\n\n"},"lang":{"kind":"string","value":"code"}}},{"rowIdx":151,"cells":{"text":{"kind":"string","value":"भ्रष्ट इंडिया : सेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ रुपया कहां से दिया ?\nसेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ रुपया कहां से दिया ?\nनई दिल्ली। सेना द्वारा एक आरटीआई के जवाब ने यह प्रश्न खड़ा कर दिया है कि सेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ रुपया कहां से दिया ?\nदरअसल, सेना का कहना है कि उसने सैनिकों के वेतन से कोई चंदा नहीं दिया। ऐसे में उस चेक पर सवाल उठ रहा है, जिस पर लिखा गया था कि रकम सेना के वेतन से ली गई है।\nचार महीने पहले आर्मी चीफ दलबीर सिंह सुहाग ने प्रधानमंत्री राहत कोष के लिए १०० करोड़ रुपये का चेक दिया था। यह चेक उन्होंने खुद पीएम नरेंद्र मोदी को दिया। मगर अब सेना की तरफ से जारी बयान में कहा गया है कि सैनिकों की सैलरी से या अन्य तरीके से ऐसा कोई दान नहीं दिया गया है।\nदेहरादून के रहने वाले प्रभु डंडरियाल की तरफ से डाली गई आरटीआई के जवाब में आर्मी के सीपीआईओ लेफ्टिनेंट कर्नल राजीव गुलेरिया ने लिखा है संबंधित एजेंसी ने सूचित किया है कि सेना के जवानों के वेतन से प्रधानमंत्री राहत कोष में कोई अनुदान नहीं दिया गया है। यह मामला अभी विचाराधीन है।\nगौरतलब है कि प्रधानमंत्री नरेद्र मोदी के ऑफिस की वेबसाइट पर डाले गए ६७वें सेना दिवस समारोह की तस्वीरों में एक चेक दिख रहा था। जनरल सुहाग इस चेक को पीएम मोदी को सौंप रहे हैं। इस चेक में लिखा है, भारतीय सेना के सभी रैंक्स का एक दिन का वेतन।\nडंडरियाल ने अब प्रधानमंत्री कार्यालय से आरटीआई के जरिए इस चेक के बारे में जानकारी मांगी है। डंडरियाल ने कहा, जब २० मार्च तक दान के लिए वेतन से कुछ नहीं लिया गया था, तो आर्मी चीफ ने १०० करोड़ रुपये का चेक कैसे दे दिया? इसीलिए मैंने आरटीआई के जरिए पीएमओ से जवाब मांगा है।\nसेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ ...\nजानियेनेपाल जैसे भूकंप से ब्रज वसुंधरा का क्या...\nयूपी: आईपीएस अमिताभ ठाकुर के खिलाफ गवर्नर ने दिए ज..."},"lang":{"kind":"string","value":"hindi"}}},{"rowIdx":152,"cells":{"text":{"kind":"string","value":"रनवे पर उतरते समय दुर्घटनाग्रस्त हुआ प्रशिक्षु विमान, पायलट सुरक्षित - ट्रेनी प्लान क्रैश इन अमेठी\nअमेठीः इंदिरा गांधी राष्ट्रीय उड़ान अकादमी पर उतरते समय एक प्रशिक्षु विमान दुर्घटनाग्रस्त हो गया और उसमें आग लग गई। गनीमत रही कि ट्रेनी पायलट बाल-बाल बच गया।\nफुरसतगंज स्थित अकादमी के मुख्य प्रशासनिक अधिकारी संदीप पुरी ने बताया कि एक विमान रनवे पर उतरते समय फिसलकर घास के मैदान में चला गया और उसमें आग लग गई। विमान का कुछ हिस्सा क्षतिग्रस्त हो गया। विमान पायलट ने कूदकर अपनी जान बचाई।\nअधिकारियों के मुताबिक यह विमान चेकस्लोवाकिया का बना हुआ था और प्रशिक्षु पायलट की यह दूसरी अकेली उड़ान थी।"},"lang":{"kind":"string","value":"hindi"}}},{"rowIdx":153,"cells":{"text":{"kind":"string","value":"پادَر سٕہہ ( کٲشُر : /paːdar sɨh/ ) یا شیرِ بَبَر ( کٲشُر : /ʃeːri babar/ ) چھُ اَکھ جَنٛگلی جانوَر۔"},"lang":{"kind":"string","value":"kashmiri"}}},{"rowIdx":154,"cells":{"text":{"kind":"string","value":"مگر ٲخٕر کر ہے تہٕ کَرِ کیٛاہ"},"lang":{"kind":"string","value":"kashmiri"}}},{"rowIdx":155,"cells":{"text":{"kind":"string","value":"#!/bin/sh\nset -e\n[ -d \"build\" ] || mkdir \"build\"\ncd \"build\"\nfind .. -type f -name \"*.c\" | grep -Fxv \"../demo/demo.c\" | sort -R | C_INCLUDE_PATH=\"/usr/include/libxml2\" xargs -rd '\\n' \\\n\t\tgcc -Wall -Wextra -pedantic -std=c99 -O2 -shared -fPIC -lcrypto -lcurl -lxml2 -o \"libdecrypt.so\"\ngcc -Wall -Wextra -pedantic -std=c99 -O2 -L. -ldecrypt -o \"demo\" \"../demo/demo.c\"\ncp -t . \"../demo/test.\"{rsdf,ccf,dlc}\nLD_LIBRARY_PATH=\"`readlink -f .`\" ./demo\n"},"lang":{"kind":"string","value":"code"}}},{"rowIdx":156,"cells":{"text":{"kind":"string","value":"import { DocumentUnknown20 } from \"../../\";\n\nexport = DocumentUnknown20;\n"},"lang":{"kind":"string","value":"code"}}},{"rowIdx":157,"cells":{"text":{"kind":"string","value":"\\begin{document}\n\n\\begin{frontmatter}[classification=text]\n\n\\author[rk]{Robert Kleinberg }\n\\author[ds]{David E Speyer}\n\\author[ws]{Will Sawin}\n\n\\begin{abstract}\nLet $G$ be an abelian group. A tri-colored sum-free set in $G$ is a collection of triples $(\\vctr{a}_i, \\vctr{b}_i, \\vctr{c}_i)$ in $G$ such that $\\vctr{a}_i+\\vctr{b}_j+\\vctr{c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be the cyclic group of order $q$. Let $\\theta = \\min_{\\rho>0} (1+\\rho+\\cdots + \\rho^{q-1}) \\rho^{-(q-1)/3}$. Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (building on previous work of Croot, Lev and Pach, and of Ellenberg and Gijswijt) showed that a tri-colored sum-free set in $C_q^n$ has size at most $3 \\theta^n$. Between this paper and a paper of Pebody, we will show that, for any $\\delta > 0$, and $n$ sufficiently large, there are tri-colored sum-free sets in $C_q^n$ of size $(\\theta-\\delta)^n$. Our construction also works when $q$ is not prime. \n\\end{abstract}\n\\end{frontmatter}\n\n\\section{Introduction}\n\nLet $G$ be an abelian group. Let $\\vctr{t} \\in G^n$. We make the following slightly nonstandard definition: a \\newword{sum-free set in $G^n$ with target $\\vctr{t}$} is a collection of triples $(\\vctr{a}_i, \\vctr{b}_i, \\vctr{c}_i)$ in $G^n \\times G^n \\times G^n$ such that $\\vctr{a}_i+\\vctr{b}_j+\\vctr{c}_k=\\vctr{t}$ if and only if $i=j=k$.\nWe may always replace $(\\vctr{a}_i, \\vctr{b}_i, \\vctr{c}_i)$ by $(\\vctr{a}_i, \\vctr{b}_i, \\vctr{c}_i - \\vctr{t})$ to make the target $\\vctr{0}$ (as we did in the abstract, and as is more standard), but allowing an arbitrary target will simplify our notation.\nThe usual terminology is ``tri-colored sum-free set\", but we omit the reference to the coloring as we never consider any other kind.\n\nIf $X \\subset G^n$ is a set with no three-term arithmetic progressions, then $\\{ (\\vctr{x}, \\vctr{x}, -2 \\vctr{x}) : \\vctr{x} \\in X \\}$ is sum-free with target $\\vctr{0}$, so lower bounds on sets without three-term arithmetic progressions are also bounds on sum-free sets. The reverse does not hold: the largest known three-term arithmetic progression free subsets of $C_3^n$ (where $C_q$ is the cyclic group of order $q$) are of size $2.217^n$~\\cite{Edel}. Before this paper, the largest known sum-free sets in $C_3^n$ were of size $2.519^n$~\\cite{Alon-Shpilka-Umans}; this paper will raise the bound to $2.755^n$ and show that this bound is tight. \n\nLetting $r_3(G^n)$ denote the largest subset of $G^n$ with no three-term arithmetic progressions, the question of \nwhether $\\lim \\sup_{n \\to \\infty} r_3(G^n)^{1/n} < |G|$ was open, until recently, for every abelian $G$ containing elements of order greater than two.\nThe breakthrough work of Croot, Lev, and Pach~\\cite{CrootLP} introduced a polynomial method to \nprove that strict inequality holds when $G$ is cyclic of order 4,\nand Ellenberg and Gijswijt~\\cite{EllenbergG} built upon their ideas to prove it for cyclic groups of odd prime order.\nBlasiak et al.~\\cite{BCCGNSU} applied the same method to prove upper bounds for sum-free sets in $G^n$ for any fixed finite abelian group $G$.\n\nWe recall here one case of their bound. Let $C_q$ be the cyclic group of order $q$. Let $\\theta = \\min_{\\beta>0} (1+\\beta+\\cdots + \\beta^{q-1}) \\beta^{-(q-1)/3}$ and let $\\rho$ be the value of $\\beta$ at which the minimum is attained. We note that the minimum is attained at a unique point which belongs to $(0,1)$ because $(1+\\beta+\\cdots + \\beta^{q-1}) \\beta^{-(q-1)/3}$ approaches $\\infty$ as $\\beta$ goes to $0$ from above, is increasing on the interval $[1,\\infty)$, and has increasing first derivative on the interval $(0,1)$. \n\nThe following result of~\\cite{BCCGNSU} is closely related to the results of~\\cite{EllenbergG} (for primes) and~\\cite[Theorem 4]{Petrov} (for prime powers). (What we denote $\\theta$ is called $q J(q)$ in~\\cite{BCCGNSU}.)\n \n\\begin{theorem}[{\\cite[Theorem~4.14]{BCCGNSU}}] \\label{UpperBound}\nIf $q$ is a prime power, then sum-free sets in $C_q^n$ have size at most $3 \\theta^n$. \n\\end{theorem}\n\nPrior to this paper, it was not clear whether any of these applications of the polynomial method yielded tight\nbounds. In fact, Theorem \\ref{UpperBound} is tight to within a subexponential factor. \n\n\\begin{theorem} \\label{LowerBound}\nFix an integer $q \\geq 2$. Define $\\theta$ as above. For $n$ sufficiently large, there are sum-free sets in $C_q^n$ with size $\\geq \\theta^n e^{- 2 \\sqrt{(2 \\log 2 \\log \\theta ) n} - O_q(\\log n)}$.\n\\end{theorem}\n\nIn this paper, we show Theorem \\ref{LowerBound} except for a hypothesis on the existence of a probability distribution satisfying certain properties (Theorem \\ref{DistributionExists}). In~\\cite{Pebody}, Pebody will verify Theorem \\ref{DistributionExists}, completing the proof of Theorem \\ref{LowerBound}.\\footnote{\\cite{Pebody} was written in response to a preprint version of the paper which stated Theorem \\ref{DistributionExists} as a conjecture, and we have chosen to preserve the chronology here, though otherwise updating the paper to reflect his result.}\n\nThe question of whether Theorem \\ref{UpperBound} also yields a tight bound for $\\lim \\sup r_3(G^n)^{1/n}$ remains open.\n\nSum-free sets have applications in theoretical\ncomputer science, especially the circle of ideas surrounding fast matrix multiplication algorithms. The\n$O(n^{2.41})$ algorithm of Coppersmith and Winograd~\\cite{Coppersmith} rests on a combinatorial\nconstruction that can, in hindsight, be interpreted\\footnote{This interpretation was made explicit by Fu and\nKleinberg~\\cite{Fu}.} as a large sum-free set in $\\mathbb{F}_2^n$. In the same paper they presented a\nconjecture in additive combinatorics that, if true, would imply that \nthe exponent of matrix multiplication is 2, i.e., that there exist\nmatrix multiplication algorithms with running time $O(n^{2+\\epsilon})$\nfor any $\\epsilon>0$. This conjecture, along with another conjecture by\nCohn et al.~\\cite{CohnKSU} that also implies the exponent of matrix\nmultiplication is 2, was shown \nby Alon, Shpilka, and Umans~\\cite{Alon-Shpilka-Umans}\nto necessitate the existence of sum-free \nsets of size $3^{n-o(n)}$ in $\\mathbb{F}_3^n$. The upper bound on sum-free\nsets by Blasiak et al.~\\cite{BCCGNSU} thus refutes both of these conjectures.\nFurthermore, Blasiak et al.~\\cite{BCCGNSU} show that a more general family \nof proposed fast matrix multiplication algorithms based on the ``simultaneous\ntriple product property'' (STPP)~\\cite{CohnKSU} in an abelian group $H$ necessitates\nthe existence of sum-free sets of size $|H|^{1-o(1)}$. Their upper bound\non sum-free sets in abelian groups of bounded exponent thus precludes\nachieving matrix multiplication exponent 2 using STPP constructions in such\ngroups. \n\nA second application of sum-free sets in theoretical computer science concerns \nproperty testing, the study of randomized algorithms for distinguishing functions \n$f$ having a specified property from those which have large Hamming\ndistance from every function that satisfies the property. A famous example\nis the Blum-Luby-Rubinfeld (BLR) linearity tester~\\cite{BlumLR}, which\nqueries the function value at only $O(1/\\delta)$ points and\nsucceeds, with error probability less than $1/3$, in \ndistinguishing linear functions\non $\\mathbb{F}_2^n$ from those that have distance $\\delta \\cdot 2^n$ from any linear function.\nTesters which can distinguish low-degree polynomials on $\\mathbb{F}_2^n$\nfrom those that are far from any low-degree polynomial constitute an \nimportant ingredient in the celebrated PCP Theorem~\\cite{ALMSS}.\nBhattacharya and Xie~\\cite{BX} demonstrated that constructions\nof large sum-free sets in $\\mathbb{F}_2^n$ could be used to derive lower\nbounds on the complexity of testing certain linear-invariant properties\nof Boolean functions. \n\nFinally, sum-free sets have applications to removal lemmas in \nadditive combinatorics, a topic that is heavily intertwined with property testing.\nIn particular, Green~\\cite{Green05} proved an ``arithmetic removal lemma'' for\nabelian groups which implies that for every $\\epsilon>0$, there\nis a $\\delta>0$ such that for any abelian group $G$ and three subsets\n$A, B, C$, either there are at least $\\delta |G|^2$ distinct triples\n$(a,b,c) \\in A \\times B \\times C$ satisfying $a+b+c=0$, or one can\neliminate all such triples by deleting at most $\\epsilon |G|$ elements\nfrom each of $A,B,$ and $C$. Green's argument yields an upper bound\nfor $\\delta^{-1}$ which is a tower of twos of height polynomial in \n$\\epsilon^{-1}$. This bound can be improved using\ncombinatorial\\footnote{See~\\cite{FoxTR}, building upon the \ncombinatorial proof of Green's result in~\\cite{KralSV}.}\nor Fourier analytic\\footnote{See~\\cite{HatamiST}, which pertains to the\ncase $G=\\mathbb{F}_2^n$ and adapts the\nproof idea of~\\cite{FoxTR} to the analytic setting.} techniques,\nbut for general abelian groups $G$ the value of $\\delta$ is not bounded\nbelow by any polynomial function of $\\epsilon$. However,\nwhen $G$ is the group $\\mathbb{F}_q^n$, Fox and Lovasz~\\cite{FoxL} have applied our \nnearly-tight construction of sum-free sets in $G$ to obtain bounds of the form \n$$\n \\epsilon^{-C_q + o(1)} \\, < \\, \\delta^{-1} \\, <(\\epsilon/3)^{-C_q} ,\n$$\nwhere $C_q$ is a constant depending on $q$ but not $n$, and where $o(1)$ goes to $0$ as $\\epsilon$ goes to $0$ for any fixed $q$.\n\n\\section{Notation}\n\nThroughout this paper, we will use the following conventions: Lower case Roman letters denote integers, elements of cyclic groups (denoted $C_q$), of finite fields (denoted $\\mathbb{F}_q$), or general finite sets. Lower case Roman letters \nin boldface denote elements of $\\mathbb{Z}_{\\geq 0}^m$ (for any $m$), $C_q^m$ or $\\mathbb{F}_q^m$. Capital Roman letters denote subsets of $\\mathbb{Z}_{\\geq 0}^m$, $C_q^m$ or $\\mathbb{F}_q^m$. Lower case Greek letters denote real numbers; lower case Greek letters \n\nin boldface denote elements of $\\mathbb{R}^m$. \n\nA notation such as $\\alpha(x)$ or $\\vctr{\\alpha}(x)$ refers to a function of $x$ valued in real numbers, or real vectors.\nFor any sets $U$ and $V$, we write $U^V$ for the set of $U$-valued functions on $V$. All logarithms are to base $e$.\n\nWe fix a positive integer $q$. In section $4$, we will fix $n$ to be a positive integer divisible by $3$. The notation $O_q( \\ )$ will always refer to bounds as $n \\to \\infty$ through integers divisible by $3$, with $q$ fixed. Let $\\vctr{t} = (q-1, q-1, \\ldots, q-1) \\in \\mathbb{Z}_{\\geq 0}^n$.\n\nWe define the following sets of lattice points:\n\\[ \\begin{array}{rcl}\nI &= & \\{ 0,1,\\ldots, q-1 \\} \\subset \\mathbb{Z}_{\\geq 0} \\\\\nT &=& \\{ (a,b,c) \\in I^3 : a+b+c = q-1 \\} \\\\\n\n\\end{array} \\]\n\n\\section{Entropy}\n\nLet $A$ be a finite set and let $\\vctr{e} = (e_1, e_2, \\ldots, e_n) \\in A^n$. We define the probability distribution $\\vctr{\\sigma}(\\vctr{e})$ on $A$ by $\\vctr{\\sigma}_a(\\vctr{e}) = \\# \\{ r : e_r = a \\}/n$. In other words, $\\vctr{\\sigma}(\\vctr{e})$ is the probability distribution of uniformly randomly selecting an element of $\\vctr{e}$. \n\nLet $A$ be a finite set and $\\vctr{\\lambda} \\in \\mathbb{R}_{\\geq 0}^A$ a probability distribution on $A$. The \\newword{entropy}, $\\eta(\\vctr{\\lambda})$, is defined by\n\\[ \\eta(\\vctr{\\lambda}) = - \\sum_{a \\in A} \\vctr{\\lambda}_a \\log(\\vctr{\\lambda}_a) \\]\nwhere $0 \\log 0$ is considered to be $0$. \nThe importance of the entropy function in our situation is the following:\n\n\\begin{lemma} \\label{Histogram}\nLet $A$ be a finite set, and let $\\vctr{e}_0 \\in A^n$.\nThen \n\\[\nn \\eta(\\vctr{\\sigma}(\\vctr{e}_0)) - O_{|A|}(\\log n)\\leq \\log \\left( \\# \\left\\{ \\vctr{e} \\in A^n : \\vctr{\\sigma}(\\vctr{e}) = \\vctr{\\sigma}(\\vctr{e}_0) \\right\\} \\right) \\leq n \\eta(\\vctr{\\sigma}(\\vctr{e}_0)). \\]\n\\end{lemma}\n\nThe implied constant in $O$ depends only on $|A|$ and not on $n$ or $\\vctr{e}_0$.\n\n\\begin{proof}\nFor $a \\in A$, let $n_a = n \\vctr{\\sigma}_a(\\vctr{e}_0)$ be the number of times $a$ appears in $\\vctr{e}_0$. \n\nThe number of $\\vctr{e} \\in A^n$ such that $\\vctr{\\sigma}(\\vctr{e}) = \\vctr{\\sigma}(\\vctr{e}_0)$ is \nequal to the multinomial coefficient\n\\[\n\\binom{n}{(n_a)_{a \\in A}} : = \\frac{n!}{\\prod_{a \\in A} n_a!}.\n\\]\n\nFor the upper bound, we take one term from the multinomial formula\n\\[ n^n = \\left( \\sum_{a\\in A} n_a\\right)^n \\geq \\binom{n}{(n_a)_{a\\in A}} \\prod_{a \\in A} n_a^{n_a}, \\]\nso\n\\[ \\binom{n}{(n_a)_{a\\in A}} \\leq \\prod_{a \\in A} \\left(\\frac{n}{n_a}\\right)^{n_a} = \\exp(n \\eta(\\vctr{\\sigma}(e_0))).\\]\n\nFor the lower bound, we use the\nfollowing version of Stirling's formula.\n(See, e.g.,~\\cite{Robbins}.)\n\\[\n (n + \\tfrac12) \\log(n) - n + \\tfrac12 \\log(2 \\pi) \\;<\\; \\log(n!) \\;<\\; (n + \\tfrac12) \\log(n) - n + \\tfrac12 \\log(2 \\pi)+\\tfrac{1}{12}\n\\]\nApplying this estimate to each of the factorial terms, and using $\\sum_{a \\in A} n_a = n$\nwe find that\n\\[\n\n \\left|\n\\log \\binom{n}{(n_a)_{a \\in A}} - \n \\sum_{a \\in A} n_a \\log \\left( \\frac{n}{n_a} \\right) \n \\right|\n \\leq\n|A| \\left[ \\log(n) + \\log(2 \\pi) + \\frac{1}{6} \\right].\n\\]\nNote that $\\eta(\\vctr{\\sigma}(\\vctr{e}_0)) = \\sum_{a \\in A} \\frac{n_a}{n} \\log \\left( \\frac{n}{n_a} \\right)$, so this gives\n\\[\n \\left|\n\\log \\binom{n}{(n_a)_{a \\in A}} - \n n \\eta(\\vctr{\\sigma}(\\vctr{e}_0)) \n \\right|\n \\leq\n|A| \\left[ \\log(n) + \\log(2 \\pi) + \\frac{1}{6} \\right]. \\qedhere\n\\]\n\\end{proof}\n\nIf $A$ and $B$ are finite sets, $f: A \\to B$ is a map and $\\vctr{\\lambda}$ is a probability distribution on $A$, then we define the probability distribution $f_{\\ast} \\vctr{\\lambda}$ on $B$ by\n\\[ (f_{\\ast} \\vctr{\\lambda})_b = \\sum_{a \\in f^{-1}(b)} \\vctr{\\lambda}_a . \\]\n\nIt is well known that $\\eta(f_{\\ast} \\vctr{\\lambda}) \\leq \\eta(\\vctr{\\lambda})$, with strict inequality if there are distinct elements $a_1$ and $a_2 \\in A$ with $f(a_1) = f(a_2)$ and $\\vctr{\\lambda}_{a_1}$, $\\vctr{\\lambda}_{a_2} > 0$.\n\nWith $\\rho$ and $\\theta$ as defined before, \ndefine a probability distribution $\\vctr{\\psi}$ on $I$ by\n\\[ \\vctr{\\psi}_k = \\frac{\\rho^k}{1+\\rho+\\cdots + \\rho^{q-1}}. \\]\nLet $f: T \\to I$ be the map $f((i,j,k)) = k$. \nThe following is proved in ~\\cite{Pebody}.\\footnote{A proof was also claimed in a preprint \\cite{Norin}, but we are unable to confirm all the steps in the argument.}\n\n\\begin{theorem}[{\\cite[Theorem 4]{Pebody}}] \\label{DistributionExists}\nThere is an $S_3$-symmetric probability distribution $\\vctr{\\pi}$ on $T$ with $f_{\\ast} (\\vctr{\\pi}) = \\vctr{\\psi}$.\n\\end{theorem}\n\nMore precisely, \\cite{Pebody} proves that $\\vctr{\\psi},\\vctr{\\psi},\\vctr{\\psi}$ are compatible in the sense that there are random variables $X_1,X_2,X_3$ whose distributions are each $\\vctr{\\psi}$ and such that $X_1+X_2+X_3$ is constant. As each variable has expectation $(p-1)/3$, that constant is certainly $p-1$, so $(X_1,X_2,X_3)$ is a random $T$-valued variable. Its probability distribution is a probability distribution on $T$ whose three projections are each $\\vctr{\\psi}$. Symmetrizing it, we obtain an $S_3$-symmetric probability distribution on $T$ whose projection under $f$ is $\\vctr{\\psi}$, as stated in Theorem \\ref{DistributionExists}.\n\nWe will need to compute:\n\\begin{lemma} \\label{EntropyOfPsi}\nWith notation as above, $\\eta(\\vctr{\\psi}) = \\log \\theta$.\n\\end{lemma}\n\n\\begin{proof}\nNote that\n\\[ \\vctr{\\psi}_k = \\frac{\\rho^{k-(q-1)/3}}{\\theta}. \\]\n\nWe have\n\\begin{equation} \\label{eq:EntropyOfPsi.1}\n \\eta(\\vctr{\\psi}) =- \\sum_{k \\in I} \\vctr{\\psi}_k \\log \\frac{\\rho^{k-(q-1)/3}}{\\theta} = \\left( \\sum_{k \\in I} \\vctr{\\psi}_k \\right) \\log \\theta - \\left( \\sum_{k \\in I} (k-(q-1)/3) \\vctr{\\psi}_k \\right) \\log \\rho.\n\\end{equation}\nThe result follows by substituting \n\\begin{align*} \n & \\sum_{k \\in I} \\vctr{\\psi}_k = 1 \\\\\n\n & \\sum_{k \\in I} (k-(q-1)/3) \\vctr{\\psi}_k = \n \\frac{\\rho}{\\theta} \\cdot \n \\frac{d}{d \\beta} \\left[\n (1 + \\beta + \\cdots + \\beta^{q-1}) \\beta^{-(q-1)/3}\n \\right]_{\\beta = \\rho} = 0,\n\\end{align*}\n\ninto~\\eqref{eq:EntropyOfPsi.1}.\n\\end{proof}\n\n\\begin{remark}\nIf $\\vctr{\\pi}$ is any $S_3$-symmetric probability distribution on $T$ then $f_{\\ast} (\\vctr{\\pi})$ has expected value $\\tfrac{q-1}{3}$. Of all probability distributions on $I$ with expected value $\\tfrac{q-1}{3}$, the distribution $\\vctr{\\psi}$ has the greatest entropy. \n\\end{remark}\n\n\\section{The construction}\n\nLet $\\vctr{\\pi}$ be the probability distribution on $T$ guaranteed by Theorem~\\ref{DistributionExists}. Fix $n$ divisible by $3$, so that when $S_3$ acts on the lattice $\\mathbb{Z}^T$ by permuting the coordinates according to the $S_3$ action on $T$, the fixed point set of the action includes lattice vectors whose coordinates sum up to $n$. We can approximate $\\vctr{\\pi}$ to within $O_q(1/n)$ by an $S_3$-symmetric distribution $\\vctr{\\pi'}$ where the probability of each element is an integer multiple of $1/n$; such a $\\vctr{\\pi}'$ can be found by scaling down $\\mathbb{Z}^T$ by $1/n$, taking the set of $S_3$-fixed points that belong to the probability simplex, and selecting the closest such point to $\\vctr{\\pi}$. Then the marginal distribution $\\vctr{\\psi'}$ will be within $O_q(1/n)$ of $\\vctr{\\psi}$. The entropy function of a probability distribution, viewed as function of the vector of the probabilities of the elements, is a differentiable function on the open set of probability distributions assigning positive probability to every element. Thus, because $\\vctr{\\psi}$ assigns positive probability to each element, the entropy is Lipschitz in a neighborhood of $\\vctr{\\psi}$. For large enough $n$, $\\vctr{\\psi'}$ is in that neighborhood, so \n\\begin{equation} \\label{EntropyOfPsiPrime}\n \\eta(\\vctr{\\psi'}) = \\eta(\\vctr{\\psi}) - O_q(1/n) = \\log \\theta - O_q(1/n) .\n\\end{equation} \n(The second equality is~\\Cref{EntropyOfPsi}.)\n\nDefine the following sets:\n\\begin{align*}\n W &= \\{ \\vctr{a} \\in I^n : \\vctr{\\sigma}(\\vctr{a}) = \\vctr{\\psi'} \\} \\\\\n V &= \\{ (\\vctr{a}, \\vctr{b}, \\vctr{c}) \\in W^3 : \\vctr{a}+\\vctr{b}+\\vctr{c} = \\vctr{t} \\} .\n\\end{align*}\nWe will show in Lemma~\\ref{LowerBound0} that $|V|$ and $|W|$ grow exponentially in $n$, with $|V|$ having the faster growth rate. \nOur sum-free set in $C_q^n$ will be a subset of $V$.\n\nLet $p$ be a prime number between $4 |V|/|W|$ and $8 |V|/|W|$ (such a prime exists by Bertrand's postulate). Since $|V|$ grows faster than $|W|$, the prime $p$ goes to $\\infty$ as $n$ does. Let $S$ be a subset of $\\mathbb{F}_p$ having no three distinct elements in arithmetic\nprogression. \nBehrend's construction~\\cite{Behrend}, with Elkin's improvement~\\cite{Elkin}, implies that, for $p$ sufficiently large one can \nchoose such a set whose cardinality is at least $p \\cdot e^{-2\\sqrt{2 \\log 2 \\log p}}$.\n\nLet ${h}: \\mathbb{Z}^{n+2} \\to \\mathbb{F}_p$ be a linear map, chosen uniformly at random from all such linear maps.