le \\left( 10(10^{m}+1) m C (2m C \\hat C)^{m} \\right)^{-1}$, \n\t$P_1,P_2 \\in \\mathcal{P}({N},m)$ and $S=\\Delta(y_0,\\dots,y_{m}) \\subset P_1$ \n\tan $(m,\\frac{t}{C})$-simplex with $S \\subset B(x,\\hat C t)$, $x \\in \\mathbb{R}^{{N}}$\n\tand $d(y_i,P_2) \\le t \\sigma$ for all $i \\in \\{0,\\dots,m\\}$. \n\tIt follows that \n\t\\[\\varangle(P_1,P_2)\\le 4m (10^m + 1)\\left(2m C (2m C\\hat C)^{m } \\right)\\sigma.\\] \n\\end{lem}\n\\begin{proof}\n\tUse Lemma \\ref{30.05.12.1}, to get some orthonormal basis of \n\t\\mbox{$\\operatorname{span}(y_1-y_0, \\dots, y_{m} - y_0) $} and $\\gamma_{l,r} \\in \\mathbb{R}$.\n\tWe set $\\hat y_{0}:=\\pi_{P_{2}}(y_{0})$ and we obtain for $1 \\le l\\le m$\n\t\\begin{align*}\n\t\td(o_l,P_2-\\hat y_{0})\n\t\t& \\le \\sum_{r=1}^{l} |\\gamma_{l,r}| (d(y_r,P_2)+d(y_{0},P_{2}))\n\t\t \\le 2m C(2m C\\hat C)^{m } \\sigma.\n\t\\end{align*}\n\tSetting $\\tilde \\sigma = 2m C(2m C \\hat C)^{m}\\sigma \\le \\frac{1}{10(10^{m}+1)} $ \n\tthe assertion follows with Corollary \\ref{24.10.11.1} ($G_{1}=P_{1}-y_{0}$, $G_{2}=P_{2}-\\hat y_{0})$.\n\\end{proof}\n\n\n\\begin{lem} \\label{23.03.2012.1}\n\tLet $\\sigma > 0$, $t \\ge 0$, $P_1,P_2 \\in \\mathcal{P}({N},m)$ with $\\varangle(P_1,P_2)\\le \\sigma$ and \n\tassume that \n\tthere exists $p_1 \\in P_1$, \n\t$p_2 \\in P_2$ with $d(p_1,p_2) \\le t\\sigma$. Then $d(w,P_2) \\le \\sigma (d(w,p_1)+t)$\n\tholds for every $w\\in P_1$.\n\\end{lem}\n\\begin{proof}\n\tFor $w\\in P_1$, set $\\tilde w:= w - p_1 \\in P_1 - p_1$.\n\tWe obtain\n\t\\begin{align*}\n\t\td(w,P_2)\n\t\t& \\le |\\tilde w| \\Bigl|\\frac{\\tilde w}{|\\tilde w|}\n\t\t\t-\\pi_{P_2-p_2}\\left(\\frac{\\tilde w}{|\\tilde w|}\\right)\\Bigr| + d(p_1,p_2)\n\t\t \\le |\\tilde w| \\varangle(P_1-p_1,P_2-p_2) + t\\sigma.\n\t\\end{align*}\n\\end{proof}\n\n\n\\setcounter{equation}{0}\n\\section{Integral Menger curvature and rectifiability}\n\\subsection{Main result}\\label{1.11.2014.2}\nLet $n,{N} \\in \\mathbb{N}$ with $1 \\le n < {N}$. We start with some definitions. \n\\newcommand{proper}{proper}\n\\begin{dfn}[Proper integrand] \\label{4.10.12.1} \\label{muproper}\t\n\tLet $\\mathcal{K} : \\left(\\mathbb{R}^{{N}}\\right)^{n+2} \\to [0,\\infty)$ and $p>1$. \n\tWe say that $\\mathcal{K}^{p}$ is a \\textit{proper{} integrand} if it fulfils the following four conditions:\n\t\\begin{itemize}\n\t\t\\item $\\mathcal{K}$ is $\\left(\\mathcal{H}^{n}\\right)^{n+2}$-measurable, where $\\left(\\mathcal{H}^{n}\\right)^{n+2}$\n\t\t\tdenotes the $n+2$-times product measure of $\\mathcal{H}^{n}$.\n\t\t\\item There exists some constants $c=c(n,\\mathcal{K},p) \\ge 1$ and $l=l(n,\\mathcal{K},p) \\ge 1$ so that, for all\n\t\t\t$t>0$, $C \\ge 1$, $x \\in \\mathbb{R}^{{N}}$ and all $(n,\\frac{t}{C})$-simplices $\\Delta(x_{0},\\dots,x_{n}) \\subset B(x,Ct)$, \n\t\t\twe have\n\t\t\\[\\left(\\frac{d(w,\\textnormal{aff}(x_0,\\dots,x_{n}))}{t}\\right)^{p} \\le c C^{l} t^{n(n+1)} \\mathcal{K}^{p}(x_0,\\dots,x_{n},w)\\]\n\t\tfor all $w \\in B(x,Ct)$.\n\t\t\\item For all $t > 0$, we have\n\t\t$t^{n(n+1)}\\mathcal{K}^{p}(tx_{0},\\dots,tx_{n+1})=\\mathcal{K}^{p}(x_{0},\\dots,x_{n+1})$.\n\t\t\\item For every $b \\in \\mathbb{R}^{{N}}$, we have $\\mathcal{K}(x_{0}+b,\\dots,x_{n+1}+b)=\\mathcal{K}(x_{0},\\dots,x_{n+1})$.\n\t\\end{itemize}\n\n\\end{dfn}\n\n\\begin{rem}\n\tIf instead of the first condition, we have that $\\mathcal{K}$ is $\\left(\\mu\\right)^{n+2}$-measurable for some\n\tBorel measure $\\mu$ on $\\mathbb{R}^{{N}}$ we call $\\mathcal{K}$ \\textit{$\\mu$-proper{}}.\n\\end{rem}\n\n\n\n\\begin{dfn}\n\t(i) We call a Borel set $E \\subset \\mathbb{R}^{{N}}$ \\textit{purely $n$-unrectifiable} if for every \n\t\tLipschitz continuous \\index{purely $n$-unrectifiable}\n\t\tfunction $\\gamma : \\mathbb{R}^{n} \\rightarrow \\mathbb{R}^{{N}}$, we have\n\t\t$\\mathcal{H}^{n}(E \\cap \\gamma( \\mathbb{R}^{n})) = 0$.\\\\\n\t(ii) A Borel set $E \\subset \\mathbb{R}^{{N}}$ is \\textit{$n$-rectifiable} if there exists some countable\n\t\tfamily of Lipschitz continuous functions\n\t\t\\index{$n$-rectifiable}\n\t\t$\\gamma_{i} : \\mathbb{R}^{n} \\rightarrow \\mathbb{R}^{{N}}$ so that\n\t\t$ \\mathcal{H}^{n}(E \\setminus \\bigcup_{i=1}^{\\infty} \\gamma_{i}( \\mathbb{R}^{n})) = 0$.\n\\end{dfn}\n\n\\begin{dfn}[Integral Menger curvature]\n\tLet $E \\subset \\mathbb{R}^{{N}}$ be a Borel set and $\\mu$ be a Borel measure on $\\mathbb{R}^{{N}}$. \n\tWe define the \\textit{integral Menger curvature} of $E$ and $\\mu$\n\twith integrand $\\mathcal{K}^{p}$ by \n\t$\\mathcal{M}_{\\mathcal{K}^{p}}(E):= \\mathcal{M}_{\\mathcal{K}^{p}}(\\mathcal{H}^{{N}}\\big|_{E})$\n\tand\n\t\\[\\mathcal{M}_{\\mathcal{K}^{p}}(\\mu):=\\int \\dots \\int \\mathcal{K}^{p}(x_{0},\\dots,x_{n+1}) \\ \\mathrm{d} \\mu(x_{0}) \\dots \\mathrm{d} \\mu(x_{n+1}).\\]\n\\end{dfn}\n\nNow we can state our main result.\n\n\\begin{thm} \\label{maintheorem}\n\tLet $E \\subset \\mathbb{R}^{{N}}$ be a borel set with \n\t$\\mathcal{M}_{\\mathcal{K}^{2}}(E) < \\infty$, where $\\mathcal{K}^{2}$ is some proper{} integrand.\n\tThen $E$ is $n$-rectifiable.\n\\end{thm}\n\n\\subsection{Examples of admissible integrands}\\label{22.10.2014.2}\nWe start with flat simplices. \n\\begin{dfn} \\label{30.09.2013.1}\n\tWe define the $(\\mathcal{H}^{n})^{n+2}$-measurable set\n\t\\[X_{0}:=\\left\\{(x_{0},\\dots,x_{n+1}) \\in (\\mathbb{R}^{{N}})^{n+2} \\big|\n\t\t\\text{Gram}(x_1 - x_0, \\dots, x_{n+1}-x_0)=0\\right\\}\\]\n\t(the Gram determinant is defined in Definition \\ref{30.09.2013.2})\n\twhich is the set of all simplices with $n+2$ vertices\n\tin $\\mathbb{R}^{{N}}$ which \n\tspan at most an $n$-dimensional affine subspace.\n\\end{dfn}\n\nThe following lemma is helpful to prove that a given integrand fulfils the second condition of a proper{} integrand.\n\\begin{lem} \\label{25.09.2013}\n\tLet $t>0$, $C \\ge 1$, $x \\in \\mathbb{R}^{{N}}$, $w \\in B(x,Ct)$ and let\n\t$S=\\Delta(x_{0},\\dots,x_{n}) \\subset B(x,Ct)$ be some\n\t$(n,\\frac{t}{C})$-simplex. \n\tSetting $S_{w}=\\Delta(x_0,\\dots,x_{n},w)$,\n\t$A(S_{w})$ as the surface area of the simplex $S_{w}$ \n\tand choosing\n\t$i,j \\in \\{0, \\dots, n \\}$ with $j \\neq i$ we have the following statements:\n\t\\begin{itemize}\n\t\t\\item $\\frac{t}{C} \\le d(x_i,x_{j}) \\le \\operatorname{diam}(S_{w}) \\le 2Ct$,\n\t\t\\item $d(x_{i},w) \\le 2Ct$,\n\t\t\\item $\\frac{t^{n}}{C^{n} n!} \\le \\mathcal{H}^{n}(S) \\le \\frac{(2C)^{n}}{n!}t^{n}$,\n\t\t\\item $\\mathcal{H}^{n}(S) \\le A(S_{w}) \\le [(n +1 )2C^2+1]\\mathcal{H}^{n}(S)$,\n\t\t\\item $d(w,\\textnormal{aff}(x_{0},\\dots,x_{n})) =n \\frac{\\mathcal{H}^{n+1}(S_{w})}{\\mathcal{H}^{n}(S)}$.\n\t\\end{itemize}\n\\end{lem}\n\\begin{proof}\n\tSince $S$ is an $(n,\\frac{t}{C})$-simplex, we have\n\t\\begin{align} \\label{18.10.12.1}\n\t\t\\frac{t}{C} \n\t\t&\\le \\height{i}(S) \n\t\t\\le d(x_{i},x_{j}) \\le \\operatorname{diam}(S_{w}) \n\t\t= \\max_{l,m \\in \\{0,\\dots,n\\}} \\left\\{d(x_l,x_m),d(x_l,w) \\right\\} \n\t\t\\le 2Ct\n\t\\end{align}\n\tand because of $x_i,w \\in B(x,Ct)$, we get\n\t$d(x_{i},w) \\le 2Ct$.\n\tNow, with Remark \\ref{23.08.12.2}, we conclude that \n\t$ \\mathcal{H}^{n}(S)=\\frac{1}{n!} \\prod_{l=0}^{n-1} d(x_{l},\\textnormal{aff}(x_{l+1},\\dots,x_{n}))$ \n\twhich implies with Remark \\ref{25.09.2013.10}\n\t\\begin{align*}\n\t\t\\frac{t^{n}}{C^{n} n!} \n\t\t\\stackrel{\\eqref{18.10.12.1}}{\\le} \\frac{1}{n!} \\prod_{l=0}^{n-1} \\height{l}(S)\n\t\t\\le \\mathcal{H}^{n}(S) \n\t\t\\le \\frac{1}{n!} \\prod_{l=0}^{n-1} d(x_{l},x_{n}))\n\t\t\\stackrel{\\eqref{18.10.12.1}}{\\le} \\frac{(2C)^{n}}{n!}t^{n}.\n\t\\end{align*}\n\tUsing Remark \\ref{23.08.12.2} and $\\height{w}(\\face{i}(S_w)) \\le d(w,x_j) \\le 2C t$, we obtain \n\t\\begin{align*}\n\t\t\\mathcal{H}^{n}(\\face{i}(S_w)) \n\t\t& \\stackrel{\\ref{23.08.12.2}}{=} \\frac{1}{n} \\height{w}(\\face{i}(S_w)) \\mathcal{H}^{n-1}(\\face{i,w}(S_w))\n\t\t\\stackrel{\\substack{\\hphantom{\\ref{23.08.12.2}} \\\\ \\eqref{18.10.12.1}}}{\\le} \\frac{1}{n} 2C^2 \\height{i}(S) \\mathcal{H}^{n-1}(\\face{i}(S))\n\t\t\\stackrel{\\ref{23.08.12.2}}{=} 2C^2 \\mathcal{H}^{n}(S),\n\t\\end{align*}\n\t\tso that with \n\t\t$A(S_{w})=\\sum_{i=0}^{n} \\mathcal{H}^{n}(\\face{i}S_w) + \\mathcal{H}^{n}(\\face{w}S_w)$\n\t\tand $\\face{w}(S_w)=S$, we get\n\t\\begin{align*}\n\t\t\\mathcal{H}^{n}(S) \\le A(S_w) & \\le [(n +1 )2C^2+1] \\mathcal{H}^{n}(S).\n\t\\end{align*}\n\tFinally, with Remark \\ref{23.08.12.2} and using that $S=\\face{w}(S_w)$, we deduce \n\t\\begin{align*}\n\t\td(w,\\textnormal{aff}(x_0,\\dots,x_{n})) &= \\height{w}(S_{w})\n\t\t= \\frac{\\height{w}(S_w) \\cdot \\mathcal{H}^{n}(\\face{w}(S_w))}{\\mathcal{H}^{n}(S)}\n\t\t= \\frac{n \\mathcal{H}^{n+1}(S_{w})}{\\mathcal{H}^{n}(S)}.\n\t\\end{align*}\n\\end{proof}\n\nNow we can state some examples of proper{} integrands. \nUse the previous lemma to verify the second condition.\nWe define all following examples to be $0$ on $X_{0}$ and will only give\nan explicit definition on $(\\mathbb{R}^{{N}})^{n+2} \\setminus X_{0}$.\nWe mention that our main result is only valid for all\nintegrands which are proper{} for integrability exponent $p=2$.\n\n\\subsubsection*{Proper Integrands with exponent $2$}\nWe start with the one used in the introduction of this work.\nLet $x_{0}, \\dots, x_{n+1} \\in (\\mathbb{R}^{{N}})^{n+2} \\setminus X_{0} $ and set\n\\[ \\mathcal{K}_{1}(x_{0},\\dots,x_{n+1}):= \n\t\\displaystyle{\\frac{\\mathcal{H}^{n+1}(\\Delta(x_{0},\\dots,x_{n+1}))}{\\Pi_{0 \\le i < j \\le n+1}d(x_{i},x_{j})}}, \\]\nthen $\\mathcal{K}_{1}^{2}$ is proper{}.\nThe next proper{} integrand is used by Lerman and Whitehouse in \\cite{MR2848529,MR2558685},\n\\[ \\mathcal{K}_{2}^{2}(x_{0},\\dots,x_{n+1})\n\t:= \\frac{1}{n+2} \\cdot \\frac{\\textnormal{Vol}_{n+1}(\\Delta(x_{0},\\dots,x_{n+1}))^{2}}{\\operatorname{diam}(\\Delta(x_{0},\\dots,x_{n+1}))^{n(n+1)}}\n\t\\sum_{i=0}^{n+1} \\frac{1}{\\prod_{\\genfrac{}{}{0pt}{}{j=0}{j\\neq i}}^{n+1}|x_{j}-x_{i} |^{2}},\\]\nwhere $\\textnormal{Vol}_{n+1}$ is $(n+1)!$ times the volume of the simplex\n$\\Delta(x_{0}, \\dots,x_{n+1})$, which is equal to the volume of the parallelotope spanned by \nthis simplex, cf. Definition \\ref{30.09.2013.2}.\nThe following proper integrand, $\\mathcal{K}_{3}^{2}$, is mentioned among others in \\cite[section 6]{MR2558685}:\n\\[ \\mathcal{K}_{3}(x_{0},\\dots,x_{n+1}):= \n\t\\displaystyle{\\frac{\\mathcal{H}^{n+1}(\\Delta(x_{0},\\dots,x_{n+1}))}{\\operatorname{diam} \\Delta(x_{0},\\dots,x_{n+1})^{\\frac{(n+1)(n+2)}{2}}}}.\\]\n\n\n\\subsubsection*{Proper Integrands with exponents different from $2$}\n\nNow we present some integrands for integral Menger curvature used in several papers, where the scaling behaviour\nimplies that our main result can not be applied. Nevertheless, most of our partial results\nare valid also for these integrands.\nThe first integrand we consider was introduced for $n=2, {N}=3$ in \\cite{SvdMsurface},\n\\[ \\mathcal{K}_{4}(x_{0},\\dots,x_{n+1}):= \n\t\\displaystyle{\\frac{V(T)}{A(T) (\\operatorname{diam} T)^2}},\\]\nwhere $V(T)$ is the volume of the simplex $T=\\Delta(x_0,\\dots,x_{n+1})$ and $A(T)$ \nis the surface area of $T$. $\\mathcal{K}_{4}^{p}$ is a proper{} integrand with $p=n(n+1)$.\nThe next one, $\\mathcal{K}_{5}^{p}$, is a proper{} integrand with $p=n(n+1)$ and is used, for example, in \\cite{MR2921162,MR3061777},\n\\[ \\mathcal{K}_{5}(x_{0},\\dots,x_{n+1}):= \n\t\\displaystyle{\\frac{\\mathcal{H}^{n+1}(\\Delta(x_{0}, \\dots, x_{n+1}))}{\\operatorname{diam} (\\Delta(x_{0}, \\dots, x_{n+1}))^{n+2}}}.\\]\nFinally, L\\'{e}ger suggested the following integrand in \\cite{Leger} for a higher dimensional analogue of \nhis theorem. Unfortunately, we can not confirm his suggestion. This one, $\\mathcal{K}_{6}^{p}$, is a proper{} integrand with $p=(n+1)$\nwhere\n\\[ \\mathcal{K}_{6}(x_{0},\\dots,x_{n+1}):= \n\t\\displaystyle{\\frac{d(x_{n+1},\\textnormal{aff}(x_0,\\dots,x_{n}))}{d(x_{n+1},x_0) \\dots d(x_{n+1},x_{n})}}.\\]\nHence our main result does \\textit{not} apply for $n \\neq 1$.\nFor $n=1$ up to a factor of $2$, this integrand gives the inverse of the circumcircle of \nthe three points $x_{0}, x_{1}, x_{2}$.\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{\\texorpdfstring{$\\beta$-numbers}{{\\ss}-numbers}}\\label{beta}\nIn this chapter, let $C_0 \\ge 10$ and $\\mu$ a Borel measure on $\\mathbb{R}^{{N}}$ with compact \nsupport $F$ that is upper Ahlfors regular, i.e., \n\\begin{enumerate} \\label{GrundeigenschaftenBetaBeta}\n\t\\renewcommand{\\labelenumi}{(\\Alph{enumi})}\n\t\\setcounter{enumi}{1}\n\t\\item for every ball $B$ we have $\\mu(B) \\le C_{0} (\\operatorname{diam} B)^{n}$.\n\\end{enumerate}\nIf $B=B(x,r)$ is some ball in $\\mathbb{R}^{{N}}$ with centre $x$ and radius $r$ and $t \\in (0,\\infty)$, then\nwe set $tB:=B(x,tr)$. Distinguish this notation from the case $t\\Upsilon=\\{tz|z \\in \\Upsilon\\}$ \nwhere $\\Upsilon \\subset \\mathbb{R}^{{N}}$ is some arbitrary set.\nFurthermore, in this and the following chapters, we assume that every ball is closed. We need this to apply\nVitali's and Besicovitch's covering theorems. By $C$, we denote a generic constant with a fixed value which\nmay change from line to line.\n\n\\subsection{Measure quotient}\n\\begin{dfn}[Measure quotient]\\label{Definitionvondeltaschlange}\n\tFor a ball $B=B(x,t)$ with centre $x \\in \\mathbb{R}^{{N}}$, radius $t > 0$ and \n\ta $\\mu$-measurable set $\\Upsilon \\subset \\mathbb{R}^{{N}}$, we define the \\textit{measure quotient}\n\t\\begin{align*}\n\t\t\\delta(B \\cap \\Upsilon) = \\delta_{\\mu}(B \\cap \\Upsilon) &:= \\frac{\\mu(B(x,t) \\cap \\Upsilon)}{t^{n}}.\n\t\\end{align*}\n\tIn most instances, we will use the special case $\\Upsilon = \\mathbb{R}^{{N}}$ and write $\\delta(B)$ instead of\n\t$\\delta(B \\cap \\mathbb{R}^{{N}})$.\n\\end{dfn}\nThis measure quotient compares the amount of the support $F$ contained in a ball with the size of this ball.\nThe following lemma states that if we have a lower control on the measure quotient of some ball, then we can find\na not too flat simplex contained in this ball, where at each vertex we have a small ball with a lower control \non its quotient measure.\n\n\\begin{lem} \\label{lem2.3}\n\tLet $0 < \\lambda \\le 2^{n}$ and $N_{0}=N_{0}({N})$ be the constant from \n\tBesicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans} depending only on the dimension ${N}$. There exist constants \n\t$C_{1} := \\frac{4\\cdot120^n n^{n+1} N_{0}C_{0}}{\\lambda}>3$\n\tand $ C_{2} := \\frac{2^{n+2}N_{0}C_{1}^n}{\\lambda}>1$\n\tso that for a given \n\tball $B(x,t)$ and some $\\mu$-measureable set $\\Upsilon$ \n\twith $ \\delta(B(x,t)\\cap \\Upsilon) \\ge \\lambda$, there exists \n\tsome $T=\\Delta(x_0,\\dots,x_{n+1}) \\in F \\cap B(x,t) \\cap \\Upsilon$ \n\tso that \n\t$\\face{i}(T)$ is an $(n,10n \\frac{t}{C_1})$-simplex and\n\t$\\mu\\left( B\\left(x_i,\\frac{t}{C_1}\\right) \\cap B(x,t) \\cap \\Upsilon\\right) \\ge \\frac{t^{n}}{C_{2}}$\n\tfor all $i \\in \\{0,\\dots,n+1 \\}$.\n\\end{lem}\n\\begin{proof}\n\tLet $B(x,t)$ be the ball with $\\delta(B(x,t) \\cap \\Upsilon) \\ge \\lambda $ and\n\t$\\mathcal{F} := \\{ B(y,\\frac{t}{C_{1}}) | y \\in F \\cap B(x,t) \\cap \\Upsilon \\}$.\n\tWith Besicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans} we get $N_{0}=N_{0}(n)$ families \n\t$\\mathcal{B}_{m} \\subset \\mathcal{F}$, $m=1,...,N_{0}$ of disjoint balls so that\n\t$ F \\cap B(x,t) \\cap \\Upsilon \\subset \\bigcup_{m=1}^{N_{0}} \\ \\dot{\\bigcup_{B \\in \\mathcal{B}_{m}}} B$.\n\tWe have\n\t\\begin{align*}\n\t\t\\lambda & \\le \\frac{1}{t^{n}} \\mu \\left(\\bigcup_{m=1}^{N_{0}}\n\t\t\t \\bigcup_{B \\in \\mathcal{B}_{m}} (B \\cap B(x,t) \\cap \\Upsilon)\\right) \n\t\t \\le \\frac{1}{t^{n}} \\sum_{m=1}^{N_{0}} \\sum_{B\\in \\mathcal{B}_{m}} \\mu(B \\cap B(x,t) \\cap \\Upsilon)\n\t\\end{align*}\n\tand hence there exists a family $\\mathcal{B}_{m}$ with\n\t\\begin{eqnarray} \\label{sterneinss}\n\t \t\\sum_{B \\in \\mathcal{B}_{m}} \\mu(B \\cap B(x,t) \\cap \\Upsilon) \\ge \\frac{\\lambda t^{n}}{N_{0}}.\n\t\\end{eqnarray}\n\tWe assume that for every $S=\\Delta(y_0,\\dots,y_{n +1}) \\in F \\cap B(x,t) \\cap \\Upsilon$, \n\tthere exists some $i \\in \\{0,\\dots,n +1 \\}$ so that either\n\t$\\face{i}(S)$ is no $(n,10n\\frac{t}{C_1})$-simplex or \n\t$\\mu( B(y_i,\\frac{t}{C_1}) \\cap B(x,t) \\cap \\Upsilon) < \\frac{t^{n}}{C_{2}}$. \n\tWe define\n\t$\\mathcal{G} := \\left\\{ B \\in \\mathcal{B}_{m} \\Big| \\mu(B \\cap B(x,t)\\cap \\Upsilon) \\ge \\frac{t^{n}}{C_{2}} \\right\\}$\n\tand mention that $\\mathcal{G}$ is a finite set since Lemma \\ref{22.2.2012.1} implies that \n\t$\\#\\mathcal{B}_{m} \\le (2C_{1})^{n}$. \n\tWith Lemma \\ref{29.02.2012.1} (where we set $G$ as the set of centres of balls in $\\mathcal{G}$ and $C=10n\\frac{t}{C_1}$),\n\twe know that there exists some \n\t$T_z=\\Delta(z_0,\\dots,z_n)$ so that for every ball $B(y,\\frac{t}{C_1}) \\in \\mathcal{G}$, \n\tthere exists some $i \\in \\{0,\\dots,n \\}$ so that $d(y,\\textnormal{aff}(\\face{i}(T_z))) \\le 20n\\frac{t}{C_1}$.\n\tWe define for $i \\in \\{0,\\dots,n \\}$\n\t\\begin{align*}\n\t\tT_i & := \\textnormal{aff}(\\face{i}(T_z)) \\cap B(\\pi_{\\textnormal{aff}(\\face{i}(T_z))}(x),2t), \\\\\n\t\t\\mathcal{S}_i & := \\left\\{ y \\in \\mathbb{R}^n | d(y,\\textnormal{aff}(\\face{i}(T_z))) \\le {\\textstyle \\frac{30n t}{C_1}}, \\pi_{\\textnormal{aff}(\\face{i}(T_z))}(y) \\in T_i \\right\\}\n\t\\end{align*}\n\tand we know that\n\t$B \\in \\mathcal{G}$ implies $B \\subset S_i$ for some $i \\in \\{0,\\dots,n \\}$.\n\tWith Lemma \\ref{23.2.2012.1} applied to $B(x,r)=T_i$, $s=\\frac{4}{C_1}t < 2t=r$ and $m=n-1$, there exists a family \n\t$\\mathcal{E} $ of disjoint closed balls\n\twith $\\operatorname{diam} B = \\frac{8}{C_1}t$ for all $B \\in \\mathcal{E}$,\n\t$T_i \\subset \\bigcup_{B \\in \\mathcal{E}} 5B$ and\n\t$\\# \\mathcal{E}\\le C_1^{n-1}$.\n\tLet $y \\in S_i$. We have $d(y,\\textnormal{aff}(\\face{i}(T_z))) \\le \\frac{30n}{C_1}t$ and $\\pi_{\\textnormal{aff}(\\face{i}(T_z))}(y) \\in T_i$.\n\tSo, there exists some $B=B(z,\\frac{4}{C_1}t) \\in \\mathcal{E}$ with $\\pi_{\\textnormal{aff}(\\face{i}(T))}(y) \\in 5B$ and we have\n\t$d(y,z) \\le \\frac{30 n}{C_1}t + 5\\frac{4}{C_1}t < \\frac{60n}{C_1}t$.\n\tThis proves $S_i \\subset \\bigcup_{B \\in \\mathcal{E}} 15n B$.\n\tWe therefrom derive with (B) (see page \\pageref{GrundeigenschaftenBetaBeta})\n\t\\begin{align} \\label{18.2.11.1}\n\t \\mu(S_i) &\\le \\sum_{B \\in \\mathcal{E}} \\mu\\left(15 n B\\right)\n\t \\stackrel{\\text{(B)}}{\\le} \\sum_{B \\in \\mathcal{E}} C_0\\left(15n \\operatorname{diam} B\\right)^{n} \n\t \\le \\# \\mathcal{E} C_0 \\frac{(120 n)^n t^{n}}{C_1^{n}} \n\t \\le (120 n)^n C_0 \\frac{t^{n}}{C_1}.\n\t\\end{align}\n\tWe define for $i \\in \\{1,\\dots, n \\}$\n\t\\begin{align*}\n\t\t\\mathcal{G}_0 &:= \\left\\{ B \\in \\mathcal{G} | B \\subset S_0 \\right\\}, \\hspace{9mm} \\text{and} \\hspace{10mm}\n\t\t\\mathcal{G}_i := \\left\\{ B \\in \\mathcal{G} | B \\subset S_i \\text{ and } B \\notin {\\textstyle \\bigcup_{j=0}^{i-1}} \\mathcal{G}_i \\right\\}\n\t\\end{align*}\n\tas a partition of $\\mathcal{G}$ (compare the remark after the definition of $\\mathcal{S}_{i}$).\n\tNow we have\n\t\\begin{align*}\n\t \\sum_{B \\in \\mathcal{G}} \\mu(B \\cap B(x,t)\\cap \\Upsilon) \n\t & \\le \\sum_{i=0}^{n}\\mu(S_i) \\stackrel{\\eqref{18.2.11.1}}{\\le} n ( 120 n)^n C_0 \\frac{t^{n}}{C_1}.\n\t \\end{align*}\n\t Moreover, we have\n\t \\[ \\sum_{B \\in \\mathcal{B}_{m} \\setminus \\mathcal{G}} \\mu(B \\cap B(x,t)\\cap \\Upsilon) < \n\t\t\\sum_{B \\in \\mathcal{B}_{m} \\setminus \\mathcal{G}} \n\t\t\\frac{t^{n}}{C_{2}} \\stackrel{\\#\\mathcal{B}_m\\le (2C_1)^n}{\\le} (2 C_{1})^{n} \\frac{t^{n}}{C_{2}}.\\]\n\t All in all, we get with (\\ref{sterneinss}) and the definition of $C_{1}$ and $C_{2}$\n\t \\[ \\lambda \\le N_{0} \\frac{1}{t^{n}}\\left(2^{n}t^{n} \\frac{C_{1}^{n}}{C_{2}} + 120^{n} n^{n +1} t^{n} C_{0} \\frac{1}{C_{1}}\\right) \n\t \t= N_{0}\\left(2^{n} \\frac{C_{1}^{n}}{C_{2}} \n\t\t+ 120^{n} n^{n +1} C_{0} \\frac{1}{C_{1}}\\right) \\le \\frac{\\lambda}{2}, \\]\n\t thus in contradiction to $\\lambda > 0$. This completes the proof of Lemma \\ref{lem2.3}.\n\\end{proof}\n\nIn most instances, we will use a weaker version of Lemma \\ref{lem2.3}:\n\\begin{kor} \\label{04.09.12.1}\n\tLet $0 < \\lambda \\le 2^{n}$. There exist constants $C_{1}=C_{1}({N},n,C_{0},\\lambda)>3$ and \n\t$ C_{2}=C_{2}({N},n,C_{0},\\lambda)>1$ so that for a given ball $B(x,t)$ and some $\\mu$-measurable set \n\t$\\Upsilon$ with $ \\delta(B(x,t)\\cap \\Upsilon) \\ge \\lambda$, there exists \n\tsome $(n,10n \\frac{t}{C_1})$-simplex $T=\\Delta(x_0,\\dots,x_{n}) \\in F \\cap B(x,t) \\cap \\Upsilon$ \n\tso that $\\mu\\left( B\\left(x_i,\\frac{t}{C_1}\\right) \\cap B(x,t) \\cap \\Upsilon\\right) \\ge \\frac{t^{n}}{C_{2}}$\n\tfor all $i \\in \\{0,\\dots, n \\}$.\n\\end{kor}\n\n\\subsection{\\texorpdfstring{$\\beta$-numbers and integral Menger curvature}{\\ss -numbers and integral Menger curvature}}\\label{1.11.2014.1}\n\\begin{dfn}[$\\beta$-numbers]\n\tLet $k > 1$ be some fixed constant, $x \\in \\mathbb{R}^{{N}}$, $t>0$, $B=B(x,t)$, $p\\ge 1$, $ \\mathcal{P}({N},n) $ the set of all \n\t$n$-dimensional planes in $\\mathbb{R}^{{N}}$ \n\tand $ P \\in \\mathcal{P}({N},n)$. We define\n\t\\begin{align*}\n\t\t \\beta_{p;k}^{P}(B) = \\beta_{p;k}^{P}(x,t) = \\beta_{p;k;\\mu}^{P}(x,t) & := \\left( \\frac{1}{t^{n}} \\int_{B(x,kt)} \\left( \\frac{d(y,P)}{t} \\right)^{p} \n\t\t \t\\mathrm{d} \\mu (y) \\right)^{\\frac{1}{p}}, \\\\\n\t\t \\beta_{p;k} (B) = \\beta_{p;k} (x,t) = \\beta_{p;k;\\mu} (x,t) & := \\inf_{P \\in \\mathcal{P}({N},n)} \\beta_{p;k}^{P}(x,t).\n\t\\end{align*}\n\\end{dfn}\n\nThe $\\beta$-numbers measure how well the support of the measure $\\mu$ can be approximated by some plane. A small \n$\\beta$-number of some ball implies either a good approximation of the support by some plane or a low \nmeasure quotient $\\delta$ (cf. Definition \\ref{Definitionvondeltaschlange}).\nHence, since we are interested in good approximations by planes, we will use the $\\beta$-numbers mainly for balls \nwhere we have some lower control on the measure quotient.\n\n\n\\begin{dfn}[Local version of $\\mathcal{M}_{\\mathcal{K}^{p}}$]\n\tFor $\\kappa >1$, $x \\in \\mathbb{R}^{{N}}$, $t>0$, $p>0$, we define\n\t\\[ \\mathcal{M}_{\\mathcal{K}^{p};\\kappa}(x,t):= \\idotsint_{\\mathcal{O}_{\\kappa}(x,t)}\\mathcal{K}^{p}(x_0,\\dots,x_{n+1}) \n\t\t\\mathrm{d}\\mu(x_0) \\dots \\mathrm{d} \\mu(x_{n+1}),\\]\n\twhere $\\mathcal{K}^{p}$ is a $\\mu$-proper{} integrand (cf. Definition \\ref{muproper} on page \\pageref{muproper}) and\n\t\\[ \\mathcal{O}_{\\kappa}(x,t) := \\left\\{ (x_0,\\dots,x_{n+1}) \\in (B(x,\\kappa t))^{n+2} \n\t\t\\Big| d(a,b) \\ge \\frac{t}{\\kappa}, \\forall \\ a ,b \\in \\{x_0,\\dots,x_{n+1} \\},a \\neq b \\right\\}. \\]\n\\end{dfn}\n\n\\begin{thm} \\label{lem2.5}\n\tLet $\\mathcal{K}^{p}$ be a symmetric $\\mu$-proper{} integrand and let\n\t$0<\\lambda < 2^{n}$, $k > 2$, $k_{0} \\ge 1$.\n\tThere exist constants $k_{1}=k_{1}({N},n,C_{0},k,k_{0},\\lambda)> 1$ and \n\t$C=C({N},n,\\mathcal{K},p,C_{0},k,k_{0},\\lambda) \\ge 1$\n\tsuch that if $x \\in \\mathbb{R}^{{N}}$ and $t > 0$ with $\\delta(B(x,t)) \\ge \\lambda$ for every \n\t$y \\in B(x,k_{0}t) $, we have\n\t\\[ \\beta_{p;k}(y,t)^{p} \\le C \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{n}} \n\t\t\\le C \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}+k_{0}}(y,t)}{t^{n}}.\\]\n\\end{thm}\n\\begin{proof}\n\tWith Lemma \\ref{lem2.3} for $\\Upsilon=\\mathbb{R}^{{N}}$, \n\tthere exists some $T=\\Delta(x_0,\\dots,x_{n+1}) \\in F \\cap B(x,t)$ \n\tso that $\\face{i}(T)$ is an $(n,10n \\frac{t}{C_1 })$-simplex and\n\t$\\mu\\left( B\\left(x_i,\\frac{t}{C_1 }\\right) \\cap B(x,t) \\right) \\ge \\frac{t^{n}}{C_{2} }$\n\tfor all $i \\in \\{0,\\dots,n+1 \\}$\n\twhere $C_1, C_{2}$ are the constants from Lemma \\ref{lem2.3} depending on the present constant $\\lambda > 0$,\n\tthe constant $C_0$ determined in (B) on page \\pageref{GrundeigenschaftenBetaBeta}, as well as ${N}$ and $ n$.\n\tWe set $B_i:=B\\left(x_i,\\frac{t}{C_1}\\right)$, \n\t$k_{1} := \\mathrm{max}(C_{1},(2+k + k_{0})) > 1$ \\label{19.10.12.10} and go on with some intermediate results.\n\n\tI. \\, Let $z_i \\in B_i$ for all $i \\in \\{0,\\dots,n+1\\}$, \n\t\t\t$w \\in B(x,(k+k_0)t) \\setminus \\bigcup_{\\genfrac{}{}{0pt}{}{l=0}{l\\neq j}}^{n+1} 2B_l$\n\t\tor $w \\in 2B_{j}$\n\t\tfor some fixed $j \\in \\{0,\\dots, n+1\\}$.\n\t\tSince $\\face{i}(T)$ is an $(n,10n \\frac{t}{C_1 })$-simplex we obtain\n\t\t$(z_{0},\\dots,\\hat z_{j}, \\dots, z_{n+1},w) \\in \\mathcal{O}_{k_{1}}(x,t)$,\n\t\twhere $(z_{0},\\dots,\\hat z_{j}, \\dots, z_{n+1},w)$ denotes the $(n+2)$-tuple \n\t\t$(z_{0},\\dots,z_{j-1},z_{j+1}, \\dots, z_{n+1},w)$.\n\t\n\tII. \\, Let $z_{i} \\in B_{i}=B(x_{i},\\frac{t}{C_{1}})$ for all $i \\in \\{0,\\dots,n+1\\}$.\n\t\tThen Lemma \\ref{17.11.11.2} implies that\n\t\t$\\face{i}(\\Delta(z_0,\\dots,z_{n+1}))$ is an $\\left(n,(9n-1) \\frac{t}{C_1}\\right)$-simplex for all \n\t\t\t $i \\in \\{0,\\dots,n+1 \\}$.\n\t\n\tIII. \\, Let $z_{i} \\in B_{i}=B(x_{i},\\frac{t}{C_{1}})$ for all $i \\in \\{0,\\dots,n+1\\}$,\n\t\t$w \\in B(x,(k+k_{0})t)$.\n\t\tSince $\\mathcal{K}^{p}$ is a $\\mu$-proper{} integrand with II. there exists some constant \n\t\t$\\tilde C=\\tilde C({N},n,\\mathcal{K},p,C_{0},k,k_{0},\\lambda)$ so that for all $j\\in \\{0,\\dots,n+1\\}$, we have\n\t\t\\begin{align*}\n\t\t\t\\left( \\frac{d(w,\\textnormal{aff}(z_{0},\\dots,\\hat z_{j},\\dots,z_{n+1}))}{t} \\right)^{p}\n\t\t\t& \\le \\tilde C t^{n(n+1)}\\mathcal{K}^{p}(z_{0},\\dots,\\hat z_{j},\\dots,z_{n+1},w).\n\t\t\\end{align*}\n\t\n\tIV. \\, There exist some constant $C=C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda)$ and $z_i \\in F \\cap B_i \\cap B(x,t)$, \n\t\t$i \\in \\{0,\\dots, n+1\\}$, so that for all $l \\in \\{0,\\dots,n+1 \\}$,\n\t\twe have\n\t\t\\begin{equation} \\label{24.08.12.1}\n\t\t\t\\int \\Eins_{\\{ (z_0,\\dots,\\hat z_l, \\dots,z_{n+1},w) \\in \\mathcal O _{k_{1}}(x,t)\\}} \n\t\t\t\t\\mathcal{K}^{p}(z_0,\\dots,\\hat z_l, \\dots,z_{n+1},w) \\mathrm{d} \\mu(w) \n\t\t\t\t\\le C \\ \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{(n+1)n}}\n\t\t\\end{equation}\n\t\tand with $P_{n+1}:=\\textnormal{aff}(z_0,\\dots,z_n)$\n\t\t\\begin{equation} \\label{24.08.12.2}\n\t\t\t\\left( \\frac{d(z_{n+1},P_{n+1})}{t} \\right)^{p} \n\t\t\t\\le C \\ \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{n}}.\n\t\t\\end{equation}\n\t\\begin{proof}\n\tFor $E \\subset \\mathbb{R}^{{N}}$ with $\\# E=m+1$, $E=\\{e_0,\\dots,e_m \\}$, $0 \\le m \\le n$, we set\n\t\\begin{multline*} \n\t\t\\mathcal{R}(E):= \\int_{\n\t\tF^{n-m+1}}\\Eins_{\\{(e_0,\\dots,e_m,w_{m+1},\\dots,w_{n+1}) \\in \\mathcal{O}_{k_1}(x,t)\\}} \\\\\n\t\t\\mathcal{K}^{p}(e_0,\\dots,e_m,w_{m+1},\\dots,w_{n+1}) \n\t\\mathrm{d} \\mu(w_{m+1}) \\dots \\mathrm{d} \\mu(w_{n+1}).\n\t\\end{multline*}\n\tThe integrand $\\mathcal{K}$ is symmetric, hence\n\tthe value $\\mathcal{R}(E)$ is well-defined because it does not depend on the\n\tnumbering of the elements of $E$. In the following part, we use the convention that\n\t$\\{0,\\dots,-1\\}=\\emptyset$ and $\\{z_{0},\\dots,z_{-1}\\}=\\emptyset$.\n\tAt first, we show by an inductive construction that, for all $m \\in \\mathbb{N}$ with $0 \\le m \\le n+1$,\n\tthere holds:\n\n\tFor all $j \\in \\{0,\\dots, m\\}$ and $ i \\in \\{j,\\dots,n+1\\}$, there \n\texist constants $C^{(j)}>1$,\n\tsets $Z_{i}^{j} \\subset F \\cap B_{i} \\cap B(x,t)$ \n\tand, for all $l \\in \\{0,\\dots,m-1\\}$, there exist $z_{l} \\in Z_{l}^{l}$\n\twith \n\t\\begin{align} \\label{28.08.12.1}\n\t\t\\mu(Z_{i}^{j}) &> \\frac{t^{n}}{2^{j+1}C_{2}},\n\t\\end{align}\n\tand, for all $u \\in \\{0,\\dots, m\\}$, \n\tfor all $E \\subset\\{z_{0},\\dots,z_{u-1}\\}$ and\n\t$z \\in Z_{r}^{u}$, where $r \\in \\{u,\\dots,n+1\\}$, we have\n\t\\begin{align} \\label{23.08.12.1}\n\t\t\\mathcal{R}(E \\cup \\{z\\}) \\le C^{(u)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{(\\#E+1)n}}.\n\t\\end{align}\n\n\tWe start with $m=j=0$ and choose the constant $C^{(0)} := 2 C_{2} $, set $\\Upsilon_{i}:=F \\cap B_i \\cap B(x,t)$\n\tand define for every $i \\in \\{0,\\dots,n+1 \\}$\n\t\\begin{eqnarray} \\label{2.5;1}\n\tZ_{i}^{0} := \\left\\{ z \\in \\Upsilon_{i} \\Big| \\mathcal{R}(\\{z\\}) \\le C^{(0)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{n}} \\right\\}.\n\t\\end{eqnarray}\n\tWe have \n\t$\\mu(Z_{i}^{0}) \\ge \\mu(\\Upsilon_{i}) - \\mu(\\Upsilon_{i}\\setminus Z_{i}^{0})> \\frac{t^{n}}{2C_{2}}$\n\tbecause\n\t$\\mu(\\Upsilon_{i}) \\stackrel{\\text{(ii)}}{\\ge} \\frac{t^{n}}{C_{2} }$,\n\tand with \\eqref{2.5;1}, Chebyshev's inequality and \n\t$\\int \\mathcal{R}(\\{z\\}) \\mathrm{d} \\mu (z)=\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)$ we obtain\n\t$\\mu(\\Upsilon_{i}\\setminus Z_{i}^{0}) < \\frac{t^{n}}{C^{(0)}}$.\n\tIf $u=0$, $E \\subset \\{z_{0},\\dots,z_{-1}\\}=\\emptyset$ and \n\t$z \\in Z_{r}^{0}$, where $r \\in \\{0,\\dots,n+1\\}$, the definition \\eqref{2.5;1} implies \\eqref{23.08.12.1}\n\tin this case.\n\t\n\tNow let $m \\in \\{0,\\dots,n \\}$ and we assume that \n\tfor all $j \\in \\{0,\\dots, m\\}$ and $ i \\in \\{j,\\dots,n+1\\}$, there \n\texist constants $C^{(j)}>1$,\n\tsets $Z_{i}^{j} \\subset F \\cap B_{i} \\cap B(x,t)$ \n\tand for all $l \\in \\{0,\\dots,m-1\\}$ there exist $z_{l} \\in Z_{l}^{l}$ with \n\t\\begin{align} \\label{16.7.12.1}\n\t\t\\mu(Z_{i}^{j}) &> \\frac{t^{n}}{2^{j+1}C_{2}},\n\t\\end{align}\n\tand for all $u \\in \\{0,\\dots, m\\}$, \n\tfor all $E \\subset\\{z_{0},\\dots,z_{u-1}\\}$ and\n\t$z \\in Z_{r}^{u}$ where $r \\in \\{u,\\dots,n+1\\}$, we have\n\t\\begin{align} \\label{6.5.2013.2}\n\t\t\\mathcal{R}(E \\cup \\{z\\}) \\le C^{(u)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{(\\#E+1)n}}.\n\t\\end{align}\n\n\tNext we start with the inductive step.\n\tFrom the induction hypothesis, we already have the constants $C^{(j)}$ and the sets $Z_{i}^{j}$\n\tfor $j \\in \\{0,\\dots, m\\}$ and $ i \\in \\{j,\\dots,n+1\\}$ as well as $z_{l} \\in Z_{l}^{l}$ \n\tfor $l \\in \\{0,\\dots,m-1\\}$. Since $\\mu(Z_{m}^{m})>0$, we can choose $z_{m} \\in Z_{m}^{m}$.\n\tWe define \n\t$C^{(m+1)}:= 2^{2m+2} C^{(m)} C_{2}$\n\tand, for $ i \\in \\{m+1, \\dots, n+1 \\}$, we define\n\t\\begin{align} \\label{2.5;3.1}\n\t\tZ_{i}^{m+1} \n\t\t& := \\bigcap_{\\genfrac{}{}{0cm}{}{E \\subset \\{z_0,\\dots,z_m\\}}{z_m \\in E}} \\underbrace{\n\t\t\\left\\{ z \\in Z_{i}^{m} \\Big| \\mathcal{R}(E \\cup \\{z \\}) \\le \n\t\tC^{(m+1)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{(\\#E+1)n}} \\right\\}}_{=:D_{i,E}^{m}}.\n\t\\end{align}\n\tWe have $\\mu(Z_i^{m+1}) \\ge \\mu(Z_{i}^{m}) - \\mu \\left( Z_{i}^{m} \\setminus Z_{i}^{m+1} \\right) \\ge \\frac{t^{n}}{2^{m+2}C_{2}}$ \n\tfor all $i \\in \\{m+1,\\dots,n+1 \\}$ because if\n\t$E \\subset \\{z_{0}, \\dots, z_{m}\\}$ with $z_{m} \\in E$, we get, using \\eqref{2.5;3.1}, Chebyshev's inequality,\n\t$\\int \\mathcal{R}(E \\cup \\{ z\\}) \\mathrm{d} \\mu (z) =\\mathcal{R}((E \\setminus \\{z_{m}\\}) \\cup \\{z_{m}\\})$\n\tand \\eqref{6.5.2013.2} that\n\t\\begin{align*}\n\t\t\\mu \\left( Z_{i}^{m} \\setminus D_{i,E}^{m} \\right) \n\t\t& < \\left( C^{(m+1)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{(\\#E+1)n}} \\right)^{-1} \n\t\t\t\\mathcal{R}((E \\setminus \\{z_{m}\\}) \\cup \\{z_{m}\\})\n\t\t= \\frac{C^{(m)}}{C^{(m+1)}} t^{n} \n\t\\end{align*}\n\twhich implies\n\t\\begin{align*}\n\t\t\\mu(Z_i^m \\setminus Z_i^{m+1}) \n\t\t& \\le \\sum_{\\genfrac{}{}{0cm}{}{E \\subset \\{z_0,\\dots,z_m\\}}{z_m \\in E}} \\mu\\left( Z_i^m \\setminus D_{i,E}^{m} \\right)\n\t\t< \\frac{1}{2^{m+2}C_{2} }t^{n}.\n\t\\end{align*}\n\tNow let $u \\in \\{0,\\dots,m+1\\}$ and $E \\subset \\{z_0,\\dots,z_{u-1}\\}$ \n\tand $z \\in Z_{r}^{u}$ where $r \\in \\{u,\\dots,n+1\\}$.\n\tWe have to show that \\eqref{23.08.12.1} is valid.\n\tDue to the induction hypothesis and $z \\in Z_{r}^{m+1} \\subset Z_{r}^{v}$ for all $v \\in \\{0,\\dots,m+1\\}$,\n\twe only have to consider the case $u=m+1$ and $z_{m} \\in E$. Then the inequality follows from \\eqref{2.5;3.1}. \n\t\\hfill\n\tEnd of induction.\n\n\tNow we construct $z_{n+1}$.\t\n\tWe set $P_{n+1}:=\\textnormal{aff}(z_{0},\\dots,z_{n})$, $\\hat C^{(n+1)} := \\tilde C \\ C^{(n)} 2^{n+3} C_{2}$,\n\twhere $\\tilde C$ is the constant from III,\n\tand define\n\t\\begin{eqnarray} \\label{2.5;6}\n\t\t\\hat Z_{n+1}^{n+1} := \\left\\{ z \\in Z_{n+1}^{n+1} \\Big| \n\t\t\\left( \\frac{d(z,P_{n+1})}{t} \\right)^{p} \\le \n\t\t\\hat C^{(n+1)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{n}} \\right\\}.\n\t\\end{eqnarray}\n\tNext we show $\\mu \\left( \\hat Z_{n+1}^{n+1} \\right) \\ge \\frac{t^{n}}{2^{n+3}C_{2} } > 0 $.\n\tLet $u \\in Z_{n+1}^{n+1} \\setminus \\hat Z_{n+1}^{n+1} \\subset B_{n+1} \\subset B(x,(k+k_{0})t)$. \n\tWith III applied on $w=u$ and $j=n+1$, we get\n\t\\begin{align}\\label{10.2.11.11}\n\t\t\\left(\\frac{d(u,P_{n+1})}{t}\\right)^{p} \n\t\t\t\\le \\tilde C t^{n(n+1)} \\mathcal{K}^{p}(z_0,\\dots,z_{n},u).\n\t\\end{align}\n\tNow we get with \\eqref{2.5;6}, Chebyshev's inequality and \\eqref{10.2.11.11} that\n\t\\begin{align*}\n\t\t\\mu\\left( Z_{n+1}^{n+1} \\setminus \\hat Z_{n+1}^{n+1} \\right) \n\t\t& \\le \\left( \\hat C^{(n+1)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{n}} \\right) ^{-1} \n\t\t\t\\tilde C t^{n(n+1)}\n\t\t\t\\int_{Z_{n+1}^{n+1} \\setminus \\hat Z_{n+1}^{n+1}} \\mathcal{K}^{p}(z_{0},\\dots,z_n,u) \\mathrm{d} \\mu (u).\n\t\\end{align*}\n\tBy using I. we see that the integral on the RHS is equal to $\\mathcal{R}(\\{z_{0},\\dots,z_{n-1}\\} \\cup \\{z_{n}\\})$.\n\tHence with \\eqref{28.08.12.1} and \\eqref{23.08.12.1} we obtain\n\t\\[ \\mu(\\hat Z_{n+1}^{n+1}) \\ge \\mu(Z_{n+1}^{n+1}) - \\mu(Z_{n+1}^{n+1} \\setminus \\hat Z_{n+1}^{n+1}) > 0,\\]\n\tand we are able to choose $ z_{n+1} \\in \\hat Z_{n+1}^{n+1} \\subset Z_{n+1}^{n+1}$.\n\tLet $l\\in \\{0,\\dots, n+1\\}$ and $E=\\{z_{0},\\dots,z_{n+1}\\} \\setminus \\{z_{l}\\}$.\n\tSet $z:=z_{n}$ if $l=n+1$ or $z:=z_{n+1}$ otherwise. Now set $E^{'}:=E \\setminus \\{z\\}$ and use \\eqref{23.08.12.1}\n\tto obtain \n\t$\\mathcal{R}(E)=\\mathcal{R}(E^{'}\\cup \\{z\\}) \\le C^{(n+1)} \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{(n+1)n}}$\n\t\n\tAll in all, there exists some constant $C=C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda)$ such that \n\t\\begin{align*}\n\t\t\\int \\Eins_{\\{ (z_0,\\dots,\\hat z_l, \\dots,z_{n+1},w) \\in \\mathcal O _{k_{1}}(x,t)\\}} \n\t\t\t\\mathcal{K}^{p}(z_0,\\dots,\\hat z_l, \\dots,z_{n+1},w) \\mathrm{d} \\mu(w)\n\t\t& = \\mathcal{R}(E)\n\t\t\\le C \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{(n+1)n}}\n\t\\end{align*}\n\tfor all $l \\in \\{0,\\dots,n+1 \\}$. This ends the proof of IV.\n\t\\end{proof}\n\n\tWith IV, there exist some $z_{i} \\in F \\cap B_{i} \\cap B(x,t)$, $i \\in \\{0,\\dots,n+1\\}$\n\tfulfilling \\eqref{24.08.12.1} and \\eqref{24.08.12.2}.\n\tLet $ w \\in \\left(F \\cap B\\left(x,(k + k_{0})t \\right) \\right) \\setminus \\bigcup_{j=0}^{n} 2 B_{j} $.\n\tHence we get with III ($P_{n+1}=\\textnormal{aff}(z_0,\\dots,z_n)$), I and \\eqref{24.08.12.1}\n\t\\begin{align}\\label{2.5;13}\n\t\t\\int_{B(x,(k+k_{0})t) \\setminus \\bigcup_{j=0}^{n}2B_j} \n\t\t\t\\left( \\frac{d(w,P_{n+1})}{t} \\right)^{p} \\mathrm{d} \\mu(w) \n\t\t& < C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda) \\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t). \n\t\\end{align}\n\n\tNow we prove this estimate on the ball $2B_{j}$, where $j \\in \\{0,\\dots,n\\}$.\n\tWe define the plain $P_{j}:=\\textnormal{aff}(\\{z_0,\\dots,z_{n+1} \\} \\setminus \\{z_j\\})$ and get analogously with III, I\n\tand \\eqref{24.08.12.1}\n\t\\begin{align}\\label{2.5;8}\n\t\t\\int_{2B_{j}} \\left( \\frac{d(w,P_{j})}{t} \\right)^{p} \\mathrm{d} \\mu(w) \n\t\t& < C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda) \\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t). \n\t\\end{align}\n\tNow we have an estimate on the ball $2B_{j}$ but with plane $P_{j}$ instead of $P_{n+1}$. \n\tIf $z_{n+1} \\in P_{n+1}$, we have $P_{n+1}=P_{j}$ for all $j \\in \\{0,\\dots,n+1\\}$\n\tand hence we get estimate \\eqref{2.5;8} for $P_{n+1}$.\n\tFrom now on, we assume that $z_{n+1} \\notin P_{n+1}$. Let $w \\in 2B_{j}$,\n\tset $w':=\\pi_{P_j}(w)$, $w^{''}:=\\pi_{P_{n+1}}(w')$\n\tand deduce by inserting the point $w'$ with triangle inequality\n\t\\begin{align}\\label{2.5;9}\n\t\td(w,P_{n+1})^{p} & \\le d(w,w^{''})^{p} \n\t\t\\le 2^{p-1} \\left (d(w,P_{j})^{p} + d(w',P_{n+1})^{p} \\right). \n\t\\end{align}\n\tIf $d(w',P_{n+1}) >0$, i.e., $w' \\notin P_{n+1}$, we gain\n\twith Lemma \\ref{20.2.2012.4} ($P_{1}=P_{j}$, $P_{2}=P_{n+1}$, $a_{1}=w'$, $a_{2}=z_{n+1}$) \n\twhere $P_{j,n+1}:=P_j \\cap P_{n+1}$\n\t\\begin{align}\\label{10.2.11.3}\n\t\td(w',P_{n+1}) \n\t\t\t& = d(z_{n+1},P_{n+1})\\frac{d(w',P_{j,n+1})}{d(z_{n+1},P_{j,n+1})}. \n\t\\end{align}\n\tWith $l \\in \\{0,\\dots,n \\}$, $l\\neq j$ ($k_1$ is defined on page \\pageref{19.10.12.10}), we get \n\t\\begin{align*}\n\t\td(w',P_{j,n+1}) & \\le d\\bigl(w,P_{j,n+1}\\bigr) \\le d(w,x)+d(x,x_l)+d(x_l,z_l) \\le k_{1}t.\n\t\\end{align*}\n\tWith II. we get that\n\t$\\face{j}(\\Delta(z_{0},\\dots,z_{n +1}))$ is an $(n,(9n-1) \\frac{t}{C_1})$-simplex and we obtain\n\t\\begin{align}\\label{2.5;10}\n\t\t\\left( \\frac{d(w',P_{n+1})}{t} \\right)^{p} \n\t\t& \\stackrel{\\eqref{10.2.11.3}}{\\le} \\left( \\frac{d(z_{n+1},P_{n+1})}{t} \\frac{k_{1}tC_{1}}{(9n-1) t} \\right)^{p}\n\t\t\\stackrel{\\eqref{24.08.12.2}}{\\le} C \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{n}} \n\t\\end{align}\n\twhere $C=C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda)$.\n\tIf $d(w',P_{n+1})=0$, this inequality is trivially true.\n\n\tFinally, applying \\eqref{2.5;8}, \\eqref{2.5;8}, \\eqref{2.5;10} and\n\t$\\mu(2B_{j}) \\stackrel{\\text{(B)}}{\\le} C_0 (\\operatorname{diam}(2B_{j}))^n \\le C_0 \\left( \\frac{4t}{C_1} \\right)^n $\n\t((B) from page \\pageref{GrundeigenschaftenBetaBeta}), we obtain\n\t\\begin{align*}\n\t\t\\int_{2B_{j}} \\left( \\frac{d(w,P_{n+1})}{t} \\right)^{p} \\mathrm{d} \\mu (w)\n\t\t& \\le C\\left({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda \\right) \\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t).\n\t\\end{align*}\n\tGiven that $B(y,kt) \\subset B(x,(k+k_{0})t)$, it follows with \\eqref{2.5;13} that\n\t\\begin{align*}\n\t\\beta_{p;k}(y,t)^{p} \n\t& \\le \\frac{1}{t^{n}} \\int_{B(x,(k+k_{0})t)} \\left( \\frac{d(w,P_{n+1})}{t} \\right)^{p} \\mathrm{d} \\mu (w)\n\t \\le C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda) \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{1}}(x,t)}{t^{n}}.\n\t\\end{align*}\n\tTo obtain the main result of this theorem, the only thing left to show is\n\t$ \\mathcal{O}_{k_{1}}(x,t) \\subset \\mathcal{O}_{k_{1}+k_{0}}(y,t)$\n\tLet $(z_{0},\\dots,z_{n+1}) \\in \\mathcal{O}_{k_{1}}(x,t)$. It follows that\n\t$z_{0},\\dots,z_{n+1} \\in B(x,k_{1}t) \\subset B(y,(k_{0}+k_{1})t)$ and\n\t$d(z_{i},z_{j}) \\ge \\frac{t}{k_{1}} \\ge \\frac{t}{k_{1}+k_{0}}$ with $i\\neq j$ and $ i,j=0,\\dots,n$.\n\tThus $(z_{0},\\dots,z_{n+1}) \\in \\mathcal{O}_ {k_{1}+k_{0}}(y,t)$.\n\\end{proof}\n\n\\begin{thm}\\label{13.11.2014.4}\n\tLet $ 0 < \\lambda < 2^{n}$, $k > 2$, $k_{0} \\ge 1$ and $\\mathcal{K}^{p}$ be some $\\mu$-proper{} symmetric integrand \n\t(see Definition \\ref{muproper}). There exists a constant\n\t$C=C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda) $ such that\n\t\\[\\int \\int_{0}^{\\infty} \\beta_{p;k}(x,t)^{p}\\Eins_{\\left\\{ \\tilde{\\delta}_{k_{0}}(B(x,t)) \\ge \\lambda \\right\\}} \\frac{\\mathrm{d} t}{t} \n\t\t \\mathrm{d} \\mu(x) \\le C\\mathcal{M}_{\\mathcal{K}^{p}}(\\mu),\\]\n\twhere $\\tilde{\\delta}_{k_{0}}(B(x,t)):= \\sup_{y \\in B(x,k_{0}t)}\\delta( B(y,t))$.\n\\end{thm}\n\\begin{proof}\n\tAt first, we prove some intermediate results.\\\\\n\tI. \\, \tLet $x \\in F$, $t>0$ and $ \\tilde{\\delta}_{k_{0}}(B(x,t)) \\ge \\lambda$. \n\t\tThere exists some $z \\in B(x,k_{0}t)$ with $\\delta(B(z,t)) \\ge \\frac{\\lambda}{2}$. Now with \n\t\tTheorem \\ref{lem2.5} there exist some constants $k_{1}$ and $C$\n\t\tso that with $k_{2}:=k_{1}+k_{0}$, we obtain\n\t\t$\\beta_{p;k}(x,t)^{p} \\le C \\frac{\\mathcal{M}_{\\mathcal{K}^{p};k_{2}}(x,t)}{t^{n}}$.\\\\\n\tII.\\,\tLet $(x,t) \\in \\mathcal{D}_\\kappa(u_0,\\dots,u_{n+1}):=\\{(y,s)\\in F \\times (0,\\infty)| (u_0,\\dots,u_{n+1})\n\t\t\t\\in \\mathcal{O}_\\kappa(y,s)\\}$ where $u_0,\\dots,u_{n+1} \\in F$.\n\t\tWe have $(u_0,\\dots,u_{n+1}) \\in \\mathcal{O}_\\kappa(x,t)$ and so \n\t\t$\\frac{d(u_0,u_1)}{2\\kappa} \\le t \\le \\kappa d(u_0,u_1)$ as well as $x \\in B(u_0,\\kappa t)$.\\\\\n\tIII.\\,\tWith Fubini's theorem \\cite[1.4, Thm. 1]{Evans} and condition (B) from page \n\t\t\\pageref{GrundeigenschaftenBetaBeta} we get\n\t\t\\begin{align*}\n\t\t\t\\int_{F} \\int_{0}^{\\infty} \\chi_{\\mathcal{D}_{k_2}(u_0, \\dots,u_{n+1})}(x,t) \\frac{1}{t^{n}} \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mu(x)\n\t\t\t& \\stackrel{\\hidewidth\\text{II}\\hidewidth}{\\le} \\int_{\\frac{d(u_0,u_1)}{2k_{2}}}^{k_{2}d(u_0,u_1)} \\frac{1}{t^{n}} \n\t\t\t\t\\int_{B(u_0,k_{2}t)} 1 \\ \\mathrm{d} \\mu(x) \\frac{\\mathrm{d} t}{t}\n\t\t\t\\stackrel{\\hidewidth\\text{(B)} \\hidewidth}{=} C.\n\t\t\\end{align*}\n\tNow we deduce with Fubini's theorem \\cite[1.4, \\mbox{Thm. 1}]{Evans} \n\t\\begin{align*}\n\t\t& \\ \\ \\ \\int_{F} \\int_{0}^{\\infty} \\beta_{p;k}(x,t)^{p}\\Eins_{\\{ \\tilde{\\delta}_{k_{0}}(B(x,t)) \\ge \\lambda \\}} \n\t\t\t\\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mu(x)\\\\\n\t\t& \\stackrel{\\text{I}}{\\le} C \\int_{F} \\int_{0}^{\\infty} \\idotsint_{\\mathcal{O}_{k_2}(x,t)}\n\t\t\t \\frac{\\mathcal{K}^{p}(u_0,\\dots,u_{n+1})}{t^{n}} \\mathrm{d} \\mu(u_0) \\dots \\mathrm{d} \\mu(u_{n+1}) \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mu(x)\n\t\t \\stackrel{\\text{III}}{\\le} C \\mathcal{M}_{\\mathcal{K}^{p}}(\\mu).\n\t\\end{align*}\n\\end{proof}\n\n\\begin{kor} \\label{thm2.4}\n\tLet $ 0 < \\lambda < 2^{n}$, $k > 2$, $k_{0} \\ge 1$ and $\\mathcal{K}^{p}$ be some symmetric \n\t$\\mu$-proper{} integrand (see Definition \\ref{muproper}). \n\tThere exists a constant\n\t$C=C({N},n,\\mathcal{K},p,C_0,k,k_0,\\lambda) $ such that\n\t\\[\\int \\int_{0}^{\\infty} \\beta_{1;k}(x,t)^{p}\\Eins_{\\left\\{ \\tilde{\\delta}_{k_{0}}(B(x,t)) \\ge \\lambda \\right\\}} \\frac{\\mathrm{d} t}{t} \n\t\t \\mathrm{d} \\mu(x) \\le C\\mathcal{M}_{\\mathcal{K}^{p}}(\\mu).\\]\n\\end{kor}\n\\begin{proof}\n\tThis is a direct consequence of the previous Theorem and H\\\"older's inequality.\n\\end{proof}\n\n\n\\subsection{\\texorpdfstring{$\\beta$-numbers, approximating planes and angles}{\\ss -numbers, approximating planes and angles}}\nThe following lemma states, that if two balls are close to each other and if \neach part of the support of $\\mu$ contained in those balls is well approximated by\nsome plane, then these planes have a small angle.\n\\begin{lem} \\label{lem2.6}\n\tLet $ x,y \\in F$, $c\\ge1$, $\\xi \\ge 1$ and $ t_{x},t_{y}>0$ with $c^{-1}t_{y} \\le t_{x} \\le c t_{y}$. \n\tFurthermore, let $ k \\ge 4c$ and $ 0 < \\lambda < 2^{n}$ with\n\t$ \\delta(B(x,t_{x})) \\ge \\lambda$, $\\delta(B(y,t_{y})) \\ge \\lambda$ and $d(x,y) \\le \\frac{k}{2c}t_{x}$.\n\tThen there exist some constants $C_{3}=C_{3}({N},n,C_0,\\lambda,\\xi,c) >1$ \n\tand $\\varepsilon_{0}=\\varepsilon_{0}({N},n,C_{0},\\lambda, \\xi ,c) > 0$ so \n\tthat for all $\\varepsilon < \\varepsilon_{0}$ and all planes $P_{1}, P_{2} \\in \\mathcal{P}({N},n)$ with \n\t$\\beta_{1;k}^{P_{1}}(x,t_{x}) \\le \\xi\\varepsilon$ and $\\beta_{1;k}^{P_{2}}(y,t_{y}) \\le \\xi\\varepsilon$\n\twe get:\n\tFor all $w \\in P_{1}$, we have $d(w,P_{2}) \\le C_{3}\\varepsilon(t_{x} + d(w,x))$, for all \n\t\t$w \\in P_{2}$, we have $d(w,P_{1}) \\le C_{3} \\varepsilon(t_{x} + d(w,x))$ and we have\n\t$\\varangle(P_{1},P_{2}) \\le C_{3} \\varepsilon$.\n\\end{lem}\n\\begin{proof}\n\tDue to $ \\delta(B(x,t_{x})) \\ge \\lambda$ and Corollary \\ref{04.09.12.1}, there exist \n\tsome constants $C_{1}>3$ and $C_{2}$ depending on ${N},n,C_0,\\lambda$, and some\n\tsimplex $T=\\Delta(x_0,\\dots,x_{n}) \\in F \\cap B(x,t_{x})$ \n\tso that \n\t$T$ is an $(n,10n \\frac{t_x}{C_1})$-simplex and\n\t$\\mu( B(x_i,\\frac{t_x}{C_1}) \\cap B(x,t_{x})) \\ge \\frac{t_x^{n}}{C_{2}}$\n\t for all $i \\in \\{0,\\dots,n\\}$.\n\tFor $B_i:=B(x_i,\\frac{t_x}{C_1})$ and $i \\in \\{0,\\dots,n \\}$, we have \n\t$ \\mu(B_{i}) \\ge \\mu(B_{i} \\cap B(x,t_{x})) \\ge \\frac{t_{x}^{n}}{C_{2}} \\ge \\frac{t_{y}^{n}}{c^{n}C_{2}}$.\n\tSince $ B_{i} \\cap B(x,t_{x}) \\ne \\emptyset$ and $k \\ge 4c \\ge 4$ we obtain\n\t$B_{i} \\subset B(x,kt_{x})$ and $B_{i} \\subset B(y,kt_{y})$.\n\tNow we see for $i \\in \\{0,\\dots,n\\}$ \n\t\\begin{align*}\n\t\\frac{1}{\\mu(B_{i})} \\int_{B_{i}} d(z,P_{1})+d(z,P_{2}) \\mathrm d \\mu(z) \n\t& = C_{2} t_{x} \\beta_{1;k}^{P_{1}}(x,t_{x}) + c^{n} C_{2} t_{y} \\beta_{1;k}^{P_{2}}(y,t_{y})\n\t \\le 2c^{n+1}C_{2} xi t_{x} \\varepsilon .\n\t\\end{align*}\n\tWith Chebyshev's inequality, there exists $z_{i} \\in B_{i}$ \n\tso that\n\t\\begin{align}\\label{lem2.6;d} \n\t\td(z_{i},P_{j}) \\le d(z_{i},P_{1}) + d(z_{i},P_{2}) \\le 2c^{n+1} C_{2} \\xi t_{x} \\varepsilon \n\t\\end{align}\n\tfor $ i\\in \\{0,\\dots,n\\}$ and $j = 1,2$. \n\tWe set $y_{i}:=\\pi_{P_1}(z_i)$ and with \n\t\\[\\varepsilon < \\varepsilon_{0}:=\\frac{1}{2c^{n+1} C_{2} \\xi}\\min {\\textstyle \\left\\{\\frac{1}{C_{1}},\n\t\t\\left(10(10^{n}+1)\\frac{C_1}{6} \\left(2\\frac{C_{1}}{3} \\right)^{n} \\right)^{-1} \\right\\} }\\]\n\twe deduce\n\t\\begin{align*}\n\t\td(y_i,x_i) & \\le d(y_i,z_i) +d(z_i,x_i)\n\t\t \\le d(z_i,P_1) + { \\textstyle \\frac{t_x}{C_1} }\n\t\t \\le 2c^{n+1} C_{2} \\xi \\ t_{x} \\ \\varepsilon + { \\textstyle \\frac{t_x}{C_1} \\le 2\\frac{t_x}{C_1}},\n\t\\end{align*}\n\tso, with Lemma \\ref{17.11.11.2}, $S:=\\Delta(y_0,\\dots,y_n)$ is an $(n,6n\\frac{t_x}{C_1})$-simplex and\n\t$S \\subset B(x,\\frac{2t_{x}}{C_{1}}+t_{x})\\subset B(x,2t_x)$.\n\tFurthermore, with \\eqref{lem2.6;d} we have\n\t$ d(y_i,P_2) \\le d(y_i,z_i) + d(z_i,P_2) \\le 2c^{n+1}C_{2} \\xi t_{x} \\varepsilon$. \n\tNow, with Lemma \\ref{21.11.11.2} ($C=\\frac{C_1}{6n}$, $\\hat C = 2$, $t=t_{x}$,\n\t\t$\\sigma = 2c^{n+1} C_{2} \\xi \\varepsilon$, $m=n$) we obtain\n\t\\[\\varangle(P_1,P_2) \\le 4n (10^n + 1)2\\frac{C_1}{6}\\left(2\\frac{C_1}{3}\\right)^{n} \n\t\t\t2c^{n+1} C_{2} \\xi \\varepsilon \n\t\t= C({N},n,C_0,\\lambda,\\xi,c)\\varepsilon.\\]\n\tMoreover, we have\n\t$d(y_{0},\\pi_{P_{2}}(z_{0})) \\le d(z_{0},P_{1}) + d(z_{0},P_{2}) \n\t\t\\stackrel{\\eqref{lem2.6;d}}{\\le} 2c^{n+1} C_{2} \\xi t_{x} \\varepsilon$,\n\tso finally, with Lemma \\ref{23.03.2012.1} ($\\sigma=C \\varepsilon$, $t=t_{x}$, \n\t$p_1=y_0$. $p_2=\\pi_{P_2}(z_{0})$), we get for $w \\in P_1$ that\n\t$ d(w,P_2) \\le C(d(w,y_0)+t_{x}) \\varepsilon \\le C(d(w,x) + t_{x}) \\varepsilon$\n\tand for $w \\in P_2$ we obtain\n\t$d(w,P_1) \\le C (d(w,\\pi_{P_2}(z_0))+t_{x}) \\le C(d(w,x) + t_{x}) \\varepsilon$, \n\twhere $C=C({N},n,C_0,\\lambda,\\xi,c)$.\n\\end{proof}\n\nThe next lemma describes the distance from a plane to a ball if the plain approximates the support of $\\mu$ contained\nin the ball.\n\\begin{lem} \\label{nachlem2.6}\n\tLet $\\sigma>0$, $x \\in \\mathbb{R}^{{N}}$, $t >0$ and $\\lambda >0$ with $ \\delta (B(x,t)) \\ge \\lambda$. \n\tIf $P \\in \\mathcal{P}({N},n)$ with\n\t$\\beta_{1;k}^{P}(x,t) \\le \\sigma$, there exists some $ y \\in B(x,t) \\cap F$ so that \n\t$d(y,P) \\le \\frac{t}{\\lambda} \\sigma$.\n\tIf additionally $\\sigma \\le \\lambda$, we have $B(x,2t) \\cap P \\neq \\emptyset$.\n\\end{lem}\n\\begin{proof}\n\tWith the requirements, we get $ \\mu(B(x,t)) \\ge t^{n} \\lambda$, and so\n\t\\begin{align*}\n\t\t\\frac{1}{\\mu(B(x,t))} \\int_{B(x,t)} d(z,P) \\mathrm{d} \\mu(z)\n\t\t& \\le \\frac{t}{\\lambda} \\frac{1}{t^{n}}\\int_{B(x,kt)} \\frac{d(z,P)}{t} \\mathrm{d} \\mu(z)\n\t\t = \\frac{t}{\\lambda} \\beta_{1;k}^{P}(x,t)\n\t\t \\le \\frac{t}{\\lambda} \\sigma.\n\t\\end{align*}\n\tWith Chebyshev's inequality, we get some $y \\in B(x,t) \\cap F$ with $ d(y,P) \\le \\frac{t}{\\lambda} \\sigma$.\n\tIf $ \\sigma \\le \\lambda$, it follows that $B(x,2t) \\cap P \\neq \\emptyset$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Proof of the main result}\\label{27.10.2014.2}\nAt the end of this section (page \\pageref{BeweisdesHptsatzes}), we will give a proof of our \nmain result Theorem \\ref{maintheorem} under the assumption that\nthe forthcoming Theorem \\ref{16.10.2013.1} is correct. We start with a few lemmas helpful for this proof.\n\n\n\\subsection{Reduction to a symmetric integrand}\n\\begin{lem} \\label{6.5.2013.1}\n\tLet $\\mathcal{K}^{p}$ be some proper{} integrand (see Definition \\ref{4.10.12.1}). There exists some proper{} integrand\n\t$\\tilde{\\mathcal{K}}^{p}$, which is symmetric in all components and fulfils \n\t$\\mathcal{M}_{\\mathcal{K}^{p}}(E)=\\mathcal{M}_{\\tilde{\\mathcal{K}}^{p}}(E)$ for all Borel sets $E$.\n\\end{lem}\n\\begin{proof}\n\tWe set\n\t$\\tilde{\\mathcal{K}}^{p}(x_{0},\\dots,x_{n+1}) \n\t\t:= \\frac{1}{\\# S_{n+2}} \\sum_{\\phi \\in S_{n+2}} \\mathcal{K}^p(\\phi(x_{0},\\dots,x_{n+1}))$,\n\twhere $S_{n+2}$ is the symmetric group of all permutations of $n+2$ symbols.\n\tDue to\n\t$\\mathcal{K}^{p} \\le \\# S_{n+2} \\ \\tilde{\\mathcal{K}}^{p}$, the integrand\n\t$\\tilde{\\mathcal{K}}^{p}$ fulfils the conditions of a proper{} integrand.\n\tNow Fubini's theorem \\cite[1.4, Thm. 1]{Evans} implies\n\t$\\mathcal{M}_{\\tilde{\\mathcal{K}}^{p}}(E) = \\mathcal{M}_{\\mathcal{K}^{p}}(E)$.\n\\end{proof}\n\n\\subsection{Reduction to finite, compact and more regular sets with small curvature}\n\n\n\\begin{lem} \\label{19.04.2013.1}\n\tLet $E$ be a Borel set with $\\mathcal{M}_{\\mathcal{K}^{p}}(E)< \\infty$, where $\\mathcal{K}^{p}$ is some proper{} integrand. \n\tThen we have $\\mathcal{H}^{n}(E \\cap B) < \\infty$ for every ball $B$.\n\\end{lem}\n\\begin{proof}\n\tLet $B$ be some ball and set $F:=E \\cap B$. We prove the contraposition so we assume that $\\mathcal{H}^{n}(F)= \\infty$.\n\tWith Lemma \\ref{16.04.2013.1}, there exists some constant $C>0$ and some $(n+1,(n+3)C)$-simplex \n\t$T=\\Delta(x_{0},\\dots,x_{n+1}) \\in B$ with $\\mathcal{H}^{n}(B(x_{0},C)\\cap F)=\\infty$ and\n\t$\\mathcal{H}^{n}(B(x_{i},C)\\cap F)> 0$ for all $i \\in \\{1,\\dots,n+1 \\}$. With Lemma \\ref{17.11.11.2},\n\twe conclude that $S=\\Delta(y_{0},\\dots,y_{n+1})$ is an $(n+1,C)$-simplex for all\n\t$y_{i} \\in B(x_{i},C)$, $i\\in \\{0,\\dots,n+1\\}$.\n\tFor $t=C\\sqrt{\\frac{\\operatorname{diam} B}{2C}+1}$ and $\\bar C =\\sqrt{\\frac{\\operatorname{diam} B}{2C}+1}$, \n\twe get $S \\in B(x,t \\bar C)$, where $x$ is the centre of the ball $B$, \t\n\tand $S$ is an $(n+1,\\frac{t}{\\bar C})$-simplex.\n\tHence we are in the right setting for using the second condition of a proper{} integrand.\n\tWe obtain \n\t\\[\\mathcal{M}_{\\mathcal{K}^{p}}(E) \\ge \\int_{B(x_{n+1},C)\\cap F} \\dots \\int_{B(x_{0},C)\\cap F} \\mathcal{K}^{p}(y_{0},\\dots,y_{n+1}) \n\t\t\\mathrm{d} \\mathcal{H}^{n}(y_{0}) \\dots \\mathrm{d} \\mathcal{H}^{n}(y_{n+1}) = \\infty.\\]\n\\end{proof}\n\n\n\\begin{lem} \\label{satz1.1}\n\tIn this lemma, the integrand $\\mathcal{K}$ of $\\mathcal{M}_{\\mathcal{K}^{p}}$ only needs to be an $(\\mathcal{H}^{n})^{n+2}$-integrable \n\tfunction. Let $p>0$, $n < {N}$ and\n\t$E \\subset \\mathbb{R}^{{N}}$ be a Borel set with $0 < \\mathcal{H}^{n}(E) < \\infty$ and \n\t$\\mathcal{M}_{\\mathcal{K}^{p}}(E) < \\infty$. For all $ \\zeta > 0$, there exists some compact $E^{*} \\subset E$ with\n\t\\begin{enumerate}\n\t\\renewcommand{\\labelenumi}{\\textup{(\\roman{enumi})}} \n\t\\item $ \\mathcal{H}^{n}(E^{*}) > \\frac{(\\operatorname{diam} E^{*})^n \\omega_{\\N}}{2^{2n+1}}$,\n\t\\item $ \\forall x \\in E^{*}, \\forall t > 0, \\ \\mathcal{H}^{n}(E^{*} \\cap B(x,t)) \\le 2\\omega_{\\N} t^{n}$,\n\t\\item $\\mathcal{M}_{\\mathcal{K}^{p}}(E^{*}) \\le \\zeta \\ (\\operatorname{diam} E^{*})^n $,\n\t\\end{enumerate}\n\twhere $\\omega_{\\N}=\\mathcal{H}^{n}(B(0,1))$ is the $n$-dimensional volume of the $n$-dimensional unit ball.\n\\end{lem}\n\\begin{proof}\n\tDue to $ 0 < \\mathcal{H}^{n}(E) < \\infty$ and \\cite[2.3, Thm. 2]{Evans}, for $\\mathcal{H}^{n}$-almost all $x \\in E $ we have\n\t\\begin{equation} \\label{evans}\n\t\t\\frac{1}{2^{n}} \\le \\limsup_{t \\rightarrow 0^{+}} \\frac{\\mathcal{H}^{n}(E \\cap B(x,t))}{\\omega_{\\N} t^{n}} \\le 1.\n\t\\end{equation}\n\tFor $ l \\in \\mathbb{N}$, we define the $\\mathcal{H}^{n}$-measurable set\n\t\\begin{align} \\label{21.2.11.1}\n\t\tE_{m} &:= \\left\\{ x \\in E \\ \\Big| \\ \\forall t \\in \\left(0,\\frac{1}{m} \\right), \n\t\t\\mathcal{H}^{n}(E \\cap B(x,t)) \\le 2\\omega_{\\N} t^{n} \\right\\}.\n\t\\end{align}\n\tDue to $E_{l} \\subset E_{l+1}$, \\cite[1.1.1, Thm. 1, (iii)]{Evans} and \\eqref{evans} we get that\n\t\\[\\lim_{l \\rightarrow \\infty} \\mathcal{H}^{n}(E_{l})\n\t\t=\\mathcal{H}^{n}\\left({\\textstyle \\bigcup_{l=1}^{\\infty}} E_{l}\\right)=\\mathcal{H}^{n}(E)\\]\n\tHence there exists some $m \\in \\mathbb{N}$ with $ \\mathcal{H}^{n}(E_{m}) \\ge \\frac{1}{2} \\mathcal{H}^{n}(E)$ and\n\t$\\mathcal{M}_{\\mathcal{K}^{p}}(E_{m}) \\le \\mathcal{M}_{\\mathcal{K}^{p}}(E) < \\infty $.\n\tDefine for $ \\tau > 0$\n\t\\begin{align} \\label{21.2.11.2} \n\t\t\\mathcal{I}(\\tau) := \\int_{A(\\tau)} \\mathcal{K}^{p}(x_0, \\dots, x_{n+1}) \n\t\t\\mathrm{d} \\mathcal{H}^{n}(x_0) \\dots \\mathrm{d} \\mathcal{H}^{n}(x_{n+1}),\n\t\\end{align}\n\twhere $A(\\tau) := \\left\\{ (x_0, \\dots, x_{n+1}) \\in E_{m}^{n+2} \\Big| \n\t\t\td(x_0,x_i) < \\tau \\text{ for all } i \\in \\{ 1,\\dots, n+1 \\} \\right\\}$.\n\tUsing \\eqref{21.2.11.1} we obtain $\\left(\\mathcal{H}^{n}\\right)^{n+2}(A(\\tau)) \\rightarrow 0$ for $\\tau \\rightarrow 0$.\n\tWith $ \\mathcal{M}_{\\mathcal{K}^{p}}(E_{m}) < \\infty $, we conclude \n\t$\\lim_{\\tau \\rightarrow 0} \\mathcal{I}(\\tau) = 0$,\n\tand so we are able to pick some $ 0 < \\tau_{0} \\le \\frac{1}{2m}$ with\n\t\\begin{align} \\label{21.2.11.3}\n\t\t\\mathcal{I}(2\\tau_{0}) \\le \\frac{\\zeta \\mathcal{H}^{n}(E_{m})}{2\\omega_{\\N} \\cdot 2^{n+3}}.\n\t\\end{align}\n\tWe set\n\t\\[ \\mathcal{V} := \\left\\{ B(x,\\tau) \\Big| x \\in E_{m}, 0 < \\tau <\\tau_{0}, \\mathcal{H}^{n}(E_{m} \\cap B(x,\\tau))\n\t\t \\ge \\frac{\\tau^n \\omega_{\\N}}{2^{n+1}} \\right\\}. \\]\n\tSince $0 < \\mathcal{H}^{n}(E_{m}) < \\infty$, we get (\\ref{evans}) with $E_m$ instead of $E$, \\cite[2.3, Thm. 2]{Evans}.\n\tThis implies \n\t$\\inf \\left\\{ \\tau \\big| B(x,\\tau) \\in \\mathcal{V} \\right\\} = 0$ for $\\mathcal{H}^{n}$-almost every $x \\in E_{m}$. \n\tAccording to \\cite[1.3]{Falconer}, $\\mathcal{V}$ is a Vitali class.\n\tFor every countable, disjoint subfamily $\\{B_{i}\\}_{i}$ of $\\mathcal{V}$, we have\n\t$\\sum_{i \\in \\mathbb{N}} (\\operatorname{diam} B_{i})^{n} \\le \\frac{2^{2n+1}}{\\omega_{\\N}} \\mathcal{H}^{n}(E_{m})< \\infty$.\n\tApplying Vitali's Covering Theorem \\cite[1.3, Thm. 1.10]{Falconer}, we get a countable subfamily of\n\t$\\mathcal{V}$ with disjoint balls $B_{i} = B(x_{i},\\tau_{i})$ fulfilling\n\t$\\mathcal{H}^{n}\\left(E_{m} \\setminus \\bigcup_{i \\in \\mathbb{N}} B_{i}\\right) = 0$.\n\tTherefore, using \\eqref{21.2.11.1}, we have \n\t$\\mathcal{H}^{n}(E_{m}) \\stackrel{\\hphantom{\\eqref{21.2.11.1}}}{\\le} \\sum_{i\\in \\mathbb{N}} \\mathcal{H}^{n}(E_{m} \\cap B_{i}) \n\t\t\\le \\sum_{i\\in \\mathbb{N}} 2\\omega_{\\N} \\tau_{i}^n$,\n\tso that\n\t\\begin{equation} \\label{summetaui}\n\t\t\\sum_{i \\in \\mathbb{N}} \\tau_{i}^n \\ge \\frac{\\mathcal{H}^{n}(E_{m})}{2\\omega_{\\N}}.\n\t\\end{equation}\n\tFurthermore, with $ (B_{i} \\cap E_{m})^{n+2} \\subset A(2 \\tau_{0}) \\cap B_{i}^{n+2} $, we obtain\n\t\\begin{align}\n\t\t\\sum_{i \\in \\mathbb{N}} \\mathcal{M}_{\\mathcal{K}^{p}}(B_{i} \\cap E_{m}) \n\t\t& \\stackrel{\\eqref{21.2.11.2}}{\\le} \\mathcal{I}(2\\tau_{0}) \n\t\t\\stackrel{\\eqref{21.2.11.3}}{\\le} \\frac{\\zeta \\mathcal{H}^{n}(E_{m})}{2\\omega_{\\N} \\cdot 2^{n+3}}. \\label{21.2.11.4}\n\t\\end{align}\n\tWe define\n\t\\[ I_{b} := \\left\\{ i \\in \\mathbb{N} \\Big| \\mathcal{M}_{\\mathcal{K}^{p}}(B(x_{i},\\tau_{i}) \\cap E_{m}) \\ge \n\t\t\\zeta {\\textstyle \\frac{ \\tau_{i}^n}{2^{n+2}}} \\right\\}\\]\n\tand so\n\t\\[ \\sum _{i \\in I_{b}} \\mathcal{M}_{\\mathcal{K}^{p}}(B(x_{i},\\tau_{i})\\cap E_{m}) \n\t\t\\ge \\zeta\\frac{ \\sum_{i \\in I_{b}} \\tau_{i}^{n}}{2^{n+2}}.\\]\n\tWe have $ \\sum_{i \\in I_{b}} \\tau_{i}^n \\le \\frac{\\mathcal{H}^{n}(E_{m})}{4\\omega_{\\N}}$\n\tsince assuming the converse would imply\n\t\\begin{align*}\n\t\t\\sum_{i \\in \\mathbb{N}} \\mathcal{M}_{\\mathcal{K}^{p}}(B(x_{i},\\tau_{i}) \\cap E_{m}) \n\t\t& \\stackrel{\\eqref{21.2.11.4}}{<} \\zeta\\frac{ \\sum_{i \\in I_{b}} \\tau_{i}^n}{2^{n+2}} \n\t\t\\stackrel{\\hphantom{\\eqref{21.2.11.4}}}{\\le} \\sum_{i \\in I_{b}} \\mathcal{M}_{\\mathcal{K}^{p}}(B(x_{i},\\tau_{i})\\cap E_{m}).\n\t\\end{align*}\n\tUsing \\eqref{summetaui}, we obtain $ I_b \\neq \\mathbb{N} $.\n\tNow we choose some $i \\in \\mathbb{N} \\setminus I_b$ and the regularity of the Hausdorff measure \n\t\\cite[1.2, Thm. 1.6]{Falconer} implies the existence of some compact set $E^{*} \\subset B(x_{i},\\tau_{i}) \\cap E_{m}$\n\twith\n\t\\begin{enumerate}\n\t\t\\renewcommand{\\labelenumi}{\\textup{(\\roman{enumi})}} \n\t\t\\item $\\mathcal{H}^{n}(E^{*}) > \\frac{1}{2}\\mathcal{H}^{n}(B(x_{i},\\tau_{i})\\cap E_{m})\n\t\t\t\\ge \\frac{\\tau_{i}^{n}\\omega_{\\N}}{2^{n+1}} \\ge \\frac{(\\operatorname{diam} E^{*})^{n} \\omega_{\\N}}{2^{2n+1}}$\n\t\t\\item $ \\forall x \\in E^{*}, \\forall t > 0$, we have \n\t\t\t$\\mathcal{H}^{n}(E^{*} \\cap B(x,t)) \\le \\mathcal{H}^{n}(B(x_{i},\\tau_{i}) \\cap E_{m} \\cap B(x,t)) \\le 2\\omega_{\\N} t^{n}$ \n\t\t\tsince if $t < \\frac{1}{m}$ \\eqref{21.2.11.1} implies $\\mathcal{H}^{n}(E \\cap B(x,t))\\le 2\\omega_{\\N} t^{n}$\n\t\t\tand if $\\tau_{i}<\\frac{1}{m} 2$ such that for every\n\t$C_{0} \\ge 10$, there exists some \n\t$ \\eta=\\eta({N},n,\\mathcal{K},C_{0},k) \\in (0,\\vol2^{-(2n+2)}]$\n\tso that if \n\t$\\mu$ is a Borel measure on $\\mathbb{R}^{{N}}$ with\n\tcompact support $F$ such that $\\mathcal{K}^{2}$ is a symmetric $\\mu$-proper{} integrand (cf. Definition \\ref{muproper})\n\tand $\\mu$ fulfils \n\t\\begin{enumerate}\n\t\t\\renewcommand{\\labelenumi}{\\textup{(\\Alph{enumi})}}\n\t\t\\item $\\mu(B(0,5)) \\ge 1, \\ \\mu(\\mathbb{R}^{{N}} \\setminus B(0,5)) = 0$,\n\t\t\\item $\\mu(B) \\le C_{0} \\left(\\operatorname{diam} B\\right)^{n}$ for every ball $B$,\n\t\t\\item $\\mathcal{M}_{\\mathcal{K}^{2}}(\\mu) \\le \\eta$,\n\t\t\\item $\\beta_{1;k;\\mu}^{P_{0}}(0,5) \\le \\eta$ for some plane $P_{0} \\in \\mathcal{P}({N},n)$\n\t\t\twith $0 \\in P_{0}$,\n\t\\end{enumerate}\n\tthen there exists some Lipschitz function $A: P_{0} \\to P_{0}^{\\perp} \\subset \\mathbb{R}^{{N}}$ so that\n\tthe graph $G(A) \\subset \\mathbb{R}^{{N}}$ fulfils\n\t$ \\mu(G(A)) \\ge {\\textstyle \\frac{99}{100}} \\mu(\\mathbb{R}^{{N}})$.\n\t($P_0^{\\perp}:=\\{x \\in \\mathbb{R}^{{N}}|x\\cdot v=0 \\text{ for all } v \\in P_{0}\\}$ denotes the \n\torthogonal complement of $P_{0}$.)\n\\end{thm}\n\nAt first, we show that, under the assumption that the previous theorem is correct,\nwe can prove Theorem \\ref{maintheorem}. The remaining proof of Theorem \\ref{16.10.2013.1} is then given by \nthe following chapters \\ref{construction}, \\ref{gamma} and \\ref{notanullset}.\nWe will use the notation $sE:=\\{x \\in \\mathbb{R}^{{N}}|s^{-1}x \\in E\\}$ for $s > 0 $ and some set $E \\subset \\mathbb{R}^{{N}}$.\nDistinguish this notation from $sB(x,t)=B(x,st)$, where the centre stays unaffected and only the radius is scaled.\n\n\\begin{proof}[Proof of Theorem \\textup{\\ref{maintheorem}}] \\label{16.10.2013.2} \\label{BeweisdesHptsatzes}\n\tLet $\\mathcal{K}^{2}$ be some proper{} integrand (see Definition \\ref{muproper}), $E \\subset \\mathbb{R}^{{N}}$ \n\tsome Borel set with $\\mathcal{M}_{\\mathcal{K}^{2}}(E) < \\infty$ and let $C_{0}= 2^{2n+2}$.\n\tFurthermore, let $k>2$ and $0< \\eta \\le \\omega_{\\N} 2^{-(2n+2)}$ be the constants given by Theorem \\ref{16.10.2013.1}.\n\tUsing Lemma \\ref{6.5.2013.1}, we can assume that $\\mathcal{K}$ is symmetric.\n\n\tWe start with a countable covering of $\\mathbb{R}^{{N}}$ with balls $B_{i}$ so that \n\t$\\mathbb{R}^{{N}} \\subset \\bigcup_{i \\in \\mathbb{N}} B_{i}$. We will show that for all $i\\in \\mathbb{N}$ \n\tthe sets $E \\cap B_{i}$ are $n$-rectifiable, which\n\timplicates that $E$ is $n$-rectifiable.\n\n\tLet $i \\in \\mathbb{N}$ with $\\mathcal{H}^{n}(E \\cap B_{i})>0$. With Lemma \\ref{19.04.2013.1}, we conclude \n\tthat $\\mathcal{H}^{n}(E \\cap B_{i})< \\infty$.\n\tThen, using \\cite[Thm. 3.3.13]{Federer}, we can decompose \n\t$E \\cap B_{i}=E_{\\text{r}}^{i} \\ \\dot \\cup \\ E_{\\text{u}}^{i}$ into two disjoint subsets,\n\twhere $E_{\\text{r}}^{i}$ is $n$-rectifiable and $E_{\\text{u}}^{i}$ is purely $n$-unrectifiable.\n\t\n\tNow we assume that $E \\cap B_{i}$ is not $n$-rectifiable, so $ \\mathcal{H}^{n}(E_{\\text{u}}^{i}) > 0$. \n\tThe set $E_{\\text{u}}^{i}$ is a Borel set and\n\tfulfils $0 < \\mathcal{H}^{n}(E_{\\text{u}}^{i}) \\le \\mathcal{H}^{{N}}(E \\cap B_{i}) < \\infty$ and \n\t$\\mathcal{M}_{\\mathcal{K}^{2}}(E_{\\text{u}}^{i}) \\le \\mathcal{M}_{\\mathcal{K}^{2}}(E) < \\infty$. \n\tNow we apply Lemma \\ref{satz1.1} with $\\zeta = \\eta \\frac{1}{\\hat C \\tilde C}$ where the constants $\\hat C$ and\n\t$\\tilde C$ are given in this passage\n\tand get some compact set $E^{*}\\subset E_{\\text{u}}^{i}$\n\twhich fulfils condition (i),(ii) and (iii) from Lemma \\ref{satz1.1}.\n\tWe set $a:= (\\operatorname{diam} E^{*})^{-1}$ and $\\tilde \\mu = \\mathcal{H}^{n} \\textsf{ L } aE^{*}$.\n\tLet $\\tilde B$ be a ball with $aE^{*}\\subset \\tilde B$ and $\\operatorname{diam} \\tilde B = 2$.\n\tUsing (i), we get $\\delta_{\\tilde \\mu}(\\tilde B) \\ge \\frac{\\omega_{\\N}}{2^{2n+1}}$. \n\tSo, Theorem \\ref{lem2.5} ($p=2$, $x=y\\mathrel{\\hat=}\\text{centre of } \\tilde B$, $t=1$, \n\t$\\lambda = \\frac{\\omega_{\\N}}{2^{3n+1}}$, $k_{0}=1$) implies \n\t$\\beta_{2;k;\\tilde \\mu}(\\tilde B)^{2}\n\t\t< \\hat C \\mathcal{M}_{\\mathcal{K}^{2}}(\\tilde \\mu) \\le \\eta^{2},$\n\tfor some constant $\\hat C= \\hat C({N},n,\\mathcal{K},C_{0},k) \\ge 1$.\n\tUsing H\\\"older's inequality there exists some $n$-dimensional plane \n\t$\\tilde{P_{0}} \\in \\mathcal{P}({N},n) \\index{Subsets of $\\mathbb{R}^{{N}}$ ! $\\tilde{P_{0}}$}$ with\n\t$ \\beta_{1;k;\\tilde \\mu}^{\\tilde{P_{0}}}(\\tilde B) \\le \\eta$.\n\tNow we define a measure $\\mu$ by $\\mu(\\cdot):= \\frac{2^{2n +1}}{\\omega_{\\N}}\\tilde \\mu (\\ \\cdot \\ + \\pi_{\\tilde P_{0}}(b))$,\n\twhere $b$ is the centre of $\\tilde B$.\n\tThis is also a Borel measure with compact support and Lemma \\ref{nachlem2.6} \n\t($\\sigma = \\eta$, $B(x,t)=\\tilde B$, $\\lambda=\\frac{\\omega_{\\N}}{2^{2n+1}}$) \n\timplies that the support fulfils \n\t$F:=aE^{*} -\\pi_{\\tilde P_{0}}(b) \\subset B(0,2)$.\n\tThis measure fulfils condition (D) from Theorem \\ref{16.10.2013.1} ($P_{0}=\\tilde P_{0}-\\pi_{\\tilde P_{0}}(b)$)\n\tand (i) implies condition (A). To get condition (B) for some arbitrary ball,\n\tcover it by some ball with centre on F, double diameter and apply (ii). Use \n\t$\\mathcal{M}_{\\mathcal{K}^{2}}(\\mu) = \\tilde C(n) a^{n} \\ \\mathcal{M}_{\\mathcal{K}^{2}}(E^{*})$ and (iii) to obtain (C).\n\tFinally we mention that $\\mathcal{K}^{2}$ is $\\mu$-proper, since $\\mu$ is an adapted version of $\\mathcal{H}^{n}$.\n\tHence we can apply Theorem \\ref{16.10.2013.1} and after some scaling and translation we \n\tobtain some Lipschitz function which covers a part of positive Hausdorff measure of $E_{u}^{i}$\n\twhich is in contrast to $E_{u}^{i}$ being purely $n$-unrectifiable. Hence $E \\cap B_{i}$ is $n$-rectifiable.\n\\end{proof}\n\n\n\\setcounter{equation}{0}\n\\section{Construction of the Lipschitz graph} \\label{construction}\n\\subsection{\\texorpdfstring{Partition of the support of the measure $\\mu$}{Partition of the support of the measure u}}\\label{04.02.2014.1}\nNow we start with the proof of Theorem \\ref{16.10.2013.1}.\nLet $\\mathcal{K} : \\left(\\mathbb{R}^{{N}}\\right)^{n+2} \\to [0,\\infty)$ \nand let $C_{0} \\ge 10$ be some fixed constant.\nThere is one step in the proof which only works for integrability exponent \n$p=2$. ($p=2$ is used in Lemma \\ref{25.09.2014.1} so that the results of Theorem \\ref{2.9.2014.1} \nand Theorem \\ref{thm4.1} fit together.)\nSince most of the proof can be given with less constraints to $p$,\nwe start with $p \\in (1,\\infty)$ and restrict to $p=2$ only if needed.\nFurthermore, let $k>2$, \\mbox{$0<\\eta \\le \\vol2^{-(2n+2)}$}, \n$P_{0} \\in \\mathcal{P}({N},n)$ \\index{Subsets of $\\mathbb{R}^{{N}}$ ! $P_{0}$}\nwith $0 \\in P_{0}$ and $\\mu$ be a Borel measure on $\\mathbb{R}^{{N}}$ with\ncompact support $F$ \\index{Subsets of $\\mathbb{R}^{{N}}$ ! $F$} \\label{Grundeigenschaften} \nsuch that $\\mathcal{K}^{p}$ is a symmetric $\\mu$-proper{} integrand (cf. Definition \\ref{muproper}) and \\label{3.12.2013.3}\n\\begin{enumerate}\n\t\\renewcommand{\\labelenumi}{\\textup{(\\Alph{enumi})}}\n\t\\item $\\mu(B(0,5)) \\ge 1, \\ \\mu(\\mathbb{R}^{{N}} \\setminus B(0,5)) = 0$,\n\t\\item $\\mu(B) \\le C_{0} \\left(\\operatorname{diam} B\\right)^{n}$ for every ball $B$,\n\t\\item $\\mathcal{M}_{\\mathcal{K}^{p}}(\\mu) \\le \\eta$,\n\t\\item $\\beta_{1;k;\\mu}^{P_{0}}(0,5) \\le \\eta$.\n\\end{enumerate}\n\nIn this chapter, we will prove that if $k$ is large and $\\eta$ is small enough, \nwe can construct some function $A: P_{0} \\to P_{0}^{\\perp}$ which covers some part of the \nsupport $F$ of $\\mu$.\nFor this purpose, we will\ngive a partition of the support of $\\mu$ in four parts,\n$\\operatorname{supp}(\\mu)=\\mathcal{Z} \\dot{\\cup} F_{1} \\dot{\\cup} F_{2} \\dot{\\cup} F_{3}$, and\nconstruct the function $A$ so that the graph of $A$ covers $\\mathcal{Z}$, i.e., $\\mathcal{Z} \\subset G(A)$.\n\nThe following chapters \\ref{gamma} and \\ref{notanullset} will give a proof of\n$\\mu(F_{1} \\cup F_{2} \\cup F_{3}) \\le {\\textstyle\\frac{1}{100}}$,\nhence with (A) we will obtain $\\mu(G(A)) \\ge {\\textstyle\\frac{99}{100}} \\mu(\\mathbb{R}^{{N}})$,\nwhich is the statement of Theorem \\ref{16.10.2013.1}.\n\nFrom now on, we will only work with the fixed measure $\\mu$, so we can simplify the expressions by\nsetting $\\beta_{1;k}:=\\beta_{1;k;\\mu}$ and $ \\delta(\\cdot):=\\delta_{\\mu}(\\cdot)$.\nFurthermore, we fix the constant \\label{WahlderConstants}\n\\begin{align}\n\t\\delta &:= \\min\\left\\{\\frac{10^{-10}}{600^{n}N_{0}},\\frac{2}{50^{n}}\\right\\}, \n\t\t\\index{Constants ! $\\delta$} \\label{Wahlvondelta}\n\\end{align}\nwhere $N_{0}=N_0({N})$ \\index{Constants ! $N_{0}$} \nis the constant from Besicovitch's Covering Theorem \\cite[1.5.2, Thm. 2]{Evans}.\n\n\\begin{dfn}\\label{12.07.13.1} \\label{Definitionvonh} \nLet $\\alpha, \\varepsilon > 0$.\nWe define the set \\index{$S_{total}$} \n\\[ S_{total}^{\\varepsilon,\\alpha} := \\left\\lbrace (x,t) \\in F \\times (0,50) \\ \\\n\t\\begin{array}{|ll}\n\t (i) & \\delta(B(x,t)) \\ge \\frac{1}{2}\\delta \\\\\n\t (ii) & \\beta_{1;k}(x,t) < 2\\varepsilon \\\\\n\t (iii) & \\exists \\ P_{(x,t)}\\in \\mathcal{P}({N},n) \\index{Subsets of $\\mathbb{R}^{{N}}$ ! $P_{(x,t)}$} \\ \\ \\text{s.t.} \\ \n\t \t\\begin{cases}\n \t\t\t\\beta_{1;k}^{P_{(x,t)}}(x,t) \\le 2\\varepsilon \\\\\n \t\t\t \\ \\ \\text{and} \\\\\n\t\t\t\\varangle(P_{(x,t)},P_{0}) \\le \\alpha\n\t\t\\end{cases} \\\\\n\\end{array} \n\\right\\rbrace. \\]\nHaving in mind that the definition of $S_{total}^{\\varepsilon,\\alpha}$ depends on the choice of $\\varepsilon$\nand $\\alpha$, we will normally skip these and write $S_{total}$ instead.\nIn the same manner, we will handle the following definitions of $H,h$ and $S$.\nFor $x \\in F$ we define \n\\[H(x):=\\Big\\lbrace t \\in (0,50) \\ \\Big| \\ \\exists \\ y \\in F, \\ \\exists \\ \\tau \\text{ with } \\\\ \\frac{t}{4} \n\t \\le \\tau \\le \\frac{t}{3}, \\ d(x,y) < \\frac{\\tau}{3} \\ \\text{and} \\ (y,\\tau) \\notin S_{total} \\Big\\rbrace,\\]\n\\[ h(x) := \\sup (H(x) \\cup \\{0\\}) \\quad \\quad \\text{ and } \\quad \\quad S := \\left\\lbrace (x,t) \\in S_{total} \\ | \\ t \\ge h(x) \\right\\rbrace.\\]\nSometimes, we identify a ball $B=B(x,t)$ with the tuple $(x,t)$ and write to simplify matters $B \\in S$ instead of\n$(x,t) \\in S$. In the same manner we use the notation $\\beta_{1;k}(B)$.\n\\end{dfn}\n\n\\begin{lem} \\label{rem3.1}\n\tLet $\\alpha, \\varepsilon > 0$.\n\tIf $\\eta \\le 2\\varepsilon$, we have that $S_{total}\\neq \\emptyset$ and\n\t\\begin{enumerate}\n\t\\renewcommand{\\labelenumi}{(\\roman{enumi})} \n\t\\item $F \\times [40,50) \\subset \\{(x,t) \\in F \\times (0,50)|t\\ge h(x)\\}=S$,\n\t\\item If $ (x,t) \\in S$ and $t \\le t^{'} < 50$, we have $(x,t^{'}) \\in S$.\n\t\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\t(i)\\, If $x \\in F \\subset B(0,5)$ and $10\\le t < 50$, we have\n\t$F \\subset B(x,t)$. Using (A),(D) and $P_{(x,t)}:=P_0$ we get $(x,t) \\in S_{total}$, which implies that\n\t$F \\times [10,50) \\subset S_{total}$. Now if $x \\in F$ and $t \\in [40,50)$ we deduce for arbitrary \n\t$y\\in F$ and $\\tau \\in [\\frac{t}{4},\\frac{t}{3}]$ that $(y,\\tau)\\in S_{total}$, \n\twhich implies that $H(x) \\subset (0,40)$, $h(x) \\le 40$ and hence the first inclusion.\n\tFor the equality it is enough to prove that the central set is contained in $S$.\n\tLet $ x \\in F $ and $t \\in (0,50)$ with $h(x) \\le t < 50$.\n\tAssume that $(x,t) \\notin S$. Due to $h(x)\\le t $, we obtain $(x,t) \\notin S_{total}$,\n\twhich implies that $t < 10$.\n\tHence with $y=x$ and $\\tau=t$ we get $3t \\in H(x)$.\n\tThis implies $h(x) \\ge 3t > t$ and hence a contradiction to $t \\ge h(x)$. So, we obtain $(x,t) \\in S$. \\\\\n\t(ii)\\, We have $x \\in F$ and $h(x) \\le t \\le t^{'} < 50 $ so with (i) we conclude that $(x,t^{'}) \\in S$ .\n\\end{proof}\n\nRemember that the function $h$ depends on the set $S_{total}$, which depends on \nthe choice of $\\varepsilon$ and $\\alpha$. Hence the sets defined in the following definition \ndepend on $\\alpha$ and $\\varepsilon$ as well.\n\\begin{dfn}[Partition of $F$] \\label{def3.2} \\index{Partition of $F$! $\\mathcal{Z}$}\n\t\\index{Partition of $F$! $F_{1}$} \\index{Partition of $F$! $F_{2}$} \\index{Partition of $F$! $F_{3}$}\n\tLet $\\alpha, \\varepsilon > 0$. We define\n\t\\[ \\mathcal{Z} := \\left\\lbrace x \\in F \\ | \\ h(x) = 0 \\right\\rbrace, \\]\n\t\\[ F_{1} := \\left\\lbrace x \\in F \\setminus \\mathcal{Z} \\ \n\t\t\\begin{array}{|ll}\n\t\t & \\exists y \\in F, \\exists \\tau \\in \\left[\\frac{h(x)}{5},\\frac{h(x)}{2}\\right],\n\t\t \t\\text{ with } d(x,y) \\le \\frac{\\tau}{2} \\\\\n\t\t & \\ \\text{and} \\\\\n\t\t & \\delta(B(y,\\tau)) \\le \\delta \\ \n\t\t\\end{array} \n\t\t\\right\\rbrace, \\]\n\t\\[ F_{2} := \\left\\lbrace x \\in F \\setminus (\\mathcal{Z} \\cup F_{1}) \\ \n\t\t\\begin{array}{|ll}\n\t\t & \\exists y \\in F, \\exists \\tau \\in \\left[\\frac{h(x)}{5},\\frac{h(x)}{2}\\right],\n\t\t \t\\text{ with } d(x,y) \\le \\frac{\\tau}{2} \\\\\n\t\t & \\ \\text{and} \\\\\n\t\t & \\beta_{1;k}(y,\\tau) \\ge \\varepsilon \\ \n\t\t\\end{array} \n\t\t\\right\\rbrace, \\]\n\t\\[ F_{3} := \\left\\lbrace x \\in F \\setminus (\\mathcal{Z} \\cup F_{1} \\cup F_{2}) \\ \n\t\t\\begin{array}{|ll}\n\t\t & \\exists y \\in F, \\exists \\tau \\in \\left[\\frac{h(x)}{5},\\frac{h(x)}{2}\\right],\n\t\t \t\\text{ with } d(x,y) \\le \\frac{\\tau}{2} \\\\\n\t\t & \\text{and for all planes } P \\in \\mathcal{P}({N},n)\\text{ with} \\\\\n\t\t & \\beta_{1;k}^{P}(y,\\tau) \\le \\varepsilon \\text{ we have } \\varangle(P,P_{0}) \\ge \\frac{3}{4}\\alpha \\ \n\t\t\\end{array} \n\t\t\\right\\rbrace. \\]\n\\end{dfn}\nIn this chapter, we prove that $\\mathcal{Z}$ is rectifiable by constructing a function $A$ such that the\ngraph of $A$ will cover $\\mathcal{Z}$. This is done by inverting the orthogonal projection \n$\\pi|_{\\mathcal{Z}} : \\mathcal{Z} \\to P_{0}$.\nAfter that, to complete the proof, it remains to show that $\\mathcal{Z}$ constitutes the major part of $F$.\nRight now, we can prove that $\\mu(F_{2}) \\le 10^{-6}$ (cf. section \\ref{F2issmall}, $F_2$ is small) where\nthe control of the other sets need some more preparations.\n\n\\begin{lem}\n\tLet $\\alpha, \\varepsilon > 0$.\n\tDefinition \\ref{def3.2} gives a partition of $F$, i.e.\n\t$F = \\mathcal{Z} \\ \\dot \\cup \\ F_{1} \\ \\dot \\cup \\ F_{2} \\ \\dot \\cup \\ F_{3}$.\n\\end{lem}\n\\begin{proof}\n\tFrom the definition we see that the sets are disjoint. We show\n\t$F \\setminus \\mathcal{Z} \\subset F_{1} \\cup F_{2} \\cup F_{3}$.\n\tLet $x \\in F \\setminus \\mathcal{Z} $, so we have $h(x) > 0$.\n\tThere exist some sequences $(y_{l})_{l \\in \\mathbb{N}} \\in F^{\\mathbb{N}}$, $(t_{l})_{l \\in \\mathbb{N}}$ and \n\t$(\\tau_{l})_{l \\in \\mathbb{N}}$ so that for all $l \\in \\mathbb{N}$, we have $0 < t_{l} \\le h(x)$, \n\t$ t_{l} \\rightarrow h(x)$, $\\frac{t_{l}}{4} \\le \\tau_{l} \\le \\frac{t_{l}}{3}$, $ d(x,y_{l}) < \\frac{\\tau_{l}}{3}$ \n\tand $ (y_{l},\\tau_{l}) \\notin S_{total}$. Due to\n\t$\\tau_{l} \\le \\frac{t_{l}}{3} \\le \\frac{h(x)}{3} \\le \\frac{50}{3}$, we have for every $l \\in \\mathbb{N}$ either\n\t$\\delta(B(y_{l},\\tau_{l}))= \\frac{\\mu(B(y_{l},\\tau_{l}))}{\\tau_{l}^{n}} < \\frac{1}{2}\\delta$ or\n\t$\\delta(B(y_{l},\\tau_{l})) \n\t\t\t\t\\ge \\frac{1}{2} \\delta \\text{ and } \\beta_{1;k}(y_{l},\\tau_{l}) \\ge 2 \\varepsilon$ or \n\t$\\delta(B(y_{l},\\tau_{l})) \n\t\t\t\t\\ge \\frac{1}{2} \\delta \\text{ and } \\beta_{1;k}(y_{l},\\tau_{l}) < 2 \\varepsilon$,\n\t\t\tand for every plane $P \\in \\mathcal{P}({N},n)$ with \n\t\t\t\\mbox{$\\beta_{1;k}^{P}(y_{l},\\tau_{l}) \\le 2 \\varepsilon$}, we have\n\t\t\t$\\varangle(P, P_{0}) > \\alpha.$\n\n\tChoose $l$ so large that $\\frac{4h(x)}{5} \\le t_{l}$. We obtain\n\t$\\frac{h(x)}{5} \\le \\frac{t_{l}}{4} \\le \\tau_{l} \\le \\frac{t_{l}}{3} \\le \\frac{h(x)}{2}$.\n\tFurthermore, we have $y_{l} \\in F$ and $d(x,y_{l}) \\le \\frac{\\tau_{l}}{3} < \\frac{\\tau_{l}}{2}$.\n\tSince $(y_{l},\\tau_{l})$ fulfils one of this tree cases, it follows\n\t$ x \\in F_{1} \\cup F_{2} \\cup F_{3}$.\n\\end{proof}\n\n\nThe following lemma is for later use (cf. Lemma \\ref{25.09.2014.2} and Lemma \\ref{25.09.2014.1}).\n\\begin{lem} \\label{rem3.3}\n\tLet $\\alpha >0$. There exists some constant \n\t$ \\bar \\varepsilon = \\bar \\varepsilon({N},n,C_{0},\\alpha)$ so that if $\\eta < 2 \\bar \\varepsilon$\n\tand $k \\ge 2000$, \n\tthere holds for all $\\varepsilon \\in [\\frac{\\eta}{2},\\bar \\varepsilon)$:\n\tIf $x \\in F_{3}$ and $ h(x) \\le t \\le \\min\\{100 h(x),49\\}$,\n\twe get $ \\varangle(P_{(x,t)},P_{0}) > \\frac{1}{2} \\alpha$, where $P_{(x,t)}$ is the plane granted \n\tsince $(x,t) \\in S_{total}$ (cf. Definition \\ref{12.07.13.1}).\n\\end{lem}\n\\begin{proof}\n\tLet $\\alpha > 0$ and $k \\ge 400$. We set \n\t$\\bar \\varepsilon:= \\min\\{\\varepsilon_{0},\\varepsilon_{0}',\\alpha(5C_{3})^{-1}\\}$,\n\twhere $\\varepsilon_{0}$, $\\varepsilon_{0}'$, $C_{3}$ and $C_{3}^{'}$ depend only on ${N},n$ and $C_{0}$\n\twill be chosen during this proof. Furthermore, let $\\eta \\le 2 \\varepsilon < 2 \\bar \\varepsilon$.\n\n\tSince $x \\in F_{3}$ and $x \\notin (F_1 \\cap F_2)$, there exists some $y \\in F$, \n\t$\\tau \\in \\left[ \\frac{h(x)}{5},\\frac{h(x)}{2} \\right]$ and $\\bar P \\in \\mathcal{P}({N},n)$ \n\twith $d(x,y) \\le \\frac{\\tau}{2}$, $\\beta_{1;k}^{\\bar P}(y,\\tau) \\le \\varepsilon$ and\n\t$\\varangle(\\bar P,P_{0}) \\ge \\frac{3}{4}\\alpha$.\n\tFurthermore $h(x) \\le t$ implies $(x,t) \\in S \\subset S_{total}$ and hence $\\delta(B(x,t)) \\ge \\frac{1}{2}\\delta$\n\tand $\\beta_{1;k}^{P_{(x,t)}}(x,t) \\le 2\\varepsilon.$\n\tNow with Lemma \\ref{lem2.6} ($c=500$, $\\xi =2$, $t_{x}=t$, $t_{y}=\\tau$, $\\lambda = \\frac{\\delta}{2}$), \n\tthere exist some constants $C_{3}=C_{3}({N},n,C_{0}) > 1$ and \n\t$\\varepsilon_{0}=\\varepsilon_{0}({N},n,C_{0})>0$ so that \n\t$\\varangle(\\bar P,P_{(x,t)}) \\le C_{3} \\varepsilon$.\n\tDue to $\\varangle(\\bar P,P_{0}) \\ge \\frac{3}{4}\\alpha$ and $\\varepsilon < \\frac{\\alpha}{4C_{3}}$ this gives\n\t$\\varangle(P_{(x,t)},P_{0}) > \\frac{1}{2} \\alpha$.\n\\end{proof}\n\n\n\\subsection{The distance to a well approximable ball}\nWe recall that the set $S$ depends on the choice of $\\alpha$ and $\\varepsilon $. Hence the functions $d$ and $D$\ndefined in the next definition depend on $\\alpha$ and $\\varepsilon$ as well.\nWe introduce $\\pi:=\\pi_{P_0}: \\mathbb{R}^{{N}} \\to P_0$ \\index{Functions ! $\\pi$}, the orthogonal \nprojection on $P_0$.\n\n\\begin{dfn}[The functions $d$ and $D$] \\index{Functions ! $d(x)$} \\index{Functions ! $D(x)$}\n\tLet $\\alpha, \\varepsilon >0$.\n\tIf $\\eta \\le 2 \\varepsilon$, we get with Lemma \\ref{rem3.1} (i) that $S \\neq \\emptyset$.\n\tWe define $d : \\mathbb{R}^{{N}} \\rightarrow [0,\\infty)$ and $D : P_{0} \\rightarrow [0,\\infty)$ with\n\t\\[ d(x) := \\inf_{(X,t) \\in S} (d(X,x) + t) \\hspace{20mm} D(y) := \\inf_{x \\in \\pi^{-1}(y)} d(x).\\]\n\\end{dfn}\n\nLet us call a ball $B(X,t)$ with $(X,t) \\in S$ a good ball.\nThen the function $d$ measures the distance from the given point $x$ to the nearest good ball,\nusing the furthermost point in the ball. This implies that a ball $B(x,d(x))$\nalways contains some good ball.\nThe function $D$ does something similar. Consider the projection of all good balls to the plane $P_{0}$.\nThen $D$ measures the distance to the nearest projected good ball in the same sense as above\n(cf. next lemma).\n\n\\begin{lem}\\label{remnachdefD}\n\tLet $\\alpha, \\varepsilon > 0$. If $\\eta \\le 2 \\varepsilon$ and $y \\in P_{0}$ we have\n\t$D(y) = \\inf_{(X,t) \\in S}(d(\\pi(X),y) + t)$.\n\\end{lem}\n\\begin{proof}\n\tDue to $d(X,x) \\ge d(\\pi(X),\\pi(x))$ we have $D(y) \\ge \\inf_{(X,t) \\in S} (d(\\pi(X),y) + t)$.\n\tAssume that $\\lim_{l \\rightarrow \\infty} (d( \\pi(X_{l}),y) + t_{l}) > \\inf_{(X,t) \\in S} (d(\\pi(X),y) + t)$\n\tfor some sequence $(X_{l},t_{l}) \\in S$.\n\tNow there exists some $ l \\in \\mathbb{N}$ so that \n\t\\begin{align*}\n\t\tD(y) > d\\big(\\pi(X_{l}) + X_{l} - \\pi(X_{l}) ,y + X_{l} - \\pi(X_{l})\\big) + t_{l} \n\t\t& \\ge \\inf_{x \\in \\pi^{-1}(y)} d(X_{l} ,x) + t_{l} \\ge D(y)\n\t\\end{align*}\n\twhich is a contradiction.\n\\end{proof}\n\n\\begin{lem}\\label{rem3.7}\n\tThe functions $d$ and $D$ are Lipschitz functions with Lipschitz constant 1.\n\\end{lem}\n\\begin{proof}\n\tLet $x,y \\in \\mathbb{R}^{{N}}$. We get with the triangle inequality $d(x) \\le d(y) + d(x,y)$ and\n\t$d(y) \\le d(x) + d(x,y)$. This implies $|d(x)-d(y)| \\le d(x,y)$. Using the previous lemma,\n\twe can use the same argument for the function $D$.\n\\end{proof}\n\n\\begin{lem} \\label{mengebeschraenkt}\n\tWe have \n\t\t$\\left\\{ x \\in \\mathbb{R}^{{N}} \\big| d(x) < 1 \\right\\} \\subset B(0,6)$\n\tand\n\t\t$d(x) \\le 60$\n\tfor all $x \\in B(0,5)$.\n\\end{lem}\n\\begin{proof}\n\tLet $x \\in \\mathbb{R}^{{N}}$ with $ \\inf_{(X,t) \\in S} (d(X,x) + t) = d(x) < 1$.\n\tHence there exists some $X \\in F \\subset B(0,5)$ with\n\t$ d(0,x) \\le d(0,X) + d(X,x) < 6.$\n\tIf $x \\in B(0,5)$, we have $d(x) \\le 10 + 50$.\n\\end{proof}\n\n\\begin{lem} \\label{rem3.8} \\label{7.2.10;1}\n\tLet $\\alpha, \\varepsilon >0$. If $\\eta \\le 2 \\varepsilon$,\n\twe have $d(x) \\le h(x)$ for all $x \\in F$ and\n\t \\[ \\mathcal{Z} = \\left\\{ x \\in F | d(x)=0 \\right\\}, \\quad\n\t \\pi(\\mathcal{Z})=\\{y \\in P_{0} \\ | \\ D(y)=0\\}.\\]\n\tFurthermore, both sets $\\mathcal{Z}$ and $\\pi(\\mathcal{Z})$ are closed. \n\tWe recall that $\\pi$ denotes the orthogonal projection on the plane $P_{0}$. \n\\end{lem}\n\\begin{proof}\n\tLet $x \\in F$. With Lemma \\ref{rem3.1} (i), we have $(x,h(x)) \\in S$ and hence $d(x) \\le h(x)$. \n\tThis implies $\\mathcal{Z} \\subset \\left\\{ x \\in F | d(x) = 0 \\right\\}$.\n\n\tNow let $x \\in F$ with $h(x) > 0$. We prove $d(x)>0$.\n\tThere exist some sequences $t_{l} \\rightarrow h(x)$ and\n\tsome sequence $(X_{i},s_{i}) \\in S$ with $d(X_{i},x) + s_{i} \\to d(x)$.\n\tIf on the one hand there exists some subsequence with $X_{i} \\to x$ we obtain for another subsequence \n\t$s_{i} \\ge h(X_{i}) \\ge t_{i}>0$ for sufficiently large $i$ and hence $d(x)>0$.\n\tIf on the other hand $d(X_{i},x)$ has an positive lower bond, we conclude\n\t$d(x) \\ge \\lim_{l \\rightarrow \\infty}d(X_{l},x) > 0$.\n\t\n\tNow we prove the second equality.\n\tIf $y \\in \\pi(\\mathcal{Z})$, there exists some $x_{0} \\in \\mathcal{Z}$ with $\\pi(x_{0})=y$ and $d(x_{0})=0$. \n\tNow we get $0 \\le D(y) \\le d(x_{0})=0.$\n\n\tIf $y \\in P_{0}$ with $D(y)=0$, since $d$ is continuous, we get with Lemma \\ref{mengebeschraenkt}\n\tthat there exists some $a \\in \\pi^{-1}(y)$ with $d(a)=0$. This implies $a \\in F$ and hence \n\t$a \\in \\mathcal{Z}$. Thus $y \\in \\pi(\\mathcal{Z})$.\n\n\tAccording to Lemma \\ref{rem3.7}, $d$ and $D$ are continuous and hence these sets are closed.\n\\end{proof}\n\n\\begin{lem} \\label{lem3.9}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. There exists some \n\t$\\bar \\varepsilon = \\bar \\varepsilon ({N},n,C_{0})$ so that if $\\eta < 2 \\bar \\varepsilon$ and $k \\ge 4$\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\bar \\varepsilon)$, there holds:\n\tFor all $x,y \\in F$ we have\n\t\\begin{align*}\n\t\td(x,y) &\\le 6(d(x)+d(y)) + 2d(\\pi(x),\\pi(y)),\\\\\n\t\td(\\pi^\\perp(x),\\pi^\\perp(y)) &\\le 6(d(x)+d(y)) + 2\\alpha d(\\pi(x),\\pi(y)).\n\t\\end{align*}\n\\end{lem}\n\\begin{proof}\n\tLet $0< \\alpha < \\frac{1}{4}$ and $k \\ge 4$. During this proof, there occur several smallness conditions on \n\t$\\varepsilon$. The minimum of those will give us the constant $ \\bar \\varepsilon$. Let \n\t$\\eta \\le 2 \\varepsilon < 2\\bar \\varepsilon$.\n\n\tThe first estimate is an immediate consequence of the second estimate. So we focus on this one.\n\tDue to $F \\subset B(0,5)$ the LHS is always less than 10. Hence we can assume that $d(x) + d(y)<2$.\n\tWe choose some arbitrary $r_{x} \\in (d(x),d(x)+1)\\subset (0,3)$. There exists some $(X,t) \\in S$ with\n\t$ d(x) \\le d(X,x) + t < r_{x} $. According to Lemma \\ref{rem3.1} (ii), \n\tit follows that $(X,r_{x}) \\in S$. Analogously, for all $r_{y} \\in (d(y),d(y)+1)$, \n\twe can choose some $Y \\in F$ with $d(Y,y) < r_{y}$ and $(Y,r_{y}) \\in S$.\n\tNow it is enough to prove $d(\\pi^{\\perp}(x),\\pi^{\\perp}(y)) \\le 6(r_{x}+r_{y}) + 2\\alpha d(\\pi(x),\\pi(y))$\n\tsince $r_{x} \\ge d(x)$ and $r_{y} \\ge d(y)$ were arbitrarily chosen.\n\tWe can assume $d(X,Y) > 2( r_{x}+r_{y})$ since otherwise $d(x,y) \\le d(x,X) + d(X,Y) + d (Y,y)$\n\timmediately implies the desired estimate.\n\n\tWe define $B_{1} := B(X, \\frac{1}{2}d(X,Y))$ and $ B_{2} := B(Y, \\frac{1}{2}d(X,Y))$. \n\tWith Lemma \\ref{rem3.1} (i) we obtain $B_{1}, B_{2} \\in S$.\n\tLet $P_{1} $ and $ P_{2}$ be the associated planes to $B_{1}$ and $B_{2}$\n\t(see Definition \\ref{12.07.13.1}). \n\tWith Lemma \\ref{lem2.6} ($x=X$, $y=Y$, $c=1$, $\\xi = 2$, $t_x=t_y=\\frac{1}{2}d(X,Y)$, $\\lambda = \\frac{1}{2}\\delta$)\n\tthere exist some constants $C_{3}=C_{3}({N},n,C_{0})>1$ and \n\t$\\varepsilon_{0}=\\varepsilon_{0}({N},n,C_{0})>0$ so that if $\\varepsilon < \\varepsilon_{0}$\n\tfor $w \\in P_{1}$, we obtain\n\t\\begin{align} \\label{2.12.09;1}\n\t\t d(w,P_{2}) \\le C_{3}({N},n,C_{0},\\delta) \\varepsilon\\left(\\textstyle{\\frac{1}{2}}d(X,Y)+d(w,X)\\right).\n\t\\end{align}\n\n\tLet $B_{1}^{'} := B(X, \\textstyle{\\frac{1}{2}}\\varepsilon^{\\frac{1}{2n}}d(X,Y)+r_{x})$ and \n\t$B_{2}^{'} := B(Y, \\textstyle{\\frac{1}{2}}\\varepsilon^{\\frac{1}{2n}}d(X,Y)+r_{y})$. \n\tLemma \\ref{rem3.1} (i) implies that these balls are in $S$.\n\tNow we conclude using $\\delta(B_{i}^{'}) \\ge \\frac{\\delta}{2}$, $B_{i}^{'} \\subset kB_{i},$ and \n\t$\\beta_{1;k}^{P_{i}}(B_{i}) \\le 2 \\varepsilon$ for $i \\in \\{1,2\\}$ that\n\t\\begin{align*}\n\t\t\\frac{1}{\\mu(B_{i}^{'})} \\int_{B_{i}^{'}}\\frac{d(X^{'},P_{i})}{d(X,Y)} \\mathrm d\\mu(X^{'})\n\t\t& \\le \\frac{1}{\\delta \\varepsilon^{\\frac{1}{2}}} \\frac{1}{\\left(\\textstyle{\\frac{1}{2}}d(X,Y)\\right)^{n}} \n\t\t\t\\int_{kB_{i}} \\frac{d(X^{'},P_{i})}{\\textstyle{\\frac{1}{2}}d(X,Y)} \\mathrm d\\mu(X^{'}) \n\t\t\\le \\frac{2}{\\delta } \\varepsilon^{\\frac{1}{2}}.\n\t\\end{align*}\n\tWith Chebyshev's inequality, we deduce that there exists some $X^{'} \\in B_{1}^{'}$ and some $Y^{'} \\in B_{2}^{'}$ \n\tso that \n\t$d(X^{'},P_{1}) \\le \\frac{2}{\\delta } \\varepsilon^{\\frac{1}{2}} d(X,Y)$ and \n\t$d(Y^{'},P_{2}) \\le \\frac{2}{\\delta } \\varepsilon^{\\frac{1}{2}} d(X,Y)$.\n\t\n\tNow let $X_{1}^{'}:= \\pi_{P_1}(X^{'}) $ be the orthogonal projection of $X^{'}$ on $P_{1}$, \n\t$Y_{2}^{'}:=\\pi_{P_2}(Y^{'})$ the \n\torthogonal projection of $Y^{'}$ on $P_{2}$, and \n\t$X_{12}^{'}:=\\pi_{P_2}(X_1^{'})$ the orthogonal projection of $X_{1}^{'}$ on $P_{2}$. \n\tIf $\\varepsilon$ is small enough, we have with $\\varrho \\in \\{\\pi,\\pi^{\\perp} \\}$\n\t\\begin{align*}\n\t\td(\\varrho(X),\\varrho(X^{'}))\\le d(X,X^{'}) \n\t\t& \\le \\textstyle{\\frac{1}{2}}\\varepsilon^{\\frac{1}{2n}}d(X,Y) + r_{x}, \\\\\n\t\td(\\varrho(Y),\\varrho(Y^{'}))\\le d(Y,Y^{'}) \n\t\t& \\le \\textstyle{\\frac{1}{2}}\\varepsilon^{\\frac{1}{2n}}d(X,Y) + r_{y}, \\\\ \n\t\td(\\varrho(X^{'}),\\varrho(X_{1}^{'}))\\le d(X^{'},X_{1}^{'}) \n\t\t& = d(X^{'},P_{1}) \\le \\frac{2}{\\delta }\\varepsilon^{\\frac{1}{2}}d(X,Y), \\\\ \n\t\td(\\varrho(Y^{'}),\\varrho(Y_{2}^{'}))\\le d(Y^{'},Y_{2}^{'}) \n\t\t& = d(Y^{'},P_{2}) \\le \\frac{2}{\\delta }\\varepsilon^{\\frac{1}{2}}d(X,Y), \\\\\n\t\td(\\varrho(X_{1}^{'}),\\varrho(X_{12}^{'}))\\le d(X_{1}^{'},X_{12}^{'}) \n\t\t& = d(X_{1}^{'},P_{2}) \\stackrel{\\eqref{2.12.09;1}}{<} 2C_{3}\\varepsilon d(X,Y). \n\t\\end{align*}\n\tAccording to Definition \\ref{12.07.13.1}, we have $\\varangle(P_{2},P_{0}) \\le \\alpha$ and \n\twe get with Lemma \\ref{6.9.11.1} ($X_{12}^{'},Y_{2}^{'} \\in P_{2}$) using $\\alpha \\le \\frac{1}{4}$ \n\t\\begin{align}\\label{07.10.2013.4}\n\t\td(X_{12}^{'},Y_{2}^{'}) \n\t\t&\\le \\frac{1}{1-\\alpha}d(\\pi(X_{12}^{'}),\\pi(Y_{2}^{'})) \\le 2d(\\pi(X_{12}^{'}),\\pi(Y_{2}^{'})), \\\\\n\t\td(\\pi^{\\perp}(X_{12}^{'}),\\pi^{\\perp}(Y_{2}^{'})) \n\t\t& \\le \\frac{\\alpha}{1-\\alpha} d(\\pi(X_{12}^{'}),\\pi(Y_{2}^{'}))\n\t\t\t\\le \\frac{4}{3}\\alpha d(\\pi(X_{12}^{'}),\\pi(Y_{2}^{'})).\\label{13.7.11.1}\n\t\\end{align}\n\tInserting the intermediate points $X^{'}$, $X_{1}^{'}$, $X_{12}^{'}$, $Y_{2}^{'}$, $Y^{'}$\n\tusing triangle inequality twice and using the previous inequalities, there exists some constant $C$ so that\n\t\\begin{align*}\n\t\td(X,Y) \n\t\t& \\le C\\textstyle{\\frac{1}{\\delta}}\\varepsilon^{\\frac{1}{2n}} d(X,Y) + r_{x}+r_{y} + 2d(\\pi(X_{12}^{'}),\\pi(Y_{2}^{'}))\\\\\n\t\t& \\le C\\textstyle\\frac{1}{\\delta}\\varepsilon^{\\frac{1}{2n}} d(X,Y) + 3(r_{x}+r_{y}) + 2d(\\pi(X),\\pi(Y))\n\t\\end{align*}\n\tand hence if $\\varepsilon$ fulfils $C \\frac{1}{\\delta} \\varepsilon^{\\frac{1}{2n}} \\le \\frac{1}{2}$,\n\twe get\n\t\\begin{align} \\label{07.10.2013.3}\n\t\td(X,Y) \\le 6(r_{x}+r_{y}) + 4d(\\pi(X),\\pi(Y)).\n\t\\end{align}\n\tAs for $d(X,Y)$, we estimate $d\\left(\\pi^{\\perp}(X),\\pi^{\\perp}(Y)\\right)$ by repeated use\n\tof the triangle inequality and \\eqref{13.7.11.1}.\n\tWith \\eqref{07.10.2013.3}, we deduce \n\t\\begin{align*} \n\t\t& \\ \\ \\ \\ \\ \\ \\ \\ d\\big(\\pi^{\\perp}(X),\\pi^{\\perp}(Y)\\big) \\\\\n\t\t&\\stackrel{\\hphantom{\\eqref{07.10.2013.3}}}{\\le} C\\textstyle{\\frac{1}{\\delta}}\\varepsilon^{\\frac{1}{2n}} d(X,Y) + 3(r_{x}+r_{y}) + \\frac{4}{3}\\alpha d(\\pi(X),\\pi(Y))\\\\\n\t\t&\\stackrel{\\eqref{07.10.2013.3}}{\\le} C\\textstyle{\\frac{1}{\\delta}}\\varepsilon^{\\frac{1}{2n}} [6(r_{x}+r_{y}) + 4d(\\pi(X),\\pi(Y))] + 3(r_{x}+r_{y}) + \\frac{4}{3}\\alpha d(\\pi(X),\\pi(Y))\\\\\n\t\t&\\stackrel{\\hphantom{\\eqref{07.10.2013.3}}}{\\le} 4(r_{x}+r_{y}) + 2\\alpha d(\\pi(X),\\pi(Y)).\n\t\\end{align*}\n\tThis implies using $d(\\pi^{\\perp}(x),\\pi^{\\perp}(X)) \\le d(x,X) \\le r_{x}$ \n\tand $d(\\pi^{\\perp}(Y),\\pi^{\\perp}(y)) \\le d(Y,y) \\le r_{y}$ that\n\t\\begin{align*}\n\t\td(\\pi^\\perp(x),\\pi^\\perp(y)) & \\le 5(r_{x}+r_{y}) + 2\\alpha d(\\pi(X),\\pi(Y))\n\t\t\\le 6(r_{x}+r_{y}) + 2\\alpha d(\\pi(x),\\pi(y)).\n\t\\end{align*}\n\\end{proof}\n\n\n\\subsection{\\texorpdfstring{A Whitney-type decomposition of $P_{0} \\setminus \\pi(\\mathcal{Z})$}{A Whitney-type decomposition}}\n\nIn this part, \nwe show that $P_{0} \\setminus \\pi(\\mathcal{Z})$ can be decomposed as a union of disjoint cubes $R_{i}$,\nwhere the diameter of $R_{i}$ is proportional to $D(x)$ for all $x \\in R_{i}$. This result is \na variant of the Whitney decomposition for open sets in $\\mathbb{R}^{n}$, cf. \\cite[Appendix J]{FourierAnalysis}.\n\n\\begin{dfn}[Dyadic primitive cells]\n\t1.\\, We set $\\mathcal{D}$ to be the set of all dyadic primitive cells on $P_0$. We recall that \n\tthe plane $P_0$ is an $n$-dimensional linear subspace of $\\mathbb{R}^{{N}}$.\\\\\n\t2.\\, Let $r \\in (0,\\infty)$ and $Q$ be some cube in $\\mathbb{R}^{{N}}$.\n\t\tBy $rQ$, we denote the cube\n\t\twith the same centre and orientation as $Q$ but $r$-times the diameter.\n\\end{dfn}\n\nWe mention that the function $D$ depends on the choice of $\\alpha$ and $\\varepsilon$ because $D$ depends\non the set $S \\subset S_{total}^{\\varepsilon,\\alpha}$. Hence the family of cubes given by the following \nlemma depends on the choice of $\\alpha$ and $\\varepsilon$ as well.\n\n\\begin{lem} \\label{inneredisjunkt} \\label{Riueberdeckung} \\label{rem3.10} \\label{lem3.11} \n\tLet $\\alpha, \\varepsilon >0$. If $\\eta \\le 2 \\varepsilon$, then\n\tthere exists a countable family of cubes $\\{R_{i}\\}_{i \\in I} \\subset \\mathcal{D}$ such that\n\t\\begin{enumerate} \\renewcommand{\\labelenumi}{(\\roman{enumi})} \n\t\\item $10\\operatorname{diam} R_{i} \\le D(x) \\le 50 \\operatorname{diam} R_{i}$ for all $x \\in 10 R_{i}$,\n\t\\item $P_{0} \\setminus \\pi(\\mathcal{Z}) = \\bigcup_{i \\in I} R_{i} = \\bigcup_{i \\in I} 2R_{i}$ \n\t\tand cubes $R_{i}$ have disjoint interior,\n\t\\item for every $ i,j \\in I$ with $10R_{i} \\cap 10R_{j} \\neq \\emptyset$, we have\n\t\t$ \\frac{1}{5} \\operatorname{diam} R_{j} \\le \\operatorname{diam} R_{i} \\le 5 \\operatorname{diam} R_{j},$\n\t\\item for every $ i \\in I$, there are at most $180^{n}$ cells $R_{j}$ with \n\t\t$10R_{i} \\cap 10 R_{j} \\neq \\emptyset$.\n\t\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\tFor $z \\in P_{0}$, $D(z)>0$, we define\n\t$Q_{z} \\in \\mathcal{D}$ \\index{Subsets of $P_{0}$ ! $R_{i}$}\n\tas the largest dyadic primitive cell that contains $z$ and fulfils \n\t$\\operatorname{diam} Q_{z} \\le \\frac{1}{20} \\inf_{u \\in Q_{z}} D(u)$.\n\tFor such a given $z$ the cell $Q_{z}$ exists because\n\tthe function $D$ is continuous and $D(z)>0$. Hence if we choose a small enough dyadic primitive \n\tcell $Q$ that contains $z$, we get $\\operatorname{diam} Q \\le \\frac{1}{20} \\inf_{u \\in Q} D(u)$. Due to the \n\tdyadic structure, there can only be one largest dyadic primitive cell that contains $z$ and \n\tfulfils the upper condition.\n\tWe choose $ R_{i} \\in \\mathcal{D}$ such that\n\t$\\{R_{i}| i \\in I\\} = \\{Q_{z} \\in \\mathcal{D} | z \\in P_{0}, D(z)>0\\}$ and $R_i=R_j$ is equivalent to $i=j$.\\\\\n\t(i)\\, \n\tLet $x \\in 10 R_{i}$ and $u \\in R_{i}$. We get $20 \\operatorname{diam} R_{i} \\le D(u) < D(x) + 10 \\operatorname{diam} R_{i}$,\n\tand hence $10 \\operatorname{diam} R_{i} \\le D(x)$.\n\tLet $J_{i}\\in \\mathcal{D}$ be the smallest cell in $\\mathcal{D}$ with $R_{i} \\subsetneq J_{i}$\n\tand choose $u \\in J_{i}$ so that\n\t$D(u) < 20 \\operatorname{diam} J_{i} = 40 \\operatorname{diam} R_{i}$. This is possible because otherwise $R_{i}$ is not maximal\n\trelating to $\\operatorname{diam} R_{i} \\le \\frac{1}{20} \\inf_{v \\in R_{i}} D(v)$.\n\tWe obtain $D(x) \\le D(u) + d(u,x) < 50 \\operatorname{diam} R_{i}$.\\\\\n\t(ii) \\, \n\tIf the interior of some cells $R_{i}$ and $R_{j}$ were not disjoint, because of the dyadic structure,\n\tone cell would be contained in the other. But then one of those would not be the maximal cell. \n\tHence the $R_{i}$'s have disjoint interior.\n\tFor all $x \\in 2R_{i}$, we obtain using (i) and Lemma \\ref{7.2.10;1} that $x \\notin \\pi(\\mathcal{Z})$. \n\tNow let $x \\in P_{0} \\setminus \\pi(\\mathcal{Z})$. With Lemma \\ref{7.2.10;1}, we get $D(x)>0$.\n\tSo there exists the cube $Q_{x} \\in \\mathcal{D}$ with $x \\in Q_{x}$ and hence \n\t$x \\in \\bigcup_{i \\in I} R_{i}$.\\\\\n\t(iii) \\,\n\tIf $10 R_{i} \\cap 10R_{j} \\neq \\emptyset$ we can apply (i) for some $x \\in 10 R_{i} \\cap 10R_{j}$ \n\tand obtain the assertion.\n\t(iv)\\,\n\tLet $i \\in I$ and $R_{j}$ with $10 R_{i} \\cap 10R_{j} \\neq \\emptyset$. We conclude with (iii) that\n\t$d(R_{i},R_{j}) \\le 30 \\operatorname{diam} R_{i}$ and so $R_{j} \\subset (1+30+5) R_{i} $. \n\tFurthermore, we have $\\operatorname{diam} R_{j} \\ge \\frac{1}{5} \\operatorname{diam} R_{i}$. Since the cells $R_{j}$ are disjoint, \n\tthere exist at most $\\frac{\\mathcal{H}^{n}(36R_{i})}{\\mathcal{H}^{n}(R_{j})} \\le (180)^{n}$\n\tcells $R_{j}$ with $10 R_{i} \\cap 10R_{j} \\neq \\emptyset$.\n\\end{proof}\n\nNow we set \\label{17.1.10;1} $ U_{12} := B(0,12) \\cap P_{0}$ \\index{Subsets of $P_{0}$ ! $U_{12}$}\nand $I_{12} := \\{ i \\in I | R_{i} \\cap U_{12} \\neq \\emptyset \\}$ \\index{Index sets ! $I_{12}$}.\n\n\\begin{lem} \\label{vor3.12}\n\tLet $\\alpha, \\varepsilon >0$. If $\\eta \\le 2 \\varepsilon$, \n\tfor every $ i \\in I_{12}$, there exists some ball $B_{i}=B(X_{i},t_{i}) $ \n\t\\index{Subsets of $\\mathbb{R}^{{N}}$ ! $B_{i}$} \n\twith $(X_{i},t_{i}) \\in S$, \n\t$\\operatorname{diam} R_{i} \\le \\operatorname{diam} B_{i} \\le 200 \\operatorname{diam} R_{i}$ and\n\t$d(\\pi(B_{i}),R_{i}) \\le 100 \\operatorname{diam} R_{i}$.\n\\end{lem}\n\\begin{proof}\n\tLet $ i \\in I_{12}$ and $x \\in R_{i}$. Use Lemma \\ref{remnachdefD}, Lemma \\ref{7.2.10;1} \n\tand Lemma \\ref{rem3.10} (i), (ii) to get some $(X,t) \\in S$ with \n\t$d(\\pi(X),x) + t \\le 2D(x) \\le 100 \\operatorname{diam} R_{i}$.\n\tChoose $B_{i} :=B(X_{i},t_{i}) := B(X,r)$ with $r = \\max \\{ t,\\frac{\\operatorname{diam} R_{i}}{2} \\} \\le 100 \\operatorname{diam} R_{i}$. \n\tNow we have $d(\\pi(B_{i}),R_{i}) \\le 100 \\operatorname{diam} R_{i}$ and\n\t$\\operatorname{diam} R_{i} \\le \\operatorname{diam} B_{i} \\le 200 \\operatorname{diam} R_{i}$.\n\tYou can show that $r < 50$ and hence with Lemma \\ref{rem3.1} (ii), we get $(X,r) \\in S$.\n\\end{proof}\n\n\\subsection{\\texorpdfstring{Construction of the function $A$}{Construction of the function A}} \\label{17.05.2013.1}\n\nWe recall that $\\pi:=\\pi_{P_0}: \\mathbb{R}^{{N}} \\to P_0$ is\nthe orthogonal projection on $P_0$\nand introduce\n$\\pi^{\\perp}:=\\pi_{P_0}^{\\perp}: \\mathbb{R}^{{N}} \\to P_0^{\\perp}$, \\index{Functions ! $\\pi^{\\perp}$}\nthe orthogonal projection on $P_0^{\\perp}$, where\n$P_0^{\\perp}:=\\{x \\in \\mathbb{R}^{{N}}|x\\cdot v=0 \\text{ for all } v \\in P_{0}\\}$ \nis the orthogonal complement of $P_{0}$.\nTo define the function $A$, we want to invert the projection $\\pi|_{\\mathcal{Z}}$ on $\\mathcal{Z}$.\n\n\\begin{lem}\\label{12.11.2013.1}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. There exists some \n\t$\\bar \\varepsilon = \\bar \\varepsilon ({N},n,C_{0})$ so that if $\\eta < 2 \\bar \\varepsilon$\n\tand $k \\ge 4$\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\bar \\varepsilon)$, \n\tthe orthogonal projection\n\t$\\pi|_{\\mathcal{Z}} : \\mathcal{Z} \\rightarrow P_{0}$ is injective.\n\\end{lem}\n\\begin{proof}\n\tThe assertion follows directly from Lemma \\ref{rem3.8} and Lemma \\ref{lem3.9}.\n\\end{proof}\nSince $\\pi|_{\\mathcal{Z}} : \\mathcal{Z} \\rightarrow P_{0}$ is injective, we are able to define the desired Lipschitz function \n$A$ on \\label{DefvonAonpiZ}\n$\\pi(\\mathcal{Z})$ by \\index{Functions ! $A$}\n\\[A(a):= \\displaystyle\\pi^{\\perp}\\left( \\pi|_{ \\mathcal{Z}}^{-1}(a) \\right)\\]\nwhere $a \\in \\pi(\\mathcal{Z})$.\n\n\\begin{lem} \\label{ALipschitz1}\n\tUnder the conditions of the previous lemma,\n\tthe map $A\\big|_{\\pi(\\mathcal{Z})}$ is $2 \\alpha$-Lipschitz.\n\\end{lem}\n\\begin{proof}\n\tDue to Lemma \\ref{12.11.2013.1} for $ a, b \\in \\pi(\\mathcal{Z})$, there exist distinct \n\t$ X,Y \\in \\mathcal{Z}$ with $\\pi(X)=a$ and $ \\pi(Y) = b$.\n\tWe have $A(a) = \\pi^{\\perp}(X)$, $A(b)= \\pi^{\\perp}(Y)$ and Lemma \\ref{rem3.8} implies that $d(X)=d(Y)=0$. \n\tSo, with Lemma \\ref{lem3.9}, we get $d(A(a),A(b)) \\le 2\\alpha d(a,b)$.\n\\end{proof}\n\nNow we have a Lipschitz function $A$ defined on $\\pi(\\mathcal{Z})$. By using \nKirszbraun's theorem \\cite[Thm 2.10.43]{Federer}, we would obtain a Lipschitz extension of $A$ \ndefined on $P_{0}$ with the same Lipschitz constant $2\\alpha$, where the graph of the extension covers $\\mathcal{Z}$.\nBut until now, we do not know that $\\mathcal{Z}$ is a major part of $F$. We cannot even be sure \nthat $\\mathcal{Z}$ is not a null set. So we do not use Kirszbraun's theorem here, but we will \nextend $A$ by an explicit construction. This will help us to show that the other parts of $F$, \nin particular $F_{1}, F_{2}, F_{3}$, are quite small.\n\n\\begin{dfn}\\label{3.12.2013.10}\n\tLet $\\alpha, \\varepsilon > 0$. If $\\eta \\le 2 \\varepsilon$,\n\tfor all $i \\in I_{12}$, we set $P_{i} := P_{(X_{i},t_{i})}$,\n\twhere $P_{(X_{i},t_{i})}$ is the $n$-dimensional plane, which is,\n\tin the sense of Definition \\ref{12.07.13.1}, associated to the ball \n\t$B(X_{i},t_{i})=B_{i}$ given by Lemma \\ref{vor3.12}.\n\\end{dfn}\n\n\\begin{lem} \\label{AiLipschitz}\n\tLet $0< \\alpha \\le \\frac{1}{2}$ and $\\varepsilon >0$.\n\tIf $\\eta \\le 2 \\varepsilon$, then for all $i \\in I_{12}$, there exists \n\tsome affine map $A_{i}: P_{0} \\rightarrow P_{0}^{\\perp}$ \n\twith graph $G(A_i)=P_i$ and $A_{i}$ is $2 \\alpha$-Lipschitz.\n\\end{lem}\n\\begin{proof}\n\tUse $\\varangle(P_{i},P_{0})\\le \\alpha \\le \\frac{1}{2}$ (cf. definition of $S_{total}$) \n\tand apply Corollary \\ref{24.04.2012.1}.\n\\end{proof}\n\n\n\\label{14.12.09;1}\n\nIn the following, we use differentiable functions defined on subsets of $P_{0}$. For the definition of the\nderivative see section \\ref{diff_on_lin_subspace} on page \\pageref{diff_on_lin_subspace}.\n\n\\begin{lem}\\label{15.10.2013.1}\n\tLet $\\alpha, \\varepsilon > 0$. If $\\eta \\le 2 \\varepsilon$, then \n\tthere exists some partition of unity $\\phi_{i} \\in C^{\\infty}(U_{12},\\mathbb{R})$, $i \\in I_{12}$,\n\twith $0 \\le \\phi_{i} \\le 1$ on $U_{12}$, $\\phi_{i} \\equiv 0$ on the exterior of $3R_{i}$ and\n\t$\\sum_{i \\in I_{0}} \\phi_{i}(a)= 1$ for all $a \\in U_{12}$. Furthermore there exists some constant $C=C(n)$ with\n\t$| \\partial^{\\omega} \\phi_{i}(a) | \\le \\frac{C(n)}{ (\\operatorname{diam} R_{i})^{|\\omega|}}$ where $\\omega$ \n\tis some multi-index with $1 \\le |\\omega|\\le 2$.\n\\end{lem}\n\\begin{proof}\n\tFor every $i \\in I_{12}$, we choose some function \n\t$\\tilde{\\phi_{i}} \\in \\mathcal{C}^{\\infty}(P_{0},\\mathbb{R})$ with \\label{Defphii}\n\t$0 \\le \\tilde{\\phi_{i}} \\le 1, \\tilde{\\phi_{i}} \\equiv 1 $ on $2R_{i}$, $\\tilde{\\phi_{i}} \\equiv 0$ \n\ton the exterior of $3R_{i}$, $ | \\partial^{\\omega} \\tilde{\\phi_{i}} | \\le \\frac{C}{ \\operatorname{diam} R_{i}}$ \n\tfor all multi-indices $\\omega$ with $|\\omega|=1$ and \n\t$| \\partial^{\\kappa}\\tilde{\\phi_{i}} | \\le \\frac{C}{(\\operatorname{diam} R_{i}) ^{2}} $ for all multi-indices $\\kappa$ \n\twith $|\\kappa|=2$. Now on $V := \\bigcup_{i \\in I_{12}} 2R_{i}$, we can define the partition of unity\n\t$\\phi_{i}(a) := \\frac{\\tilde{\\phi_{i}}(a)}{\\sum_{j \\in I_{12}} \\tilde{\\phi_{j}}(a)}$.\n\tFor all $a \\in V$, there exists some $i \\in I_{12}$ with $a \\in 2R_{i}$ and hence \n\t$\\sum_{j \\in I_{12}} \\tilde{\\phi}_{j}(a) \\ge 1$. \n\tMoreover, due to Lemma \\ref{lem3.11} (iv), there are only finitely many $j \\in I_{12}$ such that\n\t$\\tilde \\phi_{j}(a) \\neq 0$.\n\tDue to the control we have on the derivatives of $\\tilde{\\phi_{i}}$, we obtain with\n\tLemma \\ref{lem3.11} (iv) the desired estimates of the derivatives of $\\phi_{i}$.\n\\end{proof}\n\n\\begin{dfn}[Definition of $A$ on $U_{12}$]\\index{Functions ! $A$} \\label{04.11.2013.1}\n\tLet $\\alpha, \\varepsilon >0$. If $\\eta \\le 2 \\varepsilon$ and $k \\ge 4$,\n\twe extend the function $A: \\pi(\\mathcal{Z}) \\to P_{0}^{\\perp} \\subset \\mathbb{R}^{{N}}$, \n\t$a \\mapsto \\pi^{\\perp}\\left( \\pi|_{ \\mathcal{Z}}^{-1}(a) \\right)$ (see page \\pageref{DefvonAonpiZ}) to \n\tthe whole set $U_{12}$ by setting for $a \\in U_{12}$ \n\t\\[A(a):= \\begin{cases}\n\t\t\t{\\displaystyle\\pi^{\\perp}\\left( \\pi|_{ \\mathcal{Z}}^{-1}(a) \\right)} & ,a \\in \\pi(\\mathcal{Z}) \\\\ \\vspace{-2mm}\\\\\n\t\t\t{\\displaystyle \\sum_{i \\in I_{12}} \\phi_{i}(a) A_{i}(a)} &, a \\in U_{12} \\cap \\bigcup_{i \\in I_{12}}2R_{i}.\n\t \\end{cases}\\]\n\tWith $\\mathcal{Z} \\subset F \\subset B(0,5)$, we get $\\pi(\\mathcal{Z}) \\subset U_{12}$ and, \n\twith Lemma \\ref{lem3.11} (ii), we obtain \\\\\n\t$ \\bigcup_{i \\in I_{12}}2R_{i} \\cap \\pi(\\mathcal{Z}) = \\emptyset$, hence\n\twe have defined $A $ on the whole set \\\\ $U_{12}=(U_{12} \\cap \\bigcup_{i \\in I_{12}}2R_{i}) \\ \\dot{\\cup} \\ \\pi(\\mathcal{Z})$. \n\t\\label{Aeindeutigdefiniert}\n\\end{dfn}\n\n\\subsection{\\texorpdfstring{$A$ is Lipschitz continuous}{A is Lipschitz continuous}}\nIn this section, we show that $A$ is Lipschitz continuous.\nWe start with some useful estimates.\n\n\\begin{lem} \\label{lem3.12}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. \n\tThere exists some $\\bar k \\ge 4$ and some\n\t$\\bar \\varepsilon = \\bar \\varepsilon ({N},n,C_{0})$ so that if $k \\ge \\bar k$ and\n\t$\\eta < 2 \\bar \\varepsilon$\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\bar \\varepsilon)$, \n\tthere exist some constants $C>1$ and $\\bar C=\\bar C({N},n,C_{0})>1$ \n\tso that for all $i,j \\in I_{12}$ with $i \\neq j$ and \n\t$10R_{i} \\cap 10R_{j} \\neq \\emptyset$, we get\n\t\\begin{enumerate}\n\t\\renewcommand{\\labelenumi}{\\textup{(\\roman{enumi})}} \n\t\t\\item $d(B_{i},B_{j}) \\le C \\operatorname{diam} R_{j}$,\n\t\t\\item $d(A_{i}(q),A_{j}(q)) \\le \\bar C\n\t\t\t\\varepsilon \\operatorname{diam} R_{j}$ for all $ q \\in 100R_{j}$,\n\t\t\\item the Lipschitz constant of the map $(A_{i}-A_{j}): P_{0} \\rightarrow P_{0}^{\\perp}$ fulfils\n\t\t\t$ \\operatorname{Lip}_{A_{i}-A_{j}} \\le \\bar C \\varepsilon $,\n\t\t\\item $d(A(u),A_{j}(u)) \\le \\bar C \\varepsilon \\operatorname{diam} R_{j}$ for all $u \\in 2R_{j} \\cap U_{12}$.\n\t\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. We set \n\t$\\bar \\varepsilon = \\min \\big\\{\\frac{\\delta}{2},\\bar \\varepsilon',\\varepsilon_{0}\\big\\}$, where \n\t$\\delta=\\delta({N},n)$ is defined on page \\pageref{Wahlvondelta}, $\\bar \\varepsilon'$ is the \n\tupper bound for $\\varepsilon$ given by Lemma \\ref{lem3.9} and $\\varepsilon_{0}$ is the constant from\n\tLemma \\ref{lem2.6}.\n\tLet $\\eta < 2 \\bar\\varepsilon$ and choose $\\varepsilon$ such that\n\t$\\eta \\le 2\\varepsilon < 2\\bar\\varepsilon$.\\\\\n\t\t(i)\\, \n\t\tLet $B_{i}= B(X_{i},t_{i})$ and $B_{j}= B(X_{j},t_{j})$.\n\t\tLemma \\ref{lem3.11} and Lemma \\ref{vor3.12} imply $d(\\pi(X_{i}),\\pi(X_{j})) \\le C \\operatorname{diam} R_{j}$,\n\t\tand, using $(X_{l},t_{l}) \\in S$ we have\n\t\t$d(X_{l}) \\le 500 \\operatorname{diam} R_{j}$ for $l \\in \\{ i,j\\}$.\n\t\tNow Lemma \\ref{lem3.9} implies the assertion.\\\\\n\t\t(ii)\\,\n\t\tAt first, we show for $q \\in 100 R_{j}$ that $ d(A_{i}(q) + q, X_{i}) \\le C \\operatorname{diam} R_{j}$.\n\t\tSince $(X_{i},t_{i}) \\in S \\subset S_{total}$, $\\varepsilon \\le \\frac{\\delta}{4}$, \n\t\tand Lemma \\ref{nachlem2.6} \n\t\t($\\sigma= 2\\varepsilon$, $x=X_{i}$, $t=t_{i}$, $\\lambda = \\frac{1}{2} \\delta$, $P=P_{i}$) \n\t\twe get $B(X_{i},2t_{i}) \\cap P_{i} \\neq \\emptyset$. Thus there exists some \n\t\t$a \\in P_{0}$ with $A_{i}(a)+a \\in B(X_{i},2t_{i}) \\cap P_{i}$ and \n\t\t$a \\in \\pi(2B_{i})$. Since $A_{i}$ is $2\\alpha$-Lipschitz and $ \\alpha < \\frac{1}{2}$, using \n\t\tLemma \\ref{lem3.11} and \\ref{vor3.12} we obtain by inserting $A_{i}(a)+a$ with triangle inequality\n\t\t\\begin{align}\\label{13.1.10;1}\n\t\t\td(A_{i}(q)+q,X_{i}) \\le |A_{i}(q)-A_{i}(a)| + d(q,a) + \\operatorname{diam} B_{i} \n\t\t\t\\le C \\operatorname{diam} R_{j}.\n\t\t\\end{align}\n\t\tWith Lemma \\ref{lem3.11} and \\ref{vor3.12}, there exists some constant $C>2$ so that\n\t\t$\\frac{1}{C}t_{j} \\le t_{i} \\le C t_{j}$.\n\t\tMoreover, we have $(X_{i},t_{i}),(X_{j},t_{j}) \\in S\\subset S_{total}$ \n\t\tWith $k \\ge \\bar k :=2C^{2} \\ge 4C$, Lemma \\ref{lem2.6} \n\t\t($x=X_{j}$, $y=X_{i}$, $c=C$, $\\xi =2$, $t_{x}=t_{j}$, $t_{y}=t_{i}$\n\t\t$\\lambda = \\frac{\\delta}{2}$) implies that there exists some $\\varepsilon_{0} >0 $\n\t\tand some constant $C_{3}=C_{3}({N},n,C_{0}) >1$ so that,\n\t\tfor $\\varepsilon < \\bar \\varepsilon \\le \\varepsilon_{0}$ with the already shown (i),\n\t\t\\eqref{13.1.10;1} and Lemma \\ref{vor3.12}, we get\n\t\t\\begin{align}\\label{9.4.2013.1}\n\t\t\td(A_{i}(q)+q,P_{j}) \\le C_{3} \\varepsilon \\left(t_{j}+d(A_{i}(q)+q,X_{j}) \\right)\n\t\t\t\\le C \\varepsilon \\operatorname{diam} R_{j}.\n\t\t\\end{align}\n\t\tFurthermore, there exists some $o \\in P_{0}$ so that $A_{j}(o)+o = \\pi_{P_{j}}(A_{i}(q)+q)$. \n\t\tNow, since $A$ is $2\\alpha$-Lipschitz, we have \n\t\t$d(A_{j}(o)+o,A_{j}(q)+q) \\le 2d(o,q) \\le 2d(A_{i}(q)+q,A_{j}(o)+o)$ and hence\n\t\twith Lemma \\ref{lem3.11} and Lemma \\ref{vor3.12} we obtain for some $C=C({N},n,C_{0})$\n\t\t\\[d(A_{i}(q) + q, A_{j}(q)+ q) \\le d(A_{i}(q)+q , P_{j}) + d(A_{j}(o)+o, A_{j}(q)+q)\n\t\t \\stackrel{\\eqref{9.4.2013.1}}{\\le} C \\varepsilon \\operatorname{diam} R_{j}.\\]\n\t\t(iii)\\, \n\t\tWithout loss of generality, we assume $ \\operatorname{diam} R_{i} \\le \\operatorname{diam} R_{j}$. \n\t\tWe have $B(y,2\\operatorname{diam} R_{i}) \\cap P_{0} \\subset 20 R_{i} \\cap 20 R_{j}$\n\t\tfor some $y \\in 10 R_{i} \\cap 10 R_{j} \\neq \\emptyset$.\n\t\tWe choose arbitrary $a,b \\in B(y,2\\operatorname{diam} R_{i})\\cap P_{0}$ with $d(a,b) \\ge \\operatorname{diam} R_{i}$. \n\t\tNow, with (ii), we get\n\t\t\\[ |(A_{i}-A_{j})(a)-(A_{i}-A_{j})(b) | \\le C \\varepsilon \\operatorname{diam} R_{i} \\le C({N},n,C_{0}) \\varepsilon d(a,b).\\]\n\t\tSince $A_{i}-A_{j}$ is an affine map, this implies $ \\operatorname{Lip}_{A_{i}-A_{j}} \\le C({N},n,C_{0}) \\varepsilon $.\\\\\n\t\t(iv)\\,\n\t\tWe get the estimate using Definition \\ref{04.11.2013.1}, $\\sum_{l \\in I_{12}}\\phi_{l}(u)=1$, \n\t\tLemma \\ref{lem3.11} (iv) and (ii) of the current Lemma.\n\\end{proof}\n\n\n\\begin{lem}\\label{4.5.2012.1}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. \n\tThere exists some $\\bar k \\ge 4$ and some\n\t$\\bar \\varepsilon = \\bar \\varepsilon ({N},n,C_{0},\\alpha) < \\alpha$ so that if $k \\ge \\bar k$ and\n\t$\\eta < 2 \\bar \\varepsilon$\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\bar \\varepsilon)$, \n\tthe function $A$ is Lipschitz continuous on $2R_{j} \\cap U_{12}$ \n\tfor all $j \\in I_{12}$ with Lipschitz constant $3 \\alpha$.\n\\end{lem}\n\\begin{proof}\n\tLet $0 < \\alpha \\le \\frac{1}{4}$. We set \n\t$\\bar \\varepsilon := \\min\\big\\{\\bar \\varepsilon', \\frac{\\alpha}{\\tilde C}\\big\\}$,\n\twhere $\\bar \\varepsilon'$ is the upper bound for $\\varepsilon$ given by Lemma \\ref{lem3.12} and \n\t$\\tilde C({N},n,C_{0})$ is some constant presented at the end of this proof. Let $\\eta < 2 \\bar \\varepsilon$ and choose\n\t$\\varepsilon >0$ such that $\\eta \\le 2 \\varepsilon < 2 \\bar \\varepsilon$.\n\tLet $a,b \\in 2R_{j} \\cap U_{12}$. We obtain \n\t\\[|A(a)-A(b)| \\le \\sum_{i \\in I_{12}} \\phi_{i}(a) |A_{i}(a)-A_{i}(b)| \n\t\t+ \\sum_{i \\in I_{12}} | \\phi_{i}(a) - \\phi_{i}(b) | |A_{i}(b)-A_{j}(b)|. \\]\n\tIf $\\phi_{i}(a) - \\phi_{i}(b) \\neq 0 $, we get \\label{18.12.09;3}\n\t$3 R_{i} \\cap 2 R_{j} \\neq \\emptyset$ and so we can apply Lemma \\ref{lem3.11} (iii), (iv) and \n\tLemma \\ref{lem3.12} (ii). \n\tSince $\\varepsilon < \\bar \\varepsilon \\le \\frac{\\alpha}{\\tilde C}$, \n\twe obtain with Lemma \\ref{AiLipschitz} and Lemma \\ref{15.10.2013.1} that $A$ is $3 \\alpha$ Lipschitz.\n\\end{proof}\n\n\\begin{lem}\\label{29.3.10;1}\n\tUnder the conditions of the previous lemma for\n\t$a,b \\in U_{12} \\setminus \\pi(\\mathcal{Z})$ \n\twith $[a,b] \\subset U_{12} \\setminus \\pi(\\mathcal{Z})$,\n\twe have that $d(A(a),A(b)) \\le 3\\alpha d(a,b)$.\n\\end{lem}\n\\begin{proof}\n\tLemma \\ref{Riueberdeckung} (ii) implies that for all $v \\in [a,b]$, there exists some\n\t$j \\in I_{12}$ with $v \\in R_{j}$ and, with Lemma \\ref{rem3.10} (i), we get $D(v) >0$.\n\tAssume that the set $\\tilde{I}_{12} := \\left\\{i \\in I_{12} | R_{i} \\cap [a,b] \\neq \\emptyset \\right\\}$ \n\tis infinite. The cubes $R_{i}$ have disjoint interior, so there exists some sequence \n\t$(R_{i_{l}})_{l \\in \\mathbb{N}}$, $i_{l} \\in \\tilde{I}_{12}$ with $ \\operatorname{diam} R_{i_{l}} \\rightarrow 0$.\n\tHence there exists some sequence $(v_{l})_{l \\in \\mathbb{N}}$ with $v_{l} \\in R_{i_{l}} \\cap [a,b]$ \n\tand, with Lemma \\ref{rem3.10} (i), we obtain\n\t$D(v_{l}) \\le 50 \\operatorname{diam} R_{i_{l}} \\rightarrow 0$.\n\tLet $\\overline{v} \\in [a,b]$ be an accumulation point of $(v_{l})_{l \\in \\mathbb{N}}$.\n\tSince $D$ is continuous (Lemma \\ref{rem3.7}), we deduce $D(\\overline{v})=0$, which is \n\taccording to Lemma \\ref{rem3.8} equivalent to\n\t$\\overline{v} \\in \\pi(\\mathcal{Z})$. \n\tThis is in contradiction to $[a,b] \\subset P_{0} \\setminus \\pi(\\mathcal{Z})$ and so the set $\\tilde{I}_{12}$ has to be finite.\n\tWith Lemma \\ref{4.5.2012.1} and $[a,b] \\subset \\bigcup_{i \\in \\tilde I_{12}} R_{i}$, we get \n\t$d(A(a),A(b)) \\le 3 \\alpha d(a,b)$.\n\\end{proof}\n\nNow we show that $A$ is Lipschitz continuous on $U_{12}$ with some large Lipschitz constant. \nAfter that, using the continuity of $A$,\nwe are able to prove that $A$ is Lipschitz continuous with Lipschitz constant $3\\alpha$.\n\n\\begin{lem} \\label{AstetigaufU0}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. \n\tThere exists some $\\bar k \\ge 4$ and some\n\t$\\bar \\varepsilon = \\bar \\varepsilon ({N},n,C_{0},\\alpha) < \\alpha$ so that if $k \\ge \\bar k$ and\n\t$\\eta < 2 \\bar \\varepsilon$\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\bar \\varepsilon)$, $A$ is Lipschitz continuous on $U_{12}$.\n\\end{lem}\n\\begin{proof}\n\tLet $0 < \\alpha \\le \\frac{1}{4}$, $k \\ge \\bar k \\ge 4$, where $\\bar k$ is the constant from Lemma\n\t\\ref{4.5.2012.1}, and let $\\bar \\varepsilon = \\bar \\varepsilon({N},n,C_{0},\\alpha) \\le \\frac{\\delta}{4}$\n\tbe so small that we can apply Lemma \\ref{lem3.9}, \\ref{ALipschitz1}, \\ref{lem3.12} and Lemma \\ref{29.3.10;1}.\n\tFurthermore, let $\\varepsilon >0$ such that $\\eta \\le 2 \\varepsilon < 2 \\bar \\varepsilon$.\n\tLet $a, b \\in U_{12}$ with $a \\in \\pi(\\mathcal{Z})$ and $b \\in 2R_{j}$ for some $j\\in I_{12}$.\n\tWe estimate $d(A(a),A(b)) \\le d(A(a)+a,X_{j}) + d(X_{j},A(b)+b)$\n\twhere $X_{j}$ is the centre of the ball $B_{j}=B(X_{j},t_{j})$ (see Lemma \\ref{vor3.12}).\n\n\tAt first, we consider $d(A(a)+a,X_{j})$.\n\tSince $A(a)+a \\in \\mathcal{Z}$, Lemma \\ref{rem3.8} implies $d(A(a)+a) = 0$.\n\tMoreover, with Lemma \\ref{vor3.12} and $(X_{j},t_{j}) \\in S$, we deduce $d(X_{j}) \\le 100 \\operatorname{diam} R_{j} $\n\tand \n\t\\begin{align*}\t\n\t\td(\\pi(A(a)+a) , \\pi(X_{j})) \n\t\t& \\le d(a,b) + d(b,\\pi(X_{j})) \n\t\t\\le d(a,b) + C \\operatorname{diam} R_{j}.\n\t\\end{align*}\n\tUsing those estimates, Lemma \\ref{lem3.9} implies\n\t$d(A(a)+a,X_{j}) \\le 2d(a,b) + C\\operatorname{diam} R_{j}$.\n\n\tNow we consider $d(X_{j},A(b)+b)$.\n\tWe have $(X_{j},t_{j}) \\in S \\subset S_{total}$ and hence, with Lemma \\ref{nachlem2.6} \n\tusing $\\varepsilon < \\bar \\varepsilon \\le \\frac{\\delta}{4}$,\n\tthere exists some\n\t$y \\in B(X_{j},2t_{j})\\cap P_{j}$, where $P_{j}$ is the associated plane to $B_{j}$\n\t(see Definition \\ref{3.12.2013.10}). \n\tSince $\\varangle(P_{j},P_{0}) \\le \\alpha \\le \\frac{1}{4}$, \n\twe deduce with Lemma \\ref{12.7.11.1}, Lemma \\ref{vor3.12} and Lemma \\ref{lem3.12} (iv) that\n\t\\[ d(X_{j},A(b)+b) \\le d(X_{j},y) + d(y,A_{j}(b)+b) + d(A_{j}(b)+b,A(b)+b)\n\t\t\\le C (\\operatorname{diam} R_{j} + d(a,b)).\\]\n\tWith Lemma \\ref{lem3.11}, Lemma \\ref{7.2.10;1} and using that $D$ is $1$-Lipschitz (Lemma \\ref{rem3.7}) we obtain\n\t$\\operatorname{diam} R_{j} \\le D(b)-D(a) \\le d(a,b)$ and hence $d(A(a),A(b)) \\le C d(a,b)$.\n\tDue to Lemma \\ref{ALipschitz1} and Lemma \\ref{29.3.10;1} it remains to handle the case were\n\t$a,b \\notin \\pi(\\mathcal{Z})$ and $[a,b] \\cap \\pi(\\mathcal{Z}) \\neq \\emptyset$.\n\tThis follows immediately from the just proven case and triangle inequality.\n\\end{proof}\n\n\\begin{lem} \\label{29.3.10;2}\n\tUnder the conditions of Lemma \\ref{AstetigaufU0} for some\n\t$a \\in \\pi(\\mathcal{Z})$, $i \\in I_{12}$ and $b \\in 2R_{j}$, we get\n\t$d(A(a),A(b)) \\le 3 \\alpha d(a,b)$.\n\\end{lem}\n\\begin{proof}\n\tWe set $c:= \\inf_{x \\in [a,b]\\cap \\pi(\\mathcal{Z})} d(x,b)$. Due to Lemma \\ref{7.2.10;1}, \n\tthere exists some $v \\in [a,b]\\cap \\pi(\\mathcal{Z})$ with $d(v,b) = c$. \n\tFurthermore, there exists some sequence $(v_{l})_{l} \\subset [v,b]$ \n\twith $v_{l} \\rightarrow v$ where $l \\rightarrow \\infty$.\n\tWith Lemma \\ref{Riueberdeckung}, we deduce $([v,b] \\setminus\\{v\\}) \\subset \\bigcup_{j \\in I_{12}} 2R_{j}$.\n\tFor every $l \\in \\mathbb{N}$ we obtain with Lemma \\ref{29.3.10;1}\n\t$d(A(v),A(b)) \\le d(A(v),A(v_{l})) + 3\\alpha d(v,b)$.\n\tand, since $A$ is continuous (Lemma \\ref{AstetigaufU0}) we conclude with $l \\rightarrow \\infty$ that\n\t$d(A(v),A(b)) \\le 3 \\alpha d(v,b)$.\n\tThe assertion follows since we already know that $A$ is $2\\alpha$-Lipschitz on $\\pi(\\mathcal{Z})$.\n\\end{proof}\n\n\\begin{lem}\n\tUnder the conditions of Lemma \\ref{AstetigaufU0} we have \n\t$d(A(a),A(b)) \\le 3 \\alpha d(a,b)$\n\tfor $a, b \\in \\bigcup_{j \\in I_{12}} 2R_{j} \\cap U_{12}$.\n\\end{lem}\n\\begin{proof}\n\tThis is an immediate consequence of Lemma \\ref{4.5.2012.1}, Lemma \\ref{29.3.10;1} and Lemma \\ref{29.3.10;2}.\n\\end{proof}\n\n\n\\begin{lem}\\label{ALipschitz}\n\tUnder the conditions of Lemma \\ref{AstetigaufU0}, the function \n\t$A$ is Lipschitz continuous on $U_{12}$ with Lipschitz constant $3 \\alpha$.\n\\end{lem}\n\\begin{proof}\nThis follows directly from the previous Lemma and Lemma \\ref{ALipschitz1}.\n\\end{proof}\n\n\nThe following estimate is for later use.\n\\begin{lem} \\label{abschaetzungableitungvonA}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. \n\tThere exists some $\\bar k \\ge 4$ and some\n\t$\\bar \\varepsilon = \\bar \\varepsilon ({N},n,C_{0})$ so that if $k \\ge \\bar k$ and\n\t$\\eta < 2 \\bar \\varepsilon$\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\bar \\varepsilon)$, \n\tthere exists some constant $C=C({N},n,C_{0})$ so that for all $ j \\in I_{12}$, $a \\in 2R_{j}$ and\n\tfor all multi-indices $\\kappa$ with $|\\kappa|=2$ we have\n\t$\\partial^{\\kappa} A(a)| \\le \\frac{C \\varepsilon}{\\operatorname{diam} R_{j}}$.\n\\end{lem}\n\\begin{proof}\n\tChoose $\\bar k$ and $\\bar \\varepsilon$ as in Lemma \\ref{lem3.12}.\n\tLet $\\kappa$ be some multi-index with $|\\kappa|=2$.\n\tFor $i \\in I_{12}$, the function\n\t$A_{i}$ is an affine map and hence \n\tfor some suitable $l_1,l_2 \\in \\{1,\\dots,n\\}$ we have\n\t\\begin{align} \\label{15.10.2013.3}\n\t\t\\partial^{\\kappa} A &= \\partial^{\\kappa} \\Bigl( \\sum_{i \\in I_{12}} \\phi_{i}A_{i} \\Bigr) \n\t\t= \\sum_{i \\in I_{12}} \\left( \\partial^{\\kappa} \\phi_{i} \\right) A_{i} \n\t\t\t+ \\sum_{i \\in I_{12}} \\left(\\partial_{l_1} \\phi_{i} \\partial_{l_2} A_{i} +\n\t\t\t\t\\partial_{l_2} \\phi_{i} \\partial_{l_1} A_{i}\\right).\n\t\\end{align}\n\tLet $j \\in I_{12}$ and $a \\in 2R_{j}$. \n\tLemma \\ref{lem3.11} implies that there exist at most $180^{n}$\n\tcells $R_{i}$ so that $\\partial^{\\kappa} \\phi_{i}(a) \\neq 0$ or $\\partial^{\\omega} \\phi_{i}(a) \\neq 0$,\n\twhere $\\omega$ is a multi-index with $|\\omega|=1$. \n\tSo only finite sums occur in the following estimates.\n\tWe have \n\t$ \\sum_{i \\in I_{12}} \\partial^{\\omega} \\phi_{i}= \\partial^{\\omega} \\sum_{i \\in I_{12}} \\phi_{i} \n\t\t= \\partial^{\\omega} \\ 1 =0$\n\tso that we get\n\t\\begin{align*}\n\t\t |\\partial^{\\kappa} A| \n\t\t \\stackrel{\\eqref{15.10.2013.3}}{\\le} \n\t\t\t\\sum_{i \\in I_{12}} |\\partial^{\\kappa} \\phi_{i}| \\ | A_{i} - A_{j}| \n\t\t+ \\sum_{i \\in I_{12}}| \\partial_{l_1} \\phi_{i} | \\ |\\partial_{l_2} (A_{i} - A_{j})|\n\t\t+ \\sum_{i \\in I_{12}}| \\partial_{l_2} \\phi_{i} | \\ |\\partial_{l_1} (A_{i} - A_{j})|.\n\t\\end{align*}\n\tTo estimate these sums, we only have to consider the case when $a$ is in the support of $\\phi_{i}$\n\tfor some $i\\in I_{12}$. This implies \n\t$3R_{i} \\cap 2R_{j} \\neq \\emptyset$.\n\tNow use Lemma \\ref{lem3.12} (ii), (iii), Lemma \\ref{15.10.2013.1}, and Lemma \\ref{lem3.11} (iii), (iv)\n\tto obtain the assertion.\n\\end{proof}\n\n\n\\setcounter{equation}{0}\n\\section{\\texorpdfstring{$\\gamma$-functions}{y-functions}} \\label{gamma}\n\\newcommand{g}{g}\nIn this chapter, we introduce the $\\gamma$-function of some function $g:P_{0} \\to P_{0}^{\\perp}$. \nThis function measures how well $g$ \ncan be approximated in some ball by some affine function. \nThe main results of this chapter are Theorem \\ref{2.9.2014.1} on page \\pageref{2.9.2014.1} and \nTheorem \\ref{thm4.1} on page \\pageref{thm4.1}.\nWe will use these statements in section \\ref{F3issmall} to prove that $\\mu(F_{3})$ is small.\n\n\n\\begin{dfn}\n\tLet $U \\subset P_{0}$, $ q \\in U$ and $ t > 0$ so that $B(q,t) \\cap P_{0} \\subset U$.\n\tFurthermore, let $\\mathcal{A}=\\mathcal{A}(P_0,P_0^{\\perp})$ be the set of all affine functions \n\t$a:P_{0} \\rightarrow P_{0}^{\\perp}$ and let $g:U \\to P_{0}^{\\perp}$ be some function. \n\tWe define \n\t\\begin{align*}\n\t\t\\gamma_{g}(q,t)&:= \\inf_{a \\in \\mathcal{A}} \\frac{1}{t^{n}} \\int_{B(q,t)\\cap P_{0}} \n\t\t\\frac{d(g(u),a(u))}{t} \\mathrm{d} \\mathcal{H}^{n}(u).\n\t\\end{align*}\n\\end{dfn}\n\n\\begin{lem} \\label{bem4.2}\n\tLet $U \\subset P_{0}$, $q \\in U$ and $t > 0$ so that $B(q,t) \\cap P_{0} \\subset U$. \n\tFurthermore, let \\\\\n\t$g: U \\to P_{0}^{\\perp}$ be a Lipschitz continuous function such that the Lipschitz constant fulfils\n\t$60n(10^{n}+1)\\left(8n \\frac{\\vo{n-1}}{\\omega_{\\N}} \\right)^{n+1} \\le \\operatorname{Lip}_{g}^{-1},$\n\twhere $\\omega_{\\N}$ denotes the $n$-dimensional volume of the $n$-dimensional unit ball.\n\tThen we have\n\t\\[\\gamma_{g}(q,t) \\le 3 \\ \\tilde \\gamma_{g}(q,t):=3 \\inf_{P \\in \\mathcal{P}({N},n)} \\frac{1}{t^{n}} \n\t\t\\int_{B(q,t)\\cap P_{0}} \\frac{d(u+g(u),P)}{t} \\mathrm{d} \\mathcal{H}^{n}(u),\\]\n\twhere $\\mathcal{P}({N},n)$ is the set of all $n$-dimensional affine planes in $\\mathbb{R}^{{N}}$.\n\\end{lem}\n\\begin{proof}\n\tLet $g$ be a Lipschitz continuous function with an appropriate Lipschitz constant.\n\tBy using $a : u \\to g(q) \\in \\mathcal{A}$ as a constant map and by using that g is $1$-Lipschitz,\n\twe deduce $\\gamma_{g}(q,t) \\le \\operatorname{Lip}_{g} \\omega_{\\N}.$\n\tIt follows, since for every $a \\in \\mathcal{A}$ the graph $G(a)$ of $a$ is in $\\mathcal{P}({N},n)$, that\n\t$\\tilde \\gamma_{g}(q,t) \\le \\gamma_{g}(q,t) \\le \\operatorname{Lip}_{g} \\omega_{\\N}$.\n\tLet $0<\\xi < \\operatorname{Lip}_{g} \\omega_{\\N}$ and choose some $P \\in \\mathcal{P}({N},n)$ so that\n\t\\begin{align} \\label{5.12.11.1}\n\t \\frac{1}{t^{n}} \\int_{B(q,t)\\cap P_{0}} \\frac{d(u+g(u),P)}{t} \\mathrm{d} \\mathcal{H}^{n}(u)\\le \\tilde \\gamma_{g}(q,t)+\\xi\n\t\t\\le 2 \\operatorname{Lip}_{g} \\omega_{\\N}. \n\t\\end{align}\n\tWe set $D_{1}:=\\left\\{ v \\in B(q,t) \\cap P_0 | d(v+g(v),P) \\le 4 \\operatorname{Lip}_{g} t \\right\\}$,\n\t$D_{2}:=(B(q,t) \\cap P_{0}) \\setminus D_{1}$ and obtain using Chebyshev's inequality and \\eqref{5.12.11.1}\n\t\\begin{align}\\label{21.11.11.1}\n\t\t\\mathcal{H}^{n}(D_1) & \\ge \\omega_{\\N} t^{n} - \\mathcal{H}^{n}(D_{2}) \\ge \\frac{\\omega_{\\N}}{2}t^{n} \n\t\\end{align}\n\tAssume that every simplex $\\triangle(u_0,\\dots,u_{n}) \\in D_{1}$ is\n\tnot an $(n,H)$-simplex, where $H=\\frac{\\omega_{\\N}}{4\\vo{n-1}}t $.\n\tWith Lemma \\ref{18.11.11.1} ($m=n$, $D=D_{1}$), there exists some plane $\\hat P \\in \\mathcal{P}({N},n-1)$ such that \n\t$D_1 \\subset U_{H}(\\hat P) \\cap B(q,t) \\cap P_0$.\n\tWe get\n\t\\begin{align*}\n\t\t\\mathcal{H}^{n}(D_1)\n\t\t& \\le \\mathcal{H}^{n}(U_{H}(\\hat P) \\cap B(q,t) \\cap P_0)\n\t\t \\le 2H \\vo{n-1}t^{n-1}\n\t\t = \\frac{\\omega_{\\N} }{2} t^{n}.\n\t\\end{align*}\n\tThis is in contradiction to \\eqref{21.11.11.1}, so there exists some $(n,H)$-simplex\n\t$\\triangle(u_0,\\dots,u_{n}) \\in D_{1}$.\n\tWe set $\\hat P_0 := P_0 + g(u_0)$, $y_i:=u_i+g(u_0) \\in \\hat P_0$ for all $i \\in \\{0,\\dots,n\\} $\n\tand $S:=\\Delta(y_0,\\dots,y_{n}) \\subset \\hat P_0 \\cap B(q+g(u_{0}),t)$. \n\tWe recall that $P$ is the plane satisfying \\eqref{5.12.11.1}.\n\tWe obtain for all $i \\in \\{0,\\dots,n\\}$\n\t\\begin{align*}\n\t\td(y_i,P) & \\le d(u_i + g(u_0), u_i + g(u_i)) + d(u_i + g(u_i),P)\n\t\t \\le \\operatorname{Lip}_{g} d(u_0,u_i) +4 \\operatorname{Lip}_{g} t\n\t\t \\le 6 \\operatorname{Lip}_{g} t.\n\t\\end{align*}\n\tWith Lemma \\ref{21.11.11.2}, $C=4\\frac{\\vo{n-1}}{\\omega_{\\N}} > 1$\\footnote{As the volume of the unit\n\tsphere is strictly monotonously decreasing when the dimension $n \\ge 5$ increases, we get \n\t$\\frac{\\vo{n-1}}{\\omega_{\\N}} > 1$ for all $n \\ge 6$. With the factor $4$ we have that $4\\frac{\\vo{n-1}}{\\omega_{\\N}} > 1$\n\tfor all $n \\in \\mathbb{N}$.}, \n\t$\\hat C=1$, $m=n$, $\\sigma = 6 \\operatorname{Lip}_{g}$,\n\t$P_1=\\hat P_{0}$, $P_2=P$ and $x=q+g(u_{0})$, we get $\\varangle(P_{0},P)=\\varangle(\\hat P_{0},P) < \\frac{1}{2}$,\n\tand, with Corollary \\ref{24.04.2012.1},\n\tthere exists some affine map $\\bar a:P_0 \\to P_0^{\\perp}$ with graph $G(\\bar a)=P$.\n\tNow we obtain with Lemma \\ref{6.9.11.1} ($P_{1}=P$, $P_{2}=P_{0}$), $u,v \\in P_0$ and \n\t$\\varangle(P_{0},P)< \\frac{1}{2}$ that\n\t\\begin{align}\\label{21.11.11.3}\n\t\td(v +\\bar a(v),u+\\bar a(u)) \n\t\t &\\le 2d(\\pi_{P_0}(v +\\bar a(v)),\\pi_{P_0}(u +g(u))). \n\t\\end{align}\n\tThat yields for $u \\in B(q,t) \\cap P_{0}$ and some suitable $v \\in P_{0}$ with $v+\\bar a(v)=\\pi_P(u+g(u))$\n\t\\begin{align*}\n\t\td(g(u),\\bar a(u)) \n\t\t& \\stackrel{\\hphantom{\\eqref{21.11.11.3}}}{\\le} d(u+g(u),P)+d(\\pi_P(u+g(u)),u+\\bar a(u))\\\\\n\t\t& \\stackrel{\\eqref{21.11.11.3}}{\\le}\n\t\t d(u+g(u),P) + 2d(\\pi_{P_0}(v+\\bar a(v)),\\pi_{P_0}(u+g(u)))\n\t\t\\stackrel{\\hphantom{\\eqref{21.11.11.3}}}{=} 3 d(u+g(u),P).\n\t\\end{align*}\n\tFinally, using $\\bar a \\in \\mathcal{A}$ and the last estimate, we get \n\t$\\gamma_{g}(q,t)\\stackrel{\\eqref{5.12.11.1}}{\\le} 3 (\\tilde \\gamma_{g}(q,t)+\\xi)$,\n\tand $0<\\xi < \\alpha \\omega_{\\N}$ was arbitrarily chosen.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{\\texorpdfstring{$\\gamma$-functions and affine approximation of Lipschitz functions}{y-functions and affine approximation of Lipschitz functions}}\\label{27.10.2014.3}\nIn this and the following subsections, we use the notation $U_{l}:=B(0,l)\\cap P_{0}$ for $l\\in \\{6,8,10\\}$.\n\\begin{thm} \\label{2.9.2014.1}\n\tLet $1 < p < \\infty$ and let $g:P_{0} \\to P_{0}^{\\perp}$ be a Lipschitz continuous function with Lipschitz constant $\\operatorname{Lip}_{g}$\n\tand compact support.\n\tFor all $\\theta >0$, there exists some set $H_{\\theta} \\subset U_{6}$ and some constants \n\t$C=C(n,p)$ and $\\hat C=\\hat C(n,{N})$ with\n\t\\[\\mathcal{H}^{n}(U_{6} \\setminus H_{\\theta}) \n\t\t\\le \\frac{C}{\\theta^{p(n+1)} \\operatorname{Lip}_{g}^{p}} \\int_{U_{10}} \\left( \\int_{0}^{2} \\gamma_{g}(x,t)^{2} \\frac{\\mathrm{d} t}{t} \\right)^{\\frac{p}{2}}\n\t\t\\mathrm{d} \\mathcal{H}^{n}(x)\\]\n\tso that, for all $y \\in P_{0}$, there exists some affine map $a_{y}:P_{0} \\to P_{0}^{\\perp}$ \n\tso that if $ r \\le \\theta$ and $ B(y,r) \\cap H_{\\theta} \\neq \\emptyset$, we have \n\t\\[\\|g-a_{y}\\|_{L^{\\infty}(B(y,r)\\cap P_{0},P_{0}^{\\perp})} \\le \\hat C r \\theta \\operatorname{Lip}_{g}, \\]\n\twhere $\\|\\cdot\\|_{L^{\\infty}(E)}$ denotes the essential supremum on $E \\subset P_{0}$\n\twith respect to the $\\mathcal{H}^{n}$-measure.\n\\end{thm}\n\nTo prove this theorem, we need the following lemma.\nIf $\\nu$ is some map, we use the notation \n$\\nu_{t}(x):=\\frac{1}{t^{n}}\\nu \\left(\\frac{x}{t} \\right)$. \\index{$\\nu_{t}(x)$}\n\n\\begin{lem}\\label{22.02.2013.01}\n\tThere exists some radial function $\\nu \\in C_{0}^{\\infty}(P_0,\\mathbb{R})$ with\n\t\\begin{enumerate}\n\t\\item\t$\\operatorname{supp}(\\nu) \\subset B(0,1)\\cap P_{0}$ and $\\widehat{\\nu}(0)=0$,\n\t\\item for all $x \\in P_0 \\setminus \\{0\\}$ and $i \\in \\{1,\\dots, n \\}$, we have\n\t\t\\begin{align} \\label{26.3.10;1}\t\n\t\t\t\\int_{0}^{\\infty}|\\widehat{\\nu}(t x)|^{2}\\frac{\\mathrm{d} t}{t}=1 \\ \\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\ \\\n\t\t\t0 < \\int_{0}^{\\infty} |\\widehat{(\\partial_{i} \\nu)_{t}}(x)|^{2} \\ \\frac{\\mathrm{d} t}{t} < \\infty,\n\t\t\\end{align}\n\t\\item for all $i \\in \\{1,\\dots, n \\}$, the function $\\partial_{i} \\nu$ has mean value zero \n\t\tand, for all $a \\in \\mathcal{A}(P_{0},P_{0}^{\\perp})$ (affine functions), the function $a \\nu$\n\t\thas mean value zero as well.\n\t\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\tLet $\\nu_{1} : P_{0} \\rightarrow \\mathbb{R}$ be some non harmonic ($\\Delta \\nu_{1} \\neq 0$), \n\tradial $C^{\\infty}$ function with support in $B(0,1)\\cap P_{0}$. \n\tWe set $\\nu_2 := \\Delta \\nu_{1} \\in C^{\\infty}(P_{0}) \\cap C_0^{\\infty}(B(0,1)\\cap P_{0})$ and\n\t$0 < c_{1}:= \\int_{0}^{\\infty}|\\widehat{\\nu_{2}}(te) |^{2}\\frac{\\mathrm{d} t}{t}$,\n\twhere $e$ is some normed vector in $P_{0}$. With Lemma \\ref{22.02.2013.02}, \n\twe get $\\nu_2$ is radial as well.\n\tUsing Lemma \\ref{10.12.12.2}, we obtain $|\\widehat{\\nu_{2}}(te)| = 4\\pi^2 t^2 |\\widehat{\\nu_1}(te)|$\n\tand hence\n\t\\begin{align*}\n\t\t0 < c_{1} &= \\int_{0}^{\\infty}|\\widehat{\\nu_{2}}(te) |^{2}\\frac{\\mathrm{d} t}{t}\n\t\t= 16\\pi^{4}\\int_{0}^{\\infty} t^3 |\\widehat{\\nu_{1}}(te) |^{2} \\mathrm{d} t < \\infty\n\t\\end{align*}\n\tbecause $\\nu_{1}$ is in the Schwarz space and therefore $\\widehat \\nu_{1}$ \n\tas well \\cite[2.2.15, 2.2.11 (11)]{FourierAnalysis}.\n\tThe previous equality also implies $\\widehat{\\nu_{2}}(0) = 0$.\n\tNow we set $\\nu:= \\sqrt{\\frac{1}{c_{1}}} \\nu_{2}$,\n\twhich is a radial $C_{0}^{\\infty}(P_{0},\\mathbb{R})$ function that fulfils 1. \n\tWe have for all $x \\in P_{0} \\setminus \\{0\\}$ \n\t(use substitution with $t=r\\frac{1}{|x|}$ and the fact that $\\widehat \\nu$ is radial)\n\t$\\int_{0}^{\\infty}|\\widehat{\\nu}(t x)|^{2}\\frac{\\mathrm{d} t}{t}\n\t\t=\\int_{0}^{\\infty}|\\widehat{\\nu}(re)|^{2}\\frac{\\mathrm{d} r}{r}=1.$\n\tIn a similar way, we deduce for $i \\in \\{1,\\dots,n\\}$ with Lemma \\ref{10.12.12.2} \n\t(using $|(\\phi^{-1}(tx))^{\\kappa}|\\le|\\phi^{-1}(tx)|=|tx|$ where $\\kappa$ is some multi-index\n\twith $|\\kappa|=1$)\n\t\\begin{align*}\n\t\t\\int_{0}^{\\infty} |\\widehat{(\\partial_{i} \\nu)_{t}}(x)|^{2} \\ \\frac{\\mathrm{d} t}{t}\n\t\t& \\le |2\\pi i|^{2} \\int_{0}^{\\infty} |tx|^{2} \\left| \\widehat{\\nu}(tx) \\right|^{2} \\ \\frac{\\mathrm{d} t}{t} \n\t\t = 4\\pi^{2} \\int_{0}^{\\infty} r \\left| \\widehat{\\nu}\\left(r\\frac{x}{|x|}\\right) \\right|^{2} \\ \\mathrm{d} r < \\infty,\n\t\\end{align*}\n\twhere we use that the Fourier transform of a Schwartz function is a Schwartz function as well\n\t\\cite[2.2.15]{FourierAnalysis}.\n\tThe left hand side of the previous inequality can not be zero, because this would implicate that \n\t$\\partial_{i}\\nu(x)=0$ for all $x \\in P_{0}$, which is in contradiction to \n\t$0 \\neq \\nu \\in C_{0}^{\\infty}(P_{0},\\mathbb{R})$.\n\tHence $\\nu$ fulfils 2.\n\tUsing partial integration and $\\Delta a=0$ for all $a \\in \\mathcal{A}(P_{0},P_{0}^{\\perp})$ implies\n\tthat $\\partial_{i} \\nu$ and $a \\nu$ have mean value zero.\n\\end{proof}\nFor some function $f : P_{0} \\rightarrow P_{0}^{\\perp}$ and $x \\in P_{0}$, we define the convolution\n\tof $\\nu_{t}$ and $f$ by\n\\[ (\\nu_{t} * f)(x) := \\int_{P_{0}} \\nu_{t}(x-y)f(y) \\mathrm{d} \\mathcal{H}^{n}(y).\\]\n\n\\begin{lem}[Calder\\'on's identity] \\label{calderon}\n\tLet $\\nu$ be the function given by Lemma \\ref{22.02.2013.01} and let \n\t$u \\in P_{0} \\setminus \\{0\\}$ and $f\\in L^{2}(P_{0},P_{0}^{\\perp})$ or let $f \\in \\mathscr{S}^{'}(P_{0})$ \n\tbe a tempered distribution and $u \\in \\mathscr{S}(P_{0})$ (Schwartz space) with $u(0)=0$.\n\tThen we have\n\t\\begin{align}\n\t\tf(u) = \\int_{0}^{\\infty} (\\nu_{t}*\\nu_{t}*f)(u) \\frac{\\mathrm{d} t}{t}.\n\t\\end{align}\n\tL\\'eger calls this identity ``Calder\\'on's formula'' \\cite[p. 862, 5. Calder\\'on's formula and the size of $F_3$]{Leger}.\n\tGrafakos presents a similar version called ``Calder\\'on reproducing formula'' \n\t\\cite[p.371, Exercise 5.2.2]{FourierAnalysis}.\n\\end{lem}\n\n\\begin{proof}\n\tAt first, let $f \\in L^2(P_{0},P_{0}^{\\perp})$ and $u \\in P_{0} \\setminus \\{0\\}$. \n\tWe have with Lemma \\ref{10.12.12.2} that $\\widehat{(\\nu_{t})}(u)= \\widehat{\\nu}(t u)$\n\tand, with Fubini's theorem and Lemma \\ref{FourFal}, we obtain \n\t\\begin{align*}\n\t\t\\left( \\int_{0}^{\\infty} (\\nu_{t}*\\nu_{t}*f)(u) \\frac{\\mathrm{d} t}{t}\\right)^{\\widehat{}} \n\t\t& = \\int_{0}^{\\infty} \\widehat{(\\nu_{t})}(u)\\widehat{(\\nu_{t})}(u)\\widehat{f}(u) \\frac{\\mathrm{d} t}{t}\n\t\t\\stackrel{\\eqref{26.3.10;1}}{=} \\widehat{f}(u).\n\t\\end{align*}\n\tThe Fourier inversion holds on $L^{2}(P_{0},P_{0}^{\\perp})$ \n\t\\cite[2.2.4 The Fourier Transform on $L^1+L^2$]{FourierAnalysis}, which gives the statement.\n\tUse the same idea to get this result for tempered distributions.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{2.9.2014.1}]\nLet $g \\in C_{0}^{0,1}(P_{0},P_{0}^{\\perp})$ and let $\\nu$ be the function given by Lemma \\ref{22.02.2013.01}.\nWe define\n\\begin{align*}\n\tg_{1}(u) &:= \\int_{2}^{\\infty} (\\nu_{t}* \\nu_{t}*g)(u) \\frac{\\mathrm{d} t}{t}\n\t\t + \\int_{0}^{2} (\\nu_{t}* (\\Eins_{P_{0} \\setminus U_{10}}\\cdot(\\nu_{t}*g)))(u)\n\t\t\t \\frac{\\mathrm{d} t}{t},\\\\\n\tg_{2}(u) & := \\int_{0}^{2} (\\nu_{t}* (\\Eins_{U_{10}}\\cdot(\\nu_{t}*g)))(u)\n\t\t\t \\frac{\\mathrm{d} t}{t}\n\\end{align*}\nand the previous lemma implies that $g = g_{1} + g_{2}$.\nWe recall the notation $U_{l}=B(0,l)\\cap P_{0}$ for $l\\in \\{6,8,10\\}$ and \ncontinue the proof of Theorem \\ref{2.9.2014.1} with several lemmas.\n\n\\begin{lem}\\label{lem5.2}\n\t$g_{1} \\in C^{\\infty}(U_{8})$ and there exists some constant $C = C(\\nu)$ so that\n\tfor all multi-indices $\\kappa$ with $|\\kappa|\\le 2$ we have\n\t$\\| \\partial^{\\kappa} g_{1}\\|_{L^{\\infty}(U_{8},P_{0}^{\\perp})} \\le C \\operatorname{Lip}_{g}$.\n\n\t$g_{2}$ is Lipschitz continuous on $U_{8}$ with Lipschitz constant $C(\\nu)\\operatorname{Lip}_{g}$.\n\\end{lem}\n\\begin{proof}\n\tFor $x \\in P_0$ we set\n\t\\[ g_{11}(x) := \\int_{2}^{\\infty} (\\nu_{t} * \\nu_{t} * g)(x) \\frac{\\mathrm{d} t}{t}, \\ \\ \\\n\t\tg_{12}(x) := \\int_{0}^{2} (\\nu_{t} * (\\Eins_{P_{0} \\setminus U_{10}} \\cdot(\\nu_{t} * g)))(x)\n\t\t\t \\frac{\\mathrm{d} t}{t}\\]\n\tso that $g_{1} = g_{11} + g_{12}$ and we set\n\t$\\varphi(x) := \\int_{2}^{\\infty} (\\nu_{t} * \\nu_{t})(x) \\frac{\\mathrm{d} t}{t}$.\n\n\tAt first, we look at some intermediate results:\n\t\\begin{enumerate}\n\t\t\\renewcommand{\\labelenumi}{\\Roman{enumi}.}\n\t\t\\item $g_{12}(x) = 0$ for all $x \\in U_{8}$, due to the support of $\\nu_{t}$ and \n\t\t$\\Eins_{P_{0} \\setminus U_{10}}\\cdot(\\nu_{t} * g)$.\n\t\t\\item For every multi-index $\\kappa$, there exists some constant $C=C(n,\\nu,\\kappa)$ such that\n\t\t\t$|\\partial^{\\kappa} \\varphi| \\le C$, where \n\t\t\t$\\partial^{\\kappa} \\varphi(y) := \\int_{2}^{\\infty} \\partial^{\\kappa}(\\nu_{t} * \\nu_{t})(y) \\frac{\\mathrm{d} t}{t}$.\n\t\t\tThis is given by \n\t\t\t$\\partial^{\\kappa} (\\nu_{t}(y)) = \\frac{1}{t^{|\\kappa|}}(\\partial^{\\kappa} \\nu)_{t}(y)$,\n\t\t\tand $|\\partial^{\\kappa}(\\nu_{t} * \\nu_{t})(y)| \n\t\t \\le \\| \\nu\\|_{L^{\\infty}(P_{0},\\mathbb{R})} \\| \\partial^{\\kappa}\\nu\\|_{L^{\\infty}(P_{0},\\mathbb{R})} \\frac{\\omega_{\\N}}{t^{n+|\\kappa|}}$.\n\t\t\\item\n\t\tFor every multi-index $\\kappa$, the function $\\partial^{\\kappa}\\varphi$ has bounded support in $B(0,4) \\cap P_{0}$.\n\t\t\\begin{proof}\n\t\tLet $0 0$. We define\n\t$H_{\\theta} := \\left\\{ x \\in U_{6} | N(g_{2})(x) \\le \\theta^{n+1} \\operatorname{Lip}_{g} \\right\\}$.\n\\end{dfn}\n\n\\begin{lem}\\label{3.9.2014.1}\n\tLet $\\theta > 0$. There exists some constant $C=C(n,p,\\nu)$ so that\n\t\\[\\mathcal{H}^{n}(U_{6} \\setminus H_{\\theta}) \n\t\t\\le \\frac{C}{\\theta^{p(n+1)} \\operatorname{Lip}_{g}^{p}} \\int_{U_{10}} \\left( \\int_{0}^{2} \\gamma_{g}(x,t)^{2} \\frac{\\mathrm{d} t}{t} \\right)^{\\frac{p}{2}}\n\t\t\\mathrm{d} \\mathcal{H}^{n}(x).\\]\n\\end{lem}\n\\begin{proof}\n\tWith Lemma \\ref{lem5.3}, Lemma \\ref{lem5.1} and \n\t$\\|Dg_{2} \\|_{L^{p}(P_{0},P_{0}^{\\perp})}^{p} \\le n^{p-1} \\sum_{i=1}^{n} \\|\\partial_{i} g_{2}\\|_{L^{p}(P_{0},P_{0}^{\\perp})}^{p}$, \n\tthere exists some constant $C=C(n, p,\\nu)$ with\n\t\\[ \\|N(g_{2})\\|_{L^{p}(P_{0},P_{0}^{\\perp})}^{p} \n\t\t\\le Csum_{i=1}^{n} \\|\\partial_{i} g_{2}\\|_{L^{p}(P_{0},P_{0}^{\\perp})}^{p}\n\t\t\\le C\\int_{U_{10}} \\left( \\int_{0}^{2} \\gamma_{g}(x,t)^{2} \\frac{\\mathrm{d} t}{t} \\right)^{\\frac{p}{2}}\n\t\t\t\\mathrm{d} \\mathcal{H}^{n}(x).\\]\n\tHence, using Chebyshev's inequality, we get the assertion.\n\\end{proof}\n\n\n\n\n\\begin{lem} \\label{lem5.4}\n\tLet $B$ be a ball with centre in $P_{0}$.\n\tIf $ (B \\cap P_{0}) \\subset U_{8}$, then there exists some constant $C=C({N},n,\\nu)$ with \n\t\\[\\operatorname{osc}_{B} (g_{2}) \\le C \\operatorname{diam} B \\left( \t\t\n\t\t\t\\frac{1}{\\operatorname{diam} B}\\Avg{B}\\Bigl(|g_{2} -\\Avg{B}(g_{2})|\\Bigr) \\right)^{\\frac{1}{n+1}}\\operatorname{Lip}_{g}^{\\frac{n}{n+1}}.\\]\n\\end{lem}\n\\begin{proof}\n\tLet $(B \\cap P_{0}) \\subset U_{8}$ and $\\lambda := \\operatorname{osc}_{B} (g_{2})$.\n\tThe function $g_{2}$ is Lipschitz continuous on $U_{8}$ with $\\operatorname{Lip}_{g_{2}}=C(\\nu)\\operatorname{Lip}_{g}$ \n\t(see Lemma \\ref{lem5.2} on page \\pageref{lem5.2}) and $B \\cap P_{0}$ is closed.\n\tHence there exists some $y \\in B \\cap P_{0}$ with\n\t$\\lambda = |g_{2}(y)-\\Avg{}_{B}g_{2}|$ and\n\twe get for $x \\in B$ with \n\t$d(x,y) \\le \\frac{\\lambda}{2\\operatorname{Lip}_{g_{2}}}$ using triangle inequality\n\t$|g_{2}(x)-\\Avg{B}(g_{2})| \\ge \\frac{\\lambda}{2}$.\n\tFurthermore, using that $g_{2}$ is continuous on $U_{8}$ \n\tfor all $l \\in \\{1,\\dots,{N}\\}$, there exists some $z_{l} \\in B \\cap P_{0}$, with \n\t$g_{2}^{l}(z_{l}) =\\Avg{B}(g_{2}^{l})$ (where $g_{2}^{l}(z_{l}) \\in \\mathbb{R}$ means the $l$-th component\n\tof $g_{2}(z_{l}) \\in \\mathbb{R}^{{N}}$).\n\tWith $g_{2}^{l}(y)-\\Avg{B}(g_{2}^{l}) \\le \\operatorname{Lip}_{g_{2}} d(y,z_{l})$ for all $l \\in \\{1,\\dots, {N}\\}$ we get\n\t$\\lambda^{2} \\le {N} \\left(\\operatorname{Lip}_{g_{2}} \\operatorname{diam} B \\right)^{2}$,\n\twhich implies $\\frac{\\lambda}{\\sqrt{{N}}\\operatorname{Lip}_{g_{2}}} \\le \\operatorname{diam} B$.\n\tSince $y \\in B$, there exists some ball $\\hat B \\subset B \\cap B\\left(y,\\frac{\\lambda}{2\\operatorname{Lip}_{g_{2}}}\\right)$\n\twith $ \\operatorname{diam} \\hat B \\ge \\frac{\\lambda}{2\\sqrt{{N}}\\operatorname{Lip}_{g_{2}}} $\n\tand hence with $|g_{2}(x) -\\Avg{}_{B}(g_{2})| \\ge \\frac{\\lambda}{2}$ we obtain\n\t\\begin{align*}\n\t\t(\\operatorname{diam} B)^{n} \\Avg{B} \\ |g_{2}(x) -\\Avg{B}(g_{2})| \n\t\t\t& \\ge \\omega_{\\N} \\left(\\textstyle{\\frac{\\lambda}{4\\sqrt{{N}}\\operatorname{Lip}_{g_{2}}}}\\right)^{n} \\frac{\\lambda}{2} .\n\t\\end{align*}\n\tUsing $\\operatorname{Lip}_{g_{2}}=C(\\nu)\\operatorname{Lip}_{g}$, this implies the assertion.\n\\end{proof}\n\n\\begin{lem} \\label{lem5.5}\n\tLet $\\theta > 0$ and $y \\in P_{0}$. There exists some constant $C=C({N},n,\\nu)$ and some affine map $a_{y}:P_{0} \\to P_{0}^{\\perp}$ \n\tso that if $ r \\le \\theta$ and $ B(y,r) \\cap H_{\\theta} \\neq \\emptyset$, we have \\\\\n\t$\\|g-a_{y}\\|_{L^{\\infty}(B(y,r)\\cap P_{0},P_{0}^{\\perp})} \\le C r \\theta \\operatorname{Lip}_{g}$. \n\\end{lem}\n\\begin{proof}\n\tLet $y \\in P_{0}$.\n\tIf $\\theta \\ge 1$, we can choose $a_y(y'):=g(y)$ as a constant and get the desired result directly from the\n\tLipschitz condition.\n\tNow let $0 < \\theta < 1$ and $y' \\in B(y,r) \\cap P_{0}$. We set $a_{y}(y'):=g(y) + D g_{1}(y)\\phi^{-1}(y'-y)$.\n\tWe have $d(y',U_{6}) \\le d(y',H_{\\theta}) \\le d(y',y) + d(y,H_{\\theta}) \\le 2$. So we get $y',y \\in U_{8}$.\n\tUsing Taylor's theorem and Lemma \\ref{lem5.2} we obtain\n\t\\[|g_{1}(y')-[g_{1}(y) + D g_{1}(y)\\phi^{-1}(y'-y) ]| \n\t\t\\le \\sum_{|\\kappa|=2}\\|\\partial^{\\kappa} g_{1}\\|_{L^{\\infty}(U_{8})} |y'-y|^{2}\n\t\t\\le C(n,\\nu) \\operatorname{Lip}_{g}r^{2}\\]\n\n\tSince $r \\le \\theta <1$, $B(y,r) \\cap H_{\\theta} \\neq \\emptyset$ and $H_{\\theta} \\subset U_{6}$, \n\twe obtain $B(y,r)\\cap P_{0} \\subset U_{8}$ and we can apply Lemma \\ref{lem5.4}.\n\tTogether with the definition of $H_{\\theta}$ this leads to\n\t\\[\\operatorname{osc}_{B(y,r)}g_{2} + \\operatorname{Lip}_{g}r^{2} \\le C({N},n,\\nu)r \\theta \\operatorname{Lip}_{g}.\\]\n\tNow by using $g=g_{1}+g_{2}$ and $|g_{2}(y') - g_{2}(y)| \\le 2 \\operatorname{osc}_{B(y,r)}g_{2}$\n\twe get for every $y' \\in B(y,r) \\cap P_{0}$\n\t\\[|g(y')-[g(y) + D g_{1}(y)\\phi^{-1}(y'-y) ]| \\le C({N},n,\\nu)r \\theta \\operatorname{Lip}_{g}. \\]\n\\end{proof}\n\n \nLemma \\ref{3.9.2014.1} and Lemma \\ref{lem5.5} complete the proof of Theorem \\ref{2.9.2014.1}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{\\texorpdfstring{The $\\gamma$-function of $A$ and integral Menger curvature}{The y-function of A and integral Menger curvature}}\\label{3.11.2014.2}\nIn this section, we prove the following Theorem \\ref{thm4.1}.\nIt states that we get a similar control on the $\\gamma$-functions applied to our function $A$\nas we get in Corollary \\ref{thm2.4} on the $\\beta$-numbers.\n\nFor $\\alpha, \\varepsilon > 0$, $\\eta \\le 2 \\varepsilon $ and $k \\ge 4$, we defined $A$ on $U_{12}$\n(cf. Definition \\ref{04.11.2013.1} on page \\pageref{04.11.2013.1}).\nSince in this section we only apply the $\\gamma$-functions to $A$, we set\n$\\gamma(q,t):=\\gamma_{A}(q,t)$ and we recall the notation $U_{10}:=B(0,10)\\cap P_{0}$.\n\n\n\\begin{thm} \\label{thm4.1}\n\tThere exists some $\\tilde k \\ge 4$ and some $\\tilde \\alpha =\\tilde \\alpha (n) > 0$\n\tso that, for all $\\alpha$ with $0 < \\alpha \\le \\tilde \\alpha$, there exists some\n\t$\\tilde \\varepsilon = \\tilde \\varepsilon({N},n,C_{0},\\alpha)$ \n\tso that, if $k \\ge \\tilde k$ and $\\eta \\le \\tilde \\varepsilon^{p}$, we have for all\n\t$\\varepsilon \\in [\\eta^{\\frac{1}{p}},\\tilde \\varepsilon]$ \n\tthat there exists some constant $C= C({N},n,\\mathcal{K},p,C_{0},k)$ so that\n\t\\begin{align*}\n\t\t\\int_{U_{10}} \\int_{0}^{2} \\gamma(q,t)^{p} \\frac{\\mathrm{d} t}{t} \\mathrm{d}\\mathcal{H}^{n}(q) \n\t\t&\\le C\\varepsilon^{p} + C\\mathcal{M}_{\\mathcal{K}^{p}}(\\mu)\n\t\t \\le C\\varepsilon^{p}.\n\t\\end{align*}\n\\end{thm}\n\\begin{proof}\nLet $\\bar k \\ge 4$ be the maximum of all thresholds for $k$ given in chapter \\ref{construction} and let \n$\\tilde \\alpha=\\tilde \\alpha(n) \\le \\frac{1}{4}$ \\label{alphafuergamma}\nbe the upper bound for the Lipschitz constant given by Lemma \\ref{bem4.2}. \nWe set $\\tilde k :=\\max\\{\\bar k,\\tilde C+1,\\hat C\\}$ \\label{22.11.2013.1}\nwhere the constants $\\tilde C$ and $\\hat C$ are fixed constants which will be set during this section\\footnote{\n$\\tilde C$ is given in Lemma \\ref{2.1.10;10}, $\\hat C$ is given in Lemma \\ref{lem4.4} V}.\nLet $0 \\le \\alpha \\le \\tilde \\alpha$. Let $\\bar \\varepsilon=\\varepsilon({N},n,C_{0},\\alpha) \\le \\alpha$ \nbe the minimum of all thresholds for $\\varepsilon$ given in chapter \\ref{construction}.\nWe set $\\tilde \\varepsilon := \\min\\{\\bar \\varepsilon,(2C^{'}C_{1})^{-1}\\}<1$\\footnote{\n$C^{'},C_{1}$ are given in Lemma \\ref{lem4.3}}\n\\label{epsilonfuergamma} and \nassume that $k \\ge \\tilde k$ and $\\eta \\le \\tilde \\varepsilon^{p}$. Now let $\\varepsilon > 0$ with \n$\\eta \\le \\varepsilon^{p} \\le \\tilde \\varepsilon^{p}$. \\label{3.12.2013.4}\nFor the rest of this section, we fix the parameters $k,\\eta,\\alpha,\\varepsilon$ and mention that\nthey meet all requirements of the lemmas in Chapter \\ref{construction}.\n\nWe start the proof of Theorem \\ref{thm4.1} with several lemmas.\nAt first, we prove\n\\begin{lem}\\label{24.2.10;17} There exists some constant $C=C({N},n,p,C_{0})$ so that\n\t\\[\\sum_{i \\in I_{12}}\\int_{R_{i}\\cap U_{10}} \\int_{0}^{\\frac{\\operatorname{diam} R_{i}}{2}} \\gamma(q,t)^{p}\\frac{\\mathrm{d} t}{t} \n\t\t\\mathrm{d} \\mathcal{H}^{n}(q) \\le C \\varepsilon^{p}.\\]\n\\end{lem}\n\\begin{proof}\n\tLet $i \\in I_{12}$, $q \\in R_{i}$, $0 < t < \\frac{\\operatorname{diam} R_{i}}{2}$ and $u \\in B(q,t) \\cap P_{0} \\subset 2R_i$.\n\tThe function $A$ is in $C^{\\infty}(2R_{i},P_{0}^{\\perp})$ \n\t(see definition of $A$ on page \\pageref{04.11.2013.1}).\n\tTaylor's theorem implies \n\t\\[\\inf_{a \\in \\mathcal{A}}d(A(u),a(u)) \\le t^{2}\\frac{C({N},n,C_{0}) \\varepsilon}{\\operatorname{diam} R_{i}}\\]\n\tsince the infimum over all affine functions cancels out the \n\tlinear part and the second order derivatives of the remainder can be estimated using \n\tLemma \\ref{abschaetzungableitungvonA}.\n\tNow we have \n\t\\[\\gamma(q,t) \\le \\frac{\\omega_{\\N}}{t} \\sup_{u \\in B(q,t)\\cap P_{0}} \\inf_{a \\in \\mathcal{A}} d(A(u),a(u))\n\t\t\\le t \\frac{C({N},n,C_{0})\\varepsilon}{\\operatorname{diam} R_{i}}.\\]\n\tHence, Lemma \\ref{inneredisjunkt} (ii) implies the statement.\n\\end{proof}\n\nThe previous lemma implies that, due to Lemma \\ref{inneredisjunkt} (ii), it remains to handle \nthe two terms in the following sum to prove Theorem \\ref{thm4.1}.\nIf $q_{1} \\in R_{i}$, we get with Lemma \\ref{rem3.10} that $\\frac{D(q_{1})}{100} \\le\\frac{\\operatorname{diam} R_{i}}{2}$\nand, if $q_{2} \\in \\pi(\\mathcal{Z})$, we obtain with Lemma \\ref{rem3.8} $D(q_{2})= 0$.\nHence we conclude using Lemma \\ref{inneredisjunkt} (ii)\n\\begin{align}\n\t& \\ \\ \\ \\sum_{i \\in I_{12}} \\int_{R_{i} \\cap U_{10}} \\int_{\\frac{\\operatorname{diam} R_{i}}{2}}^{2}\n\t\\gamma(q,t)^{p} \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mathcal{H}^{n}(q) +\n\t\\int_{\\pi(\\mathcal{Z})\\cap U_{10}} \\int_{0}^{2} \\gamma(q,t)^{p} \n\t \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mathcal{H}^{n}(q)\n\t\\nonumber \\\\\n\t& = \\int_{U_{10}}\\int_{\\frac{D(q)}{100}}^{2}\n\t\\gamma(q,t)^{p} \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mathcal{H}^{n}(q). \\label{10.1.2010;b}\n\\end{align}\n\nIn the following, we prove some estimate for $\\gamma(q,t)$ \nwhere $q \\in U_{10}$ and $ \\frac{D(q)}{100} < t < 2$. To get this estimate, we need\nthe next lemma.\n\\begin{lem} \\label{28.11.2013.1}\n\tFor all $q \\in U_{10}$ and for all $t$ with $\\frac{D(q)}{100} < t < 2$,\n\tthere exists some $\\tilde X=\\tilde X(q) \\in F$ and some $T=T(t)> 0$ with\n\t\\begin{align} \\label{2.1.10;27}\n\t\t(\\tilde X,T) &\\in S,&\n\t\td(\\pi(\\tilde X),q) &\\le T &\n\t\t&\\text{ and }&\n\t\t20t &\\le T \\le 200t.\n\t\\end{align}\n\\end{lem}\n\\begin{proof}\n\tWe have\n\t$ D(q) = \\inf_{(X,s) \\in S} (d(\\pi(X),q) + s)$,\n\tand hence there exists some $(\\tilde X, \\tilde s) \\in S$ \\label{DefvonXSchlange} with\n\t$ d(\\pi(\\tilde X),q) + \\tilde s \\le D(q)+100t \\le 200t$.\n\tWe set $T:= \\min\\{40,200t\\}$ which fulfils $20t \\le T \\le 200t$ as $t < 2$. \n\tUsing Lemma \\ref{rem3.1} (i), (ii) and $200t \\ge \\tilde s$, we obtain $(\\tilde X,T) \\in S$.\n\n\tWith $d(\\pi(\\tilde X),q) \\le d(\\pi(\\tilde X),0) + d(0,q) \\le 5 + 10$ we get $d(\\pi(\\tilde X),q) \\le T$.\n\\end{proof}\n\nNow let $q,t, \\tilde X$ and $T$ \\label{2.1.10;25} as in Lemma \\ref{28.11.2013.1}.\nFurthermore, let $X \\in B(\\tilde X,200t)\\cap F \\label{16.3.10;1}$. \n\\newcommand{\\hat P}{\\hat P}\nWe choose some $n$-dimensional plane named $\\hat P=\\hat P(q,t,X)$ with \n\\begin{align} \\label{2.1.10;30}\n\t\\beta_{1;k}^{\\hat P}(X,t) \\le 2 \\beta_{1;k}(X,t)\n\\end{align}\nand define\n\\[ \\mathcal{I}(q,t):= \\left\\{ i \\in I_{12} \\big| R_{i} \\cap B(q,t) \\neq \\emptyset \\right\\}.\\label{2.1.10;21}\\]\nWith Lemma \\ref{Riueberdeckung}, we have \n$(B(q,t)\\cap P_{0}) \\subset U_{12} \\subset \\pi(\\mathcal{Z}) \\cup \\bigcup_{i \\in I_{12}}R_{i}$. We set\n\\[K_{0}:= \\int_{B(q,t)\\cap \\pi(\\mathcal{Z})} \\frac{d(u+A(u),\\hat P)}{t^{n+1}} \\mathrm{d} \\mathcal{H}^{n}(u), \\ \\ \\\n\tK_{i}:= \\int_{B(q,t)\\cap R_{i}} \\frac{d(u+A(u),\\hat P)}{t^{n+1}} \\mathrm{d} \\mathcal{H}^{n}(u)\\]\nand get with Lemma \\ref{bem4.2} that\n\\begin{align}\n\t\\gamma(q,t)\n\t& \\le 3 \\ K_{0} + 3 \\sum_{i \\in \\mathcal{I}(q,t)} K_{i}. \\label{2.1.10;1}\n\\end{align}\nAt first, we consider $K_{0}$.\n\\begin{lem} \\label{2.1.10;10}\n\tThere exists some constant $\\tilde C>1$ so that\n\t\\[ \\int_{B(q,t) \\cap \\pi(\\mathcal{Z})} d(u + A(u),\\hat P) \\mathrm{d} \\mathcal{H}^{n}(u)\n\t\t\\le \\int_{B(X,\\tilde Ct)\\cap \\mathcal{Z}} d(x,\\hat P) \\mathrm{d} \\mathcal{H}^{n}(x).\\]\n\\end{lem}\n\\begin{proof}\n\tLet $g: \\pi(\\mathcal{Z}) \\rightarrow \\mathcal{Z}, u \\mapsto u+A(u)$. \n\tThis function is bijective, continuous ($A$ is $2 \\alpha$-Lipschitz on $\\pi(Z)$) and \n\t$g^{-1}=\\pi|_{\\mathcal{Z}}$ is Lipschitz continuous with Lipschitz constant 1. \n\tWith $f(x) = d(x,\\hat P)$ and $s=n$,\n\twe apply \\cite[Lem. A.1]{Sch2012} and get\n\t\\[\\int_{B(q,t) \\cap \\pi(\\mathcal{Z})} d(u + A(u),\\hat P) \\mathrm{d} \\mathcal{H}^{n}(u)\n\t\t \\le \\int_{g(B(q,t) \\cap \\pi(\\mathcal{Z}))} d(x,\\hat P) \\mathrm{d} \\mathcal{H}^{n}(x).\\]\n\tNow it remains to show that there exists some constant $C$ so that\n\t$g(B(q,t) \\cap \\pi(\\mathcal{Z})) \\subset B(X,Ct)\\cap \\mathcal{Z}$. \n\tLet $ x \\in g(B(q,t) \\cap \\pi(\\mathcal{Z}))$. This implies $x \\in \\mathcal{Z}$\n\tand so, using Lemma \\ref{7.2.10;1}, we get $d(x)=0$.\n\tWith \\eqref{2.1.10;27}, we conclude \n\t$d(\\tilde{X}) \\le d(\\tilde X, \\tilde X) + T \\le 200t$,\n\tand we obtain with \\eqref{2.1.10;27}\n\t$d(\\pi(x),\\pi(\\tilde X)) \\le 201t$.\n\tSo, with Lemma \\ref{lem3.9}, we have\n\t$d(x,\\tilde X) \\le 1602t$.\n\tWe deduce with $\\tilde C=1802$ that\n\t$d(x,X) \\le d(x, \\tilde X) + d(\\tilde X,X) \\le \\tilde Ct$\n\tand so\n\t$g(B(q,t) \\cap \\pi(\\mathcal{Z})) \\subset B(X,\\tilde Ct)\\cap \\mathcal{Z}$.\n\\end{proof}\n\n\\begin{lem} \\label{2.1.10;11}\n\tThere exists some constant $C=C({N},n,C_{0})>1$ so that\n\t\\[ \\int_{B(X,\\tilde Ct)\\cap \\mathcal{Z}}d(x,\\hat P)\\mathrm{d} \\mathcal{H}^{n}(x)\n\t\t\\le C \\int_{B(X,(\\tilde C +1)t)}d(x,\\hat P)\\mathrm{d} \\mu(x).\\]\n\\end{lem}\n\\begin{proof}\n\tAt first, we prove for an arbitrary ball $B$ with centre in $\\mathcal{Z}$\n\t\\begin{align} \\label{23.2.10;a}\n\t\t\\mathcal{H}^{n}(\\mathcal{Z} \\cap B) \\le C({N},n,C_{0}) \\mu(B).\n\t\\end{align}\n\tWith \\cite[Dfn. 2.1]{Evans}, we get\n\t$\\mathcal{H}^{n}(\\mathcal{Z} \\cap B) = \\lim_{\\tau \\to 0}\n\t\t\\mathcal{H}^{n}_{\\tau}(\\mathcal{Z} \\cap B)$.\n\tLet $0<\\tau_{0} < \\min\\left\\{\\frac{\\operatorname{diam} B}{2},50\\right\\}$.\n\tWe define\n\t$\\mathcal{F}:= \\left\\{ B(x,s) | x \\in \\mathcal{Z}\\cap B, s \\le \\tau_{0} \\right\\}$.\n\tWith Besicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans}, there exist $N_{0}=N_{0}({N})$ \n\tcountable families $\\mathcal{F}_{j} \\subset \\mathcal{F}$, $j=1,...,N_{0}$, of disjoint balls \n\twhere the union of all those balls covers $\\mathcal{Z} \\cap B$.\n\tFor every ball $\\tilde B = B(x,s) \\in \\mathcal{F}_{j}$, we have $x \\in \\mathcal{Z}$ and hence, \n\tusing the definition of $\\mathcal{Z}$ (see page \\pageref{def3.2}), we deduce $h(x)=0$.\n\tWith $h(x)=0 < s < 50$ and Lemma \\ref{rem3.1} (i), we get\n\t$(x,s) \\in S \\subset S_{total}$ and so\n\t$\\left(\\frac{\\operatorname{diam} \\tilde B}{2}\\right)^{n} \\le 2\\frac{\\mu(\\tilde B)}{\\delta}$.\n\tThe centre of $B$ is also in $\\mathcal{Z}$ and hence, analogously, we conclude\n\t$\\left(\\frac{\\operatorname{diam} B}{2}\\right)^{n} \\le 2\\frac{\\mu(B)}{\\delta}$.\n\tWith (B) from page \\pageref{Grundeigenschaften}, we get\n\t$\\mu(2B) \\le4^{n} C_{0} \\frac{2}{\\delta} \\mu(B)$.\n\tSince $x \\in B$ and $s \\le \\tau_{0} < \\frac{\\operatorname{diam} B}{2} $, we obtain\n\t$\\tilde B = B(x,s) \\subset 2B$.\n\tNow, by definition of $\\mathcal{H}^{n}_{\\tau_{0}}$ \\cite[Dfn. 2.1]{Evans} and \n\tbecause $\\delta =\\delta({N},n)$\n\t(see \\eqref{Wahlvondelta} on page \\pageref{Wahlvondelta}),\n\twe deduce\n\t\\begin{align*}\n\t\t\\mathcal{H}^{n}_{\\tau_{0}}(\\mathcal{Z}\\cap B) \n\t\t& \\le\t 2\\sum_{j=1}^{N_{0}} \\sum_{\\tilde B \\in \\mathcal{F}_{j}} \\omega_{\\N}\\frac{\\mu(\\tilde B)}{\\delta} \n\t\t \\le 2\\frac{\\omega_{\\N}}{\\delta} \\sum_{j=1}^{N_{0}} \\mu(2B) \n\t\t \\le C({N},n,C_{0}) \\mu(B).\n\t\\end{align*}\n\tSo, with $\\tau_{0} \\rightarrow 0$, the inequality \\eqref{23.2.10;a} is proven.\n\t\n\tLet $\\tilde C$ be the constant from Lemma \\ref{2.1.10;10}. For an arbitrary $0 < \\sigma \\le t$, we define\n\t\\[ \\mathcal{G}_{\\sigma}:= \\left\\{ B(x,s) \\Big| x \\in \\mathcal{Z} \\cap B(X,\\tilde Ct), s \\le \\sigma \\right\\}.\\]\n\tWith Besicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans}, there exist $N_{0}=N_{0}({N})$ families \n\t$\\mathcal{G}_{\\sigma, j} \\subset \\mathcal{G}_{\\sigma}$ of disjoint balls, where $j= 1,..,N_{0}$ and those\n\tballs cover $\\mathcal{Z} \\cap B(X,\\tilde Ct)$.\n\tWe denote by $p_{B}$ the centre of the ball $B$ and conclude\n\t\\begin{align*}\n\t\t& \\hphantom{\\stackrel{\\hphantom{\\eqref{23.2.10;a}}}{=}}\n\t\t\t \\int_{\\mathcal{Z} \\cap B(X,\\tilde Ct)} d(x,\\hat P)\\mathrm{d} \\mathcal{H}^{n}(x) \\\\\n\t\t& \\stackrel{\\hphantom{\\eqref{23.2.10;a}}}{\\le}\n\t\t\t \\sum_{j=1}^{N_{0}} \\sum_{B \\in \\mathcal{G}_{\\sigma, j}} \\int_{\\mathcal{Z} \\cap B} \n\t\t\t\\sigma + d(p_{B},\\hat P)\n\t\t\t\\mathrm{d} \\mathcal{H}^{n}(x)\\\\ \\displaybreak[1]\n\t\t& \\stackrel{\\eqref{23.2.10;a}}{\\le} \n\t\t\tC({N},n,C_{0}) \\sum_{j=1}^{N_{0}} \\sum_{B \\in \\mathcal{G}_{\\sigma, j}} \\int_{B}\n\t\t\t\\left(\\sigma + d(p_{B},\\hat P)\\right) \n\t\t\t\\mathrm{d} \\mu(x)\\\\ \\displaybreak[1]\n\t\t& \\stackrel{\\hphantom{\\eqref{23.2.10;a}}}{\\le}\n\t\t\t C({N},n,C_{0}) \\left( \\mu(B(X,(\\tilde C +1)t)) 2\\sigma \n\t\t\t \t+ \\int_{B(X,(\\tilde C +1)t)} d(x,\\hat P) \\mathrm{d} \\mu(x)\\right). \n\t\\end{align*}\n\tWith $ \\sigma \\rightarrow 0$, the assertion holds.\n\\end{proof}\nWith Lemma \\ref{2.1.10;10} and Lemma \\ref{2.1.10;11}, we get for $K_{0}$ using that \n$k \\ge \\tilde k \\ge \\tilde C+1$, where $\\tilde k$ is defined on page \\pageref{22.11.2013.1}\n\\begin{align}\n\tK_{0} \n\t& \\stackrel{\\hphantom{\\eqref{2.1.10;30}}}{\\le} C({N},n,C_{0}) \\ \\beta_{1;k}^{\\hat P}(X,t)\n\t \\stackrel{\\eqref{2.1.10;30}}{\\le} C({N},n,C_{0}) \\ \\beta_{1;k}(X,t). \\label{2.1.10;2}\n\\end{align}\n\nTo estimate $K_{i}$, we need the following lemma.\n\\begin{lem} \\label{29.12.09;1}\n\tThere exists some constant $C_{4}=C_{4}({N},n,C_{0}) > 1$ so that, for all $i \\in I_{12}$ \n\tand $ u \\in R_{i}$, we have \n\t$d(\\pi_{P_{i}}(u+ A(u)),B_{i}) \\le C_{4} \\operatorname{diam} R_{i}$.\n\tWe recall that $P_{i}$ is the $n$-dimensional plane, which is,\n\tin the sense of Definition \\ref{12.07.13.1}, associated to the ball \n\t$B(X_{i},t_{i})=B_{i}$ given by Lemma \\ref{vor3.12} (cf. Definition \\ref{3.12.2013.10}).\n\\end{lem}\n\\begin{proof}\n\tFor every $i \\in I_{12}\\subset I$, we have with Lemma \\ref{vor3.12} that $B_i=B(X_i,t_i)$ \n\tand $(X_i,t_i) \\in S \\subset S_{total}$. Hence we can use Lemma \\ref{nachlem2.6} \n\t($\\sigma=2 \\varepsilon$, $x=X_{i}$, $t=t_{i}$, $\\lambda = \\frac{\\delta}{2}$, $P=P_{i}$)\n\tto get some $y \\in 2B_i \\cap P_{i}$, where $P_i=P_{(X_i,t_i)}$.\n\tWe obtain with Lemma \\ref{6.9.11.1} ($P_{1}=P_{j}$, $P_{2}=P_{0}$), $\\alpha \\le \\tilde \\alpha < \\frac{1}{2}$ \n\t($\\tilde \\alpha$ is defined on page \\pageref{alphafuergamma})\n\tand Lemma \\ref{vor3.12}\n\t\\begin{align*}\n\t\td(u+A_{i}(u),y) \n\t\t& \\le \\frac{1}{1-\\alpha}d(u,\\pi(y))\n\t\t < 2[d(u, \\pi(X_i))+ d(\\pi(X_i),\\pi(y))]\n\t\t \\le C \\operatorname{diam} R_i.\n\t\\end{align*}\n\tMoreover, with Lemma \\ref{lem3.12} (iv) and $\\varepsilon \\le \\tilde \\varepsilon \\le 1$ \n\t($\\tilde \\varepsilon$ is defined on page \\pageref{epsilonfuergamma}), we get\n\t\\[d(\\pi_{P_i}(u+A(u)),u+A_i(u)) \\le d(u+A(u),u+A_i(u)) \\le C\\operatorname{diam} R_i\\]\n\tfor some $C=C({N},n,C_{0})$. Using these estimates, $u+A_i(u)=\\pi_{P_i}(u+A_i(u))$ and triangle inequality,\n\twe obtain the assertion.\n\\end{proof}\n\nNow, with Lemma \\ref{29.12.09;1} and $K_{i}$ from \\eqref{2.1.10;1}, we obtain for $i \\in \\mathcal{I}(q,t) \\subset I_{12}$\n\\begin{align}\n\tK_{i} \n\t& \\stackrel{\\hphantom{\\text{L. \\ref{29.12.09;1}}}}{\\le} \n\t\t\\frac{1}{t^n} \\int_{B(q,t)\\cap R_{i}} \\frac{d(u+A(u),P_{i})}{t}\\mathrm{d} \\mathcal{H}^{n}(u)\\nonumber \\\\ \n\t&\\hphantom{\\stackrel{\\text{L. \\ref{29.12.09;1}}}{\\le}}\n\t\t + \\frac{1}{t^n} \\sup \\left\\{ \\frac{d(\\pi_{P_{i}}(v + A(v)),\\hat P)}{t} \\Big| v \\in B(q,t) \\cap R_{i} \\right\\}\n\t\t\\mathcal{H}^{n}(B(q,t)\\cap R_{i})\\nonumber \\displaybreak[1] \\\\ \n\t& \\stackrel{\\text{L. \\ref{29.12.09;1}}}{\\le} \\frac{1}{t^n} \n\t\t\\int_{B(q,t)\\cap R_{i}} \\frac{d(u+A(u),P_{i})}{t}\\mathrm{d} \\mathcal{H}^{n}(u)\\nonumber \\\\\n\t& \\hphantom{\\stackrel{\\text{L. \\ref{29.12.09;1}}}{\\le}}\n\t\t + \\omega_{\\N} \\left( \\frac{\\operatorname{diam} R_{i}}{t}\\right)^n \\sup \\left\\{ \\frac{d(w,\\hat P)}{t} \n\t\t\\Big| w \\in P_{i}, d(w,B_{i}) \\le C_{4} \\operatorname{diam} R_{i} \\right\\}.\n\t\t\\label{2.1.10;3}\n\\end{align}\nSince $P_{i}$ is the graph of $A_{i}$, we get for any $u \\in B(q,t)\\cap R_{i}$ with Lemma \\ref{lem3.12} (iv) \nthat there exists some $\\bar C= \\bar C({N},n,C_{0})$ with\n\\begin{align*}\n\td(u+ A(u),P_{i}) & \\le d(u + A(u),u+A_{i}(u))\n\t = d(A(u),A_{i}(u)) \n\t \\le \\bar C\\varepsilon \\operatorname{diam} R_{i},\n\\end{align*}\nand so, using Lemma \\ref{24.10.12.1},\n\\begin{align}\n\t\\frac{1}{t^n} \\int_{B(q,t)\\cap R_{i}} \\frac{d(u+A(u),P_{i})}{t}\\mathrm{d} \\mathcal{H}^{n}(u) \n\t\t& \\le \\varepsilon \\ C({N},n,C_{0}) \\left( \\frac{\\operatorname{diam} R_{i}}{t} \\right)^{n+1}. \\label{2.1.10;4}\n\\end{align}\n\n\\begin{lem} \\label{lem4.3}\n\tThere exists some constant $C=C({N},n,C_{0})$ so that for all $i \\in \\mathcal{I}(q,t)$\n\t\\begin{align*}\n\t\t & \\ \\ \\ \\sup \\left\\{ \\frac{d(w,\\hat P)}{t} \\Big| w \\in P_{i}, d(w,B_{i}) \n\t\t\t\\le C_{4} \\operatorname{diam} R_{i} \\right\\} \\\\\n\t\t &\\le C \\varepsilon \\frac{\\operatorname{diam} R_{i}}{t} + C \\frac{1}{t} \\left( \\frac{1}{(\\operatorname{diam} R_{i})^n}\n\t\t \\int_{2B_{i}} d(z,\\hat P)^{\\frac{1}{3}} \\mathrm{d} \\mu(z) \\right)^{3}.\n\t\\end{align*}\n\\end{lem}\n\\begin{proof} \n\tLet $i \\in \\mathcal{I}(q,t)$.\n\tDue to the construction of $B_{i}=B(X_{i},t_{i})$ (see Lemma \\ref{vor3.12}), we have \n\t$(X_{i},t_{i}) \\in S \\subset S_{total}$\n\tand so\n\t$\\delta(X_{i},t_{i}) \\ge \\frac{\\delta}{2}$.\n\tWith Corollary \\ref{04.09.12.1} ($\\lambda=\\frac{\\delta}{2}$, $B(x,t)=B(X_{i},t_{i})$, $\\Upsilon=\\mathbb{R}^{{N}}$), \n\tthere exist constants $C_{1}=C_{1}({N},n,C_{0})>3$, $C_{2}=C_{2}({N},n,C_{0})>1$ \n\tand some $(n,10n\\frac{t_i}{C_1})$-simplex $T=\\Delta(x_0,\\dots,x_{n}) \\in F \\cap B_i$\n\twith\n\t\\begin{align}\\label{11.03.2013.1}\n\t\t\\mu\\left( B\\left(x_\\kappa,\\textstyle{\\frac{t_i}{C_1}}\\right) \\cap B_{i}\\right) \n\t\t\\ge \\textstyle{\\frac{t_{i}^{n}}{C_{2}}} \\ \\ \\text{and} \\ \\\n\t\tB\\left(x_{\\kappa},\\frac{t_i}{C_1}\\right) \\subset 2B_{i} \\subset kB_{i}=B(X_{i},kt_{i}).\n\t\\end{align}\n\tfor all $\\kappa = 0, \\dots, n$ and we used that $C_{1} > 3$ and $k \\ge \\tilde k \\ge 2$ \n\t($\\tilde k$ is defined on page \\pageref{22.11.2013.1})., we have\n\tWe set $C^{'} := 400C_{2}$,\n\t$\\tilde B_{\\kappa}:=B\\left(x_{\\kappa},\\frac{t_i}{C_1}\\right)$\n\tand define for all $\\kappa = 0,\\dots,n$\n\t\\begin{align} \\label{29.12.09;e}\n\t\tZ_{\\kappa}:= \\left\\{ z \\in \\tilde B_{\\kappa} \\cap F \\big| d(z,P_{i}) \\le C^{'} \\varepsilon \\operatorname{diam} R_{i} \\right\\}.\n\t\\end{align}\n\tWe have $(X_{i},t_{i}) \\in S_{total}$ and hence $\\beta_{1;k}^{P_{i}}(X_i,t_i) \\le 2 \\varepsilon$. Using this\n\tand Lemma \\ref{vor3.12}, we obtain with Chebyshev's inequality\n\t\\[\\mu(\\tilde B_{\\kappa}\\setminus Z_{\\kappa}) \n\t\t < \\frac{t_{i}^{n+1}}{C^{'}\\varepsilon \\operatorname{diam} R_{i}} \\beta_{1;k}^{P_{i}}(X_i,t_i)\n\t\t\\le \\frac{t_{i}^{n+1} \\ 100}{C^{'}\\varepsilon t_{i}} 2\\varepsilon\n\t\t = \\frac{t_{i}^n}{2C_{2}}.\\]\n\tUsing Lemma \\ref{vor3.12} again, we get\n\t\\begin{align}\n\t\t\\mu(Z_{\\kappa}) \n\t\t&\\stackrel{\\hphantom{\\eqref{11.03.2013.1}}}{\\ge}\n\t\t\t \\mu(\\tilde B_{\\kappa}) - \\mu(\\tilde B_{\\kappa}\\setminus Z_{\\kappa}) \n\t\t \\stackrel{\\eqref{11.03.2013.1}}{\\ge} \\frac{t_i^{n}}{C_{2}} \n\t\t\t- \\frac{t_{i}^{n}}{2C_{2}} \n\t\t \\stackrel{\\hphantom{\\eqref{11.03.2013.1}}}{=} \\frac{t_{i}^{n}}{2C_{2}} \n\t\t \\stackrel{\\hphantom{\\eqref{11.03.2013.1}}}{\\ge} \\frac{\\operatorname{diam} R_{i}^{n}}{2^{n+1}C_{2}} >0. \\label{29.12.09;g}\n\t\\end{align}\n\tFor all $\\kappa \\in \\{0,\\dots,n\\}$, let $z_{\\kappa} \\in Z_{\\kappa} \\subset \\tilde B_{\\kappa}$ \n\tand set $y_{\\kappa}:=\\pi_{P_i}(z_{\\kappa})$. \n\tSince $\\varepsilon \\le \\tilde \\varepsilon \\le \\frac{1}{2C^{'} C_{1}}$ ($\\tilde \\varepsilon$ was chosen on\n\tpage \\pageref{epsilonfuergamma}), we deduce \n\t\\begin{align*} \n\t\td(y_{\\kappa},x_{\\kappa}) \n\t\t& \\le d(y_{\\kappa},z_{\\kappa}) + d(z_{\\kappa},x_{\\kappa})\n\t\t \\le d(z_{\\kappa},P_i) + \\frac{t_i}{C_1}\n\t\t \\stackrel{\\eqref{29.12.09;e}}{\\le} C^{'}\\varepsilon \\operatorname{diam} R_i + \\frac{t_i}{C_1}\n\t\t \\le 2\\frac{t_i}{C_1}.\n\t\\end{align*}\n\tDue to Lemma \\ref{17.11.11.2},\n\tthe simplex $S=\\Delta(y_0, \\dots,y_{n})$ is an $(n,6n\\frac{t_i}{C_1})$-simplex and, \n\tusing the triangle inequality, we obtain $S \\subset 2B_i$.\n\tNow, with Lemma \\ref{30.05.12.1}, ($C=\\frac{C_1}{6n}$, $\\hat C=2$, $t=t_{i}$, $m=n$, $x=X_{i}$) \n\tthere exists some orthonormal basis $(o_1, \\dots, o_n)$ of \n\t$P_{i}-y_{0} $ and there exists $\\gamma_{l,r} \\in \\mathbb{R}$ with\n\t$o_l = \\sum_{r=1}^{l} \\gamma_{l,r} (y_r-y_0)$\n\tand\n\t$|\\gamma_{l,r}| \\le \\left(\\frac{2C_1}{3}\\right)^{n} \\frac{C_1}{6n t_i}$\n\tfor all $1 \\le l \\le n$ and $1 \\le r \\le l$.\n\n\tNow let $ w \\in P_{i}$ with $d(w,B_{i}) \\le C_{4} \\operatorname{diam} R_{i}$. \n\tWe obtain\n\t\\begin{align} \\label{10.12.2013.1}\n\t\tw -y_0 \n\t\t& = \\sum_{\\kappa = 1}^{n} \\langle w-y_0,o_{\\kappa} \\rangle o_{\\kappa} \n\t\t = \\sum_{\\kappa = 1}^{n} \\langle w-y_0,o_{\\kappa} \\rangle \\sum_{r=1}^{\\kappa} \\gamma_{\\kappa,r} (y_r-y_0)\n\t\\end{align}\n\tand so, with Remark \\ref{12.4.11} ($b=w$, $P=\\hat P$)\n\tand $|w-y_{0}|\\le d(w,B_i) + \\operatorname{diam} B_i + d(B_i,y_0) \\le Ct_i$, we get\n\t\\begin{align}\n\t\td(w,\\hat P) \n\t\t& \\stackrel{\\eqref{10.12.2013.1}}{\\le} \n\t\t\tn C C_{1}^{n+1} \\sum_{r=1}^{n} \\left(d(y_r,z_r) + d(z_r,\\hat P) \\right) \\nonumber \\\\\\displaybreak[1]\n\t\t& \\stackrel{\\eqref{29.12.09;e}}{\\le} \n\t\t\tn^{2} CC_1^{n+1}C^{'} \\varepsilon \\operatorname{diam} R_{i}\n\t\t\t+ n CC_1^{n+1} \\sum_{r=0}^{n} d(z_{r},\\hat P). \\label{29.12.09;f}\n\t\\end{align}\n\tThe previous results are valid for arbitrary $z_{\\kappa} \\in Z_{\\kappa}$, hence we get\n\t\\begin{align*}\n\t\t& \\hphantom{\\eqref{29.12.09;g} \\eqref{11.03.2013.1} }\n\t\t\td(w,\\hat P) -n^{2} CC_1^{n+1}C^{'} \\varepsilon \\operatorname{diam} R_{i}\\\\\n\t\t& \\stackrel{\\substack{\\hphantom{\\eqref{29.12.09;g} \\eqref{11.03.2013.1}} \\\\ \\eqref{29.12.09;f}}}{\\le} \n\t\t\t\\left(\\frac{1}{\\prod_{r=0}^{n} \\mu(Z_r)} \\int_{Z_{0}} \\dots \\int_{Z_{n}} \n\t\t\t\\left(n CC_1^{n+1} \\sum_{r=0}^{n} d(z_{r},\\hat P) \\right)^{\\frac{1}{3}} \\\n\t\t\t\\mathrm{d} \\mu(z_{n}) \\dots \\mathrm{d} \\mu(z_{0})\\right)^{3} \\\\ \\displaybreak[1]\n\t\t& \\stackrel{\\hphantom{\\eqref{29.12.09;g} \\eqref{11.03.2013.1}}}{\\le} n CC_1^{n+1} \n\t\t\t\\left( \\sum_{r=0}^{n} \\frac{1}{\\mu(Z_r)} \\int_{Z_{r}} \n\t\t\t d(z_{r},\\hat P)^{\\frac{1}{3}} \\\n\t\t\t\\mathrm{d} \\mu(z_{r})\\right)^{3} \\\\ \\displaybreak[1]\n\t\t& \\stackrel{\\eqref{29.12.09;g} \\eqref{11.03.2013.1}}{\\le} n CC_1^{n+1}\n\t\t\t\\left( \\frac{2^{n+1}C_{2}}{\\operatorname{diam} R_{i}^{n}} \\int_{2B_{i}}\n\t\t\td(z,\\hat P)^{\\frac{1}{3}} \\mathrm{d} \\mu(z) \\right)^{3},\n\t\\end{align*}\n\twhere we used that the sets $Z_{r}$ are disjoint.\n\tSince $w \\in P_{i}$ was arbitrarily chosen with $d(w,B_{i}) \\le C_{4} \\operatorname{diam} R_{i}$, we get the statement.\n\\end{proof}\n\\begin{lem} \\label{lem4.4}\n\tThere exists some constant $C=C(n,C_{0})$ so that\n\t\\begin{align*}\n\t\t\\sum_{i \\in \\mathcal{I}(q,t)} \\left(\\frac{\\operatorname{diam} R_{i}}{t}\\right)^{n} \\frac{1}{t} \n\t\t\t\\left( \\frac{1}{(\\operatorname{diam} R_{i})^{n}} \\int_{2B_{i}}\n\t\t\td(z,\\hat P)^{\\frac{1}{3}} \\mathrm{d} \\mu(z) \\right)^{3} \n\t\t& \\le C \\beta_{1;k}(X,t).\n\t\\end{align*}\n\\end{lem}\n\\begin{proof}\n\tLet $i \\in \\mathcal{I}(q,t)$ ($\\mathcal{I}(q,t)$ is defined on page \\pageref{2.1.10;21})\n\tand $x \\in 2B_{i}$. \n\tWe define \\label{28.2.10;2}\n\t\\[ J(i) := \\left\\{ j \\in \\mathcal{I}(q,t) \\big| \\operatorname{diam} B_{j} \\le \n\t\t\t\\operatorname{diam} B_{i}, 2B_{i} \\cap 2B_{j} \\neq \\emptyset \\right\\}, \\ \\ \\text{and} \\ \\\n\t\t\\Xi_{i}(x) := \\sum_{j \\in J(i)} \\chi_{_{2B_{j}}}(x).\\]\n\tAt first, we prove some intermediate results:\\\\\n\tI.\\,\tFor all $i \\in \\mathcal{I}(q,t)$, we have\n\t\t$\\int_{2B_{i}}\\Xi_{i}(x) \\mathrm{d} \\mu(x) \\le C(n,C_{0}) (\\operatorname{diam} R_{i})^{n}$.\n\t\tThis implies that $\\Xi_{i}(x)< \\infty$ for $\\mu$-almost all $x \\in 2B_{i}$.\n\t\t\\begin{proof}\n\t\t\tLet $i \\in \\mathcal{I}(q,t)$ and $j \\in J(i)$. With Lemma \\ref{vor3.12} applied to $j$ \n\t\t\tand the definition of $J(i)$, we deduce\n\t\t\t$\\operatorname{diam} R_{j} \\le 200 \\operatorname{diam} R_{i}$. Using Lemma \\ref{vor3.12} and $j \\in J(i)$, we get\n \t\t\t$ d(R_{i},R_{j}) \\le C \\operatorname{diam} R_{i}$.\n\t\t\tThis implies for some large enough constant $C>1$ that $R_{j} \\subset C R_{i}$.\n\t\t\tSince the cubes $ \\mathring{R}_{j}$ are disjoint\n\t\t\t(see Lemma \\ref{inneredisjunkt} (ii)), \n\t\t\twe get with Lemma \\ref{24.10.12.1}\n\t\t\t\\begin{align*}\n\t\t\t\t\\sum_{j\\in J(i)} (\\operatorname{diam} R_{j})^{n} \n\t\t\t\t& = \\sum_{j\\in J(i)} (\\sqrt{n})^{n} \\mathcal{H}^{n}(R_{j})\n\t\t\t\t \\le (\\sqrt{n})^{n} \\mathcal{H}^{n}\\big( C R_{i}\\big)\n\t\t\t\t = C(n) (\\operatorname{diam} R_{i})^{n}.\n\t\t\t\\end{align*}\n\t\t\tIn the following, we apply Fatou's Lemma \\cite[1.3, Thm.1]{Evans} to \n\t\t\tinterchange the integration with the summation.\n\t\t\tWith (B) from page \\pageref{Grundeigenschaften}\n\t\t\tand Lemma \\ref{vor3.12}, we obtain\n\t\t\t\\[\\int_{2B_{i}}\\Xi_{i}(x) \\mathrm{d} \\mu(x) \\le \\sum_{j \\in J(i)} \\mu(2B_{j})\n\t\t\t\t \\stackrel{\\text{(B)}}{\\le} C(n,C_{0}) \\sum_{j\\in J(i)} (\\operatorname{diam} R_{j})^{n}\n\t\t\t\t \\le C(n,C_{0}) (\\operatorname{diam} R_{i})^{n}.\\]\n\t\t\\end{proof} \\noindent\n\tII.\\, Let $x \\in \\mathbb{R}^{{N}}$ and $m \\in \\mathbb{N}$. There exists some $C=C(n)>1$ with\n\t\t$\\sum_{\\genfrac{}{}{0pt}{}{i \\in \\mathcal{I}(q,t) }{ \\Xi_{i}(x)=m}} \\chi_{_{2B_{i}}}(x) \\le C$.\n\t\t\\begin{proof}\n\t\t\tLet $l,o \\in \\mathcal{I}(q,t)$ with $x \\in 2B_{l} \\cap 2B_{o}$ and\n\t\t\t$\\Xi_{l}(x) = m = \\Xi_{o}(x)$.\n\t\t\tWithout loss of generality, we have $\\operatorname{diam} B_{l} \\le \\operatorname{diam} B_{o}$.\n\n\t\t\tAssume that $ \\operatorname{diam} B_{l} < \\operatorname{diam} B_{o}$.\n\t\t\tWe define\n\t\t\t$J(l,x) := \\left\\{ \\iota \\in J(l) \\big| x \\in 2B_{\\iota}\\right\\}$.\n\t\t\tLet $j \\in J(l,x)$. By definition of $J(l)$, we get\n\t\t\t$ \\operatorname{diam} B_{j} \\le \\operatorname{diam} B_{l} < \\operatorname{diam} B_{o}$ \n\t\t\tand $x \\in 2B_{j}$. \n\t\t\tSince $x \\in 2B_{o}$, it follows $2B_{o} \\cap 2B_{j} \\neq \\emptyset$ and,\n\t\t\tbecause $\\operatorname{diam} B_{j} < \\operatorname{diam} B_{o}$, \n\t\t\twe get $j \\in J(o,x)$. \n\t\t\tFurthermore, we have $o \\in J(o,x)$, but $o \\notin J(l,x)$ because by our assumption\n\t\t\twe have $ \\operatorname{diam} B_{l} < \\operatorname{diam} B_{o}$. So we get\n\t\t\t$J(l,x) \\subsetneq J(o,x)$.\n\t\t\tNow we obtain a contradiction\n\t\t\t\\[ m=\\Xi_{l}(x)= \\sum_{j \\in J(l)} \\chi_{_{2B_{j}}}(x)=\\sum_{j \\in J(l,x)} \\chi_{_{2B_{j}}}(x)<\n\t\t\t\t\\sum_{j \\in J(o,x)} \\chi_{_{2B_{j}}}(x)=\\Xi_{o}(x)=m.\\]\n\t\t\tHence there exists some $\\lambda=\\lambda(x,m) \\in (0,\\infty)$ so that,\n\t\t\tfor $l \\in \\mathcal{I}(q,t)$ with $x \\in 2B_{l}$ and\n\t\t\t$\\Xi_{l}(x) = m$, we have \n\t\t\t$\\operatorname{diam} B_{l} = \\lambda$, \n\t\t\tand, we obtain with Lemma \\ref{vor3.12} that\n\t\t\t$\\lambda \\le 200 \\operatorname{diam} R_{l} \\le 200 \\lambda$ and $d(R_{l},\\pi(B_{l})) \\le 100 \\lambda$.\n\t\t\tUsing\n\t\t\t$d(R_{l},\\pi(x)) \\le d(R_{l},\\pi(B_{l}))+2\\operatorname{diam} B_{l} \\le 102 \\lambda$,\n\t\t\twe get $R_{l} \\subset B(\\pi(x),103 \\lambda) \\cap P_{0}$.\n\t\t\tWith Lemma \\ref{24.10.12.1}, we have \n\t\t\t$\\mathcal{H}^{n}(R_{l}) \\ge (\\sqrt{n})^{-n}(\\textstyle{\\frac{1}{200}}\\lambda)^{n}$\n\t\t\tand, according to Lemma \\ref{inneredisjunkt} (ii) the cubes $R_{l}$ \n\t\t\thave disjoint interior. This implies that there exists some constant $C(n)$ so that\n\t\t\tthere are at most $C(n)$ indices $l \\in \\mathcal{I}(q,t)$ with $\\Xi_{l}(x)=m$ and $x \\in 2B_{l}$.\n\t\t\tThis implies the assertion.\n\t\t\\end{proof} \\noindent\n\tIII.\\, We have $i \\in J(i)$ and so $\\Xi_{i}(x) \\neq 0$ for all $x \\in 2B_{i}$. \n\t\tHence, with $x \\in \\mathbb{R}^{{N}}$, the term\n\t\t\\[ \\chi_{_{2B_{i}}}(x)\\Xi_{i}(x)^{-2}:= \\begin{cases}\n\t\t\t\t\t\t\t\t\\Xi_{i}(x)^{-2} &\\text{if } x \\in 2B_{i}\\\\\n\t\t\t\t\t\t\t\t0\t\t& \\text{otherwise}\n\t\t\t\t\t\t\t\\end{cases} \\]\n\t\tis well-defined.\n\t\tNow there exists some constant $C(n)$ so that, for all $x \\in \\mathbb{R}^{{N}}$, we get\n\t\t\\begin{align*}\n\t\t\t\t\\sum_{i \\in \\mathcal{I}(q,t)} \\chi_{_{2B_{i}}}(x)\\Xi_{i}(x)^{-2} \n\t\t\t\t& = \\sum_{m=1}^{\\infty} \\sum_{\n\t\t\t\t\t\\genfrac{}{}{0pt}{}{i \\in \\mathcal{I}(q,t) }{ \\Xi_{i}(x)=m}} \\chi_{_{2B_{i}}}(x)\n\t\t\t\t\\frac{1}{m^{2}} \\stackrel{\\text{II}}{\\le} C(n).\n\t\t\\end{align*} \\noindent\n\tIV.\\, Let $i \\in \\mathcal{I}(q,t)$. Since $i \\in J(i)$, we have $\\Xi_{i}(x) \\neq 0$ for $x \\in 2B_{i}$. \n\t\tWe obtain with H\\\"older's inequality\n\t\t\\begin{align*}\n\t\t\t& \\ \\ \\ \\left( \\frac{1}{(\\operatorname{diam} R_{i})^{n}} \\int_{2B_{i}} d(z,\\hat P)^{\\frac{1}{3}} \\Xi_{i}(z)^{\\frac{-2}{3}}\n\t\t\t\t\\Xi_{i}(z)^{\\frac{2}{3}} \\mathrm{d} \\mu(z) \\right)^{3}\\\\ \\displaybreak[1]\n\t\t\t& \\stackrel{\\text{I}}{\\le} C(n,C_{0}) \\frac{1}{(\\operatorname{diam} R_{i})^{n}} \n\t\t\t\t\\int_{2B_{i}} d(z,\\hat P) \\Xi_{i}(z)^{-2} \\mathrm{d} \\mu(z).\n\t\t\\end{align*}\\noindent\n\tV.\\, We have\n\t\t\\[ \\frac{1}{t^{n+1}} \\int_{\\bigcup_{i\\in \\mathcal{I}(q,t)}2B_{i}} d(z,\\hat P) \n\t\t\t\\mathrm{d} \\mu(z) \\le 2 \\beta_{1;k}(X,t),\\]\n\t\twhere $X \\in B(\\tilde X(q),200t)$ (cf. page \\pageref{2.1.10;25}).\n\t\t\\begin{proof}\n\t\t\tAt first, we prove that there exists some constant $\\hat C>1$ so that \n\t\t\tfor $i \\in \\mathcal{I}(q,t)$ we have\n\t\t\t$ 2B_{i} \\subset B(X,\\hat Ct)$. \n\t\t\tLet $i \\in \\mathcal{I}(q,t)$. \n\t\t\tBy definition of $\\mathcal{I}(q,t)$ (see page \\pageref{2.1.10;21}), we obtain\n\t\t\t$R_{i} \\cap B(q,t) \\neq \\emptyset$.\n\t\t\tLet $\\tilde{u} \\in R_{i} \\cap B(q,t)$.\n\t\t\tSince $ \\frac{D(q)}{100} < t$ (see page \\pageref{2.1.10;25}), we get,\n\t\t\tusing the triangle inequality,\n\t\t\t$D(\\tilde{u}) \\le D(q)+ d(q,\\tilde{u}) < 101t$.\n\t\t\tIt follows with Lemma \\ref{lem3.11} (i) that\n\t\t\t\\begin{align} \\label{2.1.10;c}\n\t\t\t\t\\operatorname{diam} R_{i} \\le \\textstyle{\\frac{1}{10}} D(\\tilde u) < 11t.\n\t\t\t\\end{align}\n\t\t\tWith Lemma \\ref{vor3.12} and \\eqref{2.1.10;27} from page \\pageref{2.1.10;27}, we\n\t\t\tget ($X \\in B(\\tilde{X},200t)$, see page \\pageref{16.3.10;1})\n\t\t\t\\begin{align} \\label{2.1.10;d}\n\t\t\t\td(\\pi(B_{i}),\\pi(X)) &\\stackrel{\\phantom{\\eqref{2.1.10;27}}}{\\le} d(\\pi(B_{i}),\\tilde u) + d(\\tilde u,q)+ \n\t\t\t\t\td(q,\\pi(\\tilde X)) + d(\\pi(\\tilde X),\\pi(X))\\nonumber \\\\ \n\t\t\t\t& \\stackrel{\\eqref{2.1.10;27}}{\\le} \n\t\t\t\t\td(\\pi(B_{i}),R_{i}) + \\operatorname{diam} R_{i} + t + 200t + d(\\tilde X,X) \\stackrel{\\eqref{2.1.10;c}}{\\le} C t.\n\t\t\t\\end{align}\n\t\t\tNow let $x \\in 2B_{i}=B(X_{i},2t_{i})$. Since $(X_{i},t_{i}) \\in S$,\n\t\t\tusing Lemma \\ref{vor3.12} and \\eqref{2.1.10;c}, we get\n\t\t\t$d(x) < 4400t$.\n\t\t\tDue to $X \\in B(\\tilde X,200t) \\cap F$ and \\eqref{2.1.10;27}, we deduce $d(X) \\le 400 t$.\n\t\t\tWith Lemma \\ref{vor3.12} and estimates \\eqref{2.1.10;c} and \\eqref{2.1.10;d}, we obtain\n\t\t\twith triangle inequality $d(\\pi(x),\\pi(X)) \\le Ct$.\n\t\t\tNow there exists some constant $\\hat C > 1$ so that, we get with Lemma \\ref{lem3.9}\n\t\t\t$d(x,X) \\le \\hat C t$.\n\t\t\tAll in all we have proven that, for all $i \\in \\mathcal{I}(q,t)$, we have\n\t\t\t$2B_{i} \\subset B(X,\\hat C t)$.\n\t\t\tSince $k \\ge \\tilde k \\ge \\hat C$ (see page \\pageref{22.11.2013.1}), we get the assertion\n\t\t\twith condition \\eqref{2.1.10;30} from page \\pageref{2.1.10;30}.\n\t\t\\end{proof}\n\tNow, Lemma \\ref{lem4.4} can be proven by applying IV, III, and V and using \n\tthe monotone convergence theorem \\cite[1.3, Thm. 2]{Evans} to interchange the summation and the integration\n\\end{proof}\n\nNow we can give some estimate for $\\gamma(q,t)$, where $q \\in U_{10}$ and $\\frac{D(q)}{100} < t < 2.$\nUsing the inequalities \\eqref{2.1.10;1}, \\eqref{2.1.10;2}, \\eqref{2.1.10;3}, \\eqref{2.1.10;4}, \nLemma \\ref{lem4.3} and Lemma \\ref{lem4.4}, we get using $T \\le 200t$ \n(cf. Lemma \\ref{28.11.2013.1}) for every \n$X \\in B(\\tilde X,T) \\cap F \\subset B(\\tilde X,200t) \\cap F$ \n\\[\\gamma(q,t) \\le C({N},n,C_{0})\\ \\beta_{1;k}(X,t) + C({N},n,C_{0}) \\\n\t\t \\varepsilon \\sum_{i \\in \\mathcal{I}(q,t)} \\left( \\frac{\\operatorname{diam} R_{i}}{t} \\right)^{n+1}.\\] \nWith Lemma \\ref{28.11.2013.1}, we get $(\\tilde X, T) \\in S \\subset S_{total}$ and $20t \\le T \\le 200t$. Using this,\nthe previous estimate, the definition of $\\delta=\\delta(n)$ on page \\pageref{Wahlvondelta} and\n(B) from page \\pageref{Grundeigenschaften}, we get\n\\begin{align*}\n\t\\gamma(q,t)^{p} \n\t& \\stackrel{\\hphantom{\\text{(B)}}}{\\le} \\frac{2}{\\delta T^{n}}\\int_{B(\\tilde X,T)} \\gamma(q,t)^{p} \\mathrm{d} \\mu(X)\\\\ \\displaybreak[1]\n\t& \\stackrel{\\hphantom{\\text{(B)}}}{\\le} C\\frac{1}{t^{n}}\\int_{B(\\tilde X,200t)} \n\t\\beta_{1;k}(X,t)^{p}\\mathrm{d} \\mu(X) \n\t +C\\varepsilon^{p}\\left( \\sum_{i \\in \\mathcal{I}(q,t)} \\left( \\frac{\\operatorname{diam} R_{i}}{t} \\right)^{n+1} \\right)^{p} ,\n\\end{align*}\nwhere $C=C({N},n,p,C_{0})$.\nWe recall that for every $q \\in U_{10}$ there exists some $\\tilde X=\\tilde X(q)$ (cf. Lemma \\ref{28.11.2013.1}) such that\nthe previous inequality is valid. This implies\n\\begin{align}\n\t\\int_{U_{10}} \\int_{\\frac{D(q)}{100}}^{2} \\gamma(q,t)^{p} \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mathcal{H}^{n}(q) \n\t& \\le C({N},n,p,C_{0}) \\ a + C({N},n,p,C_{0}) \\ \\varepsilon^{p} \\ b, \\label{10.1.2010;c}\n\\end{align}\nwhere\n\\[a:= \\int_{U_{10}} \\int_{\\frac{D(q)}{100}}^{2} \\frac{1}{t^{n}}\\int_{B(\\tilde X(q),200t)} \n\t\t\\beta_{1;k}(X,t)^{p}\\mathrm{d} \\mu(X) \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mathcal{H}^{n}(q),\\]\n\\[b:= \\int_{U_{10}} \\int_{\\frac{D(q)}{100}}^{2} \\left( \\sum_{i \\in \\mathcal{I}(q,t)} \n\t\t\\left( \\frac{\\operatorname{diam} R_{i}}{t} \\right)^{n+1} \\right)^{p} \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mathcal{H}^{n}(q).\\]\nTo estimate $a$ and $b$, we need the following lemma.\n\\begin{lem}\\label{06.12.2013.1}\n\tLet $q \\in U_{10}$, $\\frac{D(q)}{100} \\le t \\le 2$ and $X \\in B(\\tilde X(q),200t) \\cap F$,\n\twhere $\\tilde X(q)$ is given by Lemma \\ref{28.11.2013.1} on page \\pageref{DefvonXSchlange}.\n\tThen \n\t$d(\\pi(X),q) \\le 400t $\n\tand there exists some $\\tilde \\lambda = \\tilde \\lambda({N},n,C_{0})>0$ so that, with $k_{0}=401$, we have\n\t$\\tilde{\\delta}_{k_{0}} (B(X,t))=\\sup_{y \\in B(X,k_{0}t)} \\frac{\\mu(B(y,t))}{t^{n}} \\ge \\tilde \\lambda$,\n\twhere $\\tilde{\\delta}_{k_{0}} (B(X,t))$ was defined on page \\pageref{Definitionvondeltaschlange}. \n\tFurthermore, there holds for all $i \\in \\mathcal{I}(q,t)$ that\n\t\\begin{align}\\label{06.12.2013.2}\n\t\td(q,R_{i}) &\\le t, &\\operatorname{diam} R_{i} &< 11t,\n\t\\end{align}\n\tand there exists some constant $C=C(n)$ with\n\t\\begin{align}\n\t\t\\sum_{i \\in \\mathcal{I}(q,t)}\\left( \\frac{\\operatorname{diam} R_{i}}{t} \\right)^{n+1} &\\le C, \\ \\ \\ \\ \\ \\ \\ \\ \n\t\t\\sum_{i \\in I_{12}} (\\operatorname{diam} R_{i})^{n} \\le C. \\label{2.1.10;101}\n\t\\end{align}\n\\end{lem}\n\\begin{proof}\n\tLet $q \\in U_{10}$, $\\frac{D(q)}{100} \\le t \\le 2$ and $X \\in B(\\tilde X(q),200t) \\cap F$.\n\tWe have $d(X, \\tilde X(q)) \\le 200t$ and, with \\eqref{2.1.10;27}, we get $d(\\pi(\\tilde X(q)),q) \\le 200t$. \n\tThis implies $d(\\pi(X),q) \\le 400t$ by using triangle inequality.\n\tWith \\eqref{2.1.10;27}, we obtain\n\t$(\\tilde X(q), T) \\in S \\subset S_{total}$ and, by definition of $S_{total}$, we conclude\n\t$\\delta(B(\\tilde X(q), T)) \\ge \\frac{\\delta}{2}$. We have\n\t$B(\\tilde X(q), T) \\subset B(X,400t)$ and hence with \\eqref{2.1.10;27} we get\n\t$\\delta(B(X,400t)) \\ge \\frac{\\delta}{2\\cdot20^{n}}$.\n\tApplying Corollary \\ref{04.09.12.1} (ii) with $\\lambda = \\frac{\\delta}{2\\cdot 20 ^{n}}$ \n\ton $B(X,400t)$, we get constants $C_{1}=C_{1}({N},n,C_{0})$, $C_{2}=C_{2}({N},n,C_{0})$ and\n\tin particular one ball $B(x,s)$ with $ s = \\frac{400t}{C_{1}}$ and\n\t\\begin{align} \\label{24.2.10;10}\n\t\t\\mu(B(x,s) \\cap B(X,400t)) \\ge {\\textstyle \\frac{(400t)^{n}}{C_{2}}}.\n\t\\end{align}\n\tWe have $\\delta \\le \\frac{2}{50^{n}}$ (cf. \\eqref{Wahlvondelta} on page \\pageref{Wahlvondelta}), \n\tand Lemma \\ref{lem2.3} gives us $C_{1} > 400.$\n\tThat yields $s < t$. From \\eqref{24.2.10;10}, we get $B(x,s) \\cap B(X,400t) \\neq \\emptyset$\n\twhich implies $d(x,X) < 401t$ and with \\eqref{24.2.10;10} we get\n\t$\\sup_{y \\in B(X,401t)} \\delta(B(y,t)) \\ge \\frac{400^{n}}{C_{2}}=:\\tilde \\lambda$.\n\tLet $i \\in \\mathcal{I}(q,t)$. \n\tDue to the definition of $\\mathcal{I}(q,t)$ (see page \\pageref{2.1.10;21}), we have $d(q,R_{i}) \\le t$\n\tand we can choose some $\\tilde u \\in R_{i} \\cap B(q,t)$.\n\tWith Lemma \\ref{lem3.11} (i), we obtain $ 10\\operatorname{diam} R_{i} \\le (D(q) + d(q,\\tilde u)) < 11t$.\n\tThe intervals $ R_{i}$ have disjoint interior (see Lemma \\ref{inneredisjunkt} (ii)) and, from\n\t$R_{i} \\cap B(q,t) \\neq \\emptyset$ for all $i \\in \\mathcal{I}(q,t)$, we get \n\t$R_i \\subset B(q,12t)$. With Lemma \\ref{24.10.12.1}, this implies\n\t\\[\\sum_{i \\in \\mathcal{I}(q,t)} \\left( \\frac{\\operatorname{diam} R_{i}}{t} \\right)^{n+1} \n\t\t\\stackrel{\\eqref{06.12.2013.2}}{<} \\frac{11}{t^{n}} \\sum_{i \\in \\mathcal{I}(q,t)} (\\operatorname{diam} R_{i})^n \n\t\t= \\frac{11}{t^{n}} \\sum_{i \\in \\mathcal{I}(q,t)} (\\sqrt{n})^{n} \\mathcal{H}^{n}(R_i)\\\\ \\nonumber\n\t\t=C(n).\\]\n\tNow let $i \\in I_{12}$. We have $R_{i} \\cap B(0,12) \\neq \\emptyset$. \n\tIf $(Y,r) \\in S \\subset S_{total}$, we get $Y \\in F \\subset B(0,5)$ (cf. (A) on page \\pageref{3.12.2013.3})\n\tand hence we obtain $d(\\pi(Y),0) \\le 5$ as well as $r\\le 50$.\n\tWith $\\tilde v \\in R_{i} \\cap B(0,12)$ and\n\tLemma \\ref{lem3.11} (i), we get\n\t\\[ \\operatorname{diam} R_{i} \\le \\frac{1}{10} D(\\tilde v) \n\t\t= \\frac{1}{10} \\inf_{(Y,r) \\in S}(d(\\pi(Y),\\tilde v) + r) \n\t\t\\le \\frac{1}{10} (5+12 + 50) < 7. \\]\n\tHence, for all $i \\in I_{12}$, we have $R_i \\subset B(0,19)$ and the cubes $R_{i}$ have disjoint interior\n\t(cf. Lemma \\ref{inneredisjunkt} (ii)). With Lemma \\ref{24.10.12.1}, we deduce\n\t$ \\sum_{ i \\in I_{12}} (\\operatorname{diam} R_{i})^{n} = C(n)$.\n\\end{proof}\nTo control the terms $a$ and $b$ we will use \nFubini's Theorem \\cite[1.4, Thm. 1]{Evans}, in the following abbreviated by (F).\nNow, using Lemma \\ref{06.12.2013.1} and Corollary \\ref{thm2.4} ($\\lambda = \\tilde \\lambda$, $k_{0}=401$), we conclude\n\\begin{align*}\n\ta & \\stackrel{\\text{(F)}}{\\le} \n\t\t\\int_{F} \\int_{0}^{2} \\frac{1}{t^{n}} \\int_{U_{10}} \n\t\t\\Eins_{\\left\\{d(\\pi(X),q)\\le 400t\\right\\}} \\mathrm{d} \\mathcal{H}^{n}(q) \\\n\t\t\\Eins_{\\left\\{ \\tilde{\\delta}_{k_{0}}(B(X,t)) \\ge \\tilde \\lambda \\right\\}} \n\t\t\\beta_{1;k}(X,t)^{p} \\ \\frac{\\mathrm{d} t}{t} \\mathrm{d} \\mu(X) \\\\\n\t& \\stackrel{\\hphantom{\\text{(F)}}}{\\le} C({N},n,\\mathcal{K},p,C_0,k) \\ \\mathcal{M}_{\\mathcal{K}^{p}}(\\mu).\n\\end{align*}\nNow we consider the integral $b$. \nWe get using Fatou's Lemma \\cite[1.3, Thm.1]{Evans} to interchange the summation with the integration\n\\begin{align*}\n\tb \\ \\ & \\stackrel{\\eqref{2.1.10;101} \\eqref{06.12.2013.2}}{\\le} C \\ \\int_{U_{10}} \\int_{\\frac{D(q)}{100}}^{2} \\sum_{i \\in I_{12}} \n\t\t\\Eins_{\\left\\{t > \\frac{\\operatorname{diam} R_{i}}{11},d(q,R_{i}) \\le t \\right\\}}\n\t\t\\left( \\frac{\\operatorname{diam} R_{i}}{t} \\right)^{n+1} \\frac{\\mathrm{d} t}{t}\\mathrm{d} \\mathcal{H}^{n}(q) \\\\ \\displaybreak[1]\n\t& \\stackrel{\\substack{\\hphantom{\\eqref{2.1.10;101} \\eqref{06.12.2013.2}}\\\\(F)}}{\\le} C \\ \\sum_{i \\in I_{12}} (\\operatorname{diam} R_{i})^{n+1} \\int_{\\frac{\\operatorname{diam} R_{i}}{11}}^{\\infty} \n\t\t\\int_{U_{10}} \\Eins_{\\left\\{d(q,R_{i}) \\le t \\right\\}} \\mathrm{d} \\mathcal{H}^{n}(q) \n\t\t \\frac{\\mathrm{d} t}{t^{n+2}}\n\t\\stackrel{\\eqref{2.1.10;101}}{\\le} C(n,p).\n\\end{align*}\nDue to Lemma \\ref{inneredisjunkt} (ii) the proof of Theorem \\ref{thm4.1} is completed by applying \nLemma \\ref{24.2.10;17}, \\eqref{10.1.2010;b}\nand with (C) from page \\pageref{Grundeigenschaften}\nbecause $ \\mathcal{M}_{\\mathcal{K}^{p}}(\\mu) \\stackrel{\\text{(C)}}{\\le} \\eta < \\varepsilon^{p}$ \n(see page \\pageref{3.12.2013.3} and page \\pageref{3.12.2013.4}).\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{\\texorpdfstring{$\\mathcal{Z}$ Is not too Small}{Z Is not too Small}} \\label{notanullset}\nOur aim is to prove Theorem \\ref{16.10.2013.1}. \nIn Definition \\ref{def3.2}, we defined a partition of the support $F$ of our measure $\\mu$ in four parts, namely \n$\\mathcal{Z}$, $F_{1}$, $F_{2}$, $F_{3}$. Then, in section \\ref{17.05.2013.1}, we constructed some function $A$, \nthe graph $\\Gamma$ of which\ncovers the set $\\mathcal{Z}$. To get our main result, we need to know that we covered a major part of $F$.\nIn this last part of the proof of Theorem \\ref{16.10.2013.1}, we show that the $\\mu$-measure of \n$F_{1}$, $F_{2}$, $F_{3}$\nis quite small. In particular, we deduce $\\mu(F_{1} \\cup F_{2} \\cup F_{3}) \\le \\frac{1}{100}$. As stated at the\nbeginning of section \\ref{04.02.2014.1} on page \\pageref{04.02.2014.1}, this completes the proof of \nTheorem \\ref{16.10.2013.1}.\n\n\\subsection{\\texorpdfstring{Most of $F$ is close to the graph of $A$}{Most of F lies near the graph of A}}\nWith $K := 2 \\left(104 \\cdot 10 \\cdot 6 + 214 \\right)$, \\label{WahlvonK}\nwe define the set $G$ by\n\\begin{align*}\n\t\\left\\{ x \\in F \\setminus \\mathcal{Z} \\ | \\ \n\t\t\\forall i \\in I_{12} \\text{ with } \\pi(x) \\in 3R_{i} \\text{, we have } x \\notin KB_{i} \\right\\} \n\t \\cup \n\t\\left\\{ x \\in F \\setminus \\mathcal{Z} \\ | \\ \n\t\t\\pi(x) \\in \\pi (\\mathcal{Z}) \\right\\}. \n\\end{align*}\nAt first, we show that the $\\mu$-measure of $G$ is small.\n\\begin{lem} \\label{lem3.14}\n\tLet $0< \\alpha \\le \\frac{1}{280}$. There exist some \n\t$ \\tilde \\varepsilon=\\tilde \\varepsilon({N},n,C_{0},\\alpha)$ so that, if \n\t$\\eta < 2 \\tilde \\varepsilon$ and $k \\ge 4$, \n\tthere exists some constant $C=C({N},n,\\mathcal{K},p,C_{0})$ so that,\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\tilde \\varepsilon)$, we have\n\t\\[\\mu(G) \\le C \\mathcal{M}_{\\mathcal{K}^{p}}(\\mu) \\stackrel{\\text(C)}{\\le} C \\eta,\\]\n\twhere the condition (C) was given on page \\pageref{Grundeigenschaften}.\n\\end{lem}\n\\begin{proof}\n\tLet $0< \\alpha \\le \\frac{1}{280}$ and\n\t$\\tilde \\varepsilon := \\min\\{\\bar \\varepsilon,\\frac{\\alpha}{\\bar C}\\} $ where \n\t$\\bar \\varepsilon$ is given by Lemma \\ref{lem3.9} and $\\bar C = \\bar C({N},n,C_{0})$ is a fixed \n\tconstant defined \n\tin this proof on page \\pageref{12.06.2014.1}. Furthermore let $ \\eta < 2 \\tilde \\varepsilon$, $k \\ge 4$ and\n\t$\\eta \\le 2 \\varepsilon < 2 \\tilde \\varepsilon$.\n\n\tLet $x \\in G$. \n\tIf $x \\in G\\setminus \\pi^{-1}(\\pi(\\mathcal{Z})) \\subset F \\subset B(0,5)$,\n\twith Lemma \\ref{Riueberdeckung} (ii), there exists some $i \\in I_{12}$ with $\\pi(x) \\in R_{i} \\subset 2R_{i}$.\n\tLet $X_{i}$ be the centre of $B_{i}$ (cf. Lemma \\ref{vor3.12}).\n\tWe set\n\t\\[X(x):= \\begin{cases}\n\t\t\tX_i & \\text{if } x \\in G \\setminus \\pi^{-1}(\\pi(\\mathcal{Z}))\\\\\n\t\t\t\\pi(x)+A(\\pi(x)) & \\text{if } x \\in G \\cap \\pi^{-1}(\\pi(\\mathcal{Z})).\n\t\t\\end{cases}\\]\n\t\\claim{1} For all $x \\in G$ and $X=X(x)$ defined as above, we have\n\t\\begin{align}\n\t\td(x,X) &< 7 d(x), \\label{23.10.12.1}\n\t\t&d(\\pi(x),\\pi(X)) \\le \\textstyle{\\frac{d(x)}{10}}, \n\t\t& &\\textstyle{\\frac{d(x)}{2}} \\le d(X,x), \n\t\t& &\\left(X,\\textstyle{\\frac{d(x)}{10}}\\right) \\in S. \n\t\\end{align}\n\t\\proofofclaim{1}\n\n\t1. Case: $x \\in G \\setminus \\pi^{-1}(\\pi(\\mathcal{Z}))$.\\\\\n\tDue to the definition of $G$ and $\\pi(x) \\in 2R_{i} \\subset 3R_{i}$, we have $ x \\notin KB_{i}$.\n\tBy adding some $q \\in R_{i}$ with triangle inequality and using Lemma \\ref{vor3.12} \n\twe get $d(\\pi(x),\\pi(X_{i})) \\le 104 \\operatorname{diam} B_{i} \\label{31.1.10;41}$.\n\tWith Lemma \\ref{vor3.12}, we know $\\bigl( X_{i}, \\frac{\\operatorname{diam} B_{i}}{2} \\bigr) \\in S$ and hence we get\n\t$d(X_{i}) < \\operatorname{diam} B_{i}$.\n\tUsing $x \\notin KB_{i}$ and Lemma \\ref{lem3.9}, we get\n\t$K \\cdot \\frac{\\operatorname{diam} B_i}{2} < d(x,X_i) < 6d(x)+ 214 \\operatorname{diam} B_i $\n\twhich yields by definition of $K$ (cf.~the beginning of this subsection) $104\\operatorname{diam} B_{i} < \\frac{d(x)}{10}$.\n\tFrom the previous two estimates, we get\n\t$d(x,X_i) < 7 d(x)$, i.e., the first inequality holds in this case.\n\tFurthermore, we have the second one since $d(\\pi(x),\\pi(X_{i})) \\le 104 \\operatorname{diam} B_{i} < \\frac{d(x)}{10}$.\n\tWe have $\\bigl( X_{i}, \\frac{\\operatorname{diam} B_{i}}{2} \\bigr) \\in S$, so we get\n\t$d(x) \\le d(X_{i},x) + \\frac{\\operatorname{diam} B_{i}}{2} < d(X_{i},x) + \\frac{d(x)}{2}$,\n\tand hence the third inequality holds in this case.\n\tDue to Lemma \\ref{mengebeschraenkt}, we have \n\t$\\frac{\\operatorname{diam} B_{i}}{2} < \\frac{d(x)}{10} < \\frac{60}{10}<50$ so that \n\twith Lemma \\ref{rem3.1} (ii) we deduce $\\bigl(X,\\frac{d(x)}{10}\\bigr) \\in S$.\n\n\t2. Case: $x \\in G \\cap \\pi^{-1}(\\pi(\\mathcal{Z}))$.\\\\\n\tWe have $\\pi(x) \\in \\pi(\\mathcal{Z})$ and hence $X=\\pi(x)+A(\\pi(x)) \\in \\mathcal{Z}$ \n\t(cf. Definition \\ref{04.11.2013.1}).\n\tBy definition of $\\mathcal{Z}$ and Lemma \\ref{rem3.1} (i), \n\twe obtain $(X,\\sigma) \\in S$ for all $\\sigma \\in (0,50)$ and hence\n\t$\\frac{d(x)}{2} \\le d(X,x)+\\sigma$, which establishes the third estimate.\n\tMoreover, we have\n\t$ d(\\pi(X),\\pi(x)) = d(\\pi(x),\\pi(x))=0$.\n\tUsing Lemma \\ref{rem3.8}, we obtain $d(X)=0$ and hence\n\twe get with Lemma \\ref{lem3.9} $ d(x,X) \\le 6d(x)$.\n\tFurthermore, we have with Lemma \\ref{mengebeschraenkt} that \n\t$\\frac{d(x)}{10} \\le 6 < 50$\n\tso that by definition of $\\mathcal{Z}$, we get\n\t$\\bigl(X,\\frac{d(x)}{10}\\bigr) \\in S$.\n\t\\proofofclaimend{1} \\indent\n\tLet $P_{x}:=P_{\\bigl(X,\\frac{d(x)}{10}\\bigr)}$ be the plane associated to $B(X,\\frac{d(x)}{10})$ \n\t(cf. Definition \\ref{12.07.13.1}). \n\tWe define the set\n\t\\begin{align}\\label{31.1.10;46}\n\t\t\\Upsilon := \\left\\{ u \\in B\\left(X,\\textstyle{\\frac{d(x)}{10}} \\right) \\Big| d(u,P_{x}) \n\t\t\\le \\textstyle{\\frac{8}{\\delta}} \\textstyle{\\frac{d(x)}{10}} \\varepsilon \\right\\}.\n\t\\end{align}\n\tDue to Definition \\ref{12.07.13.1} we have \n\t$\\beta_{1;k}^{P_{x}}\\bigl(X,\\frac{d(x)}{10}\\bigr) \\le 2\\varepsilon$\n\tand hence we get using Chebyshev's inequality\n\t\\begin{align*}\n\t\t\\mu\\left(B\\left(X,\\textstyle{\\frac{d(x)}{10}} \\right) \\setminus \\Upsilon \\right) \n\t\t\\le \\textstyle{\\frac{\\delta}{8 \\varepsilon}} \\left(\\textstyle{\\frac{d(x)}{10}}\\right)^{n} \n\t\t\t \\beta_{1;k}^{P_{x}}\\left(X, \\textstyle{\\frac{d(x)}{10}}\\right) \n\t\t\\le \\textstyle{\\frac{\\delta}{4}} \\left(\\textstyle{\\frac{d(x)}{10}}\\right)^{n}\n\t\\end{align*}\t\n\tSince $\\Upsilon \\subset B\\left(X,\\frac{d(x)}{10}\\right)$ and \n\t$\\delta \\big(B\\big(X,\\frac{d(x)}{10}\\big)\\big) \\ge \\frac{1}{2}\\delta$ (cf.~Definition \\ref{12.07.13.1} of \n\t$S_{total}$), we obtain\n\t\\begin{align*}\n\t\t\\mu\\left(B\\left(X,\\textstyle{\\frac{d(x)}{10}}\\right) \\cap \\Upsilon\\right)\n\t\t \\ge \\mu\\left(B\\left(X,\\textstyle{\\frac{d(x)}{10}} \\right)\\right) \n\t\t\t- \\mu\\left(B\\left(X,\\textstyle{\\frac{d(x)}{10}} \\right) \\setminus \\Upsilon \\right) \n\t\t \\ge \\textstyle{\\frac{\\delta}{4}} \\left(\\textstyle{\\frac{d(x)}{10}}\\right)^{n}.\n\t\\end{align*}\n\tWith Corollary \\ref{04.09.12.1} ($\\lambda=\\frac{\\delta}{4}$, $t=\\frac{d(x)}{10}$), there exist constants\n\t$C_{1}=C_{1}({N},n,C_{0})$, $C_{2}=C_{2}({N},n,C_{0})$ and\n\tan $\\left(n,10n\\frac{d(x)}{10 C_1}\\right)$-simplex \n\t$T=\\Delta(x_0,\\dots,x_{n}) \\in F \\cap B\\left(X,\\frac{d(x)}{10}\\right) \\cap \\Upsilon$\n\tso that for all $j \\in \\{0,\\dots,n \\}$\n\t\\begin{align} \\label{21.12.09;5}\n\t\t\\mu\\left( B\\left(x_{j},\\textstyle{\\frac{d(x)}{10C_1}}\\right) \\cap B\\left(X,\\textstyle{\\frac{d(x)}{10}}\\right) \\cap \\Upsilon\\right) \n\t\t& \\ge \\left(\\textstyle{\\frac{d(x)}{10}}\\right)^{n} \\textstyle{\\frac{1}{C_{2}}}.\n\t\\end{align}\n\tLet $y_j \\in B\\left(x_{j},\\frac{d(x)}{10C_1}\\right)\\cap \\Upsilon$ for all $j \\in \\{0,\\dots,n \\}$.\n\tBy applying Lemma \\ref{17.11.11.2} $(n+1)$ times, we find that \n\t$\\Delta(y_0,\\dots,y_{n})$ is an $\\Bigl(n,8n\\frac{d(x)}{10C_{1}}\\Bigr)$-simplex.\n\n\t\\claim{2}\n\tFor all $x \\in G$, we have $d(x,\\textnormal{aff}(y_{0},\\dots,y_{n})) \\ge \\frac{d(x)}{4}$.\n\n\t\\proofofclaim{2}\n\t\tLet $P_{y}:=\\textnormal{aff}(y_{0},\\dots,y_{n})$ be the plane through $y_0, \\dots,y_n$.\n\t\tApplying Lemma \\ref{21.11.11.2} ($C=\\frac{C_1}{8n}$, $\\hat C=1$, $t=\\frac{d(x)}{10}$,\n\t\t$\\sigma = \\frac{8}{\\delta}\\varepsilon$, $P_{1}=P_{y}$, $P_{2}=P_{x}$,\n\t\t$S=\\Delta(y_{0}, \\dots,y_{n})$, $x=X$, $m=n$) yields $\\varangle(P_{y},P_{x}) \\le \\alpha$,\n\t\twhere we use that $\\varepsilon \\le \\tilde \\varepsilon \\le \\frac{\\alpha}{\\bar C}$ and $\\bar C$ is\n\t\tgiven by Lemma \\ref{21.11.11.2} \\label{12.06.2014.1}.\n\t\tSo, with Definition \\ref{12.07.13.1}, we obtain $\\varangle(P_{y},P_{0}) \\le 2\\alpha$.\n\t\tLet $\\hat P_y \\in \\mathcal{P}({N},n)$ be the $n$-dimensional plane parallel to $P_y$ \n\t\twith $X \\in \\hat P_y$, and \n\t\t$\\hat P_0 \\in \\mathcal{P}({N},n)$ be the plane parallel to $P_0$ with $X \\in \\hat P_0$.\n\t\tWe have $\\alpha \\le \\frac{1}{280}$ and hence\n\t\t\\[ d(\\pi_{\\hat P_{y}}(x) , \\pi_{\\hat P_{0}}(x)) \n\t\t\t = |\\pi_{\\hat P_{y}-X}(x-X) - \\pi_{\\hat P_{0}-X}(x-X)|\n\t\t\t \\le d(x,X) \\ \\varangle(\\hat P_{y},\\hat P_{0})\n\t\t\t \\stackrel{\\eqref{23.10.12.1}}{<} \\frac{d(x)}{20}.\\]\n\t\tFurthermore, with \\eqref{23.10.12.1}, we get\n\t\t$d(\\pi_{\\hat P_{0}}(x),X) = d(\\pi(x),\\pi(X)) \\le \\frac{d(x)}{10}$.\n\t\tUsing triangle inequality, the previous two estimates imply\n\t\t\t$d(\\pi_{\\hat P_{y}}(x),X) \n\t\t\t \\le \\frac{d(x)}{20} + \\frac{d(x)}{10}$.\n\t\tSince $y_{0} \\in \\Upsilon \\subset B(X,\\frac{d(x)}{10})$ we have \n\t\t$d(P_{y},\\hat P_{y})=d(X,P_{y})\\le d(X,y_{0})\\le \\frac{d(x)}{10}$ and hence\n\t\t\\begin{align*}\n\t\t\t\\frac{d(x)}{2} \n\t\t\t&\\stackrel{\\eqref{23.10.12.1}}{\\le} d(x,P_{y})+ d(P_{y},\\hat P_{y})+ d(\\pi_{\\hat P_{y}}(x),X)\n\t\t\t\\le d(x,P_{y})+ \\frac{d(x)}{4}\n\t\t\\end{align*}\n\t\tand gain $d(x,P_{y}) \\ge \\frac{d(x)}{4}$.\n\t\\proofofclaimend{2} \\indent\n\tWith \\eqref{23.10.12.1} and\n\t$ d(y_{j},X) \\le d(y_{j},x_{j}) + d(x_{j},X) \\le \\frac{d(x)}{10 C_{1}} + \\frac{d(x)}{10}$,\n\twe obtain $y_{0}, \\dots y_{n},x \\in B(X,7d(x))$ which is a subset of $B(X, \\frac{C_{1}}{8n} \\frac{d(x)}{10})$,\n\twhere we used the explicit characterisation of $C_{1}$ given in Lemma \\ref{lem2.3}.\n\tDue to the second property of a $\\mu$-proper{} integrand (see Definition \\ref{muproper}),\n\tthere exists some $\\tilde C = \\tilde C({N},n,\\mathcal{K},p,C_{0}) \\ge 1$ so that we get with Claim 2\n\t\\begin{align*}\n\t\t\\mathcal{K}^{p}(y_{0},\\dots,y_{n},x) \n\t\t&\\ge \\frac{1}{\\left( \\frac{d(x)}{10} \\right)^{n(n+1)} \\tilde C} \n\t\t\\left(\\frac{d(x,\\textnormal{aff}(y_{0},\\dots,y_{n}))}{\\frac{d(x)}{10}}\\right)^{p}\n\t\t> \\tilde C^{-1} \\left( \\frac{10}{d(x)} \\right)^{n(n+1)}.\n\t\\end{align*}\n\tThis estimate holds for all $y_{i} \\in B\\big( x_{i},\\frac{d(x)}{10C_{1}}\\big) \\cap \\Upsilon$.\n\tBy restricting the integration to the balls $B\\big( x_{i},\\frac{d(x)}{10C_{1}}\\big)$ and using\n\tthe previous estimate as well as estimate \\eqref{21.12.09;5}, we get \n\t\\begin{align*}\n\t\t\\int \\dots \\int \\mathcal{K}^{p}(y_0,\\dots,y_{n},x) \\mathrm{d} \\mu(y_0) \\dots \\mathrm{d}\\mu(y_{n})\n\t\t& \\ge \\tilde C^{-1} C_{2}^{-(n+1)}. \n\t\\end{align*}\n\tWe have proven the previous inequality for all $x \\in G$, so\n\tfinally we deduce with (C) from page \\pageref{Grundeigenschaften} that there exists some constant\n\t$C=C({N},n,\\mathcal{K},p,C_{0})$ so that\n\t\\[\\mu(G) \\le \\tilde C C_{2}^{(n+1)}\n\t\t\t\\int_{G} \\int \\dots \\int \\mathcal{K}^{p}(y_0,\\dots,y_{n},x) \\mathrm{d} \\mu(y_0) \\dots \\mathrm{d}\\mu(y_{n})\\mathrm{d} \\mu(x) \n\t\t \\stackrel{\\text{(C)}}{\\le} C \\eta.\\]\n\\end{proof}\n\n\\begin{lem} \\label{lem3.15}\n\tLet $\\alpha, \\varepsilon >0$. If $\\eta \\le 2\\varepsilon$, \n\twe have \n\t$(20K)^{-1}d(x) \\le D(\\pi(x)) \\le d(x)$ for all $x \\in F \\setminus G$,\n\twhere $K$ is the constant defined on page \\pageref{WahlvonK} at the beginning of this subsection.\n\\end{lem}\n\\begin{proof}\n\tLet $x \\in F \\setminus G$.\n\tWe have $D(\\pi(x)) = \\inf_{y \\in \\pi^{-1}(\\pi(x))}d(y) \\le d(x).$\n\tIf $x \\in \\mathcal{Z}$, Lemma \\ref{rem3.8} implies $d(x)=0$, so the statement is trivial.\n\tNow we assume $x \\notin \\mathcal{Z}$.\n\tSince $x \\notin G \\cup \\mathcal{Z}$, by definition of $G$, \n\tthere exists some $i \\in I_{12}$ with $\\pi(x) \\in 3R_{i}$ and \n\t$x \\in KB_{i}$. \n\tWe have $B_{i} = B(X_{i},t_{i})$ where $(X_{i},t_{i}) \\in S$ (see Lemma \\ref{vor3.12}) \n\tand $K > 1$ (see page \\pageref{WahlvonK}) so we obtain\n\t$d(x) \\le d(X_{i},x)+t_{i} \\le < K \\operatorname{diam} B_{i}$.\n\tNow, with Lemma \\ref{rem3.10} (i) and \\ref{vor3.12}, we deduce\n\t$ D(\\pi(x)) \\ge \\frac{1}{20K}d(x)$.\n\\end{proof}\n\n\\begin{lem} \\label{lem3.16}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. \n\tThere exists some \n\t$\\bar \\varepsilon = \\bar \\varepsilon ({N},n,C_{0})$ and some $\\tilde k \\ge 4$ so that, if\n\t$\\eta < 2 \\bar \\varepsilon$ and $k \\ge \\tilde k$,\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2}, \\bar \\varepsilon)$ we have that the following is true.\n\tThere exists some constant $C=C(n)$ so that, for all $x \\in F$ with $t \\ge \\frac{d(x)}{10}$, we have\n\t\\[ \\int_{B(x,t)\\setminus G} d\\big(u,\\pi(u)+A(\\pi(u))\\big) \\mathrm{d} \\mu(u) \\le C \\varepsilon t^{n+1}.\\]\n\\end{lem}\n\\begin{proof}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. We choose some $\\varepsilon$ with \n\t$\\eta \\le 2 \\varepsilon < 2 \\bar \\varepsilon$ and some $k \\ge \\tilde k := \\max\\{\\bar k, \\tilde C\\}$, \n\twhere $\\bar \\varepsilon$ and $\\bar k$ are given by\n\tLemma \\ref{lem3.12} and $\\tilde C$ is a fixed constant introduced in step VI of this proof.\n\tLet $x \\in F$ and $t \\ge \\frac{d(x)}{10}$. We define\n\t\\[I(x,t):= \\left\\{ i \\in I_{12} | (3R_{i} \\times P_{0}^{\\perp}) \\cap B(x,t) \\cap (F \\setminus G) \\neq \\emptyset \\right\\}\\]\n\twhere $3R_{i} \\times P_{0}^{\\perp}:=\\{ x \\in \\mathbb{R}^{{N}} | \\pi(x) \\in 3R_{i}\\}$.\n\tAt first, we prove some intermediate results:\\\\\n\tI.\\, Due to the definition of $G$ we have\n\t\t$(B(x,t)\\cap F)\\setminus (G \\cup \\mathcal{Z}) \\subset \n\t\t\\bigcup_{i \\in I(x,t)}(3R_{i} \\times P_{0}^{\\perp}) \\cap KB_{i}$.\\\\\n\tII.\\, Let $u \\in 3R_{i} \\times P_{0}^{\\perp}$. Using Lemma \\ref{lem3.11} (iv) implies that \n\t\t$\\sum_{j\\in I_{12}} \\phi_{j}(\\pi(u))$ is a finite sum.\\\\\n\tIII.\\, Let $i \\in I(x,t)$ and $j \\in I_{12}$.\n\t\tWe define $\\phi_{i,j}$ to be $0$ if $3R_{i}$ and $3R_{j}$ are disjoint and $1$ if they are not\n\t\tdisjoint. We have $\\phi_{j}(\\pi(u)) \\le 1 = \\phi_{i,j}$ for all \n\t\t$u \\in (3R_{i} \\times P_{0}^{\\perp}) \\cap KB_{i}$, since\n\t\tif $\\phi_{j}(\\pi(u)) \\neq 0$ the definition of $\\phi_{j}$ (see page \\pageref{Defphii}) \n\t\tgives us $\\pi(u) \\in 3R_{j}$ and, because $\\pi(u) \\in 3R_{i}$, we deduce\n\t\t$ 3R_{i}\\cap 3R_{j} \\neq \\emptyset$. \\\\\n\tIV.\\, If $\\phi_{i,j} \\neq 0$, we can apply Lemma \\ref{rem3.10} (iii) and Lemma \\ref{lem3.12} (i).\n\t\tHence, using Lemma \\ref{vor3.12}, the size of $B_{i}$ as well as the distance of \n\t\t$B_{i}$ to $B_{j}$ are comparable to the size of $B_{j}$.\n\t\tConsequently, there exists some constant $\\tilde C$ so that \n\t\t$KB_{i} \\subset \\tilde C B_{j} \\subset k B_{j}$.\\\\\n\tV.\\,\tIf $u \\in kB_{j}$, we have\n\t\t$ |\\pi^{\\perp}(u)- A_{j}(\\pi(u))| < 2d(u,P_{j})$.\n\t\tWe recall that $P_{j}$ is the graph of the affine map $A_{j}$ \n\t\t(cf. Definition \\ref{3.12.2013.10} and Lemma \\ref{AiLipschitz}).\n\t\t\\begin{proof}\n\t\t\tWe set $\\hat P_{0}:= P_{0} + A_{j}(\\pi(u))$ and \n\t\t\t$v:= \\pi(u) + A_{j}(\\pi(u)) = \\pi_{\\hat P_{0}}(u)$.\n\t\t\tRemark \\ref{24.04.2013.1} implies\n\t\t\t\\[|\\pi_{P_j}(u) - v| = |\\pi_{P_j-v}(u-v)- \\pi_{\\hat P_{0}-v}(u-v)|\n\t\t\t\t \\le |u-v| \\ \\varangle(P_{j},P_{0}). \\]\n\t\t\tUsing this and $\\varangle(P_{j},P_{0}) \\le \\alpha < \\frac{1}{2}$ \n\t\t\t(cf. Definition \\ref{3.12.2013.10})\n\t\t\twe obtain\n\t\t\t$|u-v| < d(u,P_{j}) + \\frac{1}{2}|u-v|$\n\t\t\tand hence\n\t\t\t$|\\pi^{\\perp}(u) - A_{j}(\\pi(u))|=|u-v| < 2 d(u,P_{j})$. \n\t\t\\end{proof}\n\tIf $ u \\in \\mathcal{Z}$, the definition of $A$ (see page \\pageref{04.11.2013.1})\n\tyields $d(u,\\pi(u)+ A(\\pi(u)))=0$.\n\tUsing Lemma \\ref{15.10.2013.1} and Definition \\ref{04.11.2013.1}, we get\n\t\\[\\int_{B(x,t)\\setminus G} d(u,\\pi(u)+ A(\\pi(u))) \\mathrm{d} \\mu(u) \n\t\t\\le \\int_{B(x,t)\\setminus (G \\cup \\mathcal{Z})} \\sum_{j\\in I_{12}}\\phi_{j}(\\pi(u)) \n\t\t\\left|\\pi^{\\perp}(u) - A_{j}(\\pi(u))\\right| \\mathrm{d} \\mu(u). \\]\n\tUsing I to V we obtain\n\t\\[\\int_{B(x,t)\\setminus G} d(u,\\pi(u)+ A(\\pi(u))) \\mathrm{d} \\mu(u) \\le\n\t\t2 \\sum_{i \\in I(x,t)}\\sum_{j\\in I_{12}} \\phi_{i,j} t_{j}^{n+1} \n\t\t\t\\frac{1}{t_{j}^{n}} \\int_{kB_{j}}\n\t\t \t\\frac{d\\left(u,P_{j}\\right)}{t_{j}} \\mathrm{d} \\mu(u).\\]\n\tNow we get the statement by using the following five results.\\\\\n\tVI.\\, Lemma \\ref{lem3.12} and the definition of $S_{total}$ imply\n\t\t$\\beta_{1;k}^{P_{j}}(B_{j}) \\le 2\\varepsilon$.\\\\\n\tVII.\\, Let $i \\in I(x,t)$ and $j \\in I_{12}$. If $\\phi_{i,j} \\neq 0$, we conclude that \n\t\t$ 3R_{i}\\cap3R_{j} \\neq \\emptyset$. \n\t\tHence, with Lemma \\ref{lem3.11} (iii) and Lemma \\ref{vor3.12}, we deduce\n\t\t$2t_{j}=\\operatorname{diam} B_{j} \\le 1000 \\operatorname{diam} R_{i}$.\\\\\n\tVIII.\\, For $i \\in I(x,t)$, we have with Lemma \\ref{lem3.11} (iv) that \n\t\t$ \\sum_{j\\in I_{12}} \\phi_{i,j} \\le (180)^{n}$.\\\\\n\tIX.\\, For $i \\in I(x,t)$, there exists some $y \\in B(x,t) \\cap (F\\setminus G)$ with \n\t\t$ \\pi(y) \\in 3R_{i}$. We obtain with Lemma\n\t\t\\ref{rem3.10}, Lemma \\ref{lem3.15} and our assumption $t \\ge \\frac{d(x)}{10}$ that\n\t\t$10\\operatorname{diam} R_{i} \\le d(x)+d(x,y) \\le 11t$.\\\\\n\tX.\\, Let $ i \\in I(x,t)$. With XI we obtain \n\t\t$ \\operatorname{diam} R_{i}< 2t$ and, because $(3R_{i} \\times P_{0}^{\\perp}) \\cap B(x,t) \\neq \\emptyset$,\n\t\twe get $ R_{i} \\subset B(\\pi(x),t+\\operatorname{diam} 3R_{i})\\cap P_{0}\\subset B(\\pi(x),7t)\\cap P_{0}$.\n\t\tMoreover, with Lemma \\ref{inneredisjunkt} (ii), the primitive cells $R_{i}$ have disjoint interior\n\t\tand hence we get with Lemma \\ref{24.10.12.1}\n\t\t(we recall that $\\omega_{\\N}$ denotes the volume of the $n$-dimensional unit sphere)\n\t\t\\begin{align*}\\sum_{i \\in I(x,t)} (\n\t\t\t\\operatorname{diam} R_{i})^{n} \n\t\t\t& \\le \\sqrt{n}^{n} \\mathcal{H}^{n}(B(\\pi(x),7t)\\cap P_{0})\n\t\t\t = \\sqrt{n}^{n} \\omega_{\\N} (7t)^{n}.\n\t\t\\end{align*}\n\\end{proof}\n\n\\begin{dfn} \\label{DefinitionvonFtilde}\n\tWe define\n\t$\\tilde{F} := \\big\\{ x \\in F \\setminus G \\ | \\ d(x,\\pi(x) + A(\\pi(x))) \\le \\varepsilon^{\\frac{1}{2}} d(x) \\big\\}$.\n\\end{dfn}\n\n\\begin{thm} \\label{thm3.18}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. \n\tThere exists some \n\t$\\hat \\varepsilon = \\hat \\varepsilon ({N},n,C_{0})\\le \\frac{1}{4}$ and some $\\tilde k \\ge 4$ so that, if\n\t$\\eta < 2 \\hat \\varepsilon$ and $k \\ge \\tilde k$,\n\tthere exists some constant $C_{5} = C_{5}({N},n,\\mathcal{K},p,C_{0})$ so that,\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\hat \\varepsilon)$, we have\n\t$ \\mu(F \\setminus \\tilde{F}) \\le C_{5} \\varepsilon^{\\frac{1}{2}}$. \n\\end{thm}\n\\begin{proof}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. We choose some $\\varepsilon$ with \n\t$\\eta \\le 2 \\varepsilon < 2 \\hat \\varepsilon:= \\min\\{2\\tilde \\varepsilon, 2\\bar \\varepsilon,\\frac{1}{2}\\}$ \n\tand some $k \\ge \\tilde k$ \n\twhere $\\tilde \\varepsilon$ is given by Lemma \\ref{lem3.14} and \n\t$\\bar \\varepsilon$ and $\\tilde k$ are given by Lemma \\ref{lem3.16}.\n\n\tAt first, we prove some intermediate results:\\\\\n\tI.\\,\tWe have $\\mathcal{Z} \\subset \\tilde F$ because for $ x \\in \\mathcal{Z}$ \n\t\tthe definition of $A$ on $\\mathcal{Z}$ (see Definition \\pageref{04.11.2013.1}) implies that\n\t\t$ d(x,\\pi(x)+A(\\pi(x)))=d(x,x)=0$.\\\\\n\tII.\\,\tIf $x \\in F \\setminus(\\tilde{F} \\cup G)$, we conclude with I that $x \\notin \\mathcal{Z}$ \n\t\tand, with Lemma \\ref{rem3.8}, we deduce\n\t\t$d(x) \\neq 0$. So \n\t\t$\\mathcal{G} = \\left\\{ B\\left(x,\\frac{d(x)}{10}\\right) \\Big| x \\in F \\setminus(\\tilde{F} \\cup G) \\right\\}$\n\t\tis a set of nondegenerate balls.\n\t\tFor $x \\in F \\subset B(0,5)$, we have $ d(x) \\le 60 $ (see Lemma \\ref{mengebeschraenkt})\n\t\tso that we can apply the Besicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans} to $\\mathcal{G}$\n\t\tand get $N_{0}=N_{0}({N})$ families \n\t\t$\\mathcal{B}_{m} \\subset \\mathcal{G}, m=1,...,N_{0}$\n\t\tof disjoint balls with\n\t\t\\[ F \\setminus(\\tilde{F} \\cup G) \\subset \\bigcup_{m=1}^{N_{0}} \\bigcup_{B \\in \\mathcal{B}_{m}}B.\\]\n\tIII.\\,\tSince $d$ is 1-Lipschitz (Lemma \\ref{rem3.7}), for all $u \\in B\\big(x,\\frac{d(x)}{10}\\big)$\n\t\t$d(x)-d(u) \\le d(x,u) \\le \\frac{d(x)}{10}$\n\t\tand hence\n\t\t$\\frac{1}{d(u)} \\le \\frac{10}{9}\\frac{1}{d(x)} < \\frac{2}{d(x)}$.\\\\\n\tIV.\\,\tLet $1 \\le m \\le N_{0}$ and\n\t\tlet $B_{x} = B\\left(x,\\frac{d(x)}{10}\\right)$ and $B_{y} = B\\left(y,\\frac{d(y)}{10}\\right)$ \n\t\tbe two balls in \n\t\t$\\mathcal{B}_{m}$. Then we either have\n\t\t\\begin{enumerate}[a)]\n\t\t\t\\item $\\pi\\left(\\frac{1}{40K}B_{x}\\right) \\cap \\pi\\left(\\frac{1}{40K}B_{y}\\right) = \\emptyset $ \n\t\t\tor\n\t\t\t\\item if $2d(x) \\ge d(y)$, we have $B_{y} \\subset 200 B_{x}$ \n\t\t\tand $ \\operatorname{diam} B_{y} > (40K)^{-1} \\operatorname{diam} B_{x}$,\n\t\t\\end{enumerate}\n\t\twhere $K$ is the constant from page \\pageref{WahlvonK}.\n\t\t\\begin{proof}\n\t\tLet $\\pi\\left(\\frac{1}{40K}B_{x}\\right) \\cap \\pi\\bigl(\\frac{1}{40K}B_{y}\\bigr) \\neq \\emptyset $ \n\t\tand $2d(x) \\ge d(y)$. We deduce with Lemma \\ref{lem3.9}\n\t\t$d(x,y) <19 d(x)$, which implies\n\t\t$B_{y} \\subset B\\big(x,19d(x)+\\textstyle{\\frac{d(y)}{10}}\\big) = 200 B_{x}$.\n\t\tWith Lemma \\ref{lem3.15}, we get\n\t\t$ \\frac{d(x)}{20K} \\le D(\\pi(y))+d(\\pi(x),\\pi(y)) < d(y) + \\frac{d(x)}{40K},$\n\t\tand hence $ d(y) > (40K)^{-1}d(x)$.\n\t\tAll in all, we have proven that either case a) or case b) is true.\n\t\t\\end{proof} \\noindent\n\tV.\\,\tThere exists some constant $C=C(n)$ so that\n\t\t$\\sum_{B \\in \\mathcal{B}_{m}} (\\operatorname{diam} B)^{n} \\le C$ for all $1\\le m \\le N_{0}$.\n\t\t\\begin{proof}\n\t\t\tLet $1 \\le m \\le N_{0}$.\n\t\t\tWe recursively construct for every $m$ some sequence of numbers, \n\t\t\tsome sequence of balls and some sequence of sets.\n\t\t\tAt first, we define the initial elements. Let $d_{m}^{1} := \\sup_{B \\in \\mathcal{B}_{m}} \\operatorname{diam} B$.\n\t\t\tWe have $d_{m}^{1} < \\infty$ because, for all $x \\in F \\subset B(0,5)$, we have with \n\t\t\tLemma \\ref{mengebeschraenkt} that\n\t\t\t$ d(x) \\le 60$. \n\t\t\tNow we choose $B_{m}^{1} \\in \\mathcal{B}_{m}$ with $ \\operatorname{diam} B_{m}^{1} \\ge \\frac{d_{m}^{1}}{2}$ \n\t\t\tand define\n\t\t\t\\[ \\mathcal{B}_{m}^{1} := \\left\\{ B \\in \\mathcal{B}_{m} \n\t\t\t\t\\Big| \\pi\\left(\\textstyle{\\frac{1}{40K}}B_{m}^{1}\\right) \\cap \\pi\\left(\\textstyle{\\frac{1}{40K}}B\\right) \\neq \\emptyset \\right\\}.\\]\n\t\t\tWe continue this sequences recursively. We set\n\t\t\t$d_{m}^{i+1}= \\sup_{B^{'} \\in \\mathcal{B}_{m} \\setminus \\bigcup_{j=1}^{i}\\mathcal{B}_{m}^{j}} \\operatorname{diam} B^{'}$,\n\t\t\tchoose\n\t\t\t$B_{m}^{i+1} \\in \\mathcal{B}_{m} \\setminus \\bigcup_{j=1}^{i}\\mathcal{B}_{m}^{j}$\n\t\t\twith $ \\operatorname{diam} B_{m}^{i+1} \\ge \\frac{d_{m}^{i+1}}{2}$ and define\n\t\t\t\\[ \\mathcal{B}_{m}^{i+1} := \\Bigg\\{ B \\in \\mathcal{B}_{m} \\setminus \\bigcup_{j=1}^{i}\\mathcal{B}_{m}^{j}\n\t\t\t\t\\Big| \\pi\\left(\\textstyle{\\frac{1}{40K}}B_{m}^{i+1}\\right) \\cap \\pi\\left(\\textstyle{\\frac{1}{40K}}B\\right) \\neq \\emptyset \\Bigg\\}.\\]\n\t\t\tIf there exists some $l \\in \\mathbb{N}$ so that eventually\n\t\t\t$\\mathcal{B}_{m} \\setminus \\bigcup_{j=1}^{l} \\mathcal{B}_{m}^{j} = \\emptyset$, \n\t\t\twe set for all $i \\ge l$\n\t\t\t$\\mathcal{B}_{m}^{i} := \\emptyset$, and interrupt the sequences $(d_{m}^{i})$ and $(B_{m}^{i})$.\n\t\t\tWe have the following results:\\\\\n\t\t\t(i)\\, For all $l \\in \\mathbb{N}$ and \n\t\t\t$B_{m}^{l}=B\\big(x_{m}^{l},\\frac{d(x_{m}^{l})}{10}\\big)$, \n\t\t\twe have with Lemma \\ref{mengebeschraenkt} and\n\t\t\t$x_{m}^{l} \\in F \\subset B(0,5)$ that $\\frac{d(x_{m}^{l})}{10} \\le 6$. Hence we get\n\t\t\t$B_{m}^{l} \\subset B(0,11)$.\\\\\n\t\t\t(ii)\\, For all $1 \\le m \\le N_{0}$, we have\n\t\t\t$\\bigcup_{i=1}^{\\infty}\\mathcal{B}_{m}^{i} = \\mathcal{B}_{m}$.\\\\\n\t\t\t\\textit{Proof.}\n\t\t\t\tIf there exist only finitely many $d_{m}^{l}$, the construction\n\t\t\t\timplies $\\mathcal{B}_{m} \\subset \\bigcup_{j=1}^{\\infty} \\mathcal{B}_{m}^{j}$.\\\\\n\t\t\t\tNow we assume that there exist infinitely many $d_{m}^{l}$.\n\t\t\t\tSince $\\mathcal{B}_{m}$ is a family of disjoint balls, the set\n\t\t\t\t$\\{B_{m}^{l} |l \\in \\mathbb{N}\\}$ is also a family of disjoint balls.\n\t\t\t\tDue to (i), all of those balls are contained in $B(0,11)$. If there exists some $c >0$ \n\t\t\t\twith $\\operatorname{diam} B_{m}^{l} > c$ for all $l \\in \\mathbb{N}$,\n\t\t\t\tthere can not be infinitely many of such balls. Hence we deduce\n\t\t\t\t$\\operatorname{diam} B_{m}^{l} \\rightarrow 0$ if $l \\rightarrow \\infty.$\n\t\t\t\tLet $B \\in \\mathcal{B}_{m}$. If\n\t\t\t\t$B \\notin \\bigcup_{i=1}^{\\infty}\\mathcal{B}_{m}^{i}$,\n\t\t\t\twe obtain $2\\operatorname{diam} B_{m}^{l} \\ge d_{m}^{l} \\ge \\operatorname{diam} B$ for all $l \\in \\mathbb{N}$\n\t\t\t\twhere we used the definition of $d_{m}^{l}$.\n\t\t\t\tThis is in contradiction to $\\operatorname{diam} B_{m}^{l} \\rightarrow 0$. So we get\n\t\t\t\t$B \\in \\bigcup_{i=1}^{\\infty}\\mathcal{B}_{m}^{i}$.\n\t\t\t\tAll in all, we have proven\n\t\t\t\t$ \\bigcup_{i=1}^{\\infty}\\mathcal{B}_{m}^{i} \\supset \\mathcal{B}_{m}$.\n\t\t\t\tThe inverse inclusion follows by definition of $\\mathcal{B}_{m}^{i}$.\n\t\t\t\\hfill$\\square$\\\\\n\t\t\t(iii)\\, Let $1 \\le m \\le N_{0}$, $l \\in \\mathbb{N}$ and \n\t\t\t$B_{y}=B\\left(y,\\frac{d(y)}{10}\\right) \\in \\mathcal{B}_{m}^{l}$, \n\t\t\t$B_{m}^{l}=B\\left(x_{m}^{l},\\frac{d(x_{m}^{l})}{10}\\right) \\in \\mathcal{B}_{m}^{l}$.\n\t\t\tWe have\n\t\t\t$\\pi\\left(\\frac{1}{40K}B_{m}^{l}\\right) \\cap \\pi\\left(\\frac{1}{40K}B_{y}\\right) \\neq \\emptyset$\n\t\t\tand \n\t\t\t$2d(x_{m}^{l}) = 10 \\operatorname{diam} B_{m}^{l} \\ge 10 \\frac{d_{m}^{l}}{2} \\ge 10 \\frac{ \\operatorname{diam} B_{y}}{2} = d(y)$.\n\t\t\tHence IV implies $B_{y} \\subset 200B_{m}^{l}$ and \n\t\t\t$\\operatorname{diam} B_{y} > (40K)^{-1} \\operatorname{diam} B_{m}^{l}$.\n\t\t\tThe balls in $\\mathcal{B}_{m}^{l}$ are disjoint, so, with Lemma \\ref{22.2.2012.1} \n\t\t\t($s=\\frac{\\operatorname{diam} B_{m}^{l}}{80K}$, $r=200\\frac{\\operatorname{diam} B_{m}^{l}}{2}$),\n\t\t\twe deduce $\\# \\mathcal{B}_{m}^{l} \\le (200 \\cdot 80K)^{{N}}$.\\\\\n\t\t\t(iv)\\,\t$\\{\\frac{1}{40K}B_{m}^{l}\\}_{l\\in \\mathbb{N}}$ is a family of disjoint balls and\n\t\t\twith (i) we get $\\pi\\left(\\textstyle{\\frac{1}{40K}}B_{m}^{l}\\right) \\subset \\pi(B(0,11))$\n\t\t\tfor all $l \\in \\mathbb{N}$.\n\t\t\tHence we obtain\n\t\t\t$\\sum_{l=1}^{\\infty} \\left(\\operatorname{diam} \\pi\\left( \\textstyle{\\frac{1}{40K}}B_{m}^{l} \\right)\\right)^{n}\n\t\t\t\t \\le \\frac{2^{n}}{\\omega_{\\N}} \\mathcal{H}^{n}\\left(\\pi\\left( B(0,11) \\right)\\right)\n\t\t\t\t = 22^{n}.$\n\t\t\t\t\n\t\t\tNow we are able to prove V by using (ii),(iii) and (iv):\n\t\t\t\\[\\sum_{B \\in \\mathcal{B}_{m}} \\left(\\operatorname{diam} B \\right)^{n}\n\t\t\t\t \\le \\sum_{l=1}^{\\infty} \\sum_{B \\in \\mathcal{B}_{m}^{l}} \\left(\\textstyle d_{m}^{l}\\right)^{n}\n\t\t\t\t = C(n) \\sum_{l=1}^{\\infty} \\left(\\operatorname{diam} \\pi\\left( \\textstyle \\frac{1}{40K}B_{m}^{l} \\right)\\right)^{n} \n\t\t\t\t \\le C(n).\\]\n\t\t\\end{proof}\n\tFinally, we can finish the proof of Theorem \\ref{thm3.18}.\n\tLet $p_{B}$ denote the centre of some ball $B$.\n\tUsing the definition of $ \\tilde{F}$ and Lemma \\ref{lem3.16}, there exists some constant $C=C(n)$ \n\tso that we obtain\n\t\\begin{align*}\n\t\t\\varepsilon^{\\frac{1}{2}} \\mu(F \\setminus(\\tilde F \\cup G)) \n\t\t& \\stackrel{\\hphantom{\\text{III}}}{<} \n\t\t\t\\int_{F \\setminus(\\tilde F \\cup G)}\\frac{d(u,\\pi(u) + A(\\pi(u)))}{d(u)} \\mathrm{d} \\mu(u)\\\\ \\displaybreak[1]\n\t\t& \\stackrel{\\substack{{\\hphantom{\\text{III}}}\\\\{\\text{II}}}}{\\le} \n\t\t\t\\sum_{m=1}^{N_{0}} \\sum_{B \\in \\mathcal{B}_{m}} \n\t\t\t\\int_{B \\setminus (\\tilde F \\cup G)}\\frac{d(u,\\pi(u) + A(\\pi(u)))}{d(u)} \\mathrm{d} \\mu(u)\\\\ \\displaybreak[1]\n\t\t& \\stackrel{\\text{III}}{<} \\sum_{m=1}^{N_{0}} \\sum_{B \\in \\mathcal{B}_{m}} \\frac{2}{d(p_{B})}\n\t\t\tC \\varepsilon \\left(\\frac{\\operatorname{diam} B}{2}\\right)^{n+1}\\\\\n\t\t& \\stackrel{\\substack{{\\hphantom{\\text{III}}}\\\\{\\text{V}}}}{\\le} \n\t\t\tC({N},n) \\varepsilon.\n\t\\end{align*}\n\tThis leads to \n\t$\\mu(F \\setminus(\\tilde F \\cup G)) \\le C({N},n) \\varepsilon^{\\frac{1}{2}}$.\n\tWith $\\eta < 2\\varepsilon \\le \\varepsilon^{\\frac{1}{2}}$ and Lemma \\ref{lem3.14}\n\tthe assertion holds.\n\\end{proof}\n\\subsection{\\texorpdfstring{$F_1$ is small}{F1 is small}}\nNow we are able to estimate $\\mu(F_{1})$. \nWe recall that $\\eta$ and $k$ are fixed constants (cf. the first lines of section \\ref{04.02.2014.1}),\nand that $F_{1}$ depends on the choice of $\\alpha, \\varepsilon >0$ (cf.~Definition \\ref{def3.2}).\n\\begin{thm}\n\tLet $0 < \\alpha \\le \\frac{1}{4}$. There exist some $\\varepsilon^{*}=\\varepsilon^{*}({N},n,C_{0})$\n\tand some $\\tilde k \\ge 4$ so that, if $\\eta < 2 \\varepsilon^{*}$ and $k \\ge \\tilde k$, for all\n\t$\\varepsilon \\in [\\frac{\\eta}{2},\\varepsilon^{*})$, we have $\\mu(F_{1}) < 10^{-6}$.\n\\end{thm}\n\\begin{proof}\n\tLet $0<\\alpha \\le \\frac{1}{4}$ and let $\\hat \\varepsilon$, $C_{5}$ and $\\tilde k$ \n\tbe the constants given by Theorem \\ref{thm3.18}.\n\tWe set $\\varepsilon^{*}:= \\min\\big\\{\\hat \\varepsilon, \\frac{10^{-14}}{C_{5}^2}\\big\\}$\n\tand choose some $k \\ge \\tilde k$ and some $\\varepsilon \\in [\\frac{\\eta}{2},\\varepsilon^{*})$.\n\tAt first, we prove some intermediate results:\\\\\n\t\tI.\\, \tLet \n\t\t$\\mathcal{G} = \\left\\{ B\\big(x,\\frac{h(x)}{10}\\big) \\Big| x \\in F_{1} \\cap \\tilde F \\right\\}$.\n\t\tThis is a set of nondegenerate balls because $\\mathcal{Z}\\cap F_{1} = \\emptyset$ \n\t\tand, by definition of $h(\\cdot)$ (see page \\pageref{Definitionvonh}), we get $h(x) \\le 50$ for all $x \\in F$.\n\t\tWith Besicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans}, there exist $N_{0}=N_{0}({N})$ \n\t\tfamilies $\\mathcal{B}_{m} \\subset \\mathcal{G}$, $m=1,...,N_{0}$, \n\t\tcontaining countably many disjoint balls with\n\t\t\\[ F_{1} \\cap \\tilde F \\subset \\bigcup_{m=1}^{N_{0}} \\bigcup_{B \\in \\mathcal{B}_{m}}B.\\]\n\t\tII.\\,\tLet $1 \\le m \\le N_{0}$ and $B=B\\big(x,\\frac{h(x)}{10}\\big)$\n\t\t\twhere $x \\in F_{1} \\cap \\tilde F$.\n\t\t\tDue to the definition of $F_{1}$, there exists some $y \\in F$ and some \n\t\t\t$\\tau \\in \\left[ \\frac{h(x)}{5}, \\frac{h(x)}{2} \\right]$\n\t\t\twith $d(x,y) \\le \\frac{\\tau}{2}$ and $ \\delta(B(y,\\tau)) \\le \\delta$.\n\t\t\tFor any $z \\in B$, we get\n\t\t\t$d(z,y) \\le \\frac{h(x)}{10} + \\frac{\\tau}{2} \\le \\tau$.\n\t\t\tHence we obtain $B \\subset B(y,\\tau)$ and conclude\n\t\t\t$\\mu(B) \\le \\delta \\tau^{n} < 3^{n} \\delta (\\operatorname{diam} B)^{n}$.\\\\\n\t\tIII.\\,\n\t\tFor all $1 \\le m \\le N_{0}$, we have $\\sum_{B \\in \\mathcal{B}_{m}} (\\operatorname{diam} B)^{n} \\le 192^{n}$.\n\t\\begin{figure}[ht]\n\t\\begin{center}\n\t\\begin{tikzpicture}[scale=1]\n\t\t\\draw[-] (3,3) -- (15,3)node [below] {$P_{0}$};\n\t\t\\draw[-] (4,2.8) -- (4,3.2);\n\t\t\\draw[-] (14,2.8) -- (14,3.2);\n\t\t\\path [draw] (8,5.3) circle (2);\n\t\t\\path [fill] (8,5.3) circle (1pt)node [below] {$x$};\n\t\t\\draw[dashed] (8,3.3) -- (11,3.3);\n\t\t\\draw[decoration={brace,amplitude=5},decorate,thick] (11.1,5.3) -- (11.1,3.3);\n\t\t\\path [fill] (11.3,4.35) circle (0pt)node [right] {$ \\frac{h(x)}{10}$};\n\n\t\t\\path [fill] (8,6) circle (1.3pt);\n\t\t\\draw [-] (7.9,6.1) -- (6.2,7.3) node [left] {$x_{0}=\\tilde A (\\pi(x))$};\n\t\t\\path [draw] (8,6) circle (0.5);\n\t\t\\path [fill] (8,3) circle (1pt) node [below] {$\\pi(x)$};\n\t\t\\draw plot [smooth] coordinates {(4,6) (5,5.5) (6.11,5.95) (7,6.3) (8,6) (8.862,7.1) (9.5,7.8) (11,7.5) (13,8) (14,7.2)};\n\t\t\\path [fill] (12.8,7.9) circle (0pt)node [below] {$\\tilde A(U_0)$};\n\t\t\\draw[thick] plot [smooth] coordinates {(6.11,5.95) (7,6.3) (8,6) (8.862,7.1)};\n\t\t\\path [fill] (4,1) circle (0pt)node [right] {$\\underbrace{\\hspace{9.8cm}}_{U_{12}}$};\n\t\t\\draw[dotted] (6,2.8) -- (6,5.3);\n\t\t\\draw[dotted] (10,2.8) -- (10,5.3);\n\t\t\\draw[dashed] (7.5,2.8) -- (7.5,7.7);\n\t\t\\draw[dashed] (8.5,2.8) -- (8.5,7.7);\n\n\t\t\\draw[dashed] (6.11,2.8) -- (6.11,5.95);\n\t\t\\draw[dashed] (8.862,2.8) -- (8.862,7.1);\n\t\t\\path [fill] (5.8,2) circle (0pt)node [right] {$\\underbrace{\\hspace{2.7cm}}_{\\pi\\left( B\\left(x,\\frac{h(x)}{10} \\right) \\cap \\tilde A(U_{12}) \\right)}$};\n\n\t\t\\draw[dashed] (8,6) -- (12,6);\n\t\t\\draw[dashed] (8,5.3) -- (12,5.3);\n\t\t\\path [fill] (11.8,5.63) circle (0pt)node [right] {$\\Big\\} \\le \\frac{h(x)}{20}$};\n\t\t\\draw[dashed] (8,6.5) -- (11,6.5);\n\t\t\\path [fill] (10.9,6.27) circle (0pt)node [right] {$\\big\\} \\frac{h(x)}{40}$};\n\t\\end{tikzpicture}\n\t\\end{center}\n\t\\caption[bla]{$\\pi\\left( B\\left(x_{0},\\frac{h(x)}{40} \\right) \\right) \\subset\\pi\\left( B\\left(x,\\frac{h(x)}{10} \\right) \\cap \\tilde A(U_{12}) \\right)$}\n\t\\end{figure}\n\t\t\\begin{proof}\n\t\t\tWe define the function\n\t\t\t$\\tilde{A}: U_{12} \\rightarrow \\mathbb{R}^{{N}}, u \\mapsto u + A(u)$,\n\t\t\twhere $U_{12}=B(0,12) \\cap P_{0}$.\n\t\t\t$\\tilde{A}$ is Lipschitz continuous with Lipschitz constant less than $2$ \n\t\t\tbecause $A$ is defined on $U_{12}$ (see page \\pageref{Aeindeutigdefiniert}), \n\t\t\t$3\\alpha$-Lipschitz continuous (see Lemma \\ref{ALipschitz}) \n\t\t\tand $\\alpha \\le \\frac{1}{4}$. \n\t\t\tLet $ B=B\\left(x,\\frac{h(x)}{10}\\right) \\in \\mathcal{B}_{m}$. \n\t\t\tWe have $F \\subset B(0,5)$ (see (A) on page \\pageref{Grundeigenschaften})\n\t\t\tand so $\\pi(F) \\subset P_{0} \\cap B(0,5)$ because $\\pi$ is the orthogonal projection\n\t\t\ton $P_{0}$ and $0 \\in P_{0}$.\n\t\t\tWith the definition of $\\tilde F$, Lemma \\ref{rem3.8} and \n\t\t\t$\\varepsilon^{\\frac{1}{2}}<\\frac{1}{20}$, we obtain $d(x,x_{0}) < \\frac{h(x)}{20}$\n\t\t\twhere $x_{0} := \\tilde A(\\pi(x))$.\n\t\t\tLet $z \\in \\pi\\left( B\\left(x_{0},\\frac{h(x)}{40} \\right) \\right) \\subset U_{12}$.\n\t\t\tUsing triangle inequality with the point $\\tilde A(\\pi(x_{0}))=x_{0}$ and\n\t\t\t$\\tilde A$ is $2$-Lipschitz, we get\n\t\t\t$d(\\tilde A(z),x)\\le \\frac{h(x)}{10}$. This implies \n\t\t\t$\\tilde A (\\pi(B(x_{0},\\frac{h(x)}{40}))) \\subset B \\cap \\tilde A (U_{12})$, and hence\n\t\t\twe gain $\\pi\\left( B\\left(x_{0},\\frac{h(x)}{40} \\right) \\right) \n\t\t\t\t\\subset\\pi\\left( B \\cap \\tilde A(U_{12}) \\right)$.\n\t\t\tNow we have with \\cite[2.4.1, Thm. 1]{Evans}\n\t\t\t\\begin{align}\n\t\t\t\t\\frac{\\omega_{\\N}}{8^{n}} \\left( \\operatorname{diam} B \\right)^{n} \n\t\t\t\t = \\omega_{\\N} \\left( \\textstyle \\frac{h(x)}{40} \\right)^{n}\n\t\t\t\t = \\mathcal{H}^{n} \\left( \\textstyle \\pi\\left( B\\left(x_{0},\\frac{h(x)}{40} \\right) \\right) \\right)\n\t\t\t\t& \\le \\mathcal{H}^{n} (B \\cap \\tilde A(U_{12})). \\label{1.2.10;1}\n\t\t\t\\end{align}\n\t\t\tThe balls in $\\mathcal{B}_{m}$ are disjoint, so we conclude using \\cite[2.4.1, Thm. 1]{Evans}\n\t\t\tfor the last estimate\n\t\t\t\\[ \\sum_{B \\in \\mathcal{B}_{m}} (\\operatorname{diam} B)^{n} \\stackrel{\\eqref{1.2.10;1}}{\\le} \n\t\t\t\t\\frac{8^{n}}{\\omega_{\\N}} \\sum_{B \\in \\mathcal{B}_{m}} \\mathcal{H}^{n}(B \\cap \\tilde{A}(U_{12})) \n\t\t\t\t\\le \\frac{8^{n}}{\\omega_{\\N}} \\mathcal{H}^{n}(\\tilde{A}(U_{12})) \\le 192^{n}.\\]\n\t\t\\end{proof}\n\tNow we have\n\t$\\mu(F_{1} \\cap \\tilde F) \\stackrel{\\text{I}}{\\le} \\sum_{m=1}^{N_{0}} \\sum_{B \\in \\mathcal{B}_{m}} \\mu(B) \n\t\t \\stackrel{\\text{II, III}}{\\le} \\delta N_{0} \\cdot 576^{n}.$\n\tSince $\\delta \\le \\frac{10^{-10}}{600^{n} N_{0}}$ (see \\eqref{Wahlvondelta} on page \\pageref{Wahlvondelta}) \n\tand $\\varepsilon^{\\frac{1}{2}} < \\frac{10^{-7}}{C_{5}}$, we deduce together with Theorem \\ref{thm3.18} that\n\t$\\mu(F_{1}) < 10^{-6}$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{\\texorpdfstring{$F_2$ is small}{F2 is small}}\\label{F2issmall}\nWe recall that $0< \\eta \\le 2^{-(n+1)}$ and $k \\ge 1$ are fixed constants (cf. the first lines of \nsection \\ref{04.02.2014.1}) and that $F_{2}$ depends on the choice of $\\alpha, \\varepsilon >0$\n(cf.~Definition \\ref{def3.2}).\n\\begin{thm}\n\tLet $\\alpha, \\varepsilon > 0$. There exists some constant $C=C({N},n,\\mathcal{K},p,C_{0},k)$ so that,\n\tif $\\eta \\le \\frac{\\varepsilon^{p}}{C}10^{-6}$, we have $\\mu(F_{2}) \\le 10^{-6}$.\n\\end{thm}\n\\begin{proof}\n\tLet $x \\in F_{2}$ and $ t \\in (h(x),2h(x))$. \n\tIt follows that $x \\notin F_{1} \\cup \\mathcal{Z}$ and hence,\n\tfor all $ y \\in F$ and for all $ \\tau \\in \\left[ \\frac{h(x)}{5},\\frac{h(x)}{2} \\right]$ \n\twith $d(x,y) \\le \\frac{\\tau}{2}$, we obtain\n\t$\\delta(B(y,\\tau)) > \\delta$. So, in particular, we get\n\t$\\delta\\big(B\\big( x,\\frac{h(x)}{2}\\big) \\big) > \\delta$ for $x=y$ and $\\tau = \\frac{h(x)}{2}$.\n\tIf $k_{0}=1$, this implies\n\t$\\tilde{\\delta}_{k_{0}}(B(x,t)) \\ge \\delta(B(x,t)) > \\frac{\\delta}{4^{n}}$,\n\twhere we used $\\frac{h(x)}{2} < t < 2 h(x)$.\n\tLet $(y,\\tau)$ as in the definition of $F_{2}$. \n\tThen we have\n\t$ d(x,y) + \\tau < 2\\tau \\le h(x) < t$\n\tand hence $B(y,\\tau) \\subset B(x,t)$. We conclude\n\t$\\beta_{1;k}(x,t) \\ge \\left( \\frac{\\tau}{t} \\right)^{n+1} \\beta_{1;k}(y,\\tau)\n\t\t\\ge \\frac{\\varepsilon}{10^{n+1}}$.\n\tNow, with Corollary \\ref{thm2.4} ($\\lambda=\\frac{\\delta}{4^{n}}$, $k_{0}=1$), there exists some \n\tconstant $C=C({N},n,\\mathcal{K},p,C_{0},k)$ so that\n\t\\begin{align*}\n\t\t\\mathcal{M}_{\\mathcal{K}^p}(\\mu) \n\t\t& \\ge \\frac{1}{C} \\int_{F_{2}} \\int_{h(x)}^{2h(x)} \n\t\t\t\\beta_{1;k}(x,t)^{p}\n\t\t\t\\Eins_{\\{ \\tilde{\\delta}_{k_{0}}(B(x,t))\\ge \\frac{\\delta}{4^{n}} \\}}\n\t\t\t\\frac{\\mathrm{d}t}{t} \\mathrm{d}\\mu(x) \\\\\n\t\t& \\ge \\frac{1}{C} \\int_{F_{2}}\\int_{h(x)}^{2h(x)} \n\t\t\t\\left( \\frac{\\varepsilon}{10^{n+1}} \\right)^{p} \\frac{\\mathrm{d}t}{t} \\mathrm{d}\\mu(x) \\\\\n\t\t& \\ge \\frac{1}{C} \\left( \\frac{\\varepsilon}{10^{n+1}} \\right)^{p} \\mu(F_{2}) \\ln(2).\n\t\\end{align*}\n\tFinally, using the previous inequality, condition (C) from page \\pageref{Grundeigenschaften} and\n\t$\\eta \\le \\frac{\\ln(2)}{10^{p(n+1)}C}\\varepsilon^{p}10^{-6}$, we get the assertion.\n\\end{proof}\n\\subsection{\\texorpdfstring{$F_3$ is small}{F3 is small}}\\label{F3issmall}\nWe mention for review that $\\tilde F$ is defined on page \\pageref{DefinitionvonFtilde} and set\n\\[ \\tilde{\\tilde{F}} := \\left\\{x \\in \\tilde{F} \\ \\Big| \\ \n\t\\mu(\\tilde{F} \\cap B(x,t)) \\ge \\frac{99}{100} \\mu(F \\cap B(x,t)) \\text{ for all } t \\in (0,2) \\right\\}.\\]\n\n\\begin{lem}\\label{lem5.8;1}\n\tLet $0<\\alpha \\le \\frac{1}{4}$. \n\tThere exists some \n\t$\\hat \\varepsilon = \\hat \\varepsilon ({N},n,C_{0})\\le \\frac{1}{4}$ and some $\\tilde k \\ge 4$ so that, if\n\t$\\eta < 2 \\hat \\varepsilon$ and $k \\ge \\tilde k$,\n\tthere exists some constant $C=C({N},n,\\mathcal{K},p,C_{0})$ so that,\n\tfor all $\\varepsilon \\in [\\frac{\\eta}{2},\\hat \\varepsilon)$, we have \n\t$\\mu(F \\setminus \\tilde{\\tilde{F}}) \\le C \\varepsilon^{\\frac{1}{2}}$.\n\\end{lem}\n\\begin{proof}\n\tLet $0<\\alpha \\le \\frac{1}{4}$ and choose $\\hat \\varepsilon$, $\\tilde k$ to be the constants given by \n\tTheorem \\ref{thm3.18} and let $k \\ge \\tilde k$, $\\eta \\le 2 \\varepsilon < 2 \\hat \\varepsilon$.\n\tDue to Theorem \\ref{thm3.18}, we only have to consider $\\mu(\\tilde{F} \\setminus \\tilde{\\tilde{F}})$. \n\tFor all $ x \\in \\tilde{F} \\setminus \\tilde{\\tilde{F}}$ using the definition of $\\tilde F$, \n\tthere exists some $t_{x} \\in (0,2)$ with\n\t$\\mu(\\tilde F \\cap B(x,t_{x})) \\le 99 \\mu((F \\setminus \\tilde F)\\cap B(x,t_{x}))$.\n\tHence $\\tilde{F} \\setminus \\tilde{\\tilde{F}}$ is covered by balls $B(x,t_{x})$ with centre in \n\t$\\tilde{F} \\setminus \\tilde{\\tilde{F}}$.\n\tSo with Besicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans} \n\tthere exist $N_{0}=N_{0}({N})$ families $\\mathcal{B}_{m}$,\n\t$m = 1,.. , N_{0}$, of disjoint balls $B(x,t_{x})$ so that\n\t\\[\\mu(\\tilde{F} \\setminus \\tilde{\\tilde{F}}) \n\t\t\\le \\sum_{m=1}^{N_{0}} \\sum_{B \\in \\mathcal{B}_{m}} \\mu( \\tilde{F} \\cap B)\n\t\t \\le\n\t\t\t 99 \\sum_{m=1}^{N_{0}} \\sum_{B \\in \\mathcal{B}_{m}} \\mu( (F \\setminus \\tilde{F}) \\cap B)\n\t\t \\le 99 N_{0} \\ \\mu(F \\setminus \\tilde{F}),\\]\n\tand with Theorem \\ref{thm3.18} the assertion holds.\n\\end{proof}\n\n\\begin{lem} \\label{lem5.6}\n\tLet $\\theta,\\alpha > 0$. There exist some constant $C=C({N},n,C_{0},\\theta) > 1$ and some constant\n\t$\\varepsilon_{0}=\\varepsilon_{0}({N},n,C_{0},\\theta) > 0$ so that, if\n\t$\\eta < 2 \\varepsilon_{0}$ and $k \\ge 4$, we have for all \n\t$\\varepsilon \\in [\\frac{\\eta}{2},\\varepsilon_{0})$ \n\tthat the following is true. If $(x,t) \\in S$ and $100 t \\ge \\theta$, then we have\n\t$\\varangle(P_{(x,t)},P_{0}) \\le C \\varepsilon$.\n\\end{lem}\n\\begin{proof}\n\tLet $\\theta, \\alpha > 0$, $k\\ge 4$ and $\\eta < 2 \\varepsilon < 2\\varepsilon_{0}$ where the constant \n\t$\\varepsilon_{0}$ is given by Lemma \\ref{lem2.6}.\n\tLet $t \\ge \\frac{\\theta}{100}$ and $(x,t) \\in S$.\n\tWe get with (A) and (D) (see page \\pageref{Grundeigenschaften})\n\t$\\beta_{1;k}^{P_{0}}(x,t) \\le \\left( \\frac{500}{\\theta} \\right)^{n+1} 2\\varepsilon$.\n\tFurthermore, we have with Definition \\ref{12.07.13.1} that $ \\beta_{1;k}^{P_{(x,t)}}(x,t) \\le 2 \\varepsilon$\n\tand with $(x,t) \\in S \\subset S_{total}$ we obtain $ \\delta(B(x,t)) \\ge \\frac{\\delta}{2}$.\n\tNow, with Lemma \\ref{lem2.6} \n\t($y=x$, $c=1$, $\\xi=2\\left( \\frac{500}{\\theta} \\right)^{n+1}$, $t_{x}=t_{y}=t$, $\\lambda=\\frac{\\delta}{2}$),\n\tthere exists some\n\tconstant $C_{3}=C_{3}({N},n,C_{0},\\theta)$ so that \n\t$\\varangle(P_{(x,t)},P_{0}) \\le C_{3}\\varepsilon$. \n\\end{proof}\n\n\\begin{lem}\\label{25.09.2014.2}\n\tLet $\\theta, \\alpha >0$. If $k \\ge 400$, there exists some constant \n\t$ \\varepsilon^{*} = \\varepsilon^{*}({N},n,C_{0},\\alpha,\\theta)$ so that, if $\\eta < 2 \\varepsilon^{*}$,\n\twe have for all $\\varepsilon \\in [\\frac{\\eta}{2},\\varepsilon^{*})$ that \n\tfor all $x \\in F_{3}$ we have $h(x) < \\frac{\\theta}{100}$.\n\\end{lem}\n\\begin{proof}\n\tLet $\\theta, \\alpha > 0$ and $k \\ge 400$.\n\tWe set $\\varepsilon^{*}:= \\min \\{\\bar \\varepsilon, \\varepsilon_{0},\\frac{\\alpha}{2C}\\}$ where \n\t$\\bar \\varepsilon$ is given by Lemma \\ref{rem3.3} and $\\varepsilon_{0}$ as well as \n\t$C$ are given by Lemma \\ref{lem5.6}.\n\tLet $\\eta \\le 2 \\varepsilon < 2 \\varepsilon^{*}$ and $x \\in F_{3}$.\n\tWith Lemma \\ref{rem3.1} (i), we have $(x,h(x)) \\in S$ and,\n\twith Lemma \\ref{rem3.3}, we get $\\varangle(P_{(x,h(x))},P_{0}) > \\frac{1}{2} \\alpha$.\n\tHence we obtain $h(x) < \\frac{\\theta}{100}$ with Lemma \\ref{lem5.6}.\n\\end{proof}\n\n\n\\begin{lem}\\label{25.09.2014.1}\n\tLet $p=2$.\n\tThere exists some $\\hat k \\ge 400$, some $\\tilde \\alpha =\\tilde \\alpha (n) > 0$\n\tand some $\\hat \\theta = \\hat \\theta({N},n,C_{0}) \\in (0,1)$\n\tso that for all $\\alpha \\in (0, \\tilde \\alpha]$ and $\\theta \\in (0,\\hat \\theta]$ there exists some\n\t$\\hat \\varepsilon = \\hat \\varepsilon({N},n,C_{0},\\alpha,\\theta)$ \n\tso that, if $k \\ge \\hat k$ and $\\eta < \\hat \\varepsilon^{2}$, we have for all\n\t$\\varepsilon \\in [\\sqrt{\\eta},\\hat \\varepsilon)$ that\n\tthere exists some set $H_{\\theta} \\subset U_{6}$ and some constant\n\t$C=C({N},n,\\mathcal{K},C_{0},k)$ with\n\t$\\mathcal{H}^{n}(U_{6} \\setminus H_{\\theta}) h(x)$.\n\\end{lem}\n\\begin{proof}\n\tLet $\\tilde k$ and $\\tilde \\alpha(n)$ be the thresholds given by Theorem \\ref{thm4.1} and let\n\t$\\hat C=\\hat C({N},n)$ be the constant given by Theorem \\ref{2.9.2014.1}.\n\tMoreover, let $C_{1}=C_{1}({N},n,C_{0})$ and $C_{2}=C_{2}({N},n,C_{0})$ be the constants given by \n\tCorollary \\ref{04.09.12.1} applied with $\\lambda = \\frac{\\delta}{4}$, and $\\delta=\\delta({N},n)$ is the value fixed on\n\tpage \\pageref{Wahlvondelta}. We set \n\t$\\hat \\theta :=\\frac{1}{400}\\left[ 18n(10^{n}+1)\\left(\\frac{C_{1}}{4}\\right)^{n+1} \\hat C \\right]^{-1}$\n\tand choose $\\theta \\in (0,\\hat \\theta]$.\n\tLet $\\alpha \\in (0, \\tilde \\alpha]$, and let \n\t$\\bar \\varepsilon_{1} = \\bar \\varepsilon({N},n,C_{0},\\alpha)$, \n\t$\\bar \\varepsilon_{2} = \\bar \\varepsilon({N},n,C_{0},\\alpha)$, \n\t$\\tilde \\varepsilon=\\tilde \\varepsilon({N},n,C_{0},\\alpha)$, \n\t$\\varepsilon_{0}=\\varepsilon_{0}({N},n,C_{0},\\theta)$,\n\t$\\varepsilon^{*}=\\varepsilon^{*}({N},n,C_{0},\\alpha,\\theta)$\n\tbe the thresholds given by Lemma \\ref{rem3.3}, \\ref{AstetigaufU0}, Theorem \\ref{thm4.1}, Lemma \\ref{lem5.6} and\n\tLemma \\ref{25.09.2014.2} respectively.\n\tFinally, let $C$ be the constant from Lemma \\ref{lem5.6}.\n\tWe set\n\t$\\hat \\varepsilon:= \\min\\left\\{ \\bar \\varepsilon_{1},\\bar \\varepsilon_{2},\\tilde \\varepsilon, \n\t\t\\varepsilon_{0}, \\varepsilon^{*}, (\\hat C \\theta \\alpha)^{2},\n\t\t\\frac{\\alpha}{400} \\left[4n(10^{n}+1)\\left(\\frac{C_{1}}{4}\\right)^{n+1}2C_{2}\\right]^{-1},\n\t\t\\frac{\\alpha}{100C}\\right\\}$\n \tand assume that $k \\ge \\hat k:=\\max\\{\\tilde k,400\\}$ and $\\eta \\le {\\hat \\varepsilon}^{2}$. \n\tNow let $\\varepsilon > 0$ with\n\t$\\eta \\le \\varepsilon^{2} < {\\hat \\varepsilon}^{2}$.\n\n\tUntil now, we defined the map $A$ only on $U_{12}=B(0,12)\\cap P_{0}$\n\t(see page \\pageref{Aeindeutigdefiniert}).\n\tFurthermore, we have shown that $A$ is Lipschitz continuous with Lipschitz constant\n\t$3\\alpha$ (see Lemma \\ref{ALipschitz} on page \\pageref{ALipschitz}).\n\tWith Lemma \\ref{8.11.12.1}, an application of Kirszbraun's Theorem,\n\tthere exists an extension $\\tilde A : P_{0} \\to \\mathbb{R}^{{N}}$ of $A$ with compact support, the same Lipschitz \n\tconstant $3\\alpha$ and $A = \\tilde{A}$ on $U_{12}$.\n\tIf one wants to omit Zorn's lemma, used for the proof of Lemma \\ref{8.11.12.1}, \n\tone can get the same result with a slightly larger Lipschitz constant (see the\n\tremark after Lemma \\ref{8.11.12.1} for details).\n\t\\label{FortsetzungvonA}\n\tWe denote this extension of $A$ also by $A$.\n\n\tUsing Theorem \\ref{2.9.2014.1} with $g = A$, $p=2$ and Theorem \\ref{thm4.1}, there exist some set \n\t$H_{\\theta} \\subset U_{6}$ and some constant $C=C({N},n,\\mathcal{K},C_{0},k)$ with\n\t$\\mathcal{H}^{n}(U_{6} \\setminus H_{\\theta}) \n\t\t\\le \\frac{C(n)}{\\theta^{2(n+1)} \\operatorname{Lip}_{A}^{2}} C \\varepsilon^{2}$.\n\tFurthermore, we get for all $y \\in P_{0}$ some affine map $a_{y}:P_{0} \\to P_{0}^{\\perp}$ so that,\n\tif $r \\le \\theta$ and $B(y,r) \\cap H_{\\theta} \\neq \\emptyset$, we have\n\t$\\|A-a_{y}\\|_{L^{\\infty}(B(y,r)\\cap P_{0},P_{0}^{\\perp})} \\le \\hat C r \\theta \\operatorname{Lip}_{A}$.\n\tWe recall that $\\operatorname{Lip}_{A}=3 \\alpha$ (cf. Lemma \\ref{ALipschitz}).\n\tFor $x \\in F_{3} \\cap \\tilde{\\tilde F} \\subset F_{3} \\cap \\tilde F$, \n\twe have with the previous lemma that $h(x) < \\frac{\\theta}{100}$.\n\tLet $t \\in [h(x),\\frac{\\theta}{100}]$.\n\tIf $x' \\in B(x,2t) \\cap \\tilde{F}$, we obtain with Lemma \\ref{rem3.8} and the definition of $\\tilde F$\n\t$d(x',\\pi(x')+ A(\\pi(x'))) \\le \\varepsilon^{\\frac{1}{2}} \\left( d(x) + d(x,x') \\right)\n\t\t \\le 3 \\varepsilon^{\\frac{1}{2}}t$.\n\tLet $P_{\\pi(x)}$ denote the $n$-dimensional plane, which is the graph of the affine map $a_{\\pi(x)}$.\n\tNow we assume, contrary to the statement of this lemma, that $d(\\pi(x),H_{\\theta}) \\le h(x)$. This implies\n\t$\\pi(B(x,2t)) \\cap H_{\\theta} \\neq \\emptyset$, and so we have \n\t$d(\\pi(x')+ A(\\pi(x')),P_{\\pi(x)})\n\t\t \\le \\|A-a_{\\pi(x)}\\|_{L^{\\infty}(B(\\pi(x),2t)\\cap P_{0},P_{0}^{\\perp})}\n\t\t \\le 6 \\hat C \\theta \\alpha t$ for all $x' \\in B(x,2t) \\cap \\tilde F$.\n\tWe combine those estimates and obtain using $ 3\\varepsilon^{\\frac{1}{2}} \\le 3\\hat C \\theta \\alpha$\n\t\\begin{align}\n\t\td(x',P_{\\pi(x)}) & \\le d(x',\\pi(x')+A(\\pi(x'))) + d(\\pi(x')+ A(\\pi(x')),P_{\\pi(x)}) \n\t\t \\le 9 \\hat C \\theta \\alpha t. \\label{19.1.10;11}\n\t\\end{align}\n\tSince $h(x) \\le t$, we get $(x,t) \\in S \\subset S_{total}$ with Lemma \\ref{rem3.1} (i) so that\n\twe have $\\delta(B(x,t)) \\ge \\frac{\\delta}{2}$. If $x \\in \\tilde{\\tilde{F}}$, this estimate and\n\tthe definition of $\\tilde{\\tilde{F}}$ implies $\\delta(\\tilde{F} \\cap B(x,t)) > \\frac{1}{4} \\delta$.\n\t\n\tNow we apply Corollary \\ref{04.09.12.1} ($\\Upsilon=\\tilde F$, $\\lambda = \\frac{\\delta}{4}$),\n\tand so there exist constants $C_{1}({N},n,C_{0})$, $C_{2}({N},n,C_{0})$ \n\tand an $(n,10n \\frac{t}{C_1})$-simplex \n\t$T=\\Delta(x_0,\\dots,x_{n}) \\in F \\cap B(x,t) \\cap \\tilde F$ \n\tso that \\mbox{$\\mu(\\tilde B_{i}) \\ge \\frac{t^{n}}{C_{2}}$} for all $i \\in \\{0,\\dots, n \\}$\n\twhere $\\tilde B_{i}:=B\\left(x_i,\\frac{t}{C_1}\\right) \\cap B(x,t) \\cap \\tilde F$.\n\tWith $(x,t) \\in S \\subset S_{total}$, we get for all $i\\in \\{0,\\dots,n\\}$\n\t\\begin{align*}\n\t\t\\frac{1}{\\mu(\\tilde B_{i})} \\int_{\\tilde B_{i}} d(z,P_{(x,t)}) \\mathrm{d} \\mu(z) \n\t\t& \\le C_{2} t \\beta_{1;k}^{P_{(x,t)}}(x,t) \n\t\t \\le 2C_{2} t \\varepsilon.\n\t\\end{align*}\n\tThis implies for all $i\\in \\{0,\\dots,n\\}$ the existence of $y_{i} \\in \\tilde B_{i}$ with\n\t$d(y_{i},P_{(x,t)}) \\le 2C_{2}t \\varepsilon$.\n\tWith Lemma \\ref{17.11.11.2}, we deduce that $S:=\\Delta(y_{0},\\dots,y_{n}) \\subset B(x,t)$ is \n\tan $(n,8n\\frac{t}{C_{1}})$-simplex.\n\tNext, we apply Lemma \\ref{21.11.11.2} ($m=n$, $C=\\frac{C_{1}}{8n}$,$\\hat C =1$, $\\sigma=2C_{2} \\varepsilon$) and get\n\t$\\varangle(P_{(x,t)},P_{y_{0},\\dots,y_{n}}) \\le\\frac{\\alpha}{400}$.\n\tWe have $y_{i} \\in \\tilde B_{i} \\subset B(x,2t) \\cap \\tilde F$ and hence \n\twe get with \\eqref{19.1.10;11} and Lemma \\ref{21.11.11.2}\n\t($C=\\frac{C_{1}}{8n}$, $\\hat C = 1$, $\\sigma= 9 \\hat C \\theta \\alpha$)\n\t$\\varangle(P_{y_{0},\\dots,y_{n}},P_{\\pi(x)}) \\le \\frac{\\alpha}{400}$.\n\tWe combine those two angel estimates and conclude \n\t$ \\varangle(P_{(x,t)},P_{\\pi(x)}) \\le \\frac{\\alpha}{200}$,\n\twhich is true for all $x \\in F_{3} \\cap \\tilde{\\tilde F}$ with $d(\\pi(x),H_{\\theta}) \\le h(x)$ \n\tand all $t \\in [h(x),\\frac{\\theta}{100}]$.\n\tNow we use this result for $t=h(x)$ and for $t= \\frac{\\theta}{100}$ and obtain\n\t$\\varangle(P_{(x,h(x))},P_{(x,\\frac{\\theta}{100})}) \\le \\frac{\\alpha}{100}$.\n\tTogether with Lemma \\ref{lem5.6} we get $\\varangle(P_{(x,h(x))},P_{0}) \\le \\frac{\\alpha}{50}$.\n\tThis is in contradiction to Lemma \\ref{rem3.3} hence our assumption that\n\t$d(\\pi(x),H_{\\theta}) \\le h(x)$ is invalid for all $x \\in F_{3} \\cap \\tilde{\\tilde{F}}$.\n\\end{proof}\n\n\\begin{thm}\n\tLet $p =2$.\n\tThere exists some constants $\\bar {\\bar k} \\ge 4$, \n\t$0 < \\bar {\\bar \\alpha} = \\bar {\\bar \\alpha}(n) < \\frac{1}{6}$ and \n\t$0 < \\bar { \\bar \\theta} = \\bar {\\bar \\theta}({N},n,C_{0})$ so that, for all \n\t$\\alpha \\in (0,\\bar {\\bar \\alpha}]$ and all $\\theta \\in (0,\\bar {\\bar \\theta}]$,\n\tthere exists some\n\t$0<\\bar {\\bar \\varepsilon}=\\bar {\\bar \\varepsilon}({N},n,C_{0},\\alpha,\\theta)< \\frac{1}{8}$ so that,\n\tif $ k \\ge \\bar {\\bar k}$ and $\\eta < \\bar {\\bar \\varepsilon}^{2}$, we obtain for all \n\t$\\varepsilon \\in [\\sqrt{\\eta},\\bar{\\bar \\varepsilon})$ \n\t\\[ \\mu(F_{3}) \\le 10^{-6}.\\]\n\\end{thm}\n\\begin{proof}\n\tLet $\\bar {\\bar k}$ be the maximum and $\\bar {\\bar \\alpha}<\\frac{1}{6}$ be the minimum of all thresholds for\n\t$k$ and $\\alpha$ given by Lemma \\ref{ALipschitz}, \\ref{lem5.8;1}, \\ref{25.09.2014.2} and \\ref{25.09.2014.1}. \n\tFurthermore, we set $\\bar {\\bar \\theta}:=\\hat \\theta$, where $\\hat \\theta=\\hat \\theta({N},n,C_{0})$ \n\tis given by Lemma \\ref{25.09.2014.1}. \n\tLet $0 < \\alpha \\le \\bar {\\bar \\alpha}$ and $0 < \\theta \\le \\bar {\\bar \\theta}$. We define \n\t$\\bar {\\bar \\varepsilon}=\\bar {\\bar \\varepsilon}({N},n,C_{0},\\alpha,\\theta)$ \n\tas the minimum of $\\frac{1}{16}$, a small constant depending on ${N},n,\\mathcal{K},C_{0},\\alpha, \\theta$ given by \n\tthe last lines of this proof, and of all upper bounds for $\\varepsilon$ stated in \n\tLemma \\ref{ALipschitz}, \\ref{lem5.8;1}, \\ref{25.09.2014.2} and \\ref{25.09.2014.1}.\n\tLet $k \\ge \\bar {\\bar k}$ and $\\eta \\le \\varepsilon^{2} < \\bar { \\bar \\varepsilon}^{2}$.\n\tWe have\n\t$ \\mu(F_{3}) \\le \\mu(F_{3} \\cap \\tilde{\\tilde{F}}) + \\mu(F_{3}\\setminus \\tilde{\\tilde{F}})$.\n\tWith Lemma \\ref{lem5.8;1} ($p=2$), there exists some constant $C=C({N},n,\\mathcal{K},C_{0})$ so that\n\t$\\mu(F_{3}\\setminus \\tilde{\\tilde{F}}) \\le \\mu(F\\setminus \\tilde{\\tilde{F}}) \n\t\t\\le C \\varepsilon^{\\frac{1}{2}}$.\n\tHence we only have to consider $\\mu(F_{3} \\cap \\tilde{\\tilde{F}})$.\n\tWe set\n\t$\\mathcal{G} := \\left\\{B(x,2h(x)) | x \\in F_{3} \\cap \\tilde{\\tilde{F}}) \\right\\}$.\n\tThis is a set of nondegenerate balls because $x \\in F_{3} \\subset F \\setminus \\mathcal{Z}$. \n\tFurthermore, we have $h(x) \\le 50$ for all $x \\in F$ (see Definition of $h$ on page \\pageref{Definitionvonh}).\n\tWith Besicovitch's covering theorem \\cite[1.5.2, Thm. 2]{Evans} there exist\n\t$N_{0}$ families $\\mathcal{B}_{l} \\subset \\mathcal{G}$, $l=1,...,N_{0}$, of disjoint balls such that\n\twe conclude with property (B) from page \\pageref{Grundeigenschaften}\n\t\\begin{align*}\n\t\t\\mu(F_{3} \\cap \\tilde{\\tilde{F}}) \n\t\t& \\le \\sum_{l=1}^{N_{0}} \\sum_{B \\in \\mathcal{B}_{l}} \\mu(B \\cap \\tilde{\\tilde{F}})\n\t\t \\stackrel{\\text{(B)}}{\\le} C_{0} \\sum_{l=1}^{N_{0}} \\sum_{B \\in \\mathcal{B}_{l}} (\\operatorname{diam} B)^{n}.\n\t\\end{align*}\n\tLet $ 1 \\le l \\le N_{0}$ and let\n\t$B_{1} = B(x_{1},2h(x_{1})), B_{2} = B(x_{2},2h(x_{2})) \\in \\mathcal{B}_{l}$ with $B_{1}\\neq B_{2}$.\n\tSince the balls in $\\mathcal{B}_{l}$ are disjoint, we deduce \n\t$2h(x_{1})+2h(x_{2}) \\le d(x_{1},x_{2})$ and, because of the definition of $\\tilde F$ and\n\tLemma \\ref{rem3.8}, we get \n\t$d(x_{i},\\pi(x_{i})+A(\\pi(x_{i}))) \\le \\varepsilon^{\\frac{1}{2}}d(x_{i}) \\le \\varepsilon^{\\frac{1}{2}}h(x_{i})$\n\tfor $i=1,2$.\n\tSince $ \\varepsilon^{\\frac{1}{2}} < \\frac{1}{4}$, \n\t$\\alpha < \\frac{1}{6}$ and $A$ is $3\\alpha$ Lipschitz continuous,\n\tthe former two estimates imply $h(x_{1}) + h(x_{2}) < d(\\pi(x_{1}),\\pi(x_{2}))$.\n\tThus $\\pi(\\frac{1}{2}B_{1})$ and $\\pi(\\frac{1}{2}B_{2})$ are disjoint. \n\tWe have $x_{i} \\in \\left( \\tilde{\\tilde{F}} \\cap F_{3}\\right) \\subset F \\subset B(0,5)$ \n\tfor $i=1,2$. With Lemma \\ref{25.09.2014.2}, we conclude \n\tthat $h(x_{i}) \\le \\frac{\\theta}{100} < \\frac{1}{2}$.\n\tThis implies $\\pi(\\frac{1}{2}B_{i}) \\subset U_{6}$.\n\tUsing Lemma \\ref{25.09.2014.1}, there exists some set $H_{\\theta} \\subset U_{6}$\n\tand some constant $C=C({N},n,\\mathcal{K},C_{0},k)$ with\n\t$\\mathcal{H}^{n}(U_{6} \\setminus H_{\\theta}) h(x)$ for all $x \\in F_{3} \\cap \\tilde{\\tilde{F}}$,\n\tin particular for $x=x_{i}$.\n\tWe conclude that\n\t$ \\pi(\\frac{1}{2}B_{i}) \\cap H_{\\theta} = \\emptyset$, and hence\n\t\t\\[\\sum_{B \\in \\mathcal{B}_{l}} (\\operatorname{diam} B)^{n} \n\t\t= 4^{n}\\sum_{B \\in \\mathcal{B}_{l}} \\left({\\textstyle\\frac{1}{2}}\\operatorname{diam} \\pi\\left({\\textstyle\\frac{1}{2}}B\\right)\\right)^{n}\n\t\t= 4^{n}\\sum_{B \\in \\mathcal{B}_{l}} \\frac{1}{\\omega_{\\N}} \\mathcal{H}^{n}\\left(\\pi\\left({\\textstyle\\frac{1}{2}}B\\right)\\right)\n\t\t\\le \\frac{4^{n}}{\\omega_{\\N}} \\mathcal{H}^{n}(U_{6} \\setminus H_{\\theta}).\\]\n\tNow we obtain\n\t\\[ \\mu(F_{3} \\cap \\tilde{\\tilde{F}}) \\le C_{0}N_{0} \\frac{4^{n}}{\\omega_{\\N}} \\mathcal{H}^{n}(U_{6} \\setminus H_{\\theta})\n\t\t\\le C \\left(\\frac{\\varepsilon}{\\theta^{n+1}\\alpha}\\right)^{2}.\\]\n\tand we have already shown that $\\mu(F_{3} \\setminus \\tilde{\\tilde{F}}) \\le C \\varepsilon^{\\frac{1}{2}}$.\n\tUsing $\\varepsilon < \\bar{\\bar \\varepsilon}$, we finally get\n\t\\mbox{$\\mu(F_{3}) < 10^{-6}$}.\n\\end{proof}\n\n\n\\renewcommand{\\theequation}{\\Alph{section}.\\arabic{equation}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSince the experimental discovery of the positron in 1932~\\cite{Anderson}, positron beams have played a major role in experimental physics, not only for fundamental studies but also for their wide range of practical applications, which include medicine and material science. Sub-relativistic or mildly relativistic positrons are naturally emitted by elements undergoing $\\beta^+$-decay, a widely used source for positron annihilation spectroscopy and positron emission tomography. On the other hand, ultra-relativistic positron beams are widely used in experimental particle and nuclear physics, predominantly in electron-positron colliders. The current record for the highest centre-of-mass energy in an electron-positron collider belongs to the recently dismissed LEP at CERN, able to achieve up to 209 GeV. Recent proposals for the next generation of particle colliders aim at increasing this energy, eventually reaching the TeV-scale centre-of-mass energy. For example, the International Linear Collider (ILC)~\\cite{ILC} and the Compact Linear Collider (CLIC)~\\cite{CLIC} aim, in their first stage of implementation, to achieve 250 GeV and 380 GeV, respectively. \n\n\nIncreasing collider energies to hundreds of GeV would allow precision studies of the Higgs boson and the top quark, such as the possibility of the Higgs boson being a composite particle \\cite{Higgs1} and its coupling with the top quark and itself \\cite{Higgs2,Higgs3}, besides providing hints as to why it condenses giving a non-zero vacuum expectation value \\cite{Higgs4}. Breaking the TeV barrier, on the other hand, would allow for searches beyond the Standard Model of particle physics. Besides the challenging requirement of accessing TeV-scale centre-of-mass energies, it is also necessary to create collisions at extremely high luminosity, since the typical cross sections involved scale as the inverse of the energy squared. For this reason, ILC and CLIC are aiming for luminosities greater than $10^{34}$ cm$^{-2}$s$^{-1}$, well beyond the $10^{31}$ cm$^{-2}$s$^{-1}$ previously achieved at LEP.\n\nConventional electron-positron colliders rely on established radio-frequency technology, which guarantees a maximum accelerating gradient at the level of tens to a $100\\,$MV\/m. This field is limited by the dielectric breakdown of the materials involved, implying that TeV energies can only be reached after tens to hundreds of km of acceleration. The sheer scale of these machines is thus a fundamental limiting factor in the development of lepton colliders justifying research into alternative acceleration schemes that can provide similar performance but in a more compact configuration. Among many, plasma-based wakefield acceleration is arguably one of the most promising, since it can sustain accelerating fields well beyond the GeV\/m \\cite{LWFAreview}, with experimentally demonstrated fields up to 100 GeV\/m \\cite{Liu}. Recent promising results in this area include, for instance, the demonstration of energy doubling of a 42 GeV electron beam in less than one meter of plasma \\cite{Blumenfeld}, a 2 GeV energy gain of a positron beam in one metre of plasma \\cite{Corde}, and the laser-driven acceleration of electrons up to 4.2 GeV in only 10 cm of plasma \\cite{Leemans}. Thanks to its fast-paced development, particle-driven wakefield acceleration is now actively pursued in large-scale projects, including AWAKE at CERN \\cite{AWAKE} and FACET at SLAC \\cite{FACET} (together with its current upgrade, FACET-II \\cite{FACETII}). On the other hand, laser-driven wakefield is currently studied only in relatively small University-scale laboratories even though the physics of laser-driven wakefield acceleration is in principle sufficiently mature to widespread its use to larger-scale applications. One example in this direction is the European funded project EuPRAXIA \\cite{EuPRAXIA}, which aims at building a plasma-based electron accelerator of industrial quality able to accelerate, in a stable and consistent manner, narrow-band and high-current electron beams with a maximum energy of 5 GeV.\n\nDeveloping plasma based accelerator technology to the level required by TeV-scale colliders represents a major challenge requiring progress in many fields as outlined in the recently published US roadmap for advanced accelerators~\\cite{USroadmap}. In a nutshell, a possible proposal is to use multi-staged wakefields, following promising results in proof-of-principle experiments of double-staged electron acceleration~\\cite{Steinke}.\n\nEven though the field of plasma-based electron acceleration is advancing fast, experimental studies of positron acceleration in a plasma is of a more challenging nature, due to the difficulty of providing an injector of suitable quality that can be, for instance, synchronised with the positron-accelerating region of a wakefield. \nTo date, only the proposed upgrade of FACET \\cite{FACETII} will be able, in the near future, to provide a source of ultra-relativistic positrons suitable for test studies of subsequent wakefield acceleration. In preliminary studies, including those in FACET-I~\\cite{FACET}, two alternative methods have been studied. The first option uses an electron beam as a driver and a positron beam as a witness~\\cite{Lotov}, both propagating inside a hollow channel to avoid the positron-defocussing fields~\\cite{Schroeder}. The temporal synchronisation is ensured by propagating the electron drive through a thin converter to generate the positron beam~\\cite{Wang}. Alternatively, the self-loaded plasma wakefield acceleration has been proposed~\\cite{Corde, doche}, in which a high-charge positron beam is propagated through a hollow channel. The front of the positron bunch is in charge of generating the wakefield, whose accelerating fields are experienced by the back of the bunch.\n\nProviding positron beams of suitable quality to study these acceleration schemes is indeed extremely challenging, with effectively only FACET-II as a viable facility proposed to date. However, recent experimental results on laser-based generation of high-quality ultra-relativistic positrons are suggesting an alternative pathway towards this goal. For instance, G. Sarri and co-authors have recently reported on the generation of fs-scale and narrow divergence positron beams in a plasma-based configuration \\cite{SarriPRL,SarriNCOMM}. In a nutshell, the positrons are generated as a result of a quantum cascade initiated by a laser-driven electron beam propagating through a high-Z solid target. For sufficiently high electron energy and thin converter targets, the generated positrons present properties that resemble those of the parent electron beam, hence the fs-scale duration~\\cite{Lundh}, mrad-scale divergence~\\cite{Osterhoff}, and small source size~\\cite{Kneip, Brunetti}. The maximum positron energy attainable in this scheme is naturally dictated by the peak energy of the parent electron beam. Positrons with energy up to 0.5 GeV were produced in recent experiments~\\cite{SarriNCOMM, SarriPPCF2}. These beams present unique advantages for being injected in further wakefields, when compared to more conventional sources. For instance, they have durations comparable to the positron-accelerating region of a wakefield~\\cite{Wang} and are naturally synchronised with a high-power laser. However, other characteristics still need to be carefully optimised, such as their non-negligible normalised emittance and the relatively low charge. Providing a high-quality source of ultra-relativistic positrons is however of paramount importance for the development of plasma-based accelerators of positrons, to date an area of research predominantly of a theoretical nature (see, for instance, Ref. \\cite{positrons_wakefield}). \n\nIn this paper, we report on an extensive numerical study devoted to define precisely the main properties of laser-driven positrons and assess their suitability to be further transported and manipulated in order to act as a seed for further wakefield-based acceleration stages. In this study, we assume three main ranges of parameters for the primary electron beam. In the first range, we consider typical electron parameters obtainable by 100\\,TW commercially-available laser systems (maximum electron energy of 1 GeV). In the second, we assume state-of-the-art electron beam properties, similar to the characteristics reported in Ref. \\cite{Leemans} and planned to be achieved by EuPRAXIA \\cite{EuPRAXIA} (maximum electron energy of 5 GeV). Finally, we consider near-term beam properties expected to be achieved by the next generation of ultra-high intensity laser systems, such as the Extreme Light Infrastructure Nuclear Pillar \\cite{ELI-NP} (maximum electron energy of 20 GeV). Our simulations show that high current (exceeding the kA) positron beams with sufficiently low emittance and high density can be generated, providing a firm baseline for the design of a test facility dedicated to optimising positron beam manipulation and further acceleration. \n\n\n\\section{Spatial and spectral properties of the positrons at source}\n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Figure1.eps}\n\t\\caption{\\textbf{Positron spectra}. Spectra of the positrons escaping a 1cm lead target traversed by an electron beam carrying 100 pC of charge and with an energy bandwidth of 5\\%. Different primary electron energies are considered: 1 GeV (blue), 5 GeV (red), and 20 GeV (black). The spectra show the number of positrons as a function of energy in a 5\\% bandwidth.}\n\t\\label{spectra}\n\\end{figure}\n\nIn order to assess the main characteristics of the positron beam, we simulated the propagation of an ultra-relativistic electron beam through a high-Z solid target, using the Monte-Carlo scattering code FLUKA \\cite{FLUKA1, FLUKA2}. For its practical simplicity of use, we simulate lead as a converter target and we assume a length of the order of two radiation lengths ($2L_{RAD}\\approx 1$ cm for lead). This length is the one that maximises the number of escaping positrons, for any defined set of parameters of the incoming electron beam \\cite{SarriPRL,SarriPPCF}. In order to make the obtained characteristics of the positron beam independent from those of the parent electrons, we assume a pencil-like electron beam with a point-like source and zero temporal duration. For a realistic electron beam, it is thus only necessary to convolute the parameters of the parent electrons with those of the positrons. For each simulation, $10^6$ primary electrons are assumed, and consider the positrons exiting from the rear side of the converter. The main properties of the positron beams that we consider are: source size $d$, divergence $\\theta_{div}$, temporal duration $\\tau$, Lorentz factor $\\gamma$, geometrical emittance $\\epsilon$, and normalised emittance $\\bar{\\epsilon}$. As mentioned in the introduction, three different electron energies are chosen: $E_0 =$ 1, 5, and 20 GeV. For each energy, we simulated a primary electron beam with a Gaussian spectrum centred around $E_0$ and a standard deviation $\\sigma_E = 5\\%E_0$. These three energies are chosen in order to represent the typical performance of a laser-driven electron accelerator employing a commercial laser system, state-of-the art high-intensity lasers \\cite{Leemans}, and the next generation of high-intensity lasers (such as ELI \\cite{ELI-NP}), respectively. For the 5 GeV case, it is also worth noticing that this is the baseline energy of the EuPRAXIA project \\cite{EuPRAXIA}. For the number of positrons escaping the solid target, we generally assume that the primary electron beam carries a charge of 100 pC. However, the number of positrons is directly proportional to the number of electrons in the parent beam, allowing for a simple rescaling of the results for different initial electron charges.\n\nThe spectra of the positrons escaping the converter are depicted in Fig. \\ref{spectra}. As expected from the quantum cascade in the solid, the spectra present a monotonic shape with a maximum energy corresponding to the energy of the parent electron beam \\cite{SarriPPCF}. However, it is worth noticing that a 100 pC electron beam, as routinely obtainable in a laser wakefield accelerator, is able to generate a significant number of positrons, even in a narrow energy slice. For instance, assuming EuPRAXIA-like parameters, up to $10^{11}$ positrons with energy exceeding 100 MeV can be generated, with up to $6\\times10^6$ positrons in a 5\\% bandwidth around 1 GeV, corresponding to a charge of approximately 1 pC. This number raises to $1.8\\times10^7$ (charge of 3 pC) if an initial electron beam with an energy of 20 GeV is considered. As a further comment, it must be noted that the spectrum of the generated positrons is virtually insensitive to the spectrum of the parent electron beam. It is thus preferable to use high-charge broadband electron beams, rather than focussing on narrow-band laser-wakefield acceleration. If we assume a 1 nC electron beam with a broadband spectrum up to 5 GeV, we can then obtain up to 10 pC of positrons in a 5\\% bandwidth around 1 GeV. \n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Figure2.eps}\n\t\\caption{\\textbf{Energy-dependent source-size of the positron at the exit of the converter target}. \\textbf{(a)} Spatial distribution of the entire positron beam at the rear surface of the converter. \\textbf{(b)} Transverse distribution of the positrons for different positron energies, generated by a $5\\pm5\\%\\,$GeV electron beam, fitted by Lorentzian distributions of different width. \\textbf{(c)} Extracted energy-dependent size of the real source of the positrons for different parent electron beam energies. The source size of the positrons is seen to scale approximately as the cubic root of their energy (as fitted by the red and black dashed lines).}\n\t\\label{source_size}\n\\end{figure}\n\nThis is a sizeable positron beam that could in principle be injected in further stages of laser-driven or particle-driven wakefield acceleration, provided that the beam possesses sufficient spatial quality. In particular, it is necessary to assess the source size, the divergence, and the emittance of the beam. Due to the generation of the positrons throughout the converter, the definition of the source is somewhat ambiguous. In our case, we define the real source size as the size of the positron beam at the rear surface of the converter and a virtual source size corresponding to the waist of the beam obtained by back-tracking the generated positrons. \n\nThe positron beam profile at the exit of the converter target is depicted in Fig.~\\ref{source_size}(a) for the case of a 5 GeV initial electron beam, accounting for all the positrons emitted independent of their energy or time of arrival. As it can be seen, the positron beam exhibits a smooth profile, in agreement with previous experimental observations~\\cite{SarriPPCF2}. The source size of the overall beam can be estimated to be $\\sim33\\upmu\\mbox{m}$, given by the radius or Half-Width-at-Half-Maximum of the distribution. However, the source size is energy-dependent with the spatial distribution being well-approximated by a Lorentzian distribution: \n$N_{e^+}\\simeq k\/\\left[\\pi\\sigma_x\\left(1+\\left(x\/\\sigma_x\\right)^2\\right)\\right]$ (see Fig. \\ref{source_size}(b)), with smaller sizes ($\\sigma_x$) for higher energies (see Fig. \\ref{source_size}(c)). As an example, 1 GeV positrons have a source size of around 20 microns (10 microns) if a parent 5 GeV (20 GeV) electron beam is assumed. The virtual source size is found to be about 16 micron (7.5 micron), placed inside the converter at a distance of 1.6mm (0.8mm) from its rear surface. This source size arises from assuming a pencil-like electron beam with a point-like source. In order to compute the realistic positron source size one would then need to convolute these results with the intrinsic beam size of the specific electron beam used. However, laser-driven electron beams typically have a source size in the micron range \\cite{Kneip} and typical divergences in the mrad range \\cite{Osterhoff}, inducing only small corrections to the results reported here.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Figure3.eps}\n\t\\caption{\\textbf{Energy-dependent divergence of the positrons at the exit of the converter target.} Different colours correspond to different energies of the primary electron beam. At multi-GeV, the divergence of the positrons scales approximately as the inverse of their energy (fit shown by the black dashed line), in good qualitative agreement with the Rossi-Greisen formula (red dashed line).}\n\t\\label{divergence}\n\\end{figure}\n\nSimilarly, the divergence of the positrons is seen to decrease for increasing positron energies, as depicted in Fig.~\\ref{divergence}. At 1 GeV the positrons have a divergence of the order of 10 mrad, which decreases further down to 3 mrad at 5 GeV. The divergence is seen to scale with the positron energy as $\\propto E^{-0.87}$, in good qualitative agreement with the Rossi-Greisen formula~\\cite{Greisen} routinely used in accelerator physics. Again this divergence should be convoluted with the divergence of the primary electron beam, inducing a small correction, in the mrad range.\n\n\\section{Temporal properties of the positrons at source}\n\n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Figure4.eps}\n\t\\caption{\\textbf{Temporal properties of the positron beam at source.} \\textbf{(a)} Temporal profile of the positrons of energy 250, 500, 1000 and 2500 MeV, produced from the interaction of a 5 GeV electron beam with a 1cm-thick Pb target. The points are well fitted by a log-normal distribution. \\textbf{(b)} Duration of the positron beam as a function of their energy. \\textbf{(c)} Peak current of the positrons contained within each 5\\% energy bin.}\n\t\\label{current}\n\\end{figure}\n\nA defining feature of laser-driven electron beams is their short temporal duration, down to even the femtosecond-scale~\\cite{Lundh}. Such a short duration is in principle achievable by radiofrequency accelerators, but at the cost of implementing complex beam slicing devices~\\cite{beam_slicing}. It is thus to be expected that positrons generated from such electron beams will present similarly short durations. In order to extract the temporal duration of the positron beam at source, a specific add-on code was written in the source files of FLUKA. Obtained beam durations are shown in Fig. \\ref{current}. Frame a) depicts the temporal distributions of positrons of different energy. Each distribution resembles a log-normal distribution with different variances. \n\nAs intuitively expected, higher positron energies correspond to shorter beam durations, of the order of approximately 1 fs (longitudinal size of 300 nm) at 1 GeV for a 5 GeV parent electron beam. This effectively implies that the 1 GeV positron beams will have virtually the same duration as the primary electron beam. Assuming a realistic electron beam with a duration of 5fs, we obtain a longitudinal size of the positron beam of the order of $\\sigma_z\\approx 1.5\\,\\upmu$m. \n\nThis value is much smaller than the one proposed for FACET-II \\cite{FACETII} (10 $\\upmu$m) and indeed extremely useful for high-energy applications. For instance, a short longitudinal size is highly desirable for injection in additional wakefield acceleration stages. The extremely small positron-accelerating region, of a few microns, in plasma wakefield accelerators have led to alternative schemes being investigated, such as self-loaded PWFA~\\cite{corde, doche}. In SL-PWFA, the front of a long, high-charge positron beam drives a wakefield, which accelerates the positrons contained in the back of the bunch, in a positron-accelerating region larger than in the regular electron-driven wakefield. However, such a mechanism requires very large positron charges in order to generate the wakefield, while only a small fraction of these positrons are actually accelerated. Furthermore, the extension of the accelerating region comes at the cost of a significant reduction of the accelerating field. The intrinsically-short durations of laser-driven positron beams would allow the positron acceleration in the original wakefield produced by a laser or an electron drive beam, reducing the number of stages required for high-energy applications.\n\nMoreover, a short beam duration is recommended in order to minimise beam disruption in high-energy electron-positron collisions, as caused by beam focussing induced by the electromagnetic fields of the other beam. The beam disruption parameter is in fact directly proportional to the beam longitudinal size~\\cite{chen1988}: \n\\begin{equation}\nD_{x,y} \\approx \\frac{2r_eN\\sigma_z}{\\gamma \\sigma_{x,y},(\\sigma_x+\\sigma_y)},\n\\end{equation}\nwhere $r_e$ is the classical electron radius, $\\gamma$ is the particle Lorentz factor, $N$ is the number of particles in a beam, and $\\sigma_{x,y}$ are the transverse dimensions of the beam.\n\nMoreover, a short longitudinal size of the beam is also desirable in order to minimise beamsstrahlung, one of the main sources of background noise in high-energy colliders~\\cite{schroeder2012}. In its quantum regime, the number of photons generated via beamsstrahlung scales as $n_{phot}\\propto\\sigma_z^{1\/3}$~\\cite{delahaye1999} and the number of pairs produced coherently $n_{coh}\\propto\\sigma_z^{2\/3}$~\\cite{yokoya1990}\n\nFinally, a short beam duration naturally translates into a high peak current of the beam (Fig. \\ref{current}.c). In a 5\\% slice centered around 1 GeV, a 5 GeV, 100 pC electron beam can generate a positron current exceeding the kA. Even though one must note that a high current is not explicitly required in liner colliders, this characteristic is still of great interest for other applications in material science.\n\n\\section{Emittance of the positrons at source}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Figure5.eps}\n\t\\caption{\\textbf{Positron emittance}. \\textbf{(a)} Phase-space diagram of 1 GeV positrons exiting the converter target. \\textbf{(b)} Energy-dependent normalised emittance of the positron at the exit of a 1-cm thick Pb target for different initial electron energies. The emittance of the positrons scales approximately as $E^{-2\/3}$.} \n\t\\label{norm_emittance}\n\\end{figure}\n\nThe positrons generated inside the converter are emitted in a beam-like fashion. Such a beam can be described, in a first approximation, by its source size and divergence. However, the most important parameter to characterise the spatial quality of a beam is arguably its emittance.\nThe emittance is a measure of the volume of the particle beam in phase-space, and defines the ease for beam transportation using magnetic fields, as well as being the main parameter in conjunction with the current defining the brightness of the beam. The emittance is mathematically defined as $\\epsilon_x = \\sqrt{\\langle x^2\\rangle \\langle x'^2 \\rangle - \\langle xx' \\rangle^2}$, where $x$ is the position of the particles and $x'$ its divergence. Typically, the emittance is defined for particles within a narrow energy range (monoenergetic approximation) and low angular divergence (paraxial approximation), which allows to define the normalised emittance as $\\bar{\\epsilon} = \\gamma \\beta \\epsilon$, where $\\gamma$ is the particle Lorentz factor and $\\beta$ its velocity normalised to the speed of light in vacuum. \nAs an example, the phase-space of positrons of energy $1.00\\pm0.05\\,$GeV leaving the converter is shown in Fig.~\\ref{norm_emittance}(a), as generated from the interaction of a $5\\,$GeV electron beam. The emittance in this case corresponds to the area of the ellipse defined by the rms of the distribution, depicted in Fig.~\\ref{norm_emittance}(a) by the black, dash-dotted ellipse. With that, a geometrical emittance of $\\epsilon_x=0.1\\pi\\,$mm mrad is obtained (normalised emittance $\\bar{\\epsilon}_x=200\\pi\\,$mm mrad). \nGiven the strong dependence of both source size and divergence on the positron energy considered, the emittance is expected to follow a similar trend. The scaling of the normalised emittance with the positron energy bin is shown in Fig.~\\ref{norm_emittance}(b), showing the data points to close follow a power-law function.\n\n\\section{Discussion}\n\nThe main properties of positrons generated from laser-driven electrons have been shown in the previous sections. These characteristics can be directly compared with those obtainable with more conventional generation schemes and with the requirements for injection in laser-driven and particle-driven wakefield acceleration stages. We take FACET-I and FACET-II as representative examples of state-of-the-art particle-driven wakefield facilities and this comparison is shown in table~\\ref{tab:Comparison}.\n\n\n\\begin{table}[b!]\n\\centering\n\\begin{tabular}[b!]{ccccc}\n\\hhline{=====}\n\t\t & Units & FACET-I & FACET-II & LWFA \\\\\n\\hline\\hline\n\t\t$E$ & GeV & 21 & 10 & 1 \\\\\n\t $P$ & W & 7.4 & 9.6 & 3 \\\\\n\t\t$Q_e$ & pC & 350 & 500 & 2 \\\\\n\t\t$\\sigma_x$ & $\\upmu$m& 30 & 4 & 16\\\\\n\t\t$\\sigma_y$ & $\\upmu$m& 30 & 4 & 16\\\\\n\t\t$\\sigma_z$ & $\\upmu$m& 50 & 6.4 & 0.6\\\\\n\t\t$\\bar{\\epsilon}_x$ & mm\\,mrad & 200 & 7 & 500\\\\\n\t\t$\\bar{\\epsilon}_y$ & mm\\,mrad & 50 & 3 & 500\\\\\n\t\t$\\Delta E$ & \\% & 1.5 & 1 & 5 \\\\\n\t\n\t\t$ f $ & Hz & 1 & 1 & $10$ - $10^{3}$ \\\\\n\t\t$\\ell$ & cm$^{-2}$s$^{-1}$ & $5\\times10^{23}$ & $6\\times10^{25}$ & $10^{22 - 24}$ \\\\\n\\hhline{=====}\n\\end{tabular}\n\\caption{\\textbf{Comparison with FACET-I and FACET-II} the main positron beam parameters obtained in FACET-I and expected for FACET-II \\cite{Joshi} are compared with typical parameters at source from a laser wakefield scheme as discussed in this manuscript. The latter assumes a 100 pC 5 GeV primary electron beam. $E$ denotes the beam energy, $W$ the average power, $Q_e$ the total charge (in a 5\\% bandwidth centered around 1 GeV in the LWFA case), $\\sigma_{x,y}$ the transverse sizes, $\\sigma_z$ the longitudinal size, $\\bar{\\epsilon}_{x,y}$ the normalised emittance, $\\Delta E$ the energy spread, $f$ the frequency of operation, and $\\ell$ the luminosity.}\n\\label{tab:Comparison}\n\\end{table}\n\n\nAs one can see, the main drawback for the use of laser-driven sources is given by the lower number of positrons in the beam, limited to a few pC of charge for realistic charges of the parent electron beam (100pC-1nC). This charge is two orders of magnitude lower than that attainable with FACET, and up to three orders of magnitude lower than the charges in conventional accelerators, such as the Stanford Linear Collider (SLC, 1.6nC), Large Electron Positron collider (LEP, 240pC), or estimated for the International Linear Collider (ILC, 3.2nC). The main reason for the lower charge is the significantly lower electron charge attainable via LWFA compared to the 10s-100s nC of charge in conventional accelerators in the other cases. However, the positron beams from conventional accelerators typically form bunches significantly longer than those for the case of laser-driven sources. Thus, the positron densities for a ns-long from a conventional accelerator would be of the order of a few Coulomb per second, lower than the hundreds of Coulombs per second in the case of beams with a few pC in a few fs for the laser-driven sources. Furthermore, it should be noted that few pC charges is sufficient for most applications (including cooling and acceleration studies for positrons towards colliders), particularly with the advances in high power lasers towards higher repetition rates.\n\n\nAs one can see, the main drawback for the use of laser-driven sources is given by the lower number of positrons in the beam, limited to a few pC of charge for realistic charges of the parent electron beam (100pC-1nC). This charge is two orders of magnitude lower than that attainable with FACET, and up to three orders of magnitude lower than the charges in conventional accelerators, such as the Stanford Linear Collider (SLC, 1.6nC), Large Electron Positron collider (LEP, 240pC), or estimated for the International Linear Collider (ILC, 3.2nC). The main reason for the lower charge is the significantly lower electron charge attainable via LWFA compared to the 10s-100s nC of charge in conventional accelerators in the other cases. However, it should be noted that few pC charges is sufficient for most applications (including cooling and acceleration studies for positrons towards colliders), particularly with the advances in high power lasers towards higher repetition rates.\nThe main bottleneck to reach higher repetition rates in high power lasers has been the cooling of the lasing medium, due to the use of white flashlamps as pumps. The use of diodes with wavelength in the absorption band for the laser emission will allow for higher repetition rates. For instance, the DiPOLE project has recently demonstrated a stable system delivering 10 J laser pulses at 10 Hz. The design is scalable and demonstration of 100 J laser pulse energy at 10 Hz is forthcoming, eventually aiming for a 1kJ 10 Hz laser. The need for the development of 1kHz high power lasers is also identified in the US roadmap for future novel accelerators.\nSuch high repetition rates will help compensating the lower single-shot charge by providing a greater time-averaged charge. \n\nFurthermore, the total charge in the beam can be increased significantly simply by considering a larger energy bandwidth. For example, the final stage of LEP used positrons accelerated by the pre-linac with energy in the range $90\\pm 7\\%$\\,MeV, and SLC could accelerate up to $20\\%$ of the positrons initially contained in the energy range $11\\pm 80\\%$\\,MeV. Considering both solutions, the time-averaged charge of LWFA-driven positrons would reach values of hundreds of pC per second, comparable to that of the FACET projects.\n\nAs shown in table~\\ref{tab:Comparison}, the normalised emittance of the LWFA-based positrons is already comparable to that of FACET-I. Such an emittance would thus allow for a direct injection in a particle-based wakefield accelerator, without the need of a linear accelerator and emittance damp to produce the positron beam. In fact, recent studies~\\cite{doche} have shown that a positron beam with degraded emittance can still be significantly accelerated using PWFA in FACET-I, at the price of achieving a lower energy gain. With respect to conventional accelerators, the emittance of the LWFA-based positrons is significantly lower than that in their initial acceleration stages. For example, the initial stages of SLC presented normalised emittances of $6500\\pi$ mm\\,mrad (geometrical emittance of 500 mm\\,mrad), and $11200\\pi$ mm\\,mrad (geometrical emittance of $62\\pi$ mm\\,mrad) for the case of LEP. The largely reduced emittance would allow for the injection in conventional accelerators without the need of emittance damps, that would reduce the charge in the beam and increase the cost and size of the system. \n\nFinally, the temporal duration of the positrons by laser-driven sources is significantly shorter than the other sources. Conventional accelerators naturally produce beams with durations of the order of a few ns, which can then be treated to reduce it down to hundreds of fs, at the cost of a reduction in the charge in the bunch. The shorter duration, automatically given by the intrinsically short duration of the electron beam, presents direct advantages for applications in material science. Additionally, as mentioned earlier, the reduced longitudinal dimension of the beam is smaller than the positron-accelerating region of the wakefield in a plasma~\\cite{Wang}. The region of positron accelerating and focussing fields in an electron- or laser-driven plasma wakefield is limited to a sub-10 micron region along the propagation axis ($\\sim35\\,$fs).\nAlthough future studies, both in particle-driven or laser-driven wakefields in FACET-II and EuPRAXIA, might open up new possibilities for a more robust and flexible positron acceleration scheme, currently only short beams fitting that region, such as the laser-driven positrons, can be efficiently accelerated by the wakefield. Furthermore, the laser-driven positrons present additional advantages. For instance, the wakefield structure can be significantly modified if a high-charge positron beam is used as a witness~\\cite{Lotov}, reducing the peak accelerating field. Although high charges are desirable for schemes such as self-loaded PWFA, in which the positron beam both generates the wakefield and is accelerated~\\cite{Corde}, lower charges may be preferred in more conventional plasma wakefield accelerators to maximise the electric field and reduce the number of stages required. Finally, a laser-driven source would also present the advantage of being jitter-free, allowing a stable synchronisation on a fs-scale. This problem is partially solved by using the drive electron beam to generate the positrons by propagating through a high-Z material~\\cite{Wang}. However, such a procedure results in a slight worsening of the electron drive beam emittance, as well as adding the technological complication of ensuring the appropriate temporal separation between the electron drive and the positron witness beams, to ensure the synchronisation between the positron beam and the wakefield rather than the electron beam. A laser-driven source, on the other hand, would be intrinsically synchronised with a high-power laser. The lack of jittering, thanks to the laser beams being produced from the same oscillators, ensures a stable synchronisation of the different laser-driven wakefield stages, with temporal matching to the fs-level already achieved experimental~\\cite{Corvan}. It should be noted, however, that in the case of TeV accelerators, beam cooling might be required in order to focus the beams to the nanometer-scale, boosting the overall luminosity. Further studies would be required regarding the cooling needs, which would come at the cost of extending the bunch longitudinal size (duration), but would allow for a broader bandwidth accepted, significantly increasing the number of positrons in the beam.\n\nA final note must be made regarding the transport and energy selection of the laser-driven positrons. As shown earlier, the $1.00\\pm0.05\\,$GeV positrons generated exhibit a low emittance. However, it should be noted that this energy band has to be selected from the broadband positron emission from the converter. \nAlso, even the positrons contained in the reduced bandwidth present a significant level of divergence, rapidly expanding the beam to sizes greater than the extension of the positron-focussing fields in the wakefield, of the order of tens of microns. Therefore, the positron beam requires collimation and re-focussing in order to be further accelerated. Both collimation and energy-selection appear feasible using magnetic systems. Laser-driven schemes are compact and produce lower levels of radiation, allowing for closely-coupled experiments. Thanks to that, a compact quadrupole triplet or a plasma lens can be used to collimate the positron emission. Given the low divergence and narrow energy band considered, a large portion of the positron beam can be collected by the system, greater than the $20\\%$ collected at SLC, in which a broader energy range was accepted. Once the beam has been collimated, well-known energy selection system based on magnetic chicanes, such as those in conventional accelerators, can be implemented to select the positron energies to be further transported.\n\n\\section{Conclusions}\nWe have simulated the main properties of laser-driven positrons in order to test the feasibility of their injection in a multi-stage, plasma-based accelerator. From the interaction of the LWFA-driven electrons with the high-Z converter, a broadband positron beam can be generated, whose characteristics significantly improve with energy. For an electron beam similar to that proposed by the EuPRAXIA project (5 GeV, 100pC), up to $5\\times10^6$ positrons are generated in a energy range $1.00\\pm0.05\\,$GeV, with a normalised emittance of $190\\pi$\\,mm\\,mrad. Such an emittance is compatible with the further injection on secondary acceleration stages of plasma-based accelerators. Additionally, the naturally short duration of the positron beam, similar to that of the parent electron beam, can present additional advantages towards the further acceleration in a wakefield structure, as well as being beneficial for future colliders thanks to the reduced beamsstrahlung. Laser-driven positron beams can thus constitute an appealing alternative as a witness beam for beam-driven and laser-driven wakefield test facilities and, potentially, as injector of reduced cost and size for future high energy accelerators and lepton colliders.\n\n\\section{Acknowledgments}\nThe authors acknowledge financial support from EPSRC (Grant No: EP\/N027175\/1 and EP\/P010059\/1). This work was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 653782. The simulation data are available at \\emph{(URL to be inserted)}\n\n\\section{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzebti b/data_all_eng_slimpj/shuffled/split2/finalzzebti new file mode 100644 index 0000000000000000000000000000000000000000..83d61fdc87d665265cc008f23f1d9537ea9440cb --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzebti @@ -0,0 +1,5 @@ +{"text":"\\section*{Results for ISTAT data}%\n\\label{sect:ISTATresults}\n\nLifetimes beyond 105 years are highly unusual and the application of extreme value models \\citep{dehaanferreira2006} is warranted. We use the generalized Pareto distribution,\n\\begin{equation}\n\\label{eq:GP}\nF(x) = \\begin{cases}\n 1-(1+{\\gamma}x\/{\\sigma})_+^{-1\/{\\gamma}}, & x \\geq 0, \\gamma \\neq 0, \\\\\n 1-e^{-{x}\/{\\sigma}}, & x \\geq 0, \\gamma = 0,\n \\end{cases}\n\\end{equation}\nto model $x$, the excess lifetime above $u$ years. In~\\eqref{eq:GP}, $a_+=\\max(a,0)$ and $\\sigma > 0$ and $\\gamma \\in \\mathbb{R}$ are scale and shape parameters. For negative shape parameter $\\gamma$ the distribution has a finite upper endpoint at $-\\sigma\/\\gamma$, whereas $\\gamma\\geq 0$ yields an infinite upper endpoint.\n\nThe corresponding hazard function, often called the ``force of mortality'' in demography, is the density evaluated at excess age $x$, conditional on survival to then, i.e., \n\\begin{equation}\n\\label{eq:GPhazard}\nh(x)=\\dfrac{f(x)}{1-F(x)} = \\dfrac{1}{(\\sigma + \\gamma x)_+}, \\quad x\\geq 0, \n\\end{equation}\nwhere $f(x)=\\mathrm{d}F(x)\/\\mathrm{d}x$ is the generalized Pareto density function. If $\\gamma <0$, the hazard function tends to infinity at the finite upper limit for exceedances. When $\\gamma = 0$, $F$ is exponential and the hazard function is constant, meaning that the likelihood that a living individual dies does not depend on age beyond the threshold. In this case, mortality can be said to have plateaued at age $u$. \n\n\\begin{figure\n\\centering\n\\includegraphics[width=0.65\\linewidth]{figure\/Fig2.pdf}\n\\includegraphics[width=0.65\\linewidth]{figure\/Fig5.pdf}\n\\caption{Parameter stability plots for the \\textsf{ISTAT}{} data (top) and for the \\textsf{France 2019}{} data (bottom), showing the shape $\\gamma$ of the generalized Pareto distribution (left) and the scale $\\sigma_e$ of the exponential distribution (right) based on lifetimes that exceed the age threshold on the $x$-axis. The plots give maximum likelihood estimates with 95\\% confidence intervals derived using a likelihood ratio statistic. The horizontal lines in the right-hand panels correspond to the estimated scale for excess lifetimes over 108 years for the \\textsf{ISTAT}{} data.}\n\\label{fig:parameterstability}\n\\end{figure}\n\nThe choice of a threshold $u$ such that Eq.~[\\ref{eq:GP}] models exceedances appropriately is a basic problem in extreme value statistics and is surveyed by Scarrott \\& MacDonald~\\cite{scarrott\/macdonald:2012}. If $u$ is high enough for Eq.~[\\ref{eq:GP}] to provide an adequate approximation to the distribution of exceedances, then the shape parameter $\\gamma$ is approximately unchanged if a higher threshold $u'$ is used, and the scale parameters for $u$ and $u'$ have a known relationship, so a simple and commonly-used approach to the choice of threshold is to plot the parameters of the fitted distributions for a range of thresholds \\citep{davison+s:1990} and to use the lowest threshold above which parameter estimates stabilise. This choice balances the extrapolation bias arising if the threshold is too low with the increased variance incurred when taking $u$ too high to retain an adequate number of observations.\n\nThe upper left-hand panel of Figure~\\ref{fig:parameterstability} shows that for age thresholds close to 105 years the estimated shape parameters for excess life lengths are negative, with 95\\% confidence intervals barely touching zero, but there is no systematic indication of non-zero shape above 107 years. The upper right-hand panel displays the estimated scale parameter of the exponential model fitted to life lengths exceeding the threshold. The scale parameters decrease for ages 105--107 but show no indication of change after age 107, where the scale parameter estimate is 1.45. Parameter stability plots suggest an exponential model and hence a constant hazard after age 107 or so, where a mortality plateau seems to be attained.\n\n\n\n\n\n\n\n\nThe upper part of \\Cref{tab:ISTAT-MLE} shows results from fitting Eq.~[\\ref{eq:GP}] and the exponential distribution to the \\textsf{ISTAT}{}\\ for a range of thresholds. The exponential model provides an adequate fit to the exceedances over a threshold at 108 years, above which the hypothesis that $\\gamma=0$, i.e., the exponential model is an adequate model simplification, is not rejected. \n\n\n\\begin{table*}[t]\n\\centering\n\n\\begingroup\\footnotesize\n\\begin{tabular}{lrrrrrrrr}\n \\toprule\n\\textsf{ISTAT}{} & threshold & $105$ & $106$ & $107$ & $108$ & $109$ & $110$ & $111$\\\\\n \\midrule\n& $n_u$ & $3836$ & $1874$ & $947$ & $415$ & $198$ & $88$ & $34$ \\\\ \n& $\\sigma$ & $1.67\\; (0.04)$ & $1.7\\; (0.06)$ & $1.47\\; (0.08)$ & $1.47\\; (0.11)$ & $1.33\\; (0.15)$ & $1.22\\; (0.23)$ & $1.5\\; (0.47)$ \\\\ \n& $\\gamma$ & $-0.05\\; (0.02)$ & $-0.07\\; (0.03)$ & $-0.02\\; (0.04)$ & $-0.01\\; (0.06)$ & $0.03\\; (0.09)$ & $0.12\\; (0.17)$ & $0.06\\; (0.30)$ \\\\ \n& $\\sigma_e$ & $1.61\\; (0.03)$ & $1.6\\; (0.04)$ & $1.45\\; (0.05)$ & $1.45\\; (0.08)$ & $1.36\\; (0.11)$ & $1.35\\; (0.17)$ & $1.58\\; (0.32)$ \\\\ \n& $p$-value & $0.04$ & $0.01$ & $0.70$ & $0.82$ & $0.74$ & $0.44$ & $0.84$ \\\\ \n& $p_\\infty$ & $0.02$ & $0.01$ & $0.35$ & $0.41$ & $0.63$ & $0.78$ & $0.58$ \\\\ \n \\bottomrule\n\\toprule\n\\textsf{France 2019}{} & threshold & $105$ & $106$ & $107$ & $108$ & $109$ & $110$ & $111$\\\\\n \\midrule\n& $n_u$ & $9835$ & $5034$ & $2472$ & $1210$ & $550$ & $240$ & $106$ \\\\ \n& $\\sigma$ & $1.69\\; (0.02)$ & $1.59\\; (0.03)$ & $1.54\\; (0.04)$ & $1.43\\; (0.06)$ & $1.36\\; (0.08)$ & $1.34\\; (0.13)$ & $1.26\\; (0.18)$ \\\\ \n& $\\gamma$ & $-0.06\\; (0.01)$ & $-0.04\\; (0.01)$ & $-0.04\\; (0.02)$ & $-0.02\\; (0.03)$ & $0.02\\; (0.05)$ & $0.05\\; (0.07)$ & $0.09\\; (0.11)$ \\\\ \n& $\\sigma_e$ & $1.62\\; (0.02)$ & $1.53\\; (0.03)$ & $1.49\\; (0.03)$ & $1.41\\; (0.05)$ & $1.38\\; (0.07)$ & $1.39\\; (0.11)$ & $1.36\\; (0.16)$ \\\\ \n& $p$-value & $4 \\times 10^{-7}$ & $0.01$ & $0.05$ & $0.60$ & $0.72$ & $0.48$ & $0.32$ \\\\\n& $p_\\infty$ & $2 \\times 10^{-7}$ & $2 \\times 10^{-3}$ & $0.02$ & $0.30$ & $0.64$ & $0.76$ & $0.84$ \\\\ \n \\bottomrule\n\\end{tabular}\n\\caption{Estimates (standard errors) of scale and shape parameters ($\\sigma$, $\\gamma$) for the generalized Pareto distribution and of the scale parameter ($\\sigma_e$) for the exponential model for the \\textsf{ISTAT}{} and \\textsf{France 2019}{} datasets as a function of threshold, with number of threshold exceedances ($n_u$), $p$-value for the likelihood ratio test of $\\gamma=0$ and probability that $\\gamma \\geq 0$ based on the profile likelihood ratio test under the generalized Pareto model ($p_\\infty$).} \n\\label{tab:ISTAT-MLE}\n\\endgroup\n\\end{table*}\n\n\nThe estimated scale parameter obtained by fitting an exponential distribution to the \\textsf{ISTAT}{}\\ data for people older than 108 is 1.45 (years) with 95\\% confidence interval $(1.29, 1.61)$. Hence the hazard is estimated to be 0.69 (1\/years) with 95\\% confidence interval $(0.62, 0.77)$; above 108 years the estimated probability of surviving at least one more year at any given age is 0.5 with 95\\% confidence interval $(0.46, 0.54)$. \n\n\nWe investigated birth cohort effects, but found none; see {Appendix}~\\ref{subsect:cohorteffect}.\n\n\n\\section*{Results for \\textsf{France 2019}{} data}%\n\\label{sect:Frenchresults}\n\n\n\\begin{table*}[t!]\n\t\\centering\n\t\t\n\t\\begin{tabular}{l rl l rl l rl}\n\t\t\\toprule\n & \\multicolumn{2}{c}{\\textsf{ISTAT}{}}& & \\multicolumn{2}{c}{\\textsf{France 2019}{}}&&\\multicolumn{2}{c}{\\textsf{IDL 2016}{}} \\\\\n \\cline{2-3}\\cline{5-6}\\cline{8-9}\\\\\n\t\t & \\multicolumn{1}{c}{$n$} &\\multicolumn{1}{c}{$\\sigma_e$ (95\\% CI)}& & \\multicolumn{1}{c}{$n$} & \\multicolumn{1}{c}{$\\sigma_e$ (95\\% CI)} & &\\multicolumn{1}{c}{$n$} & \\multicolumn{1}{c}{$\\sigma_e$ (95\\% CI)} \\\\\n\t \\midrule\n\t\t{\\footnotesize women }& $375$ &$1.45~(1.23, 1.62)$ && $1116$ & $1.46~(1.36, 1.56)$&& $507$ & $1.39~(1.25, 1.54)$\\\\\n\t\t{\\footnotesize men } &$40$ &$1.41~(0.86, 1.98)$&& $94$ & $0.90~(0.70, 1.11)$ && $59$& $1.68~(1.16, 2.20)$ \\\\\n {\\footnotesize All}& $415$ &$1.45~(1.29, 1.61)$ && $1210$ & $1.41~(1.32, 1.51)$ && $566$ & $1.42~(1.28, 1.56)$ \\\\\n \\bottomrule\n\t\t\\end{tabular}\n\t\t\\caption{Estimates of the scale, $\\sigma_e$, of the exponential distribution, with 95\\% confidence intervals (CI). This distribution is fitted to exceedances of 108 years in the \\textsf{ISTAT}{}\\ and \\textsf{France 2019}{} data and of 110 years in the \\textsf{IDL 2016}{}\\ data analysed in \\cite{rootzen-zholud:2017}. }\n\\label{table:women-men}\n\t\t\\end{table*}\t\t\n\nEstimation for the \\textsf{France 2019}{} data was done as described in Rootz\\'en \\& Zholud \\citep{rootzen-zholud:2017}, taking into account the left- and right truncation of the lifetimes. Both the parameter stability plots in the lower panels of \\Cref{fig:parameterstability} and the results given in the lower part of Table~\\ref{tab:ISTAT-MLE} indicate that the exponential model is adequate above 108 years. For persons older than 108 the exponential scale parameter is estimated to be 1.41 (years) with 95\\% confidence interval $(1.32, 1.51)$, the hazard is estimated to be 0.71 (years$^{-1}$) with 95\\% confidence interval $(0.66, 0.76)$ and the estimated probability of surviving at least one more year is 0.49 with 95\\% confidence interval $(0.47, 0.52)$.\n \nTable~\\ref{table:women-men} shows that estimates of the scale parameter for the exponential distribution for women and men for the \\textsf{France 2019}{} data differ. If men are excluded, then the estimated scale parameter increases from 1.41 to 1.46 years, and if the oldest woman, Jeanne Calment, is also excluded, the estimate for women drops to 1.44 years. Similarly to the \\textsf{ISTAT}{} data, survival for ages 105--107 was lower in earlier cohorts. \n\n\\section*{Power}\n\n\\section*{Power}\n\nOur analysis above suggests that there is no upper limit to human lifetimes and that constant hazard adequately models excess liftime if one considers only those persons whose lifetimes exceed 108 years: there is no evidence that the force of mortality above this age is other than constant. One might wonder whether increasing force of mortality would be detectable, however, as the number of persons attaining such ages is relatively small. To assess this we performed a simulation study described in the~{Appendix}~\\ref{subsect:power}, mimicking the sampling schemes of the \\textsf{ISTAT}{}, \\textsf{France 2019}{} and \\textsf{IDL 2016}{} datasets as closely as possible and generating samples from the generalized Pareto distribution with $-0.25\\leq \\gamma \\leq 0$. To remove overlap between the last two datasets, we dropped France from \\textsf{IDL 2016}{}.\n\n\nAny biological limit to their lifespan should be common for all humans, whereas differences in mortality rates certainly arise due to social and medical environments and can be accommodated by letting hazards vary by factors such as country or sex. With the overlap dropped we can treat the datasets as independent and compute the power for a combined likelihood ratio test of $\\gamma=0$ (infinite lifetime) against alternatives with $\\gamma<0$ (finite lifetime). For concreteness of interpretation we express the results in terms of the implied upper limit of lifetime $\\iota=u-\\sigma\/\\gamma$. The left-hand panel of \\Cref{fig:powerendpoint} shows the power curves for the three datasets individually and pooled. The power of the likelihood ratio test for the alternatives $\\iota \\in \\{125, 130, 135\\}$ years, for example, is $0.46\/0.32\/0.24$ for the \\textsf{ISTAT}{} data above 108, $0.82\/0.61\/0.46$ for the \\textsf{France 2019}{} data above 108, and $0.62\/0.40\/0.29$ for the \\textsf{IDL 2016}{} data above 110. The power for $\\iota=125\/130\/135$ years based on all three datasets is $0.96\/0.80\/0.64$, so it appears to be rather unlikely that an upper limit to the human lifespan, if there is such a limit, is below 130 years or so.\n\nSimilar calculations give the power for testing the null hypothesis $\\gamma=0$ against alternatives $\\gamma<0$. Forcing all datasets to have the same shape parameter would allow them to have different endpoints so we reject the overall null hypothesis if we reject the exponential hypothesis any of the three datasets. The power of this procedure is shown with a dashed black line in \\Cref{fig:powerendpoint}. The resulting combined power exceeds $0.8$ for $\\gamma < -0.07$ and equals $0.97$ for the alternative $\\gamma=-0.1$, giving strong evidence against a sharp increase in the hazard function after 108 years. \n\n\n\n\n\\begin{figure*\n\\centering\n\\includegraphics[width=0.95\\linewidth]{figure\/Fig9b.pdf}\n\\caption{Power functions based on the \\textsf{IDL 2016}{} (excluding French records), \\textsf{France 2019}{} and \\textsf{ISTAT}{} databases and combined datasets, with rugs showing the lifetimes above 115. Left: power for the alternative of a finite endpoint $\\iota$ against the null hypothesis of infinite lifetime based on the likelihood ratio statistic. The endpoint cannot be lower than the largest observations in each database. Right: power of the Wald statistic for the null hypothesis $ \\gamma = 0$ against the one-sided alternative $ \\gamma < 0$ as a function of $\\gamma$; the dashed line represents the power obtained by rejecting the null of exponentiality when any of the three one-sided test rejects. The curves are obtained by conditioning on the birthdates and left-truncated values in the databases, then simulating generalized Pareto data whose parameters are the partial maximum likelihood estimates $(\\widehat{\\sigma}_\\gamma, \\gamma)$. The simulated records are censored if they fall outside the sampling frame for the \\textsf{ISTAT}{} data and are simulated from a doubly truncated generalized Pareto distribution for \\textsf{IDL 2016}{} and \\textsf{France 2019}{}. See {Appendix}~\\ref{subsect:power} for more details.}\n\\label{fig:powerendpoint}\n\\end{figure*}\n\n\n\\section*{Gompertz model}\n\n\nThe hazard function of the generalized Pareto distribution cannot model situations in which the hazard increases to infinity, but the upper limit to lifetimes is infinite. This possibility is encompassed by the Gompertz distribution \\citep{Gompertz:1825}, which has long been used for modelling lifetimes and often provides a good fit to data at lower ages \\citep[e.g.,][]{Thatcher:1999}. When the Gompertz model is expressed in the form\n\\begin{align*}\nF(x) = 1 - \\exp\\left\\{-(e^{\\beta x\/\\sigma }-1)\/\\beta\\right\\}, \\quad x>0,\\quad \\sigma, \\beta>0, \n\\end{align*}\n $\\sigma$ is a scale parameter with the dimensions of $x$, and the dimensionless parameter $\\beta$ controls the shape of the distribution. Letting $\\beta\\to 0$ yields the exponential distribution with mean $\\sigma$; small values of $\\beta$ correspond to small departures from the exponential model. The fact that $\\beta$ cannot be negative affects statistical comparison of the Gompertz and exponential models; see~{Appendix}~\\ref{subsect:gompertz}.\n\nThe Gompertz distribution has infinite upper limit to its support, so it cannot be used to assess whether there is a finite upper limit to the human lifespan. Its hazard function, $\\sigma^{-1}\\exp(\\beta x\/\\sigma)$, is finite but increasing for all $x$ ($\\beta>0$) or constant ($\\beta=0$). The limiting distribution for threshold exceedances of Gompertz variables is exponential, and this limit is attained rather rapidly, so a good fit of the Gompertz distribution for lower $x$ would be compatible with good fit of the exponential distribution for threshold exceedances at higher values of $x$. \n\nComputations summarised in~{Appendix}~\\ref{subsect:gompertz} show that the exponential model, and hence also the Gompertz model with very small $\\beta$, give equally good fits to the Italian and the French datasets above age 107, and that the Gompertz and generalised Pareto models fit equally well above age 105.\n\n\n\n\n\t\t\n\n\\section*{Conclusions}%\n\\label{sect:concluskons}\nNone of the analyses of the \\textsf{ISTAT}{}, \\textsf{IDL 2016}{} or \\textsf{France 2019}{} data, for women and men separately or combined, indicates any deviation from exponentially distributed residual lifetimes, or equivalently from constant force of mortality, beyond 108 years.\n\nTable~\\ref{table:women-men} shows no differences between survival after age 108 in the \\textsf{ISTAT}{} data and survival after age 110 in the \\textsf{IDL 2016}{} data for women, for men, or for women and men combined, so we merged these estimates by taking a weighted average with weights inversely proportional to the estimated variances. The resulting estimates also show no significant differences in survival between men and women, and we conclude that survival times in years after age 108 in the \\textsf{ISTAT}{} data and after age 110 in the \\textsf{IDL 2016}{} data are exponentially distributed with estimated scale parameter $1.43$ and 95\\% confidence interval $(1.33, 1.52)$. The corresponding estimated probability of surviving one more year is $0.5$ with 95\\% confidence interval $(0.47, 0.52)$. \n\nThere was no indication of differences in survival for women or the whole of the \\textsf{France 2019}{} data and in the combined \\textsf{ISTAT}{} and \\textsf{IDL 2016}{} data, but survival for men was lower in the \\textsf{France 2019}{} data. A weighted average of the estimates for the \\textsf{ISTAT}{} data, the \\textsf{France 2019}{} data and the \\textsf{IDL 2016}{} data with France removed gives an exponential scale parameter estimate of $1.42$ years with 95\\% confidence interval $(1.34, 1.49)$, and estimated probability $0.49 (0.47, 0.51)$ of surviving one more year. \n\nDeleting the men from the \\textsf{France 2019}{} data or dropping Jeanne Calment changes estimates and confidence intervals by at most one unit in the second decimal.\n\nThere is high power for detection of an upper limit to the human lifespan up to around 130 years, based on fits of the generalized Pareto model to the three datatbases. Moreover there is no evidence that the Gompertz model, with increasing hazard, fits better than the exponential model, constant hazard, above 108 years.\n\t\t\n\n\n\n\n\n\n\n\n\n\\section*{Discussion}\n\\label{sec:discussion}\n\nThe results of the analysis of the newly-available \\textsf{ISTAT}{}\\ data agree strikingly well with those for the \\textsf{IDL}{}\\ supercentenarian database and for the women in the \\textsf{France 2019}{}\\ data. Once the effects of the sampling frame are taken into account by allowing for truncation and censoring of the ages at death, a model with constant hazard after age 108 fits all three datasets well; it corresponds to a constant probability of 0.49 that a living person will survive for one further year, with 95\\% confidence interval (0.48, 0.51). The power calculations make it implausible that there is an upper limit to the human lifespan of 130 years or below.\n\nAlthough many fewer men than women reach high ages, no difference in survival between the sexes is discernible in the \\textsf{ISTAT}{} and the \\textsf{IDL 2016}{} data. Survival of men after age 108 is lower in the \\textsf{France 2019}{} data, but it seems unlikely that this reflects a real difference between France and Italy and between France and the other countries in the \\textsf{IDL}{}. It seems more plausible that this effect is due to some form of age bias or is a false positive caused by multiple testing. \n\nIf the \\textsf{ISTAT}{} and \\textsf{France 2019}{} data are split by birth cohort, then we find roughly constant mortality from age 105 for those born before the end of 1905, whereas those born in 1906 and later have lower mortality for ages 105--107; this explains the cohort effects detected by \\cite{Barbi:2018}. Possibly the mortality plateau is reached later for later cohorts. The plausibility of this hypothesis could be weighed if further high-quality data become available.