texts
sequence | meta
dict | scores
sequence | avg_score
float64 0
0.25
| num_sents
int64 5
5
|
|---|---|---|---|---|
[
"<!",
"DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/html; charset=utf-8\" />\n<title>企业商城 - [.$emmm_web.website.]</title>\n<meta name=\"keywords\" content=\"[.$emmm_web.webkeywords.]\" /",
">\n<meta name=\"description\" content=\"[.$emmm_web.webdescriptions.]\"/>\n<meta name=\"Author\" content=\"vidar.club\" />\n<LINK href=\"[.$webpath.]function/plugs/YIQI-UI/YIQI-UI.min.css\" rel=stylesheet>\n<link href=\"[.$webpath.]function/plugs/YIQI-UI/font/YIQI-UI-iconfont.min.css\" rel=\"stylesheet\">\n<LINK href=\"[.$templatepath.]shop/css/emmm.css?2\" rel=stylesheet>\n<script type=\"text/javascript\" src=\"[.$webpath.]function/plugs/jquery/1.7.2/jquery-1.7.2.min.js\"></script>\n<script type=\"text/javascript\" src=\"[.$templatepath.]shop/js/emmm.js\"></script>\n<script type=\"text/javascript\" src=\"[.$webpath.]function/plugs/lazyload/jquery.lazyload.js\"></script>\n<script type=\"text/javascript\" src=\"[.$templatepath.]shop/js/banner1.js\"></script>\n<script type=\"text/javascript\" src=\"[.$templatepath.]shop/js/rightnav.js\"></script>\n<script type=\"text/javascript\">\n$(document).ready(function(){\n\t$(\"img\").lazyload({ \n\t\t effect: \"fadeIn\"\n\t}); \n\ttcjs('wx');\n\ttcjs('sj');\n\ttcjs('wxkf');\n\tnavjs('navleft');\n});\n</script>\n</head>\n<body>\n[.include file=\"cn/cn_shoptop.html\".]",
"\n<div class=\"clear\"></div>\n<div class=\"banner\">\n <div class=\"b-img\">\n [.banner lang=\"cn\" row=\"5\" name=\"banner\" id=\"0\" type=\"0\".]",
"\n <a href=\"[.$banner.url.]\" ",
"style=\"background:url([.$banner.img.]) ",
"center no-repeat;\"></a> \n [./banner.]",
"\n </div>\n <div class=\"b-list\"></div>\n <a class=\"bar-left\" href=\"#\"><em></em></a>\n <a class=\"bar-right\" href=\"#\"><em></em></a>\n</div>\n<div class=\"clear\"></div>\n<div class=\"center\">\n\n\t<div class=\"emmm_content\">\n \t<div class=\"left\">\n \t<h1>促销/资讯</h1>\n <ul>\n [.emmm form=\"article\" row=\"8\" lang=\"cn\" id=\"0\" type=\"op\" name=\"emmm\".]",
"\n <li><a href=\"[.$emmm.url.]\" ",
"title=\"[.$emmm.title.]\">[.$emmm.title|truncate:15:\"...\".]</a></li>\n [./emmm.]",
"\n </ul>\n </div>\n <div class=\"right\">\n \t<h1>特价秒杀</h1>\n <ul>\n [.emmm form=\"product\" row=\"4\" lang=\"cn\" id=\"0\" type=\"3\" name=\"emmm\".]",
"\n <li><a href=\"[.$emmm.url.]\" ",
"title=\"[.$emmm.title.]\"><img src=\"[.$emmm.minimg.]\" /",
">\n <p class=\"title\">[.$emmm.title|truncate:30:\"...\".]</p>\n <p class=\"jg\">促销价：<span>￥[.$emmm.webmarket.]</span></p>\n </a></li>\n [./emmm.]",
"\n </ul>\n </div>\n \n <div class=\"clear\"></div>\n \n <div class=\"f\">\n \t<h1 class=\"h1\">数码产品<span><a href=\"\">手机配件</a> <a href=\"\">手机卡套餐</a> <a href=\"\">更多</a></span></h1>\n <ul>\n [.emmm form=\"product\" row=\"5\" lang=\"cn\" id=\"0\" type=\"op\" name=\"emmm\".]",