\nFor any $(\\vctr{a},\\vctr{b},\\vctr{c}) \\in V$, the sequence \n\\[\n {h}(0,1,\\vctr{a}), \\;\\; \\tfrac12 {h}(1,1,\\vctr{t}- \\vctr{b}), \\;\\; {h}(1,0,\\vctr{c})\n\\]\nconstitutes a (possibly degenerate) arithmetic progression in $\\mathbb{F}_p$. Thus,\nthis arithmetic progression is contained in $S$ if and only if its three terms are\nall equal to one another and lie in $S$. Define $V'$ to be the subset of $V$ given by\n\\[\nV' = {\\Big \\{} (\\vctr{a},\\vctr{b},\\vctr{c}) \\in W^3 : \\begin{array}{l} \\vctr{a}+\\vctr{b}+\\vctr{c} = \\vctr{t} \\\\ {h}(0,1,\\vctr{a}) = \\tfrac{1}{2} {h}(1,1,\\vctr{t}- \\vctr{b}) =\n{h}(1,0,\\vctr{c}) \\in S \\end{array} {\\Big \\}} .\n\\]\n\nDefine \n$V''$ to be the set of all $(\\vctr{a},\\vctr{b},\\vctr{c}) \\in V'$ \nsuch that every other $(\\vctr{a}',\\vctr{b}',\\vctr{c}') \\in V'$ obeys\n$\\vctr{a'} \\neq \\vctr{a}, \\vctr{b'} \\neq \\vctr{b}, \\vctr{c'} \\neq \\vctr{c}$.\n\n\\begin{remark}\nFor this remark, assume $q$ is odd. Define a tri-colored $3$-AP-free set in $C_q^n$ to be a set of triples $(\\vctr{a}_i, \\vctr{b}'_i, \\vctr{c}_i)$ in $(C_q^n)^3$ such that $\\vctr{a}_i + \\vctr{c}_k = 2 \\vctr{b}'_j$ if and only if $i=j=k$. Replacing $(\\vctr{a}_i, \\vctr{b}_i, \\vctr{c}_i)$ with $(\\vctr{a}_i, \\tfrac{1}{2}(\\vctr{t} - \\vctr{b}) \\bmod q, \\vctr{c}_j)$ turns any tri-colored sum-free set into a tri-colored $3$-AP-free set. \nIn our set $V''$, each of $\\vctr{a}$, $\\vctr{b}$ and $\\vctr{c}$ has entries distributed over $I$ with probability distribution $\\vctr{\\psi}$. Therefore in the tri-colored $3$-AP free set, the entries of $\\vctr{a}$ and $\\vctr{c}$ will be distributed with probability $\\vctr{\\psi}$, but the entries of $\\vctr{b}$ will be distributed with the different distribution $g_{\\ast} \\vctr{\\psi}$ where $g: I \\to I$ is the map $g(b) = \\tfrac{1}{2} (q-1-b) \\bmod q$. By contrast, if $X \\subset C_q^n$ is a $3$-AP-free set in the standard sense, then $\\{ (\\vctr{x}, \\vctr{x}, \\vctr{x}) : \\vctr{x} \\in X \\}$ is a tri-colored $3$-AP-free set but, for this tri-colored $3$-AP-free set, each of the three components has the same distribution. This discrepancy suggests that it may be hard to lift our constructions out of the colored setting.\n\\end{remark}\n\nThe set $V''$ will be our sum-free set. We verify that it is sum-free in~\\Cref{LowerBound1}.\n\n\\begin{lemma} \\label{PsiExpectation}\nFor any $\\vctr{a} = (a_1,a_2, \\ldots, a_n) \\in W$, we have $\\sum a_i = n(q-1)/3$.\n\\end{lemma}\n\n\\begin{proof}\nBy definition, $\\vctr{\\sigma}(\\vctr{a})=\\vctr{\\psi}'$, so we want to show the expected value of the distribution $\\vctr{\\psi}'$ is $(q-1)/3$. But $\\vctr{\\psi}'$ is the marginal of the $S_3$ symmetric distribution $\\vctr{\\pi}'$ on $T$. As $\\vctr{\\pi'}$ is a symmetric distribution for a triple of random variables summing to $q-1$, the expectation of each variable must be $(q-1)/3$.\n\\end{proof}\n\n\\begin{lemma} \\label{LowerBound1}\nFor any choice of the map ${h}$, the set $V''$ is a sum-free set\nwith target $\\vctr{t}$ in $C_q^n$. \n\\end{lemma}\n\\begin{proof}\nSuppose that we have three (not necessarily distinct) triples \n$(\\vctr{a}_i,\\vctr{b}_i,\\vctr{c}_i) \\, (i=0,1,2)$ in $V''$ such that\n$\\vctr{a}_0 + \\vctr{b}_1 + \\vctr{c}_2 = \\vctr{t}$ in $C_q^n$. \n\nWe claim that we also have $\\vctr{a}_0 + \\vctr{b}_1 + \\vctr{c}_2 = \\vctr{t}$ in $\\mathbb{Z}^n$.\nBy~\\Cref{PsiExpectation}, the entries of $\\vctr{a}_0$, $\\vctr{b}_1$ and $\\vctr{c}_2$ each sum to $n(q-1)/3$ (in $\\mathbb{Z}$) so the sum of all the entries of $\\vctr{a}_0 + \\vctr{b}_1 + \\vctr{c}_2$ (with the sum taken in $\\mathbb{Z}$) must be $n(q-1)$.\nNow the sum $\\vctr{a}_0 + \\vctr{b}_1 + \\vctr{c}_2$ in $\\mathbb{Z}^n$ has each entry congruent to $q-1$ mod $q$, by the assumption $\\vctr{a}_0 + \\vctr{b}_1 + \\vctr{c}_2 = \\vctr{t}$ in $C_q^n$, and each entry is nonnegative, because the entries of $\\vctr{a}_0,\\vctr{b}_1,$ and $\\vctr{c}_2$ are nonnegative. So each entry is at least $q-1$. We just saw that the sum of all the entries is $n(q-1)$, so each entry is exactly $q-1$, as claimed.\n\nNow that we know $\\vctr{a}_0 + \\vctr{b}_1 + \\vctr{c}_2 = \\vctr{t}$ in $\\mathbb{Z}^n$, we deduce that \n$\\left( {h}(0,1,\\vctr{a}_0), \\tfrac{1}{2} {h}(1,1,\\vctr{t}-\\vctr{b}_1), {h}(1,0,\\vctr{c}_2) \\right)$ is an arithmetic progression in $\\mathbb{F}_p$. \nSince $(\\vctr{a}_0,\\vctr{b}_0,\\vctr{c}_0) \\in V'$, we have $\\vctr{a}_0 \\in W$ and ${h}(0,1,\\vctr{a}_0) \\in S$. Similarly, \n$\\vctr{b}_1, \\, \\vctr{c}_2 \\in W$ and $\\tfrac{1}{2} {h}(1,1,\\vctr{t}-\\vctr{b}_1), \\, {h}(1,0,\\vctr{c}_2) \\in S$.\nSo $\\left( {h}(0,1,\\vctr{a}_0), \\tfrac{1}{2} {h}(1,1,\\vctr{t}-\\vctr{b}_1), {h}(1,0,\\vctr{c}_2) \\right)$ is a (possibly degenerate) arithmetic progression in $S$.\nAs $S$ is arithmetic-progression-free, we must have $ {h}(0,1,\\vctr{a}_0)= \\tfrac{1}{2} {h}(1,1,\\vctr{t}-\\vctr{b}_1)= {h}(1,0,\\vctr{c}_2) \\in S$. \nWe have now checked that $(\\vctr{a}_0, \\vctr{b}_1, \\vctr{c}_2)$ obeys all the conditions to be an element of $V'$.\n\nNow, recalling the definition of $V''$ and the fact that $(\\vctr{a}_i,\\vctr{b}_i,\\vctr{c}_i)\\in V'$ for $i=0$, $1$, $2$,\nwe may conclude that $(\\vctr{a}_i,\\vctr{b}_i,\\vctr{c}_i) = (\\vctr{a}_0, \\vctr{b}_1, \\vctr{c}_2)$ for\n$i=0$, $1$, $2$. In other words, the three triples $(\\vctr{a}_0,\\vctr{b}_0,\\vctr{c}_0)$, \n$(\\vctr{a}_1,\\vctr{b}_1,\\vctr{c}_1)$ and $(\\vctr{a}_2,\\vctr{b}_2,\\vctr{c}_2)$ are\nall equal to one another. \n\\end{proof}\n\nWe will now begin to estimate the expected value of $|V''|$.\n\n\\begin{lemma} \\label{LowerBound0}\nWe have\n\\[\n |V| \\; \\geq \\; \\exp(\\eta(\\vctr{\\pi}') n - O_q(\\log n)) \n\\]\nand\n\\[\n \\exp(\\eta(\\vctr{\\psi'}) n) \\; \\geq \\; |W| \n \\; \\geq \\; \\exp(\\eta(\\vctr{\\psi}')n - O_q(\\log n)) .\n\\]\n\\end{lemma}\n\nSince $\\vctr{\\psi}' = f_{\\ast} \\vctr{\\pi}'$, we have $\\eta(\\vctr{\\pi}') \\geq \\eta(\\vctr{\\psi}')$. Moreover, if $n$ is large enough that the distribution $\\vctr{\\pi}'$ is not a point-mass on $(\\frac{q-1}{3},\\frac{q-1}{3},\\frac{q-1}{3})$, then we have strict inequality since $\\vctr{\\pi}'$ is $S_3$-symmetric, so $\\vctr{\\pi}'_{ijk}>0$ implies $\\vctr{\\pi}'_{jik}>0$. This establishes the previous claim that $|V|$ and $|W|$ grow exponentially, with $|V|$ having the faster rate.\n\\begin{proof}\nSince $W= \\{ \\vctr{e} \\in I^n : \\vctr{\\sigma}(\\vctr{e}) = \\vctr{\\psi'} \\} $, the lower and upper bounds for $|W|$ follow from \\Cref{Histogram}.\nWe now need to establish the lower bound for $V$.\n\nLet $V_0 = \\{ \\vctr{f} \\in T^n : \\vctr{\\sigma}(\\vctr{f}) = \\vctr{\\pi'} \\}$. An element of $T^n$ is an $n$-tuple of triples of integers $((a_1, b_1, c_1), (a_2, b_2, c_2), \\ldots, (a_n, b_n, c_n))$ with $a_i+b_i+c_i = q-1$. Reorganizing these integers as $((a_1, a_2, \\ldots, a_n), (b_1, b_2, \\ldots, b_n), (c_1, c_2, \\ldots, c_n))$, we obtain a triple of length $n$ vectors $\\vctr{a}$, $\\vctr{b}$ and $\\vctr{c}$ with $\\vctr{a}+\\vctr{b}+\\vctr{c}= \\vctr{t}$. Let us apply this construction to some $\\vctr{f}$ in $V_0$ to get some $\\vctr{a}$, $\\vctr{b}$ and $\\vctr{c}$. Since $\\vctr{\\pi}'$ is $S_3$ symmetric, we have $\\vctr{\\sigma}(\\vctr{a}) = \\vctr{\\sigma}(\\vctr{b}) = \\vctr{\\sigma}(\\vctr{c}) = \\vctr{\\psi}'$ so $\\vctr{a}$, $\\vctr{b}$ and $\\vctr{c}$ lie in $W$ and $(\\vctr{a}, \\vctr{b}, \\vctr{c}) \\in V$. This construction gives an injection from $V_0$ into $V$, so $|V| \\geq |V_0|$.\n\nBy~\\Cref{Histogram}, $|V_0| = \\exp(\\eta(\\vctr{\\pi}') n - O_q(\\log n))$, so $|V| \\geq \\exp(\\eta(\\vctr{\\pi}') n - O_q(\\log n))$ as desired.\n\\end{proof}\n\n\\begin{lemma} \\label{LinIndep}\n\nSuppose $p > q$.\nFor any two distinct elements $(\\vctr{a},\\vctr{b},\\vctr{c})$,\n$(\\vctr{a}',\\vctr{b}',\\vctr{c}') \\in V$, the $(n+2) \\times 6$-matrix over \n$\\mathbb{F}_p$ given by\n\\[\n M = \\begin{pmatrix}\n\n 0 & 0 & 1/2 & 1/2 & 1 & 1 \\\\\n1 & 1 & 1/2 & 1/2 & 0 & 0 \\\\\n \\vctr{a} & \\vctr{a}' & (\\vctr{t}-\\vctr{b})/2 & (\\vctr{t}- \\vctr{b}')/2 &\n\\vctr{c} &\\vctr{c}' \\\\\n \\end{pmatrix}\n\\]\nhas rank at least $3$.\n\\end{lemma}\n\n\\begin{proof}\nThe first two rows already have rank $2$, so we simply must show that the bottom $n$ rows are not all in the span of the first two. \nIf the bottom $n$ rows were in the span of the first two, then modulo $p$ the first column would equal the second, the third column equal the fourth, and the fifth column equal the sixth. Since the entries of the matrix are between $0$ and $q-1$, and $p > q$, equality of columns modulo $p$ implies outright equality. This gives $\\vctr{a} = \\vctr{a}'$, $\\vctr{b} = \\vctr{b}'$ and $\\vctr{c}=\\vctr{c}'$, contrary to our assumption that $(\\vctr{a},\\vctr{b},\\vctr{c})$ and\n$(\\vctr{a}',\\vctr{b}',\\vctr{c}')$ are distinct.\n\\end{proof}\n\n\\begin{lemma} \\label{LowerBound2}\n\nWhen $p >q $ and ${h}$ is a uniformly random homomorphism of $\\mathbb{Z}^{n+2}$ to $\\mathbb{F}_p$,\nthe expected cardinality of $V''$ is at least $\\frac{1}{32}e^{-2\\sqrt{2 \\log 2 \\log p}} \\cdot |W|$.\n\\end{lemma}\n\\begin{proof}\nFor any $(\\vctr{a},\\vctr{b},\\vctr{c}) \\in V$, we want to compute the probability that\nthere exists $s \\in S$ such that \n\\begin{equation} \\label{eq:LB2.1}\n {h}(0,1,\\vctr{a}) = \\tfrac{1}{2} {h}(1,1, \\vctr{t}-\\vctr{b}) = {h}(1,0,\\vctr{c}) = s.\n\\end{equation}\nFurthermore, since ${h}(0,1,\\vctr{a}), \\, \\frac12 {h}(1,1, \\vctr{t}-\\vctr{b}), \\,\n{h}(1,0,\\vctr{c})$ always form a (possibly degenerate) arithmetic\nprogression, if any two of these values are equal to $s$ then the third one \nequals $s$ as well. The vectors $(0,1,\\vctr{a})$ and $(1,0,\\vctr{c})$ are \nlinearly independent modulo $p$, so the pair \n$({h}(0,1,\\vctr{a}), {h}(1,0,\\vctr{c}))$ is \nuniformly distributed in $\\mathbb{F}_p^2$ and the probability\nthat~\\eqref{eq:LB2.1} is satisfied for a fixed $s \\in S$ is $p^{-2}$. Summing\nover all $(\\vctr{a},\\vctr{b},\\vctr{c}) \\in V$ and $s \\in S$ we obtain\n\\begin{equation} \\label{eq:LB2.2}\n \\mathbb{E} ( |V'| ) = \\frac{|V| |S| }{ p^2}.\n\\end{equation}\nAn element $(\\vctr{a},\\vctr{b},\\vctr{c}) \\in V'$ belongs to $V''$ unless\nthere exists some other $(\\vctr{a}', \\vctr{b}', \\vctr{c}') \\in V'$ such that\none of the equations $\\vctr{a}=\\vctr{a}', \\, \\vctr{b}=\\vctr{b}'$, or\n$\\vctr{c}=\\vctr{c}'$ holds. In order for any such equation to hold, it\nmust be the case that there is a single element $s \\in S$ such that \n\\begin{equation} \\label{eq:LB2.3}\n s = {h}(0,1,\\vctr{a}) = {h}(0,1,\\vctr{a}') =\n \\tfrac12 {h}(1,1,\\vctr{t}-\\vctr{b}) = \\tfrac12 {h}(1,1,\\vctr{t}-\\vctr{b}') =\n {h}(1,0,\\vctr{c}) = {h}(1,0,\\vctr{c}').\n\\end{equation}\nBy \\Cref{LinIndep}, the six-tuple $({h}(0,1,\\vctr{a}), \\, {h}(0,1,\\vctr{a}'), \\,\n \\tfrac12 {h}(1,1,\\vctr{t}-\\vctr{b}) , \\, \\tfrac12 {h}(1,1,\\vctr{t}-\\vctr{b}') , \\,\n {h}(1,0,\\vctr{c}) , \\, {h}(1,0,\\vctr{c}'))$\nis uniformly distributed on a subspace of $\\mathbb{F}_p^6$ of dimension\nat least 3. \nHence, for any $(\\vctr{a},\\vctr{b},\\vctr{c}),(\\vctr{a}',\\vctr{b}',\\vctr{c}')\\in V$ and for a fixed $s$, the probability that~\\eqref{eq:LB2.3} holds is \nat most $p^{-3}$. \nThe probability that there exists some $s$ for which~\\eqref{eq:LB2.3} holds is thus bounded by $|S| p^{-3}$. \n\nFor any $(\\vctr{a},\\vctr{b},\\vctr{c}) \\in V$, the number of elements $(\\vctr{a}',\\vctr{b}',\\vctr{c}') \\in V$ such that $\\vctr{a}' = \\vctr{a}$ is equal to $|V|/|W|$.\n(To see this, note that the group $S_n$ acts on $V$ and $W$ by \npermuting the coordinates of vectors. These actions are compatible with\nthe projection map $V \\to W$ defined by \n$(\\vctr{a},\\vctr{b},\\vctr{c}) \\mapsto \\vctr{a}$.\nThe fibers of this projection map must be equinumerous\nbecause the action of $S_n$ on $W$ is transitive.)\nThus, for any $(\\vctr{a},\\vctr{b},\\vctr{c}) \\in V$\nthe probability that $(\\vctr{a},\\vctr{b},\\vctr{c})$ belongs \nto $V'$ but not $V''$\nbecause it ``collides'' with another ordered triple of the form $(\\vctr{a},\\vctr{b}',\\vctr{c}')$\nin $V'$ is bounded above by $\\frac{|V|}{|W|} |S| p^{-3}$. \nThe analogous counting argument applies to collisions with triples of the form\n$(\\vctr{a}',\\vctr{b},\\vctr{c}')$ and $(\\vctr{a}',\\vctr{b}',\\vctr{c})$.\nSumming over $|V|$ choices of $(\\vctr{a},\\vctr{b},\\vctr{c})$,\n\nwe find that the expected cardinality\nof $V' \\setminus V''$ is bounded above by \n\\[\n 3 |V| \\frac{|V|}{|W|} |S| p^{-3} =\n \\frac{3 |V|}{p|W|} \\cdot \\frac{|V| |S|}{p^2} <\n \\frac{3}{4} \\cdot \\mathbb{E} ( |V'| ).\n\\]\nThus,\n\\[\n \\mathbb{E}( |V''|) \\geq \\frac14 \\mathbb{E} (|V'|) = \\frac{|V| |S|}{4 p^2} = \\frac14 \\cdot \\frac{|V|}{p} \\cdot \\frac{|S|}{p}\n > \\frac{e^{-2\\sqrt{2 \\log 2 \\log p}}}{32} \\cdot |W|.\n\\]\n\\end{proof}\n\nWe now prove our main theorem.\n\n\\begin{theorem} \nIf $n$ is sufficiently large then \nthere exists a sum-free set in $C_q^n$ with target $\\vctr{t}$ \nwhose size is greater than $\\theta^n e^{ - 2 \\sqrt{2\\log 2 \\log \\theta \\ n } - O_q(\\log n)}$.\n\\end{theorem}\n\\begin{proof}\nThe random set $V''$ constructed above is a sum-free set in $C_q^n$\nwith target $\\vctr{t}$ (\\Cref{LowerBound1}) and its expected size\nis greater than $\\frac{1}{32} e^{-2\\sqrt{2 \\log 2 \\log p}}\\cdot |W|$ (\\Cref{LowerBound2}), because we may take $n$ large enough that $p>q$. \nUsing \\Cref{LowerBound0} we have \n\\[\n |W| \\geq \\exp(\\eta(\\vctr{\\psi}') \\, n - O_q(\\log n) ) \\geq \\exp((\\log \\theta - O_q(1/n)) \\, n - O_q(\\log n)) \\geq \\theta^n \\exp( - O_q(\\log n))\n\\]\nfor all sufficiently large $n$. \nThe inequality $|V| \\leq |W|^2$ holds because \nthe projection map $V \\to W^2$ defined by \n$(\\vctr{a},\\vctr{b},\\vctr{c}) \\mapsto (\\vctr{a},\\vctr{b})$\nis one-to-one. This justifies the second inequality in\n\\[\n p < 8 \\frac{|V|}{|W|} \\leq 8 |W| < 8 \\exp(\\eta(\\vctr{\\psi}') \\, n)\n \\leq 8 \\exp ( (\\log \\theta + O_q(1/n) ) \\, n ),\n\\]\nwhile the third inequality follows from \\Cref{LowerBound0}.\nTaking logarithms of both sides, we deduce that \n$\\log p < n \\log \\theta + O_q(1)$, and hence\n\\[\ne^{- 2 \\sqrt{2 \\log 2 \\log p} }> e^{- 2 \\sqrt{2 \\log 2 (n \\log \\theta + O_q(1))} } > e^{- 2 \\sqrt{2 \\log 2 \\log \\theta \\ n} -O_q(1/\\sqrt{n})} .\n\\]\nHence,\n\\[\n \\mathbb{E} (|V''|) > \\frac{1}{32} e^{- 2 \\sqrt{2 \\log 2 \\log \\theta \\ n} -O_q(1/\\sqrt{n})}\n \\cdot |W| \n \\geq \\theta^n e^{ - 2 \\sqrt{2 \\log 2 \\log \\theta \\ n} - O_q(\\log n)}\n\\]\nfor sufficiently large $n$. The theorem follows because there must exist at least one choice of \n${h}$ for which the cardinality of the random set $V''$ is at least as large as its expected value.\n\\end{proof}\n\nIt follows from Roth's theorem that our construction produces sum-free sets $V'' \\subseteq V$ of size $\\mathbb{E}(|V''|) \\leq\\mathbb{E}(V') = \\frac{V |S|}{p^2}=o(|W|)$ regardless of how we choose $S$. We do not know if an arbitrary sum-free set contained in $V$ must have size $o(|W|)$, only the trivial bound $ |W|$. It would be interesting to improve this situation. \n\n\\begin{dajauthors}\n\\begin{authorinfo}[rk]\nRobert Kleinberg \\\\\nDepartment of Computer Science \\\\ Cornell University \\\\ Ithaca, NY 14853, USA \\\\\nrobert\\imagedot{}kleinberg\\imageat{}cornell\\imagedot{}edu \\\\\n\\end{authorinfo}\n\\begin{authorinfo}[ws]\nWill Sawin \\\\\nETH Institute for Theoretical Studies \\\\ ETH Zurich \\\\ 8092 Z\\\"{u}rich, Switzerland \\\\\nwilliam\\imagedot{}sawin\\imageat{}math\\imagedot{}ethz\\imagedot{ch} \\\\\n\\end{authorinfo}\n\\begin{authorinfo}[ds]\nDavid E Speyer \\\\\nDepartment of Mathematics \\\\ University of Michigan \\\\ Ann Arbor, MI 48109, USA \\\\\nspeyer\\imageat{}umich\\imagedot{}edu \\\\\n\\end{authorinfo}\n\\end{dajauthors}\n\n\\end{document}"},"lang":{"kind":"string","value":"math"}}},{"rowIdx":158,"cells":{"text":{"kind":"string","value":"Sidney Carlow [Parents] was born in 1869 in Birmingham, Warwickshire, England. He died in 1900 in Birmingham, Warwickshire, England. He married Annie Hardle on 20 Dec 1891 in St Davids, Birmingham, Warwickshire, England.\nAnnie Hardle [Parents] was born in 1870 in Birmingham, Warwickshire, England. She died in 1944 in Birmingham, Warwickshire, England. She married Sidney Carlow on 20 Dec 1891 in St Davids, Birmingham, Warwickshire, England.\nM ii Henry Carlow was born in 1895 in Birmingham, Warwickshire, England.\nM iv Edward Carlow was born in 1899 in Birmingham, Warwickshire, England.\nHenry Hardle was born in 1844 in Restbury, Gloucestershire, England.\nM ii Thomas H. Hardle was born in 1874 in Birmingham, Warwickshire, England.\nSidney Thomas Carlow [Parents] was born in 1893 in Birmingham, Warwickshire, England. He died on 10 May 1951 in Birmingham, Warwickshire, England. He married Amy Pamela Foster in 1921 in Birmingham, Warwickshire, England.\nAmy Pamela Foster was born on 13 Jun 1903 in Birmingham, Warwickshire, England. She died in 1975 in Birmingham, Warwickshire, England. She married Sidney Thomas Carlow in 1921 in Birmingham, Warwickshire, England.\nSamuel Nathaniel Skipp [Parents] was born in 1871 in Birmingham, Warwickshire, England. He died in 1944 in Birmingham, Warwickshire, England. He married Annie Hardle on 5 Apr 1906 in Bishop Ryder, Birmingham, Warwickshire,England.\nAnnie Hardle [Parents] was born in 1870 in Birmingham, Warwickshire, England. She died in 1944 in Birmingham, Warwickshire, England. She married Samuel Nathaniel Skipp on 5 Apr 1906 in Bishop Ryder, Birmingham, Warwickshire,England.\nM i Samuel Skipp was born in 1907 in Birmingham, Warwickshire, England.\nM ii Nathaniel Skipp was born in 1908 in Birmingham, Warwickshire, England.\nWilliam Lester was born in 1807 in Wolstan, Warwickshire, England. He married Maria about 1832.\nMaria was born in 1808 in Coventry, Warwickshire, England. She married William Lester about 1832.\nM i William Lester was born in 1834 in Coventry, Warwickshire, England.\nM ii John Lester was born in 1837 in Coventry, Warwickshire, England.\nF iii Catherine Lester was born in 1838 in Coventry, Warwickshire, England.\nF iv Phoebe Lester was born in 1841 in Coventry, Warwickshire, England.\nF vi Maria Lester was born in 1845 in Birmingham, Warwickshire, England.\nM vii George Lester was born in 1847 in Birmingham, Warwickshire, England.\nF viii Mary Ann Lester was born in 1849 in Birmingham, Warwickshire, England.\nM ix Samuel Lester was born in 1849 in Birmingham, Warwickshire, England.\nJoseph William Wootton [Parents] was born in 1888 in Birmingham, Warwickshire, England. He married Amelia Maria Hardiker on 7 Mar 1909 in St Saviours, Birmingham, Warwickshire, England.\nAmelia Maria Hardiker [Parents] was born in Birmingham, Warwickshire, England. She was christened on 4 May 1890 in Birmingham, Warwickshire, England. She married Joseph William Wootton on 7 Mar 1909 in St Saviours, Birmingham, Warwickshire, England.\nM i William Wootton was born in 1909 in Birmingham, Warwickshire, England."},"lang":{"kind":"string","value":"english"}}},{"rowIdx":159,"cells":{"text":{"kind":"string","value":"अमरनाथ यात्रा की तैयारी से महबूबा मुफ्ती को परेशानी जताई नाराज़गी\nअमरनाथ यात्रा सालों से होती आ रही है,लेकिन दुर्भाग्य से जो इंतजाम इस साल किए गए हैं, वह कश्मीर के लोगों के खिलाफ हैं-महबूबा मुफ्ती\nपीडीपी नेता महबूबा मुफ्ती ने अमरनाथ यात्रा के लिए सरकार द्वारा किए गए इंतजामों पर नाराजगी जताई है और कहा है कि ये कश्मीर के लोगों के खिलाफ हैं। उन्होंने कहा, अमरनाथ यात्रा सालों से होती आ रही है। लेकिन दुर्भाग्य से जो इंतजाम इस साल किए गए हैं, वह कश्मीर के लोगों के खिलाफ हैं। इससे स्थानीय लोगों को अपनी रोजमर्रा की जिंदगी में परेशानी का सामना करना पड़ा रहा है। मैं राज्यपाल से इस मामले पर संज्ञान लेने के लिए अनुरोध करती हूं।\nघाटी के पहलगाम और बालटाल आधार शिविरों में अमरनाथ यात्रियों की संख्या ज्यादा हो जाने के कारण जम्मू के भगवती नगर आधार शिविर से जाने वाला यात्रियों का जत्था सोमवार को यहां से नहीं भेजा जाएगा। हालांकि देर रात तक इस बारे में कोई आधिकारिक सूचना जारी नहीं की गई थी।\nये भी पढ़े:खेत में मिला युवक का शव,लाश के पास मिला तमंचा और पांच कारतूस\nपुलिस और प्रशासनिक सूत्रों के अनुसार, भगवती नगर आधार शिविर से प्रतिदिन तीन से चार हजार अमरनाथ यात्री विभिन्न जत्थों में रवाना किए जाते हैं। इसके अलावा बड़ी संख्या में श्रद्धालु निजी तौर पर सीधे घाटी पहुंच जाते हैं। इससे घाटी के आधार शिविरों में श्रद्धालुओं की संख्या अचानक काफी बढ़ जाती है। ऐसी स्थिति में मौसम अथवा कानून व्यवस्था बिगड़ने पर यात्रियों की सुरक्षा एक बड़ी चुनौती हो जाती है। हालांकि आधिकारिक रूप से कोई खुलकर बोलने को तैयार नहीं है। परंतु यात्रा व्यवस्था से जुड़े सूत्र इस बात की पुष्टि कर रहे हैं।"},"lang":{"kind":"string","value":"hindi"}}},{"rowIdx":160,"cells":{"text":{"kind":"string","value":"أمِس جاناوارَس وُچھِتھ سَمَے ؤلۍ ؤلی اَتِکۍ واریٚہہ بٔسکٟن اَتہِ أنٛدۍ أنٛدۍ تہٕ لٲگِکھ أمِس واریَہہ سوال پرٛژھٕنؠ"},"lang":{"kind":"string","value":"kashmiri"}}},{"rowIdx":161,"cells":{"text":{"kind":"string","value":"#------------------------------------------------------------------------------\n#\n# Copyright (c) 2007, Enthought, Inc.\n# All rights reserved.\n#\n# This software is provided without warranty under the terms of the BSD\n# license included in /LICENSE.txt and may be redistributed only\n# under the conditions described in the aforementioned license. The license\n# is also available online at http://www.enthought.com/licenses/BSD.txt\n#\n# Thanks for using Enthought open source!\n#\n#------------------------------------------------------------------------------\n\"\"\" Test whether HasTraits objects with cycles can be garbage collected.\n\"\"\"\n\nfrom __future__ import absolute_import\n\nimport gc\nimport time\nfrom traits.testing.unittest_tools import unittest\n\n# Enthought library imports\nfrom ..api import HasTraits, Any, DelegatesTo, Instance, Int\n\n\nclass TestCase(unittest.TestCase):\n def _simple_cycle_helper(self, foo_class):\n \"\"\" Can the garbage collector clean up a cycle with traits objects?\n \"\"\"\n\n # Create two Foo objects that refer to each other.\n first = foo_class()\n second = foo_class(child=first)\n first.child = second\n\n # get their ids\n foo_ids = [id(first), id(second)]\n\n # delete the items so that they can be garbage collected\n del first, second\n\n # tell the garbage collector to pick up the litter.\n gc.collect()\n\n # Now grab all objects in the process and ask for their ids\n all_ids = [id(obj) for obj in gc.get_objects()]\n\n # Ensure that neither of the Foo object ids are in this list\n for foo_id in foo_ids:\n self.assertTrue(foo_id not in all_ids)\n\n def test_simple_cycle_oldstyle_class(self):\n \"\"\" Can the garbage collector clean up a cycle with old style class?\n \"\"\"\n class Foo:\n def __init__(self, child=None):\n self.child = child\n\n self._simple_cycle_helper(Foo)\n\n def test_simple_cycle_newstyle_class(self):\n \"\"\" Can the garbage collector clean up a cycle with new style class?\n \"\"\"\n class Foo(object):\n def __init__(self, child=None):\n self.child = child\n\n self._simple_cycle_helper(Foo)\n\n def test_simple_cycle_hastraits(self):\n \"\"\" Can the garbage collector clean up a cycle with traits objects?\n \"\"\"\n class Foo(HasTraits):\n child = Any\n\n self._simple_cycle_helper(Foo)\n\n def test_reference_to_trait_dict(self):\n \"\"\" Does a HasTraits object refer to its __dict__ object?