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec:introduction}}\n\nPublic key cryptography is a critical component of many widely-used cryptosystems, and forms the basis for much of our ecommerce transaction security infrastructure. Unfortunately, the most common public key schemes are known to be insecure against quantum computers. In 1994, Peter Shor developed a quantum algorithm for efficient factorization and discrete logarithms~\\cite{shor:factor2}; the (supposed) hardness of these two problems formed the basis for RSA and DSA, respectively. Sufficiently powerful quantum computers do not yet exist, but the possibility of their existence in the future already poses problems for those with significant forward security requirements.\n\nA more secure replacement for public key cryptography is needed. Ideally, this replacement would offer information-theoretic security, and would possess most or all of the favorable qualities of public key cryptography. At present, no complete replacement exists, but quantum key distribution (QKD)---in conjunction with one-time pad (OTP) or other symmetric ciphers---appears promising.\n\nQKD---first developed by Bennett and Brassard~\\cite{bennett:BB84}---is a key distribution scheme that relies upon the uncertainty principle of quantum mechanics to guarantee that any eavesdropping attempts will be detected. In a typical QKD setup, individual photons are sent through optical fiber or through free space from the sender to the receiver. The receiver performs measurements on the photons, and sender and receiver communicate via an authenticated (but not necessarily private) classical channel.\n\nOptical attenuation of these single photon pulses limits the maximum transmission distance for a single QKD link to about 200 km over fiber with present technology~\\cite{takesue:qkd}, and significantly less through air. Unlike optically-encoded classical information, the ``signal strength'' of these photons cannot be amplified using a conventional optical amplifier; the No Cloning Theorem~\\cite{wootters:cloning} prohibits this. We refer to this challenge as the \\emph{relay problem}.\n\nTwo classes of quantum repeaters have been proposed to resolve the distance limitations of QKD. The first makes use of quantum error correction to detect and rectify errors in specially-encoded pulses. Unfortunately, the extremely low error thresholds for such schemes ($\\sim 10^{-4}$) make this impractical for use in a realistic quantum repeater. The second class of quantum repeaters uses entanglement swapping and distillation~\\cite{briegel:5932,duan:6862} to establish entanglement between the endpoints of a chain of quantum repeaters, which can then be used for QKD~\\cite{ekert:661}. This method is much more tolerant of errors, and offers resource costs that scale only polynomially with the number of repeaters (i.e., polynomially with distance). However, such repeaters do have one major drawback: they require quantum memories with long decoherence times~\\cite{duan:6862}.\n\nIn order to be useful for practical operation, a quantum repeater must possess a quantum memory that meets the following three requirements:\n\\begin{enumerate}\n\\item Long coherence times: at a minimum, coherence times must be comparable to the transit distance for the entire repeater chain (e.g., $\\sim 10\\ \\mathrm{ms}$ for a trans-Atlantic link).\n\\item High storage density: the bandwidth for a quantum repeater is limited by the ratio of its quantum memory capacity to the transit time for the entire repeater chain~\\cite{simon:190503}.\n\\item Robustness in extreme environments: practical quantum repeaters must be able to operate in the range of environments to which telecom equipment is exposed (e.g., on the ocean floor, in the case of a trans-oceanic link).\n\\end{enumerate}\nThese requirements are so demanding that it is possible that practical quantum repeaters will not be widely available until after large-scale quantum computers have been built---in other words, not until too late.\n\nThe distance limitations of QKD and the issues involved in developing practical quantum repeaters make it challenging to build secure QKD networks that span a large geographic area. The na\\\"{\\i}ve solution of classical repeaters leads to exponentially decaying security with transmission distance if each repeater has some independent probability of being compromised. If large QKD networks are to be built in the near future (i.e., without quantum repeaters), an alternative method of addressing the single-hop distance limitation must be found. We refer to this as the \\emph{relay problem}.\n\nGiven an adversary that controls a randomly-determined subset of nodes in the network, we have developed a solution to the relay problem that involves encoding encryption keys into multiple pieces using a secret sharing protocol~\\cite{shamir:secret,blakley:313}. These shares are transmitted via multiple multi-hop paths through a QKD network, from origin to destination. Through the use of a distributed re-randomization protocol at each intermediate stage, privacy is maintained even if the attacker controls a large, randomly-selected subset of all the nodes. \n\nWe note that authenticated QKD is information-theoretic secure~\\cite{renner:012332}, as is OTP; in combination, these two cryptographic primitives provide information-theoretic security on the level of an individual link. Our protocol makes use of many such links as part of a network that provides information-theoretic security with very high probability. In particular, with some very small probability $\\delta$, the protocol fails in such a way as to allow a sufficiently powerful adversary to perform undetected man-in-the-middle (MITM) attacks. The failure probability $\\delta$ can be made arbitrarily small by modest increases in resource usage. In all other cases, the network is secure. We describe the level of security of our protocol as \\emph{probabilistic information-theoretic}.\n\nIn analyzing our protocol, we consider a network composed of a chain of ``cities'', where each city contains several parties, all of whom are linked to all the other parties in that city. We assume intracity bandwidth is cheap, whereas intercity bandwidth is expensive; intercity bandwidth usage is the main resource considered in our scaling analysis. For the sake of simplicity, we consider communication between two parties (Alice and Bob) who are assumed to be at either end of the chain of cities. A similar analysis would apply to communication between parties at any intermediate points in the network.\n\n\\section{Adversary and Network Model}\n\nIt is convenient to model networks with properties similar to those described above by using undirected graphs, where each vertex represents a node or party participating in the network, and each edge represents a secure authenticated private channel. Such a channel could be generated by using QKD in conjunction with a shared secret key for authentication, or by any other means providing information-theoretic security.\n\nWe describe below an adversary and network model similar in some ways to one we proposed earlier\\footnote{Pre-print available at www.arXiv.org as arXiv:0803.2717} \nin the context of a protocol for authenticating mutual strangers in a very large QKD network, which we referred to as the \\emph{stranger authentication protocol}. In that protocol, edges represented shared secret keys, whereas here they represent physical QKD links. Network structure in the previous model was assumed to be random (possibly with a power law distribution, as is common in social networks), whereas here the network has a specific topology dictated by geographic constraints, the distance limitations of QKD, and the requirements of the protocol.\n\n\\subsection{Adversarial Capabilities and Limitations\\label{sec:ad_cap}}\nWe call the following adversary model the \\emph{sneaky supercomputer}:\n\\begin{enumerate}[(i)]\n\\item \\label{it:adcap1}The adversary is computationally unbounded.\n\\item \\label{it:adcap2}The adversary can listen to, intercept, and alter any message on any public channel.\n\\item \\label{it:adcap3}The adversary can compromise a randomly-selected subset of the nodes in the network. Compromised nodes are assumed to be under the complete control of the adversary. The total fraction of compromised nodes is limited to $(1-t)$ or less.\n\\end{enumerate}\n\nSuch an adversary is very powerful, and can successfully perform MITM attacks against public key cryptosystems (using the first capability) and against unauthenticated QKD (using the second capability), but not against a QKD link between two uncompromised nodes that share a secret key for authentication (since quantum mechanics allows the eavesdropping to be detected) \\cite{renner:012332}. The adversary can always perform denial-of-service (DOS) attacks by simply destroying all transmitted information; since DOS attacks cannot be prevented in this adversarial scenario, we concern ourselves primarily with security against MITM attacks. Later, we will briefly consider variants of this adversarial model and limited DOS attacks.\n\nThe third capability in this adversarial model---the adversary's control of a random subset of nodes---simulates a network in which exploitable vulnerabilities are present on some nodes but not others. As a first approximation to modeling a real-world network, it is reasonable to assume the vulnerable nodes are randomly distributed throughout the network.\n\nAn essentially equivalent adversarial model is achieved if we replace the third capability as follows: suppose the adversary can attempt to compromise any node, but a compromise attempt succeeds only with probability $(1-t)$, and the adversary can make no more than one attempt per node. In the worst case where the adversary attempts to compromise all nodes, the adversary will control a random subset of all nodes, with the fraction of compromised nodes being roughly $(1-t)$.\n\n\\subsection{The Network}\nFor the relay problem, let us represent the network as a graph~$G$, with~$V(G)$ being the set of vertices (nodes participating in the network) and $E(G)$ being the set of edges (secure authenticated channels, e.g. QKD links between parties who share secret keys for authentication). $N = |V(G)|$ is the number of vertices (nodes). $V_d$ is the set of compromised nodes, which are assumed to be under the adversary's control; $|V_d| \\leq N (1-t)$. Furthermore, let us assume that the network has the following structure: nodes are grouped into $m$ clusters---completely connected sub-graphs containing $n$ nodes each. There are thus $N=mn$ nodes in the network. We label the nodes as $v_{i,j}$, $i\\in \\left\\{1,\\dots,n\\right\\}$, $j\\in \\left\\{1,\\dots,m\\right\\}$. Each node is connected to one node in the immediately preceding cluster and one node in the cluster immediately following it. \n\nMore formally, let $E_\\ell(G) \\equiv \\{(v_{i,j},v_{i,j+1}) : v_{i,j}, v_{i,j+1} \\in V(G)\\}$ and $E_\\sigma(G) \\equiv \\{(v_{i,j},v_{k,j}) : v_{i,j}, v_{k,j} \\in V(G)\\}$. Then, $E(G) \\equiv E_\\ell(G) \\cup E_\\sigma(G)$.\n\nThis network structure models a chain of $m$ cities (a term which we use interchangeably with ``cluster''), each containing $n$ nodes. The cities are spaced such that the physical distance between cities allows QKD links only between adjacent cities. To realistically model the costs of communication bandwidth, we assume that use of long distance links (i.e., those represented by $E_\\ell(G)$) is expensive, whereas intracity links (i.e., $E_\\sigma(G)$) are cheap.\n\nNext, we consider two additional nodes---a sender and a receiver. The sender (hereafter referred to as Alice or simply $A$) has direct links to all the nodes in city 1, while the receiver (Bob, or $B$) has a link to all nodes in city $m$. We assume Alice and Bob to be uncompromised. An example is shown in Fig. \\ref{fig:relay_graph}.\n\n\\section{The Relay Protocol\\label{sec:relay}}\nIn the relay problem, Alice wishes to communicate with Bob over a distance longer than that possible with a single QKD link, with quantum repeaters being unavailable. As described above, Alice and Bob are separated by $m$ ``cities'', each containing $n$ participating nodes. (In the case where different cities contain different numbers of participating nodes, we obtain a lower bound on security by taking $n$ to be the minimum over all cities.) \n\n\\begin{figure} \\centering\n\\includegraphics[width=3.25 in, keepaspectratio=true]{relay_graph}\n\\caption{\\label{fig:relay_graph} White vertices represent honest parties, whereas shaded vertices represent dishonest parties. Double vertical lines represent secure communication links between all joined vertices (i.e., all parties within a given city can communicate securely). In the graph shown above, $40\\%$ of the parties in cities between Alice and Bob are dishonest, but Alice and Bob can still communicate securely using the method described in Sec. \\ref{sec:relay} and Fig. \\ref{fig:protocol}.}\n\\end{figure}\n\nTo achieve both good security and low intercity bandwidth usage, we can employ a basic secret sharing scheme with a distributed re-randomization of the shares \\cite{ben-or:distributed} performed by the parties in each city. This re-randomization procedure is similar to that used in the mobile adversary proactive secret sharing scheme \\cite{ostrovsky:112605,herzberg:339}. Note that in the following protocol description, the second subscript labels the city, while the first subscript refers to the particular party within a city.\n\n\\begin{enumerate}[(i)]\n \\item Alice generates $n$ random strings $r_{i,0}, i\\in\\{1,\\ldots,n\\}$ of length $\\ell$, $r \\in \\{0,1\\}^\\ell$. $\\ell$ is chosen as described in Sec. \\ref{sec:verify_protocol}.\n \\item Alice transmits the strings to the corresponding parties in the first city: $v_{i,1}$ receives $r_{i,0}$.\n \\item \\label{it:party_rec}When a party $v_{i,j}$ receives a string $r_{i,j-1}$, it generates $n-1$ random strings $q_{i,j}^{(k)}, k\\neq i$ of length $\\ell$, and transmits each string $q_{i,j}^{(k)}$ to party $v_{k,j}$ (i.e., transmission along the vertical double lines shown in Fig. \\ref{fig:relay_graph}).\n \\item \\label{it:party_gen}Each party $v_{i,j}$ generates a string $r_{i,j}$ as follows: \n \\[r_{i,j} \\equiv r_{i,j-1} \\oplus \\left(\\bigoplus_{k,k\\neq i} q_{i,j}^{(k)} \\right) \\oplus \\left( \\bigoplus_{k,k\\neq i} q_{k,j}^{(i)} \\right),\\]\n where the symbols ($\\oplus$ and $\\bigoplus$) are both understood to mean bitwise XOR. Note that the string $r_{i,j-1}$ is received from a party in the previous city, the strings $q_{i,j}^{(k)}$ are generated by the party $v_{i,j}$, and the strings $q_{k,j}^{(i)}$ are generated by other parties in the same city as $v_{i,j}$. The string $r_{i,j}$ is then transmitted to party $v_{i,j+1}$ (i.e., transmission along the horizontal lines shown in Fig. \\ref{fig:relay_graph}).\n \\item Steps (\\ref{it:party_rec}) and (\\ref{it:party_gen}) are repeated until the strings reach the parties in city $m$. All the parties $v_{i,m}$ in city $m$ forward the strings they receive to Bob.\n \\item Alice constructs $s \\equiv \\prod_{i} r_{i,0}$ and Bob constructs $s^\\prime \\equiv \\prod_{i} r_{i,j-1}$.\n \\item Alice and Bob use the protocol summarized in Fig. \\ref{fig:protocol} and described in detail in Section \\ref{sec:verify_protocol} to determine if $s=s^\\prime$. If so, they are left with a portion of $s$ (identified as $s_3$), which is their shared secret key. If $s \\neq s^\\prime$, Alice and Bob discard $s$ and $s^\\prime$ and repeat the protocol.\n\\end{enumerate}\n\n\\begin{figure} \\centering\n \\includegraphics[width=3.50 in, keepaspectratio=true]{alice_bob_verify}\n \\caption[]{\\label{fig:protocol} Alice and Bob perform a verification sub-protocol to check that their respective secret keys, $s=(s_1,s_2,s_3)$ and $s^\\prime=(s_1^\\prime,s_2^\\prime,s_3^\\prime)$, are in fact the same. Alice generates a random number $r$, concatenates it with the hash $H[s_3]$ of $s_3$, XORs this with $s_1$, and sends the result to Bob. Bob decodes with $s_1^\\prime$, verifies that $H[s_3] = H[s_3^\\prime]$, then sends back to Alice the result of bit-wise XORing the hash of $r$, $H[r]$, with $s_2^\\prime$. Finally, Alice decodes with $s_2$ and checks to see that the value Bob has computed for $H[r]$ is correct. Alice and Bob now know $s_3 = s_3^\\prime$ and can store $s_3$ for future use. Note that with this protocol, the adversary can fool Alice and Bob into accepting $s \\neq s^\\prime$ with 100 \\% probability if the adversary knows $s$ and $s^\\prime$. }\n \\end{figure}\n\n\\subsection{Key Verification \\label{sec:verify_protocol}}\nIn the last step of the protocol described above, Alice and Bob must verify that their respective keys, $s$ and $s^\\prime$, are the same and have not been tampered with. We note that there are many ways\\footnote{See for example pp. 13--14 of the SECOQC technical report D-SEC-48, by L. Salvail \\cite{salvail:qkd}.} to accomplish this; we present one possible method here (summarized in Fig. \\ref{fig:protocol}) for definiteness, but make no claims as to its efficiency.\n\nWe consider Alice's key $s$ to be composed of three substrings, $s_1$, $s_2$, and $s_3$, with lengths $\\ell_1$, $\\ell_2$, and $\\ell_3$, respectively (typically, $\\ell_3 \\gg \\ell_1,\\ell_2$). Bob's key $s^\\prime$ is similarly divided into $s_1^\\prime$, $s_2^\\prime$, and $s_3^\\prime$. If Alice and Bob successfully verify that $s_3^\\prime = s_3$, they can use $s_3$ as a shared secret key for OTP encryption or other cryptographic purposes.\n\nThe verification is accomplished as follows:\n\\begin{enumerate}[(i)]\n\\item Alice generates a random nonce $r$, and computes the hash $H[s_3]$ of $s3$. She then sends $(r,H[s_3]) \\oplus s_1$ to Bob.\n\\item Bob receives the message from Alice, decrypts by XORing with $s_1^\\prime$, and verifies that the received value of $H[s_3]$ matches $H[s_3^\\prime]$. If so, he accepts the key, and sends Alice the message $H[r] \\oplus s_2^\\prime$. If not, Bob aborts. \n\\item Alice decrypts Bob's message by XORing with $s_2$, and verifies that the received value of $H[r]$ is correct. If so, Alice accepts the key, and verification is successful. If not, Alice aborts.\n\\end{enumerate}\n\nWe now outline a proof of the security of this verification process, and discuss requirements for the hash function $H$. We begin with the assumption that Eve does not know $s$ or $s^\\prime$; if she does, the relay protocol has failed, and Eve can perform MITM attacks without detection (conditions under which the relay protocol can fail are analyzed in Sec. \\ref{sec:security}). Our goal is to show that Alice and Bob will with very high probability detect any attempt by Eve to introduce errors in $s_3^\\prime$ (i.e., any attempt by Eve to cause $s_3^\\prime \\neq s_3$), and that the verification process will also not reveal any information about $s_3$ to Eve.\n\nWe note that any modification by Eve of the messages exchanged by Alice and Bob during the verification process is equivalent to Eve introducing errors in $s_1^\\prime$ and $s_2^\\prime$ during the main part of the relay protocol. If she controls at least one intermediate node, Eve can introduce such errors by modifying one or more of the strings transmitted by a node under her control. We can thus completely describe Eve's attack on the protocol by a string $e=(e_1,e_2,e_3)$, where $s^\\prime = s \\oplus e$, and the three substrings $e_1$, $e_2$, and $e_3$ have lengths $\\ell_1$, $\\ell_2$, and $\\ell_3$, respectively (with $\\ell = \\ell_1+\\ell_2+\\ell_3$).\n\nIt is clear that Eve cannot gain any information about $s_3$ from the verification process, since the only information ever transmitted about $s_3$ (the hash $H[s_3]$) is encrypted by the OTP $s_1$, and $s_1$ is never re-used.\n\nBefore proceeding, let us further partition $s_1$ into two strings $s_{1a}$ and $s_{1b}$, where $s_{1a}$ is the portion of $s_1$ used to encrypt $r$, and $s_{1b}$ is the portion used to encrypt $H[s_3]$. Let $\\ell_{1a}$ and $\\ell_{1b}$ be the lengths of $s_{1a}$ and $s_{1b}$. We similarly partition $s_1^\\prime$ and $e_1$.\n\nEve's only hope of fooling Bob into accepting a tampered-with key (i.e., accepting even though $s_3^\\prime \\neq s_3$) is for her to choose $e_{1b}$ and $e_3$ such that the expression $H[s_3]\\oplus H[s_3 \\oplus e_3] = e_{1b}$ is satisfied. Random guessing will give her a $\\sim2^{-\\ell_{1b}}$ chance of tricking Bob into accepting; for Eve to do better, she must be able to exploit a weakness in the hash function $H$ that gives her some information as to the correct value of $e_{1b}$ for some choice of $e_3$. Note that Eve's best strategy for this attack is to choose $e_{1a}$ and $e_2$ to be just strings of zeroes.\n\nFrom this observation, we obtain the following condition on the hash function: for a random $s_3$ (unknown to Eve), there exists no choice of $e_3$ such that Eve has any information about the value of $e_{1b}$ she should choose to satisfy $H[s_3]\\oplus H[s_3 \\oplus e_3] = e_{1b}$. In practice, it would be acceptable for Eve to gain a very small amount of information, as long as the information gained did not raise Eve's chances much beyond random guessing. This is a relatively weak requirement on $H$, and is likely satisfied by any reasonable choice of hash function.\n\nTo fool Alice into falsely accepting, Eve can either fool Bob via the aforementioned method, or Eve can attempt to impersonate Bob by sending Alice a random string of length $\\ell_2$, in the hopes that it happens to be equal to $s_2 \\oplus H[r]$. Clearly, her chances for the latter method are no better than $2^{-\\ell_2}$. The latter method of attack only fools Alice and not Bob; it is thus of limited use to Eve.\n\nWe note that the security of the verification protocol depends on the choice of $\\ell_1$ and $\\ell_2$ (as described above); these parameters should be chosen so as to provide whatever degree of security is required. Alice and Bob choose $\\ell_3$ so as to obtain whatever size key they desire. Since the security of the verification process does not depend on $\\ell_3$, the communication cost of key verification is negligible in the limit of large $\\ell_3$ (i.e., in the limit of large final key size).\n\n\\section{Security of the Relay Protocol\\label{sec:security}}\nIn order for the secret to be compromised, there must be some $j \\in \\{1, \\ldots, m-1\\}$ such that, for all $i \\in \\{1, \\ldots, n\\}$, at least one of $v_{i,j}$ and $v_{i,j+1}$ is dishonest (i.e., such that, for some $j$, every string $r_{i,j}$ is either sent or received by a compromised party). If this happens, we say the protocol has been compromised at stage $j$. For a given $j$, the probability of compromise is $(1-t^2)^n$, but the probability for $j$ is not entirely independent of the probabilities for $j-1$ and $j+1$. Thus, we can bound from below the overall probability of the channel between Alice and Bob being secure, $p_s$, by (\\ref{eq:relay_bounds}):\n\\begin{eqnarray}\np_s & \\geq & \\left[1- (1-t^2)^n\\right]^{m-1}. \\label{eq:relay_bounds}\n\\end{eqnarray}\nFrom this result, we see that, if we wish to ensure our probability of a secure channel between Alice and Bob is at least $p_s$, it is sufficient to choose $n = \\log \\left( 1- p_s^{1\/(m-1)} \\right) \/ \\log \\left( 1- t^2 \\right)$. Intercity bandwidth consumed is proportional to $n$, so we see that we have good scaling of resource consumption with communication distance. Alternatively, we can re-write the equation for choosing $n$ in terms of a maximum allowed probability of compromise, $\\delta = 1 - p_s$. For $\\delta \\ll 1$, we obtain the following relation:\n\\begin{eqnarray*}\nn & \\simeq & \\frac{\\log{(m-1)} - \\log {\\delta}}{-\\log {(1 -t^2)}}.\n\\end{eqnarray*}\nTotal resource usage (intercity communication links required) scales as $\\mathcal{O}(mn)$, or $\\mathcal{O}(m \\log{m})$ for fixed $\\delta$, $t$. While intracity communication requirements scale faster (as $\\mathcal{O}(mn^2)$), it is reasonable to ignore this because of the comparatively low cost of intracity communication and the finite size of the earth (which effectively limits $m$ to a maximum of 100 or so for a QKD network with single link distances of $\\sim100\\ \\mathrm{km}$).\n\nIf each party in the network simultaneously wished to communicate with one other party (with that party assumed to be $m\/2$ cities away on average), total intercity bandwidth would scale as $\\mathcal{O}(m^2n^2)$. By comparison, the bandwidth for a network of the same number of parties employing public key cryptography (and no secret sharing) would scale as $\\mathcal{O}(m^2n)$. Since $n$ scales relatively slowly (i.e., with $\\log m$), this is a reasonable penalty to pay for improved security.\n\n\\section{Alternative Adversary Models}\nWe now briefly consider a number of alternative adversary models. First, let us consider replacing adversary capability (\\ref{it:adcap3}) with the following alternative, which we term (\\ref{it:adcap3}$^\\prime$): the adversary can compromise up to $k-1$ nodes of its choice. Compromised nodes are assumed to be under the complete control of the adversary, as before. In this scenario, the security analysis is trivial. If $k > n$, the adversary can compromise Alice and Bob's communications undetected. Otherwise, Alice and Bob can communicate securely. \n\nWe could also imagine an adversary controls some random subset of nodes in the network---as described by (\\ref{it:adcap3})---and wishes to disrupt communications between Alice and Bob (i.e., perform a DOS attack), but does not have the capability to disrupt or modify public channels. Alice and Bob can modify the protocol to simultaneously protect against both this type of attack and also the adversary mentioned in Section \\ref{sec:ad_cap}. To do so, they replace the simple secret sharing scheme described above with a Proactive Verifiable Secret Sharing (PVSS) scheme~\\cite{darco:vss}. In this scenario, nodes can check at each stage to see if any shares have been corrupted, and take corrective measures. This process is robust against up to $n\/4 - 1$ corrupt shares, which implies that PVSS yields little protection against DOS attacks unless $t > t_{\\mathrm{thresh}} \\approx \\sqrt{3}\/2$.\n\n\\section{Conclusion\\label{sec:conclusion}}\n\nWe have shown a protocol for solving the relay problem and building secure long-distance communication networks with present-day QKD technology. The protocol proposed employs secret sharing and multiple paths through a network of partially-trusted nodes. Through the choice of moderately large $n$ in the relay problem, one can make the possibility of compromise vanishingly small. For fixed probability of compromise of each of the intermediate nodes, the number of nodes per stage required to maintain security scales only logarithmically with the number of stages (i.e., with distance).\n\nGiven that QKD systems are already commercially available, our methods could be implemented today. \n\n\n\\section{Acknowledgments}\nWe wish to thank Louis Salvail, Aidan Roy, Rei Safavi-Naini, Douglas Stebila, Hugh Williams, Kevin Hynes, and Renate Scheidler for valuable discussions. TRB acknowledges support from a US Department of Defense NDSEG Fellowship. BCS acknowledges support from iCORE and CIFAR.\n\n\\bibliographystyle{splncs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOn numerous social media platforms, such as YouTube, Facebook, or Instagram, people share their\nopinions on all kinds of topics in the form of posts, images, and video clips. \nWith the proliferation of smartphones and tablets, which has greatly boosted content sharing, \npeople increasingly share their opinions on newly released products or on other topics in form of video reviews or comments. \nThis is an excellent opportunity for large companies to capitalize on, by extracting\nuser sentiment, suggestions, and complaints on their products from these video reviews.\nThis information also opens new horizons to improving our quality of life by making informed\ndecisions on the choice of products we buy, services we use, places we visit, or movies we watch\nbasing on the experience and opinions of other users.\n\nVideos convey information through three channels: audio, video, and text (in\nthe form of speech). Mining opinions from this plethora of multimodal data calls for\na solid multimodal sentiment analysis technology. One of the major problems\nfaced in multimodal sentiment analysis is the fusion of features pertaining to\ndifferent modalities. For this, the majority of the recent works in multimodal sentiment\nanalysis have simply concatenated the feature vectors of different\nmodalities. However, this does not take into account that different modalities\nmay carry conflicting information. We hypothesize that the fusion method we\npresent in this paper deals with this issue better, and present experimental\nevidence showing improvement over simple concatenation of feature vectors. Also,\nfollowing the state of the art~\\citep{porcon}, we employ recurrent\nneural network (RNN) to propagate contextual information between utterances in a\nvideo clip, which significantly improves the classification results and outperforms the\nstate of the art by a significant margin of 1--2\\% for all the modality\ncombinations.\n\nIn our method, we first obtain unimodal features for each utterance for all\nthree modalities. Then, using RNN we extract context-aware utterance features. \nThus, we transform the context-aware utterance vectors to the\nvectors of the same dimensionality. We assume that these transformed vectors contain\nabstract features representing the attributes relevant to sentiment\nclassification. Next, we compare and combine each bimodal combination of these\nabstract features using fully-connected layers. This yields fused bimodal\nfeature vectors. Similarly to the unimodal case, we use RNN to generate\ncontext-aware features. Finally, we combine these bimodal vectors into a\ntrimodal vector using, again, fully-connected layers and use a RNN to pass\ncontextual information between them. We empirically show that the feature vectors\nobtained in this manner are more useful for the sentiment classification task.\n\nThe implementation of our method is publicly available in the form of open-source code.\\footnote{\\url{http:\/\/github.com\/senticnet}}\n\nThis paper is structured as follows: \\cref{sec:related-work-1} briefly discusses important previous work in multimodal feature fusion; \\cref{sec:model} describes our method in details; \\cref{sec:experiments} reports the results of our experiments and discuss their implications; finally, \\cref{sec:conclusions} concludes the paper and discusses future work.\n\n\\section{Related Work}\n\\label{sec:related-work-1}\nIn recent years, sentiment analysis has become increasingly popular for processing social media data on online communities, blogs, wikis, microblogging platforms, and other online collaborative media~\\citep{camacsa}. Sentiment analysis is a branch of affective computing research~\\citep{porrev} that aims to classify text -- but sometimes also audio and video~\\citep{hazcon} -- into either positive or negative -- but sometimes also neutral~\\citep{chadis}. Most of the literature is on English language but recently an increasing number of works are tackling the multilinguality issue~\\citep{loomul,dashtipour2016multilingual}, especially in booming online languages such as Chinese~\\citep{penlea}.\nSentiment analysis techniques can be broadly categorized into symbolic and sub-symbolic approaches: the former include the use of lexicons~\\citep{banlex}, ontologies~\\citep{draont}, and semantic networks~\\citep{camnt5} to encode the polarity associated with words and multiword expressions; the latter consist of supervised~\\citep{onesta}, semi-supervised~\\citep{hussem} and unsupervised~\\citep{liilea} machine learning techniques that perform sentiment classification based on word co-occurrence frequencies. Among these, the most popular recently are algorithms based on deep neural networks~\\citep{yourec} and generative adversarial networks~\\citep{liigen}.\n\nWhile most works approach it as a simple categorization problem, sentiment analysis is actually a suitcase research problem~\\citep{camsui} that requires tackling many NLP tasks, including word polarity disambiguation~\\citep{xiawor}, subjectivity detection~\\citep{chasub}, personality recognition~\\citep{majdee}, microtext normalization~\\citep{satpho}, concept extraction~\\citep{dhegra}, time tagging~\\citep{zhotem}, and aspect extraction~\\citep{maatar}.\n\nSentiment analysis has raised growing interest both within the scientific community, leading to many exciting open challenges, as well as in the business world, due to the remarkable benefits to be had from financial~\\citep{xinfin} and political~\\citep{ebrcha} forecasting, e-health~\\citep{campat} and e-tourism~\\citep{valsen}, user profiling~\\citep{mihwha} and community detection~\\citep{cavlea}, manufacturing and supply chain applications~\\citep{xuuada}, human communication comprehension~\\citep{zadatt} and dialogue systems~\\citep{youaug}, etc.\n\nIn the field of emotion recognition, early works by~\\citet{de1997facial} and\n\\citet{chen1998multimodal} showed that fusion of audio and visual\nsystems, creating a bimodal signal, yielded a higher accuracy than any unimodal\nsystem. Such fusion has been analyzed at both feature\nlevel~\\citep{kessous2010multimodal} and decision\nlevel~\\citep{schuller2011recognizing}.\n\nAlthough there is much work done on audio-visual fusion for emotion recognition,\nexploring contribution of text along with audio and visual modalities in\nmultimodal emotion detection has been little\nexplored. \\citet{wollmer2013youtube} and~\\citet{rozgic2012ensemble} fused\ninformation from audio, visual and textual modalities to extract emotion and\nsentiment. \\citet{metallinou2008audio} and~\\citet{eyben2010line} fused audio and\ntextual modalities for emotion recognition. Both approaches relied on a\nfeature-level fusion. \\citet{wu2011emotion} fused audio and textual clues at\ndecision level. \\citet{pordee} uses convolutional neural network (CNN) to\nextract features from the modalities and then employs multiple-kernel learning\n(MKL) for sentiment analysis. The current state of the art, set forth by\n\\citet{porcon}, extracts contextual information from the surrounding utterances\nusing long short-term memory (LSTM). \\citet{porrev} fuses different\nmodalities with deep learning-based tools. \\citet{zadten} uses tensor\nfusion. \\citet{porens} further extends upon the ensemble of CNN and MKL.\n\nUnlike existing approaches, which use simple concatenation based early fusion~\\citep{pordee,pordep} and non-trainable tensors based fusion~\\citep{zadten}, this work proposes a hierarchical fusion capable of learning the bimodal and trimodal correlations for data fusion using deep neural networks. The method is end-to-end and, in order to accomplish the fusion, it can be plugged into any deep neural network based multimodal sentiment analysis framework. \n\n\\section{Our Method}\n\\label{sec:model}\n\nIn this section, we discuss our novel methodology behind solving the sentiment\nclassification problem. First we discuss the overview of our method and then we\ndiscuss the whole method in details, step by step.\n\n\\subsection{Overview}\n\\label{sec:overview}\n\n\\subsubsection{Unimodal Feature Extraction}\nWe extract utterance-level features for three modalities. This step is discussed\nin \\cref{UFE}.\n\n\\subsubsection{Multimodal Fusion}\n\n\\paragraph{Problems of early fusion}\nThe majority of the work on multimodal data use concatenation, or early fusion\n(\\cref{fig:early_fusion}), as their fusion strategy. The problem with this\nsimplistic approach is that it cannot filter out and conflicting or redundant\ninformation obtained from different modalities. To address this major issue, we\ndevise an hierarchical approach which proceeds from unimodal to bimodal vectors\nand then bimodal to trimodal vectors.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.46]{.\/hfusion-concatenation-trimmed.pdf}\n \\caption{Utterance-level early fusion, or simple concatenation}\n \\label{fig:early_fusion}\n\\end{figure}\n\n\\paragraph{Bimodal fusion}\nWe fuse the utterance feature vectors for each bimodal combination, i.e., T+V,\nT+A, and A+V. This step is depicted in \\cref{fig:hfusion-bimodal} and discussed\nin details in \\cref{sec:bimodal}.\nWe use the penultimate layer for \\cref{fig:hfusion-bimodal} as bimodal features.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.46]{.\/hfusion-2-modal-trimmed.pdf}\n \\caption{Utterance-level bimodal fusion}\n \\label{fig:hfusion-bimodal}\n\\end{figure}\n\n\\paragraph{Trimodal fusion}\nWe fuse the three bimodal features to obtain trimodal feature as depicted in\n\\cref{fig:hfusion-trimodal}. This step is discussed in details in \\cref{sec:trimodal}.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.46]{.\/hfusion-Gelbukh-trimmed.pdf}\n \\caption{Utterance-level trimodal hierarchical fusion.\\protect\\footnotemark}\n \\label{fig:hfusion-trimodal}\n\\end{figure}\n\n\\paragraph{Addition of context}\nWe also improve the quality of feature vectors (both unimodal and multimodal) by\nincorporating information from surrounding utterances using RNN. We model the\ncontext using gated recurrent unit (GRU) as depicted in \\cref{fig:architecture}.\nThe details of context modeling is discussed in \\cref{sec:context} and the\nfollowing subsections.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=\\textwidth]{.\/chfusion.pdf}\n \\caption{Context-aware hierarchical fusion}\n \\label{fig:architecture}\n\\end{figure}\n\n\\paragraph{Classification}\nWe classify the feature vectors using a softmax layer.\n\n\n\\subsection{Unimodal Feature Extraction}\n\\label{UFE}\n\nIn this section, we discuss the method of feature extraction for three different\nmodalities: audio, video, and text.\n\n\\subsubsection{Textual Feature Extraction}\n\\label{text}\n\nThe textual data is obtained from the transcripts of the videos. We apply a deep\nConvolutional Neural Networks (CNN)~\\citep{karpathy2014large} on each utterance\nto extract textual features. Each utterance in the text is represented as an\narray of pre-trained 300-dimensional {\\tt word2vec}\nvectors~\\citep{mikolov2013efficient}. Further, the utterances are truncated or\npadded with null vectors to have exactly 50 words.\n\nNext, these utterances as array of vectors are passed through two different\nconvolutional layers; first layer having two filters of size 3 and 4\nrespectively with 50 feature maps each and the second layer has a filter of size\n2 with 100 feature maps. Each convolutional layer is followed by a max-pooling\nlayer with window $2\\times 2$.\n\nThe output of the second max-pooling layer is fed to a fully-connected layer\nwith 500 neurons with a rectified linear unit (ReLU)~\\citep{whyeteh2001rate}\nactivation, followed by softmax output. The output of the penultimate\nfully-connected layer is used as the textual feature. The translation of\nconvolution filter over makes the CNN learn abstract features and with each\nsubsequent layer the context of the features expands further.\n\n\n\n\n\\subsubsection{Audio Feature Extraction}\n\\label{audio}\n\nThe audio feature extraction process is performed at 30 Hz frame rate with 100\nms sliding window. We use openSMILE~\\citep{eyben2010opensmile}, which is capable\nof automatic pitch and voice intensity extraction, for audio feature\nextraction. Prior to feature extraction audio signals are processed with voice\nintensity thresholding and voice normalization. Specifically, we use\nZ-standardization for voice normalization. In order to filter out audio segments\nwithout voice, we threshold voice intensity. OpenSMILE is used to perform both\nthese steps. Using openSMILE we extract several Low Level Descriptors (LLD)\n(e.g., pitch , voice intensity) and various statistical functionals of them\n(e.g., amplitude mean, arithmetic mean, root quadratic mean, standard deviation,\nflatness, skewness, kurtosis, quartiles, inter-quartile ranges, and linear\nregression slope). ``IS13-ComParE'' configuration file of openSMILE is used to\nfor our purposes. Finally, we extracted total 6392 features from each input\naudio segment.\n\n\n\n\\subsubsection{Visual Feature Extraction}\n\\label{visual}\n\nTo extract visual features, we focus not only on feature extraction from each\nvideo frame but also try to model temporal features across frames. To achieve\nthis, we use 3D-CNN on the video. 3D-CNNs have been successful in the past,\nspecially in the field of object classification on 3D data~\\citep{ji20133d}. Its\nstate-of-the-art performance on such tasks motivates its use in this paper.\n\nLet the video be called $vid \\in \\mathbb{R}^{3\\times f\\times h\\times w}$, where\n$3$ represents the three RGB channels of an image and $f,\\ h,\\text{ and }w$\ndenote the cardinality, height, and width of the frames, respectively. A 3D\nconvolutional filter, named $f_{lt}\\in \\mathbb{R}^{f_m\\times 3\\times f_d\\times\nf_h\\times f_w}$, is applied to this video, where, similar to a 2D-CNN, the\nfilter translates across the video and generates the convolution output\n$conv_{out} \\in \\mathbb{R}^{f_m\\times 3\\times (f-f_d+1)\\times (h-f_h+1)\\times\n(w-f_w+1)}$. Here, $f_m,\\ f_d,\\ f_h,\\text{ and }f_w$ denote number of feature\nmaps, depth of filter, height of filter, and width of filter,\nrespectively. Finally, we apply max-pooling operation to the $conv_{out}$, which\nselects the most relevant features. This operation is applied only to the last\nthree dimensions of $conv_{out}$. This is followed by a dense layer and softmax\ncomputation. The activations of this layer is used as the overall video features\nfor each utterance video.\n\nIn our experiments, we receive the best results with filter dimensions $f_m =\n32$ and $f_d,f_h,f_w = 5$. Also, for the max-pooling, we set the window size as\n$3\\times 3\\times 3$ and the succeeding dense layer with $300$ neurons.\n\n\n\n\\footnotetext{Figure adapted from~\\citep{mastersthesis} with permission.}\n\n\\subsection{Context Modeling}\n\\label{sec:context}\n\nUtterances in the videos are semantically dependent on each other. In other\nwords, complete meaning of an utterance may be determined by taking preceding\nutterances into consideration. We call this the context of an utterance.\nFollowing~\\citet{porcon}, we use RNN, specifically GRU\\footnote{LSTM does not\n perform well} to model semantic\ndependency among the utterances in a video.\n\nLet the following items represent unimodal features:\n\\begin{align*}\n f_A \\in \\mathbb{R}^{N\\times d_A}&\\quad\\text{(acoustic features)},\\\\\n f_V \\in \\mathbb{R}^{N\\times d_V}&\\quad\\text{(visual features)},\\\\\n f_T \\in \\mathbb{R}^{N\\times d_T}&\\quad\\text{(textual features)},\n\\end{align*}\nwhere $N=$ maximum number of utterances in a video. We pad the shorter videos\nwith dummy utterances represented by null vectors of corresponding length.\nFor each modality, we feed the unimodal utterance features $f_m$ (where $m \\in\n\\{A,V,T\\}$) (discussed in \\cref{UFE}) of a video to $GRU_m$ with\noutput size $D_m$, which is defined as\n\\begin{flalign*}\n z_m&=\\sigma(f_{mt}U^{mz}+s_{m(t-1)}W^{mz}),\\\\\n r_m&=\\sigma(f_{mt}U^{mr}+s_{m(t-1)}W^{mr}),\\\\\n h_{mt}&=\\tanh(f_{mt}U^{mh}+(s_{m(t-1)}*r_m)W^{mh}),\\\\\n F_{mt}&=\\tanh(h_{mt}U^{mx}+u^{mx}),\\\\\n s_{mt}&=(1-z_m)*F_{mt}+z_m*s_{m(t-1)},\n\\end{flalign*}\nwhere $U^{mz} \\in \\mathbb{R}^{d_m\\times D_m}$, $W^{mz} \\in \\mathbb{R}^{D_m\\times\nD_m}$, $U^{mr} \\in \\mathbb{R}^{d_m\\times D_m}$, $W^{mr} \\in\n\\mathbb{R}^{D_m\\times D_m}$, $U^{mh} \\in \\mathbb{R}^{d_m\\times D_m}$, $W^{mh}\n\\in \\mathbb{R}^{D_m\\times D_m}$, $U^{mx} \\in \\mathbb{R}^{d_m\\times D_m}$,\n$u^{mx} \\in \\mathbb{R}^{D_m}$, $z_m \\in \\mathbb{R}^{D_m}$, $r_m \\in\n\\mathbb{R}^{D_m}$, $h_{mt} \\in \\mathbb{R}^{D_m}$, $F_{mt} \\in \\mathbb{R}^{D_m}$,\nand $s_{mt} \\in \\mathbb{R}^{D_m}$. This yields hidden outputs $F_{mt}$ as\ncontext-aware unimodal features for each modality. Hence, we define\n$F_m=GRU_m(f_m)$, where $F_m \\in \\mathbb{R}^{N\\times D_m}$. Thus, the\ncontext-aware multimodal features can be defined as\n\\begin{flalign*}\n \n F_A &= GRU_A(f_A),\\\\\n F_V &= GRU_V(f_V),\\\\\n F_T &= GRU_T(f_T).\n\\end{flalign*}\n\n\\subsection{Multimodal Fusion}\n\\label{sec:mul_fusion}\n\nIn this section, we use context-aware unimodal features $F_A, F_V,$ and $F_T$ to\na unified feature space.\n\nThe unimodal features may have different dimensions, i.e., $D_A\\neq D_V\\neq\nD_T$. Thus, we map them to the same dimension, say $D$ (we obtained best\nresults with $D=400$), using fully-connected layer as follows:\n\\begin{flalign*}\n g_A &= \\tanh(F_AW_A+b_A),\\\\\n g_V &= \\tanh(F_VW_V+b_V),\\\\\n g_T &= \\tanh(F_TW_T+b_T),\n\\end{flalign*}\nwhere $W_A \\in \\mathbb{R}^{D_A \\times D}$, $b_A\\in \\mathbb{R}^D$, $W_V \\in\n\\mathbb{R}^{D_V\\times D}$, $b_V\\in \\mathbb{R}^D$, $W_T \\in\n\\mathbb{R}^{D_T\\times D}$, and $b_T\\in \\mathbb{R}^D$. We can represent\nthe mapping for each dimension as\n\\[\n g_x=\\left[\n \\begin{array}{ccccc}\n c^x_{11} & c^x_{21} & c^x_{31} & \\cdots & c^x_{D1}\\\\\n c^x_{12} & c^x_{22} & c^x_{32} & \\cdots & c^x_{D2}\\\\\n \\vdots & \\vdots & \\vdots & \\cdots & \\vdots\\\\\n c^x_{1N} & c^x_{2N} & c^x_{3N} & \\cdots & c^x_{DN}\\\\\n \\end{array}\n\\right],\n\\]\nwhere $x \\in \\{V,A,T\\}$ and $c^x_{lt}$ are scalars for all $l=1,2,\\dots,D$ and\n$t=1,2,\\dots,N$. Also, in $g_x$ the rows represent the utterances and the\ncolumns the feature values. We can see these values $c^x_{lt}$ as more abstract\nfeature values derived from fundamental feature values (which are the components\nof $f_A$, $f_V$, and $f_T$). For example, an abstract feature can be the\nangriness of a speaker in a video. We can infer the degree of angriness from\nvisual features ($f_V$; facial muscle movements), acoustic features ($f_A$,\nsuch as pitch and raised voice), or textual features ($f_T$, such as the language and choice of\nwords). Therefore, the degree of angriness can be represented by $c^x_{lt}$,\nwhere $x$ is $A$, $V$, or $T$, $l$ is some fixed integer between $1$ and $D$, and $t$ is some\nfixed integer between $1$ and $N$.\n\nNow, the evaluation of abstract feature values from all the modalities may not\nhave the same merit or may even contradict each other. Hence, we need the network\nto make comparison among the feature values derived from different modalities to\nmake a more refined evaluation of the degree of anger. To this end, we take\neach bimodal combination (which are audio--video, audio--text, and video--text) at\na time and compare and combine each of their respective abstract feature values\n(i.e. $c^V_{lt}$ with $c^T_{lt}$, $c^V_{lt}$ with $c^A_{lt}$, and $c^A_{lt}$\nwith $c^T_{lt}$) using fully-connected layers as follows:\n\\begin{align}\n i^{VA}_{lt}&=\\tanh(w^{VA}_{l}.[c_{lt}^V,c_{lt}^A]^\\intercal+b^{VA}_{l}),\\label{bimodal:1}\\\\\n i^{AT}_{lt}&=\\tanh(w^{AT}_{l}.[c_{lt}^A,c_{lt}^T]^\\intercal+b^{AT}_{l}),\\label{bimodal:2}\\\\\n i^{VT}_{lt}&=\\tanh(w^{VT}_{l}.[c_{lt}^V,c_{lt}^T]^\\intercal+b^{VT}_{l}),\\label{bimodal:3} \n\\end{align}\nwhere $w^{VA}_l \\in \\mathbb{R}^2$, $b^{VA}_l$ is scalar, $w^{AT}_l \\in\n\\mathbb{R}^2$, $b^{AT}_l$ is scalar, $w^{VT}_l \\in \\mathbb{R}^2$, and $b^{VT}_l$\nis scalar, for all $l=1,2,\\dots,D$ and $t=1,2,\\dots,N$. We hypothesize that it\nwill enable the network to compare the decisions from each modality against the\nothers and help achieve a better fusion of modalities.\n\n\\paragraph{\\textbf{Bimodal fusion}}\n\\label{sec:bimodal}\n\n\\crefrange{bimodal:1}{bimodal:3} are used for bimodal fusion. The bimodal\nfused features for video--audio, audio--text, video--text are defined as\n\\begin{flalign*}\n f_{VA}= (f_{VA1},f_{VA2},\\dots,f_{VA(N)}), \\text{ where } f_{VAt}&=(i^{VA}_{1t},i^{VA}_{2t},\\dots,i^{VA}_{Dt}), \\\\\n f_{AT}= (f_{AT1},f_{AT2},\\dots,f_{AT(N)}), \\text{ where } f_{ATt}&=(i^{AT}_{1t},i^{AT}_{2t},\\dots,i^{AT}_{Dt}), \\\\\n f_{VT}= (f_{VT1},f_{VT2},\\dots,f_{VT(N)}), \\text{ where } f_{VTt}&=(i^{VT}_{1t},i^{VT}_{2t},\\dots,i^{VT}_{Dt}).\n\\end{flalign*}\n\nWe further employ $GRU_m$(~\\cref{sec:context}) ($m \\in \\{VA, VT, TA\\}$), to\nincorporate contextual information among the utterances in a video with\n\\begin{flalign*}\n F_{VA} = (F_{VA1},F_{VA2},\\dots,F_{VA(N)}) = GRU_{VA}(f_{VA}),\\\\\n F_{VT} = (F_{VT1},F_{VT2},\\dots,F_{VT(N)}) = GRU_{VT}(f_{VT}),\\\\\n F_{TA} = (F_{TA1},F_{TA2},\\dots,F_{TA(N)}) = GRU_{TA}(f_{TA}),\n\\end{flalign*}\nwhere\n\\begin{flalign*}\n F_{VAt}= (I^{VA}_{1t},I^{VA}_{2t},\\dots,I^{VA}_{D_2t}),\\\\\n F_{VTt}= (I^{AT}_{1t},I^{AT}_{2t},\\dots,I^{AT}_{D_2t}),\\\\\n F_{TAt}= (I^{VT}_{1t},I^{VT}_{2t},\\dots,I^{VT}_{D_2t}),\n\\end{flalign*}\n$F_{VA}$, $F_{VT}$, and $F_{TA}$ are context-aware bimodal features\nrepresented as vectors and $I^m_{nt}$ is scalar for $n=1,2,\\dots,D_2$,\n$D_2=500$, $t=1,2,\\dots,N$, and $m=\\text{VA,VT,TA}$.\n\n\\paragraph{Trimodal fusion}\n\\label{sec:trimodal}\n\nWe combine all three modalities using fully-connected layers as follows:\n\\begin{flalign*}\n z_{lt}=\\tanh(w^{AVT}_l.[I^{VA}_{lt},I^{AT}_{lt},I^{VT}_{lt}]^\\intercal+b^{AVT}_l),\n\\end{flalign*}\nwhere $w^{AVT}_l \\in \\mathbb{R}^3$ and $b^{AVT}_l$ is a scalar for all $l=1,2,\\dots,D_2$\nand $t=1,2,\\dots,N$.\nSo, we define the fused features as\n\\begin{flalign*}\n f_{AVT}=(f_{AVT1},f_{AVT2},\\dots,f_{AVT(N)}),\n\\end{flalign*}\nwhere\n$f_{AVTt}=(z_{1t},z_{2t},\\dots,z_{D_2t})$,\n$z_{nt}$ is scalar for $n=1,2,\\dots,D_2$ and $t=1,2,\\dots,N$.\n\nSimilarly to bimodal fusion (\\cref{sec:bimodal}), after trimodal fusion we pass\nthe fused features through $GRU_{AVT}$ to incorporate contextual information in\nthem, which yields\n\\begin{flalign*}\n F_{AVT} = (F_{AVT1},F_{AVT2},\\dots,F_{AVT(N)}) = GRU_{AVT}(f_{AVT}),\n\\end{flalign*}\nwhere $F_{AVTt}= (Z_{1t},Z_{2t},\\dots,Z_{D_3t})$, $Z_{nt}$ is scalar for $n=1,2,\\dots,D_3$, $D_3=550$, $t=1,2,\\dots,N$,\nand $F_{AVT}$ is the context-aware trimodal feature vector.\n\n\\subsection{Classification}\n\\label{sec:classification}\n\nIn order to perform classification, we feed the fused features $F_{mt}$ (where\n$m=AV,VT,TA,\\text{ or } AVT$ and $t=1,2,\\dots,N$) to a softmax layer with $C=2$\noutputs. The classifier can be described as follows:\n\\begin{flalign*}\n \\mathcal{P} &=\n \\text{softmax}(W_{\\mathit{softmax}}F_{mt}+b_{\\mathit{softmax}}),\\\\\n \\hat{y}&=\\underset{j}{\\text{argmax}}(\\mathcal{P}[j]),\n \n\\end{flalign*}\nwhere $W_{\\mathit{softmax}}\\in \\mathbb{R}^{C\\times D}$,\n$b_{\\mathit{softmax}}\\in \\mathbb{R}^C$, $\\mathcal{P}\\in \\mathbb{R}^C$, $j=$\nclass value ($0$ or $1$), and $\\hat{y}=$ estimated class value.\n\n\\subsection{Training}\n\\label{training}\nWe employ categorical cross-entropy as loss function ($J$) for training,\n\\begin{flalign*}\n J=-\\frac{1}{N}\\sum_{i=1}^N{\\sum_{j=0}^{C-1}{y_{ij}\\log{\\mathcal{P}_i[j]}}},\n \n\\end{flalign*}\nwhere $N=$ number of samples, $i=$ index of a sample, $j=$ class value, and\n\\[\n y_{ij}=\n \\begin{cases}\n 1, & \\text{if expected class value of sample }i\\text{ is }j\\\\\n 0, & \\text{otherwise.}\n \\end{cases}\n\\]\n\n Adam~\\citep{DBLP:journals\/corr\/KingmaB14} is used as optimizer due to its\n ability to adapt learning rate for each parameter individually. We train the\nnetwork for 200 epochs with early stopping, where we optimize the parameter set\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \\begin{flalign*}\n\\theta=&\\bigcup_{m\\in M}\\left (\\bigcup_{j\\in \\{z,r,h\\}}\\{U^{mj},W^{mj}\\}\\cup \\{U^{mx},u^{mx}\\}\\right\n)\\\\\n&\\cup \\bigcup_{m\\in M_2}\\bigcup_{i=1}^{D_2}\\{w^m_i\\} \\cup\n\\bigcup_{i=1}^{D_3}\\{w^{AVT}_i\\}\\cup \\bigcup_{m\\in M_1}\\{W_m,b_m\\}\\\\\n&\\cup \\{W_{softmax},b_{softmax}\\},\\\\\n \\end{flalign*}\nwhere $M=\\{A,V,T,VA,VT,TA,AVT\\}$, $M_1=\\{A,V,T\\}$, and\n$M_2=\\{VA,VT,\\allowbreak TA\\}$. \\cref{algorithm} summarizes our method.\\footnote{Implementation of this algorithm is available at\n \\url{http:\/\/github.com\/senticnet}}\n\n\n\\begin{algorithm}[!ht]\n \\small\n \\caption{Context-Aware Hierarchical Fusion Algorithm}\\label{algorithm}\n \\begin{algorithmic}[1]\n \n \n \\vspace{2mm}\n \\Procedure{TrainAndTestModel}{$U$, $V$}\\Comment{\\footnotesize{$U$ = train set, $V$ = test set}}\n \\vspace{2mm}\n \\State \\textbf{Unimodal feature extraction:}\n \\For{\\texttt{i:[1,$N$]}}\\Comment{\\footnotesize{extract baseline features}}\n \\State \\texttt{$f_{A}^{i} \\gets AudioFeatures(u_{i})$ }\n \\State \\texttt{$f_{V}^{i} \\gets VideoFeatures(u_{i})$ }\n \\State \\texttt{$f_{T}^{i} \\gets TextFeatures(u_{i})$ }\n \\EndFor\n \\For{\\texttt{m $\\in \\{A, V, T\\}$}}\n \\State $F_m$ = $GRU_m$($f_m$)\n \\EndFor\n \\vspace{2mm}\n \\State \\textbf{Fusion}:\n \\State \\texttt{$g_{A} \\gets MapToSpace(F_A)$ }\\Comment{\\footnotesize{dimensionality equalization}}\n \\State \\texttt{$g_{V} \\gets MapToSpace(F_V)$ }\n \\State \\texttt{$g_{T} \\gets MapToSpace(F_T)$ }\n\n \\vspace{2mm}\n \\State \\texttt{$f_{VA} \\gets BimodalFusion(g_V, g_A)$}\\Comment{\\footnotesize{bimodal fusion}}\n \\State \\texttt{$f_{AT} \\gets BimodalFusion(g_A, g_T)$}\n \\State \\texttt{$f_{VT} \\gets BimodalFusion(g_V, g_T)$}\n \\For{\\texttt{m $\\in \\{VA, AT, VT\\}$}}\n \\State $F_m$ = $GRU_m$($f_m$)\n \\EndFor\n \n \\vspace{2mm}\n \\State $f_{AVT} \\gets TrimodalFusion(F_{VA}, F_{AT}, F_{VT})$\n \\Comment{\\small{trimodal fusion}}\n \\State $F_{AVT}$ = $GRU_{AVT}$($f_{AVT}$)\n\n \\vspace{2mm}\n \\For{\\texttt{i:[1,$N$]}}\\Comment{\\footnotesize{softmax classification}}\n \\State $\\hat{y}^i =\\underset{j}{\\text{argmax}}(softmax(F_{AVT}^i)[j])$ \n \\EndFor\n \n\n \\State $TestModel(V)$\n \\EndProcedure\n \n \\vspace{2mm}\n \\Procedure{MapToSpace}{$x_z$} \\Comment{\\footnotesize{for modality $z$}}\n \\State $ g_z \\gets \\tanh(W_zx_z+b_z) $\n \\State \\textbf{return} $g_z$\n \\EndProcedure\n\n \\vspace{2mm}\n \\Procedure{BimodalFusion}{$g_{z_1}$, $g_{z_2}$} \\Comment{\\footnotesize{for modality\n $z_1$ and $z_2$, where $z_1\\neq z_2$}}\n \\For{\\texttt{i:[1,$D$]}}\n \\State $f_{z_1z_2}^i \\gets \\tanh(w_i^{z_1z_2}.[g^i_{z_1},\n g^i_{z_2}]^\\intercal+b_i^{z_1z_2})$\n \\EndFor\n \\State $f_{z_1z_2} \\gets (f_{z_1z_2}^1, f_{z_1z_2}^2,\\dots,f_{z_1z_2}^{D})$\n \\State \\textbf{return} $f_{z_1z_2}$\n \\EndProcedure\n \n \n \\vspace{2mm}\n \\Procedure{TrimodalFusion}{$f_{z_1}$, $f_{z_2}$, $f_{z_3}$} \\Comment{\\footnotesize{for\n modality combination $z_1$, $z_2$, and $z_3$, where $z_1\\neq z_2\\neq z_3$}}\n \\For{\\texttt{i:[1,$D$]}}\n \\State $f^i_{z_1z_2z_3} \\gets \\tanh(w_i.[f^i_{z_1}, f^i_{z_2}, f^i_{z_3}]^\\intercal+b_i)$\n \\EndFor\n \\State $f_{z_1z_2z_3} \\gets (f_{z_1z_2z_3}^1, f_{z_1z_2z_3}^2,\\dots,f_{z_1z_2z_3}^{D})$\n \\State \\textbf{return} $f_{z_1z_2z_3}$\n \\EndProcedure\n \n \\vspace{2mm}\n \\Procedure{TestModel}{$V$}\n \\State \\footnotesize{Similarly to training phase, $V$ is passed through the learnt models\n to get the features and classification outputs. \\cref{training}\n mentions the trainable parameters ($\\theta$).}\n \\EndProcedure\n \\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Dataset Details}\n\\label{datasets}\nMost research works in multimodal sentiment analysis are performed on datasets\nwhere train and test splits may share certain speakers. Since, each individual\nhas an unique way of expressing emotions and sentiments, finding generic and\nperson-independent features for sentiment analysis is\ncrucial. \\cref{tab:dataset} shows the train and test split for the datasets\nused.\n\n\\begin{table}[h]\n\t\\small\n\t \\addtolength\\tabcolsep{-5pt}\n\t\\begin{center}\n\t\t\\begin{tabular}{|*{14}{c|}}\n\t\t\t\\hline\n\t\t\t\\multicolumn{2}{|c|}{\\multirow{2}{*}{Dataset}} & \\multicolumn{6}{c|}{Train} & \\multicolumn{6}{c|}{Test}\\\\ \\cline{3-14}\n\t\t\t\\multicolumn{2}{|c|}{}& \\emph{pos.}&\\emph{neg.}&\\emph{happy}&\\emph{anger}&\\emph{sad}&\\emph{neu.}&\\emph{pos.}&\\emph{neg.}&\\emph{happy}&\\emph{anger}&\\emph{sad}&\\emph{neu.}\\\\ \\hline\n\t\t\t\\multicolumn{2}{|c|}{MOSI}&709&738&-&-&-&-&467&285&-&-&-&-\\\\ \\hline\n\t\t\t\\multicolumn{2}{|c|}{IEMOCAP}&-&-&1194&933&839&1324&-&-&433&157&238&380\\\\ \\hline\n\t\t\t\\multicolumn{14}{l}{\\scriptsize{pos. = positive, neg. = negative, neu. = neutral}}\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace{-4.5mm}\n\t\\caption {Class distribution of datasets in both train and test splits. }\n\t\\label{tab:dataset}\n\\end{table}\n\n\\subsubsection{CMU-MOSI}\n\\label{sec:mosi}\nCMU-MOSI dataset~\\citep{zadeh2016multimodal} is rich in sentimental\nexpressions, where 89 people review various topics in English. The videos are\nsegmented into utterances where each utterance is annotated with scores between\n$-3$ (strongly negative) and $+3$ (strongly positive) by five annotators. We\ntook the average of these five annotations as the sentiment polarity and\nconsidered only two classes (positive and negative). Given every individual's\nunique way of expressing sentiments, real world applications should be able to\nmodel generic person independent features and be robust to person variance. To\nthis end, we perform person-independent experiments to emulate unseen\nconditions. Our train\/test splits of the dataset are completely disjoint with\nrespect to speakers. The train\/validation set consists of the first 62\nindividuals in the dataset. The test set contains opinionated videos by rest of\nthe 31 speakers. In particular, 1447 and 752 utterances are used for training\nand test respectively.\n\n\\subsubsection{IEMOCAP}\n\\label{sec:iemocap}\nIEMOCAP~\\citep{iemocap} contains two way conversations\namong ten speakers, segmented into utterances. The utterances are tagged with\nthe labels anger, happiness, sadness, neutral, excitement, frustration, fear,\nsurprise, and other. We consider the first four ones to compare with the\nstate of the art~\\citep{porcon} and other works. It contains 1083\nangry, 1630 happy, 1083 sad, and 1683 neutral videos. Only the videos by the\nfirst eight speakers are considered for training.\n\n\\subsection{Baselines}\nWe compare our method with the following strong baselines.\n\n\\paragraph{Early fusion}\n\\label{early-fusion}\nWe extract unimodal features (\\cref{UFE}) and simply concatenate them to\nproduce multimodal features. Followed by support vector machine (SVM)\nbeing applied on this feature vector for the final sentiment\nclassification.\n\n\\paragraph{Method from~\\citep{pordee}}\nWe have implemented and compared our method with the approach proposed by\n\\citet{pordee}. In their approach, they extracted visual features using\nCLM-Z, audio features using openSMILE, and textual features using CNN. MKL was then applied to the features obtained from\nconcatenation of the unimodal features. However, they did not conduct speaker\nindependent experiments.\n\nIn order to perform a fair comparison with~\\citep{pordee}, we\nemploy our fusion method on the features extracted by~\\citet{pordee}.\n\n\n\n\n\n\n\\paragraph{Method from~\\citep{porcon}}\nWe have compared our method with~\\citep{pordep}, which takes\nadvantage of contextual information obtained from the surrounding\nutterances. This context modeling is achieved using LSTM. We reran the\nexperiments of~\\citet{pordep} without using SVM for classification since using\nSVM with neural networks is usually discouraged. This provides a fair comparison\nwith our model which does not use SVM.\n\n\\paragraph{Method from~\\citep{zadten}}\nIn~\\citep{zadten}, they proposed a trimodal fusion method based on the tensors. We have also compared our method with their. In particular, their dataset configuration was different than us so we have adapted their publicly available code ~\\footnote{\\url{https:\/\/github.com\/A2Zadeh\/TensorFusionNetwork}} and employed that on our dataset.\n\\subsection{Experimental Setting}\n\\label{sec:exp_set}\n\nWe considered two variants of experimental setup while evaluating our model.\n\n\\paragraph{HFusion} In this setup, we evaluated hierarchical fusion\nwithout context-aware features with CMU-MOSI dataset. We removed all the GRUs\nfrom the model described in \\cref{sec:mul_fusion,sec:context} forwarded\nutterance specific features directly to the next layer. This setup is depicted\nin \\cref{fig:hfusion-trimodal}.\n\n\\paragraph{CHFusion} This setup is exactly as the model described in\n\\cref{sec:model}.\n\n\\subsection{Results and Discussion}\n\nWe discuss the results for the different experimental settings discussed in\n\\cref{sec:exp_set}.\n\n\\begin{table*}[t]\n \\centering\n \\small\n \\caption{Comparison in terms of accuracy of Hierarchical Fusion\n (HFusion) with other fusion methods for CMU-MOSI dataset; bold font\n signifies best accuracy for the corresponding feature set and\n modality or modalities, where T stands for text, V for video, and A for audio. $SOTA^1$ = Poria et al.~\\citep{pordee}, $SOTA^2$ = Zadeh et al.~\\citep{zadten}}\n \\resizebox{\\textwidth}{!}{\n \\addtolength\\tabcolsep{-3pt}\n \\begin{tabular}[t]{@{\\extracolsep{5pt}}ccccccc}\n \\hline\n \\multirow2*{\\begin{tabular}{c}Modality\\\\ Combination\\end{tabular}}&\n \\multicolumn{3}{c}{\\citep{pordee} feature set}&\n \\multicolumn{3}{c}{Our feature set}\\\\\n \\cline{2-4}\\cline{5-7}\n & $SOTA^{1}$&\n $SOTA^2$&\n HFusion&\n Early fusion &\n $SOTA^2$&\n HFusion\\\\\n \\cline{1-1}\\cline{2-4}\\cline{5-7}\n T & \\multicolumn{3}{c}{N\/A} & \\multicolumn{3}{c}{75.0\\%} \\\\\n V & \\multicolumn{3}{c}{N\/A} & \\multicolumn{3}{c}{55.3\\%} \\\\\n A & \\multicolumn{3}{c}{N\/A} & \\multicolumn{3}{c}{56.9\\%} \\\\\n \\hline\n T+V & 73.2\\% & 73.8\\% & \\textbf{74.4\\%} & 77.1\\% & 77.4\\% & \\textbf{77.8\\%}\\\\\n T+A & 73.2\\% & 73.5\\% & \\textbf{74.2\\%} & 77.1\\% & 76.3\\% & \\textbf{77.3\\%}\\\\\n A+V & 55.7\\% & 56.2\\% & \\textbf{57.5\\%} & 56.5\\% & 56.1\\% & \\textbf{56.8\\%}\\\\\n \\hline\n A+V+T & 73.5\\% & 71.2\\% & \\textbf{74.6\\%} & 77.0\\% & 77.3\\% & \\textbf{77.9\\%}\\\\\n \\hline\n \\end{tabular}\n }\n \\label{table:hfusion}\n\\end{table*}\n\n\\subsubsection{Hierarchical Fusion (HFusion)}\n\\label{hfusion}\n\nThe results of our experiments are presented in \\cref{table:hfusion}. We\nevaluated this setup with CMU-MOSI dataset (\\cref{sec:mosi}) and two\nfeature sets: the feature set used in~\\citep{pordee}\nand the set of unimodal features discussed in \\cref{UFE}.\n\nOur model outperformed~\\citep{pordee}, which employed MKL, for all bimodal\nand trimodal scenarios by a margin of 1--1.8\\%. This leads us to present two\nobservations. Firstly, the features used in~\\citep{pordee} are inferior to\nthe features extracted in our approach. Second, our hierarchical\nfusion method is better than their fusion method.\n\nIt is already established in the literature\n\\citep{pordee,perez2013utterance} that multimodal analysis outperforms\nunimodal analysis. We also observe the same trend in our experiments where\ntrimodal and bimodal classifiers outperform unimodal classifiers. The textual\nmodality performed best among others with a higher unimodal classification\naccuracy of 75\\%. Although other modalities contribute to improve the\nperformance of multimodal classifiers, that contribution is little in compare to\nthe textual modality.\n\nOn the other hand, we compared our model with early fusion\n(\\cref{early-fusion}) for aforementioned feature sets\n(\\cref{UFE}). Our fusion mechanism consistently outperforms early fusion for\nall combination of modalities. This supports our\nhypothesis that our hierarchical fusion method captures the\ninter-relation among the modalities and produce better performance vector than\nearly fusion. Text is the strongest individual modality, and we observe that\nthe text modality paired with remaining two modalities results in consistent\nperformance improvement.\n\nOverall, the results give a strong indication that the comparison among the\nabstract feature values dampens the effect of less important modalities, which\nwas our hypothesis. For example, we can notice that for early fusion T+V and T+A\nboth yield the same performance. However, with our method text with video\nperforms better than text with audio, which is more aligned with our\nexpectations, since facial muscle movements usually carry more emotional nuances\nthan voice.\n\nIn particular, we observe that our model outperformed all the strong baselines mentioned above. The method by~\\citep{pordee} is only able to fuse using concatenation. Our proposed method outperformed their approach by a significant margin; thanks to the power of hierarchical fusion which proves the capability of our method in modeling bimodal and trimodal correlations. However on the other hand, the method by~\\citep{zadten} is capable of fusing the modalities using a tensor. Interestingly our method also outperformed them and we think the reason is the capability of bimodal fusion and use that for trimodal fusion. Tensor fusion network is incapable to learn the weights of the bimodal and trimodal correlations in the fusion. Tensor Fusion is mathematically formed by an outer product, it\nhas no learn-able parameters. Wherein our method learns the weights automatically using a neural network (Equation 1,2 and 3).\n\n\\begin{table*}[t]\n \\centering\n \\caption{Comparison of Context-Aware Hierarchical Fusion (CHFusion) in terms of accuracy ($\\text{CHFusion}_{acc}$) and f-score (for IEMOCAP: $\\text{CHFusion}_{fsc}$) with the state of the art for CMU-MOSI\n and IEMOCAP dataset; bold font signifies best accuracy for the corresponding dataset and\n modality or modalities, where T stands text, V for video, A for audio. $SOTA^1$ = Poria et al.~\\citep{pordee}, $SOTA^2$ = Zadeh et al.~\\citep{zadten}. $\\text{CHFusion}_{acc}$ and $\\text{CHFusion}_{fsc}$ are the accuracy and f-score of CHFusion respectively.}\n \\resizebox{\\textwidth}{!}{\n \\addtolength\\tabcolsep{-6pt}\n \\begin{tabular}[t]{@{\\extracolsep{4pt}}cccccccc}\n \\hline\n \\multirow2*{\\begin{tabular}{c}Modality\\end{tabular}} & \\multicolumn{3}{c}{CMU-MOSI} & \\multicolumn{4}{c}{IEMOCAP} \\\\\n \\cline{2-4}\\cline{5-8}\n & $SOTA^1$ & $SOTA^2$ & $\\text{CHFusion}_{acc}$ & $SOTA^1$ & $SOTA^2$ & $\\text{CHFusion}_{acc}$ & $\\text{CHFusion}_{fsc}$\\\\\n \\cline{1-1}\\cline{2-4}\\cline{5-7}\\cline{8-8}\n T & \\multicolumn{3}{c}{76.5\\%} & \\multicolumn{3}{c}{73.6\\%} & -\\\\\n V & \\multicolumn{3}{c}{54.9\\%} & \\multicolumn{3}{c}{53.3\\%} & -\\\\\n A & \\multicolumn{3}{c}{55.3\\%} & \\multicolumn{3}{c}{57.1\\%} & -\\\\\n \\hline\n T+V & 77.8\\% & 77.1\\% & \\textbf{79.3\\%} & 74.1\\% & 73.7\\% & \\textbf{75.9\\%} & 75.6\\%\\\\\n T+A & 77.3\\% & 77.0\\% & \\textbf{79.1\\%} & 73.7\\% & 71.1\\% & \\textbf{76.1\\%} & 76.0\\%\\\\\n A+V & 57.9\\% & 56.5\\% & \\textbf{58.8\\%} & 68.4\\% & 67.4\\% & \\textbf{69.5\\%} & 69.6\\% \\\\\n \\hline\n A+V+T & 78.7\\% & 77.2\\% & \\textbf{80.0\\%} & 74.1\\% & 73.6\\% & \\textbf{76.5\\%} & 76.8\\%\\\\\n \\hline\n \\end{tabular}\n }\n \\label{table:chfusion}\n\\end{table*}\n\n\\subsubsection{Context-Aware Hierarchical Fusion (CHFusion)}\n\\label{chfusion}\n\nThe results of this experiment are shown in \\cref{table:chfusion}. This setting\nfully utilizes the model described in \\cref{sec:model}. We applied this\nexperimental setting for two datasets, namely CMU-MOSI~(\\cref{sec:mosi}) and\nIEMOCAP~(\\cref{sec:iemocap}). We used the feature set discussed in \\cref{UFE},\nwhich was also used by~\\citet{porcon}. As expected our method outperformed the simple early fusion based fusion by~\\citep{pordee}, tensor fusion by~\\citep{zadten}. The method by~\\citet{porcon} used a scheme to learn contextual features from the surrounding features. However, as a method of fusion they adapted simple concatenation based fusion method by~\\citep{pordee}. As discussed in Section \\ref{sec:context}, we employed their contextual feature extraction framework and integrated our proposed fusion method to that. This has helped us to outperform~\\citet{porcon} by significant margin thanks to the hierarchical fusion (HFusion).\n\n\\paragraph{CMU-MOSI}\nWe achieve 1--2\\% performance improvement over the state of the art\n\\citep{porcon} for all the modality combinations having textual\ncomponent. For A+V modality combination we achieve better but similar\nperformance to the state of the art. We suspect that it is due to both audio and\nvideo modality being significantly less informative than textual modality. It is\nevident from the unimodal performance where we observe that textual modality on\nits own performs around 21\\% better than both audio and video modality. Also,\naudio and video modality performs close to majority baseline. On the other hand,\nit is important to notice that with all modalities combined we achieve about\n3.5\\% higher accuracy than text alone.\n\n\nFor example, consider the following utterance: \\emph{so overall new moon even with the bigger better budgets huh it was still too long}.\nThe speaker discusses her opinion on the movie Twilight New Moon. Textually the\nutterance is abundant with positive words however audio and video comprises of a\nfrown which is observed by the hierarchical fusion based model.\n\n\\paragraph{IEMOCAP}\nAs the IEMOCAP dataset contains four distinct emotion categories, in the last layer of the network we used a softmax classifier whose output dimension is set to 4. \nIn order to perform classification on IEMOCAP dataset we feed the fused features $F_{mt}$ (where\n$m=AV,VT,TA,\\text{ or } AVT$ and $t=1,2,\\dots,N$) to a softmax layer with $C=4$\noutputs. The classifier can be described as follows:\n\\begin{flalign*}\n\\mathcal{P} &=\n\\text{softmax}(W_{\\mathit{softmax}}F_{mt}+b_{\\mathit{softmax}}),\\\\\n\\hat{y}&=\\underset{j}{\\text{argmax}}(\\mathcal{P}[j]),\n\\end{flalign*}\nwhere $W_{\\mathit{softmax}}\\in \\mathbb{R}^{4\\times D}$,\n$b_{\\mathit{softmax}}\\in \\mathbb{R}^4$, $\\mathcal{P}\\in \\mathbb{R}^4$, $j=$\nclass value ($0$ or $1$ or $2$ or $3$), and $\\hat{y}=$ estimated class value.\n\n\\begin{table}[t]\n \\centering\n \\caption{Class-wise accuracy and f-score for IEMOCAP dataset for trimodal scenario.}\n \n \n \\begin{tabular}[t]{ccccc}\n \\hline\n \\multirow{2}{*}{Metrics} & \\multicolumn{4}{c}{Classes}\\\\\n \\cline{2-5} & Happy & Sad & Neutral & Anger\\\\\n \\hline\n Accuracy & 74.3 & 75.6 & 78.4 & 79.6 \\\\\n F-Score & 81.4 & 77.0 & 71.2 & 77.6 \\\\\n \\hline\n \\end{tabular}\n \n \\label{table:iemocap-classwise}\n\\end{table}\nHere as well, we achieve performance improvement consistent with CMU-MOSI. This\nmethod performs 1--2.4\\% better than the state of the art for all the modality\ncombinations. Also, trimodal accuracy is 3\\% higher than the same for textual\nmodality. Since, IEMOCAP dataset imbalanced, we also present the f-score for each modality combination for a better evaluation. One key observation for IEMOCAP dataset is that its A+V modality\ncombination performs significantly better than the same of CMU-MOSI dataset. We\nthink that this is due to the audio and video modality of IEMOCAP being richer than\nthe same of CMU-MOSI. The performance difference with another strong baseline~\\citep{zadten} is even more ranging from 2.1\\% to 3\\% on CMU-MOSI dataset and 2.2\\% to 5\\% on IEMOCAP dataset. This again confirms the superiority of the hierarchical fusion in compare to~\\citep{zadten}. We think this is mainly because of learning the weights of bimodal and trimodal correlation (representing the degree of correlations) calculations at the time of fusion while Tensor Fusion Network (TFN) just relies on the non-trainable outer product of tensors to model such correlations for fusion.\nAdditionally, we present class-wise accuracy and f-score for IEMOCAP for trimodal (A+V+T) scenario in \\cref{table:iemocap-classwise}.\n\n\\subsubsection{HFusion vs.\\ CHFusion}\n\nWe compare HFusion and CHFusion models over CMU-MOSI dataset. We observe that\nCHFusion performs 1--2\\% better than HFusion model for all the modality\ncombinations. This performance boost is achieved by the inclusion of\nutterance-level contextual information in HFusion model by adding GRUs in\ndifferent levels of fusion hierarchy.\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nMultimodal fusion strategy is an important issue in multimodal sentiment analysis. \nHowever, little work has been done so far in this direction. \nIn this paper, we have presented a novel and comprehensive fusion strategy. \nOur method outperforms the widely used early fusion on both datasets typically used to test multimodal sentiment analysis methods.\nMoreover, with the addition of context modeling with GRU, \nour method outperforms the state of the art in multimodal sentiment analysis and emotion detection by significant margin. \n\nIn our future work, we plan to improve the quality of unimodal features, especially textual features, which will further improve the accuracy of classification.\nWe will also experiment with more sophisticated network architectures.\n\n\\section*{Acknowledgement}\nThe work was partially supported by the Instituto Polit\\'ecnico Nacional via grant SIP 20172008 to A.~Gelbukh.\n\n\\bibliographystyle{elsarticle-num-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn complex geometry, as a generalization of holomorphic and totally real immersions, slant immersions were defined by Chen \\cite{chen}. Cabrerizo et al \\cite{cabrerizo} defined bi-slant submanifolds in almost contact metric manifolds. In \\cite{uddin} Uddin et al. studied warped product bi-slant immersions in Kaehler manifolds. They proved that there do not exist any warped product bi-slant submanifolds of Kaehler manifolds other than hemi-slant warped products and CR-warped products.\n\nThe theory of Riemannian submersions as an analogue of isemetric immersions was initiated by O'Neill \\cite{oneill} and Gray\\cite{gray}. The Riemannian submersions are important in physics owing to applications in the Yang-Mills theory, Kaluza-Klein theory, robotic theory, supergravity and superstring theories. In Kaluza-Klein theory, the general solution of a recent model is given in point of harmonic maps satisfying Einstein equations (see \\cite{yang1,kaluza 1,kaluza2,kaluza3,yang2,supergravity1,supergravity2}). Altafini \\cite{altafini} expressed some applications of submersions in the theory of robotics and \\c{S}ahin \\cite{sahin} also investigated some applications of Riemannian submersions on redundant robotic chains. On the other hand Riemannian submersions are very useful in studying the geometry of Riemannian manifolds equipped with differentiable structures.\nIn \\cite{watson} Watson introduced the notion of almost Hermitian submersions between almost complex manifolds. He investigated some geometric properties between base manifold and total manifold as well as fibers. \\c{S}ahin \\cite{sahinanti} introduced anti-invariant Riemannian submersions from almost Hermitian manifolds. He showed that such maps have some geometric properties. Also he studied slant submersions from almost Hermitian manifolds onto a Riemannian manifolds \\cite{sahinslant}. Recently, considering different conditions on Riemannian submersions many studies have been done (see \\cite{sezin,sayar2,sayar,sayar3,sahinsemi,hakan,Tastan2}).\n\nAs a special horizontally conformal maps which were introduced independently by Fuglede and Ishihara, horizontally conformal submersions are defined as follows $(M_{1},g_{1})$ and $(M_{2},g_{2})$ are Riemannian manifolds of dimension $m_{1}$ and $m_{2}$, respectively. A smooth submersion $f:(M_{1},g_{1})\\rightarrow (M_{2},g_{2})$ is called a horizontally conformal submersion if there is a positive function $\\lambda$ such that\n\\begin{align*}\n\\lambda^{2}g_{1}\\left(X_{1},Y_{1}\\right)=g_{2}\\left(f_{*}X_{1},f_{*}Y_{1}\\right)\n\\end{align*}\nfor all $X_{1}, Y_{1}\\in \\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$.\nHere a horizontally conformal submersion $f$ is called horizontally homothetic if the $grad \\lambda$ is vertical i.e.\n\\begin{align*}\n\\mathcal{H}\\left(grad\\lambda\\right)=0.\n\\end{align*}\n We denote by $\\mathcal{V}$ and $\\mathcal{H}$ the projections on the vertical distributions $\\left(ker f_{*}\\right)$ and horizontal distributions $\\left(ker f_{*}\\right)^{\\perp}$. It can be said that Riemannian submersion is a special horizontally conformal submersion with $\\lambda=1$.\nRecently, Akyol and \\c{S}ahin introduced conformal anti-invariant submersions \\cite{akyolantiinv2}, conformal semi-invariant submersion\\cite{akyolsemiinv}, conformal slant submersion \\cite{akyol1} and conformal semi-slant submersions\\cite{akyolsemislant}. Also the geometry of conformal submersions have been studied by several authors \\cite{gunduzalpconf,kumar}.\\\\\nIn section 2 we review basic formulas and definitions needed for this paper. In section 3, we define the new conformal bi-slant submersion from almost Hermitian manifolds onto Riemannian manifolds and present a example. We investigate the geometry of the horizontal distribution and the vertical distribution. Finally we obtain necessary and sufficient conditions for a conformal bi-slant submersion to be totally geodesic.\n\n\\section{Preliminaries}\n\nLet $\\left(M_{1},g_{1},J\\right)$ be an almost Hermitian manifold. Then this means that $M_{1}$ admits a tensor field $J$ of type $(1,1)$ on $M_{1}$ which satisfy\n\\begin{align}\nJ^{2}=-I, \\ \\ g_{1}\\left(JE_{1},JE_{2}\\right)=g_{1}\\left(E_{1},E_{2}\\right)\n\\end{align}\nfor $E_{1},E_{2}\\in \\Gamma(TM_{1})$. An almost Hermitian manifold $M_{1}$ is called Kaehlerian manifold if\n\\begin{align*}\n\\left(\\nabla_{E_{1}}J\\right)E_{2}=0, \\ \\ E_{1},E_{2}\\in\\Gamma\\left(TM_{1}\\right)\n\\end{align*}\nwhere $\\nabla$ is the operator of Levi-Civita covariant differentiation.\n\nNow, we will give some definitions and theorems about the concept of (horizontally) conformal submersions.\n\\begin{definition}\n\tLet $(M_{1},g_{1})$ and $(M_{2},g_{2})$ are two Riemannian manifolds with the dimension $m_{1}$ and $m_{2}$, respectively. A smooth map $f:(M_{1},g_{1})\\rightarrow (M_{2},g_{2})$ is called horizontally weakly conformal or semi conformal at $q\\in M$ if, either\n\t\\begin{enumerate}[i.]\n\t\t\\item $df_{q}=0$, or\n\t\t\\item $df_{q}$ is surjective and there exists a number $\\Omega(q)\\neq 0$ satisfying\n\t\t\\begin{align*}\n\t\tg_{2}\\left(df_{q}X,df_{q}Y\\right)=\\Omega(q)g_{1}\\left(X,Y\\right)\n\t\t\\end{align*}\n\t\tfor $X,Y\\in \\Gamma\\left(\\ker(df)\\right)^{\\perp}$.\n\t\\end{enumerate}\nHere the number $\\Omega(q)$ is called the square dilation. Its square root $\\lambda(q)=\\sqrt{\\Omega(q)}$ is called the dilation. The map $f$ is called horizontally weakly conformal or semi-conformal on $M_{1}$ if it is horizontally weakly conformal at every point of $M_{1}$. it is said to be a conformal submersion if $f$ has no critical point.\n\\end{definition}\nLet $f:M_{1}\\rightarrow M_{2}$ be a submersion. A vector field $X_{1}$ on $M_{1}$ is called a basic vector field if $X_{1}\\in \\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$ and $f$-related with a vector field $X_{2}$ on $M_{2}$ i.e $f_{*}(X_{1q})=X_{2f(q)}$ for $q\\in M_{1}$.\n\nThe two $(1,2)$ tensor fields $\\mathcal{T}$ and $\\mathcal{A}$ on $M$ are given by the formulas\n\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray}\n\\mathcal{T}(E_{1},E_{2})&=&\\mathcal{T}_{E_{1}}E_{2}=\\mathcal{H}\\nabla_{\\mathcal{V}E_{1}}\\mathcal{V}E_{2}+\\mathcal{V}\\nabla_{\\mathcal{V}E_{1}}\\mathcal{H}E_{2} \\label{2.2} \\\\\n\\mathcal{A}(E_{1},E_{2})&=&\\mathcal{A}_{E_{1}}E_{2}=\\mathcal{V}\\nabla_{\\mathcal{H}E_{1}}\\mathcal{H}E_{2}+\\mathcal{H}\\nabla_{\\mathcal{H}E_{1}}\\mathcal{V}E_{2} \\label{2.3}\n\\end{eqnarray}\nfor $E_{1},E_{2}\\in \\Gamma\\left(TM\\right)$ \\cite{falcitelli}.\\\\\n\nNote that a Riemannian submersion $f:M_{1}\\longrightarrow M_{2}$ has totally geodesic fibers if and only if $\\mathcal{T}$ vanishes identically.\n\nConsidering the equations (2.3) and (2.4), one can write\n\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray}\n\\nabla_{U_{1}}U_{2}&=&\\mathcal{T}_{U_{1}}U_{2}+\\bar{\\nabla}_{U_{1}}U_{2} \\label{2.4}\\\\\n\\nabla_{U_{1}}X_{1}&=&\\mathcal{H}\\nabla_{U_{1}}X_{1}+\\mathcal{T}_{U_{1}}X_{1} \\label{2.5}\\\\\n\\nabla_{X_{1}}U_{1}&=&\\mathcal{A}_{X_{1}}U_{1}+\\mathcal{V}\\nabla_{X_{1}}U_{1} \\label{2.6}\\\\\n\\nabla_{X_{1}}X_{2}&=&\\mathcal{H}\\nabla_{X_{1}}X_{2}+\\mathcal{A}_{X_{1}}X_{2} \\label{2.7}\n\\end{eqnarray}\nfor $X_{1},X_{2}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$ and $U_{1},U_{2}\\in\\Gamma\\left(\\ker f_{*}\\right)$, where $\\bar{\\nabla}_{U_{1}}U_{2}=\\mathcal{V}\\nabla_{U_{1}}U_{2}$. Then we easily seen that $\\mathcal{T}_{U_{1}}$ and $\\mathcal{A}_{X_{1}}$ are skew-symmetric i.e $\ng_{1}\\left(\\mathcal{A}_{X_{1}}E_{1},E_{2}\\right)=-g_{1}\\left(E_{1},\\mathcal{A}_{X_{1}}E_{2}\\right)$ and\n$g_{1}\\left(\\mathcal{T}_{U_{1}}E_{1},E_{2}\\right)=-g_{1}\\left(E_{1},\\mathcal{T}_{U_{1}}E_{2}\\right)$ for any $E_{1},E_{2}\\in\\Gamma\\left(TM_{1}\\right)$. For the special case where $f$ as the horizontal, the following Proposition be given:\n\\begin{proposition}\n\tLet $f:\\left(M_{1},g_{1}\\right)\\rightarrow\\left(M_{2},g_{2}\\right)$ be a horizontally conformal submersion with dilation $\\lambda$ and $X_{1},X_{2}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$, then\n\t\\begin{align}\n\t\\mathcal{A}_{X_{1}}X_{2}=\\frac{1}{2}\\left(\\mathcal{V}\\left[X_{1},X_{2}\\right]-\\lambda^{2}g_{1}\\left(X_{1},X_{2}\\right)grad_{\\mathcal{V}}\\left(\\frac{1}{\\lambda^{2}}\\right)\\right)\n\t\\end{align}\n \\end{proposition}\n\nLet $f:\\left(M_{1},g_{1}\\right)\\rightarrow\\left(M_{2},g_{2}\\right)$ be a smooth map between $\\left(M_{1},g_{1}\\right)$ and $\\left(M_{2},g_{2}\\right)$ Riemannian manifolds. Then the second fundamental form of $f$ is given by\n\\begin{align}\\label{2.9}\n\\left(\\nabla f_{*}\\right)\\left(E_{1},E_{2}\\right)=\\nabla^{f}_{E_{1}}f_{*}(E_{2})-f_{*}\\left(\\bar{\\nabla}_{E_{1}}E_{2}\\right)\n\\end{align}\nfor any $E_{1},E_{2}\\in\\Gamma\\left(TM_{1}\\right)$. It is known that the second fundamental form $f$ is symmetric \\cite{baird}.\n\\begin{lemma}\n\tSuppose that $f:M_{1}\\rightarrow M_{2}$ is a horizontally conformal submersion. Then for $X_{1},X_{2}\\in \\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$ and $U_{1},U_{2}\\in \\Gamma\\left(\\ker f_{*}\\right)$ we have\n\t\\begin{enumerate}[i.]\n\t\t\\item $\\left(\\nabla f_{*}\\right)\\left(X_{1},X_{2}\\right)=X_{1}\\left(\\ln\\lambda\\right)f_{*}X_{2}+X_{2}\\left(\\ln\\lambda\\right)f_{*}X_{1}-g_{1}\\left(X_{1},X_{2}\\right)f_{*}\\left(\\nabla\\ln\\lambda\\right)$\n\t\t\\item $\\left(\\nabla f_{*}\\right)\\left(U_{1},U_{2}\\right)=-f_{*}\\left(\\mathcal{T}_{U_{1}}U_{2}\\right)$\n\t\t\\item $\\left(\\nabla f_{*}\\right)\\left(X_{1},U_{1}\\right)=-f_{*}\\left(\\bar{\\nabla}_{X_{1}}U_{1}\\right)=-f_{*}\\left(\\mathcal{A}_{X_{1}}V_{1}\\right)$.\n\t\\end{enumerate}\n\\end{lemma}\n The smoooth map $f$ is called a totally geodesic map if $\\left(\\nabla f_{*}\\right)\\left(E_{1},E_{2}\\right)=0$ for $E_{1},E_{2}\\in\\Gamma(TM)$ \\cite{baird}.\n \nWe assume that $g$ is a Riemannian metric tensor on the manifold $M=M_{1}\\times M_{2}$ and the canonical foliations $D_{M_{1}}$ and $D_{M_{2}}$ intersect vertically everywhere. Then $g$ is the metric tensor of a usual product of Riemannian manifold if and only if $D_{M_{1}}$ and $D_{M_{2}}$ are totally geodesic foliations.\n\n\\section{Conformal Bi-Slant Submersions}\n\n\\begin{definition}\n\tLet $\\left(M_{1},g_{1},J\\right)$ be an almost Hermitian manifold and $\\left(M_{2},g_{2}\\right)$ a Riemannian manifold. A horizontal conformal submersion $f:M_{1}\\longrightarrow M_{2}$ is called a conformal bi-slant submersion if $D$ and $\\bar{D}$ are slant distributions with the slant angles $\\theta$ and $\\bar{\\theta}$, respectively, such that $\\ker f_{*}=D\\oplus \\bar{D}$. $f$ is called proper if its slant angles satisfy $\\theta,\\bar{\\theta}\\neq 0,\\frac{\\pi}{2}$.\n\\end{definition}\n We now give a example of a proper conformal bi-slant submersion.\n \\begin{example}\n We consider the compatible almost complex structure $J_{\\omega}$ on $\\mathbb{R}^{8}$ such that\n \\begin{align*}\n J_{\\omega}=\\left(\\cos\\omega\\right) J_{1}+\\left(\\sin\\omega\\right) J_{2}, \\ 0<\\omega\\leq\\frac{\\pi}{2}\n \\end{align*}\n where\n \\begin{align*}\n J_{1}\\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\right)=\\left(-x_{2},x_{1},-x_{4},x_{3},-x_{6},x_{5},-x_{8},x_{7}\\right)\\\\\n J_{2}\\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\right)=\\left(-x_{3},x_{4},x_{1},-x_{2},-x_{7},x_{8},x_{5},-x_{6}\\right)\n \\end{align*}\n Consider a submersion $f:\\mathbb{R}^{8} \\rightarrow \\mathbb{R}^{4}$ defined by\n \\begin{align*}\n f\\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\right)=\\pi^{5}\\left(\\frac{x_{1}-x_{3}}{\\sqrt{2}},x_{4}, \\frac{x_{5}-x_{6}}{\\sqrt{2}},x_{7}\\right)\n \\end{align*}\n Then it follows that\n \\begin{align*}\n D=span\\{U_{1}=\\frac{1}{\\sqrt{2}}\\left(\\frac{\\partial}{\\partial x_{1}}+\\frac{\\partial}{\\partial x_{3}}\\right),U_{2}=\\frac{\\partial}{\\partial x_{2}}\\}\\\\\n \\bar{D}=span\\{U_{3}=\\frac{1}{\\sqrt{2}}\\left(\\frac{\\partial}{\\partial x_{5}}+\\frac{\\partial}{\\partial x_{6}}\\right),U_{4}=\\frac{\\partial}{\\partial x_{8}}\\}\n \\end{align*}\n Thus $f$ is conformal bi-slant submersion with $\\theta$ and $\\bar{\\theta}$ such that $\\cos\\theta=\\frac{1}{\\sqrt{2}}\\cos\\omega$ and $\\cos\\bar{\\theta}=\\frac{1}{\\sqrt{2}}\\sin\\omega$.\n \\end{example}\n\n\nSuppose that $f$ is a conformal bi-slant submersion from a almost Hermitian manifold $\\left(M_{1},g_{1},J_{1}\\right)$ onto a Riemannian manifold $(M_{2},g_{2})$. For $U_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)$, we have\n\\begin{equation}\\label{3.1}\nU_{1}=\\alpha U_{1}+\\beta U_{1}\n\\end{equation}\nwhere $\\alpha U_{1}\\in\\Gamma\\left(D_{1}\\right)$ and $\\beta U_{1}\\in\\Gamma\\left(D_{2}\\right)$.\\\\\nAlso, for $U_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)$, we write\n\\begin{equation}\\label{3.2}\nJU_{1}=\\xi U_{1}+\\eta U_{1}\n\\end{equation}\nwhere $\\xi U_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)$ and $\\eta U_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)^\\perp$.\\\\\nFor $X_{1}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^\\perp\\right)$, we have\n\\begin{equation}\\label{3.3}\nJX_{1}=\\mathcal{B}X_{1}+\\mathcal{C}X_{1}\n\\end{equation}\nwhere $\\mathcal {B}X_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)$ and $\\mathcal{C}X_{1}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^\\perp\\right)$.\\\\\nThe horizontal distribution $(\\ker f_{*})^{\\perp}$ is decompesed as\n\\begin{align*}\n(\\ker f_{*})^{\\perp}=\\eta D_{1}\\oplus\\eta D_{2}\\oplus\\mu\n\\end{align*}\nwhere $\\mu$ is the complementary distribution to $\\eta D_{1}\\oplus\\eta D_{2}$ in $(\\ker f_{*})^{\\perp}$.\\\\\n\nConsidering Definition 3.1 we can give the following result that we will use throughout the article.\n\n\\begin{theorem}\n\tSuppose that $f$ is a conformal bi-slant submersion from an almost Hermitian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$. Then we have\n\t\\begin{enumerate}[i)]\n\t\t\\item $\\xi^{2}U_{1}=-\\left(\\cos^{2}\\theta\\right)U_{1} \\ \\text{for} \\ U_{1}\\in\\Gamma\\left(D\\right)$\n\t\t\\item $\\xi^{2}{V}_{1}=-\\left(\\cos^{2}\\bar{\\theta}\\right)V_{1} \\ \\text{for} \\ V_{1}\\in\\Gamma\\left(\\bar{D}\\right)$\n\t\\end{enumerate}\n\n\\end{theorem}\n\\begin{proof}\n\tThe proof of this theorem is similar to slant immersions \\cite{chen}.\n\\end{proof}\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$. Then\n\t\\begin{enumerate}\n\t\t\\item[\\textit{i)}] the distribution $D$ is integrable if and only if\n\t\\begin{align*}\n\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{1},\\eta U_{2}\\right),f_{*}\\eta V_{1}\\right)\n=&g_{1}\\left(\\mathcal{T}_{U_{2}}\\eta\\xi U_{1}-\\mathcal{T}_{U_{1}}\\eta\\xi U_{2},V_{1}\\right)\\\\&+g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta U_{2}-\\mathcal{T}_{U_{2}}\\eta U_{1},\\xi V_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{2},\\eta U_{1}\\right),f_{*}\\eta V_{1}\\right).\n\\end{align*}\n\t\t\\item[\\textit{ii)}] the distribution $\\bar{D}$ is integrable if and only if\n\t\t\t\\begin{align*}\n\t\t\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(V_{1},\\eta V_{2}\\right),f_{*}\\eta U_{1}\\right)\n\t\t=&g_{1}\\left(\\mathcal{T}_{V_{2}}\\eta\\xi V_{1}-\\mathcal{T}_{V_{1}}\\eta\\xi V_{2},U_{1}\\right)\\\\&+g_{1}\\left(\\mathcal{T}_{V_{1}}\\eta V_{2}-\\mathcal{T}_{V_{2}}\\eta V_{1},\\xi U_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(V_{2},\\eta V_{1}\\right),f_{*}\\eta U_{1}\\right).\n\t\t\\end{align*}\n\t\\end{enumerate}\n\twhere $U_{1},U_{2}\\in \\Gamma\\left(D\\right)$, $V_{1},V_{2}\\in \\Gamma\\left(\\bar{D}\\right)$.\n\\end{theorem}\n\n\\begin{proof}\t\n$i)$ From $U_{1},U_{2} \\in \\Gamma\\left(D\\right)$ and $V_{1} \\in \\Gamma\\left(\\bar{D}\\right)$ we have\n\t\\setlength\\arraycolsep{2pt}\n\t\\begin{align*}\n\tg_{1}\\left([U_{1},U_{2}],V_{1}\\right)=&g_{1}\\left(\\nabla_{U_{1}}\\xi U_{2},J V_{1}\\right)+g_{1}\\left(\\nabla_{U_{1}}\\eta U_{2},J V_{1}\\right)\\\\&-g_{1}\\left(\\nabla_{U_{2}}\\xi U_{1},J V_{1}\\right)-g_{1}\\left(\\nabla_{U_{2}}\\eta U_{1},J V_{1}\\right).\n\t\\end{align*}\n\tConsidering Theorem 3.1 we arrive\n\t\\begin{align*}\n\t\\sin^{2}\\theta g_{1}\\left([U_{1},U_{2}],V_{1}\\right)\n\t=&-g_{1}\\left(\\nabla_{U_{1}}\\eta\\xi U_{2},V_{1}\\right)+g_{1}\\left(\\nabla_{U_{1}}\\eta U_{2}, JV_{1}\\right)\\\\&+g_{1}\\left(\\nabla_{U_{2}}\\eta\\xi U_{1},V_{1}\\right)-g_{1}\\left(\\nabla_{U_{2}}\\eta U_{1},JV_{1}\\right).\n\t\\end{align*}\n\tBy using the equation \\eqref{2.5} we obtain\n\t\\begin{align*}\n\t\\sin^{2}\\theta g_{1}\\left([U_{1},U_{2}],V_{1}\\right)\n\t=&g_{1}\\left(\\mathcal{T}_{U_{2}}\\eta\\xi U_{1}-\\mathcal{T}_{U_{1}}\\eta\\xi U_{2},V_{1}\\right)+g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta U_{2}-\\mathcal{T}_{U_{2}}\\eta U_{1},\\xi V_{1}\\right)\\\\&-\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{1},\\eta U_{2}\\right),f_{*}\\eta V_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{2},\\eta U_{1}\\right),f_{*}\\eta V_{1}\\right).\n\t\\end{align*}\n\tThe proof of $ii)$ can be made by applying similar calculations.\n\\end{proof}\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$. Then the distribution $D$ defines a totally geodesic foliation if and only if\n\t\\small{\n\t\\begin{align}\n\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(\\eta U_{2},U_{1}\\right),f_{*}\\eta V_{1}\\right)=-g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta\\xi U_{2},V_{1}\\right)+g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta U_{2},\\xi V_{1}\\right).\n\\end{align}}\\normalsize\n\tand\n\t\\begin{align}\n\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta U_{1},f_{*}\\eta U_{2}\\right)=&-\\sin^{2}\\theta g_{1}\\left(\\left[U_{1},X_{1}\\right],U_{1}\\right)+g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta\\xi U_{1},U_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad(\\ln\\lambda),X_{1}\\right)g_{1}\\left(\\eta U_{1},\\eta U_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad(\\ln\\lambda),\\eta U_{1}\\right)g_{1}\\left(X_{1},\\eta U_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad(\\ln\\lambda),\\eta U_{2}\\right)g_{1}\\left(X_{1},\\eta U_{1}\\right)\\nonumber\\\\&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta U_{1},\\xi U_{2}\\right)\n\t\\end{align}\n\twhere $U_{1},U_{2}\\in \\Gamma\\left(D\\right)$, $V_{1}\\in \\Gamma\\left(\\bar{D}\\right)$ and $X_{1}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$.\n\\end{theorem}\n\n\\begin{proof}\n\t\n\tFor $U_{1},U_{2}\\in \\Gamma\\left(D\\right)$ and $V_{1}\\in\\Gamma\\left(\\bar{D}\\right)$ we have\n\t\\begin{align*}\n\tg_{1}\\left(\\nabla_{U_{1}}U_{2},V_{1}\\right)=&-g_{1}\\left(\\nabla_{U_{1}}\\xi^{2}U_{2},V_{1}\\right)-g_{1}\\left(\\nabla_{U_{1}}\\eta\\xi U_{2},V_{1}\\right)+g_{1}\\left(\\nabla_{U_{1}}\\eta U_{2},JV_{1}\\right).\n\t\\end{align*}\n\tThus we can write\n\t\\begin{align*}\n\t\\sin^{2}\\theta g_{1}\\left(\\nabla_{U_{1}}U_{2},V_{1}\\right)=&-g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta\\xi U_{2},V_{1}\\right)+g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta U_{2},\\xi V_{1}\\right)\\\\&+g_{1}\\left(\\mathcal{H}\\nabla_{U_{1}}\\eta U_{2},\\eta V_{1}\\right).\n\t\\end{align*}\n\tUsing \\eqref{2.9} we obtain \n\t\\begin{align*}\n\t\\sin^{2}\\theta g_{1}\\left(\\nabla_{U_{1}}U_{2},V_{1}\\right)=&-g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta\\xi U_{2},V_{1}\\right)+g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta U_{2},\\xi V_{1}\\right)\\\\&-\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(\\eta U_{2},U_{1}\\right),f_{*}\\eta V_{1}\\right).\n\t\\end{align*}\n\twhich is first equation in Theorem 3.3.\n\t\n\tOn the other hand any $U_{1},U_{2} \\in \\Gamma(D) $ and $X_{1}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$ we can write\n\t\\setlength\\arraycolsep{2pt}\n\t\\begin{align*}\n\tg_{1}\\left(\\nabla_{U_{1}}U_{2},X_{1}\\right)=&-g_{1}\\left(\\left[U_{1},X_{1}\\right],U_{2}\\right)-g_{1}\\left(\\nabla_{X_{1}}U_{1},U_{2}\\right)\\\\\n\t=&-g_{1}\\left(\\left[U_{1},X_{1}\\right],U_{2}\\right)+g_{1}\\left(\\nabla_{X}J\\xi U_{1},U_{2}\\right)-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},J U_{2}\\right).\n\t\\end{align*}\n\tUsing Theorem 3.1, we arrive following equation\n\t\\begin{align*}\n\tg_{1}\\left(\\nabla_{U_{1}}U_{2},X_{1}\\right)=&-g_{1}\\left(\\left[U_{1},X_{1}\\right],U_{2}\\right)-\\cos^{2}\\theta g_{1}\\left(\\nabla_{X_{1}}U_{1},U_{2}\\right)\\\\&+g_{1}\\left(\\nabla_{X_{1}}\\eta\\xi U_{1},U_{2}\\right)-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},JU_{2}\\right)\\nonumber\n\t\\end{align*}\n\tFrom \\eqref{2.7} and Lemma 2.1 we have\n\t\\begin{align*}\n\t\\sin^2\\theta g_{1}\\left(\\nabla_{U_{1}}U_{2},X_{1}\\right)=&-\\sin^{2}\\theta g_{1}\\left(\\left[U_{1},X_{1}\\right],U_{1}\\right)+g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta\\xi U_{1},U_{2}\\right)\\\\&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta U_{1},\\xi U_{2}\\right)-\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta U_{1},f_{*}\\eta U_{2}\\right)\\\\&+g_{1}\\left(grad(\\ln\\lambda),X_{1}\\right)g_{1}\\left(\\eta U_{1},\\eta U_{2}\\right)\\\\&+g_{1}\\left(grad(\\ln\\lambda),\\eta U_{1}\\right)g_{1}\\left(X_{1},\\eta U_{2}\\right)\\\\&+g_{1}\\left(grad(\\ln\\lambda),\\eta U_{2}\\right)g_{1}\\left(X_{1},\\eta U_{1}\\right)\\nonumber\n\t\\end{align*}\n\tThis completes the proof.\n\\end{proof}\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$. Then the distribution $\\bar{D}$ defines a totally geodesic foliation if and only if\n\t\\small{\n\t\\begin{align}\n\t\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(\\eta V_{2},V_{1}\\right),f_{*}\\eta U_{1}\\right)=-g_{1}\\left(\\mathcal{T}_{V_{1}}\\eta\\xi V_{2},U_{1}\\right)+g_{1}\\left(\\mathcal{T}_{V_{1}}\\eta V_{2},\\xi U_{1}\\right).\n\t\\end{align}}\\normalsize\n\tand\n\t\\begin{align}\n\t\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta V_{1},f_{*}\\eta V_{2}\\right)=&-\\sin^{2}\\bar{\\theta} g_{1}\\left(\\left[V_{1},X_{1}\\right],V_{1}\\right)+g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta\\xi V_{1},V_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad(\\ln\\lambda),X_{1}\\right)g_{1}\\left(\\eta V_{1},\\eta V_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad(\\ln\\lambda),\\eta V_{1}\\right)g_{1}\\left(X_{1},\\eta V_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad(\\ln\\lambda),\\eta V_{2}\\right)g_{1}\\left(X_{1},\\eta V_{1}\\right)\\nonumber\\\\&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta V_{1},\\xi V_{2}\\right)\n\t\\end{align}\n\twhere $U_{1}\\in \\Gamma\\left(D\\right)$, $V_{1}, V_{2}\\in \\Gamma\\left(\\bar{D}\\right)$ and $X_{1}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$.\n\\end{theorem}\n\n\\begin{proof}\n\tThe proof of this theorem is similar to the proof of Theorem 3.3.\n\\end{proof}\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$.Then, the vertical distribution $\\left(\\ker f_{*}\\right)$ is a locally product $M_{D}\\times M_{\\bar{D}}$ if and only if the equations (3.4), (3.5), (3.6) and (3.7) are hold where $M_{D}$ and $M_{\\bar{D}}$ are integral manifolds of the distributions $D$ and $\\bar{D}$, respectively. \t\n\\end{theorem}\n\n\\begin{theorem}\nSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$. Then the distribution $\\left(\\ker f_{*}\\right)^\\perp$ defines a totally geodesic foliation if and only if\n\\begin{align}\n\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta U_{1},f_{*}CX_{2}\\right)=&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta U_{1},BX_{2}\\right)\n+\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta\\xi U_{1},f_{*}X_{2}\\right)\\nonumber\\\\&-g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta\\xi U_{1},X_{2}\\right)\\nonumber\\\\&-g_{1}\\left(grad\\ln\\lambda,\\eta\\xi U_{1}\\right)g_{1}\\left(X_{1},X_{2}\\right)\\nonumber\\\\&+g_{1}\\left(X_{1},\\eta\\xi U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,X_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta U_{1}\\right)g_{1}\\left(X_{1},CX_{2}\\right)\\nonumber\\\\&-g_{1}\\left(X_{1},\\eta U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,CX_{2}\\right).\n\\end{align}\nand\n\\begin{align}\n\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta V_{1},f_{*}CX_{2}\\right)=&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta V_{1},BX_{2}\\right)\n+\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta\\xi V_{1},f_{*}X_{2}\\right)\\nonumber\\\\&-g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta\\xi V_{1},X_{2}\\right)\\nonumber\\\\&-g_{1}\\left(grad\\ln\\lambda,\\eta\\xi V_{1}\\right)g_{1}\\left(X_{1},X_{2}\\right)\\nonumber\\\\&+g_{1}\\left(X_{1},\\eta\\xi V_{1}\\right)g_{1}\\left(grad\\ln\\lambda,X_{2}\\right)\\nonumber\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta V_{1}\\right)g_{1}\\left(X_{1},CX_{2}\\right)\\nonumber\\\\&-g_{1}\\left(X_{1},\\eta V_{1}\\right)g_{1}\\left(grad\\ln\\lambda,CX_{2}\\right).\n\\end{align}\n\twhere $X_{1}, X_{2} \\in\\Gamma\\left(\\ker f_{*}\\right)^\\perp$, $U_{1}\\in\\Gamma\\left(D\\right)$ and $V_{1}\\in\\Gamma\\left(\\bar{D}\\right)$.\n\\end{theorem}\n\n\\begin{proof}\n\tFor $X_{1},X_{2} \\in\\Gamma\\left(\\ker \\pi_{*}\\right)^\\perp$ and $U_{1}\\in\\Gamma\\left(D\\right)$ we can write\n\t\\setlength\\arraycolsep{2pt}\n\t\\begin{eqnarray*}\n\tg_{1}\\left(\\nabla_{X_{1}}X_{2},U_{1}\\right)=-g_{1}\\left(\\nabla_{X_{1}}\\xi U_{1}, JX_{2}\\right)-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},JX_{2}\\right)\n\t\\end{eqnarray*}\n\tFrom Theorem 3.1 we have\n\t\\begin{align*}\n\tg_{1}\\left(\\nabla_{X_{1}}X_{2},U_{1}\\right)=&-\\cos^{2}\\theta g_{1}\\left(\\nabla_{X_{1}}U_{1},X_{2}\\right)+g_{1}\\left(\\nabla_{X_{1}}\\eta\\xi U_{1},X_{2}\\right)\\\\&-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},JX_{2}\\right)\n\t\\end{align*}\n\tBy using the equation (2.7) we derive\n\t\\begin{align*}\n\t\\sin^{2}\\theta g\\left(\\nabla_{X_{1}}X_{2},U_{1}\\right)=&g\\left(\\mathcal{H}\\nabla_{X_{1}}\\eta\\xi U_{1},X_{2}\\right)\n\t-g\\left(\\mathcal{H}\\nabla_{X_{1}}\\eta U_{1},CX_{2}\\right)\\\\&-g\\left(\\nabla_{X_{1}}\\eta U_{1},BX_{2}\\right).\n\t\\end{align*}\n\tThen it follows from Lemma 2.1 that\n\t\\begin{align*}\n\t\\sin^{2}\\theta g_{1}\\left(\\nabla_{X_{1}}X_{2},U_{1}\\right)=&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta U_{1},BX_{2}\\right)\n\t+\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta\\xi U_{1},f_{*}X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta\\xi U_{1},X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,\\eta\\xi U_{1}\\right)g_{1}\\left(X_{1},X_{2}\\right)\\\\&+g_{1}\\left(X_{1},\\eta\\xi U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,X_{2}\\right)\\\\&-\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta U_{1},f_{*}CX_{2}\\right)\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta U_{1}\\right)g_{1}\\left(X_{1},CX_{2}\\right)\\\\&-g_{1}\\left(X_{1},\\eta U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,CX_{2}\\right).\n\t\\end{align*}\n\tThus we have the first desired equation.\n\tSimilarly for $X_{1},X_{2} \\in\\Gamma\\left(\\left(\\ker \\pi_{*}\\right)^\\perp\\right)$ and $V_{1}\\in\\left(\\bar{D}\\right)$ we find\n\t\\begin{align*}\n\t\\sin^{2}\\bar{\\theta} g_{1}\\left(\\nabla_{X_{1}}X_{2},V_{1}\\right)=&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\eta V_{1},BX_{2}\\right)\n+\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta\\xi V_{1},f_{*}X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta\\xi V_{1},X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,\\eta\\xi V_{1}\\right)g_{1}\\left(X_{1},X_{2}\\right)\\\\&+g_{1}\\left(X_{1},\\eta\\xi V_{1}\\right)g_{1}\\left(grad\\ln\\lambda,X_{2}\\right)\\\\&-\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta V_{1},f_{*}CX_{2}\\right)\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta V_{1}\\right)g_{1}\\left(X_{1},CX_{2}\\right)\\\\&-g_{1}\\left(X_{1},\\eta V_{1}\\right)g_{1}\\left(grad\\ln\\lambda,CX_{2}\\right).\n\t\\end{align*}\n\tHence the proof is completed.\n\\end{proof}\n\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$.Then the distribution $\\left(\\ker f_{*}\\right)$ defines a totally geodesic foliation on $M_{1}\n\t$ if and only if\n\t\t\\setlength\\arraycolsep{2pt}\n\t\t\\small{\n\t\\begin{align}\n\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\omega U_{1},f_{*}\\omega V_{1}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right)g_{1}\\left(\\nabla_{X_{1}}QU_{1},V_{1}\\right)-g_{1}\\left(\\mathcal{A}_{X_{1}}V_{1},\\eta\\xi U_{1}\\right)\\nonumber\\\\&-g_{1}\\left(\\mathcal{A}_{X_{1}}\\xi V_{1},\\eta U_{1}\\right)-\\sin^{2}\\theta g_{1}\\left(\\left[U_{1},X_{1}\\right],V_{1}\\right)\\nonumber\\\\&-g_{1}\\left(X_{1},\\eta U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,\\eta V_{1}\\right)\\nonumber\\\\&+g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta U_{1},\\eta V_{1}\\right)\\nonumber\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta U_{1}\\right)g_{1}\\left(X_{1},\\eta V_{1}\\right)\n\\end{align}}\\normalsize\n\twhere $X_{1}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^\\perp\\right)$ and $U_{1},V_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)$.\n\\end{theorem}\n\n\\begin{proof}\n\tGiven $X_{1} \\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^\\perp\\right)$ and $U_{1},V_{1}\\in\\left(\\ker f_{*}\\right)$. Then we obtain\n\t{\\small\n\t\t\\begin{align*}\n\t\tg_{1}\\left(\\nabla_{U_{1}}V_{1},X_{1}\\right)=&-g_{1}\\left(\\left[U_{1},X_{1}\\right],V_{1}\\right)+g_{1}\\left(J\\nabla_{X_{1}}\\xi U_{1},V_{1}\\right)-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},JV_{1}\\right)\n\t\t\\end{align*}}\n\tBy using Theorem 3.1 we have\n\t\\begin{align*}\n\tg_{1}\\left(\\nabla_{U_{1}}V_{1},X_{1}\\right)=&-g_{1}\\left(\\left[U_{1},X_{1}\\right],V_{1}\\right)-\\cos^{2}\\theta g_{1}\\left(\\nabla_{X_{1}} PU_{1},V_{1}\\right)\\\\&-\\cos^{2}\\bar{\\theta}g_{1}\\left(\\nabla_{X} QU_{1},V_{1}\\right)+g_{1}\\left(\\nabla_{X_{1}}\\eta\\xi U_{1},V_{1}\\right)\\\\&-g_{1}\\left(\\nabla_{X_{1}}\\omega U_{1},\\xi V_{1}\\right)-g_{1}\\left(\\nabla_{X_{1}}\\omega U_{1},\\eta V_{1}\\right).\n\t\\end{align*}\n Then we arrive\n\t\\begin{align*}\n\t\\sin^{2}\\theta g_{1}\\left(\\nabla_{U_{1}}V_{1},X_{1}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right)g_{1}\\left(\\nabla_{X_{1}}QU_{1},V_{1}\\right)\\\\&+g_{1}\\left(\\nabla_{X_{1}}\\eta\\xi U_{1},V_{1}\\right)-\\sin^{2}\\theta g_{1}\\left(\\left[U_{1},X_{1}\\right],V_{1}\\right)\\\\&-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},\\xi V_{1}\\right)-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},\\eta V_{1}\\right)\n\t\\end{align*}\n\tFrom the equation \\eqref{2.6} and Lemma 2.1 we obtain\n\t\t\\begin{align*}\n\t\\sin^{2}\\theta g_{1}\\left(\\nabla_{U_{1}}V_{1},X_{1}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right)g_{1}\\left(\\nabla_{X_{1}}QU_{1},V_{1}\\right)-g_{1}\\left(\\mathcal{A}_{X_{1}}V_{1},\\eta\\xi U_{1}\\right)\\\\&-\\sin^{2}\\theta g_{1}\\left(\\left[U_{1},X_{1}\\right],V_{1}\\right)-g_{1}\\left(\\mathcal{A}_{X_{1}}\\xi V_{1},\\eta U_{1}\\right)\\\\&+g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta U_{1},\\eta V_{1}\\right)\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta U_{1}\\right)g_{1}\\left(X_{1},\\eta V_{1}\\right)\\\\&-g_{1}\\left(X_{1},\\eta U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,\\eta V_{1}\\right)\\\\&-\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{X_{1}}f_{*}\\eta U_{1},f_{*}\\eta V_{1}\\right)\n\t\\end{align*}\n\tUsing above equation the desired equality is achieved.\n\\end{proof}\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$. Then, the total space $M_{1}$ is a locally product $M_{1D}\\times M_{\\bar{1D}}\\times M_{1\\left(\\ker f_{*}\\right)^{\\perp}} $ if and only if the equations (3.4), (3.5), (3.6), (3.7), (3.8) and (3.9) are hold where $M_{1D}$, $M_{1\\bar{D}}$ and $M_{1\\left(\\ker f_{*}\\right)^{\\perp}}$ are integral manifolds of the distributions $D$, $\\bar{D}$ and $\\left(\\ker f_{*}\\right)^{\\perp}$, respectively. \t\n\\end{theorem}\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$. Then, the total space $M_{1}$ is a locally product $M_{1\\ker f_{*}}\\times M_{1\\left(\\ker f_{*}\\right)^{\\perp}} $ if and only if the equations (3.8), (3.9) and (3.10)are hold where $M_{1\\ker f_{*}}$ and $M_{1\\left(\\ker f_{*}\\right)^{\\perp}}$ are integral manifolds of the distributions $\\ker f_{*}$ and $\\left(\\ker f_{*}\\right)^{\\perp}$, respectively. \t\n\\end{theorem}\n\n\\begin{theorem}\n\tSuppose that $f$ is a proper conformal bi-slant submersion from a Kaehlerian manifold $(M_{1},g_{1},J)$ onto a Riemannian manifold $(M_{2},g_{2})$ with slant functions $\\theta,\\bar{\\theta}$. Then $f$ is totally geodesic if and only if\n\n\\begin{align*}\n-\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{\\eta V_{1}}f_{*}\\eta U_{1},f_{*}JCX_{1}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right) g_{1}\\left(\\mathcal{T}_{U_{1}}QV_{1},X_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(\\xi U_{1},\\eta V_{1}\\right),f_{*} JCX_{1}\\right)\\\\&-g_{1}\\left(\\eta U_{1},\\eta V_{1}\\right)g_{1}\\left(grad\\ln\\lambda,JCX_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{1},\\eta\\xi V_{1}\\right),f_{*}X_{1}\\right)\\\\&-g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta V_{1},BX_{1}\\right)\n\\end{align*}\n\tand\n\t\\begin{align*}\n\\lambda^{-2}g_{2}\\left(\\nabla_{X_{1}}^{f}f_{*}\\eta U_{1},f_{*}CX_{2}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right)g_{1}\\left(\\mathcal{A}_{X_{1}}QU_{1},X_{2}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla_{X_{1}}^{f}f_{*}\\eta \\xi U_{1},f_{*}X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta\\xi U_{1},X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,\\eta\\xi U_{1}\\right)g_{1}\\left(X_{1},X_{2}\\right)\\\\&+g_{1}\\left(X_{1},\\eta\\xi U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,X_{2}\\right)\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta U_{1}\\right)g_{1}\\left(X_{1},CX_{2}\\right)\\\\&-g_{1}\\left(X_{1},\\eta U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,CX_{2}\\right)\\\\&+g_{1}\\left(\\mathcal{A}_{X_{1}}BX_{2},\\eta U\\right).\n\t\\end{align*}\n\n\twhere $X_{1},X_{2}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^\\perp\\right)$ and $U_{1},V_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)$.\n\\end{theorem}\n\n\\begin{proof}\nGiven $U_{1},V_{1}\\in \\Gamma\\left(\\ker f_{*}\\right)$ and $X_{1}\\in\\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$ Then we write\n\t\\begin{equation*}\n\t\\lambda^{-2}g_{2}\\left(\\nabla f_{*}(U_{1},V_{1}),f_{*}X\\right)=-\\lambda^{-2}g_{2}\\left(f_{*}\\nabla_{U_{1}}V_{1},f_{*}X\\right).\n\t\\end{equation*}\n\tFrom Theorem 3.1 we obtain\n\t\\setlength\\arraycolsep{2pt}\n\t\\begin{align*}\n\\left(\\sin^{2}\\theta\\right) \\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{1},V_{1}\\right),f_{*}X_{1}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right) g_{1}\\left(\\nabla_{U_{1}}QV_{1},X_{1}\\right)\\\\&-g_{1}\\left(\\nabla_{U_{1}}\\eta V_{1},JX_{1}\\right)+g_{1}\\left(\\nabla_{U_{1}}\\eta\\xi V_{1},X_{1}\\right)\n\t\\end{align*}\n\tConsidering \\eqref{2.4}, \\eqref{2.5} and Lemma 2.1 we find \n\t\\begin{align*}\n\t\\left(\\sin^{2}\\theta\\right) \\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{1},V_{1}\\right),f_{*}X_{1}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right) g_{1}\\left(\\mathcal{T}_{U_{1}}QV_{1},X_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(\\xi U_{1},\\eta V_{1}\\right),f_{*} JCX_{1}\\right)\\\\&-g_{1}\\left(\\eta U_{1},\\eta V_{1}\\right)g_{1}\\left(grad\\ln\\lambda,JCX_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla^{f}_{\\eta V_{1}}f_{*}\\eta U_{1},f_{*}JCX_{1}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{1},\\eta\\xi V_{1}\\right),f_{*}X_{1}\\right)\\\\&-g_{1}\\left(\\mathcal{T}_{U_{1}}\\eta V_{1},BX_{1}\\right).\n\t\\end{align*}\n\tTherefore we obtain the first equation of Theorem 3.6.\\\\\n\tOn the other hand, for $X_{1},X_{2}\\in \\Gamma\\left(\\left(\\ker f_{*}\\right)^{\\perp}\\right)$ and $U_{1}\\in\\Gamma\\left(\\ker f_{*}\\right)$ we can write\n\t\\begin{align*}\n\t\\left(\\sin^{2}\\theta\\right) \\lambda^{-2} g_{2}\\left(\\nabla f_{*}\\left(U_{1},X_{1}\\right),f_{*}X_{2}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right)g_{1}\\left(\\nabla_{X_{1}}QU_{1},X_{2}\\right)\\\\&+g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},BX_{2}\\right)-g_{1}\\left(\\nabla_{X_{1}}\\eta U_{1},CX_{2}\\right).\n\t\\end{align*}\n\tBy using the equation (2.6) and Lemma 2.1, we arrive\n\t\\setlength\\arraycolsep{2pt}\n\t\\begin{align*}\n\t\\left(\\sin^{2}\\theta\\right) \\lambda^{-2}g_{2}\\left(\\nabla f_{*}\\left(U_{1},X_{1}\\right),f_{*}X_{2}\\right)=&\\left(\\cos^{2}\\theta-\\cos^{2}\\bar{\\theta}\\right)g_{1}\\left(\\mathcal{A}_{X_{1}}QU_{1},X_{2}\\right)\\\\&+\\lambda^{-2}g_{2}\\left(\\nabla_{X_{1}}^{f}f_{*}\\eta \\xi U_{1},f_{*}X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,X_{1}\\right)g_{1}\\left(\\eta\\xi U_{1},X_{2}\\right)\\\\&-g_{1}\\left(grad\\ln\\lambda,\\eta\\xi U_{1}\\right)g_{1}\\left(X_{1},X_{2}\\right)\\\\&+g_{1}\\left(X_{1},\\eta\\xi U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,X_{2}\\right)\\\\&-\\lambda^{-2}g_{2}\\left(\\nabla_{X_{1}}^{f}f_{*}\\eta U_{1},f_{*}CX_{2}\\right)\\\\&+g_{1}\\left(grad\\ln\\lambda,\\eta U_{1}\\right)g_{1}\\left(X_{1},CX_{2}\\right)\\\\&-g_{1}\\left(X_{1},\\eta U_{1}\\right)g_{1}\\left(grad\\ln\\lambda,CX_{2}\\right)\\\\&+g_{1}\\left(\\mathcal{A}_{X_{1}}BX_{2},\\eta U_{1}\\right).\n\t\\end{align*}\n\tThis concludes the proof.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nPlanetary systems are formed from rotating protoplanetary disks, which are\nthe evolved phase of circumstellar disks produced during the collapse of\na protostellar cloud with some angular momentum.\n\nA standard model of such a protoplanetary disk, is that\nof a steady-state disk in vertical hydrostatic equilibrium, with gas and \ndust fully mixed and thermally coupled (Kenyon \\& Hartmann 1987). Such\na disk is flared, not flat, but still geometrically thin in the sense defined\nby Pringle (1981). The disk intercepts a significant\namount of radiation from the central star, but other heating sources (e.g.\nviscous dissipation) can be more important. If dissipation due to mass\naccretion is high, it becomes the main source of heating. Such are the\nprotoplanetary disks envisioned by Boss (1996, 1998), which have relatively\nhot (midplane temperature $T_{\\rm m}>$ 1200~K) inner regions due to mass\naccretion rates of $\\sim 10^{-6}$ to $10^{-5} M_{\\odot} {\\rm yr}^{-1}$.\nHowever, typical T Tauri disks of age $\\sim$1~Myr\nseem to have much lower mass accretion rates\n($\\leq 10^{-8} M_{\\odot} {\\rm yr}^{-1}$) with all other characteristics\nof protoplanetary disks (Hartmann et al. 1998, D'Alessio et al. 1998).\nFor disks of such low accretion rates stellar irradiation becomes increasingly\nthe dominant source of heating, to the limit of a passive disk modeled by\nChiang \\&\nGoldreich (1997). In our paper we will confine our attention to the latter\ncase, without entering the discussion about mass accretion rates.\n\nThe optical depth in the midplane of the disk is very high in the radial\ndirection, hence the temperature structure there is governed by the\nreprocessed irradiation of the disk surface. This is the case of a passive\ndisk (no accretion). At some point along the radial direction the temperature in the\nmidplane would drop below the ice sublimation level -- Hayashi (1981)\ncalled it the \"snow line\". \n\nIn this paper we revisit the calculation of the \"snow line\" for a protosolar\nprotoplanetry disk, given its special role in the process of planet formation.\nWe pay particular emphasis on the issues involved in treating the radiative\ntransfer and the dust properties.\n\n\\section{The Model} \nOur model is that of a star surrounded by a flared disk. In this paper, we\nhave chosen two examples -- a passive disk and a disk with a \n$10^{-8} M_{\\odot} {\\rm yr}^{-1}$ accretion rate. Both have the same \ncentral star of effective\ntemperature $T_*$= 4000~K, mass $M_*$= 0.5~$M_{\\odot}$, and radius\n$R_*$= 2.5~$R_{\\odot}$. Thus they correspond to the examples used by\nChiang \\& Goldreich (1997) and D'Alessio et al. (1998), respectively.\nOur disk has a surface gas mass density\n$\\Sigma$= $r^{-3\/2}{\\Sigma}_0$, with $r$ in AU and \n${\\Sigma}_0$= 10$^3$g~cm$^{-2}$ for our standard minimum-mass solar\nnebula model; we varied ${\\Sigma}_0$ between 10$^2$ and 10$^4$g~cm$^{-2}$\nto explore the effect of disk mass on the results.\n\nThe emergent\nspectrum of the star is calculated with a stellar model atmosphere code\nwith Kurucz (1992) line lists and opacities. The disk intercepts the\nstellar radiation $F_{irr}(r)$ at a small grazing angle $\\phi(r)$ (defined in \\S 5).\nThe emergent stellar spectrum\nis input into a code which solves the continuum radiative transfer problem\nfor a dusty envelope. The solution is a general spherical geometry solution\nwith a modification of the equations to a section corresponding to a flared\ndisk (see Menshchikov \\& Henning (1997) for a similar approach). \nIn that sense, the radiative transfer is solved essentially in 1D (vertically),\nas opposed to a full-scale consistent 2D case. The appeal of our approach\nis in the detailed radiative transfer allowed by the 1D scheme.\n\nThe continuum radiative transfer problem for a dusty envelope is solved\nwith the method developed by Ivezi\\'c \\& Elitzur (1997).\nThe scale invariance applied in the method is\npractically useful when the absorption coefficient is independent of\nintensity, which is the case of dust continuum radiation. The energy\ndensity is computed for all radial grid points through matrix inversion,\ni.e. a direct solution to the full scattering problem. This is both very\nfast and accurate at high optical depths.\nNote that in our calculations (at $r \\geq 0.1$~AU) the temperatures never\nexceed 1500-1800K in the disk and we do not consider dust sublimation;\nthe dust is present at all times and is the dominant opacity source. As in\nthe detailed work by Calvet et al. (1991) and more recently by D'Alessio\net al. (1998), the frequency ranges of scattering and emission can be treated\nseparately.\n\nFor the disk with mass accretion, the energy rate per unit volume\ngenerated locally by viscous stress is given by\n$2.25 \\alpha P(z){\\Omega}(r)$, where the turbulent viscosity coefficient\nis ${\\nu}= {\\alpha}c_{\\rm s}^2\\Omega^{-1}$, $\\Omega$ is the Keplerian angular\nvelocity, ${c_{\\rm s}}^2=P{\\rho}^{-1}$ is the sound speed, and a standard value\nfor $\\alpha =0.01$ is used. The net flux produced by viscous dissipation, $F_{vis}$, is\nthe only term to balance $F_{rad}$ $-$ unlike D'Alessio et al. (1998) we\nhave ignored the flux produced by energetic particles ionization.\nThen we have the standard relation (see Bell et al. 1997), which holds true for the interior of the\ndisk where accretion heating occurs:\n$$\n\\sigma T_{vis}^4 = {{3\\dot{M} GM_*}\\over {8\\pi r^3}}\\left[1-({R_*\\over r})^{1\/2}\\right] ,\n$$\nwhere $\\dot{M}$ ~is the mass accretion rate, and $M_*$ and $R_*$ are the stellar mass\nand radius.\n\n\\section{The Dust}\n\nThe properties of the dust affect the wavelength dependence of scattering\nand absorption efficiencies. The temperature in the midplane is sensitive to\nthe dust scattering albedo (ratio of scattering to total opacity) -- a higher \nalbedo would reduce the absorbed stellar flux.\nAs with our choice of mass accretion rates, we will use dust grains with\nproperties which best describe the disks of T Tauri stars. \n\nThe modelling of circumstellar disks has always applied dust grain \nproperties derived from the interstellar medium. Most commonly used have\nbeen the grain parameters of the Mathis et al. (1977) distribution with\noptical constants from Draine \\& Lee (1984). However, recent work on\nspectral distributions (Whitney et al. 1997) and high-resolution images\n(Wood et al. 1998) of T Tauri stars has favored a grain mixture which\nKim, Martin, \\& Hendry (1994) derived from detailed fits to the\ninterstellar extinction law (hereafter KMH). Important grain properties\nare the opacity, $\\kappa$, the scattering albedo, $\\omega$, \nand the scattering asymmetry parameter, $g$.\nThe latter defines the forward throwing properties of the\ndust and ranges from 0 (isotropic scattering) to 1. What sets the KMH grains\napart is that they are more forward throwing ($g$= 0.40($R$), 0.25($K$)), \nand have higher albedo ($\\omega$= 0.50($R$), 0.36($K$)) at\neach wavelength (optical to near-IR). They are also less polarized, but\nthat is a property we do not use here. The grain size distribution has the\nlower cutoff of KMH (0.005$\\mu$m) and a smooth exponential falloff, instead\nof an upper cutoff, at 0.25$\\mu$m. Since the dust settling time is\nproportional to (size)$^{-1}$, we performed calculations with upper\ncutoffs of 0.05$\\mu$m and 0.1$\\mu$m. None of these had any significant\neffect on the temperatures.\n\n\\section{The Temperature Structure and the Snow Line}\nWe are interested in planet formation and therefore want to find the\nice condensation line (\"snow line\") in the midplane of the disk. \nTemperature inversions in the disk's vertical structure (see D'Alessio et al. \n1998) may lead to lower temperatures above the midplane, but ice condensation\nthere is quickly destroyed upon crossing the warmer disk plane.\nWe define the snow line simply in terms of the local gas-dust temperature\nin the midplane, and at a value of 170~K.\n\nIn our passive disk, under hydrostatic and radiative equilibrium;\nthe vertical and radial temperature profiles are similar to those of\nChiang \\& Goldreich (1997) and $T(r) \\propto r^{-3\/7}$. Here is why.\nThe disk has a concave upper surface (see\nHartmann \\& Kenyon 1987) with pressure scale height of the gas at the\nmidplane temperature, $h$:\n$$\n{{h}\\over r} = \\left[{rkT}\\over {GM_* \\mu m_H}\\right] ^{1\/2},\n$$\nwhere $G$ is the gravitational constant, $\\mu$ and $m_H$ are the molecular\nweight and hydrogen mass,\n$r$ is radius in the disk, and $T$ is the midplane temperature at that\nradius. For the inner region (but $r \\gg R_*$) of a disk with such concave\nshape the stellar\nincident flux $F_{irr}(r) \\propto \\phi(r) \\sigma T_*^4 r^{-2}$, where\n$\\phi(r) \\propto r^{2\/7}$ (see next section). Here $T_*$ is the effective\ntemperature of the central star. Then our calculation makes use of\nthe balance between heating by irradiation and radiative cooling:\n$\\sigma T^4(r) = F_{irr}(r)$. Therefore our midplane temperature will scale\nas $T(r)$= $T_0 r^{-3\/7}$~K. This is not surprising, given our standard\ntreatment of the vertical hydrostatic structure of the disk irradiated\nat angles $\\phi(r)$. Only the scaling coefficient, $T_0 = 140$, will be\ndifferent.\nThe difference with the Chiang \\& Goldreich model is our treatment \nof the dust\ngrains $-$ less energy is redistributed inwards in our calculation\nand the midplane temperature is lower (Figure 1). The model\nwith accretion heating is much warmer inwards of 2.5$AU$ where it joins\nthe no-accretion (passive) model $-$ stellar irradiation dominates.\n\nThe result above is for our model with $M_*=0.5M_{\\odot}$ and $T_*=4000$~K,\nwhich is standard for T~Tauri stars. It is interesting to see how the snow\nline changes for other realistic initial parameters. By retaining the same\ndust properties, this can be achieved using scaling relations rather than\ncomplete individual models as shown by Bell et al. (1997) for the pre-main\nsequence mass range of 0.5 to 2~$M_{\\odot}$. An important assumption at\nthis point is that we have still retained the same (minimum-mass solar nebula)\ndisk. With our set of equations,\nthe midplane temperature coefficient, $T_0$, will be proportional to the stellar mass:\n$T_0 \\propto M_*^{3\/(10-k)}$, where $k$ is a function of the total opacity in\nthe disk and $0 \\leq k < 2$.\n\nOn the other hand, different disk masses for a fixed central star \n($M_*=0.5M_{\\odot}$ and $T_*=4000$~K) can be modeled for zero accretion\nrate, by changing $\\Sigma_0$ by a factor of 10 in each direction (defined\nin \\S 2). Here a remaining assumption is the radial dependence of\n$\\Sigma \\propto r^{-3\/2}$; the latter could certainly be $\\propto r^{-1}$\n(Cameron 1995), or a more complex function of $r$, but it is beyond the\nintent of our paper to deal with this. Moreover that we find minimal change\nin the midplane temperature in the $r$ range of interest to us (0.1$-$5~AU).\nThe reason is a near cancellation that occurs between the amount of heating\nand increased optical depth to the midplane. One could visualize the vertical\nstructure of a passive disk for $r = 0.1-5.0$~AU as consisting of three zones:\n(1)~optically thin heating and cooling region (dust heated by direct starlight),\n(2)~optically thin cooling, but optically thick heating layer, and (3)~the\nmidplane zone, where both heating and cooling occur in optically thick\nconditions. The rate of stellar heating of the disk per unit volume is\ndirectly proportional to the density, and affects the location and temperature\nof the irradiation layer. That is nearly cancelled (except for second order\nterms) in the mean intensity which reaches the midplane. Therefore we find\nthat $T(r)$ changes within $\\pm 10K$ for a change in disk density ($\\Sigma_0$)\nof a factor of 10. Note that $T(r)$ is only approximately $\\propto r^{-3\/7}$\neven for $r = 0.1-5.0$~AU; the small effect of density on $T(r)$ has an\n$r$ dependence. However, for the purposes of this paper, $i.e.$ our chosen\nvolume of parameter space, the effect of disk mass on $T(r)$ and the ``snow line\" \nis insignificant, and we do not pursue the issue in more detail.\nNote that for a disk with a heat source in the midplane, $i.e.$ with an\naccretion rate different from zero, the midplane $T(r)$ will be strongly\ncoupled to the density, roughly $\\propto {\\rho}^{1\/4}$, and will increase\nat every $r$ for higher disk masses (e.g. Lin \\& Papaloizou 1985).\n\n\\section{The Shape of the Upper Surface of the Disk}\n\nThe \"snow line\" calculation in the previous section is made under the\nassumption that the upper surface of the disk is perfectly concave and\nsmooth at all radii, $r$. This is a very good description of such\nunperturbed disks, because thermal and gravitational instabilities are\ndamped very efficiently (D'Alessio et al. 1999).\nObviously this is not going to be the case when an\nalready formed planet core distorts the disk. But even \na small distortion of the disk's surface may affect the\nthermal balance. The distortion need only be large enough compared to\nthe grazing angle at which the starlight strikes the disk, $\\phi(r)$:\n$$\n\\phi(r) = {{0.4R_*}\\over {r}}+r{{d}\\over {dr}}({{h}\\over {r}}),\n$$\nwhere $h$ is the local scale height.\nThis small angle has a minimum\nat 0.4$AU$ and increases significantly only at very large distances:\n$\\phi(r) \\approx 0.005r^{-1} + 0.05r^{2\/7}$ (e.g. see Chiang \\& Goldreich\n1997).\n\nThe amount of compression due to the additional mass of the planet, $M_p$, \nwill depend on the Hill radius, $R_H = r ({{M_p}\\over {M_*}})^{1\/3}$,\nand how it compares to the local scale height, $h$. The depth of the depression\nwill be proportional to $(R_H\/h)^3$. The resulting depressions\n(one on each side) will be in the shadow from the central star, with a\nshade area dependent on the grazing angle, $\\phi(r)$. The solid angle\nsubtended by this shade area from the midplane determines the amount of cooling\nand the new temperature in the sphere of influence of the planet core.\nThe question then arises,\nif during the timescale preceeding the opening of the gap the midplane\ntemperature in the vicinity of the accreting planet core could drop\nbelow the ice condensation limit even for orbits with $r$ much shorter\nthan the \"snow line\" radius in the undisturbed disk. \nThe answer appears to be affirmative and a runaway develops whereby local\nice condensation leads to rapid growth of the initial rocky core, which\nin turn deepens the depression in the disk and facilitates more ice\ncondensation inside the planet's sphere of influence.\nDetails about the\ninstability which develops in this case will be given in a separate paper.\n\n\\section{Conclusion}\nWhen the large fraction of close-in extrasolar giant planets became\napparent, we thought of questioning the standard notion of a distant \"snow\nline\" beyond 3$AU$ in a protoplanetary disk. Thence comes this paper.\nWe revisited the issue by paying attention to the stellar irradiation\nand its radiative effects on the disk, thus limiting ourselves to\npassive or low accretion rate disks.\n\nWe find a snow line as close as 0.7$AU$ in a passive disk, and not\nmuch further away than 1.3$AU$ in a disk with 10$^{-8}M_{\\odot} {\\rm yr}^{-1}$\naccretion rate for $M_*=0.5M_{\\odot}$. The result is robust regardless\nof different reasonable model assumptions $-$ similar values\ncould in principle be inferred from existing disk\nmodels (Chiang \\& Goldreich 1997; D'Alessio et al. 1998). For more massive\n(and luminous) central stars, the snow line shifts outwards: to 1.0$AU$ \n(1$M_{\\odot}$) and 1.6$AU$ (2$M_{\\odot}$). The effect of different disk\nmass is much smaller for passive disks $-$ the snow line shifts inwards\nby 0.08$AU$ for ${\\Sigma}_0$= 10$^4$g~cm$^{-2}$. Our results\ndiffer from existing calculations (in that they bring the\nsnow line even closer in), because the dust grains properties we\nused have higher albedo and more forward throwing. The dust grains\nand the disk models we used are typical of T Tauri stars of age $\\sim$1~Myr.\nSo our conclusion is, that if such T Tauri disks are typical of\nprotoplanetry disks, then the snow line in them could be as close-in\nas 0.7 AU.\n\nOur estimate of the snow line is accurate to within 10\\%, once the model\nassumptions are made. These assumptions are by no means good or obvious,\nand can change the numbers considerably. For a passive disk model, the assumptions\nthat need to be justified are: the equilibrium of the disk, the lack of\ndust settling (i.e. gas and dust are well mixed), the used KMH properties \nof the dust grains, and the choice of molecular opacities. For a low\naccreting disk model, one has to add to the above list: the choice of\nviscous dissipation model (and $\\alpha$=0.01).\n\nFinally, we note that these estimates reflect a steady-state disk in\nhydrostatic equilibrium. The disk will get disturbed as planet formation\ncommences, which may affect the thermal balance locally given the small\nvalue of the grazing angle, $\\phi$. For a certain planet core mass,\nan instability can develop at orbits smaller than 1 AU which can lead \nto the formation of giant planets in situ. What is then the determining\nfactor for the division between terrestrial and giant planets in our\nSolar System remains unexplained (as it did even with a snow line at 2.7$AU$).\n\n\\acknowledgements{\nWe thank N. Calvet, B. Noyes, S. Seager, and K. Wood for reading the\nmanuscript and helpful discussions, and the referee for very thoughtful\nquestions.\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzehxv b/data_all_eng_slimpj/shuffled/split2/finalzzehxv new file mode 100644 index 0000000000000000000000000000000000000000..c8a2bcd26c20df0c05db81a262031c034076fe75 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzehxv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nWith the commissioning of new powerful facilities \nwith high production rates, low- and medium-energy meson physics will \nexperience renewed interest.\n$B$ physics at higher energies and $K$ physics at \nlower energies have stimulated large efforts on the \nexperimental as well as the theoretical side; \nthe field of $\\eta$ and $\\eta'$ physics \nprovides a considerable amount of open questions, and the new facilities are\nexpected to address them from the experimental side. \nThe anticipated numbers of $10^{8}$ - $10^9$ observed etas per year at \nCELSIUS ($\\sim 2.2 \\cdot 10^9$), \nITEP ($\\sim (0.27 - 2.7)\\cdot 10^9$) and DA$\\Phi$NE \n($\\sim 3.2 \\cdot 10^8$) \\cite{DAFNE1} \nwill allow for experiments which on the one hand \nsupply precise figures on the more frequent eta decays and which on the other\nhand focus on rare eta decays. \nSuch rare decays can supply valuable information on anomalous processes or a \npossible $C$ violation in eta decays. \nAnother interesting question is whether it will be feasible to observe some \nof the rare eta decays at the new laser backscattering facility GRAAL at \nGrenoble ($\\sim 10$ decay events per second)\nor even at the $cw$ electron facilities at Mainz (MAMI), \nBonn (ELSA) and Newport News (CEBAF), where \nmany eta photo- and electroproduction experiments are scheduled or already \nbeing carried through. \nOn the \ntheoretical side the development of chiral perturbation theory \n\\cite{Weinberg,GL} as an \neffective theory for the confinement phase of QCD has supplied a \nconsistent framework for the calculation of low-energy processes in the \n$SU\\!\\left(2\\right)$ as well as the $SU\\!\\left(3\\right)$ flavor \nsector of QCD. Chiral\nperturbation theory has turned out to be a valuable tool in the investigation \nof meson interactions at low energies. However, \nthe theory seems to work much better in the $SU\\!\\left(2\\right)$ sector \nthan in the \n$SU\\!\\left(3\\right)$ sector due to the comparatively large mass of the \nstrange quark. In particular, processes involving the eta meson have \nconfronted chiral perturbation theory with various problems \nwhich could only be solved partly by considering next-to-leading-order terms \nand electromagnetic corrections or by going even beyond next-to-leading order \n(see, e.g., \\cite{GLe,Pich}). \nThese problems are also related to \nthe large \\mbox{$\\eta$ - $\\eta'$} mixing angle and the \n$U_A(1)$ problem. \nIn pure chiral perturbation theory up to ${\\cal{O}}(p^4)$, \nthe eta singlet field is integrated out \\cite{GL}, \nbecause the mass of the corresponding physical $\\eta'$, $m_{\\eta'}=\n957.7\\,{\\mathrm{MeV}}$, \nis larger than the low-energy scale \n$\\Lambda_{CPT} \\approx m_{\\rho}$. \nOn the other hand, current-algebra-like calculations with phenomenologically \ndetermined decay constants and $\\eta$ - $\\eta'$ mixing, which keep the eta \nsinglet field as an explicit degree of freedom, \nyield reasonable results. \nSince we will restrict our calculation of the process \n$\\eta \\rightarrow \\pi\\pi\\gamma\\gamma$ to tree level,\nwe will choose such a phenomenologically inspired approach. \nThe charged mode of this \nprocess is of particular interest, because it can supply information on a\nWess-Zumino-Witten contact term \\cite{WZ,Witten} \ninvolving the interaction of \nthree mesons and two photons (for a review of anomalous processes see, e.g.,\n\\cite{bijnensrev}). \nThe eta meson is the only particle of the pseudoscalar octet which decays \nthrough such a mechanism. \nAnother possibility of testing this special kind of contact\nterm in a scattering experiment is the process $\\gamma\\gamma\\rightarrow\n\\pi^+\\pi^-\\pi^0$ which has recently \nbeen treated in chiral perturbation theory \n\\cite{Bos}. Due to the lack of meson collision facilities, it\nis difficult to extract information on this vertex from a scattering experiment\nwith a meson in the initial state, because in practice such experiments \ninvolve virtual particles and necessitate uncertain\nextrapolations. \nOn the experimental side a data analysis dating back to \nthe sixties with rather low\nstatistics gives \nupper bounds on the branching ratios of the charged \nmode $\\eta \\rightarrow \\pi^+\\pi^-\\gamma\\gamma$ \\cite{PDG94,PC,bal}: \n\\begin{eqnarray}\n\\protect{\\cite{PC}}:& \n\\Gamma\\left(\\eta\\rightarrow\n\\pi^+\\pi^-\\gamma\\gamma\\right)\/\\Gamma_{tot} & < 2.1\\times 10^{-3}, \n\\label{e1} \\\\\n\\protect{\\cite{bal}}:& \\Gamma\\left(\\eta\\rightarrow\n\\pi^+\\pi^-\\gamma\\gamma\\right)\/\\Gamma_{tot} & < 3.7\\times 10^{-3}.\n\\label{e2}\n\\end{eqnarray}\nIn the analyses \\cite{PC} and \\cite{bal} the upper limits for the branching\nratios were derived for a missing mass of neutral particles larger than \n$195 \\, \\mathrm{MeV}$. Hopefully, the situation will improve when future\nexperiments will be carried through. \n\n\\section{Kinematics and Observables}\nThe four-momenta and polarization vectors for the charged\ndecay mode $\\eta\\rightarrow\\pi^+\\pi^-\\gamma\\gamma$ are defined in Fig.~1. \nFor the neutral decay mode we use analogous\ndescriptors. \nThe full kinematics of a decay process with four particles in the final \nstate requires five independent kinematical variables (see, e.g., \n\\cite{CaMa,DAFNE}). \nFor the definition of these variables we \nwill consider three reference frames: the rest system of the eta meson \n$\\Sigma_{\\eta}$, the dipion center-of-mass system $\\Sigma_{\\pi\\pi}$, and the \ndiphoton center-of-mass system $\\Sigma_{\\gamma\\gamma}$. Our kinematical \nvariables are (see Fig.~2)\n\\begin{itemize}\n\\item $s_{\\pi}$, the square of the center-of-mass energy of the pions,\n\\item $s_{\\gamma}$, the square of the center-of-mass energy of the photons,\n\\item $\\theta_{\\pi_1}$, the angle of the pion with momentum $k_1$ in \n$\\Sigma_{\\pi\\pi}$ with respect to the direction of flight of the dipion in \n$\\Sigma_{\\eta}$,\n\\item $\\theta_{\\gamma_1}$, the angle of the photon with momentum $q_1$ in \n$\\Sigma_{\\gamma\\gamma}$ with respect to the direction of flight of the \ndiphoton in \n$\\Sigma_{\\eta}$, \n\\item $\\phi$, the angle between the plane formed by the pions in \n$\\Sigma_{\\eta}$ and the corresponding plane formed by the photons.\n\\end{itemize}\nIn order to define these variables more precisely we \nintroduce a unit vector ${\\hat{v}}$ along the direction of flight of the \ndipion in $\\Sigma_{\\eta}$, and unit vectors ${\\hat{c}}$ and \n${\\hat{d}}$ along \nthe projections of $\\vec k_1$ perpendicular to ${\\hat{v}}$ and \nof $\\vec q_1$ perpendicular to $-{\\hat{v}}$, respectively,\n\\begin{eqnarray}\n{\\hat{c}} & = & \\left( \\vec k_1 - {\\hat{v}} {\\hat{v}} \\cdot \\vec k_1 \\right)\n\/ \\left[\\vec k_1^{\\,2} - \\left(\\vec k_1 \\cdot {\\hat{v}} \\right)^2 \n\\right]^{1\/2},\\\\\n{\\hat{d}} & = & \\left( \\vec q_1 - {\\hat{v}} {\\hat{v}} \\cdot \\vec q_1 \\right)\n\/ \\left[\\vec q_1^{\\,2} - \\left(\\vec q_1 \\cdot {\\hat{v}} \\right)^2 \n\\right]^{1\/2}.\n\\end{eqnarray}\nWith these definitions the five kinematical variables are defined as \nfollows:\n\\begin{eqnarray}\ns_{\\pi} & = & \\left(k_1+k_2\\right)^2,\\\\\ns_{\\gamma} & = & \\left(q_1+q_2\\right)^2,\\\\\n\\cos \\theta_{\\pi_1} & = & {\\hat{v}} \\cdot \\vec k_1 \/ \\mid\\! \\vec k_1 \\!\\mid,\\\\\n\\cos \\theta_{\\gamma_1} & = & - {\\hat{v}} \\cdot \\vec q_1 \/ \\mid\n\\! \\vec q_1 \\!\\mid,\\\\\n\\cos \\phi & = & {\\hat{c}} \\cdot {\\hat{d}}.\n\\end{eqnarray}\nThe physical region of the decay process is reflected in the range of the \nkinematical variables:\n\\begin{eqnarray}\n0 \\leq & s_{\\gamma} & \\leq \\left( m_{\\eta} - 2 m_{\\pi} \\right)^2,\\\\\n4 m_{\\pi}^2 \\leq & s_{\\pi} & \\leq \\left( m_{\\eta} - \\sqrt{s_{\\gamma}}\n\\right)^2,\\\\\n0 \\leq & \\theta_{\\pi_1} & \\leq \\pi,\\\\\n0 \\leq & \\theta_{\\gamma_1} & \\leq \\pi,\\\\\n0 \\leq & \\phi & \\leq 2 \\pi.\n\\end{eqnarray}\nThe invariant matrix element squared, \n$\\mid\\!{\\cal{\\overline{M}}}\\!\\mid^2$, will be expressed in terms of \nLorentz scalar products of the five momenta $k_1$,$k_2$,$q_1$, \n$q_2$ and $p$. \nOne of these momentum vectors can be eliminated because of momentum\nconservation. In order to express the Lorentz scalar products in terms of the \nkinematical variables specified above we now introduce adequate \nlinear combinations of the momenta:\n\\begin{eqnarray}\nK & = & k_1 + k_2,\\\\\nL & = & k_1 - k_2,\\\\\nQ & = & q_1 + q_2,\\\\\nR & = & q_1 - q_2.\n\\end{eqnarray} \nFor further reference we need the expressions\n\\begin{eqnarray}\nK \\cdot K & = & s_{\\pi},\\\\\nQ \\cdot Q & = & s_{\\gamma},\\\\\nK \\cdot Q & = & \\frac{1}{2} \\left( m_{\\eta}^2 - s_{\\pi} - s_{\\gamma} \\right)\n,\\\\\nK \\cdot R & = & x \\cos \\theta_{\\gamma_1},\\\\\nL \\cdot Q & = & \\sigma_{\\pi} x \\cos \\theta_{\\pi_1}, \\\\\nL \\cdot R & = & \\sigma_{\\pi} \\left[ K \\cdot Q \\cos \\theta_{\\pi_1} \n\\cos \\theta_{\\gamma_1} \\right.\\nonumber \\\\\n& & \\left.- \\left( s_{\\pi} s_{\\gamma} \\right)^{1\/2} \n\\sin \\theta_{\\pi_1} \\sin \\theta_{\\gamma_1} \\cos \\phi \\right],\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nx & = & \\sqrt{\\left( K \\cdot Q \\right)^2 - s_{\\pi} s_{\\gamma}},\\\\\n\\sigma_{\\pi} & = & \\sqrt{1 - 4 m_{\\pi}^2\/s_{\\pi}}.\n\\end{eqnarray}\nThe ten Lorentz scalar products in \n$\\mid \\!{\\overline{\\cal{M}}}\\!\\mid^2$ can now be expressed as follows:\n\\begin{eqnarray}\nk_1 \\cdot k_2 & = & \\frac{1}{2} \\left( K \\cdot K - 2 m_{\\pi}^2\\right),\\\\\nq_1 \\cdot q_2 & = & \\frac{1}{2} Q \\cdot Q,\\\\\nk_1 \\cdot q_1 & = & \\frac{1}{4} \\left(K \\cdot Q + L \\cdot Q + K \\cdot R + L \n\\cdot R \\right),\\\\\nk_2 \\cdot q_1 & = & \\frac{1}{4} \\left(K \\cdot Q - L \\cdot Q + K \\cdot R - L \n\\cdot R \\right),\\\\\nk_1 \\cdot q_2 & = & \\frac{1}{4} \\left(K \\cdot Q + L \\cdot Q - K \\cdot R - L \n\\cdot R \\right),\\\\\nk_2 \\cdot q_2 & = & \\frac{1}{4} \\left(K \\cdot Q - L \\cdot Q - K \\cdot R + L \n\\cdot R \\right),\\\\\np \\cdot k_1 & = & k_1 \\cdot q_1 + k_1 \\cdot q_2 +k_1 \\cdot k_2 + m_{\\pi}^2,\\\\\np \\cdot k_2 & = & k_2 \\cdot q_1 + k_2 \\cdot q_2 +k_1 \\cdot k_2 + m_{\\pi}^2,\\\\\np \\cdot q_1 & = & q_1 \\cdot q_2 + k_1 \\cdot q_1 + k_2 \\cdot q_1,\\\\\np \\cdot q_2 & = & q_1 \\cdot q_2 + k_1 \\cdot q_2 + k_2 \\cdot q_2.\n\\end{eqnarray}\nThe differential decay rate can be written as\n\\begin{eqnarray}\n\\label{width}\n{\\mathrm{d}}^5\\Gamma \\left(\\eta \\rightarrow \\pi\\pi\\gamma\\gamma\\right) \n& = &\n2^{-14} \\pi^{-6} m_{\\eta}^{-3} C^{-1} \\sigma_{\\pi} x \n\\mid\\! {\\overline{\\cal{M}}} \n\\!\\mid^2 \n\\nonumber \\\\\n& & {\\mathrm{d}}s_{\\pi} {\\mathrm{d}}s_{\\gamma} \n{\\mathrm{d}}\\!\\cos\\theta_{\\pi_1} \n{\\mathrm{d}}\\!\\cos\\theta_{\\gamma_1} {\\mathrm{d}}\\phi,\n\\end{eqnarray}\nwhere the symmetry factor $C$ is equal to $4$ in the decay \n$\\eta \\rightarrow \\pi^0\\pi^0\\gamma\\gamma$ because of two pairs of identical \nparticles in the final state, and equal to $2$ in the decay \n$\\eta \\rightarrow \\pi^+\\pi^-\\gamma\\gamma$\nbecause of one identical particle pair in the final state. Now we will proceed\nto investigate how chiral dynamics manifests itself in the Lorentz-invariant\nmatrix element ${\\cal{M}}$.\n\n\\section{Chiral Dynamics of the decay $\\eta \\rightarrow \\pi\\pi\\gamma\\gamma$}\nWe will restrict our calculation of ${\\cal{M}}$ to the leading-order \ncontributions. \nSince the process $\\eta \\rightarrow \\pi\\pi\\gamma\\gamma$ \ninvolves the electromagnetic interaction of an odd number of pseudoscalar\nmesons, \nthe leading contributions must contain a vertex of odd intrinsic parity. Such a\nvertex is at least of ${\\cal{O}}\\!\\left(p^4\\right)$ in the momentum expansion,\nand thus, according to Weinberg's power counting \\cite{Weinberg}, \nwe expect the leading\ncontribution to be of ${\\cal{O}}\\!\\left(p^4\\right)$. \nConsequently, the \ninteraction Lagrangian we will use for our tree-level calculation \ncontains the standard \n${\\cal{O}}\\!\\left(p^2\\right)$ piece \\cite{Weinberg} \nand the anomalous Wess-Zumino-Witten Lagrangian \\cite{WZ,Witten}, \nbut no terms from the Gasser-Leutwyler Lagrangian \\cite{GL} of \n${\\cal{O}}\\!\\left(p^4\\right)$:\n\\widetext\n\\begin{eqnarray}\n{\\cal{L}} & = & {\\cal{L}}^{(2)}+{\\cal{L}}^{(4)}_{WZW} \\nonumber \\\\\n& = & \n\\frac{F_{\\pi}^2}{4} {\\mathrm{tr}} ((D^{\\mu} U)^{\\dagger} D_{\\mu} U)\n+ \\frac{F_{\\pi}^2}{4} {\\mathrm{tr}}(\\chi^{\\dagger} U + \\chi U^{\\dagger})\n\\nonumber\\\\\n&&\n+\\frac{eN_c}{48\\pi^2}\\varepsilon^{\\mu\\nu\\alpha\\beta}A_\\mu {\\mathrm{tr}}\n(Q\\partial_\\nu U \\partial_\\alpha U^\\dagger\\partial_\\beta U U^\\dagger\n-Q\\partial_\\nu U^\\dagger\\partial_\\alpha U\\partial_\\beta U^\\dagger U)\n\\nonumber\\\\ \n&& \n- \\frac{i e^2 N_c}{24 \\pi^2} \\varepsilon^{\\mu\\nu\\alpha\\beta}\\partial_{\\mu}\nA_{\\nu} A_{\\alpha} {\\mathrm{tr}}(Q^2(U \\partial_{\\beta} U^{\\dagger} \n+ \\partial_{\\beta} U^{\\dagger} U )\n-\\frac{1}{2}Q U^{\\dagger} Q \\partial_{\\beta} U + \\frac{1}{2} Q U Q\n\\partial_{\\beta} U^{\\dagger}), \\label{genlag}\n\\end{eqnarray}\n\\narrowtext\nwhere $\\varepsilon_{0123}=1$.\nIn Eq.\\ (\\ref{genlag}) we have only listed those terms of \n${\\cal{L}}^{(4)}_{WZW}$ \nwhich actually give a contribution to the invariant amplitude. \nThe covariant derivative is defined as \n\\begin{equation}\nD_{\\mu} U = \n\\partial_{\\mu} U + i e\nA_{\\mu} \\left[ Q, U \\right], \n\\end{equation}\nwhere the matrix $Q$ represents the electromagnetic charges \nof the three flavors \nin $SU\\!\\left(3\\right)$, \n\\begin{equation}\nQ={\\mathrm{diag}} \\left(2,-1,-1 \\right)\/3.\n\\end{equation}\nThe matrix\n\\begin{equation}\n\\chi = 2 B_0 m\n\\end{equation}\ncontains the quark masses, \n\\begin{equation}\nm = {\\mathrm{diag}}\\left(m_u,m_d,m_s\\right),\n\\end{equation} \nwhere $B_0$ is related to the quark condensate and is\ngiven by the relation $\\left(m_u+m_d\\right) B_0 = m_{\\pi}^2$. \nThe meson field operators are represented by the matrix \n$U = \\mathrm{exp}\\left(i \\Phi\/F_{\\Phi} \\right)$, where \nthe nonet field matrix $\\Phi$ can be decomposed into an octet and a singlet \npart, $\\Phi = \\Phi_8 + \\Phi_1$, with\n\\begin{equation}\n\\Phi_8 = \n\\left(\n\\begin{array}{ccc}\n\\pi_3 + \\frac{1}{\\sqrt{3}} \\eta_8 & \\sqrt{2} \\pi^+ & 0 \\\\\n\\sqrt{2} \\pi^- & - \\pi_3 + \\frac{1}{\\sqrt{3}} \\eta_8 & 0 \\\\\n0 & 0 & - \\frac{2}{\\sqrt{3}} \\eta_8 \n\\end{array}\n\\right)\n\\end{equation}\nand\n\\begin{equation}\n\\Phi_1 = \\sqrt{\\frac{2}{3}} \\eta_0 \\,\n{\\mathrm{diag}} \\left( 1, 1, 1 \\right).\n\\end{equation}\nIn the expansion of $U$ the decay constants \n$F_{\\pi}$, $F_8$ or ${\\overline{F}}_0$ will be inserted \nfor the constant $F_{\\Phi}$, depending on whether the constant belongs to a \n$\\pi$, $\\eta_8$ or $\\eta_0$ field. We will use $F_{\\pi}=93\\,{\\mathrm{MeV}}$, \n$F_8=1.25 F_{\\pi}$ and ${\\overline{F}}_0=1.06 F_{\\pi}$ \nas numerical values \\cite{GK}. \nOur calculation in chiral perturbation theory will be carried out with the \ngroup theoretical octet and singlet eta states, $\\mid\\!\\eta_8 \\rangle$ \nand $\\mid\\!\\eta_0 \\rangle$. We will introduce $\\eta$-$\\eta'$ mixing via the \nphenomenological mixing angle $\\theta = -20^\\circ$ \\cite{GK}: \n\\begin{eqnarray}\n|\\eta\\rangle & = & \\cos \\theta |\\eta_8 \\rangle - \\sin \\theta | \\eta_0 \\rangle\n, \\\\ \n|\\eta' \\rangle & = & \\sin \\theta |\\eta_8 \\rangle + \\cos \\theta |\\eta_0 \\rangle.\n\\end{eqnarray} \nThe Feynman diagrams contributing to the charged decay mode \n$\\eta\\rightarrow\\pi^+\\pi^-\\gamma\\gamma$ \nof \n${\\cal{O}}\\!\\left(p^4\\right)$ are \ndisplayed in Fig.~\\ref{feyndiagcharged}. \nThere are three different classes of \ndiagrams at tree level: diagrams with a four-meson vertex of \n${\\cal{O}}\\!\\left(p^2\\right)$, a propagating \nneutral meson, and a decay vertex into two photons of \n${\\cal{O}}\\!\\left(p^4\\right)$ \n(class 1), \nWess-Zumino-Witten contact terms \nof ${\\cal{O}}\\!\\left(p^4\\right)$ \n(class 2) and \ninternal bremsstrahlung diagrams, where one photon is emitted off a charged \npion line (classes 3.1 and 3.2). \nThe first class of diagrams is gauge invariant \nby itself, whereas the amplitudes corresponding to the diagrams from the \nsecond and third class have to be added in order to obtain gauge \ninvariance. In the neutral decay mode $\\eta\\rightarrow\\pi^0\\pi^0\\gamma\\gamma$ \nonly the first class \nof diagrams is relevant (Fig.~\\ref{feyndiagneutral}). \n\nStarting from the general chiral Lagrangian of Eq.\\ (\\ref{genlag}), we now \nlist the interaction terms \nrelevant for the process $\\eta \\rightarrow \n\\pi^+\\pi^-\\gamma\\gamma$:\n\\begin{eqnarray}\n{\\cal{L}}^{(4),2\\gamma3\\phi}_{WZW} & = & \n\\frac{e^2 N_c}{12 \\sqrt{3} \\pi^2 F_{\\pi}^2}\n\\varepsilon^{\\mu\\nu\\alpha\\beta}\n\\partial_{\\mu}A_{\\nu} A_{\\alpha} \n\\pi^+ \\pi^-\n\\nonumber \\\\\n& & \n\\times\n\\left(\\frac{1}{F_8} \\partial_{\\beta} \\eta_8 + \n\\frac{\\sqrt{2}}{{\\overline{F}}}_0\n\\partial_{\\beta} \\eta_0\n\\right), \\label{lanf} \\label{li} \\\\\n{\\cal{L}}^{(4),1\\gamma3\\phi}_{WZW} & = & \n\\frac{i e N_c}{12 \\sqrt{3} \\pi^2 F_{\\pi}^2}\n\\varepsilon^{\\mu\\nu\\alpha\\beta}\nA_{\\mu}\n\\partial_{\\nu}\\pi^+\\partial_{\\alpha}\\pi^-\n\\nonumber \\\\\n& & \n\\times\n\\left(\\frac{1}{F_8} \\partial_{\\beta} \\eta_8 + \n\\frac{\\sqrt{2}}{{\\overline{F}}}_0\n\\partial_{\\beta} \\eta_0\n\\right), \\\\\n{\\cal{L}}^{(4),2\\gamma1\\phi}_{WZW} & = & \n-\\frac{ e^2 N_c}{24 \\pi^2 }\n\\varepsilon^{\\mu\\nu\\alpha\\beta}\n\\partial_{\\mu} A_{\\nu} A_{\\alpha} \n\\nonumber \\\\\n& & \n\\times\n\\partial_{\\beta}\\left(\n\\frac{1}{F_{\\pi}}\\pi^0+\n\\frac{1}{\\sqrt{3}F_8} \\eta_8 + \\frac{2\\sqrt{2}}{\\sqrt{3}\n\\,{\\overline{F}}_0}\n\\eta_0\n\\right), \\\\\n{\\cal{L}}^{(2),4\\phi} & = & \n\\frac{ F_{\\pi}^2 B_0}{24}\n\\left(\n\\frac{4 \\left(m_u+m_d\\right)}{F_{\\pi}^2F_8^2}\\eta_8\\eta_8\\pi^+\\pi^-\n\\right. \\nonumber \\\\\n& & \n\\left.\n+\n\\frac{8 \\sqrt{2} \\left(m_u+m_d\\right)}{F_{\\pi}^2 F_8 {\\overline{F}}_0}\n\\eta_8\\eta_0\\pi^+\\pi^-\\right.\\nonumber\\\\\n& & \\left.+\n\\frac{8 \\left(m_u+m_d\\right)}{F_{\\pi}^2 {\\overline{F}}_0^2}\n\\eta_0\\eta_0\\pi^+\\pi^-\n+\n\\right. \\nonumber \\\\\n& & \n\\left.\n\\frac{8 \\left(m_u-m_d\\right)}{\\sqrt{3}F_{\\pi}^3 F_8}\n\\eta_8\\pi^+\\pi^-\\pi^0\n\\right. \\nonumber \\\\\n& & \n\\left.\n+\n\\frac{8 \\sqrt{2}\\left(m_u-m_d\\right)}{\\sqrt{3}F_{\\pi}^3 {\\overline{F}}_0}\n\\eta_0\\pi^+\\pi^-\\pi^0\n\\right), \\label{vpvc} \\\\\n{\\cal{L}}^{(2),1\\gamma 2\\phi} & = & \ni e A^{\\rho}\\left(\\partial_{\\rho}\\pi^-\\pi^+-\\partial_{\\rho}\\pi^+\\pi^-\\right).\n\\label{lend}\n\\end{eqnarray}\nThe invariant amplitude for $\\eta\\rightarrow\\pi^+\\pi^-\\gamma\\gamma$ is then\ngiven as the sum of the amplitudes from the three classes of Feynman\ndiagrams (Fig.~\\ref{feyndiagcharged}),\n\\begin{equation}\n{\\cal{M}}={\\cal{M}}_1+{\\cal{M}}_2+{\\cal{M}}_3,\n\\end{equation}\nwhere\n\\widetext\n\\begin{eqnarray}\n{\\cal{M}}_1 & = & - \\frac{i e^2 N_c}{12 \\sqrt{3} \\pi^2} \n\\varepsilon^{\\mu\\nu\\alpha\\beta} \\varepsilon_{1,\\mu}\n\\varepsilon_{2,\\alpha} q_{1,\\nu} q_{2,\\beta} \n\\left\\{\n\\frac{B_0\\left(m_u+m_d\\right)}{3 \\left( 2 q_1 \\cdot q_2 - m_{\\eta_8}^2 \n\\right)}\n\\left(\n\\frac{\\cos \\theta}{F_8^3}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_8^2 {\\overline{F}}_0}\n\\right)\n\\right.\\nonumber\\\\\n& & +\n\\frac{4 B_0\\left(m_u+m_d\\right)}{3 \\left( 2 q_1 \\cdot q_2 - m_{\\eta_0}^2\n\\right)}\n\\left(\n\\frac{\\cos \\theta}{{\\overline{F}}_0^2 F_8}\n-\\frac{\\sqrt{2}\\sin \\theta}{{\\overline{F}}_0^3}\n\\right)\n\\nonumber\\\\\n&&\\left. +\n\\frac{B_0\\left(m_u-m_d\\right)}{3 \\left( 2 q_1 \\cdot q_2 - m_{\\pi_3}^2\n\\right)}\n\\left(\n\\frac{\\cos \\theta}{F_{\\pi}^2 F_8}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_{\\pi}^2 {\\overline{F}}_0}\n\\right)\n\\right\\},\\\\\n{\\cal{M}}_2 & = &\n-\\frac{i e^2 N_c}{12 \\sqrt{3} \\pi^2} \n\\varepsilon^{\\mu\\nu\\alpha\\beta}\np_{\\beta} \\varepsilon_{1,\\mu} \\varepsilon_{2,\\alpha} \\left(q_1 - q_2 \n\\right)_{\\nu}\n\\left(\n\\frac{\\cos \\theta}{F_{\\pi}^2 F_8}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_{\\pi}^2 {\\overline{F}}_0}\n\\right),\\\\\n{\\cal{M}}_3 & = &\n\\frac{i e^2 N_c}{12 \\sqrt{3} \\pi^2} \n\\varepsilon^{\\mu\\nu\\alpha\\beta}\n\\left(\n\\frac{\\cos \\theta}{F_{\\pi}^2 F_8}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_{\\pi}^2 {\\overline{F}}_0}\n\\right)\n\\left\\{\n\\varepsilon_{1,\\mu}\nq_{1,\\alpha} k_{2,\\nu} p_{\\beta} \n\\frac{\\varepsilon_2 \\cdot k_1}{q_2 \\cdot k_1}\n+\n\\varepsilon_{2,\\mu}\nq_{2,\\alpha} k_{2,\\nu} p_{\\beta} \n\\frac{\\varepsilon_1 \\cdot k_1}{q_1 \\cdot k_1}\n\\right.\\nonumber \\\\\n& & \\left. \n+\n\\varepsilon_{1,\\mu}\nq_{1,\\alpha} k_{1,\\nu} p_{\\beta} \n\\frac{\\varepsilon_2 \\cdot k_2}{q_2 \\cdot k_2}\n+\n\\varepsilon_{2,\\mu}\nq_{2,\\alpha} k_{1,\\nu} p_{\\beta} \n\\frac{\\varepsilon_1 \\cdot k_2}{q_1 \\cdot k_2}\n\\right\\}.\n\\end{eqnarray}\n\\narrowtext\nThe masses in the propagators can be expressed in terms of the physical \nmasses using the relations \n\\begin{eqnarray}\nm_{\\eta_8}^2 & = & m_{\\eta}^2 \\cos^2 \\theta + m_{\\eta'}^2 \\sin^2 \\theta, \\\\\nm_{\\eta_0}^2 & = & m_{\\eta}^2 \\sin^2 \\theta + m_{\\eta'}^2 \\cos^2 \\theta, \\\\\nm_{\\pi_3}^2 & = & m_{\\pi^0}^2. \n\\end{eqnarray}\nAs we will not be concerned with\nphoton polarizations in the final state, \nwe carry out the sum over the polarizations of\nthe real photons, \n\\begin{equation}\n\\mid\\! {\\overline{\\cal{M}}} \\!\\mid^2\n=\n\\sum_{\\kappa_1,\\kappa_2} \n{\\cal{M}}\\left(\\kappa_1,\\kappa_2\\right)\n{\\cal{M}}^*\\left(\\kappa_1,\\kappa_2\\right).\n\\end{equation}\nFor that purpose we exploit the completeness \nrelation \n\\begin{equation}\n\\sum_{\\kappa} \\varepsilon_{\\gamma}\\left(\\kappa\\right) \n\\varepsilon_{\\delta}^* \\left(\\kappa\\right) \\rightarrow - g_{\\gamma\\delta},\n\\end{equation}\nwhich is based on current conservation (see, e.g., \\cite{HM}): \n\\widetext\n\\begin{eqnarray}\n\\sum_{\\kappa_1,\\kappa_2}\n{\\cal{M}}_i\\left(\\kappa_1, \\kappa_2\\right)\n{\\cal{M}}^*_j\\left(\\kappa_1,\\kappa_2\\right) \n& = & \\sum_{\\kappa_1,\\kappa_2} \n{\\cal{M}}_i^{\\lambda\\sigma}\n{{\\cal{M}}_j^{\\rho\\tau}}^*\n\\varepsilon_{1,\\lambda}\\left(\\kappa_1\\right)\n\\varepsilon_{1,\\rho}^*\\left(\\kappa_1\\right)\n\\varepsilon_{2,\\sigma}\\left(\\kappa_2\\right)\n\\varepsilon_{2,\\tau}^*\\left(\\kappa_2\\right)\n\\nonumber \\\\\n& = & \ng_{\\lambda\\rho} g_{\\sigma\\tau}\n{\\cal{M}}_i^{\\lambda\\sigma}\n{{\\cal{M}}_j^{\\rho\\tau}}^*\n= \n{\\cal{M}}_i^{\\lambda\\sigma}\n{{\\cal{M}}_{\\lambda\\sigma,j}}^* \\,\\,\\,\\,\\left(i,j \\in\n\\left\\{1,2,3\\right\\}\\right).\n\\end{eqnarray}\nThe result for $\\mid \\!{\\overline{\\cal{M}}}\\! \\mid^2$ is too complicated \nto be displayed in this contribution. \nIt depends on all Lorentz scalar products which can be constructed from the \nfour final-state momentum vectors $k_1$, $k_2$, $q_1$ and $q_2$.\n\nFor the neutral decay mode $\\eta \\rightarrow \\pi^0\\pi^0\\gamma\\gamma$ we \nobtain the interactions\n\\begin{eqnarray}\n{\\cal{L}}^{(4),2\\gamma1\\phi}_{WZW} & = & - \\frac{e^2 N_c}{24 \n\\pi^2} \\varepsilon^{\\mu\\nu\\alpha\\beta} \\partial_{\\mu} A_{\\nu} \nA_{\\alpha} \\left(\n\\frac{1}{F_{\\pi}}\\partial_{\\beta}\\pi^0\n+\n\\frac{1}{\\sqrt{3} F_8}\\partial_{\\beta}\\eta_8\n+\n\\frac{\\sqrt{2}}{3 \\sqrt{3} \\,{\\overline{F}}_0}\\partial_{\\beta}\\eta_0\n\\right),\\\\\n{\\cal{L}}^{(2),4\\phi} & = & \n-\\frac{ F_{\\pi}^2 B_0}{24}\n\\left(\n\\frac{4 \\left(m_u-m_d\\right)}{\\sqrt{3}F_{\\pi}^3 F_8}\\eta_8\\pi^0\\pi^0\\pi^0\n+\n\\frac{2 \\left(m_u+m_d\\right)}{F_{\\pi}^2 F_8^2}\n\\eta_8\\eta_8\\pi^0\\pi^0 \\right.\\nonumber\\\\\n&&\n+\n\\frac{4 \\sqrt{2}\\left(m_u+m_d\\right)}{F_{\\pi}^2 F_8 {\\overline{F}}_0}\n\\eta_8\\eta_0\\pi^0\\pi^0\n+\n\\frac{4 \\sqrt{2} \\left(m_u-m_d\\right)}{\\sqrt{3}F_{\\pi}^3 {\\overline{F}}_0}\n\\eta_0\\pi^0\\pi^0\\pi^0\\nonumber\\\\\n&&\\left.\n+\n\\frac{4 \\sqrt{2}\\left(m_u+m_d\\right)}{F_{\\pi}^2 F_8 {\\overline{F}}_0}\n\\eta_8\\eta_0\\pi^0\\pi^0\n+\n\\frac{4 \\left(m_u+m_d\\right)}{F_{\\pi}^2 {\\overline{F}}_0^2}\n\\eta_0\\eta_0\\pi_0\\pi_0\n\\right),\n\\label{masseq}\n\\end{eqnarray}\nwhich lead to the diagrams in Fig.~\\ref{feyndiagneutral} and \nresult in the Lorentz-invariant matrix element\n\\begin{eqnarray}\n{\\cal{M}} & = & \n\\frac{i e^2 N_c B_0}{96\\pi^2} \n\\varepsilon^{\\mu\\nu\\alpha\\beta} \\varepsilon_{1,\\nu}\n\\varepsilon_{2,\\alpha}q_{1,\\mu}q_{2,\\beta}\n\\left\\{\n\\frac{8\\left(m_u-m_d\\right)}{\\sqrt{3} \\left( \n2 q_1 \\cdot q_2 - m_{\\pi_3}^2\n\\right)}\n\\left(\n\\frac{\\cos \\theta}{F_{\\pi}^2 F_8}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_{\\pi}^2 {\\overline{F}}_0}\n\\right)\n\\right.\\nonumber\\\\\n& & +\n\\frac{8 \\left(m_u+m_d\\right)}{3\\sqrt{3} \\left(\n2 q_1 \\cdot q_2 - m_{\\eta_8}^2\n\\right)}\n\\left(\n\\frac{\\cos \\theta}{F_8^3}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_8^2{\\overline{F}}_0}\n\\right)\n\\nonumber\\\\\n&&\\left. +\n\\frac{32\\left(m_u+m_d\\right)}{3\\sqrt{3}\n\\left(\n2 q_1 \\cdot q_2 - m_{\\eta_0}^2\n\\right)\n}\n\\left(\n\\frac{\\cos \\theta}{F_8 {\\overline{F}}_0^2}\n-\\frac{\\sqrt{2}\\sin \\theta}{{\\overline{F}}_0^3}\n\\right)\n\\right\\}.\n\\nonumber\\\\\n&&\n\\end{eqnarray}\nSumming over the possible photon polarizations $\\kappa_1$ and $\\kappa_2$ \nwe obtain a compact result \nfor the invariant matrix element squared,\n\\begin{eqnarray}\n\\mid\\!{\\overline{\\cal{M}}}\\!\\mid^2 & = & \n\\frac{e^4 N_c^2 B_0^2}{72 \n\\pi^4}\n\\left( q_1 \\cdot q_2 \\right)^2 \n\\left\\{\n\\frac{m_u-m_d}{\\sqrt{3} \\left(2 q_1 \\cdot q_2 - m_{\\pi_3}^2\\right)}\n\\left(\n\\frac{\\cos \\theta}{F_{\\pi}^2 F_8}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_{\\pi}^2 {\\overline{F}}_0}\n\\right)\\right. \n\\nonumber\\\\\n&& + \n\\frac{m_u+m_d}{3\\sqrt{3}\\left( 2 q_1 \\cdot q_2 - m_{\\eta_8}^2\\right)}\n\\left(\n\\frac{\\cos \\theta}{F_8^3}\n-\\frac{\\sqrt{2}\\sin \\theta}{F_8^2{\\overline{F}}_0}\n\\right) \n\\nonumber\\\\\n&&\\left.\n+\\frac{4\\left(m_u+m_d\\right)}{3\\sqrt{3}\\left( 2 q_1 \\cdot q_2 - m_{\\eta_0}^2\n\\right)}\n\\left(\n\\frac{\\cos \\theta}{F_8 {\\overline{F}}_0^2}\n-\\frac{\\sqrt{2}\\sin \\theta}{{\\overline{F}}_0^3}\n\\right)\n\\right\\}^2.\n\\end{eqnarray}\nWe note that the \nfinal result only depends on one Lorentz scalar product, namely $q_1 \\cdot \nq_2 = s_{\\gamma}\/2$. \n\\narrowtext\n\n\\section{Results and Discussion}\n\nHaving determined $\\mid\\!{\\overline{\\cal{M}}}\\!\\mid^2$ we proceed to\ninvestigate the decay spectra by integrating Eq.\\ (\\ref{width}) numerically.\nFirst we will discuss the Dalitz plot ${\\mathrm{d}}^2\n\\Gamma\/{\\mathrm{d}}s_{\\pi}\n{\\mathrm{d}}s_{\\gamma}$ for the charged decay mode. \nWe note that the soft-photon limit for any of the two photons implies \n$s_{\\gamma} \\rightarrow 0$.\nThe contour plot for the\nfull chiral perturbation theory \ncalculation (Fig.~\\ref{dalfull}) clearly shows the infrared bremsstrahlung\nsingularity for $s_{\\gamma} \\rightarrow 0$. \nThe Dalitz plot also reflects a pole at $s_{\\gamma} = m_{\\pi^0}^2 = \n18219 \\, {\\mathrm{MeV}}^2$ due to the\nclass 1 Feynman diagram with a propagating $\\pi^0$ \n(see Fig.~\\ref{feyndiagcharged}). \nIn the neighborhood of this pole it is impossible to distinguish between the\nprocesses $\\eta\\rightarrow\\pi^+\\pi^-\\gamma\\gamma$ and $\\eta\\rightarrow\\pi^+\n\\pi^-\\pi^0$. If we switch off the diagrams of class 1, \nthe invariant amplitude is\nstill gauge invariant, and the corresponding Dalitz plot \n(Fig.~\\ref{dalwithoutpole}) is very similar to the \nfull calculation (Fig.~\\ref{dalfull}) except \nfor the region around the $\\pi^0$ pole. \nWe conclude that the diagrams of class 1 do not contribute significantly to the\nprocess which also becomes evident from the diphoton spectrum \n${\\mathrm{d}} \\Gamma\/{\\mathrm{d}}z$ (Fig.~\\ref{dipion}), where \n$z = s_{\\gamma}\/m_{\\eta}^2$. \nThe effect of the $\\pi^0$ pole is confined to a very small region around \n$s_{\\gamma}=m_{\\pi^0}^2$. \nThe calculation without class 1 diagrams almost coincides with the full \ncalculation except for this region. \nThe reaction mechanisms of class 2 \nand class 3 diagrams dominate the spectrum over a wide energy range. \nWe conclude that the detection of this decay mode should be a good indication \nfor the presence of a Wess-Zumino-Witten contact term (class 2). \nTowards small values of $s_{\\gamma}$ the bremsstrahlung diagrams\n(class 3) will be responsible for a divergence in the spectrum. A measurement\nof the steep rise in the spectrum for vanishing $s_{\\gamma}$ depends on\nthe resolution of the detector facilities or, more accurately speaking, on the\nminimum photon energy detectable. Hence, in both experiment and theory it\nis only possible to determine the partial decay rate as a function of an \nenergy cut $\\delta m_{brems}$ applied around $\\sqrt{s_{\\gamma}} = 0$. \nWe obtained $\\Gamma\\left(\\eta\\rightarrow\\pi^+\\pi^-\\gamma\\gamma\\right)$ by \nintegrating the diphoton spectrum for the calculation without class 1 \ndiagrams. \nAccording to Fig.~\\ref{dipion} the error introduced by this approximation\nshould be negligible. \nIn Fig.~\\ref{partialwidthcharged} we show the result as a\nfunction of $\\delta m_{brems}$. \nComparing our absolute numbers for the partial decay rate with the total eta \ndecay rate $\\Gamma_{tot} = 1.2 \\times 10^{-3}\\, {\\mathrm{MeV}}$ \\cite{PDG94}, \nwe note that the resulting branching ratio is well within the reach of \nthe facilities mentioned in the introduction.\nUsing an energy cut at $\\delta m_{brems} = 195\\,{\\mathrm{MeV}}$,\nour result for the partial decay rate is smaller than the experimental\nupper limits in Eqs.\\ (\\ref{e1}) and (\\ref{e2}) by about three\norders of magnitude.\n\nLet us now turn to the neutral decay mode \n$\\eta \\rightarrow \\pi^0\\pi^0\\gamma\\gamma$.\nIn contrast to the charged decay mode, where the diagrams of class 1 can be\nneglected, these diagrams generate the only tree-level contributions in the\ncase of neutral mesons. \nFor this reason we want to analyse the structure of the vertices\nin the diagrams more closely. \nWhereas the two-photon-one-meson vertices have been investigated\nextensively in the two-photon decays of the $\\pi^0$, $\\eta$ and $\\eta'$, the\nfour-meson vertices, in particular the $\\eta_8\\eta_8\\pi^0\\pi^0$ interaction,\nare not known with comparable precision. \nHowever, these vertices are of great\ninterest, because they are directly related to the sum or difference of the\nlight quark masses (see Eq.\\ (\\ref{masseq})). While the $\\eta\\pi\\pi\\pi$ and \n$\\eta'\\eta\\pi\\pi$ interaction can be directly investigated in the decays \n$\\eta\\rightarrow\\pi\\pi\\pi$ and $\\eta'\\rightarrow\\eta\\pi\\pi$, this is not the\ncase for the $\\eta\\eta\\pi\\pi$ interaction. Moreover, the minimum center-of-mass\nenergy for the scattering process\n$\\pi\\pi\\rightarrow\\eta\\eta$ is beyond the convergence radius of chiral\nperturbation theory, and $\\pi\\eta\\rightarrow\\pi\\eta$ scattering is not\na realistic alternative either. \nAs a consequence, the only possibility to get information on the \n$\\eta\\eta\\pi\\pi$ interaction is the investigation of a composite process \nwith at least one additional vertex. \nAt first glance the decay $\\eta\\rightarrow\\pi^0\\pi^0\\gamma\\gamma$ \nseems to be a good candidate since $G$ parity is conserved\nat the $\\eta\\eta\\pi^0\\pi^0$ vertex, whereas in the diagram with a propagating\n$\\pi^0$ the corresponding vertex violates $G$ parity and vanishes in the\nisospin limit. \nHowever, as is well-known from the decay $\\eta\\rightarrow\\pi\\pi\\pi$, \n$G$ parity is violated significantly and isospin breaking is reflected by the \nfact that the $\\eta\\pi\\pi\\pi$ vertex function is directly proportional to the \nquark mass difference (see Eqs. (\\ref{vpvc}) and (\\ref{masseq})). \nFor this reason the diagram with the propagating neutral pion cannot\nbe neglected. \nSince $m_{\\pi^0}^2$ lies within the range of integration for $s_{\\gamma}$, \nthe diagram with a propagating $\\pi^0$ will cause a pole in the amplitude \n${\\cal{M}}$. \nThus, despite $G$ parity and isospin violation, this diagram is strongly \nenhanced in comparison with the diagrams with a propagating $\\eta_8$ and \n$\\eta_0$. \nThe diphoton spectrum clearly demonstrates the fact \nthat the $\\eta_8\\eta_8\\pi^0\\pi^0$ interaction plays a minor role in the \nprocess (see Fig.~\\ref{diphoton00}).\n\nIn the pole region it is impossible to distinguish\nthe decay mode $\\eta\\rightarrow \\pi^0\\pi^0\\gamma\\gamma$ from $\\eta \\rightarrow\n\\pi^0\\pi^0\\pi^0$. \nThus, in order to obtain new information on chiral dynamics from the decay \n$\\eta\\rightarrow \\pi^0\\pi^0\\gamma\\gamma$, one should choose an energy regime \nfor $s_{\\gamma}$ which is sufficiently far away from this pole.\n\nAt ${\\cal O}(p^2)$ effective chiral Lagrangians do not reproduce \nthe experimentally determined interaction strength of the \n$\\eta\\pi^0\\pi^0\\pi^0$ and $\\eta'\\eta\\pi^0\\pi^0$ vertices \n(for the correct treatment beyond tree level see \\cite{GLe}).\nTherefore, we have also performed a phenomenological calculation, \nwhere we fixed \nthese interactions using the experimentally observed branching ratios\n\\cite{PDG94} \n$\\Gamma\\left(\\eta\\rightarrow\\pi^0\\pi^0\\pi^0\\right)\/\\Gamma_{tot} = 0.319$ and \n$\\Gamma\\left(\\eta'\\rightarrow\\eta\\pi^0\\pi^0\\right)\/\n\\Gamma_{tot} = 0.208$. \nWhereas experimental data agree very well with a {\\em constant} tree-level \nprediction for $\\eta \\rightarrow \\pi^0\\pi^0\\pi^0$,\nsuch an assumption does not seem to work so well for \n$\\eta' \\rightarrow \\eta\\pi^0\\pi^0$ \n(see, e.g., the discussion in \\cite{OW}). \nHowever, this is of only minor importance in our case, because the pole at \n$m_{\\eta'}^2$ in the amplitude is far outside the physical region of \n$s_{\\gamma}$. \nIt turns out that the result with the vertices fixed by experiment is larger \nthan the prediction of chiral perturbation theory by about one order of \nmagnitude. \nFurthermore, we also found that $\\pi$-$\\eta$ and $\\pi$-$\\eta'$ mixing\naccording to \\cite{bagchi} is negligible in this process. Future work\nwill focus on higher-order corrections to our tree-level calculation.\n\nThe question whether it is realistic to measure the decay \n$\\eta\\rightarrow\\pi^0\\pi^0\\gamma\\gamma$ must be decided from the partial decay\nrate $\\Gamma\\left(\\eta\\rightarrow\\pi^0\\pi^0\\gamma\\gamma\\right)$, which we\nobtain by integration of the diphoton spectrum applying a cut of the width \n$2\\delta m$ around the pole at $m_{\\pi^0}$ (Fig.~\\ref{neutraltot}). \nFrom this figure it becomes obvious that this partial decay rate is rather \nsmall, but possibly within the reach of the new eta facilities. \nHowever, it will probably be impossible to draw any precise \ninformation on the $G$-parity-conserving \n$\\eta^8\\eta^8\\pi^0\\pi^0$ coupling from measuring this\nprocess, because the corresponding Feynman diagram gives only a small\ncontribution to the complete amplitude. Finally, we want to\nmention that the calculation in \\cite{ST} based on old experimental data \nyields\n$\\Gamma\\left(\\eta\\rightarrow\\pi^0\\pi^0\\gamma\\gamma\\right)=\n1.38\\times10^{-3}\\,{\\mathrm{eV}}$.\n\nWe conclude that the rare eta decay $\\eta\\rightarrow\\pi^+\\pi^-\\gamma\\gamma$ is\nan interesting test of the anomalous Lagrangian, because it is a simple way to\naccess the three-meson-two-photon vertex predicted by \\cite{WZ} and \n\\cite{Witten} and allows for a consistency check with the results from\n$\\gamma\\gamma\\rightarrow\\pi^+\\pi^-\\pi^0$. \nThe neutral decay mode also investigated in this contribution will be much \nharder to detect. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nNowadays the sequential hypothesis test has been widely applied in many applications because it generally requires smaller sample size on average compared to its fixed-sample-size counterpart. Notably, \\cite{Wald48} proved that the sequential probability ratio test (SPRT) yields the minimum expected sample size under both null and alternative hypotheses given the error probabilities. Since this pioneering work, a rich body of studies on the sequential test have emerged under different circumstances \\cite{SeqA_book}. One of the most important applications of sequential test is found in sensor networks \\cite{Veeravalli93,Tsitsiklis93,Blum97,Mei08,LiLi15}. In this work, we consider the sequential hypothesis test when sensor access at the fusion center is restricted, and efficient sensor scheduling\/selection is of interest. That is, the sensor network with different types of sensors (i.e., heterogenous sensors) and a fusion center aims to test between two hypotheses; however, only one of the available sensors can take samples and communicate with the fusion center at each sampling instant. Such a setup often arises when the fusion center possessses limited processing capability\/resources, or the sensors contradict\/exclude one another. For instance, the echo-based sensors like sonar sensors can interfere with each other \\cite{Gupta06}. In practice, the heterogenous sensors could also refer to multiple information resources, and the processing unit (i.e., fusion center) can only analyze one at a time. This model well describes, for example, the human decision process. As such, in order to reach a quick and reliable decision, strategically selecting the ``most informative'' sensor, which often depends on the parameter values or the true hypothesis that is unknown, has become the pivotal problem. \n\n\nIn the context of fixed-sample-size statistical inference, sensor selection has been well studied, mainly from the optimization standpoint. In particular, \\cite{Gupta06} proposed a random selection scheme to minimize the error covariance of a process tracking problem; for the Kalman filter, \\cite{Mo11} devised a multi-stage strategy to select a subset of sensors so that an objective function related to the error covariance matrix was minimized; \\cite{Joshi09} put forth a convex-optimization-based approach to select multiple sensors for the parameter estimation in linear system. For the fixed-sample-size hypothesis test, \\cite{Bajovic11} investigated sensor scheduling based on information-metric criteria such as Kullback-Leibler and Chernoff distances. \n\nThe studies on the sensor selection for sequential hypothesis test have mainly branched into the offline (a.k.a. open-loop) and online (a.k.a. closed-loop) approaches. The former category essentially involves independent random selection over time, with the probability preassigned to each sensor. Along this direction, \\cite{Srivastava11,Srivastava11_2} introduced random sensor selection to the multi-hypothesis sequential probability ratio test (MSPRT), and designed the selection probability such that its approximate decision delay was minimized. They concluded that the optimal random selection strategy involve at most two sensors for binary-hypothesis test. Namely, the fusion center should either always use one sensor, or randomly switch between two sensors, and disregard the rest. Similar teniques were later applied to the quickest detection with stochastic surveillance control \\cite{Srivastava13}. Recently, focusing on the binary-hypothesis test, \\cite{Bai15} further imposed constraints on the sensor usages, i.e., sensors, on average, cannot be selected more than their prescribed limits, and obtained the selection probabilities for SPRT with random sensor selection. \n\n\nDespite their simple implementations, the open-loop approaches do not make use of the accumulating sample information, thus are suboptimal in general. On the contrary, the online approaches take all previous samples into account at each step for sensor selection, and generally yield superior performance. As a matter of fact, dynamic sensing control is one of the major advantages of sequential processing. To this end, \\cite{Chernoff59} selected the sensor that was most informative under the most likely true hypothesis at each step. \\cite{Javidi10,Javidi13,Javidi13_2} investigated the sequential multi-hypothesis test with observation control, and provided lower and upper bound for its asymptotic performance. Two asymptotically optimal algorithms were proposed there. The variant of sequential hypothesis test---changepoint detection with observation control were considered by \\cite{Banerjee12,Banerjee13} based on Bayesian and non-Bayesian settings respectively. Meanwhile, \\cite{Kumar08} assumed identical sensors, and studied the Bayesian changepoint detection with control on the number of active sensors. Most of the above online approaches are based on heuristics and perform well in the asymptotic regime, where error probabilities are extremely low. \nOn the other hand, focusing on the non-asymptotic regime, \\cite{Bai14} considered the online sensor selection strategy for the SPRT. However, it aimed to minimize the decision delay given that SPRT was used. \nInstead, the recent work \\cite{Xiaoou15} jointly solved a Bayesian hypothesis testing problem for both the optimal sequential test and online selection strategy. \n\nIn this work, we also aim for the optimal sequential test and online sensor selection simultaneously. Moreover, we further introduce the constraints on the sensor usages into the formulation, which would potentially embrace a much wider range of practical problems. That is, certain sensors in the network are not allowed to be selected more than a prescribed number of times on average. The usage constraints naturally arise when one intends to restrain the sensors from being overused due to their limited battery\/lifetime, or if the fairness for all sensors in the network is important \\cite{Bai15}. We summarize the contributions as follows:\n\\begin{itemize}\n\\item To the best of our knowledge, this is the first work that jointly solves for the optimal sequential test and online sensor selection when sensor usage constraints are considered. Moreover, this work distinguishes from \\cite{Bai15}, where the usage-contrained sensor selection is also studied, in terms of its online\/closed-loop setup.\n\n\n\\item Note that most of the existing works on sensor selection for sequential test only apply to infinite-horizon, where sample size (or decision delay) at a specific realization can go to infinity if necessary. This may not be realistic in some applications. In contrast, we consider {\\it both} the infinite-horizon and finite-horizon scenarios. In the later case, a fixed upper bound is imposed on the random sample size at every realizations.\n\n\\item We propose practical algorithm to systematically evaluate the parameters in the optimal sequential test and selection strategy. As long as the test performance constraints and the sensor usage constraints remain the same, this algorithm only needs to be run once offline. That is, once the parameters are calculated, they can be stored at the fusion center, based on which, the sequential test can be easily implemented.\n\\end{itemize}\n\nThe reminder of the paper is organized as follows. We first formulate the usage-constrained sequential hypothesis test in Section II. Then the optimal sequential test and sensor selection strategy are derived in Section III. In Section IV, we propose practical algorithms to design the parameters in the optimal scheme. Section V provides numerical results to illustrate the theoretical results, and to compare with the offline random selection scheme. Finally, Section VI concludes this paper.\n\\section{Problem Formulation}\n\nConsider a system consisting of $K$ sensors and a fusion center that aims to test between two hypotheses, whose priors are given as $\\mathbb{P}\\left(\\mathcal{H}=i\\right)=\\pi_i, \\, i=0, 1$. At each time instant, the fusion center selects one sensor to take a sample that is sent to the fusion center. This process continues until a reliable decision can be made. It is assumed that the fusion center possesses the statistical characterization of all sensors. That is, the conditional probability density functions $f_{\\mathcal{H}}^\\ell(x)$ of the random samples collected by sensor $\\ell, \\,\\ell=1, 2, \\ldots, K$ are known to the fusion center. Without loss of generality, we assume that the sensor network is heterogenous, i.e., there are no two sensors with identical $f_{\\mathcal{H}}^\\ell(x)$'s. In addition, the random samples are assumed to be independent and identically distributed (i.i.d.) over time for the same sensor $\\ell$, and independent across different sensors. \n\nOn one hand, if there is a dominant sensor that always outperforms all other sensors, the fusion center should always use it in the absence of usage constraint. Then the problem reduces to a single-sensor sequential hypothesis test, and the SPRT yields the quickest decision. One such example is the test between zero ($\\mathcal{H}_0$) and non-zero Gaussian means ($\\mathcal{H}_1$), where the sensor with the largest mean shift under $\\mathcal{H}_1$ should prevail. On the other hand, the efficiency of a sensor generally depends on the true hypothesis. For example, some sensors can be more informative under $\\mathcal{H}_0$ and less so under $\\mathcal{H}_1$, thus accelerating the decision speed when $\\mathcal{H}_0$ is true, and slowing down the decision speed otherwise. \nMoreover, even the dominant sensor cannot be used all the time if its usage is restrained. \nIn general, the online sensor selection procedure is performed based on the accumulated sample information, which is explained as follows.\n\nThere are three essential operations in the online procedure:\n\\begin{enumerate}\n\\item Sensor selection strategy: Let $\\Pi\\triangleq\\{1, 2, \\ldots, K\\}$ be the set of all sensors, and $\\{X_1, \\ldots, X_t\\}$ denote the sequence of samples received at the fusion center. Then the sensor selected at time $t$ can be defined as $\\delta_t: \\{X_1, \\ldots, X_{t-1}\\}\\to j\\in \\Pi$. In addition, we denote the sequence of sensor selections from time $i$ to time $j$ as ${\\boldsymbol\\delta}_{i:j}$, and ${\\boldsymbol\\delta}_{i:j}\\triangleq \\emptyset$ if $i>j$. Note that since at any time, the distribution of the next sample depends on the selection function, the fusion center observes dependent random samples $\\{X_t\\}$.\n\\item Stopping rule: The random sample size is characterized by the stopping time $\\mathsf{T}$. In specific, the event $\\{\\mathsf{T}=t\\}$ means that the sample size is equal to $t$, which depends on $\\{X_1, \\ldots, X_t\\}$. In this work, we focus on the deterministic stopping rule, i.e., $\\mathbb{P}\\left(\\mathsf{T}=t|X_1, \\ldots, X_t\\right)$ is either zero or one.\n\\item Decision function: Upon stopping at $\\mathsf{T}=t$, a final decision between the two hypotheses is made, $D_t: \\{X_1, \\ldots, X_t\\}\\to \\{0, 1\\}$.\n\\end{enumerate}\nAs such, the fusion center is faced with the following hypothesis testing problem:\n\\begin{align*}\n\\begin{array}{ll}\n\\mathcal{H}_0: &X_t\\sim f^{\\delta_t}_0(x), \\quad t=1, 2, \\ldots \\\\\n\\mathcal{H}_1: & X_t\\sim f^{\\delta_t}_1(x), \\quad t=1, 2, \\ldots.\n\\end{array}\n\\end{align*}\nThe performance indicators for sequential hypothesis test include the expected sample size and the error probabilities. In particular, the expected sample size $\\mathbb{E}\\mathsf{T}=\\pi_0\\mathbb{E}_0\\left(\\mathsf{T}\\right)+\\pi_1\\mathbb{E}_1\\left(\\mathsf{T}\\right)$ is the weighted sum of the conditional expected sample sizes, and the type-I and type-II error probabilities are $\\mathbb{P}_0\\left( D_\\mathsf{T}=1\\right)$ and $\\mathbb{P}_1\\left( D_\\mathsf{T}=0\\right)$ respectively\\footnote{One can also use the weighted sum of type-I and type-II error rates as the error probability. Here we adopt the formulation in \\cite{Bai15}, and consider them individually. Nevertheless, the method developed in this work can be applied to the former case.}. \nHere the expectation $\\mathbb{E}\\left(\\cdot\\right)$ is taken over the joint distribution of $\\mathcal{H}$ and $X_t$, and $\\mathbb{E}_i\\left(\\cdot\\right)$ is taken over the distribution of $X_t$ conditioned on $\\{\\mathcal{H}=i\\}$. \n\nMoreover, we also impose constraints on the usage of sensors. Denote $\\Omega$ as the set of sensors whose usages are restrained. Then for each sensor $\\ell\\in \\Omega$, the average number of times that sensor $\\ell$ is selected, $\\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T} \\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right)$, is constrained to be no greater than $T^\\ell\\in \\mathbb{R}^+$. As such, we arrive at the following constrained sequential problem:\n\\begin{align}\\label{P1}\n\\begin{array}{ll}\n\\min_{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}} & \\mathbb{E} \\mathsf{T} \\\\\n\\text{subject to} & \\ignore{\\mathbb{E}\\left(\\mathbbm{1}_{\\{D_\\mathsf{T}\\neq \\mathcal{H}\\}}\\right)}\\mathbb{P}_0\\left( D_\\mathsf{T}=1\\right)\\le \\alpha,\\, \\mathbb{P}_1\\left( D_\\mathsf{T}=0\\right)\\le \\beta,\\\\ \\phantom{\\text{subject to}} & \\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T} \\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right)\\le T^\\ell, \\quad \\ell\\in \\Omega.\n\\end{array}\\tag{P1}\n\\end{align}\nIn the following sections, we will solve \\eqref{P1} under both the finite-horizon and infinite-horizon setups. The finite-horizon setup imposes an upper bound on $\\mathsf{T}$ for any realization, beyond which no sample can be taken; whereas the infinite-horizon setup allows the sequential test to continue as long as the termination condition is not met. In addition to its relevance in many applications, the finite-horizon case can also be used as a building block for the infinite-horizon problem. For notational convenience, we define the class of infinite-horizon procedures:\n\\begin{align}\n{\\bf C}\\left( \\alpha, \\beta, \\{T^\\ell\\}_{\\ell\\in \\Omega}\\right)\\triangleq &\\Big\\{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}: \\, \\mathbb{P}_0\\left( D_\\mathsf{T}=1\\right)\\le \\alpha,\\, \\nonumber\\\\& \\mathbb{P}_1\\left( D_\\mathsf{T}=0\\right)\\le \\beta, \\;\\text{and}\\;\\; \\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T} \\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right)\\le T^\\ell, \\ell\\in \\Omega\\Big\\},\n\\end{align}\nand the class of finite-horizon procedures:\n\\begin{align}\n{\\bf C}_N\\left( \\alpha, \\beta, \\{T^\\ell\\}_{\\ell\\in \\Omega}\\right)\\triangleq \\Big\\{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}\\in {\\bf C}\\left( \\alpha, \\beta, \\{T^\\ell\\}_{\\ell\\in \\Omega}\\right): \\mathsf{T}\\le N\\Big\\}.\n\\end{align}\nOur goal is to find the optimal triplets $\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, \\mathsf{T}, D_\\mathsf{T}\\}$ that yield the smallest expected sample sizes $\\mathbb{E}\\mathsf{T}$ in the classes ${\\bf C}_N\\left( \\alpha, \\beta, \\{T^\\ell\\}_{\\ell\\in \\Omega}\\right)$ and ${\\bf C}\\left( \\alpha, \\beta, \\{T^\\ell\\}_{\\ell\\in \\Omega}\\right)$ respectively. \n\n\\section{Optimal Sequential Test with Constrained Online Sensor Selection}\nIn this section, we first recast \\eqref{P1} into an unconstrained optimal stopping problem, which we then solve under both finite-horizon and infinite-horizon setups. The solutions lead us to the optimal sequential solutions to the original constrained problem \\eqref{P1}. \n\nBy introducing Lagrange multipliers to \\eqref{P1}, we arrive at the following Bayes objective function:\n\\begin{align}\\label{Bayes_obj}\n\\mathcal{R}({\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T})&\\triangleq \\mathbb{E}\\mathsf{T}+\\mu_0\\pi_0\\mathbb{P}_0\\left({D_\\mathsf{T}=1}\\right)+\\mu_1\\pi_1\\mathbb{P}_1\\left({D_\\mathsf{T}=0}\\right)+\\sum_{\\ell\\in\\Omega}\\lambda_\\ell \\,\\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right)\\nonumber\\\\&=\\mathbb{E}\\left( \\mathsf{T}+\\mu_0\\mathbbm{1}_{\\{D_\\mathsf{T}=1; \\mathcal{H}=0\\}}+\\mu_1\\mathbbm{1}_{\\{D_\\mathsf{T}=0; \\mathcal{H}=1\\}}+\\sum_{\\ell\\in\\Omega}\\lambda_\\ell \\left( \\sum_{t=1}^\\mathsf{T}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right)\\rb\\nonumber\\\\&={\\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T} \\underbrace{\\left( 1+\\mathbbm{1}_{\\{\\delta_t\\in \\Omega\\}}\\lambda_{\\delta_t}\\right)}_{\\mathcal{C}_{\\delta_t}}+\\underbrace{\\mu_0\\mathbbm{1}_{\\{D_\\mathsf{T}=1; \\mathcal{H}=0\\}}+\\mu_1\\mathbbm{1}_{\\{D_\\mathsf{T}=0; \\mathcal{H}=1\\}}}_{\\mu\\left( D_\\mathsf{T}, \\mathcal{H}\\right)}\\right)}.\n\\end{align}\nNote that $\\mathcal{C}_j\\triangleq 1+\\lambda_j$ and $\\lambda_j\\ge 0$ for $j\\in \\Omega$, and $\\mathcal{C}_j\\triangleq 1$ for $j\\notin \\Omega$. \n\n\n\\subsection{Finite-Horizon Solution to the Bayes Problem}\nIn this subsection, under the finite-horizon setup, we aim to find the optimal sensor selection, stopping time and decision rule such that the Bayes risk in \\eqref{Bayes_obj} is minimized, i.e., \n\\begin{align}\\label{P1_Bayes}\n\\min_{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}, \\mathsf{T}\\le N}\\quad \\mathcal{R}\\left({\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\right)=\\mathbb{E}\\left(\\sum_{t=1}^\\mathsf{T}\\mathcal{C}_{\\delta_t}+\\mu\\left( D_\\mathsf{T}, \\mathcal{H}\\right)\\rb.\n\\end{align}\nDefine the cumulative log-likelihood ratio (LLR)\n\\begin{align}\nL_n\\triangleq \\sum_{t=1}^n\\underbrace{\\log \\frac{f_1^{\\delta_t}(X_t)}{f_0^{\\delta_t}(X_t)}}_{l_{\\delta_t}(X_t)},\n\\end{align}\nand the posterior probabilities $\\pi_i(t)\\triangleq\\mathbb{P}\\left(\\mathcal{H}=i|X_{1:t}, {\\boldsymbol\\delta}_{1:t}\\right)$, $i\\in\\{0, 1\\}$ with $\\pi_i(0)=\\pi_i$. These two statistics relate to each other as follows\n\\begin{align}\\label{posterior}\n\\pi_1(n)=\\frac{\\pi_1e^{L_n}}{\\pi_0+\\pi_1e^{L_n}}=\\frac{\\pi_1({n-1})e^{l_{\\delta_n}}}{\\pi_0({n-1})+\\pi_1({n-1})e^{l_{\\delta_n}}},\\quad L_n=\\log\\frac{\\pi_0\\pi_1(n)}{\\pi_1\\pi_0(n)}.\n\\end{align}\n\\subsubsection{Decision Function} We begin with solving the terminal decision function. Since\n\\begin{align}\n\\mathcal{R}\\left({\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\right) =&\\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T}{\\mathcal{C}_{\\delta_t}}\\right)+\\sum_{t=1}^\\infty \\mathbb{E}\\left[\\mathbbm{1}_{\\{\\mathsf{T}= t\\}}\\left(\\mu_0\\mathbbm{1}_{\\{D_\\mathsf{T}=1; \\mathcal{H}=0\\}}+\\mu_1\\mathbbm{1}_{\\{D_\\mathsf{T}=0; \\mathcal{H}=1\\}}\\right)\\right]\\nonumber\\\\ =&\\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T}{\\mathcal{C}_{\\delta_t}}\\right)+\\sum_{t=1}^\\infty \\mathbb{E}\\left(\\mathbb{E}_\\mathcal{H}\\left(\\left.\\mu_0\\mathbbm{1}_{\\{D_t=1; \\mathcal{H}=0\\}}+\\mu_1\\mathbbm{1}_{\\{D_t=0; \\mathcal{H}=1\\}}\\right|X_{1:t},{\\boldsymbol\\delta}_{1:t}\\right)\\mathbbm{1}_{\\{\\mathsf{T}=t\\}}\\right)\\nonumber\\\\ =&\\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T}{\\mathcal{C}_{\\delta_t}}\\right)+\\sum_{t=1}^\\infty \\mathbb{E}\\left[\\left(\\mu_0\\pi_0(t)\\mathbbm{1}_{\\{D_t\\neq 0\\}}+\\mu_1\\pi_1(t)\\mathbbm{1}_{\\{D_t\\neq 1\\}}\\right)\\mathbbm{1}_{\\{\\mathsf{T}=t\\}}\\right],\\label{opt_decision}\n\\end{align}\nwe have $D_t^\\star=\\mathbbm{1}_{\\{\\mu_0\\pi_0(t)\\le \\mu_1\\pi_1(t)\\}}$ given $\\mathsf{T}=t$, i.e., \n\\begin{align}\\label{DecRule}\nD_\\mathsf{T}^\\star=\\mathbbm{1}_{\\{\\mu_0\\pi_0(\\mathsf{T})\\le \\mu_1\\pi_1(\\mathsf{T})\\}}.\n\\end{align}\n\\subsubsection{Selection Strategy and Stopping Rule} For notational convenience, define the class \n\\begin{align}\\label{classA}\n\\mathcal{A}_n^N\\triangleq \\left\\{\\{{\\boldsymbol\\delta}_{n+1:\\mathsf{T}}, \\mathsf{T}\\}: n\\le \\mathsf{T}\\le N\\right\\},\n\\end{align} \nin which the procedures do not stop before $n$ and can not go beyond $N$. By substituting $D_\\mathsf{T}$ with \\eqref{DecRule}, \\eqref{P1_Bayes} becomes\n\\begin{align}\\label{Opt_stopping}\n\\min_{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, \\mathsf{T}\\}\\in \\mathcal{A}^N_0}\\; \\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T} \\mathcal{C}_{\\delta_t} + \\underbrace{\\min\\left\\{\\mu_0\\pi_0(\\mathsf{T}), \\mu_1\\pi_1(\\mathsf{T})\\right\\}}_{\\phi(\\pi_1(\\mathsf{T}))}\\right),\n\\end{align}\nwhere $\\phi(x)\\triangleq \\min\\{\\mu_1x, \\mu_0(1-x)\\}$. We next solve \\eqref{Opt_stopping} to obtain the optimal sensor selection strategy and stopping rule.\n\nDefine the optimal cost of the procedures that do not stop before $t=n$, i.e., the ``cost-to-go'' function\n\\begin{align}\\label{V_n_N}\n\\mathcal{V}^N_{n}\\left( X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)&\\triangleq\\min_{\\{{\\boldsymbol\\delta}_{n+1:\\mathsf{T}}, \\mathsf{T}\\}\\in \\mathcal{A}^N_n} \\mathbb{E}\\left(\\left. \\sum_{t=1}^\\mathsf{T}\\mathcal{C}_{\\delta_t}+\\phi\\left(\\pi_1(\\mathsf{T})\\right)\\right|X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right)\n\\end{align}\nNote that $\\mathcal{V}_0^N$ (which is not a function of any samples) is equal to \\eqref{Opt_stopping} by definition and $\\mathcal{V}_N^N(X_{1:N},{\\boldsymbol\\delta}_{1:N})=\\phi\\left( \\pi_1(N)\\right)+\\sum_{t=1}^N \\mathcal{C}_{\\delta_t}$ since the test has to stop at $N$ if not before it. Invoking the technique of dynamic programming, the cost-to-go \\eqref{V_n_N} can be recursively solved by the following backward recursion \\cite{SeqA_book}:\n\\begin{align}\\label{Backward}\n&{\\mathcal{V}}_n^N(X_{1:n}, {\\boldsymbol\\delta}_{1:n})=\n\\min\\left\\{\\underbrace{\\phi\\left( \\pi_1(n)\\right)+\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}}_{r_s\\left( X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right)},\\; \\underbrace{\\min_{\\delta_{n+1}}\\left[\\mathbb{E}\\left(\\left.\\mathcal{V}_{n+1}^N\\left( X_{1:n+1}, {\\boldsymbol\\delta}_{1:n+1}\\right)\\right|X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)\\right]}_{r_c\\left( X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right)}\\right\\},\n\\end{align}\nwith $n=N-1, N-2, \\ldots, 1, 0$. According to the principle of optimality, the optimal stopping time happens when the cost of stopping at the present instant is lower than the expected cost of continuing \\cite{Wald48,Ferguson_book}, i.e., $\\mathsf{T}^\\star=\\min\\{n: g_n(X_{1:n}, {\\boldsymbol\\delta}_{1:n})\\triangleq r_s\\left( X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right)-r_c\\left( X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right) \\le 0\\}$, where\n\\begin{align}\\label{stopping_rule}\ng_n&\\left( X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)= \\phi\\left( \\pi_1(n)\\right)+\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}-\\min_{\\delta_{n+1}}\\left[\\mathbb{E}\\left(\\left.\\mathcal{V}_{n+1}^N\\left( X_{1:n+1}, {\\boldsymbol\\delta}_{1:n+1}\\right)\\right|X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)\\right]\\nonumber\\\\&=\\phi\\left( \\pi_1(n)\\right)-\\min_{\\delta_{n+1}} \\left\\{\\mathcal{C}_{\\delta_{n+1}}+\n\\min_{\\{{\\boldsymbol\\delta}_{n+2:\\mathsf{T}}, \\mathsf{T}\\}\\in \\mathcal{A}^N_{n+1}}\\left[\\mathbb{E}\\left( \\left.\\phi(\\pi_1(\\mathsf{T}))+\\sum_{t=n+2}^\\mathsf{T} \\mathcal{C}_{\\delta_t}\\right|X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)\\right]\\right\\},\n\\end{align}\nwhere the second equality is due to the definition of $\\mathcal{V}_n^N$ in \\eqref{V_n_N}.\n\nIn theory, \\eqref{Backward} and $\\mathsf{T}^\\star$ fully characterize the optimal stopping rule and selection strategy from the first to the $N$-th steps. However, this result is of limited practical value due to the high complexity brought by the high-dimensional quantities (i.e., $X_{1:n}$ and ${\\boldsymbol\\delta}_{1:n}$). To this end, the following lemma significantly simplifies $\\mathsf{T}^\\star$ and \\eqref{stopping_rule}, since it states that the hypothesis posterior (or equivalently, the LLR) is the sufficient statistic for the optimal stopping rule. \n\\begin{lemma}\\label{lemma:1}\nThe optimal stopping rule for \\eqref{P1_Bayes} is a function of time and hypothesis posterior, i.e., a time-variant function of the posterior, $\\mathsf{T}^\\star=\\min\\{n: g_n(\\pi_1(n))\\le0\\}$.\n\\end{lemma}\n\n\\proof\nSee Appendix.\n\\endproof\nThe important implication of Lemma \\ref{lemma:1} is that the selection strategy, which depends on all previous samples, can be summarized into a more compact form. \n\\begin{lemma}\nThe optimal selection strategy for \\eqref{P1_Bayes} is characterized by a time-variant function of the hypothesis posterior (or equivalently, the LLR), i.e., $\\delta_{n+1}^\\star=\\psi_{n+1}(\\pi_1(n))$. \n\\end{lemma}\n\\proof\nFrom \\eqref{Backward}, the optimal selection strategy for $t=n+1$ is\n\\begin{align}\\label{Lemma2_1}\n\\delta_{n+1}^\\star&=\\arg\\,\\min_{\\delta_{n+1}}\\, \\mathbb{E}\\left(\\left.\\mathcal{V}_{n+1}^N\\left( X_{1:n+1}, {\\boldsymbol\\delta}_{1:n+1}\\right)\\right|X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right),\n\\end{align}\nand, by its definition, we have\n\\begin{align}\\label{Lemma2_2}\n\\mathcal{V}_{n+1}^N\\left( X_{1:n+1},{\\boldsymbol\\delta}_{1:n+1}\\right)&=\\min \\left\\{r_s\\left( X_{1:n+1},{\\boldsymbol\\delta}_{1:n+1}\\right), r_c\\left( X_{1:n+1},{\\boldsymbol\\delta}_{1:n+1}\\right)\\right\\}\n\\nonumber\\\\&=\\min\\left\\{0, -g_{n+1}(\\pi_1(n+1))\\right\\}+r_s\\left( X_{1:n+1},{\\boldsymbol\\delta}_{1:n+1}\\right)\\nonumber\\\\&=\\phi\\left(\\pi_1(n+1)\\right)+\\sum_{t=1}^{n+1}\\mathcal{C}_{\\delta_t}-\\max\\left\\{g_{n+1}(\\pi_1(n+1)),0\\right\\}. \n\\end{align}\nSubstituting \\eqref{Lemma2_2} into \\eqref{Lemma2_1} and neglecting the term $\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}$ that is independent of $\\delta_{n+1}$, we arrive at\n\\begin{align}\n\\delta_{n+1}^\\star&=\\text{arg}\\,\\min_{\\delta_{n+1}}\\left\\{\\mathcal{C}_{\\delta_{n+1}}+ {\\mathbb{E} \\left.\\Big[ \\phi\\left( \\pi_1(n+1)\\right)-\\max\\left\\{g_{n+1}\\left( \\pi_1(n+1)\\right), 0\\right\\}\\right|X_{1:n},{\\boldsymbol\\delta}_{1:n}\\Big]} \\right\\}\\nonumber\\\\&=\\text{arg}\\,\\min_{\\delta_{n+1}}\\left\\{\\mathcal{C}_{\\delta_{n+1}}+ \\underbrace{\\mathbb{E} \\left.\\Big[ \\phi\\left( \\pi_1(n+1)\\right)-\\max\\left\\{g_{n+1}\\left( \\pi_1(n+1), n+1\\right), 0\\right\\}\\right|\\pi_1(n)\\Big]}_{u_n\\left( \\pi_1(n), \\delta_{n+1}\\right)} \\right\\}.\n\\end{align}\nNote that the fact that the expectation term in the bracket is a time-variant function of $\\pi_1(n)$ and $\\delta_{n+1}$ (i.e., $u_n\\left( \\pi_1(n), \\delta_{n+1}\\right)$) follows from the relation between $\\pi_1(n)$ and $\\pi_1(n+1)$ given by \\eqref{posterior}. Then $\\delta_{n+1}^\\star=\\arg\\min_{\\delta} \\;\\widetilde u_n\\left(\\pi_1{(n)},\\delta\\right)\\triangleq \\mathcal{C}_{\\delta}+u_n\\left( \\pi_1(n),\\delta\\right)$ which implies that the optimal selection is a time-variant function of the posterior, i.e., $\\delta_{n+1}^\\star=\\psi_{n+1}\\left( \\pi_1(n)\\right)$. \n\\endproof\nThis result agrees with the intuition. Since the sensor efficiency depends on the actual hypothesis, it is reasonable to base the sensor selection upon the present belief (i.e., posterior) on the hypothesis.\n\nNext we continue to study the stopping rule $\\mathsf{T}^\\star$ in more details. \nDefine\n\\begin{align}\n\\mathcal{G}^N_n(X_{1:n}, {\\boldsymbol\\delta}_{1:n})&\\triangleq\\mathcal{V}_n^N\\left( X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)-\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}\\nonumber\\\\&=\\min_{\\{{\\boldsymbol\\delta}_{n+1:\\mathsf{T}}, \\mathsf{T}\\}\\in \\mathcal{A}_n^N} \\mathbb{E}\\left(\\left. \\sum_{t=n+1}^\\mathsf{T}\\mathcal{C}_{\\delta_t}+\\phi\\left(\\pi_1(\\mathsf{T})\\right)\\right|X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right).\\label{def_G}\n\\end{align}\nMeanwhile, $\\mathcal{G}_n^N(X_{1:n}, {\\boldsymbol\\delta}_{1:n})$ can be written as a function of $\\pi_1(n)$ by using \\eqref{Lemma2_2} as\n\\begin{align}\\label{G_pi}\n\\mathcal{G}_n^N(X_{1:n}, {\\boldsymbol\\delta}_{1:n})\n&=\\phi\\left(\\pi_{1}(n)\\right)-\\max\\left\\{g_n(\\pi_1(n)),0\\right\\}=\\mathcal{G}^N_n\\left(\\pi_1(n)\\right),\n\\end{align}\nwhere $\\mathcal{G}_N^N(X_{1:N}, {\\boldsymbol\\delta}_{1:N})=\\phi\\left(\\pi_1(N)\\right)$. \n\n\nThen, by substracting $\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}$ on both sides of \\eqref{Backward}, we obtain\n\\begin{align}\nr_s-\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}&=\\phi(\\pi_1(n)),\\label{G_0}\\\\\n\\text{and}\\quad r_c\\left( X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right)-\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}&=\\min_{\\delta_{n+1}}\\;\\mathbb{E}\\left[\\left.\\mathcal{V}_{n+1}^N\\left( X_{1:n+1}, {\\boldsymbol\\delta}_{1:n+1}\\right)-\\sum_{t=1}^n\\mathcal{C}_{\\delta_t}\\right|X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right]\\nonumber\\\\&=\\min_{\\delta_{n+1}}\\;\\mathbb{E}\\left[\\left.\\mathcal{C}_{\\delta_{n+1}}+\\mathcal{G}^N_{n+1}\\left( \\pi_1(n+1)\\right)\\right|X_{1:n},{\\boldsymbol\\delta}_{1:n}\\right]\\label{G_1}\\\\&=\\min_{\\delta_{n+1}}\\;\\mathcal{C}_{\\delta_{n+1}}+\\mathbb{E}\\left[\\left.\\mathcal{G}^N_{n+1}\\left(\\pi_1(n+1)\\right)\\right|\\pi_1(n)\\right],\\label{G_2}\n\\end{align}\nwhere \\eqref{G_1} follows from the definition of ${\\cal G}_{n}^N$, and \\eqref{G_2} holds since ${\\cal C}_{\\delta_{n+1}}$ is constant given $\\{X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\}$ and $\\mathbb{E}\\left[\\left.\\mathcal{G}^N_{n+1}\\left(\\pi_1(n+1)\\right)\\right|X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right]=\\mathbb{E}\\left[\\left.\\mathcal{G}^N_{n+1}\\left(\\pi_1(n+1)\\right)\\right|\\pi_1(n)\\right]$. Substituting \\eqref{G_0}-\\eqref{G_2} into \\eqref{Backward}, \nthe backward recursion is significantly simplified to the following\n \\begin{align}\\label{G_BI}\n{\\mathcal{G}}_n^N(\\pi_1(n))&\\!=\\!\n\\min\\!\\left\\{\\phi\\left( \\pi_1(n)\\right), \\min_{\\delta_{n+1}}\\left[\\underbrace{\\mathcal{C}_{\\delta_{n+1}}\\!+\\!\\mathbb{E}\\left(\\left.\\mathcal{G}_{n+1}^N\\left(\\frac{\\pi_1(n)\\exp\\left( l_{\\delta_{n+1}}\\right)}{\\pi_0(n)+\\pi_1(n)\\exp\\left( l_{\\delta_{n+1}}\\right)}\\right)\\right|\\pi_1(n)\\right)}_{\\overline{\\mathcal{G}}_n^N(\\pi_1(n),\\delta_{n+1})}\\right]\\right\\},\n\\end{align}\nwith $n=N-1, N-2, \\ldots, 1, 0$. Obviously, we have \n\\begin{align}\\label{V0N}\n\\mathcal{G}_0^N(\\pi_1)=\\mathcal{V}_0^N(\\pi_1) \n\\end{align}\ndue to the definition in \\eqref{def_G}. \n\nWith the lemma below, we can further analyze the optimal stopping rule given in Lemma \\ref{lemma:1}.\n\\begin{lemma}\\label{lemma:3}\n$\\overline{\\mathcal{G}}_n^N(\\pi_1(n),\\delta_{n+1})$ is a concave function of $\\pi_1(n)$. Moreover, the function \n\\begin{align}\\label{def:tildeG}\n\\widetilde{\\cal G}_n^N(\\pi_1(n))\\triangleq \\min_{\\delta_{n+1}}\\;\\overline{\\cal G}_n^N(\\pi_1(n),\\delta_{n+1})\n\\end{align} \nis concave with $\\widetilde{\\mathcal{G}}_n^N(0)>0, \\widetilde{\\mathcal{G}}_n^N(1)>0$, for $n=0, 1, \\ldots, N$.\n\\end{lemma}\n\\proof First, $\\mathcal{G}_N^N(\\pi_1(N))=\\phi(\\pi_1(N))=\\min\\{\\mu_1\\pi_1(N), \\mu_0(1-\\pi_1(N))\\}$ is concave. Second, the recursion \\eqref{G_BI} suggests that, if $\\mathcal{G}_{n+1}^N(\\pi_1(n+1))$ is concave, $\\mathcal{G}_{n}^N(\\pi_1(n))$ is concave as well. This can be shown as follows:\n\nAssume that $\\mathcal{G}_{n+1}^N(x)$ is concave, since $\\frac{x\\exp\\left( l_{\\delta_{n+1}}\\right)}{1-x+x\\exp\\left( l_{\\delta_{n+1}}\\right)}$ is an increasing function of $x$ and the expectation operation preserves the concavity, the compound function $\\mathbb{E}\\left(\\left. \\mathcal{G}_{n+1}^N\\left(\\frac{x\\exp\\left( l_{\\delta_{n+1}}\\right)}{1-x+x\\exp\\left( l_{\\delta_{n+1}}\\right)}\\right)\\right|\\pi_1(n)=x\\right)$ is concave, which further leads to the concavity of $\\overline{\\cal G}_n^N(\\pi_1(n),\\delta_{n+1})$ in terms of $\\pi_1(n)$; in addition, regarding $\\overline{\\cal G}_n^N(\\pi_1(n),\\delta_{n+1})$ as a series of concave functions indexed by $\\delta_{n+1}$, since the point-wise minimum preserves the concavity, $\\widetilde{\\mathcal{G}}_{n}^N(\\pi_1(n))$ is a concave function; due to the same argument, the point-wise minimum of $\\widetilde{\\cal G}_n^N(\\pi_1(n))$ and $\\phi(\\pi_1(n))$, i.e., $\\mathcal{G}_n^N(\\pi_1(n))$, is concave as well. \n\nTherefore, by induction, we conclude that $\\mathcal{G}_n^N(\\pi_1(n)),\\; n=0, 1, \\ldots, N$ are concave functions. Furthermore, from the proof above, we know that the concavity of $\\mathcal{G}_n^N(\\pi_1(n))$ leads to the concavities of $\\overline{\\cal G}_n^N(\\pi_1(n),\\delta_{n+1})$ and $\\widetilde{\\mathcal{G}}_{n}^N(\\pi_1(n))$. Thus $\\overline{\\cal G}_n^N(\\pi_1(n),\\delta_{n+1})$ and $\\widetilde{\\mathcal{G}}_{n}^N(\\pi_1(n))$ for $n=0, 1, \\ldots, N$ are concave functions. \n\\endproof\nTogether with Lemma \\ref{lemma:1}, Lemma \\ref{lemma:3} reveals the following optimal stopping rule. \n\\begin{lemma}\n$\\mathsf{T}^\\star=\\min\\{n: \\pi_1(n)\\notin (a_n,b_n)\\}$, where $a_n$ and $b_n$ are roots for \n\\begin{align}\\label{Lemma4_Eq}\n\\mu_0(1- x)= \\widetilde{\\mathcal{G}}_{n}^N(x)\\; \\; \\text{and}\\;\\; \\mu_1 x= \\widetilde{\\mathcal{G}}_{n}^N(x),\n\\end{align}\nrespectively. Moreover, $a_0b_1>\\ldots>b_N=\\frac{\\mu_0}{\\mu_0+\\mu_1}$.\n\\end{lemma}\n\\proof\nSee Appendix.\n\\endproof\nNow we have obtained the optimal solution $\\{{\\boldsymbol\\delta}^\\star_{1:\\mathsf{T}^\\star}, D^\\star_{\\mathsf{T}^\\star}, \\mathsf{T}^\\star\\}$ to \\eqref{P1_Bayes}, which is summarized in the theorem below. Note that we have changed the sufficient statistic $\\pi_1(n)$ to its equivalent form, i.e., LLR $L_n$ to draw parallel to the well-known SPRT, and with an abuse of notation, the selection function is also denoted as $\\psi_{t+1}(L_t)$.\n\n\\begin{theorem}\\label{thm1}\nThe optimal sequential procedure that solves \\eqref{P1_Bayes} features a sequential probability ratio test with curved stopping boundary, and time-variant sensor selection strategy, i.e.,\n\\begin{enumerate}\n\\item The optimal sensor selection rule is a time-variant function of LLR: $\\delta_{t+1}^\\star\\triangleq \\psi_{t+1}(L_t)$;\n\\item The optimal stopping rule is in the form of a truncated SPRT, i.e.,\n\\begin{align}\n&\\mathsf{T}^\\star=\\min\\{t: L_t\\notin (-A_t, B_t)\\},\\quad \\text{with}\\\\\n&B_0>B_1>\\ldots>B_N=\\log\\frac{\\mu_0\\pi_0}{\\mu_1\\pi_1},\\; \\text{and}\\quad A_0>A_1>\\ldots>A_N=-\\log\\frac{\\mu_0\\pi_0}{\\mu_1\\pi_1};\n\\end{align}\n\\item The optimal decision rule $D^\\star_{\\mathsf{T}^\\star}$ decides $\\mathcal{H}_0$ if $L_{\\mathsf{T}^\\star}\\le -A_{\\mathsf{T}^\\star}$, and decides $\\mathcal{H}_1$ if $L_{\\mathsf{T}^\\star}\\ge B_{\\mathsf{T}^\\star}$.\n\\end{enumerate}\n\\end{theorem}\n\nFor the scheme given in Theorem \\ref{thm1}, $\\mathsf{T}^\\star\\le N$ is guaranteed by noting that $-A_N=B_N=\\log \\frac{\\mu_0\\pi_0}{\\mu_1\\pi_1}$, and $(-A_N, B_N)$ is an empty set. In other words, any value of $L_N$ results in stopping. In specific, $L_N\\ge B_N$ gives decision $\\delta_N=1$, and $L_N\\le -A_N$ gives decision $\\delta_N=0$. Since $L_N=-A_N=B_N=\\log \\frac{\\mu_0\\pi_0}{\\mu_1\\pi_1}$ holds with zero probability, the equality situation for decision can be ignored in this case. \nTheorem \\ref{thm1} reveals the important structure of the optimal solution to \\eqref{P1_Bayes}, while the specific values of $A_t, B_t$ and $\\psi_{t+1}(L_t)$ need to be evaluated by solving the dynamic program \\eqref{G_BI}. In specific, in the posterior domain, the continuation region (i.e., the sequential test stops if the posterior goes beyond this region) and the selection region for sensor $\\ell$ are given respectively by\n\\begin{align}\n&{\\cal R}_t\\triangleq \\{\\pi_1(t): \\phi(\\pi_1(t))\\ge \\widetilde {\\cal G}_t^N(\\pi_1(t))\\},\\label{cont_reg}\\\\\n& {\\cal D}_t^\\ell\\triangleq\\left\\{\\pi_1(t): \\ell=\\arg\\;\\min_{\\delta}\\;{\\overline{\\cal G}_t^N(\\pi_1(t),\\delta)}\\right\\}, \\;\\ell=1, \\ldots, K.\\label{selection_reg}\n\\end{align}\nTransforming ${\\cal R}_t$ and ${\\cal D}^\\ell_t$ into the LLR domain according to \\eqref{posterior}, which we denote as $\\widetilde{\\cal R}_t$ and $\\widetilde {\\cal D}^\\ell_t$, then the thresholds in Theorem \\ref{thm1} are evaluated as \n\\begin{align}\\label{thres_evaluation}\nA_t=-\\min \\{L_t:L_t\\in \\widetilde{\\cal R}_t\\}, \\quad B_t=\\max \\{L_t:L_t\\in \\widetilde{\\cal R}_t\\}.\n\\end{align}\nMoreover, Lemma 3 and \\eqref{selection_reg} indicate that the selection strategy boils down to finding the minimum of $K$ concave functions, i.e., $\\overline{\\cal G}_n^N(\\pi_1(t),\\delta), \\; \\delta=1,\\ldots, K$, in the domain of posterior. Since concave functions are nicely behaved functions, the resulting selection scheme essentially partitions the domain of posterior into a finite number of intervals (assuming $K$ is finite) and assign each interval with the sensor index, whose value of $\\overline{\\cal G}_n^N$ is minimum within that interval. This observation suggests that, once computed offline, the sensor selection strategy can be easily stored in the fusion center. In practice, the recursion \\eqref{G_BI}, the sensor selection function \\eqref{selection_reg}, and the stopping rule \\eqref{cont_reg} and \\eqref{thres_evaluation} are implemented by discretizing the domain of posterior $\\pi_1(t)$. We summarize this procedure in Algorithm \\ref{DP}, where ${\\boldsymbol\\nu}$ and ${\\boldsymbol{L}}$ are vectors containing the discrete values of $\\pi_1(t)$ and $L_t$ respectively, ${\\cal G}({\\boldsymbol\\nu},t)$ and $\\psi({\\boldsymbol\\nu}, t+1)$ and $\\psi({\\boldsymbol{L}}, t+1)$ are vectors formed by evaluating the function for each element of ${\\boldsymbol\\nu}$ and $\\boldsymbol{L}$, representing the functions ${\\cal G}_t^N(\\pi_1(t))$ and $\\psi_{t+1}(\\pi_1(t))$, and $\\psi_{t+1}(L_t)$ respectively. The expectation $\\mathbb{E}(\\cdot)=\\pi_0\\mathbb{E}(\\cdot|{\\cal H}_0)+\\pi_1\\mathbb{E}(\\cdot|{\\cal H}_1)$ therein is taken w.r.t. the distribution of random sample $X$, and is evaluated by numerical integration. The output $\\psi({\\boldsymbol{L}}, t+1), t=0, 1, \\ldots, N-1$ (i.e., a sequence of vectors) and $\\{A(t), B(t)\\}$ give the selection function and decision thresholds respectively, and ${\\cal G}(\\pi_1, 0)$ gives the optimal cost ${\\cal G}_0^N(\\pi_1)$ (or equivalently, ${\\cal V}_0^N(\\pi_1)$), which will be used in Section IV.\n\\begin{algorithm}\n\\caption{\\bf : Procedure for computing $A_t, B_t$ and $\\psi_{t+1}(L_t)$ in Theorem \\ref{thm1}}\n\\begin{algorithmic}[1]\\label{DP}\n\\STATE {\\bf Input:} $N, \\pi_1, \\mu_0, \\mu_1, \\{\\lambda_j\\}_{j\\in\\Omega}$, the distributions of $X$ under ${\\cal H}_0$ and ${\\cal H}_1$\\\\\n\\STATE {\\bf Initialization:} \n\\\\${\\cal G}({\\boldsymbol\\nu},N)\\leftarrow \\min\\left(\\mu_1{\\boldsymbol\\nu},\\mu_0(1-{\\boldsymbol\\nu})\\right)$, $\\psi({\\boldsymbol\\nu},N) \\leftarrow 0$, $\\boldsymbol{L}\\leftarrow \\log \\frac{\\pi_0\\boldsymbol\\nu}{\\pi_1(1-\\boldsymbol\\nu)}$\n\\STATE {\\bf for } $t=N-1$ to $0$ {\\bf do}\n\\STATE Evaluate selection function at $t+1$: \\\\$\\psi({\\boldsymbol\\nu},t+1)\\leftarrow\\arg \\min_{\\delta}\\left\\{\\mathcal{C}_{\\delta}+\\mathbb{E}\\left[ {\\cal G}(\\frac{{\\boldsymbol\\nu} e^{l_{\\delta}(X)}}{1-{\\boldsymbol\\nu}+{\\boldsymbol\\nu} e^{l_{\\delta}(X)}},t+1)\\right]\\right\\}$\n\\STATE Update ``cost-to-go'': \\\\${\\cal G}({\\boldsymbol\\nu},t)\\leftarrow \\min\\left\\{\\min\\left(\\mu_1{\\boldsymbol\\nu},\\mu_0(1-{\\boldsymbol\\nu})\\right), \\mathcal{C}_{\\psi({\\boldsymbol\\nu},t+1)}+\\mathbb{E}\\left[ {\\cal G}(\\frac{{\\boldsymbol\\nu} e^{l_{\\psi(\\pi,t+1)}(X)}}{1-{\\boldsymbol\\nu}+{\\boldsymbol\\nu} e^{l_{\\psi(\\pi,t+1)}(X)}},t+1)\\right]\\right\\}$\n\\STATE Evaluate stopping thresholds:\\\\\n$a(t)\\leftarrow \\min\\left\\{\\nu\\in {\\boldsymbol\\nu}: \\min\\left(\\mu_1\\nu,\\mu_0(1-\\nu)\\right)\\ge\\mathcal{C}_{\\psi(\\nu,t+1)}+\\mathbb{E}\\left[ {\\cal G}(\\frac{\\nu e^{l_{\\psi(\\nu,t+1)}(X)}}{1-\\nu+\\nu e^{l_{\\psi(\\nu,t+1)}(X)}},t+1)\\right]\\right\\}$\\\\\n$b(t)\\leftarrow \\max\\left\\{\\nu\\in{\\boldsymbol\\nu}: \\min\\left(\\mu_1\\nu,\\mu_0(1-\\nu)\\right)\\ge\\mathcal{C}_{\\psi(\\nu,t+1)}+\\mathbb{E}\\left[ {\\cal G}(\\frac{\\nu e^{l_{\\psi(\\nu,t+1)}(X)}}{1-\\nu+\\nu e^{l_{\\psi(\\nu,t+1)}(X)}},t+1)\\right]\\right\\}$\n\n\\STATE Transform to the domain of LLR:\\\\\n$A(t)\\leftarrow -\\log\\frac{\\pi_0 a(t)}{\\pi_1(1-a(t))}$\\\\\n$B(t)\\leftarrow \\log\\frac{\\pi_0 b(t)}{\\pi_1(1-b(t))}$\\\\\n$\\psi({\\boldsymbol{L}},t+1)\\leftarrow \\psi(\\frac{\\pi_1e^{\\boldsymbol{L}}}{\\pi_0+\\pi_1e^{\\boldsymbol{L}}},t+1)$ (which is evaluated in step 4)\\\\\n\\STATE {\\bf end}\\\\\n\\STATE {\\bf Output:}\\\\ ${\\cal G}(\\pi_1, 0)$, $\\psi({\\boldsymbol{L}}, t+1), A(t), B(t)$ for $t=0, 1, \\ldots, N$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Infinite-Horizon Solution to the Bayes Problem}\n\nNext, by building on the finite-horizon results developed in the last subsection, we consider the infinite-horizon version of the problem in \\eqref{P1_Bayes}. \n\nThe essential step of bridging the two problems is to show that the finite-horizon case approaches the infinite-horizon case as $N\\to \\infty$ \\cite{Ferguson_book,SeqA_book,Xiaoou15}. Then the results in the last subsection can be readily generalized to the infinite-horizon scenario. Defining the optimal cost of the infinite-horizon Bayesian problem:\n\\begin{align}\\label{P1_Bayes_InfH}\n\\widetilde{\\mathcal{V}}(\\pi_1)&\\triangleq \\min_{\\{\\mathsf{T}, D_\\mathsf{T}, {\\boldsymbol\\delta}_{1:\\mathsf{T}}\\}} \\;\\mathcal{R}\\left({\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\right)\n\\end{align}\nwhere $\\pi_1$ is the prior on $\\mathcal{H}_1$. First, note that the optimal decision function derived in \\eqref{opt_decision} is independent of the horizon limit, thus $D^\\star_\\mathsf{T}$ in \\eqref{DecRule} can be substituted into \\eqref{P1_Bayes_InfH}, which gives the similar optimal stopping problem as that in \\eqref{Opt_stopping}:\n\\begin{align}\n\\widetilde{\\mathcal{V}}(\\pi_1)=\\min_{\\{\\mathsf{T}, {\\boldsymbol\\delta}_{1:\\mathsf{T}}\\}\\in {\\cal A}_0^\\infty} \\mathbb{E}\\left(\\sum_{t=1}^\\mathsf{T}{\\cal C}_{\\delta_t}+\\phi(\\pi_1(\\mathsf{T}))\\right).\n\\end{align}\nRecalling that ${\\cal V}_0^N(\\pi_1)= \\min_{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T} \\}, \\mathsf{T}\\le N}{\\cal R}\\left( {\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\right)$ according to \\eqref{V_n_N}, we have the following lemma.\n\\begin{lemma}\\label{lemma:FtoInF}\n$\\lim_{N\\to \\infty}\\mathcal{V}_0^N(\\pi_1)=\\widetilde{\\mathcal{V}}(\\pi_1)$ for all $\\pi_1\\in [0,1]$.\n\\end{lemma}\n\\proof\nLet $\\{{\\boldsymbol\\delta}^\\star_{1:{\\mathsf{T}^\\star}}, D^\\star_{\\mathsf{T}^\\star}, \\mathsf{T}^\\star\\}$ be the optimal solution to the infinite-horizon problem \\eqref{P1_Bayes_InfH}. Define the auxiliary procedure $\\{{\\boldsymbol\\delta}^\\star_{1:{\\widehat{\\mathsf{T}}_N}}, D^\\star_{\\widehat{\\mathsf{T}}_N}, \\widehat{\\mathsf{T}}_N\\}$ where $\\widehat{\\mathsf{T}}_N=\\min\\{\\mathsf{T}^\\star, N\\}$, then we have\n\\begin{align}\n&\\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{\\widehat{\\mathsf{T}}_N}}, D^\\star_{\\widehat{\\mathsf{T}}_N}, \\widehat{\\mathsf{T}}_N\\right)-\\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{{\\mathsf{T}}^\\star}}, D^\\star_{{\\mathsf{T}}^\\star}, {\\mathsf{T}}^\\star\\right)\\nonumber\\\\=&\\;\\mathbb{E}\\left(\\mathbbm{1}_{\\{\\mathsf{T}^\\star\\ge N\\}}\\left(\\phi\\left( \\pi_1(\\widehat{\\mathsf{T}}_N)\\right)-\\phi\\left( \\pi_1(\\mathsf{T}^\\star)\\right)-\\sum_{t=N+1}^\\infty\\mathcal{C}_{\\delta_t}\\right)\\rb\\nonumber\\\\\\le &\\; \\;\\mathbb{E}\\left(\\mathbbm{1}_{\\{\\mathsf{T}^\\star\\ge N\\}}\\left(\\phi\\left( \\pi_1(\\widehat{\\mathsf{T}}_N)\\right)\\rb\\right)\n\\label{fin-to-inf_1}\\\\=&\\;\\mathbb{E}\\left(\\phi\\left(\\pi_1(N)\\right) \\mathbbm{1}_{\\{\\widehat{\\mathsf{T}}_N=N\\}}\\right),\\label{fin-to-inf_2}\n\\end{align}\nwhere \\eqref{fin-to-inf_1} follows from the fact that $\\phi(\\pi_1(\\mathsf{T}^\\star))$ and ${\\cal C}_{\\delta_t}$ are positive, and \\eqref{fin-to-inf_2} is true because $\\widehat{\\mathsf{T}}_N=N$ holds with probability one given that $\\mathsf{T}^\\star\\ge N$ due to the definition of $\\widehat{\\mathsf{T}}_N$. \nUsing \\eqref{fin-to-inf_2} and the fact that ${\\cal V}_0^N(\\pi_1)$ is the optimal cost for all $\\mathsf{T}\\le N$ whereas $\\{{\\boldsymbol\\delta}^\\star_{1:{\\widehat{\\mathsf{T}}_N}}, D^\\star_{\\widehat{\\mathsf{T}}_N}, \\widehat{\\mathsf{T}}_N\\}$ is a constructed scheme for $\\mathsf{T}\\le N$, we arrive at the following inequalities\n\\begin{align}\n\\mathcal{V}_0^N(\\pi_1)\\le \\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{\\widehat{\\mathsf{T}}_N}}, D^\\star_{\\widehat{\\mathsf{T}}_N}, \\widehat{\\mathsf{T}}_N\\right)\\le \\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{{\\mathsf{T}}^\\star}}, D^\\star_{{\\mathsf{T}}^\\star}, {\\mathsf{T}}^\\star\\right)+ \\mathbb{E}\\left(\\phi\\left(\\pi_1(N)\\right) \\mathbbm{1}_{\\{\\widehat{\\mathsf{T}}_N=N\\}}\\right).\\label{limit}\n\\end{align}\nBy the strong law of large number, we know that $L_N\\to \\infty, \\;\\text{a.s.}$ as $N\\to \\infty$, thus $\\phi\\left(\\pi_1(N)\\right)=\\min\\{\\mu_0\\pi_0(N),\\mu_1\\pi_1(N)\\}\\to 0\\; \\text{a.s.}$ as $N\\to\\infty$ \\cite{Ferguson_book}. Taking $N\\to \\infty$ on both sides of \\eqref{limit}, we have\n\\begin{align}\\label{lemma5_1}\n\\lim_{N\\to \\infty}\\mathcal{V}_0^N(\\pi_1)\\le \\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{{\\mathsf{T}}^\\star}}, D^\\star_{{\\mathsf{T}}^\\star}, {\\mathsf{T}}^\\star\\right)=\\widetilde{\\cal V}(\\pi_1).\n\\end{align}\nOn the other hand, ${\\cal V}_0^N(\\pi_1)\\ge \\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{{\\mathsf{T}}^\\star}}, D^\\star_{{\\mathsf{T}}^\\star}, {\\mathsf{T}}^\\star\\right)$, since ${\\cal V}_0^N(\\pi_1)$ is the minimal cost for the finite-horizon problem, i.e., $\\mathsf{T}\\le N$, whereas $\\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{{\\mathsf{T}}^\\star}}, D^\\star_{{\\mathsf{T}}^\\star}, {\\mathsf{T}}^\\star\\right)$ is the minimal cost for the infinite-horizon problem, where no bound on $\\mathsf{T}$ is imposed. Thus, we have $\\lim_{N\\to \\infty} {\\cal V}_0^N(\\pi_1)\\ge \\mathcal{R}\\left( {\\boldsymbol\\delta}^\\star_{1:{{\\mathsf{T}}^\\star}}, D^\\star_{{\\mathsf{T}}^\\star}, {\\mathsf{T}}^\\star\\right)=\\widetilde{\\cal V}(\\pi_1)$ that, together with \\eqref{lemma5_1}, completes the proof.\n\\endproof\n\nMeanwhile, in the finite-horizon solution \\eqref{G_BI}, since $\\mathcal{G}_n^N(\\pi_1(n))$ is a function of the homogenous Markov chain $\\pi_1(n)$, we have $\\mathcal{G}_n^N(x)=\\mathcal{G}_0^{N-n}(x)=\\mathcal{V}_0^{N-n}(x)$. The first equality follows from the homogeneity property, and second equality follows from definitions. Therefore, the backward induction \\eqref{G_BI} can be equivalently expressed as the recursion\n\\begin{align}\n&{\\mathcal{V}}_0^{N-n}(x)=\n\\min\\left\\{\\phi\\left( x\\right), \\min_{\\delta_{n+1}}\\left[\\mathcal{C}_{\\delta_{n+1}}+\\mathbb{E}\\left(\\mathcal{V}_{0}^{N-n-1}\\left(\\frac{x\\exp\\left( l_{\\delta_{n+1}}\\right)}{1-x+x\\exp\\left( l_{\\delta_{n+1}}\\right)}\\right)\\rb\\right]\\right\\},\n\\end{align}\n\\ignore{\n \\begin{align}\n{\\mathcal{V}}_0^{N}(x)&=\n\\min\\left\\{\\phi\\left( x\\right), \\min_{\\delta}\\left[\\mathcal{C}_{\\delta}+\\mathbb{E}\\left(\\left.\\mathcal{V}_{0}^{N-1}\\left(\\frac{x\\exp\\left( l_{\\delta}\\right)}{1-x+x\\exp\\left( l_{\\delta}\\right)}\\right)\\right|x\\right)\\right]\\right\\},\n\\end{align}}\nwith $\\mathcal{V}_0^0(x)=\\phi(x)$.\nBy letting $N\\to \\infty$, and invoking Lemma \\ref{lemma:FtoInF}, we arrive at\n\\begin{align}\\label{Bellman}\n&\\widetilde{\\mathcal{V}}(x)=\n\\min\\left\\{\\phi\\left( x\\right), \\min_{\\delta}\\left[\\mathcal{C}_{\\delta}+\\mathbb{E}\\left(\\widetilde{\\mathcal{V}}\\left(\\frac{x\\exp\\left( l_{\\delta}\\right)}{1-x+x\\exp\\left( l_{\\delta}\\right)}\\right)\\rb\\right]\\right\\}.\n\\end{align}\nThis is the Bellman equation for the infinite-horizon Bayesian problem \\eqref{P1_Bayes_InfH}. Note that, thanks to Lemma \\ref{lemma:FtoInF}, $\\widetilde{\\cal V}\\left( x\\right)$ preserves the concavity of $\\mathcal{V}_0^N$. Therefore, \\eqref{Bellman} reveals that the stopping boundaries under infinite-horizon are constants. Moreover, the sensor selection function $\\delta_{t+1}$ depends only on the posterior\/LLR, and is independent of time. We summarize the optimal solution to the infinite-horizon problem in the theorem below. \n\\begin{theorem}\\label{thm2}\nThe optimal procedure that solves \\eqref{P1_Bayes} features an SPRT with stationary sensor selection strategy, i.e.,\n\\begin{enumerate}\n\\item The optimal sensor selection rule is a time-invariant function of the likelihood raito, i.e., $\\delta^\\star_{t+1}=\\psi(L_t)$. \n\\item The stopping rule is in the form of the SPRT $\\mathsf{T}^\\star=\\min\\{t: L_t\\notin (-A, B)\\}$.\n\\item The optimal decision rule $D^\\star_{\\mathsf{T}^\\star}$ decides $\\mathcal{H}_0$ if $L_{\\mathsf{T}^\\star}\\le -A$, and decides $\\mathcal{H}_1$ if $L_{\\mathsf{T}^\\star}\\ge B$.\n\\end{enumerate}\nThe function $\\psi(L_t)$ and the thresholds $A, B$ can be evaluated numerically by solving the Bellman equation \\eqref{Bellman}.\n\\end{theorem}\nThe proof for Theorem \\ref{thm2} follows similarly to that of Theorem \\ref{thm1} by using the Bellman equation \\eqref{Bellman}. In brief, $\\widetilde {\\cal V}(x)$ and ${\\cal E}(x) \\triangleq \\min_\\delta \\left[{\\cal C}_\\delta+\\mathbb{E}\\left( \\widetilde{\\cal V}\\left( \\frac{x \\exp(l_\\delta)}{\\left( 1-x+x\\exp(l_\\delta)\\right)}\\right)\\rb\\right]$ can be proved to be concave functions with ${\\cal E}(0)>0$ and ${\\cal E}(1)>0$ by letting $N\\to \\infty$ in Lemma 3; then the operation $\\min_\\delta$ in ${\\cal E}(x)$ indicates that the selection rule is a time-invariant function of the posterior, leading to Theorem 2-(1); moreover, analogous to \\eqref{Lemma4_Eq} in Lemma 3, the stopping thresholds are given by the roots for $\\mu_0(1-x)={\\cal E}(x)$ and $\\mu_1x={\\cal E}(x)$ which are constants, leading to Theorem 2-(2). The key difference here is that ${\\cal E}(x)$ is independent of $n$ in contrast with $\\widetilde {\\cal G}_n^N(x)$ in the proof of Theorem 1. Interestingly, Theorem \\ref{thm2} implies that the stopping thresholds and selection strategy of the infinite-horizon Bayesian problem converge to a sequential procedure that, in essence, is a combination of the SPRT and stationary sensor selection function $\\psi(L_t)$. Several approaches are available to solve the Bellman equation for $\\psi(L_t)$ and $A, B$. In this work, by virtue of Lemma \\ref{lemma:FtoInF}, we solve a finite-horizon problem with sufficiently large $N$ to approximately obtain them, which will be explained in Section IV.\n\\subsection{Optimal Solution to the Usage-Constrained Problem}\n\nNow that the Bayesian optimal stopping problem is solved in the previous subsections, we are ready to establish the optimal sequential procedure for \\eqref{P1} as follows. \n\\begin{theorem}\\label{thm3}\nLet ${\\boldsymbol\\mu}\\triangleq [\\mu_0, \\mu_1]$ be chosen such that the reliability constraints are satisfied with equalities; let ${\\boldsymbol\\lambda}\\triangleq \\{\\lambda_j\\}_{j\\in \\Omega}$ be chosen such that all usage constraints are satisfied, and moreover, the usage constraints for the sensors in $\\Omega_c\\triangleq\\{\\ell:\\lambda_\\ell>0\\}$ are satisfied with equalities. Then the optimal sequential procedure given by Theorems \\ref{thm1} and \\ref{thm2} give the optimal triplets $\\{\\mathsf{T}^\\star, D^\\star_{\\mathsf{T}^\\star}, {\\boldsymbol\\delta}^\\star_{1:\\mathsf{T}^\\star}\\}$ that solve the constrained problem \\eqref{P1} in finite-horizon and infinite-horizon scenarios, respectively.\n\\end{theorem}\n\\proof\nThe proofs are the same for finite-horizon and inifite-horizon problems, thus we only show the latter for conciseness. \n\nConsidering the results in Section III-A\\&B, we have $\\mathcal{R}\\left( {\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\right)\\ge \\mathcal{R}\\left( {{\\boldsymbol\\delta}^\\star}_{1:{\\mathsf{T}^\\star}}, D^\\star_{\\mathsf{T}^\\star}, \\mathsf{T}^\\star\\right)$ for any procedure $\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}$. That is\n\\begin{align}\n\\mathbb{E}\\mathsf{T}+&\\mu_0\\pi_0\\mathbb{P}_0\\left( D_\\mathsf{T}=1\\right)+\\mu_1\\pi_1\\mathbb{P}_1\\left( D_\\mathsf{T}=0\\right) +\\sum_{\\ell\\in\\Omega_c}\\lambda_\\ell \\mathbb{E}\\left(\\sum_{t=1}^\\mathsf{T}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right)\\nonumber\\\\&\\ge\\mathbb{E}\\mathsf{T}^\\star+\\mu_0\\pi_0\\mathbb{P}_0\\left( D^\\star_{\\mathsf{T}^\\star}=1\\right)+\\mu_1\\pi_1\\mathbb{P}_1\\left( D^\\star_{\\mathsf{T}^\\star}=0\\right)+\\sum_{\\ell\\in\\Omega_c}\\lambda_\\ell\\mathbb{E}\\left(\\sum_{t=1}^{\\mathsf{T}^\\star}\\mathbbm{1}_{\\{\\delta^\\star_t=\\ell\\}}\\right)\\nonumber\\\\&=\\mathbb{E}\\mathsf{T}^\\star+\\mu_0\\pi_0\\alpha+\\mu_1\\pi_1\\beta+\\sum_{\\ell\\in\\Omega_c} \\lambda_\\ell T^\\ell.\n\\end{align}\nNote that $\\mu_0\\ge 0$, $\\mu_1\\ge 0$ and $\\lambda_\\ell> 0$ for $\\ell\\in\\Omega_c$, thus $\\mathbb{E}\\mathsf{T}\\ge \\mathbb{E}\\mathsf{T}^\\star$ must hold true for any procedure $\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}\\in {\\bf C}\\left( \\alpha, \\beta, \\{T^\\ell\\}_{\\ell\\in \\Omega}\\right)$.\n\\endproof\nThe insight for Theorem \\ref{thm3} is intuitive. The sensors in $\\Omega_c$ (referred to as the effective set henceforth) will be overused without imposing the constraint, thus additional sampling cost $\\lambda_\\ell>0$ is assigned to penalize their usages (recall the definition of $\\mathcal{C}_{\\delta_t}$ in \\eqref{Bayes_obj}). Nevertheless, in order to optimize the test performance, they should be used at full capacity, i.e., usage constraints are satisfied with equalities. Section IV will address how we obtain $\\Omega_c$ from a general set $\\Omega$ that are under usage constraints in the formulation \\eqref{P1}.\n\n\nNext, we investigate the performance of the optimal sequential procedure under infinite-horizon. The challenge stems from the fact that random samples are no longer i.i.d., and the typical method based on Wald's identity fails to given valid performance analysis. However, by capitalizing on the optimal structures revealed in Theorems 2 and 3, we can derive an insightful bound to approximately characterize the performance. Define the Kullback-Leibler divergence (KLD):\n\\begin{align}\n\\mathcal{D}^\\ell_i\\left( f^{\\ell}_i||f^{\\ell}_j\\right)\\triangleq \\mathbb{E}_i\\left(\\log\\frac{f_i^\\ell(X)}{f_j^\\ell(X)}\\right).\n\\end{align}\n\\begin{proposition}\\label{cor1}\nBased on the Wald's approximation \\cite{SeqA_book} (i.e., $L_{\\mathsf{T}^\\star}\\approx -A$ given $D^\\star_{\\mathsf{T}^\\star}=0$ or $L_{\\mathsf{T}^\\star}\\approx B$ given $D^\\star_{\\mathsf{T}^\\star}=1$), the expected sample size for the optimal procedure for the infinite-horizon problem of \\eqref{P1} is lower bounded by\n\\begin{align}\\label{Bound}\n&\\mathbb{E}\\mathsf{T}^\\star\\ge\\nonumber\\\\\n&\\pi_0\\frac{\\mathcal{D}\\left( \\alpha||1-\\beta\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} \\mathcal{D}_0^\\ell}+\\pi_1\\frac{\\mathcal{D}\\lb1-\\beta||\\alpha\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} \\mathcal{D}_1^\\ell}-\\sum_{\\ell\\in\\Omega_c}\\left(\\max\\left\\{\\frac{\\mathcal{D}^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} \\mathcal{D}_1^\\ell},\\frac{\\mathcal{D}^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} \\mathcal{D}_0^\\ell}\\right\\}-1\\right) T^\\ell,\n\\end{align}\nwhere $\\mathcal{D}\\left( p||q\\right) \\triangleq p\\log \\frac{p}{q}+(1-p)\\log\\frac{1-p}{1-q}$ is the KLD of binary distribuitons, and $\\overline{\\Omega}_c\\triangleq \\Pi\\backslash_ {\\Omega_c}$ contains all sensors except those in $\\Omega_c$.\n\\end{proposition}\n\\proof\nSee Appendix.\n\\endproof\nThe performance characterization agrees with intuition. The first two terms on right-hand side of \\eqref{Bound} characterize the asymptotic performance of the optimal sequential procedure as $\\alpha$ and $\\beta$ go to zero, or $\\mathcal{D}\\left( \\alpha||1-\\beta\\right)$ and $\\mathcal{D}\\lb1-\\beta||\\alpha\\right)$ go to infinity. It is seen that the asymptotic expected sample size is determined by the KLDs of the sensors in $\\overline{\\Omega}_c$, i.e., the free sensors that do not reach their full usage. This result is consistent with that in \\cite{Javidi10}, where all sensors are constraint-free. Meanwhile, the third term on the right-hand side of \\eqref{Bound} accounts for the effect of the fully used sensors, which depends on their KLDs compared to that of the free sensors. If $\\max\\left\\{\\frac{\\mathcal{D}^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} \\mathcal{D}_1^\\ell},\\frac{\\mathcal{D}^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} \\mathcal{D}_0^\\ell}\\right\\}> 1$, then sensor $\\ell$ decreases the expected sample size due to its larger KLDs; otherwise, sensor $\\ell$ increases the expected sample size.\n\\section{Parameters Design for the Optimal Sequential Test}\\label{sec:algorithm}\n\nIn previous sections, we derived the optimal solutions to \\eqref{P1} under both finite-horizon and infinite-horizon setups, given that ${\\boldsymbol\\mu}$ and ${\\boldsymbol\\lambda}$ are set to satisfy certain conditions as given in Theorem \\ref{thm3}. These multipliers determine the parameters in the optimal sequential test and selection function, i.e., $A_t, B_t, \\psi_{t+1}(L_t)$ for finite-horizon, and $A, B, \\psi(L_t)$ for infinite-horizon. In practice, one can choose the multipliers by manually refining their values according to the simulation results; however, it is not an efficient approach, especially when the number of constraints is large. In this section, we propose a systematic approach to approximately evaluate the multipliers, which involves minimizing a concave function. \n\nBy drawing on the idea of the recent work \\cite{Fauss15}, we evaluate the multipliers by introducing the dual problem of \\eqref{P1}:\n\\begin{align}\\label{DualProb}\n\\max_{\\{{\\boldsymbol\\lambda}, {\\boldsymbol\\mu}\\}\\in\\mathbb{R}^+} \\min_{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}}\\mathcal{L}(\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu}),\n\\end{align}\nwhere the Lagrangian admits\n\\begin{align}\n\\mathcal{L}(\\{{\\boldsymbol\\delta}_1^\\mathsf{T}, D_\\mathsf{T}, \\mathsf{T}\\}, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu})\\triangleq \\mathbb{E}\\mathsf{T}&+\\mu_0\\pi_0\\left(\\mathbb{P}_0\\left({D_\\mathsf{T}=1}\\right)-\\alpha\\right)\\nonumber\\\\&+\\mu_1\\pi_1\\left(\\mathbb{P}_1\\left({D_\\mathsf{T}=0}\\right)-\\beta\\right)+\\sum_{\\ell\\in\\Omega}\\lambda_\\ell \\left( \\sum_{t=1}^\\mathsf{T}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}-T^\\ell\\right)\\nonumber\\\\\n=\\mathcal{R}&({\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T})-\\mu_0\\pi_0\\alpha-\\mu_1\\pi_1\\beta-\\sum_{\\ell\\in\\Omega}\\lambda_\\ell T^\\ell.\n\\end{align}\nThe reason is that if there exist multipliers such that the constraints hold as equalities, they must reside in the saddle point as expressed in \\eqref{DualProb}. \n\nWe first begin with the $N$-horizon problem. Since the Bayesian problem is solved in Section III, \\eqref{DualProb} becomes \n\\begin{align}\\label{Dual2}\n\\max_{\\{{\\boldsymbol\\lambda}, {\\boldsymbol\\mu}\\}\\in \\mathbb{R}^+}\\widetilde{\\mathcal{L}}_N({\\boldsymbol\\lambda}, {\\boldsymbol\\mu})&\\triangleq \\underbrace{\\min_{\\{D, \\mathsf{T}, {\\boldsymbol\\delta}_{1:\\mathsf{T}}\\}}\\; \\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T} \\mathcal{C}_{\\delta_t}+\\mu\\left( D_\\mathsf{T}, \\mathcal{H}\\right)\\rb}_{{\\mathcal{V}}_0^N(\\pi_1, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu})}-\\sum_{\\ell\\in\\Omega}\\lambda_\\ell T^\\ell-\\mu_0\\pi_0\\alpha-\\mu_1\\pi_1\\beta,\n\\end{align}\nwhere $\\widetilde{\\mathcal{L}}_N({\\boldsymbol\\lambda}, {\\boldsymbol\\mu})$ is a concave function of ${\\boldsymbol\\lambda}$ and ${\\boldsymbol\\mu}$. Note that ${\\cal V}_0^N(\\pi_1, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu})$ is the same function as defined in \\eqref{V0N} while we explicitly show the variables ${\\boldsymbol\\lambda}$ and ${\\boldsymbol\\mu}$ here for clarity. \n\nNote that \\eqref{Dual2} is a constrained concave problem that still requires complex solving process, for example, the interior-point method \\cite{Boyd_book}. In this work, we propose a simple procedure based on gradient ascent. In brief, we first assume that the effective set of constraints $\\Omega_c$ is known, based on which, \\eqref{Dual2} can be recast into an unconstrained optimization problem; we then give the scheme for evaluating $\\Omega_c$. The detailed procedure includes the following steps:\n\\begin{itemize}\n\\item Given any $\\Omega_c$, it is known that the optimal multipliers $\\mu_0>0$, $\\mu_1>0$, $\\lambda_j> 0$ for $j\\in \\Omega_c$ and $\\lambda_j=0$ for $j\\in \\overline{\\Omega}_c$ (cf. Theorem \\ref{thm3}). Consequently, the original problem \\eqref{Dual2} can be reduced to an unconstrained problem by removing $\\lambda_j, \\;j\\in\\overline{\\Omega}_c$:\n\\begin{align}\\label{Dual2_unconstrained}\n\\max_{{\\boldsymbol\\lambda}_{\\Omega_c}, {\\boldsymbol\\mu}}\\widetilde{\\mathcal{L}}_N({\\boldsymbol\\lambda}_{\\Omega_c}, {\\boldsymbol\\mu})&\\triangleq {{\\mathcal{V}}_0^N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}, {\\boldsymbol\\mu})}-\\sum_{\\ell\\in\\Omega_c}\\lambda_\\ell T^\\ell-\\mu_0\\pi_0\\alpha-\\mu_1\\pi_1\\beta,\n\\end{align}\nwith ${\\boldsymbol\\lambda}_{\\Omega_c}\\triangleq \\{\\lambda_j\\}_{j\\in \\Omega_c}$, since the optimal values of $\\lambda_j, \\; j\\in\\Omega_c$ and ${\\boldsymbol\\mu}$ reside in the interior of the positiveness constraint. Now \\eqref{Dual2_unconstrained} can be solved with the gradient ascent algorithm. To this end, note that $\\mathcal{V}_0^N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}, {\\boldsymbol\\mu})$ can be obtained efficiently given any value of the variables ${\\boldsymbol\\mu}, {\\boldsymbol\\lambda}_{\\Omega_c}$ through the dynamic programming \\eqref{G_BI}, i.e., Algorithm 1. This allows us to approximate the gradients at the $t$th iteration by using small shifts $\\Delta_{\\boldsymbol\\lambda}$ and $\\Delta_{\\boldsymbol\\mu}$ for ${\\boldsymbol\\lambda}_{\\Omega_c}$ and ${\\boldsymbol\\mu}$ respectively.\n\\ignore{\\begin{align*}\n&\\nabla_{\\boldsymbol\\lambda}\\widetilde{\\mathcal{L}}_N({\\boldsymbol\\lambda}^{(t)},{\\boldsymbol\\mu}^{(t)})=\\nabla_{\\boldsymbol\\lambda}\\mathcal{V}^N({\\boldsymbol\\lambda}^{(t)},{\\boldsymbol\\mu}^{(t)})-[N_1;\\ldots; N_{|\\Omega_c|}],\\\\\n&\\nabla_{\\boldsymbol\\mu}\\widetilde{\\mathcal{L}}_N({\\boldsymbol\\lambda}^{(t)},{\\boldsymbol\\mu}^{(t)})=\\nabla_{\\boldsymbol\\lambda}\\mathcal{V}^N({\\boldsymbol\\lambda}^{(t)},{\\boldsymbol\\mu}^{(t)})-[\\alpha; \\beta],\n\\end{align*}\nwith\n\\begin{align*}\n\\nabla_{\\boldsymbol\\lambda}\\mathcal{V}_0^N({\\boldsymbol\\lambda}^{(t)},{\\boldsymbol\\mu}^{(t)})\\triangleq&\\left.\\frac{\\partial \\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu}^{(t)})}{\\partial {\\boldsymbol\\lambda}}\\right|_{{\\boldsymbol\\lambda}={\\boldsymbol\\lambda}^{(t)}}\\\\\\approx \\Big[&\\frac{\\mathcal{V}^N(\\pi, [\\lambda^{(t)}_1+\\Delta_\\lambda, \\lambda^{(t)}_2, \\ldots, \\lambda^{(t)}_L], {\\boldsymbol\\mu}^{(t)})-\\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, {\\boldsymbol\\mu}^{(t)})}{\\Delta_\\lambda};\\\\&\\frac{\\mathcal{V}^N(\\pi, [\\lambda^{(t)}_1, \\lambda^{(t)}_2+\\Delta_\\lambda, \\ldots, \\lambda^{(t)}_L], {\\boldsymbol\\mu}^{(t)})-\\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, {\\boldsymbol\\mu}^{(t)})}{\\Delta_\\lambda};\\nonumber\\\\&\\quad\\vdots\\nonumber\\\\& \\frac{\\mathcal{V}^N(\\pi, [\\lambda^{(t)}_1, \\ldots, \\lambda^{(t)}_L+\\Delta_\\lambda], {\\boldsymbol\\mu}^{(t)})-\\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, {\\boldsymbol\\mu}^{(t)})}{\\Delta_\\lambda}\\Big],\\\\\n\\nabla_{\\boldsymbol\\mu}\\mathcal{V}_0^N({\\boldsymbol\\lambda}^{(t)}, {\\boldsymbol\\mu}^{(t)})\\triangleq &\\left.\\frac{\\partial \\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, {\\boldsymbol\\mu})}{\\partial {\\boldsymbol\\mu}}\\right|_{{\\boldsymbol\\mu}={\\boldsymbol\\mu}^{(t)}}\\\\\\approx \\Big[&\\frac{\\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, [\\mu_0^{(t)}+\\Delta_\\mu; \\mu^{(t)}_1])-\\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, {\\boldsymbol\\mu}^{(t)})}{\\Delta_\\mu};\\nonumber\\\\& \\frac{\\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, [\\mu_0^{(t)}; \\mu_1^{(t)}+\\Delta_\\mu])-\\mathcal{V}^N(\\pi, {\\boldsymbol\\lambda}^{(t)}, {\\boldsymbol\\mu}^{(t)})}{\\Delta_\\mu}\\Big].\n\\end{align*}}\nMoreover, since ${\\boldsymbol\\mu}$ and ${\\boldsymbol\\lambda}_{\\Omega_c}$ are typically at different scales, for example, ${\\boldsymbol\\mu}$ are usually in the order of hundreds, while ${\\boldsymbol\\lambda}_{\\Omega_c}$ are fractional numbers, we apply the alternating minimization to speed up the convergence. Algorithm \\ref{Alg_la1} summarizes the procedure for evaluating the multipliers and the resulting parameters (i.e., $A_t, B_t, \\psi_{t+1}(L_t)$) for the finite-$N$ optimal sequential test, where $\\text{Alg}_1(\\cdot)$ invokes Algorithm 1. In addition, $p_t$ and $q_t$ are step-sizes obtained by backtracking line search \\cite{Boyd_book}, ${\\boldsymbol\\mu}_\\text{int}, {\\boldsymbol\\lambda}_\\text{int}$ are initial values to begin the iterations.\n\n\\item To obtain the effective set $\\Omega_c$, we add an outer iteration to Algorithm \\ref{Alg_la1}. In particular,\n\\begin{enumerate}\n\\item Begin with an empty set of effective usage constraints (i.e., $\\Omega_c=\\emptyset$).\n\\item Solve the problem \n\\begin{align}\\label{temp_problem}\n\\min_{\\{{\\boldsymbol\\delta}_{1:\\mathsf{T}}, D_\\mathsf{T}, \\mathsf{T}\\}\\in{\\bf C}_N\\left( \\alpha, \\beta, \\{T^\\ell\\}_{\\ell\\in \\Omega_c}\\right)}\\;\\; \\mathbb{E} \\mathsf{T}.\n\\end{align}\n\\item Evaluate the sensor usages based on the solution to \\eqref{temp_problem}, and find the set of sensors in $\\Omega$ whose constraints are violated (denoted as $\\Lambda$). Update the effective set $\\Omega_c\\leftarrow \\Omega_c\\cup\\Lambda$.\n\\item Go to step 2) and solve \\eqref{temp_problem} for the updated $\\Omega_c$.\n\\end{enumerate}\nThis loop of 2)-4) continues until no inequality constraints are violated. Upon termination, $\\Omega_c$ is effective set of constraints, whose associated multipliers are positive, whereas the rest of constraints are naturally satisfied with zero multipliers. \n\\end{itemize}\n\\begin{algorithm}\n\\caption{\\bf : Procedure for solving \\eqref{Dual2_unconstrained}}\n\\begin{algorithmic}[1]\\label{Alg_la1}\n\\STATE Initialization: $t\\leftarrow 0, {\\boldsymbol\\mu}^{(0)}\\leftarrow{{\\boldsymbol\\mu}_\\text{int}}, {\\boldsymbol\\lambda}_{\\Omega_c}^{(0)}\\leftarrow{\\boldsymbol\\lambda}_\\text{int}$\n\\STATE {\\bf while} $\\lVert\\nabla_{\\boldsymbol\\lambda}\\widetilde{\\mathcal{L}}^N({\\boldsymbol\\lambda}_{\\Omega_c}^{(t)},{\\boldsymbol\\mu}^{(t)})\\rVert_2>\\epsilon_0\\; \\text{or}\\;\\lVert\\nabla_{\\boldsymbol\\lambda}\\widetilde{\\mathcal{L}}^N({\\boldsymbol\\lambda}_{\\Omega_c}^{(t)},{\\boldsymbol\\mu}^{(t)})\\rVert_2>\\epsilon_1$ {\\bf do}\\\\\n\\;\\;{\\bf update ${\\boldsymbol\\mu}$:}\\\\\n\\STATE \\quad ${\\cal V}_0^N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)})\\leftarrow{\\cal G}(\\pi_1,0)\\leftarrow\\text{Alg}_1(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)})$\n\\STATE \\quad ${\\cal V}_0^N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)}+\\Delta_{\\boldsymbol\\mu})\\leftarrow{\\cal G}(\\pi_1,0)\\leftarrow\\text{Alg}_1(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)}+\\Delta_{\\boldsymbol\\mu})$ \n\\STATE \\quad Evaluate $\\widetilde {\\cal L}_N({\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)})$ and $\\widetilde {\\cal L}_N({\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)}+\\Delta_{\\boldsymbol\\mu})$ by its definition in \\eqref{Dual2_unconstrained}\n\\STATE \\quad Approximate the gradient $\\nabla_{{\\boldsymbol\\mu}}\\widetilde{\\mathcal{L}}_N(\\pi, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)})$ \n\\STATE \\quad Update ${\\boldsymbol\\mu}^{(t+1)}={\\boldsymbol\\mu}^{(t)}+p_t\\nabla_{{\\boldsymbol\\mu}}\\widetilde{\\mathcal{L}}_N(\\pi, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t)})$, where $p_t$ is the step-size computed by \\\\ \\quad backtracking line search\\\\\n\\;\\;{\\bf update ${\\boldsymbol\\lambda}$:}\\\\\n\\STATE \\quad ${\\cal V}_0^N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t+1)})\\leftarrow{\\cal G}(\\pi_1,0)\\leftarrow\\text{Alg}_1(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t+1)})$\n\\STATE \\quad ${\\cal V}_0^N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}+\\Delta_{\\boldsymbol\\lambda}, {\\boldsymbol\\mu}^{(t+1)})\\leftarrow{\\cal G}(\\pi_1,0)\\leftarrow\\text{Alg}_1(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}+\\Delta_{\\boldsymbol\\lambda}, {\\boldsymbol\\mu}^{(t+1)})$ \n\\STATE \\quad Evaluate $\\widetilde {\\cal L}_N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t+1)})$ and $\\widetilde {\\cal L}_N(\\pi_1, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}+\\Delta_{\\boldsymbol\\lambda}, {\\boldsymbol\\mu}^{(t+1)})$ by its definition in \\eqref{Dual2_unconstrained}\n\\STATE \\quad Approximate the gradient $\\nabla_{{\\boldsymbol\\lambda}}\\widetilde{\\mathcal{L}}_N(\\pi, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t+1)})$\n\\STATE \\quad Update ${\\boldsymbol\\lambda}_{\\Omega_c}^{(t+1)}={\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}+q_t\\nabla_{{\\boldsymbol\\lambda}}\\widetilde{\\mathcal{L}}_N(\\pi, {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}, {\\boldsymbol\\mu}^{(t+1)})$ where $q_t$ is the step-size computed by \\\\ \\quad backtracking line search\n\\STATE \\quad $t \\leftarrow t+1$\n\\STATE {\\bf end while}\n\\STATE Output:\\\\ ${\\boldsymbol\\lambda}_{\\Omega_c}^\\star\\leftarrow {\\boldsymbol\\lambda}_{\\Omega_c}^{(t)}$, ${\\boldsymbol\\mu}^\\star\\leftarrow {\\boldsymbol\\mu}^{(t)}$, \n$\\left\\{\\psi(\\boldsymbol{L}, t), A(t), B(t)\\right\\}_{t=0}^N\\leftarrow \\text{Alg}_1(\\pi_1,{\\boldsymbol\\lambda}_{\\Omega_c}^{\\star},{\\boldsymbol\\mu}^{\\star})$\n\\end{algorithmic}\n\\end{algorithm}\n\\ignore{\\begin{algorithm}\n\\caption{\\bf : Evaluating the Effective Set of Usage Constraints}\n\\begin{algorithmic}[1]\\label{Alg_la2}\n\\STATE Initialization: $\\Omega_c\\leftarrow \\emptyset, \\Lambda\\leftarrow \\Omega, {\\boldsymbol\\lambda}_{\\Omega_c}\\leftarrow {\\bf 0}, {\\boldsymbol\\mu}\\leftarrow {\\boldsymbol\\mu}_\\text{int}$\n\\STATE {\\bf while} $\\Lambda\\neq \\emptyset$ {\\bf do}\n\\STATE\\quad Obtain the sequential test using Algorithm 1 with the current ${\\boldsymbol\\mu}$ and ${\\boldsymbol\\lambda}_{\\Omega_c}$\n\\STATE \\quad Evaluate the usages for all sensors and find $\\Lambda$ to be the set of sensors that do not meet \\\\\\quad usage constraints\n\\STATE \\quad Update the effective set: $\\Omega_c\\leftarrow \\Omega_c\\cup \\Lambda$\n\\STATE \\quad Solve \\eqref{Dual2_unconstrained} using Algorithm \\ref{Alg_la1} to update ${\\boldsymbol\\mu}$ and ${\\boldsymbol\\lambda}_{\\Omega_c}$\n\\STATE {\\bf end while}\n\\end{algorithmic}\n\\end{algorithm}}\n\nNext we consider the infinite-horizon scenario, whose evaluation of multipliers boils down to the following optimization problem:\n\\begin{align*}\n&\\max_{\\{{\\boldsymbol\\lambda}, {\\boldsymbol\\mu}\\}\\in \\mathbb{R}^+} \\; \\widetilde{\\mathcal{V}}(\\pi_1, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu})-\\mu_0\\pi_0\\alpha-\\mu_1\\pi_1\\beta-\\sum_{\\ell\\in\\Omega}\\lambda_\\ell T^\\ell\\\\\n&\\text{s.t.}\\; \\; \\widetilde{\\mathcal{V}}(x, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu})\\!=\\!\\min\\!\\left\\{\\!\\mu_0(1-x),\\mu_1x, \\min_{\\delta}\\left( \\!1\\!+\\!\\lambda_{\\delta}\\!+\\!\\mathbb{E}\\left(\\widetilde{\\mathcal{V}}(\\frac{xe^{l_\\delta}}{1-x+xe^{l_\\delta}}, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu})\\right)\\rb\\right\\}, \\; x\\in [0, 1].\n\\end{align*}\nOne option is to adopt the method in \\cite{Fauss15} (only SPRT and ${\\boldsymbol\\mu}$ were of interest there), which discretizes $x, {\\boldsymbol\\lambda}, {\\boldsymbol\\mu}$, and recasts the above problem into a linear program. However, this approach becomes computationally infeasible due to the high-dimensional variables in our problem. To that end, by the virtue of Lemma 5, we propose to approximate the infinite-horizon problem through finite-horizon approach \\eqref{Dual2}, i.e., $\\widetilde{\\mathcal{V}}\\approx\\mathcal{V}_0^N$ with sufficiently large $N$. \nMoreover, we obtain the multipliers and the resulting test parameters (i.e., $A, B, \\psi(L_t)$) for the optimal infinite-horizon sequential test by setting $A\\leftarrow A(0)$, $B\\leftarrow B(0)$, $\\psi(\\boldsymbol{L})\\leftarrow \\psi(\\boldsymbol{L},1)$, where $A(0), B(0)$ and $\\psi(\\boldsymbol{L},1)$ are the thresholds and selection function respectively evaluated for the finite-horizon problem with large $N$.\n\n\n\\ignore{\\begin{algorithm}\n\\caption{\\bf : Solving for Multipliers for Infinite-Horizon Problem}\n\\begin{algorithmic}[1]\\label{Alg_la_gen}\n\\STATE Initialization: $\\Omega_c\\leftarrow \\emptyset, \\Lambda\\leftarrow \\Omega, {\\boldsymbol\\lambda}\\leftarrow {\\bf 0}, {\\boldsymbol\\mu}\\leftarrow {\\boldsymbol\\mu}_\\text{int}$\n\\STATE Choose a sufficient large $N$ (the performance bound is helpful)\n\\STATE Run Algorithm \\ref{Alg_la1}-\\ref{Alg_la2}\n\\STATE Output: \n\\end{algorithmic}\n\\end{algorithm}}\n\\section{Numerical Results}\nIn this section, we provide numerical results to illustrate the theoretical findings in previous sections, and also to compare with the existing methods. Our experiments focus on the following hypotheses \n\\begin{align*}\n&\\mathcal{H}_0: X_t\\sim \\text{exp}\\left(\\eta_0^\\ell\\right), \\quad t=1, 2, \\ldots, \\quad \\ell\\in\\{1, 2, \\ldots, 4\\},\\\\\n&\\mathcal{H}_1: X_t\\sim \\text{exp}\\left(\\eta_1^\\ell\\right), \\quad t=1, 2, \\ldots, \\quad \\ell\\in\\{1, 2, \\ldots, 4\\}. \n\\end{align*}\n\\begin{table*}\n\\centering\n\\caption{}\n \\begin{tabular}{ | c | c | c | c | c |}\n \\hline\n \\phantom{1} & $\\eta^\\ell_0$ & $\\eta^\\ell_1$ & $D^\\ell_0$ &$D^\\ell_1$ \\\\ \\hline\n \n \n Sensor 1 & $0.5$ & $1$& $0.2692$ & $0.1739$ \\\\ \\hline\n Sensor 2 & $1$ & $0.5$ & $0.1739$ & $0.2692$ \\\\ \\hline \n Sensor 3 & $0.52$ & $1$ & $0.3069$ & $0.1931$\\\\ \\hline\n Sensor 4 & $1$ & $0.52$ & $0.1931$ & $0.3069$ \\\\ \\hline\n \n \n\n \\end{tabular}\\label{Table:models}\n\\end{table*} \nIn particular, the LLR at sensor $\\ell$ is \n\\begin{align}\nl^\\ell(X_t)=X_t\\left(\\eta^\\ell_0-\\eta^\\ell_1\\right)+\\log\\left( \\frac{\\eta^\\ell_1}{\\eta^\\ell_0}\\right)\n\\end{align}\nand the KLDs are expressed respectively as\n\\begin{align}\n&\\mathcal{D}_1^\\ell=\\mathbb{E}_0\\left( l^\\ell\\right)= \\frac{\\eta^\\ell_0}{\\eta^\\ell_1}-1-\\log\\left(\\frac{\\eta_0^\\ell}{\\eta^\\ell_1}\\right),\\\\\n&\\mathcal{D}_0^\\ell=\\mathbb{E}_0\\left( -l^\\ell\\right)= \\frac{\\eta^\\ell_1}{\\eta^\\ell_0}-1-\\log\\left(\\frac{\\eta^\\ell_1}{\\eta_0^\\ell}\\right).\n\\end{align}\nTable \\ref{Table:models} lists the distribution parameters and KLD for each sensor. Throughout the experiment, the domain of posterior $[0, 1]$ is discretized into $8000$ points to implement Algorithm 1. \n\\subsection{Finite-Horizon Scenario}\n\nWe first consider a finite-horizon problem with sample size limit $N=100$. \n\n\\begin{figure}\n\\centering\n\\subfigure[Unconstrained]{\\includegraphics[width=0.49\\textwidth]{Unconstrained_decision_region}}\n\\subfigure[$T^1=7, T^2=7$]{\\includegraphics[width=0.49\\textwidth]{Constrained_decision_region}}\n\\subfigure[$T^1=6, T^2=9$]{\\includegraphics[width=0.49\\textwidth]{Constrained_decision_region_2}}\n\\caption{The stopping boundaries and selection region for $N=100$. We set $\\alpha\\approx0.01, \\beta\\approx 0.01$. Black: sensor 1. Blue: sensor 2. Red: sensor 3. Green: sensor 4.\n\\label{fig:decision_region}\n\\end{figure}\n\nFig. \\ref{fig:decision_region} illustrates the decision region of the $N$-horizon sequential test, including the stopping boundaries (i.e., $[-A_t, B_t]$) and selection function (i.e., $\\psi_{t+1}(L_t)$). Note that, hereafter, we represent the results in terms of the sufficient statistic LLR, which is equivalent to the posterior given the prior. The black, blue, red, and green colors represent the intervals within which Sensor 1, 2, 3, and 4 should be selected respectively. The following observations are made:\n\\begin{itemize}\n\\item The curved stopping boundaries comply with the result in Theorem 1-(b). \n\\item The selection function $\\psi_{t+1}(L_t)$ in Theorem 1-(a) is represented by simple partitions of the LLR domain. In specific, the fusion center decides the selected sensor at $t+1$ based on the region that $L_t$ resides in. Interestingly, the selection function from $t=1\\to N$ is highly structured, and does not require large memory for storage.\n\\item The sensor usages are equal to the discrete time that LLR spends in the corresponding region before stopping. Thus the selection strategy controls the sensor usages by altering these selection regions. In Fig. \\ref{fig:decision_region}-(a), if all sensors are constraint-free, then Sensor 1 and Sensor 2 are always preferred over the other two. Intuitively Sensor 1 dominates sensor 3, Sensor 2 dominates Sensor 4, since their KLDs under both hypotheses are larger. In Fig. \\ref{fig:decision_region}-(b), if we impose the usage constraints on Sensors 1 and 2, then Sensors 3 and 4 are used more, thus the partition of LLR domain is reassigned to comply with the constraints. That is, the selection region for Sensor 1 is split mainly by Sensor 3, while that of Sensor 2 by Sensor 2. Fig. \\ref{fig:decision_region}-(c) shows that the selection regions alter as the usage constraints change from $T^1=6, T^2=9$ to $T^1=7, T^2=7$. In specific, the selection region of Sensor 1 shrinks while that of Sensor 2 expands. \n\\end{itemize}\n\nFrom Section III, we know that the selection regions, and thus the sensor usages, are governed by the multipliers, which are the parameters one can choose to meet the usage constraints. Bearing this in mind, Fig. \\ref{fig:UsagevsLas} illustrates how we can control the sensor usages by setting the values of multipliers. \nIn particular, Fig. \\ref{fig:UsagevsLas}-(a) shows that the usage of Sensor 1 decreases from the full usage to zero as $\\lambda_1$ increases, while other sensors increase their usages. Fig. \\ref{fig:UsagevsLas}-(b) shows that the usage of Sensor 2 decreases to zero as $\\lambda_2$ increases with fixed $\\lambda_1=0.15$. \\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.75\\textwidth]{Nhorizon_UsagevsLambda1}}\n\\subfigure[]{\n\\includegraphics[width=0.75\\textwidth]{Nhorizon_UsagevsLambda2}}\n\\caption{The sensor usage decreases as its associated multiplier increases. The error probabilities are set as $\\alpha\\approx0.0018, \\beta\\approx 0.0025$. (a) $\\lambda_2=0$; (b) $\\lambda_1=0.15$.}\\label{fig:UsagevsLas}\n\\end{figure}\n\nFinally, in Fig. \\ref{fig:ETvsErr}, we compare the proposed finite-$N$ sequential test with the existing method in \\cite{Bai15}, which is an offline random selection algorithm. \n\\ignore{The offline algorithm proposed in \\cite{Bai15} build on solving a relaxation of the sequential hypothesis testing problem:\n\\begin{align*}\n&\\min \\quad g({\\bf q})\\triangleq\\pi_0 \\frac{\\alpha B-(1-\\alpha)A}{{\\bf D}^0{\\bf q}}+\\pi_1 \\frac{(1-\\beta) B-\\beta A}{{\\bf D}^1{\\bf q}}\\\\\n&\\;\\text{s.t.}\\quad \\,\\; q_\\ell\\, g({\\bf q})\\le T^\\ell, \\;\\;\\ell\\in \\Omega_c\n\\end{align*}\nOne issue about this problem is that it is formulated from a relaxation of the original problem \\eqref{P1}, and in practice, it is usually the case that error rates depend on both $A,B$ and the sensor selection scheme. }\nThe comparison is carried out at varying error probabilities $\\alpha=\\beta$, and fixed sensor usage constraints for Sensor 1 and 2 ($T^1=6, T^2=9$, and Sensor 3 and 4 are free sensors). The corresponding multipliers are evaluated using the algorithm in Section IV. It is seen that the proposed online algorithm consistently outperforms the offline scheme with the same usage constraints and error probabilities. The improvement becomes more significant as the error probabilities decrease. Furthermore, Fig. \\ref{fig:UsagevsErr} depicts the sensor usages of the proposed scheme in this experiment. When error probabilities are moderate ($\\alpha=0.1 \\to 0.06$ in Fig. \\ref{fig:UsagevsErr}), Sensors 1 and 2 operate in free mode, and Sensors 3 and 4 are idle, which corresponds to the unconstrained scenario (i.e., the effective set of constraints are empty $\\Omega_c=\\emptyset$). This is similar to the case in Fig. \\ref{fig:decision_region}-(a). As error rates decrease ($\\alpha=0.04 \\;\\text{and}\\;0.02$), Sensor 1 reaches the usage constraint first, while Sensor 2 still operates in free mode (i.e., $\\Omega_c=\\{1\\}$). After $\\alpha\\le 0.01$, both Sensor 1 and 2 reach their usage limit and are under constraints (i.e., $\\Omega_c=\\{1, 2\\}$). In this regime, we find multipliers such that constraints are satisfied with equalities. As error rates further decrease, free sensors like Sensors 3 and 4 are used more often, while Sensor 1 and 2 remain maximum usages at $T^1=6$ and $T^2=9$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{ET_err}\n\\caption{Comparison of the proposed sequential test and the SPRT with offline random selection strategy.}\\label{fig:ETvsErr}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{Usage_err}\n\\caption{Sensor usages of the proposed scheme corresponding to the experiment in Fig. \\ref{fig:ETvsErr}.}\\label{fig:UsagevsErr}\n\\end{figure}\n\\subsection{Infinite-Horizon}\nIn this subsection, the performance of the proposed scheme in the infinite-horizon setup is examined. We use a finite-horizon problem with sufficiently large $N=200$ to approximately evaluate the parameters (i.e., $A$, $B$ and selection regions) of the optimal sequential test. \n\nAgain, Fig. \\ref{fig:Selection_region_200} depicts the decision regions for the finite-horizon problem with $N=200$. Since a larger $N$ is used, compared to Fig. \\ref{fig:decision_region}, Fig. \\ref{fig:Selection_region_200} shows that the stopping boundaries and section strategy converge to the stable one at $t=0$, which is approximately the infinite-horizon solution according to Lemma 5. Unlike in the finite-horizon scenario, the fusion center only needs to store stopping boundaries and selection regions at $t=0$, which is depicted in Fig. \\ref{fig:Inf_SelectionIntervals}, and use it for any $t$. This further lowers the storage demand. In specific, the selected sensor at $t+1$ is decided by which interval the LLR resides in at time $t$ within the stopping boundaries. We clearly see that the selection functions in Fig. \\ref{fig:Inf_SelectionIntervals}-(a) change to that in Fig. \\ref{fig:Inf_SelectionIntervals}-(b) as the usage constraints alter. \n\nFinally, in Fig. \\ref{fig:comparison}, we compare the proposed scheme with the existing offline random selection scheme in \\cite{Bai15}. Compared to Fig. \\ref{fig:ETvsErr}, the expected sample size slightly decreases due to the removal of the hard limit on horizon $N$. Again, the proposed online scheme increasingly outperforms the offline selection scheme as the error probabilities become small. In addition, we also plot the close-form approximation for the optimal performance, which is given by \\eqref{Bound}. Note that this analytical result (i.e., the red solid line) lies parallel to the performance curve of the proposed scheme (i.e., the black line with circle marks), indicating its accurate characterization for the asymptotical performance. The constant gap in between is largely caused by the inequality \\eqref{bound_2} that lower bounds the constant term (i.e., independent of $\\alpha$ and $\\beta$) in \\eqref{bound_1}, which ultimately leads to \\eqref{Bound}. Therefore, the constant gap can be small if \\eqref{bound_2} is tight, depending on the specific model. To see this, assuming that we derive the performance formula directly based on \\eqref{bound_1} (specifically, $T_0^\\ell$ and $T_1^\\ell$ in \\eqref{bound_1} need to be evaluated through simulation), it is shown in Fig. \\ref{fig:comparison} that the resulting lower bound (i.e., the green dash line) aligns closely to the performance of the proposed scheme. \n\\begin{figure}\n\\centering\n\\subfigure[$T^1=6, T^2=9$]{\\includegraphics[width=0.49\\columnwidth]{Constrained_decision_region_3}}\n\\subfigure[$T^1=7, T^2=7$]{\\includegraphics[width=0.49\\columnwidth]{Constrained_decision_region_77}}\n\\caption{The stopping boundaries and selection function for $N=200$. We set $\\alpha\\approx0.01, \\beta\\approx 0.01$. Black: sensor 1. Blue: sensor 2. Red: sensor 3. Green: sensor 4.}\\label{fig:Selection_region_200}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\subfigure[$T^1=6, T^2=9$]{\\includegraphics[width=0.49\\columnwidth]{SelectionFunction}}\n\\subfigure[$T^1=7, T^2=7$]{\\includegraphics[width=0.49\\columnwidth]{SelectionFunction77}}\n\\caption{The stopping boundaries and selection intervals for the infinite-horizon problem. }\\label{fig:Inf_SelectionIntervals}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.76\\textwidth]{ET_err_infH_bound}\n\\caption{Comparison of the proposed sequential test and the SPRT with offline random selection strategy.}\\label{fig:comparison}\n\\end{figure}\n\n\n\n\\section{Conclusions}\nIn this work, we have studied the sequential hypothesis testing with online sensor selection and sensor usage constraints. The optimal sequential test and selection strategy are obtained for both the finite-horizon and infinite-horizon scenarios. We have also proposed algorithms to approximately evaluate the parameters in the optimal sequential procedure. Finally, extensive numerical results have been provided to illustrate the theoretical findings and comparison with the existing method. Future works may include applying the same framework to address the usage-constrained sensor selection in other sequential problems, for example, change-point detection. Instead of the average sample size, other objective can also be studied, for example, the worst-case sample size. The applications in distributed sensor networks can be considered as well. For example, dynamic selection of quantization mode in the sequential detection \\cite{Nguyen06,Mei08}. \n\\section*{Appendix}\n\\proof[Proof of Lemma 1]\nWe want to prove that $g_n\\left( X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)=g_n\\left( \\pi_1(n)\\right)$.\nIt suffices to prove that for any realizations of $\\{X_{1:n},{\\boldsymbol\\delta}_{1:n}\\}$, i.e., $\\{x_{1:n}, s_{1:n}\\}$ and $\\{\\bar{x}_{1:{n}}, \\bar{s}_{1:n}\\}$, that lead to equal posteriors $\\pi_1(n)=\\bar{\\pi}_1({{n}})$, we have $g_n\\left( x_{1:n}, s_{1:n} \\right)=g_{{n}}\\left( \\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)$.\n\nConditioned on the event $\\{\\mathsf{T}=n\\}$, by \\eqref{stopping_rule}, it is obvious that $g_n\\left( x_{1:n}, s_{1:n} \\right)=g_{n}\\left( \\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)=\\phi\\left(\\pi_1(n)\\right)$. Conditioned on the event $\\{n<\\mathsf{T}\\le N\\}$, we will prove by contradiction. On one hand, assume that $g_n\\left( x_{1:n}, s_{1:n}\\right)>g_{n}\\left( \\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)$, then there exists a procedure $\\left\\{\\tilde{\\delta}_{n+1}, \\{\\tilde{{\\boldsymbol\\delta}}_{n+2:\\widetilde{\\mathsf{T}}}, \\widetilde{\\mathsf{T}}\\}\\in \\mathcal{A}_{n+1}^N\\right\\}$ (given $\\{{x}_{1:n}, {s}_{1:n}\\}$) such that\n\\begin{align}\\label{contradiction_N}\ng_n(x_{1:n},s_{1:n})&\\ge \\underbrace{\\phi\\left( \\pi_1(n)\\right) - \\left[\\mathbb{E}\\left( \\left.\\phi\\left(\\pi_1(\\widetilde{\\mathsf{T}})\\right)+\\sum_{t=n+1}^{\\widetilde{\\mathsf{T}}} \\mathcal{C}_{\\widetilde{\\delta}_t}\\right|X_{1:n}=x_{1:n}, {\\boldsymbol\\delta}_{1:n}=s_{1:n}\\right)\\right]}_{\\widetilde{g}_n\\left( x_{1:n}, s_{1:n}\\right)}\\nonumber\\\\&>{g_n\\left( \\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)},\n\\end{align}\ndue to the definition of $g_n$ in \\eqref{stopping_rule}.\n\n\nOn the other hand, we construct the following procedure $\\left\\{\\widehat{\\delta}_{n+1}, \\{\\widehat{{\\boldsymbol\\delta}}_{n+2:{\\widehat{\\mathsf{T}}}}, \\widehat{\\mathsf{T}}\\}\\in \\mathcal{A}^N_{n+1}\\right\\}$ (given $\\{\\bar{x}_{1:n}, \\bar{s}_{1:n}\\}$). Let \n\\begin{align}\\label{construction_1}\n\\widehat \\delta_{n+1}(\\bar{x}_1, \\ldots, \\bar{x}_n)=\\widetilde \\delta_{n+1}({x}_1, \\ldots, {x}_n),\n\\end{align} \nand, given the same samples after time $n$ (denoted as $x_{n+1}, x_{n+2}, \\ldots$), \n\\begin{align}\\label{construction_2}\n\\widehat \\delta_{t}(\\bar{x}_1, \\ldots, \\bar{x}_n, x_{n+1}, \\ldots, x_{t-1})=\\widetilde \\delta_{t}({x}_1, \\ldots, {x}_n, x_{n+1}, \\ldots, x_{t-1}), \\quad t=n+2, \\ldots, N. \n\\end{align}\nMoreover, let $\\widehat{\\mathsf{T}}$ stop if $\\widetilde{\\mathsf{T}}$ stops given the same samples $\\{x_{n+1}, x_{n+2},\\ldots\\}$, and the decision rule $$\\widehat D(\\bar{x}_1, \\ldots, \\bar{x}_n, x_{n+1}, \\ldots, x_{\\widehat\\mathsf{T}})=\\widetilde D(\\bar{x}_1, \\ldots, \\bar{x}_n, x_{n+1}, \\ldots, x_{\\widetilde\\mathsf{T}}).$$ { In short, the procedure $\\left\\{\\widehat{{\\boldsymbol\\delta}}_{n+1:{\\widehat{\\mathsf{T}}}}, \\widehat{\\mathsf{T}}\\right\\}$ is designed to yield the exact same actions as that of the procedure $\\left\\{\\widetilde{{\\boldsymbol\\delta}}_{n+1:{\\widetilde{\\mathsf{T}}}}, \\widetilde{\\mathsf{T}}\\right\\}$ given the same samples at time $n$, i.e., $\\{x_{n+1}, x_{n+2}, \\ldots\\}$.\nNote that, according to the above construction process, $\\left\\{\\widehat{{\\boldsymbol\\delta}}_{n+1:{\\widehat{\\mathsf{T}}}}, \\widehat{\\mathsf{T}}\\right\\}$ and $\\left\\{\\widetilde{{\\boldsymbol\\delta}}_{n+1:{\\widetilde{\\mathsf{T}}}}, \\widetilde{\\mathsf{T}}\\right\\}$ are not identical procedures since $\\{{x}_{1:n}, {s}_{1:n}\\}\\neq \\{\\bar{x}_{1:n}, \\bar{s}_{1:n}\\}$.}\n\nAgain, due to the definition of $g_n$ in \\eqref{stopping_rule}, we also have\n\\begin{align}\\label{proof_pre0}\ng_n\\left( \\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)\\ge\n\\underbrace{\\phi\\left( \\pi_1(n)\\right) - \\left[\\mathbb{E}\\left( \\left.\\phi\\left(\\pi_1(\\hat{\\mathsf{T}})\\right)+\\sum_{t=n+1}^{\\hat{\\mathsf{T}}} \\mathcal{C}_{\\hat{\\delta}_t}\\right|{X}_{1:n}=\\bar{x}_{1:n}, {{\\boldsymbol\\delta}}_{1:n}=\\bar{s}_{1:n}\\right)\\right]}_{\\widehat{g}_n\\left( \\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)}.\n\\end{align}\n\nNext, we prove that \n\\begin{align}\\label{proof_0}\n\\widehat{g}\\left(\\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)=\\widetilde{g}\\left({x}_{1:n}, {s}_{1:n}\\right),\n\\end{align}\nwhich requires \n\\begin{align}\\label{proof_1}\n\\mathbb{E}\\!\\!\\left(\\!\\! \\left.\\phi\\left(\\pi_1(\\widetilde{\\mathsf{T}})\\right)\\!\\!+\\!\\!\\sum_{t=n+1}^{\\widetilde{\\mathsf{T}}} \\mathcal{C}_{\\widetilde{\\delta}_t}\\right|{X}_{1:n}\\!=\\!{x}_{1:n}, {{\\boldsymbol\\delta}}_{1:n}\\!=\\!{s}_{1:n}\\right)\\!=\\!\\mathbb{E}\\!\\!\\left(\\!\\! \\left.\\phi\\left(\\pi_1(\\widehat{\\mathsf{T}})\\right)\\!\\!+\\!\\!\\sum_{t=n+1}^{\\widehat{\\mathsf{T}}}\\mathcal{C}_{\\widehat{\\delta}_t}\\right|{X}_{1:n}\\!=\\!\\bar{x}_{1:n}, {{\\boldsymbol\\delta}}_{1:n}\\!=\\!\\bar{s}_{1:n}\\right).\n\\end{align}\nFirst, due to the construction of $\\left\\{\\widehat{{\\boldsymbol\\delta}}_{n+1:{\\widehat{\\mathsf{T}}}}, \\widehat{\\mathsf{T}}\\right\\}$, we have \n\\begin{align}\\label{proof_2}\n\\left.\\widetilde{\\mathsf{T}}-n\\right|\\{X_{1:n}=\\bar{x}_{1:n}, {\\boldsymbol\\delta}_{1:n}=\\bar{s}_{1:n}\\}=\\left.\\widehat{\\mathsf{T}}-n\\right|\\{X_{1:n}={x}_{1:n}, {\\boldsymbol\\delta}_{1:n}={s}_{1:n}\\},\\quad \\text{a.s.}.\n\\end{align}\nTo show that the first terms on both sides of \\eqref{proof_1} are equal, i.e., \n\\begin{align}\n\\mathbb{E}\\left( \\left.\\phi\\left(\\pi_1(\\widetilde{\\mathsf{T}})\\right)\\right|X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)=\\mathbb{E}\\left( \\left.\\phi\\left(\\pi_1(\\bar{\\mathsf{T}})\\right)\\right|\\bar{X}_{1:n}, \\bar{{\\boldsymbol\\delta}}_{1:n}\\right),\n\\end{align}\nnotice that\n\\begin{align}\n\\pi_1(\\widetilde{\\mathsf{T}})=\\frac{\\pi_1(n)e^{\\sum_{t=n+1}^{\\widetilde{\\mathsf{T}}} l_{\\widetilde{\\delta}_t}}}{\\pi_0(n)+\\pi_1(n)e^{\\sum_{t=n+1}^{\\widetilde{\\mathsf{T}}} l_{\\widetilde{\\delta}_t}}}\n\\end{align}\nhas the same distribution conditioned on $\\{\\bar{x}_{1:n}, \\bar{s}_{1:n}\\}$ as that of \n\\begin{align}\n\\pi_1({\\widehat{\\mathsf{T}}})=\\frac{\\bar{\\pi}_1(n)e^{{\\sum_{t=n+1}^{{\\widehat\\mathsf{T}}} l_{\\widehat{\\delta}_t}}}}{\\bar{\\pi}_0(n)+\\bar{\\pi}_1(n)e^{\\sum_{t=n+1}^{\\widehat{\\mathsf{T}}} l_{\\widehat{\\delta}_t}}}\n\\end{align}\nconditioned on $\\{{x}_{1:n}, {s}_{1:n}\\}$. This is true because $\\pi_1(n)=\\bar{\\pi}_1(n)$ and $\\sum_{t=n+1}^{\\widetilde\\mathsf{T}} l_{\\widetilde\\delta_t}$ has the same posterior distribution as $\\sum_{t=n+1}^{\\widehat{\\mathsf{T}}} l_{\\widehat{\\delta}_t}$ due to \\eqref{construction_1}-\\eqref{construction_2} and \\eqref{proof_2}. In addition, the second terms on both sides of \\eqref{proof_1} are also equal by combining \\eqref{construction_1}-\\eqref{construction_2} and \\eqref{proof_2}.\n\nUsing \\eqref{proof_pre0}-\\eqref{proof_0}, we arrive at\n\\begin{align}\ng_{n}\\left( \\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)\\ge \\widehat{g}_{n}\\left(\\bar{x}_{1:n}, \\bar{s}_{1:n}\\right)=\\widetilde{g}_n\\left( x_{1:n}, s_{1:n}\\right)\n\\end{align}\nwhich contradicts with \\eqref{contradiction_N}. \n\nSimilar contradiction appears if we assume $g_n\\left( X_{1:n}, {\\boldsymbol\\delta}_{1:n}\\right)b_{n}$. At $t=N$, the procedure has to stop and make decision, thus $a_N=b_N$. $\\mu_1\\pi_1(N)\\,\\substack{>\\\\<}\\,\\mu_0\\pi_0(N)$ which gives $\\pi_1(N)\\,\\substack{>\\\\<}\\,a_N=\\mu_0\/(\\mu_0+\\mu_1)$.\n\\endproof\n\n\\proof[Proof of Proposition \\ref{cor1}]\nNote that for the LLR statistic, we have\n\\begin{align}\\label{Bound_0}\n\\mathbb{E}_0\\left( L_\\mathsf{T}\\right)=\\mathbb{E}_0\\left[\\sum_{t=1}^\\mathsf{T}\\left(\\sum_{\\ell\\in\\Omega_c}l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}+l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\right)\\right].\n\\end{align}\nThe first term of \\eqref{Bound_0} can be expressed as\n\\begin{align}\\label{Bound_0_term1}\n\\mathbb{E}_0\\left[\\sum_{t=1}^\\mathsf{T}\\sum_{\\ell\\in\\Omega_c}l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right]=&\\sum_{\\ell\\in\\Omega_c}\\mathbb{E}_0\\left( \\sum_{t=1}^\\infty l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\\nonumber\\\\=&\\sum_{\\ell\\in\\Omega_c}\\mathbb{E}_0\\left(\\sum_{t=1}^\\infty\\mathbb{E}_0\\left(\\left. l_{\\delta_t}\\right|X_{1:(t-1)}, {\\boldsymbol\\delta}_{1:t-1}\\right)\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\\nonumber\\\\=&-\\sum_{\\ell\\in\\Omega_c}D^\\ell_0\\,\\mathbb{E}_0\\left(\\sum_{t=1}^\\infty\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\\nonumber\\\\=&-\\sum_{\\ell\\in\\Omega_c} D_0^\\ell T_0^\\ell,\n\\end{align}\nwhere $D_0^\\ell\\triangleq \\mathbb{E}_0\\left( -l_\\ell \\right)$ is the KL divergence of sensor $\\ell$ and $T^\\ell_0\\triangleq \\mathbb{E}_0\\left(\\sum_{t=1}^\\mathsf{T} \\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right)$ is the mean usage under $\\mathcal{H}_0$. Furthermore, the second term of \\eqref{Bound_0} can be bounded as follows\n\\begin{align}\n\\mathbb{E}_0\\left[\\sum_{t=1}^\\mathsf{T} l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\right]=&\\mathbb{E}_0\\left(\\sum_{t=1}^\\infty l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\\nonumber\\\\=&\\mathbb{E}_0\\left(\\sum_{t=1}^\\infty \\mathbb{E}_0\\left(\\left. l_{\\delta_t}\\right|X_{1:(t-1)}, {\\boldsymbol\\delta}_{1:t-1}\\right)\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\\label{Bound_0_ineq}\\\\\\ge&- \\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell \\,\\mathbb{E}_0\\left(\\sum_{t=1}^\\infty \\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\\nonumber\\\\=&-\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell \\left(\\mathbb{E}_0\\mathsf{T}-\\sum_{\\ell\\in\\Omega_c}T_0^\\ell \\right),\\label{Bound_0_term2}\n\\end{align}\nwhere inequality \\eqref{Bound_0_ineq} holds because $\\mathbb{E}_0\\left(\\left. l_{\\delta_t}\\right|X_{1:(t-1)}, {\\boldsymbol\\delta}_{1:t-1}\\right)\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\ge \\min_{\\ell\\in\\overline{\\Omega}_c} \\mathbb{E}_0\\left( l_\\ell\\right)\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}$.\nApplying \\eqref{Bound_0_term1} and \\eqref{Bound_0_term2} to \\eqref{Bound_0} results in\n\\begin{align}\n\\mathbb{E}_0\\left( L_\\mathsf{T}\\right)\\ge -\\sum_{\\ell\\in \\Omega_c}D_0^\\ell T_0^\\ell -\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell \\left(\\mathbb{E}_0\\mathsf{T}-\\sum_{\\ell\\in\\Omega_c}T_0^\\ell\\right),\n\\end{align}\nwhich leads to the bound for mean sample size under $\\mathcal{H}_0$:\n\\begin{align}\n\\mathbb{E}_0\\mathsf{T}&\\ge \\left[-\\mathbb{E}_0\\left( L_\\mathsf{T}\\right)-\\sum_{\\ell\\in \\Omega_c}D_0^\\ell T_0^\\ell +\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell\\sum_{\\ell\\in\\Omega_c}T_0^\\ell\\right]\\frac{1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}\\nonumber\\\\&=\\frac{-\\mathbb{E}_0\\left( L_\\mathsf{T}\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}+\\sum_{\\ell\\in\\Omega_c}\\left( 1-\\frac{D^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}\\right) T^\\ell_0.\\label{Bound_H0}\n\\end{align}\nUnder $\\mathcal{H}_1$, similarly as in \\eqref{Bound_0_term1} and \\eqref{Bound_0_term2}, we have\n\\begin{align}\n\\mathbb{E}_1\\left[\\sum_{t=1}^\\mathsf{T}\\sum_{\\ell\\in\\Omega_c}l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right]&=\\sum_{\\ell\\in\\Omega_c} D_1^\\ell T_1^\\ell,\\\\\n\\text{and}\\quad \\mathbb{E}_1\\left[\\sum_{t=1}^\\mathsf{T} l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\right]&=\\mathbb{E}_1\\left(\\sum_{t=1}^\\infty l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\n\\nonumber\\\\&=\\mathbb{E}_1\\left(\\sum_{t=1}^\\infty \\mathbb{E}_1\\left( l_{\\delta_t}|\\mathcal{F}_{t-1}\\right)\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\n\\nonumber\\\\&\\le \\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell \\,\\mathbb{E}_1\\left(\\sum_{t=1}^\\infty \\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\mathbbm{1}_{\\{\\mathsf{T}\\ge t\\}}\\right)\n\\nonumber\\\\&=\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell \\left(\\mathbb{E}_1\\mathsf{T}-\\sum_{\\ell\\in\\Omega_c}T_1^\\ell \\right),\n\\end{align}\nthat lead to\n\\begin{align}\n\\mathbb{E}_1\\left( L_\\mathsf{T}\\right)&=\\mathbb{E}_1\\left[\\sum_{t=1}^\\mathsf{T}\\sum_{\\ell\\in\\Omega_c}l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t=\\ell\\}}\\right]+\\mathbb{E}_1\\left[\\sum_{t=1}^\\mathsf{T} l_{\\delta_t}\\mathbbm{1}_{\\{\\delta_t\\in \\overline{\\Omega}_c\\}}\\right]\\nonumber\\\\&\\le \\sum_{\\ell\\in \\Omega_c}D_1^\\ell T_1^\\ell +\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell \\left(\\mathbb{E}_1\\mathsf{T}-\\sum_{\\ell\\in\\Omega_c}T_1^\\ell\\right) .\n\\end{align}\nAs a result, we can bound the mean sample size under $\\mathcal{H}_1$ by\n\\begin{align}\n\\mathbb{E}_1\\mathsf{T}&\\ge \\left[\\mathbb{E}_1\\left( L_\\mathsf{T}\\right)-\\sum_{\\ell\\in \\Omega_c}D_1^\\ell T_1^\\ell +\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell\\sum_{\\ell\\in\\Omega_c}T_1^\\ell\\right]\\frac{1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell}\\nonumber\\\\&=\\frac{\\mathbb{E}_1\\left( L_\\mathsf{T}\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell}+\\sum_{\\ell\\in\\Omega_c}\\left( 1-\\frac{D^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell}\\right) T^\\ell_1.\\label{Bound_H1}\n\\end{align}\nFinally, the expected mean sample size, i.e., $\\mathbb{E}\\mathsf{T}=\\pi_0\\mathbb{E}_0\\mathsf{T}+\\pi_1\\mathbb{E}_1\\mathsf{T}$, can be bounded below as follows:\n\\begin{align}\n\\mathbb{E}\\mathsf{T}&\\ge \\pi_0\\frac{-\\mathbb{E}_0\\left( L_\\mathsf{T}\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}+\\pi_1\\frac{\\mathbb{E}_1\\left( L_\\mathsf{T}\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell}+\\sum_{\\ell\\in\\Omega_c}T^\\ell-\\sum_{\\ell\\in\\Omega_c}\\left( \\frac{\\pi_0D^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}T^\\ell_0+\\frac{\\pi_1D^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell}T^\\ell_1\\right)\\label{bound_1}\\\\&\\ge\\pi_0\\frac{-\\mathbb{E}_0\\left( L_\\mathsf{T}\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}+\\pi_1\\frac{\\mathbb{E}_1\\left( L_\\mathsf{T}\\right)}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell}+\\sum_{\\ell\\in\\Omega_c}\\lb1-\\max\\left\\{\\frac{D^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell},\\frac{D^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}\\right\\}\\right) T^\\ell,\\label{bound_2}\n\\end{align}\nwhere the second inequality is obtained by noting that $\\pi_0T^\\ell_0+\\pi_1T^\\ell_1=T^\\ell$, thus \n\\begin{align}\\label{bound_inq}\n\\frac{\\pi_0D^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}T^\\ell_0+\\frac{\\pi_1D^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell}T^\\ell_1\\le\\max\\left\\{\\frac{D^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell},\\frac{D^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}\\right\\}T^\\ell,\n\\end{align}\nwith equality holds if $T^\\ell=\\pi_iT_i^\\ell, \\; i=\\arg \\max \\left\\{\\frac{D^\\ell_1}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_1^\\ell},\\frac{D^\\ell_0}{\\max_{\\ell\\in\\overline{\\Omega}_c} D_0^\\ell}\\right\\}$.\n\nNext, by drawing on the Wald's approximation \\cite{SeqA_book}, i.e., $L_{\\mathsf{T}^\\star}\\approx -A$ given $D^\\star_{\\mathsf{T}^\\star}=0$ or $L_{\\mathsf{T}^\\star}\\approx B$ given $D^\\star_{\\mathsf{T}^\\star}=1$, we obtain\n\\begin{align}\n\\mathbb{E}_0\\left( L_{\\mathsf{T}^\\star}\\right)&=\\alpha \\mathbb{E}_0\\left( L_{\\mathsf{T}^\\star}|D^\\star_{\\mathsf{T}^\\star}=1\\right) +(1- \\alpha) \\mathbb{E}_0\\left( L_{\\mathsf{T}^\\star}|D^\\star_{\\mathsf{T}^\\star}=0\\right) \\nonumber\\\\&=\\alpha B-(1-\\alpha) A,\\label{appendix1}\\\\\n\\mathbb{E}_1\\left( L_{\\mathsf{T}^\\star}\\right)&=(1-\\beta) \\mathbb{E}_1\\left( L_{\\mathsf{T}^\\star}|D^\\star_{\\mathsf{T}^\\star}=1\\right) +\\beta \\mathbb{E}_1\\left( L_{\\mathsf{T}^\\star}|D^\\star_{\\mathsf{T}^\\star}=0\\right) \\nonumber\\\\&=(1-\\beta)B-\\beta A.\\label{appendix2}\n\\end{align}\nMoreover, invoking the change of measure technique and the Wald's approximation, we have\n\\begin{align}\n&\\alpha=\\mathbb{E}_0\\left(\\mathbbm{1}_{\\{D^\\star_{\\mathsf{T}^\\star}=1\\}}\\right)=\\mathbb{E}_1\\left(\\mathbbm{1}_{\\{D^\\star_{\\mathsf{T}^\\star}=1\\}}e^{-L_{\\mathsf{T}^\\star}}\\right)\\approx e^{-B}\\left( 1-\\beta\\right),\\\\&\\beta=\\mathbb{E}_1\\left(\\mathbbm{1}_{\\{D^\\star_{\\mathsf{T}^\\star}=0\\}}\\right)=\\mathbb{E}_0\\left(\\mathbbm{1}_{\\{D^\\star_{\\mathsf{T}^\\star}=0\\}}e^{L_{\\mathsf{T}^\\star}}\\right)\\approx e^{-A}\\left( 1-\\alpha\\right),\n\\end{align}\nwhich lead to\n\\begin{align}\\label{appendix3}\nB\\approx \\log \\frac{1-\\beta}{\\alpha}, \\quad A\\approx \\log \\frac{1-\\alpha}{\\beta}.\n\\end{align} \nSubstituting \\eqref{appendix3} into \\eqref{appendix1}-\\eqref{appendix2} gives\n$\\mathbb{E}_0(L_{\\mathsf{T}^\\star})\\approx -\\mathcal{D}(\\alpha||1-\\beta)$ and $\\mathbb{E}_1(L_{\\mathsf{T}^\\star})\\approx \\mathcal{D}(1-\\beta||\\alpha)$. \n\\endproof\n\n\\ignore{\\begin{table*}\n\\centering\n\\caption{{\\color{red} This is table is for online experiment recording Fig. 2}}\n \\begin{tabular}{| c | c | c | c | c | c | c | c | c |}\n \\hline\n Fig. & $\\alpha=\\beta$ & $\\{\\mu_0, \\mu_1\\}$ &$\\{T^\\ell\\}$ & ${\\boldsymbol\\lambda}$ & $\\{-A, B\\}$& $\\{\\alpha_\\text{exp}, \\beta_\\text{exp}\\}$ & $\\mathbb{E}\\mathsf{T}$ & $\\mathbb{E}_i\\mathsf{T}^\\ell$ \\\\ \\hline\n 2-a & $0.01$ & $\\substack{383\\\\400}$ & $\\{{\\bf 0}\\}$& $\\{{\\bf 0}\\}$ & $\\substack{-4.06\\\\ 4.01}$ & $\\substack{0.010\\\\0.010}$ & $17.32$ & $\\substack{\\{12.05, 4.73, {\\bf 0}\\} \\\\\\{6.15, 11.70, {\\bf 0}\\}}$ \\\\ \\hline\n 2-b & $0.01$ & $\\substack{443\\\\450}$ & $\\{7, 7, {\\bf 0}\\}$ & $\\{0.1438, 0.137, {\\bf 0}\\}$ & $\\substack{-4.02\\\\ 4.11}$ & $\\substack{0.010\\\\ 0.010}$ & $17.59$ & $\\substack{\\{9.31, 4.49, 2.97, 0.64\\} \\\\\\{4.95, 9.33, 0.78, 2.71\\}}$ \\\\ \\hline\n \\end{tabular}\n\\end{table*}\n\\ignore{\n\\begin{table*}\n\\centering\n\\caption{{\\color{red} This is table is for online experiment recording Fig. 3 (Unconstrained)}}\n \\begin{tabular}{| c | c | c | c | c | c | c |}\n \\hline\n Fig. & $\\alpha=\\beta$ & $\\{\\mu_0, \\mu_1\\}$ & $\\{-A, B\\}$& $\\{\\alpha_\\text{exp}, \\beta_\\text{exp}\\}$ & $\\mathbb{E}\\mathsf{T}$ & $\\mathbb{E}_i\\mathsf{T}^\\ell$ \\\\ \\hline\n \n \n 3 & $0.1$ & $\\substack{60\\\\60}$ & $\\substack{-1.86\\\\ 1.87}$ & $\\substack{0.100\\\\ 0.100}$ & $8.23$ & $\\substack{\\{4.45, 3.785, {\\bf 0}\\}\\\\\\{4.35, 3.70, {\\bf 0}\\} \\\\\\{4.55, 3.87, {\\bf 0}\\}}$ \\\\ \\hline\n 3 & $0.08$ & $\\substack{72\\\\72}$ & $\\substack{-2.07\\\\ 2.07}$ & $\\substack{0.079\\\\ 0.080}$ & $9.32$ & $\\substack{\\{4.99, 4.33, {\\bf 0}\\}\\\\\\{5.28, 3.92, {\\bf 0}\\} \\\\\\{4.70, 4.74, {\\bf 0}\\}}$ \\\\ \\hline\n 3 & $0.06$ & $\\substack{88\\\\88}$ & $\\substack{-2.33\\\\ 2.31}$ & $\\substack{0.060\\\\ 0.060}$ & $10.51$ & $\\substack{\\{5.56, 4.955, {\\bf 0}\\}\\\\\\{6.24, 4.2, {\\bf 0}\\} \\\\\\{4.88, 5.71, {\\bf 0}\\}}$ \\\\ \\hline\n \n 3 & $0.04$ & $\\substack{120\\\\125}$ & $\\substack{-2.68\\\\ 2.59}$ & $\\substack{0.039\\\\ 0.040}$ & $12.07$ & $\\substack{\\{6.44, 5.63, {\\bf 0}\\}\\\\\\{7.71, 4.22, {\\bf 0}\\} \\\\\\{5.17, 7.04, {\\bf 0}\\}}$ \\\\ \\hline\n\n 3 & $0.02$ & $\\substack{210\\\\220}$ & $\\substack{-3.37\\\\ 3.36}$ & $\\substack{0.02\\\\ 0.02}$ & $15.02$ & $\\substack{\\{7.855, 7.16, {\\bf 0}\\}\\\\\\{10.08, 4.86, {\\bf 0}\\} \\\\\\{5.63, 9.46, {\\bf 0}\\}}$ \\\\ \\hline\n 3 & $0.01$ & $\\substack{383\\\\400}$ & $\\substack{-4.06\\\\ 4.01}$ & $\\substack{0.010\\\\0.010}$ & $17.32$ & $\\substack{\\{9.10, 8.215, {\\bf 0}\\}\\\\\\{12.05, 4.73, {\\bf 0}\\} \\\\\\{6.15, 11.70, {\\bf 0}\\}}$ \\\\ \\hline\n \n 3 & $0.008$ & $\\substack{488\\\\520}$ & $\\substack{-4.28\\\\ 4.33}$ & $\\substack{0.079\\\\ 0.081}$ & $18.53$ & $\\substack{\\{13.48, 5.03, {\\bf 0}\\} \\\\\\{6.16, 12.39, {\\bf 0}\\}}$ \\\\ \\hline\n 3 & $0.006$ & $\\substack{640\\\\688}$ & $\\substack{-4.56\\\\ 4.55}$ & $\\substack{0.006\\\\ 0.006}$ & $19.15$ & $\\substack{\\{14.01, 4.92, {\\bf 0}\\} \\\\\\{5.93, 13.43, {\\bf 0}\\}}$ \\\\ \\hline\n3 & $0.004$ & $\\substack{915\\\\1010}$ & $\\substack{-4.92\\\\ 5.05}$ & $\\substack{0.0039\\\\ 0.0041}$ & $20.75$ & $\\substack{\\{15.59, 4.99, {\\bf 0}\\} \\\\\\{6.35, 14.57, {\\bf 0}\\}}$ \\\\ \\hline\n3 & $0.002$ & $\\substack{1845\\\\2220}$ & $\\substack{-5.62\\\\ 5.94}$ & $\\substack{0.0019\\\\ 0.0021}$ & $23.33$ & $\\substack{\\{18.11, 4.73, {\\bf 0}\\} \\\\\\{6.78, 17.04, {\\bf 0}\\}}$ \\\\ \\hline\n \\end{tabular}\n\\end{table*}\n}\n\\begin{table*}\n\\centering\n\\caption{{\\color{red} This is table is for online experiment recording (constrained $N_1=6, N_2=9$)}}\n \\begin{tabular}{| c | c | c | c | c | c | c | c |}\n \\hline\n Fig. & $\\alpha=\\beta$ & $\\{\\mu_0, \\mu_1\\}$ & ${\\boldsymbol\\lambda}$ & $\\{-A, B\\}$& $\\{\\alpha_\\text{exp}, \\beta_\\text{exp}\\}$ & $\\mathbb{E}\\mathsf{T}$ & $\\mathbb{E}_i\\mathsf{T}^\\ell$ \\\\ \\hline\n \n \n3 & $0.1$ & $\\substack{60\\\\60}$ & $\\{{\\bf 0}\\}$ & $\\substack{-1.86\\\\ 1.87}$ & $\\substack{0.100\\\\ 0.100}$ & $8.23$ & $\\substack{\\{4.45, 3.785, {\\bf 0}\\}\\\\\\{4.35, 3.70, {\\bf 0}\\} \\\\\\{4.55, 3.87, {\\bf 0}\\}}$ \\\\ \\hline\n 3 & $0.08$ & $\\substack{72\\\\72}$ & $\\{{\\bf 0}\\}$ & $\\substack{-2.07\\\\ 2.07}$ & $\\substack{0.079\\\\ 0.080}$ & $9.32$ & $\\substack{\\{4.99, 4.33, {\\bf 0}\\}\\\\\\{5.28, 3.92, {\\bf 0}\\} \\\\\\{4.70, 4.74, {\\bf 0}\\}}$ \\\\ \\hline\n 3 & $0.06$ & $\\substack{88\\\\88}$ & $\\{{\\bf 0}\\}$ & $\\substack{-2.33\\\\ 2.31}$ & $\\substack{0.060\\\\ 0.060}$ & $10.51$ & $\\substack{\\{5.56, 4.95, {\\bf 0}\\}\\\\\\{6.24, 4.2, {\\bf 0}\\} \\\\\\{4.88, 5.71, {\\bf 0}\\}}$ \\\\ \\hline\n\n3 & $0.04$ & $\\substack{120\\\\123}$ & $\\{0.005, {\\bf 0}\\}$ & $\\substack{-2.68\\\\ 2.59}$ & $\\substack{0.039\\\\ 0.040}$ & $12.26$ & $\\substack{\\{5.82, 6.44, {\\bf 0}\\}\\\\\\{7.15, 5.16, {\\bf 0}\\} \\\\\\{4.50, 7.73, {\\bf 0}\\}}$ \\\\ \\hline\n\n 3 & $0.02$ & $\\substack{210\\\\230}$ & $\\{0.085, {\\bf 0}\\}$ & $\\substack{-3.39\\\\ 3.30}$ & $\\substack{0.02\\\\ 0.02}$ & $14.86$ & $\\substack{\\{6.01, 8.83, 0.01, 0\\},\\\\ \\{8.565, 6.976, 0.0006, 0\\}\\\\\\{3.465, 10.687, 0.0147, 0\\}}$ \\\\ \\hline\n\n 3 & $0.01$ & $\\substack{440\\\\480}$ & $\\{0.15, 0.1115, {\\bf 0}\\}$ & $\\substack{-4.09\\\\ 4.11}$ & $\\substack{0.010\\\\0.010}$ & $17.93$ & $\\substack{\\{5.97, 8.93, 2.55, 0.484\\}\\\\\\{7.99, 5.80, 4.23, 0.29\\} \\\\\\{3.94, 12.07, 0.874, 0.68\\}}$ \\\\ \\hline\n 3 & $0.008$ & $\\substack{546\\\\598}$ & $\\{0.153, 0.1205, {\\bf 0}\\}$ & $\\substack{-4.31\\\\ 4.35}$ & $\\substack{0.080\\\\ 0.080}$ & $18.96$ & $\\substack{\\{6.096, 8.976, 3.08, 0.80\\}\\\\\\{8.176, 5.59, 5.08, 0.42\\} \\\\\\{4.016, 12.362, 1.087, 1.185\\}}$ \\\\ \\hline\n\n 3 & $0.006$ & $\\substack{700\\\\800}$ & $\\{0.1562,0.127, {\\bf 0}\\}$ & $\\substack{-4.60\\\\ 4.63}$ & $\\substack{0.0062\\\\ 0.0061}$ & $19.77$ & $\\substack{\\{6.00, 8.94, 3.75,1.09\\}\\\\\\{8.30, 5.34, 6.02, 0.42\\} \\\\\\{3.69, 12.53, 1.47, 1.76\\}}$ \\\\ \\hline\n \n 3 & $0.004$ & $\\substack{1050\\\\1220}$ & $\\{0.16125, 0.1365, {\\bf 0}\\}$ & $\\substack{-4.99\\\\ 5.11}$ & $\\substack{0.0039\\\\ 0.0039}$ & $21.43$ & $\\substack{\\{5.95, 8.98, 4.85, 1.65\\}\\\\\\{8.30, 5.34, 6.02, 0.42\\} \\\\\\{3.69, 12.53, 1.47, 1.76\\}}$ \\\\ \\hline\n\n3 & $0.002$ & $\\substack{2035\\\\2695}$ & $\\{0.17158, 0.1445, {\\bf 0}\\}$ & $\\substack{-5.69\\\\ 5.94}$ & $\\substack{0.0019\\\\ 0.0020}$ & $24.265$ & $\\substack{\\{6.03, 8.95, 6.95, 2.33\\}\\\\\\{8.68, 4.78, 10.32 0.22\\} \\\\\\{3.39, 13.11, 3.58 4.45\\}}$ \\\\ \\hline\n\n \\end{tabular}\n\\end{table*}\n\\ignore{\n\\begin{table*}\n\\centering\n\\caption{{\\color{red} This is table is for offline experiment recording (Unconstrained)}}\n \\begin{tabular}{ | c | c | c | c | c | c | c |}\n \\hline\n $\\alpha\\, (\\text{or}\\,\\beta)$ & $\\{-A, B\\}$& $\\{\\alpha_\\text{exp}, \\beta_\\text{exp}\\}$ & ${\\bf p}$ & $\\mathbb{E}\\mathsf{T}$ & $\\mathbb{E}\\mathsf{T}^\\ell (\\mathbb{E}_0\\mathsf{T}^\\ell)$ \\\\ \\hline\n \n \n $0.1$ & $\\{-1.8, 1.8\\}$ & $\\{0.10, 0.10\\}$ & $\\{0.5, 0.5, {\\bf 0}\\}$ & $8.52$ & $\\{4.256, 4.259, 0, 0\\}$ \\\\ \\hline\n $0.08$ & $\\{-2, 2\\}$ & $\\{0.080, 0.080\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$9.47$& $\\{4.736, 4.736, {\\bf 0}\\}$ \\\\ \\hline\n $0.06$ & $\\{-2.32, 2.31\\}$ & $\\{0.060, 0.060\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$11.00$& $\\{5.5, 5.5, {\\bf 0}\\}$ \\\\ \\hline\n $0.04$ & $\\{-2.7, 2.7\\}$ & $\\{0.040, 0.040\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$12.88$& $\\{6.44, 6.44, {\\bf 0}\\}$ \\\\ \\hline\n $0.02$ & $\\{-3.4, 3.4\\}$ & $\\{0.020, 0.020\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$16.00$& $\\{8.00, 8.00, {\\bf 0}\\}$ \\\\ \\hline\n $0.01$ & $\\{-4.1, 4.08\\}$ & $\\{0.01, 0.01\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$18.982$& $\\{9.4833, 9.4991, {\\bf 0}\\}$ \\\\ \\hline\n $0.008$ & $\\{-4.3, 4.3\\}$ & $\\{0.0082, 0.0080\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$19.91$& $\\{9.95, 9.955, {\\bf 0}\\}$ \\\\ \\hline\n $0.006$ & $\\{-4.6, 4.6\\}$ & $\\{0.0056, 0.0060\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$21.22$& $\\{10.62, 10.61, {\\bf 0}\\}$ \\\\ \\hline\n $0.004$ & $\\{-5.05, 5.05\\}$& $\\{0.0039, 0.0036\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$& $23.06$&$\\{11.52, 11.53, {\\bf 0}\\}$ \\\\ \\hline\n $0.002$ & $\\{-5.83, 5.83\\}$& $\\{0.0019, 0.0020\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$& $26.22$&$\\{13.10, 13.12, {\\bf 0}\\}$ \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n}\n\\begin{table*}\n\\centering\n\\caption{{\\color{red} This is table is for offline experiment recording (Constrained: $N_1=6, N_2=9$)}}\n \\begin{tabular}{ | c | c | c | c | c | c | c |}\n \\hline\n $\\alpha\\, (\\text{or}\\,\\beta)$ & $\\{-A, B\\}$& $\\{\\alpha_\\text{exp}, \\beta_\\text{exp}\\}$ & ${\\bf p}$ & $\\mathbb{E}\\mathsf{T}$ & $\\mathbb{E}\\mathsf{T}^\\ell (\\mathbb{E}_0\\mathsf{T}^\\ell)$ \\\\ \\hline\n \n \n $0.1$ & $\\{-1.8, 1.8\\}$ & $\\{0.10, 0.10\\}$ & $\\{0.5, 0.5, {\\bf 0}\\}$ & $8.52$ & $\\{4.256, 4.259, 0, 0\\}$ \\\\ \\hline\n $0.08$ & $\\{-2, 2\\}$ & $\\{0.080, 0.080\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$9.47$& $\\{4.736, 4.736, {\\bf 0}\\}$ \\\\ \\hline\n $0.06$ & $\\{-2.32, 2.31\\}$ & $\\{0.060, 0.060\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$11.00$& $\\{5.5, 5.5, {\\bf 0}\\}$ \\\\ \\hline\n $0.04$ & $\\{-2.7, 2.7\\}$ & $\\{0.041, 0.041\\}$&$\\{0.475, 0.5, 0.015, 0\\}$ &$12.88$& $\\{5.99, 6.44, 0.453, 0\\}$ \\\\ \\hline\n $0.02$ & $\\{-3.4, 3.4\\}$ & $\\{0.020, 0.020\\}$&$\\{0.372, 0.5, 0.128, 0\\}$ &$16.23$& $\\{6.036, 8.1253, 2.073, 0\\}$ \\\\ \\hline\n $0.01$ & $\\{-4.1, 4.08\\}$ & $\\{0.01, 0.01\\}$&$\\{0.31, 0.465, 0.16, 0.065\\}$ &$19.4280$& $\\{6.0259, 9.0397, 3.0985, 1.2639\\}$ \\\\ \\hline\n $0.008$ & $\\{-4.3, 4.3\\}$ & $\\{0.0081, 0.0081\\}$&$\\{0.29, 0.435, 0.15, 0.125\\}$ &$20.4767$& $\\{5.994, 9.00, 4.51, 0.971\\}$ \\\\ \\hline\n $0.006$ & $\\{-4.6, 4.6\\}$ & $\\{0.006, 0.0060\\}$&$\\{0.275, 0.4125, 0.18, 0.1325\\}$ &$21.88$& $\\{6.0176, 9.0265, 3.941,2.9\\}$ \\\\ \\hline\n $0.004$ & $\\{-5.05, 5.05\\}$& $\\{0.0040, 0.0041\\}$&$\\{0.25, 0.375, 0.18, 0.195{\\bf 0}\\}$& $23.9722$&$\\{5.992, 8.994, 4.313, 4.673\\}$ \\\\ \\hline\n $0.002$ & $\\{-5.83, 5.83\\}$& $\\{0.0020, 0.0020\\}$&$\\{0.218, 0.327, 0.22, 0.235\\}$& $27.537$&$\\{6.01, 9.002, 6.0494, 6.4762\\}$ \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{{\\color{red} Infinite-Horizon: This is table is for online experiment recording (constrained $N_1=6, N_2=9$)}}\n \\begin{tabular}{| c | c | c | c | c | c | c | c |}\n \\hline\n $N$ & $\\alpha=\\beta$ & $\\{\\mu_0, \\mu_1\\}$ & ${\\boldsymbol\\lambda}$ & $\\{-A, B\\}$& $\\{\\alpha_\\text{exp}, \\beta_\\text{exp}\\}$ & $\\mathbb{E}\\mathsf{T}$ & $\\mathbb{E}_i\\mathsf{T}^\\ell$ \\\\ \\hline\n \n \n150 & $0.1$ & $\\substack{60\\\\60}$ & $\\{{\\bf 0}\\}$ & $\\substack{-1.86\\\\ 1.87}$ & $\\substack{0.100\\\\ 0.100}$ & $8.23$ & $\\substack{\\{4.45, 3.785, {\\bf 0}\\}\\\\\\{4.35, 3.70, {\\bf 0}\\} \\\\\\{4.55, 3.87, {\\bf 0}\\}}$ \\\\ \\hline\n 150 & $0.08$ & $\\substack{72\\\\72}$ & $\\{{\\bf 0}\\}$ & $\\substack{-2.07\\\\ 2.07}$ & $\\substack{0.079\\\\ 0.080}$ & $9.32$ & $\\substack{\\{4.99, 4.33, {\\bf 0}\\}\\\\\\{5.28, 3.92, {\\bf 0}\\} \\\\\\{4.70, 4.74, {\\bf 0}\\}}$ \\\\ \\hline\n 150 & $0.06$ & $\\substack{88\\\\88}$ & $\\{{\\bf 0}\\}$ & $\\substack{-2.31\\\\ 2.33}$ & $\\substack{0.060\\\\ 0.061}$ & $10.42$ & $\\substack{\\{5.5, 4.92, {\\bf 0}\\}\\\\\\{6.15, 4.05, {\\bf 0}\\} \\\\\\{4.855, 5.78, {\\bf 0}\\}}$ \\\\ \\hline\n\n150 & $0.04$ & $\\substack{120\\\\123}$ & $\\{0.005, {\\bf 0}\\}$ & $\\substack{-2.70\\\\ 2.69}$ & $\\substack{0.040\\\\ 0.040}$ & $12.25$ & $\\substack{\\{5.88, 6.375, {\\bf 0}\\}\\\\\\{7.23, 5.054, {\\bf 0}\\} \\\\\\{4.533, 7.70, {\\bf 0}\\}}$ \\\\ \\hline\n\n 200 & $0.02$ & $\\substack{210\\\\230}$ & $\\{0.085, {\\bf 0}\\}$ & $\\substack{-3.39\\\\ 3.30}$ & $\\substack{0.02\\\\ 0.02}$ & $14.82$ & $\\substack{\\{5.975, 8.834, 0.0141, 0\\},\\\\ \\{8.50, 7.00, 0.0011, 0\\}\\\\\\{3.451, 10.670, 0.0271, 0\\}}$ \\\\ \\hline\n\n 200 & $0.01$ & $\\substack{420\\\\460}$ & $\\{0.1470, 0.105, {\\bf 0}\\}$ & $\\substack{-4.04\\\\ 4.06}$ & $\\substack{0.010\\\\0.010}$ & $17.62$ & $\\substack{\\{5.973, 9.026, 2.320, 0.302\\}\\\\\\{8.1758, 6.0010, 3.840, 0.166\\} \\\\\\{3.770, 12.043, 0.800, 0.438\\}}$ \\\\ \\hline\n 200 & $0.008$ & $\\substack{520\\\\570}$ & $\\{0.1512, 0.1160, {\\bf 0}\\}$ & $\\substack{-4.2637\\\\ 4.3}$ & $\\substack{0.080\\\\ 0.081}$ & $18.57$ & $\\substack{\\{5.93, 8.96, 3.026, 0.65\\}\\\\\\{7.941, 5.659, 5.036, 0.3146\\} \\\\\\{3.911, 12.293, 1.016, 0.983\\}}$ \\\\ \\hline\n\n 200 & $0.006$ & $\\substack{690\\\\780}$ & $\\{0.1535,0.1265, {\\bf 0}\\}$ & $\\substack{-4.58\\\\ 4.62}$ & $\\substack{0.0062\\\\ 0.0061}$ & $20.066$ & $\\substack{\\{6.0321, 9.0176, 3.8684, 1.1476\\}\\\\\\{8.2862, 5.5553, 6.2202, 0.4541\\} \\\\\\{3.778, 12.480, 1.5165, 1.841\\}}$ \\\\ \\hline\n \n 200 & $0.004$ & $\\substack{1000\\\\1180}$ & $\\{0.1562, 0.1346, {\\bf 0}\\}$ & $\\substack{-4.97\\\\ 5.05}$ & $\\substack{0.0039\\\\ 0.0039}$ & $21.36$ & $\\substack{\\{6.06, 9.026, 4.57, 1.65\\}\\\\\\{8.66, 5.402, 7.2165, 0.399\\} \\\\\\{3.562, 12.651, 1.926, 2.905\\}}$ \\\\ \\hline\n\n200 & $0.002$ & $\\substack{1860\\\\2580}$ & $\\{0.1615, 0.1373, {\\bf 0}\\}$ & $\\substack{-5.65\\\\ 5.81}$ & $\\substack{0.0019\\\\ 0.0019}$ & $24.217$ & $\\substack{\\{6.011, 9.02, 6.913, 2.264\\}\\\\\\{9.1554, 4.7665, 10.178, 0.253\\} \\\\\\{2.867, 13.2926, 3.649, 4.274\\}}$ \\\\ \\hline\n\n \\end{tabular}\n\\end{table*}\n\\begin{table*}\n\\centering\n\\caption{{\\color{red} Infinite-Horizon: This is table is for offline experiment recording (Constrained: $N_1=6, N_2=9$)}}\n \\begin{tabular}{ | c | c | c | c | c | c | c |}\n \\hline\n $\\alpha\\, (\\text{or}\\,\\beta)$ & $\\{-A, B\\}$& $\\{\\alpha_\\text{exp}, \\beta_\\text{exp}\\}$ & ${\\bf p}$ & $\\mathbb{E}\\mathsf{T}$ & $\\mathbb{E}\\mathsf{T}^\\ell (\\mathbb{E}_0\\mathsf{T}^\\ell)$ \\\\ \\hline\n \n \n $0.1$ & $\\{-1.8, 1.8\\}$ & $\\{0.10, 0.10\\}$ & $\\{0.5, 0.5, {\\bf 0}\\}$ & $8.52$ & $\\{4.256, 4.259, 0, 0\\}$ \\\\ \\hline\n $0.08$ & $\\{-2, 2\\}$ & $\\{0.080, 0.080\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$9.47$& $\\{4.736, 4.736, {\\bf 0}\\}$ \\\\ \\hline\n $0.06$ & $\\{-2.32, 2.31\\}$ & $\\{0.060, 0.060\\}$&$\\{0.5, 0.5, {\\bf 0}\\}$ &$11.00$& $\\{5.5, 5.5, {\\bf 0}\\}$ \\\\ \\hline\n $0.04$ & $\\{-2.7, 2.7\\}$ & $\\{0.041, 0.041\\}$&$\\{0.472, 0.5, 0.028, 0\\}$ &$12.838$& $\\{6.0575, 6.425, 0.3557, 0\\}$ \\\\ \\hline\n $0.02$ & $\\{-3.4, 3.35\\}$ & $\\{0.020, 0.020\\}$&$\\{0.373, 0.6, 0.027, 0\\}$ &$15.98$& $\\{5.960, 9.5873, 0.4333, 0\\}$ \\\\ \\hline\n $0.01$ & $\\{-4.1, 4.1\\}$ & $\\{0.01, 0.01\\}$&$\\{0.31, 0.465, 0.16, 0.065\\}$ &$19.430$& $\\{6.048, 9.080, 3.123, 1.270\\}$ \\\\ \\hline\n $0.008$ & $\\{-4.3, 4.3\\}$ & $\\{0.0081, 0.0081\\}$&$\\{0.29, 0.435, 0.15, 0.125\\}$ &$20.477$& $\\{5.994, 9.00, 4.51, 0.971\\}$ \\\\ \\hline\n $0.006$ & $\\{-4.6, 4.6\\}$ & $\\{0.0061, 0.0060\\}$&$\\{0.274, 0.411, 0.250, 0.065\\}$ &$21.74$& $\\{5.963, 8.9243, 5.4415, 1.4113\\}$ \\\\ \\hline\n $0.004$ & $\\{-5.00, 5.00\\}$& $\\{0.0040, 0.0041\\}$&$\\{0.253, 0.3795, 0.210, 0.1575\\}$& $23.71$&$\\{5.992, 9.002, 4.983, 3.734\\}$ \\\\ \\hline\n $0.002$ & $\\{-5.72, 5.72\\}$& $\\{0.0019, 0.002\\}$&$\\{0.223, 0.3345, 0.330, 0.1125\\}$& $27.06$&$\\{6.0317, 9.0467, 8.934, 3.0477\\}$ \\\\ \\hline\n\\end{tabular}\n\\end{table*}}\n\\ignore{\n\\section{Optimal Adaptive Scheme with Usage Constraints---Scenario II}\n{\\color{red} Start with this problem. Prove the structure with a general theorem. Interpret the theorem in a more general application background (general cost). }\n\\begin{align}\\label{P2}\n\\begin{array}{cl}\n\\min_{\\{{\\bf s}_1^\\infty, D, \\mathsf{T}\\}} & \\mathbb{E}^{{\\bf s}_1^\\infty}\\left( c\\mathsf{T} +\\mathbbm{1}_{\\{D\\neq H\\}}\\right)\\\\\n\\text{subject to} & \\mathbb{E}^{{\\bf s}_1^\\infty}\\left( \\sum_{t=1}^\\mathsf{T} {\\bf s}_t(\\ell)\\right)\\le \\xi_\\ell \\,\\mathbb{E}^{{\\bf s}_1^\\infty}\\mathsf{T}, \\quad \\ell=1, \\ldots, L.\\tag{P2}\n\\end{array}\n\\end{align}\n\n{\\color{red} Shall we consider a general theorem here, or just consider a specific problem. At least, the SPRT structure remains the same for general problem.\nThe optimal structure holds for any selection strategy. We then can discuss the particular case where one sensor is selected at a time, that could correspond to one mode is switched on at each time. }\n\nNote that the utilization constraints are on average sense. For instance, the sensor system can be resused and the average battery life imposes the constraint on each sensor. {\\color{red} The constraint on the utilization percentage. } Here $\\sum_{\\ell=1}^L \\xi_\\ell\\ge 1$ is necessary because we assume that at each time at least one sensor must be selected, i.e., $\\sum_{\\ell=1}^L{\\bf s}_t(\\ell)\\ge 1$ for $t=1, \\ldots, \\mathsf{T}$, thus\n\\begin{align}\\label{eq1}\n\\sum_{\\ell=1}^L\\xi_\\ell\\,\\mathbb{E}^{{\\bf s}_1^\\infty}\\mathsf{T}\\ge \\mathbb{E}^{{\\bf s}_1^\\infty}\\left(\\sum_{t=1}^\\mathsf{T}\\sum_{\\ell}^L{\\bf s}_t(\\ell)\\right)\\ge\\mathbb{E}^{{\\bf s}_1^\\infty}\\left(\\mathsf{T}\\right),\n\\end{align}\nwhere the second equality holds true when $\\sum_{\\ell}^L{\\bf s}_t(\\ell)=1$, only one sensor is used at each time. If $\\sum_{\\ell=1}^L\\xi_\\ell=1$, then only one sensor is active at each time, and all constraints are satisfied with equality. This is true because $$\\mathbb{E}^{{\\bf s}_1^\\infty}\\mathsf{T}\\ge \\mathbb{E}^{{\\bf s}_1^\\infty}\\left(\\sum_{t=1}^\\mathsf{T}\\sum_{\\ell=1}^L{\\bf s}_t(\\ell)\\right)\\ge \\mathbb{E}^{{\\bf s}_1^\\infty}\\left(\\mathsf{T}\\right)\\to \\mathbb{E}^{{\\bf s}_1^\\infty}\\left(\\sum_{t=1}^\\mathsf{T}\\sum_{\\ell=1}^L{\\bf s}_t(\\ell)\\right)= \\mathbb{E}^{{\\bf s}_1^\\infty}\\left(\\mathsf{T}\\right)$$\ni.e., the second inequality in \\eqref{eq1} holds as equality, thus $\\sum_{\\ell}^L{\\bf s}_t(\\ell)=1$ (one sensor is active at each time). If there exists a sensor $\\ell$ such that $\\mathbb{E}^{{\\bf s}_1^\\infty}\\sum_{t=1}^\\mathsf{T}{\\bf s}_t(\\ell)<\\xi_\\ell\\, \\mathbb{E}^{{\\bf s}_1^\\infty}\\mathsf{T}$, then\n$$\\sum_{\\ell=1}^L\\mathbb{E}^{{\\bf s}_1^\\infty}\\sum_{t=1}^\\mathsf{T}{\\bf s}_t(\\ell)<\\mathbb{E}^{{\\bf s}_1^\\infty}\\mathsf{T},$$ and contradiction occurs. By introducing the multipliers, \\eqref{P1} becomes\n\\begin{align}\n\\min_{\\{{\\bf s}_1^\\infty, D, \\mathsf{T}\\}}\\quad \\mathbb{E}\\left( c\\mathsf{T} +\\mathbbm{1}_{\\{D\\neq H\\}}\\right)+ c\\sum_{\\ell=1}^L\\lambda_\\ell\\,\\mathbb{E}\\left( \\sum_{t=1}^\\mathsf{T} \\left({\\bf s}_t(\\ell)-\\xi_\\ell\\right)\\rb, \\quad \\lambda_\\ell\\ge 0.\n\\end{align}\nHere the factor $c$ in the second term is introduced for presentation conciseness in the forthcoming derivations. We can further write the objective function as\n\\begin{align}\n &c\\,\\mathbb{E}\\mathsf{T}+\\mathbb{E}\\left( \\mathbbm{1}_{\\{D\\neq H\\}}\\right)+c\\,\\mathbb{E}\\left(\\sum_{t=1}^\\mathsf{T}\\sum_{\\ell=1}^L\\lambda_\\ell\\,({\\bf s}_t(\\ell)-\\xi_\\ell)\\right)\\nonumber\\\\=&\\,\\mathbb{E}\\left( c \\sum_{t=1}^\\mathsf{T} \\left( 1+\\sum_{\\ell=1}^L\\lambda_\\ell\\,{\\bf s}_t(\\ell)-\\sum_{\\ell=1}^L\\lambda_\\ell\\xi_\\ell\\right)+\\mathbbm{1}_{\\{D\\neq H\\}}\\right)\n\\end{align}\n\n{\\color{red} Any insight compared to (P1)?}\n}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Methods}\n\n\n\\noindent\\textbf{Calculation of binding energy.} \nFirst, the geometry of each molecule is optimized at the Hartree-Fock level, using the standard 6-311++G($d$,$p$) Gaussian basis for the electronic wave functions. The positron-molecule interaction is modelled using the potential\n\\begin{equation}\\label{eq:potential}\nV(\\vec{r}) = V_\\text{st}(\\vec{r}) + V_\\text{cor}(\\vec{r}),\n\\end{equation}\nwhere $V_\\text{st}({\\bf r})$ is the electrostatic potential of the molecule calculated at the Hartree-Fock level, and $V_\\text{cor}({\\bf r})$ is a model potential that describes long-range polarization of the molecule by the positron, viz.,\n\\begin{equation}\nV_\\text{cor}(\\vec{r}) = - \\sum_A \\frac{\\alpha_A}{2\\lvert \\vec{r}-\\vec{r}_A\\rvert^4} \\left[ 1 - \\exp\\left( -\\lvert \\vec{r}-\\vec{r}_A\\rvert^6\/\\rho_A^6\\right)\\right] .\n\\end{equation}\nHere, the sum is over all atoms $A$ within the molecule, $\\vec{r}$ is the position of the positron, $\\vec{r}_A$ is the position of atom $A$ (relative to an arbitrary origin), and $\\alpha_A$ is the hybrid dipole polarizability of atom $A$, i.e., the effective dipole polarizability of atom $A$ in the environment of the molecule \\cite{Miller90}. The expression in brackets is a cutoff function that prevents unphysical growth of $V_\\text{cor}({\\bf r})$ as $\\lvert \\vec{r}-\\vec{r}_A\\rvert\\to0$, with $\\rho_A$ a cutoff radius specific to atom $A$.\n\nTo obtain values of the hybrid polarizabilities $\\alpha_\\text{C}$ and $\\alpha_\\text{H}$,\nwe used the experimental values of the molecular polarizabilities $\\alpha_{\\text{C}_n\\text{H}_{2n+2}}$ for the $n$-alkanes (with $n=1$--12) from ref.~\\cite{Miller90}. We assumed that they could be obtained by adding the hybrid polarizabilities of the constituent atoms:\n\\begin{align}\n\\alpha_{\\text{C}_n\\text{H}_{2n+2}} &= n\\alpha_\\text{C} + (2n+2) \\alpha_\\text{H} \\notag\\\\\n&= (\\alpha_\\text{C} + 2\\alpha_\\text{H})n + 2\\alpha_\\text{H}.\n\\end{align}\nPerforming a linear least-squares fit for $\\alpha_{\\text{C}_n\\text{H}_{2n+2}}$ as a function of $n$, we obtained values of $\\alpha_\\text{C}=7.096$~a.u. and $\\alpha_\\text{C}=2.650$~a.u., which are used throughout our calculations. As for the cutoff radii, a value of $\\rho_\\text{C}=\\rho_\\text{H}=2.25$~a.u. was used throughout (to match the calculated binding energy for $n=12$ with the experimental value).\n\nThe Schr\\\"odinger equation for a positron moving in the potential given by equation (\\ref{eq:potential}) is solved to obtain the binding energy and the corresponding wave function. In practice, this is done using the standard quantum-chemistry package \\textsc{gamess} with the \\textsc{neo} plugin \\cite{Schmidt93,Gordon05,Webb02,Adamson08}, which we have modified to include the model potential $V_\\text{cor}$; see ref.~\\onlinecite{Swann18} for details. \nA Gaussian basis is used for the positron wave function.\nMatrix elements of $V_\\text{cor}$ between these basis functions are evaluated analytically by expanding the potential $V_\\text{cor}$ in a separate set of Gaussians, consisting of 25 $s$-type functions centred on each of the atoms.\nThe positron basis consists of 12 $s$-type Gaussian functions centred on each carbon atom and 8 $s$-type Gaussians centred on each hydrogen atom. Explicitly, the Gaussian basis functions have the form $\\exp(-\\zeta_k \\lvert \\vec{r}-\\vec{r}_A\\rvert^2)$, where the $\\zeta_k$ are chosen according to the even-tempered scheme\n\\begin{align}\n\\zeta_k &= \\zeta_1 \\times 3^{k-1},\n\\end{align}\nwith $\\zeta_1=0.0001$~a.u. and $k=1,\\dotsc,12$ for centres on C atoms, and $\\zeta_1=0.0081$~a.u. and $k=1,\\dotsc,8$ for centres on H atoms. We deemed the inclusion of higher-angular-momentum-type Gaussian functions in the basis to be unnecessary, as $s$-type functions placed on multiple centres effectively generate higher-angular-momentum-type functions \\cite{Swann18}$^\\text{,}$ \\cite{Whitten63,Whitten66,Petke69}. To check this, we calculated the binding energy for $n$-dodecane using a basis set with six additional $p$- and $d$-type functions (with $\\zeta_1=0.0081$~a.u.) on each of the C atoms, and observed that the binding energy increased by less than 1\\%.