"\n <li><a href=\"[.$emmm.url.]\" ",
"title=\"[.$emmm.title.]\"><img src=\"/skin/noimage.png\" data-original=\"[.$emmm.minimg.]\" /",
">\n <p class=\"title\">[.$emmm.title|truncate:30:\"...\".]</p>\n <p class=\"scj\">市场价：<s>￥[.$emmm.market.]</s></p>\n <p class=\"bzj\">本站价：<span>￥[.$emmm.webmarket.]</span></p>\n </a></li>\n [./emmm.]",
"\n </ul>\n </div>\n <div class=\"clear\"></div>\n <div class=\"f\">\n \t<h1 class=\"h2\">母婴用品<span><a href=\"\">奶瓶</a> <a href=\"\">瓶刷</a> <a href=\"\">电动车</a> <a href=\"\">更多</a></span></h1>\n <ul>\n [.emmm form=\"product\" row=\"5\" lang=\"cn\" id=\"0\" type=\"op\" name=\"emmm\".]",
"\n <li><a href=\"[.$emmm.url.]\" ",
"title=\"[.$emmm.title.]\"><img src=\"/skin/noimage.png\" data-original=\"[.$emmm.minimg.]\" /",
">\n <p class=\"title\">[.$emmm.title|truncate:30:\"...\".]</p>\n <p class=\"scj\">市场价：<s>￥[.$emmm.market.]</s></p>\n <p class=\"bzj\">本站价：<span>￥[.$emmm.webmarket.]</span></p>\n </a></li>\n [./emmm.]",
"\n </ul>\n </div>\n <div class=\"clear\"></div>\n <div class=\"f\">\n \t<h1 class=\"h3\">男装女装\t<span><a href=\"\">休闲装</a> <a href=\"\">耐克</a> <a href=\"\">正装</a> <a href=\"\">更多</a></span></h1>\n <ul>\n [.emmm form=\"product\" row=\"5\" lang=\"cn\" id=\"0\" type=\"op\" name=\"emmm\".]",
"\n <li><a href=\"[.$emmm.url.]\" ",
"title=\"[.$emmm.title.]\"><img src=\"/skin/noimage.png\" data-original=\"[.$emmm.minimg.]\" /",
">\n <p class=\"title\">[.$emmm.title|truncate:30:\"...\".]</p>\n <p class=\"scj\">市场价：<s>￥[.$emmm.market.]</s></p>\n <p class=\"bzj\">本站价：<span>￥[.$emmm.webmarket.]</span></p>\n </a></li>\n [./emmm.]",
"\n </ul>\n </div>\n <div class=\"clear\"></div>\n <div class=\"f\">\n \t<h1 class=\"h4\">珠宝饰品\t<span><a href=\"\">玉器</a> <a href=\"\">古玩</a> <a href=\"\">9999黄金</a> <a href=\"\">更多</a></span></h1>\n <ul>\n [.emmm form=\"product\" row=\"5\" lang=\"cn\" id=\"0\" type=\"op\" name=\"emmm\".]",
"\n <li><a href=\"[.$emmm.url.]\" ",
"title=\"[.$emmm.title.]\"><img src=\"/skin/noimage.png\" data-original=\"[.$emmm.minimg.]\" /",
">\n <p class=\"title\">[.$emmm.title|truncate:30:\"...\".]</p>\n <p class=\"scj\">市场价：<s>￥[.$emmm.market.]</s></p>\n <p class=\"bzj\">本站价：<span>￥[.$emmm.webmarket.]</span></p>\n </a></li>\n [./emmm.]",
"\n </ul>\n </div>\n <div class=\"clear\"></div>\n <div class=\"f\">\n \t<h1 class=\"h5\">化妆品\t<span><a href=\"\">美宝莲</a> <a href=\"\">口红</a> <a href=\"\">更多</a></span></h1>\n <ul>\n [.emmm form=\"product\" row=\"5\" lang=\"cn\" id=\"0\" type=\"op\" name=\"emmm\".]",
"\n <li><a href=\"[.$emmm.url.]\" ",
"title=\"[.$emmm.title.]\"><img src=\"/skin/noimage.png\" data-original=\"[.$emmm.minimg.]\" /",
">\n <p class=\"title\">[.$emmm.title|truncate:30:\"...\".]</p>\n <p class=\"scj\">市场价：<s>￥[.$emmm.market.]</s></p>\n <p class=\"bzj\">本站价：<span>￥[.$emmm.webmarket.]</span></p>\n </a></li>\n [./emmm.]",
"\n </ul>\n </div>\n <div class=\"clear\"></div>\n </div>\n\n</div>\n<div class=\"clear\"></div>\n[.include file=\"cn/cn_shopfoot.html\".]",