\n\n This test may point to why the previous one fails. Even if it\n doesn't, the functionality is needed for detecting problems\n with memory in debug.memory_tracker\n \"\"\"\n\n class Foo(HasTraits):\n child = Any\n\n foo = Foo()\n\n # It seems like foo sometimes has not finished construction yet, so\n # the frame found by referrers is not _exactly_ the same as Foo(). For\n # more information, see the gc doc: http://docs.python.org/lib/module-\n # gc.html\n #\n # The documentation says that this (get_referrers) should be used for\n # no purpose other than debugging, so this is really not a good way to\n # test the code.\n\n time.sleep(0.1)\n referrers = gc.get_referrers(foo.__dict__)\n\n self.assertTrue(len(referrers) > 0)\n self.assertTrue(foo in referrers)\n\n def test_delegates_to(self):\n \"\"\" Tests if an object that delegates to another is freed.\n \"\"\"\n class Base(HasTraits):\n \"\"\" Object we are delegating to. \"\"\"\n\n i = Int\n\n class Delegates(HasTraits):\n \"\"\" Object that delegates. \"\"\"\n\n b = Instance(Base)\n\n i = DelegatesTo('b')\n\n # Make a pair of object\n b = Base()\n d = Delegates(b=b)\n\n # Delete d and thoroughly collect garbage\n del d\n for i in range(3):\n gc.collect(2)\n\n # See if we still have a Delegates\n ds = [obj for obj in gc.get_objects() if isinstance(obj, Delegates)]\n self.assertEqual(ds, [])\n\nif __name__ == '__main__':\n unittest.main()\n"},"lang":{"kind":"string","value":"code"}}},{"rowIdx":162,"cells":{"text":{"kind":"string","value":"आकाश विजयवर्गीय पर बोले मोदी, 'ऐसे लोगों को पार्टी से निकाल देना चाहिए' - इना न्यूज\nहोम / नई दिल्ली / राजनीती / आकाश विजयवर्गीय पर बोले मोदी, 'ऐसे लोगों को पार्टी से निकाल देना चाहिए'\nआकाश विजयवर्गीय पर बोले मोदी, 'ऐसे लोगों को पार्टी से निकाल देना चाहिए'\nनई दिल्ली, २ जुलाई- प्रधानमंत्री नरेंद्र मोदी ने एक सरकारी कर्मचारी पर बल्ले से हमला करने को लेकर पार्टी नेता कैलाश विजयवर्गीय के बेटे आकाश विजयवर्गीय की निंदा करते हुए मंगलवार को कहा, \"बेटा किसी का भी हो, ऐसे लोगों को पार्टी से निकाल देना चाहिए। मोदी ने यह टिप्पणी संसद में भाजपा संसदीय दल की बैठक के दौरान की।\nमोदी ने कहा, \"हम ऐसा कोई नेता नहीं चाहते जो पार्टी की छवि को खराब करे। बेटा किसी का भी हो, ऐसे नेताओं को पार्टी से निकाल देना चाहिए।मोदी इंदौर के एक भाजपा विधायक आकाश विजयवर्गीय का जिक्र कर रहे थे, जिन्होंने २६ जून को नगर निगम के एक अधिकारी पर मकान गिराने के मामले में हमला किया था।\nमोदी ने जेल से छूटने के बाद आकाश विजयवर्गीय का जोरदार स्वागत करने को लेकर भी पार्टी नेताओं की आलोचना की और कहा, \"जिन्होंने उनका स्वागत किया, ऐसे नेताओं को भी पार्टी से बर्खास्त किया जाना चाहिए।बल्ले से पीटने के मामले में आकाश विजयवर्गीय को गिरफ्तार कर लिया गया था और बाद में उन्हें जमानत दे दी गई।\nनई दिल्ली राजनीती"},"lang":{"kind":"string","value":"hindi"}}},{"rowIdx":163,"cells":{"text":{"kind":"string","value":"बिहार के बाद अब इस राज्य में होगी पूर्ण शराबबंदी! | डन न्यूज न्यूज़\nहोम न्यूज़ बिहार के बाद अब इस राज्य में होगी पूर्ण शराबबंदी!\nबिहार के बाद अब इस राज्य में होगी पूर्ण शराबबंदी!\nपटना.न्यूज़डेस्क. नीतीश सरकार द्वारा बिहार में पूर्ण शराबबंदी के बाद कई अन्य राज्यों में भी पूर्ण शराबबंदी की मांग उठने लगी है. उत्तर प्रदेश, राजस्थान, झारखंड आदि राज्यों में शराब बंद करने की मांग उठ रही है. मांग करने वालों में महिलाएं शामिल है.\nनहीं आएगा बिहार में तूफान मगर मौसम विभाग ने कहा, बिहार में इस बार की गर्मी होगी\nझारखंड की गुलबी गैंग की महिलाओं ने राजभवन के सामने मांग की कि झारखंड में भी पूर्ण शराबबंदी लागू होनी चाहिए. विशेष कर ग्रामीण क्षेत्र की महिलाएं इस पर काफी जोर दे रही है कि झारखंड में भी शराब बंद होनी चाहिए.\nऐसा नही है कि महिलाओं के इस मांग की आवाज सिर्फ सड़कों तक सीमित है. झारखंड की राजनीतिक गलियारों तक भी उनकी आवाज पहुंची है. इस मामने पर बीजेपी के प्रदेश अध्यक्ष ने कहा कि शराबंदी को लेकर उनकी सकारात्मक है. सरकार उचित समय आने पर इसके लिए सही कदम उठाया जाएगा.\nबीजेपी के इस बयान के बाद यह संभावना जताई जा रही है कि बिहार के बाद अब झारखंड में भी पूर्ण शराबबंदी लागू होगी. सरकार ने भी साफ कर दिया है कि वह सही समय का इंतजार कर रही है.\nलालू परिवार पर इट की कार्रवाई के बाद नित्यानंद राय का आया अजीबोगरीब बयान\nबिहार में मचे सियासी घमासान के बीच लालू और नीतीश पर बरस पड़े जीतन राम मांझी\nकांवड़ियों के नारों से भड़की धार्मिक हिंसा, १२ से ज्यादा घायल"},"lang":{"kind":"string","value":"hindi"}}},{"rowIdx":164,"cells":{"text":{"kind":"string","value":"Usually the speaker is referring to an amount significant enough to affect a person’s decision to buy something or participate in an activity. This is similar to a texting abbreviation, and is very informal. The use of three dollar signs serves two purposes: to make sure it isn't a typo and there should be an amount written next to it, and to emphasize a large expense.\nEtymology : From the dollar sign $ which we use to symbolize the dollar (at least in the United States).\nThis is similar to a texting abbreviation, and is very informal. The use of three dollar signs serves two purposes: to make sure it isn't a typo and there should be an amount written next to it, and to emphasize a large expense."},"lang":{"kind":"string","value":"english"}}},{"rowIdx":165,"cells":{"text":{"kind":"string","value":"मैरी कॉम ने जीता एशियाई मुक्केबाज़ी चैंपियनशिप में सोना | हल्लाबोल तोडे\nहोम खेल मैरी कॉम ने जीता एशियाई मुक्केबाज़ी चैंपियनशिप में सोना\nमैरी कॉम ने जीता एशियाई मुक्केबाज़ी चैंपियनशिप में सोना\nभारतीय मुक्केबाजी की वंडर गर्ल एम सी मैरी कॉम ( ४८ किलो ) ने एशियाई मुक्केबाजी में पांचवीं बार गोल्ड मेडल अपने नाम कर लिया। पांच बार की विश्व चैंपियन और ओलिंपिक ब्रांज मेडल विजेता मैरी कॉम ने उत्तर कोरिया की किम ह्यांग मि को ५-० से हराया।\nयह २०१४ एशियाई खेलों के बाद मैरी कॉम का पहला अंतरराष्ट्रीय गोल्ड मेडल है और एक साल में उनका पहला मेडल है। ३५ बरस की मैरी कॉम का सामना किम ह्यांग मि के रूप में सबसे आक्रामक प्रतिद्वंद्वी से था, लेकिन वह इस चुनौती के लिए तैयार थीं। अब तक पहले तीन मिनट एक दूसरे को आंकने में जाते रहे थे, लेकिन इस मुकाबले में शुरूआती पलों से ही खेल आक्रामक रहा।\nमैरी कॉम ने अपनी प्रतिद्वंद्वी के हर वार का माकूल जवाब दिया, दोनों ओर से तेज पंच लगाए गए। मैरी कॉम उनके किसी भी वार से विचलित नहीं हुईं और पूरे सब्र के साथ खेलते हुए जीत दर्ज की।\nबॉक्सिंग फेडरेशन ऑफ इंडिया की और से जारी विज्ञाप्ति में अध्यक्ष अजय सिंह ने इस जीत पर मैरी कॉम की तारीफ करते हुए कहा, ३५ साल की उम्र में तीन बच्चों की मां होने के बावजूद मैरी कॉम की यह उपलब्धि शानदार है। उन्होंने खुद को साबित किया है। भारतीय टीम में १० बॉक्सरों में से ७ बॉक्सर मेडल के सात स्वदेश लौटेंगे, यह वाकई बड़ी कामयाबी है। मैं इसके लिए टीम कोच को और स्पोर्टिंग स्टॉफ को भी बधाई देता हूं।\nएशियाई मुक्केबाज़ी चैंपियनशिप\nप्रेवियस आर्टियलफिटनेस का बेस्ट फंडा चाहिए तो हर्षित गुप्ता से ज़रुर मिलिए\nनेक्स्ट आर्टियलपुलिस आपकी दोस्त क्यों नहीं?"},"lang":{"kind":"string","value":"hindi"}}},{"rowIdx":166,"cells":{"text":{"kind":"string","value":"\\begin{document}\n\n\\title[]{\nIterative methods for $k$-Hessian equations\n}\n\n\\author{Gerard Awanou}\n\\address{Department of Mathematics, Statistics, and Computer Science, M/C 249. University of Illinois at Chicago, Chicago, IL 60607-7045, USA}\n\n\\maketitle\n\n\\begin{abstract}\nOn a domain of the $n$-dimensional Euclidean space, and for an integer $k=1,\\ldots,n$, the $k$-Hessian equations are fully nonlinear elliptic equations for $k >1$ and consist of the Poisson equation for $k=1$ and the Monge-Amp\\`ere equation for $k=n$. We analyze for smooth non degenerate solutions a 9-point finite difference scheme. We prove that the discrete scheme has a locally unique solution with a quadratic convergence rate.\n\nIn addition we propose new iterative methods which are numerically shown to work for non smooth solutions. A connection of the latter with a popular Gauss-Seidel method for the Monge-Amp\\`ere equation is established and new Gauss-Seidel type iterative methods for $2$-Hessian equations are introduced. \n\n\\end{abstract}\n\n\\section{Introduction}\n\nLet $\\Omega$ be a bounded, connected open subset of $\\mathbb{R}^n, n\\geq 2$ with boundary denoted $\\partial \\Omega$. \n\nLet $u \\in C^2(\\Omega)$ and for $x \\in \\Omega$, let $D^2 u(x)=\\bigg( (\\partial^2 u(x))(\\partial x_i \\partial x_j)\\bigg)_{i,j=1,\\ldots, n} $ \ndenote its Hessian. We denote the eigenvalues of $D^2 u(x)$ by $\\lambda_i(x), i=1,\\ldots, n$. \nFor $1 \\leq k \\leq n$, the $k$-Hessian operator is defined as\n\\begin{align*}\nS_k(D^2 u) = \\sum_{i_1 < \\cdots < i_k} \\lambda_{i_1} \\cdots \\lambda_{i_k}.\n\\end{align*}\n We note that $S_1(D^2 u) = \\Delta u$ is the Laplacian operator and $S_n(D^2 u) = \\det D^2 u$ is the Monge-Amp\\`ere operator. For $k \\geq 2$, we are interested in the numerical approximation of \n solutions of the Dirichlet problem for the $k$-Hessian equation \n\\begin{equation}\nS_k(D^2 u) = f \\, \\text{in} \\, \\Omega, u=g \\, \\text{on} \\, \\partial \\Omega, \\label{k-H1}\n\\end{equation}\nwith $f$ and $g$ given and $f \\geq 0$. \n\n\\subsection{Local existence, uniqueness and quadratic convergence rate for a finite difference discretization}\n\n Let $u^0$ be a sufficiently close initial guess to the smooth solution $u$ of \\eqref{k-H1}. Consider the iterative method\n \\begin{align} \\label{broyden}\n\\begin{split}\n\\operatorname{div} \\bigg( \\{S_k^{ij}(D^2 u^0) \\} D u^{m+1} \\bigg)& = \\operatorname{div} \\bigg( \\{S_k^{ij}(D^2 u^0) \\} D u^{m} \\bigg) +f-S_k (D^2 u^m) \\, \\text{in} \\, \\Omega \\\\\n u^{m+1} & = g \\, \\text{on} \\, \\partial \\Omega, \n \\end{split}\n\\end{align}\nwhere $ \\{S_k^{ij}(D^2 u^0) \\}$ is a matrix which generalizes the cofactor matrix of $D^2 u^0$. \n\nWe prove the convergence of \\eqref{broyden} at the continuous level in H$\\ddot{\\text{o}}$lder spaces. A discrete version of \\eqref{broyden} is also shown to converge to a solution of a 9-point stencil discretization of \\eqref{k-H1}. This establishes the local existence and uniqueness of a discrete solution. In addition the convergence rate of the discretization is shown to be quadratic.\n\nIt is reasonable to expect that the discrete version of the iterative method \\eqref{broyden} will retrieve the correct solution when it is smooth and non degenerate. As with Newton's method it is not effective for non smooth and degenerate solutions. For these, we advocate iterative methods like the subharmonicity preserving iterations described below. The discrete version of \\eqref{broyden} is used in this paper to prove the local solvability of the 9-point scheme when $u$ is smooth and non degenerate. These results form a building block of a theory which explains why standard discretizations work for non smooth solutions \\cite{Awanou-Std-fd-jsc}. In addition results for smooth solutions are also needed for the analysis of hybrid schemes where the 9 point scheme is used in part of the region where the solution is smooth and a monotone scheme elsewhere \\cite{AwanouHybrid}.\n\n\\subsection{Newton's method}\n\nIf one is only interested in smooth solutions, Newton's method is the most appropriate method. \nWe analyze the convergence of Newton's method for solving \\eqref{k-H1} when it has a smooth solution.\n \n\n\\subsection{Numerical work for subharmonicity preserving iterations}\n\nA smooth function $u$ is said to be $k$-convex if $S_l (D^2 u) \\geq 0, 1 \\leq l \\leq k$. Convexity of a function can be shown to be equivalent to $n$-convexity, Lemma \\ref{n-convexity}. \nIt is of interest in some applications to be able to handle \\eqref{k-H1} when it has a non smooth $k$-convex solution. It has only been recently understood, c.f. \\cite{Awanou-Std-fd-jsc} for the Monge-Amp\\`ere equation, that what is needed is a numerical method provably convergent for smooth solutions and numerically robust to handle non smooth solutions. The approach in \\cite{Awanou-Std-fd-jsc} is to regularize the data and use approximation by smooth functions. The key to numerically handle non smooth solutions of \\eqref{k-H1} is to preserve $k$-convexity in the iterations. For discrete $k$-convexity we simply require discrete analogues of the condition $S_l (D^2 u) \\geq 0$ with a natural discretization of $D^2 u$. We refer to \\cite{Aguilera2008} where this approach was first used for the discretization of $n$-convexity.\n \nConsider the iterative method\n\\begin{align}\n\\begin{split}\n \\Delta u^{m+1} & = \\bigg( (\\Delta u^{m})^k + \\frac{1}{c(k,n)}(f-S_k (D^2 u^m)) \\bigg)^{\\frac{1}{k}} \\, \\text{in} \\, \\Omega, \n u^{m+1} = g \\, \\text{on} \\, \\partial \\Omega, \\label{k-H-iterative}\n \\end{split}\n\\end{align}\nwith \n$c(k,n) = \\binom{n}{k}/n^k$.\n\nIf $D^2 u$ has positive eigenvalues, we have the inequality\n\\begin{equation} \\label{G-am}\nS_k(D^2 u) \\leq c(k,n) (\\Delta u)^{k}, \n\\end{equation}\nwhich follows from the Maclaurin inequalities, \n\\cite[Proposition 1.1 (v i)]{Gavitone2009}.\n\nFor $k=2$, \\eqref{G-am} also holds with {\\it no convexity assumption} on $u$, \\cite[Lemma 15.11]{Lieberman96}. Explicitly \n$c(2,3)=1/3$. Also, $c(n,n)=1/n^n$ which gives\n$$\n\\det D^2 u \\leq \\frac{1}{n^n} (\\Delta u)^n, \n$$\na direct consequence of the arithmetic mean - geometric mean inequality.\n\nIf one starts with an initial guess $u^0$ such that $\\Delta u^0 \\geq 0$, \\eqref{k-H-iterative} enforces $\\Delta u^m \\geq 0$ for all $m$. Indeed recall that $f \\geq 0$ and assume that $\\Delta u^m \\geq 0$. Then by \\eqref{G-am}\n$1/c(k,n) S_k (D^2 u^m) \\leq (\\Delta u^{m})^k$, and using \\eqref{k-H-iterative} it follows that $(\\Delta u^{m+1})^k \\geq 0$. In other words, starting with an initial guess $u^0$ with $\\Delta u^0 \\geq 0$, \\eqref{k-H-iterative} enforces subharmonicity \n\nin arbitrary dimension for smooth convex solutions and subharmonicity for 2-Hessian equations with no convexity assumption on $u$. In addition for 2-Hessian equations, the limit solution solves $S_2(D^2 u)=f \\geq 0$. That is, the sequence $u^{m+1}$ defined by \\eqref{k-H-iterative} has a formal limit which solves $\\Delta u \\geq 0$ and $S_2(D^2 u)\\geq 0$. Thus \\eqref{k-H-iterative} enforces 2-convexity in arbitrary dimension for 2-Hessian equations. \n\n \n Another class of iterative methods we introduce \n in this paper are Gauss-Seidel type iterative methods. \n The Gauss-Seidel methods are more efficient than \\eqref{k-H-iterative} for large scale problems.\n \n The simplicity of the methods discussed in this paper and the facility with which they can be implemented, make them attractive to researchers interested in Monge-Amp\\`ere equations. The other major motivation to study the subharmonicity preserving iterations is that they can be adapted to the finite element context and have been numerically shown in that context to be robust for non smooth solutions.\n \n\n \n\nIn two dimension, \\eqref{k-H-iterative} appears to perform well in the degenerate case $f \\geq 0$ as discrete $k$-convexity is enforced in the iterations. The situation is different in three dimension with $k=2$. \nWe were not able to reproduce the solution $u(x,y,z)=|x-1/2|$ by solving \\eqref{k-H1} with $k=2$ and using \\eqref{k-H-iterative}. Here, since $u$ does not depend on $z$, we have $f(x,y,z)=0$ as in the two dimensional case.\nHowever, for $n=3$ and $k=3$, we can preserve convexity in the degenerate case by using the sequence of nonlinear $2$-Hessian equations\n\\begin{align} \\label{sigma2k}\nS_2(D^2 u^{m+1}) = 3 \\bigg(\\bigg(\\frac{1}{3} S_2 (D^2 u^m)\\bigg)^{\\frac{3}{2}} + f - \\det D^2 u^m\n\\bigg)^{\\frac{2}{3}},\n\\end{align}\nwith $u^{m+1}=g$ on $\\partial \\Omega$. Each of these equations is solved iteratively by \\eqref{k-H-iterative} with $k=2, n=3$. \n\nWe note that $\\bigg(\\frac{1}{3} S_2 (D^2 u^m)\\bigg)^{\\frac{3}{2}} - \\det D^2 u^m \\geq 0$ when $S_2 (D^2 u^m)>0$, \\cite[Lemma 15.12]{Lieberman96}. Starting with an initial guess which satisfies $S_2 (D^2 u^0) >0$ and setting $\\det D^2 u^m=0$ in \\eqref{sigma2k} whenever $S_2 (D^2 u^m)=0$, we obtain a double sequence iterative method which at the limit enforce $\\Delta u \\geq 0, S_2 (D^2 u) \\geq 0$, and $ \\det D^2 u = f \\geq 0$. \n\nThe reason for setting $\\det D^2 u^m=0$ in \\eqref{sigma2k} whenever $S_2 (D^2 u^m)=0$ is motivated by the observation that in the case $f=0$, if $S_2 (D^2 u^m)=0$, $S_2 (D^2 u^{m+1})$ is ill-defined or complex valued if $\\det D^2 u^m>0$. While \\eqref{k-H-iterative} may be inexact for degenerate 2-Hessian equations, its use inside a double iterative method appears effective. This is reminiscent of inexact Uzawa algorithms. \n\n\\subsection{Relation with other work}\nThe $k$-Hessian equations have mainly applications in conformal geometry and physics. The Monge-Amp\\`ere operator has received recently a lot of interest from numerical analysts. For $n=3$ and $k=2$, \nthe numerical resolution of \\eqref{k-H1} has been considered in \\cite{Sorensen10}, where it was referred to as the $\\sigma_2$ problem. \n\nThe iterative method \\eqref{k-H-iterative} generalizes an iterative method introduced in \\cite{Benamou2010} for the two dimensional Monge-Amp\\`ere equation. The latter corresponds to the choice $ k=n=2$ and the constant $c(2,2)=1/4$ replaced by 1/2. The $2$-Hessian equation has also been considered recently in \\cite{FroeseObermanSalvago} from the point of view of monotone schemes.\n\n We will see that if the central finite difference discretization of \\eqref{k-H-iterative} is solved by a Gauss-Seidel iterative method, one recovers a Gauss-Seidel iterative method which has been used by many authors to solve the two dimensional Monge-Amp\\`ere equation. We will refer to the latter method as the 2D Gauss-Seidel method for Monge-Amp\\`ere equation. It has been used in the numerical simulation of Ricci flow \\cite{Headrick05}, as a smoother in multigrid methods for the balance vortex model in meteorology, \\cite{Chen2010b,Chen2010c} and has been recently shown numerically to capture the viscosity solution of the 2D Monge-Amp\\`ere equation \\cite{Benamou2010}. The connection between \\eqref{k-H-iterative} and the 2D Gauss-Seidel method for the Monge-Amp\\`ere equation is what enables us to introduce new Gauss-Seidel type iterative methods for $k$-Hessian equations. \n \n The ingredients of our proof of the convergence rate for the finite difference discretization are discrete Schauder estimates and a suitable generalization of the combined fixed point iterative method used in \\cite{Feng2009}.\n Schauder estimates were also used in the proof of convergence of Newton's method at the continuous level \\cite{Loeper2005}.\n\n\\subsection{Organization of the paper}\nThe paper is organized as follows: In the next section, we \ngive some notations, recall the Schauder estimates and their discrete analogues. In section \\ref{elliptic} we prove our main results on the quadratic convergence rate of a finite difference discretization of \\eqref{k-H1} and in section \\ref{newton-sec} we prove the convergence of Newton's method. In section \\ref{convexity} we \nintroduce new Gauss-Seidel type iterative methods and their connections with the subharmonicity preserving iterations \\eqref{k-H-iterative}.\n\nSection \\ref{num} is devoted to numerical results. We conclude with some remarks. \n\nThe reader interested only in the Monge-Amp\\`ere equation, or for a first reading, may assume that $k=n$.\n\n\\section{Notation and preliminaries} \\label{notation}\n\n\\subsection{H$\\ddot{\\text{o}}$lder spaces and Schauder estimates} \\label{notation1}\nFor a nonnegative integer $r$ or for $r=\\infty$, we denote by $C^r(\\Omega)$ the set of all functions having all derivatives of order $\\leq r$ continuous on $\\Omega$ \n and by $C^r(\\tir{\\Omega})$, the set of all functions in $C^r(\\Omega)$\nwhose derivatives of order $\\leq r$ have continuous\nextensions to $\\tir{\\Omega}$. For a multi-index $\\beta=(\\beta_1,\\ldots,\\beta_n) \\in \\mathbb{N}^n$, put $|\\beta|=\\beta_1+\\ldots+\\beta_n$. We use the notation $D^{\\beta} u(x)$ for the partial derivative\n$(\\partial /\\partial x_1)^{\\beta_1} \\ldots (\\partial /\\partial x_n)^{\\beta_n} u(x)$.\n\nThe norm in $C^r(\\Omega)$ is given by\n$$\n||u||_{r;\\Omega} = \\sum_{j=0}^r \\, |u|_{j;\\Omega}, \\quad |u|_{j;\\Omega} = \\text{sup}_{|\\beta|=j} \\text{sup}_{\\Omega} |D^{\\beta}u(x)|.\n$$\nWe denote by $|x|$ the Euclidean norm of $x \\in \\mathbb{R}^n$. A function $u$ is said to be uniformly H$\\ddot{\\text{o}}$lder continuous with exponent $\\alpha, 0 <\\alpha \\leq 1$ in $\\Omega$ if\nthe quantity\n$$\n \\text{sup}_{x \\neq y} \\frac{|u(x)-u(y)|}{|x-y|^{\\alpha}},\n$$\nis finite. \nThe space $C^{r,\\alpha}(\\tir{\\Omega})$\nconsists of functions whose $r$-th order derivatives are uniformly H$\\ddot{\\text{o}}$lder\ncontinuous with exponent $\\alpha$ in $\\Omega$. It is a Banach space with norm\n$$\n||u||_{r,\\alpha;\\Omega} = ||u||_{r;\\Omega} + [u]_{r,\\alpha;\\Omega},\n$$\nwhere\n$$\n[u]_{r,\\alpha;\\Omega} = \\text{sup}_{|\\beta|=r} \\text{sup}_{x \\neq y} \\frac{|D^{\\beta}u(x)-\nD^{\\beta} u(y)|}{|x-y|^{\\alpha}}. \n$$\nThe norms $|| \\, ||_{r;\\Omega}$ and $|| \\, ||_{r,\\alpha;\\Omega}$ are naturally extended to vector fields and matrix fields by taking the supremum over all components. We make the standard convention of using $C$ for a generic constant. For $A=(a_{ij})_{i,j=1,\\ldots,n}$ and $B=(b_{ij})_{i,j=1,\\ldots,n}$ \nwe recall that \n$A:B=\\sum_{i,j=1}^n a_{i j} b_{ij}$. \nWe will often use the following property\n\\begin{equation} \\label{alpha-prod1}\n|| f g ||_{0,\\alpha;\\Omega} \\leq C || f ||_{0,\\alpha;\\Omega} || g ||_{0,\\alpha;\\Omega}, \\, \\text{for} \\, f,g \\in C^{0,\\alpha}(\\tir{\\Omega}),\n\\end{equation}\nfrom which it follows that if $A, B$ are matrix fields \n\\begin{equation} \\label{alpha-prod2}\n||A:B||_{0,\\alpha;\\Omega} \\leq C \\sum_{i,j=1}^n || a_{ij} ||_{0,\\alpha;\\Omega} || b_{ij} ||_{0,\\alpha;\\Omega}.\n\\end{equation}\n\nWe first state a global regularity result for the solution of strictly elliptic equations, which follows from \\cite[Theorems 6.14, 6.6 and Corollary 3.8 ]{Gilbarg2001}.\n\\begin{thm} \\label{SchauderPoisson}\nAssume $0< \\alpha < 1$. Let $\\Omega$ be a $C^{2,\\alpha}$ domain in $\\mathbb{R}^n$ and $f, a^{ij} \\in C^{\\alpha}(\\tir{\\Omega})$, $\\phi \\in C^{2,\\alpha}(\\tir{\\Omega})$. We consider the strictly elliptic operator\n\\begin{equation} \\label{st-elliptic}\nL u = \\sum_{i,j=1}^n a^{ij}(x) \\frac{\\partial^2}{\\partial x_i \\partial x_j} u(x),\n\\end{equation}\nwith coefficients satisfying for positive constants $\\lambda, \\Lambda$, \n$$\n\\sum_{i,j=1}^na^{ij}(x) \\zeta_i \\zeta_j \\geq \\lambda \\sum_{l=1}^n \\zeta_l^2, \\zeta_l \\in \\mathbb{R}, \\, \\text{and} \\, |a^{i,j}|_{0,\\alpha;\\Omega} \\leq \\Lambda.\n$$\nThen the solution $u$\nof the equation\n$$\nL u =f \\, \\text{in} \\, \\Omega, u = \\phi \\, \\text{on} \\, \\partial \\Omega,\n$$\nsatisfies\n$$\n||u||_{2,\\alpha;\\Omega} \\leq C(||\\phi||_{2,\\alpha;\\Omega}+ ||f||_{0,\\alpha;\\Omega}),\n$$\nwhere $C$ depends on $n, \\alpha, \\lambda, \\Lambda, \\Omega, \\sup_{\\partial \\Omega} |\\phi|$, and $\\sup_{\\Omega} |f|/\\lambda$.\n\n\\end{thm}\n\nWe will make the slight abuse of language of also denoting by $S_k(x), x=(x_1,\\ldots,x_n)$ the $k$th elementary symmetric polynomial of the variable \n$x$, i.e.\n$$\nS_k(\\lambda) = \\sum_{i_1 < \\cdots < i_k} \\lambda_{i_1} \\cdots \\lambda_{i_k}.\n$$\nA function $u \\in C^2(\\Omega) \\cap C^0(\\tir{\\Omega})$ with Hessian $D^2 u$ having eigenvalues $\\lambda_i, i=1,\\ldots,n$ is said to be $k$-admissible if\n$S_j(\\lambda) > 0, j=1,\\ldots,k$. Solutions of the $k$-Hessian equation will be required to be $k$-admissible, thus requiring $f>0$.