\n\n\\noindent\\textbf{Annihilation rate.}\nThe 2$\\gamma$ annihilation rate for the positron from the bound state, averaged over the electron and positron spins, is given by \\cite{Gribakin10}\n\\begin{equation}\\label{eq:ann_rate}\n\\Gamma = \\pi r_0^2 c \\delta_{ep},\n\\end{equation}\nwhere $r_0$ is the classical electron radius, $c$ is the speed of light, and $\\delta_{ep}$ is the average electron density at the positron. In the independent-particle approximation, $\\delta_{ep}$ is given by\n\\begin{equation}\n\\delta_{ep} = 2 \\sum_{i=1}^{N\/2} \\int \\lvert \\varphi_i(\\vec{r}) \\rvert^2 \\lvert \\psi(\\vec{r}) \\rvert^2 \\, d\\tau,\n\\end{equation}\nwhere the $N$-electron molecule is assumed to be closed shell, with $N\/2$ doubly occupied electronic orbitals, $\\varphi_i$ is the wave function of molecular orbital $i$, and $\\psi$ is the wave function of the bound positron.\n\nShort-range electron-positron correlations increase the probability of finding an electron at the positron with respect to the independent-particle approximation. Many-body theory calculations for atoms show that this effect can be described by introducing the so-called \\textit{enhancement factors} $\\gamma _i$ \\cite{Green18}, and that they are to a good approximation functions of the electron orbital energy $\\epsilon _i$ \\cite{Green15,Swann18},\n\\begin{equation}\n\\gamma_i = 1 + \\sqrt{\\frac{1.31}{-\\epsilon_i}} + \\left( \\frac{0.834}{-\\epsilon_i} \\right)^{2.15}.\n\\end{equation}\nThe enhanced value of $\\delta_{ep}$ is calculated thus:\n\\begin{equation}\\label{eq:delta_ep}\n\\delta_{ep} = 2 \\sum_{i=1}^{N\/2} \\gamma_i\\int \\lvert \\varphi_i(\\vec{r}) \\rvert^2 \\lvert \\psi(\\vec{r}) \\rvert^2 \\, d\\tau.\n\\end{equation}\n\nAnother many-body correction concerns the \\textit{normalization} of the positron wave function. The true correlation potential that describes the interaction of a positron with a many-electron system is a nonlocal, energy-dependent potential $\\Sigma _E({\\bf r},{\\bf r}')$, equal to the self-energy of the positron Green's function \\cite{Gribakin04,Green14}. When using it in the Shr\\\"odinger-like Dyson equation, the negative-energy eigenvalue that describes a bound state, becomes of a function of $E$, i.e., $\\epsilon _0(E)$, and the binding energy is to be found self-consistently, as\n$\\epsilon _b=-\\epsilon _0(-\\epsilon _b)$. The corresponding positron wave function is in fact a \\textit{quasiparticle} wave function (or Dyson orbital), normalized as follows \\cite{Chernysheva88,Ludlow10}:\n\\begin{equation}\\label{eq:a}\n\\int |\\psi ({\\bf r})|^2d\\tau =\\left( 1-\\left.\\dfrac{\\partial \\epsilon _0}{\\partial E}\\right|_{E=-\\epsilon _b}\\right)^{-1}\\equiv a<1.\n\\end{equation}\nThe energy dependence of the correlation potential can be examined by looking at the dimensionless strength parameter $S(E)$ \\cite{Green18a}. Many-body theory calculations for $s$-wave positrons interacting with noble-gas atoms show that its can be parameterised by $S(E)=A\/(B-E)$, and that the value of $B$ scales with the ionization potential of the target, e.g., $B=1.163$~a.u. for Ar, 0.9428~a.u. for Kr and 0.7215~a.u. for Xe. Given that the ionization potential of alkanes is closest to that of Xe, we use $B=0.7215$~a.u. to estimate the relative energy dependence of the correlation potential at $E\\approx 0$ from $(dS\/dE)\/S(E)=1\/B$. Applying this to our model potential $V_\\text{cor}({\\bf r})$, we see that a 1\\% increase in this potential corresponds to an effective energy difference of $\\delta E=0.01B$. Hence, we perform an additional calculation of the binding energy using $1.01V_\\text{cor}({\\bf r})$ and evaluate the normalisation constant from $a=\\{1+[\\ensuremath{\\epsilon_b} (1.01 V_\\text{cor})-\\ensuremath{\\epsilon_b} (V_\\text{cor})]\/\\delta E\\}^{-1}$. The values we thus obtain range from $a=0.992$ for C$_3$H$_8$ to 0.933 for C$_{16}$H$_{34}$ for the first bound state, and from $a=0.967$ for C$_{12}$H$_{26}$ to 0.946 for C$_{16}$H$_{34}$, for the second bound state. We use them to multiply the enhanced contact density from equation (\\ref{eq:delta_ep}).\n\nOnce $\\delta_{ep}$ has been calculated, the annihilation rate [eq.~(\\ref{eq:ann_rate})] is found as\n\\begin{equation}\n\\Gamma[\\text{ns}^{-1}] = 50.470 \\times \\delta_{ep}[\\text{a.u.}]\n\\end{equation}\n\n\\end{comment}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec:Intro}}\nStrongly correlated $d$- and $f$-electron systems support a rich variety of low-temperature phases, including magnetism and superconductivity~\\cite{Tokura1998,Dagotto2005,Moore2009,Pfleiderer2009}. Among these phases, long-range order of multipoles, namely high-rank electric or magnetic moments, has great interest~\\cite{Kuramoto2009,Santini2009,Cameron2016,Suzuki2018}. For example, second-rank quadrupolar moments could lead to novel phenomena including the quadrupolar Kondo effect~\\cite{Cox1987} and quadrupole-fluctuation-mediated superconductivity~\\cite{Kotegawa2003}. In $d$-electron systems, the orbital angular momentum is usually quenched by large crystal-field (CF) splitting, hindering multipolar moments. $f$-electron systems, on the other hand, are suitable choices to study multipolar interactions and ordering phenomena by virtue of the interplay of the spin and orbital degrees of freedom. Indeed, the actinide dioxides, in which $5f$-electrons play an important role, serve as a paradigm for understanding the physics of multipolar interactions~\\cite{Santini2009}. Quadrupolar orderings have also been discovered in a number of $4f$-electron compounds~\\cite{Tm1993,Tm1998,Dy2000,Np2002,Pr2014,Cameron2016}.\n\nCeB$_{6}$, with its simple chemical composition, lattice structure, and electronic configuration, is considered a prototypical example of heavy-fermion metal with quadrupolar ordering. This material has a cubic structure (space group Pm$\\overline{3}$m, No. 221; point group O$_{h}$) composed of cerium ions and boron octahedrons [Fig.~\\ref{fig:Intro1}(a)]. Every Ce$^{3+}$ ion has only one electron in its $4f$ orbital and O$_{h}$ site symmetry. CeB$_{6}$ undergoes a second-order phase transition into a non-magnetic phase at T$_{Q}$\\,=\\,3.2\\,K, before developing an antiferromagnetic (AFM) order below T$_{N}$\\,=\\,2.3\\,K~\\cite{Takase1980,Fujita1980}. The AFM phase has a double-Q commensurate magnetic structure with Q$_1$=(0.25, 0.25, 0) and Q$_2$=(0.25, 0.25, 0.5)~\\cite{Burlet1982,Zaharko2003}. As for the non-magnetic phase, neutron scattering shows no structural transition at T$_{Q}$~\\cite{Zaharko2003}. Resonant X-ray diffraction determines that this non-magnetic phase involves an orbital ordering with wavevector (0.5, 0.5, 0.5)~\\cite{Nakao2001}, and the C$_{44}$ elastic constant, related to $\\epsilon_{xy}$-type strains, shows an anomaly at T$_{Q}$~\\cite{Nakamura1994}. Based on these results, it is generally believed that the non-magnetic phase is a two-sublattice arrangement of Ce$^{3+}$ $O_{xy}$-type electric quadrupole moments, with a wavevector (0.5, 0.5, 0.5)~\\cite{Cameron2016}. This proposed antiferroquadrupolar (AFQ) model is consistent with experimental data in the presence of magnetic field~\\cite{Hanzawa2000,Kunimori2012,Matsumura2009,Matsumura2012,Schlottmann2012}, but to our knowledge, up to now there is no direct evidence demonstrating the $O_{xy}$-type AFQ order in zero field. A sketch of field-temperature phase diagram for CeB$_{6}$ is shown in Fig.~\\ref{fig:Intro1}(b). \n\nAll experimental results reported in this study correspond to the zero-field paramagnetic (PM) phase, namely, the data is acquired at T\\,$>$\\,T$_{Q}$.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig1.pdf}\n\\caption{\\label{fig:Intro1}(a) Crystal structure of CeB$_{6}$. (b) A sketch of field-temperature phase diagram for CeB$_{6}$.}\n\\end{figure}\n\nIn the recent years multiple experimental studies have revealed the importance of unexpected ferromagnetic (FM) correlations in the low-temperature ordering phenomena of CeB$_{6}$. In the AFQ phase with finite magnetic field, electron spin resonance (ESR) with narrow linewidth was uncovered, pointing to existence of FM correlations~\\cite{Demishev2009}. Theoretical study suggested that such FM correlations result from AFQ ordering~\\cite{Schlottmann2012}. A zone-center excitation at the (110) point, following the energy of ESR, was found by inelastic neutron scattering (INS)~\\cite{Portnichenko2016}. In the AFQ phase at zero magnetic field, this finite-energy mode collapses into a quasi-elastic peak~\\cite{Jang2014}. Moreover, intense FM fluctuations were uncovered in the AFM phase, suggesting propensity to FM instability~\\cite{Jang2014}.\n\nBoth the AFQ and AFM phases are closely related to the CF ground state~\\cite{Cameron2016}. In CeB$_{6}$, 6-fold degenerate $^2F_{5\/2}$ is the ground multiplet, and 8-fold $^2F_{7\/2}$ is the lowest-energy excited multiplet [Fig.~\\ref{fig:Intro2}]. These two multiplets were identified in photoemission spectroscopy studies~\\cite{Takahashi1995,Koitzsch2016} by the self-energy effects~\\footnote{Photoemission spectroscopy probes energy states below the Fermi level, thus, ARPES cannot directly access the $^2F_{7\/2}$ multiplet. However, virtual excitations to the narrow $^2F_{7\/2}$ multiplet contribute to the self energy of the spectral function, making the $^2F_{7\/2}$ multiplet identifiable in the photoemission spectra. Details of this mechanism can be found in Refs.~\\cite{Gunnarsson1983,Coleman1984,Patthey1987}}. From group theory analysis~\\cite{Koster1963}, the cubic CF potential splits the $^2F_{5\/2}$ multiplet into quartet $\\Gamma_8$ and doublet $\\Gamma_7$ states, and the $^2F_{7\/2}$ multiplet into doublet $\\Gamma_6^*$, doublet $\\Gamma_7^*$, and quartet $\\Gamma_8^*$ \nstates~\\footnote{Asterisks are used to distinguish the CF states of the $^2F_{7\/2}$ multiplet ($\\Gamma_6^*$, $\\Gamma_7^*$ and $\\Gamma_8^*$) from those of the $^2F_{5\/2}$ multiplet ($\\Gamma_7$ and $\\Gamma_8$).}. For the $^2F_{5\/2}$ multiplet, the $\\Gamma_8$ state is the ground state~\\cite{Sato1984,Zirngiebl1984,Sakai1997,Sundermann2017} and the $\\Gamma_7$ state has an energy of 372\\,cm$^{-1}$ at room temperature~\\cite{Zirngiebl1984,Loewenhaupt1985ccf}. For the $^2F_{7\/2}$ multiplet, the energy of the CF levels has not been determined experimentally.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.48\\textwidth]{Fig2.pdf}\n\\caption{\\label{fig:Intro2}\nSchematic energy diagram illustrating the splitting of 4f orbital by spin-orbital coupling and cubic crystal field. The same color scheme is used in Figs.~\\ref{fig:Intro2},~\\ref{fig:CF1},~\\ref{fig:CF2}, and~\\ref{fig:CF4} to identify the four crystal-field transitions.} \n\\end{figure}\n\nIn order to better understand the low-temperature ordering phenomena \nin CeB$_{6}$, a more detailed study of the interplay of CF \nexcitations, lattice dynamics and the FM correlations is required. \nRaman spectroscopy is a suitable technique providing \nsymmetry-resolved excitation spectra of electronic, magnetic, and \nphononic degrees of freedom. As a photon-in-photon-out inelastic \nscattering process, polarization-resolved Raman scattering has the \nunique advantage of high energy-resolution and the ability to \ndisentangle the excitation spectra into individual symmetry channels. \nThe symmetry of a particular excitation can be identified by \ncontrolling the polarization of the incident and scattered \nlight~\\cite{Hayes2004}. \nThis experimental method has been \nsuccessfully used to study CF excitations \n\\cite{Cardona2000,Guntherodt1987}; \nit is a well-fitted choice of \ninvestigating the intra- and inter-multiplet CF excitations of \nCeB$_{6}$. \nMoreover, Raman scattering makes it possible to study the \nexcitations in the magnetic dipolar (T$_{1g}$ of O$_{h}$ group) and \nelectric quadrupolar (E$_{g}$ and T$_{2g}$ of O$_{h}$ group) channels \nseparately. Thus, the relationship between the quadrupolar \ncorrelations and FM correlations can be clarified. Notice that \nquadrupolar excitations involve a change of the component of angular \nmomentum along the quantization axis by two quantum units. Among \nconventional experimental probes, only photons can induce quadrupolar \nexcitations. \n\nIn this paper, we present a comprehensive study of CeB$_{6}$ using \noptical secondary-emission spectroscopy. We identify an intense \nphoto-luminescence feature corresponding to $5d-4f$ recombination \nprocess. We analyze the temperature-dependence of both intra- and \ninter-multiplet CF excitations, and illustrate the interaction \nbetween light and CF states by a model Hamiltonian calculation. We \ndraw information about the electron-phonon interaction by studying \nlattice dynamics. We observe dynamical magnetic fluctuations related \nto the ordered broken-symmetry states. Especially, we demonstrate two \nvirtues of Raman scattering which have not been generally \nappreciated: first, the temperature dependence of the parameters of \nCF excitations reveals the interaction between $f$-electrons and \nitinerant electrons; and second, the low-energy Raman response probes \ndynamical fluctuations related to exotic multipolar \nordering. \n\nThe rest of this paper is organized as follows. In Sec.~\\ref{sec:Exp} we describe the sample preparation and experimental setup. In Sec.~\\ref{sec:Res} we present and discuss the experimental results; in this section, we first show an overview of the main spectral features in SubSec.~\\ref{subsec:OV} and then discuss them separately in the following subsections. In SubSec.~\\ref{subsec:PL} we show the high-energy photo-luminescence (CF) feature. In SubSec.~\\ref{subsec:CF} we discuss the CF excitations. Specifically, in \\ref{subsubsec:Iden} we present the four lowest-energy CF excitations of Ce$^{3+}$ ions, and identify the symmetry of the CF states; in \\ref{subsubsec:Tem}, we analyze the temperature dependence of the CF parameters, and explain the observed anomaly on the basis of Kondo effect; in \\ref{subsubsec:Model}, we build a single-ion Hamiltonian, and fit the measured CF energies with this Hamiltonian to evaluate the SOC and CF strength, and to obtain the wavefunctions of eigenstates. In SubSec.~\\ref{subsec:P} we discuss lattice dynamics. The asymmetric lineshape, and relatively large full-width-at-half-maximum (FWHM) of the optical phonon modes point to electron-phonon interaction. In SubSec.~\\ref{subsec:QE} we discuss quasi-elastic excitations. We find that quasi-elastic fluctuations in the symmetry channel containing magnetic excitations develops below 20\\,K, and that the temperature dependence of the corresponding Raman susceptibility is consistent with the previously-reported static magnetic susceptibility data. Finally, in Sec.~\\ref{sec:Con} we provide a summary of our \nobservations and their implications. \n\n\\section{Experimental\\label{sec:Exp}}\nSingle crystals of CeB$_{6}$ were grown in Al flux by slow cooling from 1450\\,$^\\circ$C. The crystals were removed from the Al flux by leaching in NaOH solution~\\cite{Foroozani2015,Canfield1992}. The sample measured in this study was cleaved in ambient condition to expose its (001) crystallographic plane; the cleaved surface was then examined under a Nomarski microscope to find a strain-free area.\n\n\\begin{table}[b]\n\\caption{\\label{tab:Exp1}The relationship between the scattering geometries and the symmetry channels. Every scattering geometry is represented by E$_{i}$E$_{s}$, where E$_{i}$ and E$_{s}$ are the polarizations of incident and scattered light; X, Y, X' and Y' are the [100], [010], [110] and [1$\\overline{1}$0] crystallographic directions; R and L are right and left circular polarizations. A$_{1g}$, E$_{g}$, T$_{1g}$ and T$_{2g}$ are the irreducible representations of the O$_{h}$ group.}\n\\begin{ruledtabular}\n\\begin{tabular}{ll}\nScattering Geometry&Symmetry Channel\\\\\n\\hline\nXX&A$_{1g}$+4E$_g$\\\\\nXY&T$_{1g}$+T$_{2g}$\\\\\nX'X'&A$_{1g}$+E$_g$+T$_{2g}$\\\\\nX'Y'&3E$_g$+T$_{1g}$\\\\\nRR&A$_{1g}$+E$_g$+T$_{1g}$\\\\\nRL&3E$_g$+T$_{2g}$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nRaman-scattering measurements were performed in a quasi-back scattering geometry from sample placed in a continuous helium-gas-flow cryostat. A set of lines from a Kr$^+$ ion laser, 476, 531, 647, 676 and 752\\,nm, were used for excitation. Incident light with less than 10\\,mW power was focused into a 50$\\times$100\\,$\\mu$m$^{2}$ spot. The temperature points reported in this paper were corrected for laser heating, which was estimated to be \n0.5\\,K$\\slash$\\,mW~\\footnote{We mainly followed the procedure discussed in Ref.~\\cite{Maksimov1992} to estimate the laser heating. The optical absorption coefficient data were extracted from the optical data~\\cite{Kimura1990,Kimura1992,Kimura1994}, while the thermal conductivity data were taken from Ref.~\\cite{Sera1996}.}.\n\nSix polarization configurations were employed to probe excitations in different symmetry channels. The relationship between the scattering geometries and the symmetry channels~\\cite{Hayes2004} is given in Table.~\\ref{tab:Exp1}. The algebra used to decompose measured spectra into four symmetry channels is shown in Table.~\\ref{tab:Exp2}. \n\nWe used a custom triple-grating spectrometer with a liquid-nitrogen-cooled charge-coupled device (CCD) detector for analysis and collection of the scattered light. Low-resolution gratings with 150 lines per mm were used to measure the broad PL feature, while high-resolution gratings with 1800 lines per mm were used for measurements of the sharp Raman features. The data were corrected for the spectral response of the system.\n\nFor first-order scattering processes, the measured secondary-emission intensity $I(\\omega,T)$ is related to the Raman response $\\chi''(\\omega,T)$ by $I(\\omega,T)=[1+n(\\omega,T)]\\chi''(\\omega,T)+L(\\omega,T)$, where $n$ is the Bose factor, $\\omega$ is excitation energy, $T$ with temperature, and $L(\\omega,T)$ represents \nphoto-luminescence~\\cite{SM}. \nFor the second-order acoustic-phonon scattering process to be \ndiscussed in SubSec.~\\ref{subsec:P}, assuming the two constitute \nexcitations have the same energy, $I(\\omega,T)$ and \n$\\chi''(\\omega,T)$ are related by \n$I(\\omega,T)=[1+n(\\omega\/2,T)]^2\\chi''(\\omega,T)+L(\\omega,T)$~\\cite{SM}. \n\n\\begin{table}\n\\caption{\\label{tab:Exp2}The algebra used in this study to decompose the data into four symmetry channels.}\n\\begin{ruledtabular}\n\\begin{tabular}{lc}\nSymmetry Channel&Expression\\\\\n\\hline\nA$_{1g}$&$(1\/3)(XX+X'X'+RR-X'Y'-RL)$\\\\\nE$_g$&$(1\/6)(X'Y'+RL-XY)$\\\\\nT$_{1g}$&$(1\/2)(XY+RR-X'X')$\\\\\nT$_{2g}$&$(1\/2)(XY+RL-X'Y')$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\section{Results and Discussion\\label{sec:Res}}\n\n\\subsection{Overview\\label{subsec:OV}}\n\nIn Fig.~\\ref{fig:OV} we present a typical secondary-emission spectrum over a large energy range, covering Raman features of distinct origins. Among the Raman features, quasi-elastic excitations have the lowest-energy. Second-order acoustic phonon excitations are at around 200\\,cm$^{-1}$, while first-order optical phonon excitations are near 1000\\,cm$^{-1}$. The energy of the intra-multiplet CF excitation is around 400\\,cm$^{-1}$, while that of the inter-multiplet CF excitations is more than 2000\\,cm$^{-1}$. The PL continuum arises from a broad PL peak at around 2.0\\,eV. In the following subsections we will discuss every spectral feature separately in details.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig3.pdf}\n\\caption{\\label{fig:OV} \nAn overview of the low-temperature secondary-emission intensity measured in XY geometry at 20\\,K with 476\\,nm excitation in log-log scale. The top scale is the absolute energy of the secondary-emission photons in electron-Volts. The bottom scale is the energy loss, the laser-photon energy minus the scattered-photon energy, also called the Raman shift, in spectroscopic units cm$^{-1}$. The Raman features are superposed on a strong photo-luminescence continuum. Different Raman features are schematically represented by different colors: cyan, quasi-elastic (QE) Raman excitations; blue, the continuum of electronic Raman excitations; orange, second-order acoustic-phonon (AP) excitations and first-order optical-phonon (OP) excitations; red, crystal-field (CF) excitations; and green: the continuum of the photo-luminescence (PL).} \n\\end{figure}\n\n\\subsection{Photo-Luminescence\\label{subsec:PL}}\n\nIn Fig.~\\ref{fig:PL1}(a) we show the excitation dependence of the PL feature at room temperature. The PL peak has 2.0\\,eV excitation threshold, and excitations below 2.0\\,eV threshold show predominantly Raman features. The PL feature is centered at 1.95\\,eV, just below the threshold energy, and has about 0.4\\,eV full width at half maximum (FWHM). Upon cooling the peak shifts slightly to higher energy [Fig.~\\ref{fig:PL1}(b)].\n\n\\begin{figure}\n\\includegraphics[width=0.44\\textwidth]{Fig4.pdf}\n\\caption{\\label{fig:PL1}\n(a) Excitation dependence of the secondary-emission intensity I($\\omega$,300\\,K) measured in XY geometry at 300\\,K. For clarity, each spectrum is vertically shifted by a factor proportional to the excitation energy. The broad peak which does not change in the absolute emission energy with excitation energy is a photo-luminescence feature, while the sharp modes which follow the excitation energy are the Raman features. (b) Temperature dependence of the photo-luminescence feature measured in XY geometry with 476\\,nm excitation.}\n\\end{figure}\n\nThe optical conductivity shows a shoulder at around 2.0\\,eV~\\cite{Kimura1990,Kimura1992,Kimura1994}, suggesting an optical gap. Band-structure calculations further indicate a 2.0\\,eV gap between the Ce dispersive $5d$-band bottom and flat $4f$-band~\\cite{Kitamura1994,Suvasini1996,Neupane2015}. We therefore attribute the PL peak to the recombination of the electron-hole excitations between the $5d$- and $4f$-bands. Transitions between $d$- and $f$-states are dipole allowed, and the energy separation of the $5d$-band bottom and the $4f$-band is consistent with the energy of this PL peak. The enhancement of PL intensity for excitations above the 2\\,eV threshold results from the increase of the density of states (DOS) for the $4f$ to $5d$ interband transition.\n\n\\subsection{Crystal-Field Excitations\\label{subsec:CF}}\n\n\\subsubsection{Identification\\label{subsubsec:Iden}}\n\nIn total, there are four CF excitations from the $\\Gamma_8$ ground state to the higher states within the $^2F_{5\/2}$ and $^2F_{7\/2}$ multiplets: one intra-multiplet excitation and three inter-multiplet excitations [Fig.~\\ref{fig:Intro2}]. In Fig.~\\ref{fig:CF1} we present the spectrum of the four CF excitations measured at 15\\,K. Four peaks at 380\\,cm$^{-1}$, 2060\\,cm$^{-1}$, 2200\\,cm$^{-1}$ and 2720\\,cm$^{-1}$ are observed. The 380\\,cm$^{-1}$ excitation is the intra-multiplet $\\Gamma_8\\rightarrow\\Gamma_7$ transition. Among the three inter-multiplet excitations, only the $\\Gamma_8\\rightarrow\\Gamma_8^*$ transition can have a finite $A_{1g}$ component~\\cite{Koster1963}. In the inset of Fig.~\\ref{fig:CF1} we show that among the inter-multiplet excitations only the one at 2200\\,cm$^{-1}$ contains an $A_{1g}$ component. The 2200\\,cm$^{-1}$ excitation is therefore assigned to the $\\Gamma_8\\rightarrow\\Gamma_8^*$ transition. The CF excitation at 2720\\,cm$^{-1}$, in turn, can only be a transition between the $\\Gamma_8$ ground state and the $\\Gamma_6^*$ or $\\Gamma_7^*$ states. Raman scattering cannot distinguish between $\\Gamma_8\\rightarrow\\Gamma_6^*$ and $\\Gamma_8\\rightarrow\\Gamma_7^*$ transitions because they both contain the same irreducible representations~\\cite{Koster1963}: $\\Gamma_8\\otimes\\Gamma_6^*$\\,=\\,$\\Gamma_8\\otimes\\Gamma_7^*$\\,=\\,$E_{g}\\oplus T_{1g}\\oplus T_{2g}$. However, we will show in \\ref{subsubsec:Model} that the electron-cloud distribution of the $\\Gamma_6^*$ state has the smallest overlap with the boron octahedrons, the $\\Gamma_8^*$ state has intermediate overlap, and the $\\Gamma_7^*$ state has the largest overlap. Because of the Coulomb repulsion between cerium and boron electrons, the $\\Gamma_7^*$ state has the highest energy while the $\\Gamma_6^*$ state has the lowest energy. Indeed, within the $^2F_{5\/2}$ multiplet because the $\\Gamma_7$ state has more overlap with the boron octahedrons it has a higher energy than the $\\Gamma_8$ state. Therefore, the 2720\\,cm$^{-1}$ excitation is assigned to the $\\Gamma_8\\rightarrow\\Gamma_7^*$ transition, and the 2060\\,cm$^{-1}$ excitation is assigned to the $\\Gamma_8\\rightarrow\\Gamma_6^*$ transition.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig5.pdf}\n\\caption{\\label{fig:CF1}Raman response $\\chi^{\\prime\\prime}$($\\omega$,15\\,K) of the CF excitations measured in XY scattering geometry (T$_{1g}$+T$_{2g}$) with 476\\,nm excitation at 15\\,K. Three axis breakers are used on the horizontal axis in order to show the four excitations together. The spectral resolution is 3.5\\,cm$^{-1}$. Inset: $\\chi^{\\prime\\prime}$($\\omega$,15\\,K) measured in XX scattering geometry (A$_{1g}$+4E$_g$) at 15\\,K. The spectral resolution of the inset is about 30\\,cm$^{-1}$.}\n\\end{figure}\n\n\\subsubsection{Temperature Dependence\\label{subsubsec:Tem}}\n\nIn Fig.~\\ref{fig:CF2} we present the temperature dependence of the energy and FWHM of three CF excitations. The spectral parameters of the CF excitations were obtained by fitting the measured spectral peaks with a Lorentzian lineshape.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig6.pdf}\n\\caption{\\label{fig:CF2} Temperature dependence of the energy (a-c) and FWHM (d-f) of the $\\Gamma_8\\rightarrow\\Gamma_7$, $\\Gamma_8\\rightarrow\\Gamma_8^*$ and $\\Gamma_8\\rightarrow\\Gamma_7^*$ CF excitations shown in Fig.~\\ref{fig:CF1}. The line-joined square labels in (a-c) represent the excitation energies calculated by our model Hamiltonian calculation. The error bars represent one standard deviation of the Lorentzian fit.}\n\\end{figure}\n\nOn cooling from 304\\,K to 15\\,K the lattice contraction strengthens the electrostatic potential at the Ce sites resulting in increase of the $\\Gamma_8\\rightarrow\\Gamma_7$, $\\Gamma_8\\rightarrow\\Gamma_8^*$, and $\\Gamma_8\\rightarrow\\Gamma_7^*$ transition energy by 7\\,cm$^{-1}$, 5\\,cm$^{-1}$, and 18\\,cm$^{-1}$ respectively~\\footnote{The energy of the $\\Gamma_8\\rightarrow\\Gamma_7$ transition shows monotonic temperature dependence. The earlier reports, Ref.~\\cite{Zirngiebl1984,Loewenhaupt1985ccf}, showed no temperature dependence of the transition energy on cooling from 300\\,K to 20\\,K, followed by a rapidly hardening on cooling below 20\\,K.}. A discussion of the change of the energy of the CF states with increasing CF potential will be given in \\ref{subsubsec:Model}. \n\nAt room temperature, the CF spectral lines of CeB$_{6}$ are broader than those measured from Ce$^{3+}$ ions embedded in insulators, e.g. Ce-doped Y$_2$O$_3$~\\cite{Nolas1994} or Ce-doped LuPO$_4$~\\cite{Williams1989}. The broadening could be caused by two factors: first, lattice of Ce$^{3+}$ ions leads to small dispersion of the narrow $4f$-bands; and second, hopping of conduction electrons among the boron sites induces fluctuations of the electrostatic potential at the Ce sites, which broadens FWHM.\n\nOn cooling, the FWHM of the $\\Gamma_8\\rightarrow\\Gamma_7$ and $\\Gamma_8\\rightarrow\\Gamma_7^*$ CF transitions decrease from 300\\,K to 80\\,K, but anomalously increases below 80\\,K [Fig.~\\ref{fig:CF2}~(d) and (f)]. The decrease of FWHM is expected because lattice vibrations, causing fluctuations of the electrostatic potential at Ce sites, diminish with cooing. In order to understand the anomalous increase of FWHM below 80\\,K, it is important to notice that the electrical resistivity of CeB$_{6}$ has its local minimum at 80\\,K. The resistivity upturn below 80\\,K results from the Kondo effect~\\cite{Takase1980} due to increase in the rate of conduction electron scattering from the local moments at the Ce sites on cooling~\\cite{Hewson1993,Amusia2015}. The Kondo effect shortens the lifetime of the $\\Gamma_7$ and $\\Gamma_7^*$ CF states, so the FWHM of the $\\Gamma_8\\rightarrow\\Gamma_7$ and $\\Gamma_8\\rightarrow\\Gamma_7^*$ CF transitions increases below 80\\,K. Nevertheless, the FWHM of the $\\Gamma_8\\rightarrow\\Gamma_8^*$ CF transition does not show an upturn below 80\\,K [Fig.~\\ref{fig:CF2}~(e)]. This is because the $\\Gamma_8^*$ state has smaller overlap with the boron octahedrons than the $\\Gamma_7$ and $\\Gamma_7^*$ states, therefore, it is less influenced by the increased conduction electron scattering rate. \n\nOur data do not display directly the splitting of the $\\Gamma_8$ CF ground state. However, the minimum FWHM of the $\\Gamma_8\\rightarrow\\Gamma_7$ is around 33\\,cm$^{-1}$ [Fig.~\\ref{fig:CF2}~(d)]: if the splitting of the CF ground state is small, it would not be resolved. The previous studies suggested a splitting of 20\\,cm$^{-1}$~\\cite{Zirngiebl1984,Terzioglu2001}, which does not contradict our data.\n\n\\subsubsection{Model Hamiltonian Calculation\\label{subsubsec:Model}}\n\nTo shed light on the nature of the CF transitions, we perform a model Hamiltonian calculation. We use the following single-ion Hamiltonian\n\\begin{equation}\nH=E_0+H_{SOC}+H_{CF}~.\n\\label{eq:H}\n\\end{equation}\nThe first term $E_0$ represents the energy of unperturbed $4f$ shell. The value $E_0$ is chosen to put the $\\Gamma_8$ ground state at zero energy.\nThe second term\n\\begin{equation}\nH_{SOC}=\\xi\\hat{\\mathbf{L}}\\cdot\\hat{\\mathbf{\\sigma}}\n\\label{eq:HSOC}\n\\end{equation}\ndescribes the effect of SOC. Here $\\xi$ is the SOC coefficient, $\\hat{\\mathbf{L}}$ is the orbital angular momentum operator and $\\hat{\\mathbf{\\sigma}}$ are Pauli matrices.\nThe third term\n\\begin{equation}\nH_{CF}=B_4(\\hat{O}^0_4+5\\hat{O}^4_4)+B_6(\\hat{O}^0_6-21\\hat{O}^4_6)\n\\label{eq:HCF}\n\\end{equation}\nis the general expression for a CF potential of cubic site symmetry~\\cite{Lea1962}, where $\\hat{O}^0_4$, $\\hat{O}^4_4$, $\\hat{O}^0_6$ and $\\hat{O}^4_6$ are Stevens operators~\\cite{Stevens1952}, and $B_4$ and $B_6$ are the CF coefficients~\\footnote{When comparing the CF coefficients across different literature, additional constants are needed~\\cite{Hutchings1964,Kassman1970}.}: \n\\begin{equation}\nB_4=A_4\\beta~,\n\\label{eq:B4}\n\\end{equation}\n\\begin{equation}\nB_6=A_6\\gamma~, \n\\label{eq:B6}\n\\end{equation}\n$A_4$ and $A_6$ are the geometrical coordination factors determined by the charge configuration around the Ce sites. Regardless of the specific configuration, $A_4\\,\\sim\\,a^{-5}$ and $A_6\\,\\sim\\,a^{-7}$, where $a$ is the lattice constant; $$ and $$ are the mean fourth and sixth powers of the radii of the Ce$^{3+}$ $4f$-orbital, and $\\beta$ and $\\gamma$ are the Stevens multiplicative factors~\\cite{Stevens1952}.\n\n\\begin{figure*}\n\\includegraphics[width=0.98\\textwidth]{Fig7.pdf}\n\\caption{\\label{fig:CF4}Eigenenergies and eigenstates derived from the model Hamiltonian calculation. (a) Evolution of the $4f$-orbital energy with CF potential and SOC strength. [from left to center] Increasing CF potential in the absence of SOC (a1), and then increasing SOC strength in the existence of full CF potential (a2); [from left to center] increasing SOC in the absence of CF potential (a3), and then increasing CF potential in the existence of full SOC (a4). In this panel, the full SOC strength is $\\xi$=610\\,cm$^{-1}$, and the full CF potential strengths are $B_4$=-0.758\\,cm$^{-1}$ and $B_6$=-0.0165\\,cm$^{-1}$. (b) The wavefunctions and the angular electron-cloud distribution of the eigenstates. [left] The wavefunctions of the eigenstates when only CF potential is present. Red denotes positive value while blue denotes negative value; [middle] the angular electron-cloud distribution of the eigenstates when both SOC and CF potential are present; [right] the angular electron-cloud distribution of the eigenstates when only SOC is present.}\n\\end{figure*}\n\nThe effects of SOC and CF potential on the energy and angular electron-cloud distribution of the CF levels are illustrated in Fig.~\\ref{fig:CF4}. In the absence of the SOC, the CF eigenfunctions could be classified by the irreducible representations (IRs) of O$_h$ double group. The relevant IRs are the one-dimensional $A_{2u}$, three-dimensional $T_{2u}$, and three-dimensional $T_{1u}$ for the orbital part of the wavefunction, and two-dimensional $\\Gamma_6$ for the spin part. The 14-fold degenerate 4f orbital would be split into 2-fold $A_{2u}\\otimes\\Gamma_6$, 6-fold $T_{2u}\\otimes\\Gamma_6$, and 6-fold $T_{1u}\\otimes\\Gamma_6$ orbitals. Finite SOC splits further these orbitals and results in mixing of wavefunctions derived from different orbitals. The symmetry of the split states is given by the decomposition of the direct products into direct sums of IRs of O$_h$ double group~\\cite{Koster1963}: $A_{2u}\\otimes\\Gamma_6\\,=\\,\\Gamma_7$, $T_{2u}\\otimes\\Gamma_6\\,=\\,\\Gamma_8\\oplus\\Gamma_7$, and $T_{1u}\\otimes\\Gamma_6\\,=\\,\\Gamma_6\\oplus\\Gamma_8$.\n\nOn the other hand, if cubic CF were absent, the 4f orbital would be \nsplit into 8-fold $^2F_{7\/2}$ ($J\\,=\\,L+S$) and 6-fold $^2F_{5\/2}$ \n($J\\,=\\,L-S$) multiplets. Finite CF potential splits the two \nmultiplets and induces mixing of wavefunctions derived from different \nmultiplets~\\cite{SM}. \nThe symmetry of the split states is given by the compatibility table showing the mapping of IRs of the full rotational group into IRs of O$_h$ double group~\\cite{Koster1963}: $^2F_{7\/2}\\,=\\,\\Gamma_8\\oplus\\Gamma_7\\oplus\\Gamma_6$, and $^2F_{5\/2}\\,=\\,\\Gamma_8\\oplus\\Gamma_7$. With both SOC and CF present, the CF eigenfunctions should be classified by the IRs of the double group, namely two-dimensional $\\Gamma_6$, two-dimensional $\\Gamma_7$, and four-dimensional $\\Gamma_8$.\n\nWe diagonalize the Hamiltonian (\\ref{eq:H}) in the basis of $|L,m_l\\rangle|S,m_s\\rangle$, where $L, m_l, S, m_s$ are quantum numbers corresponding to $\\hat{\\mathbf{L}}, \\hat{L}_z, \\hat{\\mathbf{S}}, \\hat{S}_z$ operators, respectively. After diagonalization, the CF transition energies can be expressed in terms of $\\xi$, $B_4$ and $B_6$. We obtain these three parameters by fitting the energies of three CF transitions to the data at 15\\,K (the weakest $\\Gamma_8\\rightarrow\\Gamma_6^*$ transition is not accounted in this procedure). The obtained set of parameters comprises $\\xi$=610\\,cm$^{-1}$, $B_4$=-0.758\\,cm$^{-1}$ and $B_6$=-0.0165\\,cm$^{-1}$. The same values automatically render the energy of weakest transition at 2070\\,cm$^{-1}$, which is close to the observed value at 2060\\,cm$^{-1}$. The value of $\\xi$ (610\\,cm$^{-1}$) is also consistent with the estimated value for the Ce$^{3+}$ ion embedded in LuPO$_4$ (614\\,cm$^{-1}$)~\\cite{Williams1989}. Such consistency demonstrates the reliability of the model (\\ref{eq:H}).\n\nWe can further use this single-ion model to describe the temperature dependence of the CF excitation energy. Here we assume that $\\xi$ is temperature-independent, and that the temperature dependence of the $B_4$ and $B_6$ coefficients comes from the temperature dependence of the lattice constant $a(T)$. We therefore rewrite $B_4$ and $B_6$ as $B_4(T)$=$C_4a(T)^{-5}$ and $B_6(T)$=$C_6a(T)^{-7}$, where $C_4$ and $C_6$ are temperature-independent factors. The temperature dependence of the lattice constant $a(T)$ is obtained from the Refs.~\\cite{Tanaka2002,Zaharko2003}. Then, we determine the values of $\\xi$, $C_4$ and $C_6$ by matching the calculated values with the measured data at 300\\,K. Finally, we use the determined $\\xi$, $C_4$ and $C_6$ to calculate CF excitation energies below 300\\,K. The results are shown in Fig.~\\ref{fig:CF2}~(a-c). The discrepancy between the measured data and the calculated values below 200\\,K results from unaccounted terms in the model Hamiltonian [Eq.~(\\ref{eq:H})]; for an example, interactions between localized $f$-electrons and the itinerant conduction electrons.\n\nBy virtue of the obtained eigenfunctions, the Raman intensity of the four CF transitions can be calculated. For non-resonant scattering, the Raman response $\\chi^{\\prime\\prime}(\\omega)$ has the following expression~\\cite{Devereaux2007}:\n\\begin{equation}\n\\chi^{\\prime\\prime}(\\omega)\\sim\\frac{1}{Z}\\sum_{i,f}|\\langle f|\\hat{R}_{\\mu\\nu}|i\\rangle|^2e^{-E_i\/kT}\\delta(E_f-E_i-\\hbar\\omega)~,\n\\label{eq:I}\n\\end{equation}\nwhere $Z$ is the partition function, $|i\\rangle$, $|f\\rangle$ are the initial and final state with energy $E_i$ and $E_f$, $\\omega$ is the Raman shift, and $\\hat{R}_{\\mu\\nu}$ is the effective Raman operator. In our case, $|i\\rangle$ is the CF ground state and $|f\\rangle$ is one of the excited CF states. For nonresonant Raman scattering, $\\hat{R}_{\\mu\\nu}$ is a quadrupolar operator depending on the crystallographic symmetry and scattering geometry $\\mu\\nu$~\\cite{Axe1964,Kiel1969,Williams1989}. For XY scattering geometry in a cubic crystal, $\\hat{R}_{XY}$ transforms in the same way as quadrupole $xy$ under the symmetry operations of O$_h$ point group:\n\\begin{equation}\n\\hat{R}_{XY}=\\frac{1}{2}(\\hat{L}_x\\hat{L}_y+\\hat{L}_y\\hat{L}_x)=\\frac{1}{4i}(\\hat{L}_+^2-\\hat{L}_-^2)~,\n\\label{eq:R}\n\\end{equation}\nwhere $\\hat{L}_+$ and $\\hat{L}_-$ are the ladder operators of the orbital angular momentum. We note that because light only couples to the electron's orbital degree of freedom, the effective Raman operator should be written in terms of the orbital angular momentum operators, rather than the total angular momentum operators. Expression~(\\ref{eq:R}) should accordingly be evaluated in the basis of $|L,m_l\\rangle|S,m_s\\rangle$.\n\nIn Fig.~\\ref{fig:CF5} we compare the calculated and measured CF transition intensity. Because the 476\\,nm excitation is resonant with interband transitions (see SubSec.~\\ref{subsec:PL}) but the expression~(\\ref{eq:R}) is only valid for non-resonant scattering, we expect discrepancy between the calculated and measured results. Nevertheless, the relative intensity of the three inter-multiplet transitions is reproduced.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig8.pdf}\n\\caption{\\label{fig:CF5}Normalized intensity of the four CF transitions in XY scattering geometry at 15\\,K, measured (in yellow) and calculated (in blue). The measured\/calculated intensity of the four transitions is normalized by their respective largest value.}\n\\end{figure}\n\n\\subsection{Phononic Excitations\\label{subsec:P}}\n\nAn overview of the phonon modes is presented in Fig.~\\ref{fig:P1}(a). From group-theory analysis, CeB$_{6}$ has three Raman-active optical phonon modes: A$_{1g}$, E$_{g}$ and T$_{2g}$. Their respective energies are 1271, 1143 and 681.7\\,cm$^{-1}$ at 300\\,K, consistent with previous results~\\cite{Zirngiebl1986,Ogita2003}. Their lineshapes at 300\\,K and 4\\,K are presented in Fig.~\\ref{fig:P1}(b); no anomaly is observed on cooling. The E$_{g}$ and T$_{2g}$ optical phonon modes exhibits asymmetric lineshape. The underlying electronic continuum likely results from electronic interband transitions: according to the calculated and measured band structure~\\cite{Kitamura1994,Suvasini1996,Neupane2015}, many direct interband transitions are allowed and in turn can contribute to the nearly flat continuum below 1500\\,cm$^{-1}$ ($\\sim$0.2\\,eV).\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig9.pdf}\n\\caption{\\label{fig:P1}(a) Symmetry-decomposed Raman response $\\chi^{\\prime\\prime}$($\\omega$,300\\,K), measured with 532\\,nm excitation at 300\\,K. Inset in (a): Symmetry-decomposed Raman spectrum of the second-order acoustic phonon scattering peak, measured with 752\\,nm excitation at 300\\,K. Thermal factor $[1+n(\\omega\/2,T)]^2$ is used to derive this particular inset; the other Raman spectra presented in this paper are obtained with the normal thermal factor $[1+n(\\omega,T)]$. (b), (c) and (d): Raman spectra of the T$_{2g}$, A$_{1g}$ and E$_{g}$ optical phonon modes, measured with 532\\,nm excitation at 300\\,K and 4\\,K. In (b), (c) and (d), the spectral resolution is 2.8\\,cm$^{-1}$ for the high temperature data and 1.3\\,cm$^{-1}$ for the low temperature data. (e) The schematic vibration patterns for the three optical phonon modes. Because the cerium ions are at the inversion centers, Raman-active phonon modes only involve vibrations of the boron octahedrons.}\n\\end{figure}\n\nThe peak at 194\\,cm$^{-1}$ is not fully polarized. It originates from second-order scattering of acoustic branches at the Brillouin-zone boundary~\\cite{Ogita2003}, where the flat dispersion gives rise to a large density of states. From this peak, we infer that the maximum of the acoustic phonon frequency is around 100\\,cm$^{-1}$, which is consistent with the INS data~\\cite{Kunii1997}. Another feature at 373\\,cm$^{-1}$ shows larger T$_{2g}$ contribution and smaller E$_{g}$ contribution. It is the $\\Gamma_8\\rightarrow\\Gamma_7$ CF excitation discussed in SubSec.~\\ref{subsec:CF}. The peak at 1400\\,cm$^{-1}$ has strong A$_{1g}$ contribution and very weak E$_{g}$ contribution. It results from second-order scattering of the T$_{2g}$ phonon mode~\\cite{Ogita2003}. The symmetry-decomposed spectra further reveal an A$_{1g}$ peak at 1158\\,cm$^{-1}$, which was not reported previously. This peak might correspond to the summation mode of the 373\\,cm$^{-1}$ CF excitation and the T$_{2g}$ phonon mode. Such coupling has been observed in another $f$-electron system UO$_2$~\\cite{Livneh2008}.\n\nIn Fig.~\\ref{fig:P2} we show the temperature dependence of the energy and FWHM of the A$_{1g}$ contribution of the second-order acoustic mode, and the A$_{1g}$ optical mode. The spectral parameters of the phonon modes were obtained by fitting the measured spectral peaks with a Lorentzian lineshape.\n\n\\begin{figure}[b]\n\\includegraphics[width=0.48\\textwidth]{Fig10.pdf}\n\\caption{\\label{fig:P2} Temperature dependence of the energy (in red) and FWHM (in black) of (a) the A$_{1g}$ component of the second-order acoustic phonon scattering peak, and (b) the A$_{1g}$ optical phonon mode. The solid lines are fitting curves of an anharmonic decay model assuming decay into two optical modes, or an optical plus an acoustic modes~\\cite{Wallis1966,Wallis1983}. The error bars represent one standard deviation of the Lorentzian fit.}\n\\end{figure}\n\nTemperature dependence of the phonon energy and FWHM is usually described by anharmonic effects. In most cases, the three-phonon processes renders the fastest relaxation, and higher-order processes can be neglected. Furthermore, the A$_{1g}$ optical mode at $\\Gamma$ point has the highest frequency among all the phonon branches of CeB$_{6}$~\\cite{Gurel2010}; hence we only need to consider processes in which one A$_{1g}$ optical mode at $\\Gamma$ point decays into two phonon modes satisfying conservation of energy and momentum~\\footnote{There are other constraints which the decay processes must satisfy. For example, the spontaneous decay of a phonon by anharmonic processes of any order into a set of phonons of higher phase velocity is impossible~\\cite{Lax1981}.}. \nWe use an generalized anharmonic decay model assuming multiple decay channels; for every channel, the decay products can be two acoustic modes, an optical plus an acoustic modes, or two acoustic modes~\\cite{Wallis1966,Wallis1983}~\\footnote{Because in CeB$_{6}$ the maximum acoustic phonon frequency is around 100\\,cm$^{-1}$, for the high-frequency A$_{1g}$ mode decay into two acoustic modes is impossible.}:\n\\begin{equation}\n\\omega(T)=\\omega_0-\\sum_{i}\\omega_{\\delta (i)}[1+\\frac{1}{e^{\\hbar\\omega_{1(i)}\/k_BT}-1}+\\frac{1}{e^{\\hbar\\omega_{2(i)}\/k_BT}-1}]~,\n\\label{energyTwoDiff}\n\\end{equation}\n\\begin{equation}\n\\Gamma(T)=\\Gamma_0+\\sum_{i}\\Gamma_{\\delta (i)}[1+\\frac{1}{e^{\\hbar\\omega_{1(i)}\/k_BT}-1}+\\frac{1}{e^{\\hbar\\omega_{2(i)}\/k_BT}-1}]~,\n\\label{gammaTwoDiff}\n\\end{equation}\nwhere the subscript $(i)$ indicates the decay channel. $\\omega_{\\delta (i)}$ and $\\Gamma_{\\delta (i)}$ are factors reflecting the relative importance of the various decay channels. $\\hbar\\omega_{1(i)}$ and $\\hbar\\omega_{2(i)}$ are the energy of the decay products in the decay channel labelled by $(i)$. $\\hbar(\\omega_0-\\sum_{i}\\omega_{\\delta (i)})$ and $\\Gamma_0+\\sum_{i}\\Gamma_{\\delta (i)}$ \ncorrespond to the zero-temperature phonon energy and the FWHM, \nrespectively. $\\Gamma_0$ accounts for the temperature-independent part of the FWHM originating not from anharmonic decay processes, but from, for example, imperfection of the sample. \n\nBoth $\\omega_{\\delta (i)}$ and $\\Gamma_{\\delta (i)}$ are proportional to\n\\begin{equation}\n\\sum_{\\mathbf{k}_{1(i)},\\mathbf{k}_{2(i)}}|\\alpha(\\mathbf{k}_{1(i)},\\mathbf{k}_{2(i)})|^2\\,\\delta[\\omega_{A_{1g}}-\\omega_{1(i)}(\\mathbf{k}_{1(i)})-\\omega_{2(i)}(\\mathbf{k}_{2(i)})]~,\n\\label{decayC}\n\\end{equation}\nwhere $\\alpha$ is the anharmonic coefficient; $\\mathbf{k}_{1(i)}$ and $\\mathbf{k}_{2(i)}$ are the wavevector of the decay products in the decay channel labelled by $(i)$; $\\delta$ represents the Dirac $\\delta$-function.\n\nReferring to the calculated phonon dispersion~\\cite{Gurel2010}, we expect two decay channels for the 1278\\,cm$^{-1}$ A$_{1g}$ phonon: (1) decay into one 684\\,cm$^{-1}$ optical phonon and one 594\\,cm$^{-1}$ optical phonon with opposite momenta; (2) decay into one 1178\\,cm$^{-1}$ optical phonon near +R point and one 100\\,cm$^{-1}$ acoustic phonon near -R point.\n\nThe two phonon branches involved in the decay channel (1) is essentially flat over the whole Brillouin zone; hence a large number of states are available for the decay to happen. On the contrary, for the two phonon branches of the decay channel (2), only states near R point simultaneous satisfy the requirements of energy and momentum conservation. Therefore, the decay channel (1) would dominate if the anharmonic coefficient is not significantly different for the two channels.\n\n\\begin{table}[b]\n\\caption{\\label{tab:P1}The fitting parameters for the energy and FWHM of the A$_{1g}$ optical phonon mode. Units are cm$^{-1}$.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\n$\\omega_0$&$\\omega_{\\delta (1)}$&$\\omega_{\\delta (2)}$\\\\\n1309.0$\\pm$0.1&28.12$\\pm$0.05&2.664$\\pm$0.003\\\\\n\\hline\n$\\Gamma_0$&$\\Gamma_{\\delta (1)}$&$\\Gamma_{\\delta (2)}$\\\\\n1.07$\\pm$0.08&8.4$\\pm$0.2&1.76$\\pm$0.01\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nThe fitting results of the anharmonic decay model are summarized in Table~\\ref{tab:P1}. Indeed, $\\omega_{\\delta (1)}$\\,\\textgreater\\,$\\omega_{\\delta (2)}$ and $\\Gamma_{\\delta (1)}$\\,\\textgreater\\,$\\Gamma_{\\delta (2)}$. The temperature-independent $\\Gamma_0$ is much smaller than $\\Gamma_{\\delta (1)}+\\Gamma_{\\delta (2)}$, indicating not only that the lineshape broadening mainly results from the anharmonic decay, but also that the sample is of excellent quality. In contrast to the behavior of the A$_{1g}$ optical mode, the second-order scattering of acoustic modes in the A$_{1g}$ channel shows decreasing energy on cooling [Fig.~\\ref{fig:P2}(a)]. The 2\\% softening might be a prelude to the AFQ ordering.\n\nWe attribute the apparent asymmetric lineshape of the T$_{2g}$ and \nE$_{g}$ optical phonon modes to the coupling between these phonons \nand the low-frequency fluctuations [SubSection.~\\ref{subsec:QE}]. \nThe observed spectral lineshapes are resulted from convolution of the \nphononic Lorentzian and Drude-like function describing the low-lying \nfluctuations. \nWe use the following expression to fit modes' lineshape at 4\\,K: \n\\begin{multline}\n\\chi^{\\prime\\prime}(\\omega,4\\,K)=\n\\sum_i\\{\\frac{A^2_{(i)}\\gamma_{L(i)}}{(\\omega-\\omega_{L(i)})^2+\\gamma_{L(i)}^2} +\\\\\n\\frac{A^2_{(i)}v_{(i)} \\theta(\\omega-\\omega_{L(i)}) \n(\\omega-\\omega_{L(i)})[1+n(\\omega-\\omega_{L(i)},4\\,K)]}{(\\omega-\\omega_{L(i)})^2+(\\gamma_{L(i)}+\\gamma_{D(i)})^2} +\\\\ \n\\frac{A^2_{(i)}v_{(i)} \\theta(\\omega_{L(i)}-\\omega) \n(\\omega_{L(i)}-\\omega) n(\\omega_{L(i)}-\\omega,4\\,K)}{(\\omega_{L(i)}-\\omega)^2+(\\gamma_{L(i)}+\\gamma_{D(i)})^2}\\}~.\n\\label{eq:Asym}\n\\end{multline}\nIn Eq.~(\\ref{eq:Asym}), the first term describes the bare phonon \npart, while the second and third terms correspond to the Stokes and \nanti-Stokes of the phonon assisted electronic scattering. \nThe summation runs over all \nthe $k$-points in the Brillouin zone. \nReferring to the calculated phonon dispersion~\\cite{Gurel2010}, the \nT$_{2g}$ mode belongs to a \nflat branch over the Brillouin zone, while the E$_{g}$ mode belongs a \ndispersive branch which has high DOS at $\\Gamma$ and $R$ \npoints~\\cite{Gurel2010}. \nTherefore, for the latter case we only consider coupling at $\\Gamma$ \nand $R$ points. \nIn this equation, $A_{(i)}$ is the phonon light-scattering vertex; \n$\\omega_{L(i)}$ is the phonon frequency; \n$2\\gamma_{L(i)}$ is the FWHM of the bare phonon Lorentzian function; \n$\\gamma_{Di}$ measures the relaxation rate of the Drude function; \n$v_{(i)}$ represents the electron-phonon coupling strength; \nand $\\theta(\\omega)$ is the Heaviside step function. \n\nFor the T$_{2g}$ mode, we choose $\\gamma_{D(\\Gamma)}$ to be \n3.0\\,cm$^{-1}$, which is consistent with the measured value of the \nT$_{1g}$ quasi-elastic fluctuations at 16\\,K. \nFor the E$_{g}$ mode, \nwe choose both $\\gamma_{D(\\Gamma)}$ and $\\gamma_{D(R)}$ to be \n11\\,cm$^{-1}$, which is consistent with the measured value of the A$_{1g}$ \nquasi-elastic fluctuations at 16\\,K. \nWe further require that \n$v_{(\\Gamma)}$ and $v_{(R)}$ are the same. \n\nThe fitting results of the T$_{2g}$ and E$_{g}$ composite modes are \nshown in Fig.~\\ref{fig:Asym} and summarized in Table~\\ref{tab:P2}. \nThe dip of the fitting curve in Fig.~\\ref{fig:Asym}(b) results from \nthe negligence of the contributions at $k$-points between $\\Gamma$ \nand $R$ points.\nThe FWHM of the bare T$_{2g}$ phonon mode \n($\\sim$11\\,cm$^{-1}$) is similar to that of the A$_{1g}$ phonon mode \n($\\sim$12\\,cm$^{-1}$), while the FWHM of the bare E$_{g}$ phonon mode \n($\\sim$17\\,cm$^{-1}$) is larger. \nThis large E$_{g}$ FWHM, \nagain, is an artifact caused by negligence of the contributions \nfrom remaining $k$-points. \nThe energy difference between the E$_{g}$ mode \nat $\\Gamma$ and $R$ points is $\\sim$17\\,cm$^{-1}$, which is \ncomparable to the calculated difference of \n$\\sim$30\\,cm$^{-1}$~\\cite{Gurel2010}. \n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig11.pdf}\n\\caption{\\label{fig:Asym}\nThe measured with 532\\,nm excitation at 4\\,K Raman response function \n(black points with one standard deviation error bars) fitted with the model of Eq.~(\\ref{eq:Asym}) for \n(a) the T$_{2g}$ and (b) the E$_{g}$ optical phonons coupled to \nlow-frequency electronic excitations.}\n\\end{figure}\n\n\\begin{table}[b]\n\\caption{\\label{tab:P2}The fitting parameters for the T$_{2g}$ and \nE$_{g}$ composite modes by Eq.~(\\ref{eq:Asym}). Units are given \nin the brackets.} \n\\begin{ruledtabular}\n\\begin{tabular}{lcc}\nParameter (Units)&T$_{2g}$ Mode&E$_{g}$ Mode\\\\\n\\hline\n$A_{(\\Gamma)}$ (a.u.)&20.29$\\pm$0.03&7.4$\\pm$0.4\\\\\n$\\gamma_{L(\\Gamma)}$ (cm$^{-1}$)&5.67$\\pm$0.02&8.5$\\pm$0.5\\\\\n$\\omega_{L(\\Gamma)}$ (cm$^{-1}$)&682.73$\\pm$0.02&1138.4$\\pm$0.3\\\\\n$A_{(R)}$ (a.u.)&&7.0$\\pm$0.8\\\\\n$\\gamma_{L(R)}$ (cm$^{-1}$)&&10$\\pm$1\\\\\n$\\omega_{L(R)}$ (cm$^{-1}$)&&1155.0$\\pm$0.5\\\\\n$v$ (cm$^{-1}$)&0.691$\\pm$0.005&2.2$\\pm$0.2\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Quasi-Elastic Excitations\\label{subsec:QE}}\n\nIn Fig.~\\ref{fig:QE1} we show the symmetry-decomposed Raman response measured with 752\\,nm excitation at 300\\,K and 16\\,K. The low-energy Raman response shows quasi-elastic features which can be described by a Drude lineshape:\n\\begin{equation}\n\\chi^{\\prime\\prime}(\\omega,T)\\propto\\frac{\\alpha^2\\omega}{\\omega^2+\\gamma^2}~,\n\\label{eq:Drude}\n\\end{equation}\nwhere $\\alpha$ is the light-scattering vertex and $\\gamma$ measures the fluctuation rate.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.48\\textwidth]{Fig12.pdf}\n\\caption{\\label{fig:QE1}Raman response $\\chi^{\\prime\\prime}(\\omega,T)$ in the four Raman-active symmetry channels measured with 752\\,nm excitation at (a) 300\\,K and (b) 16\\,K. The solid lines are Drude fits [Eq.~(\\ref{eq:Drude})]. The error bars represent one standard deviation.}\n\\end{figure}\n\nThe Raman response gets enhanced in all the channels on cooling. Especially, the T$_{1g}$ Raman response changes qualitatively and develops into a strong quasi-elastic feature at low temperature. The basis functions of the T$_{1g}$ representation in O$_{h}$ group transform as the three components of angular momentum, which behave as a pseudovector~\\cite{Koster1963}. This transformation property indicates that the observed quasi-elastic peak in T$_{1g}$ channel may have a magnetic origin.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Fig13.pdf}\n\\caption{\\label{fig:QE2}(a) Temperature dependence of the Raman response $\\chi^{\\prime\\prime}(\\omega,T)$ in the T$_{1g}$ symmetry channel measured with 752\\,nm excitation. The solid lines are Drude fits [Eq.~(\\ref{eq:Drude})]. (b) Comparison between the temperature dependence of the static Raman susceptibility $\\chi(0,T)$ (black) and that of the magnetic susceptibility $\\chi_{mag}$ (purple)~\\cite{Kawakami1980}. (c) Comparison between the temperature dependence of the inverse static Raman susceptibility (black) and that of the inverse magnetic susceptibility (purple)~\\cite{Kawakami1980}. The blue arrow indicates the magnetic ordering temperature while the red one indicates the orbital ordering temperature. The error bars represent one standard deviation.}\n\\end{figure}\n\nWe measured the temperature dependence of Raman response in the XY \nscattering geometry, in which T$_{1g}$+T$_{2g}$ symmetry components \nare probed. Since T$_{2g}$ signal at low-temperature is nearly \nconstant [Fig.~\\ref{fig:QE1}(b)], we fit the Raman response with the \nsum of Drude and constant terms, and then remove the constant part to \nobtain the desired T$_{1g}$ component~\\cite{SM}.\nThe T$_{1g}$ Raman response obtained this way is shown in Fig.~\\ref{fig:QE2}(a). The quasi-elastic excitation in T$_{1g}$ symmetry channel becomes significant below 20\\,K, and its intensity increases on further cooling. The static Raman susceptibility, $\\chi(0,T)$, plotted in Fig.~\\ref{fig:QE2}(b) is obtained from the Raman response by virtue of Kramers-Kronig relation: $\\chi(0,T)=\\frac{2}{\\pi}\\int_{0}^{50\\,cm^{-1}} \\frac{\\chi^{\\prime\\prime}(\\omega,T)}{\\omega} d\\omega$. Drude function in Eq.~(\\ref{eq:Drude}) is used to extrapolate $\\chi^{\\prime\\prime}(\\omega,T)$ below 4\\,cm$^{-1}$. In Fig.~\\ref{fig:QE2}(b) and (c), the temperature dependence of the static Raman susceptibility is compared with that of the magnetic susceptibility~\\cite{Kawakami1980}. The fact that the temperature dependence of both quantities follows the same trend further supports the magnetic origin of the quasi-elastic peak in T$_{1g}$ symmetry channel~\\cite{Kung2015}.\n\nIn zero magnetic field, Raman scattering data cannot determine whether the observed T$_{1g}$ quasi-elastic response is of FM or AFM origin. Nevertheless, the Raman-measured T$_{1g}$ quasi-elastic response is consistent with the FM correlations studied by INS: without external magnetic field and above T$_{Q}$, the magnitude of the INS-measured zone-center quasi-elastic peak decreases on warming~\\cite{Jang2014}. We note by passing that a first-principle calculation for CeB$_{6}$ indicates that the expected values of both $4f$-orbital occupancy and total angular momentum exhibit an obvious anomalies around 20\\,K~\\cite{Lu2017}. This is the same temperature around which the T$_{1g}$ quasi-elastic Raman response starts to develop.\n\nThe mechanism responsible for the FM correlations can be understood as follows~\\cite{Schlottmann2012}. Consider the two electrons at neighboring Ce$^{3+}$ sites. In the staggered orbital-ordering phase, the orbital part of the total wavefunction of these two electrons is antisymmetric. Due to the resulting exchange interaction, the spins at neighboring Ce$^{3+}$ sites are FM correlated.\n\nThe $\\Gamma_8$ CF ground state of O$_{h}$ group has zero quadrupole moment. If the site symmetry is reduced from O$_{h}$ group to D$_{4h}$ group, the $\\Gamma_8$ state of O$_{h}$ group would be split into the $\\Gamma_6$ and $\\Gamma_7$ states of D$_{4h}$ group. The $\\Gamma_6$ and $\\Gamma_7$ states can only have quadrupole moments of $x^2-z^2$ or $y^2-z^2$ type, rather than the proposed $xy$, $yz$, and $zx$ type. Hence, only when the site symmetry is reduced to D$_{2h}$ group, and the $\\Gamma_8$ state of O$_{h}$ group is split into two $\\Gamma_5$ states of D$_{2h}$ group, can the CF ground state carries finite quadrupole moments of $xy$, $yz$, and $zx$ type. However, in a continuous second-order phase transition, the symmetry of the system cannot be directly reduced from cubic to orthorhombic, which violates Landau theory~\\cite{Landau1980}. Theories which claim an AFQ phase with O$_{xy}$-type moments using a localized picture should address this difficulty. Inconsistency of the AFQ description has also been suggested based on magnetic-susceptibility anisotropy and magnetostriction measurements~\\cite{Amara2012}.\n\n\\section{Conclusion\\label{sec:Con}}\n\nIn summary, we have employed optical secondary-emission spectroscopy to study the spin-orbital coupling (SOC), electronic crystal-field (CF) excitations, electron-phonon interaction and long-wavelength magnetic fluctuations in the heavy-fermion metal CeB$_{6}$. \n\nCe$^{3+}$ ions have a single electron in the $4f$-shell. The SOC splits the degenerate 4f levels into a lower-energy $^2F_{5\/2}$ multiplet and a higher-energy $^2F_{7\/2}$ multiplet, with a separation of around 2000\\,cm$^{-1}$, from which we estimate the SOC strength $\\xi$=610\\,cm$^{-1}$.\n\nThe two multiplets are further split into five Kramers-degenerate CF states by the cubic CF potential. The $^2F_{5\/2}$ multiplet is composed of one quartet $\\Gamma_8$ ground state and one doublet $\\Gamma_7$ excited state, and the $^2F_{7\/2}$ multiplet consists of $\\Gamma_6^*$ and $\\Gamma_7^*$ doublets, and a $\\Gamma_8^*$ quartet states. We resolve all four electronic CF transitions: 380\\,cm$^{-1}$ for the intra-multiplet excitation, and 2060, 2200 and 2720\\,cm$^{-1}$ for the three inter-multiplet transitions. \n\nOn cooling, the FWHM for the $\\Gamma_8\\rightarrow\\Gamma_7$ and $\\Gamma_8\\rightarrow\\Gamma_7^*$ transitions first decreases from 300\\,K to 80\\,K, but then increases below 80\\,K. We relate the decrease of the FWHM to lattice vibration driven fluctuations of the electrostatic potential at Ce sites, which diminish on cooling. The increase of the FWHM below 80\\,K results from the Kondo effect, an electron-correlation effect which increases the self-energy of the excited CF states. \nWe apply a single-ion Hamiltonian model to obtain the eigenvalues and eigenfunctions of the 4f-electron CF states. Using the Fermi Golden Rule, we also calculate the intensity of the four Raman active CF transitions and compare the calculation to the experimental data. \n\nWe study the lattice dynamics of CeB$_{6}$ and analyze the temperature dependence of all Raman active phonon modes. In the phonon spectra, we interpret the asymmetric lineshape of E$_{g}$ and T$_{2g}$ optical phonons as manifestation of electron-phonon interaction. We also identify a composite CF plus phonon excitation at 1158\\,cm$^{-1}$. \n\nWe acquire temperature dependence of the low-energy Raman response for all Raman-allowed symmetry channels, and uncover the development of a quasi-elastic Raman response in the magnetic-dipolar T$_{1g}$ symmetry channel below 20\\,K. The corresponding static Raman susceptibility shows similar temperature dependence as the magnetic susceptibility data, which supports the interpretation of its magnetic origin. By comparing the quasi-elastic Raman scattering data with electron spin resonance and inelastic neutron scattering results, we relate this T$_{1g}$ spectral feature to ferromagnetic correlations.\n\nAdditionally, we detect photo-luminescence emission centered at 1.95\\,eV at room temperature. We relate this emission to recombination of the electron-hole excitations between the 5d- and 4f-bands.\n\nThe experimental methods, models, and analyses demonstrated in this study can be applied to a range of systems, especially for rare-earth materials containing localized f-electrons of Ce$^{3+}$ or Yb$^{3+}$ ions at high-symmetry crystallographic sites~\\cite{Ye2019}. The approach could enable us to probe ferroquadrupolar (FQ) fluctuations in TmAg$_{2}$ (T$_{FQ}$\\,=\\,5.0\\,K)~\\cite{Tm1993} or TmAu$_{2}$ (T$_{FQ}$\\,=\\,7.0\\,K)~\\cite{Tm1998} systems, to name a few examples. Also, magnetic correlation induced by quadrupolar ordering could be probed in antiferroquadrupolar (AFQ) systems, for instance in UPd$_{3}$ (multiple AFQ phases, with the highest T$_{AFQ}$\\,=\\,7.6\\,K)~\\cite{U1999}, NpO$_{2}$ (T$_{AFQ}$\\,=\\,25.0\\,K)~\\cite{Np2002}, or DyB$_{2}$C$_{2}$ (T$_{AFQ}$\\,=\\,24.7\\,K)~\\cite{Dy2000}.\n\n\\begin{acknowledgments}\nWe are grateful to K. Haule and P. Coleman for discussions. We thank \nA. Lee for participating in the early data acquisition. \nThe spectroscopic work at Rutgers (M.Y., H.-H.K, G.B.) was supported \nby NSF Grant No. DMR-1709161. \nSample synthesis at Los Alamos was performed under the auspices of \nthe U.S. Department of Energy, Office of Basic Energy Sciences, \nDivision of Materials Science and Engineering. \nG.B. also acknowledges \nthe QuantEmX grant from ICAM, the Gordon and Betty Moore Foundation \nthrough Grant No. GBMF5305 allowing G.B. to make a collaborative \nvisit to Stanford. \nWork at NICBP was supported by IUT23-3 grant. \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\nAdvances in fabrication of materials at the nanoscale have made quantum computers a realistic possibility. The quest to create quantum computers has been likened to the Space Race in the 20th-century, since encryption of government data is based on a problem that quantum computers can solve with much more ease than classical computers \\cite{2018NatureRev}.\n\nThe qubit forms the basis of a quantum computer and experimentally realizing a system of multiple qubits is necessary for quantum computation. \nNumerous qubit systems have been proposed in the last two decades \\cite{Bogdanov2011}. Superconducting qubits based on Josephson junction circuits have been heavily studied and are considered one of the major tracks towards quantum computation \\cite{SuperQbit2020}.\n Another auspicious path is based on topological qubits \\cite{Kitaev}. One of the main problems facing superconducting qubits is the scaling problem, as they are sensitive to noise which will cause accumulation of error with increased number of qubits \\cite{SuperQbit2019}. Quantum computers of this type have therefore been called \"Noisy intermediate scale quantum computers\" \\cite{SuperQbit2020}. Topological qubits have the potential to solve this issue, \n with topological quantum computers being fault-tolerant \\cite{Kitaev}.\n \nA promising candidate for a topological qubit are Majorana zero modes\n(MZM), also known as Majorana bound states \\cite{MajoranaQubit}. They\nhave been predicted to have non-Abelian exchange statistics, meaning\nthat the order of their spatial exchange matters, making them suitable\nto encode fault-tolerant quantum computation, by braiding their\nworld lines \\cite{AnyonComp, QmTopComp}. \n\nMajorana zero modes get their name from\nthe Italian physicist Ettore Majorana. In the year 1937 he published a\npaper with a solution of the Dirac equation describing neutral particles\nwhich are their own antiparticles,\n\n\\begin{equation}\n\\gamma=\\gamma^{\\dagger},\n\\label{Majorana}\n\\end{equation}\n thus annihilating each other on contact \\cite{EttoreMajorana}. Particles with this property have since been called Majorana fermions \\cite{NeutrinoMajorana}.\n\n\nIn superconductors, mixed states of electrons and holes can emerge as quasiparticle excitations with similar behavior \\cite{TudorBok}. MZM are mid-gap excitations at zero energy, that mimic the Majorana fermion and are localized near defects or boundaries in topological superconductors \\cite{Alicea_2012,Topological}.\nTopological superconductors can be engineered by combining standard superconductors with semiconductors \\cite{FuKane}. Currently one of the most promising platforms for realizing MZM are hybrid superconductor-semiconductor devices \\cite{schapersGroup2019, Hybrid, Stanescu_2013}. When a metal or semiconductor is paired with a superconductor, the superconductivity can penetrate into the metal, making it superconducting up to a certain depth. This process of proximity-induced superconductivity is one of three necessary ingredients, along with Zeeman splitting and strong spin-orbit coupling, for engineering topological superconductors \\cite{TudorBok}.\nThe most common model of a system with MZM is a semiconductor nanowire in \n\tcontact with a metal superconductor \\cite{Lutchyn2010,Oreg2010}, which is a quasi-one-dimensional system. From the superconducting property, the Hamiltonian of a proximitized nanowire obtains an implicit particle-hole symmetry. With the application of an external magnetic field, time-reversal symmetry is broken and the system can support a $\\mathbb{Z}_2$ topological invariant corresponding to the topological class $D$ \\cite{Stanescu2011,TopoClass}.\n\t The parameter space of such systems has been thoroughly explored with detailed calculations in order to predict the key conditions \n\tfor experimental detection \\cite{Stanescu2011,Stanescu2013}.\n\nTubular nanowires of core-shell type provide an experimental platform to include all required aspects for hosting MZM \\cite{Hybrid}. Prismatic core-shell nanowires from various semiconductors have been fabricated and continue to be an active field of research \\cite{TriHex,2015TriHex,2019ProxCoreShell,Haas_2018, ZhangAruni2017,ZhangAruni2019, Sonner2019}.\nRecent numerical simulations with contributions from two of the present authors have\nindicated that several MZM's can be hosted in prismatic core-shell nanowires, where the \nelectrons with low energies tend to localize around the prism edges \\cite{Andrei,Stanescu2018}. \nThis experimentally available system can be a host for previously discussed\nmultichain ladder models \\cite{Poyhonen2014,Wakatsuki2014,Sedlmayr2016}.\n\nIn this paper we present computational results for nanowires with core geometry different from the shell geometry. We take the cases of a triangular wire with a hexagonal core (Tri-Hex), inspired by recently fabricated structures \\cite{TriHex}, and the inverse system, a hexagonal wire with a triangular core (Hex-Tri) which has also been obtained \\cite{HexTri_Dick2010}. We show that the separation of energy levels of the Tri-Hex structure is significantly larger than in the case of a triangular core. With the Hex-Tri structure, the number of phase boundaries in the topological phase diagram is reduced, since the triangular core results in the ground states being localized at the three sides, rather than the six corners as is the case for a hexagonal core. The localization of MZMs is shown to correspond to the single particle localization of both configurations. The results suggest that these structures are worthy of further experimental investigation. To finalize, we discuss the possibility braiding MZM at the nanowire end planes.\n\\section{QUANTUM MECHANICAL MODEL AND METHODS}\nIn the numerical calculations we use cylindrical coordinates, with\nthe $z$ axis along the nanowire. \nThe geometry of the wire cross-section is defined by applying appropriate boundaries to \na discretized disc in polar coordinates $(\\phi,r)$ \\cite{Daday2011}.\nThe Hamiltonian of the cross-section describes the transverse modes of the nanowire\n\\begin{equation}\nH_t= \\frac{(p_{\\phi}+eA_{\\phi})^2}{2m_e} \n-\\frac{\\hbar^2}{2m_er} \\frac{\\partial}{\\partial r} \n\\left(r\\frac{\\partial}{\\partial r}\\right)\\ ,\n\\end{equation}\n where $p_{\\phi}=-i\\frac{\\hbar}{r}\\frac{\\partial}{\\partial\\phi}$ \nis the momentum in the $\\hat\\phi$ direction,\nand $A_{\\phi}=\\frac{1}{2}Br$ the vector potential\nassociated with a magnetic field of strength $B$ oriented along the nanowire, in the symmetric gauge. The\neigenstates of $H_t$ can be written in terms of the lattice sites\n\\begin{equation}\n|a \\rangle = \\sum_\\kappa c_a |r_\\kappa \\phi_\\kappa\\rangle.\n\\end{equation}\nThe nanowire length is incorporated with longitudinal modes which are given by\n\\begin{equation}\nH_l=\\frac{p_z^2}{2m_e},\n\\end{equation}\nwith the corresponding eigenstates\n\\begin{equation}\n|k \\rangle =L^{-1\/2}\\exp(ikz) \\ ,\n\\label{infbase}\n\\end{equation}\nfor an infinite nanowire, $L\\to\\infty$. In the case of a finite wire, with $z=0$ set in the middle of the nanowire,\n\\begin{equation}\n|n \\rangle =L^{-1\/2}\\sqrt{2} \\sin\\left(n\\pi \\left(\\frac{z}{L_z}+\\frac{1}{2}\\right)\\right) \\ .\n\\end{equation}\n\nThe Zeeman effect due to the applied external magnetic field is given in terms of the effective Land\\'{e} g-factor, Bohr magneton $\\mu_B$, spin $|\\sigma \\rangle $ and magnetic field strength $B$,\n\\begin{equation}\nH_Z = -g^* \\mu_B \\sigma B \\ .\n\\end{equation}\n\nIn order to satisfy the Majorana property, Eq.\\ \\eqref{Majorana}, the system needs to be effectively \ndegenerated. \nAlong with the Zeeman splitting, materials such as InAs or InSb are normally used in experiments due to the possibility of obtaining strong Rashba spin-obit coupling which lifts the spin degeneracy by coupling the spin to the momentum. \n\nFor a thin cylindrical shell the spin-orbit interaction can be modeled with the Hamiltonian\n\\begin{equation}\nH_{SOI}= \\frac{\\alpha}{\\hbar}( \\sigma_{\\phi} p_z -\\sigma_z p_{\\phi}) \\ ,\n\\label{HSOI}\n\\end{equation}\nwhich corresponds to the regular planar model transformed in cylindrical coordinates\n\\cite{Bringer2011}, where $\\alpha$ is the Rashba coupling constant. \nFor a prismatic shell, a more elaborated model is in principle necessary, \nthat should take into account the details of the nonuniform interface between the core and \nthe shell. Recent calculations based on the \n$k\\cdot p$ method indicate that the spin-orbit coupling inside the core can significantly \nincrease due to the effective interface field, in the hexagonal geometry \\cite{Wojcik2019}.\nHowever, to our knowledge, no such study has been performed for the electrons situated \nin the shell. Since we shall deal mostly with electrons localized within narrow\nangular regions of the shell, on the corners or on the sides, we will assume they are \nexperiencing a local effective electric field in the radial direction only, and we will \nuse Eq.\\ \\eqref{HSOI} as if that field does not depend on the angle $\\phi$. This model\nhas been already used for describing Majorana states in core-shell nanowires with \na simpler geometry \\cite{Andrei,Stanescu2018}, and should be reasonable for a qualitative\napproach.\n\nOur model of the core-shell nanowire is based on the composite Hamiltonian\n\\begin{equation}\nH_w= H_t + H_l + H_Z + H_{SOI} \\ .\n\\label{wireHamiltonian}\n\\end{equation}\nThe eigenstates and energy values for a finite wire are obtained by diagonalizing the matrix\n\\begin{equation}\n\\langle an\\sigma|H_w|a'n'\\sigma'\\rangle \\ .\n\\end{equation}\n\nThe final ingredient needed to model MZM in the system is superconductivity. Due to the proximity effect, superconductivity is induced in the nanowire shell. The first hybrid superconductor-semiconductor nanowire systems consisted of nanowires lying on superconducting substrates \\cite{DelftEvidence2012}. The proximity effect was assumed to make the whole wire superconducting given that the thickness of the nanowire is much smaller than the coherence length, $\\xi$. More recently, nanowires covered with a superconductor have been fabricated with in-situ methods \\cite{FluxInducedMajorana2018}. In such systems, the proximity effect can more rightly be assumed to be homogeneous in the system. However, a complete superconductor shell will invoke the Little-Parks effect which has to be taken into account in the transport through the system \\cite{evenodd}.\n\n The current analysis essentially describes a core-shell nanowire with an insulating or hollow core and a fully proximitized semiconductor shell, which is justified in the light of experimental results of Ref.\\ \\cite{ABandreevXi}. Nonetheless, the authors are aware that the proximity effect deserves a more thorough treatment \\cite{TudorSarma} as the temperature and sample geometry can both significantly influence the coherence length \\cite{Stenuit_2004}.\n The superconducting property is incorporated by an order parameter $\\Delta$ which couples two Schr\\\"{o}dinger equations for electrons and holes with opposite spin in the Bogoliubv-DeGennes (BdG) Hamiltonian \\cite{Bogoliubov:1958km}. This has been extensively used to describe quasiparticle excitations in superconductors \\cite{Jianxin},\n\\begin{equation}\nH_{BdG}= \n\\begin{pmatrix}\nH_w&\\Delta\\\\\n-\\Delta^*&-H^*_{w}\n\\end{pmatrix} \\ .\n\\end{equation}\n\nThe eigenstates of the BdG Hamiltonian have both electron and hole components and are written in the basis \n\\begin{equation}\n|q\\rangle =|\\eta a n \\sigma \\rangle =|\\eta g \\rangle \\ ,\n\\end{equation}\nwhere $|g\\rangle$ denotes the basis for the finite wire Hamiltonian in Eq.\\ \\eqref{wireHamiltonian} with the added electron-hole degree of freedom described with the isospin quantum number $\\eta$. In the case of a finite wire, the longitudinal eigenstates are as in Eq.\\ \\eqref{infbase}. The matrix elements are obtained for $\\eta=\\eta'$ with\n\\begin{align}\n\\begin{split}\n\\langle g \\eta |H_{BdG}|g'\\eta' \\rangle&=\n\t\\eta[\\text{Re}\\langle g|H_w|g'\\rangle \\\\&+ i\\eta \\langle g|H_w|g'\\rangle - \\mu \\delta_{gg'}],\n\\end{split}\n\\label{Diag}\n\\end{align}\nand for $\\eta \\neq \\eta'$,\n\\begin{equation}\n\\langle g \\eta |H_{BdG}|g'\\eta' \\rangle= \\eta \\sigma \\delta_{\\sigma,-\\sigma'} \\delta_{aa'} \\delta_{nn'} \\Delta \\ .\n\\end{equation}\nNote that the excitation energies are evaluated relative\nto the chemical potential $\\mu$, Eq.\\ \\eqref{Diag} which gives the BdG Hamiltonian an implicit particle-hole symmetry.\n\n By calculating the spectra of both the finite and infinite cases for increasing magnetic field strength, a closing and reopening of the quasiparticle energy gap is observed with the emergence of topological edge states in the finite length spectrum \\cite{Sato_2017,HallTopo1985}.\n\n It has been shown that a pair of MZM will emerge for each corner in a prismatic core-shell nanowire but due to wavefunction overlap or tunneling caused by the finite width of the sides, some pairs will be shifted symmetrically above and below zero energy. \nIn the case of three edges, one pair will be at precisely zero energy. The other two hybrid MZM's, slightly above and below zero energy, have been termed pseudo MZM's and will be referred to as such \\cite{Andrei}. \n\n\n Throughout this work, our chosen parameters correspond to InSb with $\\gamma=\\frac{1}{2} g^* m_e =0.393$, SOI parameter $\\alpha=1 \\text{ meV nm}$ and a superconducting gap parameter $\\Delta= 0.50 \\text{ meV}$.\nThe number of sites used to describe the nanowire cross section were between 1700-2400.\n\n\n\\section{SINGLE-PARTICLE ENERGIES}\n\nTriangular nanowires with hexagonal cores have been fabricated with both side-matched (SM) cores \\cite{TriHex}, Figs.\\ \\ref{TriLoc}(c,d) and corner-matched (CM) cores \\cite{2015TriHex}, Figs.\\ \\ref{TriLoc}(e,f). \nTo compare with earlier analysis \\cite{Anna2015} we explore the single-particle localization and the energy level separation of these nanowires, in conjunction with triangular cores, Figs.\\ \\ref{TriLoc}(a,b). \nThe hexagonal core geometry enlarges the area of the corner localized peaks, whilst the side localization is suppressed, compared to the case of the triangular core. The side states of the triangular core are split by the hexagonal core geometry, resulting in two peaks per corner. \n\n\\begin{figure\n\t\\centering\n\t\\includegraphics[width=0.44\\textwidth]{lLocalization_02Shell_3-eps-converted-to.pdf}\\\\\n\t\\includegraphics[width=0.44\\textwidth]{lLocalization_02Shell_4-eps-converted-to.pdf}\n\t\\caption{Single-particle cross-sectional localization of triangular nanowires with a triangular core (a,b), side-matched hexagonal core (c,d) and corner-matched hexagonal core (e,f). The upper row shows the corner localization of the first three quasi-degenerate energy states. The lower row shows the states of the adjacent energy level. The minimal shell thickness is 10 nm for the first two cases but slightly less in the corner matched case since the core is rotated with respect to the central case.}\n\t\\label{TriLoc}\n\\end{figure}\n\nThe hexagonal core geometry further results in a larger separation between of the lowest (corner) states, Fig.\\ \\ref{TriHexEnergyt}, which is favourable for Majorana physics \\cite{Andrei}, as it provides more robust subspace of corner states. The energy separation decreases with increasing shell thickness - as the hexagonal core gets smaller, the wavefunction overlap at the sides becomes larger.\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.42\\textwidth]{Comp_Energies_12states_D02-eps-converted-to.pdf} \n\t\\caption{Single-particle energy states for the three triangular nanowire configurations. The three lowest-energy states are nearly degenerate and localized in the corners of the shell.}\n\\label{TriHexEnergyt}\n\\end{figure}\n\\FloatBarrier\n\nIn the case of a hexagonal wire, Fig.\\ \\ref{HexTriLoc}, the triangular core outline three sides with an enlarged area, compared to the hexagonal core. To the best of our knowledge, the corner-matched configuration (e,f) has not yet been fabricated, but is included in the analysis for the sake of completeness. The separation of energy levels is larger for the case of a hexagonal wire with a triangular core compared to a hexagonal core, Fig.\\ \\ref{HexTriEnergyt} but not as large as in the case of triangular wires. This is due to the overlap between localization peaks, which is clearly visible in Fig.\\ \\ref{HexTriLoc}(f).\n We say that the three states are quasi-degenerate as the degeneracy pattern is 1-2, 2-1 which becomes more evident for higher states in both Fig.\\ \\ref{TriHexEnergyt} and Fig.\\ \\ref{HexTriEnergyt}.\n\\FloatBarrier\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{lHex_Localization_02Shell_3-eps-converted-to.pdf}\\\\[0.2cm]\n\t\\includegraphics[width=0.45\\textwidth]{lHex_Localization_02Shell_5-eps-converted-to.pdf}\n\t\\caption{Single-particle cross-sectional localization of hexagonal nanowires with a hexagonal core (a,b), side-matched triangular core (c,d) and corner-matched triangular core (e,f). The upper row shows the first three quasi-degenerate energy states, which also describes the localization of the second energy level. The lower row shows the adjacent higher states. The minimal shell thickness is 10 nm for the first two cases but slightly more in the corner matched case since the core is only rotated with respect to the central case.}\n\\label{HexTriLoc}\n\\end{figure}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{Hex_Comp_Energies_12states_D02-eps-converted-to.pdf} \n\t\\caption{Single-particle energy states for the three hexagonal nanowire configurations.}\n\t\\label{HexTriEnergyt}\n\\end{figure}\n\\FloatBarrier\n\\section{MAJORANA ZERO MODES AND TOPOLOGICAL PHASE DIAGRAMS}\nFirst we show a set of results for a nanowire with the \n\tcross section illustrated in Fig.\\ \\ref{BdGspectra}(a). The radius of the \n\tnanowire i.e.\\ the distance between the center and the external corners, is $R=50$ nm,\n\tand the side thickness is 10 nm. In Panel (b) we show the energy dispersion in\n\tsuch a nanowire with infinite length, vs.\\ the longitudinal wavevector times the nanowire\n\tradius, showing the Zeeman splitting due to the external magnetic field applied parallel to the nanowire. Next, in \n\tFigs.\\ \\ref{BdGspectra}(c,d), we show the energy spectra for the BdG Hamiltonian,\n\tfor the the nanowire of infinite and finite length respectively. In the later case the \n\tlength was 200 nm. Three values of the longitudinal magnetic field are shown, to indicate\n\tthe onset of the MZM. Whereas in Panel (c) we can only observe the vanishing \n\tgap (at $k=0$), in Panel (d) we have the complementary information on the \n\tsix states created around zero energy, one for each corner of the nanowire section,\n\ttimes two due to the electron-hole symmetry.\n\n \\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{2x2plotRev-eps-converted-to.pdf}\n\t\\caption{(a) Discretized nanowire cross section with Tri-Hex geometry, radius of 50 nm and 10 nm minimal shell thickness. \\newline (b) Corresponding energy dispersion for an infinite wire in the presence of an external magnetic field of magnitude $0.55$ T. \\newline (c) BdG quasiparticle spectra for an infinite wire showing the closing of the energy gap as the longitudinal external magnetic field strength is increased. (d) Corresponding energy spectra of a finite wire of length 200 nm, showing the emergence of Majorana Zero Modes. The line colors in figure (c) correspond to the magnetic field values shown in figure (d).}\n\t\\label{BdGspectra}\n\\end{figure}\n\nBy calculating the BdG spectra for a range of magnetic field strengths and values of the chemical potential, we can find for which set of parameters the BdG spectra closes. The closing of the BdG energy gap signifies a phase transition of the system from a topologically trivial state to a non-trivial one, with the emergence of a pair of edge states. The system can not be adiabatically deformed back to the original trivial state, just as a sphere can not be continuously deformed into a doughnut, therefore the states are considered topologically distinct \\cite{TopoPhase_Shou-Cheng,AdiaTopoPhase2013}. \n By plotting the ratio of the BdG energy and the superconducting gap parameter $\\Delta$ at $k=0$, as a function of the magnetic field strength and the chemical potential, a phase diagram is obtained \\cite{Stanescu2011,AkhmerovTopoPhase}. Since there are three states at the lowest energy, for both the Tri-Hex and Hex-Tri case, the system can host three pairs of edge states and there will be a closing of the BdG spectra for the emergence of each one. We will therefore have three curves on our phase diagrams which are boundaries between phases with different number of pairs of edge states \\cite{TopoPhase2017,Exp_phase_diag_Chene1701476}.\nTopological phase diagrams of prismatic core-shell nanowires with uniform geometry have been recently studied \\cite{Stanescu2018}. Here we explore the effect of the core geometry on the phase diagrams. Since a realistic nanowire is not strictly symmetric, from now on we choose to slightly break the cross-sectional symmetry of our nanowires with a weak transverse electric field. \n\n\n\\subsection{Tri-Hex nanowire geometry}\n\nFor a minimal shell thickness of 10 nm, the phase diagram of the Tri-Hex systems consists of three lateral parabolas, which is the same phase diagram as for three isolated one dimensional nanowires \\cite{Stanescu2018}. In order to observe signs of wavefunction overlap between corners, the minimal shell thickness is increased to 12.5 nm, correspondingly the chemical potential is shifted to lower values compared to Fig.\\ \\ref{BdGspectra}.\nContrary to the phase diagram of three isolated wires, the tunneling removes the crossing of the phase boundaries, Fig.\\ \\ref{TopoTriHex125nm}.\n The same phase diagram is observed for all three configurations of core geometry so only the side-matched case is presented.\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.42\\textwidth]{Pd_TriHex_SiM_paperRev}\n\t\\caption{Phase diagram for a side-matched Tri-Hex structure with 12.5 nm shell thickness in a weak transverse electric field. The colourscale represents the ratio of the energy to the superconducting gap parameter $\\Delta$ at $k=0$ on a $\\log_{10}$ scale.}\n\t\\label{TopoTriHex125nm}\n\\end{figure}\n\\FloatBarrier \nWe conclude that the effect of the hexagonal core is minimal for shell thickness up to 12.5 nm. Larger values will decrease the separation of energy levels, which is undesirable for hosting MZM \\cite{Andrei} and are thus not further elaborated on here. \n\\subsection{Hex-Tri nanowire geometry}\n\\FloatBarrier\n\nFor a hexagonal wire with a triangular core, we can expect to see more signs of wavefunction overlap between the sides of the wire, as the separation of energy levels is much smaller than in the case of the triangular wire, Fig.\\ \\ref{HexTriEnergyt}. As the differences between the side-matched and corner-matched cases is small, the phase diagram for the side-matched structure is presented only.\nIn the topological phase diagram the presence of a threefold phase boundary signifies that the particles form three channels induced by the triangular core geometry, Fig.\\ \\ref{Pd_HexTri_SiM_B01p5}. The topological phase is entered at a lower value of both the chemical potential and magnetic field strength, compared to the Tri-Hex case, Fig.\\ \\ref{TopoTriHex125nm}.\nAs shown in Ref.\\ \\cite{Andrei}, orbital Zeeman effects both skew the phase boundaries and lower the magnetic field strength threshold for the topological phase. Furthermore, combined with the spin Zeeman effect, they are responsible for the curious elliptic island, Fig.\\ \\ref{Pd_HexTri_SiM_B01p5}, around $\\mu=61.5 \\text{ meV}$ and $B=0.6 \\text{ T}$. \n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{Pd_HexTri_SiM_paper2Rev}\n\t\\caption{Phase diagram for a side-matched Hex-Tri structure with 10 nm shell thickness. Three topological phases, corresponding to the three sides of the triangular core are distinguished.\n}\n\t\\label{Pd_HexTri_SiM_B01p5}\n\\end{figure}\nIn Fig.\\ \\ref{ZeemanOsc}, the lowest energy transverse states are shown to oscillate with increasing strength of the external magnetic field due to the orbital Zeeman effect. The oscillations are more pronounced for the Hex-Tri geometry, which is echoed in the corresponding phase diagram. The phase boundaries are three in number but a hexagonal core results in six phase boundaries \\cite{Andrei}. The effect of the triangular core geometry is therefore quite significant in this case, mainly in that the number of phase boundaries is halved, compared to a hexagonal core.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{ZeemanOsc_spinsplit-eps-converted-to.pdf}\n\t\\caption{Oscillations of the lowest energy transverse\n\t\tstates with increasing the magnetic field, compared for a side-matched Tri-Hex and Hex-Tri wire geometries.}\n\t\\label{ZeemanOsc}\n\\end{figure}\n\\FloatBarrier\nThe possibility of gap closing at non-zero values of the wave vector in the continuous spectrum was also considered. In all of the studied cases, the same phase diagram was obtained, meaning that these topological phases are stable \\cite{Andrei}.\n\\FloatBarrier\n\\section{TRANSVERSE AND LONGITUDINAL MAJORANA ZERO MODE LOCALIZATION}\n To explore the correspondence between the single-particle localization and MZM localization, the BdG probability density of the nanowire cross section at the end of the wire is calculated. \nWe find that for both the Tri-Hex and Hex-Tri structures, the localization of MZM's, Fig.\\ \\ref{TriHex_Maj}, coincides with the single-particle localization, Figs.\\ \\ref{TriLoc} and \\ref{HexTriLoc}, in that we have states localized at the largest area. The ideal MZM Fig.\\ \\ref{TriHex_Maj}(c,d) at $E=0$ differ from the pseudo MZM (a,b,e,f) by different weights of the lowest (corner) states, controllable with the electric field. The degeneracy pattern 1-2, 2-1 is reflected in the localization.\n\n Only the side-matched cases are presented, as the difference to the corner-matched cases is negligible.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.425\\textwidth]{Maj_Loc_TH-eps-converted-to.pdf}\\\\[0.25cm]\n\t\t\\includegraphics[width=0.425\\textwidth]{Maj_Loc_HT-eps-converted-to.pdf}\n\t\\caption{ Localization of Majorana zero modes at the nanowire ends for side matched Tri-Hex and Hex-Tri structures. The arrow shows the directionality of the applied transverse electric field for all instances, that breaks the localization symmetry.}\n\t\\label{TriHex_Maj}\n\\end{figure}\n\nTo confirm that the MZM are localized at the ends of the nanowire, the BdG probability density of the corresponding state is calculated as a function of the nanowire length, for a given point on the nanowire cross section.\nThe longitudinal localization for the corresponding top corner of the side-matched Tri-Hex, Fig.\\ \\ref{TriHex_Maj} can be seen in Fig.\\ \\ref{Longiloc}. As expected, we observe strong edge localization which is characteristic of the MZM and topological edge states in general \\cite{FuKane, Liu_2017}.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.43\\textwidth]{Longiloc-eps-converted-to.pdf}\n\t\\caption{Single site lengthwise localization of an MZM. The probability density is greatest at the nanowire ends and falls off exponentially towards the nanowire center.}\n\t\\label{Longiloc}\n\\end{figure}\n\\FloatBarrier\n\n\\section{FUTURE OUTLOOK}\nIn the light of the zero bias conductance peaks not being conclusive evidence \\cite{Zhang2019}, there is now a general consensus that braiding or fusing MZM is required as conclusive evidence to confirm experimentally their realization \\cite{Zhang2019, 2018NatureRev}. Solely demonstrating topological phases as we have done here, is only the necessary foundation. Various schemes for braiding have been proposed \\cite{BraidingBeenakker}. In particular, braiding protocols for Y-junctions have been heavily studied \\cite{BraidYjuncError,BraidError2}. In such a system, one of the emerging problems is the misalignment of the longitudinal magnetic field, as the Y-junction does not lie on a single axis \\cite{BraidingYjunctionTuningFork}. Braiding MZM's in the plane of the nanowire end will circumvent this problem. However, this can not be done in a single core nanowire system as the MZM's would always meet up along the shell and annihilate. \n\nBy fabricating a core-shell nanowire with two insulating cores or effectively so via doping, Fig.\\ \\ref{2core}, this problem can be overcome.\nThe next problem is then how the MZM's can be moved around in the nanowire end plane. Even though the MZM can be manipulated with an external electric field, that may not provide sufficient control to perform a braiding operation.\nLastly, a readout is needed, based on a coupling of the fermion parity to an observable \\cite{BraidingBeenakker} in order to confirm that $\\sigma_{AB}\\sigma_{BC}\\neq\\sigma_{BC}\\sigma_{AB}$. \n\\begin{figure}[h!]\n\t\\centering\n\\includegraphics[width=0.28\\textwidth]{TriHexG2s-eps-converted-to.pdf}\n\\caption{End plane of a hypothetical dual core-shell nanowire. The genus 2 surface provides sufficient motional degrees of freedom for the braiding operations $\\sigma_{AB},\\sigma_{BC}$.}\n\\label{2core}\n\\end{figure}\n\\FloatBarrier\nFormulation of an exact braiding scheme is beyond the scope of this article. Here, we only intend to stimulate the discussion about braiding in the nanowire end planes, as the nanowire edges model three strands, needed to demonstrate the simplest non-commutative braiding operation.\n\n\n\n\\section{SUMMARY AND CONCLUSIONS}\nBy combining hexagonal and triangular geometry in nanowires of core-shell type, both energy levels and topological phases can be influenced. With the polygonal geometry, a well separated group of corner states is obtained. Here we show that\nthe separation of corner-localized energy levels is much larger for triangular nanowires with hexagonal cores, compared to those with triangular cores of the same minimal shell thickness, whilst the effect on the topological phase is minimal. On the contrary, for the complimentary configuration of a hexagonal wire with a triangular core, the effect on energy level separation is minimal whilst the effect on the topological phase boundaries is significant. We find for both configurations that the localization of the Majorana zero modes coincides with the single-particle localization. The problem of braiding Majorana zero modes in a nanowire end plane is addressed and a split\/dual core structure is proposed as a system with the necessary motional degrees of freedom for a three strand braiding operation. \n\\begin{acknowledgments}\nThis research is supported by the Reykjavik University Research Fund, project no.\\ 218043. We are grateful to Kristof Moors and Pujitha Perla for fruitful discussion about split\/dual core-shell nanowires and braiding. \n\\end{acknowledgments}\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzeoeg b/data_all_eng_slimpj/shuffled/split2/finalzzeoeg new file mode 100644 index 0000000000000000000000000000000000000000..f0589af91fc1214b49da558af0ab7bac03c391c1 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzeoeg @@ -0,0 +1,5 @@ +{"text":"\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nNeural networks have outperformed earlier \\gls{GMM} based acoustic models in terms of modeling power and increased robustness to acoustic distortions.\nDespite that, speech enhancement has been shown to deliver additional \\gls{WER} improvements, if multi-channel data is available.\nThis is due to their ability to exploit spatial information, which is reflected by phase differences of microphone channels in the \\gls{STFT} domain. This information is not accessible by the \\gls{ASR} system, at least not if it operates on the common log mel spectral or cepstral feature sets.\nAlso, dereverberation algorithms have been shown to consistently improve ASR results, since the temporal dispersion of the signal caused by reverberation is difficult to capture by an ASR acoustic model~\\cite{Delcroix2014LINEARPD}.\n\nHowever, there has been a long debate whether it is advisable to apply speech enhancement on data used for ASR training, because it is generally agreed upon that the recognizer should be exposed to as much acoustic variability as possible during training, as long as this variability matches the test scenario~\\cite{Bippus_is99, Baker_asru2015, Vincent_csl2016}.\nMulti-channel speech enhancement, such as acoustic \\gls{BF} or source separation, would not only reduce the acoustic variability, it would also result in a reduction of the amount of training data by a factor of $M$, where $M$ is the number of microphones~\\cite{Menne_chime4_2016}. Previous studies have shown the benefit of training an ASR on matching enhanced speech~\\cite{Deng_isclp2000, Delcroix_is2013} or on jointly training the enhancement and the acoustic model~\\cite{Li_is2016}.\nAlternatively, the training data is often artificially increased by adding even more degraded speech to it.\nFor instance, Ko et al.~\\cite{Ko_icassp17} found that adding simulated reverberated speech improves accuracy significantly on several large vocabulary tasks.\nSimilarly, Manohar et al.~\\cite{Manohar_icassp19} \nimproved the \\gls{WER} of the baseline CHiME-5 system by relative \\SI{5.5}{\\%} by augmenting the training data with approx. \\SI{160}{hrs} of simulated reverberated speech. \nHowever, not only can the generation of new training data be costly and time consuming, the training process itself is also prolonged if the amount of data is increased.\n\nIn this contribution we advocate for the opposite approach. Although we still believe in the argument that ASR training should see sufficient variability, instead of adding degraded speech to the training data, we clean up the training data. We make, however, sure that the remaining acoustic variability is at least as large as on the test data. By applying a beamformer to the multi-channel input, we even reduce the amount of training data significantly. Consequently, this leads to cheaper and faster acoustic model training.\n\nWe perform experiments using data from the CHiME-5 challenge which focuses on distant multi-microphone conversational ASR in real home environments~\\cite{Barker2018CHiME5}.\nThe CHiME-5 data is heavily degraded by reverberation and overlapped speech. As much as \\SI{23}{\\%} of the time more than one speaker is active at the same time~\\cite{Zorila_icassp19}.\nThe challenge's baseline system poor performance (about~\\SI{80}{\\%}~\\gls{WER}) is an indication that ASR training did not work well.\nRecently, \\Gls{GSS} enhancement on the test data was shown to significantly improve the performance of an acoustic model, which had been trained with a large amount of unprocessed and simulated noisy data~\\cite{Kanda2019}.\n\\Gls{GSS} is a spatial mixture model based blind source separation approach which exploits the annotation given in the CHiME-5 database for initialization and, in this way, avoids the frequency permutation problem~\\cite{Boeddecker2018chime5}.\n\nWe conjectured that cleaning up the training data would enable a more effective acoustic model training for the CHiME-5 scenario. \nWe have therefore experimented with enhancement algorithms of various strengths, from relatively simple beamforming over single-array \\gls{GSS} to a quite sophisticated multi-array \\gls{GSS} approach, and tested all combinations of training and test data enhancement methods.\nFurthermore, compared to the initial \\Gls{GSS} approach in~\\cite{Boeddecker2018chime5}, we describe here some modifications, which led to improved performance.\nWe also propose an improved neural acoustic modeling structure compared to the CHiME-5 baseline system described in \\cite{Manohar_icassp19}. It consists of initial \\gls{CNN} layers followed by \\gls{TDNN-F} layers, instead of a homogeneous \\gls{TDNN-F} architecture.\n\nUsing a single acoustic model trained with \\SI{308}{hrs} of training data, which resulted after applying multi-array GSS data cleaning and a three-fold speed perturbation, we achieved a \\gls{WER} of \\SI{41.6}{\\%} on the development (DEV) and \\SI{43.2}{\\%} on the evaluation (EVAL) test set of CHiME-5, if the test data is also enhanced with multi-array GSS. \nThis compares very favorably with the recently published top-line in \\cite{Kanda2019}, where the single-system best result, i.e., the WER without system combination, was \\SI{45.1}{\\%} and \\SI{47.3}{\\%} on DEV and EVAL, respectively, using an augmented training data set of \\SI{4500}{hrs} total.\n\nThe rest of this paper is structured as follows. Section~\\ref{sec:c5} describes the CHiME-5 corpus, Section~\\ref{sec:gss} briefly presents the guided source separation enhancement method, Section~\\ref{sec:exp} shows the ASR experiments and the results, followed by a discussion in Section~\\ref{sec:disc}. Finally, the paper is concluded in Section~\\ref{sec:concl}.\n\n\n\\section{CHiME-5 corpus description}\n\\label{sec:c5}\n\nThe CHiME-5 corpus comprises twenty dinner party recordings (sessions) lasting for approximately \\SI{2}{hrs} each. A session contains the conversation among the four dinner party participants. \nRecordings were made in kitchen, dining and living room areas with each phase lasting for a minimum of \\SI{30}{mins}.\n16 dinner parties were used for training, 2 were used for development, and 2 were used for evaluation. \n\nThere were two types of recording devices collecting CHiME-5 data: distant 4-channels (linear) Microsoft Kinect arrays (referred to as units or `U') and in-ear Soundman OKM II Classic Studio binaural microphones (referred to as worn microphones or `W'). Six Kinect arrays were used in total and they were placed such that at least two units were able to capture the acoustic environment in each recording area. Each dinner party participant wore in-ear microphones which were subsequently used to facilitate human audio transcription of the data.\nThe devices were not time synchronized during recording. Therefore, the W and the U signals had to be aligned afterwards using a correlation based approach provided by the organizers.\nDepending on how many arrays were available during test time, the challenge had a single (reference) array and a multiple array track. For more details about the corpus, the reader is referred to~\\cite{Barker2018CHiME5}.\n\n\\section{Guided source separation}\n\\label{sec:gss}\n\n\\gls{GSS} enhancement is a blind source separation technique originally proposed in \\cite{Boeddecker2018chime5}\\footnote{\\url{https:\/\/github.com\/fgnt\/pb_chime5}} to alleviate the speaker overlap problem in CHiME-5.\nGiven a mixture of reverberated overlapped speech, \\gls{GSS} aims to separate the sources using a pure signal processing approach. An \\gls{EM} algorithm estimates the parameters of a spatial mixture model and the posterior probabilities of each speaker being active are used for mask based beamforming.\n\nAn overview block diagram of this enhancement by source separation is depicted in \\cref{fig:enhancement_block}. It follows the approach presented in~\\cite{Kanda2019}, which was shown to outperform the baseline version.\nThe system operates in the \\gls{STFT} domain and consists of two stages: (1) a dereverberation stage, and (2) a guided source separation stage. For the sake of simplicity, the overall system is referred to as GSS for the rest of the paper.\nRegarding the first stage, the multiple input multiple output version of the \\gls{WPE} method was used for dereverberation ($M$ inputs and $M$ outputs)~\\cite{Yoshioka2012GWPE,Drude2018naraWPE}\\footnote{\\url{https:\/\/github.com\/fgnt\/nara_wpe}} and, regarding the second stage, it consists of a spatial \\gls{MM} \\cite{Ito2016cACGMM} and a source extraction (SE) component.\nThe model has five mixture components, one representing each speaker, and an additional component representing the noise class. \n\n\\begin{figure}[t]\n\t\\centering\n\n\t\\input{images\/block_ali_2}\n\n\t\\caption{Overview of speech enhancement system with \\acrfull{WPE} dereverberation, \\acrfull{MM} estimation, Source Extractor (SE) and \\acrfull{ASR}.}\n\t\\label{fig:enhancement_block}\n\n\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\input{images\/activity_example}\n\t\\caption{\n\t\tVisualization of time annotations on a fragment of the CHiME-5 data.\n\t\tThe grey bars indicate source activity,\n\t\tthe inner vertical blue lines denote the utterance boundaries of a segment of speaker P01,\n\t\tand the outer vertical red lines the boundaries of the extended utterance, consisting of the segment and the ``context'',\n\t\ton which the mixture model estimation algorithm operates.\n\t}\n\t\\label{fig:activity}\n\n\\end{figure}\n\nThe role of the \\gls{MM} is to support the source extraction component for estimating the target speech.\nThe class affiliations computed in the E-step of the EM algorithm are employed to estimate spatial covariance matrices of target signals and interferences, from which the coefficients of an \\gls{MVDR} beamformer are computed~\\cite{Souden2010MVDR}.\nThe reference channel for the beamformer is estimated based on an SNR criterion\\cite{Erdogan2016MVDR}. \nThe beamformer is followed by a postfilter to reduce the remaining speech distortions ~\\cite{Warsitz2007GEVBAN}, which in turn is followed by an additional (optional) masking stage to improve crosstalk suppression.\nThose masks are also given by the mentioned class affiliations.\nFor the single array (CHiME-5) track, simulations have shown that multiplying the beamformer output with the target speaker mask improves the performance on the U data, but the same approach degrades the performance in the multiple array track~\\cite{Boeddecker2018chime5}.\nThis is because the spatial selectivity of a single array is very limited in CHiME-5: the speakers' signals arrive at the array, which is mounted on the wall at some distance, at very similar impinging angles, rendering single array beamforming rather ineffective.\nConsequently, additional masking has the potential to improve the beamformer performance.\nConversely, the \\gls{MM} estimates are more accurate in the multiple array case since they benefit from a more diverse spatial arrangement of the microphones, and the signal distortions introduced by the additional masking rather degrade the performance.\nConsequently, for our experiments we have used the masking approach for the single array track, but not for the multiple array one.\n\nGSS exploits the baseline CHiME-5 speaker diarization information available from the transcripts (annotations) to determine when multiple speakers talk simultaneously (see~\\cref{fig:activity}). This crosstalk information is then used to guide the parameter estimation of the \\gls{MM} both during EM initialization (posterior masks set to one divided by the number of active speakers for active speakers' frames, and zero for the non-active speakers) and after each E-step (posterior masks are clamped to zero for non-active speakers).\n\nThe initialization of the \\gls{EM} for each mixture component is very important for the correct convergence of the algorithm.\nIf the \\gls{EM} initialization is close enough to the final solution, then it is expected that the algorithm will correctly separate the sources and source indices are not permuted across frequency bins. This has a major practical application, since frequency permutation solvers like \\cite{Sawada2004Perm} become obsolete.\n\nTemporal context also plays an important role in the EM initialization. Simulations have shown that a large context of 15 seconds left and right of the considered segment improves the mixture model estimation performance significantly for CHiME-5~\\cite{Boeddecker2018chime5}. However, having such a large temporal context may become problematic when the speakers are moving, because the estimated spatial covariance matrix can become outdated due to the movement~\\cite{Kanda2019}. Alternatively, one can run the EM first with a larger temporal context until convergence, then drop the context and re-run it for some more iterations. \nAs shown later in the paper, this approach did not improve ASR performance. Therefore, the temporal context was only used for dereverberation and the mixture model parameter estimation, while for the estimation of covariance matrices for beamforming the context was dropped and only the original segment length was considered~\\cite{Kanda2019}.\n\nAnother avenue we have explored for further source separation improvement was to refine the baseline CHiME-5 annotations using ASR output (see \\cref{fig:enhancement_block}).\nA first-pass decoding using an \\gls{ASR} system is used to predict silence intervals. Then this information is used to adjust the time annotations, which are used in the EM algorithm as described above. When the ASR decoder indicates silence for a speaker, the corresponding class posterior in the \\gls{MM} is forced to zero.\n\nDepending on the number of available arrays for CHiME-5, two flavours of GSS enhancement were used in this work.\nIn the single array track, all 4 channels of the array are used as input ($M = 4$), and the system is referred to as GSS1.\nIn the multi array track, all six arrays are stacked to form a 24 channels super-array ($M = 24$), and this system is denoted as GSS6.\nThe baseline time synchronization provided by the challenge organizers was sufficient to align the data for GSS6.\n\n\\section{Experiments}\n\\label{sec:exp}\n\n\\subsection{General configuration}\\label{ssec:setup}\n\nExperiments were performed using the CHiME-5 data. Distant microphone recordings (U data) during training and\/or testing were processed using the speech enhancement methods depicted in Table~\\ref{tab:enh_meths}.\nSpeech was either left unprocessed, enhanced using a weighted delay-and-sum beamformer (BFIt)~\\cite{Anguera_ieeetaslp2007} with or without dereverberation (WPE), or processed using the guided source separation (GSS) approach described in Section~\\ref{sec:gss}. In Table~\\ref{tab:enh_meths}, the strength of the enhancement increases from top to bottom, i.e., GSS6 signals are much cleaner than the unprocessed ones.\n\nThe standard CHiME-5 recipes were used to: \n(i) train GMM-HMM alignment models, \n(ii) clean up the training data, and \n(iii) augment the training data using three-fold speed perturbation.\nThe acoustic feature vector consisted of 40-dimensional MFCCs appended with 100-dimensional i-vectors.\nBy default, the acoustic models were trained using the \\gls{LF-MMI} criterion and a 3-gram language model was used for decoding~\\cite{Barker2018CHiME5}.\nDiscriminative training (DT)~\\cite{Ghoshal_is2013} and an additional RNN-based language model (RNN-LM)~\\cite{Mikolov_is2010} were applied to improve recognition accuracy for the best performing systems.\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{Naming of the speech enhancement methods.}\n\\begin{center}\n\\begin{tabular}{lcl}\n\t\\hline\n\tEnhancement & Array & Label \\\\\n\t\\hline\n\t Unprocessed \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& Single\/Multi & None \\\\\n\t BeamformIt~\\cite{Anguera_ieeetaslp2007} \t& Single\t\t\t\t& BFIt \\\\\n\t WPE + BeamformIt~\\cite{Manohar_icassp19} & Single \t\t\t\t& WPE+BFIt \\\\\n\t WPE + GSS1 + BF w\/o Context~\\cite{Boeddecker2018chime5}\t& Single & GSS1 \\\\\n\t WPE + GSS6 + BF w\/o Context~\\cite{Boeddecker2018chime5} \t& Multi & GSS6 \\\\\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:enh_meths}\n\\end{table}\n\n\\subsection{Acoustic model}\n\nThe initial baseline system~\\cite{Barker2018CHiME5} of the CHiME-5 challenge uses a \\gls{TDNN} \\gls{AM}.\nHowever, recently it has been shown that introducing factorized layers into the TDNN architecture facilitates training deeper networks and also improves the ASR performance~\\cite{Povey_is2018}.\nThis architecture has been employed in the new baseline system for the challenge~\\cite{Manohar_icassp19}.\nThe \\gls{TDNN-F} has 15 layers with a hidden dimension of 1536 and a bottleneck dimension of 160; each layer also has a resnet-style bypass-connection from the output of the previous layer, and a ``continuous dropout'' schedule~\\cite{Manohar_icassp19}.\nIn addition to the \\gls{TDNN-F}, the newly released baseline\\footnote{\\tiny\\url{https:\/\/github.com\/kaldi-asr\/kaldi\/tree\/master\/egs\/chime5\/s5b}} also uses simulated reverberated speech from worn microphone recordings for augmenting the training set, it employes front-end speech dereverberation and beamforming (WPE+BFIt), as well as robust i-vector extraction using 2-stage decoding.\n\n\\Glspl{CNN} have been previously shown to improve ASR robustness~\\cite{AbdelHamid_ieeetaslp2014}. Therefore, combining \\gls{CNN} and \\gls{TDNN-F} layers is a promising approach to improve the baseline system of~\\cite{Manohar_icassp19}.\nTo test this hypothesis, a CNN-TDNNF \\gls{AM} architecture\\footnote{\\tiny\\url{https:\/\/github.com\/kaldi-asr\/kaldi\/tree\/master\/egs\/swbd\/s5c}}\nconsisting of 6 CNN layers followed by 9 TDNN-F layers was compared against an AM having 15 TDNN-F layers.\nAll \\gls{TDNN-F} layers have the topology described above.\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{Comparison of baseline \\gls{TDNN-F}~\\cite{Manohar_icassp19} and proposed CNN-TDNNF~\\gls{AM}s in terms of \\gls{WER} for the DEV (EVAL) set.}\n\\begin{center}\n\t\\setlength{\\tabcolsep}{5pt} \n\\begin{tabular}{llccc}\n\t\\hline\n\t AM \t& \tEnh. in trng \/ \\si{hrs} \t& \tEnh. in test \t& \tWER (\\si{\\%}) \\\\\n\t\\hline\n\tTDNNF~\\cite{Manohar_icassp19} & None \/ $1416$ & WPE+BFIt & $69.6$ ($61.7$) \\\\\n\tCNN-TDNNF \t\t\t\t\t\t\t\t\t\t& None \/ $1416$ \t& WPE+BFIt & $\\textbf{67.2}$ ($\\textbf{58.7}$) \\\\\n\t\\hline\n\tCNN-TDNNF\t \t\t\t\t\t\t\t\t\t& None \/ $316$ \t& BFIt & $68.7$ ($61.3$) \\\\\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:WERs_AM}\n\\end{table}\n\nASR results are given in Table~\\ref{tab:WERs_AM}.\nThe first two rows show that replacing the \\gls{TDNN-F} with the CNN-TDNNF AM yielded more than \\SI{2}{\\%} absolute \\gls{WER} reduction.\nWe also trained another CNN-TDNNF model using only a small subset (worn + 100k utterances from arrays) of training data (about \\SI{316}{hrs} in total) which has produced slightly better \\gls{WER}s compared with the baseline \\gls{TDNN-F} trained on a much larger dataset (roughly \\SI{1416}{hrs} in total). For consistency, 2-stage decoding was used for all results in Table~\\ref{tab:WERs_AM}.\nWe conclude that the CNN-TDNNF model outperforms the TDNNF model for the CHiME-5 scenario and, therefore, for the remainder of the paper we only report results using the CNN-TDNNF AM.\n\n\n\\subsection{Enhancement effectiveness for ASR training and test}\n\nAn extensive set of experiments was performed to measure the WER impact of enhancement on the CHiME-5 training and test data. \nWe test enhancement methods of varying strengths, as described in Section~\\ref{ssec:setup}, and the results are depicted in Table~\\ref{tab:WERs_enh}. \nIn all cases, the (unprocessed) worn dataset was also included for AM training since it was found to improve performance (supporting therefore the argument that data variability helps ASR robustness).\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{\\gls{WER} results on the DEV (EVAL) set and various combinations of speech enhancement for ASR training and test (CNN-TDNNF AM). Amount of training data (\\si{hrs}) is also specified.}\n\\begin{center}\n\t\\setlength{\\tabcolsep}{5pt} \n\\begin{tabular}{lcccc}\n\t\\hline\n\t \\multirow{2}{*}{\\shortstack{Enh. in trng \\\\{ (\\si{hrs})}}} \t& \t\\multicolumn{4}{c}{Enhancement in test} \\\\ \\cline{2-5}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& None & BFIt & GSS1 & GSS6 \\\\\n\t\\hline\n\tNone ($2046$)\t\t\t\t& $69.3$ ($59.9$) & $69.1$ ($59.7$) & $62.2$ ($58.2$) & $51.8$ ($51.6$) \\\\\n\tBFIt ($680$)\t\t\t\t& $68.9$ ($59.1$) & $68.5$ ($58.5$) & $59.9$ ($57.3$) & $48.8$ ($49.9$) \\\\\n\tGSS1 ($791$)\t\t\t\t& $74.3$ ($67.5$) & $73.7$ ($66.4$) & $53.0$ ($49.6$) & $48.0$ ($47.5$) \\\\\n\tGSS6 ($308$)\t\t\t\t& $78.5$ ($73.1$) & $76.9$ ($69.2$) & $58.0$ ($56.1$) & $\\textbf{45.4}$ ($\\textbf{45.7}$) \\\\\n\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:WERs_enh}\n\\vspace{-10pt}\n\\end{table}\n\nIn Table~\\ref{tab:WERs_enh}, in each row the recognition accuracy improves monotonically from left to right, i.e., as the enhancement strategy on the test data becomes stronger. Reading the table in each column from top to bottom, one observes that accuracy improves with increasing power of the enhancement on the training data, however, only as long as the enhancement on the training data is not stronger than on the test data.\nCompared with unprocessed training and test data (None-None), GSS6-GSS6 yields roughly \\SI{35}{\\%} (\\SI{24}{\\%}) relative WER reduction on the DEV (EVAL) set, and \\SI{12}{\\%} (\\SI{11}{\\%}) relative WER reduction when compared with the None-GSS6 scenario. Comparing the amount of training data used to train the acoustic models, we observe that it decreases drastically from no enhancement to the GSS6 enhancement.\n\n\\subsection{State-of-the-art single-system for CHiME-5}\n\nTo facilitate comparison with the recently published top-line in~\\cite{Kanda2019} (H\/UPB), we have conducted a more focused set of experiments whose results are depicted in Table~\\ref{tab:WERs_best}.\nAs explained in Section~\\ref{sec:tempCtxtConfig}, we opted for~\\cite{Kanda2019} instead of~\\cite{Boeddecker2018chime5} as baseline because the former system is stronger.\nThe experiments include refining the GSS enhancement using time annotations from ASR output (GSS w\/ ASR), performing discriminative training on top of the AMs trained with LF-MMI and performing RNN LM rescoring.\nAll the above helped further improve ASR performance. We report performance of our system on both single and multiple array tracks. To have a fair comparison, the results are compared with the single-system performance\nreported in\\cite{Kanda2019}.\n\n\nFor the \\emph{single array track}, the proposed system without RNN LM rescoring achieves \\SI{16}{\\%} (\\SI{11}{\\%}) relative WER reduction on the DEV (EVAL) set when compared with System8 in~\\cite{Kanda2019} (row one in Table~\\ref{tab:WERs_best}). \nRNN LM rescoring further helps improve the proposed system performance. \n\nFor the \\emph{multi array track}, the proposed system without RNN LM rescoring achieved \\SI{6}{\\%} (\\SI{7}{\\%}) relative WER reduction on the DEV (EVAL) set when compared with System16 in~\\cite{Kanda2019} (row six in Table~\\ref{tab:WERs_best}). \n\n\\begin{table}[t]\n\\footnotesize\n\\caption{Comparison of reference~\\cite{Kanda2019} and proposed (single) systems in terms of \\gls{WER} for the DEV (EVAL) set. Test data enhancement was refined using ASR alignments or oracle alignments.}\n\\begin{center}\n\t\\setlength{\\tabcolsep}{1.2pt} \n\\begin{tabular}{llC{1.0cm}C{1.8cm}ccc}\n\t\\hline\n\t\n\tTrack & System & Enh. in trng & Enh. in \\newline test & DT & RNN-LM & WER (\\si{\\%}) \\\\\n\t\\hline\n\t\n\t\\multirow{5}{*}{Single} & H\/UPB~\\cite{Kanda2019} & None & GSS1 w\/ ASR & $\\checkmark$ & & $58.3$ ($53.1$) \\\\\n\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& GSS1 & GSS1 w\/ ASR & & & $50.2$ ($48.4$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& GSS1 & GSS1 w\/ ASR & $\\checkmark$ & & $49.1$ ($47.3$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& GSS1 & GSS1 w\/ ASR & $\\checkmark$ & $\\checkmark$ & $\\bf{48.6}$ ($\\bf{46.7}$) \\\\ \\cline{2-7}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& \\textcolor{gray}{Proposed} & \\textcolor{gray}{GSS1} & \\textcolor{gray}{GSS1 w\/ oracle} & \\textcolor{gray}{$\\checkmark$} & \\textcolor{gray}{$\\checkmark$} & \\textcolor{gray}{$47.3$ ($46.1$)} \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\hline\n\t\n\t\\multirow{5}{*}{Multiple} & H\/UPB~\\cite{Kanda2019} & None & GSS6 w\/ ASR & $\\checkmark$ & & $45.1$ ($47.3$) \\\\\n\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& GSS6 & GSS6 w\/ ASR & & & $43.2$ ($44.2$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& GSS6 & GSS6 w\/ ASR & $\\checkmark$ & & $42.3$ ($43.9$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& GSS6 & GSS6 w\/ ASR & $\\checkmark$ & $\\checkmark$ & $\\bf{41.6}$ ($\\bf{43.2}$) \\\\ \\cline{2-7}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& \\textcolor{gray}{Proposed} \t& \\textcolor{gray}{GSS6} & \\textcolor{gray}{GSS6 w\/ oracle} & \\textcolor{gray}{$\\checkmark$} & \\textcolor{gray}{$\\checkmark$} & \\textcolor{gray}{$39.9$ ($42.0$)} \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:WERs_best}\n\\end{table}\n\nWe also performed a test using GSS with the oracle alignments (GSS w\/ oracle) to assess the potential of time annotation refinement (gray shade lines in Table~\\ref{tab:WERs_best}). It can be seen that there is some, however not much room for improvement.\n\nFinally, cleaning up the training set not only boosted the recognition performance, \nbut managed to do so using a fraction of the training data in~\\cite{Kanda2019}, as shown in Table~\\ref{tab:WERs_hrs}.\nThis translates to significantly faster and cheaper training of acoustic models, which is a major advantage in practice.\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{Comparison of the reference~\\cite{Kanda2019} and proposed systems in terms of amount of training data.}\n\n\\begin{center}\n\t\\setlength{\\tabcolsep}{3pt} \n\\begin{tabular}{llcc}\n\t\\hline\n\t\n\tTrack & System & Amount trng data (\\si{hrs}) & WER (\\si{\\%}) \\\\\n\t\\hline\n\t\n\t\\multirow{2}{*}{Single} & H\/UPB~\\cite{Kanda2019} & $4500$ & $58.3$ ($53.1$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& $791$ & $\\bf{48.6}$ ($\\bf{46.7}$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\hline\n\t\n\t\\multirow{2}{*}{Multiple} & H\/UPB~\\cite{Kanda2019} & $4500$ & $45.1$ ($47.3$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t& Proposed \t& $308$ & $\\bf{41.6}$ ($\\bf{43.2}$) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:WERs_hrs}\n\\vspace{-10pt}\n\\end{table}\n\n\\section{Discussion}\n\\label{sec:disc}\n\n\\subsection{Temporal context configuration for GSS}\n\\label{sec:tempCtxtConfig}\n\nOur experiments have shown that the temporal context of some GSS components has a significant effect on the WER. Two cases are investigated: (i) partially dropping the temporal context for the EM stage, and (ii) dropping the temporal context for beamforming.\nThe evaluation was conducted with an acoustic model trained on unprocessed speech and the enhancement was applied during test only.\nResults are depicted in Table~\\ref{tab:WERs_context}.\n\nThe first row corresponds to the GSS configuration in~\\cite{Boeddecker2018chime5} while the second one corresponds to the GSS configuration in~\\cite{Kanda2019}. First two rows show\nthat dropping the temporal context for estimating statistics for beamforming improves ASR accuracy.\nFor the last row, the EM algorithm was run 20 iterations with temporal context, followed by another 10 without context. Since the performance decreased, we concluded that the best configuration for the \\gls{GSS} enhancement in CHiME-5 scenario is using full temporal context for the EM stage and dropping it for the beamforming stage. Consequently, we have chosen system~\\cite{Kanda2019} as baseline in this study since is using the stronger GSS configuration.\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{WER results using CNN-TDNNF AM trained on unprocessed (None) when some \\gls{GSS} enhancement (test) components ignore the temporal context.}\n\n\\begin{center}\n\\begin{tabular}{llc}\n\t\\hline\n\tEM iterations & BF & WER (\\si{\\%})\\\\\n\t\\hline\n\t20 w\/ context~\\cite{Boeddecker2018chime5} & w\/ context & $54.7$ ($52.3$) \\\\\n\t20 w\/ context~\\cite{Kanda2019} \t\t\t\t\t\t& w\/o context& $\\bf{51.8}$ ($\\bf{51.6}$) \\\\\n\t20 w\/ + 10 w\/o context \t\t\t\t\t\t\t\t\t\t& w\/o context & $52.2$ ($52.5$) \\\\\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:WERs_context}\n\\vspace{-10pt}\n\\end{table}\n\n\n\\subsection{Analysis of speaker overlap effect on WER accuracy}\n\nThe results presented so far were overall accuracies on the test set of CHiME-5.\nHowever, since speaker overlap is a major issue for these data, it is of interest to investigate the methods' performance as a function of the amount of overlapped speech. Employing the original CHiME-5 annotations, the word distribution of overlapped speech was computed for DEV and EVAL sets (silence portions were not filtered out).\nThe five-bin normalized histogram of the data is plotted in Fig.~\\ref{fig:data_distr}.\nInterestingly, the percentage of segments with low overlapped speech is significantly higher for the EVAL than for the DEV set, and, conversely, the number of words with high overlapped speech is considerably lower for the EVAL than for the DEV set.\nThis distribution may explain the difference in performance observed between the DEV and EVAL sets.\n\n\\begin{figure}[b!]\n\t\\begin{centering}\n\t\t\\input{images\/tikz_data_overlap_distrib.tex}\n\t\t\\caption{Word distribution of overlapped speech for the DEV and EVAL sets of CHiME-5.}\n\t\t\\label{fig:data_distr}\n\t\\end{centering}\n\\end{figure}\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{Breakdown of absolute \\gls{WER} results on the DEV (EVAL) set for the same training and test enhancement (matched case, CNN-TDNNF AM).}\n\\begin{center}\n\t\\setlength{\\tabcolsep}{1.5pt} \n\\begin{tabular}{lccccc}\n\t\\hline\n\t \\multirow{2}{*}{\\shortstack{Enh. \\\\(trng+test)}} \t& \t\\multicolumn{5}{c}{Amount of overlap (\\si{\\%})} \\\\ \\cline{2-6}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& $0-20$ & $20-40$ & $40-60$ & $60-80$ & $80-100$ \\\\\n\t\\hline\n\tNone\t\t\t\t& $48.3$ ($47.2$) & $49.0$ ($49.5$) & $56.9$ ($57.4$) & $64.5$ ($67.4$) & $89.5$ ($84.1$) \\\\\n\tBFIt\t\t\t\t& $46.6$ ($45.6$) & $47.8$ ($48.4$) & $54.9$ ($55.0$) & $63.6$ ($66.7$) & $89.5$ ($84.2$) \\\\\n\tGSS1\t\t\t\t& $42.2$ ($43.3$) & $41.6$ ($43.4$) & $44.8$ ($47.8$) & $50.6$ ($55.3$) & $69.0$ ($67.6$) \\\\\n\tGSS6\t\t\t\t& $36.5$ ($40.1$) & $36.4$ ($40.8$) & $41.0$ ($44.6$) & $43.8$ ($49.9$) & $58.8$ ($62.0$) \\\\\n\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:WER_overlap_matched}\n\\end{table}\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{Breakdown of absolute \\gls{WER} results on the DEV (EVAL) set for unprocessed training data and various test enhancements (mismatched case, CNN-TDNNF AM).}\n\\begin{center}\n\t\\setlength{\\tabcolsep}{1pt} \n\\begin{tabular}{L{1.5cm}ccccc}\n\t\\hline\n\t \\multirow{2}{*}{\\shortstack{Enh. (test)}} \t& \t\\multicolumn{5}{c}{Amount of overlap (\\si{\\%})} \\\\ \\cline{2-6}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& $0-20$ & $20-40$ & $40-60$ & $60-80$ & $80-100$ \\\\\n\t\\hline\n\tNone\t\t\t\t& $48.3$ ($47.2$) & $49.0$ ($49.5$) & $56.9$ ($57.4$) & $64.5$ ($67.4$) & $89.5$ ($84.1$) \\\\\n\tBFIt\t\t\t\t& $47.5$ ($47.0$) & $48.4$ ($49.7$) & $56.5$ ($56.6$) & $64.3$ ($66.9$) & $89.3$ ($83.7$) \\\\\n\tGSS1\t\t\t\t& $48.8$ ($51.1$) & $49.2$ ($51.4$) & $53.4$ ($55.3$) & $58.5$ ($63.5$) & $78.3$ ($76.2$) \\\\\n\tGSS1 \\newline ~~ w\/o Mask& $44.0$ ($44.9$) & $45.8$ ($46.8$) & $51.5$ ($52.9$) & $57.7$ ($62.4$) & $82.4$ ($78.2$) \\\\\n\tGSS6\t\t\t\t& $40.3$ ($45.5$) & $41.2$ ($45.1$) & $45.1$ ($50.0$) & $48.2$ ($54.9$) & $66.7$ ($68.9$) \\\\\n\tGSS6 \\newline ~~ w\/ ASR\t\t\t& $38.8$ ($44.5$) & $39.8$ ($43.8$) & $43.3$ ($49.2$) & $46.4$ ($53.4$) & $63.5$ ($67.1$) \\\\\n\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:WER_overlap_mismatched}\n\\vspace{-15pt}\n\\end{table}\n\n\nBased on the distributions in Fig.~\\ref{fig:data_distr}, the test data was split. Two cases were considered: (a) same enhancement for training and test data (matched case, Table~\\ref{tab:WER_overlap_matched}), and (b) unprocessed training data and enhanced test data (mismatched case, Table~\\ref{tab:WER_overlap_mismatched}). As expected, the WER increases monotonically as the amount of overlap increases in both scenarios, and the recognition accuracy improves as the enhancement method becomes stronger.\n\n\\begin{figure*}[t]\n\t\\begin{subfigure}{0.5\\textwidth}\\centering\n\t\t\\input{images\/tikz_analysis_matched_dev.tex}\n\t\t\\caption{DEV}\n\t\t\\label{fig:matched_dev}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.5\\textwidth}\\centering\n\t\t\\input{images\/tikz_analysis_matched_eval.tex}\n\t\t\\caption{EVAL}\n\t\t\\label{fig:matched_eval}\n\t\\end{subfigure}\n\\caption{Relative WER gain for the matched case vs unprocessed, Table~\\ref{tab:WER_overlap_matched} row one (CNN-TDNNF AM).}\n\\label{fig:matched}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\t\\begin{subfigure}{0.5\\textwidth}\\centering\n\t\t\\input{images\/tikz_analysis_mismatched_A_dev.tex}\n\t\t\\caption{DEV}\n\t\t\\label{fig:mmatchedA_dev}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.5\\textwidth}\\centering\n\t\t\\input{images\/tikz_analysis_mismatched_A_eval.tex}\n\t\t\\caption{EVAL}\n\t\t\\label{fig:mmatchedA_eval}\n\t\\end{subfigure}\n\\caption{Relative WER gain for the mismatched case vs unprocessed, Table~\\ref{tab:WER_overlap_mismatched} row one (CNN-TDNNF AM trained on unprocessed).}\n\\label{fig:mmmatchedA}\n\\end{figure*}\n\nGraphical representations of WER gains (relative to the unprocessed case) in Tables~\\ref{tab:WER_overlap_matched} and \\ref{tab:WER_overlap_mismatched} are given in Figs.~\\ref{fig:matched} and \\ref{fig:mmmatchedA}.\nThe plots show that as the amount of speaker overlap increases, the accuracy gain (relative to the unprocessed case) of the weaker signal enhancement (BFIt) drops. \nThis is an expected result since BFIt is not a source separation algorithm.\nConversely, as the amount of speaker overlap increases, the accuracy gain (relative to None) of the stronger GSS enhancement improves quite significantly.\nA rather small decrease in accuracy is observed in the mismatched case (Fig.~\\ref{fig:mmmatchedA}) for GSS1 in the lower overlap regions.\nAs already mentioned in Section~\\ref{sec:gss}, this is due to the masking stage. It has previously been observed that using masking for speech enhancement without a cross talker decreases ASR recognition performance. \nWe have also included in Fig.~\\ref{fig:mmmatchedA} the GSS1 version without masking (GSS w\/o Mask), which indeed yields significant accuracy gains on segments with little overlap. \nHowever, since the overall accuracy of GSS1 with masking is higher than the overall gain of GSS1 without masking, GSS w\/o mask was not included in the previous experiments.\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nIn this paper we performed an extensive experimental evaluation on the acoustically very challenging CHiME-5 dinner party data showing that:\n(i) cleaning up training data can lead to substantial word error rate reduction, and (ii) enhancement in training is advisable as long as enhancement in test is at least as strong as in training.\nThis approach stands in contrast and delivers larger accuracy gains at a fraction of training data than the common data simulation strategy found in the literature.\nUsing a CNN-TDNNF acoustic model topology along with GSS enhancement refined with time annotations from ASR, discriminative training and RNN LM rescoring, we achieved a new single-system state-of-the-art result on CHiME-5, which is \\SI{41.6}{\\%} (\\SI{43.2}{\\%}) on the development (evaluation) set, which is a \\SI{8}{\\%} relative improvement of the word error rate over a comparable system reported so far.\n\n\\section{Acknowledgments}\n\nParts of computational resources required in this study were provided by the Paderborn Center for Parallel Computing.\n\n\\newpage\n\n\\balance\n\\bibliographystyle{myIEEEbib}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\ignorespaces}{\\unskip\\bibn@me}\n \\bigbreak\\bgroup\n 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\\par\\vfill\\supereject\\end}\n\n\n\\magnification=1200\n\n\n\n\\font\\tenbfit=cmbxti10\n\\font\\sevenbfit=cmbxti10 at 7pt\n\\font\\sixbfit=cmbxti5 at 6pt\n\n\\newfam\\mathboldit\n\n\\textfont\\mathboldit=\\tenbfit\n \\scriptfont\\mathboldit=\\sevenbfit\n \\scriptscriptfont\\mathboldit=\\sixbfit\n\n\\def\\bfit\n{\\tenbfit\n \\fam\\mathboldit\n}\n\n\\def{\\bf {Q}}{{\\bf {Q}}}\n\\def{\\bf {K}}{{\\bf {K}}}\n\\def {O_{\\K}}} \\def\\U{{U_K}{ {O_{{\\bf {K}}}}} \\def\\U{{U_K}}\n\\def{\\bf N}} \\def\\L{{\\bf L}{{\\bf N}} \\def\\L{{\\bf L}}\n\\def{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}{{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}}\n\\def{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}{{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}}\n\\def{{\\tilde{\\cal {K}}} 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{AD}}}}{{\\bfit Bad}}\n\\def{\\overline U}{{\\overline U}}\n\\def{\\overline V}{{\\overline V}}\n\\def{\\overline W}{{\\overline W}}\n\\def{\\overline X}{{\\overline X}}\n\\def{\\overline A}{{\\overline A}}\n\\def\\bf N{{\\bf N}}\n\n\\def\\noindent {\\bf{D\\'emonstration : }}{\\noindent {\\bf{D\\'emonstration : }}}\n\\def\\noindent {\\bf{Remarque : }}{\\noindent {\\bf{Remarque : }}}\n\\def{\\it cf. }} \\def\\resp{{\\it resp. }{{\\it cf. }} \\def\\resp{{\\it resp. }}\n\\def{\\thinspace}} \\def\\diam{{\\rm diam}{{\\thinspace}} \\def\\diam{{\\rm diam}}\n\n\n\\def\\house#1{\\setbox1=\\hbox{$\\,#1\\,$}%\n\\dimen1=\\ht1 \\advance\\dimen1 by 2pt \\dimen2=\\dp1 \\advance\\dimen2 by 2pt\n\\setbox1=\\hbox{\\vrule height\\dimen1 depth\\dimen2\\box1\\vrule}%\n\\setbox1=\\vbox{\\hrule\\box1}%\n\\advance\\dimen1 by .4pt \\ht1=\\dimen1\n\\advance\\dimen2 by .4pt \\dp1=\\dimen2 \\box1\\relax}\n\n\n\\def{\\rm M}} \\def\\h{{\\rm h}} \\def\\J{{\\rm J}{{\\rm M}} \\def\\h{{\\rm h}} \\def\\J{{\\rm J}}\n\\def{\\rm Norm}} \\def\\de{\\delta _{\\K}} \\def\\res{{\\rm Res}{{\\rm Norm}} \\def\\de{\\delta _{{\\bf {K}}}} \\def\\res{{\\rm Res}}\n\\def{\\rm N}_{\\K \/ \\Q}} \\def\\NL{{\\rm N}_{\\L \/ \\Q}{{\\rm N}_{{\\bf {K}} \/ {\\bf {Q}}}} \\def\\NL{{\\rm N}_{\\L \/ {\\bf {Q}}}}\n\\def{\\rm N}_{\\K(s) \/ \\Q}} \\def\\NLL{{\\rm N}_{\\L(e_1) \/ \\Q}{{\\rm N}_{{\\bf {K}}(s) \/ {\\bf {Q}}}} \\def\\NLL{{\\rm N}_{\\L(e_1) \/ {\\bf {Q}}}}\n\\def{\\Delta_f}} \\def\\NLK{{\\rm N}_{\\L \/ \\K}{{\\Delta_f}} \\def\\NLK{{\\rm N}_{\\L \/ {\\bf {K}}}}\n\n\\def{\\Gamma}} \\def\\al{{\\alpha}{{\\Gamma}} \\def\\al{{\\alpha}}\n\\def{\\kappa}} \\def\\eps{{\\varepsilon}{{\\kappa}} \\def\\eps{{\\varepsilon}}\n\\def{\\gamma}} \\def\\La{{\\Lambda}{{\\gamma}} \\def\\La{{\\Lambda}}\n\\def{\\beta}} \\def\\de{{\\delta}{{\\beta}} \\def\\de{{\\delta}}\n\\def{\\lambda}} \\def\\th{{\\theta}{{\\lambda}} \\def\\th{{\\theta}}\n\n\\def{\\cal H}{{\\cal H}}\n\\def{\\underline{v}}} \\def\\ux{{\\underline{x}}} \\def\\uy{{\\underline y}{{\\underline{v}}} \\def\\ux{{\\underline{x}}} \\def\\uy{{\\underline y}}\n\n\\def\\displaystyle} \\def\\scr{\\scriptstyle{\\displaystyle} \\def\\scr{\\scriptstyle}\n\\def\\smallskip} \\def\\ens{\\enspace} \\def\\noi{\\noindent{\\smallskip} \\def\\ens{\\enspace} \\def\\noi{\\noindent}\n\n\\def\\build#1_#2^#3{\\mathrel{\\mathop{\\kern 0pt#1}\\limits_{#2}^{#3}}}\n\n\\def\\date {le\\ {\\the\\day}\\ \\ifcase\\month\\or janvier\n\\or fevrier\\or mars\\or avril\\or mai\\or juin\\or juillet\\or\nao\\^ut\\or septembre\\or octobre\\or novembre\n\\or d\\'ecembre\\fi\\ {\\oldstyle\\the\\year}}\n\n\\font\\fivegoth=eufm5 \\font\\sevengoth=eufm7 \\font\\tengoth=eufm10\n\n\\newfam\\gothfam \\scriptscriptfont\\gothfam=\\fivegoth\n\\textfont\\gothfam=\\tengoth \\scriptfont\\gothfam=\\sevengoth\n\\def\\fam\\gothfam\\tengoth{\\fam\\gothfam\\tengoth}\n\n\\def{\\goth p}} \\def\\aa{{\\goth a}} \\def\\bb{{\\goth b}{{\\fam\\gothfam\\tengoth p}} \\def\\aa{{\\fam\\gothfam\\tengoth a}} \\def\\bb{{\\fam\\gothfam\\tengoth b}}\n\\def{\\goth c}} \\def\\qq{{\\goth q}} \\def\\PP{{\\goth P}{{\\fam\\gothfam\\tengoth c}} \\def\\qq{{\\fam\\gothfam\\tengoth q}} \\def\\PP{{\\fam\\gothfam\\tengoth P}}\n\n\n\\def\\noindent {\\it Proof. }{\\noindent {\\it Proof. }}\n\n\\def\\vbox{\\hrule\\hbox{\\vrule height 1 ex\\kern 1 ex\\vrule}\\hrule}{\\vbox{\\hrule\\hbox{\\vrule height 1 ex\\kern 1 ex\\vrule}\\hrule}}\n\\def\\hfill \\smallsquare\\vskip 3mm{\\hfill \\vbox{\\hrule\\hbox{\\vrule height 1 ex\\kern 1 ex\\vrule}\\hrule}\\vskip 3mm}\n\n\n\n\\def\\bf N{\\bf N}\n\\def\\bf Q{\\bf Q}\n\\def\\bf R{\\bf R}\n\\def\\bf N{\\bf N}\n\n\n\n\n\n\n\\centerline{}\n\n\\vskip 4mm\n\n\n\n\n\\centerline{\n{\\bf On the complexity of a putative counterexample}}\n\\smallskip\n\\centerline{\\bf to the $p$-adic Littlewood conjecture}\n\n\n\\vskip 8mm\n\\centerline{Dmitry B{\\sevenrm ADZIAHIN},\nYann B{\\sevenrm UGEAUD\\footnote{}{\\rm\n2000 {\\it Mathematics Subject Classification : } 11J04; 11J61, 11J83, 37A35, 37A45, 37D40.\nKeywords: Diophantine approximation, Littlewood conjecture, complexity, \ncontinued fractions, measure rigidity.}},\nManfred E{\\sevenrm INSIEDLER}\\footnote{}{\\rm\nD.B. was supported by grant EP\/L005204\/1 from EPSRC.},\n\\& Dmitry K{\\sevenrm LEINBOCK}\\footnote{}{\\rm\nD.K. was supported in part by NSF grant DMS-1101320.}\n}\n\n\n\n{\\narrower\\narrower\n\\vskip 12mm\n\n\n\\proclaim Abstract. {\nLet $\\Vert \\cdot \\Vert$ denote the distance to the nearest integer\nand, for a prime number $p$, let $| \\cdot |_p$ denote the $p$-adic absolute\nvalue. In 2004, de Mathan and Teuli\\'e asked whether\n$\\inf_{q \\ge 1} \\, q \\cdot \\Vert q \\alpha \\Vert \\cdot \\vert q \\vert_p = 0$\nholds for every badly approximable real number $\\alpha$ and every\nprime number $p$. Among other results, we establish that, if the\ncomplexity of the sequence of partial quotients of a real number\n$\\alpha$ grows too rapidly or too slowly, then their conjecture is true\nfor the pair $(\\alpha, p)$ with $p$ an arbitrary prime. }\n\n}\n\n\n\\vskip 11mm\n\n\n\\centerline{\\bf 1. Introduction}\n\n\\vskip 6mm\n\n\nA famous open problem in simultaneous\nDiophantine approximation is\nthe Littlewood conjecture which claims that,\nfor every given pair $(\\alpha, \\beta)$ of real numbers, we have\n$$\n\\inf_{q \\ge 1} \\, q \\cdot \\Vert q \\alpha \\Vert \\cdot\n\\Vert q \\beta \\Vert = 0, \\eqno (1.1)\n$$\nwhere $\\Vert \\cdot \\Vert$ denotes the distance to the nearest integer.\nThe first significant contribution on this question\ngoes back to Cassels and Swinnerton-Dyer \\cite{CaSw} who showed that\n(1.1) holds when $\\alpha$ and $\\beta$ belong to the same cubic field.\nDespite some recent remarkable progress\n\\cite{PoVe,EKL} the Littlewood conjecture remains an open problem.\n\n\n\nLet ${\\cal D}=(d_{k} )_{k \\ge 1} $ be a sequence of integers greater\nthan or equal to $2$. Set $e_0 = 1$ and, for any $n \\ge 1$,\n$$\ne_{n} = \\prod _{1 \\le k \\leq n} d_{k}.\n$$\nFor an integer $q$, set\n$$\nw_{{\\cal D}} (q)=\\sup \\{n \\ge 0 : q\\in e_{n} {\\bf Z}\\}\n$$\nand\n$$\n\\vert q\\vert _{{\\cal D}} =1\/e_{w_{{\\cal D}} (q)} =\\inf\n\\{1\/e_{n} : q\\in e_{n} {\\bf Z}\\}.\n$$\nWhen $ {\\cal D}$ is the constant sequence equal to $ p$, where $ p$ is a\nprime number, then $ \\vert \\cdot \\vert _{{\\cal D}}$ is the usual\n$ p$-adic value $| \\cdot |_p$, normalized by $|p|_p = p^{-1}$.\nIn analogy with the Littlewood conjecture,\nde Mathan and Teuli\\'e \\cite{BdMTe} proposed in 2004 the following conjecture.\n\n\\proclaim Mixed Littlewood Conjecture. For every real number\n$\\alpha$ and every sequence ${\\cal D}$ of integers greater than or\nequal to $2$, we have\n$$\n\\inf_{q \\ge 1} \\, q \\cdot \\Vert q\\alpha \\Vert\n\\cdot \\vert q \\vert _{\\cal D} =0 \\eqno (1.2)\n$$\nholds for every real number $ \\alpha $.\n\n\nObviously, (1.2) holds if $\\alpha$ is rational or\nhas unbounded partial quotients. Thus, we only consider the case\nwhen $\\alpha$ is an element of the set ${\\hbox{B{\\sevenrm {AD}}}} $ of badly approximable\nnumbers, where\n$$\n{\\hbox{B{\\sevenrm {AD}}}} = \\{ \\alpha \\in {\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w} : \\inf_{q \\ge 1} \\,\nq \\cdot \\Vert q \\alpha \\Vert > 0\\}.\n$$\nDe Mathan and Teuli\\'e proved that (1.2) and even the stronger\nstatement\n$$\n\\liminf_{q \\to + \\infty} \\, q \\cdot \\log q \\cdot \\Vert q\\alpha \\Vert\n\\cdot \\vert q \\vert _{\\cal D} < + \\infty \\eqno (1.3)\n$$\nholds for every quadratic irrational $\\alpha$ when the sequence\n${\\cal D}$ is bounded.\n\nWe highlight the particular case when\n${\\cal D}$ is the constant sequence equal to a prime number.\n\n\\proclaim $p$-adic Littlewood Conjecture. For every real number\n$\\alpha$ and every prime number $p$, we have\n$$\n\\inf_{q \\ge 1} \\, q \\cdot \\Vert q \\alpha \\Vert \\cdot\n\\vert q \\vert_p = 0. \\eqno (1.4)\n$$\n\n\nEinsiedler and Kleinbock \\cite{EiKl07} established that, for every given prime number $p$,\nthe set of real numbers $\\alpha$ such that the pair $(\\alpha, p)$\ndoes not satisfy (1.4) has zero Hausdorff dimension.\nThey also explained how to modify their proof to get an analogous result\nwhen ${\\cal D}$ is the constant sequence equal to $d \\ge 2$ (not necessarily prime).\n\n\nIn an opposite direction, by means of a subtle\nCantor-type construction, Badziahin and Velani \\cite{BaVe11}\nestablished that, for every sequence ${\\cal D}$ of integers greater\nthan or equal to $2$, the set of real numbers $\\alpha$ such that\n$$\n\\inf_{q \\ge 3} \\, q \\cdot \\log q \\cdot \\log \\log q \\cdot \\Vert q\\alpha \\Vert\n\\cdot \\vert q \\vert _{\\cal D} > 0\n$$\nhas full Hausdorff dimension. Moreover,\nthey showed that, for ${\\cal D} = (2^{2^n})_{n \\ge 1}$,\nthe set of real numbers $\\alpha$ such that\n$$\n\\inf_{q \\ge 16} \\, q \\cdot \\log \\log q \\cdot \\log \\log \\log q \\cdot \\Vert q\\alpha \\Vert\n\\cdot \\vert q \\vert _{\\cal D} > 0\n$$\nhas full Hausdorff dimension.\n\nRegarding explicit examples of real numbers $\\alpha$\nsatisfying (1.4), it was proved in \\cite{BDM} that, if the sequence of partial quotients\nof the real number $\\alpha$ contains arbitrarily long concatenations\nof a given finite block, then the pair $(\\alpha, p)$ satisfies (1.4) for any prime number $p$.\nA precise statement is as follows.\n\n\n\\proclaim Theorem BDM.\nLet $\\alpha = [a_0; a_1, a_2, \\ldots ]$ be in ${\\hbox{B{\\sevenrm {AD}}}} $.\nLet $T \\ge 1$ be an integer and $b_1, \\ldots, b_T$\nbe positive integers.\nIf there exist two sequences $(m_k)_{k \\ge 1}$ and\n$(h_k)_{k \\ge 1}$ of positive integers with $(h_k)_{k \\ge 1}$\nbeing unbounded and\n$$\na_{m_k +j + n T} = b_j, \\quad\n\\hbox{for every $j=1, \\ldots, T$ and every $n=0, \\ldots, h_k-1$},\n$$\nthen the pair $(\\alpha, p)$ satisfies (1.4) for any prime number $p$.\n\nThe main purposes of the present note is to give new\ncombinatorial conditions ensuring that a real number satisfies\nthe $p$-adic (Theorem 2.1) and the mixed (Corollary 2.4) Littlewood conjectures\nand to study the complexity of the continued fraction expansion of a\nputative counterexample to (1.2) or (1.4); see Theorem 2.1 and\nCorollary 2.4 below. Furthermore, in Section 3 we make a connection\nbetween the mixed Littlewood conjecture and a problem on the\nevolution of the sequence of the Lagrange constants of the multiples\nof a given real number. Proofs of our results are given in Sections\n4 to 6.\n\nThroughout the paper,\nwe assume that the reader is familiar with the classical results from the theory\nof continued fractions.\n\n\n\n\\vskip 5mm\n\n\\goodbreak\n\n\\centerline{\\bf 2. New results on the mixed and\nthe $p$-adic Littlewood conjectures}\n\n\\vskip 6mm\n\nTo present our results, we adopt a point of view from\ncombinatorics on words. We look at the continued fraction\nexpansion of a given real number $\\alpha$\nas an infinite word.\n\nFor an infinite word ${\\bf w} = w_1 w_2 \\ldots$\nwritten on a finite alphabet\nand for an integer $n \\ge 1$, we denote\nby $p(n, {\\bf w})$ the number of distinct blocks of $n$ consecutive\nletters occurring in ${\\bf w}$, that is,\n$$\np(n, {\\bf w}) := {\\rm Card} \\{w_{\\ell+1} \\ldots w_{\\ell + n} : \\ell\n\\ge 0\\}.\n$$\nThe function $n \\mapsto p(n, {\\bf w})$ is called the {\\it complexity\nfunction} of ${\\bf w}$. For\na badly approximable real number $\\alpha =\n[a_0; a_1, a_2, \\ldots]$, we set\n$$\np(n, \\alpha) := p(n, a_1 a_2 \\ldots), \\quad n \\ge 1,\n$$\nand we call $n \\mapsto p(n, \\alpha)$ the {\\it complexity\nfunction} of $\\alpha$.\nObserve that, for all positive integers $n, n'$, we have\n$$\np(n + n', \\alpha) \\le p(n, \\alpha) \\cdot p(n', \\alpha),\n$$\nthus, the sequence $(\\log p(n, \\alpha))_{ n \\ge 1}$ is subadditive\nand the sequence $((\\log p(n, \\alpha))\/n)_{ n \\ge 1}$ converges.\n\nIn the present paper we show that if the real number $\\alpha$ is a\ncounterexample to the $p$-adic Littlewood conjecture, then its\ncomplexity function $n \\mapsto p(n, \\alpha)$ can neither increase\ntoo slowly nor too rapidly as $n$ tends to infinity.\n\n\\bigskip\n\\goodbreak\n\n\\centerline{\\bf 2.1. High complexity case}\n\n\n\\vskip 4mm\n\nFor a positive integer $K$, set\n$$\n{\\hbox{B{\\sevenrm {AD}}}}_K:= \\{\\alpha = [a_0; a_1, a_2\\ldots]\\;:\\; a_i\\le K, i \\ge 1 \\}\n$$\nand observe that the set of badly approximable numbers is the union over\nall positive integers $K$ of the sets ${\\hbox{B{\\sevenrm {AD}}}}_K$.\nIt immediately follows from the definition of the\ncomplexity function $n \\mapsto p(n, \\alpha)$ that, for\nevery $\\alpha$ in ${\\hbox{B{\\sevenrm {AD}}}}_K$ and every $n \\ge 1$, we have\n$$\np(n, \\alpha) \\le K^n.\n$$\nConsequently, the complexity function of the continued fraction of any number\n$\\alpha$ in ${\\hbox{B{\\sevenrm {AD}}}}$ grows at most exponentially fast. Our first result shows that a\nputative counterexample to the $p$-adic Littlewood conjecture must\nsatisfy a much more restrictive\ncondition.\n\n\\proclaim Theorem 2.1.\nLet $\\alpha$ be a real number satisfying\n$$\n\\lim_{n\\to\\infty}{\\log p(n,\\alpha) \\over n} >0. \\eqno (2.1)\n$$\nThen, for every prime number $p$, we have\n$$\n\\inf_{q \\ge 1} q\\cdot ||q\\alpha||\\cdot |q|_p=0.\n$$\n\nIn other words the complexity of the continued fraction expansion of every potential\ncounterexample to the $p$-adic Littlewood conjecture must grow subexponentially.\n\nOur proof relies on a~$p$-adic generalisation of the measure classification\nresult in~\\cite{Lindenstrauss} (provided by~\\cite{EinLin}), the connection\nbetween such dynamical results and the Diophantine approximation problem\nas was used before in~\\cite{EKL,EiKl07}, and the observation that one counterexample\nactually gives rise to many more counterexamples (see Proposition 4.1).\n\n\\bigskip\n\n\\centerline{\\bf 2.2. Low complexity case}\n\n\\vskip 4mm\n\nA well-known result of Morse and Hedlund \\cite{MoHe38,MoHe40}\nasserts that $p(n, {\\bf w}) \\ge n + 1$ for $n \\ge 1$, unless ${\\bf\nw}$ is ultimately periodic (in which case there exists a constant\n$C$ such that $p(n, {\\bf w}) \\le C$ for $n \\ge 1$). Infinite words\n${\\bf w}$ satisfying $p(n, {\\bf w}) = n + 1$ for every $n \\ge 1$ do\nexist and are called {\\it Sturmian words}. In the present paper we show\nthat if $\\alpha$ is a counterexample to the $p$-adic (or, even, to the mixed)\nLittlewood conjecture, then the lower bound for the\ncomplexity function of $\\alpha$ must be stronger than this estimate.\nBefore stating our result we give a classical definition (see e.g. \\cite{AlSh}).\n\n\\proclaim Definition 2.2.\nAn infinite word ${\\bf w}$ is\nrecurrent if every finite block occurring in ${\\bf w}$ occurs\ninfinitely often.\n\nClassical examples of recurrent infinite words include periodic words, Sturmian\nwords, the Thue--Morse word, etc.\n\n\n\n\\proclaim Theorem 2.3.\nLet $(a_k)_{k \\ge 1}$ be a sequence of\npositive integers. If there exists an integer $m \\ge 0$\nsuch that the infinite word $a_{m+1} a_{m+2} \\ldots $ is recurrent,\nthen, for every sequence ${\\cal D}$ of integers greater\nthan or equal to $2$, the real number $\\alpha := [0; a_1, a_2, \\ldots ]$\nsatisfies\n$$\n\\inf_{q \\ge 1} \\, q \\cdot \\Vert q\\alpha \\Vert\n\\cdot \\vert q \\vert _{\\cal D} =0.\n$$\n\nAs a particular case, Theorem 2.3 asserts that (1.2) holds for every\nquadratic number $\\alpha$ and every sequence ${\\cal D}$ of integers\ngreater than or equal to $2$, including unbounded sequences (unlike\nin \\cite{BdMTe}, where ${\\cal D}$ is assumed to be bounded). Unlike\nin \\cite{BdMTe}, our proof does not use $p$-adic analysis.\n\n\n\nTheorem 2.3 implies a non-trivial lower bound for the complexity\nfunction of the continued fraction expansion of a putative\ncounterexample to (1.2).\n\n\\proclaim Corollary 2.4. Let ${\\cal D}$ be a sequence of integers\ngreater than or equal to $2$ and $\\alpha$ be a real number such that\nthe pair $(\\alpha, {\\cal D})$ is a counterexample to the mixed\nLittlewood conjecture (i.e., does not satisfy (1.2)). Then, the\ncomplexity function of $\\alpha$ satisfies\n$$\n\\lim_{n \\to + \\infty} \\, p(n, \\alpha) - n = + \\infty.\n$$\n\n\nThe next corollary highlights a special family of infinite recurrent words.\nA finite word $w_1 \\ldots w_n$ is called a {\\it palindrome}\nif $w_{n+1-i} = w_i$ for $i=1, \\ldots , n$.\n\n\\proclaim Corollary 2.5.\nLet $(a_k)_{k \\ge 1}$ be a sequence of\npositive integers. If there exists an increasing sequence $(n_j)_{j\n\\ge 1}$ of positive integers such that $a_1 \\ldots a_{n_j}$ is a\npalindrome for $j \\ge 1$, then, for every sequence ${\\cal D}$ of\nintegers greater than or equal to $2$, the real number $\\alpha :=\n[0; a_1, a_2, \\ldots ]$ satisfies\n$$\n\\inf_{q \\ge 1} \\, q \\cdot \\Vert q\\alpha \\Vert\n\\cdot \\vert q \\vert _{\\cal D} =0.\n$$\n\n\nAs shown in Section 6, our approach allows us to give an alternative\nproof to (1.3) when $\\alpha$ is quadratic irrational and ${\\cal D}$\nis bounded. Furthermore, we are able to quantify Theorem 2.3 for a\nspecial class of recurrent words.\n\n\n\\proclaim Definition 2.6.\nWe say that an infinite word ${\\bf w}$ is linearly\nrecurrent if there exists $C > 1$ such that the\ndistance between two consecutive occurrences of any finite block $W$\noccurring in ${\\bf w}$ is bounded by $C$ times the length of $W$.\n\n\nWe obtain the following quantitative result.\n\n\n\\proclaim Theorem 2.7.\nLet $(a_k)_{k \\ge 1}$ be a bounded sequence of positive integers.\nIf there exists an integer $m \\ge 0$\nsuch that the infinite word $a_{m+1} a_{m+2} \\ldots $ is linearly recurrent,\nthen, for every sequence ${\\cal D}$ of integers greater\nthan or equal to $2$, the real number $\\alpha := [0; a_1, a_2, \\ldots ]$\nsatisfies\n$$\n\\liminf_{q \\to + \\infty} \\, q \\cdot (\\log \\log q)^{1\/2} \\cdot \\Vert q\\alpha \\Vert\n\\cdot \\vert q \\vert _{\\cal D} < + \\infty.\n$$\n\n\\bigskip\n\n\\goodbreak\n\n\\centerline{\\bf 2.3. Comparison with the Littlewood conjecture}\n\n\\vskip 4mm\n\nAccording to Section 5 of \\cite{BdMTe}, the initial motivation of the introduction of the mixed\nLittlewood conjecture was the study of a problem quite close to the \nLittlewood conjecture, but seemingly a little simpler,\nwith the hope to find new ideas suggesting a possible approach towards the\nresolution of the Littlewood conjecture itself.\n\nWe are not aware of any relationship between both conjectures. For instance, a real\nnumber $\\alpha$ being given, we do \nnot know any connection between the two statements \n`(1.2) holds for every sequence ${\\cal D}$' and \n`(1.1) holds for every real number $\\beta$'.\n\nThe interested reader is directed to \\cite{Bu14} for a survey of \nrecent results and developments on and around the Littlewood conjecture\nand its mixed analogue. He will notice that the state-of-the-art regarding the\nLittlewood and the $p$-adic Littlewood conjectures is essentially, but\nnot exactly, the same.\n\nFor instance, Theorem 5 in \\cite{Lin10} asserts that for every real number $\\alpha$\nwith (2.1), we have\n$$\n\\inf_{q \\ge 1} \\, q \\cdot \\| q \\alpha \\| \\cdot\n\\| q \\beta \\| = 0,\n$$\nfor every real number $\\beta$. This is the exact analogue to Theorem 2.1 above. \nHowever, the low complexity case remains very mysterious for the\nLittlewood conjecture, since we even do not know whether or not it holds\nfor the pair $(\\sqrt{2}, \\sqrt{3})$.\n\n\n\\vskip 5mm\n\n\\goodbreak\n\n\\centerline{\\bf 3. On the Lagrange constants of the multiples of a real number}\n\n\\vskip 6mm\n\nOur main motivation was the study of the $p$-adic and the mixed Littlewood\nconjectures. However, the proofs of Theorems 2.3 and 2.8\nactually give us much stronger results on the behaviour of the Lagrange\nconstants of the multiples of certain real numbers.\n\n\\proclaim Definition 3.1.\nThe Lagrange constant $c(\\alpha)$\nof an irrational real number $\\alpha$ is the quantity\n$$\nc(\\alpha) := \\liminf_{q \\to + \\infty} \\, q \\cdot ||q \\alpha||.\n$$\n\nClearly, $\\alpha$ is in ${\\hbox{B{\\sevenrm {AD}}}}$ if and only if $c(\\alpha) > 0$. A\nclassical theorem of Hurwitz (see \\cite{Per,BuLiv}) asserts that\n$c(\\alpha) \\le 1\/ \\sqrt{5}$ for every irrational real number\n$\\alpha$.\n\nFor any positive\ninteger $n$ and any badly approximable number $\\alpha$ we have\n$$\n{c(\\alpha) \\over n} \\le c(n \\alpha ) \\le n c(\\alpha). \\eqno (3.1)\n$$\nTo see this, note that\n$$\n\\Bigl| n \\alpha - {np \\over q} \\Bigr| = n \\Bigl| \\alpha - {p \\over q} \\Bigr|\n$$\nand\n$$\n\\Bigl| \\alpha - {p \\over n q} \\Bigr| = {1 \\over n} \\, \\Bigl| n \\alpha - {n p \\over n q} \\Bigr|.\n$$\n\nThe first general result on the behaviour of the sequence $(c(n \\alpha))_{n \\ge 1}$\nis Theorem 1.11 of Einsiedler, Fishman, and Shapira \\cite{EFS}, reproduced below.\n\n\\proclaim Theorem EFS.\nEvery badly approximable real number $\\alpha$ satisfies\n$$\n\\inf_{n \\ge 1} \\, c(n \\alpha) = 0.\n$$\n\nTheorem EFS motivates the following question.\n\n\\proclaim Problem 3.2.\nProve or disprove that every badly approximable real number $\\alpha$ satisfy\n$$\n\\lim_{n \\to + \\infty} \\, c(n \\alpha) = 0. \\eqno (3.2)\n$$\n\nThere is a clear connection between Problem 3.2 and the mixed\nLittlewood conjecture. Indeed, if $\\alpha$ satisfies (3.2) and\nif ${\\cal D}$ is as in Section 1, then, keeping the notation from this\nsection, for every $\\eps > 0$, there exists a positive integer $n$\nsuch that $c(e_n \\alpha) < \\eps$. Consequently, there are\narbitrarily large integers $q$ with the property that\n$$\nq \\cdot || q e_n \\alpha || < \\eps\n$$\nthus,\n$$\nq e_n \\cdot || q e_n \\alpha || \\cdot |q e_n|_{\\cal D} < \\eps ,\n$$\nsince $|q e_n|_{\\cal D} \\le 1\/e_n$. This proves that (1.2)\nholds for the pair $(\\alpha, {\\cal D})$.\n\nOur proof of Theorem 2.3 actually gives the following\nstronger result.\n\n\\proclaim Theorem 3.3.\nLet $(a_k)_{k \\ge 1}$ be a sequence of positive integers.\nIf there exists an integer $m \\ge 0$\nsuch that the infinite word $a_{m+1} a_{m+2} \\ldots $ is recurrent,\nthen the real number $\\alpha := [0; a_1, a_2, \\ldots ]$ satisfies (3.2) and, moreover,\n$$\nc(n \\alpha) \\le {8 q_m^2 \\over n}, \\quad \\hbox{for $n \\ge 1$,}\n$$\nwhere $q_m$ denotes the denominator of the rational\nnumber $[0; a_1, \\ldots , a_m]$.\n\n\n\nIn view of the left-hand inequality of (3.1), the conclusion\nof Theorem 3.3 is nearly best possible. \n\nUsing the same arguments as for the proof of Corollary 2.4, we\nestablish that the complexity function of a real number which does\nnot satisfy (3.2) cannot be too small.\n\n\\proclaim Corollary 3.4. \nLet $\\alpha$ be a real number such that\n$$\n\\sup_{n \\ge 1} \\, n \\, c(n \\alpha) = + \\infty.\n$$\nThen, the complexity function of $\\alpha$ satisfies\n$$\n\\lim_{n \\to + \\infty} \\, p(n, \\alpha) - n = + \\infty.\n$$\n\n\n\\vskip 5mm\n\n\\goodbreak\n\n\\centerline{\\bf 4. High complexity case}\n\n\\vskip 5mm\n\nWe follow the interpretation of the $p$-adic Littlewood conjecture used\nby Einsiedler and Kleinbock in~\\cite{EiKl07} and consider the\nfollowing more general problem:\n\n\\proclaim Generalized $p$-adic Littlewood Conjecture.\nFor every prime\nnumber $p$ and for every pair $(u,v)\\in {\\bf R_{>0}}\\times {{\\bf Q}_p}$ we have\n$$\n\\inf_{a\\in{\\bf N}} \\def\\L{{\\bf L},b\\in {\\bf N}} \\def\\L{{\\bf L}\\cup\\{0\\}} \\max\\{|a|,|b|\\}\\cdot |au-b|\\cdot |av-b|_p\n=0. \\eqno(4.1)\n$$\n\nClearly~$\\alpha$ satisfies the $p$-adic Littlewood conjecture\n(i.e.~(1.4)) if and only if~$-\\alpha$ satisfies the $p$-adic\nLittlewood conjecture. For that reason we restrict our attention to\npositive numbers. Moreover, one can check (see for\nexample~\\cite{EiKl07}, a discussion after Theorem 1.2) that if\n$\\alpha$ is a counterexample to the $p$-adic Littlewood conjecture\nthen $(\\alpha^{-1},0)$ is a counterexample to the above generalized\n$p$-adic Littlewood conjecture. The next proposition goes further\nand shows that one counterexample $\\alpha$ to the $p$-adic\nLittlewood conjecture provides a countable collection of\ncounterexamples to the generalized $p$-adic Littlewood conjecture.\n\n\n\\proclaim Proposition 4.1. Let $p$ be a prime number and $\\alpha>0$\nan irrational number. Let $\\eps$ be in $(0, 1\/2]$ and assume that\n$$\n\\inf_{q\\ge 1} q\\cdot ||q\\alpha||\\cdot |q|_p>\\eps.\n$$\nThen, we have\n$$\n\\inf_{a\\in{\\bf N}} \\def\\L{{\\bf L},b\\in{\\bf N}} \\def\\L{{\\bf L}\\cup\\{0\\}}\\;\\; \\max\\{a,b\\} \\cdot \\left|a\\cdot\n{||q_n\\alpha||\\over ||q_{n-1}\\alpha||}-b\\right|\\cdot \\left|a\\cdot\n\\left({q_n \\over q_{n-1}}\\right)+b\\right|_p>{\\eps^2 \\over 4},\n\\eqno(4.2)\n$$\nwhere $(q_k)_{k \\ge 1}$ is the sequence of the\ndenominators of the convergents to $\\alpha$.\n\nNote that, writing $\\alpha = [a_0; a_1, a_2, \\ldots]$, we have\n$$\n{q_n \\over q_{n-1}} = [0;a_n,a_{n-1},\\ldots,a_1]\\quad{\\rm and}\\quad\n{||q_n\\alpha||\\over ||q_{n-1}\\alpha||} = [0;a_{n+1},a_{n+2},\\ldots]\\in(0,1),\n$$\nfor every $n \\ge 1$.\n \n \n \n\\noi {\\it Proof of Proposition~4.1.} We assume that\n$||q\\alpha||\\cdot |q|_p>\\eps \/ q$ for every integer $q \\ge 1$. We\nuse the classical estimate from the theory of continued fractions\n$$\n||q_n\\alpha||0$ the set of $(u,v)\\in [0,1]\\times {\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p$ which satisfy\n$$\n\\inf_{a\\in {\\bf N}} \\def\\L{{\\bf L},b\\in{\\bf N}} \\def\\L{{\\bf L}\\cup\\{0\\}} \\max\\{a,b\\}\\cdot |au-b|\\cdot |av-b|_p \\ge\n\\eps.\\eqno(4.4)\n$$\nhas box dimension zero.\n\nThe proof of Theorem 4.2 is postponed to Section 5.\n\nRoughly speaking, this theorem together with Proposition~4.1 implies that for every\ncounterexample $\\alpha = [0; a_1, a_2, \\ldots ]$ to the $p$-adic Littlewood conjecture the set\n$$\n\\{[0;a_m,a_{m+1},\\ldots]\\;:\\;m \\ge 1 \\}\n$$\nhas box dimension zero. \nLet us now show that this in turn implies the\nstatement of Theorem~2.1.\n\n\n\\vskip 4mm\n\n\\noi {\\it Proof of Theorem~2.1.}\nWe will prove the contrapositive of the theorem.\nSo assume that~$\\alpha$ is a counterexample to (1.4).\nBy the homogeneity of (1.4)\n we may assume that~$\\alpha=[a_0;a_1,\\ldots]$ is positive.\nBy Proposition~4.1 this leads\nto a countable collection\n$$\n B=\\Bigl\\{ (\\alpha_n,\\beta_n)\n = \\Bigl( [0;a_{n+1},a_{n+2},\\ldots],{q_n \\over q_{n-1}}\\Bigr)\n : n\\geq 1\\Bigr\\}\n$$\nof pairs in~$[0,1]\\times{\\bf {Q}}_p$ that all satisfy (4.2). By Theorem~4.2,\nthe set\n$$\nB\\cap [0,1]\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p=\\{(\\alpha_n,\\beta_n):n\\geq 1 \\enspace\n\\hbox{and~$|q_{n-1}|_p=1$}\\}\n$$\nhas box dimension zero.\n\nLet~$N\\geq 1$ be such that~$a_n\\in\\{1,\\ldots,N\\}$ for all~$n\\geq 1$.\nWe set~\n$$\nS'=\\{\\ell\\geq 1: |q_{\\ell-1}|_p=1\\}.\n$$\nLet~$\\delta>0$ be arbitrary and let~$\\pi_\\infty:[0,1]\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p\\to[0,1]$ denote the projection\nto the real coordinate.\nThen, the definition of box dimension shows that, for all sufficiently large~$n$, the set\n$$\nB'=\\{\\alpha_\\ell:\\ell\\in S'\\}=\\pi_\\infty(B\\cap [0,1]\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p)\n$$\ncan be covered by~$s_n\\leq e^{\\delta n}$ intervals~$I_1,\\ldots,I_{s_n}$\nof size~$(1+N)^{-2n}$.\n\nWe also define another disjoint collection of intervals.\nTo any word~$w = w_1 \\ldots w_n$ in $\\{1,\\ldots,N\\}^n$,\nwe associate the interval $[w]$ composed of the real\nnumbers in $(0, 1)$ whose first $n$ partial quotients are $w_1, \\ldots , w_n$. \nThe basic\nproperties of continued fractions show that the length of~$[w]$ is at most~$2^{-n+2}$\nand at least~$(1+N)^{-2n}$. It follows that a given interval~$I_j$ from the above list\ncan intersect at most two (neighbouring) intervals of the form~$[w]$ for~$w\\in\\{1,\\ldots,N\\}^n$.\nThis implies that\n$$\n{\\rm Card} \\{a_{\\ell+1}\\ldots a_{\\ell+n}:\\ell \\in S'\\} \\leq 2 s_n\\leq 2 e^{\\delta n}.\n$$\nTo remove the restriction~$\\ell\\in S'$ in the above counting, we note that~$\\ell\\notin S'$\nimplies~$\\ell+1\\in S'$ since~$q_{\\ell-1}$ and~$q_\\ell$ are coprime, by the properties\nof continued fractions. Therefore,\n$$\np(n,\\alpha)= {\\rm Card} \\{a_{\\ell+1}\\ldots a_{\\ell+n}:\\ell \\geq 0\\}\n\\leq 2 s_n+2 N s_{n-1}\\leq 2(1+N) e^{\\delta n}.\n$$\nAs~$\\delta>0$ was arbitrary, the theorem follows.\n \\hfill \\smallsquare\\vskip 3mm\n\n\n\n\n\n\n\\vskip 4mm\n\n\\centerline{\\bf 5. Measure Rigidity and the Proof of Theorem 4.2.}\n\n\n\n\\vskip 4mm\n\n \nWe follow the strategy\noutlined in \\cite{EiKl07} (which in turn generalizes the\nargument from \\cite{EKL}). For this,\nwe set\n$$\nG={\\rm SL}_2({\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w})\\times {\\rm SL}_2({\\bf {Q}}_p), \\ \\Gamma={\\rm SL}_2({\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}[1\/p]),\n\\ X =G\/\\Gamma\\,,\\eqno(5.1)\n$$\nwhere ${\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}[1\/p]$ is embedded diagonally via~$a\\mapsto (a,a)$ in ${\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}\\times{\\bf {Q}}_p$.\n In other words, for $(A,B) \\in G$,\npoints $x=(A,B)\\Gamma\\in X$ are identified with\nunimodular \nlattices $(A,B){\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}[1\/p]$ in ${\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}^2\\times{\\bf {Q}}_p^2$ that are\ngenerated by the column vectors of $A$ and $B$.\n\nWe also set\n$$\n\\psi(t,n)=\\left (\\left(\\matrix{ e^{-t}&0\\cr 0&e^t }\\right),\n\\left(\\matrix{p^n&0\\cr 0&p^{-n }}\\right)\\right)\\eqno(5.2)\n$$\nfor~$(t,n)\\in{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}$, and define the cone\n$$\nC=\\{(t,n)\\mid n\\geq 0, e^t p^{-n}\\geq 1\\}\\,.\\eqno(5.3)\n$$\nFurthermore, for~$(u,v)\\in{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}\\times{\\bf {Q}}_p$,\nwe define the coset (which we will think of as a point)\n$$\nx_{u,v}=\\left (\\left(\\matrix{ 1&0\\cr u&1 }\\right),\n\\left(\\matrix{1&0\\cr v&1}\\right)\\right)\\Gamma.\n$$\n\nCompact subsets of $X$ can be characterized by the analogue\nof Mahler's compactness criterion\n(see \\cite{EiKl07}, Theorem\\ 2.1) so that a subset $K\\subset X$\nhas compact closure if and only if\nthere exists some~$\\delta>0$ so that $K\\subset K_\\delta$, with\n$$\n K_\\delta=\\left\\{ g\\Gamma\\in K : g{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}[1\/p]^2\\cap B_\\delta^{{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}^2\\times{\\bf {Q}}_p^2}=\\{0\\}\\right\\},\n$$\nwhere $B_\\delta$ denotes the ball of radius $\\delta$ centered at\nzero.\n\nIn \\cite{EiKl07}, Proposition\\ 2.2, a connection between unboundedness of the\ncone orbit\n$$\n\\psi(C)x_{u,v}=\\{\\psi(t,n)x_{u,v} : (t,n)\\in C\\}\n$$\nand (4.3) is given. However, we will need to show the following\nrefinement.\n\n\\proclaim Proposition 5.1.\nLet $(u,v)\\in \n(0,1)\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p$ and $0 < \\varepsilon<1$ be arbitrary. If $(u,v)$ satisfies (4.4),\nthen \n$\\psi(C) x_{u,v}\\subset K_\\delta$ for~$\\delta =(\\eps\/2)^{1\/3}$.\n\\vskip 4mm\n\n\nWe note that in \\cite{EiKl07} the converse of the above implication,\nin a slightly different form, has also been claimed (without a proof).\nBut the other direction is not clear and luckily is also not needed for the proof of our results (or\nthe results of~\\cite{EiKl07}).\n\\vskip 4mm\n\n\\noi {\\it Proof of Proposition~5.1.}\nTake $\\delta =(\\eps\/2)^{1\/3}$ and suppose that $\\psi(C) x_{u,v}$\nis not contained in $K_\\delta$; that is, there exists a pair $(t,n)$ with $n\\ge 0$\n and $e^tp^{-n}\\geq 1$ such that $\\psi(t,n)x_{u,v}{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}[1\/p]$\n contains a nonzero element in $B_\\delta^{{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}^2\\times{\\bf {Q}}_p^2}$.\n\n Clearly, $\\psi{(t,n)}x_{u,v}$ is generated by\n$$\n \\left(\\pmatrix {e^{-t}\\cr e^tu},\n \\pmatrix {p^{n}\\cr p^{-n}v}\\right){ \\rm and }\n \\left(\\pmatrix{ 0\\cr e^t},\n \\pmatrix{ 0\\cr p^{-n}}\\right).\n$$\n However, since $\\psi{(t,n)}x_{u,v}$ is a\n ${\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}[1\/{p}]$-module, the vectors\n$$\n \\left(\\pmatrix{ e^{-t}p^{-n}\\cr e^tp^{-n}u},\n \\pmatrix{ 1\\cr p^{-2n}v}\\right){ \\rm and }\n \\left(\\pmatrix{ 0\\cr e^tp^{-n}},\n \\pmatrix{ 0\\cr p^{-2n}}\\right)\n$$\n are also generators. Therefore, there exists some nonzero\n $(a,b)\\in{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}[1\/{p}]^2$ such that\n$$\n \\left(\\pmatrix{e^{-t}p^{-n}a\\cr e^tp^{-n}(au-b)},\n \\pmatrix{a\\cr p^{-2n}(av-b)}\\right)\\in{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}^2\\times{\\bf {Q}}_p^2\n$$\n is $\\delta$-small. In particular,\n$\n|a|_p$ is less than $\\delta$,\nwhich \nimplies that $a\\in{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}$. Since\n $n\\geq 0$ and $v\\in{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p$, the inequality\n$$\np^{2n}|av-b|_p = |p^{-2n}(av-b)|_p <\\delta \\eqno (5.4)\n$$\nshows that $b\\in{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}$ as well. \nAlso, since $|u| < 1$, the inequalities $t \\ge 0$,\n$|e^{-t}p^{-n}a| <\\delta$\nand\n$$\n|e^{t}p^{-n}(au-b)| <\\delta \\eqno (5.5)\n$$\nimply that\n$$\ne^{-t}p^{-n}\\max\\{ |a|,|b| \\} <2\\delta. \\eqno (5.6)\n$$\nBy taking the product of the inequalities (5.4), (5.5) and (5.6),\nwe arrive at\n$$\n\\max \\{ |a|,|b| \\} \\cdot|au-b|\\cdot|av-b|_p<2\\delta^3 = \\eps.\n$$\nAlso note that~$u>0$,~(5.5) and~$e^tp^{-n}\\geq 1$ imply that~$a$ and~$b$\n have the same sign (in the sense that~$ab\\geq 0$).\n Without loss of generality we may assume~$a,b\\geq 0$.\n Similarly,~$b=0$ implies~$a=0$ and contradicts our choice of~$(a,b)$.\n However,~$a\\geq 1$ and~$b\\geq 0$ contradicts (4.4).\nConsequently, $\\psi(C) x_{u,v}$\nis contained in $K_\\delta$.\n\\hfill \\smallsquare\\vskip 3mm\n\\vskip 4mm\n\nWe also need the following partial measure classification result.\n\n\\proclaim Theorem 5.2.\n The Haar measure is the only~$\\psi$-invariant and ergodic probability measure $\\mu$\n on~$X$ for which some~$(t,n)\\in{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}$ has positive entropy $h_\\mu(\\psi(t,n))>0$.\n \\vskip 4mm\n\n\nTheorem 5.2 follows from Theorem 1.3 of \\cite{EinLin}, recalled below.\nWe write~$\\infty$ for the archimedean place of~${\\bf {Q}}$ and~${\\bf {Q}}_\\infty={\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}$. Moreover,\nfor a finite set~$S$ of places we define~${\\bf {Q}}_S=\\prod_{v\\in S}{\\bf {Q}}_v$\nfor the corresponding product of the local fields.\n\n\\proclaim Theorem EL.\n Let $\\bf G$ be a ${\\bf {Q}}$-almost simple linear algebraic group, let~$S$\n be a finite set of places containing the archimedean place~$\\infty$,\n let $\\Gamma0$ for some $a\\in A$.\n Then there is a finite index subgroup $L0$, \nlet $\\delta=\\varepsilon^{1\/3}$ and $Y=K_\\delta\\subset X$.\nAlso pick~$t > 0$ such that $(t,1)\\in C$.\n Note that~$\\{x_{u,v} : (u,v)\\in{\\bf R}} \\def\\M{{\\bf M}} \\def{\\cal {K}}} \\def\\cC{{\\cal {C}}} \\def\\cS{{\\cal {S}}{{\\bf B}} \\def\\tw{{\\tilde w}\\times{\\bf {Q}}_p\\}$\n is the unstable\n horospherical subgroup\n for~$\\psi(t,1)$.\n By Theorem 5.2\n and since~$Y$ is a proper closed subset of~$X$,\n there is no~$\\psi$-invariant and ergodic probability measure supported on~$Y$,\n which is precisely the assumption \n of Proposition EK.\n By that result we obtain that the set of~$(u,v)\\in [0,1]\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p$\n with~$\\psi(C)x_{u,v}\\subset Y$ has box dimension zero.\n However, by Proposition 5.1, this implies\n that\n the set of $(u,v)\\in (0,1)\\times{\\bf Z}} \\def\\A{{\\bf A}} \\def\\tA{{{\\tilde\\A}}} \\def\\tN{{{\\tilde N}}_p$ satisfiying (4.4) has box dimension zero, \n and Theorem~4.2 follows.\n\\hfill \\smallsquare\\vskip 3mm\n\\vskip 4mm\n\n\n\n\n\\vskip 4mm\n\n\n\\centerline{\\bf 6. Low complexity case}\n\n\\bigskip\n\n\\centerline{\\bf 6.1. Auxiliary results}\n\n\\vskip 6mm\n\nWe begin with two classical lemmata on continued fractions, whose\nproofs can be found for example in Perron's book \\cite{Per}.\n\n\nFor positive integers $a_1, \\ldots, a_n$, denote\nby $K_n (a_1, \\ldots, a_n)$ the denominator of the rational number\n$[0; a_1, \\ldots, a_n]$. It is commonly called a {\\it continuant}.\n\n\n\\proclaim Lemma 6.1. For any positive integers $a_1, \\ldots, a_n$\nand any integer $k$ with $1 \\le k \\le n-1$, we have\n$$\nK_n (a_1, \\ldots, a_n) = K_n (a_n, \\ldots, a_1)\n$$\nand\n$$\n\\eqalign{\nK_k (a_1, \\ldots, a_k) \\cdot K_{n-k} (a_{k+1}, \\ldots, a_n)\n& \\le K_n (a_1, \\ldots , a_n) \\cr\n& \\le 2 \\, K_k (a_1, \\ldots, a_k) \\cdot K_{n-k} (a_{k+1}, \\ldots, a_n). \\cr}\n$$\n\n\n\\proclaim Lemma 6.2. Let $\\alpha = [0; a_1, a_2, \\ldots]$ and $\\beta\n= [0; b_1, b_2, \\ldots]$ be real numbers. Assume that there exists a\npositive integer $n$ such that $a_i = b_i$ for any $i=1, \\ldots, n$.\nWe then have $|\\alpha - \\beta| \\le K_n (a_1, \\ldots , a_n)^{-2}$.\n\n\n\n\n\nA homogeneous linear recurrence sequence with constant\ncoefficients ({\\it recurrence sequence} for short) is a\nsequence $(u_n)_{n \\ge 0}$ of complex numbers such that\n$$\nu_{n+d} = v_{d-1} u_{n+d-1} + v_{d-2} u_{n+d-2} +\n\\ldots + v_0 u_n \\quad (n \\ge 0),\n$$\nfor some complex numbers $v_0, v_1, \\ldots , v_{d-1}$\nwith $v_0 \\not=0$ and with initial values $u_0, \\ldots , u_{d-1}$\nnot all zero. The positive integer $d$ is called the\n{\\it order} of the recurrence.\n\n\n\n\\proclaim Lemma 6.3. Let $(u_n)_{n \\ge 1}$ be a recurrence sequence\nof order $d$ of rational integers. Then, for every prime number $p$\nand every positive integer $k$, the period of the sequence $(u_n)_{n\n\\ge 1}$ modulo $p^k$ is at most equal to $(p^d - 1) p^{k-1}$.\n\n\\noindent {\\it Proof. }\nSee Everest et al. \\cite{EPSW03}, page 47. \\hfill \\smallsquare\\vskip 3mm\n\n\n\n\\proclaim Lemma 6.4.\nLet $\\alpha = [a_0; a_1, \\ldots , a_{r-1}, b_0, b_1, \\ldots , b_{s-1}, b_0, \\ldots , b_{s-1}, \\ldots]$\nbe a quadratic irrational number and denote by $(p_n \/ q_n)_{n \\ge 0}$\nthe sequence of its convergents. Then, there exists an integer $t$ such that\n$$\nq_{n + 2s} - t q_{n+s} + (-1)^s q_n = 0\n$$\nfor $n \\ge r$. In particular, the sequence $(q_n)_{n \\ge 0}$ satisfies\na linear recurrence with constant integral coefficients.\n\n\\noindent {\\it Proof. }\nThis result is included in the proof of Theorem 1 in \\cite{LeSh93}. \\hfill \\smallsquare\\vskip 3mm\n\n\n\n\n\\vskip 5mm\n\n\\goodbreak\n\n\\centerline{\\bf 6.2. Proofs}\n\n\\vskip 6mm\n\n\\noi {\\it Preliminaries.}\n\nWithout any loss of generality, we consider real numbers in $(0,\n1)$. We associate to every real irrational number $\\alpha := [0;\na_1, a_2, \\ldots]$ the infinite word ${\\bf a} := a_1 a_2 \\ldots $\nformed by the sequence of the partial quotients of its fractional\npart. Set\n$$\np_{-1} = q_0 = 1, \\quad p_0 = q_{-1} = 0,\n$$\nand\n$$\n{p_n \\over q_n} = [0; a_1, \\ldots, a_n], \\quad\n\\hbox{for $n \\ge 1$.}\n$$\nBy the theory of\ncontinued fractions, we have\n$$\n{q_n \\over q_{n-1}} = [a_n; a_{n-1}, \\ldots , a_1].\n$$\nThis is one of the key tools of our proofs.\n\n\\bigskip\n\n\\noi {\\it Proof of Theorems 2.3 and 3.3.}\n\nAssume that the infinite word $a_{m+1} a_{m+2} \\ldots $ is recurrent.\nThen, there exists an increasing sequence of positive integers $(n_j)_{j \\ge 1}$\nsuch that\n$$\n\\hbox{$a_{m+1} a_{m+2} \\ldots a_{m+n_j}$\nis a suffix of $a_{m+1} a_{m+2} \\ldots a_{m+n_{j+1}}$, for $j \\ge 1$.}\n$$\nSay differently, there are finite words $V_1, V_2, \\ldots $ such that\n$$\n\\hbox{$a_{m+1} a_{m+2} \\ldots a_{m+n_{j+1}} =\nV_j a_{m+1} a_{m+2} \\ldots a_{m+n_{j}}$, for $j \\ge 1$.}\n$$\nActually, these properties are equivalent.\n\nLet $\\ell \\ge 2$ be an integer.\nLet $k \\ge \\ell^2 + 1$ be an integer.\nBy Dirichlet's {\\it Schubfachprinzip}, there exist integers $i, j$\nwith $1 \\le i < j \\le k$ such that\n$$\nq_{m + n_i} \\equiv q_{m + n_j} \\pmod \\ell , \\quad\nq_{m + n_i - 1} \\equiv q_{m + n_j - 1} \\pmod \\ell\n$$\nand $j$ is minimal with this property.\n\nSetting\n$$\nQ := |q_{m + n_i} q_{m+n_j-1} - q_{m+n_i -1} q_{m + n_j}|,\n$$\nwe observe that\n$$\n\\hbox{$\\ell$ divides $Q$,} \\eqno (6.1)\n$$\nand we derive from Lemma 6.2 that\n$$\n\\eqalign{\n0 < Q & = q_{m + n_i} q_{m + n_j} \\Bigl|\n{q_{m + n_j - 1} \\over q_{m + n_j}} - {q_{m + n_i -1} \\over q_{m + n_i}} \\Bigr| \\cr\n& \\le q_{m + n_i} q_{m + n_j} K(a_{m+n_i}, \\ldots , a_{m+1})^{-2}, \\cr}\n$$\nsince the $n_i$ first partial\nquotients of $q_{m + n_j - 1} \/ q_{m + n_j}$ and $q_{m + n_i -1}\/q_{m + n_i}$\nare the same, namely $a_{m+n_i}, \\ldots , a_{m+1}$. Furthermore, we have\n$$\n||Q \\alpha|| \\le || q_{m+n_i} (q_{m+n_j-1}\\alpha)|| + ||\nq_{m+n_i-1}(q_{m+n_j}\\alpha)|| \\le 2 q_{m + n_i} q_{m + n_j}^{-1}.\n$$\nUsing that\n$$\nq_{m + n_i} \\le 2 q_m K(a_{m+n_i}, \\ldots , a_{m+1}),\n$$\nby Lemma 6.1, we finally get\n$$\nQ \\cdot ||Q \\alpha|| \\le 8 q_m^2. \\eqno (6.2)\n$$\nIt then follows from (6.1) and (6.2) that\n$$\nQ \\cdot ||Q \\alpha || \\cdot |Q|_{\\ell} \\le 8 q_m^2 \\ell^{-1}, \\eqno (6.3)\n$$\nwhere $|Q|_{\\ell}$ is equal to $\\ell^{-a}$ if $\\ell^a$ divides $Q$ but $\\ell^{a+1}$\ndoes not.\nSince $\\ell$ can be an arbitrary prime power, this proves Theorem 2.3.\n\nOur proof shows that there are arbitrarily large integers $q$ such that\n$$\nq \\ell \\cdot ||q (\\ell \\alpha) || \\le 8 q_m^2,\n$$\nwhich implies that\n$$\nc(\\ell \\alpha) \\le { 8 q_m^2 \\over \\ell},\n$$\nand establishes Theorem 3.3. \\hfill \\smallsquare\\vskip 3mm\n\n\n\n\\bigskip\n\n\\noi {\\it Proof of Corollary 2.4.}\n\nLet ${\\bf a}$ be an infinite Sturmian word.\nWe first claim that every prefix of finite length of ${\\bf a}$ occurs\ninfinitely often in ${\\bf a}$. Indeed, otherwise, there would exist a positive\ninteger $n$, a finite word $W$ and an infinite word ${\\bf a}'$\nsuch that ${\\bf a} = W {\\bf a}'$ and $p(n, {\\bf a}') \\le n$, which would imply\nthat ${\\bf a'}$ is ultimately periodic, a contradiction with the assumption\nthat ${\\bf a}$ is Sturmian.\n\nLet ${\\bf a}$\nbe an infinite word on a finite alphabet ${\\cal A}$ such\nthat there are positive integers $k$ and $n_0$ with\n$$\np(n, {\\bf a}) = n + k, \\quad\n\\hbox{for $n \\ge n_0$}.\n$$\nThen, by a result of Cassaigne \\cite{Cassa98},\nthere exist finite words $W, W_0, W_1$\non ${\\cal A}$ and a Sturmian word\n${\\bf s}$ on $\\{0, 1\\}$ such that\n$$\n{\\bf a} = W \\phi({\\bf s}),\n$$\nwhere $\\phi({\\bf s})$ denotes the infinite word obtained by\nreplacing in ${\\bf s}$ every $0$ by $W_0$ and every $1$ by $W_1$. We\nconclude by applying Theorem 2.3 with $m$ being the length of $W$.\n\\hfill \\smallsquare\\vskip 3mm\n\n\n\\bigskip\n\n\\noi {\\it Proof of Corollary 2.5.}\n\nIt is sufficient to note that, if $a_1 \\ldots a_n$ and $a_1 \\ldots\na_{n'}$ are palindromes with $n' > 2 n$, then $a_{n'-n+1} \\ldots\na_{n'} = a_n \\ldots a_1 = a_1 \\ldots a_n$. The corollary then\nfollows from Theorem 2.3 applied with $m=0$. \\hfill \\smallsquare\\vskip 3mm\n\n\\bigskip\n\n\\noi {\\it Proof of (1.3) when $\\alpha$ is a quadratic irrationality\nand ${\\cal D}$ is bounded.}\n\nSince ${\\cal D}$ is bounded, every product $e_n = \\prod_{1\\le k\\le n} d_k$\nis divisible by a finite collection of prime numbers. Let $p_1,\n\\ldots , p_h$ be these primes and denote by $S$ the set of integers\nwhich are divisible only by primes from $\\{p_1, \\ldots , p_h\\}$. Let\n$\\alpha$ be a quadratic real number. By Lemma~6.4, the sequence\n$(q_n)_{n \\ge 0}$ of denominators of convergents to $\\alpha$ is\neventually a recurrence sequence of positive integers. \nBy Lemma 6.3, there exists a positive integer $C_1$\nsuch that, for $i=1 , \\ldots , h$ and $v \\ge 1$,\nthe sequence $(q_n)_{n \\ge 0}$ is eventually periodic modulo\n$p_i^v$, with period length at most equal to $C_1 p_i^v$.\n\n\nConsequently, there exists a positive integer $C_2$ such that,\nfor every positive integer $\\ell$\nin $S$, the sequence $(q_n)_{n \\ge 1}$ modulo $\\ell$\nis eventually periodic of period at most $C_2 \\ell$.\n\n\nWe need to slightly modify the proof of Theorem 2.3. Take $\\ell =\ne_n\\in S$. Denote by $m$ the length of the preperiod of $(q_n)_{n\n\\ge 1}$ and by $d$ the length of the period of $(q_n)_{n \\ge 1}$\nmodulo $\\ell$. Observe that\n$$\nq_{m} \\equiv q_{m + d} \\pmod \\ell , \\quad\nq_{m + 1} \\equiv q_{m + d + 1} \\pmod \\ell .\n$$\nWe then set\n$$\nQ := |q_m q_{m+ d + 1} - q_{m + 1} q_{m + d}|\n$$\nand proceed exactly as in the proof of Theorem 2.3 to get that\n$$\nQ \\cdot ||Q \\alpha|| \\le 2 q_{m+1}^2.\n$$\nNoticing that $|Q|_{\\cal D}\\le \\ell^{-1}$ and $Q \\le C_3^{\\ell}$, for\nsome integer $C_3$ depending only on $p_1, \\ldots , p_h$, this\nestablishes (1.3). \\hfill \\smallsquare\\vskip 3mm\n\n\n\n\\bigskip\n\n\\goodbreak\n\n\\noi {\\it Proof of Theorem 2.7.}\n\nWe keep the notation of the proof of Theorem 2.3.\nBy assumption, we can select a suitable sequence $(n_j)_{j \\ge 1}$\nwith the property that $n_j < C_1^j$ for some integer $C_1 \\ge 2$ and every $j \\ge 1$.\nThen, there are positive constants $C_2, C_3$, depending only\non $C_1$, such that\n$$\nQ \\le C_2^{n_i + n_j},\n$$\nthus,\n$$\n\\log \\log Q \\le C_3 \\ell^2,\n$$\nsince $i$ and $j$ are at most equal to $\\ell^2+1$.\nCombined with (6.3), this proves the theorem. \\hfill \\smallsquare\\vskip 3mm\n\n\\bigskip\n\n \n\n\\vskip 7mm\n\n\\vfill\\eject\n\n\\centerline{\\bf References}\n\n\\vskip 7mm\n\n\\beginthebibliography{999}\n\n\n\\bibitem{AlSh}\nJ.-P. Allouche and J. Shallit,\nAutomatic Sequences: Theory, Applications, Generalizations.\nCambridge University Press, Cambridge, 2003.\n\n\\bibitem{BaVe11}\nD. Badziahin and S. Velani,\n{\\it Multiplicatively badly approximable numbers\nand the mixed Littlewood conjecture},\nAdv. Math. 228 (2011), 2766--2796.\n\n\n\\bibitem{BuLiv}\nY. Bugeaud,\nApproximation by algebraic numbers,\nCambridge Tracts in Mathematics 160,\nCambridge, 2004.\n\n\\bibitem{Bu14}\nY. Bugeaud,\n{\\it Around the Littlewood conjecture in Diophantine approximation},\nPubl. Math. Besan\\c con Alg\\`ebre Th\\'eorie Nr. (2014), 5--18. \n\n\n\n\n\\bibitem{BDM}\nY. Bugeaud, M. Drmota, and B. de Mathan,\n{\\it On a mixed Littlewood conjecture in Diophantine approximation},\nActa Arith. 128 (2007), 107--124.\n\n\n\\bibitem{Cassa98}\nJ. Cassaigne,\n{\\it Sequences with grouped factors}.\nIn: DLT'97, Developments in\nLanguage Theory III, Thessaloniki,\nAristotle University of Thessaloniki, 1998, pp. 211--222.\n\n\\bibitem{CaSw}\nJ. W. S. Cassels and H. P. F. Swinnerton-Dyer,\n{\\it On the product of three homogeneous linear forms and indefinite\nternary quadratic forms}, Philos. Trans. Roy. Soc. London,\nSer. A, 248 (1955), 73--96.\n\n\n\\bibitem{EFS}\nM. Einsiedler, L. Fishman, and U. Shapira,\n{\\it Diophantine approximation on fractals},\nGeom. Funct. Anal. 21 (2011), 14--35.\n\n\n\\bibitem{EKL}\nM. Einsiedler, A. Katok, and E. Lindenstrauss,\n{\\it Invariant measures and the set of exceptions to the\nLittlewood conjecture},\nAnn. of Math. 164 (2006), 513--560.\n\n\n\\bibitem{EiKl07}\nM. Einsiedler and D. Kleinbock,\n{\\it Measure rigidity and $p$-adic Littlewood-type problems},\nCompositio Math. 143 (2007), 689--702.\n\n\n\\bibitem{EinLin}\nM. Einsiedler and E. Lindenstrauss,\n{\\it On measures invariant under tori on quotients of semi-simple groups},\nAnn. of Math. To appear. \n\n\n\\bibitem{EPSW03}\nG. Everest, A. van der Poorten, I. Shparlinski, and T. Ward,\nRecurrence sequences. Mathematical Surveys and Monographs, 104.\nAmerican Mathematical Society, Providence, RI, 2003.\n\n\n\\bibitem{LeSh93}\nH. W. Lenstra and J. O. Shallit,\n{\\it Continued fractions and linear recurrences},\nMath. Comp. 61 (1993), 351--354.\n\n\\bibitem{Lindenstrauss}\nE. Lindenstrauss,\n{\\it Invariant measures and arithmetic quantum unique\nergodicity}, Ann. of Math. (2) 163 (2006), 165--219.\n\n\\bibitem{Lin10}\nE. Lindenstrauss,\n{\\it Equidistribution in homogeneous spaces and number theory}.\nIn: Proceedings of the International Congress of Mathematicians. \nVolume I, 531--557, Hindustan Book Agency, New Delhi, 2010.\n\n\n\n\n\\bibitem{BdMTe}\nB. de Mathan et O. Teuli\\'e,\n{\\it Probl\\`emes diophantiens simultan\\'es},\nMonatsh. Math. 143 (2004), 229--245.\n\n\\bibitem{MoHe38}\nM. Morse and G. A. Hedlund,\n{\\it Symbolic dynamics},\nAmer. J. Math. 60 (1938), 815--866.\n\n\n\\bibitem{MoHe40}\nM. Morse and G. A. Hedlund,\n{\\it Symbolic dynamics II},\nAmer. J. Math. 62 (1940), 1--42.\n\n\\bibitem{Per}\nO. Perron,\nDie Lehre von den Ketterbr\\\"uchen.\nTeubner, Leipzig, 1929.\n\n\\bibitem{PoVe}\nA. D. Pollington and S. Velani,\n{\\it On a problem in simultaneous Diophantine approximation:\nLittlewood's conjecture},\nActa Math. 185 (2000), 287--306.\n\n\n\n\n\\vskip 4mm\n\n\n\\egroup\\par\n\n\\goodbreak\n\n\\vskip 6mm\n\n\n\n\n\\noindent Dmitry Badziahin \\hfill Yann Bugeaud\n\n\\noindent University of Durham \\hfill Universit\\'e de Strasbourg\n\n\\noindent Department of Mathematical Sciences \\hfill Math\\'ematiques\n\n\\noindent South Rd \\hfill 7, rue Ren\\'e Descartes\n\n\\noindent Durham DH1 3LE \\hfill 67084 Strasbourg Cedex\n\n\\noindent UK \\hfill France\n\n\\medskip\n\n\\noindent {\\tt dzmitry.badziahin@durham.ac.uk} \\hfill {\\tt bugeaud@math.unistra.fr}\n\n\n\\goodbreak\n\n\\bigskip\n\n\n\\noindent Manfred Einsiedler \\hfill Dmitry Kleinbock\n\n\\noindent ETH Z\\\"urich, Departement Mathematik \\hfill Brandeis University\n\n\\noindent R\\\"amistrasse 101 \\hfill Department of Mathematics\n\n\\noindent CH-8092 Z\\\"urich \\hfill Waltham, MA 02454\n\n\\noindent Switzerland \\hfill USA\n\n\\medskip\n\n\\noindent {\\tt manfred.einsiedler@math.ethz.ch} \\hfill {\\tt kleinboc@brandeis.edu}\n\n\n\n \\par\\vfill\\supereject\\end\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{S1}\n\n\n\\IEEEPARstart Multiple-input multiple-output (MIMO) technology has been successfully applied to many communication systems, such as the 4th generation (4G) cellular system LTE-A~\\cite{Dongwoon12}, IEEE 802.11n wireless LAN system~\\cite{Skordoulis08}, etc. It is widely recognized as a promising key technology for future wireless communications~\\cite{federico14}. Unlike the traditional small-scale MIMO (e.g., at most 8 antennas in LTE-A), large-scale MIMO, which equips a very large number of antennas (e.g., 128 antennas or even more) at the base station (BS) to simultaneously serve multiple user equipments (UEs), is recently proposed~\\cite{marzetta10}. It has been theoretically proved that large-scale MIMO can achieve orders of increase in spectrum and energy efficiency~\\cite{ngo11}.\n\n\nHowever, realizing the attractive benefits of large-scale MIMO in practice faces some challenging problems, one of which is the practical signal detection algorithm in the uplink~\\cite{rusek13} due to the increased multi-user interferences. The optimal detector is the maximum likelihood (ML) detector, but its complexity increases exponentially with the number of transmit antennas, which makes it impractical for large-scale MIMO systems. Some non-linear detection algorithms such as fixed-complexity sphere decoding (FSD)~\\cite{barbero08} and tabu search (TS)~\\cite{datta10} are proposed to achieve close-optimal performance with reduced complexity. However, their complexity is still unaffordable when the dimension of the MIMO systems is large or the modulation order is high~\\cite{rusek13} (e.g., 128 antennas at the BS with 64 QAM modulation). To make a tradeoff between the performance and complexity, one can resort to low-complexity linear detection algorithms such as zero-forcing (ZF) and minimum mean square error (MMSE) with near-optimal performance for uplink multi-user large-scale MIMO systems~\\cite{rusek13}, but these algorithms involve unfavorable matrix inversion with high complexity. Recently, the Neumann series approximation algorithm was proposed to convert the matrix inversion into a series of matrix-vector multiplications~\\cite{yin13} to reduce the complexity. However, only marginal reduction in complexity can be achieved.\n\nIn this paper, we propose a low-complexity near-optimal signal detection algorithm based on the Gauss-Seidel (GS) method~\\cite{bjorck1996numerical} for large-scale MIMO systems. Firstly, based on the special property that the MMSE filtering matrix of large-scale MIMO systems is Hermitian positive definite, we propose a low-complexity signal detection algorithm, which utilizes GS method to iteratively realize the MMSE estimate without matrix inversion. Then, based on the fact that the MMSE filtering matrix is diagonally dominant for uplink large-scale MIMO systems, we propose to use the diagonal component of the MMSE filtering matrix to obtain a diagonal-approximate initial solution to the GS method, which can accelerate the convergence rate. After that, we propose a approximated method to calculate the channel gain and the noise-plus-interference (NPI) variance for log-likelihood ratios (LLRs) computation, which also utilizes the diagonal dominant property of the MMSE filtering matrix.\nWe verify through simulation results that the proposed GS-based algorithm with the approximated method for LLRs computation can attain the near-optimal performance of the classical MMSE algorithm with a small number of iterations. To the best of our knowledge, this work is the first one to utilize the GS method for signal detection in uplink large-scale MIMO systems.\n\n\nThe rest of the paper is organized as follows. Section~\\ref{S2} briefly introduces the system model. Section~\\ref{S3} specifies the proposed low-complexity signal detection algorithm based on the GS method. The simulation results of the bit-error rate (BER) performance are shown in Section~\\ref{S4}. Finally, conclusions are drawn in Section~\\ref{S5}.\n\n\n{\\it Notation}: We use lower-case and upper-case boldface letters to denote vectors and matrices, respectively; ${( \\cdot )^T}$, ${( \\cdot )^H}$, ${( \\cdot )^{ - 1}}$, and $\\left| \\cdot \\right|$ denote the transpose, conjugate transpose, matrix inversion, and absolute operators, respectively; ${{\\mathop{\\rm Re}\\nolimits} \\{ \\cdot \\} }$ and ${{\\mathop{\\rm Im}\\nolimits} \\{ \\cdot \\} }$ denote the real part and imaginary part of a complex number, respectively; Finally, ${{\\bf{I}}_N}$ represents the $ N \\times N $ identity matrix.\n\n\\section{System Model}\\label{S2}\nWe consider a uplink large-scale MIMO system employing ${N}$ antennas at the BS to simultaneously serve ${K}$ selected single-antenna UEs for communications, where we usually have ${N \\gg K}$, e.g., ${N = 128}$ and ${K = 16}$ have been considered in~\\cite{Hoydis}. The parallel transmitted bit streams from ${K}$ different users are first separately encoded by the channel encoder, and then mapped to constellation symbols by taking values from a energy-normalized modulation constellation ${{\\cal Q}}$. Let ${{\\bf{s}}}$ denote the ${K \\times 1}$ transmitted signal vector containing the transmitted symbols from all ${K}$ users, and ${{\\bf{H}} \\in {\\mathbb{C}^{N \\times K}}}$ denote the flat Rayleigh fading channel matrix whose entries are independent and identically distributed (i.i.d.) with zero mean and unit variance~\\cite{ngo11}. Then the ${N \\times 1}$ received signal vector ${{\\bf{y}}}$ at the BS can be presented as\n\\begin{equation}\\label{eq1}\n{\\bf{y}} = {\\bf{Hs}} + {\\bf{n}},\n\\end{equation}\nwhere ${{\\bf{n}}}$ is a ${N \\times 1}$ additive white Gaussian noise (AWGN) vector whose entries follow ${{\\cal CN}(0,\\sigma ^2)}$.\n\nThe task of multi-user signal detection at the BS is to estimate the transmitted signal vector ${{\\bf{s}}}$ from the received noisy signal vector ${{\\bf{y}}}$ (note that the channel matrix ${\\bf{H}}$ can be usually obtained through time-domain and\/or frequency-domain training pilots~\\cite{dai13, Gao14b}). The estimate of the transmitted signal vector ${{\\bf{\\hat s}}}$ achieved by the MMSE linear detection algorithm can be presented as\n\\begin{equation}\\label{eq2}\n{\\bf{\\hat s}} = {\\left( {{{\\bf{H}}^H}{\\bf{H}} + {\\sigma ^2}{{\\bf{I}}_K}} \\right)^{ - 1}}{{\\bf{H}}^H}{\\bf{y}} = {{\\bf{W}}^{ - 1}}{\\bf{\\bar y}},\n\\end{equation}\nwhere ${{\\bf{\\bar y}} = {{\\bf{H}}^H}{\\bf{y}}}$ can be interpreted as the matched-filter output of ${{\\bf{y}}}$, and the MMSE filtering matrix ${{\\bf{W}}}$ is denoted by\n\\begin{equation}\\label{eq3}\n{\\bf{W}} = {\\bf{G}} + {\\sigma ^2}{{\\bf{{I}}}_{K}},\n\\end{equation}\nwhere ${{\\bf{G}} = {{\\bf{{H}}}^H}{\\bf{{H}}}}$ presents the Gram matrix. After the estimation of the transmitted signal vector, the soft information LLRs can be extracted from the estimated results for soft-input channel decoding. Let ${{\\bf{E}} = {{\\bf{W}}^{ - 1}}{\\bf{G}}}$ denote the equivalent channel matrix and ${{\\bf{U}} = {{\\bf{W}}^{ - 1}}{{\\bf{H}}^H}{({{\\bf{W}}^{ - 1}}{{\\bf{H}}^H})^H} = {{\\bf{W}}^{ - 1}}{\\bf{G}}{{\\bf{W}}^{ - 1}}}$. Then, by combining (1) and (2), the MMSE estimate ${{\\bf{\\hat s}}}$ can be rewritten as\n${{\\bf{\\hat s}} = {\\bf{Es}} + {{\\bf{W}}^{ - 1}}{{\\bf{H}}^H}{\\bf{n}} }$.\nThe estimate of the transmitted symbol for the ${k}$th user (i.e., the ${k}$th element of ${{\\bf{\\hat s}}}$) can be presented as ${{\\hat s_k} = {\\mu _k}{s_k} + {\\nu _k}}$, where ${s_k}$ denotes the ${k}$th element of the transmitted signal vector ${{\\bf{s}}}$, ${{\\mu _k} = {E_{kk}}}$ is the equivalent channel gain, and ${\\nu _k^2 = \\sum\\limits_{m \\ne k}^K {{{\\left| {{E_{mk}}} \\right|}^2}} + {U_{kk}}{\\sigma ^2}}$ denotes the NPI variance, ${{{E_{mk}}}}$ and ${{U_{mk}}}$ present the element of matrix ${{\\bf{E}}}$ and ${\\bf{U}}$ in the ${m}$th row and ${k}$th column, respectively. Then the max-log approximated LLR ${L_{k,b}}$ of bit ${b}$ for the ${k}$th user can be obtained by~\\cite{Wu14}\n\\begin{equation}\\label{eq4}\n{L_{k,b}}\\! =\\! {\\gamma _k}\\left( {\\mathop {\\min }\\limits_{q \\in S_b^0} {{\\left| {\\frac{{{{\\hat s}_k}}}{{{\\mu _k}}}\\! -\\! q} \\right|}^2}\\! -\\! \\mathop {\\min }\\limits_{q' \\in S_b^1} {{\\left| {\\frac{{{{\\hat s}_k}}}{{{\\mu _k}}}\\! -\\! q'} \\right|}^2}} \\right),\n\\end{equation}\nwhere ${{\\gamma _k} = \\mu_k^2\/\\nu_k^2}$ is the signal-to-interference-plus-noise ratio (SINR) for the ${k}$th user, ${S_b^0}$ and ${S_b^1}$ are the sets containing the symbols from the modulation constellation ${{\\cal Q}}$, where the ${b}$th bit of the symbol is 0 and 1, respectively.\n\nIt has been proved that MMSE linear detection algorithm is near-optimal for uplink multi-user large-scale MIMO systems~\\cite{rusek13}. However, the MMSE algorithm inevitably involves complicated matrix inversion ${{{\\bf{W}}^{{\\bf{ - 1}}}}}$ to achieve the MMSE estimate, the channel gain, and the NPI variance, all of which are required to calculate the final LLRs for soft-input channel decoding. The computational complexity of matrix inversion is ${{\\cal O}({K^3})}$, which is high since ${K}$ is usually large in uplink large-scale MIMO systems~\\cite{Hoydis}.\n\n\n\\section{Low-complexity Soft-output Signal Detection For Uplink Large-scale MIMO}\\label{S3}\nIn this section, We first propose a low-complexity signal detection algorithm which utilizes GS method to iteratively realize the MMSE estimate without matrix inversion. To further accelerate the convergence rate and reduce the complexity, we also propose a diagonal-approximate initial solution to the GS method. Then we propose a approximated method to compute the channel gain and the NPI variance for LLRs computation, which does not need to compute the exact matrix inversion. Finally, the complexity analysis of the proposed GS-based algorithm is provided to show its advantages over conventional algorithms.\n\n\\subsection{Signal detection algorithm based on Gauss-Seidel method}\\label{S2.1}\nFor uplink large-scale MIMO systems, the channel matrix ${\\bf{H}}$ is column full-rank and column asymptotically orthogonal~\\cite{rusek13}, which guarantees that the MMSE filtering matrix ${\\bf{W}}$ is Hermitian positive definite.\nThis special property inspires us to exploit the GS method~\\cite{bjorck1996numerical} to iteratively solve (2) without matrix inversion. The GS method is used to solve the ${N}$-dimension linear equation ${{\\bf{Ax}} = {\\bf{b}}}$, where ${{\\bf{A}}}$ is the ${N \\times N}$ Hermitian positive definite matrix, ${{\\bf{x}}}$ is the ${N \\times 1}$ solution vector, and ${{\\bf{b}}}$ is the ${N \\times 1}$ measurement vector. Unlike the traditional method that directly computes ${{{\\bf{A}}^{ - 1}}{\\bf{b}}}$ to obtain ${{\\bf{x}}}$, the GS method can iteratively solve the equation ${{\\bf{Ax}} = {\\bf{b}}}$ with low complexity. Since the MMSE filtering matrix ${{\\bf{W}}}$ is also Hermitian positive definite as mentioned above, we can decompose ${{\\bf{W}}}$ as\n\\begin{equation}\\label{eq9}\n{\\bf{W}} = {\\bf{D}} + {\\bf{L}} + {{\\bf{L}}^H},\n\\end{equation}\nwhere ${{\\bf{D}}}$, ${{\\bf{L}}}$, and ${{{\\bf{L}}^H}}$ denote the diagonal component, the strictly lower triangular component, and the strictly upper triangular component of ${{\\bf{W}}}$, respectively. Then we can exploit the GS method to estimate the transmitted signal vector ${{\\bf{s}}}$ as below\n\\begin{equation}\\label{eq10}\n{{\\bf{s}}^{(i)}} = {({\\bf{D + L}})^{ - 1}}({\\bf{\\bar y}} - {{\\bf{L}}^H}{{\\bf{s}}^{(i-1)}}),\\quad i = 1,2, \\cdot \\cdot \\cdot\n\\end{equation}\nwhere ${i}$ is the number of iterations, and ${{{\\bf{s}}^{(0)}}}$ denotes the initial solution which will be discussed later in Section~\\ref{S3}-B. Since ${({\\bf{D + L}})}$ is a lower triangular matrix, we can obtain ${{{\\bf{s}}^{(i)}}}$ with low complexity as will be addressed in Section~\\ref{S3}-D. It is worth noting that the proposed GS-based algorithm is convergent for any initial solution since the MMSE filtering matrix ${{\\bf{W}}}$ is Hermitian positive definite~\\cite[Theorem 7.2.2]{bjorck1996numerical}.\n\n\n\\subsection{Diagonal-approximate initial solution}\\label{S2.3}\nTraditionally, due to no priori information of the final solution is available, the initial solution ${{{\\bf{s}}^{(0)}}}$ in (6) is set as a zero-vector~\\cite{bjorck1996numerical}, which is simple but usually far away from the final solution. Although the initial solution doesn't influence the convergence, it plays an important role in the convergence rate and affects both computational complexity and detection accuracy when the number of iterations is limited. In this subsection, we propose a diagonal-approximate initial solution to the GS-based algorithm to achieve a faster convergence rate.\n\n\nFor uplink large-scale MIMO systems, the channel matrix ${\\bf{H}}$ is asymptotically orthogonal when ${N \\gg K}$~\\cite{rusek13}, so we have\n\\begin{equation}\\label{eq10}\n\\frac{{\\bf{h}}_m^H{{\\bf{h}}_k}}{N} \\to 0,\\quad m \\ne k,\\quad m,k = 1,2, \\cdot \\cdot \\cdot , K,\n\\end{equation}\nwhere ${{{\\bf{h}}_m}}$ denotes the ${m}$th column vector of the channel matrix ${\\bf{H}}$. This indicates that the MMSE filtering matrix ${{\\bf{W}} = {{\\bf{H}}^H}{\\bf{H}} + {\\sigma ^2}{{\\bf{I}}_{K}}}$ is diagonally dominant for uplink large-scale MIMO systems.\nBased on this principle, we can conclude that the matrix ${{{\\bf{W}}^{ - 1}}}$ is also diagonally dominant.\nFig. 1 shows the normalized entries of the matrix ${{{\\bf{W}}^{ - 1}}}$ for different values of ${N}$ when ${K}$ is fixed to 16, where ${W_{mk}^{ - 1}}$ and ${W_{\\max }^{ - 1}}$ denote the ${m}$th row and ${k}$th column entry and the maximum entry of ${{{\\bf{W}}^{ - 1}}}$, respectively. We can observe that the domination of the diagonal elements of ${{{\\bf{W}}^{ - 1}}}$ becomes more obvious with the increasing value of ${N\/K}$, and the difference between the diagonal matrix ${{{\\bf{D}}^{ - 1}}}$ and the non-diagonal matrix ${{{\\bf{W}}^{ - 1}}}$ becomes smaller. This special property inspires us to utilize ${{{\\bf{D}}^{ - 1}}}$ to approximate ${{{\\bf{W}}^{ - 1}}}$ with small error~\\cite{yin13,Wu14,Wu13}. Then, the initial solution ${{{\\bf{s}}^{(0)}}}$ in (6) can be approximately selected as\n\\begin{equation}\\label{eq24}\n{{\\bf{s}}^{(0)}} = {{\\bf{D}}^{ - 1}}{\\bf{\\hat y}}.\n\\end{equation}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth]{diagonal_total1}\n\\end{center}\n\\vspace{-3mm}\n\\caption{Normalized entries of ${{{\\bf{W}}^{ - 1}}}$ for different values of ${N}$ when ${K}$ is fixed to 16.} \\label{FIG1}\n\\end{figure}\n\nSince the approximation error of ${{{\\bf{s}}^{(0)}}}$ in (8) should be small as shown in Fig. 1, it is expected that the proposed diagonal-approximate initial solution will be closer to the final MMSE estimate ${{\\bf{\\hat s}}}$ compared to the traditional zero-vector initial solution. Therefore, a faster convergence rate can be achieved. Besides, note that the computational complexity to compute ${{{\\bf{D}}^{ - 1}}}$ (or equivalently ${{{\\bf{s}}^{(0)}}}$) is very low, since ${{\\bf{D}}}$ is a diagonal matrix.\n\n\\vspace*{0mm}\n\\subsection{Approximated method to compute LLRs}\\label{S2.4}\n\\emph{1) Exact method}: Although the GS-based algorithm is originally designed to obtain the MMSE estimate ${{\\bf{\\hat s}}}$, it can be also utilized to obtain the estimate of the matrix inversion ${{{\\bf{W}}^{ - 1}}}$. Combining (2) and (6), we set ${{\\bf{\\bar y}} = {{\\bf{e}}_m}}$ where ${{{\\bf{e}}_m}}$ denotes the ${m}$th ${K \\times 1}$ unit vector, then the result of the GS-based algorithm ${\\left( {{{{\\bf{\\hat w}}}_{{\\rm{inv}}}}} \\right)_m^{(i)}\\! =\\! {({\\bf{D}}\\! +\\! {\\bf{L}})^{\\! -\\! 1}}\\left( {{{\\bf{e}}_m}\\! -\\! {{\\bf{L}}^H}\\left( {{{{\\bf{\\hat w}}}_{{\\rm{inv}}}}} \\right)_m^{(i\\! -\\! 1)}} \\right)}$ for ${i = 1,2, \\cdot \\cdot \\cdot }$ will be the ${m}$th column of the estimate of the matrix ${{{\\bf{W}}^{ - 1}}}$ in the ${i}$th iteration, where ${{\\left( {{{{\\bf{\\hat w}}}_{\\rm{inv}}}} \\right)_m^{(0)}}}$ can be selected as the diagonal-approximate initial solution addressed in Section~\\ref{S3}-B, i.e., ${\\left( {{{{\\bf{\\hat w}}}_{{\\rm{inv}}}}} \\right)_m^{(0)} = {{\\bf{D}}^{ - 1}}{{\\bf{e}}_m}}$. Thus, the estimate ${{\\left( {{{{\\bf{\\hat W}}}_{{\\rm{inv}}}}} \\right)^{(i)}}}$ of the matrix ${{{\\bf{W}}^{ - 1}}}$ in the ${i}$th iteration can be achieved by\n\\begin{equation}\\label{eq24}\n\\begin{array}{l}\n{\\left( {{{{\\bf{\\hat W}}}_{{\\rm{inv}}}}} \\right)^{(i)}} = {({\\bf{D}} + {\\bf{L}})^{ - 1}}\\left( {{{\\bf{I}}_K} - {{\\bf{L}}^H}{{\\left( {{{{\\bf{\\hat W}}}_{{\\rm{inv}}}}} \\right)}^{(i - 1)}}} \\right),\\quad \\\\\n\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad i = 1,2, \\cdot \\cdot \\cdot\n\\end{array}\n\\end{equation}\nThen, by replacing the matrix ${{{\\bf{W}}^{ - 1}}}$ by the estimated matrix ${{\\left( {{{{\\bf{\\hat W}}}_{\\rm{inv}}}} \\right)^{(i)}}}$, we have ${{{\\bf{\\hat E}}^{(i)}} = {\\left( {{{{\\bf{\\hat W}}}_{\\rm{inv}}}} \\right)^{(i)}}{\\bf{G}}}$ and ${{{\\bf{\\hat U}}^{(i)}} = {\\left( {{{{\\bf{\\hat W}}}_{\\rm{inv}}}} \\right)^{(i)}}{\\bf{G}}{\\left( {{{{\\bf{\\hat W}}}_{\\rm{inv}}}} \\right)^{(i)}}}$, and the equivalent channel gain ${\\hat \\mu _k^{(i)}}$ and NPI variance ${{\\left( {\\hat \\nu _k^{(i)}} \\right)^2}}$ achieved by the GS-based algorithm in the ${i}$th iteration can be presented as\n\\begin{equation}\\label{eq24}\n\\hat \\mu _k^{(i)} = \\hat E_{kk}^{(i)},\n\\end{equation}\n\\begin{equation}\\label{eq24}\n{\\left( {\\hat \\nu _k^{(i)}} \\right)^2} = \\sum\\limits_{m \\ne k}^K {{{\\left| {\\hat E_{mk}^{(i)}} \\right|}^2}} + \\hat U_{kk}^{(i)}{\\sigma ^2},\n\\end{equation}\nSubstituting (6), (10), and (11) into (4), we can obtain the exact mag-log LLRs for soft-input channel decoding.\n\n\n\n\n\n\n\n\\emph{2) Approximated method}: The exact method above can compute the exact max-log LLRs to produce a good BER performance, but it inevitably involves the calculation of ${{{{\\left( {{{{\\bf{\\hat W}}}_{\\rm{inv}}}} \\right)}^{(i)}}}}$, which requires ${K}$ times of the GS method with the complexity ${{\\cal O}({K^2})}$ for each time. Therefore, although the proposed GS-based algorithm can obtain the MMSE estimate ${{\\bf{\\hat s}}}$ with low complexity ${{\\cal O}({K^2})}$, the exact method to compute LLRs still suffers from the complexity as high as ${{\\cal O}({K^3})}$. To solve this problem, we propose\nan approximated method inspired by~\\cite{Wu13} to calculate the channel gain and NPI variance for LLRs computation, which can avoid the complicated matrix inversion.\n\nSince ${{{\\bf{W}}^{ - 1}}}$ is diagonal dominant for uplink large-scale MIMO systems as we have verified in Section~\\ref{S3}-B, we can utilize the diagonal matrix ${{{\\bf{D}}^{ - 1}}}$ to approximate ${{{\\bf{W}}^{ - 1}}}$ with small error~\\cite{yin13,Wu14,Wu13}. Then the approximated channel gain ${\\tilde \\mu _k}$ and the approximated NPI variance ${\\tilde \\nu _k^2}$ can be achieved by\n\\begin{equation}\\label{eq24}\n{\\tilde \\mu _k} = {\\tilde E_{kk}},\n\\end{equation}\n\\begin{equation}\\label{eq24}\n\\tilde \\nu _k^2 = \\sum\\limits_{m \\ne k}^K {{{\\left| {{{\\tilde E}_{mk}}} \\right|}^2}} + {\\tilde U_{kk}}{\\sigma ^2},\n\\end{equation}\nwhere ${{\\bf{\\tilde E}} = {{\\bf{D}}^{ - 1}}{\\bf{G}}}$ and ${{\\bf{\\tilde U}} = {{\\bf{D}}^{ - 1}}{\\bf{G}}{{\\bf{D}}^{ - 1}}}$. Substituting (6), (12), and (13) in (4), we can obtain the approximated max-log LLRs. Since ${{{\\bf{D}}^{ - 1}}}$ is a diagonal matrix, the computation of ${{\\bf{\\tilde E}}}$ and ${{\\bf{\\tilde U}}}$ involves low complexity. Besides, since the MMSE estimate ${{\\bf{\\hat s}}}$ can be obtained without matrix inversion, the overall computational complexity to compute LLRs can be significantly reduced as will be quantified in the following subsection.\n\nIt is worth pointing out that the method proposed in~\\cite{Wu13} also utilizes ${{{\\bf{D}}^{ - 1}}}$ to approximate ${{{\\bf{W}}^{ - 1}}}$, but it simplifies the LLRs computation by first computing a conjugate gradient matrix with low complexity, which is then used to compute LLRs, while our method directly utilizes ${{{\\bf{D}}^{ - 1}}}$ to obtain LLRs.\n\n\n\\subsection{Computational complexity analysis }\\label{S2.4}\nSince both the MMSE algorithm and the proposed GS-based algorithm need to compute the Gram matrix ${{\\bf{G}} = {{\\bf{H}}^H}{\\bf{H}}}$ (or equivalently ${{\\bf{W}} = {\\bf{G}} + {\\sigma ^2}{{\\bf{{I}}}_{K}}}$) and the matched-filter output ${{\\bf{\\bar y}}}$, we focus on the complexity of the LLRs computation, and evaluate it in terms of the required number of (complex) multiplications~\\cite{gao2014low}.\nIt can be found from (4) that the computational complexity of the proposed GS-based algorithm to obtain LLRs comes from three parts:\n\n1) The first one comes from the diagonal-approximate initial solution (8) addressed in Section III-B, which involves the computation of ${{{\\bf{D}}^{ - 1}}}$ and a multiplication of the ${K \\times K}$ diagonal matrix ${{{\\bf{D}}^{ - 1}}}$ and the ${K \\times 1}$ vector ${{\\bf{\\bar y}}}$. Therefore the required number of complex multiplications is ${2K}$.\n\n2) The second one originates from solving the linear equation (6). Considering the definition of ${{\\bf{D}}}$ and ${{\\bf{L}}}$ in (5), the solution can be presented as\n\\begin{equation}\\label{eq25}\n\\begin{array}{l}\ns_m^{(i)} = \\frac{1}{{{W_{mm}}}}({{\\bar y}_m} - \\sum\\limits_{k < m} {{W_{mk}}s_k^{(i)} - \\sum\\limits_{k > m} {{W_{mk}}s_k^{(i-1)}} } ), \\\\\n\\quad \\quad \\quad \\quad \\quad \\quad m,k = 1,2, \\cdot \\cdot \\cdot, K,\n\\end{array}\n\\end{equation}\nwhere ${s_m^{(i)}}$, ${s_m^{(i-1)}}$, and ${{\\bar y_m}}$ denote the ${m}$th element of ${{{\\bf{s}}^{(i)}}}$, ${{{\\bf{s}}^{(i-1)}}}$, and ${{\\bf{\\bar y}}}$, respectively, and ${{W_{mk}}}$ denotes the element of ${{\\bf{W}}}$ in the ${m}$th row and ${k}$th column. It is clear that the required number of complex multiplications to compute ${s_m^{(i)}}$ is ${K}$. Since there are ${K}$ elements in vector ${{{\\bf{s}}^{(i)}}}$, solving the equation (6) requires ${{iK^2}}$ times of complex multiplications.\n\n3) The third one is from the computation of channel gain and NPI variance. It can be found from (12) and (13) that for the proposed approximated method to compute LLRs, it requires to calculate two parts, i.e., all the elements of the matrix ${{\\bf{\\tilde E}}}$ and the diagonal elements of the matrix ${{\\bf{\\tilde U}}}$. Due to ${{\\bf{\\tilde E}} = {{\\bf{D}}^{ - 1}}{\\bf{G}}}$, and the diagonal matrix ${{{\\bf{D}}^{ - 1}}}$ has been obtained when we use the diagonal-approximate initial solution, the required number of complex multiplications of the first part is ${{K^2}}$. For the second part, we only need the diagonal elements of the matrix ${{\\bf{\\tilde U}}}$, which can be presented as ${{\\tilde U_{kk}} = D_{kk}^{ - 2}{G_{kk}}}$ for ${k = 1,2, \\cdot \\cdot \\cdot, K}$, where ${D_{kk}^{ - 1}}$ and ${{G_{kk}}}$ denote the ${k}$th diagonal element of ${{{\\bf{D}}^{ - 1}}}$ and ${{\\bf{G}}}$, respectively. Thus, the required number of complex multiplications of the second part is as small as ${2K}$.\n\n\nTo sum up, the overall required number of complex multiplications by the proposed GS-based algorithm is ${(i + 1){K^2} + 4K}$, so the computational complexity is ${{\\cal O}({K^2})}$ for arbitrary number of iterations.\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{complexity1}\n\\end{center}\n\\vspace{-3mm}\n\\caption{Complexity comparison against number of users.} \\label{FIG2}\n\\end{figure}\n\nFig. 2 compares the complexity of the Neumann-based algorithm~\\cite{Wu14} and the proposed GS-based algorithm, whereby the MMSE algorithm with Cholesky decomposition is also included as a baseline for comparison~\\cite{Wu14}. Note that all these three algorithms utilize the approximated method to compute the LLRs as described in Section~\\ref{S3}-C. It shows that the Neumann-based algorithm has lower complexity than the MMSE algorithm with Cholesky decomposition when ${i \\le\\ 3}$, especially when ${i=2}$ with the complexity ${{\\cal O}({K^2})}$. However, when ${i \\ge 4}$, the complexity of the Neumann-based algorithm is ${{\\cal O}\\left( {(i - 2){K^3}} \\right)}$~\\cite{Wu14}, which is even higher than that of the MMSE algorithm. Since usually large value of ${i}$ is required to ensure the final approximation performance as will be verified later by simulation results in Section~\\ref{S4}, the reduction in complexity achieved by the Neumann-based algorithm is marginal. By contrast, since ${K}$ is usually large for large-scale MIMO systems (e.g., ${K=16}$ in~\\cite{rusek13}), we can observe that the proposed GS-based algorithm can evidently reduce the complexity from ${{\\cal O}({K^3})}$ to ${{\\cal O}({K^2})}$ for arbitrary number of iterations. Even for ${i=2}$, the proposed algorithm enjoys a lower complexity than the Neumann-based algorithm. Moreover, as shown in Fig. 2, the proposed GS-based algorithm will lead to more significant reduction in complexity when the dimension of MIMO system becomes larger, which means that the proposed algorithm with low complexity is quite suitable for large-scale MIMO systems.\n\n\n\nAdditionally, we can observe from (14) that the computation of ${s_m^{(i)}}$ utilizes ${s_k^{(i)}}$ for ${k = 1,2, \\cdot \\cdot \\cdot ,m - 1}$ in the current ${i}$th iteration and ${s_l^{(i-1)}}$ for ${l = m + 1,m+2, \\cdot \\cdot \\cdot ,K}$ in the previous ${(i-1)}$th iteration. This characteristic of the GS method will lead to the GS-based algorithm hard to be parallelized, however it can bring two other benefits. Firstly, after ${s_m^{(i)}}$ has been obtained, we can use it to overwrite ${s_m^{(i-1)}}$ which is useless in the next computation of ${s_{m + 1}^{(i)}}$. Consequently, only one storage vector of size ${K \\times 1}$ is required; Secondly, when ${i}$ increases, the solution to (6) becomes closer to the final MMSE estimate ${{\\bf{\\hat s}}}$. Thus, ${s_m^{(i)}}$ can exploits the elements of ${s_k^{(i)}}$ for ${k = 1,2, \\cdot \\cdot \\cdot ,m - 1}$ that have already been computed in the current iteration to produce more reliable result than the conventional algorithm, which only utilizes all the elements of ${{{\\bf{s}}^{(i-1)}}}$ in the previous iteration. Thus, a faster convergence rate can be expected, and the required number of iterations to achieve a certain estimate accuracy becomes smaller. Based on these facts, the overall complexity of the proposed algorithm can be reduced further.\n\n\\section{Simulation Results}\\label{S4}\nThe simulation results of BER performance against the signal-to-noise ratio (SNR) are provided to compare the GS-based algorithm with the recently proposed Neumann-based algorithm~\\cite{Wu14}. The BER performance of the classical MMSE algorithm with Cholesky decomposition is also included as the benchmark for comparison. In all simulations, we consider the modulation scheme of 64 QAM, and the rate-1\/2 industry standard convolutional code with ${[{133_o}\\;{171_o}]}$ polynomial. At the receiver, LLRs are extracted from the detected signal for soft-input Viterbi decoding. Note that the SNR is defined at the receiver~\\cite{Wu14}.\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth]{comparsionLLR_128_161}\n\\end{center}\n\\vspace{-3mm}\n\\caption{BER performance comparison between the exact method and the proposed approximated method to compute LLRs.} \\label{FIG3}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth]{total_128_16_initial1}\n\\end{center}\n\\vspace{-3mm}\n\\caption{BER performance comparison between the conventional zero initial solution and the proposed initial solution.} \\label{FIG4}\n\\end{figure}\n\n\n\n\n\nFirstly, we consider the uncorrelated Rayleigh fading channel. Fig. 3 shows the BER performance comparison between the GS-based algorithm with the exact method and the approximated method to compute LLRs, when ${N \\times K = 128 \\times 16}$. Note that ${i}$ denotes the number of iterations and we choose the diagonal-approximate initial solution. We can observe that compared to the exact method to compute LLRs involving high complexity, the proposed approximated method can achieve a satisfying performance when the number of iterations ${i}$ is relatively large (e.g., ${i \\ge 3}$). For example, when ${i=3}$, the difference between the exact method and the approximated method is within 0.1 dB.\n\nFig. 4 compares the BER performance between the GS-based algorithm with the conventional zero-vector initial solution and the proposed diagonal-approximate initial solution, when ${N \\times K = 128 \\times 16}$ and the approximated method to compute LLRs is employed. It is clear that the proposed diagonal-approximate initial solution can accelerate the convergence rate. When ${i = 3}$, the GS-based algorithm with diagonal-approximate initial solution has almost the same performance as that with the conventional zero-vector initial solution when ${i=4}$, which means the overall complexity of the proposed algorithm can be reduced further.\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth]{total_128_161}\n\\end{center}\n\\vspace{-3mm}\n\\caption{BER performance comparison between the conventional Neumann-based algorithm and the proposed GS-based algorithm.} \\label{FIG5}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth]{total_Nr1}\n\\end{center}\n\\vspace{-3mm}\n\\caption{BER performance comparison against the number of antennas at BS.} \\label{FIG6}\n\\end{figure}\n\nFig. 5 shows the BER performance comparison between the conventional Neumann-based algorithm~\\cite{Wu14} and the proposed GS-based algorithm, when ${N \\times K = 128 \\times 16}$. Note that we choose the diagonal-approximate initial solution and the proposed approximated method to compute LLRs for the GS-based algorithm. It is clear that with the increased number of iterations, the BER performance of both algorithms becomes closer to the MMSE algorithm. However, the GS-based algorithm outperforms the conventional one when the same number of iterations is used. As we can observe from Fig. 5, when ${i=3}$, the SNR required by the GS-based algorithm to achieve the BER of ${{10^{ - 4}}}$ is 14 dB, while for the Neumann-based algorithm, the required SNR is 15 dB.\n\nIn addition, in Fig. 6 we also provide the simulation results about the BER performance of the proposed GS-based algorithm against the number of antennas at BS (${N}$) when a fixed number of users ${K=16}$ is considered. Note that SNR = 13 dB is adopted.\nWe can observe that the performance of the MMSE algorithm improves when ${N}$ increases, and the GS-based algorithm can achieve the exact performance of the MMSE algorithm with a small number of iterations (i.e., ${i=4}$) regardless of the value of ${N}$. By contrast, although the performance of the Neumann-based algorithm also improves with the increasing value of ${N}$, it still suffers a non-negligible performance loss, which further verifies that the proposed GS-based algorithm outperforms the conventional Neumann-based algorithm in large-scale MIMO systems. More importantly, we can also observe from Fig. 6 that the proposed GS-based algorithm is near-optimal compared to the optimal ML algorithm, since when ${N \\gg K}$ (e.g., ${N\/K=8}$) the performance of the GS-based algorithm with ${i=4}$ is close to that of the optimal ML algorithm.\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth]{corr_128_161}\n\\end{center}\n\\vspace{-3mm}\n\\caption{BER performance comparison for different values of correlated magnitude.} \\label{FIG7}\n\\end{figure}\n\n\n\nFinally, as the spatial correlation of MIMO channels plays a crucial role in the performance of realistic MIMO systems, we show in Fig. 7 how the channel correlation affects the performance of the proposed GS-based algorithm. Note that we adopt the exponential correlation model described in~\\cite{Godana13}, and ${\\xi }$ (${0 \\le \\xi \\le 1}$) denotes the correlated factor between two adjacent antennas.\nWe can observe that the performance of the classical MMSE algorithm degrades when the channel correlation becomes serious, which is consistent with the theoretical analysis in~\\cite{Godana13}, and the GS-based algorithm can still converge to the MMSE algorithm without obvious performance loss. However, the required number of iterations by the proposed GS-based algorithm to converge becomes larger with an increasing value of ${\\xi }$ (e.g., ${i = 7}$ when ${\\xi=0.5}$, but ${i=10}$ when ${\\xi=0.7}$), which means more serious channel correlation will lead to slower convergence rate. However, the GS-based algorithm can still enjoy a lower complexity than the Neumann-based algorithm and the MMSE algorithm with Cholesky decomposition.\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{S5}\nIn this paper, by fully exploiting the special characteristics of uplink large-scale MIMO systems, we propose a low-complexity near-optimal signal detection algorithm based on the GS method. To reduce the complexity further, we propose a diagonal-approximate initial solution to the GS-based algorithm which is close to the final solution to accelerate the convergence rate. We also propose a approximated method to compute LLRs with low complexity for soft-input channel decoding. Analysis shows that the proposed algorithm can reduce the complexity from ${{\\cal O}({K^3})}$ to ${{\\cal O}({K^2})}$. It is verified that the proposed algorithm outperforms the conventional method, and achieves the near-optimal performance of the classical MMSE algorithm with a small number of iterations. Additionally, the idea of using the GS method to efficiently solve the complicated matrix inversion can be applied to other signal processing problems involving matrix inversion of large size in wireless communications, such as the downlink precoding in large-scale MIMO systems.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\label{intro}\nAs one of the most important manipulation skills, vision-based grasping has been studied intensely in the field of robotics for decades.\nAlthough robots performing pick and place tasks have been applied successfully in industry, creating autonomous robots for grasping objects in unstructured real-world scenes remains an open question.\nIn recent years, deep reinforcement learning (DRL) has attracted increasing research attention in robotics since its success in video game.\nCombining with CNNs, DRL maps the features of visual observations directly into control policies by trial-and-error.\nThis provides a general way for robots to learn manipulation skills by using information acquired from cameras \\cite{simulation-robotic}, \\cite{dexterous-manipulation}.\n\nThe typical way to train a visual-based DRL agent is in an end-to-end fashion, in which the reward signal of reinforcement learning is used to train both CNNs and policy networks synchronously.\nHowever, in order to achieve satisfying performance, large amounts of interaction data are required for training of a DRL agent.\nFor example, to collect enough training data, \\cite{hand-eye} executed 800,000 robotic grasping attempts in several months with 14 robots,\nand \\cite{self-super} collected 700 hours robot grasping data with 50,000 grasping attempts.\nMoreover, DRL methods try to map raw visual observation into a lower dimensional latent space of control that preserves various types of information about manipulated objects.\nHowever, tangled and uninterpretable latent representation restricts the generalization across object and environment and further leads to poor control policy.\nMost works have to evaluate the trained DRL agents with the similar objects and environments as that of training \\cite{robotic-autoencoder}, \\cite{dexterous-manipulation}, since the networks trained before have to be fine tuned to adapt to change and hundreds of thousands of robotic manipulation experiences may be needed once more when transferring to a new environment.\nThese limitations will definitely prohibit the use of DRL method in real-world robotic application. \n\nA feasible way to alleviate data requirements is to train DRL agents in simulation, in which the interaction could be speed up easily by using programming techniques, such as multithreading. \nWith this approach, large volumes of experiences can be captured efficiently than that of real world interactions and meanwhile many variants of environments could be constructed for generalization concern.\nHowever, there is a huge gap between the simulation and the real-world, which causes the agents trained in simulation to be hardly applied in real world conditions, especially in the context of vision-based robotic manipulation, where illumination changes and varying textures can have significant effects on the quality of the results.\nTherefore, some DRL based robotic control approaches are only verified in simulation due to difficulties in transferring from simulation to real robots \\cite{no-real-robot2}, \\cite{no-real-robot1}.\nTo alleviate the problem some techniques are proposed to allow for automatic adaptation to the real world environments \\cite{simulation2real2}, \\cite{simulation2real}.\n\nIn this paper, we propose a DRL-based visual grasping system aiming at improving generalization performance with the least cost of the acquisition of real world experiences. \nFollowing the typical visual based DRL paradigm, our framework consists of two major components: a CNN based visual \\emph{perception} and a DRL based control \\emph{policy}.\nThe perception module extracts features from visual observation (i.e. raw images) and then the features are mapped into the action space by the policy module.\n\nWe train the perception and the policy separately instead of end-to-end.\nThe perception is trained in a supervised setting to produce the semantic and spatial information of the grasped objects.\n In the meantime, the control policy is trained in a simulation environment where the class and pose of the object to be grasped can be read automatically.\n Training the policy with the quantitative description of manipulated objects can be beneficial to both generalization and transferability since the information irrelevant for control decision is discarded.\n In our work, after roughly \\textbf{30 minutes} of training in simulation, the policy is directly transferred to a real robotic system \\textbf{without any additional training}.\n The performance of our system is evaluated on challenging tasks including semantic grasping, clustered object and moving object grasping.\n The experimental results demonstrate the robustness and generalization of our approach.\n\n\\section{Related work}\n\\label{relatedwork}\nSimulation environments can provide experiences data much more effective \nsince the simulation could be accelerated by programming. \nAnd many robotic manipulation DRL algorithms are verified in simulation environments \\cite{simulation-robotic}, \\cite{model-base}.\nUnfortunately, the gap between simulations and real-world makes the the agent trained in simulation can hardly use in physical robot. \nMany works had tried to bridge reality gap \\cite{simulation2real}, \\cite{simulation2real2}.\nIn \\cite{simulation2real}, the images came from simulations were rendered in randomization, and while the visual perception had seen enough variability over simulations, the real-world images may appear to the model just as another variation.\nSuch randomization made a successful visual grasping model in real-word.\n\\cite{simulation2real2} unified the visual inputs from simulation and real-world using an adaptation network.\nThe adaptation network was trained to generalize canonical images from randomized images from simulation.\nAnd because of the randomization, the trained network could also generalize canonical images from real-world.\nThe generalized canonical images which had been mapped into the same space were used to train the visual grasping DRL agents.\nBenefited from the adaptation network, DRL agents could be trained in simulation and used in real-world.\n\nMany researches have tried to relieve data inefficiency by improving the efficiency of DRL training process and experiences data generation.\nGuided policy search (GPS) algorithm \\cite{guided-policy-search} converts reinforcement learning into supervised learning,\nwhere a trained a local linear controller provided with full state observation (i.g., object poses) served as supervisor.\nAnd a global policy parameterized by neural networks derived from supervision. \nThis allows a CNNs policy with 92,000 parameters to be trained in tens of minutes of real-world robotic manipulation time \nand in test stage the full state is no longer available that the policy trained in a supervised setting could handle several novel, unknown configurations.\nAnother direction to improve sample efficiency is to accelerate model-free reinforcement learning with a learned dynamics models \\cite{model-base}.\nThe learned models can generate synthetic sample data to enrich the agent experiences efficiently \nwhich has no need to execute the physical robot, though it needs additional efficient model learning algorithms \\cite{model-learn}, \\cite{model-learn-3}.\nHowever, the learned model would quite differ from the true dynamics and the induced error would weaken performance of learned policy.\n\nThese methods tried to train an optimal policy and visual perception simultaneously in an end-to-end style.\nHowever, it can hardly be generalized to different manipulated objects and different execution environments.\nSince the generalization is relied on the distribution of training data,\nit requires a huge experience data to achieve usable generalization ability \\cite{e2eGeneralize}.\nIt is impractical in a robotic grasping task to acquire enough data.\nAn intuitive alternative is to train image representation and reinforcement agent separately \\cite{auto-encoder}, \\cite{robotic-autoencoder}.\nWith an auto encoder pretrained by an auxiliary reconstruct loss, the high dimension of image input is embedded into a low dimension, latent space and aggregate useful features before interacting with environment. \nThis way the training of the reinforcement agent networks would be more easily with much less interaction experiences \nfor there is no need to learn the state representation and the training would significantly speed up. \nHowever, the latent feature representation has no exact physical meanings and would be lack of interpretability as well the trained policy. \nFrom this perspective, meaningful feature representation would significantly improve generalization ability.\n\n\n\n\\section{Framework}\n\\label{framework}\n\nWe propose a robotic grasping framework based on deep reinforcement learning. \nReinforcement learning enables agents (e.g., robots) to learn an optimal policy through interaction with environments by trial-and-error. \nIn doing so, we formulate a robotic grasping problem as a Markov decision process: \nat time $t$, the robot receives the state of target objects and constructs environment state $s_t$ accordingly. \nAfter that, the robot chooses an action $a_t$ to move itself based on the current policy $\\pi \\left(a_{t}|s_{t}\\right)$. \nThen the environment transits to a new state $s_{t+1}$ reacting to $a_t$ and an immediate reward $R_t \\left(s_t,a_t,s_{t+1}\\right)$ is offered by the environment. The goal of the robot is to find an optimal policy $\\pi ^{*}\\left(a_{t}|s_{t}\\right)$ that maximizes the discounted future rewards\n\n$$G_t=\\sum _{i=t}^{\\infty}\\gamma ^{i-t}R_{i}\\left(s_i,a_i,s_{i+1}\\right)$$ \nwhere $0 < \\gamma < 1$ is the discounted factor which reduces the weight of future rewards.\n\nSimilar to recent works \\cite{robotic-autoencoder}, \\cite{guided-policy-search}, our framework is composed of two stages, as shown in Fig.\\ref{framework_figure}. \n\\begin{figure*}\n\n\n\t \\includegraphics[width=1\\textwidth]{framework.pdf}\n\n\t\\caption{Our framework for robotic prehensile control.} \n\t\\label{framework_figure} \n\\end{figure*}\nRaw RGB images from camera are input into the perception, where object semantic segmentation and pose estimation are made by Mask R-CNN and PCA respectively. \nThe policy is a PPO \\cite{ppo} agent which receives the control quantities of desired objects and decides which action will be taken to execute grasping. \nThe pseudo code of grasping a single object with our framework is presented in Algorithm \\ref{framework_pseudo}. \nThe details of each component are discussed in the following subsection.\n\nThe perception and policy are trained separately. \nIn particular, Mask R-CNN is trained in a supervised way, in which the labels are constructed manually using a tool LabelMe \\cite{labelme}. \nThere is no training stage for PCA as it is an unsupervised method. \nThe PPO is trained in simulation for fast experience data acquisition. \nSince the semantic and position information can be read by an interface provided by the simulation environment, the training of PPO could proceed in parallel with that of Mask R-CNN. \n\n\\begin{algorithm}\n\t\\caption{Separate Perception and Policy}\n\t\\label{framework_pseudo}\n\t\\begin{algorithmic}[1]\n\t\t\n\t\t\\Require $image$\n\t\t\\Statex\n\t\t\\Statex \\textbf{\\emph{Perception}}\n\t\t\\State \\Comment detect object class and mask from raw images\n\t\t\\State $mask, class$ $\\gets$ \\textbf{mask-rcnn}($image$)\n\t\t\\State \\Comment get physical quantities from mask\n\t\t\\State $center, direction$ $\\gets$ \\textbf{PCA}($mask$)\n\t\t\\Statex\n\t\t\\Statex \\textbf{\\emph{Policy}}\n\t\t\\Repeat\n\t\t\t\\State \\Comment decide action via PPO\n\t\t\t\\State $action$ $\\gets$ \\textbf{PPO}($center, direction$)\n\t\t\\Until{$|action| < \\epsilon$}\n\t\t\\Statex\n\t\t\\State \\Comment grasp specific object and success check\n\t\t\\State $success$ $\\gets$ \\textbf{grasp}($class$)\n\t\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Perception}\n\\label{perception}\n\nThe perception plays a sensor-like role that transforms raw image inputs into physical quantities binding with object semantic information (i.e., object class and its corresponding pose). \nWe should note that this work focuses on a 3DOF grasp \\cite{grasp-pose} given that the workspace of a robot is constrained to a tabletop. \n\n\\subsubsection{Semantic Segmentation} \n\nTo grasp a target object, the robot must know where the target is. \nTo achieve this, we leverage a popular semantic segmentation method Mask R-CNN \\cite{maskrcnn} as the front part of the perception to detect and segment objects from raw images. \n\nBased on Faster R-CNN \\cite{fastrcnn}, Mask R-CNN introduces a mask branch at the end of the original network for segmentation tasks.\nIt proceeds in two stages: first, the region proposal network (RPN) \\cite{fasterrcnn} is applied to scan the image to find the area where the target exists; \nsecondly, the object class and its bounding box coordinates are predicted simultaneously. \nFinally, a pixel-wise binary mask for each bounding box is generated by a fully convolutional network (FCN) indicating whether each pixel in bounding box is the point of the detected object. \nAs a result, masks on the original image exactly cover the areas where the objects exist. \nThe class and mask of an object provide us with a good starting point for pose estimation.\n\n\\subsubsection{Object Pose}\n\nSince an object mask produced carries information about object pose, \nlearning a DRL policy from masks is in principle possible, as what most DRL based approaches do. \nIn realistic robotic applications however, we can not afford to collect such huge interaction data required by a policy learning algorithm. \nTo avoid this difficulty, we further infer pose for an object instance based on the object mask obtained. \nIn a 3DOF grasp setting, object pose can be represented by a 2-dimensional position coordinates of the object center and the direction of the object. \nHere, we develop a Principal Component Analysis (PCA) based method to estimate 3D object pose from a pixel-wise binary mask output by Mask R-CNN. \nIn general, PCA is an unsupervised method that could identify the main components of data with largest variances from a big dataset. \nFor our purpose, the center and main direction of a set of pixel points are inferred by using PCA.\n\nThe output of Mask R-CNN is an object with its covered mask which contains all pixel points formalized as \n$$mask=\\left \\{ \\left(x_0,y_0\\right), \\left(x_1,y_1\\right),...,\\left(x_n,y_n\\right)\\right \\} $$\nwhere $n$ is the number of pixel points in the mask, i.e the number of samples in PCA.\nFirstly, we calculate the mean point of $mask$ as the center point of the mask:\n$$\nc=(x,y)=\\frac{1}{n} \\sum_{i=1}^n (x_i, y_i)\n$$ \nNote that, the mean point $c=(x,y)$ is the geometric center of a mask. \nAfter that, all the points in the $mask$ are subtracted by the $c$ resulting residual coordinates\n$$Res = mask - c$$\nThus, the covariance matrix of $Res$ and its corresponding eigenvalues and eigenvectors are calculated:\n$$\n\\lambda _1, \\lambda _2 = eigenvalues \\left ( covMat\\left(Res\\right) \\right )\n$$\n$$\n\\alpha _1, \\alpha _2 = eigenvectors \\left ( covMat\\left(Res\\right) \\right )\n$$ \nSince the pixel points are in two dimensions, there are totally two eigenvalues and eigenvectors of $Res$ matrix. \nThe $\\lambda$ and $\\alpha \\in \\mathbb{R}^{2\\times 1}$ are sorted by the magnitude of eigenvalues in descending order. \nFinally, the main component with largest variance is calculated: \n$$\nM = Res \\cdot \\alpha _1 \\cdot \\alpha _1^\\top + c\n$$\n$M \\in \\mathbb{R}^{n\\times 2}$ contains $n$ points on a straight line. \nWe take two points from $M$ randomly to construct a straight line. \n$\\theta$ represents the angle of the straight line respect to the horizontal axis. \n\nFig.\\ref{PCA-result} shows the results of PCA on a number of objects with various shapes. \nWith the help of a calibrated camera, the position and orientation in pixel coordinates can be mapped into that of a physical coordinate system.\n\\begin{figure*}\n\t\\centering \n\t\\includegraphics[scale=1]{multi-obj-pca.pdf}\n\t\\caption{\n\tThe example results of PCA pose estimation. \n\tThe black shadows represent the masks produced by Mask R-CNN, and the green points are the center points of the masks, i.e. \n\tthe position coordinate of the object, and the red lines indicate the main component of the masks, \n\ti.e. the orientation of the object in the plane. \n\tThe range of $\\theta$ is $\\left [-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right ]$, where the green arrow indicates a positive $\\theta$, while the red arrow is negative. }\n\t\\label{PCA-result}\n\\end{figure*}\n\n\\subsection{Policy}\n\\label{policy}\n\nThe policy is a deep reinforcement agent that receives the physical quantities from the perception and decides an optimal action to move the robot. For our framework, \nwe adopt a policy gradient method called Proximal Policy Optimization (PPO) \\cite{ppo} which is favorable for high dimension continuous robotic control problem. \nPPO significantly improves data efficiency over other policy gradient methods by updating multiple gradient steps with the same trajectory. \nMoreover, to avoid an increase in data variance, PPO introduces a clipping coefficient which stops the gradient when the difference between the updated and original policy is too large. \n\n\\subsubsection{State Representation} \n\\label{state-representation}\n\nWe concatenate the output from the perception $x, y, \\theta$ (the pose of a target object) and the robotic configuration $x_r, y_r, \\theta _r$ ( the pose of the end effector of a robot) to form the environment state $s_{t}$ as the input of the policy network PPO:\n$$\ns_{t} = \\left(x,y,\\theta,x_r, y_r, \\theta _r\\right)\n$$\nThrough several fully connected layers, PPO finally outputs an action distribution over current state $s_t$.\nThereafter, the optimal action in the current state $s_t$ at time step $t$ is sampled from the action distribution:\n$$\na_t \\sim \\pi\\left (a|s_t \\right )\n$$\nThe action is a three-dimensional continuous variable instructing the robot's next moving direction and magnitude. \nThe execution of the action would lead to a new environment state $s_{t+1}$ and an immediate reward $R_{t}$ offered by environment.\n\n\\subsubsection{Reward} \n\\label{reward}\n\nThe reward function for learning the policy $R_{t}$ is defined as: \n$$\nR_{t} = \\left\\{\\begin{matrix}\n\t\t-d_t-0.1 & away\\\\ \n\t\t-d_t+0.1 & approaching\\\\ \n\t\t1 & grasp~success\\\\\n\t\t-1 & grasp~failed\\\\\n\t\\end{matrix}\\right.\n$$\nwhere $d_t$ is the distance between the end effector's current and target position in time step $t$.\n\nIf $d_t$ is decreasing compared to the previous $d_{t-1}$, the end effector is $approaching$ the target and will receive a slightly positive reward addition, \nand otherwise, it is $away$ with a negative reward. \nIn this way, we encourage the end effector to approach and trace the target object as soon as possible.\nWhen the policy decides actions bounded in a very small magnitude for several time steps, \nthe policy will decide to execute the grasping, i.e., the end effector moves down on $z$ coordinate and closes the gripper. \nA grasping is counted as a success if the gripper is not fully closed. \nWith a larger reward for a successful grasping, the policy could learn the tracing target policy and grasping policy simultaneously.\n\n\\subsubsection{Training Loss} \n\\label{traing-loss}\n\nPPO is an Actor-Critic style \\cite{a3c} algorithm and typically contains a value function and a policy function.\nThe value function $V_{s_t}$ which estimate the expected reward from a state $s_t$ is trained to minimize TD error \\cite{rl-introduction}, whose loss function is defined as:\n$$\nL_{V} = \\left ( V\\left(s_{t}\\right)- \\left (R_{t}+\\gamma V\\left(s_{t+1}\\right)\\right )\\right )^{2}\n$$\nwhere $\\gamma $ is the discounted factor.\n\nThe policy function $\\pi(a_t|s_t)$ which decides an optimal action over a state $s_t$ is trained to maximize a novel surrogate objective \\cite{ppo} according to the value function:\n$$\nL_{a} = \\textbf{min}\\left ( r_t A_t, clip\\left ( r_t, 1-\\epsilon, 1+\\epsilon \\right ) A_t \\right )\n$$\nwhere $r_t = \\frac{\\pi \\left (a_t|s_t \\right )}{\\pi_{old}\\left (a_t|s_t \\right )}$ is the importance sampling coefficient in which $\\pi_{old}\\left (a_t|s_t\\right )$ is the behavior policy whose parameters are frozen during one update epoch. \n$A_t$ is the advantage function \\cite{a3c} which indicates if the reward of current action is above average. \nAnd it could be estimated easily by $A_{t} =R_{t} + \\gamma V\\left(s_{t+1}\\right)- V\\left(s_{t}\\right) $ or GAE method \\cite{gae} according to the value function $V(s_t)$.\nAnd a \\emph{clip} is a function that limits the importance sampling value between $1-\\varepsilon$ and $1+\\varepsilon$ \nin order to avoid a large step update where $\\varepsilon$ is the clipping coefficient which usually equals to $0.2$ \\cite{ppo}. \n\nTherefore, the final loss function $L$ becomes\n$$\nL = L_V - L_a\n$$\nand the parameters of network are updated through gradient descent method according to $L$.\n\n\\section{Experimental Evaluation}\n\\label{experiments}\n\\subsection{Implementation}\n\\label{implementation}\n\nTo evaluate our approach, we implemented a visual-based grasping system based on the framework shown in Fig.\\ref{framework_figure}. \nThe perception consists of a Mask R-CNN and a PCA procedure, which are executed in pipeline. \nThe input images resized into $600\\times 600$ are fed into the Mask R-CNN and a number of object instances covered with their masks are produced. \nFor each mask produced, PCA is invoked to compute its position and orientation as the output of the perception. \nThe implementation of the Mask R-CNN is based on \\cite{maskrcnn_implementation}. \nInstead of using a pre-trained Mask R-CNN model on general objects datasets such as MSCOCO \\cite{coco}, we train the Mask R-CNN on our own dataset considering detection accuracy. \nTo this end, 1000 images of 21 classes of objects are collected and labeled with their mask ground truth manually by the label tool LabelMe \\cite{labelme}.\n\n\nFor the policy, three fully-connected layers are stacked together to form a PPO agent. \nThe first layer takes as input a six dimension vector concatenating the object position and the robot position and transforms the input into a 512 dimension latent vector.\nAnd the second layer transforms the 512 dimension vector into two streams: a 512 dimension action vector and a 512 dimension value vector. \nThen one stream is transformed into two 3-dimension vectors, representing the parameters of action distribution $\\mu$ and $\\sigma$, while another stream is transformed into a scalar representing the value of current environment state. \nFor efficient training of PPO, we setup a simulation environment in V-REP \\cite{vrep}, as shown in Fig.\\ref{simulation-environment}. \n\\begin{figure}\n\t\\centering \n\t\\includegraphics[scale=1]{simulation.pdf} \n\t\\caption{Simulation environment set up in V-REP \\cite{vrep}. }\n\t\\label{simulation-environment} \n\\end{figure} \nSeven classes of objects from a robotic manipulation benchmarks YCB \\cite{YCB} are used for PPO training in simulation, including a detergent bottle, an orange, a round can, a rectangular can, a cup, a pudding box and an electric drill. \nSince the class and pose of an object in the simulation environment can be obtained directly through software interfaces, PPO could be trained separately, without the help of the perception. \nThe parameters of PPO are learned in a learning rate of $1e-5$ using Adam optimization method \\cite{adam}. \n\n\nBoth the training of PPO and Mask R-CNN is done on a PC with a RTX 2080Ti GPU. \nThe average rewards of PPO training in simulation over 5 runs are shown in Fig.\\ref{train-result}. \nVery impressive results are obtained after about \\textbf{30 minutes} training of PPO, indicating by the red arrow in Fig.\\ref{train-result}. \n\\begin{figure}\n\t\\centering \n\t\\includegraphics[scale=0.5]{train_process.pdf} \n\t\\caption{The process of policy training in simulation. \n\tThe rewards converged very quickly and the model trained has achieved a good performance after about 30 minutes.\n\t The policy model used in all experiments is trained for 200 episodes, as indicated by the red arrow.} \n\t\\label{train-result} \n\\end{figure} \nBy contrast, the interactions with the same number of episodes would take tens of hours for a real physical robot. \nThe training of Mask R-CNN on our own dataset takes about 10 hours. \nThis training may be not necessary since a pre-trained model on general dataset usually works well in many cases. \nIt is worth noting that all the training above does not require a real-world robot and the trained networks will be transferred into a real-world robotic grasping system directly. \n\n\\subsection{Real-world Evaluation} \n\\label{real-world}\n\nThe overall goal of our evaluation is to determine whether the trained networks can enable a real world robot to perform various grasping tasks without any further training. \nTo this end, a number of grasp tasks commonly used in our daily life are designed to evaluate the ability to perform grasping skills and generalization over objects and situations.\nWe use an industrial UR5 robot arm with an RG2 gripper to achieve two-finger rigid grip. A RealSense camera \\cite{realsense} is located 100 cm above the work surface, producing RGB images for input. \nA laptop with a RTX 2080 GPU acceleration is used for real-time robotic control and communication with UR5 via TCP\/IP protocol. \nThe experimental hardware platform is shown in Fig.\\ref{real-world-environment}. \nIt is worth note that the objects used in the experiments are totally different from that of PPO training in simulation. \n\\begin{figure}\n\t\\centering \n\t\\includegraphics[scale=1]{real-world.pdf} \n\t\\caption{The hardware setup of our system. }\n\t\\label{real-world-environment} \n\\end{figure} \n\n\\subsubsection{Sim-to-Real Transfer}\nAs mentioned before, the trained networks including Mask R-CNN and PPO are transferred into our robotic grasping system without any further training. \nWe first examine the behavior of the system in a controlled manner. \nAs shown in Fig.\\ref{corn}, a target object (a corn) is placed on the work surface in various positions and orientations. \nThe robot grasps the target successfully for 20 randomly chosen object locations.\n\\begin{figure}\n\t\\centering \n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.85]{corn.jpg}\n\t}\n\t\\quad\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.85]{corn_grasp.png}\n\t}\n\t\\caption{Sim-to-Real transferring test of the trained policy. \n\t(a) Picking up a corn in various position and orientation. \n\t(b) Picking up a corn with different robotic configurations.}\n\t\\label{corn}\n\\end{figure}\n\nFurthermore, in order to test the robustness of the control policy, we manually introduce external disturbances. \nAs shown in Fig.\\ref{trajectory}, the control policy could find its correct trajectory again and grasp the target successfully after a sudden change on the robot configuration during the robot's execution, exhibiting good stability and robustness.\n\\begin{figure}\n\t\\centering \n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.8]{trajectory-a.png}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.8]{trajectory-b.png}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.8]{trajectory-c.png}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.8]{trajectory-d.png}\n\t}\n\t\\caption{Robustness test of the trained policy. \n\t(a) The robot starts to perform a task. \n\t(b) We forced it manually to an unseen configuration in training. \n\t(c) The robot finds a proper path back to the right way. \n\t(d) The robot grasps the target successfully.}\n\t\\label{trajectory}\n\\end{figure}\n\n\\subsubsection{Multi-object Grasping}\n\\label{multiobj-grasp}\n\n Multi-object grasping is a common task used to measure the performance for a vision based robotic grasping system. \n In our test setting, 10-13 objects are placed randomly on the table and the UR5 robot is requested to pick up all objects sequentially and then put them out of the workspace.\nIn addition, the background color of the work surface is shifted from white into brown or green. Two example test settings in different backgrounds are shown in Fig.\\ref{multiobj}. \nA grasp is successful if an object is grasped and threw aside, while a remove completion means no objects are left on the table. \nWe perform 10 tests for each background and grasp success rate and remove completion rate are presented in Table \\ref{multiobj-result}.\n\\begin{figure}\n\t\\centering \n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.90]{single_green.png}\n\t}\n\t\\quad\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.90]{single_yellow.png}\n\t}\n\t\\caption{Multiple object removing up. \n\t(a) 11 objects in a green background. \n\t(b) 12 objects in a brown background.}\n\t\\label{multiobj}\n\\end{figure}\n\n\\begin{table}\n\n\n\t\\caption{The results of multiple objects remove up. }\n\t\\label{multiobj-result} \n\n\t\\begin{tabular}{ccc}\n\t\n\t\\hline\\noalign{\\smallskip}\n\tscenarios & grasp success & remove completion \\\\\n\t\n\t\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\tbrown & 100\\%(112\/112) & 100\\%(10\/10)\\\\\n\tgreen & 100\\%(120\/120) & 100\\%(10\/10)\\\\\n\tdense & 93.7\\%(104\/111) & 80\\%(8\/10)\\\\\n\t\\noalign{\\smallskip}\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{Clustered object Grasping}\n\\label{in-cluster}\n\nA challenge task in robotic manipulation is to grasp objects clustered closely together. \nAs shown in Fig.\\ref{dense-cluster}, in a cluster scenario, the objects would block each other and some objects may be completely invisible. \nIn such a task, the order of manipulations really matters if we want to remove all the objects in sequence. \nTo decide the ordering of picking up, we define a mask ratio $r$ for each object recognized as follows:\n$$\nr = \\frac{m}{M}\n$$\nwhere $m$ is the recognized mask with possible occlusion and $M$ is the full mask of the object which is pre-determined. \nThe larger the ratio, the more likely the object is to be picked up firstly as it is less occluded by others. \nFor dense cluster scenarios, we perform 10 tests and grasp success rate and remove completion rate are presented in Table \\ref{multiobj-result}. \nThe failure cases occur due to misidentifications by Mask R-CNN because of partially visible objects. \n\n\\begin{figure}\n\t\\centering \n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.90]{cluster_2.png}\n\t}\n \\quad\n \\subfigure[]{\n\t\t\\includegraphics[scale=0.90]{cluster.png}\n }\n \\caption{Dense clustered object removing up. \n (a) Objects blocked each other but every object is visible. \n (b) The red pepper is completely invisible. }\n\t\\label{dense-cluster}\n\\end{figure}\n\n\\subsubsection{Semantic Grasping}\n\\label{semantic-grasp}\n\nIn a semantic grasping task, a robot is instructed to grasp a specified object among a set of candidates. \nThe capability of semantic grasping is essential to allow autonomous robots to perform manipulations in an unstructured environment. \nBenefiting from the power of Mask R-CNN to detect objects, our system first identifies the class of an object before deciding how to pick up.\nSimilar to the experiment setting in multi-object grasping, 10-13 objects are randomly placed on the table for each trial. For each run, we randomly specify one object and simply count the number of successful grasps. \nWe perform five trails and the success rate achieves \\textbf{100\\% (60\/60)}.\n\n\n\\begin{figure*}\n\t\\centering \n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.35]{mouse_1.png}\n\t}\n\t\\quad\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.35]{mouse_2.png}\n\t}\n\t\\quad\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.35]{mouse_3.png}\n\t}\n\t\\quad\n\t\\subfigure[]{\n\t\t\\includegraphics[scale=0.35]{mouse_4.png}\n\t}\n\t\\caption{Moving object grasping. \n\t(a) The fake mouse is in the initial position. \n\t(b) The mouse is moving forward and the robot is changing its control strategy. \n\t(c) The mouse keeps moving and the robot keeps tracing it while gripper moving down simultaneously. \n\t(d) The robot grasps the mouse successfully.}\n\t\\label{moving-mouse}\n\\end{figure*}\n\n\\subsubsection{Moving Object Grasping}\n\\label{semantic-tracking}\n\nGrasping a moving object is still a challenging task in visual-based robotic manipulation \\cite{moving-grasp}. \nOur learning based approach provides a promising way to approach this challenge. \nTo demonstrate the effectiveness of our approach, a case study is conducted in a scenario where a small fake mouse is moving and the robot is ordered to pick up the mouse in motion. \n\n\nTo pick up the moving mouse, our robot needs to be able to track the target continuously and decide to execute a grasping once the action outputted by the control policy is smaller than the preset threshold. \nIn order to further reduce the time between making a grasp decision and closing the gripper, \nwe add a fixed movement in $z$ direction simultaneously with the movement in the $x-y$ plane, instead of moving down in $z$ direction after the decision making. \nThis minor modification significantly improves the successful rate of picking up the mouse in our experiments. \nThe example pictures of robot's execution on picking up a moving mouse are shown in Fig.\\ref{moving-mouse}. \nHowever, due to the computation cost of the system and communication delay between the laptop and the robot, the delay time in our current implementation is about $200ms$, which limits the speed of moving objects in our experiments. \n\n\n\\section{Conclusion and future work}\n\\label{conclusion}\nWe presented a robotic grasping approach that combines visual perception and a DRL based control policy. \nComparing with other alternatives, training on a real robot is avoided by decoupling the control from visual perception with the help of a physical representation of objects, which makes them easier to be trained. \nMoreover, the policy trained in simulation could be transferred to a real system without any further training. Real world experiments on UR5 demonstrate the robustness and generalization over a wide variation in challenging grasping tasks.\nHowever, in this work, we only consider 3DOF grasping in which objects are placed on a table and the grasping height is fixed. \nIn future work, we would like to extend this work to a 6DOF grasping. \nTo do so, it will be important to investigate the pose of gripper in 3D shape perception. \n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{A brief introduction of Tao diagram} \\label{IntroTao}\nA Tao diagram is a generalization of 5d $(p,q)$ web diagram with the critical number of flavors that gives the UV completion as 6d theory. A typical example is 5d $SU(2)$ superconformal theory of $N_f=8$ flavors, which is understood as a circle compactification of 6d E-string theory. Note that the theories of $N_f\\le 7$ have the 5d UV fixed point \\cite{Seiberg:1996bd} while the $N_f=8$ case has 6d UV fixed point. The corresponding $(p,q)$ web configuration has different features than the cases with $N_f\\le7$. Salient features are that the web diagram form as periodic web configuration in a spiral whose period is identified as the instanton factor (squared) of the theory which is in turns identified as the KK mode of the circle compactified 6d theory. Infinite spirals correspond to an infinite KK spectrum coming from the compactification on a circle. The shape resembles a Taoism symbol which is why it is named to be Tao diagram. Suitable 7-brane monodromies give various yet equivalent diagrams, one of which can be made out of $(1,0), (0,1)$ and $(1,1)$ 5-brane charges, which enables one to compute the partition function via the topological vertex technique. As for $SU(2)$ gauge theory of $N_f=8$ flavors, an explicit computation \\cite{Kim:2015jba} was done and reproduced the result of the E-string elliptic genus computation \\cite{Kim:2014dza}. This confirms that the Tao diagram indeed describes a circle compactification of the 6d E-string theory. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=14cm]{taoweb.pdf}\n\\end{center}\n\\caption{An example of Tao diagram for 5d $SU(2)$ theory of $N_f=8$. There are twelve 7-branes of various charges which are the end point of 5-branes of the same charge. The colored dots denote 7-branes and different colors means different charges. Middle: Six branch cuts are chosen such that other remaining 7-branes are supposed to cross as being pulled to infinity. Right: The charges of 7-branes are altered as they crossed the branch cut. A repeated process generates an infinite spiral web. Right: The last step is to pull out 7-branes associated with the branch cuts. In so doing, the Hanany-Witten transition creates 5-branes along the direction that the 7-branes are pulled out such that the newly created 5-branes are bound by one 7-brane. The resulting diagram makes a Tao diagram which possesses a constant period giving rise to KK mode, and infinite web diagram corresponding to infinite KK spectrum so that the Tao diagram describes a circle compactification of 6d SCFT.} \n\\label{Fig:Taodiag}\n\\end{figure}\n\nTo see how infinite spiral shape arises, one first chooses some of the 7-brane branch cuts, and then pulls out other 7-branes along the geodesic direction given by the charge of 7-brane in the $(p,q)$ web plane. As 7-branes cross the branch cuts, the charges of the 7-branes change according to the monodromy associated with the branch cut. For instance, see Figure \\ref{Fig:Taodiag}. When the number of flavors is critical, all the pulled-out 7-branes inevitably cross all the chosen branch cuts so that the charge of a pulled-out 7-brane becomes the same after one revolution making a spiral shape of a constant separation, or a constant period. The resulting diagram is a web diagram of infinite spiral. \n\nThe constant period is expressed in terms of the instanton factor of the 5d theory. It is expected for a circle compactification of a 6d theory, and the constant period can be identified with the KK mode of the compactification. Hence, it possesses a compactified circle, more precisely, the radius of the T-dual circle. \n\n\n\nFor higher rank gauge theory or quiver theories, the configuration is basically same as shown in the Tao diagrams in the main text.\n\n\nTao web diagrams also give a computational tool using topological vertex methods.\n\n\n\\section{Discussions}\n\\label{sec:discussion}\n\nWe have started with 6d ${\\cal N}=(1, 0)$ superconformal field theories whose Type IIA brane representations are made of NS5-branes, D6-branes, D8-branes and a single $O8^-$-plane. \nOn their tensor branch, they are in general written as a linear quiver diagram with various matter hypermultiplets. \nAfter a circle compactification and T-duality, we have found a very rich group of 5d ${\\cal N}=1$ gauge theories that are dual to each other. \nThis diversity of the 5d theories for a given 6d theory originates from the choice in the resolution of two $O7^-$ planes and in the distribution of D5 and D7-branes as well as a part of the $SL(2,\\mathbb{Z})$ duality. \nA decoupling of the same flavors from the dual 5d theories leads to another duality between 5d gauge theories which would have the identical 5d superconformal field theory. \n\n\nWhile we provided the argument for the duality between 5d gauge theories by starting with the brane picture of their 6d mother theory, one can have different class of 6d superconformal field theories whose brane picture is not in the class considered here. There are also some 6d superconformal field theories with no obvious brane picture. \nIt would be interesting to consider their 5d counterparts and the corresponding 5d dualities. \n\n\n\n\\section{Introduction and Summary}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n6d $\\mathcal{N}=(1, 0)$ superconformal field theories (SCFTs) possess various mysterious and interesting features. For example, they may not have an ultraviolet (UV) Lagrangian description. They may have tensionless self-dual strings as fundamenal degrees of freedom. Still, they are expected to be well-defined local qunatum field theories. Therefore, a better understanding of the 6d SCFTs will deepen our understandings of quantum field theories. Various aspects of 6d $\\mathcal{N} = (1,0)$ SCFTs have been analyzed recently, for example, the classification from F-theory \\cite{Heckman:2013pva, Gaiotto:2014lca, DelZotto:2014hpa, Heckman:2014qba, Heckman:2015bfa, Bhardwaj:2015xxa}, the elliptic genus computation of the self-dual strings \\cite{Haghighat:2013gba, Haghighat:2013tka, Haghighat:2014pva, Kim:2014dza, Haghighat:2014vxa, Gadde:2015tra, Haghighat:2015ega}, the anomaly polynomials \\cite{Ohmori:2014pca, Ohmori:2014kda, Intriligator:2014eaa, Heckman:2015ola, Heckman:2015axa, Cordova:2015fha, Bobev:2015kza} and also their torus compactifications \\cite{Ohmori:2015pua, DelZotto:2015rca, Ohmori:2015pia} as well as compactifications on other Riemann surfaces \\cite{Gaiotto:2015usa, Franco:2015jna, Hanany:2015pfa}. \n\nAnother important way of analyzing a 6d $\\mathcal{N}=(1, 0)$ SCFT is to use a 5d $\\mathcal{N}=1$ supersymmetric quantum field theory whose UV completion is the 6d SCFT. From a 6d SCFT, we first move to a tensor branch where scalars in tensor multiplets acquire a vacuum expectation value (VEV), and then perform a circle compactification with gauge or flavor Wilson lines along the $S^1$. After the compactification, we obtain a certain 5d $\\mathcal{N}=1$ supersymmetric quantum field theory. From this point of view, the Kaluza-Klein (KK) modes become instantons in the 5d theory. In other words, the KK modes are dynamically generated in the 5d theory, and they becomes massless at the UV fixed point, namely we recover the 6d SCFT at the UV fixed point of the 5d theory. Although this picture is useful for the analysis of 6d SCFTs, it is typically difficult to identify what is a 5d description of a 6d SCFT on $S^1$. \n\nA new way to see a direct connection between a 6d SCFT and a 5d theory was discovered in \\cite{Hayashi:2015fsa}. It starts from a brane cofiguration in Type IIA string theory developed in \\cite{Brunner:1997gf, Hanany:1997gh}, realizing a 6d $Sp(N)$ gauge theory with $N_f = 2N+8$ hypermultiplets in the fundamental representation with one tensor multiplet coupled. The fixed point of the 6d theory is the $(D_{N+4}, D_{N+4})$ minimal conformal matter theory. We then compactify the 6d theory on $S^1$ with Wilson lines, and perform T-duality along the $S^1$. Then the brane configuration in Type IIA string theory becomes a 5-brane web in Type IIB string theory. Moreover, it possible to see the 5-brane web diagram yields a 5d $SU(N+2)$ gauge theory with $N_f = 2N+8$ hypermultiplets in the fundamental representation. The web diagram has a particular feature of infinitely expanding spiral shape. Hence, it is in a class of so-called Tao web diagram introduced in \\cite{Kim:2015jba}, which has been conjectured to give a 5d theory whose UV completion is a 6d SCFT. This way gives us a direct connection between the 6d $(D_{N+4}, D_{N+4})$ minimal conformal matter theory and the 5d $SU(N+2)$ gauge theory with $2N+8$ flavors\\footnote{The same claim that the UV completion of the 5d $SU(N+2)$ gauge theory with $2N+8$ flavors is the 6d $(D_{N+4}, D_{N+4})$ minimal conformal matter was also obtained in \\cite{Yonekura:2015ksa} by the analysis of the instanton operator in the 5d theory. The instanton operator analysis is also a useful way to explore 6d SCFTs from 5d theories \\cite{Tachikawa:2015mha, Zafrir:2015uaa, Yonekura:2015ksa, Gaiotto:2015una}. The quantization under a one-instanton background can yield instanton states. We may construct a Dynkin diagram of an enhanced flavor symmetry from the instanton states. If the Dynkin diagram is affine, then the 5d theory is conjectured to have a 6d UV completion.}. \n\nThe main aim of this paper is to generalize the connection between the 6d SCFT and the 5d theory to a broader class of 6d SCFTs realized by a brane configuration in Type IIA string theory constructed in \\cite{Brunner:1997gf, Hanany:1997gh}. The system consists of NS5-branes, D6-branes, D8-branes and an $O8^-$-plane. The D6-branes are divided into pieces by NS5-branes and D8-branes introduce some hypermultiplets in the fundamental representation. The worldvolume theory on the D6-branes yield a 6d theory on a tensor branch of a 6d SCFT. It is possible to apply the new method to various 6d SCFTs that can be constructed by the brane system with an $O8^{-}$-plane in Type IIA string theory. The 6d theories on the tensor branch of the 6d SCFTs we will consider are in the following two classes\\footnote{In our notation, $[k]_{R}$ means $k$ hypermultiplets in the representation $R$. When $R$ is the fundamental representation, we will omit it for simplicity.}\n\\begin{equation}\n6d \\; Sp(N) - SU(2N+8) - SU(2N+16) - \\cdots - SU(2N+8(n-1)) - [2N+8n], \\label{6dquiver1}\n\\end{equation} \nand \n\\begin{equation}\n6d \\; [1]_{A} - SU(N) - SU(N+8) - SU(N+16) - \\cdots - SU(N+8(n-1)) - [N+8n], \\label{6dquiver2}\n\\end{equation}\nwhere $n$ is a positive integer and $A$ stands for the anti-symmetric representation. With the sufficient number of the flavors at the end node of the quiver theories, it is possible to Higgs the theories as in \\cite{Gaiotto:2014lca}. The renormalization group (RG) flow triggered by the Higgs vev yields a different 6d theory at low energies. The gauge theories may have different ranks and fundamental hypermultiplets are attached to various nodes, depedning on the Higgsing. We will consider such two large families of 6d SCFTs realized by Type IIA branes with an $O8^-$-plane. As in the previous case, we find that a circle compactification of the system with T-duality along the $S^1$ always yields a Tao web diagram leading to a certain 5d theory which have the 6d UV fixed point. Furthermore, the Tao web diagram enables us to read off a gauge theory description of the 5d theory.\n\nIn the process of going down to 5d from 6d, we find that there are several ambiguities and different choices yield 5d theories with different gauge groups and matter. We claim that those 5d theories are dual to each other in the sense that they have the same UV completion as a 6d SCFT. The connection between the brane configuration in Type IIA string theory and that in Type IIB string theory by T-duality gives us a direct way to see how the dual 5d theories arise from a 6d SCFT. We find that there are three different origins for the 5d dualities. \n \n The first type of the 5d duality that we argue comes from an ambiguity of whether we resolve two $O7^-$-planes or one $O7^-$-plane after the T-duality. The starting 6d brane setup includes an $O8^-$-plane, and hence the T-duality induces two $O7^-$-planes. An $O7^-$-plane can be split into two 7-branes by the non-perturbative effects in the string coupling \\cite{Sen:1996vd}. Since we have two $O7^-$-planes,we may split the two $O7^-$-planes or only one of the two $O7^-$-planes\\footnote{Keeping two $O7^-$-planes will not give a 5d description due to the explicit exisitence of the $S^1$.}. In fact, the two descriptions express two different 5d theories. However, since the two web diagrams are essentially the same diagrams,the two theories are dual to each other with the same 6d UV completion. The difference is simply a matter of how we read off the 5d gauge theory description from the 5-brane web diagram. From the dual 5d theories which have the same 6d UV fixed point, we can decouple flavors one by one by sending the masses to $\\pm \\infty$, leading to 5d theories which have the 5d UV fixed point. In particular, if we decouple the flavors in the dual 5d theories exactly in the same way, the resulting 5d theories should be also dual to each and have the same 5d UV fixed point. Hence, we can also obtain 5d dualities in the sense that the dual 5d theories have the same 5d UV fixed point. The 5d $Sp-SU$ duality proposed in \\cite{Gaiotto:2015una} is a very particular example in the class of dual pairs that we consider in this paper. \n\nThe second type of the 5d duality that we claim arises from an ambiguity of how we allocate D5-branes after the T-duality. In the Type IIA brane configuration with an $O8^-$-plane, the fundamental region is either the left side or the right side of the $O8^-$-plane since the transverse space to the $O8^-$-plane is a real one-dimensional space. After the T-duality, the $O8^-$-plane becomes two $O7^-$-planes and the transverse to the $O7^-$-planes is a real two-dimensional space. Each $O7^-$-plane induces a point-symmetric 5-brane web configuration with respect to the point where the $O7^-$-plane is located in the two-dimensional plane. Then, the fundamental region is a region bounded by two parallel infinite mirror lines with each line passing through a different $O7^-$-plane. In this case, it is possible to allocate different numbers of D5-branes between the left side and the right side of a line passing through the two $O7^-$-planes. After splitting the two $O7^-$-planes, the different distributions give a different 5d quiver theory with different ranks of the gauge groups. Again, the different 5d theories have the same 6d UV fixed point and they are dual to each other. Although this allocation may sound weird, it is also possible to show the duality at the level of the 5-brane web diagrams. In fact.the different distributions may be understood by certain mass deformations in their S-dual pictures. Again, decoupling matter in the same way in the dual 5d theories yield another 5d dualities in the sense that the dual 5d theories have the same 5d UV fixed point. \n\nThe third type of the 5d dualities is a different from the previous two types since we will utilize the $SL(2,\\mathbb{Z})$ duality in Type IIB string theory. The rotation of a 5-brane web in a real two-dimensional space is a part of the $SL(2,\\mathbb{Z})$ duality. In parcular, S-duality corresponds to the rotation by $90$ degrees. Therefore, the $90$ degrees rotatoin of any 5-brane webs obtained after the T-duality from the 6d brane configuration leads to another dual picture. We also find that there is another dual frame corresponding to a rotation by $45$ degrees or the $TST$-duality. After the $45$ degrees rotation, the 5-brane web may still admit a 5d gauge theory description. Therefore, this rotation gives another class of dual 5d theories. \n\nFrom the 6d quiver theories of \\eqref{6dquiver1} and \\eqref{6dquiver2} and their Higgsed theories, we find that combinations of the three types of the 5d dualities can give rise to various dual pairs of 5d theories which have the same 6d UV fixed point. Furthermore, it is interesting to note that we will find a sort of exotic examples after a chain of the dualities. Namely, those exotic 5d theories do not show an enhancement of the flavor symmetry to an affine type when one applies the instanton operator analysis performed in \\cite{Yonekura:2015ksa}. However, this is not a contradiction since the analysis in \\cite{Yonekura:2015ksa} only includes instanton states whose total instanton number is $1$. Therefore, we argue that those exotic 5d theories should also show the enhancement of the flavor symmetry to an affine type when we include instanton states with the higher instanton number.\n\nAmong the 6d theories \\eqref{6dquiver1} and \\eqref{6dquiver2} with various Higgsings, we find that it is particularly interesting to focus on a case where the Type IIA brance configuration includes $8$ D8-branes with an $O8^-$-plane on top of each other after the Higgsing. An important point in this case is that the $8$ D8-branes with the $O8^-$-plane become two groups of $4$ D7-branes with an $O7^-$-plane after the T-duality. The combination of $4$ D7-branes with an $O7^-$-plane is special in the sense that the S-dual of the group gives the same 7-brane configuration. Namely, we can still interpret it as $4$ D7-branes and an $O7^-$-plane even after the S-duality. Due to this feature, we find that the resulting S-dualized 5d quiver may have an $SU$ gauge node with an anti-symmetric hypermultiplet or an $Sp$ gauge node at either or both ends of the quiver, depending on the splitting type of an $O7^-$-plane. Namely, these special 6d theories yield another class of dual 5d theories which were not obtained in other cases. \n\nAs described above, we can relate various brane configurations in Type IIA string theory to Tao web diagrams in Type IIB string theory. We also note that there is another type of Tao web diagrams that do not arise from the T-duality from a brane system in Type IIA string theory. The 5d theories from that class of the Tao web diagrams can be obtained by adding some flavors at the end nodes of the 5d linear quiver theory realizing the 5d $T_N$ theory \\cite{Kim:2015jba}. We will call the 5d theory as 5d ``$T_N$ Tao'' theory. Although the Tao web diagram for a general 5d $T_N$ Tao theory does not have a direct connection to a Type IIA brane system, we will propose a 6d description of the 5d $T_N$ Tao theory by matching the global symmetry as well as the number of certain multiplets between the two theories. \n\nThe organization of this paper is as follows. In section \\ref{sec2:6dspN}, we first start the discussion of the 5d dualities that arise from the simplest 6d setup corresponding to \\eqref{6dquiver1} with $n=1$. We then move onto the next simplest example that is \\eqref{6dquiver2} with $n=1$ in section \\ref{sec:6dSUNA}. In this case, we will see the apperance of the three types of the 5d dualities as well as an exotic example. In section \\ref{sec:general}, we extend our analysis to 5d dualities for general 6d quiver theories of \\eqref{6dquiver1} and \\eqref{6dquiver2} and also the ones after various Higgsings. In section \\ref{sec:speicalcases}, we focus on a special case where the Type IIA brane configuration has $8$ D8-branes and an $O8^-$-plane on top of each other. In this case, we find that the dual 5d theories can become another class that has not been obtained in the other sections.\nIn section \\ref{sec:TNTao}, we propose a 6d description of the 5d $T_N$ Tao theory and we find that it is not generically written by a Type IIA brane setup.\nAppendix \\ref{IntroTao} gives a brief introduction of Tao web diagrams. \n\n\n\nWhile we are writing this paper, a related work \\cite{Zafrir:2015rga} has appeared recently.\n\n\n\n\n\n\n\n \n\n\n\\section{6d \\texorpdfstring{\\boldmath $Sp(N)$}{Sp(N)} gauge theory with\n \\texorpdfstring{\\boldmath $N_f = 2N+8$}{Nf} and one tensor multiplet}\\label{sec2:6dspN}\nVarious brane constructions in Type IIA string theory for 6d anomaly free theories were done in \\cite{Hanany:1997gh,Brunner:1997gf} in terms of an O8 orientifold, D8-branes, D6-branes, and NS5-branes. In this section, we analyze a circle compactification of a 6d $Sp(N)$ theory with $2N+8$ hypermultiplets in the fundamental representation and a tensor multiplet whose brane configuration has an $O8^-$ orientifold. The T-duality along the $S^1$ can give a 5d $SU(N+2)$ with $2N+8$ flavors as done in\n\\cite{Hayashi:2015fsa}, However, we also claim that the the picture also yields\na 5d $Sp$ gauge theory with the same rank as well as the same number of flavors. There was an interesting observation that the superconformal indices for a 5d $SU(N+2)$ gauge theory with $N_f$ flavors and the Chern-Simons (CS) level $\\kappa = N+4 - \\frac{N_f}{2}$ and a 5d $Sp(N+1)$ gauge theory with $N_f$ flavors are equivalent, and the two theories are dual to each other \\cite{Gaiotto:2015una}. We attempt to give an account for the equivalence\nfrom the perspective of different pattern of the resolution of $O7^-$ orientifold planes in Type IIB string theory. As the\ndecoupling limit of flavors yields another 5d theories of 5d UV fixed point, such a dual picture still holds for resulting 5d theories\nwith less flavors if one decouples exactly the same flavors in the both pictures. \nDepicted in the $(p,q)$ 5-brane web diagram, the resulting 5d theories also show S-dual picture.\n\nConsider the 6d $\\mathcal{N}=(1,0)$ $Sp(N)$ gauge theory with $N_f = 2N+8$ flavors and one tensor multiplet. The UV fixed point for this theory is known as 6d $(D_{N+4}, D_{N+4})$ minimal conformal matter theory. We compactify this theory on a circle to obtain 5d theories. Along different Wilson lines, we will see that we either get 5d $SU(N+2)_0$ theory with the same number of flavors and the zero CS level\\footnote{From here on, we use the following short hand notation for denoting the CS level of the $SU(n)$ theory: $SU(n)_\\kappa$ is an $SU(n)$ theory of the CS level $\\kappa$.}, or 5d $Sp(N+1)$ with the same number of flavors. Although the introduction of the Wilson line breaks the global symmetry, in the symmetric phase, both have the same global symmetry as that of its 6d mother theory, $SO(4N+16)$. We note that as discuss in \\cite{Hayashi:2015fsa}, the 5d global symmetry has additional $U(1)_I$ factor\nassociated to the KK modes\nWe discuss two cases on $(p,q)$ 5-brane web diagrams in details.\n\n\n\\subsection{5d \\texorpdfstring{$SU(N+2)_0$}{SU(N+2)} theory with \\texorpdfstring{$N_f=2N+8$}{Nf=2N+8} flavors}\n\\label{subsec:su(N+2)Nf}\nA circle compactification of 6d $\\mathcal{N}=(1,0)$ $Sp(N)$ gauge theory with $N_f = 2N+8$ flavors and a tensor multiplet, leading to 5d $SU(N+2)_0$ gauge theory with $N_f = 2N+8$, is discussed with great detail in \\cite{Hayashi:2015fsa}. We here review its brane configuration. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{HZ1.pdf}\n \\end{center}\n\\caption{Left: Type IIA brane realization of the $(D_{N+4}, D_{N+4})$ minimal conformal matter in the tensor branch. Right: The quiver diagram of the 6d theory.} \n\\label{Fig:6dspN}\n\\end{figure}\nIn Type IIA description, 6d brane configuration \\cite{Hanany:1997gh, Brunner:1997gf} for $\\mathcal{N}=(1,0)$ $Sp(N)$ gauge theory with $N_f = 2N+8$ flavors and a tensor multiplet consists of $2N$ D6 branes (its mirror images included) suspended between a NS5 brane and its mirror NS5 brane through an orientifold plane $O8^-$ together with $(2N+8)$ D8 branes on top of $O8^-$, as depicted in Figure \\ref{Fig:6dspN}. The brane configuration in the ten-dimensional space time in Type IIA string theory is summarized in Table \\ref{Tb:TypeIIAbrane}. Therefore, the horizontal direction of the two-dimensional plane in Figure \\ref{Fig:6dspN} is $x^6$ and the vertical direction is one of $x^7, x^8, x^9$. \n\\begin{table}[t]\n\\begin{center}\n\\begin{eqnarray}\n \\begin{array}{c|c c c c c c |c| c c c}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\\\\n \\hline\n \\text{D6-brane} & \\times & \\times & \\times & \\times & \\times & \\times & \\times &&& \\\\\n \\text{NS5-brane} & \\times & \\times & \\times & \\times & \\times & \\times & &&& \\\\\n \\text{D8-brane} & \\times & \\times & \\times & \\times & \\times &\\times& &\\times&\\times&\\times \\\\\n \\text{O8-plane} & \\times & \\times & \\times & \\times & \\times &\\times & &\\times&\\times&\\times \n \\end{array} \\nonumber\n \\end{eqnarray}\n\\end{center}\n\\caption{The brane configuration that yields a 6d theory on a tensor branch of a 6d SCFT in Type IIA string theory.}\n\\label{Tb:TypeIIAbrane}\n\\end{table}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{O7ToSU.pdf}\n \\end{center}\n\\caption{Type IIB brane descriptions for 6d ${\\mathcal N}=(1,0)$ $Sp(N)$ gauge theory with one tensor multiplet and $2N+8$ flavors in the fundamental representation, which yields ${\\mathcal N}=1$ $SU(N+2)$ gauge theory with the number of flavors. The horizontal direction is $x^6$ and the vertical direction is the T-dualized direction. For simplicity, $N=1$. Left: The brane configuration with two $O7^-$ planes. A dashed line denotes the branch cut of 7-brane, while a dash-dot line denotes the mirror reflection cut of an $O7^-$ plane. \nMiddle: The brane configuration with two $O7^-$ planes which are non-perturbatively resolved. Here we denote $\\mathbf{A}$ for a D7 brane, $\\mathbf{B}$ for a $[1,-1]$ 7-brane, and $\\mathbf{C}$ for a $[1,1]$ 7-brane. Right: Pulling out 7-branes gives rise to a web configuration for $SU(3)$ gauge theory with $N_f=10$ flavors and zero CS level.} \n\\label{Fig:IIB}\n\\end{figure}\n\nTo have the 5d picture, we compactify one of the direction, for example $x^5$, parallel to the world volume of D6-branes on $S^1$ with a Wilson line, and take T-dual along the compactified direction. The resulting 5-brane configuration involves $N$ D5-branes, two NS5-branes, The $(2N+8)$ D7-branes and two $O7^-$-planes are maximally separated apart along the T-dualized circle. The Wilson line putting $N$ D5-branes away from the $O7^-$-planes may break the 6d $Sp(N)$ gauge group to 5d $U(N)$.\nThe non-perturbative effect in string theory can resolve the $O7^-$-planes into a pair of two $[p,q]$ 7-branes subject to the monodromy condition that the monodromy of the $O7^-$-plane with four D7-branes is the minus of the identity\n\\cite{Sen:1996vd}. A suitable choice would be a pair of $[1,1]$ and $[1,-1]$ 7-branes. \nAs one resolves the $O7^-$ planes, the fundamental domain of the 5-brane web diagram that is confined by ``the mirror reflection\nlines'' is expanded to the two-dimensional plane while and the boundary made by NS5 branes with the mirror reflection\nlines forms ``5-brane loops'' \\cite{DeWolfe:1999hj}. (See Figure \\ref{Fig:IIB}.)\n\nGiven such a $(p, q)$ 5-brane configuration with 7-branes and 5-brane loops, one has a 5-dimensional gauge theory description by pulling all the 7-branes to infinity, while keeping the gauge coupling finite as pulling out all the 7-branes. 5-branes are naturally generated as a result of the Hanany-Witten effect such that a 7-brane is attached to the end of an external 5-brane. Hence, $2N+8$ D7-branes give rise to semi-infinite $2N+8$ D5-branes yielding the flavor branes of the theory. Due to the charge conservation at a junction of the web diagram, 5-brane loops induce two more color D5-branes. This means that the resulting 5d gauge theory in this process has $N+2$ color branes, implying that the rank of the resulting 5d gauge theory is increased by one. For this, one can also argue that the 6d tensor multiplet contribution is\nconverted to a 5d vector multiplet giving one more rank to the 5d gauge group. It is worth noting that the obtained 5d theory has zero CS level. As the CS level can be measured as the angle difference of two vertically inclined 5-branes playing a role of NS5-brane in the web configuration, the angle between the two upper external 5-branes and the angle between the two lower external 5-branes are\nthe same\nas the one between as shown in Figure \\ref{Fig:IIB}, confirming the zero CS level of the 5d theory. Therefore, the resultant 5d theory is $\\mathcal{N}=1$ $SU(N+2)_0$ gauge theory with $N_f=2N+8$ flavors.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{su3tao.pdf}\n\\end{center}\n\\caption{Tao diagram for 5d $\\mathcal{N}=1$ $SU(3)$ gauge theory with $N_f=10$ flavors which is a circle compactification of 6d $\\mathcal{N}=(1,0)$ $Sp(1)$ gauge theory with one tensor multiplet and $10$ flavors. The rightmost brane configuration on Figure \\ref{Fig:IIB} can be reorganized to be the the leftmost web configuration given here, using 7-brane monodromies. Pulling out 7-branes passing through the branch cut of other 7-branes gives a Tao wed diagram on the rightmost.}\n\\label{Fig:su3tao}\n\\end{figure}\n\n\nThe web diagram for this case was already studied by the authors in \\cite{Hayashi:2015fsa} and is named as a Tao web diagram. Such Tao diagram obtained by pulling all the 7-branes outside has a necessary structure to be interpreted as a circle compactification of the 6d theory. Its shape is of a spiral infinitely rotating, but\nwith a constant period, as given in Figure \\ref{Fig:su3tao}. This period can be expressed in terms of the parameter of the 5d theory, instanton factor, which is an expected property for a circle compactification of 6d theory. The spectrum coming from infinitely\nexpanding 5-branes are naturally identified as the KK spectrum of the compactification. For a short introduction, see Appendix \\ref{IntroTao} or \\cite{Kim:2015jba, Hayashi:2015fsa}.\n\n\n\\subsection{5d \\texorpdfstring{ $Sp(N+1)$}{Sp(N+1)} theory with \\texorpdfstring{ $N_f=2N+8$}{Nf=2N+8} flavors}\n\\label{subsec:sp(N+1)Nf}\nWhile the quantum resolution of both $O7^-$ planes leads to the 5d $SU(N+2)$ gauge theory with $2N+8$ flavors, one may resolve only one of the $O7^-$ planes. In this process of reducing the 6d theory to 5d, the Wilson line is introduced such that all the D5-branes are located\nabove the unresolved $O7^-$ plane, to retain an $Sp$ gauge group. As discussed earlier, a resolution of the $O7^-$ plane into a pair of two 7-branes induces an extra color D5 brane when 7-branes are pulled out to infinity.\nHence, the resulting 5d theory is $\\mathcal N=1$ $Sp(N+1)$ gauge theory with the same number of flavors, $N_f=2N+8$. The increase of the rank of the gauge group can be again due to the contribution of one 6d tensor multiplet.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{O7ToSp.pdf}\n\\end{center}\n\\caption{Type IIB brane descriptions for 6d ${\\mathcal N}=(1,0)$ $Sp(N)$ gauge theory with one tensor multiplet and $2N+8$ flavors in the fundamental representation, which yields ${\\mathcal N}=1$ $Sp(N+1)$ gauge theory with the number of flavors. For simplicity, $N=1$. Left: The brane configuration with two $O7^-$ planes. Middle: The brane configuration with only one of two $O7^-$ planes is resolved. Right: The resulting $Sp(2)$ gauge theory with $N_f=10$ flavors.\n}\n\\label{Fig:O7ToSp}\n\\end{figure}\nA brane setup for the 5d $Sp(N+1)$ gauge theory in the Coulomb branch is given in Figure \\ref{Fig:O7ToSp}.\n\n\nAs $\\pi_4(Sp(n))=\\mathbb{Z}_2$, there is a discrete theta parameter for $Sp(n)$ gauge theory. This parameter can be interpreted as two inequivalent resolutions of the $O7^-$ plane if the $Sp(n)$ gauge theory has no flavors. When the $Sp(n)$ gauge theory has flavors, the effect of the discrete theta parameter can be absorbed by a redefinition of mass parameters and the difference is not physical \\cite{Intriligator:1997pq}. Therefore, one can choose whatever splitting of the $O7^-$-plane. For example, one can resolve the $O7^-$ plane with a pair of $[1,1]$ and $[1,-1]$ 7-branes as chosen earlier, and one can also resolve it with a pair of $[2,1]$ and $[0,1]$ 7-branes.\n\n\n\n\n\n\\subsection{Equivalence between Sp an SU theories and their flavor decoupling limits}\n\\label{subsec:equivalence.ch2}\n\nIn the preceding subsections, we have discussed a circle compactification of 6d $\\mathcal N=(1, 0)$ $Sp(N)$ gauge theory with $N_f=2N+8$ flavor and one tensor multiplet in two ways with respect to the resolution of the $O7^-$-planes as well as the Wilson lines: one of which leads to the 5d $SU(N+2)$ theory involving the resolution of two $O7^-$-planes, and the other leads to the 5d $Sp(N+1)$ theory involving the resolution of only one $O7^-$-plane. Of course, both have the same number of flavors as the 6d theory, as the flavors are associated with D7-branes which come from a T-dual of D8-branes in the brane picture of the 6d theory. Note also that global symmetry for both theories is $SO(4N+16)\\times U(1)_I$ where $SO(4N+16)$ is the global symmetry of 6d theory and $U(1)_I$ comes from the circle compactification.\n\nAs both have the same UV completion as the 6d SCFT, $(D_{N+4}, D_{N+4})$ minimal conformal matter theory, and the same global symmetry, we claim that these two theory are dual to each other at the UV. Some physical observables like index functions would be equivalent. For instance, it would be interesting to see the partition functions, or elliptic genera, of both theories do agree or not.\n\nInterestingly, the superconformal index for the 5d $SU(N+2)$ theory and the $Sp(N+1)$ theory with less flavors $N_f\\le 2N+7$ which have the 5d UV fixed point,\nwas computed \\cite{Gaiotto:2015una}, showing the equivalence of the two indices. We also claim that this equivalence can be easily explained if one considers the flavor decoupling of the 5d theories with the critical flavors. In the web diagram, one can take the mass decoupling limit by deforming the mass of a flavor to an extreme value of either $\\infty$ or $-\\infty$. \n\nWe note that in order to hold the 5d dualities for the less flavor cases, the CS level of $SU(N+2)$ theory should be either maximum or minimum. This comes from the fact that we need to decouple the flavors exactly in the same way between the 5d $SU(N+2)$ gauge theory and the 5d $Sp(N+1)$ gauge theory. For example, from the Figure \\ref{Fig:O7ToSp}, we can take the mass of the flavors only to $+\\infty$ for the $Sp(N+2)$ gauge theory. Therefore, we should also take the same decoupling limit for the 5d $SU(N+2)$ gauge theory.\n\n\n\n\n\\subsection{S-dualities}\n\\label{subsec:Sdual.ch2}\nIn the $(p, q)$ 5-brane web diagram, the S-duality structure of the theory is manifest as one rotates the $(p, q)$ 5-brane web\nby 90 degrees, namely, D5-branes become NS5-branes and NS5-branes become D5-branes. A simplest example would be the S-duality of the $SU(3)$ theory which yields the quiver theory of $SU(2)\\times SU(2)$. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=13cm]{su2su2.pdf}\n\\end{center}\n\\caption{The S-dual of $SU(3)$ theory with $N_f=10$ flavor is $SU(2)\\times SU(2)$ quiver theory with $N_f=4$ flavors to each $SU(2)$ theory of the quiver.\n}\n\\label{Fig:su2su2}\n\\end{figure}\nFor instance, the brane configuration on the right of Figure \\ref{Fig:IIB} describes a 5d $SU(3)$ theory with $N_f=10$ flavors, and its S-dual theory is the $[4]-SU(2)\\times SU(2)-[4]$ quiver theory which has four flavors\n\\footnote{The middle configuration of Figure \\ref{Fig:su2su2} has two ``$SU(1)$''s linked to each $SU(2)$ instead of four flavors. Such $SU(1)$ has the instanton factor which can be used to express the period of the Tao web diagram. One the other hand, using 7-brane monodromy analysis, it can be shown that such $SU(1)$ factor together with a bi-fundamental hypermultiplet connecting between the $SU(1)$ and $SU(2)$ also represents two flavors.}\n(denoted in the square bracket ``$[~]$'') \non each gauge group as in Figure \\ref{Fig:su2su2}. In such Tao diagrams of Figure \\ref{Fig:IIB}, as discussed, the constant period of spiral shape is expressed in terms of the instanton factor of the gauge theory. It holds for the S-dual picture, the constant period is expressed in terms of instantons factors of the quiver theory \\cite{Hayashi:2015fsa}. This S-duality generalizes to higher rank cases so that the S-dual of the 5d $SU(N+2)$ gauge theory with $N_f = 2N+8$ flavors is given by the $SU(2)$ quiver theory\n\\begin{align}\n[4] - SU(2) - \\cdots - SU(2) - [4],\n\\end{align}\nwhere there are $N+1$ $SU(2)$ gauge nodes.\nThe period for these Tao diagram is expressed in terms of a product of the instanton factors. \n\nWe summarize the 5d theories when compactifying the 6d $Sp(N)$ gauge theory with $N_f = 2N+8$ and one tensor multiplet, on a circle:\n\\begin{enumerate}[(i)]\n\\item $SU(N+2)-[\\,2N+8\\,]$, ~$SU(N+2)$ gauge theory with $N_f = 2N+8$ flavors,\n\\item $Sp(N+1)-[\\,2N+8\\,]$, ~$Sp(N+1)$ gauge theory with $N_f = 2N+8$ flavors,\n\\item $[4] - SU(2) - \\cdots - SU(2) - [4]$ with $N+1$ $SU(2)$ gauge nodes.\n\\end{enumerate}\nThe (i) and (ii) are obtained by resolving either both two $O7^-$ planes or only one $O7^-$ plane when reducing to 5d, and (iii) is S-dual of (i). Therefore, (iii) is also (S-)dual to (ii).\n\n\n\\paragraph{Decoupling flavors}\nAs discussed in \\cite{Hayashi:2015fsa}, one can decouple flavors of a given Tao diagram, which yields 5d supersymmetric gauge theory which has the 5d UV fixed point. In $Sp(N+1)$ or $SU(N+2)$ gauge theories, we can decouple the the same number of flavors. The same idea applies to the $SU(2)$ quiver theory. \nThe resulting flavor\ndecoupled theories from $SU(N+1)$, $Sp(N+1)$, and the $SU(2)$ quiver theories are all dual to each other, and have the same 5d UV fixed point.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=13cm]{massdef9F.pdf}\n\\end{center}\n\\caption{Mass decoupling limit. (a) $SU(3)_\\frac{1}{2}$ theory with $N_f=9$ flavors and the CS level $\\frac12$; (b) $Sp(2)$ theory with $N_f=9$ flavors and $k=0$; (c) $[3]-SU(2)-SU(2)-[4]$ quiver theory.\n}\n\\label{Fig:massdef9F}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{6d \\texorpdfstring{$SU(N)$}{SU} gauge theory with \\texorpdfstring{$N_f = N+8$}{Nf} and \\texorpdfstring{$N_a = 1$}{Na}}\n\\label{sec:6dSUNA}\n\n\nIn this section, we consider six dimensional $SU(N)$ gauge theory with $N_f = N+8$ flavor and $N_a = 1$ hypermultiplet in antisymmetric tensor representation together with one tensor multiplet. \nThis theory has $SU(N+8) \\times U(1)$ anomaly free global symmetry.\n\nWhen $N$ is small, there can be the enhancement. \nWhen $N=3$, the anti-symmetric representation is equivalent to the (anti-)fundamental representation. \nTherefore, this theory is interpreted as SU(3) gauge theory with 12 flavors.\nIn this case, the global symmetry $SU(11) \\times U(1)$ actually enhances to $SU(12)$.\nWhen $N=4$, anti-symmetric representation is the real representation.\nTherefore, we can consider the $SU(2)$ global symmetry acting on two half hypermultiplets.\nIn this case, the global symmetry $SU(12) \\times U(1)$ actually enhances to $SU(12) \\times SU(2)$.\n \n\n\nDiscussion analogous to that in section \\ref{sec2:6dspN} is possible also in this case. \nWe discuss various 5d description corresponding to this 6d theory.\n\n\n\\subsection{5d \\texorpdfstring{$SU\\times SU$}{SUSU} gauge theory with flavors}\\label{subsec:SUSU}\n\n\n6d $SU(N)$ gauge theory with $N_f = N+8$, $N_a = 1$ and with one tensor multiplet\nis realized by the brane set up depicted in Figure \\ref{Fig:6dSUa}.\nCompared to Figure \\ref{Fig:6dspN}, there is an additional NS5-brane on top of the $O8^-$ plane.\nMoreover, odd number of D6-branes are also allowed in this case.\n\n\n\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=5cm]{IIA6dSUa.pdf}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=5cm]{6dSUA.pdf}\n\\end{minipage}\n\\caption{Left: Type IIA brane set up for 6d \\texorpdfstring{$SU(N)$}{SU} gauge theory with \\texorpdfstring{$N_f = N+8$}{Nf} and \\texorpdfstring{$N_a = 1$}{Na}. Right: Corresponding quiver diagram.} \n\\label{Fig:6dSUa}\n\\end{figure}\n\nWe compactify one of the direction which NS5-brane extends and take T-duality along this direction.\nThen, we obtain the IIB brane setup depicted in Figure \\ref{Fig:5SUSU},\nwhere we chose the fundamental region to be the half of the compactified $S^1$ direction.\nCompared to \\ref{Fig:IIB}, we have additional $(p,q)$ 5-brane connecting two $O7^-$ planes at the middle.\nWe have $n$ D5-branes at the left hand side \nand the $N-n$ D5-branes at the right hand side of this 5-brane.\nThis ``distribution'' labeled by $n$ depend on the Wilson line of the $SU(N)$ gauge group \nwhich we introduce when we compactify on $S^1$.\nThe charge of these 5-branes which are attached to the $O7^-$ planes depends on this distribution.\nThe D7-branes are also distributed in analogous way into $n'$ and $N+8-n'$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{SUaTdual.pdf}\n\\caption{T dual description of Figure \\ref{Fig:6dSUa}.} \n\\label{Fig:5SUSU}\n\\end{figure}\n\n\nHere, we would like to consider the quantum resolution for one of the $O7^-$ plane based on \\cite{Sen:1996vd}.\nHowever, the decomposition in this case is more non-trivial than in section \\ref{sec2:6dspN}\nbecause the $O7^-$ planes are attached to the 5-branes.\nThe decomposition of the $O7^-$ plane which is attached to the 5-brane is proposed in \\cite{Bergman:2015dpa}.\nWe should choose the charge of the two 7-branes in such a way that one of the 7-brane is attached to the 5-brane to which the $O7^-$ plane was originally attached.\nFor example, when the $O7^-$ plane is attached to $(1,-1)$ 5-brane as depicted in the left of Figure \\ref{O7dec},\n the $O7^-$ plane should be decomposed into $B=[1,-1]$ 7-brane and $C=[1,1]$ 7-brane,\n where $B$ is attached to the (1,-1) 5-brane.\nSuppose that the D5-brane is attached to this $(1,-1)$ 5-brane from the right side as in Figure \\ref{O7dec}.\nBy moving $B$ along this $(1,-1)$ 5-brane and by using Hanay-Witten transition,\nit is also possible to transform into the brane configuration, \nwhere $B$ and $C$ are separated by the NS5-brane as depicted in the right of Figure \\ref{O7dec}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{O7dec.pdf}\n\\caption{Fig:Resolution of the $O7^-$ plane attached to $(1,-1)$ 5-brane. $[1,-1]$ 7-brane is denoted as $B$ and $[1,1]$ 7-brane is denoted as $C$.} \n\\label{O7dec}\n\\end{figure}\n\n\n\n\nNow we assume that the new 5-branes are generated due to the quantum resoludtion of $O7^-$ plane\n and the 5-brane loop is closed analogous to the discussion in section \\ref{sec2:6dspN}.\nThe resulting brane configuration is depicted in the left of Figure \\ref{Fig:5dquiv}.\nAfter moving one of the D7-branes from each side, respectively, across one of the 7-branes\ngenerated from the $O7^-$ plane as depicted in the arrow in the figure,\nwe move all the 7-branes to outside.\nThen we obtain the 5-brane web diagram depicted in the right of Figure \\ref{Fig:5dquiv}.\nOnly when the newly generated 5-branes are D5-brane, \nwe obtain the five dimensional field theory description.\nThis happens only when the two parameters $n$ and $n'$,\nwhich parametrize the distribution of the color D5-branes \nand flavor D7-branes, respectively, are related as $n'=3n+4-N$.\nIn the following, we assume that the Wilson line for the flavor $SU(N_f)$ is \nintroduced in such a way that this distribution is realized.\nIn this case, we find that the brane configuration in the right of Figure \\ref{Fig:5dquiv} is for \nthe quiver gauge theory,\n\\begin{eqnarray}\n[3n+3-N] - SU(n+1)_0 - SU(N+1-n)_0 - [2N+3-3n].