"\n</body>\n</html>\n"
] | {
"pile_set_name": "Github"
} | [
0,
0.00872093023255814,
0,
0.007352941176470588,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.011494252873563218,
0,
0.00303951367781155,
0,
0.011494252873563218,
0,
0,
0,
0.011494252873563218,
0,
0.003003003003003003,
0,
0.011494252873563218,
0,
0.0032679738562091504,
0,
0.011494252873563218,
0,
0,
0
] | 0.002302 | 5 |
[
"// Spinning Icons\n// --------------------------\n\n.#{$fa-css-prefix}-spin {\n -webkit-animation: fa-spin 2s infinite linear;\n animation: fa-spin 2s infinite linear;\n}\n\n@-webkit-keyframes fa-spin {\n 0% {\n -webkit-transform: rotate(0deg);\n transform: rotate(0deg);\n }\n 100% {\n -webkit-transform: rotate(359deg);\n transform: rotate(359deg);\n }\n}\n\n@keyframes fa-spin {\n 0% {\n -webkit-transform: rotate(0deg);\n transform: rotate(0deg);\n }\n 100% {\n -webkit-transform: rotate(359deg);\n transform: rotate(359deg);\n }\n}\n"
] | {
"pile_set_name": "Github"
} | [
0.015437392795883362
] | 0.015437 | 5 |
[
"Leaching of seven pesticides currently used in cotton crop in Mato Grosso State-Brazil.",
"\nThis study aimed to evaluate the leaching of pesticides and the applicability of the Attenuation Factor (AF) Model to predict their leaching. ",
"The leaching of carbofuran, carbendazim, diuron, metolachlor, alpha and beta endosulfan and chlorpyrifos was studied in an Oxisol using a field experiment lysimeter located in Dom Aquino-Mato Grosso. ",
"The samples of percolated water were collected by rain event and analyzed. ",
"Chemical and physical soil attributes were determined before pesticide application to the plots. ",
"The results showed that carbofuran was the pesticide that presented a higher leaching rate in the studied soil, so was the one representing the highest contamination potential. ",
"From the total carbofuran applied in the soil surface, around 6% leached below 50 cm. ",
"The other pesticides showed lower mobility in the studied soil. ",
"The calculated values to AF were 7.06E-12 (carbendazim), 5.08E-03 (carbofuran), 3.12E-17 (diuron), 6.66E-345 (alpha-endosulfan), 1.47E-162 (beta-endosulfan), 1.50E-06 (metolachlor), 3.51E-155 (chlorpyrifos). ",
"AF Model was useful to classify the pesticides' potential for contamination; however, that model underestimated pesticide leaching."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0.006993006993006993,
0.02,
0,
0.010309278350515464,
0,
0,
0,
0,
0
] | 0.00373 | 5 |
[
"1715, from Latin lichen, from Greek leichen \"tree-moss, lichen,\" originally \"what eats around itself,\" probably from leichein \"to lick\" (see lick (v.1)). ",
"Used earlier (c. 1600) of liverwort, which was thought to be related. ",
"Also in English, as in Greek, of eczema and certain other skin diseases. ",
"Related: Lichenic; licheniform; lichenous; lichenaceous."