\n\nMoreover, let $\\kappa=(\\kappa_1,\\ldots,\\kappa_{n-1})$ denote the principal curvatures of $\\partial \\Omega$. \n\n\\begin{defn} \\label{k-convexity-domain}\nThe domain $\\Omega$ is said to be $(k-1)$-convex if there exists $c_0 >0$ such that\n$$\nS_{k-1}(\\kappa) \\geq c_0 >0 \\, \\text{on} \\, \\partial \\Omega.\n$$\n\\end{defn}\nWe then have, (\\cite[Theorems 3.3 and 3.4 ]{WangXJ09}) \n\n\\begin{thm} \\label{k-Hessian}\nAssume that $\\Omega$ is $(k-1)$-convex, $\\partial \\Omega \\in C^{3,1}$, $f \\in C^{1,1}(\\tir{\\Omega})$, inf $f >0$, $g \\in C^{3,1}(\\tir{\\Omega})$. Then there is a unique $k$-admissible solution $u \\in C^{3,1}(\\tir{\\Omega})$ to the Dirichlet problem \\eqref{k-H1}. \n\n\\end{thm}\n\nWe will need some identities for the $k$-Hessian operator $S_k(D^2 u)$ which are derived explicitly for example in \\cite[p. 5--6]{Gavitone2009}. See also \\cite{WangXJ09}.\nFor a symmetric matrix $A=(a_{ij})_{i,j}=1,\\ldots,n$ with eigenvalues $\\lambda_i, i=1,\\ldots,n$, let us also denote by $S_k(A)$ the $k$-th elementary symmetric polynomial of $\\lambda$. This is equivalent to say that $S_k(A)$ is the sum of all $k \\times k$ principal minors of $A$. Using the permutation definition of the determinant, we have\n\\begin{align} \\label{k-minor}\nS_k(A) = \\frac{1}{k!}\\sum_{1 \\leq i_1,\\cdots,i_k\\leq n} \\delta^{j_1,\\cdots,j_k}_{i_1,\\cdots,i_k} a_{i_1 j_1} \\cdots a_{i_k j_k},\n\\end{align}\nwhere $\\delta^{j_1,\\cdots,j_k}_{i_1,\\cdots,i_k}$ is the generalized Kronecker delta which takes the value +1 if $i_1,\\cdots,i_k$ differs from $j_1,\\cdots,j_k$ by an even permutation and the value -1 in the case of an odd permutation. In other words, for a choice of $i_1,\\ldots,i_k$, $\\delta^{j_1,\\cdots,j_k}_{i_1,\\cdots,i_k}$ is the signature of the permutation $\\sigma$ defined by $\\sigma(i_l)=j_l, l=1,\\ldots,k$. This implies that we only consider the case where the sets $\\{i_1,\\ldots,i_k\\}$ and $\\{j_1,\\ldots,j_k\\}$ are identical. Moreover we define $\\delta^{j_1,\\cdots,j_k}_{i_1,\\cdots,i_k}$ to be 0 if $\\{i_1,\\ldots,i_k\\} \\neq \\{j_1,\\ldots,j_k\\}$. Note also that $\\{i_1,\\ldots,i_k\\}$ is a subset of $k$ elements of $\\{ 1, \\ldots, n \\}$.\n\nWe have\n\\begin{align*}\nS_k^{ij}(A)\\coloneqq \\frac{\\partial}{\\partial a_{ij}} S_k(A) = \\frac{1}{(k-1)!} \n\\sum_{1 \\leq i, i_1,\\cdots,i_{k-1}\\leq n} \\delta^{j,j_1,\\cdots,j_{k-1}}_{i,i_1,\\cdots,i_{k-1}} a_{i_1 j_1} \\cdots a_{i_{k-1} j_{k-1}}, \n\\end{align*}\nand so\n$\nS_k(A) = \\frac{1}{k} \\sum_{i,j=1}^n S_k^{ij}(A) a_{i j}\n$\nby the $k$-homogeneity of $S_k$ and Euler's theorem for homogeneous functions. Here $\\{j_1,\\ldots,j_{k-1}\\}$ is the image of the set of $k-1$ elements $\\{i_1,\\ldots,i_{k-1}\\}$ not containing $i$ by a permutation.\n\nLet us denote by $\\{S_k^{ij}(A) \\}$ the symmetric matrix with entries $S_k^{ij}(A)$. We can write $S_k(A)=1/k \\, \\{S_k^{ij}(A) \\}: A $, that is\n\n$S_k(D^2 v) = \\frac{1}{k} \\{S_k^{ij}(D^2 v) \\} : D^2 v$.\n\nUsing \\eqref{k-minor} and observing that the expression of $S_k(A) $ can be written in terms of a multilinear map, we obtain\n\\begin{align} \\label{k-Hdiv0}\nS_k'(D^2 v) D^2 w = \\{S_k^{ij}(D^2 v) \\}: D^2 w.\n\\end{align}\n\nLet us denote by $ \\{S_k^{ij}(A) \\}' $ the Fr\\'echet derivative of the mapping $A \\to \\{S_k^{ij}(A) \\}$. Since $\\{S_k^{ij}(A) \\}' (B)$ is a sum of terms each of which is a product of $k-2$ terms from $A$ and is linear in $B$, we have\n\\begin{equation} \\label{Sij-der}\n|| \\{S_k^{ij}(D^2 v) \\}' D^2 w ||_{0;\\Omega} \\leq C |v|_{2;\\Omega}^{k-2} |w|_{2;\\Omega}.\n\\end{equation}\nUsing \\eqref{alpha-prod2} and \\eqref{Sij-der} we also have\n\\begin{equation} \\label{cof-estimate}\n|| \\{S_k^{ij}(D^2 v) \\}' D^2 w ||_{0,\\alpha;\\Omega} \\leq C |v|_{2,\\alpha;\\Omega}^{k-2} |w|_{2,\\alpha;\\Omega}.\n\\end{equation}\n\nFinally we note that\n\\begin{lemma} \\label{close}\nLet $v$ be a $C^2$ strictly convex function with Hessian having smallest eigenvalue uniformly bounded below by a constant $a >0$. Then for $\\eta=a/(2 n)$, we have $w$ strictly convex, whenever $||w-v||_{C^2(\\Omega)} < \\eta$. \n\\end{lemma}\n\\begin{proof}\n\nIt follows from \\cite[Theorem 1 and Remark 2 p. 39]{Hoffman53} that for two symmetric $n \\times n$ matrices $A$ and $B$, \n\\begin{equation} \\label{cont-eig}\n|\\lambda_l(A) - \\lambda_l(B)| \\leq n \\max_{i,j} |A_{ij} - B_{ij}|, l=1, \\ldots, n.\n\\end{equation}\nIt follows that for $u, v \\in C^2(\\Omega)$, \n\\begin{align}\n|\\lambda_1( D^2 u(x)) - \\lambda_1( D^2 v(x))| & \\leq n ||w-v||_{C^2(\\Omega)} \\label{lambda1}.\n\n\\end{align}\nThe result then follows.\n\\end{proof}\n\nWe conclude this section with the equivalence of $n$-convexity and convexity in the usual sense.\n\\begin{lemma} \\label{n-convexity}\nA $C^2$ function $u$ is convex if and only if it is $n$-convex.\n\\end{lemma}\n\n\\begin{proof}\nIf $u$ is $C^2$, $\\lambda_i \\geq 0$ on $\\Omega$ for all $i$ and thus $S_l(D^2 u) \\geq 0, l=1,\\ldots,n$. \n\nConversely let us assume that $A$ is a symmetric matrix with\n$S_l(A) \\geq 0, l=1,\\ldots,n$. We show that its eigenvalues $\\lambda_i$ are all positive. Let\n$$\np(\\lambda)= \\lambda^n + c_1 \\lambda^{n-1} + \\ldots + c_n,\n$$\ndenote the characteristic polynomial of $A$. It can be shown \\cite[Theorem 1.2.12]{Horn85} that\n$$\nc_l = (-1)^l S_l(A), l=1,\\ldots,n.\n$$\nWe show that if $\\lambda_i <0$ then $p(\\lambda_i) \\neq 0$. We have\n\\begin{align*}\np(\\lambda_i) & = \\lambda_i^n + c_1 \\lambda_i^{n-1} + \\ldots + c_n \\\\\n & = \\lambda_i^n + \\sum_{l=1}^n (-1)^{l} S_l(A) \\lambda_i^{n-l} \\\\\n & = (-1)^n \\bigg( (-\\lambda_i)^n + \\sum_{l=1}^{n} (-1)^{l-n} S_l(A) \\lambda_i^{n-l}\\bigg) \\\\\n & = (-1)^n \\bigg( (-\\lambda_i)^n + \\sum_{l=1}^{n} S_l(A) (-\\lambda_i)^{n-l}\\bigg). \n\\end{align*}\nSince $-\\lambda_i >0$ and $S_l(A) \\geq 0$ for all $l$, we have $(-1)^n p(\\lambda_i) \\geq 0$. Moreover since \n$\\sum_{l=1}^{n} S_l(A) (-\\lambda_i)^{n-l} \\geq 0$ and $-\\lambda_i > 0$ we have $(-1)^n p(\\lambda_i) \\neq 0$.\nWe conclude that $\\lambda_i \\geq 0$ for all $i$. This completes the proof.\n\\end{proof}\n\n\\subsection{Discrete Schauder estimates and related tools} \\label{disc-schauder}\nWe will study the numerical approximation of \\eqref{k-H1}--\\eqref{k-H-iterative} by standard finite difference discretizations. \nFor simplicity, we consider a cuboidal domain $\\Omega = (0,1)^n \\subset \\mathbb{R}^n$. Let $0 < h < 1 \\, \\text{with} \\, 1/h \\in \\mathbb{Z}$. Put\n\\begin{align*}\n\\mathbb{Z}_h & = \\{x=(x_1,\\ldots,x_n)^T \\in \\mathbb{R}^n: x_i/h \\in \\mathbb{Z} \\}\\\\\n\\Omega^h_0 &= \\Omega \\cap \\mathbb{Z}_h, \\Omega^h = \\tir{\\Omega} \\cap \\mathbb{Z}_h, \\partial \\Omega^h = \\partial \\Omega \\cap \\mathbb{Z}_h= \\Omega^h \\setminus \\Omega^h_0.\n\\end{align*}\nLet $e^i, i=1,\\ldots,n$ denote the $i$-th unit vector of $\\mathbb{R}^n$. We define the following first order difference operators on the space $\\mathcal{M}(\\Omega^h)$ of grid functions $v^h(x), x \\in \\mathbb{Z}_h$,\n\\begin{align*}\n\\partial^i_{+} v^h(x) & \\coloneqq \\frac{v^h(x+he^i)-v^h(x)}{h}, \\\\\n\\partial^i_{-} v^h(x) & \\coloneqq \\frac{v^h(x)-v^h(x-he^i)} {h},\\\\\n\\partial^i_h v^h(x) & \\coloneqq \\frac{v^h(x+he^i)-v^h(x-he^i)}{2 h}.\n\\end{align*}\nHigher order difference operators are obtained by combining the above difference operators. For a multi-index $\\beta=(\\beta_1,\\ldots,\\beta_n) \\in \\mathbb{N}^n$, we define\n$$\n\\partial^{\\beta}_{+} v^h \\coloneqq \\partial^{\\beta_1}_{+} \\cdots \\partial^{\\beta_n}_{+}v^h.\n$$\nThe operators $\\partial^{\\beta}_{-}$ and $\\partial^{\\beta}_{h}$ are defined similarly. Note that\n\\begin{align}\n\n\\partial^i_{+} \\partial^i_{-} v^h(x) & = \\frac{v^h(x+he^i)-2v^h(x)+v^h(x-he^i)}{h^2}, \\label{second-disc1}\n\\end{align}\n\\begin{align}\n\\begin{split}\n\\partial^i_h \\partial^j_h v^h(x) & = \\frac{1}{4 h^2} \\bigg\\{v^h(x+he^i+h e^j)+v^h(x-he^i-h e^j) \\\\\n& \\qquad \\qquad \\qquad -v^h(x+he^i-h e^j)-v^h(x-he^i+ he^j)\\bigg\\}, i \\neq j. \\label{second-disc2}\n\\end{split}\n\\end{align}\nThe second order derivatives $\\partial^2 v/\\partial x_i \\partial x_j$ are discretized using \\eqref{second-disc1} and \\eqref{second-disc2} for $i \\neq j$. This gives a discretization of the Hessian $D^2 u$ which we denote by $\\mathcal{H}_d(u^h)$. \n\nThus the discrete version of \\eqref{k-H1} takes the form\n\\begin{align} \\label{k-H1h}\nS_k (\\mathcal{H}_d \\, u^h(x)) = f(x), x \\in \\Omega^h_0, u^h(x) = g(x)\\, \\text{on} \\, \\partial \\Omega^h. \n\\end{align}\nThe discrete Laplacian takes the form\n\\begin{align} \\label{second-disc3}\n\\Delta_d (u^h) = \\sum_{i=1}^n \\partial^i_{+} \\partial^i_{-} u^h.\n\\end{align}\nWe consider a discrete uniformly elliptic linear operator \nwith low order terms\n\\begin{align*}\nL_d v^h(x) = \\sum_{i,j=1}^n a^{ij}(x) \\partial^i_- \\partial^j_+ v^h(x) + \\sum_{i=1}^n b^{i}(x) \\partial^i_+ v^h(x), x \\in \\Omega_0^h,\n\\end{align*}\ni.e. the matrix $(a^{ij}(x))_{i,j=1,\\ldots,n}$ is uniformly positive definite. \nWe now define discrete analogues of the H$\\ddot{\\text{o}}$lder norms and semi-norms following \\cite{Johnson74}. Let $[\\xi,\\eta]$ denote the set of points $\\zeta \\in \\Omega^h$ such that $\\xi_j \\leq \\zeta_j \\leq \\eta_j, j=1,\\ldots,n$. Then for $v^h \\in \\mathcal{M}(\\Omega^h), 0 < \\alpha < 1$, we define\n\\begin{align*}\n|v^h|_{j;\\Omega_0^h} & = \\, \\text{max} \\, \\{\\, |\\partial^{\\beta}_{+}v^h (\\xi)|, |\\beta|=j, [\\xi,\\xi+\\beta] \\subset \\Omega^h \\, \\}, \\\\\n[v^h]_{j,\\alpha;\\Omega_0^h} & = \\, \\text{max} \\, \\bigg\\{\\, \\frac{|\\partial^{\\beta}_{+}v^h (\\xi)- \\partial^{\\beta}_{+}v^h (\\eta)|}{( |\\xi-\\eta|)^{\\alpha}}, \n|\\beta|=j, \\xi \\neq \\eta, [\\xi,\\xi+\\beta] \\cup [\\eta,\\eta+\\beta] \\subset \\Omega^h \\, \\bigg\\}, \\\\\n||v^h||_{p;\\Omega_0^h} & = \\, \\text{max}_{j \\leq p} \\, |v^h|_{j;\\Omega_0^h}, \\\\\n||v^h||_{p,\\alpha;\\Omega_0^h} & = ||v^h||_{p;\\Omega_0^h}+ [v^h]_{p,\\alpha;\\Omega_0^h}. \n\\end{align*}\n\nThe above norms are extended canonically to vector fields and matrix fields by taking the maximum over all components. For $j=0$, we have discrete analogues of the maximum and $C^{0,\\alpha}$ norms. \n\nFor a domain $O \\subset \\mathbb{R}^n$, we denote by $\\mathcal{D}_h(O)$ the set of mesh functions on $\\mathbb{R}^n$ which vanish outside $O$. If $v^h=0$ on $\\partial \\Omega^h$, extending $v^h$ by 0 to $\\mathbb{Z}_h$, we obtain $v^h \\in \\mathcal{D}_h(\\Omega)$. The following theorem then follows from \n\n\\cite[Lemma 3.4]{Thomee1970}. \n\n\\begin{thm} \\label{discShauderPoisson}\nAssume $ 0<\\alpha<1$ and $v^h=0$ on $\\partial \\Omega^h$. Then there are constants $C$ and $h_0$ such that for $v^h \\in \\mathcal{M}(\\Omega^h), h \\leq h_0$\n\\begin{align} \\label{discShauderPoisson0}\n||v^h||_{2,\\alpha;\\Omega_0^h} \\leq C ||L_d \\, v^h||_{0,\\alpha;\\Omega_0^h}, \n\\end{align}\nwith the constant $C$ independent of $h$.\n\n\\end{thm}\n\nSince\n\\begin{align*}\n\\partial^i_{+} \\partial^i_{-} v^h(x) & =\\partial^i_{+} \\partial^i_{+} v^h (x-h e^i) \\, \\text{and} \\, \\\\\n\\partial^j_{h} \\partial^i_{h} v^h(x) & = \\frac{1}{4}\\bigg(\\partial^j_{+} \\partial^i_{+} v^h (x) + \\partial^j_{+} \\partial^i_{+} v^h (x- h e^i) + \\partial^j_{+} \\partial^i_{+} v^h (x- h e^j) \\\\\n& \\qquad \\qquad \\qquad +\\partial^j_{+} \\partial^i_{+} v^h (x- h e^i-h e^j) \\bigg), \n\\end{align*}\nwe have max $\\{||\\partial^i_{+} \\partial^i_{-} v^h ||_{0,\\alpha;\\Omega_0^h}, ||\\partial^j_{h} \\partial^i_{h} v^h||_{0,\\alpha;\\Omega_0^h}, i,j=1,\\ldots,n \\} \\leq ||v^h||_{2,\\alpha;\\Omega_0^h}$ and hence the above theorem also applies when the second order derivatives \\eqref{second-disc1}\nand \\eqref{second-disc2} are used in the definition of $|| . ||_{2,\\alpha;\\Omega_0^h}$.\n\n \nBy Taylor series expansions, it is not difficult to verify that for $v \\in C^2(\\Omega)$\n\\begin{align*}\n|v |_{j;\\Omega_0^h} \\leq |v|_{2;\\Omega}, j \\leq 2.\n\\end{align*}\nMoreover, for $v \\in C^{4,\\alpha}(\\Omega)$,\n\\begin{align} \\label{consistent}\n||D^2 v - \\mathcal{H}_d(v)||_{0;\\Omega_0^h} \\leq C h^2 |v|_{4;\\Omega},\n\\end{align}\nand \n\\begin{align*} \n[D^2 v - \\mathcal{H}_d( v) ]_{0,\\alpha;\\Omega_0^h} \\leq C h^{2} [v]_{4,\\alpha;\\Omega}.\n\\end{align*}\nTo see that the last inequality holds, it is enough to consider a function of one variable $v \\in C^{4,\\alpha}(-1,1)$ and estimate \n$[v''(x)-(v(x+h)-2v(x)+v(x-h))/h^2]_{0,\\alpha}$. Now,\n$$\nv''(x)-\\frac{v(x+h)-2v(x)+v(x-h)}{h^2} = -\\frac{h^2}{24} (v^{(4)}(x+ t_1 h) + v^{(4)}(x- t_2 h)), t_1, t_2 \\in [0,1].\n$$\nNext we note that, using the definition, the $C^{0,\\alpha}$ norm of $v^{(4)}(x+ t_1 h)$ is bounded above by the $C^{0,\\alpha}$ norm of $v^{(4)}$. The result then follows.\n\nWe have for $v \\in C^{4,\\alpha}(\\Omega)$,\n\\begin{align} \\label{consistent2}\n|| D^2 v - \\mathcal{H}_d( v)||_{0,\\alpha;\\Omega_0^h} \\leq C h^{2} ||v||_{4,\\alpha;\\Omega}.\n\\end{align}\n\n\\begin{lemma} \\label{est-d2}\nWe have for $u \\in C^{4,\\alpha}(\\Omega)$\n$$\n||S_k (D^2 u)-S_k (\\mathcal{H}_d( u)) ||_{0,\\alpha;\\Omega_0^h} \\leq C h^{2} |u|_{2;\\Omega}^{k-1} ||u||_{4,\\alpha;\\Omega}.\n$$\n\\end{lemma}\n\\begin{proof}\nBy the mean value theorem, using \\eqref{k-Hdiv0}, we have for some $t$ in $[0,1]$, and $x\\in \\Omega_0^h$, \n\\begin{align*}\nS_k (D^2 u) (x) - S_k (\\mathcal{H}_d( u))(x) & = S_k'(t D^2(u)(x) + (1-t) \\mathcal{H}_d( u)(x)): (D^2 u (x) \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\mathcal{H}_d( u)(x)) \\\\\n& = \\sum_{i,j=1}^n S_k^{ij} (t D^2(u)(x) + (1-t) \\mathcal{H}_d( u)(x)) (D^2 u (x) \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\mathcal{H}_d( u)(x))_{ij}.\n\\end{align*}\nUsing \\eqref{alpha-prod2}, it follows that\n\\begin{align*}\n||S_k (D^2 u)-S_k (\\mathcal{H}_d( u) ) ||_{0,\\alpha;\\Omega_0^h} & \\leq C (|u|_{2;\\Omega} + | u|_{2;\\Omega_0^h} )^{k-1} ||D^2 u - \\mathcal{H}_d( u)||_{0,\\alpha;\\Omega_0^h} \\\\\n& \\leq C h^{2} |u|_{2;\\Omega}^{k-1} ||u||_{4,\\alpha;\\Omega}. \n\\end{align*}\n\\end{proof}\n\n\\section{Approximations by linear elliptic problems} \\label{elliptic}\nIn this section, we prove the convergence of the iterative method \\eqref{broyden} and its discrete version. As indicated in the introduction, we also obtain the existence and uniqueness of the solution of the discrete version of \\eqref{k-H1}, i.e. \\eqref {k-H1h}, as well as error estimates.\n\n\\subsection{Convergence at the operator level} \n\nWe assume that \nthere is a unique $k$-admissible solution $u \\in C^{2,\\alpha}(\\tir{\\Omega})$ of \\eqref{k-H1} for $0 < \\alpha < 1$. Let $u^0 \\in C^{2,\\alpha}(\\tir{\\Omega})$ such that \n\\begin{equation} \\label{is-delta}\n||u-u^0||_{2,\\alpha;\\Omega} < \\delta.\n\\end{equation}\nFor $k=n$, using an eigenvalue argument, it is not difficult to prove that the cofactor matrix is uniformly positive definite under the assumption $f \\geq f_0 >0$ for a constant $f_0$. We assume that the matrix $\\{S_k^{ij}(D^2 u) \\}$ is uniformly positive definite. We claim that this holds if $u \\in C^2(\\tir{\\Omega})$ and there is $c_3 >0$ such that\n $$\n c_3 \\leq S_l(D^2 u), 1 < l \\leq k.\n $$\n We then have \n \\begin{equation} \\label{c4}\n c_3 \\leq S_l(D^2 u) \\leq c_4, 1 < l \\leq k,\n \\end{equation}\n for a constant $c_4$. The proof is essentially given as \\cite[Theorem 1.3 ]{Gavitone2009}. We define\n $$S_k^i(\\lambda):= \\frac{\\partial}{\\partial \\lambda_i} S_k(\\lambda).$$\n First we note from the proof of \\cite[Theorem 1.3 ]{Gavitone2009} that the eigenvalues of $\\{S_k^{ij}(D^2 u) \\}$ are given by $S_k^i(\\lambda(D^2 u)), 1 \\leq i \\leq n$. \nOn the other hand, since $S_l(D^2 u) \\geq c_3 >0, 1 < l \\leq k$, we have by \\cite[Proposition 1.1]{Caffarelli1985} \n$$\n\\frac{\\partial}{\\partial \\lambda_i} S_k(\\lambda)^{\\frac{1}{k}} >0 \\text{ for } \\lambda=\\lambda(D^2 u).\n$$\nFinally, as $S_k(D^2 u) \\leq c_4$ and $u \\in C^2(\\tir{\\Omega})$, the result follows. \n\nBy the continuity of the smallest eigenvalue of a matrix as a function of its entries, $\\{S_k^{ij}(D^2 u^0) \\}$ is also uniformly positive definite for $|u-u^0|_{2;\\Omega}$ sufficiently small. \n\nNext, $\\{S_k^{ij}(D^2 u^0)\\}$ is a symmetric matrix and divergence free by \\cite[ Formula 1.10 ]{Gavitone2009}. Thus we obtain \n \\begin{equation} \\label{div-prop}\n \\operatorname{div}\\bigg( \\{S_k^{ij}(D^2 u^0) \\} D v) \\bigg) = \\{S_k^{ij}(D^2 u^0) \\}: D^2 v.\n \\end{equation}\n\nWe have\n\\begin{thm} \\label{contT-broyden}\n\nUnder the assumption that there is a unique $k$-admissible solution $u \\in C^{2,\\alpha}(\\tir{\\Omega})$ of \\eqref{k-H1} for $0 < \\alpha < 1$, the sequence defined by \\eqref{broyden} converges to $u$\nfor $u^0$ sufficiently close to $u$. \n\\end{thm}\n\\begin{proof}\nWe define the operator $R: C^{2,\\alpha}(\\tir{\\Omega}) \\to C^{2,\\alpha}(\\tir{\\Omega})$ by\n\\begin{align*}\n-\\operatorname{div}\\bigg( \\{S_k^{ij}(D^2 u^0) \\} D (v-R v)\\bigg) &= - S_k (D^2 v) + f \\, \\text{in} \\, \\Omega \\\\\nR(v) & = g \\, \\text{on} \\, \\partial \\Omega.\n\\end{align*}\nBy Theorem \\ref{SchauderPoisson}, the operator $R$ is well defined. We show that for $\\rho>0$ sufficiently small, $R$ is a strict contraction in the ball\n$B_{\\rho}(u) = \\{v \\in C^{2,\\alpha}(\\tir{\\Omega}), ||u-v||_{2,\\alpha;\\Omega} < \\rho \\}$. \n\nFor $v, w \\in B_{\\rho}(u)$ we have using \\eqref{div-prop}\n\\begin{multline*}\n\\operatorname{div}\\bigg( \\{S_k^{ij}(D^2 u^0) \\} D (R v-R w) \\bigg) =\\operatorname{div}\\bigg(\\{S_k^{ij}(D^2 u^0) \\} D ( v- w) \\bigg) + S_k (D^2 w) - S_k (D^2 v)\\\\\n = -\\{S_k^{ij}(D^2 u^0) \\} : (D^2 w - D^2 v) + S_k (D^2 w) - S_k (D^2 v).\n\\end{multline*}\nNext, by the mean value theorem and using \\eqref{k-Hdiv0}, we have for some $t$ in $[0,1]$, \n\\begin{multline*}\nS_k(D^2 w)-S_k(D^2 v) = \\{S_k^{ij}(t D^2 w + (1-t) D^2 v) \\} : D^2 (w-v) \\\\\n= \\{S_k^{ij}(t (D^2 w -D^2 u^0)+ (1-t) (D^2 v-D^2 u^0) + D^2 u^0) \\} : D^2 (w-v).\n\\end{multline*}\n\nWe use \\eqref{cof-estimate} to estimate the $C^{0,\\alpha}$ norm of \n$$\nA=\\{S_k^{ij}(t (D^2 w -D^2 u^0)+ (1-t) (D^2 v-D^2 u^0) + D^2 u^0) \\} - \\{S_k^{ij}(D^2 u^0) \\}.\n$$\nFor $0 \\leq s \\leq 1$ to be specified below, put\n$$\n\\alpha_{s t} = s t (D^2 w -D^2 u^0)+ s (1-t) (D^2 v-D^2 u^0) + D^2 u^0.\n$$\nWe have\n\\begin{equation} \\label{a-st}\n|\\alpha_{s t}|_{0,\\alpha;\\Omega} \\leq ||u^0-v||_{2,\\alpha;\\Omega}+||u^0-w||_{2,\\alpha;\\Omega} +||u^0||_{2,\\alpha;\\Omega}.\n\\end{equation}\nBy the mean value theorem, for some $s \\in [0,1]$ we have\n$$\nA= \\{S_k^{ij}(\\alpha_{s t} ) \\}' (t (D^2 w -D^2 u^0)+ (1-t) (D^2 v-D^2 u^0)),\n$$\nand thus by \\eqref{cof-estimate}\n\\begin{equation} \\label{a-st2}\n||A||_{0,\\alpha;\\Omega} \\leq C |\\alpha_{s t}|_{0,\\alpha;\\Omega}^{k-2} (||u^0-v||_{2,\\alpha;\\Omega}+||u^0-w||_{2,\\alpha;\\Omega} ).\n\\end{equation}\nBy Schauder estimates (Theorem \\ref{SchauderPoisson}), \\eqref{alpha-prod2}, \\eqref{a-st} and \\eqref{a-st2} we obtain\n\\begin{align} \\label{contraction-cont-level}\n\\begin{split}\n||R(v) - R(w)||_{2,\\alpha;\\Omega} & \\leq C ||A||_{0,\\alpha;\\Omega} ||D^2(v-w)||_{0,\\alpha;\\Omega} \\\\\n& \\leq C (||u^0-v||_{2,\\alpha;\\Omega}+||u^0-w||_{2,\\alpha;\\Omega} +||u^0||_{2,\\alpha;\\Omega} )^{k-2} \\\\\n& \\qquad \\qquad (||u^0-v||_{2,\\alpha;\\Omega}+||u^0-w||_{2,\\alpha;\\Omega} ) ||v-w||_{2,\\alpha;\\Omega} \\\\\n& \\leq C (\\rho+\\delta+||u^0||_{2,\\alpha;\\Omega})^{k-2} (\\rho+\\delta)||v-w||_{2,\\alpha;\\Omega},\n\\end{split}\n\\end{align}\nwhere $\\delta$ measures how close $u^0$ is to $u$ \\eqref{is-delta}. \nThus, for $\\rho$ and $\\delta$ sufficiently small, $R$ is a strict contraction in $B_{\\rho}(u)$. \n\nIt remains to show that $R$ maps $B_{\\rho}(u)$ into itself. We note by the definition of $R$ and unicity of the solution of \\eqref{k-H1}, a fixed point of $R$ solves \\eqref{k-H1}. Let $v \\in B_{\\rho}(u)$,\n\\begin{align*}\n||u-R v||_{2,\\alpha;\\Omega} & =||R u-R v||_{2,\\alpha;\\Omega} \\leq ||u-v||_{2,\\alpha;\\Omega} \\leq \\rho,\n\\end{align*}\nwhich shows that $R$ maps $B_{\\rho}(u)$ into itself. The existence of a fixed point follows from the Banach fixed point theorem. Moreover, the sequence defined by $u^{m+1}=R(u^m)$, i.e. the sequence defined by \n \\eqref{broyden}, converges for $\\rho$ and $\\delta$ sufficiently small to $u$.\n\\end{proof}\n\\subsection{Finite difference discretization}\n\nNext \nwe consider the following discrete version of \\eqref{broyden}\n\\begin{align} \\label{broyden-D}\n\\begin{split}\n \\{S_k^{ij}(\\mathcal{H}_d \\, u^{0,h}) \\} : \\mathcal{H}_d u^{m+1,h} & = \\{S_k^{ij}(\\mathcal{H}_d \\, u^{0,h}) \\} : \\mathcal{H}_d u^{m,h} \\\\\n& \\qquad \\qquad \\qquad \\qquad +f-S_k (\\mathcal{H}_d \\, u^{m,h}) \\, \\text{in} \\, \\Omega^h_0 \\\\\n u^{m+1,h} & = g \\, \\text{on} \\, \\partial \\Omega^h.\n \\end{split}\n\\end{align}\nUnder the assumptions of Theorem \\ref{disc-thm} below, we show that \\eqref{k-H1h} has a unique solution to which the above sequence converges. Moreover, the convergence rate is O($h^{2}$). \nDefine\n\\begin{equation} \\label{ball-h}\nB_{\\rho} ( u) = \\{v^h \\in \\mathcal{M}(\\Omega^h), ||v^h- u||_{2,\\alpha;\\Omega_0^h} \\leq \\rho \\}.\n\\end{equation}\n\\begin{lemma} \\label{sum-lem}\nLet $S^h: \\mathcal{M}(\\Omega^h) \\to \\mathcal{M}(\\Omega^h)$ be a strict contraction with contraction factor less than 1/2, i.e. for $v^h, w^h \\in \\mathcal{M}(\\Omega^h)$\n$$\n||S^h(v^h) - S^h(w^h)||_{2,\\alpha;\\Omega_0^h} \\leq\\frac{1}{2} ||v^h - w^h||_{2,\\alpha;\\Omega_0^h}. \n$$\nLet us also assume that $S^h$ does not move the center $u$ of the ball $B_{\\rho} ( u)$ too far, i.e.\n$$\n||S^h( u) - u||_{2,\\alpha;\\Omega_0^h} \\leq C_0 h^2. \n$$\nThen $S^h$ maps $B_{\\rho} ( u)$ into itself for $\\rho=2 C_0 h^{2}$. Moreover $S^h$ has a unique fixed point $u^h$ in $B_{\\rho} ( u)$ with the error estimate\n$$\n|| u - u^h ||_{2,\\alpha;\\Omega_0^h} \\leq 2 C_0 h^2.\n$$ \n\\end{lemma}\n\\begin{proof}\n\nFor $v^h \\in B_{\\rho} ( u)$,\n\\begin{align*}\n||S^h(v^h)- u||_{2,\\alpha;\\Omega_0^h} & \\leq ||S^h(v^h)- S^h( u)||_{2,\\alpha;\\Omega_0^h}+ ||S^h( u)- u||_{2,\\alpha;\\Omega_0^h}\\\\\n& \\leq \\frac{1}{2} ||v^h- u||_{2,\\alpha;\\Omega_0^h} + C_0 h^2 \\\\\n& \\leq \\frac{\\rho}{2}+C_0 h^2 \\leq \\frac{\\rho}{2} + \\frac{\\rho}{2}=\\rho.\n\\end{align*}\n\nThis proves that $S^h$ maps $B_{\\rho} ( u)$ into itself. \nThe existence of a fixed point follows from the Banach fixed point theorem. \nThe convergence rate follows from the observation that\n\\begin{align*}\n|| u - u^h ||_{2,\\alpha;\\Omega_0^h} & \\leq || u - S^h( u) ||_{2,\\alpha;\\Omega_0^h} + ||S^h( u) - S^h(u^h) ||_{2,\\alpha;\\Omega_0^h} \\\\\n & \\leq C_0 h^2 + \\frac{1}{2} ||u^h- u||_{2,\\alpha;\\Omega_0^h}.\n\\end{align*}\n\n\\end{proof}\n\n\\begin{rem} \\label{u0h-rem}\nFor $h$ sufficiently small, $\\mathcal{H}_d (u)$ is sufficiently close to $D^2 u$ and hence $\\{S_k^{ij}(\\mathcal{H}_d u)\\}$ is positive definite, a property which also holds for $\\{S_k^{ij}(\\mathcal{H}_d \\,u^{0,h})\\}$ for $u^{0,h}$ sufficiently close to $u$. The arguments are similar to the ones of Lemma \\ref{close}. See also Lemma \\ref{lboundDelta} below.