\n\\label{eq:5dSUSU}\n\\end{eqnarray}\nThe integer $n$ should be chosen in such a way that\nall the flavors and ranks of the gauge groups of this theory should be positive.\nAs long as $n$ satisfies such condition,\nwe claim that the 5d theory with any $n$ is a good 5d description of the \n6d $SU(N)$ gauge theory with $N_f = N+8$, $N_a = 1$ and with one tensor multiplet compactified on $S^1$.\nIn other words, we claim that the theory with different $n$ have identical 6d UV fixed point.\nWe denote this equivalence as ``distribution duality''.\nWe will discuss this duality more in detail in section \\ref{subsec:distribution}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=14cm]{SUa7in-1.pdf}\n\\caption{Left: The two $O7^-$ planes in Figure \\ref{Fig:5SUSU} is resolved. Right: Web-diagram obtained by moving all the 7-branes outside.} \n\\label{Fig:5dquiv}\n\\end{figure}\n\n\n\n\n\\subsection{5d \\texorpdfstring{$SU(N+1)$}{SU} gauge theory with \\texorpdfstring{$N_f = N+8$}{Nf} and \\texorpdfstring{$N_a = 1$}{Na}}\\label{5dNa=1}\n\nAnalogous to section \\ref{subsec:sp(N+1)Nf}, we can decompose only one of the $O7^-$ plane out of the two\nin Figure \\ref{Fig:5SUSU}.\nIn this case, we obtain the brane configuration depicted in the left of Figure \\ref{Fig:5dSUa}.\nAfter the analogous process done in Figure \\ref{Fig:5dquiv}, \nwe obtain the web diagram in the right of Figure \\ref{Fig:5dSUa}.\nThis configuration corresponds to adding the flavor branes to the one discussed in \\cite{Bergman:2015dpa}.\nThis diagram is interpreted as the 5d $SU(N+1)_0$ gauge theory with $N_f = N+8$ flavor and $N_a = 1$ antisymmetric tensor.\nTherefore, we conclude that \n6d $SU(N)$ gauge theory with $N_f = N+8$, $N_a = 1$ and with one tensor multiplet compactified on $S^1$ is also described by 5d $SU(N+1)_0$ gauge theory with $N_f = N+7$ flavor and $N_a = 1$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=14cm]{5dSUa1-1.pdf}\n \\caption{5d $SU(N+1)_0$ gauge theory with $N_f = N+7$ and $N_a = 1$} \n\\label{Fig:5dSUa}\n\\end{figure}\n\n\n\n\n\\subsection{Equivalence of the two 5d description and their flavor decoupling limits}\\label{subsec:mass dec}\n\nIn the previous two subsections, we studied two ways of 5d description for the 6d $SU(N)$ gauge theory with $N_f = N+8$, $N_a = 1$ and with one tensor multiplet. That is\n\\begin{enumerate}[(i)]\n\\item 5d $[3n+3-N] - SU(n+1)_0 - SU(N+1-n)_0 - [2N+3-3n]$.\n\\item 5d $SU(N+1)_0$ gauge theory with $N_f = N+7$ and $N_a = 1$. \n\\end{enumerate}\nWe interpret these two types of 5d description to be dual theory in a sense that they have the same UV fixed point.\n\nFurthermore, by considering the flavor decoupling limit by taking some of the mass parameters to be infinity, we will again obtain the dual theories, which have again the identical five dimensional fixed point.\n\nOn one hand, we can give heavy masses to $n_1$ out of $3n+3-N$ and $n_2$ out of $2N+3-3n$ flavors coupling to $SU(n+1)_0$ and $SU(N+1-n)_0$, respectively. \nThen, the flavors reduces to $3n+3-N-n_1$ and $2N+3-3n-n_2$, respectively.\nSuppose that all these masses are positive infinity, which corresponds to moving these $n_1$ and $n_2$ flavor branes upward in Figure \\ref{Fig:5dquiv}.\nIn this case, the CS level of each gauge group, which were original zero, change to $n_1$ and $n_2$, respectively.\n\nOn the other hand, \nwhen we consider \nthe corresponding flavor decoupling limit in Figure \\ref{Fig:5dSUa},\nthe flavor reduces to $N_f = N+7-(n_1+n_2)$.\nMoreover, the CS level changes to $n_1-n_2$.\nIn summary, we obtain the following 5d duality:\n\\begin{enumerate}[(i)]\n\\item 5d $[3n+3-N-n_1] - SU(n+1)_{n_1} - SU(N+1-n)_{n_2} - [2N+3-3n-n_2]$.\n\\item 5d $SU(N+1)_{n_1-n_2}$ gauge theory with $N_f = N+7-(n_1+n_2)$ and $N_a = 1$. \n\\end{enumerate}\n\n\n\\subsection{S-dualities}\\label{subsec:SUA-Sdual}\n\nBy using S-duality, we can obtain further 5d description which have the identical UV fixed point.\nIn this subsection, we consider the S-dual description of the theory discussed in section \\ref{subsec:SUSU}.\nAs an example, we consider $N=6$ and $n=3$, which is 5d quiver gauge theory\n\\begin{eqnarray}\n[6] - SU(4)_{0} - SU(4)_{0} - [6].\n\\end{eqnarray}\nThe corresponding web diagram is written in the left of Figure \\ref{Fig:Sdual90}.\n\nParallel to section \\ref{subsec:Sdual.ch2}, we first consider the S-duality transformation \nexchanging D5-branes and NS5-branes to each other,\nwhich corresponds to to 90 degree rotation of the diagram.\nWe denote this type of S-duality as ``S-transformation''.\nActing S-transformation to the left diagram in Figure \\ref{Fig:Sdual90},\nwe obtain the diagram written in the middle of Figure \\ref{Fig:Sdual90}.\nAfter moving $[1,1]$ 7-branes and $[1,-1]$ 7-branes and by using the Hanany-Witten transition,\nwe obtain the following 5d quiver gauge theory\n\\begin{eqnarray}\n[5] - SU(3)_{0} - SU(3)_{0} - SU(3)_{0} - [5]\n\\label{333}\n\\end{eqnarray}\nas depicted in the right of \\ref{Fig:Sdual90}.\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{0.3\\hsize}\n\\includegraphics[width=5cm]{S1.pdf}\n\\end{minipage}\n\\begin{minipage}{0.36\\hsize}\n\\includegraphics[width=6cm]{S2.pdf}\n\\end{minipage}\n\\begin{minipage}{0.30\\hsize}\n\\includegraphics[width=5cm]{S3.pdf}\n\\end{minipage}\n\\caption{S-transformation of the 5d quiver gauge theory $[6] - SU(4)_{0} - SU(4)_{0} - [6]$.}\n\\label{Fig:Sdual90}\n\\end{figure}\n\n\n\n\nHere, we can consider another type of S-duality,\nwhich transform D5 branes, NS5 branes, and $(1,1)$ 5-branes \ninto $(1,-1)$ 5-branes, NS5 branes, and D5 branes, respectively.\nWe denote this as ``STS transformation'', which roughly corresponds to 45 degree rotation of the diagram.\nWe start from the same theory, which is \nexpressed as the left of Figure \\ref{Fig:Sdual45}.\nIt is related to the left of Figure \\ref{Fig:Sdual90} by \na simple Hanany-Witten transition to move D7-branes.\nAfter taking the STS transformation and by moving $[0,1]$ 7-branes,\nwe obtain the 5d quiver gauge theory\n\\begin{eqnarray}\n[3] - SU(2)\n- {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(3)_0}} \n- {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(3)_0}}\n - SU(2) - [3].\n \\label{2332}\n \\end{eqnarray}\nIn this way, we obtain several 5d description by different types of S-duality transformation.\n\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=5cm]{S4.pdf}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=5cm]{S5.pdf}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=5cm]{S6.pdf}\n\\end{minipage}\n\\caption{STS-transformation of the 5d quiver gauge theory $[6] - SU(4)_{0} - SU(4)_{0} - [6]$.}\n\\label{Fig:Sdual45}\n\\end{figure}\n\n\nHere, we comment a dependence of the resulting theory on the mass parameters.\nInstead of starting from the left of Figure \\ref{Fig:Sdual90} or Figure \\ref{Fig:Sdual45},\nwe could have started from the diagram in Figure \\ref{Fig:shifted}.\nThis is the diagram of the same theory with different mass parameters.\nAlthough the original 5d description is identical up to the value of the mass parameters,\nwe will obtain the different result under the S-duality transformation.\nIn this case, it is straightforward to see that S-transformation gives \n5d quiver gauge theory (\\ref{2332}) rather than (\\ref{333}).\nIt is worth emphasizing that \ndepending on the value of the mass parameters of the flavors,\nthe same 5d gauge theory can be mapped to different gauge theories \nunder the identical S-duality transformation.\nAnalogously, \nthe STS-transformation gives the theory (\\ref{333})\nrather than (\\ref{2332}). \n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4cm]{Shifted.pdf}\n\\caption{Same theory with different mass parameter}\n\\label{Fig:shifted}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Distribution duality}\\label{subsec:distribution}\n\nIn section \\ref{subsec:SUSU}, we discussed that 5d quiver gauge theory (\\ref{eq:5dSUSU})\nwith different value of $n$ have identical 6d UV fixed point.\nIn this subsection, we discuss that more in detail on this distribution duality\nfrom the viewpoint of type IIB 5-brane web diagram.\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=15cm]{deriv1.pdf}\n\\caption{Derivation of the distribution duality.}\n\\label{Fig:Distr}\n\\end{figure}\n\nWe start from the left diagram in Figure \\ref{Fig:Distr},\nwhich describes the 5d quiver gauge theory (\\ref{eq:5dSUSU}).\nThen, we move one of the $[0,1]$ 7-brane along the arrow.\nIn this process, lower part of the 5-brane web changes their $(p,q)$ charges because they pass through \nthe monodromy cut generated by this $[0,1]$ 7-brane. \nThis changes the structure of the web diagram and the resulting \ndiagram actually corresponds to \n\\begin{eqnarray}\n[3n+6-N]-SU(n+2)_0-SU(N-n)_0-[2N-3n].\n\\label{eq:5dSUSU2}\n\\end{eqnarray}\nWe find that this theory is the one obtained by shifting $n$ by 1 in the theory (\\ref{eq:5dSUSU}).\nBy repeating this procedure, we can show the quiver gauge theory (\\ref{eq:5dSUSU}) or (\\ref{eq:5dSUSU2})\nwith arbitrary $n$ are dual as long as all the numbers of flavor and the rank of the gauge groups are positive.\n\n\nBy considering the the flavor decoupling limit discussed in section \\ref{subsec:mass dec}, \nit is also straightforward to show that this distribution duality holds also for the \ntheories with 5 dimensional UV fixed points.\nSince the process depicted in Figure \\ref{Fig:Distr} does not depend on \nthe upper part of the diagram, the discussion is parallel even after we \ndecouple $n_1$ and $n_2$ flavors coupling to the two gauge groups.\nWe see that the following two theories are again related by the analogous procedure\n\\begin{itemize}\n\\item 5d $[3n+3-N-n_1]-SU(n+1)_{n_1}-SU(N+1-n)_{n_2}-[2N-3n+3-n_2]$ \n\\item 5d $[3n+6-N-n_1]-SU(n+2)_{n_1}-SU(N-n)_{n_2}-[2N-3n-n_2]$ \n\\end{itemize}\nWe can further consider moving the flavor branes upward.\nThus, this class of theories with all the possible value $n$ are related by the distribution duality.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{deriv2.pdf}\n\\caption{Derivation of the distribution duality 2.}\n\\label{Fig:Distr2}\n\\end{figure}\n\n\n\n\nUp to here, we have discussed the flavor decoupling limit, which corresponds to moving the flavor branes upward in Figure \\ref{Fig:Distr}.\nWe can also consider another type of flavor decoupling limit corresponding to moving flavor branes downward.\n\nAs an example, suppose that we move one of the flavor branes coupled to $SU(N+1-n)$ downward in the left of Figure \\ref{Fig:Distr} to downward.\nThen, we obtain the left diagram in Figure \\ref{Fig:Distr2}.\nBy considering the analogous procedure, we find the following two are related \n\\begin{itemize}\n\\item 5d $[3n+3-N] - SU(n+1)_{0} - SU(N+1-n)_{-\\frac{1}{2}} - [2N-3n+2]$ \n\\item 5d $[3n+5-N] - SU(n+2)_{-\\frac{1}{2}} - SU(N-n)_{0} - [2N-3n]$ \n\\end{itemize}\nContrary to the previous case, they are not related by shifting $n$ any more.\nHowever, this duality keeps the total number of flavors, total number of gauge ranks,\nand total number of CS levels.\nIn this sense, this is also the analogue of the distribution duality.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{deriv3.pdf}\n \\end{center}\n \\caption{Derivation of the distribution duality 3.} \n\\label{Fig:Distr3}\n\\end{figure}\n\nIt would be also straightforward to see the following duality from Figure \\ref{Fig:Distr3} :\n\\begin{itemize}\n\\item 5d $[3n+3-N] - SU(n+1)_{0} - SU(N+1-n)_{-1} - [2N-3n+1]$ \n\\item 5d $[3n+4-N] - SU(n+2)_{-1} - SU(N-n)_{0} - [2N-3n]$ \n\\end{itemize}\n\n\nIt would be also possible to combine them with the flavor decoupling limit\n to move $n_1$ and $n_2$ flavor branes upward.\nTherefore, the discussion here can be summarized as the following duality:\n\\begin{itemize}\n\\item 5d $[3n+3-N-n_1] - SU(n+1)_{n_1} - SU(N+1-n)_{n_2-\\frac{k}{2}} - [2N-3n+3-k-n_2]$ \n\\item 5d $[3n+6-N-k-n_1] - SU(n+2)_{n_1-\\frac{k}{2}} - SU(N-n)_{n_2} - [2N-3n-n_2]$ \n\\end{itemize}\nfor $k=0,1,2$.\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=10cm]{dualmass.pdf}\n\\caption{Distribution duality in the S-dual frame}\n\\end{figure}\n\nAlthough the processes to move the $[0,1]$ 7-brane as in Figure \\ref{Fig:Distr}, \\ref{Fig:Distr2}, or \\ref{Fig:Distr3}\nmay look exotic, it looks often quite natural when we see this in the S-dual frame.\nTake Figure \\ref{Fig:Distr3} as an example and consider the S-transformation for both.\nThen, we observe that both diagrams correspond to the identical 5d theory in the form\n\\begin{eqnarray}\n\\cdots \n- {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(3)}} \n-SU(2)-[1].\n\\end{eqnarray}\n\nThe deformation of moving the $[0,1]$ 7-brane in Figure \\ref{Fig:Distr3}\nis translated to moving the D7-brane.\nThis D7-brane corresponds to the flavor charged under the $SU(3)$ gauge group\nand moving it simply corresponds to changing the mass of this flavor.\nTherefore, the ``distribution duality'', which change the distribution of the rank of the gauge group in the original frame,\nis actually just the mass deformation in the S-dual frame.\nThis is essentially the same observation seen in section \\ref{subsec:SUA-Sdual} that \ndepending on the value of the mass parameters of the flavors,\nthe same 5d gauge theory can be mapped to different gauge theories \nunder the identical S-duality transformation.\n\n\n\n\\subsection{Chain of duality and exotic example}\\label{subsec:chain}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=12cm]{Chain.pdf}\n\\caption{Chain of dualities.}\n\\label{chain}\n\\end{figure}\n\nCombining what we studied in section \\ref{subsec:distribution}\nand section \\ref{subsec:distribution},\nwe demonstrate that varous 5d quiver gauge theories have identical 6d UV fixed point.\n\nWe again consider the 5d theories corresponding to \n6d $SU(6)$ with $N_f=14$ and $N_a=1$ and with a tensor multiplet.\nIn Figure \\ref{chain}, we write various web diagrams,\nwhich are related to each other either by mass deformation \nor the distribution duality, which is sometimes interpreted also as the mass deformation in the S-dual frame.\n\n\nIn the original frame, we see that \n\\begin{eqnarray}\nI, \nI\\hspace{-.15em}I,\nI\\hspace{-.15em}I\\hspace{-.15em}I\n&:& [6] - SU(4)-SU(4) -[6]\n\\nonumber \\\\\nI\\hspace{-.15em}V, \nV,\nV\\hspace{-.15em}I\n&:& [9] - SU(5)-SU(3) -[3]\n\\nonumber \\\\\nV\\hspace{-.15em}I\\hspace{-.15em}I, \nV\\hspace{-.15em}I\\hspace{-.15em}I\\hspace{-.15em}I, \nI\\hspace{-.15em}X,\nX\n&:& [12] - SU(6)-SU(2) \n\\nonumber \\\\\nX\\hspace{-.15em}I,\nX\\hspace{-.15em}I\\hspace{-.15em}I\n&:& [12] - SU(7) - ``SU(1)'' \n\\end{eqnarray}\nThe last one includes $``SU(1)''$ factor and thus, does not have the standard field theory description\nand moreover, does not look fit with the distribution duality discussed previously.\nHowever, by using the analogous process to move one of the $[0,1]$ 7-brane\nas in Figure \\ref{Fig:Distr}, \\ref{Fig:Distr2}, or \\ref{Fig:Distr3},\nwe can show that diagram X and diagram XI are actually connected by such process.\nWe note that $I\\hspace{-.15em}I \\leftrightarrow I\\hspace{-.15em}V$\nand $V\\hspace{-.15em}I \\leftrightarrow V\\hspace{-.15em}I \\hspace{-.15em}I \\hspace{-.15em}I $\nare also related by the distribution dualities.\n\nIn the S-dual frame, we see that \n\\begin{eqnarray}\nI &:& [5] - SU(3)_0 - SU(3)_0 - SU(3)_0 - [5]\n\\nonumber \\\\\nI\\hspace{-.15em}I, I\\hspace{-.15em}V\n&:& [3] - SU(2)\n- {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(3)_0}} \n- {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(3)_0}} \n-SU(2) -[3]\n\\nonumber \\\\ \nI\\hspace{-.15em}I\\hspace{-.15em}I \n&:& [3] - SU(2)-SU(2)-SU(3)_0-SU(2)- SU(2) - [3]\n\\nonumber \\\\ \nV &:& [3] - SU(2)-SU(2)-SU(3)_{\\frac{1}{2}}-SU(3)_{0} - [5]\n\\nonumber \\\\\nV\\hspace{-.15em}I\\hspace{-.15em}I&:& [3] - SU(2)-SU(2)-SU(3)_{1}-SU(2)- SU(2) - [3]\n\\nonumber \\\\\nV\\hspace{-.15em}I\\hspace{-.15em}I\\hspace{-.15em}I&:& [3] - SU(2)-SU(2)-SU(2)-SU(2)- SU(3)_{1} - [3]\n\\nonumber \\\\\nX\\hspace{-.15em}I\\hspace{-.15em}I &:& [3] - SU(2)-SU(2)-SU(2)-SU(2)\n- {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(2)}} \n-SU(2)\n\\end{eqnarray}\nWe omitted the case where the diagram does not have clear field theory interpretation.\nIn this way, we obtain various 5d quiver gauge theories which have identical 6d UV fixed point.\n\nHere, we comment on the global symmetry.\nThese theories are all expected to have $SU(14)$ global symmetry at UV fixed point.\nIndeed, we can explicitly check that 7-brane monodromy analysis for these diagrams all show this expected global symmetry.\nWhen we apply the method \\cite{Yonekura:2015ksa} using instanton operators,\nsome of these nicely shows affine $SU(14)$ symmetry and thus,\nconsistent with the claim that these 5d theories have 6d UV fixed point with $SU(14)$ global symmetry.\nHowever, strange to say, application of \\cite{Yonekura:2015ksa} to some of these 5d quiver gauge theories \n fails to reproduce the expected affine structure.\nFor example, the S-dual description of diagram III, \nthe central gauge node $SU(3)$ has flavor $N_f < 2N$ and thus, does not show affine symmetry.\nWe claim that one-instanton analysis is not enough in this case.\nIt would be interesting to generalize the method with instanton operators\nto include the higher instanton contribution for these case to check the \nexpected affine global symmetry is obtained.\n\n\n\n\n\n\n\n\n\\section{5d dualities from 6d: Generalization}\n\\label{sec:general}\n\nIn this section, we consider a general 6d quiver theory constructed by NS5-branes, D6-branes, D8-branes an $O8^-$-plane introduced in \\cite{Brunner:1997gf, Hanany:1997gh}. It is straightforward to repeat essentially the same analysis done in section \\ref{sec2:6dspN} and \\ref{sec:6dSUNA} for the general 6d quiver theories. We will see that the 5d reduction yields various 5d dualities. \n\n\\subsection{6d \\texorpdfstring{$Sp- \\prod SU$}{sp-su} quiver theories}\n\\label{sec:6dSpSUquiver}\n\nWe first focus on 6d quiver theories with one $Sp$ gauge node, which is a generalization of the 6d quiver considered in section \\ref{sec2:6dspN}. The simple generalization of the setup in section \\ref{sec2:6dspN} realizes a 6d linear quiver \n\\begin{equation}\n6d \\; Sp(N) - SU(2N+8) - SU(2N+16) - \\cdots - SU(2N+8(n-1)) - [2N+8n], \\label{6dcanonical1}\n\\end{equation}\nand its Type IIA brane configuration is given in Figure \\ref{Fig:6dquiver-1}. The case with $n=1$ falls in to the case in section \\ref{sec2:6dspN}. Hence, we concentrate on the cases with $n > 1$. The global symmetry is generically $SU(2N+8n) \\times U(1)$. The number of the tensor multiplets is $n$ and the total rank of the gauge groups is $N(2n-1) + 4n^2-5n+1$.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{O8SUquiver.pdf}\n\\end{center}\n\\caption{Type IIA brane configuration for the 6d linear quiver theory \\eqref{6dcanonical1}.} \n\\label{Fig:6dquiver-1}\n\\end{figure}\n\n\nWe can further generalize the 6d quiver theoy by Higgsing. The flavor symmetry of the brane construction of the canonical 6d quiver \\eqref{6dcanonical1} is associated to a symmetry on semi-infinite $2N+8n$ D6-branes at the end. It is also possible to let one semi-infinite D6-brane become finite and end on one D8-brane. Since each D6-brane end on one D8-brane, we have $2N+8$ D8-branes in total. In this picture, each D6-brane at the end connect an NS5-brane with a D8-brane. A Higgs branch of the 6d theory arises by attaching one D6-brane on two or more D8-branes. Since the condition of preserving supersymmetry implies that only one D6-brane can connect an NS5-brane with a D8-brane \\cite{Hanany:1996ie}, the D6-brane should connect to other D6-brane between NS5-branes by jumping some NS5-branes as in Figure \\ref{Fig:Higgs}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{Higgs.pdf}\n\\end{center}\n\\caption{A process of the 6d Higgsing.} \n\\label{Fig:Higgs}\n\\end{figure}\nThen, there appears fractionated D6-branes between D8-branes. The motion of the D6-branes between D8-branes describes the vev of the corresponding hypermultiplets. After sending the fractionated D6-branes into infinity, we have decoupled D8-branes and hence the flavor symmetry is reduced. Due to the appearance of the jumping of D6-branes over NS5-banres, the ranks of some gauge groups are also reduced. \n\n\nIn general, such a Higgsing is classified by a Young diagram as in \\cite{Gaiotto:2014lca}. When one represents a Young diagram by a vector with non-increasing numbers, each component represent the number of D6-branes ending on D8-brane. The total number of the boxes in the Young diagram is $2N+8n$. Therefore, the brane configuration in Figure \\ref{Fig:6dquiver-1} corresponds to a Young diagram $[1, \\cdots, 1]$ with $2N+8n$ entries of $1$'s. A general Higgsing corresponds to a Young diagram $[n, \\cdots, n, n-1, \\cdots, n-1, \\cdots, 2, \\cdots, 2, 1, \\cdots, 1]$ where the number of $l$ is $k_l$ with a condition $\\sum_{l=1}^{n}lk_l = 2N+8n$. The 6d gauge theory content can be read off by moving D8-branes to the left in the brane configuration until no D6-branes end on the D8-branes. Depending on the Young diagram, various fundamental hypermultiplets are introduced to some middle gauge nodes in the 6d quiver. Hence, after a Higgsing corresponds to the Young diagram $[n, \\cdots, n, n-1, \\cdots, n-1, \\cdots, 2, \\cdots, 2, 1, \\cdots, 1]$, we obtain the following 6d quiver theory at low energies \n\\begin{equation}\n6d \\; {\\overset{\\overset{\\text{\\large$[k_{n}]$}}{\\textstyle\\vert}}{Sp(N) }} - {\\overset{\\overset{\\text{\\large$[k_{n-1}]$}}{\\textstyle\\vert}}{SU(2N+8-k_n)}} - \n\\cdots - \n{\\overset{\\overset{\\text{\\large$[k_1]$}}{\\textstyle\\vert}}{SU(2N+8(n-1) - \\sum_{l=1}^{n-1}lk_{l+1})}}. \\label{6dHiggs1}\n\\end{equation}\nType IIA brane configuration is depicted in Figure \\ref{Fig:6dquiver-2}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{O8SUquiverhiggs.pdf}\n\\end{center}\n\\caption{Type IIA brane configuration for the 6d linear quiver theory \\eqref{6dHiggs1}.} \n\\label{Fig:6dquiver-2}\n\\end{figure}\n\n\nWe will consider a circle compactification of the 6d quivers of \\eqref{6dcanonical1} and \\eqref{6dHiggs1}, and see the 5d descriptions of the theories as well as the 5d dualities. \n\n\\subsubsection{5d \\texorpdfstring{$SU$}{SU} quviers}\n\\label{subsubsec:5dsuquiver1}\n\nWe first focus on an $S^1$ compactification of the canonical 6d quiver theory \\eqref{6dcanonical1}. The T-duality along the $S^1$ gives us a 5-brane configuration with two $O7^-$-planes as in Figure \\ref{Fig:Brane6dcanonical1}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{6dsuquiver1.pdf}\n\\end{center}\n\\caption{Type IIB brane configuration after performing a T-duality to Figure \\ref{Fig:6dquiver-1}.} \n\\label{Fig:Brane6dcanonical1}\n\\end{figure}\nAfter the T-duality, we have the distribution ambiguity discussed in section \\ref{subsec:SUSU}. The number of D5-branes in the middle column is always $N$, which corresponds to the first gauge node of the 6d quiver \\ref{6dcanonical1}.\nThe $N+8$ color D5 branes originated from the 2nd gauge node in 6d quivers \\ref{6dcanonical1} \nare distributed into the next columns, which are left and right to the center, respectively.\nThe number of the D5-branes in these columns can change by the distribution ambiguity. \nWe label the ambiguity by a non-negative number $m$, and set the number of the D5-branes in the left column to the middle to be $N+4+m$ and the number of the D5-branes in the right column to the middle to be $N+4-m$. \n\nThe $N+8i$ color D5 branes corresponding to the $(i+1)$-th gauge node in 6d quivers \\ref{6dcanonicalA} are distributed into the $i$-th left columns and $i$-th right columns from the center.\nHere, we need to care about the condition that the resulting 5-brane web diagram after pulling all the 7-branes outside has a 5d gauge theory interpretation. In fact, this condition may determine the number of D5-branes in the column next to the column which is next to the middle one. By repeating this analysis, it turns out that there is only one degree of freedom for the ambiguity parameterized by $m$. The resulting distribution is depicted in Figure \\ref{Fig:Brane6dcanonical1}. \n\nNext task is the resolution of the $O7^-$-planes. As in section \\ref{sec2:6dspN}, we can consider either splitting both $O7^-$-planes or splitting only one of the two $O7^-$-planes. We consider the former case first. The condition that the final 5-brane web after pulling out all the 7-braness should have a 5d gauge theory interpretation also constrains the splitting type of the $O7^-$-planes. The relative difference between the two splitting type is not important, and hence we fix the splitting type of the upper $O7^-$-plane to be ${\\bf B}\\; {\\bf C}$. Then, the splitting type of the lower $O7^-$-plane is determined to be ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ in order that the resulting 5-brane web diagram after pulling all the 7-branes outside has a 5d gauge theory interpretation. The resolution creates the $n$ 5-brane loops as in Figure \\ref{Fig:5braneloop}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{6dsuquiver2.pdf}\n\\end{center}\n\\caption{The 5-brane web diagram after splitting two $O7^-$-planes compared to Figure \\ref{Fig:Brane6dcanonical1}. Note that the slope of the lines is schematically depicted and does not represent the corresponding 5-brane charge precisely.} \n\\label{Fig:5braneloop}\n\\end{figure}\nThen, we move the branch cuts of ${\\bf B}, {\\bf C}$ and ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ so that some of the D5-branes and D7-branes cross the branch cuts as in Figure \\ref{Fig:5braneloop2}. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{6dsuquiver3.pdf}\n\\end{center}\n\\caption{The motion of the branch cuts of ${\\bf B}, {\\bf C}$ and ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$. Relatively, it can be realized by moving some amount of D5-branes and D7-branes. The D5-branes and the D7-branes that move are indicated by arrows in this figure.} \n\\label{Fig:5braneloop2}\n\\end{figure}\nIn the upper part of the diagram, D7-branes and D5-branes cross the branch cuts of the {\\bf B} and {\\bf C} 7-branes in a counterclockwise and clockwise direction respectively. Then the D7-branes become $[0,1]$ 7-branes, and the D5-branes become NS5-branes. On the other hand, in the lower part of the diagram, D7-branes and D5-branes cross the branch cuts of the ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ 7-branes in a counterclockwise and clockwise direction respectively. Then, the D7-branes and the D5-branes become $[m, -1]$ 7-brane and $(m, -1)$ 7-brane respectively. After crossing the branch cuts, the 5-brane loops are divided by $2n$ vertical lines as in Figure \\ref{Fig:5braneloop3}. This is the origin of 5d quiver theories with $2n-1$ gauge nodes. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{6dsuquiver4.pdf}\n\\end{center}\n\\caption{The schematic diagram after moving the branch cuts of ${\\bf B}, {\\bf C}$ and ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ 7-branes from Figure \\ref{Fig:5braneloop}. The red lines arise from D5-branes which cross the branch cuts of ${\\bf B}, {\\bf C}$ and ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ 7-branes. The $[0,1]$ 7-branes denoted by the yellow circles in the upper part of the diagram arise from D7-branes which cross the branch cuts of ${\\bf B}, {\\bf C}$ 7-branes. On the other hand, the $[m,-1]$ 7-branes denoted by the black circles in the lower part of the diagram arise from D7-branes which cross the branch cuts of ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ 7-branes. } \n\\label{Fig:5braneloop3}\n\\end{figure}\n\n\nAfter pulling out all the 7-branes, the 5-brane web diagram becomes the one in Figure \\ref{Fig:Brane6dcanonical2}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{6dsuquivern2.pdf}\n\\end{center}\n\\caption{The 5-brane web diagram after pulling out all the 7-branes outside of the 5-brane loops from the one in Figure \\ref{Fig:5braneloop2}. For simplicity, we write a web diagram of $n=2$.} \n\\label{Fig:Brane6dcanonical2}\n\\end{figure}\n The 5d gauge theory realized by the 5-brane web is then\n \\begin{eqnarray}\n{\\overset{\\overset{\\text{\\large$[L_1]$}}{\\textstyle\\vert}}{SU(N_1) }} - SU(N_{2}) - \\cdots - SU(N_{n-1}) - SU(2N+2n) - SU(M_{n-1}) - \\cdots - SU(M_{2}) - {\\overset{\\overset{\\text{\\large$[R_1]$}}{\\textstyle\\vert}}{SU(M_1) }}\\nonumber\\\\\n\\label{5dsuquiverfromSpquiver}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nN_l &=& N+2n+(n-l)m,\\qquad \\text{for}\\quad 1 \\leq l \\leq n-1,\\\\\nM_l &=& N+2n -(n-l)m,\\qquad \\text{for}\\quad 1 \\leq l \\leq n-1,\\\\\nL_1 &=& N+2n+2+nm,\\\\\nR_1&=&N+2n+2-nm.\n\\end{eqnarray}\nThe parameters are constrained such that the rank of each gauge group or the number of the flavors should be greater than zero at least. \n\nThe number of the Coulomb branch moduli can be easily counted and it is $(2n-1)(N+2n-1)$ which is the sum of the number of the tensor multiplets and the number of the vector multiplets in the Cartan subalgebra of the 6d quiver theory \\eqref{6dcanonical1}. The global symmetry analysis by 7-branes should recover $SU(2N+8n)$ since the very first 5-brane web gives $2N+8n$ D7-branes on top of each other\\footnote{Note that a D7-brane is mutually local to an $O7^-$-plane. D7-branes do not change its change when it crosses the branch cut of an $O7^-$-plane.}.\n\nThe different choice of $m$ in \\eqref{5dsuquiverfromSpquiver} gives a different-looking 5d gauge theory. However, we claim that they are dual to each other, and it is the distribution duality. \n\n\n\n\n\\subsubsection{5d \\texorpdfstring{$Sp- \\prod SU$}{sp-su} quivers}\n\\label{subsubsec:5dspsuquiver}\n\nWe then consider the case of splitting only one of the two $O7^-$-planes. The quantum resolution splits the $O7^-$-plane into {\\bf B} and {\\bf C} 7-branes. The analysis is essentially the same as in section \\ref{subsubsec:5dsuquiver1}. About the motion of the branch cuts of 7-branes, we can only consider the one corresponding to the upper transformations in Figure \\ref{Fig:5braneloop2}, the resulting 5-brane configuration is depicted in \\ref{Fig:5braneloop4}. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{6dsuquiver5.pdf}\n\\end{center}\n\\caption{The schematic diagram after moving the branch cuts of ${\\bf B}, {\\bf C}$ and ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ 7-branes in the case when one resolves one of the $O7^-$-planes in Figure \\ref{Fig:Brane6dcanonical1}. The motion corresponds to the upper ones depicted in Figure \\ref{Fig:5braneloop2}.} \n\\label{Fig:5braneloop4}\n\\end{figure}\n\n\nAfter pulling out all the 7-branes, the final 5-barne web diagram is given by Figure \\ref{Fig:Brane6dcanonical3}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{6dspsuquivern2.pdf}\n\\end{center}\n\\caption{The 5-brane web diagram realizing the 5d $Sp(N+2) - SU(2N+8) - [2N+14]$. We only write the web diagram for the case with $n=2$ for simplicity. This web diagram can be obtained by pulling all the 7-branes outside the 5-brane loops from the configuration depicted in Figure \\ref{Fig:5braneloop4} with $n=2$.} \n\\label{Fig:Brane6dcanonical3}\n\\end{figure}\nNote that in this case the distribution ambiguity does not matter. Due to the presence of the $O7^-$-plane, the column which originally has $N+k(4+m)$ D5-branes is connected to the column which originally has $N+k(4-m)$ D5-branes for $1 \\leq k \\leq n-1$. Therefore, the resulting 5d theory is \n\\begin{equation}\nSp(N+n) - SU(2N+2n+4) - SU(2N+2n+8) - \\cdots - SU(2N+6n-4) - [2N+6n+2].\\label{5dspsuquiverfromSpquiver}\n\\end{equation}\nDue to the process of obtaining (\\ref{5dspsuquiverfromSpquiver}), the 7-brane analysis should give the $SU(2N+8n)$ flavor symmetry. The number of the Coulomb branch moduli is again $(2n-1)(N+2n-1)$, which is consistent with the sum of the number of the tensor multiplets and the number of the vector multiplets in the Cartan subalgebra of the 6d quiver theory \\eqref{6dcanonical1}.\n\n\\subsubsection{5d dualities}\n\\label{subsubsec:5ddualsuquiver}\n\nFrom the same 6d quiver theory of \\eqref{6dcanonical1}, we have obtained two types of 5d quiver theories. One type has only $SU$ gauge nodes and the other type has one $Sp$ gauge node with other $SU$ gauge nodes. Those two theories are described by essentially the same 5-brane web diagrams and we claim that they are dual to each other. This is the generalization of the claim in section \\ref{subsec:equivalence.ch2}. \n\nAs in section \\ref{subsec:Sdual.ch2}, it is possible to generate further dual 5d theories by acting the $SL(2, \\mathbb{Z})$ duality on \\eqref{5dsuquiverfromSpquiver}. For example, let us consider a case where $m=0$, which yields, \n\\begin{equation}\n[N+2n+2] - SU(N+2n) - \\cdots - SU(N+2n) - \\cdots - SU(N+2n) - [N+2n+2], \\label{5dsuspecialquiverfromSpquiver}\\\\\n\\end{equation}\nwhere it has $2n-1$ gauge nodes. The S-duality or the $90$ degrees rotation gives \n\\begin{equation}\n[2n+2] - SU(2n) - \\cdots - SU(2n) - \\cdots - SU(2n) - [2n+2], \\label{5dsuspecialquiverfromSpquiver.d1}\\\\\n\\end{equation}\n where it has $N+2n-1$ gauge nodes. It is in fact more interesting to see the TST-duality or the $45$ degrees rotation. The resulting theory is \n \\begin{equation}\n{\\overset{\\overset{\\text{\\large$[3]$}}{\\textstyle\\vert}}{SU(2) }} - SU(3) - \\cdots - {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(N+2n) }}- SU(N+2n) - \\cdots - SU(N+2n)- {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(N+2n) }}- \\cdots - SU(3) - {\\overset{\\overset{\\text{\\large$[3]$}}{\\textstyle\\vert}}{SU(2) }}, \\label{5dsuspecialquiverfromSpquiver.d2}\\\\\n\\end{equation}\nwhere it has $2n-3$ $SU(N+2n)$ gauge nodes. In parcitular, when $n=2$, the 5d quiver \\eqref{5dsuspecialquiverfromSpquiver.d2} has a peculiar form \n \\begin{equation}\n{\\overset{\\overset{\\text{\\large$[3]$}}{\\textstyle\\vert}}{SU(2) }} - SU(3) - \\cdots - {\\overset{\\overset{\\text{\\large$[2]$}}{\\textstyle\\vert}}{SU(N+2n) }}- \\cdots - SU(3) - {\\overset{\\overset{\\text{\\large$[3]$}}{\\textstyle\\vert}}{SU(2) }}, \\label{5dsuspecialquiverfromSpquiver.d3}\\\\\n\\end{equation}\n which can be regarded as gauging $SU(N+2n)$ in the $SU(N+2n+2)$ flavor symmetry of two the $T_{N+2n}$ Tao theories which we will discuss in detail in section \\ref{sec:TNTao}.\n \n As in section \\ref{sec2:6dspN}, it is possible to decouple some flavors from the 5d theories whose UV completion is the 6d SCFT. After decoupling some flavors, the 5d theory has a 5d UV fixed point. Suppose we have dual 5d theories whose UV completion is the same 6d SCFT, decoupling the same flavors in the both theories leads to another dual 5d theories whose UV completion is a same 5d SCFT. Working on explicit examples is quite straightforward and can be done in parallel to the analysis in section \\ref{subsec:equivalence.ch2} and \\ref{subsec:Sdual.ch2}. \n \n \n\\subsubsection{Higgsed cases}\\label{subsubsec:genHiggsSp}\n\nLet us then move on to a circle compactification of the Higgsed 6d quiver theories \\eqref{6dHiggs1}. After performing the T-duality along the $S^1$, we again obtain a brane configuration with 5-branes and two $O7^-$-planes. The difference from the cases in section \\ref{subsubsec:5dspsuquiver} is that we now have D7-branes in various columns, depending on the number of the fundamental hypermultiplets attached to some gauge nodes in the 6d quiver theory \\eqref{6dHiggs1}. In this case, there is another ambiguity of distributing D7-branes in addition to the distribution of D5-branes which we saw in section \\ref{subsubsec:5dsuquiver1}. Due to this distribution ambiguity of D7-branes, one can allocate $k_{n-l}$ D7-branes for $1 \\leq i \\leq n-1$, which originate from the $k_{n-l}$ D8-branes in the Type IIA brane configuration in Figure \\ref{Fig:6dquiver-2}, to the $i$-th left column and the $i$-th right column from the center column. Again, the number of D7-branes in the middle column is fixed to be $k_n$. We choose the branch cuts of D7-branes in the columns left from the center extend in the left direction, and the branch cuts of D7-branes in the columns right from the center extend in the right direction. Furthermore, we also assume that the branch cuts of $k_n$ D7-branes extend in the left direction. \n\nThe requirement that the final 5-brane web diagram admits a 5d gauge theory interpretation constrains the number of D5-branes in each column except for the center. The number of D5-branes in the middle column is again always $N$. Compared to the case in section \\ref{subsubsec:5dspsuquiver}, we have the branch cuts of D7-branes in the columns that extend in either left or right direction. The branch cuts also affect the number of D5-barnes in each column which gives a web diagram admitting a 5d gauge theory description at the final stage after pulling all the 7-branes outside. It turns out that the 5-brane configuration depicted in Figure \\ref{Fig:Brane6dHiggsed1} yields a 5-brane web which admits a 5d gauge theory interpretation. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{6dsuquiverH1.pdf}\n\\end{center}\n\\caption{Type IIB brane configuration after performing a T-duality to Figure \\ref{Fig:6dquiver-2}. We defined non-negative numbers $i_l, j_l$ such that they satisfy $i_l + j_l = k_l$ for $l=1, \\cdots, n-1$.} \n\\label{Fig:Brane6dHiggsed1}\n\\end{figure}\n\nIn order to go from the brane configuration in Figure \\ref{Fig:Brane6dHiggsed1} to a 5-brane web yielding a 5d gauge theory, we take two steps as in section \\ref{subsubsec:5dsuquiver1} and \\ref{subsubsec:5dspsuquiver}. The first step is we resolve either two or one of the $O7^-$-planes. When we split two $O7^-$-planes, we fix the splitting type of the upper $O7^-$-plane into {\\bf B} and {\\bf C} 7-branes, and the splitting type of the lower $O7^-$-plane is chosen to be ${\\bf X}_{[1+m, -1]}\\; {\\bf X}_{[1-m,1]}$ so that the final 5-brane web admits a 5d gauge theory interpretation. When we split only one of the two $O7^-$-planes, the splitting type may be generically {\\bf B} and {\\bf C} 7-branes. The second step is moving the branch cuts of 7-branes which arise by the quantum resolution of two or one $O7^-$-plane. After the second setp, the 5-brane loops are divided by vertical lines and the structure eventually leads to a 5d quiver theory. The procedure is completely prallel to the one in section \\ref{subsubsec:5dsuquiver1} and \\ref{subsubsec:5dspsuquiver}. \n\nWhen we split the two $O7^-$-planes, we obtain the 5-brane configuration in Figure \\ref{Fig:5braneloopH1} after the two steps.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{6dsuquiverH4.pdf}\n\\end{center}\n\\caption{Type IIB brane configuration after the two setps from the one in Figure \\ref{Fig:Brane6dHiggsed1} when two $O7^-$-planes are resolved.} \n\\label{Fig:5braneloopH1}\n\\end{figure}\nAfter pulling all the 7-branes from Figure \\ref{Fig:5braneloopH1}, the final 5-brane web configuration gives rise to a 5d quiver theory with $SU$ gage nodes with various flavors attached to each node. The explicit expression is \n \\begin{eqnarray}\n{\\overset{\\overset{\\text{\\large$[L_1]$}}{\\textstyle\\vert}}{SU(N_1) }} -{\\overset{\\overset{\\text{\\large$[L_2]$}}{\\textstyle\\vert}}{SU(N_2) }} - \\cdots - {\\overset{\\overset{\\text{\\large$[L_{n-1}]$}}{\\textstyle\\vert}}{SU(N_{n-1}) }} - {\\overset{\\overset{\\text{\\large$[k_n]$}}{\\textstyle\\vert}}{SU(2N+2n) }}- {\\overset{\\overset{\\text{\\large$[R_{n-1}]$}}{\\textstyle\\vert}}{SU(M_{n-1}) }} - \\cdots -{\\overset{\\overset{\\text{\\large$[R_2]$}}{\\textstyle\\vert}}{SU(M_2) }} - {\\overset{\\overset{\\text{\\large$[R_1]$}}{\\textstyle\\vert}}{SU(M_1) }}\\nonumber\\\\\n\\label{5dsuquiverhiggsed}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nN_p &=& N + 2n + (n-p)(m - k_n) - \\sum_{l=p+1}^{n-1}(l-p)i_l\\qquad \\text{for} \\quad 1 \\leq p \\leq n-1,\\\\\nM_p &=& N + 2n - (n-p)m - \\sum_{l=p+1}^{n-1}(l-p)j_l \\qquad \\text{for} \\quad1 \\leq p \\leq n-1,\\\\\nL_p &=& i_p, \\qquad \\text{for} \\quad 2 \\leq p \\leq n-1,\\\\\nL_1 &=& N+2n+n(m-k_n)-\\sum_{l=2}^{n-1}l i_l +2,\\\\\nR_p&=& j_p, \\qquad \\text{for} \\quad 2 \\leq p \\leq n-1,\\\\\nL_1 &=& N+2n-nm-\\sum_{l=2}^{n-1}l j_l +2\n\\end{eqnarray}\n where $(i_l + j_l) = k_l$ for $l=1, \\cdots, n-1$, and the rank of each gauge group as well as the number of the flavors should be larger than zero at least.\n \nWhen we split only one of the $O7^-$-planes, we obtain the 5-brane configuration in Figure \\ref{Fig:5braneloopH2} after the two steps.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{6dsuquiverH5.pdf}\n\\end{center}\n\\caption{Type IIB brane configuration after the two setps from the one in Figure \\ref{Fig:Brane6dHiggsed1} when one $O7^-$-plane is resolved.} \n\\label{Fig:5braneloopH2}\n\\end{figure}\nAfter pulling all the 7-branes from Figure \\ref{Fig:5braneloopH2}, the final 5-brane web configuration gives rise to a 5d quiver theory with one $Sp$ gauge node and other $SU$ gage nodes with various flavors attached to each node. The explicit expression is \n\\begin{eqnarray}\n{\\overset{\\overset{\\text{\\large$ [k_{n}] $}}{\\textstyle\\vert}}{ Sp(N+n) }}- \n{\\overset{\\overset{\\text{\\large$ [k_{n-1}]$}}{\\textstyle\\vert}}{SU(N_1) }} &-& \n{\\overset{\\overset{\\text{\\large$[k_{n-2}]$}}{\\textstyle\\vert}}{SU(N_2) }} - \\cdots-\n{\\overset{\\overset{\\text{\\large$[k_1-2n+2]$}}{\\textstyle\\vert}}{SU(N_{n-1}) }},\n\\end{eqnarray}\nwhere \n\\begin{equation}\nN_p = 2N+2n+4p-\\sum_{l=n-p}^{n-1}(l-n+p+1)k_{l+1},\\qquad \\text{for} \\quad 1 \\leq p \\leq n-1.\n\\end{equation}\nIn this case, the distribution ambiguity of D5-branes and D7-branes does not matter since $i$-th left column from the center and the $i$-th right column from the center are the same column due to the orientifold action. \n\nSince the two types of the theories have the same UV completion as the 6d SCFT, they are dual to each other, which is a further generalization of the claim in section \\ref{subsubsec:5ddualsuquiver}. Furthermore, the 5d $SU$ quiver theory \\eqref{5dsuquiverhiggsed} has parameters associated to the distribution of D5-branes and D7-branes. Again, all the combinations descend from the same 6d theory, we argue that they are dual to each other. The $90$ degrees or $45$ degrees rotation of the 5d theories also give various 5d dual theories. \n\n\nIt is also possible to decouple some flavors from the 5d theories. Then, each 5d theory will have a5d UV fixed point. Decoupling exactly the same flavors from the dual 5d theories should give another dualities between 5d theories which has the same UV completion as a 5d SCFT. \n\n\n\n\\subsection{6d \\texorpdfstring{$SU$}{SU} quivers with an antisymmetric hypermultiplet}\n\\label{subsec:6dSUquivgen}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{O8SUquiverA.pdf}\n\\end{center}\n\\caption{Type IIA brane configuration for the 6d linear quiver theory \\eqref{6dcanonicalA}.} \n\\label{Fig:6dquiverA}\n\\end{figure}\n\n\nIn this subsection, we consider 6d $SU$ quiver gauge theories with the gauge node at the edge having\none hypermultiplet in the antisymmetric tensor representation, which is a generalization of what we studied in section \\ref{sec:6dSUNA}.\nWe consider the generalization analogous to what we did in section \\ref{sec:6dSpSUquiver}.\nThe simple generalization is the 6d linear quiver\n\\begin{eqnarray}\n6d \\; [1]_A- SU(N) - SU(N+8) - \\cdots - SU(N+8(n-1)) - [N+8n], \\label{6dcanonicalA}\n\\end{eqnarray}\nThe type IIA brane configuration is depicted in Figure \\ref{Fig:6dquiverA}. \nCompared to Figure \\ref{Fig:6dquiver-1}, we have an extra NS5-brane on top of $O8^-$-plane.\nThe global symmetry is generically $SU(N+8n) \\times U(1)$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{O8SUquiverhiggsA.pdf}\n\\end{center}\n\\caption{Type IIA brane configuration for the 6d linear quiver theory \\eqref{6dHiggsA}.} \n\\label{Fig:6dquiverA-2}\n\\end{figure}\n\nWe can again further generalize this quiver by Higgsing which is induced exactly by the same mechanism discussed in section \\ref{sec:6dSpSUquiver}.\nBy considering the Higgsing specified by the Young diagram $[n, \\cdots, n, n-1, \\cdots, n-1, \\cdots, 2, \\cdots, 2, 1, \\cdots, 1]$ where the number of $l$ is $k_l$ with a condition $\\sum_{l=1}^{n}lk_l = 2N+8n$,\nwe obtain the 6d quiver gauge theory\n\\begin{equation}\n6d \\; \n{\\overset{\\overset{\\text{\\large$[k_{n}]$}}{\\textstyle\\vert}}{SU(N) }} \n- {\\overset{\\overset{\\text{\\large$[k_{n-1}]$}}{\\textstyle\\vert}}{SU(N+8-k_n)}} - \n\\cdots - \n{\\overset{\\overset{\\text{\\large$[k_1]$}}{\\textstyle\\vert}}{SU(N+8(n-1) - \\sum_{l=1}^{n-1}lk_{l+1})}}. \\label{6dHiggsA}\n\\end{equation}\nThe brane setup for this theory is given in Figure \\ref{Fig:6dquiverA-2}.\nWe will study these 6d quivers compactified on $S^1$ and consider their 5d descriptions \nas well as the 5d dualities.\n\n\n\n\n\\subsubsection{5d \\texorpdfstring{$SU$}{SU} quviers}\n\\label{subsubsec:5dSUquivgen}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{6dsuquiverA1.pdf}\n\\end{center}\n\\caption{Type IIB brane configuration after performing a T-duality to Figure \\ref{Fig:6dquiverA}.} \n\\label{Fig:Brane6dcanonicalA}\n\\end{figure}\n\nFirst, we start with the 6d quiver gauge theory (\\ref{6dcanonicalA}) on $S^1$,\nT-duality along the $S^1$ gives type IIB brane configuration.\nAnalogous to the brane setup discussed in section \\ref{subsubsec:5dsuquiver1},\nthe distribution of the D5-branes should be considered.\nThere is no center column in this case and \nthe $N+8(i-1)$ color D5 branes originated from the $i$-th gauge node in 6d quivers (\\ref{6dcanonicalA})\nare distributed into $i$-th left column and $i$-th right column from the center.\nBy imposing the condition that we should obtain a 5d gauge theory interpretation in the end, \nit turns out that the distribution ambiguity is again labeled by the single non-negative integer $m$.\nThe resulting distribution is depicted in Figure \\ref{Fig:Brane6dcanonicalA}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{6dsuquiver3A.pdf}\n\\end{center}\n\\caption{The diagram after resolving two $O7^-$ planes in Figure \\ref{Fig:Brane6dcanonicalA}.\nThe branch cuts of ${\\bf B}, {\\bf C}$ and ${\\bf X}_{[N+2m-1, 1]}\\; {\\bf X}_{[N+2m+1,1]}$\nare moved in such a way that the D5-branes and the D7-branes indicated by the arrows go across the cut.} \n\\label{Fig: 5loopA}\n\\end{figure}\n\n\nNext, we consider the resolution of the two $O7^-$-planes attached to 5-branes.\nThe charges of the split 7-branes are determined by the non-negative integer $m$ introduced above\ndue to the condition that one of such 7-branes should be attached to the 5-brane\nto which originally $O7^-$-plane was attached.\nIf we fix the splitting type of the upper $O7^-$-plane to be \n{\\bf B}, {\\bf C}.\nThen, the splitting type of the lower $O7^-$-plane is determined to be ${\\bf X}_{[N-2m-1,1]}$, ${\\bf X}_{[N-2m+1,1]}$. \nThe resolution of the $O7^-$-planes create $n$ 5-brane loops as in Figure \\ref{Fig: 5loopA}.\nThen, we move the branch cut of these four 7-branes in such a way that \nsome of the D5-branes and D7-branes go across the cut as dictated in the arrow \\ref{Fig: 5loopA}.\nContrary to the case in Figure \\ref{Fig:5braneloop2}, \nthe number of D5-branes at upper left and lower right is one more \nthan the ones at upper right and lower left.\nApart from this small asymmetry, the procedure is quite parallel to section \\ref{subsubsec:5dsuquiver1}.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{6dsuquiver4A.pdf}\n\\end{center}\n\\caption{Diagram after the motion in Figure \\ref{Fig: 5loopA}.} \n\\label{Fig:5loopA2}\n\\end{figure}\n\n\nAfter this motion, we obtain the diagram in Figure \\ref{Fig:5loopA2}.\nThe 5-branes depicted as red lines are the ones \ncoming from the D5-branes which went across the 7-brane monodromy cut.\nThese 5-branes becomes part of the walls spliting each gauge nodes.\nMoreover, it turns out that\nthe newly generated 5-brane loop gives\nextra color D5-branes for each column\ndue to the tuned distribution parametrized by $m$ mentioned previously.\nThen, from Figure \\ref{Fig:5loopA2}, we see that \n$2n$ color branes are added to the two columns at the center\nsince all the $n$ 5-branes loops contribute as color D5-branes to these.\nWhen we move to the next gauge node, the extra color D5-branes reduce by two\nand there are only two additional color D5 branes at the columns at the both edges.\n\n\nTherefore, after moving all the 7-branes outside,\nwe interpret this diagram as the following 5d quiver gauge theory:\n\\begin{eqnarray}\n[N_0] - SU(N_1) - SU(N_2) - \\cdots - SU(N_{2n}) - [N_{2n+1}]\n\\label{eq:gen5dquivA}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nN_0 &=& -nN+(2n+1)(m+1), \\nonumber \\\\\nN_\\ell &=& 2n-1 + (-n+\\ell) N + (2n-2\\ell+1) m, \\qquad \\ell=1,\\cdots, 2n, \\\\\nN_{2n+1} &=& (n+1) N- (2n+1)(m-1). \\nonumber \n\\end{eqnarray}\nThe parameters are constrained such that the rank of each gauge group or the number of the flavors should be positive. \nIn summary, we see that the 6d quiver theory (\\ref{6dcanonicalA}) on $S^1$ is described by \nthe 5d quiver gauge theory (\\ref{eq:gen5dquivA}).\n\n\n\\subsubsection{5d \\texorpdfstring{$SU$}{SU} quivers with an antisymmetric hypermultiplet}\n\\label{subsubsec:5dSUAgen}\n\nHere, we consider the case of resolving only one of the two $O7^-$-planes in the diagram in Figure \\ref{Fig:Brane6dcanonicalA}. After we resolve the upper $O7^-$-plane into {\\bf B} and {\\bf C} 7-branes,\nwe consider the same procedure, namely we move the branch cuts of the {\\bf B} and {\\bf C} 7-branes so that some of the D7-branes and D5-branes cross them. The explicit motion of the branch cuts or equivalently the motion of D7-branes and D5-branes is exactly the same one that we performed for the upper part of the diagram in Figure \\ref{Fig: 5loopA},\nwhile we keep the lower half part as it is.\nAfter moving the branch cuts, it is straightforward to see that we obtain the diagram depicted in Figure \\ref{Fig:5AoneO7}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{6dsuquiver5A.pdf}\n\\end{center}\n\\caption{The diagram for the case where only one of the two $O7^-$-planes is resolved.} \n\\label{Fig:5AoneO7}\n\\end{figure}\nIn this case, the distribution ambiguity does not matter as the $i$-th left column from the center is actually the same column as the $i$-th right column from the center by the orientifold action. In other words, we have $n$ $SU$ gauge nodes. \n\n\nAfter moving all the 7-branes outside, we obtain a 5-brane web diagram. It is possible to can read off from the diagram that this theory is \nthe following 5d quiver gauge theory:\n\\begin{eqnarray}\n[1]_A - SU(N+2n-1) - SU(N+2n+3) - \\cdots - SU(N+6n-5) - [N+6n+1]. \\nonumber \\\\\n\\label{eq:gen5dA}\n\\end{eqnarray}\nNote that we have an anti-symmetric hypermultiplet at the left end node as a 5-brane is attached to the lower unresolved $O7^-$-plane \\cite{Bergman:2015dpa}. \n\n\n\n\n\\subsubsection{5d dualities}\n\nIn section \\ref{subsubsec:5dSUquivgen} and in section \\ref{subsubsec:5dSUAgen},\n we have discussed two different types of 5d description for the 6d theory (\\ref{6dcanonicalA}),\n which are 5d quivers (\\ref{eq:gen5dquivA}) and (\\ref{eq:gen5dA}).\nSince their type IIB diagrams come from the identical type IIA diagram for the 6d theory,\nwe claim that these two types of 5d theory are dual to each other,\nwhich means that they have the identical 6d UV fixed point.\n\n\nWhen we combine mass deformation and S-duality as discussed in section \\ref{subsec:chain},\nwe will be able to obtain various dual 5d theories.\nFurthermore, by considering the flavor decoupling limit,\nall these dualities can be also reduced to the dualities for another set of theories \nwhich have an identical 5d UV fixed point.\nClassifying all these dual 5d theories would be interesting future problem.\n\n\n\n\n\\subsubsection{Higgsed cases}\nNow, we go on to the 6d theory (\\ref{6dHiggsA}), which is obtained by Higgising (\\ref{6dcanonicalA}).\nDiagramatic derivation of the 5d description is quite parallel to what we did in section \\ref{subsubsec:5dSUquivgen}\nand in section \\ref{subsubsec:5dSUAgen}.\nThe different point is that some of flavor D7-branes exist at internal columns.\nThe effect of such D7-branes are again quite parallel to what we discussed in section \\ref{subsubsec:genHiggsSp}.\nHere, we summarize the resulting 5d description.\n\nWhen we resolve two $O7^-$ planes, we obtain the 5d quivers \n\\begin{eqnarray}\n{\\overset{\\overset{\\text{\\large$[L]$}}{\\textstyle\\vert}}{SU(N_1) }} \n- {\\overset{\\overset{\\text{\\large$[i_2]$}}{\\textstyle\\vert}}{SU(N_2) }} \n- \\cdots \n- {\\overset{\\overset{\\text{\\large$[i_n]$}}{\\textstyle\\vert}}{SU(N_n) }} \n- {\\overset{\\overset{\\text{\\large$[j_n]$}}{\\textstyle\\vert}}{SU(M_{n}) }} \n- \\cdots \n- {\\overset{\\overset{\\text{\\large$[j_2]$}}{\\textstyle\\vert}}{SU(M_{2}) }} \n- {\\overset{\\overset{\\text{\\large$[R]$}}{\\textstyle\\vert}}{SU(M_{1}) }} \n\\label{eq:gen5dquivA2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nL &=& -nN+(2n+1)(m+1) - \\sum_{l=2}^{n}li_l ,\n \\nonumber \\\\\nN_p &=& 2n-1 + (-n+p) N + (2n-2p+1) m - \\sum_{l=p+1}^n (l-p) i_l , \\qquad p=1,\\cdots, n, \n \\nonumber \\\\\nM_p &=& 2n-1 + (n-p+1) N - (2n-2p+1) m - \\sum_{l=p+1}^n (l-p) j_l, \\qquad p=1,\\cdots, n,\n \\nonumber \\\\\nR &=& (n+1) N- (2n+1)(m-1) - \\sum_{l=2}^{n}l j_l. \\nonumber \n\\end{eqnarray}\nwhere $i_l + j_l = k_l$. \n\nWhen we resolve only one out of the two $O7^-$ planes, we obtain the 5d quivers \n\\begin{eqnarray}\n{\\overset{\\overset{\\text{\\large$[k_{n}, 1_A]$}}{\\textstyle\\vert}}{SU(N_1) }} \n- {\\overset{\\overset{\\text{\\large$[k_{n-1}]$}}{\\textstyle\\vert}}{SU(N_2)}} \n- {\\overset{\\overset{\\text{\\large$[k_{n-2}]$}}{\\textstyle\\vert}}{SU(N_3) }} \n - \\cdots \n- {\\overset{\\overset{\\text{\\large$[k_{2}]$}}{\\textstyle\\vert}}{SU(N_{n-1}) }} \n- {\\overset{\\overset{\\text{\\large$[k_1-2n+1]$}}{\\textstyle\\vert}}{SU(N_n) }} \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nN_s=N+2n-5 + 4s - \\sum_{l=1}^{s-1}lk_{l+1+n-s}.\n\\end{eqnarray}\n\n\n\nWe claim that these two types of the theories have the same 6d UV fixed point.\nAlso, the 5d SU quiver theory (\\ref{eq:gen5dquivA2}) with any possible value for $m$, $i_l$, and $j_l$ \ngive the set of dual theories.\nS-duality of these 5d theories will also give various dual theories.\nFlavor decoupling limit will give other set of dual theories which have the same 5d UV fixed point.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Special cases: the \\texorpdfstring{$O8^- + 8 D8$'s; $[1_A,8] - SU(N) - \\cdots - [N]$}{O8+8D8}}\n\\section{Special cases: the \\texorpdfstring{6d quiver gauge theories of $O8^- + 8 D8$'s configuration}{O8+8D8}}\\label{sec:speicalcases}\n\nIn this section, we consider the 6d SCFT configurations in tensor branch which are composed of an $O8^-$ plane together with eight D8 branes, leading to indefinite sequence of quiver type made of D6 branes stretched between two NS5 branes. These cases can be, in principle, obtained through the Higgsing of general 6d brane configuration discussed in the previous section. It is however non-trivial to get 5d description following the procedure described in the previous sections. It is partial because, to have Lagrangian description in 5d, many D7 branes are converted to other 7-branes resulting in lack of flavor 7-branes. On the other hand, we discuss below that one can still have an Lagrangian description in the S-dual frame or by implementing the $SL(2,\\mathbb{Z})$ transformation on the $(p,q)$ plane. These special cases also give intriguing dual pictures depending the resolution of the $O7^-$ plane into a pair of 7-brane. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7cm]{A8SUSU.pdf}\n\\end{center}\n\\caption{6d brane configuration for the quiver $[1_A,8] - SU(N) - SU(N) - \\cdots - SU(N) - [N]$ which has an antisymmetric hypermultiplet and eight fundamental flavors of the leftmost gauge group of the quiver also coupled to tensor multiplets.}\n\\label{Fig:A8SUSU}\n\\end{figure}\n\\subsection{\\texorpdfstring{6d $[1_A,8]-SU(N)-SU(N)-\\cdots-[N]$ quiver}{[A,8]-SU(N)-SU(N)-}}\\label{subsec:18sususun}\nWe first consider the case where an $O8^-$ with eight D8 branes and a NS5 brane being on top of the $O8^-$ plane, \n\\begin{align}\n{\\rm 6d} ~[1_A,8] - \\underbrace{SU(N) - SU(N) - \\cdots - SU(N)}_\\text{$n$ nodes} - [N].\n\\end{align}\nThe corresponding 6d brane configuration is given in Figure \\ref{Fig:A8SUSU}.\n\n\n\n\\paragraph{\\texorpdfstring{$\\mathbf{SL(2,\\mathbb{Z})}$}{SL2Z} invariant 7-brane combinations.}\nWe will discuss various dualities along the reduction of the 6d theories to 5d theories by analyzing the 7-brane monodromies with or without the $O7^-$ planes. In order to see the dual picture, it is useful to consider combinations of 7-branes which are invariant under the $SL(2,\\mathbb{Z})$ transformation on the $(p,q)$ 5-brane web plane. An $O7^-$ plane and four D7 branes would be an obvious example among many $SL(2,\\mathbb{Z})$ invariant 7-brane combinations, as the total monodromy of them is proportional to minus identity matrix.\nIt follows immediately that a pair of 7-branes as a resolution of the $O7^-$ plane, together with four D7 branes, is thus $SL(2,\\mathbb{Z})$ invariant. For instance, $(\\mathbf{B},\\mathbf{C})$ or $(\\mathbf{N},\\mathbf{X})$ 7-branes with four D7 branes (four $\\mathbf{A}$ 7-branes). We will show a few distinctive examples of such combinations involving 7-branes from different resolutions of the $O7^-$ plane. In particular, the cases where one of the 7-branes is attached to a 5-brane of the same charge will give rise to fruitful dualities among the resulting 5d quiver gauge theories. We consider such combinations frequently appearing through T-duality when reducing a 6d theory to 5d. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{sl2zcom1.pdf}\n\\end{center}\n\\caption{(i) An $SL(2,\\mathbb{Z})$ invariant 7-brane combination where $\\bf B$ 7-brane is attached to 5-brane. A sequence of monodromy analysis makes $\\bf A^2BCA^4$ into $\\bf N^2CBN^2$ which will be converted into $\\bf A^2BCA^4$ again via an S-dual action.} \n\\label{Fig:SL2Zcom1}\n\\end{figure}\nAs discussed earlier, when going from 6d to 5d, one can distribute D7 branes and D5 branes with a suitable Wilson line, which enables us to allocate any number of D7 branes close to $O7^-$ planes. Regarding resolution of $O7^-$ planes, we restrict ourselves to two distinctive resolutions of $O7^-$ planes, which is either ($\\bf B, C$) or ($\\bf N, X_{[2,1]}$) 7-brane pair. \n\n(i) Consider the 7-brane configuration\n$\\bf A^2BCA^2$ where $\\bf B$ is attached to a $(1,-1)$ 5-brane as depicted in Figure \\ref{Fig:SL2Zcom1}. It is straightforward to show that it can be expressed as the $\\bf N^2CBN^2$ 7-brane configuration where $\\bf B$ is still attached to the 5-brane\\footnote{Using the 7-brane monodromies, one finds that\n\\begin{align}\n\\bf CA = AN = NC, \\quad NA = AB= BN, \\quad {\\rm or}\\quad \n\\bf N^2C= CA^2 = A^2B = BN^2, \n\\end{align}\nwhich yield\n\\begin{align}\n{\\bf A^2 \\bar{B}C A^2} = {\\bf A^2 \\bar{B} A^2B} = {\\bf A^2 \\bar{B} BN^2 }= {\\bf C A^2 \\bar{B} N^2}= {\\bf N^2 C \\bar{B} N^2},\n\\end{align}\nwhere $\\bf \\bar{B}$ 7-brane is attached to $(1,-1)$ 5-brane. \nA pictorial version of this monodromies is given in Figure \\ref{Fig:SL2Zcom1}.} \n(See Figure \\ref{Fig:SL2Zcom1}).\nBy performing an S-duality transformation $(p,q)\\to (-q,p)$, one sees that \n ${\\bf A^2 {B}C A^2} $ is an S-dual invariant combination\n\\begin{align}\n{\\bf A^2 \\bar{B}C A^2} = {\\bf N^2 C \\bar{B} N^2} \\quad \\overset{\\rm S-dual}{\\longrightarrow}\\quad {\\bf A^2 \\bar{B}C A^2},\n\\end{align}\nwhere we used a bar ``$\\bar{}$'' to denote the 7-brane is attached to a 5-brane of the same charge.\n\n\n\\vskip 0.5cm\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=11cm]{sl2zcom2.pdf}\n\\end{center}\n\\caption{(ii) An $SL(2,\\mathbb{Z})$ invariant 7-brane combination where $\\bf N$ 7-brane is attached to 5-brane. A sequence of monodromy analysis makes $\\bf ANX_{[2,1]}A^3$ into $\\bf NCBN^3$}\n\\label{Fig:SL2Zcom2}\n\\end{figure}\n\\noindent (ii) Another example is ${\\bf A {N}X_{[]2,1]} A^3}$ with $\\bf N$ 7-brane being attached to $(0,1)$ 5-brane. See Figure \\ref{Fig:SL2Zcom2}. Again a little monodromy calculation \n\\footnote{The relevant monodromy relation for this case is \n\\begin{align}\n\\bf X_{[2,1]} A = AC , \\quad CA = AN=NC, \n\\end{align}\nyielding\n\\begin{align}\n{\\bf A {N}X_{[2,1]} A^3} = {\\bf CA^3 B} = {\\bf NCBN^3} .\n\\end{align}\n}\nleads that ${\\bf A {N}X_{[]2,1]} A^3}$ is an $SL(2,\\mathbb{Z})$ invariant 7-brane combination\n\\begin{align}\n{\\bf A \\bar{N}X_{[2,1]} A^3} = {\\bf \\bar{N} CB N^3} \\quad \\overset{\\rm S-dual}{\\longrightarrow}\\quad {\\bf \\bar{A}BC A^3},\n\\end{align}\nas $\\bf NX_{[2,1]}$ and $\\bf BC$ are related by a successive application $T$-action of the $SL(2,\\mathbb{Z})$ transformation. \nNotice that before taking an S-duality, the $\\bf N$ 7-brane is attached to $(0,1)$ 5-brane, but after the S-duality and a manipulation of monodromies as well as the Hanany-Witten transition, the $\\bf A$ 7-brane is instead attached to a D5 brane. This behavior plays an important role in showing a dual description of 5d quiver theories which we discuss below.\n\\vskip 0.5cm\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{sl2zcom3.pdf}\n\\end{center}\n\\caption{(iii) An $SL(2,\\mathbb{Z})$ invariant 7-brane combination where no 7-brane is attached to 5-brane. A sequence of monodromy analysis makes $\\bf A^2BCA^4$ into $\\bf N^2CBN^2$.\n} \n\\label{Fig:sl2zcom3}\n\\end{figure}\n\\noindent (iii) When no 5-brane is attached to 7-branes, it is easier for one to see $SL(2,\\mathbb{Z})$ invariance. For instance, consider $\\bf A^2 O7^- A^2$ or $\\bf A^2 BC A^2$. (See also Figure \\ref{Fig:sl2zcom3}.)\n Using the monodromy, one can easily find that ${\\bf A^2 BC A^2}$ is S-dual invariant \n\\begin{align}\n{\\bf A^2 BC A^2} = {\\bf C A^2A^2B} = {\\bf N^2 C BN^2} \\quad \\overset{\\rm S-dual}{\\longrightarrow}\\quad {\\bf A^2BC A^2}.\n\\label{a2bca2no5bra}\n\\end{align}\n\nWith all these $SL(2,\\mathbb{Z})$ invariant 7-brane combinations, (i), (ii), and (iii), we study special brane configurations which lead various 5d quiver theories.\n\n\n\\bigskip\n\\paragraph{\\texorpdfstring{6d $[1_A,8]-SU(1)-SU(1)-\\cdots-[1]$ quiver and 5d $Sp$ theory with $Nf=8$ and $N_a=1$.}{[1_A,8]-SU(1)-SU(1)-}}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{A85dconf.pdf}\n\\end{center}\n\\caption{Left: 5d brane configuration for the quiver $[1_A,8] - SU(1) - SU(1) - \\cdots - SU(1) - [1]$. ~Right:\nSplitting the $O7^-$ plane gives 5-brane loops which are denoted as circular loops. \n}\n\\label{Fig:A85dconf}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{SpnwithA.pdf}\n\\end{center}\n\\caption{6d $[1_A,8] - SU(1) - SU(1) - \\cdots - SU(1) - [1]$ giving 5d $Sp(n)$ with $N_f=8$ and $N_a=1$. From the figure on the right of Figure \\ref{Fig:A85dconf}, one moves $\\bf B$ 7-brane along the direction of the charge $(1,-1)$ so that the $\\bf B$ brane comes a freely floating 7-brane. This procedure gives a configuration on the left here. Using the 7-brane monodromy, one can put all the 7-brane on one side like the figure on the right. \n}\n\\label{Fig:SpnwithA}\n\\end{figure}\nFor the $N=1$ case with $n$ quiver nodes, we claim that the 6d brane configuration gives rise to 5d $Sp(n)$ gauge theory with $N_f=8$ and $N_a=1$ flavors:\n\\begin{align}\n{\\rm 6d~} [1_A,8]-\\underbrace{SU(1)-\\cdots-SU(1)}_\\text{$n$ nodes}-[1] ~\\Rightarrow~ {\\rm 5d}~ Sp(n)~{\\rm theory~ with}~ N_f=8~ \\& ~N_a=1.\n\\end{align}\nIt can be seen as follows. A circle compactification followed by T-duality makes an $O8^-$ into two $O7^-$'s separated maximally along the T-dual circle. NS5 brane stuck on the $O8^-$ is now a brane junction of NS5 brane connecting two $O7^-$s as well as a D5 brane. Because of this junction, when resolving the $O7^-$ planes into two pairs of two 7-branes, each $O7^-$ is resolved differently, for instance, \n($[-1,1]$ and $[1,1]$) 7-branes for an $O7^-$ plane, and ($[0,1]$ and $[2,1]$) 7-branes for the other. See Figure \\ref{Fig:A85dconf}. From 7-brane monodromies for this configuration, one finds that it leads to the configuration for 5d $Sp(n)$ gauge theory with $N_f=8$ and $N_a=1$ flavors, which describes a circle compactification of 6d higher rank E-string theory. We note that the web diagram for higher rank E-string theory of massless antisymmetric hypermultiplet was already discussed in \\cite{Kim:2015jba} which is made out of eight D7 branes, and its flavor decoupling limit was also discussed in \\cite{Kim:2014nqa} as a limit taking masses of D7 branes to infinite.\n Notice however that we now have {\\it nine} D7 branes in the original setup, and as a consequence of 7-brane monodromy, one of D7 branes is now converted to a $[0,1]$-brane. Hence the remaining D7 branes account eight flavors, but as we have the $[0,1]$ 7-brane which does not exit for the web configuration for the massless antisymmetric hypermultiplet, this 7-brane is thus associated with a non-zero mass of antisymmetric hypermultiplet. See Figure \\ref{Fig:SpnwithA}. Taking the flavor decoupling limit for the fundamental flavors, one would find 5d web configuration for $Sp(n)$ gauge theory with $N_f\\le 7$ and $N_a=1$ flavors, given in \\cite{Bergman:2015dpa}. By pulling out all the 7-branes, it is not so difficult for one to find that it makes a Tao web diagram. \n\n\n\n\\bigskip\n\\paragraph{\\texorpdfstring{$N=2$ and $n$ nodes: duality between 5d $SU(2n+1)$ with $N_a=2$ and $Sp(n)\\times Sp(n)$}{N=2}.}\n\nUsing an aforementioned $SL(2,\\mathbb{Z})$ invariant combination with ($\\mathbf{B}, \\mathbf{C}$) 7-branes (a resolution of the $O7^-$ plane) and four D7 brane, one finds that the $N=2$ case of $n$ quiver nodes \n\\begin{align}\n{\\rm 6d~} [1_A,8]-\\underbrace{SU(2)-\\cdots-SU(2)}_\\text{$n$ nodes}-[2]\n\\end{align}\ngives rise to two 5d theories. Firstly, reducing it to 5d, we can take the S-duality in a $SL(2,\\mathbb{Z})$ invariant way that the 7-branes are rearranging themselves to respect the original 7-brane structure. As this case can be understood as the case when $N$ is even, which we will discuss in what follows. We state that the resulting 5d theories. This case leads to 5d $SU(2n+1)$ gauge theory with {\\it two} antisymmetric hypermultiplets\n\\begin{align}\n{\\rm 5d~} [1_A,4] - SU(2n+1) -[1_A,4].\n\\end{align}\nOn the other hand, a different resolution of the $O7^-$ plane, e.g., $O7^-\\to \\bf N,\\, X_{[2,1]}$, yields the quiver of $Sp(n)\\times Sp(n)$ theory only with fundamental hypermultiplets\n\\begin{align}\n{\\rm 5d~}[4] - Sp(n) - Sp(n) - [4].\n\\end{align}\nThis suggests that these two 5d theories are dual to each other in the sense that they have the same UV fixed points. \n\n\n\n\n\\bigskip\n\\bigskip\n\\paragraph{\\texorpdfstring{$N=$ even: Duality between $Sp$ quiver and $SU$ quiver.}{N=even}}\nFor $N=2m$ and $n$ quiver nodes, 6d theory is \n\\begin{align}\n{\\rm 6d~} [1_A,8]-\\underbrace{SU(2m)-\\cdots-SU(2m)}_\\text{$n$ nodes}-[2m].\n\\end{align}\nAgain, one uses $SL(2,\\mathbb{Z})$ invariant 7-brane combinations introduced earlier and S-duality to obtain the following 5d theories:\n\\begin{align}\n{\\rm 5d~} & [1_A,4] - \\underbrace{SU(2n+1) - SU(2n+1) - \\cdots - SU(2n+1)}_\\text{$m$ nodes} - [1_A,4]\\\\\n{\\rm 5d~} & [3] - \\underbrace{Sp(n) - {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(2n+1)}} - SU(2n+1) - \\cdots - SU(2n+1) - {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(2n+1)}} - Sp(n)}_\\text{$m+1$ nodes} - [3].\n\\end{align}\n\n\\noindent The first case is realized when we split the $O7^-$ into $\\mathbf{B}$ and $\\mathbf{C}$ 7-branes. The second case is realized when we split the $O7^-$ into $\\bf N$ and $\\bf X_{[2,1]}$ 7-branes.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{AABCAA.pdf}\n\\end{center}\n\\caption{A T-dual version of 6d $[1_A, 8]- SU(4)-SU(4)-SU(4)-[4]$ yielding 5d\n$[1_A, 4]- SU(7)-SU(7)-[1_A,4]$ implementing S-dual invariant 7-brane combinations. \n}\n\\label{Fig:AABCAA}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=16cm]{ANXA3.pdf}\n\\end{center}\n\\caption{A T-dual version of 6d $[1_A, 8]- SU(4)-SU(4)-SU(4)-[4]$ yielding 5d\n$[3]- Sp(3)-( SU(7) - [2] )-Sp(3)-[3]$ implementing S-dual invariant 7-brane combinations. \n}\n\\label{Fig:ANXA3}\n\\end{figure}\n\nAs a representative example, the brane configuration for 6d $[1_A, 8]- SU(4)-SU(4)-SU(4)-[4]$ ($m=2$ and $n=3$) yielding \n\\begin{align}\n{\\rm 5d~}[1_A, 4]- SU(7)-SU(7)-[1_A,4]\n\\end{align}\nis depicted in Figure \\ref{Fig:AABCAA}. As there are eight D7 branes, one can relocate and arrange D7 branes with suitable Wilson lines, such that four D7 branes are located close to each the $O7^-$ plane so to make an $SL(2,\\mathbb{Z})$ invariant 7-brane combination including the resolution of the $O7^-$ plane into the $\\bf B, C$ 7-branes. Performing an S-duality and also taking a weak coupling limit to obtain the $O7^-$ plane out of two suitable 7-branes, one finds that the resulting web configuration gives rise to the quiver gauge theory with an antisymmetric hypermultiplet at the edge gauge node. \nIn a different way of resolving the $O7^-$ planes, 6d brane configuration yields a seemingly different 5d quiver theory \n\\begin{align}\n{\\rm 5d~} [3]- Sp(3)-{\\overset{\\overset{\\text{\\large$[2]$}}{\\textstyle\\vert}}{SU(7)}}-Sp(3)-[3]\n\\end{align}\nis also depicted in Figure \\ref{Fig:ANXA3}. Like the previous case, one has another $SL(2,\\mathbf{Z})$ invariant 7-brane combination except for a different resolution of the $O7^-$ plane, $O7^-\\to {\\bf N, X_{[2,1]}}$. When performing an S-duality, the web configuration becomes completely different from the previous one as a D7 brane is being attached to a D5 brane which makes a floating 7-brane pair which can be converted into an $O7^-$ plane in the weak coupling limit. One can further move this D7 brane across 5-brane junction, and as the Hanany-Witten transition, one has freely floating D7 branes, which play the role of the fundamental flavors. The resulting configuration yields a quiver theory made out of $Sp$ and $SU$ gauge groups. See Figure \\ref{Fig:ANXA3}. Therefore, we claim that two different quiver theories have the same 6d origin at UV and thus they are two dual description at IR. We note also that taking the flavor decoupling limit, this duality would still hold for less flavor cases. \n\n\n\\bigskip\n\\paragraph{\\texorpdfstring{$N=$ odd: $Sp-SU$ quiver.}{N=odd}}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=16cm]{Noddcase.pdf}\n\\end{center}\n\\caption{A T-dual version of 6d $[1_A, 8]- SU(5)-SU(5)-SU(5)-[5]$ yielding 5d\n$[1_A, 4]- SU(7)-( SU(7) - [1] )-Sp(3)-[3]$ implementing S-dual invariant 7-brane combinations. \n}\n\\label{Fig:Noddcase}\n\\end{figure}\nIn a similar fashion, one can obtain 5d theory for $N=2m+1$. In this case, the resolution of each $O7^-$ brane is different, one does get dual description, rather one finds the resulting 5d theory is a hybrid of the $N=2m$ cases: \n\\begin{align}\n{\\rm 5d~}[3] - \\underbrace{Sp(n) - {\\overset{\\overset{\\text{\\large$[1]$}}{\\textstyle\\vert}}{SU(2n+1)}} \n- SU(2n+1) - \\cdots - SU(2n+1)}_\\text{$m+1$ nodes} - [1_A,4].\n\\end{align}\nAs an example, 5d brane configuration for 6d $[1_A, 8]- SU(5)-SU(5)-SU(5)-[5]$ quiver theory is depicted in Figure \\ref{Fig:Noddcase}, yielding\n\\begin{align}\n{\\rm 5d~}[1_A, 4]- SU(7)-{\\overset{\\overset{\\text{\\large$[2]$}}{\\textstyle\\vert}}{SU(7)}}-Sp(3)-[3].\n\\end{align}\n\n\n\n\n\n\n\n\n\\bigskip\n\\subsection{\\texorpdfstring{6d $[8] - Sp(N) - SU(2N) - SU(2N) - \\cdots - SU(2N) - [2N]$ quiver}{[8]-Sp-SU}}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7cm]{A8SpSU.pdf}\n\\end{center}\n\\caption{6d brane configuration for the $[8] - Sp(N) - SU(2N) - SU(2N) - \\cdots - SU(2N) - [2N]$ quiver theory.}\n\\label{Fig:A8SpSU}\n\\end{figure}\n\nNext we consider another special case:\n\\begin{align}\n{\\rm 6d~}[8] - \\underbrace{Sp(N) - SU(2N) - SU(2N) - \\cdots - SU(2N)}_\\text{$n$ nodes} - [2N],\n\\end{align}\nwhose brane configuration is given in Figure \\ref{Fig:A8SpSU}, consisting of an $O8^-$ plane, and D8 branes, D6 branes and NS5 branes. We claim that from a circle compactification and T-dual, one obtains the 5d theory given by \n\\begin{align}\n{\\rm 5d~} [4] - \\underbrace{Sp(n) - SU(2n) - \\cdots - SU(2n) - Sp(n)}_\\text{$N+1$ nodes} - [4].\n\\end{align}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{8sp2su4su4.pdf}\n\\end{center}\n\\caption{T-dual version of 6d brane configuration for the $[8] -Sp(2) - SU(4) - SU(4) -[4]$ quiver theory.}\n\\label{Fig:8sp2su4su4}\n\\end{figure}\nAs an example, consider \n\\begin{align}\n{\\rm 6d~} [8] -Sp(2) - SU(4) - SU(4) -[4].\n\\end{align}\nReducing it to 5d, one gets a brane configuration of $O7^-$ planes, D7 branes and D5 branes given in the web configuration on the left of Figure \\ref{Fig:8sp2su4su4}. By resolving two $O7^-$ planes, one obtains the configuration on the right of Figure \\ref{Fig:8sp2su4su4}.\nAs discussed in the beginning of this section, using the $SL(2,\\mathbb{Z})$ invariant 7-brane combination \\eqref{a2bca2no5bra}, one sees that S-duality leads to \n\\begin{align}\n{\\rm 5d~} [4] -Sp(3) - SU(6) - Sp(3) -[4].\n\\end{align}\nThe procedure is depicted in Figure \\ref{Fig:4sp3su6}. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=11cm]{4sp3su6-1.pdf}\n\\end{center}\n\\caption{The first figure is a simplified version of the right of Figure \\ref{Fig:8sp2su4su4} for\n6d $[8]-Sp(2) - SU(4) - SU(4) - SU(4) -[4]$ quiver. The 7-branes are distributed and combined to form an S-dual invariant 7-brane combination. The second figure is the rearrangement of these 7-branes via monodromies explained in section \\ref{subsec:18sususun}. The third one is the result of S-dual action of the second one. The last one is the corresponding brane configuration showing \n5d $[4]-Sp(3)-SU(6)-Sp(3)-[4]$.}\n\\label{Fig:4sp3su6}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{6d description of 5d \\texorpdfstring{$T_N$}{TN} Tao theory}\n\\label{sec:TNTao}\n\nThere is another important class of 5d theories obtained by Tao diagrams. It has been known that the 5d $T_N$ theory can be realized by a 5-brane web diagram \\cite{Benini:2009gi}. By turning on certain mass deformations, the 5d $T_N$ theory flows to a 5d linear quiver theory $[N] - SU(N-1) - SU(N-2) - \\cdots - SU(2) - SU(1)$ \\cite{Hayashi:2013qwa, Aganagic:2014oia, Bergman:2014kza, Hayashi:2014hfa}. The last $SU(1)$ node can be understood as two flavors coupled to the $SU(2)$ gauge node, and hence the quiver is equivalently written as $[N] - SU(N-1) - SU(N-2) - \\cdots - SU(2)- [2]$ \\cite{Bergman:2014kza, Hayashi:2014hfa, Tachikawa:2015mha}. It is possible to construct a Tao web diagram by adding flavors to the two end nodes of the quiver \\cite{Kim:2015jba}. An example of a Tao web diagram arising by adding flavors to the 5d $T_5$ theory is depicted in Figure \\ref{Fig:T5Tao}. The 5d quiver theory realized by the Tao diagram is then \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{T5Tao.pdf}\n\\end{center}\n\\caption{The Tao web diagram which arises by adding flavors at the end nodes of the 5d quiver corresponding to the 5d $T_5$ theory.} \n\\label{Fig:T5Tao}\n\\end{figure}\n\\begin{equation}\n[N+2] - SU(N-1) - SU(N-2) - \\cdots - SU(3) - SU(2) - [3]. \\label{quiver.TNTao}\n\\end{equation}\nWe will call the 5d quiver theory as 5d $T_N$ Tao theory. Since the 5d $T_N$ Tao theory is realized by the Tao diagram, the theory is expected to have a 6d UV completion. In this section, we propose a 6d description of the 5d $T_N$ Tao theory \\eqref{quiver.TNTao} realized by the $T_N$ Tao diagram by examining the global symmetry as well as the number of the Coulomb branch moduli.\n\n\\subsection{Global symmetry from 7-branes}\n\nThe global symmetry of a 5d theory realized by a 5-brane web can be understood from the symmetry on 7-branes attached to external 5-branes in the web diagram \\cite{DeWolfe:1999hj}. Namely, 7-branes can determine the global symmetry realized at the UV fixed point. In order to see the flavor symmetry, we pull all the 7-branes inside the 5-brane loops and try to put 7-branes on top of each other. The symmetry realized on 7-branes gives the non-Abelian part of the flavor symmetry of the 5d theory. Abelian part can be recovered by counting the number of parameters of the theory since the number of the parameters of the 5d theory should agree with the rank of the total global symmetry. The method was applied to the Tao diagram realizing the 5d $SU(n)$ gauge theory with $N_f = 2n+4$ flavors in \\cite{Hayashi:2015fsa}, and the global symmetry was determined to be $SO(4n+8)$. We repeat the same procedure to determine the global symmetry of the 5d $T_N$ Tao theory. \n\nWe first pull all the 7-branes inside the 5-brane loops. Since $[p, q]$ 7-branes mutually commute with $(p, q)$ 5-branes, it is possible to collect the 7-branes into three chambers, the top-left one (1st chamber), the bottom-left one (2nd chamber) and the bottom-right one (3rd chamber) as in Figure \\ref{Fig:T5Tao2}. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{T5Tao2.pdf}\n\\end{center}\n\\caption{The 5-branes web after pulling all the 7-branes inside the 5-brane loops. We write an example of the 5d $T_5$ Tao theory as an example.} \n\\label{Fig:T5Tao2}\n\\end{figure}\nThe 7-branes in each chamber are schematically summarized as\n\\begin{equation}\n{\\bf ANCA}^{N-2}|| {\\bf NCAN}^{N-2} || {\\bf CANC}^{N-2} || \\label{TN7branes}\n\\end{equation}\nwhere ${\\bf A}, {\\bf C}$ and ${\\bf N}$ represent $[1,0]$, $[1,1]$ and $[0,1]$ 7-branes respectively. We will also express $[1,-1]$ 7-brane by ${\\bf B}$. In this expression of \\eqref{TN7branes}, the branch cuts are assumed to extend in the lower direction, and the charge of a 7-brane change by the monodromy matrix\n\\begin{eqnarray}\n\\left(\n\\begin{array}{cc}\n1-pq & p^2\\\\\n-q^2 & 1+pq\n\\end{array}\n\\right)\n\\end{eqnarray}\nwhen the 7-brane crosses the branch cut of a $[p, q]$ 7-brane counterclockwise. Also, the first, the second and the third partition lines in \\eqref{TN7branes} represent $(1,0)$ 5-branes, $(0,1)$ 5-branes and $(1,1)$ 5-branes respectively. Hence, 7-branes with the same charge as the 5-brane corresponding to the partition line can crosse its line. Therefore, we can move 7-branes in \\eqref{TN7branes}, and obtain\n\\begin{equation}\n{\\bf C}^{N-2} {\\bf ANC}|| {\\bf A}^{N-2}{\\bf NCAN}^{N-2} || {\\bf CAN}|| \\label{TN7branes1}\n\\end{equation}\nBy rearrangement of 7-branes inside each chamber, the 7-brane configuration becomes \n\\begin{equation}\n{\\bf X}_{[2,1]}{\\bf A}^N|| {\\bf A}^{2N-2}{\\bf B} || {\\bf N}^2{\\bf X}_{[2,1]}|| \\label{TN7branes2}\n\\end{equation}\nBy moving 7-branes between the chambers with the rearrangemnt in the second chamber, we finally obtain\n\\begin{equation}\n{\\bf X}_{[2,1]}|| {\\bf A}^{3N}{\\bf B} || {\\bf X}_{[2,1]}|| \\label{TN7branes3}\n\\end{equation}\nThe presence of $3N$ A-branes on top of each other means that the non-Abelian part of the global symmetry of the 5d $T_N$ Tao theory is $SU(3N)$ symmetry. \n\nIt is possible to count the number of the parameters of the 5d $T_N$ Tao theory. The number of mass parameters of the fundamental hypermultiplets is $N+5$. The number of mass parameters of the bi-fundamental multiplets is $N-3$. Also, the number of the gauge couplings or the mass parameters for the instantons is $N-2$. The total number is $3N$. Therefore, the full global symmetry of the 5d theory is $SU(3N) \\times U(1)_I$. As expected, we have the $U(1)_I$ symmety associated to the Kaluza-Klein modes from the $S^1$ compactification of a 6d theory. \n\nThe dimension of the Coulomb branch moduli space can be determined easily by counting the rank of the gauge groups, and it is $\\frac{(N-1)(N-2)}{2}$.\n\n\\subsection{5d \\texorpdfstring{$T_4, T_5$}{T4, T5} and \\texorpdfstring{$T_6$}{T6} Tao theories}\n\nLet us move on to a candidate for a 6d description of the 5d $T_N$ Tao theory. We first focus on the 5d $T_4, T_5, T_6$ theories, which will be basic examples for the general 5d $T_N$ Tao theory. \n\nIn fact, the 5d $T_4$ Tao theory is a very special case in the examples considered in section \\ref{sec:6dSUNA}. The 6d description in the tensor branch is \n\\begin{equation}\n[1]_{A} - SU(3) - [11]. \\label{T4}\n\\end{equation}\nWe also have one tensor multiplet corresponding to the number of the gauge nodes. The theory has one tensor multiplet and two vector multiplets in the Cartan subalgebra, and hence the sum of them reproduces the correct number of the 5d vector multiplets in the Cartan subalgebra of the 5d $T_4$ theory.\n\nLet us then determine the flavor symetery of the 6d theory \\eqref{T4}. It is important to note that the anti-symmetric representation of the $SU(3)$ is equivalent to the anti-fundamental representation of the $SU(3)$. Although a 6d hypermultiplet contains a 6d Weyl spinor, the complex conjugation of the 6d Weyl spinor does not change its chilarity. Therefore, the hypermultiplet in the anti-symmetric representation of $SU(3)$ is equivalent to a hypermultiplet in the fundamental representation of $SU(3)$. Namely, the 6d theory \\eqref{T4} is equivalent to a 6d $SU(3)$ gauge theory with $12$ flavors with one tensor multiplet coupled. From this point of view, we have at least an $SU(12)$ flavor symmetry.\n\nThere can be a potential $U(1)$ flavor symmetry that may come from the overall part of $U(12)$. However, we argue that the $U(1)$ symmetry is broken by the anomaly $U(1)_{\\text{global}} - SU(3)_{\\text{gauge}}^3$, and does not appear as a global symmetry of the 6d theory. Since the third-order Casimir invariant for $SU(3)$ is non-trivial, it is impossible to cancel the anomaly if there is only one representation. Therefore, the 6d theory does not have an Abelian symmetry. \n\nAfter the circle compactification, the KK mode provides a $U(1)_I$ symmetry. Hence, the total flavor symmetry is $SU(12) \\times U(1)_I$, which agrees with the flavor symmetry of the 5d $T_4$ Tao theory.\n\nLet us then consider the 5d $T_5$ Tao theory. In this case, it is not included in the examples we have discussed so far. This means that it may not admit a brane configuration in Type IIA string theory. However, it has been conjectured that all the 6d SCFTs may be realized by F-theory comapctifications \\cite{Heckman:2013pva, DelZotto:2014hpa, Heckman:2014qba, Heckman:2015bfa}. Then it is possible to find a candidate for its 6d description in the 6d SCFTs classified by the F-theory compactifications, by comparing the global symmetry as well as the dimension of the Coulomb branch moduli space. \n\nIn order to realize a 6d SCFT from F-theory, we consider an F-theory compactification on a Calab-Yau threefold which is given by an elliptically fibration over a non-compact complex two dimensional Kahler manifold $B$. In the base $B$, there is a bunch of $\\mathbb{P}^1$'s. Each size of the compact $\\mathbb{P}^1$ corresponds to a vev of a scalar in a tensor multiplet. The elliptic fibration over the $\\mathbb{P}^1$ can be singular, meaning that 7-branes wrap the $\\mathbb{P}^1$ and yielding a gauge symmetry. When all the compact $\\mathbb{P}^1$'s vanish simultaneously, the 6d theory is at the superconformal fixed point. This in fact constrains the shape of the sequence of the $\\mathbb{P}^1$'s significantly \\cite{Heckman:2013pva, Heckman:2015bfa}.\n\nAs for the 5d $T_5$ Tao theory, the dimension of the Coulomb branch moduli space is $6$. Therefore, the sum of the number of the tensor multiplets and the number of the vector multiplets in the Cartan subalgebra of the corresponding 6d theory must be $6$. Since the expected 6d flavor symmetry is $SU(15)$, we need one non-compact $\\mathbb{P}^1$\\footnote{We call non-compact $\\mathbb{P}^1$ as a $\\mathbb{P}^1$ whose size is taken to be infinity.} with a singular elliptic fiber giving the $SU(15)$ algebra on it. We denote the non-compact 2-cycle by $C_{SU(15)}$. Due to the strong contraint of the shape of the base geometry, only pairwise intersections are allowed. Furthermore, the non-trivial gauge algebra on the adjacent $\\mathbb{P}^1$ should be either $SU$ or $Sp$ \\cite{Heckman:2015bfa}. \n\nLet us then consider the case where one compact $\\mathbb{P}^1$ is attached to $C_{SU(15)}$. The theory has one tensor multiplet. Hence, the rank of the $SU$ or $Sp$ gauge group should be $5$. Then, the 6d anomaly cancellation condition with the $15$ flavors gives us a unique possibility \n\\begin{equation}\n\\left[\\frac{1}{2}\\right]_{\\Lambda^3} - SU(6) - [15], \\label{T5}\n\\end{equation}\nwhere $\\left[\\frac{1}{2}\\right]_{\\Lambda^3}$ stands for one half-hypermultiplet in the rank three anti-symmetric representation. This 6d theory on $S^1$ reproduces the correct number of the 5d vector multiplets in the Cartan subalgbera of the 5d $T_5$ Tao theory as well as the $SU(15)$ flavor symmetry.\nThe potential $U(1)$ flavor symmetry which may come from the overall part of the $U(15)$ is again broken by the anomaly $U(1) - SU(6)^3$.\n\nWhen one increases the number of compact $\\mathbb{P}^1$'s attached to $C_{SU(15)}$, then one cannot find a candidate for a non-trivial gauge group on the compact $\\mathbb{P}^1$'s that satisfies the 6d gauge anomaly cancellation condition with the $15$ flavors. If we increase the number of the compact $\\mathbb{P}^1$'s, the total rank of the gauge groups should be less than or equal to $4$. Then, there is no possible choice for the gauge group on the compact $\\mathbb{P}^1$ next to $C_{SU(12)}$ by the 6d anomaly cancellation condition. If there is no gauge symmetry on the compact $\\mathbb{P}^1$ next to $C_{SU(15)}$, then the $15$ flavors on $C_{SU(15)}$ may give a $U(15)$ flavor symmetry at least\\footnote{For example, the E-string theory on the tensor branch consists of $8$ hypermultiplets and one tensor multiplet, The flavor symmetry is $SO(16)$ which is further enhanced to $E_8 \\supset SO(16)$.}. Therefore, it contradicts the flavor symmetry of the 5d $T_5$ Tao theory.\n\nHence, we propose that the 6d description of the 5d $T_5$ Tao theory is given by the 6d quiver of \\eqref{T5}.\n\nThe last case is the 5d $T_6$ Tao theory. This is also a very special case in the examples considered in section \\ref{subsec:6dSUquivgen}. The 6d description in the tensor branch is \n\\begin{equation}\nSU(1) - SU(9) - [17].\n\\end{equation}\nNote that there is no hypermultiplet in the anti-symmetric representation for the $SU(1)$. Furthermore, the $SU(1)$ does not a dynamical vector multiplet. Therefore, the 6d theory is effectively described by the $SU(9)$ gauge theory with $18$ hypermultiplets in the fundamental representation with two tensor multiplets coupled. Therefore, after the circle compactification, it gives $2+8 = 10$ 5d vector multiplets in the Cartan subalgebras, which reproduces the correct number for the 5d $T_6$ Tao theory. Furthermore, the flavor symmetry is $SU(18) \\times U(1)_I$, which also agrees with the flavor symmetry of the 5d $T_6$ Tao theory. The potential $U(1)$ symmetry of the overall part of $U(18)$ is again broken by the anomaly $U(1) - SU(9)^3$. \n\n\n\n\n\n\\subsection{Generalization}\n\nWe move on to the analysis for a 6d description of the 5d $T_N$ Tao theory. Given the 6d description of the 5d $T_4, T_5, T_6$ Tao theory, there is a very natural generalization to a general $N$. \n\nThe 6d description of the 5d $T_4$ Tao theory is described by the 6d $SU(3)$ gauge theory with $12$ flavors, and it has the $SU(12)$ flavor symmetry. We then gauge the $SU(12)$ flavor symmetry. Due to the 6d anomaly cancellation condition, the $SU(12)$ gauge node needs to have $12$ more flavors. Namely, a 6d anomaly free quiver theory by gauging the $SU(12)$ flavor symmetry is \n\\begin{equation}\nSU(3) - SU(12) - [21]. \\label{T7}\n\\end{equation}\nIt has two tensor multiplets and $13$ vector multiplets in the Cartan subalgebra. The total number is $15$, and this number in fact agrees with the number of the Coulomb branch moduli of the 5d $T_7$ theory. Furthermore, the flavor symmetry of the 6d theory \\eqref{T7} after the $S^1$ compactification is $SU(21) \\times U(1)_I$, which also agrees with the global symmetry of the 5d $T_7$ Tao theory. In this case, there can be potential $U(1) \\times U(1)$ flavor symmetries. One may be associated to the one bi-fundamental hypermultiplet between $SU(3)$ and $SU(12)$, and the other may be associated to the $21$ fundamental hypermultiplets. However, any linear combination of the two $U(1)$'s has non-zero anomaly either from $U(1) - SU(3)^3$ or $U(1) - SU(12)^3$. Therefore, the 6d theory \\eqref{T7} is a natural candidate for the 5d $T_7$ theory. \n\nIt is easy to generalize this consideration by gauging the $SU(21)$ flavor symmetry. The general quiver theory by the successive gauging is \n\\begin{equation}\nSU(3) - SU(12) - SU(21) - \\cdots - [9N+3].\\label{T3N+1}\n\\end{equation}\nIt has $N$ tensor multiplets and $\\frac{N(9N-5)}{2}$ vector multiplets in the Cartan subalgbera. The total number is $\\frac{3N(3N-1)}{2}$, which exactly agrees with the number of the Coulomb branch moulid of the 5d $T_{3N+1}$ Tao theory. The flavor symmetry after the $S^1$ compactification is $SU(9M+3) \\times U(1)_I$, which also agrees with that of the 5d $T_{3N+1}$ Tao theory. The potential $U(1)^{N}$ flavor symmetries are all broken by the [global - gauge$^3$] anomalies. Hence, the 6d quiver theory \\eqref{T3N+1} is a candidate for the 6d description of the 5d $T_{3N+1}$ Tao theory with $N \\geq 1$. \n\nIt is interesting to note that the rank of the gauge group of the 6d quiver \\eqref{T3N+1} increases by $9$. Hence, this 6d theory is not realized by a brane configuration with an $O8^-$-plane in Type IIA string theory. However, it may admit a F-theory realization, and a possible configuration may be\n\\begin{eqnarray}\n\\begin{array}{ccccc}\n \\mathfrak{su}(3)&\\mathfrak{su}(12) & \\cdots & \\mathfrak{su}(9N-6) & [\\mathfrak{su}(9N+3)]\\nonumber\\\\\n 1 & 2 & \\cdots & 2 & \\nonumber\n\\end{array} \n\\end{eqnarray}\nEach columen represents a $\\mathbb{P}^1$ and the last one stands for a non-compact $\\mathbb{P}^1$ in the base $B$. The uppper row indicates the singular type of the elliptic fibration over the $\\mathbb{P}^1$. The lower row gives the self-intersection number of each compact $\\mathbb{P}^1$ inside $B$. \n\nIn fact, the proposed 6d descriptions of the 5d $T_5$ and $T_6$ Tao theories also admit the same generalization. By the succesive gauging of the flavor symmetries of \\eqref{T5}, one arrives at \n\\begin{equation}\n\\left[\\frac{1}{2}\\right]_{\\Lambda^3} - SU(6) - SU(15) - \\cdots - SU(9N-3) - [9N+6]. \\label{T3N+2}\n\\end{equation}\nThe 6d quiver theory has $N$ tensor multiplets and $\\frac{N(9N+1)}{2}$ vector multiplets in the Cartan subalgbera. The sum of the numbers give $\\frac{3N(3N+1)}{2}$, which is exactly the dimension of the Coulomb branch moduli space of the 5d $T_{3N+2}$ Tao theory with $N \\geq 1$. The potential $U(1)^{N}$ global symmetries are all broken by the [global - gauge$^3$] anomaly of some gauge group. Hence, the flavor symmetry after the circle compactification is $SU(9N+6) \\times U(1)_I$. and it agrees with that of the 5d $T_{3N+2}$ Tao theory. Therefore, the 6d quiver theory \\eqref{T3N+2} is a natural candidate of the 6d description of the 5d $T_{3N+2}$ Tao theory. \n\nAgain, the 6d quiver theory \\eqref{T3N+2} does not have a Type IIA bran realization, but it may have a F-theory realization. The possible configuration is \n\\begin{eqnarray}\n\\begin{array}{ccccc}\n \\mathfrak{su}(6)&\\mathfrak{su}(15) & \\cdots & \\mathfrak{su}(9N-3) & [\\mathfrak{su}(9N+6)]\\nonumber\\\\\n 1 & 2 & \\cdots & 2 & \\nonumber\n\\end{array} \n\\end{eqnarray}\nwhere the first $\\mathbb{P}^1$ can support the half-hypermultiplet in the rank three anti-symmetric representation \\cite{Heckman:2015bfa}. \n\nLet us finally turn to the generalization of the 5d $T_6$ theory by gauging the flavor symmetry. The 6d theory is a quiver theory described by \n\\begin{equation}\nSU(0) - SU(9) - SU(18) - \\cdots - SU(9N) - [9N+9]. \\label{T3N+3}\n\\end{equation}\nThe first $SU(0)$ indicates an additional tensor multiplet without any gauge dynamics. The 6d theory has $N+1$ tensor multiplets and $\\frac{N(9N+7)}{2}$. The sum of the numbers is $\\frac{(3N+2)(3N+1)}{2}$, which agrees with the number of the Coulomb branch moduli of the 5d $T_{3N+3}$ theory. Due to the [global - gauge$^3$] anomalies, the potential $U(1)^N$ global symmetries are all broken. Hence, the flavor symmetry after the $S^1$ compactification is $SU(9N+9) \\times U(1)_I$, and it correctly reproduces the global symmetry of the 5d $T_{3N+3}$ Tao theory. The possible F-theory realization of the 6d theory is \n\\begin{eqnarray}\n\\begin{array}{ccccc}\n \\emptyset &\\mathfrak{su}(9) & \\cdots & \\mathfrak{su}(9N) & [\\mathfrak{su}(9N+9)]\\nonumber\\\\\n 1 & 2 & \\cdots & 2 & \\nonumber\n\\end{array} \n\\end{eqnarray}\nwhere the first $\\mathbb{P}^1$ does not support a gauge algebra. \n\nSo far we have considered the 5d $T_{3N+3}$ Tao theory with $N \\geq 1$. It is interesting to think of the case where $N=0$. This case admits a Type IIA brane configuration that was considered in section \\ref{sec:speicalcases}. The 6d quiver description is \n\\begin{equation}\nSU(1) - [9].\n\\end{equation}\n In fact, this theory has been shown to be equivalent to the rank $1$ E-string theory in section \\ref{sec:speicalcases}. This is perfect agreement with the formal substitution of $N=0$ into the 5d $T_{3N+3}$ Tao theory since the 5d $T_3$ Tao theory is nothing but the rank $1$ E-string theory \\cite{Kim:2015jba}.\n \n \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzexcl b/data_all_eng_slimpj/shuffled/split2/finalzzexcl new file mode 100644 index 0000000000000000000000000000000000000000..98b00c2fd410df4c2780a2133c017432b701d2b0 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzexcl @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nRecently discovered topologically nontrivial phases have attracted many researchers and offered a new direction to modern physics\\cite{Hasan2010,Qi2011}.\nTopologically nontrivial phase and trivial phase, in the presence of time-reversal symmetry, are distinguished by the $Z_2$ invariant\\cite{Fu2007,Fu2007a}.\nStrong spin-orbit coupling is known to be essential to realize topological phases, since topological phases originate in the parity change in the lowest unoccupied band from even to odd induced by spin-orbit coupling.\nTopological phases are characterized by the gapless edge (surface) states which are protected by time-reveral symmetry.\nIn 3D topological insulators, the surface states are described by the two-component massless Dirac fermions.\nThe bulk states in such as Bi$_2$Se$_3$ are described by the four-component anisotropic massive Dirac fermions\\cite{Zhang2009}.\nIt is known that the surface states are robust against perturbation and disorder\\cite{Bardarson2007,Nomura2007}.\nWhat about against electron correlation, i.e. Coulomb interaction?\nThis is a natural question, because it has been revealed that strong electron correlation is important in many systems and may induce novel phenomena.\n\nA novel Mott-insulating phase was found recently in an iridate\\cite{Kim2008}, a $5d$-electron system, and has gathered much attention.\nRemarkably, the phase is induced by the cooperation of strong spin-orbit coupling and strong electron correlation.\nEvolved by this discovery, many studies have been done intensively in systems where both spin-orbit coupling and electron correlation exist, for the search for novel phases induced by them.\nEspecially, it is of interest that topological phases such as the quantum spin Hall insulator\\cite{Shitade2009} and the Weyl semimetal\\cite{Wan2011} are predicted in iridates.\nThese results suggest that topological phases may emerge in strongly correlated $d$-electron systems.\nPreceding studies mainly focus on the competition between the spin or charge ordered phase and the topological phase in Hubbard-like models on honeycomb lattices\\cite{Raghu2008,Meng2010,Rachel2010,Varney2010,Hohenadler2011,Yamaji2011,Ruegg2011,Zheng2011,Yu2011}, other 2D lattices\\cite{Sun2009,Wen2010,Yoshida2012,Yoshida2012a,Hohenadler2012} and 3D lattices\\cite{Zhang2009a,Pesin2010,Mong2010,Kurita2011}.\nAnother study on the surface Dirac fermions shows that the Dirac fermions become massive with finite correlation strength due to the spotaneous magnetization\\cite{Baum2012}.\n\nOn the other hand, the electron correlation effect in graphene, a two-dimensional Dirac fermion system, has been studied widely.\nIn graphene in vacuum, the coupling constant becomes effectively large due to the small Fermi velocity. It has been predicted that a finite band gap is induced in charge neutral graphene in vacuum.\nIn such a case, the strong coupling lattice gauge theory is applied\\cite{Hands2008,Drut2009,Drut2009a,Armour2010,Drut2010,Araki2010,Araki2011,Araki2012,Buividovich2012}.\nThe chiral condensate is the order parameter for the insulator-semimetal transition in the lattice gauge theory.\nIt is noteworthy that lattice Monte Carlo studies show quantitatively correct critical value of the coupling strength below which the system becomes gapless\\cite{Drut2009,Drut2009a,Buividovich2012} (graphene on a SiO$_2$ substrate is conducting).\nThese results motivated us to do this study.\n\nIn this paper, we focus on the strong electron correlation effect in a 3D Dirac fermion system on a lattice which is a simple model describing a topologically nontrivial state.\nWe adopt $1\/r$ long-range Coulomb interaction as an interaction between the bulk electrons, because the screening effect in Dirac fermion systems is considered to be weak due to the vanishing of the density of states.\nThis situation is nothing but what is described by the U(1) lattice gauge theory.\nTherefore, we can perform the strong coupling expansion of the lattice gauge theory by assuming that the effective coupling constant is large.\nThe procedure is as follows.\nFirst we derive the effective action by the strong couling expansion.\nNext we calculate the effective potential (the free energy per unit volume at zero temperature) with the use of the Hubbard-Stratonovich transformation and the mean-field approximation.\nFinally we obtain the value of the chiral condensate as the stationary point of the effective potential.\nOur model, the Wilson fermions, breaks chiral symmetry by itself, and thus we cannot use the chiral condensate as the order parameter for the the insulator-semimetal transition.\nWe regard the chiral condensate as a correction to the bare mass.\n\nThe main purpose of this study is devided into two parts:\n(I) answer the question that whether the topological insulator phase survives at the limit of infinitely strong Coulomb interaction between the bulk electrons, or not. \nTo do this, we have to obtain the value of the chiral condensate, which corresponds to a correction to the bare mass, in the strong coupling limit.\n(II) search for the phase in which time-reversal and inversion symmetries are spontaneously broken due to electron correlation.\nSuch a phase, \"Aoki phase\" has been confirmed in the lattice quantum chromodynamics (QCD) with Wilson fermions\\cite{Aoki1984,Aoki1986,Sharpe1998} and was suggested recently in a mean-field study of Wilson fermions with the short-range interaction\\cite{Sekine2012}.\n\n\\section{Model}\nIt is known that the effective Hamiltonian of 3D topological insulators such as Bi$_2$Se$_3$ is described by the Wilson fermion\\cite{Zhang2009}:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{H}_{0}(\\bm{k})={\\sum}_j\\sin k_j\\cdot\\alpha_j+m(\\bm{k})\\beta,\\label{H_0}\n\\end{aligned}\n\\end{equation}\nwhere $m(\\bm{k})=m_0+r\\sum_j\\left(1-\\cos k_j\\right)$, $r>0$, $j\\ (=1,2,3)$ denotes spacial axis, and $\\alpha_j$, $\\beta$ are the Dirac gamma matrices given by\n\\begin{equation}\n\\begin{aligned}\n\\alpha_j=\n\\begin{bmatrix}\n0 & \\sigma_j\\\\\n\\sigma_j & 0\n\\end{bmatrix},\\ \\ \\ \\ \\\n\\beta=\n\\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix}.\n\\end{aligned}\n\\end{equation}\nThe energy of this system is measured in units of $v_{\\mathrm{F}}\/a$ with $v_{\\mathrm{F}}$ and $a$ being the Fermi velocity and the lattice constant, respectively.\nThe Hamiltonian (\\ref{H_0}) has time-reversal ($\\mathcal{T}$) symmetry and inversion ($\\mathcal{I}$) symmetry, i.e., $\\mathcal{T}\\mathcal{H}_{0}(\\bm{k})\\mathcal{T}^{-1}=\\mathcal{H}_{0}(-\\bm{k})$ and $\\mathcal{I}\\mathcal{H}_{0}(\\bm{k})\\mathcal{I}^{-1}=\\mathcal{H}_{0}(-\\bm{k})$ are satisfied, where $\\mathcal{T}=\\bm{1}\\otimes(-i\\sigma_2)\\mathcal{K}$ ($\\mathcal{K}$ is the complex conjugation operator) and $\\mathcal{I}=\\sigma_3\\otimes\\bm{1}$.\nIn the Hamiltonian (\\ref{H_0}), the spinor is written in the basis of $\\left[c^\\dag_{\\bm{k}A\\uparrow},c^\\dag_{\\bm{k}A\\downarrow},c^\\dag_{\\bm{k}B\\uparrow},c^\\dag_{\\bm{k}B\\downarrow}\\right]$, where $c^\\dag$ is the creation operator of an electron, $A$, $B$ denote two orbitals, and $\\uparrow$ ($\\downarrow$) denotes up- (down-) spin\\cite{Zhang2009}.\n\nIn the presence of time-reversal symmetry and inversion symmetry, the $Z_2$ invariant of the system is given by\\cite{Fu2007,Fu2007a}\n\\begin{equation}\n\\begin{aligned}\n(-1)^\\nu=\\prod_{i=1}^8\\left\\{-\\mathrm{sgn}\\left[m\\left(\\bm{\\Lambda}_i\\right)\\right]\\right\\}, \\label{Z2invariant}\n\\end{aligned}\n\\end{equation}\nwhere $\\bm{\\Lambda}_i$ are the eight time-reversal invariant momenta.\nIt is easily shown that if $0>m_0>-2r$ or $-4r>m_0>-6r$ ($m_0>0$, $-2r>m_0>-4r$, or $-6r>m_0$), the system is topologically nontrivial (trivial).\n\nLet us consider a strongly correlated topological insulator in the Euclidean spacetime, which is described by the Wilson fermions with $1\/r$ Coulomb interaction between the bulk electrons.\nWe start from the Euclidean action of (3+1)D Wilson fermion interacting with electromagnetic field on a lattice, which is given by\n\\begin{equation}\n\\begin{aligned}\nS_{F}=&-\\sum_{n,\\mu}\\left[\\bar{\\psi}_nP^-_\\mu U_{n,\\mu}\\psi_{n+\\hat{\\mu}} + \\bar{\\psi}_{n+\\hat{\\mu}}P^+_\\mu U^\\dag_{n,\\mu}\\psi_n\\right]\\\\\n&+(m_0+4r)\\sum_{n}\\bar{\\psi}_n \\psi_n,\\label{ActionQED}\n\\end{aligned}\n\\end{equation}\nwhere $P^\\pm_\\mu=(r\\pm \\gamma_\\mu)\/2$.\nHere $n=(n_0,n_1,n_2,n_3)$ denotes a site on a spacetime lattice and $\\hat{\\mu}$ ($\\mu=0,1,2,3$) denotes the unit vector along $\\mu$-direction.\n$U_{n,\\mu}$ is the link variable, which is defined by $U_{n,\\mu}=e^{iagA_{\\mu}(n+\\hat{\\mu}\/2)}$, where $A_\\mu=(A_0,\\bm{A})$ is the four-vector potential, $a$ is the lattice constant, and $g^2=e^2\/\\epsilon$ with $e$ and $\\epsilon$ being electric charge and the permittivity of the system, respectively.\nAlthough the timelike Wilson term (the term proportional to $r$) is introduced artificially to eliminate fermion doublers, the spatial Wilson terms have a physical meaning (arise due to strong spin-orbit coupling).\nIn this paper, according to the Hamiltonian (\\ref{H_0}), we adopt the Dirac representation in the Euclidean spacetime ($\\{\\gamma_\\mu,\\gamma_\\nu\\}=2\\delta_{\\mu\\nu}$):\n\\begin{equation}\n\\begin{aligned}\n\\gamma_0=\n\\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix},\\ \\ \\ \n\\gamma_j=\n\\begin{bmatrix}\n0 & -i\\sigma_j\\\\\ni\\sigma_j & 0\n\\end{bmatrix},\\ \\ \\ \n\\gamma_5=\n\\begin{bmatrix}\n0 & 1\\\\\n1 & 0\n\\end{bmatrix},\n\\end{aligned}\\label{gamma-matrices}\n\\end{equation}\nwhere $j=1,2,3$ and $\\sigma_j$ are the Pauli matrices.\n\nIn the case of 3D topological insulators, the Fermi velocity $v_{\\rm F}$ is about $3\\times 10^{-3}c$ where $c$ is the speed of light in vacuum.\nThen the interactions between the bulk electrons can be regarded as only the instantaneous Coulomb interaction ($A_j=0$) like in the case of graphene\\cite{Hands2008,Drut2009,Drut2009a,Armour2010,Drut2010,Araki2010,Araki2011,Araki2012,Buividovich2012}, so the action (\\ref{ActionQED}) is rewritten as\n\\begin{equation}\n\\begin{aligned}\nS_{F}=S_F^{(\\tau)}+S_F^{(s)}+(m_0+4r)\\sum_{n}\\bar{\\psi}_n \\psi_n,\\label{ActionTI}\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\nS_F^{(\\tau)}&=-\\sum_{n}\\left[\\bar{\\psi}_nP^-_0 U_{n,0}\\psi_{n+\\hat{0}} + \\bar{\\psi}_{n+\\hat{0}}P^+_0 U^\\dag_{n,0}\\psi_n\\right]\\\\\nS_F^{(s)}&=-\\sum_{n,j}\\left[\\bar{\\psi}_nP^-_j\\psi_{n+\\hat{j}} + \\bar{\\psi}_{n+\\hat{j}}P^+_j\\psi_n\\right],\n\\end{aligned}\n\\right.\n\\end{equation}\nand $U_{n,0}=e^{i\\theta_n}\\ (-\\pi\\leq\\theta_n\\leq\\pi)$.\nThe Wilson fermions breaks chiral symmetry by itself (the terms proportional to $r$ and $m_0$), i.e., the action (\\ref{ActionTI}) is not invariant under the chiral transformation $\\psi\\rightarrow e^{i\\theta\\gamma_5}\\psi$.\nIn our model, chiral symmetry is equivalent to the symmetry of the pseudospin for two $p$-orbitals $A$ and $B$.\nThe pure U(1) gauge action on a lattice is given by\n\\begin{equation}\n\\begin{aligned}\nS_G=\\beta\\sum_n\\sum_{\\mu>\\nu}\\left[1-\\frac{1}{2}\\left(U_{n,\\mu\\nu}+U^\\dag_{n,\\mu\\nu}\\right)\\right],\n\\end{aligned}\n\\end{equation}\nwhere $\\beta=v_{\\rm F}\/g^2$. The plaquette contribution $U_{n,\\mu\\nu}$ is defined by\n\\begin{equation}\n\\begin{aligned}\nU_{n,\\mu\\nu}=U_{n,\\mu}U_{n+\\hat{\\mu},\\nu}U^\\dag_{n+\\hat{\\nu},\\mu}U^\\dag_{n,\\nu},\n\\end{aligned}\n\\end{equation}\nwhere $U_{n,j}=1$ in our case.\nThe total action on a lattice is written as\n\\begin{equation}\n\\begin{aligned}\nS=S_F+S_G.\n\\end{aligned}\n\\end{equation}\nThe dielectric constant $\\epsilon_r$ of Bi$_2$Se$_3$ is rather large\\cite{dielectricconstant-Bi2Se3} ($\\epsilon_r=\\epsilon\/\\epsilon_0\\approx 100$).\nThis means that the Coulomb interaction between the bulk electrons in Bi$_2$Se$_3$ is considered to be weak.\nIn fact, the value of $\\beta$ is approximated as\n\\begin{equation}\n\\begin{aligned}\n\\beta=\\frac{v_{\\rm F}\\epsilon_r}{4\\pi c}\\cdot\\frac{4\\pi\\epsilon_0\\hbar c}{e^2}\\approx 3,\n\\end{aligned}\n\\end{equation}\nand we cannot perform the strong coupling expansion in Bi$_2$Se$_3$.\nHowever, we think it would be important from a theorerical viewpoint to examine the strong electron correlation effect in Dirac fermion systems which describe topologically nontrivial states.\n\n\n\\section{Effective Action}\nLet us perform the strong coupling expansion.\nThe strong coupling expansion has been often used in QCD\\cite{Kawamoto1981,Hoek1982,Drouffe1983,Nishida2004,Miura2009} where the coupling between fermions (quarks) and gauge fields (gluons) are strong.\nWe can carry out the $U_0$ integral by using the SU($N_c$) group integral formulae:\n\\begin{equation}\n\\begin{aligned}\n\\int dU1=1,\\ \\ \\int dUU_{ab}=0,\\ \\ \\int dUU_{ab}U_{cd}^\\dag=\\frac{1}{N_c}\\delta_{ad}\\delta_{bc}.\n\\end{aligned}\n\\end{equation}\nOur case corresponds to the case of $N_c=1$.\nIn the following, we derive the effective action $S_{\\rm eff}[\\psi,\\bar{\\psi}]$ by carrying out the $U_0$ integral:\n\\begin{equation}\n\\begin{aligned}\nZ=\\int \\mathcal{D}[\\psi,\\bar{\\psi},U_0]e^{-S_F-S_G}=\\int \\mathcal{D}[\\psi,\\bar{\\psi}]e^{-S_{\\rm eff}}.\n\\end{aligned}\n\\end{equation}\n\nFirst we consider the strong coupling limit ($\\beta=0$). In this case, the timelike partition function is given by\n\\begin{equation}\n\\begin{aligned}\nZ^{(\\tau)}_{\\mathrm{SCL}}[\\psi,\\bar{\\psi}]=\\int \\mathcal{D}U_0e^{-S^{(\\tau)}_F}.\n\\end{aligned}\n\\end{equation}\nIntegration with respect to $U_0$ is carried out to be\n\\begin{equation}\n\\begin{aligned}\n&Z^{(\\tau)}_{\\mathrm{SCL}}=\\exp\\left[\\sum_n\\bar{\\psi}_nP^-_0 \\psi_{n+\\hat{0}}\\bar{\\psi}_{n+\\hat{0}}P^+_0 \\psi_n\\right].\\label{Z_SCL}\n\\end{aligned}\n\\end{equation}\nHere we have used the fact that the grassmann variables $\\psi$'s and $\\bar{\\psi}$'s satisfy $\\psi^2=\\bar{\\psi}^2=0$. We can rewrite this term as\n\\begin{equation}\n\\begin{aligned}\n\\bar{\\psi}_nP^-_0 \\psi_{n+\\hat{0}}\\bar{\\psi}_{n+\\hat{0}}P^+_0 \\psi_n= -\\mathrm{tr}\\left[M_nP^+_0M_{n+\\hat{0}}P^-_0\\right],\n\\end{aligned}\n\\end{equation}\nwhere we have defined $(M_n)_{\\alpha\\beta}=\\bar{\\psi}_{n,\\alpha}\\psi_{n,\\beta}$ and used $(P^\\pm_0)_{\\alpha\\beta}=(P^\\pm_0)_{\\beta\\alpha}$. The subscripts $\\alpha$ and $\\beta$ denote the component of spinors.\n\nNext we evaluate the term of the order of $\\beta$. \nIn order to evaluate the plaquette contributions from $S_G$, we use the cumulant expansion\\cite{Miura2009,Kubo1962}.\nLet us define an expectation value:\n\\begin{equation}\n\\begin{aligned}\n\\left\\langle A\\right\\rangle&\\equiv \\frac{1}{Z^{(\\tau)}_{\\mathrm{SCL}}}\\int \\mathcal{D}U_0A[U_0]e^{-S^{(\\tau)}_F}.\n\\end{aligned}\n\\end{equation}\nThen using this definition, the full timelike partition function can be expressed as \n\\begin{equation}\n\\begin{aligned}\nZ^{(\\tau)}=\\int \\mathcal{D}U_0e^{-S^{(\\tau)}_F-S_G}=Z^{(\\tau)}_{\\mathrm{SCL}}\\left\\langle e^{-S_G}\\right\\rangle.\\label{Z-tau-full}\n\\end{aligned}\n\\end{equation}\nThe contribution from $S_G$ is given by\n\\begin{equation}\n\\begin{aligned}\n\\Delta S\\equiv -\\log\\left\\langle e^{-S_G}\\right\\rangle=-\\sum_{n=1}^{\\infty}\\frac{(-1)^n}{n!}\\left\\langle S_G^n\\right\\rangle_c,\n\\end{aligned}\n\\end{equation}\nwhere $\\left\\langle \\cdots\\right\\rangle_c$ is a cumulant.\nThe correction to the action up to $\\mathcal{O}(\\beta)$ is given by\n\\begin{equation}\n\\begin{aligned}\n\\Delta S&=\\left\\langle S_G\\right\\rangle_c=\\left\\langle S_G\\right\\rangle\\\\\n&=-\\frac{\\beta}{2}\\sum_n\\sum_{\\mu>\\nu}\\left\\langle U_{n,\\mu\\nu}+U^\\dag_{n,\\mu\\nu}\\right\\rangle.\\label{S-NLO}\n\\end{aligned}\n\\end{equation}\nThe expectation value of $U_{n,\\mu\\nu}$ is evaluated as follows\\cite{Miura2009}:\n\\begin{equation}\n\\begin{aligned}\n\\left\\langle U_{n,\\mu\\nu}\\right\\rangle\\simeq \\int dU_{n,0} U_{n,\\mu\\nu}e^{-s^{(\\tau)}_P},\n\\end{aligned}\n\\end{equation}\nwhere $s^{(\\tau)}_P$ is the plaquette-related part of $S^{(\\tau)}_F$.\nWe see that the terms with $(\\mu,\\nu)=(i,j)$ become constant and find only $(\\mu,\\nu)=(j,0)$ terms to survive:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\left\\langle U_{n,j0}\\right\\rangle&=-\\mathrm{tr}\\left[V^+_{n,j}P^+_0V^-_{n+\\hat{0},j}P^-_0\\right],\\\\\n\\left\\langle U^\\dag_{n,j0}\\right\\rangle&=-\\mathrm{tr}\\left[V^-_{n,j}P^+_0V^+_{n+\\hat{0},j}P^-_0\\right],\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere we have defined $\\left(V^+_{n,j}\\right)_{\\alpha\\beta}=\\bar{\\psi}_{n,\\alpha}\\psi_{n+\\hat{j},\\beta}$ and $\\left(V^-_{n,j}\\right)_{\\alpha\\beta}=\\bar{\\psi}_{n+\\hat{j},\\alpha}\\psi_{n,\\beta}$.\n\nFinally, substituting Eqs. (\\ref{Z_SCL}) and (\\ref{S-NLO}) to Eq. (\\ref{Z-tau-full}), we obtain the effective action up to $\\mathcal{O}(\\beta)$:\n\\begin{equation}\n\\begin{aligned}\nS_{\\mathrm{eff}}=&(m_0+4r)\\sum_{n}\\bar{\\psi}_n\\psi_n-\\sum_{n,j}\\left[\\bar{\\psi}_nP^-_j\\psi_{n+\\hat{j}} + \\bar{\\psi}_{n+\\hat{j}}P^+_j\\psi_n\\right]\\\\\n&+\\sum_n\\mathrm{tr}\\left[M_nP^+_0M_{n+\\hat{0}}P^-_0\\right]\\\\\n&+\\frac{\\beta}{2}\\sum_{n,j}\\left\\{\\mathrm{tr}\\left[V^+_{n,j}P^+_0V^-_{n+\\hat{0},j}P^-_0\\right]+(V^+\\longleftrightarrow V^-)\\right\\}.\\label{Eff-Act}\n\\end{aligned}\n\\end{equation}\n\n\\section{Effective Potential and Chiral Condensate}\nIn this section, we derive the effective potential with the use of the extended Hubbard-Stratonovich transformation (EHS)\\cite{Miura2009,Araki2010,Araki2011,Araki2012}, and then we obtain the value of the chiral condensate as the stationary point of the effective potential.\nWe apply the EHS to the trace of arbitrary two matrices. Introducing two auxiliary fields $R$ and $R'$, we obtain\n\\begin{equation}\n\\begin{aligned}\n&e^{\\kappa\\mathrm{tr}AB}\\propto\\\\\n&\\hspace{0.3cm}\\int \\mathcal{D}[R,R'] \\exp\\left\\{-\\kappa\\sum_{\\alpha\\beta}\\left[(R_{\\alpha\\beta})^2+(R'_{\\alpha\\beta})^2\\rule{0pt}{3ex}\\right.\\right.\\\\\n&\\hspace{0.5cm}\\left.\\left.-(A_{\\alpha\\beta}+B^T_{\\alpha\\beta})R_{\\beta\\alpha}-i(A_{\\alpha\\beta}-B^T_{\\alpha\\beta})R'_{\\beta\\alpha}\\rule{0pt}{3ex}\\right]\\rule{0pt}{5ex}\\right\\},\\label{EHS}\n\\end{aligned}\n\\end{equation}\nwhere $\\kappa$ is a positive constant and the superscript $T$ denotes the transpose of a matrix.\nTwo auxiliary fields take the saddle point values $R_{\\alpha\\beta}=\\left\\langle A+B^T\\right\\rangle_{\\beta\\alpha}$\/2 and $R'_{\\alpha\\beta}=i\\left\\langle A-B^T\\right\\rangle_{\\beta\\alpha}$\/2, respectively. \nDefining $Q=R+iR'$ and $Q'=R-iR'$, Eq. (\\ref{EHS}) is rewritten as\n\\begin{equation}\n\\begin{aligned}\n&e^{\\kappa\\mathrm{tr}AB}\\propto\\\\\n&\\int \\mathcal{D}[Q,Q'] \\exp\\left\\{-\\kappa\\left[Q_{\\alpha\\beta}Q'_{\\alpha\\beta}-A_{\\alpha\\beta}Q_{\\beta\\alpha}-B^T_{\\alpha\\beta}Q'_{\\beta\\alpha}\\right]\\right\\},\\label{EHS-Q}\n\\end{aligned}\n\\end{equation}\nwith the saddle point values $Q_{\\alpha\\beta}=\\left\\langle B^T\\right\\rangle_{\\beta\\alpha}$ and $Q'_{\\alpha\\beta}=\\left\\langle A\\right\\rangle_{\\beta\\alpha}$.\n\n\\subsection{Effective Potential in the Strong Coupling Limit}\nWe consider to decouple the third term in the effective action (\\ref{Eff-Act}) to fermion bilinear form.\nTo do this, we set $(\\kappa, A, B)=(1, M_nP^+_0, -M_{n+\\hat{0}}P^-_0)$ in Eq. (\\ref{EHS-Q}). \nIn this case, the saddle point values are given by $Q_{\\alpha\\beta}=-\\left\\langle M_{n+\\hat{0}}P^-_0\\right\\rangle_{\\alpha\\beta}$ and $Q'_{\\alpha\\beta}=\\left\\langle M_nP^+_0\\right\\rangle_{\\beta\\alpha}$.\nHere let us assume that\n\\begin{equation}\n\\begin{aligned}\n\\left\\langle M_n\\right\\rangle&=\\sigma e^{i\\theta\\gamma_5}=\\sigma (\\cos\\theta I+i\\sin\\theta\\gamma_5)\\\\\n&=\\sigma\n\\begin{bmatrix}\n\\cos\\theta & i\\sin\\theta\\\\\ni\\sin\\theta & \\cos\\theta\n\\end{bmatrix},\\label{M_n}\n\\end{aligned}\n\\end{equation}\nbecause we are now interested in the phase structure of the Wilson fermions interacting via the long-range Coulomb interaction in the strong coupling limit, i.e., the mass term (the terms proportional to the identity matrix) is important when determining the phase is whether topologically trivial or nontrivial (see Eq. (\\ref{Z2invariant})).\nWe are also interested in the possibility of the existence of the symmetry broken phase (\"Aoki phase\") in this model.\nThus the pseudoscalar modes $i\\gamma_5$ should be taken into account.\nThen it follows that\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\left\\langle \\bar{\\psi}\\psi\\right\\rangle&=\\sigma\\cos\\theta\\equiv \\phi_\\sigma\\\\\n\\left\\langle \\bar{\\psi}i\\gamma_5\\psi\\right\\rangle&=\\sigma\\sin\\theta\\equiv \\phi_\\pi.\n\\end{aligned}\n\\right.\n\\end{equation}\nThe terms $\\left\\langle \\bar{\\psi}\\psi\\right\\rangle$ and $\\left\\langle \\bar{\\psi}i\\gamma_5\\psi\\right\\rangle$ decribe the chiral condensate and the condensate of pseudoscalar mode, respectively.\n\nThe Wilson fermions breaks chiral symmetry by itself (the terms proportional to $r$ and $m_0$).\nHence we cannot use the value of $\\left\\langle \\bar{\\psi}\\psi\\right\\rangle$ to determine the system is whether insulating or semimetallic, unlike in the case of graphene where chiral symmetry is not broken in the noninteracting limit. \nWe regard the value of the chiral condensate $\\left\\langle \\bar{\\psi}\\psi\\right\\rangle$ as a correction to the bare mass.\n\nSubstituting $(\\kappa, A, B)=(1, M_nP^+_0, -M_{n+\\hat{0}}P^-_0)$ to Eq. (\\ref{EHS-Q}), we obtain\n\\begin{equation}\n\\begin{aligned}\n&\\exp\\left\\{-\\sum_n\\mathrm{tr}\\left[M_nP^+_0M_{n+\\hat{0}}P^-_0\\right]\\right\\}\\\\\n&\\sim\\exp\\left\\{-\\sum_{n}\\left[(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2\\rule{0pt}{3ex}\\right.\\right.\\\\\n&\\left.\\left.\\hspace{0.35cm}+\\frac{1}{2}\\bar{\\psi}_n\\left[-(1-r^2)\\phi_\\sigma+i\\gamma_5^T(1+r^2)\\phi_\\pi\\right]\\psi_n\\rule{0pt}{3ex}\\right]\\rule{0pt}{4ex}\\right\\},\n\\end{aligned}\n\\end{equation}\nwhere we have applied the mean-field approximation for the chiral condensate and the condensate of pseudoscalar mode.\nThus the effective action in the strong coupling limit expressed by the two auxiliary fields $\\phi_\\sigma$ and $\\phi_\\pi$ is given by\n\\begin{equation}\n\\begin{aligned}\nS_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=&N_sN_\\tau\\left[(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2\\right]\\\\\n&+\\sum_k \\bar{\\psi}_k\\mathcal{M}(\\bm{k};\\phi_\\sigma,\\phi_\\pi)\\psi_k, \\label{S_eff_SCL}\n\\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{M}=&{\\sum}_j i\\gamma_j\\sin k_j+m_0+r\\left(4-{\\sum}_j\\cos k_j\\right)\\\\\n&-\\frac{1}{2}(1-r^2)\\phi_\\sigma+i\\gamma_5^T\\frac{1}{2}(1+r^2)\\phi_\\pi.\\label{M-SCL}\n\\end{aligned}\n\\end{equation}\nHere $N_s=V$ and $N_\\tau=1\/T$ with $V$ and $T$ being the volume and the temperature of the system, respectively and we have done the Fourier transform from $n=(n_0,\\bm{n})$ to $k=(k_0,\\bm{k})$.\n\nThe effective potential at zero temperature per unit spacetime volume is given by\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=-\\frac{1}{N_sN_\\tau}\\log Z(\\phi_\\sigma,\\phi_\\pi).\n\\end{aligned}\n\\end{equation}\nIntegration with respect to $\\psi$ and $\\bar{\\psi}$ is carried out by the formula $\\int D[\\psi,\\bar{\\psi}]e^{-\\bar{\\psi}\\mathcal{M}\\psi}=\\mathrm{det}\\mathcal{M}$. \nTherefore we need to calculate the determinant of $\\mathcal{M}$.\nFrom Eq. (\\ref{M-SCL}), the matrix $\\mathcal{M}$ is written explicitly as\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{M}&=\n\\begin{bmatrix}\n\\tilde{m}(\\bm{k})+r & \\sigma_j\\sin k_j+i\\frac{1+r^2}{2}\\phi_\\pi\\\\\n-\\sigma_j\\sin k_j+i\\frac{1+r^2}{2}\\phi_\\pi & \\tilde{m}(\\bm{k})+r\n\\end{bmatrix}\\\\\n&\\equiv\n\\begin{bmatrix}\nA & B\\\\\nC & D\n\\end{bmatrix},\n\\end{aligned}\\label{M}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\tilde{m}(\\bm{k})=m_0-\\frac{1-r^2}{2}\\phi_\\sigma+r{\\sum}_j\\left(1-\\cos k_j\\right). \\label{m_eff}\n\\end{aligned}\n\\end{equation}\nAs we see from Eq. (\\ref{m_eff}), the chiral condensate $\\phi_\\sigma$ corresponds to a correction to the bare mass $m_0$ in the original Hamiltonian (\\ref{H_0}).\nThat is, $m(\\bm{k})$ in the noninteracting Hamiltonian (\\ref{H_0}) changes to $\\tilde{m}(\\bm{k})$ in the strong coupling limit.\nThe term \"$r$\" of $\\tilde{m}(\\bm{k})+r$ in Eq. (\\ref{M}) originates in the timelike components of the action.\nAfter a straightforward calculation, we have\n\\begin{equation}\n\\begin{aligned}\n&\\mathrm{det}\\mathcal{M}=\\mathrm{det}A\\cdot\\mathrm{det}\\left(D-CA^{-1}B\\right)\\\\\n&=\\left[{\\sum}_j\\sin^2k_j+\\left[\\tilde{m}(\\bm{k})+r\\right]^2+\\frac{(1+r^2)^2}{4}\\phi_\\pi^2\\right]^2.\\label{detM}\n\\end{aligned}\n\\end{equation}\nThe same result can be derived by the formula $\\mathrm{det}\\mathcal{M}=\\sqrt{\\mathrm{det}(\\mathcal{M}\\mathcal{M}^\\dag)}$.\nFinally we arrive at the effective potential in the strong coupling limit:\n\\begin{equation}\n\\begin{aligned}\n&\\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2-2\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\\\\n&\\times\\log\\left[{\\sum}_j\\sin^2k_j+\\left[\\tilde{m}(\\bm{k})+r\\right]^2+\\frac{(1+r^2)^2}{4}\\phi_\\pi^2\\right].\\label{Feff-SCL}\n\\end{aligned}\n\\end{equation}\n\nThe values of $\\phi_\\sigma$ and $\\phi_\\pi$ are obtained by the stationary conditions $\\partial \\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)\/\\partial \\phi_\\sigma=\\partial \\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)\/\\partial \\phi_\\pi=0$.\nWhen $r=1$, Eq. (\\ref{Feff-SCL}) does not depend on $\\phi_\\sigma$.\nIn this case, the stationary point is obtained by the following equation:\n\\begin{equation}\n\\begin{aligned}\n\\phi_\\sigma&=\\frac{1}{4N_sN_\\tau}\\frac{\\int \\mathcal{D}[\\psi,\\bar{\\psi},U_0]\\sum_n\\bar{\\psi}_n\\psi_n e^{-S}}{\\int \\mathcal{D}[\\psi,\\bar{\\psi},U_0]e^{-S}}\\\\\n&=-\\frac{1}{4N_sN_\\tau}\\frac{1}{Z}\\frac{d Z}{d m_0}\\\\\n&=\\frac{1}{4}\\frac{d\\mathcal{F}_{\\mathrm{eff}}}{d m_0}.\n\\end{aligned}\n\\end{equation}\nWhen $r>1$, the coefficient of the first term in Eq. (\\ref{Feff-SCL}), $1-r^2$, becomes negative and thus Eq. (\\ref{Feff-SCL}) does not have the stationary point.\nThis is because the logarithmic term is doninant when $\\phi_\\sigma$ is small and then $\\phi_\\sigma^2$ term becomes dominant as $\\phi_\\sigma$ gets larger.\nTherefore the condition that the coefficient of $\\phi_\\sigma^2$ must be positive is needed for Eq. (\\ref{Feff-SCL}) to have the stationary point.\nThis fact is consistent with the requirement of the reflection positivity of lattice gauge theories with Wilson fermions\\cite{Menotti1987}.\n\nIn the chiral limit ($r=m_0=0$), the effective potential is a function of only $\\sigma$, reflecting the chiral symmetry of the action.\nThis is understood as follows: in the chiral limit, the action is invariant under the chiral transformation $\\psi\\rightarrow e^{i\\theta\\gamma_5}\\psi$.\nThis transformation doesn't depend on the value of $\\theta$, and thus the effective potential also doesn't depend on it.\nNote that this effective potential corresponds to that of the staggered fermion (SF) model for graphene\\cite{Araki2010,Araki2011} except for an additional factor 4 by setting $r=0$ and changing from (3+1)D to (2+1)D:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=4\\mathcal{F}^{\\mathrm{SF}}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi).\n\\end{aligned}\n\\end{equation}\nThis result is reasonable, because the two cases describes the same system where the $2^3=8$ fermion doublers appear.\n\n\\subsection{Effective Potential Up to $\\bm{\\mathcal{O}(\\beta)}$}\nLet us evaluate the $\\mathcal{O}(\\beta)$ contribution to the effective potential.\nWe write the fourth term in the effective action (\\ref{Eff-Act}) as $\\Delta S_1+\\Delta S_2(\\equiv \\Delta S)$.\nThen we should choose such that $(\\kappa, A, B)=(\\beta\/2, V^+_{n,j}P^+_0, -V^-_{n+0,j}P^-_0)$ in Eq. (\\ref{EHS-Q}) for $\\Delta S_1$:\n\\begin{equation}\n\\begin{aligned}\ne^{-\\Delta S_1}\n\\propto&\\exp\\left\\{-\\frac{\\beta}{2}\\sum_{n,j}\\left[S_{\\alpha\\beta}S'_{\\alpha\\beta}-A_{\\alpha\\beta}S_{\\beta\\alpha}-B^T_{\\alpha\\beta}S'_{\\beta\\alpha}\\right]\\right\\}, \\label{Delta S_1}\n\\end{aligned}\n\\end{equation}\nwith the saddle point values $S_{\\alpha\\beta}=\\left\\langle B^T\\right\\rangle_{\\beta\\alpha}=-\\left\\langle V^-_{n+0,j}P^-_0\\right\\rangle_{\\alpha\\beta}$ and $S'_{\\alpha\\beta}=\\left\\langle A\\right\\rangle_{\\beta\\alpha}=\\left\\langle V^+_{n,j}P^+_0\\right\\rangle_{\\beta\\alpha}$.\nSimilarly, setting $(\\kappa, A, B)=(\\beta\/2, V^-_{n,j}P^+_0, -V^+_{n+0,j}P^-_0)$ in Eq. (\\ref{EHS-Q}) for $\\Delta S_2$, we obtain\n\\begin{equation}\n\\begin{aligned}\ne^{-\\Delta S_2}\n\\propto&\\exp\\left\\{-\\frac{\\beta}{2}\\sum_{n,j}\\left[T_{\\alpha\\beta}T'_{\\alpha\\beta}-A_{\\alpha\\beta}T_{\\beta\\alpha}-B^T_{\\alpha\\beta}T'_{\\beta\\alpha}\\right]\\right\\}, \\label{Delta S_2}\n\\end{aligned}\n\\end{equation}\nwith the saddle point values $T_{\\alpha\\beta}=\\left\\langle B^T\\right\\rangle_{\\beta\\alpha}=-\\left\\langle V^+_{n+0,j}P^-_0\\right\\rangle_{\\alpha\\beta}$ and $T'_{\\alpha\\beta}=\\left\\langle A\\right\\rangle_{\\beta\\alpha}=\\left\\langle V^-_{n,j}P^+_0\\right\\rangle_{\\beta\\alpha}$.\n\nNext we decompose $\\left\\langle V^+_{n,j}\\right\\rangle$ and $\\left\\langle V^-_{n,j}\\right\\rangle$ into spinor components as follows:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\langle V_{n,j}^+ \\rangle & \\equiv v_s^+ + i \\gamma_5 v_p^+ + \\sum_\\mu \\gamma_\\mu v_{v \\mu}^+ + \\sum_\\mu i \\gamma_5 \\gamma_\\mu v_{a \\mu}^+,\\\\\n\\langle V_{n,j}^- \\rangle & \\equiv v_s^- + i \\gamma_5 v_p^- + \\sum_\\mu \\gamma_\\mu v_{v \\mu}^- + \\sum_\\mu i \\gamma_5 \\gamma_\\mu v_{a \\mu}^-,\n\\end{aligned}\n\\right. \\label{V^+-V^-}\n\\end{equation}\nwhere the first, second, third and fourth terms are the components of scalar, pseudoscalar, vector and pseudovector (axial vector) mode, respectively.\nThe terms $\\left\\langle V^+_{n,j}\\right\\rangle$ and $\\left\\langle V^-_{n,j}\\right\\rangle$ are equivalent to the propagator from a point to another point.\nOnly the scalar and vector modes appear when parity is not broken, and the pseudoscalar and pseudovector modes may also appear when parity is broken.\nTherefore these four modes should be considered in Eq. (\\ref{V^+-V^-}).\n\nAfter the calculation in the appexdix, we obtain the $\\mathcal{O}(\\beta)$ contribution to the action as\n\\begin{widetext}\n\\begin{equation}\n\\begin{aligned}\n\\Delta S=&\\beta\\sum_{n,j}\\left[(1-r^2) v_s^- v_s^+ +(1+r^2) v_p^- v_p^+ +(1-r^2) v_{v0}^- v_{v0}^+ - (1+r^2) \\sum_l v_{vl}^- v_{vl}^+ -(1+r^2) v_{a0}^- v_{a0}^+ + (1-r^2) \\sum_l v_{al}^- v_{al}^+ \\right]\\\\\n&+\\sum_{n,j} \\left[ \\bar{\\psi}_n \\mathcal{A}_- \\psi_{n+\\hat{j}} + \\bar{\\psi}_{n+\\hat{j}} \\mathcal{A}_+ \\psi_n \\right], \\label{Delta S}\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\langle \\mathcal{A}_- \\rangle &=\\frac {\\beta}{4}\\left[\n-(1-r^2) v_s^- + (1+r^2) i \\gamma_5 v_p^- - (1-r^2)\\gamma_0 v_{v0}^- \n+ (1+r^2)\\sum_l \\gamma_l v_{vl}^- + (1+r^2)i \\gamma_5 \\gamma_0 v_{a0}^- - (1-r^2)\\sum_l i \\gamma_5 \\gamma_l v_{al}^-\\right]^T,\\\\\n\\langle \\mathcal{A}_+ \\rangle &=\\frac {\\beta}{4}\\left[\n-(1-r^2) v_s^+ + (1+r^2) i \\gamma_5 v_p^+ - (1-r^2)\\gamma_0 v_{v0}^+ \n+ (1+r^2)\\sum_l \\gamma_l v_{vl}^+ + (1+r^2)i \\gamma_5 \\gamma_0 v_{a0}^+ - (1-r^2)\\sum_l i \\gamma_5 \\gamma_l v_{al}^+\\right]^T.\n\\end{aligned}\n\\end{equation}\nThen doing the Fourier transform and combining Eqs. (\\ref{S_eff_SCL}) and (\\ref{Delta S}), we get the effective action up to $\\mathcal{O}(\\beta)$ with auxiliary fields:\n\\begin{equation}\n\\begin{aligned}\nS_{\\mathrm{eff}}=S_{\\mathrm{eff}}^{\\mathrm{aux}}\\left(\\phi_\\sigma,\\phi_\\pi, v^\\pm_s, v^\\pm_p, v^\\pm_{v\\mu}, v^\\pm_{a\\mu}\\right)+\\sum_k \\bar{\\psi}_k\\mathcal{M}\\left(\\bm{k};\\phi_\\sigma,\\phi_\\pi, v^\\pm_s, v^\\pm_p, v^\\pm_{v\\mu}, v^\\pm_{a\\mu}\\right)\\psi_k.\n\\end{aligned}\n\\end{equation}\nFor the explicit forms of $S_{\\mathrm{eff}}^{\\mathrm{aux}}$ and $\\mathcal{M}$, see the appendix.\n\nFinally, after eliminating the auxiliary fields $v$'s by the stationary conditions, we arrive at the effective potential up to $\\mathcal{O}(\\beta)$ given by\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}_{\\mathrm{eff}}\\left(\\phi_\\sigma,\\phi_\\pi\\right)=&(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2-2\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\log X_0\n-\\frac{\\beta}{3}(1+r^2)\\sum_j\\left[\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\frac{\\sin^2 k_j}{X_0}\\right]^2\\\\\n&-\\frac{\\beta}{3}(1+r^2)\\left[\\frac{1+r^2}{2}\\phi_\\pi\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\frac{\\sum_j\\cos k_j}{X_0}\\right]^2\n-\\frac{\\beta}{3}(1-r^2)\\left[\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\frac{\\tilde{m}(\\bm{k})+r}{X_0}\\sum_j\\cos k_j\\right]^2+\\mathcal{O}(\\beta^2), \\label{F_eff-final}\n\\end{aligned}\n\\end{equation}\nwhere we have defined $X_0= {\\sum}_j\\sin^2k_j+\\left[\\tilde{m}(\\bm{k})+r\\right]^2+\\frac{(1+r^2)^2}{4}\\phi_\\pi^2$.\n\\end{widetext}\n\\section{Numerical Results}\nAt first, we found that the value of $\\phi_\\pi$ is zero at the stationary point for any set of $(r,m_0)$.\nHence in the following, we set $\\phi_\\sigma=-\\sigma$ and $\\phi_\\pi=0$ in Eq. (\\ref{Feff-SCL}) to calculate the value of the chiral condensate $\\sigma$.\nThe term $i\\bar{\\psi}\\gamma_5\\psi$ is odd under both time-reversal and inversion.\nTherefore, this means that the phase with spontaneously broken time-reversal and inversion symmetries does not arise in the strong coupling (electron correlation) limit.\nA mean-field study of Wilson fermions with the short-range interaction from the weak coupling\\cite{Sekine2012} and a lattice strong coupling expansion study of the Kane-Mele model on a honeycomb lattice\\cite{Araki2013} suggest the existence of this phase.\nSuch a phase, \"Aoki phase\" (where parity and flavor symmetry are spotaneously broken) has also confirmed in the lattice QCD with Wilson fermions\\cite{Aoki1984,Aoki1986,Sharpe1998}.\nWe mention the main difference between this analysis and lattice QCD\nexcept for the gauge group as follows.\nOur effective model has only temporal (timelike) link variablies in contrast with lattice QCD.\nSpatial link variables are absent, like in the case of free fermions. \nParity-flavor symmetry is not spontaneously broken in free fermions. \nThis is one of the reasons why the parity broken phase does not appear in this analysis.\n\nThe $m_0$-dependence of the chiral condensate $\\sigma$ is shown in Fig. \\ref{fig1}(a).\nThe value of $\\sigma$ is expected to be quantitatively correct, based on the fact that the result of a strong coupling expansion study in graphene\\cite{Araki2010,Araki2011} is in good agreement with that of lattice Monte Carlo studies\\cite{Hands2008,Drut2009,Drut2009a,Armour2010}.\nAs mentioned above, in the noninteracting limit (i.e. at $\\beta=\\infty$), the system with $0>m_0>-2r$ ($m_0>0$) is identified as a topological (normal) insulator.\nThe chiral condensate is equivalent to a correction to the bare mass.\nHence it is natural to define the effective mass in Eq. (\\ref{m_eff}):\n\\begin{equation}\n\\begin{aligned}\nm_{\\rm{eff}}=m_0+(1-r^2)\\sigma\/2.\n\\end{aligned}\n\\end{equation}\nThe phase diagram with $r=0.5$ in the strong coupling limit calculated by the $Z_2$ invariant (Eq. (\\ref{Z2invariant})) is shown in Fig \\ref{fig1}(b).\nIn the strong coupling limit, the system with $0>m_{\\rm eff}>-2r$ ($m_{\\rm eff}>0$) is identified as a topological (normal) insulator.\nFrom this phase diagram, we see that the effect of the long-range Coulomb interaction is to shift the region of the topological insulator phase.\nThis result doesn't contradict that of a mean-field analysis from the weak coupling\\cite{Sekine2012}.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\columnwidth,clip]{fig1.eps}\n\\caption{(Color online) (a) $m_0$-dependence of the chiral condensate $\\sigma$ in the strong coupling limit ($\\beta=0$).\n(b) Phase diagram with $r=0.5$ in the strong coupling limit. The phase boundaries are determined by the condition $m_{\\rm eff}=0$ or $m_{\\rm eff}=-2r$.}\\label{fig1}\n\\end{center}\n\\end{figure}\n\nThe $\\beta$-dependence of the chiral condensate $\\sigma$ is shown in Fig. \\ref{fig2}.\nWe see that $\\sigma$ is a monotonically decreasing function of the coupling strength $\\beta$.\nThis behavior is consistent with a mean-field analysis from the weak coupling\\cite{Sekine2012}.\nOur result shows that the mass gap remains finite, in contrast to the mean-field analysis in which the mass gap becomes infinity in the strong coupling limit.\nWe see also that as $r$ becomes smaller, the rate of decrease of $\\sigma$ becomes notable.\nNamely, as the original mass of doublers becomes smaller, the energy gap of the system becomes smaller, as is understood intuitively.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.83\\columnwidth,clip]{fig2.eps}\n\\caption{(Color online) $\\beta$-dependence of the chiral condensate $\\sigma$ at $m_0=-r$.}\\label{fig2}\n\\end{center}\n\\end{figure}\nFrom Fig \\ref{fig2}, it is concluded that the gapped phases (normal or topological insulator phases) are stable in the strong coupling region.\nThis contrasts with the result of the strong coupling expansion in graphene\\cite{Araki2010,Araki2011}.\nIn graphene, the rate of decrease of $\\sigma$ from $\\beta=0$ to $\\beta=0.5$ is about 60\\%\\cite{Araki2011}, whereas that of our model is about 3\\% at $r=0.2$.\nNamely, in our model, the topological insulator phase survives in the strong coupling limit, although graphene undergoes the semimetal-insulator transition in the strong coupling region.\n\n\n\\section{Discussion and Summary}\nSo far we have obtained the value of the effective mass up to of the order of the coupling strength $\\beta$.\nWe can connect the phase boundary in the noninteracting limit and that in the strong coupling region.\nA possible phase diagram of the Wilson fermions interacting via the long-range Coulomb interaction is shown in Fig. \\ref{fig3}.\nA similar behavior of the phase boundary between the topological insulator phase and the normal insulator phase have been obtained in a mean-field analysis of the Wilson fermions with the short-range interaction\\cite{Sekine2012}.\nOne might wonder why the topological insulator phase survives at infinite coupling.\nIf the interaction is short-range, i.e., Hubbard-like, the antiferromagnetic phase is considered to be dominant.\nHowever, in the present case, the interaction is pure $1\/r$ Coulomb interaction.\nThis difference may affect the phase structure.\nA lattice strong coupling expansion study of the Kane-Mele model on a honeycomb lattice shows a similar result that when spin-orbit coupling is sufficiently strong, the topological insulator phase survives in the strong coupling limit.\n\nTo summarize, we have studied the strong electron correlation effect in a 3D topological insulator which effective Hamiltonian can be described by the Wilson fermions.\nBased on the U(1) lattice gauge theory, we have performed the strong coupling expansion.\nIt was found that the effect of long-range Coulomb interaction corresponds to the renormalization of the bare mass.\nThe values of the chiral condensate, which is regarded as a correction to the bare mass in the strong coupling limit, are expected to be correct quantitatively.\nThe behavior of the chiral condensate in our model is similar to that of the lattice QCD with Wilson fermions.\nThe phase where time-reversal and inversion symmetries are spontaneously broken (\"Aoki phase\") was not found in the strong coupling region, in contrast to the case of lattice QCD.\nIt was also found that the gapped phase is stable in the strong coupling region.\nThis suggests that the topological insulator phase survives in the strong coupling limit.\nIn this study, the bulk property of a 3D topological insulator was examined.\nIt will be interesting to examine the strong correlation effect in the surface Dirac fermions.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.90\\columnwidth,clip]{fig3.eps}\n\\caption{(Color online) A possible phase diagram of the Wilson fermions interacting via the long-range Coulomb interaction ($r=0.5$).}\\label{fig3}\n\\end{center}\n\\end{figure}\n\n\\begin{acknowledgments}\nT. Z. N. is thankful to H. Iida, D. Satow, and S. Gongyo for fruitful discussions.\nY. A. is thankful to T. Kimura for valuable discussions.\nThis work was supported by the Grants-in-Aid for Scientific Research (No. 24740211 and No. 10J03314) from the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT).\nA. S. is supported by the global COE program \"Weaving Science Web beyond Particle-Matter Hierarchy\" from MEXT.\nT. Z. N. is supported by the Fellowship for Young Scientists (No. 22-3314) from Japan Society for the Promotion of Science (JSPS) and the global COE program \"The Next Generation of Physics, Spun from Universality and Emergence\" from MEXT.\nY. A. is supported by JSPS Postdoctoral Fellowship for Research Abroad (No.25.56).\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{The Appelquist-Terning One Family Walking TC Model (Model A) }\n\nWe take this model as a typical example of the improved TC models\nwithout assisted by topcolor. To reduce the value of the oblique correction \nparameter $S$, this model is designed such that the techniquark ($Q$) sector \nrespects the custodial $SU(2)$ symmetry, while the technilepton ($L$) sector \nis custodial $SU(2)$ violating, and the vacuum expectation values (VEV's)\nof $\\bar{Q}Q$ and $\\bar{L}L$ are further designed to be $~F_Q\\gg F_L$ \\cite{AT}. \nThe color-singlet would-be Goldstone bosons eaten by $W$ and $Z$\nare mainly composed of techiquarks. There are 36 PGBs in this model \\cite{AT},\nin which the color-singlet PGBs are mainly composed of technileptons which \nare irrelevant to the $s$-channel $t\\bar{t}$ production. At the hadron \ncolliders, the color-octet PGBs $\\Pi ^{0a}~(a=1,...,8)$ composed of \ntechniquarks can contribute to the $s$-channel $t\\bar{t}$ productions via the\ntechniquark and top quark triangle loops [cf. Fig. 1]. This is the main \ndifference between the present case and the $\\gamma\\gamma\\to t\\bar{t}$ case in \nRef.\\cite{gamgamtt}. The decay constant of the color-octet PGBs is\n$F_\\Pi=123$ GeV \\cite{AT}. The masses of $\\Pi^{0a}$ are model-dependent.\nFollowing Ref.\\cite{AT}, we take $M_{\\Pi^{0a}}$ in the range \n$400~{\\rm GeV}\\alt M_{\\Pi^{0a}} \\alt 500$ GeV.\n\nSince the techniquark $Q$ is very heavy, the triangle loop in Fig. 1(a) can be\nsimply evaluated by the Adler-Bell-Jackiw anomaly \\cite{anomaly}, and the\ngeneral form of which is \\cite{Lubicz,DE}\n\\begin{eqnarray} \n\\frac{S_{\\Pi^aB_1B_2}}{4\\pi^2 F_{\\pi}} \\epsilon_{\\mu\\nu\\lambda\\rho}\nk_1^{\\lambda}k_2^{\\rho}\\,,\n\\label{ABJ}\n\\end{eqnarray}\nwhere $B_1$ and $B_2$ denote the two gauge fields which, in our case, are the\ntwo gluons $g_b$ and $g_c$ with the color indices $b$ and $c$, respectively.\nThe factor $S_{\\Pi^a g_bg_c}$ can be easily obtained from the formulae\nin Ref.\\cite{Lubicz,DE,hadtt}, and it is\\footnote{Here, and in (\\ref{t-loop}),\n(\\ref{J-Pia}) and (\\ref{S(B)}), we have corrected some typos in \nRef.\\cite{hadtt}.}\n\\begin{eqnarray} \nS_{\\Pi^{0a}g_bg_c} =\\frac{1}{\\sqrt{2}}g_s^2 N_{TC}d_{abc}\\,,\n\\label{S_Piagg}\n\\end{eqnarray}\nwhere $d_{abc}$ is the symmetric tensor in the color $SU(3)_c$ group.\n\nThe evaluation of the triangle loop in Fig. 1(b) needs more consideration. The \ntop quark is not heavy enough for the validity of simply using the \nAdler-Bell-Jackiw anomaly. Correction of the $m_t$ effect has to be\ntaken into account. This has been calculated in Ref.\\cite{J}, and the\nresult is\n\\begin{eqnarray}\n-i\\frac{C_t g_s^2}{8\\pi^2 F_{\\pi}}\\frac{d_{abc}}{2} J(R_{\\hat{s}})\n\\epsilon_{\\mu \\nu \\lambda \\rho} k_1^{\\lambda} k_2^{\\rho}\\,, \n\\label{t-loop}\n\\end{eqnarray}\nwhere $C_t$ is a model-dependent coupling constant which is expected to be\nof order 1 \\cite{TC2-2,Lubicz,RS}, $\\hat{s}$ is the center-of mass energy of \nthe $t\\bar t$ system and $J(R_{\\hat{s}})$ is defined as \\cite{J}\n\\begin{eqnarray}\nJ(R_{\\hat{s}})&\\equiv& -\\frac{1}{R^2_{\\hat{s}}}\n\\int_0^1 \\frac{dx}{x(1-x)}\\nonumber\\\\\n&&\\times \\ln[1 - R^2_{\\hat{s}} x(1-x) ]\\,, \n\\label{J-Pia}\n\\end{eqnarray}\nwith $R_{\\hat{s}} \\equiv \\sqrt{\\hat s}\/m_t$.\n\nCombining (\\ref{ABJ}), (\\ref{S_Piagg}) and (\\ref{t-loop}), we obtain the \nproduction amplitude for Fig. 1\n\\begin{eqnarray} \n\\displaystyle\n {\\cal M}^{(A)}_{\\Pi^{0a}}&=& \\frac{C_tm_tg_s^2[N_{TC}+\\frac{1}{2\\sqrt{2}}\n C_tJ(R_{\\hat s})]d_{abc}}{4\\sqrt{2}\\pi^2F^2_{\\Pi}[\\hat{s}-M^2_{\\Pi^{0a}}\n+iM_{\\Pi{0a}}\\Gamma_{\\Pi^{0a}}]}\n\\nonumber\\\\\n&&\\times(\\bar{t}\\gamma_5\\frac{\\lambda_a}{2}t)\\epsilon_{\\mu\\nu\\lambda\\rho}\n\\varepsilon_1^{\\mu}\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\n\\end{eqnarray}\nwhere $\\Gamma_{\\Pi^{0a}}$ is the total width of $\\Pi^{0a}$ which\nhas been given in Ref.\\cite{hadtt}. The total production amplitude is then\n\\begin{eqnarray} \n{\\cal M}^{(A)}={\\cal M}^{SM}_{tree}+ {\\cal M}^{(A)}_{\\Pi^{0a}}\\,.\n\\label{ATamplitude}\n\\end{eqnarray}\n\n\\subsection{The Original Topcolor-Assisted Technicolor Model (Model B)}\n\nSince the original topcolor-assisted technicolor model (Model B) was\nproposed \\cite{TC2-1}, there have been refinements of the model to make\nit more realistic \\cite{TC2-1'}. In the present study, we are only\ninterested in the characteristic PGB effects of this kind of model in \n$t\\bar t$ productions which do not concern the subtleties of the\nrefinements, so that we simply take the original Model B as a typical\nexample of this kind of model in our calculation.\nIn this model, the TC sector is taken to be the standard extended\ntechnicolor model in which there are 60 TC PGBs with the decay\nconstant $F_\\Pi\\approx 120$ GeV\\footnote{This is slightly smaller than\nthe usual value $F_\\Pi=123$ GeV in the extended technicolor model since, in \nthe topcolor-assisted technicolor model, the total vacuum expectation value is \nalso contributed by the topcolor sector.}, and both the \nthe color-octet PGB $\\Pi^{0a}$ and the color-singlet PGB $\\Pi^0$ contribute to \nthe $t\\bar t$ production. As in Model A, we take \n$400\\alt M_{\\Pi^{0a}}\\alt 500$ GeV. The mass of $\\Pi^0$\nis lighter, say around $150$ GeV \\cite{ETC}. The coupling of $\\Pi^0$ to\ngluons via the techniquark and top quark triangle loops is\ndecscribed by \\cite{Lubicz,hadtt}\n\\begin{eqnarray} \n\\displaystyle\nS^{(B)}_{\\Pi^0 g_b g_c}&=&\\frac{1}{2\\sqrt{3}}g^2_s\\delta_{bc}N_{TC}\\nonumber\\\\\n&&+\\frac{1}{\\sqrt{2}}g^2_sJ(R_{\\hat s})\\delta_{bc}\\,,\n\\label{S(B)}\n\\end{eqnarray}\nwhere the first term is from the techniquark loop and the second term\nis from the top quark loop.\n\nThere is a topcolor sector in this model responsible for causing the main part \nof the top quark mass. In the topcolor sector, there is a PGB called top-pion \n$\\Pi_t$ with decay constant $F_{\\Pi_t}=50$ GeV. The mass $M_{\\Pi_t}$\nwas first estimated as around $200$ GeV in the original paper \\cite{TC2-1}. \nHowever, recent phenomenological analyses up to one-loop calculations show \nthat the LEP\/SLD precision data of $R_b$ give severe constraint on the value \nof $M_{\\Pi_t}$ due to the large negative contribution to $R_b$ from the \ncorrections related to $\\Pi_t$, and it requires $M_{\\Pi_t}$ to be of the order \nof 1 TeV \\cite{BK,LT}. However, as is pointed out in Ref.\\cite{BK}, such a \nconstraint can only be regarded as a rough estimate since higher order\ncorrections related to $\\Pi_t$ may be substantial due to the large \n$\\Pi_t-t-\\bar b$ coupling. Furthermore, the extended technicolor gauge boson \ncontribution to $R_b$, which has been shown to be positive \\cite{R_b}, is not \ntaken into account in the analyses in Refs.\\cite{BK,LT}, and the actual \nconstraint on $M_{\\Pi_t}$ may be relaxed when such a positive contribution is \ntaken into account.\nTherefore, to see the $M_{\\Pi_t}$-dependence of the cross \nsection, we take $M_{\\Pi_t}$ to vary in the range $~500~{\\rm GeV}\\alt\nM_{\\Pi_t}\\alt 1~{\\rm TeV}~$ in our calculation. \n\nThe top quark mass $m_t$ comes from two sources in this model. The TC\nsector gives rise to a small portion of it, and we call this portion \n$m^\\prime_t$. The value of $m^\\prime_t$ is model-dependent.\nLow energy data, especially the $b\\to s\\gamma$ experiment, give constraints on \n$m^\\prime_t$, and the reasonable range of $m^\\prime_t$ is about \n$5~{\\rm GeV}\\alt m^\\prime_t\\alt 20~{\\rm GeV}$ \\cite{TC2-1,Balaji}. The rest \npart of $m_t$, say $m_t-m^\\prime_t$ comes from the topcolor sector. Thus the \ncouplings of the technipions to the top quark can be written as \n\\cite{EL}\\cite{Lubicz}\n\\begin{eqnarray}\n\\displaystyle\n\\frac{C_t m_t^\\prime}{\\sqrt{2} F_\\Pi} \\Pi^0 (\\bar{q} \\gamma^5 q) \n\\end{eqnarray}\n\\begin{eqnarray} \n\\frac{C_t m_t^{\\prime}}{F_\\Pi} \\Pi^{0a} (\\bar{q} \\gamma^5 \\frac{\\lambda^a}{2} \nq)\n\\end{eqnarray}\nwhere $\\lambda^a$ is the Gell-Mann matrix of the color group. \nThe interactions of the top-pions with the top quark is \\cite{TC2-1,TC2-1'}\n\\begin{eqnarray}\n\\displaystyle\n\\frac{m_t - m_t^{\\prime}}{\\sqrt{2} F_{\\Pi_t}} [\\bar{t} \\gamma_5 t \\Pi_t^0 +\n\\frac{i}{\\sqrt{2}} \\bar{t} (1 - \\gamma_5) b \\Pi_t^+ \\nonumber\\\\\n+ \\frac{i}{\\sqrt{2}}\\bar{b} (1 + \\gamma_5) t \\Pi_t^-]\\,. \n\\end{eqnarray}\n\nWith these couplings, the PGB contributed production amplitudes in this\nmodel described in Fig. 1 are\n\\begin{eqnarray} \n\\displaystyle\n {\\cal M}^{(B)}_{\\Pi^{0a}}&=& \\frac{C_tm_t^\\prime g_s^2[N_{TC}+\\frac{1}\n{2\\sqrt{2}}C_t J(R_{\\hat s})]\nd_{abc}}{4\\sqrt{2}\\pi^2F^2_{\\Pi}[\\hat{s}-M^2_{\\Pi^{0a}}\n+iM_{\\Pi{0a}}\\Gamma_{\\Pi^{0a}}]}\n\\nonumber\\\\\n&&\\times(\\bar{t}\\gamma_5\\frac{\\lambda_a}{2}t)\\epsilon_{\\mu\\nu\\lambda\\rho}\n\\varepsilon_1^{\\mu}\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\\label{MPi0a}\n\\end{eqnarray}\n\\begin{eqnarray} \n\\displaystyle\n{\\cal M}^{(B)}_{\\Pi^0}&=&\\frac{C_tm_t^{\\prime}g_s^2[N_{TC}+\\frac{\\sqrt{6}}{2}\nC_t J(R_{\\hat s})]\\delta_{bc}}\n{8\\sqrt{6}\\pi^2F^2_{\\Pi}[\\hat{s}-M^2_{\\Pi^0}+iM_{\\Pi^0}\\Gamma_{\\Pi^0}]}\n\\nonumber\\\\\n&&\\times (\\bar{t}\\gamma_5t)\\epsilon_{\\mu\\nu\\lambda\\rho}\\varepsilon_1^\n{\\mu}\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\\label{MPi0}\n\\end{eqnarray}\n\\begin{eqnarray} \n\\displaystyle\n{\\cal M}^{(B)}_{\\Pi_t}&=&\\frac{(m_t-m_t^{\\prime})g_s^2J(R_{\\hat s})\\delta_{bc}}\n{8\\sqrt{6}\\pi^2F^2_{\\Pi_t}[\\hat{s}-M^2_{\\Pi_t^0}+iM_{\\Pi_t^0}\\Gamma_{\\Pi_t^0}]}\n\\nonumber\\\\\n&&\\times (\\bar{t}\\gamma_5t)\\epsilon_{\\mu\\nu\\lambda\\rho}\\varepsilon_1^{\\mu}\n\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\\label{M^B_Pi_t}\n\\end{eqnarray}\nwhere $\\Gamma_{\\Pi^0}$ and $\\Gamma_{\\Pi_t}$ are, respectively, the total \nwidths of $\\Pi^0$ and $\\Pi_t$ given in Ref.\\cite{hadtt}.\nThe total production amplitude in this model is then\n\\begin{eqnarray} \n{\\cal M}^{(B)}={\\cal M}^{SM}_{tree}+{\\cal M}^{(B)}_{\\Pi^{0a}}\n+{\\cal M}^{(B)}_{\\Pi^0}+{\\cal M}^{(B)}_{\\Pi_t}\\,.\n\\end{eqnarray}\nCompared with ${\\cal M}^{(A)}$, the amplitude ${\\cal M}^{(B)}$ contains two \nextra terms ${\\cal M}^{(B)}_{\\Pi^0}$ and ${\\cal M}^{(B)}_{\\Pi_t}$. As we shall \nsee later that this makes Model A and Model B experimentally distinguishable at\nthe LHC.\n\n\\subsection{The Topcolor-Assisted Multiscale Technicolor Model (Model C)}\n\nThe topcolor-assisted multiscale technicolor model (Model C) \n\\cite{TC2-2,EL,hadtt} is different from Model B by its extended technicolor \nsector which is taken to be the multiscale technicolor model \\cite{MTC}.\nIn this model, the value of the decay constant $F_\\Pi$ is $F_\\Pi=40$ GeV \nrather than $120$ GeV, and the technipion $\\Pi^0$ is almost \ncomposed of pure techniquarks (ideal mixing) \\cite{TC2-2} which leads to\n\\begin{eqnarray} \n\\displaystyle\nS^{(C)}_{\\Pi^0 g_b g_c}=\\frac{1}{\\sqrt{3}}N_{TC}\\delta_{bc}\\,.\n\\label{S(C)}\n\\end{eqnarray}\nThen the production amplitudes in Model C is\n\\begin{eqnarray} \n{\\cal M}^{(C)}={\\cal M}^{SM}_{tree}+{\\cal M}^{(C)}_{\\Pi^{0a}}\n+{\\cal M}^{(C)}_{\\Pi^0}+{\\cal M}^{(C)}_{\\Pi_t}\\,,\n\\end{eqnarray}\nwith\n\\begin{itemize}\n\\item ${\\cal M}^{(C)}_{\\Pi_t}={\\cal M}^{(B)}_{\\Pi_t}$;\n\\item the formula for ${\\cal M}^{(C)}_{\\Pi^0}$ differs from that for\n${\\cal M}^{(B)}_{\\Pi^0}$ by a factor of $2$;\n\\item the value of $F_\\Pi$ in ${\\cal M}^{(C)}_{\\Pi^{0a}}$ and \n${\\cal M}^{(C)}_{\\Pi^0}$ is $F_\\Pi=40$ GeV rather than $123$ GeV.\n\\end{itemize}\nThe smallness of the value of $F_\\Pi$ and the ideal mixing of $\\Pi^0$\nin model C enhance the technicolor PGB contributions in $t\\bar t$ production\nrelative to the top-pion contribution. As we shall see later that this\nmakes Model C experimentally distinguishable from Model B and Model A.\n\n\\null\\vspace{0.4cm}\n\\begin{center}\n{\\bf III. CROSS SECTIONS AND NUMERICAL RESULTS}\n\\end{center}\n\nWe take the method in Ref.\\cite{helicity} to do the numerical calculation. \nOnce the elementary cross section $\\hat \\sigma$ is calculated at the \nparton-level, the total cross section $\\sigma$ can be obtained by folding \n$\\hat \\sigma$ with the parton distribution functions $f^{p(\\bar p)}_i(x_i,Q)$ \n\\cite{EHLQ}\n\\begin{eqnarray} \n\\sigma(pp(\\bar{p})\\to t\\bar{t})&=&\\sum\\limits_{ij}\\int dx_idx_jf_i^{(p)}(x_i,Q)\nf_j^{(p(\\bar{p}))}(x_j,Q)\\nonumber\\\\\n&&\\times\\hat{\\sigma}(ij\\to t\\bar{t})\n\\end{eqnarray}\nwhere $i$ and $j$ stand for the partons $g$, $q$ and $\\bar{q}$; $x_i$ is the \nfraction of longitudinal momentum of the proton (antiproton) carried by the \n$i$th parton; $Q^2\\approx \\hat s$; and $f_i^{(p(\\bar{p}))}$ is the parton \ndistribution function in the proton (antiproton). In this paper, we take the \nMRS setA$^\\prime$ parton distribution for $f_i^{p(\\bar{p})}$ \\cite{MRS}. To take \naccount of the QCD corrections, we shall multiply the obtained cross section \nby a factor of 1.6 \\cite{LSN} as what was done in Ref.\\cite{hadtt}. The\nvalues of the tree-level SM cross section $\\sigma_0$ at the $\\sqrt{s}=2$ TeV\nTevatron and the $\\sqrt{s}=14$ TeV LHC are, respectively\n\\begin{eqnarray} \n&&{\\rm Tevatron}:~~~~~~~~~~~~~~\\sigma_0=8.02~{\\rm pb}\\,,\\nonumber\\\\\n&&{\\rm LHC}:~~~~~~~~~~~~~~~~~~~~\\sigma_0=826~{\\rm pb}\\,.\n\\label{sigma_0}\n\\end{eqnarray}\nIn the numerical calculations, we take $\\alpha_s(\\sqrt{\\hat s})$ the same as\nthat in the MRS set A$^\\prime$ parton distributions, $m_t=174$ GeV,\nand we simply take the technicolor model parameter $C_t=1$. In the\nfollowing analysis, we consider the one-year-run integrated luminosities\nfor the Tevatron Run II and the LHC\n\\begin{eqnarray} \n{\\rm Tevatron}:~~~~~~~~\\int {\\cal L}dt&=&2~{\\rm fb}^{-1}\\,,\\nonumber\\\\\n{\\rm LHC}:~~~~~~~~~~~~~\\int {\\cal L}dt&=&100~{\\rm fb}^{-1}\\,,\n\\label{luminosity}\n\\end{eqnarray}\nand assume a $10\\%$ detecting efficiency. \n\nThe obtained total production\ncross sections can be compared with the recently measured $t\\bar t$ \nproduction cross sections by the CDF Collaboration and the D0 Collaboration \n\\cite{Heinson}\n\\begin{eqnarray} \n{\\rm CDF}:~~~~~~~~~~\\sigma(p\\bar p\\to t\\bar t)&=&10.1\\pm 1.9^{+4.1}_{-3.1}\n~{\\rm pb}\\,,\\nonumber\\\\\n{\\rm D0}:~~~~~~~~~~~\\sigma(p\\bar p\\to t\\bar t)&=&7.1\\pm 2.8\\pm 1.5~{\\rm pb}\\,.\n\\label{sigmatt}\n\\end{eqnarray}\nThe data in (\\ref{sigmatt}) can serve as a constraint on the parameters in the \nTC models.\n\n\\subsection*{A. Results of Model A}\n\nIn Table I, we list the results of the cross sections at the Tevatron Run II \nand the LHC in Model A with $M_{\\Pi^{0a}}$ varying from 400 GeV to 500 GeV. We \nsee from Table I that the values of $\\sigma^{(A)}_{t\\bar t}$ for the Tevatron \nare consistent with the recent CDF and D0 measurements (\\ref{sigmatt}). The \nrelative technicolor corrections to the SM tree-level cross section $\\sigma_0$\nare $\\Delta\\sigma^{(A)}\/\\sigma_0\\approx (10-36)\\%$ for the Tevatron\n\n\\null\\vspace{0.4cm}\n\\noindent\n{\\small Table I. Cross sections in Model A at the $\\sqrt{s}=2$ TeV Tevatron\nand the $\\sqrt{s}=14$ TeV LHC with $M_{\\Pi^{0a}}$ varying from 400 GeV to 500 \nGeV. $\\sigma_0$ denotes the SM tree-level cross section,\n$\\Delta\\sigma^{(A)}$ denotes the correction to $\\sigma_0$,\nand $\\sigma^{(A)}_{t\\bar{t}}=\\sigma_0+\\Delta\\sigma^{(A)}$ is the total cross \nsection. All masses are in GeV.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 0.5pt\n\\begin{tabular}{c cc cc}\n\\hline\\hline\n& $~~~~~~~~~~$Tevatron& &$~~~~~~~$LHC& \\\\\n\\hline\n$M_{\\Pi^{0a}}$&$\\Delta\\sigma^{(A)}(pb)$&$\\sigma^{(A)}_{t\\bar{t}}(pb)$ &\n$~~~~\\Delta\\sigma^{(A)}(nb)$ & $~~\\sigma^{(A)}_{t\\bar{t}}(nb)~~$\\\\\n\\hline\n400 & 2.92 & 10.94 ~~& 1.36 & 2.19 \\\\\n450 & 1.54 & 9.56 ~~& 1.04 & 1.87 \\\\\n500 & 0.84 & 8.86 ~~& 0.81 & 1.63 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{0.4cm}\n\n\\noindent\nand $\\Delta\\sigma^{(A)}\/\\sigma_0\\approx (98-165)\\%$ for the LHC, which are \nquite large due to the $\\Pi^{0a}$ resonance effects. The relative corrections \nare much larger than those in the $\\gamma\\gamma\\to t\\bar t$ process given in \nRef.\\cite{gamgamtt} because of the existence of the $\\Pi^{0a}$ contribution\nat the hadron colliders.\nWith the integrated luminosities in (\\ref{luminosity}) and assuming a $10\\%$ \ndetecting efficiency, we see from Table I that Model A predicts around\n2000 $t\\bar t$ events at the Tevatron and around $2\\times 10^7~ t\\bar t$ events\nat the LHC. The statistical uncertainty at the $95\\%$ C.L. in the case of the \nTevatron is then around $4\\%$ which is about the same level as the expected \nsystematic error of the $t\\bar t$ cross section measurement ($\\sim 5\\%$ \n\\cite{Liss}), and the statistical uncertainty in the case of the LHC is \naround $4\\times 10^{-4}$ which is much smaller than the expected systematic \nerror ($\\sim {\\rm few}\\%$ \\cite{Liss}). The relative corrections \n$\\Delta\\sigma^{(A)}\/\\sigma_0$ from Table I are all larger than the above\nuncertainties and thus {\\it these events are all experimentally\ndetectable at both the Tevatron and the LHC}. To illustrate the resonances, \nwe further plot the $t\\bar t$ invariant mass distributions for\n$M_{\\Pi{0a}}=400$ GeV at the Tevatron and the LHC in Fig. 2(a) and Fig. 2(b), \nrespectively. The resonance effects at $M_{\\Pi^{0a}}$ can be clearly\nseen. Comparing Fig. 2(a) with the new vector resonances (with\nthe width about $20\\%$ of the mass) shown in Ref.\\cite{HP}, we see that the \n$\\Pi^{0a}$ resonance is sharper.\n\n\\subsection*{B. Results of Model B}\n\nThe results of the cross sections in Model B at the Tevatron are listed in \nTable II. Since $M_{\\Pi^0}$ is much lower than the $t\\bar t$ threshold, there \nis almost no $\\Pi^0$ resonance effect, so that we simply take a typical value \n$M_{\\Pi^0}=150$ GeV in the calculation. To see the resonance effects of \n$\\Pi^{0a}$ and $\\Pi_t$ with various values of $M_{\\Pi^{0a}}$ and $M_{\\Pi_t}$, \nwe take their masses varying in the ranges $~400~{\\rm GeV}\\alt M_{\\Pi^{0a}}\n\\alt 500~{\\rm GeV}~$ and $~500~{\\rm GeV}\\alt M_{\\Pi_t}\\alt 1~{\\rm TeV}$, \nrespectively. For the parameter $m^\\prime_t$, we take two typical values \n$m^\\prime_t=5$ GeV (denoted by the superscript $i=1$) and $m^\\prime_t=15$ GeV \n(denoted by the superscript $i=2$), and the cross sections with these\ntwo values of $m^\\prime_t$ are denoted by $\\sigma^{(B1)}_{t\\bar t}$ \n\n\\null\\vspace{0.4cm}\n\\noindent\n{\\small Table II. Cross sections in Model B at the $\\sqrt{s}=2$ TeV Tevatron. \n$\\Delta\\sigma^{(Bi)}$ denotes the correction to the SM tree-level cross\nsection $\\sigma_0$, and $\\sigma^{(Bi)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Bi)}$ is the total cross section. The superscript $i$\ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$ GeV\n($i=2$). All masses are in GeV, and all cross sections are in pb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\\hline\\hline\n$M_{\\Pi_t^0}$ & $M_{\\Pi^{0a}}$ & $\\Delta\\sigma^{(B1)}$ &\n$\\sigma^{(B1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{(B2)}$ & $\\sigma^{(B2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.13 & 8.15 & 0.47 & 8.49 \\\\\n500 & 450 & 0.11 & 8.13 & 0.29 & 8.31 \\\\\n500 & 500 & 0.09 & 8.11 & 0.18 & 8.20 \\\\\n600 & 400 & 0.10 & 8.12 & 0.44 & 8.46 \\\\\n600 & 450 & 0.08 & 8.09 & 0.26 & 8.28 \\\\\n600 & 500 & 0.06 & 8.08 & 0.15 & 8.17 \\\\ \n700 & 400 & 0.08 & 8.10 & 0.42 & 8.44 \\\\\n700 & 450 & 0.06 & 8.08 & 0.24 & 8.26 \\\\\n700 & 500 & 0.05 & 8.06 & 0.14 & 8.15 \\\\\n800 & 400 & 0.07 & 8.09 & 0.42 & 8.43 \\\\\n800 & 450 & 0.05 & 8.07 & 0.24 & 8.25 \\\\\n800 & 500 & 0.04 & 8.06 & 0.13 & 8.15 \\\\\n900 & 400 & 0.07 & 8.08 & 0.41 & 8.43 \\\\\n900 & 450 & 0.05 & 8.06 & 0.23 & 8.25 \\\\\n900 & 500 & 0.03 & 8.05 & 0.13 & 8.14 \\\\\n1000 & 400 & 0.06 & 8.08 & 0.41 & 8.42 \\\\\n1000 & 450 & 0.04 & 8.06 & 0.23 & 8.24 \\\\\n1000 & 500 & 0.03 & 8.05 & 0.12 & 8.14 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{0.4cm}\n\\noindent\nand $\\sigma^{(B2)}_{t\\bar t}$, respectively. From the values of \n$\\sigma^{(B1)}_{t\\bar t}$ and $\\sigma^{(B2)}_{t\\bar t}$ in Table II, we\nsee that they are consistent with the recent CDF and D0 measurements \n(\\ref{sigmatt}). We know that the width of a heavy $\\Pi_t$ is \nrather large due to the largeness of $(m_t-m^\\prime_t)\/F_{\\Pi_t}$ (the\nsmallness of $F_{\\Pi_t}$), thus \nthe cross sections depend more sensitively on $M_{\\Pi^{0a}}$ than on \n$M_{\\Pi_t}$ as we see in Table II. Moreover, the $\\Pi^{0a}$ couplings are \nproportional to $m^\\prime_t$, while the $\\Pi_t$ couplings are\nproportional to $m_t-m^\\prime_t$. The former is much sensitive to\n$m^\\prime_t$ than the latter does since $m_t\\gg m^\\prime_t$. Thus the cross \nsections with $m^\\prime_t=15$ GeV are all larger than those with \n$m^\\prime_t=5$ GeV in Table II. From Table II we see that all relative \ncorrections $\\Delta\\sigma^{(B1)}\/\\sigma_0$ and $\\Delta\\sigma^{(B2)}\/\\sigma_0$ \nare at most $6\\%$ which is of the same order as the expected systematic error \n($\\sim 5\\%$). Hence {\\it Model B can hardly be detected at the Tevatron}.\n\nThe obtained cross sections in Model B at the LHC are listed in Table III.\nNow the relative corrections $|\\Delta\\sigma^{(B1)}|\/\\sigma_0$\nin Table III are around $(3\\--7)\\%$ depending on the value of $\\Pi_t$.\nSince the statistical uncertainty is of the order of $10^{-4}$ as can\nbe seen from Table III and\n\\newpage\n{\\small Table III. Cross sections in Model B at the $\\sqrt{s}=14$ TeV LHC. \n$\\Delta\\sigma^{(Bi)}$ denotes the correction to the SM tree level cross \nsections $\\sigma_0$, and $\\sigma^{(Bi)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Bi)}$ is the total cross section. The superscript $i$ \ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$ GeV\n($i=2$). All masses are in GeV, and all cross sections are in nb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$M_{\\Pi_t^0}$ & $M_{\\Pi^{0a}}$ & $\\Delta\\sigma^{(B1)}$ &\n$\\sigma^{(B1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{(B2)}$ & $\\sigma^{(B2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.06 & 0.89 & 0.23 & 1.05 \\\\\n500 & 450 & 0.06 & 0.88 & 0.19 & 1.02 \\\\\n500 & 500 & 0.05 & 0.88 & 0.15 & 0.98 \\\\\n600 & 400 & 0.05 & 0.87 & 0.21 & 1.04 \\\\\n600 & 450 & 0.05 & 0.87 & 0.18 & 1.01 \\\\\n600 & 500 & 0.04 & 0.87 & 0.14 & 0.97 \\\\ \n700 & 400 & 0.04 & 0.87 & 0.20 & 1.03 \\\\\n700 & 450 & 0.04 & 0.86 & 0.17 & 1.00 \\\\\n700 & 500 & 0.03 & 0.86 & 0.13 & 0.96 \\\\\n800 & 400 & 0.04 & 0.86 & 0.20 & 1.03 \\\\\n800 & 450 & 0.03 & 0.86 & 0.17 & 0.99 \\\\\n800 & 500 & 0.03 & 0.86 & 0.13 & 0.95 \\\\\n900 & 400 & 0.03 & 0.86 & 0.20 & 1.02 \\\\\n900 & 450 & 0.03 & 0.86 & 0.16 & 0.99 \\\\\n900 & 500 & 0.03 & 0.85 & 0.12 & 0.95 \\\\\n1000 & 400 & 0.03 & 0.86 & 0.19 & 1.02 \\\\\n1000 & 450 & 0.03 & 0.85 & 0.16 & 0.99 \\\\\n1000 & 500 & 0.02 & 0.85 & 0.12 & 0.95 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{0.4cm}\n\\noindent\neq.(\\ref{sigmatt}), {\\it the PGB effects for\n$m^\\prime_t=5$ GeV in Model B can be marginally detected at the LHC (at\nleast for $M_{\\Pi_t}\\alt 800$ GeV and $M_{\\Pi^{0a}}\\alt 450$ GeV)}.\nFor $m^\\prime_t=15$ GeV, the relative corrections\n$\\Delta\\sigma^{(B2)}\/\\sigma_0$ are in the range of $(15\\--27)\\%$\nwhich are larger than the systematic error and the statistical uncertainty.\nThus {\\it the PGB effects for $m^\\prime_t=15$ GeV in Model B can be\nclearly detected at the LHC}. Comparing the cross sections in Table I and \nTable III, we see that the relative differences between Model A and Model B \nat the LHC are $R^{(1)}_{AB}\\equiv (\\sigma^{(A)}_{t\\bar t}\n-\\sigma^{(B1)}_{t\\bar t})\/\\sigma^{(A)}_{t\\bar t}\\approx (46\\--61)\\%$ and\n$R^{(2)}_{AB}\\equiv (\\sigma^{(A)}_{t\\bar t}-\\sigma^{(B2)}_{t\\bar t})\/\n\\sigma^{(A)}_{t\\bar t}\\approx (40\\--53)\\%$. These are all much larger\nthan the systematic error and the statistical uncertainty, so that\n{\\it Model A and Model B can be clearly distinguished at the LHC}.\n\nAs an illustration, the $t\\bar t$ invariant mass distributions in Model B for\n$M_{\\Pi^{0a}}=400$ GeV and $M_{\\Pi_t}=500$ GeV at the Tevatron and\nthe LHC are shown in Fig. 3. Since the width of $\\Pi^{0a}$ in Model B depends \non $m^\\prime_t\/F_\\Pi$ rather than on $m_t\/F_\\Pi$, {\\it the resonance of \n$~\\Pi^{0a}$ in Model B is much sharper than that in Model A}. This is a clear \ndistinction between Model B and Model A. The width of $\\Pi_t$ is very wide due \nto the largeness of $(m_t-m^\\prime_t)\/F_{\\Pi_t}$ (the smallness of \n$F_{\\Pi_t}$). Because of the large width of $\\Pi_t$, no resonance peak\nof $\\Pi_t$ can be seen, and the contribution of $\\Pi_t$ is just a\nslight enhancement of the $M_{t\\bar t}$ distribution in a certain region. In \nFig. 3(b), the solid curve and the dotted curve denote the $M_{t\\bar{t}}$ \ndistribution with and without the $\\Pi_t$ contribution, respectively. From the\ndifference of these two curves, we can see the effect of the $\\Pi_t$ \ncontribution.\nWe see that both the $\\Pi^{0a}$ and the $\\Pi_t$ contributions look very\ndifferent from those of the new heavy vector resonances (with the \nwidth about $20\\%$ of the mass) shown in Ref.\\cite{HP}.\n\n\n\\subsection*{C. Results of Model C}\n\nThe obtained cross sections in Model C at the Tevatron and the LHC are\nlisted in Table IV and Table V, respectively. The cross sections\nin Table IV are consistent with the CDF and D0 data. In Model C, the decay\nconstant $F_\\Pi$ is much smaller than that in Model B, so that the\n$\\Pi^{0a}$ and $\\Pi^0$ contributions are enhanced\\footnote{In this\npaper, we have considered the effect of ideal mixing of $\\Pi^0$ in\nmodel C, while this effect is not considered in Ref.\\cite{hadtt}.}, and\nthus the cross sections in Tables IV and V are larger than those in Tables\nII and III. \n\n\\null\\vspace{0.4cm}\n{\\small Table IV. Cross sections in Model C at the $\\sqrt{s}=2$ TeV Tevatron. \n$\\Delta\\sigma^{(Ci)}$ denotes the correction to the SM tree-level cross\nsection $\\sigma_0$, and $\\sigma^{(Ci)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Ci)}$ is the total cross section. The superscript $i$\ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$\nGeV ($i=2$). All masses are in GeV, and all cross sections are in pb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$M_{\\Pi_t^0}$ & $M_{\\Pi^{0a}}$ & $\\Delta\\sigma^{(C1)}$ &\n$\\sigma^{(C1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{C(2)}$ & $\\sigma^{(C2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.50 & 8.52 & 3.46 & 11.48 \\\\\n500 & 450 & 0.33 & 8.35 & 1.91 & 9.93 \\\\\n500 & 500 & 0.21 & 8.23 & 0.99 & 9.00 \\\\\n600 & 400 & 0.48 & 8.49 & 3.43 & 11.45 \\\\\n600 & 450 & 0.30 & 8.32 & 1.88 & 9.90 \\\\\n600 & 500 & 0.18 & 8.20 & 0.96 & 8.98 \\\\\n700 & 400 & 0.46 & 8.47 & 3.42 & 11.44 \\\\\n700 & 450 & 0.28 & 8.30 & 1.87 & 9.89 \\\\\n700 & 500 & 0.16 & 8.18 & 0.95 & 8.96 \\\\\n800 & 400 & 0.45 & 8.47 & 3.42 & 11.43 \\\\\n800 & 450 & 0.28 & 8.29 & 1.86 & 9.88 \\\\\n800 & 500 & 0.16 & 8.17 & 0.94 & 8.96 \\\\\n900 & 400 & 0.44 & 8.46 & 3.41 & 11.43 \\\\\n900 & 450 & 0.27 & 8.29 & 1.86 & 9.38 \\\\\n900 & 500 & 0.15 & 8.17 & 0.94 & 8.95 \\\\\n1000 & 400 & 0.44 & 8.46 & 3.41 & 11.43 \\\\\n1000 & 450 & 0.27 & 8.29 & 1.86 & 9.87 \\\\\n1000 & 500 & 0.15 & 8.16 & 0.93 & 8.95 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\nFrom Table IV we see that, at the tevatron, the relative correction\n$\\Delta\\sigma^{(C1)}\/\\sigma_0$ is about $(2\\--6)\\%~$ which is at most\nof the same order as the expected systematic error, and \n$\\Delta\\sigma^{(C2)}\/\\sigma_0$ is around $~(12\\--43)\\%~$ which is larger \nthan the systematic error and the statistical uncertainty. So that, at the \nTevatron, {\\it the PGB effects in Model C ~for $m^\\prime_t=5$ GeV can hardly \nbe detected, while those for $m^\\prime_t=15$ GeV can be clearly detected}. \nThe relative differences \n$R^{(2)}_{CB}\\equiv (\\sigma^{(C2)}_{t\\bar t}-\\sigma^{(B2)}_{t\\bar t})\/\n\\sigma^{(C2)}_{t\\bar t}\\approx (9\\--26)\\%$, so that, for\n$m^\\prime_t=15$ GeV, {\\it Model C can be distinguished from Model B at the \nTevatron}. However, the relative difference \n$R^{(2)}_{CA}\\equiv (\\sigma^{(C2)}_{t\\bar t}-\\sigma^{(A)}_{t\\bar t})\/\n\\sigma^{(C2)}_{t\\bar t}$ is at most $5\\%$, therefore, {\\it even for \n$m^\\prime_t=15$ GeV, Model C can hardly be distinguished from Model A at the \nTevatron}.\n\n\n\\vspace{0.4cm}\n{\\small Table V. Cross sections in Model C at the $\\sqrt{s}=14$ TeV LHC. \n$\\Delta\\sigma^{(Ci)}$ denotes the correction to the SM tree level cross \nsections $\\sigma_0$, and $\\sigma^{(Ci)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Ci)}$ is the total cross section. The superscript $i$ \ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$ GeV\n($i=2$). All masses are in GeV, and all cross sections are in nb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$M_{\\Pi_t^0}$ & $M_{\\Pi^{0a}}$ & $\\Delta\\sigma^{(C1)}$ &\n$\\sigma^{(C1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{(C2)}$ & $\\sigma^{(C2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.22 & 1.04 & 1.64 & 2.47 \\\\\n500 & 450 & 0.20 & 1.02 & 1.35 & 2.18 \\\\\n500 & 500 & 0.16 & 0.99 & 1.02 & 1.85 \\\\\n600 & 400 & 0.20 & 1.03 & 1.63 & 2.45 \\\\\n600 & 450 & 0.19 & 1.01 & 1.34 & 2.17 \\\\\n600 & 500 & 0.15 & 0.98 & 1.01 & 1.84 \\\\\n700 & 400 & 0.19 & 1.02 & 1.62 & 2.45 \\\\\n700 & 450 & 0.18 & 1.00 & 1.33 & 2.16 \\\\\n700 & 500 & 0.14 & 0.97 & 1.00 & 1.83 \\\\\n800 & 400 & 0.19 & 1.02 & 1.62 & 2.44 \\\\\n800 & 450 & 0.17 & 1.00 & 1.33 & 2.15 \\\\\n800 & 500 & 0.14 & 0.96 & 1.00 & 1.82 \\\\\n900 & 400 & 0.19 & 1.01 & 1.61 & 2.44 \\\\\n900 & 450 & 0.17 & 1.00 & 1.33 & 2.15 \\\\\n900 & 500 & 0.13 & 0.96 & 1.00 & 1.82 \\\\\n1000 & 400 & 0.18 & 1.01 & 1.61 & 2.44 \\\\\n1000 & 450 & 0.17 & 0.99 & 1.32 & 2.15 \\\\\n1000 & 500 & 0.13 & 0.96 & 0.99 & 1.82 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\null\\vspace{0.4cm}\n\nFrom Table V we see that, at the LHC, the relative corrections \n$\\Delta\\sigma^{(C1)}\/\\sigma_0\\approx (16\\--26)\\%$,\n$\\Delta\\sigma^{(C2)}\/\\sigma_0\\approx (120\\--200)\\%$. These are all much\nlarger than the systematic error and the statistical uncertainty. So that \n{\\it the PGB effects in Model C, for both $m^\\prime_t=5$ Gev and \n$m^\\prime_t=15$ GeV, can be clearly detected at the LHC}.\nComparing the cross sections in Table V with those in Table I and Table III, we\nsee that the relative differences are~~ $R^{(1)}_{AC}\\equiv \n(\\sigma^(A)_{t\\bar t}-\\sigma^{(C1)}{t\\bar t})\/\n\\sigma^{(A)}_{t\\bar t}\\approx (40\\--54)\\%$,~~ $R^{(1)}_{CB}\\equiv\n(\\sigma^{(C1)}_{t\\bar t}-\\sigma^{(B1)}_{t\\bar t})\/\\sigma^{(C1)}_{t\\bar t}\n\\approx (11\\--15)\\%$,~~ $R^{(2)}_{CA}\\equiv (\\sigma^{(C2)}_{t\\bar t}\n-\\sigma^{(A)}_{t\\bar t})\/\\sigma^{(C2)}_{t\\bar t}\\approx (11\\--16)\\%$, \n~~ $R^{(2)}_{CB}\\equiv (\\sigma^{(C2)}_{t\\bar t}-\\sigma^{(B2)}_{t\\bar\nt})\/\\sigma^{(C2)}_{t\\bar t}\\approx (47\\--58)\\%$. These are all much\nlarger than the systematic error and the statistical uncertainty. So\nthat for both $m^\\prime_t=5$ GeV and $m^\\prime_t=15$ GeV, {\\it Model C\ncan be clearly distinguished from Model A and Model B at the LHC}.\n\nFor comparison with Fig. 2 and Fig. 3, the corresponding $t\\bar t$ invariant \nmass distributions at the Tevatron and the LHC in Model C are illustrated in \nFig. 4. We see that {\\it the resonances of $\\Pi^{0a}$ are significantly wider \nthan those in Model B, and clearly narrower than those in Model A} \nbecause the width of $\\Pi^{0a}$ depends on $m^\\prime_t\/F_\\Pi$, and the\nvalues of $F_\\Pi$ are very different in Model B and Model C. This character \nshows the clear distinction of the three kinds of TC models. Here we\nsee again that the $\\Pi_t$ contribution does not show up as a resonance\npeak, and its effect can be seen from the difference between the curve\nwith its contribution (the solid curve) and the curve without its\ncontribution (the dotted curve). The shapes of the $\\Pi^{0a}$ and $\\Pi_t$\ncontributions in Model C all look very different from those of the new\nheavy vector reaonaces shown in Ref.\\cite{HP}.\n\n\\null\\vspace{0.4cm}\n\\begin{center}\n{\\bf IV. CONCLUSIONS}\n\\end{center}\n\nIn this paper, we have studied the pseudo-Goldstone boson contributions\nto the $t\\bar t$ production cross sections at the Fermilab Tevatron Run\nII and the CERN LHC in various technicolor models, and have examined the\npossiblity of testing and distinguishing different technicolor models in the \nexperiments. We take the Appilquist-Terning one-family walking\ntechnicolor model (Model A), the original topcolor-assisted technicolor\nmodel (Model B), and the topcolor-assisted multiscale technicolor model\n(Model C) as three typical examples of technicolor models with and\nwithout topcolor. At the hadron colliders, the $s$-channel\npseudo-Goldstone boson contrbutions described in Fig. 1 dominate. In the\ncalculation, the MRS set A$^\\prime$ parton distribution functions are used to\nobtain the $p(\\bar p)\\to t\\bar t$ cross sections, and the\npseudo-Goldstone boson masses $M_{\\Pi^{0a}}$ and $M_{\\Pi_t}$ are taken\nto vary in certain ranges (as discussed in Sec. II) to see the dependence of \nthe cross sections on them. The obtained results are compared with the recent \nCDF and D0 data on the $t\\bar t$ production cross sections at the Tevatron \n[cf. eq.(\\ref{sigmatt})]. It is shown that all the obtained cross\nsections at the Tevatron are consistent with the CDF and D0 data.\n\nThe results of the calculated cross sections are listed in Table I to\nTable V. Considering the expected systematic error at the Tevatron and\nthe LHC, and assuming a $10\\%$ detecting efficiency, we have the following \nconclusions:\n\n\\begin{enumerate}\n\\item Model A can be clearly detected both at the Tevatron and the LHC.\n\\item In Model B and Model C, the $\\Pi^{0a}$ couplings are proportional\nto $m^\\prime_t$ ($m^\\prime_t\\ll m_t$) rather than to $m_t$ as in Model A.\nTherefore the $\\Pi^{0a}$ contributions in Model B and Model C are\nsignificantly reduced relative to Model A. This causes the fact that,\nconsidering the expected systematic error and the statistic uncertainty,\nModel B can hardly be detected at the Tevatron, and model C can be\ndetected at the Tevatron only for large $m^\\prime_t$, say $m^\\prime_t=15$ GeV. \nThe situation is much better for the LHC. Model B with $m^\\prime_t=15$ GeV and \nModel C (with $m^\\prime_t=5~{\\rm GeV~and}~m^\\prime_t=15$ GeV) can all \nbe clearly detected, and Model B with\n$m^\\prime_t=5$ GeV can be marginally detected at the LHC.\n\\item Due to the smallness of $F_{\\Pi_t}$ (the largeness of \n$(m_t-m^\\prime_t)\/F_{\\Pi_t}$), the width of the $\\Pi_t$ resonance is\nvery large which causes the fact that the cross sections are not so sensitive \nto the variation of $M_{\\Pi_t}$.\n\\item For the detectable cases, all the three kinds of models can be\nexperimentally distinguished by the significant differences of their\ncross sections. Furthermore, the $\\Pi^{0a}$ resonance peaks in the invariant \nmass $M_{t\\bar t}$ distributions for the three kinds of models are also very \ndifferent. \nThe width of the $\\Pi^{0a}$ resonance in the three models are: \n$\\Gamma^{(A)}_{\\Pi^{0a}}>\\Gamma^{(C)}_{\\Pi^{0a}}>\\Gamma^{(B)}_{\\Pi^{0a}}$.\nThis can serve as a clear distinction between the three kinds of \nmodels.\n\\item Comparing the present results with the heavy vector \nresonances (with the width about $20\\%$ of the mass) shown in Ref.\\cite{HP}, \nwe see that the $\\Pi^{0a}$ resonances are much sharper and the $\\Pi_t$ \ncontributions do not show up as resonances. The behavior of the \npresent resonances are very different from those heavy vector \nresonances studied in Ref\\cite{HP}. \n \n\n\\end{enumerate}\n\nIn summary, the PGB effects in $t\\bar t$ productions at the LHC provide \nfeasible tests of technicolor models including distinguishing different \ntypical models. It is complementary to other tests such as the tests studied in\nRefs.\\cite{HP,TC2-1,DE,RS,test}.\n\n\\vspace{0.4cm}\n\n\\begin{center}\n{\\bf Acknowledgment}\n\\end{center}\n\nThis work is supported by the National Natural Science Foundation of China,\nthe Fundamental Research Foundation of Tsinghua University, and a\nspecial grant from the Ministry of Education of China.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe theory of rough path analysis has been developed from the initial paper\nby Lyons \\ \\cite{L}. The aim of this theory is to analyze dynamical\nsystems $dx_{t}=f(x_{t})dy_{t}$, where the control function $y$ is not\ndifferentiable but has finite $p$-variation for some $p> 1$.\nThere is a wide literature on rough path analysis (see, for instance,\nLyons and Qian \\cite{L-Q}, Friz and Victoir \\cite{FV2}, Lejay \\cite{Le}, Lyons \\cite{LCL} or Gubinelli \\cite{Gu}). \n\n\n A path-wise approach to classical stochastic\ncalculus has been one of the motivations to build rough path analysis theory. A nice application of the rough path\nanalysis is the stochastic calculus with respect to the fractional Brownian\nmotion with Hurst parameter $H\\in (0,1)$. We\nrefer, for instance to Coutin and Lejay \\cite{CA}, Friz and Victoir \\cite{Fr}, Friz \\cite{FV} and Ledoux {\\it et al.} \\cite{LQZ} for some\napplications of rough path analysis to the stochastic calculus.\n\n Nualart and R\\u{a}\\c{s}canu in \\cite{NR} developed an alternative approach to the study of dynamical\nsystems $dx_{t}=f(x_{t})dy_{t}$, where the control function $y$ is H\\\"{o}lder\ncontinuous of order $\\beta >\\frac{1}{2}$.\nIn this case the Riemann-Stieltjes integral $\\int_{0}^{t}f(x_s)dy_s$ can be expressed as a Lebesgue integral\nusing fractional derivatives following the ideas of Z\\\"{a}hle \\cite{Z}.\nLater, Hu and Nualart \\cite{H-N} extended this approach to the case $\\beta \\in (\\frac{1}{3},\\frac{1}{2})$ . In this work they give an explicit expression for the integral $\\int_{0}^{t}f(x_s)dy_s$\nthat depends on the functions $x$, $y$ and a quadratic multiplicative functional \n$x\\otimes y$. Using this formula, the authors have established the existence and uniqueness of a solution for the dynamical system $dx_{t}=f(x_{t})dy_{t}$ driven by a H\\\"{o}lder\ncontinuous function $y$ of order $\\beta \\in (\\frac{1}{3},\\frac{1}{2})$. Finally, using the same approach, Besal\\'u and Nualart \\cite{B-N} got estimates\nfor the supremum norm of the solution.\n\n\n The purpose of this paper is to study a differential delay equation with non-negativity constraints \n driven by a H\\\"older continuous function $y$ of order \n\\beta\\in\\left(\\frac{1}{3},\\frac{1}{2}\\right)$ using the methodology introduced in \\cite{H-N}.\n We will consider the problems of existence, uniqueness and boundedness of the solutions. As an\napplication we will study a\nstochastic delay differential equations with non-negativity constraints driven by a fractional Brownian motion\nwith Hurst parameter $H\\in\\left(\\frac{1}{3},\\frac{1}{2}\\right)$. \nThese results extend the work by Besal\\'u and Rovira \\cite{B-R}, where is considered the case $H>\\frac 12$.\n\nMore precisely, we consider a delay differential equation with positivity constraints on $\\R^d$ of the form:\n\\begin{eqnarray}\nx(t)&=&\\eta(0)+\\int_0^t b(s,x)ds+\\int_0^t \\si(x(s-r))dy_s+z(t),\\quad t\\in(0,T],\\nonumber\\\\\nx(t)&=& \\eta(t),\\qquad t\\in[-r,0], \\nonumber\n\\end{eqnarray} \nwhere $r$ denotes a strictly positive time delay, $y$ is a $m$-dimensional $\\beta$-H\\\"older continuous function with $\\frac13<\\beta<\\frac{1}{2}$, $b(s,x)$ the hereditary term, depends on the path $\\left\\{x(u),-r\\leq u\\leq s\\right\\}$, while $\\eta:[-r,0]\\rightarrow\\R^d_+$ is a non negative smooth function, with $\\R^d_+=\\left\\{u\\in\\R^d;\\,u_i\\geq 0\\;\\mathrm{for}\\;i=1,\\ldots,d\\right\\}$ and $z$ is a vector-valued non-decreasing process which ensures that the non-negativity constraints on $x$ are enforced.\n\n\n\n\n\n\n\\vskip 7pt\n\nThen, we will apply pathwise our deterministic result to a stochastic delay differential equation with positivity constraints on $\\R^d$ of the form:\n\\begin{eqnarray}\nX(t)&=&\\eta(0)+\\int_0^t b(s,X)ds+\\int_0^t \\si(X(s-r))dW_s^H+Z(t),\\quad t\\in(0,T],\\nonumber\\\\\nX(t)&=& \\eta(t),\\qquad t\\in[-r,0], \\nonumber\n\\end{eqnarray} \nwhere $W^H=\\left\\{W^{H,j},\\,j=1,\\ldots,m\\right\\}$ are independent fractional Brownian motions with Hurst parameter $\\frac13\\frac12$ (\\cite{B-R}). Furthermore, \nthe literature about stochastic delay differential equations driven by a fractional Brownian\nmotion is scarce. For the case $H > \\frac12$ has been studied the existence and uniqueness of solution (\\cite{FR}, \\cite{LT}),\nthe existence and regularity of the density (\\cite{LT}) and the convergence when the delay goes to zero (\\cite{F-R}). For $H<\\frac12$ we can find the\nresults about the existence and uniqueness of solution (\\cite{NNT}, \\cite{TT}). Actually, in \\cite{NNT} the authors consider a similar equation to our case but without reflection. Moreover, they use another approach in order to define the stochastic integral based on L\\'evy area. In any case,\nwe will use some results on fractional Brownian motion taken from this paper.\n\n\n\n\\vskip 4pt\n\nAnyway, as it has been described in this paper of Kinnally and Williams \\cite{K-W} there are some models afected by some type of noise where the dynamics are related to propagation delay and some of them are naturally non-negative quantities. So, it is natural to continue the study of the stochastic delay differential equations and non-negativity constraints driven by a fractional Brownian motion.\n\n\\vskip 7pt\n\nIn our work, we will make use of the techniques introduced by Hu and Nualart \\cite{H-N} with some ideas borrowed from Besal\\'u and Rovira \\cite{B-R}. In this framework, let us point out again that one novelty of our paper is the non-negative constraints dealing with equations driven by a H\\\"older continuous function of order $\\beta \\in (\\frac13,\\frac12)$. We have used the\nSkorohod's mapping. Let us recall now the Skorokhod problem.\nSet $$\\cc_+(\\R_+,\\R^d):=\\left\\{x\\in\\cc(\\R_+,\\R^d): x(0)\\in\\R^d_+\\right\\}.$$\n\\vskip 5pt\n\\noindent\n \n\n\\begin{definition}\nGiven a path $z\\in\\cc_+(\\R_+,\\R^d)$, we say that a pair $(x,y)$ of functions in $\\cc_+(\\R_+,\\R^d)$ solves the Skorokhod problem for $z$ with reflection if\n\\begin{enumerate}\n\\item $x(t)=z(t)+y(t)$ for all $t\\geq 0$ and $x(t)\\in\\R_+^d$ for each $t\\geq 0$,\n\\item for each $i=1,\\ldots, d$, $y^i(0)=0$ and $y^i$ is nondecreasing,\n\\item for each $i=1,\\ldots, d$, ${\\displaystyle \\int_0^t x^i(s)dy^i_s=0}$ for all $t\\geq 0$, so $y^i$ can increase only when $x^i$ is at zero.\n\\end{enumerate}\n\\end{definition}\n\\noindent\nIt is known that we have an explicit formula for $y$ in terms of $z$: for each $i=1,\\ldots,d$\n\\begin{equation*}\ny^i(t)=\\max_{s\\in[0,t]} \\left(z^i(s)\\right)^-.\n\\end{equation*}\n\nThe path $z$ is called the reflector of $x$ and the path $y$ is called the regulator of $x$. We use the Skorokhod mapping for constraining a continuous real-valued function to be non-negative by means of reflection at the origin. \n\n\\vskip 7pt\n\n\n\nThe structure of the paper is as follows:\nin the next section we give some preliminaries, our hypothesis and we state the main results of our paper.\nIn Section 3, we give some basic facts about fractionals integrals. \nSection 4 is devoted to prove our main result: the existence and\nuniqueness for the solution for deterministic equations, while Section 5 deals with the problem of the boundedness.\nIn Section 6 we apply the deterministic results to the stochastic case. Finally, Section 7\nis devoted to give some technical results, as a fixed point theorem, and some properties related to the Skorohod problem.\n\n\n\\renewcommand{\\theequation}{2.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\section{Main results}\n\\vskip 5pt\n\\noindent\nFix a time interval $[0,T]$. For any function $x:[0,T]\\rightarrow \\mathbb{R\n{}^{n}$, the $\\gamma $-H\\\"{o}lder norm of $x$ on the interval $[s,t] \\subset \\lbrack 0,T]$, where $0< \\gamma \\le 1$, will be denoted by \n\\begin{equation*}\n\\left\\| x\\right\\| _{\\gamma(s,t) }=\\sup_{s< u\\frac{1}{\\beta}-2$.\n\\item[\\bfseries(H2)] $b:[0,T]\\times C(-r,T;\\R^d)\\rightarrow \\R^d$ is a measurable function such that for every $t>0$ and $f\\in C(-r,T;\\R^d),\\, b(t,f)$ depends only on $\\left\\{f(s);-r\\leq s\\leq t\\right\\}$. Moreover, there exists $b_0\\in L^\\rho(0,t;\\R^d)$ with $\\rho\\geq 2$ and $\\forall N\\geq 0$ there exists $L_N>0$ such that:\n\\begin{itemize}\n\\item[{\\bf (1)}] $\\left|b(t,x)-b(t,y)\\right|\\leq L_N \\left\\|x-y\\right\\|_{\\infty(-r,t)},\\; \\forall x,y\\;\\mathrm{such\\,that} \\left\\|x\\right\\|_{\\infty(r)}\\leq N,$ \n$\\left\\|y\\right\\|_{\\infty(r)}\\leq N,\\;\\forall t\\in[0,T]$,\n\\item[{\\bf (2)}] $\\left|b(t,x)\\right|\\leq L_0 \\left\\|x\\right\\|_{\\infty(-r,t)}+b_0(t),\\quad \\forall t\\in[0,T]$.\n\\end{itemize}\n\\end{itemize}\n\n\\vskip 5pt\n\\noindent\nThe result of existence and uniqueness states as follows:\n\n\\begin{theorem}\\label{tt1}\nAssume that $\\sigma$ and $b$ satisfy the hypothesis {\\upshape \\bfseries(H1)} and {\\upshape \\bfseries(H2)} respectively with $\\rho\\geq \\frac{1}{1-\\beta}$. Assume also that $\\eta\\geq 0$, $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,r)$ and $(y_{\\cdot-r},y,y_{\\cdot-r}\\otimes y)\\in M^\\beta_{m,m}(r,T)$. Then the equation (\\ref{det}) has a unique solution $x\\in \\cc(-r,T;\\R^d_+)$.\n\\end{theorem}\n\n\\vskip 5pt\n\n\\noindent {\\bf Remark}. If we assume that $\\eta\\geq 0$ is a differentiable continuous function with positive derivative, then the assumptions on $\\eta$ of this theorem are satisfied.\n\n\\medskip\n\n\\noindent\nIn order to study the boundedness of the solutions we need to stronger our hyphotesis. Consider now:\n\\begin{itemize}\n\\item[\\bfseries(H3)] $b$ and $\\sigma'$ are bounded function.\n\\end{itemize}\nThen, the result is as follows:\n\\begin{theorem}\\label{tt2}\nAssume that $\\sigma$ and $b$ satisfy the hypothesis {\\upshape \\bfseries(H1)}, {\\upshape \\bfseries(H2)} and {\\upshape \\bfseries(H3)}. Also assume that $\\eta\\geq 0$ satisfies $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,r)$ and finally that $(y_{\\cdot-r},y,y_{\\cdot-r}\\otimes y)\\in M^\\beta_{m,m}(r,T)$. Set\n$$\\mu=\\left\\|b\\right\\|_{\\infty}+\\left\\|\\sigma\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\gamma}.$$\nThen, the solution of (\\ref{det}) is bounded as follows\n\\begin{equation}\n\\|x\\|_{\\infty}\\le 2+ \\eta(0) + T \\left\\{K\\left(\\left\\|\\eta\\right\\|_{\\beta}\n+\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n+\\mu (d^\\frac12 +1) \\left[\\left\\|y\\right\\|_{\\beta}\n+\\left\\|y\\right\\|_{\\beta}^2+ \\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\right]\\right)\\right\\}^\\frac{1}{\\beta},\\label{e5.1}\\end{equation}\nwhere $K$ is a universal constant depending only on $\\beta$ and $\\gamma$,\nand\n\\begin{eqnarray*}\n\\left\\|\\eta\\right\\|_{\\beta}&:=&\\left\\|\\eta\\right\\|_{\\beta(-r,0)},\\\\\n\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n&:=&\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta(0,r)},\\\\\n\\left\\|y\\right\\|_{\\beta}&:=&\\left\\|y\\right\\|_{\\beta(0,T)},\\\\\n\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n&:=&\n\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta(r,T)}.\n\\end{eqnarray*}\n\\end{theorem}\n\n\\vskip 5pt\n\\noindent\nOur last result is an application of the above theorems to stochastic delay differential equations. More precisely, let us consider a\nstochastic delay differential equation with positivity constraints on $\\R^d$ of the form:\n\\begin{eqnarray}\nX(t)&=&\\eta(0)+\\int_0^t b(s,X)ds+\\int_0^t \\si(X(s-r))dW_s^H+Z(t),\\quad t\\in(0,T],\\nonumber\\\\\nX(t)&=& \\eta(t),\\qquad t\\in[-r,0], \\label{eqstoc}\n\\end{eqnarray} \nwhere $W^H=\\left\\{W^{H,j},\\,j=1,\\ldots,m\\right\\}$ are independent fractional Brownian motions with Hurst parameter $\\frac130$. The\nleft-sided and right-sided fractional Riemann-Liouville integrals of $f$ of\norder $\\alpha $ are defined for almost all $t\\in (a,b)$ by \n\\begin{equation*}\nI_{a+}^{\\alpha }f(t)=\\frac{1}{\\Gamma (\\alpha )}\\int_{a}^{t}(t-s)^{\\alpha\n-1}f(s)ds\n\\end{equation*\nand \n\\begin{equation*}\nI_{b-}^{\\alpha }f(t)=\\frac{(-1)^{-\\alpha }}{\\Gamma (\\alpha )\n\\int_{t}^{b}(s-t)^{\\alpha -1}f(s)ds,\n\\end{equation*\nrespectively, where $(-1)^{-\\alpha }=e^{-i\\pi \\alpha }$ and $\\Gamma (\\alpha\n)=\\int_{0}^{\\infty }r^{\\alpha -1}e^{-r}dr$ is the Euler gamma function. For any $p\\ge 1$, let \nI_{a+}^{\\alpha }(L^{p})$ (resp. $I_{b-}^{\\alpha }(L^{p})$) be the image of \nL^{p}(a,b)$ by the operator $I_{a+}^{\\alpha }$ (resp. $I_{b^{-}}^{\\alpha }\n). If $f\\in I_{a+}^{\\alpha }(L^{p})$ (resp. $f\\in I_{b-}^{\\alpha }(L^{p})$)\nand $0<\\alpha <1$, then the Weyl derivatives are defined as \n\\begin{eqnarray}\nD_{a+}^{\\alpha }f(t) &=&\\frac{1}{\\Gamma (1-\\alpha )}\\left( \\frac{f(t)}\n(t-a)^{\\alpha }}+\\alpha \\int_{a}^{t}\\frac{f(t)-f(s)}{(t-s)^{\\alpha +1}\nds\\right) , \\nonumber \\\\\nD_{b-}^{\\alpha }f(t) &=&\\frac{(-1)^{\\alpha }}{\\Gamma (1-\\alpha )}\\left( \n\\frac{f(t)}{(b-t)^{\\alpha }}+\\alpha \\int_{t}^{b}\\frac{f(t)-f(s)}\n(s-t)^{\\alpha +1}}ds\\right) , \\nonumbe\n\\end{eqnarray\nwhere $a\\leq t\\leq b$ (the convergence of the integrals at the singularity \ns=t$ holds point-wise for almost all $t\\in (a,b)$ if $p=1$ and moreover in\nthe $L^{p}$-sense if $11$, it is proved in \\cite{Z} that the Riemman-Stieltjes integral \n\\int_{a}^{b}fdg$ exists. The following proposition provides an explicit\nexpression for the integral $\\int_{a}^{b}fdg$ in terms of fractional\nderivatives (see \\cite{Z}).\n\n\\begin{proposition}\nSuppose that $f\\in C^{\\lambda }(a,b)$ and $g\\in C^{\\mu }(a,b)$ with $\\lambda\n+\\mu >1$. Let $1-\\mu <\\alpha <\\lambda $. Then the Riemann-Stieltjes integral \n$\\int_{a}^{b}fdg$ exists and it can be expressed as \n\\begin{equation}\n\\int_{a}^{b}fdg=(-1)^{\\alpha }\\int_{a}^{b}D_{a+}^{\\alpha\n}f(t)D_{b-}^{1-\\alpha }g_{b-}(t)dt, \\label{forpart}\n\\end{equation\nwhere $g_{b-}(t)=g(t)-g(b).$\n\\end{proposition}\n\\noindent\nBut if $x,y \\in C^{\\beta} (a,b)$ with $\\beta \\in (\\frac13,\\frac12)$ we can not use Equation (\\ref{forpart}) to define the integral $\\int_{a}^{b}f(x(t))dy_{t}$, so\nwe need to recall the construction of the integral $\\int_{a}^{b}f(x(t))dy_{t}$\ngiven by Hu and Nualart in \\cite{H-N} using fractional derivatives.\n\\begin{definition}\n\\label{defhn} Let $(x,y,x\\otimes y)\\in M_{d,m}^{\\beta }(0,T)$. Let $f\n\\mathbb{R}{}^{d}\\rightarrow \\mathbb{R}{}^{m}\\otimes \\mathbb{R}{}^{d}$ be a\ncontinuously differentiable function such that $f^{\\prime }$ is locally \n\\lambda $-H\\\"{o}lder continuous, where $\\lambda >\\frac{1}{\\beta }-2$. Fix \n\\alpha >0$ such that $1-\\beta <\\alpha <2\\beta $, and $\\alpha <\\frac{\\lambda\n\\beta +1}{2}$. Then, for any $0\\leq a \\frac{1}{\\beta} -2.$ Then for any\n$0 \\le a < b \\le T$ we have\n\\begin{eqnarray*}\n\\Vert \\int f(x(r)) dy_r \\Vert_{\\beta(a,b)} &\\le & k \\vert f(x(a)) \\vert \\Vert y \\Vert_{\\beta(a,b)} + k \\Phi_{a,b,\\beta} (x,y)\n\\\\\n&& \\times \\left( \\Vert f' \\Vert_\\infty + \\Vert f' \\Vert_\\gamma \\Vert x \\Vert^\\gamma_{\\beta(a,b)} (b-a)^{\\gamma \\beta} \\right) (b-a)^\\beta,\n\\end{eqnarray*}\nwhere\n$$ \\Phi_{a,b,\\beta} (x,y)= \\Vert x \\otimes y \\Vert_{2\\beta(a,b)} + \\Vert x \\Vert_{\\beta(a,b)} \\Vert y \\Vert_{\\beta(a,b)}.$$\n\\end{proposition}\n\n\\begin{proposition}\\label{prop24}\nSuppose that $(x,y,x \\otimes y)$ and $(y,z,y \\otimes z)$ belong to $M_{d,m}^\\beta (0,T)$. Let $f: \\R^d \\longrightarrow \\R^m$ be a continuously\ndifferentiable function such that $f'$ is $\\gamma$-H\\\"older continuous and bounded, where $\\gamma > \\frac{1}{\\beta} -2.$ \nFix $\\alpha >0$ such that $1 - \\beta < \\alpha <2 \\beta, \\alpha < \\frac{\\gamma \\lambda +1}{2}.$ Then the following estimate holds:\n\\begin{eqnarray*}\n\\vert \\int_a^b f(x(r)) d (y\\otimes z)_{\\cdot,b}(r) \\vert & \\le & k \\vert f(x(a)) \\vert\\Phi_{a,b,\\beta}(y,z) (b-a)^{2\\beta}\n\\\\\n&& + k \\left( \\Vert f' \\Vert_\\infty + \\Vert f' \\Vert_\\gamma \\Vert x \\Vert^\\gamma_{\\beta(a,b)} (b-a)^{\\gamma \\beta} \\right) \\Phi_{a,b,\\beta}(x,y,z) (b-a)^{3\\beta},\n\\end{eqnarray*}\nwhere\n$$ \\Phi_{a,b,\\beta} (x,y,z)= \\Vert x \\Vert_{\\beta(a,b)} \\Vert y \\Vert_{\\beta(a,b)} \\Vert z \\Vert_{\\beta(a,b)} + \\Vert z \\Vert_{\\beta(a,b)}\\Vert x \\otimes y \\Vert_{2\\beta(a,b)} + \\Vert x \\Vert_{\\beta(a,b)} \\Vert y \\otimes z \\Vert_{2\\beta(a,b)}.$$\n\\end{proposition}\n\n\\renewcommand{\\theequation}{4.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\section{Existence and uniqueness for deterministic integral equations}\n\\vskip 5pt\n\\noindent\nThe aim of this section is the proof of Theorem \\ref{tt1}. For simplicity let us assume $T=Mr$.\n\\vskip 5pt\n\\noindent\n{\\bf Proof of Theorem \\ref{tt1}:}\nIn order to prove that equation (\\ref{det}) admits a unique continuous solution on $[-r,T]$, we will use an induction argument.\nWe shall prove that if the equation (\\ref{det}) admits a unique solution $x^{(n)}$ on $[-r,nr]$ we can prove that there is a unique solution $x^{(n+1)}$ on $[-r,(n+1)r]$. More precisely, \nour induction hypothesis is the following:\\\\\n$\\bf{(H_n)}$ The equation\n\\begin{eqnarray*}\nx^{(n)}(t)&=&\\eta(0)+\\int_0^t b(s,x^{(n)})ds+\\int_0^t\\sigma(x^{(n-1)}(s-r))dy_s+ z^{(n)}(t),\\quad t\\in (0,nr],\\nonumber\\\\\nx^{(n)}(t) &=& \\eta(t),\\quad t\\in[-r,0],\\label{sn}\n\\end{eqnarray*}\nwhere for $i=1,\\dots,d$, $(z^{(n)})^{i}(t)=\\max_{s\\in[0,t]}((\\xi^{(n)})^{i}(s))^-$ with\n\\[\\xi^{(n)}(t)=\\eta(0)+\\int_0^t b(s,x^{(n)})ds+\\int_0^t\\sigma(x^{(n-1)}(s-r))dy_s,\\]\nhas a unique solution $x^{(n)}\\in \\cc(-r,nr,\\R_+^d)$ and moreover $(x^{(n)}_{\\cdot-r},y,x^{(n)}_{\\cdot-r}\\otimes y)\\in M^\\beta_{d,m}(0,(n+1)r)$.\n\\vskip 5pt\n\\noindent\nActually, when we want to check $\\bf{(H_{n+1})}$ assuming $\\bf{(H_{n})}$, we can write the equation of $\\bf{(H_{n+1})}$\n as\n\\begin{eqnarray}\nx^{(n+1)}(t)&=&\\eta(0)+\\int_0^t b(s,x^{(n+1)})ds+\\int_0^t\\sigma(x^{(n)}(s-r))dy_s+ z^{(n+1)}(t),\\quad t\\in (0,(n+1)r],\\nonumber\\\\\nx^{(n+1)}(t) &=& \\eta(t),\\quad t\\in[-r,0].\\label{sn1}\n\\end{eqnarray}\nSince $(x^{(n)}_{\\cdot-r},y,x^{(n)}_{\\cdot-r}\\otimes y)\\in M^\\beta_{d,m}(0,(n+1)r)$ we know that we can use Definition {\\ref{defhn}} to define the integral $\\int_0^t\\sigma(x^{(n)}(s-r))dy_s$ appearing in equation (\\ref{sn1}).\nThen, \nthe proof will consist in checking the following steps:\n\\begin{enumerate}\n\\item Existence of a solution of the equation (\\ref{sn1}) in the space $\\cc(-r,(n+1)r;\\R_+^d)$.\n\\item Uniqueness of a solution of the equation (\\ref{sn1}) in the space $\\cc(-r,(n+1)r;\\R_+^d)$.\n\\item The solution $x^{(n+1)}$ satifies that $(x^{n+1}_{\\cdot-r},y,x^{(n+1)}_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,(n+2)r)$.\n\\end{enumerate}\n\nActually, we will only proof the first case, that is $\\bf{(H_{1})}$. Notice that \nthe induction step, that is the proof of $\\bf{(H_{n+1})}$ assuming that $\\bf{(H_{n})}$ is true, can be done repeating the computations of this initial case.\n\\vskip 5pt\n\\noindent So, let us check $\\bf{(H_{1})}$. We will deal with the equation\n\\begin{eqnarray}\nx^{(1)}(t)&=&\\eta(0)+\\int_0^t b(s,x^{(1)})ds+\\int_0^t\\sigma(\\eta(s-r))dy_s+ z^{(1)}(t),\\quad t\\in (0,r],\\nonumber\\\\\nx^{(1)}(t) &=& \\eta(t),\\quad t\\in[-r,0],\\label{s1}\n\\end{eqnarray}\nwhere for $i=1,\\;\\ldots,\\;d$, $(z^{(1)})^i(t)=\\max_{s\\in[0,t]}((\\xi^{(1)})^i(s))^-$ and\n\\[\\xi^{(1)}(t)=\\eta(0)+\\int_0^t b(s,x^{(1)})ds+\\int_0^t\\sigma(\\eta(s-r))dy_s.\\]\nNote that since $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,r)$ we can use Definition \\ref{defhn} in order to define the integral $\\int_0^t\\sigma(\\eta(s-r))dy_s$ appearing in (\\ref{s1}). The proof of this initial case will be divided en 3 steps:\n\\begin{enumerate}\n\\item Existence of a solution in the space $\\cc(-r,r;\\R^d_+)$.\n\\item Uniqueness of a solution in the space $\\cc(-r,r;\\R^d_+)$.\n\\item The solution $x^{(1)}$ satisfies that $(x^{(1)}_{\\cdot-r},y,x^{(1)}_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,2r)$\n\\end{enumerate}\n\\noindent To simplify the proof we will assume $d=m=1$.\n\n\\vskip 5pt\n\\noindent\n{\\underline{Step 1:}} In order to prove the existence of solution we will use Lemma \\ref{puntfix}, a fixed point argument on $\\cc(-r,r,\\R_+)$.\n\\vskip 5pt\n\\noindent\nLet us consider the operator \n\\[\\cl: \\cc(-r,r;\\R_+)\\rightarrow \\cc(-r,r;\\R_+)\\]\nsuch that\n\\begin{eqnarray*}\n\\cl(u)(t)&=&\\eta(0)+\\int_0^t b(s,u)ds+\\int_0^t\\si(\\eta(s-r))dy_s+z(t),\\qquad t\\in[0,r],\\\\\n\\cl(u)(t)&=&\\eta(t),\\qquad t\\in[-r,0].\n\\end{eqnarray*}\nwhere setting \n\\[\\xi(t)=\\eta(0)+\\int_0^t b(s,u)ds+ \\int_0^t \\si(\\eta(s-r))dy_s,\\]\nthen ${\\displaystyle z(t)=\\max_{s\\in[0,t]}(\\xi(s))^-}$.\n\\vskip 5pt\n\\noindent \nClearly\n$\\cl$ is well defined. Let us use the notation $u^*=\\cl(u)$.\\\\\nNow, we need to introduce a family of norms in the space $\\cc(-r,r;\\R_+)$. That is, for any $\\la\\geq 1$, let us consider\n\\[\\left\\|f\\right\\|_{\\infty,\\la(-r,r)}:= \\sup_{t\\in[-r,r]} e^{-\\la t}|f(t)|.\\]\nIt is easy to check that these norms are equivalent to $\\left\\|f\\right\\|_{\\infty(-r,r)}$.\n\\vskip 5pt\n\\noindent\nUsing standard arguments (see for instance \\cite{B-R} for similar computations) we obtain that\n\\begin{eqnarray}\n\\left\\|u^*\\right\\|_{\\infty,\\lambda(-r,r)}&\\leq&\\left\\|\\eta\\right\\|_{\\infty,\\la(-r,0)}+2|\\eta(0)|\n+2 \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\int_0^t b(s,u)ds\\right|\\nonumber\\\\\n&&+2 \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\int_0^t \\si(\\eta(s-r))dy_s\\right|.\\label{cotaE1}\n\\end{eqnarray}\nWe obtain easily (see again \\cite{B-R}) that\n{\\begin{eqnarray}\n\\sup_{t\\in[0,r]} e^{-\\la t}\\left|\\int_0^t b(s,u)ds\\right|&\\leq& \\frac{L_0}{\\la} \\left\\|u\\right\\|_{\\infty,\\la(-r,r)}+\\frac{C_\\rho}{\\la^{1-\\rho}}\\left\\|b_0\\right\\|_{L^{\\rho}}. \\label{c1}\n\\end{eqnarray}}\n\\vskip 5pt\n\\noindent \nIt only remains the study of the term with the fractional integral. Using the bound appearing on the proof of Proposition \\ref{prop23}, we get for any $\\lambda \\ge 1$,\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle\\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\int_0^t\\sigma(\\eta({s-r}))dy_s\\right|\\\\\n\\qquad\\quad\\displaystyle\\leq k|\\sigma(\\eta({-r}))|\\left\\|y\\right\\|_{\\beta(0,r)}\\sup_{t\\in[0,r]}e^{-\\la t}t^\\beta+\\\\\\qquad\\qquad\n\\displaystyle+k\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}\\sup_{t\\in[0,r]}e^{-\\la t}t^{2\\beta}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\sup_{t\\in[0,r]}e^{-\\la t}t^{(\\gamma+2)\\beta}\\right)\\\\\n\\qquad\\quad\\displaystyle\\leq k|\\sigma(\\eta({-r}))|\\left\\|y\\right\\|_{\\beta(0,r)} \\left(\\frac{\\beta}{\\lambda}\\right)^\\beta e^{-\\beta}+\\\\\n\\qquad\\qquad\\displaystyle k\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}\\left(\\frac{2\\beta}{\\lambda}\\right)^{2\\beta} e^{-2\\beta}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\left(\\frac{(\\gamma+2)\\beta}{\\lambda}\\right)^{(\\gamma+2)\\beta} e^{(\\gamma+2)\\beta}\\right)\\\\\n\\qquad\\quad\\displaystyle\\leq C_{\\beta,\\gamma}\\frac{1}{\\la^{\\beta}}\\left(|\\sigma(\\eta({-r}))|\\left\\|y\\right\\|_{\\beta(0,r)}+\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\right)\\right),\n\\end{array} \\label{a2} \n\\end{equation}\nwhere in the last inequality we have used that\n\\[\\sup_{t\\in[0,r]} t^\\mu e^{-\\la t} \\leq \\left(\\frac{\\mu}{\\la}\\right)^\\mu e^{-\\mu}\\]\nand $C_{\\beta,\\gamma}$ is a constant depending on $\\beta$ and $\\gamma$.\n\\vskip 5pt\n\\noindent\nSo putting together (\\ref{cotaE1}), (\\ref{c1}) and (\\ref{a2}) we have\n\\[\\left\\|u^*\\right\\|_{\\infty,\\lambda(-r,r)}\\leq M_1(\\lambda)+M_2(\\lambda)\\left\\|u\\right\\|_{\\infty,\\lambda(-r,r)},\\]\nwhere\n\\begin{eqnarray*}\nM_1(\\lambda)&=&\\left\\|\\eta\\right\\|_{\\infty,\\la(-r,0)}+2|\\eta(0)|+\\frac{2C_\\rho}{\\la^{1-\\rho}}\\left\\|b_0\\right\\|_{L^{\\rho}}\\\\\n&&+C_{\\beta,\\gamma}\\frac{2}{\\la^{\\beta}}\\left(|\\sigma(\\eta(-r))|\\left\\|y\\right\\|_{\\beta(0,r)}+\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\right)\\right),\\\\\nM_2(\\la)&=&2L_0\\frac{1}{\\la}.\n\\end{eqnarray*}\nNow, we can choose $\\la=\\la_0$ large enough such that $M_2(\\la_0) \\leq \\frac{1}{2}$, Then, ${\\displaystyle\\left\\|u\\right\\|_{\\infty,\\la_0(-r,r)}\\leq 2M_1(\\la_0)}$ yields that $${\\displaystyle\\left\\|u^*\\right\\|_{\\infty,\\la_0(-r,r)}\\leq 2M_1(\\la_0)}$$ and ${\\displaystyle \\cl(B_0)\\subseteq B_0}$ for\n\\[B_0=\\left\\{u\\in \\cc(-r,r;\\R^d_+); \\left\\|u\\right\\|_{\\infty,\\la_0(-r,r)}\\leq 2M_1(\\la_0)\\right\\}.\\]\nThe first hypothesis in Lemma \\ref{puntfix} is now satisfied with the metric $\\rho_0$ associated to the norm $\\left\\|\\cdot\\right\\|_{\\infty,\\la_0(-r,r)}$.\nTo finish the proof it suffices to find a metric $\\rho_1$ satisfying the second hypothesis in Lemma \\ref{puntfix}.\n\\vskip 5pt \\noindent\n Notice first that if $u\\in B_0$ then $\\left\\|u\\right\\|_{\\infty(-r,r)}\\leq 2e^{\\la_0r}M_1(\\la_0):=N_0$.\nConsider $u,u'\\in B_0$ and $\\la\\geq 1$. Then\n\\begin{equation}\n \\left\\|\\cl(u)-\\cl(u')\\right\\|_{\\infty,\\la(-r,r)}\\leq \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\xi(t)-\\xi'(t)\\right|\n +\\sup_{t\\in[0,r]}e^{-\\la t}\\left|z(t)-z'(t)\\right|.\\label{laa}\n\\end{equation} \nFrom Lemma \\ref{le2} notice that given $t \\in [0,r]$ there exists $t_2 \\le t$ such that\n$$ \n\\left|z(t)-z'(t)\\right| \\le K_l \\left|\\xi(t_2)-\\xi'(t_2)\\right| .$$\nSo\n$$ \ne^{-\\lambda t}\\left|z(t)-z'(t)\\right| \\le K_l e^{-\\lambda t_2} \\left|\\xi(t_2)-\\xi'(t_2)\\right|$$\nand it follows easily that\n\\begin{equation}\n\\sup_{t\\in[0,r]}e^{-\\la t}\\left|z(t)-z'(t)\\right| \\le\nK_l \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\xi(t)-\\xi'(t)\\right|.\\label{lab}\\end{equation}\nFrom (\\ref{laa}) and (\\ref{lab}) we can write\n\\begin{eqnarray*}\n\\left\\|\\cl(u)-\\cl(u')\\right\\|_{\\infty,\\la(-r,r)}&\\leq &(1+K_l) \\sup_{t\\in[0,r]}e^{-\\la t}\\int_0^t\\left|b(s,u)-b(s,u')\\right|ds\\\\\n&\\leq & L_{N_0} (1+K_l) \\sup_{t\\in[0,r]}e^{-\\la t}\\int_0^t\\sup_{0\\leq v\\leq s}\\left|u(v)-u'(v)\\right|ds\\\\\n&\\leq & L_{N_0}(1+K_l)\\sup_{t\\in[0,r]}\\int_0^t e^{-\\la( t-s)} e^{-\\la s} \\sup_{-r\\leq v\\leq s}\\left|u(v)-u'(v)\\right|ds\\\\\n&\\leq & L_{N_0}(1+K_l )\\frac{1}{\\la}\\left\\|u-u'\\right\\|_{\\infty,\\la(-r,r)} .\n\\end{eqnarray*}\n\\vskip 6pt \\noindent\nSo, choosing $\\la=\\la_1$ such that ${\\displaystyle \\frac{L_{N_0}(1+K_l)}{\\la_1}\\leq \\frac{1}{2}}$, the second hypothesis is satisfied for the metric $\\rho_1$ associated with the norm $\\left\\|\\cdot\\right\\|_{\\infty,\\la_1(-r,r)}$ and $a=\\frac12$.\n\\vskip 7pt\n\\noindent\n{\\underline{Step 2:}} We deal now with the uniqueness problem. \\\\\nLet $x$ and $x'$ be two solutions of (\\ref{s1}) in the space $\\cc(-r,r;\\R_+)$ and choose $N$ large enough such that $\\left\\|x\\right\\|_{\\infty(-r,r)}\\leq N$ and $\\left\\|x'\\right\\|_{\\infty(-r,r)}\\leq N$.\n\\vskip 5pt\n\\noindent\nFor any $t\\in[0,r]$,\n$$\n\\sup_{s\\in[0,t]} \\left|x(s)-x'(s)\\right| \\le \\sup_{s\\in[0,t]}\\left|\\xi(s)-\\xi'(s)\\right|+ \\sup_{s\\in[0,t]} \\left|z(s)-z'(s)\\right|.$$\nMoreover, using Lemma \\ref{le2} we have\n$$\n\\sup_{s\\in[0,t]} \\left|z(s)-z'(s)\\right|\\leq K_l \\sup_{s\\in[0,t]} \\left|\\xi(t)-\\xi'(t)\\right|.$$\nSo, putting together the last two inequalities we get that\n\\begin{eqnarray*}\n\\sup_{s\\in[0,t]} \\left|x(s)-x'(s)\\right|&\\le & (1+K_l) \\sup_{s\\in[0,t]} \\left|\\xi(s)-\\xi'(s)\\right| \\\\&\\leq&\n(1+K_l) \\sup_{s\\in[0,t]}\n\\left|\\int_0^s \\left(b(\\tau,x)-b(\\tau,x')\\right)d\\tau\\right|\\\\&\\leq&\n(1+K_l) L_N\\sup_{s\\in[0,t]}\\left|\\int_0^s\\sup_{0\\leq v\\leq \\tau}|x(v)-x'(v)|d\\tau\\right|\\\\&\\leq& L_N(1+K_l)\\int_0^t\\sup_{v\\in[0, \\tau]}|x(v)-x'(v)|d\\tau.\n\\end{eqnarray*}\nApplying now Gronwall's inequality, we have that for all $t\\in[0,r]$\n\\[\\sup_{s\\in[0,t]}|x(s)-x'(s)|= 0.\\]\nSo\n\\[\\left\\|x-x'\\right\\|_{\\infty(-r,r)}= 0 \\]\nand the uniqueness has been proved.\n\\vskip 5pt\n\\noindent\n{\\underline{Step 3:}} \nWe have to prove that $(x^{(1)}_{\\cdot-r},y,x^{(1)}_{\\cdot-r}\\otimes y)\\in M_{1,1}^{\\beta}(0,2r)$.\\\\\n\nWe have to check the three conditions appearing in Definition \\ref{d2.1}:\n\\begin{enumerate}\n\\item $y:[0,2r]\\rightarrow\\R$ is $\\beta$-H\\\"older continuous. This condition is one of the hypothesis of our theorem.\n\\item $x^{(1)}_{\\cdot-r}:[0,2r]\\rightarrow \\R$ is $\\beta$-H\\\"older continuous.\\\\\nWe can write that\n\\begin{eqnarray*}\n\\left\\|x^{(1)}_{\\cdot-r}\\right\\|_{\\beta(0,2r)}&=&\\left\\|x^{(1)}\\right\\|_{\\beta(-r,r)}=\\sup_{-r\\leq v\\leq w\\leq r} \\frac{|x^{(1)}(w)-x^{(1)}(v)|}{(w-v)^\\beta}\\\\\n&\\leq& \\sup_{-r\\leq v\\leq w<0} \\frac{|\\eta(w)-\\eta(v)|}{(w-v)^\\beta}+\\sup_{\\substack{-r\\leq v\\leq 0\\\\ 0\\leq w\\leq r}}\\frac{|x^{(1)}(w)-\\eta(v)|}{(w-v)^\\beta}\\\\\n&&+ \\sup_{0\\leq v\\leq w\\leq r} \\frac{|x^{(1)}(w)-x^{(1)}(v)|}{(w-v)^\\beta}.\n\\end{eqnarray*}\nNote that \n\\[\\frac{|x^{(1)}(w)-\\eta(v)|}{(w-v)^\\beta}\\leq\\frac{|x^{(1)}(w)-\\eta(0)|}{(w-0)^\\beta}+\\frac{|\\eta(0)-\\eta(v)|}{(0-v)^\\beta}. \\]\nSo\n\\begin{equation}\n\\left\\|x^{(1)}_{\\cdot-r}\\right\\|_{\\beta(0,2r)}\\leq 2\\left\\|\\eta\\right\\|_{\\beta(-r,0)}+2\\left\\|x^{(1)}\\right\\|_{\\beta(0,r)}. \\label{a3}\n\\end{equation}\n\n\nMoreover\n\\begin{equation}\n\\left\\|x^{(1)}\\right\\|_{\\beta(0,r)} \\leq \\left\\|\\int_0^\\cdot b(s,x^{(1)})ds\\right\\|_{\\beta(0,r)}+\\left\\|\\int_0^\\cdot \\sigma(\\eta(s-r))dy_s\\right\\|_{\\beta(0,r)}+\\left\\|z^{(1)}\\right\\|_{\\beta(0,r)}. \\label{a4}\n\\end{equation}\nUsing Lemma \\ref{le2} we also get that\n\\begin{equation}\n\\left\\|z^{(1)}\\right\\|_{\\beta(0,r)}\\leq \\left\\|\\xi^{(1)}\\right\\|_{\\beta(0,r)}. \\label{a5}\n\\end{equation}\n\nFurthermore, putting together (\\ref{a3}), (\\ref{a4}) and (\\ref{a5}) and using again Proposition \\ref{prop23} we obtain that\n\\begin{eqnarray*}\n\\left\\|x^{(1)}\\right\\|_{\\beta(-r,r)}&\\leq&4\\left\\|\\eta\\right\\|_{\\beta(-r,0)}+4\\left\\|\\int_0^\\cdot b(s,x^{(1)} )ds\\right\\|_{\\beta(0,r)}\n+4\\left\\|\\int_0^\\cdot \\sigma(\\eta (s-r))dy_s\\right\\|_{\\beta(0,r)}\\\\\n&\\leq&4\\left\\|\\eta\\right\\|_{\\beta(-r,0)}+4 C\\left(1+\\left\\|x^{(1)}\\right\\|_{\\infty(-r,r)}\\right)\n+4(k|\\sigma(\\eta (-r))|\\left\\|y\\right\\|_{\\beta(0,r)}\\\\\n&&+\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)(\\left\\|\\sigma'\\right\\|_\\infty+\\left\\|\\sigma'\\right\\|_\\gamma\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}r^{\\gamma\\beta}))r^\\beta.\n\\end{eqnarray*}\nSo we can conclude that $x^{(1)}_{\\cdot-r}$ is $\\beta$-H\\\"older continuous.\n\n\\item Let us define $(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}$ for $s,t\\in \\Delta_{2r}$. For completeness, we will give this definition for any dimensions $d$ and $m$, unless we will still consider $d=m=1$ in the proofs. For any $k \\in \\{1,\\cdots,d\\}$ and $l \\in \\{1,\\cdots,m\\}$, set:\n\\begin{itemize}\n\\item\nif $s,\\,t\\in[0,r]$, \n\\[(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}^{k,l}=(\\eta_{\\cdot-r}\\otimes y)_{s,t}^{k,l},\\]\n\\item if $s,\\,t\\in [r,2r]$, set\n\\begin{eqnarray*}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}^{k,l}&=&\\int_s^t (y^l(t)-y^l(v))b^k(v-r,x^{(1)})dv+\\sum_{j=1}^m \\int_{s}^{t} \\si^k_j(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot, t}^{j,l}(v)\\\\\n&&+\\int_s^t (y^l(t)-y^l(v))d(z^{(1)})_{v-r}^k,\n\\end{eqnarray*}\n\\item if $s\\in[0,r]$ and $t\\in[r,2r]$, \n\\[(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}^{k,l}=(\\eta_{\\cdot-r}\\otimes y)_{s,r}^{k,l}+(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}^{k,l}+(\\eta^k(0)-\\eta^k(s-r))\\otimes (y^l(t)-y^l(r)).\\]\n\\end{itemize}\n\\vskip 5pt\n\\noindent\nLet us check that the multiplicative property (let us recall that we consider again $d=m=1$ for simplicity) is satisfied, that is, \nfor any $ 0 \\le s \\le u \\le t \\le 2r$ it holds that\n\\begin{equation}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}+(x^{(1)}(u-r)-x^{(1)}(s-r))\\otimes (y(t)-y(u))=(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}. \\label{s2}\n\\end{equation}\n\nWe have to distinguish several cases:\n\\begin{itemize}\n\\item[a)] Case $0\\leq s\\leq u\\leq t\\leq r$.\n \nSince on $\\Delta_r$ it holds that\n$$(x^{(1)}_{\\cdot-r},y,x^{(1)}_{\\cdot-r}\\otimes y)=(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y),$$ \nthe multiplicative property follows from the fact that we are assuming that $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)$ \n is a $\\beta-$ H\\\"older continuous functional.\n\n\\item[b)] Case $r\\leq s\\leq u\\leq t\\leq 2r$. \n\nNotice first that,\n\\begin{equation*}\n\\begin{array}{l}\n\\displaystyle (x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}=\\int_s^u (y(u)-y(v))b(v-r,x^{(1)})dv\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_{s}^{u} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)+\\int_s^u(y(u)-y(v))dz^{(1)}_{v-r}\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_u^t (y(t)-y(v))b({v-r},x^{(1)})dv+\\int_{u}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_u^t(y(t)-y(v))dz^{(1)}_{v-r}\\\\[4mm]\n\\displaystyle \\qquad =\\int_s^t (y(t)-y(v))b({v-r},x^{(1)})dv+\\int_{s}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\\\[4mm]\n\\displaystyle \\qquad\\quad +\\int_s^t(y(t)-y(v))dz^{(1)}_{v-r}\\\\[4mm]\n\\displaystyle \\qquad\\quad+(y(u)-y(t))\\left(\\int_s^u b({v-r},x^{(1)})dv+z^{(1)}(u-r)-z^{(1)}(s-r)\\right)\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_{s}^{u} \\si(\\eta(v-2r))(d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)-d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)).\n\\end{array}\n\\end{equation*}\nSo\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle (x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}\\\\[4mm]\n\\displaystyle \\qquad =(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}+(y(u)-y(t))\\left(\\int_s^u b({v-r},x^{(1)})dv+z^{(1)}(u-r)-z^{(1)}(s-r)\\right)\n\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_{s}^{u} \\si(\\eta(v-2r))(d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)-d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v))\n\\end{array}\\label{xxa}\n\\end{equation}\nOn the other hand, from Definition \\ref{d2.1} we obtain that\n\\begin{equation}\n\\int_{s}^{u} \\si(\\eta(v-2r))(d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)-d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v))=(y(u)-y(t))\\int_{s}^{u} \\si(\\eta(v-2r))dy_{v-r}.\n\\label{xxb}\n\\end{equation}\nFinally, using that\n\\begin{eqnarray*}\n\\int_s^u b({v-r},x^{(1)})dv&=&\\int_{s-r}^{u-r} b({v},x^{(1)})dv,\\\\\n\\int_{s}^{u} \\si(\\eta(v-2r))dy_{v-r}&=&\\int_{s-r}^{u-r} \\si(\\eta(v-r))dy_{v},\n\\end{eqnarray*}\nand putting together (\\ref{xxa}) and (\\ref{xxb}) we get the multiplicative property (\\ref{s2}).\n\n\\item[c)] Case $0\\leq s\\leq r$ and $r\\leq u\\leq t\\leq 2r$.\\\\\nNotice first that from the definition of $(x^{(1)}_{\\cdot-r}\\otimes y)$ it follows that\n\\begin{equation}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}=(\\eta_{\\cdot-r}\\otimes y)_{s,r}+(x^{(1)}_{\\cdot-r}\\otimes y)_{r,u}+(\\eta(0)-\\eta(s-r))\\otimes (y(u)-y(r)).\n\\label{bba}\n\\end{equation}\nOn the other hand, we have seen in the case b) (choosing $s=r$) that\n\\begin{equation}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}=(x^{(1)}_{\\cdot-r}\\otimes y)_{r,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}+(x^{(1)}(u-r)-\\eta(0))\\otimes (y(t)-y(u)). \n\\label{bbb}\n\\end{equation}\nSo, putting together (\\ref{bba}) and (\\ref{bbb}) we can write\n\\begin{equation*}\n\\begin{array}{l}\n\\displaystyle (x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}+(x^{(1)}(u-r)-\\eta(s-r))\\otimes (y(t)-y(u))\\\\[4mm]\n\\displaystyle \\qquad =(\\eta_{\\cdot-r}\\otimes y)_{s,r}+(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}+(\\eta_0-\\eta_{s-r})\\otimes (y_t-y_r)\\\\[4mm]\n\\displaystyle \\qquad=(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t},\n\\end{array}\n\\end{equation*}\nwhere the last equality follows for the definition of $(x^{(1)}_{\\cdot-r}\\otimes y)$. The proof of this case is now finished.\n\\item[d)] Case $0\\leq s\\leq u\\leq r$ and $r\\leq t\\leq 2r$.\n\nThis case can be done following the same ideas that the case c).\n\\end{itemize}\n\n\\item Now only remains to prove that $|(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}|\\leq C|t-s|^{2\\beta}$. We will distinguish again three cases:\\\\\n\n\\begin{enumerate}\n\n\\item Assume that $s,t\\in[r,2r]$. Then\n\\begin{eqnarray*}\n|(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}|&\\leq& \\left|\\int_s^t(y(t)-y(v))b({v-r},x^{(1)})dv\\right|+\\left|\\int_{s}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\right|\\\\ &&+\\left|\\int_s^t(y(t)-y(v))dz^{(1)}_{v-r}\\right|.\n\\end{eqnarray*}\nSince $y$ is $\\beta-$H\\\"older continuous function, we have that\n\\[\\left|\\int_s^t (y(t)-y(v))dz^{(1)}_{v-r}\\right|\\leq K|t-s|^\\beta|z^{(1)}(t-r)-z^{(1)}(s-r)|,\\]\nfor a constant $K$. \nThen, using Lemma \\ref{le2} we get\n\\begin{equation}\n\\left|\\int_s^t (y(t)-y(v))dz^{(1)}_{v-r}\\right|\\leq K|t-s|^{2\\beta}. \\label{s3}\n\\end{equation}\nOn the other hand, using the hypothesis on $b$ we have\n\\begin{eqnarray}\n& &\\left|\\int_s^t(y(t)-y(v))b(v-r,x^{(1)})dv\\right| \\leq K|t-s|^\\beta\\left|\\int_s^t(L_0\\sup_{-r\\leq u\\leq v-r}|x^{(1)}(u)|+b_0(v))dv\\right|\\nonumber\\\\\n& & \\qquad \\leq K|t-s|^{\\beta+1}\\left\\|x^{(1)}\\right\\|_{\\infty(-r,r)}+|t-s|^{\\beta+1-\\frac{1}{\\rho}}\\left\\|b_0\\right\\|_{L^\\rho}. \\label{s4}\n\\end{eqnarray}\nFinnally using Proposition \\ref{prop24} we get\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle \n\\left|\\int_{s}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\right|\\leq k|\\sigma(\\eta(s-2r))|\\Phi_{s,t,\\beta}(y_{\\cdot-r},y)(t-s)^{2\\beta}\\\\[4mm]\n\\displaystyle \\qquad\\quad+k\\left(\\left\\|\\sigma'\\right\\|_\\infty+\\left\\|\\sigma'\\right\\|_\\gamma\\left\\|\\eta_{.-2r}\\right\\|^\\gamma_{\\beta(s,t)}(t-s)^{\\gamma\\beta}\\right)\\Phi_{s,t,\\beta}(\\eta_{\\cdot-2r},y_{\\cdot-r},y)(t-s)^{3\\beta},\n\\end{array} \\label{s5}\n\\end{equation}\nwhere \n\\begin{eqnarray*}\n\\Phi_{a,b,\\beta}(x,y,z)&=&\\left\\|y\\right\\|_{\\beta(a,b)}\\left\\|z\\right\\|_{\\beta(a,b)} \\left\\|x\\right\\|_{\\beta(a,b)} \\\\\n&& +\\left\\|z\\right\\|_{\\beta(a,b)}\\left\\|x\\otimes y\\right\\|_{\n2\\beta(a,b)}+\\left\\|x\\right\\|_{\\beta(a,b)}\\left\\|y\\otimes z\\right\\|_{2\\beta(a,b)}.\n\\end{eqnarray*}\n\\vskip 5pt\n\\noindent\nNow putting together (\\ref{s3}), (\\ref{s4}) and (\\ref{s5}) we finish the proof.\n\\vskip 5pt\n\\noindent\n\\item If $s\\in[0,r]$ and $t\\in[r,2r]$,\n\\begin{eqnarray*}\n|(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}|&\\leq&|(\\eta_{\\cdot-r}\\otimes y)_{s,r}|+|(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}|+|(\\eta(0)-\\eta(s-r))\\otimes (y(t)-y(r))|\\\\\n&\\leq& K|r-s|^{2\\beta}+K|t-r|^{2\\beta}+K|s-r|^\\beta|t-r|^\\beta\\leq K|t-s|^{2\\beta}.\n\\end{eqnarray*}\n\\item If $s,\\,t\\in[0,r]$ then $x^{(1)}_{\\cdot-r}=\\eta_{\\cdot-r}$ and the result is already true.\n\\end{enumerate}\n\\end{enumerate}\n\\hfill $\\Box$\n\n\\renewcommand{\\theequation}{5.\\arabic{equation}} \\setcounter{equation}{0}\n\n\\section{Boundedness for deterministic integral equations}\n\nThe aim of this section is the proof of Theorem \\ref{tt2}. For simplicity let us assume $T=Mr$.\n\n\n\\vskip 5pt\n\\noindent\n{\\bf Proof of Theorem \\ref{tt2}:}\nThe proof will be done in several steps.\n\\vskip 5pt\n\\noindent\n{\\underline{Step 1:}} Assuming that $(x,y,x\\otimes y)\\in M^\\beta_{d,m}(0,T)$, let us define \n$\\left(x_{\\cdot-r}\\otimes y_{\\cdot-r}\\right)_{s,t}$. Set\n\\begin{equation}\\label{e5.1biss}\\left(x_{\\cdot-r}\\otimes y_{\\cdot-r}\\right)_{s,t}:=\\left(x\\otimes y\\right)_{s-r,t-r}.\\end{equation}\nIt clearly t belongs to $M^\\beta_{d,m}(r,T)$. Notice that the functions $x_{\\cdot-r}$ and $y_{\\cdot-r}$ are $\\beta$-H\\\"older\ncontinuous and $x_{\\cdot-r}\\otimes y_{\\cdot-r}$ is a continuous functions satisfying the multiplicative property. Indeed, we have, for \n$s\\leq u\\leq t$, \n\\begin{equation*}\\begin{array}{l}\n\\displaystyle (x_{\\cdot-r}\\otimes y_{\\cdot-r})_{s,u}+(x_{\\cdot-r}\\otimes y_{\\cdot-r})_{u,t}+(x(u-r)-x(s-r))\\otimes\n(y(t-r)-y(u-r))\\\\\n\\qquad \\displaystyle \n=(x\\otimes y)_{s-r,u-r}+(x\\otimes y)_{u-r,t-r}+(x(u-r)-x(s-r))\\otimes\n(y(t-r)-y(u-r))\\\\\n\\qquad \\displaystyle =(x\\otimes y)_{s-r,t-r} =(x_{\\cdot-r}\\otimes y_{\\cdot-r})_{s,t}.\n\\end{array}\\end{equation*}\nFinally, we also have that,\nfor all $(s,t)\\in \\{(s,t): r\\le s0$, $x_0\\in X$ such that if $B_0=\\left\\{x\\in X; \\rho_0(x_0,x)\\leq r_0\\right\\}$ then $\\cl(B_0)\\subseteq B_0$,\n\\item There exists $a\\in (0,1)$ such that $\\rho_1\\left(\\cl(x),\\cl(y)\\right)\\leq a\\rho_1(x,y)$ for all $x,y\\in B_0$.\n\\end{enumerate}\nThen there exists $x^*\\in\\cl(B_0)\\subseteq X$ such that $x^*=\\cl(x^*)$.\n\\end{lemma}\n\nWe also need a result with some properties of the solution of Skorohod's problem.\n\n\\begin{lemma} \\label{le2}\nFor each path $\\xi\\in \\cc(\\R_+,\\R^d)$, there exists a unique solution $(x,z)$ to the Skorokhod problem for $\\xi$. Thus there exists a pair of functions ${\\displaystyle (\\phi,\\varphi): \\cc_+(\\R_+,\\R^d)\\rightarrow\\cc_+(\\R_+,\\R^{2d})}$ defined by $\\left(\\phi(\\xi),\\varphi(\\xi)\\right)=(x,z)$. The pair $\\left(\\phi,\\varphi\\right)$ satisfies the following:\n\\vskip 5pt\nThere exists a constant $K_l>0$ such that for any $\\xi_1,\\xi_2\\in\\cc_+(\\R_+,\\R^d)$ we have for each $t\\geq 0$,\n\\begin{eqnarray*}\n\\left\\|\\phi(\\xi_1)-\\phi(\\xi_2)\\right\\|_{\\infty(0,t)}&\\leq& K_l\\left\\|\\xi_1-\\xi_2\\right\\|_{\\infty(0,t)},\\\\\n\\left\\|\\varphi(\\xi_1)-\\varphi(\\xi_2)\\right\\|_{\\infty(0,t)}&\\leq& K_l\\left\\|\\xi_1-\\xi_2\\right\\|_{\\infty(0,t)}.\n\\end{eqnarray*}\nMoreover for each $0\\leq sz^i(u)$, let us define\n\\begin{eqnarray*}\nu^*&:=&\\sup\\{ u'\\ge u; z^i(u)=z^i(u') \\},\\\\\nv^*&:=&\\inf\\{ v' \\le v; z^i(v)=z^i(v') \\}.\n\\end{eqnarray*}\nThen, $u \\le u^* < v^* \\le v$ and $z^i(u)=z^i(u^*), z^i(v)=z^i(v^*)$. So\n$$ \\frac{\\vert z^i(v)-z^i(u) \\vert}{(v-u)^\\beta} \\le \\frac{\\vert z^i(v^*)-z^i(u^*) \\vert}{(v^*-u^*)^\\beta} =\n\\frac{\\vert \\xi^i(v^*)-\\xi^i(u^*) \\vert}{(v^*-u^*)^\\beta} $$\nwhere the last equality follows from the fact that\n$\\xi^i$ and $z^i$ coincides whenever $z^i$ is not constant.\n\nThen, note that\n$$\n\\sup_{s < u 10^{10^6}$ and $k$ is even. His result has since then been improved on.\nButske et al.~\\cite{graphs} have shown by computing, rather\nthan estimating, certain quantities in Moser's original proof that $m>1.485\\cdot 10^{\\,9\\,321\\,155}$.\nBy proceeding along these lines this bound cannot be improved on substantially.\nButske et al.~\\cite[p. 411]{graphs} expressed\nthe hope that new insights will eventually make it possible to reach the more natural benchmark\n$10^{10^7}$.\n\nUsing that $\\Sigma_k(m)=1^k+2^k+\\dots+m^k\\le\\int_1^m t^k \\d t$\nand $\\Sigma_k(m+1)>\\int_0^m t^k\\d t$ we obtain that $k+15.7462\\cdot 10^{427}.\n\\end{align*}\nFor some further references and info on the Erd\\H{o}s--Moser equation we refer to the book\nby Guy~\\cite[D7]{Guy}.\n\nIn this note we attack \\eqref{EME} using the theory of\ncontinued fractions. This approach was first explored in~1976 by Best and\nte Riele \\cite{Best} in their attempt to solve the related conjecture of\nErd\\H{o}s~\\cite{E} that there are infinitely many pairs $(m,k)$ such that\n$\\Sigma_k(m)\\ge m^k$ and $2(m-1)^k0$ and \\emph{real} $k>0$ satisfying equation~\\eqref{EME},\nwe have the asymptotic expansion\n\\begin{equation}\n\\label{eq:2}\nk=\\log2\\biggl(m-\\frac32-\\frac{c_1}m+O\\biggl(\\frac 1{m^2}\\biggr)\\biggr)\n\\quad\\text{as}\\; m\\to\\infty,\n\\end{equation}\nwith $c_1=\\frac{25}{12}-3\\log 2\\approx 0.00389\\dots$\\,.\nMoreover, if $m>10^9$ then\n\\begin{equation}\n\\frac km=\\log2\\biggl(1-\\frac3{2m}-\\frac {C_m}{m^2}\\biggr),\n\\qquad\\text{where}\\quad 01$ is odd}; \\\\\n-\\sum_{p\\mid l, \\, p-1\\mid r}\\frac1p\\pmod1 &\\text{otherwise}.\n\\end{cases}\n\\end{equation}\nThis identity can be proved using the Von Staudt--Clausen theorem; for\nalternative proofs see, e.g., Carlitz \\cite{Carlitz} or Moree \\cite{Canada}.\nIts relevance for the study of \\eqref{EME} was first pointed out by Moree~\\cite{Oz}.\n\nGiven $N\\ge 1$, put\n$$\n{\\mathcal{P}}(N)=\\{p:p-1\\mid N\\}\\cup\\{p:\\text{$3$ is a primitive root modulo $p$}\\}.\n$$\nBy a classical result of Hooley \\cite{Hooley} it follows, assuming the\nGeneralized Riemann Hypothesis (GRH), that ${\\mathcal{P}}(N)$ has\na natural density $A$, with $A=0.3739558136\\dots$ the Artin constant, in the set of primes.\nIf $2k\/(2m-3)=p_j\/q_j$ is a convergent of $\\log 2$ arising in Corollary~\\ref{cor:1}, then it can be shown that\n$(q_j,6)=1$ and,\nif $p\\in {\\mathcal{P}}(N_2)$ and $p$ divides $q_j$, then\n$$\n\\nu_p(q_j)=\\nu_p(3^{p-1}-1)+1\\ge 2,\n$$\nwhere we write $\\nu_p(n)=a$ if $p^a\\mid n$ and $p^{a+1}\\nmid n$.\nAll primes $p\\le 2017$ are in ${\\mathcal{P}}(N_2)$. For $p\\ne 3$ we have $\\nu_p(3^{p-1}-1)=1$ unless\n$3^{p-1}\\equiv 1\\pmod{p^2}$, that is, $p$~is a Mirimanoff prime. (It is known that the only\nMirimanoff primes $p<10^{14}$ are $11$ and $1006003$.)\n\nThe main idea of this paper is, in essence, to make use of the fact that the convergents\n$p_j\/q_j$ of $\\log 2$ have no reason to also satisfy $N_2\\mid p_j$. The first piece of information\ncomes from asymptotic analysis and the latter piece from arithmetic. Analysis and\narithmetic give rise to conditions on the solutions that `do not feel\neach other' and this is exploited in our main result:\n\n\\begin{Thm}\n\\label{main}\nLet $N\\ge 1$ be an arbitrary integer. Let\n$$\\frac{\\log 2}{2N} = [a_0,a_1,a_2,\\dots] = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cdots}}$$\nbe the (regular) continued fraction of\n$({\\log 2})\/(2N)$, with $p_i\/q_i = [a_0,a_1,\\dots,a_i]$ its $i$-th partial convergent.\n\nSuppose that the integer pair $(m,k)$ with $k\\ge 2$ satisfies \\eqref{EME}\nwith $N\\mid k$.\nLet $j=j(N)$ be the smallest integer such that:\n\\begin{itemspec}\n\n\\itemindent18pt\n\\item\n\\label{ca}\n$j$ is even;\n\n\\item\n\\label{cb}\n$a_{j+1}\\ge 180N-2$;\n\n\\item\n\\label{cc}\n$(q_j,6)=1$; and\n\n\\item\n\\label{cd}\n$\\nu_p(q_j)=\\nu_p(3^{p-1}-1)+\\nu_p(N)+1$\nfor all primes $p\\in {\\mathcal{P}}(N)$ dividing $q_j$.\n\\end{itemspec}\nThen $m>q_j\/2$.\n\\end{Thm}\n\nComputing many partial quotients (that is, continued fraction digits)\nof $\\log 2$ is closely related to computing $\\log 2$ with\nmany digits of accuracy. Indeed, it is a well-known result of Lochs that for a generic number knowing\nit accurately up to $n$ decimal digits implies that we can compute about $0.97n$\n(where $0.97\\approx6(\\log 2)(\\log 10)\/\\pi^2$)\ncontinued fraction digits accurately. For example, knowing 1000 decimal digits of~$\\pi$ allows one\nto compute 968 continued fraction digits.\n\nIt seems a hopeless problem to prove anything about ${\\mathsf E}(\\log q_{j(N)})$, the\nexpected value of $\\log q_{j(N)}$ produced by the result. However, metric theory\nof continued fractions offers some hope of proving a non-trivial lower bound for\n${\\mathsf E}(\\log q_{j(N)}(\\xi))$,\nwhere we require conditions \\ref{ca}, \\ref{cb}, \\ref{cc} and~\\ref{cd} to be satisfied but replace\n$(\\log 2)\/(2N)$ by a `generic' $\\xi\\in [0,1]\\setminus \\mathbb Q$.\nIn this context recall the result of L\\'evy~\\cite{L} that, for such a $\\xi$,\n\\begin{equation}\n\\lim_{j\\to\\infty}\\frac{\\log q_j(\\xi)}j=\\frac{\\pi^2}{12\\log 2}\\approx1.18.\n\\label{Levy}\n\\end{equation}\nThe Gauss--Kuz'min statistics asserts that,\nfor a generic~$\\xi$, the probability that a given term\nin its continued fraction expansion is at least~$b$,\nequals $\\log_2(1+1\/b)$. This allows one to deal with the case where we only have\ncondition~\\ref{cb}. Likewise a result of Moeckel~\\cite{Moeckel},\nreproved in a very different way a few years later by Jager and\nLiardet \\cite{JL}, allows one to deal with the case\nwhere we only focus on condition~\\ref{cc}. Their result says\nthat for a generic $\\xi\\in[0,1]\\setminus \\mathbb Q$ we have\n$$\n\\lim_{n\\to\\infty}\\frac{\\{1\\le m\\le n:q_m(\\xi)\\equiv a\\pmod{d}\\}}{n}\n=\\frac d{J(d)}\\,\\frac{\\varphi((a,d))}{(a,d)},\n$$\nwhere $\\varphi$ denotes Euler's totient function, $J(m)=m^2\\prod_{p\\mid m}(1-1\/p^2)$ Jordan's\ntotient and $(a,m)$ the greatest common divisor of $a$ and~$m$.\nThis result shows that $(q_j,6)=1$ with probability $1\/2$\n(note that a natural number is coprime to~6 with probability $1\/3$).\nP.~Liardet communicated\nto us that methods of his paper \\cite{Liardet} can be used to take into account\nboth conditions \\ref{ca} and~\\ref{cc};\nalso the authors of \\cite{Harman} claim that this can be done.\nWe expect that there is a positive constant $c_1$ such that for a generic\n$\\xi$ satisfying conditions\n\\ref{ca}, \\ref{cb} and \\ref{cc}, we have ${\\mathsf E}(\\log q_{j(N)}(\\xi))\\sim c_1N$ as $N$ tends to\ninfinity. Furthermore, we expect that for a generic $\\xi$ satisfying conditions\n\\ref{ca}, \\ref{cb}, \\ref{cc} and \\ref{cd},\n${\\mathsf E}(\\log q_{j(N)}(\\xi))\\sim c_2 N\\log^{\\beta} N$\nfor some positive constants $c_2$ and $\\beta$; condition~\\ref{cb}\nis responsible for $N$, condition~\\ref{cd} for $\\log^{\\beta}N$, while conditions \\ref{ca} and~\\ref{cc}\naffect $c_2$. We are definitely not experts in metric aspects of number theory,\nthus leave this problem to the interested reader acquainted with the subject.\nIndeed, we even expect that going beyond computing the expected value of $\\log q_{j(N)}(\\xi)$ is\npossible, and a probability distribution for $\\log q_{j(N)}(\\xi)$ can be obtained.\n\nUsing the above results from metric theory of continued fractions and some heuristics we are led\nto believe that roughly speaking we can get\n$$\nm>10^{257N}\n$$\nfrom Theorem~\\ref{main}. Being able to compute the convergents of $(\\log 2)\/(2N)$ arbitrarily far,\nwe would expect (taking $N=N_2$) to show that $m>10^{10^{400}}$.\nWith the current computer technology computing sufficiently many convergents is the bottleneck.\nTaking this into consideration we would expect to get\n$$\nm>10^{0.515 r},\n$$\nfrom Theorem~\\ref{main}, where $r$ is the number of convergents we can compute accurately\nand $0.515$ is the base~10 logarithm of L\\'evy's constant~\\eqref{Levy}.\nNote that the fact that $N_2$ has many divisors gives us some flexibility and increases\nthe likelihood of the heuristics to be applicable. Indeed, our numerical experimenting agrees well\nwith our heuristic considerations (see Section~\\ref{s4}). Early 2009, A.~Yee and R.~Chan \\cite{AJY}\nreached $r>31\\cdot10^9$ for~$\\log 2$. On the other hand,\nY.~Kanada and his team~\\cite{Kanada} computed $\\pi$ to over 1.24 trillion decimal\ndigits already in 2002, using formulae of the same complexity as those used\nfor the computation of $\\log2$ (see \\cite[Chapter~3]{BB} for details).\nThus, given the present computer (im)possibilities,\none could hope to show (with a lot of effort!) that $m>10^{10^{12}}$.\n\nApplying Theorem~\\ref{main} with $N=2^8\\cdot 3^5\\cdot 5^3$\nor $N=2^8\\cdot 3^5\\cdot 5^4$, and invoking the result of\nMoree et al.~\\cite{MRU} that $N\\mid k$, we obtain the following\n\n\\begin{Thm}\n\\label{YVES}\nIf an integer pair $(m,k)$ with $k\\ge 2$ satisfies \\eqref{EME}, then\n$$m > 2.7139 \\cdot 10^{\\,1\\,667\\,658\\,416}.$$\n\\end{Thm}\n\nAs an application we can show that $\\omega(m-1)\\ge 33$,\nthis improves on the result of Brenton and Vasiliu \\cite{BV},\nwho have shown that $\\omega(m-1)\\ge 26$, where $\\omega$\ndenotes the number of distinct prime divisors; see Section~\\ref{s5.1} for further details.\n\nThe fact $N_2\\mid k$ naively implies that $k$~is of size $10^{427}$ (at least),\nwhich is much smaller than Moser's $10^{10^6}$. However, in this\npaper we show that the fact actually yields that $k>10^{10^9}$ (and likely even\n$k>10^{10^{400}}$)\\,---\\,a modestly {\\small small} number dividing~$k$\nleads to a huge lower bound for~$k$.\nThus, on revisiting \\cite{MRU} after 16~years, its main result is seen\nto be far more powerful than the second author thought at that time.\n\nIn the three following sections we prove Theorems~\\ref{th:As},\n\\ref{main} and \\ref{YVES}, respectively.\nOur final Section~\\ref{s5} is devoted to discussing some problems\nrelated to the Erd\\H os--Moser equation.\n\n\\section{Asymptotic dependence of $k$ in terms of $m$}\n\\label{sec:As}\n\nOur proof of Theorem~\\ref{th:As} makes use of the following lemma.\n\\begin{Lem}\n\\label{lem:As}\nFor any real $k>0$, we have\n\\begin{equation}\n(1-y)^k=e^{-ky}\\biggl(1-\\frac k2y^2-\\frac k3y^3\n+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5+O(y^6)\\biggr)\n\\quad \\text{as}\\; y\\to0.\n\\label{E02}\n\\end{equation}\nMoreover, for $k>8$ and $02$ and $03$ and $01-kx+\\frac{k(k-1)}2x^2-\\frac{k(k-1)(k-2)}6x^3.\n\\label{E07}\n\\end{equation}\n\nUsing the right inequality in~\\eqref{E04} and taking $x=x_1$ in~\\eqref{E06} we obtain, for $k>2$,\n\\begin{align}\n(1-y)^ke^{ky}\n&<1-k\\biggl(\\frac{y^2}2+\\frac{y^3}3+\\frac{y^4}8\\biggr)\n+\\frac{k(k-1)}2\\biggl(\\frac{y^2}2+\\frac{y^3}3+\\frac{y^4}8\\biggr)^2\n\\nonumber\\\\\n&=1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4\n\\nonumber\\\\ &\\qquad\n+k(k-1)y^5\\biggl(\\frac16+\\frac{17}{144}y+\\frac1{24}y^2+\\frac1{128}y^3\\biggr)\n\\nonumber\\\\\n&<1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4\n+\\frac{385}{1152}k(k-1)y^5\n\\label{E08}\n\\end{align}\nimplying the upper estimate in~\\eqref{E03}.\nIn the same vein, the application of the left identity in~\\eqref{E04} and of~\\eqref{E07}\nwith $x=x_2$ results, for $k>3$, in\n\\begin{align}\n(1-y)^ke^{ky}\n&>1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5\n\\nonumber\\\\ &\\qquad\n-ky^6\\sum_{n=0}^{12}(a_nk^2+b_nk+c_n)y^n,\n\\label{E09}\n\\end{align}\nwhere the polynomials $p_n(k)=a_nk^2+b_nk+c_n$, $n=0,1,\\dots,12$, all have\npositive leading coefficients $a_n$; moreover, $p_n(k)>0$ for $k>3$ and\n$n=2,3,\\dots,12$, \\ $p_1(k)=\\frac1{24}k^2-\\frac{11}{60}k+\\frac{17}{120}>0$\nfor $k>4$, and $p_0(k)=\\frac1{48}k^2-\\frac{13}{72}k+\\frac{121}{720}>0$\nfor $k>8$. Using this positivity of the polynomials we can continue\nthe inequality in~\\eqref{E09} for $k>8$ as follows:\n\\begin{align}\n(1-y)^ke^{ky}\n&>1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5\n\\nonumber\\\\ &\\qquad\n-ky^6\\sum_{n=0}^{12}(a_nk^2+b_nk+c_n)\n\\nonumber\\\\\n&=1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5\n\\nonumber\\\\ &\\qquad\n-ky^6\\biggl(\\frac16k^2-\\frac{17}{24}k+\\frac{11}{20}\\biggr),\n\\label{E10}\n\\end{align}\nfrom which we deduce the left inequality in~\\eqref{E03}, and the lemma\nfollows.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~{\\rm\\ref{th:As}}]\nThe original equation~\\eqref{EME} is equivalent to\n\\begin{equation}\n\\label{eq:1}\n1=\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k.\n\\end{equation}\nApplying to each term on the right-hand side the inequality\nfrom~\\eqref{E03} we obtain\n\\begin{align}\n&\nS_0-\\frac k{2m^2}S_2-\\frac k{3m^3}S_3+\\frac{k(k-2)}{8m^4}S_4+\\frac{k(5k-6)}{30m^5}S_5-\\frac{k^3}{6m^6}S_6\n\\nonumber\\\\ &\\qquad\n<\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k\nm\/2>30$, using our equation \\eqref{eq:1} we can write\nthe estimates~\\eqref{E12} as\n\\begin{align}\n&\n\\frac{k(5k-6)}{30m^5}S_5'-\\frac{k^3}{6m^6}S_6'-\\frac1{m^3}\n\\nonumber\\\\ &\\qquad\n<1-S_0'+\\frac k{2m^2}S_2'+\\frac k{3m^3}S_3'-\\frac{k(k-2)}{8m^4}S_4'\n<\\frac{k^2}{2m^5}S_5'+\\frac1{m^3}.\n\\label{E13}\n\\end{align}\nNoting that $e^{1\/2}0.005m^{-2}-100m^{-3}\n\\quad\\text{and}\\quad\nf_m(0.004)<-0.00015m^{-2}+100m^{-3}\n$$\nfor $m\\ge100$. Therefore,\n$f_m(0)>110000\/m^3$ for $m>2202\\cdot10^4$ and\n$f_m(0.004)<-110000\/m^3$ for $m>734\\cdot10^6$,\nso that $|f_m(C)|<110000\/m^3$ is possible only\nif $010^9$ satisfying \\eqref{eq:1}, we necessarily\nhave\n$$\n\\frac km=\\log2\\biggl(1-\\frac3{2m}-\\frac{C_m}{m^2}\\biggr)\n$$\nwith $010^9$. It follows from Theorem~\\ref{th:As} that\n\\begin{equation}\n\\label{drie}\n0<\\log 2-\\frac{2k}{2m-3} < \\frac{0.0111}{(2m-3)^2}.\n\\end{equation}\nBy Legendre's theorem, $|\\log2-p\/q|<1\/(2q^2)$ implies that $p\/q$ is a convergent\nof~$\\log2$, while $\\log2>p\/q$ insures that the index of the convergent is even.\nThus, $2k\/(2m-3)$ is a convergent $p_j\/q_j$ of the continued fraction of $\\log 2$\nwith $j$~even.\n\\end{proof}\n\n\\section{The proof of the main theorem}\n\\label{s3}\n\nIn this section we prove Theorem~\\ref{main}. The restrictions on the prime factorization\nof~$q_j$ in that result are established using an argument in the style of Moser given\nin the proof of the following lemma.\n\n\\begin{Lem}\n\\label{twee}\nLet $(m,k)$ be a solution of {\\rm\\eqref{EME}} with $k\\ge 2$. Let $p$ be a prime\ndivisor of $2m-3$. If $p-1\\mid k$, then\n$$\n\\nu_p(2m-3)=\\nu_p(3^{p-1}-1)+\\nu_p(k)+1\\ge 2.\n$$\nIf $3$~is a primitive root modulo $p$, then $p-1\\mid k$.\n\\end{Lem}\n\n\\begin{proof}\nUsing that $k$ must be even, we find that\n\\begin{align*}\n\\sum_{j=1}^{2m-4}j^k\n& \\equiv \\sum_{j=1}^{m-1}j^k+\\sum_{j=1}^{m-3}(2m-3-j)^k\n\\equiv \\sum_{j=1}^{m-1}j^k+\\sum_{j=1}^{m-3}j^k\\pmod{2m-3}\n\\\\\n& \\equiv m^k+m^k-(m-1)^k-(m-2)^k\n\\equiv 2(3^k-1)(m-1)^k\\pmod{2m-3},\n\\end{align*}\nwhere we used that $m^k\\equiv (2m-3+m)^k\\equiv 3^k(m-1)^k\\pmod{2m-3}$ and\n$(m-2)^k\\equiv (2m-3-m+1)^k\\equiv (m-1)^k\\pmod{2m-3}$.\nOn applying \\eqref{staudt} with $l=2m-3$ and $r=k$ we then obtain that\n\\begin{equation}\n\\label{sta}\n\\frac{2(3^k-1)(m-1)^k}{2m-3}\\equiv -\\sum_{\\substack{p\\mid 2m-3\\\\p-1\\mid k}}\\frac1p\\pmod1.\n\\end{equation}\nIf $p\\mid 2m-3$ and $p-1\\mid k$, the $p$-order of the right-hand side is~$-1$. The $p$-order of\nthe left-hand side must also be $-1$, that is, we must have\n$$\n\\nu_p(2m-3)=\\nu_p(3^k-1)+k\\nu_p(m-1)+1=\\nu_p(3^{p-1}-1)+\\nu_p(k)+1,\n$$\nwhere we used that $m-1$ and $2m-3$ are coprime.\nNow suppose that $p\\mid 2m-3$ and $3$ is a primitive root modulo $p$ (thus $p\\mid 3^k-1$ implies\n$p-1\\mid k$). If $p-1\\nmid k$, the $p$-order of the left-hand side is $\\le -1$ and $>-1$ on the\nright-hand side. Thus, we infer that $p-1\\mid k$.\n\\end{proof}\n\nThis completes the required ingredients needed in order to prove the main result.\n\n\\begin{proof}[Proof of Theorem {\\rm\\ref{main}}]\nSince by assumption $N\\mid k$, we can write $k=Nk_1$ and thus rewrite \\eqref{drie} as\n\\begin{equation}\n\\label{E}\n0<\\frac{\\log 2}{2N}-\\frac{k_1}{2m-3}<\\frac{0.0111}{2N(2m-3)^2}.\n\\end{equation}\nWe infer that $k_1\/(2m-3)=p_j\/q_j$ is a convergent to\n$(\\log 2)\/(2N)$ with $j$~even.\nSince $p\\mid m$ implies $p-1\\nmid k$ (see, e.g., Moree \\cite[Proposition 9]{Oz}),\nwe have $(6,q_j)=1$.\nWe rewrite~\\eqref{E} as\n$$\n0<\\frac{\\log 2}{2N}-\\frac{p_j}{q_j}<\\frac{0.0111}{2Nd^2q_j^2},\n$$\nwith $d$ the greatest common divisor of $k_1$ and $2m-3$.\nOn the other hand,\n$$\n\\frac{\\log 2}{2N}-\\frac{p_j}{q_j}>\\frac1{(a_{j+1}+2)q_j^2},\n$$\nhence $(a_{j+1}+2)^{-1}<0.0111\/(2Nd^2)$, from which the\nresult follows on also noting that $2m-3\\ge q_j$ and invoking Lemma~\\ref{twee}\n(note that if $\\nu_p(q_j)\\ge 1$, then $\\nu_p(q_j)=\\nu_p(2m-3)-\\nu_p(k_1)$).\n\\end{proof}\n\nTo prove that $p\\mid m$ implies $p-1\\nmid k$ one uses that $k$ must be even\nand takes $l=m$ in~\\eqref{staudt}, showing that $\\sum_{p\\mid m,\\,p-1\\mid k}\\frac1p$ must\nbe an integer. Since a sum of reciprocals of distinct primes can never be an integer,\nthe result follows.\n\n\\section{Computation of the continued fractions}\n\\label{s4}\n\nWe make use of conditions \\ref{ca}, \\ref{cb}, \\ref{cc} of Theorem~\\ref{main}. We recall that\nwe expect ${\\mathsf E}(\\log q_{j(N)}(\\xi))\\sim c_1N$ for a generic $\\xi\\in [0,1]$ satisfying\nthese conditions.\nIndeed, on the basis of theoretical results, heuristics and numerical experiments, we conjecture\nthat $c_1=60\\pi^2$.\n\n\\begin{table}[ht\n\\begin{center}\n\\begin{tabular}{|c|r|r|l|c|c|}\n\\hline\n\\multicolumn{1}{|c|}{\\T $N$} & \\multicolumn{1}{|c|}{$j=j(N)$} & \\multicolumn{1}{|c|}{$a_{j+1}$} &\n\\multicolumn{1}{|c|}{$q_j$ \\ (rounded down)} & \\multicolumn{1}{|c|}{\\small $q_j \\bmod 6$} & \\multicolumn{1}{|c|}{$p = p(q_j)$} \\\\\n\\hline\n\\T $1$ & $642$ & $764$ & $2.383153 \\cdot 10^{\\,330}$ & $-1$ & $149$ \\\\\n\\hline\n\\T $2$ & $664$ & $1\\,529$ & $2.383153 \\cdot 10^{\\,330}$ & $-1$ & $149$ \\\\\n\\hline\n\\T $2^2$ & $1\\,254$ & $21\\,966$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^3$ & $1\\,264$ & $43\\,933$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^4$ & $1\\,280$ & $87\\,866$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^5$ & $1\\,294$ & $175\\,733$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^6$ & $8\\,950$ & $26\\,416$ & $3.458446 \\cdot 10^{\\,4\\,589}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^7$ & $8\\,926$ & $52\\,834$ & $3.458446 \\cdot 10^{\\,4\\,589}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8$ & $119\\,476$ & $122\\,799$ & $1.374540 \\cdot 10^{\\,61\\,317}$ & $+1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3$ & $119\\,008$ & $368\\,398$ & $1.374540 \\cdot 10^{\\,61\\,317}$ & $+1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^2$ & $139\\,532$ & $782\\,152$ & $9.351282 \\cdot 10^{\\,71\\,882}$ & $+1$ & $56\\,131$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^3$ & $6\\,168\\,634$ & $1\\,540\\,283$ & $8.220719 \\cdot 10^{\\,3\\,177\\,670}$ & $+1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^4$ & $22\\,383\\,618$ & $5\\,167\\,079$ & $5.128265 \\cdot 10^{\\,11\\,538\\,265}$ & $+1$ & $17$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5$ & $155\\,830\\,946$ & $31\\,664\\,035$ & $2.257099 \\cdot 10^{\\,80\\,303\\,211}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5$ & $351\\,661\\,538$ & $85\\,898\\,211$ & $9.729739 \\cdot 10^{\\,181\\,214\\,202}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5^2$ & $1\\,738\\,154\\,976$ & $1\\,433\\,700\\,727$ & $1.594940 \\cdot 10^{\\,895\\,721\\,905}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $$ & $1\\,977\\,626\\,256$ & $853\\,324\\,651$ & $1.196828 \\cdot 10^{\\,1\\,019\\,133\\,881}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5^3$ & $2\\,015\\,279\\,170$ & $4\\,388\\,327\\,617$ & $5.565196 \\cdot 10^{\\,1\\,038\\,523\\,018}$ & $-1$ & $19$ \\\\\n\\hline\n\\T $$ & $3\\,236\\,170\\,820$ & $2\\,307\\,115\\,390$ & $5.427815 \\cdot 10^{\\,1\\,667\\,658\\,416}$ & $+1$ & \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5^4$ & $2\\,015\\,385\\,392$ & $21\\,941\\,638\\,090$ & $5.565196 \\cdot 10^{\\,1\\,038\\,523\\,018}$ & $-1$ & $19$ \\\\\n\\hline\n\\T $$ & $3\\,236\\,257\\,942$ & $11\\,535\\,576\\,954$ & $5.427815 \\cdot 10^{\\,1\\,667\\,658\\,416}$ & $+1$ & $$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Smallest integers $j$ satisfying conditions \\ref{ca}, \\ref{cb} and \\ref{cc} of Theorem~\\ref{main}}\n\\label{tab:j}\n\\end{table}\n\nThe computation of $(\\log 2)\/(2N)$ is done in two steps. First, we generate $d$~digits\nof~$\\log2$. For this we use the $\\gamma$-cruncher~\\cite{AJY}. With\nthis program, A.~Yee and R.~Chan computed 31~billion decimal digits\nof~$\\log2$ in about 24~hours. Second, we set a rational approximation\nof $(\\log 2)\/(2N)$ with a relative error bounded by~$10^{-d}$. Then\npartial quotients of the continued fraction of $(\\log 2)\/(2N)$ are\ncomputed: about $0.97 d$ of them can be evaluated, with safe error\ncontrol~\\cite{BPR96} (cf.\\ the result of Lochs mentioned in Section~\\ref{s1}).\nWe maintain a floating point approximation\nof numbers~$q_j$ (rounded down) and residues of $q_j \\pmod 6$\nby the formula $q_{i+1} = a_{i+1} q_i + q_{i-1}$ for $i \\geq 0$,\nwhere $q_0 = 1$ and $q_{-1} = 0$.\n\nTable~\\ref{tab:j} was created with the `basic method' of~\\cite{BPR96}\nfor $N \\leq 2^8 \\cdot 3^4$. It was fast enough to reach the benchmark\n$m > 10^{10^7}$ in four days with $50\\cdot 10^6$ digits of~$\\log2$.\nBit-complexity of this algorithm (or of the indirect or direct\nmethods~\\cite{BPR96}) is quadratic and reaching the $m > 10^{10^{10}}$ milestone\nwould take centuries.\n\nSome subquadratic GCD algorithms were discovered that have\nasymptotic running time $O(n(\\log n)^2\\log\\log n)$~\\cite{MOL08}.\nA faster version of the program was written: this time a recursive\nHGCD method is applied. It is adapted for computing a continued\nfraction by using Lemma~3 of~\\cite{BPR96} (which is similar to\nAlgorithm~1.3.13 of~\\cite{Cohen93}) for error control. With it\nthe program leaps over $10^{10^8}$ in just about one hour.\nFinally, the new benchmark $m > 10^{10^9}$ is established in no more\nthan 10~hours with $3\\cdot 10^9$ digits of~$\\log2$, $N = 1555200$\nand condition~\\ref{cd}: the first found solution fits\nconditions \\ref{ca}--\\ref{cc}, but not \\ref{cd}. With $N = 7776000$, $m > 10^{10^9}$\nis achieved for the smallest $j$. See Table~\\ref{tab:j}: in the\nlast column, $p$~is a prime such that $p \\in \\mathcal{P}(N)$ and\n$\\nu_p(q_j) = 1$, that is, such that condition~\\ref{cd} of Theorem~\\ref{main}\nis violated.\n\nNow, computation time is not a problem to achieve the $m > 10^{10^{10}}$\nmilestone, a few days will be sufficient on a computer with a large\namount of memory. We remark that the complexity and\nhardware requirement for computation of the digits of~$\\log2$, respectively\nfor computation of its continued fraction expansion, are similar.\n\n\\section{Miscellaneous}\n\\label{s5}\n\n\\subsection{{\\tt The number of distinct prime factors of $m-1$}}\n\\label{s5.1}\n\nThere is a different application of Theorem~\\ref{YVES} suggested\nby the work of Brenton and Vasiliu~\\cite{BV}, to factorization\nproperties of the number $m-1$ coming from a non-trivial solution $(m,k)$ of~\\eqref{EME}.\nA result of Moser~\\cite{Moser} (which can also be deduced from the key identity~\\eqref{staudt},\ncf.\\ the proof of Lemma~\\ref{twee} above) asserts that\n\\begin{equation}\n\\sum_{p\\mid m-1}\\frac1p+\\frac1{m-1}\\in\\mathbb Z;\n\\label{pm}\n\\end{equation}\nin particular, the number $m-1$ is square-free. Since the sum of reciprocals\nof the first 58~primes is less than~2, we conclude that either\n$\\omega(m-1)\\ge 58$ or the integer\nin~\\eqref{pm} is equal to~1. In the latter case,\nwe can apply Curtiss' bound~\\cite{Cur} for positive integer solutions\nof Kellogg's equation\n$$\n\\sum_{i=1}^n\\frac1{x_i}=1,\n$$\nnamely, $\\max_i\\{x_i\\}\\le A_n-1$, where the Sylvester sequence\n$\\{A_n\\}_{n\\ge1}=\\{2,3,7,43,\\dots\\}$ is defined by the recurrence\n$A_n=1+\\prod_{i=1}^{n-1}A_i$ (for some further info, see e.g. Odoni \\cite{Odoni}).\n{}From this result and the estimate\n$A_n<(1.066\\cdot10^{13})^{2^{n-7}}$, we infer\n$$\nm<(1.066\\cdot10^{13})^{2^{\\omega(m-1)-6}},\n$$\nwhich together with the lower bound on~$m$ from Theorem~\\ref{YVES} yields\n$\\omega(m-1)\\ge 33$.\nA similar estimate on the basis of another \\eqref{pm}-like identity of Moser implies\nthat $\\omega(m+1)\\ge 32$.\n\n\\subsection{{\\tt Generalized EM equation}}\n\\label{s5.2}\n\nThe method we use in Section~\\ref{sec:As} for deriving the asymptotics of $k$ in terms of~$m$\nworks for the more general equation\n\\begin{equation}\n\\label{EMEt}\n1^k+2^k+\\dots+(m-1)^k=tm^k,\n\\end{equation}\nwith $t\\in\\mathbb N$ fixed, as well. Indeed, the coefficients in the Taylor series expansion\n\\begin{equation}\n\\label{TS}\n(1-y)^ke^{ky}\n=1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\dotsb\n=\\sum_{n=0}^\\infty g_n(k)y^n\n\\end{equation}\nare polynomials satisfying\n\\begin{equation}\n\\label{TSa}\ng_0(k)=1, \\quad g_1(k)=0,\n\\qquad\\text{and}\\qquad\n\\deg_kg_n(k)=\\biggl[\\frac n2\\biggr], \\quad\ng_n(0)=0 \\quad\\text{for $n\\ge2$};\n\\end{equation}\nthe latter follows from raising the series $(1-y)e^y=1-y^2\/2-y^3\/3-\\dotsb$\nto the power~$k$. In these settings, equation~\\eqref{EMEt} becomes\n\\begin{align*}\nt\n&=\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k\n=\\sum_{j=1}^{m-1}e^{-kj\/m}\\sum_{n=0}^\\infty g_n(k)\\biggl(\\frac jm\\biggr)^n\n\\\\\n&=\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}\\sum_{j=1}^{m-1}j^ne^{-jk\/m}\n\\\\ \\intertext{(since $\\sum_{j=m}^\\infty j^ne^{-jk\/m}=O(m^ne^{-k})$)}\n&\\sim\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}\\sum_{j=1}^\\infty j^ne^{-jk\/m}\n=\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}\n\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac z{1-z}\\biggr)\\bigg|_{z=e^{-k\/m}}\n\\\\\n&=\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}(-1)^n\n\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac1{z-1}\\biggr)\\bigg|_{z=e^{k\/m}},\n\\end{align*}\nhence in the notation $\\lambda=k\/m$ and $x=1\/m$ we have\n\\begin{equation}\n\\label{ast}\nt=\\sum_{n=0}^\\infty g_n\\biggl(\\frac\\lambda x\\biggr)(-x)^n\n\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac1{z-1}\\biggr)\\bigg|_{z=e^\\lambda}.\n\\end{equation}\nSearching $\\lambda$ in the form $\\lambda=c_0+c_1x+c_2x^2+\\dotsb$,\nwe find successively\n\\begin{gather*}\nc_0=c(t)=\\log\\biggl(1+\\frac1t\\biggr)=\\log\\frac{t+1}t,\n\\qquad c_1=-\\biggl(t+\\frac12\\biggr)c,\n\\\\\nc_2=\\biggl(t+\\frac12\\biggr)^3c^2-\\biggl(t+\\frac12\\biggr)^2c\n-\\frac14\\biggl(t+\\frac12\\biggr)c^2+\\frac c6,\n\\end{gather*}\nand so on. Note that $c_n(-(t+1))=(-1)^{n+1}c_n(t)$ for $n=0,1,2,\\dots$;\nthis reflects the equivalence of equation \\eqref{EMEt} and\n\\begin{equation}\n\\label{EMEtdual}\n1^k+2^k+\\dots+(m-1)^k+m^k=(t+1)m^k.\n\\end{equation}\n\n{}From this asymptotics we see that\n\\begin{align}\n\\frac{2k}{2m-t_1}\n&=c+\\frac{{t_1^3}c^2-2{t_1^2}c-t_1c^2+4c\/3}{2(2m-t_1)^2}+O\\biggl(\\frac1{(2m-t_1)^3}\\biggr),\n\\label{CFt}\n\\end{align}\nwhere $t_1=2t+1$ and $c=\\log(1+1\/t)$.\nIt can be checked that for all positive integers $t$ we have the inequality\n$$\n-0.22<{t_1^3}c^2-2{t_1^2}c-t_1c^2+\\frac{4c}3<0,\n$$\nand hence $2k\/(2m-2t-1)$ is a convergent (with even index) of this logarithm\n$c=\\log(1+1\/t)$ for $m$~large enough.\n\n\\subsection{{\\tt Saddle-point method}}\n\\label{s5.3}\n\nA different approach to treat the asymptotic behaviour of $k$ in terms of $m$\nfor $k$ and $m$ satisfying \\eqref{EME} (or, more generally, \\eqref{EMEt})\nis based on the integral representation\n$$\n1^k+2^k+\\dots+(m-1)^k\n=\\frac{\\Gamma(k)}{2\\pi i}\\int_{C-i\\infty}^{C+i\\infty}\n\\frac{e^{mz}}{(e^z-1)z^{k+1}}\\,\\d z,\n$$\nwhere $C$~is an arbitrary positive real number (cf.~\\cite[p.~273]{Delange}).\nOn noting that\n$$\n\\frac{e^{mz}}{e^z-1}\n=\\frac{e^{(m-1)z}}{1-e^{-C}}\\biggl(1+\\frac{1-e^{z-C}}{e^z-1}\\biggr)\n$$\none obtains, on taking $C=(k+1)\/(m-1)$ and after invoking some rather\ntrivial estimates, that\n\\begin{equation}\n\\label{hubert}\n1^k+2^k+\\dots+(m-1)^k=\\frac{(m-1)^k}{1-e^{-(k+1)\/(m-1)}}\\bigl(1+\\rho_k(m)\\bigr),\n\\end{equation}\nwith\n$$\n|\\rho_k(m)|<\\frac{\\sqrt{2(k+1)}C}{\\sqrt\\pi(k-1)(e^C-1)}.\n$$\n(This part of the argument is due to Delange; for more details see\n\\cite[pp.~273--274]{Delange}.) By~\\eqref{ineq},\n$C$~is bounded and we infer that $|\\rho_k(m)|=O(k^{-1\/2})=O(m^{-1\/2})$.\nOn putting $m^k$ on the left-hand side of~\\eqref{hubert}\nand using $(1-1\/m)^m=\\exp(-1+O(m^{-1}))$, we immediately conclude\nthat, as $m\\to\\infty$,\n$$\n\\frac km=\\log2+O\\biggl(\\frac1{\\sqrt{m}}\\biggr),\n$$\nwhere the implied constant is absolute. A more elaborate analysis,\nusing the saddle-point method, will very likely allow one as many terms\nin the latter expansion as required.\n\n\\subsection{{\\tt Experimental asymptotics}}\n\\label{s5.4}\n\nIt is worth mentioning a fast experimental approach of doing asymptotics\nlike~\\eqref{km}. Given numerically a few hundred terms of a sequence\n$s=\\{s_n\\}_{n\\ge1}$ that one believes has an asymptotic expansion\nin inverse powers of $n$, one can try to apply the $\\mathtt{asymp}_k$\ntrick, a simple but often powerful method to numerically determine the\ncoefficients in the ansatz\n$$\ns_n\\sim c_0+\\frac{c_1}{n}+\\frac{c_2}{n^2}+\\dotsb.\n$$\nAs a second step one tries to identify the so-found coefficients\nwith (linear combinations of) known constants. Thus, one arrives\nat a conjecture that hopefully can be turned into a proof.\nFor more details and some `victories' achieved by\nthe $\\mathtt{asymp}_k$ method, see Gr\\\"unberg and Moree \\cite{GM}.\n\nD.~Zagier has applied this trick to the sequence of $k=k(m)$ obtained from\n\\eqref{EME} on letting $m$ run through the first thousand values.\nExcellent agreement with our theoretical results was obtained in this way.\n\n\\medskip\n{\\tt Acknowledgements}. The second author is very indebted to Jerzy Urbanowicz\nfor involving him in the early nineties in his EM research (with \\cite{MRU} as\nvisible outcome). The second and third author would like to thank D.~Zagier\nfor verifying some of our results using the $\\mathtt{asymp}_k$ trick and for some\ninformative discussions regarding the saddle-point method (reflected in the final section). H.~te Riele provided us\nwith the unpublished report\n\\cite{Best}, which became the `initial spark' for the current project. C.~Baxa pointed out the\nrelevance of \\cite{Harman} to us. Further thanks are due to T.~Agoh and I.~Shparlinski.\n\nThis research was carried out whilst the third author was visiting in the Max Planck Institute for Mathematics\n(MPIM) and the Hausdorff Center for Mathematics (HCM) financially supported by these institutions.\nHe and the second author thank the MPIM and HCM for providing such a nice research environment.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfgqo b/data_all_eng_slimpj/shuffled/split2/finalzzfgqo new file mode 100644 index 0000000000000000000000000000000000000000..62c222641f3a54bf287a65854cd0efa0bf785d46 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfgqo @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\n\\subsection{Generalized diagonals for rational billiards}\\label{subsec:gen:diag} Let $P$ be a Euclidean polygon with angles in $\\pi\\bb Q$. We call such a polygon \\emph{rational}. A classical dynamical system is given by the idealized motion of a billiard ball on $P$: the (frictionless) motion of a point mass at unit speed with elastic collisions with the sides. \n\nA \\emph{generalized diagonal} for the polygon $P$ is a trajectory for the billiard flow that starts at one vertex of $P$ and ends at another vertex. Since the group $\\Delta_P$ generated by reflections in the sides of $P$ is finite, the \\emph{angle} of a trajectory is well defined in $S^1 \\cong S^1\/\\Delta_P$. A motivating question for our paper is the following: \\emph{how close in angle can two generalized diagonals of (less than) a given length be} (in terms of the length)?\n\nMasur~\\cite{Masur} showed that the number of generalized diagonals of length at most $R$ grows quadratically in $R$. We will show, for some families of billiards, that the smallest gap $\\gamma^P_R$ between two generalized diagonals on $P$ of length at most $R$ satisfies \\begin{equation}\\label{eq:billiards:smallgap} \\lim_{R \\rightarrow \\infty}R^2\\gamma^P_R =0,\\end{equation} and for other specific billiard tables that\\begin{equation}\\label{eq:billiards:nogap}\\liminf_{R \\rightarrow \\infty} R^2 \\gamma^P_R >0.\\end{equation}\n\n\\subsection{Organization of the paper}\\label{subsec:organization} In the rest of \\S\\ref{sec:intro}, we describe the moduli space $\\Omega_g$ of holomorphic differentials, and state the general versions of (\\ref{eq:billiards:smallgap}) and (\\ref{eq:billiards:nogap}). In \\S\\ref{sec:sl2:strata}, we recall the definition and properties of the $SL(2, \\bb R)$-action on $\\Omega_g$ and the decomposition of $\\Omega_g$ into strata, and discuss the connection to billiards in \\S\\ref{subsubsec:billiards:translation}. We state our main technical results and important applications in \\S\\ref{sec:results}. We prove these results in \\S\\ref{sec:axiom}, following the approach in~\\cite{EskinMasur}. In \\S\\ref{sec:measure:bounds}, we calculate bounds for measures of certain sets in $\\mathcal H$ and give examples of flat surfaces $\\omega \\in \\mathcal H$ with different types of gap behavior. We study a family of flat surfaces arising from rational billiards in \\S\\ref{sec:billiards}. \n\n\\subsection{Translation surfaces}\\label{subsec:holo} Let $\\Sigma_g$ be a compact surface of genus $g \\geq 2$. Let $\\Omega_g$ be the moduli space of holomorphic differentials on $\\Sigma_g$. That is, a point $\\omega \\in\\Omega_g$ is a equivalence class of pairs $(M, \\omega)$, where $M$ is a genus $g$ Riemann surface, and $\\omega$ is a holomorphic differential on $M$, i.e., a tensor with the form $f(z)dz$ in local coordinates, such that $\\frac{i}{2}\\int_{\\Sigma_g} \\omega \\wedge \\bar{\\omega} = 1$. \n\nTwo pairs $(M_1, \\omega_1)$ and $(M_2, \\omega_2)$ are equivalent if there is a biholomorphism $f:M_1 \\rightarrow M_2$ such that $f_{*} \\omega_1 = \\omega_2$. For notational purposes, we will simply refer to the pair $(M, \\omega)$ by $\\omega$. Given $\\omega \\in \\Omega_g$, one obtains (via integration of the form) an atlas of charts on $\\Sigma_g$ (away from the finite set of zeros of $\\omega$) to $\\bb C \\cong \\bb R^2$, with transition maps of the form $z \\mapsto z + c$. For this reason, we will refer to $\\omega \\in \\Omega_g$ as a \\emph{translation surface} of genus $g$.\n\nThese charts determine a unique flat metric on $M$ with conical singularities at the zeros of the differential $\\omega$. Geometrically, a zero of the form $z^k (dz)$ corresponds to a cone angle of order $(2k+2)\\pi$. Zeroes of $\\omega$ are \\emph{singular} points for the flat metric. We refer to non-singular points as \\emph{regular points}. The space $\\Omega_g$ can be decomposed naturally into \\emph{strata} $\\mathcal H$ (see \\S\\ref{sec:sl2:strata} for details), each carrying a natural measure $\\mu_{\\mathcal H}$. \n\n\\subsection{Saddle connections and cylinders}\\label{subsec:sc} Fix $\\omega \\in \\Omega_g$. A \\emph{saddle connection} on $\\omega$ is a geodesic segment in the flat metric connecting two singular points (that is, zeros of $\\omega$) with no singularities in its interior. Given a regular point $p$, a \\emph{regular closed geodesic} through $p$ is a closed geodesic not passing through any singular points. Regular closed geodesics appear in families of parallel geodesics of the same length, which fill a cylindrical subset of the surface.\n\n\\subsubsection{Holonomy vectors}\\label{subsubsec:holonomy} Let $\\gamma$ be an (oriented) saddle connection or regular closed geodesic. Define the associated holonomy vector \n\\begin{equation}\\label{eq:sc:def} \\mathbf v_{\\gamma} : = \\int_{\\gamma} \\omega.\\end{equation}\n\n\\noindent Note that if $\\gamma$ is a closed geodesic, $\\mathbf v_{\\gamma}$ only depends on the cylinder it is contained in, since regular closed geodesics appearing in a fixed cylinder all have the same length and direction. View $\\mathbf v_{\\gamma}$ as an element of $\\bb R^2$ by identifying $\\bb C$ with $\\bb R^2$. \n\nLet \\begin{eqnarray}\\label{eq:la:def}\\Lambda^{sc}_{\\omega} &=& \\{ \\mathbf{v}_{\\gamma}: \\gamma \\mbox{ a saddle connection on } \\omega\\}\\\\ \\Lambda^{cyl}_{\\omega} &=& \\{ \\mathbf{v}_{\\gamma}: \\gamma \\mbox{ a cylinder on } \\omega\\} \\nonumber\\end{eqnarray} \n\n\\noindent be the set of holonomy vectors of saddle connections and cylinders respectively. For $\\Lambda_{\\omega} = \\Lambda^{sc}_{\\omega}$ or $\\Lambda^{cyl}_{\\omega}$, we have that $\\Lambda_{\\omega}$ is discrete in $\\bb R^2$ (see, e.g., ~\\cite[Proposition 3.1]{Vorobets96}), but Masur~\\cite{Masur:billiards} showed that associated set of directions\n$$\\Theta^{\\omega} : = \\{\\arg(\\mathbf v): \\mathbf v \\in \\Lambda_{\\omega}\\}$$\n\\noindent is dense in $[0, 2\\pi)$ for any $\\omega \\in \\Omega_g$.\n\n\n\n\\subsection{Decay of gaps}\\label{subsec:gap:decay}\nIn this paper, we give a measure of the quantitative nature of this density by considering fine questions about the \\emph{distribution} of saddle connection directions. Given $R >0$, let \n\\begin{equation}\\label{eq:Theta:def} \\Theta^{\\omega}_R : = \\{\\arg(\\mathbf v): \\mathbf v \\in \\Lambda_{\\omega} \\cap B(0, R)\\}\\end{equation} denote the set of directions of saddle connections (or cylinders) of length at most $R$. We write $\\Theta^{\\omega}_R : = \\{0 \\le \\theta_1 <\\theta_2 < \\ldots < \\theta_n<2\\pi\\}$, where $n = \\tilde{N}(\\omega, R)$ is the cardinality of $\\Theta^{\\omega}_R$, and we view $\\theta_{n+1}$ as $\\theta_1$. Note that if we define $$N(\\omega, R) : = |\\Lambda_{\\omega} \\cap B(0, R)|,$$ we have $$\\tilde{N}(\\omega, R) \\le N(\\omega, R).$$ Let $\\gamma^{\\omega}(R)$ be the size of the smallest gap, that is $\\gamma^{\\omega}(R) = \\min_{\\theta_i \\in \\Theta_R} |\\theta_i - \\theta_{i+1}|$, Masur~\\cite{Masur} showed that the counting function $N(\\omega, R)$ grows quadratically in $R$ for any $\\omega$. Since there are at most finitely many (at most $4g-4$) saddle connections in a given direction, this shows that $\\tilde{N}(\\omega, R)$ also has quadratic growth. Thus, one would expect the $\\gamma^{\\omega}(R)$ to decay quadratically. Our main theorem addresses the asymptotic behavior of the rescaled quantity $R^2 \\gamma^{\\omega}(R)$. Let $\\mathcal H$ be a stratum of $\\Omega_g$, and let $\\mu = \\mu_{\\mathcal H}$.\n\n\\begin{Theorem}\\label{theorem:main:gap} For $\\mu$-almost every $\\omega \\in \\mathcal H$,\n\\begin{equation}\\label{eq:main:gap}\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega}(R) = 0.\n\\end{equation}\n\\noindent Moreover, for any $\\epsilon >0$, the proportion of gaps less than $\\epsilon\/R^2$ is positive. \n\\begin{equation}\\label{eq:main:proportion}\n\\lim_{R \\rightarrow \\infty} \\frac{ |\\{1 \\le i \\le \\tilde{N}(\\omega, R): (\\theta_{i+1} - \\theta_i) \\le \\epsilon\/R^2\\}|}{\\tilde{N}(\\omega, R)} >0.\n\\end{equation}\n\\end{Theorem}\n\n\\bigskip\n\n\\noindent Theorem~\\ref{theorem:main:gap} cannot be extended to \\emph{all} $\\omega \\in \\mathcal H$, since for any stratum $\\mathcal H$ there are many examples $\\omega \\in \\mathcal H$ for which \n\\begin{equation}\\label{eq:gap:lower}\\liminf_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega}(R) >0.\\end{equation}\n\n\\noindent We say that $\\omega$ has \\emph{no small gaps} (NSG) if (\\ref{eq:gap:lower}) holds. An important motivating example of a surface with NSG is the case of the square torus $(\\bb C\/\\bb Z^2, dz)$. Since there are no singular points, there are no saddle connections, but cylinders are given by integer vectors, and $\\Theta^{\\omega_0}$ then corresponds to rational slopes. It can be shown that $3\/\\pi^2$ is a lower bound for $R^2 \\gamma^{\\omega_0}(R)$ (see, for example~\\cite{BCZ1}). \n\nThe torus is an example of a lattice surface. Recall that $\\omega$ is said to be a \\emph{lattice surface} if the group of derivatives of affine diffeomorphisms of $\\omega$ is a lattice in $SL(2, \\bb R)$ (see \\S\\ref{sec:sl2:strata} for more details). We have:\n\n\\begin{Theorem}\\label{theorem:lattice:nsg} $\\omega$ is a lattice surface if and only if it has no small gaps. \\end{Theorem}\n\\medskip\n\nWe prove Theorem~\\ref{theorem:lattice:nsg} in \\S\\ref{subsec:lattice}, using a result of Smillie-Weiss~\\cite{nosmalltri} which characterizes lattice surfaces using the \\emph{no small triangles} (NST) property defined by Vorobets~\\cite{Vorobets96}. \n\nTheorem~\\ref{theorem:main:gap} will follow from a precise statement about the asymptotic distribution of saddle connection directions. This generalizes work of Vorobets~\\cite{Vorobets}, who showed that the sets $\\Theta^{\\omega}_R$ become uniformly distributed (as $R \\rightarrow \\infty$) in $[0, 2\\pi)$ for almost every $\\omega$ (in particular, for those with exact quadratic asymptotics of saddle connections). Our techniques are inspired by those of Marklof-Strombergsson~\\cite{MS}, who studied the distribution of affine lattice points in Euclidean spaces by reducing them to equidistribution problems in homogeneous spaces. Much of the technical machinery is drawn from~\\cite{EskinMasur}, in which Eskin-Masur give precise quadratic asymptotics of $N(\\omega, \\bb R)$ using equidstribution of translate of orbits under the $SL(2, \\bb R)$-action on $\\Omega_g$.\n\n\\subsubsection{Quadratic differentials}\\label{subsubsec:qd} For notational convenience, we work with the space $\\Omega_g$ instead of the space of quadratic differentials $Q_g$. A quadratic differential determines a flat metric, and so saddle connections and cylinders are well defined. For a saddle connection or cylinder curve $\\gamma$, the holonomy vector $\\mathbf v_{\\gamma}$ is given by integrating a square root of the differential, and are thus defined up to a choice of sign. The set of directions can then be viewed as a subset of $[0, \\pi)$. Our results apply, \\emph{mutatis mutandis}, except when explicitly indicated, to the setting of quadratic differentials.\n\n\\subsection{Hyperbolic angle gaps} Higher-genus translation surfaces can be viewed as an intermediate setting betwen flat tori and hyperbolic surfaces. Recently, Boca-Pasol-Popa-Zaharescu~\\cite{BPPZ} study a spiritually similar problem in the hyperbolic setting. They calculate the limiting gap distribution for the angles (measured from the vertical geodesic) of hyperbolic geodesics connecting $i \\in \\bb H^2$ to points in its $SL(2, \\bb Z)$-orbit. In this setting, they show the limiting distribution does have support at $0$, similar to the case of a generic translation surface.\n\n\n\n\n\n\\subsection{Acknowledgements}\\label{subsec:ack} This paper was inspired by \nthe beautiful paper~\\cite{MS} on the distribution of affine \nlattice points. We thank Alex Eskin, Jens Marklof and William Veech for \nuseful discussions. Howard Masur not only patiently answered many technical questions about this paper but more generally taught us much of what we know about the subject. The initial discussions for the project took place while the authors were attending the Hausdorff Institute of Mathematics `Trimester Program on Geometry and Dynamics of Teichm\\\"uller Spaces' in Bonn. We would like to thank the Hausdorff Institute and the organizers of this program for their hospitality. The second author would like to thank the University of Illinois at Urbana-Champaign for its hospitality. The second author was supported in part by an NSF postdoc.\n\\section{The $SL(2, \\bb R)$-action and strata}\\label{sec:sl2:strata}\n\nIn this section, we describe the $SL(2, \\bb R)$ action and stratification of $\\Omega_g$ (\\S\\ref{subsec:polygons} - \\S\\ref{subsec:sl2:affine}), and the construction of an $SL(2, \\bb R)$-invariant measure (\\S\\ref{subsec:coord:meas}). We also describe (\\S\\ref{subsubsec:billiards:translation}) the connection between rational billiards and translation surfaces. This is standard background material in the subject, and our exposition is brief, and drawing on~\\cite{EMM, EMZ}. Excellent general references are~\\cite{Zorich:survey, MasurTab}.\n\n\\subsection{Translation surfaces and polygons}\\label{subsec:polygons} A more geometric description of a translation surface can be given by a union of polygons $P_1 \\cup \\dots \\cup P_n$ where each $P_i \\subset \\bb C$, and the $P_i$ are glued along parallel sides, such that each side is glued to exactly one other, and the total angle in each vertex is an integer multiple of $2 \\pi$. Since translations are holomorphic, and preserve $dz$, we obtain a complex structure and a holomorphic differential on the identified surface. The zeroes of the differential will be at the identified vertices with total angle greater than $2\\pi$. The sum of the excess angles (that is, the orders of the zeros) will be $2g-2$, where $g$ is the genus of the identified surface.\n\n\n\\subsection{Combinatorics of flat surfaces}\\label{subsec:comb:flat} The space $\\Omega_g$ can be stratified by integer partitions of $2g-2$. If $\\alpha =\n(\\alpha_1, \\dots, \\alpha_k)$ is a partition of $2g-2$, we denote by\n$\\mathcal H(\\alpha) \\subset \\Omega_g$ the moduli space of translation surfaces $(M,\\omega)$\nsuch that the multiplicities of the zeroes of $\\omega$ are given by\n$\\alpha_1, \\dots, \\alpha_n$ (or equivalently such that the orders of\nthe conical singularities are $2 \\pi (\\alpha_1 + 1), \\dots, 2\n\\pi(\\alpha_n + 1)$). For technical reasons, the\nsingularities of $(M,\\omega)$ should be labeled; thus, an element of\n$\\mathcal H(\\alpha)$ is a tuple $(M,\\omega, p_1,\\ldots,p_n)$, where\n$p_1,\\ldots,p_n$ are the singularities of~$M$, and the multiplicity\nof~$p_i$ is~$\\alpha_i$. The moduli space of translation surfaces is\nnaturally stratified by the spaces $\\mathcal H(\\alpha)$; each is called a\n{\\em stratum}. Strata are not always connected, but Kontsevich-Zorich~\\cite{KZ} (and Lanneau~\\cite{Lanneau} in the setting of quadratic differentials) have classified the connected components. Most strata are connected, and there are never more than three connected components.\n\n\n\n\n\n\\subsection{$SL(2, \\bb R)$ and affine diffeomorphisms}\\label{subsec:sl2:affine}\n\nThere is an action of $SL(2,\\bb R)$ on the moduli space of\ntranslation surfaces that preserves the stratification. Since $SL(2,\\bb R)$ acts on $\\bb C$ via linear maps on $\\bb R^2$, given a surface $P_1 \\cup \\dots \\cup P_n$, we can define $g S = g P_1 \\cup \\dots \\cup\ng P_n$, where all identifications between the sides of the polygons for\n$gS$ are the same as for $S$. This action generalizes the action of\n$SL(2,\\bb R)$ on the space of (unit-area) flat tori $SL(2,\\bb R)\/SL(2,\\bb Z)$. Note that $SL(2, \\bb R)$ preserves the area of the surface $\\omega$.\n\n\n\\subsubsection{Lattice surfaces}\\label{subsubsec:lattice}\nFor $\\omega \\in \\mathcal H(\\alpha)$, let $\\Gamma(\\omega) \\subset SL(2,\\bb R)$\ndenote the stabilizer of $\\omega$. The group $\\Gamma(\\omega)$ is called the\n{\\em Veech group} of $S$. If $\\Gamma(\\omega)$ is a lattice in $SL(2,\\bb R)$\nthen $\\omega$ is called a {\\em lattice surface}.\n\nEquivalently, let $\\mbox{Aff}(\\omega)$ denote the set of affine (area-preserving) diffeomorphisms of $\\omega$. The derivative of any $f \\in \\mbox{Aff}(\\omega)$ will be a matrix in $SL(2, \\bb R)$, and the collection $\\{Df: f \\in \\mbox{Aff}(\\omega)\\}$ coincides with $\\Gamma(\\omega)$ (up to a finite index subgroup). Thus $\\Gamma(\\omega)$ is a lattice if and only if $D(\\mbox{Aff}(\\omega))$ is.\n\n\\subsubsection{Billiards and translation surfaces}\\label{subsubsec:billiards:translation} An important motivation for studying translation surfaces is their relationship to rational billiards. Recall that a polygon $P \\subset \\bb C$ is called rational if all angles of $P$\nare rational multiples of $\\pi$. The \\emph{unfolding} procedure in ~\\cite{Zelmjakov:Katok} describes how to associate a translation surface $\\omega_P$ so that the billiard flow on $P$ is described by the geodesic flow on $\\omega_P$. \n\nLet $\\Delta_P \\subset O(2)$ denote the group generated by reflections in the sides of the polygon $P$. Since $P$ is rational, $\\Delta_P$ is finite. $\\omega_P$ consists of $|\\Delta_P|$ copies of $P$, with each copy glued to each of its mirror images along the reflecting side.\n\nFor example, if $P$ is the unit square, then $\\omega_P \\in \\mathcal H(\\emptyset)$ is the torus\n$\\bb C\/2 \\bb Z \\oplus 2 \\bb Z$, and if $P$ is the $(\\pi\/8, 3\\pi\/8)$ right triangle, $\\omega_P \\in \\mathcal H(2)$ is a regular octagon with opposite sides identified.\n\n\\subsection{Coordinates and measure on strata}\\label{subsec:coord:meas}\n\nLet $\\alpha = (\\alpha_1, \\ldots, \\alpha_k)$ be an integer partition of $2g-2$. We describe how to put a topology and measure on $\\mathcal H(\\alpha)$. Our exposition is drawn from~\\cite{EMZ}. For a flat surface\n$\\omega_0 \\in\\mathcal H(\\alpha)$ with zero set $\\Sigma = \\{p_1,\\dots, p_k\\}$, choose a\nbasis for the relative homology\n$H_1(\\Sigma_g, \\Sigma;\\bb Z)$. We can pick a basis consisting of saddle connections, since we can choose saddle connections that cut $\\omega_0$ into a\nunion of polygons. For any $\\omega$ near $\\omega_0$ holonomy vectors $\\{\\mathbf v_{\\gamma_i}\\}$ yield local coordinates. That is, we view $\\omega$ as an element of the relative cohomology\n$H^1(\\Sigma_g,\\Sigma;\\bb C) \\cong \\bb R^{4g+2k-2}$, and a domain in this vector space gives us a local coordinate chart. We write $n = 4g+2k-2$. We normalize Lebesgue measure on $\\bb R^n$ so that the integer lattice $\\bb Z^n \\cong H^1(\\Sigma_g,\\Sigma; \\bb Z[i])$ has covolume $1$. Our measure $\\mu(S)$ on $\\mathcal H(\\alpha)$ is given by pulling back this measure via our coordinate maps. This is well-defined, the choice of volume element on $H^1(\\Sigma_g,\\Sigma;\\bb C)$ is independent of choice of basis.\n\nWe will work with \\emph{unit-area} surfaces. Let $\\mathcal H_1(\\alpha)\\subset\\mathcal H(\\alpha)$ be the subset of unit area translation surfaces. Let $a_i,b_i$, $i=1,\\ldots,g$ be a symplectic basis for homology $H_1(\\Sigma_g, \\bb Z)$. The area of the translation surface in the flat metric given by $\\omega$ is given by\n\n$$\\int_{\\Sigma_g} |\\omega|^2 dx dy=\\frac{i}{2}\\int_{\\Sigma_g}\n\\omega\\wedge\\bar\\omega=\n\\frac{i}{2}\\sum_i\\left(\\int_{A_i}\\omega\\int_{B_i}\\bar\\omega-\n\\int_{A_i}\\bar\\omega\\int_{B_i} \\omega\\right).\n$$\n\\noindent This can be viewed as an (indefinite) quadratic form in our local coordinates, and so the level set $\\mathcal H_1(\\alpha)$ can be thought of as a `hyperboloid'. The measure on $\\mathcal H(\\alpha)$ induces a measure on the hypersurface $\\mathcal H_1(\\alpha)$. We can represent any\n$\\omega \\in\\mathcal H(\\alpha)$ as $\\omega = r \\omega'$, where $r\\in\\bb R_+$, and $\\omega' \\in \\mathcal H_1(\\alpha)$.\nHolonomy vectors of saddle connections and cylinders on $\\omega'$ are\nmultiplied by $r$ to give vectors associated to corresponding\nsaddle connections on $\\omega$, and $\\mbox{area}(\\omega) =\nr^2\\cdot\\mbox{area}(\\omega')=r^2$. The measure $\\mu_1$ on\n$\\mathcal H_1(\\alpha)$ is given by disintegration of the volume element $\\mu$\non $\\mathcal H(\\alpha)$:\n %\n$$\nd\\mu(\\omega) = r^{n-1} \\, dr\\, d\\mu_1(\\omega').\n$$\n\n\\noindent In the sequel, by abuse of notation, we will fix a connected component $\\mathcal H$ of $\\mathcal H_1(\\alpha)$ and denote the Lebesgue measure on it by $\\mu$. We note that in any stratum, the set of surfaces arising from billiards as in \\S\\ref{subsubsec:billiards:translation} has measure zero. Thus, statements about almost every translation surface do not yield results about billiard flows, in particluar, Theorem~\\ref{theorem:main:gap} does not apply to billiard flows. In \\S\\ref{sec:billiards} we discuss some special classes of billiards for which we can prove a version of Theorem~\\ref{theorem:main:gap}.\n\n\\subsubsection{$SL(2,\\bb R)$-invariance and ergodicity}\\label{subsubsec:ergodic} Since the $SL(2,\\bb R)$-action on $\\mathcal H(\\alpha)$ preserves the area, it acts on the level set $\\mathcal H_1(\\alpha)$. It also preserves connected components, so we can consider it acting on $\\mathcal H$. The measure $\\mu$ constructed above is invariant under $SL(2,\\bb R)$. The following theorem is due (independently) to Veech~\\cite{Veech:gauss} and Masur~\\cite{Masur:IET}.\n\n\\begin{theorem*}[Veech~\\cite{Veech:gauss}, Masur~\\cite{Masur:IET}] $\\mu$ is a finite, ergodic, $SL(2,\\bb R)$-invariant measure on $\\mathcal H$. \\end{theorem*}\n\n\\subsubsection{Short saddle connections}\\label{subsubsec:short:sc} We record here a crucial measure estimate on the set of surfaces with short saddle connections. It is originally due to Masur-Smillie~\\cite{MasurSmillie} we recall it as it is quoted in~\\cite[Lemma 7.1]{EMZ}\n\n\\begin{lemma}\n[H.~Masur, J.~Smillie]\n\\label{lemma:short:saddle:connections}\n %\nThere is a constant $M$ such that for all $\\epsilon,\\kappa>0$ the\nsubset of $\\mathcal H$ consisting of those flat surfaces,\nwhich have a saddle connection of length at most $\\epsilon$, has\nvolume at most $M\\epsilon^2$. The volume of the set of flat\nsurfaces with a saddle connection of length at most $\\epsilon$\nand a nonhomologous saddle connection with length at most\n$\\kappa$ is at most $M\\epsilon^2\\kappa^2$.\n %\n\\end{lemma}\n\n\\noindent\\textbf{Remark:} Note that by the construction of the measure, the volumes of these sets will be \\emph{at least} $m \\epsilon^2$ and $m \\epsilon^2 \\kappa^2$ for some possibly smaller $m$, since we can construct local coordinates using a basis given by our short saddle connections.\n\n\\section{Saddle connections}\\label{sec:results}\n\n\\noindent This section contains statements of our main results. In \\S \\ref{subsec:windows}, we give our main distribution result Theorem~\\ref{theorem:wedge:ae} for the directions of saddle connections and cylinders. In \\S\\ref{subsec:proof:main:gap} we show how to use this result to derive Theorem~\\ref{theorem:main:gap}. Theorem~\\ref{theorem:wedge:ae} relies on results on limit measures for certain subsets of $SL(2, \\bb R)$-orbits, which we describe in \\S\\ref{subsec:limit:circles}, and show how to use these limit theorems to obtain results for billiards. In \\S\\ref{subsec:lattice}, we describe how lattice surfaces yield exceptional behavior in our context. In \\S\\ref{subsec:limit:dist} we state results on measure bounds and gap distribution.\n\n\n\\subsection{Counting points in thinning segments}\\label{subsec:windows} \n\nWe recall notation: $\\mathcal H$ is a connected component of a stratum $\\mathcal H_1(\\alpha)$ of unit area differentials in $\\Omega_g$, and $\\mu$ is Lebesgue measure on $\\mathcal H$, normalized to be a probability measure. Given $\\omega \\in \\mathcal H$, $\\Lambda_{\\omega}$ denotes set of holonomy vectors of either saddle connections or periodic cylinders in the flat metric determined by $\\omega$. \n\n\\subsubsection{The Siegel-Veech transform}\\label{subsubsec:siegel:veech} We recall the defintion of the \\emph{Siegel-Veech transform} from~\\cite[\\S2.1]{EskinMasur}. Given a compactly supported function $f: \\bb R^2 \\rightarrow \\bb R$, define $\\hat{f}: \\mathcal H \\rightarrow \\bb R$ by\n\\begin{equation}\\label{eq:sv:transform}\\hat{f}(\\omega) = \\sum_{\\mathbf v \\in \\Lambda_{\\omega}} f(\\mathbf v).\n\\end{equation}\n\nVeech~\\cite{Veech} formulated the following seminal result, now known as the \\emph{Siegel-Veech formula}:\n\n\\begin{Theorem}\\label{theorem:sv:formula} Let $\\eta$ be an ergodic $SL(2, \\bb R)$ invariant probability measure on $\\mathcal H$. There is a $b = b(\\eta)$ so that for all $f \\in C^{\\infty}_0(\\bb R^2)$, $$\\int_{\\mathcal H} \\hat{f} d\\eta = b \\int_{\\bb R^2} f dm$$\n\\end{Theorem}\n\n\n\\subsubsection{Thinning annular regions}\\label{subsec:annular} Following \\cite[\\S 2.3]{MS}, we consider a family of thinning annular regions in $\\bb R^2$ (see Figure~\\ref{polar} below): given $\\theta \\in [0, 2\\pi)$, $\\sigma, R >0$, and $0 \\le c < 1$ define the annular region\n\\begin{equation}\\label{eq:c:theta:sigma} \nA^{\\theta}_R(c, \\sigma) : = \\{\\mathbf v \\in \\bb R^2: cR \\le ||\\mathbf v|| \\le R, \\arg(\\mathbf v) \\in (\\theta - \\sigma R^{-2}, \\theta + \\sigma R^{-2})\\}.\n\\end{equation}\n\\makefig{The region $A = A^{\\theta}_R(c, \\sigma)$}{polar}{\\input{wedge.pstex_t}} \n\n\n\n\\noindent As $R \\rightarrow \\infty$, this gives a narrowing wedge of directions around the angle $\\theta$. For $\\omega \\in \\mathcal H$ define the counting function \n\n\\begin{equation}\\label{eq:n:theta:sigma} N^{\\theta}_R(\\omega, \\sigma, c) : = |\\Lambda_{\\omega} \\cap A^{\\theta}_R(c, \\sigma) |.\\end{equation}\n\n\\noindent We think of $N^{\\theta}_R(\\omega, \\sigma, c)$ as the number of saddle connections in a small neighborhood of the direction $\\theta$. Note that $N^{\\theta}_R(\\omega, \\sigma, c)$ can be viewed as the Siegel-Veech transform of the indicator function of $A^{\\theta}_R(c, \\sigma)$. Given that $|\\Lambda_{\\omega} \\cap B(0, R)|$ has quadratic asymptotics, one would expect $N^{\\theta}_R(\\omega, \\sigma)$ to be proportional to $\\sigma$. We frame the following question. Fixing an integer $k$, if we choose $\\theta$ uniformly in $[0, 2\\pi)$, what is the probability that $N^{\\theta}_R(\\omega, \\sigma, c) = k$?\n\n\\begin{Theorem}\\label{theorem:wedge:ae} Fix $\\sigma >0$ and $c \\in [0, 1)$, $k \\in \\bb Z_{\\geq 0}$. Let $\\mathcal H$ be a (connected component of) stratum $\\mathcal H_1(\\alpha)$ and let $\\mu = \\mu_{\\mathcal H}$ denote the natural $SL(2,\\bb R)$-invariant probability measure on $\\mathcal H$. For $\\mu$-a.e. $\\omega_0 \\in \\mathcal H$, \n\\begin{equation}\\label{eq:wedge:ae}\n\\lim_{R \\rightarrow \\infty} \\lambda(\\theta: N^{\\theta}_R(\\omega_0, \\sigma, c) = k) = \\mu(\\omega: \\Lambda_{\\omega} \\cap T(c, \\sigma) = k)\\end{equation}\n\n\\noindent where $\\lambda$ denotes the Lebesgue probability measure on $[0, 2\\pi)$ and $T(c, \\sigma)$ is the trapezoid with vertices $(c, \\pm c\\sigma), (1, \\pm \\sigma)$.\n\n\\end{Theorem}\n\n\\medskip\n\n\\noindent In \\S\\ref{sec:axiom}, we will see that this theorem will hold with $\\Lambda_{\\omega}$ replaced by other sets of holonomy vectors of special trajectories on $\\omega$.\n\n\\subsection{Proof of Theorem~\\ref{theorem:main:gap}}\\label{subsec:proof:main:gap} We show how Theorem~\\ref{theorem:main:gap} follows from Theorem~\\ref{theorem:wedge:ae}. Given $k \\in \\bb Z_{\\geq 0}, \\sigma >0$, let\n\\begin{equation}\\label{eq:pksigma:def} p_k(\\sigma) := \\mu(\\omega: \\Lambda_{\\omega} \\cap T(\\sigma) = k)\\end{equation}\n\\noindent where $T(\\sigma) = T(0, \\sigma)$. We require the following lemma.\n\n\\begin{lemma}\\label{lemma:p2sigma} For any $\\sigma>0$\n\\begin{equation}\\label{eq:p2sigma} p_{2}(\\sigma)>0. \\end{equation}\n\n\\end{lemma}\n\n\\noindent\\textbf{Proof:} As in the remark following Lemma~\\ref{lemma:short:saddle:connections}, we note that we can construct in $\\mathcal H$ a set of measure at least $m \\sigma^4$ ($m$ depending on $\\mathcal H$) with two saddle connections with holonomy vectors in $T(\\sigma)$.\n\\qed\n\\medskip\n\n\\noindent Let $n \\in \\bb N$, let $\\mathcal H_{n} \\subset \\mathcal H$ be the full measure set of $\\omega \\in \\mathcal H$ so that (\\ref{eq:wedge:ae}) holds for $\\sigma = 1\/n$, $c =0$. Then $\\mathcal H_{\\infty} = \\bigcap_{n=1}^{\\infty} \\mathcal H_n$ is also a full measure set. We claim that for any $\\omega_0 \\in \\mathcal H_{\\infty}$, (\\ref{eq:main:gap}) holds, that is, \n$$\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega_0}(R) = 0.$$\n\\noindent Let $n \\in \\bb N$. Since $\\omega_0 \\in \\mathcal H_{\\infty}$, (\\ref{eq:wedge:ae}), there is an $R_n$ such that \nfor all $R> R_n$, \n\\begin{equation}\\label{eq:lambda:positive}\n\\lambda(\\theta: N^{\\theta}_R(\\omega_0, 1\/n, 0) \\geq 2) \\geq p_2(\\sigma)\/2 >0. \\end{equation}\n\\noindent The last inequality follows from Lemma~\\ref{lemma:p2sigma}. By construction of $N^{\\theta}_R(\\omega_0, 1\/n, 0)$, (\\ref{eq:lambda:positive}) implies that for $R > R_n$, \n$$\\gamma^{\\omega_0}(R) \\le \\frac{1}{nR^2}.$$\n\\noindent Since $n$ was arbitrary (\\ref{eq:main:gap}) follows. To see (\\ref{eq:main:proportion}), note that $\\lambda(\\theta: N^{\\theta}_R(\\omega_0, \\epsilon, 0) \\geq 2)$ gives a lower bound for the limiting proportion of gaps of size less than $\\epsilon\/R^2$. Theorem \\ref{theorem:main:gap} now follows from Theorem~\\ref{theorem:wedge:ae} and Lemma~\\ref{lemma:p2sigma}. \\qed\\medskip\n\n\\subsection{Limit measures}\\label{subsec:limit:circles} The main weakness of Theorem~\\ref{theorem:wedge:ae} is that it does not give us information about any particular surface $\\omega_0 \\in \\mathcal H$. To obtain such information, we must have further knowledge about the limiting behavior (as $t \\rightarrow \\infty$) of the orbits $\\{g_t r_{\\theta} \\omega_0: 0 \\le \\theta < 2\\pi\\}$, where \n\n \\begin{equation}\n \\label{eq:matrices}\ng_t = \\left(\\begin{array}{cc} e^{-t\/2} & 0 \\\\ 0 & e^{t\/2}\n\\end{array}\\right),\nr_{\\theta} = \\left(\\begin{array}{cc} \\cos\\theta& -\\sin\\theta \\\\ \\sin\\theta & \\cos\\theta\n\\end{array}\\right).\n\\end{equation}\n\n\\noindent Let $\\nu_{t, \\omega_0}$ denote the Lebesgue probability measure supported on $\\{g_t r_{\\theta} \\omega_0: 0 \\le \\omega_0 < 2\\pi\\}$. Suppose $\\lim_{t \\rightarrow \\infty} \\nu_{t, \\omega_0} = \\mu_0$, and $\\mu_0$ is $SL(2, \\bb R)$-invariant. In this case we say $\\mu_0$ is the \\emph{circle limit measure} associated to $\\omega_0$. By~\\cite[Theorem 5.2]{EskinMasur}, $\\mu_0$ is a probability measure. \n\n\\begin{Theorem}\\label{theorem:wedge:limit} Suppose $\\omega_0 \\in \\mathcal H$ has circle limit measure $\\mu_0$. Then \\begin{equation}\\label{eq:wedge:limit}\n\\lim_{R \\rightarrow \\infty} \\lambda(\\theta: N^{\\theta}_R(\\omega_0, \\sigma, c) = k) = \\mu_0(\\omega: \\Lambda_{\\omega} \\cap T(c, \\sigma) = k).\\end{equation}\n\\end{Theorem}\n\n\\medskip \n\n\\noindent Letting $\\gamma^{0}(R) = \\gamma^{\\omega_0}(R)$, the proof of Theorem~\\ref{theorem:main:gap} yields the following corollary to Theorem~\\ref{theorem:wedge:limit}. Let $p^0_2(\\sigma) : = \\mu_0(\\omega: \\Lambda_{\\omega} \\cap T(\\sigma) = 2)$.\n\n\\begin{Cor}\\label{cor:gap:limit} Fix notation as in Theorem~\\ref{theorem:wedge:limit}. Suppose for all $\\sigma >0$, $p^0_2(\\sigma)>0$. Then \n\\begin{equation}\\label{eq:gap:limit}\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{0}(R) = 0.\n\\end{equation}\n\\noindent Moreover, for any $\\epsilon >0$, the proportion of gaps less than $\\epsilon\/R^2$ is positive. That is, writing $\\Theta^{\\omega_0}_R : = \\{0 \\le \\theta_1 \\le \\theta_2 \\le \\ldots \\le \\theta_n\\}$, we have\n\\begin{equation}\\label{eq:limit:proportion}\n\\lim_{R \\rightarrow \\infty} \\frac{ |\\{1 \\le i \\le \\tilde{N}(\\omega_0, R): (\\theta_{i+1} - \\theta_i) \\le \\epsilon\/R^2\\}|}{\\tilde{N}(\\omega_0, R)} >0.\n\\end{equation}\n\\end{Cor}\n\n\\medskip\n\n\\subsubsection{Billiards with barriers}\\label{subsubsec:billiards:barriers} We describe how Theorem~\\ref{theorem:wedge:limit} and Corollary~\\ref{cor:gap:limit} can be used to give information about specific families of billiards. Following~\\cite{EMS}, we consider the following family of billiards. Given $\\alpha \\in \\bb R$, consider the polygon $P_{\\alpha}$ whose boundary is the boundary of the square $[0, 1] \\times [0, 1]$ together with a barrier given by the vertical segment $\\{1\/2\\} \\times [0, \\alpha]$. \n\n\\medskip\n\n\\noindent\\textbf{Remark:} In fact, slits based at any rational point $p\/q$ are considered in~\\cite{EMS}. We restrict to $1\/2$ for notational convenience and ease of exposition.\n\n\\medskip\n\nThe associated surface, which we denote by $\\omega_{\\alpha}$ is (after rescaling) an element of $\\mathcal H(1,1)$. A crucial observation is that $\\omega_{\\alpha}$ is a double cover of the torus. That is, there is a covering map (branched at the zeros) $\\pi: \\omega_{\\alpha} \\rightarrow \\bb C\/\\Lambda$ where $\\Lambda$ is a lattice in $\\bb C$ and that $\\omega_{\\alpha}$ is obtained by pulling back the form $dz$. Using this construction, we observe that the set $\\{\\omega_{\\alpha}: \\alpha \\in \\bb R\\}$ is contained in an $SL(2,\\bb R)$-invariant subvariety $\\mathcal M \\subset \\mathcal H$ which can be identified with the moduli space of tori with marked points, $(SL(2,\\bb R)\\ltimes\\bb R^2)\/(SL(2,\\bb Z)\\ltimes\\bb Z^2)$. \n\nLet $\\mu_{\\mathcal M}$ denote the natural $SL(2,\\bb R)$-invariant probability measure supported on $\\mathcal M$. Applying a theorem of Shah~\\cite{Shah:SL2} (which uses Ratner's measure classification), it is shown in~\\cite{EMS} that for any irrational $\\alpha$, $\\mu_{\\mathcal M}$ is a circle limit measure for $\\omega_{\\alpha}$. We will see in \\S\\ref{subsec:translate} that for any $\\sigma >0$,\n$$p^{\\mathcal M}_2(\\sigma) := \\mu_{\\mathcal M}(\\omega: \\Lambda_{\\omega} \\cap T(\\sigma) = 2)>0,$$\n\n\\noindent and thus, Corollary~\\ref{cor:gap:limit} applies in this situation.\n\n\n\n\\subsection{Lattice surfaces}\\label{subsec:lattice} In this section we prove Theorem~\\ref{theorem:lattice:nsg}. Fix $\\gamma^{\\omega}_R$ to denote the smallest gap for saddle connections of length at most $R$. We split the proof into two lemmas. The first shows that any lattice surface has NSG.\n\n\\begin{lemma}\\label{lemma:lattice:gap} For any lattice surface $\\omega$ there exists a constant $\\epsilon>0$ such that $\\gamma^{\\omega}_R\\geq \\frac{\\epsilon}{R^2}$ for all $R>1.$\n\\end{lemma}\n\\begin{proof}We want to show that if $\\omega \\in \\mathcal H$ is a lattice surface, then $\\omega$ has no small gaps. By \\cite[Proposition 6.1]{Vorobets96} if $\\omega$ is a lattice surface there exists a constant $s$ such that any two saddle connections in the same direction have the ratio of their lengths at most $s$.\n Given a periodic direction $\\theta$ there must be a cylinder of area at least $\\frac 1 {4g-4}$ in that direction. Let $R$ be its length. By Lemma \\ref{cyl flow} points in this cylinder are not in another cylinder of length at most $R$ in a direction within $\\frac{1}{(8g-8)R^2}$. Any other saddle connection in this direction must have length $\\frac{R} s$ which implies that a direction $\\psi \\in B(\\theta,\\frac{1}{(8g-8)R^2})$ can have no periodic cylinders of length less than $\\frac R s$. \\end{proof}\n \n \\medskip\n \n\\noindent For the converse, we recall that $\\omega$ has \\emph{no small triangles} if there is a $\\delta >0$ so that all triangles on $\\omega$ with vertices at singularities, and no singularities in the interior have area at least $\\delta$. Smillie-Weiss~\\cite{nosmalltri} showed:\n\n\\begin{theorem*} $\\omega$ is a lattice surface if and only if it has no small triangles. \\end{theorem*}\n\n\\medskip\n\n\\noindent Combining this theorem with the following lemma completes the proof of Theorem~\\ref{theorem:lattice:nsg}. \n\n\\begin{lemma}\\label{lemma:nst:nsg} If $\\omega$ has no small gaps then it has no small triangles.\\end{lemma}\n\\begin{proof} Let $\\epsilon >0$ be such that $R^2 \\gamma^{\\omega}_R > \\epsilon$. Let $T$ be a triangle on $\\omega$ with vertices at singularities and with no singularities in the interior. Without loss of generality we can assume that the sides of $T$ are saddle connections, since if not it can be decomposed into triangles which are. Let $R$ be the length of the longest side. Dropping a perpendicular from the opposite vertex, we decompose the side into two segments, at least one of which has length at least $R\/2$. Consider the right triangle formed by the perpendicular and this segment. The angle opposite the perpendicular is an angle between saddle connections of length at most $R$, so it is at least $\\frac{\\epsilon}{R^2}$. See Figure~\\ref{triangle} below.\n\n \\makefig{$\\theta \\geq \\frac{\\epsilon}{R^2}$}{triangle}{\\input{triangle.pstex_t}} \n\n\n\\noindent The length $L$ of the perpendicular is at least $\\frac{R}{2}\\tan(\\frac{\\epsilon}{R^2})$, so $L > \\frac{R}{2}\\frac{\\delta}{R^2} = \\frac{\\delta}{2R}$ for some $\\delta > 0$. Thus the area of the triangle is bounded below by $\\frac{\\delta}{4}$, and so is the area of $T$. Since $T$ was arbitrary, we have the $\\omega$ has no small triangles.\n\\end{proof}\n\n \n \n\n\n\nIf we replace saddle connections with cylinders, then, as pointed out to us by Barak Weiss, we can construct a non-lattice surface with no small cylinder gaps as follows: take a branched cover of a torus by varying relative periods (see \\S\\ref{sec:billiards}). This surface has absolute holonomy in\n$\\bb Z^2$ and therefore has no small gaps for vectors which are holonomies of\ncylinder core curves. \n\n\\subsection{Measure bounds and gaps}\\label{subsec:limit:dist}\nLet $\\hat{G}_2(\\Theta_R^{\\omega})=\\left\\{R^2 (\\theta_{i+1}-\\theta_i)\\right\\}_{i=1}^{|\\Theta_R^{\\omega}|}$.\n Let $\\nu_2^{\\omega}(R)$ be the probability measure obtained by normalizing the measure given by delta mass at\n each element of $\\hat{G}_2(\\Theta_R^{\\omega})$.\n\\begin{Prop}\\label{lim distr} For almost every surface $\\omega$ the measure $\\nu_2^{\\omega}(R)$ converges (as $R \\rightarrow \\infty$) in the weak-* topology.\n\\end{Prop}\n\\begin{proof} The existence of $p_0(\\sigma)$ and the quadratic growth of saddle connections\nimplies this by \\cite[Theorem 2.1]{MS}. In particular, after \nunwinding the definitions there we have \n$\\nu_{\\infty}([a,b])=\\frac {d} {d\\sigma} p_0(\\sigma)|_{a}-\\frac d {d\\sigma}p_0(\\sigma)|_b$. \nNote that this makes sense for almost every $\\sigma$ because $p_0(\\sigma)$ is a decreasing function of $\\sigma$.\n\\end{proof}\n\n\n\n\n\\section{Equidistribution on strata}\\label{sec:axiom}\n\nIn this section, we prove Theorem~\\ref{theorem:wedge:ae} and and Theorem~\\ref{theorem:wedge:limit}. We follow the strategy outlined in~\\cite{EskinMasur} for proving results on the asymptotics of $N(\\omega, R)$ and modify the techniques to our situation.\n\n\\subsection{Cones in $\\bb R^2$}\\label{subsec:cones}\n\nThe crucial geometric observation in the proof of Theorems~\\ref{theorem:wedge:ae} and~\\ref{theorem:wedge:limit} is the following. Let $R>>0$, and $t = 2 \\log R$. Then \n$$r_{\\theta} g_{-t} T(c, \\sigma) \\approx A^{\\theta}_R(c, \\sigma)$$\n\\noindent This can be seen as follows: $g_{-t} T(c, \\sigma)$ is a trapezoid with vertices at $(cR, \\pm c \\sigma \/R), (R, \\pm \\sigma\/ R)$. Rotating it by angle $\\theta$, we have that $r_{\\theta} g_{-t} T(c, \\sigma)$ is a thin trapezoidal wedge around the set line $\\{\\mathbf v \\in \\bb R^2: \\arg(\\mathbf v) = \\theta\\}$, with vectors of length roughly between $cR$ and $R$ (there is an error of up to $ \\frac 1 R$ because $r_{\\theta}g_{-t}T(c,\\sigma)$ is a trapezoid and not a wedge of an annulus). Finally, the angular width of this wedge is roughly $c\/R^2$, since for small angles $\\phi$ the slope $\\tan(\\phi) \\approx \\phi$. Thus, for any $\\omega \\in \\mathcal H$, $R >>0$,\n\\begin{equation}\\label{eq:lambda:wedge}|\\Lambda_{\\omega} \\cap A^{\\theta}_R(c, \\sigma)| \\approx |g_{t} r_{-\\theta} \\Lambda_{\\omega} \\cap T(c, \\sigma)|.\\end{equation}\n\n\\noindent For $k \\in \\bb N$, let $f_k: \\mathcal H \\rightarrow [0, 1]$ be the indicator function of the set \n\\begin{equation}\\label{eq:level:set}\\mathcal H_{c, \\sigma, k} : = \\{\\omega \\in \\mathcal H: |\\Lambda_{\\omega} \\cap T(c, \\sigma)| = k\\}.\\end{equation}\n\n\\noindent Using (\\ref{eq:lambda:wedge}), we can write\n\\begin{equation}\\label{eq:circle:relation}\n \\lambda(\\theta: N^{\\theta}_R(\\omega, \\sigma, c) = k) \\approx \\int_0^{2\\pi} f_k(g_t r_{-\\theta} \\omega) d\\lambda(\\theta) = \\int_{\\mathcal H} f_k d\\nu_{t,\\omega}.\n\\end{equation}\n\\noindent Here, $\\approx$ denotes that the difference goes to $0$ as $R \\rightarrow \\infty$. We will rigorously justify (\\ref{eq:circle:relation}) below.\n\\subsection{Equidistribution}\\label{subsec:equi} Equation (\\ref{eq:circle:relation}) reduces proving Theorem~\\ref{theorem:wedge:ae} and Theorem~\\ref{theorem:wedge:limit} to understanding \n$$\\lim_{t \\rightarrow \\infty} \\int_{\\mathcal H} f_k d\\nu_{t,\\omega}.$$We first prove Theorem~\\ref{theorem:wedge:limit} assuming (\\ref{eq:circle:relation}):\n\n\\noindent\\textbf{Proof of Theorem~\\ref{theorem:wedge:limit}:} While there is no general pointwise ergodic theorem known for the measures $\\nu_{t,\\omega}$, our functions $f_k$ are indicator functions of level sets of the Siegel-Veech transform of the indicator function $h$ of $T(c, \\sigma)$ (see \\S\\ref{subsubsec:SV} below). Following~\\cite[\\S4]{EskinMasur}, there is an approximation argument that allows us to conclude that if $\\nu_{t,\\omega_0} \\rightarrow \\mu_0$, we have, as desired\n$$\\lim_{t \\rightarrow \\infty} \\int_{\\mathcal H} f_k d\\nu_{t, \\omega_0} = \\int_{\\mathcal H} f_k d\\mu_0 = \\mu_0(\\omega \\in \\mathcal H: \\Lambda_{\\omega} \\cap T(c, \\sigma) = k).$$\n\\noindent\\qed\\medskip\n\n\n\\noindent\\textbf{Proof of Theorem~\\ref{theorem:wedge:ae}:} Combine Theorem~\\ref{theorem:wedge:limit} with~\\cite[Proposition 3.3]{EskinMasur}.\n\\qed\\medskip\n\n\\subsubsection{Proof of (\\ref{eq:circle:relation})}\\label{sec:circle:relation}\n\nLet $F_t$ denote the indicator function of $g_{t} r_{-\\theta} A^{\\theta}_R(c, \\sigma)$. By construction, this does \\emph{not} depend on $\\theta$, as it is the $g_{t}$ image of a horizontal cone. As above, let $h$ denote the indicator function of $T(c, \\sigma)$. We abbreviate $\\nu_{t, \\omega_0}$ by $\\nu_t$. Denote the symmetric difference of $g_t r_{-\\theta} A^{\\theta}_R(c, \\sigma)$ and $T(c, \\sigma)$ by $E_t$, and note that $E_s \\subset E_t$ for $s > t$, and the volume of $E_t$ tends to zero uniformly in $\\theta$ as $t \\rightarrow \\infty$. Also note that by construction, $$\\widehat{F_t}(g_{t} r_{-\\theta} \\omega) = N^{\\theta}_R(\\omega, \\sigma, c),$$ and $f_k$ is the indicator function of the level set $\\{\\omega: \\widehat{h}(\\omega) = k\\}$. Thus we would like to show that, given any basepoint $\\omega_0$, $$\\lim_{t \\rightarrow \\infty} \\nu_t (\\omega: \\widehat{F_t} (\\omega) = k) = \\lim_{t \\rightarrow \\infty} \\nu_t(\\omega: \\widehat{h}(\\omega) = k).$$ Recall that for any set $L \\subset \\mathcal H$, $$\\nu_t(\\omega: \\omega \\in L) = \\lambda(\\theta: g_{-t} r_{-\\theta} \\omega_0 \\in L).$$\n\n\\noindent Let $I_t = \\{\\omega: \\widehat{F_t}(\\omega) \\neq \\widehat{g}(\\omega)\\}$. Letting $b_0$ denoting the Siegel-Veech constant (see Theorem~\\ref{theorem:sv:formula}) of $\\mu_0$, we have $$\\mu_0(I_t) < b_0 m(E_t),$$where $m$ denotes Lebesgue measure on $\\bb R^2$. Also note that $I_s \\subset I_t$ for $s > t$. Fix $\\epsilon >0$. Let $t_0 >0$ be such that $m(E_t) < \\epsilon\/C_0$ for all $t > t_0, \\theta \\in [0, 2\\pi)$, so $$\\mu_0(I_t) < \\epsilon.$$\n\n\\noindent Since $\\nu_t \\rightarrow \\mu_0$, we can pick $t_1 >t_0$ so that for all $t > t_1$, $$\\nu_t (I_t) \\le \\nu_t(I_{t_0}) \\le 2 \\epsilon.$$\n\n\\noindent Write $$\\nu_t(\\omega: \\widehat{F_t}(\\omega) = k ) = \\nu_t(\\omega \\notin I_t: \\widehat{F_t} = k) + \\nu_t(\\omega \\in I_t: \\widehat{F_t}(\\omega) = k)$$\n\n\\noindent The second term can be bounded above by $\\nu_t(I_t)$ and thus by $2 \\epsilon$. The first term can be rewritten as \n$$\\nu_t(\\omega: \\widehat{g} (\\omega) = k) - \\nu_t(\\omega \\in I_t: \\widehat{g}(\\omega) = k)$$\n\n\\noindent Again using our bound on $\\nu_t(I_t)$, the difference $$|\\nu_t(\\omega: \\widehat{F_t}(\\omega) = k ) -\\nu_t(\\omega: \\widehat{g} (\\omega) = k)|$$ can be bounded by $4 \\epsilon$. Passing to the limit, we obtain that the limits can differ by no more than $4 \\epsilon$. Since $\\epsilon$ was arbitrary, we have that the limits must be equal. \\qed\\medskip\n\n\n\\subsubsection{The Siegel-Veech formula}\\label{subsubsec:SV} Using the Siegel-Veech transform, we can obtain results on the \\emph{expected} number of points in a thinning wedge. We fix some notation. Recall that for a bounded compactly supported function $f: \\bb R^2 \\backslash \\{0\\} \\rightarrow \\bb R$, we define $\\widehat{f}(\\omega) = \\sum_{\\mathbf v \\in \\Lambda_{\\omega}} f(\\mathbf v)$. Fixing $R, c, \\sigma$, let $h_{\\theta} = \\chi_{A^{\\theta}_R(c, \\sigma)}$, that is, it is the indicator function of the thinning wedge. The expected number of lattice points in a random thinning wedge can be written as\n\n$$\\sum_{k=1}^{\\infty} k \\lambda(\\theta: N^{\\theta}_R(\\omega, \\sigma, c) = k) = \\int \\widehat{h_{\\theta}}d\\lambda(\\theta).$$\n\n\\noindent Writing $h$ for the indicator function of $T(c, \\sigma)$, and using (\\ref{eq:lambda:wedge}) we can write this (for $t>>0$) as \n$$\\int_{\\mathcal H} \\widehat{h} d\\nu_{t,\\omega}.$$\n\n\\noindent If $\\omega_0$ has a circle limit measure $\\mu_{0}$ satisfying a certain technical condition~\\cite[Theorem 8.2(D)]{EMM}, then \n$$\\lim_{t \\rightarrow \\infty} \\int_{\\mathcal H} \\widehat{h} d\\nu_{t,\\omega} = \\int_{\\mathcal H} \\widehat{h} d\\mu_0.$$\n\n\\noindent As above, let $b_0$ be the Siegel-Veech constant for $\\mu_0$. Then we have\n\\begin{equation}\\label{eq:siegel:veech}\\int_{\\mathcal H} \\widehat{h} d\\mu_0 = b_0 \\int_{R^2} h.\\end{equation}\n\n\\noindent That is, the expected number of points in a thinning wedge is proportional to the volume of the trapezoid $T(c, \\sigma)$. We will see in \\S\\ref{subsec:second:moment} that the \\emph{second moment} of the limiting distribution is \\emph{not} well-defined.\n\n\\subsubsection{Fiber bundles and special trajectories}\\label{subsubsec:fiber} In this paper, we focus on the sets of holonomy vectors of oriented saddle connections or cylinders. Our results will also apply to the sets of holonomy vectors connecting a fixed point on the surface to singular points or the set of vectors connecting two marked points. These can be obtained by considering the corresponding equidistribution results for the spaces $Y_i$ of translations surfaces with $i$ marked points, $i=1, 2$. These spaces can be viewed as fiber bundles over $\\mathcal H$ with fiber over $\\omega \\in \\mathcal H$ given by $(M, \\omega)^i$. We refer the interested reader to~\\cite[\\S2, \\S9]{EskinMasur} for details.