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0,
0,
0
] | 0 | 5 |
[
"Q:\n\njQuery-Ui Ajax Tabs bug?",
"\n\nI have a JSON that returns from the server which tabs to build,\nso I init them in my JS like this:\n$('#tabs').tabs( 'ajaxOptions', { \n timeout: 20000, \n error: function(xhr, status, index, anchor){ \n console.log( status, index, anchor );\n }\n})\n.tabs('add', item.",
"CategoryLink, item.",
"CategoryName);\n\nThing is, when I click a tab, and before it is done loading I click another tab,\nthe previous request is aborted and never called again when I click that first one again!",
"\nthis is very bad, because it obviously didn't fetch the request, so what gives?",
"\nI tried bypassing this by setting:\n.tabs({ cache: false })\nbut this is a bad thing to do, because I don't want to have a request each time again...\nit should be cachced if response was sent.",
"\nusing jquery-ui 1.8.1\n\nA:\n\nWorkaround from the bug tracker:\nuse this inside the tabs load event:\nload: function(event, ui){\n $(ui.tab).data(\"cache.tabs\",($(ui.panel).html() == \"\") ? ",
"false : true);\n}\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0,
0,
0.05263157894736842,
0.005376344086021506,
0,
0,
0.005376344086021506,
0
] | 0.007923 | 5 |
[
"---\nabstract: |\n A simplified model of the tumor angiogenesis can be described by a Keller-Segel equation [@FrTe; @Le; @Pe]. ",
"The stability of traveling waves for the one dimensional system has recently been known by [@JinLiWa; @LiWa]. ",
"In this paper we consider the equation on the two dimensional domain $ (x, y) \\in \n {\\mathbb R}\\times {\\mathbf S^{{\\lambda}}}$ for a small parameter ${\\lambda}>0$ where $ \\mathbf S^{{\\lambda}}$ is the circle of perimeter ${\\lambda}$. Then the equation allows a planar traveling wave solution of invading types. ",
"We establish the nonlinear stability of the traveling wave solution if the initial perturbation is sufficiently small in a weighted Sobolev space without a chemical diffusion. ",
"When the diffusion is present, we show a linear stability. ",
"Lastly, we prove that any solution with our front conditions eventually becomes planar under certain regularity conditions. ",
"The key ideas are to use the Cole-Hopf transformation and to apply the Poincaré inequality to handle with the two dimensional structure.",
"\naddress:\n- 'Department of Mathematics, Hankyung University, Anseong-si, Gyeonggi-do, Republic of Korea'\n- 'Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea'\n- 'Department of mathematics, Yonsei University, Seoul, Republic of Korea'\n- 'Department of Mathematics, Chung-Ang University, Seoul, Republic of Korea '\nauthor:\n- Myeongju Chae\n- Kyudong Choi\n- Kyungkeun Kang\n- Jihoon Lee\ntitle: |\n Stability of planar traveling waves\\\n in a Keller-Segel equation\\\n on an infinite strip domain\n---\n\nIntroduction and main theorems\n==============================\n\n[The formation of new blood vessels (angiogenesis) is the essential mechanism for tumour progression and metastasis.]{} ",
"A simplified model of the tumor angiogenesis can be described by a Keller-Segel equation [@FrTe; @Le; @Pe]. ",