\n\\end{rem}\n \n\\begin{lemma}\\label{lboundDelta} Let $u$ be a $k$-admissible solution of \\eqref{k-H1}. Assume that inf $f >0$ and $u \\in C^4(\\Omega)$. Then for $h$ sufficiently small, $\\Delta_d (u) \\geq c_0 >0$ where \n$c_0= 1/2 ((\\text{inf} \\, f)/c(k,n))^{1/k}$. Moreover, if $u$ is a strictly convex function, then for $\\rho= O(h^{2})$, $\\mathcal{H}_d( u)$ is a positive matrix and $v^h$ is a discrete convex function, when $v^h \\in B_{\\rho} ( u)$. \n\\end{lemma} \n\\begin{proof}\nSince the eigenvalues of a matrix are continuous functions of its entries (as roots of the characteristic polynomial), for a matrix $A=(a_{ij})$ with \n$S_k (A) >0$, we have for $\\operatorname{\\epsilon}ilon >0$, the existence of $\\gamma >0$ depending only on the space dimension $n$ such that $|S_k ( B) - S_k ( A)| < \\operatorname{\\epsilon}ilon$ when $\\text{sup}_{ij} |b_{ij}-a_{ij}| <\\gamma$. This implies $S_k ( B)> S_k ( A) - \\operatorname{\\epsilon}ilon$. Thus with $\\operatorname{\\epsilon}ilon=S_k( A)/2$, we have $S_k ( B) > S_k ( A)/2$. \n\nFor $h$ sufficiently small we have $C h^2 |u|_{4;\\Omega} < \\gamma$ and thus since $S_k(D^2 u) =f > \\text{inf} \\, f >0 $, by \\eqref{consistent} $S_k(\\mathcal{H}_d( u)) \\geq 1/2 \\, \\text{inf} \\, f $. \nBy \\eqref{G-am} \n$$\n\\Delta_d (u) \\geq \\frac{1}{2} ((\\text{inf} \\, f)/c(k,n))^{1/k}.\n$$\n\nLet $v^h \\in B_{\\rho} ( u)$. Then by definition of $ B_{\\rho} ( u)$ and \\eqref{consistent}\n\n\\begin{align*}\n||\\mathcal{H}_d(v^h) - \\mathcal{H}_d( u)||_{0,\\alpha;\\Omega_0^h} & \\leq ||\\mathcal{H}_d(v^h) - D^2 u||_{0,\\alpha;\\Omega_0^h} + ||D^2 u- \\mathcal{H}_d( u)||_{0,\\alpha;\\Omega_0^h} \\\\\n& \\leq \\rho + C h^2 |u|_{4;\\Omega},\n\\end{align*}\nwhich can be made smaller than $\\gamma$ for $h$ and $\\rho$ sufficiently small. Thus given that $\\mathcal{H}_d( u)$ is positive definite, the same holds for $\\mathcal{H}_d(v^h)$. \n\\end{proof}\n\n\\begin{thm}\\label{disc-thm}\nAssume that $u \\in C^{4,\\alpha}(\\tir{\\Omega})$ is $k$-admissible. \nChoose $u^{0,h}$ such that \n$||u^{0,h}- u||_{2,\\alpha;\\Omega_0^h} = O(h^2)$.\nFor $h$ sufficiently small, \\eqref{k-H1h} has a locally unique solution $u^h$ which satisfies $\\Delta_d (u^h) \\geq 0$ and \n\n$u^h$ converges to the unique solution $u$ of \\eqref{k-H1} as $h \\to 0$ at the rate O$(h^{2})$. \n\\end{thm}\n\n\\begin{proof}\nIt follows from the assumptions that inf $f >0$. We define the operator\n$R^h: \\mathcal{M}(\\Omega^h) \\to \\mathcal{M}(\\Omega^h)$ by\n\\begin{align*} \n- \\{S_k^{ij}(\\mathcal{H}_d \\, u^{0,h}) \\} : \\mathcal{H}_d (v^h-R^h v^h) &= - S_k (\\mathcal{H}_d \\, v^h) + f \\, \\text{in} \\, \\Omega^h_0 \\\\\nR^h(v^h) & = g \\, \\text{on} \\, \\partial \\Omega^h,\n\\end{align*}\nand show that $R^h$ has a unique fixed point in $B_{\\rho}( u)$ for $\\rho=O(h^{2})$. By Remark \\ref{u0h-rem} the above problem is then well defined. \nIt follows from \\eqref{div-prop} that the operator $R^h$ is a discrete version of the operator $R$ used in the proof of Theorem \\ref{contT-broyden}. Thus, as in the proof of Theorem \\ref{contT-broyden} \nwe obtain \n\\begin{align*}\n \\{S_k^{ij}(\\mathcal{H}_d \\, u^{0,h}) \\} : \\mathcal{H}_d (R^h v^h - R^h w^h) & =\nS_k (\\mathcal{H}_d \\, w^h) - S_k (\\mathcal{H}_d \\, v^h) \\\\\n& \\qquad + \\{S_k^{ij}(\\mathcal{H}_d \\, u^{0,h}) \\}:\\mathcal{H}_d \\,(v^h-w^h). \n\\end{align*}\nAnd thus by the mean value theorem and discrete Schauder estimates, as in the proof of Theorem \\ref{contT-broyden}\n\\begin{multline} \\label{strict-step-01}\n||R^h(v^h) - R^h(w^h)||_{2,\\alpha;\\Omega_0^h} \\leq \\\\ C (\\rho+\\delta_h+||u^{0,h}||_{2,\\alpha;\\Omega_0^h})^{k-2} (\\rho+\\delta_h)||v^h-w^h||_{2,\\alpha;\\Omega_0^h}.\n\\end{multline}\nNext, note that with \\eqref{consistent2} applied to $u$ one has $|u|_{2,\\alpha;\\Omega_0^h} \\leq C ||u||_{4,\\alpha;\\Omega}$. \nIt follows that $||u^{0,h}||_{2,\\alpha;\\Omega_0^h}\\leq ||u^{}||_{2,\\alpha;\\Omega_0^h} + \\delta_h \\leq C ||u||_{4,\\alpha;\\Omega} + \\delta_h$. We recall that by assumption \n$||u^{0,h}- u||_{2,\\alpha;\\Omega_0^h} = O(h^2)$. \nThus\n$R^h$ is a strict contraction in $B_{\\rho}( u)$ for $\\rho=$O$(h^{2})$ and $h$ sufficiently small. \nMoreover, the contraction factor can be made smaller than 1/2 by choosing $h$ sufficiently small. \n\nSince $f=S_k(D^2 u)$, by the discrete Schauder estimates Theorem \\ref{discShauderPoisson} and Lemma \\ref{est-d2} \n\\begin{align*}\n||R^h( u) - u||_{2,\\alpha;\\Omega_0^h} \\leq C ||S_k (D^2 u)-S_k (\\mathcal{H}_d( u)) ||_{0,\\alpha;\\Omega_0^h} \\leq C h^2. \n\\end{align*}\nBy Lemma \\ref{sum-lem} we conclude that $R^h$ has a fixed point $u^h$ in $B_{\\rho}( u)$ with the claimed convergence rate.\n\nThe claimed property of $u^h$ follows from the fact that $u^h \\in B_{\\rho}( u)$ and Lemma \\ref{lboundDelta}.\n\n\\end{proof}\n\n\\section{Newton's method} \\label{newton-sec}\n\nAs in the previous section, we assume that $ \\{S_k^{ij}(D^2 u) \\}$ is uniformly positive definite. By Remark \\ref{u0h-rem}, \nfor $h$ sufficiently small, there exists $m' >0$ such that for $v^h \\in B_{\\rho}( u)$, $ \\{S_k^{ij}(\\mathcal{H}_d v^h) \\}$ has smallest eigenvalue greater than $m'$. We consider for $u^{0,h} \\in B_{\\rho}( u)$ the sequence of iterates\n\\begin{align} \\label{newton}\n\\begin{split}\n\\{S_k^{ij}(\\mathcal{H}_d u^{m,h}) \\}: (\\mathcal{H}_d u^{m+1,h} - \\mathcal{H}_d u^{m,h}) & = f- S_k(\\mathcal{H}_d u^{m,h}) \\ \\text{in} \\ \\Omega_0^h \\\\\n u^{m+1,h} & = g \\ \\text{in} \\ \\partial \\Omega^h.\n\\end{split}\n\\end{align} \nWe note that \\eqref{newton} defines $u^{m+1,h}$ as the solution of a discrete second order elliptic equation in non divergence form, which is uniformly elliptic for $u^{m,h} \\in B_{\\rho}( u)$ for $h$ sufficiently small. \n\n\\begin{thm} \\label{newton-th}\nThe sequence defined by \\eqref{newton} satisfies \n\\begin{equation} \\label{newton-quad}\n||u^{m+1,h} - u^h||_{2,\\alpha;\\Omega_0^h} \\leq C ||u^{m,h} - u^h||_{2,\\alpha;\\Omega_0^h}^2,\n\\end{equation}\nfor $\\rho$ and $h$ sufficiently small and where $u^h$ denotes the solution of \\eqref{k-H1h} in $B_{\\rho}( u), \\rho=O(h^{2})$. \n\\end{thm}\n\n\\begin{proof}\nPut\n\\begin{equation} \\label{B0}\nB = \\{S_k^{ij}(\\mathcal{H}_d u^{m,h}) \\}: (\\mathcal{H}_d u^{m+1,h} - \\mathcal{H}_d u^{h}) .\n\\end{equation}\nWe have by \\eqref{k-H1h}\n\\begin{align} \\label{B}\n\\begin{split}\nB & = \\{S_k^{ij}(\\mathcal{H}_d u^{m,h}) \\}: (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}) +S_k (\\mathcal{H}_d u^{h} )- S_k(\\mathcal{H}_d u^{m,h}) \\\\\n & = \\bigg( \\{S_k^{ij}(\\mathcal{H}_d u^{m,h}) \\} - \\{S_k^{ij}(\\mathcal{H}_d u^{h} ) \\} \\bigg): (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}) \\\\\n& \\quad \\quad + \\{S_k^{ij}(\\mathcal{H}_d u^{h}) \\}: (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}) +S_k (\\mathcal{H}_d u^{h} )- S_k(\\mathcal{H}_d u^{m,h}).\n\\end{split}\n\\end{align}\nPut\n\\begin{equation} \\label{B1}\nB_1 = \\bigg( \\{S_k^{ij}(\\mathcal{H}_d u^{m,h}) \\} - \\{S_k^{ij}(\\mathcal{H}_d u^{h} ) \\} \\bigg): (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}),\n\\end{equation}\nand\n\\begin{equation} \\label{B2}\nB_2 = \\{S_k^{ij}(\\mathcal{H}_d u^{h}) \\}: (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}) +S_k (\\mathcal{H}_d u^{h} )- S_k(\\mathcal{H}_d u^{m,h}).\n\\end{equation}\nBy the mean value theorem, \\eqref{k-Hdiv0} and \\eqref{cof-estimate}, we have\n$$\nB_1 =\\big( \\{ S_k^{ij}(t \\mathcal{H}_d u^{m,h} + (1-t) \\mathcal{H}_d u^{h} ) \\}'(\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}) \\big): (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}),\n$$\nfor $t \\in [0,1]$ and thus\n\\begin{align} \\label{B1-est}\n\\begin{split}\n||B_1||_{0,\\alpha;\\Omega_0^h} & \\leq C (||u^h||_{2,\\alpha;\\Omega_0^h} + ||u^{m,h}||_{2,\\alpha;\\Omega_0^h} )^{k-2} ||u^{m,h} - u^h||_{2,\\alpha;\\Omega_0^h}^2 \\\\\n& \\leq C (|| u||_{2,\\alpha;\\Omega_0^h} +\\rho )^{k-2} ||u^{m,h} - u^h||_{2,\\alpha;\\Omega_0^h}^2 \\\\\n& \\leq C (|| u||_{2,\\alpha;\\Omega_0^h} +\\rho )^{k-2} ||u^{m,h} - u^h||_{2,\\alpha;\\Omega_0^h}^2.\n\\end{split}\n\\end{align}\nWe also have by the mean value theorem \n\\begin{align} \\label{B2-est}\n\\begin{split}\nB_2 & = \\{S_k^{ij}(\\mathcal{H}_d u^{h}) \\}: (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}) \\\\\n & \\qquad + \\{S_k^{ij}( t \\mathcal{H}_d u^{h} + (1-t) \\mathcal{H}_d u^{m,h}) \\}: (\\mathcal{H}_d u^{h} - \\mathcal{H}_d u^{m,h})\\\\\n & = \\bigg(\\{S_k^{ij}(\\mathcal{H}_d u^{h}) \\} - \\{S_k^{ij}( t \\mathcal{H}_d u^{h} + (1-t) \\mathcal{H}_d u^{m,h}) \\} \\bigg) : (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}) \\\\\n & = \\bigg( \\{S_k^{ij}( (1-s) \\mathcal{H}_d u^{h} +s t \\mathcal{H}_d u^{h} + s(1-t) \\mathcal{H}_d u^{m,h}) \\}' \\\\\n & \\qquad \\qquad \\qquad \\big((1-t)(\\mathcal{H}_d u^{h} - \\mathcal{H}_d u^{m,h})\\big) \\bigg) : (\\mathcal{H}_d u^{m,h} - \\mathcal{H}_d u^{h}),\n\\end{split}\n\\end{align}\nfor $s,t \\in [0,1]$. As for $B_1$ we obtain\n\\begin{align}\n\\begin{split}\n||B_2||_{0,\\alpha;\\Omega_0^h} & \\leq C (|| u||_{2,\\alpha;\\Omega_0^h} +\\rho )^{k-2} ||u^{m,h} - u^h||_{2,\\alpha;\\Omega_0^h}^2.\n\\end{split}\n\\end{align}\nCombining \\eqref{B0}--\\eqref{B2-est} \nand using Schauder estimates, we obtain \\eqref{newton-quad}. \n\\end{proof}\n\nChoosing $\\rho=O(h^{2})$ we have $C \\rho < 1$ for $h$ sufficiently small. We conclude that $u^{m+1,h} \\in B_{\\rho}( u)$ when $u^{m,h} \\in B_{\\rho}( u)$ and the quadratic convergence rate of Newton's method.\n\\begin{rem}\n\nHaving established that the discrete problem has a locally unique solution and that $v^h$ is a discrete convex function for $v^h$ sufficiently close to $u$, the convergence of Newton's method also follows from the verification of standard assumptions given in \\cite[p. 68]{Kelley95}. See \\cite{Oberman2010a} for an example of verification of the standard assumptions for a wide stencil discretization.\n\n\\end{rem}\n\n\\section{Gauss-Seidel iterative methods} \\label{convexity}\n\nIt is a natural idea to solve \\eqref{k-H1h} by a nonlinear Gauss-Seidel method, that is solve \\eqref{k-H1h} for $u^h(x)$ and solve the resulting nonlinear equations by a Gauss-Seidel method. Although this seems a daunting task for arbitrary $k$, we show that for $k=2$, this takes a very elegant form. We then establish a connection between the resulting nonlinear Gauss-Seidel iterative method for $2$-Hessian equations and the discrete version of \\eqref{k-H-iterative}, i.e.\n\\begin{align} \\label{k-H1-iterativeD}\n\\begin{split}\n\\Delta_d \\, u^{m+1,h} & = \\bigg( (\\Delta_d \\, u^{m,h})^k + \\frac{1}{c(k,n)}( f-S_k (\\mathcal{H}_d \\, u^{m,h} ) )\\bigg)^{\\frac{1}{k}} \\ \\text{in} \\ \\Omega_0^h \\\\\nu^{m+1,h} & = g \\, \\text{on} \\, \\partial \\Omega^h,\n\\end{split}\n\\end{align}\nwhen the Gauss-Seidel method is used to solve the Poisson equations. \n\n\\subsection{Nonlinear Gauss-Seidel method for 2-Hessian equations}\n\nWe start with the identity\n\\begin{align} \\label{identity}\n\\Delta_d \\, u^{h} = \\bigg( (\\Delta_d \\, u^{h})^2 + \\frac{1}{c(2,n)}(f-S_2 (\\mathcal{H}_d \\, u^{h} ) ) \\bigg)^{\\frac{1}{2}},\n\\end{align}\nand show that the right hand side is independent of $u^h(x)$. Note that by \\eqref{second-disc2}, $\\partial^i_h \\partial^j_h u^h(x), i \\neq j$ is independent of $u^h(x)$ and \nby \\eqref{second-disc3},\n$$\n\\frac{\\partial (\\Delta_d \\, u^{h}(x))}{\\partial (u^h(x))} = \\sum_{i=1}^n -\\frac{2}{h^2} = -\\frac{2 n}{h^2}.\n$$\nSince $\\partial S_k(A)/\\partial z = \\sum_{i,j=1}^n( \\partial S_k(A)/\\partial a_{ij}) ( \\partial a_{ij}/\\partial z)$,\nwe conclude that\n\\begin{align*}\n\\frac{\\partial }{\\partial (u^h(x))} S_2 (\\mathcal{H}_d \\, u^{h}(x) ) & = \\sum_{i,j=1 \\atop i \\neq j}^n S_2^{ij}(\\mathcal{H}_d \\, u^{h}(x)) \\frac{\\partial}{\\partial (u^h(x))} \\partial^i_h \\partial^j_h u^h(x) \\\\\n& \\qquad + \\sum_{i=1}^n S_2^{i i}(\\mathcal{H}_d \\, u^{h}(x)) \\frac{\\partial}{\\partial (u^h(x))} \\partial^i_+ \\partial^i_- u^h(x) \\\\\n& = -\\frac{2}{h^2} \\sum_{i=1}^n S_2^{ii}(\\mathcal{H}_d \\, u^{h}(x)) = -\\frac{2}{h^2} \\sum_{i=1}^n \\sum_{1 \\leq p \\leq n \\atop p \\neq i} \\delta_{i p}^{i p} \\,\n\\partial^p_+ \\partial^p_- u^h(x)\\\\\n& = -\\frac{2}{h^2} \\sum_{i=1}^n \\sum_{p \\neq i} \\partial^p_+ \\partial^p_- u^h(x)= -\\frac{2}{h^2} (n-1) \\Delta_d \\, u^{h}(x)\\\\\n& = -\\frac{2}{h^2} (2 n) \\, c(2,n) \\Delta_d \\, u^{h}(x) = -\\frac{4 n}{h^2} c(2,n) \\Delta_d \\, u^{h}(x),\n\n\\end{align*}\nand we recall that the definition of $\\delta_{i p}^{i p}$ was given in section \\ref{notation1}.\nThis gives\n$$\n\\frac{\\partial }{\\partial (u^h(x))} \\bigg( (\\Delta_d \\, u^{h}(x))^2 + \\frac{1}{c(2,n)}(f-S_2 (\\mathcal{H}_d \\, u^{h}(x) )) \\bigg) = 0.\n$$\nWe can therefore rewrite \\eqref{identity} as \n\\begin{align} \\label{identity2}\n\\begin{split}\nu^h(x) & = \\frac{h^2}{2 n} \\bigg[\\sum_{i=1}^n \\frac{u^h(x+he^i) + u^h(x-h e^i)}{h^2} \\\\\n& \\qquad \\qquad \\qquad \\quad - \\bigg( (\\Delta_d \\, u^{h}(x))^2 + \\frac{1}{c(2,n)}(f-S_2 (\\mathcal{H}_d \\, u^{h}(x) ) \\bigg)^{\\frac{1}{2}} \\bigg],\n\\end{split}\n\\end{align}\nwhere the solution with $\\Delta_d \\, u^{h} \\geq 0$ has been selected. For $n=2$, this is the identity which was solved in \\cite{Headrick05,Chen2010b,Chen2010c,Benamou2010} by a Gauss-Seidel iterative method, as indicated in the introduction. For $n \\geq 3$, this provides new iterative methods for the $2$-Hessian equations. \n\nHenceforth, we shall assume that a row ordering of the elements of $\\Omega^h$ is chosen. Note that if we apply the Gauss-Seidel method to the problem \\eqref{k-H1-iterativeD}, we obtain a double sequence $u^{m,p,h}$ defined by\n\\begin{align*} \n\\begin{split}\nu^{m+1,p+1,h}(x) & = \\frac{h^2}{2 n} \\bigg[\\sum_{i=1}^n \\frac{u^{m+1,p,h}(x+he^i) + u^{m+1,p+1,h}(x-h e^i)}{h^2} \\\\\n& \\qquad \\qquad \\qquad \\quad - \\bigg( (\\Delta_d \\, u^{m,h}(x))^2 + \\frac{1}{c(2,n)}(f-S_2 (\\mathcal{H}_d \\, u^{m,h}(x) ) \\bigg)^{\\frac{1}{2}} \\bigg],\n\\end{split}\n\\end{align*}\nThis leads us to consider the double sequence $u^{m,h}_p$ defined by\n\\begin{align*} \n\\begin{split}\nu^{m+1,h}_{p+1}(x) & = \\frac{h^2}{2 n} \\bigg[\\sum_{i=1}^n \\frac{u^{m+1,h}_p(x+he^i) + u^{m+1,h}_{p+1}(x-h e^i)}{h^2} \\\\\n& \\qquad \\qquad \\qquad \\quad - \\bigg( (\\Delta_d \\, u^{m,h}_{p*}(x))^2 + \\frac{1}{c(2,n)}(f-S_2 (\\mathcal{H}_d \\, u^{m,h}_{p*}(x) ) \\bigg)^{\\frac{1}{2}} \\bigg],\n\\end{split}\n\\end{align*} \nwhere $\\Delta_d \\, u^{m,h}_{p*}(x)$ and $S_2 (\\mathcal{H}_d \\, u^{m,h}_{p*}(x) )$ are the actions of the discrete Laplace and $2$-Hessian operators on $u^{m,h}_p$ updated with the most recently computed values.\n\nFormally, as $m \\to \\infty$, this gives the nonlinear Gauss-Seidel method \n\\begin{align} \\label{Gauss} \n\\begin{split}\nu^{h}_{p+1}(x) & = \\frac{h^2}{2 n} \\bigg[\\sum_{i=1}^n \\frac{u^{h}_p(x+he^i) + u^{h}_{p+1}(x-h e^i)}{h^2} \\\\\n& \\qquad \\qquad \\qquad \\quad - \\bigg( (\\Delta_d \\, u^{h}_{p*}(x))^2 + \\frac{1}{c(2,n)}(f-S_2 (\\mathcal{H}_d \\, u^{h}_{p*}(x) ) \\bigg)^{\\frac{1}{2}} \\bigg],\n\\end{split}\n\\end{align}\nwhere as above $\\Delta_d \\, u^{h}_{p*}(x)$ and $S_2 (\\mathcal{H}_d \\, u^{h}_{p*}(x) )$ are the actions of the discrete Laplace and $2$-Hessian operators on $u^{h}_p$ updated with the most recently computed values of $u^{h}_{p+1}$.\nIn particular, the right hand side of \\eqref{Gauss} does not depend on $u^h_{p+1}$ since as shown above, the right hand side of \\eqref{identity2} does not depend on $u^h(x)$.\n\n\\section{Numerical results} \\label{num}\n\nWe give numerical results for the $\\sigma_2$ problem, i.e. for $k=2, n=3$ using the subharmonicity preserving iterations. \nAlthough our theoretical results only cover smooth solutions, as indicated in the abstract and in the introduction, the subharmonicity preserving iterations appear able to handle non smooth solutions. The initial guess in all of our numerical experiments is taken as the finite difference approximation of the solution of the Poisson equation $\\Delta u = 2 \\sqrt{f}$ in $\\Omega$ with $u=g$ on $\\partial \\Omega$.\n\nWe use the following test functions on the unit cube $[0,1]^3$:\n\nTest 1: A smooth solution which is strictly convex, $u(x,y,z)=e^{x^2+y^2+z^2}$ so that \n$f(x,y,z)=4(3+x^2+y^2+z^2)e^{2(x^2+y^2+z^2)}$ and $g(x,y,z)=e^{x^2+y^2+z^2}$ on $\\partial \\Omega$.\n\nTest 2: A smooth solution which is $2$-convex but not convex. It is known that for a radial function $u(x)=\\phi(r), r=|x|, x \\in \\mathbb{R}^n$ the eigenvalues of $D^2 u$ are given by\n$\\lambda_1=\\phi''(r)$ with multiplicity 1 and $\\lambda_2=\\phi'(r)/r$ with multiplicity $n-1$. See for example \\cite[Lemma 2.1]{Felmer2003}. It follows that with $u(x,y,z)=\\ln(a+x^2+y^2+z^2)$, we have $\\phi(r)= \\ln(a+r^2)$ and we get \n\n$\\Delta u = \\frac{6 a + 2 r^2}{ (a+r^2)^2} \\geq 0, \\, S_2(D^2 u)= 4\\frac{3 a-r^2}{ (a+r^2)^3} \\geq 0, \\, \\det D^2 u = 2\\frac{a-r^2} {(a+r^2)^2},$\n\nin $[0,1]^3$. With $a=2$, $\\det D^2 u$ takes negative values in $[0,1]^3$.\n\nTest 3: A solution not in $H^2(\\Omega)$, $u(x,y,z)=-\\sqrt{3-x^2-y^2-z^2}$ \nso that $f(x,y,z)=-(x^2+y^2+z^2-9)/(-3+x^2+y^2+z^2)^2$ and $g(x,y,z)=-\\sqrt{3-x^2-y^2-z^2}$ on $\\partial \\Omega$.\n\nTest 4: No exact solution is known. Here\n$f(x,y,z)=1$ and $g(x,y,z)=0$. \n\nTest 5: A degenerate three dimensional Monge-Amp\\`ere equation. We take $f(x,y,z)=0$ and $g(x,y,z)=|x-1/2|$. \nWe use the double iterative method based on \\eqref{sigma2k}. \n\nNumerically, the solution computed may not satisfy $S^2 D^2 u^{m} \\geq 0$. At those points we set both $S_2 (D^2 u^{m})$ and $\\det D^2 u^m$ to 0 in \n\\eqref{sigma2k}. If the numerical value of $S_2(D^2 u^{m})$ is negative, then 0 is a better approximate value. Since $S_2( D^2 u^{m})$ is computed from $u^m$, the numerical value of $\\det D^2 u^m$ would also be inaccurate. Since $u^m$ is expected to be an approximate solution of $u$ for which $\\det D^2 u \\geq 0$, a better approximation of $\\det D^2 u^m$ at any stage where the latter is negative is also 0. It would be interesting to analyze the effect of these rounding off errors on the overall numerical convergence of the method. For example, one may analyze the convergence of the inexact double iteration. Similar situations appear with inexact Newton's methods and inexact Uzawa algorithms. \n\nThe right hand side $f(x,y,z)$ can be computed from the exact solution $u(x,y,z)$ using the definition of $S_2(D^2 u)$ as the sum of the $2 \\times 2$ principal minors.\n\nFor all tests but Test 3, we used the direct solver \\eqref{k-H1-iterativeD}. For $h=2^6$, we run out of memory with \\eqref{k-H1-iterativeD}. For Test 3, the Gauss-Seidel method was used since there is no memory issue for the latter with $h=2^6$. As expected, we have quadratic convergence (as $h \\to 0$) for the smooth solutions of Tests 1 and 2 while enough data is not available to give the convergence rate for the singular solution of Test 3.\n\n\\begin{table} \n\\begin{tabular}{c|ccccc} \n \\multicolumn{6}{c}{$h$}\\\\\n & $1/2^1$ & $1/2^2$ & $1/2^3$& $1/2^4$ & $1/2^5$ \\\\%& $1/2^{10}$\\\\ \\hline\nError & 6.2328 $10^{-2}$ & 2.6556 $10^{-2}$ & 7.7836 $10^{-3}$ & 2.0616 $10^{-3}$ & 5.2449 $10^{-4}$ \\\\\nRate & &1.23 & 1.77&1.92 &1.97 \n\\end{tabular} \n\n\\caption{Maximum error with Test 1.}\n\\end{table}\n\n\\begin{table} \n\\begin{tabular}{c|ccccc} \n \\multicolumn{6}{c}{$h$}\\\\\n & $1/2^1$ & $1/2^2$ & $1/2^3$& $1/2^4$ & $1/2^5$ \\\\%& $1/2^{10}$\\\\ \\hline\nError & 6.5241 $10^{-4}$ & 5.0653 $10^{-4}$ & 1.3850 $10^{-4}$ & 3.5587 $10^{-5}$ & 9.1276 $10^{-6}$ \\\\\nRate & &0.36 & 1.87&1.96 &1.96 \n\\end{tabular} \n\n\\caption{Maximum error with Test 2.}\n\\end{table}\n\n\\begin{table} \n\\begin{tabular}{c|ccc} \n \\multicolumn{4}{c}{$h$}\\\\\n & $1/2^4$ & $1/2^5$ & $1/2^6$ \\\\%& $1/2^7$ & $1/2^8$ \\\\%& $1/2^{10}$\\\\ \\hline\nError & 1.1084 $10^{-3}$ & 9.7971 $10^{-4}$ & 7.6618 $10^{-4}$ \\\\&& & \\\\\nRate & &0.18 &0.35 \\\\%& &\n\\end{tabular} \n\n\\caption{Maximum error with Test 3.}\n\\end{table}\n\n\\begin{figure}\n\\caption{Test 4, $h=1/2^5$. Graph and contour in plane $z=1/2$.}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Test 5, $h=1/2^4$. Graph in the plane $z=1/2$.}\n\\end{figure}\n\nIn \n\\cite{Benamou2010}, it was argued based on numerical evidence that the Gauss-Seidel method \\eqref{Gauss} is faster than a certain variant of the direct solver \\eqref{k-H1-iterativeD} for singular solutions. In our implementation we saw evidence of the contrary, that is, the Gauss-Seidel method is less efficient. We note that the Gauss-Seidel method requires much more loops which are not efficient in MATLAB. \n\n\\section{Concluding Remarks}\n\n\\begin{rem}\nAlthough the pseudo-transient and time marching methods introduced in \\cite{AwanouPseudo10} work as well for $k$-Hessian equations, and apply to more general fully nonlinear equations, the subharmonicity preserving iterative methods introduced in this paper are parameter free. \nAll these type of methods can be accelerated with fast Poisson solvers and multigrid methods.\n\\end{rem}\n\n\\begin{rem}\nWhen it comes to numerical methods for fully nonlinear equations, there are two types of convergence to study. Since the equations are nonlinear, they must be solved iteratively. One must then address the convergence to the discrete solution of the iterative methods used. The second type of convergence is the convergence of the numerical solution to the exact solution as the discretization parameter converges to 0. \n\nWe have addressed both types of convergence in this paper. \n\n\\end{rem}\n\n\\begin{rem}\nExistence of a discrete solution and convergence (as the mesh size $h \\to 0$), for finite difference discretization of smooth solutions of fully nonlinear equations, are not often discussed. \nIt is clear that convergence does not simply follow from the consistency of standard finite difference discretization of the second order derivatives. For viscosity solutions, convergence of monotone, stable and consistent schemes follows immediately from the theory of Barles and Souganidis.\n\\end{rem}\n\n\\begin{rem}\nThe iterative method \\eqref{k-H-iterative} can be viewed as a linearization of the fully nonlinear equation \\eqref{k-H1}. It is possible to linearize \\eqref{k-H1} in ways different from \\eqref{broyden} and \\eqref{k-H-iterative}. See for example the methods described in \\cite{AwanouPseudo10}. The iterative method \\eqref{k-H-iterative} has been shown numerically to select discrete solutions which converge to non smooth solutions. Since \\eqref{k-H-iterative} consists of a sequence of Poisson equations, the numerical solution of \\eqref{k-H1} can now be tackled with any good numerical method. \n\\end{rem}\n\n{\\bf Acknowledgments.}\n\nThe author would like to thank the referees for a careful reading of the manuscript. The author is grateful to M. Neilan for many useful discussions. The author was supported in part by NSF grants DMS-0811052, DMS-1319640 and the Sloan Foundation.\n\n\\end{document}"},"lang":{"kind":"string","value":"math"}}},{"rowIdx":167,"cells":{"text":{"kind":"string","value":"یہِ کیٛاہ چھُ"},"lang":{"kind":"string","value":"kashmiri"}}},{"rowIdx":168,"cells":{"text":{"kind":"string","value":"Little Sonny is now approaching the eve of his 86th birthday. Being blessed with a longevity gene courtesy of his mother Pasqualina (she lived to almost 100), he is a force of nature and amazement to anyone who has been lucky enough to know him.