\n\n\\section{Measure bounds and gap distribution}\\label{sec:measure:bounds}\nThis section provides a variety of results on the gaps between saddle connection and cylinder directions. It includes results on the likelihood of finding many saddle connections in a small region. In particular, Lemma \\ref{k asym} says that having $k$ saddle connections in a small interval is proportional to having two saddle connections in a small interval. As a corollaries we show Corollary \\ref{cor:second:moment} which says that the limiting distribution of $p_k(\\sigma)$ does not have finite second moments and Corollary \\ref{SV not L2} which says that the Siegel-Veech transform does not send continuous compactly supported functions to $L_2$ (though it is norm preserving for positive functions as a map from $L_1$ to $L_1$). We also show Theorem \\ref{p sig 0 lower} which says that some gaps between cylinder directions are larger than one would expect .\n\\subsection{Notation}\\label{subsec:not}\n\nIn this section, we will need to distinguish between holonomy vectors of saddle connections and cylinders. Let $\\omega \\in \\mathcal H$, let $\\Lambda^{sc}_{\\omega}$ and $\\Lambda^{cyl}_{\\omega}$ be as in (\\ref{eq:la:def}), and let $\\mu$ be Lebesgue measure on $\\mathcal H$. We define\n\n\\begin{eqnarray}\\label{eq:tildepksigma:def} p_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{sc}_{\\omega} \\cap T(\\sigma) = k) \\\\ \\nonumber \\tilde{p}_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{sc}_{\\omega} \\cap T(\\sigma) \\geq k)\\\\ \\nonumber p^{cyl}_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{cyl}_{\\omega} \\cap T(\\sigma) = k) \\\\ \\nonumber \\tilde{p}^{cyl}_k(\\sigma) :&=& \\mu(\\omega: \\Lambda^{cyl}_{\\omega} \\cap T(\\sigma) \\geq k).\\\\ \\nonumber \\end{eqnarray}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\sigma \\rightarrow 0$ asymptotics}\\label{subsec:sigma:0}\n\\begin{Prop}$ \\underset{\\sigma \\to 0}{\\lim}\\, p_{\\sigma,1}=0$. \n\\end{Prop}\n\n\\begin{proof} This follows from the fact that the volume of $T(\\sigma) \\rightarrow 0$ as $\\sigma \\rightarrow 0$.\\end{proof}\n\n\\begin{Theorem}\\label{sig to 0} As $\\sigma \\rightarrow 0$,\n$\\sigma^{-2} p_{\\sigma,2}$ is bounded away from 0 and infinity.\n\\end{Theorem}\n\\begin{proof}\nConsider $g_{-\\sqrt{\\sigma}} T(c,\\sigma)$. Because the action of\n$g_t$ preserves $\\mu$ on $\\mathcal{H}$ it follows that the measure\nof surfaces with two saddle connections $T(c, \\sigma)$ is equal to\nthe measure of surfaces with two saddle connections in\n$g_{-\\sqrt{\\sigma}}T(c,\\sigma)$. It follows from Lemma\n\\ref{lemma:short:saddle:connections} and the remark following it that $$\\mu(\\{\\omega:\n\\Lambda_{\\omega}\\cap T(c,\\sigma)\\geq 2\\}) \\sim \\sqrt{\\sigma}^2\\sqrt{\\sigma}^2=\\sigma^2,$$ where $\\sim$ denotes proportionality.\n\\end{proof}\n\n\nThis result states that given a saddle connection appearing in a wedge the probability of having another one is roughly independent. In general this is false, as the next section shows the probability of having many saddle connections in a small wedge decays slowly.\n\n \n\n\n\\subsection{More on $\\sigma \\rightarrow 0, k \\rightarrow \\infty$ }\nFirst,\n\\begin{lemma} \\label{2k above} For any fixed $\\sigma>0$ we have $\\underset{k \\to \\infty}{\\limsup} \\, k^2 \\tilde{p}^{\\text{cyl}}_k(\\sigma)<\\infty$.\n\\end{lemma}\n\nThis is similar to the proof of Theorem \\ref{sig to 0}. We say a surface has an $\\epsilon$\\emph{-thin neck} if there exists a pair of saddle connections $\\mathbf v$\n and $\\mathbf w$ such that $|\\mathbf v| \\leq \\epsilon$ and $\\mathbf w$, $\\mathbf v$, $\\mathbf w$\n are adjacent saddle connections. See Figure~\\ref{thin neck} below. \n \\makefig{An $\\epsilon$-thin neck, i.e., $|\\mathbf v| < \\epsilon$.}{thin neck}{\\input{thin_neck.pstex_t}} \n\n\n If $\\mathbf v$, $\\mathbf w$ define a $\\frac{\\sigma}{3k}$ thin neck in $\\omega$ and \n $\\frac 1 3 \\leq |\\mathbf w| \\leq \\frac 2 3$ then $N_1^{\\arg (\\mathbf w)}(\\omega,\\sigma,0)\\geq k$.\n To see this notice that a saddle connection that goes once in the $\\mathbf w$ direction and $l$ times in the $\\mathbf v$\n direction has its associated vector in $\\Lambda_{\\omega}$ contained in $A_1^{\\arg(\\mathbf w)}(0,\\sigma)$. \n\\begin{lemma}\\label{k asym} For any fixed $\\sigma >0$ we have $ k^2 \\tilde{p}_k(\\sigma)$ is bounded away from 0 and $\\infty$.\n\\end{lemma}\n\\begin{lemma} \\label{sig asym} For any fixed $k>1$ we have $\\sigma^{-2} \\tilde{p}_k(\\sigma)$ is bounded away from $0$ and $\\infty$.\n\\end{lemma}\nThese two lemmas follow from the previous paragraph by noticing that the measure of surfaces with an $\\epsilon$-thin neck is proportional to $\\epsilon^2$ (see \\S \\ref{subsec:coord:meas}). Showing that it is bounded away from $\\infty$ follows from Theorem \\ref{sig to 0} and Lemma \\ref{2k above}.\n\n\\subsubsection{Non-existence of second moments}\\label{subsec:second:moment} Lemma~\\ref{k asym} has the following corollary.\n\n\\begin{Cor}\\label{cor:second:moment} For any $\\sigma >0$, $$\\sum_{k =0}^{\\infty} k^2 p_k(\\sigma) \\mbox{ diverges }.$$ That is, the limiting distribution $p_k(\\sigma)$ does not have finite second moment.\\end{Cor}\n\n\\medskip\n\\noindent The corollary follows from Lemma~\\ref{k asym} and the general lemma below:\n\n\\begin{lemma}\\label{lemma:prob} Let $X$ be a positive integer valued random variable, with $P(X = k) = p_k$. Suppose there is a $c >0$ so that $P(X \\geq k ) \\geq \\frac{c}{k^2}$. Then\n$$\\sum_{k =0}^{\\infty} k^2 p_k \\mbox{ diverges}.$$\n\\end{lemma}\n\n\\begin{proof} We are interested in calculating $E(X^2) = \\sum_{k=0}^{\\infty} k^2 p_k$. We can write\n$$E(X^2) = \\sum_{k=0}^{\\infty} P(X^2 \\geq k)$$ For $i^2 \\le k < (i+1)^2$, $P(X^2 \\geq k) = P(X \\geq i)$. Thus $$\\sum_{k=0}^{\\infty} P(X^2 \\geq k) = \\sum_{i=1}^{\\infty} 2i P(X \\geq i) \\geq \\sum_{i=1}^{\\infty} \\frac{c}{i}.$$\\end{proof}\n\n\n\n\\subsubsection{Glued-in tori} For the remaining estimates we consider gluing in a small torus.\n We say a surface has a \\emph{glued in\ntorus} with parameters $(a,b)$ and gluing slit $s$ there is a portion of the surface where the points travel as if they are in a torus with basis lengths $a,b$ except if they cross a saddle connection of length at most $s$. See Figure~\\ref{almost torus} below.\n\n \\makefig{The slit torus is glued to the rest of the surface along the saddle connection of length at most $s$.}{almost torus}{\\input{almost_torus.pstex_t}} \n\n\\begin{Prop}\n The measure of unit volume surfaces that have a glued in torus with\nparameters $(a,b)$ and gluing slit $s$ where $a, b \\in\n[\\sqrt{c},2\\sqrt{c}]$ and $s \\in [c,2c]$ is at least proportional to $c^{-4}$\nas $c $ goes to zero.\n\\end{Prop}\nThis follows from the main result in \\S \\ref{subsec:coord:meas}. If we glue in a torus with parameters comparable to $(\\sqrt{\\sigma}$,\n$\\sqrt{\\sigma})$ with gluing slit $\\sigma$ it has quadratic growth of\nsaddle connections or periodic cylinders. Because a torus is a\nlattice surface, the saddle connection directions are completely\nperiodic. If the length of the periodic cylinder is less than $t$\nthen the trajectory in the torus crosses the direction of the slit\nat most $c\\frac{t}{\\sqrt{\\sigma}}=ct\\sqrt{\\sigma}^{-1}$ times. \nSo some points do not leave the torus before\nclosing up. Therefore all saddle connection directions of the torus\nwith length less than $\\frac{\\sigma}{c}$ are cylinder directions for the\nsurface.\n\nIt follows that there exists $C>0$ such that for any $\\sigma>0$ small\nenough a surface that has a torus with gluing parameters\n$\\sqrt{\\sigma}$, $\\sqrt{\\sigma}$, not too small angle between these sides and gluing slit $\\sigma$ has at\nleast $C \\sigma^{-1}$ periodic directions whose cylinders have\nlength less than or equal to 1. It follows from the pigeonhole\nprinciple that almost half of these directions are separated by at\nmost $4\\pi C \\sigma^{-1}$.\n\n\n\n\\begin{lemma}\nFor any fixed $k>0$ we have\n$$\\underset{\\sigma \\to 0}{\\liminf}\\,\n\\sigma^{-4}\\, \\tilde{p}^{cyl}_k(\\sigma)>0.$$\n\\end{lemma}\n\n\n\\begin{proof}Glue in a torus of parameters comparable to $\\sqrt{\\sigma}$,$\\sqrt{\\sigma}$ with gluing slit $\\sigma$.\\end{proof}\n\\begin{lemma}For any fixed $\\sigma>0$ we have\n$$\\underset{k \\to \\infty}{\\liminf}\\, k^{4} \\tilde{p}^{cyl}_k>0.$$\n\\end{lemma}\n\\begin{proof}Glue in a torus of parameters comparable to $\\sqrt{2 \\sigma\nk}^{-1}$, $\\sqrt{2\\sigma k}^{-1}$ with gluing slit $(2\\sigma k)^{-1}$.\\end{proof}\n\n\n\\begin{Cor}\\label{SV not L2} Let $f: \\mathbb{R}^2 \\backslash \\{0\\} \\to \\mathbb{R}$ by $f(x)=1$ if $x \\in B(0,1) \\backslash \\{0\\}$ and 0 otherwise. Its Siegel-Veech transform $\\hat{f}$ is not in $L^{2}(\\mathcal H, \\mu)$. However $\\hat{f}$ is in $L^{2-\\epsilon}(\\mathcal H,\\mu)$ for any $\\epsilon>0$.\n\\end{Cor}\n\\medskip\n\n\\noindent This is an immediate consequence of Lemma \\ref{k asym}. Notice that $f$ is in $L^{\\infty}(\\mathbb{R}^2)$. The above corollary fails if we take the analogue of the Siegel-Veech transform for the directions of cylinders. That is, let \n$$\\tilde{f}(\\omega)= \\sum_{\\mathbf v \\in \\Lambda_{\\omega}^{cyl}} f(\\mathbf v) .$$ In this case the function is not in $L^{4}(\\mathcal H, \\mu)$. However, if we fix the minimal volume of cylinders we consider then by Lemma \\ref{cyl flow} this variant of the Siegel-Veech transform sends $L^{\\infty}(\\mathbb{R}^2\\backslash \\{0\\})$ to $L^{\\infty}(\\mathcal H,\\mu)$. The $L^{\\infty}$ norm may increase and that this increase can be bounded by the genus of the surfaces parametrized by $\\mathcal H$ and the lower bound on the volume of the cylinders. Thus while the different versions of the Siegel-Veech transform are all $L^1$ norm preserving on positive functions they have different behavior in $L^p$ in general.\n\n\\subsection{$\\sigma \\rightarrow \\infty$ asymptotics}\\label{subsec:sigma:infty}\n\\begin{lemma}\\label{cyl flow} Suppose $x$ is in a periodic cylinder of length $L$, area $a$ in direction $\\theta$. Then $x$ is not in a periodic cylinder of length less than $R$ and direction $\\theta'$ where $ \\theta' \\neq \\theta$ and $|\\theta'-\\theta|<\\min\\{\\frac 1 {2LRa}, \\frac{\\pi}{3}\\}$.\n\\end{lemma}\n\\begin{proof} Consider the periodic cylinder in the hypothesis of the lemma.\n It has a width vector $\\mathbf v$ where $|\\mathbf v|=\\frac{a}{L}$ and let $x \\in \\mathbf v$. If $x+L\\tan(\\epsilon)< \\frac a L$ then $F_{\\theta+\\epsilon}|_{\\mathbf v}(x)=x+ L \\tan(\\epsilon)$, where $F_{\\theta+\\epsilon}|_{\\mathbf v}$ denotes the induced map of $F_{\\theta+\\epsilon}$ on $\\mathbf v$. It follows that if $\\theta+\\epsilon$ is the direction of a periodic cylinder $x$ lies in we have $F_{\\theta+\\epsilon}|_{\\mathbf v}^k(x)=x$ and so $kL\\tan(\\epsilon)\\geq |\\mathbf v|.$ The length of this cylinder is at least $kL\\sec(\\epsilon)$. Therefore if $kL\\frac{|\\mathbf v|}{R}=\\frac{a}{RL}.$ Noticing that $\\epsilon< \\frac{\\pi}{3}$ implies $\\tan(\\epsilon)<2\\epsilon$ completes the lemma.\n\\end{proof}\n\n \n \\subsubsection{Absence of cylinder gaps}\\label{subsec:cylinder:gap}\nLet $\\Theta_{\\omega}^{*}(R)$ be the directions $\\theta$ such that the volume of the periodic cylinders in direction $\\theta$ with length than or equal to $R$ is at least $\\frac{3}{5}$. In the rest\n\\begin{lemma}\\label{farey} Let $\\theta \\in \\Theta_{\\omega}^{*}(L)$ then\n $$\\left(\\theta - \\frac{1}{20(4g-4)},\\theta+ \\frac{1}{20(4g-4)RL}\\right)\n\\cap \\Theta_{\\omega}^{*}(R)=\\theta.$$\n\\end{lemma}\n\\begin{proof} One can easily see that the number of \nsaddle connections in a given direction is at most $4g-4$.\n Therefore $\\theta \\in \\Theta_{\\omega}^{*}(L)$ \nthen at least half the points in the surface must lie in \ncylinders of area at least $\\frac{1}{10(4g-4)}$.\n If $\\phi \\in \\Theta_{\\omega}^{\\frac3 5}(R)$ then some points of the surface\n must lie in periodic cylinders in direction $\\phi$ and\n periodic cylinders of area at least $\\frac 1 {10(4g-4)}$\n in direction $\\theta.$\nThe result follows from Lemma \\ref{cyl flow}.\n\\end{proof}\nIn the square torus, periodic directions correspond to rational slopes. Notice that if $\\frac a L \\neq \\frac b R $ are rational numbers and $CL\\frac{C}{R^2}$. This implies that for any $C'>0$ a positive proportion of periodic directions of length less than $R$ are at least $\\frac{C'}{R^2}$ separated from the closest periodic direction of length less than $R$. The next theorem generalizes this fact for periodic cylinders of substantial area.\n\n\\begin{Theorem}\\label{p sig 0 lower}\nIf the cardinality of $\\Theta_{\\omega}^{*}(R)=\\{\\theta_1 \\leq \\theta_2 \\leq \\dots \\theta_N\\}$\n grows quadratically then \n $$\\underset{\\sigma \\to \\infty}{\\liminf} \\, \\underset{R \\to \\infty}{\\liminf} \\,\\sigma^2 \\frac{|\\{\\theta_i \\in \\Theta_{\\omega}^{*}(R): \\theta_{i+1}-\\theta_i \\geq \\frac{\\sigma}{R}\\}|}{|\\Theta_{\\omega}^{\\frac 3 5 }(R)|}>0.$$\n\\end{Theorem}\n\\begin{proof} By the fact that $\\omega$ has quadratic growth there exists $c>0$ such that $$\\frac{|\\Theta_{\\omega}^{*}(L)|}{|\\Theta_{\\omega}^{*}(R)|}\\geq c \\left(\\frac L R\\right)^2.$$ Lemma \\ref{farey} implies that if\n $\\theta \\in \\Theta_{\\omega}^{*}(L)$ then \nit has a gap of size at least $\\frac 1 {20(4g-4)RL}$ to the closest\ndirection in $\\Theta_{\\omega}^{*}(R)$. Let $L=\\frac {R}{\\sigma 20(4g-4)}$.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Billiards}\\label{sec:billiards} In this section we use Theorem~\\ref{theorem:wedge:limit} to obtain results about special trajectories for billiards, as discussed in \\S\\ref{subsubsec:billiards:barriers}. We follow closely the exposition in~\\cite{EMM, EMS}, particularly focusing on the examples studied in the latter paper.\n\n\n\\subsection{Rectangles with barriers}\\label{subsec:barrier} We recall notation from \\S\\ref{subsubsec:billiards:barriers}: given $\\alpha \\in \\bb R$, consider the polygon $P_{\\alpha}$ whose boundary is the boundary of the square $[0, 1] \\times [0, 1]$ together with a barrier given by the vertical segment $\\{1\/2\\} \\times [0, \\alpha]$. \n\nWe recall the `unfolding' procedure from~\\cite{EMS}: to obtain a translation surface $\\omega_{\\alpha}$ from $P_{\\alpha}$, take four copies of $P= P_{\\alpha}$ which are images of $P$\nunder reflection in the two coordinate axes and reflection in the origin. \n\nIdentifying the interior sides, we obtain a square of area 4 with two vertical double lines, corresponding to the interval $1\/2 \\times [0,\\alpha]$. Identify the top and bottom of the square, and the left and right sides. The glue the left side of the\nright line to the right side of the left line, and the right side of\nthe right line to the left side of the left line. See Figure~\\ref{unfold} below.\n\n\n\\makefig{Unfolding $P_{\\alpha}$ to $\\omega_{\\alpha}$.}{unfold}{\\input{developwall.pstex_t}} \n\n\nRescaling by $1\/4$, we obtain an area $1$ translation surface $\\omega_{\\alpha} \\in \\mathcal H(1,1)$, with the $2$ zeroes located at the endpoints of the vertical lines. A billiard trajectory $\\lambda$ on $P_{\\alpha}$ corresponds to a straight line on $\\omega_{\\alpha}$.\n\n\\subsection{Branched covers}\\label{subsec:branch} As mentioned in \\S\\ref{subsubsec:billiards:barriers}, the crucial property of the surface $\\omega_{\\alpha}$ is that it is a double (branched) cover of the torus. That is, there is a covering map (branched at the zeros) $\\pi: \\omega_{\\alpha} \\rightarrow \\bb C\/\\Lambda$ where $\\Lambda$ is a lattice in $\\bb C$ and that $\\omega_{\\alpha}$ is obtained by pulling back the form $dz$. \n\nThe set of all $\\omega \\in \\mathcal H(1, 1)$ satisfying this property is a closed, $SL(2, \\bb R)$-subvariety $\\mathcal M$~\\cite[Lemma 2.1]{EMS}. $\\mathcal M$ is a finite cover of $\\mathcal T = (SL(2, \\bb R) \\ltimes \\bb R^2)\/(SL(2, \\bb Z) \\ltimes \\bb Z^2)$ and the covering map commutes with the $SL(2,\\bb R)$-action~\\cite[Lemma 2.2]{EMS}. Recall that $\\mathcal T$ is the space of tori with two marked points (assuming that one marked point is always at the origin). The covering map $\\Pi: \\mathcal M \\rightarrow \\mathcal T$ is given by \n$$\\Pi(\\omega) = (\\Lambda, \\pi(z_1), \\pi(z_2)),$$ \n\\noindent where $\\Lambda \\subset \\bb C$ is the lattice so that $\\omega$ covers $\\bb C\/\\Lambda$ and $z_1, z_2$ are the zeros of $\\omega$. \n\nLet $\\alpha \\in \\bb R$ be irrational. For $\\omega_{\\alpha} \\in \\mathcal M$, let $\\mathcal M(\\alpha)$ denote the connected component of $\\mathcal M$ containing $\\omega_{\\alpha}$. Let $\\bar{\\mu}$ denote the pullback of the Haar probability measure on $\\mathcal T$. It is an ergodic $SL(2, \\bb R)$ invariant measure on $\\mathcal M(\\alpha)$. The circle limit measure for $\\omega_{\\alpha}$ is $\\bar{\\mu}$~\\cite[Lemma 2.4]{EMS}. Thus, we obtain the following corollary to Theorem~\\ref{theorem:wedge:limit}.\n\n\\begin{Cor}\\label{cor:wedge:billiard} For any irrational $\\alpha$,\n$$\\lim_{R \\rightarrow \\infty} \\lambda(\\theta: N^{\\theta}_R(\\omega_{\\alpha}, \\sigma, c) = k) = \\bar{\\mu}(\\omega \\in \\mathcal M(\\alpha): \\Lambda^{sc}_{\\omega} \\cap T(c, \\sigma) = k).$$\n\\end{Cor}\n\n\\subsubsection{Branched covers of lattice surfaces}\\label{subsubsec:branch:lattice} A similar calculation of circle limit measures was carried out for branched covers of lattice surfaces in~\\cite{EMM}. Thus, we can obtain a version of Corollary~\\ref{cor:wedge:billiard} for these surfaces as well. This will yield results for triangular billiards $P_n$ with angles\n $$ \\frac{n-2}{2n} \\pi, \\ \\frac{n-2}{2n} \\pi, \\ \\frac{4}{2n} \\pi\n ,$$ where $n \\ge 5$, $n$ odd. For details, see~\\cite[\\S9]{EMM}. \n\n\\subsection{Lattice translates}\\label{subsec:translate} To obtain a version of Theorem~\\ref{theorem:main:gap} for $\\omega \\in \\mathcal M$, we have to understand the set of holonomy vectors of saddle connections $\\Lambda^{sc}_{\\omega}$. Suppose $\\omega$ is a cover of $\\bb C\/\\Lambda$, with covering map $\\pi$. Note that under the projection to the torus, the saddle connections must connect either $\\pi(z_1) = 0$ or $\\pi(z_2)$ to themselves or to each other. That is, they must be primitive vectors in the lattice $\\Lambda$ or in the translate $\\Lambda + \\mathbf v$ where $\\mathbf v$ is a choice of vector connecting the two marked points on the torus (defined up to $\\Lambda$, so $\\Lambda + \\mathbf v$ is well-defined). Thus, if we have $\\Pi(\\omega) = (\\Lambda, \\mathbf v)$ (viewing $\\mathcal T$ as the space of marked tori, or equivalently, lattices and a choice of vector), we have $$\\Lambda^{sc}_{\\omega} = \\Lambda_{prim} \\cup ( \\Lambda_{prim} + \\mathbf v),$$ where $\\Lambda_{prim}$ denotes the set of primitive vectors in $\\Lambda$. \n\n$\\mathcal T$ is a fiber bundle over the modular surface $SL(2,\\bb R)\/SL(2, \\bb Z)$. It can be broken up into a Haar measure on $SL(2, \\bb R)\/SL(2,\\bb Z)$ together with Lebesgue measure on the torus fibers. While a lattice $\\Lambda$ will never have two points in $T(0, \\sigma)$ for $\\sigma < <1$, we can construct a positive measure set of pairs $(\\Lambda, \\mathbf v) \\in \\mathcal T$ so that $\\Lambda_{prim} \\cup (\\Lambda_{prim} + \\mathbf v)$ does intersect $T(\\sigma)$ (at least) twice for all $\\sigma >0$ by considering the lattices $\\Lambda$ so that $\\Lambda \\cap T(\\sigma) \\neq \\emptyset$ and $\\mathbf v \\in T(\\sigma)$ ($\\mathbf v \\notin \\Lambda$). Thus for all $\\alpha$ irrational, all $\\sigma >0$, we have\n$$\\bar{\\mu}(\\omega \\in \\mathcal M(\\alpha): \\Lambda^{sc}_{\\omega} \\cap T(\\sigma) \\geq 2) >0.$$\n\\noindent (In fact, we will obtain a set of measure proportional to $\\sigma^4$) (see~\\cite[Remark 2.3]{MS} for an explicit description of the distribution of gaps for the set $\\mathbf v + \\Lambda$). Thus, we obtain:\n\n\n\\begin{Cor}\\label{cor:main:gap} For all irrational $\\alpha$, \n\\begin{equation}\\label{eq:cor:gap}\\lim_{R \\rightarrow \\infty} R^2 \\gamma^{\\omega_{\\alpha}}(R) = 0.\n\\end{equation}\n\\noindent Moreover, for any $\\epsilon >0$, the proportion of gaps less than $\\epsilon\/R^2$ is positive. That is, writing $\\Theta^{\\omega_{\\alpha}}_R : = \\{0 \\le \\theta_1 \\le \\theta_2 \\le \\ldots \\le \\theta_n\\}$, we have\n\\begin{equation}\\label{eq:cor:proportion}\n\\lim_{R \\rightarrow \\infty} \\frac{ |\\{1 \\le i \\le \\tilde{N}(\\omega_{\\alpha}, R): (\\theta_{i+1} - \\theta_i) \\le \\epsilon\/R^2\\}|}{\\tilde{N}(\\omega_{\\alpha}, R)} >0\n\\end{equation}\n\\end{Cor}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Newton-Puiseux Theorem states that, if $f\\left(x,y\\right)$ is\nan analytic germ in two variables, then the solutions $y=\\varphi\\left(x\\right)$\nof the equation $f=0$ can be expanded as Puiseux series that are\nconvergent in a neighbourhood of the origin (see for example \\cite{brieskorn_knorrer;plane_algebraic_curve}).\nA multivariable version of this result in the real case states that,\nif $f\\left(x_{1},\\ldots,x_{m},y\\right)$ is a real analytic germ,\nthen, after a finite sequence of blow-ups with centre a real analytic\nmanifold, the solutions $y=\\varphi\\left(x_{1},\\ldots,x_{m}\\right)$\nof the equation $f=0$ are analytic in a neighbourhood of the origin\n(see for example \\cite[Theorem 4.1]{parusinski:preparation}). An\nequivalent formulation states that the solutions $y=\\varphi\\left(x_{1},\\ldots,x_{m}\\right)$\nin a neighbourhood of the origin are obtained, piecewise, as finite\ncompositions of analytic functions, taking $n^{\\text{th}}$ roots\nand quotients (see for example \\cite[Corollary 2.15]{dmm:exp} and\n\\cite[Theorem 1]{lr:prep}).\n\n\n\\paragraph{{}}\n\nHere we extend this result to functions belonging to a \\emph{generalised\nquasianalytic class} (see Definition \\ref{def:qa class}). Roughly,\na generalised quasianalytic class is a collection of algebras of continuous\nreal-valued functions together with an \\emph{injective} $\\mathbb{R}$-algebra\nmorphism $\\mathcal{T}$ which, given the germ at zero $f$ of a function\nin the collection, associates to $f$ a formal power series $\\mathcal{T}\\left(f\\right)$\nwith natural or real exponents. Given a generalised quasianalytic\nclass, we already have a local uniformisation result \\cite{rsw,martin_sanz_rolin_local_monomialization,rolin_servi:qeqa}\nwhich allows to parametrise the zero set of a function in the class.\nOur aim here is to refine this procedure, in the spirit of the elimination\nresult in \\cite{vdd:d}, in the following way: given a function $f\\left(x,y\\right)$\nin the class under consideration, we provide a uniformisation algorithm\nwhich ``respects'' the variable $y$ and hence allows to solve the\nequation $f=0$ with respect to $y$. \n\n\n\\paragraph{{}}\n\nExamples of generalised quasianalytic classes are the following (see\nRemark \\ref{rem: thm abcde}).\n\n\n\\paragraph{{}}\n\n\\noindent a) Let $M=\\left(M_{0},M_{1},\\ldots\\right)$ be an increasing\nsequence of positive real numbers (with $M_{0}\\geq1$) and $B\\subseteq\\mathbb{R}^{m}$\nbe a compact box. We assume that $M$ is strongly log-convex and we\nconsider the Denjoy-Carleman algebra of functions $\\mathcal{C}_{B}\\left(M\\right)$\ndefined in \\cite{rsw}. This is an algebra of functions $f:B\\to\\mathbb{R}$\nwhich each extend to a $\\mathcal{C}^{\\infty}$ function on some open\nneighbourhood $U\\supseteq B$ and whose derivatives satisfy a certain\ntype of bounds depending on $M$ (see \\cite[p. 751]{rsw}). The functions\nin $\\mathcal{C}_{B}\\left(M\\right)$ are not analytic in general, however,\nif $\\sum_{i\\in\\mathbb{N}}\\frac{M_{i}}{M_{i+1}}=\\infty$, then $\\mathcal{C}_{B}\\left(M\\right)$\nis \\emph{quasianalytic}, i.e. for every $x\\in B$, the algebra morphism\nwhich associates to $f\\in\\mathcal{C}_{B}\\left(M\\right)$ its (divergent)\nTaylor expansion at $x$ is injective. The quasianalytic Denjoy-Carleman\nclass $\\mathcal{C}\\left(M\\right)$ is the union of the collection\n$\\left\\{ \\mathcal{C}_{B}\\left(M\\right):\\ m\\in\\mathbb{N},\\ B\\subseteq\\mathbb{R}^{m}\\ \\text{compact\\ box}\\right\\} $. \n\n\n\\paragraph{{}}\n\nb) Let $H=\\left(H_{1},\\ldots,H_{r}\\right):\\left(0,\\varepsilon\\right)\\to\\mathbb{R}^{r}$\nbe a $\\mathcal{C}^{\\infty}$ solution of a system of first order singular\nanalytic differential equations of the form $x^{p+1}y'\\left(x\\right)=A\\left(x,y\\right)$,\nwhere $A$ is real analytic in a neighbourhood of $0\\in\\mathbb{R}^{p+1}$,\nsatisfying conditions a) and b) in \\cite[p. 413]{rss}, and $A\\left(0,0\\right)=0$.\nSuppose furthermore that $H$ admits an asymptotic expansion for $x\\rightarrow0^{+}$\nas in \\cite[2.2]{rss}. As in \\cite[Section 3]{rss}, we let $\\mathcal{A}_{H}$\nbe the smallest collections of real germs containing the germ at zero\nof the $H_{i}$ and closed under composition, monomial division and\ntaking implicit functions. A function $f$, defined on an open set\n$U\\subseteq\\mathbb{R}^{m}$, is said to be $\\mathcal{A}_{H}$-\\emph{analytic\n}if for every $a\\in U$ there exists a germ $\\varphi_{a}\\left(x\\right)\\in\\mathcal{A}_{H}$\nsuch that the germ of $f\\left(x\\right)$ at $a$ is equal to the germ\n$\\varphi_{a}\\left(x-a\\right)$. It is proven in \\cite{rss} that the\ncollection of all $\\mathcal{A}_{H}$-analytic functions forms a quasianalytic\nclass of $\\mathcal{C}^{\\infty}$ functions.\n\n\n\\paragraph{{}}\n\n\\noindent c) A (formal) generalised power series in $m$ variables\n$X=\\left(X_{1},\\ldots,X_{m}\\right)$ is a series $F\\left(X\\right)=\\sum_{\\alpha}c_{\\alpha}X^{\\alpha}$\nsuch that $\\alpha\\in[0,\\infty)^{m}$, $c_{\\alpha}\\in\\mathbb{R}$ and\nthere are well-ordered subsets $S_{1},\\ldots,S_{m}\\subseteq[0,\\infty)$\nsuch that the support of $F$ is contained in $S_{1}\\times\\ldots\\times S_{m}$\n(see \\cite{vdd:speiss:gen}). The series $F$ is convergent if there\nis a polyradius $r=\\left(r_{1},\\ldots,r_{m}\\right)\\in\\left(0,\\infty\\right)^{m}$\nsuch that $\\sum_{\\alpha}|c_{\\alpha}|r^{\\alpha}<\\infty$. A convergent\ngeneralised power series gives rise to a real-valued function $F\\left(x\\right)=\\sum c_{\\alpha}x^{\\alpha}\\in\\mathbb{R}\\left\\{ x^{*}\\right\\} _{r}$,\nwhich is continuous on $[0,r_{1})\\times\\ldots\\times[0,r_{m})$ and\nanalytic on the interior of its domain. We denote by $\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $\nthe algebra of all formal generalised power series and consider the\nalgebra $\\mathbb{R}\\left\\{ x^{*}\\right\\} =\\bigcup_{r\\in\\left(0,\\infty\\right)^{m}}\\mathbb{R}\\left\\{ x^{*}\\right\\} _{r}$\nof all convergent generalised power series. Examples of convergent\ngeneralised power series are the function $\\zeta\\left(-\\log x\\right)=\\sum_{n=1}^{\\infty}x^{\\log n}$\n(where $\\zeta$ is the Riemann zeta function) and the solution $f\\left(x\\right)=\\sum_{n,i=0}^{\\infty}\\frac{1}{2^{i}}x^{2+n-\\frac{1}{2^{i}}}$\nof the functional equation $\\left(1-x\\right)f\\left(x\\right)=x+\\frac{1}{2}x\\left(1-\\sqrt{x}\\right)f\\left(\\sqrt{x}\\right)$.\n\n\n\\paragraph{{}}\n\n\\noindent d) For $R=\\left(R_{1},\\ldots,R_{m}\\right)\\in\\left(0,\\infty\\right)^{m}$\na polyradius, we consider the algebra $\\mathcal{G}\\left(R\\right)$\nof functions defined in \\cite[Definition 2.20]{vdd:speiss:multisum}\nby means of sums of multisummable formal series in the real direction.\nIts elements are $\\mathcal{C}^{\\infty}$ functions defined on $\\left[0,R_{1}\\right]\\times\\ldots\\times\\left[0,R_{m}\\right]$\nand their derivatives satisfy a Gevrey condition. By a known result\nin multisummability theory, these algebras satisfy the following quasianalyticity\ncondition: the morphism, which associates to the germ at zero of a\nfunction in $\\mathcal{G}\\left(R\\right)$ its (divergent) Taylor expansion\nat the origin, is injective (see \\cite[Proposition 2.18]{vdd:speiss:multisum}).\nWe let $\\mathcal{G}$ be the union of the collection $\\left\\{ \\mathcal{G}\\left(R\\right):\\ m\\in\\mathbb{N},\\ R\\in\\left(0,\\infty\\right)^{m}\\right\\} $.\nThis collection contains the function $\\psi\\left(x\\right)$ appearing\nin Binet's second formula, i.e. such that $\\log\\Gamma\\left(x\\right)=\\left(x-\\frac{1}{2}\\right)\\log\\left(x\\right)+\\frac{1}{2}\\log\\left(2\\pi\\right)+\\psi\\left(\\frac{1}{x}\\right)$,\nwhere $\\Gamma$ is Euler's Gamma function (see \\cite[Example 8.1]{vdd:speiss:multisum}). \n\n\n\\paragraph{{}}\n\n\\noindent e) For $r\\in\\left(0,\\infty\\right)^{m+n}$ a polyradius,\nwe consider the algebra $\\mathcal{Q}_{m,n,r}$ defined in \\cite[Definition 7.1]{krs}.\nIts elements are continuous real-valued functions which have a holomorphic\nextension to some ``quadratic domain'' $U\\subseteq\\mathbf{L}^{m+n}$,\nwhere $\\mathbf{L}$ is the Riemann surface of the logarithm. One can\ndefine a morphism $T$ which associates to the germ $f$ of a function\nin $\\mathcal{Q}_{m,n,r}$ an \\emph{asymptotic expansion} $T\\left(f\\right)\\in\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $.\nIt is shown in \\cite[Proposition 2.8]{krs}, using results of Ilyashenko's\nin \\cite{ilyashenko:dulac}, that the morphism $T$ is injective (quasianalyticity).\nWe let $\\mathcal{Q}$ be the collection $\\left\\{ \\mathcal{Q}_{m,n,r}:\\ m,n\\in\\mathbb{N},\\ r\\in\\left(0,\\infty\\right)^{m+n}\\right\\} $.\nThe motivation for looking at this type of algebras is that they contain\nthe Dulac transition maps of real analytic planar vector fields in\na neighbourhood of hyperbolic non-resonant singular points. \n\n\n\\paragraph{{}}\n\nBefore stating our main result, we need to give a definition. \n\\begin{defn}\n\\label{def: cell}Let $\\mathcal{A}$ be a collection of real-valued\nfunctions. An $\\mathcal{A}$\\emph{-term} is defined inductively as\nfollows. An $\\mathcal{A}$-term of depth zero is an element of $\\mathcal{A}$.\nLet $x=\\left(x_{1},\\ldots,x_{m}\\right)$. A function $f\\left(x\\right)$\nis an $\\mathcal{A}$-term of depth $\\leq k$ if there exist $m\\in\\mathbb{N},\\ g\\in\\mathcal{A}$\nand $\\mathcal{A}$-terms $t_{1}\\left(x\\right),\\ldots,t_{m}\\left(x\\right)$\nof depth $\\leq k-1$ such that $\\text{Im}\\left(t_{1}\\right)\\times\\ldots\\times\\text{Im}\\left(t_{m}\\right)\\subseteq\\text{dom}\\left(g\\right)$\nand $f\\left(x\\right)=g\\left(t_{1}\\left(x\\right),\\ldots,t_{m}\\left(x\\right)\\right)$.\n\nA connected set $C\\subseteq\\mathbb{R}^{m}$ is an $\\mathcal{A}$\\emph{-base}\nif there are a polyradius $r\\in\\left(0,\\infty\\right)^{m}$ and $\\mathcal{A}$-terms\n$t_{0},t_{1},\\ldots,t_{q}$ defined on $(0,r_{1})\\times\\ldots\\times(0,r_{m})$,\nsuch that\n\\[\nC=\\left\\{ x\\in(0,r_{1})\\times\\ldots\\times(0,r_{m}):\\ t_{0}\\left(x\\right)=0,\\ t_{1}\\left(x\\right)>0,\\ \\ldots,t_{q}\\left(x\\right)>0\\right\\} .\n\\]\nA set $D\\subseteq\\mathbb{R}^{m+1}$ is an $\\mathcal{A}$\\emph{-cell\n}if there are an $\\mathcal{A}$-base $C\\subseteq\\mathbb{R}^{m}$ and\nterms $t_{1}\\left(x\\right),t_{2}\\left(x\\right)$ in $m$ variables\nsuch that $D$ is of either of the following forms:\n\\begin{align*}\n\\left\\{ \\left(x,y\\right):\\ x\\in C,\\ y=t_{1}\\left(x\\right)\\right\\} , & \\ \\left\\{ \\left(x,y\\right):\\ x\\in C,\\ t_{1}\\left(x\\right)0\\\\\n0 & \\text{if}\\ x\\leq0\n\\end{cases}$ (for all $p\\in\\mathbb{N}$). \n\nWe can now state our main result.\n\\begin{namedthm}\n{Main Theorem}\\label{thm abcd}Let $\\mathcal{C}$ a generalised quasianalytic\nclass, as in Definition \\ref{def:qa class}. Let $\\mathcal{A}=\\mathcal{C}\\cup\\left\\{ \\left(\\cdot\\right)^{-1}\\right\\} \\cup\\left\\{ \\sqrt[p]{\\cdot}:\\ p\\in\\mathbb{N}\\right\\} $\nand $x=\\left(x_{1},\\ldots,x_{m}\\right)$. Let $y$ be a single variable\nand let $f\\left(x,y\\right)\\in\\mathcal{C}$. Then there exist a neighbourhood\n$W\\subseteq\\mathbb{R}^{m+1}$ of the origin and an $\\mathcal{A}$-cell\ndecomposition of $W\\cap\\text{dom}\\left(f\\right)$ which is compatible\nwith the set $\\left\\{ \\left(x,y\\right)\\in W\\cap\\text{dom}\\left(f\\right):\\ f\\left(x,y\\right)=0\\right\\} $.\n\\end{namedthm}\nThe Main Theorem immediately implies that the solutions of the equation\n$f\\left(x,y\\right)=0$ have the form $\\varphi:C\\to\\mathbb{R}$, where\n$C\\subseteq\\mathbb{R}^{m}$ is an $\\mathcal{A}$-base and $\\varphi\\left(x\\right)$\nis an $\\mathcal{A}$-term. \n\n\n\\paragraph{{}}\n\nWe now briefly illustrate the strategy of proof. In analogy with the\nreal analytic case, we define a class of blow-up transformations adapted\nto the functions under consideration. We show that, after a finite\nsequence of such transformations, the germ at zero of $f$ is normal\ncrossing.\n\nWe stress that the monomialisation algorithm we exhibit here differs\nfrom the ones in \\cite{rsw,martin_sanz_rolin_local_monomialization,bm_semi_subanalytic}.\nIn fact, the transformations we use \\emph{respect} the variable $y$\nin the following way: if $\\rho:\\mathbb{R}^{m+1}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in\\mathbb{R}^{m+1}$\nis one of such transformations and the Main Theorem holds for $f\\circ\\rho\\left(x',y'\\right)$,\nthen it also holds for $f\\left(x,y\\right)$. Moreover, such transformations\nare bijective outside a set of small dimension and the components\nof the inverse map, when defined, are $\\mathcal{A}$-terms. \n\nIt is worth pointing out that our algorithm does not use the Weierstrass\nPreparation Theorem, since this theorem does not always hold in generalised\nquasianalytic classes (see for example \\cite{parusinski_rolin:weierstrass_quasianalytic}).\n\n\n\\paragraph{{}}\n\nThe desingularisation procedure which allows to reduce to the case\nwhen $f$ is normal crossing exploits the fundamental property of\nquasianalyticity, which allows to deduce the wanted result for $f$\nfrom a formal monomialisation algorithm for the series $\\mathcal{T}\\left(f\\right)$.\n\n\n\\paragraph{{}}\n\nThe Main Theorem could also be deduced from a general quantifier elimination\nresult in \\cite{rolin_servi:qeqa}. However, the solving process described\nin \\cite{rolin_servi:qeqa} is not algorithmic, since it uses a highly\nnonconstructive result, namely an o-minimal Preparation Theorem in\n\\cite{vdd:speiss:preparation_theorems}. Here instead we deduce the\nexplicit form of the solutions of $f=0$ solely from the analysis\nof the Newton polyhedron of $\\mathcal{T}\\left(f\\right)$. \n\nAlthough all known generalised quasianalytic classes generate o-minimal\nstructures (see \\cite{vdd:tame} for the definition and basic properties\nof o-minimal structures), the proof of our main result does not use\no-minimality.\n\n\n\\section{Generalised quasianalytic classes}\n\nIn this section we establish our setting.\n\nWe recall the definition and main properties of generalised power\nseries (see \\cite{vdd:speiss:gen} for more details). \n\nLet $m\\in\\mathbb{N}$. A set $S\\subset[0,\\infty)^{m}$ is called \\emph{good}\nif $S$ is contained in a cartesian product $S_{1}\\times\\ldots\\times S_{m}$\nof well ordered subsets of $[0,\\infty)$. If $S$ is a good set, define\n$S_{\\mathrm{min}}$ as the set of minimal elements of $S$ with respect\nto the following order: let $s=\\left(s_{1},\\ldots,s_{m}\\right),\\ s'=\\left(s_{1}',\\ldots,s_{m}'\\right)\\in S;$\nthen $s\\leq s'$ iff $s_{i}\\leq s_{i}'$ for all $i=1,\\ldots,m$.\nBy \\cite[Lemma 4.2]{vdd:speiss:gen}, $S_{\\mathrm{min}}$ is finite. \n\nA \\emph{formal generalised power series }has the form\n\\[\nF(X)=\\sum_{\\alpha}c_{\\alpha}X^{\\alpha},\n\\]\n where $\\alpha=(\\alpha_{1},\\ldots,\\alpha_{m})\\in[0,\\infty)^{m},\\ c_{\\alpha}\\in\\mathbb{R}$\nand $X^{\\alpha}$ denotes the formal monomial $X_{1}^{\\alpha_{1}}\\cdot\\ldots\\cdot X_{m}^{\\alpha_{m}}$,\nand the \\emph{support of }$F$ $\\text{Supp}\\left(F\\right):=\\left\\{ \\alpha:\\ c_{\\alpha}\\not=0\\right\\} $\nis a good set. These series are added the usual way and form an $\\mathbb{R}$-algebra\ndenoted by $\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $. \n\nLet $\\mathcal{G}\\subseteq\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket $\nbe a family of series such that the \\emph{total support} $\\text{Supp}\\left(\\mathcal{G}\\right):=\\bigcup_{F\\in\\mathcal{G}}\\text{Supp}\\left(F\\right)$\nis a good set. Then $\\text{Supp}\\left(\\mathcal{G}\\right)_{\\text{min}}$\nis finite and we denote by $\\mathcal{G}_{\\text{min}}:=\\left\\{ X^{\\alpha}:\\ \\alpha\\in\\text{Supp}\\left(\\mathcal{G}\\right)_{\\text{min}}\\right\\} $\nthe \\emph{set of minimal monomials} of $\\mathcal{G}$.\n\n\n\\paragraph{{}}\n\nLet $m,n\\in\\mathbb{N}$ and $(X,Y)=(X_{1},\\ldots,X_{m},Y_{1},\\ldots,Y_{n})$.\nWe define $\\mathbb{R}\\llbracket X^{*},Y\\rrbracket$ as the subring\nof $\\mathbb{R}\\llbracket(X,Y)^{*}\\rrbracket$ consisting of those\nseries $F$ such that $\\text{Supp}(F)\\subset[0,\\infty)^{m}\\times\\mathbb{N}^{n}$.\nSince $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket \\subseteq\\mathbb{R}\\left\\llbracket X^{*}\\right\\rrbracket \\left\\llbracket Y\\right\\rrbracket $,\nwe say that the variables $X$ are \\emph{generalised} and that the\nvariables $Y$ are \\emph{standard}.\n\\begin{void}\n\\label{vuoto: functions}For every $m,n\\in\\mathbb{N}$ and polyradius\n$r=\\left(s_{1},\\ldots,s_{m},t_{1},\\ldots,t_{n}\\right)\\in\\left(0,\\infty\\right)^{m+n}$,\nwe let $\\mathcal{C}_{m,n,r}$ be an algebra of real functions, which\nare defined and continuous on the set\n\\[\nI_{m,n,r}:=[0,s_{1})\\times\\ldots\\times[0,s_{m})\\times\\left(-t_{1},t_{1}\\right)\\times\\ldots\\times\\left(-t_{n},t_{n}\\right),\n\\]\nand $\\mathcal{C}^{1}$ on $\\mathring{I}_{m,n,r}$. We denote by $x=\\left(x_{1,}\\ldots,x_{m}\\right)$\nthe \\emph{generalised} variables and by $y=\\left(y_{1},\\ldots,y_{n}\\right)$\nthe \\emph{standard} variables. We require that the algebras $\\mathcal{C}_{m,n,r}$\nsatisfy the following list of conditions:\\end{void}\n\\begin{itemize}\n\\item The coordinate functions of $\\mathbb{R}^{m+n}$ are in $\\mathcal{C}_{m,n,r}$. \n\\item If $r'\\leq r$ (i.e. if $s_{i}'\\leq s_{i}$ for all $i=1,\\ldots,m$\nand $t_{j}'\\leq t_{j}$ for all $j=1,\\ldots,n$) and $f\\in\\mathcal{C}_{m,n,r}$,\nthen $f\\restriction I_{m,n,r'}\\in\\mathcal{C}_{m,n,r'}$.\n\\item If $f\\in\\mathcal{C}_{m,n,r}$ then there exists $r'>r$ and $g\\in\\mathcal{C}_{m,n,r'}$\nsuch that $g\\restriction I_{m,n,r}=f$.\n\\item Let $k,l\\in\\mathbb{N}$, $s_{1}',\\ldots,s_{k}',t_{1}',\\ldots,t_{l}'\\in\\left(0,\\infty\\right)$\nand $r'=\\left(s_{1},\\ldots,s_{m},s_{1}',\\ldots,s_{k}',t_{1},\\ldots,t_{n},t_{1}',\\ldots,t_{l}'\\right)$.\nThen $\\mathcal{C}_{m,n,r}\\subset\\mathcal{C}_{m+k,n+l,r'}$ in the\nsense that if $f\\in\\mathcal{C}_{m,n,r}$ then the function\n\\[\n\\xyC{0mm}\\xyL{0mm}\\xymatrix{F\\colon & I_{m+k,n+l,r'}\\ar[rrrr] & \\ & \\ & \\ & \\mathbb{R}\\\\\n & \\left(x_{1},\\ldots,x_{m},x_{1}',\\ldots,x_{k}',y_{1},\\ldots,y_{n},y_{1}',\\ldots,y_{l}'\\right)\\ar@{|->}[rrrr] & & & & f\\left(x_{1},\\ldots,x_{m},y_{1},\\ldots,y_{n}\\right)\n}\n\\]\nis in $\\mathcal{C}_{m+k,n+l,r'}$.\n\\item $\\mathcal{C}_{m,n,r}\\subset\\mathcal{C}_{m+n,0,r}$, in the sense that\nif $f\\in\\mathcal{C}_{m,n,r}$ then $f\\restriction I_{m+n,0,r}\\in\\mathcal{C}_{m+n,0,r}$. \\end{itemize}\n\\begin{defn}\n\\label{def: quasi-analyticity}We denote by $\\mathcal{C}_{m,n}$ the\nalgebra of germs at the origin of the elements of $\\mathcal{C}_{m,n,r}$,\nfor $r$ a polyradius in $\\left(0,\\infty\\right)^{m+n}$. We say that\n$\\left\\{ \\mathcal{C}_{m,n}:\\ m,n\\in\\mathbb{N}\\right\\} $ is a collection\nof \\emph{quasianalytic algebras of germs} if, for all $m,n\\in\\mathbb{N}$,\nthere exists an \\textbf{injective} $\\mathbb{R}$-algebra morphism\n\\[\n\\mathcal{T}_{m,n}:\\mathcal{C}_{m,n}\\to\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket ,\n\\]\nwhere $X=\\left(X_{1},\\ldots,X_{m}\\right)=\\mathcal{T}\\left(x\\right),\\ Y=\\left(Y_{1},\\ldots,Y_{n}\\right)=\\mathcal{T}\\left(y\\right)$.\nMoreover, for all $m'\\geq m,\\ n'\\geq n$ we require that the morphism\n$\\mathcal{T}_{m',n'}$ extend $\\mathcal{T}_{m,n}$, hence, from now\non we will write $\\mathcal{T}$ for $\\mathcal{T}_{m,n}$.\n\nA number $\\alpha\\in[0,\\infty)$ is an \\emph{admissible exponent} if\nthere are $m,n\\in\\mathbb{N},$ $f\\in\\mathcal{C}_{m,n},\\ \\beta\\in\\text{Supp}\\left(\\mathcal{T}\\left(f\\right)\\right)\\subset\\mathbb{R}^{m}\\times\\mathbb{N}^{n}$\nsuch that $\\alpha$ is a component of $\\beta$. If $\\mathbb{A}$ is\nthe set of all admissible exponents and $\\mathbb{A}\\not=\\mathbb{N}$,\nthen we let $\\mathbb{K}$ be the set of nonnegative elements of the\nfield generated by $\\mathbb{A}$. Otherwise, we set $\\mathbb{K}=\\mathbb{A}=\\mathbb{N}$.\n\\end{defn}\nWe require the collection $\\left\\{ \\mathcal{C}_{m,n}:\\ m,n\\in\\mathbb{N}\\right\\} $\nto be closed under certain operations, which we now define.\n\\begin{defn}\n\\label{def:elem transf}Let $m,n\\in\\mathbb{N},\\ \\left(x,y\\right)=\\left(x_{1},\\ldots,x_{m},y_{1},\\ldots,y_{n}\\right)$.\nFor $m',n'\\in\\mathbb{N}$ with $m'+n'=m+n$, we set $\\left(x',y'\\right)=\\left(x_{1}',\\ldots,x_{m'}',y_{1}',\\ldots,y_{n'}'\\right)$.\nLet $r,r'$ be polyradii in $\\mathbb{R}^{m+n}$. An \\emph{elementary\ntransformation} is a map $I_{m',n',r'}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in I_{m,n,r}$\nof either of the following forms.\n\\begin{itemize}\n\\item A \\emph{ramification}: let $m=m',n=n'$, $\\gamma\\in\\mathbb{K}^{>0}$\nand $1\\leq i\\leq m$, and set\n\\begin{align*}\nr_{i}^{\\gamma} & \\left(x',y'\\right)=\\left(x,y\\right),\\ \\ \\ \\mathrm{where}\\ \\begin{cases}\nx_{k}=x_{k}' & 1\\leq k\\leq m,\\ k\\not=i\\\\\nx_{i}=x{}_{i}'^{\\gamma}\\\\\ny_{k}=y_{k} & 1\\leq k\\leq n\n\\end{cases}.\n\\end{align*}\n\n\\item A \\emph{Tschirnhausen translation}: let $m=m',n=n'$ and $H\\in\\mathcal{C}_{m,n-1,s}$\n(where $s\\in\\left(0,\\infty\\right)^{m+n-1}$ is a polyradius), with\n$H\\left(0\\right)=0$, and set\n\\[\n\\tau_{H}\\left(x',y'\\right)=\\left(x,y\\right),\\ \\ \\ \\mathrm{where}\\ \\begin{cases}\nx_{k}=x{}_{k}' & 1\\leq k\\leq m\\\\\ny_{n}=y_{n}'+H\\left(x',y_{1}',\\ldots,y_{n-1}'\\right)\\\\\ny_{k}=y_{k}' & 1\\leq k\\leq n-1\n\\end{cases}.\n\\]\n\n\\item A \\emph{linear transformation}: let $m=m',n=n'$, $1\\leq i\\leq n\\mathrm{\\ and\\ }c=\\left(c_{1},\\ldots,c_{i-1}\\right)\\in\\mathbb{R}^{i-1}$,\nand set\n\\[\nL_{i,c}\\left(x',y'\\right)=\\left(x,y\\right),\\ \\ \\ \\mathrm{where}\\ \\begin{cases}\nx_{k}=x_{k}' & 1\\leq k\\leq m\\\\\ny_{k}=y_{k}' & i\\leq k\\leq n\\\\\ny_{k}=y_{k}'+c_{k}y_{i}' & 1\\leq k1$ and $m_{i}=m_{i-1}'$\nfor all $i=1,\\ldots,k$, then we say that $\\rho:=\\nu_{1}\\circ\\ldots\\circ\\nu_{k}$\nis an \\emph{admissible transformation}.\n\nAn \\emph{elementary family }is either of the following collections\nof elementary transformations: $\\left\\{ r_{i}^{\\gamma}\\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i\\leq m\\text{)},$ $\\left\\{ \\sigma_{m+i}^{+},\\sigma_{m+i}^{-}\\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i\\leq n\\text{)},$ $\\left\\{ \\tau_{H}\\right\\} ,$\n$\\left\\{ L_{i,c}\\right\\} $ $\\text{(for\\ some}\\ 1\\leq i\\leq n\\text{)},$\n$\\left\\{ \\pi_{i,j}^{\\lambda}:\\ \\lambda\\in\\left[0,\\infty\\right]\\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i,j\\leq m\\text{)},$ or $\\left\\{ \\pi_{m+i,j}^{\\lambda}:\\ \\lambda\\in\\mathbb{R}\\cup\\left\\{ \\pm\\infty\\right\\} \\right\\} $\n$\\text{(for\\ some}\\ 1\\leq i\\leq n,\\ 1\\leq j\\leq m\\text{)}$. An \\emph{admissible\nfamily} is defined inductively. An admissible family of length $1$\nis an elementary family. An admissible family $\\mathcal{F}$ of length\n$\\leq q$ is obtained from an elementary family $\\mathcal{F}_{0}$\nin the following way: for all $\\nu\\in\\mathcal{F}_{0}$, let $\\mathcal{F}_{\\nu}$\nbe an admissible family of length $\\leq q-1$ such that $\\forall\\rho'\\in\\mathcal{F}_{\\nu},\\ \\nu\\circ\\rho'$\nis an admissible transformation and define $\\mathcal{F}=\\left\\{ \\nu\\circ\\rho':\\ \\nu\\in\\mathcal{F}_{0},\\ \\rho'\\in\\mathcal{F}_{\\nu}\\right\\} $.\n\nFinally, we say that a series $F\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nhas a certain property $P$ \\emph{after admissible family} if there\nexists ad admissible family $\\mathcal{F}$ such that for every $\\rho\\in\\mathcal{F}$\nthe series $F\\circ\\rho\\left(X',Y'\\right)$ has the property $P$.\nThe same notation extends to elements of $\\mathcal{C}$.\n\\end{defn}\nWe fix a generalised quasianalytic class $\\mathcal{C}$ and we let\n$\\widehat{\\mathcal{C}}_{m,n}$ be the image of $\\mathcal{C}_{m,n}$\nunder the morphism $\\mathcal{T}$ and $\\widehat{\\mathcal{C}}=\\bigcup\\widehat{\\mathcal{C}}_{m,n}$.\nIt follows from the conditions in \\ref{emp:properties of the morph}\nthat, if $\\rho:I_{m',n',r'}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in I_{m,n,r}$\nis an admissible transformation and $F\\left(X,Y\\right)\\in\\widehat{\\mathcal{C}}_{m,n}$,\nthen $F\\left(X',Y'\\right)\\in\\widehat{\\mathcal{C}}_{m',n'}$. \n\nMoreover, it is easy to verify that if $\\mathcal{G}\\subseteq\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nis a collection with good total support, then the collection $\\left\\{ F\\circ\\rho:\\ F\\in\\mathcal{G}\\right\\} $\nhas good total support. For example, let $F\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nand $H\\in\\mathbb{R}\\left\\llbracket X^{*},Y_{1},\\ldots,Y_{n-1}\\right\\rrbracket $;\nsuppose $\\mathrm{Supp}\\left(F\\right)\\subseteq S_{1}\\times\\ldots\\times S_{m}\\times\\mathbb{N}^{n}$\nand $\\mathrm{Supp}\\left(H\\right)\\subseteq S'_{1}\\times\\ldots\\times S'_{m}\\times\\mathbb{N}^{n-1}$,\nwhere $S_{i},S'_{i}\\subset[0,\\infty)$ are well ordered sets. Then\nwe have $\\mathrm{Supp}\\left(F\\circ L_{i,c}\\right)\\subseteq S_{1}\\times\\ldots\\times S_{m}\\times\\mathbb{N}^{n}$\nand $\\mathrm{Supp}\\left(F\\circ\\tau_{H}\\right)\\subseteq\\tilde{S}_{1}\\times\\ldots\\times\\tilde{S}_{m}\\times\\mathbb{N}^{n}$,\nwith $\\tilde{S}_{k}=\\left\\{ a+lb:\\ a\\in S_{k},\\ b\\in S'_{k},\\ l\\in\\mathbb{N}\\right\\} $.\nMoreover, $\\mathrm{Supp}\\left(F\\circ r_{i}^{\\gamma}\\right)\\subseteq\\tilde{S}_{1}\\times\\ldots\\times\\tilde{S}_{m}\\times\\mathbb{N}^{n}$,\nwith $\\tilde{S}_{i}=\\left\\{ \\gamma a:\\ a\\in S_{i}\\right\\} $ and $\\tilde{S}_{k}=S_{k}$\nfor $k\\not=i$. Finally, for $1\\leq i,j\\leq m$ with $i\\not=j$, we\nhave $\\mathrm{Supp}\\left(F\\circ\\pi_{i,j}^{0}\\right)\\subseteq\\tilde{S}_{1}\\times\\ldots\\times\\tilde{S}_{m}\\times\\mathbb{N}^{n}$,\nwith $\\tilde{S}_{j}=\\left\\{ a+b:\\ a\\in S_{j},\\ b\\in S_{i}\\right\\} $\nand $\\tilde{S}_{k}=S_{k}$ for $k\\not=j$. The argument for the other\ntypes of blow-up transformation and for reflections is similar.\n\\begin{void}\n\\label{vuoto: normal}A series $F\\in\\widehat{\\mathcal{C}}_{m,n}$\nis \\emph{normal }if there are $\\alpha\\in[0,\\infty)^{m},\\ \\beta\\in\\mathbb{N}^{n}$\nand a unit $U\\in\\left(\\widehat{\\mathcal{C}}_{m,n}\\right)^{\\times}$\nsuch that $F\\left(X,Y\\right)=X^{\\alpha}Y^{\\beta}U\\left(X,Y\\right)$. \\end{void}\n\\begin{notation}\nThroughout this section, we let $m,n\\in\\mathbb{N},\\ \\left(x,y\\right)=\\left(x_{1},\\ldots,x_{m},y_{1},\\ldots,y_{n}\\right)$\nand $z$ be a single variable. We let $\\mathcal{C}_{m,n,1}$ be either\n$\\mathcal{C}_{m,n+1}$ (i.e. $z$ is considered as a standard variable)\nor $\\mathcal{C}_{m+1,n}$ (i.e. $z$ is considered as a generalised\nvariable). The same convention applies to the formal variables $X,Y,Z$\nand to $\\widehat{\\mathcal{C}}$.\n\\end{notation}\nLet $f\\left(x,y,z\\right)\\in\\mathcal{C}_{m,n,1}$. Our first aim is\nto show that, after a family of admissible transformations \\textquotedblleft{}respecting\\textquotedblright{}\\emph{\n}$Z$, the series $\\mathcal{T}\\left(f\\right)\\left(X,Y,Z\\right)$ is\nnormal. This motivates the next definition.\n\\begin{defn}\n\\label{def: vertical}Let $\\nu:I_{m',n'+1,r'}\\ni\\left(x',y',z'\\right)\\mapsto\\left(x,y,z\\right)\\in I_{m,n+1,r}$\nbe an elementary transformation. Let $\\nu_{0},r'_{0},r_{0}$ denote\nthe first $m+n$ components of $\\nu,r',r$ respectively. We say that\n$\\nu$ \\emph{respects} \\emph{the variable} $z$ if $\\nu_{0}$ does\nnot depend on $z'$. Hence $\\nu_{0}:I_{m',n',r'_{0}}\\ni\\left(x',y'\\right)\\mapsto\\left(x,y\\right)\\in I_{m,n,r_{0}}$\nis an elementary transformation. Analogously, we extend this definition\nto the case when $z'$ and$\\slash$or $z$ are generalised variables\nby requiring that the components of $\\nu$ which correspond to the\nvariables $\\left(x,y\\right)$ depend only on $\\left(x',y'\\right)$\nand not on $z'$.\\end{defn}\n\\begin{lem}\n\\label{lem: singular set}Suppose that $\\nu$ respects $z$, as in\nthe above definition. Then there exists a set $S\\subseteq I_{m',n',r'_{0}}$\n(which is either empty or the zeroset of some variable) such that\nthe maps $\\nu\\restriction I_{m',n'+1,r'}\\setminus\\left(S\\times\\mathbb{R}\\right)$\nand $\\nu_{0}\\restriction I_{m',n',r'_{0}}\\setminus S$ are bijections\nonto their image and for all $\\left(x',y'\\right)\\in I_{m',n',r'_{0}}\\setminus S$\nthe map $z'\\mapsto z=\\nu_{m+n+1}\\left(x',y',z'\\right)$ is a monotonic\nbijection onto its image. Moreover, the components of the inverse\nmaps $\\left(x,y\\right)\\mapsto\\left(x',y'\\right)$ and $\\left(x',y',z\\right)\\mapsto z'$\nare $\\mathcal{A}$-terms. Finally, if $S\\neq\\emptyset$ then $\\nu$\nis a blow-up chart and $\\nu\\left(S\\times\\mathbb{R}\\right)$ is the\ncommon zeroset of two variables.\\end{lem}\n\\begin{proof}\nWe only give the details for $\\nu:\\left(x',y',z'\\right)\\mapsto\\left(x',y',x_{1}'\\left(\\lambda+z'\\right)\\right)$,\nfor some $\\lambda\\in\\mathbb{R}$. In this case, $\\nu_{0}$ is the\nidentity map, $S=\\left\\{ x_{1}'=0\\right\\} $ and $\\nu\\left(S\\times\\mathbb{R}\\right)=\\left\\{ x_{1}=z=0\\right\\} $.\nFor all $\\left(x',y'\\right)\\not\\in S$, the inverse function $z\\mapsto z'=\\frac{z}{x_{1}'}-\\lambda$\nis an $\\mathcal{A}$-term. $ $\\end{proof}\n\\begin{defn}\nWe say that an admissible family $\\mathcal{F}$ of transformations\n$\\left(x',y',z'\\right)\\mapsto\\left(x,y,z\\right)$ \\emph{respects }$z$\nif all the elementary transformations appearing in $\\mathcal{F}$\nrespect $z$ (with the obvious convention that if, for example, $\\mathcal{F}\\ni\\rho=\\nu_{1}\\circ\\nu_{2}:\\left(x',y',z'\\right)\\mapsto\\left(x'',y'',z''\\right)\\mapsto\\left(x,y,z\\right)$,\nthen $\\nu_{1}$ respects $z$ and $\\nu_{2}$ respects $z''$). We\nsay that $\\mathcal{F}$ \\emph{almost} \\emph{respects} $z$ if for\nall $\\rho=\\nu_{1}\\circ\\ldots\\circ\\nu_{k}$ the elementary transformations\n$\\nu_{1},\\ldots,\\nu_{k-1}$ respect $z$ and either $\\nu_{k}$ respects\n$z$ or $\\nu_{k}$ is a blow-up chart at infinity involving $z$ and\nsome other variable (i.e. $\\nu_{k}$ is either $\\pi_{m+1,j}^{\\infty}$\nor $\\pi_{m+n+1,j}^{\\pm\\infty}$, for some $j\\in\\left\\{ 1,\\ldots,m\\right\\} $).\n\\end{defn}\nWe prove the following monomialisation result.\n\\begin{thm}\n\\label{thm: vertical monomialisation}Let $F\\left(X,Y,Z\\right)\\in\\widehat{\\mathcal{C}}_{m,n,1}$.\nThen, after admissible family almost respecting $Z$, we have that\n$F$ is normal. \n\\end{thm}\nBefore proving the above theorem, we show how it implies the Main\nTheorem. Since we want to keep track of standard and generalised variables,\nwe will change the notation and prove the Main Theorem for a germ\n$f\\left(x,y,z\\right)\\in\\mathcal{C}_{m,n,1}$, where $y$ is now an\n$n$-tuple of variables and $z$ is a single variable.\n\\begin{proof}\n[Proof of the Main Theorem]Let $f\\left(x,y,z\\right)\\in\\mathcal{C}_{m,n,1}$.\nBy Theorem \\ref{thm: vertical monomialisation} and the quasianalyticity\nproperty, after some admissible family almost respecting $z$, the\ngerm of $f$ is normal (i.e. it is the product of a monomial by a\nunit of $\\mathcal{C}$). The proof is by induction on the pairs $\\left(d,l\\right)$,\nwhere $d=m+n+1$ is the total number of variables and $l$ is the\nminimal length of an admissible monomialising family for $f$. \n\nIf $d=0$ or $l=0$ then there is nothing to prove. So we may suppose\n$d,l>0$.\n\nLet $\\mathcal{F}$ be a monomialising family for $f$ of length $l$.\nNote that, for every $\\rho\\in\\mathcal{F}$, we may partition the domain\nof $\\rho$ (which is either of the form $I_{m_{\\rho}+1,n_{\\rho},r_{\\rho}}$\nor $I_{m_{\\rho},n_{\\rho}+1,r_{\\rho}}$, for some $m_{\\rho},n_{\\rho}$\nsuch that $m_{\\rho}+n_{\\rho}=m+n$) into a finite union of sub-quadrants\n$Q_{\\rho,j}$ (i.e. sets of the form $B_{1}\\times\\ldots\\times B_{m+n+1}$,\nwhere $B_{i}$ is either $\\left\\{ 0\\right\\} $, or $\\left(-r_{\\rho,i},0\\right)$,\nor $\\left(0,r_{\\rho,i}\\right)$) such that $f\\circ\\rho$ has constant\nsign on $Q_{\\rho,j}$. By a classical compactness argument (see for\nexample \\cite[p. 4406]{vdd:speiss:gen}), there exists a finite subfamily\n$\\mathcal{F}_{0}\\subseteq\\mathcal{F}$ and an open neighbourhood $W\\subseteq\\mathbb{R}^{m+n+1}$\nof the origin such that $W\\cap\\text{dom}\\left(f\\right)=\\bigcup_{\\rho\\in\\mathcal{F}_{0}}\\bigcup_{j\\leq J}\\rho\\left(Q_{\\rho,j}\\right)$\n, for some $J\\in\\mathbb{N}$. Notice that, if $A,B$ are $\\mathcal{A}$-cells,\nthen $A\\cap B$ and $A\\setminus B$ are finite disjoint unions of\n$\\mathcal{A}$-cells. \n\nLet $\\mathcal{F}_{1}$ be an elementary family and $\\mathcal{F}_{2}$\nbe an admissible family of length $\\ldots>\\alpha_{d}\\in\\mathbb{K}$,\n$H_{1},\\ldots,H_{d}\\in\\widehat{\\mathcal{C}}_{m,n}$, which are normal,\nand units $U_{1},\\ldots,U_{d}\\in\\left(\\widehat{\\mathcal{C}}_{m,n,1}\\right)^{\\times}$\nsuch that $F\\left(X,Y,Z\\right)=H_{1}\\left(X,Y\\right)G\\left(X,Y,Z\\right)$,\nwhere \n\\[\nG\\left(X,Y,Z\\right)=Z^{\\alpha_{1}}U_{1}\\left(X,Y,Z\\right)+H_{2}\\left(X,Y\\right)Z^{\\alpha_{2}}U_{2}\\left(X,Y,Z\\right)+\\ldots+H_{d}\\left(X,Y\\right)Z^{\\alpha_{d}}U_{d}\\left(X,Y,Z\\right).\n\\]\n\\end{defn}\n\\begin{prop}\n\\label{prop: finite pres}Suppose that the Inductive Hypothesis \\ref{empty: ind hyp}\nholds. Then $F$ admits a finite presentation of some order $d\\in\\mathbb{N}$,\nafter admissible family respecting the variable $Z$ (in fact, the\nadmissible transformations required act as the identity on $Z$).\n\\end{prop}\nThe ring $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $ is\nclearly not Noetherian. However, the next lemma provides a finiteness\nproperty which is enough for our purposes. The proof takes inspiration\nfrom \\cite[Theorem 6.3.3]{horm}. \n\\begin{lem}\n\\label{lem: quasi noeth}Let $\\mathcal{G}=\\left\\{ F_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} \\subseteq\\widehat{\\mathcal{C}}_{m,n}$\nbe a family with good total support. Then,\\smallskip{}\n\n\n\\noindent \\begin{flushleft}\na) after admissible family, there are $\\beta\\in[0,\\infty)^{m}$ and\na collection $\\left\\{ G_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} \\subseteq\\widehat{\\mathcal{C}}_{m,n}$\nsuch that $\\forall\\alpha\\in A,\\ F_{\\alpha}\\left(X,Y\\right)=X^{\\beta}G_{\\alpha}\\left(X,Y\\right)$\nand $G_{\\alpha_{0}}\\left(0,Y\\right)\\not\\equiv0$, for some $\\alpha_{0}\\in A$;\n\\par\\end{flushleft}\n\n\\noindent \\begin{flushleft}\nb) for every $d\\in\\mathbb{N}$, after admissible family, the $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $-module\ngenerated by the tuples $\\left\\{ \\left(F_{\\alpha_{1}},\\ldots,F_{\\alpha_{d}}\\right):\\ \\alpha_{1},\\ldots,\\alpha_{d}\\in A\\right\\} $\nis finitely generated.\n\\par\\end{flushleft}\n\nThe numbers $m,n$ may change after admissible transformation.\\end{lem}\n\\begin{proof}\nFor the proof of a), we view $\\mathcal{G}$ as a subset of $\\mathbb{B}\\left\\llbracket X^{*}\\right\\rrbracket $,\nwith $\\mathbb{B}=\\mathbb{R}\\left\\llbracket Y\\right\\rrbracket $. In\n\\cite[4.11]{vdd:speiss:gen} the authors define the \\emph{blow-up\nheight} of a finite set of monomials, denoted by $b_{X}$. It follows\nfrom the definition of $b_{X}$ that if $b_{X}\\left(\\mathcal{G}_{\\mathrm{min}}\\right)=\\left(0,0\\right)$,\nthen there exists $\\beta\\in[0,\\infty)^{m}$ such that $\\mathcal{G}_{\\mathrm{min}}=\\left\\{ X^{\\beta}\\right\\} $,\nwhich is what we want. The proof of this step is by induction on the\npairs $\\left(m,b_{X}\\left(\\mathcal{G}_{\\text{min}}\\right)\\right)$,\nordered lexicographically. If $m=0$, there is nothing to prove. If\n$m=1$, then $b_{X}\\left(\\mathcal{G}_{\\mathrm{min}}\\right)=\\left(0,0\\right)$. \n\nHence we may assume that $m>1$ and $b_{X}\\left(\\mathcal{G}_{\\mathrm{min}}\\right)\\not=\\left(0,0\\right)$.\nIt follows from the proof of \\cite[Proposition 4.14]{vdd:speiss:gen}\nthat there are $i,j\\in\\left\\{ 1,\\ldots,m\\right\\} $ and suitable ramifications\n$r_{i}^{\\gamma},\\ r_{j}^{\\delta}$ of the variables $X_{i}$ and $X_{j}$\nsuch that, after the admissible transformations $\\rho_{0}:=r_{i}^{\\gamma}\\circ r_{j}^{\\delta}\\circ\\mathfrak{\\pi}_{i,j}^{0}$\nand $\\rho_{\\infty}:=r_{i}^{\\gamma}\\circ r_{j}^{\\delta}\\circ\\pi_{i,j}^{\\infty}$,\nthe blow-up height $b_{X}$ of $\\mathcal{G}_{\\text{min}}$ has decreased\n(to see this, consider $\\alpha_{i},\\beta_{j}$ in the proof of \\cite[Lemma 4.10]{vdd:speiss:gen}\nand perform the mentioned ramifications with $\\gamma=\\beta_{j}$ and\n$\\delta=\\alpha_{i}$). Moreover, for every $\\lambda\\in\\left(0,\\infty\\right)$,\nafter the admissible transformation $\\rho_{\\lambda}:=r_{i}^{\\gamma}\\circ r_{j}^{\\delta}\\circ\\pi_{i,j}^{\\lambda}$,\nthe series in the family $\\mathcal{G}$ have one less generalised\nvariable and one more standard variable, so $m$ has decreased. Since\nadmissible transformations preserve having good total support, the\ninductive hypothesis applies and we obtain the required conclusion.\\bigskip{}\n\n\n\nThe proof of b) is by induction on the pairs $\\left(m+n,d\\right)$,\nordered lexicographically. Arguing by induction on $d$ as in \\cite[Lemma 6.3.2]{horm},\nit is enough to prove the case $d=1$. If $m+n=1$ then, since $\\mathcal{G}$\nhas good total support, the ideal generated by $\\mathcal{G}$ is principal.\nHence suppose that $m+n>1$. Recall that, by part a) of this lemma,\nthere are $\\beta\\in[0,\\infty)^{m}$ and a collection $\\left\\{ G_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} \\subseteq\\widehat{\\mathcal{C}}_{m,n}$\nsuch that $\\forall\\alpha\\in A,\\ F_{\\alpha}\\left(X,Y\\right)=X^{\\beta}G_{\\alpha}\\left(X,Y\\right)$\nand $G_{\\alpha_{0}}\\left(0,Y\\right)\\not\\equiv0$, for some $\\alpha_{0}\\in A$.\nAfter a linear transformation $L_{n,c}$, we may suppose that $G_{\\alpha_{0}}$\nis regular of some order $d$ in the variable $Y_{n}$.\n\n\nLet $\\hat{Y}=\\left(Y_{1},\\ldots,Y_{n-1}\\right)$. By the formal Weierstrass\nDivision for generalised power series (see \\cite[4.17]{vdd:speiss:gen}),\nfor every $\\alpha\\in A$ there are $Q_{\\alpha}\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nand $B_{\\alpha,0},\\ldots,B_{\\alpha,d-1}\\in\\mathbb{R}\\left\\llbracket X^{*},\\hat{Y}\\right\\rrbracket $\nsuch that $G_{\\alpha}=G_{\\alpha_{0}}Q_{\\alpha}+R_{\\alpha}$, where\n$R_{\\alpha}\\left(X,Y\\right)=\\sum_{i=0}^{d-1}B_{\\alpha,i}\\left(X,\\hat{Y}\\right)Y_{n}^{i}$.\nIt is proven in \\cite[p. 4390]{vdd:speiss:gen} that the total support\nof the collection $\\left\\{ B_{\\alpha,j}:\\ \\alpha\\in A,\\ j=0,\\ldots,d-1\\right\\} $\nis contained in the good set $\\Sigma\\text{Supp}\\left(\\mathcal{G}\\right)$\nof all finite sums (done component-wise) of elements of $\\text{Supp}\\left(\\mathcal{G}\\right)$.\nHence, by the inductive hypothesis on the total number of variables,\nafter admissible family acting on $\\left(X,\\widehat{Y}\\right)$, the\n$\\mathbb{R}\\left\\llbracket X^{*},\\hat{Y}\\right\\rrbracket $-module\ngenerated by $\\mathcal{B}=\\left\\{ B_{\\alpha}=\\left(B_{\\alpha,0},\\ldots,B_{\\alpha,d-1}\\right):\\ \\alpha\\in A\\right\\} $\nis finitely generated. Therefore, there are $\\alpha_{1},\\ldots,\\alpha_{q}\\in A$\nand for all $\\alpha\\in A$ there are $C_{\\alpha,1},\\ldots,C_{\\alpha,q}\\in\\mathbb{R}\\left\\llbracket X^{*},\\hat{Y}\\right\\rrbracket $\nsuch that $B_{\\alpha}=\\sum_{j=1}^{q}C_{\\alpha,j}B_{\\alpha_{j}}$.\nPutting everything together, we obtain that, for every $\\alpha\\in A$,\n\\[\nF_{\\alpha}=\\left(Q_{\\alpha}-\\sum_{j=1}^{q}C_{\\alpha,j}Q_{\\alpha_{j}}\\right)F_{\\alpha_{0}}+\\sum_{j=1}^{q}C_{\\alpha,j}F_{\\alpha_{j}}.\n\\]\n\n\n\\end{proof}\n\n\\begin{proof}\n[Proof of Proposition \\ref{prop: finite pres}]Write $F\\left(X,Y,Z\\right)=\\sum_{\\alpha\\in A}F_{\\alpha}\\left(X,Y\\right)Z^{\\alpha}$\nand consider the family $\\mathcal{G}=\\left\\{ F_{\\alpha}\\left(X,Y\\right):\\ \\alpha\\in A\\right\\} $,\nwhich is contained in $\\widehat{\\mathcal{C}}_{m,n}$ by Conditions\n2 and 5 in \\ref{emp:properties of the morph}. Note that $A\\subseteq[0,\\infty)$\nis a well ordered set and $\\mathcal{G}$ has good total support.\n\nBy Lemma \\ref{lem: quasi noeth}, after admissible family acting on\n$\\left(X,Y\\right)$, the $\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $-ideal\ngenerated by $\\mathcal{G}$ is finitely generated. Hence we can apply\nthe Inductive Hypothesis \\ref{empty: ind hyp} simultaneously to the\ngenerators and obtain that, after admissible family acting on $\\left(X,Y\\right)$,\nthe generators are normal and linearly ordered by division. Hence,\nthere is $\\alpha_{1}\\in A$ and for all $\\alpha\\in A$ there is $Q_{\\alpha}\\in\\mathbb{R}\\left\\llbracket X^{*},Y\\right\\rrbracket $\nsuch that $F_{\\alpha}=F_{\\alpha_{1}}\\cdot Q_{\\alpha}$. Notice that,\nsince $F_{\\alpha_{1}}$ is normal, by monomial division $Q_{\\alpha}\\in\\widehat{\\mathcal{C}}_{m,n}$\n(by Remark \\ref{rem: taylor}, the inverse of a unit belonging to\n$\\hat{\\mathcal{C}}$ also belongs to $\\hat{\\mathcal{C}}$). This allows\nus to write \n\\[\nF\\left(X,Y,Z\\right)=\\sum_{\\alpha<\\alpha_{1}}F_{\\alpha}\\left(X,Y\\right)Z^{\\alpha}+F_{\\alpha_{1}}\\left(X,Y\\right)Z^{\\alpha_{1}}U\\left(X,Y,Z\\right),\n\\]\nwhere $U\\left(X,Y,Z\\right)=1+\\sum_{\\alpha>\\alpha_{1}}Q_{\\alpha}\\left(X,Y\\right)Z^{\\alpha-\\alpha_{1}}$.\nThe series $G\\left(X,Y,Z\\right)=\\sum_{\\alpha<\\alpha_{1}}F_{\\alpha}\\left(X,Y\\right)Z^{\\alpha}$\nbelongs to $\\widehat{\\mathcal{C}}_{m,n,1}$ by Condition 5 in \\ref{emp:properties of the morph},\nhence $U\\in\\left(\\widehat{\\mathcal{C}}_{m,n,1}\\right)^{\\times}$.\nWe repeat the above argument for $G$. This procedure will provide,\nafter admissible family acting on $\\left(X,Y\\right)$, a decreasing\nsequence $\\alpha_{1}>\\alpha_{2}>\\ldots$ which is necessarily finite\n(say, of length $d$), since $A$ is well-ordered. Now it is enough\nto rename $H_{i}:=Q_{\\alpha_{i}}$ for $i=1,\\ldots,d$ and factor\nout $H_{1}$ to obtain the required finite presentation.\n\n\\end{proof}\nWe can now finish the proof of Theorem \\ref{thm: vertical monomialisation}\nby showing how to reduce the order of a finite presentation for $F$.\n\\begin{proof}\n[Proof of theorem \\ref{thm: vertical monomialisation}]In what follows,\nup to suitable reflections, there is no harm in considering the variables\n$\\left(X,Y\\right)$ as generalised, hence, to simplify the notation,\nwe will suppose $Y=\\emptyset$.\n\n\\bigskip{}\n\n\nSuppose first that $F\\in\\widehat{\\mathcal{C}}_{m,1}$, i.e. $Z$ is\na standard variable. By Proposition \\ref{prop: finite pres}, we may\nsuppose that $F$ admits a finite presentation as in Definition \\ref{def: finite pres}.\nSince the exponents $\\alpha_{i}$ are in $\\mathbb{N}$, we have that\n$G$ is regular of order $\\alpha_{1}$ in the variable $Z$.\n\n\nIf $\\alpha_{1}=1$, then we perform the Tschirnhausen transformation\ntranslating $Z$ by the solution to the implicit function problem\n$G=0$, and obtain that $F$ is normal. \n\n\nSuppose that $\\alpha_{1}>1$. We follow, up to suitable reflections\nand ramifications, the algorithm for decreasing the order of regularity\nin the proof of \\cite[Theorem 2.5]{rsw}, which we briefly summarise\n(the details can be found in \\cite[Section 4.2.2]{martin_sanz_rolin_local_monomialization}).\nBy the Taylor formula, there are series $A_{1},\\ldots,A_{d}\\in\\widehat{\\mathcal{C}}_{m}$,\nwith $A_{i}\\left(0\\right)=0$, and a unit $U\\in\\left(\\widehat{\\mathcal{C}}_{m,1}\\right)^{\\times}$\nsuch that\n\\[\nG\\left(X,Z\\right)=A_{d}\\left(X\\right)+\\ldots+A_{1}\\left(X\\right)Z^{\\alpha_{1}-1}+U\\left(X,Z\\right)Z^{\\alpha_{1}}.\n\\]\nAfter a Tschirnhausen translation, we may assume that $A_{1}=0$.\nWe apply the Inductive Hypothesis \\ref{empty: ind hyp} simultaneously\nto the $A_{i}$ in such a way that, after admissible family acting\non $X$, the $A_{i}$ are normal, i.e. $A_{i}\\left(X\\right)=X^{\\beta_{i}}U_{i}\\left(X\\right)$\nfor some $\\beta_{i}\\in\\mathbb{K}^{m}$, $U_{i}\\in\\left(\\widehat{\\mathcal{C}}_{m}\\right)^{\\times}$,\nand for some $l\\in\\left\\{ 2,\\ldots,d\\right\\} $ the series $A_{l}^{1\/l}$\ndivides all the series $A_{i}^{1\/i}$. Let $j\\in\\left\\{ 1,\\ldots m\\right\\} $\nbe such that the variable $X_{j}$ appears with a nonzero exponent\nin the monomial $X^{\\beta_{l}}$ and consider the family of blow-up\ntransformations $\\left\\{ \\pi_{m+1,j}^{\\lambda}:\\ \\lambda\\in\\mathbb{R}\\cup\\left\\{ \\pm\\infty\\right\\} \\right\\} $. \n\n\nAfter the transformations $\\pi_{m+1,j}^{\\pm\\infty}$, the series $G$\nhas the form $Z^{\\alpha_{1}}V\\left(X,Z\\right)$, where $V\\in\\left(\\widehat{\\mathcal{C}}_{m,1}\\right)^{\\times}$,\nso in this case $F$ is normal, and we are done. \n\n\nAfter the transformation $\\pi_{m+1,j}^{0}$, the exponent of $X_{j}$\nin the monomial $X^{\\beta_{l}}$ has decreased by the quantity $l$.\nBy repeating the procedure and applying it to the other variables\nappearing with a nonzero exponent in the monomial $X^{\\beta_{l}}$\n, we can reduce the order of regularity of $G$ to $\\alpha_{1}-l$.\n\n\nFor $\\lambda\\in\\mathbb{R}\\setminus\\left\\{ 0\\right\\} $, after the\ntransformation $\\pi_{m+1,j}^{\\lambda}$, thanks to the fact that $A_{1}=0$,\nthe order of $G$ is at most $\\alpha_{1}-1$. \n\n\nThis shows that, in the case when $Z$ is a standard variable, after\nadmissible family almost respecting $Z$, the series $F$ is normal.\n\\bigskip{}\n\n\n\nNow suppose that $F\\in\\widehat{\\mathcal{C}}_{m+1,0}$, i.e. $Z$ is\na generalised variable. By Proposition \\ref{prop: finite pres}, we\nmay suppose that $F$ admits a finite presentation as in Definition\n\\ref{def: finite pres}. We can apply the Inductive Hypothesis \\ref{empty: ind hyp}\nsimultaneously to $H_{1},\\ldots,H_{d}$ in such a way that, after\nadmissible family, we have\n\\[\nG\\left(X,Z\\right)=Z^{\\alpha_{1}}\\tilde{U}_{1}\\left(X,Z\\right)+X^{\\Gamma_{2}}Z^{\\alpha_{2}}\\tilde{U}_{2}\\left(X,Z\\right)+\\ldots+X^{\\Gamma_{d}}Z^{\\alpha_{d}}\\tilde{U}_{d}\\left(X,Z\\right),\n\\]\nfor some units $\\tilde{U}_{i}\\in\\left(\\widehat{\\mathcal{C}}_{m+1,0}\\right)^{\\times}$,\nand the exponents $\\Gamma_{i}=\\left(\\gamma_{i}^{\\left(1\\right)},\\ldots,\\gamma_{i}^{\\left(m\\right)}\\right)$\nare such that the monomials $\\left\\{ X^{\\frac{\\Gamma_{i}}{\\alpha_{1}-\\alpha_{i}}}:\\ i=2,\\ldots,d\\right\\} $\nare linearly ordered by division. Let $i_{0}\\in\\left\\{ 2,\\ldots,d\\right\\} $\nbe smallest with the property that\n\\[\n\\forall i\\in\\left\\{ 2,\\ldots,d\\right\\} ,\\ \\forall j\\in\\left\\{ 1,\\ldots,m\\right\\} ,\\ \\ \\frac{\\gamma_{i_{0}}^{\\left(j\\right)}}{\\alpha_{1}-\\alpha_{i_{0}}}\\leq\\frac{\\gamma_{i}^{\\left(j\\right)}}{\\alpha_{1}-\\alpha_{i}}.\\tag{\\#}\n\\]\nSuppose $\\gamma_{i_{0}}^{\\left(1\\right)}\\not=0$ and perform a ramification\nof the variable $X_{1}$ with exponent $\\gamma:=\\frac{\\gamma_{i_{0}}^{\\left(1\\right)}}{\\alpha_{1}-\\alpha_{i_{0}}}$.\nWe consider the family of blow-up transformations $\\left\\{ \\pi_{m+1,1}^{\\lambda}:\\ \\lambda\\in\\left[0,\\infty\\right]\\right\\} $. \n\n\nAfter the transformation $\\pi_{m+1,1}^{\\infty}$, we can write\n\\[\nG\\left(X,Z\\right)=Z^{\\alpha_{1}}\\left[\\tilde{U}_{1}\\left(X,Z\\right)+X^{\\Gamma_{2}}Z^{\\beta_{2}}\\tilde{U}_{2}\\left(X,Z\\right)+\\ldots+X^{\\Gamma_{d}}Z^{\\beta_{d}}\\tilde{U}_{d}\\left(X,Z\\right)\\right],\n\\]\nwhere $\\beta_{i}:=\\frac{\\gamma_{i}^{\\left(1\\right)}}{\\gamma_{i_{0}}^{\\left(1\\right)}}\\left(\\alpha_{1}-\\alpha_{i_{0}}\\right)+\\alpha_{i}-\\alpha_{1}$\nis nonnegative, thanks to (\\#). Notice that, since by (\\#) every $\\gamma_{i}^{\\left(1\\right)}$\nis nonzero, the expression between square brackets is a unit. Hence\nin this case $F$ has a finite presentation of order $1$, i.e. $F$\nis normal, and we are done. \n\n\nAfter the transformation $\\pi_{m+1,1}^{0}$, we can write\n\\[\nG\\left(X,Z\\right)=X_{1}^{\\gamma\\alpha_{1}}\\left[Z^{\\alpha_{1}}\\tilde{U}_{1}\\left(X,Z\\right)+X^{\\Delta_{2}}Z^{\\alpha_{2}}\\tilde{U}_{2}\\left(X,Z\\right)+\\ldots+X^{\\Delta_{d}}Z^{\\alpha_{d}}\\tilde{U}_{d}\\left(X,Z\\right)\\right],\n\\]\nwhere $\\Delta_{i}=\\left(\\delta_{i}^{\\left(1\\right)},\\ldots,\\delta_{i}^{\\left(m\\right)}\\right):=\\left(\\gamma_{i}^{\\left(1\\right)}-\\gamma_{i_{0}}^{\\left(1\\right)}\\frac{\\alpha_{1}-\\alpha_{i}}{\\alpha_{1}-\\alpha_{i_{0}}},\\gamma_{i}^{\\left(2\\right)},\\ldots,\\gamma_{i}^{\\left(m\\right)}\\right)$\n. Remark that, by (\\#), the exponents $\\delta_{i}^{\\left(1\\right)}$\nare nonnegative and $\\delta_{i_{0}}^{\\left(1\\right)}=0$. Hence, up\nto factoring out by a power of $X_{1}$, the variable $X_{1}$ does\nnot appear any more in the $i_{0}^{\\text{th}}$ term of the above\nfinite presentation. By repeating this step with the other variables\n$X_{j}$ such that $\\gamma_{i_{0}}^{\\left(j\\right)}\\not=0$, we obtain\n\\[\nG\\left(X,Z\\right)=X^{\\Delta}\\left[Z^{\\alpha_{i_{0}}}V\\left(X,Z\\right)+X^{\\Delta'_{i_{0}+1}}Z^{\\alpha_{i_{0}+1}}\\tilde{U}_{i_{0}+1}\\left(X,Z\\right)+\\ldots+X^{\\Delta_{d}'}Z^{\\alpha_{d}}\\tilde{U}_{d}\\left(X,Z\\right)\\right],\n\\]\nwhere $V\\in\\left(\\widehat{\\mathcal{C}}_{m+1,0}\\right)^{\\times}$,\nthe components of $\\Delta$ are $\\frac{\\alpha_{1}\\gamma_{i_{0}}^{\\left(j\\right)}}{\\alpha_{1}-\\alpha_{i_{0}}}$\nand the components of $\\Delta_{i}'$ are $\\gamma_{i}^{\\left(j\\right)}-\\gamma_{i_{0}}^{\\left(j\\right)}\\frac{\\alpha_{1}-\\alpha_{i}}{\\alpha_{1}-\\alpha_{i_{0}}}$.\nHence $F$ has a finite presentation of order $d-i_{0}+1$.\n\n\nIf $\\lambda\\in\\left(0,\\infty\\right)$, then after the transformation\n$\\pi_{m+1,1}^{\\lambda}$, the variable $Z$ is standard and we have\nreduced to the case $F\\in\\widehat{\\mathcal{C}}_{m,1}$. \n\n\nFinally, notice that if $F\\in\\widehat{\\mathcal{C}}_{0,m+1}$, i.e.\nall the variables are standard, then we can start the proof by first\nramifying the variables $X$ with exponent $d!$, in order to ensure\nthat only natural exponents appear in the series $A_{l}^{1\/l}$.\n\n\\end{proof}\n\\begin{rem}\n\\label{rem: no flatness}In the case when the set of admissible exponents\nis $\\mathbb{N}$ the proof of Theorem \\ref{thm: vertical monomialisation}\ncan be simplified. In fact, by Noetherianity of $\\mathbb{R}\\left\\llbracket X,Y\\right\\rrbracket $,\nthe $\\mathbb{R}\\left\\llbracket X,Y\\right\\rrbracket $-ideal generated\nby the family $\\mathcal{G}$ is finitely generated and one obtains\nimmediately a \\textquotedblleft{}formal\\textquotedblright{} finite\npresentation for $F$, where the units are formal power series, not\nnecessarily belonging to $\\widehat{\\mathcal{C}}$. After monomialising\nthe generators and factoring out an appropriate monomial, this automatically\nimplies that $F$ is regular of some order in the variable $Z$. Hence\nwe can dispense with Proposition \\ref{prop: finite pres} and implement\ndirectly the last part of the proof of Theorem \\ref{thm: vertical monomialisation}. \n\nThis argument also implies that in the real analytic setting, in order\nto obtain regularity in a chosen variable $Z$, there is no need to\nprove a convergent version of the finite presentation in Definition\n\\ref{def: finite pres}. In their proof of quantifier elimination\nfor the real field with restricted analytic functions and the function\n$x\\mapsto1\/x$, Denef and van den Dries prove such a convergent version\n(see \\cite[Lemma 4.12]{vdd:d}), by invoking a consequence of faithful\nflatness in \\cite[(4C)(ii)]{matsumura:commutative_algebra}. Our remark\nimplies that this is not necessary.\n\\end{rem}\n\\bibliographystyle{amsalpha}\n\\def$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1}\n\\setcounter{equation}{0}}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{prop}[theorem]{Proposition}\n\n\\theoremstyle{definition}\n\\newtheorem{remark}[theorem]{Remark}\n\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\n\\theoremstyle{definition}\n\\newtheorem{assumption}[theorem]{Assumption}\n\n\\newcommand{{\\int\\hspace*{-4.3mm}\\diagup}}{{\\int\\hspace*{-4.3mm}\\diagup}}\n\\makeatletter\n\\def\\dashint{\\operatorname%\n{\\,\\,\\text{\\bf-}\\kern-.98em\\DOTSI\\intop\\ilimits@\\!\\!}}\n\\makeatother\n\n\\newcommand{\\WO}[2]{\\overset{\\scriptscriptstyle0}{W}\\,\\!^{#1}_{#2}}\n\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\def\\textit{\\textbf{c}}{\\textit{\\textbf{c}}}\n\\def\\textit{\\textbf{u}}{\\textit{\\textbf{u}}}\n\\def\\textit{\\textbf{v}}{\\textit{\\textbf{v}}}\n\\def\\textit{\\txfextbf{w}}{\\textit{\\txfextbf{w}}}\n\\def\\textit{\\textbf{f}}{\\textit{\\textbf{f}}}\n\\def\\textit{\\textbf{g}}{\\textit{\\textbf{g}}}\n\\def\\textit{\\textbf{h}}{\\textit{\\textbf{h}}}\n\\def\\textit{\\textbf{P}}{\\textit{\\textbf{P}}}\n\\def\\textit{\\textbf{\\phi}}{\\textit{\\textbf{\\phi}}}\n\\def\\\\det{\\text{det}}\n\n\\def\\tilde{\\mathcal{L}_0^\\sigma}{\\tilde{\\mathcal{L}_0^\\sigma}}\n\\def\\hat{\\mathcal{L}_0^\\sigma}{\\hat{\\mathcal{L}_0^\\sigma}}\n\n\\def\\alpha'+\\sigma{\\alpha'+\\sigma}\n\\def\\alpha'\/\\sigma{\\alpha'\/\\sigma}\n\n\n\\defa{a}\n\\defb{b}\n\\defc{c}\n\n\\def{\\sf A}{{\\sf A}}\n\\def{\\sf B}{{\\sf B}}\n\\def{\\sf M}{{\\sf M}}\n\\def{\\sf S}{{\\sf S}}\n\\def\\mathrm{i}{\\mathrm{i}}\n\n\\def\\.5{\\frac{1}{2}}\n\\def\\mathbb{A}{\\mathbb{A}}\n\\def\\mathbb{O}{\\mathbb{O}}\n\\def\\mathbb{R}{\\mathbb{R}}\n\\def\\mathbb{Z}{\\mathbb{Z}}\n\\def\\mathbb{E}{\\mathbb{E}}\n\\def\\mathbb{N}{\\mathbb{N}}\n\\def\\mathbb{H}{\\mathbb{H}}\n\\def\\mathbb{Q}{\\mathbb{Q}}\n\\def\\mathbb{C}{\\mathbb{C}}\n\n\\def\\tilde{G}{\\tilde{G}}\n\n\\def\\textsl{\\textbf{a}}{\\textsl{\\textbf{a}}}\n\\def\\textsl{\\textbf{x}}{\\textsl{\\textbf{x}}}\n\\def\\textsl{\\textbf{y}}{\\textsl{\\textbf{y}}}\n\\def\\textsl{\\textbf{z}}{\\textsl{\\textbf{z}}}\n\\def\\textsl{\\textbf{w}}{\\textsl{\\textbf{w}}}\n\n\\def\\mathfrak{L}{\\mathfrak{L}}\n\\def\\mathfrak{B}{\\mathfrak{B}}\n\\def\\mathfrak{O}{\\mathfrak{O}}\n\\def\\mathfrak{R}{\\mathfrak{R}}\n\\def\\mathfrak{S}{\\mathfrak{S}}\n\\def\\mathfrak{T}{\\mathfrak{T}}\n\\def\\mathfrak{q}{\\mathfrak{q}}\n\n\\def\\text{Re}\\,{\\text{Re}\\,}\n\\def\\text{Im}\\,{\\text{Im}\\,}\n\n\\def\\mathcal{A}{\\mathcal{A}}\n\\def\\mathcal{B}{\\mathcal{B}}\n\\def\\mathcal{C}{\\mathcal{C}}\n\\def\\mathcal{D}{\\mathcal{D}}\n\\def\\mathcal{E}{\\mathcal{E}}\n\\def\\mathcal{F}{\\mathcal{F}}\n\\def\\mathcal{G}{\\mathcal{G}}\n\\def\\mathcal{H}{\\mathcal{H}}\n\\def\\mathcal{P}{\\mathcal{P}}\n\\def\\mathcal{M}{\\mathcal{M}}\n\\def\\mathcal{O}{\\mathcal{O}}\n\\def\\mathcal{Q}{\\mathcal{Q}}\n\\def\\mathcal{R}{\\mathcal{R}}\n\\def\\mathcal{S}{\\mathcal{S}}\n\\def\\mathcal{T}{\\mathcal{T}}\n\\def\\mathcal{L}{\\mathcal{L}}\n\\def\\mathcal{U}{\\mathcal{U}}\n\\def\\mathcal{I}{\\mathcal{I}}\n\\newcommand\\frC{\\mathfrak{C}}\n\n\\def\\bar{P}{\\bar{P}}\n\n\\newcommand{\\RN}[1]{%\n \\textup{\\uppercase\\expandafter{\\romannumeral#1}}%\n}\n\\newcommand{\\ip}[1]{\\left\\langle#1\\right\\rangle}\n\\newcommand{\\set}[1]{\\left\\{#1\\right\\}}\n\\newcommand{\\norm}[1]{\\lVert#1\\rVert}\n\\newcommand{\\Norm}[1]{\\left\\lVert#1\\right\\rVert}\n\\newcommand{\\abs}[1]{\\left\\lvert#1\\right\\rvert}\n\\newcommand{\\tri}[1]{|\\|#1|\\|}\n\\newcommand{\\operatorname{div}}{\\operatorname{div}}\n\\newcommand{\\text{dist}}{\\text{dist}}\n\\newcommand{\\operatornamewithlimits{argmin}}{\\operatornamewithlimits{argmin}}\n\\renewcommand{\\epsilon}{\\varepsilon}\n\n\\newcounter{marnote}\n\\newcommand\\marginnote[1]{\\stepcounter{marnote}$^{\\bullet\\,\\themarnote}$\\marginpar{\\tiny$\\bullet\\,\\themarnote$:\\,#1}}\n\n\n\n\\begin{document}\n\\title[Asymptotics for the perfect conductivity problem ]{Asymptotics for the electric field when $M$-convex inclusions are close to the matrix boundary}\n\n\n\\author[Z.W. Zhao]{Zhiwen Zhao}\n\n\\address[Z.W. Zhao]{1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. }\n\\address{2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands.}\n\\email{zwzhao@mail.bnu.edu.cn.}\n\n\n\n\n\n\n\n\\date{\\today}\n\n\n\n\n\\begin{abstract}\nIn the perfect conductivity problem of composites, the electric field may become arbitrarily large as $\\varepsilon$, the distance between the inclusions and the matrix boundary, tends to zero. The main contribution of this paper lies in developing a clear and concise procedure to establish a boundary asymptotic formula of the concentration for perfect conductors with arbitrary shape in all dimensions, which explicitly exhibits the singularities of the blow-up factor $Q[\\varphi]$ introduced in \\cite{LX2017} by picking the boundary data $\\varphi$ of $k$-order growth. In particular, the smoothness of inclusions required for at least $C^{3,1}$ in \\cite{LLY2019} is weakened to $C^{2,\\alpha}$, $0<\\alpha<1$ here.\n\\end{abstract}\n\n\n\\maketitle\n\n\n\n\\section{Introduction and main results}\n\nIt is well known that field concentrations appear widely in nature and industrial applications. These fields include extreme electric, heat fluxes and mechanical loads. Motivated by the issue of material failure initiation, in this paper we are devoted to the investigation of blow-up phenomena arising from high-contrast fiber-reinforced composites with the densely packed fibers. The key feature of the concentrated fields is that the blow-up comes from the narrow regions between fibers and the thin gaps between fibers and the matrix boundary. It is worth emphasizing that the latter is more interesting due to the interaction from the boundary data. Although there has made great progress in the engineering and mathematical literature since Babu\\u{s}ka et al's famous work \\cite{BASL1999} over the past two decades, accurate numerical computation of the concentrated field are still very hard for lack of fine characterization to develop an efficient numerical scheme. So, it is significantly important from a practical point of view to precisely describe the singular behavior of such high concentration.\n\nIn the context of electrostatics, the field is the gradient of a solution to the Laplace equation and the blow-up rate of the gradient were captured accurately. Denote the distance between two inclusions or between inclusions and the matrix boundary by $\\varepsilon$. It has been proved that for the perfect conductivity problem, the blow-up rate of the gradient is $\\varepsilon^{-1\/2}$ in two dimensions \\cite{AKLLL2007,BC1984,BLY2009,AKL2005,Y2007,Y2009,K1993}, while it is $|\\varepsilon\\ln\\varepsilon|^{-1}$ in three dimensions \\cite{BLY2009,LY2009,BLY2010,L2012}.\n\nBesides these foregoing estimates of the singularities for the field, there is another direction of investigation to establish the asymptotic formula of $\\nabla u$ in the thin gap of electric field concentration. In two dimensions, consider the following conductivity problem\n\\begin{align}\\label{per001}\n\\begin{cases}\n\\Delta u=0,&\\hbox{in}\\;\\mathbb{R}^{2}\\setminus\\overline{D_{1}\\cup D_{2}},\\\\\nu=C_{j}, &\\hbox{on}\\;\\partial D_{j},\\;j=1,2,\\\\\nu(\\mathbf{x})-H(\\mathbf{x})=O(|\\mathbf{x}|^{-1}),&\\mathrm{as}\\;|\\mathbf{x}|\\rightarrow\\infty,\\\\\n\\int_{\\partial D_{j}}\\frac{\\partial u}{\\partial\\nu}\\big|_{+}=0,&j=1,2,\n\\end{cases}\n\\end{align}\nwhere $H$ is a given harmonic function in $\\mathbb{R}^{2}$ and\n$$\\frac{\\partial u}{\\partial\\nu}\\Big|_{+}:=\\lim_{\\tau\\rightarrow0}\\frac{u(x+\\nu\\tau)-u(x)}{\\tau}.$$\nHere and throughtout this paper $\\nu$ is the unit outer normal of $D_{j}$ and the subscript $\\pm$ shows the limit from outside and inside the domain, respectively. For problem (\\ref{per001}), Kang, Lim and Yun \\cite{KLY2013} obtained a complete characterization of the singularities of $\\nabla u$ with $D_{1}$ and $D_{2}$ being disks as follows\n\\begin{align}\\label{singular}\n\\nabla u(\\mathbf{x})=\\frac{2r_{1}r_{2}}{r_{1}+r_{2}}(\\mathbf{n}\\cdot\\nabla H)(\\mathbf{p})\\nabla h(\\mathbf{x})+\\nabla g(\\mathbf{x}),\n\\end{align}\nwhere $h(\\mathbf{x})=\\frac{1}{2\\pi}(\\ln|\\mathbf{x}-\\mathbf{p}_{1}|-\\ln|\\mathbf{x}-\\mathbf{p}_{2}|)$ with $\\mathbf{p}_{1}\\in D_{1}$ and $\\mathbf{p}_{2}\\in D_{2}$ being the fixed point of $R_{1}R_{2}$ and $R_{2}R_{1}$ respectively, $R_{j}$ is the reflection with respect to $\\partial D_{j}$, $\\mathbf{n}$ is the unit vector in the direction of $\\mathbf{p}_{2}-\\mathbf{p}_{1}$, $\\mathbf{p}$ is the middle point of the shortest line segment connecting $\\partial D_{1}$ and $\\partial D_{2}$, and $|\\nabla g|$ is bounded independently of $\\varepsilon$ on any bounded subset of $\\mathbb{R}^{2}\\setminus\\overline{D_{1}\\cup D_{2}}$. Obviously $\\nabla h$ characterizes the singular behavior of $\\nabla u$ explicitly. Ammari, Ciraolo, Kang, Lee, Yun \\cite{ACKLY2013} extended the characterization (\\ref{singular}) to the case when inclusions $D_{1}$ and $D_{2}$ are strictly convex domains in $\\mathbb{R}^{2}$ by utilizing disks osculating to convex domains. In three dimensions, Kang, Lim and Yun \\cite{KLY2014} derived an asymptotic formula of $\\nabla u$ for two spherical perfect conductors with the same radii. The asymptotics for perfectly conducting particles with the different radii can be seen in \\cite{LWX2019}. Recently, a great work on establishing an asymptotic formula in dimensions two and three for two arbitrarily $2$-convex inclusions has been completed by Li, Li and Yang in \\cite{LLY2019}. It is worth mentioning that for high-contrast composites with the matrix described by nonlinear constitutive laws such as $p$-Laplace, Gorb and Novikov \\cite{G2012} captured the stress concentration factor. Additionally, the asymptotics of the eigenvalues of the Poincar\\'{e} variational problem for two close-to-touching inclusions were obtained by Bonnetier and Triki in \\cite{BT2013}. More related work can be seen in \\cite{ABTV2015,AKLLZ2009,BCN2013,BLL2015,BLL2017,BT2012,BV2000,DL2019,G2015,KY2019,KLY2015,LLBY2014,LY2015,M1996,MMN2007,BJL2017,LX2017}.\n\nHowever, to the best of our knowledge, previous investigations on the asymptotics of the field concentration only focused on the narrow region between inclusions. This paper, by contrast, aims at deriving a completely asymptotic characterization for the perfect conductivity problem (\\ref{con002}) with $m$-convex inclusions close to the matrix boundary and the boundary data of $k$-order growth in all dimensions. The asymptotic results in this paper also provide an efficient way to compute the electrical field numerically.\n\nTo state our main works in a precise manner, we first describe our domain and notations. Let $D\\subset\\mathbb{R}^{n}\\,(n\\geq2)$ be a bounded domain with $C^{2,\\alpha}~(0<\\alpha<1)$ boundary, which has a $C^{2,\\alpha}$-subdomain $D_{1}^{\\ast}$ touching matrix boundary $\\partial D$ only at one point. That is, by a translation and rotation of the coordinates, if necessary,\n\\begin{align*}\n\\partial D_{1}^{\\ast}\\cap\\partial D=\\{0'\\}\\subset\\mathbb{R}^{n-1}.\n\\end{align*}\nThroughout the paper, we use superscript prime to denote ($n-1$)-dimensional domains and variables, such as $\\Sigma'$ and $x'$. After a translation, we set\n\\begin{align*}\nD_{1}^{\\varepsilon}:=D_{1}^{\\ast}+(0',\\varepsilon),\n\\end{align*}\nwhere $\\varepsilon>0$ is a sufficiently small constant. For the sake of simplicity, denote\n\\begin{align*}\nD_{1}:=D_{1}^{\\varepsilon},\\quad\\mathrm{and}\\quad\\Omega:=D\\setminus\\overline{D}_{1}.\n\\end{align*}\n\nThe conductivity problem with inclusions close to touching matrix boundary can be modeled by the following scalar equation with piecewise constant coefficients\n\\begin{align}\\label{con001}\n\\begin{cases}\n\\mathrm{div}{(a_{k}(x)\\nabla u)}=0,&\\hbox{in}\\;D,\\\\\nu=\\varphi, &\\hbox{on}\\;\\partial{D},\n\\end{cases}\n\\end{align}\nwhere\n\\begin{align*}\na_{k}(x)=\n\\begin{cases}\nk\\in[0,1)\\cup(1,\\infty],&\\hbox{in}\\;D_{1},\\\\\n1,&\\hbox{on}\\;D\\setminus D_{1}.\n\\end{cases}\n\\end{align*}\nActually, equation (\\ref{con001}) can also be used to describe more physical phenomenon, such as dielectrics, magnetism, thermal conduction, chemical diffusion and flow in porous media.\n\nWhen the conductivity of $D_{1}$ degenerates to be infinity, problem (\\ref{con001}) turns into the perfect conductivity problem as follows\n\\begin{align}\\label{con002}\n\\begin{cases}\n\\Delta u=0,&\\hbox{in}\\;D\\setminus D_{1},\\\\\nu=C_{1}, &\\hbox{in}\\;\\overline{D}_{1},\\\\\n\\int_{\\partial D_{1}}\\frac{\\partial u}{\\partial\\nu}\\big|_{+}=0,\\\\\nu=\\varphi, &\\mathrm{on}\\;\\partial D,\n\\end{cases}\n\\end{align}\nwhere the free constant $C_{1}$ is determined later by the third line of (\\ref{con002}). There has established the existence, uniqueness and regularity of weak solutions to (\\ref{con002}) in \\cite{BLY2009} with a minor modification. We further assume that there exists a small constant $R>0$ independent of $\\varepsilon$, such that the portions of $\\partial D$ and $\\partial D_{1}$ near the origin can be written as\n\\begin{align*}\nx_{n}=\\varepsilon+h_{1}(x')\\quad\\mathrm{and}\\quad x_{n}=h(x'),\\quad\\quad x'\\in B_{2R}',\n\\end{align*}\nwhere $h_{1}$ and $h$ satisfy that for $m\\geq2$,\n\\begin{enumerate}\n{\\it\\item[(\\bf{\\em H1})]\n$h_{1}(x')-h(x')=\\lambda|x'|^{m}+O(|x'|^{m+1}),$\n\\item[(\\bf{\\em H2})]\n$|\\nabla_{x'}^{i}h_{1}(x')|,\\,|\\nabla_{x'}^{i}h(x')|\\leq \\kappa_{1}|x'|^{m-i},\\;\\,i=1,2,$\n\\item[(\\bf{\\em H3})]\n$\\|h_{1}\\|_{C^{2,\\alpha}(B'_{2R})}+\\|h\\|_{C^{2,\\alpha}(B'_{2R})}\\leq \\kappa_{2},$}\n\\end{enumerate}\nwhere $\\lambda$ and $\\kappa_{j},j=1,2$, are three positive constants independent of $\\varepsilon$.\n\nTo explicitly uncover the effect of boundary data $\\varphi$ on the singularities of the field, we classify $\\varphi\\in C^{2}(\\partial D)$ according to its parity as follows. Denote the bottom boundary of $\\Omega_{R}$ by $\\Gamma^{-}_{R}=\\{x\\in\\mathbb{R}^{n}|\\,x_{n}=h(x'),\\,|x'|0$ and $k>1$ is a positive integer.\n\nFor $z'\\in B'_{R},\\,0n+i-1,\\\\\n|\\ln\\varepsilon|,&m=n+i-1,\\\\\n1,&mn+i-1,\\\\\n1,&m=n+i-1,\n\\end{cases}\n\\end{align*}\nwhere $\\Gamma(s)=\\int^{+\\infty}_{0}t^{s-1}e^{-t}\\,dt$, $s>0$ is the Gamma function. Denote by $\\omega_{n-1}$ the area of the surface of unit sphere in $(n-1)$-dimension. For $(z',z_{n})\\in\\Omega_{2R}$, denote\n\\begin{align}\\label{ZZW666}\n\\delta(z'):=\\varepsilon+h_{1}(z')-h(z').\n\\end{align}\nLet $\\Omega^{\\ast}:=D\\setminus\\overline{D^{\\ast}_{1}}$. We define a linear functional with respect to $\\varphi$,\n\\begin{align}\\label{linear001}\nQ^{\\ast}[\\varphi]:=\\int_{\\partial D_{1}}\\frac{\\partial v_{0}^{\\ast}}{\\partial\\nu},\n\\end{align}\nwhere $v_{0}^{\\ast}$ is a solution of the following problem:\n\\begin{align}\\label{con003}\n\\begin{cases}\n\\Delta v_{0}^{\\ast}=0,\\quad\\quad\\;\\,&\\mathrm{in}\\;\\Omega^{\\ast},\\\\\nv_{0}^{\\ast}=0,\\quad\\quad\\;\\,&\\mathrm{on}\\;\\partial D_{1}^{\\ast},\\\\\nv_{0}^{\\ast}=\\varphi(x),\\quad\\;\\,&\\mathrm{on}\\;\\partial D.\n\\end{cases}\n\\end{align}\nNote that the definition of $Q^{\\ast}[\\varphi]$ is valid under case ({\\bf{\\em S2}}) but only valid for $m1$ are all positive integers in the following.\n\\begin{theorem}\\label{thm001}\nAssume that $D_{1}\\subset D\\subseteq\\mathbb{R}^{n}\\,(n\\geq2)$ are defined as above, conditions ({\\bf{\\em H1}})--({\\bf{\\em H3}}) and ({\\bf{\\em S1}}) hold. Let $u\\in H^{1}(D;\\mathbb{R}^{n})\\cap C^{1}(\\overline{\\Omega};\\mathbb{R}^{n})$ be the solution of (\\ref{con002}). Then for a sufficiently small $\\varepsilon>0$ and $x\\in\\Omega_{R}$,\n\n$(i)$ for $m\\geq n+k-1$,\n\\begin{align*}\n\\nabla u=\\frac{\\eta\\Gamma^{n+k}_{m}}{\\lambda^{\\frac{k}{m}}\\Gamma^{n}_{m}}(1+O(r_{\\varepsilon}))\\rho_{k;0}(n,m;\\varepsilon)\\nabla\\bar{u}+\\nabla\\bar{u}_{0}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}}\\|\\varphi\\|_{C^{2}(\\partial D)};\n\\end{align*}\n\n$(ii)$ for $n-1\\leq mn+k,\\\\\n\\varepsilon^{\\frac{1}{m}}|\\ln\\varepsilon|,&m=n+k,\\\\\n|\\ln\\varepsilon|^{-1},&m=n+k-1,\\\\\n\\varepsilon^{\\frac{n+k-1-m}{(n+k-1)(m+1)}},&n-1n+k-1$ or $|\\ln\\varepsilon|$ if $m=n+k-1$ for the boundary data $\\varphi$ with $k$-order growth. In addition, when $m>2$, the remainder of order $O(\\varepsilon^{1-2\/m})$ in the shortest line segment between the conductors and the matrix boundary provides a more precise characterization on the asymptotic behavior of the concentration than that of $m=2$. Finally, the concisely main terms $\\nabla\\bar{u}$ and $\\nabla\\bar{u}_{0}$ together with their coefficients can completely describe the singular effect of the geometry, which will greatly reduce the complexity of numerical computation for $\\nabla u$.\n\\end{remark}\n\n\\begin{remark}\nThe asymptotics of $\\nabla u$ in Theorem \\ref{thm001} indicate that\n\n$(1)$ if $m\\leq n+k-1$, then its maximum achieves only at $\\{x'=0'\\}\\cap\\Omega$;\n\n$(2)$ if $m>n+k-1$, then the maximum achieves at both $\\{x'=0'\\}\\cap\\Omega$ and $\\{|x'|=\\varepsilon^{\\frac{1}{m}}\\}\\cap\\Omega$.\n\\end{remark}\n\n\n\\begin{remark}\nIn order to further reveal the effect of principal curvatures of the geometry, we take $n=3$ relevant to physical dimension for example. Consider\n\\begin{align*}\n\\varphi=\\eta_{1}|x_{1}|^{k}+\\eta_{2}|x_{2}|^{k},\\quad x\\in\\{\\lambda_{1}|x_{1}|^{m}+\\lambda_{2}|x_{2}|^{m}0$,\n\n$(i)$ for $m\\geq n-1$,\n\\begin{align*}\n\\nabla u=\\frac{m\\lambda^{\\frac{n-1}{m}}Q^{\\ast}[\\varphi]}{(n-1)\\omega_{n-1}\\Gamma^{n}_{m}}\\frac{1+O(\\tilde{r}_{\\varepsilon})}{\\rho_{0}(n,m;\\varepsilon)}\\nabla\\bar{u}+\\nabla\\bar{u}_{0}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}}\\|\\varphi\\|_{C^{2}(\\partial D)};\n\\end{align*}\n\n$(ii)$ for $mn-1,\\\\\n|\\ln\\varepsilon|^{-1},&m=n-1,\\\\\n\\max\\{\\varepsilon^{\\frac{m+n-2}{(m+1)(2m+n-2)}},\\varepsilon^{\\frac{1}{6}}\\}.&mn$, then the maximum attains at $\\{|x'|=\\varepsilon^{\\frac{1}{m}}\\}\\cap\\Omega$.\n\\end{remark}\n\n\\begin{remark}\nIf (\\ref{Geometry}) holds in Theorem \\ref{coro002}, we can obtain that the coefficient of the main term $\\nabla\\bar{u}$ has an explicit dependence of $\\sqrt[m]{\\lambda_{1}\\lambda_{2}}$.\n\\end{remark}\n\nThe organization of this paper is as follows. In section 2, we carry out a linear decomposition of the solution $u$ to problem (\\ref{con002}) as $v_{0}$ and $v_{1}$, defined by (\\ref{con005}) and (\\ref{con006}) below, and we prove the correspondingly main terms $\\bar{u}_{0}$ and $\\bar{u}$ constructed by (\\ref{con009}) and (\\ref{con016}), respectively, in Lemma \\ref{lem001} and Theorem \\ref{thm002}. Based on the results obtained in section 2, we give the proofs of Theorem \\ref{thm001} and Theorem \\ref{coro002} consisting of the asymptotics of blow-up factor $Q[\\varphi]$ and $a_{11}$ in section 3.\n\n\n\n\n\n\n\n\n\\section{Preliminary}\n\n\\subsection{Solution split}\nAs in \\cite{LX2017}, we decompose the solution $u$ of (\\ref{con002}) as follows\n\\begin{align}\\label{con0033}\nu(x)=C_{1}v_{1}(x)+v_{0}(x),\\quad\\;\\,\\mathrm{in}\\;D\\setminus\\overline{D}_{1},\n\\end{align}\nwhere $v_{i}$, $i=0,1$, verify\n\\begin{align}\\label{con005}\n\\begin{cases}\n\\Delta v_{0}=0,\\quad\\quad\\;\\,&\\mathrm{in}\\;\\Omega,\\\\\nv_{0}=0,\\quad\\quad\\;\\,&\\mathrm{on}\\;\\partial D_{1},\\\\\nv_{0}=\\varphi(x),\\quad\\;\\,&\\mathrm{on}\\;\\partial D,\n\\end{cases}\n\\end{align}\nand\n\\begin{align}\\label{con006}\n\\begin{cases}\n\\Delta v_{1}=0,\\quad\\quad\\;\\,&\\mathrm{in}\\;\\Omega,\\\\\nv_{1}=1,\\quad\\quad\\;\\,&\\mathrm{on}\\;\\partial D_{1},\\\\\nv_{1}=0,\\quad\\;\\,&\\mathrm{on}\\;\\partial D,\n\\end{cases}\n\\end{align}\nrespectively. Similarly as (\\ref{linear001}) and (\\ref{con003}), we define a linear functional of $\\varphi$ as follows\n\\begin{align}\\label{linear002}\nQ[\\varphi]=\\int_{\\partial D_{1}}\\frac{\\partial v_{0}}{\\partial\\nu},\n\\end{align}\nwhere $v_{0}$ is defined by (\\ref{con005}). Denote\n\\begin{align*}\na_{11}:=\\int_{\\Omega}|\\nabla v_{1}|^{2}dx.\n\\end{align*}\nThen, it follows from the third line of (\\ref{con002}) and the decomposition (\\ref{con0033}) that\n\\begin{align*}\nC_{1}\\int_{\\partial D_{1}}\\frac{\\partial v_{1}}{\\partial\\nu}+\\int_{\\partial D_{1}}\\frac{\\partial v_{0}}{\\partial\\nu}=0.\n\\end{align*}\nRecalling the definition of $v_{1}$ and making use of integration by parts, we have\n\\begin{align}\\label{con007}\n\\nabla u=\\frac{Q[\\varphi]}{a_{11}}\\nabla v_{1}+\\nabla v_{0}.\n\\end{align}\n\n\n\\subsection{A general boundary value problem}\n\nTo obtain the asymptotic expansion for $\\nabla u$, we first consider the following general boundary value problem:\n\\begin{equation}\\label{con008}\n\\begin{cases}\n\\Delta v=0,\\quad\\;\\,&\\mathrm{in}\\;\\,\\Omega,\\\\\nv=\\psi,&\\mathrm{on}\\;\\,\\partial D_{1},\\\\\nv=0,&\\mathrm{on}\\;\\,\\partial D,\n\\end{cases}\n\\end{equation}\nwhere $\\psi\\in C^{2}(\\partial D_{1})$ is a given scalar function. Note that if $\\psi=1$ on $\\partial D_{1}$, then $v_{1}=v$. Extend $\\psi\\in C^{2}(\\partial D_{1})$ to $\\psi\\in C^{2}(\\overline{\\Omega})$ such that $\\|\\psi\\|_{C^{2}(\\overline{\\Omega\\setminus\\Omega_{R}})}\\leq C\\|\\psi\\|_{C^{2}(\\partial D_{1})}$. Construct a cutoff function $\\rho\\in C^{2}(\\overline{\\Omega})$ satisfying $0\\leq\\rho\\leq1$, $|\\nabla\\rho|\\leq C$ on $\\overline{\\Omega}$, and\n\\begin{align}\\label{con011}\n\\rho=1\\;\\,\\mathrm{on}\\;\\,\\Omega_{\\frac{3}{2}R},\\quad\\rho=0\\;\\,\\mathrm{on}\\;\\,\\overline{\\Omega}\\setminus\\Omega_{2R}.\n\\end{align}\nFor $x\\in\\Omega$, we define\n\\begin{align*}\n\\bar{v}(x)=[\\rho(x)\\psi(x',\\varepsilon+h_{1}(x'))+(1-\\rho(x))\\psi(x)]\\bar{u}(x),\n\\end{align*}\nwhere $\\bar{u}$ is defined by (\\ref{con009}). Specially,\n\\begin{align*}\n\\bar{v}(x)=\\psi(x',\\varepsilon+h_{1}(x'))\\bar{u}(x),\\quad\\;\\,\\mathrm{in}\\;\\Omega_{R}.\n\\end{align*}\nDue to (\\ref{con009}), we have\n\\begin{align}\\label{KK6}\n\\|\\bar{v}\\|_{C^{2}(\\Omega\\setminus\\Omega_{R})}\\leq C\\|\\psi\\|_{C^{2}(\\partial D_{1})}.\n\\end{align}\n\nSimilarly as in \\cite{LX2017}, we can obtain an asymptotic expansion of the gradient for problem (\\ref{con006}).\n\\begin{theorem}\\label{thm002}\nAssume as above. Let $v\\in H^{1}(\\Omega)$ be a weak solution of (\\ref{con008}). Then, for a sufficiently small $\\varepsilon>0$,\n\\begin{align}\\label{con013}\n|\\nabla(v-\\bar{v})(x)|\\leq C\\delta^{1-\\frac{2}{m}}(|\\psi(x',\\varepsilon+h_{1}(x'))|+\\delta^{\\frac{1}{m}}\\|\\psi\\|_{C^{2}(\\partial D_{1})}),\\quad\\mathrm{in}\\;\\,\\Omega_{R}.\n\\end{align}\nConsequently, (\\ref{con013}), together with choosing $\\psi=1$ on $\\partial D_{1}$, yields that\n\\begin{align}\\label{con015}\n\\nabla v_{1}=\\nabla\\bar{u}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}},\\quad\\;\\,\\mathrm{in}\\;\\Omega_{R},\n\\end{align}\nand\n\\begin{align*}\n\\|\\nabla v\\|_{L^{\\infty}(\\Omega\\setminus\\Omega_{R})}\\leq C\\|\\psi\\|_{C^{2}(\\partial D_{1})}.\n\\end{align*}\nwhere $v_{1}\\in H^{1}(\\Omega)$ is a weak solution of (\\ref{con006})\n\n\\end{theorem}\nNote that when $m>2$, the remainder of order $O(1)$ in \\cite{LX2017} is improved to that of order $O(\\varepsilon^{1-2\/m})$ for $x\\in\\{x'=0'\\}\\cap\\Omega_{R}$ here. For readers' convenience, the detailed proof of Theorem \\ref{thm002} is left in the Appendix. Similarly, by applying Theorem \\ref{thm002}, we can find that the leading term of $\\nabla v_{0}$ is $\\nabla\\bar{u}_{0}$ in the following.\n\\begin{lemma}\\label{lem001}\nAssume as above. Let $v_{0}$ be the weak solution of (\\ref{con005}). Then, for a sufficiently small $\\varepsilon>0$,\n\\begin{align}\\label{con018}\n\\nabla v_{0}=\\nabla\\bar{u}_{0}+O(\\mathbf{1})\\delta^{1-\\frac{2}{m}}(|\\varphi(x',h(x'))|+\\delta^{\\frac{1}{m}}\\|\\varphi\\|_{C^{2}(\\partial D)}),\\quad\\;\\,\\mathrm{in}\\;\\Omega_{R},\n\\end{align}\nand\n\\begin{align}\\label{con01818}\n\\|\\nabla_{x'}v_{0}\\|_{L^{\\infty}(\\Omega_{R})}\\leq C\\|\\varphi\\|_{C^{2}(\\partial D)},\\;\\,\\|\\nabla v_{0}\\|_{L^{\\infty}(\\Omega\\setminus\\Omega_{R})}\\leq C\\|\\varphi\\|_{C^{2}(\\partial D)},\n\\end{align}\nwhere $\\bar{u}_{0}$ is defined by (\\ref{con016}).\n\\end{lemma}\n\nTherefore, recalling the decomposition (\\ref{con007}) and in view of (\\ref{con015}) and (\\ref{con018}), for the purpose of deriving the asymptotic of $\\nabla u$, it suffices to establish the following two aspects of expansions:\n\n(i) Expansion of $Q[\\varphi]$;\n\n(ii) Expansion of $a_{11}$.\n\n\n\n\n\n\n\n\n\\section{Proofs of Theorem \\ref{thm001} and Theorem \\ref{coro002}}\n\n\\subsection{Expansion of $Q[\\varphi]$}\nBefore proving Theorem \\ref{thm001} and Theorem \\ref{coro002}, we first give an expansion of $Q[\\varphi]$ with respect to $\\varepsilon$.\n\\begin{lemma}\\label{lem002}\nAssume as above. Then, for a sufficiently small $\\varepsilon>0$,\n\n$(a)$ if ({\\bf{\\em S1}}) holds for $m\\geq n+k-1$ in Theorem \\ref{thm001},\n\\begin{align*}\nQ[\\varphi]=&\\frac{(n-1)\\omega_{n-1}\\eta\\Gamma^{n+k}_{m}}{m\\lambda^{\\frac{n+k-1}{m}}}\\rho_{k}(n,m;\\varepsilon)\n\\begin{cases}\n1+O(1)\\varepsilon^{\\frac{1}{m}},&m>n+k,\\\\\n1+O(1)\\varepsilon^{\\frac{1}{m}}|\\ln\\varepsilon|,&m=n+k,\\\\\n1+O(1)|\\ln\\varepsilon|^{-1},&m=n+k-1;\n\\end{cases}\n\\end{align*}\n\n$(b)$ if ({\\bf{\\em S1}}) holds for $mn+k,\\\\\n\\varepsilon^{-\\frac{1}{m}}+O(1)|\\ln\\varepsilon|\\|\\varphi\\|_{C^{2}(\\partial D)},&m=n+k,\\\\\n|\\ln\\varepsilon|+O(1)\\|\\varphi\\|_{C^{2}(\\partial D)},&m=n+k-1.\n\\end{cases}\n\\end{align*}\n\n{\\bf Step 2.} Proofs of $(b)$ and $(c)$. In view of the definitions of $Q[\\varphi]$ and $Q^{\\ast}[\\varphi]$, it follows from integration by parts that\n\\begin{align*}\nQ[\\varphi]=\\int_{\\partial D}\\frac{\\partial v_{1}}{\\partial\\nu}\\varphi(x),\\quad\\quad Q^{\\ast}[\\varphi]=\\int_{\\partial D}\\frac{\\partial v_{1}^{\\ast}}{\\partial\\nu}\\varphi(x),\n\\end{align*}\nwhere $v_{1}$ and $v_{1}^{\\ast}$ are defined by (\\ref{con006}) and (\\ref{con022}). Thus,\n\\begin{align*}\nQ[\\varphi]-Q^{\\ast}[\\varphi]=\\int_{\\partial D}\\frac{\\partial(v_{1}-v_{1}^{\\ast})}{\\partial\\nu}\\cdot\\varphi(x).\n\\end{align*}\n\nTo estimate $v_{1}-v_{1}^{\\ast}$, we first introduce a scar auxiliary functions $\\bar{u}^{\\ast}$ satisfying $\\bar{u}^{\\ast}=1$ on $\\partial D_{1}^{\\ast}\\setminus\\{0\\}$, $\\bar{u}^{\\ast}=0$ on $\\partial D$, and\n$$\\bar{u}^{\\ast}=\\frac{x_{n}-h(x')}{h_{1}(x')-h(x')},\\quad\\mathrm{in}\\;\\,\\Omega_{2R}^{\\ast},\\quad\\;\\,\\|\\bar{u}^{\\ast}\\|_{C^{2}(\\Omega^{\\ast}\\setminus\\Omega_{R}^{\\ast})}\\leq C,$$\nwhere $\\Omega^{\\ast}_{r}:=\\Omega^{\\ast}\\cap\\{|x'|0$,\n\n$(i)$ for $m\\geq n-1$,\n\\begin{align*}\na_{11}=&\n\\frac{(n-1)\\omega_{n-1}\\Gamma^{n}_{m}}{m\\lambda^{\\frac{n-1}{m}}}\\rho_{0}(n,m;\\varepsilon)\n\\begin{cases}\n1+O(1)\\varepsilon^{\\frac{1}{m}},&m>n,\\\\\n1+O(1)\\varepsilon^{\\frac{1}{m}}|\\ln\\varepsilon|,&m=n,\\\\\n1+O(1)|\\ln\\varepsilon|^{-1},&m=n-1;\n\\end{cases}\n\\end{align*}\n\n$(ii)$ for $m\\frac{n-1}{2},\\\\\n\\varepsilon|\\ln\\varepsilon|,&m=\\frac{n-1}{2},\\\\\n\\varepsilon,&m<\\frac{n-1}{2}.\n\\end{cases}\n\\end{align}\nBy picking $\\gamma=\\frac{1}{2m}$ in {\\bf Step 2.1} of the proof of Lemma \\ref{lem002}, it follows from (\\ref{con029})--(\\ref{con031}) and the maximum principle that\n\\begin{align*}\n|v_{1}-v_{1}^{\\ast}|\\leq C\\varepsilon^{\\frac{1}{2}},\\quad\\;\\,\\mathrm{in}\\;\\,D\\setminus\\big(\\overline{D_{1}\\cup D_{1}^{\\ast}\\cup\\mathcal{C}_{\\varepsilon^{\\frac{1}{2m}}}}\\big).\n\\end{align*}\nSimilarly as before, utilizing the standard interior and boundary estimates, we derive that\n\\begin{align}\\label{con035}\n|\\nabla(v_{1}-v_{1}^{\\ast})|\\leq C\\varepsilon^{\\frac{1}{6}},\\quad\\;\\,\\mathrm{in}\\;\\,D\\setminus\\big(\\overline{D_{1}\\cup D_{1}^{\\ast}\\cup\\mathcal{C}_{\\varepsilon^{\\frac{1}{3m}}}}\\big).\n\\end{align}\nThen combining (\\ref{con027}) and (\\ref{con035}), we obtain that\n\\begin{align}\\label{con036}\n|\\mathrm{II}_{2}|\\leq&C\\varepsilon^{\\frac{1}{6}}.\n\\end{align}\n\nAs for $\\mathrm{II}_{3}$, it follows from (\\ref{con026}) and (\\ref{con027}) that\n\\begin{align*}\n\\mathrm{II}_{3}=&\\int_{\\Omega^{\\ast}_{R}\\setminus\\Omega^{\\ast}_{\\varepsilon^{\\bar{\\gamma}}}}|\\nabla\\bar{u}^{\\ast}|^{2}+2\\int_{\\Omega^{\\ast}_{R}\\setminus\\Omega^{\\ast}_{\\varepsilon^{\\bar{\\gamma}}}}\\nabla\\bar{u}^{\\ast}\\cdot\\nabla(v_{1}^{\\ast}-\\bar{u}^{\\ast})+\\int_{\\Omega^{\\ast}_{R}\\setminus\\Omega^{\\ast}_{\\varepsilon^{\\bar{\\gamma}}}}|\\nabla(v_{1}^{\\ast}-\\bar{u}^{\\ast})|^{2}\\\\\n=&\\int_{\\varepsilon^{\\bar{\\gamma}}<|x'|n,\\\\\n\\varepsilon^{-\\frac{1}{m}}+O(1)|\\ln\\varepsilon|,&m=n,\\\\\n|\\ln\\varepsilon|+O(1),&m=n-1;\n\\end{cases}\n\\end{align*}\n\n(ii) For $m0$,\n\\begin{align}\\label{ADE009}\nF(t_{0})\\leq C\\varepsilon^{n+2-\\frac{4}{m}}\\left(|\\psi(z',\\varepsilon+h_{1}(z'))|^{2}+\\varepsilon^{\\frac{2}{m}}\\|\\psi\\|^{2}_{C^{2}}\\right).\n\\end{align}\n\n{\\bf Case 2.} If $\\varepsilon^{\\frac{1}{m}}\\leq|z'|\\leq R$ and $0 0.8$, were placed in the ETG bin. The\nclassifier algorithm in \\citet{Huertas-Company:2011} assigned a probability to\neach galaxy for belonging to four different morphological types: E\n(elliptical); S0 (lenticular); Sab (early LTG); and Scd (late LTG).\n\\citet[][in their Section 6]{Sahu:2019:II} argued that $p(\\rm E-S0) > 0.8$\nimplied that at least 10 per cent of the SDSS-DR7 ETGs selected by\n\\citetalias{Shankar:2016} using the above criteria were misclassified and may\nbe LTGs (Sab or Scd). \\citet{Sahu:2019:II} noted that the apparent offset of\nthe SMBH sample from the SDSS-DR7 ETG $\\sigma_{\\rm HL}$--$M_{\\rm *,gal}$ curve\nseems higher at the low-mass end ($10 \\lesssim \\log(M_{\\rm *,gal}\/{\\rm\n M}_{\\odot}) \\lesssim 10.5$) than at the high masses due to at least 10 per cent\ncontamination by LTGs in the supposed ETG sample of \\citetalias{Shankar:2016}.\nThis is because the LTGs (spiral) define a different \n$\\sigma_{\\rm HL}$--$M_{\\rm *,gal}$ relation than ETGs and reside below ETGs in the\n$\\sigma_{\\rm HL}$--$M_{\\rm *,gal}$ diagram \\citep[see Figure~15\n in][]{Sahu:2019:II}. This may be a reason behind a fraction of the offset \nbetween the ETG SMBH sample and the SDSS-DR7 ETG sample. However,\n\\citet{Sahu:2019:II} did not thoroughly investigate this offset which is now\ndone here.\\footnote{In 2019, we were aware of the results presented herein; \n however, pandemic-related delays have meant that we are only reporting them\n now.}\n\nImportantly, from the SDSS-DR7 spectroscopic galaxy sample without a\ndirect-dynamical SMBH mass measurement, \\citetalias{Shankar:2016} used the\nETGs within the redshift range of $0.05 1$, with $K\\to\\infty$ in the limit of weak\ninteractions, where a Gross-Pitaevskii or Bogoliubov approximation\napplies. As shown by Kane and Fisher \\cite{KaneFisher} \n(see also \\cite{Enss} for a recent discussion), the\ninteraction of a single impurity with a LL depends crucially on the\nvalue of $K$: for $K>1$, the impurity is irrelevant for the low energy\nproperties. A 1D Bose liquid is therefore effectively superfluid,\nalthough there is no true condensate \\cite{Popov,KaneFisher}. For\n$K<1$ the impurity changes the ground state of the liquid in a\nnon-perturbative way, effectively cutting it into two disconnected\nparts. In this case, we will see that the induced interaction between\ntwo impurities is essentially a Casimir-like effect. Indeed, at low\nenergies, two impurities at distance $r$ define a box with reflecting\nboundary conditions for the phonon modes of the quantum liquid, which\nleads to an attractive Casimir interaction energy proportional to\n$u\/r$ with $u$ the sound velocity. The case $K=1$ is marginal and\ncorresponds to a non-interacting Fermi gas in 1D or - equivalently -,\na system of hard core bosons, the Tonks-Girardeau gas \\cite{Girardeau,\nWeiss, Paredes}. In the following we will study the interactions\nmediated by the 1D quantum liquid between two impurities for the\nvarious cases, including fermions with spin. We focus our analysis on\nthe case of {\\it static} impurities, while the situation of a slow\ntime dependence, relevant for atomic quantum dots, where the\ninteractions depend on the internal states is only discussed\nqualitatively at the end of the paper.\n\n\n\\section{One-dimensional fermionic liquid}\n\\subsection{Non-interacting fermions}\\label{nonint}\nBefore considering the generic situation of impurities embedded in a\nsea of interacting particles, we first address the marginal case $K=1$\nof non-interacting fermions. For simplicity we start from a gas of\n$N$ non-interacting spinless fermions in the presence of two localized\nimpurities separated by a distance $r$. Considering cold gases in\natomic quantum wires, the solution of this problem is not just an\nacademic exercise. Indeed, since fermions in a single hyperfine state\nhave no s-wave interactions due to the Pauli principle, they realize\nan ideal Fermi gas at sufficiently low temperatures. We assume that\nthe particles are contained in a periodic box of length $L$ with\naverage density $\\rho_0\\equiv N\/L$. The (grand canonical) partition\nfunction of the liquid at a given temperature $T$ may be expressed in\nterms of a functional integral over the Grassman fields\n$(\\bar{\\psi},\\psi)$ representing the fermions:\n\\begin{equation} \nZ=\\int \\mathrm{D} \\bar{\\psi} \\mathrm{D}\\psi\n \\exp(-S_\\mathrm{FL}-S_{i}).\n\\label{eq:statsum}\n\\end{equation}\nThe corresponding action of an ideal gas is\n\\begin{equation}\nS_\\mathrm{FL}=\\int_{0}^{\\beta} \\!\\!\\mathrm{d} x\\,\\mathrm{d}\\tau\n \\left[\\bar{\\psi}\\,\\partial_{\\tau}\\psi-\\left(\\frac{1}{2m} \\nabla\n \\bar{\\psi} \\nabla\\psi -\\mu\\,\\bar{\\psi}\\psi \\right)\\right],\n\\label{eq:SFL}\n\\end{equation} \nwhere $\\tau$ is the imaginary time running from $0$ to $1\/T=\\beta$,\nand $\\mu$ is the chemical potential (we use units such that\n$\\hbar=k_B=1$). The fields are anti-periodic in imaginary time. For\nshort range interactions, appropriate for cold atoms, the interaction\nof the impurities with the liquid can be described by an additional\ncontribution\n\\begin{equation}\nS_{i}=\\int_{0}^{\\beta}\\mathrm{d}\\tau\\!\\sum_{\\alpha=1,2}g_{\\alpha}\\bar{\\psi}\n(x_{\\alpha})\\psi(x_{\\alpha}),\n\\label{eq:Sint}\n\\end{equation} \nproportional to the local density at the impurity positions\n$x_{\\alpha}$. Here, the index $\\alpha=1,2$ labels the impurities,\nwhile the coupling constants $g_{\\alpha}$ describe the strength of\ncollisions between the atoms in the liquid and the impurities. The\nexpression for $S_{i}$ is based on assuming an effective\npseudopotential for the interaction between the impurity and the\nquantum liquid. More precisely, the interaction should be replaced by\na spatial integral of the detailed impurity potential with the\nmicroscopic density operator of the liquid. \nIn the present section dealing with non-interacting fermions, there is no need of \na high energy cutoff, as one can directly work with the well behaved microscopic \ntheory. However, in order to discuss the low energy behavior and to make contact \nwith the following sections dealing with interacting fermions using the Luttinger \nliquid phenomenology, we introduce a high energy cutoff $\\omega_c$. Its value can be\nestimated as $\\omega_{c}\\sim \\textrm{Min}\\{u\/l_{0},\\mu\\}$, where $u$\nis the characteristic velocity of excitations ($u=v_F$ in an ideal Fermi gas), \n$l_0$ is the impurity size and $\\mu=p_{F}^2\/2m$ is the chemical potential (at zero \ntemperature) with $p_{F}=mv_F \\equiv \\pi \\rho_0$, the Fermi momentum. \nAs will become clear from our results\nbelow, the coupling constants $g_{\\alpha}$ are then - up to a factor\n$v_{F}$ - identical with the dimensionless backscattering amplitudes\n$f_{1,2}(\\pi)$ for fermions at the impurities. Microscopically, they\nare thus determined by the solution of the single particle scattering\nproblem off a single impurity. In practice, an appreciable value of\nthe backscattering amplitude requires the impurity size to be smaller\nor of the order of the interparticle spacing, since otherwise the\nFourier component of the potential at $2p_{F}$ is close to zero, and\nhence the dimensionless coupling constants $\\gamma_{\\alpha}\\equiv\ng_{\\alpha}\/v_F$ vanish. Therefore, in the following, we will take\n$\\omega_c \\sim \\mu$.\n\nThe Grassman fields $(\\bar{\\psi},\\psi)$ are free everywhere apart from\nthe points $x=x_{1,2}$ and hence can be easily integrated out by the\nfollowing standard trick: first we formally introduce four\n$\\delta$-functions into the integrand\n\\begin{eqnarray}\nZ&=& \\int \\mathrm{D}\\bar{\\psi} \\mathrm{D}\\psi \\prod_{\\alpha=1,2}\n\\mathrm{D}\\bar{\\eta}_{\\alpha} \\mathrm{D}\\eta_{\\alpha} \\,\n\\delta[\\psi(x_{\\alpha},\\tau)-\\eta_{\\alpha}(\\tau)]\\nonumber \\\\ &\\times&\n\\delta[\\bar{\\psi}(x_{\\alpha},\\tau)-\\bar{\\eta}_{\\alpha}(\\tau)]\n\\,e^{-S_\\mathrm{FL}-S_{i}},\n\\end{eqnarray}\nwhere $(\\bar{\\eta}_{\\alpha},\\eta_{\\alpha})$ are the new Grassman\nvariables describing the fermions at the location of the individual\nimpurities. Then we introduce a set of auxiliary fields\n$(\\bar{\\kappa}_{\\alpha},\\kappa_{\\alpha})$ using the identity\n$\\delta(f)\\sim\\int \\mathrm{D}\\kappa\\exp(i\\int\\kappa f\\mathrm{d}\\tau)$\nto rise the $\\delta$-functions into the action. Finally, we integrate\nout the fermionic fields $(\\bar{\\psi},\\psi)$, which appear only\nquadratically, by Fourier transformation:\n\\begin{equation} \nZ=Z_\\mathrm{FL}^0 \\int\\prod_{\\alpha=1,2} \\mathrm{D}\\bar{\\eta}_{\\alpha}\n\\mathrm{D}\\eta_{\\alpha} \\mathrm{D}\\bar{\\kappa}_{\\alpha}\n\\mathrm{D}\\kappa_{\\alpha} e^{-S'-S_{i}(\\bar{\\eta},\\eta)},\n\\end{equation} \nwhere\n\\begin{equation}\nS_{i}(\\bar{\\eta},\\eta)=\\int_{0}^{\\beta}\\mathrm{d}\\tau\\sum_{\\alpha=1,2}\ng_{\\alpha}\\bar{\\eta}_{\\alpha}\\eta_{\\alpha},\n\\end{equation}\nand\n\\begin{eqnarray}\nS'&=&\\frac{L}{\\beta}\\sum_{n}\\int\\frac{\\mathrm{d}p}{2\\pi}\n\\frac{\\sum_{\\alpha,\\beta}\\bar{\\kappa}_{\\alpha}\n\\kappa_{\\beta}e^{ip(x_{\\alpha}-x_{\\beta})}}\n{-i\\omega_{n}+\\xi_{p}}\\nonumber \\\\ &-&i \\sum_{\\alpha,n}\n(\\kappa_{\\alpha}\\eta_{\\alpha} +\\bar{\\kappa}_{\\alpha}\n\\bar{\\eta}_{\\alpha}),\n\\end{eqnarray}\nwhere $\\xi_{p}=p^{2}\/2m-\\mu$, and the summation is over the fermionic\nMatsubara frequencies $\\omega_{n}$. The trivial prefactor\n$Z_\\mathrm{FL}^0$ arises from the integration over the fermionic\nfields in the absence of impurities, giving the grand partition\nfunction of the homogeneous liquid. The fields\n$(\\bar{\\kappa}_{\\alpha},\\kappa_{\\alpha})$ depend only on imaginary\ntime $\\tau$, or frequency $\\omega_{n}$ in the Fourier representation\nand thus the integral over $p$ can be easily calculated. Since for a\nsufficiently large separation $\\vert x_1-x_2\\vert =r\\gg p_{F}^{-1}$,\nthe interaction energy is small, the relevant frequencies are small\ncompared to the Fermi energy $\\omega_c\\sim \\mu\\sim v_{F} p_F$. An\nexpansion to leading order in $\\omega_{n}\\ll\\mu$ then gives\n\\begin{eqnarray}\nS'&=&-\\frac{i L}{\\beta v_{F}}\\sum_{n} \\sum_{\\alpha,\\alpha'} s_n\n\\bar{\\kappa}_{\\alpha} \\kappa_{\\alpha'}\ne^{ip_{F}|x_{\\alpha}-x_{\\alpha'}|s_n} \\nonumber \\\\ &\\times&\ne^{-|\\omega_{n}|(1-\\delta_{\\alpha,\\alpha'})\/\\omega_{r}}\n-i\\sum_{\\alpha,n} (\\kappa_{\\alpha}\\eta_{\\alpha}+ \\bar{\\kappa}_{\\alpha}\n\\bar{\\eta}_{\\alpha}),\n\\end{eqnarray}\nwhere $\\omega_{r}\\equiv u\/r\\ll\\mu$ and $s_n\\equiv\n\\mathrm{sign}(\\omega_n)$. The characteristic frequency $\\omega_{r}$\nwill play an important role in our subsequent discussions. Physically\nit represents the inverse flight time for a characteristic excitation\nin the liquid between the locations of the two impurities, which\nnaturally obeys the inequality $\\omega_{r}\\ll\\omega_{c}$ provided the\nimpurities are much further apart than the average distance between\ntwo fermions in the liquid. It is also the quantization energy\nbetween the two impurities. In order to obtain the impurity\ninteraction directly from the partition function, we integrate out the\nauxiliary fields $(\\bar{\\kappa}_{\\alpha},\\kappa_{\\alpha})$. This\nresults in an action\n\\begin{equation}\nS'(\\bar{\\eta},\\eta)=\\frac{\\beta v_{F}}{i L} \\sum_{n}\n (\\bar{\\eta}_{1},\\bar{\\eta}_{2})\\left(\n\\begin{array}{cc}\nf_{n} & -f_{n}e_n\\\\ -f_{n}e_n & f_{n}\n\\end{array}\n\\right)\\left(\n\\begin{array}{c}\n\\eta_{1}\\\\ \\eta_{2}\n\\end{array}\n\\right),\n\\end{equation}\nwhich only depends on the four time dependent Grassman fields\n$(\\bar{\\eta}_{\\alpha},\\eta_{\\alpha})$ which describe the Fermi field\nat the impurity positions. The coefficients $f_{n}$ and $e_{n}$ are\ndefined by\n\\begin{equation}\nf_{n}\\equiv \\frac{s_n}{1-e_n^2},\\,\\, e_n\\equiv e^{i s_n p_{F} r -\n|\\omega_{n}|\/\\omega_{r}}.\n\\end{equation}\nIncluding the contribution (\\ref{eq:Sint}) due to the interaction\nbetween the impurities and the liquid, the complete expression for the\nstatistical sum (\\ref{eq:statsum}) can now be written as\n\\begin{equation} \nZ=Z_\\mathrm{FL}^0 Z_{\\kappa} \\int \\prod_{\\alpha=1,2}\n\\mathrm{D}\\bar{\\eta}_{\\alpha} \\mathrm{D}\\eta_{\\alpha} e^{-S'-S_{i}},\n\\end{equation}\nwhere $Z_{\\kappa}$ comes from the integration of the auxiliary fields:\n\\begin{equation}\nZ_{\\kappa}=\\prod_n \\left(\\frac{i L}{\\beta v_{F}} \\right)^2 (1-e_n^2).\n\\end{equation}\nThe total effective action $S'+S_{i}$ is quadratic in the Grassman\nfields $(\\bar{\\eta}_{\\alpha},\\eta_{\\alpha})$ and thus the integration\ncan be done exactly to yield $Z=Z_\\mathrm{FL}^0 Z_{\\kappa} Z_{\\eta}$,\nwith\n\\begin{equation}\nZ_{\\eta}=\\prod_n \\left(\\frac{\\beta v_{\\mathrm{\\scriptscriptstyle\n F}}}{i L} \\right)^2 \\left[ (f_n+i\\gamma_1)(f_n+i\\gamma_2)-(f_n\n e_n)^2\\right]\\ .\n\\end{equation}\nHere, the $\\gamma_{\\alpha}=g_{\\alpha}\/v_F$ are the dimensionless\nbackscattering amplitudes, characterizing the interaction of the\nimpurities and the liquid. We can now obtain the free energy of the 1D\nFermi gas in the presence of the impurities from $F=-\\log Z\/\\beta$ as\nfollows\n\\begin{align}\nF =& F^0-\\frac{1}{\\beta} \\sum_{n} \\log \\{(1-e_n^2) \\nonumber \\\\\n&\\times [(f_n+i\\gamma_{1})(f_n+i\\gamma_{2})-(f_{n}e_n)^2]\\},\n\\label{F12}\n\\end{align}\nwhere $F^0=-\\log Z_\\mathrm{FL}^0\/\\beta$ is the free energy of the\nundisturbed, homogeneous liquid. The expression (\\ref{F12}) is ill\ndefined as it stands, since it contains both the energy of zero-point\nfluctuations in the gas, as well as the formally divergent\nself-energies of the separate impurities. The relevant interaction\nenergy associated with a change of the separation of the two\nimpurities is given by:\n\\begin{eqnarray}\nV_{12} &\\equiv& F (\\gamma_{\\alpha},r)-F(0,r)- [F\n(\\gamma_{\\alpha},\\infty)-F (0,\\infty)]\\nonumber \\\\ &=&F\n(\\gamma_{\\alpha},r)- F (\\gamma_{\\alpha},\\infty).\n\\label{renorm}\n\\end{eqnarray}\nThe renormalization thus requires subtracting first the free energy of\nthe liquid without the impurities ($\\gamma_{\\alpha}=0$, vacuum energy)\nand then the free energy of the system when the impurities are very\nfar apart ($r\\rightarrow \\infty$, self-energy of the\nimpurities). While both the vacuum energy and the individual\nself-energies are infinite in the absence of a cutoff, the\nrenormalized interaction (\\ref{renorm}) is finite and independent of\nthe cutoff (see also the discussion below in section \\ref{classical}).\n\nAt low temperature $T\\ll\\omega_{r}$ we can switch from summation to\nintegration according to $\\mathrm{d}\\omega=2\\pi T\\mathrm{d}n$, so that\nthe effective interaction energy between the impurities can be\nexpressed as:\n\\begin{equation}\nV_{12} =-\\int_{0}^{\\infty} \\frac{\\mathrm{d}\\omega}{\\pi}\\log\n\\left|1+\\frac{\\gamma_{1}\\gamma_{2}e^{-2\\omega\/\\omega_{r}+ 2 i p_{F}r}}\n{1+i(\\gamma_{1}+ \\gamma_{2})- \\gamma_{1}\\gamma_{2}}\\right|.\n\\label{eq:Vfreefermionsfin}\n\\end{equation}\nThe integral can be performed analytically to yield our final result\nfor the impurity interaction at $T=0$:\n\\begin{equation}\nV_{12}=\\frac{v_{F}}{2\\pi r}\\, \\Re\\, \\mathrm{Li}_2 \\left(-\\frac{\n \\gamma_{1}\\gamma_{2} e^{2i p_{F} r}}{1+ i (\\gamma_{1} + \\gamma_{2})-\n \\gamma_{1} \\gamma_{2}}\\right) \\ ,\n\\label{eq:Dilog}\n\\end{equation}\nwhere $\\mathrm{Li}_2$ is the di-logarithmic function \\cite{dilog} and\n$\\Re$ is the real part. Obviously the interaction quite generally\nfalls off very slowly like $1\/r$ with an amplitude, which is a\nstrictly periodic function. Its period $\\pi\/p_F=\\rho_0^{-1}$ is equal\nto the average inter-particle distance. This is a characteristic\nproperty of degenerate fermions, essentially reflecting the well known\nFriedel oscillations of the density (see below). Trivially, the\ninteraction vanishes, if one of the scattering amplitudes\n$\\gamma_{1,2}$ is zero.\n\nA simple expression for the renormalized interaction energy $V_{12}$\nis obtained in two limiting cases. First, if the interaction of the\nimpurities with the liquid is weak $\\gamma_{\\alpha}\\ll 1$, we can\nexpand the di-logarithm in Eq.~(\\ref{eq:Dilog}) to obtain:\n\\begin{equation}\nV_{12}=-\\gamma_{1} \\gamma_{2} \\frac{v_F}{2\\pi r} \\cos(2p_{F}r)\\, .\n\\label{eq:V12idealweak}\n\\end{equation} \nIn the limit of strong impurities $\\gamma_{\\alpha}\\gg1$, we find in\nturn the result\n\\begin{equation}\nV_{12}=\\frac{v_{F}}{2\\pi r}\\, \\Re\\, \\mathrm{Li}_2 \\left(e^{i 2 p_{F}\n r} \\right) \\ ,\n\\label{eq:V12limits}\n\\end{equation} \nwhich is completely independent of the scattering amplitudes. In this\ncase, the interaction energy $V_{12}$ can be represented as\n$V_{12}=\\frac{v_{\\mathrm{\\scriptscriptstyle F}}}{2 \\pi\nr}f(2p_{\\mathrm{\\scriptscriptstyle F}}r)$, where $f(x)\\equiv \\Re\\,\n\\mathrm{Li}_2 (e^{i x})$ is a periodic function bounded as follows\n$f_{{\\min}}\\leq f\\leq f_{{\\rm max}}$ where:\n\\begin{equation}\nf_{{\\rm max},{\\rm min}}= \\mathrm{Li}_2 (\\pm 1)\n=\\frac{\\pi^2}{6},-\\frac{\\pi^2}{12}\\ .\n\\end{equation}\nA simple way of understanding the slow $1\/r$-decay and the\noscillations with period $\\pi\/p_F$ may be obtained in the weak\nscattering limit Eq.~(\\ref{eq:V12idealweak}). Indeed, the density\nperturbation created by a single impurity of strength $\\gamma_1$ at\nposition $x_1$ is asymptotically given by:\n\\begin{equation} \n\\rho_1(x) \\approx \\rho_0 - \\frac{\\gamma_1}{2} \\frac{\\cos(2p_{F}\n |x-x_1|)}{2\\pi |x-x_1|}\\ .\n\\end{equation}\nThis expression for the Friedel oscillations in a spinless 1D\nnon-interacting Fermi gas is valid in the limit where $\\gamma_1 \\ll 1$\nand $|x-x_1|\\gg \\rho_0^{-1}$ \\cite{Matveev}. Since the impurities\ncouple to the local density, the interaction energy (excluding\nself-energies) of the system of two weak impurities is simply obtained\nfrom $U_{12}(\\gamma_{\\alpha},r)= g_2\\rho_1(x_2)+g_1\\rho_2(x_1)$ where\n$\\gamma_{\\alpha}\\ll 1$. When renormalized\n$V_{12}=U_{12}(\\gamma_{\\alpha},r)- U_{12}(\\gamma_{\\alpha},\\infty)$,\nthis interaction energy coincides with\nEq.~(\\ref{eq:V12idealweak}). Alternatively, the result may be derived\nby using the random-phase approximation (RPA) as shown in the Appendix\n\\ref{app:RPA}.\n\nThe analysis in this section is readily generalized to the case of a\nFermi gas with spin. In fact for non-magnetic impurities, such as\nconsidered in the current article, the two spin modes are decoupled\nand therefore the energy due to the presence of the two impurities is\nsimply multiplied by a factor of two.\n\nIt should be emphasized that the calculation above, can be immediately\nextended to the case of noninteracting fermions in two or three\ndimensions $d=2,3$, giving rise to an interaction energy for weak\ncoupling of the form\n\\begin{equation}\nV_{12}\\sim f_{1}(\\pi)f_{2}(\\pi) \\frac{p_F v_F}{(p_{F} r)^d} \\cos(2p_{F}r)\\, ,\n\\label{23d}\n\\end{equation}\nwhere $f_{\\alpha}(\\pi)$ are the dimensionless backscattering\namplitudes of the impurities. This result follows most simply by\nconsidering the density fluctuations $\\delta\\rho_{1}(\\vec x)$ induced\nby a single impurity at position $\\vec x_{1}$. As discussed, e.g., in\nRef.~\\cite{Zwerger2} they exhibit Friedel oscillations proportional to\nthe dimensionless backscattering amplitude $f_{1}(\\pi)$ at the Fermi\nenergy. The resulting interaction energy is then simply given by\n$V_{12}\\propto f_{2}(\\pi)\\delta\\rho_{1}(\\vec x_2)$. In fact, this is a\nspecial case of a general result \\cite{FerrellLuther}, that the\nasymptotic interaction between two impurities is determined by the\nproduct of their backscattering amplitudes. In the presence of short\nrange repulsive interactions between the fermions, we expect that the\nresult (\\ref{23d}) remains qualitatively correct in the case of two\nand three dimensions. This is based on the existence of a Fermi liquid\ndescription in $d=2,3$, which guarantees that the low energy\nproperties are qualitatively unchanged from those of a Fermi gas. For\nexample, assuming that the static density response function at $2p_F$\nis given by the particle-hole bubble \\cite{Negele}, the\nrenormalization factor $Z<1$ in the single-particle Green function\nwill give rise to a Fermi liquid correction factor $Z^{2}$ in\n$V_{12}$. This argument, however, neglects possible vertex corrections\nin the density response which may lead to an enhancement rather than a\nsuppression of the amplitude of the Friedel oscillations. In fact this\neffectively happens in the one-dimensional case, where the vanishing\n$Z\\,$-factor gives rise to Friedel oscillations, which decay more\nslowly than in the noninteracting case (see below). While we are not\naware of a quantitative calculation of the $2p_F\\,$-density response\nin Fermi-liquids, it is very likely that they will give rise only to\nfinite, multiplicative corrections to Eq.~(\\ref{23d}). As we will see\nbelow, however, the situation in one dimension is quite different from\nthat in $d=2,3$ in the sense, that even qualitatively the asymptotic\nform of the interaction is \\emph{not} given by the Friedel oscillation\npicture, even for very weak impurities.\n\nFinally, we mention a recent work dealing with neutron matter. Bulgac\n\\emph{et al.} \\cite{Bulgac} consider a neutron star crust, which is\nmodeled as a degenerate non-interacting neutron gas (i.e. an ideal 3D\nFermi gas) containing various kinds of defects or inhomogeneities\n(such as nuclei or bubbles) immersed in it. These authors compute the\ninteraction energy between two defects resulting from the quantum\nfluctuations of the Fermi sea of neutrons. They obtain expressions\nsimilar to Eq.~(\\ref{23d}), which can be interpreted as RKKY-like\ninteractions between defects. In addition, they discuss the influence\nof the shape of the defects and consider situations with more than two\ndefects.\n\n\n\\subsection{Spinless Fermi Luttinger liquid}\\label{less}\nRealistic Fermi systems consist of interacting particles. In three and\nalso in two dimensions, it is possible to describe even strong\ninteractions by Landau's Fermi liquid theory. As is well known,\nhowever, this concept fails in one dimension. Here we consider\nfermions with repulsive short-range interaction. At low-energy such a\nsystem exhibits a gapless excitation spectrum with a linear\ndispersion, characteristic for the universality class of Luttinger\nliquids (LL) \\cite{Haldane1,Haldane2,Giamarchi}.\n\nFor simplicity, we start by considering spinless fermions, for which\nthe low-energy description is given by the following hydrodynamic\naction:\n\\begin{equation}\n S_\\mathrm{LL}=\\frac{1}{2\\pi K}\\int \\!\\!\\mathrm{d} x\\,\\mathrm{d}\\tau\n \\left[ u\n (\\partial_{x}\\theta)^{2}+\\frac{1}{u}(\\partial_{\\tau}\\theta)^{2}\n \\right]. \\label{eq:LLaction}\n\\end{equation} \nHere $u$ is the sound velocity and $K$ the Luttinger parameter. In a\ntranslationally invariant system, they obey the relation $u K= v_F$\n\\cite{Haldane2}, with $v_F=p_F\/m=\\pi \\rho_0\/m$ the Fermi velocity of\nthe associated non interacting spinless Fermi gas. We consider repulsive \ninteractions for which $K<1$. Since the Luttinger liquid description only\napplies at low energies, the fields have to be cutoff at energy\n$\\omega_c \\sim \\mu$, where $\\mu$ is the chemical potential. \nThe associated cutoff length $a\\equiv u\/\\omega_c$\nis of order $1\/\\rho_0$. Of course, for a quantitative calculation of\nthe scale at which the low energy description applies, a microscopic\nmodel is needed, which allows to determine nonuniversal\nproperties. For single impurities in Luttinger liquids this problem\nhas only recently been discussed, see \\cite{Meden}. Since we are\nconcerned with the interaction at distances much longer that the\naverage interparticle separation, only the low energy properties are\nrelevant, which are well described by the hydrodynamic action\n(\\ref{eq:LLaction}). The corresponding field $\\theta$ is related to\nthe density of the liquid by\n\\begin{equation}\n\\rho(x)\\approx \\left(\\rho_{0}+ \\frac{\\partial_x \\theta}{\\pi}\\right)\n\\left[1+ 2\\cos(2\\theta+ 2 p_{F} x)\\right],\n\\label{density}\n\\end{equation}\nwhere $\\rho_{0}$ is the equilibrium density and only the first\nharmonics are taken into account \\cite{Haldane2,KaneFisher}.\n\nThe interaction between the impurities and the Luttinger liquid is\ntaken to be of the form\n\\begin{equation}\nS_{i}=\\int_{0}^{\\beta}\\mathrm{d}\\tau\\!\\sum_{\\alpha=1,2}\\tilde{g}_{\\alpha} \n\\rho(x_{\\alpha})\\, ,\n\\label{eq:Sidens}\n\\end{equation}\ni.e. a coupling to the local density with phenomenological \nscattering amplitudes $\\tilde{g}_{\\alpha}$. Inserting the\nexpansion (\\ref{density}) into this interaction, gives rise to four\ndifferent terms. The first term is just the constant Hartree\nself-energy of the impurities, which - of course - does not contribute\nto the renormalized interaction energy $V_{12}$. In addition, there\nare terms containing $\\partial_{x}\\theta$ due to forward scattering.\nThey describe quantum corrections to the self energies but again are\nirrelevant for the interaction $V_{12}$ between two widely separated\nimpurities. The dominant term for this interaction is the\ncontribution proportional to $\\cos(2\\theta+2p_{F}x)$, which is due to\nbackward scattering. In addition there are higher order terms like\n$\\partial_{x}\\theta \\cos(2\\theta+2p_{F}x)$, however these are less\nrelevant in the renormalization group (RG) sense \\cite{NoteRG}.\nTaking only the most relevant part of the interaction, the coupling\nbetween the impurities and the LL leads to the following nonlinear\ncontribution to the action\n\\begin{equation}\nS_{i}[\\Theta]\\approx \\int_0^{\\beta} \\mathrm{d}\\tau \\sum_{\\alpha=1,2} 2\n\\tilde{g}_{\\alpha} \\rho_0 \\cos[2\\sqrt{\\pi K}\\Theta(x_\\alpha)+ 2\np_{F}x_{\\alpha}]\\ ,\n\\label{eq:SiTheta}\n\\end{equation}\nwhere we introduced the renormalized field $\\Theta$ by\n$\\theta=\\Theta\\sqrt{\\pi K}$. The complete statistical sum of the\nsystem can again be represented by a functional integral. To perform\nthe integration over the field $\\Theta$ we use the same approach as\nfor the ideal Fermi gas: first we introduce the new variables\n$\\Theta(x_{1},\\tau)=\\Theta_{1}(\\tau)$ and\n$\\Theta(x_{2},\\tau)=\\Theta_{2}(\\tau)$ and then insert the two\n$\\delta$-functions into the functional integral:\n\\begin{eqnarray}\nZ&=& \\int \\mathrm{D}\\Theta \\prod_{\\alpha=1,2}\n\\mathrm{D}\\Theta_{\\alpha} \\delta[\\Theta(x_{\\alpha})-\\Theta_{\\alpha}]\ne^{-S_\\mathrm{LL}-S_{i}}.\n\\end{eqnarray}\nWe then transform the $\\delta$ functions into the functional integrals\nover auxiliary fields and perform the Gaussian integration\n\\begin{equation} \nZ= Z_\\mathrm{LL}^0 Z_{\\kappa} \\int\n\\mathrm{D}\\Theta_{1}\\mathrm{D}\\Theta_{2}e^{-S_{{\\rm\neff}}-S_{i}[\\Theta_\\alpha]}\\label{eq:smallKZ}\\ ,\n\\end{equation}\nwhere the effective action for the real fields $\\Theta_{1,2}$ is\n\\begin{equation} \nS_{{\\rm eff}}=\\sum_{n} (\\Theta_{1,-n},\\Theta_{2,-n}) \\left(\n\\begin{array}{cc}\nf_{n} & -f_{n}e_n\\\\ -f_n e_n & f_{n}\n\\end{array}\n\\right)\\left(\n\\begin{array}{c}\n\\Theta_{1,n}\\\\ \\Theta_{2,n}\n\\end{array}\n\\right),\n\\end{equation} \nwith the summation occurring over the bosonic Matsubara frequencies\n$\\omega_{n}$, where $f_{n}\\equiv \\beta|\\omega_{n}|\/(1-e_n^2)$ and\n$e_n\\equiv e^{-|\\omega_{n}|\/\\omega_r}$ with $\\omega_r\\equiv u\/r$. The\nfactor $Z_\\mathrm{LL}^0$ comes from the integration over\n$\\Theta(x,\\tau)$ and is independent of the impurities, describing the\nhomogeneous Luttinger liquid. By contrast, the factor\n\\begin{equation}\nZ_{\\kappa}=\\prod_{n} \\left(1-e_n^2\\right)^{-1\/2}.\n\\end{equation}\nwhich comes from the integration over the auxiliary fields, describes\nthe change in the phonon modes due to the constraint on the Fermi\nfields at the positions of the two impurities. Similar to the\nnoninteracting situation, this contribution depends on the associated\ncharacteristic frequency $\\omega_{r}$ and is crucial in obtaining a\nfinite interaction energy $V_{12}$ which is independent of the cutoff.\n\nThe remaining and now non-trivial functional integral over the\ntime-dependent fields $\\Theta_{\\alpha}$ is of the same form as the one\nwhich appears in the context of quantum Brownian motion in a periodic\npotential \\cite{Zwerger}. Indeed the effective action (29) basically\ndescribes two quantum particles subject to ohmic dissipation of\ndimensionless strength $1\/K$ which move in a periodic potential\ngenerated by the backscattering amplitude. As has been shown in\n\\cite{Zwerger} this problem leads to a localized ground state if\n$1\/K>1$ with small fluctuations in the field $\\Theta$. For a quantum\nliquid with sufficiently strong interactions between the fermions\n$K\\ll 1$ and strong impurities \n$\\tilde{\\gamma}_{\\alpha}\\equiv \\tilde{g}_{\\alpha}\/v_F\\gg 1$, the functional\nintegral (\\ref{eq:smallKZ}) over the time-dependent fields\n$\\Theta_{\\alpha}$ can thus be calculated using the stationary phase\napproximation (SPA): expanding the functions $\\cos(2\\sqrt{\\pi\nK}\\Theta_{\\alpha}+2p_{F}x_{\\alpha})$ from Eq.~(\\ref{eq:SiTheta}) to\nsecond order in the fields around one of its minimum, we approximate\nthe interaction Lagrangian in the form\n\\begin{equation}\n\\tilde S_{i}=\\int_0^{\\beta}\n\\mathrm{d}\\tau\\sum_{\\alpha=1,2}E_{\\alpha}\\Theta_{\\alpha}^{2}\\ ,\n\\label{eq:statphaseint}\n\\end{equation}\nwhere $E_{\\alpha}=4\\pi K \\tilde{g}_{\\alpha}\\rho_0$. Physically this means that\nthe interaction between each of the impurities and the liquid is\nsufficiently strong to pin the local phase $\\Theta$ near the value\nminimizing the potential energy. The quantities $E_{\\alpha}$ play the\nrole of effective frequencies for the evolution of the fields\n$\\Theta_{\\alpha}$. The approximation of the original Lagrangian\n(\\ref{eq:SiTheta}) by the quadratic form (\\ref{eq:statphaseint}) is\nequivalent to an adiabatic approximation which describes physical\nprocesses occurring slower than a time scale given by\n$E_{\\alpha}^{-1}$. As the typical frequency of interest is $\\omega_r$,\nthe stationary phase approximation is valid when $E_{\\alpha}\\gg\n\\omega_r$. It is thus applicable in the case of strong impurities or -\nequivalently - long distances (see below) in a strongly repulsive\nliquid: $r\\rho_0 \\tilde{\\gamma}_{\\alpha}\\gg 1\/K^2\\gg 1$.\n\nWithin the quadratic approximation (\\ref{eq:statphaseint}), the full\neffective action in Eq.~(\\ref{eq:smallKZ}) is quadratic in\n$\\Theta_{\\alpha}$ and hence can be evaluated exactly\n$Z=Z_\\mathrm{LL}^0 Z_{\\kappa} Z_{\\Theta}$, where:\n\\begin{equation}\nZ_{\\Theta}=\\prod_{n}\\left[(f_n+\\beta E_{1}) (f_n+\\beta\nE_{2})-(f_{n}e_n)^2 \\right]^{-1\/2}\\ .\n\\end{equation}\nThe associated free energy $F=-\\log Z\/\\beta $ is given by\n\\begin{align} \nF=&F^0+\\frac{1}{2\\beta} \\sum_{n} \\log \\{(1-e_n^2)\\nonumber \\\\ &\\times\n[(f_n+\\beta E_{1})(f_n+\\beta E_{2})-(f_{n}e_n)^2 ]\\}\\ ,\n\\end{align} \nwhere $F^0=-\\log Z_\\mathrm{LL}^0 \/\\beta$ again describes the\nundisturbed, homogeneous liquid. From the free energy we obtain the\nrenormalized interaction energy between the two impurities $V_{12}$ in\nprecisely the same manner as in Eq.~(\\ref{renorm}):\n\\begin{equation}\nV_{12}= \\frac{1}{\\beta} \\sum_{n>0} \\log\\left(1-\\frac{E_{1} E_{2}\n e^{-2\\omega_n\/\\omega_{r}}}{\\omega_n^{2} + \\omega_n\n (E_{1}+E_{2})+E_{1}E_{2}} \\right) \\ .\n\\end{equation}\nAt sufficiently low temperatures $T\\ll\\omega_{r}$, the summation may\nbe replaced by an integral over the real frequency $\\omega$:\n\\begin{equation}\nV_{12}=\\frac{1}{2\\pi}\\int_0^\\infty\n\\mathrm{d}\\omega\\log\\left(1-\\frac{E_{1}E_{2}e^{-2\\omega\/\\omega_{r}}}\n{\\omega^{2}+\\omega(E_{1}+E_{2})+E_{1}E_{2}}\\right).\n\\end{equation}\n\nIn the limit of long distances (or strong impurities), i.e.\n$E_{\\alpha}\\gg \\omega_{r}$, the integral converges at\n$\\omega\\sim\\omega_{r}$ and hence $\\omega^{2}\\ll\\omega E_{\\alpha}\\ll\nE_{\\alpha}^{2}$. For sufficiently large separations, we thus obtain\nthe simple universal interaction\n\\begin{equation} \nV_{12}=\\frac{u}{2\\pi r}\\int_0^\\infty\n \\mathrm{d}y\\log(1-e^{-2y})=-\\frac{\\pi}{24}\\frac{u}{r}\\ ,\n\\label{eq:V12bosefin}\n\\end{equation} \nwhich decays inversely with distance. The short distance regime,\nwhere $E_{\\alpha}\\ll \\omega_{r}$, can, however, not be considered\nwithin the stationary phase approximation\n(\\ref{eq:statphaseint}). Indeed, the latter is only justified if the\ncharacteristic energy scale $\\omega\\lesssim\\omega_{r}$ of excitations\ninvolved in the interaction does not exceed the effective frequencies,\ni.e. if $E_{\\alpha}\\gg\\omega_{r}$. As a result, for intermediate and\nshort distances, the interaction energy $V_{12}$ does not follow a\nsimple $1\/r$ -behavior. In particular, it is impossible to describe\nthe limit $r\\to 0$ without properly including a cutoff or working in a\nmicroscopic model from the beginning. It is only within such a more\ncomplete calculation, that the full energy $F\n(\\tilde{\\gamma}_{\\alpha},r)-F(0,r)$ of the two impurity problem approaches the\nself energy of the doubled single impurity case, as expected on\nphysical grounds. For a single scalar field in 1D, as described by\nour hydrodynamic action (23), this calculation has recently been done\nby Jaffe \\cite{Jaffe}.\n\nThe validity of the quadratic expansion (\\ref{eq:statphaseint}) may be\nextended to the whole relevant range $K< 1$ of the Luttinger parameter\nwith the help of the self-consistent harmonic approximation (SCHA)\n\\cite{Saito,Zwerger}. In the context of Friedel oscillations around a\nsingle impurity in a Luttinger liquid, this has been used by Egger and\nGrabert \\cite{Egger}. It is based on making a quadratic approximation\n(\\ref{eq:statphaseint}) for the backscattering term, however with\nfrequencies $E_{\\alpha}$ which are determined from Feynman's\nvariational principle. Using Eq.~(\\ref{eq:statphaseint}) as the trial\naction, one has to minimize the free energy\n\\begin{equation}\\label{Fvar}\nF_\\mathrm{var} = -\\frac{1}{\\beta} \\log \\tilde{Z} +\n\\frac{1}{\\beta}\\langle S - \\tilde{S} \\rangle_{\\tilde{S}} \\ ,\n\\end{equation}\nwhere $S= S_\\mathrm{e f f }+S_i$ and $\\tilde{S}= S_\\mathrm{e f f} +\n\\tilde{S}_i$. Taking $E_{\\alpha}$ as variational parameters, we obtain\n\\begin{equation} \n\\frac{E_{\\alpha}}{4\\pi K \\tilde{g}_{\\alpha} \\rho_0}=\\left( 1+\n\\frac{\\omega_{c}}{E_{\\alpha}}\\right)^{-K},\n\\end{equation} \nwhere $\\omega_{c}$ is the high-energy cutoff. Following \\cite{Egger},\nwe define the crossover scale $r_0$ by $E_{\\alpha}\\equiv u\/r_0$ when\nboth impurities have approximatively the same strength $\\tilde{\\gamma}_1\n\\approx \\tilde{\\gamma}_2 \\approx \\tilde{\\gamma}$. The SCHA is a good approximation\nwhen $K<1$ and $E_{\\alpha} \\gg \\omega_{r}$, i.e. at long distances\n$r> r_0$.\n\nIn the limit of strong impurities $\\tilde{\\gamma}_{\\alpha}\\gg 1$, the SCHA\nfrequencies $E_{\\alpha}\\approx 4\\pi K \\tilde{g}_{\\alpha}\\rho_0$ are the same\nas those obtained within the stationary phase approximation and the\ncrossover scale is given by $r_0\\rho_0 \\sim 1\/K^2\\tilde{\\gamma}$. In the\nopposite limit $\\tilde{\\gamma}_{\\alpha}\\ll 1$, they are given by\n\\begin{equation}\nE_{\\alpha}\\approx 4\\pi K \\tilde{g}_{\\alpha}\\rho_0\\left(\\frac{4\\pi K\n\\tilde{g}_{\\alpha}\\rho_0}{\\omega_{c}}\\right)^{\\frac{K}{1-K}}\n\\label{crossoverenergy}\n\\end{equation} \nand the crossover scale is\n\\begin{equation}\nr_0 \\rho_0 \\sim (K^2\\tilde{\\gamma})^{-1\/(1-K)}(\\rho_0 a)^{-K\/(1-K)}\\ ,\n\\label{crossoverscale}\n\\end{equation}\nwhere $\\rho_0 a \\sim 1$ for repulsive fermions. Note the singular\nbehavior of the crossover scale $r_0\\to \\infty$ in the limit $K\\to 1$\nof non-interacting fermions. This implies that the regime of validity\nof the SCHA is moved out to extremely long scales $r_0$. Quite\ngenerally, therefore, the long distance behavior $r\\gg r_0$ of the\ninteraction energy is always given by the Casimir like expression\n(\\ref{eq:V12bosefin}) whenever $K<1$. The scale, however, beyond\nwhich this simple result applies, strongly depends on the strength of\nthe backscattering amplitude and the repulsive interaction.\n\nIn order to study the limit of weak impurities and weak interactions\n($\\tilde{\\gamma}_{\\alpha}\\ll 1$ and $K<1$ close to 1) in more detail, we use\nperturbation theory. At second order in $\\tilde{\\gamma}_{\\alpha}$, and when\n$r\\gg\\rho_0^{-1}$, we find\n\\begin{equation}\nV_{12} = - \\tilde{\\gamma}_1 \\tilde{\\gamma}_2 (\\rho_0 a \\pi)^2 \n\\left(\\frac{r}{a}\\right)^{2(1-K)} \n\\frac{v_F}{2\\pi r}\\cos (2 p_F r) h(K)\\ ,\n\\label{eq:genFriedel}\n\\end{equation} \nwhere $h(K)\\equiv \\frac{K}{\\sqrt{\\pi}}\\frac{\\Gamma(K-1\/2)}{\\Gamma(K)}$ and \n$\\Gamma(z)$ is the Gamma function. The function $h(K)$ diverges as $K\\to 1\/2$ \nand approaches one at $K= 1$. The above equation was obtained under the assumption \nthat $1\/21\/2$ when \nconsidering contact interactions between fermions, see e.g. \\cite{Recati3}. The action term that\ndescribes the interaction with the impurities is still given by\nEq.~(\\ref{eq:Sidens}) where the density is now simply the sum of the\ndensities of the two spin modes that have the same form as\nEq.~(\\ref{density}). As stated in the previous section we only keep\nthe most relevant term, which corresponds to backscattering by the\nimpurity, and hence obtain:\n\\begin{align}\nS_i[\\theta_\\mu] = \\sum_{\\alpha=1,2} 2 \\tilde{g}_{\\alpha} \\rho_0 \\int_0^\\beta\n\\!\\!\\mathrm{d}\\tau \\, & \\cos [\\sqrt{2} \\theta_\\rho (x_{\\alpha}) + 2\np_F x_{\\alpha} ] \\nonumber \\\\ & \\times \\cos [ \\sqrt{2} \\theta_\\sigma\n(x_{\\alpha}) ] \\ .\n\\end{align}\n\nNow we follow the same procedure as in previous sections and obtain\n\\begin{equation}\nZ= Z^0_\\mathrm{LL} Z_{\\kappa} \\int \\! \\prod_{\\alpha=1,2}\n\\prod_{\\mu=\\rho,\\sigma}\\! \\mathrm{D}\\Theta_{\\mu \\alpha}\\\ne^{-S_\\mathrm{e f f}[\\Theta_{\\mu \\alpha} ] - S_i[\\Theta_{\\mu \\alpha}]}\n\\ ,\n\\end{equation}\nwhere we have rescaled the fields at the impurity positions by\n$\\theta_{\\mu \\alpha}= \\sqrt{\\pi K_\\mu} \\Theta_{\\mu \\alpha}$. The\neffective action $S_{\\mathrm{e f f}}$ is given by\n\\begin{equation}\nS_\\mathrm{e f f} [\\Theta_{\\mu \\alpha}] = \\sum_{n}\n\\sum_{\\mu,\\alpha,\\delta} I^\\mu_{\\alpha \\delta}\\ \\Theta_{\\mu \\alpha,-n}\n\\, \\Theta_{\\mu \\delta,n} \\ ,\n\\end{equation}\nwhere the $n$ refers to the Matsubara frequencies $\\omega_n$ and\n$I^\\mu_{\\alpha \\delta}$ are the elements of the matrix\n\\begin{equation}\nI^\\mu = f_{\\mu n}\n\\begin{pmatrix}\n1 & -e_{\\mu n} \\\\ -e_{\\mu n} & 1\n\\end{pmatrix} \\ ,\n\\end{equation}\nwith $f_{\\mu n}= \\beta |\\omega_n|\/(1-e_{\\mu n}^2)$, $e_{\\mu n}=\ne^{-|\\omega_n|\/\\omega_\\mu}$, and $\\omega_\\mu = u_\\mu\/r$.\n\nIn order to calculate the partition function, we again use the\nstationary phase approximation, which corresponds to expanding $S_i$\naround its minima to second order in the fields, assuming that the\nimpurities are strong, i.e.~$\\tilde{\\gamma}_{\\alpha}=\\tilde{g}_{\\alpha}\/v_{F}\\gg 1$.\nThe nonlinear action $S_i$ is thus replaced by a quadratic\napproximation\n\\begin{equation}\\label{Sitilde}\n\\tilde{S}_i= \\beta \\sum_{\\mu \\alpha} \\sum_n E_{\\mu \\alpha}\n|\\Theta_{\\mu \\alpha,n}|^2 \\ ,\n\\end{equation}\nwhere $E_{\\mu \\alpha}= 2 \\pi K_\\mu\\,\\tilde{g}_{\\alpha} \\rho_0$.\n\nSince the charge and spin fields are now completely decoupled, we have\n$\\tilde{Z}= Z_{\\rho} Z_{\\sigma}$, with:\n\\begin{equation}\n Z_{\\mu} = \\prod_n \\left[ \\left( f_{\\mu n } +\\beta E_{\\mu 1} \\right)\n \\left( f_{\\mu n}+ \\beta E_{\\mu 2} \\right) -(f_{\\mu n} e_{\\mu\n n})^2 \\right]^{-1\/2} \\ .\n\\end{equation}\nThe total free energy $F=F_\\rho+F_\\sigma$ is then simply the sum of\nthe charge and spin contribution.\n\nAfter renormalization the free energy is given by:\n\\begin{equation}\nV_{12} = \\frac{1}{\\beta} \\sum_{\\mu} \\sum_{n>0} \\log \\left[1-\n\\frac{E_{\\mu 1} E_{\\mu 2} \\ e^{-2 \\omega_n\/\\omega_\\mu}}{(\\omega_n^2 +\nE_{\\mu 1}) (\\omega_n^2 + E_{\\mu 2})} \\right] \\ .\n\\end{equation}\nAt low temperature $T\\ll\\omega_\\mu$ we can again replace the sum over\nMatsubara frequencies by an integral. In the limit of strong\nimpurities $\\omega_\\mu\\ll E_{\\mu a}$ we obtain\n\\begin{equation}\nV_{12} = \\sum_{\\mu} u_\\mu \\int_0^\\infty \\! \\frac{\\mathrm{d} y}{2 \\pi}\n\\log (1- e^{-2 y}) = -\\frac{\\pi}{24}\\frac{u_\\rho+u_\\sigma}{r}.\n\\label{eq:CasSpin}\n\\end{equation}\nThis is the straightforward generalization for spin $1\/2$ fermions of\nthe result obtained in the previous section, see\nEq.~(\\ref{eq:V12bosefin}). As discussed there, the SPA is valid only\nif $\\omega_\\mu\\ll E_{\\mu a}$. Hence we are unable to calculate the\ninteraction at shorter distances, where $\\omega_\\mu\\gg E_{\\mu a}$.\n\nAs in the case of spinless fermions, we can go beyond the SPA regime,\nusing the SCHA. In particular we use $S= S_\\mathrm{e f f }+S_i$ and\n$\\tilde{S}= S_\\mathrm{e f f} + \\tilde{S}_i$ in Eq.~(\\ref{Fvar}); and\nthen we minimize $F_\\mathrm{var}$ with respect to $E_{\\mu a}$. In the\ncase of identical impurities, i.e.~$\\tilde{\\gamma}_1=\\tilde{\\gamma}_2=\\tilde{\\gamma}$, the\nvalues of $E_{\\mu \\alpha}$ that minimize $F_\\mathrm{var}$ are such\nthat $E_{\\mu 1} = E_{\\mu 2} = E_{\\mu}=\\pi K_\\mu E$. For large\ndistances, i.e., for $\\omega_\\mu \\ll E_\\mu$, the SCHA is valid and $E$\nis given by\n\\begin{equation}\nE = 2 \\tilde{g} \\rho_0 \\left( 1+ \\frac{\\omega_c}{\\pi K E}\\right) ^{-K\/2}\n\\left( 1+ \\frac{\\omega_c}{\\pi K E}\\right)^{-1\/2} \\ ,\n\\end{equation}\nwhere $\\omega_c$ is the high energy cutoff. As in the spinless case,\nwe can define the crossover scale as $r_0=\\max(u_\\mu\/E_\\mu)=v_{F}\/\\pi\nK^2 E$, since $K<1$. In the limit of very strong impurity\nbackscattering $E_\\mu\\gg \\omega_c$, we recover the SPA result, i.e.\n$E_\\mu=2 \\pi K_\\mu \\tilde{g} \\rho_0$, and the crossover scale is $r_0\n\\rho_0\\sim 1\/K^2 \\tilde{\\gamma}$. For intermediate impurity strength\n$\\omega_\\mu \\ll E_\\mu \\ll \\omega_c$, we obtain\n\\begin{equation}\nE_\\rho = E_\\sigma\/K = 2 \\pi \\tilde{g} \\rho_0 K^\\frac{1}{1-K} \\left(\\frac{2 \\pi\n\\tilde{g} \\rho_0}{\\omega_c} \\right)^{\\frac{1+K}{1-K}} \\ ,\n\\end{equation}\nand the crossover scale in this case is given by\n\\begin{equation} \\label{spincross}\n\\rho_0 r_0 \\sim \\tilde{\\gamma}^{-\\frac{2}{1-K}} K^{-\\frac{3}{1-K}}\n(a\\rho_0)^{-\\frac{1+K}{1-K}} \\ ,\n\\end{equation}\nwhere $a\\equiv v_{F} \/K\\omega_c$ is the short distance cutoff and $a\\rho_0\n\\sim 1$. Note that, as in the spinless case, the crossover scale\ndiverges $r_0\\to \\infty$ when $K\\to 1$. For weak interactions,\ntherefore, the result (\\ref{eq:CasSpin}) is only valid at very\nlarge distances.\n\nThe regime of weak impurity strength can be studied using perturbation\ntheory. At second order in $\\tilde{\\gamma}_{\\alpha}$, we find\n\\begin{equation}\\label{spinpert}\nV_{12} = - \\tilde{\\gamma}_1 \\tilde{\\gamma}_2 (\\rho_0 a \\pi)^2 \\left(\\frac{r}{a}\\right)^{1-K} \n\\frac{v_F}{\\pi r}\\cos (2 p_F r) h(K)\\ ,\n\\end{equation}\nwhere $h(K)\\equiv K^K\n\\frac{\\Gamma(K\/2)}{\\sqrt{\\pi}\\Gamma(\\frac{K+1}{2})} {}_2F_{1} \\left(\n\\frac{K}{2},\\frac{K}{2}; \\frac{K+1}{2};1-K^2 \\right)$ and ${}_2F_{1}$\nis the hyper\\-geometric function \\cite{hypergeo}. The function $h(K)$\nis a smooth, monotonically decreasing function of $K$ which diverges\nlike $h(K)\\sim 2\/\\pi K$ as $K\\to 0$ and approaches one at $K= 1$. \nWhen $K=1$, the preceding result reproduces exactly Eq.~(\\ref{eq:V12idealweak}) \n(with an extra factor of two due to the spin degeneracy) obtained for the non-interacting\n(spinless) Fermi gas, provided that we choose the following relation between the microscopic and \nphenomenological impurity coupling constants $\\gamma_{\\alpha}=\\tilde{\\gamma}_{\\alpha} \\rho_0 a \\pi$. \nThe perturbative result (\\ref{spinpert})\nis valid for $r \\ll r_0$, with $r_0$ given in\nEq.~(\\ref{spincross}). At the crossover scale $r=r_0$ and for\n$K\\lesssim 1$, $|V_{12}|\\sim \\omega_r$ as in\nEq.~(\\ref{eq:CasSpin}). In conclusion, therefore, the interaction\nbetween two impurities follows the behavior given by\nEq.~(\\ref{spinpert}) only for intermediate distances and weak\ninteractions. By contrast, for large distances, there is a crossover\nto the universal Casimir-type interaction Eq.~(\\ref{eq:CasSpin}) which\ndepends only on the velocities $u_{\\rho}$ and $u_{\\sigma}$.\n\n\n\\subsection{Discussion}\\label{sec:discussion}\nAs our main result, we have shown that for separations much larger\nthan the interparticle spacing, the interaction energy of two\nimpurities in a Luttinger liquid of repulsive fermions ($K<1$) is a\nCasimir-type interaction, given by a very simple universal relation,\nEq.~(\\ref{eq:V12bosefin}) (resp. Eq.~(\\ref{eq:CasSpin})) for spinless\n(resp. spin $1\/2$) fermions. In contrast to the result\nEq.~(\\ref{eq:V12limits}) obtained for strong impurities in a\nnon-interacting Fermi gas it does not contain Friedel oscillations and\nis independent of both the impurity strengths and the interaction\nparameter $K$. The physical origin of this long range force is thus\nquite different from the $K=1$ case. In the non-interacting gas, the\nlong range force comes from the polarization of the ground state. In\nthe strongly interacting case, the Friedel oscillations of the ground\nstate density around each of the independent impurity still exist\n\\cite{Egger} but they are not relevant for the impurity interaction at\nlong distances. Instead the result Eq.~(\\ref{eq:V12bosefin}) is best\nunderstood as being the Casimir interaction energy of two mirrors,\ni.e. impenetrable impurities, in a phononic bath \\cite{Casimir}. This\ninterpretation is supported by the direct calculation of the\ninteraction energy of two mirrors in the vacuum fluctuations of a 1D\nscalar field which represents the density modes of the intervening\nquantum liquid, see, e.g., Ref.~\\cite{Zee}. In fact, a similar result\nhas previously been found for the force between two infinite mass\nbeads on a string \\cite{DHokerSikivie}. The Friedel oscillations are\nrelevant for the interaction between two impurities only in the\nnon-interacting case or at intermediate distances in the interacting\nLuttinger liquid ($K<1$), see Eq.~(\\ref{eq:genFriedel}) and\n(\\ref{spinpert}).\n\nThe resulting picture is consistent with the RG calculation of Kane\nand Fisher for a single impurity \\cite{KaneFisher}: when $K<1$ and\n$\\tilde{\\gamma}>0$, the backscattering amplitude is renormalized to strong\ncoupling in the low energy (or long distance $r\\gg r_0$) limit. The\nliquid is thus effectively cut into pieces, with the impurities acting\nlike perfect mirrors for the acoustic modes, resulting in a Casimir\nforce between them. The scale on which the impurities flow to strong\ncoupling depends on: i) the initial strength of the impurities\n$\\tilde{\\gamma}$ and ii) the flow velocity given by $1-K$, see below. When the\nimpurities are strong and the liquid is strongly interacting, the\nimpurities flow quickly to strong coupling. The associated crossover\nscale $r_0$ is thus of the order of the interparticle distance. By\ncontrast, when the impurities are weak and the liquid is almost\nnon-interacting, it takes very long distances to reach the asymptotic\nregime. Qualitatively, the crossover scale $r_0$ for two weak\nimpurities can already be obtained from the scaling theory for a\nsingle impurity. Indeed, the perturbative flow equation of Kane and\nFisher \\cite{KaneFisher} gives as the running impurity strength:\n\\begin{equation} \n\\tilde{\\gamma}_{\\mathrm{eff}}\\approx \\tilde{\\gamma} (r\/a)^{1-K}\\ .\n\\end{equation}\nFor spin $1\/2$ fermions, the preceding equation holds provided that \nthe exponent $1-K$ is replaced by $(1-K)\/2$. \nThe crossover scale $r_0$ then corresponds to the distance at which\nthe running impurity strength $\\tilde{\\gamma}_{\\mathrm{eff}}$ becomes of order\none, i.e. $r_0 \\rho_0 \\sim \\tilde{\\gamma}^{-1\/(1-K)}$, in agreement with\nEq.~(\\ref{crossoverscale}), because $\\rho_0 a \\sim 1$ and $1\/21$, the effective\ncoupling of a single impurity is renormalized to zero in the\nlow-energy limit. The RG flow thus starts at high-energy\n$\\omega_c=u\/a\\sim \\mu$ (corresponding to a short distance cutoff\n$a\\sim \\xi$) and ends at a much lower energy $\\omega_r=u\/r$ which is\nset by the separation $r$ of the two impurities. Similar to the case\nof fermions, we define a crossover scale $r_0$ by the condition that\nthe effective dimensionless impurity strength at this scale is of\norder one:\n\\begin{equation}\n\\gamma_{\\mathrm{eff}}\\approx \\gamma (a\/r)^{K-1}\\sim 1 \\Leftrightarrow\nr_0\\approx \\xi \\gamma^{1\/(K-1)} \\lesssim \\xi \\ .\n\\end{equation}\nSince $\\gamma \\ll 1$ for weak impurities and $K=\\pi\/\\sqrt{\\gamma_B}\\to\n\\infty$ due to the weak interaction condition, we find $r_0 \\approx\n\\xi$. This confirms that there is no interaction between two weak\nimpurities embedded in a weakly interacting Bose gas when they are\nfurther apart than a distance of the order of the healing length. More\ngenerally, the scaling theory indicates that there is no long range\ninteraction between weak impurities even in a strongly interacting\nBose gas, in which $K\\gtrsim 1$. Of course the limiting case $K=1$ of\nhard-core bosons is special as is the case $K=\\infty$ of no\ninteraction at all. The latter case is treated in Appendix\n\\ref{app:nibosons} and shows, that there is no interaction between the\nimpurities whatever the impurity strength. In strong contrast to that,\nthe limit of a Tonks-Girardeau gas of hard-core bosons is equivalent\nto the case of non-interacting fermions for properties depending only\non the modulus of the ground state wavefunction like the density\ndistribution. On the basis of the calculations in section\n\\ref{nonint}, one thus expects a long-range interaction of the form\n(\\ref{eq:Dilog}) between impurities in a Tonks-Girardeau gas which\nexhibits Friedel-like oscillations. In view of the fact that the\nmomentum distribution of hard-core bosons is quite different from that\nof a free Fermi gas, showing no jump at $p_F$, this is a quite\nremarkable result \\cite{Caza}. Another singular limit, where\nlong-range interactions appear in a Bose liquid is that of\nimpenetrable impurities $\\gamma = \\infty$. For arbitrary values\n$\\infty> K>1$ of the interactions, the interacting Bose liquid is then\ncut in three disconnected pieces. The impurities thus act as perfect\nmirrors for the low energy phonon excitations of the intervening Bose\nliquid, giving rise to a Casimir interaction energy precisely as in\nEq.~(\\ref{eq:V12bosefin}) for spinless fermions. It appears, however,\nthat the limit of impenetrable impurities at arbitrary energies is\nnonphysical, imposing strict Dirichlet boundary conditions on a scalar\nfield \\cite{Jaffe}. The Casimir force is thus expected to be\nrestricted to $\\gamma=\\infty$, while for any finite $\\gamma$ only\nshort range interactions should survive. Describing these crossovers\nin detail, clearly requires a quantitative theory of impurity\ninteractions in 1D Bose liquids at arbitrary values of $K$ and\n$\\gamma$, an interesting problem for further study.\n\n\n\\section{Experimental realization and detection of the Casimir-like force}\nThe recent realization of one-dimensional ultracold Fermi gases in a\nstrong 2D optical lattice \\cite{Esslinger} provides a novel\nopportunity to study Luttinger liquid effects in a setup with cold\ngases, e.g. spin-charge separation \\cite{Recati2}. In order to study\nwhether the Casimir interactions discussed here might be observed in\nthese systems, we consider an atomic gas of fermions in two hyperfine\nstates. These two internal states play the role of (iso)spin 1\/2\nstates. In principle both the sign and the strength of the interaction\ncan be controlled using scattering resonances, e.g. a confinement\ninduced resonance as shown by Olshanii \\cite{Olshanii}. For\nsimplicity, we assume that the two spin states are equally populated\n$N_{\\uparrow}=N_{\\downarrow}=N\/2$. We note that in order to have $K<1$, \none needs to consider a two-component atomic Fermi gas. Indeed, in a single \ncomponent Fermi gas, Pauli's principle forbids $s$-wave collisions implying \nthat $K=1$ for spinless fermions interacting via a contact potential.\n\nFollowing several recent ideas \\cite{Recati,KleinFleishhauer,Raizen}\nwhich involve trapping single atoms in ultracold gases, we consider an\natomic quantum dot (AQD) like configuration, which consists of single\natoms confined in a tight trap created either magnetically or\noptically, e.g. by an additional optical lattice. We assume that the\nconfining potential can be adjusted in such a way, that it does not\naffect the atoms of the bath. The impurity atom, which is trapped in\na certain internal state $|a\\rangle$ interacts with the atoms of the\nbath through $s$-wave collisions. In the case where two such AQDs are\nembedded in the bath and both impurity atoms are in state $|a\\rangle$,\nthe system precisely realizes the situation of two localized\nimpurities interacting via a 1D quantum liquid. Provided the liquid\nconsists of spin-1\/2 repulsive fermions, we expect that for distance\n$r$ much larger than the average interparticle spacing, the\ninteraction is of the Casimir form given in Eq.~(\\ref{eq:CasSpin}).\nIn principle, using a scattering resonance may allow to reach the\nstrong impurity regime $\\tilde{\\gamma}\\gg 1$ where the crossover scale $r_0$\nis even smaller than the inter-particle distance $1\/\\rho_0$.\n\\begin{figure}[ptb]\n\\includegraphics[height=6cm]{scheme2.eps}\n\\caption{Schematic setup of 2 AQDs coupled to a 1D atomic reservoir.\nThe impurity atoms (see text) in tightly confining potential interact\nwith the bath when their internal level is $|a>$. Here $\\delta$ is the\nrenormalized detuning and $\\Omega$ is the Rabi frequency coming from a\nlaser induced coupling, see section \\ref{conclusion}.}\n\\label{scheme}\n\\end{figure}\n\nA possible way to detect the interaction energy $V_{12}(r)$, is to do\nspectroscopy of a single trapped atom as a function of the distance\n$r$ to a neighboring trapped atom. In addition to the mean field line\nshifts modifying the internal levels of the impurity atom, the Casimir\ninteraction produces a line shift depending on distance as $1\/r$. For\na quantitative estimate of this effect, we compute the energy\n(\\ref{eq:CasSpin}) for the experimental situation realized in\nRef. \\cite{Esslinger}. There, about $N\\sim 100$ $^{40}$K atoms (per\ntube) form an atomic wire of length $L\\sim 10$~$\\mu$m. The temperature\ncan be as low as $T\\sim 50$~nK, which is about one tenth of the Fermi\ntemperature. The Fermi velocity is of order $v_F \\sim 2.10^{-2}$~m\/s\nand we take $u_{\\sigma}+u_{\\rho} \\sim 2v_F$. As the tube length is of\nthe order of 10~$\\mu$m and the inter-particle distance $1\/\\rho_0 \\sim\n0.1$~$\\mu$m, we assume the inter-impurity distance to be $r \\sim\n1$~$\\mu$m, which is larger than the crossover length for strong\nimpurities. This gives a Casimir-related line shift of the order of\n$1$~kHz, which is in an experimentally accessible range. With the\nparameters given above, the characteristic frequency $\\omega_r$ is of\nthe order of $\\sim 3T$, which is not much larger than $T$ as required\nfor the validity of the zero temperature limit in which the Casimir\nforce is obtained.\n\n\n\\section{Conclusion}\\label{conclusion}\nIn conclusion, we studied the long-range interaction between two\nimpurities mediated by the 1D quantum liquid in which they are\nembedded. We found that for repulsive fermions, the impurities\ninteract via a RKKY-like interaction at intermediate distances and via\na Casimir-like force at large distances. The crossover scale\nseparating these two regimes depends on the strength of the impurities\nand on the interactions between fermions. We proposed an experimental\nrealization of such a system with atomic quantum dots in an ultra-cold\natomic gas and suggested a way to detect the Casimir-type interaction\nby spectroscopy of a single atom in an AQD.\n\nAn issue which is still open is to understand the interaction between\nimpurities in a strongly interacting Bose liquid. In particular, how\ndoes the short-range interaction (on the scale of the healing length)\nturn into a long-range interactions featuring Friedel oscillations in\nthe Tonks-Girardeau limit? Another issue is to assess the validity of\nthe self-consistent harmonic approximation (SCHA) used to discuss spin\n$1\/2$ fermions in section \\ref{spinful}. Indeed, it is not obvious\nthat the variational ansatz, which assumes decoupling of the charge\nand spin modes in presence of the impurities, is a valid starting\npoint.\n\nIn this paper, we studied static impurities. The situation becomes\neven more interesting if one has dynamic impurities, as in an\nAQD. First, we discuss the possibility of \\emph{internal} dynamics for\nthe AQD. We have seen that the characteristic frequencies of vacuum\nmodes (excitations) responsible for the long-range interactions\nbetween AQDs are limited by $\\omega_{r}$. This means that the\neffective interaction potential between static impurities can be used\n(in an adiabatic approximation) for time-dependent impurity strengths\n$\\tilde{\\gamma}_{\\alpha}(t)$ provided that the interaction properties change\nslowly compared with the time scale $\\omega_{r}^{-1}$. Consider, for\ninstance, the configuration with two AQDs described previously, see\nFigure \\ref{scheme} (a similar scheme for two impurities in a 3D\nBose-Einstein condensate is discussed in\nRef.~\\cite{KleinFleishhauer}). The two-level impurity atoms can be\ndescribed as isospins $1\/2$ or qubits. A laser can drive transitions\n(equivalent to single-qubit gate) between the two internal levels\n$|a\\rangle$ and $|b\\rangle$. In the adiabatic approximation, the AQD\nvariables are slow and can be taken out of the integrals so that our\nprevious treatment to calculate the interaction between two impurities\napplies. Thus one can easily write an effective Hamiltonian for the 2\nAQDs in the form\n\\begin{eqnarray}\nH_{{\\rm\neff}}&=&\\sum_{\\alpha=1,2}\\left(-\\frac{\\delta}{2}\\sigma_{z}^{(\\alpha)}+\\Omega\\sigma_{x}^{(\\alpha)}\\right)\n\\nonumber\\\\\n&+&\\frac{1}{2}V_{12}(\\sigma_{z}^{(1)}+1)(\\sigma_{z}^{(2)}+1)\\ ,\n\\label{eq:Hent}\n\\end{eqnarray}\nwhere $\\delta$ is the (renormalized) detuning, $\\Omega$ is the\n(effective) Rabi frequency coming from the laser induced coupling\n\\cite{Recati} and $\\sigma_x^{(\\alpha)}$, $\\sigma_y^{(\\alpha)}$,\n$\\sigma_z^{(\\alpha)}$ are the Pauli matrices describing the isospin\n$1\/2$ of each AQD $\\alpha=1,2$. The long-range potential $V_{12}$\ndepends, as we have seen, strongly on the characteristics of the bath\nand on the distance between the AQDs. In addition, the case of\nimpurities with internal dynamics embedded in a spin $1\/2$ fermionic\nliquid is of course also relevant for discussing the RKKY interaction\nin Luttinger liquids, as discussed perturbatively in\n\\cite{EggerSchoeller}.\n\nIt is also possible to imagine \\emph{external} motion or dynamics for\nthe impurities. The argument of adiabaticity also holds in the case\nwhere the distance between the impurities changes sufficiently\nslowly. This means that the Casimir-like interaction could be used,\nfor example, to create long range attractive forces in mixtures of 1D\nfermionic gases. However, when the external dynamics of the impurities\nare taken into account on the same footing as the bath dynamics other\neffects could reduce, if not wash out, the Casimir force\n\\cite{CastroNeto}.\n\n\n\n\n\\section*{Acknowledgments}\nThis work was started in collaboration with Piotr Fedichev, whose\nsubstantial contributions to the ideas presented here are gratefully\nacknowledged. It is a pleasure also to thank Prof. Alan Luther for\nhelpful remarks on his unpublished work and Mauro Antezza, Iacopo\nCarusotto, Peter Kopietz, Walter Metzner, In\\`es Safi and Peter Zoller\nfor useful discussions. Laboratoire de Physique des Solides is a mixed\nresearch unit (UMR 8502) of the CNRS and the Universit\\'e Paris-Sud\n$11$ in Orsay. A.R. has been also supported by the European Commission\nthrough contract IST-2001-38863 (ACQP).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Overview Blockchain and Its Role in Healthcare Apps}\nThis section gives an overview of blockchain and the open-source Ethereum blockchain that provides additional support for smart contracts. It then outlines key challenges in healthcare with respect to interoperability and how the properties of blockchain-based apps can help address these challenges\n\n\\subsection{Overview of Blockchain}\nA blockchain is a decentralized computing architecture that maintains a growing list of ordered transactions grouped into blocks that are continually reconciled to keep information up-to-date, as shown in Figure ~\\ref{blockchain}. Only one block can be added to the blockchain at a time and each block is mathematically verified (using cryptography) to ensure it follows in sequence from the previous block to maintain consensus across the entire decentralized network. The verification process is also called \"mining\" or Proof of Work (PoW)~\\cite{nakamoto2012bitcoin}, which allows network nodes (also called \"miners\") to compete to be the first to have their block be the next one added to the blockchain by solving a computationally expensive puzzle. The winner then announces the solution to the entire network to gain some mining rewards in cryptocurrency. This mechanism combines game theory, cryptography, and incentive engineering to ensure that the network reaches consensus regarding each block in the blockchain and that no tampering occurs with the transaction history. All transaction records are kept in the blockchain and are shared with all network nodes, thereby ensuring properties of transparency, incorruptibility, and robustness (since there is no single point of failure).\n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{images\/blockchain}\n\\caption{Blockchain Structure: a Continuously Growing List of Ordered and Validated Transactions}\n\\label{blockchain}\n\\end{figure}\n\nIn the Bitcoin application, a blockchain serves as a public ledger for all transactions of cryptocurrency in bitcoins to promote trustless finance between individual users, securing all their interactions with cryptography. The Bitcoin blockchain has limitations, however, when supporting different types of applications involving contracts, equity, or other information, such as crowdfunding, identity management, and democratic voting registry~\\cite{buterin2014ethereum}. To address the needs for a more flexible framework, Ethereum was created as an alternative blockchain, giving users a general, trustless platform that can run smart contracts, which are computer protocols that enable different types of decentralized applications beyond cryptocurrencies. \n\nThe Ethereum public blockchain is essentially a distributed state transition system, where state is made up of accounts and state transitions are direct transfers of value and information between accounts. Two types of accounts exist in Ethereum: (1) \\textit{externally owned accounts} (EOAs), which are controlled via private keys and only store Ethereum's native value-token \"ether\" and (2) \\textit{smart contract accounts} (SCAs) that are associated with contract code and can be triggered by transactions or function calls from other contracts~\\cite{buterin2014ethereum}.\n\nTo protect the blockchain from malicious attacks and abuse (such as distributed denial of service attacks in the network or hostile infinite loops in the contract code), Ethereum also enforces a payment protocol, whereby a fee is charged for memory storage and each computational step that is executed in a contract or transaction. These fees are collected by miners who verify, execute, propagate transactions, and then group them into blocks. Just like in the Bitcoin network, the mining rewards provide an economic incentive for users to dedicate powerful hardware and electricity to the public Ethereum network.\n\n\\subsection{Addressing Healthcare Interoperability via Blockchain-based Apps}\n\n\n\t\n\nMany research and engineering ideas have been proposed to apply blockchain to healthcare and implementation attempts underway~\\cite{ekblaw2016case,peterson2016blockchain,porru2017blockchain,bartoletti2017empirical}, but few published studies have addressed the software design considerations needed to implement blockchain-based healthcare apps effectively. While it is crucial to understand the fundamental properties that blockchain possesses, it is also important to apply solid software engineering practices when programming the blockchain to minimize software maintenance effort via code modularity and extensibility, especially in the fast-growing and highly demanded healthcare domain. The remainder of this section summarizes key interoperability challenges in healthcare and how blockchain technologies can provide assistance.\n\n\\subsubsection{Challenge 1: Maintaining Evolvability While Minimizing Integration Complexity}\n \nMany applications are written with the assumption that data is easy to change. In a blockchain application, however, data is immutable and difficult to modify in mass. A critical design consideration when building blockchain applications for healthcare is ensuring that the data and contracts written into the blockchain are designed in a way to facilitate evolution where needed. Although evolution must be facilitated, healthcare data must often be accessible from a variety of deployed systems that cannot easily be changed over time. Therefore, the evolvability should be designed in a way that minimizes the impact of evolution on the clients that interact with data in the blockchain. Section 4.2.1 shows how the Abstract Factory pattern can be applied in Ethereum contracts to facilitate evolution while minimizing the impact on dependent healthcare application clients.\n \n\\subsubsection{Challenge 2: Minimizing Data Storage Requirements}\n \nHealthcare applications can serve thousands or millions of participants, which can potentially place an enormous burden when all of this data is stored in the blockchain -- particularly if data normalization and denormalization techniques are not carefully thought through. An important design consideration is maximizing data sharing while ensuring that sufficient flexibility exists to manage individual health concerns. As we describe in Section 4.2.2, the Flyweight pattern can be applied to help ensure that common intrinsic data is shared across Ethereum contracts while still allowing extrinsic data to vary across the contracts specific to individuals.\n \n\\subsubsection{Challenge 3: Balancing Integration Ease with Security Concerns}\n Basic technical requirements for achieving interoperability, defined by the Office of the National Coordination for Health Information Technology (ONC), include identifiability and authentication of all participants, ubiquitous and secure infrastructure to store and exchange data, authorization and access control of data sources, and the ability to handle data sources of various structures~\\cite{onc2014vision}. Blockchain technologies are emerging as promising and cost-effective means to meet some of these requirements due to their inherent design properties, such as secure cryptography and a resilient peer-to-peer network. Likewise, blockchain-based apps can benefit the healthcare domain via their properties of asset sharing, audit trails of data access, and pseudonymity for user information protection, which are essential for solving interoperability issues in healthcare. Although there are substantial potential benefits to interoperability and availability of storing patient data in the blockchain, it carries significant risks -- even when encryption is applied. In Section 4.2.3, we discuss how the Proxy pattern can be applied to aid in facilitating interoperability through the blockchain while keeping sensitive patient data from being directly encoded in the blockchain. \n \n\\subsubsection{Challenge 4: Tracking Relevant Health Changes Across Large Patient Populations}\n \nCommunication gaps and information sharing challenges are a serious impediment to healthcare innovation and the quality of patient care. Providers, hospitals, insurance companies, and departments within health organizations experience disconnectedness caused by delayed or lack of information flow. It is common for patients to be cared for by various sources like private clinics, regional urgent care centers, and enterprise hospitals. As we discuss in Section 4.2.4, a provider may have hundreds or more patients that have associated contracts that they need to track. Determining how to scalably detect when changes of relevance in an individual's contract have occurred is not easy but can be addressed by implementing the Publisher Subscriber pattern in Ethereum contracts. \n\n\\section{Case Study of the DApp for Smart Health (DASH)}\nThis section presents the structure and functionality of the DASH app we are developing to explore the efficacy of applying blockchain technology to the healthcare domain. It also discusses the challenges associated with developing blockchain-based apps to improve healthcare interoperability.\n\n\\subsection{Structure and Functionality of DASH}\nDASH provides a web-based portal for patients to access and update their medical records, as well as submit prescription requests. Likewise, the app allows providers to review patient data and fulfill prescription requests based on permissions given by patients. Figure~\\ref{dash} gives an overview of the structure of DASH.\n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{images\/dash}\n\\caption{Structure and Workflow of DASH}\n\\label{dash}\n\\end{figure}\n\nDASH is implemented on an Ethereum test blockchain, with a SMART (Substitutable Medical Apps, Reusable Technology)~\\cite{mandel2016smart} on FHIR (Fast Healthcare Interoperability Resources)~\\cite{bender2013hl7} schema as the standard data format for stored patient data. Using the smart contract support in Ethereum, a \\textit{Patient Registry} contract is created to store a mapping between unique patient identifiers and their associated \\textit{Patient Account} contract addresses. Each \\textit{Patient Account} contract also contains a list of health providers that are granted read\/write access to the patient's medical records. At its initial state, DASH can provide basic access services to the two types of users: patients and providers. \n\n\\subsection{Key Implementation Challenges of DASH}\nAlthough our prototype of DASH was effective for its initial purposes, the following challenges arose when we attempted to extend it to support new types of users from other departments within the same organization:\n\\begin{enumerate}\n\\item \\textbf{Tightly coupled designs.} Our initial design created tight coupling between a lot of variable components, resulting in rewriting of many contracts that propagated to changes in client code. For instance, metadata can be stored in the smart contract internal states as member variables or event logs, but when better design choices are available, new contracts must be instantiated, leaving existing contracts obsolete due to blockchain's immutability property. Unfortunately, the tight coupling between (1) storage and access of storage and (2) entities and entity creation mechanism caused significant overhead in terms of implementation effort as well as computational and storage costs. \n\\item \\textbf{Duplicated resources.} Despite the diverse functioning entities in healthcare, much commonality may be shared across multiple entities. For example, patients assigned to different care management teams may have the same insurance plans, which could be duplicated many times if their care teams belong to different departments, consuming more than necessary storage to maintain. Duplicated resources are also difficult and costly to manage because when an update occurs, all the corresponding copies have to be updated correctly and timely to avoid confusion. In the likely event of failing to update certain copies, all copies will have to be reevaluated or recollected to ensure data integrity. \n\\item \\textbf{Lack of scalability.} Another challenge in blockchain-based healthcare apps stems from the ease of broadcasting events to separate functioning groups, such as patients, providers, and billing departments. For instance, to update providers on a particular patient's activities, a back-end server must perform computationally-intense tasks to constantly watch for events associated with that patient's \\textbf{Patient Account} contract. This approach, unfortunately, does not scale when a large number of parties need to receive notifications regarding activities of other parties with which they interact.\n\\end{enumerate}\n\nTo resolve these implementation challenges while still providing solutions to address the interoperability challenges in healthcare as aforementioned, we redesigned the DASH app with better software engineering practice by applying foundational software patterns as we discuss in Section 4. \n\n\\section{Concluding Remarks}\n \tThis paper provided an overview of the blockchain platform and described the motivations for applying blockchain technology to solve healthcare interoperability issues, focusing on (1) maintaining evolvability while minimizing integration complexity, (2) minimizing data storage requirements, (3) balancing integration ease with security concerns, and (4) tracking relevant health changes across large patient populations. We presented a case study of our DApp for Smart Health (DASH) to highlight the implementation challenges that rose when we attempted to extended the app, such as tightly coupled designs, duplicated resources, and lack of scalability. To address interoperability challenges while providing solutions to the implementation challenges, we identified and applied four software patterns to DASH to improve its design and functionality. \n \n\tFrom this study we learned that the public, immutable, and verifiable properties of the blockchain allow for a more interoperable environment that is not easily achieved using traditional approaches that mostly rely on a centralized server or data storage. Important design decisions need to be made in advance to better take advantage of the smart contract support while avoiding much computation and storage overhead. Combining good software design practice with these unique properties of the blockchain can create apps that are more modular and easier maintenance of the smart contracts. In particular, we applied Abstract Factory, Flyweight, Proxy, and the Publisher-Subscriber patterns to DASH to decouple the creation and access of entities, maximize sharing of resources, and improve application scalability.\n \n\tOur future work will extend the app described in Sections 3 and 4 to delve into the challenges and pinpoint the most practical design process in designing a healthcare blockchain architecture. In addition, we will explore the potential application of other software patterns to handle various challenges, such as security, privacy, dependability, and performance. One approach is to conduct experiments to evaluate the efficacy of applying software patterns to this architecture compared to other alternative designs. We will also investigate extensions of the blockchain from a healthcare perspective, such as creating an alternative health chain that exclusively serves the healthcare sector.\n\n\\section{Introduction}\n\nMany techniques for 2D flow visualization have been developed and\napplied. These include grids of little arrows, still the most common\nfor many applications, equally spaced streamlines\n\\cite{Turk1996,Jobard1997}, and line integral convolution (LIC)\n\\cite{Cabral1993}. But which is best and why? \\citeN{Laidlaw2001}\nshowed that the ``which is best'' question can be answered by means\nof user studies in which participants are asked to carry out tasks\nsuch as tracing advection pathways or finding critical points in the\nflow field. (Note: An advection pathway is the same as a streamline\nin a steady flow field.) \\citeN{Ware2008} proposed that the ``why''\nquestion may be answered through the application of recent theories\nof the way contours in the environment are processed in the visual\ncortex of the brain. But Ware only provided a descriptive sketch\nwith minimal detail and no formal expression. In the present paper,\nwe show, through a numerical model of neural processing in the\ncortex, how the theory predicts which methods will be best for an\nadvection path tracing task.\n\n\\subsection{The IBQ Approach in Image Quality Estimation}\n\nThe IBQ approach combined with psychometric methods has proven suitable,\nespecially for testing the performance of imaging devices or their\ncomponents and then returning this quality information to the product\ndevelopment or evaluation stages. When the subjective changes in image\nquality are multivariate, the technical parameters changing in the test\nimage are unknown or difficult to compute. However, the IBQ approach can be\nused to determine the subjectively important quality dimensions with a wide\nrange of natural image material related to changes caused by different\ndevices or their components. In order to tune the image-processing\ncomponents for optimal performance, it is important to know what the\nsubjectively crucial characteristics that change in the perceived image\nquality are as a function of the tuning parameters, or simply for different\ncomponents. Table I describes the problems caused by multivariate changes in\nimage quality and offers suggestions of how to approach them by using\ndifferent measurement methods that complement each other. The IBQ approach\ncan complement the psychometric approaches and objective measurements by\ndefining the subjective meaning of image quality attributes and\ncharacteristics; in other words, it reveals how important they are for the\noverall perceived quality. This information can then be used as guidance in\ntuning, and no complex models are needed in order to understand the relation\nbetween objective measures and subjective quality ratings.\n\n\\begin{table}[t]\n\\tbl{Multivariate Changes in Image Quality Attributes, the Relationship\nof Psychometric and Objective Image Quality Estimations and the IBQ Approach}{%\n\\begin{tabular}{|l|p{8pc}|p{8pc}|p{12pc}|}\n\\hline\n~PROBLEM & \\multicolumn{3}{l|}{{Estimating the performance when image\n quality changes are multivariate}}\\\\\\hline\n{APPROACH} & {Objective measurements} & \\multicolumn{2}{|{c}|}{Subjective measurements}\\\\\\cline{3-4}\n & & IBQ approach & Psychometric approach\\\\\\hline\nGOAL & Objective and computational\n measures for describing the\n changes in the images & Definition of\n subjectively\n crucial image quality\n characteristics & The amount of\n change in either\n the overall quality\n or a single attribute\\\\\\hline\nQUESTION & What changes physically? & What matters for the\n observer? & How big is the perceived\n change?\\\\\\hline\n\\end{tabular}}\n\\begin{tabnote}\nThe IBQ approach can help to determine the subjectively crucial\ncharacteristics of an image and therefore to give weights to objective and\ncomputational measures.\n\\end{tabnote}\n\\label{tab1}\n\\end{table}\n\n\nOur basic rational is as follows. Tracing an advection pathway for a\nparticle dropped in a flow field is a perceptual task that can be\ncarried out with the aid of a visual representation of the flow.\nThe task requires that an individual attempts to trace a continuous\ncontour from some designated starting point in the flow until some\nterminating condition is realized. This terminating condition might\nbe the edge of the flow field or the crossing of some designated\nboundary. If we can produce a neurologically plausible model of\ncontour perception then this may be the basis of a rigorous theory of\nflow visualization efficiency.\n\\begin{description}\n \\item[Identify] Characteristics of an object.\n \\item[Locate] Absolute or relative position.\n \\item[Distinguish] Recognize as the same or different.\n \\item[Categorize] Classify according to some property (e.g., color, position, or shape).\n \\item[Cluster] Group same or related objects together.\n \\item[Distribution] Describe the overall pattern.\n \\item[Rank] Order objects of like types.\n \\item[Compare] Evaluate different objects with each other.\n \\item[Associate] Join in a relationship.\n \\item[Correlate] A direct connection.\n\\end{description}\n\n\\subsection{Conditions}\nThe reproduction of the gestures was performed in the presence or\nabsence of visual and auditory feedback, resulting in four (2 $\\times$ 2) conditions.\n\\begin{enumerate}\n\\item Visual and auditory feedback (V\\,$+$\\,A).\n\\item Visual feedback, no auditory feedback (V).\n\\item Auditory feedback, no visual feedback (A).\n\\item No visual or auditory feedback (None).\n\\end{enumerate}\nThe order of the four conditions was randomized across participants.\n\\begin{itemize}\n \\item \\textit{when} $+$ \\textit{where} $\\Rightarrow$\n \\textit{what}: State the properties of an object or objects at a\n certain ~time, or set of times, and a certain place, or set of places.\n \\item \\textit{when} $+$ \\textit{what} $\\Rightarrow$\n \\textit{where}: State the location or set of locations.\n \\item \\textit{where} $+$ \\textit{what} $\\Rightarrow$\n \\textit{when}: State the time or set of times.\n\\end{itemize}\nWhen conducting a user study, the goal for the study is to measure\nthe suitability of the visualization in some sense. What is actually\nmeasured is a fundamental question that we believe can be handled by\nusing the concepts of {effectiveness}, {efficiency},\nand {satisfaction}. These three concepts are derived from the\nISO standard of usability 9241-11.\n\\begin{quote}\n Extent to which a product can be used by specified users to\n achieve specified goals with \\textit{effectiveness},\n \\textit{efficiency}, and \\textit{satisfaction} in a specified context of use.\n\\end{quote}\n\nThe mechanisms of contour perception have been studied by\npsychologists for at least 80 years, starting with the Gestalt\npsychologists. A major breakthrough occurred with the work of Hubel\nand Wiesel \\citeyear{Hubel1962,Hubel1968} and from that time,\nneurological theories of contour perception developed. In this\narticle, we show that a model of neural processing in the visual\ncortex can be used to predict which flow representation methods will\nbe better. Our model has two stages. The first is a contour\nenhancement model. Contour enhancement is achieved through lateral\nconnections between nearby local edge detectors. This produces a\nneural map in which continuous contours have an enhanced\nrepresentation. The model or cortical processing we chose to apply is\nadapted from \\citeN{Li1998a}. The second stage is a contour\nintegration model. This represents a higher level cognitive process\nwhereby a pathway is traced.\n\\begin{theorem}\nFor a video sequence of $n$ frames, an optimal approach based on\ndynamic programming can retrieve all levels of key frames together\nwith their temporal boundaries in O($n^4$) times.\n\\end{theorem}\n\nWe apply the model to a set of 2D flow visualization methods that\nwere previously studied by \\citeN{Laidlaw2001}. This allows us to\ncarry out a qualitative comparison between the model and how humans\nactually performed. We evaluated the model against human performance\nin an experiment in which humans and the model performed the same task.\n\nOur article is organized as follows. First we summarize what is\nknown about the cortical processing of contours and introduce Li's\n\\citeyear{Li1998a} model of the cortex. Next we show how a slightly\nmodified version of Li's model differentially enhances various flow\nrendering methods. Following this, we develop a perceptual model of\nadvection tracing and show how it predicts different outcomes for an\nadvection path-tracing task based on the prior work of\n\\citeN{Laidlaw2001}. Finally we discuss how this work relates to\nother work that has applied perceptual modeling to data visualization\nand suggest other uses of the general method.\n\n\\begin{figure}[tp]\n\\centering\n\\includegraphics{..\/acmlarge-mouse}\n\\caption{Neurons are arranged in V1 in a column architecture. Neurons\nin a particular column respond preferentially to the same edge\norientation. Moving across the cortex (by a minute amount) yields\ncolumns responding to edges having different orientations. A\nhypercolumn is a section of cortex that represents a complete set of\norientations for a particular location in space.}\n\\label{corticalarchitecturefig}\n\\end{figure}\n\n\\section{Cortical Processing of Contours}\nVisual information passes along the optic nerve from the retina of\nthe eye where it is relayed, via a set of synaptic junctions in the\nmidbrain lateral geniculate nucleus, to the primary visual cortex at\nthe back or the brain (Visual Area 1 or V1). It has been known since\nthe Hubel and Wiesel's work in the 60s that the visual cortex\ncontains billions of neurons that are sensitive to oriented edges and\ncontours in the light falling on the retina. Such neurons have\nlocalized receptive fields each responding to the orientation\ninformation contained within the light imaged in a small patch of\nretina. A widely used mathematical model of a V1 neuron's receptive\nfield is the Gabor function \\cite{Daugman1985}:\n\\begin{equation}\n\\label{gaboreqn}\nGabor(u,v,\\lambda,\\theta,\\phi,\\sigma,\\gamma)=e^{-\\frac{u'^{2}+\n\\gamma^{2}v'^{2}}{2\\sigma^{2}}}cos(2\\pi\\frac{u'}{\\lambda}+\\phi).\n\\end{equation}\n\nHubel and Wiesel \\citeyear{Hubel1962,Hubel1968} found that neurons\nresponding to similar orientations were clustered together in a\nstructure they called a column which extended from the surface of the\nvisual cortex to the white matter (see Figure\n\\ref{corticalarchitecturefig}). Later, they and other researchers\ndiscovered hypercolumn structures consisting of thousands of neurons\nall responding to the same area of visual space and selecting for a\nrange of orientations. Overall, V1 contains a topographic map of the\nvisual field having the property that every part of the retinal image\nis processed in parallel for all orientations. These orientation\nselective neurons have provided the basis for all subsequent theories\nof contour and edge detection.\n\nThere remains the problem of how the output of orientation sensitive\nneurons, each responding to different parts of a visual contour,\nbecomes combined to represent the whole contour. Part of the solution\nappears to be a contour enhancement mechanism. \\citeN{Field1993}\nexamined the human's ability to perceive a contour composed of\ndiscrete oriented elements. They placed a contour composed of\nseparated Gabor patches, among a field of randomly orientated Gabor\npatches. Contours were detected when the patches were smoothly\naligned. They were not detected when there was misalignment. This\nwork suggests that there is some manner of lateral coupling among the\nvisual elements involved in perceiving the Gabor patches in the\ncontour. These researchers have suggested that similarly oriented\naligned contours mutually excite one another, while they inhibit\nother neurons that are nearby (Figure~\\ref{neuronalignmentfig}).\n\n\\begin{figure}[tp]\n\\centering\n\\includegraphics{..\/acmlarge-mouse}\n\\caption{Neurons whose receptive fields are aligned along a\ncontinuous contour mutually reinforce each other. They inhibit nearby\nneurons with a similar orientation sensitivity.}\n\\label{neuronalignmentfig}\n\\end{figure}\n\n\n\\section{Li's V1 Model}\nBased on the observed organization of the neurons in the visual\ncortex by Hubel and Wiesel \\citeyear{Hubel1962,Hubel1968} and the\nexperimental evidence by \\citeN{Field1993}, Zhaoping Li constructed a\nsimplified model of the behavior of V1 neurons and examined the\nmodel's ability to integrate contours across multiple V1 neurons.\nThe model is introduced briefly here, and described in more detail in\n\\citeN{Li1998a}. In Li's model, the cortex is approximated by a set\nof hypercolumns arranged in a hexagonal grid. Each hexagonal cell has\n12 orientation-selective neuron pairs oriented in 15-degree\nincrements. One of the main simplifications embodied in Li's model is\nthat it fails to incorporate the way the mammalian visual systems\nscales with respect to the fovea. Real neural architectures have much\nsmaller receptive fields near the fovea at the center of vision than\nat the edges of the visual field.\nThe neurons in each hex cell were grouped into excitatory and\ninhibitory pairs responding to an edge of a particular orientation at\nthat location. Thus there were a total of 24 neurons per cell. The\nfiring rates of both the inhibitory and excitatory neurons were\nmodeled with real values. The neuron pairs affected neighboring\nneuron pairs via a transfer function that depended on the alignment\nof the edge selectivity orientations. Neuron pairs that were aligned\nwith one another exhibited an excitatory effect on each other, while\npairs that were not aligned inhibited each other. Finally, Li's model\nalso contains feedback pathways for higher-level visual areas to\ninfluence individual neurons.\n\nIn our implementation, the mapping of the hexagonal grid to the image\nspace was such that the hex centers were separated by 10 pixels. For\nthe V1 neuron response, we used the Gabor function (Eq.\n(\\ref{gaboreqn})) with a wavelength, $\\lambda$, of 21 pixels, a\n$\\sigma$ of 7 pixels, and an aspect ratio, $\\gamma$, of 1.\n\n\\section{Streamline Tracing Algorithm}\n\\citeN{Laidlaw2001} compared the effectiveness of\nvisualization techniques by presenting test subjects with the task of\nestimating where a particle placed in the center of a flow field\nwould exit a circle. Six different flow-field visualization methods\nwere assessed by comparing the difference between the actual exit\nnumerically calculated and the estimation of the exit by the human\nsubjects. Laidlaw et~al.'s experiment was carried out on humans but,\nin our work, we apply this evaluation technique to humans as well as\nto our model of the human visual system and use a streamline tracing\nalgorithm to trace the path of the particle.\n\nWe use the term streamline tracing to describe the higher level\nprocess that must exist for people to judge a streamline pathway.\nWe call it streamline tracing because the task seems to require the\nuser to make a series of judgments, starting at the center, whereby\nthe path of a particle dropped in the center is integrated in a\nstepwise pattern to the edge of the field. Though many algorithms\nexist in the machine vision literature for contour tracing, we found\nthese to be inappropriate for use in this application. Contour\ntracing algorithms are generally designed to trace out the boundary\nof some shape but a streamline tracing algorithm must also be able\nable to produce a streamline in a field of disconnected contours,\nsuch as is the case with the regular arrows. The streamline to be\ntraced will often not follow a visible contour but instead be locate\nbetween contours, and will sometimes pass through areas devoid of\nvisual elements. Thus we developed a specialized algorithm that is\ncapable of tracing streamlines that do not necessarily correspond to\nthe boundary of any shape but can pass between visual contours.\n\nPerception is a combination of top-down and bottom-up processes.\nBottom-up processes are driven by information on the retina and are\nwhat is simulated by Li's model \\citeyear{Li1998a}. Top-down\nprocesses are much more varied and are driven in the brain by\nactivation from regions in the frontal and temporal cortex that are\nknown to be involved in the control of pattern identification and\nattention \\cite{Lund2001}. All of the flow visualizations evaluated\nby \\citeN{Laidlaw2001}, except for LIC, contain symbolic information\nregarding the direction of flow along the contour elements (e.g. an\narrowhead). In a perpetual\/cognitive process this would be regarded\nas a top-down influence. At present our model does not deal with\nsymbolic direction information but it does do streamline tracing once\nset in the right general direction.\n\nStreamline tracing is a combination of top-down and bottom-up\nprocesses. Broadly speaking, top-down processes reflect task demands\nand the bottom-up processes reflect environmental information. In our\ncase, the bottom-up information comes from the different types of\nvisualization, while the top-down information is an attempt to model\nthe cognitive process of streamline pathway tracing. Contour\nintegration was modeled using the following iterative algorithm.\n\n\\medskip\n\\begin{algorithm}[H]\n\\SetAlgoNoLine\n$current\\_position \\leftarrow$ center \\\\\n$current\\_direction \\leftarrow$ up \\\\\n$current\\_position$ is inside circle \\\\\n\\While{$current\\_position$ is inside circle,}{\n $neighborhood \\leftarrow$ all grid hexes within two hexes from $current\\_position$ \\\\\n \\For{ each $hex$ in $neighborhood$, }{\n \\For{each $neuron$ in $hex$}{\n convert $neuron\\_orientation$ to $vector$ \\\\\n scale $vector$ by $neuron\\_excitation$ \\\\\n $vector\\_sum \\leftarrow vector\\_sum + vector$}}\n\n normalize $vector\\_sum$ \\\\\n $current\\_position \\leftarrow current\\_position + vector\\_sum$ \\\\\n $current\\_direction \\leftarrow vector\\_sum$ \\\\\nreturn $current\\_position$ \\\\\n}\n \\caption{Iterative Algorithm}\n \\label{alg:one}\n \\end{algorithm}\n\\medskip\n\nThe algorithm maintains a context that contains a current position\nand direction. Initially, the position is the center, and the\ndirection set to upward. This context models the higher-order,\ntop-down influence on the algorithm that results from the task\nrequirements (tracing from the center dot) and the directionality\nwhich in our experiment was set to be always in an upwardly trending direction.\n\nThe algorithm traces the contour by repeatedly estimating the flow\ndirection at the $current\\_position$ and moving the position a small\ndistance (.5 hex radii) in that direction. The flow direction is\ncalculated from the neural responses in the local neighborhood of the\n$current\\_position$. The excitation of each neuron is used to\ngenerate a vector whose length is proportional to the strength of the\nresponse and whose orientation is given by the receptive field\norientation. Because receptive field orientations are ambiguous as to\ndirection (for any vector aligned with the receptive field, its\nnegative is similarly aligned). The algorithm chose the vector most\nclosely corresponding to the vector computed on the previous\niteration. Vectors are computed for all neurons in hypercolumns\nwithin a 2-hexes radius of the current position; they are summed and\nnormalized to generate the next $current\\_direction$.\n\nSome changes were made from the method published by\n\\citeN{Pineo2008}. Previously, the algorithm considered only a single\nhex cell at each iteration of the algorithm. We found that this would\noccasionally cause unrealistically large errors in streamline\ntracing. For example, on visualizations with arrowheads, the neural\nnetwork might yield a very strong edge orthogonal to the flow field\npositioned at the back of an arrowhead. If the algorithm considered\nonly the edges at this point, it may make a significant error,\ndespite the edges in nearby positions indicating the correct\ndirection. We felt that creating an average over $neighborhood$ was\nthe more correct approach, and we found closer agreement with human\nperformance with this change.\n\n\\subsection{Qualitative Evaluation}\nFour different flow visualization methods were used in our evaluation\nof the theory. These were implementations of four of the six used by\n\\citeN{Laidlaw2001}. We chose to investigate a regular arrow grid\nbecause it is still the most commonly used in practice and a jittered\narrow grid because of the arguments that have been made that this\nshould improve perceptual aliasing problems \\cite{Turk1996}. We added\nLine Integral Convolution (LIC) because of its widespread advocation\nby the visualization community \\cite{Cabral1993} and head-to-tail\naligned streaklets because of Laidlaw et al.'s finding that is was\nthe best and the theoretical arguments in support of this method\n\\cite{Ware2008}. Note that Laidlaw et al. used Turk and Banks\nalgorithm to achieve aligned arrows on equally spaced streamlines\nwhile we used Jobard and Lefer's \\citeyear{Jobard1997} method to\nachieve the same effect and we used streaklets without an arrowhead\n\\cite{Fowler1989}.\n\n\\begin{figure}[tp]\n \\begin{minipage}[t]{0.45\\linewidth}\n \\centering\n \\includegraphics{..\/acmlarge-mouse}\n \\caption{Regular arrows.}\n \\label{regularfig}\n \\end{minipage}\n \\hspace{0.1\\linewidth}\n \\begin{minipage}[t]{0.45\\linewidth}\n \\centering\n \\includegraphics{..\/acmlarge-mouse}\n \\caption{Jittered arrows.}\n \\label{jitteredfig}\n \\end{minipage}\n\\end{figure}\n\n\\begin{figure}[tp]\n \\begin{minipage}[t]{0.45\\linewidth}\n \\centering\n \\includegraphics{..\/acmlarge-mouse}\n \\caption{Closeup of neural response to arrowheads.}\n \\label{ortharrowheadfig}\n \\end{minipage}\n \\hspace{0.1\\linewidth}\n \\begin{minipage}[t]{0.45\\linewidth}\n \\centering\n \\includegraphics{..\/acmlarge-mouse}\n \\caption{Closeup of neural response to aligned streaklets.}\n \\label{alignedcloseupfig}\n \\end{minipage}\n\\end{figure}\n\n\nV1 is known to have detectors at different scales. However, to make\nthe problem computationally tractable we chose only a single scale\nfor the V1 and designed the data visualizations with elements scaled\nsuch that they were effectively detected by the gabor filter used by\nthe model. The widths of the arrows and streaklets were chosen to be\nsmaller than the central excitatory band of the gabor filter. This\nallowed the edge to be detected even if not precisely centered on the\nreceptive field of the neuron. The spatial frequency of the LIC\nvisualization is defined by the texture over which the vector field\nis convoluted. Our texture was created by generating a texture of\nrandom white noise of one-third the necessary size and scaling it up\nvia. interpolation. The resulting spacial frequency of the LIC\nvisualization was of a scale that was effectively detected by the\ngabor filters of the model.\n\n\\subsubsection{Regular Arrows (Figure \\ref{regularfig})} This\nvisualization is produced by placing arrow glyphs at regular\nspacings. The magnitude of the vector field is indicated by the arrow\nlength, and the flow direction by the arrow head. The grid underlying\nthe regular arrows is apparent to humans, but the edge weights of the\nmodel show no obvious signs of being negatively affected. In fact,\nthe regularity ensures that the arrows are well spaced, preventing\nany false edge responses that might be produced by the interference\nof multiple arrows. We can expect that nontangential edge responses\nwill be produced by the arrowheads and these will lead to errors in\nthe streamline advection task.\n\n\\paragraph{Jittered arrows (Figure \\ref{jitteredfig})}\nThis visualization is similar to the regular arrows, but the arrows\nare moved a small random distance from the regular locations. While\ncomposed of the same basic elements as the regular grid, we see\ninstances where nearby arrows interfere with each other and produce\nedge responses nontangential to the flow direction. Also, as with\ngridded arrows, the arrowheads will excite neurons with orientation\nselectivity nontangential to the flow. This can be seen in\nFigure~\\ref{ortharrowheadfig}. In this figure, we can see orthogonal\nneural excitation to each side of the upper arrow, caused by the back\nedge of the arrowhead (blue circles). We can also see excitation\ncaused by the interference of two arrows at the bottom right (green\ncircle). These nontangential responses are much stronger than those\nfound in the aligned streaklets visualization (Figure \\ref{alignedcloseupfig}).\n\n\n\\section{Discussion}\nThe overall agreement between the pattern of results for human\nobservers and the V1-based model provides strong support of the\nperceptual theory we outlined in the introduction. The aligned arrows\nstyle of visualization produced clear chains of mutually reinforcing\nneurons along the flow path in the representation, making the flow\npathway easy to trace as predicted by theory.\n\nThe fact that LIC produced results as good as the equally spaced\nstreamlines was something of a surprise, and this lends support to\nits popularity within the visualization community. While it did not\nproduce as much neuron excitation as the aligned arrows method, this\nwas offset by the lack of nontangential edge responses produced by\nglyph-based visualizations. However, its good performance was\nachieved only because our evaluation method ignored the directional\nambiguity inherent in this method. \\citeN{Laidlaw2001} found this\nmethod to be the worst and there is little doubt that had we allowed\nflow in any direction, up or down, human observers would have found\npathways with close to 180 degrees of error half of the time.\n\nThe performance of both the model and the human test subjects is\nlikely to be highly dependent on the underlying vector field used.\nAs described in Section 5.1.6, the vector field was generated by\ninterpolating between an 8x8 grid of random, but generally upward\npointing vectors. A consequence of this is that when adjacent vectors\nin this grid point somewhat toward each other, the vector field forms\nan area of convergence. This convergence area tends to funnel\nneighboring streamline paths together, reducing error in streamline\ntracing (Figure \\ref{regularfig} is an example of this). Thus, the\noverall accuracies of both the model and human subjects may be higher\nthan might be might be observed using a vector field without such convergence zones.\n\nWe were surprised that the computer algorithm actually did better at\nthe task than human observers. One reason for this may have been that\nhumans would have to make saccadic eye movements to trace a path,\nwhereas the computer did not. For the patterns we used, it is likely\nthat the observers had to make fixations on several successive parts\nof a path, and errors may have accumulated as they resumed a trace\nfrom a previous fixation. Nevertheless, we feel that the algorithm\ncould easily be adjusted to make it give results closer to human\nsubjects. A more sophisticated approach would be to simulate eye fixations.\n\nThe model we applied is a considerable simplification over what\nactually occurs. It only uses the simplest model of the simplest\norientation sensitive neurons, and fails to include cortical\nmagnification, among other shortcomings. Real cortical receptive\nfields are not arranged in a rigid hexagonal grid as they are in Li's\nmodel. Furthermore, the neurons of V1 respond to many frequencies,\nhowever our model only uses one in its present form. In addition,\nbesides the so-called simple cells modeled by \\citeN{Li1998a}, other\nneurons in V1 and V2 called complex and hypercomplex cells all have\nimportant functions. For example, end-stopped cell respond best to a\ncontour that terminates in the receptive field and understanding\nthese may be important in showing how the direction of flow along a\ncontour can be unambiguously shown. Moreover, visual information is\nprocessed through several stages following the primary cortex,\nincluding V2, V4 and the IT cortex. Each of these appears to abstract\nmore complex, less localized patterns. Researchers are far from\nhaving sufficient information to model the operations of these stages\nall of which may have a role in tracing contours. Nevertheless, the\nresults are compelling and there are advantages in having a\nrelatively simple model. We have plans to add some of these more\ncomplex functions in future versions of the model.\n\n\n\n\n\\section{Introduction}\nOver the past several years blockchain technology has attracted interest from computer scientists and domain experts in various industries, including finance, real estate, healthcare, and transactive energy. This interest initially stemmed from the popularity of Bitcoin~\\cite{nakamoto2012bitcoin} and the Bitcoin platform, which is a cryptographic currency framework that was the first application of blockchain. Blockchain possesses certain properties, such as decentralization, transparency, and immutability, that have allowed Bitcoin to become a viable platform for \"trustless\" transactions, which can occur directly between any parties without the intervention of a centralized intermediary. \n\nAnother blockchain platform, Ethereum, extended the capabilities of the Bitcoin blockchain by adding support for \"smart contracts.\" Smart contracts are computer programs that directly control exchanges or redistributions of digital assets between two or more parties according to certain rules or agreements established between involved parties. Ethereum's programmable smart contracts enable the development of decentralized apps (DApps)~\\cite{johnston2014general} , which are autonomously operated services cryptographically stored on the blockchain that enable direct interaction between end users and providers. \n \nThis paper focuses on a previously unexplored topic related to blockchain, namely, the application of software patterns to modularize and facilitate extensibility of blockchain-based apps that focus on addressing the interoperability challenges in healthcare. In the healthcare context, interoperability is defined as the ability for different information technology systems and software apps to communicate, exchange data, and use the information that has been exchanged. The high (and growing) cost of healthcare in many countries motivates our focus on applying blockchain technologies to help bridge the gap in communication and information exchange~\\cite{desalvo2015connecting}.\n\nThe remainder of this paper is organized as follows: Section 2 gives an overview of blockchain and outlines how blockchain-based apps can help address key challenges in healthcare interoperability; Section 3 provides an end-to-end case study of a blockchain-based healthcare app we are developing and the implementation challenges we encountered when extending the app; Section 4 describes foundational software patterns that can be applied to address interoperability requirements in the blockchain app covered in Section 3 and discusses key concerns when realizing these patterns in healthcare-focused blockchain apps; Section 5 compares our work with existing work on potential pros and cons of a health blockchain; and Section 6 presents concluding remarks and summarizes our future work on applying blockchain technologies in the healthcare domain.\n\n\n\\subsubsection{Applying Abstract Factory to Address Interoperability Challenge 1}\\mbox{}\\\\\n\n\\textbf{Motivation.} Smart contracts allow code to be executed on the blockchain by the EVM. The incorruptibility property of blockchain ensures that interfaces (e.g. methods and properties) of already instantiated contracts cannot be modified or upgraded. Each new version of a smart contract has to be created as a new contract object on the blockchain and distributed among all the network nodes to be executed on demand. Therefore, it is important to design a contract class to modularize code and minimize changes to its interface over time.\n\nFor example, if a blockchain is used to enable some interactions between different departments in a hierarchical health organization, without a well thought out design, a lot of decisions have to be made by clients based on the specific department or sub-department type (a large number of if-else statements) encountered. As new departments are introduced, the decision-making code will likely have to change more than once, and each obsolete version of contract discarded.\n\nThe Abstract Factory pattern can facilitate this scenario because its \"factory\" object (the factory itself is a contract instance) is then responsible for providing creation services of concrete departmental contracts for the entire health organization. Each factory object can create contracts for a group of departments or subdivisions that are related or always interact, and it is easy to instantiate another factory object when new interactions take place. As a more concrete example, the billing department within a hospital organization may have smaller subdivisions that assist different sets of patients depending on what insurance they carry. When the hospital decides to accept insurance plans from a new provider, the interaction between the hospital subdivision and patients with the new provider can simply be created by a factory instead of changing the existing code to include this new interaction. \n\nWithout this factory contract, the client interested in creating department accounts for the entire organization has to access each of the specific account creation factories and make a lot of if-else decisions at runtime. The tight coupling makes it cumbersome for the client since it needs to know all the implementation details in order to use each factory's methods correctly. As shown in Figure~\\ref{noabsfac}, to create an account for a care provider and insurance departments, the client imports both the ProviderFactory and the InsuranceFactory and invokes the createAccount() and createOrganization() methods for each account creation. As an organization scales out and up with more departments and subdivisions, the client will be overwhelmed with these detailed implementations. \n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{images\/noabstractfactory}\n\\caption{Structure of an Application Without Applying the Abstract Factory Pattern}\n\\label{noabsfac}\n\\end{figure}\n\n\n\\textbf{Intent.} \n\\begin{itemize}\n\\item Defines a contract \"interface\" for creating a family of related or dependent contracts without having to explicitly specify their class types\n\\item Enables encapsulation through the hierarchical construction structure\n\\end{itemize}\n \n\\textbf{Applicability.} The Abstract Factory pattern should be applied in the following scenarios:\n\\begin{itemize}\n\\item When the system needs to be independent from how different contracts in the system are created, such as creations of various multi-level user account contracts\n\\item When the system can be configured to work with families of related contracts, such as allowing interactions between a ProviderAccount from the AbstractAccount type and a ProviderOrganization from the AbstractOrganizationAccount type\n\n\\item When a library of contracts needs to be created that is relevant to the interface but not concrete contract implementations \n\\end{itemize}\n\nFor example, in a healthcare blockchain app, a \"super\" contract Account could be used to define some common logic used by all types of user accounts (e.g. pharmacies, physicians' offices, insurance companies). Some examples of such common logic may be patient data access, payment handling, and prescription lookup. At creation time, the Account contract may not be able to anticipate all user contract types, so applying this pattern can allow future derived contracts to be created at runtime. Specific and concrete implementations of the logic functions can be deferred to individual derived entity contracts. \nIn the case of payment handling logic, for instance, a derived pharmacy contract could bill a pharmacy benefit management organization for a patient's prescriptions purchase; a physician's office derived contract may send bills associated with the patient visit to a medical insurance group; while an insurance group derived contract would then calculate the payment amount based on patients' benefit packages. \n \n\\textbf{Contract Structure.} As shown in Figure~\\ref{absfac}, functions createAccount() and createOrganization() in the AbstractAccountFactory contract interface instantiate account contracts of a type created by the concrete implementation that handles the method call and returns the address to the newly created contract object. \n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{images\/abstractfactory}\n\\caption{Structure of an Application with the Abstract Factory Pattern}\n\\label{absfac}\n\\end{figure}\n\n\\textbf{Consequences.} The Abstract Factory pattern introduces a weak coupling between the application and concrete contract implementations, facilitating future contract extensions that share some commonality and also supporting customizations to the concrete contracts. The downside of applying this pattern in the blockchain, however, is the extra cost incurred by an added layer of indirection in terms of storage for the interface contract and computation of instantiating the interface contract. \n \n\\textbf{Sample Code.} The following code (Figure~\\ref{code1}) from DASH provides an example of using the Abstract Factory pattern to create a phyisician account contract and a pharmacist account contract.\n\n\\begin{figure}[tp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{images\/p1}\n\\label{code1}\n\\end{figure}\n\n\\subsubsection{Applying Flyweight to Address Interoperability Challenge 2}\\mbox{}\n \n\\textbf{Motivation.} In order to achieve blockchain's transparency and immutability properties, all of the data and transaction records are maintained in the blockchain by replicating and distributing to every node in the network. It is important to limit the amount of data stored in the blockchain to avoid high cost of data storage and unattended data when it is no longer needed.\n\nFor example, if a blockchain is being used to store patient billing data, there will be millions of patients stored in the blockchain. Moreover, each patient will require storage of their insurance information, which will include details, such as their ID\\#, insurance contact information, coverage details, and other aspects that the provider needs to bill for services. Capturing this huge amount of information for every patient can generate a large amount of data in the blockchain.\n\nThe Flyweight pattern can aid in resolving the tension in this example between needing to store detailed insurance and billing information and the desire to minimize the information stored in the blockchain. Since most patients are covered by one of a relatively small subset of insurers (in comparison to the total number of patients), there is a substantial amount of intrinsic non-varying information that is common across patients, such as the policies on what procedures are covered by their insurance policy. Each insurance policy may cover 10,000s or 100,000s of patients and this information can potentially be reused and shared across patients. The details on what a particular policy covers are common across every patient with that policy. However, in order to bill for a service, this common intrinsic information needs to be combined with extrinsic information that is unique to each patient, such as the patient's ID\\#.\n\nUsing the Flyweight pattern in smart contracts, intrinsic data shared by patients is stored in the common contract, while extrinsic data is stored in a separate contract representing that specific patient. The contract with intrinsic data can also store references to specific patient contracts sharing the data. When retrieving complete billing information, a common method call can be invoked to return the combined intrinsic and extrinsic data. \n \n\\textbf{Intent.}\n\\begin{itemize}\n\\item Reuses existing similar contracts by storing them and create new contracts when no matching contract is found \n\\item Supports a large number of contracts, such as account contracts, that have some common states where other parts vary\n\\end{itemize}\n \n\\textbf{Applicability.}\nThis pattern applies to a program that needs to handle a lot of contract objects that share some common states that can be externalized while other internal states vary. It is also often used in combination with a factory that checks a pool of flyweights to determine if a flyweight object already exists in the pool. In a healthcare system, each of many account types would contain a large number of detailed accounts that share some common metadata and\/or accessing methods. By applying the Flyweight pattern, shared representations can then be de-encapsulated in the Account \"flyweight\" factory only once, avoiding an exorbitant amount of memory usage from saving repeated data in all accounts. In addition, a flyweight factory can act as a Registry that maintains a mapping between unique user identifiers and the referencing addresses to the each account contract, preventing user account duplications. At account creation, only if no account with the provided identifier exists in the registry does the factory create a new Account and return its address; otherwise the address associated with the existing Account contract will be returned. A flyweight can have its own states and operations not shared with other objects. \n\tA further substantial advantage of the flyweight pattern in the context of blockchains is that data is immutable once written. If insurance policy details are stored in each patient's contract directly, the cost to change a policy detail will be immense, since it will require rewriting a huge number of impacted contracts. The flyweight pattern also helps to minimize the cost of changes to the intrinsic state in blockchain applications.\n \n\\textbf{Contract Structure.}\nAs shown in Figure~\\ref{flyweight}, AbstractPatientAccount acts as a flyweight interface that defines intrinsic data structure and functions for concrete flyweights. PatientAccountFactory, the flyweight factory keeps a records of created patient account contracts of all account types and creates a new flyweight when it does not already exist in the accounts mapping.\n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{images\/flyweight}\n\\caption{Structure of Applying the Flyweight Pattern}\n\\label{flyweight}\n\\end{figure}\n\n\\textbf{Consequences.}\nApplying the Flyweight pattern can provide better management to the large object pool, such as user accounts in the example above. It minimizes redundancy in similar objects and, in the meantime, maximizes data and operation sharing. Similar to the drawback of applying the Abstract Factory pattern, however, Flyweight incurs additional overhead to the client because of the extra layer of complexity. Factory operation that returns the flyweight contract may also take longer to execute because it becomes another transaction to verify and included in the blockchain before a valid address can be available. However, the computation delay may still be outweighed by the efficiency in management provided by Flyweight. \n \n\\textbf{Sample Code.}\nThe following code (Figure~\\ref{code2}) from DASH uses a Flyweight to maximize sharing of data between different patients and act as a registry to store each unique patient's account contract address.\n\n\\begin{figure}[tp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{images\/p2}\n\\label{code2}\n\\end{figure}\n\n\\subsubsection{Applying Proxy to Address Interoperability Challenge 3}\\mbox{}\n \n\\textbf{Motivation.} A fundamental aspect of a blockchain is that all data stored in the blockchain is public, immutable, and verifiable. For financial transactions that are focused on proving that transfer of an asset occurred, these properties are critical. However, when the goal is to store data in the blockchain, it is important to understand how these properties will impact the use case.\n\t\nFor example, storing patient data in a blockchain can be problematic, since it requires the data to be public and immutable. Although data can be encrypted before being stored in the blockchain, should all patient data be publicly distributed in the blockchain to all other parties for verification? Even if encryption is used, it is possible that the encryption technique may be broken in the future or that bugs in the implementation of the encryption algorithms or protocols used may lead to the information potentially being decryptable in the future. The immutability, however, prevents owners of the data from removing the data from the blockchain if a security flaw is ever found. Many other scenarios, ranging from discovery of medical mistakes in the data to changing standards may necessitate the need to change the data over time.\n\nIn scenarios where the data may need to be changed, the public and immutable nature of the blockchain creates a fundamental tension that needs to be resolved. On the one hand, the healthcare providers would like the data to be incorruptible so that it cannot be tampered with. At the same time, providers want the data changeable and private to protect patient privacy and account for possible errors. \n\nThe Proxy pattern is a well-known software pattern that can be applied to blockchain-based data storage to resolve the tension created by the public and immutable aspects of the blockchain. Using the proxy pattern with a blockchain, a proxy contract is created to provide some lightweight representation or placeholder for the data with more intensive computation (such as acquiring data from off-blockchain storage via an Oracle~\\cite{oraclize2017}). The proxy contract can expose some simpler metadata of a patient and later refer to the heavyweight implementation on demand to obtain the real data object. Each read request and modification operation of the data store can be logged in an audit trail that is transparent to the entire blockchain network for verification against data corruption. When the proxified contract (with heavyweight implementation) is updated with new storage option, for instance, interface to the proxy contract can remain the same, encapsulating detailed implementation variations. \n \n\\textbf{Intent.}\n\\begin{itemize}\n\\item Provides a surrogate or placeholder contract for a health data storage object for another object to control access to it\n\\item Supports distributed or controlled access to sensitive information using an additional level of indirection\n\\item Creates a wrapper to protect data storage from undue complexity\n\\end{itemize}\n\n \n\\textbf{Applicability.}\nRemote proxies can be useful in an Ethereum healthcare application as they provide a local representation for an object that is in a different address space. In the storage case, patient medical data could be stored either inside contracts on the blockchain as contract states or event logs or externally off the blockchain as encrypted file objects. In this scenario, a remote proxy can be applied on the application server side to create a reference or wrapper to the actual data storage and later retrieve the data when needed. Instead of directly applying a remote proxy inside a smart contract in Solidity, it would be applied in the healthcare application's web server to create a wrapper for the contract object containing relevant data if it is used as storage.\n\nA protective proxy can also be used to control access to the original sensitive object. For instance, to prevent unauthorized users on the blockchain to change the states of a patient data object, it can set up a proxy to do some permission checking prior to forwarding the change request.\n \n\\textbf{Contract Structure.}\nAs shown in Figure~\\ref{proxy}, Proxy and RealPatientData implement the same interface, with Proxy being the lightweight surrogate to RealPatientData, the heavyweight implementation.\n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{images\/proxy}\n\\caption{Structure of Applying the Proxy Pattern}\n\\label{proxy}\n\\end{figure}\n\n\\textbf{Consequences.}\nA proxy object can perform simple housekeeping or auditing tasks by storing some commonly used metadata in its internal states without having to perform heavy operations such as decrypting a file. It follows the same interface as the real object and can execute the original heavyweight function implementations on demand. It can also hide information about the real object from the client to protect patient data privacy. Since Proxy introduces another layer of indirection, however, it could cause disparate behavior when the real object is accessed directly by some client while the surrogate is accessed by others. In addition, contracts in Solidity are instantiated by other contracts or transactions, meaning that the proxy can only reference the real contract with its address on the blockchain, which defeats the purpose of using a protection proxy to hide the real object from the client. Nevertheless, both remote and protective proxies can be used on the server side of the application that accesses these contracts. \n \n\\textbf{Sample Code.}\nThe following code (Figure~\\ref{code3}) from our DASH app uses a protective Proxy to control access to the smart contract implementation of data acquisition. \n\n\\begin{figure}[tp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{images\/p3}\n\\label{code3}\n\\end{figure}\n\n\\subsubsection{Applying Publisher-Subscriber to Address Interoperability Challenge 4}\\mbox{}\n \n\\textbf{Motivation.} \n\tThe Ethereum blockchain maintains a public record of contract creations and operation executions along with regular cryptocurrency transactions. The availability of information makes blockchain a more autonomous approach to improve the coordination of patient care across different teams (e.g. physicians, pharmacists, insurance agents, etc) who would normally communicate through various channels with a lot of manual effort, such as through telephoning or faxing. Although, from a continually growing list of records, directly capturing any specific health-related topic of occurred events would require a lot of transaction receipt lookups and topic filtering, which requires non-trivial computation and may result in delayed responses. \n\nFor instance, when a blockchain application is used to support coordinated care through patient self reporting of illness or prescription request, it is important to relay the report and any follow up procedure to and from the associated care provider offices in a timely manner. In this case, the Publisher-Subscriber pattern can assist in broadcasting the information only to care providers that subscribe to events relating to this patient. It solves the issue of constant information filtering by actively monitoring patient activities and sending notifications to the patient's care team as the activities take place. To avoid computation overhead on the blockchain, the actual processing of patient activities data can be done off-chain by a back-end server. When receiving the events of interest, the subscribers can then pass the heavy computation tasks to the server.\n\n \n\\textbf{Intent.}\n\\begin{itemize}\n\\item Creates a messaging infrastructure using separation of concerns by allowing publishers to create messages and subscribers to receive messages of their interests.\n\n\\item Enables interoperability within a healthcare ecosystem so that participating departments can be notified of health-related events they are concerned with.\n\n\\end{itemize}\n\n \n\\textbf{Applicability.}\nThis pattern should be applied when a state change in one object must be reflected in another object without keeping the objects tightly coupled and when the system is to be enhanced with new topics or subscribers with minimal changes to the implementation.\n\nIn a health blockchain application, a care provider may prescribe a patient a list of medications during a visit, sending (publishing) a message of the prescription update. Involved parties, the patient's pharmacy and insurance company may subscribe to messages related to this patient's prescription activities in order to receive this update from the provider's office. Other parties that do not have direct associations with this patient, such as a hospital in a different geographic area, may choose not to subscribe to this patient's activities to avoid the overhead of receiving irrelevant information. \n\nThe Publisher-Subscriber pattern can be realized in a smart contract because the publisher itself can handle event notifications to subscribed parties, but it is more cost-efficient to be implemented by the health application server. \n \n\\textbf{Contract Structure.}\nAs shown in Figure~\\ref{pubsub}, publisher keeps track of a list of subscribers. When an event occurs, publisher notifies all the subscribers of the event, and subscribers can then process the event update..\n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{images\/pubsub}\n\\caption{Structure of Applying the Publisher Subscriber Pattern}\n\\label{pubsub}\n\\end{figure}\n\n\\textbf{Consequences.}\nThe limitations of realizing the Publisher-Subscriber pattern on the blockchain include (1) delays in updates received by subscribers due to the extra step of validation required by the blockchain infrastructure, (2) high computational costs associated with filtering very fine-grained topic subscriptions on-chain, and (3) high executional costs associated with notifying events. With a much faster mining process on the Ethereum blockchain compared to Bitcoin, each block can be mined and added to the blockchain roughly every 12 seconds in Ethereum. The order of which transactions are to be executed is determined by the miners based on the transaction fees paid by the senders and how long transactions have been in the transaction pool. Transactions with the highest priority will be executed first and added to the blockchain sooner, which could cause delays in publishers sending out messages and subscribers receiving messages of their interest. Every computation occurred on the blockchain will be charged some amount of \"gas\", so the more intensive the computations are, the more costs to realize them. If subscribers want to receive very fine-grained message topics, the amount of computation for filtering out the messages sent out by publishers may cost an exorbitant amount, resulting in either failure to publish messages due to gas shortage or unacceptable implementation costs. A workaround could be to have broader topics with fewer filter requirements on-chain and handle more detailed message filtering off Blockchain. \n \n\\textbf{Sample Code.}\nIn the following example from our DASH app (Figure~\\ref{code4}), 5 observers are subscribed to the Subject contract. After the execute method is invoked, each subscriber will receive an update that is reflected by the change of value in the variable \"state\". \n\n\\begin{figure}[tp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{images\/p4}\n\\label{code4}\n\\end{figure}\n\n\\section{Applying Foundational Software Patterns to Blockchain-based Health Apps}\nThis section describes how foundational software patterns~\\cite{gamma1995design,buschmann2007pattern} can be applied in blockchain-based health apps (such as DASH) to address the interoperability challenges described in Section 2 while providing solutions to the implementation concerns highlighted in Section 3. \n\n\\subsection{Overview of Solidity}\nEthereum smart contracts are built on a Turing-complete programming language, called Solidity~\\cite{solidity2017ethereum}. This contract language has allowed the Ethereum blockchain to become a platform for creating decentralized applications (DApps), thereby providing possible solutions to healthcare interoperability challenges. \n\nThe Solidity language has an object-oriented flavor and is intended primarily for writing contracts in Ethereum. A \"class\" in Solidity is realized through a \"contract\", which is a prototype of an object that lives on the blockchain. Just like an object-oriented class can be instantiated into a concrete object at runtime, a contract may be instantiated into a concrete SCA by a transaction or a function call from another contract. At instantiation, it is given a uniquely identifying address similar to a reference or pointer in C\/C++-like languages, with which it can then be called. Contracts also contain persistent state variables that can be used as data storage. Although one contract can be instantiated into many SCAs, it should be treated as a singleton to avoid storage overhead. After a contract is created, its associated SCA address is typically stored at some place (e.g., a configuration file or a database) and used by the application to access its internal states and functions. \n\nSolidity supports multiple inheritance including polymorphism. When a contract inherits from one or more other contracts, a single contract is created by copying all the base contracts code into the created contract instance. Abstract contracts in Solidity allow declaration headers to be defined without concrete implementations. They cannot be compiled into a SCA but can be used as base contracts. Due to their similarity to C\/C++-like classes, many foundational software patterns can be applied to smart contracts to address various design challenges, as described next.\n\n\\subsection{Applying Software Patterns to Improve DASH Design}\nThe remainder of this section focuses on four software patterns--Abstract Factory, Flyweight, Proxy, and Publisher-Subscriber--that we applied to DASH to address the healthcare interoperability challenges in Section 2, but they are not the only patterns relevant in this domain. Figure~\\ref{patterns} shows how these patterns can interact in the DASH app's ecosystem. For instance, \\textit{Abstract Factory} can assist with user account creations based on user types; whereas \\textit{Flyweight} ensures accounts on the blockchain are unique and maximizes commonality sharing, and \\textit{Publisher-Subscriber} can be used to notify collaborating users when events of interest occur on the blockchain. In addition, to abstract away storage implementation detail, the Proxy pattern can be applied to allow seamless interactions between separate components in the system while still supporting variations in data storage options.\n\n\\begin{figure}[bp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{images\/patterns}\n\\caption{Applying Software Patterns to Improve DASH Design and Extensibility}\n\\label{patterns}\n\\end{figure}\n\n\\input{pattern1}\n\\input{pattern2}\n\\input{pattern3}\n\\input{pattern4}\n\n\nDespite the drawbacks of added complexity and computation costs, it is considered good practice to modularize contract code and server-side code that accesses contracts with appropriate software patterns to allow easier extensibility and flexible customizations to each major component in a healthcare blockchain application. Software patterns also provide separation of concerns through loose coupling between components, allowing components to access shared methods and artifacts while still encapsulating variabilities in detailed implementations. An interoperable healthcare chain should enable seamless communications and interactions between components, which can be made robust and customizable with careful evaluations of overall workflow and applicable software patterns. \n\nOn the application's server end, we applied the Publisher-Subscriber pattern to allow health providers to subscribe to patient account activities. For instance, when a patient requests a prescription drug through the web portal, all the subscribed providers of that patient will be notified of the event, but only permissioned providers will be able to respond to the prescription request via an update to the patient data. \n\nWhen writing data to a patient's account, a permissioned care provider first queries the patient registry for the location of the patient account contract. If the patient does not yet exist in the registry mapping, a new Patient Account contract will be automatically created and registered in Patient Registry with the patient ID, using the Flyweight and Abstract Factory patterns. The registry then returns the address pointing to the specified patient account. With the contract address, the provider will then be able able to write new data to the patient health records by adding the new data entry in the contract internal data array. Similarly, when reading a patient record, a provider must first query Patient Registry with a patient ID. If the requested patient exists, Patient Registry returns the address of Patient Account contract from which patient data can be retrieved; otherwise, it creates a contract account for the patient and returns the new address where the data field is empty. \n\nFor testing and backup purposes, we have also created a separate patient record database in MongoDB. To decouple our data store choices from other operations, on the application's server end, we applied the Proxy pattern to create a wrapper for a generic database object and refer to the Proxy until we need to obtain the actual data.\n\n\n\\subsection{Related Work}\nAlthough relatively few papers focus on realizing software patterns in blockchains, many papers relate to healthcare blockchain solutions and applying software design principles in this space. This section gives an overview of related research on (1) the challenges of applying blockchain-based technology in the healthcare space and innovative implementations of blockchain-based healthcare systems and (2) design principles and recommended practice for blockchain application implementations.\n\n\\subsection{Challenges of healthcare blockchain and proposed solutions.}\nEkblaw et. al proposed MedRec as an innovative, working healthcare blockchain implementation for handling EHRs, based on principles of existing blockchains and Ethereum's smart contracts~\\cite{ekblaw2016case}. The MedRec system uses database \"Gatekeepers\" for accessing a node's local database governed by permissions stored on the MedRec blockchain. Peterson et. al presented a healthcare blockchain with a single centralized source of trust for sharing patient data, introducing \"Proof of Interoperability\" based on conformance to the Fast Healthcare Interoperability Resources (FHIR) protocol as a means to ensure network consensus~\\cite{peterson2016blockchain}.\n\n\\subsection{Applying software design practice to blockchain.} Porru et. al highlighted evident challenges in state-of-the-art blockchain-oriented software development by analyzing open-source software repositories and addressed future directions for developing blockchain-based software, focusing on macro-level design principles such as improving collaboration, integrating effective testing, and evaluations of adopting the most appropriate software architecture~\\cite{porru2017blockchain}. Bartoletti et. al surveyed the usage of smart contracts and identified nine common software patterns shared by the studied contracts, e.g. using \"Oracles\" to interface between contracts and external services and creating \"Polls\" to vote on some question. These patterns summarize the most frequent solutions to handle some repeated scenarios~\\cite{bartoletti2017empirical}. ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\\IEEEPARstart{C}{ausal} inference refers to the process of inferring causal relationships from data. Randomized controlled trials (RCTs) remain the gold standard for causal inference in most fields of science. However, RCTs cannot distinguish between causes and \\textit{root causes} of disease, or the initial perturbations that ultimately induce a diagnostic label. Randomization also introduces a myriad of ethical, financial and logistical issues -- such as withholding potentially lifesaving treatments from patients. We therefore instead focus on identifying root causes from \\textit{observational data}, where patients are not subject to randomization.\n\nConsider for example the causal process depicted by the directed graph in Figure \\ref{fig_root_cause}, where nodes represent random variables and directed edges their direct causal relations. The blue lightning bolt depicts an exogenous ``shock'' to the causal process, such as the effect of a somatic mutation or a virus, at $X_2 \\in \\bm{X}$. The shock is felt by many downstream variables $X_3,\\dots, X_6$ and ultimately induces a diagnostic label $D=1$. We focus on identifying $X_2$ from data because it corresponds to the initial perturbation and therefore the root cause. The problem is challenging because the root cause may lie arbitrarily far from $D$, and we must differentiate it from the other variables in $\\bm{X}$ that may be causes but not necessarily the root cause of the diagnosis.\n\nThe problem is further complicated by the fact that complex diseases may have \\textit{multiple} root causes that even \\textit{differ} between patients within the same diagnostic category. As a result, simply identifying root causes at the group level can lead to many statistically significant variables with clinically insignificant effect sizes. We instead focus our efforts on identifying \\textit{patient-specific} root causes in order to make complex diseases more tractable.\n\nWe identify patient-specific root causes by first defining a causal process using a structural equation model (SEM), where variables are related by a series of deterministic equations and stochastic error terms. Patient-specific root causes then correspond to the predictivity of the exogenous errors as assessed by sample-specific Shapley values (see Section \\ref{sec_root} for details). Authors have thus far only implemented this idea in the linear case \\cite{Strobl22}. However, real datasets frequently contain non-linear relations, and running linear algorithms on data sampled from a non-linear SEM can lead to large errors in causal inference.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.0, shorten >=1pt,auto,node distance=2.8cm, semithick,\n inj\/.pic = {\\draw (0,0) -- ++ (0,2mm) \n node[minimum size=2mm, fill=red!60,above] {}\n node[draw, semithick, minimum width=2mm, minimum height=5mm,above] (aux) {};\n \\draw[thick] (aux.west) -- (aux.east); \n \\draw[thick,{Bar[width=2mm]}-{Hooks[width=4mm]}] (aux.center) -- ++ (0,4mm) coordinate (-inj);\n }]\n \n\\tikzset{vertex\/.style = {inner sep=0.4pt}}\n\\tikzset{edge\/.style = {->,> = latex'}}\n \n\\node[vertex] (1) at (0,0) {$X_1$};\n\\node[vertex] (2) at (1.5,0) {$X_2$};\n\\node[vertex] (3) at (3,0.5) {$X_3$};\n\\node[vertex] (4) at (3,-0.5) {$X_4$};\n\\node[vertex] (5) at (4.5,0.5) {$X_5$};\n\\node[vertex] (6) at (4.5,-0.5) {$X_6$};\n\\node[vertex] (7) at (6,0) {$D$};\n\n\n\\fill [blue, decoration=lightning bolt, decorate] (1.5,0.25) -- ++ (0.75,0.75);\n\n\\draw[edge] (1) to (2);\n\\draw[edge,blue] (2) to (3);\n\\draw[edge,blue] (2) to (4);\n\\draw[edge,blue] (3) to (5);\n\\draw[edge,blue] (4) to (6);\n\\draw[edge,blue] (5) to (7);\n\\draw[edge,blue] (6) to (7);\n\\end{tikzpicture}\n\\caption{Intuitive illustration of the difference between a cause and patient-specific root cause.} \\label{fig_root_cause}\n\\end{figure}\n\n\\begin{tcolorbox}[breakable,enhanced,frame hidden]\nIn this paper, we generalize to the non-linear setting as follows:\n\\begin{enumerate}[leftmargin=*,label=(\\arabic*)]\n \\item We identify patient-specific root causes using the \\textit{heteroscedastic noise model} (HNM) given by $Y = m(X) + \\varepsilon \\sigma(X)$ with arbitrary non-linear functions $m$ and $\\sigma$ representing the conditional mean and conditional mean absolute deviation (MAD), respectively (Section \\ref{sec_HNM}).\n \\item We prove identifiability of the full causal graph under HNM (Section \\ref{sec_identify}).\n \\item We introduce a principled algorithm called Generalized Root Causal Inference (GRCI) that extracts the error terms of an SEM satisfying HNM and computes sample-specific Shapley values (Section \\ref{sec_GRCI}).\n \\end{enumerate}\n\n\\end{tcolorbox}\n\\noindent Experiments highlight considerable improvements in accuracy compared to prior methods because GRCI correctly identifies the exogenous errors by flexibly accounting for nonlinear causal relations. \n\n\n\\section{Background}\n\n We define a causal process using a \\textit{structural equation model} (SEM), or a series of equations in the form:\n\\begin{equation} \\label{eq_SEM_gen}\n Z_i = f_i(\\textnormal{Pa}(Z_i),E_i), \\hspace{2mm}\\forall Z_i \\in \\bm{Z},\n\\end{equation}\nwhere $\\textnormal{Pa}(Z_i)$ denotes the \\textit{parents}, or direct causes, of $Z_i$. The set $\\bm{E}$ contains mutually independent error terms. We assume $\\mathbb{E}(\\bm{E}) = 0$ without loss of generality. A linear SEM admits the more specific form:\n\\begin{equation} \\label{eq_SEM_linear}\n Z_i = \\textnormal{Pa}(Z_i) \\beta_{\\textnormal{Pa}(Z_i)} + E_i, \\hspace{2mm}\\forall Z_i \\in \\bm{Z},\n\\end{equation}\nwhere $\\beta$ denotes a matrix of coefficients. \n\n A \\textit{directed graph} $\\mathbb{G}$ is a graph with a directed edge $\\rightarrow$ or $\\leftarrow$ between any two vertices in $\\bm{Z}$. We have $Z_i \\rightarrow Z_j$ in $\\mathbb{G}$ if $Z_i \\in \\textnormal{Pa}(Z_j)$ or, equivalently, $Z_j$ is a \\textit{child} or direct effect of $Z_i$: $Z_j \\in \\textnormal{Ch}(Z_i)$. The \\textit{neighbors} of $Z_i$ unify the parents and children: $\\textnormal{Ne}(Z_i) = \\textnormal{Pa}(Z_i) \\cup \\textnormal{Ch}(Z_i)$. We more specifically write $\\textnormal{Pa}_{\\mathbb{G}}(Z_i)$ to emphasize the underlying graph -- likewise for the children and neighbors. A \\textit{sink node} is a vertex without children. A \\textit{directed path} from $Z_i$ to $Z_j$ refers to a sequence of directed edges from $Z_i$ to $Z_j$. $Z_i$ is an \\textit{ancestor} of $Z_j$, denoted by $Z_i \\in \\textnormal{Anc}_{\\mathbb{G}}(Z_j)$, when there exists a directed path from $Z_i$ to $Z_j$; we likewise say $Z_j$ is a \\textit{descendant} of $Z_i$. The set $\\textnormal{Nd}_{\\mathbb{G}}(Z_i)$ corresponds to the non-descendants of $Z_i$. A \\textit{cycle} occurs when $Z_i \\in \\textnormal{Anc}_{\\mathbb{G}}(Z_j)$, and we have $Z_j \\rightarrow Z_i$. A directed graph is called a \\textit{directed acylic graph} (DAG), if it does not contain cycles. An \\textit{augmented graph} $\\mathbb{G}^\\prime$ is a DAG over $\\bm{Z} \\cup \\bm{E}$ such that $E_i \\in \\textnormal{Pa}_{\\mathbb{G}^\\prime}(Z_i)$ and $\\textnormal{Pa}_{\\mathbb{G}^\\prime}(E_i) = \\emptyset$ for all $E_i \\in \\bm{E}$. We provide an example of a directed graph in Figure \\ref{fig_root_cause} and its corresponding augmented graph in Figure \\ref{fig_graphs}.\n \n The triple $\\langle Z_i, Z_j, Z_k \\rangle$ forms a \\textit{collider} in $\\mathbb{G}$, if we have $Z_i \\rightarrow Z_j \\leftarrow Z_k$, and $Z_i$ and $Z_k$ are non-adjacent. $Z_i$ and $Z_j$ are \\textit{d-connected} given $\\bm{W} \\subseteq \\bm{Z} \\setminus \\{Z_i, Z_j\\}$ if there exists a path between the two vertices such that any collider on the path is an ancestor of $\\bm{W}$ and no non-collider on the path is in $\\bm{W}$. Otherwise, $Z_i$ and $Z_j$ are \\textit{d-separated} given $\\bm{W}$. \n \nA density $p(\\bm{Z})$ associated with a DAG $\\mathbb{G}$ factorizes according to the product of the conditional densities of each variable in $\\bm{Z}$ given its parents:\n\\begin{equation} \\nonumber\np(\\bm{Z})=\\prod_{i=1}^{p} p(Z_i | \\textnormal{Pa}_{\\mathbb{G}}(Z_i)).\n\\end{equation}\nAny distribution which factorizes according to the above equation also satisfies the \\textit{global Markov property} where d-separation between $Z_i$ and $Z_j$ given $\\bm{W}$ in $\\mathbb{G}$ implies conditional independence (CI) between $Z_i$ and $Z_j$ given $\\bm{W}$ \\cite{Lauritzen90}. We refer to the converse as \\textit{d-separation faithfulness}. The density $p(\\bm{X})$ is \\textit{causally minimal} if no proper subset of $\\mathbb{G}$ also obeys the global Markov property. D-separation faithfulness implies causal minimality \\cite{Peters14}.\n\nThe \\textit{Kolmogorov complexity} of a finite binary string $x$, denoted by $K(x)$, is the length of the shortest self-delimiting binary program that generates $x$ on a universal Turing machine and then halts. The universal Turing machine is not unique, but the Kolmogorov complexity between any two such machines only differs by at most a constant. Most equalities and inequalities in algorithmic information theory are therefore only understood up to a constant; the notation $\\stackrel{+}{=}$ means equality up to a constant and likewise $\\stackrel{+}{\\leq}$ for inequality.\n\n To prevent cluttering of notation with too many parentheses, we write $p(Y)$ as $p_Y$ when referring to the entire density. We keep the standard notation $p(y) = p(Y=y)$ when referring to a specific value of the density.\n\n\\section{The Heteroscedastic Noise Model} \\label{sec_HNM}\n\n\\subsection{Definition}\n We set $\\bm{Z} = \\bm{X} \\cup D$, where $D$ denotes a binary diagnostic label. We will have more to say about $D$ in Section \\ref{sec_root} and focus on $\\bm{X}$ for now. We can generalize the linear SEM in Equation \\eqref{eq_SEM_linear} to an HNM SEM as follows:\n\\begin{definition} (Heteroscedastic noise model) \\label{assump_HNM}\nAn SEM obeys the heteroscedastic noise model (HNM) if the following holds for each $X_i \\in \\bm{X}$:\n\\begin{equation} \\label{eq_HNM}\n X_i = m_i(\\textnormal{Pa}_\\mathbb{G}(X_i)) + E_i \\sigma_i(\\textnormal{Pa}_\\mathbb{G}(X_i)),\n\\end{equation}\nfor non-linear functions $m_i$ and $\\sigma_i > 0$.\n\\end{definition}\n\\noindent We assume that $\\mathbb{E}\\big(\\bm{E}\\big|\\textnormal{Pa}_\\mathbb{G}(X_i)\\big) = 0$ and $\\mathbb{E}\\big(|\\bm{E}|\\big|\\textnormal{Pa}_\\mathbb{G}(X_i)\\big) = 1$ without loss of generality. HNM thus generalizes the linear SEM in Equation \\eqref{eq_SEM_linear} by allowing the expectation and MAD (of the mean) to change as arbitrary non-linear functions of the parents.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.6]{Ex_HNM.png}\n \\caption{Example of an HNM with $Y=X^2 + EX$ plus 10.}\n \\label{fig:HNM}\n\\end{figure}\n\nConsider for example the bivariate HNM in Figure \\ref{fig:HNM}. The conditional expectation in solid red and conditional MAD in dashed red (at 95\\% prediction intervals) change as functions of $X$. In contrast, the best linear SEM erroneously fits a linear conditional expectation and assumes a constant variance. HNM thus increases modeling flexibility considerably.\n\n\\subsection{Error-Term Extraction}\n We can extract the error terms $\\bm{E}$ from an HNM model using the Partial-Out algorithm summarized in Algorithm \\ref{alg_PO}. The error term $E_i \\in \\bm{E}$ corresponds to:\n\\begin{equation} \\label{eq_errHNM}\n E_i = \\frac{X_i - m_i(\\textnormal{Pa}_\\mathbb{G}(X_i))}{\\sigma_i(\\textnormal{Pa}_\\mathbb{G}(X_i))}.\n\\end{equation}\nThe call Partial-Out($\\textnormal{Pa}_\\mathbb{G}(X_i),X_i$) first estimates the conditional expectation $m_i(\\textnormal{Pa}_\\mathbb{G}(X_i))$ by regressing $X_i$ on $\\textnormal{Pa}_\\mathbb{G}(X_i)$ in Line \\ref{alg_PO:expectation}. Partial-Out optimizes all regression hyperparameters by cross-validation. The residuals correspond to:\n\\begin{equation} \\nonumber\n X_i - \\widehat{m}_i(\\textnormal{Pa}_\\mathbb{G}(X_i)) = E_i \\sigma_i(\\textnormal{Pa}_\\mathbb{G}(X_i)) + o_p(1),\n\\end{equation}\nwhere $\\widehat{m}_i(\\textnormal{Pa}_\\mathbb{G}(X_i))$ denotes the estimate of the conditional expectation using a non-linear regression method. Let $\\ddot{m}_i(\\textnormal{Pa}_\\mathbb{G}(X_i))$ denote the estimates of the conditional expectation on the validation folds with the best hyperparameter set. The algorithm then estimates the conditional MAD in Line \\ref{alg_PO:variance} by regressing $|X_i - \\ddot{m}_i(\\textnormal{Pa}_\\mathbb{G}(X_i))|$ on $\\textnormal{Pa}_\\mathbb{G}(X_i)$ using the \\textit{same folds} as in Line \\ref{alg_PO:expectation} because:\n\\begin{equation} \\nonumber\n\\begin{aligned}\n &\\mathbb{E}\\big(|X_i - m_i(\\textnormal{Pa}_\\mathbb{G}(X_i))|\\big|\\textnormal{Pa}_\\mathbb{G}(X_i)\\big)\\\\ = \\hspace{1mm} &\\sigma_i(\\textnormal{Pa}_\\mathbb{G}(X_i)) \\cancelto{1}{\\mathbb{E}\\big(|E_i|\\big|\\textnormal{Pa}_\\mathbb{G}(X_i)\\big)}.\n\\end{aligned}\n\\end{equation}\nCare must be taken to regress on $|X_i - \\ddot{m}_i(\\textnormal{Pa}_\\mathbb{G}(X_i))|$ and not $|X_i - \\widehat{m}_i(\\textnormal{Pa}_\\mathbb{G}(X_i))|$ so that the training folds from Line \\ref{alg_PO:expectation} do not influence the validation folds in Line \\ref{alg_PO:variance}. We use the conditional MAD instead of the conditional standard deviation because we can directly estimate the conditional MAD and divide by it. Squaring the residuals in Step \\ref{alg_PO:variance} to estimate the conditional variance and then taking its square root to obtain the conditional standard deviation can lead to large estimation errors in practice. \n\nPartial-Out finally computes the error estimate in Line \\ref{alg_PO:error} as:\n\\begin{equation} \\label{eq_errHNM_est}\n \\widehat{E}_i = \\frac{X_i - \\widehat{m}_i(\\textnormal{Pa}_\\mathbb{G}(X_i))}{\\widehat{\\sigma}_i(\\textnormal{Pa}_\\mathbb{G}(X_i))},\n\\end{equation}\nper Equation \\eqref{eq_errHNM}. We \\textit{partial out} $\\textnormal{Pa}_\\mathbb{G}(X_i)$ from $X_i$, when we compute $\\widehat{E}_i$ by running Partial-Out($\\textnormal{Pa}_\\mathbb{G}(X_i),X_i$) under HNM. \n\nWe implement all univariate regressions in practice using linear splines due to their relative robustness to overfitting and their ability to admit fast leave one out cross-validation using the Sherman\u2013Morrison\u2013Woodbury formula. We normalize all variables to $[0,1]$ and then use $m$ equispaced knots on $[0,1]$ (always including $1$ and replacing it with an offset). We choose $m$ by leave one out cross-validation from 10 equispaced points between $2$ and $\\sqrt{n\/10}$ inclusive, where $n$ denotes the sample size. We generalize to multivariate regression by randomly projecting $t>1$ variables onto $[0,1]$ using $\\sum_{i=1}^t w_i X_i$ with the vector $\\bm{w}$ obeying a Dirichlet distribution with alpha vector equal to all ones. This process ensures that we sample all weights uniformly from the $t-1$ simplex, since we have no prior knowledge about the sparsity level. \n\n\\begin{algorithm}[t]\n \\nonl \\textbf{Input:} $\\bm{V},X_i$\\\\\n \\nonl \\textbf{Output:} $\\widehat{E}_i$\\\\\n \\BlankLine\n\n$\\widehat{m}_i(\\bm{V}), \\ddot{m}_i(\\bm{V})\\leftarrow$ Regress $X_i$ on $\\bm{V}$ with cross-validation\\\\ \\label{alg_PO:expectation}\n$\\widehat{\\sigma}_i(\\bm{V}) \\leftarrow$ Regress $|X_i - \\ddot{m}_i(\\bm{V})|$ on $\\bm{V}$ with cross-validation\\\\ \\label{alg_PO:variance}\nCompute $\\widehat{E}_i$ per Equation \\eqref{eq_errHNM_est} \\label{alg_PO:error}\n\\caption{Partial-Out} \\label{alg_PO}\n\\end{algorithm}\n\n\\section{Identifiability} \\label{sec_identify}\n\nWe will use Partial-Out to extract the error terms from an SEM obeying HNM. The increased flexibility of accounting for heteroscedastic noise fortunately preserves \\textit{identifiability} of the model, or the ability to pinpoint the exact DAG when given the joint distribution.\n\nWe assume strictly positive densities throughout. We first have the following result in the bivariate case:\n\\begin{theorem} \\label{thm_DE}\nAssume the forward model $X \\rightarrow Y$ obeys HNM so that $p(x,y) = p\\big(\\frac{y-m(x)}{\\sigma(x)}\\big) p(x)$ with $m(X)$ and $\\sigma(X)$ once differentiable. If there is a backward model $Y \\rightarrow X$ also obeying HNM so that $p(x,y) = p\\big(\\frac{x-n(y)}{t(y)}\\big) p(y)$, then the following differential equation holds:\n\\begin{equation} \\label{eq_DE}\n\\begin{aligned}\n &-\\frac{\\sigma(x)}{Q(x,y)}\\frac{\\partial ^2}{\\partial x \\partial y}r(x,y)-\\frac{\\partial ^2}{\\partial y^2}r(x,y) +\\\\ &\\frac{\\sigma^\\prime(x)}{Q(x,y)}\\frac{\\partial }{\\partial y}r(x,y) = q^{\\prime \\prime}(y) - \\frac{\\sigma^\\prime(x)}{Q(x,y)} q^\\prime(y),\n \\end{aligned}\n\\end{equation} \nwhere $r(x,y) = \\textnormal{log } p\\big(\\frac{x-n(y)}{t(y)}\\big), q(y) = \\textnormal{log } p(y)$ both twice differentiable and $Q(x,y) = \\sigma(x)m^\\prime(x) + (y-m(x))\\sigma^\\prime(x)$. Moreover, if there exists a quadruple $(x_0,m(x_0),\\sigma(x_0),p(x_0|y))$ such that $Q(x_0,y) \\not = 0$ for all but countably many $y$, then $p_Y$ is completely determined by $(y_0, q^\\prime(y_0))$ -- i.e., the set of all $p_Y$ satisfying the differential equation is contained in a two dimensional affine space.\n\\end{theorem}\n\\noindent We delegate the longer proofs to the Supplementary Materials.\n\nEquation \\eqref{eq_DE} expresses a very specific relationship and suggests that finding a backward model satisfying the relation is like finding a needle in the haystack; we will almost never encounter this needle in practice. The statement that $p_Y$ lies in a two dimensional space formalizes this intuition. It implies that the forward model cannot be inverted in general because the space of all possible $p_Y$ is infinite dimensional \\textit{a priori}.\n\nWe recover the differential equation $-\\frac{\\sigma(x)}{Q(x,y)}\\frac{\\partial ^2}{\\partial x \\partial y}r(x,y)-\\frac{\\partial ^2}{\\partial y^2}r(x,y) = q^{\\prime \\prime}(y)$ in the special case of an ANM with $\\sigma^\\prime(x) = 0$ -- thus replicating Lemma 1 in \\cite{Janzing10_2}. We can see that this relation holds when $p_{XY}$ is Gaussian, a well-known case where we \\textit{cannot} identify the causal direction. We can of course just work out the equations with $Y=X\\beta + \\varepsilon_Y$: $-\\frac{\\sigma(x)}{Q(x,y)}\\frac{\\partial ^2}{\\partial x \\partial y}r(x,y) \\stackrel{+}{=} -\\frac{1}{\\beta}\\frac{\\beta}{\\sigma^2_Y} = -\\frac{1}{\\sigma^2_Y}$, $\\frac{\\partial ^2}{\\partial y^2}r(x,y) \\stackrel{+}{=} -\\frac{1}{\\sigma^2_Y}$ and $q^{\\prime \\prime}(y) = \\frac{2}{\\sigma^2_Y}$, so that Equation \\eqref{eq_DE} holds in the Gaussian case. But more intuitively, Theorem 1 says that, if we are given information about $X$ in terms of $(\\textcolor{blue}{x_0,x_0\\beta},\\textcolor{red}{\\sigma_X},p(x_0|y))$, then we can recover $p_Y$ with two points $(\\textcolor{Green}{y_0, q^\\prime(y_0)})$ when HNM holds in both directions. This of course holds in the Gaussian case because we can recover the entire (centered) bivariate density by only knowing $(\\textcolor{blue}{\\beta},\\textcolor{red}{\\sigma_X}, \\textcolor{Green}{\\sigma_Y})$.\n\nThe fact that $p_Y$ is completely determined by just two parameters of $Y$ when both directions hold suggests that $p_{X|Y}$ provides a substantial amount of information about $p_Y$. This conflicts with past work postulating that nature implements an \\textit{independence of causal mechanisms}, whereby $p_{X|Y}$ provides almost no information about $p_Y$ \\cite{Janzing10,Janzing10_2}. Authors rigorously define this information as follows:\n\\begin{definition} (Algorithmic mutual information)\nLet $s$ and $t$ denote two binary strings. The algorithmic mutual information between $s$ and $t$ is:\n\\begin{equation} \\nonumber\n I(s:t) = K(t) - K(t|s^\\star),\n\\end{equation}\nwhere $s^\\star$ denotes the shortest program that computes $s$.\n\\end{definition}\n\\noindent If $p_{X|Y}$ provides information about $p_Y$, then $K(p_Y|p^\\star_{X|Y})$ is small, so we expect $I(p_Y:p_{X|Y}) \\gg 0$. \n\nThe two parameter conclusion from Theorem \\ref{thm_DE} implies that $I(p_Y:p_{X|Y}) \\gg 0$ in general under HNM. We can alternatively interpret Theorem \\ref{thm_DE} as follows: we must choose $p_Y$ in a contrived fashion once we know $p_{X|Y}$, so that Equation \\eqref{eq_DE} holds. The following theorem formalizes this intuition by showing that the complexity of $p_Y$ indeed lower bounds $I(p_Y:p_{X|Y})$; in other words, if $I(p_Y:p_{X|Y}) \\gg 0$, then $p_Y$ is likely complex.\n\\begin{theorem} \\label{thm:k_py}\nConsider the same assumptions as Theorem \\ref{thm_DE}. If both the forward and backward models follow HNM, then we have:\n\\begin{equation} \\nonumber \\label{eq_alg_MI}\n\\begin{aligned}\n &I(p_Y:p_{X|Y})\\\\ &\\stackrel{+}{\\geq} K(p_Y) - \\inf_{(x_0,y_0)} K(x_0,m(x_0),\\sigma(x_0),y_0,q^\\prime(y_0)), \n\\end{aligned}\n\\end{equation}\nassuming of course that all inputs are computable.\n\\end{theorem}\nThe above theorem suggests that $p_Y$ likely has high Kolmogorov complexity because $I(p_Y:p_{X|Y}) \\gg 0$. This conclusion also dovetails nicely with complexity based approaches which posit that $K(p_X) + K(p_{Y|X}) \\stackrel{+}{\\leq} K(p_Y) + K(p_{X|Y})$ when $X \\rightarrow Y$ \\cite{Janzing10,Stegle10}. If both the forward and backward directions admit HNM, then the inequality is still likely to hold because $K(p_Y)$ is large. Finally, Theorem \\ref{thm:k_py} connects with the main idea of the Information Geometric Causal Inference (IGCI) algorithm, where we can determine the causal direction $X \\rightarrow Y$ when we can replace $p_X$ with a simple density, such as the uniform or Gaussian density, but preserve the correlation between $p_X$ and an arbitrary property of $p_{Y|X}$ \\cite{Janzing12,Janzing15}. GRCI will go a step further than IGCI by determining both causal direction \\textit{and} the values of the error terms in order to compute patient-specific statistics.\n\nGRCI will in particular extract the values of \\textit{all} of the error terms by partialing out the parents of each variable in $\\bm{X}$. The algorithm thus requires identifiability of the entire causal graph $\\mathbb{G}$, but Theorem \\ref{thm_DE} only applies to the bivariate case. We can fortunately extend Theorem \\ref{thm_DE} to the multivariate setting by considering the following definition:\n\\begin{definition} (Restricted HNM) Equation \\eqref{eq_HNM} is a restricted HNM if, for all $Y \\in \\bm{X}$, $X \\in \\textnormal{Pa}_{\\mathbb{G}}(Y)$ and $\\bm{S}$ such that $(\\textnormal{Pa}_{\\mathbb{G}}(Y) \\setminus X) \\subseteq \\bm{S} \\subseteq (\\textnormal{Nd}_{\\mathbb{G}}(Y) \\setminus X)$, there exists $\\bm{S} = \\bm{s}$ where $p(\\bm{s})>0$\nand $p(x,y|\\bm{s})$ does \\underline{not} satisfy Equation \\eqref{eq_DE}.\n\\end{definition}\n\\noindent In other words, Equation \\eqref{eq_DE} does not hold when we condition on some value of the non-descendants of $Y$ -- notice that this is a very weak assumption. Let $\\mathcal{G}$ denote the space of all causally minimal DAGs obeying a restricted HNM. We have the following result:\n\\begin{theorem} \\label{thm:full}\nAssume Equation \\eqref{eq_HNM} is a restricted HNM according to $\\mathbb{G}$. Then, $\\mathbb{G}$ is uniquely identified from $\\mathcal{G}$.\n\\end{theorem}\n\\noindent The proof reduces the multivariate model to a bivariate one and then applies a contradiction using Theorem \\ref{thm_DE}. We conclude that the HNM model uniquely identifies the entire DAG as required for GRCI.\n\n\n\\section{Patient-Specific Root Causes of Disease} \\label{sec_root}\n\n\\begin{figure}\n\\centering \n\\begin{tikzpicture}[scale=1.0, shorten >=1pt,auto,node distance=2.8cm, semithick]\n \n\\tikzset{vertex\/.style = {inner sep=0.4pt}}\n\\tikzset{edge\/.style = {->,> = latex'}}\n \n\\node[vertex] (1) at (0,0) {$X_1$};\n\\node[vertex] (2) at (1.5,0) {$X_2$};\n\\node[vertex] (3) at (3,0.5) {$X_3$};\n\\node[vertex] (4) at (3,-0.5) {$X_4$};\n\\node[vertex] (5) at (4.5,0.5) {$X_5$};\n\\node[vertex] (6) at (4.5,-0.5) {$X_6$};\n\\node[vertex] (7) at (6,0) {$D$};\n\n\\node[vertex] (8) at (-0.5,1) {$E_1$};\n\\draw[edge] (8) to (1);\n\\node[vertex] (9) at (1,1) {\\textcolor{blue}{$E_2=e_2$}};\n\\draw[edge,blue] (9) to (2);\n\\node[vertex] (10) at (2.5,1.5) {$E_3$};\n\\draw[edge] (10) to (3);\n\\node[vertex] (11) at (4,1.5) {$E_5$};\n\\draw[edge] (11) to (5);\n\\node[vertex] (13) at (2.5,-1.5) {$E_4$};\n\\draw[edge] (13) to (4);\n\\node[vertex] (14) at (4,-1.5) {$E_6$};\n\\draw[edge] (14) to (6);\n\\node[vertex] (12) at (5.5,1) {$E_7$};\n\\draw[edge] (12) to (7);\n\n\\draw[edge] (1) to (2);\n\\draw[edge,blue] (2) to (3);\n\\draw[edge,blue] (2) to (4);\n\\draw[edge,blue] (3) to (5);\n\\draw[edge,blue] (4) to (6);\n\\draw[edge,blue] (5) to (7);\n\\draw[edge,blue] (6) to (7);\n\\end{tikzpicture}\n\n\\caption{The augmented graph of Figure \\ref{fig_root_cause}. We represent the shock as a perturbation to the exogenous error term $E_2$.} \\label{fig_graphs}\n\\end{figure}\n\nWe want GRCI to compute patient-specific statistics to more specifically identify \\textit{patient-specific root causes of disease}, so we need to rigorously define the term. Consider a binary variable $D$ denoting a diagnostic label of disease when $D=1$ and healthy when $D=0$. We assume that $D$ is a sink node in $\\mathbb{G}$; this is a reasonable assumption because scientists who seek to identify the causes of $D$ frequently measure phenomena like genomic levels or environmental exposures that are believed to precede the diagnosis in time.\n\nA patient-specific root cause then corresponds to an exogenous shock to an otherwise healthy causal process that increases the probability that $D=1$ as a downstream effect; we provided an example in Figure \\ref{fig_root_cause}. We model this initial shock as a perturbation of the exogenous error term from a ``healthy'' value $\\widetilde{e}_i$ to an ``unhealthy'' one $e_i$:\n\\begin{equation} \\nonumber\n X_i = g_i(\\textnormal{Pa}_{\\mathbb{G}}(X_i), \\textcolor{blue}{E_i=\\cancelto{e_i}{\\widetilde{e}_i}}).\n\\end{equation}\nThe unhealthy value affects downstream variables and increases the probability that $D=1$ (Figure \\ref{fig_graphs}). \n\n We quantify the \\textit{marginal} contribution of $E_i$ to the increase in probability that $D=1$ using the following difference:\n\\begin{equation} \\label{eq_gamma}\n \\gamma_{E_i \\cup \\bm{W}} = q[\\mathbb{P}(D=1|E_i,\\bm{W})] -\n q[\\mathbb{P}(D=1|\\bm{W})],\n\\end{equation}\nfor some monotonic function $q$ and $\\bm{W} = \\emptyset$. We set $q$ to the identity function for GRCI. If $E_i$ alone increases the probability that $D=1$, then $\\gamma_{E_i} > 0$ because $\\mathbb{P}(D=1|E_i) > \\mathbb{P}(D=1)$.\n\n Complex diseases may however have \\textit{multiple} root causes that induce disease only when present in combination. For example, a single genetic perturbation may not lead to cancerous growth, but multiple perturbations often do. We therefore generalize Equation \\eqref{eq_gamma} to average over all possible \\textit{joint} effects of $E_i$ with the other error terms $\\bm{E} \\setminus E_i$:\n\\begin{equation} \\label{eq_shap}\n S_i = \\frac{1}{p}\\hspace{-13mm}\\underbrace{\\sum_{\\bm{W} \\subseteq (\\bm{E} \\setminus E_i)} \\frac{1}{\\binom{p-1} {|\\bm{W}|}}}_{\\textnormal{Average over all possible combinations of } \\bm{E} \\setminus E_i} \\hspace{-12mm}\\gamma_{E_i \\cup \\bm{W}}.\n\\end{equation}\nThis is precisely the \\textit{Shapley value} because we average over $\\gamma_{E_i \\cup \\bm{W}}$ for all possible combinations of the errors \\cite{Lundberg18}. $S_i$ can thus be greater than zero, when the joint effect $\\gamma_{E_i \\cup \\bm{W}}$ is greater than zero but the marginal effect $\\gamma_{E_i}$ equals zero. The Shapley value is also a \\textit{sample-specific} random variable because we average over the error terms but not the samples; we specifically have $S_i=s_i$ for any given patient $i$ when $E_i \\cup \\bm{W} = e_i \\cup \\bm{w}_i$ for all $\\bm{W} \\subseteq (\\bm{E} \\setminus E_i)$.\n\n We are now ready to define a patient-specific root cause:\n\\begin{definition} \\label{def_root}\n$X_i \\in \\bm{X}$ is a patient-specific root cause of $D$ if $X_i \\in \\textnormal{Anc}_{\\mathbb{G}^\\prime}(D)$ and $S_i > 0$.\n\\end{definition}\nIn other words, $X_i \\in \\bm{X}$ is a patient-specific root cause if it is a cause of $D$ and its error predictably induces $D=1$, where predictivity is defined using the Shapley value $S_i$. Moreover, $S_i$ is defined for each variable in $X_i \\in \\bm{X}$, so a patient may have multiple root causes that lead to disease.\n\n\\section{Generalized Root Causal Inference} \\label{sec_GRCI}\n\nWe now detail the GRCI algorithm that recovers patient-specific root causes of disease from data. We summarize GRCI in Algorithm \\ref{alg_GRCI}.\n\n\\begin{algorithm}[b]\n \\nonl \\textbf{Input:} $\\bm{X}$, test set $\\mathcal{T}$\\\\\n \\nonl \\textbf{Output:} $\\mathcal{S}$\\\\\n \\BlankLine\n\n$\\widehat{\\mathbb{G}} \\leftarrow$ Skeleton-Stable($\\bm{X}$) \\label{alg_GRCI:PC} \\\\\n$\\bm{E},\\bm{N} \\leftarrow$ Extract-Errors($\\bm{X},\\widehat{\\mathbb{G}}$) \\label{alg_GRCI:error}\\\\\nCompute the matrix $\\mathcal{S}$ containing the estimated sample-specific Shapley values of each patient in $\\mathcal{T}$ \\label{alg_GRCI:S}\n\n\\caption{Generalized Root Causal Inference (GRCI)} \\label{alg_GRCI}\n\\end{algorithm}\n\n\\subsection{Skeleton Discovery}\n\nNon-linear regressors can easily overfit in high dimensions. GRCI therefore first reduces the dimensionality of the necessary regressions in Step \\ref{alg_GRCI:PC} by identifying the \\textit{skeleton} of $\\bm{X}$, or the presence and absence of the directed edges in $\\mathbb{G}$. GRCI uses an algorithm called Skeleton-Stable -- the skeleton discovery procedure of the well-known PC-Stable algorithm which identifies the skeleton using a series of CI tests \\cite{Colombo14}. An edge is present between any two variables $X_i$ and $X_j$ in $\\mathbb{G}$ if and only if $X_i \\not \\ci X_j | \\bm{W}$, where $\\bm{W} \\subseteq \\textnormal{Pa}_\\mathbb{G}(X_i)$ or $\\bm{W} \\subseteq \\textnormal{Pa}_\\mathbb{G}(X_j)$, under d-separation faithfulness \\cite{Spirtes00}. Skeleton-Stable therefore tests whether $X_i$ and $X_j$ are conditionally independent given dynamically adjusted supersets of the parents. We skip further details of the algorithm, since they are not important for this paper.\n\n\\subsection{Global Error Term Extraction}\nIn Step \\ref{alg_GRCI:error}, GRCI uses the skeleton identified by Skeleton-Stable to extract the error terms of $\\bm{X}$ with the Extract-Errors algorithm.\n\nWe have summarized Extract-Errors in Algorithm \\ref{alg_EE}. Extract-Errors initializes $\\bm{M}$ to the set of all variables in $\\bm{X}$. The algorithm then iteratively removes a member from $\\bm{M}$ in Line \\ref{alg_EE:remove} and places it into $\\bm{N}$ in Line \\ref{alg_EE:add} so that $\\bm{N}$ ultimately contains a reverse partial-order of $\\bm{X}$. Extract-Errors identifies the variable to remove from $\\bm{M}$ in Step \\ref{alg_EE:sink} using the Find-Sink algorithm.\n\nWe summarize Find-Sink in Algorithm \\ref{alg_sink}. Find-Sink identifies the variable whose parents are most independent of its residuals due to the following result:\n\\begin{lemma} \\label{lem_base}\nIf $X_i \\in \\bm{M}$ is a sink node, then $E_i \\ci X_j$ for all $X_j \\in \\bm{M} \\setminus X_i$. \n\\end{lemma}\n\\begin{proof}\n$E_i$ and $X_j$ are d-separated in $\\mathbb{G}^\\prime$ for all $X_j \\in \\bm{M} \\setminus X_i$. The conclusion follows by the global Markov property. \n\\end{proof}\nThe algorithm in particular runs Partial-Out on each variable $X_i \\in \\bm{M}$ given its neighbors to recover the residuals $R_i$. Find-Sink then computes the mutual information score $\\max_{X_j \\in \\textnormal{Ne}_{\\mathbb{G}}(X_i)} I(X_j;R_i)$ using the nearest neighbor technique proposed in \\cite{Kraskov04}. A lower mutual information score indicates a higher degree of independence. Find-Sink logs all of the mutual information scores associated with $\\bm{M}$ in $\\bm{T}$, and then identifies the sink node in Line \\ref{alg_sink:min} as the variable in $\\bm{M}$ associated with the smallest score in $\\bm{T}$. \n\nExtract-Errors then partials out the sink node $S$ identified by Find-Sink in Line \\ref{alg_EE:partial}. The algorithm also removes edges adjacent to $S$ in $\\widehat{\\mathbb{G}}$. The neighborhoods of some of the variables in $\\bm{M}$ change due to this step -- denote these variables in $\\bm{M}$ by $\\bm{U}$. Extract-Errors updates the scores in $\\bm{T}$ for $\\bm{U}$ in the next iteration. Repeating this process of identifying a sink node in $\\bm{M}$, partialing out its errors and placing it into $\\bm{N}$ until $\\bm{M}$ is empty results in (1) a reverse partial order in $\\bm{N}$ and (2) all of the error terms collected in $\\bm{E}$. More formally:\n\\begin{lemma} \\label{lem_errors}\nExtract-Errors recovers all of the error terms of $\\bm{X}$.\n\\end{lemma}\n\\begin{proof}\nWe prove this induction. The base case follows by Lemma \\ref{lem_base}. For the induction step, assume that Extract-Errors recovers all error terms when $|\\bm{M}| = n$. We need to show that the statement holds for $n+1$. We can recover a sink node from $\\bm{M}$ when $|\\bm{M}| = n+1$ by Lemma \\ref{lem_base}. The conclusion follows by the inductive hypothesis. \n\\end{proof}\n\n\\subsection{Shapley Values}\nGRCI finally computes the matrix $\\mathcal{S}$ containing the Shapley values in Step \\ref{alg_GRCI:S}. The $i^\\textnormal{th}$ column and $j^\\textnormal{th}$ row of $\\mathcal{S}$ contains the Shapley value of $X_i \\in \\bm{X}$ for the patient $j$ in test set $\\mathcal{T}$. We compute these values in practice by predicting $D$ with XGBoost using the error terms recovered by Extract-Errors and then applying the TreeSHAP algorithm \\cite{Lundberg18,Chen16}. We certify GRCI with the following theorem:\n\\begin{theorem}\n(Fisher consistency) Under d-separation faithfulness and HNM over $\\bm{X}$, GRCI recovers the true Shapley values and therefore the patient-specific root causes of $D$ for all samples in $\\mathcal{T}$.\n\\end{theorem}\n\\begin{proof}\nSkeleton-stable recovers a superset of the skeleton of $\\mathbb{G}$ under d-separation faithfulness \\cite{Colombo14}. Extract-Errors recovers all of the error terms of $\\bm{X}$ by Lemma \\ref{lem_errors}. The mutual independence of the error terms and the exactness of the TreeSHAP algorithm ensure that the matrix $\\mathcal{S}$ contains the true Shapley values \\cite{Lundberg18}. The conclusion follows for the entries in $\\mathcal{S}$ greater than zero by Definition \\ref{def_root}. \n\\end{proof}\n\n\\subsection{Time Complexity}\nGRCI is composed of three steps as summarized in Algorithm \\ref{alg_GRCI}. Skeleton-Stable in Step \\ref{alg_GRCI:PC} calls a CI test at most $O(p^r)$ times, where $r$ denotes the maximum number of neighbors of a vertex in $\\mathbb{G}$. The CI test we implement performs a fixed number of multivariate adaptive spline regressions (MARS) each requiring $O(nrm^4)$ time, where $n$ denotes the sample size and $m$ the maximum number of basis functions \\cite{Friedman91}.\\footnote{We replace random Fourier regression with MARS regression and use a fixed number of non-linear transformations similar to \\cite{Strobl19}.} Skeleton-Stable therefore requires $O(nrp^rm^4)$ time. The Extract-Errors function in Step \\ref{alg_GRCI:error} iterates twice over the variables, so it requires on the order of $p^2$ iterations. Each iteration is dominated by Partial-Out which requires $O(n^2b + b^3)$ time, where $b$ denotes the maximum number of basis functions used during cross-validation. Extract-Errors therefore requires $O(n^2p^2b + p^2b^3)$ time. Finally, the TreeSHAP algorithm computes in $O(ntld^2)$ time, where $t$ refers to the number of trees, $l$ the maximum number of leaves, and $d$ the maximum tree depth. Repeating this process for each of the $p$ variables requires $O(nptld^2)$ time. GRCI thus ultimately requires $O(nrp^rm^4) + O(n^2p^2b + p^2b^3) + O(nptld^2)$ time. We conclude that GRCI scales quadratically with respect to sample size and polynomially $O(p^r)$ with respect to the number of variables if $r \\geq 2$.\n\n\n\\begin{algorithm}[t]\n \\nonl \\textbf{Input:} $\\bm{X}, \\widehat{\\mathbb{G}}$\\\\\n \\nonl \\textbf{Output:} $\\bm{E},\\bm{N}$\\\\\n \\BlankLine\n\n$\\bm{M}, \\bm{U} \\leftarrow \\bm{X}$\\\\\n$\\bm{N} \\leftarrow \\emptyset$\\\\\n$\\bm{T} \\leftarrow \\infty$\\\\\n\\Repeat{$\\bm{M} = \\emptyset$}{\n $S \\leftarrow$ Find-Sink($\\bm{M},\\bm{U},\\bm{T},\\widehat{\\mathbb{G}}$) \\\\ \\label{alg_EE:sink}\n $\\bm{M} \\leftarrow \\bm{M} \\setminus S$ \\label{alg_EE:remove}\\\\\n $\\bm{N} \\leftarrow \\bm{N} \\cup S$ \\label{alg_EE:add} \\\\\n $\\bm{E}_S \\leftarrow \\textnormal{Partial-Out}(\\textnormal{Ne}_{\\widehat{\\mathbb{G}}}(S),S)$\\label{alg_EE:partial}\\\\\n $\\bm{U} \\leftarrow $ members of $\\bm{M}$ adjacent to $S$ in $\\widehat{\\mathbb{G}}$\\\\\n Remove edges adjacent to $S$ in $\\widehat{\\mathbb{G}}$\n }\n\\caption{Extract-Errors} \\label{alg_EE}\n\\end{algorithm}\n\n\n\\begin{algorithm}[t]\n \\nonl \\textbf{Input:} $\\bm{M},\\bm{U},\\bm{T},\\widehat{\\mathbb{G}}$\\\\\n \\nonl \\textbf{Output:} sink $S$\\\\\n \\BlankLine\n\n\\textbf{return} $\\bm{M}$ if $|\\bm{M}| = 1$\\\\\n\n\\For{$X_i \\in \\bm{U}$}{\n $R_i \\leftarrow \\textnormal{Partial-Out}(\\textnormal{Ne}_{\\widehat{\\mathbb{G}}}(X_i),X_i)$\\\\\n $T_i \\leftarrow \\max_{X_j \\in \\textnormal{Ne}_{\\mathbb{G}}(X_i)} I(X_j;R_i)$ \\label{alg_sink:D}\n}\n$S \\leftarrow \\bm{M}[\\argmin_{X_i \\in \\bm{U}} T_i]$ \\label{alg_sink:min}\n\\caption{Find-Sink} \\label{alg_sink}\n\\end{algorithm}\n\n\\section{Related Work}\nAuthors have proposed to identify causal direction using functional forms more restrictive than HNM. For example, LiNGAM considers a linear SEM with non-Gaussian errors, while ANM considers a nonlinear SEM with additive noise \\cite{Shimizu06,Hoyer08}. The post-nonlinear model (PNL) assumes that the error can be made homoscedastic under a monotonic transformation of the response \\cite{Zhang09}. All of these models therefore only consider additive errors, whereas HNM allows both additive and multiplicative forms.\n\nRecently, \\cite{Xu22} also considered HNM and proposed an algorithm called HEC for determining causal direction in the bivariate case. HEC divides the range of the predictor variable into a finite set of bins and then fits an additive model in each bin. The authors additionally assume that the error terms follow a Gaussian distribution in order to optimize the number of bins using the BIC score. Another algorithm called Fourth Order Moment (FOM) assumes approximately Gaussian errors but allows the conditional variance to change in a smooth, rather than in a piece-wise, fashion \\cite{Cai20}. GRCI in contrast admits a smooth conditional variance \\textit{and} allows the error term to admit an arbitrary, potentially non-Gaussian distribution.\n\nOther methods, such as those proposed in \\cite{Tagasovska20,Mitrovic18,Liu17}, also allow heteroscedastic noise but determine causal direction \\textit{without} recovering the error terms. We therefore cannot use these algorithms to compute the Shapley values necessary for identifying patient-specific root causes of disease.\n\nA third set of algorithms attempt to identify root causes rather than just determine causal direction. The RCI algorithm for example identifies patient-specific root causes of disease but assumes the LiNGAM model \\cite{Strobl22}. Unfortunately, we cannot simply substitute LiNGAM with HNM in RCI because \\textit{indirect} causal relations may not follow HNM even if Equation \\eqref{eq_HNM} holds -- i.e., HNM is not closed under marginalizatin. Other authors defined patient-specific root causes as conditional outliers, but not all root causes are outliers and not all outliers induce disease \\cite{Janzing19}. We therefore instead define patient-specific root causes using Shapley values. A third algorithm identifies root causes by substituting certain causal conditionals into an SEM, but this method struggles to scale beyond several variables and identifies root causes at the population level rather than at the desired patient-specific level \\cite{Budhathoki21}. The root causes of complex diseases likely differ dramatically between patients, so we must identify \\textit{patient-specific} root causes in order to make complex diseases tractable.\n\n\\begin{tcolorbox}[breakable,enhanced,frame hidden]\nIn summary, GRCI improves upon previous work because it:\n\\begin{enumerate}[leftmargin=*,label=(\\arabic*)]\n \\item adopts the identifiable heteroscedastic noise model which includes both LiNGAM and ANM as special cases;\n \\item generalizes RCI to models that are not closed under marginalization, such as HNM and ANM;\n \\item identifies root causes at the patient-specific level in order to make complex diseases tractable.\n\\end{enumerate}\n\\end{tcolorbox}\n\n\\section{Experiments}\n\n\\noindent\\textbf{Hyperparameters. } GRCI requires two hyperparameters: the $\\alpha$ value for Skeleton-Stable and the $k$ value for the nearest neighbor mutual information estimator. We set $\\alpha$ to the liberal threshold of 0.1, which in practice causes Skeleton-Stable to output a superset of the true skeleton. This superset represents a small subset of the fully connected graph that greatly reduces the dimensionality of the regressions performed in Step \\ref{alg_GRCI:error}.\n\nWe fixed $k=10$ for the mutual information estimator for three reasons. First, the entropy estimate is consistent for any fixed value of $k$. The standard deviation of the estimator also stabilizes at $k=10$ for most sample sizes according to Figure 4 of \\cite{Kraskov04}. Moreover, the estimate is near exact when independence truly holds as shown in Figure 2 of \\cite{Kraskov04}. Both of these experimental results hold for nearly all cases tested by the authors.\\\\${}$\\vspace{-3mm}\\\\\n\\noindent \\textbf{Reproducibility. } All R code needed to replicate experimental results is available at github.com\/ericstrobl\/GRCI.\n\n\\subsection{Causal Direction}\n\nGRCI computes a (reverse) partial ordering $\\bm{N}$, so we can use the algorithm to recover causal direction in the bivariate setting after assuming that an edge exists between $X$ and $Y$. We compared GRCI against four algorithms on their ability to identify causal direction in the bivariate setting:\n\\begin{enumerate}[label=(\\arabic*)]\n\\item HEteroscedastic noise Causal model (HEC): bins $X$ and fits a polynomial regressor in each bin while assuming intra-bin homoscedasticity. The algorithm chooses the causal direction as the one minimizing the BIC score \\cite{Xu22}.\n\\item Fourth Order Moment (FOM): estimates the fourth-order moment of the residuals using a heteroscedastic Gaussian process. The algorithm chooses the causal direction as the one minimizing the fourth-order moment \\cite{Cai20}.\n\\item REgression and Subsequent Independence Test (RESIT): assumes an ANM, regresses out the conditional mean using a Gaussian process and determines causal direction using a reproducing kernel-based conditional independence test \\cite{Peters14}.\n\\item Direct LiNGAM (DL): assumes variables are linearly related with non-Gaussian errors \\cite{Shimizu11}. The algorithm decides causal direction using the differential entropy measure proposed in \\cite{Hyvarinen13}.\n\\end{enumerate}\nThe first two algorithms cover state of the art methods that handle heteroscedastic noise. The other two algorithms are state of the art for the additive noise and linear non-Gaussian acyclic models. Other algorithms in the literature utilize information theoretic measures and do not impose functional forms. We however only compare against methods which can extract the values of the error terms, since we are ultimately interested in performing patient-specific root causal inference rather than just determining causal direction. \n\n\\subsubsection{Synthetic Data} \\label{sec:pair_synth}\n\nWe generated data using three different functional models:\n\\begin{enumerate}[label=(\\arabic*)]\n \\item LiNGAM: $Y=X\\beta + E$\n \\item ANM: $Y=f(X) + E$\n \\item HNM: $Y=f(X) + E g(X)$,\n\\end{enumerate}\nwith $f(X)$ and $g(X)-1$ uniformly sampled from the set $\\{ \\sqrt{X^2 + 1} -1 , X\\Psi(X), 1\/(1+\\textnormal{exp}(-X))\\}$; we subtracted one from $g(X)$ to ensure non-zero variance. We sampled the distribution of $E$ uniformly from the following possibilities: uniform distribution on $[-1,1]$, t-distribution with five degrees of freedom, chi square distribution with three degrees of freedom. Note that LiNGAM requires at least one non-Gaussian error, whereas ANM and HNM do not. We therefore also included the centered Gaussian distribution with variance $1\/9$ as one of the possibilities for the error term of ANM and HNM. We repeated the above procedure 200 times for LiNGAM with non-Gaussian errors, 200 times for ANM with non-Gaussian errors and another 200 times with Gaussian errors, 200 times for HNM with non-Gaussian errors and another 200 times with Gaussian errors. We therefore generated a total of 1000 independent datasets.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.6]{Synth_pairs_results1.png}\n \\caption{Results on causal direction with synthetic data under different conditions. Rows correspond to functional model and columns to Gaussianity. Error bars denote 95\\% confidence intervals of the mean. GRCI performs well across all conditions while the other algorithms only perform well in some cases.}\n \\label{fig:synth_pair}\n\\end{figure}\n\nWe report the results in Figure \\ref{fig:synth_pair}. LiNGAM, GRCI and RESIT performed well under LiNGAM. Of course, LiNGAM and RESIT outperform GRCI in this case, because they are specifically designed for the homoscedastic setting. Only GRCI and RESIT performed well under ANM with non-Gaussian errors because LiNGAM assumes linear conditional expectations. GRCI, HEC and FOM all performed equivalently with Gaussian error terms under both ANM and HNM. However, GRCI outperformed the other two -- sometimes by a very large margin -- with non-Gaussian errors. Recall that HEC and FOM make a variety of Gaussian approximations which unfortunately do not work well in the non-Gaussian setting. Overall, GRCI achieves the best performance with an accuracy of 83.5\\% as compared to 71.0\\% for HEC, 72.8\\% for FOM, 69.9\\% for RESIT and 49.0\\% for DL. We conclude that GRCI maintains good performance across all conditions while other algorithms only perform well in certain cases. Timing results are located in the Supplementary Materials; GRCI completed within 0.4 seconds on average.\n\n\\subsubsection{Real Data}\n\nThe T\\\"{u}bingen cause-effect pairs benchmark contains 108 datasets of real cause-effect pairs \\cite{Mooij16}. We summarize the results for the 108 pairs in Figure \\ref{fig:Tuebin}. As is standard in the literature, we exclude pairs containing multivariate vectors or binary variables; this includes pair numbers 47, 52-55, 70, 71, 105 and 107. We evaluate accuracy using the suggested weighted average in order to account for the potential bias introduced by pairs derived from the same multivariable dataset. The x-axis in in Figure \\ref{fig:Tuebin} corresponds to the cause-effect pair number (1-108), and the y-axis to the moving weighted accuracy. An ideal algorithm should achieve the highest weighted accuracy at any pair number. GRCI obtained an overall weighted accuracy of 81.6\\%, as opposed to 71.2\\% for FOM, 70.5\\% for HEC, 64.0\\% for RESIT and 51.5\\% for LiNGAM. Interestingly, these results match the overall accuracy for the synthetic data to within a few percentage points. GRCI also maintained the best weighted accuracy at any pair number. We conclude that GRCI accurately discovers causal direction using real data. In general, algorithms that account for heteroscedasticity (GRCI, FOM, HEC) perform better than those that only account for homoscedasticity (RESIT, LiNGAM), and algorithms that account for non-linear relations (GRCI, FOM, HEC, RESIT) perform better than those that only account for linear relations (LiNGAM). Timing results are located in the Supplementary Materials; GRCI completed within 5 seconds on average.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.65]{Tuben_pairs.png}\n \\caption{Results on the 108 T\\\"{u}bingen cause-effect pairs. GRCI maintains the highest weighted accuracy at any pair number.}\n \\label{fig:Tuebin}\n\\end{figure}\n\n\\begin{tcolorbox}[breakable,enhanced,frame hidden]\nSummarizing the results of the causal direction experiments:\n\\begin{enumerate}[leftmargin=*,label=(\\arabic*)]\n \\item GRCI maintains good performance across LiNGAM, ANM and HNM, regardless of whether the errors are Gaussian or not.\n \\item HEC and FOM do not perform well when error terms deviate from Gaussianity.\n \\item GRCI obtains and maintains the best performance with real data. \n \\item The overall accuracy on the synthetic data almost exactly mimics the final accuracy on the real data.\n\\end{enumerate}\n\\end{tcolorbox}\n\n\\subsection{Root Causal Inference}\n\nWe next investigate the performance of GRCI in discovering patient-specific root causes of disease. We compare against four other algorithms:\n\\begin{enumerate}[label=(\\arabic*)]\n \\item Root Causal Inference (RCI): recovers patient-specific root causes assuming that the joint distribution obeys the LiNGAM model \\cite{Strobl22}. \n \\item Prediction with ICA (ICA): runs ICA and then ranks the identified sources using a local variable importance measure of random forest \\cite{Lasko19}. \n \\item Conditional Outliers (CO): learns a causal graph $\\widehat{\\mathbb{G}}$ and then identifies patient-specific root causes as conditional outliers according to the score $\\frac{|X_i-m_i(\\textnormal{Pa}_{\\widehat{\\mathbb{G}}}(X_i))|}{\\sigma_i(\\textnormal{Pa}_{\\widehat{\\mathbb{G}}}(X_i))}$ \\cite{Janzing19}. \n \\item Model Substitution (MS): learns a causal graph and then identifies \\textit{group level} root causes by substituting causal conditional distributions into the joint distribution \\cite{Budhathoki21}. \n\\end{enumerate}\nWe equipped CO and MS with Partial-Order because the methods can adapt to different functional assumptions. Using linear methods as in \\cite{Strobl22} results in worse performance. \n\nCO and MS also require a method for identifying the underlying causal graph. We tested RESIT and GDS as proposed in \\cite{Peters14}, but they did not scale even after substituting a fast non-parametric conditional independence test \\cite{Strobl19}. We therefore instead ran Steps \\ref{alg_GRCI:PC} and \\ref{alg_GRCI:error} of GRCI to recover a partial order. The parents of a variable must precede it in the partial order. So we next ran Skeleton-Stable with conditioning sets restricted to preceding variables according to the partial order and then oriented directed edges according to the partial order. This process recovers a unique DAG. We fixed the alpha threshold to 0.05 because it led to the best results in our experiments. \n\nComputing the ground truth Shapley values requires an exponential number of summations per Equation \\eqref{eq_shap}. We therefore instead estimated the ground truth to negligible error by (1) feeding XGBoost fifty thousand samples of the \\textit{ground truth} error terms and (2) running the TreeSHAP algorithm on the learned model. We reran all applicable algorithms (RCI and ICA) using XGBoost and TreeSHAP in order to prevent GRCI from achieving an unfair advantage due to possible biases introduced during ground truth estimation.