"In this paper we consider the equation on the two dimensional cylindrical domain $ (x, y) \\in {\\Omega}= {\\mathbb R}\\times [0, {\\lambda}]$ with a front boundary condition in $x$ and the periodic condition in $y$, both specified later, $$\\begin{aligned}\n\\label{KS} \\begin{aligned}\n{\\partial}_t n - {\\Delta}n& = - {\\nabla}\\cdot (n \\chi(c) {\\nabla}c ), \\\\\n{\\partial}_t c - {\\epsilon}{\\Delta}c & = - c^m n\n\\end{aligned} \\end{aligned}$$ with $m \\ge 1$, where $n(x, y, t) \\ge 0 $ denote the density of endothelial cells, $c(x,y, t) \\ge 0$ stands for the concentration of the chemical substance or the protein known as the vascular endothelial growth factor (VEGF) and $\\chi(\\cdot): {\\mathbb R}^+ \\to {\\mathbb R}^+ $ is a decreasing chemosensitivity function, reflecting that the chemosensitivity is lower for higher concentration of the chemical. ",
"The system includes the both zero chemical diffusion $({\\epsilon}=0)$ and non-zero chemical diffusion $({\\epsilon}>0)$ cases.\\\nEndothelial cells forming the linings of the blood vessels are responsible for extending and remodeling the network of blood vessels, tissue growth and repair. ",
"Tumors or cancerous cells are also dependent on bloods supply by newly generated capillaries formed toward them, which process is called the endothelial angiogenesis. ",
"In the modeling the endothelial angiogenesis, the biological implication is that the endothelial cells behaves as a invasive species, responding to signals produced by the tissue. [",
"^1] Accordingly the system is given the front condition at left-right ends such that $$\\begin{aligned}\n\\label{nfront}\n\\lim_{x\\to -\\infty} n (x, y, t) = n_- >0 , \\quad \\lim_{x\\to \\infty} n(x, y, t) = 0,\\\\\n\\lim_{x\\to -\\infty} c (x, y, t) = 0, \\quad \\lim_{x\\to \\infty} c(x, y, t) = c_+ > 0. ",
"\\label{cfront}\\end{aligned}$$ We choose the $x$-axis by the propagating direction.",
"\n\nA *planar* traveling wave solution of is a traveling wave solution independent of the transversal direction $y$ such that $$\\begin{aligned}\n\\label{twave}\n n(x,y,t) =N(x-st), \\quad c(x, y, t) = C(x-st)\\end{aligned}$$ with a wave speed $s >0$. From now on, we consider only planar traveling waves $(N, C)$ satisfying the above boundary conditions and , and moreover we assume $$\\label{derivative}\nN'(\\pm \\infty) = C' (\\pm \\infty) =0.$$ (We denote $\\displaystyle \\lim_{x\\to \\infty} N(x)$ by $N(\\infty)$ in short.)\\\nIn this paper we study the stability of a planar traveling wave solution $(N, C)$ of for ${\\epsilon}\\geq0$ case assuming $\\chi(c)= c^{-1}$, $m =1$, $$\\begin{aligned}\n\\label{eq:main}\n \\begin{aligned}\n{\\partial}_t n - {\\Delta}n& = - {\\nabla}\\cdot \\left(n \\frac{ {\\nabla}c}{c} \\right), \\\\\n{\\partial}_t c-{\\epsilon}{\\Delta}c & = - cn, \\quad (x, y) \\in {\\mathbb R}\\times [0, {\\lambda}].",
"\n\\end{aligned} \\end{aligned}$$\n\n[Main Theorems]{} Let us introduce the weight function: $$w(x) = 1 +e^{sx}, \\quad x\\in {\\mathbb R}$$ and define the Sobolev spaces $H^k$ and $H^k_w$ by their norms: $$\\begin{aligned}\n&\\| \\varphi\\|_{H^k}^2 = \\sum_ { i+ j \\le k}\\int_{{\\mathbb R}\\times [0, {\\lambda}]} | {\\partial}_x^i {\\partial}_y^j \\varphi (x, y) |^2 dxdy, \\quad\\\\\n& \\| \\varphi\\|_{H^k_w}^2 = \\sum_ { i+ j \\le k}\\int_{{\\mathbb R}\\times [0, {\\lambda}]} | {\\partial}_x^i {\\partial}_y^j \\varphi (x, y) |^2 w(x) dxdy, \\quad k = 0, 1, 2 \\dots.",