\nHe loves to cook in the old world ways just as his mother and father taught him, and he loves to share that gift with all around him. A more generous spirit of a man would be hard to find.\nUpon entering the Rossi kitchen, his son Giorgio would ask, “What’s cooking, Pops?” and the answer shot back was always “You eat what I cook you!”. Picky eaters with dietary disciplines were never an interest in Alfredo and Pasqualina’s house, and as the head chef and bottle washer in his own kitchen, Poppa Nick wasn’t about to change tradition.\nWhich was fine. Anything he cooks, he cooks with love and a sense of history, along with a sense of artistry as well. “Eating what He Cooks You” is a real authentic experience.\nA couple of years ago, Giorgio approached both Nick and Linda on the idea of recording the family cuisine on video for family use; both as tutorials for the third and fourth generations of the Rossi family, but also as a way to archive and preserve those old world culinary customs and tricks to ensure that they wouldn’t be immediately lost in the march of time.\nGiorgio posted a few of Poppa Nick’s kitchen hi-jinks on YouTube, and well... here we are.\nWhat started as an insignificant family portrait project through food preparation has grown into a fairly large “revolving virtual dinner party” with the advent of social networking; where if the recipes are followed to specification, anybody can now eat what Nick cooks for them. And people have started cooking them a lot!\nHe gets to potentially throw this party for anybody that wants to come, and conveniently avoid having to clean a mountain of dirty dishes in the aftermath.\nAs a direct descendant of Neopolitan fishermen and pasta manufacturers, Nick knows a thing or two about Italian \"Peasant Cuisine\".\nThis is the cuisine that he has taught his children and grandchildren how to prepare, and that he still prepares, serving it up and sharing it with friends and family in his home for over 85 years on planet earth. Its the family voodoo stew and secret family alchemical formulas.\nIn the end, the food and the preparation of it in tutorial form is just a delivery system of the love and history of a family.\nAssisted rather clumsily by his son Giorgio “Secondo” Rossi, Poppa “Primo” knows YOU EAT WHAT I COOK YOU is about love; giving and sharing unconditionally, spreading warmth and good cheer whenever you can recognize an opportunity to do so.\nThis is just a big internet version of what anyone who crosses over the family’s threshold would experience when they are arriving for dinner.\n“Siddown... have a glass of Prosecco and some Peppers!"},"lang":{"kind":"string","value":"english"}}},{"rowIdx":169,"cells":{"text":{"kind":"string","value":"Boulder traps hiker for nine hours in Northern Idaho's Selkirk Mountains, results in Air Force rescue via helicopter airlift.\nAmmi Midstokke is airlifted out of harm’s way by a UH-1N Iroquois helicopter from the Fairchild Air Force Base in Washington.\nAmmi Midstokke after the accident at Northern Idaho’s Selkirk Mountains.\nAmmi Midstokke’s foot after the boulder collapsed on it.\nThe 2010 film 127 Hours depicts every hiker’s worst nightmare: getting hurt and being trapped for an extended amount of time with nobody around to help.\nLuckily for 36-year-old Ammi Midstokke, she was only trapped for nine hours and had the help of her hiking partner after a 1.5 ton boulder came crashing down on her right foot in Northern Idaho’s Selkirk Mountains.\nMidstokke and her hiking partner, Jason Luthy, had successfully reached the summit of Chimney Rock on Sept. 19. The duo was climbing out when a boulder moved on Midstokke, hitting her in the head and then pinning her, landing on her foot.\nLuthy, a trained paramedic, tried to move the boulder with no luck.\n“We called search and rescue and tried to stabilize me with heat and good stories,” Midstokke wrote on Facebook after the ordeal.\nLuthy called search and rescue around 5:30 p.m. By 8 p.m., more than 20 rescuers from Priest Lake Search and Rescue and the Bonner County EMS Wilderness Response Team set out to look for Midstokke, according to the Spokesman-Review.\nThe team reached Midstokke and Luthy around 1 a.m., after navigating steep and narrow terrain in the dark. Once there, it took less than an hour for a web and pulley system to lift the boulder off of Midstokke’s “very deformed, very dead looking foot,” as she said on Facebook.\nHiking out of the area, however, wasn’t an option. At 7:10 a.m., a four-member crew from the 36th Rescue Flight at Fairchild Air Force Base in Washington was contacted. The team used a UH-1N Iroquois helicopter to fly in at 7:45 a.m. and take Midstokke out of the area.\nAfter securing Midstokke to a harness, the team airlifted her off of the mountain at 8:35 a.m.\nThe result of her injury?\n“Initial X rays show tarsal breakage, but a remarkably whole foot,” Midstokke said.\nDonations can be made to Priest Lake Search and Rescue here."},"lang":{"kind":"string","value":"english"}}},{"rowIdx":170,"cells":{"text":{"kind":"string","value":"مٔلِکانہِ ییٚلہِ یہِ وُچھ تَس گۄو یہِ تیوٗت خۄش زِ تَمہِ تُج نہٕ بادشاہَس تھپھٕے زِ سُہ انناوِ ہے بییٚہِ تہِ یِہوے کَپُر"},"lang":{"kind":"string","value":"kashmiri"}}},{"rowIdx":171,"cells":{"text":{"kind":"string","value":"\\begin{document}\n\n\\title{ THE ASYMPTOTIC BEHAVIOR OF QUSI-HARMONIC FUNCTIONS AND EIGENFUNCTIONS OF DRIFT LAPLACIAN AT INFINITY}\n\\keywords{Quasi-Laplacian, singularity, asymptotic behavior.}\n\\thanks{\\noindent \\textbf{MR(2010)Subject Classification} 47F05 58C40}\n\\author{min chen}\n\\author{jiayu Li}\n\\author{yuchen Bi}\n\\address[Corresponding author] {University of Science and Technology of China, No.96, JinZhai Road Baohe District,Hefei,Anhui, 230026,P.R.China.} \n\n\\email{cmcm@mail.ustc.edu.cn}\n\n\\thanks{The research is supported by the National Nature Science Foudation of China No. 11721101 No. 11526212 }\n\n\\pagestyle{fancy}\n\\fancyhf{}\n\\renewcommand{0pt}{0pt}\n\\fancyhead[CE]{}\n\\fancyhead[CO]{\\leftmark}\n\\fancyhead[LE,RO]{\\thepage}\n\n\\begin{abstract}\nNote that $\\mathbb{R}^m$ with the metric $g=e^{-\\frac{|x|^2}{2(m-2)}}ds_0^2$ is actually a Riemannian manifold with a singularity at $\\infty.$ The metric is quite singular at infinity and it is not complete. Colding-Minicozzi \\cite{11} pointed out that the Ricci curvature of this metric does not have a sign and goes to negative infinity at infinity and thus there is no way to smoothly extend the metric to a neighborhood of infinity. Chen-Li \\cite{7} proved that any non-constant quasi-harmonic function or eigenfunction of drift Laplacian is discontinuous at infinity. In this paper, we show expansions of quasi-harmonic functions and of eigenfunctions of drift-Laplacian in terms of spherical harmonics. Using these expansions, we have a more precise description of the asymptotic behavior of quasi-harmonic functions and of eigenfunctions of drift-Laplacian at infinity. Moreover, we improve the Liouville theorem of quasi-harmonic functions and eigenfunctions of drift-Laplacian by reducing the requirement of the conditions.\n\\end{abstract}\n\n\\maketitle\n\n\\numberwithin{equation}{section}\n\\section{Introduction}\n\\setcounter{equation}{0}\nChen-Li \\cite{7} studied eigenfunctions of Quasi-Laplacian $\\Delta_g = e^{\\frac{|x|^2}{2(m-2)}}(\\Delta_{g_0} - \\nabla_{g_0} h \\cdot \\nabla_{g_0}) = e^{\\frac{|x|^2}{2(m-2)}} \\Delta_h $ for $h = \\frac{|x|^2}{4}$ and proved that any non-constant quasi-harmonic function is discontinuous at infinity in Corollary 3.3 and any non-constant eigenfunction of drift Laplacian $\\Delta_h = \\Delta_{g_0} - \\nabla_{g_0} h \\cdot \\nabla_{g_0}$ is discontinuous at infinity in Theorem 1.4, which means that any quasi-harmonic function or eigenfunction of drift Laplacian could not converge to a constant at infinity. \n\nRecently, Colding-Minicozzi defined a frequency function $U(r) = \\frac{r}{2} (\\log{I})'$ where $I(r) = r^{1-n} \\int_{\\partial B_r} u^2$ and used the mean value $\\sqrt{I}$ of $u$ to measure the rate of the growth of eigenfunctions of drift Laplacian $\\mathcal{L} = \\Delta_{h}$ in Theorem 4.8 in \\cite{5} and Theorem 1.1 in \\cite{6}. \n\\begin{theorem}\\label {thm:1.4}\n\\cite{6} Given $\\epsilon > 0$ and $\\delta > 0$, there exist $r_1 > 0$ such that if $\\mathcal{L}u = -\\lambda u$ and $U(\\bar{r}_1) \\ge \\delta + 2 \\sup\\{0, \\lambda\\}$ for some $\\bar{r}_1 \\ge r_1$, then for all $r \\ge R(\\bar{r}_1)$\n\\begin{align*}\n\tU(r) > \\frac{r^2}{2} - n - 2\\lambda - \\epsilon.\n\\end{align*}\n\\end{theorem}\n\n$U(r) > \\frac{r^2}{2} - n - 2\\lambda - \\epsilon$ implies that $\\sqrt{I} \\ge Ce^{\\frac{r^2}{4}} r^{-(n + 2\\lambda - \\epsilon)}$. Theorem 1.4 shows that there is a sharp dichotomy for the growth of eigenfunctions of $\\mathcal{L}$: either $\\sqrt{I} \\le Cr^{2(\\delta + 2\\lambda)}$, $u$ grows at most polynomially; or $\\sqrt{I} \\ge Ce^{\\frac{r^2}{4}} r^{-(n + 2\\lambda - \\epsilon)}$, $u$ grows at least like $Ce^{\\frac{r^2}{4}} r^{-(n + 2\\lambda - \\epsilon)}$. They use $\\sqrt{I}$ to describe the asymptotic behavior of $u$. In this paper, we give the expansion of $u$ in terms of spherical harmonics in Lemma 3.2 to see its asymptotic behavior directly.\n\nTo know the sharp dichotomy phenomenon of the growth rate, we compare the expansion of quasi-harmonic function in Lemma 2.3 with the expansion harmonic function. \n\nAssume $\\varphi_k(\\theta)$ is an eigenfunction of $L^2(S^{m-1})$ corresponding to the eigenvalue $\\lambda_k$ and $C(N)$ is an Euclidean Cone $(0,\\infty)\\times N^{m-1}.$ \n\\begin{theorem}\\label {thm:1.5}\n\\cite{10} If $u$ is a harmonic function on $C(N)$, then\n\\[ u(r,\\theta) = \\sum c_k r^{p_k} \\varphi_k(\\theta),\\]\nwhere the $c_k$ are constants and $p_k = \\frac{-(m-2) + \\sqrt{(m-2)^2 + 4\\lambda_k}}{4}$ increases strictly from $0$ to $+\\infty$ as $k\\rightarrow +\\infty$. Furthermore, $u$ has polynomial growth if and only if this is a finite sum.\n\\end{theorem}\nWe know that the property of quasi-harmonic and harmonic functions are quiet different.\nLet us recall some basic results about the metric $g=e^{-\\frac{|x|^2}{2(m-2)}}ds_0^2.$ Lin-Wang \\cite{1} introduced the quasi-harmonic sphere, which is a harmonic map from $M=\\big(\\mathbb{R}^m, e^{-\\frac{|x|^2}{2(m-2)}}ds_0^2\\big)$ to $N$ with finite energy when they study the regularity of the heat flow of harmonic maps(c.f.\\cite{12}). Here $ds_0^2$ is Euclidean metric in $\\mathbb{R}^m$. Colding-Minicozzi \\cite{11} also pointed out self-shrinkers in $\\mathbb{R}^{m-1}$ are minimal hypersurfaces for the metric $g=e^{-\\frac{|x|^2}{2(m-2)}}ds_0^2.$ Note that $\\mathbb{R}^m$ with this metric is actually a Riemannian manifold with a singularity at $\\infty.$ The compactification of $\\mathbb{R}^m$ provided by this metric is a topological $m-$sphere. Colding-Minicozzi \\cite{11} mentioned that the Ricci curvature of this metrics does not have a sign and goes to negative infinity at infinity and thus there is no way to smoothly extend the metric to a neighborhood of infinity. The metric $g=e^{-\\frac{|x|^2}{2(m-2)}}ds_0^2$ is quite singular at infinity and it is not complete. Ding-Zhao \\cite{2} showed that if the target $N$ is a sphere, any equivariant quasi-harmonic sphere is discontinuous at infinity and conjectured that any non-constant quasi-harmonic sphere is discontinuous at infinity. \n\nIn this paper, we will give a more precise description of the behavior of quasi-harmonic function and eigenfunctions of $\\Delta_h$ near the infinity.\n\nAssume $u_0 = \\frac{1}{2} \\int_{0}^{r} e^{\\frac{r^2}{4}} r^{1-m} d r,$ which is an radially symmetric solution of $\\Delta_g u = 0$, we will show that $\\frac{u(r,\\theta)}{u_0(r)}$ could be asymptotic to any given function $g(\\theta) \\in H^{[\\frac{m}{2}]+2}(S^{m-1})$ from the following result.\n\\begin{theorem}\\label {thm:1.6}\nAssume $\\bar{M} = (\\mathbb{R}^m, g)$, for any given function $g(\\theta) \\in H^{[\\frac{m}{2}]+2}(S^{m-1})$, there exists a quasi-harmonic function $u(r,\\theta)$ (i.e., $\\Delta_g u = 0$) on $\\mathbb{R}^m\\backslash \\{0\\}$ such that $\\lim\\limits_{r \\to +\\infty} \\frac{u(r,\\theta)}{u_0(r)} = g(\\theta)$. Moreover, if $\\frac{u(r,\\theta)}{u_0(r)}$ and $\\frac{\\bar{u}(r,\\theta)}{u_0(r)}$ are asymptotic to the same function $g(\\theta)$, then $u(r,\\theta) = \\bar{u}(r, \\theta) + c.$\n\\end{theorem}\n\nSimilarly, assume $u_0 = e^{\\frac{r^2}{4}} r^{-(m+2\\lambda)}$, we have\n\\begin{theorem}\\label {thm:1.7}\nIf $ 2\\lambda$ is not an integer, for any given function $g(\\theta) \\in H^{[\\frac{m}{2}]+2}(S^{m-1})$, there exists an eigenfunction $u(r,\\theta)$ of $\\Delta_h$ on $\\mathbb{R}^m\\backslash \\{0\\}$ such that $\\lim\\limits_{r \\to +\\infty} \\frac{u(r,\\theta)}{u_0(r)} = g(\\theta)$. Moreover, if $\\frac{u(r,\\theta)}{u_0(r)}$ and $\\frac{\\bar{u}(r,\\theta)}{u_0(r)}$ are asymptotic to the same function $g(\\theta)$, then $u(r,\\theta) = \\bar{u}(r, \\theta)+p(r)$, where $p(r) \\sim r^{2\\lambda}$. If $\\lambda$ is an integer, for any given function $g(\\theta) \\in H^{[\\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\\langle g, \\varphi_k \\rangle = 0$ when $k \\in \\{0, m-2, m, m+2, \\cdots, m+2\\lambda\\} = A$. Then there exists an eigenfunction $u(r, \\theta)$ of $\\Delta_h$ on $\\mathbb{R}^m\\backslash \\{0\\}$ such that $\\lim\\limits_{r \\to +\\infty} \\frac{u(r,\\theta)}{u_0(r)} = g(\\theta)$. If $2\\lambda$ is an integer and $\\lambda$ is not an integer, for any given function $g(\\theta) \\in H^{[\\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\\langle g, \\varphi_k \\rangle = 0 $ when $k \\in \\{m-2, m, m+2, \\cdots, m+2[\\lambda]\\} = B$. Then there exists an eigenfunction $u(r, \\theta)$ of $\\Delta_h$ on $\\mathbb{R}^m\\backslash \\{0\\}$ such that $\\lim\\limits_{r \\to +\\infty} \\frac{u(r,\\theta)}{u_0(r)} = g(\\theta)$.\n\\end{theorem}\n\nFinally, we consider Liouville theorem of quasi-harmonic function with these expansions.One may wonder whether the quasi-harmonic functions still possess the basic properties of harmonic functions. Cheng-Yau \\cite{3} proved that any harmonic function with sub-linear growth on manifolds with non-negative Ricci curvature must be constant. Li-Wang \\cite{4} showed that there is no non-constant positive quasi-harmonic function on $\\mathbb{R}^m$ with polynomial growth in Theorem 4.2. In this paper, we can improve this result by replacing the condition of polynomial growth with exponential growth.\n \n\\begin{theorem}\\label {thm:1.1}\nLet $u$ be a quasi-harmonic function (i.e, $\\Delta_gu=0$) in $\\mathbb{R}^m$. If there exists a sequence $r_i \\rightarrow +\\infty$ such that $\\big(\\int_{S^{m-1}} ( u(r_i,\\theta))^2\\big)^{\\frac{1}{2}} \\le Ce^{\\frac{r_i^2}{4}}r_i^{-(m+\\epsilon)}$ for some $\\epsilon > 0$, then $u$ is a constant.\n\\end{theorem}\n\nColding-Minicozzi mentioned that if $\\Delta_hu=0$ and $||u||_{L^2(\\mathbb{R}^m)}^2=\\int_{\\mathbb{R}^m}u^2e^{-h} < \\infty$ in \\cite{6}, then $u$ must be constant. More generally, they showed the following result in \\cite{5}.\n\\begin{lemma}\n\\cite{5} If $\\Delta_{h}u=-\\lambda u$ on $\\mathbb{R}^m$ and $\\int_{\\mathbb{R}^m}u^2e^{-h} < \\infty$, then $\\lambda$ is a half-integer and $u$ is a polynomial of degree $2\\lambda$.\n\\end{lemma}\n\\begin{theorem}\\label {thm:1.3}\nIf $\\Delta_{h}u=-\\lambda u$ on $\\mathbb{R}^m\\backslash \\{0\\}.$ If there exists a sequence $r_i \\rightarrow +\\infty$ such that $\\big(\\int_{S^{m-1}} ( u(r_i,\\theta))^2\\big)^{\\frac{1}{2}} \\le C e^{\\frac{r_i^2}{4}} r_i^{-(m+2\\lambda + \\epsilon)}$ for some $\\epsilon > 0$, then $u$ is a polynomial of degree $2 \\lambda$ when $2 \\lambda$ is an integer or $u=p(r)$ satisfying $p(r) \\sim r^{2\\lambda}$ when $2 \\lambda$ is not an integer.\n\\end{theorem}\nSince $\\int_{\\mathbb{R}^m}u^2e^{-h} =\\int^{+\\infty}_0\\int_{S^{m-1}}u^2e^{-\\frac{r^2}{4}}< \\infty$ implies that there exists a sequence $r_i \\rightarrow +\\infty$ such that $\\big(\\int_{S^{m-1}} ( u(r_i,\\theta))^2\\big)^{\\frac{1}{2}} \\le Ce^{\\frac{r_i^2}{8}}$. Theorem 1.5 and Theorem 1.7 can also be see as a generalization of Lemma 1.6.\n\n\\hspace{0.4cm}\n\n\\section{The asymptotic behavior of quasi-harmonic functions at infinity}\n\\setcounter{equation}{0}\n\nAssume that $u$ is quasi-harmonic function , i.e.,\n\\begin{equation}\n\\Delta_g u = 0.\n\\end{equation}\nWe rewrite it in the following form\n\\begin{equation}\n\\Delta u - (\\nabla h, \\Delta u) = 0,\n\\end{equation}\nwhere $h = \\frac{r^2}{4}$. We know that the Euclidean metric of $\\mathbb{R}^m$ can be written in spherical coordinates $(r, \\theta)$ as\n\\[ ds^2 = dr^2 + r^2d\\theta^2,\\]\nwhere $d\\theta^2$ is the standard metric on $S^{m-1}$. Then\n\\[\\Delta = \\Delta_r + \\frac{1}{r^2}\\Delta_\\theta,\\]\nwhere $\\Delta_\\theta$ is the Laplacian on the standard $S^{m-1}$. It is clear that\n\\[ \\nabla h \\cdot \\nabla = \\frac{r}{2} \\frac{\\partial}{\\partial r}.\\]\nIt follows from (2.2) that\n\\[ u_{rr} + \\frac{m-1}{r} u_r + \\frac{1}{r^2} \\Delta_\\theta u - \\frac{r}{2} \\frac{\\partial u}{\\partial r} = 0.\\]\n\nLet $\\varphi_k$ be the orthonormal basis on $L^2(S^{m-1})$ corresponding to the eigenvalues,\n\\[ 0 = \\lambda_0 < \\lambda_1 \\le \\lambda_2 \\le \\cdots \\le \\lambda_k \\to \\infty \\]\nwe have\n\\[ \\Delta_\\theta \\varphi_k = - \\lambda_k \\varphi_k. \\]\nLet $\\langle \\cdot, \\cdot \\rangle$ denote $L^2$ inner product of $L^2 (S^{m-1})$. Then we have\n\\begin{align*}\n\\langle \\Delta_r u, \\varphi_k \\rangle &= \\Delta_r \\langle u, \\varphi_k \\rangle,\\\\\n\\langle \\Delta_\\theta u, \\varphi_k \\rangle &= \\langle u, \\Delta_\\theta \\varphi_k \\rangle = -\\lambda_k \\langle u, \\varphi_k \\rangle,\\\\\n\\langle \\frac{\\partial u}{\\partial r}, \\varphi _k \\rangle &= \\frac{\\partial \\langle u, \\varphi_k \\rangle}{\\partial r}.\n\\end{align*}\nLet $f_k(r) = \\langle u(r,\\cdot), \\varphi_k \\rangle$ for $k \\ge 0$. Then we see that $f_k$ satisfies\n\\begin{equation}\n(f_k)_{rr} + (\\frac{m-1}{r} - \\frac{r}{2})(f_k)_r = \\frac{\\lambda_k f_k}{r^2}.\n\\end{equation}\nLet \n\\begin{align*}\n&f_k(r) = w(z)z^{l_k}, \\\\\n&z = r^2, \\\\\n&l_k = \\frac{-(m-2) + \\sqrt{(m-2)^2 + 4\\lambda_k}}{4}.\n\\end{align*}\nThen\n\\begin{align*}\n(f_k)_r &= 2(w'(z)z^{l_k + \\frac{1}{2}} + l_kz^{l_k - \\frac{1}{2}}w(z)), \\\\\n(f_k)_{rr} &= 4w''(z)z^{l_k+1} + 2(4l_k + 1)z^{l_k}w'(z) + l_k(4l_k - 2)z^{l_k -1}w(z).\n\\end{align*}\nThen by (2.3), we have\n\\begin{equation}\nzw''(z) + ((2l_k + \\frac{m}{2}) - \\frac{z}{4})w'(z) - \\frac{l_k}{4} w(z) = 0.\n\\end{equation}\nAssume\n\\[ w(z) = e^{\\frac{z}{4}}y(-\\frac{z}{4}).\\]\nThen\n\\begin{align*}\nw_z &= \\frac{1}{4} e^{\\frac{z}{4}}(y(-\\frac{z}{4}) - y'(-\\frac{z}{4})), \\\\\nw_{zz} &= \\frac{1}{16} e^{\\frac{z}{4}}(y(-\\frac{z}{4}) - 2y'(-\\frac{z}{4}) + y''(-\\frac{z}{4})).\n\\end{align*}\nBy (2.4), we have\n\\begin{equation}\n\\frac{z}{4} y''(-\\frac{z}{4}) + (-\\frac{z}{4} - (2l_k + \\frac{m}{2}))y'(-\\frac{z}{4}) + (l_k + \\frac{m}{2})y(-\\frac{z}{4}) = 0.\n\\end{equation}\nLet $x = -\\frac{z}{4}$, then we have\n\\begin{equation}\nxy''(x) + ((2l_k + \\frac{m}{2}) -x)y'(x) - (l_k + \\frac{m}{2})y(x) = 0.\n\\end{equation}\n\\begin{lemma}\nFor $k\\ge 1,$ the general solution of (2.3) is\n\\[ f_k(r) = c_1 e^{\\frac{r^2}{4}} r^{2l_k} F[l_k + \\frac{m}{2}, 2l_k + \\frac{m}{2}; -\\frac{r^2}{4}],\\]\nwhere $F[a,b;x]$ is a Kummer's function in \\cite{9} (see Page 2).\n\\end{lemma}\n\\begin{proof}\nSet $b = 2l_k + \\frac{m}{2}, a = l_k + \\frac{m}{2}$. Assume that one solution of the Kummer's equation (2.6) is\n\\[ y = a_0 x^c + a_1 x^{c+1} + a_2 x^{c+2} + \\cdots + a_n x^{c+n} + \\cdots .\\]\nIf we substitute this series and its first two derivatives in the differential equation, and then equate to zero the coefficients of powers of $x$, we find that\n\\begin{align*}\na_0 c(c + b - 1) &= 0, \\\\\na_1 (c + 1)(c + b) &= a_0(c+a), \\\\\na_2(c+2)(c + b + 1) &= a_1 (c + a + 1), \\\\\n&\\cdots\n\\end{align*}\nSince $a_0\\ne 0,$ it has two roots:\\\\\n(i) $c=0$, we have\n\\[ a_n = \\frac{a\\cdots (a+n-1)}{b\\cdots (b+n-1)n!},\\]\nwhich gives one solution in terms of Kummer's series\n\\[ y = a_0 F [a,b;x].\\]\n(ii) $c = 1 - b$, which leads to a second solution\n\\begin{align*}\na_1 (2 - b)& = a_0 (1 - b + a), \\\\\n&\\cdots \\\\\na_{n-1} (n - b)(n-1) &= a_{n-2} (1-b+a+n-2), \\\\\na_n (n + 1 - b)n & = a_{n-1} (1 - b + a + n -1), \\\\\na_{n+1}(n+2-b)(n + 1)& = a_n(1-b+a+n), \\\\\n&\\cdots \n\\end{align*}\n\nIf $b = n $ for some integer $n$, since $b > a > 1$, then $1 - b + a \\le 0$ and $1 - b + a +n - 2 > 0$. It implies that the solution is a polynomial of some degree $i\\le n$\n\\[ y = x^{1-b} (a_0 + a_1x + \\cdots + a_i x^i).\\]\nIf $b \\neq n $ for any integer $n > 0$, then\n\\[a_k = \\frac{(1-b+a)\\cdots(1-b+a + k -2)}{((k+1)-b)\\cdots (2-b)k!},\\]\nand\n\\[y = x^{1-b}(a_0 + a_1x + \\cdots + a_k x^k+ \\cdots).\\]\nThen the solution of (2.3) is\n\\begin{align*}\nf_k(r) &= c_1 e^{\\frac{r^2}{4}} r^{2l_k} y(-\\frac{r^2}{4}) \\\\\n&= c_1 e^{\\frac{r^2}{4}} r^{2l_k} (-\\frac{r^2}{4})^{1-(2l_k + \\frac{m}{2})}(a_0 + a_1(-\\frac{r^2}{4}) + \\cdots) \\\\\n&= (-4)^{2l_k + \\frac{m}{2} - 1} c_1 e^{\\frac{r^2}{4}} r^{2 - (2l_k + m)}(a_0 + a_1(-\\frac{r^2}{4}) + \\cdots),\n\\end{align*}\nwhich implies that $\\lim\\limits_{r \\to 0} f_k(r) \\neq 0$. This contradicts with the initial condition that\n\\[ f_k(0) = \\lim_{r \\to 0} f_k(r) = \\langle \\lim_{r \\to 0} u(r,\\theta), \\varphi_k \\rangle = \\langle u(0), \\varphi_k \\rangle = 0.\\]\nHence the second solution should be ruled out and the solution of (2.3) has the\nfollowing form\n\\[ f_k(r) = c_k e^{\\frac{r^2}{4}} r^{2l_k} F[l_k + \\frac{m}{2}, 2l_k + \\frac{m}{2}; -\\frac{r^2}{4}].\\]\n\\end{proof}\n\n\\begin{lemma}\n\nFor any fixed $\\delta>0$ and $k\\ge 1,$ we set $a=l_k + \\frac{m}{2}, b=2l_k + \\frac{m}{2}, c=-a+\\delta$. Assume $L> \\delta$ and $x$ is a positive real number, the following\nasymptotic relation holds:\n\\[ F[a, b; -x] = x^{-a} \\frac{\\Gamma(b)}{\\Gamma(b-a)}\\Big(1 +\\sum_{n=1}^{[L-\\delta]} \\frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \\frac{\\Gamma(b-a)}{\\Gamma(a)}J_k(x)x^a\\Big), \\]\n where $(a)_n=a(a+1)\\cdots(a+n-1)$ and $J_k(x)= \\int_{c-i\\infty}^{c+i\\infty} \\frac{\\Gamma(L-s)\\Gamma(a - L +s)}{2\\pi i \\Gamma(b-L+s)} x^{s-L}ds$. Moreover, $\\left| J_k(x)\\right|\\le C(m,L,\\delta)l_k^{2(L-\\delta)+\\frac{m}{2}} |x|^{c-L}$ for $k$ sufficient large.\nAssume $g_k(x)=\\sum_{n=1}^{[L-\\delta]} \\frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \\frac{\\Gamma(b-a)}{\\Gamma(a)}J_k(x)x^a$, in particular, if $k$ is sufficient large, we set $L=2\\delta,$ then\n\\[ |g_k(x)| \\le C(m,\\delta)\\frac{l_k^{2\\delta}}{x^{\\delta}},\\] \n and if $k\\le K$ for some $K>0$, we set $L=\\delta+1,$ then \n\\[|g_k(x)| \\le C(m,K,\\delta)\\frac{1}{x}.