\n\n\\subsubsection{Synthetic Data}\n\nWe generated data from a DAG with an expected neighborhood size of two, $\\mathbb{E}(N)=2$. We assigned adjacencies using independent realizations of a Bernoulli$\\Big(\\frac{\\mathbb{E}(N)}{p-1}\\Big)$ random variable in an upper triangular matrix. We then replaced the binary variables twice with samples from $\\textnormal{Uniform}([-1,-0.25] \\cup [0.25, 1])$. Let $\\beta^1$ denote the first resultant coefficient matrix, and $\\beta^1_{ji}$ to the $j^\\textnormal{th}$ row and $i^\\textnormal{th}$ column; likewise for $\\beta^2$. We generated the non-Gaussian error terms using the same procedure described in Section \\ref{sec:pair_synth}. The HNM model corresponds to $X_i=f_i(\\sum_{X_j \\in \\textnormal{Pa}_{\\mathbb{G}}(X_i)} X_j\\beta^1_{ji}) + E_i g_i(\\sum_{X_j \\in \\textnormal{Pa}_{\\mathbb{G}}(X_i)} X_j\\beta^2_{ji})$ for each $X_i \\in \\bm{X}$ with functions $f_i,g_i$ drawn randomly as in Section \\ref{sec:pair_synth}. We finally permuted the variable order. Repeating the above procedure 200 times for sample sizes of $n=500, 1000, 2000$ and dimensions $p=10, 30, 50$ generated a total of $200 \\times 3 \\times 3 = 1800$ datasets.\n\n\\textbf{Metrics. } Comparing the algorithms is not straightforward because the algorithms have different outputs. GRCI returns sample-specific Shapley values for all of the variables. RCI returns sample-specific Shapley values only for some of the variables, since it performs variable selection. ICA outputs sample-specific scores according to a random forest metric, but it can be modified to return sample-specific Shapley values for all of the variables. MS outputs \\textit{group-level} Shapley values, and CO outputs sample-specific conditional outlier scores both only for some of the variables. We need a method that compares the algorithms on a common footing and accounts for outputs of different lengths. \n\nAll algorithms fortunately can return a ranked list of variables. The top ranked variables ideally should correspond to the root causes with the largest effect on $D$. We therefore evaluated the algorithms using ranked-biased overlap (RBO) \\cite{Webber10}, a well-established metric that compares two ranked lists. Let $\\mathcal{R}^k$, correspond to the ground truth ranking of the root causes for patient $k$ according to the true Shapley values. Similarly let $\\widehat{\\mathcal{R}}^k$ denote the estimate of the ranking given by an algorithm. The RBO corresponds to:\n\\begin{equation} \\label{eq:RBO_shapley}\n \\frac{1}{n} \\sum_{k=1}^n \\sum_{i=1}^{q_k} \\widetilde{s}_i^k | \\widehat{\\mathcal{R}}_{1:i}^k \\cap \\mathcal{R}_{1:i}^k|\/i,\n\\end{equation}\nwhere $s_i^k$ denotes the true Shapley value of $X_i$ for patient $k$, $\\widetilde{s}_i^k = \\frac{s_i^k}{\\sum_{i=1}^{q_k} s_i^k}$ the version normalized to sum to one, and $q_k$ the total number of root causes for patient $k$. RBO can compare ranked lists of potentially varying lengths and weighs top variables more heavily than bottom ones.\nThe metric takes values between zero and one; it equals one when the top ranked variables coincide exactly between the two lists, and zero when there is no overlap. A higher RBO is therefore better.\n\nGRCI, RCI and ICA also attempt to estimate the values of the error terms which directly impact the quality of the estimated Shapley values. We therefore evaluated these algorithms using the mean squared error (MSE) to the ground truth error values as well:\n\\begin{equation} \\nonumber\n \\frac{1}{nw} \\sum_{k=1}^n \\sum_{i=1}^w (\\widehat{s}_i^k - s_i^k)^2,\n\\end{equation}\nwhere $w$ denotes number of ground truth ancestors of $D$. We set $\\widehat{s}_i^k = 0$, if the algorithm does not output a score for variable $i$. A lower MSE is better.\n\n\\textbf{Results. } We summarize the accuracy results with the synthetic data using RBO and MSE in Tables \\ref{exp_synth:RBO} and \\ref{exp_synth:MSE}, respectively. Recall that we implemented two versions of RCI and ICA - the original ones and the modified forms using TreeSHAP as labeled using the subscript $t$. We therefore compared GRCI against a total of six algorithms. Bolded values in each row of the tables correspond to the best performing algorithms according to paired two-tailed t-tests each at a Bonferonni corrected threshold of 0.05\/6. \n\n\\begin{table}[t]\n\\begin{subtable}{0.45\\textwidth} \n\\centering\n\\captionsetup{justification=centering,margin=2cm}\n\\begin{tabular}{cc|ccccccc}\n\\hhline{=========}\n\\textit{p} & \\textit{n} & GRCI & RCI & RCI$_t$ & ICA & ICA$_t$ &CO & MS \\\\ \\hline\n10 & 500 & \\textbf{0.735} & \\textbf{0.706} & 0.690 & 0.579 & 0.639 & 0.616 & 0.508\\\\\n & 1000 & \\textbf{0.773} & 0.699 & 0.689 & 0.603 & 0.669 & 0.623 & 0.502\\\\\n & 2000 & \\textbf{0.809} & 0.710 & 0.708 & 0.614 & 0.695 & 0.631 & 0.503\\\\ \\hline\n30 & 500 & \\textbf{0.653} & \\textbf{0.622} & 0.616 & 0.477 & 0.519 & 0.496 & 0.392 \\\\\n & 1000 & \\textbf{0.711} & 0.654 & 0.647 & 0.537 & 0.593 & 0.463 & 0.347\\\\\n & 2000 & \\textbf{0.745} & 0.682 & 0.673 & 0.573 & 0.641 & 0.485 & 0.379\\\\ \\hline\n50 & 500 & \\textbf{0.639} & 0.569 & 0.580 & 0.327 & 0.345 & 0.432 & 0.348 \\\\\n & 1000 & \\textbf{0.685} & 0.613 & 0.609 & 0.506 & 0.556 & 0.402 & 0.338\\\\\n & 2000 & \\textbf{0.741} & 0.642 & 0.636 & 0.555 & 0.615 & 0.383 & 0.311\\\\ \n\\hhline{=========}\n\\end{tabular}\n\\caption{RBO} \\label{exp_synth:RBO}\n\\end{subtable}\n\n\\vspace{5mm}\\begin{subtable}{0.45\\textwidth} \n\\centering\n\\captionsetup{justification=centering,margin=2cm}\n\\begin{tabular}{cc|ccccccc}\n\\hhline{=========}\n\\textit{p} & \\textit{n} & GRCI & RCI & ICA \\\\ \\hline\n10 & 500 & \\textbf{0.160} & 0.650 & 3.044 \\\\\n & 1000 & \\textbf{0.113} & 0.659 & 3.362 \\\\\n & 2000 & \\textbf{0.104} & 0.620 & 3.435 \\\\ \\hline\n30 & 500 & \\textbf{0.183} & 0.756 & 3.455 \\\\\n & 1000 & \\textbf{0.138} & 0.700 & 3.556 \\\\\n & 2000 & \\textbf{0.111} & 0.635 & 3.355 \\\\ \\hline\n50 & 500 & \\textbf{0.186} & 0.791 & 3.558 \\\\\n & 1000 & \\textbf{0.170} & 0.702 & 3.632 \\\\\n & 2000 & \\textbf{0.108} & 0.643 & 3.361 \\\\ \n\\hhline{=========}\n\\end{tabular}\n\\caption{MSE} \\label{exp_synth:MSE}\n\\end{subtable}\n\n\\vspace{5mm}\\begin{subtable}{0.45\\textwidth} \n\\centering\n\\captionsetup{justification=centering,margin=2cm}\n\\begin{tabular}{cc|ccccccc}\n\\hhline{=========}\n\\textit{p} & \\textit{n} & GRCI & RCI & RCI$_t$ & ICA & ICA$_t$ &CO & MS \\\\ \\hline\n10 & 500 & \\cellcolor{Gray!25}1.613 & 0.003 & 0.651 & 0.182 & 0.776 & \\cellcolor{Gray!25}1.208 &\\cellcolor{Gray!25}1.248\\\\\n & 1000 & \\cellcolor{Gray!25}4.075 & 0.004 & 0.832 & 0.404 & 1.073 & \\cellcolor{Gray!25}3.585 &\\cellcolor{Gray!25}3.686\\\\\n & 2000 & \\cellcolor{Gray!25}13.85 & 0.009 & 1.186 & 0.914 & 1.659 & \\cellcolor{Gray!25}13.43 &\\cellcolor{Gray!25}13.95\\\\ \\hline\n30 & 500 & \\cellcolor{Gray!25}9.199 & 0.011 & 0.720 & 0.383 & 1.075 & \\cellcolor{Gray!25}10.51 &\\cellcolor{Gray!25}10.64\\\\\n & 1000 & \\cellcolor{Gray!25}22.56 & 0.020 & 0.946 & 0.925 & 1.644 & \\cellcolor{Gray!25}24.40 & \\cellcolor{Gray!25}24.80\\\\\n & 2000 & \\cellcolor{Gray!25}108.2 & 0.043 & 1.375 & 2.285 & 2.830 & \\cellcolor{Gray!25}111.5 & \\cellcolor{Gray!25}113.6\\\\ \\hline\n50 & 500 & \\cellcolor{Gray!25}32.90 & 0.033 & 0.850 & 0.650 & 1.477 & \\cellcolor{Gray!25}39.32 &\\cellcolor{Gray!25}39.61\\\\\n & 1000 & 8\\cellcolor{Gray!25}3.21 & 0.058 & 1.135 & 1.603 & 2.398 & \\cellcolor{Gray!25}91.39 & \\cellcolor{Gray!25}92.28\\\\\n & 2000 & \\cellcolor{Gray!25}222.0 & 0.125 & 1.806 & 4.145 & 4.708 & \\cellcolor{Gray!25}235.7 & \\cellcolor{Gray!25}240.0\\\\ \n\\hhline{=========}\n\\end{tabular}\n\\caption{Time in seconds} \\label{exp_synth:time}\n\\end{subtable}\n\\caption{GRCI obtains the highest mean RBO values in (a) and lowest mean MSE values in (b) in every situation tested with the synthetic data. All HNM-based algorithms take approximately the same amount of time to complete as highlighted in gray in (c). } \n\\end{table}\n\nGRCI achieved the highest mean RBO in every situation (Table \\ref{exp_synth:RBO}). The original version of RCI came in second place and TreeSHAP did not improve its performance. TreeSHAP improved ICA, but both versions of ICA performed much worse than GRCI and RCI. ICA frequently got stuck in local optima as evidenced by the terribly inaccurate error values when compared to RCI (Table \\ref{exp_synth:MSE}). GRCI recovered the error terms about two to six times more accurately than RCI. MS and CO had the worst performances because the algorithms either recovered conditional outliers that did not induce disease or failed to output sample-specific scores. We conclude that GRCI performs the most accurately across all tested sample sizes, dimensions and metrics even after incorporating TreeSHAP into applicable alternatives.\n\nWe summarize timing results in Table \\ref{exp_synth:time}. Algorithms that search over the space of HNMs -- including GRCI, CO and MS highlighted in light gray -- take about the same amount of time. These methods also expectedly take longer than the linear algorithms RCI and ICA.\n\n\\subsubsection{Real Data} We compared all seven algorithms on their ability to discover patient-specific root causes using two real datasets. Note that we do not have access to the ground truth Shapley values with real data, so we use the following modified RBO metric:\n\\begin{equation} \\nonumber\n \\frac{1}{n} \\sum_{k=1}^n \\sum_{i=1}^{q_k} \\frac{1}{q_k} | \\widehat{\\mathcal{R}}_{1:i}^k \\cap \\mathcal{R}_{1:i}^k|\/i,\n\\end{equation}\nwhere we no longer weight the score by Shapley values.\n\n\\begin{figure}\n\\begin{subfigure}{0.45\\textwidth}\n \\centering\n \\includegraphics[scale=0.6]{PBC_RBO_HNM.png}\n \\caption{Primary Biliary Cirrhosis}\n \\label{fig:real_PBC}\n\\end{subfigure}\n\n\\begin{subfigure}{0.45\\textwidth}\n \\centering\n \\includegraphics[scale=0.6]{Diabetes_RBO_HNM.png}\n \\caption{Pima Indians Diabetes}\n \\label{fig:real_Diabetes}\n\\end{subfigure}\n\\caption{Sorted accuracy results with the real datasets.}\n\\end{figure}\n\n\\textbf{Primary Biliary Cholangitis. } \nThe Mayo Clinic Primary Biliary Cholangitis (PBC) dataset contains samples from 258 patients with PBC who entered into a randomized clinical trial assessing the effects of medication called D-penicillamine \\cite{Fleming11}. PBC is an autoimmune disease that slowly destroys the small bile ducts of the liver, eventually causing liver cirrhosis, liver decompensation and then death \\cite{Hirschfield13}. The dataset contains the following continuous variables: age, bilirubin, albumin, alkaline phosphatase, copper, cholesterol, platelets, AST and pro-thrombin time.\n\nWe sought to identify the patient-specific root causes of mortality. We know that age and bilirubin cause death because older patients pass away and increased bilirubin leads to neurotoxicity \\cite{Lopez14}. Intervening on the other variables does not consistently change mortality, so they are likely non-ancestors of death. High levels of bilirubin increase the frequency of death more than old age. We set the gold standard ranking as bilirubin then age if the bilirubin is at or above 2 mg\/dL -- in accordance with the classic Child-Turcotte cut-off \\cite{Child64} -- and bilirubin then age otherwise.\n\nWe ran the algorithms on 1000 bootstrapped draws of the dataset. We report accuracy results among patients who passed in Figure \\ref{fig:real_PBC}. GRCI achieved the best accuracy compared to all other methods. RCI came in second place in accordance with the synthetic data results. GRCI took 8 seconds on average (see the Supplementary Materials for full timing results).\n\n\\textbf{Pima Indians Diabetes. } The Pima Indians Diabetes Database is a observational dataset containing samples from females in the Pima Indian population near Pheonix, Arizona \\cite{Smith88}. The dataset contains the following variables: number of pregnancies, plasma glucose concentration at two hours in an oral glucose tolerance test, diastolic blood pressure, triceps skinfold thickness, two-hour serum insulin, body mass index, diabetes pedigree function, age, and presence or absence of diabetes. \n\nWe sought to identify the patient-specific root causes of diabetes. Recall that the incidence of diabetes increases with age, and clinicians can diagnose diabetes if the blood glucose reaches at least 200 mg\/dL with a two hour oral glucose tolerance test. We therefore set the gold standard as age and glucose ranked according to their z-score in decreasing order.\n\nWe ran the algorithms again using 1000 bootstrapped draws. We reports the results for patients with diabetes in Figure \\ref{fig:real_Diabetes}. GRCI again achieved the best accuracy compared to all other methods. The results with the Pima Indians Diabetes Database also mimic those seen with the PBC and synthetic datasets. GRCI took 15.2 seconds on average (Supplementary Materials). \n\n\\begin{tcolorbox}[breakable,enhanced,frame hidden]\nSummarizing the results of the patient-specific root causal inference experiments:\n\\begin{enumerate}[leftmargin=*,label=(\\arabic*)]\n \\item GRCI achieves the best performance -- in terms of both RBO or MSE -- across all sample sizes and dimensions with the synthetic data.\n \\item GRCI also achieves the best performance in two real datasets with known root causes, and the real data results mimic the synthetic ones.\n \\item GRCI, MS and CO take longer than the linear algorithms but still complete within about 4 minutes on average with $n=2000, p=50$.\n\\end{enumerate}\n\\end{tcolorbox}\n\n\\section{Conclusion}\n\nWe presented GRCI, the first method that generalizes the original RCI algorithm to the non-linear setting. GRCI accommodates both non-linear expectations and heteroscedastic noise under the HNM model. We proved identifiability of HNM in general and described a procedure that partials out both the conditional mean and MAD in a two-step regression process. We then rigorously defined patient-specific root causes using sample-specific Shapley values of the error terms. We introduced GRCI as an efficient method that recovers these errors by combining error extraction in functional causal models with constraint-based skeleton discovery. Experiments with both synthetic and real data highlighted considerable improvements in accurately recovering both causal direction and patient-specific root causes of disease. GRCI even outperformed other methods based on HNM engineered specifically for causal direction because GRCI does not make any Gaussian approximations. \n\n\\bibliographystyle{vancouver}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThe growth of gravitationally bound haloes of collisionless cold dark matter (CDM) through accretion and mergers is one of the primary physical processes of interest in the hierarchical structure formation paradigm. The stochasticity inherent in the initial cosmological seed fluctuations, coupled with the nonlinearity of gravitational evolution of a collisionless fluid, renders this problem analytically challenging, although considerable insights may be gained through simplified models of structure formation \\citep{bcek91,lc93,ms14-markov}. Since dark haloes are expected to be the cradles of \\emph{galaxy} formation and evolution \\citep{wr78}, understanding the evolving nature of halo mass accretion and its dependence on the local and large-scale environment of haloes is expected to yield important clues into corresponding correlations in the observed distribution and evolution of galaxies in the Universe.\n\nThe understanding that accretion rates in the late Universe are sensitive to the shape of the \\emph{initial} power spectrum \\citep[][see also below]{lc93} suggests a useful tool for investigating the nature of mass accretion, in the form of CDM-like power spectra which are suppressed at small scales (large $k$), mimicking the effects of a thermally produced warm dark matter (WDM) particle. The steep cut-off in power in such models creates an extreme situation where haloes forming close to the corresponding mass scale experience dramatically enhanced growth, and haloes below this scale simply do not exist \\citep[e.g.,][]{angulo}. Simulations performed with WDM-like power spectra, therefore, offer ideal test-beds for the environment dependence of mass accretion: any physical model that purports to explain the nature of mass accretion in CDM must also do so for WDM, since the physics of a self-gravitating collisionless fluid is common to both \\citep{hp14}. Of course, WDM models are physically interesting in their own right, from the point of view of small-scale challenges for the CDM framework \\citep[see][for a review]{bb-k17}, although this is not the focus of the present work.\n\n\nThere has been considerable work to date studying mass accretion by haloes via mergers and smooth (or diffuse) accretion \\citep{fm10, genel, benson+13, elahi}, as well as its dependence on halo environment \\citep[defined typically in terms of halo-centric dark matter density using different smoothing schemes, see][see also \\citealp{mdk15}]{genel,fm09, fm10,maulbetsch+07,bprg17, lee}. The overall understanding that has emerged from these studies, regardless of the exact choice of definition of halo-centric density, is that denser environments tend to promote mergers while underdense environments are more conducive to diffuse accretion. There are also stark differences between accretion in CDM and WDM, particularly at low mass, as expected from the discussion above, with low-mass WDM haloes accreting rapidly and primarily through diffuse accretion.\n\nA related line of study is that of \\emph{halo assembly bias} or secondary bias, i.e., the correlation (at fixed halo mass) between secondary halo properties other than mass and halo-centric density (or bias) measured at cosmological scales \\citep{st04,gsw05}. Although assembly bias has been studied over a wide range of halo mass and redshift using many choices of secondary variables such as age \\citep{st04,gsw05,jsm07}, concentration \\citep{wechsler+06,abl08}, shape \\citep{fw10, vDaw12}, angular momentum or spin \\citep{gw07} and velocity dispersion structure \\citep{fw10}, the correlation between halo bias and mass accretion rate has thus far been limited to massive objects at $z=0$ \\citep{lazeyras}. Local halo environment has been found to influence halo assembly bias of various secondary variables \\citep{Wang07, Jung14, lee, Yang17, Musso18}. Moreover, recent work on assembly bias at $z=0$ has revealed the importance of the \\emph{local tidal environment} of haloes \\citep{hpdc09, bprg17,phs18a} \nin explaining the assembly bias of many secondary variables, including concentration, spin, shape and velocity dispersion structure over a wide range of halo mass \\citep[][see also \\citealp{han+19}]{rphs19}. It is therefore very interesting to ask whether the halo tidal environment plays a similar role in explaining any assembly bias trends with mass accretion.\n\n\nIn this work, we perform a detailed study of the nature of the environment dependence of mass accretion by haloes, segregated into contributions due to mergers and diffuse mass, as a function of redshift and for a range of halo masses. We will do so using $N$-body simulations of both CDM and WDM cosmologies; as mentioned above, the latter will allow us to better resolve the multi-scale environment dependence of mass accretion due to its sensitivity to the shape of the initial matter power spectrum. We will connect these results to the assembly bias literature by performing, for the first time, a comprehensive study of assembly bias due to mass accretion across cosmic time, along with its connection to the evolving local halo environment, for both CDM and WDM. Our results reproduce previously observed trends while extending these to new local environmental variables such as the tidal anisotropy, and are expected to place useful constraints on semi-analytical models of halo formation and growth \\citep[e.g.,][]{sk99,pch08,jvdb14}, which in turn form the bedrock of several semi-analytical models of galaxy formation and evolution \\citep[e.g.,][]{bb10,benson12,barausse12,dfdp14,birrer+14,spt15,yung+19}.\n\nThe paper is organised as follows. We describe our simulations and analysis techniques in section~\\ref{sec:sims}. We present our results along with a discussion in section~\\ref{sec:results}, with section~\\ref{subsec:mergvsdiff} focusing on environment-independent evolution, sections~\\ref{subsec:medianenv} and~\\ref{subsec:mdot<->env} focusing on trends with local environment and section~\\ref{subsec:assemblybias} devoted to assembly bias. We discuss some of these trends in the framework of the excursion set approach in section~\\ref{sec:analytical} and conclude with a summary of our main results in\nsection~\\ref{sec:conclude}. Throughout, the base-10 logarithm is denoted by `log' and the natural logarithm by `ln'.\n\n\\section{Simulations and Techniques}\n\\label{sec:sims}\nHere we describe our simulations and analysis tools for identifying haloes and measuring their accretion rates and local as well as large-scale environments.\n\n\n\\subsection{Simulations}\nWe have used $N$-body simulations of CDM and WDM performed using the tree-PM code \\textsc{gadget-2} \\citep{springel}\\footnote{\\href{http:\/\/www.mpa-garching.mpg.de\/gadget\/}{http:\/\/www.mpa-garching.mpg.de\/gadget\/}} in a cubic, periodic box of comoving length $150\\ensuremath{h^{-1}{\\rm Mpc}}$ sampled with $1024^3$ particles, corresponding to a particle mass $m_{\\rm p} = 2.4\\times 10^8 \\ensuremath{h^{-1}M_{\\odot}}$, with a $2048^3$ PM grid and comoving force resolution $4.9 \\ensuremath{h^{-1}{\\rm kpc}}$ corresponding to $1\/30$ of the Lagrangian inter-particle spacing. \n\nThe CDM transfer function $T_{\\rm cdm}(k)$ for generating initial conditions was computed using the code \\textsc{camb} \\citep{camb}\\footnote{\\href{http:\/\/camb.info}{http:\/\/camb.info}} with a spatially flat $\\Lambda$CDM cosmology having total matter density parameter $\\Omega_{\\rm m}=0.276$, baryonic matter density $\\Omega_{\\rm b}=0.045$, Hubble constant $H_0=100h\\,{\\rm kms}^{-1}{\\rm Mpc}^{-1}$ with $h=0.7$, primordial scalar spectral index $n_{\\rm s}=0.961$ and r.m.s. linear fluctuations in spheres of radius $8\\ensuremath{h^{-1}{\\rm Mpc}}$, $\\sigma_8=0.811$, consistent with the 7-year results of the \\emph{Wilkinson Microwave Anisotropy Probe} \\citep[WMAP7,][]{Komatsu2010}.\n\nFor the WDM model, we additionally suppress small-scale power in the linear transfer function as appropriate for the free-streaming of a thermally produced WDM particle with mass $m_{\\rm dm}=0.4\\,{\\rm keV}$ according to the fitting function of \\citet{Bode2001} \\citep[with parameters taken from][]{Viel2005}\n\\begin{equation}\n T_{\\rm wdm}(k) = T_{\\rm cdm}(k) \\left[1 + (\\alpha k)^{2\\mu}\\right]^{-5\/\\mu},\n\\label{eq:Tk-wdm}\n\\end{equation}\nwith $\\mu=1.12$ and\n\\begin{equation}\n\\alpha \\equiv 0.049 \\left(\\frac{\\Omega_{\\rm m}}{0.25}\\right)^{0.11} \\left(\\frac{h}{0.7}\\right)^{1.22}\n\\left( \\frac{m_{\\rm dm}}{1\\,{\\rm keV}} \\right)^{-1.11}\\,\\ensuremath{h^{-1}{\\rm Mpc}}\\,.\n\\label{alpha}\n\\end{equation}\nThe resulting ``half-mode'' mass-scale \\citep[c.f., e.g.,][]{Schneider2012} of $M_{\\rm hm}\\simeq 3\\times10^{11}\\ensuremath{h^{-1}M_{\\odot}}$ is resolved with $\\sim1200$ particles (see below).\\footnote{The collisionless WDM fluid is assumed to be in the perfectly cold limit, i.e., we ignore the small thermal velocity dispersion of a real WDM fluid. This is expected to be accurate at the epochs of our interest, well after perturbations have been suppressed below the largest free-streaming scale in linear theory \\citep{angulo}.} Although a WDM particle with $m_{\\rm dm}=0.4\\,{\\rm keV}$ is completely ruled out by Lyman-alpha forest observations as being the dominant component of dark matter \\citep{Viel2013,irsic+17,pd+20}, it allows us to resolve the entire initial power spectrum up to the truncation scale with sufficient particles, thus providing a useful test-bed for studying mass accretion and its environment dependence in the absence of small-scale perturbations.\n\nInitial conditions for both CDM and WDM were generated at $z=99$ with the \\emph{same random seed} using $2^{\\rm nd}$ order Lagrangian perturbation theory \\citep{scoccimarro98} with the code \\textsc{music} \\citep{hahn11-music}.\\footnote{\\href{https:\/\/www-n.oca.eu\/ohahn\/MUSIC\/}{https:\/\/www-n.oca.eu\/ohahn\/MUSIC\/}} Snapshots were stored starting from $z=12$ to $z=0$ at time intervals equally spaced in the scale factor with $\\Delta a = 0.004615$ (leading to 201 snapshots), which provides sufficient time resolution for the accretion and merger analysis described below. The simulations and analysis were performed on the Perseus and Pegasus clusters at IUCAA.\\footnote{\\href{http:\/\/hpc.iucaa.in}{http:\/\/hpc.iucaa.in}}\n\n\n\n\\subsection{Halo identification and masses}\n\\label{subsec:massfn}\nHaloes were identified using the code \\textsc{rockstar} \\citep{behroozi-a}\\footnote{\\href{https:\/\/bitbucket.org\/gfcstanford\/rockstar}{https:\/\/bitbucket.org\/gfcstanford\/rockstar}} which implements a Friends-of-Friends algorithm in 6-dimensional phase space and provides information on gravitationally bound haloes as well as their substructure. Merger trees were constructed using the 201 snapshots in each simulation (CDM and WDM) using the code \\textsc{consistent-trees} \\citep{behroozi-b}.\\footnote{\\href{https:\/\/bitbucket.org\/pbehroozi\/consistent-trees}{https:\/\/bitbucket.org\/pbehroozi\/consistent-trees}} Since we wish to focus on the accretion rates of well-resolved haloes in this work, we exclude all subhaloes and further consider only those objects for which the virial energy ratio $\\eta = 2T\/|U|$ satisfies $0.5<\\eta<1.5$, which mitigates the effects of unrelaxed objects and numerical artefacts \\citep{bett07}. We exclude $\\sim 2\\%(6\\%)$ objects of low mass halo population which do not satisfy this virial ratio condition in CDM(WDM). For the high mass bin these objects form $\\sim 4\\%$ of the halo population in either cosmology.\n\nWe further exclude splashback objects, which spatially mimic genuine haloes at the redshift of interest but have passed through a larger host in the past and hence are physically closer to subhaloes. We do so using the value of the `first accretion scale' reported by \\textsc{consistent-trees}, which records the earliest epoch at which the main progenitor of a given descendant passed through the virial radius of a larger object; we exclude all descendants whose first accretion scale is earlier than the epoch of interest.\\footnote{To avoid effects of numerical precision in comparing scale factor values, we define splashbacks as objects whose first accretion epoch is more than 100 Myr in the past of the epoch of interest. We have checked that varying this threshold by factor 2 on either side makes no difference to our results.} For completeness, we report that splashback haloes at $z=0$ form $\\lesssim 6\\% \\,(4.5\\%)$ by number of our low-mass halo populations in CDM (WDM) and $\\lesssim0.7\\%$ of high-mass objects in either cosmology (see below for the corresponding mass ranges), with the fractions becoming vanishingly small at high redshifts. With all the selection criteria, we report that, we select, $\\sim 80\\% (\\sim 75\\%)$ of low mass ROCKSTAR objects in CDM(WDM) and $\\sim 87 \\%$ of high mass objects in the either cosmology. We have checked that the selected low mass haloes in CDM and WDM are \\emph{relaxed} by plotting the distributions of the positions and velocities of the particles forming haloes. The distributions are found to show single well defined peaks.\n\n\n\n\nThroughout, we will quote halo and progenitor masses using the mass definition $M_{\\rm vir}$ as reported by \\textsc{rockstar}. This corresponds to the bound mass contained in a halo-centric sphere of virial radius $R_{\\rm vir}$ which encloses a density equal to $\\Delta_{\\rm vir}$ times the critical density of the Universe, where $\\Delta_{\\rm vir}$ is the spherical collapse overdensity and is taken from the fitting function provided by \\citet{bn98} (at $z=0$, $\\Delta_{\\rm vir}\\simeq98$ for our cosmology, while $\\Delta_{\\rm vir}\\to18\\pi^2\\simeq178$ at high-$z$ during matter domination). Additionally, we use the \\textsc{rockstar} values of $M_{\\rm 200b}$ to infer the halo-centric radius $R_{\\rm 200b}$ which encloses a density equal to 200 times the mean density of the Universe at each redshift; this will be useful when characterising local halo environments below.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.485\\textwidth,trim= 2 4 20 20,clip]{.\/images\/massfunction.png}\n\\caption{{\\bf Halo mass function:} Dashed (solid) lines show the halo mass function in CDM (WDM) at $z=0.0$ (thick curves) and $z=1.5$ (thin curves) as a function of virial mass $M_{\\rm vir}$. The horizontal axis starts at $M_{\\rm vir}= {1.4\\times10^{10}\\ensuremath{h^{-1}M_{\\odot}}}$, corresponding to a 60 particle cut which we use for identifying progenitors. The first vertical dotted line from the left indicates $M_{\\rm vir}=1.2\\times10^{11}\\ensuremath{h^{-1}M_{\\odot}}$ or 500 particles, which we use as the threshold for identifying descendants at any redshift (see text for other criteria applied in selecting a clean halo sample). The low-mass and high-mass bins we use are indicated in blue and red colours, respectively, demarcated by the remaining vertical lines and with the corresponding ranges mentioned in the labels.\n\\underline{\\emph{Highlights:}} In the WDM mass function, we clearly see the suppression of halo counts below the half-mode mass $M_{\\rm hm}\\simeq3\\times10^{11}\\ensuremath{h^{-1}M_{\\odot}}$, and (at low redshift) the spurious upturn at very low masses due to numerical artefacts. \n} \n\\label{massfunction}\n\\end{figure}\n\nFigure \\ref{massfunction} shows the mass functions for haloes in CDM (dashed) and WDM (solid) at $z=0$ and $z=1.5$. The vertical lines demarcate the mass bins we will use below; the lowest mass threshold corresponds to 500 particles and the mass functions in the low-mass (high-mass) bin we employ are coloured blue (red). \nThe CDM mass function shows the well-studied power law rise towards low masses and exponential decline at high masses \\citep{ps74,bcek91,st99,Tinker08}. \n\nThe WDM mass function is identical to the CDM one at high masses for each redshift, but turns over around the half-mode mass (see above), leading to a suppression of halo abundances at low masses as expected from the lack of small-scale perturbations (\\citealp{hp14}; see also \\citealp{benson+13,ssr13}). At masses smaller than $\\sim10^{10.4}\\ensuremath{h^{-1}M_{\\odot}}$, however, we see an \\emph{up-turn} in the $z=0$ WDM mass function, which is a well-known consequence of numerical artefacts in the $N$-body technique applied to initial conditions with suppressed small-scale power \\citep{W&W,angulo,Lovell2014,A&C}. Note that the virial cut of $0.5<\\eta<1.5$ mentioned earlier already removes many spurious objects at masses close to the half-mode mass \\citep{A&C}; removing the lower-mass spurious objects would require a more sophisticated study of the \\emph{initial} proto-patches from which these objects evolve \\citep{Lovell2014}, which we have not performed here.\n\nOur choice of the lowest mass bin for \\emph{descendant} haloes is sufficiently far above the mass scale where spurious objects start becoming numerically relevant. We have checked this by applying the correction suggested by \\citet{ssr13} by fitting a power law to the mass function below the up-turn scale (not displayed) and subtracting it from the measured mass function; the result agrees with the measured mass function in the mass bins of interest at better than $\\sim2\\%$ at all redshifts. Spurious objects therefore do not contribute to our chosen populations of descendant haloes at any redshift. However, when identifying \\emph{progenitor} haloes in the merger trees, we will employ a lower mass cut of 60 particles (the left edge of the horizontal axis in figure~\\ref{massfunction}). This affects the quantification of mass accretion rates in the WDM case at low redshifts, by artificially enhancing the accretion due to mergers and correspondingly decreasing the accretion of diffuse mass. We return to this point below.\n\n\n\n\\subsection{Measuring specific mass accretion rates}\n\\label{subsec:Mdot-measure}\nFor the analysis below, we use haloes in the two mass bins discussed above, all of which are well-resolved with more than 500 particles.\nWe used merger trees generated by \\textsc{consistent-trees} to find all the progenitors of a given halo within the previous 2 dynamical times\\footnote{We have repeated our main analysis for accretion rates calculated over 1 dynamical time and find qualitatively similar but noisier results. We therefore focus on results using accretion rates over 2 dynamical times.} at any chosen redshift. The values of the dynamical time $T_{\\rm dyn}$ and the redshift interval $\\Delta z\\equiv z_{\\rm i}-z_{\\rm f}$ corresponding to $2T_{\\rm dyn}$ at different redshifts are given in table \\ref{tab:table1}. Our snapshot resolution of $\\Delta a\\simeq0.005$ (see above) provides us with excellent sampling of the required $\\Delta z$ at all the redshifts we probe (see table~\\ref{tab:table1}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth,trim= 2 0 20 20,clip]{.\/images\/M_ratesdistri.png}\n\\caption{\n{\\bf Accretion rate distributions:}\nHistograms in the \\emph{left (right) panel} show the distributions of the specific mass accretion rates \\ensuremath{\\Gamma_{\\rm dif}}\\ (\\ensuremath{\\Gamma_{\\rm mer}}) due to diffuse mass (mergers) at $z=0$ (see equations~\\ref{eq:Mdot-def}). Dashed (solid) lines show results for CDM (WDM) haloes, with the red (blue) curves showing results for the high-mass (low-mass) bin. The legend in the \\emph{left panel} indicates the number of haloes that contributed to each mass bin, while the arrows mark the median values of \\ensuremath{\\Gamma_{\\rm dif}}\\ for the respective distributions. \\underline{\\emph{Highlights:}} Low-mass WDM haloes have the highest \\ensuremath{\\Gamma_{\\rm dif}}\\ and lowest \\ensuremath{\\Gamma_{\\rm mer}}\\ values on average. \n} \n\\label{Mrates-distri}\n\\end{figure*}\n\nThroughout, we retain progenitors that are resolved with more than 60 particles ($M_{\\rm vir}>1.4 \\times 10^{10}\\ensuremath{h^{-1}M_{\\odot}}$). \nThis leads to a different dynamic range in progenitor-to-descendant mass ratio $\\chi$ for the low-mass ($\\chi\\gtrsim0.04$) and high-mass bin ($\\chi\\gtrsim6\\times10^{-5}$). To the extent that accretion rates are self-similar with mass \\citep{fg84,Bertschinger85b}, this would lead to accretion by `mergers' being systematically enhanced in high-mass haloes simply because relatively low-mass objects are counted as progenitors as compared to the low-mass bin.\nProvided accretion rates are accurately estimated without double-counting due to (artificial) fragmentation, however, the contribution to genuine mergers from very low mass ratios (say $\\chi<10^{-2}$) should be small \\citep{genel}. \n\\textsc{consistent-trees} improves the consistency of progenitor\/descendant assignment across time steps by solving differential equations for the expected locations of potential descendants \\citep{behroozi-b}, so that effects of artificial fragmentation are reduced.\nAlso, since we define accretion rates over $2T_{\\rm dyn}$, any residual effects of instantaneous fragmentation should typically be averaged over. \nMoreover, environmental trends have been found to be relatively insensitive to whether progenitors are defined using a fixed minimum particle count or minimum $\\chi$ \\citep{fm10}.\nWe therefore proceed with our analysis using a fixed threshold of 60 particles on progenitor mass and comment below on analyses using other choices. \n\nWe follow \\citet{fm10} and divide the mass accreted by a halo into two parts: (i) via other resolved haloes as mergers and (ii) in the form of diffuse mass, which includes bound structures below the chosen mass resolution, actually unbound particles which may have been tidally stripped from other objects, or genuinely diffuse mass which has never been part of a bound structure earlier.\nIn practice, we calculate the mass accreted in a given time interval $z_{\\rm i} > z > z_{\\rm f}$ due to mergers as the total mass of all progenitors except the main progenitor. Correspondingly, diffuse mass accreted in the same time interval is the difference between the descendant halo's mass and the total mass of all the progenitors. Finally, the total mass accreted in this time interval is the sum of the masses accreted via mergers and diffuse accretion, i.e., the difference between the descendant halo's mass and the mass of the main progenitor. Symbolically, if we define\n\n$M_0$ : virial mass of descendant halo at $z=z_{\\rm f}$,\n\n$M_j$ : virial mass of the $j^{th}$ progenitor halo at $z=z_{\\rm i}$,\n\n$M_1$ : virial mass of the main\nprogenitor at $z=z_{\\rm i}$,\n\n\\noindent\nthen the \\emph{dimensionless specific accretion rates} \\ensuremath{\\Gamma_{\\rm mer}}\\ (merger), \\ensuremath{\\Gamma_{\\rm dif}}\\ (diffuse) and \\ensuremath{\\Gamma_{\\rm tot}}\\ (total) -- where $\\Gamma\\sim\\ensuremath{{\\rm d}}\\ln M\/\\ensuremath{{\\rm d}}\\ln a$ \\citep[e.g.,][]{dk14} -- can be written as \n\\begin{align}\n \\ensuremath{\\Gamma_{\\rm mer}} &\\equiv \\frac{\\left(\\sum_{j\\ge2}M_j\\right)}{M_0 (z_{\\rm i}-z_{\\rm f})}\\times(1+z_{\\rm f})\\,, \\notag\\\\\n \\ensuremath{\\Gamma_{\\rm dif}} &\\equiv \\frac{\\left(M_0 - \\sum_{j\\ge1}M_j\\right)}{M_0 (z_{\\rm i}-z_{\\rm f})}\\times(1+z_{\\rm f})\\,, \\notag\\\\\n \\ensuremath{\\Gamma_{\\rm tot}} &\\equiv \\frac{\\left(M_0 - M_1\\right)}{M_0 (z_{\\rm i}-z_{\\rm f})}\\times(1+z_{\\rm f}) \\,.\n \\label{eq:Mdot-def}\n\\end{align}\nWe have chosen a convention such that positive values of the rates correspond to increase in mass from $z_{\\rm i}$ to $z_{\\rm f}$.\n\n\n\\begin{table}\n\\centering\n \\caption{Value of dynamical time $T_{\\rm dyn}$ in Gyr, the redshift interval $\\Delta z$ corresponding to $2\\,T_{\\rm dyn}$ in the past, and the corresponding number of simulation snapshots $N_{\\rm snap}$ tracked, at each redshift studied in this work.}\n \\label{tab:table1}\n \\begin{tabular}{|c | c | c | c | c | c | c | c |} \n \\hline\\hline\n $z$ & 0.0 & 0.25 & 0.5 & 0.8 & 1.0 & 1.5 & 2.0 \\\\\n \\hline\\hline\n $T_{\\rm dyn}$ (Gyr) & 3.14 & 2.55 & 2.08 & 1.67 & 1.46 & 1.08 & 0.83\\\\\n \\hline\n $\\Delta z\\,(2 T_{\\rm dyn})$ & 0.68 & 0.80 & 0.95 & 1.11 & 1.20 & 1.50 & 2.23 \\\\\n \\hline\n $N_{\\rm snap}$ & 89 & 70 & 57 & 47 & 42 & 34 & 32\\\\\n \\hline\\hline\n \\end{tabular}\n\\end{table}\n\nUsing the progenitors selected as above, we computed \\ensuremath{\\Gamma_{\\rm mer}}, \\ensuremath{\\Gamma_{\\rm dif}}\\ and \\ensuremath{\\Gamma_{\\rm tot}}\\ within the previous 2 dynamical times at seven redshifts $z=0.0$, 0.25, 0.5, 0.8, 1.0, 1.5 and 2.0 for the CDM and WDM simulations.\\footnote{As a check, we also explicitly calculated \\ensuremath{\\Gamma_{\\rm tot}}\\ over $100$ My, finding that these are within $\\sim10\\%$ of the values reported by \\textsc{consistent-trees}. The differences are likely due to different choices of progenitor mass resolution between our values and the default settings of \\textsc{consistent-trees}, and are not expected to alter our conclusions qualitatively.}\nThese rates are found to show distributions similar to those shown in \\citet{fm10}. Figure \\ref{Mrates-distri} shows the normalised probability distributions for $\\ensuremath{\\Gamma_{\\rm dif}}$ \\emph{(left panel)} and $\\ensuremath{\\Gamma_{\\rm mer}}$ \\emph{(right panel)} for $z=0$ haloes in the low-mass (blue) and high-mass (red) bin, for CDM (dashed lines) and WDM (solid lines). Arrows in the left panel (with identical colour-coding and line styles) indicate the corresponding median values of $\\ensuremath{\\Gamma_{\\rm dif}}$. These arrows show that the median \\ensuremath{\\Gamma_{\\rm dif}}\\ is the highest for WDM low-mass haloes. For the high-mass bin, CDM and WDM haloes show very similar distributions of the two accretion rates. $\\ensuremath{\\Gamma_{\\rm mer}}$ is always positive except if there were no recent mergers, in which case $\\ensuremath{\\Gamma_{\\rm mer}}=0$ exactly. $\\ensuremath{\\Gamma_{\\rm dif}}$ can be negative if the total mass of progenitors is more than the mass of the descendant halo, which can happen due to tidal stripping; this is evidently a significant effect only for low-mass CDM haloes. Below, we will investigate in detail the redshift evolution and environmental trends of both \\ensuremath{\\Gamma_{\\rm dif}}\\ and \\ensuremath{\\Gamma_{\\rm mer}}.\n\nAs mentioned earlier, the dominant presence of spurious objects in the WDM simulation at low redshifts and masses $\\lesssim10^{10.4}\\ensuremath{h^{-1}M_{\\odot}}$ affects the measurement of WDM accretion rates. In particular, \\ensuremath{\\Gamma_{\\rm mer}}\\ is overestimated and \\ensuremath{\\Gamma_{\\rm dif}}\\ correspondingly underestimated due to the fact that mass is locked up in spuriously identified objects. However, the \\emph{amount} of mass in these low-mass spurious objects is not dramatically large. \nBy integrating the measured and power-law-corrected (see section~\\ref{subsec:massfn}) mass functions, weighted by mass, over the range $60\\,m_{\\rm p} \\leq M_{\\rm vir} \\leq 500\\,m_{\\rm p}$, we find that the contribution of spurious objects in this range is $\\simeq40\\%$ by mass at $z=0$ and falls to $\\lesssim4\\%$ by $z=1.5$. Moreover, we will see below that accretion due to mergers is subdominant in WDM in any case. We therefore tentatively conclude that, although the WDM merger accretion rates are likely to be systematically overestimated due to the presence of spurious objects, accounting for this spurious contribution is not expected to alter any of our qualitative conclusions. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth,trim= 2 0 20 20,clip]{.\/images\/alpha-distri.png}\n\\caption{{\\bf Local tidal anisotropy:} Normalised distributions of the tidal anisotropy $\\alpha$ (equation~\\ref{eq:alpha-def}) for low-mass \\emph{(left panel)} and high-mass \\emph{(right panel)} haloes in CDM (dashed curves) and WDM (solid curves) at four redshifts between $0\\leq z\\leq1.5$ (indicated by colours). Vertical line in each panel indicates $\\alpha=0.5$, with higher (lower) values corresponding to filamentary (node-like) environments. \n\\underline{\\emph{Highlights:}} Distributions for low-mass WDM haloes are systematically narrower than their CDM counterparts.}\n\\label{cdmwdm-alpha-4z}\n\\end{figure*}\n\nTo assess the effects of the potential caveats discussed above, namely, the effect of a fixed particle threshold versus a threshold in mass ratio for defining progenitors, and the effect of spurious WDM haloes, we have repeated our entire analysis using four different selection criteria. In the first modification, we replaced the fixed cut of $>60$ particles per progenitor with a lower threshold of $>30$ particles, which should emphasize the effects of spurious WDM haloes. In the second case, we repeated the analysis for low mass WDM only, with $>100$ particle progenitors, which should completely remove the contribution from spurious WDM haloes. In the third, we used $>60$ particles and imposed relaxation criteria $0.5 < \\eta < 1.5$ (which was not done in the default analysis) on the progenitors at $2T_{dyn}$. In the fourth, we used the 60 particle threshold for progenitors of the low-mass descendants, but a threshold of $M_{\\rm vir} > 4.84\\times10^{11}\\ensuremath{h^{-1}M_{\\odot}}$ for progenitors of high-mass descendants. This corresponds to $\\chi\\simeq0.08$ with respect to the median descendant mass in the high-mass bin, which is the same as the ratio between the 60 particle threshold and the median mass of the low-mass bin, and should emphasize any differences between environmental trends due to the different progenitor selection. It has been studied that \\citep{genel, WangJ11} with changing the mass ratio threshold of most massive progenitor to mergers, the merger accretion rates are found to change significantly. We find that \\emph{all our qualitative results remain unchanged in each case}, with only a few minor differences which we comment on later. This indicates that (a) the environmental trends we study are indeed largely insensitive to the choice of progenitor threshold mass and (b) spurious haloes in WDM \\emph{do not affect} the inference of the trends. All our results below will be quoted for the (more conservative) fixed threshold of $>60$ particles per progenitor (without imposing $0.5 < \\eta < 1.5$ criteria) for each mass bin.\n\n\n\n\\subsection{Quantifying local halo environment}\n\nWe quantify the \\emph{local} environment of haloes using scalars constructed from the smoothed halo-centric tidal tensor $T_{ij}(x) = \\partial_i \\partial_j \\Psi_R (x)$, where the Newtonian potential $\\Psi_R$ satisfies the normalised Poisson equation $\\nabla^2\\Psi_R = \\delta_R$ with $\\delta_R$ being the halo-centric dark matter overdensity Gaussian-smoothed on comoving scale $R$. \nIn practice, for any smoothing scale $R$, we invert the Poisson equation in Fourier space using cloud-in-cell (CIC) interpolation for the unsmoothed density on a $512^3$ grid. Namely, we Fourier transform the CIC overdensity to obtain $\\delta(\\mathbf{k})$, using which the tidal tensor is the inverse Fourier transform of $(k_ik_j\/k^2)\\delta(\\mathbf{k})\\e{-k^2R^2\/2}$.\n\n\nAs the two scalars of choice, we use the overdensity $\\delta_R$ itself, and the tidal anisotropy $\\alpha_R$ introduced by \\citet{phs18a}.\nIf the eigenvalues of $T_{ij}$ are denoted $\\lambda_1 \\leq \\lambda_2 \\leq \\lambda_3$, then we have\n\\begin{equation}\n\\delta_R = \\lambda_1 + \\lambda_2 + \\lambda_3 \n\\label{eq:delta-def}\n\\end{equation}\nand\n\\begin{equation}\n\\alpha_R = \\sqrt{q_{R}^2} \/ (1+\\delta_R)\n\\label{eq:alpha-def}\n\\end{equation}\nwhere $q_{R}^2$ is the tidal shear \\citep{heavens, catelan} defined as,\n\\begin{equation}\nq_{R}^2 = \\frac{1}{2} [(\\lambda_3 - \\lambda_1)^2 + (\\lambda_3 - \\lambda_2)^2 + (\\lambda_2 - \\lambda_1)^2]\\,.\n\\end{equation}\nFor the reasons discussed by \\citet{phs18a} and \\citet{rphs19}, we define the local halo environment at scales $R = 4R_{\\rm 200b}\/\\sqrt{5}$ for each halo. (In practice, we first evaluate the tidal tensor on the grid for a fixed set of smoothing scales and then interpolate to the adaptive scale and spatial position corresponding to each halo.) This adaptive filtering choice maximises the correlation between the local halo tidal environment and large-scale halo bias, useful for studies of assembly bias. Hereafter, for brevity we will write $\\alpha\\equiv\\alpha_{R}$ and $\\delta\\equiv\\delta_{R}$.\n\nA common approach to classifying cosmic web environment is by counting the signs of the eigenvalues of the tidal tensor or density Hessian defined at some fixed smoothing scale; with the tidal tensor we would have \\citep[e.g.,][]{hpcd07} $\\lambda_1 > 0$ : node, $\\lambda_1 < 0$ and $\\lambda_2 > 0$ : filament, $\\lambda_2 < 0$ and $\\lambda_3 > 0$ : sheet, $\\lambda_3 < 0$ : void.\nThe use of $\\alpha$ and $\\delta$ provides an alternative, continuous definition of halo environment adapted to the halo size. These variables, although correlated, are not completely degenerate \\citep[][see also below]{rphs19}. Such a continuous measure of environment is much better suited for studies of large-scale environmental correlations or assembly bias \\citep{phs18a,rphs19} as well as environmental trends in galaxy evolution \\citep{zphp20}. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth,trim= 2 0 16 20,clip]{.\/images\/delta-distri.png}\n\\caption{{\\bf Local overdensity:} Same as figure~\\ref{cdmwdm-alpha-4z}, showing results for halo-centric local overdensity $\\delta$ (equation~\\ref{eq:delta-def}).\n\\underline{\\emph{Highlights:}} As with $\\alpha$, the distributions of $\\delta$ for low-mass WDM haloes are systematically narrower than their CDM counterparts, and also have higher median values.}\n\\label{cdmwdm-delta-4z}\n\\end{figure*}\n\nPhysically, small values of the adaptively defined $\\alpha$ correspond to isotropic environments (nodes), while large values correspond to very anisotropic environments (filaments). As discussed by \\citet{phs18a} and \\citet{paranjape20}, the value $\\alpha=0.5$ forms a rather sharp threshold separating haloes in node-like and filament-like environments. We have checked that this is true at all the redshifts we study in this work. \n\nFigure \\ref{cdmwdm-alpha-4z} shows the distribution of $\\alpha$ at four redshifts for CDM (dashed) and WDM (solid) haloes in the low-mass \\emph{(left panel)} and high-mass bin \\emph{(right panel)}. The distribution of $\\alpha$ at fixed redshift depends on halo mass, with massive haloes having preferentially lower values of $\\alpha$ and low-mass haloes spanning a wide range of $\\alpha$. Haloes at higher redshifts also have preferentially lower values (as well as narrower distributions) of $\\alpha$, which is possibly mainly a consequence of mass resolution (since haloes at fixed mass are rarer in the past), but might also be partially reflecting a genuine evolution of the local cosmic web environment of haloes. \nFigure~\\ref{cdmwdm-delta-4z} is formatted identically to figure~\\ref{cdmwdm-alpha-4z} and shows the corresponding distributions of $\\delta$. At fixed redshift, we see that the distributions for low-mass haloes have wider tails than for high-mass haloes, at both low and high density. At fixed mass, high-redshift haloes span substantially narrower ranges of $\\delta$ than those at lower redshift.\nFor both $\\alpha$ and $\\delta$, the results for WDM and CDM at each redshift are very similar to each other for high-mass haloes, while the distributions for low-mass WDM haloes are narrower than their CDM counterparts (with the median $\\delta$ also being systematically higher for WDM). This extends the results of \\citet{phs18a}, who studied CDM haloes at $z=0$, to significantly higher redshift as well as WDM cosmologies.\n\n\n\\subsection{Large-scale environment: halo bias}\n\\label{subsec:bias}\nAs an indicator of the \\emph{large-scale} halo environment, we estimate the linear bias $b_1$ for each halo using the technique outlined by \\citet{phs18a} and \\citet{pa20}. This is essentially an object-by-object version of the usual cross-correlation definition of bias in Fourier space -- $P_{\\rm hm}(k)\/P_{\\rm mm}(k)$ -- averaged over low-$k$ modes using the weights discussed by \\citet{pa20}, where $P_{\\rm hm}$ and $P_{\\rm mm}$ are the halo-matter cross power spectrum and matter auto-power spectrum, respectively. \n\nThe box size of $150\\ensuremath{h^{-1}{\\rm Mpc}}$ leads to small-volume systematic effects in the absolute value of $b_1$ measured for each object, due to missing long-wavelength modes. Here, however, we are only interested in the correlation between $b_1$ and other halo properties like accretion rates and halo environment. As demonstrated by \\citet{rphs19}, these correlations are relatively insensitive to volume effects and we therefore expect our results to be robust to such systematics.\n\n\n\\section{Results and discussion}\n\\label{sec:results}\nWe now turn to our main results, starting with the environment-independent evolution of mass accretion, followed by trends with local environment and finally assembly bias.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth,trim= 2 0 20 20,clip]{.\/images\/fractionevolution.png}\n\\caption{{\\bf Mergers versus diffuse accretion:}\nEvolution with redshift of the fraction $f_{\\rm no\\,merge}$ of haloes with zero merger rate in the previous 2 dynamical times. Results for the low-mass (high-mass) bin are shown in blue (red), with the dashed (solid) curves showing results for CDM (WDM). \\underline{\\emph{Highlights:}} Low-mass WDM haloes accrete mainly diffuse mass at all epochs, while essentially every high-mass CDM halo has accreted some mass through mergers in the previous 2 dynamical times, except at very low redshift. } \n\\label{fraction}\n\\end{figure}\n\n\n\\subsection{Mass accretion: mergers vs. diffuse accretion}\n\\label{subsec:mergvsdiff}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth,trim= 2 0 20 20,clip]{.\/images\/MAH.png}\n\\caption{{\\bf Accretion history:} Evolution with redshift of the median values of the specific accretion rates \\ensuremath{\\Gamma_{\\rm dif}}\\ \\emph{(left panel)}, \\ensuremath{\\Gamma_{\\rm mer}} \\emph{(middle panel)} and \\ensuremath{\\Gamma_{\\rm tot}}\\ \\emph{(right panel)}, for low-mass (blue) and high-mass (red) haloes in CDM (dashed) and WDM (solid). Error bars were calculated by bootstrap sampling at each redshift.\n\\underline{\\emph{Highlights:}} \\ensuremath{\\Gamma_{\\rm dif}}\\ for low-mass WDM haloes is the highest of all categories. In the high-mass bin, although \\ensuremath{\\Gamma_{\\rm tot}}\\ is nearly identical between CDM and WDM, the relative contribution to this total from mergers versus diffuse accretion is higher in CDM than in WDM. \n} \n\\label{MAH}\n\\end{figure*}\n\n\nFigure \\ref{fraction} shows the evolving fraction $f_{\\rm no\\,merge}$ of haloes which do not accrete any mass through mergers in the last 2 dynamical times (i.e., haloes without any secondary progenitor larger than the 60 particle threshold, having $\\ensuremath{\\Gamma_{\\rm mer}}=0$), for CDM and WDM. High-mass CDM haloes essentially always had mergers, except at very low redshifts where a small fraction have evolved with $\\ensuremath{\\Gamma_{\\rm mer}}=0$. At fixed mass, the value of $f_{\\rm no\\,merge}$ is always larger in WDM than in CDM. This can be easily understood as being due to the lack of small-scale structure and hence fewer low-mass bound structures in WDM. In fact, we see that haloes in the low-mass WDM bin accrete mainly diffuse mass at all redshifts, with $f_{\\rm no\\,merge}\\gtrsim0.9$ at all $z$. This is consistent with corresponding results from \\citet{elahi}. \n\nFigure \\ref{MAH} shows the evolution of the median \\ensuremath{\\Gamma_{\\rm dif}}\\ and \\ensuremath{\\Gamma_{\\rm mer}}\\ (\\emph{left} and \\emph{middle panels}, respectively) with the total accretion rate \\ensuremath{\\Gamma_{\\rm tot}}\\ shown in the \\emph{right panel}.\\footnote{The trend in the evolution is observed to change near $z\\sim1.5$. This trend at high redshift could not be verified with the analytical model and deserves further study.} Error bars on the measurements were calculated by bootstrap sampling: at each redshift and for each mass bin, the accretion rate data are sampled with repetition a number of times and the standard deviation in the median values of each sample gives the value of error. \nWe see that WDM high-mass haloes have nearly the same \\emph{total} accretion rate as CDM high-mass haloes, at all epochs \\citep[consistent with the earlier results by][]{knebe02, benson+13}. WDM low-mass haloes, on the other hand, have higher total accretion rates than, both, CDM low-mass haloes and also WDM (and CDM) high-mass haloes.\n\nUpon splitting the accretion rate between mergers and diffuse accretion, we see that \ndiffuse accretion dominates the accretion budget at all redshifts, in each mass bin for both CDM and WDM.\nComparing low-mass and high-mass results for diffuse accretion, in CDM we see that low-mass haloes have lower \\ensuremath{\\Gamma_{\\rm dif}}\\ than high-mass haloes, while the opposite is true for WDM (see also figure~\\ref{Mrates-distri}). For accretion by mergers, on the other hand, high-mass haloes have higher \\ensuremath{\\Gamma_{\\rm mer}}\\ than low-mass haloes in both CDM and WDM.\n\nAnd, interestingly, while the high-mass total accretion rates in CDM and WDM are nearly identical, the relative contribution to this total from mergers versus diffuse accretion is higher in CDM than in WDM.\n\nThese results are all consistent with an overall picture in which the arrested growth of small-scale structure in WDM inhibits mass accretion through mergers, as compared to CDM. The fact that differences between WDM and CDM accretion rates are dramatic at low masses is not surprising, considering that our low-mass bin is at slightly smaller mass than the half-mode mass for our WDM cosmology (see section~\\ref{sec:analytical} for analytical insights into this behaviour). The fact that \\emph{high-mass} haloes also show differences between WDM and CDM can be attributed to the difference in available substructure for such haloes in the two cosmologies. In the following sections, we explore the relation between these accretion rates and the local environment of haloes.\n\n\n\n\\subsection{Environments of haloes accreting with and without mergers}\n\\label{subsec:medianenv}\n\nTo start with, we focus on whether the environmental variables $\\alpha$ and $\\delta$ evolve differently on average for haloes that add mass with and without mergers, in both CDM and WDM cosmologies. \nFigure \\ref{median-evoln} shows the median $\\alpha$ \\emph{(left panel)} and $\\delta$ \\emph{(right panel)} for haloes having zero and non-zero merger rates in the low-mass (blue, cyan) and high-mass (red, orange) bins. The solid (dashed) lines show results for WDM (CDM). Error bars on the measurements were calculated by bootstrap sampling at each redshift. Several interesting trends are apparent, as we discuss next.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.85\\textwidth,trim= 2 2 10 20,clip]{.\/images\/med_alphaevolution.png}\n\\caption{{\\bf Local environments with and without mergers:} Evolution of median local tidal anisotropy $\\alpha$ \\emph{(left panel)} and local overdensity $\\delta$ \\emph{(right panel)} of haloes selected to have grown without mergers (blue, red) and with mergers (cyan, orange) in the previous 2 dynamical times, for the low-mass (cooler colours) and high-mass (warmer colours) bin. Results for CDM (WDM) are shown using dashed (solid) curves. Error bars were calculated by bootstrap sampling at each redshift.\n\\underline{\\emph{Highlights:}} While most of these trends conform to expectations based on the preponderance of small-scale bound structures in CDM as compared to WDM, two trends are noteworthy: (i) low-mass CDM haloes growing without mergers are in slightly more \\emph{anisotropic} environments compared to low mass CDM haloes growing with mergers and (ii) low-mass WDM haloes are in denser environments than other categories (see also figure~\\ref{cdmwdm-delta-4z}). \n} \n\\label{median-evoln}\n\\end{figure*}\n\n\nOverall, we see that the tidal environments of objects at any fixed mass are typically more anisotropic at later times, while most local densities evolve moderately or decrease at later times. This is broadly consistent with the growth of the cosmic web such that a larger fraction of haloes at fixed mass find themselves in filamentary or sheet-like environments at late times.\n\nComparing the trends for low-mass and high-mass objects, we see for CDM that high-mass objects (dashed, warm colours) are in relatively denser and more isotropic environments than low-mass objects (dashed, cool colours), at any redshift. This is fully consistent with the standard hierarchical picture in which massive haloes are more clustered and dominate their tidal environments more than low-mass haloes. \n\nIn WDM, while massive haloes do live in more isotropic environments than their low-mass counterparts (solid warm versus solid cool in left panel), unlike CDM, now the \\emph{low-mass} haloes are in denser environments than high-mass ones (same in right panel, see also figure~\\ref{cdmwdm-delta-4z}). These low-mass WDM environments are also more overdense than the corresponding low-mass CDM environments.\\footnote{We have also checked that the median $b_1$ values for various halo categories are nearly identical between CDM and WDM, and conform to the usual expectation of low-mass objects being less biased than high-mass ones.} We return to this point later.\n\n\nWe focus next on the \\emph{high-mass bin} where one might expect similarities between CDM and WDM haloes (although see section~\\ref{subsec:mergvsdiff}). Indeed, we see that haloes with mergers occupy nearly identical median density and tidal environments at all redshifts in CDM and WDM (compare the dashed versus solid orange curves in each panel). And, within errors, this is also true for haloes without recent mergers (red curves).\n\nThese high-mass environments tend to be the most isotropic of all (c.f. figure~\\ref{cdmwdm-alpha-4z}) while having intermediate densities, at almost any redshift.\nThus, most of the differences between high-mass CDM and WDM haloes are related to the lack of substructure and corresponding dominance of diffuse accretion in WDM seen in figure~\\ref{MAH}. \n\nIn the \\emph{low-mass bin}, the results are more nuanced, with several differences between zero and non-zero merger environments in CDM and WDM which can be summarized as follows:\n\\begin{itemize}\n \\item The environments of low-mass CDM haloes with and without mergers have similar densities at all redshifts (dashed cyan versus dashed blue in the right panel), but are systematically more anisotropic at low redshift for haloes without mergers at any redshift (same in the left panel). We discuss this further below. \n \\item In contrast, low-mass WDM haloes without mergers (solid blue) live in systematically less dense and more \\emph{isotropic} environments than those with mergers (solid cyan). This is sensible, since isolated, isotropic environments would allow diffuse accretion to dominate over mergers. \n \\item Low-mass WDM haloes without mergers (solid blue) live in more dense environments (with similar tidal anisotropy) than their CDM counterparts (dashed blue).\n This is consistent with an enhancement of diffuse accretion in WDM; the lack of substructure in WDM compared to CDM means that environments conducive to purely diffuse accretion (i.e., no recent mergers) can be denser in WDM than in CDM. \n \\item In contrast, low-mass WDM haloes with mergers (solid cyan) live in substantially more dense and more \\emph{anisotropic} environments than their CDM counterparts (dashed cyan). We discuss this below. \n\\end{itemize}\n\nThese low-mass results can be mostly understood keeping in mind the preponderance of small objects in WDM as compared to CDM, as well as the facts that a higher local tidal anisotropy would typically inhibit the accretion of diffuse mass \\citep[e.g., by redirecting local flows towards nodes; see][]{bprg17}, while higher densities would enhance mergers. Some of these trends, however, deserve more careful consideration. For example,\nthe CDM haloes without mergers are in more \\emph{anisotropic} environments than those with mergers, contrary to what is suggested by the argument above. Similarly, the higher anisotropy of low-mass WDM haloes with mergers as compared to their CDM counterparts is intriguing as well.\nOn the other hand, the higher \\emph{density} of low-mass WDM haloes with mergers as compared to all other categories might be understood as being due to the substantially decreased occurrence of mergers in the smooth WDM cosmic web, with only the highest density environments being capable of sustaining merger events.\nIn the next section, we will explore in more detail the correlations between environment and mass accretion rate, which will shed further light on the nature and origin of these trends. \n\n\n\n\\subsection{Correlation between mass accretion rates and local halo environment}\n\\label{subsec:mdot<->env}\n\nTo quantify the dependence of mass accretion rates on environment, in this section we study the evolving correlations between our environmental proxies $\\alpha$ and $\\delta$, and the mass accretion rates due to diffuse accretion (\\ensuremath{\\Gamma_{\\rm dif}}) and mergers (\\ensuremath{\\Gamma_{\\rm mer}}). \n\n\nFollowing \\citet{rphs19}, we calculate Spearman rank correlation coefficients between the environmental variables and accretion rates; namely, in each mass bin and for every redshift, we calculate the coefficients $\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}}$, $\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}}$, $\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}}$, $\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}}$, as well as the coefficient $\\alpha\\leftrightarrow\\delta$ for reference \\citep[which was also studied by][for $z=0$ CDM haloes]{rphs19}. In order to assess which, if any, of the two environmental proxies is more important, we also compute \\emph{conditional coefficients}. For stochastic variables $a$, $b$ and $c$, the coefficient between $a$ and $b$, conditioned on $c$, is given by \n\\begin{equation}\n(a\\leftrightarrow b)|c = (a\\leftrightarrow b) - (a\\leftrightarrow c)(b\\leftrightarrow c)\\,. \n\\label{eq:conditional-cc}\n\\end{equation}\nIf the variables were Gaussian distributed with a joint distribution that had the structure $p(a,b,c) = p(a|c)p(b|c)p(c)$, then the variables $a$ and $b$ would be conditionally independent and we would have $(a\\leftrightarrow b)|c = 0$ although $(a\\leftrightarrow b)$ need not be zero. We point the reader to \\citet{rphs19} for a detailed justification for constructing conditional coefficients using \\eqn{eq:conditional-cc} with Spearman rank coefficients, even for variables that are \\emph{not} Gaussian distributed, as is the case here (see also below).\n\nFigure \\ref{CDM-spear} (figure \\ref{WDM-spear}) shows the correlation coefficients for CDM (WDM). In each case, in the \\emph{left (right) panel} we display correlations between the environment and \\ensuremath{\\Gamma_{\\rm dif}}\\ (\\ensuremath{\\Gamma_{\\rm mer}}) as a function of redshift.\\footnote{For this correlation analysis, we only evaluate \\ensuremath{\\Gamma_{\\rm mer}}\\ using those haloes that had mergers in the previous 2 dynamical times, i.e., objects having $\\ensuremath{\\Gamma_{\\rm mer}}\\neq0$. \\ensuremath{\\Gamma_{\\rm dif}}\\ on the other hand, is evaluated using all haloes in the bin.} Additionally, the \\emph{left panel} in each case also shows the evolving correlation $(\\alpha\\leftrightarrow\\delta)$. Results for the low-mass (high-mass) bin are displayed using cool (warm) colours. The solid curves show the primary correlations such as $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ while the dotted curves show conditional coefficients such as $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})|\\delta$. Error bars on the measurements were calculated by bootstrap sampling at each redshift. This represents the first detailed study in the literature, of the evolving correlation between local environment and mass accretion rates in either CDM or WDM cosmologies.\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=0.85\\textwidth,trim= 2 2 30 20,clip]{.\/images\/spearman_cdm.png}\n\\caption{{\\bf Correlations between accretion rates and local environment (CDM haloes):} Spearman rank correlation coefficients between local environmental variables $\\alpha$, $\\delta$ and specific accretion rates \\ensuremath{\\Gamma_{\\rm dif}}\\ \\emph{(left panel)} and \\ensuremath{\\Gamma_{\\rm mer}}\\ \\emph{(right panel)}, as a function of redshift. In each panel, cool (warm) colours indicate results for low-mass (high-mass) haloes. Solid curves show the correlations $(\\Gamma\\leftrightarrow\\alpha)$ (red, blue) and $(\\Gamma\\leftrightarrow\\delta)$ (orange, cyan) and, in the \\emph{left panel}, $(\\alpha\\leftrightarrow\\delta)$ (brown, black). Dotted curves show conditional coefficients $(\\Gamma\\leftrightarrow\\alpha)|\\delta$ and $(\\Gamma\\leftrightarrow\\delta)|\\alpha$, calculated using \\eqn{eq:conditional-cc}. \n\\underline{\\emph{Highlights:}} \n(i) There are significant anti-correlations $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\delta)$ and $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\alpha)$ in both mass bins (left panel, solid curves). \n(ii) \\ensuremath{\\Gamma_{\\rm mer}}\\ correlates positively (negatively) with $\\delta$ ($\\alpha$) in the high-mass bin (right panel, solid orange and red), and does not correlate with either in the low-mass bin (right panel, solid cyan and blue).\n(iii) The conditional coefficient $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\alpha)|\\delta$ for high-mass haloes (left panel, dotted red) almost vanishes at all $z$, indicating that $\\delta$ almost fully explains the environment dependence of diffuse accretion in high-mass CDM haloes. \n} \n\\label{CDM-spear}\n\\end{figure*}\n\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=0.85\\textwidth,trim= 2 2 30 20,clip]{.\/images\/spearman_wdm.png}\n\\caption{Same as figure \\ref{CDM-spear}, for {\\bf WDM haloes}. \n\\underline{\\emph{Highlights:}} (i) The coefficient $(\\ensuremath{\\Gamma_{\\rm mer}}\\leftrightarrow\\alpha)$ for low-mass haloes is, surprisingly, positive (right panel, solid blue) and the corresponding conditional coefficient $(\\ensuremath{\\Gamma_{\\rm mer}}\\leftrightarrow\\delta)|\\alpha$ (right panel, dotted cyan) is close to zero. \n(ii) The conditional coefficient $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\delta)|\\alpha$ for low-mass haloes nearly vanishes (left panel, dotted cyan), even though the primary correlation $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\delta)$ (solid cyan) is significantly non-zero. \nThe same is true for the conditional coefficient $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\alpha)|\\delta$ for high-mass haloes (left panel, dotted red).\nThus, $\\alpha$ ($\\delta$) almost completely explains the environment dependence of diffuse accretion for low-mass (high-mass) WDM haloes. \n} \n\\label{WDM-spear}\n\\end{figure*}\n\n\nWe see that $\\alpha$ and $\\delta$ are always positively correlated for both CDM and WDM, with a strength that is relatively independent of halo mass and that only weakly depends on redshift and dark matter type. This extends the results of \\citet{rphs19} (who studied CDM haloes at $z=0$) to significantly higher redshift, as well as to WDM cosmologies. The remaining correlations, which depend only weakly on redshift, can be summarized as follows.\n\n\n\n\\begin{itemize}\n \\item \\underline{\\emph{High-mass bin:}} \n \\begin{itemize}\n \\item \n For both CDM and WDM, we see a strong negative correlation $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$, and a weaker but significant positive correlation $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$, at all redshifts (solid orange). This is consistent with the expectation that overdense environments enhance mergers while underdense regions allow for more diffuse accretion \\citep{fm10}. \n \n \\item The corresponding primary correlations $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ and $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ (solid red) are all negative, with magnitudes comparable to those in the case of $\\delta$, for both CDM and WDM. Interestingly, however, the \\emph{conditional} coefficient $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\delta$ (dotted red in left panels) for both CDM and WDM almost vanishes at all $z$, while the corresponding conditional coefficient $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})|\\delta$ does not vanish (dotted red in right panels).\\footnote{When using a mass threshold $M_{\\rm vir}>4.84\\times10^{11}\\ensuremath{h^{-1}M_{\\odot}}$ for defining progenitors (see section~\\ref{subsec:Mdot-measure}), the high-mass negative $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ correlation becomes weaker, and in fact slightly positive at low redshift. This is the only qualitative difference we find between different progenitor definitions.} \\emph{Thus, for high-mass haloes, $\\alpha$ and $\\delta$ are equally important for accretion via mergers, while $\\delta$ almost completely explains the environment dependence of diffuse accretion.}\n \n \\end{itemize}\n \\item \\underline{\\emph{Low-mass bin:}}\n \\begin{itemize}\n \\item In low-mass CDM haloes, \\ensuremath{\\Gamma_{\\rm mer}}\\ shows almost no correlation with either $\\alpha$ or $\\delta$ at any redshift, consistent with similar weak correlations seen previously using other environmental proxies \\citep[c.f. figure 4 of][]{fm10}.\n \n \\item The situation changes in WDM: we now see \\emph{positive} correlations $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ and $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$, comparable in strength to each other and increasing at higher redshift. Moreover, at least at low redshift, we see that the conditional coefficient $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})|\\alpha$ (dotted cyan) is close to zero. In fact, this is the case within errors at all $z\\lesssim1.5$. \\emph{Thus, $\\alpha$ mostly accounts for the environment dependence of \\ensuremath{\\Gamma_{\\rm mer}}\\ at nearly all redshifts for low-mass WDM haloes, with \\ensuremath{\\Gamma_{\\rm mer}}\\ being enhanced in more anisotropic tidal environments.} \n \n \n \\item For \\ensuremath{\\Gamma_{\\rm dif}}\\ in low-mass CDM haloes, both $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ and $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ are negative, with comparable magnitudes (solid cyan and blue). Also, neither of the conditional coefficients $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\delta$ and $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\alpha$ is close to zero (dotted cyan and blue). Thus, both $\\alpha$ and $\\delta$ play a role for \\ensuremath{\\Gamma_{\\rm dif}}, consistent with the expectation that diffuse accretion should be more efficient far from crowded regions, i.e. in underdense, isotropic environments in CDM.\n \n \\item For \\ensuremath{\\Gamma_{\\rm dif}}\\ in low-mass WDM haloes, on the other hand, we see a stronger $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ anticorrelation (solid blue), which almost completely explains the $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ (solid cyan) anticorrelation, seen as the near-vanishing of $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\alpha$ (dotted cyan) at all redshifts. \\emph{Thus, the environment dependence of \\ensuremath{\\Gamma_{\\rm dif}}\\ in low-mass haloes is related to both $\\alpha$ and $\\delta$ for CDM and almost completely explained by $\\alpha$ for WDM.}\n The dominant role played by the tidal environment for low-mass WDM haloes is consistent with previous simulation results \\citep{angulo} which indicate that these objects form in a well-established tidal environment which can then further affect their mass accretion.\n \n \\end{itemize}\n\\end{itemize}\n\n\nThe nature of the environmental correlations discussed above reveals a complex interplay between mass accretion rates and local environment. While this is perhaps not surprising, considering the intimate connection between mass accretion in the halo outskirts and the internal structure of haloes \\citep[e.g.,][]{dk14,mdk15}, our results above show several interesting aspects. In section~\\ref{subsec:medianenv}, for example, we saw that low-mass WDM haloes with mergers live in systematically more anisotropic environments than their CDM counterparts, a trend consistent with the correlations seen above in which $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ is positive for WDM but negative for CDM. This is counter-intuitive, considering that accretion due to mergers is enhanced in overdense, isotropic environments in CDM, and one might expect this trend to be even more pronounced in WDM which has less small-scale structure available in all environments. \nMoreover, the small magnitude of $(\\delta\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})|\\alpha$ for these low-mass WDM haloes shows that $\\alpha$, in fact, dominates the environmental trends for \\ensuremath{\\Gamma_{\\rm mer}}. This connection between mergers and the local tidal environment of low-mass WDM haloes deserves further study.\n\n\n\nAnother puzzling result from section~\\ref{subsec:medianenv} was that the environments of low-mass CDM haloes without mergers are more anisotropic than of those with mergers. In this case as well, the correlation results above do not particularly clarify the situation, since they are consistent with \\ensuremath{\\Gamma_{\\rm dif}}\\ being enhanced in underdense, isotropic environments (with both $\\delta$ and $\\alpha$ playing comparable roles) and \\ensuremath{\\Gamma_{\\rm mer}}\\ being nearly uncorrelated with environment. We have checked that these trends are not restricted to the median $\\alpha$ alone; rather, the entire distributions of $\\alpha$ in such haloes are shifted relative to each other. \n\n\n\nAs a check on systematic errors due to our use of correlation coefficients and \\eqn{eq:conditional-cc}, we repeated the analysis by measuring the median accretion rates \\ensuremath{\\Gamma_{\\rm dif}}\\ and \\ensuremath{\\Gamma_{\\rm mer}}\\ in narrow bins of $\\alpha$ and $\\delta$. The results, although noisy, are fully consistent with the conclusions above. \n\n\n \n\n\\subsection{Accretion rate assembly bias}\n\\label{subsec:assemblybias}\nIn this section, we explore the nature of the correlations between mass accretion rate and the \\emph{large-scale} halo environment (characterised by halo bias $b_1$; see section~\\ref{subsec:bias}) in fixed mass bins, also known as halo assembly bias or secondary bias. \n\nPrevious studies have largely focused on secondary halo variables such as age, concentration, shape, angular momentum and velocity dispersion structure (see the Introduction for references). To our knowledge, the only previous work that studied the assembly bias of mass accretion rates was by \\citet{lazeyras}, who focused on the total mass accretion rates of high-mass CDM haloes at $z=0$. Our analysis below therefore substantially extends these results in terms of halo mass, redshift range and dark matter type.\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=0.85\\textwidth,trim= 2 2 20 20,clip]{.\/images\/spearman-bias_cdm.png}\n\\caption{\n{\\bf Accretion rate assembly bias (CDM haloes):} Spearman rank correlation coefficients between large-scale halo bias $b_1$ and specific accretion rates \\ensuremath{\\Gamma_{\\rm dif}}\\ \\emph{(left panel)} and \\ensuremath{\\Gamma_{\\rm mer}}\\ \\emph{(right panel)} for low-mass (blue) and high-mass (red) CDM haloes, as a function of redshift. Solid curves show the correlations $(\\Gamma\\leftrightarrow b_1)$ and, in the \\emph{left [right] panel}, green curves show $(b_1\\leftrightarrow\\alpha)$ [$(b_1\\leftrightarrow\\delta)$], with light (dark) green for high-mass (low-mass) haloes. Dashed and dotted curves respectively show conditional coefficients $(\\Gamma\\leftrightarrow b_1)|\\alpha$ and $(\\Gamma\\leftrightarrow b_1)|\\delta$, calculated using \\eqn{eq:conditional-cc}. Error bars were estimated using bootstrap sampling at each redshift. \\underline{\\emph{Highlights:}} There is a significant negative correlation $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow b_1)$ for low-mass haloes at all redshifts (left panel, solid blue), which is largely explained by the $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\alpha)$ correlation (conditional coefficient shown by the dashed blue curve is close to zero). \n} \n\\label{CDM-spearbias}\n\\end{figure*}\n\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=0.85\\textwidth,trim= 2 2 20 20,clip]{.\/images\/spearman-bias_wdm.png}\n\\caption{Same as figure \\ref{CDM-spearbias}, for {\\bf WDM haloes}. \\underline{\\emph{Highlights:}} There is a significant negative correlation $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow b_1)$ for low-mass haloes (left panel, solid blue), which increases in magnitude at low redshift, is stronger than the corresponding correlation for CDM haloes and is mostly explained by the $(\\ensuremath{\\Gamma_{\\rm dif}}\\leftrightarrow\\alpha)$ correlation (conditional coefficient shown by the dashed blue curve is close to zero).\n} \n\\label{WDM-spearbias}\n\\end{figure*}\n \n\nWe define the assembly bias of mass accretion as the two correlations $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ and $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ measured using Spearman rank correlation coefficients in different mass bins for all redshifts. The results of \\cite{rphs19} indicate that, at $z=0$ in CDM, the assembly bias of each of the internal properties halo concentration, spin, shape and velocity dispersion can be mostly attributed to two fundamental correlations, one between $b_1$ and $\\alpha$ and the other between $\\alpha$ and the halo internal property. Considering that there are relatively strong correlations between the mass accretion rates \\ensuremath{\\Gamma_{\\rm mer}}, \\ensuremath{\\Gamma_{\\rm dif}}\\ and the local environmental proxies $\\alpha$ and $\\delta$ (see figures~\\ref{CDM-spear} and~\\ref{WDM-spear}), it is interesting to ask (a) what is the overall strength of assembly bias with mass accretion rate in comparison to other internal halo properties, and (b) what role, if any, do the local environmental proxies play in explaining these trends?\n\n\nFigure \\ref{CDM-spearbias} (figure \\ref{WDM-spearbias}) shows the results for CDM (WDM) haloes. As with figures~\\ref{CDM-spear} and~\\ref{WDM-spear}, the \\emph{left (right) panels} show results for \\ensuremath{\\Gamma_{\\rm dif}}\\ (\\ensuremath{\\Gamma_{\\rm mer}}), with the solid (dotted) curves showing primary (conditional) coefficients. Additionally, the \\emph{left (right) panels} show the correlation $(b_1\\leftrightarrow\\alpha)$ ($(b_1\\leftrightarrow\\delta)$) for comparison.\n\nSimilarly to the results for $(\\alpha\\leftrightarrow\\delta)$ in figures~\\ref{CDM-spear} and~\\ref{WDM-spear}, we see that the correlations $(b_1\\leftrightarrow\\alpha)$ and $(b_1\\leftrightarrow\\delta)$ are significantly positive, very similar across halo mass as well as between CDM and WDM, and evolve moderately between $z=2$ and $z=0$. The $(b_1\\leftrightarrow\\alpha)$ correlation at any redshift is $\\sim50\\%$ larger than the corresponding $(b_1\\leftrightarrow\\delta)$ correlation, for both CDM and WDM. This extends the $z=0$ CDM results of \\citet{phs18a} and \\citet{rphs19} to significantly higher redshifts and to WDM cosmologies.\n\nIn the \\emph{high-mass bin}, the correlations $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ and $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ are relatively weak at nearly all redshifts for both CDM and WDM. The exception is at the highest redshift $z=2$ for WDM, where $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ is negative while $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ is positive, but with large errors. Overall, therefore, we conclude that high-mass objects in CDM and WDM do not show significant assembly bias for either mass accretion rate.\n\nIn the \\emph{low-mass bin}, we see more interesting results. Although the $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ correlation is essentially zero across all redshifts for both CDM and WDM, the same is not true for the $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ correlation, which remains significantly negative at nearly all redshifts, in both CDM and WDM. In fact, the strength of the correlation at $z=0$, namely $|(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\gtrsim 0.05\\, (0.1)$ for CDM (WDM), is large and comparable to the strongest correlations seen for any secondary variable, at this halo mass. \\citep[see, e.g., in the middle panel of figure 2 in ][]{rphs19}. If we extrapolate the results in \\citep{rphs19} to the low masses $(1.2 \\times 10^{11} \\ensuremath{h^{-1}M_{\\odot}} < M_{vir} < 3.85 \\times 10^{11} \\ensuremath{h^{-1}M_{\\odot}})$ we study in this paper, we find $b_1 \\leftrightarrow c$ correlation to be $ \\sim 0.07(0.09)$ for CDM(WDM).\nThe magnitude of the correlation is nearly constant with redshift for CDM while showing an increasing trend with time for WDM.\n\\emph{Thus, the assembly bias of low-mass accretion rates, especially at low redshift, is a comparatively large effect, entirely driven by diffuse accretion.} \n\nMore interestingly, the conditional coefficient $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\alpha$ for low-mass haloes is close to zero at all redshifts, for both CDM and WDM. (For CDM, the coefficient $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\delta$ also has a small magnitude, but is not as close to zero as $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})|\\alpha$.) \n\n\\emph{Thus, the local tidal anisotropy plays a key role in explaining the accretion rate assembly bias of both CDM and WDM haloes}, with local overdensity also playing a significant role in CDM. \nThese findings are consistent with previous results in the literature \\citep[e.g.,][]{dalal, hpdc09,bprg17} which show that low-mass haloes which are in the vicinity of massive haloes, and are hence highly clustered at large-scales while residing in anisotropic local tidal environments, show suppressed (diffuse) accretion rates.\n\n\n\n\\section{Analytical insights}\n\\label{sec:analytical}\nIn this section, we discuss the numerical results presented above in the language of the excursion set approach \\citep{bcek91,lc93,lc94,bm96}. Below, we briefly recapitulate the basic concepts underlying the excursion set approach and its modern variants, before discussing our results in this context.\n\n\n\\subsection{Recap of the excursion set (peaks) approach}\n\nIn the excursion set framework, halo abundances, large-scale clustering and accretion rates are estimated by identifying and counting likely locations of virialisation identified in the \\emph{initial} stochastic density field (linearly extrapolated to present epoch). A key ingredient in the traditional excursion set approach is a choice of collapse threshold or `barrier' $B$ which must be crossed by random walks in the linearly extrapolated density field $\\delta_R$ as a function of Lagrangian smoothing scale $R$. The barrier $B$ is usually adopted from spherical \\citep{gg72} or ellipsoidal \\citep{smt01} collapse models, and depends on the redshift $z$ of interest and possibly additional stochastic variables \\citep[such as those related to tidal effects in the ellipsoidal model, see][]{cphs17}. \n\n\nThe statistical properties of the random walks, on the other hand, are determined by the choice of initial matter power spectrum and a smoothing filter, which fixes the functional relation $\\sigma^2(m)$ where $\\sigma^2$ is the variance of the linearly extrapolated density contrast smoothed on a Lagrangian scale corresponding to mass $m\\sim R^3$. \nIf a random walk first upcrosses the barrier $B(z)$ at scale $R(m)$ (starting from $R\\to\\infty$ or $\\sigma^2\\to0$), then a halo of mass $m$ is declared to form at redshift $z$.\nWhile early work on the subject focused on a filter that is sharp in Fourier space (for reasons of analytical simplicity), later developments have shown how to efficiently analyze the effects of more realistic filters that are compact in real space and lead to random walks with correlated steps as $R$ is varied (\\citealp{ms12}; see also \\citealp{bcek91,zentner07,pls12}). Finally, the recognition that the sites of virialisation are special \\citep{smt01} -- e.g., peaks in the linearly extrapolated density field \\citep{bbks86} -- leads to an `excursion set peaks' approach \\citep[ESP,][]{ps12b,aj90} which has been shown to agree with simulated halo mass functions and large-scale clustering in CDM cosmologies at the $\\sim10\\%$ level \\citep{psd13,cphs17}. \n\nThe ESP framework has also been studied in the context of WDM cosmologies by \\citet[][hereon, HP14]{hp14}, who demonstrated that the suppression of small-scale power in these models is an excellent diagnostic tool for testing the assumptions underlying the excursion set (peaks) approach. In particular, HP14 argued that a single-barrier framework is insufficient to properly explain halo abundances due to small but systematic errors in the ellipsoidal collapse model \\citep{Monaco99,gmst07,lbp14}, which become dramatically amplified in WDM models as compared to CDM. Here, however, we are interested in the prediction \\citep{lc93,lc94} that mass accretion rates are intimately connected with the slope of the initial power spectrum near the scales of interest, with steeper spectra leading to higher accretion rates. The consequences of the latter effect for WDM can be understood without delving into the details of any specific excursion set model and only depend on the correlated nature of the random walks, as we discuss next.\n\n\n\\subsection{Accretion rates in the ESP framework}\n\nIn the following, in addition to the barrier $B$ for linearly extrapolated density fluctuations and the $\\sigma(m)$ relation mentioned above, we will need two more quantities from the excursion set lexicon. The first is the proto-halo `significance' $\\nu(m,z) = \\delta_{\\rm c}(z)\/\\sigma(m)$, where $\\delta_{\\rm c}(z) \\propto 1\/D(z)$ is the collapse threshold from the spherical model, with $D(z)$ being the linear theory growth factor. The second is the typical peak curvature $\\avg{x|\\nu}$ at scale $\\nu(m,z)$, where $x=-\\nabla^2\\delta_R\/\\sqrt{{\\rm Var}(\\nabla^2\\delta_R)}$ \\citep[see figure 6 of][]{bbks86}.\n\nAs mentioned above, accretion rates in CDM and WDM are expected to be very different due to the comparative steepness of the \\emph{initial} matter power spectrum in WDM around the half-mode mass scale. To see what this implies, we must mainly keep in mind that the $\\sigma(m)$ relation becomes `stretched out' for a truncated power spectrum such as WDM, with $\\sigma(m)\\to$ a constant below the half-mode mass (see, e.g., figure 2 of HP14). In CDM, on the other hand, $\\sigma(m)$ continues to increase down to very small masses. A simple calculation now shows that the diffuse mass accretion rate can be written as \\citep[e.g.,][]{lazeyras}\n\n\\begin{align}\n \\Gamma = \\frac{\\ensuremath{{\\rm d}}\\ln M}{\\ensuremath{{\\rm d}}\\ln a} &= \\frac{1}{\\sigma}\\,\\frac{1}{\\ensuremath{{\\rm d}} B\/\\ensuremath{{\\rm d}}\\sigma} \\frac{\\ensuremath{{\\rm d}} B\/\\ensuremath{{\\rm d}}\\ln a}{\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M} \\notag\\\\\n &\\approx \\frac{1}{\\sigma}\\,\\frac{\\gamma}{\\avg{x|\\nu}}\\,\\frac{|\\ensuremath{{\\rm d}} B\/\\ensuremath{{\\rm d}}\\ln a|}{|\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M|}\\,,\n \\label{eq:hp14-accrate}\n\\end{align}\nwhere the first line is a manipulation of variables and the second line relates the slope at barrier crossing, $\\ensuremath{{\\rm d}} B\/\\ensuremath{{\\rm d}} \\sigma$, to the peak curvature $\\avg{x|\\nu}$ \\citep{ms12,ps12b}, with $\\gamma$ being a spectral variable of order unity in both CDM and WDM ($\\gamma\\simeq0.5$ for our cosmology).\n\nAt high masses, for both CDM and WDM, $\\avg{x|\\nu}$ is large, $\\sigma$ is small and $|\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M|$ is finite. At masses smaller than the half-mode mass for WDM, $\\avg{x|\\nu}$ is smaller than at high masses, $\\sigma$ is \\emph{constant} and $|\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M|$ approaches zero. This last feature makes the accretion rates very high. At low masses for CDM, on the other hand, $\\avg{x|\\nu}$ is similar to that in WDM, $\\sigma$ is significantly \\emph{larger} than at high masses and $|\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M|$ is finite, so that low-mass CDM accretion rates remain small. This qualitatively explains the overall trends seen in the right panel of figure~\\ref{MAH}.\n\nQuantitatively, in our high-mass bin at $z=0$, for both CDM and WDM we have $\\sigma\\lesssim1.5$, $\\avg{x|\\nu}\\gtrsim3$ and $|\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M|\\lesssim0.2$. Assuming that the time dependence of the barrier is completely determined by $\\delta_{\\rm c}(z)$, we also have $|\\ensuremath{{\\rm d}} B\/\\ensuremath{{\\rm d}}\\ln a|\\sim1$ at $z=0$. Equation~\\eqref{eq:hp14-accrate} then predicts high-mass accretion rates of $\\Gamma\\sim0.5$ at $z=0$. \nIn the low-mass bin, for CDM, we have $\\sigma\\sim2.5$, $\\avg{x|\\nu}\\sim2.5$ and $|\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M|\\lesssim0.15$. For WDM, on the other hand, while $\\sigma$ and $\\avg{x|\\nu}$ are not very different from CDM for our cosmology and choice of mass bin, we have $|\\ensuremath{{\\rm d}}\\ln\\sigma\/\\ensuremath{{\\rm d}}\\ln M|\\lesssim0.075$, a factor 2 smaller than in CDM. Thus low-mass accretion rates at $z=0$ are predicted to be about a factor 2 higher for our WDM low-mass bin than the corresponding CDM values, which agrees with the trend seen in the right panel of figure~\\ref{MAH}. The values predicted, $\\Gamma\\sim0.5\\,(1.0)$ for CDM (WDM), are higher than the measured ones, which is perhaps not surprising considering the approximate nature of our calculation.\\footnote{We have also ignored the issue of mass reassignment discussed by HP14, in which the $\\sigma(m)$ relation must effectively be modified for collapsed objects, with the effect becoming more prominent in WDM at scales substantially smaller than the half-mode mass.}\n\n\nThe discussion of environmental trends for mass accretion rates requires a prediction of the cross-correlation between the slopes of random walks at barrier crossing and the values attained by these walks at larger smoothing scales. This is conceptually easiest for the density environment, which is the natural variable used in excursion set calculations. \\citet{lazeyras} have applied such arguments to show that an overall assembly bias trend, in which slowly accreting haloes are strongly clustered as compared to rapid accretors, i.e. a negative correlation $(b_1\\leftrightarrow\\Gamma)$, is a natural prediction of the excursion set peaks framework. The distinction between diffuse accretion and that via mergers, on the other hand, requires a higher level of sophistication in the simultaneous prediction of continuous and discrete accretion rates, an aspect which excursion set models have only recently begun to explore \\citep{ms14-markov}. And the trends we have noted with tidal anisotropy $\\alpha$, especially the result that $\\alpha$ largely explains low-mass assembly bias trends with mass accretion, are currently not predictable by any excursion set (peaks) model that we are aware of \\citep[although see][for some initial steps in this direction]{cphs17}. We therefore defer a discussion of analytical predictions for the environment dependence of mass accretion to future work, where we hope to develop a versatile excursion set framework that can address these issues. \n\n\n\\section{Summary \\& Conclusions}\n\\label{sec:conclude}\n\nWe have investigated the evolving correlations between mass accretion rates and environment in $N$-body simulations of CDM and WDM cosmologies. The latter are characterised by a strong suppression of small-scale power (we deliberately chose a somewhat extreme case with $m_{\\rm dm}=0.4\\,{\\rm keV}$), which creates dramatic differences in the nature of mass accretion between haloes smaller and larger than the half-mode mass scale. \n\nWhile accretion rates in WDM models have been compared with their CDM counterparts previously in the literature \\citep{knebe02, benson+13, elahi, khimey20}, and environmental trends of CDM accretion rates have also been studied before (see the Introduction), our analysis represents the first systematic comparison of environmental trends of accretion rates between CDM and WDM haloes, at masses above and below the WDM half-mode mass, using multiple proxies for the local as well as large-scale environment, and over a wide range of redshift $2\\geq z\\geq 0$.\n\nMost of our results (section~\\ref{sec:results}) regarding the evolving nature and environment dependence of specific accretion rates due to mergers (\\ensuremath{\\Gamma_{\\rm mer}}) and diffuse mass (\\ensuremath{\\Gamma_{\\rm dif}}; see equations~\\ref{eq:Mdot-def}) can be understood in terms of the lack of small-scale structure below the half-mode mass scale in WDM, which affects not only the low-mass haloes but also the accretion histories of massive objects. We summarize these below and also highlight a few puzzling findings that deserve further study.\n\n\\vskip 0.1in\n\\noindent\n\\underline{\\it Environment-independent trends:}\n\\begin{itemize}\n \\item Specific accretion rates of low-mass CDM haloes are lower than those of high-mass CDM haloes, while low-mass WDM haloes have higher accretion rates than their high-mass counterparts (figure~\\ref{MAH}). This is a straightforward prediction of the excursion set approach with correlated steps (section~\\ref{sec:analytical}). \n \n \\item Mass accretion in WDM haloes is dominated by \\ensuremath{\\Gamma_{\\rm dif}}, consistent with previous work \\citep{benson+13,elahi}. \\ensuremath{\\Gamma_{\\rm dif}}\\ (\\ensuremath{\\Gamma_{\\rm mer}}) in WDM haloes is also higher (lower) than that in CDM haloes of the same mass, as expected from the lack of small-scale bound structures in WDM. \n\\end{itemize}\n\n\\noindent\n\\underline{\\it Trends with local environment:}\n\\begin{itemize}\n \\item The evolving median local density and tidal anisotropy of haloes with and without mergers (figure~\\ref{median-evoln}) are also largely consistent with expectations based on the lack of small-scale structure in WDM. The only exceptions, which deserve further study, are\n \\begin{itemize}\n \\item the environments of low-mass WDM haloes are denser than those of all other categories, and\n \\item the environments of low-mass CDM haloes without mergers are more anisotropic than of those with mergers. \n \n \\end{itemize}\n \\item The correlations between environment and accretion rates in figures~\\ref{CDM-spear} and~\\ref{WDM-spear} add further detail to these results. For both CDM and WDM, at all redshifts in the \\emph{high-mass bin},\n \\begin{itemize}\n \\item there is a strong negative correlation between $\\delta$ and \\ensuremath{\\Gamma_{\\rm dif}}, a weaker positive correlation between $\\delta$ and \\ensuremath{\\Gamma_{\\rm mer}}, and comparable negative correlations between $\\alpha$ and both \\ensuremath{\\Gamma_{\\rm dif}}\\ and \\ensuremath{\\Gamma_{\\rm mer}},\n \\item while $\\alpha$ and $\\delta$ are equally important for \\ensuremath{\\Gamma_{\\rm mer}}, $\\delta$ almost completely explains the environment dependence of \\ensuremath{\\Gamma_{\\rm dif}}.\n \\end{itemize}\n \\item In the \\emph{low-mass bin} at all $z$, \n for CDM haloes, $\\alpha$ and $\\delta$ are equally important in explaining the environment dependence of \\ensuremath{\\Gamma_{\\rm dif}}\\ and show no correlation with \\ensuremath{\\Gamma_{\\rm mer}}. In WDM, on the other hand,\n the environment dependence of both \\ensuremath{\\Gamma_{\\rm mer}}\\ and \\ensuremath{\\Gamma_{\\rm dif}}\\ is almost fully explained by $\\alpha$. In fact, the correlation $(\\alpha\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ is \\emph{positive}, which is counter-intuitive and deserves further study. \n \n \n\\end{itemize}\n\\emph{In summary, at all $z$, the local tidal anisotropy $\\alpha$ plays a completely dominant role for both \\ensuremath{\\Gamma_{\\rm mer}}\\ and \\ensuremath{\\Gamma_{\\rm dif}}\\ in low-mass WDM haloes, and a bigger role than the local density $\\delta$ for \\ensuremath{\\Gamma_{\\rm mer}}\\ in high-mass haloes in both CDM and WDM. In contrast, $\\delta$ plays the dominant role for \\ensuremath{\\Gamma_{\\rm dif}}\\ in high-mass CDM haloes.\n}\nIn the other cases, $\\delta$ and $\\alpha$ play equally important roles. These trends are mostly insensitive to the definition of progenitor (see section~\\ref{subsec:Mdot-measure} and \\ref{subsec:mdot<->env}) for details.\n\n\n\n\n\\vskip 0.1in\n\\noindent\n\\underline{\\it Assembly bias:}\n\\vskip 0.05in\n\\noindent\nWe defined assembly bias using the Spearman rank correlation coefficients $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ and $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm mer}})$ in each mass bin \\citep[c.f.,][]{rphs19}.\nWe find that the following holds for both CDM and WDM haloes (figures~\\ref{CDM-spearbias} and~\\ref{WDM-spearbias}):\n\\begin{itemize}\n \\item $(b_1\\leftrightarrow\\alpha)$ is always higher than $(b_1\\leftrightarrow\\delta)$ at any $z$. This extends the $z=0$ CDM results of \\citet{phs18a} and \\citet{rphs19} to higher redshifts and WDM cosmologies.\n \\item \\emph{We detect assembly bias in the low-mass bin, driven entirely by \\ensuremath{\\Gamma_{\\rm dif}}, with a strong $(b_1\\leftrightarrow\\ensuremath{\\Gamma_{\\rm dif}})$ correlation at $z=0$ with the strength that is comparable with the highest known correlations of all the secondary variables at this halo mass for CDM and is higher in WDM than in CDM} \\citep[solid blue curves in the left panels of figures~\\ref{CDM-spearbias} and~\\ref{WDM-spearbias}, see section \\ref{subsec:assemblybias} for details and compare the middle panel of figure 2 in][]{rphs19}. In the high-mass bin, there is no significant assembly bias at any but the highest redshifts we study.\n \n \\item The tidal anisotropy $\\alpha$ plays a dominant role in explaining the low-mass assembly bias, especially for WDM haloes.\n\\end{itemize}\n\n\n\nOur results place important constraints on (semi-) analytical excursion set models of halo formation and growth and, by extension, on galaxy evolution models built upon such approximate techniques (see the Introduction for references). As argued by HP14, any such model which purports to explain environmental trends of evolving haloes must logically work equally well for CDM and WDM, since the physics of collisionless self-gravitating systems is common to both. As we have seen, however, the suppression of small-scale power in WDM leads to an intricate dependence of the mass accretion on local (and large-scale) halo environment, particularly at masses smaller than the WDM half-mode mass. As with the mass function at these scales discussed by HP14, producing an accurate model of mass accretion with the correct environment dependence is likely to reveal interesting features of collisionless dynamics in the shell-crossed regime. It will be very interesting to confront our results above with excursion set models of mergers and mass accretion \\citep{lc94,mkby11,ms14-markov}, as well as excursion set-inspired semi-analytical algorithms tuned to reproduce CDM results \\citep{sk99,pch08,jvdb14}. We leave this to future work.\n\n\\section*{Acknowledgements}\nWe thank Oliver Hahn and Sujatha Ramakrishnan for useful discussions.\nPD thanks IUCAA for hospitality and working facilities, Akhilesh Peshwe (Principal, DMPDM Science College) for his kind support, Isha Pahwa for useful discussions and Dhairyashil Jagadale for constant support and discussion.\nThe research of AP is supported by the Associateship Scheme of ICTP, Trieste and the Ramanujan Fellowship awarded by the Department of Science and Technology, Government of India. \nThis work used the open source computing packages NumPy \\citep{vanderwalt-numpy},\\footnote{\\href{http:\/\/www.numpy.org}{http:\/\/www.numpy.org}} SciPy \\citep{scipy}\\footnote{\\href{http:\/\/www.scipy.org}{http:\/\/www.scipy.org}} and the plotting software Veusz.\\footnote{\\href{https:\/\/veusz.github.io\/}{https:\/\/veusz.github.io\/}}\nWe gratefully acknowledge the use of high performance computing facilities at IUCAA, Pune.\n\n\n\\section*{Data availability}\nThe data underlying this work will be shared upon reasonable request to the authors.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\nWe study the evolution of a compact region of turbulence in an otherwise unbounded domain. Our interest in this novel flow started in seeking a simple validation case for a large-eddy simulation (LES) model in free space. In past work, validations have involved canonical flows in wall-bounded or periodic domains. In free space, and without artificial forcing, there are no mechanisms to sustain turbulence, and it will decay in time. Of possible initial flow fields, two of relevance here are a collection of one or more vortex rings and a random initial condition.\n\nInitial conditions comprising a collection of vortex rings are readily created in both experiments and simulations. Recent experiments have studied the generation of turbulence through vortex-ring collisions\n\\citep{matsuzawa2019realization} and have evaluated hypothesized mechanisms of turbulence self-sustenance \\citep{mckeown2018cascade}.\nThe resulting turbulence occurs through a complex process of instabilities, vortex interaction, and reconnection \\citep{lim1992instability}. Simulations are computationally intensive since distinct turbulent and laminar regions will occur, and the associated vortex-ring Reynolds numbers must be sufficiently high for transition to occur. In contrast, a random initial condition is computationally\nsimple and turbulent Reynolds numbers can easily be reached (even in DNS), but a disadvantage is that there will be an initial transient period that, while governed by the Navier-Stokes equations, will not be associated with physical turbulence. As a compromise, we manufacture an initial condition by first generating isotropic homogeneous turbulence (IHT) in a periodic domain, and initializing a free-space {\\it cloud} of turbulence by tiling the periodic solution in space and windowing it with an indicator function that falls to zero outside a sphere of radius $R$, which can be varied compared to the initial scales in the IHT.\n\nSeveral features of the evolution of such a spherical region of turbulence in free space are of theoretical interest. In IHT, the evolution of the largest scales is governed by the initial conditions, or, in the case of forced IHT, by the forcing scheme. For example, spectra with low wavenumber that asymptotes as $k^2$ \\citep{saffman1967large} and $k^4$ \\citep{batchelor1956large} can be contrived \\citep{chasnov1995decay,ishida2006decay,davidson2010decay}. Indeed, the same is true for the spherical turbulence cloud, where, unlike IHT, the largest scales can subsequently grow and, as we will show, different behavior is obtained.\nThe spherical region of turbulence also exemplifies the localized turbulence region introduced in \\citet{phillips1956final}, where the final viscous stage of the evolution was studied theoretically.\n\nA second motivation concerns the emergence of coherent structures. IHT is devoid of large-scale instabilities (typically associated with shear, buoyancy, or other imposed forces) that give rise to important classes of coherent structures. However, as we show, the same is not the case for the spherical cloud --- we observe the formation of coherent vortex rings being ejected near the cloud edge.\n\nA related issue is the interaction of the turbulent flow with the outer irrotational fluid at the turbulent\/nonturbulent interface (TNTI). In recent work, TNTIs have been experimentally and numerically studied in shear layers and in numerically-constructed shear-free interfaces \\citep{ wolf2013,de2013multiscale,da2014characteristics}. The spherical cloud of turbulence also exhibits a TNTI, and may prove a useful source of data for further study, though in the present paper we do not investigate its behavior in detail.\n\nIn this paper, we present DNS and LES simulations for the spherical region of turbulence and examine the resulting energy spectrum and its decay, and the ejection of vortex rings from its periphery.\nIn \\S~\\ref{sec:numerical_method} we present the numerical method used to solve the incompressible Navier-Stokes equations in free space, and introduce the turbulence model used in the LES.\nIn \\S~\\ref{sec:initial_conditions}, the initial conditions are discussed in detail, including recipes for generating $k^2$ and $k^4$ spectra.\nIn \\S~\\ref{sec:viz}, DNS and LES results are used to visualize the evolution of the turbulence field.\nIn \\S~\\ref{sec:statistics}, we use statistical measures to characterize the decay. We also show LES calculations agree well with the DNS in these measures. In \\S~\\ref{sec:longterm_statistics}, LES is used to study the long-term evolution of the turbulence cloud.\nFinally, in \\S~\\ref{sec:vortex_rings}, we discuss one distinctive feature in the long-term evolution, the ejection of vortex rings, and we conjecture about the relationship between the initial condition and the scale of the ejections. A brief summary of the main conclusions is given in \\S~\\ref{sec:conclusions}.\n\\section{Numerical method}\\label{sec:numerical_method}\n\n\n\\subsection{The fast lattice Green's function method}\\label{sec:lgf}\n\nWe solve the incompressible (constant density and viscosity) Navier-Stokes equations in an unbounded three-dimensional space. A second-order mimetic finite volume scheme based on the fast lattice Green's function (LGF) method recently developed by \\citet{liska2016fast}\nis applied. The method enforces the divergence-free constraint by solving the associated (discrete) Poisson equation on a formally infinite grid using the LGF technique.\n\nFor flows with a compact vorticity field, the infinite lattice can be truncated to a finite region that adapts to the local flow according to a threshold value on the vorticity where sources to the Poisson equation are finite. The solution can be reconstructed at any position, but is only done at those lattice points needed for the next time step.\nThis method is ideal for incompressible external flow simulations for the following two reasons: first, it yields more accurate solutions since the exact far-field boundary condition is embedded in LGFs and no artificial outflow boundary condition is imposed; second, it is efficient because the computation domain is snug around the vortical region.\n\nOther building blocks of the scheme include: a third-order Runge-Kutta scheme for the time integration; an analytical integrating factor technique for the viscous term which has the advantage of neither introducing discretization errors nor imposing stability constraints on the time step. The scheme is parallelized and has been extensively validated by comparison with exact solutions and grid refinement studies \\citep{liska2014parallel}. The DNS simulation reported here uses a maximum of around $2\\times 10^9$ computational cells running on $1,500$ cores.\n\nFigure \\ref{fig:IC} shows the setup for the problem at hand. Initially only a spherical region in free space is filled with turbulence (fluid with non-zero vorticity). The spherical region then starts to deform, evolve and decay. This method utilizes fixed local cell size but is able to spatially adapt with the vortical areas by adding or removing blocks of computation cells (black contour lines in figure \\ref{fig:IC}). All simulations here are conducted with a spatial adaptive threshold $\\epsilon_{\\text{supp}}$ equal to $10^{-5}$ defined in \\citet{liska2016fast}. Because of the spatial adaptivity, the total number of computation cells varies through one simulation.\n\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=.95\\textwidth]{figs\/fig1.png}\n \\put(0,37) {\\small $t\/t_\\ell=0$}\n \\put (53,37) {\\small $t\/t_\\ell=8.6$}\n \\end{overpic}\n \\caption{\\small\n Vorticity magnitude in a cross-section through the center at $t\/t_\\ell=0$ and $t\/t_\\ell=8.6$, corresponding to the case DNS\\_0 from table \\ref{tab:simulation_summary}. Black contour lines indicate the spatially adaptive computational domain. The smallest adaptivity unit is a block of $16^3$ computational cells. }\\label{fig:IC}\n\n\\end{figure}\n\n\n\n\\subsection{Stretched vortex sub-grid stress (SGS) model }\\label{sec:stretched_vortex}\n\nThe LES solutions we report rely on the stretched vortex SGS model (SVM) \\citep{chung2009large}. The LES solves the filter-averaged Navier-Stokes equations\n\\begin{align}\\label{eq:N-S-LES}\n \\frac{\\partial \\widetilde{\\boldsymbol{u}}}{\\partial t}+\n \\widetilde{\\widetilde{\\boldsymbol{u}} \\cdot \\nabla \\widetilde{\\boldsymbol{u}}}\n =- \\nabla \\widetilde{P}+\\nu \\nabla^{2} \\boldsymbol{\\widetilde{\\boldsymbol{u}}} - \\nabla \\cdot \\widetilde{\\mathsfbi{T}},\n \\\\\n \\nabla \\cdot \\boldsymbol{\\widetilde{\\boldsymbol{u}}}=0,\n\\end{align}\nwhere $\\widetilde{\\mathsfbi{T}} = \\widetilde{\\boldsymbol{u}\\otimes \\boldsymbol{u}} - \\widetilde{\\widetilde{\\boldsymbol{u}} \\otimes \\widetilde{\\boldsymbol{u}}}$ is the\nSGS tensor.\nThe model assumes\nthat the subgrid motions for a single computational cell are dominated by small vortices in\na direction $\\boldsymbol{e^v}$ that is aligned with the principle eigenvector of the\nresolved strain rate tensor. Then the SGS tensor is given by\n\\begin{align}\n \\widetilde{T}_{i j}=K \\left(\\delta_{i j}-e_{i}^{v} e_{j}^{v}\\right), \\\\\n K=\\int_{k_{c}}^{\\infty} E(k) \\mathop{}\\!\\mathrm{d} k=\n \\mathcal{K}_{0}' \\Gamma\\left[-1 \/ 3, \\kappa_{c}^{2}\\right] \/ 2, \\label{eg:sgs-kinetic}\n\\end{align}\nwhere $K$ is the subgrid kinetic energy, $k_c = \\pi \/ \\Delta_x= \\pi \/ \\Delta_y= \\pi \/ \\Delta_z$ with $\\Delta_{(\\cdot)}$ being the cell length, and $\\Gamma$ is the incomplete gamma function. The second equality in equation (\\ref{eg:sgs-kinetic}) assumes that SGS vortices are of the stretched-spiral type with spectrum \\citep{lundgren1982strained}\n\\begin{align}\n E(k)=\\mathcal{K}_{0} \\epsilon^{2 \/ 3} k^{-5 \/ 3} \\exp \\left[-2 k^{2} \\nu \/(3|\\widetilde{a}|)\\right] \\label{eq:lundgren},\n\\end{align}\nwhere $\\nu$ is the fluid viscosity\n\\begin{align}\n\\widetilde{a}=e_{i}^{v} e_{j}^{v} \\widetilde{S}_{i j}, \\quad\n \\mathcal{K}_{0}'=\\mathcal{K}_{0} \\epsilon^{2 \/ 3} \\lambda_{v}^{2 \/ 3}, \\quad\n \\lambda_{v}=(2 \\nu \/ 3|\\widetilde{a}|)^{1 \/ 2}, \\quad \\kappa_{c}=k_{c} \\lambda_{v},\n\\end{align}\nand $\\widetilde{S}_{i j}$ is the resolved strain rate tensor. Finally the constant $\\mathcal{K}_{0}'$ in equation (\\ref{eg:sgs-kinetic}) is determined by matching the resolved second-order velocity structure function with the prediction from the energy spectrum given by equation~(\\ref{eq:lundgren}). Details regarding the efficient evaluation of the aforementioned SGS stress can be found in \\citet{voelkl2000physical,chung2009large}.\n\nThe SVM is structure based and not of the eddy-viscosity type. All model parameters are calculated dynamically using only local information from the resolved-scale field surrounding the grid cell or point where sub-grid stresses are calculated. The SVM keeps track of the actual fluid viscosity and also the subgrid kinetic energy, and will automatically become subdominant to real viscous stresses when the flow is locally resolved. It has proven robustness and has been successfully used for studies of decaying turbulence \\citep{misra1997vortex}, and wall-resolved LES of channel flow \\citep{voelkl2000physical,chung2010direct}, bluff-body flows \\citep{cheng2017large,cheng2018large} and Taylor-Couette flow \\citep{cheng2020large}.\n\n\\subsection{Initial condition}\\label{sec:initial_conditions}\n\nThe initial condition is generated by spherically windowing a turbulence field from a separate IHT computation with periodic boundary conditions. This field is then tiled in all directions to fill the free space and the velocity field is multiplied by a smooth window function of the form\n\\begin{align}\\label{eq:window}\n \\Phi(r) & = \\frac{1}{2}\\left[1-\\tanh \\left( \\frac{2 ( r-R)}{\\sigma}\\right) \\right],\n\\end{align}\nwhere $R$ is the radius of the sphere and $\\sigma$ is the width of the transition, whose impact on the results will be assessed. The forced periodic IHT field is generated using a simple 3D pseudo-spectral code and we define the domain size to be $B^3$.\nA low wavenumber forcing method is applied \\citep{huang1994}. The forcing is restricted to modes with wavenumbers $|\\boldsymbol{k}|<2.5$ and the magnitude of the forcing is chosen to keep the energy input rate constant, which would equal to the dissipation rate $\\epsilon$ after the forced turbulence becomes stationary. To make sure all IHT flows are fully resolved, $\\epsilon$ is\ndetermined such that $\\eta k_{\\mathrm{max}} \\sim 1.5$, where $k_{\\mathrm{max}} = N_s \/ 2$ is\nthe maximum wavenumber and $\\eta = \\left({\\nu^{3}}\/{\\epsilon}\\right)^{1 \/ 4}$ is the Kolmogorov length scale with $\\nu$ being the viscosity. We also confirmed the isotropy of the IHT field by verifying that $ \\frac{E_{ii}(k)}{E(k)} - \\frac{1}{3} \\approx 0, \\ i=1,2,3$.\n\nFigure \\ref{fig:IC} visualizes the initial vorticity field in a cross-section through the center. This corresponds to the case DNS\\_0 defined in table \\ref{tab:simulation_summary}. Note that black contour lines in figure \\ref{fig:IC} are the spatially adaptive computational domain which encompasses the initial voricity field as discussed in \\S \\ref{sec:lgf}. For LES, the IHT field is spectrally filtered before tiling and windowing. More about initial conditions for LES is discussed in \\S~\\ref{sec:DNSLES_visualization} when results from LES are presented.\n\n\nOnce an IHT field and resolution are selected (which give an initial turbulence Reynolds number, $\\Rey_\\lambda$), two non-dimensional parameters characterize the initial condition: $B \/ R$ and $\\sigma \/ R$. Table \\ref{tab:simulation_summary} summarizes the parameters for all runs studied in this work. The simulation parameters used in the pseudo-spectral code to generate the IHT fields and their statistical characteristics are given in table \\ref{tab:IHT}.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrrrrr}\nName \\qquad& $\\Rey_\\lambda$ \\qquad& $ \\sigma \/ R$ \\qquad& $B \/ R$ \\qquad & Spectrum Type\\\\\n\\\\\nDNS\\_0 & 122.4 & 0.10 & 1.0 & $2$ \\\\\nLES\\_0 & 122.4 & 0.10 & 1.0 & $2$ \\\\\nLES\\_IC2 & 122.4 & 0.10 & 1.0 & $4$ \\\\\nLES\\_D1 & 122.4 & 0.05 & 1.0 & $2$ \\\\\nLES\\_D2 & 122.4 & 0.20 & 1.0 & $2$ \\\\\nLES\\_B1 & 122.4 & 0.10 & 0.5 & $2$ \\\\\nLES\\_B2 & 122.4 & 0.10 & 2.0 & $2$ \\\\\nLES\\_R1 & 76.9 & 0.10 & 1.0 & $2$ \\\\\nLES\\_R2 & 45.0 & 0.10 & 1.0 & $2$ \\\\\n\n\\end{tabular}\n\\caption{\\small Simulation parameters. The spectrum type refers to the leading non-zero order in the low wavenumber limit. \n}\\label{tab:simulation_summary}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrrr}\nRun Name & $\\Rey_\\lambda$ & $\\ell\/B$ & $\\eta k_{\\text{max}}$ & Resolution \\\\\n\\\\\nLES\\_R1 & 76.9 & 0.17 & 1.52 & $128^3$ \\\\\nLES\\_R2 & 45.0 & 0.19 & 1.58 & $64^3$\\\\\nAll others & 122.4 & 0.16 & 1.53 & $256^3$ \\\\\n\\end{tabular}\n\\caption{\\small Summary of the simulation parameters used in the pseudo-spectral code and the resulting IHT fields. $\\ell$ is the integral scale, $\\eta$ is the Kolmogorov length scale, and $k_\\text{max}$ is the maximum wavenumber.\n}\\label{tab:IHT}\n\\end{table}\n\n\\subsection{Initial spectrum and low wavenumber limit}\\label{sec:ic}\n\nAs we only expect the turbulence cloud to remain homogeneous deep within the sphere, ambiguities arise in interpreting the energy spectrum: it can be viewed as the expectation of a random process, or merely as the Fourier transform of a deterministic function. In order to reach the broadest conclusions possible (i.e. ones not limited to the specific initial condition), we show in Appendix \\ref{sec:appendix_total_spectrum} that by invoking local homogeneity deep within the spherical region, we can estimate the total spectrum through a single realization of this flow. The estimated spectrum approximates the true one in the limit of large $R\/ \\ell$, which may only be barely reached in our simulations, but in principle could be improved upon in future. Thus we take\n\\begin{align}\n \\widetilde{E}(\\boldsymbol{k})\n & =\\frac{1}{16 \\pi^3} \\int_{\\mathbb{R}^3} \\int_{\\mathbb{R}^3} \\boldsymbol{u}(\\boldsymbol{x})\\cdot \\boldsymbol{u}(\\boldsymbol{x}')\\, e^{-i \\boldsymbol{k} \\cdot (\\boldsymbol{x}'-\\boldsymbol{x})} \\mathop{}\\!\\mathrm{d} \\boldsymbol{x} \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}' \\nonumber\\\\\n & =\\frac{1}{16 \\pi^3} |\\mathcal{F}\\{\\boldsymbol{u}\\}|^2 =\\frac{1}{16 \\pi^3} \\frac{1}{|\\boldsymbol{k}|^2}|\\mathcal{F}\\{\\boldsymbol{ \\omega}\\}|^2 \\nonumber\\\\\n & =\\frac{1}{16 \\pi^3} \\int_{\\mathbb{R}^3} \\int_{\\mathbb{R}^3} \\frac{1}{|\\boldsymbol{k}|^2} \\boldsymbol{ \\omega}(\\boldsymbol{x})\\cdot \\boldsymbol{ \\omega}(\\boldsymbol{x}')\\, e^{-i \\boldsymbol{k} \\cdot (\\boldsymbol{x}'-\\boldsymbol{x})} \\mathop{}\\!\\mathrm{d} \\boldsymbol{x} \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}' \\label{eq:Ek_vel},\\\\\nE(k) &=\\frac{1}{(2 \\pi)^{2}}\n \\int_{\\mathbb{R}^{3}} \\int_{\\mathbb{R}^{3}}\n \\frac{\\sin \\left(k\\left|\\boldsymbol{x}'-\\boldsymbol{x}\\right|\\right)}{k\\left|\\boldsymbol{x}'-\\boldsymbol{x}\\right|} \\boldsymbol{ \\omega}\\left(\\boldsymbol{x}'\\right) \\cdot \\boldsymbol{ \\omega} (\\boldsymbol{x}) \\mathop{}\\!\\mathrm{d} \\boldsymbol{x} \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}', \\label{eq:greg}\n\\end{align}\nwhere we expressed the spectrum in terms of the vorticity field \\citep{phillips1956final, leonard1985computing, winckelmans1993contributions, winckelmans1995some}. The spherical symmetry of the problem is used in the last step where the 3-D energy spectrum $\\widetilde{E}(\\boldsymbol{k})$ is integrated over a spherical shell to produce a scalar spectrum $E(k)$. Expanding $E(k)$ for the low wavenumber, the odd powers vanish, giving\n\\begin{align}\n \n E(k) &= \\frac{k^2}{4 \\pi^2} L + \\frac{k^4}{24 \\pi^2} I + O(k^6) ,\\label{eq:Ek_lowwavenumber}\n\\end{align}\nwhere\n\\begin{align} \\label{eq:impulse}\nL&=-\\frac{1}{6} \\int_{\\mathbb{R}^3}\\int_{\\mathbb{R}^3} |\\boldsymbol{x}'-\\boldsymbol{x}|^{2} \\, \\boldsymbol{ \\omega}(\\boldsymbol{x}')\\cdot\\boldsymbol{ \\omega}(\\boldsymbol{x})\\mathop{}\\!\\mathrm{d} \\boldsymbol{x}' \\mathop{}\\!\\mathrm{d}\\boldsymbol{x}\n= \\int_{\\mathbb{R}^3} \\int_{\\mathbb{R}^3} \\boldsymbol{u}(\\boldsymbol{x}')\\cdot\\boldsymbol{u}(\\boldsymbol{x}) \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}' \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}\n\\end{align}\nis the Saffman integral and\n\\begin{align} \\label{eq:loit}\n I=\\frac{1}{20} \\int_{\\mathbb{R}^3}\\int_{\\mathbb{R}^3} |\\boldsymbol{x}'-\\boldsymbol{x}|^{4} \\,\\boldsymbol{ \\omega}(\\boldsymbol{x}')\\cdot\\boldsymbol{ \\omega}(\\boldsymbol{x})\\mathop{}\\!\\mathrm{d} \\boldsymbol{x}' \\mathop{}\\!\\mathrm{d}\\boldsymbol{x}\n=-\\int_{\\mathbb{R}^3}\\int_{\\mathbb{R}^3} |\\boldsymbol{x}'-\\boldsymbol{x}|^2 \\,\\boldsymbol{u}(\\boldsymbol{x}') \\cdot \\boldsymbol{u}(\\boldsymbol{x}) \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}'\\mathop{}\\!\\mathrm{d}\\boldsymbol{x}\n\\end{align}\nis the Loitsyansky integral \\citep{loitsyansky1939some}.\nNote that even though Eq.~(\\ref{eq:Ek_vel}-\\ref{eq:Ek_lowwavenumber}) are well defined for the flows with finite energy, using the velocity forms from Eq.~(\\ref{eq:impulse}, \\ref{eq:loit}) requires certain decay rates of the velocity field. Thus we have used the vorticity formula for the calculation of the low wavemnumber spectra.\nDetails of the expansion and the calculation method are given in Appendix \\ref{sec:appendix_total_spectrum_B}.\n\nThe Saffman integral $L$ is related to the total momentum impulse which is an invariant of the motion and remains constant for all time. When $L \\ne 0 $ the cloud will exhibit a small-wavenumber, $k^2$ Saffman limit \\citep{saffman1967large}, whereas when $L=0$ the spectrum is of the $k^4$ Batchelor type \\citep{batchelor1956large}. The ramifications of a Saffman or Batchelor spectrum have been widely explored in IHT, but less so in other (inhomogeneous) flows. As we discuss below, the low wavenumber spectrum can be used to derive asymptotic energy decay and integral-scale growth rates, which can be compared to those obtained for the spherical cloud.\n\nIn order to investigate this issue, we develop a procedure by which we control the value of $L$ in the initial condition. Two factors contribute to the linear momentum. Firstly the IHT field generated from the pseudo-spectral code is continuously divergence-free but not necessarily discrete divergence-free as required by the finite volume FLGF scheme. Secondly, the windowing process will introduce extra non-solenoidality. i.e., given a divergence-free velocity field $\\boldsymbol{u}$ and a scalar window function $\\Phi(r)$, $\\Delta \\cdot (\\Phi(r) \\boldsymbol{u}(\\boldsymbol{x}))\\neq 0$ in general. Both of these non-solenoidal components are projected out at the very first time step and this projection will introduce an impulse. The result of this impulse as mentioned in \\citet{batchelor1967introduction} is a $1\/|\\boldsymbol{x}|^3$ decaying velocity field of the form\n\\begin{align} \\label{eq:helmholtz}\n \\lim_{x\\rightarrow \\infty}\\mathbf{u}(\\boldsymbol{x}) =\n \\frac{1}{8 \\pi} \\nabla \\left[\n \\nabla \\left(\\frac{1}{|\\boldsymbol{x}|}\\right) \\cdot \\int_{\\mathbb{R}^3} \\boldsymbol{x}' \\times \\boldsymbol{ \\omega}(\\boldsymbol{x}')\n \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}'\\right]\n .\n\\end{align}\nThis suggests a way to cancel the impulse in order to have a $k^4$ type spectrum: one can add a vortex ring with an opposite impulse to the initial velocity field. More specifically, we add a Stokes vortex ring with velocity \\citep{kambe1975generation, cantwell1986viscous}\n\\begin{align}\\label{eq:cancel}\n\\boldsymbol{u}(r, \\theta) = \\frac{1}{(2\\pi)^{3\/2}\\zeta^3}\\left[W\\left(\\frac{r}{\\zeta};1 \\right) \\boldsymbol{\\gamma} - W\\left(\\frac{r}{\\zeta};3 \\right) \\gamma \\cos \\theta \\, \\hat{\\mathbf{e}}_{r} \\right],\n\\end{align}\nwhere $\\boldsymbol{\\gamma}$ is the impulse of the vortex ring, $\\theta$ is the angle between $\\boldsymbol{\\gamma}$ and the unit vector $\\mathbf{e}_{r}$, $\\zeta$ controls the size of the ring and\n\\begin{align}\n W(\\rho;b)\n &=e^{-\\rho^2\/2} - \\frac{b}{\\rho^3} \\left[ \\sqrt{\\frac{\\pi}{2}}\\operatorname{erf}\\left(\\frac{\\rho}{\\sqrt{2}}\\right)-\\rho e^{-\\rho^{2} \/ 2}\\right]\n .\n\\end{align}\nWe chose $\\zeta\/R = 0.19$ and performed an LES computation with this cancellation, referred to as case LES\\_IC2 in table \\ref{tab:simulation_summary}. Except for LES\\_IC2, all other cases are conducted without the cancellation.\n\nWhile our method of manipulating the initial condition in order to cancel the finite impulse is arbitrary, it is effective in the sense that the added vortex ring quickly interacts with the turbulence leading. This was verified by monitoring the difference between simulations initialized with and without the cancellation. The results showed that in less than one initial large-eddy turnover time, the difference field was decorrelated with the added vortex ring. Thus we conclude that the two simulations can be regarded as representing (different random realizations of) locally homogeneous turbulence that differ significantly only in their low wavenumber spectrum.\n\nFigure \\ref{fig:intial_Ek} shows the resulting initial energy spectrum $E_0(k)$ of a spherical region of turbulence field corresponding to the condition of DNS\\_0, superposed on another spectrum where equation~\\eqref{eq:cancel} was used to cancel the impulse (i.e. the initial condition, after filter, for the case `LES\\_IC2' in table \\ref{tab:simulation_summary}). Also plotted is the energy spectrum of the original IHT field scaled by the ratio between the volume of the sphere and the cubic domain size $B^3$. We see that the $k^{-{5\/ 3}}$ portion of the spectrum from IHT is retained in the spherical cloud, whereas the low wavenumber behavior is controlled by the resulting impulse (or its absence).\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=.75\\textwidth]{figs\/fig2.eps}\n \\end{overpic}\n \\caption{ \\small Energy spectrum of\n (1) the initial condition of simulation DNS\\_0. (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black,solid,line width = 1pt](0,0) -- (5mm,0);}}), where the low wavenumber limit (left of the dotted region) is calculated through an expansion method (Appendix \\ref{sec:appendix_total_spectrum_B});\n (2) DNS\\_0 with the initial impulse cancelled using equation~\\eqref{eq:cancel} (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,brown!40!gray,dashdotdotted,line width = 1.0pt](0,2) -- (5mm,2);}});\n (3) the corresponding IHT field multiplied by the ratio between the spherical region volume $\\frac{4}{3} \\pi R^3$ and the cubic domain volume $B^3$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,orange,dashed,line width = 1.0pt](0,0) -- (5mm,0);}});\n (4) a guide line for $1.6 \\varepsilon^{2\/3} k^{-5\/3}$ scaled with the same ratio, where $\\varepsilon$ is the dissipation rate in the original IHT field (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}});\n (4) a slope of $k^2$ (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dashdotted,line width = 1.0pt](0,2) -- (5mm,2);}}) and $k^4$ (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,densely dashdotdotted,line width = 1.0pt](0,2) -- (5mm,2);}}) for the low wavenumber limit.\n }\\label{fig:intial_Ek}\n\\end{figure}\n\n\\subsection{Resolution}\\label{sec:resolution}\n\nFor DNS\\_0, the IHT field used to generate the initial condition has $\\Rey_\\lambda=122.4$ and uses a computational domain of $256^3$ in the pseudo-spectral code with $\\eta k_{\\text{max}}>1.5$ to ensure that it is fully resolved. The same resolution (same number of points used for every length scale $B$) is used in the LGF solver for the turbulence cloud. To guarantee this resolution is also sufficient for the finite volume solver, another DNS simulation of $3\/2$ times the resolution is performed up to $1.3$ initial large eddy turnover time. The difference in the total kinetic energy is about $0.23\\%$ and the maximum relative difference in the spectra for all wavenumber $kR$ is about $1\\%$ which is shown in figure \\ref{fig:res_check}.\n\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=.75\\textwidth]{figs\/fig3.eps}\n \\end{overpic}\n \\caption{ \\small Energy spectrum of\n (1) DNS\\_0 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black,solid,line width = 1pt](0,0) -- (5mm,0);}}) and\n (2) a DNS calculation at $3\/2$ times the resolution as that in the case DNS\\_0 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,cyan,dashed,line width = 1.0pt](0,0) -- (5mm,0);}}), at $t\/t_\\ell=1.3$.\n }\\label{fig:res_check}\n\\end{figure}\n\n\n\\section{Qualitative evolution}\n\\label{sec:viz}\n\\subsection{DNS}\\label{sec:DNS}\nFirst we perform DNS of the spherical cloud of turbulence corresponding to the case `DNS\\_0' in table \\ref{tab:simulation_summary}.\nThe flow evolution is shown in figure \\ref{fig:DNS}a. Instantaneous vorticity magnitude iso-surfaces\nat $t\/t_{\\ell}=0,1.7, 4.0, 8.6, 17.5$ are given, where $t_\\ell$ is the large eddy turnover time of the original IHT field. The iso-surface of the lowest vorticity magnitude represents the TNTI.\nThis interface is sufficiently thin \\citep{mathew2002some} that using a lower minimum vorticity magnitude would not affect the boundary envelope noticeably.\nAt $t\/t_{\\ell}=0$ the turbulence is contained within a spherical region defined by the window function.\nAs the turbulence evolves, the transition region is mixed with the turbulence inside and becomes gradually indistinguishable around $t\/t_{\\ell}\\sim 1.5$. At $t\/t_{\\ell}\\sim 4.0$ more fine features have developed near the boundary while the general spherical shape is still maintained. Around $t\/t_{\\ell}\\sim 8.6$, small features start to merge and create protrusions. Meanwhile the general shape has also become more ellipsoidal. The DNS flow evolution is simulated up to $t\/t_{\\ell}=17.5$. From $t\/t_{\\ell}=8.6$ to $17.5$ the cloud of turbulence becomes more irregular, and finer scales are less evident as the turbulence decays.\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=.79\\textwidth]{figs\/fig4.jpg}\n \\put(-1, 97) {\\small(a)}\n \\put(30, 97) {\\small(b)}\n \\end{overpic}\n \\caption{\\small Vorticity magnitude iso-surface of (a) DNS\\_0 and (b) LES\\_0 at $t\/t_\\ell=0, 1.72, 4.02, 8.60, 17.47$ from top to bottom, where $t_\\ell$ is the large eddy turnover time of the initial IHT field. }\\label{fig:DNS}\n\\end{figure}\n\n\\subsection{Comparison between DNS and LES}\\label{sec:DNSLES_visualization}\nDNS is applied to study the more active early stage evolution of a turbulence cloud but it is computationally expensive to reach late times. To study the long-term behavior we turn to LES calculations of the same setup. To ensure LES calculations are able to accurately capture the abiding features of the flow, we qualitatively compare the evolution for DNS and LES of the same case.\n\nThe initial condition for the LES run is created in the following way: first, the same IHT field from case DNS\\_0 is spectrally filtered from $256^3$ to $32^3$, keeping only $(1\/8)^3$ of its original spectrum; second, the same recipe (tiling and spherical windowing) is used with the filtered turbulence field to create a spherical region of under-resolved turbulence. This field is then given to the LGF finite-volume solver with the SGS model turned on. This simulation corresponds to the case `LES\\_0' in table \\ref{tab:simulation_summary}.\n\nFigure \\ref{fig:DNS} also compares DNS\\_0 (\\ref{fig:DNS}a) and LES\\_0 (\\ref{fig:DNS}b) at $t\/t_\\ell=0, 1.72, 4.02, 8.60, 17.47$. To ensure that the difference in grid resolution between DNS and LES would not affect the visualization, all iso-surfaces are re-sampled to the same grid. The LES captures the general shape and most of the large-scale features such as the radius, the ellipticity, the sizes and locations of the protrusions. On the other hand, some small-scale features near the boundary are missed. We also noticed that the vorticity is less intense in the LES run (the crimson regions) especially towards the early stage. All of these differences are to be expected, as LES is designed to capture the statistical properties of the turbulence (and specifically their influence on the largest scales). Nevertheless, over the time range displayed in figure \\ref{fig:DNS} there is little decorrelation of the large scales in DNS and LES originating from the same initial condition. To further quantify the comparison, in \\S~\\ref{sec:statistics}, four statistical measures are introduced and applied to both cases DNS\\_0 and LES\\_0.\n\n\n\\section{Quantitative evolution}\\label{sec:statistics}\n\\subsection{Statistical measures for DNS and LES}\\label{sec:stats_DNSLES}\n\nIn this section we quantify the initial evolution of the cloud of turbulence using statistical measures. DNS results are compared with LES during the initial decay period up to about $t \/ t_\\ell \\simeq 20$.\n\nFirstly the kinetic energy ${\\cal E}(t)$ decay is studied. Results from three simulations are compared in figure \\ref{fig:Ek_decay}: (1) DNS\\_0; (2) LES\\_0 and (3) an under-resolved DNS (the same setup as LES\\_0 but with SGS model turned off).\nDNS\\_0 should be regarded as the most accurate case among all three and its value at $t=0$ is used to normalize all results.\nFor LES\\_0 we show both the kinetic energy resolved by the grid and a `total kinetic energy' which is the sum of the resolved energy and the estimated subgrid energy predicted by the SGS model. The total kinetic energy in LES\\_0 compares well with DNS\\_0. The initial resolved energy in LES\\_0 is smaller than that in DNS\\_0 owing to the spectral filtering process discussed in \\S~\\ref{sec:initial_conditions}.\nOn the other hand the under-resolved DNS shows evident energy pile-up due to the lack of the SGS model.\nAfter $t\/t_\\ell \\simeq 8$, the resolution of case LES\\_0 is high enough to resolve all flow scales due to the decay and it is effectively `DNS' after this point.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{figs\/fig5.eps}\n \\caption{\\small Decay of the kinetic energy ${\\cal E}(t)$ for different simulations: DNS\\_0 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black,solid,line width = 1pt](0,0) -- (5mm,0);}}); LES\\_0 resolved kinetic energy (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,red,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}); LES\\_0 total kinetic energy (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,red,dashed,line width = 1.0pt](0,0) -- (5mm,0);}}); an under-resolved DNS (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,blue,dashdotted,line width = 1.0pt](0,2) -- (5mm,2);}}). The initial kinetic energy in DNS\\_0 is used to normalize all simulations.}\n \\label{fig:Ek_decay}\n\\end{figure}\n\nSecondly the total energy spectrum defined in \\S~\\ref{sec:ic} (equation~\\eqref{eq:greg}) is applied again here. Results for DNS\\_0 and LES\\_0 are compared in figure \\ref{fig:total_spectrum} at three time instants $t\/t_\\ell=0,4.02,17.47$, corresponding to the visualizations in figure \\ref{fig:DNS}. The agreement is significant and LES seems to only suppress the high wavenumbers slightly. For both cases, the initial condition features about a decade of inertial-range turbulence with a $k^{-{5 \/ 3}}$ spectrum; as expected, this region shrinks (from the high wavenumbers) as the turbulence decays.\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=.75\\textwidth]{figs\/fig6.eps}\n \\put(15,73) {\\small(a)}\n \\put(15,42) {\\small(b)}\n \\put(15,12) {\\small(c)}\n \\end{overpic}\n \\caption{\\small Total energy spectrum at (a) $t\/t_\\ell=0$, (b) $t\/t_\\ell=4.02$ and (c) $t\/t_\\ell=17.47$ for DNS\\_0 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black,solid,line width = 1pt](0,0) -- (5mm,0);}}) and LES\\_0 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,red,dashed,line width = 1.0pt](0,0) -- (5mm,0);}}). The same guide line for $k^{-{5 \/ 3}}$ as in figure \\ref{fig:intial_Ek} is also given (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}).}\n \\label{fig:total_spectrum}\n\\end{figure}\n\nThe third statistical measure we consider is the energy spectrum on a spherical shell. The flow is designed to be homogeneous in the azimuthal and polar directions but is only so in the radial direction deep within the cloud. As shown in \\S~\\ref{sec:DNS} visually the cloud also undergoes radial growth over time. This non-uniformity in the radial direction suggests that one should further characterize the energy spectrum as a function of the radius $r$ and time $t$. A special spectrum defined on a spherical shell of a given radius is applied \\citep{Lombardini2014}. Similar to the total energy spectrum $E(k)$, it seeks a relationship between the energy and the wavenumber, where the wavenumber on a spherical shell is defined using the spherical harmonics. The spherical shell wavenumber and the classical wavenumber defined using Fourier transform can be related via the Laplace operator. This relation also connects the spherical shell spectrum to the classical energy spectrum.\n\nFollowing the detailed derivation given by \\citet{Lombardini2014}, we acquire the shell spectrum for a field $f_r(\\theta, \\phi)$ defined for a given raidus $r$ by expanding the field using spherical harmonics:\n\\begin{align}\\label{eq:harmonicsExpansion}\n f_r(\\theta, \\phi)=\\sum_{\\ell=0}^{\\infty} \\sum_{m=-\\ell}^{\\ell} f_{\\ell m} Y_{\\ell m}(\\theta, \\phi),\n\\end{align}\nwhere\n\\begin{align}\n Y_{\\ell m}(\\theta, \\phi)=\\left\\{\\begin{array}{ll}{N_{(\\ell, m)} P_{\\ell}^{m}(\\cos \\theta) \\cos (m \\phi)} & {m \\geqslant 0} \\\\ {N_{(\\ell,|m|)} P_{\\ell}^{m |}(\\cos \\theta) \\sin (|m| \\phi)} & {m<0}\\end{array}\\right. ,\n\\end{align}\nwith $P_{\\ell}^{m}$ being the associated Legendre polynomials, $N_{(\\ell, m)}$ being the normalization constant and $\\ell$ being the equivalent wavenumber. The wavenumber $\\ell$ is then related to the classic wavenumber $k$ defined through the Fourier transform by,\n\\begin{align}\n k^{2}=\\ell(\\ell+1) \/ r^{2}.\n\\end{align}\nAssuming a power law for the energy spectrum $E(k) \\sim k^{-\\alpha}$ one has the following relationship between the energy spectrum and the shell spectrum,\n\\begin{align}\\label{eq:Cl}\n E(k)\\sim k^{-\\alpha} \\sim\\ell C_{\\ell} ,\n\\end{align}\nwhere\n\\begin{align}\n C_{\\ell}=\\frac{1}{2 \\ell+1} \\sum_{m=-\\ell}^{\\ell} f_{\\ell m}^{2}.\n\\end{align}\nEquation \\eqref{eq:Cl} suggests that one can understand the shell spectrum $C_\\ell$ in a similar way as the classical energy spectrum $E(k)$.\n\nThe shell spectra for DNS\\_0 and LES\\_0 are shown in figure \\ref{fig:sphericalShell}. Results for various radii ($r\/B=0.32,0.48,0.64,0.80,0.96$) and time instants ($t\/t_\\ell=0, 4.02, 17.47$) are given. At $t\/t_\\ell=0$ (figure \\ref{fig:sphericalShell}: a, d) all 5 curves collapse together as expected since they represent the original IHT field. The energy decays over time, but the dependence of the shell spectrum on the radius $r$ is weak. It seems that the boundary does not have a strong effect on the turbulence decay. This evidence further supports the assumption of local homogeneity that underpins our definition of the total spectra used above, as discussed in the appendix, at least up through the times considered here.\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=.95\\textwidth]{figs\/fig7.eps}\n \\put(12,59) {\\small(a)}\n \\put(12,34) {\\small(b)}\n \\put(12,9) {\\small(c)}\n \\put(58,59) {\\small(d)}\n \\put(58,34) {\\small(e)}\n \\put(58,9) {\\small(f)}\n \\end{overpic}\n \\caption{\\small\n Spherical-shell spectrum at different radii and times.\n The left column (a-c) are results from DNS\\_0 at $t\/t_\\ell=0, 4.02, 17.47$ respectively. The right column (d-f) are results from LES\\_0 at the same times. For each figure, the gradation in color corresponds to radii $r\/B=(0.32,0.48,0.64,0.80,0.96)$ from darkest to lightest shade. An equivalent guide line for $k^{-{5 \/ 3}}$ as in figure \\ref{fig:intial_Ek} is given (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}).\n }\n \\label{fig:sphericalShell}\n\\end{figure}\n\n\\subsection{Long-term statistics and low wavenumber behavior}\\label{sec:longterm_statistics}\nThe long-term evolution of a turbulence cloud is studied through LES. Two cases LES\\_0 and LES\\_IC2 from table \\ref{tab:simulation_summary} are simulated up to $t\/t_\\ell=400$ where LES\\_0 has a $k^2$-type initial spectrum while LES\\_IC2 has a $k^4$-type. The evolution is visualized in figure \\ref{fig:longterm_LES}. The spread of the cloud is similar in both cases, but details of the large-scale structures are different.\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=0.75\\textwidth]{figs\/fig8.jpg}\n \\put(0,98) {\\small(a)}\n \\put(28,98) {\\small(b)}\n \\end{overpic}\n \\caption{\\small Long-term evolution of (a) LES\\_0 and (b) LES\\_IC2 at $t\/t_\\ell=0, 4, 17, 66, 143, 263$ from top to bottom. }\\label{fig:longterm_LES}\n\\end{figure}\n\nFigure \\ref{fig:longterm_Ek}a shows the long-term evolution of the kinetic energy decay. In the case of IHT, \\citet{saffman1967large} predicts an asymptotic decay rate of $t^{-6\/5}$ for the $k^2$ turbulence and \\citet{kolmogorov1962refinement} predicts a decay rate of $t^{-10\/7}$ for the $k^4$ spectrum (both guide lines are indicated in the figure). However in both the case of LES\\_0 and LES\\_IC2, the decay is similar and closer to the Saffman scaling. Figure \\ref{fig:longterm_Ek}b shows the evolution of the integral scale over time for LES\\_0 and LES\\_IC2, compared to the theoretical asymptotic growth rate for Saffman IHT ($t^{2\/5}$) and Batchelor IHT ($t^{2\/7}$). As in the energy decay, both cases are closer to the Saffman type.\n\nThis apparent discrepancy with the theory can be clarified by examining the long-term decay of the total energy spectrum depicted in figure \\ref{fig:longterm-spectrum}, which shows results for both LES\\_0 ($k^2$) and LES\\_IC2 ($k^4$) cases. The $k^4$ spectrum is similar to that reported in \\citet{ishida2006decay}. As expected the coefficient of the limiting $k^2$ spectrum for LES\\_0 is invariant, but the coefficient of the $k^4$ term for LES\\_IC2 is increasing over time first rapidly up to about $t \/ t_{\\ell} \\simeq 20$, and then more slowly, as shown in figure~\\ref{fig:loitsyansky}. The coefficient of the $k^4$ term is proportional to the Loitsyansky integral $I$ given by equation~\\eqref{eq:loit} which is assumed constant in the theory under the assumption that remote points be statistically independent \\citep{loitsyansky1939some}.\n\nSuperposition of the energy spectra for cases LES\\_0 and LES\\_IC2 shows they are similar for $k R > 1$, corresponding to a wavelength $\\lambda = 2 \\pi R$. Apart from the very largest scales, which cannot be seen in visualizations like figure~\\ref{fig:longterm_LES}, the two simulations are otherwise statistically similar. The weak vortex ring and vortex ring dipole associated with the $k^2$ and $k^4$ terms would only become evident as $t \\rightarrow \\infty$, after which all turbulence will decay. However at the same time, the properties are entirely predictable from their initial conditions. Indeed, figure~\\ref{fig:meander} shows that the entire cloud of LES\\_0 meanders in space but eventually attains a trajectory that is associated with the initial impulse.\n\nThe predictability of the long-term evolution from the initial condition argues against any universality of the very largest scales of the spherical cloud of turbulence. While we expect the wavenumber spectrum for $k R > 1$ is approximately universal, the low wavenumber behavior is always an artifact of initial and boundary conditions.\n\nWhile this lack of universality is perhaps unsurprising, the veracity of the Saffman-type decay-rate predictions even in the absence of a $k^2$ spectrum is interesting. Consider the process by which the initial $k^4$ spectrum is created for the case LES\\_IC2, whereby a weak vortex ring is added to offset the initial impulse associated with windowing the IHT field. While we superposed this ring at the center of our cloud, we could have cancelled the impulse by adding a ring at any position, even one very far from the cloud. Over the timescale simulated here, the results would be identical to those of the $k^2$ cloud, and one would have to go to even lower values of $k R$ in order to see the ultimate $k^4$ behavior.\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=0.75\\textwidth]{figs\/fig9.eps}\n \\put(18,62) {\\small(a)}\n \\put(18,45) {\\small(b)}\n \\end{overpic}\n \\caption{ \\small Long-term evolution of (a) the kinetic energy decay compared with asymptotic behavior of Saffman IHT $\\mathcal{E}(t) \\sim t^{-6\/5}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}) and Bathelor IHT $\\mathcal{E}(t) \\sim t^{-10\/7}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dashdotdotted,line width = 1.0pt](0,2) -- (5mm,2);}}); (b) the integral scale growth for case LES\\_0 (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,red,solid,line width = 1pt](0,0) -- (5mm,0);}}) and LES\\_IC2 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,blue,dashed,line width = 1.0pt](0,0) -- (5mm,0);}}) up to $t\/t_\\ell = 400$ compared with asymptotic behavior of Saffman IHT $\\ell \\sim t^{2\/5}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}) and Bathelor $\\ell \\sim t^{2\/7}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dashdotdotted,line width = 1.0pt](0,2) -- (5mm,2);}}). }\\label{fig:longterm_Ek}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=0.75\\textwidth]{figs\/fig10.eps}\n \\put(18,93) {\\small(a)}\n \\put(18,46) {\\small(b)}\n \\end{overpic}\n \\caption{ \\small Long-term evolution of the total spectrum for (a) LES\\_0 and (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,red,solid,line width = 1pt](0,0) -- (5mm,0);}}) (b) LES\\_IC2 (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,red,blue,line width = 1pt](0,0) -- (5mm,0);}}), up to $t\/t_\\ell=500$ with $\\Delta t\/t_\\ell=15$ between each line. In figure (a) guide lines for $k^{-5\/3}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}) and $k^2$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dashdotted,line width = 1.0pt](0,2) -- (5mm,2);}}) are given. In figure (b) guide lines for $k^{-5\/3}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}) and $k^4$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dashdotted,line width = 1.0pt](0,2) -- (5mm,2);}}) are given. }\\label{fig:longterm-spectrum}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=0.75\\textwidth]{figs\/fig11.eps}\n \\end{overpic}\n \\caption{ \\small\n Long-term evolution of the normalized Loitsyansky integral $I(t)\/I(0)$ of LES\\_IC2 ($k^4$ type), up to $t\/t_\\ell=500$. }\\label{fig:loitsyansky}\n\\end{figure}\n\n\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=0.9\\textwidth]{figs\/fig12.eps}\n \\end{overpic}\n \\caption{ \\small Trajectory of the center of the turbulence cloud from $t\/t_\\ell=0$ to $400$ for LES\\_0 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,red,solid,line width = 1pt](0,0) -- (5mm,0);}}) and LES\\_IC2 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,red,blue,line width = 1pt](0,0) -- (5mm,0);}}) with every marker sepearted by $\\Delta t\/t_\\ell=30$. For LES\\_0 the direction of the initial impulse is indicated with the arrow at the end of the trajectory. For LES\\_IC2 the impulse is zero.}\\label{fig:meander}\n\\end{figure}\n\nLastly we consider the radial growth of the turbulence cloud over time. Because the cloud does not hold its sphericity we define the radius by a statistical moment\n\\begin{align}\n\\overline{r} &=\\left(\\frac{\\int u^{2}|\\boldsymbol{x} -\\boldsymbol{x}_c|^{p} \\mathop{}\\!\\mathrm{d} \\boldsymbol{x} }{\\int u^{2} \\mathop{}\\!\\mathrm{d} \\boldsymbol{x} }\\right)^{1 \/ p},\n\\end{align}\nwhere $u$ is the velocity magnitude and $\\mathbf{x_c}$ is the center of the turbulence cloud, defined using\n\\begin{align}\n\\mathbf{x_c}=\\frac{\\int \\mathbf{x}\\, u^2 \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}}{\\int u^2 \\mathop{}\\!\\mathrm{d} \\boldsymbol{x}}.\n\\end{align}\nA definition of the center is necessary because, as discussed in \\S~\\ref{sec:initial_conditions}, the final stage of a turbulence cloud is a large vortex ring drifting in the direction of the impulse.\nAlso $p \\leq 2$ is needed for $\\overline{r} $ to exist as the velocity field $\\boldsymbol{u}(\\mathbf{x}) \\sim 1\/|\\mathbf{x}|^3$ as $|\\mathbf{x}|\\rightarrow \\infty$. Here we only consider the case when $p = 2$ for simplicity. The results between LES\\_0 and LES\\_IC2 are shown in figure \\ref{fig:longterm_radius}. The mean radius growth in time is almost the same for both cases, and approaches a power-law behavior that is similar to the growth of the integral scale.\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=0.75\\textwidth]{figs\/fig13.eps}\n \n \n \\end{overpic}\n \\caption{ \\small Comparison of the long-term mean radius $\\overline{r}$ for $p=2$ between LES\\_0 (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,red,solid,line width = 1pt](0,0) -- (5mm,0);}}) and LES\\_IC2 (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,blue,dashed,line width = 1.0pt](0,0) -- (5mm,0);}}) up to $t\/t_\\ell \\sim 400$. Results are normalized with their initial mean radii $\\bar{r}_o$. Guide lines for $t^{2\/5}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dotted,line width = 1.0pt](0,2) -- (5mm,2);}}) and $t^{2\/7}$ (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,green!40!gray,dashdotdotted,line width = 1.0pt](0,2) -- (5mm,2);}}) are also given.}\\label{fig:longterm_radius}\n\\end{figure}\n\n\n\n\\section{Vortex ring ejections}\\label{sec:vortex_rings}\n\nOne of the most distinctive features in the late-stage evolution of a turbulence cloud (figure \\ref{fig:longterm_LES}) is that it ejects vortex rings of roughly the same size from its boundary. In this section, we investigate the relation between the size of the vortex rings and properties associated with the turbulence.\n\nWe create LES simulations which independently vary the three independent nondimensional parameters that control the initial conditions. Long-term evolution for all three pairs at $t=260$ are provided in Figure \\ref{fig:ejectionComparison}. The first parameter is the width of the transition region associated with the windowing function, $\\sigma\/R$, which is varied from [0.05,0.1,0.2] in three cases [LES\\_D1, LES\\_0, LES\\_D2]. We see that the width of the transition region has little influence on the number or scale of the ejections. Next, we consider varying the microscale Reynolds number $\\Rey_\\lambda$, which is varied from [45.0, 76.9, 122.4] in three cases [LES\\_R2, LES\\_R1, LES\\_0]. Again, though each cloud has a different range of scales present, the vortex ejections occur again at roughly the same scale. Finally, we vary initial integral scale $\\ell\/R$, by changing the size of the initial periodic box to the sphere radius, $B\/R$ over the range [0.5,1.0,2.0] for cases [LES\\_B1, LES\\_0, LES\\_B2]. Quite evidently, the size of the ejections is halved for case LES\\_B1 and doubled for case LES\\_B2, compared to the baseline LES\\_0. Therefore we conjecture that the vortex rings are generated by the integral-scale structures in the original IHT field.\n\n\n\\begin{figure}\n \\centering\n \\begin{overpic}[width=.95\\textwidth]{figs\/fig14.png}\n \\put(-2, 91) {\\small $\\sigma\/R=[0.05,0.1,0.2]$}\n \\put(-2, 59) {\\small $\\Rey_\\lambda= [45.0, 76.9, 122.4]$}\n \\put(-2, 30) {\\small $B\/R=[0.05,0.1,0.2]$}\n \\end{overpic}\n \\caption{\\small Long-term turbulence cloud evolution with vortex ring ejections for cases defined in table \\ref{tab:simulation_summary}. First row: LES\\_D1, LES\\_0, LES\\_D2; second row: LES\\_R2, LES\\_R1, LES\\_0; third row: LES\\_B1, LES\\_0, LES\\_B2.\n }\\label{fig:ejectionComparison}\n\\end{figure}\nWe hypothesize that the ejections occur due to a local imbalance of impulse associated with the IHT field. Consider a Gaussian weighted impulse centered at point $\\boldsymbol{x}$, with a `width' $\\varsigma$\n\\begin{align}\n \\boldsymbol{I}(\\boldsymbol{x};\\varsigma) = \n \\int_{\\mathbb{R}^3} e^{-\\frac{|\\boldsymbol{x}-\\boldsymbol{x}'|^2}{2\\varsigma^2}} \\boldsymbol{u}(\\boldsymbol{x}')\\mathop{}\\!\\mathrm{d} \\boldsymbol{x}'.\n\\end{align}\nFigure \\ref{fig:impulse} shows the maximum impulse $\\boldsymbol{I}(\\boldsymbol{x};\\varsigma)$ over $\\boldsymbol{x}$, as a function of the width $\\varsigma\/ \\ell_o$ at $t\/t_\\ell=0$, where $\\ell_o$ is the initial integral scale. The maximum Gaussian weighted impulse reaches its maximum when $\\varsigma \/ \\ell_o$ is around $1.8$.\n\nFor points deep within the cloud, imbalance of the locally filtered impulse would simply result in complicated local vortex dynamics. However, near the edge of the cloud, this imbalance, when pointed outwards, would eject vorticity out of the cloud. In some sense, this process is universal as the scale is a property of the IHT field itself, and the net imbalance would create ejections near the edge of any region of IHT. This result also agrees with studies of TNTIs. It was discussed in \\citet{townsend1980structure} that, while a wide range of turbulence scales affect the evolution of the turbulence boundary, the largest distortion at the TNTI is from the largest eddies in the turbulence.\n\n \\begin{figure}\n \\centering\n \\begin{overpic}[width=0.75\\textwidth]{figs\/fig15.eps}\n \\end{overpic}\n \\caption{ \\small Maximum Gaussian weighted impulse over $\\boldsymbol{x}$, $\\max_{\\mathbf{x}} I(\\mathbf{x};\\varsigma)$ as a function of `width' $\\varsigma$ for LES\\_0 at $t=0$ (\\protect \\raisebox{2pt}{\\tikz{\\draw[-,red,solid,line width = 1pt](0,0) -- (5mm,0);}}).}\\label{fig:impulse}\n \\end{figure}\n\n\n\\section{Concluding remarks}\n\\label{sec:conclusions}\n\nWe used DNS and LES to study a novel turbulent flow representing an isolated spherical region of turbulence evolving in free space. This flow is created by tiling a periodic IHT field in space and windowing it to be zero outside a spherical region. The DNS is used to validate the LES, which is in turn used to study the long-time evolution of the turbulence.\n\nThe flow exhibits aspects of both homogeneous turbulence, deep within the sphere, as well as inhomoengoues turbulence near the TNTI.\nFor strictly homogeneous turbulence, a spectrum of either the Saffman $k^2$ type or the Batchelor $k^4$ type determines the kinetic energy decay rate and the the integral scale growth rate. For spherical region of turbulence we showed that both types of initial conditions can also be created. For the cloud, we confirm, by comparing spectra on spherical shells of different radii from the initial center, that the turbulence remains locally homogeneous deep within the cloud. However, the resulting long-term decay of the kinetic energy and the growth of the integral scale are similar in both cases, and closer to the predictions of the Saffman theory. This may be related to an observed growth in the Loitsyansky integral, but which is assumed constant in the Batchelor characterization of the turbulence. At least through about 400 eddy turnover times, there is little difference in the shape of the respective spectra between the two cases for $k$ values near the inertial scale, and it appears that the integral scale is relatively unaffected by the behavior at very low $k$, whether $k^2$ or $k^4$. In any event, the spectrum at these wavenumbers is controlled by the initial conditions and may not be universal. Finally, we defined a mean radius of the turbulence cloud in terms of its velocity moments, and showed the turbulence gives rise to a similar growth of radius as of the integral scale.\n\nThe spherical region of turbulence is bounded by a TNTI that evolves into distinct large-scale features. By varying each of the three independent nondimensional parameters controlling the cloud, we find that the structures are related to the (initial) integral scale of the IHT field. The TNTI features include vortex rings that are ejected from the cloud. We hypothesize that this evolution is associated with an imbalance in specific impulse over the integral scale, which, near the TNTI, gives rise to the vortex rings.\n\n\n\n\\section*{Acknowledgements}\nThis work was supported by the ONR grant No. N00014-16-1-2734 and the AFOSR\/UCLA grant No. FA9550-18-1-0440.\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfxsj b/data_all_eng_slimpj/shuffled/split2/finalzzfxsj new file mode 100644 index 0000000000000000000000000000000000000000..acbc963b70f8323297576e7ec2f8ac08725d9714 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfxsj @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe mechanisms through which massive (stellar mass M$_*$$\\approx$$10^{11}$ M$_\\odot$) \nspheroidal galaxies assemble and shape their stellar mass and then quench \ntheir star formation are still unclear and represent central\ntopics in galaxy evolution.\nLinking the population of galaxies at different redshifts to constrain \ntheir evolution, it is not always a reliable technique, since it is affected \nby the well known progenitor bias problem \n\\citep[e.g.][]{vandokkum01a,carollo13}.\nThe study of spheroids, compared to other types of galaxies, is potentially less \naffected by this bias, as once an high-density bulge is formed, it is unlikely \nthat it is disrupted, or assembles efficiently a surrounding disc \\citep[e.g.][]{brooks16}.\nTherefore, even if not all the local spheroidal galaxies may have a spheroid \nas progenitor, high-redshift spheroidal galaxies are most probably the progenitors\nof some of the local ones.\n\nSimulations suggest that an early intense burst\nof star formation followed by quenching, is required to reproduce \nthe detailed structural properties \nof ellipticals and to match the observed tight scaling relations\n\\citep{ciotti07,Naab07,oser12,porter14a,brooks16}.\nObservations show that only a minor fraction of local\nspheroids has accreted newly stellar mass through secondary events of \nstar formation \\cite[e.g.][]{thomas10,gargiulo16}.\nTherefore, the stellar populations of high-redshift spheroidal \ngalaxies keep the information of the early-phases of their formation. \nIf their evolution is mostly characterized by passive ageing, then\ntheir stellar population properties hold on nearly unchanged till now.\nConsequently, stellar chemical composition and age are powerful tools to link \nspheroids across time and, most importantly, to constrain their formation \nand their past evolution. \n\nCorrelations between stellar population parameters and structural parameters of \nearly-type galaxies have been extensively studied in the past.\nThe line strengths of early-type galaxies in the optical spectral range are found to be\ncorrelated or anticorrelated to the velocity dispersion, depending whether\nthe lines are more sensitive to metallicity or age effects \n\\cite[e.g.][]{bender93,fisher95,colless99,jorgensen99,trager00,\nbernardi03, bernardi06, harrison11, mcdermid15, jorgensen17}.\nThese relationships are observed both for field and cluster early-type galaxies \nin the local universe and it seems that environment does not affect them significantly\n \\citep[e.g.][]{bernardi06,mcdermid15}.\nFor cluster early-types these relations are observed and well defined up to $z$$\\sim$0.9\n\\citep{jorgensen17}. \n\nGiven the dependence of line-strengths on metallicity and age,\nit is expected a correlation between age, metallicity\nand velocity dispersion.\nIndeed, the metallicity [Z\/H] of local spheroidal galaxies, tipically solar or super-solar \n\\cite[e.g.][]{buzzoni92,jorgensen99,thomas05}, \nis found to correlate with central velocity dispersion \n\\citep{greggio97,jorgensen99,trager00,thomas05,thomas10,harrison11,mcdermid15}.\nContrary to metallicity, the correlation between age and velocity \ndispersion is much less evident and it is more uncertain.\nSome authors find weak but significant correlation, with older galaxies \ncharacterized by higher velocity dispersions \n\\cite[e.g.][]{thomas05,gallazzi05,gallazzi14,harrison11,mcdermid15}.\nOther authors, find shallower correlations and slopes consistent\nwith zero \\cite[e.g.][]{trager00, jorgensen17}, especially when age \nis estimated over an aperture large enough to enclose most of the galaxy \nlight \\citep{labarbera10a}.\nOverall, considering velocity dispersion a proxy of galaxy mass,\nthe above relations suggest that more massive galaxies are more \nmetal rich and, possibly, older than their lower mass counterparts.\n\nActually, the mass-age and the mass-metallicity relations are observed \n\\citep[e.g.][]{thomas10,gallazzi14,mcdermid15} and both appear well defined \nwhen dynamical mass is considered, with the slope of the mass-age \nrelation being rather flat.\nThus, most massive early-type galaxies in the local universe are also the \nmost metal rich, which appears counter-intuitive if they are also the oldest \n\\citep{greggio11}.\nHowever, this could be explained by a deep gravitational potential \n(i.e. a large dark matter halo) that, on one hand, modulates an intense star formation \nand, on the other hand, efficiently retains the metals rapidly produced by \nhigh-mass stars.\nAnother possibility is that, metal rich stars are added later \nto the stellar population through later episodes of star formation. \n\nWhether and which of the above scenarios is the right one \ncan be assessed by studing the stellar population properties of high mass \nspheroids in the past, by establishing whether the relations between stellar \npopulation properties and physical and dynamical properties were already \nin place early on in the evolution of spheroids or not, and whether\nthe local age-velocity dispersion and age-mass relations hold on\nat high-redshift once evolved back in time.\n\nTo investigate these issues, we have performed new spectroscopic observations \nfor 14 spheroids candidate\nmembers of the cluster XLSSJ0223-0436 at $z\\sim1.22$.\nThe whole data set will be presented in a forthcoming paper.\nHere, we present the analysis of the seven spheroids whose spectra \nhave S\/N high enough to allow for the study of their stellar population\nproperties.\nThe paper is organized as follows.\nSection 2 briefly describes the sample, the observations and the data reduction.\nSection 3 describes the measurement of redshift and absorption line indices.\nIn Section 4, we describe the constraints obtained from absorption line strengths on\nage and metallicity of the seven spheroids.\nIn Section 5 we make use of full spectral fitting to derive age and metallicity\nof the galaxy stellar populations and to constrain their star formation histories (SFHs).\nIn Section 6 we study the relationships of age and metallicity with\nphysical and structural parameters, velocity dispersion, dynamical mass\nand stellar mass density.\nSection 7 presents a summary of results and conclusions.\n\nThroughout this paper we use a cosmology with\n$H_0=70$ Km s$^{-1}$ Mpc$^{-1}$, $\\Omega_m=0.3$, and $\\Omega_\\Lambda=0.7$.\nAll the magnitudes are in the Vega system, unless otherwise specified.\n\n\n\\section{Spectroscopic observations and data reduction}\n\\label{sec:observations}\nThe seven spheroidal galaxies belong to a sample of 23 spheroids candidate\ncluster members,\nphotometrically selected in the field around the cluster XLSSJ0223-0436 at $z=1.22$\n\\citep{andreon05,bremer06}.\nBriefly, the target galaxies were selected among all the galaxies brighter \nthan $z_{850}<24$ \nwithin a projected radius D$\\le1$ Mpc from the cluster centre.\nThen, according to their $i_{775}-z_{850}$ colour, galaxies were selected \nwithin $\\pm$0.2 mag from the peak of the color distribution centered\nat the color of the brightest cluster members.\nFinally, a visual classification on the F850LP images was used to select \nonly those galaxies with elliptical\/spheroidal morphology.\nA detailed description of the whole sample, the structural parameters of galaxies\nand their scaling relations are given in \\cite{saracco17}.\n\n\\begin{table*}\n\\begin{minipage}[t]{1\\textwidth}\n\\caption{List of observed galaxies in the XLSSJ0223 field and\nspectroscopic redshift measurements. }\n\\label{tab:sample}\n\\centerline{\n\\begin{tabular}{rccccccccccr}\n\\hline\n\\hline\n ID & RA & Dec & F850LP & $i_{775}-z_{850}$& log$\\mathcal{M}_*$& log$\\mathcal{M}_{dyn}$& $z_{spec}$ & Em$^a$& F([OII])$^b$ & SFR& S\/N$^c$\\\\\n & (h:m:s)& (d:p:s)& (mag) & (mag) &(M$_\\odot$) &(M$_\\odot$) & & & (erg cm$^{-2}$ s$^{-1}$)& (M$_\\odot$ yr$^{-1}$)&\\\\\n\\hline\n 651& 02:23:05.759& -04:36:10.27& 21.62$\\pm$0.01 & 1.09$\\pm$0.02 & 10.94&11.29& 1.2192& [NeV]& 6$\\pm$2& 0.7$\\pm0.2$ \t & 11.3 \\\\ \n 972& 02:23:04.718& -04:36:13.47& 22.61$\\pm$0.03 & 0.92$\\pm$0.03 & 10.54&11.32& 1.2153& ...& $<$6 &$<$0.8 \t & 6.2 \\\\ \n1142& 02:23:03.262& -04:36:14.60& 21.30$\\pm$0.01 & 1.00$\\pm$0.02 & 11.50&11.68& 1.2204& ...& $<$1.5 &$<$0.2 \t & 12.7 \\\\ \n1370& 02:23:02.021& -04:36:43.26& 23.10$\\pm$0.03 & 0.96$\\pm$0.04 & 10.08&10.10& 1.2249& ...& $<$7 &$<$0.9 \t & 5.6 \\\\ \n1442& 02:22:57.980 & -04:36:22.31& 21.88$\\pm$0.01 & 0.91$\\pm$0.02 & 10.80&10.67& 1.2250& [OII] & 27$\\pm$4 & 3.0$\\pm0.4$ & 11.3 \\\\ \n1630& 02:23:00.929& -04:36:50.19& 21.48$\\pm$0.01 & 1.04$\\pm$0.02 & 10.75&11.18& 1.2109& [OII]& 34$\\pm$6 & 4.0$\\pm0.6$ & 12.2 \\\\ \n1711& 02:22:59.990& -04:36:02.53& 20.92$\\pm$0.01 & 1.02$\\pm$0.02 & 11.20&11.52& 1.2097& ... & ... &... \t\t & 18.2 \\\\ \n\\hline\n\\end{tabular}\n}\n{$^a$ Emission lines detected: dots - no emission line $>$1$\\sigma$.\\\\\n$^b$ Flux is in units of $10^{-18}$ erg cm$^{-2}$ s$^{-1}$.\\\\\n$^c$ S\/N per \\AA ngstrom in the rest-frame of the galaxy, estimated in the\ninterval $\\sim$4000-4150 \\AA.}\n\\end{minipage}\n\\end{table*}\n\nSpectroscopic observations of the targets \nwere performed in multi-object spectroscopic\nmode (MOS) with the Multi-Object Double Spectrograph (MODS, 1 and 2) \n\\citep{pogge10} mounted at the Large Binocular Telescope (LBT).\nObservations confirmed 13 spheroidal galaxies\nmembers of the cluster XLSSJ0223.\nThe whole specroscopic data will be presented in a forthcoming paper\n(Saracco et al. 2018, in preparation).\nHere, we focus the analysis on the seven spheroids having the highest S\/N,\nsuited to stellar population study.\n\nObservations were carried out \nwith filter GG495 coupled with the grim G670L sampling the wavelength range \n0.5$\\mu$m$<\\lambda<1.0\\mu$m at 0.85 \\AA\/pix.\nWe adopted a slit width of 1.2'' resulting in a spectral resolution \nR$\\simeq 1150$, corresponding to a FWHM$\\simeq$7.4 \\AA\\ at 8500\\AA.\nA bright star was put in a slit to accurately\nmeasure the offsets in the observing sequence and perform the\ncorrection for telluric absorption lines (see below).\nObservations, consisting in a sequence of exposures (ABBA) \nof 900 sec each at dithered positions offset by $\\sim5$\",\nwere collected both with MODS1 and MODS2\nfor a total effective integration time of 8 hours.\n\nRaw data were first pre-processed by the MODS pipeline that applies \ncorrection for bias-subtraction, flat-field (pixel-to-pixel) variation \nand for optical distortions, providing also the wavelength calibration \nthrough the inverse dispersion solution (0.08\\AA\\ rms accuracy).\nThen, standard reduction was performed with IRAF tasks.\nA first sky-subtraction was applied by subtracting from each frame\nthe following one in the dithering observing sequence.\nThe sky-subtracted frames were aligned to sub-pixel scale \nusing {\\it drizzle} resampling algorithm, and than co-added.\nThe shifts to align the images were estimated using the \npositions of the peak of the spectrum collapsed along the wavelengths \nof the bright star present in the masks.\nBefore co-adding, MODS1 and MODS2 \n2D spectra were corrected for the relative spectral slit flats, \nobtained using quarz-halogen flat fields taken through the MOS masks, \nand for the relative sensitivity function.\nThe sensitivity function was derived from the spectrum of the same \nspectrophotometric standard star obtained with both cameras.\n\nThe 1D spectrum of each galaxy was extracted using the Iraf task $apall$.\nAfter extraction, an additional residual sky-subtraction was\napplied to remove \nsky residuals \ndue to sky intensity variations between subsequent images in the ABBA \nsequence. \nTo this end, we subtracted from the 1D object spectrum, \nthe 1D spectrum of sky residuals obtained by averaging the pixels \nabove and below the object, along the spatial direction.\nFinally, we applied the software MOLECFIT \n\\citep{kausch15} to perform telluric absorption correction, by fitting \nspectral regions of the bright star spectrum with prominent telluric\nabsorption. \nThe resulting telluric absorption model provided by MOLECFIT \nwas applied to correct all galaxy spectra.\n\n\n\\section{Measurement of redshift and absorption line indices}\n\\subsection{Redshift measurement}\nMeasurements of spectroscopic redshifts and velocity dispersion were \nperformed using stellar absorption features, \nby fitting the observed spectra with \nMILES simple stellar population (SSP) models \\citep{vazdekis10}.\nThese models are primarily based on the MILES \\citep{sanchez06,falcon11} \nand \nIndo-U.S \\citep{valdes04} stellar libraries and have a spectral \nresolution of 2.5\\AA\\ \n\\cite[][]{beifiori11}, close to the rest-frame resolution\n($\\sim$3.3\\AA) of our spectra.\nSpectral fitting was performed using the penalized\nPiXel-Fitting method \\cite[pPXF,][]{cappellari04,cappellari17}.\n\n\\begin{figure*}\n\t\\includegraphics[width=11.5truecm]{spec_651.ps}\n\t\\includegraphics[width=2.2truecm]{651.ps}\n\t\\includegraphics[width=11.5truecm]{spec_972.ps}\n\t\\includegraphics[width=2.2truecm]{972.ps}\n\t\\includegraphics[width=11.5truecm]{spec_1142.ps}\n\t\\includegraphics[width=2.2truecm]{1142.ps}\n\t\\includegraphics[width=11.5truecm]{spec_1370.ps}\n\t\\includegraphics[width=2.2truecm]{1370.ps}\n\t\\includegraphics[width=11.5truecm]{spec_1442.ps}\n\t\\includegraphics[width=2.2truecm]{1442.ps}\n\t\\includegraphics[width=11.5truecm]{spec_1630.ps}\n\t\\includegraphics[width=2.2truecm]{1630.ps}\n\t\\includegraphics[width=11.5truecm]{spec_1711.ps}\n \t\\includegraphics[width=2.2truecm]{1711.ps}\n \\caption{\\label{fig:7spectra} MODS spectra of the seven cluster members spheroidal galaxies for which we\n derived the velocity dispersion. The black curve is the observed spectrum \n arbitrarily normalized, binned to 3.4 \\AA\/pixel. \n These spectra have S\/N=[5-18] per Angstrom in the restframe of the galaxy, in the range \n 4000$<\\lambda_{rest}<$4150\\AA. \n The red curve is the best-fitting MILES model resulting from the pPXF spectral fitting. \n The dotted vertical lines mark the main spectral features labelled on top\n of the figure. To the right of each spectrum, the ACS-F850LP image\n (2$\\times$2 arcsec) of the target galaxy is shown.}\n\\end{figure*}\n\nVelocity dispersion measurements for the whole set of spectroscopic\nobservations \n will be presented in a forthcoming paper (Saracco et al. 2018, in preparation).\n{ Here, we summarize the method used.\nFor each spectrum, we derived the broadening of its absorption lines,\n $\\sigma_{obs}$, by performing spectral fitting in the range 3550--4200 \\AA. \nThe galaxy velocity dispersion, $\\sigma_\\star$ is then derived by correcting the \n$\\sigma_{obs}$ for the instrumental broadening of $\\sim$111 km\/s, resulting from the \ninstrumental resolution R$\\simeq 1150$ (see \\S 2).\n\nThe uncertanties on $\\sigma_*$ have been derived \nby repeating the measurements on 100 simulated spectra obtained by\nsumming to the best-fitting template the 1D sky residuals extracted \nfrom the real 2D spectra, randomly shuffled in wavelength.\nThe typical uncertanties are in the range 10\\%-25\\%, with the exception of \ngalaxy \\#1370 affected by a much larger uncertaninty because of the low\nS\/N (see Tab. \\ref{tab:sample}).\nWe have tested the robustness of the measurements against the library of template used \nand the wavelength range considered by repeating the fitting with a set of synthetic \nstars extracted from the library of \\cite{munari05} \nand a set of SSPs of \\cite{bruzual03},\nand by slightly varying the fitted spectral range.\nIn all the cases, we obtained measurements well within the estimated errors. \n}\n \nThe seven spheroids cluster members here analysed have S\/N$\\sim$5-18 per \\AA\\ \nin the rest-frame interval 4000-4150\\AA, that turned out to be sufficient \nto perform the stellar population analysis, as detailed below.\nIn Tab. \\ref{tab:sample} we report the main properties of the seven spheroidal \ngalaxies while their spectra are shown in Fig. \\ref{fig:7spectra}.\nIn each panel, the black curve plots the 8-hours MODS spectrum binned\nto 3.4 \\AA\/pix and the red curve is the best-fitting pPXF template.\nFor each galaxy we also show the $2\\times2$ arcsec ACS-F850LP image.\n\n\\subsection{Emission lines: AGN and star formation}\nTwo galaxies, namely \\#1442 and \\#1630 (see Tab. \\ref{tab:sample}), \nshow clearly [OII] line emission, while\ngalaxy \\#651 shows a weak [OII] emission accompanied by the presence of [NeV] \nline emission.\nThe high-ionization [NeV]$\\lambda3426$ emission line is considered a signature\n of nuclear activity \\citep{schmidt98,gilli10} since stars do not reach such high \nionization potential \\cite[e.g.][]{haehnelt01}, therefore galaxy \\#651 most probably \nhosts an AGN.\nApart from \\#651, there is no evidence for the presence of AGN\nin any other galaxy.\n\nAssuming that [OII] emission is due to star formation (also for \\#651), we derived \nthe star formation rate (SFR) from the relation \n\\begin{equation}\n SFR=1.4\\cdot 10^{-41}L([OII])\n\\end{equation}\nwhere the SFR is expressed in M$_\\odot~yr^{-1}$ and the luminosity L([OII])\nin erg$~s^{-1}$ \\citep{kennicutt98}.\nThe luminosity L([OII]) has been derived from the line flux F([OII]) measured on the \ncalibrated spectra, $L([OII])=4\\pi d^2_L\\cdot F([OII])$, where $d_L=8449.9$ Mpc is \nthe luminosity distance of the cluster at $z=1.22$ for the adopted cosmology.\nThe measured [OII] fluxes and the resulting SFR are reported in Tab. \\ref{tab:sample}\nfor those galaxies for which we detect an [OII] line flux $>$1$\\sigma$.\nOur spectra show that at least two spheroids out of the seven show evidence of \nweak star formation, at a rate $<$4 M$_\\odot~yr^{-1}$, and one hosts an AGN.\n\nWe notice that, early-type galaxies with similar values of SFR \n(few M$_\\odot~yr^{-1}$ or lower), are not rare and are observed \nboth in the field and in cluster.\nSFRs of about 2-3 M$_\\odot~yr^{-1}$ are observed, for instance, in some early-types \nin cluster RDCSJ0848 at $z\\sim1.27$ \\citep[e.g.][]{jorgensen14} as well\nas in the field at similar redshift \\cite[e.g.][]{cimatti08,belli17}.\nAlso at lower redshift, some spheroidal galaxies exhibit low levels\nof star formation, both in clusters \\citep[e.g.][]{jorgensen17} and\nin the field \\cite[e.g.][]{fukugita04,huang09,yates14,george17}.\nThis star formation activity in field early-types is thought to be fueled by \ninflowing gas and\/or gas-rich minor mergers \\citep{belli17,george17}.\nAs to our seven spheroids, it is unlikely that star formation is fueled\nby inflowing gas, since gas and galaxies should be in thermal\nequilibrium in a cluster.\nMoreover, minor mergers are disfavoured in the\ncluster center regions given the large relative velocities of galaxies\n\\citep[e.g.][]{harrison11,treu03}. \nThus, the low levels of star formation we observe could be the sign\nof a more complex or longer star formation history or the quenching\nphase of the main burst recently occurred.\nWe furthur discuss the presence of star formation below.\n \n \n\\subsection{Absorption line indices}\nWe measured absorption line Lick indices, following the definition by \\cite{worthey97}\nand \\cite{trager98}, and the indices by \\cite{rose94} in the rest-frame wavelength range\n covered by the observations.\nTo perform the measurements we made use of the software \n\\texttt{LECTOR}\\footnote{http:\/\/www.iac.es\/galeria\/vazdekis\/vazdekis\\_software.html}.\nBesides these absorption indices, we measured the strength of the 4000\\AA\\ break\naccording to the D4000 definition by \\cite{bruzual83} \\citep[see also][]{gorgas99}\nand the D$_n$ index by \\cite{balogh99}.\nThe uncertainties on the spectral indices have been derived through simulations \naccording to the following procedure.\nFor each observed 1D spectrum, we constructed a set of 100 simulated spectra\nhaving the same S\/N of the real one.\nThe S\/N was evaluated in the rest-frame wavelength range 4000-4150 \\AA.\nThe simulated spectra were obtained by summing to the best-fitting \nmodel template the 1D sky residual extracted from the final 2D spectra.\nIn each simulated spectrum, the values of the residuals within the \nrest-frame wavelength range 3700-4400 \\AA\\ were randomly shuffled.\nThen, for each simulated spectrum, we measured the indices using the same procedure\nused for the real spectra.\nWe adopted as uncertainty on the measured index the median absolute deviation \n(MAD) resulting from the distribution of the 100 measurements.\n\nThe measured indices were corrected for galaxy velocity dispersion.\nThe corrections were obtained for each galaxy by comparing the\nindices measured on the best-fitting model (smoothed to the $\\sigma$ of the galaxy),\nand those of the same model at the nominal resolution of the MILES\nspectral library. \nThe corrected measured Lick indices and Rose's indices with their errors are \nsummarized in Tab. \\ref{tab:lick} and Tab. \\ref{tab:rose} respectively.\n \n\\begin{table*}\n\\begin{minipage}[t]{1\\textwidth}\n\\caption{\\label{tab:lick} Lick spectral indices.}\n\\centerline{\n\\begin{tabular}{rrcccrrrrrr}\n\\hline\n\\hline\n ID & CN3883 & CaHK& D4000 & D$_{n}$ & H$\\delta_A$ & H$\\delta_F$&CN1 &CN2 &\n Ca4227& G4300 \\\\\n &(mag) &(\\AA) & \t& &(\\AA) & (\\AA) & (mag) &(mag) & (\\AA) &(\\AA) \\\\\n\\hline\n 651& 0.27$\\pm$0.05 & 23.6$\\pm$2.3 &1.86$\\pm$0.05 & 1.51 & 3.1$\\pm$2.0 & 3.7$\\pm$1.0 & -0.21$\\pm$0.05 & -0.20$\\pm$0.06 & 4.2 $\\pm$0.8 & 4.9$\\pm$1.4 \\\\ \n 972& 0.32$\\pm$0.05 & 22.4$\\pm$2.6 &2.13$\\pm$0.05 & 2.00 & 1.3$\\pm$2.2 & 4.6$\\pm$1.2 & 0.17$\\pm$0.05 & 0.25$\\pm$0.07 & 3.9 $\\pm$0.9 & 1.2$\\pm$1.6 \\\\ \n1142& 0.20$\\pm$0.04 & 30.6$\\pm$2.0 &2.19$\\pm$0.04 & 1.75 & 0.7$\\pm$1.7 & 4.1$\\pm$0.9 & 0.03$\\pm$0.04 & 0.07$\\pm$0.05 & 0.2 $\\pm$0.7 &10.4$\\pm$1.2 \\\\ \n1370& 0.04$\\pm$0.06 & 19.8$\\pm$3.0 &1.52$\\pm$0.06 & 1.30 & -2.8$\\pm$2.5 & 0.2$\\pm$1.3 & 0.09$\\pm$0.06 & 0.13$\\pm$0.08 & 3.3 $\\pm$1.0 & 6.2$\\pm$1.8 \\\\ \n1442& 0.15$\\pm$0.05 & 17.7$\\pm$2.3 &1.77$\\pm$0.05 & 1.33 & -2.3$\\pm$2.0 & 2.0$\\pm$1.0 & -0.17$\\pm$0.05 & 0.17$\\pm$0.06 &-0.9 $\\pm$0.8 &12.6$\\pm$1.4 \\\\ \n1630& -0.01$\\pm$0.04 & 10.0$\\pm$2.0 &1.82$\\pm$0.04 & 1.48 & 7.9$\\pm$1.7 & 5.9$\\pm$0.9 & -0.19$\\pm$0.04 & -0.15$\\pm$0.05 &-1.6 $\\pm$0.7 & 2.7$\\pm$1.2 \\\\ \n1711& 0.15$\\pm$0.03 & 18.5$\\pm$1.8 &1.84$\\pm$0.03 & 1.54 & 3.1$\\pm$1.5 & 3.1$\\pm$0.8 & 0.01$\\pm$0.03 & 0.01$\\pm$0.04 & 3.3 $\\pm$0.6\t& 6.8$\\pm$1.0 \\\\ \n\\hline\n\\end{tabular}\n}\n{The indices are corrected for galaxy velocity dispersion (see Sec. 3.3).}\n\\end{minipage}\n\\end{table*}\n\n\\begin{table*}\n\\begin{minipage}[t]{1\\textwidth}\n\\caption{\\label{tab:rose} Same as Tab. \\ref{tab:lick} but for Rose indices.}\n\\centerline{\n\\begin{tabular}{rcccccccc}\n\\hline\n\\hline\n ID & 3888\/3859& CaII(H\/K)& H$\\delta$\/FeI[4045] &H$\\delta$\/FeI[4063] &p[Fe\/H] \\\\\n & & & \t& & \\\\\n\\hline\n 651& 1.1$\\pm$0.4& 0.8$\\pm$0.4 & 1.1$\\pm$0.2 & 1.1$\\pm$0.2 & 1.0$\\pm$0.2 \\\\ \n 972& 0.3$\\pm$0.4& 0.9$\\pm$0.4 & 0.3$\\pm$0.2 & 0.4$\\pm$0.2 & 1.4$\\pm$0.2 \\\\ \n1142& 0.8$\\pm$0.3& 1.2$\\pm$0.3 & 0.9$\\pm$0.2 & 0.9$\\pm$0.2 & 1.3$\\pm$0.1 \\\\ \n1370& 0.7$\\pm$0.5& 0.9$\\pm$0.5 & 0.8$\\pm$0.4 & 0.8$\\pm$0.4 & 1.3$\\pm$0.3 \\\\ \n1442& 0.8$\\pm$0.4& 0.7$\\pm$0.4 & 1.1$\\pm$0.2 & 1.1$\\pm$0.2 & 1.0$\\pm$0.2 \\\\ \n1630& 0.5$\\pm$0.3& 1.0$\\pm$0.3 & 0.6$\\pm$0.2 & 0.6$\\pm$0.2 & 1.0$\\pm$0.1 \\\\ \n1711& 0.7$\\pm$0.2& 0.9$\\pm$0.2 & 0.7$\\pm$0.1 & 0.7$\\pm$0.1 & 1.1$\\pm$0.1 \\\\ \n\\hline\n\\end{tabular}\n}\n\\end{minipage}\n\\end{table*}\n\n\n\n\\section{Stellar population parameters}\nThe effective rest-frame wavelength range covered by our spectroscopic\nobservations (3500$<$$\\lambda_{rest}$$<$4400\\AA), does not provide us \nwith enough spectral features suited to constrain the abundance ratios \nof $\\alpha$-elements (Ca, Mg, Si, Ti etc.).\nIn particular, prominent tracers of Magnesium abundance, such\nas the Mgb spectral feature at $\\lambda \\sim 5270 \\, \\AA$, usually considered the best\ntracer of $\\alpha$-elements, fall outside our spectral range.\nFor this reason, in the whole analysis, we limit ourselves to constrain the total \nmetallicity [Z\/H] without considering the effect of [$\\alpha$\/Fe].\n\n\\label{sec:lines}\n\\begin{figure*}\n\\includegraphics[width=16truecm]{lick_lick1.ps}\n\\vskip -0.7truecm\n\\includegraphics[width=16truecm]{rose_rose1.ps}\n\\caption{\\label{fig:lick} Absorption line strengths versus each other. \nUpper panels - Absorption Lick indices. \nThe seven spheroidal galaxies belonging to the cluster \nXLSSJ0223 whose measurements are reported in Tab. \\ref{tab:lick}, are represented\nby the different symbols reported in the inset.\nGreen filled square are measurements of galaxies in the cluster RDCS0848 at $z\\sim1.27$ \n\\citep{jorgensen14}.\nOpen (cyan) triangles and (magenta) squares are galaxies in the clusters \nMS0451.6 at $z\\simeq0.54$ and RXJ1226.9 at $z\\simeq0.89$ respectively \\citep{jorgensen13}.\nSolid curves are predictionis from MILES SSP models \\citep{vazdekis10} for the three different \nmetallicity values \nreported in the figure legend and plotted in the age interval 0.1-5 Gyr.\nThe cyan and blue thin curves are the Z$_\\odot$ SSP in the interval 0.1-5 Gyr with the addition \nof a young (0.06 Gyr) component (0.5\\% of the mass) with Z=0.2Z$_\\odot$ and Z=1.7Z$_\\odot$,\nrespectively.\nThe dashed (fuchsia) curve show predictions from BC03 SSP model \\citep{bruzual03} with \nChabrier IMF and solar metallicity in the same age interval.\nThe arrows indicates the direction in which the age increases following the models.\nObserved indices are corrected for velocity dispersion.\nLower panels - Rose's \\citep{rose94,rose94b} absorption line indices. \nThe values are reported in Tab. \\ref{tab:rose}.\nSymbols are as in the upper panels. \n} \n\\end{figure*}\n\\subsection{Constraints from absorption line indices}\nIn Fig. \\ref{fig:lick} the main Lick absorption indices (upper panel) and the main\nRose's indices (lower panel) are plotted versus each other.\nThe upper panels show the strength of CN3883 versus CaHK and D4000, and the strength \nof CaHK versus H$\\delta_F$.\nThe three plots are arranged to show two different metal lines, one\nagainst the other (CN3883 vs CaHK), and against two different age sensitive indices\n(D4000 and H$\\delta_F$). \nThe index CN3883 has been derived according to the definition of \\cite{davidge94} \n\\citep[see also][]{pickles85}. \nThe lower panels follow the same criterion of the upper ones showing\nthe strength of p[Fe\/H] versus CaII(H\/K) and H$\\delta$\/Fe4045, and the strength of \nCaII(H\/K) versus D4000.\nThe index p[Fe\/H] is metal-abundance sensitive, H$\\delta$\/Fe4045 increases with\nthe stellar spectral type and the index CaII(H\/K) is a good tracer of \nthe presence of very young populations, being sensitive to A-type stars \n\\citep{rose94,longhetti99}. \n\nThe seven spheroidal galaxies here analyzed are shown\nwith different (filled) symbols according to the legend in the inset.\nGalaxies in cluster RDCS0848 (LinxW) at $z=1.27$, whose measurements \nhave been obtained by \\cite{jorgensen14} (green filled squares) are also shown.\nFor this cluster, we considered the galaxies belonging to samples \\#5 and \\#4 \nof Tab. 7 in \\cite{jorgensen14} for which measurements are available.\nFor comparison with lower redshift data, we show the measured indices for galaxies in \ncluster MS0451.6 at $z\\simeq0.54$ (open cyan triangles) and in cluster RXJ1226.9 at \n$z\\simeq0.89$ (open magenta squares) by \\cite{jorgensen13}.\n\nSolid curves are model predictions based on MILES SSPs \\citep{vazdekis10} with Chabrier initial \nmass function (IMF) \\citep{chabrier03}.\nThe models shown are plotted in the age range [0.1,5] Gyr and refer to three different values of \nmetallicity [Z\/H]: -0.71 (dark green curve), 0.0 (orange line) and +0.22 (dark red curve).\nWe also considered the two cases of a very young (0.06 Gyr) population representing 0.1\\% of the\nstellar mass with [Z\/H]=-0.71 (cyan curve) and [Z\/H]=0.22 (blue curve) superimposed to \nan underlying dominant [Z\/H]=0 population. \nDashed (fuchsia) curve represents the prediction based on BC03 SSPs \\citep{bruzual03} with solar \nmetallicity and Chabrier IMF, in the same age range.\nNo significant differences arise among the different models considered.\nTherefore, the results of the following analysis are independent of the adopted models. \n\nIn spite of the degeneracy between age and metallicity affecting most of the indices,\nthe diagnostic diagrams in Fig. \\ref{fig:lick}, can be used to constrain relative \ndifferences in age and metallicity among galaxies in our sample. \nThe deviation from SSPs could suggest a more complex history of star formation than the \none described by a SSP as found, for instance, by \\cite{lonoce14} for field early-type galaxies \nat $\\sim1$.\nWhether this is the case, it will be explored in the next section through full spectral fitting.\n\nGalaxies \\#1370, \\#1442 and \\#1630 lie on or below the lowest metallicity track in the \nCaHK vs H$\\delta_F$ diagram, the one less affected by the degeneracy, while galaxies \n\\#651, \\#972 and \\#1142 lie above the highest metallicity track and galaxy \\#1711 lies in between.\nThis behaviour is due to the fact that the strength of CN3883 in galaxies \n\\#1370, \\#1442, \\#1630 and \\#1711 is significantly \nweaker than in galaxies \\#651, \\#972 and \\#1142, as visible also from the CN3883 vs CaHK diagram.\nAlso the p[Fe\/H] index tends to be weaker for galaxies \\#1442, \\#1630 and \\#1711\nwith respect to the other galaxies in the sample.\nThese differences are consistent with galaxies \\#1370, \\#1442, \\#1630 and \\#1711 having lower \nmetallicity than the remaining three galaxies.\n\nAs to the age, we notice that the D4000 is stronger in galaxies \\#651, \\#972 and \\#1142 \nthan in the other galaxies, in agreement with the CaHK index being stronger in \nthese three galaxies, and with the CaII(H\/K) close to unity for \\#972 and \\#1142.\nThe possible presence of weak star formation (see \\S 3.2) superimposed to an older population,\ncould justify the lower value of this index in galaxy \\#651.\nThese three galaxies tend to occupy the end at oldest ages of the SSP tracks in the diagrams \nshowing metal-sensitive index vs age-sensitive index (e.g. CN3883\/CaHK and CN3883\/D4000), \nat odds with the other galaxies.\nThus, the picture is consistent with an age for \\#651, \\#972 and \\#1142 older than for the\nother galaxies.\n\nGalaxies \\#1630 and \\#1442 are among the youngest galaxies in our sample. \nThey both show star formation (see Tab. \\ref{tab:sample}) but are characterized by \ndifferent values of the indices, with the exception of the D4000, not significantly different.\nGalaxy \\#1630 occupies the lowest-age end of the SSP tracks in most diagrams.\nIt is characterized by the highest H$\\delta$ indices and by the lowest CaHK.\nThe Rose's index CaII(H\/K), whose CaII(H) line is sensitive to young stars, \nis close to one suggesting that the whole stellar population of this galaxy is genuinely young. \nFor comparison, galaxy \\#1442 has a higher CaHK and lower H$\\delta$ indices,\naccompanied by a CaII(H\/K) lower than unity. \nThis would suggest that the newly formed stars are superimposed to an older population. \n\n\n\\subsection{Absorption lines fitting}\nAs a first step to characterize stellar population properties of the\nseven spheroidal galaxies in XLSSJ0223, we have derived their age and\nmetallicity, [Z\/H], by comparing observed line-strengths to\npredictions from SSP models with varying age and metallicity. \nTo this effect, we use the same set of MILES SSP models \\citep{vazdekis10}\nas for spectral fitting (see Sec. 5.1), namely SSPs based on PADOVA\nisochrones (Girardi et al. 2000), with a Chabrier IMF, ages in the\nrange from 0.06 to 4.5~Gyr, and metallicity in the range\n$-2.32$--$0.22 \\, Z_{\\odot}$.\n\n\nThe fitting is performed by minimizing the expression:\n\n\\begin{equation}\n \\chi^2 (age,[Z\/H]) = \\sum_j \\frac{(O_j-M_j)^2}{s_j^2}\n\\end{equation}\n\nwhere the index $\\rm j$ runs over the selected set of Lick spectral\nindices in the rest-frame range 3500--4400 \\AA\\ (see Tab. \\ref{tab:lick}), \n$\\rm O_j$ and $\\rm M_j$ are observed and model index values (the latter\ndepending on age and $\\rm [Z\/H]$), and $\\rm s_j$ are uncertainties on\nobserved indices. \nThe resulting best-fitting age and metallicity are\nreported in Tab. \\ref{tab:parameters}. \nThe quoted errors are obtained by running the\nfitting procedure on a set of simulated spectra, having the same S\/N\nas the observed spectra.\n\nIt is worth noting that, as expected, the age and metallicity\nvalues in Tab. \\ref{tab:parameters} confirm the general trends \nobtained from the diagnostic diagrams in Fig. \\ref{fig:lick} (see Sec.~4.1), \nnamely that galaxy \\#1630\nis the youngest object in our sample, galaxies \\#651, \\#972, and \\#1142\nare the oldest, while galaxies \\#1370 and \\#1442 tend to have lower\nmetallicities than the remaining systems.\n\n \n\n\\begin{table}\n\\caption{\\label{tab:parameters} Age and metallicity estimates resulting from absorption lines fitting}\n\\centerline{\n\\begin{tabular}{rrr}\n\\hline\n\\hline\n ID & Age& [Z\/H] \\\\\n & (Gyr) & \\\\\n\\hline\n 651& 2.2$_{-0.4} ^{+0.8}$ & -0.31$_{-0.17}^{+0.44}$ \\\\ \n 972& 1.9$_{-0.7} ^{+1.3}$ & +0.02$_{-0.51}^{+0.19}$ \\\\ \n1142& 2.8$_{-0.6} ^{+1.6}$ & +0.16$_{-0.31}^{+0.06}$ \\\\ \n1370& 1.3$_{-0.3} ^{+1.0}$ & -0.05$_{-0.57}^{+0.18}$ \\\\ \n1442& 1.9$_{-0.8} ^{+1.5}$ & -0.52$_{-0.77}^{+0.50}$ \\\\ \n1630& 0.9$_{-0.1}^{+0.2}$ & +0.01$_{-0.39}^{+0.20}$ \\\\ \n1711& 1.5$_{-0.4}^{+0.7}$ & -0.10$_{-0.52}^{+0.32}$ \\\\ \n\\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\\section{Constraints on star formation history}\n\n\\begin{table*}\n\\caption{\\label{tab:fitting} Age and metallicity estimates resulting from spectral fitting}\n\\centerline{\n\\begin{tabular}{rlrrrrrcc}\n\\hline\n\\hline\n ID & Age$_{L}$& [Z\/H]$_{L}$& Age$_{M_*}$& [Z\/H]$_{M_*}$ & A$_V$& $z_{f}$&n(SSPs)& {$\\chi^2_\\nu$}\\\\\n & (Gyr) & & (Gyr) & &(mag)& & &\\\\\n\\hline\n 651& 2.7$_{-0.7} ^{+1.0}$ & +0.09$^{+0.13}_{-0.42}$ & 3.0$_{-0.7}^{+0.8}$ & +0.09$^{+0.13}_{-0.42}$ & 0.0 & 3.4 & 4 &{ 0.89}\\\\ \n 972& 3.4$_{-0.6} ^{+0.9}$ & +0.16$^{+0.06}_{-0.16}$ & 4.4$_{-0.5}^{+0.1}$ & +0.21$^{+0.01}_{-0.21}$ & 0.3 &$>$6.5& 2&{ 0.82}\\\\ \n1142& 2.3$_{-0.2} ^{+1.2}$ & +0.21$^{+0.01}_{-0.21}$ & 2.4$_{-0.2}^{+1.4}$ & +0.22$^{+0.00}_{-0.22}$ & 0.8 & 2.6 & 2&{ 0.93}\\\\ \n1370& 1.3$_{-0.2} ^{+1.8}$ & -0.27$^{+0.49}_{-0.12}$ & 1.9$_{-0.5}^{+0.8}$ & -0.10$^{+0.32}_{-0.61}$ & 1.0 & 2.0 & 2&{ 0.80}\\\\ \n1442& 1.7$_{-0.5} ^{+0.7}$ & -0.18$^{+0.44}_{-0.21}$ & 2.7$_{-0.5}^{+0.2}$ & -0.01$^{+0.23}_{-0.38}$ & 0.9 & 2.9 & 4&{ 0.90}\\\\ \n1630& 1.2$_{-0.3}^{+0.6}$ & -0.13$^{+0.39}_{-0.26}$ & 1.4$_{-0.2}^{+0.7}$ & -0.22$^{+0.44}_{-0.17}$ & 1.2 & 1.9 & 3&{ 0.88}\\\\ \n1711& 1.7$_{-0.5}^{+0.2}$ & +0.05$^{+0.17}_{-0.05}$ & 2.1$_{-0.5}^{+0.2}$ & +0.18$^{+0.04}_{-0.18}$ & 0.7 & 2.4 & 4&{ 0.90}\\\\ \n\\hline\n\\end{tabular}\n}\n\\end{table*}\n\\subsection{Full spectral fitting}\nTo recover the main properties of the stellar populations of the seven galaxies, \nthe mean stellar age and metallicity, and to constrain their star formation\nhistory (SFH), we also used full spectral fitting.\nIn this approach, one assumes that the observed spectrum is the \nsuperposition of few simple stellar populations (SSPs), \nwith age Age$_i$ and metallicity Z$_i$, each one contributing \nwith a different weight,\nextracted from a larger base of models of different ages and metallicities. \nTo perform the spectral synthesis and find the best-fitting linear\ncombination of SSPs, we used the code \\texttt{STARLIGHT} \n\\citep{cid05,cid07, mateus07, asari07}.\n\nWe adopted as reference base of models the SSPs from the MILES-based \nlibrary (v 11.0) of \\cite{vazdekis10},\ncovering the wavelength range 3540-7400 \\AA\\ at a spectral resolution of 2.5\\AA.\nWe considered Chabrier IMF and two different sets of {\\it Base} SSPs: \nthe first one based on the PADOVA00 isochrones \\citep{girardi00} and the second \none on the BaSTI isochrones \\citep{pietrinferni04}.\nFor both these sets of models, the ages considered are $\\sim$30 in the range [0.06; 4.5] Gyr \nwhile the metallicities are 7 for the PADOVA set in the range [-2.23; 0.22], and\n9 for the BaSTI set in the range [-2.27; 0.26].\nWe allowed for internal reddening in the \\texttt{STARLIGHT} fits, by considering\nboth the Cardelli \\citep[CCM;][]{cardelli89} and the Calzetti \n\\citep[HZ5;][]{calzetti00} extinction laws in the fit. \n{ The fitting was perfomed in the wavelength range 3600-4400\\AA.}\n\nTo assess the robustness of the derived stellar population properties \nwith respect to the models adopted in the fitting, we also considered\n a base composed of BC03 SSPs \n\\citep{bruzual03} with Chabrier IMF, 20 ages in the range [0.06; 4.5] Gyr \nand 5 metallicities in the range [-1.7; 0.4], and a base composed\nof \\cite{maraston11} MILES-based models (MS11 hereafter) with Chabrier \nIMF and including 20 ages in the range [0.06; 4.5] Gyr and 5 metallicities \nin the range [-2.3; 0.3].\nThe results of the fitting with these two sets of models are summarized\nin Tables \\ref{tab:bc03} and \\ref{tab:m11}, respectively.\n\n\\subsection{Ages and metallicities derivation}\nWe defined the luminosity-weighted age $\\langle$Age$\\rangle_{L}$ and metallicity\n[Z\/H]$_{L}$ as\n\\begin{equation}\n\\begin{split}\n&{\\rm Age}_{L}=\\sum_i x_i Age_i\\\\\n&{\\rm [Z\/H]}_{L}=log\\sum_i x_i Z_i\/Z_\\odot\n\\end{split}\n\\end{equation}\nwhere $x_i$ is the fractional contribution of the $i$-th SSP\nconsidered in the synthesis, while $Age_i$ and $Z_i$ are its age and metallicity.\nAnalogously, we defined the stellar mass-weighted age and metallicity as\n\\begin{equation}\n\\begin{split}\n &{\\rm Age}_{M}=\\sum_i m_i Age_i\\\\\n&{\\rm [Z\/H]}_{M}=log\\sum_i m_i Z_i\/Z_\\odot\n\\end{split}\n\\end{equation}\nwhere $m_i$ is the fraction of stellar mass of the $i$-th SSP.\n\n\\begin{figure*}\n\\includegraphics[width=15truecm]{plot1_miles_ord.ps}\n\\caption{\\label{fig:star1} Results of the spectral synthesis for galaxies\n\\#651, \\#972 and \\#1142.\nLeft panels - The best-fitting composite spectrum (blue curve) resulting from\nthe spectral synthesis performed with \\texttt{STARLIGHT} is superimposed\nto the observed spectrum (dark gray curve) binned to 3.4\\AA\/pix.\nThe green curve in the bottom panel represents the typical sky residuals extracted from the image\nof galaxy \\#1142. \nRight panels - Star formation history of galaxies. Fraction of stellar mass of each \nSSP contributing to the best-fitting spectrum as a function of lookback time \nand formation redshift (top x-axis). \nDifferent colours encode different values of metallicity:\ncyan Z$<$Z$_\\odot$ (cyan down arrows in the legend), \nyellow Z=Z$_\\odot$, magenta Z$>$Z$_\\odot$ (magenta upward arrows in the legend).\nThe bar code on the bottom represents the grid of ages considered\nin the base of SSPs adopted to run \\texttt{STARLIGHT} (see the text).\n} \n\\end{figure*}\n\\renewcommand{\\thefigure}{\\arabic{figure} (Cont.)}\n\\addtocounter{figure}{-1}\n\n\\begin{figure*}\n\\includegraphics[width=15truecm]{plot2_miles_ord.ps}\n\\caption{\\label{fig:star2} Results of the spectral synthesis for galaxies \n \\#1370, \\#1442 and \\#1630 and \\#1711. \n} \n\\end{figure*}\n\\renewcommand{\\thefigure}{\\arabic{figure}}\n\nIn Tab. \\ref{tab:fitting} the luminosity-weighted and the mass-weighted\nage and metallicity values resulting from the spectral fitting \nare reported for each galaxy, { together with the reduced $\\chi^2$ of\nthe best-fitting model}. \nThe values refer to results obtained with the set of SSPs based on PADOVA \nisochrones.\nThe corresponding best-fitting\nsynthetic spectra are shown in Figure \\ref{fig:star1}.\nResults based on BaSTI isochrones are similar to those obtained with PADOVA models, \nand are not shown here for brevity reasons, with differences in age and metallicity \nsmaller than 20\\%, and 0.15dex, respectively.\nThe two extinction laws produce negligible differences.\nThe values of metallicity and age obtained through the spectral\nfitting confirm those previously derived in \\S 4:\ngalaxies \\#651, \\#972, and \\#1142 are the oldest in our sample while galaxy \\#1620 \nis the youngest; galaxies \\#1370 and \\#1442 tend to have lower\nmetallicities than the others.\n\nThe errors on age and metallicity are\nthe confidence intervals at 68\\% confidence level of the solutions in the Age-[Z\/H] space.\nThey have been obtained considering all the fitting \nsolutions within 1$\\sigma$ from the best-fitting one. \nTo this end, we compared the reduced-$\\chi^2$ of all the solutions \nwith the reduced-$\\chi^2$ of the best-fitting one making use of the F-test.\nDue to the fact that errors on age and metallicity are anti-correlated,\n \\citep[see, e.g.,][]{thomas05}, the confidence intervals \nshould be read jointly in the sense that, older (higher) values of age (metallicity) \ncorrespond to lower (younger) values of metallicity (age) with respect to the best-fitting values.\n{ It is worth noting that predictions of MILES-SSPs for a certain range of very young ages and \nvery low metallicities, (age$<$0.1 Gyr and [Z\/H]<-1.5) are flagged as \nunsafe\\footnote{http:\/\/www.iac.es\/proyecto\/miles\/pages\/ssp-models\/safe-ranges.php} \nby \\cite{vazdekis10}.\nWe note, however, that there are no SSPs within the unsafe range\ncontributing to the fitting solutions.\n}\n\nAs can be seen from Tab. \\ref{tab:fitting}, luminosity-weighted and mass-weighted \nmetallicities do not differ significantly (considering the errors), \nsince the M\/L ratio plays a secondary role in the derivation of this quantity.\nThe best fitting metallicity values never reach the extremes of the\nrange considered, with the exception of the galaxy \\#1142, \nclose to the maximum metallicity value allowed in the reference grid of SSPs.\nFinally, we emphasize that despite to the well-known age-metallicity deneracy (whereby \nthe effect of an older age is alsmost exactly compensated by that of a lower metallicity \nin the spectral synthesis), our best-fitting age and metallicity are not anti-correlated,\nshowing again the robustness of the results.\n\nThe resulting metallicity estimates for the 7 spheroids lie within 0.2 dex from the solar value\nwith a median value <[Z\/H]$_M$>=0.09$\\pm$0.16.\nThis is in agreement with the results obtained in \\S \\ref{sec:lines} \nbased on absorption line indices (see Tab. \\ref{tab:parameters}).\nThe above results and considerations also hold for the estimates of stellar age, \nwhose median value is =2.4$\\pm$0.6 Gyr.\n\n\\subsection{Metallicity variation and evolution with redshift}\nIn the upper panel of Fig. \\ref{fig:metal}, the metallicities of the 7 cluster\nspheroids at $z=1.22$, as reported in Tab. \\ref{tab:fitting}, are compared with the \nmetallicities of galaxies in clusters at lower redshift and with the metallicities \nof field galaxies of comparable mass at lower and higher redshifts.\nThe figure shows representative median metallicity values of galaxies\nin clusters in the redshift range $0.211$ M$_\\odot$) galaxy from\n\\citet{lonoce15}, \nthe cyan point at $z\\sim1.6$ has been derived from the stacked spectra of 24 galaxies \\citep[][]{onodera15}, \nthe brown point is the measurement for a massive galaxy at $z=2.1$ \\citep[brown,][]{kriek16}, \nthe two open circles at $z\\sim2$ \n(upper panel only) are measurements of two lensed galaxies \\citep[][]{morishita18}. \nThe dashed line in the upper panel is the median value [Z\/H]=0.24 at $0 < z < 0.9$ \n\\citep[from][]{jorgensen17}.\nThe curves in the lower panel are predictions for passive evolution for\ndifferent values of formation redshift $z_f$, as labelled in the figure. \n}\n\\end{figure}\n\n\n\\subsection{Stellar age and star formation history}\nThe median stellar age of the seven spheroids, =2.4$\\pm$0.6 Gyr, implies a \nmedian formation redshift $<$$z_f$$>$$\\sim2.6_{-0.5}^{+0.7}$.\nThis value is slightly higher (not significantly, given the uncertainties) than \nthe median formation redshift ($z_f$$\\sim$2) derived \nfrom the stellar ages and from the fundamental plane for early-type galaxies \nin clusters at $z$$<$0.9 by \\cite{jorgensen17} and \\cite{jorgensen13}.\nIt is consistent with the median formation redshift $z_f$$\\sim$2.2 derived \nby \\cite{woodrum17} for field early-type galaxies at $z\\sim1$ with velocity \ndispersion larger \nthan 170 km\/s (as it is the case for six out of our seven spheroids; see Fig. \\ref{fig:age_m}).\n\nThe lower panel of Fig. \\ref{fig:metal} shows the stellar age of galaxies \nas a function of redshift for the same data samples considered in the upper panel.\nThe different curves show the prediction of stellar age for different values of \nthe formation redshift in the case of passive ageing. \nThe median stellar age of early-type galaxies in clusters at $z<0.9$ varies \nfrom cluster to cluster and, for some of them, it is consistent with formation \nredshifts even higher than our result ($z_f$$>$3).\n\nContrary to metallicity, there is a significant scatter in\nthe ages of the seven spheroids, the largest difference being $\\Delta Age\\sim3$ Gyr,\nindependently of considering mass-weighted or light-weighted values. \n{ To estimate the significance of the scatter, we constructed N=1000 random \nrealizations of the age distribution of our sample, assuming a constant mean \nage for all galaxies. \nFor each galaxy, we assumed a normal distribution with lower (upper) $\\sigma$ \ngiven by the lower (upper) relative error on its age, and the same central \nvalue of 2.4Gyr, the median age of our sample. \nFor each realization, we estimated the standard deviation of the age \ndistribution $\\sigma_{age}$. \nFrom the distribution of $\\sigma_{age}$ among different realizations, we obtained\nan expected scatter of measurement $\\sigma_m=0.57^{+0.33}_{-0.23}$ Gyr for \nour sample. \nSubtracting in quadrature the $\\sigma_m$ to the observed standard deviation of \nage estimates for our sample ($\\sim 1$~Gyr) we obtained an intrinsic age \nscatter of $\\sigma_I=0.78^{+0.24}_{-0.17}$ Gyr, \nwhich is significantly larger than zero at $\\sim 4$ sigma level. \n}\n\nThe scatter in stellar age corresponds to a significant spread of the formation redshift\n(see Tab. \\ref{tab:fitting} and Fig. \\ref{fig:metal}).\nThe seven spheroids at $z\\sim1.3$ are on the ageing tracks \nthat would lead them to join the present-day population of cluster early-type galaxies.\nHowever, this does not mean that they all have to evolve passively to $z$=0.\nThe large scatter of their ages, indeed, leaves room to possible minor \nepisodes of star formation that some of them could \nexperience during the following 9 billion years.\n\nTable \\ref{tab:fitting} reports also the number of SSPs, n(SSPs), contributing to the \nsynthesis of the spectrum. \nThe contribution of each component, in terms of stellar mass fraction, metallicity and age, \nthat is the star formation history (SFH) of each galaxy,\nis shown in the right panels of Figure \\ref{fig:star1}.\nThe histograms represent the stellar mass fraction of each component, \nthe colour identifies the metallicity (green sub-solar, yellow solar, red above solar), \nthe bar-code on the bottom of Fig. \\ref{fig:star1} shows the ages of the SSPs\nconsidered.\nThe lookback time and the formation redshift corresponding to the ages of the SSPs \nare also shown on the top x-axis of the figures. \n\nGiven the small number of galaxies, we are not in position to study the \ndependence of the SFH of early-type galaxies at $z\\sim1.3$ on galaxy mass\nas done, e.g., by \\cite{thomas05,thomas10}, \\cite{delarosa11} or \\cite{mcdermid15},\nsince our results would be dominated by the intrinsic scatter among the galaxies.\nHence, we consider here only the overall features of the sample.\n\nIn most cases, the bulk of the stellar mass is formed in \na single main episode of star formation, as for galaxies \\#972, \\#1142, \\#1370, \\#1711,\nand also for galaxy \\#1630, the youngest galaxy most likely in quenching phase\n(see \\S 4.1).\nHence, these galaxies are characterized by a similar SFH.\nHowever, their mass-weighted ages, that represent the epoch of \nformation of most of their stellar content, differ significantly,\npointing to different formation epochs.\nIn the remaining cases, the SFH is protracted over longer time or distributed among\ndifferent episodes of star formation, as in the case of \\#651, and still not ceased,\nas for \\#1442.\nThe formation redshift of the bulk of galaxy stellar mass, is \nconsistent with the mean value found by \\cite{delarosa11} and \\cite{mcdermid15} \n(lookback time $>$10 Gyr) for galaxies with similar dynamical mass\n(log(M$_{dyn}$)$>$11 M$_\\odot$), even if the SFH of some of our spheroids \nimplies a larger lookback time. \n\nOverall, our results show that the population of massive spheroids, in this cluster at \nleast, is not coeval and their stellar populations have been formed at significant \ndifferent epochs. \n\n \n\\section{Age and metallicity scaling relations}\n\n\\begin{figure*}\n \\includegraphics[width=15truecm]{age_m_sigma.ps}\n\\caption{\\label{fig:age_m} Age (upper panel) and stellar metallicity (lower panel) of galaxies \nversus velocity dispersion $\\sigma_e$ (left), dynamical mass M$_{dyn}$ (middle) and\nstellar mass density $\\Sigma_e$ (right).\nEach panel reports the probability P$_{Sp}$ based on the Spearmen rank test that\nthe light-weighted Age and [Z\/H] are uncorrelated with each quantity on the x-axis.\nThe red filled (open) circles are the light-weighted (mass-weighted)\nvalues obtained for the 7 spheroidal galaxies in the cluster XLSS0223 at $z=1.22$ \n(see Tab. \\ref{tab:fitting}); the black triangle is the median value.\nThe dark gray dashed curves are the relations of light-weighted Age and [Z\/H]\nwith velocity dispersion and dynamical mass found by \\citet{thomas10} for local galaxies.\nThe black solid curves are the local relations we derived using a sampe\nextrated from the SPIDER sample \\citep[see \\S 6;][]{labarbera10}. \nThese relations are evolved back in time to $z=1.22$ in the upper panel \nunder the assumption of passive evolution. \nThe green dot-dashed lines are the relations found by \\citet{jorgensen17}\nfor cluster early-types at $z$$\\sim$0.9.\nThe thick blue lines in the lower panels are the least-squares fit\nto the data. \n}\n\\end{figure*}\n\nIn Fig. \\ref{fig:age_m}, the age and the stellar metallicity of the seven\ncluster spheroidal galaxies are plotted versus their stellar velocity \ndispersion $\\sigma_e$ [km $s^{-1}$], dynamical mass \nM$_{dyn}=5\\sigma^2R_e\/G$ [M$_\\odot$] and stellar mass density within the \neffective radius $\\Sigma_e=0.5M^*\/\\pi R_e^2$ [${\\rm M_\\odot\\ pc^{-2}}$]. \nVelocity dispersion measurements are from Saracco et al. (2018, in preparation).\nGiven the small number of galaxies of our sample, it is beyond the\nscope of our analysis to establish the relationships between stellar\npopulations properties and structural properties in early-type \ngalaxies at $z$$\\sim$1.3.\nOur aim is to assess whether correlations can be detected in our data,\nhence whether they are already present at this redshift, and how the properties\nof these spheroidal galaxies compare with the local relations.\n\nIn Fig. \\ref{fig:age_m}, the light-weighted parameters instead\nof the mass-weighted ones, are highlighted by red filled\ncircles to make the comparison with local relations, based on \nlight-weighted values, easier.\nThe age and metallicity versus $\\sigma$ relations found by \\cite{jorgensen17} \nfor cluster early-type galaxies in the range 0.2$<$$z$$<$0.9 are shown (green \ndot-dashed curves).\n{ The local relations, age and metallicity versus $\\sigma_e$ and dynamical mass,\nfrom \\cite{thomas10} (dashed curves) are also resported, for sake of completeness.\nThese relations, indeed, cannot be used as direct comparison for our sample,\nsince they have been derived from a sample that includes galaxies\nat $z\\sim0$ with ages younger than the lookback time ($t_{LB}\\sim 8.5$~Gyr) \ncorresponding to $z\\sim1.2$ (especially at the lowest end of our velocity \ndispersion range), whose progenitors are likely not included in our sample.\nTo overcome this problem, we constructed a local sample of spheroids with properties\nsimilar to our sample, from which we derived the local reference relations.\nWe extracted from the SPIDER sample \\citep{labarbera10}\nat $z\\sim0.07$, a sample of galaxies with $\\sigma_e>190$ km\/s (i.e. the range\nwhere 6 out of our 7 spheroids lie)\nand ages older than 7 Gyr.\\footnote{Notice that we allowed for an age\nyounger than $t_{LB}$, to account for the progenitor bias, \nyet leaving room to spheroids that may have experienced a secondary minor \nburst of star formation at later times.}\nThe derived relations, age versus $\\sigma_e$, M$_{dyn}$ and $\\Sigma_e$,\npassively evolved back in time, are shown in Fig. \\ref{fig:age_m} as black solid curves.\nThe stellar mass densities $\\Sigma_e$ are derived using Sersi\\'c effective radius and \nstellar masses based on \\cite{vazdekis10} models and Chabrier IMF \\cite[see][]{swindle11}. \n}\n\nIn each panel, the probability based on the Spearmen rank test that the \ntwo parameters shown are mutually uncorrelated is also reported. \nAs expected, the relations involving the age and\/or the \nvelocity dispersion are those presenting the largest scatters.\nWe do not detect a significant correlation between age and the other\nparameters of galaxies, such as $\\sigma_e$, M$_{dyn}$ and $\\Sigma_e$.\n\n{ The seven spheroidal galaxies agree with what expected from the extrapolation\nof the local relations, when they are passively evolved back in time.\nThe data at $z\\sim1.2$ follow the same relations between age and mass,\nvelocity dispersion and mass density, than those at $z\\sim0$.}\nIt is worth to note, that the trend of age and metallicity \nwith velocity dispersion may be affected by the aperture within \nwhich measurements are done \n\\citep[e.g.][]{labarbera10,mcdermid15}, \nbecause of the presence of metallicity and age gradients \nboth in low-redshift \\cite[e.g.][]{saglia00, wu05, labarbera09,marian18} \nand in high-redshift spheroids \\cite[e.g.][]{guo11,gargiulo11,gargiulo12,chan16,ciocca17}.\nHowever, it has been shown that measurements within different regions affect \nthe offsets of the relations but not (significantly) their slope \n\\citep[e.g.][]{mcdermid15,gargiulo17}.\n{ The agreement of our data with the local relations and with\nthose found by \\cite{jorgensen17} over the range $0.2$190 km s$^{-1}$, are also shown for comparison.\nThe first one, extracted from the sample of \\cite{thomas10}, has\nstellar mass densities derived using DeVaucouleur effective radii and the stellar \nmasses of \\cite{comparat17},\nbased on MS11 models \\citep{maraston11} with Charbier IMF and MILES stellar library.\nThe second one, extracted from the SPIDER sample \\citep{labarbera10}, has\nstellar mass densities derived using Sersi\\'c effective radius and stellar masses \n based on \\cite{vazdekis10} models and Chabrier IMF of \\cite[see][]{swindle11}. \nThis fundamental relation is shown here for clarity and it will be discussed in \ndetail in a forthcoming paper.\n\nThe relationships involving metallicity, dynamical mass and \nstellar mass density, are rather robust and do not depend either on the models\nused or on the fitting method.\nFig. \\ref{fig:labarb} shows, as example, the relations as obtained through line strengths \nfitting reported in Tab. \\ref{tab:parameters} (see \\S 4).\nThe same results are obtained using the parameters obtained with MS11 or \nBC03 models.\nIndeed, as shown in Tab. \\ref{tab:bc03} and \\ref{tab:m11}, within each set of results, \ngalaxies keep nearly the same ranking with respect to a given parameter.\n\nThe most clear correlation we detect in our data is between metallicity \n[Z\/H] and dynamical mass M$_{dyn}$.\nIn this case the correlation is statistically significant and becomes much \nweaker when the velocity dispersion is considered.\nOverall, we find that the basic trends observed in the local universe, were already \nestablished at $z\\sim1.3$: { massive spheroids were characterized by higher stellar \nmetallicity, lower mass density and stellar populations tendentially older than \nthe lower mass spheroids.} \nIt is worth to recall that, here, the mass is the total mass of the galaxy,\nincludes both dark and barionic matter.\n\n\n\\begin{figure}\n\\includegraphics[width=8truecm]{mdyn_sigma.ps}\n\\caption{\\label{fig:mdyn_sigma} Dynamical mass versus stellar mass density\nof cluster spheroidal galaxies.\nThe dynamical mass M$_{dyn}=5\\sigma^2 R_e\/G$ of early-type galaxies in \nclusters at 1.2$<$$z$$<$1.4 is plotted versus their stellar mass density\n$\\Sigma_e=0.5M^*\/\\pi R_e^2$.\nThe red filled circles are the 7 spheroidal galaxies in the cluster XLSSJ0223 at $z=1.22$, \nthe green squares are spheroidal galaxies in the cluster RDCS0848 at $z$=1.27 from\n\\citet{jorgensen14}, the blue triangles are spheroidal galaxies in the cluster\nXMMUJ2235 at $z$=1.39 from \\citet{beifiori17}.\nBlack crosses are the spheroidal galaxies at $z$$\\sim$0.05 \nwith $\\sigma_e>$190 km s$^{-1}$ from the sample of \\citet{thomas10}. \nLight-gray filled circles are the spheroids at similar redshift\nwith $\\sigma_e>$190 km s$^{-1}$ extracted from the SPIDER sample \\citet{labarbera10}.\nThe black line represents the orthogonal fit to the high-z cluster data\nlog(M$_{dyn}$\/M$_\\odot$)=(-0.7$\\pm$0.2) log($\\Sigma_e$)+(13.3$\\pm$0.5). \n}\n\\end{figure}\n\\begin{figure}\n \\includegraphics[width=8.5truecm]{labarb.ps}\n\\caption{\\label{fig:labarb} Same as Fig. \\ref{fig:age_m} but for\nthe stellar population parameters of Tab. \\ref{tab:parameters}\nresulting from the fitting of spectral line indices (see \\S 4).\n}\n\\end{figure}\n\n\\section{Summary and conclusions}\nIn this paper we have presented an analysis of the stellar populations\nin seven spheroidal galaxies in the cluster XLSSJ0223 at $z$$\\sim$1.22.\nThe analysis, based on our optical spectra collected at LBT, is aimed at \nconstraining the epoch of formation and the evolutionary path of these seven \nspheroids through the derivation of their main stellar population parameters, \nage and metallicity, and the reconstruction of their SFHs.\n\nWe have measured absorption line strengths in the rest-frame 3400-4300 \\AA\\\nand used them to constrain the age and the metallicity of the stellar \npopulations through the comparison with measurements at lower redshift\nfrom the literature and predictions from stellar population models.\nWe find that the metallicity of these spheroids is not different from the \nmetallicity measured in early-type galaxies in clusters at lower redshift.\n\nWe then derived the age and the metallicity of the stellar populations\nfrom spectral fitting.\nThe resulting median metallicity of the seven galaxies is [Z\/H]=0.09$\\pm$0.16,\nwith all the values within 0.2 dex from the solar value.\nThese values agree with those of early-type galaxies\nin clusters at 0$<$$z$<0.9 and with the few single measurements of\nfield early-types at higher redshift. \nThe constant metallicity value over the redshift range 0$<$$z$$<$1.3 \nshows that no significant metallicity evolution for the population\nof cluster early-types has taken place over the last 9 billion years.\nThis implies also that no significant additional star formation and chemical \nenrichment are required for these seven spheroids to join the present-day \npopulation of cluster early-type galaxies. \n\nThe median mass-weighted age of the seven spheroids is =2.4$\\pm$0.6 Gyr,\ncorresponding to a formation redshift $<$$z_f$$>$$\\sim2.6_{-0.5}^{+0.7}$,\nor a median Lookback Time = 11$_{-1.0}^{+0.6}$ Gyr. \nWe find a significant scatter in galaxy ages and, consequently, in \nformation redshifts, showing that the population of massive spheroids, at \nleast in the cluster XLSSJ0223, is not coeval.\nOn the other hand, galaxy ages agree with those measured for cluster early-types \nat lower redshift and passive evolution would lead our sample to join the \npresent-day population.\n\nWe compared the relations between age and metallicity with velocity\ndispersion and dynamical mass of the seven spheroids with those\nat lower redshift and in the local universe.\nWe do not detect any significant correlation \nbetween age and the structural parameters \nconsidered, velocity dispersion $\\sigma_e$, dynamical mass M$_{dyn}$ and effective \nstellar mass density $\\Sigma_e$. \nThis is possibly because of the large intrinsic scatter of the age\nof a stellar population, which is affected by many physical\/local parameters,\nand the small number of galaxies.\n{ The age-velocity dispersion, age-mass and age-mass density relations of the \nseven spheroids agree with those derived for local spheroids, once \npassively evolved back in time and progenitor\nbias is taken into account.}\n\nOn the contrary, we find that the metallicity [Z\/H] \nis correlated to dynamical mass M$_{dyn}$, according to a relation very \nsimilar to the one obtained for local spheroidal galaxies.\nWe also show that the metallicity [Z\/H] is anticorrelated to the stellar mass \ndensity $\\Sigma_e$ because of the anticorrelation between M$_{dyn}$\nand $\\Sigma_e$.\n\nThe data show that no significant metallicity evolution for the population \nof cluster galaxies has taken place in the last 9 billion years.\nThis suggests that no additional major episodes of star formation\/chemical \nenrichment or minor mergers (that would dilute the metallicity) are experienced\nby cluster spheroidal galaxies at $z\\sim1.3$ \nto join the local population.\n\nWe find that the basic trends observed in the local universe, were already \nestablished at $z\\sim1.3$, with more massive spheroids having higher metallicity, \nlower stellar mass density and older ages than their lower mass counterparts.\n\n\\section*{Acknowledgements}\nWe thank the referee for the usefull and constructive comments \nthat improved the presentation of the results.This work is based on observations carried out at the Large Binocular Telescope\n(LBT) under program ID 2015\\_2016\\_28. \nThe LBT is an international collaboration among institutions in the US, Italy \nand Germany. \nWe acknowledge the support from the LBT-Italian Coordination Facility\nfor the execution of the observations, the data distribution and for \nsupport in data reduction. \nWe thank D. Thomas and J. Lian for having provided us with the median \nmetallicity values of eBOSS galaxies from Comparat et al. (2017).\nPS would like to thank T. Durden for the useful suggestions.\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and motivation}\\label{sec:intro}\n\nSymmetries have always been an indisputable phenomenon in the real world, considered often as an aspect of perfection, balance and harmony. Being symmetric is a potentially very useful feature which many real shapes exhibit, and thus symmetries in the natural world have significantly inspired people to incorporate them when producing tools, buildings or artwork. However, investigation of symmetries has also become a powerful and effective tool of theoretical research in various disciplines. For instance, the notion of symmetry is a basic concept in Felix Klein's Erlangen Program devoted to the characterization of classes of geometry based on the underlying symmetry groups, see \\cite{Kl1893}. Nowadays, symmetry detection belongs among standard problems in geometry, computer graphics, computer vision, geometric modelling, pattern recognition etc. It is very important for many applications to know how to detect symmetries in geometrical models and a number of subsequent algorithms is based on this.\n\nLet us recall that an object possesses a symmetry if there is a geometric transformation that maps this object onto itself, i.e., the object is invariant under this transformation. Symmetry is global, when it concerns the whole object or local, when only part of the object has this property. Symmetry can also be defined according to the type of transformation used (reflectional, rotational, central,\\, \\ldots). Symmetry can be exact (strong) or approximate (weak) with a given tolerance. Especially in many applications the considered geometric shapes are only approximate models of real objects (described often in a floating point representation) and therefore the symmetries are not perfect but only approximate. To develop symmetry detection methods, able to work with artificial as well as real-life data is a challenge, regardless whether the research is just theoretical or it concentrates on a certain application area. \n\nIn this paper we will focus on global exact symmetries of investigated objects, in particular of planar curves. This research area has recently (or relatively recently) experienced a renaissance in geometric modelling and related disciplines and we can find many papers focused on the detection and computation of symmetries or equivalences of curves given implicitly or by their (most often) rational parameterization. We recall e.g. \\cite{BrKn04,LeRG08}, or recent series of papers \\cite{Al14,AlHeMu14,AlHeMu14b,AlHeMu18}. In \\cite{HaJu18}, the authors present a method more general than the previous ones as they study general affine and projective equivalences in an arbitrary space dimension. The problem of deterministically computing the symmetries of a given planar algebraic curve, implicitly defined, and the problem of deterministically checking whether or not two implicitly given, planar algebraic curves are similar, i.e., equal up to a similarity transformation, was investigated in \\cite{AlLaVr19}. Projective equivalence of special algebraic varieties, including projective (and other) equivalences of rational curves were studied and presented in \\cite{BiLaVr20a}. A simple algorithm for an approximate reconstruction of an inexact planar curve which is assumed to be a perturbation of some unknown planar curve with symmetry was formulated in \\cite{BiLaVr20}. \nExact and approximate similarities of non-necessarily rational planar, parameterized curves, using centers of gravity and inertia tensors were studied in \\cite{AlQu21}.\n\nAnother problem is detection of symmetries of objects in 2D or 3D which are represented discretely as sets of points.\nSo the goal is to find the symmetry transformation (if any) under which a certain point set is (exactly or approximately) mapped onto itself. This problem is addressed mainly by computer scientists as they typically process set of points or meshes. It is interesting that some approaches suggest to compute first an implicit equation associated to the given set and then to apply mathematical methods formulated for algebraic curves and surfaces, see e.g. \\cite{TaCo98}, \\cite{TaCo00}. We recall once more that methods for detecting symmetries of point sets and objects have been widely studied in computer graphics, computer vision and pattern recognition. For the sake of brevity we recall at least some algorithms for detecting exact global symmetries from this point of view, see \\cite{Wolter1985,Alt1988,Aguilar2015}. Of course, much more papers are devoted to approximate symmetries as the mentioned scientific disciplines work mainly with real-world, i.e., non-perfect, data. \n\n\nIn this paper, we continue with our previous research on symmetries of geometric shapes form the mathematical point of view, i.e., we are mainly interested in objects described by equations. On the other hand, we must admit that motivation for our approach originated in computer graphics. We have noticed that the iterative process of Laplacian smoothing of polygonal mesh used for removing rough features of input shapes preserves symmetries of the original non-smoothed object. And this is only a small step to signal processing, discrete Fourier transform and thus also to trigonometric curves. We show that trigonometric interpolation makes perfect sense when one wants to determine exact global symmetries of closed discrete curves, as the symmetries of trigonometric curves can be easily and directly determined from their trigonometric parameterization without any need to switch to their implicit or rational description. A~special role is played by the so called higher cycloid curves considered as generalizations of standard cycloids. \n\n\nThe structure of the paper is the following. Section \\ref{sec:prelim} provides some general results, including some background on isometries and symmetries, trigonometric curves and trigonometric interpolation, to be used later in the paper. Symmetry computation of trigonometric curves is addressed in Section \\ref{sec:trigcurves_sym}, first for rotations including central symmetries, then for the axial symmetries. The next Section \\ref{sec:sym_polyline} is devoted to the application on discrete curves in plane. By unique assigning to each discrete curve a certain trigonometric parameterization we transform the problem of computing symmetries of discrete curves to the previous case. In Section~\\ref{sec:convexhull} we show that the formulated approach can be immediately used also for unorganized clouds of points. Our conclusions and some lines for future research are presented in Section \\ref{sec:sum}.\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\n\\subsection{Isometries and symmetries}\\label{sec:iso_sym}\n\nAn~\\emph{isometry} of the plane $\\mathbb{R}^2$ is a~mapping $\\phi: \\mathbb{R}^2\\rightarrow\\mathbb{R}^2$ which preserves distances. It can be always written in the form\n\\begin{equation}\\label{eq isometry}\n \\f x\\mapsto \\f A\\cdot \\f x+\\f b,\n\\end{equation}\nwhere $\\f A\\in\\mathbf{O}(2)$ is an orthogonal matrix and $\\f{b}\\in\\mathbb{R}^2$. The group of isometries in $\\mathbb{R}^2$ will be denoted by $\\mathrm{Iso}\\,(\\mathbb{R}^2)$. The isometries preserving also the orientation of the plane are called \\emph{direct} isometries and it holds $\\det(\\f A)=1$ in this case. For $\\det(\\f A)=-1$ we speak about {\\em indirect} isometries. Identity, translation and rotation are examples of direct isometries, whereas reflection (i.e., axial symmetry) belongs among indirect isometries.\n\n\nTwo point sets $X,X'\\subset \\mathbb{R}^2$ are isometric (or {\\em congruent}\\\/) if there exists $\\phi\\in \\mathrm{Iso}\\,(\\mathbb{R}^2)$ such that $X'=\\phi(X)$. Furthermore, $X$ is said to possess a \\emph{symmetry} if there exists a non-identical isometry $\\phi$ such that $X=\\phi(X)$. The {\\em group of symmetries} of $X$ will be denoted by $\\mathrm{Sym}\\,(X)$. The subgroup of orientation-preserving (i.e., direct) symmetries is called the {\\em proper symmetry group}. \n\n\\medskip\nIt is well known that any finite subgroup of $\\mathrm{Iso}\\,(\\mathbb{R}^2)$ falls into the following classes:\n{\\em cyclic groups} $C_m$ consisting of all rotations about the same center by multiples of the angle $\\frac{2\\pi}{m}$; and {\\em dihedral groups} $D_m$ consisting of all the rotations in $C_m$ together with all reflections in $m$ axes that go through the same point, the center. Especially, $C_1$ is the trivial group containing only the identity;\n$D_1$ is the group containing the identity and one reflection; \n$C_2$ is the symmetry group of the letter `S' consisting of the identity and one central symmetry; $D_2$ is the symmetry group of a rectangle which is not a square containing the identity, one central symmetry and two reflections with perpendicular axes. And for $m\\geq 3$ we have the symmetry groups ($D_m$) or the proper symmetry groups ($C_m$) of some non-oriented or oriented regular $m$-gon, respectively. \n\n\\smallskip\nHence, to determine such a symmetry group it is sufficient to find:\n\\begin{enumerate}\n \\vspace{-1ex}\n \\item the center of symmetry (barycenter),\n \\item the rotation angle $\\frac{2\\pi}{m}$ (where $m=1$ corresponds to the object without any rotational symmetry),\n \\item the direction of one symmetry axis (if the $m$-gon is non-oriented, i.e., the symmetry group $D_m$).\n\\end{enumerate}\n\n\\medskip\n\nThe strategy used in the next sections for determining symmetries is based on application of a suitable operator $\\Psi$ which commutes with isometries. Then the unique assigning of some object to a given geometric shape $X$ via $\\Psi$ enables to study symmetries of $\\Psi(X) $ instead of symmetries of $X$. This is formalized in the following lemma. Of course, it is essential to find $\\Psi$ wisely \nin order to simplify the situation, i.e., $\\mathrm{Sym}\\,(\\Psi(X))$ must be easier to determine than direct computing $\\mathrm{Sym}\\,(X)$.\n\n\n\\begin{lemma}\\label{lem:komut}\nLet be given $\\mathcal{X}\\subset\\mathcal{P}(\\mathbb{R}^2)$, where $\n\\mathcal{P}(\\mathbb{R}^2)$ is the set of all subsets of $\\mathbb{R}^2$, and $\\Psi:\\mathcal{X}\\rightarrow\\mathcal{P}(\\mathbb{R}^2)$ such that\nfor all $X\\in\\mathcal{X}$ and for all $\\phi\\in\\mathrm{Iso}\\,(\\mathbb{R}^2)$ it holds\n\\begin{equation}\n\\Psi(\\phi(X))=\\phi(\\Psi(X)),\n\\end{equation}\nthen $\\mathrm{Sym}\\,(X)\\subset\\mathrm{Sym}\\,(\\Psi(X))$.\n\\end{lemma}\n\n\\begin{proof}\n Let $\\phi\\in\\mathrm{Sym}\\,(X)$, then $\\phi(\\Psi(X))=\\Psi(\\phi(X))=\\Psi(X)$.\n\\end{proof}\n\n\n\\begin{example}\\rm\nWe recall \\cite{AlLaVr19} where $\\mathcal{X}$ is a set of real planar algebraic curves and the mapping $\\Psi$ is the Laplace operator $\\Delta$. As known this operator commutes with isometries. This property was used in the referred paper for finding symmetries of an algebraic curve, say $f$, via the chain of symmetry groups \n\\begin{equation}\n\\mathrm{Sym}\\,(f)\\subset\\mathrm{Sym}\\,(\\Delta\\,f)\\subset\\mathrm{Sym}\\,(\\Delta^2\\, f)\\subset\\cdots\\subset\\mathrm{Sym}\\,(\\Delta^{\\ell}\\, f)=\\mathrm{Sym}\\,(h),\n\\end{equation}\nwhere $\\mathrm{Sym}\\,(h)$ is finite and easier to determine. \n\\end{example}\n\n\\begin{example}\\rm\\label{ex:Discrete_Laplace}\nIn this paper, we are interested in discrete curves, see Section~\\ref{sec:sym_polyline}. So we mention also a discrete analogy of $\\Delta$. Consider a closed planar polyline $X$ formed by a sequence of ordered points $\\f v_0,\\f v_1,\\ldots, \\f v_n=\\f v_0$ connected with line segments. \n\nA~\\emph{discrete Laplacian operator} is a map associating to each $\\f v_i$ the vector \n\\begin{equation}\n L(\\f v_i) = \\f v_i-\\frac{1}{2}(\\f v_{i-1}+\\f v_{i+1}),\n\\end{equation}\nThis is a linear operator which can be represented by the circular matrix \n\\begin{equation}\n L=\n \\begin{pmatrix}\n 1 &-\\frac{1}{2}&0&\\dots&\\dots&0&-\\frac{1}{2}\\\\\n -\\frac{1}{2}&1&-\\frac{1}{2}&0&\\dots&\\dots&0\\\\\n \\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\\\\n 0&\\dots&\\dots&0&-\\frac{1}{2}&1&-\\frac{1}{2}\\\\\n -\\frac{1}{2}&0&\\dots&\\dots&0&-\\frac{1}{2}&1\n \\end{pmatrix}.\n\\end{equation}\nNext, a~\\emph{smoothing operator} is defined as \n\\begin{equation}\n S=I-\\lambda L,\n\\end{equation}\nwhere $\\lambda\\in (0,1)$, i.e., each vertex $\\f v_i$ is replaced by the affine combination $(1-\\lambda)\\f v_i+\\frac{\\lambda}{2}(\\f v_{i-1}+\\f v_{i+1})$. The smoothing operator commutes with the affine (and thus also Euclidean) transformations and thus we arrive to the conclusion that it holds $\\mathrm{Sym}\\,(X)\\subset\\mathrm{Sym}\\,(S(X))$, cf. Fig.~\\ref{fig:smoothing}.\n\\end{example}\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=0.23\\textwidth]{smoothingB_1.png}\n\\hspace{3ex}\n\\includegraphics[height=0.23\\textwidth]{smoothingB_2.png}\n\\hspace{3ex}\n\\includegraphics[height=0.23\\textwidth]{smoothingB_3.png}\n\\hspace{3ex}\n\\includegraphics[height=0.23\\textwidth]{smoothingB_4.png}\n\\caption{Laplace smoothing and preservation of symmetries. From left to right: The original polyline and the polylines after Laplace smoothing w.r.t. $10$, $100$ and $1000$ step. All the polylines are symmetric w.r.t. the red axis.}\\label{fig:smoothing}\n\\end{center}\n\\end{figure}\n\n\\begin{remark}\\rm \nIn fact, the content of Example~\\ref{ex:Discrete_Laplace} was our first step to the topic of this paper. We have noticed that \nthe procedure of smoothing used in computer graphics for removing rough features of a given shape exhibits the property that it preserves symmetries of the original non-smoothed version. And from the Laplacian smoothing and its analysis is a short way to signal processing and discrete Fourier transform and thus to trigonometric curves, see \\cite{Co-Or_etal15} and the following sections.\n\\end{remark}\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[height=0.2\\textwidth]{trig_curve_1.png}\n\\hspace{1ex}\n\\includegraphics[height=0.2\\textwidth]{trig_curve_2b.png}\n\\hspace{1ex}\n\\includegraphics[height=0.2\\textwidth]{trig_curve_3.png}\n\\hspace{1ex}\n\\includegraphics[height=0.2\\textwidth]{trig_curve_4.png}\n\\caption{Examples of trigonometric curves. From left to right: a generic non-symmetric trigonometric curve (all $\\f p_k$'s are ellipses), a trigonometric curve with a central symmetry (all $\\f p_k$'s are ellipses), a non-symmetric higher cycloid curve, and a higher cycloid curve with rotational and axial symmetries.}\\label{fig:trig_curves}\n\\end{center}\n\\end{figure}\n\n\\subsection{Trigonometric curves}\\label{sec:trigcurves}\n\nBy a~\\emph{trigonometric curve} in $\\mathbb{R}^2$ we understand a real curve possessing a parameterization in the form\n\\begin{equation}\\label{eq:C_m}\n \\f p(t)=\\sum_{k=0}^N\\big[\\f a_k \\cos(k t)+\\f b_k\\sin(k t)\\big]=\n \\f a_0+\\sum_{k=1}^N\\big[\\f a_k \\cos(k t)+\\f b_k\\sin(k t)\\big]=\\f a_0+\\sum_{k=1}^{N}\\f p_k(t),\n\\end{equation}\nwhere $\\f a_k,\\f b_k\\in\\mathbb{R}^2$ and at least one of $\\f a_N,\\f b_N$ is non-zero, i.e., the coordinates are given by trigonometric polynomials of degree at most $N$ (truncated Fourier series). In short we will call $N$ the {\\em trigonometric degree} of $\\f p(t)$. If the $\\gcd$ of all $k\\in\\{0,\\ldots,N\\}$ for which $\\f a_k\\neq 0$ or $\\f b_k\\neq 0$ is one then $\\f p(t)$ is called \\emph{primitive} (otherwise it can be reparameterized to obtain a primitive trigonometric curve). Notice that $\\f p_k(t)$, where $t\\in[0,2\\pi]$, is a parameterization of a $k$-times traced ellipse.\nIt can be shown that \\eqref{eq:C_m} parameterizes an~algebraic irreducible curve of degree at most 2N and thus one can apply a suitable tool from algebraic geometry to get useful information about these shapes. For more details about properties of trigonometric curves see e.g. \\cite{Wu47,HoSch98,RoJuSchHo09}. Some examples of trigonometric curves are shown in Fig.~\\ref{fig:trig_curves}.\n\n\n\\smallskip\nWe recall that special instances of trigonometric curves also appear in the literature under the name \\emph{higher cycloid curves} or {\\em cycloids of $N$th stage}, see e.g. \\cite{Wu47}. They belong among kinematic curves generated by the trace of a fixed point on a circle that rolls without slipping around another circle that rolls without slipping around further circle etc. (i.e., they are generalizations of well knowns hypo-\/epicycloids). The condition on \\eqref{eq:C_m} being a higher cycloid curve has the form\n\\begin{equation}\n\\f (b_{k1},b_{k2})=\\sigma_k(-a_{k2},a_{k1}),\\quad \\mbox{where } \\sigma_k\\in\\{-1,+1\\},\n\\end{equation}\ni.e., considered $\\f a_k,\\f b_k$ as vectors in $\\mathbb{R}^2$ then they are perpendicular. Thus for $\\f p_k$ we arrive at\n\\begin{equation}\\label{eq:circle_cond}\n\\f p_k=(a_{k1},a_{k2})\\cos(kt)+\\sigma_k(-a_{k2},a_{k1})\\sin(kt).\n\\end{equation}\nFor the sake of simplicity, we naturally identify $\\mathbb{R}^2$ with $\\mathbb{C}$ and then \\eqref{eq:circle_cond} can be rewritten as\n\\begin{equation}\\label{eq:circle_cond2}\n\\f p_k=(a_{k1}\\cos(kt)-\\sigma_ka_{k2}\\sin(kt),a_{k2}\\cos(kt)+\\sigma_ka_{k1}\\sin(kt))=(a_{k1}+a_{k2}\\mathrm{i})\\cdot[\\cos(\\sigma_k kt)+\\mathrm{i}\\sin(\\sigma_k kt)].\n\\end{equation}\nWhen setting $\\cos(\\sigma_k kt)+\\mathrm{i}\\sin(\\sigma_k kt)=\\mathrm{e}^{\\sigma_k kt\\mathrm{i}}$ and $a_{k1}+a_{k2}\\mathrm{i}=\\lambda_k\\mathrm{e}^{\\psi_k\\mathrm{i}}$ we obtain the parameterization of the higher cycloid curve in the form\n\\begin{equation}\\label{eq:complex_param_higher_cycloid}\n \\f p(t) = \\f a_0+\\sum_{k=1}^N \\lambda_k \\mathrm{e}^{(\\sigma_k kt+\\psi_k)\\mathrm{i}},\n\\end{equation}\nwhere $\\lambda_k\\in\\mathbb{R}^{\\geq 0}$ are the radii of the circles, $\\psi_k\\in[0,2\\pi)$ are the ``phase shifts'' and $\\sigma_k\\in\\{-1,+1\\}$ determine the \\emph{orientation} of the parameterization $\\f p_k$, \n\n\nWunderlich discussed a lot of geometric properties of higher cycloids, e.g., he proved the generalization of Euler's theorem: {\\em Every higher cycloid of $N$th stage can be generated in $N$ different ways by rolling of two higher cycloids of stage $N-1$}. Wunderlich also used higher cycloids for curve approximation of planar curves, see \\cite{Hu04}. Some examples of higher cycloid curves are shown in Fig.~\\ref{fig:trig_curves} (the last two instances).\n\n\\subsection{Periodic data and trigonometric interpolation}\\label{sec:trig_interpol}\n\nFor later use we shortly recall the trigonometric interpolation problem as a very useful method for periodic data processing. For the sake of brevity we do not go into details and mention only the computational part of this problem -- the readers more interested in this topic are kindly referred e.g. to \\cite{Br19,Zy02,StBu13}. The connection to the determination of the symmetries of discrete curves will be discussed in Section~\\ref{sec:sym_polyline}. \n\nLet be given a set of $n$ ordered data $(t_j,x_j)$ that is periodic. In addition, we assume that $\\{t_j\\}$ are rescaled such that they lie in the interval $[0,2\\pi)$, i.e.,\nwe have\n\\begin{equation}\n0\\leq t_0 < t_1 <\\ldots < t_{n-1}< 2\\pi.\n\\end{equation}\nOur goal is to construct a trigonometric interpolation of these data, i.e., to find an interpolant in the form\n\\begin{equation}\n p(t)= \\sum_{k=0}^N\\big[a_k \\cos(k t)+b_k\\sin(k t)\\big]=a_0+\\sum_{k=1}^N\\big[a_k \\cos(k t)+b_k\\sin(k t)\\big],\n\\end{equation}\nwhere $a_k,b_k\\in\\mathbb{R}$, going through $(t_j,x_j)$. In other words it is necessary to determine $2N+1$ real coefficients from $n$ input points. The existence of solution is guaranteed by the condition $n\\leq 2N+1$. Specially, for $n=2N+1$ the interpolant is determined uniquely. If we have even number of input data, say $n=2N$, we arrive at one one-parametric solution. Then an additional requirement that the term $\\sin(Nt)$ vanishes is often considered, which also results in the unique solution. The interpolation trigonometric polynomial has in this separate case the form\n\\begin{equation}\n p(t)= a_0+\\sum_{k=1}^{N-1}\\big[a_k \\cos(k t)+b_k\\sin(k t)\\big]+a_N\\cos(Nx).\n\\end{equation}\n\n\\smallskip\nFor non-uniformly spaced data $(t_j,x_j)$ one can just solve the associated system of linear equations. However when the nodes are uniformly spaced, i.e.,\n\\begin{equation}\nt_j=j\\frac{2\\pi}{n}, \\ j=0,\\ldots,n-1.\n\\end{equation}\nwe obtain for odd number of input data $n=2N+1$ the closed-form expressions\n\\begin{equation}\\label{eq:interpol_coef_odd}\n\\begin{array}{ll}\n\\displaystyle \na_0 = \\frac{\\sum_{j=0}^{n-1} x_j}{n}, &\n\\displaystyle \na_k = \\frac{2}{n}\\sum_{j=0}^{n-1} x_j \\cos(kt_j),\\quad k=1,\\dots,N,\\\\[3ex]\n&\n\\displaystyle \nb_k =\\frac{2}{n}\\sum_{j=0}^{n-1} x_j \\sin(kt_j),\\quad k=0,1,\\dots,N.\n\\end{array}\n\\end{equation}\nThe coefficients $a_k,b_k$ are called {\\em discrete Fourier coefficients}. \n \nFor even number of input data $n=2N$ and assuming that the term $\\sin(Nt)$ vanishes, \nthe coefficients are computed as\n\\begin{equation}\\label{eq:interpol_coef_even}\n\\begin{array}{ll}\n\\displaystyle \na_0 = \\frac{\\sum_{j=0}^{n-1} x_j}{n}, \\quad \na_N = \\frac{\\sum_{j=0}^{n-1} x_j \\cos(Nt_j)}{n},\\,\n&\n\\displaystyle \na_k = \\frac{2}{n}\\sum_{j=0}^{n-1} x_j \\cos(kt_j),\\quad k=1,\\dots,N-1, \\\\[3ex]\n&\n\\displaystyle \nb_k = \\frac{2}{n}\\sum_{j=0}^{n-1} x_j \\sin(kt_j),\\quad k=0,1,\\dots,N.\n\\end{array}\n\\end{equation}\n\n\n \n\n\\section{Symmetries of trigonometric curves}\\label{sec:trigcurves_sym}\n\nFirst we would like to emphasize that any trigonometric curve $C$ is algebraic and thus it has a~finite group of symmetries unless it is a~circle or a line. In addition, these curves are also rational which suggests to apply for detecting their equivalences (including symmetries) approaches formulated for general rational curves. In this context we recall that special properties of rational parameterizations of trigonometric curves were exploited e.g. in \\cite{AlQu20} for formulating a computational method for determining affine equivalences of trigonometric curves.\nAs our plan leads to a different goal (symmetries of discrete curves via trigonometric interpolation) we formulate a simple algorithm for determining symmetries of trigonometric curves directly from their trigonometric parameterizations. \n\n\n\\smallskip\nLet us recall that there are infinitely many trigonometric parameterizations of the curve $C$, when one exists. However Theorem 4.6 from \\cite{HoSch98} says that these parameterizations are `essentially' the same. For subsequent use we recall this theorem; for its proof the readers are kindly referred to the original source.\n\n\\begin{theorem}\\label{thm:schicho}\n A trigonometric parameterization of a curve is unique up to a linear parameter change.\n\\end{theorem}\n\nIf $\\phi\\in\\mathrm{Iso}\\,(\\mathbb{R}^2)$ is a symmetry of a curve with trigonometric parameterization $\\f p(t)$ then by Theorem~\\ref{thm:schicho} there exists a linear reparameterization $t\\mapsto \\alpha t+\\beta$, $\\alpha,\\beta \\in \\mathbb{R}$, such that \n\\begin{equation}\\label{eq:reparam}\n \\phi(\\f p(t))=\\f p(\\alpha t+\\beta).\n\\end{equation}\nCorollaries of this observation will allow us to determine the group of symmetries of any~trigonometric curve easily.\n\n\n\\begin{lemma}\\label{lem:center}\n Let a trigonometric curve parameterized by $\\f p(t)$ possesses a rotational symmetry with center $\\f c$. Then $\\f c=\\f a_0$.\n\\end{lemma}\n\n\\begin{proof}\nIf $\\f c =\\f o$ is the origin, then the rotation $\\rho$ is a~linear map an thus $\\rho(\\sum\\f p_k(t))=\\sum\\rho(\\f p_k(t))$. In particular, \\eqref{eq:reparam} implies that $\\rho(\\f a_0)=\\f a_0$ and thus $\\f a_0=\\f o$. The statement with general center is an immediate consequence.\n\\end{proof}\n\nIn the same way we can prove an analogous lemma dealing with axial symmetries.\n\n\\begin{lemma}\\label{lem:point on L}\n Let a trigonometric curve parameterized by $\\f p(t)$ possesses an axial symmetry with axis $L$. Then $\\f a_0\\in L$.\n\\end{lemma}\n \nFor the sake of simplicity we will assume in what follows that $\\f a_0=\\f o$ and thus in order to determine the symmetries of the curve it is enough to find the rotation angle and\/or the direction of the axis. Let us start with the rotational symmetries.\n\n\\begin{lemma}\\label{lem:rot_sym}\nLet $\\rho_{\\frac{2\\pi}{m}}:\\mathbb{R}^2\\rightarrow\\mathbb{R}^2$ be the~rotation around the origin by the angle $\\frac{2\\pi}{m}$. Then it is a symmetry of $\\f p(t)$ iff \nthere exists $d\\in\\mathbb{Z}$ such that \n\\begin{equation}\n\\rho_{\\frac{2\\pi}{m}}(\\f p(t))=\\f p\\left(t+\\frac{2\\pi d}{m}\\right).\n\\end{equation}\\end{lemma}\n\n\\begin{proof}\nBy~\\eqref{eq:reparam} we have $\\rho_{\\frac{2\\pi}{m}}(\\f p(t))=\\f p(\\alpha t+\\beta)$. Since $\\rho_{\\frac{2\\pi}{m}}$ is a linear orientation preserving mapping then $\\alpha$ must equal to $1$. Applying the rotation $m$ times we arrive at the original curve, i.e., $\\f p(t)=\\f p(t+m\\beta)$. As $\\f p(t)$ is $2\\pi$-periodic we can conclude that $m\\beta = 2\\pi d$.\n\\end{proof}\n\n\n\n\n\n\\begin{remark}\\rm\nIn the case $m=2$, the curve is centrally symmetric. If the center is at the point $\\f o$ then Lemma~\\ref{lem:rot_sym} says that $-\\f p(t)=\\f p(t+\\pi)$. Moreover, this identity must hold for every $\\f p_k(t)=\\f a_k \\cos(kt)+\\f b_k\\sin(kt)$ which implies $\\f p_k(t)=0$ for all even $k$. A particular example of a centrally symmetric trigonometric curve \nis shown in Fig.~\\ref{fig:trig_curves} (second from the left). \n\\end{remark}\n\n\nSince a general ellipse does not possess an $m$-fold rotational symmetry for $m>2$, the~requirement for the existence of such a symmetry of the whole curve $\\f p(t)$ forces each ellipse $\\f p_k(t)$ to become a circle. In this case it is possible to write $\\f p(t)$ (centered at $\\f o$) in the form\n\\begin{equation}\\label{eq:complex_param}\n \\f p(t) = \\sum_{k=1}^N \\lambda_k \\mathrm{e}^{(\\sigma_k kt+\\psi_k)\\mathrm{i}},\n\\end{equation}\nsee \\eqref{eq:complex_param_higher_cycloid}.\n\nBy Lemma~\\ref{lem:rot_sym} the curve parameterized by $\\f p(t)$ has an $m$-fold rotational symmetry if and only if\n\\begin{equation}\\label{eq:complex_sym}\n \\mathrm{e}^\\frac{2\\pi\\mathrm{i}}{m}\\cdot \\f p_k(t)=\\f p_k\\left(t+\\frac{2\\pi d}{m}\\right) \\quad \\mbox{for all } k.\n\\end{equation}\nCombining \\eqref{eq:complex_param} with \\eqref{eq:complex_sym} we arrive at\n\\begin{equation}\n\\lambda_k\\mathrm{e}^{\\frac{2\\pi\\mathrm{i}}{m}}\\cdot\\mathrm{e}^{(\\sigma_k kt+\\psi_k)\\mathrm{i}}=\\lambda_k\\mathrm{e}^{(\\sigma_k k(t+\\frac{2\\pi d}{m})+\\psi_k)\\mathrm{i}}\n\\end{equation}\nwhich yields\n\\begin{equation}\\label{eq:symmetry_cyclic_curves}\n \\frac{2\\pi}{m}(1-\\sigma_k\\cdot k\\cdot d)\\equiv 0 \\mod 2\\pi.\n\\end{equation}\nThis forces some $\\f p_k$'s to disappear. In particular the only non-zero frequencies occur when $\\sigma_k\\cdot k\\cdot d -1$ is divisible by $m$. Let us formulate this as a lemma.\n\n\\begin{lemma}\\label{lem:rot_symmetry}\n Let $\\f p(t)$ be a primitive trigonometric curve. Then it possesses an $m$-fold rotational symmetry ($m>2$) with the center $\\f c$ if and only if $\\f a_0=\\f c$ and there exists an integer $d$ such that $\\f p(t)$ can be written in the form~\\eqref{eq:complex_param}, where the only non-zero terms $\\lambda_k$ are those fulfilling the condition $\\sigma_k\\cdot k\\cdot d = j\\cdot m +1$ for some $j\\in\\mathbb{Z}$.\n\\end{lemma}\n\n\n\\begin{remark}\\rm\nWe recall that \\eqref{eq:complex_param} are higher cycloid curves, cf. Section~\\ref{sec:trigcurves}. They are non-symmetric in general, and Lemma~\\ref{lem:rot_symmetry} is a condition for their rotational symmetry.\n\\end{remark}\n\n\n\\medskip\nBased on Lemma~\\ref{lem:rot_symmetry} we define \\emph{$(m,d)$-sequence} $\\vartheta^{m,d}:\\mathbb{N}\\longrightarrow \\mathcal{P}(\\{-1,1\\})$ associating $\\{\\pm 1\\}$ to $k$ if $m$ divides $\\pm kd-1$ and $\\emptyset$ otherwise. We can observe that the following properties hold for all $k\\in\\mathbb{N}$:\n\\begin{itemize}\n \\item $\\vartheta^{m,d}(k)=\\emptyset$ whenever $\\gcd(m,d)\\neq 1$ (in particular $\\vartheta^{m,0}(k)=\\emptyset$),\n \\item $\\vartheta^{m,d+m}(k)=\\vartheta^{m,d}(k)$,\n \\item $\\vartheta^{m,-d}(k)=-\\vartheta^{m,d}(k)$,\n \\item $\\vartheta^{m,d}(k)=\\vartheta^{m,d}(k+m)$.\n\\end{itemize}\nEspecially for $m=1,2$ we arrive at $\\vartheta^{1,d}(k)=\\{\\pm 1\\}$, $\\vartheta^{2,1}(2k)=\\emptyset$ and $\\vartheta^{2,1}(2k+1)=\\{\\pm 1\\}$. \n\n\\medskip\nLet be given two sequences $\\alpha,\\beta:\\mathbb{N}\\longrightarrow\\mathcal{P}(\\{-1,1\\})$. \nWe define \n\\begin{equation}\n \\alpha\\preccurlyeq\\beta\\text{ if and only if for all }k\\in\\mathbb{N}\\text{ it holds }\\alpha(k)\\subseteq\\beta(k).\n\\end{equation}\nNext, we assign to given parameterization \\eqref{eq:complex_param} a sequence by the formula \n\\begin{equation}\n\\sigma_\\f p(k)=\n\\begin{cases}\n\\{\\sigma_k\\} & \\text{if } \\lambda_k\\neq 0,\\\\[1ex]\n\\ \\emptyset & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nThe parameterized curve possesses an $m$-fold rotational symmetry if and only if there exists $d\\in\\mathbb{Z}$ such that $\\sigma_\\f p\\preccurlyeq\\vartheta^{m,d}$. Among these sequences we are looking for the one with the maximal $m$. Such sequence is called the {\\em maximal $(m,d)$-sequence} associated to $\\f p(t)$. Let us emphasize that if two trigonometric parameterizations are related by a reparameterization $\\f q(t)=\\f p(\\pm t+\\beta)$ then $\\sigma_\\f q = \\pm\\sigma_\\f p$.\n\n\n\n\\begin{remark}\\rm\nBoth considered types of sequences $\\vartheta^{m,d}$ and $\\sigma_{\\f p}$ are infinite. Especially, $\\vartheta^{m,d}$ is $m$-periodic whose elements repeat over and over. For the sake of simplicity we will write it down by just stating the period, i.e., the listing will end with the element $\\vartheta^{m,d}(m)$. As concerns the sequence $\\sigma_{\\f p}$ then the last non-zero element is $\\sigma_{\\f p}(N)$, where $N$ is the trigonometric degree of the curve, and all the following elements are~$\\emptyset$. Hence our convention is that its listing will end with the element $\\sigma_{\\f p}(N)$. Then it makes sense to write for instance \n$$\n\\big(\\{1\\},\\{-1\\},\\emptyset,\\emptyset,\\{-1\\}\\big)\\preccurlyeq \\big(\\{1\\},\\{-1\\},\\emptyset,\\ldots\\big)=\\vartheta^{3,1}.\n$$\n\\end{remark}\n\n\n\n\n\\begin{example}\\rm\nSet $m=7$ then there exist $3$ different non-trivial types of $(7,d)$-sequences. Since all of them are $7$-periodic, they are determined by their first 7 terms, in particular\n\\begin{align*}\n \\vartheta^{7,1} &= (\\{1\\}, \\emptyset, \\emptyset, \\emptyset, \\emptyset, \\{-1\\}, \\emptyset,\\ldots),\\\\\n \\vartheta^{7,2} &= (\\emptyset, \\emptyset, \\{-1\\}, \\{1\\}, \\emptyset, \\emptyset, \\emptyset,\\ldots),\\\\\n \\vartheta^{7,3} &= (\\emptyset, \\{-1\\}, \\emptyset, \\emptyset, \\{1\\}, \\emptyset, \\emptyset,\\ldots).\n\\end{align*}\nSome particular instances of curves with these maximal $(m,d)$-sequences are depicted in Fig.~\\ref{fig:7fold}. Notice that since $\\vartheta^{7,i}=-\\vartheta^{7,7-i}$ for $i=1,2,3$, the curves with sequence $\\vartheta^{7,i}$ are related to those with sequence $\\vartheta^{7,7-i}$ by the~reparameterization $t\\mapsto -t$ and thus they describe the same curves.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{7fold1.png}\n\\hspace{5ex}\n\\includegraphics[width=0.25\\textwidth]{7fold2.png}\n\\hspace{5ex}\n\\includegraphics[width=0.25\\textwidth]{7fold3.png}\n\\caption{Examples of rotationally symmetric curves with maximal sequences (from left to right) $\\vartheta^{7,1}$, $\\vartheta^{7,2}$ and $\\vartheta^{7,3}$.}\\label{fig:7fold}\n\\end{center}\n\\end{figure}\n\\end{example}\n\n\n\n\\begin{remark}\\rm\nWe emphasize the importance of finding the maximal $(m,d)$-sequence associated to the given trigonometric curve. For instance, if the sequence is of the form\n$$\n\\sigma_{\\f p}=(\\{1\\}, \\emptyset, \\emptyset, \\emptyset,\\{-1\\}),\n$$\nthen it holds $\\sigma_{\\f p}\\preccurlyeq \\vartheta^{3,1}=(\\{1\\}, \\{-1\\},\\emptyset,\\ldots)$. However, the maximal sequence which is to be found is $\\vartheta^{6,1}=(\\{1\\}, \\emptyset, \\emptyset, \\emptyset, \\{-1\\},\\emptyset,\\ldots)$ and hence the curve possesses rotational symmetries of 6-gon.\n\\end{remark}\n\n\n\n\n\n\n\nNow let us move to reflections. We can formulate a lemma analogous to Lemma~\\ref{lem:rot_sym}.\n\n\\begin{lemma}\\label{lem:refl_sym}\nLet $\\omega_{L}:\\mathbb{R}^2\\rightarrow\\mathbb{R}^2$ be a reflection across a line $L$. Then it is a symmetry of $\\f p(t)$ iff there exists a constant $t_0$ such that\n\\begin{equation}\\label{eq:refl_sym}\n\\omega_{L}(\\f p(t_0+t))=\\f p(t_0-t).\n\\end{equation}\n\\end{lemma}\n\n\n\nBy Lemma~\\ref{lem:point on L} we already know one point on the axis $L$. Hence it remains to determine its direction, or equivalently we need to find the point $\\f p(t_0)$. Assume first that $t_0=0$ and the axis $L$ is identical to the $x$-axis, then by Lemma~\\ref{lem:refl_sym} we can write\n\\begin{equation}\n \\omega_L(\\f p_k(t))=\\f p_k(-t)\\quad \\mbox{for all } k.\n\\end{equation}\nSince $\\f p_k(t)=\\f a_k \\cos(k t)+\\f b_k\\sin(k t)$ and $\\omega_L:[x,y]\\mapsto[ x,-y]$ we arrive at the conditions $\\f a_k=(a_{k1},0)$ and $\\f b_k=(0,b_{k2})$. Geometrically, this says that all the ellipses $\\f p_k(t)$ are equally aligned with respect to their axes and in addition they are coherently parameterized, which means that $\\f p_k(0)$ is for all of them a vertex lying on the symmetry axis. Let us summarize this in a lemma\n\n\n\\begin{lemma}\\label{lem:ellipse_syzygy}\n Let $\\f p(t)$ be a primitive trigonometric curve. Then it possesses a reflectional symmetry with axis $L$ if and only if the line $L$ passes through the point $\\f a_0$ and there exists $t_0\\in[0,2\\pi)$ such that for all $k$ the point $\\f p_k(t_0)$ is the vertex of the ellipse and if $\\f p_k(t_0)-\\f a_0\\neq\\f o$ then it is the direction of $L$. \n\\end{lemma}\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{overpic}[height=0.3\\textwidth]{axis_sym_1.png\n\\put(35,40){$\\f o$}\n\\put(90,38){$L$}\n\\put(65,10){$\\f p$}\n\\end{overpic}\n\\hspace{5ex}\n\\begin{overpic}[height=0.3\\textwidth]{axis_sym_2.png}\n\\put(45,42){$\\f o$}\n\\put(95,40){$L$}\n\\put(25,15){$\\f p_1$}\n\\put(12,38){$\\f p_2$}\n\\put(52,25){$\\f a_1$}\n\\put(52,70){$\\f b_1$}\n\\put(55,50){$\\f b_2$}\n\\put(40,35){$\\f a_2$}\n\\put(83,50){$\\f p_2(t_0)$}\n\\put(54,41){$\\f p_1(t_0)$}\n\\end{overpic}\n\\caption{Left: Symmetric trigonometric curve $\\f p$ w.r.t. axis $L: y=0$. Right: Its decomposition into two ellipses $\\f p_1$ and $\\f p_2$. The conjugated half-diameters $\\f a_1$, $\\f b_1$ and $\\f a_2$, $\\f b_2$ of $\\f p_1$ and $\\f p_2$, respectively, together with the vertices $\\f p_1(t_0)$ and $\\f p_2(t_0)$ determining the axis of symmetry are shown.}\\label{fig:axis_sym}\n\\end{center}\n\\end{figure} \n\n \n\\begin{remark}\\rm\nRecall that the {\\em Rytz axis construction} is an elementary method from descriptive geometry that can be used to find the axes and the vertices of an ellipse (in our case $\\f p_k$) starting from its two conjugated half-diameters (in our case determined by $\\f a_k,\\f b_k$). Based on this construction the parameter value corresponding to the vertex is computed via the formula\n\\begin{equation}\\label{eq:t_k0}\n t_{k,0} = \\frac{1}{2k}\\mathrm{arccot}\\left(\\frac{\\f a_k\\cdot\\f a_k-\\f b_k\\cdot \\f b_k}{2\\f a_k\\cdot \\f b_k}\\right).\n\\end{equation}\nThe remaining vertices of $\\f p_k(t)$ are obtained for the parameter values $\\{t_{k,0}+\\frac{\\pi}{2k}j\\}$, $j=1,2,3$. If there exists non-empty intersection of these sets for all non-zero $\\f p_k$'s then this value is the sought $t_0$. \n\\end{remark}\n\n\n\\begin{example}\\rm\nConsider a trigonometric curve given by the following parameterization \n$$\n\\f p(t)=[\\sin (t)+2 \\sin (2 t)+\\cos (t),2 \\sin (t)-2 \\cos (t)-\\cos (2 t)].\n$$\nIt is a sum of two ellipses\n$$\n\\f p_1(t) = [1,-2] \\cos (t) + [1,2] \\sin (t) \\quad \\mathrm{and} \\quad \\f p_2(t)= [0,-1] \\cos (2 t) + [2,0] \\sin (2 t).\n$$\nFrom \\eqref{eq:t_k0}, all vertices of $\\f p_1(t)$ occur at the following values of the parameter $\\left\\{\\frac{\\pi }{4},\\frac{3 \\pi }{4},\\frac{5 \\pi }{4},\\frac{7 \\pi }{4}\\right\\}$, whereas the vertices of $\\f p_2(t)$ correspond to $\\left\\{0,\\frac{\\pi }{4},\\frac{\\pi }{2},\\frac{3 \\pi }{4}\\right\\}$. Hence $t_0 = \\frac{\\pi }{4}$ yields the sough-after axis of symmetry, see Fig.~\\ref{fig:axis_sym}. Note, that $t_0 = \\frac{3\\pi }{4}$ corresponds to the same line (axis of symmetry) but with the opposite directional vector.\n\\end{example}\n\n\nOf course, when an ellipse becomes a circle then any line through the center is its axis and we have infinitely many candidates for the vertex. Hence if the parameterization $\\f p(t)$ consists of some circles and at least one ellipse then it is most reasonable to ignore the circles, find the tentative axis and the parameter $t_0$ for this simplified parameterization and finally test whether the found symmetry is the symmetry of the original curve or not.\n\n\\medskip\nThe above discussed simplification does not help when all the $\\f p_k$'s are circles. In this case we must determine the parameter values $t_0$ such that all $\\f p_k(t_0)$ are linearly dependent. Motivated by the notion from astronomy, we will call this straight-line configuration of three or more points a {\\em syzygy}, see Fig.~\\ref{fig:syzygy}.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{overpic}[height=0.35\\textwidth]{cardioid_1.png}\n\\put(88,45){$\\f p(t_0^1)$}\n\\put(50,13){$\\f p_1(t)$}\n\\put(42,66){$\\f p_2(t)+\\f p_1(t_0^1)$}\n\\put(35,42){$\\f o$}\n\\put(72,10){$\\f p(t)$}\n\\put(90,70){\\fcolorbox{gray}{white}{\\includegraphics[width=0.1\\textwidth]{cardioid_2.png}}}\n\\end{overpic}\n\\hspace{15ex}\n\\begin{overpic}[height=0.35\\textwidth]{deltoid_1.png}\n\\put(90,50){$\\f p(t_0^1)$}\n\\put(58,62){$\\f p(t_0^2)$}\n\\put(15,92){$\\f p(t_0^3)$}\n\\put(60,13){$\\f p_1(t)$}\n\\put(75,25){$\\f p_2(t)+\\f p_1(t_0^1)$}\n\\put(50,92){$\\f p_2(t)+\\f p_1(t_0^2)$}\n\\put(-10,55){$\\f p_2(t)+\\f p_1(t_0^3)$}\n\\put(45,41){$\\f o$}\n\\put(50,25){$\\f p(t)$}\n\\put(90,70){\\fcolorbox{gray}{white}{\\includegraphics[width=0.1\\textwidth]{deltoid_2.png}}}\n\\end{overpic}\n\\caption{The syzygy of the points on the~circles $\\f p_k$ corresponds to the directions of the axes (framed) of symmetry. A generic configuration (non-syzygy) is also shown in light gray. Left: A cycloidal curve (cardioid) with one symmetry axis. Right: A cuspidal curve (deltoid) with three axes of symmetry.}\\label{fig:syzygy}\n\\end{center}\n\\end{figure}\n\n\nConsider the simplest example $\\f p(t) = \\lambda_{k_1}\\mathrm{e}^{\\mathrm{i}(\\sigma_{k_1}k_1t+\\psi_{k_1})}+\\lambda_{k_2}\\mathrm{e}^{\\mathrm{i}(\\sigma_{k_2}k_2t+\\psi_{k_2})}$, where $\\gcd(k_1,k_2)=1$. Then the condition that $\\f p_{k_1}(t)$ and $\\f p_{k_2}(t)$ are linearly dependent can be rewritten as\n\\begin{equation}\n \\sigma_{k_1}k_1t+\\psi_{k_1} = \\sigma_{k_2}k_2t+\\psi_{k_2}+ j\\pi\n\\end{equation}\nfor some $j\\in\\mathbb{Z}$, i.e., we arrive at\n\\begin{equation}\\label{eq:volba_tecek}\n t_0= \\frac{\\psi_{k_2}-\\psi_{k_1}}{\\sigma_{k_1}k_1-\\sigma_{k_2}k_2}+j\\frac{\\pi}{\\sigma_{k_1}k_1-\\sigma_{k_2}k_2}.\n\\end{equation}\nThere exist $\\sigma_{k_1}k_1-\\sigma_{k_2}k_2$ possibilities for choosing $t_0$ providing different axis, see Fig.~\\ref{fig:syzygy} , left. If $\\f p(t)$ contains more than two circles then the possible values of $t_0$ must be found as the common values arising from all the~pairs of the~circles $\\f p_k(t)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{example}\\rm\nConsider a deltoid (a hypocycloid known also as a tricuspoid curve) given by the trigonometric parameterization \n$$\n\\f p(t)=[2\\cos t + \\cos(2t),2\\sin t - \\sin(2t)],\\qquad \\text{or}\\qquad\n\\f p(t)=2\\mathrm{e}^{t\\mathrm{i}}+\\mathrm{e}^{-2t\\mathrm{i}}.\n$$\nIt follows from \\eqref{eq:volba_tecek} that we have three possible choices for $t_0$, in particular \n$$\nt_0=\\frac{\\pi}{3}j,\\quad j=0,1,2,\n$$\nand thus there exist three potential syzygy configurations, see Fig.~\\ref{fig:syzygy}, right. The cycloidal curve (cardioid) possessing one axis of symmetry is shown in see Fig.~\\ref{fig:syzygy}, left.\n\\end{example}\n\n\n\\begin{remark}\\rm\nWe can return to the situations where $\\f p_k$'s are ellipses and also in this case we will refer to the configuration guaranteeing the existence of the symmetry axis (given by a suitable value $t_0$, cf. Lemma~\\ref{lem:ellipse_syzygy}) as a syzygy. We recall that for ellipses in this configuration not only all $\\f p_k(t_0)$ are linearly dependent but in addition they are vertices for all $\\f p_k$'s which are ellipses.\n\\end{remark}\n\n\\medskip\nFinally, let us summarize the results of this section in a decision tree, see Diagram~\\ref{fig:symmetry_groups}.\n\n\n\n\\tikzstyle{intt}=[draw,text centered,minimum size=6em,text width=5.25cm]\n\\tikzstyle{intl}=[draw,text centered,text width=4cm,text height=0.34cm]\n\\tikzstyle{int}=[draw,minimum size=2.5em,text centered,text width=4cm]\n\\tikzstyle{intg}=[draw,shape=rectangle,minimum size=3em,text centered,text width=5cm]\n\\tikzstyle{intb}=[draw,shape=rectangle,rounded corners=1.5ex,minimum size=3em,text centered,text width=4cm]\n\\tikzstyle{sum}=[draw,shape=circle,text centered]\n\\tikzstyle{summ}=[drawshape=circle,inner sep=4pt,text centered,node distance=3.cm]\n\n{\\renewcommand\\figurename{Diagram}\n\\begin{figure}[!htb]\n\\centering\n\\begin{tikzpicture}[\n >=latex',\n auto\n ]\n \\node [intg] (kp) {$\\exists k$ such that $\\f p_k$ is an ellipse?};\n \\node [int] (ki1) [node distance=1.2cm and -0.2cm,below left=of kp,xshift=2mm] {Do all $\\f p_{2k}$'s vanish?};\n \\node [intb] (ki2) [node distance=1.2cm and -0.2cm,below right=of kp,xshift=-2mm] {Find the maximal $(m,d)$-sequence.};\n \\node [intl] (ki11)[node distance=3cm and 1cm,below left of=ki1,xshift=-2mm] {$\\exists$ a syzygy configuration?};\n \\node [intl] (ki12)[node distance=3cm and 1cm,below right of=ki1,xshift=2mm] {$\\exists$ a syzygy configuration?};\n \\node [intl] (ki21)[node distance=2.05cm and 0.6cm,below of=ki2] {$\\exists$ a syzygy configuration?};\n \\node [sum] (ki111)[node distance=2.8cm and 0cm,below left of=ki11,xshift=1cm] {$D_2$};\n \\node [sum] (ki112)[node distance=2.8cm and 0cm,below right of=ki11,xshift=-1cm] {$C_2$};\n \\node [sum] (ki121)[node distance=2.8cm and 0cm,below left of=ki12,xshift=1cm] {$D_1$};\n \\node [sum] (ki122)[node distance=2.8cm and 0cm,below right of=ki12,xshift=-1cm] {$C_1$};\n \\node [sum] (ki211)[node distance=2.8cm and 0cm,below left of=ki21,xshift=1cm] {$D_m$};\n \\node [sum] (ki212)[node distance=2.8cm and 0cm,below right of=ki21,xshift=-1cm] {$C_m$};\n\n \\draw[->] (kp) -- ($(kp.south)+(0,-0.75)$) -| (ki1) node[above,pos=0.25] {Yes} ;\n \\draw[->] (kp) -- ($(kp.south)+(0,-0.75)$) -| (ki2) node[above,pos=0.25] {No};\n \\draw[->] (ki1) -- ($(ki1.south)+(0,-0.75)$) -| (ki11) node[above,pos=0.25] {Yes} ;\n \\draw[->] (ki1) -- ($(ki1.south)+(0,-0.75)$) -| (ki12) node[above,pos=0.25] {No};\n \\draw[->] (ki2) -- ($(ki2.south)+(0,-0.75)$) -| (ki21) node[above,pos=0.25] {} ;\n \\draw[->] (ki11) -- ($(ki11.south)+(0,-0.75)$) -| (ki111) node[above,pos=0.25] {Yes} ;\n \\draw[->] (ki11) -- ($(ki11.south)+(0,-0.75)$) -| (ki112) node[above,pos=0.25] {No};\n \\draw[->] (ki12) -- ($(ki12.south)+(0,-0.75)$) -| (ki121) node[above,pos=0.25] {Yes} ;\n \\draw[->] (ki12) -- ($(ki12.south)+(0,-0.75)$) -| (ki122) node[above,pos=0.25] {No};\n \\draw[->] (ki21) -- ($(ki21.south)+(0,-0.75)$) -| (ki211) node[above,pos=0.25] {Yes} ;\n \\draw[->] (ki21) -- ($(ki21.south)+(0,-0.75)$) -| (ki212) node[above,pos=0.25] {No};\n\\end{tikzpicture}\n\\caption{Symmetry groups of curves with trigonometric parameterization \\eqref{eq:C_m}.}\n \\label{fig:symmetry_groups}\n\\end{figure}\n}\n\n\n\\section{Application to the symmetries of discrete curves}\\label{sec:sym_polyline}\n\nBy a~ \\emph{discrete curve} in $\\mathbb{R}^2$ we understand a closed polyline defined by the set of its vertices $\\f v_0,\\f v_1,\\ldots,\\f v_n=\\f v_0$. If $C$ is the discrete curve then its symmetry $\\phi\\in\\mathrm{Sym}\\,(C)$ is the isometry of $\\mathbb{R}^2$ such that $\\phi(C)=C$ and $\\phi(V)=V$, where $V$ is the ordered set of its vertices. Notice that if any three consecutive vertices are not collinear (which can always be ensured by removing the middle vertex) then the latter condition is fulfilled automatically.\n\n\nIn what follows we will replace the discrete curve by a suitable trigonometric curve such that the conditions of Lemma~\\ref{lem:komut} are satisfied in order to detect the symmetries of $C$. A natural candidate is the trigonometric curve interpolating the vertices $\\f v_i$. Let us fix the uniform distribution of parameter $t_j=\\frac{2\\pi}{n}j$, $j=0,\\dots,n-1$. Then for odd $n$ we obtain the unique interpolant $\\f p(t)$, see Section~\\ref{sec:trigcurves} and \\eqref{eq:interpol_coef_odd}. However when $n$ is even then there exists a one-parametric set of solutions, see again Section~\\ref{sec:trigcurves}. Because of the choice of the uniformly spaced $t_j$, we have $\\sin(Nt_j)=0$ and thus $b_N$, where $N=\\frac{n}{2}$, is the~free parameter. Following the convention from Section~\\ref{sec:trigcurves} we set $b_N=0$ and compute the coefficients of the interpolant $\\f p(t)$ via \\eqref{eq:interpol_coef_even}. In both cases let $T_C$ denote the curve parameterized by $\\f p(t)$. For later use it is essential to prove that the curve parameterized by $\\f p(t)$ does not depend on the choice of the first vertex being interpolated and neither on the orientation.\n\n\n\\begin{lemma}\\label{lem:TC}\n Let be given a trigonometric curve $C$, then $T_C$ is well defined.\n\\end{lemma}\n\n\\begin{proof}\nWe start with the odd case. Let $\\f p(t)$ be the interpolant corresponding to the ordered vertex set $\\f v_0,\\f v_1,\\ldots$ and let $\\f q(t)$ be another interpolant associated to the shifted vertex set $\\f v_i,\\f v_{i+1},\\ldots$. Then $\\f p(t+\\frac{2\\pi i}{n})$ interpolates the shifted vertex in the sense that $\\f p(t_j)=\\f v_{i+j}$. Because of the uniqueness of the interpolants we see that $\\f q(t)$ is a reparameterization of $\\f p(t)$ and thus they define the same curve $T_C$. The argument for the change of the orientation is the same. \n\n\nFor $n$ being even, we have to realize first that because of the uniform distribution of $t$ it holds\n$$\n\\cos\\left[N\\left(t+\\frac{2\\pi i}{n}\\right)\\right]=\\cos\\left[\\frac{n}{2}\\left(t+\\frac{2\\pi i}{n}\\right)\\right]=\n\\cos(Nt+\\pi i)=(-1)^{i-1}\\cos(Nt).\n$$\nHence, the type of the trigonometric parameterization (distinguished by the property $b_N=0$) is preserved by this reparameteriation. The next argumentation is analogous to the odd case, i.e., we again obtain the same parameterizations $\\f p\\left(t+\\frac{2\\pi i}{n}\\right)$ and $\\f q(t)$. \n\\end{proof}\n\n\n\\begin{corollary}\\label{cor:subset}\nLet $C$ be a discrete curve then $\\mathrm{Sym}\\,(C)\\subset\\mathrm{Sym}\\,(T_C)$.\n\\end{corollary}\n\n\\begin{proof}\n It follows from Lemma~\\ref{lem:TC} and Lemma~\\ref{lem:komut}.\n\\end{proof}\n\n\nIt can happen that $\\mathrm{Sym}\\,(T_C)$ is strictly bigger then $\\mathrm{Sym}\\,(C)$, see Fig.~\\ref{fig:narust_a_pokles_symetrie}, left. In general, both groups are finite, and thus comparable, unless the curve $T_C$ is a straight line or a circle. Since we assume in what follows that the vertices $\\f v_i$ are not collinear and the curve $T_C$ interpolates them, this can never be a straight line. \n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\hspace{-10ex}\n\\begin{overpic}[height=0.2\\textwidth]{C4.png}\n\\put(81,81){\\fcolorbox{gray}{white}{\\includegraphics[width=0.05\\textwidth]{sym_C4.png}}}\n\\end{overpic}\n\\hspace{5ex}\n\\begin{overpic}[height=0.2\\textwidth]{D4_2.png}\n\\end{overpic}\n\\hspace{3ex}\n\\vrule width1pt\n\\hspace{3ex}\n\\begin{overpic}[height=0.13\\textwidth]{bodiky.png}\n\\put(106,100){\\fcolorbox{gray}{white}{\\includegraphics[width=0.05\\textwidth]{bodiky_2.png}}}\n\\end{overpic}\n\\hspace{10ex}\n\\begin{overpic}[height=0.13\\textwidth]{domecek.png}\n\\put(106,100){\\fcolorbox{gray}{white}{\\includegraphics[width=0.05\\textwidth]{domecek_2.png}}}\n\\end{overpic}\n\\caption{Increasing and decreasing symmetries. Left: A discrete curve $C$ with $\\mathrm{Sym}\\,(C)\\cong C_4$ and the associated curve $T_C$ with the~bigger symmetry group $D_4$. Right: The given set of points with the symmetry group $D_4$ and the interpolating discrete curve with the symmetry group $D_1$.}\\label{fig:narust_a_pokles_symetrie}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\begin{lemma}\\label{lem:infinite sym}\n $T_C$ is a circle if and only if the discrete curve $C$ is a~regular $n$-gon.\n\\end{lemma}\n\n\\begin{proof}\n If $C$ is a~regular $n$-gon then the circle clearly interpolates the vertices. Because of the uniqueness of the interpolant the curve $T_C$ is forced to be a circle. Conversely if $T_C$ is a circle then the uniform distribution of the parameter implies that the points $\\f v_i$ form the consecutive vertices of a~regular $n$-gon. \n\\end{proof}\n\nFortunately if $T_C$ is not a circle or a straight line then $\\mathrm{Sym}\\,(T_C)$ is finite and thus $\\mathrm{Sym}\\,(C)$ can be identified with a subgroup of $\\mathrm{Iso}\\,({\\mathbb{R}^2})$. The whole method for determining symmetries of closed discrete curves is summarized in Algorithm~\\ref{algoritmus1}. \n\n\n\n\\begin{algorithm}[ht]\n\\caption{Symmetries of closed discrete curves.}\n\\label{algoritmus1}\n\\begin{algorithmic}[1]\n\\medskip\n\\Require\nA discrete curve $C$ given by the ordered set of its vertices $V=\\{\\f v_0,\\ldots,\\f v_{n-1}\\}$.\n\n\\smallskip\n\\State\nCompute the trigonometric interpolant $\\f p(t)$ of $C$ via \\eqref{eq:interpol_coef_odd} or \\eqref{eq:interpol_coef_even}, which parameterizes $T_C$. \n\n\\smallskip\n\\State\nDetermine $\\mathrm{Sym}\\,(T_C)$ via the decision tree presented in Diagram~\\ref{fig:symmetry_groups}.\n\n\\smallskip\n\\State\nInclude into the set $\\mathrm{Sym}\\,(C)$ only such $\\phi\\in\\mathrm{Sym}\\,(T_C)$ that satisfy $\\phi(C)=C$ (and $\\phi(V)=V$).\n\n\\medskip\n\\Ensure\n$\\mathrm{Sym}\\,(C)$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{remark}\\rm\\label{rem:alg_step3}\nIt remains to comment on how to decide which symmetries from $\\mathrm{Sym}\\,(T_C)$ to choose to $\\mathrm{Sym}\\,(C)$, i.e., how to perform {\\tt Step 3} of the algorithm. We will use the following simple test as the set of vertices $V$ is ordered. First we identify $\\f v_j$ such that $\\phi(\\f v_1)=\\f v_j$. If such $\\f v_j$ does not exist then $\\phi$ is not a symmetry of $C$. In the affirmative case we continue to compare the next vertices and their images, i.e., to include $\\phi$ into $\\mathrm{Sym}\\,(C)$ it has to hold either $\\phi(\\f v_2)=\\f v_{j+1},\\phi(\\f v_3)=\\f v_{j+2},\\phi(\\f v_4)=\\f v_{j+3},\\ldots $ for direct isometries, or $\\phi(\\f v_2)=\\f v_{j-1},\\phi(\\f v_3)=\\f v_{j-2},\\phi(\\f v_4)=\\f v_{j-3},\\ldots $ for indirect isometries (all indices are considered modulo $n$).\n\\end{remark}\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.22\\textwidth]{interpolant_1.png}\n\\hspace{2ex}\n\\includegraphics[width=0.23\\textwidth]{interpolant_3.png}\n\\hfill\n\\vrule width1pt\n\\hfill\n\\includegraphics[width=0.22\\textwidth]{interpolant_D2_1.png}\n\\hspace{2ex}\n\\includegraphics[width=0.23\\textwidth]{interpolant_D2_3.png}\n\\caption{Left: Discrete curve (gray) with the axial symmetry $D_1$ and trigonometric interpolant (blue) of their vertices (possessing the same symmetry). Right: Discrete curve (gray) with the symmetry group $D_2$ and the corresponding trigonometric interpolant (blue) with the same symmetry group. The first three ellipses $\\f p_1, \\f p_2, \\f p_3$\/ $\\f p_1, \\f p_3, \\f p_5$ (green) and the axis\/axes of symmetry (red) are also shown.}\\label{fig:interpolant_axis}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\hspace{-2ex}\n\\begin{overpic}[height=0.22\\textwidth]{interpolant_rot_1.png}\n\\end{overpic}\n\\hspace{0ex}\n\\begin{overpic}[height=0.22\\textwidth]{interpolant_rot_2.png}\n\\put(90,80){\\fcolorbox{gray}{white}{\\includegraphics[width=0.05\\textwidth]{interpolant_rot_3.png}}}\n\\end{overpic}\n\\hfill\n\\vrule width1pt\n\\hfill\n\\begin{overpic}[height=0.22\\textwidth]{interpolant_rotB_1.png}\n\\end{overpic}\n\\hspace{0ex}\n\\begin{overpic}[height=0.22\\textwidth]{interpolant_rotB_4.png}\n\\end{overpic}\n\\caption{Discrete curves (gray) with the symmetry groups $C_5$ and $D_3$ (red), and the trigonometric interpolants (blue) of their vertices with the same symmetry groups.}\\label{fig:interpolant_rot}\n\\end{center}\n\\end{figure}\n\n\n\\begin{example}\\rm\nConsider four discrete curves $K_1$, $K_2$, $K_3$ and $K_4$. Following the steps of Algorithm~\\ref{algoritmus1} we interpolate their vertices by the trigonometric curves $T_{K_1}$, $T_{K_2}$, $T_{K_3}$ and $T_{K_4}$ and then determine their symmetry groups. \n\nThe parameterizations of $T_{K_1}$ and $T_{K_2}$ contain $\\f p_k$'s which are ellipses and moreover among them there are terms of even degree only in the case of $T_{K_1}$. In addition, since in both cases there exist syzygy configurations we conclude that $T_{K_1}$ and $T_{K_2}$ possess the symmetry groups $D_1$ and $D_2$, respectively, see Fig.~\\ref{fig:interpolant_axis}. Finally, it is verified that $\\mathrm{Sym}\\,(K_1)=\\mathrm{Sym}\\,(T_{K_1})$ and $\\mathrm{Sym}\\,(K_2)=\\mathrm{Sym}\\,(T_{K_2})$.\n\n\nThe curves $T_{K_3}$ and $T_{K_4}$ are higher cycloids with the associated sequences\n$$\n\\begin{array}{rcl}\n\\sigma_{T_{K_3}} & = & \n\\left( \n\\{1\\}, \\emptyset, \\emptyset, \\{-1\\}, \\emptyset, \\{1\\}, \\emptyset, \n\\emptyset, \\{-1\\}, \\emptyset, \\{1\\}, \\emptyset, \\emptyset, \\{-1\\},\n\\emptyset, \\{1\\}, \\emptyset, \\emptyset, \\{-1\\}, \\emptyset, \\{1\\}, \n\\emptyset, \\emptyset, \\{-1\\}, \\emptyset, \\{1\\}, \\emptyset\n \\right)\\\\[0.7ex]\n& \\preccurlyeq & \\vartheta^{5,1}=\\left(\\{1\\}, \\emptyset, \\emptyset, \\{-1\\}, \\emptyset, \\ldots\n \\right),\n\\end{array} \n$$\nand\n$$\n\\begin{array}{rcl}\n\\sigma_{T_{K_4}} & = & \\left( \n\\{1\\}, \\{-1\\}, \\emptyset, \\{1\\}, \\{-1\\}, \\emptyset, \\{1\\}, \\{-1\\},\n\\emptyset, \\{1\\}, \\{-1\\}, \\emptyset, \\{1\\}, \\{-1\\}, \\emptyset, \\{1\\},\n\\{-1\\}, \\emptyset, \\{1\\}, \\{-1\\}, \\emptyset, \\right. \\\\ \n& & \\left. \\{1\\}, \\{-1\\}, \\emptyset, \\{1\\}, \\{-1\\}, \\emptyset,\n\\{1\\}, \\{-1\\}, \\emptyset, \\{1\\}\n \\right)\\\\[0.7ex]\n& \\preccurlyeq & \\vartheta^{3,1}=\\left(\\{1\\}, \\{-1\\}, \\emptyset, \\ldots\n \\right).\n\\end{array} \n$$\nHence we have arrived at rotational symmetries of a regular 5- and 3-gon, respectively. \n\n \nIn the first case there does not exist any syzygy configuration, whereas in the second one we find three such configurations. Hence we conclude that the symmetry groups are $C_5$ and $D_3$, respectively, see Fig. \\ref{fig:interpolant_rot}. Finally, it is confirmed that the discrete curves $K_3$ and $K_4$ have the same symmetry groups.\n\\end{example}\n\n\n\n\n\n\n\n\n\\begin{remark}\\rm\nThe curve $T_C$ might possess too high degree with regard to practical purposes and further computations. Then it is advisable to work with suitable alternative curves instead. These curves can be obtained via {\\em filtering} high harmonics of the original parameterization. Write $T^\\ell_C$ for the {\\em filtered curve} parameterized by $\\f a_0+\\sum_{k=1}^{N-\\ell}\\f p_k(t)$, i.e., the last $\\ell$ terms of $\\f p(t)$ are omitted. Analogously to Corollary~\\ref{cor:subset} it can be shown that it holds\n\\begin{equation}\n \\mathrm{Sym}\\,(T_C)\\subset\\mathrm{Sym}\\,(T^1_C)\\subset\\mathrm{Sym}\\,(T^2_C)\\subset \\cdots \\subset \\mathrm{Sym}\\,(T^{\\ell}_C).\n\\end{equation}\nThis is a way how to deal with curves of manageable degrees. Nevertheless one has to be aware of a possible growth of the symmetry group when the filtering process is applied, see Fig.~\\ref{fig:filtrace}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.22\\textwidth]{filter1.png}\n\\includegraphics[width=0.22\\textwidth]{filter2.png}\n\\includegraphics[width=0.22\\textwidth]{filter3.png}\n\\includegraphics[width=0.22\\textwidth]{filter4.png}\n\\caption{The sequence of filtered curves demonstrating the growth of symmetry. From left to right: Curve $T_C$ with maximal trigonometric degree 20 and $\\mathrm{Sym}\\,(T_C)\\cong D_3$, $\\mathrm{Sym}\\,(T_C^{10})\\cong D_3$, $\\mathrm{Sym}\\,(T_C^{14})\\cong D_9$, and $\\mathrm{Sym}\\,(T_C^{16})\\cong \\mathbf{O}(2)$.}\\label{fig:filtrace}\n\\end{center}\n\\end{figure}\n \n\\end{remark}\n\n\n\n\n\\section{Clouds of points}\\label{sec:convexhull}\n\nThe previous section was devoted to discrete curves determined by their vertices, i.e., by ordered sets of points.\nIn what follows we address the problem where input data in plane is unorganized. Since these data might not be suitable for interpolation by a curve, or the interpolation can lead to loosing symmetries, see Fig.~\\ref{fig:narust_a_pokles_symetrie} (right), we use an alternative approach based on the construction of the convex hull. By this we can efficiently solve the problem when symmetries of unorganized finite sets of points are to be find.\n\n\n\\begin{lemma}\\label{lem:CHsubset}\nConsider $\\mathcal{X}$ as the set of all finite subsets of $\\mathbb{R}^2$ and $\\Psi$ as the map assigning to each set $X$ its convex hull $\\mathrm{CH}(X)$. Then $\\mathrm{Sym}\\,(X)\\subset \\mathrm{Sym}\\,(\\mathrm{CH}(X))$.\n\\end{lemma}\n\n\\begin{proof}\nClearly, the convex hull is the affine, and thus also Euclidean invariant, so by Lemma~\\ref{lem:komut} the property holds.\n\\end{proof}\n\nThe boundary of the convex hull of a finite set is a discrete curve. In addition, it obviously holds\n\n\\begin{lemma}\\label{lem:polygonsubset}\nLet $P$ be a polygon in plane and $\\partial\\, P$ be its boundary. Then $\\mathrm{Sym}\\,(P)\\subset\\mathrm{Sym}\\,(\\partial\\, P)$.\n\\end{lemma}\n\n\nLet be given a finite subset $X$ of $\\mathbb{R}^2$. Then combination of Lemmas \\ref{lem:CHsubset}, \\ref{lem:polygonsubset} and Corollary \\ref{cor:subset} yields the following chain \n\\begin{equation}\n\\mathrm{Sym}\\,(X)\\subset \\mathrm{Sym}\\,(\\mathrm{CH}(X))\\subset \\mathrm{Sym}\\,(\\partial\\,\\mathrm{CH}(X))\\subset \\mathrm{Sym}\\,(T_{\\partial\\,\\mathrm{CH}(X)}).\n\\end{equation}\nHence, we assign to the unorganized finite set of points the boundary of its convex hull and then we determine all symmetries of this closed discrete curve by the previously presented methods.\n\n\\smallskip\nLet us emphasize again that the obtained symmetry group $\\mathrm{Sym}\\,(T_{\\partial\\,\\mathrm{CH}(X)})$ can be strictly larger then the sought group $\\mathrm{Sym}\\,(X)$. However, in this case we cannot apply the approach discussed in Remark \\ref{rem:alg_step3}, which is suitable for ordered point sets only. Instead, we determine the \\emph{Hausdorff distance} between two finite sets $X,Y$ given by the formula\n\\begin{equation}\n \\delta_H(X,Y)=\\max_{x\\in X}\\{ \\min_{y\\in Y}\\{\\|xy\\|\\}\\}.\n\\end{equation}\nThen $\\phi\\in\\mathrm{Sym}\\,(T_{\\partial\\,\\mathrm{CH}(X)})$ belongs to $\\mathrm{Sym}\\,(X)$ if and only if $\\delta_H(X,\\phi(X))=0$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.2\\textwidth]{cloud_1.png}\n\\hspace{2ex}\n\\includegraphics[width=0.2\\textwidth]{cloud_2.png}\n\\hspace{2ex}\n\\includegraphics[width=0.23\\textwidth]{cloud_3.png}\n\\hspace{2ex}\n\\includegraphics[width=0.23\\textwidth]{cloud_4.png}\n\\caption{Detection of symmetries of a point cloud. From left to right: The point cloud (with the symmetry group $C_4$), its convex hull and the trigonometric interpolant of the vertices of the convex hull having the symmetry group $D_4$.}\\label{fig:cloud}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.2\\textwidth]{cloudB_1.png}\n\\hspace{2ex}\n\\includegraphics[width=0.2\\textwidth]{cloudB_2.png}\n\\hspace{2ex}\n\\includegraphics[width=0.23\\textwidth]{cloudB_3.png}\n\\hspace{2ex}\n\\includegraphics[width=0.23\\textwidth]{cloudB_4.png}\n\\caption{Detection of symmetries of a point cloud (with the symmetry group $D_1$). From left to right: The point cloud, its convex hull, the trigonometric interpolant of the vertices of the convex hull having the symmetry group $D_2$.}\\label{fig:cloud2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\hspace{-5ex}\n\\includegraphics[width=0.21\\textwidth]{cloudE_1.png}\n\\hspace{2ex}\n\\includegraphics[width=0.21\\textwidth]{cloudE_2.png}\n\\hspace{2ex}\n\\includegraphics[width=0.21\\textwidth]{cloudE_3.png}\n\\hspace{2ex}\n\\begin{overpic}[height=0.21\\textwidth]{cloudE_4.png}\n\\put(95,75){\\fcolorbox{gray}{white}{\\includegraphics[width=0.05\\textwidth]{sym_C7.png}}}\n\\end{overpic}\n\\caption{Detection of symmetries of a point cloud (with the symmetry group $C_7$). From left to right: The point cloud, its convex hull, the trigonometric interpolant of the vertices of the convex hull also having the symmetry group $C_7$.}\\label{fig:cloud3}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\begin{example}\\rm\nConsider three point clouds $X_1, X_2$ and $X_3$. We determine their convex hulls $\\mathrm{CH}(X_i)$ with \nthe boundaries $\\partial\\,\\mathrm{CH}(X_i)$ being closed discrete curves and then interpolate their vertices by the trigonometric curves $T_{\\partial\\,\\mathrm{CH}(X_i)}$. The interpolant $T_{\\partial\\,\\mathrm{CH}(X_1)}$ possesses the symmetry group $D_4$, however the original point cloud $X_1$ has only $C_4$, see Fig.~\\ref{fig:cloud}. The trigonometric curve $T_{\\partial\\,\\mathrm{CH}(X_2)}$ has the symmetry group $D_2$, whereas the symmetry group of the point cloud $X_2$ is only $D_1$, see Fig.~\\ref{fig:cloud2}. Finally, both $T_{\\partial\\,\\mathrm{CH}(X_3)}$ and $X_3$ have the same symmetry group $C_7$, see Fig.~\\ref{fig:cloud3}.\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\hspace{-8ex}\n\\includegraphics[width=0.2\\textwidth]{rot_all_1.png}\n\\hspace{2ex}\n\\begin{overpic}[height=0.2\\textwidth]{rot_all_2.png}\n\\put(95,75){\\fcolorbox{gray}{white}{\\includegraphics[width=0.05\\textwidth]{sym_C5.png}}}\n\\end{overpic}\n\\hspace{7ex}\n\\vrule width1pt\n\\hspace{1ex}\n\\includegraphics[width=0.2\\textwidth]{rot_all_3.png}\n\\hspace{2ex}\n\\begin{overpic}[height=0.2\\textwidth]{rot_all_4.png}\n\\put(95,75){\\fcolorbox{gray}{white}{\\includegraphics[width=0.05\\textwidth]{sym_C5.png}}}\n\\end{overpic}\n\\caption{Detection of symmetries of a discrete curve using both presented method. \nLeft: We use the method for discrete curves. Right: The method for point clouds is employed. We emphasize that both method yields the symmetry group $C_5$.}\\label{fig:all}\n\\end{center}\n\\end{figure}\n\n\n\\begin{example}\\rm\nAs our final example we consider a discrete curve $C$. First, employing Algorithm~\\ref{algoritmus1} we conclude that $C$ has the symmetry group $C_5$, see Fig.~\\ref{fig:all} (left). Second, we treat $C$ via its vertices as a point cloud $X$, see Fig.~\\ref{fig:all} (right), and compute the symmetry group $\\mathrm{Sym}\\,(X)$. Since this symmetry group can be bigger, in general, it is always necessary to apply the confirmation step, see Remark~\\ref{rem:alg_step3}. Nonetheless, in this particular example the symmetry group of the point cloud is also $C_5$.\n\\end{example}\n\n\n\\section{Summary}\\label{sec:sum}\n\nIn this paper we have presented a novel, efficient method to compute the symmetries of planar trigonometric curves.\nThe axial and rotational symmetries were determined directly from their trigonometric parameterizations. The whole algorithm was summarized in a decision tree. Based on this a simple method for computing global exact symmetries of closed discrete curves in plane was formulated. Taking advantage of the fact that trigonometric interpolation of a given ordered set of point commutes with isometries, the studied problem dealing with discrete curves was transformed to determining symmetry groups of trigonometric curves. After a suitable modification we also applied the formulated approach on unorganized clouds of points. It is natural to wonder if the method can be suitably reformulated for 3D data. This is a question that we would like to investigate in the future. In addition we would like to extend the method for perturbed input data and approximate symmetries in our further research.\n\n\n\n\\bigskip\n\\section*{Acknowledgments}\n\nThe authors were supported by the grant 21-08009K of the Grant Agency of the Czech Republic.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Preliminaries}\n From the three-manifold theorist's point of view, hyperbolic and relatively hyperbolic groups are generalizations of Kleinian groups. Here we highlight some deep connections between the two theories. All groups are assumed to be finitely generated and all manifolds irreducible and orientable, unless otherwise specified.\n\n Trees, $\\mathbb{H}^2$, and $\\mathbb{H}^n$ are all examples of hyperbolic metric spaces. Similarly, free groups, the fundamental groups of closed hyperbolic surfaces and the fundamental groups of closed hyperbolic three-manifolds, are examples of hyperbolic groups. The fundamental groups of cusped hyperbolic 3-manifolds, such as hyperbolic knot groups, are not hyperbolic groups. They are however, relatively hyperbolic groups, as are all geometrically finite Kleinian groups. We recall the definition of a hyperbolic metric spaces and groups (the reader should verify the examples above).\n\n\\begin{defn}\n\n\n\nA \\textit{hyperbolic metric} space is a geodesic metric space such that geodesic triangles are slim. That is, there is a global constant $\\delta > 0$ such that for all geodesic triangles the third side is contained in the $\\delta$ neighborhood of the union of the other two. A group acts {\\it geometrically} on a proper metric space if the action is properly discontinuous, isometric and co-compact.\nA {\\it hyperbolic group} is a group which acts geometrically on {\\it some} proper hyperbolic metric space $X$.\n\\end{defn}\n\n A canonical example is a {\\it co-compact Kleinian group}, a group which acts geometrically on $\\mathbb{H}^3$. Also free groups, surface groups of genus $\\geq 2$, and in general convex co-compact Kleinian groups are hyperbolic groups.\n\nHyperbolic groups are often called {\\it Gromov} hyperbolic groups after \\cite{Gromov87}. We also direct the reader to the several excellent surveys: \\cite{ABC91}, \\cite{BowditchGGTnotes}, \\cite{CDP90}, among others.\n\n\\begin{notation} Throughout these notes we will discuss several boundaries for metric spaces, and use notation as follows.\n\\begin{itemize}\n\\item $\\partial X$: We will denote the Gromov boundary of a hyperbolic space by $\\partial X$. Similarly, we will denote the visual boundary of a $\\cat(0)$ space by $\\partial X$. We hope no confusion arises here. When a geodesic metric space is both $\\cat(0)$ and hyperbolic, the boundaries are homeomorphic. For definitions of these boundaries and this fact see \\cite{BH}.\n\n\\item $\\partial G$: We denote the Gromov boundary of a hyperbolic group by $\\partial G$, which is the boundary of any hyperbolic space on which $G$ acts geometrically. This is well-defined since any two spaces $X$ and $Y$ that $G$ acts upon geometrically are quasi-isometric, which implies that $\\partial X$ and $\\partial Y$ are homeomorphic. When the boundary of a $\\cat(0)$ group is well-defined, we will use the same notation $\\partial G$. While it is known that there are many examples of $\\cat(0)$ groups that do not have well-defined boundaries \\cite{Croke-Kleiner}, we will entirely restrict ourselves to $\\cat(0)$ groups with isolated flats (see Definition \\ref{IsolatedFlats}), which do have well-defined boundaries \\cite{HK05}.\n\n\\item $\\partial(G, \\mathcal{P})$: This will denote the Bowditch boundary of the relatively hyperbolic pair $(G, \\mathcal{P})$. For more elaboration, see Definition \\ref{RelHypBnd}.\n\\end{itemize}\n\\end{notation}\n\n\\noindent {\\bf Plan of paper:} In Section \\ref{day1} we discuss relatively hyperbolic groups and their boundaries, making relations with the hyperbolic and $\\cat(0)$ visual boundaries. We also give several examples.\nIn Section \\ref{day2} we discuss equivalent definitions of relative hyperbolicity, and some spaces that are useful. In section \\ref{day3} we discuss the connection with Kleinian groups, and also some algebraic information that can be gleaned from these boundaries.\n\n\\section{Relatively hyperbolic groups and their boundaries}\n\\label{day1}\n\n A {\\it geometrically finite Kleinian group} is a Kleinian group which acts geometrically finitely on the convex hull of its limit set. See Section \\ref{day3} for more detailed definitions. More generally, a {\\it relatively hyperbolic group pair $(G, \\mathcal{P})$} is a group pair that acts geometrically finitely on a proper hyperbolic metric space $X$.\n\n\n\n\n\nThere are many equivalent definitions of geometrically finite Kleinian groups, (see Bowditch \\cite{BowGF}). Similarly, there are many equivalent definitions of relatively hyperbolic groups. We will take as our definition \\cite[Def 2]{Bowrelhyp}. Other, equivalent definitions are discussed in Section \\ref{day2}.\n\nFirst, we define conical limit points and bounded parabolic points.\n\n\\begin{defn}\n\t\tLet $X$ be a proper, hyperbolic geodesic metric space where $G$ acts on $X$ properly discontinuously by isometries.\n\n\t\\begin{itemize}\n\n\n\t\t\\item Conical limit point: A point $m\\in \\partial X$ is a \\textit{conical limit point} if there is a geodesic $\\gamma\\to m$, a point $x\\in X$, and a sequence of elements $\\{g_n\\} \\subset G$ such that $g_nx \\to m$ and $d(g_nx,\\gamma) < r$ for some $r>0$. See Figure \\ref{fig:conical}.\n\n\t\t\\item Parabolic: $P\\leq G$ is a \\textit{parabolic subgroup} if it is infinite, contains no loxodromic elements, and fixes a point $x_P\\in \\partial X$. In this case, $x_P$ is called a \\textit{parabolic point}.\n\n\t\t\\item Bounded parabolic: A parabolic point $x_P\\in \\partial X$ is \\textit{bounded} if $\\partial X \\setminus \\{x_P\\} \/ P$ is compact.\n\t\\end{itemize}\n\\end{defn}\n\nNote that if an infinite subgroup fixes more than one point it must contain a loxodromic element, so a parabolic subgroup must fix exactly 1 point.\n\n\\begin{defn}[\\cite{Bowrelhyp}] \\label{def:original rel hyp} A group pair is a group $G$ and a family $\\P$ of infinite subgroups consisting of finitely many conjugacy classes. The pair $(G,\\P)$ is \\textit{relatively hyperbolic} if $G$ acts on $X$ properly discontinuously and by isometries, where $X$ is a proper hyperbolic geodesic metric space such that:\n\n\\begin{enumerate}\n\t\\item each point of $\\partial X$ is either a conical limit point or a bounded parabolic point.\n\t\\item $\\P$ is exactly the collection of maximal parabolic subgroups.\n\\end{enumerate}\n\nIn the case that we have a properly discontinuous action by isometries and these two conditions are satisfied, we say $(G, \\mathcal{P})$ acts \\textit{geometrically finitely} on $X$.\n\nThe elements of $\\P$ are called \\textit{peripheral} subgroups.