\\]\n\n\\end{lemma}\n\n\\begin{proof}\nUsing the results in \\cite{9} (see Page 36), we have\n\\begin{equation}\n|z^s| = |z|^{\\text{Re}(s)} e^{-\\text{Im}(s)\\arg{z}},\n\\end{equation}\nand\n\\begin{equation}\n\\frac{\\Gamma (a)}{\\Gamma (b)} F[a, b ; -x] = \\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c + i\\infty}\\frac{\\Gamma(-s)\\Gamma(a+s)}{\\Gamma(b+s)} x^s ds,\n\\end{equation}\nprovided that $|\\arg{x}| < \\frac{1}{2} \\pi$ and $b \\neq 0, -1, -2, \\cdots$. Now we will deduce the asymptotic expansion in $x$ for Kummer's function. Let us consider the integral\n\\[ I_k = \\frac{1}{2\\pi i} \\int_{ADEF} \\frac{\\Gamma(-s)\\Gamma(a+s)}{\\Gamma(b+s)} x^s ds\\]\nround the rectangular contour\n\\[ A(c - iN),\\ D(c + iM), \\ E(c - L + iM),\\ F(c - L - iN)\\]\nin the $s-$ space.\n\\begin{align*}\nI_k &= \\int_{AD} + \\int_{DE} + \\int_{EF} + \\int_{FA} \\\\\n&=I_{M,N} + J_4 + J_5 + J_6.\n\\end{align*}\nAs $M$ and $N \\rightarrow \\infty,$\n\\begin{align*}\n&J_4 \\to 0, \\\\\n&J_6 \\to 0, \\\\\n&-J_5 \\to J_k = \\int_{c-i\\infty}^{c+i\\infty} \\frac{\\Gamma(L-s)\\Gamma(a - L +s)}{2\\pi i \\Gamma(b-L+s)} x^{s-L}ds, \\\\\n&I_{M,N} \\to I_1 = \\int_{c-i\\infty}^{c+i\\infty} \\frac{\\Gamma(-s)\\Gamma(a + s)}{2\\pi i \\Gamma(b+s)} x^s ds \\\\\n&= \\frac{\\Gamma(a)}{\\Gamma(b)}F[a,b;-x],\n\\end{align*}\nand\n\\[ I_k = -\\sum_{n=0}^{[L]} \\frac{\\Gamma(a+n)}{\\Gamma(b-a-n)} \\frac{(-1)^{n-1}}{n!} x^{-a-n},\\]\nfrom the residues at the poles $s=-a, -a-1, \\cdots, -a - [L - \\delta]$. Then we have\n\\begin{align*}\nF[a,b;x] &= \\frac{\\Gamma (b)}{\\Gamma (a)}(I_k + J_k) \\\\\n&= x^{-a} \\frac{\\Gamma(b)}{\\Gamma(b-a)}(\\sum_{n=0}^{[L-\\delta]} \\frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \\frac{\\Gamma(b-a)}{\\Gamma(a)}J_k(x)x^a).\n\\end{align*}\nAssume $s= it + c$, then\n\\[ J_k = \\int_{-\\infty}^{+\\infty} \\frac{\\Gamma(L-it-c)\\Gamma(a-L+it+c)}{2\\pi i \\Gamma(b-L+it+c)}x^{it+c-L} idt.\\]\nBy (2.7), we have\n\\[|x^{it+c-L}| = |x|^{c-L} e^{-(\\arg{x})t}, \\]\nif $x$ is a positive real number, we have\n\\[|x^{it+c-L}| = x^{c-L} . \\]\nWe know that the classical stirling's approximate formula for the Gamma-Function\nin the form\n\\[ \\Gamma(z) = \\sqrt{2\\pi} e^{-z} z^{z-\\frac{1}{2}}(1 + O(\\frac{1}{|z|})),\\]\nfor $|\\arg{z}| < \\pi$ as $|z| \\to \\infty$. In absolute value:\n\\[ |\\Gamma(x + iy)| = \\sqrt{2\\pi}e^{-x}(x^2 + y^2)^{\\frac{x-\\frac{1}{2}}{2}}e^{-y\\arg{(x+iy)}}, \\ as \\ \\sqrt{x^2 + y^2} \\to +\\infty. \\]\nIf we set $a = l_k + \\frac{m}{2}$, $b = 2l_k + \\frac{m}{2}$ and $ c = -a + \\delta,$ then\\begin{align*}\n&L-c = l_k + \\frac{m}{2} + L - \\delta, \\\\\n&a +c -L = \\delta -L,\\\\\n&b + c - L = l_k + \\delta - L.\n\\end{align*}\nThus\n\\begin{align*}\n|\\Gamma(L - it -c)| &\\sim ((L -c)^2 + t^2)^{\\frac{L-c-\\frac{1}{2}}{2}} \\exp({t \\arg{(L-it-c)}}) e^{c-L}, \\\\\n|\\Gamma(a -L + it + c)| &\\sim ((a + c - L)^2 + t^2)^{\\frac{a-L-c-\\frac{1}{2}}{2}} \\\\\n&\\exp(-t\\arg{(a - L + c+ it)})\\exp(-(a-L+c)),\\\\\n|\\Gamma(b -L + it + c)| &\\sim ((b - L + c)^2 + t^2)^{\\frac{b-L+ c-\\frac{1}{2}}{2}} \\\\\n&\\exp(-t\\arg{(b - L + c+ it)}) \\exp(-(b-L+c)).\n\\end{align*}\nIf $k$ is large enough, we have\n\\begin{align*}\n&\\frac{((L-c)^2+t^2)^{\\frac{L-c-\\frac{1}{2}}{2}}((a+c -L)^2 + t^2)^{\\frac{a-L+c-\\frac{1}{2}}{2}}}{((b - L +c)^2 + t^2 )^{\\frac{b-L+c - \\frac{1}{2}}{2}}} \\\\\n&=\\frac{((L-\\delta + l_k + \\frac{m}{2})^2 + t^2)^{\\frac{l_k + \\frac{m}{2} + L -\\delta -\\frac{1}{2}}{2}}((L-\\delta)^2 + t^2)^{\\frac{\\delta - L - \\frac{1}{2}}{2}}}{((l_k +\\delta - L)^2 + t^2)^{\\frac{l_k - L +\\delta - \\frac{1}{2}}{2}}} \\\\\n&=(1 + \\frac{(m + 4(L-\\delta))(l_k - L + \\delta) + (\\frac{m}{2} + 2(L-\\delta))^2}{t^2 + (l_k - L +\\delta)^2})^{\\frac{l_k - L + \\delta - \\frac{1}{2}}{2}} \\\\\n&((l_k + \\frac{m}{2} + L - \\delta)^2 + t^2)^{\\frac{2(L-\\delta) + \\frac{m}{2}}{2}}((L-\\delta)^2 + t^2)^{\\frac{\\delta -L - \\frac{1}{2}}{2}} \\\\\n&\\le \\frac{\\exp(\\frac{m}{2} + 2(L - \\delta))}{(L-\\delta)^{L-\\delta}}((l_k + \\frac{m}{2} + L -\\delta)^2 + t^2)^{\\frac{2(L-\\delta) + \\frac{m}{2}}{2}}\\\\\n&\\le C(m,L,\\delta)((l_k + \\frac{m}{2} + L -\\delta)^2 + t^2)^{\\frac{2(L-\\delta) + \\frac{m}{2}}{2}}.\n\\end{align*}\nFor $t > 0$, we have\n\\begin{align*}\n&\\frac{\\exp(t\\arg(L-it-c))\\exp(-t\\arg(a-L+c+it))}{\\exp(-t\\arg(b-L+c+it)} \\\\\n&=\\exp\\big(-t \\arctan \\frac{t}{L-c} - t(\\pi + \\arctan \\frac{t}{a-L+c}) + t\\arctan \\frac{t}{b-L+c}\\big) \\\\\n\\end{align*}\n\\begin{align*}\n&=\\exp(t \\arctan \\frac{\\frac{t}{b-L+c} - \\frac{t}{L-c} }{1+ \\frac{t^2}{(b-L+c)(L-c)}})\\exp(-t(\\pi + \\arctan \\frac{t}{a-L+c}))\\\\\n&\\le \\exp(t\\arctan \\frac{2L-2c -b}{t + \\frac{(b-L+c)(L-c)}{t} })\\exp(-\\frac{\\pi}{2}t) \\\\\n&= \\exp(t\\arctan \\frac{2L-2\\delta +\\frac{m}{2}}{t + \\frac{(l_k + \\delta -L)(l_k + \\frac{m}{2} + L -\\delta)}{t} })\\exp(-\\frac{\\pi}{2}t) \\\\\n&\\le \\exp(-\\frac{\\pi}{4}t).\n\\end{align*}\nFor $t < 0$, we similarly have\n\\begin{align*}\n&\\frac{\\exp(t\\arg(L-it-c)) \\exp(-t \\arg(a-L+c+it))}{\\exp(-t \\arg(b-L+c+it)} \\\\\n&=\\exp(-t \\arctan \\frac{t}{L-c} -t(\\arctan \\frac{t}{a-L+c} - \\pi) + t\\arctan \\frac{t}{b-L+c} \\\\\n&\\le \\exp(\\frac{\\pi}{4}t).\n\\end{align*}\nAnd \n\\begin{align*}\n&\\exp (c-L) \\exp(-(a-L+c)) \\exp(b-L+c)\\\\\n&=\\exp(-a+b-L+c) \\\\\n&=\\exp(-\\frac{m}{2} - L + \\delta).\n\\end{align*}\nThen we can get\n\\begin{align*}\n&|J_k| \\le \\int_{-\\infty}^{+\\infty}\\left |\\frac{\\Gamma(L-it-c)\\Gamma(a-L+it+c)}{2\\pi i\\Gamma(b-L+it+c)} x^{c-L}\\right |dt \\\\\n&=C(m,L,\\delta) \\int_{0}^{+\\infty} \\big((l_k + \\frac{m}{2} + L -\\delta)^2 + t^2\\big)^{\\frac{2(L-\\delta)+\\frac{m}{2}}{2} } \\exp(-\\frac{\\pi}{4}t)dtx^{c-L} \\\\\n&\\le C(m,L,\\delta)l_k^{2(L-\\delta)+\\frac{m}{2}} x^{c-L}.\n\\end{align*}\nSince\n\\[ \\frac{\\Gamma(b-a)}{\\Gamma(a)} \\sim \\frac{\\sqrt{2\\pi l_k}(\\frac{l_k}{e})^{l_k}}{\\sqrt{2\\pi(l_k + \\frac{m}{2})}(\\frac{l_k + \\frac{m}{2}}{e})^{l_k + \\frac{m}{2}}},\\]\nthen\n\\[ \\left | \\frac{\\Gamma(b-a)}{\\Gamma(a)} \\right| \\le \\frac{C(m)}{(l_k + \\frac{m}{2})^{\\frac{m}{2}}}.\\]\nWe set $L=2\\delta$, then\n\\begin{align*}\n&\\left | \\frac{\\Gamma(b-a)}{\\Gamma (a)} J_k(x) x^a \\right| \\\\\n& \\le C(m,\\delta) l_k^{2\\delta} x^{-\\delta}.\n\\end{align*}\nThus\n\\begin{align*}\n|g_k(x)| &= \\left | \\sum_{n=1}^{[L-\\delta]} \\frac{a_n(1+a-b)_n}{n!}x^{-n} + \\frac{\\Gamma(b-a)}{\\Gamma(a)}J_k(x) x^a \\right | \\\\\n&\\le C(m,\\delta) l_k^{2\\delta}x^{-\\delta}.\n\\end{align*}\nIf $k\\le K$ for some $K>0$, $a,b,c$ are all bounded. We set $L=\\delta+1$, then\n\\[|g_k(x)| \\le C(m,K,\\delta)\\frac{1}{x}.\\]\n\\end{proof}\nIf $k\\ge 1,$ we can use the asymptotic relation in Lemma 2.2, then\n\n\\begin{align*}\n f_k(r)& = c_k e^{\\frac{r^2}{4}} r^{2l_k} F[l_k + \\frac{m}{2}, 2l_k + \\frac{m}{2}; -\\frac{r^2}{4}]\\\\\n&=c_k\\frac{\\Gamma(2l_k + \\frac{m}{2})}{\\Gamma(l_k)}e^{\\frac{r^2}{4}}r^{-m}\\Big(1+g_k(\\frac{r^2}{4})\\Big) \\\\\n&= C_ke^{\\frac{r^2}{4}}r^{-m}\\Big(1+g_k(\\frac{r^2}{4})\\Big),\n\\end{align*}\nwhere $C_k$ are constants.\nIf $k=0,$ the asymptotic relation of Kummer's function does not hold, we solves the equation directly.\n\\begin{equation}\n(f_0)_{rr} + (\\frac{m-1}{r} - \\frac{r}{2})(f_0)_r = 0.\n\\end{equation}\nThen\n\\begin{equation}\nf_0 = c + C_0\\int_0^r e^{\\frac{r^2}{4}} r^{-m} dr\n\\end{equation}\nis the general solution of (2.4).\nThen we can easily get that:\n\\begin{lemma}\nThe general solution of $\\Delta_g u=0$ on $\\mathbb{R}^{m}\\backslash \\{0\\}$ has the following form:\n\\[u(r,\\theta)=c + C_0\\int_0^r e^{\\frac{r^2}{4}} r^{-m} dr+\\sum_{k=1}^{\\infty}C_ke^{\\frac{r^2}{4}}r^{-m}\\Big(1+g_k(\\frac{r^2}{4})\\Big)\\varphi_k.\\]\n\\end{lemma}\n\n\\begin{lemma}\nAssume $u\\in H^{2n}(S^{m-1}),$ the fourier coefficients $|\\langle u, \\varphi_k\\rangle|\\le \\frac{C}{\\lambda_k^n},$ where $C$ is independent of $k.$\n\\end{lemma}\n\n\\begin{proof}\n\\begin{align*}\n\\langle u,\\varphi_k \\rangle &=\\int_{S^{m-1}}u\\varphi_k\\\\\n=&-\\frac{1}{\\lambda_k}\\int_{S^{m-1}}u\\Delta\\varphi_k\\\\\n=&-\\frac{1}{\\lambda_k}\\int_{S^{m-1}}\\Delta u\\varphi_k\\\\\n=&\\frac{1}{\\lambda_k^2}\\int_{S^{m-1}}\\Delta u\\Delta \\varphi_k\\\\\n=&\\frac{1}{\\lambda_k^2}\\int_{S^{m-1}}\\Delta^2 u \\varphi_k\\\\\n&\\vdots \\\\\n&=(-\\frac{1}{\\lambda_k})^n\\int_{S^{m-1}}\\Delta^n u \\varphi_k.\n\\end{align*}\n\\[|\\langle u, \\varphi_k\\rangle| \\le \\frac{1}{\\lambda_k^n}\\int_{S^{m-1}}(\\Delta^n u)^2\\le \\frac{C}{\\lambda_k^n}.\\]\n\\end{proof}\n \\begin{proof}[Proof of Theorem \\ref{thm:1.1}]\nFor any fixed $k\\ge 1,$ note that \n\\begin{align*}\n\\frac{1}{2}|C_k|e^{\\frac{r_i^2}{4}}r_i^{-m}& \\le |C_k||1+g_k(\\frac{r_i^2}{4})|e^{\\frac{r_i^2}{4}}r_i^{-m}\\\\\n&=|\\int_{S^{m-1}}u(r_i,\\theta)\\varphi_k|\\\\\n&\\le \\big(\\int_{S^{m-1}} ( u(r_i,\\theta))^2\\big)^{\\frac{1}{2}} (\\int_{S^{m-1}} \\varphi_k^2)^{\\frac{1}{2}}\\\\\n&\\le C\\sqrt{\\omega_n}e^{\\frac{r_i^2}{4}}r_i^{-(m+\\epsilon)},\n\\end{align*}\n which implies that \n\\[C_k0$, where $C$ is independent of $k$. Assume $L=1+\\delta,$ similarly we have\n\\[|\\bar f_k \\varphi_k|\\le Ce^{\\frac{r^2}{4}}r^{-m}\\frac{1}{k^{2}}\\Big(1+C(m,\\delta)\\frac{1}{r^{2}}\\Big)\\]\nfor $k$ finite, where $C$ is independent of $k$.\nIf we set $0<\\delta<\\frac{1}{2},$ the series $\\sum_{k=0}^{\\infty} \\bar f_k\\varphi_k$ is uniformly convergent on $\\overline {B_{A}(0)} \\setminus B_{\\epsilon}(0)$ for any $\\epsilon,A>0$, where $B_{\\epsilon}(0)$ is an open ball. Assume $\\bar u=\\sum_{k=0}^{\\infty} \\bar f_k\\varphi_k,$ then we have \n\\begin{align*}\n\\Delta_g(\\bar u)&=\\bar u_{rr}+\\frac{m-1}{r}\\bar u_r+\\frac{1}{r^2}\\Delta_{\\theta}\\bar u-\\frac{r}{2}\\frac{\\partial \\bar u}{\\partial r}\\\\\n&=\\sum_{k=1}^{\\infty} \\Big((\\bar f_k)_{rr}+(\\frac{m-1}{r}-\\frac{r}{2})(\\bar f_k)_r-\\frac{\\lambda_k\\bar f_k}{r^2}\\Big)\\varphi_k+(\\bar f_0)_{rr}+(\\frac{m-1}{r}-\\frac{r}{2})(\\bar f_0)_r\\\\\n&=0.\n\\end{align*}\nThus $\\bar u$ is a solution of $\\Delta_g u=0$ on $\\mathbb{R}^m\\backslash \\{0\\}.$\\\\\nSince $\\sum_{k=0}^{\\infty} \\frac{\\bar f_k}{u_0}\\varphi_k$ is uniformly convergent on $\\mathbb{R}^m\\setminus B_{\\epsilon}(0)$, we can obtain that \n\\begin{align*}\n\\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}=\\lim\\limits_{r\\to+\\infty}\\sum_{k=0}^{\\infty} \\frac{\\bar f_k}{u_0}\\varphi_k=\\sum_{k=0}^{\\infty}\\lim\\limits_{r\\to + \\infty}\\frac{\\bar f_k}{u_0}\\varphi_k=\\sum_{k=0}^{\\infty} \\bar C_k\\varphi_k=g(\\theta).\n\\end{align*}\nHence $\\bar u$ is the quasi-harmonic function which satisfies that $\\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}=g(\\theta)$ for the given function $g(\\theta)\\in H^{[\\frac{m}{2}]+2}(S^{m-1}).$\nMoreover, assume $\\frac{\\bar u(r,\\theta)}{u_0(r)}$ and $\\frac{ u(r,\\theta)}{u_0(r)}$ are asymptotic to the same function $g(\\theta)$, for $k\\ge 1,$ we have\n\\begin{align*}\n&C_k=\\lim\\limits_{r\\to+\\infty}\\frac{ f_k}{u_0}=\\lim\\limits_{r\\to+\\infty}\\langle \\frac{ u(r,\\theta)}{u_0(r)}, \\varphi_k \\rangle \\\\\n&=\\langle \\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}, \\varphi_k \\rangle=\\lim\\limits_{r\\to + \\infty}\\frac{\\bar f_k}{u_0}=\\bar C_k.\n\\end{align*}\nBy (2.5), we also have \n\\[C_0=\\bar C_0.\\]\nHence \n\\begin{align*}\n&u(r,\\theta)-\\bar u(r, \\theta)\\\\\n&=c-\\bar c+(C_0-\\bar C_0)\\int^r_0e^{\\frac{r^2}{4}}r^{1-m}dr +\\sum_{k=0}^{\\infty} (C_k-\\bar C_k)e^{\\frac{r^2}{4}}r^{-m}\\Big(1+g_k(\\frac{r^2}{4})\\Big)\\\\\n&=c-\\bar c.\n\\end{align*}\n\\end{proof}\n\n\\hspace{0.4cm}\n\n\\section{The asymptotic behavior of eigenfunctions of the drift laplacian at infinity}\n\\setcounter{equation}{0}\n\nAssume that $u$ is an eigenfunction of $\\Delta_h$ , i.e.,\n\\begin{equation}\n\\Delta_h u = -\\lambda u.\n\\end{equation}\nWe rewrite it in the following form\n\\begin{equation}\n\\Delta u - (\\nabla h, \\Delta u) = -\\lambda u,\n\\end{equation}\nwhere $h = \\frac{r^2}{4}$. \nLet $f_k(r)=\\langle u(r,\\cdot),\\varphi_k\\rangle$ for $k\\ge 0,$ we have\n\\begin{equation}\n(f_k)_{rr} + (\\frac{m-1}{r} - \\frac{r}{2})(f_k)_r = (\\frac{\\lambda_k }{r^2}-\\lambda)f_k.\n\\end{equation}\nLet \n\\begin{align*}\n&f_k(r) = w(z)z^{l_k} ,\\\\\n&z = r^2 ,\\\\\n&l_k = \\frac{-(m-2) + \\sqrt{(m-2)^2 + 4\\lambda_k}}{4}.\n\\end{align*}\nThen by (3.3), we have\n\\begin{equation}\nzw''(z) + ((2l_k + \\frac{m}{2}) - \\frac{z}{4})w'(z) - \\frac{l_k-\\lambda}{4} w(z) = 0.\n\\end{equation}\nAssume\n\\[ w(z) = e^{\\frac{z}{4}}y(-\\frac{z}{4}).\\]\nThen by (3.4), we have\n\\begin{equation}\n\\frac{z}{4} y''(-\\frac{z}{4}) + (-\\frac{z}{4} - (2l_k + \\frac{m}{2}))y'(-\\frac{z}{4}) + (l_k + \\frac{m}{2}+\\lambda)y(-\\frac{z}{4}) = 0.\n\\end{equation}\nLet $x = -\\frac{z}{4}$, then we have\n\\begin{equation}\nxy''(x) + ((2l_k + \\frac{m}{2}) -x)y'(x) - (l_k + \\frac{m}{2}+\\lambda)y(x) = 0.\n\\end{equation}\nSet $b = 2l_k + \\frac{m}{2}, a = l_k + \\frac{m}{2}+\\lambda$. The first solution of (3.6) in terms of Kummer's series is \n\\[y=c_kF[a,b;x].\\]\nThe first solution of (3.3) is \n\\[f_k(r)=c_ke^{\\frac{r^2}{4}}r^{2l_k}F[ l_k + \\frac{m}{2}+\\lambda,2l_k + \\frac{m}{2};-\\frac{r^2}{4}].\\]\nIf $k\\ge 1,$ the second solution of (3.3) should be ruled out with the same reason as in the proof of Lemma 2.1.\nWe can use the same method as in the proof of Lemma 2.2 to get the asymptotic relation of the Kummer's functions.\n\n\\begin{lemma}\nFor any fixed $\\delta>0$ and $k$ satisfying that $l_k-\\lambda\\neq-i$ for any nonnegative integer $i$, i.e., $\\frac{1}{\\Gamma(l_k-\\lambda)}\\neq 0,$ we set $a=l_k + \\frac{m}{2}+\\lambda, b=2l_k + \\frac{m}{2}, c=-a+\\delta$. Assume $L> \\delta$ and $x$ is a positive real number, the following\nasymptotic relation holds:\n\\[ F[a, b; -x] = x^{-a} \\frac{\\Gamma(b)}{\\Gamma(b-a)}\\Big(1 +\\sum_{n=1}^{[L-\\delta]} \\frac{a_n(1+a-b)_n}{n!} x^{-n} + \\frac{\\Gamma(b-a)}{\\Gamma(a)}J_k(x)x^a\\Big), \\]\n where $(a)_n=a(a+1)\\cdots(a+n-1)$ and $J_k(x)= \\int_{c-i\\infty}^{c+i\\infty} \\frac{\\Gamma(L-s)\\Gamma(a - L +s)}{2\\pi i \\Gamma(b-L+s)} x^{s-L}ds$. Moreover, $\\left| J_k(x)\\right|\\le C(m,L,\\delta,\\lambda)l_k^{2(L-\\delta)+\\frac{m}{2}} |x|^{c-L}$ for $k$ sufficient large. Assume $g_k(x)=\\sum_{n=1}^{[L-\\delta]} \\frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \\frac{\\Gamma(b-a)}{\\Gamma(a)}J_k(x)x^a$, in particular, if $k$ is sufficient large, we set $L=2\\delta$, then\n\\[ |g_k(x)| \\le C(m,\\delta,\\lambda)\\frac{l_k^{2\\delta}}{x^{\\delta}}\\] \nand if $k\\le K$ for some $K>0$, we set $L=\\delta+1,$ then\n\\[ |g_k(x)| \\le C(m,\\delta,\\lambda,K)\\frac{1}{x}.\\] \n\\end{lemma}\n\nNote that the points of nonpositive integer are the zero points of $\\frac{1}{\\Gamma(z)}.$ When $l_k-\\lambda=-i$ for any nonnegative integer $i,$ the Kummer's functions $F[l_k+\\frac{m}{2}+\\lambda,2l_k+\\frac{m}{2};-x]$ do not have the same asymptotic relation as in Lemma 3.1. We will deal with the Kummer's series directly.\nAssume $l_k-\\lambda=-i$ for some nonnegative integer $0\\le i\\le \\lambda.$ Then the first solution of (3.6) is\n\\begin{align*}\ny(x)&=\\sum_{n=0}^{\\infty}\\frac{(l_k+\\frac{m}{2}+\\lambda)\\cdots(l_k+\\frac{m}{2}+\\lambda+n-1)}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+n+1+\\lambda-i)n!}x^n\\\\\n&=\\sum_{n=0}^{\\infty}\\frac{(l_k+\\frac{m}{2}+\\lambda-i+1+n-1)\\cdots(l_k+\\frac{m}{2}+\\lambda+n-1)}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+\\lambda-1)n!}x^n\n\\end{align*}\nMultiply $x^{l_k+\\frac{m}{2}+\\lambda-i-1}$ on both sides, we have\n\\begin{align*}\n&y(x)x^{l_k+\\frac{m}{2}+\\lambda-i-1}=\\\\\n&\\sum_{n=0}^{\\infty}\\frac{(l_k+\\frac{m}{2}+\\lambda-i+1+n-1)\\cdots(l_k+\\frac{m}{2}+\\lambda+n-1)}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+\\lambda-1)n!}x^{n+l_k+\\frac{m}{2}+\\lambda-i-1}.\n\\end{align*}\nIntegrating from $0$ to $x$ for $i$ times, then we have \n\\[\\int_0^x\\cdots \\int_0^x y(x)x^{l_k+\\frac{m}{2}+\\lambda-i-1}=\\sum_{n=0}^{\\infty}\\frac{x^{n+l_k+\\frac{m}{2}+\\lambda-1}}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+\\lambda-1)n!}.\\]\nThus \n\\[\\int_0^x\\cdots \\int_0^x y(x)x^{l_k+\\frac{m}{2}+\\lambda-i-1}=\\frac{x^{l_k+\\frac{m}{2}+\\lambda-1}}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+\\lambda-1)}\\sum_{n=0}^{\\infty}\\frac{x^{n}}{n!}.\\]\nDifferentiate the equation on both sides for $i$ times, we have\n\\[y(x)x^{l_k+\\frac{m}{2}+\\lambda-i-1}=e^x\\Big(x^{l_k+\\frac{m}{2}+\\lambda-i-1}+\\cdots+\\frac{x^{l_k+\\frac{m}{2}+\\lambda-1}}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+\\lambda-1)}\\Big).\\]\nThen \n\\[y(x)=e^x\\Big(1+\\cdots+\\frac{x^{i}}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+\\lambda-1)}\\Big).\\]\nThus \n\\begin{align*}\n(f_k)_1(r)&=e^{\\frac{r^2}{4}}r^{2l_k}y(-\\frac{r^2}{4})\\\\\n&=c_k\\Big(r^{2l_k}+\\cdots+(\\frac{1}{4})^{2i}\\frac{r^{2\\lambda}}{(l_k+\\frac{m}{2}+\\lambda-i)\\cdots(l_k+\\frac{m}{2}+\\lambda-1)}\\Big)\\\\\n&=c_k q_k(r).\n\\end{align*}\nHere $q_k(r)=r^{2l_k}+\\cdots+(\\frac{1}{4})^{2(l_k-\\lambda)}\\frac{r^{2\\lambda}}{(2l_k+\\frac{m}{2})\\cdots(l_k+\\frac{m}{2}+\\lambda-1)}$ is a polynomial of degree $2\\lambda.$\n\\begin{lemma}\nAssume $A=\\{0,m-2,m,m+2,\\cdots,m+2\\lambda\\}$ and $B=\\{m-2,m,m+2,\\cdots,m+2[\\lambda]\\}.$ The general solution of $\\Delta_h u=-\\lambda u$ on $\\mathbb{R}^m\\backslash \\{0\\}$ has the following form:\\\\\nIf $2\\lambda$ is not an integer, $u(r,\\theta)=p(r)+\\sum_{k=0}^{\\infty}C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big)\\varphi_k$ with $p(r)=ce^{\\frac{r^2}{4}}r^{-(m+2\\lambda)}\\int e^{-\\frac{r^2}{4}}r^{m+4\\lambda+1}(1+O(\\frac{1}{r^2}))$ and $p(r)\\sim r^{2\\lambda}$;\\\\\nIf $\\lambda$ is an integer, $u(r,\\theta)=q(r,\\theta)+C_0e^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+O(\\frac{1}{r^2})\\big)+\\sum_{k\\notin A}C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big)\\varphi_k$ with a polynomial $q(r,\\theta)$ of degree $2\\lambda$;\\\\\nIf $2\\lambda$ is an integer but $\\lambda$ is not an integer, $u(r,\\theta)=t(r,\\theta)+\\sum_{k\\notin B}C_k e^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big)\\varphi_k$ with a polynomial $t(r,\\theta)$ of degree $2\\lambda.$ \n \\end{lemma}\n\\begin{proof}\nConsidering that $l_k = \\frac{-(m-2) + \\sqrt{(m-2)^2 + 4\\lambda_k}}{4},$ we have\n\\begin{equation}\nl_k-\\lambda=-i \\Longleftrightarrow k=-2(\\lambda-i) \\quad \\text{or} \\quad k=m-2+2(\\lambda-i).\n\\end{equation}\nSo we divide $\\lambda$ into three cases.\n\nCase 1. $2\\lambda$ is not an integer.\n\nBy (3.7), we have $\\frac{1}{\\Gamma(l_k-\\lambda)}\\neq 0$ for any $k\\ge 0.$ Then we can use Lemma 3.1 to get the asymptotic behavior of $f_k$ at infinity.\nThe first solution of (3.3) is\n\\begin{align*}\nf_k(r)&=c_ke^{\\frac{r^2}{4}}r^{2l_k}F[ l_k + \\frac{m}{2}+\\lambda,2l_k + \\frac{m}{2};-\\frac{r^2}{4}]\\\\\n&=c_k\\frac{\\Gamma(2l_k+\\frac{m}{2})}{\\Gamma(l_k-\\lambda)}e^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big)\\\\\n&=C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big).\n\\end{align*}\nThe second solution of $f_0$ can be found by using the method of reduction of order. Let $(f_0)_2(r)=v(r)(f_0)_1(r)$ be the second linearly independent solution of (3.3), then\n\\[(f_0)_1(r)v''(r)+(2(f_0)'_1(r)+(\\frac{m-1}{r} - \\frac{r}{2})(f_0)_1)v'(r)=0.\\]\nThus \n\\begin{equation}\n(f_0)_2(r)=c_0'(f_0)_1(r)\\int\\frac{e^{\\frac{r^2}{4}}r^{1-m}}{((f_0)_1(r))^2}=ce^{\\frac{r^2}{4}}r^{-(m+2\\lambda)}\\int e^{-\\frac{r^2}{4}}r^{m+4\\lambda+1}(1+O(\\frac{1}{r^2}))=p(r)\n\\end{equation}\nand $p(r)\\sim r^{2\\lambda}.$\nThus \n\\[u(r,\\theta)=\\sum_{k=0}^{\\infty}f_k\\varphi_k=p(r)+\\sum_{k=0}^{\\infty}C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big)\\varphi_k.\\]\n\n Case 2. $\\lambda$ is an integer.\n\n$l_k-\\lambda=-i $ for some integer $0\\le i\\le \\lambda \\Longleftrightarrow k\\in A,$ where $A=\\{0,m-2,m,m+2,\\cdots,m+2\\lambda\\}.$\n \nWhen $k=0,$ which implies that $i=\\lambda,$ we have \n\\[(f_0)_1(r)=c_0q_{2\\lambda}(r).\\] \nThe second linearly independent solution can be found by using the method of reduction of order.\n\\begin{equation}\n(f_0)_2(r)=c_0'(f_0)_1(r)\\int\\frac{e^{\\frac{r^2}{4}}r^{1-m}}{((f_0)_1(r))^2}=l(r),\n\\end{equation}\nwhere $l(r)=C_0e^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+O(\\frac{1}{r^2})\\big).$\nWhen $k\\ge 1,$ $f_k(r)=(f_k)_1(r),$ where $(f_k)_1(r)$ is the first solution of $f_k.$\nThus we have \n\\begin{align*}\nu(r,\\theta)&=\\sum_{k=0}^{\\infty}f_k\\varphi_k\\\\\n&=\\sum_{k\\in A}f_k\\varphi_k+\\sum_{k\\notin A}f_k\\varphi_k\\\\\n&=\\sum_{k\\in A}(f_k)_1\\varphi_k+(f_0)_2+\\sum_{k\\notin A}(f_k)_1\\varphi_k\\\\\n&=\\sum_{k\\in A}c_ke^{\\frac{r^2}{4}}r^{2l_k}F[ l_k + \\frac{m}{2}+\\lambda,2l_k + \\frac{m}{2};-\\frac{r^2}{4}]\\varphi_k+l(r)\\\\\n&+\\sum_{k\\notin A}c_ke^{\\frac{r^2}{4}}r^{2l_k}F[ l_k + \\frac{m}{2}+\\lambda,2l_k + \\frac{m}{2};-\\frac{r^2}{4}]\\varphi_k\\\\\n&=\\sum_{k\\in A}c_{k}q_{k} \\varphi_{k}+l(r)+\\sum_{k\\notin A}C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big)\n\\varphi_k\\\\\n&=q(r,\\theta)+l(r)+\\sum_{k\\notin A}C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_k(\\frac{r^2}{4})\\big)\n\\varphi_k,\n\\end{align*}\nwhere $q(r,\\theta)=\\sum_{k\\in A}c_{k}q_{k}\\varphi_{k}$ is a polynomial of degree $2\\lambda.$\n\nCase 3. $2\\lambda$ is an integer but $\\lambda$ is not an integer.\n\n$l_k-\\lambda=-i$ for some integer $0\\le i\\le [\\lambda] \\Longleftrightarrow k\\in B,$ where $B=\\{m-2,m,m+2,\\cdots,m+2[\\lambda]\\}.$\n \nSince $0\\notin B, \\frac{1}{\\Gamma(l_k-\\lambda)}\\neq 0.$ By Lemma 3.1, we have \n\\begin{align*}\n(f_0)_1(r)&=c_0e^{\\frac{r^2}{4}}r^{2l_k}F[ l_k + \\frac{m}{2}+\\lambda,2l_k + \\frac{m}{2};-\\frac{r^2}{4}]\\\\\n&=C_0e^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\big(1+g_0(\\frac{r^2}{4})\\big).\n\\end{align*}\nThe second linearly independent solution can be found by using the method of reduction of order.\n\\[(f_0)_2(r)=p(r),\\]\nwhere $p(r)\\sim r^{2\\lambda}.$\\\\\nWhen $k\\ge 1,$ $f_k(r)=(f_k)_1(r),$ where $(f_k)_1(r)$ is the first solution of $f_k.$\nThus we have \n\\begin{align*}\nu(r,\\theta)&=\\sum_{k=0}^{\\infty}f_k\\varphi_k\\\\\n&=\\sum_{k\\in B}f_k\\varphi_k+\\sum_{k\\notin B}f_k\\varphi_k\\\\\n&=\\sum_{k\\in B}(f_k)_1\\varphi_k+(f_0)_2+\\sum_{k\\notin B}(f_k)_1\\varphi_k\\\\\n&=\\sum_{k\\in B}c_ke^{\\frac{r^2}{4}}r^{2l_k}F[ l_k + \\frac{m}{2}+\\lambda,2l_k + \\frac{m}{2};-\\frac{r^2}{4}]\\varphi_k+p(r)\\\\\n&+\\sum_{k\\notin B}c_ke^{\\frac{r^2}{4}}r^{2l_k}F[ l_k + \\frac{m}{2}+\\lambda,2l_k + \\frac{m}{2};-\\frac{r^2}{4}]\\varphi_k\\\\\n&=\\sum_{k\\in B}c_{k}q_{k} \\varphi_{k}+p(r)+\\sum_{k\\notin B}C_ke^{\\frac{r^2}{4}}r^{2\\lambda+m}\\big(1+g_k(\\frac{r^2}{4})\\big)\\varphi_k\\\\\n&=\\sum_{k\\in B}c_{k}q_{k} \\varphi_{k}+p(r)+\\sum_{k\\notin B}C_ke^{\\frac{r^2}{4}}r^{2\\lambda+m}\\big(1+g_k(\\frac{r^2}{4})\\big)\n\\varphi_k\\\\\n&=s(r,\\theta)+p(r)+\\sum_{k\\notin B}C_ke^{\\frac{r^2}{4}}r^{2\\lambda+m}\\big(1+g_k(\\frac{r^2}{4})\\big)\n\\varphi_k,\n\\end{align*}\nwhere $s(r,\\theta)=\\sum_{k\\in B}c_{k}q_{k} \\varphi_{k}$ is a polynomial of degree $2\\lambda.$\nIf we denote $t(r,\\theta)=s(r,\\theta)+p(r),$ by Lemma 1.2 in \\cite{5}, we know that $ t(r,\\theta)$ is an polynomial of degree $2\\lambda.$ Then \n\\[u(r,\\theta)=t(r,\\theta)+\\sum_{k\\notin B}C_ke^{\\frac{r^2}{4}}r^{2\\lambda+m}\\big(1+g_k(\\frac{r^2}{4})\\big)\\varphi_k.\\] \n\\end{proof}\n\n \\begin{proof}[Proof of Theorem \\ref{thm:1.3}]\nWe divide $\\lambda$ into three cases:\n\nCase1. $2\\lambda$ is not an integer.\n\nFor any fixed $k\\ge 0,$ note that \n\\begin{align*}\n\\frac{1}{2}|C_k|e^{\\frac{r_i^2}{4}}r_i^{-(m+2\\lambda)}& \\le |C_k||1+g_k(\\frac{r_i^2}{4})|e^{\\frac{r_i^2}{4}}r_i^{-(m+2\\lambda)}\\\\\n&=|\\int_{S^{m-1}}u(r_i,\\theta)\\varphi_k|\\\\\n&\\le (\\int_{S^{m-1}}( u(r_i,\\theta))^2)^{\\frac{1}{2}}(\\int_{S^{m-1}} \\varphi_k^2)^{\\frac{1}{2}}\\\\\n&\\le C\\sqrt{\\omega_n}e^{\\frac{r_i^2}{4}}r_i^{-(m+2\\lambda+\\epsilon)},\n\\end{align*}\n which implies that \n\\[C_k0$, where $C$ is independent of $k$.\nIf we set $0<\\delta<\\frac{1}{2},$ the series $\\sum_{k=0}^{\\infty} \\bar f_k\\varphi_k$ is uniformly convergent on $\\overline { B_A(0)}\\setminus B_{\\epsilon}(0)$ for any $\\epsilon,A>0.$ Assume $\\bar u=\\sum_{k=0}^{\\infty} \\bar f_k\\varphi_k,$ then we have \n\\begin{align*}\n\\Delta_g(\\bar u)&=\\bar u_{r r}+\\frac{m-1}{r}\\bar u_r+\\frac{1}{r^2}\\Delta_{\\theta}\\bar u-\\frac{r}{2}\\frac{\\partial \\bar u}{\\partial r}\\\\\n&=\\sum_{k=1}^{\\infty} \\Big((\\bar f_k)_{rr}+(\\frac{m-1}{r}-\\frac{r}{2})(\\bar f_k)_r-\\frac{\\lambda_k\\bar f_k}{r^2}\\Big)\\varphi_k+(\\bar f_0)_{r r}+(\\frac{m-1}{r}-\\frac{r}{2})(\\bar f_0)_r\\\\\n&=-\\lambda\\sum_{k=0}^{\\infty}\\bar f_k\\varphi_k\\\\\n&=-\\lambda \\bar u\n\\end{align*}\nThus $\\bar u$ is a solution of $\\Delta_g u=-\\lambda u$ on $\\mathbb{R}^m\\backslash \\{0\\}.$\\\\\nSince $\\sum_{k=0}^{\\infty} \\frac{\\bar f_k}{u_0}\\varphi_k$ is uniformly convergent on $\\mathbb{R}^m\\setminus B_{\\epsilon}(0)$ for any $\\epsilon>0$, then we can obtain that \n\\begin{align*}\n\\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}=\\lim\\limits_{r\\to+\\infty}\\sum_{k=0}^{\\infty} \\frac{\\bar f_k}{u_0}\\varphi_k=\\sum_{k=0}^{\\infty} \\lim\\limits_{r\\to + \\infty}\\frac{\\bar f_k}{u_0}\\varphi_k=g(\\theta).\n\\end{align*}\nHence $\\bar u$ is the solution of $\\Delta_h u=-\\lambda u$ which satisfies that $\\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}=g(\\theta)$ for any given function $g(\\theta)\\in H^{[\\frac{m}{2}]+2}(S^{m-1}).$\nMoreover, assume $\\frac{\\bar u(r,\\theta)}{u_0(r)}$ and $\\frac{ u(r,\\theta)}{u_0(r)}$ are asymptotic to the same function $g(\\theta)$, then \n\\begin{align*}\n&C_k=\\lim\\limits_{r\\to+\\infty} \\frac{ f_k}{u_0}=\\langle\\lim\\limits_{r\\to+\\infty} \\frac{ u(r,\\theta)}{u_0(r)}, \\varphi_k \\rangle \\\\\n&=\\langle \\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}, \\varphi_k \\rangle=\\lim\\limits_{r\\to + \\infty}\\frac{\\bar f_k}{u_0}=\\bar C_k.\n\\end{align*}\nHence \n\\begin{align*}\n&u(r,\\theta)-\\bar u(r, \\theta)\\\\\n&=p(r)-\\bar p(r)+\\sum_{k=0}^{\\infty} (C_k-\\bar C_k)e^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\Big(1+g_k(\\frac{r^2}{4})\\Big)\\varphi_k\\\\\n&=p(r)-\\bar p(r).\n\\end{align*}\n Case 2. $\\lambda$ is an integer.\n\nFor any $g(\\theta)\\in H^{[\\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\\langle g,\\varphi_k\\rangle=0$ when $k\\in \\{0,m-2,m,m+2,\\cdots,m+2\\lambda\\}=A,$ by Lemma 2.4, we have $g(\\theta)=\\sum_{k\\notin A}\\bar C_k\\varphi_k$ and $|\\bar C_k|\\le \\frac{C}{\\lambda_k^{\\frac{1}{2}[\\frac{m}{2}]+1}}.$\\\\ \nSet \\[u(r,\\theta)=\\sum_{k\\notin A}\\bar C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\Big(1+g_k(\\frac{r^2}{4})\\Big)\\varphi_k+\\bar C_0l(r).\\]\nThen $\\bar u$ is a solution of $\\Delta_h=-\\lambda u$ which satisfies that $\\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}=g(\\theta).$\n\nCase 3. $2\\lambda$ is an integer but $\\lambda$ is not an integer.\n\nFor any $g(\\theta)\\in H^{[\\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\\langle g,\\varphi_k\\rangle=0$ when $k\\in \\{m-2,m,m+2,\\cdots,m+2[\\lambda]\\}=B,$ by Lemma 2.4, we have $g(\\theta)=\\sum_{k\\notin B}\\bar C_k\\varphi_k$ and $|\\bar C_k|\\le \\frac{C}{\\lambda_k^{\\frac{1}{2}[\\frac{m}{2}]+1}}.$\\\\ \nSet \\[u(r,\\theta)=\\sum_{k\\notin B}\\bar C_ke^{\\frac{r^2}{4}}r^{-(2\\lambda+m)}\\Big(1+g_k(\\frac{r^2}{4})\\Big)\\varphi_k.\\]\nThen $\\bar u$ is a solution of $\\Delta_h u =-\\lambda u$ which satisfies that $\\lim\\limits_{r\\to+\\infty}\\frac{\\bar u(r,\\theta)}{u_0(r)}=g(\\theta).$\n\\end{proof}\n\n\\end{document}"},"lang":{"kind":"string","value":"math"}}},{"rowIdx":172,"cells":{"text":{"kind":"string","value":"var indicators = [\n { key: \"absoluteVolumeOscillatorIndicator\", text: \"絶対出来高オシレーター\" },\n { key: \"averageTrueRangeIndicator\", text: \"ATR (アベレージ トゥルー レンジ)\" },\n { key: \"accumulationDistributionIndicator\", text: \"蓄積/分配\" },\n { key: \"averageDirectionalIndexIndicator\", text: \"平均方向性指数\" },\n { key: \"bollingerBandsOverlay\", text: \"ボリンジャー バンド オーバーレイ\" },\n { key: \"bollingerBandWidthIndicator\", text: \"ボリンジャー帯域幅\" },\n { key: \"chaikinOscillatorIndicator\", text: \"チャイキン オシレーター\" },\n { key: \"chaikinVolatilityIndicator\", text: \"チャイキン ボラティリティ\" },\n { key: \"commodityChannelIndexIndicator\", text: \"商品チャネル指数 (CCI)\" },\n { key: \"detrendedPriceOscillatorIndicator\", text: \"トレンド除去価格オシレーター\" },\n { key: \"easeOfMovementIndicator\", text: \"イーズ オブ ムーブメント\" },\n { key: \"fastStochasticOscillatorIndicator\", text: \"ファスト スト キャスティクス オシレーター\" },\n { key: \"forceIndexIndicator\", text: \"フォース インデックス\" },\n { key: \"fullStochasticOscillatorIndicator\", text: \"フル ストキャスティクス オシレーター\" },\n { key: \"marketFacilitationIndexIndicator\", text: \"マーケット ファシリテーション インデックス\" },\n { key: \"massIndexIndicator\", text: \"マス インデックス\" },\n { key: \"medianPriceIndicator\", text: \"メディアンプライス\" },\n { key: \"moneyFlowIndexIndicator\", text: \"マネー フロー インデックス \" },\n { key: \"movingAverageConvergenceDivergenceIndicator\", text: \"移動平均収束拡散\" },\n { key: \"negativeVolumeIndexIndicator\", text: \"負出来高指数 (NVI)\" },\n { key: \"onBalanceVolumeIndicator\", text: \"OBV (オン バランス ボリューム) インジケーター\" },\n { key: \"percentagePriceOscillatorIndicator\", text: \"パーセンテージ価格オシレーター\" },\n { key: \"percentageVolumeOscillatorIndicator\", text: \"パーセンテージ出来高オシレーター\" },\n { key: \"positiveVolumeIndexIndicator\", text: \"PVI (ポジティブ ボリューム インデックス)\" },\n { key: \"priceChannelOverlay\", text: \"プライス チャネル オーバーレイ\" },\n { key: \"priceVolumeTrendIndicator\", text: \"価格出来高トレンド\" },\n { key: \"rateOfChangeAndMomentumIndicator\", text: \"変化率およびモメンタム\" },\n { key: \"relativeStrengthIndexIndicator\", text: \"相対力指数\" },\n { key: \"slowStochasticOscillatorIndicator\", text: \"スロー ストキャスティクス オシレーター\" },\n { key: \"standardDeviationIndicator\", text: \"標準偏差\" },\n { key: \"stochRSIIndicator\", text: \"ストキャスティクス RSI\" },\n { key: \"trixIndicator\", text: \"Trix\" },\n { key: \"typicalPriceIndicator\", text: \"代表価格\" },\n { key: \"ultimateOscillatorIndicator\", text: \"アルティメット オシレーター\" },\n { key: \"weightedCloseIndicator\", text: \"重み付きクローズ\" },\n { key: \"williamsPercentRIndicator\", text: \"ウィリアム パーセント レンジ\" }\n ];\n"},"lang":{"kind":"string","value":"code"}}},{"rowIdx":173,"cells":{"text":{"kind":"string","value":"Information on the government’s policy on the assessed and supported year in employment programme. Explains the ASYE programme for newly qualified social workers, and how organisations can register to take part.\nThese files contain information for suppliers developing software and management information systems (MIS) for local authorities and schools.\nThis includes information on learners who are studying on a course at a further education college, learners studying courses within their local community, employees undertaking an apprenticeship, and employees undertaking other qualifications in the workplace.Please note: data on providers is only published annually.\nIt has been designed to complement the main statistical releases, and act as a ‘one stop shop’ for data and information on learners, learning programmes and learner achievement.\nUse the ready reckoner to calculate level 3 value-added results.\nThe transition matrices spreadsheet shows data for the subjects included in the level 3 value-added measure.\nLevel 3 value-added is a progress measure for school sixth forms and colleges which is used in the 16 to 18 performance tables.\nYou can also read the 16 to 18 accountability technical guide to learn more about the level 3 value-added measure.\nHow to check the infant class size information in your spring 2019 school census return.\nThe Selective Schools Expansion Fund (SSEF) will provide funding of £49.3 million for 16 expansion projects. This will create over 3,000 more grammar school places.\nThis guide helps you prepare and submit your school’s 2019 school-level annual school census (SLASC) return.\nSLASC census day for registered independent schools is Thursday 17 January 2019.\nA list of COLLECT queries and explanatory notes to help schools, academies and local authorities complete the school census."},"lang":{"kind":"string","value":"english"}}},{"rowIdx":174,"cells":{"text":{"kind":"string","value":"سُہ کھوٚت ہیرِ تہٕ دِژٕن اَتھ مچہِ نظرا اتہِ اوس تِیُتھ پھَکھ یِوان زِ سُہ اوسنہٕ أکِس لحظَس تہِ اتِتھ روزُن یژھان چؠنٕچ چھَنہٕ کَتھٕے"},"lang":{"kind":"string","value":"kashmiri"}}},{"rowIdx":175,"cells":{"text":{"kind":"string","value":"Scuba diver Jimmy Roseman came face to face with a 12-foot great white shark while surveying the coral reefs at a depth of 90 feet just off the Atlantic Coast near Vero Beach, Florida.\nAll the action was caught on his GoPro camera that was mounted to his diver helmet.\nThe shark was at a distance when he took the dive but came close to Roseman really quickly.\nThe interaction lasted two minutes when finally Roseman's air tank was hit by the shark's fin. It was then that he used the spear gun to fend off the potentially deadly shark.\nIn the video it looks like the great white shark may have a baby with it.\nNever does Roseman shot or injure the shark and both are left unharmed.\nIn a video interview with Fox 35, Roseman said, \"I felt something hit me on the back and I then I seen the shark come over my head and swim off. He circled me...I had to poke him a few times. He wouldn't go away and then I finally jabbed him hard enough to where he left me long enough i could get to the surface and get out of there.\"\nThe great white shark is typically found in cold and warm-temperate waters throughout the world, although occurrences in tropical waters have been documented, according to the National Oceanic and Atmospheric Administration.\nResearch suggests that great whites migrate to Florida in a seasonal pattern with distribution limited by water temperature, food resources, or other factors.\nJimmy Roseman of West Melbourne, FL has an encounter with a Great White Shark off of the coast of Florida at Bethel Shoals. Obviously, the beginning is computer generated, but the GoPro video is 100% raw."},"lang":{"kind":"string","value":"english"}}},{"rowIdx":176,"cells":{"text":{"kind":"string","value":"\\begin{document}\n\n\\selectlanguage{english}\n\n\\title{Subvarieties of moduli spaces of sheaves via finite coverings}\n\n\\author{Markus Zowislok\\footnote{Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK}}\\date{}\\maketitle\\newtheorem{theorem}{Theorem}[section]\\newtheorem{definition}[theorem]{Definition}\\newtheorem{corollary}[theorem]{Corollary}\\newtheorem{lemma}[theorem]{Lemma}\\newtheorem{proposition}[theorem]{Proposition}\\newtheorem{example}[theorem]{Example} \\newcommand{\\textit}{\\textit} \n\n\\begin{abstract}\nGiven a finite unbranched covering of a nonsingular projective scheme we analyse the morphism between moduli spaces of sheaves induced by pullback.\nWe have a closer look at cyclic coverings and, in particular, at canonical coverings of surfaces.\nOur main application is the construction of Lagrangian subvarieties of certain irreducible holomorphically symplectic manifolds that arise from moduli spaces of sheaves on K3 or abelian surfaces.\n\\end{abstract}\n\n\\maketitle\n\n\\newcommand{\\ensuremath{\\mathbb{Z}}}{\\ensuremath{\\mathbb{Z}}}\n\\newcommand{\\ensuremath{\\mathbb{N}}}{\\ensuremath{\\mathbb{N}}}\n\\newcommand{\\ensuremath{\\mathcal{H}}}{\\ensuremath{\\mathcal{H}}}\n\\newcommand{\\ensuremath{\\mathrm{H}}}{\\ensuremath{\\mathrm{H}}}\n\\newcommand{\\ensuremath{\\mathcal{E}xt}}{\\ensuremath{\\mathcal{E}xt}}\n\\renewcommand{\\hom}{\\ensuremath{\\mathrm{hom}}}\\newcommand{\\ensuremath{\\mathrm{Hom}}}{\\ensuremath{\\mathrm{Hom}}}\n\\newcommand{\\ensuremath{\\mathrm{codim}}}{\\ensuremath{\\mathrm{codim}}}\n\\newcommand{\\ensuremath{\\mathrm{Coh}}}{\\ensuremath{\\mathrm{Coh}}}\n\\newcommand{\\ensuremath{\\mathrm{Num}}}{\\ensuremath{\\mathrm{Num}}}\n\\newcommand{\\ensuremath{\\mathrm{NS}}}{\\ensuremath{\\mathrm{NS}}}\n\\newcommand{\\ensuremath{\\mathrm{Amp}}}{\\ensuremath{\\mathrm{Amp}}}\n\\newcommand{\\ensuremath{\\Lambda'(Y)}}{\\ensuremath{\\Lambda'(Y)}}\n\\newcommand{\\ensuremath{\\mathrm{Hilb}}}{\\ensuremath{\\mathrm{Hilb}}}\n\\newcommand{\\ensuremath{\\;(\\le_0)\\;}}{\\ensuremath{\\;(\\le_0)\\;}}\n\\newcommand{\\ensuremath{\\mathrm{gcd}}}{\\ensuremath{\\mathrm{gcd}}}\n\\newcommand{\\ensuremath{\\mathrm{ord}}}{\\ensuremath{\\mathrm{ord}}}\n\\newcommand{\n\\[\\qedhere\\]}{\n\\[\\qedhere\\]}\n\\renewcommand{(\\alph{enumii})\\ }{(\\alph{enumii})\\ }\n\n\\newtheorem{question}[theorem]{Question}\\newtheorem{conjecture}[theorem]{Conjecture}\n\n\\section{Introduction}\n\nIs Gieseker stability preserved under pullback by a finite covering? How are the corresponding moduli spaces related?\nIn this article, we exhibit the behaviour of stability under pullback by a finite unbranched covering of a nonsingular projective scheme.\nThe roots of this question go back to Kim's article \\cite{Kim98a}, which is on the canonical covering of an Enriques surface and $\\mu$-stable sheaves.\n\nOur main results hold not only for the notion of Gieseker stability, but also for twisted stability and for $(H,A)$-stability. Therefore we will always write (semi)stable whenever the statement allows all three notions. These stability notions are recalled in Section \\ref{StabNot}.\n\nFor the whole article let $X$ be a nonsingular projective irreducible variety over $\\IC$, $G$ a finite group acting freely on $X$, $f\\colon X\\to Y$ the quotient of $X$ by $G$, and $H$ an ample divisor on $Y$.\nIn Section \\ref{arbdim} we show that the pullback $f^*E$ of a semistable sheaf $E$ on $Y$ is again semistable (Proposition \\ref{PbStab}).\nAfter some more precise results on the behaviour of the stability property, we apply these results to moduli spaces:\\\\\n\n\\noindent \\textbf{Theorem \\ref{MHApb}}. \\titar{Let $P$ be a polynomial, $M_Y$ a quasiprojective and nonempty subscheme of the moduli space $M_Y(P)$ of semistable sheaves on $Y$ with (twisted) Hilbert polynomial $P$, $M_X=M_X(\\deg f\\cdot P)$ the moduli space of semistable sheaves on $X$ with (twisted) Hilbert polynomial $\\deg f\\cdot P$, and $M_Y^s$ and $M_X^s$ the respective loci of stable sheaves.\nThe pullback by $f$ induces a morphism $f^*\\colon M_Y\\to M_X$ which maps closed points $[E]$ to $[f^*E]$.\nThe closed points of its image are represented by polystable $G$-sheaves, and $(f^*)^{-1}( M_X^s )\\subseteq M_Y^s\\,.$}\\\\\n\nThe restriction to a cyclic covering given by a line bundle $L$ of finite order in Section \\ref{arbCC} allows a deeper analysis of the pullback morphism. The main tool is the group action on the moduli space of sheaves on $Y$ induced by tensoring with $L$. This allows us to give a precise description of the locus of stable sheaves becoming strictly semistable in Theorem \\ref{CycStab}.\\\\\n\nSection \\ref{Surfaces} contains the application to canonical coverings of surfaces. We investigate the double covering $X\\to Y$ of an Enriques surface $Y$ by a K3 surface $X$ as well as the canonical covering $X\\to Y$ of a bielliptic surface $Y$ by an abelian surface $X$.\nThe main results are given in Theorems \\ref{CC} and \\ref{CCsst}.\nAn interesting question is in what cases there are Lagrangian subvarieties as images of $f^*$ inside the moduli spaces of semistable sheaves on $X$, and what kind of varieties these subvarieties are. \nIf $X$ is a K3 surface and $f^*u$ is primitive then the moduli space $M_X(f^*u)$ in general is an irreducible symplectic manifold, and\nwhenever there is a suitable stable sheaf on $Y$, one gets a Lagrangian subvariety as described in Proposition \\ref{CanDouSt}.\nIf $X$ is an abelian surface, the situation is different: in order to produce higher dimensional irreducible symplectic manifolds out of moduli spaces of sheaves on $X$ one has to get rid of superfluous factors in the Bogomolov decomposition by taking a fibre of the Albanese map. The last part of Section \\ref{LIHS} explains how and why this reduction reduces the Lagrangian subvarieties to (smaller) Lagrangian subvarieties in the case of a double covering, resulting in Proposition \\ref{CanDouStK}.\n\nThe most prominent Lagrangian subvarieties of irreducible holomorphically symplectic manifolds occur as fibres or sections of Lagrangian fibrations. Recently, smooth Lagrangian tori have attracted a lot of interest, sparked by the following question of Beauville \\cite{Beau10}:\nIf an irreducible holomorphically symplectic manifold $M$ contains a smooth Lagrangian torus, is this a fibre of a Lagrangian fibration on $M$? \nThis has been answered positively combining \\cite{GLR11}, \\cite{HW12}, and \\cite{Mat12}.\n\nIt would be interesting to understand better, which kind of Lagrangian subvarieties can be obtained by Propositions \\ref{CanDouSt} and \\ref{CanDouStK}. \nSome concrete results are contained in Section \\ref{example}.\nUnfortunately, our knowledge on moduli spaces of sheaves on Enriques and bielliptic surfaces is quite limited at the moment.\nHowever, we expect that complex tori do not occur as Lagrangian subvarieties in the case of sheaves of odd rank. In particular, they do not occur as image under pullback of the moduli space of sheaves of rank one on Enriques surfaces, i.e.\\ in the Hilbert scheme case, as explained in Section \\ref{example}.\nThe case of even rank seems to be more promising, as an example of Hauzer can be used to produce an elliptic curve as Lagrangian subvariety in a K3 surface, as explained in Section \\ref{example} as well. Hence there is hope that higher-dimensional examples of Lagrangian tori inside symplectic varieties may occur.\n\nOn the other side, it is interesting in its own to construct different Lagrangian subvarieties and, in particular, to study their intersection. In \\cite{BF09} the authors construct a Gerstenhaber algebra structure and a compatible Batalin-Vilkovisky module structure\ngiving rise to a de Rham type cohomology theory for Lagrangian intersections.\n\nShortly after the first version of this article appeared on arXiv.org, an independent article by Sacc\\`a \\cite{Sac12} on the case of one-dimensional sheaves on Enriques surfaces and the induced pullback by the canonical double covering appeared there as well. Her results are related to my Question \\ref{OSgen}.\n\nFinally we give an outlook in Section \\ref{Outlook} considering the case of surfaces of general type,\nand an appendix briefly recalls the notion of a general ample divisor which occurs in Section \\ref{Surfaces}.\\\\\n\n\\acks\n{\\small The author would like to express his gratitude to S\\\"onke Rollenske for fruitful discussions and helpful suggestions.\nHe thanks Arvid Perego for directing his attention to Kim's articles and Hauzer's article and Manfred Lehn for useful comments.\nMoreover, he thanks the DFG for the SFB 701, which provides a vibrant research atmosphere at Bielefeld.\nThis article was completed during a stay at Imperial College London supported by a DFG research fellowship (Az.: ZO 324/1-1).}\n\n\\section{Pullback by finite \\'etale coverings and stability}\\label{arbdim}\n\nIn this section we recall the considered notions of stability of sheaves, analyse their behaviour under pullback by $f$ and apply the results to get our general Theorem \\ref{MHApb} on the moduli spaces of sheaves.\nWe assume familiarity with the material presented in \\cite{HL10} and use the notation therein.\n\n\\subsection{Three stability notions}\\label{StabNot}\n\nOur main results hold for the notions of Gieseker stability, twisted stability and $(H,A)$-stability.\nTwisted stability and $(H,A)$-stability are two generalisations of Gieseker stability, with an overlap in the case of a surface, see \\cite[Corollary 6.2.6]{Zow10}.\nWe briefly recall the definitions.\n\n\\begin{enumerate}\n\\item \\textit{Gieseker stability.} \nA detailed treatment of this notion can be found e.g.\\ in \\cite[Section 1.2]{HL10}.\nLet $H$ be an ample divisor on $Y$ and $E$ a nontrivial coherent sheaf on $Y$.\nThe Hilbert polynomial of $E$ is $P_H(E)(n):=\\chi(E\\otimes H^{\\otimes n})$. Its leading coefficient multiplied by $(\\dim E)!$ is called multiplicity of $E$ and denoted here by $\\alpha^H(E)$. It is always positive, and $p_H(E)(n):=\\frac{\\chi(E\\otimes H^{\\otimes n})}{\\alpha^H(E)}$ is called reduced Hilbert polynomial of $E$.\nWith these at hand, one says that $E$ is $H$-(semi)stable if $E$ is pure and for all nontrivial proper subsheaves $F\\subseteq E$ one has that $p_H(F) \\lel p_H(E)$, i.e.\\ $p_H(F)(n) \\lel p_H(E)(n)$ for $n\\gg 0$. \n\nIn order to avoid case differentiation for stable and semistable sheaves we here follow the Notation 1.2.5 in \\cite{HL10} using bracketed inequality signs, e.g.\\ an inequality with $\\lel$ for (semi)stable sheaves means that one has $\\le$ for semistable sheaves and $<$ for stable sheaves.\n\\item \\textit{Twisted stability as defined in \\cite[Definition 4.1]{Yos03b}.}\nLet $H$ be an ample divisor on $Y$, $V$ a locally free sheaf on $Y$ and $E$ a nontrivial coherent sheaf on $Y$.\nThe $V$-twisted Hilbert polynomial is $P^V_H(E)(n):=\\chi(E\\otimes V\\otimes H^{\\otimes n})$, and\nthe reduced $V$-twisted Hilbert polynomial is $$p^V_H(E)(n):=\\frac{\\chi(E\\otimes V\\otimes H^{\\otimes n})}{\\alpha^H(E\\otimes V)}\\,.$$\nOne says that $E$ is $V$-twisted $H$-(semi)stable if $E$ is pure and for all nontrivial proper subsheaves $F\\subseteq E$ one has that $p^V_H(F) \\lel p^V_H(E)$.\n\n\\item \\textit{$(H,A)$-stability as defined in \\cite[Definition 7.1]{Zow12}.}\nLet $H$ and $A$ be two ample divisors on $Y$ and $E$ a nontrivial coherent sheaf on $Y$.\nWe defined\n$$\nP_{H,A}(E)(m,n):=\\chi(E\\otimes H^{\\otimes m} \\otimes A^{\\otimes n}) \\quad\\mathrm{and}\\quad\np_{H,A}(E):=\\frac{P_{H,A}(E)}{\\alpha^H(E)} \\,.\n$$\nThese are polynomials in $m$ and $n$ with degree $d:=\\dim E$ in $n$ and $m$ and total degree $d$, and one has $P_{H,A}(E)(\\bullet,0)=P_{H}(E)$ and $p_{H,A}(E)(\\bullet,0)=p_{H}(E)$.\n\nThere is a natural ordering of polynomials in one variable given by the lexicographic ordering of their coefficients.\nThis generalises to polynomials of two variables by the identification $\\IQ[m,n]=(\\IQ[m])[n]$, i.e.\\ we consider the elements as polynomials in $n$ and use the ordering of $\\IQ[m]$ for comparing coefficients.\n\nWe introduce another ordering on $\\IQ[m,n]$ by defining\n$$f\\le_0 g\\quad:\\Leftrightarrow\\quad (f(\\bullet,0),-f)\\le(g(\\bullet,0),-g)$$ for $f,g\\in\\IQ[m,n]$, where on the right hand side we use lexicographic ordering on the product $\\IQ[m]\\times\\IQ[m,n]$, i.e.\\ $f\\le_0 g$ if and only if $f(\\bullet,0)