\n\\end{defn}\n\n\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}\n\n\n \\draw[thick,red] (-2.5,2) -- (0,0);\n \\draw[thick,red] (2.5,2) -- (0,0);\n \\draw[green] (0,3) -- (0,0);\n \\draw[thick] (-3,0) -- (3,0);\n\n\n \\node[above] at (0,3) {\\scriptsize$\\gamma$};\n \\node[below] at (0,0) {\\scriptsize$m_c$};\n\n \\node[above] at (1,2) {\\scriptsize$x$};\n \\draw[fill] (1,2) circle [radius=0.025];\n\n \\node[above] at (.3,1.1) {\\scriptsize$~~~~g_1(x)$};\n \\draw[fill] (.3,1.1) circle [radius=0.025];\n\n \\node[above] at (.5,.7) {\\scriptsize$g_2(x)$};\n \\draw[fill] (.5,.7) circle [radius=0.025];\n\n \\node[above] at (-.4,.5) {\\scriptsize $g_3(x)$};\n \\draw[fill] (-.3,.5) circle [radius=0.025];\n\n \\draw[fill] (.2,.3) circle [radius=0.025];\n\n\n\n \\draw[fill] (0,0) circle [radius=0.025];\n\n \\end{tikzpicture}\n \\caption{A conical limit point}\n \\label{fig:conical}\n\\end{figure}\n\n\n\\begin{defn} \\label{RelHypBnd} The relatively hyperbolic boundary $\\partial(G, \\mathcal{P})$ (alternatively $\\partial_B(G, \\mathcal{P})$) of the group pair $(G, \\P)$ is the boundary of any hyperbolic metric space that $(G, \\mathcal{P})$ acts on geometrically finitely. This is also called the Bowditch boundary.\n\\end{defn}\n\nTwo such spaces are not necessarily equivariantly quasi-isometric \\cite{Burnsrigid}, as in the hyperbolic case. However, the relatively hyperbolic boundary is still well-defined up to homeomorphism, \\cite[Section 9]{Bowrelhyp}. In the case of a hyperbolic knot complement such as the figure-eight knot complement, the Bowditch boundary of the group pair is $S^2$, the boundary of $\\mathbb{H}^3$. It is an open question to understand relatively hyperbolic group pairs with Bowditch boundary $S^2$ or even a subset of $S^2$ (those with planar boundary). Variants of this question have been explored in \\cite{GMS, TW, HW1,MartinSkora89}. See Question \\ref{question: vgfk} and Conjecture \\ref{conjecture: vgfk} for further discussion.\n\n\n\nThere are lots of natural relatively hyperbolic groups. One good source of examples is the class of hyperbolic groups. Given an almost malnormal collection of quasiconvex subgroups (which might be the empty set), one obtains a relatively hyperbolic group pair. Let $G$ be a hyperbolic group and $\\P$ a collection of quasiconvex subgroups. We say the collection is \\textit{almost malnormal} if for every $P,P'\\in \\P$ and $g\\in G$, whenever\n\n $$|P' \\cap gPg^{-1}| = \\infty$$\n then $P=P'$ and $g\\in P$.\n\n\n\n\\begin{thm}[{\\cite[Theorem 7.11]{Bowrelhyp}}]\nLet $G$ be a non-elementary hyperbolic group and $\\P$ an almost malnormal collection of quasiconvex subgroups of $G$. Then $(G,\\P)$ is relatively hyperbolic.\n\\end{thm}\n\n\n\n\n\n\n\n\n\\begin{example}\\label{example:ap} To illustrate how the Bowditch boundary can change dramatically when the collection of peripheral groups changes, even amongst Kleinian groups, we describe three examples where the group is $F_2$. Each relatively hyperbolic pair $(G, \\mathcal{P})$ can be realized as a Kleinian group, where the collection $\\mathcal{P}$ is parabolic, but the peripheral subgroups are different, which changes the relatively hyperbolic boundary. Each has peripheral groups consisting of all the conjugates of some subset of the elements corresponding to the curves $a$, $b$ and $c$ on the one-holed torus:\n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=.2\\linewidth]{punctured-torus.png}\n \\caption{A torus with a boundary component}\n \\label{fig:sub1}\n\\end{figure}\n\n\\begin{enumerate}\n\t\\item $G=F_2 = \\langle a, b \\rangle$, $\\P= \\varnothing$, $X=$ convex hull of the limit set.\n\n\t$\\partial(F_2, \\varnothing) = \\mathcal{C} = $ Cantor set. This is the same as the Gromov boundary, since the set of peripheral subgroups is empty. Here $a, b$ are realized as isometries of $\\mathbb{H}^2$ that map the bottom black curve to the top and the left to the right in the central octagon respectively. The group acts geometrically on the convex hull of the limit set (region enclosed by the red axes of conjugates of $\\langle[a, b]\\rangle$), as shown in Figure \\ref{fig:cantor}.\n\n\t\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=.4\\linewidth]{cantor.png}\n\t\\caption{a (relatively) hyperbolic action of $(F_2, \\P)$ on $\\mathbb{H}^2$ \\\\ with quotient a torus with a boundary component}\n\t\\label{fig:cantor}\n\t\\end{figure}\n\n\t\\item $G=F_2=\\langle a,b\\rangle$, $\\P$ is the collection of conjugates of the subgroup $\\langle [a,b]\\rangle$, $X=\\H^2$.\n\n\t$\\partial (F_2,\\langle [a,b] \\rangle) = S^1$. This can be realized by putting a finite-area hyperbolic structure on the cusped torus. With such a representation, $F_2$ is a finite co-volume subgroup of $\\operatorname{Isom}(\\mathbb{H}^2)$ and the limit set and the Bowditch boundary are both $S^1$.\n\n\n\t\\begin{figure}[H]\n\t\\centering\n\t\\label{fig:ptorusgp}\n\t\\includegraphics[width=.4\\linewidth]{torus_with_cusp.png}\n\t\\caption{a (relatively) hyperbolic action of $(F_2, \\P)$ on $\\mathbb{H}^2$ \\\\ with quotient a cusped torus}\\end{figure}\n\n\t\\item $G=F_2$, $\\P$ is the collection of all conjugates of the subgroups $ \\{ \\langle a \\rangle, \\langle b \\rangle, \\langle [a,b] \\rangle \\}$.\n\n\n\t$\\partial (G,\\P) = $ Apollonian gasket, see \\cite{HPWKleinainBoundary}. Here $X$ is the convex hull of the Apollonian gasket in $\\mathbb{H}^3$ and $G$ acts as a geometrically finite Kleinian group on $X$.\n\n\n\t\t\t\\begin{figure}[h!]\n\t \\centering\n\t\\includegraphics[width=.8\\linewidth]{ap-gasket.png}\n\t \\caption{The Apollonian gasket}\n\t \\label{fig:gasket}\n\t\\end{figure}\n\n\n\\end{enumerate}\n\\end{example}\n\n\n\n\n\n\n\n\nAnother source of examples of relatively hyperbolic groups are certain $\\cat(0)$ groups. In particular, $\\cat(0)$ groups with isolated flats admit a relatively hyperbolic group structure.\n\n\\begin{defn}\nA \\textit{flat} is an isometric embedding of $\\E^n$ for $n\\geq 2$.\n\\end{defn}\n\n\\begin{defn}[Isolated Flats] \\label{IsolatedFlats}\n Let $X$ be a $\\cat(0)$ space admitting a geometric action by $G$. The space $X$ has \\textit{isolated flats} if there is a $G$-invariant collection $\\mathbb{F}$ of flats in $X$ such that:\n\n \\begin{enumerate}\n \\item There is a constant $D<\\infty$ such that each flat in $X$ lies in a $D$-tubular neighborhood of some flat $F\\in \\mathbb{F}$.\n \\item For each positive $\\rho<\\infty$, there is a constant $\\kappa=\\kappa(\\rho)$ such that for any two distinct flats $F,F'\\in \\mathbb{F}$, we have\n\n $$diam(\\mathcal{N}_\\rho(F) \\cap \\mathcal{N}_\\rho(F') ) < \\kappa$$\n \\end{enumerate}\n\\end{defn}\nThe first condition says that all flats of $X$ are close to a flat of $\\mathbb{F}$ and therefore we think of $\\mathbb{F}$ has the collection of maximal dimensional flats. The second condition says that the flats in $\\mathbb{F}$ are far apart, hence isolated.\n\n\n\\begin{thm}[\\cite{HK05}]\nIf $G$ is a $\\cat(0)$ group with isolated flats, then $(G,\\P)$ is relatively hyperbolic, where $\\P$ is the collection of flat stabilizers.\n\nWhen $G$ is $\\cat(0)$ with isolated flats, the visual boundary $\\partial G$ is well-defined.\n\\end{thm}\n\n\n\nA theorem of Tran relates the two boundaries of a $\\cat(0)$ group with isolated flats.\n\n\\begin{thm}[\\cite{Tran13}] \\label{thm:TranCAT}\nLet $G$ be a $\\cat(0)$ group acting geometrically on $X$ with isolated flats and let $\\P$ be the collection of flat stabilizers. There is a surjective map\n\n$$\\pi \\colon \\partial X \\to \\partial (G,\\P)$$\nwhich is defined by collapsing the boundary of each maximal dimensional flat to a single point.\n\\end{thm}\n\n\\begin{example} The fundamental group of a hyperbolic knot complement. Let $G=\\pi_1(S^3\\setminus \\text{figure 8-knot})$. This is not a hyperbolic group since it contains a $\\mathbb{Z} \\oplus \\mathbb{Z}$. However, $G$ acts geometrically on truncated $\\H^3$, since the peripheral subgroups in the Kleinian structure preserve a collection of horoballs. By a result of Ruane \\cite{RuaneTruncHyp} the $\\cat(0)$ visual boundary $\\partial G$ is $\\mathcal{S}$, the Sierpinski carpet. The group $G$ is $\\cat(0)$ with isolated flats. We can apply Tran's theorem above to see that $(G, \\mathcal{P})$ where $\\mathcal{P}$ is the collection of peripheral $\\mathbb{Z} \\oplus \\mathbb{Z}$ has Bowditch boundary homeomorphic to $S^2$. This follows from the fact that a decomposition which is a null-sequence is an upper semicontinuous decomposition, \\cite[page 14]{Daverman} and a theorem of Moore \\cite{Mooreusc} that the quotient of an upper semicontinuous decomposition into non-separating continua of $S^2$ is again $S^2$.\n\\end{example}\n\n\n\nTran's theorem works when $G$ is hyperbolic (see also \\cite{Jasonrelhyp} in these notes):\n\n\\begin{thm}[\\cite{Tran13}] \\label{thm:TranHyp}\nLet $(G,\\P)$ be a relatively hyperbolic group and let $G$ be hyperbolic. Then there is a surjective map\n\n$$\\pi:\\partial G \\to \\partial (G,\\P)$$\ndefined by collapsing all the boundaries of the $P\\in\\P$.\n\\end{thm}\n\n\\begin{example}\nLet $M^3$ be a hyperbolic manifold with totally geodesic boundary. Let $G = \\pi_1(M^3)$. Then the Gromov boundary is $\\partial G\\cong \\mathcal{S}$, the Sierpinski carpet, and $\\partial (G,\\P) \\cong S^2$. This can be seen by collapsing the boundaries of the circles removed from the Sierpinski carpet.\n\\end{example}\n\n\nThe next examples illustrate the use of Tran's theorem to understand the relatively hyperbolic boundaries of relatively hyperbolic group pairs.\n\n\n\\begin{example} \\label{example:TreeOfCircles} Let $G$ be the fundamental group of a genus 2 surface, realized as a Fuchsian group. We can denote by $c$ the element of $G$ corresponding to a separating curve which bounds a commutator on both sides. Since the subgroup $\\langle c \\rangle$ is quasi-convex and its conjugates (corresponding to the red separating curves in Figure \\ref{fig:bass-serre}) form a malnormal collection, $(G, \\mathcal{P})$ is a relatively hyperbolic structure on $G$, where $\\mathcal{P}$ consist of $\\langle c \\rangle$ and its conjugates. With this relatively hyperbolic structure, the Bowditch boundary of $(G,\\P)$ is a tree of circles. This boundary $\\partial(G, \\mathcal{P})$ can be realized by looking at the Bass-Serre tree for the splitting of $G$ over $\\langle c\\rangle$, where each vertex in the Bass-Serre tree corresponds to a circle in the Bowditch boundary. Two circles meet at a point exactly when there is an edge in the Bass-Serre tree between the vertices. See Figure \\ref{fig:tree-of-circles}.\n\\end{example}\n\\begin{figure}[H]\n \\begin{subfigure}{.5\\textwidth}\n \\centering\n\t \\includegraphics[scale=.2]{bass-serre-surface.png}\n\t \\caption{Bass-Serre tree in $\\mathbb{H}^2$}\n\t \\label{fig:bass-serre}\n \\end{subfigure}\n \\hspace*{-5pt}\n \\begin{subfigure}{.5\\textwidth}\n\t \\centering\n\t \\includegraphics[scale=.042]{tree_of_circles.png}\n\t \\caption{The Bowditch boundary of $(G, \\P)$}\n\t \\label{fig:tree-of-circles}\n \\end{subfigure}\n\\caption{}\n\n\\end{figure}\n\n\\begin{example}\n\nHere is an example which is not a 3-manifold group. However, the Bowditch boundary will be planar and we can understand this using Tran's theorem. Consider three surfaces (with genus at least 1) each with a boundary component. Attach the three boundary curves to the curves of the $T^2$ pictured in Figure \\ref{fig:torus 3 }. Let $G$ be the fundamental group of this 2-complex and $\\P$ the collection of abelian subgroups of rank 2. The resulting Bowditch boundary is a tree of circles and is planar. Work of Hruska-Walsh shows that the $\\cat(0)$ boundary contains a $K_{3,3}$ graph. Such a graph is an obstruction to the group acting properly discontinuously on a contractible 3-manifold by work of Bestvina-Kapovich-Kleiner \\cite{BestvinaKapovichKleiner02}.\n\nUsing Tran's Theorem \\ref{thm:TranCAT}, $\\partial (G,\\P)$ is obtained by collapsing the circles in the boundary coming from the $\\mathbb{Z} \\oplus \\mathbb{Z}$ subgroups. This boundary is planar, but has cut points. See \\cite{HW1} for details.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[scale=.3]{torus-3-curves.png}\n \\caption{3 curves on $T^2$}\n \\label{fig:torus 3 }\n\\end{figure}\n\\end{example}\n\n\\section{More definitions of relatively hyperbolic groups}\\label{day2}\n\nIn this section we introduce several more definitions of a relatively hyperbolic group pair. By work of Dahmani, Hruska, and Groves--Manning, all these are equivalent (and are equivalent to our first definition in Section \\ref{day1}) \\cite{DahmaniRelHypEquiv, HruskaRelHypEquiv, GMCusp}. This gives us multiple ways of identifying and studying relatively hyperbolic groups. Furthermore, two of the definitions are constructive in the sense that algebraic information about the group can be used to build an appropriate hyperbolic space.\n\nAs noted before, all groups we consider are finitely generated throughout this section. Many of these definitions can be adapted to non-finitely generated groups, but we do not do so here. See \\cite{HruskaRelHypEquiv} for some of these definitions. We recall the definition of a Cayley graph.\n\n\\begin{defn}\n For a group $G$ with a generating set $\\mathcal{A}$, the \\textit{Cayley Graph}, denoted $C(G,\\mathcal{A})$ is a graph with\n \\begin{enumerate}\n \\item a vertex for each $g\\in G$ and\n \\item an edge labelled by $a\\in A$ joining the vertices $g$ and $ga$.\n \\end{enumerate}\n\\end{defn}\n\nThe group $G$ acts on its Cayley graph on the left as seen in Figure \\ref{fig:cayey-action}\n\n\\vskip .2in\n\\begin{figure}[H]\n \\centering\n\\begin{tikzpicture}\n\\draw (0,0) --(1,1);\n\\node [left] at (.6,.6) {$a$};\n\\node [left] at (0,0) {$g$};\n\\node [right] at (1,1) {$ga$};\n\\draw[fill] (1,1) circle [radius=0.025];\n\\draw[fill] (0,0) circle [radius=0.025];\n\\draw (4,0) -- (5,1);\n\\node [left] at (4,0) {$bg$};\n\\node [right] at (5,1) {$bga$};\n\\draw[fill] (5,1) circle [radius=0.025];\n\\draw[fill] (4,0) circle [radius=0.025];\n\\node [left] at (4.6,.6) {$a$};\n\\draw [|->] (2,.5) -- (3,.5);\n\\node [above] at (2.5,.5) {$b$};\n\n\n\\end{tikzpicture}\n \\caption{The action of $G$ on its Cayley graph}\n \\label{fig:cayey-action}\n\\end{figure}\n\n\n\nFor hyperbolic groups, the Cayley graph, endowed with a metric where each edge has length 1, captures the hyperbolic geometry. For a relatively hyperbolic group, however, the Cayley graph does a poor job of capturing desired geometric properties of the group. Farb introduced the notion of the coned-off Cayley graph as a way of capturing these properties using a graph akin to the Cayley graph.\n\n\\begin{defn}[Coned-off Cayley Graph, \\cite{FarbCoset}]\n Let $(G,\\P)$ be a relatively hyperbolic pair and $\\mathcal{A}$ a finite, symmetric generating set for $G$. The \\textit{coned-off Cayley graph}, denoted $\\mathcal{C}(G,\\P,\\mathcal{A})$ is the Cayley graph $C(G,\\mathcal{A})$ with some additions. For each coset $gP$, where $P\\in \\P$, add a vertex $v_{gP}$. Then for each $h\\in gP$, add an edge of length $\\frac{1}{2}$ from $h$ to $v_{gP}$. The resulting graph is $\\mathcal{C}(G,\\P,\\mathcal{A})$. If $\\gamma$ is a path in $C(G,\\mathcal{A})$, then let $\\hat \\gamma$ be the path in $\\mathcal{C}(G,\\P,\\mathcal{A})$ where we replace each maximal subpath in a coset of a peripheral subgroup with two edges of length 1\/2, meeting $v_{gP}$.\n\\end{defn}\n\nIn an ideal world, we would say when this resulting graph is hyperbolic, then the group is relatively hyperbolic. Unfortunately, this would be too broad of a definition, as it would allow $\\Z^2$ to be relatively hyperbolic, as shown in the next example.\n\n\\begin{example}\\label{example:Z rel hyp}\nConsider $G=\\langle a,b| [a,b]\\rangle = \\Z\\oplus \\Z$ and let $\\P$ consist of $\\langle a \\rangle$ and its conjugates (which is just $\\langle a \\rangle$). Then the coned off Cayley graph $\\mathcal{C}(G,\\P,\\mathcal{A})$ is hyperbolic (as it is quasi-isometric to a line) where $\\mathcal{A} = \\{ a, b\\}$.\n\\end{example}\n\nIn light of this example, the following definitions are required to achieve the desired definition.\n\n\\begin{defn}[Without backtracking]\n A path in $\\mathcal{C}(G,\\P,\\mathcal{A})$ is \\textit{without backtracking} if once the path hits $v_{gP}$, it never returns. If $\\gamma$ is a path in $C(G, \\mathcal{A})$, we say it is without backtracking if $\\hat\\gamma$ is without backtracking. A path in $\\gamma$ in $C(G, \\mathcal{A})$ \\textit{penetrates} the coset $gH$ if $\\hat\\gamma$ passes through $v_{gH}$.\n\\end{defn}\n\n\\begin{defn}[Bounded coset penetration]\n Let $C(G,\\P,\\mathcal{A})$ be the coned-off Cayley graph for $(G,\\mathcal{A})$. This has \\textit{bounded coset penetration} if for each $\\lambda\\geq 1$, there is a constant $a(\\lambda)>0$ such that if $\\gamma,\\gamma'$ are two $(\\lambda,0)$-quasigeodesics in $C(G,\\mathcal{A})$ without backtracking, with the same initial vertex, and with endpoints that are no more than $1$ apart, then the following two conditions hold:\n\n \\begin{enumerate}\n \\item if $\\gamma$ penetrates $gP$ and $\\gamma'$ does not, then the entering and exiting vertices (endpoints of the subpath in $gP$) of $\\gamma$ are at most $a(\\lambda)$ from each other in $C(G,\\mathcal{A})$.\n \\item if $\\gamma$ and $\\gamma'$ both penetrate $gP$, then, in $C(G,\\mathcal{A})$, the two entering vertices of each are $a(\\lambda)$-close and the two exiting vertices are $a(\\lambda)$-close.\n \\end{enumerate}\n\\end{defn}\nExample \\ref{example:Z rel hyp} does not satisfy the bounded coset penetration, which is exactly what we wanted. Therefore, the following is one of our definitions of relatively hyperbolic:\n\n\\begin{defn}[Relatively hyperbolic]\\label{def: coned-off}\n A group pair $(G,\\P)$ is relatively hyperbolic if the coned-off Cayley graph $\\mathcal{C}(G,\\P,\\mathcal{A})$ is $\\delta$-hyperbolic and satisfies the bounded coset penetration property \\cite{FarbCoset}.\n\\end{defn}\n\nThe equivalence of Definition \\ref{def: coned-off} with Definition \\ref{def:original rel hyp} was proved by Hruska \\cite{HruskaRelHypEquiv}.\n\nSomething to note: the hyperbolic space $\\mathcal{C}(G,\\P,\\mathcal{A})$ is \\textit{not} proper because the vertices $v_{gP}$ have infinite valence. We can still define the boundary of $\\mathcal{C}$ as before, but it is not compact because $\\mathcal{C}$ is not proper. This boundary is missing the parabolic points. There is, however, a relationship between the Bowditch boundary and the boundary of the coned-off Cayley graph, see \\cite[Theorem 9.1]{Bowrelhyp} for the correct topology:\n\n$$\\partial (G,\\P) = \\partial (\\mathcal{C}(G,\\P,\\mathcal{A})) \\bigcup_{g\\in G} \\{v_{gP}\\}$$\n\n\n\\begin{example}\nLet $G=\\pi_1(S^2\\setminus K)$ be a hyperbolic knot complement. Then $\\partial (G, \\mathcal{P}) = S^2$ because $G$ is a geometrically finite Kleinian group with finite co-volume. The set of parabolic fixed points is dense in $S^2$. So we can understand the boundary of the coned-off Cayley graph $ \\partial (\\mathcal{C}(G,\\P\\mathcal{A})$ as $S^2$ with a countable dense collection of points removed. These are the parabolic fixed points.\n\\end{example}\n\n\\begin{comment}\nThe astute note taker related this to another lecture series included in this volume *******\n\\begin{thm}\\todo{cite tran}\nIf each $P\\in \\P$ have no Morse geodesics, then, as a set,\n\n$$\\text{Morse Boundary} \\hookrightarrow \\partial (\\mathcal{C}(G,\\P,\\mathcal{A}))$$\n\n\\end{thm}\n\\end{comment}\n\nThe next definition of a relatively hyperbolic pair also uses the Cayley graph to construct an appropriate hyperbolic space. But first we introduce combinatorial horoballs:\n\n\\begin{defn}[Combinatorial Horoballs, \\cite{GMCusp}]\nLet $\\Gamma$ be a graph with all edges length 1. Construct a new graph $\\mathcal{H}(\\Gamma)$ with vertex set\n$$V(\\mathcal{H}(\\Gamma)) = V(\\Gamma) \\times \\Z_{\\geq 0}.$$\nThere are two types of edges in $\\mathcal{H}(\\Gamma)$: For all $v\\in V(\\Gamma)$, there is an edge between $(v,k)$ and $(v,k+1)$. For each $k$, there is an edge between $(v,k)$ and $(w,k)$ if, in $\\Gamma$, $d(v,w)\\leq 2^k$. The first type of edges we call vertical and the second type are horizontal.\n\n\\end{defn}\n\\begin{figure}[H]\n \\centering\n\\begin{tikzpicture}\n\n \\node [below] at (0,0) {$(v,k)$};\n \\node [above] at (0,1) {$(v,k+1)$};\n \\draw (0,0) -- (0,1);\n \\draw[fill] (0,0) circle [radius=0.04];\n \\draw[fill] (0,1) circle [radius=0.04];\n\n \\node[left] at (2,.5) {$(v,k)$};\n \\node[right] at (3,.5) {$(w,k)$};\n \\draw (2,.5) -- (3,.5);\n \\draw[fill] (2,.5) circle [radius=0.04];\n \\draw[fill] (3,.5) circle [radius=0.04];\n\n\n \\end{tikzpicture}\n \\caption{Vertical and horizontal edges in the graph $\\mathcal{H}(\\Gamma)$}\n \\label{fig:horoballs}\n\\end{figure}\n\nThe motivation for this definition comes from the example of $F_2$ acting on $\\H^2$ with quotient a cusped torus. In this action, there is a collection of invariant horoballs centered at the parabolic points on the boundary. The combinatorial horoball definition and the original Gromov definition, see \\cite{Gromov87, Szcz} model this behavior.\n\nGiven a group pair $(G, \\P)$, we can construct a graph using the Cayley graph and combinatorial horoballs. If the resulting space is hyperbolic, then the group pair $(G,\\P)$ is relatively hyperbolic, and the combinatorial horoballs mimic the behavior seen above in $F_2$ acting on $\\H^2$.\n\nNote, in the definition below, $\\mathbb{P}$ is a \\textbf{finite} collection of parabolic subgroups, which differs from the definition of $(G,\\P)$ from Section \\ref{day1}. To go from $\\mathbb{P}$ to $\\P$, take the union of all conjugacy classes of $P\\in \\mathbb{P}$. To go the other way, pick a representative of each conjugacy class in $\\P$.\n\n\\begin{defn}[Cusped Cayley graph, \\cite{GMCusp}]\\label{def:cusped-off}\nLet $G$ be a group and $\\mathbb{P}$ a finite collection of subgroups. Let $\\mathcal{A}$ be a generating set for $G$ which contains a generating set for each $P\\in \\mathbb{P}$. Construct the Cayley Graph $C(G,\\mathcal{A})$ and, for each coset $gP$ of some $P\\in \\mathbb{P}$, attach a copy of $\\mathcal{H}(gP)$ to $C(G,\\mathcal{A})$. Here the $0$-level of $\\mathcal{H}(gP)$ is identified with $C(gP)$. We denote this space $X(G,\\mathbb{P})$ and call it the \\textit{cusped Cayley graph}.\n\\end{defn}\n\nNote that this construction requires a generating set for \\textbf{both} $G$ and each $P\\in \\mathbb{P}$, so each parabolic subgroup must be finitely generated. However, if $(G, \\P)$ is relatively hyperbolic and $G$ is finitely presented, then each $P\\in \\mathbb{P}$ is finitely presented as well \\cite{DahmaniPresenting} .\n\n\\begin{thm}[Groves-Manning, \\cite{GMCusp}]\nThe pair $(G,\\P)$ is relatively hyperbolic (in the sense of Definition \\ref{def:original rel hyp}) when the cusped Cayley graph $X(G,\\P)$ is hyperbolic. Furthermore, $\\partial (X(G,\\P))$ is the Bowditch boundary.\n\\end{thm}\n\nBoth Definitions \\ref{def: coned-off} and \\ref{def:cusped-off} give constructions for creating a hyperbolic space admitting an action by the relatively hyperbolic group pair $(G,\\P)$. By taking algebraic information (a group, a collection of subgroups, and a generating set), we can build a geometric model for $(G,\\P)$, which is not the case for Definition \\ref{def:original rel hyp}. Furthermore, the boundaries of the resulting spaces are either very close to the Bowditch boundary (in the coned-off Cayley graph) or exactly the Bowditch boundary (in the cusped Cayley graph). And lastly, unlike the spaces satisfying Definition \\ref{def:original rel hyp}, any two cusped Cayley graphs for the same relatively hyperbolic pair are quasi-isometric \\cite{HruskaHealyCusped}.\n\nOur final definition of relatively hyperbolic is also due to Bowditch (as was the original). As in Definitions \\ref{def: coned-off} and \\ref{def:cusped-off}, the hyperbolic space of interest will be a graph. But unlike those definitions, the graph does not come from the Cayley graph, nor is it constructive.\n\n\\begin{defn}\n A graph $K$ is \\textit{fine} if each edge of $K$ is contained in only finitely many circuits of length $n$ for each $n$.\n\\end{defn}\n\n\\begin{defn}[Relatively hyperbolic, \\cite{Bowrelhyp}]\\label{def:fine rel hyp}\nLet $G$ act on a $\\delta$-hyperbolic graph $K$ with finite edge stabilizers and finitely many orbits of edges. If $K$ is fine, then $(G,\\P)$ is \\textit{relatively hyperbolic}, where $\\P$ consists of stabilizers of infinite valance vertices.\n\\end{defn}\n\nThe equivalence of this definition with Definition \\ref{def:original rel hyp} is due (independently) to Bowditch \\cite[Theorem 7.10]{Bowrelhyp}, Dahmani \\cite{DahmaniRelHypEquiv}and Hruska \\cite{HruskaRelHypEquiv}.\n\nWe have 4 equivalent, yet separate, definitions of relatively hyperbolic, which we will summarize here. Note there are more (equivalent) definitions, for example \\cite{gerasimov, Yaman}.\n\n\\begin{enumerate}\n \\item Definition \\ref{def:original rel hyp}: When $G$ acts properly discontinuously and by isometries on a hyperbolic metric space $X$ with each $x\\in \\partial X$ either conical limit point or bounded parabolic and $\\P$ is the collection of maximal parabolic subgroups, then $(G,\\P)$ is relatively hyperbolic.\n \\item Definition \\ref{def:fine rel hyp}: If $G$ acts on a $\\delta$-hyperbolic graph $K$ with finite edge stabilizers and finitely many orbits of edges, and $K$ is fine, then $(G,\\P)$ is relatively hyperbolic, where $\\P$ is the collection of stabilizers of infinite valance vertices.\n \n\n \n \\item Definition \\ref{def: coned-off}: When the coned-off Cayley graph $\\mathcal{C}(G,\\P,\\mathcal{A})$ is hyperbolic and has bounded coset penetration, $(G,\\P)$ is relatively hyperbolic.\n \\item Definition \\ref{def:cusped-off}: When the cusped Cayley graph $X(G,\\P)$ is hyperbolic, $(G,\\P)$ is relatively hyperbolic.\n\n\\end{enumerate}\n\\section{What the boundary tells us and the relation to Kleinian groups}\\label{day3}\n\nWhat can the boundary of a relatively hyperbolic group tell you about the group?\n\nWe'll begin by examining the case of Kleinian groups. These are key examples of hyperbolic and relatively hyperbolic groups.\nMany of the results about hyperbolic and relatively hyperbolic groups in this section were inspired by known results regarding Kleinian groups and their associated manifolds. We will be discussing some manifold theory without always giving detailed definitions. The first chapter of \\cite{Kapbook} has comprehensive definitions.\nThe key idea is that a hyperbolic three-manifold has a ``characteristic submanifold\" containing the essential annuli in the manifold. This can be seen from the limit set of the associated Kleinian group. A similar and important phenomenon happens with hyperbolic and relatively hyperbolic groups. This theory was begun by Bowditch (echoing the Jaco-Shalen and Johannsen characteristic submanifold theory) and continued by many people.\n\n\\begin{defn}[Kleinian group]\nA group $\\Gamma$ is \\textit{Kleinian} if it is a discrete subgroup of $\\PSL(2,\\mathbb{C})$. Note that $\\PSL(2,\\mathbb{C}) = \\operatorname{Isom}^+(\\mathbb{H}^3)$, the orientation preserving isometries of $\\mathbb{H}^3$.\n\\end{defn}\n\n\\begin{defn}[Limit set]\nLet $\\Gamma$ be a Kleinian group and let the boundary of $\\H^3$ be $\\hat{\\mathbb{C}}$. Fix $x\\in \\H^3$, then the \\textit{limit set} of $\\Gamma$, denoted $\\Lambda_\\Gamma$, is\n\n$$\\Lambda_\\Gamma:=\\overline{\\Gamma \\cdot x} \\cap \\hat{\\mathbb{C}}$$\n\n\\end{defn}\n\n\\begin{remark}\n The choice of $x\\in \\H^3$ does not change $\\Lambda_\\Gamma$.\n\\end{remark}\n\n\n\n\\begin{defn}[Geometrically finite]\nLet $\\Gamma$ be a Kleinian group and let $C(\\Gamma)$ be the convex hull of $\\Lambda_\\Gamma$. Then $\\Gamma$ is \\textit{geometrically finite} if $C(\\Gamma)\/\\Gamma$ has finite volume.\n\n\\end{defn}\n\nA natural connection between Kleinian groups and relatively hyperbolic groups comes from the following fact: if $\\Gamma<\\PSL(2,\\mathbb{C})$ is geometrically finite, then $(\\Gamma, \\P)$ is relatively hyperbolic and $\\Lambda_\\Gamma = \\partial (G,\\P)$, where $\\P$ is the collection of parabolic elements of $\\Gamma$, see \\cite{BowGF}.\n\nAn example and a non-example of geometrically finite Kleinian groups:\n\n\\begin{example} If $\\Gamma < \\PSL(2, \\mathbb{C})$ is $\\pi_1(M^3)$ where $M^3$ is a closed hyperbolic manifold, then $\\Gamma$ is geometrically finite. Also, when $\\H^3\/\\Gamma$ is a finite-volume hyperbolic cusped 3-manifold, $\\Gamma$ is geometrically finite. More generally, if $H< \\Gamma$ is a finite index subgroup, where $\\Gamma$ is a geometrically finite Kleinian group, then $H$ is also geometrically finite. Note that when $H$ is finite index then $\\Lambda_H=\\Lambda_\\Gamma$, and $H$ acts geometrically on the convex hull of its limit set.\n\\end{example}\n\n\\begin{example}\nNon-Example: Let $\\psi$ be a pseudo-Anosov homeomorphism of a hyperbolic surface $S_g$. The mapping torus $M_\\psi^3$ is a hyperbolic 3-manifold and the limit set of its fundamental group, $\\Gamma$, is $\\hat{\\mathbb{C}}$. This manifold fibers over the circle and $\\pi_1(S_g)$ is normal in $\\Gamma$, so $\\Lambda_{\\pi_1(S_g)}=\\Lambda_\\Gamma=\\hat{\\mathbb{C}}$. Since $\\Lambda_\\Gamma$ is the entire boundary of $\\H^3$, the convex hull of the limit set is all of $\\H^3$. $\\H^3\/\\pi_1(S_g)$ is infinite volume, hence not geometrically finite.\n\\end{example}\n\n\nFor some time, lots of manifolds have been known to admit geometrically finite hyperbolic structures. By work of Thurston, Haken manifolds whose fundamental groups do not contain free abelian groups of rank 2 can be realized as hyperbolic manifolds. The proof is quite involved, see \\cite{Kapbook} and \\cite{MorganOnUniformization}, and includes the case when the manifold has boundary. A 3-manifold $M$ is \\emph{irreducible} if every 2-sphere in $M$ bounds a 3-ball in $M$ and \\emph{atoroidal} if it is irreducible and $\\pi_1(M)$ contains no $\\mathbb{Z} \\oplus \\mathbb{Z}$.\n\n\n\\begin{thm}[Theorem A, page 70 \\cite{MorganOnUniformization}]\nLet $M$ be a compact, atoroidal, Haken 3-manifold. Then there is a geometrically finite, complete hyperbolic manifold $N$ such that $C(\\pi_1(N))\/\\pi_1(N)$ is homeomorphic to $M$.\n\\end{thm}\n\nThere is a corresponding theorem for ``pared manifolds\" \\cite[pg70 Theorem B']{MorganOnUniformization}]. This theorem provides a large family of relatively hyperbolic pairs, where the peripheral subgroups are exactly the parabolic subgroups of the corresponding Kleinian group.\n\n\n\\begin{remark} \\label{remark genus 2}\n It is possible for the manifold to have annuli. For example, let $S_{2,1}$ be a genus two surface with one boundary component. Then the three manifold obtained by gluing three copies of $S_{2,1} \\times I$ along $\\partial(S_{2,1}) \\times I$ to a solid torus along three parallel annuli on the boundary of the solid torus satisfies the hypotheses above. Thus this admits a geometrically finite hyperbolic structure. The quotient of $H^3$ by this Kleinian group is infinite volume, however the quotient of the convex hull of the limit set has finite volume. Thus it act geometrically on the convex hull of its limit set, which is hyperbolic.\n\\end{remark}\nThe following definition allows us to understand the essential annuli in a 3-manifold. For geometrically finite hyperbolic 3-manifolds, the characteristic submanifold can be described from the limit set, see \\cite{walshbump}.\n\n\n\\begin{defn}[Characteristic Submanifold]\nLet $(M,\\partial M)$ be a 3-manifold with incompressible boundary (e.g. $\\pi_1(M)$ is a geometrically finite Kleinian group). Then the \\textit{characteristic submanifold}, $(X,S)$, is a submanifold with:\n\n\\begin{enumerate}\n \\item Each component is an $I$-bundle over a surface or a solid torus with a Seifert-fibered structure.\n \\item $\\partial X \\cap \\partial M = S$\n \\item The components of $\\partial X \\setminus S$ are essential annuli.\n \\item Any essential annulus or M\\\"obius band is properly homotopic into $(X,S)$.\n \\item $(X,S)$ is unique up to isotopy.\n\n\\end{enumerate}\n\n\\end{defn}\n\n\nThe characteristic submanifold allows us to detect a splitting of the fundamental group of the 3-manifold (with incompressible boundary) along infinite cyclic subgroups coming from the essential annuli. A result of Bowditch tells us that we can detect such a splitting in \\textbf{any} hyperbolic group, and this splitting can be detected through the topology of the boundary \\cite{BowHypeSplitting}.\n\n\\begin{defn}[Splitting]\nA {\\it splitting} of a group $G$ over a class of subgroups is a non-trivial finite graph of groups representation of $G$, where each edge group belongs to the class.\n\\end{defn}\n\n\\begin{thm}[\\cite{BowHypeSplitting}]\\label{thm:BowHyp}\nLet $G$ be a hyperbolic group. If $\\partial G$ has a local cut point, then $G$ splits over a virtually cyclic (i.e. 2-ended) subgroup. Furthermore, $G$ splits as a bipartite graph of groups with three types of vertices:\n\n\\begin{enumerate}\n \\item virtually cyclic\n \\item virtually Fuchsian\n \\item rigid--these contain no further splittings.\n\\end{enumerate}\n\\end{thm}\n\nTo see this in the boundary $\\partial G$ of $G$, there is a cut pair in $\\partial G$ which is the limit set of the subgroup that $G$ splits over. In fact, all the conjugates of this subgroup will have limit set consisting of a cut pair.\n\n\n\\begin{example}\nLet $A, B, C$ be three copies of $S \\times I$, where $S$ is a torus with one boundary component. Let $T$ be a solid torus $T^2 \\times I$. Glue the $\\partial(S) \\times I$ of $A, B, C$ to parallel longitudinal annuli on $T \\times \\{0\\}$ by degree 1 maps. We obtain a 3-manifold $M$ with boundary as shown in Figure \\ref{fig:surfacethickened}, where $T$ is tricolor.\n\nThe fundamental group $G$ of $M$ has a graph of groups decomposition with three vertex groups $F_2 = \\pi_1(A) = \\pi_1(B) = \\pi_1(C)$, one vertex group $\\mathbb{Z} = \\pi_1(T)$ and three edge groups $\\mathbb{Z}$ (Figure \\ref{fig:surfacegraphofgroups}).\n\n\\begin{figure}[H]\n \\begin{subfigure}{.5\\textwidth}\n \\hspace*{1cm}\n\t \\includegraphics[scale=.075]{surface_thickened.png}\n\t \\caption{$M$}\n\t \\label{fig:surfacethickened}\n \\end{subfigure}\n \\hspace*{-5pt}\n \\begin{subfigure}{.5\\textwidth}\n\t \\centering\n\t \\includegraphics[scale=.12]{surface_graph_of_groups.png}\n\t \\caption{graph of group decomposition }\n\t \\label{fig:surfacegraphofgroups}\n \\end{subfigure}\n\\caption{}\n\\label{fig:surface}\n\\end{figure}\n\nThe universal cover $\\tilde M$ of $M$ is a tree of spaces, where the tree is bipartite with two types of vertices. Vertices of type I are universal covers of $T$, and have valence 3. Vertices of type II are universal covers of $A, B, C$, as in Figure \\ref{fig:cantor}, and have valence $\\infty$. Figure \\ref{fig:surfaceuniversalcover} shows three sheets of universal covers of $A, B, C$ attached along a universal cover of $T$, the vertical thickened line. Each sheet contains infinitely many universal covers of $T$, whose fundamental groups are conjugates of $\\pi_1(T)$. Attached to each universal cover of $T$ are three sheets of universal covers of $A, B, C$.\n\nFor the relatively hyperbolic boundary of $G$, note that $G$ acts geometrically on two proper hyperbolic metric spaces: the convex hull of $\\Lambda(G)$ and $\\tilde M$, so $\\partial G \\cong \\Lambda(G) \\cong \\partial (\\tilde M)$ (Figure \\ref{fig:surfacelimitset}).\n\n\n\n\\begin{figure}[H]\n \\begin{subfigure}{.5\\textwidth}\n\t \\centering\n\t \\includegraphics[scale=.15]{surface_universal_cover.png}\n\t \\caption{the universal cover $\\tilde M$ of $M$}\n\t \\label{fig:surfaceuniversalcover}\n \\end{subfigure}\n \\hspace*{-5pt}\n \\begin{subfigure}{.5\\textwidth}\n\t \\centering\n\t \\includegraphics[scale=.03]{surface_limit_set.png}\n\t \\caption{the limit set $\\Lambda(G) \\cong \\partial(\\tilde M)$}\n\t \\label{fig:surfacelimitset}\n\n \\end{subfigure}\n \\caption{}\n\\end{figure}\n\nIn Bowditch's language (Theorem \\ref{thm:BowHyp}), the fundamental groups of the $\\infty$-valent vertices are of type 1 (virtually cyclic), and those of the 3-valent vertices of type 2 (virtually Fuchsian). All are quasiconvex subgroups of $G$, with their boundaries embedded in $\\partial G$. Here, $\\partial G$ has a tree-like structure, where each vertex is a pair of points (for type 1 vertices, of valence 3) or a Cantor set (for type 2 vertices, of valence $\\infty$), and where each edge is a pair of points (for the edge groups $\\mathbb{Z}$) that coincides with vertices of type 1. In $\\Lambda (G) = \\partial G$, we see a cut pair of valence 3 at the north and south pole, corresponding to the boundary of the 2-ended subgroup $\\pi_1(T) = \\mathbb{Z}$ of type 1. All other cut pairs of valence 3 correspond to conjugates of the 2-ended subgroup. We can see $\\partial G$ as three Cantor sets glued together along every 3-valent cut pair, together with the boundary of the bipartite tree, which corresponds to rays that keep switching sheets and thus does not belong to the boundary of any vertex or edge.\n\\end{example}\n\n\n\n\n\nKapovich and Kleiner use Bowditch's result to classify the types of 1-dimensional boundaries possible for a 1-ended hyperbolic groups.\n\n\\begin{thm}[\\cite{KapKleinerOneDimBoundary}]\\label{thm:KapKleiner}\nLet $G$ be a 1-ended hyperbolic group with $\\partial G$ 1-dimensional and $G$ does not split over a 2-ended subgroup. Then $\\partial G$ is homeomorphic to one of the following:\n\n\\begin{enumerate}\n \\item $S^1$\n \\item $\\mathcal{S}$, a Sierpinski carpet (planar)\n \\item a Menger curve (non-planar)\n\\end{enumerate}\n\\end{thm}\nThe reason that we have the two-ended hypothesis is that there are Kleinian groups with boundaries different from above (as in Figure \\ref{fig:surfacelimitset}). A hyperbolic manifold group that splits over an infinite cyclic group is not rigid in the sense that it admits many different hyperbolic structures.\nFurthermore, work of Canary and McCullough shows that there are hyperbolic 3-manifolds $M_1$ and $M_2$ with $\\pi_1(M_1)\\cong \\pi_1(M_2)$ but $M_1$ and $M_2$ are not homeomorphic. See \\cite{CanaryMcMullough}. For example, if we alter the example of Remark \\ref{remark genus 2} so that the surfaces have different genera, then changing the cyclic order around the solid torus will not change the fundamental group.\n\n\n\\begin{defn}[Peripheral splitting]\nLet $(G, \\P)$ be a relatively hyperbolic pair. A splitting is {\\it relative} to $\\P$ if each subgroup in the collection $\\P$ is conjugate into one of the vertex groups.\n\nA {\\it peripheral splitting} of $(G, \\P)$ is a bipartite splitting of $G$ relative to $\\P$, where each $P \\in \\P$ is conjugate into vertices of one color.\n\\end{defn}\n\n\\begin{defn}[Tame]\nLet $(G,\\P)$ be a relatively hyperbolic pair. A subgroup $P\\in \\P$ is \\textit{tame} if $P$ is finitely generated, 1- or 2-ended, and does not contain an infinite torsion subgroup.\n\\end{defn}\n\n\\begin{defn}[Rigid, \\cite{DahmaniRecognizing}]\nA relatively hyperbolic pair $(G, \\P)$ is {\\it rigid} if $G$ has no splitting relative to $\\P$ over virtually cyclic groups or over parabolic subgroups.\n\\end{defn}\n\nMatt Haulmark proves a theorem similar to Theorem \\ref{thm:KapKleiner} for relatively hyperbolic pairs.\n\n\n\\begin{thm}[\\cite{HaulRHBoundary}] \\label{thm:Haul}\nLet $(G, \\P)$ be a rigid relatively hyperbolic pair with $\\partial(G, \\P)$ 1-dimensional and every $P \\in \\P$ one-ended. If each $P\\in \\P$ is tame, then $\\partial(G, \\P)$ is homeomorphic to one of the following:\n\\begin{enumerate}\n \\item $S^1$\n \\item $\\mathcal{S}$, a Sierpinski carpet (planar)\n \\item a Menger curve (non-planar)\n\\end{enumerate}\n\\end{thm}\n\n\nBy work of Dasgupta and Hruska \\cite{DasguptaHruska}, the tameness condition on the peripherals can be dropped.\n\nThe theorem is based on the characterizations of global cut points in the Bowditch boundary. Just as Theorem \\ref{thm:BowHyp} allows one to see splittings of hyperbolic groups via cut pairs in the boundary, further work of Bowditch shows that cut points in relatively hyperbolic boundaries correspond to splittings of the group over a subgroup of a peripheral group.\n\n\\begin{thm}[\\cite{Bowperiph}]\nSuppose $(G, \\P)$ is a 1-ended relatively hyperbolic pair. If $G$ admits a peripheral splitting, then $\\partial(G, \\P)$ contains a global cut point.\n\\end{thm}\n\n\\begin{thm}[\\cite{HaulRHBoundary}]\nSuppose $(G, \\P)$ is a 1-ended relatively hyperbolic pair with tame peripherals. If $\\partial(G, \\P)$ has a global cut point, then there exists a peripheral splitting of $(G, \\P)$.\n\\end{thm}\n\nJust as with Theorem \\ref{thm:Haul}, tameness of peripherals can be dropped \\cite{DasguptaHruska}.\n\nThus, a 1-ended relatively hyperbolic pair admits a peripheral splitting if and only if the Bowditch boundary contains a global cut point (if and only if there is a parabolic fixed point). See Figure \\ref{fig:tree-of-circles} for an example of global cut points and a peripheral splitting.\nNote that the Gromov boundaries of hyperbolic groups do not have cut points.\n\nCut points in Bowditch boundaries can cause exotic phenomena which prevent $G$ from being a Kleinian group. There are examples of relatively hyperbolic group pairs $(G, \\mathcal{P})$ where the planar Bowditch boundary $\\partial(G, \\P)$ has cut points but where no peripheral structure on $G$ is virtually Kleinian or even virtually a manifold group \\cite{HW1}.\n\nThis leads to the following question:\n\n\\begin{question}\\label{question: vgfk}\nWhen is a relatively hyperbolic group (virtually) a geometrically finite Kleinian group?\n\\end{question}\n\nThe Bowditch boundary of such a group must be planar, but having no cut points is not necessary. There are examples of geometrically finite Kleinian groups whose relatively hyperbolic boundary has cut points, such as a surface subgroup with accidental parabolics. Example \\ref{example:TreeOfCircles} can be realized as a geometrically finite Kleinian group.\n\nWe have the following conjecture on the sufficient conditions \\cite{HW1}.\n\n\\begin{conjecture}\\label{conjecture: vgfk}\nLet $(G,\\P)$ be a relatively hyperbolic group pair. If $\\partial(G,\\P)$ is planar and has no cut points, then $G$ is virtually a geometrically finite Kleinian group.\n\\end{conjecture}\n\n\n\n\n\\printbibliography\n\n\n\\end{document}\n\n\nRemoved from preamble: \\newcommand\\form[1]{\\langle #1\\rangle}\n\\theoremstyle{theorem}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{cor}[theorem]{Corollary}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{question}[theorem]{Question}\n\\newtheorem{claim}[theorem]{Claim}\n\\newtheorem{observation}[theorem]{Observation}\n\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{defn}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem*{nonumberclaim}{Claim}\n\n\n\n\n\n\n\t\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{bass-serre-surface.png}\n \\caption{The Bass-Serre tree for the splitting visualized inside of $\\H^2$}\n \\label{fig:bass-serre}\n\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\nThe bright solar analogues 16 Cygni A (HD 186408, HR 7503) and 16 Cygni B (HD 186427, HR 7504) for many reasons represent a very interesting stellar system. While a red dwarf, 16 Cygni C, is in orbit around the first component 16 Cygni A \\citep{turner01,patience02}, the second component, 16 Cygni B, hosts a giant Jovian planet with minimum mass of 1.5 $M_{jup}$ located on an eccentric orbit (e=0.63), with an orbital period of 800.8 days \\citep{cochran97}. The two main stars are separated widely enough, with an orbital period longer than 18,000 years \\citep{hauser99}, to be studied in the same way as two isolated stars, with no common dynamical effects. \n\nThis situation allows for precise differential studies between a planet-hosting star and a non-planet-hosting star with similar birth conditions. The red dwarf around 16 Cygni A may be the reason why no accretion disk has developed around it, whereas a planetary disk remained around 16 Cygni B, including the observed giant planet, and probably smaller as yet unobserved bodies.\n\nThe two main stars of the 16 Cygni system have been studied in many ways, using spectroscopy, interferometry, and asteroseismology. The abundances of the heavy elements in these two stars are very similar. Although a very small difference has been claimed by \\citealt{ramirez11} and \\citealt{tuccimaia14}, \\cite{schuler11} found them to be indistinguishable. On the other hand, the surface lithium abundance of 16 Cygni B is lower than that of 16 Cygni A by at least a factor 4.7 \\citep{king97}. These observations lead to several open questions, which remain to be answered for a better understanding of these stars. \n\nThe interest of this study is that these stars have the same birth site and the same age, with masses of the same order, so that their past evolution is similar for most aspects. The observed differences between them must basically be due to the planetary disk around B. For this reason, the detailed study of this stellar system helps understanding the differences between stars with and without planetary disks.\n\nThe present paper is motivated by two considerations. The first one is that none of the previous modelling of these two stars\nconsidered atomic diffusion including the radiative acceleration on each element. Most stellar evolution codes currently include the atomic diffusion of helium (without radiative accelerations), some of them also include it for heavy elements, but very few consider the radiative accelerations.\n\nThe second consideration refers to the consequences of the accretion of heavy matter onto stars, which may occur when the star has a planetary disk. Many studies and past publications assumed that the accreted matter remains inside the outer stellar convective zone, so that accretion can lead to an increase of the heavy element abundances, as well as to an increase of the lithium abundance. This assumption is not valid, as shown in detail by \\citet{vauclair04}, \\citet{garaud11}, \\citet{theado12}, and \\citet{deal13}. When heavy matter falls onto the star, it creates an inverse gradient of molecular weight, which leads to a double-diffusive instability now called fingering (or thermohaline) convection. The heavy elements are mixed downwards until the mean molecular weight gradient becomes nearly flat. In most cases, no signature of the accreted heavy elements remains at the surface. Meanwhile, as computed in detail by \\citet{theado12bis}, the induced mixing may lead to an extra lithium depletion in the star. As a consequence, the accretion of heavy matter cannot lead to any increase of lithium at the surface of a star, but may conversely lead to a decrease of its observed abundance.\n\nThe observational results on the two main stars 16 Cygni A and B are presented in Sect. 2 together with a discussion of the still-unsolved questions. In Sect. 3 we present models of 16 Cygni A and B that fit the observed seismic frequencies. We compare the parameters of these models with previously published ones. The surface abundances of heavy elements and lithium that are obtained after diffusion in these models are discussed in Sect. 4 . Finally, in Sect. 5 we show that the accretion of metal rich planetary matter at the beginning of the main sequence on 16 Cygni B may explain the lithium difference between the two stars. A summary and discussion of all these results are given in Sect. 6.\n\n\\section{Observational constraints} \n\nThe position in the sky of the 16 Cygni system allowed seismic observations with the \\textit{Kepler} satellite. More than 40 modes of degree l=0, 1, 2, and 3 could be detected. Analyses with the Asteroseismic Modelling Portal (AMP) and comparisons with other seismic studies led to precise values of the masses, radii, and ages of the two stars \\citep{metcalfe12}. Moreover, the seismic observations allowed measurements of their stellar rotation periods \\citep{davies15}.\n\nThe effective temperatures and gravities of the two stars were derived from spectroscopic observations (e.g. \\citealt{ramirez11,schuler11,tuccimaia14}). We must insist, however, that asteroseismology leads to a log $g$ value with a much better precision than spectroscopy. As we show in Sect. 3, all the models that closely fit the observed seismic frequencies, even if they have different masses, radii, helium abundances, ages, and luminosities, have the same log $g$ value with a precision of 0.01.\nThe determinations of the bolometric magnitudes \\citep{torres10} and Hipparcos parallaxes \\citep{vanleeuwen07} lead to precise values of the luminosities of the two stars. They are also bright enough for their radii to be determined by interferometric techniques \\citep{white13}. The results are given in Table \\ref{table:1}.\n\n\n\n\n\\begin{table}\n\\caption{Properties of 16 Cygni A and B from the literature} \n\\label{table:1}\n\\centering\n\\begin{tabular}{c c c}\n\\hline \n & 16 Cygni A & 16 Cygni B \\\\ \n\\hline\\hline\n \n$T_{\\rm eff}$(K) & $5825\\pm 50$\\tablefootmark{a} & $5750\\pm 50$\\tablefootmark{a} \\\\%ramirez 2009 \n & $5813\\pm 18$\\tablefootmark{b} & $5749\\pm 17$\\tablefootmark{b} \\\\%ramirez 2011\n \n & $5796\\pm 34$\\tablefootmark{c} & $5753\\pm 30$\\tablefootmark{c} \\\\%schuler2011\n & $5839\\pm 42$\\tablefootmark{d} & $5809\\pm 39$\\tablefootmark{d} \\\\%white2013 \n\\smallskip\n & $5830\\pm 7$\\tablefootmark{f} & $5751\\pm 6$\\tablefootmark{f} \\\\%tucci2014 \n \nlog $g$ & $4.33\\pm 0.07$\\tablefootmark{a} & $4.34\\pm 0.07$\\tablefootmark{a} \\\\%ramirez 2009\n & $4.282\\pm 0.017$\\tablefootmark{b} & $4.328\\pm 0.017$\\tablefootmark{b} \\\\%ramirez 2011 \n & $4.38\\pm 0.12$\\tablefootmark{c} & $4.40\\pm 0.12$\\tablefootmark{c} \\\\\n \\smallskip\n & $4.30\\pm 0.02$\\tablefootmark{f} & $4.35\\pm 0.02$\\tablefootmark{f} \\\\ \n\n[Fe\/H] & $0.096\\pm 0.026$\\tablefootmark{a} & $0.052\\pm 0.021$\\tablefootmark{a} \\\\%ramirez 2009 \n & $0.104\\pm 0.012$\\tablefootmark{b} & $0.061\\pm 0.011$\\tablefootmark{b} \\\\%ramirez 2011\n & $0.07\\pm 0.05$\\tablefootmark{c} & $0.05\\pm 0.05$\\tablefootmark{c} \\\\\n\\smallskip \n & $0.101\\pm 0.008$\\tablefootmark{f} & $0.054\\pm 0.008$\\tablefootmark{f}\\\\\n\\smallskip \nA(Li) & $1.27\\pm0.05$\\tablefootmark{i} & $\\leq0.6$\\tablefootmark{i} \\\\ \n\\smallskip \nA(Be) & $0.99\\pm0.08$\\tablefootmark{j} & $1.06\\pm0.08$\\tablefootmark{j} \\\\\n\\hline\\hline \nMass ($M_{\\odot}$) & $1.05\\pm 0.02$\\tablefootmark{b} & $1.00\\pm 0.01$\\tablefootmark{b} \\\\ \n & $1.07\\pm 0.05$\\tablefootmark{d} & $1.05\\pm 0.04$\\tablefootmark{d} \\\\\n\\smallskip \n & $1.11\\pm 0.02$\\tablefootmark{g} & $1.07\\pm 0.02$\\tablefootmark{g} \\\\%metcalfe2012\n\nRadius ($R_{\\odot}$) & $1.218\\pm 0.012$\\tablefootmark{d} & $1.098\\pm 0.010$\\tablefootmark{d} \\\\ \n & $1.22\\pm 0.02$\\tablefootmark{e} & $1.12\\pm 0.02$\\tablefootmark{e} \\\\\n\\smallskip\n & $1.243\\pm 0.008$\\tablefootmark{g} & $1.127\\pm 0.007$\\tablefootmark{g} \\\\\n\\smallskip\nLuminosity ($L_{\\odot}$) & $1.56\\pm 0.05$\\tablefootmark{g} & $1.27\\pm 0.04$\\tablefootmark{g} \\\\ \n\nAge (Gyr) & $7.15_{-1.03}^{+0.04}$\\tablefootmark{b} & $7.26_{-0.33}^{+0.69}$\\tablefootmark{b} \\\\ \n\\smallskip \n & $6.9\\pm 0.3$\\tablefootmark{g} & $6.7\\pm 0.4$\\tablefootmark{g} \\\\\n\\smallskip\n$Z_{i}$ & $0.024\\pm 0.002$\\tablefootmark{g} & $0.023\\pm 0.002$\\tablefootmark{g} \\\\ \n\\smallskip\n$Y_{i}$ & $0.25\\pm 0.01$\\tablefootmark{g} & $0.25\\pm 0.01$\\tablefootmark{g} \\\\ \n\\smallskip\n$v~\\rm{sin}$ $i$ (km.s$^{-1}$) & $2.23\\pm 0.07$\\tablefootmark{h} & $1.27\\pm 0.04$\\tablefootmark{h} \\\\ \n\\smallskip\n$P_{rot}$ (days) & $23.8_{-1.8}^{+1.5}$\\tablefootmark{h} & $23.2_{-3.2}^{+11.5}$\\tablefootmark{h} \\\\ \n\\smallskip\nPlanet detected & no & yes\\tablefootmark{k} \\\\ \n\\hline\n\\end{tabular}\n\\tablefoot{\\tablefoottext{a}{\\cite{ramirez09}}; \\tablefoottext{b}{\\cite{ramirez11}}; \\tablefoottext{c}{\\cite{schuler11}}; \\tablefoottext{d}{\\cite{white13}, seismic determination}; \\tablefoottext{e}{\\cite{white13}, interferometric determination}; \\tablefoottext{f}{\\cite{tuccimaia14}}; \\tablefoottext{g}{\\cite{metcalfe12}}; \\tablefoottext{h}{\\cite{davies15}}; \\tablefoottext{i}{\\cite{king97}}; \\tablefoottext{j}{\\cite{deliyannis00}}; \\tablefoottext{k}{\\cite{cochran97}}\\\\\n}\n\\end{table}\n\n\n\nDetailed determinations of their element abundances have been given by several authors. From spectroscopy with high signal-to-noise ratio of the 10 m Keck 1 telescope and HIRES echelle spectrograph, \\citet{schuler11} determined the abundance of 15 heavy elements in both stars and found them indistinguishable. On the other hand, \\citet{ramirez11} and \\citet{tuccimaia14} claimed that 16 Cygni A is slightly more metal rich than 16 Cygni B based\non spectra with high resolution and high signal-to-noise ratio\nthat were obtained with the R.G. Hull coude spectrograph on the 2.7m Harlan Smith telescope at Mc Donald Observatory. \n\nA more striking difference between the two stars, confirmed by all spectroscopic observations, is that the star hosting a planet, 16 Cygni B, has an abundance of lithium at least four times lower than the star without a planet (\\citealt{friel93,king97}). While both stars are lithium depleted compared to F stars and to the meteoritic value, detailed measurements show that 16 Cygni A is slightly less depleted than the Sun (log $N(Li) = 1.27$ compared to 1.05 for the Sun), whereas 16 Cygni B is more depleted (log $N(Li) < 0.60$). On the other hand, \\citet{deliyannis00} found that the Be and B abundances are the same in the two stars within the limits of the uncertainties.\n\n\n\n\n\\section{Asteroseismic studies and stellar models including radiative accelerations} \n\n\\subsection{Stellar models}\n\nWe used the Toulouse Geneva Evolution Code (TGEC) to compute stellar models that fit the seismic observations of 16 Cygni A and B. This code performs complete computations of atomic diffusion, including radiative accelerations, for 21 species, namely 12 elements and their main isotopes: H, $^{3}$He, $^{4}$He, $^{6}$Li, $^{7}$Li, $^{9}$Be, $^{10}$B, $^{12}$C, $^{13}$C, $^{14}$N, $^{15}$N, $^{16}$O, $^{17}$O,$^{18}$O, $^{20}$Ne, $^{22}$Ne, $^{24}$Mg, $^{25}$Mg, $^{26}$Mg, $^{40}$Ca and $^{56}$Fe \\citep{theado12}. The diffusion coefficients used in the code are those derived by \\citet{paquette86}. \n\nThe Rosseland opacities are recalculated inside the model, at each time step and at every mesh point, using OPCD v3.3 and data from \\citet{seaton05}, to take into account the local chemical composition. In this way, the stellar structure is consistently computed all along the evolutionary tracks, as well as the individual radiative accelerations of C, N, O, Ne, Mg, Ca, and Fe. This is done by using the improved semi-analytical prescription proposed by \\citet{alecian04}. A more detailed discussion of these computations is given in \\citet{dealprep} (in prep).\n\nThe equation of state used in the code is the OPAL2001 equation \\citep{rogers02}.\nThe nuclear reaction rates are from the NACRE compilation \\citep{angulo99}. The mixing length formalism is used for the convective zones with a mixing length parameter of 1.8, as needed to reproduce solar models.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig1.eps}\n\\caption{Profiles of the radiative accelerations of three elements Fe, Mg, and C as a function of the mass fraction in 16 Cygni A. }\n\\label{grad}\n\\end{center}\n\\end{figure}\n\nAs an example, Fig. \\ref{grad} displays the $g_{rad}$ profiles for C, Mg, and Fe compared to gravity for one of the computed models of 16 Cygni A. This model best fits the asteroseismic observations, as we discuss in Sect. 3.2. The dashed vertical line represents the position of the surface convective zone. Below the convective zone $g_{rad}$ is at least 1\/3 smaller than $g$, so that the effect of the radiative accelerations is small and generally negligible inside these stars.\n\n\\subsection{Best models from asteroseismic fits}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{fig2-a.eps}\n\\includegraphics[width=0.495\\textwidth]{fig2-b.eps}\n\\caption{Evolutionary tracks for models from 1.05 to 1.14 $M_{\\odot}$ (from right to left) with $Z_i=0.024$ and $Y_i=0.25$ (left panel) and evolutionary tracks for models from 1.05 to 1.12 $M_{\\odot}$ (from right to left) with $Z_i=0.024$ and $Y_i=0.26$ (right panel) . The error boxes are those of \\cite{ramirez09} (red dashed lines), \\cite{ramirez11} (green dot-dashed lines), \\cite{schuler11} (blue dotted lines) and \\cite{tuccimaia14} (black dotted lines). The blue dots indicate models with the right large separation, taking \\cite{kjeldsen08} corrections into account. The blue squares correspond to models that also have the right small separations and best fit the Echelle diagram. The black thick segments of each line indicate the models whose radii are consistent with the interferometric determinations of \\cite{white13}.}\n\\label{trace}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{fig3-a.eps}\n\\includegraphics[width=0.49\\textwidth]{fig3-b.eps}\n\\caption{Echelle diagrams for 16 Cygni A (left panel) and 16 Cygni B (right panel). The observed frequencies are represented by crosses.The frequencies computed for the models with Y=0.26 and Z=0.024 are represented by blue dots (l=0), green triangles (l=1), red squares (l=2), and black diamonds (l=3).}\n\\label{ED}\n\\end{center}\n\\end{figure*}\n\nThe stellar oscillation modes were derived for each model using the PULSE code \\citep{brassard08}. The frequencies were corrected for surface effects in the way proposed by \\citet{kjeldsen08}.\n\n\nWe computed models with masses ranging from 1.05 to 1.14$M_{\\odot}$. The initial helium mass fraction $Y_i$ was varied from 0.245 to 0.26 and the heavy element mass fraction $Z_i$ from 0.023 to 0.025 (initial mass fraction of all elements heavier than helium). \n\nEvolutionary tracks computed for two different initial compositions $Y_i$=0.25; $Z_i$=0.024 and $Y_i$=0.26; $Z_i$=0.024 are presented in Fig. \\ref{trace}. \n\nThe observational uncertainties on the oscillation frequencies for $l=0$ to $l=2$ modes lie between 0.1 to 1.45 $\\mu$Hz \\citep{metcalfe12}. The derived large separations $\\Delta \\nu$ for 16 Cygni A and for 16 Cygni B are $\\Delta \\nu_{obs}=103.56\\pm0.10~\\mu Hz$ and $\\Delta \\nu_{obs}=117.17\\pm0.10~\\mu Hz$, respectively. The models with large separations that fit these values are represented by blue dots in Fig. \\ref{trace}. They include the corrections for surface effects as proposed by \\citet{kjeldsen08}. Neglecting this effect would lead to models slightly above the blue dots, with a small difference in age of 150 million years on average.\nWe note that all the models presented in Fig. \\ref{trace} lie on horizontal lines, which means that they all have about the same surface gravity. This is a well-known result of asteroseismology. The asymptotic treatment of the oscillations (e.g. \\citealt{tassoul80}) shows that the large separation $\\Delta \\nu$ directly gives the average stellar density. In the range of our possible models, the variations in radii are small so that log $g$ is also nearly constant. In other words, if any parameter of the model is modified, for instance, the initial Y value, the other parameters such\nas age adjust to obtain a model with the same large separation, and thus the same gravity. Spectroscopy is only used in our description to constrain the effective temperatures.\n\n The uncertainty on the log $g$ values derived from the observed seismic frequencies for each track is about $10^{-2}$. This is the uncertainty on the position of the models represented by blue dots in Fig. \\ref{trace}. The corresponding error bars would lie inside the printed symbols. We note that the seismic log $g$ values lie outside the ranges given by \\cite{ramirez11} (see Fig. \\ref{trace}), which suggests that their uncertainties are underestimated. \n\nWe then derived the best of all these models for both stars. We first did it independently for each $Y_i$ value. The best models (represented by squares) were selected first because their small separations $\\delta\\nu_{0,2}$ best fit the observed ones, second because their echelle diagrams best fit the observed ones, according to $\\chi^2$ minimisations performed between the observed and the modelled frequencies.\n\nWhen compared with spectroscopic observations (see Fig. \\ref{trace}), it is clear that the best models for $Y_i$=0.25 lie inside the\nbox\nreported by \\cite{schuler11} for 16 Cygni A alone, not for 16 Cygni B, and that they are outside the box derived by \\cite{tuccimaia14} for both stars. In contrast, for $Y_i$=0.26 lie inside all the boxes for both stars. For this reason, we find that the $Y_i$=0.26 value is more probable than that of $Y_i$=0.25.\n\nThe echelle diagrams corresponding to the best case computed with $Y_i$=0.26 are presented in Fig. \\ref{ED} for 16 Cygni A (left panel) and 16 Cygni B (right panel). The fits between the computed and observed frequencies are very good for both stars. The corresponding stellar masses are $M_A=1.10M_{\\odot}$ and $M_B=1.06M_{\\odot}$.\n\nThese values and the other parameters obtained for these best models are given in Table \\ref{table:2}. The uncertainties we\nlist in this table correspond to the possible range of parameters of all the computed models, which have large and small separations in the observational uncertainties.\n\nWe must note that the surface helium abundances derived from these computations are slightly subsolar, while the spectroscopic parameters given in the literature have been computed using a solar helium value. It would be interesting in the future to iterate with spectroscopists and see how such a helium difference could influence the derived effective temperature range.\n\n\\begin{table}\n\\caption{Properties of 16 Cygni A and B from this work} \n\\label{table:2}\n\\centering\n\\begin{tabular}{c c c}\n\\hline \n & 16 Cygni A & 16 Cygni B \\\\\n\\hline\n\\smallskip \n$T_{\\rm eff}$(K) & $5821\\pm25$ & $5747\\pm25$ \\\\\n\\smallskip \nlog $g$ & $4.29\\pm0.01$ & $4.36\\pm0.01$ \\\\ \n\\smallskip\nMass ($M_{\\odot}$) & $1.10\\pm0.01$ & $1.06\\pm0.01$ \\\\ \n\\smallskip\nRadius ($R_{\\odot}$) & $1.24\\pm0.01$ & $1.13\\pm0.01$ \\\\ \n\\smallskip\nLuminosity ($L_{\\odot}$) & $1.58\\pm0.03$ & $1.25\\pm0.03$ \\\\ \n\\smallskip\nAge (Gyr) & $6.4\\pm0.4$ & $6.4\\pm0.4$ \\\\ \n\\smallskip\n$Z_{i}$ & $0.024$ & $0.024$ \\\\ \n\\smallskip\n$Y_{i}$ & $0.26$ & $0.26$ \\\\ \n\\smallskip\n$Z_{surf}$\\tablefootmark{a} & $0.0221$ & $0.0223$ \\\\ \n\\smallskip\n$Y_{surf}$\\tablefootmark{a} & $0.2226$ & $0.2265$ \\\\ \n\\hline\n\\end{tabular}\n\\tablefoot{\\tablefoottext{a}{Values at the age of best models}\n}\n\\end{table}\n\n\\section{Heavy elements and lithium abundances in 16 Cygni A and B, observational and theoretical discussions}\n\n\\subsection{Spectroscopic observations}\n\nThe abundances of heavy elements in 16 Cygni A and B have been a subject of debate for 15 years. \\cite{deliyannis00} reported an iron overabundance higher in 16 Cygni B than in 16 Cygni A, which has not been confirmed by more recent papers.\nWhereas \\citet{schuler11} did not report any difference between 16 Cygni A and B for the abundances of heavy elements, \\citet{ramirez11} and \\cite{tuccimaia14} claimed that the heavy elements are overabundant in 16 Cygni A compared to 16 Cygni B. \nIn any case, all observers agree that lithium is more depleted in 16 Cygni B than in 16 Cygni A by a large factor. Lithium has not been detected in B, whereas its abundance in A is slightly higher than that of the Sun. This difference between these very similar stars is difficult to account for using traditional explanations of lithium depletion in G stars.\n\nIn the following, we discuss the influence of atomic diffusion and mixing on the lithium and heavy elements abundances, and we show how the lithium difference between the two stars may be explained. We also discuss the consequences of these effects for heavy elements.\n\n\n\\subsection{Influence of atomic diffusion on the abundances of heavy elements and lithium}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig4.eps}\n\\caption{Abundance evolution of C, Mg, and Fe in our best models of 16 Cygni A and B, under the influence of atomic diffusion below the convective zone, when no mixing is taken into account.}\n\\label{xx0}\n\\end{center}\n\\end{figure}\n\nOur models include the computation of detailed atomic diffusion for a large number of elements, as discussed in Sect. 3.1. \nFigure \\ref{xx0} presents the surface abundance evolution for three of these elements, C, Mg, and Fe, in both 16 Cygni A and B. As the mass of 16 Cygni A is higher than that of 16 Cygni B, the convective zone is smaller and the diffusion processes faster. As a consequence, the surface abundances predicted by the models are higher in 16 Cygni B than in 16 Cygni A by $\\sim0.002$ dex at the age of our best models (vertical line). Later on in the evolution, at around 8 Gyr, the situation is reversed because the surface convective zone sinks more quickly in the more massive star. This occurs in models that are too old to account for the observations, however.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig5.eps}\n\\caption{Abundance profiles C, Mg, and Fe inside the stars under the same conditions as for Fig. \\ref{xx0}. }\n\\label{abprofil}\n\\end{center}\n\\end{figure}\n\nFigure \\ref{abprofil} displays the abundance profiles for the same three elements C, Mg, and Fe in the best models for 16 Cygni A and B. The bottoms of the convective zones are shown as vertical dashed lines. The difference of heavy elements abundances is due to the difference in surface convective zone depth, which is deeper in 16 Cygni B than A (grey dashed line).\nAtomic diffusion coupled with nuclear destruction also has an effect on the abundance of $^7$Li (see Fig. \\ref{LiLi0}). In this case, the final abundance of $^7$Li is lower in 16 Cygni B than in 16 Cygni A. This is not enough to account for the observations. First lithium is depleted in 16 Cygni A by a factor smaller than two, while in the real star it is depleted by 100. Second the depletion ratio between the two star is only $\\sim1.4,$ while the observations show a ratio of at least 4.7. Extra mixing below the convective zones is clearly needed to account for the observations.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig6.eps}\n\\caption{Abundance profiles of lithium under the same conditions as for Fig. \\ref{xx0}.}\n\\label{LiLi0}\n\\end{center}\n\\end{figure}\n\n \n\n\\subsection{Mixing processes below the outer convective zone}\n\nMany papers have been published in the past 30 years on the subject of the mixing processes that may occur below the convective zones of solar type stars and result in lithium destruction in their outer layers. The most common one is rotation-induced mixing (e.g.\\citealt{vauclair88,pinsonneault90,charbonnel92,charbonnel94,castro09}).\n\nThe rotation periods of 16 Cygni A and B have recently been measured with high precision using asteroseismology \\citep{davies15}. They are very similar, 23.8 days for A, 23.2 days for B, with a difference smaller than the uncertainties. The authors have also determined their inclination angles, 56 degrees for A, 36 degree for B. Using the interferometric radius, the resulting $v~\\rm sin$ $i$ is 2.23 km.s$^{-1}$ for A and 1.27 km.s$^{-1}$ for B.\n\nWhen studying rotation-induced mixing, it appears that the efficiency of the mixing is related to the local linear rotation velocity in such a way that it increases with radius. The bottom of the convective zone lies at a larger radius in A than in B, but the mixing is also more efficient. Simple expressions of the mixing diffusion coefficients (e.g. \\citealt{zahn92}) include the factor ($\\Omega^2 R^3\/GM$), which is about 1.3 larger for 16 Cygni A than for 16 Cygni B. As already discussed by \\cite{deliyannis00}, it does not seem possible to account for the lithium abundance ratio between A and B by such a simple process. It would not be realistic to invoke a stronger rotation-induced mixing effect in B than in A simply to account for the observations. Another process is clearly needed. \nWe do not intend to discuss all these processes here. We only simulate turbulent mixing by using a turbulent diffusion coefficient adjusted to obtain a lithium destruction by a factor 100 in 16 Cygni A, and we used the same to deduce the lithium destruction in 16 Cygni B. As in the simulations of \\cite{richer00}, we used a simple form $D_T=\\omega D\\rm (H_e)$ $\\left(\\frac{\\rho_0}{\\rho}\\right) ^{n}$ with n = 3, $\\omega=325$, $D$(He) the helium diffusion coefficient and $\\rho_0$ the density at the bottom of the outer convective zone.\n\nFigure \\ref{Lit} displays the resulting surface lithium abundance evolution in the two stars (green solid line and blue dashed line). As expected, lithium is more destroyed in 16 Cygni B than in 16 Cygni A, but only by a factor 2.9, not enough to account for the observations of an abundance ratio higher than 4.7 at the age of the stars. \n\nMeanwhile, the abundance of heavy elements is also modified, but the abundances in 16 Cygni B may not become lower than in 16 Cygni A (see Fig. \\ref{xt}).\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig7.eps}\n\\caption{Evolution of lithium surface abundances including a mixing at the bottom of the outer convective zone needed to reproduce\nthe 16 Cygni A abundance in the two models (green solid line and blue dashed line) (Sect. 4.3). Evolution of lithium surface abundances including accretion of 0.66$~M_{\\oplus}$ for 16 Cygni B (blue dotted line) at the beginning of the main sequence, but with the same mixing as before (Sect. 5). Abundances determined using observations from \\cite{king97} are represented by black crosses.}\n\\label{Lit}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig8.eps}\n\\caption{Abundance evolution of C, Mg, and Fe in our best models of 16 Cygni A and B, under the influence of atomic diffusion below the convective zone, when mixing is taken into account. }\n\\label{xt}\n\\end{center}\n\\end{figure}\n\n\n\\section{Fingering convection and lithium destruction induced by planetary accretion}\n\nWe now return to the fact that the star 16 Cygni B hosts a giant planet while 16 Cygni A does not. As discussed in the introduction, the red dwarf that orbits 16 Cygni A may be the reason for the fact that the main star could not develop any planetary disk.\n\nIn this situation, it is quite possible that in its early period on the main sequence, 16 Cygni B was able to accrete some matter from its planetary disk, which could not occur for 16 Cygni A.\nAs discussed by \\citep{theado12bis}, the accretion of heavy planetary matter onto a stellar surface builds an unstable compositional gradient at the bottom of the surface convective zone that triggers fingering (thermohaline) convection. This mixing leads to complete dilution of the heavy matter inside the star, so that no signature appears at the surface for the heavy elements. On the other hand, it may lead to extra lithium depletion because the mixing continues\ndown to the destruction layers.\n\n\\subsection{Computations of fingering convection} \n\nFingering convection may occur in stars every time a local accumulation of heavy elements appears in the presence of a stable temperature gradient. It occurs in particular in the case of the accretion of planetary matter \\citep{vauclair04, garaud11, deal13}. \n\nFingering convection is characterised by the so-called density ratio $R_0$ , which is the ratio between the thermal and $\\mu$-gradients: \n\\[\nR_0=\\frac{\\nabla - \\nabla_{ad}}{\\nabla_{\\mu}}. \n\\] \nThe instability can only develop if this ratio is higher than one and lower than the Lewis number, which is the ratio of the thermal to the molecular diffusivities. In this case, a heavy blob of fluid falls down inside the star and continues to fall because it exchanges heat more quickly than particles with the surroundings. If $R_0$ is smaller than one, the region is dynamically convective (Ledoux criterium), and if it is larger than the Lewis number, the region is stable.\n\nVarious analytical treatments of fingering convection in stars were given in the past, leading to mixing coefficients that differed by orders of magnitude \\citep{ulrich72,kippenhahn80}. More recently, 2D and 3D numerical simulations were performed, converging on coefficients of similar orders \\citep{denissenkov10,traxler11}. \n\nHere we used the recent prescription given by \\citet{brown13}, which has been confirmed by the 3D simulations of \\citealt{zemskova14}.\n\n\\subsection{Effect of accretion-induced fingering convection on the lithium abundance of 16 Cygni B} \n\nWe computed stellar models of 16 Cygni B with the assumption of accretion of planetary matter at the beginning of the main-sequence phase. These models included the treatment of fingering convection in the case of an inversion of the mean molecular weight gradient. We tested different accretion masses to characterise their effect on the lithium surface abundance due to fingering convection. In these computations, the accreted matter was assumed to have an Earth-like chemical composition \\citep{allegre95}. Changing the relative abundances of the accreted heavy elements has a very weak effect because their contribution to the mean molecular weight may be slightly different. For example, decreasing the iron abundance by a factor two compared to other elements modifies the computed accretion mass by less than $10\\%$.\n\nAbundance profiles of lithium after the accretion of various masses of planetary matter are presented in Fig. \\ref{Li-accr}.\n\nAn accreted mass lower than or equal to 0.6$~M_{\\oplus}$ does not have any real effect on the lithium abundance because it reduces its surface value by a factor lower than 10 percent. The mixing is not efficient enough to mix the lithium down to the destruction layers. For masses higher than 0.6$~M_{\\oplus}$, the lithium destruction becomes important and reaches a factor 3.5 for an accreted mass of 0.66$~M_{\\oplus}$ and more than a factor 100 for an accreted mass of 1$~M_{\\oplus}$.\n\nThus a small amount of planetary matter is enough to significantly\nreduce the lithium surface abundance in this type of stars. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig9.eps}\n\\caption{Lithium abundance profiles after the accretion of different masses in the model of 16 Cygni B: no accretion (blue solid line), 0.6$~M_{\\oplus}$ (green dashed line), 0.66$~M_{\\oplus}$ (red dotted line), and 1$~M_{\\oplus}$ (cyan dotted-dashed line) at the beginning of the MS.}\n\\label{Li-accr}\n\\end{center}\n\\end{figure}\n\nAfter the accretion episode in 16 Cygni B, the lithium abundance continues to decrease due to other extra mixing processes in the two stars, as discussed in the previous section. We plot in Fig. \\ref{Lit} the lithium surface abundance evolution with the assumption of an accretion episode of 0.66 $M_{\\oplus}$ (blue dotted line), which is enough to account for the observations. As the lithium observation in 16 Cygni B is only an upper limit, any accretion mass higher than this value can explain the abundance difference in the two stars. \n\n\\subsection{Beryllium }\n\nBeryllium has been detected in 16 Cygni A and B by \\cite{deliyannis00}, who found that the difference between the two abundances, if any, must be smaller than 0.2 dex. \nBeryllium is destroyed by nuclear reactions at a temperature of $T\\sim3.5~10^6 K$, which is higher than the temperature needed\nto destroy lithium. The fact that it is not depleted gives an upper limit on the accreted mass that must not lead to mixing down to its destruction region.\n\nTo characterise this effect, we plot the same figure as Fig. \\ref{Li-accr} for beryllium (see Fig. \\ref{Be-accr}). An important result is that for an accretion of 0.66$~M_{\\oplus}$ (red dotted line) beryllium is not destroyed by nuclear reactions because fingering convection does not mix the stellar matter deep enough. However, for an accretion of 1$~M_{\\oplus}$ (cyan dot-dashed line) beryllium is already reduced by a factor 5. If the beryllium abundance determination of the two components of the 16 Cygni system is confirmed, it may lead to a precise determination of the accreted mass.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{fig10.eps}\n\\caption{Beryllium abundance profiles after the accretion of different masses: no accretion (blue solid line), 0.66$~M_{\\oplus}$ (red dotted line), and 1$~M_{\\oplus}$ (cyan dot-dashed line) at the beginning of the MS.}\n\\label{Be-accr}\n\\end{center}\n\\end{figure}\n\n \n\n\\section{Summary and discussion}\n\n\nThe 16 Cygni system is particularly interesting for comparative studies of stars with and without planets. \nThe two main stars of this system have been observed in several ways, leading to very precise constraints. \n\nThe most striking feature is the lithium abundance, which is slightly higher than solar for 16 Cygni A and lower by a factor of at least 4.7 in 16 Cygni B (\\citealt{friel93,king97}). The lithium value given for this later star is an upper limit, which\nmeans that it could be completely lithium depleted.\n\nWe here presented detailed computations of these two stars, leading to models that precisely fit their asteroseismic frequency determinations, as well as their radii, luminosities, effective temperatures, and gravities. These models were computed by fully taking into account atomic diffusion of helium and heavy elements. Their characteristics are given in Table \\ref{table:2} \\footnote{\\textbf{Note added in proof:}\nAfter this paper was accepted, Travis Metcalfe draw our attention to his new article accepted for publication in ApJ \"Asteroseismic modeling of 16 Cyg A \\& B using the complete Kepler data set\" by T.S. Metcalfe, O.L. Creevey \\& G.R. Davies, arXiv:1508.00946v2. The authors revise the data given in \\cite{metcalfe12}, particularly the age ($7.0 \\pm 0.3 $ Gyr) and the composition ($Z = 0.021\\pm0.002$, $Yi = 0.25\\pm0.01$). They acknowledge that the helium content they derive may be slightly underestimated due to their neglecting atomic diffusion of heavy elements in the models. It is interesting to stress that we indeed find a slightly higher helium content and a younger age than they.}. \n\nThen we discussed the importance of the lithium observations for the two stars. We first showed that the very large lithium depletion observed in 16 Cygni B cannot be accounted for by the classical means of rotation-induced mixing or similar types of extra mixing. The observable parameter difference between them is too small to explain such a large difference in lithium as observed. We suggest that the accretion of heavy matter onto the planet host star 16 Cygni B in its early main-sequence phase induced a special kind of mixing, namely fingering convection, which led to a large lithium destruction on a relatively small timescale (cf. \\citealt{vauclair04,theado12bis}).\n\nWe recall that when a star accretes heavy matter onto lighter one, the induced $\\mu$-gradient leads to a specific kind of hydrodynamical instability, called fingering convection or thermohaline convection, which mixes all the accreted matter downwards. Contrary to what is often assumed, the accreted heavy elements do not leave any signature in the stellar outer layers because of this mixing. On the other hand, the same mixing may transport lithium down to the layers where it is destroyed by nuclear reactions. \n\nWe suggest that during the evolution of the planetary disk, at the beginning of the main-sequence phase, 16 Cygni B may have swallowed a fraction of an Earth-like planet that has led to fingering convection below the convective zone. This short-time efficient extra-mixing allowed the transport of lithium down to the nuclear destruction region. We have seen that two thirds of an Earth-mass planet would be enough to account for the observed upper value of lithium in 16 Cyg B, as shown in Fig. \\ref{Lit}. \nObservations of beryllium have been reported in these two stars by \\cite{deliyannis00}. They found no difference in beryllium\nbetween the two stars. If confirmed, this means that the accretion of planetary matter could not have exceeded one Earth mass. \n\nWe also computed the detailed variations of the heavy element abundances due to atomic diffusion along the two evolutionary tracks leading to the best models for 16 Cygni A and B. Atomic diffusion alone leads to carbon depletion by about 0.05 dex and to magnesium depletion by about 0.03 dex. The resulting abundances in the two stars are very similar, as observed by \\cite{schuler11}. Differences as suggested by \\citealt{ramirez11} and \\citealt{tuccimaia14} cannot be explained in this context. \n\nIn summary, the 16 Cygni system gives evidence of fingering convection induced by the accretion of planetary matter, which is probably the unknown process invoked in the past by several authors to account for the large lithium differences between the two stars.\n \nThis result can be generalized to all planet-hosting stars, which may accrete planetary matter in a random way, with various accretion rates. As suggested by \\citet{theado12}, this could lead to an average lithium abundance that is lower in planet-hosting stars than in other stars, as claimed by \\citet{israelian09} and \\citet{delgado14}. In this framework, precise lithium abundances in seismically observed stars with or without observed planets would be very useful. Furthermore, beryllium observations in the same stars would help constraining the mass accreted onto the star. \n\n\n\\begin{acknowledgements}\nWe thank the \"Programme National de Physique Stellaire\" (PNPS) of CNRS\/INSU (France) for financial support. We also warmly thank Piercarlo Bonifacio for very useful comments on the first version of this paper.\n\\end{acknowledgements} \n \n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFor a prime number $p$ and a field $K$, let $K(p)$ be the maximal $p$-extension of $K$ ---\nnamely, $K(p)$ is the compositum of all finite Galois $p$-extensions $L\/K$.\nThe Galois group of $K(p)\/K$, called the {\\sl maximal pro-$p$ Galois group} of $K$ and denoted by $G_K(p)$,\nis the maximal pro-$p$ quotient of the absolute Galois group $G_K$ of $K$\n(note that the class of maximal pro-$p$ Galois groups includes also absolute Galois groups which are pro-$p$).\nCharacterizing pro-$p$ groups which are realizable as maximal pro-$p$ Galois groups is one of the major problems\nin current research in Galois theory.\n\nA pro-$p$ group $G$ is said to be a {\\sl one-relator} pro-$p$ group if it has a minimal presentation \nwith only one defining relation.\nI.e., $G=F\/R$, with $F$ a free pro-$p$ group (such that $G$ and $F$ have the same minimal number of generators)\nand $R$ is generated as closed normal subgroup of $F$ by one element.\nIn \\cite{wurf} T.~W\\\"urfel proved the following characterization for absolute Galois groups which are\none-relator pro-$p$ groups: if such $G_K$ is finitely generated and $K$ contains all the roots of unity\nof order a $p$-power, then $G_K$ is a free-by-Demushkin group --- i.e., one may find a free pro-$p$ normal subgroup\n$N$ of $G_K$ such that the quotient $G_K\/N$ is a Demushkin group.\n(Demushkin groups are Poincar\\'e pro-$p$ groups of cohomological dimension two,\nand they appear as maximal pro-$p$ Galois groups of certain local fields, see Section~\\ref{ssec:demushkin}).\n\nIn order to obtain such result, W\\\"urfel used the Merkurjev-Suslin theorem (cf. \\cite{ms}), which is\nthe ``degree-2 step'' of the well-known Bloch-Kato conjecture,\nproved by M.~Rost and V.~Voevodsky (cf. \\cite[Thm.~6.4.3]{nsw:cohm}).\nIn particular, the Bloch-Kato conjecture implies that if a field $K$ contains a root of unity of order $p$,\nthen the $\\mathbb{F}_p$-cohomology ring $H^\\bullet(G_K(p),\\mathbb{F}_p)=\\bigoplus_{n\\geq0}H^n(G_K(p),\\mathbb{F}_p)$,\nendowed with the (graded-commutative) cup product\n\\[ \\cup\\colon H^r(G_K(p),\\mathbb{F}_p)\\times H^s(G_K(p),\\mathbb{F}_p)\\longrightarrow H^{r+s}(G_K(p),\\mathbb{F}_p) ,\\]\nis a {\\sl quadratic $\\mathbb{F}_p$-algebra}, i.e., such algebra is generated by elements of degree one\nand its defining relations are generated by (homogeneous) relations of degree two (cf. \\cite[\\S~2]{cq:bk}).\nOur study of maximal pro-$p$ Galois group which are finitely generated one-relator pro-$p$ groups\nstarts from the following observation.\n\n\\begin{observation}\\label{obs1}\nLet $K$ be a field containing a root of unity of order $p$ (and also $\\sqrt{-1}$ if $p=2$),\nand assume that the maximal pro-$p$ Galois group $G_K(p)$ is a finitely generated one-relator pro-$p$ group.\nThen one has the decomposition as free product of quadratic $\\mathbb{F}_p$-algebras \n\\begin{equation}\\label{eq:obsA}\n H^\\bullet(G_K(p),\\mathbb{F}_p)\\simeq\\left(\\mathbb{F}_p\\oplus V_1\\oplus H^2(G_K(p),\\mathbb{F}_p)\\right)\\sqcap(\\mathbb{F}_p\\oplus V_2),\n\\end{equation}\nwith $V_1\\oplus V_2=H^1(G_K(p),\\mathbb{F}_p)$, where the cup product induces a non-degenerate pairing\n$V_1\\times V_1\\to H^2(G_K(p),\\mathbb{F}_p)$, and $V_2=H^1(G_K(p),\\mathbb{F}_p)^\\perp$.\n\\end{observation}\n\nFor a quadratic $\\Bbbk$-algebra $A_\\bullet=\\bigoplus_{n\\geq0}A_n$ one defines the {\\sl quadratic dual} $A_\\bullet^!$\nas the quadratic $\\Bbbk$-algebra generated by the dual space $A_1^*$ of $A_1$, and whose relations are\nthe orthogonal space of the generating relations of $A_\\bullet$ (see \\cite[\\S~1.2]{PoliPosi} and Definition~\\ref{defi:algebras}).\nIt is natural to ask what is the quadratic dual of the $\\mathbb{F}_p$-cohomology ring of a maximal pro-$p$ Galois group:\nfor maximal pro-$p$ Galois groups which are finitely generated one-relator pro-$p$ groups we prove the following.\n\n\\begin{theorem}\\label{thm2}\nLet $K$ be a field containing a root of unity of order $p$ such that $G_K(p)$ is a finitely generated\none-relator pro-$p$ group.\nThe quadratic dual of the $\\mathbb{F}_p$-cohomology ring of $G_K(p)$ is the graded group algebra \n\\[{\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])=\\bigoplus_{n\\geq0}I^n\/I^{n+1},\\qquad \\text{with }I^0=\\mathbb{F}_p[\\![ G_K(p)]\\!],\\]\nwhere $I$ is the augmentation ideal of the complete group algebra $\\mathbb{F}_p[\\![ G_K(p)]\\!]$. \nAlso, such graded algebra splits as free product \n\\begin{equation}\\label{eq:thmB}\n {\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])\\simeq D_\\bullet\\sqcup\\F_p\\langle X\\rangle,\n\\end{equation}\nwhere $D_\\bullet$ is a quadratic Demushkin algebra and $\\F_p\\langle X\\rangle$ is the free (graded) associative algebra\non a set of indeterminates $X=\\{X_1,\\ldots,X_m\\}$.\n\\end{theorem}\n\n\nDemushkin algebras are one-relator (non-commutative) polynomial algebras related to Demushkin groups\n(to be defined in Section~\\ref{ssec:demushkin}).\nIn \\cite{MQRTW} it is conjectured that the graded group algebra ${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])$ is the quadratic dual \nof the $\\mathbb{F}_p$-cohomology ring $H^\\bullet(G_K(p),\\mathbb{F}_p)$ for every field $K$ containing a root of unity of order $p$\nsuch that $G_K(p)$ is finitely generated.\nThus, Theorem~\\ref{thm2} provides a positive answer in the case of one-relator maximal pro-$p$ Galois groups.\n\nNote that by \\eqref{eq:obsA} and \\eqref{eq:thmB} the two algebras $H^\\bullet(G_K(p),\\mathbb{F}_p)$\nand ${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])$ split in a ``Demushkin part'' and in a ``free part''.\nIn fact, we prove a result for maximal pro-$p$ Galois groups which is analogous to W\\\"urfel's one\n(cf. Theorem~\\ref{thm:wurfel}).\n\nFor a graded $\\Bbbk$-algebra $A_\\bullet$, one defines the $\\Bbbk$-cohomology of $A_\\bullet$ as the derived\nfunctor of $\\mathrm{Hom}_{A_\\bullet}(\\Bbbk,\\Bbbk)$, which is endowed with an additional internal grading --- induced \nby the grading of $A_\\bullet$ ---, so that one gets a bigraded algebra.\nIf the non-tivial cohomology groups are concentrated on the diagonal,\n$A_\\bullet$ is called a {\\sl Koszul $\\Bbbk$-algebra} (cf. \\cite[\\S~1.1, \\S~2.1]{PoliPosi}).\nKoszul algebras are rather mysterious. \nThey were introduced by S.~Priddy in \\cite{priddy}, and they arise in various fields of mathematics,\nsuch as representation theory, noncommutative algebra, noncommutative geometry, algebraic geometry and topology.\n\nKoszul algebras have been studied in\nthe context of Galois cohomology by L.~Positselski and A.~Vishik (cf. \\cite{posivis:koszul}).\nIn particular, Positselski conjectured that if a field $K$ contains a root of unity of order $p$,\nthen the $\\mathbb{F}_p$-cohomology algebra of $G_K(p)$ is Koszul, and he showed that such conjecture is\na generalization of the Bloch-Kato conjecture (cf. \\cite{positselsky:koszul1,positselsky:koszul2}).\nMore recently, Koszul algebras have been studied in the context of Lie algebras and pro-$p$ groups by \nT.~Weigel (cf. \\cite{thomas:KoszulLie,thomas:proc}).\nIn particular, Weigel conjectured that under the same assumptions on the field $K$, the graded group algebra\n${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])$ is Koszul (cf. \\cite[Question~1]{thomas:proc},\n\nFrom Observation~\\ref{obs1} and Theorem~\\ref{thm2} we obtain a positive answer to Positselski's and Weigel's conjectures\nin the case of one-relator maximal pro-$p$ Galois groups.\n\n\\begin{theorem}\n\\label{thm3}\n Let $K$ be a field satisfying the assumptions of Theorem~\\ref{thm2}.\nThen the algebras $H^\\bullet(G_K(p),\\mathbb{F}_p)$ and ${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])$ are Koszul.\n\\end{theorem}\n\nFinally, we observe that there are finitely generated free-by-Demushkin pro-$p$ groups for which\nthe results of the present paper do hold, but which are not realizable\nas maximal pro-$p$ Galois groups of fields (cf. Example~\\ref{ex:KZ}).\nWe ask whether one may find further conditions that the defining relation of such pro-$p$ groups\nshould satisfy to make them Galois --- or obstructions which prevent them to be maximal pro-$p$ Galois groups.\n\n\n\n\\section{Quadratic algebras and Koszul algebras}\n\\label{sec:2}\n\n\\subsection{Graded algebras and Koszul algebras}\n\\label{ssec:graded}\n\nHenceforth every graded algebra $A_\\bullet$ over a field $\\Bbbk$ is assumed to be\nunitary associative and non-negatively graded of finite-type,\ni.e., $A_0=\\Bbbk$, $A_n=0$ for $n<0$ and $\\dim(A_n)<\\infty$ for $n\\geq1$.\n\nLet consider $\\Bbbk$ as a $A_\\bullet$-module via the augmentation epimorphism $\\varepsilon\\colon A_\\bullet\\to\\Bbbk$.\nThen one may define the extension spaces $\\mathrm{Ext}_{A_\\bullet}^n(\\Bbbk,\\Bbbk)$, $n\\geq0$, as derived functor of the\nfunctor $\\mathrm{Hom}_{A_\\bullet}(\\Bbbk,\\Bbbk)$ in the usual sense.\nThe cup product (or Yoneda product) induces a graded algebra structure on \n$H^\\bullet(A_\\bullet)=\\bigoplus_{n\\geq0}\\mathrm{Ext}_{A_\\bullet}^n(\\Bbbk,\\Bbbk)$\n(cf. \\cite[\\S~1]{priddy} and \\cite[\\S~1.1]{PoliPosi}).\n\nThe grading of $A_\\bullet$ induces an additional internal grading, so that \n$H^\\bullet(A_\\bullet)$ may be viewed as a bigraded $\\Bbbk$-algebra\n\\[H^\\bullet(A_\\bullet)=\\bigoplus_{i,j\\geq0}\\mathrm{Ext}_{A_\\bullet}^{i,j}(\\Bbbk,\\Bbbk),\\]\nwhere $\\mathrm{Ext}_{A_\\bullet}^{i,j}(\\Bbbk,\\Bbbk)=0$ for $i>j$ (cf. \\cite[Prop.~1.3.1]{PoliPosi}).\nA graded $\\Bbbk$-algebra is called a {\\sl Koszul} algebra if its extension spaces are cocentrated on the \ndiagonal, i.e., if $\\mathrm{Ext}_{A_\\bullet}^{i,j}(\\Bbbk,\\Bbbk)=0$ for $i\\neq j$, so that in fact\n$H^\\bullet(A_\\bullet)=\\bigoplus_{i\\geq0}\\mathrm{Ext}_{A_\\bullet}^{i,i}(\\Bbbk,\\Bbbk)$. \nFor further details on the cohomology of graded algebras we refer to \\cite[\\S~1-2]{priddy}, \\cite[\\S~1.1-1.5]{PoliPosi}\nand \\cite[\\S~2]{positselsky:koszul1}.\n\n\n\\subsection{Quadratic algebras}\n\\label{ssec:quadratic}\n\nFor a vector space $V$ over $\\Bbbk$ (of finite dimension), let $T_\\bullet(V)$ denote\nthe graded tensor $\\Bbbk$-algebra generated by $V$. I.e.,\n \\[T_\\bullet(V)=\\bigoplus_{n\\geq0}V^{\\otimes n},\\qquad \\text{with }V^{\\otimes0}=\\Bbbk,\\]\nendowed with the multiplication induced by the tensor product.\nMoreover, let $V^*=\\mathrm{Hom}_{\\Bbbk}(V,\\Bbbk)$ be the $\\Bbbk$-dual space of $V$.\nSince $V$ has finite dimension, one may identify $(V\\otimes V)^*=V^*\\otimes V^*$.\n\n\\begin{definition}\\label{defi:algebras}\n\\begin{itemize}\n \\item[(i)] A graded $\\Bbbk$-algebra $A_\\bullet$ is said to be {\\sl quadratic} if $A_\\bullet$ is a quotient \n$T_\\bullet(A_1)\/\\langle \\Omega\\rangle$, where $\\langle \\Omega\\rangle$ is the two-sided ideal\nof $T_\\bullet(A_1)$ generated by a subset $\\Omega\\subseteq A_1\\otimes A_1$.\n \\item[(ii)] For a graded $\\Bbbk$-algebra $A_\\bullet$, the {\\sl quadratic part} of $A_\\bullet$,\ndenoted by $\\mathrm{q}A_\\bullet$, is the quadratic $\\Bbbk$-algebra such that $\\mathrm{q}A_1=A_1$\nand $\\mathrm{q}A_2$ embeds in $A_2$.\n \\item[(iii)] For a quadratic $\\Bbbk$-algebra $A_\\bullet=T_\\bullet(A_1)\/\\langle \\Omega\\rangle$, let\n$\\Omega^\\perp\\subseteq (A_1\\otimes A_1)^*$ denote the orthogonal space of $\\Omega$, i.e.,\n\\[ \\Omega^\\perp=\\{\\alpha\\in(A_1\\otimes A_1)^*\\mid \\alpha(\\omega)=0\\text{ for all }\\omega\\in \\Omega\\}.\\]\nThus, we may consider $\\Omega^\\perp$ as a subspace of $A_1^*\\otimes A_1^*$.\nThe {\\sl quadratic dual} of $A_\\bullet$, denoted by $A_\\bullet^!$, is the quadratic $\\Bbbk$-algebra\nobtained via the quotient $T_\\bullet(A_1^*)\/\\langle \\Omega^\\perp\\rangle$.\n\\end{itemize}\n\\end{definition}\n\nIn particular, for a quadratic $\\Bbbk$-algebra $A_\\bullet$ one has $(A_\\bullet^!)^!=A_\\bullet$.\nMoreover, one has the following (cf. \\cite[Prop.~1.3.1 and Definition~2.1]{PoliPosi}).\n\n\\begin{proposition}\\label{prop:koszul}\nlet $A_\\bullet$ be a graded $\\Bbbk$-algebra.\n\\begin{itemize}\n \\item[(i)] The diagonal of $H^\\bullet(A_\\bullet)$ is isomorphic to the quadratic dual of $\\mathrm{q}A_\\bullet$,\ni.e., one has an isomorphism $\\bigoplus_{i\\geq0}\\mathrm{Ext}_{A_\\bullet}^{i,i}(\\Bbbk,\\Bbbk)\\simeq(\\mathrm{q}A_\\bullet)^!$.\n \\item[(ii)] If $A_\\bullet$ is Koszul, then it is quadratic.\n\\end{itemize}\n\\end{proposition}\n\nThe above proposition implies that a quadratic $\\Bbbk$-algebra $A_\\bullet$\nis Koszul if, and only if, one has $H^\\bullet(A_\\bullet)\\simeq A_\\bullet^!$.\nMoreover, a quadratic $\\Bbbk$-algebra $A_\\bullet$ is Koszul if, and only if, the quadratic dual \n$A_\\bullet^!$ is Koszul (cf. \\cite[Prop.~3.4.8]{lodval}).\nFor further details on Koszul algebras we refer to \\cite[Ch.~2]{PoliPosi}.\n\nOne has the following examples of Koszul $\\Bbbk$-algebras (cf. \\cite[Ex.s~3.2.5 and Ex.s~3.4.12]{lodval}).\n\n\\begin{example}\\label{exs:quadratic}\nLet $V$ be a vector space of finite dimension over $\\Bbbk$.\n \\begin{itemize}\n \\item[(a)] The tensor $\\Bbbk$-algebra $T_\\bullet(V)$ and the trivial quadratic $\\Bbbk$-algebra $\\Bbbk\\oplus V$\n(with $V$ the part of degree 1) are Koszul algebras, and $T_\\bullet(V)^!=\\mathbb{F}_p\\oplus V^*$ and conversely.\n \\item[(b)] The symmetric algebra $S_\\bullet(V)$ and the exterior algebra $\\Lambda_\\bullet(V)$\nare Koszul algebras, and $S_\\bullet(V)^!=\\Lambda_\\bullet(V^*)$ and conversely.\n \\end{itemize}\n\\end{example}\n\nGiven two quadratic $\\Bbbk$-algebras $A_\\bullet$ and $B_\\bullet$, the direct product and the free product\nof $A_\\bullet$ and $B_\\bullet$ are defined in the following way.\n\\begin{itemize}\n \\item[(a)] The direct product of $A_\\bullet$ and $B_\\bullet$ is the quadratic $\\mathbb{F}_p$-algebra\n$C_\\bullet=A_\\bullet\\sqcap B_\\bullet$ with $C_n=A_n\\oplus B_n$ for every $n\\geq1$.\n \\item[(b)] The free product of $A_\\bullet$ and $B_\\bullet$ is the quadratic $\\mathbb{F}_p$-algebra\n$C_\\bullet=A_\\bullet\\sqcup B_\\bullet$ with $C_1=A_1\\oplus B_1$ and \n$C_2=A_2\\oplus B_2\\oplus(A_1\\otimes B_1)\\oplus(B_1\\otimes A_1)$.\n\\end{itemize}\n\nFor such algebras one has the following (cf. \\cite[\\S~3.1]{PoliPosi}).\n\n\\begin{proposition}\\label{prop:products}\n Let $A_\\bullet$ and $B_\\bullet$ be quadratic $\\Bbbk$-algebras.\nIf both $A_\\bullet$ and $B_\\bullet$ are Koszul algebras, then also the direct product $A_\\bullet\\sqcap B_\\bullet$\nand the free product $A_\\bullet\\sqcup B_\\bullet$ are Koszul algebras.\nMoreover, one has\n\\begin{equation}\\label{eq:oproducts duals}\n (A_\\bullet\\sqcap B_\\bullet )^!=A_\\bullet^!\\sqcup B_\\bullet^! \\qquad \\text{and}\\qquad\n (A_\\bullet\\sqcup B_\\bullet)^!=A_\\bullet^!\\sqcap B_\\bullet^!.\n\\end{equation}\n\\end{proposition}\n\n\n\\subsection{Demushkin algebras}\n\\label{ssec:demushkin}\n\nFor a vector space $V$ over $\\Bbbk$ of finite dimension $d$, let $ X =\\{X_1,\\ldots,X_d\\}$ be\na set of (non-commutative) indeterminates.\nThe free associative algebra $\\Bbbk\\langle X\\rangle$ --- i.e., the $\\Bbbk$-algebra of polynomials on the\nnon commutative indeterminates $X$ --- comes endowed with the grading induced\nby the subspaces of homogeneous polynomials.\nWe may identify $ X $ with a fixed basis of $V$, and such identification induces an isomorphism\nof quadratic $\\Bbbk$-algebras $T_\\bullet(V)\\simeq\\Bbbk\\langle X\\rangle$.\n\n\\begin{lemma}\\label{lemma:onerel koszul}\n Let $f\\in\\Bbbk\\langle X\\rangle$ be a homogeneous polynomial of degree 2 such that \n$f\\neq X_1^{2\\delta_1}+\\ldots+X_d^{2\\delta_d}$, with $\\delta_i\\in\\{0,1\\}$ for every $i=1,\\ldots,d$\n--- i.e., $f$ contains a monomial $\\alpha X_iX_j$ with $\\alpha\\in\\Bbbk\\smallsetminus\\{0\\}$ and $i\\neq j$.\nThen the quadratic $\\Bbbk$-algebra $\\Bbbk\\langle X\\rangle\/\\langle f\\rangle$ is Koszul.\n\\end{lemma}\n\n\\begin{proof}\n Up to renumbering of $X$ we may assume that $f$ contains the monomial $X_dX_{d-1}$.\nFor every $n\\geq1$ let the monomial basis $\\mathcal{S}_n=\\{X_{i_1}\\cdots X_{i_n},1\\leq i_1,\\ldots,i_n\\leq d\\}$\nbe endowed with the lexicographic ordering $\\prec$.\nThen $(\\bigcup_n\\mathcal{S}_n,\\prec)$ is a totally ordered set.\nIn particular, we may write the relation $f=0$ as\n\\[X_dX_{d-1}=\\text{smaller terms w.r.t. }\\prec.\\]\n\nIn a quadratic $\\Bbbk$-algebra $A_\\bullet=\\Bbbk\\langle X\\rangle\/\\langle \\Omega\\rangle$, \nwith $\\Omega$ a set of homogeneous polynomials of degree 2,\na monomial $h=X_{i_1}X_{i_2}X_{i_3}\\in\\mathcal{S}_3$ is said to be {\\sl critical}\nif both $X_{i_1}X_{i_2}$ and $X_{i_2}X_{i_3}$ are leading terms (with respect to $\\prec$) of some relations in $\\Omega$\n(c.f. \\cite[\\S~4.1]{lodval}) --- i.e., the monomial $h$ may be rewritten in $A_\\bullet$ using lower monomials\nwith respect to $\\prec$ in at least two ways, which may eventually lead to different results.\n \nThe $\\Bbbk$-algebra $\\Bbbk\\langle X\\rangle\/\\langle f\\rangle$ has no critical monomials, as one can not have\n$X_{i_1}X_{i_2}=X_{i_2}X_{i_3}=X_dX_{d-1}$ for any $i_1,i_2,i_3\\in\\{1,\\ldots,d\\}$,\nand no monomials can be rewritten using lower monomials.\nBy \\cite[Thm.~4.1.1]{lodval}, quadratic $\\Bbbk$-algebras without critical monomials are Koszul,\nand so is $\\Bbbk\\langle X\\rangle\/\\langle f\\rangle$.\n\\end{proof}\n\nFix a prime number $p$.\nHenceforth we shall concentrate on graded algebras over the finite field $\\mathbb{F}_p$.\nA finitely generated pro-$p$ group $G$ is called a {\\sl Demushkin group} if $H^2(G,\\mathbb{F}_p)\\simeq\\mathbb{F}_p$\nand the cup product induces a non-degenerate $\\mathbb{F}_p$-pairing \n\\[H^1(G,\\mathbb{F}_p)\\times H^1(G,\\mathbb{F}_p)\\longrightarrow H^2(G,\\mathbb{F}_p)\\simeq\\mathbb{F}_p.\\]\nDemushkin groups are one-relator pro-$p$ groups, and their structure was fully described by S.P.~Demushkin,\nJ.-P.~Serre and J.P.~Labute (cf. \\cite[\\S~III.9]{nsw:cohm}).\nMoreover, they appear in Galois theory as the maximal pro-$p$ Galois groups of certain $p$-adic local fields\n(cf. \\cite[Thm.~7.5.11]{nsw:cohm}).\n\n\\begin{definition}\\label{defi:demushkin}\nA {\\sl Demushkin $\\mathbb{F}_p$-algebra} $D_\\bullet$ is a quadratic $\\mathbb{F}_p$-algebra such that its diagonal\n$\\mathbb{F}_p$-cohomology is isomorphic to the $\\mathbb{F}_p$-cohomology ring of a Demushkin group. \nI.e., $\\mathrm{Ext}_{D_\\bullet}^{1,1}\\simeq D_1^*$,\n$\\mathrm{Ext}_{D_\\bullet}^{2,2}(\\mathbb{F}_p,\\mathbb{F}_p)\\simeq\\mathbb{F}_p$ and the cup product induces a non-degenerate $\\mathbb{F}_p$-pairing \n\\[\\mathrm{Ext}_{D_\\bullet}^{1,1}(\\mathbb{F}_p,\\mathbb{F}_p)\\times\\mathrm{Ext}_{D_\\bullet}^{1,1}(\\mathbb{F}_p,\\mathbb{F}_p)\\longrightarrow \\mathbb{F}_p.\\]\n \\end{definition}\n\nBy Proposition~\\ref{prop:koszul} the diagonal cohomology of a Demushkin $\\mathbb{F}_p$-algebra is a \nquadratic $\\mathbb{F}_p$-algebra $A_\\bullet$ with $\\dim(A_1)=\\dim(D_1)=d$, with $d$ even if $p$ is odd.\nIn particular, there is a basis $\\{a_1\\,\\ldots,a_d\\}$ of $A_1$ such that \n\\begin{equation}\\label{eq:rel cohomology demushkin 1}\n a_1a_2=-a_2a_1=a_3a_4=-a_4a_3=\\ldots=a_{d-1}a_d=-a_da_{d-1}\\neq0\n\\end{equation}\n(and possibly $a_1^2=a_1a_2$ if $p=2$), and $a_ia_j=0$ in any other case, if $d$ is even; and\n\\begin{equation}\\label{eq:rel cohomology demushkin 2}\n a_1^2=a_2a_3=a_3a_2=a_4a_5=a_5a_4=\\ldots=a_{d-1}a_d=a_da_{d-1}\\neq0\n\\end{equation}\nand $a_ia_j=0$ in any other case, if $d$ is even (and thus necessarily $p=2$).\nDemushkin $\\mathbb{F}_p$-algebras are described by the following.\n\n\\begin{proposition}\\label{prop:demushkin shape}\nFor a Demushkin algebra $D_\\bullet$ set $d=\\dim(D_1)$.\nOne may present $D_\\bullet$ as a quotient $D_\\bullet=\\F_p\\langle X\\rangle\/\\langle f\\rangle$,\nwith $X=\\{X_1,\\ldots,X_d\\}$ non-commutative indeterminates and $f$ a homogeneous polynomial of degree two, such that:\n\\begin{itemize}\n \\item[(a)] if $p$ is odd then $d$ is necessarily even and \\[f=[X_1,X_2]+[X_3,X_4]+\\ldots+[X_{d-1},X_d];\\]\n \\item[(b)] if $p=2$ and $d$ is even then either $f$ is as above or \n\\[f=X_1^2+[X_1,X_2]+[X_3,X_4]+\\ldots+[X_{d-1},X_d];\\]\n \\item[(c)] if $p=2$ and $d$ is even then \\[f=X_1^2+[X_2,X_3]+[X_4,X_5]+\\ldots+[X_{d-1},X_d].\\]\n\\end{itemize}\n(Here $[X_i,X_j]$ denotes the commutator polynomial $X_iX_j-X_jX_i$.)\n\\end{proposition}\n\n\\begin{lemma}\\label{lemma:demushkin dual}\nLet $D_\\bullet$ be a quadratic $\\mathbb{F}_p$-algebra described in Proposition~\\ref{prop:demushkin shape}.\nThen the quadratic dual $D_\\bullet^!$ is a quadratic $\\mathbb{F}_p$-algebra $A_\\bullet$ described\nby \\eqref{eq:rel cohomology demushkin 1} or \\eqref{eq:rel cohomology demushkin 2}.\n\\end{lemma}\n\n\\begin{proof}\nFix a basis $\\{a_1\\,\\ldots,a_d\\}$ of $D_1^!=D_1^*$ which is dual to $X$ --- i.e., $a_i(X_j)=\\delta_{ij}$.\nLet $\\Omega\\subseteq D_1\\otimes D_1$ be the subspace generated by the homogeneous polynomial $f$.\nThen \n\\begin{equation}\\label{eq:dim omega}\n \\dim(\\Omega)=\\dim(A_1\\otimes A_1)-\\dim(\\Omega^\\perp)=\\dim(A_2)=1.\n\\end{equation}\nThe only (monic) monomials of $D_1^!\\otimes D_1^!$ whose evaluation on $f$ is not trivial are $a_{2h-1}a_{2h}$\nand $a_{2h}a_{2h-1}$, with $1\\leq h\\leq d\/2$, if $d$ is even (and possibly also $a_1^2$ if the monomial\n$X_1^2$ appears in $f$),\nor $a_1^2$, $a_{2h}a_{2h+1}$ and $a_{2h+1}a_{2h}$, with $1\\leq h\\leq (d-1)\/2$ if $d$ is odd. \nTherefore, \\eqref{eq:dim omega} implies the relations in $D_2^!$\n\\[ a_{2h-1}a_{2h}=-a_{2h}a_{2h-1}\\neq0,\\qquad1\\leq h\\leq \\frac{d}{2},\\]\nand $a_ia_j=0$ else, if $d$ is even (and possibly $a_1^2$ equal to the above monomials), and\n\\[ a_1^2=a_{2h}a_{2h+1}=a_{2h+1}a_{2h}\\neq0,\\qquad1\\leq h\\leq \\frac{d-1}{2},\\]\nand $a_ia_j=0$ else, if $d$ is odd.\n\nThese are the same relations as in \\eqref{eq:rel cohomology demushkin 1} and \\eqref{eq:rel cohomology demushkin 2},\nand the claim follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:demushkin shape}]\nLet $D_\\bullet$ be a Demushkin $\\mathbb{F}_p$-algebra, and let $A_\\bullet$ be its diagonal cohomology.\nThen by Proposition~\\ref{prop:koszul} one has $D_\\bullet=(D_\\bullet^!)^!=A_\\bullet^!$, and the claim follows by\nLemma~\\ref{lemma:demushkin dual}.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:demushkin algebra koszul}\n A Demushkin $\\mathbb{F}_p$-algebra $D_\\bullet$ is Koszul.\n\\end{proposition}\n\n\\begin{proof}\nAssume first that $p=2$ and $\\dim(D_1)=1$.\nThen $D_\\bullet=\\mathbb{F}_p[X_1]\/\\langle X_1^2\\rangle$, i.e., $D_\\bullet$ is a trivial quadratic $\\mathbb{F}_2$-algebra,\nand such algebra is Koszul by Example~\\ref{exs:quadratic}.\nIf $\\dim(D_1)>1$ then the claim follows by Proposition~\\ref{prop:demushkin shape} and Lemma~\\ref{lemma:onerel koszul}.\n\\end{proof}\n\n\\section{Pro-$p$ groups and graded algebras}\n\\label{sec:prop}\n\n\\subsection{The Zassenhaus filtration}\n\\label{ssec:zassenhaus}\n\nHenceforth, subgroups of pro-$p$ groups are assumed to be closed (in the pro-$p$ topology),\nand every generating set is to be intended in the topological sense.\nIn particular, given two (closed) subgroups $C_1$ and $C_2$ of a pro-$p$ group $G$,\nthe subgroup $[C_1,C_2]$ is the (closed) subgroup of $G$ generated by the commutators\n\\[ [x,y] = (y^{-1})^x\\cdot y = x^{-1}y^{-1}xy , \\qquad x\\in C_1 , y\\in C_2 .\\]\nMoreover, for a closed subgroup $C$ of $G$ and $n\\geq1$, $C^n$ denotes the (closed) subgroup of $G$ generated by\nthe elements $g^n$, with $g\\in C$.\n\nFor a pro-$p$ group $G$ let $\\F_p\\dbl G\\dbr$ be the {\\sl complete} group algebra.\nI.e., \\[\\F_p\\dbl G\\dbr=\\varprojlim_{U\\vartriangleleft_o G}\\mathbb{F}_p[G\/U],\\]\nwith $U$ running through the set of open normal subgroups of $G$. \nThe augmentation ideal $I\\subseteq\\F_p\\dbl G\\dbr$ is the kernel of the augmentation map $\\F_p\\dbl G\\dbr\\to\\mathbb{F}_p$,\ngiven by $g\\mapsto1$ for every $g\\in G$.\nThus, one defines the {\\sl Zassenhaus filtration} of $G$ as follows.\n\n\\begin{definition}\\label{defi:grG and zassenhaus}\nThe Zassenhaus filtration $G_{(n)}$, $n\\geq1$, is the filtration of normal subgroups of $G$ defined by \n$G_{(n)}=\\{g\\in G\\mid g-1\\in I^n\\}$ for every $n\\geq1$.\n\\end{definition}\n\nThe Zassenhaus filtration of a pro-$p$ group $G$ is the fastest filtration starting at $G$ with the following property:\nfor every $x\\in G_{(n)}$ and $y\\in G_{m}$, $n,m\\geq1$, one has\n\\begin{equation}\\label{eq:properties Zassenhaus}\n [x,y]\\in G_{(n+m)} \\qquad\\text{and}\\qquad x^p\\in G_{(np)}\n\\end{equation}\nIn particular, one has $G_{(1)}=G$, $G_{(2)}=G^p[G,G]$ --- i.e., $G_{(2)}$ is the Frattini subgroup of $G$ ---, and \n\\begin{equation}\\label{eq:G3}\n G_{(3)}=\\left\\{\\begin{array}{cc} G^p[[G,G],G] & \\text{if }p\\neq2 \\\\ G^4[G,G]^2[[G,G],G] & \\text{if }p=2\\end{array}\\right.\n\\end{equation}\n(cf. \\cite[\\S~11.1]{ddsms:analytic}).\n\n\\begin{remark}\\label{rem:free1}\n Let $F$ be a finitely generated pro-$p$ group, and let $\\{x_1,\\ldots,x_d\\}$ be a minimal set of generators for $F$.\nMoreover, let $\\F_p\\dbml X\\dbmr$ be the $\\mathbb{F}_p$-algebra of formal power series on the set of indeterminates $X=\\{X_1,\\ldots,X_d\\}$.\nThen one has an isomorphism of (topological) $\\mathbb{F}_p$-algebras $\\mathbb{F}_p[\\![ F]\\!]\\simeq\\F_p\\dbml X\\dbmr$,\ncalled the {\\sl Magnus morphism}, given by $x_i\\mapsto 1+X_i$ for every $i=1,\\ldots,d$.\nThe Magnus morphism maps the augmentation ideal of $\\mathbb{F}_p[\\![ F]\\!]$ onto the two-sided ideal\n\\[I(X)=\\langle X_1,\\ldots,X_d\\rangle=\\bigoplus_{i=1}^d\\F_p\\dbml X\\dbmr\\cdot X_i.\\]\nThus, we may identify the graded algebra ${\\rm gr}_\\bullet(\\F_p\\dbml X\\dbmr)=\\bigoplus_{n\\geq0}I(X)^n\/I(X)^{n+1}$, with $I(X)^0=\\F_p\\dbml X\\dbmr$,\nwith the free polynomial algebra $\\F_p\\langle X\\rangle$, endowed with the grading induced by the subspaces of homogeneous polynomials.\nAlso, via this identification one has $F_{(n)}=\\{g\\in F\\mid g-1\\in I(X)^n\\}$ for every $n\\geq1$.\n\\end{remark}\n\n\n\\subsection{Pro-$p$ groups and restricted Lie algebras}\n\\label{ssec:restricted}\n\nGiven an associative $\\mathbb{F}_p$-algebra $R$, let $R_L$ denote the Lie $\\mathbb{F}_p$-algebra endowed with the Lie brackets\n$(a,b)=ab-ba$ for every $a,b\\in R$.\nOne has the following definition (cf. \\cite[\\S~12.1]{ddsms:analytic}).\n\n\\begin{definition}\\label{defi:restricted}\n\\begin{itemize}\n \\item[(i)] A {\\sl restricted} Lie $\\mathbb{F}_p$-algebra $L$ over $\\mathbb{F}_p$ is a Lie $\\mathbb{F}_p$-algebra equipped with\na $p$-power map $[p]\\colon L\\longrightarrow L$ which is compatible with the Lie brackets.\nI.e., there exists an associative $\\mathbb{F}_p$-algebra $R$ and a monomorphism of Lie algebras\n$\\theta\\colon L\\to R_L$ such that $\\theta(v^{[p]})=\\theta(v)^p$ for all $v\\in L$.\n \\item[(ii)] The algebra $R$, endowed with the monomorphism $\\theta\\colon L\\to R_L$, is called the\n{\\sl universal restricted envelope} of $L$ if for any morphism of restricted Lie algebras $\\varphi\\colon L\\to B_L$,\nwith $B$ an associative $\\mathbb{F}_p$-algebra, there exists a unique homomorphism of associative $\\mathbb{F}_p$-algebras\n$\\tilde\\varphi\\colon R\\to B$ such that $\\varphi=\\tilde\\varphi\\circ\\theta$.\n\\end{itemize}\n\\end{definition}\n\nWe shall denote the universal restricted envelope of a restricted Lie $\\mathbb{F}_p$-algebra $L$ by ${\\mathcal U}(L)$.\nA restricted ideal $\\mathfrak{r}$ of a restricted Lie $\\mathbb{F}_p$-algebra is an ideal in the classic sense\nwith the further condition that $v^{[p]}\\in\\mathfrak{r}$ for every $v\\in\\mathfrak{r}$.\nThen one has the following (cf. \\cite[Prop.~2.1]{jochen:massey})\n\n\\begin{proposition}\\label{prop:presentation U}\nLet $L$ be a restricted Lie algebra over $\\mathbb{F}_p$ and let $\\mathfrak{r}\\subseteq L$ be a restricted ideal.\nLet $\\mathcal{R}$ be the left ideal of ${\\mathcal U}(L)$ generated by the image of $\\mathfrak{r}$\nvia the embedding $\\theta\\colon L\\to{\\mathcal U}(L)$.\nThen $\\mathcal{R}$ is a two-sided ideal and the epimorphism $L\\to L\/\\mathfrak{r}$ induces a short exact sequence\n\\begin{equation}\\label{eq:ses U}\n \\xymatrix{ 0\\ar[r] & \\mathcal{R}\\ar[r] & {\\mathcal U}(L)\\ar[r] & {\\mathcal U}(L\/\\mathfrak{r})\\ar[r] & 0 }\n\\end{equation}\n\\end{proposition}\n\nFor a pro-$p$ group $G$ let $L(G)$ be the graded object \\[L(G)=\\bigoplus_{n\\geq1}G_{(n)}\/G_{(n+1)}.\\]\nBy \\eqref{eq:properties Zassenhaus} every quotient $G_{(n)}\/G_{(n+1)}$ is a $\\mathbb{F}_p$-vector space, and \nthe commutators and the $p$-power induce the structure of graded restricted Lie algebra on $L(G)$.\nMoreover, the universal restricted envelope of $L(G)$ is the graded $\\mathbb{F}_p$-algebra ${\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)$\n(cf. \\cite[Thm.~12.8]{ddsms:analytic}).\nIf $g\\in G_{(n)}\\smallsetminus G_{(n+1)}$, $n\\geq1$, we shall call the image of $g$ in $G_{(n)}\/G_{(n+1)}$\nthe {\\sl initial form} of $g$ in $L(G)$.\n\n\\begin{remark}\\label{rem:free2}\nIf $F$ is a finitely generated free pro-$p$ group, with minimal generating set $\\{x_1,\\ldots,x_d\\}$,\nwe may identify the universal restricted envelope ${\\mathcal U}(L(F))$ --- and therefore the graded group algebra\n${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ F]\\!])$ --- with the free polynomial algebra $\\F_p\\langle X\\rangle$, with $X=\\{X_1,\\ldots,X_d\\}$.\nIn particular, the restricted Lie $\\mathbb{F}_p$-algebra $L(F)$ is the free restricted Lie $\\mathbb{F}_p$-algebra on $X$\n(cf. \\cite[Remark~2.3]{jochen:massey}).\n\\end{remark}\n\nLet $G=F\/R$ be a finitely generated pro-$p$ group, with $R$ a normal subgroup of $F$.\nThen the restricted Lie $\\mathbb{F}_p$-algebra $L(G)$ is a quotient $L(F)\/\\mathfrak{r}$, with $\\mathfrak{r}$\na restricted ideal of $L(F)$, and thus by Proposition~\\ref{prop:presentation U} the graded $\\mathbb{F}_p$-algebra\n${\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)$ is isomorphic to a quotient $\\F_p\\langle X\\rangle\/\\mathcal{R}$, with $\\mathcal{R}$ the two-sided\nideal of $\\F_p\\langle X\\rangle$ generated by $\\mathfrak{r}$.\n\n\\subsection{Mild pro-$p$ groups}\n\\label{ssec:mild}\n\nMoreover, such a pro-$p$ group $G$ is called {\\sl mild} (with respect to the Zassenhaus filtration)\nif there is a minimal set of defining relations $\\{r_1,\\ldots,r_m\\}$ (i.e., a minimal subset of $F$ \nwhich generates $R$ as normal subgroup of $F$) which is a {\\sl strongly free} sequence (with respect\nto the Zassenhaus filtration), see \\cite[Definition~2.7 and Definition~2.11]{jochen:massey}.\nMild pro-$p$ groups were introduced by J.~Labute in \\cite{labute:mild} to study the Galois groups\nof the extensions of number fields with restricted ramification.\nMild pro-$p$ groups have the following properties (cf. \\cite[Thm.~2.12]{jochen:massey}). \n\n\\begin{proposition}\\label{prop:mild}\n Let $G$ be a mild pro-$p$ group, with a strongly free minimal set of defining relations $\\{r_1,\\ldots,r_m\\}$.\nThen\n\\begin{itemize}\n \\item[(i)] one has $H^n(G)=0$ for all $n>2$;\n \\item[(ii)] one has an isomorphism of graded $\\mathbb{F}_p$-algebras \n\\begin{equation}\\label{eq:iso mild}\n {\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)\\simeq \\F_p\\langle X\\rangle\/\\mathcal{R},\n\\end{equation}\nwhere $\\mathcal{R}$ denotes the two-sided ideal generated by the initial forms of $r_i$'s.\n\\end{itemize}\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\\section{One-relator pro-$p$ groups}\n\\label{sec:onerel}\n\n\\subsection{Cohomology of one-relator pro-$p$ groups}\n\\label{ssec:cohom onerel}\n\nFor a pro-$p$ group $G$ we shall denote the $\\mathbb{F}_p$-cohomology groups simply by $H^n(G)$ for every $n\\geq0$.\nIf $G$ is a finitely generated one-relator pro-$p$ group, then one has the minimal presentation\n\\begin{equation}\\label{eq:presentation G}\n \\xymatrix{ 1\\ar[r] & R\\ar[r]& F\\ar[r] & G\\ar[r] & 1 }\n\\end{equation}\nwith $H^1(F)\\simeq H^1(G)$ of dimension $d=\\dim(H^1(G))$ equal to the minimal number of generators of $G$,\nand $R$ generated as normal subgroup of $F$ by an element $r\\in F$, which is called the defining relation.\nIn particular, one has $r\\in F_{(2)}$, as the presentation is minimal, and $H^2(G)\\simeq\\mathbb{F}_p$,\nas $\\dim(H^2(G))$ is the minimal number of generators of $R$ as normal subgroup of $F$\n(cf. \\cite[\\S~III.9]{nsw:cohm}).\n\n\\begin{proposition}\\label{prop:onerel}\n Let $G=F\/R$ be a finitely generated one-relator pro-$p$ group, with defining relation $r\\in F_{(2)}\\smallsetminus F_{(3)}$\n(if $p=2$ assume further that $r\\notin F_{(2)}\\smallsetminus [F,F]$).\nThen $G$ is mild, and the $\\mathbb{F}_p$-cohomology ring $H^\\bullet(G)$ is quadratic.\n\\end{proposition}\n\n\\begin{proof}\nBy hypothesis, one may find a a minimal generating set $\\{x_1,\\ldots,x_d\\}\\subseteq F$ such that\nthe commutator $[x_i,x_j]$ appears in $r$ for some $i,j\\in\\{1,\\ldots,d\\}$, $i\\neq j$.\nUp to renumbering, we may assume $i=d-1$ and $j=d$.\n\nIdentify the graded group algebra ${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ F]\\!])$ with the algebra $\\F_p\\langle X\\rangle$, with $X=\\{X_1,\\ldots,X_d\\}$.\nThe initial form $\\rho$ of $r$ is a homogeneous polynomial of degree two, with leading monomial\n$\\alpha X_dX_{d-1}$, with $\\alpha\\in\\mathbb{F}_p$.\nSince the sequence $\\{X_nX_{n-1}\\}$ is combinatorially free in the sense of \\cite[Definition~3.1]{jochen:massey},\nthe sequence $\\{\\rho\\}\\subseteq\\F_p\\langle X\\rangle$ is strongly free by \\cite[Thm.~3.5]{jochen:massey}, and $G$ is a mild.\n(In the case $p\\neq2$ this follows also from \\cite[Cor.~5.10]{jochen:massey}.)\n\nMoreover, by \\cite[Prop.~3.9.13]{nsw:cohm} the one has $\\chi_{d-1}\\cup\\chi_d\\neq0$. \nThus, $\\chi_{d-1}\\cup\\chi_d$ generates $H^2(G)\\simeq\\mathbb{F}_p$, and $H^n(G)=0$ for $n>2$ by \\cite[Thm.~2.12-(i)]{jochen:massey}.\nTherefore, $H^\\bullet(G)$ is a quadratic $\\mathbb{F}_p$-algebra.\n\\end{proof}\n\nHenceforth, we shall assume that the $\\mathbb{F}_p$-cohomology ring $H^\\bullet(G)$ is quadratic.\nThe cup product induces a (skew-commutative) $\\mathbb{F}_p$-pairing\n\\begin{equation}\\label{eq:pairing cup}\n \\xymatrix{ H^1(G)\\times H^1(G)\\ar[r]^-{\\cup} & \\mathbb{F}_p }.\n\\end{equation}\n(cf. \\cite[Prop.~3.9.12]{nsw:cohm}).\nIf the pairing \\eqref{eq:pairing cup} is perfect, then $G$ is a Demushkin group (cf. \\cite[Definition~3.9.9]{nsw:cohm}).\nOtherwise, let $V_2=H^1(G)^\\perp$ be the radical of $H^1(G)$ with respect to the cup product ---\ni.e., \n\\[V_2=H^1(G)^\\perp=\\{\\chi\\in H^1(G)\\mid \\chi\\cup\\psi=0\\text{ for all }\\psi\\in H^1(G)\\}.\\]\nThen we may decompose $H^1(G)=V_1\\oplus V_2$, so that \\eqref{eq:pairing cup} induces a perfect pairing\n$V_1\\times V_1\\to \\mathbb{F}_p$.\n\nLet $G^{ab}=G\/[G,G]$ be the abelianization of $G$.\nSince $G$ is a one-relator pro-$p$ group, one has an isomorphism of abelian pro-$p$ groups\n\\[ G^{ab}\\simeq\\mathbb{Z}_p^d\\qquad\\text{or}\\qquad G^{ab}\\simeq\\mathbb{Z}_p\/q\\mathbb{Z}_p\\times\\mathbb{Z}_p^{d-1},\\]\nwith $d=\\dim(H^1(G))$ and $q$ a power of $p$.\nIn the former case we shall say that $q=0$.\nNow one has two cases:\n\\begin{itemize}\n \\item[(i)] $p$ is odd, or $p=2$ and $q\\neq2$;\n \\item[(ii)] $p=q=2$.\n\\end{itemize}\n\n\n\\subsection{First case: $q\\neq2$}\n\nSet $n=\\dim(V_1)$ and $m=\\dim(V_2)$, and let \\eqref{eq:presentation G} be a minimal presentation of $G$.\nBy \\cite[Prop.~3.9.13]{nsw:cohm} we may choose a minimal generating set\n$\\mathcal{S}=\\{x_1,\\ldots,x_n,y_1,\\ldots,y_m\\}$\nof $F$ such that\n\\begin{equation}\\label{eq:rel case1 gross0}\n r\\equiv \\prod_{i=1}^n x_i^{a_ip}\\cdot\\prod_{i=1}^m y_i^{b_ip}\\cdot\\prod_{1\\leq i1$.\nThis is the only case where $H^k(G)$ is not trivial for $k>2$ ---\ni.e., the cohomological dimension of $G$ is more than two.\n\\end{remark}\n\n\n\\section{Main results}\n\\label{sec:main}\n\n\n\\subsection{Maximal pro-$p$ Galois groups}\n\\label{ssec:proofs}\n\nAssume that the maximal pro-$p$ Galois group $G=G_K(p)$ of a field $K$ is a finitely generated one-relator pro-$p$ group.\nNote that $G$ is finitely generated if, and only if, the quotient $K^\\times\/(K^\\times)^p$ of the multiplicative group\n$K^\\times=K\\smallsetminus\\{0\\}$ is finite, as one has the (Pontryagin) duality \n\\[K^\\times\/(K^\\times)^p=H^1(G,\\mathbb{F}_p)= (G\/G_{(2)})^*.\\]\n\nA pro-$p$ group $G$ is called a {\\sl Bloch-Kato pro-$p$ group} if the $\\mathbb{F}_p$-cohomology algebra $H^\\bullet(C)$\nis a quadratic $\\mathbb{F}_p$-algebra for every closed subgroup $C$ of $G$. \nIf $K$ contains a root of unity of order $p$, the Galois group $G_K(p)$\nis a Bloch-Kato pro-$p$ group by the Rost-Voevodsky Theorem (cf. \\cite[\\S~2]{cq:bk}).\n\nObservation~\\ref{obs1} follows directly by Proposition~\\ref{prop:case1 cohomology}\nand Proposition~\\ref{prop:case2 cohomology} (and by Remark~\\ref{rem:p2} in the case $p=q=2$ and $n=1$).\n\n\\begin{proof}[Proof of Theorem~\\ref{thm2} and Theorem~\\ref{thm3}]\nSet $d=\\dim(H^1(G_K(p)))$, and let $q$, and $n,m$ be as in Section~\\ref{sec:onerel}.\n\nAssume first that $p=q=2$ and $n=1$.\nBy \\eqref{eq:grad n1} the graded group algebra ${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])$ splits as free product\nof a trivial quadratic $\\mathbb{F}_p$-algebra (on one generator) and a free associative graded $\\mathbb{F}_p$-algebra\n(on $m$ generators).\nThe former is a Demushkin $\\mathbb{F}_p$-algebra by Proposition~\\ref{prop:demushkin shape}, and \\eqref{eq:thmB} holds.\nMoreover, by \\eqref{eq:cohom n1} the $\\mathbb{F}_p$-cohomology algebra $H^\\bullet(G_K(p))$ decomposes as direct product of the free $\\mathbb{F}_p$-algebra\non $V_1\\simeq\\mathbb{F}_p$ and the trivial quadratic $\\mathbb{F}_p$-algebra on $V_2$.\nBy Example~\\ref{exs:quadratic} such algebras are quadratic dual respectively to the $\\mathbb{F}_p$-algebras in \\eqref{eq:grad n1},\nand thus \n\\begin{equation}\\label{eq:dualityproof}\n H^\\bullet(G_K(p))\\simeq{\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])^!\n\\end{equation}\nby Proposition~\\ref{prop:products}.\nThis establishes Theorem~\\ref{thm2} in this case.\nFinally, by Example~\\ref{exs:quadratic} all the algebras appearing in \\eqref{eq:grad n1} and \\eqref{eq:cohom n1}\nare Koszul, thus both ${\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ G_K(p)]\\!])$ and $H^\\bullet(G_K(p))$ are Koszul by Proposition~\\ref{prop:products},\nand this proves Theorem~\\ref{thm3} in this case.\n\nAssume now that $p$ is odd, or $p=2$ and $q\\neq2$, so that we are in the first case.\nThe isomorphism \\eqref{eq:case1 algebra} yields \\eqref{eq:thmB}.\nMoreover, by Lemma~\\ref{lemma:demushkin dual} the Demushkin $\\mathbb{F}_p$-algebra $D_\\bullet$ in \\eqref{eq:case1 algebra}\nis quadratic dual to the algebra $A_\\bullet$ in \\eqref{eq:case1 cohomology}, and by Example~\\ref{exs:quadratic}\nthe free $\\mathbb{F}_p$-algebra in \\eqref{eq:case1 algebra} is quadratic dual to\nthe trivial quadratic $\\mathbb{F}_p$-algebra $B_\\bullet$ in \\eqref{eq:case1 cohomology},\nso that Proposition~\\ref{prop:products} yields duality \\eqref{eq:dualityproof}.\nThis establishes Theorem~\\ref{thm2} in this case.\nFinally, by Example~\\ref{exs:quadratic} and Proposition~\\ref{prop:demushkin algebra koszul} all such algebras\nare Koszul, and Proposition~\\ref{prop:products} yields Theorem~\\ref{thm3} in this case.\n\nAt last, assume that $p=q=2$ and $n>1$.\nThe isomorphism \\eqref{eq:case2 algebra} yields \\eqref{eq:thmB}.\nMoreover, by Lemma~\\ref{lemma:demushkin dual} the Demushkin $\\mathbb{F}_p$-algebra $D_\\bullet$ in \\eqref{eq:case2 algebra}\nis quadratic dual to the algebra $A_\\bullet$ in \\eqref{eq:case2 cohomology}, and by Example~\\ref{exs:quadratic}\nthe free $\\mathbb{F}_p$-algebra in \\eqref{eq:case2 algebra} is quadratic dual to\nthe trivial quadratic $\\mathbb{F}_p$-algebra $B_\\bullet$ in \\eqref{eq:case2 cohomology},\nso that Proposition~\\ref{prop:products} yields duality \\eqref{eq:dualityproof}.\nThis establishes Theorem~\\ref{thm2} in this case.\nFinally, by Example~\\ref{exs:quadratic} and Proposition~\\ref{prop:demushkin algebra koszul} all such algebras\nare Koszul, and Proposition~\\ref{prop:products} yields Theorem~\\ref{thm3} in this case.\n\\end{proof}\n\nMoreover, it is possible to generalize W\\\"urfel's theorem to Bloch-Kato pro-$p$ groups as follows.\n\n\\begin{theorem}\n \\label{thm:wurfel}\n Let $G$ be a finitely generated one-relator Bloch-Kato pro-$p$ group.\nOne has the short exact sequence of pro-$p$ groups\n\\begin{equation}\\label{ref:seswurfel}\n \\xymatrix{ 1\\ar[r] & N\\ar[r] & G\\ar[r] & \\bar G\\ar[r] & 1}\n\\end{equation}\n--- where $N$ is the normal closure of a free pro-$p$ subgroup $C\\subseteq G$, and $\\bar G$ is a Demushkin group ---,\nwhich induces the isomorphisms\n\\[ \\begin{split}\n & H^\\bullet(G)\\simeq H^\\bullet(\\bar G)\\sqcap H^\\bullet(C),\\qquad\\text{and}\\\\\n & {\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)\\simeq {\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ \\bar G]\\!])\\sqcup{\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ C]\\!]).\n \\end{split} \\]\nIn particular, this is true when $G=G_K(p)$ is a finitely generated one-relator maximal pro-$p$ Galois group,\nwith $K$ conaining a root of unity of order $p$.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $C$ be the closed subgroup of $G$ such that the restriction morphism ${\\rm res}_{G,C}^1\\colon H^1(G)\\to H^2(C)$\ninduces an isomorphism $H^1(G)^\\perp\\simeq H^1(C)$.\nIn particular, one has $\\mathrm{ker}({\\rm res}_{G,C}^1)=V_1$.\nTherefore, the commutative diagram\n\\[\\xymatrix@R=0.8truecm{ H^1(G)\\times H^1(G) \\ar[rr]^-{\\cup}\\ar@<5ex>@{->>}[d]_{{\\rm res}_{G,C}^1}\\ar@<-5ex>@{->>}[d]_{{\\rm res}_{G,C}^1}\n&& H^2(G)\\ar[d]_{{\\rm res}_{G,C}^2} \\\\ H^1(C)\\times H^1(C) \\ar[rr]^-{\\cup} && H^2(C) }\\]\nimplies that the lower horizontal arrow is trivial and thus $H^2(C)=0$,\nas $G$ is a Bloch-Kato pro-$p$ group.\nConsequently, $C$ is a free pro-$p$ group (cf. \\cite[Prop.~3.5.17]{nsw:cohm}).\n\nLet $N\\subseteq G$ be the normal closure of $C$ in $G$, and set $\\bar G=G\/N$.\nSince $H^1(N)^{\\bar G}\\simeq H^1(C)$, the five terms exact sequence induced by the quotient $G\/N$\nimplies that $H^1(\\bar G)\\simeq V_1$ and that the inflation map $\\inf_{G,N}^2\\colon H^2(\\bar G)\\to H^2(G)$\nis a monomorphism (cf. \\cite[Prop.~1.6.7]{nsw:cohm}).\nThus, in the commutative diagram\n\\[\\xymatrix@R=0.8truecm{ H^1(\\bar G)\\times H^1(\\bar G) \\ar[rr]^-{\\cup}\\ar@<5ex>@{>->}[d]_{\\inf_{G,N}^1}\\ar@<-5ex>@{>->}[d]_{\\inf_{G,N}^1}\n&& H^2(\\bar G)\\ar@{>->}[d]_{\\inf_{G,N}^2} \\\\ H^1(G)\\times H^1(G) \\ar[rr]^-{\\cup} && H^2(G) }\\]\nthe upper line is a non-degenerate pairing --- in particular, $\\bar G$ is a one relator pro-$p$ group too.\nTherefore, $\\bar G$ is a Demushkin group.\n\\end{proof}\n\n\n\\subsection{Koszul algebras of elementary type}\n\nIn analogy with the class of pro-$p$ groups of elementary type defined by I.~Efrat (cf. \\cite[\\S~3]{ido:small}),\none defines the class of {\\sl compact Koszul $\\mathbb{F}_p$-algebras of elementary type}, respectively the class\nof {\\sl discrete Koszul $\\mathbb{F}_p$-algebras of elementary type}, as the minimal class of quadratic\n$\\mathbb{F}_p$-algebras whose ``building blocks'' are free associative $\\mathbb{F}_p$-algebras and Demushkin $\\mathbb{F}_p$-algebras,\nresp. trivial quadratic $\\mathbb{F}_p$-algebras and the $\\mathbb{F}_p$-cohomology algebras of Demushkin pro-$p$ groups\n(i.e., quadratic $\\mathbb{F}_p$-algebras $A_\\bullet$ subject to \\eqref{eq:rel cohomology demushkin 1} or \n\\eqref{eq:rel cohomology demushkin 2}, with $A_n=0$ for $n\\geq3$),\nand whose elements can be assembled according to the following rules:\n\\begin{itemize}\n \\item[(a)] if the Koszul $\\mathbb{F}_p$-algebras $A_\\bullet$ and $B_\\bullet$ are compact of elementary type, then\nalso the free product $A_\\bullet\\sqcup B_\\bullet$ is compact of elementary type,\n \\item[(b)] if the compact Koszul $\\mathbb{F}_p$-algebras $A_\\bullet$ is of elementary type and $B_\\bullet$ is a free algebra\nwith $\\dim(B_n)=1$ for every $n\\geq0$, then also the quadratic algebra\n\\[A_\\bullet\\otimes^1 B_\\bullet=\\frac{A_\\bullet\\sqcup B_\\bullet}{\\langle\\Omega\\rangle},\\quad \\text{with }\n\\Omega=\\{ab-ba,a\\in A_1,b\\in B_1\\}\\]\nis compact of elementary type (the operation $\\otimes^1$ is called the {\\sl commutative tensor product} of quadratic\nalgebras, cf. \\cite[\\S~3.1]{PoliPosi}),\n\\end{itemize}\nfor the class of compact Koszul $\\mathbb{F}_p$-algebras of elementary type; and \n\\begin{itemize}\n \\item[(a')] if the Koszul $\\mathbb{F}_p$-algebras $A_\\bullet$ and $B_\\bullet$ are discrete of elementary type, then\nalso the direct product $A_\\bullet\\sqcap B_\\bullet$ is discrete of elementary type,\n \\item[(b')] if the Koszul $\\mathbb{F}_p$-algebras $A_\\bullet$ is discrete of elementary type and $B_\\bullet=\\mathbb{F}_p\\oplus B_1$\nis a trivial quadratic $\\mathbb{F}_p$-algebra with $\\dim(B_1)=1$, then also the quadratic algebra\n\\[A_\\bullet\\otimes^{-1} B_\\bullet=\\frac{A_\\bullet\\sqcup B_\\bullet}{\\langle\\Omega\\rangle},\\quad \\text{with }\n\\Omega=\\{ab+ba,a\\in A_1,b\\in B_1\\}\\]\nis discrete of elementary type (the operation $\\otimes^{-1}$ is called the\n{\\sl skew-commutative tensor product} of quadratic algebras, cf. \\cite[\\S~3.1]{PoliPosi}),\n\\end{itemize}\nfor the class of discrete Koszul $\\mathbb{F}_p$-algebras of elementary type (cf. \\cite[\\S~4.7]{MQRTW}).\n\nIn \\cite{ido:ETC}, Efrat conjectures that if $K$ contains a root of unity of order $p$ and the maximal pro-$p$\nGalois group $G_K(p)$ is finitely generated, then $G_K(p)$ is a pro-$p$ group of elementary type.\nThe results of Section~\\ref{sec:onerel} --- together with Theorem~\\ref{thm2} --- yield the following.\n\n\\begin{coro}\\label{coro:fine}\nIf $G$ is a finitely generated one-relator pro-$p$ group with a quadratic $\\mathbb{F}_p$-cohomology algebra $H^\\bullet(G)$,\nthen the graded group algebra ${\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)$ is a compact Koszul $\\mathbb{F}_p$-algebra of elementary type,\nand $H^\\bullet(G)$ is a discrete Koszul $\\mathbb{F}_p$-algebra of elementary type. \nIn particular, this is true for $G=G_K(p)$, with $K$ a field contining a root of unity of order $p$.\n\\end{coro}\n\nIn \\cite[Question~2]{MQRTW} it is conjectured that the graded group $\\mathbb{F}_p$-algebra of $G_K(p)$,\nresp. the $\\mathbb{F}_p$-cohomology algebra $H^\\bullet(G_K(p))$, is a compact Koszul $\\mathbb{F}_p$-algebra of elementary type,\nresp. is a discrete Koszul $\\mathbb{F}_p$-algebra of elementary type, for every field $K$ containing a root of unity\nof order $p$ with $G_K(p)$ finitely generated.\nThus, Corollary~\\ref{coro:fine} provides a positive answer\nto this conjecture in the case of one-relator maximal pro-$p$ Galois groups.\n\n At the end of \\cite{wurf}, W\\\"urfel asked whether every one-relator absolute Galois pro-$p$ group\n--- subject to some further cohomological restrictions, as such paper deals with the case where the base field $K$\ncontains all roots of unity of order a $p$-power --- decomposes as a free pro-$p$ product $\\bar G\\ast_{\\hat p} F$\nof a Demushkin group $\\bar G$ with a finitely generated free pro-$p$ group $F$.\nIn fact, for $G=\\bar G\\ast_{\\hat p} F$ one has\n\\[ \\begin{split}\n & H^\\bullet(G)\\simeq H^\\bullet(\\bar G)\\sqcap H^\\bullet(F)\\qquad\\text{and}\\\\\n & {\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)\\simeq {\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ \\bar G]\\!])\\sqcup {\\rm gr}_\\bullet(\\mathbb{F}_p[\\![ F]\\!]).\n \\end{split} \\]\nIt is worth observing that by Theorem~\\ref{thm:wurfel} one-relator maximal pro-$p$ Galois groups\n``behave'' like such free pro-$p$ products in terms of $\\mathbb{F}_p$-cohomology and restricted Lie algebras.\nIn other words, the algebras $H^\\bullet(G)$ and ${\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)$\ncannot ``distinguish'' between such free pro-$p$ products and other one-relator maximal pro-$p$ Galois groups.\n\n\\begin{example}\\label{ex:KZ}\n In \\cite{kz}, D.~Kochloukova and P.~Zalesskii produced the following example of free-by-Demushkin pro-$p$ group\nwhich is not such a free pro-$p$ product: let $G$ be the pro-$p$ group with minimal presentation\n\\[G=\\left\\langle x_1,x_2,x_3\\mid [x_1,x_2]=x_3^q\\right\\rangle,\\]\nwith $q\\geq0$ a $p$-power, $q\\neq2$ if $p=2$.\nSuch pro-$p$ group satisfies all the conditions in W\\\"urfel's theorem (cf. \\cite[Thm.~2]{kz}),\nand also Proposition~\\ref{prop:onerel}.\nIn particular, one has \\[H^\\bullet(G)=H^\\bullet(\\bar G)\\sqcap H^\\bullet(C),\\] with $C=\\langle x_3\\rangle$\nand $\\bar G=G\/N\\simeq\\mathbb{Z}_p^2$, with $N$ the normal closure of $C$, and \n\\[{\\rm gr}_\\bullet(\\F_p\\dbl G\\dbr)\\simeq\\mathbb{F}_p[X_1,X_2]\\sqcup\\mathbb{F}_p[X_3].\\]\nYet, the group $G$ is not realizable as maximal pro-$p$ Galois group of any field\n(see also \\cite[\\S~4.2.1]{cq:thesis}).\n\\end{example}\n\n\n\\section*{}\n\\subsection*{Acknowledgment}\n\nThe author wishes to \nexpress his gratitude to I.~Efrat and J.P.~Labute, for their precious comments and suggestions,\nto J.~Min\\'a\\v{c} and N.D.~T\\^an, for the stimulating time spent \nworking together at the University of Western Ontario,\nand to F.W.~Pasini for the inspiring discussions about quadratic algebras.\nAlso, he owes many thanks to C.~Maire, E.~Matzri, D.~Neftin, I.~Snopce and J.~Sonn\nfor their interest.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgeyo b/data_all_eng_slimpj/shuffled/split2/finalzzgeyo new file mode 100644 index 0000000000000000000000000000000000000000..c0b1ac2ff394f1414dc82dd4656545c31227894e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgeyo @@ -0,0 +1,5 @@ +{"text":"\n\n\\subsection{Experimental plan}\n\nIn this work, we present the analysis of OpenStack version 3.12.1 (release \\textit{Pike}), which was the latest version of OpenStack when we started this work. We injected bugs into the most fundamental services of OpenStack \\cite{denton2015learning,solberg2017openstack}: (i) the \\textbf{Nova} subsystem, which provides services for provisioning instances (VMs) and handling their life cycle; (ii) the \\textbf{Cinder} subsystem, which provides services for managing block storage for instances; and (iii) the \\textbf{Neutron} subsystem, which provides services for provisioning virtual networks for instances, including resources such as \\emph{floating IPs}, \\emph{ports} and \\emph{subnets}. Each subsystem includes several components (e.g., the Nova sub-system includes \\emph{nova-api}, \\emph{nova-compute}, etc.), which interact through message queues internally to OpenStack. The Nova, Cinder, and Neutron sub-systems provide external REST API interfaces to cloud users.\n\n \n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.75\\columnwidth]{testbed_architecture_v2.pdf}\n \n \\end{centering}\n \\vspace*{-4.5mm}\n \\caption{OpenStack testbed architecture.}\n \\label{fig:testbed_architecture}\n\\vspace{-0.8cm}\n\\end{figure}\n\n\\figurename{}~\\ref{fig:testbed_architecture} shows the testbed used for the experimental analysis of OpenStack. We adopted an all-in-one virtualized deployment of OpenStack, in which the OpenStack services run on the same VM, for the following reasons: (1) to prevent interferences on the tests from transient issues in the physical network (e.g., sporadic network faults, network delays caused by other user traffic in our local data center, etc.); (2) to parallelize a high number of tests on several physical machines, by using the \\emph{Packstack} installation utility \\cite{packstack} to have a reproducible installation of OpenStack across the VMs; (3) to efficiently revert any persistent effect of a fault injection test on the OpenStack deployment (e.g., file system issues), in order to assure independence among the tests. Moreover, the all-in-one virtualized deployment is a common solution for performing tests on OpenStack \\cite{evaluating_openstack,Markelov2016}. The hardware and VM configuration for the testbed includes: 8 virtual Intel Xeon CPUs (E5-2630L v3 @ 1.80GHz); 16GB RAM; 150 GB storage; Linux CentOS v7.0.\n\n\n\n\n\n\n\nIn addition to the core services of OpenStack (e.g., Nova, Neutron, Cinder, etc.), the testbed also includes our own components to automate fault injection tests. \nThe \\emph{Injector Agent} is the component that analyzes and instruments the source code of OpenStack. The \\emph{Injector Agent} can: (i) scan the source code to identify injectable locations (i.e., source-code statements where the bug types discussed in \\S{}~\\ref{subsec:fault_injection} can be applied); (ii) instrument the source code by introducing logging statements in every injectable location, in order to get a profile of which locations are covered during the execution of the workload (\\textbf{coverage analysis}); (iii) instrument the source code to introduce a bug into an individual injectable location.\n\nThe \\emph{Controller} orchestrates the experimental workflow.\nIt first commands the \\emph{Injector Agent} to perform a preliminary coverage analysis, by instrumenting the source code with logging statements, restarting the OpenStack services, and launching the \\emph{Workload Generator}, but without injecting any fault. \nThe \\emph{Workload Generator} issues a sequence of API calls in order to stimulate OpenStack services.\nThe \\emph{Controller} retrieves the list of injectable locations and their coverage from the \\emph{Injector Agent}. Then, it iterates over the list of injectable locations that are covered, and issues commands for the \\emph{Injector Agent} to perform fault injection tests. For each test, the \\emph{Injector Agent} introduces an individual bug by mutating the source code, restarts the OpenStack services, starts the workload, and triggers the injected bug as discussed in \\S{}~\\ref{subsec:fault_injection}. The \\emph{Injector Agent} collects the logs files from all OpenStack subsystems and from the \\emph{Workload Generator}, which are sent to the \\emph{Controller} for later analysis (\\S{}~\\ref{subsec:failure_analysis}).\n\n\n\n\n\n\n\nWe performed a full scan of injectable locations in the source code of Nova, Cinder, and Neutron, for a total of \\numprint{2016} analyzed source code files. We identified \\numprint{911} injectable faults that were covered by the workload. \\figurename{}~\\ref{fig:fault_injection_tests} shows the number of faults per sub-system and per type of fault. \nThe number of faults for each type and sub-system depends on the number of calls to the target functions, and on their input and output parameters, as discussed in \\S{}~\\ref{subsec:fault_injection}.\nWe executed one the test per injectable location, by injecting one fault at a time. \n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.72\\columnwidth]{fault_injection_tests.png}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Number of fault injection tests.}\n \\label{fig:fault_injection_tests}\n \\vspace{-5.3mm}\n\\end{figure}\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.72\\columnwidth]{failure_types_distribution.png}\n \\vspace{-0.5cm}\n \\end{centering}\n \n \\caption{Distribution of OpenStack failures.}\n \\label{fig:all_component_failure_types}\n \\vspace{-0.75cm}\n\\end{figure}\n\n\nAfter executing the tests, we found failures respectively in 52.6\\% (231 out of 439 tests), 46.4\\% (125 out of 269 tests), and 61\\% (124 out of 203 tests) of tests in Nova, Cinder, and Neutron, for a total of \\numprint{480}.\nIn the remaining 47.3\\% of the tests (431 out of 911 tests), instead, there were neither an API error nor assertion failures: in these cases, the fault was not activated (even if the faulty code was covered by the workload), or there was no error propagation to the component interface. The occurrence of tests not causing failures is a typical phenomenon that occurs with code mutations, which may not infect the state even when the faulty code is executed \\cite{christmansson1996generation,lanzaro2014empirical}. Yet, the injections provided us a large and diverse set of failures for our analysis.\n\n\n\n\n\n\n\n\n\\subsection{Does OpenStack show a fail-stop behavior?}\n\\label{subsec:rq1}\n\n\n\nWe first analyze the impact of failures on the service interface APIs provided by OpenStack. \nThe \\emph{Workload Generator} (which impersonates a user of the cloud management system) invokes these APIs, looks for errors returned by the APIs and performs assertion checks between API calls. A fail-stop behavior occurs when an API returns an error before any failed assertion check. In such cases, the \\emph{Workload Generator} stops on the occurrence of the API error. Instead, it is possible that an API invocation terminates without returning any error, but leaving the internal resources of the infrastructure (instances, volumes, etc.) in a failed state, which is reported by assertion checks. These cases represent violations of the fail-stop hypothesis, and represent a risk for the users as they are unaware of the failure.\nTo investigate this aspect, we initially focus on the faulty round of each test, in which fault injection is enabled (\\figurename{}~\\ref{fig:workflow}). \n\n\\figurename{}~\\ref{fig:all_component_failure_types} shows the number of tests that experienced failures, divided into \\emph{API Error only}, \\emph{Assertion Failure only}, and \\emph{Assertion Failure(s), followed by an API Error}. The figure shows the data divided with respect to the subsystem where the bug was injected (respectively in Nova, Cinder, and Neutron); moreover, \\figurename{}~\\ref{fig:all_component_failure_types} shows the distribution across all fault injection tests. \nWe can see that the cases in which the system does not exhibit a fail-stop behavior (i.e., the categories \\emph{Assertion Failure only} and \\emph{Assertion Failure followed by an API Error}) represent the majority of the failures.\n\n\n\\figurename{}~\\ref{fig:all_component_failure_assertion_types} shows a detailed perspective on the failures of assertion checks. Notice that the number of assertion is greater than the number of tests classified in the Assertion failure category (i.e., \\emph{Assertion Failure only} and \\emph{Assertion Failure followed by an API Error}) since a test can generate multiple assertion failures.\nThe most common case has been one of the instances not active because the instance creation failed (i.e., it did not move into the \\textit{ACTIVE} state \\cite{openstack_instances_states}). \nIn other cases, the instance could not be reached through the network or could not be attached to a volume, even if in the \\textit{ACTIVE} state. A further common case is the failure of the volume creation, but only the faults injected in the Cinder sub-system caused this assertion failure. \n\n\n\n\n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=.75\\columnwidth]{assertion_distribution.png}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Distribution of assertion check failures.}\n \\label{fig:all_component_failure_assertion_types}\n \\vspace{-0.1cm}\n\\end{figure}\n\n\n\\begin{figure}[t]\n \\vspace{-0.4cm}\n \\begin{centering}\n \\includegraphics[width=0.6\\columnwidth]{api_distribution.png}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Distribution of API Errors.}\n \\label{fig:all_component_api_errros_after_assertion}\n \n \\vspace{-7mm}\n\\end{figure}\n\nThese cases point out that OpenStack lacks redundant checks to assure that the state of the virtual resources after a service call is in the expected state (e.g., newly-created instances are active). Such redundant checks would assess the state of the virtual resources before and after a service invocation and would raise an error if the state does not comply with the expectation (such as a new instance could not be activated). However, these redundant checks are seldom adopted, most likely due to the performance penalty they would incur, and because of the additional engineering efforts to design and implement them. Nevertheless, the cloud management system is exposed to the risk that residual bugs can lead to non-fail-stop behaviors, where failures are notified with a delay or not notified at all. This makes not trivial to prevent data losses and to automate recovery actions.\n\n\n\\figurename{}~\\ref{fig:all_component_api_errros_after_assertion} provides another perspective on API errors. It shows the number of tests in which each API returned an error, focusing on 15 out of 40 APIs that failed at least one time. The API with the highest number of API errors is the one for adding a volume to an instance (\\textit{openstack server add volume}), provided by the Cinder sub-system. \nThis API generated errors even when faults were injected in Nova (instance management) and Neutron (virtual networking). This behavior means that the effects of fault injection propagated from other sub-systems to Cinder (e.g., if an instance is in an incorrect state, other APIs on that resource are also exposed to failures). On the one hand, this behavior is an opportunity for detecting failures, even if in a later stage. On the other hand, it also represents the possibility of a failure to spread across sub-systems, thus defeating fault containment and exacerbating the severity of the failure. We will analyze fault propagation in more detail in Section~\\ref{subsec:rq3}.\n\n\n\n\n\n\n\n\n\n\\begin{figure}[!t]\n \\vspace{-3mm}\n \\begin{centering}\n \\includegraphics[width=0.75\\columnwidth]{all_components_assertion_and_api_errors_latency.pdf}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Cumulative distribution of API Error latency.\n \\label{fig:all_components_assertion_and_api_errors_latency}\n \n \\vspace{-0.92cm}\n\\end{figure}\n\nTo understand the extent of non-fail-stop behaviors, we also analyze the period of time (\\textbf{latency}) between the execution of the injected bug and the resulting API error. \nIt is desirable that this latency is as low as possible. Otherwise, the longer the latency, the more difficult is to relate an API error with its root cause (i.e., an API call invoked much earlier, on a different sub-system or virtual resource); and the more difficult to perform troubleshooting and recovery actions. \nTo track the execution of the injected bug, we instrumented the injected code with logging statements to record the timestamp of its execution. \nIf the injected code is executed several times before a failure (e.g., in the body of a loop), we conservatively consider the last timestamp. \nWe consider separately the cases where the API error is preceded by assertion check failures (i.e., the injected bug was triggered by an API different from the one affected by the bug) from the cases without any assertion check failure (e.g., the API error arises from the same API affected by the injected bug).\n\n\n\n\n\n\\figurename{}~\\ref{fig:all_components_assertion_and_api_errors_latency} shows the distributions of latency for API errors that occurred after assertion check failures, respectively for the injections in Nova, Cinder, and Neutron. Table~\\ref{tab:all_components_all_latencies_table_last_activation} summarizes the average, the 50$^{th}$, and the 90$^{th}$ percentiles of the latency distributions. We note that in the first category (API errors after assertion checks), all sub-systems exhibit a median API error latency longer than 100 seconds, with cases longer than several minutes. This latency should be considered too long for cloud services with high-availability SLAs (e.g., four \\emph{nines} or more \\cite{Bauer:2012:RAC:2339445}), which can only afford few minutes of monthly outage. \nThis behavior points out that the API errors are due to a ``reactive'' behavior of OpenStack, which does not actively perform any redundant check on the integrity of virtual resources, but only reacts to the inconsistent state of the resources once they are requested in a later service invocation. Thus, OpenStack experiences a long API error latency when a bug leaves a virtual resource in an inconsistent state. \nThis result suggests the need for improved error checking mechanisms inside OpenStack to prevent these failures. \n\n\nIn the case of failures that are notified by API errors without any preceding assertion check failure (the second category in Table~\\ref{tab:all_components_all_latencies_table_last_activation}), the latency of the API errors was relatively small, less than one second in the majority of cases. \nNevertheless, there were few cases with an API error latency higher than one minute. In particular, these cases happened when bugs were injected in Nova, but the API error was raised by a different sub-system (Cinder). In these cases, the high latency was caused by the propagation of the bug's effects across different API calls. These cases are further discussed in \\S{}~\\ref{subsec:rq3}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table}\n \\begin{center}\n \\footnotesize\n \\caption{Statistics on API Error latency.}\n \\label{tab:all_components_all_latencies_table_last_activation}\n \\vspace{-3mm}\n \n \\begin{tabular}{ >{\\centering}m{0.8in} | >{\\centering}m{0.4in} | >{\\centering}m{0.45in} | >{\\centering}m{0.45in} | >{\\centering\\arraybackslash}m{0.45in} | }\n \\cline{2-5}\n & \n \\textbf{Subsys.} & \\textbf{Avg [s]} & \\textbf{50$^{th}$ \\%ile [s]} & \\textbf{90$^{th}$ \\%ile [s]} \\\\\n \\hline\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \\multicolumn{1}{ |c| }{\n \\multirow{3}{*}{\\textbf{\\shortstack{API Errors after\\\\ an Assertion\\\\ failure}}} }\n & Nova & 152.25 & 168.34 & 191.60 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Cinder & 74.52 & 93.00 & 110.00 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Neutron & 144.72 & 166.00 & 263.60 \\\\\n \\hline\n \n \n \n\n\n\n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \\multicolumn{1}{ |c| }{\n \\multirow{3}{*}{\\textbf{\\shortstack{API Errors\\\\ only}}} }\n & Nova & 3.73 & 0.21 & 0.55 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Cinder & 0.30 & 0.01 & 1.00 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Neutron & 0.30 & 0.01 & 1.00 \\\\\n \\hline\n \n \n \n \\end{tabular}\n \\vspace{-0.5cm}\n \\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Is OpenStack able to log failures?}\n\\label{subsec:rq2}\n\nSince failures can be notified to the end-user with a long delay, or even not at all, it becomes important for system operators to get additional information to troubleshoot these failures. In particular, we here consider log messages produced by OpenStack sub-systems.\n\n\n\n\n \n \n\n\n\n\nWe computed the percentage (\\textbf{logging coverage}) of failed tests which produced at least one high-severity log message (see also \\S{}~\\ref{subsec:failure_analysis}). Table \\ref{tab:all_components_logged_not_logged_failed_tests} provides the logging coverage for different subsets of failures, by dividing them with respect to the injected subsystem and to the type of failure. \nFrom these results, we can see that OpenStack logged at least one high-severity message (i.e., with severity level \\textit{ERROR} or \\textit{CRITICAL}) in most of the cases. The Cinder subsystem shows the best results since logging covered almost all of the failures caused by fault injection. However, in the case of Nova and Neutron, logs missed some of the failures. In particular, the failures without API errors (i.e., \\emph{Assertion Failure only}) exhibited the lowest logging coverage. This behavior can be problematic for recovery and troubleshooting since the failures without API errors lack an explicit error notification. These failures are also the ones in need of complementary sources of information, such as logs.\n\nTo identify opportunities to improve logging in OpenStack, we analyzed the failures without any high-severity log across, with respect to the bug types injected in these tests. We found that \\textit{MISSING FUNCTION CALL} and \\textit{WRONG RETURN VALUE} represent the 70.7\\% of the bug types that lead to non-logged failures (43.9\\% and 26.8 \\%, respectively). \nThe \\textit{WRONG RETURN VALUE} faults are the easiest opportunity for improving logging and failure detection since the callers of a function could perform additional checks on the returned value and record anomalies in the logs.\nFor example, one of the injected bugs introduced a \\textit{WRONG RETURN VALUE} in calls to a database API called by the Nova sub-system to update the information linked to a new instance. The bug forced the function to return a \\emph{None} instance object. The bug caused an assertion check failure, but OpenStack did not log any high-severity message. By manually analyzing the logs, we could only find one suspicious message with the only \\textit{WARNING} severity and with little information about the problem, as this message was not related to database management.\n\n\nThe non-logged failures caused by a \\textit{MISSING FUNCTION CALL} emphasize the need for redundant end-to-end checks to identify inconsistencies in the state of the virtual resources. For example, in one of these experiments, we injected a \\textit{MISSING FUNCTION CALL} in the \\textit{LibvirtDriver} class in the Nova subsystem, which allows OpenStack to interact with the \\textit{libvirt} virtualization APIs \\cite{libvirt}. Because of the injected bug, the Nova driver omits to attach a volume to an instance, but the Nova sub-system does not perform checks that the volume is indeed attached to the instance. This kind of end-to-end checks could be introduced at the service API interface of OpenStack (e.g., in \\emph{nova-api}) to test the availability of the virtual resources at the end of API service invocations (e.g., by pinging them).\n\n\n\n\n\n\n\n\n\\begin{table}\n \\begin{center}\n \\footnotesize\n \\caption{Logging coverage of high-severity log messages.}\n \\label{tab:all_components_logged_not_logged_failed_tests}\n \\vspace{-3mm}\n \n \\begin{tabular}{ >{\\centering}m{0.75in} | >{\\centering}m{0.7in} | >{\\centering}m{0.7in} | >{\\centering\\arraybackslash}m{0.6in} | }\n \n \\cline{2-4}\n & \\multicolumn{3}{c|}{\\textbf{Logging coverage}} \\\\\n \n \\hline\n \\multicolumn{1}{|c|}{\\textbf{Subsystem}} & \\textbf{API Errors\\\\only} & \\textbf{Assertion\\\\ failure only} & \\textbf{Assertion failure and API Errors} \\\\\n \\hline\n \n \\multicolumn{1}{|c|}{Nova} & 90.32\\% & 80.77\\% & 82,56\\% \\\\\n \\hline\n \\multicolumn{1}{|c|}{Cinder} & 100\\% & 95,65\\% & 100\\% \\\\\n \\hline\n \\multicolumn{1}{|c|}{Neutron} & 98.67\\% & 66.67\\% & 95\\% \\\\\n \\hline\n \\end{tabular}\n \\vspace{-0.6cm}\n \\end{center}\n\\end{table}\n\n\\subsection{Do failures propagate across OpenStack?}\n\\label{subsec:rq3}\n\n\n\n\n\nWe analyze failure propagation across sub-systems, to identify more opportunities to reduce their severity. We consider failures of both the ``faulty'' and the ``fault-free'' rounds, respectively (\\figurename{}~\\ref{fig:workflow}).\n\nIn the faulty round, we are interested in whether the injected bug impacted on sub-systems beyond the injected one. To this aim, we divide the API errors with respect to the API that raised the error (e.g., an API exposed by Nova, Neutron, or Cinder). Similarly, we divide the assertion check failures with respect to the sub-system that manages the virtual resource checked by the assertion. There is a \\textbf{spatial} fault propagation across the components if an injection on a sub-system (say, Nova) causes an assertion check failure or an API error on a different sub-system (say, Cinder or Neutron).\n\n\n\\begin{figure*}[!ht]\n \\begin{centering}\n \\begin{subfigure}{0.40\\textwidth}\n \\includegraphics[width=\\linewidth]{r1.pdf}\n \\caption{During faulty round.}\n \\label{fig:openstack_spatial_propagation_round1}\n \\end{subfigure}\n \\qquad\n \\begin{subfigure}{0.40\\textwidth}\n \\includegraphics[width=\\linewidth]{r2.pdf}\n \\caption{After removing the injected fault (fault-free round).}\n \\label{fig:openstack_spatial_propagation_round2}\n \\end{subfigure}\n \n \n \\end{centering}\n \\vspace*{-4mm} \n \\caption{Fault propagation during fault injection tests.}\n\\vspace{-5mm}\n\\end{figure*}\n\n\n\\figurename{}~\\ref{fig:openstack_spatial_propagation_round1} shows a graph with of events occurred during the faulty round of the tests with a failure. The nodes on the top of the graph represent the sub-systems where bugs were injected; the nodes on the middle represent assertion check failures; the nodes on the bottom represent API errors. The edges that originate from the nodes on the top represent the number of injections that were followed by an assertion check failure or an API error. Moreover, the edges between the middle and the bottom nodes represent the number of tests where an assertion check failure was followed by an API error. The most numerous cases are emphasized with proportionally thicker edges and annotated with the number of occurrences. We used different shades to differentiate the cases with respect to the injected sub-system.\n\n \n\n\nThe failures exhibited a propagation across OpenStack services in a significant amount of cases (37.5\\% of the failures). \nIn many cases, the propagation initiated from an injection in Nova, which caused a failure at activating a new instance; as discussed in the previous subsections, the unavailability of the instance was detected in a later stage, such as when the user attaches a volume to the instance using the Cinder API. \nEven worse, there are some cases of propagation from Neutron across Nova and Cinder.\nThese failures represent a severe issue for fault containment since an injection in Neutron not only caused a failure of their APIs but also impacted on virtual resources that were not managed by these sub-systems. Therefore, the failures are not necessarily limited to the virtual resources managed by the sub-system invoked at the time of the failure, but also to other related virtual resources. Therefore, end-to-end checks on API invocations should also include resources that are indirectly related to the API (such as, checking the availability of an instance after attaching a volume).\nFor as concerns Cinder, instead, there are no cases of error propagation from this sub-system across Nova and Neutron.\n\n\nWe further analyze the propagation of failures by considering what happens during the fault-free round of execution. The fault-free round invokes the service APIs after the buggy execution path is disabled as dead code. Moreover, the fault-free round executes on new virtual resources (i.e., instances, networks, routers, etc., are created from scratch). \nTherefore, it is reasonable to expect (and it is indeed the case) that the fault-free round executes without experiencing any failure. However, we still observe a subset of failures (7.5\\%) that propagate their effects to the fault-free round. These failures must be considered critical, since they are affecting service requests that are supposed to be independent but are still exposed to \\textbf{temporal} failure propagation through shared state and resources. \nWe remark that the failures in the fault-free round are caused by the injection in the faulty round. Indeed, we assured that previous injections do not impact on the subsequent experiments by restoring all the persistent state of OpenStack before every experiment. \n\n\n\n\n\\figurename{}~\\ref{fig:openstack_spatial_propagation_round2} shows the propagation graph for the fault-free round. The most cases, the Nova sub-system was unable to create new instances, even after the injected bug is removed from Nova. A similar persistent issue happens for a subset of failures caused by injections in Neutron. These sub-systems both manage a relational database which holds information on the virtual instances and networks, and we found that the persistent issues are solved only after that the databases are reverted to the state before fault injection. This recovery action can be very costly since it can take a significant amount of time, during which the cloud infrastructure may become unavailable. For this reason, we remark the need for detecting failures as soon as they occur, such as using end-to-end checks at the end of service API calls. Such detection would support quicker recovery actions, such as to revert the database changes performed by an individual transaction.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Discussion and lessons learned}\n\nThe experimental analysis pointed out that software bugs often cause erratic behavior of the cloud management system, hindering detection and recovery of failures. We found failures that were notified to the user only after a long delay when it is more difficult to trace back the root cause of the failure, and recovery actions are more costly (e.g., reverting the database); or, the failures were not notified at all. \nMoreover, our analysis suggests the following practical strategies to mitigate these failures. \n\n\n \n\\noindent\n$\\rhd$ \\textbf{Need for deeper run-time verification of virtual resources.} \nFault injections pointed out OpenStack APIs that leaked resources on failures, or left them in an inconsistent state, due to missing or incorrect error handlers.\nFor example, the \\emph{server-create} API failed without creating a new VM, but it did not deallocate virtual resources (e.g., instances in ``dead'' state, unused virtual NICs) created before the failure.\nThese failures can be prevented through fault injection. Moreover, residual faults should be detected and handled by means of run-time verification strategies, which perform redundant, end-to-end checks after a service API call, to assert whether the virtual resources are in the expected state. For example, these checks can be specified using temporal logic and synthesized in a run-time monitor \\cite{delgado2004taxonomy,chen2007mop,zhou2014runtime,rabiser2017comparison}, e.g., a logical predicate for a traditional OS can assert that a thread suspended on a semaphore leads to the activation of another thread \\cite{arlat2002dependability}. In the context of cloud management, the predicates should test at run-time the availability of virtual resources (e.g., volumes and connectivity), similarly to our assertion checks (\\tablename{}~\\ref{tab:assertion_and_description}).\n\n\n\n\n\n\\noindent\n$\\rhd$ \\textbf{Increasing the logging coverage.} \nThe logging mechanisms in OpenStack reported high-severity error messages for many of the failures. However, there were failures with late or no API errors that would benefit from logs to diagnose the failure, but such logs were missing. In particular, fault injection identified function call sites in OpenStack where the injected wrong return values were ignored by the caller. These cases are opportunities for developers to add logging statements and to improve the coverage of logs (e.g., by checking the outputs produced by the faulty function calls). Moreover, the logs can be complemented with the run-time verification checks. \n\n\n\n\n\n\n\n\n\\noindent\n$\\rhd$ \\textbf{Preventing corruptions of persistent data and shared state.} \nThe experiments showed that undetected failures can propagate across several virtual resources and sub-systems. Moreover, we found that these propagated failures can impact on shared state and persistent data (such as databases), causing permanent issues. Fault injection identified failures that were detected much later after their initial occurrence (i.e., with high API error latency, or no API errors at all). In these cases, it is very difficult for operators to diagnose which parts of the system have been corrupted, thus increasing the cost of recovery. Therefore, in addition to timely failure detection (using deeper run-time verification techniques, as discussed above), it becomes important to address the corruptions as soon as the failure is detected, since the scope of recovery actions can be smaller (i.e., the impact of the failure is limited specific resources involved by the failed service API call). One potential direction of research is on selectively undoing recent changes to the shared state and persistent data of the cloud management system \\cite{weber2012automatic,satyal2017rollback}.\n\n\n\n\n\\subsection{Threats to validity}\n\\label{subsec:threats_validity}\n\nThe injection of software bugs is still a challenging and open research problem. We addressed this issue by using code mutations to generate realistic run-time errors. This technique is widespread in the field of mutation testing \\cite{jia2011survey,just2014mutants,papadakis2018mutation,papadakis2019mutation} to devise test cases; moreover, it is also commonly adopted by studies on software dependability \\cite{chillarege1996:generation-error-set,voas1997:predicting,ng2001design,duraes2006emulation,giuffrida2013edfi} and on assessing bug finding tools \\cite{dolan2016lava,roy2018bug}. In our context, bug injection is meant to anticipate the potential consequences of bugs on service availability and resource integrity. \nTo strengthen the connection between the real and the experimental failures, we based our selection of code mutations on past software bugs in OpenStack. \nThe injected bug types were consistent with code mutations typically adopted for mutation testing and fault injection (e.g., the omission of statements). Moreover, the analysis of OpenStack bugs gave us insights on where to apply the injections (e.g., on method calls for controlling Nova, for performing SQL queries, etc.). \nEven if some categories of failures may have been over- or under-represented (e.g., the percentages for failures that were not detected or that propagated), our goal is to point out the existence of potential, critical classes of failures, despite possible errors in the estimates of the percentages. In our experiments, these classes were large enough to be considered a threat to cloud management platforms. \n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\\input{intro.tex}\n\n\n\\section{Overview on the research problem}\n\\label{sec:research_problem}\n\\input{research_problem.tex}\n\n\n\n\\section{Methodology}\n\\label{sec:methodology}\n\\input{methodology.tex}\n\n\n\\section{Experimental results}\n\\label{sec:experiments}\n\\input{experiments.tex}\n\n\n\\section{Related work}\n\\label{sec:related}\n\\input{related.tex}\n\n\n\\section{Experimental artifacts}\n\\label{sec:artifacts}\n\\input{artifacts.tex}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conclusion.tex}\n\n\n\\section*{Acknowledgments}\nThis work has been partially supported by the PRIN 2015 project ``GAUSS'' funded by MIUR (Grant n. 2015KWREMX\\_002) and by UniNA and Compagnia di San Paolo in the frame of Programme STAR. We are grateful to Alfonso Di Martino for his help in the early stage of this work.\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\subsection{Bug analysis}\n\\label{subsec:bug_analysis}\n\n\n\n\n\n\n\n\n\n\n \n \n \n\n\n \n \n \n\n\n\n\n\n\nA key aspect to perform software fault injection experiments is to inject representative software bugs \\cite{chillarege1996:generation-error-set,duraes2006emulation}. \nSince the body of knowledge on bugs in Python software \\cite{rodriguez2018if,orru2015curated}, the programming language of OpenStack, is relatively smaller compared to other languages, we seek for more insights about bugs in the OpenStack project. Therefore, we analyzed the OpenStack issue tracker on the \\emph{Launchpad} portal \\cite{openstack_launchpad}, by looking for bug-fixes at the source code level, in order to identify \\emph{bug patterns} \\cite{duraes2006emulation,pan2009toward,martinez2013automatically,zhong2018towards,tufano2018learning} for this project. From these patterns, we defined a set of bug types to be injected.\n\nWe went through the problem reports and inspected the related source code. We looked for reports where: (i) the root cause of the problem was a software bug, excluding build, packaging and installation issues; \n(ii) the problem had been marked with the highest severity level (i.e., the problem has a strong impact on OpenStack services); \n(iii) the problem was fixed, and the bug-fix was linked to the discussion. We manually analyzed a sample of 179 problem reports from the Launchpad, focusing on entries with importance set to ``\\emph{Critical}'', and with status set to ``\\emph{Fix Committed}'' or ``\\emph{Fix Released}'' (such that the problem report also includes a final solution shipped in OpenStack). Of these problem reports, we identified 113 reports that met all of the three criteria. We shared the full set of bug reports (see Section \\ref{sec:artifacts}).\n\nThe bugs encompass several areas of OpenStack, including: bugs that affected the service APIs exposed to users (e.g., \\emph{nova-api}); bugs that affected dictionaries and arrays, such as a wrong key used in {\\fontfamily{lmtt}\\selectfont image['imageId']}; bugs that affected SQL queries (e.g., database queries for information about instances in Nova); bugs that affected RPC calls between OpenStack subsystems (e.g., \\emph{rpc.cast} was omitted, or had a wrong topic or contents); bugs that affected calls to external system software, such as \\emph{iptables} and \\emph{dsnmasq}; bugs that affected pluggable modules in OpenStack, such as network protocol plugins and agents in Neutron.\nFigure~\\ref{fig:faults_nova_neutron} shows statistics about the bug types that we identified from the problem reports and their bug-fixes. The five most frequent bug types include the following ones.\n\n\n\n\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Wrong parameters value}: The bug was an incorrect method call inside OpenStack, where a wrong variable was passed to the method call. For example, this was the case of the Nova bug \\#1130718 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1130718}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/22431\/} by changing the exit codes passed through the parameter {\\fontfamily{lmtt}\\selectfont check\\_exit\\_code}).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Missing parameters}: A method call was invoked with omitted parameters (e.g., the method used a default parameter instead of the correct one). For example, this was the case of the Nova bug \\#1061166 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1061166}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/14240\/} by adding the parameter {\\fontfamily{lmtt}\\selectfont read\\_deleted='yes'} when calling the SQL Alchemy APIs).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Missing function call}: A method call was entirely omitted. For example, this was the case of the Nova bug \\#1039400 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1039400}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/12173\/} by adding and calling the new method \n\n\\noindent\n{\\fontfamily{lmtt}\\selectfont trigger\\_security\\_group\\_members\\_refresh}).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Wrong return value}: A method returned an incorrect value (e.g., {\\fontfamily{lmtt}\\selectfont None} instead of a Python object). For example, this was the case of the Nova bug \\#855030 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/855030}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/1930\/} by returning an object allocated through {\\fontfamily{lmtt}\\selectfont allocate\\_fixed\\_ip}).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Missing exception handlers}: A method call lacks exception handling. For example, this was the case of the Nova bug \\#1096722 ({\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1096722}}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/19069\/} by adding an exception handler for {\\fontfamily{lmtt}\\selectfont exception.InstanceNotFound}).\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fault injection}\n\\label{subsec:fault_injection}\n\nIn this study, we perform \\emph{software fault injection} to analyze the impact of software bugs \\cite{voas1997:predicting,chillarege1996:generation-error-set,natella2016assessing}. This approach deliberately introduces programming mistakes in the source code, by replacing parts of the original source code with faulty code. \nFor example, in \\figurename{}~\\ref{fig:workflow}, the injected bug emulates a missing optional parameter (a port number) to a function call, which may cause failure under certain conditions (e.g., a VM instance may not be reachable through an intended port).\nThis approach is based on previous empirical studies, which observed that the injection of code changes can realistically emulate software faults \\cite{daran1996software,chillarege1996:generation-error-set,andrews2005mutation}, in the sense that \\emph{code changes produce run-time errors that are similar to the ones produced by real software faults}.\nThis approach is motivated by the high efforts that would be needed for experimenting with hand-crafted bugs or with real past bugs: in these cases, every bug would require to carefully craft the specific conditions that trigger it (i.e., the topology of the infrastructure, the software configuration, and the hardware devices under which the bug surfaces). \nTo achieve a match between injected and real bugs, we focus the injection on the most frequent five types found by the bug analysis. These bug types allow us to cover all of the main areas of OpenStack (API, SQL, etc.), and suffice to generate a large and diverse set of faults over the codebase of OpenStack. \n\n\n\\begin{figure*}[!ht]\n \\begin{centering}\n \n \\includegraphics[width=0.8\\textwidth]{workflow_v2.pdf}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Overview of a fault injection experiment}\n \\label{fig:workflow}\n\\vspace{-5mm}\n\\end{figure*}\n\n\n\n\n\n\n\n\nWe emulate the bug types by mutating the existing code of OpenStack. \nThe \\figurename{}~\\ref{fig:workflow} shows the steps of a fault injection experiment. \nWe developed a tool to automate the bug injection process in Python code. The tool uses the \\emph{ast} Python module to generate an \\emph{abstract syntax tree} (AST) representation of the source code; then, it scans the AST by looking for relevant elements (function calls, expressions, etc.) where the bug types could be injected; it modifies the AST, by removing or replacing the nodes to introduce the bug; finally, it rewrites the modified AST into Python code, using the \\emph{astunparse} Python module. \nTo inject the bug types of Section~\\ref{subsec:fault_injection}, we modify or remove method calls and their parameters. We targeted method calls related to the bugs that we analyzed, by targeting calls to internal APIs for managing instances, volumes, and networks (e.g., which are denoted by specific keywords, such as \\emph{instance} and \\emph{nova} for the methods of the Nova subsystem). Wrong input and parameters are injected by wrapping the target expression into a function call, which returns at run-time a corrupted version of the expression based on its data type (e.g., a null reference in place of an object reference, or a negative value in place of an integer). Exceptions are raised on method calls according to a pre-defined list of exception types.\n\nThe tool inserts fault-injected statements into an \\emph{if} block, together with the original version of the same statements but in a different branch (as in step 2 in \\figurename{}~\\ref{fig:workflow}). \nThe execution of the fault-injected code is controlled by a \\emph{trigger} variable, which is stored in a shared memory area that is writable from an external program. This approach has been adopted for controlling the occurrence of failures during the tests. In the first phase (\\textbf{round 1}), we enable the fault-injected code, and we run a workload that exercises the service APIs of the cloud management system. During this phase, the fault-injected code could generate run-time errors inside the system, which will potentially lead to user-perceived failures. Afterward, in a second phase (\\textbf{round 2}), we disable the injected bug, and we execute the workload for a second time. This fault-free execution points out whether the scope of run-time errors (generated by the first phase) is limited to the service API invocations that triggered the buggy code (e.g., the bug only impacts on local session data). If failures still occur during the second phase, then the system has not able to handle the run-time errors of the first phase. Such failures point out the propagation of effects across the cloud management system (see \\S~\\ref{sec:research_problem}).\n\n\n\n\n\nWe implemented a workload generator to automatically exercise the service APIs of the main OpenStack sub-systems. The workload has been designed to cover several sub-systems of OpenStack and several types of virtual resources, in a similar way to integration test cases from the OpenStack project \\cite{openstack_tempest}. The workload creates VM instances, along with key pairs and a security group; attaches the instances to volumes; creates a virtual network, with virtual routers; and assigns floating IPs to connect the instances to the virtual network. Having a comprehensive workload allows us to point out propagation effects across sub-systems caused by bugs.\n\n\nThe experimental workflow is repeated several times. Every experiment injects a different fault, and only one fault is injected per experiment. Before a new experiment, we clean-up any potential residual effect from the previous experiment, in order to be able to relate failure to the specific bug that caused it. The clean-up re-deploys OpenStack removes all temporary files and processes and restores the database to its initial state. However, we do not perform these clean-up operations between the two workload rounds (i.e., no clean-up between the steps 6 and 8 of \\figurename{}~\\ref{fig:workflow}), since we want to assess the impact of residual side effects caused by the bug.\n\n\n\\begin{table}[t]\n \\begin{center}\n \\caption{Assertion check failures.}\n \\label{tab:assertion_and_description}\n \\vspace{-4mm}\n \n {\n \\scriptsize\n \n\n \n \\begin{tabulary}{\\columnwidth}{|L|L|}\n \n \\hline\n \\textbf{Name} & \\textbf{Description}\\\\\n \\hline\n FAILURE IMAGE ACTIVE & The created \\textit{image} does not transit into the \\textit{ACTIVE} state\\\\ \\hline\n FAILURE INSTANCE ACTIVE & The created \\textit{instance} does not transit into the \\textit{ACTIVE} state \\\\ \\hline\n FAILURE SSH & It is impossible to establish a \\textit{ssh} session to the created instance\\\\ \\hline\n FAILURE KEYPAIR & The creation of a \\textit{keypair} fails\\\\ \\hline\n FAILURE SECURITY GROUP & The creation of a \\textit{security group} and \\textit{rules} fails\\\\ \\hline\n FAILURE VOLUME CREATED & The creation of a \\textit{volume} fails\\\\ \\hline\n FAILURE VOLUME ATTACHED & Attaching a \\textit{volume} to an instance fails\\\\ \\hline\n FAILURE FLOATING IP CREATED & The creation of a \\textit{floating IP} fails\\\\ \\hline\n FAILURE FLOATING IP ADDED & Adding a \\textit{floating IP} to an instance fails\\\\ \\hline\n FAILURE PRIVATE NETWORK ACTIVE & The created \\textit{network} resource does not transit into the \\textit{ACTIVE} state\\\\ \\hline\n FAILURE PRIVATE SUBNET CREATED & The creation of a \\textit{subnet} fails\\\\ \\hline\n FAILURE ROUTER ACTIVE & The created \\textit{router} resource does not transit into the \\textit{ACTIVE} state\\\\ \\hline\n FAILURE ROUTER INTERFACE CREATED & The creation of a router interface fails\\\\ \\hline\n \n \n \n \\end{tabulary}\n }\n \n \\end{center}\n \n \\vspace{-0.65cm}\n \n\\end{table}\n\n\n\\subsection{Failure data collection}\n\\label{subsec:failure_data_collection}\n\nDuring the execution of the workload, we record inputs and outputs of service API calls of OpenStack. Any exception generated from the call (\\emph{API Errors}) is also recorded. In-between calls to service APIs, the workload also performs \\emph{assertion checks} on the status of the virtual resources, in order to point out failures of the cloud management system. \nIn the context of our methodology, assertion checks serve as \\emph{ground truth} about the occurrence of failures during the experiments. These checks are valuable to point out the cases in which a fault causes an error, but the system does not generate an API error (i.e., the system is unaware of the failure state). \nOur assertion checks are similar to the ones performed by the integration tests as test oracles \\cite{ju2013fault,openstack_instances_states}: they assess the connectivity of the instances through SSH and query the OpenStack API to check that the status of the instances, volumes and network is consistent with the expectation of the test cases. \nThe assertion checks are performed by our workload generator. For example, after invoking the API for creating a volume, the workload queries the volume status to check if it is available (\\emph{VOLUME CREATED assertion}). These checks are useful to find failures not notified through the API errors.\n\\tablename{}~\\ref{tab:assertion_and_description} describes the assertion checks.\n\n\nIf an API call generates an error, the workload is aborted, as no further operation is possible on the resources affected by the failure (e.g., no volume could be attached if the instance could not be created). In the case that the system fails without raising an exception (i.e., an assertion check highlights a failure, but the system does not generate an API error), the workload continues the execution (as a hypothetical end-user, being unaware of the failure, would do), regardless of failed assertion check(s). The workload generator records the outcomes of both the API calls and of the assertion checks. Moreover, we collect all the log files generated by the cloud management system. This data is later analyzed for understanding the behavior of the system under failure.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Failure analysis}\n\\label{subsec:failure_analysis}\n\nWe analyze fault injection experiments according to three perspectives discussed in Section~\\ref{sec:research_problem}. \nThe first perspective classifies the experiments \\emph{with respect to the type of failure that the system experiences}. The possible cases are the following ones.\n\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{API Error}: In these cases, the workload was not able to correctly execute, due to an exception raised by a service API call. In these cases, the cloud management system has been able to handle the failure in a fail-stop way, since the user is informed by the exception that the virtual resources could not be used, and it can perform recovery actions to address the failure. In our experiments, the workload stops on the occurrence of an exception, as discussed before.\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Assertion failure}: In these cases, the failure was not pointed out by an exception raised by a service API. The failure was detected by the assertion checks made by the workload in-between API calls, which found an incorrect state of virtual resources. In these cases, the execution of the workload was not interrupted, as no exception was raised by the service APIs during the whole experiment, and the service API did (apparently) work from the perspective of the user. These cases point out non-fail-stop behavior.\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Assertion failure(s), followed by an API Error}: In these cases, the failure was initially detected by assertion checks, which found an incorrect state of virtual resources in-between API calls. After the assertion check detected the failure, the workload continued the execution, by performing further service API calls, until an API error occurred in a later API call. These cases also point out issues at handling the failure, since the user is unaware of the failure state and cannot perform recovery actions.\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{No failure}: The injected bug did not cause a failure that could be perceived by the user (neither by API exceptions nor by assertion checks). It is possible that the effects of the bug were tolerated by the system (e.g., the system switched to an alternative execution path to provide the service); or, the injected source code was harmless (e.g., an uninitialized variable is later assigned before use). Since no failure occurred, these experiments are not further analyzed, as they do not allow to draw conclusions on the failure behavior of the system.\n\n\\vspace{2pt}\n\n\n\nFailed executions are further classified according to a second perspective, \\emph{with respect to the execution round in which the system experienced a failure}. The possible cases are the following ones.\n\n\n\\vspace{1pt}\n\\noindent\n$\\rhd$ \\textbf{Failure in the faulty round only}: In these cases, a failure occurred in the first (faulty) execution round (\\figurename{}~\\ref{fig:workflow}), in which a bug has been injected; and no failure is observed during the second (fault-free) execution round, in which the injected bug is disabled, and in which the workload operates on a new set of resources. This behavior is the likely outcome of an experiment since we are deliberately forcing a service failure only in the first round through the injected bug.\n\n\\vspace{1pt}\n\\noindent\n$\\rhd$ \\textbf{Failure in the fault-free round (despite the faulty round)}: \nThese cases are concerns for fault containment since the system is still experiencing failures despite the bug is disabled after the first round and the workload operates on a new set of resources. This behavior is due to residual effects of the bug that propagated through session state, persistent data, or other shared resources.\n\n\n\\vspace{1pt}\n\n\nFinally, the experiments with failures are classified from the perspective of \\emph{whether they generated logs able to indicate the failure}. In order to make more resilient a system, we are interested in whether it produces information for detecting failures and for triggering recovery actions. \nIn practice, developers are conservative at logging information for post-mortem analysis, by recording high volumes of low-quality log messages that bury the truly important information among many trivial logs of similar severity and contents, making it difficult to locate issues \\cite{zhu2015learning,li2017log,yuan2012improving}. Therefore, we cannot simply rely on the presence of logs to conclude that a failure was detected.\n\n\nTo clarify the issue, \\figurename{}~\\ref{fig:logging_stats} shows the distribution of the number of log messages in OpenStack across severity levels, \\textit{TRACE} to \\textit{CRITICAL}, during the execution of our workload generator, and \\emph{without} any failure. We can notice that all OpenStack components generate a large number of messages with severity \\textit{WARNING}, \\textit{INFO}, and \\textit{DEBUG} even when there is no failure. Instead, there are no messages of severity \\textit{ERROR} or \\textit{CRITICAL}. Therefore, even if a failure is logged with severity \\textit{WARNING} or lower, such log messages cannot be adopted for automated detection and recovery of the failure, as it is difficult to distinguish between ``informative'' messages and actual issues. Therefore, to evaluate the ability of the system to support recovery and troubleshooting through logs, we classify failures according to the presence of one or more \\emph{high-severity message} (i.e., \\emph{CRITICAL} or \\emph{ERROR}) recorded in the log files (\\textbf{logged failures}), or no such message (\\textbf{non-logged failures}). \n\n\n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.95\\columnwidth]{severity2.pdf}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Distribution of log messages severity during a fault-free execution of the workload.}\n \\label{fig:logging_stats}\n \\vspace{-6mm}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn \\cite{biha4}, B. Y. Chen established sharp inequality for a submanifold in a real space form involving intrinsic invariants of the submanifolds and squared mean curvature, the main extrinsic invariant and in \\cite{biha2}, B. Y. Chen obtained the same inequality for complex space form. After that many research articles \\cite{biha5, biha6, biha1} have been published by different authors for different submanifolds and ambient spaces in complex as well as in contact version. In this article we obtain these inequalities for submanifolds in Bochner Kaehler manifold.\n\n In \\cite{bishopwrap} Bishop and O'Neil initiated the thoery of warped product submanifold as a generalization of pseudo-Riemannian product manifold. In \\cite{chen warp} Chen introduced the notion of CR-warped products. In This paper we study the CR-warped product submanifolds of Bochner Kaehler manifolds.\n\\newline\n\\section{Preliminaries}\nLet $\\mathcal{W}$ be a $n$-dimensional submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ of dimension $2m$. Let $\\nabla$ and $\\overline{\\nabla}$ be the Levi-Civita connection on $\\mathcal{W}$ and $\\overline{\\mathcal{W}}$ respectively. Let $J$ be the complex structure on $\\overline{\\mathcal{W}}$. Then the Gauss and Weingarten formulas are given respectively by\n\\begin{eqnarray}\\label{a1}\n\\overline{\\nabla}_{X}Y = \\nabla_{X}Y + \\omega(X,Y),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{a2}\n\\overline{\\nabla}_{X}V = - B_{V}X + \\nabla_{X}^{\\perp}Y ,\n\\end{eqnarray}\nfor all $X, Y$ tangent to $\\mathcal{W}$ and vector field $V$ normal to $\\mathcal{W}$. Where $\\omega$, $\\nabla_{X}^{\\perp}$, $B_{V}$ denotes the second fundamental form, normal connection and the shape operator respectively. The second fundamental form and the shape operator are related by\n\\begin{eqnarray}\\label{a3}\ng(\\omega(X,Y), V) = g(B_{V}X, Y).\n\\end{eqnarray}\nLet $R$ be the curvature tensor of $\\mathcal{W}$, Then the Gauss equation is given by \\cite{biha4}\n\\begin{eqnarray*}\\label{a4}\n\\overline{R}(X,Y,Z,W) = R(X,Y,Z,W) + g(\\omega(X,W),\\omega(Y,Z)) - g(\\omega(X,Z),\\omega(Y,W))\n\\end{eqnarray*}\nfor any vector fields $X$, $Y$, $Z$, $W$ tangent to $\\mathcal{W}$.\n\nThe curvature tensor of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ is given by \\cite{biha7}\n\\begin{eqnarray}\\label{a5}\n\\overline{R}(X,Y,Z,W)\\nonumber &=& L(Y,Z)g(X,W) - L(X,Z)g(Y,W) + L(X,W)g(Y,Z) \\\\ \\nonumber && - L(Y,W)g(X,Z) + M(X,W)g(JX,W) - M(X,Z)g(JY,W)\\\\ \\nonumber && + M(X,W)g(JY,Z) - M(Y,W)g(JX,Z) \\\\ && - 2M(X,Y)g(JZ,W) - 2M(Z,W)g(JX,Y)\n\\end{eqnarray}\n\nwhere\n\\begin{eqnarray}\\label{a6}\nL(Y,Z) = \\frac{1}{2n+4}Ric(Y,Z) - \\frac{\\rho}{2(2n+2)(2n+4)}g(Y,Z),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{a7}\nM(Y,Z) = -L(Y,JZ),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{a8}\nL(Y,Z) = L(Z,Y),\\hspace{1cm} L(Y,Z) = L(JY, JZ), \\hspace{1cm} L(Y, JZ) = -L(JY,Z),\n\\end{eqnarray}\n$Ric$ and $\\rho$ are the Ricci tensor and scalar curvature of $\\mathcal{W}$.\n\nLet $x\\in \\mathcal{W}$ and $\\{e_{1}, ... , e_{n}\\}$ be an orthonormal basis of the tangent space $T_{x}\\mathcal{W}$ and $\\{e_{n+1}, ... , e_{2m}\\}$ be the orthonormal basis of $T^{\\perp}\\mathcal{W}$. We denote by $\\mathcal{H}$, the mean curvature vector at $x$, that is\n\\begin{eqnarray}\\label{a9}\n\\mathcal{H}(x) = \\frac{1}{n}\\sum_{i=1}^{n}\\omega(e_{i},e_{i}),\n\\end{eqnarray}\nAlso, we set\n\\begin{eqnarray*}\\label{a10}\n\\omega_{ij}^{r} = g(\\omega(e_{i},e_{j}),e_{r}), \\hspace{1cm} i,j \\in \\{ 1, ... , n\\},\\hspace{.3cm} r \\in \\{n+1, ... ,2m\\}\n\\end{eqnarray*}\nand\n\\begin{eqnarray}\\label{a11}\n\\|\\omega\\|^{2} = \\sum_{i,j=1}^{n}(\\omega(e_{i},e_{j}), \\omega(e_{i},e_{j})).\n\\end{eqnarray}\nFor any $x \\in \\mathcal{W}$ and $X \\in T_{x}\\mathcal{W}$, we put $JX = TX + FX$, where $TX$ and $FX$ are the tangential and normal components of $JX$, respectively.\n\n We denote by\n\\begin{eqnarray*}\\label{a12}\n\\|T\\|^{2} = \\sum_{i,j=1}^{n}g^{2}(Te_{i}, e_{j}).\n\\end{eqnarray*}\nLet $\\mathcal{W}$ be a Riemannian manifold. Denote by $\\mathcal{K}(\\pi)$ the sectional curvature of $\\mathcal{W}$ of the plane section $\\pi \\subset T_{x}\\mathcal{W}, x\\in \\mathcal{W}$. The scalar curvature $\\rho$ for an orthonormal basis$ \\{e_{1}, e_{2}, ..., e_{n}\\}$ of the tangent space $T_{x}\\mathcal{W}$ at $x$ is defined by\n\\begin{eqnarray*}\\label{a121}\n\\rho(x) = \\sum_{i2,\\\\\n\\omega_{ij}^{r} = 0, \\forall i \\neq j,\\hspace{.5cm}i,j = 3, ..., 2m, \\hspace{.5cm}r= n+1, ..., 2m,\\\\\n\\omega_{11}^{r}+\\omega_{22}^{r} = 0, \\forall r=n+2, ..., 2m,\\hspace{2cm}\\\\\n\\omega_{11}^{n+2}+\\omega_{22}^{n+1} = ... = \\omega_{11}^{m}+\\omega_{22}^{m}=0.\\\\\n\\end{cases}$\n\nNow, if we choose $e_{1},e_{2}$ such that $\\omega_{12}^{n+1}$= 0 and we denote by $\\alpha = \\omega_{11}^{r}, \\beta = \\omega_{22}^{r}$, $\\xi = \\omega_{33}^{n+1} = ... = \\omega_{33}^{r}$. Therefore by choosing the suitable orthonormal basis the shape operators take the desired forms.\n\\end{proof}\nWe conclude the following corollary from this theorem.\n\\begin{corollary}\nLet $\\mathcal{W}$ be a submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ which is Einstein. Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p21}\n\\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3\\|T\\|^{2}}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6\\lambda}{2(2n+4)} \\|T\\|^{2}.\n\\end{eqnarray*}\nThe equality at a point $x \\in \\mathcal{W}$ holds iff there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{t2}) and (\\ref{t3}).\n\\end{corollary}\nSimilarly, in case if $\\mathcal{W}$ is a slant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. We have the following theorem\n\\begin{theorem}\nLet $\\mathcal{W}$ be a slant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p22}\n\\nonumber \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3cos^{2}\\theta}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6}{2(2n+4)}Ric(e_{i},Je_{j})cos\\theta.\\\\\n\\end{eqnarray*}\nEquality holds if and only if there exists an orthonormal basis $\\{e_{1} ,e_{2}, ... , e_{n}\\}$ of $T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1}, e_{n+2} , ... , e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that the shape operator takes the following forms\n\n\\begin{eqnarray}\\label{p23}\nB_{n+1} =\n \\begin{pmatrix}\n \\alpha & 0 & 0 & \\cdots & 0 \\\\\n 0 & \\beta & 0 & \\cdots & 0 \\\\\n 0 & 0 & \\xi &\\cdots & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & 0 & \\cdots & \\xi\n \\end{pmatrix} , \\alpha+\\beta = \\xi\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{p24}\nB_{r} =\n \\begin{pmatrix}\n \\omega_{11}^{r} & \\omega_{12}^{r} & 0 & \\cdots & 0 \\\\\n \\omega_{12}^{r} & - \\omega_{11}^{r} & 0 & \\cdots & 0 \\\\\n 0 & 0 & 0 &\\cdots & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & 0 & \\cdots & 0\n \\end{pmatrix} , r = n+2, ..., 2m.\n\\end{eqnarray}\n\\end{theorem}\nFrom this theorem, following corollaries can be easily deduced.\n\\begin{corollary}\nLet $\\mathcal{W}$ be a slant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$, which is Einstein . Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p25}\n\\nonumber \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3cos^{2}\\theta}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6\\lambda}{2(2n+4)}cos^{2}\\theta.\\\\\n\\end{eqnarray*}\nThe equality holds at a point $x \\in \\mathcal{W}$ if and only if there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{p23}) and (\\ref{p24}).\n\\end{corollary}\n\\begin{corollary}\nLet $\\mathcal{W}$ be a invariant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ . Then, for each point $x\\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p26}\n \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6}{2(2n+4)}Ric(e_{i}, Je_{j}).\n\\end{eqnarray*}\nThe equality at a point $x\\in \\mathcal{W}$ holds iff there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{p23}) and (\\ref{p24}).\n\\end{corollary}\n\\begin{corollary}\nLet $\\mathcal{W}$ be a anti-invariant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ . Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p26}\n \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2}.\n\\end{eqnarray*}\nThe equality at a point $x \\in \\mathcal{W}$ holds iff there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{p23}) and (\\ref{p24}).\n\\end{corollary}\n\\section{Warped product of CR-submanifolds of Bochner Kaehler manifolds}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be the warped product CR-submanifolds of Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ such that the invariant distribution is $D = T\\mathcal{W}_{T}$ and anti-invariant distribution is $D^{\\perp} = T\\mathcal{W}_{\\perp}$, where $f: \\mathcal{W}_{T} \\longrightarrow \\mathbb{R}$. Then the metric $g$ on $\\mathcal{W}$ is given by \\cite{chen warp}\n\\begin{equation*}\\label{w1}\ng(X,Y) = \\langle \\pi_{*}X, \\pi_{*}Y\\rangle + (f\\circ\\pi)^{2}\\langle \\sigma_{*}X, \\sigma_{*}Y\\rangle\n\\end{equation*}\nwhere $\\pi$ and $\\sigma$ are the projection maps from $\\mathcal{W}$ onto $\\mathcal{W}_{T}$ and $\\mathcal{W}_{\\perp}$ respectively.\n\n\nIt is easy to see that\n\\begin{eqnarray}\\label{w6}\nT\\mathcal{W} = D \\oplus D^{\\perp}\\hspace{.3cm} \\text{and} \\hspace{.3cm} T^{\\perp}\\mathcal{W} = JD^{\\perp}\\oplus \\nu,\n\\end{eqnarray}\nwhere $\\nu$ is the orthogonal distribution to $JD^{\\perp}$ in the normal bundle $T^{\\perp}\\mathcal{W}.$\n\nFrom (\\ref{w6}), we can write\n\\begin{eqnarray*}\\label{w7}\n\\omega(X,Y) = \\omega_{JD^{\\perp}}(X,Y) + \\omega_{\\nu}(X,Y)\n\\end{eqnarray*}\nAlso for warped product submanifold $\\mathcal{W}$ of $\\overline{\\mathcal{W}}$, we have \\cite{chen warp}\n\\begin{eqnarray}\\label{w8}\n\\nabla_{X}Z = X(logf)Z = \\frac{X(f)}{f}Z\n\\end{eqnarray}\nfor any vector fields $X \\in D$ and $Z \\in D^{\\perp}$.\n\nFurther, we can decompose $(\\overline{\\nabla}_{X}J)Y$ into the tangential and normal components as under\n\\begin{eqnarray}\\label{w9}\n(\\overline{\\nabla}_{X}J)Y = \\mathcal{P}_{X}Y + \\mathcal{Q}_{X}Y\n\\end{eqnarray}\nwhere $\\mathcal{P}_{X}Y$ and $\\mathcal{Q}_{X}Y$ denotes the tangential and normal components of $(\\overline{\\nabla}_{X}J)Y$\n\nFirst we prove the following lemma\n\\begin{lemma}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be a CR-warped product submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. Then we have\n\\begin{equation*}\n\\omega_{JD^{\\perp}}(JX,Z) = J\\mathcal{P}_{Z}JX + X(log f)JZ\n\\end{equation*}\n\\begin{equation*}\ng(\\mathcal{P}_{Z}JX, W) = g(\\mathcal{Q}_{Z}X, JW)\n\\end{equation*}\n and\n\\begin{equation*}\ng(\\omega(JX,Z), J\\omega(X,Z)) -\\|\\omega_{\\nu}(X,Z)\\|^{2} = g(\\mathcal{Q}_{Z}X, J\\omega_{\\nu}(X,Z))\n\\end{equation*}\n for $X \\in D$ and $Z \\in D^{\\perp}.$\n\\end{lemma}\n\\begin{proof}\nFrom Gauss equation, we have\n\\begin{equation*}\\label{w10}\n\\nabla_{Z}JX + \\omega(JX,Z) -J(\\nabla_{Z}X) - J\\omega(X,Z) = \\mathcal{P}_{Z}X + \\mathcal{Q}_{Z}X\n\\end{equation*}\nUsing (\\ref{w8}), we infer\n\\begin{equation*}\\label{w11}\n\\omega(JX,Z)= \\mathcal{P}_{Z}X + \\mathcal{Q}_{Z}X + J[X(log f)Z] + J\\omega(X,Z) - JX(log f)Z .\n\\end{equation*}\nReplace $X$ by $JX$, we get\n\\begin{equation*}\\label{w12}\n-\\omega(X,Z)= \\mathcal{P}_{Z}JX + \\mathcal{Q}_{Z}JX + JX(log f)JZ + J\\omega(JX,Z) + X(log f)Z.\n\\end{equation*}\nWe can write the above equation as\n\\begin{equation}\\label{w13}\n-\\omega(X,Z)= \\mathcal{P}_{Z}JX + \\mathcal{Q}_{Z}JX + JX(log f)JZ + J\\omega_{JD^{\\perp}}(JX,Z) + J\\omega_{\\nu}(JX,Z) + X(log f)Z.\n\\end{equation}\nOn comparing the tangential components, we obtain\n\\begin{equation*}\\label{w14}\n\\mathcal{P}_{Z}JX + J\\omega_{JD^{\\perp}}(JX,Z) + X(log f)Z = 0,\n\\end{equation*}\nor\n\\begin{equation*}\\label{w15}\nJ\\omega_{JD^{\\perp}}(JX,Z) = -\\mathcal{P}_{Z}JX - X(log f)Z\n\\end{equation*}\nwhich shows that\n\\begin{equation}\\label{w16}\n\\omega_{JD^{\\perp}}(JX,Z) = J\\mathcal{P}_{Z}JX + X(log f)JZ,\n\\end{equation}\nfor $X \\in D$ and $Z \\in D^{\\perp}$.\n\nAgain on comparing the normal components in (\\ref{w13}), we have\n\\begin{equation*}\\label{w17}\n-\\omega(X,Z)= \\mathcal{Q}_{Z}JX + JX(log f)JZ + J\\omega_{\\nu}(JX,Z)\n\\end{equation*}\nfrom which we conclude that\n\\begin{equation*}\\label{w18}\n\\omega(JX,Z)= \\mathcal{Q}_{Z}X + X(log f)JZ + J\\omega_{\\nu}(X,Z)\n\\end{equation*}\nor\n\\begin{equation}\\label{w19}\n\\omega(JX,Z) - J\\omega_{\\nu}(X,Z)= \\mathcal{Q}_{Z}X + X(log f)JZ\n\\end{equation}\nBy taking the inner product (\\ref{w19}) with $JW$, we get\n\\begin{equation}\\label{w20}\ng(\\omega_{JD^{\\perp}}(JX,Z), JW) = g(\\mathcal{Q}_{Z}X,JW) + X(log f)g(JZ,JW)\n\\end{equation}\nFurther using (\\ref{w16}) in (\\ref{w20}), we have\n\\begin{equation*}\\label{w21}\ng(\\mathcal{P}_{Z}JX, W) + X(log f)g(Z, W) = g(\\mathcal{Q}_{Z}X,JW) + X(log f)g(Z, W)\n\\end{equation*}\nfrom which we conclude that\n\\begin{equation}\\label{w22}\ng(\\mathcal{P}_{Z}JX, W) = g(\\mathcal{Q}_{Z}X,JW)\n\\end{equation}\nAlso, by taking the inner product of (\\ref{w19}) with $J\\omega(X,Z)$, we find\n\\begin{equation*}\\label{w23}\ng(\\omega(JX,Z),J\\omega(X,Z)) - \\|\\omega_{\\nu}(X,Z)\\|^{2} = g(\\mathcal{Q}_{Z}X,J\\omega_{\\nu}(X,Z)).\n\\end{equation*}\n\\end{proof}\n\\begin{theorem}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be a warped product CR-submanifolds of Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ with $\\mathcal{P}_{D^{\\perp}}D \\in D$, then the squared norm of second fundamental form of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ satisfies the following inequality\n\\begin{equation*}\\label{w24}\n\\|\\omega\\|^{2} \\geq \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + q\\|grad_{D}(log f)\\|^{2}.\n\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\{X_{1}, ..., X_{p}, X_{p+1} = JX_{1}, ..., X_{2p}= JX_{p}\\}$ be a local orthonormal frame of vector fields on $N_{T}$ and $\\{Z_{1}, ..., Z_{q}\\}$ be a local orthonormal frame of vector fields on $N_{\\perp}$, where $2p+q = n$. Then we have\n\\begin{eqnarray*}\\label{w25}\n\\nonumber \\|\\omega\\|^{2} = \\sum_{i,j=1}^{2p}g(\\omega(X_{i},X_{j}), \\omega (X_{i},X_{j})) + \\sum_{\\alpha=1}^{q}\\sum_{j=1}^{2p}g(\\omega(X_{i},Z_{\\alpha}),\\omega(X_{i},Z_{\\alpha}))\\\\ +\\sum_{\\alpha,\\beta =1}^{q}g(\\omega(Z_{\\alpha},Z_{\\beta}), \\omega(Z_{\\alpha},Z_{\\beta}))\n\\end{eqnarray*}\nfrom above equation we can say that\n\\begin{eqnarray*}\\label{w26}\n \\|\\omega\\|^{2} \\geq \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}g(\\omega(X_{i},Z_{\\alpha}),\\omega(X_{i},Z_{\\alpha}))\n\\end{eqnarray*}\nNow from (\\ref{w16}), we have\n\\begin{eqnarray*}\\label{w27}\n \\|\\omega\\|^{2} \\geq \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}g(J\\mathcal{P}_{Z_{\\alpha}}X_{i} - JX_{i}(log f)JZ_{\\alpha}, J\\mathcal{P}_{Z_{\\alpha}}JX_{i}- JX_{i}(log f)JZ_{\\alpha})\n\\end{eqnarray*}\nIn view of the assumption $\\mathcal{P}_{D^{\\perp}}D \\in D$, we have\n\\begin{eqnarray*}\\label{w28}\n\\nonumber\\|\\omega\\|^{2} &\\geq& \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}\\bigg[g(J\\mathcal{P}_{Z_{\\alpha}}X_{i} , J\\mathcal{P}_{Z_{\\alpha}}X_{i}) + g(JX_{i}(log f)JZ_{\\alpha}, JX_{i}(log f)JZ_{\\alpha})\\bigg]\n\\nonumber \\\\ &=& \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}\\bigg[g(\\mathcal{P}_{Z_{\\alpha}}X_{i}, \\mathcal{P}_{Z_{\\alpha}}X_{i}) + (JX_{i}(log f))^{2}g(Z_{\\alpha},Z_{\\alpha})\\bigg]\n\\nonumber \\\\ &=& \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + \\sum_{j=1}^{2p}\\|JX_{i}(log f)\\|^{2}q\n \\\\ &=& \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + q\\|grad_{D}(log f)\\|^{2}\n\\end{eqnarray*}\nwhere $grad_{D}$ denotes the gradient of some function on the distribution $D$.\n\nThus we have\n\\begin{eqnarray*}\\label{w29}\n\\|\\omega\\|^{2} \\geq \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + q\\|grad_{D}(log f)\\|^{2}.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{theorem}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be a compact orientable warped product CR-submanifold of Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. If $\\mathcal{P}_{D^{\\perp}}D \\in D$ and $B_{\\nabla_{JX_{i}}^{\\perp}JZ}X_{i} = B_{\\nabla_{X_{i}}^{\\perp}JZ}JX_{i}$, then we have $\\rho \\leq 0$, and the equality holds iff $grad_{D}(log f) = 0.$\n\\end{theorem}\n\\begin{proof}\nLet $X \\in D$, $Z \\in D^{\\perp}$, then from (\\ref{a5}), we have\n\\begin{eqnarray}\\label{w33}\n\\overline{R} (X, JX,Z,JZ) &=& -2M(X,JX)g(Z,Z) - 2M(Z,JZ)g(X,X)\n\\end{eqnarray}\n Now Codazzi equation is\n\\begin{eqnarray*}\\label{w34}\n\\nonumber \\bigg[\\overline{R}(X,Y)Z\\bigg]^{\\perp} &=& \\bigg \\{ \\nabla_{X}^{\\perp}\\omega(Y,Z) - \\omega(\\nabla_{X}Y,Z) - \\omega(Y,\\nabla_{X}Z)\\bigg \\}\n\\\\ \\nonumber & & - \\bigg \\{ \\nabla_{Y}^{\\perp}\\omega(X,Z) - \\omega(\\nabla_{Y}X,Z) - \\omega(X,\\nabla_{Y}Z)\\bigg \\}\n\\end{eqnarray*}\nIn view of the last equation we may write\n\\begin{eqnarray}\\label{w35}\n\\nonumber \\overline{R}(X, JX, Z, JZ) &=& g( \\nabla_{X}^{\\perp}\\omega(JX,Z) - \\omega(\\nabla_{X}JX,Z) - \\omega(JX,\\nabla_{X}Z),JZ)\n\\\\ & & - g(\\nabla_{JX}^{\\perp}\\omega(X,Z) - \\omega(\\nabla_{JX}X,Z) - \\omega(X,\\nabla_{JX}Z),JZ)\n\\end{eqnarray}\nWe now compute each term of (\\ref{w35}). First we have\n\\begin{eqnarray}\\label{w36}\nXg(\\omega(JX,Z),JZ) = g(\\overline{\\nabla}_{X}\\omega(JX,Z), JZ) + g(\\omega(JX,Z),\\overline{\\nabla}_{X}JZ)\n\\end{eqnarray}\n Using Weingarten formula we have\n\\begin{eqnarray}\\label{w37}\ng(\\overline{\\nabla}_{X}^{\\perp}\\omega(JX,Z), JZ) = Xg(\\omega(JX,Z),JZ) -g(\\omega(JX,Z),\\overline{\\nabla}_{X}JZ)\n\\end{eqnarray}\nNow from (\\ref{w19})\n\\begin{eqnarray}\\label{w38}\n\\omega(JX, Z) - J\\omega_{\\nu}(X,Z) = \\mathcal{Q}_{Z}X + X(log f)JZ\n\\end{eqnarray}\nTaking the inner product of (\\ref{w38}) with $JZ$, we have\n\\begin{eqnarray}\\label{w39}\ng(\\omega(JX, Z),JZ) - g(J\\omega_{\\nu}(X,Z), JZ) = g(\\mathcal{Q}_{Z}X, JZ) + X(log f)g(JZ,JZ)\n\\end{eqnarray}\nCombining (\\ref{w22}) and (\\ref{w39}), we get\n\\begin{eqnarray}\\label{w41}\ng(\\omega(JX, Z),JZ) = g(\\mathcal{\\mathcal{P}}_{Z}JX, Z) + X(log f)\\|Z\\|^{2}\n\\end{eqnarray}\nMoreover\n\\begin{eqnarray*}\ng(\\omega(JX,Z), JZ) = X(log f)g(Z,Z)\n\\end{eqnarray*}\nHence we have\n\\begin{eqnarray}\\label{w42}\n\\nonumber Xg(\\omega(JX,Z), JZ) &=& X\\bigg\\{ X( log f) g(Z,Z) \\bigg\\}\n\\\\ \\nonumber &=& X\\big( X(log f)\\big) g(Z,Z)+ 2X(log f)g(Z,\\nabla_{X}Z)\n\\\\ \\nonumber &=& X\\big( X(log f)\\big)\\|Z\\|^{2}+ 2\\big( X(log f)\\big)^{2}\\|Z\\|^{2}\n\\\\ &=& \\bigg \\{ X (X(log f))+ 2( X(log f))^{2} \\bigg \\} \\|Z\\|^{2}\n\\end{eqnarray}\n From (\\ref{w37}) and (\\ref{w42}), we get\n\\begin{eqnarray}\\label{w43}\ng(\\nabla_{X}^{\\perp}\\omega(JX,Z), JZ) = \\bigg \\{ X (X(log f))+ 2( X(log f))^{2} \\bigg \\} \\|Z\\|^{2} - g(\\omega(JX,Z), \\overline{\\nabla}_{X}JZ)\n\\end{eqnarray}\nReplacing $X$ by $JX$ in the above equation , we find\n\\begin{eqnarray}\\label{w44}\n-g(\\nabla_{JX}^{\\perp}\\omega(X,Z), JZ) = \\bigg \\{ JX (JX(log f))+ 2( JX(log f))^{2} \\bigg \\} \\|Z\\|^{2} + g(\\omega(X,Z), \\overline{\\nabla}_{JX}JZ)\n\\end{eqnarray}\nAlso using (\\ref{w16}) and $\\mathcal{P}_{D^{\\perp}} D \\in D$, we conclude that\n\\begin{eqnarray}\\label{w45}\ng(\\omega_{JD^{\\perp}}(JX, \\nabla_{X}Z), JZ ) = g(X(log f)J\\nabla_{X}Z,JZ) = (X(log f))^{2}g(Z,Z) = (X(log f))^{2}\\|Z\\|^{2}\n\\end{eqnarray}\n Replacing $X$ by $JX$ in the above equation, we find\n\\begin{eqnarray}\\label{w46}\ng(\\omega_{JD^{\\perp}}(X, \\nabla_{JX}Z), JZ ) = -(JX(log f))^{2}\\|Z\\|^{2}\n\\end{eqnarray}\nAgain using (\\ref{w16}), we get\n\\begin{eqnarray*}\\label{w47}\n\\omega_{JD^{\\perp}}( \\nabla_{JX}X), Z ) = J\\mathcal{P}_{Z}\\nabla_{JX}X- J\\nabla_{JX}X(log f)JZ\n\\end{eqnarray*}\nor\n\\begin{eqnarray*}\\label{w48}\ng(\\omega_{JD^{\\perp}}( \\nabla_{JX}X), JZ ) = g(\\mathcal{P}_{Z}\\nabla_{JX}X, Z)- J\\nabla_{JX}X(log f)\\|Z\\|^{2}\n\\end{eqnarray*}\nThe above equation can be written as\n\\begin{eqnarray*}\\label{w49}\ng(\\omega( \\nabla_{JX}X), JZ ) = g(\\mathcal{P}_{Z}\\nabla_{JX}X, Z)- J\\nabla_{JX}X(log f))\\|Z\\|^{2}\n\\end{eqnarray*}\n But $N_{T}$ is totally geodesic in $\\overline{N}$ which implies that $\\nabla_{JX}X \\in D$. Hence $\\mathcal{P}_{Z}J\\nabla_{JX}X \\in D$. This makes the first term in the above equation zero and hence we have\n\\begin{eqnarray}\\label{w49}\ng(\\omega( \\nabla_{JX}X), JZ ) = - J\\nabla_{JX}X(log f))\\|Z\\|^{2}\n\\end{eqnarray}\nSimilarly on replacing $X$ by $JX$ in the above equation, we have\n\\begin{eqnarray*}\\label{w50}\ng(\\omega( \\nabla_{X}JX), JZ ) = - J\\nabla_{X}JX(log f))\\|Z\\|^{2}\n\\end{eqnarray*}\n Using Gauss equation, the last equation simplifies to\n\\begin{eqnarray}\\label{w51}\ng(\\omega( \\nabla_{X}JX), JZ ) = \\nabla_{X}X(log f)g(Z,Z) + \\nabla_{JX}JX(log f)g(Z,Z) - J\\nabla_{JX}X(log f)g(Z,Z)\n\\end{eqnarray}\n Putting (\\ref{w43})$\\sim$(\\ref{w51}) into (\\ref{w35}), we get\n\\begin{eqnarray}\\label{w52}\n\\nonumber \\overline{R}(X,JX,Z,JZ ) &=& \\bigg \\{ X(X(log f)) + 2(X(log f))^{2} \\bigg \\}\\|Z\\|^{2} - g(\\omega(JX,Z), \\nabla_{X}^{\\perp}JZ)\n\\nonumber \\\\ & & - \\nabla_{X}X(log f)\\|Z\\|^{2} - \\nabla_{JX}JX(log f)\\|Z\\|^{2} + J\\nabla^{JX}X(log f)\\|Z\\|^{2}\n\\nonumber \\\\ & & - (X(log f))^{2}\\|Z\\|^{2} + \\bigg \\{ JX(JX(log f)) + 2(JX(log f))^{2} \\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + g(\\omega(X,Z), \\nabla_{JX}^{\\perp}JZ) - J\\nabla_{JX}X(log f)\\|Z\\|^{2}\n\\end{eqnarray}\nFrom(\\ref{w33}) and (\\ref{w52})\n\\begin{eqnarray*}\\label{w53}\n\\noindent& &\\nonumber - 2M(X,JX)g(Z,Z) - 2M(Z,JZ)g(X,X) \\hspace{4cm}\n\\nonumber \\\\ \\text{ } = & &\\bigg \\{ X(X(log f)) + 2(X(log f))^{2} \\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & - g(\\omega(JX,Z), \\nabla_{X}^{\\perp}JZ) - \\nabla_{X}X(log f)\\|Z\\|^{2} - \\nabla_{JX}JX(log f)\\|Z\\|^{2}\n\\nonumber \\\\ & & - (X(log f))^{2}\\|Z\\|^{2} + \\bigg \\{ JX(JX(log f)) + 2(JX(log f))^{2} \\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + g(\\omega(X,Z), \\nabla_{JX}^{\\perp}JZ) - (JX(log f))^{2}\\|Z\\|^{2}\n\\end{eqnarray*}\nPutting $X = X_{i}$ and taking summation from 1 to $p$, we drive\n\\begin{eqnarray*}\\label{w54}\n\\noindent & &\\nonumber -2\\|Z\\|^{2}\\sum_{i=1}{p}M (X_{i}, JX_{i}) - 2M(Z, JZ)p\n\\nonumber \\\\ \\text{ } = & & \\sum_{i=1}^{p}\\bigg \\{ X_{i}(X_{i}(log f)) + JX_{i}(JX_{i}(log f)) - \\nabla_{X_{i}}X_{i}(log f)) - \\nabla_{JX_{i}}JX_{i}(log f)\\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + \\sum_{i=1}^{p}\\bigg \\{ (X_{i}(log f))^{2} + (JX_{i}(log f))^{2}\\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + \\sum_{i=1}{p}\\bigg[ g(\\omega(X_{i},Z), \\nabla_{JX_{i}}^{\\perp}JZ) - g(\\omega(JX_{i},Z), \\nabla_{X_{i}}^{\\perp}JZ)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\n from which we have\n\\begin{eqnarray*}\\label{w55}\n\\noindent \\nonumber & &-2\\|Z\\|^{2}\\sum_{i=1}{p} M (X_{i}, JX_{i}) - 2M(Z, JZ)p\n\\nonumber \\\\ & =& \\Delta_{D}(log f)\\|Z\\|^{2} + \\|grad_{D}(log f)\\|^{2}\\|Z\\|^{2}\n\\nonumber \\\\ & &+ \\sum_{i=1}{p}\\bigg[ g(\\omega(X_{i},Z), \\nabla_{JX_{i}}^{\\perp}JZ) - g(\\omega(JX_{i},Z), \\nabla_{X_{i}}^{\\perp}JZ)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\nUsing (\\ref{a6}) and (\\ref{a7}) in the last equation we arrive at\n\\begin{eqnarray*}\\label{w56}\n\\noindent \\nonumber \\frac{-1}{n+2}\\sum_{i=1}^{p}\\bigg[\\|Z\\|^{2}Ric(X_{i}, X_{i}) + \\|X_{i}\\|^{2} Ric(Z,Z)\\bigg] + \\frac{\\rho \\|X_{i}\\|^{2}\\|Z\\|^{2}}{2(n+1)(n+2)}\n\\nonumber \\\\ = \\Delta_{D}(log f)\\|Z\\|^{2} + \\|grad_{D}(log f)\\|^{2}\\|Z\\|^{2}\n\\nonumber \\\\+ \\sum_{i=1}^{p}\\bigg[ g(\\omega(X_{i},Z), \\nabla_{JX_{i}}^{\\perp}JZ) - g(\\omega(JX_{i},Z), \\nabla_{X_{i}}^{\\perp}JZ)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\nfrom which we have\n\\begin{eqnarray*}\\label{w58}\n\\noindent \\nonumber \\frac{-1}{n+2}\\bigg[\\|Z\\|^{2}\\sum_{i=1}^{p}Ric(X_{i}, X_{i}) + p Ric(Z,Z)\\bigg] + \\frac{\\rho p\\|Z\\|^{2}}{2(n+1)(n+2)}\n\\nonumber \\\\ = \\Delta_{D}(log f)\\|Z\\|^{2} + \\|grad_{D}(log f)\\|^{2}\\|Z\\|^{2}\n\\nonumber \\\\+ \\sum_{i=1}^{p}\\bigg[ g(B_{\\nabla_{JX_{i}}^{\\perp} JZ}X_{i}, Z) - g(B_{\\nabla_{X_{i}}^{\\perp} JZ}JX_{i}, Z)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\nSince by assumption, we have $B_{\\nabla_{JX_{i}}^{\\perp} JZ}X_{i} = B_{\\nabla_{X_{i}}^{\\perp} JZ}JX_{i} $, then (\\ref{w58}) becomes\n\\begin{eqnarray*}\\label{w59}\n\\noindent \\nonumber \\frac{-1}{n+2}\\bigg[\\sum_{i=1}^{p}Ric(X_{i}, X_{i}) +\\frac{p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg] + \\frac{p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)}\n\\nonumber \\\\ = \\Delta_{D}(log f) + \\|grad_{D}(log f)\\|^{2}\n\\end{eqnarray*}\nIntegrating both sides and using Green's equation, the last equation simplifies to\n\\begin{eqnarray}\\label{w60}\n\\nonumber\\frac{-1}{n+2}\\int\\bigg[\\sum_{i=1}^{p}Ric(X_{i}, X_{i}) +\\frac{p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv + \\int\\frac{p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)}dv\n\\\\ = \\int \\|grad_{D}(log f)\\|^{2}dv\n\\end{eqnarray}\nSimilarly we have\n\\begin{eqnarray}\\label{w61}\n\\frac{-1}{n+2}\\int\\bigg[\\sum_{i=1}^{p}Ric(JX_{i}, JX_{i}) +\\frac{p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv + \\int\\frac{p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)}dv \\nonumber \\\\ = \\int \\|grad_{D}(log f)\\|^{2}dv\n\\end{eqnarray}\nAdding (\\ref{w60}) and (\\ref{w61}), we find\n\\begin{eqnarray*}\\label{w62}\n\\frac{-1}{n+2}\\int\\bigg[\\sum_{i=1}^{p}Ric(X_{i}, X_{i})+ \\sum_{i=1}^{p}Ric(JX_{i}, JX_{i}) +\\frac{2p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv \\nonumber \\\\ + \\int\\frac{2p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)} dv = 2\\int \\|grad_{D}(log f)\\|^{2}dv\n\\end{eqnarray*}\n from which we have\n\\begin{eqnarray*}\\label{w63}\n\\frac{-1}{n+2}\\int\\bigg[\\rho_{D} +\\frac{2p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv + \\int\\frac{p \\rho \\|Z\\|^{2}}{(n+1)(n+2)} dv = 2\\int \\|grad_{D}(log f)\\|^{2} dv\n\\end{eqnarray*}\n where $\\rho _{D}$ is the scalar curvature of distribution $D$.\n Further replacing $Z$ by $Z_{\\alpha}$ and taking summation from $1$ to $q$ on both sides.\n As\n\\begin{eqnarray*}\\label{w64}\nq\\int \\|grad_{D}(log f)\\|^{2}dv \\geq 0\n\\end{eqnarray*}\nwe conclude that\n\\begin{eqnarray*}\\label{w65}\n\\frac{-1}{n+2}\\int\\bigg[q\\rho_{D} +2p\\rho_{D^{\\perp}}\\bigg]dv + \\int\\frac{pq^{2} \\rho}{(n+1)(n+2)} dv\\geq 0\n\\end{eqnarray*}\nThis shows that\n\\begin{eqnarray*}\\label{w66}\n\\frac{pq^{2}}{(n+1)(n+2)}\\int \\rho dv\\geq \\frac{1}{n+2}\\int\\bigg[q\\rho_{D} +2p\\rho_{D^{\\perp}}\\bigg]dv\n\\end{eqnarray*}\nor\n\\begin{eqnarray}\\label{w67}\n\\int \\rho dv \\geq (n+1)\\int\\bigg[\\frac{\\rho_{D}}{pq} +\\frac{2(n+1)}{q^{2}}\\rho_{D^{\\perp}}\\bigg]dv\n\\end{eqnarray}\nThus we have\n\\begin{eqnarray*}\\label{w68}\n\\int\\bigg[\\rho_{D} + \\rho_{D^{\\perp}}\\bigg]dv \\geq \\int \\bigg[\\frac{(n+1)}{pq}\\rho_{D} +\\frac{2(n+1)}{q^{2}} \\rho_{D^{\\perp}}\\bigg]dv\n\\end{eqnarray*}\nFrom we have the following observations.\nEither $(n+1) \\leq pq$ and $2(n+1) \\leq q^{2}$ or $\\rho_{D} \\leq 0$ and $\\rho_{D^{\\perp}} \\leq 0$ that id $\\rho = \\rho_{D} + \\rho_{D^{\\perp}} \\leq 0$. Equality holds if and only if either $(n+1) = pq$ and $2(n+1) = q^{2}$ or $grad_{D}(log f) = 0.$\n\\end{proof}\n\\hspace{6cm}{\\bf{References}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSelective control with resonant light pulses is an emerging route for manipulating the properties of transition metal oxides. Selectivity has been achieved through resonant excitation of magnetic dipole modes in antiferromagnets~\\cite{Kampfrath2011} and multiferroics~\\cite{Kubacka2014}, or through IR-active phonon modes, in order to drive insulator-metal transitions~\\cite{Rini2007} and superconductivity~\\cite{Fausti2011, Mitrano2016}. \n\nUltrafast control of the magnetic state, in particular, could have a strong impact on magnetic recording technology. Ultrafast demagnetization is a complex process involving strong coupling between electronic, spin, and structural degrees of freedom which is dependent on the type of magnetic order and band structure~\\cite{Koopmans2010,Kirilyuk2010}. Controlling these interactions is key for developing magnetic devices that can fully exploit femto-magnetism. Antiferromagnetic (AFM) materials are insensitive to external magnetic fields~\\cite{Marti2014a} and are more stable when miniaturized ~\\cite{Loth2012} than ferromagnets. Moreover the absence of a net magnetic moment can enable much faster control of spin dynamics ~\\cite{Kimel2004}. \n\nDespite these possibilities, spin dynamics in AFM materials are still not well understood. Conventional techniques used to study the femtosecond dynamics of magnetic materials cannot be applied to AFM materials as they lack a net magnetic moment and alternative techniques are required. The advent of free electron lasers has enabled the study of spin dynamics through resonant elastic ~\\cite{Ehrke2011a, Forst2011, Tobey2012, Kubacka2014} or inelastic magnetic scattering~\\cite{Dean2016}. However, optical techniques such as linear magnetic birefringence~\\cite{Bossini2014, Bossini2015} or second harmonic generation (SHG)~\\cite{Fiebig2005} can also be used. \n\nDue to the small band gaps in many transition metal oxides, selective control has focused on the THz or mid-IR spectral region, which may reduce the speed at which the material can be driven. This is observed in the manganites, where it has been shown that the melting of antiferromagnetism via IR-active phonons is an order of magnitude slower and less efficient than when exciting above the band gap at 1.5\\,eV~\\cite{Forst2011}.\n\nCr$_2$O$_3$ is an ideal material to demonstrate ultrafast control of demagnetization at higher speeds. The band-gap is in the near UV, enabling optical control over a broader range of wavelengths. The magnetic state has been well characterized by SHG spectroscopy~\\cite{Fiebig1994, Fiebig2005}, thus experiments can be performed with a regular laboratory setup and the static optical and electronic properties of the material are significantly better understood than those of other correlated materials, enabling an greater chance for understanding the dynamics induced by the laser pulse.\n\nIn this paper, we show that electronic visible photoexcitation can be used to selectively control the structural and magnetic properties of Cr$_2$O$_3$. By measuring the time- and polarization-resolved second harmonic signal, we show that different electronic in-gap states couple to different phonon modes. As these modes are responsible for the spin scattering process, we can control the demagnetization rate by as much as 25\\% by changing the photon energy used to excite the system.\n\n\\section{Sample Characterization and Experimental Details}\n\nA single crystal of Cr$_2$O$_3$, cut perpendicular to the trigonal $z$ axis, was grown using the flux method and was polished to a thickness of approximately 10\\,$\\mu$m and, unless otherwise stated, cooled to 77\\,K. The measured sample absorbance is shown in Fig~\\ref{fig:Static}a. Due to the small sample size and thickness it was not possible to measure attenuations greater than 2 orders of magnitude due to the dominance of scattered light around the sample edges. Optially, Cr$_2$O$_3$ can be considered a 100\\% doped ruby crystal. The absorbance shows two broad peaks ($^4T_1$ and $^4T_2$), corresponding to crystal field excitations within the 3$d$ levels of the Cr ion which preserve the spin orientation of the photo-excited electron. The resonances are broadened due to a strong electron phonon coupling. In addition, a low energy narrow resonance ($^2E$) is observed which corresponds to a dipole forbidden transition in which the spin is flipped but the spatial part of the wavefunction is the same as the ground state~\\cite{Muto1998}. The high contrast between the transmission of light on and between the resonances and the narrow Raman spectra (Fig~\\ref{fig:Static}d) indicate that the sample is of high quality. In the ruby laser, the $^4T_1$ and $^4T_2$ levels are optically pumped and the excited electrons decay non-radiatively to the $^2E$ state on sub picosecond timescales~\\cite{Fonger1975}. As this process results in a spin flip, the lifetime of the excited state is very long and is used for lasing. Cr$_2$O$_3$, unlike Ruby, is anti-ferromagnetic, so the spin flip process can lead to rapid demagnetization. In this paper we track this process using time-resolved SHG. \n\n\\begin{figure}\n\\centering\n\\includegraphics{figure1.pdf}\n\\caption{(a) The absorbance of Cr$_2$O$_3$. Black markers correspond to measured values of our 10 $\\mu$m thick sample at 77\\,K. Blue line is a guide to the eye based on the measurements found in Ref~\\cite{McClure1963}. The $^4T_1$ and $^4T_2$ states correspond to transitions within crystal field split states of the Cr 3$d$ levels which preserve the spin of the electron. The $^2E$ state corresponds to a spin-forbidden transition. The colored arrows indicate the pump wavelengths used in the time resolved experiments and the black dashed arrow corresponds to the second harmonic of the probe. (b) Polarization and temperature dependence of the SHG signal from Cr$_2$O$_3$ measured with a fundamental photon energy of 1.08\\,eV. Solid lines are fits to the data using Eq.~\\ref{eq:angle}. (c) Temperature dependence of the antiferromagnetic order parameter $l(T)$ extracted from the fits from (b) giving $T_\\mathrm{N} = 314\\pm5$\\,K. (d) Raman spectrum for two different excitation wavelengths, demonstrating coupling to $E_g$ modes is stronger for longer wavelengths at room temperature. \n\\label{fig:Static}}\n\\end{figure}\n\nCr$_2$O$_3$ is centrosymmetric with the crystallographic point group $\\bar{3}m$. The presence of inversion symmetry forbids conventional SHG of the electric field, $E$; however an axial source term $\\bm M \\propto \\bm{\\chi_m^{(2)}}\\bm{EE}$ is allowed. Below the N\\'eel temperature, $T_\\mathrm{N}= 308$\\,K, the Cr ions order antiferromagnetically along the trigonal z-axis, enabling an additional polar source term $\\bm P \\propto\\bm{\\chi_e^{(2)}}\\bm{EE}$. Both $\\bm{\\chi_e^{(2)}}$ and $\\bm{\\chi_m^{(2)}}$ satisfy the symmetry relations of the point group 32: $\\chi_{xxx}^{(2)}=-\\chi_{xyy}^{(2)}=-\\chi_{yxy}^{(2)}=-\\chi_{e\/m}^{(2)}$~\\cite{Boyd2003}. For light traveling along the trigonal $z$ axis, this gives rise to a SHG scattering pattern that depends on the linear polarization in the $(x, y)$ plane as\n\\begin{equation}\nI_{\\mathrm{SHG}}(\\theta) \\propto |\\chi_e^{(2)}\\sin(3\\theta)-\\chi_m^{(2)}\\cos(3\\theta) |^2 I_\\mathrm{f}^2,\n\\label{eq:angle}\n\\end{equation}\nwhere $I_\\mathrm{f}$ is the intensity of the fundamental beam and $\\theta$ is the polarization angle relative to the crystallographic $y$-axis. Furthermore, the non-linear susceptibilities can be expanded as a function of the antiferromagnetic order parameter, $l$, as $\\chi_e^{(2)} = c_1 l(T)$ and $\\chi_m^{(2)} = c_2 + c_3 l^2(T)$~\\cite{Muto1998, Tanabe1998, Muthukumar1996}. \n\nThe $c_i$ coefficients are temperature independent, but depend on the photon energy of the light used to probe the sample; resonant enhancement occurs for photon energies around 1.08\\,eV due to the crystal field effects in the final state~\\cite{Tanabe1998, Fiebig1994}. Thus resonant SHG is both sensitive to the magnetic order and the crystallographic structure surrounding the Cr$^{3+}$ ions. \n\nThe temperature dependence of the static SHG signal was measured with the output of an optical parametric amplifier (OPA), pumped by the output of a Ti:sapphire laser at a repetition rate of 5 kHz and tuned to the 1.08 eV (1150\\,nm) resonance. The pulse duration was approximately 60 fs. The SHG signal was measured in transmission with a high and low pass filter to separate out the 2.16\\, eV (575\\,nm) SHG signal from the fundamental and third harmonic light. A polarizer and half-waveplate set the linear polarization of the probe beam before the sample and a second polarizer after the sample selects the SHG signal at a polarization parallel to the incident beam. These were rotated to measure the polarization dependence of the SHG for the different temperatures shown in Fig.~\\ref{fig:Static}b. From this data, the complex $c_i$ coefficients and the temperature-dependent order parameter, $l$, (Figs.~\\ref{fig:Static}c) were determined, in good agreement with the literature~\\cite{Fiebig1994, Fiebig2005}. \n\nFor the time resolved experiments, a second OPA generated the pump pulse with central wavelengths at 3\\,eV (400\\,nm), 2.5\\,eV (500\\,nm), and 1.8\\,eV (700\\,nm) as indicated in Fig.~\\ref{fig:Static}a, with pulse durations of order 50\\,fs. These pump wavelengths are chosen in order to populate different excited states. 2.5\\,eV and 1.8\\,eV photons predominately populate the $d$ levels of the Cr ion, while the higher energy 3\\,eV excitation also causes charge transfer excitation between O and Cr ions. Furthermore, the photon energies are detuned from the peak absorption to ensure a greater penetration depth of the pump light. \n\nThe pump is set in a counter-propagating configuration, relative to the probe, exciting the sample from the backside and at a small angle as shown in the insert of Fig~\\ref{fig:dynamic}a. This interaction geometry is chosen because the SHG of the probe is absorbed in the crystal and thus only the SHG generated at the back surface leaves the crystal. Due to the large attenuation at 2.1\\,eV compared to the pump wavelengths chosen, the probed volume is uniformly excited. \n\n\\section{Results}\n\\begin{figure}\n\\centering\n\\includegraphics{figure2.pdf}\n\\caption{(a) Visible-pump SHG-probe signal at the three different pump photon energies, 3.0\\,eV (blue circles), 2.5\\,eV (green triangles) and 1.8\\,eV (red squares). Dashed lines correspond to the fits to Eq.~\\ref{eq:fit}. Arrows indicate the times at which the polarization dependence shown in Fig.~\\ref{fig:pol} were performed. The insert shows a schematic of the counter-propagating pump-probe setup (P - Polarizer, WP - Wave Plate, IF - Interference filters). (b) Fluence dependence of the SHG transient signal at 3\\,eV from 0.25-2 mJ\\,cm$^{-2}$ showing linear behavior. Dashed lines correspond to fits using Eq.~\\ref{eq:fit}. (c) Fit parameters of the data in (b) showing that both the amplitude of the peak and plateau signal vary linearly with fluence. The $\\tau_\\mathrm{pm}$ time-constant is independent of fluence. \\label{fig:dynamic}}\n\\end{figure}\n\nFigure~\\ref{fig:dynamic} shows the time-resolved change in the SHG signal ${\\Delta I}\/{I_0}$ for the three pump photon energies. In the traces shown, the pump, probe and SHG polarizations were all parallel to the crystallographic $x$-axis, however, other combinations were also measured. For all pump photon energies, a decrease in the SHG signal is measured. At delays $> 1$\\,ps, all three signals settle to a plateau that lasts for more than 400\\,ps. However, at shorter delays ($< 1$\\,ps) additional fast dynamics are observed, which strongly depend on the pump photon energy. Due to the differences in the absorption coefficient for each pump pulse, the pump fluences were controlled in order to give roughly the same decrease in the SHG signal at 2\\,ps. However, in all cases the transient signal was found to vary linearly with the pump fluence for all time delays as demonstrated in Figure~\\ref{fig:dynamic}b for the case of 3\\,eV excitation. Thus the different dynamics observed at early delays are not due to differences in the absorption and the response measured at 1.8\\,eV excitation cannot be recreated by changing the intensity of a 3\\,eV pump. \n\nThe dynamics can be fit well by \n\\begin{equation}\n\t\\Delta I(t)\/I_0 = \\Theta(t)[A_\\mathrm{p} e^{-t\/\\tau_\\mathrm{pm}} + A_m(1-e^{-t\/\\tau_\\mathrm{pm}})],\n\\label{eq:fit}\n\\end{equation}\nwhere $\\Theta(t)$ is the error function with a 75\\,fs rise, fixed for all wavelengths, $A_\\mathrm{p}$ and $A_\\mathrm{m}$ are the magnitudes of changes at the spike at time zero and plateau respectively, and $\\tau_\\mathrm{pm}$ is the rapid recovery time constant, which was found to vary from $300\\pm 20$\\,fs at 3.0\\,eV pumping to $400\\pm 50$\\,fs at 1.8\\,eV pumping. Figure~\\ref{fig:dynamic}c shows that the time constant obtained and the ratio between the peak and plateau changes is independent of fluence, again demonstrating the linearity of the signal in this regime. We note that similar transients were observed when measuring the probe transmission at the fundamental photon energy (data not shown). However, we can exclude pump-induced changes in the fundamental intensity as the origin of the second harmonic dynamics because this would give rise to a signal that scales with the square of the pump fluence. As a result, transient changes in the linear optical properties have a negligible effect on the SHG transients.\n\n\\begin{figure}\n\\centering\n\\includegraphics{figure3.pdf}\n\\caption{(a) Change in the polarization dependence of the SHG probe at 0\\,ps for two perpendicular pump polarizations ($x$, $y$). Lighter points are not measured and reconstructed from the symmetry of the signal. Dashed lines in the 3.0\\,eV ($A_{1g}$) and 1.8\\,eV ($E_g$) plots are fits for the $c_i$ coefficients at constant magnetization. (b) Change in the probe polarization dependence at 2 ps and corresponding fit to the scattering pattern when the AFM order parameter, $l$, is reduced by 0.75\\%. \\label{fig:pol}}\n\\end{figure}\n\nIn equilibrium, the intensity of the SHG signal can be directly related to the magnetic order. However, out of equilibrium this does not have to be the case~\\cite{Huber2015}. In order to assign an origin to the observed dynamics we measured the polarization dependence of the transient SHG signal at the peak of the change and at long times after pumping, as shown in Fig.~\\ref{fig:pol}.\n\nWhen the system is in the long lived state, the SHG signal shows the same angular dependence irrespective of the pump photon energy or polarization. In this case the data can be accurately fitted by {\\em only} reducing the antiferromagnetic order parameter, $l$, in Eq.~\\ref{eq:angle}. Fig. ~\\ref{fig:pol}b demonstrates the quality of the fit for $\\Delta l = 0.75$\\,\\%. Thus the long-lived quenching of the SHG signal results from demagnetization. However, the angular dependence at short delays (Fig. ~\\ref{fig:pol}a) shows a different behavior that strongly depends on the pump photon energy and polarization. This pattern cannot be fitted by changing $\\Delta l$. Excitation at 3\\,eV produces an isotropic change, while lowering the photon energy to 1.8\\,eV breaks the symmetry of the SHG signal. Furthermore, the direction of the asymmetry can be controlled by the polarization of the pump. In this case it is clear that the SHG dynamics near time zero are not due to the spin system alone. \n\n\\section{Discussion}\n\\begin{figure}\n\\centering\n\\includegraphics{figure4.pdf}\n\\caption{(a) Sketch of the Cr$_2$O$_3$ unit cell in the $xy$-plane showing the $A_{1g}$ and (b) $E_g$ displacements of the oxygen ions around a Cr ion (darker colors correspond to atoms at different $z$ positions). (c) DFT calculations for changes in the magnitude of the nonlinear susceptibility for $A_{1g}$ and $E_g$ displacements of {0.05\\,\\AA}. The shaded area corresponds to the measured region. The dashed line shows the calculated equilibrium spectrum.\\label{fig:theory}}\n\\end{figure}\n\nAs the excited states that are pumped are broadened due to electron-phonon coupling, we consider how crystal distortions can change the SHG signal, both in terms of amplitude and symmetry. Cr$_2$O$_3$ possesses $A_{1g}$ and $E_g$ Raman active modes, the motions of which are shown in Figs.~\\ref{fig:theory}a and~\\ref{fig:theory}b. $A_{1g}$ modes preserve the symmetry of the crystal and thus can only modify the efficiency of the SHG process through a crystal-field-induced shift in the resonance condition and should not change the polarization dependence. For small displacements the $c_i$ coefficients change as as $c_i\\rightarrow c_i (1+\\delta Q_{A_{1g}})$, where $Q_{A_{1g}}$ is the phonon amplitude and $\\delta$ is the coupling constant. Such a change can fit the polarization dependence measured for excitation at 3\\,eV as shown in Fig.~\\ref{fig:pol}a when the magnetization is kept constant and $\\delta Q_{A_{1g}} = -0.05$.\n\n$E_g$ displacements lower the crystallographic point group symmetry from $\\bar{3}m$ to $2\/m$, thus they can also change the polarization dependence as well as efficiency. As the $E_g$ mode is doubly degenerate, the direction along which the distortion occurs can be varied in the plane. Figure~\\ref{fig:theory}b depicts the case for oxygen displacements along the $y$-direction. In this case the three non-zero components of the susceptibility become independent, which can be captured by $c_i^{jkl}\\rightarrow c_i (1+\\delta^{jkl}Q_{E_g})$, where $jkl= xxx,xyy,yxy$ are the indexes of the tensor elements. Thus a minimum of three parameters are needed to describe the effect. The resulting change can fit the 1.8\\,eV experimental data, with $c_i^{xxx}=0.9881c_i ~(\\delta^{xxx}Q_{E_g}=-0.0119), c_i^{xyy}=1.0005c_i ~(\\delta^{xyy}Q_{E_g}=0.0005)$ and $c_i^{xyx}=0.9915\\,c_i ~ (\\delta^{xyx}Q_{E_g}=-0.0085)$ as shown in Fig.~\\ref{fig:pol}a. The perpendicular scattering pattern can be achieved when the symmetry breaking axis is rotated by 90$^\\circ$. In this case the original non-zero components of the susceptibility remain unchanged and the corresponding components along the $y$ axis become non-zero. The data for pumping at 2.5\\,eV can be described by a combination of the $A_{1g}$ and $E_g$ responses. \n\nThese symmetry considerations are confirmed by calculations of the SHG signal under the distorted crystal structures obtained from density functional theory (DFT). We obtained the electronic structure for the ground state of Cr$_2$O$_3$ using the Abinit code, based on plane waves and pseudopotentials~\\cite{Gonze2005,Gonze2009}. The antiferromagnetic state was taken into account in the calculations. Then the second order susceptibility was computed in Time-Dependent DFT using the 2light code based on the formalism developed in Ref.~\\cite{Luppi2010}. The second order susceptibility is split into two parts, associated with the spin-up and spin-down components. Spin-orbit coupling is not included, as it is far beyond the reach of nonlinear ab initio calculations. Therefore, as expected, see for instance Ref.~\\cite{Muto1998}, both components cancel. However, trends can be obtained for the effects of the crystal distortion by looking at one component only. The second order spin-up susceptibility for the distorted structures was obtained by moving the oxygen atoms along the $A_{1g}$ and $E_g$ modes at the level of the DFT calculations. The spectra were calculated in the independent particle approximation. Convergence was achieved with 432 off-symmetry shifted $k$ points in the full Brillouin zone and 90 unoccupied states. \n\nThe results are shown in Fig.~\\ref{fig:theory}c. The $A_{1g}$ displacement induces an equal decrease of all elements, whereas the $E_g$ distortion breaks the degeneracy of the elements, as expected from the above symmetry analysis. Furthermore, these calculations enable us to estimate the size of the induced displacements from the magnitude of our signal. The calculations were performed with a displacement of {0.05\\,\\AA} and induce changes that are larger than those measured experimentally, thus resonant SHG is extremely sensitive to highly symmetric atomic motion.\n\n\\begin{figure}\n\\centering\n\\includegraphics{figure5.pdf}\n\\caption{Part of the unit cell centered on the Cr ion and nearest oxygen neighbors. Bold arrows correspond to the spin state. Photoexcitation at 1.8\\,eV excites electrons from the $t_{2g}$ levels into the $e_g$ levels while preserving the spin state. The resulting charge redistribution is anisotropic (blue corresponds to more positively charged regions, red more negative) and couples strongly to the anisotropic $E_g$ phonon mode. The electrons are then scattered to a spin-flipped $t_{2g}$ state in 400\\,fs. Thus the distortion is relaxed and the system is demagnetized. Excitation at 3\\,eV causes charge transfer from O to Cr. This causes a symmetric charge redistribution mapping onto the $A_{1g}$ phonon mode and scattering occurring within 300\\,fs. \\label{fig:story}}\n\\end{figure}\n\nThese results enable us to build a picture for the demagnetization process, which is sketched in Fig.~\\ref{fig:story}. The pump pulse excites electrons from the occupied $^4A_2$ ground state which is composed of the $t_{2g}$ levels of the Cr ion, into final states that depend on the photon energy. Lower energy photons transfer electrons into the unoccupied T states, which mix the $3d-e_g$ and $t_{2g}$ levels of the Cr ion~\\cite{Muto1998}, changing the charge distribution in the vicinity of the Cr ion. As these orbitals are anisotropic, neighboring ions experience a different force, causing oxygen atomic motion that is along the $E_g$ phonon coordinate. Higher energy photons trigger transitions that are more of a charge transfer character, i.e. charge is redistributed between chrome and oxygen ions. In the limit that the charge transfer is complete, this produces a symmetric force on the oxygen ions which couples to $A_{1g}$ motion. The preferential coupling to the $A_{1g}$ mode for high photon energies is also confirmed by resonant Raman scattering measurements shown in Fig~\\ref{fig:Static}d. Furthermore, the timescale for this processes is in good agreement for the equivalent process that occurs in ruby~\\cite{Fonger1975}.\n\nAs both Raman modes have frequencies in the region 9 - 18\\,THz, they respond adiabatically during the excitation pulse and coherent motion of the lattice is not triggered. In the distorted state, the excited electrons in both cases can be efficiently scattered into the spin-flipped $^2E$ states of Fig.~\\ref{fig:Static}a. These states have the same spatial distribution of charge as the ground state, but the electron's spin is flipped~\\cite{Muto1998}. Therefore, the force causing the displacements is lost and the crystal relaxes, while at the same time the magnetization is reduced. Once relaxed, the equilibrium spin-lattice relaxation process dominates, which is much slower. These results differ from previous measurements of spin dynamics in Cr$_2$O$_3$~\\cite{Satoh2007, Dodge1999}, since we operate at lower temperatures and lower excitation fluences. Higher fluences and temperatures may enable different scattering processes. However, the 300-400\\,fs timescales obtained for Cr$_2$O$_3$ are very similar to the timescale observed in antiferromagnetic order melting in insulating Sr$_2$IrO$_4$ measured using time-resolved resonant diffraction~\\cite{Dean2016}, thus demonstrating that rapid control of the magnetic state is possible in antiferromagnetic insulators when exciting above the band gap. \n\n\\section{conclusion}\nIn summary, we have shown how specific lattice modes dictate the demagnetization rate in Cr$_2$O$_3$ and that the lattice modes can be controlled by resonantly tuning the light pulse in the visible spectral region, enabling a modulation of the demagnetization rate by 25\\%. This control is achieved through the presence of in-gap states that occur within the band gap of insulating materials. Through the control of the crystallographic structure, presented here, or through direct excitation of the magnetic subsystem with below gap excitation~\\cite{Bossini2014}, insulating antiferromagnets present new opportunities for light control that are not available in ferromagnetic metals. As the coupling between charge excitations and Raman active modes is direct and occurs in the optical regime, it also has the potential to be more efficient and easier to implement that selective methods that require non-linear coupling between IR and Raman active modes~\\cite{Forst2011a}.\n\nFurthermore, we have shown that, out of equilibrium, time resolved SHG is a powerful tool for monitoring {\\em both} magnetic and structural degrees of freedom and is capable of measuring small, symmetric atomic displacements. When combined with theory, the ability to observe small changes in the lattice simultaneously with the evolution of the magnetic degree of freedom, enables measurements in a standard laboratory which would otherwise require large scale user facilities. \n\n\n\\begin{acknowledgments}\n The research leading to these results has received funding from LASERLAB-EUROPE (grant agreement no. 284464, EC's Seventh Framework Program). We acknowledge GENCI (project 544) for the computational support provided. SW received financial support from Ramon y Cajal program RYC-2013-14838 and Marie Curie Career Integration Grant PCIG12-GA-2013-618487. VGS, TAM and SW acknowledge support from Severo Ochoa Excellence Grant and Fundaci\\`{o} Privada Cellex. GC acknowledges support by the European Union's Seventh Framework Program (FP7\/2007-2013) Grant No. CNECT-ICT-604391 (Graphene Flagship).\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn models of supergravity (SUGRA)~\\cite{Ref:SUGRA,WessBagger}, superweakly interacting massive fields are generally involved, such as the gravitino and modulus fields. They often cause cosmological problems, such as the moduli or Polonyi problem~\\cite{Coughlan:1983ci,Ref:Moduli}, and gravitino problems~\\cite{Ref:Gravitino,Kawasaki:2008qe}. If they are generated in an early stage of the universe, they easily spoil big bang nucleosynthesis (BBN) when they decay, or tend to overclose the universe if stable. Hence, their abundances are severely restricted. Among the SUGRA particles, gravitinos are thermally produced in hot plasma. Moreover, it has been pointed out that a large amount of gravitinos are generated via decays of \ncoherent oscillations of\nheavy scalars, such as moduli and inflatons~\\cite{MIGP,Kawasaki:2006gs,Asaka:2006bv,Endo:2007sz}. Therefore, the cosmology constrains the reheating temperature and models of the scalars.\n\nThe partial decay rate of a scalar into a pair of gravitinos depends on the structures of the scalar and supersymmetry (SUSY)-breaking sectors. The production rate has been studied in detail by Ref.~\\cite{Endo:2006tf} when the SUSY breaking is caused by a vacuum expectation value (VEV) of the F term of the SUSY breaking field. In this case, the goldstino, which is the longitudinal component of the gravitino, is composed of the spinor component of the SUSY-breaking chiral supermultiplet. \nIn the literature which studies the gravitino production, D-term contributions to the scalar potential have been discarded. If the SUSY breaking is dominated by a D-term VEV, the goldstino mainly consists of a gaugino, and the above result can significantly change. \nIn this paper, direct gravitino production by scalar decay will be studied when the D-term VEV is sizable. \n\n\nThis paper is organized as follows. In Sec.~\\ref{sec:general}, the production rate of a pair of gravitinos will be provided, taking the D-term potential into account. Scalar mixing with the fields in the SUSY-breaking sector will be investigated in the SUGRA framework. In Sec.~\\ref{sec:example}, the result will be applied to a D-term SUSY-breaking model, and its cosmological implications will be discussed. Sec.~\\ref{sec:conclusion} is devoted to a summary and discussions.\n\n\\section{Direct Gravitino Production Rate}\\label{sec:general}\n\nIn this section, we evaluate the partial decay rate of a scalar field such as the modulus or inflaton field into a pair of gravitinos. It is derived at the leading order of Planck-suppressed interactions in the framework of SUGRA.\nBoth the D-term and F-term SUSY-breaking contributions are taken into account,\nand sources of the D-term contributions are clarified.\n\n\n\n\\subsection{Gravitino Production Rate in the Mass-Eigenstate Basis}\n\nThe SUSY-breaking sector is supposed to have nonzero VEVs of the D-term potentials for gauge symmetries which are not included in the Standard Model (SM) gauge groups. The gauge symmetries can be $U(1)$ and\/or non-Abelian symmetries. \nIf the symmetry is $U(1)$, we assume that there is no genuine Fayet-Iliopoulos term, since it is difficult to embed the Fayet-Iliopoulos term in SUGRA~\\cite{FI-SUGRA}.\nThe VEVs of the D-term potentials are considered to be dynamically generated by the fields in the SUSY-breaking sector, $z^i$, which are charged under the gauge symmetries. See Sec.~\\ref{sec:example} for an example of such a model.\n\nIn gravitino production, the relevant fields are the scalar field, $\\phi$, the SUSY-breaking fields, $z^i$, and the gravitino. \nIt is assumed that the scalar $\\phi$ is much heavier than the gravitino due to the SUSY-invariant mass.\\footnote{This is the case for the inflaton in many inflation models. In the case of modulus, if the modulus mass is $m_\\phi < {\\cal O}(10 \\rm{TeV})$, its decay occurs after BBN starts and spoils the success of the standard cosmology. Thus, we assume $m_\\phi \\gg 10 \\rm{TeV} \\gtrsim m_{3\/2}$.}\nThe scalar is also assumed to be a singlet under the extra gauge symmetry, \nand couples with $z^i$ only through the terms suppressed by powers of the (reduced) Planck scale, $M_P = 2.4 \\times 10^{18} \\rm{GeV}$, which generally exist in the SUGRA Lagrangian.\\footnote{Otherwise, $\\phi$ decays into the SUSY-breaking sector much faster, which worsens the cosmological gravitino problem.}\n\nThe superpotential, K\\\"ahler potential and gauge kinetic function are represented as\\footnote{In this paper, we follow the conventions of Ref.~\\cite{WessBagger}. \nDerivatives with respect to fields are denoted by subscripts, e.g., $G_{\\phi}=\\partial G\/\\partial \\phi$.\nAlso, we omit symbols of VEV, $\\vev{\\cdots}$, if not otherwise specified.\n}\n\\begin{align}\nK &= K(\\phi ,\\bar{\\phi})+K(z^{i} ,\\bar{z}^{i}) \n+ \\sum_{n\\ge 1}\\frac{1}{M_P^n}K_{\\rm mix}^{(n)}(\\phi, \\bar{\\phi}, z^i, \\bar{z}^i), \\nonumber\\\\ \nW &= W(\\phi) + W(z^i) + \\sum_{n\\ge 1}\\frac{1}{M_P^n}W_{\\rm mix}^{(n)}(\\phi, z^i), \\nonumber\\\\\nh_{AB}&=\\left( 1+\\frac{\\theta g^{2}}{8 \\pi i} \\right) \\delta_{AB} \n+\\sum_{n\\ge 1}\\frac{1}{M_{P}^{n}}h_{AB}^{(n)}(\\phi, z^{i}),\n\\label{eq:model-base}\n\\end{align}\nwhere $K_{\\rm mix}^{(n)}$ and $W_{\\rm mix}^{(n)}$ are the interaction terms between $\\phi$ and $z^i$ whose Planck suppressions are displayed explicitly. The first term in the right-hand side of $h_{AB}$ corresponds to the gauge kinetic and $\\theta$ terms, and $g$ is the gauge coupling constant of the extra gauge symmetry.\nLet us call the basis in Eq.~\\eqref{eq:model-base} the ``model basis'' in this paper. \nNote that the VEVs of $\\phi$ and $z^i$ can induce kinetic mixings through higher-dimensional terms in the K\\\"ahler potential. In the following, we assume that those VEVs are much smaller than the Planck scale. Otherwise, the following discussion applies after the fields are shifted to absorb the large VEVs.\n\nThe decay rate is evaluated in the mass-eigenstate basis. The fields in the mass-eigenstate basis, $X^{a}$, are related to those in the model basis, $x^{\\alpha}$, as\n\\begin{eqnarray}\nX^{a}=A^{X^{a}}{}_{x^{\\alpha}} \\delta x^{\\alpha},\n\\end{eqnarray}\nwhere $\\delta x^{\\alpha}= x^{\\alpha}-\\vev{x^{\\alpha}}$ is the fluctuation around the VEV, and $A$ is a matrix to diagonalize the mass matrix and to canonicalize the kinetic terms at the potential minimum. The model basis fields, $x^{\\alpha}$, consist of $x^{\\alpha}=\\varphi^{\\alpha}$ and its Hermitian conjugate $\\bar{\\varphi}^{\\alpha}$, with $\\varphi^{\\alpha}=\\phi$ and $z^{i}$. \nOn the other hand, the mass eigenstates are generally represented by\nreal fields, $X^{a}=\\Phi_{R}, \\Phi_{I}, Z_{R}^{i}$, and $Z_{I}^{i}$, which primarily consist of the \nreal and imaginary parts of $\\phi$ and $z^i$, respectively.\nHence, $(A^{X^{a}}{}_{\\varphi^{\\alpha}})^* = A^{X^{a}}{}_{\\bar{\\varphi}^{\\alpha}}$ is satisfied.\nSince the interactions between $\\phi$ and $z^i$ are given by higher-dimensional operators,\n$A^{\\Phi_{R,I}}{}_{z^{i}}, A^{\\Phi_{R,I}}{}_{\\bar{z}^{i}}, A^{Z^{i}_{R,I}}{}_{\\phi}$, and $A^{Z^{i}_{R,I}}{}_{\\bar{\\phi}}$ are suppressed by the Planck scale.\nThe matrix $A$ will be evaluated later.\n\n\n The Lagrangian terms which are relevant for the tree-level decay of the scalar into a pair of gravitinos are found to be~\\cite{WessBagger}\n \\begin{align}\n\\mathcal{L}=& \\frac{1}{8i}\\epsilon ^{\\mu \\nu \\rho \\sigma}( G_{\\varphi^{\\alpha}} \\partial_{\\rho} \\delta \\varphi^{\\alpha} - G_{\\bar{\\varphi}^{\\alpha}}\\partial_{\\rho}\\delta \\bar{\\varphi}^{\\alpha} )\\bar{\\psi}_{\\mu}\\gamma_{\\nu}\\psi_{\\sigma} \\nonumber \\\\\n& +\\frac{1}{4}m_{3\/2} ( G_{\\varphi^{\\alpha}} \\delta \\varphi^{\\alpha} + G_{\\bar{\\varphi}^{\\alpha}}\\delta \\bar{\\varphi}^{\\alpha} ) \\bar{\\psi}_{\\mu} \\gamma ^{\\mu \\nu} \\psi_{\\nu}, \\label{Lm22g}\n\\end{align}\nwhere $\\varphi^{\\alpha}$ stands for $\\phi$ and $z^{i}$ in the model basis, while $\\psi_{\\mu}$ is the gravitino, $m_{3\/2}$ the gravitino mass, and $G=K+\\ln |W|^{2} $ the total K\\\"{a}hler potential.\nHere and hereafter, the Planck unit, $M_{P} = 1$, is used if not otherwise specified.\nThis Lagrangian is the same as that in the F-term SUSY-breaking case~\\cite{MIGP}.\nIn terms of the mass-eigenstate basis, the above Lagrangian becomes \n\\begin{align}\n\\mathcal{L}=& \\frac{1}{8i}\\epsilon ^{\\mu \\nu \\rho \\sigma} \\mathcal{G}^{\\text{(1)}}_{\\Phi_{R}} \\partial_{\\rho} \\Phi_{R} \\bar{\\psi}_{\\mu}\\gamma_{\\nu}\\psi_{\\sigma} +\\frac{1}{4}m_{3\/2} \\mathcal{G}^{\\text{(2)}}_{\\Phi_{R}} \\Phi_{R} \\bar{\\psi}_{\\mu} \\gamma ^{\\mu \\nu} \\psi_{\\nu} + (R \\rightarrow I) + \\dots,\n\\label{eq:g12RI}\n\\end{align}\nwhere the coefficients are\n\\begin{align}\n\\mathcal{G}^{\\text{(1)}}_{\\Phi_{R,I}} =2i\\, \\Im ( G_{\\varphi^{\\alpha}}(A^{-1})^{\\varphi^{\\alpha}}{}_{\\Phi_{R,I}} ),~~~\n\\mathcal{G}^{\\text{(2)}}_{\\Phi_{R,I}} =2\\, \\Re ( G_{\\varphi^{\\alpha}}(A^{-1})^{\\varphi^{\\alpha}}{}_{\\Phi_{R,I}} ),\n\\end{align}\nwhere $A^{-1}$ is the inverse matrix of $A$, and \n $\\Re(\\cdots)$ and $\\Im(\\cdots)$ represent the real and imaginary parts, respectively.\nThe omitted terms in Eq.~(\\ref{eq:g12RI}) include $Z_{R,I}^{i}$, which are irrelevant for the gravitino production by the $\\Phi_{R,I}$ decay. Note that the gravitino production rate will be derived at the tree level in this section. The rate can receive radiative corrections, which will be mentioned later.\n\nFrom the above interactions, the partial decay rate of the scalar into a pair of gravitinos is evaluated as~\\cite{MIGP}\n\\begin{align}\n\\Gamma(\\Phi_{R,I}\\to \\psi_{3\/2}\\psi_{3\/2}) =\n\\frac{|\\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}}|^{2}m_{\\Phi_{R,I}}^{5}}{288 \\pi m^{2}_{3\/2 }}, \n\\label{gravitinorate}\n\\end{align}\nwhere we have used $m_{\\Phi_{R,I}}\\gg m_{3\/2}$.\nThe effective coupling constants are defined as\n\\begin{eqnarray}\n\\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right |^{2}\n=\\frac{1}{2}\\left( \\left| \\mathcal{G}^{\\text{(1)}}_{\\Phi_{R,I}} \\right|^{2} + \\left| \\mathcal{G}^{\\text{(2)}}_{\\Phi_{R,I}} \\right|^{2} \\right) \n=2\\left| \\sum_{\\varphi^{\\alpha}=\\phi,z^i} G_{\\varphi^{\\alpha}} (A^{-1}) ^{\\varphi^{\\alpha}}{}_{ \\Phi_{R,I}} \\right|^{2}.\\label{Geffective}\n\\end{eqnarray}\nThe rate is apparently the same as that in the F-term SUSY-breaking models~\\cite{MIGP} because the relevant Lagrangian [Eq.~\\eqref{Lm22g}] is the same. The effective coupling constants [Eq.~\\eqref{Geffective}] are governed by $G_{\\varphi^{\\alpha}}$, which is related to the F term as \n\\begin{align}\nF^i = -e^{G\/2} g^{i \\bar j} G_{\\bar j},\n\\end{align}\nwhere $g^{\\bar{i}j}$ is the inverse of the K\\\"{a}hler metric, $g_{i\\bar{j}}=\\vev{G_{i\\bar{j}}}$.\nHowever, this does not mean that the D term is unimportant.\nAs we will see, the magnitudes of F terms are controlled by vacuum conditions,\nwhich are affected by the D-term potential.\nMoreover, the D term contributes to the scalar mass matrix, \nwhich determines the mixing of scalar fields with the SUSY-breaking fields.\nThese D-term contributions to the effective coupling constants are studied in the following subsection. \n\n\n\n\\subsection{Effective Coupling Constants}\n\\label{sec:EffCoupling}\n\nIn this subsection, the effective coupling constants of gravitino pair production, $\\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}}$, are evaluated in the model basis within the SUGRA framework. Since we are interested in the cosmological applications of gravitino production, it is sufficient to evaluate them at the leading order with respect to the Planck-suppressed couplings. Higher-order corrections are safely neglected. \nThe interactions are classified by powers of the inverse of the Planck scale. \nAt the zeroth order, i.e., in the global SUSY limit, $\\phi$ is secluded from the SUSY-breaking sector by the assumptions. On the other hand, SUGRA corrections which include Planck-suppressed interactions belong to higher orders of the perturbation. Turning on the corrections, the two sectors communicate with each other, and gravitino production occurs.\n\nIn Eq.~\\eqref{Geffective}, it is sufficient to evaluate $G_{z^{i}}$ and $(A^{-1})^{\\phi}{}_{ \\Phi_{R,I}}$ at the zeroth order, because $G_{\\phi}$ and $(A^{-1})^{z^{i}}{}_{ \\Phi_{R,I}}$ start from the first order of the perturbation. Let us start from $(A^{-1})^{\\phi}{}_{ \\Phi_{R,I}}$.\nIn the global SUSY limit, the mass eigenstates in the $\\phi$ sector are simply given by $\\Phi_{R}=\\sqrt{g_{\\phi\\bar{\\phi}}\/2}(\\delta\\phi+\\delta\\bar{\\phi})$ and $\\Phi_{I}=-i\\sqrt{g_{\\phi\\bar{\\phi}}\/2}(\\delta\\phi-\\delta\\bar{\\phi})$, with their masses\n\\begin{align}\nm^{2}_{\\Phi_R}=\\frac{1}{g_{\\phi \\bar{\\phi}}} \\left( V^{\\rm (g)}_{\\phi \\bar{\\phi}} + V^{\\rm (g)}_{\\phi \\phi} \\right) ,~~~~~\nm^{2}_{\\Phi_I}=\\frac{1}{g_{\\phi \\bar{\\phi}}} \\left( V^{\\rm (g)}_{\\phi \\bar{\\phi}} - V^{\\rm (g)}_{\\phi \\phi} \\right) ,\n\\end{align}\nwhere $V^{\\rm (g)}$ denotes the scalar potential in the global SUSY limit, and \n\\begin{eqnarray}\nV_{x^\\alpha x^\\beta} = \\frac{\\partial^2 V}{\\partial x^\\alpha \\partial x^\\beta}\n\\end{eqnarray}\nis the second derivative of the potential in the model basis. Here, we assume $V^{\\rm (g)}_{\\phi \\phi} = V^{\\rm (g)}_{\\bar{\\phi}\\bar{\\phi}}$ for simplicity. It is straightforward to include the phase.\nThus, the mixings of the scalar $\\phi$ itself become $(A^{-1})^{\\phi}{}_{ \\Phi_{R}}=1\/\\sqrt{2g_{\\phi\\bar{\\phi}}}$ and $(A^{-1})^{\\phi}{}_{ \\Phi_{I}}=i\/\\sqrt{2g_{\\phi\\bar{\\phi}}}$ at the zeroth order. \nThe effective coupling constants [Eq.~\\eqref{Geffective}] are approximated to be\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}}\\right|^{2}=2\\left|\\frac{1}{\\sqrt{2g_{\\phi \\bar{\\phi}}}} G_{\\phi} + G_{z^{i}} (A^{-1})^{z^{i}}{}_{\\Phi_{R}} \\right|^{2}, \\label{GeffR0} \\\\\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}}\\right|^{2}=2\\left|\\frac{i}{\\sqrt{2g_{\\phi \\bar{\\phi}}}} G_{\\phi} + G_{z^{i}} (A^{-1})^{z^{i}}{}_{\\Phi_{I}} \\right|^{2}. \\label{GeffI0}\n\\end{align}\nNext, the F terms of the SUSY-breaking fields, $G_{z^{i}}\\simeq W_{z^i}\/W$, are evaluated at the zeroth order, namely by using the field VEVs in the global SUSY limit, as discussed above. This depends on the model, and we will demonstrate it in a D-term SUSY-breaking model in the next section. Although the SUSY-breaking sector may be involved, this procedure is straightforward. \n\nThe remaining task is to evaluate $G_{\\phi}$ and $(A^{-1})^{z^{i}}{}_{\\Phi_{R,I}}$ at the first order. \nThey include contributions from the D-term potential, which is represented by the Killing potential, $D_{A}$. \nBy the gauge invariance of the action, the Killing potential satisfies\n\\begin{align}\nD_A = i X_A^i G_i = i X_A^i K_i,\n\\label{eq:D-term}\n\\end{align}\nwhere $X_A^i =-ig^{i\\bar{j}}D_{A \\bar{j}}$ is the holomorphic Killing vector, by which the gauge transformation of a chiral superfield, $\\Phi^i$, is defined as $\\delta \\Phi^i = \\Lambda^A X_A^i (\\Phi^j)$ with $\\Lambda^A$ being the gauge transformation parameters. The second equality in Eq.~\\eqref{eq:D-term} is derived by the gauge invariance of the superpotential, $\\delta W = W_i \\delta \\Phi^i = 0$. \n\nThe F term of $\\phi$ is evaluated at the minimum of the scalar potential in SUGRA. The conditions of the vanishing cosmological constant and the potential minimization are\\begin{align}\n&V = \\frac{1}{2}g^{2} D^{A}D_{A} + e^{G}(G^{i}G_{i}-3) = 0, \\label{V} \\\\\n&V_{i}= g^{2} \\left(-\\frac{1}{2}h^{R}_{ABi}D^{A}D^{B}+D^{A}D_{Ai} \\right)+e^{G}\\left( G_{i}G^{j}G_{j}-2G_{i}+G^{j}\\nabla _{i} G_{j} \\right)=0, \\label{Vi} \n\\end{align}\nwhere $V$ is the scalar potential in SUGRA, $h^{R}_{AB}\\, (h_{R}^{AB})$ is (the inverse of) the real part of the gauge kinetic function,\nand $\\nabla_{i}G_{j}=G_{ij}-\\Gamma^{k}_{ij}G_{k}$ is the covariant derivative with $\\Gamma^{k}_{ij}=g^{\\bar{l}k}G_{ij\\bar{l}}$.\nFrom Eq.~(\\ref{Vi}), $G_{\\phi}$ is obtained as\n\\begin{align}\ng^{\\phi\\bar{\\phi}} G_{\\phi} \\simeq \n\\frac{1}{\\nabla_{\\bar{\\phi}}G_{\\bar{\\phi}}} \\left[ -g^{\\bar{z}^{i}z^{j}}G_{z^{j}} \\nabla_{\\bar{\\phi}}G_{\\bar{z}^{i}} +\\frac{g^{2}}{m_{3\/2}^{2}} \\left( \\frac{1}{2}h^{R}_{AB\\bar{\\phi}}D^{A}D^{B} - D^{A}D_{A\\bar{\\phi}} \\right) \\right]\n- g^{z^{i}\\bar{\\phi}} G_{z^{i}}, \n\\label{Gphi}\n\\end{align}\nwhere the gravitino mass is from $m_{3\/2} = \\vev{e^{G\/2}}$. \nIn this expression, we have used $|\\nabla_{\\phi}G_{\\phi}| = W_{\\phi\\phi}\/W + \\cdots \\sim m_\\phi\/m_{3\/2} \\gg 1$ and $G_i \\lesssim {\\mathcal O}(1)$ from Eq.~\\eqref{V}.\nIt is found that $G_{\\phi}$ vanishes in the global SUSY limit, when $\\phi$ is secluded from the SUSY-breaking sector in the limit and is a singlet under the extra gauge symmetry.\n\nNext, let us evaluate the mixing matrix $(A^{-1})^{z^{i}}{}_{\\Phi_{R,I}}$. There are two sources of mixing between $\\phi$ and $z^{i}$. The first one is from kinetic mixing, $g_{z^{i}\\bar{\\phi}}$, which is induced by higher-dimensional operators with the field VEVs. The kinetic terms are canonicalized by redefining the fields as\n\\begin{align}\n\\phi'=\\sqrt{g_{\\phi\\bar{\\phi}}}\\;\\delta\\phi+\\frac{g_{z^{i}\\bar{\\phi}} }{\\sqrt{g_{\\phi\\bar{\\phi}}} }\\delta z^{i},~~~\nz'^{i}= (C^{-1})^{i}{}_{j}\\delta z^{j},\n\\label{non-unitary}\n\\end{align}\nat the leading order of $g_{z^{i}\\bar{\\phi}}$,\nwhere $C$ is a matrix that canonicalizes the kinetic term of the SUSY-breaking sector,\n\\begin{align}\nC^{\\dag}{}_{i}{}^{j} g_{\\bar{z}^{j}z^{k}} C^{k}{}_{l}=\\delta_{il}.\n\\end{align}\nHere, it is sufficient to evaluate $g_{z^{i}\\bar{\\phi}}$ and $g_{\\phi\\bar{\\phi}}$ \nby using the field VEVs in the global SUSY, when the first-order perturbation is considered. \nThe second source of the mixing comes from the mass term. By canonicalizing the kinetic terms, the mixings in the mass matrix become\n\\begin{align}\nV_{\\phi' \\bar{z}^{\\prime i}} &= \n\\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} (C^\\dagger)_i{}^j\n\\left[ V_{\\phi \\bar{z}^{j}} - \\frac{g_{\\phi \\bar{z}^{j}}}{g_{\\phi\\bar{\\phi}}} V^{\\rm (g)}_{\\phi\\bar{\\phi}} \\right] \n\\equiv \\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} (C^\\dagger)_i{}^j\\, \\tilde{V}_{\\phi \\bar{z}^{j}}, \n\\label{eq:mass-matrix-1} \\\\\nV_{\\phi' z^{\\prime i}} &= \n\\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} C^j{}_i\n\\left[ V_{\\phi z^{j}} - \\frac{g_{z^{j}\\bar{\\phi}}}{g_{\\phi\\bar{\\phi}}} V^{\\rm (g)}_{\\phi \\phi} \\right]\n\\equiv \\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} C^j{}_i\\, \\tilde{V}_{\\phi {z}^{j}},\n\\label{eq:mass-matrix-2}\n\\end{align}\nat the leading order of $g_{z^{i}\\bar{\\phi}}$. \nExplicit forms of $V_{\\phi z^{i}}$ and $V_{\\phi \\bar{z}^{j}}$ will be given later.\nSince the mixing terms are small, the mass matrix is diagonalized by means of the perturbation theory. At the leading order of the perturbation, $(A^{-1})^{z^{i}}{}_{\\Phi_{R,I}}$ is obtained by combining the canonicalization and the diagonalization as\n\\begin{align}\n(A^{-1})^{z^{i}}{}_{ \\Phi_{R}}&= \\frac{1}{\\sqrt{2g_{\\phi\\bar{\\phi}}}}\n\\left[ \\left( m^{2}_{\\Phi_{R}} g_{z} -m_z^{2} \\right)^{-1} \\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}+ \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right), \\\\\n(A^{-1})^{z^{i}}{}_{ \\Phi_{I}}&= \\frac{i}{\\sqrt{2g_{\\phi\\bar{\\phi}}}}\n\\left[ \\left( m^{2}_{\\Phi_{I}} g_{z} -m_z^{2} \\right)^{-1}\\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}- \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right),\n\\end{align}\nwhere the indices $\\phi$ and $z^i$ are not in the primed basis of Eq.~\\eqref{non-unitary}, but in the model one. \nHere, $\\left( m^{2}_{\\Phi} g_{z} - m_z^{2} \\right)^{-1}$ is an inverse matrix, and the index $\\tilde{z}^{j}$ runs over both $z^{j}$ and $\\bar{z}^{j}$. The matrices, $m_z^{2}$ and $g_z$, are defined in the model basis as \n\\begin{align}\nm_z^2 = &\\begin{pmatrix}\nV^{\\rm (g)}_{z^i\\bar z^j} & V^{\\rm (g)}_{z^i z^j} \\\\\nV^{\\rm (g)}_{\\bar z^i \\bar z^j} & V^{\\rm (g)}_{\\bar z^i z^j} \n\\end{pmatrix},\n~~~~~\ng_{z}=\n\\begin{pmatrix}\ng_{z^i\\bar z^j} & 0 \\\\\n0 & g_{\\bar z^i z^j}\n\\end{pmatrix}.\\label{zmatrices}\n\\end{align}\nSince the mixing \n$\\tilde{V}_{\\tilde{z}^{j} \\phi}\\pm \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}}$\nbelongs to at least the first order of the perturbation, \nthe inverse matrix is evaluated in the global SUSY.\n\nThe mixing terms $V_{\\phi \\bar{z}^{i}}$ and $V_{\\phi z^{i}}$ are obtained from the scalar potential of SUGRA. \nBy using Eqs.~\\eqref{V} and \\eqref{Vi},\nthe mass matrices are derived as\n\\begin{align}\n\\label{hmass} \nV_{i \\bar{j}} &= \n\\mathrm{e}^{G} \n\\left( \\nabla _{i} G_{k} \\nabla _{\\bar{j}} G^{k} - R_{i \\bar{j} k \\bar{l} } G^{k}G^{\\bar{l}} + g_{i \\bar{j}} \\right) \\\\\n& + g^{2} \\left( \\frac{1}{2} \\left( G_{i}G_{\\bar{j}} - g_{i \\bar{j}} \\right) D^{A}D_{A} -G_{i}D^{A}D_{A \\bar{j}} - G_{\\bar{j}}D^{A}D_{Ai} +\\frac{1}{2} \\left( h^{R}_{ABi}G_{\\bar{j}} + h^{R}_{AB \\bar{j}}G_{i} \\right) D^{A}D^{B} \\right) \\nonumber \\\\\n& + g^{2} \\left( h_{R}^{AB}D_{Ai}D_{B\\bar{j}} + h_{R}^{AB}{}_{i}D_{A}D_{B\\bar{j}}+ h_{R}^{AB}{}_{\\bar{j}}D_{A}D_{Bi} +D^{A}D_{Ai\\bar{j}} + h^{R}_{ACi}h_{R}^{CD}h^{R}_{DB\\bar{j}} D^{A}D^{B} \\right), \\nonumber \\\\\n\\label{ahmass}\nV_{ij} &= \n\\mathrm{e}^{G}\n\\left( 2\\nabla _{i} G_{j} + G^{k}\\nabla _{i} \\nabla _{j} G_{k} \\right) \\nonumber \\\\\n& + g^{2} \\left( \\frac{1}{2} \\left( G_{i}G_{j} -\\nabla _{i}G_{j} \\right)D^{A}D_{A} -G_{i}D^{A}D_{Aj} - G_{j}D^{A}D_{Ai} \\right. \\\\\n& ~~~~~~~~\n\\left. +\\frac{1}{2} \\left( h^{R}_{ABi}G_{j} + h^{R}_{ABj}G_{i} \\right)D^{A}D^{B} +\\Gamma ^{k}_{ij} \\left( -D^{A}D_{Ak}+\\frac{1}{2}h^{R}_{ABk}D^{A}D^{B} \\right) \\right) \\nonumber \\\\\n & +g^{2} \\left( h_{R}^{AB}D_{A}D_{Bij}+ h_{R}^{AB}D_{Ai}D_{Bj} \n + h_{R}^{AB}{}_iD_{A}D_{Bj} \n + h_{R}^{AB}{}_jD_{A}D_{Bi} \n + \\frac{1}{2}h_{R}^{AB}{}_{ij}D_{A}D_{B} \\right), \\nonumber\n\\end{align}\nwith\n\\begin{align}\n\\nabla_{i}\\nabla_{j}G_{k}\n&= (\\nabla_{j}G_{k})_{i} -\\Gamma^{l}_{ij}\\nabla_{l}G_{k}-\\Gamma^{l}_{ik}\\nabla_{j}G_{l} \\nonumber \\\\\n&= G_{ijk}-G_{ijk\\bar{l}}G^{\\bar{l}}-3\\Gamma^{l}_{(ij}G_{k)l}+3\\Gamma^{l}_{(ij}\\Gamma^{m}_{k)l}G_{m},\n\\end{align}\nwhere the indices in a parenthesis are totally symmetrized.\nIn each of Eqs. \\eqref{hmass} and (\\ref{ahmass}), the first parenthesis is induced by the F-term VEVs, while the others are finite when the D term contributes. \nAt the first order, the mixing terms of the mass matrices which are (potentially) relevant for the cosmology are obtained as\n\\begin{align}\nV_{\\phi \\bar{z}^{i}} &\\simeq \ne^{G} \\left( g^{\\phi\\bar{\\phi}}\\nabla_{\\phi}G_{\\phi}\\nabla_{\\bar{z}^{i}}G_{\\bar{\\phi}} \n+ g^{z^{j}\\bar{z}^{k}}\\nabla_{\\phi}G_{z^{j}}\\nabla_{\\bar{z}^{k}}G_{\\bar{z}^{i}}\n\\right. \\nonumber\\\\ &~~~~~~~~~~\\left.\n+ g^{\\phi \\bar{z}^{j}}\\nabla_{\\phi}G_{\\phi}\\nabla_{\\bar{z}^{i}}G_{\\bar{z}^{j}} \n- R_{\\phi \\bar{z}^i z^j \\bar{z}^k} g^{z^{j}\\bar{z}^{l}}g^{\\bar{z}^{k}z^{m}}G_{\\bar{z}^l} G_{z^m} \\right) \n\\nonumber \\\\ &~~~\n+ g^{2} \\left( h_{R}^{AB}D_{A\\phi}D_{B\\bar{z}^{i}}+h_{R}^{AB}{}_{\\phi}D_{A}D_{B\\bar{z}^{i}}+D^{A}D_{A\\phi\\bar{z}^{i}}\\right), \\label{Mphizbar} \\\\\nV_{\\phi z^{i}} &\\simeq\n- e^{G} \\left( G_{\\bar{\\phi}z^{i}z^{j}}G_{\\phi\\phi}+2g^{z^{k}\\bar{z}^{l}}G_{\\bar{z}^{l}\\phi (z^{i}}G_{z^{j})z^{k}} \\right) g^{z^{j}\\bar{z}^{m}} G_{\\bar{z}^{m}} \n\\nonumber \\\\ &~~~\n+ g^{2} \\left( -g^{z^{j}\\bar{z}^{k}}G_{\\phi z^{i}\\bar{z}^{k}}D^{A}D_{A z^{j}} \n+ h_{R}^{AB}D_{A}D_{B\\phi z^{i}} \n+ h_{R}^{AB}D_{A\\phi}D_{B z^{i}} \n+ h_{R}^{AB}{}_{\\phi}D_{A}D_{B z^{i}} \\right). \\label{Mphiz} \n\\end{align}\nHere, we have used $|G_\\phi| \\ll 1$ and kept only the terms which can be enhanced by\nthe following means: (i) the modulus\/inflaton is much heavier than the gravitino due to its SUSY mass, $|\\nabla_{\\phi}G_{\\phi}| \\gg 1$, (ii) in the D-term SUSY-breaking models, the fields in the SUSY-breaking sector, $z^i$, can have a large SUSY-invariant mass, $|\\nabla_{z^i}G_{z^j}| \\gg 1$, and (iii) derivatives of $D_A$ with respect to $z^i$ can be enhanced, since the VEVs of $z^i$ are much smaller than $M_P$. \n\nIn summary, the effective coupling constants are represented in the model basis as\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}}\\right|^{2}=\n\\frac{1}{g_{\\phi\\bar{\\phi}}} \\left| G_{\\phi}+ G_{z^{i}} \\left[ \\left( m^{2}_{\\phi_{R}} g_{z} - m_z^{2} \\right)^{-1} \\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}+ \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right) \\right|^{2}, \\label{GeffR} \\\\\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}}\\right|^{2}=\n\\frac{1}{g_{\\phi\\bar{\\phi}}}\\left| G_{\\phi}+ G_{z^{i}} \\left[\\left( m^{2}_{\\phi_{I}} g_{z} - m_z^{2} \\right)^{-1}\\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}- \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right) \\right|^{2}, \\label{GeffI}\n\\end{align}\nwhere $G_\\phi$ is found in Eq.~\\eqref{Gphi}, $m^{2}_{z}$ and $g_{z}$ are in Eq.~\\eqref{zmatrices}, and the mixing terms are in Eqs.~\\eqref{eq:mass-matrix-1} and \\eqref{eq:mass-matrix-2} with Eqs.~\\eqref{Mphizbar} and \\eqref{Mphiz}. They are independent of the matrix $C$ in Eq.~\\eqref{non-unitary}. The direct gravitino production rate is evaluated by substituting the above coupling constants into Eq.~\\eqref{gravitinorate}. \n\nThe D-term contributions are found in the terms proportional to $g^2$ in Eqs.~\\eqref{Gphi}, \\eqref{Mphizbar}, and \\eqref{Mphiz}. They are classified into two groups: i) contributions of $\\phi$ to the gauge kinetic term such as $h_{AB} \\sim \\phi\/M_P$, and ii) contributions to the D-term potential such as $G_{\\phi z^{i}\\bar{z}^{j}}g^{\\bar{z}^{j} z^{k}}D_{Az^{k}}$ and $D_{A\\phi}=ig_{\\phi \\bar{z}^{i}}X^{\\bar{z}^{i}}_{A}=g_{\\phi \\bar{z}^{i}}g^{\\bar{z}^{i} z^{j}}D_{A z^{j}}$, due to an effective charge which is induced by mixings with the SUSY-breaking fields from higher-dimensional operators.\nSee Eq.~\\eqref{eq:D-term} for the expression of the D term.\n Thus, all the D-term contributions vanish if the following two conditions are satisfied:\n\\begin{enumerate}\n\\item[(I)] $\\phi$ does not appear in the gauge kinetic function of the extra gauge symmetry, $h_{AB\\phi}=0$.\n\\item[(II)] $\\phi$ does not have the specific K\\\"{a}hler mixings with the fields in the SUSY-breaking sector which are charged under the extra gauge symmetry, $g_{\\phi\\bar{z}^i}=G_{\\phi z^{i}\\bar{z}^{j}}=0$.\n\\end{enumerate}\nIf these conditions are satisfied, the gravitino production is only from the F-term contributions. (See Appendix~\\ref{app:F-term} for a revision of the F-term contribution.) It is found that the rate decreases as the D-term VEV increases. This is because \nthe VEV of the F-term potential satisfies the vanishing cosmological constant condition [Eq.~\\eqref{V}]. If the D-term potential dominates the SUSY breaking, the F-term VEVs become suppressed. Then the cosmological problem of the direct gravitino production can be relaxed. \n\nFinally, let us comment on radiative corrections to pair gravitino production. The scalar field couples to a pair of gauginos via matter loops analogously to the anomaly, where the scalar interactions with the matter come from gravitational effects or higher-dimensional operators~\\cite{Endo:2007ih,Endo:2007sz}. Since the gaugino in the SUSY-breaking sector is a main component of the goldstino, the anomaly-induced decay is considered to contribute to the pair gravitino production. This decay induced by the gravitational anomalies works when the matter fields in the loop are lighter than the scalar. \n\nWhen the scalar $\\phi$ is heavier than the fields in the SUSY-breaking sector, the latter fields other than the goldstino are also produced by the scalar decay through SUGRA interactions~\\cite{Endo:2006qk,Endo:2007sz}. The decay products in the SUSY-breaking sector are considered to decay into lighter fields including the gravitino. This increases the abundance of the gravitino, and thus worsens the cosmology. \n\n\\section{Example}\\label{sec:example}\n\nIn this section, we apply the formulas in the previous section to a model of the D-term SUSY breaking.\nA fraction of the D-term SUSY breaking is defined as\n\\begin{align}\n\\delta=\\frac{V_{D}}{V_{F}+V_{D}}, \\label{delta}\n\\end{align}\nwhere the F-term SUSY breaking is represented by the F-term potential, $V_{F}=e^{G}G^{i}G_{i}$, and the D-term SUSY breaking is represented by the D-term potential, $V_{D}=(g^{2}\/2)D^{A}D_{A}$.\nIn the global SUSY, the F-term potential leads to $V_{F}=g^{i\\bar{j}}\\overline{W}_{\\bar{j}}W_{i}$.\nThe denominator of the right-hand side of Eq.~\\eqref{delta} is related to the gravitino mass through the cosmological constant condition [Eq.~\\eqref{V}], i.e.,\n\\begin{align}\n V_F + V_D = g^{i\\bar{j}}\\overline{W}_{\\bar{j}}W_{i} + \\frac{1}{2}g^{2} D^{A}D_{A}=3 m_{3\/2}^2 . \\label{V2}\n\\end{align}\nThe fraction $\\delta$ takes a value of $0\\leq \\delta <1$, where a pure D-term SUSY breaking, $\\delta=1$, is excluded because of the gauge invariance condition, $D_{A}=D_{A}^{i}G_{i}$~\\cite{WessBagger}. \n\nIn Sec.~\\ref{sec:SUSYbreaking}, a model of SUSY breaking is introduced. This model is extended to include a scalar field (such as modulus or inflaton), and the direct gravitino production rate is studied in Sec.~\\ref{sec:combined}. In particular, the conditions (I) and (II) obtained in the previous section are assumed to be satisfied. It will be explicitly shown that the rate is reduced compared to the F-term SUSY-breaking models, as the D-term SUSY-breaking effect increases. The cosmological implications are discussed in Sec.~\\ref{sec:cosmology}.\n\n\n\\subsection{D-term SUSY-Breaking Model}\n\\label{sec:SUSYbreaking}\n\nLet us consider a model of the D-term SUSY breaking explored in Ref.~\\cite{Gregoire:2005jr}.\nThe model has a $U(1)$ gauge superfield and six chiral superfields, $z_{-1}, z_{1}, z_{-1\/N}, z_{1\/N}, z_{0}$ and $z'_{0}$, with their $U(1)$ charges denoted by the subscripts.\nThe K\\\"{a}hler potential and the gauge kinetic function are minimal, while the superpotential is\n\\begin{eqnarray}\nW(z^i) = \\lambda_1 z_0 \\left( m^{1-N} z_1 z_{-1\/N}^N-m^{2} \\right) + \\lambda_2 m z_1 z_{-1}\n+ \\lambda_3 z'_0 z_{1\/N} z_{-1\/N},\n\\label{eq:modelW}\n\\end{eqnarray}\nwhere $m$ determines the SUSY-breaking scale. \nFor $\\lambda_{3}\\gg \\lambda_{1} \\gg \\lambda_{2} \\gg g$, \nthe field VEVs in the global SUSY limit are given by~\\cite{Gregoire:2005jr}\n\\begin{eqnarray}\n&& \n\\vev{z_{-1}}= \\vev{z_{1\/N}}= \\vev{z_{0}}=\\vev{z'_{0}}=0,\n\\label{eq:vev}\n\\\\\n&&\n|\\vev{z_{-1\/N}}|\\simeq m\\left( N^{3}\\hat{\\lambda}_{2}^{2} \\right) ^{\\frac{1}{2(N+2)}}, \\label{eq:vev2}\n \\\\\n&& \\vev{z_{1}}\\simeq m^{N+1}\\vev{z_{-1\/N}}^{-N}, \\label{eq:vev3}\n\\end{eqnarray}\nwhere $\\hat{\\lambda}_{2} \\equiv \\lambda_2\/g$. \nAt the vacuum, the scalar potential in the global SUSY is~\\cite{Gregoire:2005jr}\n\\begin{align}\nV \\simeq g^{2} m^{4} \\left( N^{3}\\hat{\\lambda}_{2}^{2} \\right)^{\\frac{2}{N+2}} \\left( \\frac{1}{N^{3}}+\\frac{1}{2 N^{2}} \\right).\\label{Vapprox}\n\\end{align}\nHere, $1\/N^{3}$ and $1\/2N^{2}$ in the parenthesis correspond to the F-term and D-term potentials, respectively. \nThus, the model involves the D-term SUSY-breaking effect as well as that of the F term, and the fraction of the D-term SUSY breaking is approximately given by $\\delta\\simeq N\/(N+2)$. The D-term contributions increase as $N$ becomes larger.\n\nIn the following numerical calculations, we use more precise values of the field VEVs, which are obtained by following the analysis in Appendix~\\ref{sec:potential}, and the mass spectrum, which is obtained using the mass matrices listed in Appendix~\\ref{sec:mass}.\nThe nonzero VEVs and mass spectra are shown in Table~\\ref{tab:mass_spectrum}\nfor some model points, as well as the SUSY-breaking fraction $\\delta$. The numerical results in Table~\\ref{tab:mass_spectrum} agree with the approximate ones within $\\sim 20\\%$.\n\n\nAs shown in the table, all the fields in the SUSY-breaking sector have masses scaled by $m$, which is related to the gravitino mass $m_{3\/2}$ by Eq.~\\eqref{eq:m_and_m32}, or approximately \n\\begin{align}\nm \\simeq 6^{\\frac{1}{4}}N^{\\frac{1}{2}}g^{-\\frac{1}{2}}\\sqrt{m_{3\/2}M_{P}},\n\\end{align}\nfor large $N$, where we have used Eqs.~\\eqref{V2}, \\eqref{eq:vev2}, and \\eqref{eq:vev3}.\nNote that there is no light scalar field (the so-called Polonyi field) in the SUSY-breaking sector with a mass on the order of the gravitino mass. \n\n\n\\begin{table}[t!]\n\\begin{center}\n\\caption{\nThe fractions of the D-term SUSY breaking $(\\delta )$, the field VEVs, and the mass spectra for $N=1, 10$, and 100. Here, two sets of the parameters, P1 and P2, are chosen. \nSee Appendixes~\\ref{sec:potential} and \\ref{sec:mass} for details of the analysis.\nThe field VEVs and masses are written in units of $m$, except for the gravitino mass $m_{3\/2}$.\n}\n\\begin{tabular}{|c|l|c|c|cc|l|c|}\n\\hline\n& $\\lambda_{3}, \\lambda_{1},$\n& \n& \n& &\n& \\multicolumn{2}{|c|}{mass spectrum}\n\\\\ \\cline{7-8}\n& $\\lambda_{2}, g$ \n& $N$\n& $\\delta$\n& $\\vev{z_1}$ & $\\vev{z_{-1\/N}}$\n& \\multicolumn{1}{|c|}{\\cancel{SUSY} sector}\n& $m_{3\/2}$\n\\\\ \\hline\\hline\nP1 \n& $1, 10^{-1},$ \n& $1$\n& $0.31$\n& $0.46$ & $2.2$\n& $3.7\\times 10^{-3} - 2.2$ \n& $3.2\\times 10^{-3}\\, m^2\/M_P $\n\\\\\n& $10^{-2}, 10^{-3}$\n& $10$\n& $0.83$\n& $8.3\\times 10^{-3}$ & $1.6$\n& $5.6\\times 10^{-4} - 12$ \n& $1.2\\times 10^{-4}\\, m^2 \/M_P$\n\\\\\n&\n& $100$\n& $0.97$\n& $1.4 \\times 10^{-4}$ & $1.1$\n& $1.3\\times 10^{-4} - 7.0\\times 10^{2}$ \n& $5.0 \\times 10^{-6}\\, m^2\/M_P$\n\\\\ \\hline\\hline\nP2\n& $1, 4^{-1}$, \n& $1$\n& $0.27$\n& $0.61$ & $1.6$\n& $4.0\\times 10^{-2} - 1.6$ \n& $2.6\\times 10^{-2} m^2 \/ M_P$\n\\\\\n& $4^{-2}, 4^{-3}$\n& $10$\n& $0.83$\n& $1.8 \\times 10^{-2}$ & $1.5$\n& $8.1 \\times 10^{-3} - 14$ \n& $1.6 \\times 10^{-3} m^2 \/ M_P$\n\\\\\n&\n& $100$\n& $0.98$\n& $2.9\\times 10^{-4}$ & $1.1$\n& $1.7\\times 10^{-3} - 8.5 \\times 10^{2}$ \n& $7.6 \\times 10^{-5} m^2 \/ M_P$\n\\\\ \\hline\n\\end{tabular}\n\n\\label{tab:mass_spectrum}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{Gravitino Production Rate}\n\\label{sec:combined}\n\nThe model in the previous subsection is extended to include the scalar $\\phi$. For simplicity, it is assumed that the K\\\"{a}hler potential and gauge kinetic function are minimal,\n\\begin{align}\nK=\n|\\phi |^{2}+\\left|z_{ 0} \\right|^{2}+\\left|z'_{0 } \\right|^{2}+\n\\left|z_{1 } \\right|^{2}+\\left|z_{-1 } \\right|^{2}+\n\\left|z_{1\/N } \\right|^{2}+\\left|z_{-1\/N } \\right|^{2},~~~\nh_{U(1)}=1.\n\\label{eq:modelKh}\n\\end{align}\nIt is also assumed that the mixings between $\\phi$ and $z^i$ are absent in the superpotential, $W^{(n)}_{\\rm mix}(\\phi, z^i)=0$, and the total superpotential is given by\n\\begin{align}\nW=W(z^i) +W(\\phi),\n\\end{align}\nwith $W(z^i)$ given by Eq.~\\eqref{eq:modelW}.\nIt is found that conditions (I) and (II) discussed in the previous section are satisfied. \n\nLet us now calculate the partial decay rate of the scalar $\\phi$ into gravitinos, or the effective coupling constants in Eqs.~\\eqref{GeffR} and \\eqref{GeffI}, following the discussion in the previous section. They are determined by $G_\\phi$ and the mixings between the scalar and the SUSY-breaking fields.\nIn the present model,\nfrom Eq.~\\eqref{Gphi}, $G_\\phi$ becomes\n\\begin{align}\nG_{\\phi} \\simeq - \\frac{G_{z^{i}} \\nabla_{\\bar{\\phi}}G_{\\bar{z}^{i}}}{\\nabla_{\\bar{\\phi}}G_{\\bar{\\phi}}},\n\\label{eq:Gphi_approx}\n\\end{align}\nand the mixings in Eqs.~\\eqref{eq:mass-matrix-1}, \\eqref{eq:mass-matrix-2}, \\eqref{Mphizbar}, and \\eqref{Mphiz} are\n\\begin{align}\n&\\tilde{V}_{\\phi \\bar{z}^{i}} = V_{\\phi \\bar{z}^{i}} \\simeq \ne^{G} \\left( \\nabla_{\\phi}G_{\\phi}\\nabla_{\\bar{z}^{i}}G_{\\bar{\\phi}} \n+ \\nabla_{\\phi}G_{z^{j}}\\nabla_{\\bar{z}^{j}}G_{\\bar{z}^{i}} \\right),\n\\label{eq:mix_approx}\\\\\n&\\tilde{V}_{\\phi z^{i}} = V_{\\phi z^{i}} \\simeq 0.\n\\end{align}\nIn Eqs.~\\eqref{eq:Gphi_approx} and \\eqref{eq:mix_approx}, the factors $e^G$, $\\nabla_\\phi G_\\phi$, and $\\nabla_{z^i}G_{z^j}$ are of zeroth order in the perturbation, and are given by\n\\begin{align}\ne^{G\/2} = m_{3\/2} \\simeq \\vev{W},~~~\n\\nabla_\\phi G_\\phi \\simeq \\frac{m_\\phi}{m_{3\/2}},~~~\n\\nabla_{z^i}G_{z^j} = \\frac{W_{z^i z^j}}{W} - \\frac{W_{z^i}}{W} \\frac{W_{z^j}}{W}.\n\\label{eq:nablaGzeroth}\n\\end{align}\nOn the other hand, the factor $\\nabla_{z^i} G_{\\phi} = \\nabla_{\\phi} G_{z^i}$ is suppressed by the Planck scale. Due to the absence of the $\\phi - z^i$ mixings in $K$ and $W$\nin the present model, it is given by\n\\begin{align}\n\\nabla_{\\phi} G_{z^i} =\n-\n\\frac{W_\\phi}{W}\n\\frac{W_{z^i}}{W}\n=\n(K_\\phi-G_\\phi)\\frac{W_{z^i}}{W}\n\\simeq K_\\phi\\frac{W_{z^i}}{W}.\n\\label{eq:Gzi_approx}\n\\end{align}\nHere, in the last equality, we have used $G_\\phi \\simeq K_\\phi \\times {\\cal O}(m_{3\/2} \/ m_\\phi) \\ll K_\\phi$, which can be shown by using Eq.~\\eqref{eq:Gphi_approx}, $\\nabla_\\phi G_\\phi \\simeq m_\\phi\/m_{3\/2}$, and $G_{z^i}\\simeq W_{z^i}\/W \\lesssim 1$.\nSee also Appendix~\\ref{app:F-term}.\nThe remaining task is to evaluate the F-term VEVs of the SUSY-breaking fields, $W_{z^i}$, in the global SUSY limit. From Eqs.~\\eqref{eq:modelW} -- \\eqref{eq:vev3}, the finite F terms are\n\\begin{align}\nW_{z_0} = \\lambda_{1}\\left(m^{1-N}\\vev{z_{1}}\\vev{z_{-1\/N}}^{N}-m^{2}\\right),~~~\nW_{z_{-1}} = \\lambda_2 m \\vev{z_1}.\n\\label{eq:Wderiv}\n\\end{align}\nFrom Eqs.~\\eqref{eq:vev} -- \\eqref{eq:vev3}, it is found that $W_{z_{-1}}$ dominates the F-term potential.\\footnote{Although $W_{z_0}$ seems to vanish when the VEVs in Eqs.~\\eqref{eq:vev} -- \\eqref{eq:vev3} are naively applied, it is finite as long as the D term is nonzero, because the right-hand side of $W_{z_0}$ is analytically related to the D-term VEVs via the stationary condition of the scalar potential [Eq.~\\eqref{eq:appV}]. \nThe gaugino masses of the SUSY SM can be given by $W_{z_0}$, or by the anomaly mediation.\n}\nThe other F terms are zero in the global SUSY. \n From Eqs.~\\eqref{eq:Gzi_approx} and \\eqref{eq:Wderiv}, it is found that $\\nabla_{\\phi} G_{z^i}$ \nis dominated by $\\nabla_{\\phi}G_{z_{-1}}$ as\n\\begin{align}\n\\nabla_{\\phi}G_{z_{-1}} \\simeq \\frac{\\lambda_{2} m}{m_{3\/2}} \\vev{z_{1}}\\vev{\\bar{\\phi}},\n\\end{align}\nwhere $\\vev{W} \\simeq m_{3\/2}$ is used. \nThus, the couplings are dominated by $z^i = z^j = z_{-1}$.\nPutting the above equations altogether, one obtains\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right|^{2}\n\\simeq \\left| \n\\left( 1- m^{2}_{\\phi} \\left [ \\left( m^{2}_{\\phi}-m^{2}_{z} \\right)^{-1} \\right ]^{z_{-1}\\bar{z}_{-1}} \\right)\n\\frac{m_{3\/2}}{m_\\phi} G_{z_{-1}} \\nabla_{\\bar{\\phi}} G_{\\bar{z}_{-1}}\n\\right|^2.\n\\end{align}\nFor the purpose of the cosmological application in the next subsection, we concentrate on the case in which the scalar $\\phi$ is lighter than the fields of the SUSY-breaking sector $z^{i}$. Then, the effective coupling constants are reduced to\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right|^{2}\n\\simeq\n\\left| \n\\frac{m_{3\/2}}{m_\\phi} G_{\\bar{z}_{-1}} \\nabla_{\\phi} G_{z_{-1}}\n\\right|^2\n=\n\\left| \n\\frac{\\lambda_{2}^2 m^2}{m_{3\/2} m_\\phi} \\vev{z_{1}}^2\\vev{\\bar{\\phi}}\n\\right|^2\n\\simeq\n\\frac{36}{N^2} \\frac{m_{3\/2}^2}{m_\\phi^2}\\vev{\\phi}^2.\n\\label{eq:modelGeff}\n\\end{align}\nIn the last equality we have used Eqs.~\\eqref{eq:vev2} and \\eqref{eq:vev3}, and $N\\gg 1$. Here, the effective couplings become insensitive to the model parameters of the SUSY-breaking sector except for $N$.\n\nIn Fig.~\\ref{fig:Geff}, we show the effective coupling constants as a function of the fraction of the D-term SUSY breaking $\\delta$, which is varied by changing the $N$ of the model. Here, we have used the full formulas in Eqs.~\\eqref{GeffR} and \\eqref{GeffI} with the numerically obtained field VEVs.\\footnote{We have also used Eqs.~\\eqref{eq:nablaGzeroth} and \\eqref{eq:Gzi_approx} to evaluate the derivatives of $W(\\phi)$.} The mass of the scalar $\\phi$\nis set to be smaller than the masses of the fields in the SUSY-breaking sector.\nFor the sake of comparison, we have also shown, by the star sign in the figure, the case where the SUSY is broken only by a single F-term VEV, assuming the minimal K\\\"{a}hler potential:\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right|^{2}\n\\simeq\n\\left|\n\\frac{G_z \\nabla_{\\bar{\\phi}} G_{\\bar{z}}}{\\nabla_{\\bar{\\phi}} G_{\\bar{\\phi}}} \n\\right|^2\n\\simeq\n\\left| \\frac{3m_{3\/2}}{m_\\phi} \\vev{\\phi} \\right|^2\n\\quad (\\text{pure F term}),\n\\end{align}\nwhere we have used $|G_z|\\simeq \\sqrt{3}$.\nIt is found that the effective coupling constants diminish as the D-term SUSY-breaking contribution increases. \n\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=85mm]{Geff.eps}\n \\end{center}\n \\caption{Dependence of the effective coupling squared $\\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}} \\right |^{2} \\simeq \\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}} \\right |^{2} \\equiv \\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi} \\right |^{2}$ on the fraction of D-term SUSY breaking $\\delta$. $m_{3\/2}=10^3\\rm{GeV}$, $m_{\\phi}=10^{6}\\rm{GeV}$, and $\\vev{\\phi}=10^{12}\\rm{GeV}$ are used. The lines for parameter sets P1 and P2 in Table~\\ref{tab:mass_spectrum} coincide.\nThe fraction $\\delta$ has been extrapolated to $N<1$, which is drawn by the dashed line. The star mark represents the case of pure F-term SUSY breaking.}\n \\label{fig:Geff}\n\\end{figure}\n\n\n\\subsection{Cosmological Implications}\n\\label{sec:cosmology}\n\nFinally, we estimate the total gravitino abundance.\nThe coherent oscillation of the scalar $\\phi$ is assumed to dominate the energy density of the universe and then decay to create the thermal bath.\nThere are two types of gravitino production:\none is thermal production, where the gravitinos are produced through scattering processes in the thermal bath, and the other is direct production, which is the main subject of this paper.\nIn the following, we focus on the parameter region where the scalar is lighter than the fields in the SUSY-breaking sector except for the gravitino. Otherwise, the anomaly-induced gravitino production and the direct production of the SUSY-breaking fields which are mentioned in the last section can spoil the success of the standard cosmology.\n\nFirst of all, let us consider thermal production of the gravitino.\nThe gravitino yield is defined as $Y_{3\/2}=n_{3\/2}\/s$, where $n_{3\/2}$ is the number density of the gravitino, and $s$ is the entropy density. The thermal gravitino yield is approximately given by~\\cite{Kawasaki:2008qe}\n\\begin{align}\nY_{3\/2}^{\\text{thermal}}\\simeq \n\\left( 1.3\\times 10^{-14}+8.8\\times 10^{-15} \\left( \\frac{m_{1\/2}}{m_{3\/2}}\\right)^{2} \\right)\n\\left( \\frac{T_{R}}{10^{8}\\rm{GeV}} \\right),\n\\label{Ythermal}\n\\end{align}\nwhere $T_{R}=(\\pi^{2}g_{*}(T_{R})\/90)^{-1\/4}\\sqrt{\\Gamma_\\phi}$ is the reheating temperature after the $\\phi$ decay, with the total decay rate $\\Gamma_\\phi$, and $m_{1\/2}$ is the unified gaugino mass at the grand unified theory (GUT) scale.\\footnote{The definition of $T_R$ here is different from the one in Ref.~\\cite{Kawasaki:2008qe}.\nLogarithmic corrections are omitted in Eq.~\\eqref{Ythermal}.}\n\n\nNext, let us consider the direct production of gravitinos.\nThe direct production of a pair of gravitinos provides\n\\begin{align}\nY_{3\/2}^{\\text{decay}} \n&= \\frac{3T_{R}}{4m_{\\phi}} \\times 2B_{3\/2} \n= \\frac{1}{192\\pi} \\left( \\frac{90}{\\pi^{2} g_{*}(T_{R})} \\right)^{\\frac{1}{2}} \\frac{m_{\\phi}^{4} \\left| \\mathcal{G}_\\Phi^{\\text{(eff)}} \\right|^{2}}{m_{3\/2}^{2}T_{R}}.\n\\label{eq:yield}\n\\end{align}\nHere, $m_{\\Phi_{R}} \\simeq m_{\\Phi_{I}} \\equiv m_{\\phi}$ and $\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}} \\right|^{2} \\simeq \\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}} \\right|^{2} \\equiv \\left| \\mathcal{G}_\\Phi^{\\text{(eff)}} \\right|^{2}$ are used. \n Substituting Eq.~\\eqref{eq:modelGeff} into Eq.~\\eqref{eq:yield}, the yield is reduced to be\n\\begin{align}\nY_{3\/2}^{\\text{decay}} \n&\\simeq\n\\frac{1}{192\\pi} \\left( \\frac{90}{\\pi^{2} g_{*}(T_{R})} \\right)^{\\frac{1}{2}} \n\\frac{1}{T_R} m_\\phi^2 \\vev{\\phi}^2\n\\left(\\lambda_2^4 \\frac{\\vev{z_1}^4 m^4}{m_{3\/2}^4} \\right) \\nonumber\\\\\n&\\simeq\n8.2 \\times 10^{-15} \\times\n\\frac{1}{N^2} \\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \\left( \\frac{10^{5}\\rm{GeV}}{T_{R}} \\right),\n\\label{Ydecay}\n\\end{align}\nfor large $N$, where $g_{*}(T_{R})=228.75$ is used. It is found that the yield is proportional to $m_\\phi^2, \\vev{\\phi}^2$, and $T_R^{-1}$. In particular, \nit becomes independent of $m_{3\/2}$ because $\\vev{z_1}\\propto m \\propto \\sqrt{m_{3\/2}}$. It is stressed that the abundance decreases as $N$ increases; namely, the D-term contributions dominate (See also Fig. \\ref{fig:Geff}). \n\n\\begin{figure}[thb]\n \\begin{center}\n \\includegraphics[width=85mm]{APR_typ.eps}\n \\end{center}\n \\caption{Contours of the cosmological constraints on the $m_{\\phi}$ -- $\\vev{\\phi}$ plane from the thermal and direct production of gravitinos. The upper-right regions are excluded.\nSeveral values of the D-term SUSY-breaking contributions are taken: $\\delta=0$ (the pure F-term; see Fig.~\\ref{fig:Geff} and the text), $0.27\\ (N=1)$, and $0.98\\ (N=100)$. \nThe dashed lines correspond to $m_{3\/2}=10$GeV, and the solid ones correspond to $m_{3\/2}=1$TeV.\nThe former lines end in the figure because $m_{\\phi}$ is restricted to be smaller than the smallest mass of the fields in the SUSY-breaking sector.\nThe reheating temperature is provided by $\\Gamma_\\phi=m_{\\phi}^{3}\/8\\pi$. \nThe thermal abundance is subdominant in this figure.\n }\n \\label{fig:APR_typ}\n\\end{figure}\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=85mm]{APR_opt.eps}\n \\end{center}\n \\caption{Same as Fig.~\\ref{fig:APR_typ}, except that the reheating temperature is optimized such that the total gravitino abundance is minimized. The bound on $m_\\phi$ and $\\vev{\\phi}$ is considered to be most conservative. The gravitino mass is set to be $m_{3\/2}=1\\rm{TeV}$. There are no bounds in the case of $m_{3\/2}=10$GeV in this parameter space.\n }\n \\label{fig:APR_opt}\n\\end{figure}\n\nThe cosmological constraint on the gravitino abundance depends on the gravitino mass. \nWhen the gravitino is unstable, the most stringent bound comes from BBN.\nThe precise bound depends on details of the mass spectrum including the SUSY SM.\nHere, we adopt the constraint for the model point in Case 2 of Ref.~\\cite{Kawasaki:2008qe},\\footnote{This model point is excluded by the recent SUSY search at the LHC, but we adopt it just for illustration. The main conclusion does not depend much on the details of the mass spectrum.}\n where the bounds are given by $Y_{3\/2}< 2\\times 10^{-16}$ and $Y_{3\/2}< 2\\times 10^{-12}$\nfor the gravitino masses of $m_{3\/2}=1 \\rm{TeV}$ and $30 \\rm{TeV}$, respectively.\nThus, from Eqs.~\\eqref{Ythermal} and \\eqref{Ydecay}, the allowed region of the reheating temperature is \n\\begin{align}\n3 \\times 10^6\\rm{GeV} \\times \\frac{1}{N^2}\n\\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n\\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n< &\\ T_{R} < 2\\times 10^{6}\\rm{GeV}\n& (m_{3\/2}=1\\rm{TeV}),\\\\\n5 \\times 10^2\\rm{GeV} \\times \\frac{1}{N^2}\n\\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n\\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n< &\\ T_{R} < 1\\times 10^{10}\\rm{GeV}\n& (m_{3\/2}=30\\rm{TeV}),\n\\end{align}\nwhere the upper bound comes from the thermal gravitino abundance, while the lower bound is from direct production. This is because the thermal abundance is proportional to $T_R$, while the abundance from the latter is to $1\/T_R$. \nIt is noticed that the window tends to close as the energy scale of the scalar model is higher. In order to avoid the cosmological bound, the D-term contributions must be enhanced by increasing $N$. For instance, the window opens only when $N \\gg 1$ for $m_{3\/2}=1\\rm{TeV}$, $\\vev{\\phi} = 10^{16}\\rm{GeV}$, and $m_\\phi = 10^9\\rm{GeV}$. \n\nWhen the gravitino is the lightest among the SUSY particles, the gravitino becomes stable. \nSince the massive gravitino contributes to the dark matter abundance, the gravitino abundance cannot exceed the cold dark matter abundance as\n\\begin{align}\nm_{3\/2} Y_{3\/2} < \\frac{\\rho_c}{s} \\Omega_{\\rm DM} < 4.4 \\times 10^{-10}\\rm{GeV},\n\\end{align}\nwhere $\\rho_c$ is the critical density and $\\Omega_{\\rm DM} h^2 < 0.12$ at $2\\sigma$~\\cite{Beringer:1900zz} is used. \nFor the light stable gravitino, the bound becomes\n\\begin{align}\n1.9 \\rm{GeV} \\times\\frac{1}{N^2}\\lrf{m_{3\/2}}{1 \\rm{GeV}}\n\\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n\\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n< &\\ T_{R} <\n1.4 \\times 10^{7}\\rm{GeV} \\times \\lrf{m_{3\/2}}{1\\rm{GeV}},\n\\end{align}\nwhere the gaugino mass is taken to be 600 GeV at the GUT scale, which affects the upper bound via the thermal gravitino abundance.\nAgain, the lower bound becomes milder as the D-term contributions dominate.\n\n\n\nFinally, let us show how D-term SUSY-breaking contributions enlarge the allowed parameter space spanned by $m_{\\phi}$ and $\\vev{\\phi}$. \nIn Figs.~\\ref{fig:APR_typ} and \\ref{fig:APR_opt}, contours of the cosmological bounds are drawn for various $\\delta$ in the $m_{\\phi}$ -- $\\vev{\\phi}$ plane. The region of lower $m_{\\phi}$ and $\\vev{\\phi}$ is allowed. \nIn the figures, $m_\\phi$ is restricted to be smaller than the smallest mass of the fields in the SUSY-breaking sector, for simplicity. \nIt can be seen that as the D-term SUSY-breaking contributions increase---i.e., as $\\delta$ becomes close to unity---the allowed region becomes wider. \n\n\nThe bound depends on the reheating temperature as seen in Eqs.~\\eqref{Ythermal} and \\eqref{Ydecay}. \nIn Fig.~\\ref{fig:APR_typ}, the temperature is estimated by assuming that the total decay rate is governed by a Planck-suppressed operator, $\\Gamma (\\phi \\rightarrow \\text{all}) = m_{\\phi}^{3}\/8 \\pi$.\nThe thermal production is subdominant in the figure.\nOn the other hand, the reheating temperature is taken to be most conservative in Fig.~\\ref{fig:APR_opt}. Since the thermal gravitino abundance is proportional to $T_R$, and that from the direct production is to its inverse, the total gravitino abundance becomes the minimum value by tuning $T_R$. Therefore, the bound in Fig.~\\ref{fig:APR_opt} is the most conservative one, and the model is refuted if it is excluded in Fig.~\\ref{fig:APR_opt}, unless there is an entropy production. \n\n\n\n\\section{Summary}\\label{sec:conclusion}\n\nIn this paper, we studied scalar decay into gravitinos in the presence of D-term SUSY-breaking and multiple SUSY-breaking fields, taking into account the mixing between the scalar and the fields in the SUSY-breaking sector.\nWe obtained the general formulas of the effective coupling constants for gravitino pair production, Eqs.~\\eqref{GeffR} and \\eqref{GeffI}.\nIt is found that the gravitino production is suppressed when the D-term VEV is sizable if conditions (I) and (II) in Sec.~\\ref{sec:EffCoupling} are satisfied.\n\nAs an example, we applied the formulas to the D-term SUSY-breaking model in Ref.~\\cite{Gregoire:2005jr} and showed that the D-term SUSY breaking suppresses the gravitino production.\nMoreover, an interesting feature of the model is that all the fields in the SUSY-breaking sector have masses of ${\\cal O}(\\sqrt{M_p m_{3\/2}})$.\nThis is very different from F-term SUSY-breaking models, \nin which the SUSY-breaking field tends to have a mass of $\\mathcal{O}(m_{3\/2})$, \ncausing the Polonyi problem~\\cite{Coughlan:1983ci}. \nThis may be a generic feature of D-term SUSY-breaking models. (See also\na recent work, Ref.~\\cite{Azeyanagi:2012pc}.)\nThese observations can open a new window for model building respecting cosmology.\n\n\\section*{Acknowledgments}\nThis work was supported by Grants-in-Aid for Scientific Research from\nthe Ministry of Education, Science, Sports, and Culture (MEXT), Japan,\nGrants No. 23740172 (M.E.), No. 21740164 (K.H.), and No. 22244021 (K.H.).\nThe work of T.T. was supported by an Advanced Leading Graduate Course for Photon Science grant.\nThis work was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this paper we consider a model for two-component Bose-Einstein condensates irradiated by an external electromagnetic field. The dynamics are described by two coupled nonlinear Schr\\\"odinger equations\n\\begin{equation}\\label{eq:nlss1}\ni\\d_t\\psi_j=-\\frac12\\Delta\\psi_j+V(x)\\psi_j+\\beta_{jj}|\\psi_j|^2\\psi_j+\\beta_{jk}|\\psi_k|^2\\psi_j+\\lambda\\psi_k,\n\\end{equation}\n$j, k = 1, 2$, in $(t, x)\\in\\R\\times\\R^N$, $N = 1, 2, 3$. Here $V(x)=\\frac{\\gamma^2}{2}|x|^2$ is the trapping potential, $\\beta_{jj}$ and $\\beta_{jk}$ with $\\beta_{12}=\\beta_{21}$ are the (scaled) intra- and inter-specific scattering lengths, respectively, $\\lambda$ is the Rabi frequency related to the external electric field.\n\\newline\nOur main goal in this paper is to investigate the asymptotic behavior of solutions to equation \\eqref{eq:nlss1} when the Rabi frequency $\\lambda$ becomes very large. In particular, we shall show that, after a suitable transformation of the system, the asymptotic behavior is described by two effective coupled nonlinear Schr\\\"odinger equations with the same inter- and intra-specific scattering lengths.\n\\newline\nThe equations we consider here arise in the modeling of a Bose-Einstein condensate formed of atoms in two different hyperfine states in the same harmonic trap \\cite{Ballagh}. A binary BEC of $^{87}$Rb atoms in different spin states was produced for the first time at JILA \\cite{Myatt97}. The irradiation of the condensate with an electric field induces a linear coupling in the overlap region, which causes a Josephson-type oscillation between the two species \\cite{Will99}. Such condensates are very interesting in physics, since it is possible to measure the relative phase of one component with respect to the other one \n\\cite{Hall:1998fk}. Furthermore, by controlling locally the relative phase, it is also possible to produce vortices \n\\cite{williams1999preparing}, \\cite{matthews1999vortices}. Another interesting application of this model is the creation of a stable \\emph{BEC droplet} \\cite{Saito} without using highly oscillating magnetic fields through Feshbach resonance methods: indeed the external electric field is constant but it induces an effective oscillation of the scattering lengths.\n\\newline\nThere is an extensive literature on systems of nonlinear Schr\\\"odinger equations, and giving here a comprehensive picture of all the results in different cases would be a quite hard task.\nIn particular, the case without linear coupling, i.e. $\\lambda=0$ in \\eqref{eq:nlss1}, is considered in most of the cases. \n\\cite{FanMont, LiWuLai, MaZhao} deal with the system without trapping potential and a more general class of power-type nonlinearities, giving sufficient conditions for global existence of solutions, \\cite{MR2283958} analyses the existence of ground state solutions when the nonlinearities are cubic, and \\cite{MR2449345} studies vortices in the same framework. In the presence of a trapping potential we mention \\cite{LinWei, ChenWei, Prytula}.\nOn the other hand, the case with linear coupling, i.e. $\\lambda\\neq0$ in \\eqref{eq:nlss1}, is less treated, in \n\\cite{Bao} existence and uniqueness of ground states is proved, performing also asymptotics for such solutions when \n$|\\lambda|\\to\\infty$, whereas in \\cite{Juengel11} the time-dependent system with a more general class of power-type nonlinearities is considered. Here the authors give sufficient conditions for global existence or blow up, and a semi-implicit formula shows the mass transport between the two components.\n\\newline\nMoreover, as it will be clear in the next Sections, the analysis of system \\eqref{eq:nlss1} is strictly related to the study of nonlinear Schr\\\"odinger equations with time-oscillating nonlinearities. This link is well known in physics, see \\cite{Saito}. Cazenave, Scialom \\cite{Caz10} were the first to rigorously study such problems, providing a convergence result in the highly oscillating regime, see also \\cite{Abdullaev} for numerical results, \\cite{Fang} for the case with an energy-critical nonlinearity, \n \\cite{CarGamPan} for a system of coupled nonlinear Schr\\\"odinger equations. Similar problems are also studied in some related frameworks, for example in \\cite{AS10} the authors carry out an asymptotic analysis for a \\emph{dispersion managed} NLS, i.e. when there is a highly oscillating coefficient in front of the Laplacian, or in \\cite{MR2809621} the authors consider the KdV equation with an time-dependent nonlinearity.\n \n\n\n\nIn Section \\ref{sect:prel} we introduce the notations and state the main result of our paper. We then dedicate Section \n\\ref{sect:LGWP} to the analysis of system \\eqref{eq:nlss1}, establishing the cases of global existence of solutions or possible occurrence of blow-up, depending on the different choices of the coefficients $\\beta_{jk}$'s.\nThen in Section \\ref{sect:trans} we transform \\eqref{eq:nlss1} in a way which is more suitable to study the asymptotics when \n$|\\lambda|$ becomes large. We then recover the limiting system, and in Section \\ref{sect:as} we prove rigorous convergence. \n\\section{Preliminaries and Main Result}\\label{sect:prel}\nIn what follows $C$ will denote a generic constant greater than 1, which may possibly change from line to line.\nWith $\\Re$ and $\\Im$ we denote the real and imaginary part of a complex number, respectively. By $z^*$ we denote the complex conjugate of $z$. The scalar product between two vectors $v, v'$ will be denoted by $\\langle v, v'\\rangle$.\n\\newline\nSince we are dealing with two-component Schr\\\"odinger systems, we shall indicate with capital letters the two-dimensional vector fields describing the wave-function of a two-component quantum system. For example we shall write \n$\\Psi^t=(\\psi_1, \\psi_2)$, or $\\Psi_0^t=(\\psi_{1, 0}, \\psi_{2, 0})$, $\\Psi^*=(\\psi_1^*, \\psi_2^*)$ and so on. Consequently, we may also denote\n\\begin{equation*}\n|\\Psi|^2=|\\psi_1|^2+|\\psi_2|^2.\n\\end{equation*}\nWe shall denote by $L^p(\\R^N), W^{1, p}(\\R^N)$, the usual Lebesgue and Sobolev spaces, respectively. We shall also make use of the mixed space-time Lebesgue (or Sobolev) spaces, so that for example $L^q(I;L^r(\\R^N))$ denote the space of those functions $\\Psi$ having the following norm finite,\n\\begin{equation*}\n\\|\\Psi\\|_{L^q(I;L^r(\\R^N))}:=\\pare{\\int_I\\pare{\\int_{\\R^N}|\\Psi(t, x)|^rdx}^{q\/r}dt}^{\\frac{1}{r}}.\n\\end{equation*}\nWe often shorten notation $L^q_tL^r_x(I\\times\\R^N)=L^q(I;L^r(\\R^N))$ or $L^q_tL^r_x$ if there is no source of ambiguity.\n\\newline\nWe are interested in studying system \\eqref{eq:nlss1} in the \\emph{energy space}, which is defined by\n\\begin{equation*}\n\\Sigma(\\R^N):=\\{\\Psi\\in H^1(\\R^N);\\;|\\cdot|\\Psi\\in L^2(\\R^N)\\}.\n\\end{equation*}\nLet us consider the Hamiltonian with confining potential, $H=-\\frac12\\Delta+\\frac{\\gamma^2}{2}|x|^2$. The associated propagator is $S_0(t):=e^{-itH}$, is unitary on $L^2(\\R^N)$ and from Melher's formula (see for example \\cite{Car05,Oh}) we may show it satisfies a dispersive estimate for short times, namely\n\\begin{equation}\\label{eq:disp_est}\n\\|S_0(t)f\\|_{L^\\infty(\\R^N)}\\lesssim|t|^{-\\frac{N}{2}}\\|f\\|_{L^1(\\R^N)},\\quad|t|\\leq\\delta,\n\\end{equation}\nfor some small $\\delta>0$ depending on $\\gamma$. \n\\newline\nFurthermore, in what follows we also need the following commutation formulas for $H$:\n\\begin{equation}\\label{eq:comm}\n[\\nabla, H]=\\gamma^2x,\\quad[x, H]=\\nabla.\n\\end{equation}\nAlthough \\eqref{eq:disp_est} holds only for short times, we may infer the same Strichartz estimates for $S_0(t)$ as for the free Schr\\\"odinger propagator without confining potential, but only locally in time, i.e. the constants appearing in the inequalities below depend on the length of the time interval. Here we state the results we need, for further details we address the reader to \\cite{Car05}, Section 2.\n\\begin{definition}\nWe say $(q, r)$ is an \\emph{admissible pair} if $2\\leq r\\leq\\frac{2N}{N-2}$ ($2\\leq r\\leq\\infty$ for $N=1$, $2\\leq r<\\infty$ for $N=2$), and\n\\begin{equation}\\label{eq:adm_pair}\n\\frac{1}{q}=\\frac{N}{2}\\pare{\\frac{1}{2}-\\frac{1}{r}}.\n\\end{equation}\n\\end{definition}\n\\begin{proposition}\\label{prop:strich}\nLet $(q, r), (\\tilde q, \\tilde r)$ be two arbitrary admissible pairs. Then, for any compact time interval, we have\n\\begin{equation*}\n\\begin{aligned}\n\\|S_0(t)f\\|_{L^q_tL^r_x(I\\times\\R^N)}\\leq &C(|I|, r)\\|f\\|_{L^2(\\R^N)},\\\\\n\\|\\int_0^tS_0(t-s)F(s)ds\\|_{L^q_tL^r_x(I\\times\\R^N)}\\leq &C(|I|, r, \\tilde r)\\|F\\|_{L^{\\tilde q'}_tL^{\\tilde r'}_x(I\\times\\R^N)}.\n\\end{aligned}\n\\end{equation*}\n\\end{proposition}\n We may rewrite the system of equations \\eqref{eq:nlss1} in the following compact way\n\\begin{equation}\\label{eq:nlss}\n\\left\\{\\begin{aligned}\ni\\d_t\\Psi=&-\\frac12\\Delta\\Psi+\\frac{\\gamma^2}{2}|x|^2\\Psi+\\tilde B[\\Psi]\\Psi+A\\Psi,\\\\\n\\Psi(0)=&\\Psi_0,\n\\end{aligned}\\right.\n\\end{equation}\nwhere the nonlinearity is given by the matrix\n\\begin{equation}\\label{eq:B_tilde}\n\\tilde B[\\Psi]=\\left(\\begin{array}{cc}\\beta_{11}|\\psi_1|^2&\\beta_{12}\\psi_1\\psi_2^*\\\\\n\\beta_{12}\\psi_1^*\\psi_2&\\beta_{22}|\\psi_2|^2\\end{array}\\right),\n\\end{equation}\nand $A$ determines the linear coupling,\n\\begin{equation}\\label{eq:A}\nA=\\left(\\begin{array}{cc}&\\lambda\\\\\\lambda&\\end{array}\\right).\n\\end{equation}\nThe energy and the mass associated to system \\eqref{eq:nlss} are the following\n\\begin{equation}\\label{eq:nlss_en2}\nE(t)=\\int_{\\R^N}\\Big(\\frac12|\\nabla\\Psi|^2+\\frac{\\gamma^2}{2}|x|^2|\\Psi|^2+\\frac12\\Psi^*\\tilde B[\\Psi]\\Psi\n+2\\lambda\\Re(\\psi_1^*\\psi_2) \\Big)(x,t)dx.\n\\end{equation}\n\\begin{equation}\\label{eq:nlss_mass2}\nM(t)=\\int_{\\R^N}|\\Psi(x,t)|^2dx,\n\\end{equation}\nand they are conserved quantities along the flow of solutions to \\eqref{eq:nlss}.\nIt is straightforward to see that, for any $\\Psi\\in\\Sigma(\\R^N)$, the energy in \\eqref{eq:nlss_en2} is finite.\n\\newline\nAs we already anticipated in the Introduction, in this paper we shall prove that, after a suitable transformation of the system, the asymptotic behavior is described by two effective coupled NLS equations which have the same inter- and intra-specific scattering lengths. Indeed, the two effective NLS equations are given by\n\\begin{equation}\\label{eq:u_sys_main}\ni\\d_tu_j=-\\frac12\\Delta u_j+\\frac{\\gamma^2}{2}u_j+\\chi|u_j|^2u_j+\\tilde{\\chi} |u_k|^2u_j,\\quad j, k = 1, 2,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:chi}\n\\tilde\\chi:=\\frac{\\beta_{11}+\\beta_{22}}{2} \\quad \\chi:=\\frac{\\beta_{11}+2\\beta_{12}+\\beta_{22}}{4}.\n\\end{equation}\nThe main Theorem we prove in this paper is\n\\begin{theorem}\\label{thm:main1}\nLet $\\Psi_0\\in\\Sigma(\\R^N)$. For any $\\lambda\\in\\R$, let $\\Psi^\\lambda$ be the unique maximal solution to \\eqref{eq:nlss}. Let $U^t=(u_1,u_2)$ be the solution of the system \\eqref{eq:u_sys_main} in $[0, S_{max})\\times\\R^N$, with initial datum\n\\begin{equation*}\n\\left(\\begin{array}{c}u_1(0)\\\\u_2(0)\\end{array}\\right)=\\left(\\begin{array}{c}\\frac{1}{\\sqrt2}(\\psi_{1, 0}+\\psi_{2, 0})\\\\\n\\frac{1}{\\sqrt2}(\\psi_{1, 0}-\\psi_{2, 0})\\end{array}\\right),\n\\end{equation*}\nwhere $S_{max}$ is the maximal existence time for $U$. Then for any time $00$ and a unique solution\n$\\Psi\\in\\mathcal C([0, \\delta];\\Sigma(\\R^N))$ to \\eqref{eq:nlss2}. Moreover we have\n\\begin{equation}\\label{eq:lin_pert}\n\\|\\Psi\\|_{L^\\infty([0, \\delta];\\Sigma(\\R^N))}\\leq2C\\|\\Psi_0\\|_{\\Sigma(\\R^N)}.\n\\end{equation}\nFurthermore, the solution $\\Psi$ can be extended to a maximal interval $[0, T_{max})$, and the blow-up alternative holds true, namely if $T_{max}<\\infty$, then\n$$\\lim_{t\\to T_{max}}\\|\\nabla\\Psi(t)\\|_{L^2(\\R^N)}=\\infty;$$\nFinally, for any $00$ will be chosen later, endowed with the distance\n\\begin{equation*}\nd(\\Psi, \\tilde\\Psi):=\\|\\Psi-\\tilde\\Psi\\|_{L^\\infty_tL^2_x([0, \\delta]\\times\\R^N)}\n+\\|\\Psi-\\tilde\\Psi\\|_{L^{8\/N}_tL^4_x([0, \\delta]\\times\\R^N)}.\n\\end{equation*}\nBy the hypotheses on $\\mathcal N(\\Psi)$ we have\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal N_1(\\Psi)-\\mathcal N_1(\\tilde\\Psi)\\|_{L^2(\\R^N)}\\leq &C\\|\\Psi-\\tilde\\Psi\\|_{L^2(\\R^N)}\\\\\n\\|\\mathcal N_2(\\Psi)-\\mathcal N_2(\\tilde\\Psi)\\|_{L^{4\/3}(\\R^N)}\\leq &C(\\|\\Psi\\|_{L^4(\\R^N)}^2+\\|\\tilde\\Psi\\|_{L^4(\\R^N)}^2)\\|\\Psi-\\tilde\\Psi\\|_{L^4(\\R^N)}\\\\\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_1(\\Psi)\\|_{L^2(\\R^N)}\\leq &C\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^2(\\R^N)}\\\\\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_2(\\Psi)\\|_{L^{4\/3}(\\R^N)}\n\\leq &C\\|\\Psi\\|_{L^4(\\R^N)}^2\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^4(\\R^N)}.\n\\end{aligned}\n\\end{equation*}\nLet us now consider the operator\n\\begin{equation*}\nG[\\Psi]:=S_0(t)\\Psi_0-i\\int_0^tS_0(t-s)\\mathcal N(\\Psi)(s)ds.\n\\end{equation*}\nBy using the commutation rules for the Hamiltonian $H$ and Strichartz estimates, we have on the space-time slab $[0, \\delta]\\times\\R^N$\n\\begin{equation*}\n\\begin{aligned}\n\\|G[\\Psi]\\|_{L^\\infty_t\\Sigma_x}+&\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)G[\\Psi]\\|_{L^{8\/N}_tL^4_x}\\\\\n&\\leq C\\|\\Psi_0\\|_{\\Sigma(\\R^N)}\n+C\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_1(\\Psi)\\|_{L^1_tL^2_x}\n+C\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_2(\\Psi)\\|_{L^{\\frac{8}{8-N}}_tL^{4\/3}_x}\\\\\n&\\leq C\\|\\Psi_0\\|_{\\Sigma(\\R^N)}\n+C\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^1_tL^2_x}\n+C\\|\\Psi\\|_{L^\\infty_tL^4_x}^2\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^{\\frac{8}{8-N}}_tL^4_x}.\n\\end{aligned}\n\\end{equation*}\nAnalogously we obtain\n\\begin{equation*}\n\\begin{aligned}\n\\|G[\\Psi]-G[\\tilde\\Psi]\\|_{L^\\infty_tL^2_x}+&\\|G[\\Psi]-G[\\tilde\\Psi]\\|_{L^{8\/N}_tL^4_x}\n\\leq C\\|\\mathcal N_1(\\Psi)-\\mathcal N_1(\\tilde\\Psi)\\|_{L^1_tL^2_x}+C\\|\\mathcal N_2(\\Psi)-\\mathcal N_2(\\tilde\\Psi)\\|_{L^{\\frac{8}{8-N}}_tL^{4\/3}_x}\\\\\n&\\leq C\\|\\Psi-\\tilde\\Psi\\|_{L^1_tL^2_x}+C(\\|\\Psi\\|_{L^\\infty_tL^4_x}^2+\\|\\tilde\\Psi\\|_{L^\\infty_tL^4_x}^2)\n\\|\\Psi-\\tilde\\Psi\\|_{L^{\\frac{8}{8-N}}_tL^4_x}.\n\\end{aligned}\n\\end{equation*}\nBy the Sobolev embedding $H^1\\hookrightarrow L^4$ and by using H\\\"older's inequality in time in the previous expressions, we have\n\\begin{equation*}\n\\begin{aligned}\n\\|G[\\Psi]\\|_{L^\\infty_t\\Sigma_x}\n+\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)G[\\Psi]\\|_{L^{8\/N}_tL^4_x}\n\\leq &C\\|\\Psi_0\\|_{\\Sigma}\n+C(\\delta+\\delta^{\\frac{8-2N}{8}}M^2)M\\\\\n\\|G[\\Psi_1]-G[\\Psi_2]\\|_{L^{8\/N}_tL^4_x}\\leq &\nC(\\delta+\\delta^{\\frac{8-2N}{8}}M^2)d(\\Psi, \\tilde\\Psi).\n\\end{aligned}\n\\end{equation*}\nNow, we choose $M, \\delta$ such that\n\\begin{equation*}\n\\begin{aligned}\nC\\|\\Psi_0\\|_{\\Sigma(\\R^N)}=&\\frac{M}{2}\\\\\nC(\\delta+\\delta^{\\frac{8-2N}{8}}M^2)\\leq&\\frac12,\n\\end{aligned}\n\\end{equation*}\nso that we have\n\\begin{equation}\\label{eq:21}\n\\begin{aligned}\n&\\|G[\\Psi]\\|_{L^\\infty_t\\Sigma_x}\n+\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)G[\\Psi]\\|_{L^{8\/N}_tL^4_x}\\leq M\\\\\n&d(G[\\Psi], G[\\tilde\\Psi])\\leq\\frac12d(\\Psi, \\tilde\\Psi).\n\\end{aligned}\n\\end{equation}\nThis implies $G$ is a contraction in $K$, therefore there exists a unique $\\Psi\\in K$ such that\n\\begin{equation*}\n\\Psi(t)=G[\\Psi](t)=S_0(t)\\Psi_0-i\\int_0^tS_0(t-s)\\mathcal N(\\Psi)(s)ds.\n\\end{equation*}\nHence $\\Psi\\in\\mathcal C([0, \\delta];\\Sigma(\\R^N))\\cap L^{8\/N}([0, \\delta];L^4(\\R^N))$ is a solution to \\eqref{eq:nlss2} in $[0, \\delta]$.\nFrom \\eqref{eq:21} we also see\n\\begin{equation*}\n\\|\\Psi\\|_{L^\\infty([0, \\delta];\\Sigma(\\R^N))}\\leq M=2C\\|\\Psi_0\\|_{\\Sigma(\\R^N)},\n\\end{equation*}\nwhich proves \\eqref{eq:lin_pert}. Analogously, by Strichartz estimates we also have\n\\begin{equation*}\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^q_tL^r_x([0, \\delta]\\times\\R^N)}\\leq \nC\\|\\Psi_0\\|_{\\Sigma(\\R^N)}+C\\delta^{\\frac{8-2N}{N}}(1+M^2)M\\leq M.\n\\end{equation*}\nFurthermore, from the proof in the fixed point argument above we also infer that we may extend the solution as long as the \n$L^2-$norm of the gradient of the solution remains bounded, hence the blow-up alternative holds true. This implies we can extend the solution to a maximal interval $[0, T_{max})$, and moreover for any $00$. In this case indeed the matrix\n\\begin{equation*}\nB=\\left(\\begin{array}{cc}\\beta_{11}&\\beta_{12}\\\\\\beta_{12}&\\beta_{22}\\end{array}\\right)\n\\end{equation*}\nis positive definite, it follows that \n \\begin{equation*}\n \\int_{\\R^N}\\big(\\frac{\\beta_{11}}{2}|\\psi_2|^4+\\beta_{12}|\\psi_1|^2|\\psi_2|^2+\\frac{\\beta_{22}}{2}|\\psi_2|^2\\big)(x,t)dx \\geq 0,\n \\end{equation*}\nthus using the energy conservation we get uniform bounds on the $\\Sigma$-norm of the solution $\\Psi$.\n \\begin{equation*}\n \\frac{1}{2}\\|\\nabla\\Psi(t)\\|_{L^2}^2+\\frac{\\gamma^2}{2}\\||\\cdot|\\Psi\\|_{L^2}^2\n \\leq E(t)-2\\lambda\\int_{\\R^N}\\Re(\\psi_1^*\\psi_2)(x,t)dx\n \\leq E(t)+|\\lambda| M(t)=E(0)+|\\lambda| M(0).\n \\end{equation*}\n For all other cases see \\cite{Juengel11}.\n\\end{proof}\nIn the next Theorem we show the cases in which there is possible occurrence of blow-up in finite time. We describe two situations, the first one in which the nonlinearity is negative definite. In this case we apply a method introduced by Carles \\cite{Car} for focusing NLS equations with a confining potential, by using a modified energy functional (for more discussions about this modified energy and its interpretation see \\cite{Car}). In the second case we assume that at least one of the coefficients $\\beta_{ij}$'s is focusing and that the energy is negative enough so that conditions \n(ii) or (iii) below are fulfilled. In this case we apply the method by Glassey \\cite{G} using virial identities.\n\\begin{remark}\nNote that the following sufficient condition for blow-up is valid also in the supercritical case $N=3$, whereas in \\cite{Juengel11} the authors consider only the mass-critical case $N=2$.\n\\end{remark}\n\\begin{theorem}\\label{thm:blow-up}\nLet $\\Psi\\in\\mathcal C([0, T_{max});\\Sigma(\\R^N))$ be the solution to \\eqref{eq:nlss} as in Corollary \\ref{cor:lwp1}, and let us define the virial potential \n\\begin{equation*}\nI(t)=\\int_{\\R^N}|x|^2|\\Psi(x,t)|^2dx.\n\\end{equation*}\nLet us assume $N\\geq2$ and one of the following conditions is satisfied\n\\begin{itemize}\n\\item[(i)] the nonlinearity is \\emph{negative definite}, i.e. we either have $\\beta_{12}^2-\\beta_{11}\\beta_{22}<0$ with \n$\\beta_{11}<0$ and $\\beta_{12}\\geq 0$ or $\\beta_{11}, \\beta_{12}, \\beta_{22}<0$, and we also assume that\n\\begin{equation*}\nE(0)+\\frac{|\\lambda|}{2}M(0)<\\frac{\\gamma^2}{2}I(0);\n\\end{equation*}\n\\item[(ii)] $\\min\\{\\beta_{11},\\beta_{22},\\beta_{12}\\}<0$\n$$\\frac{2N}{N+2}\\pare{E(0)+|\\lambda|M(0)}<\\frac{\\gamma^2}{2}I(0);$$\n\\item[(iii)] $\\min\\{\\beta_{11},\\beta_{22},\\beta_{12}\\}<0$; \n$I'(0)<0$ and $$\\frac{2N}{N+2}\\pare{E(0)+|\\lambda|M(0)}<-\\frac{\\gamma}{\\sqrt{2+N}}I'(0).$$\n\\end{itemize}\nThen the solution blows-up at a finite time, i.e. $\\exists\\;00$ sufficiently large. \\newline\nWe can thus repeat the same argument, starting at time $t=\\ell$. Consequently we have the solution $\\Phi^\\lambda$ exists in \n$[0, \\delta+\\ell]$ and we have a uniform bound for the $\\Sigma-$norm of $\\Phi^\\lambda$,\n\\begin{equation*}\n\\sup_{|\\lambda|\\geq\\Lambda}\\|\\Phi^\\lambda\\|_{L^\\infty((0, \\delta+\\ell); \\Sigma(\\R^N))}\\leq 2CM.\n\\end{equation*}\nThus we can apply again Lemma \\ref{lemma:14} in the time interval $[\\ell, \\delta+\\ell]$ and obtain the convergence of \n$\\Phi^\\lambda$ to $U$. Again, because of the convergence we have\n\\begin{equation*}\n\\lim_{|\\lambda|\\to\\infty}\\|\\Phi^\\lambda(\\delta+\\ell)-U(\\delta+\\ell)\\|_{\\Sigma(\\R^N)}=0,\n\\end{equation*}\nand in particular\n\\begin{equation*}\n\\sup_{|\\lambda|\\geq\\Lambda'}\\|\\Phi^\\lambda(\\delta+\\ell)\\|_{\\Sigma(\\R^N)}\\leq M.\n\\end{equation*}\nThus we repeat this argument to prove the result in the whole time interval $[0, T]$ and Theorem \\ref{thm:main} is proved.\n\\qed\\newline\n\\section*{Acknowledgements}\n The work of the second author has been supported by the Hertha-Firnberg Program \n of the Austrian Science Fund (FWF), grant T402-N13.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzziaos b/data_all_eng_slimpj/shuffled/split2/finalzziaos new file mode 100644 index 0000000000000000000000000000000000000000..92c1721ec339a5cc026942c4699ff1dfdb791227 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzziaos @@ -0,0 +1,5 @@ +{"text":"\\section{\n\\setcounter{equation}{0} \n\\@startsection {section}{1}{\\z@}{-3.5ex plus -1ex minus \n -.2ex}{2.3ex plus .2ex}{\\large\\bf}}\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\n\\def\\subsection{\\@startsection{subsection}{2}{\\z@}{-3.25ex plus -1ex minus \n -.2ex}{1.5ex plus .2ex}{\\normalsize\\bf}}\n\n\\def\\subsubsection{\\@startsection{subsubsection}{3}{\\z@}{-3.25ex plus\n -1ex minus -.2ex}{1.5ex plus .2ex}{\\normalsize\\it}}\n\n\\makeatother \n\n\n\\begin{document}\n\n \n\\title{\\bf Bulk viscosity due to kaons\\\\ in color-flavor-locked quark matter}\n\n\n\\author{\nMark G.~Alford and Matt Braby\\\\\n\\normalsize Department of Physics \\\\ \\normalsize Washington University \\\\\n\\normalsize St.~Louis, MO~63130 \\\\ USA \\\\[2ex]\nSanjay Reddy \\\\\n\\normalsize Theoretical Division \\\\\n\\normalsize Los Alamos National Laboratory \\\\\n\\normalsize Los Alamos, NM~87545\\\\ USA \\\\[2ex]\nThomas Sch\\\"afer \\\\\n\\normalsize Physics Department \\\\\n\\normalsize North Carolina State University \\\\\n\\normalsize Raleigh, NC 27695\\\\ USA \\\\[2ex]\n}\n\n\\date{24 Jan 2007\\\\\nRevised 20 Aug 2007 \\\\[1ex]\nLA-UR-07-0429}\n\n\\begin{titlepage}\n\\maketitle\n\\renewcommand{\\thepage}{} \n\n\\begin{abstract}\nWe calculate the bulk viscosity of color-superconducting\nquark matter in the color-flavor-locked (CFL) phase. We assume that\nthe lightest bosons are the superfluid mode $H$ and the kaons\n$K^0$ and $K^+$, and that there is no kaon condensate.\nWe calculate the rate of strangeness-equilibrating processes\nthat convert kaons into superfluid modes, and the resultant\nbulk viscosity. We find that for oscillations with a\ntimescale of milliseconds, at temperatures $T\\ll 1$~MeV,\nthe CFL bulk viscosity is much less than that of unpaired quark matter, \nbut at higher temperatures the bulk viscosity of CFL \nmatter can become larger.\n\\end{abstract}\n\n\\end{titlepage}\n\n\n\\section{Introduction}\nIn this paper we calculate the bulk viscosity of\nmatter at high baryon-number density (well above nuclear density) and \nlow temperature (of order 10 MeV). On theoretical grounds\nit is expected that, at\nsufficiently high baryon-number density, \nthree-flavor matter will be accurately described as a\ndegenerate Fermi liquid of weakly-interacting quarks, with\nCooper pairing at the Fermi surface (color superconductivity \\cite{Reviews})\nin the color-flavor-locked (CFL) channel \\cite{CFL}. That is the\nphase that we will study---the \nphase diagram at lower densities remains uncertain.\nThe motivation for this calculation is that\nin nature the highest baryon-number densities are attained\nin the cores of compact stars, and it is speculated that\nquark matter, perhaps in the CFL phase, may occur there. This means that\nour best chance of learning about the high-density region of the phase\ndiagram of matter is to make some connection between the physical properties of\nthe various postulated phases of dense matter and the observable behavior\nof compact stars. Calculating the bulk viscosity of CFL quark matter\nis part of that enterprise.\n\nCurrent observations of compact stars are able to give us measurements of\nquantities such as the mass, approximate size, temperature, spin and\nspin-down rate of these objects. These estimates are steadily improving,\nand other quantities, such as X-ray emission spectra, are becoming\navailable. However, it is a challenge to connect these distantly-observable\nfeatures to properties of the inner core of the star, where quark\nmatter is most likely to occur\n(for reviews, see Ref.~\\cite{Weber:2004kj,PrakashReview}).\n\nOne possible connection is via oscillations of the compact star,\nwhich on the one hand are affected by the transport properties\nof the interior, and on the other hand may have observable\neffects on the behavior of the star. The bulk viscosity is\none of the relevant transport properties, and is expected to\nplay an important role in suppressing both vibrational and rotational\noscillations. One particularly interesting application\ninvolves $r$-modes\n\\cite{Friedman:2001as,Andersson:2002ch,Kokkotas:2001ze,Madsen:1999ci}.\nIf the viscosity of the star\nis too low, unstable $r$-mode bulk flows will take place, which\nquickly spin the star down, removing its angular momentum as gravitational\nradiation. The fact that we see quickly-spinning compact stars (millisecond\npulsars) puts limits on the internal viscosity. If we can calculate\nthe viscosity of the various phases of quark matter then the observations\ncan be used to rule out some of those phases.\nThere are various complications to this simple picture. Viscosity is\nvery temperature-dependent, so to obtain useful limits\nwe need good measurements of the temperatures\nof these stars. There are also some uncertainties about additional sources\nof damping that could help to quash $r$-modes. But the essential point is\nthat viscosity calculations of the various phases are of great\npotential phenomenological importance, and in this\npaper we report on the results of such a calculation.\n\nThe phase that we choose to study is the CFL phase of quark matter.\n(The bulk viscosity for unpaired, non-interacting quark matter has\nbeen calculated previously \\cite{Madsen:1992sx,Wang:1985tg}.) \nIt is known that the mass of the strange quark induces a stress\non the CFL phase that may lead to neutral kaon condensation \n\\cite{BedaqueSchaefer,Kaplan:2001qk}, producing a ``CFL-$K^0$''\nphase. It is not known whether such condensation occurs at\nphenomenologically interesting densities, because of large uncertainties\nabout instanton effects \\cite{Schafer:2002ty},\nand in this paper we will assume that kaon condensation has not occurred: our\nresults are only applicable to the CFL phase, where there is a\nthermal population of $K^0$ and other mesons, but no condensation.\n\nBulk viscosity arises from a lag in the response of the system to an\nexternally-imposed compression-rarefaction cycle. If there are some\ndegrees of freedom that equilibrate on the same timescale as the\nperiod of the cycle, then the response will be out of phase with\nthe applied compression, and work will be done. For astrophysical\napplications, such as $r$-modes of compact stars, we are interested\nin periods of order 1~ms, which is very long compared\nto typical timescales for particle interactions.\nIn quark matter there is an obvious example of a suitably\nslowly-equilibrating quantity: flavor. Flavor is conserved\nby the strong and electromagnetic interactions, \nand only equilibrates via weak interactions.\n\nIn unpaired quark matter, the lightest degrees of freedom are\nthe quark excitations around the Fermi surface, and their\nflavor-changing weak interactions produce a bulk viscosity \n\\cite{Madsen:1992sx}.\nHowever, in the CFL phase the quark excitations are gapped\nand their contribution to thermodynamic and transport properties\nat temperatures below the gap is irrelevant.\nIgnoring the quarks, then, the lightest degrees of freedom in CFL\nquark matter are the massless superfluid ``$H$'' modes, the electrons\nand neutrinos, the (rotated) photon, and the kaons. Of these, only the\nkaons carry flavor, so this paper will focus on the their\ncontribution to flavor equilibration.\nIn order to\ncalculate the bulk viscosity of CFL matter at long timescales such as\n1~ms we must therefore calculate the production and decay rates of\nthermal kaons. We expect the dominant modes to be ones that involve the $H$,\nlike $K^0\\leftrightarrow H\\ H$, and ones that involve the leptons, like\n$K^\\pm \\leftrightarrow e^\\pm\\ \\nu$.\n\nThis paper is laid out as follows. Section~\\ref{sec:generalities}\ndescribes how the bulk viscosity is related to the production\nand decay rates of the kaons.\nSection~\\ref{sec:dynamics} describes the basic \nthermodynamics of the system including the kaons and superfluid modes. \nSection~\\ref{sec:rates}\ndescribes the overall calculation of the rates of the involved \nprocesses. Section~\\ref{sec:results} presents the results and conclusions.\n\n\n\\section{Relating bulk viscosity to microscopic processes}\n\\label{sec:generalities}\n\nThe bulk viscosity is given by \\cite{Madsen:1992sx}\n\\begin{equation}\n\\zeta = \n\\frac{2 \\bar V^2}{\\omega^2 (\\delta V)^2} \\frac{dE}{dt} \\ ,\n\\label{zeta_def}\n\\end{equation}\nwhere the system is being driven through a small-amplitude\ncompression-rarefaction cycle with volume amplitude $\\delta V$\n(see \\eqn{epsilons} below) and the driving angular frequency is $\\omega$.\nThe average power dissipated per unit volume is\n\\begin{equation}\n\\frac{dE}{dt} = -\\frac{1}{\\tau \\bar V}\\int_0^\\tau p(t)\\frac{dV}{dt}dt \\ ,\n\\label{dEdt}\n\\end{equation}\nwhere $\\tau=2\\pi\/\\omega$.\nWe can parameterize the volume oscillation by an amplitude $\\delta V \\ll \\bar V$\n(chosen to be real by convention), and the resultant\npressure oscillation $p(t)$ by a complex amplitude\n$\\delta p$ which determines its strength and phase:\n\\begin{equation}\n\\begin{array}{rcl}\nV(t) &=& \\bar{V} + {\\rm Re}(\\delta V\\, e^{i\\omega t}) \\\\[1ex]\np(t) &=& \\bar{p} + {\\rm Re}(\\delta p\\, e^{i\\omega t})\n\\end{array}\n\\label{epsilons}\n\\end{equation}\nSubstituting these into \\eqn{dEdt} and \\eqn{zeta_def}, we find that\n\\begin{equation}\n\\begin{array}{rcl}\n\\displaystyle \\frac{dE}{dt} &=& \n \\displaystyle -\\frac{1}{2}\\displaystyle \\omega \\,{\\rm Im}(\\delta p)\\,\\frac{\\delta V}{\\bar V} \\\\[2ex]\n\\zeta &=&\\displaystyle -\\frac{{\\rm Im}(\\delta p)}{\\delta V} \\frac{\\bar V}{\\omega}\n\\end{array}\n\\label{zeta_general}\n\\end{equation}\nWe therefore expect that ${\\rm Im}(\\delta p)$ will turn out to be negative.\nTo determine this quantity, we note that\nthe pressure is a function of the temperature and the chemical potentials.\nWe assume that heat arising from dissipation is conducted away quickly,\nso the whole calculation is performed at constant $T$, and\nin order to find $\\delta p$ we only need to know how the chemical potentials\nvary in response to the driving oscillation. \nWe expect the bulk viscosity to be most strongly influenced by the lightest\nexcitations that carry flavor, and for the sake of definiteness \nwe will take those to be\nkaons. Our analysis could easily be modified to treat the case where\nthe lightest bosons were pions. At this point we do not have to specify \nwhether our kaons are $K^0$ or $K^+$.\nThe relevant chemical potentials are $\\mu_d-\\mu_s$ for the\n$K^0$ and $\\mu_u-\\mu_s$ for the $K^+$. For the following generic\nanalysis we will just write the equilibrating chemical potential\nas ``$\\mu_K$''.\n\nIn thermal equilibrium, the distribution of kaons is determined by their\ndispersion relations \\eqn{Kdisp} and the Bose-Einstein distribution.\nWhen the kaon population goes slightly out of equilibrium in response to\nthe applied perturbation, strong interaction processes are still\nproceeding quickly: only weak interactions are failing to keep up.\nThis means that we can always characterize our kaon population by\nthe flavor chemical potential $\\mu_K$, so the kaon \ndistribution has the form \n$n_K(p)\\propto p^2\/(\\exp((E_K(p)-\\mu_K)\/T)-1)$\n(see Eq.~\\eqn{kaon_distribution}), and nonzero $\\mu_K$ indicates deviation\nfrom equilibrium.\n\nWe express the variations in the chemical potentials for\nquark number and strangeness in terms\nof complex amplitudes $\\delta\\mu$, and $\\delta\\mu_K$,\n\\begin{equation}\n\\begin{array}{rcl}\n\\mu(t) &=& \\bar{\\mu} + {\\rm Re}(\\delta \\mu \\, e^{i\\omega t}) \\ , \\\\\n\\mu_K(t) &=&\\phantom{\\bar{\\mu}\\, +\\,} {\\rm Re}(\\delta\\mu_K e^{i\\omega t}) \\ .\n\\end{array}\n\\label{mu_epsilons}\n\\end{equation}\nNote that the term $-m_s^2\/(2\\mu)$ which is often \ndescribed as an ``effective chemical potential'' is already included \nin the kaon dispersion relation \\eqn{Kdisp},\nso in equilibrium, $\\mu_K$ is zero.\nThe pressure amplitude is then\n\\begin{equation}\n\\delta p =\n \\frac{\\partial p}{\\partial \\mu}\\Bigr|_{\\mu_K} \\delta \\mu\n +\\frac{\\partial p}{\\partial \\mu_K}\\Bigr|_{\\mu} \\delta \\mu_K\n = n_q \\delta \\mu + n_K \\delta \\mu_K\\ ,\n\\label{dp_full}\n\\end{equation}\n(From now on all partial derivatives with respect to $\\mu$\nwill be assumed to be at constant $\\mu_K$, and vice versa.)\nIn principle one might worry that what we have called ``$n_K$'' \nis really $n_d-n_s$ (or $n_u-n_s$), but at temperatures \nbelow the gap, and in the absence of kaon condensation, \nthermal kaons make the dominant contribution to \n$n_d-n_s$ and $n_u-n_s$.\nFrom \\eqn{dp_full} and \\eqn{zeta_general} we find\n\\begin{equation}\n\\zeta = -\\frac{1}{\\omega}\\frac{\\bar V}{\\delta V}\\Bigl(\n \\bar n_q{\\rm Im}(\\delta\\mu) + \\bar n_K{\\rm Im}(\\delta\\mu_K ) \\Bigr) \\ .\n\\label{zeta}\n\\end{equation}\nTo obtain the imaginary parts of the chemical potential amplitudes,\nwe write down the rate of change of the corresponding conserved quantities,\n\\begin{equation}\n\\begin{array}{rclcl}\n\\displaystyle \\frac{dn_q}{dt} \n &=&\\displaystyle \\frac{\\partial n_q}{\\partial \\mu}\\frac{d\\mu}{dt}\n +\\frac{\\partial n_q}{\\partial \\mu_K}\\frac{d\\mu_K}{dt}\n &=&\\displaystyle -\\frac{n_q}{\\bar V} \\frac{dV}{dt} \\ , \\\\[2ex]\n\\displaystyle \\frac{dn_K}{dt} \n &=&\\displaystyle \\frac{\\partial n_K}{\\partial\\mu}\\frac{\\partial \\mu}{dt}\n + \\frac{\\partial n_K}{\\partial \\mu_K}\\frac{d\\mu_K}{dt}\n &=&\\displaystyle -\\frac{n_K}{\\bar V} \\frac{dV}{dt} - \\Gamma_{\\rm total} \\ .\n\\end{array}\n\\label{ndots1}\n\\end{equation}\nAll the partial derivatives are evaluated at equilibrium, $\\mu=\\bar\\mu$ and\n$\\mu_K=0$.\nThe right hand term on the first line expresses the fact that quark\nnumber is conserved, so when a volume is compressed, the quark density\nrises. On the second line, which gives the rate of change of kaon\nnumber, there is such a term from the compression of the existing kaon\npopulation, but there is also a net kaon annihilation rate of\n$\\Gamma_{\\rm total}$ kaons per unit volume\nper unit time, which\nreflects the fact that weak interactions will push the kaon density\ntowards its equilibrium value. \nThe annihilation rate will be calculated from the microscopic\nphysics in section \\ref{sec:rates}.\nFor small deviations from equilibrium\nwe expect $\\Gamma_{\\rm total}$ to be linear in $\\mu_K$, \nso it is convenient to write the rate in\nterms of an average kaon width $\\gamma_K$, which is defined in terms \nof the total rate by writing\n\\begin{equation}\n\\Gamma_{\\rm total}\n= \\gamma_K\\,\\frac{\\partial n_K}{\\partial \\mu_K}\\delta\\mu_K e^{i\\omega t}\n\\label{gamma_K}\n\\end{equation}\nwhere the derivatives are evaluated at $\\mu_K=0$.\n\nWe now substitute the assumed oscillations \\eqn{epsilons} and\n\\eqn{mu_epsilons} in to \\eqn{ndots1}, and solve to obtain\nthe amplitudes $\\delta\\mu$ and $\\delta\\mu_K$ in terms of the amplitude $\\delta V$\nand angular frequency $\\omega$ of the driving oscillation. Inserting their\nimaginary parts in \\eqn{zeta} we obtain the bulk viscosity\n\n\\begin{equation}\n\\zeta = C\\frac{ \\ga_{\\rm eff} }{\\omega^2 + \\ga_{\\rm eff}^2}\n\\label{zeta_K}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{array}{rcl}\n\\ga_{\\rm eff} &=&\\displaystyle \\gamma_K\\left(1 -\n \\frac{\\displaystyle\\Bigl(\\deriv{n_K}{\\mu}\\Bigr)^2}{\\displaystyle\\deriv{n_q}{\\mu}\\deriv{n_K}{\\mu_K}}\n \\right)^{-1} \\approx \\gamma_K \\\\[6ex]\nC &=&\\displaystyle \\Bigl(\\deriv{n_q}{\\mu}\\Bigr)^{-1}\\frac{\\displaystyle\n \\left( \\bar{n}_K \\deriv{n_q}{\\mu} - \\bar{n}_q \\deriv{n_K}{\\mu} \\right)^2}{\n \\displaystyle \\deriv{n_K}{\\mu_K}\\deriv{n_q}{\\mu}-\\Bigl(\\deriv{n_K}{\\mu}\\Bigr)^2}\n\\approx \n \\Bigl( \\deriv{n_K}{\\mu_K}\\Bigr)^{-1}\n \\left( \\bar{n}_K - \\bar{n}_q \\deriv{n_K}{\\mu}\n \\Bigl( \\deriv{n_q}{\\mu}\\Bigr)^{-1} \\right)^2\n\\end{array}\n\\label{C_gamma}\n\\end{equation}\nThe approximate forms on the right hand side\nare valid for $T\\ll\\mu$.\nThey follow\nfrom the fact that\nall the derivatives of the kaon free energy go to zero as $T\\to 0$, so $n_K$\nand its derivatives are suppressed relative to $\\bar n_q$ and\n$\\partial n_q\/\\partial\\mu$, which are of order $\\mu^3$ and $\\mu^2$ respectively.\n\nTo evaluate $C$ and $\\ga_{\\rm eff}$\nwe need the particle densities and their derivatives,\nwhich follow from the full free energy of the system,\n$\\Omega = \\Omega_{\\rm CFL-quarks}(\\mu) + \\Omega_K(\\mu,\\mu_K)$, where\n$\\Omega_{\\rm CFL-quarks}$ is the CFL quark free energy at zero temperature\n\\cite{Alford:2002kj} and $\\Omega_K$ is the kaon free energy \n\\eqn{kaon_distribution}. Then $n_q$ and $dn_q\/d\\mu$ come dominantly from \n$\\Omega_{\\rm CFL-quarks}$, and all the other quantities in \\eqn{C_gamma}\ncome from $\\Omega_K$. We can then see that\nthe kaon free energy depends on $E_K(p)-\\mu_K$, and we choose $m_K$ to be\nindependent of $\\mu$, so from the kaon\ndispersion relation \\eqn{Kdisp} we see that\nthe kaon free energy is a function of\n$\\mu_K + m_s^2\/(2\\mu)$. This means that\n\\begin{equation}\n\\deriv{n_K}{\\mu} = -\\frac{m_s^2}{2\\mu^2} \\deriv{n_K}{\\mu_K}\n\\end{equation}\nso terms in \\eqn{C_gamma} involving $dn_K\/d\\mu$ are suppressed \nrelative to those involving $dn_K\/d\\mu_K$.\n\n\nFrom \\eqn{zeta_K} we can already see how the bulk viscosity depends\non the angular frequency $\\omega$ of the oscillation and the equilibration rate\n$\\gamma_K$. At fixed $\\gamma_K$, the bulk viscosity decreases as the\noscillation frequency rises; it is roughly constant for\n$\\omega\\lesssim\\gamma_K$, and then drops off quickly as $1\/\\omega^2$ for\n$\\omega\\gg\\gamma_K$. At fixed $\\omega$, the bulk viscosity is dominated by\nprocesses with rate $\\gamma_K\\sim\\omega$, and their contribution is\nproportional to $1\/\\gamma_K$. If we imagine varying the rate but keeping\nother quantities fixed (e.g.~by varying the coupling constant of the\nequilibrating interaction), then for $\\gamma_K\\ll\\omega$ or $\\gamma_K\\gg\\omega$\nthe bulk viscosity tends to zero. Thus very fast processes, such as\nstrong interactions, are not an important source of bulk\nviscosity. The limit of zero equilibration rate {\\em and} zero\nfrequency is singular, and depends on the order of limits.\n\nIn this paper we will also be concerned with the temperature\ndependence of the bulk viscosity. This cannot be straightforwardly\nread off from \\eqn{zeta_K} because the rates and particle densities \ndepend on the temperature in complicated ways; however we \nexpect that as we go to higher temperatures\n($T \\gg m_K,\\mu_K$) the\nbulk viscosity will grow because the kaon density is rapidly increasing.\nIn the limit of low temperature we expect the viscosity to\nbe suppressed by $\\exp((-m_K+\\mu^{\\rm eff}_K)\/T)$ \nas the thermal kaon population disappears.\n\n\\section{Dynamics of the light modes}\n\\label{sec:dynamics}\n\nIn this section we lay out the properties of the lightest modes of the\nsystem, since they will dominate the transport properties. There is\nan exactly massless scalar Goldstone boson associated with spontaneous\nbreaking of baryon number, and some light pseudoscalars associated\nwith the spontaneous breaking of the chiral symmetry. We will ignore\nthe $\\eta'$ mode associated with the breaking of $U(1)_A$, since $U(1)_A$\nis explicitly broken in QCD at moderate densities.\n\n\\subsection{The superfluid ``$H$'' mode}\n\\label{sec:H}\nThe CFL quark condensate breaks the exact $U(1)_B$ baryon number symmetry of\nthe QCD Lagrangian, creating a superfluid with\nan exactly massless Goldstone boson $H$.\nThe Lagrangian for the superfluid mode of the CFL phase is\n\\cite{Son:2002zn}\n\\begin{equation}\nL_{\\rm eff} = \\frac{N_c N_f}{12\\pi^2}\n \\bigg[(\\partial_0\\phi - \\mu)^2 - (\\partial_i\\phi)^2\\bigg]^2\n\\end{equation}\nThis Lagrangian is correct to leading (zeroth) order in $\\alpha_s$ and\nto leading order in the derivatives of the $\\phi$ field. This can be \nrescaled to give a conventionally normalized\nkinetic term, and the total time-derivative term can be dropped\n\\cite{Manuel:2004iv}, giving\n\\begin{equation}\nL_{\\rm eff} = \\half(\\partial_0\\phi)^2 - {\\txt \\frac{1}{6}} (\\partial_i\\phi)^2 \n - \\frac{\\pi}{9\\mu^2}\\partial_0\\phi(\\partial_{\\mu}\\phi)^2 \n + \\frac{\\pi^2}{108\\mu^4} (\\partial_{\\mu}\\phi \\partial^{\\mu}\\phi)^2\n\\end{equation}\nIgnoring the interaction terms for the moment,\nthe dispersion relation for the $H$ particle is\n\\begin{equation}\nE_H(p) = v_H\\,p\n\\end{equation}\nwhere $v_H^2 = 1\/3$ is the ratio of the \nspatial and temporal derivatives above.\nIn thermal equilibrium at temperature $T$, \nthe $H$ bosons have free energy\n\\begin{equation}\n\\Omega_H = \\frac{T}{2\\pi^2}\\int_0^\\infty dk\\, k^2 \n \\ln(1-\\exp(-v_H k\/T))\n = -\\frac{\\pi^2}{90 v_H^3} T^4\n\\end{equation}\nand number density\n\\begin{equation}\nn_H = \\int \\frac{d^3 k}{(2\\pi)^3} \\frac{1}{\\exp(E_H\/T)-1} \n = \\frac{\\zeta(3)}{\\pi^2 v_H^3} T^3 \\ .\n\\end{equation}\n\nThe cubic and higher-order terms allow a single $H$ to decay into multiple\n$H$ particles.\nFor energies far below $\\mu$ the dominant process is\n$H\\to HH$ \\cite{Manuel:2004iv,Manuel:2005hu},\nwhose rate can be calculated by taking the imaginary part of the\n1-loop $H$ self energy. Higher order corrections to\nthe self energy are ignored, both in Ref.~\\cite{Manuel:2004iv} and\nin this paper. These corrections could be calculated by taking\ninto account quantum mechanical interference, the Landau-Pomeranchuk-Migdal \n(LPM) effect. This should only introduce a difference of $O(1)$ in the\ncoefficient of the self-energy but would not change the parametric result. \nThe self energy is given by Eq.~(3.7) in Ref.~\\cite{Manuel:2004iv},\n\\begin{equation}\n\\Sigma_H(p_0,p) = -\\frac{4\\pi^2}{81 \\mu^4} \\sum_{s_1,s_2 = \\pm}\n \\int \\frac{d^3 k}{(2\\pi)^3} F(p_0,p,k) \\left(\\frac{s_1 s_2}{4 E_1 E_2}\n \\, \\frac{1+f(s_1 E_1) + f(s_2 E_2)}{i\\omega - s_1 E_1 - s_2 E_2}\\right),\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{array}{c}\nF(p_0,p,k) \\equiv \\bigg[p_0^2 - v^2 p^2 - 2vk(p_0-vk)\\bigg]^2, \\\\[1ex]\nf(E) \\equiv 1\/(e^{E\/T}-1),\\qquad\nE_1 \\equiv vk,\\qquad E_2 \\equiv v|\\bm{p} - \\bm{k}|.\n\\end{array}\n\\end{equation}\nRef.~\\cite{Manuel:2004iv} showed that the real part of this self-energy\nis parametrically smaller than the imaginary part, so we will only\nconcern ourselves with the imaginary part, which we will call $\\Pi_H$.\nThere is no contribution to the imaginary part when $s_1 = s_2 = -1$ as\nthere is no pole in the integral. One can also show that the \ntwo terms where the signs of $s_1$ and $s_2$ are opposite are identical.\nWe can then rewrite this in a slightly simpler and more suggestive\nform that will be used in Section \\ref{sec:rates}.\n\\begin{equation}\n\\begin{array}{rcl}\n\\Pi_H(p_0,p) &=& \\displaystyle \\frac{2\\pi^3 p_0^2}{81 \\mu^4 v^4} \\frac{1}{1+f(p_0)} \n \\int \\frac{d^3 k}{2 k_0 (2\\pi)^3} F(p_0,p,k) G(p_0,p,k) \\ , \\\\[3ex]\n G(p_0,p,k) &=& \\displaystyle (1+f(E_1)) \n\\Bigg(\\frac{1+f(E_2)}{E_2}\\ \\delta(p_0 - E_1 - E_2)\n +\\ 2\\ \\frac{f(E_2)}{E_2}\\ \\delta(p_0 - E_1 + E_2)\\Bigg)\\ .\n\\end{array}\n\\label{PiH}\n\\end{equation}\nThe $H$ propagator can then be written as follows\n\\begin{equation}\nD_H(p_0,p) = \\frac{1}{p_0^2 - v_H^2p^2 + i\\Pi_H(p_0,p)}\n\\label{H_prop}\n\\end{equation}\nand we will use this expression in section~\\ref{sec:rates}.\nIt will be useful in the calculation of the decay rates to have the \n$H$ self-energy at momenta and energies close to mass shell ($p_0=vp$).\nThe self-energy is discontinuous at this point so there are\ntwo values $\\Pi_H^+$ and $\\Pi_H^-$ depending on whether\n$p_0$ tends to $vp$ from above or below,\n\\begin{equation}\n\\begin{array}{rclcl} \n\\Pi_H^+(p) &=&\\displaystyle \\lim_{\\varepsilon\\to 0} \\Pi_H(vp+\\varepsilon,p)\n &=& \\displaystyle-\\frac{\\pi p}{81 \\mu^4 v} \\frac{1}{1+f(vp)}\n\\int_0^p dk\\, I(p,k) \\ , \\\\[3ex]\n\\Pi_H^-(p) &=&\\displaystyle \\lim_{\\varepsilon\\to 0} \\Pi_H(vp-\\varepsilon,p)\n &=& \\displaystyle\\frac{2\\pi p}{81 \\mu^4 v} \\frac{1}{1+f(vp)}\n\\int_p^\\infty dk\\, I(p,k) \\ .\n\\end{array}\n\\label{Pi_H_pbar}\n\\end{equation}\nwhere $I(p,k) = k^2 (p-k)^2 (1+f(vk))f(vk-vp)$. \nAs $T\\to 0$, $\\Pi_H^+(p)\\propto p^6\/\\mu^4$, and $\\Pi_H^-(p)\\to 0$.\n\n\n\n\\subsection{Pions and Kaons}\n\\label{sec:pi_and_K}\nThe CFL quark condensate breaks the approximate $SU(3)$ chiral symmetry of the\nQCD Lagrangian, creating eight light pseudoscalar\npseudo-Goldstone mesons. This octet\nis just a high-density version of the pion\/kaon octet. It is described by\nan effective theory \\cite{Casalbuoni:1999zi,Son:1999cm}\n\\begin{equation}\nL_{\\rm eff} = {\\txt\\frac{1}{4}} f_\\pi^2 \\mbox{Tr}\\Bigl(\n \\nabla_0\\Sigma \\nabla_0\\Sigma^\\dagger - v_\\pi^2 \\partial_i\\Sigma \\partial_i\\Sigma^\\dagger\\Bigr)\n + \\cdots\n\\label{Sigma}\n\\end{equation}\nwhere $\\Sigma=\\exp(iP^a\\lambda^a\/f_\\pi)$, and the normalization of\nthe GellMann matrices is $\\mbox{tr}(\\lambda^a\\lambda^b)=2\\delta^{ab}$, which yields\na conventionally normalized kinetic term for the \nGoldstone boson fields $P^a$.\nAt asymptotic densities, weak-coupling calculations give \\cite{Son:1999cm}\n\\begin{equation}\nf_\\pi^2 = \\frac{21-8\\log(2)}{18}\\left(\\frac{\\mu^2}{2\\pi^2}\\right) \n \\hspace{1cm} v_\\pi^2 = \\frac{1}{3} \\ .\n\\label{fpi}\n\\end{equation}\nWhen weak interactions have equilibrated, \nthe pseudoscalars $P=\\pi^\\pm,K^\\pm,K^0,\\overline{K^0}$\nhave dispersion relations \\cite{Son:1999cm,BedaqueSchaefer}\n\\begin{equation}\nE_{P} = -\\mu^{\\rm eff}_{P} + \\sqrt{v_\\pi^2 p^2 + m_{P}^2} \\ ,\n\\label{Kdisp}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{array}{rcl}\n\\mu^{\\rm eff}_{\\pi^\\pm} &=&\\displaystyle \\pm \\frac{m_d^2-m_u^2}{2\\mu} \\ , \\\\[1ex]\n\\mu^{\\rm eff}_{K^\\pm} &=&\\displaystyle \\pm \\frac{m_s^2-m_u^2}{2\\mu}\\ , \\\\[1ex]\n\\mu^{\\rm eff}_{K^0,\\overline{K^0}} &=&\\displaystyle \\pm \\frac{m_s^2-m_d^2}{2\\mu}\\ .\n\\end{array}\n\\label{mueff}\n\\end{equation}\nBecause $m_s\\gg m_u,m_d$, the $K^0$ and $K^+$ are expected to\nhave the smallest energy gap, and so we focus on their contribution\nto the bulk viscosity.\nWe are interested in studying small departures from equilibrium, where\neach meson has an additional chemical potential $\\delta\\mu_P$.\nIt will turn out that the $K^0$ makes the dominant contribution,\nso only $\\delta\\mu_{K^0}$ is relevant (Sec.~\\ref{sec:K0_rates}).\n\nThe expression for the bulk viscosity \\eqn{zeta_K} contains terms\nof the form $\\partial n_K\/\\partial\\mu$, which take into account the fact that\nthe meson distributions depend on the meson dispersion relations, which via \n\\eqn{mueff} depend on the quark chemical potential.\nIn this paper we treat the meson masses $m_{K^0}$ etc as constants,\nbut in perturbative calculations they also depend on $\\mu$ and\nthe CFL pairing gap $\\Delta$ (see section~\\ref{sec:meson_mass}).\n\nIn thermal equilibrium at temperature $T$ and with chemical potential $\\mu_P$, \nthe free energy and number density of a meson $P$ is\n\\begin{eqnarray}\n\\Omega_{P} &=& \\frac{T}{2\\pi^2}\\int_0^\\infty dk\\, k^2\\, \n \\ln\\bigl(1-\\exp(-(E_{P}-\\delta\\mu_{P})\/T)\\bigr) \\\\\nn_{P} &=& -\\frac{\\partial \\Omega_{P}}{\\partial \\mu_{P}} \n = \\frac{1}{2\\pi^2} \\int_0^\\infty dk\\ k^2 \n \\frac{1}{\\exp((E_{P}-\\delta\\mu_{P})\/T)-1}\n\\label{kaon_distribution}\n\\end{eqnarray}\nWhen weak interactions have equilibrated $\\delta\\mu_P=0$, but when\nweak interactions are out of equilibrium the mesons may have nonzero\nchemical potentials.\n\n\\subsection{Pseudo-Goldstone-boson masses}\n\\label{sec:meson_mass}\nAlthough we treat the masses as constants, they are predicted to have\ndensity dependence in high density QCD \\cite{Schafer:2002ty}\n\\begin{equation}\n\\begin{array}{rcl}\nm_{\\pi^\\pm}^2 &=&\\displaystyle \\frac{1}{f_\\pi^2} (2A + 4Bm_s)(m_u+m_d) \\ , \\\\[2ex]\nm_{K^\\pm}^2 &=&\\displaystyle \\frac{1}{f_\\pi^2} (2A + 4Bm_d)(m_u+m_s) \\ , \\\\[2ex]\nm_{K^0,\\overline{K^0}}^2 &=&\\displaystyle \\frac{1}{f_\\pi^2} (2A + 4Bm_u)(m_d+m_s) \\ ,\n\\end{array}\n\\label{meson_mass}\n\\end{equation}\nwhere $A$ is positive and related to instantons \n\\cite{Schafer:1999fe,Manuel:2000wm}. In the limit of asymptotically large \ndensity the coefficient $A$ can be computed reliably, but at moderate \ndensity its value is quite uncertain \\cite{Schafer:2002ty}.\nAsymptotic-density QCD calculations \\cite{Son:1999cm,BedaqueSchaefer} \nalso yield \n\\begin{equation}\nB =\\displaystyle \\frac{3\\Delta^2}{4\\pi^2} \\ ,\n\\end{equation}\nalthough it is not clear how well these expressions can be trusted\nat densities of phenomenological interest.\nWe will assume that $A$ and $B$ are such that there is\nno meson condensation at zero temperature, \nwhich means that all the meson masses\nare greater than their effective chemical potentials.\n\n\\subsection{Weak interactions between light bosons}\n\\label{sec:weak_int}\n\nWe have argued above that the bulk viscosity will arise from flavor\nviolation, which will be dominated by conversion between the lightest\npseudo-Goldstone modes (neutral kaons, typically), which carry flavor,\nand the superfluid $H$ modes, which are flavorless. The dominant effect \nof the weak interaction will be to introduce mixing between the $K^0$\nand the $H$, in the form of a $K^0 \\leftrightarrow H$ vertex \\eqn{KHmixing}\nin the effective theory. \nWe now calculate the strength of that coupling.\n\nThe Lagrangian density for the $H$ modes can be written in a\nnonlinear form analogous to \\eqn{Sigma} for the pseudoscalars,\nso the leading terms in the CFL effective theory become\n\\cite{Casalbuoni:1999zi},\n\\begin{equation}\n\\label{l_cheft}\n{\\cal L}_{\\rm eff} = \\frac{f_\\pi^2}{4} {\\rm Tr}\\left[\n \\nabla_0\\Sigma\\nabla_0\\Sigma^\\dagger - v_\\pi^2\n \\partial_i\\Sigma\\partial_i\\Sigma^\\dagger \\right]\n+12 f_H^2 \\left[\n \\nabla_0 Z\\nabla_0 Z^* - v^2_H\n \\partial_iZ \\partial_i Z^* \\right]\n\\end{equation}\nwhere $Z=\\exp(iH\/2\\sqrt{6}f_H)$ is the \nfield related to the breaking of $U(1)_B$ and $f_H$ is the \ncorresponding decay constant. \nAt large density the coefficients of the CFL effective Lagrangian can \nbe determined in perturbation theory. At leading order $v_\\pi^2=v_H^2=1\/3$,\n$f_\\pi$ is given by \\eqn{fpi}, and from Ref.~\\cite{Son:1999cm} we obtain\n\\begin{equation}\nf_H^2 = \\frac{3}{4}\\left(\\frac{\\mu^2}{2\\pi^2}\\right) \\ .\n\\label{f_H}\n\\end{equation}\n\nUnder an element $(L,R,\\exp(i\\alpha))$ of the\nthe chiral flavor and baryon number symmetry group\n$SU(3)_L\\times SU(3)_R\\times U(1)_V$, the left-handed quarks,\nright-handed quarks, and bosons transform as follows:\n\\begin{equation}\n\\begin{array}{rcl}\nq_L &\\to& \\exp(i\\alpha)L\\, q_L \\ ,\\\\\nq_R &\\to& \\exp(i\\alpha)R \\,q_R \\ ,\\\\\n\\Sigma &\\to& L\\,\\Sigma \\, R^{-1} \\ ,\\\\\nZ &\\to& \\exp(i\\alpha)\\,Z \\ .\n\\end{array}\n\\label{transformation}\n\\end{equation}\nThe weak Hamiltonian breaks the approximate flavor symmetry of QCD,\nand only acts on left-handed fields.\nThe elementary process that is relevant to kaon decay is the\nconversion between strange quarks and down quarks via\nexchange of a $W^\\pm$, which can be treated at energy scales\nwell below 100 GeV as a four-fermion interaction (Ref.~\\cite{Donoghue:1992dd}, \nsect.~II-3 and II-4),\n\\begin{equation}\n{\\cal L}_{\\rm weak} = \\frac{G_F V_{ud} V_{us}}{\\sqrt{2}}\n (\\overline s \\gamma^{\\mu} u)_L(\\overline u \\gamma_{\\mu} d)_L + h.c.\n\\label{weak_H}\n\\end{equation}\nwhere $V_{ud}V_{us}\\approx 0.215(3)$.\nIn order to determine how this interaction is represented\nin the low energy effective theory of the CFL phase,\nwe introduce the spurion field $\\Lambda_{ds}$ which transforms\nas $\\Lambda_{ds}\\to L\\Lambda_{ds}L^\\dagger$. In the QCD \nvacuum we set $\\Lambda_{ds}=\\lambda_6$ using the usual\nnotation for the GellMann matrices (Ref.~\\cite{Donoghue:1992dd}, sect.~II-2), \ni.e.~$(\\Lambda_{ds})_{\\alpha\\beta}= \\delta_{2\\alpha}\\delta_{3\\beta}+\\delta_{3\\alpha}\\delta_{2\\beta}$. \nThus each time this spurion field occurs in an\ninteraction, it mediates a conversion of downness into strangeness, or\nvice versa.\nThe lowest-order terms in the effective theory that involve\nsuch a conversion are obtained by writing down the\nlowest-order terms that contain $\\Lambda_{ds}$ and are invariant under\nspatial rotations and $SU(3)_L\\times SU(3)_R\\times U(1)_V$:\n\\begin{equation}\n\\label{cfl_wk}\n{\\cal L}_{\\rm weak} = \n f_\\pi^2 f_H^2 G_{ds} {\\rm Tr}\\left[\\Lambda_{ds} \\Bigl(\n \\Sigma\\partial_0\\Sigma^\\dag\\, Z\\partial_0 Z^* \n- v_{ds}^4 \\Sigma\\partial_i\\Sigma^\\dag\\, Z\\partial_i Z^* \\Bigr)\\right] \n\\end{equation}\nwhere $G_{ds}$ and $v_{ds}$ are new couplings in the effective action.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=10cm]{new_figs\/matching}\n\\caption{Leading contribution to the $X_\\mu^6 B_\\nu$ \npolarization function in the microscopic theory with gauged\nchiral and baryon number symmetries. The shaded bar corresponds\nto the vertex of \\eqn{weak_H}, which is the low energy limit of a\n$W$-boson-mediated interaction.}\n\\label{fig:polarization}\n\\end{center}\n\\end{figure}\n\nDimensional analysis suggests that $G_{ds}\\sim G_F$ and $v_{ds}\\sim 1$. \nIf the density is large we can be more precise and determine\nthe coupling constants using a simple matching argument.\nFor this purpose we gauge the $SU(3)_L$ and $U(1)_B$ symmetries. \nWe will denote the corresponding gauge fields $X_\\mu^A$ and \n$B_\\mu$. The flavor-violating term \\eqn{cfl_wk} in the effective\naction leads to mixing between the $X_\\mu^6$ and $B_\\mu$ gauge\nbosons. By matching to a calculation of the mixing term in the\nmicroscopic theory \\eqn{weak_H}, we now proceed to\ndetermine $G_{ds}$ in terms of $G_F$.\n\nIn the effective theory, the $\\Sigma$ field has one left-handed quark index\nso $\\partial_\\mu\\Sigma \\to (\\partial_\\mu + X^A_\\mu\\lambda_A)\\Sigma$.\nFor the superfluid mode, $\\partial_\\mu Z \\to (\\partial_\\mu + B_\\mu)Z$. Substituting into\n\\eqn{cfl_wk} and evaluating in the CFL vacuum ($\\Sigma=1$)\nwe find the mixing term is\n\\begin{equation} \n{\\cal L}_{\\rm mix} = 2G_{ds}f_\\pi^2 f_H^2 (X_0^6 B_0-v_{ds}^4 X_i^6B_i) \\ .\n\\label{mixing_micro}\n\\end{equation}\nIn the microscopic theory, the corresponding calculation is the\ncomputation of the\n$X_\\mu^6 B_\\nu$ polarization function.\nAt weak coupling the dominant contribution comes from the \ntwo-loop diagram shown in Fig.~\\ref{fig:polarization}. The evaluation of\nthe Feynman diagram is described in Appendix~\\ref{weak_matching}.\nThe result \\eqn{G_ds} is\n\\begin{equation}\n\\label{g_match}\nG_{ds}= \\sqrt{2} V_{ud}V_{us}\\, G_F \\hspace{1cm} v_{ds}^2 = v^2 = 1\/3.\n\\end{equation}\nIt is now straightforward to read off the $K^0\\to H$ amplitude. \nLinearizing \\eqn{cfl_wk}, we find\n\\begin{equation}\n{\\cal L} = G_{ds} f_\\pi f_H \\left( \\partial_0 K^0 \\partial_0 H \n - v_{ds}^4 \\partial_i K^0 \\partial_i H \\right)\n\\label{KHmixing}\n\\end{equation}\nwith $G_{ds}$ and $v_{ds}$ given in (\\ref{g_match}).\nThis leads to a vertex factor for the $K$-$H$ interaction given by\n\\begin{equation}\nA = G_{ds} f_\\pi f_H (p_0^2 - v_{ds}^4\\, p^2);\n\\label{KH_amp}\n\\end{equation}\nThis is the value of the $K^0$-$H$ vertex in Feynman diagrams\nsuch as Fig.~\\ref{fig:K_decay}.\nCombining this vertex factor and the Lagrangian for the $H$, we can \ncalculate the matrix element for conversion between a kaon with\n4-momentum $p$ and two $H$s with 4-momenta $k$ and $q$,\n\\begin{equation}\n{\\it M}^2_{K^0HH}(p,k,q) =\n \\frac{G_{ds}^2f_\\pi^2 \\ (p_0^2 - v_{ds}^4\\, p^2)^2}{144 f_H^2}\n {\\bigg(p_0(k\\cdot q) + k_0 (p \\cdot q) + q_0 (p \\cdot k)\\bigg)^2}\n |D_H(p_0,p)|^2 \\ ,\n\\label{KHH_amp}\n\\end{equation}\nwhere $D_H$ is the $H$-propagator \\eqn{H_prop}.\n\n\\section{Rates of strangeness re-equilibration processes}\n\\label{sec:rates}\n\n\\subsection{Neutral Kaon Rates}\n\\label{sec:K0_rates}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{new_figs\/kzero_decay}\n\\caption{The $K^0\\to H\\ H$ diagram, including the $K^0\\to H$\nvertex (square), a full $H$ propagator (thick line) and\nthe $H \\to H\\ H$ vertex (round). All external lines are amputated.}\n\\label{fig:K_decay}\n\\end{center}\n\\end{figure}\nIn principle, the correct way to calculate the $K^0$ annihilation rate\nis as follows. As noted above, the weak interaction introduces a small\nmixing between the $K^0$ and the $H$. We rediagonalize the kinetic\nterms in the effective action, in terms of new fields $E_K$ (which is\nthe kaon with a tiny admixture of $H$) and $E_H$ (which is the $H$\nwith a tiny admixture of $K^0$). We have already seen that the $H$\nhas a width, arising from the possibility of $H\\to H\\\nH$\\footnote{There are also decays involving three or more $H$\nparticles, but we expect these to be suppressed, since the $H$ is\nderivatively coupled, and the greater the number of $H$ particles\ninvolved, the smaller the momentum carried by each of them.}.\nWhen we diagonalize, this will induce\na width for the $E_K$. Because the $E_K$ is almost the same state\nas the kaon, the $E_K$ width is a very good estimate of the $K^0$ width.\nIn terms of the original basis, this width\narises from the vertex shown in Fig.~\\ref{fig:K_decay}.\nIn the interests of brevity, we do not rediagonalize, but\nsimply calculate the contribution\nof the vertex shown in Fig.~\\ref{fig:K_decay}\nto the kaon annihilation rate. This comes via the processes\n$K^0\\leftrightarrow H\\ H$ and $H\\ K^0 \\leftrightarrow H$. \nThe net kaon annihilation rate is\n\\begin{equation}\n\\Gamma_{\\rm total} = \\Gamma_{\\rm forward} - \\Gamma_{\\rm backward}\n = (1-e^{-\\delta\\mu_{K^0}\/T}) \\Gamma_{\\rm forward}\\ \n \\approx \\frac{\\delta\\mu_{K^0}}{T} \\Gamma_{\\rm forward}\\ ,\n\\end{equation}\nwhere we have used the properties of the Bose-Einstein distributions.\nWe keep only first order in $\\delta\\mu_{K^0}$, and obtain\nthe average kaon width $\\gamma_K$ \\eqn{gamma_K}, remembering that\n$\\delta\\mu_K$ in section \\ref{sec:generalities} was the amplitude of a complex\noscillation, so $\\delta\\mu_{K^0} = \\delta\\mu_K\\exp(i\\omega t)$,\n\\begin{equation}\n\\gamma_K = \\left(\\frac{\\partial n_K}{\\partial \\mu_K}\\right)^{-1} \n \\frac{\\Gamma_{\\rm forward}(\\delta\\mu_K = 0)}{T} \\ .\n\\label{gamma_micro}\n\\end{equation}\nWe can therefore obtain the average kaon width simply from the forward\nrates $K^0\\to H\\ H$ and $H\\ K^0 \\to H$.\nThe contribution from $K^0\\to H\\ H $ is\n\\begin{equation}\n\\Gamma_{K^0 \\to H H} = \\half \\int_p \\int_{q_1} \\int_{q_2}\\, |M|^2\\, (2\\pi)^3\\, \n \\delta(\\bm{p} - \\bm{q_1} - \\bm{q_2})\\, (2\\pi)\\, \\delta(p_0 - vq_1 - vq_2)\\,\n F_{BE}(p_0,q_1,q_2)\n\\end{equation}\nwith $|M|^2$ given in \\eqn{KHH_amp} and $F_{BE}(p_0,q_1,q_2) =\nf(p_0 - \\delta\\mu_{K^0})(1+f(vq_1))(1+f(vq_2))$.\nThe rate for $K^0\\ H \\to H$ can be obtained by multiplying by $2$ for\nthe symmetry factor difference, \nswitching $q_2 \\to -q_2$ in both delta functions and turning \n$(1+f_{vq_2}) \\to f_{vq_2}$ to make that $H$ an incoming particle. \nAdding the two contributions, and performing\nthe $q_2$ integral using the momentum-conserving delta-function,\nwe obtain the total forward rate\n\\begin{equation}\n\\Gamma_{\\rm forward} = G_{ds}^2 f_\\pi^2 f_H^2 \\int_p\\, f(E_K)\\, (1+f(E_K))\\, \n\\frac{(E_K^2 - v^4\\, p^2)^2\\, \\Pi_H(E_K,p)}{\n(E_K^2 - v^2\\, p^2)^2 + \\Pi_H(E_K,p)^2}\n\\label{K_decay_rate}\n\\end{equation}\nwhere $\\Pi_H(E_K,p)$ was defined in Eq. \\eqn{PiH}.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=10cm]{new_figs\/rate_int}\n\\caption{Plot of the integrand of \\eqn{K_decay_rate}\nas a function of momentum for $p<{\\bar{p}}$. In this plot $\\mu=400~\\MeV$,\n$m_s=120~\\MeV$, and $m_K=27.92~\\MeV$, so ${\\bar{p}} = 22.15~\\MeV$ and $T_a \n\\sim 0.14~\\MeV$.\nFor $T\\ll{\\bar{p}}$ the integral is dominated by low-momentum kaons,\ni.e.~$p\\ll{\\bar{p}}$ ($T=0.05~\\MeV$ line in plot). \nBut as $T$ gets closer to ${\\bar{p}}$,\nthe near-singularity at $p={\\bar{p}}$ becomes more important, and\nat $T=0.5~\\MeV$ the integral is actually dominated by the region\nwhere $p$ is very close to ${\\bar{p}}$.\n\\label{fig:integrand}\n}\n\\end{center}\n\\end{figure}\n\nIn general this can be evaluated numerically and then combined with \n\\eqn{gamma_micro},\\eqn{zeta_K},\\eqn{C_gamma} to obtain the bulk viscosity.\nHowever for certain temperatures, the integrand is dominated by\nmomentum that make the denominator as small as possible, i.e. \nwhen $E_K = v\\,p$. \nThis corresponds to the $H\\leftrightarrow K^0$ resonance, at which\nthe kaon has a special momentum\n$\\bar p$ such that an $H$ with that momentum is also on shell,\n\\begin{equation}\n{\\bar{p}} = \\frac{m_K^2 - {\\mu^{\\rm eff}_{K^0}}^2}{2\\, v \\, {\\mu^{\\rm eff}_{K^0}}} = \\frac{\\delta m}{v}\\left(1 + \\frac{\\delta m}{2 {\\mu^{\\rm eff}_{K^0}}}\\right) \\ .\n\\label{pbar}\n\\end{equation}\nwhere $\\delta m = m_K - {\\mu^{\\rm eff}_{K^0}}$.\nNear this momentum, the virtual $H$ in Fig.~\\ref{fig:K_decay} is\nalmost on shell, so momenta close to $\\bar p$ dominate the integral.\nAs long as the numerator is slowly varying, we can approximate the\nsharp peak as a delta-function, obtaining\n\\begin{equation}\n\\Gamma_{\\rm forward} \\approx \\frac{G_{ds}^2 f_\\pi^2 f_H^2}{18\\sqrt{3}\\pi} \n (1+m_K^2\/{\\mu^{\\rm eff}_{K^0}}^2) \\bar{p}^4 \n \\frac{e^{v\\bar{p}\/T}}{(e^{v\\bar{p}\/T}-1)^2} \\ .\n\\label{rate_onshell}\n\\end{equation}\nThis expression becomes invalid at very low temperature $T \\ll T_a$, \nwhen there are very few kaons with momentum ${\\bar{p}}$ so the main\ncontribution does not come from $p\\approx {\\bar{p}}$, and at very high temperature\n$T\\gg T_b$ when there are so many thermal kaons with $p>{\\bar{p}}$ that\nthey outweigh the contribution from the resonance. One determines\n$T_a$ and $T_b$ by setting the non-resonant contribution\nequal to the resonant\nvalue from Eq.~\\eqn{rate_onshell}, giving\n$T_{a,b}$ as a function of $\\delta m$.\n$T_b$ has to be determined numerically and we found that $T_b \\sim 9~\\MeV$ for\n$\\delta m = 0.1~\\MeV$ and monotonically increases as $\\delta m$ increases, so\n$T_b$ is almost always higher than the temperature range that is of\nphysical interest. $T_a$ is given by the following condition\n\\begin{equation}\nT_a \\approx \\frac{\\delta m^2}{2{\\mu^{\\rm eff}_{K^0}}}\\left(\n \\ln\\Bigl(\\frac{\\mu^4}{\\delta m\\, (m_K\\, T_a)^{3\/2}}\\Bigr)\\right)^{-1} \\ ,\n\\label{temp_a}\n\\end{equation}\nwhere the $T_a$ dependence on the right side is logarithmically weak,\nso we expect that $T_a \\lesssim \\delta m^2\/(2{\\mu^{\\rm eff}_{K^0}})$.\n\nThe integrand of \\eqn{K_decay_rate} is plotted in Fig.~\\ref{fig:integrand},\nfor several different values of the temperature, showing that\nfor lower temperatures, $T < T_a$, the integrand is a smooth\nfunction, with a broad peak in the low-momentum region, but\nas the temperature rises, and the number of thermal\nkaon with momentum $\\bar p$ rises, the integrand develops a very\nsharp peak at $p=\\bar p$, where the intermediate $H$ is on shell.\n\n\\subsection{Charged Kaon Rates}\n\\label{sec:K+_rates}\n\nIn principle there is also a contribution to kaon number violation\nfrom the charged kaon modes, which necessarily involve charged leptons.\nWe will now show that this can be neglected compared to the contribution\nfrom the neutral kaons.\nThe lightest charged kaon is the $K^+$, and the relevant creation\/annihilation\nreactions are\n\\begin{eqnarray}\nK^+ \\leftrightarrow e^+ \\nu_e \\\\\nK^+ + e^- \\leftrightarrow \\nu_e \\\\\nK^+ + \\bar{\\nu_e} \\leftrightarrow e^+. \n\\end{eqnarray}\nThe matrix element for this process has been calculated in \nRef.~\\cite{Jaikumar:2002, Reddy:2003} and is given by\n\\begin{equation}\nA = G f_{\\pi} \\sin \\th_c p_{\\mu} \\bar{e}(k_1)\\gamma^{\\mu}(1-\\gamma_5)\\nu(k_2),\n\\end{equation}\nwhere $G$ is the appropriate coupling constant for the charged kaons in the \nmedium, which we expect is of order $G_F$.\nSumming over initial spins and averaging over final spins, we find\n\\begin{equation}\nM^2 = G^2\\, f_\\pi^2\\, \\sin^2 \\th_c\\, m_e^2\\, (k_1 \\cdot k_2)\n\\end{equation}\nThe rate of the first reaction is \n\\begin{eqnarray}\n\\Gamma &=& \\int_0^{x0} dx \\int_{y_1}^{y_2} dy x\\, y\\, \\frac{x+y+{\\mu^{\\rm eff}_{K^+}}\/T}{x+y}\n (1-z) F(x,y)\\\\\nz &=& \\frac{1}{2 v^2 x y}\\bigg((x^2+y^2)(1-v^2) + 2(x+y){\\mu^{\\rm eff}_{K^+}}\/T \n + 2xy + {({\\mu^{\\rm eff}_{K^+}}^2 - m_{K^+}^2)\/T^2}\\bigg) \\nonumber \\\\\nF(x,y) &=& \\frac{\\delta{\\mu_{K^+}}}{T}\\, \\frac{e^{x+y}}{(e^{x+y}-1)(e^x+1)(e^y+1)}.\n \\nonumber\n\\end{eqnarray}\nwhere $F(x,y)$ is the product of the distribution\nfunctions for the kaon\nand two leptons to lowest order in $\\delta {\\mu_{K^+}}$. \nThe rates for the second and third reactions, which are identical\nfor a massless electron, can be derived from a simple \nchange of $x \\rightarrow -x$ and $y \\rightarrow -y$, respectively. \nThey can then be evaluated numerically. \nNote that these calculations are done keeping only the \nlowest order term in $m_e$ and setting $\\mu_e = 0$. \n\nWe can then compare this rate to the rate for neutral kaon decay and find\nthat \n\\begin{equation}\n\\frac{\\Gamma_{K^+}}{\\Gamma_{K^0}} \\sim \\frac{T^2}{\\mu^2},\n\\label{K+suppression}\n\\end{equation}\nfor $m_{K^0} \\approx m_{K^+}$, so the contribution from charged kaons\nis suppressed by a factor of $(T\/\\mu)^2$. This is to be expected, since\nthe phase space for quarks is of order $\\mu^2 T$,\nlocalized near the quark Fermi surface, whereas the phase space for\nelectrons is of order $T^3$, since in the CFL phase there is no \nFermi sea of electrons.\n\n\\section{Results}\n\\label{sec:results}\n\nOur result for contribution\nof kaons to the bulk viscosity of CFL quark matter is given by\nequations \\eqn{zeta_K}, \\eqn{C_gamma}, \\eqn{gamma_micro}, \n\\eqn{K_decay_rate}.\nThe bulk viscosity is most sensitive to the temperature, and\nto the kaon energy gap\n\\begin{equation}\n\\delta m \\equiv m_K - {\\mu^{\\rm eff}_{K^0}} = m_K - \\frac{m_s^2-m_d^2}{2\\mu} \\ .\n\\label{energy_gap}\n\\end{equation}\nIt also depends on the orthogonal combination $m_K+{\\mu^{\\rm eff}_{K^0}}$, \nthe quark chemical potential $\\mu$ and CFL\npairing gap $\\Delta$. As discussed in section \n\\ref{sec:meson_mass}, we have no reliable way to calculate\n$m_K$ in the density range of interest for compact stars, so in\npresenting our results we will treat $\\delta m$ and $T$ as parameters.\n\nUsing the definitions of the densities of kaons and quarks, one\ncan derive the asymptotic versions of $C$ for temperatures\nfar above and far below the kaon energy gap (Table \\ref{tab:asymptotics}).\nOne can also derive the low temperature version of the rate and\nfrom that, combined with $C$, we can get the low temperature version\nof the bulk viscosity (Table \\ref{tab:asymptotics2}).\nAs one would expect, most kaon-related quantities, including the\nbulk viscosity, are suppressed by $\\exp(-\\delta m\/T)$ at low temperatures.\nThis is because the energy gap $\\delta m$ is the minimum energy required to \ncreate a $K^0$, so the population of thermal kaons is suppressed by a \nBoltzmann factor. \n\n\n\\begin{table}\n\\def\\rule[-2ex]{0em}{5ex}{\\rule[-2ex]{0em}{5ex}}\n\\[\n\\begin{array}{c@{\\qquad}c@{\\qquad}c}\n\\hline\n \\mbox{quantity} & \\multicolumn{2}{c}{\\mbox{asymptotic form}} \\\\[1ex]\n & T \\ll \\delta m & m_K \\ll T \\ll \\mu \\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} n_K & (m_K T)^{3\/2}e^{-\\delta m\/T} & T^3 \\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} \\deriv{n_K}{{\\mu^{\\rm eff}_{K^0}}} & m_K (m_K T)^{1\/2}e^{-\\delta m\/T} & T^2 \\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} \\deriv{n_K}{\\mu} & -\\frac{m_K^2}{\\mu} (m_K T)^{1\/2} e^{-\\delta m\/T} \n & -\\frac{m_K}{\\mu} T^2 \\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} n_q & \\mu^3 & \\mu^3 \\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} \\deriv{n_q}{\\mu} & \\mu^2 & \\mu^2 \\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} C & m_K^3 (m_K T)^{1\/2} e^{-\\delta m\/T} & T^4\\\\\n\\hline\n\\end{array}\n\\]\n\\caption{\nAsymptotic forms for the densities and the C parameter. Constant\nnumerical factors are not shown and it is implicitly assumed that\n$T<0.57\\Delta$ so that there is a CFL condensate, even\nwhen $T \\gg m_K$. \n}\n\\label{tab:asymptotics}\n\\end{table}\n\n\\begin{table}\n\\def\\rule[-2ex]{0em}{5ex}{\\rule[-2ex]{0em}{5ex}}\n\\[\n\\begin{array}{c@{\\qquad}c@{\\qquad}c@{\\qquad}c}\n\\hline\n \\mbox{quantity} & \\multicolumn{2}{c}{\\mbox{approximate form}} \\\\[1ex]\n & T < T_a(\\delta m) \\ll \\delta m & T_a(\\delta m) < T \\lesssim \\delta m \\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} \\Gamma_{\\rm forward} & G_F^2\\, \\sqrt{m_K^3\\,T^3}\\, \\delta m^5\\, e^{-\\delta m\/T} \n & G_F^2\\,\\mu^4\\,{\\bar{p}}^4\\,e^{-v{\\bar{p}}\/T}\\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} \\ga_{\\rm eff} & G_F^2\\, \\delta m^5 \n & G_F^2\\,\\mu^4\\,{\\bar{p}}^4\\,(m_K\\,T)^{-3\/2}\\,e^{-\\delta m^2\/(2\\,{\\mu^{\\rm eff}_{K^0}})\\,T}\\\\\n\\hline\n\\rule[-2ex]{0em}{5ex} \\zeta & G_F^2\\,\\delta m^5\\,m_K^{7\/2}T^{1\/2}\\,e^{-\\delta m\/T}\\,\\omega^{-2}\n & G_F^2\\,\\mu^4{\\bar{p}}^4\\,m_K^2\\,T^{-1}\\,e^{-v{\\bar{p}}\/T}\/{(\\ga_{\\rm eff}^2 + \\omega^2)}\\\\\n\\hline\n\\end{array}\n\\]\n\\caption{\nApproximate forms of the bulk viscosity and related quantities,\nfor small $T$. Constant numerical factors are not shown.\nThe rate has two separate ranges within the $T\\ll\\delta m$ region:\n$T < T_a\\ll \\delta m$ and $T_a < T \\ll \\delta m$, where\n$T_a\\lesssim \\delta m^2\/(2\\,{\\mu^{\\rm eff}_{K^0}})$ \\eqn{temp_a}.\nNote that ${\\bar{p}}$ is related to $\\delta m$ by \\eqn{pbar}. \nThe low temperature entry for $\\zeta$ is in general proportional\nto $(\\ga_{\\rm eff}^2 + \\omega^2)^{-1}$ rather than just $\\omega^{-2}$, but in this\ntemperature range $\\ga_{\\rm eff}$ is always much less than\nastrophysically relevant frequencies ($\\omega\\gtrsim 1$~Hz).\nThere is no third column for the higher end of the range of temperatures\nthat we study in this paper, $T\\sim m_K,\\,{\\mu^{\\rm eff}_{K^0}}$, because\nalthough we can still use\n\\eqn{rate_onshell} for $\\Gamma_{\\rm forward}$, there is no simple form for\n$\\deriv{n_K}{{\\mu^{\\rm eff}_{K^0}}}$ and hence for $\\ga_{\\rm eff}$ or $\\zeta$.\n}\n\\label{tab:asymptotics2}\n\\end{table}\n\n\nTo illustrate the likely contribution of kaons to the bulk viscosity\nof quark matter in compact stars, we now evaluate the bulk viscosity\nnumerically for a range of $\\delta m$ and $T$.\nOur calculations are performed at $\\mu=400~\\MeV$. \nWe vary $\\delta m$ by varying $m_K$ with ${\\mu^{\\rm eff}_{K^0}}$ fixed at $17.92~\\MeV$,\ncorresponding to $m_s=120~\\MeV$.\n\nCompact stars have internal temperatures\nin the MeV range immediately after the supernova, and then cool\nto temperatures in the keV range over millennia, so we explore the range\n$0.01~\\MeV \\lesssim T \\lesssim 10 ~\\MeV$.\nSince $m_K$ and\n${\\mu^{\\rm eff}_{K^0}}$ are both expected to be of order tens of MeV\n\\cite{Schafer:2002ty}, we expect $\\delta m$ to be generically of the same\norder, so we explore the range $0.1~\\MeV \\lesssim \\delta m \\lesssim 10~\\MeV$.\n\nThe bulk viscosity is determined by the kaon equilibration rate\n$\\gamma_K$ \\eqn{gamma_micro} and the coefficient $C$ \\eqn{C_gamma}, so\nwe plot these quantities separately before plotting the\nbulk viscosity.\n\n\\subsection{Proportionality constant $C$ (Fig.~\\ref{fig:c})}\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.49\\textwidth]{new_figs\/c_mKmuK}\n\\hspace{0.02\\textwidth} \n\\includegraphics[width=0.49\\textwidth,angle=0]{new_figs\/c_varyT}\n\\caption{Coefficient $C$ \\eqn{C_gamma}\nas a function of $\\delta m$ (left panel) and temperature (right panel)}\n\\label{fig:c}\n\\end{figure}\n\nIn Fig.~\\ref{fig:c} we show how $C$ depends on $\\delta m$ and $T$.\nRoughly speaking, $C$ measures how sensitive the kaon and quark number\nare to changes in $\\mu_K$ and $\\mu$.\nAt low temperatures $T\\ll\\delta m$, $C$ is suppressed\nby an exponential factor $\\exp(-\\delta m\/T)$, so the curves drop rapidly\nin the high $\\delta m$ region of the left panel, and the low $T$ region\nof the right panel. We also see that the curves for different $\\delta m$\nstart to converge at high temperature (right panel). This is because\nat high enough temperature (beyond the range that we study)\n$C$ would become proportional to $T^4$, independent of $\\delta m$\n(see table \\ref{tab:asymptotics}).\n\n\n\\subsection{Kaon width $\\ga_{\\rm eff}$ (Fig.~\\ref{fig:width})}\n\n\\begin{figure}[hbt]\n\\includegraphics[width=0.49\\textwidth]{new_figs\/rates_mKmuK}\n\\hspace{0.02\\textwidth} \n\\includegraphics[width = 0.49\\textwidth]{new_figs\/rates_varyT}\n\\caption{Plot of average $K^0$ decay width $\\gamma_K$ \\eqn{gamma_micro},\n\\eqn{K_decay_rate} as a function of $\\delta m$ \n(left panel) and temperature (right panel). The horizontal dashed line \nshows where the width is 1 kHz (${\\omega\/2\\pi} = 1~{\\rm ms}^{-1}$),\nthe fastest rotation rate of compact stars. The charged kaon width\nis also shown (dotted line) to illustrate that it is a subleading\ncontribution to strangeness equilibration \\eqn{K+suppression}.\nThe transition that occurs\nat $\\delta m \\approx T$ is where the rate becomes dominated by the $H$\nresonance (Section \\ref{sec:K0_rates}).\n}\n\\label{fig:width}\n\\end{figure}\n\nIn Fig.~\\ref{fig:width} we show how the neutral kaon effective\nwidth $\\gamma_K$ depends on $T$ and $\\delta m$.\n(We also show one charged kaon width curve \nto illustrate that it is subleading \\eqn{K+suppression}).\n\nThe $\\delta m$-dependence is shown in the left panel.\nFrom Table \\ref{tab:asymptotics2}, we expect that at a fixed\ntemperature $T$, for sufficiently\nlarge $\\delta m$, $T_a(\\delta m)$ will become greater than $T$, and\n$\\gamma_K$ will then rise as $\\delta m^5$. This is seen at the upper end of the\n$T = 0.01~\\MeV$ curve, and corresponds to the region where\nlow-momentum ($p<{\\bar{p}}$) kaons dominate the rate.\nFor the rest of the $T = 0.01~\\MeV$ curve, and\nfor all the other curves in the plot, the equilibration is dominated by\nkaons at the $H$-resonance, with momentum ${\\bar{p}}$.\nThe width shows a peak as a function of $\\delta m$, which follows from the\napproximate form for $\\ga_{\\rm eff}$ given in table \\ref{tab:asymptotics2}\n(second column). Using \\eqn{pbar} to relate $\\delta m$ to ${\\bar{p}}$, one\ncan see $\\ga_{\\rm eff} \\sim \\delta m^4 \\exp(-\\delta m^2\/(2 {\\mu^{\\rm eff}_{K^0}} T))$, which is\npeaked at $\\delta m_{\\rm peak} = 2\\sqrt{ {\\mu^{\\rm eff}_{K^0}} T}$. For our plots\n$ {\\mu^{\\rm eff}_{K^0}} \\approx 30~\\MeV$, which gives the observed positions\nof the peaks.\n\nThe $T$-dependence is shown in the right panel.\nAt the very lowest temperatures $\\ga_{\\rm eff}$ will have \na constant value which depends on $\\delta m$ (Table \\ref{tab:asymptotics2}).\nThis is clear in the curves for $\\delta m=5~\\MeV$ and $\\delta m=1~\\MeV$.\nAs with the large $\\delta m$ region of the left panel, this is where\nthe low momentum kaons are dominating the rate.\nIn the intermediate temperature region the width rises quickly\nand then peaks and drops off slowly. This comes from competition \n\\eqn{gamma_micro} between $\\Gamma_{\\rm forward}\/T$, which is\nmonotonically increasing with $T$ \\eqn{rate_onshell},\nand $(d{n_K}\/d{\\mu_K})^{-1}$, which is monotonically decreasing with $T$.\nAt high enough temperature, The expression \\eqn{rate_onshell}\nfor $\\Gamma_{\\rm forward}\/T$ rises as $T$,\nwhile $d{n_K}\/d{\\mu_K}$ rises more quickly, so the width drops.\n\nAt high enough $T$, the curve with $\\delta m = 0.1~\\MeV$ starts to bend upwards.\nThis feature actually corresponds to $T\\gtrsim T_b$ (from Section \n\\ref{sec:K0_rates}) where \\eqn{rate_onshell} becomes invalid and \nkaons with high momentum ($p>{\\bar{p}}$) dominate the rate. For the other\ncurves, $T_b$ is beyond the range that we study.\n\n\\subsection{Bulk viscosity $\\zeta$ (Fig.~\\ref{fig:bv} and \\ref{fig:bv2})}\n\\label{sec:bv}\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.49\\textwidth]{new_figs\/bv_mKmuK}\n\\hspace{0.02\\textwidth}\n\\includegraphics[width=0.49\\textwidth]{new_figs\/bv_varyT}\n\\caption{Plot of bulk viscosity as a function of $\\delta m$\n(left panel) and temperature (right panel).}\n\\label{fig:bv}\n\\end{figure}\n\nIt is useful to compare the behavior of the bulk viscosity in CFL\nquark matter with its behavior in quark matter phases whose\nre-equilibration is dominated by ungapped fermionic modes such as the\n2SC or single-flavor phases (see solid black line in\nFig.~\\ref{fig:bv}). In such phases, the bulk viscosity shows a peak\nas a function of $T$, because $\\ga_{\\rm eff}$ varies monotonically with $T$,\nwhile $C$ is determined by the phase space at the Fermi surface, and\nhence is insensitive to $T$ \\cite{Madsen:1992sx,Sa'd:2006qv,Alford:2006gy}. \nThis produces a single peak when $\\ga_{\\rm eff}$ is equal to the \nangular frequency $\\omega$ of \nthe applied compression oscillation (see \\eqn{zeta_K}). As we will\ndescribe below, our results show that in the CFL phase, the situation\nis more complicated, because $\\ga_{\\rm eff}$ is no longer a monotonic\nfunction of $T$, and also because $C$ can vary rapidly as the control\nparameters $\\delta m$ and $T$ are varied.\n\nThe dependence of the bulk viscosity (at frequency\n$\\omega\/2\\pi= 1$~kHz) on the kaon energy gap $\\delta m$\nis shown in the left panel of Fig.~\\ref{fig:bv}.\nFrom consideration of the factor of $\\ga_{\\rm eff}\/(\\ga_{\\rm eff}^2+\\omega^2)$\nin \\eqn{zeta_K} we would have expected two peaks for $T=0.01~\\MeV$\nand $0.1~\\MeV$, because from Fig.~\\ref{fig:width} we see that at these\ntemperatures $\\ga_{\\rm eff}$ passes through $\\omega=1$~kHz at two different\nvalues of $\\delta m$.\nIn fact we get one peak, close to the lower value of $\\delta m$\nat which $\\ga_{\\rm eff}=\\omega$. The higher peak is washed out by rapid variation\nof $C$ with $\\delta m$, which occurs when $\\delta m> T$ (see Fig.~\\ref{fig:c}).\nEven outside the physically relevant range of $\\delta m$ shown in our\nplots, we do not find additional peaks in the bulk viscosity.\n\nThe dependence of the bulk viscosity (again at $\\omega\/2\\pi = 1$~kHz) on the\ntemperature $T$ is shown in the right panel of Fig.~\\ref{fig:bv}.\nIt is a monotonically increasing function of $T$ for all values of $\\delta m$. \nThis is because, as is clear from the right panel of Fig.~\\ref{fig:c}, \n$C$ varies rapidly with temperature for all physically\nrelevant values of $\\delta m$. \nIn fact, the temperature-dependence of $C$ dominates the bulk viscosity,\nso the right panel of Fig.~\\ref{fig:bv} looks similar to the\nright panel of Fig.~\\ref{fig:c}. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{new_figs\/bv_varyw}\n\\end{center}\n\\caption{Bulk viscosity as a function of temperature for\noscillations of different frequencies. The curves peaked on the left\nare for unpaired 3-flavor quark matter\\cite{Madsen:1992sx} \nand the rising curves on the right are our calculation of\nthe kaonic bulk viscosity of CFL quark matter, for $\\delta m=1~\\MeV$.\n}\n\\label{fig:bv2}\n\\end{figure}\n\nFinally, Fig.~\\ref{fig:bv2} shows a plot of bulk viscosity \n$\\zeta$ as a function\nof temperature for different oscillation timescales, $\\tau=2\\pi\/\\omega$. \nWe see that for unpaired quark matter,\n$\\zeta_{\\rm unp}$ is independent of $\\omega$ at high temperatures,\nbecause $\\gamma_{\\rm unp}$ then rises far above $\\omega$, \nso, by (see \\eqn{zeta_K}), $\\zeta = C\/\\gamma_{\\rm unp}$.\nHowever, $\\zeta_{\\rm CFL}$ depends more strongly on $\\tau$ at high\ntemperatures, because $\\ga_{\\rm eff}$ is not much greater than $\\omega$\nat high temperature (Fig.~\\ref{fig:width}), so the $\\omega$-dependence in\n\\eqn{zeta_K} is not suppressed. The CFL bulk viscosity becomes larger\nas the frequency drops.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have calculated the contribution of the lightest pseudo-Goldstone\nbosons, the neutral kaons, to the bulk viscosity of CFL quark matter.\nOur results are given by equations \\eqn{zeta_K}, \\eqn{C_gamma},\n\\eqn{gamma_micro}, \\eqn{K_decay_rate}, and are displayed for\nreasonable parameter choices in Figs.~\\ref{fig:bv} and \\ref{fig:bv2}.\nThe bulk viscosity is most sensitive to the temperature, and to the\nkaon energy gap $\\delta m$ \\eqn{energy_gap}. We find that, as one would\nexpect, the kaonic bulk viscosity falls rapidly when the temperature\ndrops below the kaon energy gap, since the the kaon population is then\nheavily Boltzmann-suppressed. It is clear from the right hand panel of\nFig.~\\ref{fig:bv} that once the temperature falls below the 10 MeV\nrange (which is expected to occur in the first minutes after the\nsupernova \\cite{Burrows:1986me}) \nthe bulk viscosity of CFL quark matter at kHz frequencies is\nsuppressed by many orders of magnitude relative to that of unpaired\nquark matter.\n\nIt is noticeable that at low temperatures the suppression is less\nsevere for smaller kaon energy gaps. However, $\\delta m$ is a poorly-known\nparameter of the effective theory of the pseudo-Goldstone bosons. It\nis the difference of the kaon mass and the kaon effective chemical\npotential, both of which are expected to be roughly of the order of\n10~MeV \\cite{Schafer:2002ty}, so it is unnatural to assume that $\\delta m$\nis much smaller than an MeV or so.\nFor astrophysical applications, it is clear that CFL quark matter\ncan be sharply distinguished from quark matter by its bulk viscosity\n(as by many other transport properties) after the very earliest\ntimes in the life of a compact star. Also, a rapidly vanishing\nbulk viscosity as the temperature drops below $10~\\MeV$ could be a\npotential observable associated with core collapse supernovae.\n\nIn general, bulk viscosity arises from re-equilibration in response to\ncompression. We have calculated the dominant contribution\nto the re-equilibration of flavor in quark matter, and we believe that\nthis is the dominant contribution to the bulk viscosity as a whole\nin the range of frequencies that are of astrophysical interest,\nnamely zero to 1000 Hz. Any other contribution would have to\ncome from degrees of freedom that equilibrate on a similar timescale,\nand the only possibility that we can imagine is the thermalization\nof the low-momentum tail of the thermal distribution of $H$ particles.\n\nOur results highlight several interesting questions for future\nresearch. A natural next step would be to extend our calculation to\nthe kaon-condensed ``CFL-$K^0$'' phase, which corresponds to allowing \nthe kaon energy gap to drop to zero.\nThere are also some technical issues in our calculation of the \nflavor changing rate that remain to be addressed. The graph shown \nin Fig.~\\ref{fig:K_decay} includes the width of the $H$ boson due to the \none-loop thermal self energy. This corresponds to the resummation \nof a class of diagrams with multiple $H$ boson radiation and \nabsorption \\cite{Manuel:2004iv}. However, since the mean free path \nassociated with small angle two-body collisions is of the same order \nof magnitude as the radiation length \n(the mean free path between $H$-bremsstrahlung events)\nthis approximation is not \ncorrect. A more complete approach has to take into account \nquantum mechanical inteference, the Landau-Pomeranchuk-Migdal (LPM)\neffect, between different diagrams that have the same final \nstate \\cite{Arnold:2002zm}.\nAnother relevant improvement would be\nto include higher-order corrections to the $H$\ndispersion relation \\cite{Zarembo:2000pj} which could have\na strong effect on the collinear splitting amplitude for\n$H$ particles.\nWe do not expect that these improvements will affect our results significantly,\nbut they would change the numerical prefactors in the rate.\nFinally, it should be\nnoted that we treated the kaon mass as a numerical parameter, but it\nis expected to be density-dependent, and its $\\mu$-dependence\nwill feed into our expressions for the bulk viscosity. We expect\nthat this would only weakly affect our results, but we have not\nperformed an explicit check.\n\n\\medskip\n\\begin{center} {\\bf Acknowledgements} \\end{center}\n\n\\noindent\nWe thank Andreas Schmitt and Tanmoy Bhattacharya\nfor helpful discussions. \nMGA acknowledges the financial support of a short-term fellowship\nfrom the Japan Society for the Promotion of Science and\nthe hospitality of the Hadronic Theory Group at Tokyo University,\nwhere this work was completed.\nMGA and TS thank the Yukawa Institute for Theoretical Physics at \nKyoto University, whose YKIS2006 workshop on \"New Frontiers in QCD\" \nprovided a valuable forum for discussions.This research was\nsupported in part by the Offices of Nuclear Physics and High\nEnergy Physics of the Office of\nScience of the U.S.~Department of Energy under contracts\n\\#DE-FG02-91ER40628, \n\\#DE-FG02-05ER41375 (OJI),\n\\#DE-FG02-03ER41260,\nW-7405-ENG-36. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $\\mathcal F = \\{S_1, \\dots, S_n \\}$ be a set of sets. The \\emph{intersection graph} $G$ of $ \\mathcal F $ is the graph whose vertex-set is $\\mathcal{F}$ and two vertices $ S_i $ and $ S_j $ are adjacent in it if and only if $ i \\neq j $ and $S_i \\cap S_j \\neq \\varnothing $. If there exist a set $ S \\subseteq \\mathbb{R}^d $ such that all sets $ S_i $ are transformations of $ S $ obtained by translation and independent scaling in directions of the axis, then we say that $ G $ is an $ S $-graph.\n\n\nA \\emph{hereditary class} (or \\emph{class}, for short) of graphs is a set of graphs closed under induced subgraph and isomorphism. The class \\emph{spanned} by a set of graphs is the smallest class containing all the graphs in the set. Notice that the set of all $ S$-graphs is a hereditary class. \n\nThe \\emph{chromatic number} of a graph $G$, denoted by $\\chi(G)$, is the smallest integer $k$ such that we can\npartition the vertex-set of $G$ into $k$ stable sets. A \\emph{clique} in a graph is a set of pairwise adjacent vertices, and the size of the biggest clique in $ G $ is denoted by $ \\omega(G) $ and is called the \\emph{clique number} of $ G $. It is clear from the definition that $ \\chi(G) \\geq \\omega(G)$. \n\nAn interesting topic in the study of intersection graphs, in particular $ S $-graphs, is their chromatic number. Let $ S $ be a set in $ \\mathbb{R}^d $ and let $\\mathcal{C}_S$ be the class of $ S$-graphs. \nSince there are cliques of any arbitrary size in $ \\mathcal C_S $, the graphs in $ \\mathcal C $ have arbitrarily large chromatic number. However, it is interesting to know whether big cliques are the only reason that those graphs have bounded chromatic number. In particular, one can state the following (weaker) question for a fix $ k \\in \\mathbb{N}$:\n\\begin{question} \\label{question:chi-bounded}\n\tis there a number $ c \\in \\mathbb{N}$ such that for every $ G \\in \\mathcal C $ with $\\omega(G) \\leq k$, we have~$ \\chi(G) \\leq c $?\n\\end{question}\n\nThe case of $ k=2 $ is in particular studied more than the other cases. We call a graph \\emph{triangle-free} if $ \\omega(G) \\leq 2 $, and we say that a class is \\emph{triangle-free} if all graphs in the class are triangle-free. \n\nWe remark that one can state a more general question of whether the chromatic number of each graph in a class is bounded above by a function its clique number. However, this is not the concern of this paper. So, we refer to~\\cite{Scottsurvey} for more information regarding such studies in $\\chi$-boundedness. \n\n\n\n-- An \\emph{interval graph} is an intersection graph of intervals in $\\mathbb{R} $. It is well-known that interval graphs are perfect graphs, meaning that for every graph $ G $ in the class, one has $ \\chi(G) = \\omega(G)$. So, the answer to Question~\\ref{question:chi-bounded} is positive for interval graphs, and the constant $ c $ is equal to $k $.\n\n-- In 1960, Asplund and Gr\\\"{u}nbaum proved that this can be generalized to 2 dimensions as well. In~\\cite{Asplind60}, they showed that \nan intersection graph of axis-aligned rectangles in $ \\mathbb{R}^2 $ with clique number at most $ k $\nhas chromatic number at most $4k^2 - 3k $. \n\n-- Starting from the third dimension, however, the situation changes. \n\nIn 1965, in his Ph.D. thesis~\\cite{Burling65}, Burling studied what we can describe in graph theoretical terms as the chromatic number of intersection graph of polytopes in $ \\mathbb{R}^d $ where there are $m $ fixed lines in $ \\mathbb{R}^d $ such that the edges of the polytopes are parallel to at most $ m' $ lines out of those $ m $ lines. Among other result, he shows that for the case of $ d \\geq 3 $, i.e.\\ when we have at least three dimensions, for any $ m'$ and $m$, and for any $ k \\in \\mathbb{N}$, the answer to Question~\\ref{question:chi-bounded} is negative, i.e. the graphs have unbounded chromatic number.\n\nTo prove the mentioned result, Burling first\nreduced the problem to the case of the triangle-free intersection graphs of \\emph{axis-aligned boxes in $\\mathbb{R}^3$} (box graphs, for short). Then, he found a sequence $\\{\\mathcal G_k\\}_{k \\geq 1} $ of triangle-free box graphs such that $ \\chi(G_k) \\geq k $.\n\nThe sequence $\\{G_k\\}_{k\\geq 1}$ is known as the \\emph{sequence of Burling graphs}. The class of graphs spanned by $ \\{G_k : k\\geq 1 \\} $ is the \\emph{class of Burling graphs}. So, in particular, the class of Burling graphs is a subclass of triangle-free box graphs. \n\n-- In 2012\\footnote{even thought~\\cite{Pawlik2012} is published in 2014, the first version on arXiv is from 2012, and historically, it has appeared before their next paper~\\cite{Pawlik2013}}, in~\\cite{Pawlik2012}, Pawlik, Kozik, Krawczyk, Laso\\'n, Micek, Trotter, and Walczak showed that the answer to \nQuestion~\\ref{question:chi-bounded} is negative for triangle-free line-segment graphs, answering a question of Erd\\H{o}s. To prove this result, they found a sequence $ \\{G_k\\}_{k \\geq 1}$ of triangle-free line-segment graphs such that $ \\chi(G_k) \\geq 1 $. Surprisingly, the $ k$-th graph in their sequence is isomorphic to the $k$-th graph is the sequence of Burling graphs. So, indeed in~\\cite{Pawlik2013}, Burling graphs are rediscovered, but this time, as a subclass of triangle-free line-segment graphs. \n\n-- Later, in~\\cite{Pawlik2013}, Pawlik, Kozik, Krawczyk, Laso\\'n, Micek, Trotter, and Walczak extended their result from~\\cite{Pawlik2012}. They proved that not only for line segment graphs, but for every set $ S \\subseteq \\mathbb{R}^2 $ that is compact, path-connected, and different from an axis-aligned rectangle, the class of triangle-free $ S $-graphs have unbounded chromatic number. To do so, they introduce a sequence $\\{\\mathcal F_k \\}_{k \\geq 1}$ where each $ \\mathcal F_k $ is a collections of transformations of $ S $ (obtained by translation and independent scaling in the directions of axis), and they showed that the intersection graph of $ \\mathcal F_k $ is triangle-free and has chromatic number at least $ k$. It is easy to check that this once again, the intersection graph of $ \\mathcal F_k $ is isomorphic to $ G_k $, the $ k$-th graph in the sequence of Burling graphs. So, in other words, the class of Burling graphs is a subclass of triangle-free $ S $-graphs.\n\nSo, thanks to this result in~\\cite{Pawlik2013}, the answer to Question~\\ref{question:chi-bounded} for $ k = 2$ is known for any set $ S $ in $ \\mathbb{R}^2 $ that is compact and path-connected. \n\nIn~\\cite{Pawlik2012}, it is also explained how their result disproves a conjecture by Scott (Conjecture 8 in~\\cite{Scott97}) from 1997. This new application of Burling graph created new motivations to know this class of graphs better, in particular as intersection graphs. \n\n\n-- With this motivation, in 2016, Chalopin, Esperet, Li and Ossona de Mendez~\\cite{Chalopin2014} studied Burling graphs as \\emph{frame graphs} (a \\emph{frame} is the boundary of an axis-aligned rectangle). By setting a few restriction on how the frames can intersect, they defined the class of \\emph{restricted frame graphs}, a proper subclass of triangle-free frame graphs that contains all Burling graphs. Their work resulted in a better understanding of Burling graphs and more applications of them in solving $\\chi$-boundedness problems. \n\n-- In 2021, in~\\cite{BG1}, Trotignon and the author introduced the class of \\emph{strict frame graphs}, a subclass of triangle-free restricted frame graphs, by adding one more restriction to the set of restrictions defined in~\\cite{Chalopin2014}. They proved that the class of strict frame graphs is equal to the class of Burling graphs. In~\\cite{BG1}, they also define \\emph{strict line-segment graphs} and \\emph{strict box graphs}, subclasses of triangle-free line segment graphs and triangle-free box graphs, by setting a few restriction on how the sets can intersect. They proved that these two classes are also equal to the class of Burling graphs, thus finding Burling graphs not only as a subclass of intersection graphs, but as an exact class of intersection graphs with some restrictions. \n\n\\smallskip\n\nIn this article, we extend the mentioned result from~\\cite{BG1} to any set $ S \\subseteq \\mathbb{R}^2 $ that is compact, path-connected, and different from an axis-aligned rectangle. By setting constraints on how the sets can interact, we define \\emph{constrained $ S $-graphs} for any such set $ S $ and prove that the class of constrained $ S $-graphs is equal to the class of Burling graphs.\n\nIn Section~\\ref{sec:paths}, we introduce some topological lemmas and notions that are used in the rest of the paper. In Section~\\ref{sec:lyon-sets}, we introduce some notations concerning the sets that we work with. In Section~\\ref{sec:constrained-graphs}, we define the class of constrained $ S $-graphs as well as the class of \\emph{constrained graphs}. Finally, in Section~\\ref{sec:equality}, we prove that these two classes are equal and they are both equal to the class of Burling graphs. To do so, we use an equivalent definition of Burling graphs from~\\cite{BG1}, called \\emph{abstract Burling graphs}, which we present in Section~\\ref{sec:Burling-Graphs}.\n\n\n\\subsection*{Notation}\n\nWe use the standard notations from graph theory and topology. For any notation or term not defined in the article, we refer to~\\cite{BondyMurty} (for graph theory) and~\\cite{munkres} (for topology).\n\nAll graphs in this article are without multiple edges or loops. We denote the vertex-set and the edge-set of a graph $ G$ with $ V(G) $ and $E(G) $ respectively.\n\nFor a set $ A $ in $\\mathbb{R}^d$, we denote the interior and the closure of $ A $ respectively by $ A^\\circ$ and $ \\bar{A}$. Moreover, we denote the boundary of $ A $ by $ \\partial A $, i.e. $ \\partial A = \\bar{A} \\setminus A^\\circ $. \n\nWe consider $\\mathbb{R}^d $, and in particular $\\mathbb{R}^2$, with its usual topology. We denote the ball of radius~$ r $ and center~$ c $ in $ \\mathbb{R}^d $ by $ D(c, r) $. \nWe denote the projections on the x-axis and y-axis in $ \\mathbb{R}^2 $ respectively by $ \\pi_1$ and $\\pi_2$.\nWe denote the image of a function $ f $ by $\\im{f} $, and the restriction of $ f $ to a set $ A $ in its domain by $ f|_{A}$.\n\nWe postpone the introduction of any other notation to the sections that they are used in.\n\n\n\\section{Paths and crossings} \\label{sec:paths}\n\nIn this section, we introduce a few notions and prove some lemmas about them. These lemmas will be useful in the proofs of the next sections. \nLet us start with a lemma.\n\n\n\\begin{lemma} \\label{lem:real-border-lemma}\n\tLet $ X $ be a topological space and let $ A, B \\subseteq X$. If $ B $ is connected, $ B \\cap A^\\circ \\neq \\varnothing$, and $ B \\cap [X \\setminus \\bar{A}] \\neq \\varnothing $,\n\tthen $ B \\cap \\partial A \\neq \\varnothing $. \n\\end{lemma}\n\\begin{proof}\n\tNotice that\n\t$$\n\tB = [B \\cap A^\\circ] \\cup [B \\cap (X\\setminus \\bar{A})] \\cup [B \\cap \\partial A].\n\t$$\n\tThe sets, $ B \\cap A^\\circ $ and $ B \\cap (X\\setminus \\bar{A})$ are both open in $B $ and each is non-empty by the assumption. Moreover, their intersection is the empty set. So, if $ B \\cap \\partial A \\neq \\varnothing $, then $ B $ can be written as the union of two non-empty and non-intersecting sets that are open in $ B$, and thus $ B $ is not connected. \n\\end{proof}\n\nAn \\emph{axis-aligned rectangle} in $ \\mathbb{R}^2 $ is a set $ I_1 \\times I_2 \\in \\mathbb{R}^2 $ were $ I_1 $ and $I_2 $ are intervals in $ \\mathbb{R}$. Notice that vertical and horizontal line segments are axis-aligned rectangles. We often use the word \\emph{rectangle} to refer to axis-aligned rectangles.\nLet $ S $ be a subset of $ \\mathbb{R}^2 $. We define the following notions on $ S $:\n\\begin{align*}\n\t\\lset S &= \\inf \\{x : (x,y) \\in S \\}, \\\\\n\t\\rset S &= \\sup \\{x : (x,y) \\in S \\}, \\\\\n\t\\bset S &= \\inf \\{y : (x,y) \\in S \\}, \\\\\n\t\\tset S &= \\sup \\{y : (x,y) \\in S \\}. \n\\end{align*}\nThe letters $\\mathfrak{l} $, $ \\mathfrak{r} $, $\\mathfrak{b}$, and $ \\mathfrak{t} $ stand for \\emph{left}, \\emph{right}, \\emph{bottom}, and \\emph{top} respectively. If $ S $ is a compact set in $ \\mathbb{R}^2 $, then all the values above are finite and also, we can replace $\\inf$ and $\\sup$ by $ \\min $ and $\\max$ respectively; in other words, for each value, there exists a point in $ S $ that obtains the value. In this case, we also define $ \\wset S = \\rset S - \\lset S$ and $ \\hset{S} = \\tset{S} - \\bset{S} $. The letters $ \\mathfrak{w} $ and $\\mathfrak{h} $ stand for \\emph{width} and \\emph{height} respectively. Notice that if $ S' \\subseteq S $, we have $ \\lset{S'} \\geq \\lset{S} $, $ \\rset{S'} \\leq \\rset{S} $, $ \\bset{S'}\\geq \\bset{S} $, and $ \\tset{S'} \\leq \\tset{S}$. \n\n\n\n\nTo be clear, we recall the definitions of path and arc here. A \\emph{path} in a topological space $ X $ is a continuous function $ \\gamma: I \\rightarrow X $ where $ I $ is a closed interval in $\\mathbb{R}$. An \\emph{arc} in $ X $ is a homeomorphism $ \\delta : I \\rightarrow X $ where $ I $ is a closed interval in $ \\mathbb{R}$.\n\n\nA topological space $ X $ is \\emph{path-connected} (resp.\\ \\emph{arc-connected}) if for every $ x, y \\in X $, there exist a path (resp.\\ arc) $ \\gamma : [0,1] \\rightarrow X $ such that $ \\gamma(0) = x $ and $ \\gamma(1) = y $. By definition, an arc-connected space is also path-connected. The inverse is not true in general. However, as we will see, for the sets that we work with in this article the two notions are equivalent (see Theorem~\\ref{thm:bourbaki} and Lemma~\\ref{lem:path-connected-implies-arc-connected}).\n\n\n\nLet $ R $ be an axis-aligned rectangle. \nLet $ A \\subseteq \\mathbb{R}^2 $. We say that $ A $ \\emph{crosses $ R $} vertically (resp.\\ horizontally) if there exists a \n$ \\gamma: [0,1] \\rightarrow A \\cap R $ such that $ \\gamma(0) $ and $\\gamma(1)$ are respectively on the bottom-side and on the top-side (resp.\\ on the left-side and on the right-side) of $ R$. \n\n\n\n\\begin{lemma} \\label{lem:crossing-between-two-lines}\n\tLet $ y_0, y_1 \\in \\mathbb{R}$ such that $ y_0 \\leq y_1 $. For $ i \\in \\{0,1\\}$, let $ L_i $ denote the line $ y = y_i $ in $\\mathbb{R}^2$. Let $ \\gamma:[0,1] \\rightarrow \\mathbb{R}^2 $ be a continuous function such that for $ i \\in \\{0,1\\} $, we have $ \\gamma(i) \\in L_i$. Then, there exist $x_0, x_1 \\in \\mathbb{R} $ such that $ x_0 \\leq x_1 $ and the path $ \\gamma' = \\gamma|_{[x_0, x_1]} $ is \n\talways between or on the lines $ L_0 $ and $L_1$, i.e.\\ $\\im{\\gamma'} \\subseteq \\{ (x,y) : y_0 \\leq y \\leq y_1 \\} $. \n\\end{lemma}\n\\begin{proof}\t\n\tLet $ X_0 = \\gamma^{-1}(L_0) = \\{ x\\in [0,1] : \\gamma(x) \\in L_0 \\} $. Notice that $ X_0 $ is closed since it is the pre-image of a closed set under a continuous function, and is bounded. So, $ X_0 $ is compact. Moreover, $ 0 \\in X_0 $, so $ X_0 \\neq \\varnothing $. Thus, we can set $ x_0 = \\max X_0$. \n\t\n\tSet $ \\gamma'' = \\gamma|_{[x_0,1]} $, and let $ X_1 = \\gamma''^{-1}(L_1) = \\{x \\in [x_0, 1]: \\gamma''(x) \\in L_1 \\}$. Again, $ X_1 $ is compact, and it is non-empty since $ 1 \\in X_1 $. So, we can set $ x_1 = \\min X_1 $. \n\t\n\tSet $ \\gamma' = \\gamma''|_{[x_0, x_1]}$. We prove that $ \\im{\\gamma'} \\subseteq \\{ (x,y) : y_0 \\leq y \\leq y_1 \\} $.\n\t\n\tAssume, for the sake of contradiction, that there exists a point $ t \\in (x_0, x_1) $ such that $(\\pi_2 \\circ \\gamma'') (t) \\leq y_0 $ or $(\\pi_2 \\circ \\gamma'') (t) \\geq y_1 $. In the former case, by the intermediate value theorem, there exists $ t' \\geq t > x_0 $ such that $ (\\pi_2 \\circ \\gamma'') (t) = y_0 $. Thus $ t' \\in X_0$, contradicting the choice of $x_0 $. In the latter case, there exists $ t' \\leq t < x_1 $ such that $ (\\pi_2 \\circ \\gamma'') (t) = y_0 $. Thus $ t' \\in X_1$, contradicting the choice of $ x_1$. \n\\end{proof}\n\n\\begin{lemma} \\label{prop:crossing-a-subrectangle}\n\tLet $ R $ and $ R' $ be two axis-aligned rectangles such that: \n\t\\begin{itemize}\n\t\t\\item $ \\lset{R'} \\leq \\lset{R} \\leq \\rset{R} \\leq \\rset{R'}$, \n\t\t\\item $\\bset{R} \\leq \\bset{R'} \\leq \\tset{R'} \\leq \\tset{R} $.\n\t\\end{itemize}\n\tIf a set $ A $ crosses $ R $ vertically, then it crosses $ R'$ vertically as well. \n\\end{lemma}\n\\begin{proof}\n\tLet $ \\gamma: [0,1] \\rightarrow R \\cap A $ be the crossing path. By two times use of the intermediate theorem on the function $\\pi_2 \\circ \\gamma $, we conclude that there exist $ x_0 $ and $ x_1 $ with $ x_0 \\leq x_1 $ such that $ \\gamma(x_0) $ and $ \\gamma(x_1) $ are respectively on the bottom side-and the top-side of $ R' $. Applying Lemma~\\ref{lem:crossing-between-two-lines} to the path $ \\gamma|_{[x_0, x_1]} $ completes the proof of the lemma. \n\\end{proof}\n\n\nThe following theorem, can be found in several classical topology text-books, in particular, in~\\cite{bourbaki16} (Chapter 3, Section 2, Proposition 18).\n\n\\begin{theorem} \\label{thm:bourbaki}\n\tLet $ X $ be a Hausdorff topological space. If $ a $ and $ b $ are two points in the same path-connected component of $ X $, then there exist an injective path $\\delta : [0,1] \\rightarrow X$ such that $ \\delta(0) = a $ and $\\delta(1) = b $.\n\\end{theorem} \n\nAs a result, we have the following lemma.\n\\begin{lemma} \\label{lem:path-connected-implies-arc-connected}\n\tIf $ \\gamma: [0,1] \\rightarrow \\mathbb{R}^2 $ is a path in $ \\mathbb{R}^2 $, then there exist a arc $ \\delta: [0,1] \\rightarrow \\mathbb{R}^2 $ such that $ \\delta(0) = \\gamma(0) $, $\\delta(1) = \\gamma(1)$, and $\\im{\\delta} \\subseteq \\im{\\gamma}$. \n\\end{lemma}\n\\begin{proof}\n\tSet $ X = \\im{\\gamma} $. With the induced topology, $ X $ is a Hausdorff space. Applying Theorem~\\ref{thm:bourbaki} with $ a $ and $ b $ being $ \\gamma(0)$ and $\\gamma(1)$ implies that there exists an injective path $ \\delta: [0,1] \\rightarrow \\im{\\delta} \\subseteq X $ from $ a $ to $ b $. It is easy to show that it is indeed a homeomorphism. Since $[0,1]$ is compact, $\\delta $ is a closed bijection, and hence a homeomorphism.\n\\end{proof}\n\n\nIn the proof of the following lemma, we use the fact that $ K_5 $, the complete graph on~5 vertices, is not planar, i.e.\\ it has no planar embedding. Recall that in a planar embedding, the edges are represented by curves. \n\nWe believe that the proof that we present here is folklore, but for the sake of clarity we include it. However, the lemma can also be deduced easily from Lemma 2 of~\\cite{maehara84}.\n\n\\begin{lemma}\\label{prop:horizontal-and-vertical-crossing-intersect}\n\tLet $R$ be a rectangle in $\\mathbb{R}^2 $. Let $ A$ and $ B $ be two path-connected sets crossing $R$ vertically and horizontally respectively. Then, $ A \\cap B \\neq \\varnothing $. \n\\end{lemma}\n\\begin{proof}\n\tLet $ \\alpha: [0,1] \\rightarrow A \\cap R $ and $ \\beta: [0,1]\\rightarrow B \\cap R $ be the two crossing paths in the statement of the lemma. Assume, for the sake of contradiction, that $ \\im{\\alpha} \\cap \\im{\\beta} = \\varnothing $.\n\t\n\tIn this proof, we say that two paths (or arcs) $\\gamma$ and $\\delta $ are internally disjoint of $ \\im{\\gamma} \\cap \\im{\\delta} = $\n\tSet $ a_0 = \\alpha(0) $, $ a_1 = \\alpha(1) $, $b_0 = \\beta(0)$, and $b_1 = \\beta(1)$. \n\tBy Lemma~\\ref{lem:path-connected-implies-arc-connected}, there exist arcs \n\t$$ \\hat \\alpha: [0,1] \\rightarrow \\im{\\alpha} \\subseteq A \\cap P \\text{ and } \\hat \\beta : [0,1] \\rightarrow \\im{\\beta} \\subseteq B \\cap P $$\n\tsuch that $ \\hat{\\alpha} (0) = a_0$, $ \\hat{\\alpha} (1) = a_1$, $\\hat{\\beta} (0) = b_0$, and $ \\hat{\\beta} (1) = b_1$.\n\t\n\tFix a real number $ \\epsilon > $. Let $\\gamma_1 $, $ \\gamma_2 $, $\\gamma_3$, and $\\gamma_4$ be paths that respectively join $ b_1 $ to $ a_1 $, $ a_1 $ to $b_0$, $ b_0 $ to $ a_0 $, and $ a_0 $ to $b_1 $ such that their images are disjoint except for their beginnings and ends, and that for each $ i \\in \\{1,2,3,4\\}$, $\\im{\\gamma_i} $ is entirely outside $ R $ except for its beginning and end, and is entirely inside the rectangle\n\t$$ R' = [\\lset{R} - \\epsilon, \\rset{R}+\\epsilon] \\times [\\bset{R}-\\epsilon, \\tset{R} + \\epsilon]. $$ \n\tSee Figure~\\ref{fig:K5-planar}.\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\vspace*{-1cm}\n\t\t\\includegraphics[width=9.5cm]{fig\/crossing_proof.pdf}\n\t\t\\vspace{-1.2cm}\n\t\t\\caption{Proof of Lemma~\\ref{lem:crossing-between-two-lines}: a planar embedding of $K_5 $.} \\label{fig:K5-planar}\n\t\\end{figure}\n\t\n\tFinally, choose a point $ c $ outside $ R' $, and let $ \\delta_1 $, $ \\delta_2$, $ \\delta_3$, and $ \\delta_4 $ be four paths from $ c $ to $ b_1 $, $ a_1 $, $b_0 $, and $ a_0 $ respectively. Choose $\\delta_i$'s so that their images does not intersect except on $ c $, and such that for each $ i, j \\in \\{1,2,3,4\\} $ the sets $ \\im{\\gamma_i} $ and $ \\im{\\delta_j} $ do not intersect but possibly at the end-point of $\\delta_j$. \n\t\n\tNow, set $ V = \\{a_0,a_1, b_0, b_1 , c\\}$ and $ E = \\{ \\hat{\\alpha}, \\hat{\\beta}, \\gamma_i, \\delta_i : i \\in \\{1,2,3,4\\} \\} $, and notice that $ (V, E) $ is forms an embedding of $ K_5 $ on the plane, a contradiction.\n\\end{proof}\n\n\n\\section{Pouna sets and their territories} \\label{sec:lyon-sets}\n\nLet $ S $ be a subset of $ \\mathbb{R}^2 $.\nThe \\emph{bounding box} of $ S $, denoted by $ \\boxset S $, is defined as follows: \n$$\n\\boxset S = [\\lset S, \\rset S] \\times [\\bset S, \\tset S].\n$$\nSo, $ \\lset{\\boxset{S}} = \\lset{S} $, $ \\rset{\\boxset{S}} = \\rset{S} $, etc. \nFor a collection $ \\mathcal F $ of subsets of $\\mathbb{R}^2 $, by abuse of notation, we write $ \\boxset{\\mathcal F} $ for $ \\boxset{\\cup_{S \\in \\mathcal F} S}$.\n\nWe use the following property in some lemmas.\n\\begin{property} \\label{prop:int-box-min-S-non-empty}\n\tIf $ S $ is not an axis-aligned rectangle, then $ \\boxset{S}^\\circ \\setminus S \\neq \\varnothing$.\n\\end{property}\n\\begin{proof}\n\tFirst of all, $ S $ is not a subset of an axis-aligned line-segment. So, the closure of $\\boxset{S}^\\circ $ is equal to $\\boxset{S}$.\n\tNow, if $\\boxset{S}^\\circ \\setminus S = \\varnothing $, then $ \\boxset{S}^\\circ \\subseteq S \\subseteq \\boxset{S} $, and since $ S $ is closed, we have $ S = \\boxset{S} $, and $ S $ is an axis-aligned rectangle. \n\\end{proof}\n\n\nIn this article, we only consider transformations $ T: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2 $ which are of the following form: \n$$\nT(x,y) = (ax+c, by+d), \n$$ \nfor some $ a, b \\in \\mathbb{R}^* = \\mathbb{R} \\setminus \\{0\\} $ and $ c,d \\in \\mathbb{R}$. So, whenever we use the word \\emph{transformation}, we are referring to just such functions. Notice that the composition of two transformations of the form above is of the same form. \n\nWe say that $ T $ is a \\emph{positive} transformation if $ a > 0 $ and $ b >0 $. It is easy to see that positive transformations with composition form a group. In particular:\n\\begin{itemize}\n\t\\item the composition of two positive transformations is a positive transformation, \n\t\\item every positive transformation has an inverse.\n\\end{itemize}\n\nSeveral times, we will use the fact that if $ T:(x,y) \\mapsto (ax+c, by+d) $ is a positive transformation and $ A \\subseteq \\mathbb{R}^2$, then setting $ A' = T(A) $, we have:\n\\begin{multline*}\n\t\\lset{A'} = a. \\lset{A} + c, \\ \\rset{A'} = a. \\rset{A} +c, \\ \\bset{A'} = b. \\bset{A}+d, \\ \\text{and } \\tset{A'} = b. \\tset{A}+d. \n\\end{multline*}\nIn particular, $ \\boxset{T(A)} = T(\\boxset{A})$. \n\n\nLet $ S \\subseteq \\mathbb{R}^2 $. In this paper, we call a \\emph{transformed copy} of $ S $ any set $ S' $ of the form:\n\\begin{align*}\n\tS' = T(S) = \\{ T(x,y) : (x, y) \\in S \\},\n\\end{align*} \nfor some transformation $ T : \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$. \nWe say that $ S' $ is a \\emph{positive transformed copy} of $ S $ if $ T $ is a positive transformation. The \\emph{horizontal reflection} of $ S $ is $T(S) $ where~$ T $ is the transformation that maps~$(x,y)$ to~$(-x, y)$.\n\nIf $ \\mathcal F $ is a collection of subsets of $ \\mathbb{R}^2 $, we use the unconventional notation $ T(\\mathcal F) $ for the collection $ \\{T(S) : S \\in \\mathcal F \\} $. It is easy to see that that $ \\boxset{T(\\mathcal F)} = T(\\boxset{\\mathcal F})$. \n\n\n\\subsection{Pouna sets and their territories}\n\nA \\emph{Pouna set} is a non-empty, compact, and path-connected subset of $ \\mathbb{R}^2 $ which is not an axis-aligned rectangle. \nThe \\emph{territory} of a Pouna set $ S $, denoted by $\\Ter S $, is defined as follows:\n\\begin{equation*}\n\t\\Ter S = \\{(x,y) \\in \\boxset S \\setminus S : \\exists x'\\in \\mathbb{R} \\text{ s.t. } x' > x \\text{ and } (x',y) \\in S \\}.\n\\end{equation*}\n\nWe say that a Pouna set $ S $ is \\emph{strong} if it has a non-empty territory.\nIn Figure~\\ref{fig:territory_examples}, some examples of strong Pouna sets and their territories are represented. \n\n\\begin{figure}\n\t\\centering\n\t\\vspace*{-1cm}\n\t\\includegraphics[width=9.5cm]{fig\/TerritoryExamples.pdf}\n\t\\vspace*{-1cm}\n\t\\caption{Examples of strong Pouna sets and their territories. The Pouna sets are shown in black and their territories in gray.} \\label{fig:territory_examples}\n\\end{figure}\n\n\n\\begin{lemma} \\label{prop:strong-perkins-S-or-horizontal-reflection}\n\tFor every Pouna set $ S $, either $ S $ or its horizontal reflection is strong.\n\\end{lemma}\n\\begin{proof}\n\tLet $ S' = T(S) $ be the horizontal reflection of $ S $. Thus, $ T:(x,y) \\mapsto (-x,y)$. \n\t\n\tBy Property~\\ref{prop:int-box-min-S-non-empty}, we can choose a point $ p = (x,y) \\in \\boxset{S}^\\circ \\setminus S$. Let $ L $ be the horizontal line passing through $ p $, and set $ A $ to be the closed half-plane consisting of the points on $ L $ and under $L$. Notice that $ \\bset{S} < y < \\tset{S} $, so $ S $ has a point on the top-side of $ \\boxset{S}$, thus outside $A = \\bar{A} $ and a point on the bottom-side of $\\boxset{S}$, thus inside $ A^\\circ $. Setting $ B = S $ in the statement of Lemma~\\ref{lem:real-border-lemma}, we conclude that $ S \\cap L \\neq \\varnothing $. In other words, there is a point $ p= (x',y) \\in \\mathcal S $. \n\tIf $ x'> x $, then $ p \\in \\Ter{S} $, and $ S $ is strong. If $ x'< x $, then $ -x' > -x $. Notice that $ (-x', y) \\in S' $ and $ (-x, y) \\in \\boxset{S'}\\setminus S' $. So, $(-x, y) \\in \\Ter{S'} $, and $ S' $ is strong.\n\\end{proof}\n\nIn the next section, we will define the class of constrained $S$-graphs for strong Pouna set. Lemma~\\ref{prop:strong-perkins-S-or-horizontal-reflection} assures that focusing on Strong Lyon sets instead of Lyon sets does not reduce the generality of the definition.\n\n\nAs shown in the next Property, territories behave well under positive transformations. \n\n\\begin{property} \\label{prop:ter(T)=T(ter)}\n\tLet $ S $ be a strong Pouna set and $ T $ be a positive transformation. Then, $ \\Ter{T(S)} = T(\\Ter{S})$. In particular, $ T(S) $ is strong. \n\\end{property}\n\\begin{proof}\t\n\tLet $ T: (x,y) \\mapsto (ax+c, bx+d)$. Denote the inverse of $ T$ by $ T^{-1}$. \n\t\n\tIf $ (x,y) \\in \\boxset{S}\\setminus S $, then \n\t$$T(x,y) \\in \\boxset{S}\\setminus S = T(\\boxset{S}) \\setminus T(S) = \\boxset{T(S)} \\setminus T(S). $$\n\t\n\tMoreover, $ x'> x $ implies $ ax'+b> ax+b$.\n\tTherefore, $(x,y) \\in \\Ter{S}$ implies $ T(x,y) \\in \\Ter{T(S)} $. Hence, $ \\Ter{S} \\subseteq \\Ter{T(S)}$.\n\t\n\tTo finish the proof, notice that $S = T^{-1}(T(S)) $ and $T^{-1} $ is also a positive transformation. Thus, by what precedes, $\\Ter{T(S)} \\subseteq \\Ter{S}$. \n\\end{proof}\n\n\n\\subsection{Subterritories}\n\nThe notion of subterritory will be used in Section~\\ref{sec:equality}. \n\nLet $ B $ and $ E $ be two rectangles such that $ E \\subseteq R $. The right-extension of $ E $ in $ R $ is the rectangle $ E_r $ defined as follows:\n$$\nE_r = [\\rset{E}, \\rset{R}] \\times [\\bset{E}, \\tset{E}].\n$$\nSee Figure~\\ref{fig:right-extension}.\n\n\\begin{figure}\n\t\\centering\n\t\\vspace*{-1cm}\n\t\\includegraphics[width=6cm]{fig\/right_extension.pdf}\n\t\\vspace*{-1cm}\n\t\\caption{$E_r $ is the right extension of $ E $.} \\label{fig:right-extension}\n\\end{figure}\n\n\nA \\emph{subterritory} for a strong Pouna set $ S $ is a non-empty closed rectangle $ E $ such that\n\\begin{enumerate}\n\t\\item $ E \\subseteq \\Ter{S} $,\n\t\\item $ \\lset{E} > \\lset{S} $, $ \\rset{E} < \\rset{S}$, $\\bset{E} > \\bset{S} $, and $ \\tset{E} < \\tset{S}$,\n\t\\item $ S $ crosses the right extension of $ E $. \n\\end{enumerate}\n\n\nDespite looking too restrictive, subterritories always exist in strong Pouna sets, as we prove in the following lemma. \n\n\n\n\\begin{lemma} \\label{lem:subterritory-exists}\n\tEvery strong Pouna set has a subterritory.\n\\end{lemma}\n\\begin{proof}\n\tLet $ S $ be a strong Pouna set and let $ B = \\boxset{S} $. By Property~\\ref{prop:int-box-min-S-non-empty}, there exist a point $ p = (x_p,y_p) \\in B^\\circ \\setminus S $. So, there is $ \\epsilon > 0 $ such that $ D(p,\\epsilon) \\subseteq B^\\circ \\setminus S $. \n\t\n\tLet $L_P $ be the ray $\\{ (x,y) : y = y_p, x \\geq x_p \\}$. Notice that $ L_P \\cap S $ is non-empty and compact. Let $ s = (x_s, y_s)$ be the point in $ L_P \\cap S $ which obtains the value $ \\lset{L_P \\cap S} $. Notice that $ y_s = y_p$. Consider the following rectangle in $ B $:\n\t$$\n\tR = [x_S - \\epsilon\/2, \\rset{S}] \\times [y_s - \\epsilon , y_s + \\epsilon].\n\t$$ \n\tSee Figure~\\ref{fig:ptoof-subter-exists}. \n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\vspace*{-1cm}\n\t\t\\includegraphics[width=9cm]{fig\/proof_subterritory.pdf}\n\t\t\\vspace*{-2cm}\n\t\t\\caption{For proof of Lemma~\\ref{lem:subterritory-exists}.} \\label{fig:ptoof-subter-exists}\n\t\\end{figure}\n\t\n\tIn particular, $ s \\in R^\\circ $ and $ R = \\bar{R} $ does not intersect the border of $ B $. On the other hand there is a point $ s' $ of $ S $ on the top-side of $ B $. Since $ S $ is a path-connected set, we must have a path $\\gamma$ from $ s $ to $ s' $. By Lemma~\\ref{lem:real-border-lemma}, the image of $\\gamma$ must intersect $\\partial B $, and in particular in a point other than $(x_S - \\epsilon\/2, y_s)$ and $ (\\rset{S}, y_s)$. So, $ \\im{\\gamma} \\cap B $ is not a horizontal line. In particular, there are $ y_0, y_1 \\in \\mathbb{R} $ such that $ y_s - \\epsilon \\leq y_0 < y_2 \\leq y_s + \\epsilon$ and such that there is a path $\\delta$ in $ R $ joining a point on the line $ y = y_0 $ to a point on the line $ y = y_1 $.\n\t\n\tSo, by Lemma~\\ref{lem:crossing-between-two-lines}, applied to $\\delta $, there is a path $\\delta': [0,1] \\rightarrow \\mathbb{R}$ such that $ \\pi_2(\\delta(0)) = y_0 $, $ \\pi_2(\\delta(1)) = y_1 $, and $ \\im{\\delta'} \\subseteq [x_S - \\epsilon\/2, \\rset{S}] \\times [y_0 , y_1]$.\n\t\n\tNow, let $ E $ be a rectangle entirely inside $ D(p,\\epsilon)$ defined as follows:\n\t$$\n\tE = [x_p - \\epsilon\/2, x_p + \\epsilon\/2] \\times [(y_p+y_0)\/2, (y_p+y_1)\/2].\n\t$$\n\t\n\tNotice that by Lemma~\\ref{prop:crossing-a-subrectangle}, $ \\delta' $ stabs the right-extension of $ E $. Clearly, $ E$ satisfies all other properties of sub-territory as well. So, $ E $ is a subterritory of $ S $.\n\\end{proof}\n\n\\begin{property} \\label{prop:T(E)-subter-of-T(S)}\n\tIf $ E $ is a subterritory of a strong Pouna set $ S $, then for every positive transformation $ T $, we have that $ T(E) $ is a subterritory of $ T(S) $. \n\\end{property}\n\\begin{proof}\n\tSet $ S' = T(S)$ and $E' = T(E)$. We prove that the three items of the definition holds and $ E' $ is a subterritory of $ S' $. \n\t\n\t{\\noindent \\textbf{Claim}. \\textit{Item (1) of the definition of subterritory holds.}} \n\t\n\tBy Property~\\ref{prop:ter(T)=T(ter)}, we have that $ E'=T(E) \\subseteq T(\\Ter{S}) = \\Ter{S'} $. \n\t\n\t{\\noindent \\textbf{Claim}. \\textit{Item (2) of the definition of subterritory holds.}} \n\tSince $ a$ and $ b $ are positive, for every compact set $ A $ we have \n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\lset T(A) &= \\min\\{x: (x,y)\\in T(A) \\} \\\\\n\t\t\t&= \\min \\{x: \\big(\\frac{x-c}{a}, \\frac{y-d}{b}\\big) \\in A \\} \\\\\n\t\t\t& = \\min \\{ au+c : (u,v) \\in E \\} = a . \\lset{A} + c. \n\t\t\\end{split}\n\t\\end{equation*}\n\tIn the equations above we have again used the change of variables $ u = \\frac{x-c}{a} $ and $ v = \\frac{y-d}{b} $. \n\tSo, \n\t$$ \\lset{E'} = a. \\lset{E} + c < a. \\lset{S} + c = \\lset{S'}. $$ \n\t\n\tThe proof of the rest of the inequalities is similar. \n\t\n\t{\\noindent \\textbf{Claim}. \\textit{Item (3) of the definition of subterritory holds.}} \n\tLet $ P$ be the prob for $ \\boxset{S} $ defined by $ E $ and let $ \\gamma: [0,1] \\rightarrow S \\cap P $ be the path connecting top-side of $ P$ to bottom-side of $ P$. Denote by $P'$ the prob for $ \\boxset{S'} $ defined by $ E' $. Notice that $ P'=T(P)$. So, $T(S \\cap P)= T(S) \\cap T(P) = S' \\cap P' $. Thus, the function $ T\\circ \\gamma: [0,1] \\rightarrow S' \\cap P' $ is a path entirely inside $ S' \\cap P' $. Moreover, since $ T$ sends the top-side (resp.\\ bottom-side) of $ P$ to the top-side (resp.\\ bottom-side) of $ P' $, we have that $ (T \\circ \\gamma) (0) $ is on the top-side of $ P' $ and $ (T\\circ \\gamma)(1) $ is on the bottom-side of $ P'$, and this finishes the proof.\n\\end{proof}\n\n\\section{Constrained graphs and constrained $ S $-graphs} \\label{sec:constrained-graphs}\n\nLet $ A $ and $ B $ be two strong Pouna sets. We write \\emph{$A \\prec B$} if $ A \\subseteq \\Ter{B}$. Also, we write \\emph{$ A \\curvearrowright B $} if all the following happen: \n\\begin{itemize}\n\t\\item $\\lset B \\leq \\lset A < \\rset B < \\rset A $,\n\t\\item $\\bset B < \\bset A < \\tset A < \\tset B$,\n\t\\item $ \\{ (x,y) \\in A : x = \\lset A \\} \\subseteq \\Ter B $. \n\\end{itemize} \n\n\n\\begin{definition} \\label{def:constrained-graphs}\n\tLet $ \\mathcal F $ be a non-empty and finite collection of strong Pouna sets satisfying the following constraints:\n\t\\begin{enumerate}\t\n\t\t\\item[\\textbf{\\emph{(C1)}}] for every $ A, B \\in \\mathcal F $, if $ A \\neq B $ and $ A \\cap B \\neq \\varnothing $, then, \n\t\teither $ A \\curvearrowright B $ or $ B \\curvearrowright A $. \\label{gcond:adjacent}\n\t\t\n\t\t\\item[\\textbf{\\emph{(C2)}}] For every $ A, B \\in \\mathcal F $, if $ A \\cap B = \\varnothing $ and \n\t\t$ A \\cap \\Ter B \\neq \\varnothing$, \n\t\tthen \n\t\t$ A \\prec B$. \\label{gcond:prec}\n\t\t\n\t\t\\item[\\textbf{\\emph{(C3)}}] For every $ A, B \\in \\mathcal F $, if $ A \\neq B $ and $ A \\cap B \\neq \\varnothing $, then there exists no $ C \\in \\mathcal F $ such that $ C \\subseteq \\Ter A \\cap \\Ter B $. \\label{gcond:common-territory}\n\t\t\n\t\t\\item[\\textbf{\\emph{(C4)}}] There exist no $ A, B, C \\in \\mathcal F $ such that $ A \\prec B $, $ A \\curvearrowright C $, and $ B \\curvearrowright C $. \n\t\t\\label{gcond:strict-condition}\n\t\t\n\t\t\\item[\\textbf{\\emph{(C5)}}] The maximum number of pairwise intersecting and distinct elements in $ \\mathcal F $ is at most two. \\label{gcon:triangle-free}\n\t\\end{enumerate}\n\tThe intersection graph of $ \\mathcal F $ is a \\emph{constrained graph}.\n\\end{definition}\n\n\nLet $ S \\subseteq \\mathbb{R}^2$. We recall from the introduction that a graph is said to be an \\emph{$ S $-graph} if it is the intersection graph of $ \\{S_1, S_2, \\dots, S_n\\} $, with $ n \\in \\mathbb{N} $, where each $ S_i $ is a transformed copy of $ S $. For a Pouna set $ S$, we define a subclass of $ S$-graphs, \\emph{the constrained $S $-graphs}, by setting some constraints on how the transformed copies of $ S $ can intersect, as follows. \n\n\\begin{definition} \\label{def:constrained-S-graphs}\n\tLet $ S $ be a Pouna set, and let $ \\mathcal F $ be a non-empty and finite collection of transformed copies of $ S $ such that $ \\mathcal F $ satisfies all 5 constraints {\\emph{(C1)}}-{\\emph{(C5)}} as well as the following constraint:\n\t\\begin{enumerate} \n\t\t\\item[\\textbf{\\emph{(C6)}}] if $ S $ is strong, then all elements of $ F $ are positive transformed copies of $ S $, and otherwise, they are all positive transformed copies of the horizontal reflection of~$ S $.\n\t\\end{enumerate}\n\tThe intersection graph of $ \\mathcal F $ is a \\emph{constrained $ S$-graph}.\n\\end{definition} \n\nNotice that the set of all constrained graphs (resp.\\ constrained $ S $-graphs), i.e.\\ the set of all graphs which are isomorphic to the intersection graph of a collection $ \\mathcal F $ as in Definition~\\ref{def:constrained-graphs} (resp.\\ Definition~\\ref{def:constrained-S-graphs}), is a well-defined hereditary class of graphs, as if $ \\mathcal F $ satisfies Constraints (C1)-(C5) (resp.\\ (C1)-(C6)), then so does every non-empty subset of it.\n\nBy definition, every constrained $ S $-graph is a constrained graph. We will see, however, that the two classes are indeed equal. See Corollary~\\ref{cor:all-equal}.\n\n\\begin{property} \\label{prop:T(F)-satisfies-C1-C5}\n\tLet $ S $ be a strong Pouna set, and let $ \\mathcal F $ be a finite collection of transformed copies of $ S $ satisfying Constraints \\emph{(C1)-(C6)}, then for every positive transformation $ T $ the collection $ \\{T(S): S \\in \\mathcal F\\} $ also satisfies \\emph{(C1)-(C6)}.\n\\end{property}\n\n\\begin{proof}\n\tSet $ F' = \\{T(S): S \\in \\mathcal F\\} $. Suppose that $T: (x,y) \\mapsto (ax+c, by+d) $ where $ a > 0 $ and $ b> 0$. \n\t\n\tFist of all, notice that $ A \\cap B \\neq \\varnothing $ if and only if $ T(A) \\cap T(B) \\neq \\varnothing $. So, two sets $T(A)$ and $T(B)$ in $ \\mathcal F' $ intersect if and only if $ A $ and $ B $ intersect in $ F $. \n\t\n\tSecond, notice that for every set $ A $, $ \\lset{T(A)} = a .\\lset{A} +c $. So, since $ a> 0$, if $ \\lset{A} \\leq \\lset B$, then $\\lset{T(A)} \\leq \\lset{T(B)}$. \n\t\n\tThird, if $ A \\subseteq B $, then $ T(A) \\subseteq T(B) $, because if $ p \\in T(A) $, then $ p = (ax+c,by+d) $ for some $ (x,y) \\in A $. Now, since $ (x,y) \\in B $, we have $ p \\in T(B) $. \n\t\n\tFourth, notice that $ \\Ter{T(A)} = T(\\Ter{A}) $. This, along with the third fact implies that if $ A \\subseteq \\Ter{B} $, then $T(A) \\subseteq \\Ter{T(B)}$. \n\t\n\tWith the four facts above, it is easy to check that $ \\mathcal F' $ satisfies Constraints (C1)-(C6).\n\\end{proof}\n\n\nApplied to a specific set $ S$, the definition of constrained $ S $-graphs becomes rather intuitive. For example, when $ S $ is the boundary of a rectangle in $ \\mathbb{R}^2 $, constrained $ S $-graphs is the class of \\emph{strict frame graphs}. Also, when $ S $ is a non-vertical and non-horizontal line segment, constrained $ S $-graphs is the class of \\emph{strict line-segment graphs}. The definition of both classes are in~\\cite{BG1}, Section 6. \n\nSee Figure~\\ref{fig:examples-constrained-S-graphs} for two more examples of constrained $ S $-graphs where $ S $ is a circle and when $ S $ is a square that is not axis-aligned. In each row of the figure, from left to right, the pictures represent the following:\n\\begin{itemize}\n\t\\item The first picture shows the set $ S $ (in black) and its territory (in gray). \n\t\\item The second picture shows the way that two sets can intersect, i.e. what is described by Constraint (C1).\n\t\\item The third picture represents Constraint (C2). In other words, it shows that if two sets do not intersect but one has an intersection with the territory of the other, how they must be placed. Notice that in the first line, there are two possibilities to place a circle in the territory of another circle with no intersection. \n\t\\item The fourth picture shows the forbidden construction in Constraint (C3). \n\t\\item The fifth picture shows the forbidden construction in Constraint (C4). \n\t\\item Finally, we must keep in mind that there must not be three distinct sets that mutually intersect. \n\\end{itemize}\n\n\\begin{figure}\n\t\\centering\n\t\\vspace*{-1cm}\n\t\\includegraphics[width=12cm]{fig\/constraint-S-example.pdf}\n\t\\vspace*{-2.8cm}\n\t\\caption{Examples of constrained $ S$-graphs} \\label{fig:examples-constrained-S-graphs}\n\\end{figure}\n\n\n\\section{Burling graphs} \\label{sec:Burling-Graphs}\n\n\n\\subsection{Abstract Burling graphs}\n\nAs mentioned in the introduction, Burling~\\cite{Burling65} defined Burling graphs in 1965. Here, we do not present the definition by Burling. Instead, we recall an equivalent definitions of Burling graphs defined in~\\cite{BG1}: \\emph{abstract Burling graphs}.\n\nLet $R$ be a binary relation defined on a set $ S $. We say that $ R $ has a \\emph{directed cycle} if there exists positive integer $ k $ and elements $ x_1, \\dots, x_k \\in S $ such that $(x_1, x_2), (x_2, x_3), \\dots, (x_k, x_1) \\in R $. \n\n\\begin{definition}[\\cite{BG1}, Definition 5.1]\n\tA \\emph{Burling set}\n\tis a triple $ (\\mathcal F, \\prec, \\curvearrowright) $ where $ \\mathcal F $ is a non-empty\n\tfinite set, $\\prec$ is a strict partial order on $\\mathcal F$,\n\tand $ \\curvearrowright $ is a binary relation on $\\mathcal F$ with no\n\tdirected cycles such that the following axioms hold:\n\t\\begin{enumerate}\n\t\t\\item[\\textbf{\\emph{(A1)}}] \n\t\tif $ x \\prec y $ and $ x \\prec z $, then either\n\t\t$ y \\prec z $ or $ z \\prec y $, \\label{item:descdesc}\n\t\t\\item[\\textbf{\\emph{(A2)}}] \n\t\tif $ x \\curvearrowright y $ and $ x \\curvearrowright z $, then either\n\t\t$ y \\prec z $ or $ z \\prec y $, \\label{item:adjadj}\n\t\t\\item[\\textbf{\\emph{(A3)}}] \n\t\tif $ x \\curvearrowright y $ and $ x \\prec z $, then $ y \\prec z\n\t\t$, \\label{item:adjdesc}\n\t\t\\item[\\textbf{\\emph{(A4)}}] \n\t\tif $ x \\curvearrowright y $ and $ y \\prec z $, then either\n\t\t$ x \\curvearrowright z $ or $ x \\prec z $. \\label{item:transitiveboth}\n\t\\end{enumerate}\n\tA graph $ G $ is a \\emph{(non-oriented) abstract Burling graph} if it is obtained from a Burling set $ (\\mathcal F, \\prec, \\curvearrowright) $ by setting $V(G) = \\mathcal F $ and $ E(G) = \\{ \\{x,y\\} : x \\curvearrowright y \\} $. \n\\end{definition}\n\nEquivalently, we can say that a graph is an abstract Burling graph if it is the underlying graph of the oriented graph $ \\hat G $ obtained from a Burling set $ (\\mathcal F, \\prec, \\curvearrowright) $ by setting $V(\\hat G) = \\mathcal F $ and $ E(\\hat G) = \\{ xy : x \\curvearrowright y \\} $.\n\nFor the proof of equivalence of the two classes of abstract Burling graphs and Burling graphs as defined classically in the literature, see Theorem 5.7 of~\\cite{BG1}. \n\nThe axiomatic definition of abstract Burling graphs is useful in the proofs of the next section because, for proving that a graph is a Burling graph, we just need to define two appropriate relations $\\prec $ and $ \\curvearrowright $ on the vertex-set of the graph and prove that Axioms (A1)-(A4) hold. \n\n\n\\section{Equality of the three classes} \\label{sec:equality}\n\nIn this section, we prove that the class of Burling graphs is equal to the class of constrained graphs and to the class of constrained $ S $-graphs for every Pouna set $ S$. \n\n\n\\subsection{Constrained graphs are Burling graphs}\n\n\nWe first need a few lemmas.\n\n\\begin{lemma}\\label{lem:prec-properties}\n\tLet $ A $, $ B $ be two strong Pouna sets. If $ A \\prec B $, then \n\t\\begin{enumerate}\n\t\t\\item $\\rset A < \\rset B $,\n\t\t\\item $\\hset{A} \\leq \\hset{B} $,\n\t\t\\item $ \\Ter A \\subseteq \\Ter B $. \n\t\\end{enumerate} \n\\end{lemma}\n\\begin{proof}\n\tBy definition of $\\prec$, we have $ A \\subseteq \\Ter B $. \n\t\n\tTo prove (1), let $ r = \\rset A $. Because $ A$ is compact, there exists a point $ (r, y) $ in $ A $. Consequently, $(r,y) \\in \\Ter B $, so there exists $ r' $ such that $ r'> r $ and $ (r', y) \\in B $. Notice that, $ r' \\leq \\rset B$. Hence, $ \\rset A < \\rset B $.\n\t\n\t\n\tTo prove (2), notice that $ A \\subseteq \\Ter{B} \\subseteq \\boxset{B}$. So, $ \\bset{A} \\geq \\bset{\\boxset{B}} = \\bset{B}$ and $ \\tset{A} \\leq \\tset{\\boxset{B}} = \\tset{B} $. Therefore, $ \\hset{A} = \\tset{A} - \\bset{A} \\leq \\tset{B} - \\bset{B} = \\hset{B}$. \n\t\n\tTo prove (3), let $ p = (x,y) $ be a point in $ \\Ter A $. Notice that \n\t$$ \n\tx \\geq \\lset{\\Ter{A}}\\geq \\lset{\\boxset{A}} = \\lset A.\n\t$$\n\tAlso,\n\t$$\n\tx \\leq \\rset{\\Ter{A}} \\leq \\rset{\\boxset{A}} = \\rset{A}.\n\t$$\n\tSo, $ \\lset{A} \\leq x \\leq \\rset{A}$.\n\tThus,\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\lset{\\boxset B} & \\leq \\lset{\\Ter B} \\leq \\lset A \\leq x \\leq \\rset A \\leq \\rset{\\Ter B} \\leq \\rset{\\boxset B}.\t\t\t\n\t\t\\end{split}\n\t\\end{equation*}\n\tSimilarly, $ \\bset{\\boxset B} \\leq y \\leq \\tset{\\boxset B} $.\n\tTherefore, $ p \\in \\boxset B $. \n\t\n\tNow, by definition of territory, there exists a point $a=(x', y) \\in A $ with $ x' > x $. So, since $ A \\subseteq \\Ter B $, we have $ a \\in \\Ter B $. By definition of territory, there exists a point $ b=(x'', y) \\in B $ with $ x''> x'$, and thus $ x''> x $. This, along with the fact that $ p \\in \\boxset{B} $ implies that $ p \\in \\Ter B $. Consequently $ \\Ter A \\subseteq \\Ter B $. \n\\end{proof}\n\n\n\nWe say that two strong Pouna sets $A$ and $ B $ are \\emph{comparable} if one of the following happens: $ A \\curvearrowright B $, $ B \\curvearrowright A $, $ A \\prec B $, or $ B \\prec A $. \n\n\\begin{lemma} \\label{lem:comparable-sets}\n\tLet $ A $ and $ B $ be two strong Pouna sets in a collection $\\mathcal F $ which satisfies Constraints \\emph{(C1)} and \\emph{(C2)}. If $ \\Ter A \\cap \\Ter B \\neq \\varnothing $, then $ A$ and $ B $ are comparable. \n\\end{lemma}\n\\begin{proof}\n\tIf $ A \\cap B \\neq \\varnothing $, then by Constraint (C1), either $ A \\curvearrowright B $ or $ B \\curvearrowright A $. So, we may assume $ A \\cap B = \\varnothing $. Choose a point $ p = (x,y) \\in \\Ter A \\cap \\Ter B $. There exists $ x', x'' \\in \\mathbb{R} $, both bigger than $ x $, such that $ p'=(x', y) \\in A $ and $ p''=(x'', y) \\in B $. Since $ A $ and $ B $ are disjoint, $ x' \\neq x''$. First, assume that $ x''> x' $. Notice that $ p' \\notin B $ and that $ p'$ is on the straight line joining $ p $ and $ p'' $, which are both points in $ \\boxset B $. Therefore, $ p' \\in \\boxset B $. Consequently, $ p' \\in \\Ter B $. Therefore $ A \\cap \\Ter B \\neq \\varnothing $, and by Constraint (C2), we have $ A \\prec B $. Second, assume that $ x'' x_0 $ such that $ p' = (x', y) \\in A $. Since $ p' \\in A $, we have $ p' \\notin R $. So, in particular, $ x' \\neq x $. \n\tIf $ x' < x $, then $ x' $ is on the strait line joining $ p_0 $ and $ p $. But $ p_0, p \\in R $ and $ R $ is convex, so $ (x',y) \\in R $, a contradiction. Hence $ x'> x $. Now, to show that $ p\\in \\Ter{A}$, it is enough to show that $ p \\in \\boxset{A} \\setminus A $. But $ p $ being in $R $, is not in $ A $. On the other hand, $ p $ is in on the straight line between $ p_0 $ and $ p' $. Now because $ p_0 \\in \\Ter{A} \\subseteq \\boxset{A} $ and $ p' \\in A \\subseteq \\boxset{A} $, we have $p \\in \\boxset{ A }$. This completes the proof. \n\\end{proof}\n\n\nLet $ E $ be a rectangle in $ \\boxset{\\mathcal F} $. The prob \\emph{defined by $ E $} in $ B $ is the prob $ P$ which is obtained by extending the right side of $ E $ to reach the border of $B $, i.e.\n$\nP = \\{(x,y) \\in B: \\lset E \\leq x \\leq \\rset B, \\bset E \\leq y \\leq \\tset E \\}.\n$\nNotice that if $ E $ does not intersect any member of $ \\mathcal F $, then it is a root for $ P $. \n\n\n\\subsubsection*{The construction of Pawlik, Kozik, Krawczyk, Laso\\'n, Micek, Trotter, and Walczak}\n\nNow, we explain the construction of Pawlik, Kozik, Krawczyk, Laso\\'n, Micek, Trotter, and Walczak in~\\cite{Pawlik2013}. Our terminology is slightly different from the one of~\\cite{Pawlik2013}, but we have tried to keep the terminology as close as possible so the reader can refer to~\\cite{Pawlik2013} whenever needed. Fix a strong Pouna set $ S $ and a subterritory $ E $ of $ S $. From now on, for the transformed copy $ S'=T(S) $, we consider the subterritory $ T(E) $. \n\nLet $(\\mathcal F, \\mathcal P) $ be a tuple where $ \\mathcal F$ is a collection of transformed copies of $ S $ and $ \\mathcal P $ is a set of probs of $\\mathcal F$. We define an operation $\\Gamma$ where $ (\\mathcal F', \\mathcal P') = \\Gamma(\\mathcal F, \\mathcal P) $ is obtained as follows:\n\\begin{enumerate}\n\t\\item[\\textbf{(S$'$1)}] For every $ P \\in \\mathcal P $, let $P^\\uparrow $ and $ P^{\\downarrow} $ be respectively the top one-third and the bottom one-third of $ P $, i.e.\n\t$$\n\tP^\\uparrow = [\\lset P, \\rset P] \\times [\\frac{\\bset P + 2\\tset P}{3}, \\tset P]\n\t$$\n\tand\n\t$$\n\tP^\\downarrow = [\\lset P, \\rset P] \\times [\\bset P, \\frac{2\\bset P+ \\tset P}{3}].\n\t$$\n\t\\item[\\textbf{(S$'$2)}] Set $ S_P $ to be a transformed copy of $ S $ where we first match the boundary of $ \\boxset S $ on the boundary of $ P^\\uparrow $, and then we scale it horizontally by $ \\frac{2 \\wset{S}}{\\lset E - \\lset S} $ keeping the left-side of $\\boxset S $ fixed.\n\tFormally, the transformation described above is $ T_P = T_2 \\circ T_1 : \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2 $, where\n\t$$\n\tT_1(x,y) = \\Big( \\frac{\\wset{P^\\uparrow}}{\\wset{S}}x + \\lset{P^\\uparrow} - \\frac{\\lset{S}\\wset{P^\\uparrow}}{\\wset{S}} , \\frac{\\hset{P^\\uparrow}}{\\hset{S}}y + \\bset{P^\\uparrow} - \\frac{\\bset{S}\\hset{P^\\uparrow}}{\\hset{S}} \\Big)\n\t$$\n\tand \n\t$$\n\tT_2(x,y) = \\Big( \\frac{2 \\wset S}{\\lset E - \\lset S}x + \\lset{P^\\uparrow}(1 - \\frac{2 \\wset{S}}{\\lset E - \\lset S}) , y \\Big).\n\t$$\n\tThis transformation ensures that the subterritory of $ S_P$, i.e.\\ $ T_P(E) $, is outside~$ \\boxset{\\mathcal F} $ (See Property~\\ref{prop:prop-after-def-Gamma}). Denote $ T_P(E) $ by $ E_P$.\n\t\\item[\\textbf{(S$'$3)}] Set $ \\mathcal F' = \\mathcal F \\cup \\big( \\cup_{P \\in \\mathcal P} S_P \\big) $. \n\t\\item[\\textbf{(S$'$4)}] For $ P \\in \\mathcal P $, denote by $ P_1 $ the prob for $\\mathcal{F'}$ defined by $ E_P $, and denote by $P_2 $ the prob for $ \\mathcal{F'} $ defined by $ P^\\downarrow $.\n\t\\item[\\textbf{(S$'$5)}] Set $ \\mathcal P' = \\{ P_1, P_2 : P \\in \\mathcal P \\}$. \n\\end{enumerate} \n\n\n\nNow, inductively, we define a sequence $\\{(\\mathcal F_k, \\mathcal P_k)\\}_{k \\geq 1} $ where $ \\mathcal F_k $ is a collection of positive transformed copies of $ S $, and $ \\mathcal P_k $ is a set of probs for $ \\mathcal F_k$. \n\nFor $ k =1 $, set $ \\mathcal F_1 = \\{S\\} $ and $ \\mathcal P_1 = \\{P\\} $ where $ P$ is the prob defined by $ E $. Now, let $ k \\geq 1 $ and assume that $(\\mathcal F_k, \\mathcal P_{k})$ is defined, we define $(\\mathcal F_{k+1}, \\mathcal P_{k+1})$ as follows:\n\\begin{enumerate}\n\t\\item[\\textbf{(S1)}] Set $ (\\mathcal F, \\mathcal P) = \\Gamma(\\mathcal F_k, \\mathcal P_k)$.\n\t\\item[\\textbf{(S2)}] For every $ P \\in \\mathcal P_k $, choose a root $ R_P$. (To see that $P$ has a root, see~\\cite{Pawlik2013} or Theorem~\\ref{thm:BG-is-const-S-graph}.) Create a transformed copy $ (\\mathcal F^P, \\mathcal P^P)$ of $(\\mathcal F, \\mathcal P)$ such that $ \\boxset{\\mathcal F^P} $ is matched to $ R_P $. Formally, apply the transformation:\n\t$$\n\tT'_P(x,y) = \\Big( \\frac{\\wset{R_P}}{\\wset{B_P}}x + \\lset{R_P} - \\frac{\\lset{B_P}\\wset{R_P}}{\\wset{B_P}} , \\frac{\\hset{R_P}}{\\hset{B_P}}y + \\bset{R_P} - \\frac{\\bset{B_P}\\hset{R_P}}{\\hset{B_P}} \\Big),\n\t$$\n\twhere $ B_P = \\boxset{\\mathcal F^P} $.\n\t\\item[\\textbf{(S3)}] Set $ \\mathcal F_{k+1} = \\mathcal F_k \\cup \\big(\\cup_{P \\in \\mathcal P_k} \\mathcal F^P \\big) $.\n\t\\item[\\textbf{(S4)}] Now, for $ P \\in \\mathcal P_k $ and for $ Q \\in \\mathcal P^P $, let $ P_Q $ be the prob for $ \\mathcal F_k$ defined by $ Q $. \n\t\\item[\\textbf{(S5)}] Set $ \\mathcal P_{k+1} = \\{P_Q: P \\in \\mathcal P_k, Q \\in \\mathcal P^P \\}. $\n\\end{enumerate}\n\nThe tuple $(\\mathcal F_{k+1}, \\mathcal P_{k+1})$ is the new sentence of the sequence.\n\n\nFor the rest of this section, let $ G_k $ denote the intersection graph of $ \\mathcal F_k $. \nWe recall that the class spanned by $ \\{G_k\\}_{k\\geq 1} $ is the class of Burling graphs. \n\n\nNow, we state and prove some lemmas and properties about the construction of Pawlik, Kozik, Krawczyk, Laso\\'n, Micek, Trotter, and Walczak. \n\n\n\\begin{property} \\label{prop:prop-after-def-Gamma}\n\tAdopting the notation from the definition of $ \\Gamma $, for every $ P \\in \\mathcal P $, we have: \n\t\\begin{enumerate}\n\t\t\\item the transformation $T_P $ is positive. \n\t\t\\item $ \\lset{E_P} > \\rset{\\boxset{F}} $, so in particular, $ E_P \\cap \\boxset{F} = \\varnothing $.\n\t\\end{enumerate}\n\\end{property}\n\n\\begin{proof}\n\tThe proof of (1) is immediate from the definition of $ T_P $.\n\t\n\tTo prove (2), set $ T_P: (x,y) \\mapsto (ax+c, bx+d)$. We have \n\t$$\n\ta = \\frac{2\\wset{S}}{\\lset{E} - \\lset{S}}.\\frac{\\wset{P^\\uparrow}}{\\wset{S}},\n\t$$\n\tand\n\t$$ \n\tc = \\frac{2 \\wset S}{\\lset E - \\lset S} \\big(\\lset{P^\\uparrow} - \\frac{\\lset{S}\\wset{P^\\uparrow}}{\\wset{S}}\\big) + \\lset{P^\\uparrow}(1 - \\frac{2 \\wset{S}}{\\lset E - \\lset S}).\n\t$$ \n\t\n\tNow, notice that \n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\lset{T_P(E)} &= a.\\lset{E} + c \\\\\n\t\t\t&= \\lset{E}.\\frac{2 \\wset{S} \\wset{P^\\uparrow}}{\\wset{S}(\\lset{E}-\\lset{S})} + \\frac{2\\wset{S}\\lset{P^\\uparrow}}{\\lset{E}-\\lset{S}} \\\\ & \\ \\ \\ - \\lset{B_S}.\\frac{2 \\wset{S} \\wset{P^\\uparrow}}{\\wset{S}(\\lset{E}-\\lset{S})} + \\lset{P^\\uparrow} - \\frac{2\\wset{S}\\lset{P^\\uparrow}}{\\lset{E}-\\lset{S}} \\\\\n\t\t\t&= \\lset{P^\\uparrow} + (\\lset{E}-\\lset{S})\\frac{2 \\wset{S} \\wset{P^\\uparrow}}{\\wset{S}(\\lset{E}-\\lset{S})} \\\\\n\t\t\t& > \\lset{P^\\uparrow} + 2 \\wset{P^\\uparrow} = \\rset{P^\\uparrow} + \\wset{P^\\uparrow} > \\rset{P^\\uparrow}.\n\t\t\\end{split}\n\t\\end{equation*}\n\tTo complete the proof, notice that $ \\rset{P^\\uparrow} = \\rset{\\boxset{\\mathcal F}} $. \n\\end{proof}\n\n\\begin{property} \\label{prop:more-on-Gamma-and-disjoint-probs}\n\tLet $ \\mathcal F $ be a collection of strong Pouna sets, and let $ \\mathcal P $ be a set of probs for $ \\mathcal F $ that are mutually disjoint. Setting $(\\mathcal F',\\mathcal P') = \\Gamma(\\mathcal F, \\mathcal P)$ and adopting the notation from the definition of $\\Gamma$, we have that for every $ P \\in \\mathcal P $:\n\t\\begin{enumerate}\n\t\t\\item if $Q \\in \\mathcal P \\setminus \\{P\\}$, then $S_P \\cap Q = \\varnothing $, $ S_P \\cap S_Q = \\varnothing $, and $ \\Ter{S_P} \\cap Q = \\varnothing $,\n\t\t\\item $N_{\\mathcal F'}(P_1) = \\{S_P\\} $,\n\t\t\\item $N_{\\mathcal F'}(P_2) \\subseteq N_{\\mathcal F}(P)$ and $ N_{\\mathcal F'}(P_2) \\subseteq \\mathcal F$.\n\t\\end{enumerate}\n\\end{property}\n\n\n\n\\begin{proof}\n\tItem (1) follows from the facts that $ \\boxset{S_P} \\subseteq P $, $ S_P \\subseteq P$, $ S_Q \\subseteq Q$, and $ P \\cap Q = \\varnothing $.\n\t\n\tTo prove (2), notice that by Property~\\ref{prop:prop-after-def-Gamma}, we have $ \\lset{E_P} > \\rset{\\boxset{\\mathcal F}}$. Since $ P_1 $ is the prob defined by $ E_P$, the prob $ P_1 $ is also outside $ \\boxset{\\mathcal F}$ So, for every $ A \\in \\mathcal F$, we have $ A \\notin N_{\\mathcal F'}(P_1)$. Moreover, by item (1) of this property, for every $ Q \\in \\mathcal P \\setminus \\{P\\}$, we have $ S_Q \\notin N_{\\mathcal F'}(P_1) $. Finally, since $ E_P $ is a subterritory of $ S $, by definition of $ S_P \\cap P_1 \\neq \\varnothing $. Therefore $ N_{\\mathcal F'}(P_1) = \\{S_P\\}$. \n\t\n\t\n\tTo prove (3), assume that $ A \\in \\mathcal F' $ is of the form $ A = S_Q $ for some $ Q $. Case 1, $ Q = P$, in which case $S_Q = S_P \\subseteq P_1 $, and since $P_1 \\cap P_2 = \\varnothing $, we have $ A \\notin N_{\\mathcal F'}(P_2) $. Case 2, $ Q \\neq P $, and thus item (1) of this property implies that $ A \\notin N_{\\mathcal F'}(P_2)$. Therefore, $ N_{\\mathcal F'}(P_2) \\subseteq \\mathcal F$.\n\t\n\tHence, $ N_{\\mathcal F'}(P_2) = N_{\\mathcal F}(P_2)$. So, since $ P_2 \\cap \\boxset{F} \\subseteq P $, we have $ N_{\\mathcal F'}(P_2) \\subseteq N_{\\mathcal F}(P)$. \n\\end{proof}\n\n\n\n\\begin{lemma}\n\tLet $ S $ be a strong Pouna set. Let $ \\mathcal F $ be a collection of transformed copies of $ S $ that satisfies Constraints \\emph{(C1)-(C6)}. Let $ \\mathcal P$ be a set of mutually disjoint stable probs of $\\mathcal{F}$. If $ (\\mathcal F', \\mathcal P') = \\Gamma(\\mathcal F, \\mathcal P)$, then \n\t\\begin{enumerate}\n\t\t\\item elements of $P'$ are mutually disjoint,\n\t\t\\item every element of $P$ is a stable prob for $ \\mathcal F'$,\n\t\t\\item $ \\mathcal F' $ satisfies Constraints \\emph{(C1)-(C6)}.\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tWe adopt the notation from the definition of $\\Gamma$.\n\t\n\tSet $ \\mathfrak B = \\boxset{\\mathcal F}$ and $ \\mathfrak{B'} = \\boxset{\\mathcal F'}$. Notice that $ \\lset{\\mathfrak{B'}} = \\lset{\\mathfrak{B}}$, $ \\bset{\\mathfrak{B'}} = \\bset{\\mathfrak{B}}$, and $ \\tset{\\mathfrak{B'}} = \\tset{\\mathfrak{B}} $. However, $ \\rset{\\mathfrak{B'}} > \\rset{\\mathfrak{B}}$. For the rest of the proof, we adapt the notations in the definition of $\\Gamma$. \n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{elements of $\\mathcal P' $ are mutually disjoint.}} \t\n\t\n\tCorresponding to every $ P \\in \\mathcal P $, there are two probs in $ \\mathcal P' $, that is, $P_1 $ and $ P_2 $. Notice that $ P_1 \\cap P_2 = \\varnothing $. So, the fact that the probs in $ \\mathcal P' $ are mutually disjoint is implies directly by the same fact about $ \\mathcal P$. \n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{Elements of $\\mathcal P' $ are stable probs for $\\mathcal F' $.}}\n\t\n\tFix $ P \\in \\mathcal P $. We prove both $ P_1 $ and $P_2 $ are stable probs, and thus every prob in $ \\mathcal P' $ is stable. \n\t\n\tThe prob $ P_1 $ is defined by a subterritory $ E_P $. By Property~\\ref{prop:prop-after-def-Gamma}, $E_P \\cap \\mathfrak{B} = \\varnothing $. Therefore, for every $ A \\in \\mathcal F$, we have $ E_P \\cap A = \\varnothing $. Moreover, by definition of subterritory, $E_P \\cap S_P = \\varnothing $. Finally, since $ E_P \\subseteq P $, by Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, $ E_P \\cap S_Q = \\varnothing $ for every $ Q \\in \\mathcal P \\setminus \\{P\\} $ as well. Thus $ E_P $ does not intersect any element of $ \\mathcal F' $. So, $ E_P $ is a root for $P_1 $. \n\t\n\tNotice that by Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, we have $N(P_1) = \\{S_P\\} $, so item (2) of the definition of stable prob holds. Moreover,\n\tsince $ E_P $ is a subterritory of $ S_P$, we have \n\t\\begin{itemize}\n\t\t\\item $ E_P \\subseteq \\Ter{S_P}$, \n\t\t\\item $ \\bset{E_P} > \\bset{S_P}$ and $ \\tset{E} < \\tset{S_P}$,\n\t\t\\item $S_P $ crosses $ P_1$ vertically,\n\t\\end{itemize}\n\twhich proves item (1), (3), and (4) of the definition of stable prob, respectively. For item (3), we have used the facts that $ \\bset{P_1} = \\bset{E_P} $ and $ \\tset{P_1} = \\tset{E_P} $.\n\t\n\t\n\tBy the hypothesis, $ P $ has a root. Let $ R $ be a root of $ P $. Set $ R^\\downarrow = R \\cap P^\\downarrow $ and notice that $ R^\\downarrow $ is a root of $ P^\\downarrow $, as a prob for $\\mathcal F $. In particular, $ R^\\downarrow $ does not intersect any element of $ \\mathcal F $.\n\tNow, let $ A \\in N_{\\mathcal F'} (P_2) $.\n\tBy Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, we have $ A \\in N_{\\mathcal F}(P) $. Therefore, $ R \\subseteq \\Ter A$. Consequently, $ R^\\downarrow \\subseteq \\Ter A $. This proves item (1) of the definition of stable prob. Moreover, since $ A \\in N_{\\mathcal F}(P) $ and $ P$ is stable, we have \n\t$$\n\t\\bset{A} < \\bset{P} = \\bset{P^\\downarrow} = \\bset{P_2},\n\t\\text{ and }\n\t\\tset{A} > \\tset{P} \\geq \\tset{P^\\downarrow} = \\tset{P_2},\n\t$$\n\twhich proves item (3) of the definition. Also, since $ A $ crosses $ P$ vertically, by Property~\\ref{prop:crossing-a-subrectangle}, it crosses $ P^\\downarrow $ vertically as well, which proves item (4) of the definition. \n\t\n\tNow, assume that $ A, B \\in N_{\\mathcal F'} (P_2) $ and $ A \\neq B $. Again, by Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, we have $ A, B \\in N_{\\mathcal F}(P) $. Thus, $ A\\cap B =\\varnothing $, proving item (2) of the definition. \n\tHence, $P_2 $ is a stable prob. \n\t\n\t\n\t\n\tNow, we prove that $\\mathcal F' $ satisfies Constraints (C1)-(C6).\n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F' $ satisfies \\emph{(C1).}}} \n\t\n\tLet $ A, B \\in \\mathcal F'$ be two distinct and intersecting transformed copies of $ S $. Set $ L_A = \\{ (x,y) \\in A : x = \\lset A \\} $. Notice that $ \\lset A = \\lset{L_A}$. \n\t\n\tIf $ A, B \\in \\mathcal F $, then the result holds because $ \\mathcal F $ satisfies (C1). Furthermore, by Property~\\ref{prop:prop-after-def-Gamma}, we cannot have $ A, B \\in \\mathcal F' \\setminus \\mathcal F$. So, without loss of generality, assume $ A \\in \\mathcal F' \\setminus \\mathcal F $, so $ A = S_P $ for some $ P \\in \\mathcal P $, and $ B \\in \\mathcal F$. In particular, $ B \\subseteq \\mathfrak B $, and by construction, $ A \\cap (\\mathfrak{B} \\setminus P^\\uparrow) = \\varnothing $. Hence, $ B \\cap P^\\uparrow \\neq \\varnothing $, and therefore $ B \\in N_{\\mathcal F}(P) $. Thus, by Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, for every root $ R $ of $ P $, we have $ R \\subseteq \\Ter B $. Moreover, we have $ \\bset{B}< \\bset{P} $ and $ \\tset{B}> \\tset{P} $. Also, notice that by construction, for every $ \\mathfrak{s} \\in \\{\\mathfrak{l}, \\mathfrak{r}, \\mathfrak{b}, \\mathfrak{t} \\}$, we have $ \\mathfrak{s}(A) = \\mathfrak{s}(P^\\uparrow)$. Let $ p= (x,y) \\in L(A) $. So, $ x =\\lset{A} \\lset{P}$ and $ y \\in (\\bset{P}, \\tset{P})$. Moreover, $ \\bset P \\leq \\bset A \\leq y \\leq \\tset A \\leq \\tset P $. Therefore, $ (x, y) \\in \\{ (x',y') \\in P : x' = \\lset P \\}$. Consequently, $ (x, y) \\in R $. So, $ L_A \\subseteq R \\subseteq \\Ter B $.\n\t\n\t\n\tMoreover, we have:\n\t\\begin{multline*}\n\t\t\\lset B = \\lset{\\boxset B } \\leq \\lset{\\Ter B} \\leq \\lset{L_A} \\\\ \n\t\t= \\lset A = \\lset{P^\\uparrow} = \\lset P = \\lset R < \\rset R \\\\ \n\t\t\\leq \\rset{\\Ter B} \\leq \\rset{\\boxset B} = \\rset B \\overset{(a)}{\\leq} \\rset{\\mathfrak{B}} \\overset{(b)}{<} \\rset{A},\n\t\\end{multline*}\t\n\twhere (a) is because $ B \\in \\mathcal F $, and (b) follows from Step (S$'$2) of the construction. Therefore $ \\lset B \\leq \\lset A < \\rset B < \\rset A $.\n\t\n\tOn the other hand,\n\t\\begin{equation*}\n\t\t\\bset B < \\bset P < \\bset{P^\\uparrow} = \\bset A \\overset{(c)}{<} \\tset A = \\tset P < \\tset B, \n\t\\end{equation*}\n\twhere (c) follow from the fact that $ A $, a strong Pouna set, cannot be a subset of a horizontal line segment. Therefore $ \\bset B < \\bset A < \\tset A < \\tset B$. \n\t\n\tHence, all the items in Constraint (C1) hold and $ A \\curvearrowright B $. \n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F' $ satisfies \\emph{(C2).}}} \n\t\n\tLet $ A $ and $ B $ be two disjoint sets in $ \\mathcal F' $ such that $ A \\cap \\Ter B \\neq \\varnothing $. We prove that $ A , B \\in \\mathcal F $. For the sake of contradiction, assume that $ \\{A, B\\} \\nsubseteq \\mathcal F$. There are three cases possible. \n\t\n\tCase 1: $ A, B \\in \\mathcal F' $. So, there exists $ P, Q \\in \\mathcal P $ such that $ P \\neq Q $ and $ A=S_P $ and $ B = S_Q $. But in that case, by construction, $ \\boxset B \\subseteq Q $, and $ A \\subseteq P$. So, from $ A \\cap \\Ter B \\neq \\varnothing $, we have $ P \\cap Q \\neq \\varnothing $, a contradiction. \n\t\n\tCase 2: $ A =S_P $ for some $P \\in \\mathcal P $, and $ B \\in \\mathcal F $. Since $ A \\subseteq P $, form $ A \\cap \\Ter B \\neq \\varnothing $ we deduce that $ P \\cap \\Ter B \\neq \\varnothing$. Choose $ p = (x,y) \\in P \\cap \\Ter B $. Because by definition of Territory, there exists a point $p'= (x',y) \\in B $ with $ x' > x $. Now, because $ B \\subseteq F $, we have $ p' \\in \\mathfrak B $ and therefore $ p' \\in P $. Hence $ P \\cap B \\neq \\varnothing $, i.e.\\ $ B \\in N(P) $. Therefore, $ B $ crosses $ P$ vertically. Moreover, $A = S_P $ crosses $ P_1 $ and therefore $P$ horizontally. So, by Property~\\ref{prop:horizontal-and-vertical-crossing-intersect}, we have $ A \\cap B \\neq \\varnothing $, a contradiction.\n\t\n\tCase 3: $ A \\in \\mathcal F $ and $ B =S_P $ for some $P \\in \\mathcal P $. In this case $ \\Ter B \\subseteq P $, and therefore $ A \\cap P \\neq \\varnothing $, i.e. $A \\in N(P) $. So, $ A $ crosses $P $ vertically. On the other hand, $ B $ crosses $ P_1 $ and thus $ P$ horizontally. Therefore, by property~\\ref{prop:horizontal-and-vertical-crossing-intersect}, we have $ A \\cap B \\neq \\varnothing $, a contradiction.\n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F' $ satisfies \\emph{(C3).}}} \n\t\n\tLet $ A , B \\in \\mathcal F $ be two distinct sets with non-empty intersection. For the sake of contradiction, assume that there exists $ C \\in \\mathcal F $ such that $ C \\subseteq \\Ter A \\cap \\Ter B $. We first show that $ C \\in \\mathcal F $. Suppose not, so $ C = S_P $ for some $P \\in \\mathcal P$. Since $ C \\subset P$, neither of $ A $ and $ B $ can be some set of the form $ S_Q $. Therefore $ A, B \\in \\mathcal F $. Now, notice that $ C \\subseteq \\Ter A \\subseteq \\boxset A$. On the other hand, $ \\boxset A \\subseteq \\mathfrak B $, but $ C \\nsubseteq \\mathfrak B$, a contradiction. \n\t\n\tNow we prove that both $ A $ and $ B $ are in $ \\mathcal F $. Suppose not. Without loss of generality, assume that $A= S_P $ for some $P \\in \\mathcal P $. Since $ C \\subseteq \\Ter A $, we must have $ C \\in N(P) $. Therefore $ \\bset C < \\bset P \\leq \\bset A $. On the other hand, because $ C \\subseteq \\Ter A \\subseteq \\boxset A $, we have $ \\bset C \\geq \\bset A $, a contradiction. So, $ A, B \\in \\mathcal F $ as well, and the result follows from the fact that $\\mathcal{F}$ satisfies~(C3). \n\t\n\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F' $ satisfies \\emph{(C4).}}} \n\t\n\tFix $P \\in \\mathcal P$. Let us first prove that there exists no $ A \\in \\mathcal F; $ such that $ A \\curvearrowright S_P $ or $ S_P \\prec A $. First, if $ A \\curvearrowright S_P$, then in particular $ A \\cap S_P \\neq \\varnothing $. Thus, by Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, we have $ A\\in \\mathcal F $. Therefore, $ \\rset{A} \\leq \\rset{\\mathcal F} < \\rset{P^\\uparrow} = \\rset{S_P}$. But on the other hand, $ A \\prec S_P $ implies $ \\rset{A} > \\rset{S_P}$, a contradiction. Second, if $ S_P \\prec A $, then in particular $S_P \\subseteq \\Ter{A} $. Also, by construction $ S_P \\subseteq P $. Therefore, $ \\Ter{A} \\cap P \\neq \\varnothing $. Hence, by Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, we have $ A \\in \\mathcal F $. Therefore $ \\rset{A} \\leq \\rset{\\mathcal F} < \\rset{P^\\uparrow} = \\rset{S_P}$.\n\tOn the other hand, by Lemma~\\ref{lem:prec-properties}, $ S_P \\prec A $ implies that $ \\rset{S_P} < \\rset{A}$, a contradiction.\n\t\n\tNow, for the sake of contradiction, assume that there exists $ A, B, C \\in \\mathcal F' $ such that $ A \\prec B $, $ A \\curvearrowright C $, and $ B \\curvearrowright C $. From what we proved above, we know that $ A, C \\in \\mathcal F$. Therefore, since $ \\mathcal F $ satisfies (C4), we cannot have $ B\\in \\mathcal F $. So, $ B = S_P $ for some $ P \\in \\mathcal P $. In particular $ \\Ter{B} \\subseteq \\boxset{B} \\subseteq P^\\uparrow$. \n\t\n\tFrom $A \\prec B $, we have\n\t$\n\tA \\subseteq \\Ter B \\subset P^\\uparrow \\subseteq P\n\t$.\n\tTherefore, $ A \\in N_{\\mathcal F}(P)$. \n\t\n\tOn the other hand, from $ B \\prec C $, we have $ B \\cap C \\neq \\varnothing $, therefore $ C \\cap P^\\uparrow \\neq \\varnothing $. So, $ C \\in N_{\\mathcal F}(P)$. \n\t\n\tSo, $ A $ and $ C $ are two sets in $ N_{\\mathcal F}(P) $ that are not disjoint, which contradicts the fact that $ P$ is stable. \n\t\n\t\n\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F' $ satisfies \\emph{(C5).}}} \n\t\n\tFor the sake of contradiction, assume that $ A $, $ B $, and $ C $ are three sets in $ \\mathcal F' $ that two by two intersect. At least one of the three sets must be in $ \\mathcal F' \\setminus \\mathcal F $, because (C5) holds for $\\mathcal F $. Moreover, because of Property~\\ref{prop:more-on-Gamma-and-disjoint-probs}, at most one of the three sets is in $ \\mathcal F' \\setminus \\mathcal F $. So, without loss of generality, assume that $ A=S_P $ for some $P \\in \\mathcal P $, and that $ B, C \\in \\mathcal F $. But since $ B \\cap A \\neq \\varnothing $, we have $ B \\cap P \\neq \\varnothing$, i.e.\\ $B \\in N(P) $. Similarly, $ C \\in N(P)$. But $ B \\cap C \\neq \\varnothing $ contradicts the fact that $ P$ is stable for $ \\mathcal F $. Hence, (C5) holds for $ \\mathcal F' $. \n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F' $ satisfies \\emph{(C6).}}} \n\t\n\tBy assumption, $ S $ is strong. So, it is enough to show that $ T_P $ in Step (S$'$2) is a positive transformation for every $P \\in \\mathcal{P}$. This follows from the fact that $ T_p $ is positive, as shown in Property~\\ref{prop:prop-after-def-Gamma}. \n\t\n\tThis completes the proof of the lemma. \n\\end{proof}\n\n\n\\begin{theorem}\\label{thm:BG-is-const-S-graph}\n\tLet $ S $ be a Pouna set. Every Burling graph is a constrained $ S $-graph.\n\\end{theorem}\n\\begin{proof}\n\tFor this proof, we adopt the notations in the definition of the construction of Pawlik, Kozik, Krawczyk, Laso\\'n, Micek, Trotter, and Walczak. \n\t\n\tWe may assume that $ S $ is a strong Pouna set, otherwise, we replace every $ S $ in this proof by the horizontal reflection of $ S $.\n\t\n\tFix a subterritory $ E $ of $ S $ (which exists, by Lemma~\\ref{lem:subterritory-exists}), and apply the construction on it. For every $ k \\geq 1 $, we know that $ \\mathcal F_k $ is a collection of transformed copies of $ S $. We first prove that $ \\mathcal F_k$ satisfies Constraints (C1)-(C6). To do so, we prove the following stronger statement by induction on $ k$.\n\t\n\t\\begin{statement}\n\t\tFor every $ k \\geq 1 $, we have:\n\t\t\\begin{enumerate}\n\t\t\t\\item the elements of $ \\mathcal P_k $ are mutually disjoint,\n\t\t\t\\item $\\mathcal P_k $ is a collection of stable probs of $ \\mathcal F_k $, \n\t\t\t\\item $\\mathcal{F}_k$ satisfies constraints (C1)-(C6).\n\t\t\\end{enumerate}\n\t\\end{statement}\n\t\n\tFirst of all, for $ k=1 $, the first item of the statement follows from the fact that the fact that $ E $ is a subterritory of $ S $. Statement (2) and (3) hold trivially, as $ |\\mathcal{F}_1|=1 $.\n\t\n\t\n\tNow, assume that the statement holds for some $ k \\geq 1 $, we prove that it holds for $ k+1 $. \n\t\n\tNotice that for every $ P \\in \\mathcal P$, the transformation $ T'_P$ is positive, so the tuple $ (\\mathcal F^P, \\mathcal P^P) $ in a positive transformed copy of $ \\Gamma(\\mathcal F_{k}, \\mathcal P_{k}) $. So, by Property~\\ref{prop:T(F)-satisfies-C1-C5}, we know that \n\t\\begin{equation} \\label{eq:FP-sat-C1-6}\n\t\t\\text{for every $ P \\in \\mathcal P $, the collection $ \\mathcal F^P $ satisfies Constraints (C1)-(C6).}\n\t\\end{equation}\n\tMoreover, it is easy to check the following:\n\t\\begin{equation} \\label{eq:P-stable-mut-disj}\n\t\t\\text{for every $ P \\in \\mathcal P $, the elements of $ \\mathcal P^P $ are stable probs for $ \\mathcal F$ and are mutually disjoint.}\n\t\\end{equation}\n\t\n\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{The elements of $ \\mathcal P_k $ are mutually disjoint.}} \n\t\n\tLet $ P_Q $ and $P'_{Q'} $ be two probs in $ \\mathcal P_{k+1} $. In order to show that these two probs are disjoint, it is enough to show that $ (\\bset{Q}, \\tset{Q}) $ and $(\\bset{Q'}, \\tset{Q'}) $ are disjoint intervals. If $ P = P' $, then this follows from (\\ref{eq:P-stable-mut-disj}), and if $ P \\neq P' $ from the fact that $ Q $ and $ Q' $ are inside the roots of $P$ and $ P'$ respectively, and $ P $ and $ P' $ are disjoint by induction hypothesis. \n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{Every $ P \\in \\mathcal P_{k+1} $ is a stable prob for $\\mathcal F_{k+1}$.}} \n\t\n\tLet $ P_Q \\in \\mathcal P_{k+1} $. Notice that $ Q \\in \\mathcal P^P$ is a prob for $\\mathcal F^P $. So, by (\\ref{eq:P-stable-mut-disj}), $ Q $ has a root $ R $ such that for every $ A \\in N_{\\mathcal F^P}(Q)$, we have $ R \\subseteq \\Ter{A}$. So, item (1) of the definition of stable prob holds. \n\t\n\tSet $ N_1 = N_{\\mathcal F^P}(Q) $ and $ N_2 = N_{\\mathcal F_{k+1}}(P) $. \n\t\n\tThe elements in $ N_{\\mathcal F_{k+1}}(P_Q) $ are either the neighbors of $ Q $ as a prob for $ \\mathcal F^P $, so they are in $N_1 $, or are outside $ R_P $ and thus are in $N_2 $. The elements in $N_1 $ are mutually disjoint by (\\ref{eq:P-stable-mut-disj}) and the elements in $ N_2 $ are mutually disjoint by induction hypothesis. Finally, one element in $ N_1 $ and one element in $N_2 $ are disjoint because the former is inside $ R_P $ and the latter does not intersect $ R_P $. So, item (2) of the definition holds as well.\n\t\n\tNow, fix $ A \\in N_{\\mathcal F_{k+1}}(P_Q)$. If $ A \\in N_1 $, then\n\t$$\n\t\\bset{A} < \\bset{Q} = \\bset{P_Q},\n\t\\text{ and }\n\t\\tset{A} > \\tset{Q} = \\tset{P_Q}. \n\t$$\n\tMoreover, there is a path in $ A $ crossing $ Q $. So, the same path crosses $P_Q $ as well. \n\t\n\tIf $ A \\in N_2 $, then\n\t$$\n\t\\bset{A} < \\bset{P} = \\bset{R_P} \\leq \\bset{Q} = \\bset{P_Q},\t\n\t$$\n\tand\n\t$$\n\t\\tset{A} > \\tset{P} = \\tset{R_P} \\geq \\tset{Q} = \\tset{P_Q}.\n\t$$\n\tMoreover, there is a path in $ A $ crossing $ P$, so by Property ??, it crosses $P_Q $ as well. \n\t\n\tNow, we check that $\\mathcal{F}_{k+1}$ satisfies Constraints (C1)-(C6). In what follows, we use several times the fact that that by (\\ref{eq:FP-sat-C1-6}) and by induction hypothesis, the conditions hold when all the elements are chosen inside $ \\mathcal F_k $ or inside $ \\mathcal F^P $ for some $P \\in \\mathcal P_k$. \n\t\n\tMoreover, notice that by induction hypothesis, elements of $ \\mathcal P_k $ are disjoint. Now, because every $A \\in \\mathcal F^P $ is entirely inside $ P $, we know that \n\t\\begin{equation} \\label{eq:FP-FQ-disjoint}\n\t\t\\text{if $ P \\neq Q $, then the elements of $ \\mathcal F^P $ are disjoint from the elements of $F^Q$.}\n\t\\end{equation}\n\t\n\tFurthermore, for every $ P \\in \\mathcal P_k$, the elements of $ \\mathcal F^P $ are all inside $ R_P$. Moreover, by definition of root, no element of $\\mathcal F_k $ intersect $ R_P $, so,\n\t\\begin{equation} \\label{eq:Fk-FP-disjoint}\n\t\t\\text{for every $ P\\in \\mathcal P $, the elements of $ \\mathcal F_k $ are disjoint from the elements of $ \\mathcal F^P $. }\n\t\\end{equation}\n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F_{k+1}$ satisfies \\emph{(C1).}}} \n\t\n\t\n\tLet $ A, B \\in \\mathcal F_{k+1} $ be two distinct elements such that $ A \\cap B \\neq \\varnothing $. By (\\ref{eq:FP-FQ-disjoint}) and (\\ref{eq:FP-FQ-disjoint}), either $ A, B \\in \\mathcal F_k $ or there exists $ P \\in \\mathcal P_k$ such that $ A, B \\in \\mathcal F^P $. In the former case, by induction hypothesis, we have $A \\curvearrowright B $ or $ B \\curvearrowright A $. In the latter case, by (\\ref{eq:FP-sat-C1-6}), we have $ A\\curvearrowright B $ or $ B \\curvearrowright A $.\n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F_{k+1}$ satisfies \\emph{(C2).}}} \n\t\n\tLet $ A, B \\in \\mathcal F_{k+1} $ such that $ A \\cap B = \\varnothing $ and $ A \\cap \\Ter{B} \\neq \\varnothing $. There are four cases possible:\n\t\n\tCase 1: $ A, B \\in \\mathcal F_k $, in which case the result follows from (\\ref{eq:FP-sat-C1-6}). \n\t\n\tCase 2: $ A \\in \\mathcal F_k $ and $ B \\in \\mathcal F^P $ for some $P \\in \\mathcal P_k $. \n\t\n\tThis case is not possible, because $ \\Ter{B} \\subseteq \\boxset{B} \\subseteq R_P $. However, $ A \\in \\mathcal F_k $, so $ A $ does not intersect $ R_P $ as it is a root of a prob for $ \\mathcal F_k $. \n\t\n\t\n\tCase 3: $ A \\in \\mathcal F^P $ for some $P \\in \\mathcal P_k $ and $ B \\in \\mathcal F_k $.\n\t\n\tSince $ A \\subseteq R_P $, we have $ R_P \\cap \\Ter{B} \\neq \\varnothing $. Let $ p = (x,y) \\in R_P \\cap \\Ter{B}$. By the definition of territory, there exists $ x' > x $ such that $ p\n\t=(x', y) \\in B $. Moreover, since $ R_P $ is a root of $ P $, we have $ p' \\in P $. So, $ p' \\in B \\cap P $. Therefore, $ B \\in N_{\\mathcal F_k}(P)$. Hence, by (\\ref{eq:P-stable-mut-disj}) and using Property~\\ref{prop:every-root-stable}, we have that every root of $ P $ is inside the territory of $ B$. Hence,\n\t$\n\tA \\subseteq R_P \\subseteq \\Ter{B}\n\t$. \n\tSo, the result holds. \n\t\n\tCase 4: $ A \\in \\mathcal F^P $ and $ B \\in \\mathcal F^Q $ for $ P, Q \\in \\mathcal P_k $. Let $ p \\in A \\cap \\Ter{B} $. So, in particular $ p \\in A \\subseteq P $ and $ p \\in \\Ter{B} \\subseteq \\boxset{F^Q} \\subseteq Q $. Therefore $ P\\cap Q \\neq \\varnothing $. Hence, by the induction hypothesis, we must have $P = Q $. So, the result follows from (\\ref{eq:FP-sat-C1-6}). \n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F_{k+1} $ satisfies \\emph{(C3).}}} \n\t\n\t\n\tLet $ A, B \\in \\mathcal F_{k+1} $ be two distinct sets such that $ A \\cap B \\neq \\varnothing $. For the sake of contradiction, assume that there exists $ C \\in \\mathcal F_{k+1} $ such that $ C \\subseteq \\Ter{A} \\cap \\Ter{B} $. \n\t\n\tFirst of all, by (\\ref{eq:FP-FQ-disjoint}) and (\\ref{eq:Fk-FP-disjoint}), there are only two possible cases for $ A $ and $ B $: either $ A, B \\in \\mathcal F_{k} $ or $ A, B \\in \\mathcal F^P $ for some $ P \\in \\mathcal P_k $. \n\t\n\tCase 1: $ A, B \\in \\mathcal F_{k} $. In this case, by induction hypothesis, we cannot have $ C \\in \\mathcal F_k $. So, $ C \\in \\mathcal F^P $ for some $ P \\in \\mathcal P_k $. Consequently, $ C \\subseteq R_P $.\n\tNow, let $ p = (x,y) \\in C $. Since $ C \\subseteq \\Ter{A} $, there exists $ x'>x $ such that $ p'=(x',y) \\in A $. But also, $ p' \\in P $. Therefore $ A \\in N_{\\mathcal F_{k}}(P)$. Similarly, we can show that $ B \\in N_{\\mathcal F_k}(P)$. \n\tA contradiction with the fact that the elements in $ N_{\\mathcal F_k}(P) $ are mutually disjoint. \n\t\n\tCase 2: $ A, B \\in \\mathcal F^P $ for some $ P \\in \\mathcal P_k $. Notice that \n\t$$ \\Ter{A} \\subseteq \\boxset{A} \\subseteq \\boxset{\\mathcal F^P} \\subseteq R_P. $$\n\tSo, $ C \\subseteq R_P $. Therefore $ C \\in \\mathcal F^ P$ as well, and the result follows from (\\ref{eq:FP-sat-C1-6}).\n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F_{k+1} $ satisfies \\emph{(C4).}}} \n\t\n\tAssume, for the sake of contradiction, that there exists $ A, B, C \\in \\mathcal F_{k+1} $ such that $ A\\prec B $, $ A \\curvearrowright C $, and $ B \\curvearrowright C $. By (\\ref{eq:FP-FQ-disjoint}) and (\\ref{eq:Fk-FP-disjoint}), since $ A \\cap C \\neq \\varnothing $ and $ B \\cap C \\neq \\varnothing $, either $ A, B, C \\in \\mathcal F_k $ or $ A, B, C \\in \\mathcal F^P $ for some $P \\in \\mathcal P_k $. The former is not possible because of induction hypothesis, and the latter because of (\\ref{eq:FP-sat-C1-6}). So, there exist no such triple. \n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F_{k+1} $ satisfies \\emph{(C5).}}} \n\t\n\tFor the sake of contradiction, assume that there exist three distinct set $ A, B, C \\in \\mathcal F_{k+1}$ that are mutually intersecting. By induction hypothesis, such triple does not exists in $ \\mathcal F_{k}$. So, at least on of the sets is in $ \\mathcal F^P $ for some $P \\in \\mathcal P_k $. But then, (\\ref{eq:FP-FQ-disjoint}) and (\\ref{eq:Fk-FP-disjoint}) imply that the three sets are all in $\\mathcal{F}^P$, a contradiction with (\\ref{eq:FP-sat-C1-6}).\n\t\n\t\n\t\n\t{\\noindent \\textbf{Claim}. \\textit{$ \\mathcal F_{k+1} $ satisfies \\emph{(C6).}}} \t\n\tBy assumptions, $ S $ is strong. Thus, we only need to show that every element of $ \\mathcal F_{k+1}$ is a positive transformed copy of $ S $. This is true since the elements of $ \\mathcal F_k $ are positive transformed copies of $ S $ and the elements of each $ \\mathcal F^P $ are also positive transformed copies of $ S$, because by (\\ref{eq:FP-sat-C1-6}), the collection $\\mathcal F^P $ satisfies (C6).\n\t\n\t\n\tThis finishes the proof of the statement. \n\t\n\tTo complete the proof of the theorem, it is enough to notice that by Statement 1, the graphs in the Burling sequence are all constrained $ S $-graphs, and that the class of constrained $ S$-graphs is closed under induced subgraph. \n\\end{proof}\n\n\\subsection{The equality of the three classes}\n\n\\begin{corollary} \\label{cor:all-equal}\n\tThe class of Burling graphs is equal to the class of constrained graphs and is equal to the class of constrained $ S$-graphs for every compact path-connected subset $ S $ of $ \\mathbb{R}^2 $ that is not an axis-aligned rectangle. \n\\end{corollary}\n\\begin{proof}\n\tFollows from Theorems~\\ref{thm:1-const-is-BG} and~\\ref{thm:BG-is-const-S-graph}.\n\\end{proof}\n\n\n\\section*{Acknowledgment}\n\nThe author thanks Paul Meunier for several useful discussions and his contributions to some proofs, in particular Lemmas~\\ref{prop:horizontal-and-vertical-crossing-intersect} and~\\ref{lem:subterritory-exists}. \nThe author thanks Nicolas Trotignon for many insightful discussions.\nThe author also thanks Gael Gallot for useful discussions, in particular during his internship on this topic.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Description of the Projet}\n\nGHaFaS is an improved version of the scanning Fabry-Perot system FaNTOmM\n\\citep{he2003}, which is a resident instrument on the\nObservatoire du mont M\\'egantic (OMM) 1,6m telescope and which has also\nbeen used on the CFH and ESO La Silla 3,6m telescopes. The complete \nsystem is composed of a focal reducer, a calibration unit, a filter wheel\nfor the order sorter filters, an FP etalon and an IPCS camera. The IPCS\nis composed of an Hamamatsu intensifier MCP tube which intensifies\nevery generated electron coming from the photocatode by a factor 10$^7$.\nEach photon event, recorded on a DALSA CCD, is then analysed by a centering \nalgorithm. With this amplification, the camera has essentially no readout noise.\nBecause of this, a zero noise IPCS is to be prefered to CCDs at very low \nflux level \\citep{ga2002}, even if the GaAs IPCS has only a DQE of 25\\%.\nMoreover, because of the fast scanning capability, it can average out the\nvariation of atmospheric transmission which is not possible with the long\nintegration times needed per channel for the CCDs in order to beat the\nread-out noise.\n\nIn the last 3 years with the FaNTOmM system, around 150 galaxies were\nobserved on the OMM, CFH and ESO La Silla telescopes in the context of\n3 large surveys: the SINGS sample \\citep{da2006}, a\nsurvey of barred galaxies, the BHabar sample \\citep{he2005} and\na sample of Virgo spirals \\citep{ch2006}. While the first scientific\njustification of the Montr\\'eal group was to derive high spatial resolution\noptical rotation curves for mass modeling purposes, the data was also used \nby IAC astronomers to constrain the role of gravitational perturbations\nas well as feedback from individual HII regions on the evolution of structures\nin galaxies and by a Berkeley-Munich group and the G\\'EPI group in Paris to\ncompare those local samples to high z galaxies.\n\nGHaFaS will come with its own custom designed focal reducer developed to be\noptically and mechanically compatible with the Nasmyth focus of the WHT.\nThe system has its own control and data acquisition system. It will have \na 4x4 arcmin field with a 0.45 arcsec pixel and $\\sim$5 km\/s velocity\nresolution. Full acquisition and reduction software (mainly based on IDL\nroutines) will be provided by the Montr\\'eal group.\nThe project will be done in 3 phases. For Phase I (July 2007), the optical\nsystem (focal reducer, filter wheel \\& calibration unit) will be delivered \nto the WHT and used with the camera FaNTOmM for this first run. For Phase II\n(end of 2007), an improved GaAs IPCS will be added to the system. Phase III \n(early 2008) will provide an FP controller and possibly a monochromator to\ncalibrate the data at the observing wavelength.\n\n\\section*{Science to be done with GHaFaS}\n\nTwo-dimensional kinematics is a very powerful technique for studying the \nstructure and evolution of galaxies. The distribution of dark matter,\ncircumnuclear star formation and fuelling of active galactic nuclei, detection\nof counter-rotating and kinematically decoupled components, and the effects\nof interaction between massive stars and the interstellar medium are\namong the physical phenomena which can be studied with this technique: see e.g.\n\\citet{fa2007} and \\citet{re2007}.\n\nThe first large program for which we plan to use GHaFaS consists at observing \na sample of 46 carefully selected nearby galaxies which are all included in the\nSINGS, THINGS, GALEX, and other CO and optical archives. Due to the angular\nsize of some of the objects, 2-4 fields may be necessary to reach the 25th \nmagnitude radius. This totals to 72 fields which will require $\\sim$18 clear\nnights of observing with GHaFaS on the WHT. Priority will be given to enlarge our Virgo\nsample of galaxies. The full sample ranges from elliptical (with emission)\nto irregular galaxies, 2\/3 of which are intermediate-type objects, since\nthis is where highly star-forming regions will be observed. It will be\npossible to use this sample for many scientific projects ranging from the \nlarge scale mass modeling using the velocity fields in order to derive the dark \nmatter density profiles to the study of the internal kinematics of the\nindividual HII regions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section*{Acknowledgements}\nWe thank Andrew Ilyas and Sam Park for helpful discussions.\n\nWork supported in part by the NSF grants CCF-1553428, CNS-1815221, the Google\nPhD Fellowship, and the Microsoft Corporation. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001120C0015.\n\nResearch was sponsored by the United States Air Force Research Laboratory and\nwas accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views\nand conclusions contained in this document are those of the authors and should\nnot be interpreted as representing the official policies, either expressed or\nimplied, of the United States Air Force or the U.S. Government. The U.S.\nGovernment is authorized to reproduce and distribute reprints for Government\npurposes notwithstanding any copyright notation herein.\n\n\\section{Additional Experimental Results}\n\\label{app:res}\n\n\\subsection{Human Baselines for \\textsc{Breeds}{} Tasks}\n\\label{app:res_human}\nIn Section~\\ref{sec:humans}, we evaluate human performance on binary versions of \nour \\textsc{Breeds}{} tasks. Appendix Figures~\\ref{fig:human_pairwise_s} \nand~\\ref{fig:human_pairwise_t} show the distribution of annotator accuracy over \ndifferent pairs of superclasses for test data sampled from the source and target \ndomains respectively.\n\n\\begin{figure}[!h]\n\t\n\t\\begin{subfigure}{1\\textwidth}\n\t\t\\centering\n\t\t\t\\includegraphics[width=0.82\\textwidth]{Figures\/eval\/acc_comparison_pairwise_app_S.pdf}\n\t\t\t\\caption{Source domain (no subpopulation shift)}\n\t\t\t\\label{fig:human_pairwise_s}\n\t\\end{subfigure}\n\\begin{subfigure}{1\\textwidth}\n\t\\centering\n\t\\includegraphics[width=0.82\\textwidth]{Figures\/eval\/acc_comparison_pairwise_app_T.pdf}\n\t\\caption{Target domain (with subpopulation shift)}\n\t\\label{fig:human_pairwise_t}\n\\end{subfigure}\n\t\\caption{Distribution of annotator accuracy over pairwise superclass classification \n\t\ttasks. We observe that human annotators consistently perform \n\t\tbetter \n\t\ton tasks \n\t\tconstructed using our modified ImageNet class \n\t\thierarchy (i.e., \\textsc{Breeds}{}) as opposed to those obtained directly from \n\t\tWordNet.}\n\n\\end{figure}\n\n\\clearpage\n\n\\subsection{Model Evaluation}\n\\label{app:res_eval}\nIn Figures~\\ref{fig:perclass_a}-~\\ref{fig:perclass_d121}, we visualize model \nperformance over \\textsc{Breeds}{} superclasses for different model architectures. We observe in \ngeneral that models perform fairly uniformly over classes when the test data is drawn \nfrom the source domain. This indicates that the tasks are well-calibrated---the \nvarious superclasses are of comparable difficulty. At the same time, we see that \nmodel robustness to subpopulation shift, i.e., drop in accuracy on the target domain, \nvaries widely over superclasses. This could be either due to some superclasses\nbeing broader by construction or due to models being more sensitive to\nsubpopulation shift for some classes.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Figures\/eval\/perclass_alexnet.pdf}\n\t\\caption{Per-class source and target accuracies for AlexNet on \\textsc{Breeds}{}\n tasks.}\n\t\\label{fig:perclass_a}\n\\end{figure}\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Figures\/eval\/perclass_resnet50.pdf}\n\t\\caption{Per-class source and target accuracies for ResNet-50 on \\textsc{Breeds}{}\n tasks.}\n\t\\label{fig:perclass_r50}\n\\end{figure}\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Figures\/eval\/perclass_densenet121.pdf}\n\t\\caption{Per-class source and target accuracies for DenseNet-121 on \\textsc{Breeds}{}\n tasks.}\n\t\\label{fig:perclass_d121}\n\\end{figure}\n\n\\clearpage\n\\subsubsection{Effect of different splits}\n\\label{app:res_splits}\n\\label{app:goodbad}\n\nAs described in Section~\\ref{sec:breeds}, to create \\textsc{Breeds}{} tasks, we first identify \na set of relevant superclasses (at the chosen depth in the hierarchy), and then \npartition their subpopulations between the source and target domains. For all the \ntasks listed in Table~\\ref{tab:benchmarks}, the superclasses are balanced---each of \nthem comprise the same number of subpopulations. To ensure this is the case, the \ndesired number of subpopulations is chosen among all superclass subpopulations at \nrandom. These subpopulations are then randomly split between the source and target \ndomains.\n\nInstead of randomly partitioning subpopultions (of a given superclass) between the \ntwo domains, we could instead craft partitions to be more\/less adversarial as \nillustrated in Figure~\\ref{fig:splits_diag}. Specifically, we could control how similar \nthe subpopulations in the target domain are to those in the source domain. For \ninstance, a split would be less adversarial (\\emph{good}) if subpopulations in the \nsource and target domain share a common parent. On the other hand, we could make \na split more adversarial (\\emph{bad}) by ensuring a greater degree of separation (in \nterms of distance in the hierarchy) between the source and target domain \nsubpopulations.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{Figures\/gen\/splits.pdf}\n\t\\caption{Different ways to partition the subpopulations of a given superclass into \n\t\tthe source and target domains. Depending on how closely related the \n\t\tsubpopulations in the two domain are, we can construct splits that are more\/less \n\t\tadversarial.}\n\t\\label{fig:splits_diag}\n\\end{figure}\n\n We now evaluate model performance under such variations in the nature of the splits \n themselves---see Figure~\\ref{fig:all_splits}. \nAs expected, models perform comparably well on test data from the source domain, \nindependent of the how the subpopulations are partitioned into the two domains. \nHowever, model robustness to subpopulation \nshift varies considerably based on the nature of the split---it is lowest for the most \nadversarially chosen split. \nFinally, we observe that retraining the linear layer \non data from the target domain recovers a considerable fraction of the accuracy drop \nin all cases---indicating that even for the more adversarial splits, models do learn \nfeatures that transfer well to unknown subpopulations. \n\n\\begin{figure}[!h]\n\t\\begin{subfigure}{1.0\\textwidth}\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Figures\/eval\/ENTITY-13_splits.pdf}\n\t\\caption{\\textsc{Entity-13}{} task}\n\t\\label{fig:all3_splits}\n\t\\end{subfigure}\n\t\\begin{subfigure}{1.0\\textwidth}\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Figures\/eval\/ENTITY-30_splits.pdf}\n\t\\caption{\\textsc{Entity-30}{} task}\n\t\\label{fig:all4_splits}\n\t\\end{subfigure}\n\t\\caption{Model robustness as a function of the nature of subpopulation shift within \n\tspecific \\textsc{Breeds}{} tasks. We vary how the underlying \n\tsubpopulations of each superclass are split between the source and target \n\tdomain---we \n\tcompare random splits (used in the majority of our analysis), to ones that are more \n\t(\\emph{bad})\n\tor less adversarial (\\emph{good}).\n\tWhen models are tested on samples from the source domain, they perform equally \n\twell across different splits, as one might expect.\n\tHowever, under subpopulation shift (i.e., on samples from the target domain), \n\tmodel robustness varies drastically, and is considerably worse when the split is \n\tmore adversarial.\n\tYet, for all the splits, models have comparable target accuracy \n after retraining their final layer.\n}\n\\label{fig:all_splits}\n\\end{figure}\n\n\n\n\\clearpage\n\\subsubsection{Robustness Interventions}\n\\label{app:res_int}\nIn Tables~\\ref{tab:adv_app} and~\\ref{tab:other_rob_app}, we present the raw \naccuracies of models trained using various train-time robustness interventions.\n\\begin{table}[!h]\n \\setlength{\\tabcolsep}{1.5em}\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.05}\n\t\\begin{tabular}{llccc}\n\t\t\\toprule\n\t\t\\multicolumn{5}{c}{ResNet-18} \\\\\n\t\t\\midrule\n \\multirow{2}{*}{Task} & \\multirow{2}{*}{$\\varepsilon$} & \\multicolumn{3}{c}{Accuracy (\\%)} \\\\\n & & Source & Target & Target-RT \\\\\n\t\t\\midrule\n \\multirow{3}{*}{\\textsc{Entity-13}} & 0 & \\textbf{90.91 $\\pm$ 0.73} &\n \\textbf{61.52 $\\pm$ 1.23} & \\textbf{76.71 $\\pm$ 1.09} \\\\\n & 0.5 & 89.23 $\\pm$ 0.80 & \\textbf{61.10 $\\pm$ 1.23} & 74.92 $\\pm$ 1.04 \\\\\n & 1.0 & 88.45 $\\pm$ 0.81 & 58.53 $\\pm$ 1.26 & 73.35 $\\pm$ 1.11 \\\\\n\t\t\\midrule\n \\multirow{3}{*}{\\textsc{Entity-30}} & 0 & \\textbf{87.88 $\\pm$ 0.89} &\n \\textbf{49.96 $\\pm$ 1.31} & \\textbf{73.05 $\\pm$ 1.17}\n \\\\\n & 0.5 & 85.68 $\\pm$ 0.91 & \\textbf{48.93 $\\pm$ 1.34} & 71.34 $\\pm$ 1.14 \n\\\\\n & 1.0 & 84.23 $\\pm$ 0.91 & 47.66 $\\pm$ 1.23 & 70.27 $\\pm$ 1.17 \\\\\n\t\t\\midrule\n \\multirow{3}{*}{\\textsc{Living-17}} & 0 & \\textbf{92.01 $\\pm$ 1.30} & \n \\textbf{58.21 $\\pm$ 2.32} & \\textbf{83.38 $\\pm$ 1.79} \\\\\n & 0.5 & 90.35 $\\pm$ 1.35 & 55.79 $\\pm$ 2.44 & \\textbf{83.00 $\\pm$ 1.89} \\\\\n & 1.0 & 88.56 $\\pm$ 1.50 & 53.89 $\\pm$ 2.36 & 80.90 $\\pm$ 1.92 \\\\\n\t\t\\midrule\n \\multirow{3}{*}{\\textsc{Non-living-26}} & 0 & \\textbf{88.09 $\\pm$ 1.28} & \n \\textbf{41.87 $\\pm$ 2.01} & \\textbf{73.52 $\\pm$ 1.71} \\\\\n & 0.5 & 86.28 $\\pm$ 1.32 & \\textbf{41.02 $\\pm$ 1.91} &\n \\textbf{72.41 $\\pm$ 1.71} \\\\\n & 1.0 & 85.19 $\\pm$ 1.38 & \\textbf{40.23 $\\pm$ 1.92} & 70.61 $\\pm$ \n1.73 \\\\\n\t\t\\bottomrule \n\t\\end{tabular}\n\t\\begin{tabular}{llccc}\n\t\t\\multicolumn{5}{c}{} \\\\\n\t\\toprule\n\t\t\\multicolumn{5}{c}{ResNet-50} \\\\\n\t\t\\midrule\n \\multirow{2}{*}{Task} & \\multirow{2}{*}{$\\varepsilon$} & \\multicolumn{3}{c}{Accuracy (\\%)} \\\\\n & & Source & Target & Target-RT \\\\\n\t\\midrule\n \\multirow{3}{*}{\\textsc{Entity-13}} & 0 & \\textbf{91.54 $\\pm$ 0.64} &\n \\textbf{62.48 $\\pm$ 1.16} & \\textbf{79.32 $\\pm$ 1.01 } \\\\\n & 0.5 & 89.87 $\\pm$ 0.80 & \\textbf{63.01 $\\pm$ 1.15} & \n \\textbf{80.14 $\\pm$ 1.00} \\\\\n & 1.0 & 89.71 $\\pm$ 0.74 & 61.21 $\\pm$ 1.22 & 78.58 $\\pm$ 0.98 \\\\\n\t\\midrule\n \\multirow{3}{*}{\\textsc{Entity-30}} & 0 & \\textbf{89.26 $\\pm$ 0.78} &\n \\textbf{51.18 $\\pm$ 1.24} & 77.60 $\\pm$ 1.17 \\\\\n\n & 0.5 & 87.51 $\\pm$ 0.88 & \\textbf{50.72 $\\pm$ 1.28} & \n \\textbf{78.92 $\\pm$ 1.06} \\\\\n & 1.0 & 86.63 $\\pm$ 0.88 & \\textbf{50.99 $\\pm$ 1.27} & \n \\textbf{78.63 $\\pm$ 1.03} \\\\\n\t\\midrule\n \\multirow{3}{*}{\\textsc{Living-17}} & 0 & \\textbf{92.40 $\\pm$ 1.28} &\n \\textbf{58.22 $\\pm$ 2.42} & \\textbf{85.96 $\\pm$ 1.72} \\\\\n & 0.5 & 90.79 $\\pm$ 1.55 & \\textbf{55.97 $\\pm$ 2.38} & \n \\textbf{87.22 $\\pm$ 1.66} \\\\\n & 1.0 & 89.64 $\\pm$ 1.47 & 54.64 $\\pm$ 2.48 & \\textbf{85.63 $\\pm$ 1.73} \\\\\n\t\\midrule\n \\multirow{3}{*}{\\textsc{Non-living-26}} & 0 & \\textbf{88.13 $\\pm$ 1.30} & \n \\textbf{41.82 $\\pm$ 1.86} & 76.58 $\\pm$ 1.69 \\\\\n & 0.5 & \\textbf{88.20 $\\pm$ 1.20} & \\textbf{42.57 $\\pm$ 2.03} &\n \\textbf{78.84 $\\pm$ 1.62} \\\\\n & 1.0 & 86.17 $\\pm$ 1.36 & \\textbf{41.69 $\\pm$ 1.96} & 76.16 $\\pm$ \n1.61 \\\\\n\t\\bottomrule\n\\end{tabular}\n\\vspace{1em}\n\t\\caption{Effect of adversarial training on model robustness to subpopulation \n\tshift. All models are trained on samples from the source domain---either \n\tusing standard \n\ttraining ($\\varepsilon=0.0$) or using adversarial training. Models are then \n\tevaluated in terms of: (a) source accuracy, (b) target accuracy and (c) target \n\taccuracy after retraining the linear layer of the model with data from the \n target domain. Confidence intervals (95\\%) obtained via bootstrapping. Maximum\n task accuracy over $\\varepsilon$ (taking into account confidence interval) shown in bold.}\n\t\\label{tab:adv_app}\n\\end{table}\n\n\n\\begin{table}[!h]\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.05}\n\t\\begin{tabular}{llccc}\n\t\t\\multicolumn{5}{c}{} \\\\\n\t\\toprule\n\t\t\\multicolumn{5}{c}{ResNet-18} \\\\\n\t\t\\midrule\n \\multirow{2}{*}{Task} & \\multirow{2}{*}{Intervention} & \\multicolumn{3}{c}{Accuracy (\\%)} \\\\\n & & Source & Target & Target-RT \\\\\n\t\t\\midrule\n \\multirow{4}{*}{\\textsc{Entity-13}} & Standard & \n \\textbf{90.91 $\\pm$ 0.73} & \\textbf{61.52 $\\pm$ 1.23 } &\n 76.71 $\\pm$ 1.09 \\\\\n & Erase Noise& \\textbf{91.01 $\\pm$ 0.68} & \\textbf{62.79 $\\pm$ 1.27}\n & \\textbf{78.10 $\\pm$ 1.09} \\\\\n & Gaussian Noise & 77.00 $\\pm$ 1.04 & 47.90 $\\pm$ 1.21 & 70.37 \n$\\pm$ 1.17 \\\\\n& Stylized ImageNet & 76.85 $\\pm$ 1.00 & 50.18 $\\pm$ 1.21 & 65.91 \n$\\pm$ \n1.17 \\\\\n\t\\midrule\n \\multirow{4}{*}{\\textsc{Entity-30}} & Standard & \\textbf{87.88 $\\pm$ 0.89} & \n \\textbf{49.96 $\\pm$ 1.31 } & 73.05 $\\pm$ 1.17 \\\\\n & Erase Noise & \\textbf{88.09 $\\pm$ 0.80} & \\textbf{49.98 $\\pm$ 1.31}\n & \\textbf{74.27 $\\pm$ 1.15} \\\\\n& Gaussian Noise& 74.12 $\\pm$ 1.16 & 35.79 $\\pm$ 1.21 & 65.62 \n$\\pm$ 1.28 \\\\\n & Stylized ImageNet & 70.96 $\\pm$ 1.16 & 37.67 $\\pm$ 1.21 & 60.45 \n$\\pm$ 1.22 \\\\\n\t\\midrule\n \\multirow{4}{*}{\\textsc{Living-17}} & Standard & \\textbf{92.01 $\\pm$ 1.30} &\n \\textbf{58.21 $\\pm$ 2.32} & \\textbf{83.38 $\\pm$ 1.79} \\\\\n & Erase Noise & \\textbf{93.09 $\\pm$ 1.27} & \\textbf{59.60 $\\pm$ 2.40}\n & \\textbf{85.12 $\\pm$ 1.71} \\\\\n& Gaussian Noise & 80.13 $\\pm$ 1.99 & 46.16 $\\pm$ 2.57 & 77.31 \n$\\pm$ \n2.08 \\\\\n & Stylized ImageNet & 79.21 $\\pm$ 1.85 & 43.96 $\\pm$ 2.38 & 72.74 \n$\\pm$ \n2.09 \\\\\n\t\\midrule\n\\multirow{4}{*}{\\textsc{Non-living-26}} & \n Standard & \\textbf{88.09 $\\pm$ 1.28} & \\textbf{41.87 $\\pm$ 2.01 }\n & \\textbf{73.52 $\\pm$ 1.71} \\\\\n & Erase Noise & \\textbf{88.68 $\\pm$ 1.18} & \\textbf{43.17 $\\pm$ 2.10}\n & \\textbf{73.91 $\\pm$ 1.78} \\\\\n& Gaussian Noise & 78.14 $\\pm$ 1.60 & 35.13 $\\pm$ 1.94 & 67.79 \n$\\pm$ 1.79 \\\\\n & Stylized ImageNet & 71.43 $\\pm$ 1.73 & 30.56 $\\pm$ 1.75 & 61.83 \n$\\pm$ 1.98 \\\\\n\t\t\\bottomrule \n\t\\end{tabular}\n\t\\begin{tabular}{llccc}\n\t\t\\multicolumn{5}{c}{} \\\\\n\t\\toprule\n\t\t\\multicolumn{5}{c}{ResNet-34} \\\\\n\t\t\\midrule\n \\multirow{2}{*}{Task} & \\multirow{2}{*}{Intervention} & \\multicolumn{3}{c}{Accuracy (\\%)} \\\\\n & & Source & Target & Target-RT \\\\\n\t\t\\midrule\n \\multirow{4}{*}{\\textsc{Entity-13}} & Standard & \\textbf{91.75 $\\pm$ 0.70}\n & \\textbf{ 63.45 $\\pm$ 1.13 } & \\textbf{ 78.07 $\\pm$ 1.02} \\\\\n & Erase Noise & \\textbf{91.76 $\\pm$ 0.70}\n & \\textbf{62.71 $\\pm$ 1.25} & \\textbf{77.43 $\\pm$ 1.06} \\\\\n & Gaussian Noise & 81.60 $\\pm$ 0.97 & 50.69 $\\pm$ 1.28 & 71.50 \n$\\pm$ 1.13 \\\\\n& Stylized ImageNet & 78.66 $\\pm$ 0.94 & 51.05 $\\pm$ 1.30 & 67.38 \n$\\pm$ 1.16 \\\\\n\\midrule\n \\multirow{4}{*}{\\textsc{Entity-30}} & Standard & \\textbf{88.81 $\\pm$ 0.81} & \n \\textbf{51.68 $\\pm$ 1.28 } & \\textbf{75.12 $\\pm$ 1.11} \\\\\n & Erase Noise & \\textbf{89.07 $\\pm$ 0.82} & \\textbf{51.04 $\\pm$\n 1.27} & \\textbf{74.88 $\\pm$ 1.08} \\\\\n& Gaussian Noise & 75.05 $\\pm$ 1.11 & 38.31 $\\pm$ 1.26 & 67.47 \n$\\pm$ 1.22 \\\\\n & Stylized ImageNet & 72.51 $\\pm$ 1.10 & 38.98 $\\pm$ 1.22 & 61.65 \n$\\pm$ 1.25 \\\\\n\\midrule\n \\multirow{4}{*}{\\textsc{Living-17}} & Standard & \\textbf{92.83 $\\pm$ 1.19}\n & 59.74 $\\pm$ 2.27 & \\textbf{85.46 $\\pm$ 1.83} \\\\\n & Erase Noise & \\textbf{92.96 $\\pm$ 1.32} & \\textbf{61.13 $\\pm$\n 2.30} & \\textbf{85.66 $\\pm$ 1.78} \\\\\n& Gaussian Noise & 84.06 $\\pm$ 1.71 & 48.38 $\\pm$ 2.44 & 78.79 \n$\\pm$ 1.91 \\\\\n & Stylized ImageNet & 80.94 $\\pm$ 2.00 & 44.16 $\\pm$ 2.43 & 72.77 \n$\\pm$ 2.18 \\\\\n\\midrule\n \\multirow{4}{*}{\\textsc{Non-living-26}} & Standard & \\textbf{89.64 $\\pm$ 1.17}\n & \\textbf{43.03 $\\pm$ 1.99 } & \\textbf{74.99 $\\pm$ 1.66} \\\\\n & Erase Noise & \\textbf{89.62 $\\pm$ 1.31} & \\textbf{43.53 $\\pm$ \n 1.89} & \\textbf{75.04 \n $\\pm$ 1.70} \\\\\n & Gaussian Noise & 79.26 $\\pm$ 1.61 & 34.89 $\\pm$ 1.91 & 68.07 \n $\\pm$ 1.78\n \\\\\n & Stylized ImageNet& 71.49 $\\pm$ 1.65 & 31.10 $\\pm$ 1.80 & 62.94 \n$\\pm$ 1.90 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\vspace{1em}\n\t\\caption{Effect of various train-time interventions on model robustness to \n\tsubpopulation \n\tshift. All models are trained on samples from the the source domain. Models \n\tare then \n\tevaluated in terms of: (a) source accuracy, (b) target accuracy and (c) target \n\taccuracy after retraining the linear layer of the model with data from the \n target domain. Confidence intervals (95\\%) obtained via bootstrapping. Maximum\n task accuracy over $\\varepsilon$ (taking into account confidence interval) shown in bold.}\n\t\\label{tab:other_rob_app}\n\\end{table}\n\n\\section{The \\textsc{Breeds}{} Methodology}\n\\label{sec:breeds}\nIn this work, we focus on modeling a pertinent, yet relatively less studied,\nform of subpopulation shift: one wherein the target distribution (used for \ntesting) contains\nsubpopulations that are \\emph{entirely} absent from the source distribution \nthat the model was trained on.\nTo simulate such a shift, one needs to precisely control the data\nsubpopulations that comprise the source and target data distributions.\nOur procedure for doing this comprises two stages that are outlined\nbelow---see Figure~\\ref{fig:breeds} for an illustration.\n\n\\paragraph{Devising subpopulation structure.}\nTypical datasets do not contain \nannotations for individual subpopulations.\nSince collecting such annotations would be challenging, we take an alternative\napproach: we bootstrap the existing dataset labels to simulate \nsubpopulations.\nThat is, we group semantically similar classes into broader\nsuperclasses which, in turn, allows us to re-purpose existing class labels as \nthe desired \nsubpopulation annotations.\nIn fact, we can group classes in a hierarchical manner, obtaining superclasses\nof different specificity.\nAs we will see in Section~\\ref{sec:hierarchy}, large-scale benchmarks often\nprovide class hierarchies~\\citep{everingham2010pascal,deng2009imagenet,\nkuznetsova2018open} that aid such semantic grouping.\n\n\\paragraph{Simulating subpopulation shifts.}\nGiven a set of superclasses, we can define a classification task over them:\nthe inputs of each superclass correspond to pooling together the inputs\nof its subclasses (i.e., the original dataset classes).\nWithin this setup, we can simulate subpopulation shift \nin a relatively straightforward manner.\nSpecifically, for each superclass, we split its subclasses into two\n\\emph{random} and \\emph{disjoint} sets, and assign one of them to the source\nand the other to the target domain.\nThen, we can evaluate model robustness under subpopulation shift \nby simply training on the source domain and testing on the target domain.\nNote that the classification task remains identical between\ndomains---both domains contain the same (super)classes but the subpopulations\nthat comprise each (super)class differ.\n\\footnote{Note that this methodology can be extended to simulate milder\nsubpopulation shifts where the source and target distributions overlap but the\nrelative subpopulation frequencies vary, similar to the setting of\n\\citet{oren2019distributionally}.}\nIntuitively, this corresponds to using different dog breeds to represent the\nclass ``dog'' during training and testing---hence the name of our toolkit. \n\\newline\n\n\\begin{figure}[!t]\n\t\\centering\n \\includegraphics[width=0.9\\textwidth]{Figures\/gen\/pipeline.pdf}\n \\caption{Illustration of our pipeline to create subpopulation shift\n benchmarks. Given a dataset, we first define superclasses by \n grouping semantically similar classes together to form a hierarchy. This allows \n us to treat the dataset labels as subpopulation annotations. Then, we \n construct a \\textsc{Breeds}{} task of specified \n granularity (i.e., depth in the hierarchy) by posing the \n classification task \n in terms of superclasses at that depth and then partitioning their \n respective \n subpopulations into \n the source and target domains.\n \t}\n \\label{fig:breeds}\n\\end{figure}\n\n\\noindent\nThis methodology is quite general and can be applied to a variety of\nsetting to simulate realistic distribution shifts. \nMoreover, it has a number of additional benefits:\n\\begin{itemize}\n \\item \\textbf{Flexibility:} Different semantic groupings of a fixed set of\n classes lead to \\textsc{Breeds}{} tasks of varying granularity.\n For instance, by only grouping together classes that are quite similar one\n can reduce the severity of the subpopulation shift.\n Alternatively, one can consider broad superclasses, each having multiple\n subclasses, resulting in a more challenging benchmark.\n \\item \\textbf{Precise characterization:} The exact subpopulation shift\n between the source and target distribution is known.\n Since both domains are constructed from the same dataset, the impact\n of any external factors (e.g., differences in data collection pipelines) is\n minimized. (Note that such external factors can significantly impact the\n difficulty of the task~\\cite{ponce2006dataset,\n torralba2011unbiased,engstrom2020identifying,tsipras2020from}.)\n \\item \\textbf{Symmetry:} Since subpopulations are split into the\n source and test domains randomly, we expect the resulting tasks to have\n comparable difficulty.\n \\item \\textbf{Reuse of existing datasets:} No additional data collection or\n annotation is required other than choosing the class grouping.\n This approach can thus be used to also re-purpose other existing\n large-scale datasets---even outside the image recognition context---with\n minimal effort and cost.\n\\end{itemize}\n\n\\section{Conclusion}\nIn this work, we develop a methodology for constructing large-scale \nsubpopulation shift benchmarks.\nThe motivation behind our \\textsc{Breeds}{} benchmarks is to\ntest if models can generalize beyond the limited diversity\nof their training datasets---specifically, to novel data subpopulations.\nA major advantage of our approach is its generality.\nIt can be applied to any dataset with a \nmeaningful class structure---including tasks beyond classification \n(e.g., object detection) and domains other than computer vision \n(e.g., natural language processing).\nMoreover, the subpopulation shifts are induced in a manner that is both\ncontrolled and natural, without altering inputs synthetically or needing to \ncollect new data.\n\nWe apply this approach to the ImageNet dataset to construct\nbenchmarks of varying difficulty.\nWe then demonstrate how these benchmarks can be \nused to assess model robustness and the efficacy of various train-time \ninterventions.\nFurther, we obtain human baselines for these tasks to both put\nmodel performance in context and validate that the corresponding \nsubpopulation\nshifts do not significantly affect humans.\n\n\nOverall, our results indicate that existing models still have a long way to go\nbefore they can fully tackle the BREEDS subpopulation shifts, even with \nrobustness interventions.\nWe thus believe that our methodology provides a useful framework for studying\nmodel robustness to distribution shift---an increasingly pertinent topic for\nreal-world deployments of machine learning models.\n\n\\section{Evaluating Model Performance under Subpopulation Shift}\n\\label{sec:eval}\nWe can now use our suite of \\textsc{Breeds}{} tasks as a testbed for assessing model\nrobustness to subpopulation shift as well as gauging the effectiveness of \nvarious\ntrain-time robustness interventions. \nSpecifics of the evaluation setup and additional experimental results \nare provided in Appendices~\\ref{app:eval_setup}\nand~\\ref{app:res_eval}.\n\n\\subsection{Standard training}\nWe start by evaluating the performance of various model architectures trained\nin the standard fashion: empirical risk minimization (ERM) on the source\ndistribution (cf. Appendix~\\ref{app:models}).\nWhile these models perform well on unseen inputs from the domain they are\ntrained on, i.e., they achieve high \\emph{source accuracy}, they suffer a \nconsiderable\ndrop in accuracy under these subpopulation shifts---more than 30\\% in most cases\n(cf. Figure~\\ref{fig:core}).\nAt the same time, models that are more \\emph{accurate} on the \nsource domain also appear to be more \\emph{robust} to distribution shift.\nSpecifically, the fraction of source accuracy that is preserved in the target\ndomain is typically increasing with source accuracy. \n(If this were not the case, i.e., the accuracy of all models dropped by a \nconstant fraction under distribution shift, the target accuracy would match \nthe baseline in Figure~\\ref{fig:core}.)\nThis indicates that, while models are quite brittle to subpopulation shift,\nimprovements in source accuracy \\emph{do} correlate with models \ngeneralizing better to variations in testing conditions.\nNote that model accuracies are not directly comparable across benchmarks, \ndue to the presence of multiple conflating factors.\nOn one hand, more fine-grained tasks present a smaller subpopulation\nshift (subclasses are semantically closer).\nOn the other hand, the number of classes and training inputs\nper class changes significantly, making the task harder.\n\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figures\/eval\/drop_standard.pdf}\n\t\\caption{Robustness of standard models to \\textsc{Breeds}{} \n\t\tsubpopulation shifts. \n\t\tFor each of the four tasks, we plot the accuracy of different (source \n domain-trained) model architectures (denoted by different symbols) on\n the target domain as a function of the source accuracy (which is\n typically high).\n\t\tWe find that model accuracy drops significantly between domains\n\t\t(\\emph{orange} vs.\\ \\emph{dashed} line). Still, models\n\t\tthat are more accurate on the source domain seem to also be more \n\t\trobust\n\t\t(the improvements exceed the baseline \n\t\t(\\emph{grey}) which would correspond to a constant accuracy drop\n across models, i.e., $\\frac{source \\ acc}{target \\ acc}$ = \n\t\tconstant based on AlexNet).\n\t\tMoreover, the drop in model performance on the target domain can \n\t\tbe reduced by retraining\n\t\tthe final model layer with data from that domain (\\emph{green}). However, \n a non-trivial drop persists compared to both the original source\n accuracy, and target accuracy of models trained directly (end-to-end)\n on the target domain (\\emph{blue}).\n\t}\n\t\\label{fig:core}\n\\end{figure}\n\n\\paragraph{Models vs.\\ Humans.} We compare \nthe best performing model (DenseNet-121 in this case) to our previously \nobtained human baselines in Figure~\\ref{fig:acc_human}. To allow for a fair \ncomparison, model accuracy is measured on pairwise superclass \nclassification tasks (cf. Appendix~\\ref{app:eval_setup}). We observe \nthat models do exceedingly well on unseen samples from the source \ndomain---significantly \noutperforming annotators under our task setup. At the same time, models \nalso appear to be more brittle, performing worse than humans on the target\ndomain of these binary \\textsc{Breeds}{} tasks, despite their higher source accuracy.\n\n\n\\paragraph{Adapting models to the target domain.}\nFinally, we focus on the intermediate data representations learned by these \nmodels, aiming to assess how suitable they are\nfor distinguishing classes in the target domain.\nTo assess this, we retrain the last (fully-connected) layer of models\ntrained on the source domain with data from the target domain. \nWe find that the target accuracy of these models increases significantly after\nretraining, indicating that the learned representations indeed generalize to\nthe target domain.\nHowever, we cannot match the accuracy of models trained directly (end-to-end)\non the target domain---see Figure~\\ref{fig:core}---demonstrating that there is\nsignificant room for improvement.\n\n\n\\input{intervene}\n\n\n\\section{Simulating Subpopulation Shifts Within ImageNet}\n\\label{sec:hierarchy}\nWe now describe in more detail how our methodology can be applied to\nImageNet~\\citep{deng2009imagenet}---specifically, the ILSVRC2012\nsubset~\\citep{russakovsky2015imagenet}---to create a suite of \\textsc{Breeds}{}\nbenchmarks.\nImageNet contains a large number of classes, making it particularly\nwell-suited for our purpose.\n\n\\subsection{Utilizing the ImageNet class hierarchy}\nRecall that creating \\textsc{Breeds}{} tasks requires grouping together \nsimilar classes.\nIn the context of ImageNet, such a semantic grouping already \nexists---ImageNet classes\nare a part of the WordNet hierarchy~\\citep{miller1995wordnet}.\nHowever, WordNet is not a hierarchy of objects but rather one of \nword\nmeanings.\nTherefore, intermediate hierarchy nodes are not always well-suited\nfor object recognition due to:\n\n\\begin{itemize}\n \\item \\textbf{Abstract groupings:} WordNet nodes often correspond to\n abstract concepts, e.g., related to the functionality of an object.\n Children of such nodes might thus share little visual\n similarity---e.g., ``umbrella'' and ``roof'' are visually different,\n despite both being ``coverings''.\n \\item \\textbf{Non-uniform categorization:} The granularity of object\n categorization is vastly different across the WordNet hierarchy---e.g.,\n the subtree rooted at ``dog'' is 25-times larger than the one rooted at \n ``cat''. \n Hence, the depth of a node in this hierarchy does not always reflect\n the specificity of the corresponding object category.\n \\item \\textbf{Lack of tree structure:} Nodes in WordNet can have\n multiple parents\\footnote{In programming languages, this is known as ``the\n diamond problem'' or ``the Deadly Diamond of\n Death''~\\citep{martin1997java}.} and thus the resulting classification\n task would contain overlapping classes, making it inherently ambiguous.\n\\end{itemize}\n\n\\noindent\nDue to these issues, we cannot directly use WordNet to identify \nsuperclasses that correspond to a well-calibrated classification task.\nTo illustrate this, we present some of the superclasses\nconstructed by applying clustering algorithms directly to the WordNet \nhierarchy~\\cite{huh2016makes} in\nAppendix Table~\\ref{tab:problems}.\nEven putting the issue of overlapping classes aside, a \\textsc{Breeds}{} task based on\nthese superclasses would induce a very skewed subpopulation shift across\nclasses---e.g., varying the types of ``bread'' is very different that\ndoing the same for different ``mammal'' species.\n\n\\paragraph{Calibrating WordNet for Visual Object Recognition.}\nTo better align the WordNet hierarchy with the task of object\nrecognition in general, and \\textsc{Breeds}{} benchmarks in particular, we manually \nmodify it \naccording to the following two principles.\nFirst, nodes should be grouped together based on their visual characteristics,\nrather than abstract relationships like functionality---e.g., we eliminate\nnodes that do not convey visual information such as ``covering''.\nSecond, nodes of similar specificity should be at the same distance from the\nroot, irrespective of how detailed their categorization within WordNet is---for\ninstance, we placed ``dog'' at the same level as ``cat'' and ``flower'', even\nthough the ``dog'' sub-tree in WordNet is much larger.\nFinally, we removed a number of ImageNet classes that did not naturally fit into\nthe hierarchy.\nThe resulting hierarchy, presented in Appendix~\\ref{app:manual}, contains \nnodes of comparable granularity at the same level.\nMoreover, as a result of this process, each node ends up having a single \nparent\nand thus the resulting hierarchy is a tree.\n\n\\subsection{Creating \\textsc{Breeds}{} tasks}\n\\label{sec:tasks}\nOnce the modified version of the WordNet hierarchy is in place, \\textsc{Breeds}{} \ntasks can be\ncreated in an automated manner.\nSpecifically, we first choose the desired granularity of the task by specifying \nthe distance from the root (``entity'') and retrieving all superclasses at \nthat distance in a top-down manner.\nEach resulting superclass corresponds to a subtree of our hierarchy, with\nImageNet classes as its leaves.\nNote that these superclasses are roughly of the same specificity, due to\nour hierarchy restructuring process.\nThen, we randomly sample a fixed number of subclasses for each \nsuperclass to produce a balanced dataset (omitting superclasses with an\ninsufficient number of subclasses).\nFinally, as described in Section~\\ref{sec:breeds}, we randomly split these\nsubclasses into the source and target domain.\n\\footnote{We also consider more benign or adversarial subpopulation\nsplits for these tasks in Appendix~\\ref{app:goodbad}.}\n\nFor our analysis, we create four tasks, presented in \nTable~\\ref{tab:benchmarks}, \nbased on different levels\/parts of the hierarchy.\nTo illustrate what the corresponding subpopulation shifts look like, we also \npresent (random) image samples for a subset of \nthe tasks in Figure~\\ref{fig:samples}.\nNote that while we focus on the tasks in Table~\\ref{tab:benchmarks} in our \nstudy, our methodology readily enables us to create other \nvariants of these tasks in an automated manner.\n\n\n\\begin{table}[!ht]\n\t\\centering\n\t\\begin{tabular}{lcccr}\n\t\t\\toprule\n\t\t\\textbf{Name} & \\textbf{Subtree} & \\textbf{Level} & \n\t\t\\textbf{Subpopulations} &\n\t\t\\textbf{Examples} \\\\\n\t\t\\midrule\n\t\t\\textsc{Entity-13} & ``entity'' (root) & 3 & 20 & ``mammal'', \n\t\t``appliance'' \\\\\n\t\t\\textsc{Entity-30} & ``entity'' (root) & 4 & 8 & ``fruit'', ``carnivore''\\\\\n\t\t\\textsc{Living-17} & ``living thing'' & 5 & 4 & ``ape'', ``bear'' \\\\\n\t\t\\textsc{Non-living-26} & ``non-living thing'' & 5 & 4 & ``fence'', ``ball''\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\vspace{1em}\n\t\\caption{\\textsc{Breeds}{} benchmarks constructed using ImageNet.\n\t``Level'' indicates the depth of the \n superclasses in the class hierarchy (task granularity). The number of\n\t``subpopulations'' (per superclass) is fixed across \n\tsuperclasses to ensure a balanced \n\tdataset.\n\tWe can also construct specialized tasks, by focusing on subtrees in \n\tthe hierarchy, e.g., only living (\\textsc{Living-17}{}) \n\tor non-living (\\textsc{Non-living-26}{}) objects. \n\tDatasets are named based on the root of the subtree and the resulting number\n of superclasses they end up containing.}\n\t\\label{tab:benchmarks}\n\\end{table}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figures\/gen\/samples.pdf}\n\t\\caption{Sample images from random object categories for the \\textsc{Entity-13}{} \n and \\textsc{Living-17}{} tasks. For each task, the top and bottom row\n correspond to the source and target distributions respectively.}\n\t\\label{fig:samples}\n\\end{figure}\n\n\\paragraph{\\textsc{Breeds}{} benchmarks beyond ImageNet.} \nIt is worth nothing that the methodology we described is not restricted to ImageNet and\ncan be readily applied to other datasets as well.\nThe only requirement is that we have access to a semantic grouping of the\ndataset classes, which is the case for many popular \nvision datasets---e.g., \nCIFAR-100~\\cite{krizhevsky2009learning}, \nPascal-VOC~\\cite{everingham2010pascal}, \nOpenImages~\\cite{kuznetsova2018open}, \nCOCO-Stuff~\\cite{caesar2018cocostuff}.\nMoreover, even when a class hierarchy is entirely\nabsent, the needed semantic class grouping can be manually\nconstructed with relatively little effort (proportional \nto the number of classes, not the number of datapoints).\n\n\\input{human}\n\n\\subsection{Calibrating \\textsc{Breeds}{} benchmarks via human studies}\n\\label{sec:humans}\nFor a distribution shift benchmark to be meaningful, it is essential that the source \nand target domains capture the same high-level task---otherwise generalizing\nfrom one domain to the other would be impossible.\nTo ensure that this is the case for the \\textsc{Breeds}{} task, we assess how\nsignificant the resulting distribution shifts are for human annotators\n(crowd-sourced via MTurk).\n\n\\paragraph{Annotator task.} \nTo obtain meaningful performance estimates, it is crucial that\nannotators perform the task based {only} \\emph{on the visual content of the\nimages}, without leveraging prior knowledge of the visual world.\nTo achieve this, we design the following annotation task.\nFirst, annotators are shown images from the source domain, grouped by\nsuperclass, without being aware of the superclass name (i.e., the object \ngrouping it corresponds to).\nThen, they are presented with images from the target domain and are asked to\nassign each of them to one of the groups.\nFor simplicity, we only present two random superclasses at a time, effectively\nsimulating binary classification.\nAnnotator accuracy can be measured directly as the fraction of images that they\nassign to the superclass to which these images belong.\nWe perform this experiment for each of the \\textsc{Breeds}{} tasks constructed in\nSection~\\ref{sec:tasks}.\nAs a point of comparison, we repeat this experiment without subpopulation \nshift\n(test images are sampled from the source domain) and for the \nsuperclasses\nconstructed by~\\citet{huh2016makes} using the WordNet hierarchy directly \n(cf.\nAppendix~\\ref{app:mturk}).\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Figures\/eval\/acc_comparison.pdf}\n \\caption{Human performance on (binary) \\textsc{Breeds}{} tasks.\n\t\tAnnotators are provided with labeled \n images from the source\n\t\tdistribution for a \\emph{pair of (undisclosed) superclasses}, and asked \n\t\tto classify\n\t samples from the target domain (`T') into one of the two groups. \n\t\tAs a baseline we also measure annotator performance without \n\t\tsubpopulation shift (i.e., on test images drawn from the source \n domain, `S') and equivalent tasks created via the original WordNet\n hierarchy (cf. Appendix~\\ref{app:mturk}).\n\t\tWe can observe that across all tasks, annotators are fairly robust to \n\t\tsubpopulation shift. \n\t\tFurther, annotators consistently perform better on \\textsc{Breeds}{} task \n\t\tcompared to those based on WordNet directly---indicating that our \n\t\tmodified class hierarchy is indeed better calibrated for object \n\t\trecognition.\n\t\t(We discuss model performance in Section~\\ref{sec:eval}.)\n\t}\n\t\\label{fig:acc_human}\n\\end{figure}\n\n\n\\paragraph{Human performance.} \nWe find that, across all tasks, annotators perform well on unseen data from the\nsource domain, as expected.\nMore importantly, annotators also appear to be quite robust to subpopulation shift, \nexperiencing only a small accuracy drop between the source and target \ndomains (cf. Figure~\\ref{fig:core}).\nThis is particularly prominent in the case of \\textsc{Entity-30}{} and \\textsc{Living-17}{} where the\ndifference in source and target accuracy is within the confidence interval.\nThis indicates that the source and target domains are indeed perceptually \nsimilar for humans, making these benchmarks suitable for studying model \nrobustness.\nFinally, across all benchmarks, we observe that annotators perform better on \n\\textsc{Breeds}{} tasks, as compared to their WordNet equivalents---even on \nsamples from the source domain.\nThis indicates that our modified ImageNet class hierarchy is indeed better \naligned with the underlying visual object recognition task.\n\n\\section*{Broader Impact}\nRobustness to testing conditions is a prerequisite for safe and reliable\ndeployment of machine learning models in the real-world. \nIn this work, we expand the current model evaluation toolkit by providing\nbenchmarks for assessing robustness to subpopulation shifts.\nThis allows us to test if models generalize beyond the limited diversity of\ntheir training datasets.\n\nOn the positive side, this toolkit serves as a simple testbed that can\npinpoint ways in which our models fail to generalize.\nThis provides us with an opportunity to improve and debug these models \nbefore\ndeployment, hence avoiding catastrophic failures in real-world scenarios.\nOn the negative side, well-defined benchmarks can provide users with a false\nsense of security.\nA models that performs well on a range of benchmarks is likely to be trusted \nand deployed in the real-world, despite having blind spots that these\nbenchmarks do not capture.\nIt is thus crucial to perceive benchmark progress properly, using it as a\nguide, without trusting it blindly.\n\n\n\\subsection{Robustness interventions}\n\\label{sec:intervene}\nWe now turn our attention to existing methods for decreasing \nmodel sensitivity to specific synthetic perturbations.\nOur goal is to assess if these methods enhance model robustness to\nsubpopulation shift too.\nConcretely, we consider the following families of \ninterventions (cf. Appendix~\\ref{app:robustness} for details):\n\\begin{itemize}\n \\item \\textbf{Adversarial training}: Enhances robustness to worst-case\n $\\ell_p$-bounded perturbations (in our case $\\ell_2$) by training models \n against a projected gradient descent (PGD) adversary~\\citep{madry2018towards}.\n \\item \\textbf{Stylized Training}: Encourages models to rely more on shape\n rather than texture by training them on a stylized version of \n ImageNet~\\cite{geirhos2018imagenettrained}.\n \\item \\textbf{Random noise}: Improves model robustness to data \n corruptions by incorporating them as data augmentations during \n training---we focus on Gaussian noise and Erase\n noise~\\citep{zhong2020random}, i.e., randomly obfuscating a block of the\n image.\n\\end{itemize}\n\n\\noindent\nNote that these methods can be viewed as ways of imposing a prior on the\nfeatures that the model relies on~\\cite{heinze2017conditional,\n geirhos2018imagenettrained, engstrom2019learning}.\nThat is, by rendering certain features ineffective during training (e.g.,\ntexture) they incentivize the model to utilize alternative features\nfor its predictions (e.g., shape).\nSince different families of features may correlate differently with class labels\nin the target domain, the aforementioned interventions could significantly\nimpact model robustness to subpopulation shift.\n\n\\paragraph{Relative accuracy.}\nTo measure the impact of these interventions, we will focus on the\nmodels' \\emph{relative accuracy}---the ratio of target accuracy to source\naccuracy.\nThis metric accounts for the fact that train-time interventions can impact model\naccuracy on the source domain itself.\nBy measuring relative performance, we are able to compare different training\nmethods on an equal footing. \n\nWe find that robustness interventions \\emph{do} have a small,\nyet non-trivial, impact on the robustness of a particular model architecture to\nsubpopulation shift---see Figure~\\ref{fig:intervene}.\nSpecifically, for the case of adversarial training and erase noise, models often\nretain a larger fraction of their accuracy to the target domain compared to\nstandard training, hence lying on the Pareto frontier of a robustness-accuracy\ntrade-off.\nIn fact, for some of the models trained with these interventions, the \ntarget accuracy is slightly higher than models obtained via standard training,\neven without adjusting for their lower source accuracy (raw\naccuracies for all methods are in Appendix~\\ref{app:res_int}).\nNonetheless, it is important to note that none of these method offer\nsignificant subpopulation robustness---relative accuracy is not\nimproved by more than a few percentage points.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figures\/eval\/interventions_all_rel_bs.pdf}\n\t\\caption{Effect of train-time interventions on model robustness to \n\t\tsubpopulation shift. We measure model performance in terms of \n\t\t\\emph{relative \n\t\t\taccuracy}--i.e., the ratio between its target and source \n\t\taccuracies. \n\t\tThis allows us to visualize the accuracy-robustness trade-off along with \n\t\tthe\n\t\tcorresponding Pareto frontier (\\emph{dashed}).\n\t\t(Also shown are 95\\% confidence intervals computed via \n\t\tbootstrapping.)\n\t\tWe observe that some of these interventions do improve \n model robustness to subpopulation shift by a small\n amount---specifically, erase noise and adversarial training---albeit\n sometimes at the cost of source accuracy.\n\t}\n\t\\label{fig:intervene}\n\\end{figure}\n\\paragraph{Adapting models to the target domain.}\nThe impact of these interventions is more pronounced if we consider \nthe target accuracy of these models after their last layer has been retrained on data from the target \ndomain---see \nFigure~\\ref{fig:intervene_ft}.\nIn particular, we observe that for adversarially robust models, retraining\nsignificantly boosts accuracy on the target domain---e.g., in the case of\n\\textsc{Living-17}{} it is almost comparable to the initial accuracy on the source domain.\nThis indicates that the feature priors imposed by these interventions\nincentivize models to learn representations that generalize better to\nsimilar domains---in line with recent results\nof~\\citet{utrera2020adversarially,salman2020adversarially}.\nMoreover, we observe that models trained on the stylized version of these\ndatasets perform consistently worse, suggesting that texture might be an\nimportant feature for these tasks, especially in the presence of subpopulation\nshift.\nFinally, note that we did not perform an exhaustive exploration of the\nhyper-parameters used for these interventions (e.g., $\\ell_2$-norm)---it is\npossible that these results can be improved by additional tuning.\nFor instance, we would expect that we can tune the magnitude of the Gaussian\nnoise to achieve performance that is comparable to that of $\\ell_2$-bounded\nadversarial training~\\citep{ford2019adversarial}.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figures\/eval\/interventions_all_rel_ft_bs.pdf}\n\t\\caption{Target accuracy of models after\n\t\tthey have been retrained (only the final linear layer) on data from the \n\t\ttarget domain (with 95\\% bootstrap confidence intervals).\n\t\tModels trained with robustness interventions often\n\t have higher target accuracy than standard models post retraining.\n }\n\t\\label{fig:intervene_ft}\n\\end{figure}\n\n\\section{Introduction}\n\\label{sec:intro}\nRobustness to distribution shift has been the focus of a long line of\nwork in machine learning~\\citep{schlimmer1986beyond,widmer1993effective,\nkelly1999impact,shimodaira2000improving,sugiyama2007covariate,\nquionero2009dataset,moreno2012unifying,sugiyama2012machine}.\nAt a high-level, the goal is to ensure that models perform well not only on \nunseen\nsamples from the datasets they are trained on, but also on the diverse set of\ninputs they are likely to encounter in the real world.\nHowever, building benchmarks for evaluating such robustness is \nchallenging---it requires modeling realistic data variations in a way that is \nwell-defined, controllable, and easy to simulate.\n\nPrior work in this context has focused on building benchmarks that capture \ndistribution shifts caused by\nnatural or adversarial input\ncorruptions~\\cite{szegedy2014intriguing,fawzi2015manitest,fawzi2016robustness,\n engstrom2019rotation,ford2019adversarial,hendrycks2019benchmarking,\n kang2019testing},\ndifferences in data sources~\\cite{saenko2010adapting,torralba2011unbiased,\n khosla2012undoing,tommasi2014testbed,recht2018imagenet},\nand changes in the frequencies of data \nsubpopulations~\\cite{oren2019distributionally,sagawa2019distributionally}.\nWhile each of these approaches captures a different source of real-world\ndistribution shift, we cannot expect any single benchmark to be \ncomprehensive.\nThus, to obtain a holistic understanding of model robustness, we need \nto keep expanding our testbed to encompass more natural modes of variation.\nIn this work, we take another step in that direction by studying the following\nquestion:\n\n\\begin{center}\n\t\\emph{How well do models generalize to data subpopulations they have \n\tnot seen during training?}\n\\end{center}\n\n\\noindent\nThe notion of \\emph{subpopulation shift} this question refers to is quite\npervasive.\nAfter all, our training datasets will inevitably fail to perfectly \ncapture the diversity of the real word.\nHence, during deployment, our models are bound to encounter unseen \nsubpopulations---for instance, unexpected weather conditions in the \nself-driving car context or different diagnostic setups in medical applications.\n\n\\subsection*{Our contributions}\nThe goal of our work is to create large-scale subpopulation shift benchmarks\nwherein the data subpopulations present during model training and evaluation\ndiffer.\nThese benchmarks aim to assess how effectively models generalize beyond the\nlimited diversity of their training datasets---e.g., whether models can\nrecognize Dalmatians as ``dogs'' even when their training data for ``dogs''\n comprises only Poodles and Terriers.\nWe show how one can simulate such shifts, fairly naturally, \\emph{within}\nexisting datasets, hence eliminating the need for (and the potential biases\nintroduced by) crafting synthetic transformations or collecting additional data.\n\n\\paragraph{\\textsc{Breeds}{} benchmarks.}\nThe crux of our approach is to leverage existing dataset labels and use them to\nidentify \\emph{superclasses}---i.e., groups of semantically similar classes.\nThis allows us to construct classification tasks over such superclasses, and\nrepurpose the original dataset classes to be the subpopulations of interest.\nThis, in turn, enables us to induce a subpopulation shift by directly making\nthe subpopulations present in the training and test distributions \ndisjoint.\nBy applying this methodology to the ImageNet \ndataset~\\citep{deng2009imagenet}, we create a suite of subpopulation shift \nbenchmarks of \nvarying difficulty.\nThis involves modifying the existing ImageNet class\nhierarchy---WordNet~\\citep{miller1995wordnet}---to ensure that \nsuperclasses comprise visually coherent subpopulations.\nWe then conduct human studies to validate that the resulting \\textsc{Breeds}{}\nbenchmarks indeed capture meaningful subpopulation shifts.\n\n\\paragraph{Model robustness to subpopulation shift.} \nIn order to demonstrate the utility of our benchmarks, we employ them \nto\nevaluate the robustness of standard models to subpopulation\nshift.\nIn general, we find that model performance drops significantly on the shifted\ndistribution---even when this shift does not significantly affect humans.\nStill, models that are more accurate on the original distribution tend to also \nbe more robust to these subpopulation shifts.\nMoreover, adapting models to the shifted domain, by\nretraining their last layer on data from this domain, only partially recovers the \noriginal \nmodel\nperformance.\n\n\\paragraph{Impact of robustness interventions.}\nFinally, we examine whether various train-time interventions, designed to\ndecrease model sensitivity to synthetic data corruptions (e.g., $\\ell_2$-bounded\nperturbations) make models more robust to subpopulation shift.\nWe find that many of these methods offer small, yet non-trivial, \nimprovements to\nmodel robustness along this axis---at times, at the expense of performance on the\noriginal distribution.\nOften, these improvements become more pronounced after\nretraining the last layer of the model on the shifted distribution.\nIn the context of adversarial training, our\nfindings are in line with recent work showing that the\nresulting robust models \noften exhibit improved robustness to other data\ncorruptions~\\citep{ford2019adversarial,kang2019testing,taori2020measuring}, and transfer\nbetter to downstream\ntasks~\\citep{utrera2020adversarially,salman2020adversarially}.\nNonetheless, none of these interventions significantly alleviate\nmodel sensitivity to subpopulation shift, indicating that the \\textsc{Breeds}{} \nbenchmarks pose a challenge to current methods.\n\n\n\n\\section{Designing Benchmarks for Distribution Shift}\n\\label{sec:prior}\nWhen constructing distribution shift benchmarks, the key design choice lies in \nspecifying the \\emph{target distribution} to be used during model evaluation.\nThis distribution is meant to be a realistic variation of the \n\\emph{source distribution}, that was used for training.\nTypically, studies focus on variations due to:\n\\begin{itemize}\n \\item \\emph{Data corruptions}: The target distribution is obtained by\n modifying inputs from the source distribution via a family of\n transformations that mimic real-world corruptions.\n Examples include natural or \n adversarial forms of noise~\\cite{fawzi2015manitest,fawzi2016robustness,\n engstrom2019rotation,hendrycks2019benchmarking,ford2019adversarial,kang2019testing,\n shankar2019image}.\n \\item \\emph{Differences in data sources}: Here, the target distribution is an\n independently collected dataset for the same \n task~\\cite{saenko2010adapting,torralba2011unbiased,tommasi2014testbed,\n beery2018recognition,recht2018imagenet}---for\n instance, using PASCAL VOC~\\cite{everingham2010pascal} to evaluate\n ImageNet-trained classifiers~\\cite{russakovsky2015imagenet}. The goal is to\n test whether models are overly reliant on the idiosyncrasies of the datasets\n they are trained\n on~\\cite{ponce2006dataset,torralba2011unbiased}.\n \\item \\emph{Subpopulation shifts}: The source and target distributions \n differ in terms of how well-represented each subpopulation is.\n Work in this area typically studies whether models perform \n \\emph{equally well} across\n all subpopulations from the perspective of\n reliability~\\cite{meinshausen2015maximin, hu2016does,duchi2018learning,\n caldas2018leaf,oren2019distributionally,sagawa2019distributionally}\n or algorithmic \n fairness~\\citep{dwork2012fairness,kleinberg2017inherent,\n \tjurgens2017incorporating, \n \tbuolamwini2018gender,hashimoto2018fairness}.\n\\end{itemize} \n\nIn general, a major challenge lies in ensuring that the distribution\nshift between the source and target distributions (also referred to as \n\\emph{domains}) is caused \nsolely by the \nintended input variations.\nExternal factors---which may arise when crafting synthetic\ntransformations or collecting new \ndata---could skew the \ntarget distribution in different ways, making it hard to gauge model \nrobustness to the exact distribution shift of interest.\nFor instance, recent work~\\citep{engstrom2020identifying} demonstrates that\ncollecting a new dataset while aiming to match an existing one along a specific\nmetric (e.g., as in \\citet{recht2018imagenet}) might result in a miscalibrated\ndataset due to statistical bias.\nIn our study, we aim to limit such external influences by simulating\nshifts within existing datasets, thus avoiding any input modifications.\n\n\\section{Additional Related Work}\nIn Section~\\ref{sec:prior}, we discuss prior work that has directly focused on \nevaluating model robustness to distribution shift. We now provide an \noverview of other related work and its connections to our methodology.\n\n\\paragraph{Distributional robustness.}\nDistribution shifts that are small with respect to some $f$-divergence have been\nstudied in prior theoretical work~\\citep{ben2013robust,duchi2016statistics,\nesfahani2018data,namkoong2016stochastic}.\nHowever, this notion of robustness is typically too pessimistic to capture\nrealistic data variations~\\cite{hu2016does}.\nDistributional robustness has also been connected to\ncausality~\\citep{meinshausen2018causality}:\nhere, the typical approach is to inject spurious correlations into the dataset, \nand assess to what extent models rely on them for \ntheir predictions~\\citep{heinze2017conditional,\narjovsky2019invariant,sagawa2019distributionally}.\n\n\\paragraph{Domain adaptation and transfer learning.} \nThe goal here is to adapt models to the target domain with relatively few\nsamples from it~\\citep{ben2007analysis, saenko2010adapting,ganin2014unsupervised,\ncourty2016optimal,gong2016domain,donahue2014decaf,razavian2014cnn}.\nIn domain adaptation, the task is the same in both domains,\nwhile in transfer learning, the task itself could vary.\nIn a similar vein, the field of \\emph{domain generalization} aims to generalize\nto samples from a different domain (e.g., from ClipArt to photos) by training on\na number of explicitly annotated\ndomains~\\citep{muandet2013domain,li2017deeper,peng2019moment}.\n\n\\paragraph{Zero-shot learning.}\nWork in this domain focuses on learning to recognize previously unseen \nclasses~\\citep{lampert2009learning,xian2017zero}, typically described\nvia a semantic \nembedding~\\citep{lampert2009learning,mikolov2013distributed,socher2013zero,frome2013devise,romera2015embarrassingly}.\nThis differs from our setup, where the focus is on \ngeneralization to unseen subpopulations for the \\emph{same} set of classes.\n\n\\section{Experimental Setup}\n\n\\subsection{Dataset}\n\\label{app:datasets}\nWe perform our analysis on the ILSVRC2012\ndataset~\\citep{russakovsky2015imagenet}. This dataset contains a thousand\nclasses from the ImageNet dataset~\\cite{deng2009imagenet} with an independently\ncollected validation set. \nThe classes are part of the broader hierarchy, WordNet~\\citep{miller1995wordnet}, \nthrough which\nwords are organized based on their semantic meaning.\nWe use this hierarchy as a starting point of our investigation but modify it\nas described in Appendix~\\ref{app:manual}.\n\nFor all the \\textsc{Breeds}{} superclass classification tasks, the train and validation\nsets are obtained by aggregating the train and validation sets of the \ndescendant ImageNet classes (i.e., subpopulations). \nSpecifically, for a given subpopulation, the training and test splits from the \noriginal ImageNet dataset are used as is.\n\n\n\n\\subsection{WordNet issues}\n\\label{app:wordnet}\n\nAs discussed in Section~\\ref{sec:hierarchy}, WordNet is a semantic rather than a\nvisual hierarchy. That is, object classes are arranged based on their meaning\nrather than their visual appearance. Thus, using intermediate nodes for a\nvisual object recognition task is not straightforward. To illustrate this, we\nexamine a sample superclass grouping created by~\\citet{huh2016makes} via\nautomated bottom-up clustering in Table~\\ref{tab:problems}.\n\n\\begin{table}[htp]\n \\input{tables\/table_other_36}\n \\vspace{1em}\n \\caption{Superclasses constructed by~\\citet{huh2016makes} via\n bottom-up clustering of WordNet to obtain 36 superclasses---for\n brevity, we only show superclasses with at least 20 ImageNet classes each.}\n \\label{tab:problems}\n\\end{table}\n\nFirst, we can notice that these superclasses have vastly different\ngranularities.\nFor instance, ``organism'' contains the entire animal kingdom, hence being much\nbroader than ``produce''.\nMoreover, ``covering'' is rather abstract class, and hence its subclasses often\nshare little visual similarity (e.g., ``window shade'', ``pajama'').\nFinally, due to the abstract nature of these superclasses, a large number of\nsubclasses overlap---``covering'' and ``commodity'' share 49 ImageNet \ndescendants.\n\n\\clearpage\n\\subsection{Manual calibration}\n\\label{app:method}\nIn order to allow for efficient and automated creation of superclasses that are\nsuitable for visual recognition, we modified the WordNet hierarchy by applying\nthe following operations:\n\\begin{itemize}\n \\item \\emph{Collapse node}: Delete a node from the hierarchy and add edges\n from each parent to each child. Allows us to remove redundant or overly\n specific categorization while preserving the overall structure.\n \\item \\emph{Insert node above}: Add a dummy parent to push a node further\n down the hierarchy. Allows us to ensure that nodes of similar granularity are at \n the same level.\n \\item \\emph{Delete node}: Remove a node and all of its edges. Used to\n remove abstract nodes that do not reveal visual characteristics.\n \\item \\emph{Add edge}: Connect a node to a parent. Used to reassign the\n children of nodes deleted by the operation above.\n\\end{itemize}\nWe manually examined the hierarchy and implemented these actions in order to\nproduce superclasses that are calibrated for classification.\nThe principles we followed are outlined in Section~\\ref{sec:hierarchy} while\nthe full hierarchy can be explored using the notebooks provided with the \nhierarchy.\\footnote{\\url{https:\/\/github.com\/MadryLab\/BREEDS-Benchmarks}}\n\n\\subsection{Resulting hierarchy}\n\\label{app:manual}\n\nThe parameters for constructing the \\textsc{Breeds}{} benchmarks (hierarchy level,\nnumber of subclasses, and tree root) are given in Table~\\ref{tab:benchmarks}.\nThe resulting tasks---obtained by sampling disjoint ImageNet classes (i.e., \nsubpopulations) for the\nsource and target domain---are shown in\nTables~\\ref{tab:three},~\\ref{tab:four},~\\ref{tab:living},\nand~\\ref{tab:nonliving}.\nRecall that for each superclass we randomly sample a fixed number of \nsubclasses per superclass to ensure that the dataset is approximately \nbalanced.\n\\clearpage\n\\input{tables\/table_3_n00001740}\n\n\\clearpage\n\\input{tables\/table_4_n00001740}\n\n\\clearpage\n\\input{tables\/table_5_n00004258}\n\n\\clearpage\n\\input{tables\/table_5_n00021939}\n\n\n\\clearpage\n\\subsection{Annotator task}\n\\label{app:mturk}\nAs described in Section~\\ref{sec:humans}, the goal of our human studies is to\nunderstand whether humans can classify images into superclasses even without\nknowing the semantic grouping.\nThus, the task involved showing annotators two groups of images, each sampled\nfrom the source domain of a random superclass.\nThen, annotators were shown a new set of images from the target domain (or the\nsource domain in the case of control) and were asked to assign each of them into\none of the two groups. A screenshot of an (random) instance of our annotator\ntask is shown in Figure~\\ref{fig:screenshot}.\n\nEach task contained 20 images from the source domain of each superclass and 12\nimages for annotators to classify (the images where rescaled and center-cropped\nto size $224\\times 224$ to match the input size use for model predictions).\nThe two superclasses were randomly permuted at load time.\nTo ensure good concentration of our accuracy estimates, for every superclass, \nwe \nperformed binary classification tasks w.r.t. 3 other (randomly chosen) superclasses.\nFurther, we used 3 annotators per task. and annotators were compensated \\$0.15 \nper task.\n\n\\paragraph{Comparing with the original hierarchy.} In order to compare our\nsuperclasses with those obtained by \\citet{huh2016makes} via WordNet\nclustering,\\footnote{\\url{https:\/\/github.com\/minyoungg\/wmigftl\/tree\/master\/label_sets\/hierarchy}}\nwe need to define a correspondence between them.\nTo do so, for each of our tasks, we selected the clustering (either top-down or\nbottom-up) that had the closest number of superclasses.\nFollowing the terminology from that work, this mapping is: \\textsc{Entity-13}{} $\\to$\n\\textsc{DownUp-36}, \\textsc{Entity-30}{} $\\to$ \\textsc{UpDown-127}, \\textsc{Living-17}{} $\\to$\n\\textsc{DownUp-753} (restricted to ``living'' nodes), and \\textsc{Non-living-26}{} $\\to$\n\\textsc{DownUp-345} (restricted to ``non-living'' nodes).\n\n\\begin{figure}[htp]\n \\centering\n \\includegraphics[width=0.78\\textwidth]{Figures\/task}\n \\caption{Sample MTurk annotation task to obtain human baselines for \\textsc{Breeds}{} \n benchmarks.}\n \\label{fig:screenshot}\n\\end{figure}\n\n\n\\clearpage\n\\subsection{Evaluating model performance}\n\\label{app:eval_setup}\n\n\\subsubsection{Model architectures and training}\n\\label{app:models}\nThe model architectures used in our analysis are in\nTable~\\ref{tab:models} for which we used standard implementations\nfrom the PyTorch library \n(\\url{https:\/\/pytorch.org\/docs\/stable\/torchvision\/models.html}).\nFor training, we use a batch size of 128, weight decay of $10^{-4}$, \nand learning rates listed in Table~\\ref{tab:models}.\nModels were trained until convergence.\nOn \\textsc{Entity-13}{} and \\textsc{Entity-30}{}, this required a total of 300 epochs, \nwith 10-fold drops in learning rate every 100 epochs, while on\n\\textsc{Living-17} and \\textsc{Non-living-26}, models a total of 450 epochs, with 10-fold learning rate\ndrops every 150 epochs.\nFor adapting models, we retrained the last\n(fully-connected) layer on the train split of the target domain, starting from\nthe parameters of the source-trained model.\nWe trained that layer using SGD with a batch size of 128 for\n40,000 steps and chose the best learning rate out of \n$[0.01, 0.1, 0.25, 0.5, 1.0, 2.0, 3.0, 5.0, 7.0, 8.0, 10.0, 11.0, 12.0]$, based\non test accuracy.\n\n\n\\begin{table}[!h]\n\\begin{center}\n\t\\begin{tabular}{lcc}\n\t\t\\toprule\n\t\t\\textbf{Model} & \\phantom{x} & \\textbf{Learning Rate} \\\\\n\t\t\\midrule\n\t\t\\texttt{alexnet} && 0.01 \\\\ \n\t\t\\texttt{vgg11} && 0.01 \\\\ \n\t\t\\texttt{resnet18} && 0.1 \\\\ \n\t\t\\texttt{resnet34} && 0.1 \\\\ \n\t\t\\texttt{resnet50} && 0.1 \\\\ \n\t\t\\texttt{densenet121} && 0.1 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{center}\n\t\\caption{Models used in our analysis.} \n\t\\label{tab:models}\n\\end{table}\n\n\\subsubsection{Model pairwise accuracy}\n\\label{app:model_pairwise}\nIn order to make a fair comparison between the performance of models and human \nannotators on the \\textsc{Breeds}{} tasks, we evaluate model accuracy on\npairs of superclasses. On images from that pair,\nwe determine the model prediction to be the superclass for \nwhich the model's predicted probability is higher. A prediction is deemed correct if it \nmatches the superclass label for the image. Repeating this process over random pairs \nof superclasses allows us to estimate model accuracy on the average-case binary\nclassification task.\n\n\n\\subsubsection{Robustness interventions}\n\\label{app:robustness}\nFor model training, we use the hyperparameters provided in \nAppendix~\\ref{app:models}.\nAdditional intervention-specific hyperparameters are listed in Appendix \nTable~\\ref{tab:ri_hyperparams}. Due to computational \nconstraints, we trained a restricted set of model architectures with robustness \ninterventions---ResNet-18 and ResNet-50 for adversarial training, and ResNet-18 \nand ResNet-34 for all others.\nAdversarial training was implemented using the \\texttt{robustness}\nlibrary,\\footnote{\\url{https:\/\/github.com\/MadryLab\/robustness}} while random\nerasing using the PyTorch\n\\texttt{transforms}.\\footnote{\\url{https:\/\/pytorch.org\/docs\/stable\/torchvision\/transforms.html}}\n\n\\begin{table}[!h]\n\t\\begin{center}\n\t\\begin{minipage}{0.3\\textwidth}\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\toprule\n\t\t\t\\textbf{Eps} & \\textbf{Step size} & \\textbf{\\#Steps} \\\\\n\t\t\t\\midrule\n 0.5 & 0.4 & 3 \\\\ \n\t\t\t1 & 0.8 & 3 \\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\n \\caption*{(a) PGD-training~\\citep{madry2018towards}}\n\\end{minipage}\t\n\\hfil\n\\begin{minipage}{0.2\\textwidth}\n\t\t\t\\begin{tabular}{cc}\n\t\t\\toprule\n\t\t\\textbf{Mean} &\t\\textbf{StdDev} \\\\\n\t\t\\midrule\n 0 & 0.2 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption*{(b) Gaussian noise} \n\\end{minipage}\t\n\\hfil\n\\begin{minipage}{0.4\\textwidth}\n\t\t\t\\begin{tabular}{ccc}\n\t\t\\toprule\n \\textbf{Probability} &\t\\textbf{Scale} & \\textbf{Ratio} \\\\\n\t\t\\midrule\n 0.5 & 0.02 - 0.33 & 0.3 - 3.3 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption*{(c) Random erasing} \n\\end{minipage}\t\n\t\\end{center}\n\t\\caption{Additional hyperparameters for robustness interventions.} \n\\label{tab:ri_hyperparams}\n\\end{table}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \nOne of the main challenges in statistics is the design of a \\textit{universal} estimation procedure. Given data, a universal procedure is an algorithm that provides an estimator of the generating distribution which is simultaneously statistically consistent when the true distribution belongs to the model, and robust otherwise. Typically, a universal estimator is consistent for any model, with minimax-optimal or fast rates of convergence and is robust to small departures from the model assumptions \\cite{BickelRobust1976} such as sparse instead of dense effects or non-Gaussian errors in high dimensional linear regression. Unfortunately, most statistical procedures are based upon strong assumptions on the model or on the corresponding parameter set, and very famous estimation methods such as maximum likelihood estimation (MLE), method of moments or Bayesian posterior inference may fail even on simple problems when such assumptions do not hold. For instance, even though MLE is consistent and asymptotically normal with optimal rates of convergence in parametric estimation under suitable regularity assumptions \\cite{LeCamMLE70,VanderVaartMLE98} and in nonparametric estimation under entropy conditions, this method behaves poorly in case of misspecification when the true generating distribution of the data does not belong to the chosen model.\n\nLet us investigate a simple example presented in \\cite{Birge2006ModelSelectionTesting} that illustrates the non-universal characteristic of MLE. We observe a collection of $n$ independent and identically distributed (i.i.d) random variables $X_1,...,X_n$ that are distributed according to some mixture distribution $P^0_n = (1-2n^{-1})\\mathcal{U}([0,1\/10]) + 2n^{-1} \\mathcal{U}([1\/10,9\/10])$ where $\\mathcal{U}([a,b])$ is the uniform distribution between $a$ and $b$. We consider the parametric model of independent uniform distributions $\\mathcal{U}([0,\\theta])$, $0 \\leq \\theta < 1$, and we choose the squared Hellinger distance $h^2(\\cdot,\\cdot)$ as the risk measure. Here the maximum likelihood is the maximum of the observations $X_{(n)}:=\\max(X_1,...,X_n)$, and $\\mathcal{U}([0,1\/10])$ is a good approximation of the generating distribution $P^0_n$ as $h^2(P^0_n,\\mathcal{U}([0,1\/10])) < 5\/4n$ for $n\\geq4$. Hence, one would expect that $\\mathbb{E}[h^2(P^0_n,\\mathcal{U}([0,X_{(n)}]))]$ goes to $0$ as $n\\rightarrow +\\infty$, which is actually not the case. We do not even have consistency: $\\mathbb{E}[h^2(P^0_n,\\mathcal{U}([0,X_{(n)}]))]>0.38$. Hence, the MLE is not robust to this small deviation from the parametric assumption.\nThe same happens in Bayesian statistics: the regular posterior distribution is not always robust to model misspecification. Indeed, authors of \\cite{Barronetal1999,grunwaldmisspecifiation} show pathologic cases where the posterior does not concentrate to the true distribution.\n\nUniversal estimation is all the more important since it provides a generic approach to tackle the more and more popular problem of robustness to outliers under the i.i.d assumption, although definitions and goals involved in robust statistics are quite different from the universal estimation perspective. H\\\"uber introduced a framework that models situations where a small fraction $\\varepsilon$ of data is contaminated, and he assumes that the true generated distribution can be written $(1-\\varepsilon)P_{\\theta_0}+\\varepsilon Q$ where $Q$ is the contaminating distribution and $\\varepsilon$ is the proportion of corrupted observations \\cite{H\\\"uberRobustness}. The goal when using this approach is to estimate the true parameter $\\theta_0$ given a misspecified model $\\{P_{\\theta}\/\\theta\\in\\Theta\\}$ with $\\theta_0 \\in \\Theta$. A procedure is then said to be robust in this case if it leads to a good estimation of the true parameter $\\theta_0$. More generally, when a procedure is able to provide a good estimate of the generating distribution of i.i.d data when a small proportion of them is corrupted, whatever the values of these outliers, then such an estimator is considered as robust.\n\nInterestingly enough, none of the aforementioned works questioned the independence assumption on the observation. We believe that a universal estimation procedure should still produce sensible estimations under small deviations from this assumption.\n\n\\subsection{Related work}\n\nSeveral authors attempted to design a general universal estimation method. Sture Holm \\cite{BickelRobust1976} suggested that Minimum Distance Estimators (MDE) were the most natural procedures being robust to misspecification. Motivated by \\cite{WolfowitzMDE1957,Parr80}, MDE consists in minimizing some probability distance $d$ between the empirical distribution and a distribution in the model. The MDE $\\hat{\\theta}_n$ is defined by:\n$$\nd(\\hat{P}_n,P_{\\hat{\\theta}_n}) = \\inf_{\\theta \\in \\Theta} d(\\hat{P}_n,P_{\\theta})\n$$\nwhere $\\hat{P}_n=n^{-1}\\sum_{i=1}^n\\delta_{\\{X_i\\}}$ is the empirical measure and $\\Theta$ the parameter set associated to the model. If the minimum does not exist, then one can consider an $\\varepsilon$-approximate solution. In fact, this minimum distance estimator is used in many usual procedures. Indeed, the generalized method of moments \\cite{Hansen1982} is actually defined as minimizing the weighted Euclidean distance between moments of $\\hat{P}_n$ and $P_{\\theta}$ while the MLE minimizes the KL divergence, at least for discrete measures. When the distance $d$ is wisely chosen, e.g. when it is bounded, then MDE can be robust and consistent.\n\nA popular metric is the Total Variation (TV) distance \\cite{Yatracos1985,DevroyeL1}. \\cite{Yatracos1985} built an estimator that is uniformly consistent in TV distance and is robust to misspecification under the i.i.d assumption, but without any assumption on the true distribution of the data. The rate of convergence depends on the Kolmogorov entropy of the model. A few decades later, Devroye and Lugosi studied in details the skeleton estimate, a variant of the estimator of \\cite{Yatracos1985} that is based on the TV-distance restricted to the so-called Yatracos sets, see \\cite{DevroyeL1}. Unfortunately, the skeleton estimate and the original Yatracos estimate are not computationally tractable.\n\nIn \\cite{BaraudBirge2016RhoEstimators} and \\cite{BaraudBirgeSart2017RhoEstimators}, Baraud, Birg\\'e and Sart introduced $\\rho$-estimation, a universal method that retains some appealing properties of the MLE such as efficiency under some regularity assumptions, while being robust to deviations, measured by the Hellinger distance. This $\\rho$-estimation procedure is inspired from T-estimation \\cite{Birge2006ModelSelectionTesting}, itself inspired from earlier works of Le Cam \\cite{LeCam73,LeCam75} and Birg\\'e \\cite{Birge83}, and goes beyond the classical compactness assumption used in T-estimation. In compact models, $\\rho$-estimators can be seen as variants of T-estimators also based on robust tests, but they can be extended to noncompact models such as linear regression with fixed or random design with various error distributions. As T-estimators, they enjoy robustness properties, but involve other metric dimensions which lead to optimal rates of convergence with respect to the Hellinger distance even in cases where T-estimators can not be defined. Moreover, when the sample size is large enough, $\\rho$-estimation recovers the usual MLE in density estimation when the model is parametric, well-specified and regular enough. Hence, $\\rho$-estimation can be seen as a robust version of the MLE. Unfortunately, this strategy is also intractable.\n\n\nMore recently, \\cite{Briol2019} showed that using the Maximum Mean Discrepancy (MMD) \\cite{Gretton2012} to build a minimum distance estimator leads to both robust estimation in the i.i.d case, without any assumption on the model $\\{P_\\theta,\\theta\\in\\Theta\\}$. Moreover, this estimator is tractable as soon as the model is generative, that is, when one can sample efficiently from any $P_\\theta$. MMD, a metric based on embeddings of probability measures into a reproducing kernel Hilbert space, has been applied successfully in a wide range of problems such as kernel Bayesian inference \\cite{Song2011}, approximate Bayesian computation \\cite{Park2016}, two-sample \\cite{Gretton2012} and goodness-of-fit testing \\cite{Jit17}, and MMD GANs \\cite{Roy2015,li15} and autoencoders~\\cite{Zhao2017}, to name a few prominent examples. Such minimum MMD-based estimators are proven to be consistent, asymptotically normal and robust to model misspecification. The trade-off between the statistical efficiency and the robustness is made through the choice of the kernel. The authors investigated the geometry induced by the MMD on a finite-dimensional parameter space and introduced a (natural) gradient descent algorithm for efficient computation of the estimator. This algorithm is inspired from the stochastic gradient descent (SGD) used in the context of MMD GANs where the usual discriminator is replaced with a two-sample test based on MMD \\cite{Roy2015}. These results were extended in the Bayesian framework by~\\cite{BECA2019}.\n\nFinally, a whole branch of probability and statistics study limit theorems (LLN, CLT) under the assumptions that the data is not exactly independent, but that in some sense, the dependence between the observations is not strong. Since the seminal work of~\\cite{mixing0}, many mixing conditions, that is, restrictions on the dependence between observations, were defined. These conditions lead to limit theorems useful to analyze the asymptotic behavior of estimators computed on time series~\\cite{mixing}. Nevertheless, checking mixing assumptions is difficult in practice and many classes of processes that are of interest in statistics such as elementary Markov chains are sometimes not mixing. More recently, \\cite{DependenceDoukhan1999} proposed a new weak dependence condition for time series that is built on covariance-based coefficients which are much easier to compute than mixing ones, and that is more general than mixing as it stands for most relevant classes of processes. We believe that it is important to study robust estimators in this setting, in order to check that they are also robust from small deviations to the independence assumption.\n\n\\subsection{Contributions}\n\nIn this paper, we further investigate universality properties of minimum distance estimation based on MMD distance \\cite{Briol2019}. Inspired by the related literature, our contributions in this paper are the following:\n\\begin{itemize}\n\\item We go beyond the classical i.i.d framework. Indeed, we prove that the the estimator is robust to dependence between observations. To do so, we introduce a new dependence coefficient expressed as a covariance in some reproducing kernel Hilbert space, and which is very simple to use in practice.\n\\item We show that our oracle inequalities imply robust estimation under the i.i.d assumption in the H\\\"uber contamination model and in the case of adversarial contamination.\n\\item We propose a theoretical analysis of the SGD algorithm used to compute this estimator in \\cite{Briol2019} and \\cite{Roy2015} for some finite dimensional models. Thanks to this algorithm, we provide numerical simulations to illustrate our theoretical results.\n\\end{itemize}\n\n\n\n\nThe first result of this paper is a generalization bound in the non-i.i.d setting. It states that under a very general dependence assumption, the generalization error with respect to the MMD distance decreases in $n^{-1\/2}$ as $n\\rightarrow +\\infty$. This result extends the inequalities in \\cite{Briol2019} that are only available in the i.i.d framework, and is obtained using dependence concepts for stochastic processes. We introduce in this paper a new dependence coefficient in the wake of \\cite{DependenceDoukhan1999} which can be expressed as a covariance in some reproducing kernel Hilbert space associated with MMD. This coefficient can be easily computed in many situations and which may be related to usual mixing coefficients such as the popular $\\beta$-mixing one. We show that a weak assumption on this new dependence coefficient can relax the i.i.d assumption of \\cite{Briol2019} and can lead to valid generalization bounds even in the dependent setting.\n\nRegarding robustness, we prove that our generalization bounds for the MMD estimator implies that this estimator is robust to the presence of outliers. Note that this includes H\\\"uber's type contamination, and adversarial contamination as well. In particular, we compare the rate of convergence of the MMD estimator to the minimax estimators in the example of the estimation of the mean of a Gaussian.\n\nRegarding computational issues, we provide a Stochastic Gradient Descent (SGD) algorithm as in \\cite{Briol2019,Roy2015} involving a U-statistic approximation of the expectation in the formula of the MMD distance. We theoretically analyze this algorithm in parametric estimation using a convex parameter set. We also perform numerical simulations that illustrate the efficiency of our method, especially by testing the behavior of the algorithm in the presence of outliers.\n\nThe rest of the paper is organized as follows. Section \\ref{sec:notations} defines the MMD-based minimum distance estimator and our new dependence coefficient based on the kernel mean embedding. Section \\ref{sec:main-res} provides nonasymptotic bounds in the dependent and misspecified framework, with their implications in terms of robust parametric estimation. Section \\ref{sec:examples} illustrates the efficiency of our method in several different frameworks. We finally present an SGD algorithm with theoretical convergence guarantees in Section \\ref{sec:algo} and we perform numerical simulations in Section \\ref{sec:simu}. The proofs of the theorems of Section~\\ref{sec:main-res} are provided in Section \\ref{sec:proofs}. The supplementary material is dedicated to the remaining proofs.\n\n \\section{Background and definitions}\n\\label{sec:notations}\n \nIn this section, we first introduce some notations and present the statistical setting of the paper in Section 2.1. Then, we remind in Section 2.2 some theory on reproducing kernel Hilbert spaces (RKHS) and we define both the maximum mean discrepancy (MMD) and our minimum distance estimator based on the MMD. Finally, we introduce in Section 2.3 a new dependence coefficient expressed as a covariance in a RKHS.\n\n\\subsection{Statistical setting}\n\nWe shall consider a dependent setting throughout the paper. We observe in a measurable space $\\big( \\mathbb{X},\\mathcal{X} \\big)$ a collection of $n$ random variables $X_1$,...,$X_n$ generated from a stationary process. This implies that the $X_i$'s are identically distributed, and we will let $P^0$ denote their marginal distribution. Note that this include as an example the case where the $X_i$'s are i.i.d with generating distribution $P^0$. We introduce a statistical model $\\{ P_{\\theta}\/ \\theta \\in \\Theta \\}$ indexed by a parameter space $\\Theta$.\n\n\\subsection{Maximum Mean Discrepancy}\n\nWe consider a positive definite kernel function $k$, i.e a symmetric function $k : \\mathbb{X} \\times \\mathbb{X} \\rightarrow \\mathbb{R}$ such that for any integer $n\\geq 1$, for any $x_1,...,x_n \\in \\mathbb{X}$ and for any $c_1,...,c_n \\in \\mathbb{R}$:\n$$\n\\sum_{i=1}^n \\sum_{j=1}^n c_i c_j k(x_i,x_j) \\geq 0.\n$$\nWe then consider the reproducing kernel Hilbert space (RKHS) $({\\mathcal{H}_{k}},\\langle\\cdot,\\cdot\\rangle_{\\mathcal{H}_{k}})$ associated with the kernel $k$ which satisfies the reproducing property $f(x)=\\langle f, k(x,\\cdot)\\rangle_{\\mathcal{H}_{k}}$ for any function $f \\in {\\mathcal{H}_{k}}$ and any $x \\in \\mathbb{X}$. From now on, we assume that the kernel is bounded by some positive constant, that will be assumed to be $1$ without loss of generality. Thas is, for any $x,y \\in \\mathbb{X}$, $|k(x,y)|\\leq1$. \n\n\nNow we introduce the notion of \\textit{kernel mean embedding}, a Hilbert space embedding of a probability measure that can be viewed as a generalization of the original feature map used in support vector machines and other kernel methods. Given a probability measure $P$, we define the mean embedding $\\mu_P \\in {\\mathcal{H}_{k}}$ as:\n$$\n\\mu_P(\\cdot) := \\mathbb{E}_{X\\sim P}[k(X,\\cdot)] \\in {\\mathcal{H}_{k}} .\n$$\nAll the applications and the theoretical properties of those embeddings have been well studied \\cite{Fuku2017}. In particular, the mean embedding $\\mu_P$ satisfies the relationship $\\mathbb{E}_{X\\sim P}[f(X)] = \\langle f, \\mu_P\\rangle_{\\mathcal{H}_{k}}$ for any function $f \\in {\\mathcal{H}_{k}}$, and induces a semi-metric \\footnote{ This means that $P \\rightarrow \\| \\mu_P \\|_{\\mathcal{H}_{k}}$ satisfies the requirements of a norm besides $ \\| \\mu_P - \\mu_Q \\|_{\\mathcal{H}_{k}} = 0 $ only for $\\mu_P=\\mu_Q$. } on measures called maximum mean discrepancy (MMD), defined for two measures $P$ and $Q$ as follows:\n$$\n\\mathbb{D}_k(P,Q) = \\| \\mu_P - \\mu_Q \\|_{\\mathcal{H}_{k}}\n$$\nor equivalently\n$$\n\\mathbb{D}_k^2(P,Q) = \\mathbb{E}_{X,X' \\sim P}[k(X,X')] - 2 \\mathbb{E}_{X\\sim P,Y\\sim Q}[k(X,Y)] + \\mathbb{E}_{Y,Y'\\sim Q}[k(Y,Y')] .\n$$\nA kernel $k$ is said to be characteristic if $P\\mapsto \\mu_P$ is injective. This ensures that $\\mathcal{D}_k$ is a metric, and not only a semi-metric. Subsection 3.3.1 of the thorough survey \\cite{Fuku2017} provides a wide range of conditions ensuring that $k$ is characteristic. They also provide many examples of characteristic kernels, see their Table 3.1. Among others, when $\\mathbb{X} \\subset \\mathbb{R}^d$ equiped with the Euclidean norm $\\|\\cdot\\|$, the Gaussian kernel $k(x,y) = \\exp(-\\|x-y\\|^2\/\\gamma^2)$ and the Laplace kernel $k(x,y) = \\exp(-\\|x-y\\|\/\\gamma)$, are known to be characteristic. We actually mostly use these two kernels in our applications. From now on, we will assume that $k$ is characteristic.\n\nNote that there are many applications of the kernel mean embedding and MMD in statistics such as two-sample testing \\cite{Gretton2012}, change-point detection \\cite{Arlot2012}, detection \\cite{MONK19}, we also refer the reader to~\\cite{Vert2019} for a thorough introduction to the applications of kernels and MMD to computationnal biology .\n\nHere, we will focus on estimation of parameters based on MMD. This principle was used to train generative networks \\cite{Roy2015,li15}, it's only recently that it was studied as a general principle for estimation \\cite{Briol2019}. Following these papers we define the MMD estimator $\\hat{\\theta}_n$ such that:\n$$\n\\mathbb{D}_k(P_{\\hat{\\theta}_n},\\hat{P}_n) = \\inf_{\\theta \\in \\Theta} \\mathbb{D}_k(P_{\\theta},\\hat{P}_n)\n$$\nwhere $\\hat{P}_n=(1\/n)\\sum_{i=1}^n \\delta_{X_i}$ is the empirical measure, i.e.:\n$$\n\\hat{\\theta}_n = \\underset{\\theta \\in \\Theta}{\\arg\\min} \\, \\bigg\\{\n\\mathbb{E}_{X,X' \\sim P_\\theta}[k(X,X')] - \\frac{2}{n} \\sum_{i=1}^n \\mathbb{E}_{X\\sim P_\\theta}[k(X,X_i)] \\bigg\\}.\n$$\nIt could be that there is no minimizer, see the discussion in Theorem 1 page 9 in~\\cite{Briol2019}. In this case, we can use an approximate minimizer. More precisely, for any $\\varepsilon>0$ we can always find a $\\hat{\\theta}_{n,\\varepsilon}$ such that:\n$$\n\\mathbb{D}_k(P_{\\hat{\\theta}_{n,\\varepsilon}},\\hat{P}_n) \\leq \\inf_{\\theta \\in \\Theta} \\mathbb{D}_k(P_{\\theta},\\hat{P}_n) + \\varepsilon.\n$$\nIn what follows, we will consider the case where the minimizer exists (that is, $\\varepsilon=0$) but when this is not the case, everything can be easily extended by considering $\\hat{\\theta}_{n,1\/n}$.\n\n\\subsection{Covariances in RKHS}\n\n\nIn this subsection, we introduce and discuss a new dependence coefficient based on the kernel mean embedding. This coefficient allows to go beyond the i.i.d case in the study of the MMD estimator of \\cite{Briol2019}, and to show that it is actually robust to dependence.\n\n\\begin{dfn}\n \\label{dfn.varrho}\n We define, for any $t\\in\\mathbb{N}$,\n $$ \\varrho_t = \\left| \\mathbb{E}\\left< k(X_t,\\cdot)-\\mu_{P^0},k(X_0,\\cdot)-\\mu_{P^0} \\right>_{\\mathcal{H}_k} \\right| . $$\n\\end{dfn}\nIn the i.i.d case, note that $\\varrho_t = 0$ for any $t\\geq 1$. In general, the following assumption will ensure the consistency of our estimator:\n\\begin{asm}\n \\label{asm:our:mixing}\n There is a $\\Sigma < + \\infty$ such that, for any $n$, $\\sum_{t=1}^n \\varrho_t \\leq \\Sigma$.\n\\end{asm}\n\nOur mean embedding dependence coefficient may be seen as a covariance expressed in the RKHS $\\mathcal{H}_{k}$. We shall see throughout the paper that the kernel mean embedding coefficient $\\varrho_t$ can be easily computed in many situations, and that it is closely related to widely used mixing coefficients. In particular, we will show in Section 4.2 that our coefficient $\\varrho_t$ is upper-bounded by the popular $\\beta$-mixing coefficient. For the reader who would not be familiar with $\\beta$-mixing, we also show that any real-valued auto-regressive process $X_t = a X_{t-1} + \\varepsilon_t$ satisfies Assumption~\\ref{asm:our:mixing} as long as $|a|<1$, the $\\varepsilon_t$ are i.i.d and $\\mathbb{E}(|\\varepsilon_0|)<\\infty$. Also, we show that some special cases of such auto-regressive processes are not $\\beta$-mixing, which proves that Assumption~\\ref{asm:our:mixing} is more general than $\\beta$-mixing: an explicit example is given in Subsection~\\ref{subsec:ar}. Hence, Assumption \\ref{asm:our:mixing} may be referred to as a weak dependence condition in the wake of the concept of weak dependence introduced in \\cite{DependenceDoukhan1999}. We will show in the next section that under Assumption \\ref{asm:our:mixing}, we can obtain a nonasymptotic generalization bound of the same order than in the i.i.d case.\n\n \\section{Nonasymptotic bounds in the dependent, misspecified case}\n\\label{sec:main-res}\n \nIn this section, we provide nonasymptotic generalization bounds in MMD distance for the minimum MMD estimator. In particular, we show in Subsection~\\ref{subsec:mmd} that under a weak dependence assumption, it is robust to both dependence and misspecification, and is consistent at the same $n^{-1\/2}$ rate than in the i.i.d case. In particular, we give explicit bounds in the H\\\"uber contamination model and in a more general adversarial setting in Subsection~\\ref{subsec:robust}.\n\n\\subsection{Estimation with respect to the MMD distance}\n\\label{subsec:mmd}\n\n\nFirst, we begin with a theorem that gives an upper bound on the generalization error, i.e the expectation of $\\mathbb{D}_k(P_{\\hat{\\theta}_n},P^0)$. The rate of convergence of this error is of order $n^{-1\/2}$ independently of the dimension of the parameter space $\\Theta$. In fact, note that there is actually no assumption at all on the model $\\{P_{\\theta},\\theta\\in\\Theta\\}$ in this theorem.\n\\begin{thm}\n\\label{theorem:mmd:1}\n We have: $$ \\mathbb{E} \\left[ \\mathbb{D}_k\\left(P_{\\hat{\\theta}_n},P^0 \\right) \\right] \\leq \\inf_{\\theta\\in\\Theta} \\mathbb{D}_k\\left(P_{\\theta},P^0 \\right) + 2 \\sqrt{ \\frac{1+2\\sum_{t=1}^{n}\\varrho_t}{n}} . $$\n\\end{thm}\nAs a consequence, under Assumption~\\ref{asm:our:mixing}:\n $$ \\mathbb{E} \\left[ \\mathbb{D}_k\\left(P_{\\hat{\\theta}_n},P^0 \\right) \\right] \\leq \\inf_{\\theta\\in\\Theta} \\mathbb{D}_k\\left(P_{\\theta},P^0 \\right) + 2 \\sqrt{ \\frac{1+2\\Sigma }{n}} . $$\nWe remind that the proofs of the results in this section are deferred to Section~\\ref{sec:proofs}. It is also possible to provide a result that holds with large probability as in \\cite{Briol2019,Roy2015}. Naturally, this requires stronger assumptions, and the conditions on the dependence become more intricate in this case. Here, we use a condition introduced in \\cite{McDoRIO,McDoRIO2} for generic metric spaces that we adapt to the kernel embedding and to stationarity:\n\\begin{asm}\n\\label{asm:gamma:mixing}\nAssume that there is a family $(\\gamma_{\\ell})_\\ell$ of non-negative numbers such that, for any integer $n$, for any $\\ell\\in\\{1,\\dots,n-1\\} $ and any function $g:\\mathcal{H}_k^\\ell \\rightarrow \\mathbb{R} $ such that\n$$ |g(a_1,\\dots,a_\\ell) - g(b_1,\\dots,b_\\ell)| \\leq \\sum_{i=1}^\\ell \\|a_i - b_i\\|_{\\mathcal{H}_k} , $$\nwe have: $ |\\mathbb{E}[g(\\mu_{\\delta_{X_{\\ell+1}}},\\dots,\\mu_{\\delta_{X_{n}}})|X_{1},\\dots,X_{\\ell}]-\\mathbb{E}[g(\\mu_{\\delta_{X_{\\ell+1}}},\\dots,\\mu_{\\delta_{X_{n}}})]| \\leq \\gamma_{1} + \\dots + \\gamma_{n+\\ell-1} $, almost surely. Assume that $\\Gamma:= \\sum_{\\ell\\geq 1} \\gamma_{\\ell} < \\infty $.\n\\end{asm}\nThis assumption is more technical than Assumption~\\ref{asm:our:mixing}. The idea is quite similar: the coefficient $\\gamma_s$ is a measure of the dependence between $X_t$ and $X_{t+s}$, and the assumption will be satisfied if $X_t$ and $X_{t+s}$ are ``almost independent'' when $s$ is large -- but the sense given to ``almost independent'' is not exactly the same as in Assumption~\\ref{asm:our:mixing}. For example, we show in Subsection~\\ref{subsec:ar} that auto-regressive processes $X_{t+1}=a X_t + \\varepsilon_{t+1}$ with $|a|<1$ and i.i.d $\\varepsilon_t$ satisfy this assumption under the additional condition that the $\\varepsilon_t$ are almost surely bounded. Again, note that in the case of independence, we can take all the $\\gamma_{i}=0$ and hence $\\Gamma=0$ in addition to $\\Sigma=0$. We can now state our result in probability:\n\\begin{thm}\n\\label{theorem:mmd:briol:improved}\nAssume that Assumptions~\\ref{asm:our:mixing} and~\\ref{asm:gamma:mixing} are satisfied. Then, for any $\\delta\\in(0,1)$,\n $$ \\mathbb{P} \\left[ \\mathbb{D}_k\\left( P_{\\hat{\\theta}_n},P^0 \\right) \\leq \\inf_{\\theta\\in\\Theta} \\mathbb{D}_k\\left( P_{\\theta},P^0 \\right)\n + 2 \\frac{\\sqrt{1+2\\Sigma} + (1+\\Gamma)\\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\sqrt{n}} \\right] \\geq 1-\\delta.$$\n\\end{thm}\nAssumption \\ref{asm:gamma:mixing} is fundamental to obtain a result in probability. Indeed, the rate of convergence in Theorem \\ref{theorem:mmd:briol:improved} is characterized by some concentration inequality upper bounding the MMD distance between the empirical and the true distribution as done in \\cite{Briol2019}. Nevertheless, the proof of this inequality in \\cite{Briol2019} is based on a Hoeffding-type inequality known as McDiarmid's inequality \\cite{McDo} that is only valid for independent variables (that is, all the $\\gamma_i=0$), which makes this inequality not applicable in our dependent setting. Hence we use a version of McDiarmid's inequality for time series obtained by Rio \\cite{McDoRIO,McDoRIO2} which is available under the assumption that $\\sum_{\\ell\\geq 1} \\gamma_{\\ell} < \\infty$ (Assumption \\ref{asm:gamma:mixing}).\n\n\\begin{rmk}[The i.i.d case]\n Note that when the $X_i$'s are i.i.d, Assumptions~\\ref{asm:our:mixing} and~\\ref{asm:gamma:mixing} are always satisfied with $\\Sigma=\\Gamma=0$ and thus Theorem~\\ref{theorem:mmd:1} gives simply\n $$ \\mathbb{E} \\left[ \\mathbb{D}_k\\left(P_{\\hat{\\theta}_n},P^0 \\right) \\right] \\leq \\inf_{\\theta\\in\\Theta} \\mathbb{D}_k\\left(P_{\\theta},P^0 \\right) +\\frac{2}{\\sqrt{n}} $$\n while Theorem~\\ref{theorem:mmd:briol:improved} gives\n $$ \\mathbb{P} \\left[ \\mathbb{D}_k\\left( P_{\\hat{\\theta}_n},P^0 \\right) \\leq \\inf_{\\theta\\in\\Theta} \\mathbb{D}_k\\left( P_{\\theta},P^0 \\right)\n + 2 \\frac{1 + \\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\sqrt{n}} \\right] \\geq 1-\\delta.$$ \n\\end{rmk}\n\n\\begin{rmk}[Connection between the MMD distance and the $L^2$ norm]\n\\label{rmk:L2}\nIn Section~\\ref{sec:examples}, we study the connection between the convergence of $\\hat{P}_{\\hat{\\theta}_n}$ in terms of MMD distance, and the convergence of $\\hat{\\theta}_n$, is some classical models. However, it is also worth mentioning a connection between the MMD distance and the quadratic distance on densities. Indeed, assume $\\mathbb{X} = \\mathbb{R}^d$ and that $P$ and $Q$ have density $p$ and $q$ respectively with respect to the Lebesgue measure. Using the Gaussian kernel $k_\\gamma(x,y) = \\exp(-\\|x-y\\|^2\/\\gamma^2)$, we expect that, when $\\gamma \\rightarrow 0$, under suitable assumptions,\n$$ \\mathbb{E}_{X\\sim P, Y\\sim Q}[k(X,Y)] \\sim \\pi^{\\frac{d}{2}} \\gamma^d \\int p(x) q(x) {\\rm d}x $$\nand so that\n\\begin{equation}\n\\label{equa:lien:l2}\n\\mathbb{D}_{k_\\gamma}(P,Q) \\sim \\pi^{\\frac{d}{4}} \\gamma^{\\frac{d}{2}} \\|p-q\\|_{L^2}.\n\\end{equation}\nCorollary 4 page 1527 of~\\cite{L2} provides a formal statement of this claim. Thus, the convergence in the MMD distance has connections with the convergence of the densities (when they exist) in $L^2$.\n\nNote that~\\cite{DevroyeL1,BaraudBirge2016RhoEstimators} argue that the $L^2$-norm is not suitable for universal estimation: indeed, in some models, $P_\\theta$ does not have a density with respect to the Lebesgue measure. But~\\eqref{equa:lien:l2} allows the interpretation of the MMD distance (with the Gaussian kernel) as an approximation of the $L^2$ distance, that is however well defined for {\\it any} model $(P_\\theta)$.\n\\end{rmk}\n\n\\subsection{Robust parametric estimation}\n\\label{subsec:robust}\n\n\\subsubsection{Contamination models}\n\nAs explained in the introduction, when all observations but a small proportion of them are sampled independently from a generating distribution $P_{\\theta_0}$ ($\\theta_0 \\in \\Theta$), robust parametric estimation consists in finding estimators being both rate optimal and resistant to outliers. Two among the most popular frameworks for studying robust estimation are the so-called H\\\"uber's contamination model and the adversarial contamination model. \n\nH\\\"uber's contamination model is as follows. We observe a collection of random variables $X_1,...,X_n$. We consider a contamination rate $\\varepsilon\\in(0,1\/2)$, latent i.i.d random variables $Z_1,...,Z_n \\sim \\text{Ber}(\\varepsilon)$ and some noise distribution $Q$, such that the distribution of $X_i$ given $Z_i=0$ is $P_{\\theta_0}$, and that the distribution of $X_i$ given $Z_i=1$ is $Q$. Hence, the observations $X_i$'s are independent and sampled from the mixture $P^0=(1-\\varepsilon)P_{\\theta_0}+\\varepsilon Q$.\n\nThe adversarial model is more general. Contrary to H\\\"uber's contamination where outliers were all sampled from the contaminating distribution, we do not make any particular assumption on the outliers here. Hence, we shall adopt slightly different notations. We assume that $X_1,\\dots,X_n$ are identically distributed from $P_{\\theta_0} $ for some $\\theta_0\\in\\Theta$. However, the statistician only observes $\\tilde{X_1},\\dots,\\tilde{X_n}$ where $\\tilde{X}_i$ can be any arbitrary value for $i\\in \\mathcal{O}$, where $\\mathcal{O}$ is an arbitrary set subject to the constraint $|\\mathcal{O}| \\leq \\varepsilon n$, and $\\tilde{X}_i=X_i$ for $i\\notin \\mathcal{O}$. The estimators are built based on these observations $\\tilde{X_1},\\dots,\\tilde{X_n}$.\n\n\n\\subsubsection{Literature}\n\nOne hot research trend in robust statistics is focused on the search of both statistically optimal and computationally tractable procedures for the Gaussian mean estimation problem $\\{P_\\theta=\\mathcal{N}(\\theta,I_d)\/\\theta \\in \\mathbb{R}^d\\}$ in the presence of outliers under the i.i.d assumption, which remains a major challenge. Usual robust estimators such as the coordinatewise median and the geometric median are known to be suboptimal in this case, and there is a need to look at more complex estimators such as Tukey's median that achieves the minimax optimal rate of convergence $\\max(\\frac{d}{n},\\varepsilon^2)$ with respect to the squared Euclidean distance, where $d$ is the dimension, $n$ is the sample size and $\\varepsilon$ is the proportion of corrupted data. Unfortunately, computation of Tukey's median is not tractable and even approximate algorithms lead to an $\\mathcal{O}(n^d)$ complexity \\cite{ChanTukey2004,AmentaTukey2000}. This has led to the rise of the recent studies in robust statistics which address how to build robust and optimal statistical procedures, in the wake of the works of \\cite{Tukey1975} and \\cite{H\\\"uberRobustness}, but that are also computationally efficient.\n\nThis research area started with two seminal works presenting two procedures for the normal mean estimation problem: the \\textit{iterative filtering} \\cite{DiakonikolasRobust2016} and the \\textit{dimension halving} \\cite{LaiRaoVempala2016}. These algorithms are based upon the idea of using higher moments in order to obtain a good robust moment estimation, and are minimax optimal up to a poly-logarithmic factor in polynomial time. This idea was then used in several other problems in robust statistics, for instance in sparse functionals estimation \\cite{DuRobustFunctionals}, clustering \\cite{KothariRobustClustering}, mixtures of spherical Gaussians learning \\cite{DiakonikolasExtension1}, and robust linear regression \\cite{DiakonikolasExtension2}. In H\\\"uber's contamination model, \\cite{ColDal17a} achieves the minimax rate without any extra factor in the $\\varepsilon = \\mathcal{O}(\\min(d^{-1\/2},n^{-1\/4}))$ regime with an improved overall complexity. Meanwhile, \\cite{ChaoGaoRobustGAN2019} offers a different perspective on robust estimation and connects the robust normal mean estimation problem with Generative Adversarial Networks (GANs) \\cite{GoodfellowGAN2014,BiauGAN2018}, what enables computing robust estimators using efficient tools developed for training GANs. Hence, the authors compute depth-like estimators that retain the same appealing robustness properties than Tukey's median and that can be trained using stochastic gradient descent (SGD) algorithms that were originally designed for GANs.\n\n\nAnother popular approach for the more general problem of mean estimation under the i.i.d assumption in the presence of outliers is the study of finite-sample sub-Gaussian deviation bounds. Indeed, designing estimators achieving sub-Gaussian performance under minimal assumptions ensures robustness to outliers that are inevitably present when the generating distribution is heavy-tailed. In the univariate case, some estimators present a sub-Gaussian behavior for all distributions under first and second order moments. A simple but powerful strategy, the Median-of-Means (MOM), dates back to \\cite{Nemi1983,Jer86,Alon1999}. This method consists in randomly splitting the data into several equal-size blocks, then computing the empirical mean within each block, and finally taking the median of them. Most MOM-based procedures lead to estimators that are simultaneously statistically optimal \\cite{Lugosi2016,MOM1,Lecue2018,MONK19,chinot2019} and computationally efficient \\cite{Hopkins2019,chera2019,depersin2019}. Moreover, this approach can be easily extended to the multivariate case \\cite{Minsker2015,Hsu2016}. An important advantage is that the MOM estimator has good performance even for distributions with infinite variance. An elegant alternative to the MOM strategy is due to Catoni, whose estimator is based on PAC-Bayesian truncation in order to mitigate heavy tails \\cite{Catoni2012}. It has the same performance guarantees than the MOM method but with sharper and near-optimal constants. In \\cite{CatoniGiulini2017}, Catoni and Giulini proposed a very simple and trivial-to-compute multidimensional extension of Catoni's M-estimator defined as an empirical average of the data, with the observations with large norm shrunk towards zero, and that still satisfies a sub-Gaussian concentration using PAC-Bayes inequalities. The influence function of Catoni and Giulini has been widely used since then, see \\cite{Ilaria2017,Ilaria2018,Holland2019a,Holland2019b,Haddouche2020}. We refer the reader to the beautiful review of \\cite{Lugosi2019mean} for more details on those mean estimation procedures.\n\n\\subsubsection{Robust MMD estimation}\n\nIn this section, we show the properties of our MMD-based estimator in robust parametric estimation with outliers, both in H\\\"uber's contamination model and in the adversarial case. Our bounds are obtained by working directly in the RKHS rather than in the parameter space. the consequence of these results in terms of the Euclidean distance in the parameter space will be explored in Section~\\ref{sec:examples}. \n\nFirst we consider H\\\"uber's contamination model \\cite{H\\\"uberRobustness}. The objective is to estimate $P_{\\theta_0}$ by observing contaminated random variables $X_1$, ..., $X_n$ with actual distribution is $P^0 = (1-\\alpha) P_{\\theta_0} + \\alpha Q $ for some $Q$, and some $0\\leq \\alpha \\leq \\varepsilon$. We state the key following lemma:\n\n\\begin{lemma}\n\\label{lemma:huber}\nWe have, for any $\\theta\\in\\Theta$, $ | \\mathbb{D}_k(P_{\\theta},P^0) - \\mathbb{D}_k (P_{\\theta},P_{\\theta_0})| \\leq 2 \\varepsilon$.\n\\end{lemma}\n\nApplying Lemma~\\ref{lemma:huber} to the left-hand side, and to the right-hand side, of Theorem~\\ref{theorem:mmd:1}, we have the following result.\n\n\\begin{cor}\nAssume that $X_1,\\dots,X_n$ are identically distributed from $P^0 = (1-\\alpha) P_{\\theta_0} + \\alpha Q $ for some $\\theta_0\\in\\Theta$, some $Q$, with $0\\leq \\alpha \\leq \\varepsilon$. Then:\n $$ \\mathbb{E} \\left[ \\mathbb{D}_k\\left(P_{\\hat{\\theta}_n},P_{\\theta_0} \\right) \\right] \\leq 4\\varepsilon + 2 \\sqrt{ \\frac{1+2\\sum_{t=1}^{n}\\varrho_t}{n}} . $$\nIf moreover we assume that Assumptions~\\ref{asm:our:mixing} and~\\ref{asm:gamma:mixing} are satisfied, then for any $\\delta\\in(0,1)$,\n $$ \\mathbb{P} \\left[ \\mathbb{D}_k\\left( P_{\\hat{\\theta}_n},P_{\\theta_0}\\right) \\leq 2 \\left( 2\\varepsilon \n + \\frac{\\sqrt{1+2\\Sigma} + (1+\\Gamma)\\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\sqrt{n}} \\right) \\right] \\geq 1-\\delta.$$\n\\end{cor}\n\nWe obtain a rate $\\max(1\/\\sqrt{n},\\varepsilon)$ in MMD distance (note once again that the convergence rate with respect to more standard distances is studied in Section~\\ref{sec:examples}). When $\\varepsilon \\lesssim 1\/\\sqrt{n}$, then we recover the rate of convergence without contamination, and when $1\/\\sqrt{n} \\lesssim \\varepsilon$, then the rate is dominated by the contamination ratio $\\varepsilon$. Hence, the maximum number of outliers which can be tolerated without breaking down the rate is $n\\varepsilon \\asymp \\sqrt{n}$.\n\nThis result can also be extended to the adversarial contamination setting, where no assumption is made on the outliers.\n\n\\begin{prop}\n \\label{prop:adversarial}\n Assume that $X_1,\\dots,X_n$ are identically distributed from from $P^0 = P_{\\theta_0} $ for some $\\theta_0\\in\\Theta$. However, the statistician only observes $\\tilde{X_1},\\dots,\\tilde{X_n}$ where $\\tilde{X}_i$ can be any arbitrary value for $i\\in \\mathcal{O}$, $\\mathcal{O}$ is any arbitrary set subject to the constraint $|\\mathcal{O}| \\leq \\varepsilon n$, and $\\tilde{X}_i=X_i$ for $i\\notin \\mathcal{O}$ and builds the estimator $\\tilde{\\theta}_n$ based on these observations:\n $$ \\mathbb{D}_k\\left(P_{\\tilde{\\theta}_n},\\frac{1}{n}\\sum_{i=1}^n \\delta_{\\tilde{X}_i}\\right) = \\inf_{\\theta \\in \\Theta} \\mathbb{D}_k\\left(P_{\\theta},\\frac{1}{n}\\sum_{i=1}^n \\delta_{\\tilde{X}_i}\\right). $$\n Then:\n $$ \\mathbb{D}_k\\left(P_{\\tilde{\\theta}_n},P_{\\theta_0} \\right) \\leq \n 4\\varepsilon + 2 \\mathbb{D}_k\\left(P_{\\hat{\\theta}_n},P_{\\theta_0} \\right) . $$\nThus\n $$\n \\mathbb{E} \\left[ \\mathbb{D}_k\\left(P_{\\tilde{\\theta}_n},P_{\\theta_0} \\right) \\right] \\leq 4\\varepsilon + 4 \\sqrt{ \\frac{1+2\\sum_{t=1}^{n}\\varrho_t}{n}}\n $$\n and, under Assumptions~\\ref{asm:our:mixing} and~\\ref{asm:gamma:mixing}, for any $\\delta\\in(0,1)$,\n $$ \\mathbb{P} \\left[ \\mathbb{D}_k\\left( P_{\\hat{\\theta}_n},P_{\\theta_0}\\right) \\leq 4 \\left( \\varepsilon \n + \\frac{\\sqrt{1+2\\Sigma} + (1+\\Gamma)\\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\sqrt{n}} \\right) \\right] \\geq 1-\\delta.$$\n\\end{prop}\n\nOne can see that the rate of convergence we obtain without making any assumption on the outliers is exactly the same than in H\\\"uber's contamination model. The only difference is that the constant in the right hand side of the inequality is tighter in H\\\"uber's contamination model.\n\n \\section{Examples}\n\\label{sec:examples}\n \n\\subsection{Independent observations}\n\nIn the previous section we studied the convergence of $P_{\\hat{\\theta}_n}$ with the MMD distance. In this subsection, we show what are the consequences of these results in terms of the convergence of $\\hat{\\theta}$ in some classical models. For the sake of simplicity, we focus on i.i.d observations. That is, $\\varrho_t = 0$ for any $t\\geq 1$. Moreover, we will only use the Gaussian kernel $k_\\gamma(x,y) = \\exp(-\\|x-y\\|^2\/\\gamma^2)$.\n\n\\subsubsection{Estimation of the mean in a Gaussian model}\n\nHere, $\\mathbb{X}=\\mathbb{R}^d$ and we are interested in the estimation of the mean in a Gaussian model. For the sake of simplicity, we assume that the variance is known.\n\n\\begin{prop} \\label{prop:ex:gauss}\n Assume that $P_\\theta = \\mathcal{N}(\\theta,\\sigma^2 I_d)$ for $\\theta\\in\\Theta=\\mathbb{R}^d$. Moreover, assume that we are in an adversarial contamination model where a proportion at most $\\varepsilon$ of the observations is contaminated. Then, with probability $1-\\delta$,\n\\begin{equation} \\label{ex:gauss:3}\n\\|\\theta-\\tilde{\\theta}_n\\|^2\n\\leq - (4\\sigma^2 + \\gamma^2)\n \\log\\left\\{ 1-8 {\\rm e}^{\\frac{2\\sigma^2 d}{\\gamma^2}} \\left( \\varepsilon \n + \\frac{1 + \\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\sqrt{n}} \\right)^2 \\right\\}.\n\\end{equation}\nIn particular, the choice $\\gamma = \\sigma \\sqrt{2d}$ leads to\n$$\n\\|\\tilde{\\theta}_n - \\theta_0 \\|^2 \\leq -2\\sigma^2(d+2) \\log\\left[1-8 {\\rm e} \\left( \\varepsilon \n + \\frac{1 + \\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\sqrt{n}} \\right)^2\\right].\n$$\n\n\\end{prop}\nThe complete proof can be found in the supplementary material. Note that when $\\varepsilon$ is small and $n$ is large,\n\\begin{multline*}\n\\|\\tilde{\\theta}_n-\\theta_0\\|^2 \\leq -2\\sigma^2(d+2) \\log\\left[1-16 {\\rm e} \\left( \\varepsilon^2 \n + \\frac{\\left(1 + \\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)}\\right)^2 }{n} \\right)\\right]\n\\\\\n\\sim\n32 {\\rm e}\\sigma^2(d+2) \\left( \\varepsilon^2 \n + \\frac{\\left(1 + \\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)}\\right)^2 }{n} \\right).\n\\end{multline*}\n\nWe can see that our MMD estimator achieves a rate of convergence $d\\varepsilon^2 + d\/n$ which is the same than for several median-based estimators such as the geometric median or the coordinatewise median (see Proposition 2.1 in \\cite{ChenGao2018}). We have a quadratic dependence in $\\varepsilon$, contrary to many robust estimators such as Median-of-Means which dependence in $\\varepsilon$ is linear. Hence, as soon as the dimension is no larger than the square root of the sample size $d\\leq\\sqrt{n}$, the MMD method tolerates a larger number of outliers without affecting the usual rate of convergence (i.e. the rate with no contamination).\n\nUnfortunately, it seems that our method performs poorly compared to such estimators in large dimension. Indeed, according to Theorems 2.1 and 2.2 in \\cite{ChenGao2018}, the minimax optimal rate with respect $d$, $\\varepsilon$ and $n$ is $\\varepsilon^2 + d\/n$. Furthermore, numerical experiments and the investigation conducted for the population limit case when one has access to infinitely many samples in \\cite{MMDGANRobust} (that has been published since the first version of this paper) suggest that the MMD estimator can not match the minimax rate of convergence. Nevertheless, this non-optimality in the minimax sense does not necessarily imply inaccurate mean estimation in general, and MMD can still lead to efficient estimation in most contamination scenarios.\n\nTo understand why the MMD estimator can not match the minimax rate of convergence in high dimension, and why this is not necessarily a problem, we need to analyze the landscape of the optimization program. \n\nLet us first investigate the population limit case, where we do not work with the MMD distance to the empirical distribution $\\hat{P}_n$ but to the true distribution $(1-\\varepsilon)\\mathcal{N}(\\theta_0,\\sigma^2 I_d)+\\varepsilon Q$, as if we had access to infinitely many samples, and with a point-mass delta Dirac contamination $Q=\\delta_{\\{\\theta_c\\}}$. The optimization program is, for any value of $\\gamma$, \n\\begin{multline*}\n\\min_{\\theta\\in\\mathbb{R}^d} \\mathbb{D}_{k_\\gamma}\\left(P_{\\theta},(1-\\varepsilon)\\mathcal{N}(\\theta_0,\\sigma^2 I_d)+\\varepsilon \\delta_{\\{\\theta_c\\}}\\right)\n\\\\\n=\n\\max_{\\theta\\in\\mathbb{R}^d} \\bigg\\{\n(1-\\varepsilon) \\exp\\left(-\\frac{\\|\\theta -\\theta_0\\|^2}{\\gamma^2+4\\sigma^2}\\right) + \\varepsilon \\left(\\frac{\\gamma^2+4\\sigma^2}{\\gamma^2+2\\sigma^2}\\right)^{d\/2} \\exp\\left(-\\frac{\\|\\theta-\\theta_c\\|^2}{\\gamma^2+2\\sigma^2}\\right) \\bigg\\} .\n\\end{multline*}\n\nEven though the objective function is nonconvex in $\\theta$, it is easy to see that the solution belongs to the line between $\\theta_0$ and $\\theta_c$. More precisely, if $\\theta_0$ and $\\theta_c$ are far from each other, then the solution is simply $\\theta_0$. At the opposite, if $\\theta_0$ and $\\theta_c$ are closed, then the solution will be very close to $\\theta_0$. In the situation in between where $\\|\\theta_0-\\theta_C\\|^2\\approx d$, then it is proven in \\cite{MMDGANRobust} that the solution is at least $\\varepsilon\\sqrt{d}$ far from the true parameter $\\theta_0$, which explains the term $d\\varepsilon^2$ in the rate of convergence of the MMD estimator. Hence, we understand that the worst-case rate of the MMD estimator does not correspond to cases where $\\theta_c$ is far from $\\theta_0$ but to cases where the distance is quite large in high dimensions only (of order $\\sqrt{d}$).\n\nThe previous reasoning can be easily generalized to the MMD estimator with a finite sample. In this situation with $Q=\\delta_{\\{\\theta_c\\}}$, the optimization program can be written, denoting by $\\mathcal{O}$ the set of outliers, \n$$\n\\max_{\\theta\\in\\mathbb{R}^d} \\bigg\\{\n\\sum_{i\\notin\\mathcal{O}} \\exp\\left(-\\frac{\\|\\theta -X_i\\|^2}{\\gamma^2+2\\sigma^2}\\right) + |\\mathcal{O}| \\exp\\left(-\\frac{\\|\\theta-\\theta_c\\|^2}{\\gamma^2+2\\sigma^2}\\right) \\bigg\\} , \n$$\nand the solution belongs to the convex hull of the set of points composed of the (random) inliers in the random variables $X_1,...,X_n$ and of the contamination point $\\theta_c$. A remarkable point in high dimensional probability is that samples from a multivariate standard Gaussian distribution are concentrated on the sphere of radius $\\sqrt{d}$ centered at $\\theta_0$, which means that the typical distance $\\|X_i-\\theta_0\\|$ of a datapoint $X_i$ from the mean $\\theta_0$ is roughly $\\sqrt{d}$. Then, if the contamination is such that $\\|\\theta_0-\\theta_c\\|\n^2\\approx d$, the outliers lie at a distance $\\sqrt{d}$ from $\\theta_0$ without being detected, and thus look harmless but shift the mean by approximately $\\sqrt{d}\\varepsilon$, see Figure \\ref{shiftmean}.\n\\begin{figure}[h]\n\\caption{Illustration of the behaviour of the MMD estimator in the high-dimensional Gaussian mean estimation problem. The true parameter $\\theta_0$ and datapoints sampled from the true distribution $\\mathcal{N}(\\theta_0,I_d)$ are colored in blue. Outliers and the MMD estimator $\\hat{\\theta}_n$ are colored in red. We can see that outliers lying at a distance $\\sqrt{d}$ are not detected and shift the mean by $\\varepsilon\\sqrt{d}$.\n}\n\\label{shiftmean}\n\\includegraphics[width=7cm]{f1.pdf}\n\\centering\n\\end{figure}\n\nHence, perhaps counter-intuitively at first sight, the worst contamination does not come from a value of $\\theta_c$ that is very far away from $\\theta_0$ (in which case the estimation will simply be the mean of the inliers), but that is only $\\sqrt{d}$ away from $\\theta_0$, and hence there is mainly one \"worst-case contamination\" that explains the non-optimality in the minimax sense. Figure 1a of \\cite{MMDGANRobust} even seems to show that the error of the MMD estimator when $\\gamma$ is of order $\\sqrt{d}$ increases with $\\|\\theta_0-\\theta_c\\|$ until achieving $\\sqrt{d}$, and then decreases. The same applies to a Gaussian contamination with a small variance.\n\n\\subsubsection{Cauchy model}\n\nHere, $\\mathbb{X}=\\mathbb{R}$ and $P_\\theta=\\mathcal{C}(\\theta,1)$ where $\\mathcal{C}(\\theta,s)$ has density $1\/[\\pi s (1+(x-\\theta)^2 \/ s^2)]$.\n\n\\begin{prop} \\label{prop:ex:cauchy}\n Assume that $P_\\theta = \\mathcal{C}(\\theta,1)$ for $\\theta\\in\\Theta=\\mathbb{R}$. Moreover, assume that we are in an adversarial contamination model where a proportion at most $\\varepsilon$ of the observations is contaminated. Then, taking $\\gamma=2$ leads to, for any $\\delta>0$,\n$$\n(\\tilde{\\theta}_n - \\theta_0)^2 \\leq\n4\\left( 1 - \\frac{1}{1-96 \\pi \\left( \\varepsilon^2 + \\frac{2 + 4\\log(1\/\\delta) }{n} \\right) } \\right).\n$$\n\\end{prop}\nNote that\n$$ (\\tilde{\\theta}_n - \\theta_0)^2 \\leq\n4\\left( 1 - \\frac{1}{1-128 \\pi \\left( \\varepsilon^2 + \\frac{2 + 4\\log(1\/\\delta) }{n} \\right) } \\right) \\sim 512 \\pi \\left( \\varepsilon^2 + \\frac{2 + 4\\log(1\/\\delta) }{n} \\right) .$$\n\n\\subsubsection{Estimation with a dictionary}\n\\label{dictionary}\n\nWe consider here estimation of $P^0$ by a linear combination of measures in a dictionary. This framework actually appears in various models:\n\\begin{itemize}\n \\item first, when the dictionary contains probability distributions, this is simply a mixture of known components. In this case, the linear combination is actually a convex combination. This context is for example studied in~\\cite{Dal2017}.\n \\item assuming that $P^0$ has a density, in nonparametric density estimation, we can use this setting, the dictionary being defined by a basis of $L_2$. This is for example the point of view in~\\cite{Alquier2008,BTW1,BTW2}.\n\\end{itemize}\nWe will here focus on the first setting, but an extension to the second one is quite straightforward. Let $\\{\\Phi_1,\\dots,\\Phi_D\\}$ be a family of probability measures over $\\mathbb{X}=\\mathbb{R}^d$. For $1\\leq i\\leq D$ we remind that\n$$ \\mu_{\\Phi_i}(\\cdot) = \\int k(x,\\cdot) \\Phi_i({\\rm d}x). $$\nDefine the measure $P_\\theta=\\mathcal{D}(\\theta;\\Phi_1,\\dots,\\Phi_D)=\\sum_{i=1}^D \\theta_i \\Phi_i $, and define the model $\\{P_\\theta,\\theta\\in\\Theta\\}$ with $\\Theta \\subseteq \\mathcal{S}_D=\\{\\theta\\in\\mathbb{R}_+^D: \\sum_{i=1}^D \\theta_i = 1 \\}$. The estimator is then\n$$\n\\hat{\\theta}_n = \\underset{\\theta \\in \\Theta}{\\arg\\min} \\left\\| \\sum_{\\ell=1}^D \\theta_\\ell \\mu_{\\Phi_\\ell}(\\cdot) -\\mu_{\\hat{P}_n} \\right\\|^2_{\\mathcal{H}_k}.\n$$\nAn application of Theorem~\\ref{theorem:mmd:briol:improved} leads to:\n\n\\begin{prop}\n Assume that $P_{\\theta} = \\sum_{i=1}^D \\theta_i \\Phi_i$ where $\\Phi_i$ is a probability distribution. Define the matrix $G_\\gamma = (\\left<\\mu_{\\Phi_i},\\mu_{\\Phi_j}\\right>_{\\mathcal{H}_{k_\\gamma}} )_{1\\leq i,j\\leq D} $. Letting $\\lambda_{\\min}(\\cdot) $ denote the smallest eigenvalue of a symmetric matrix, we have:\n $$ \\mathbb{P} \\left[ \\mathbb{D}_k\\left( P_{\\hat{\\theta}_n},P^0 \\right) \\leq \\inf_{\\theta\\in\\Theta} \\mathbb{D}_k\\left( P_{\\theta},P^0 \\right)\n + 2 \\frac{1 + \\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\sqrt{n}} \\right] \\geq 1-\\delta$$\n and, in the well specified case where $P^0 = P_{\\theta_0}$,\n $$ \\mathbb{P} \\left[ \\|\\hat{\\theta}-\\theta_0\\|^2 \\leq 2 \\frac{1 + \\sqrt{2\\log\\left(\\frac{1}{\\delta}\\right)} }{\\lambda_{\\min}(G_\\gamma) \\sqrt{n}} \\right] \\geq 1-\\delta\n .$$\n\\end{prop}\n\n\\subsection{$\\beta$-mixing observations}\n\nWe now consider non-independent random variables: as in the general framework presented above, $(X_t)_{t\\in \\mathbb{Z}}$ is a strictly stationary time series, with stationary distribution $P^0$, and we observe $X_1,\\dots,X_n$. We will exhibit some condition on the dependence of the $X_i$'s ensuring that we can still estimate $P^0$ with the MMD method.\n\nThere is a very rich literature on limit theorems and exponential inequalities under conditions on various dependence coefficients. Mixing coefficients and their applications are detailed in the monographs~\\cite{mixing,mixing2}, weak dependence coefficients in~\\cite{weakd}. In this subsection, we show that our coefficient $\\varrho_t$ can be upper-bounded by the $\\beta$-mixing coefficients. So for any $\\beta$-mixing process, the estimation of $P^0$ using MMD remains possible. We also remind some examples of $\\beta$-mixing processes. Note that we will show in the next subsection that Theorem~\\ref{theorem:mmd:1} can be successfully applied to non $\\beta$-mixing processes.\n\n\\subsubsection{$\\beta$-mixing and coefficients $\\varrho_t$}\n\nWe start by a reminder of the definition of the $\\beta$-mixing coefficients, from page 4 (Chapter 1) in~\\cite{weakd}.\n\\begin{dfn}\n\\label{dfn:beta}\nGiven two $\\sigma$-algebras $\\mathcal{A}$ and $\\mathcal{B}$,\n $$ \\beta(\\mathcal{A},\\mathcal{B}) = \\frac{1}{2} \\sup_{\n \\begin{tiny}\n \\begin{array}{c}\n I,J \\geq 1\n \\\\ U_1,\\dots,U_I\n \\\\ V_1,\\dots,V_J\n \\end{array}\n \\end{tiny}\n } \\sum_{1\\leq i \\leq I} \\sum_{1\\leq j \\leq J} |\\mathbb{P}(U_i\\cap V_j) - \\mathbb{P}(U_i)\\mathbb{P}(V_j)| $$\n where $(U_1,\\dots,U_I)$ is any partition of $\\mathcal{A}$ and $V_1,\\dots,V_j$ any partition of $\\mathcal{B}$. Put:\n $$\\beta_t^{(X)} = \\beta(\\sigma(X_0,X_{-1},\\dots),\\sigma(X_t,X_{t+1},\\dots)). $$\n\\end{dfn}\nSection 1.5 in~\\cite{mixing} provides summability conditions on the $\\beta_t^{(X)}$ leading to a law of large numbers and to a central limit theorem. Examples are also discussed.\n\\begin{exm}\nAssume in this example that $(X_t)$ is an homogeneous Markov chain given by its transition kernel $P$ and $X_0\\sim \\pi$ where $\\pi P = \\pi$. Assume that there is a $0