"\n \\end{aligned}$$ We will assume the periodic boundary condition in $y$-direction for initial perturbation and use the Sobolev spaces $H^k_{p}$ and $H^k_{w, p}$ : $$\\begin{aligned}\n H^k_{p} & := {H^k} ( {\\mathbb R}\\times \\mathbf S^{{\\lambda}}) = \\{ {\\varphi}\\in {H^k} (\\Omega)\\, |\\, \n \\quad \\sum_{i+j\\le k} \n\\sum_{n\\in \\mathbb{Z}}\n \\int_{\\mathbb R}n^{2j}|{\\partial}_x^i \\varphi_n(x)|^2 dx < \\infty \\},\\end{aligned}$$ $$\\begin{aligned}\n H^k_{w,p} & := {H^k_w} ( {\\mathbb R}\\times \\mathbf S^{{\\lambda}}) = \\{ {\\varphi}\\in {H^k_w} (\\Omega)\\, |\\, \n \\quad \\sum_{i+j\\le k} \n\\sum_{n\\in \\mathbb{Z}}\n \\int_{\\mathbb R}n^{2j}|{\\partial}_x^i \\varphi_n(x)|^2 w(x) dx < \\infty \\}\\end{aligned}$$ where $ \\mathbf S^{{\\lambda}}$ is the circle of perimeter ${\\lambda}$ and $ \\varphi_n(x)$ is the $n$th Fourier coefficient of $\\varphi(x, \\cdot_y )$.\\\nIn other words, ${\\varphi}\\in {H^k_{w,p}} $ is a function on $\\mathbb{R}^2$ such that it is ${\\lambda}$-periodic in $y$ and its (weighted in $z$-direction) $H^k$ norm on ${\\Omega}=\\mathbb{R}\\times[0,{\\lambda}]$ is finite. ",
"The cylindrical domain with this periodic boundary condition fits well to the picture that blood vessels are lined with a single layer of endothelial cells. ",
"What it follows we use the equivalence of $\\| \\cdot \\|_{H^k_p}$ and $\\| \\cdot \\|_{H^k_{w,p}}$ norms to $\\| \\cdot \\|_{H^k}$ and $\\| \\cdot \\|_{H^k_{w}}$ norms for functions periodic in $y$.\\\nWe perturb the system around a planar traveling wave solution $(N, C)$ with the front conditions , and in a way that $$\\begin{aligned}\n\\label{ncp}\nn(x, y, t) &= N(x- st) + u (x-st, y, t) \\quad \\mbox{and } \n\\quad \n&c(x, y, t) = C(x- st)e^{- \\psi(x-st, y, t)} \n \\end{aligned}$$ where the traveling wave speed $s= \\sqrt{\n{\\frac{n_-}{1+{\\epsilon}}}}>0\n$. ",
"What it follows, we will use the moving frame $$z=x-st.$$ We consider the above perturbation $(u, \\psi)$ in the following function class.",
"\n\n1. ",
" There exists a vector-valued function $\\varphi\\in (H^3_{w,p})^2$ such that $u = {{\\nabla}\\cdot}\\varphi={\\partial}_z{\\varphi}^1+{\\partial}_y{\\varphi}^2$.\n\n2. ",
" $\\psi$ lies on $H^3_{p}$ with ${\\nabla}\\psi \\in H^2_{w,p}$.\n\nIn the subsection \\[hopf\\], the nonlinear perturbation $({\\varphi},{\\psi})$ is found directly related to the Cole-Hopf transformation.\\\nWe are ready to state the main results of the paper. ",
"The first result is on the nonlinear stability of the planar traveling wave solution $(N, C)$ of the system when ${\\epsilon}=0$.\n\n\\[theoremnc\\] For a given planar traveling wave solution $(N,C)$ of the system for ${\\epsilon}=0$ with , and , there exist constants $m_0>0,\\, C_0>0,$ and ${\\lambda}_0>0$ such that if $0<{\\lambda}\\le {\\lambda}_0$, then for any initial data $(n_0, c_0)$ in the form of $ N - n_0 = {{\\nabla}\\cdot}\\varphi_0$ and $c_0 /C = e^{-\\psi_0} \n $ where ${\\varphi}_0$ and ${\\psi}_0$ are ${\\lambda}-$periodic in $y$ with $M_0:=\\| {\\varphi}_0\\|^2_{H^3_w} + \\| {\\psi}_0\\|^2_{H^3} + \\|\\nabla\\psi_0\\|^2_{H^2_w} \\color{black}\\le m_0$, there exist a unique global classical solution $(n,c)$ of for ${\\epsilon}=0$ in the form of $$n(x, y, t) = N(x- st) + {{\\nabla}\\cdot}\\varphi (x-st, y, t), \n\\quad \nc(x, y, t) = C(x- st)e^{- \\psi(x-st, y, t)},$$ where ${\\varphi}$ and ${\\psi}$ are ${\\lambda}-$periodic in $y$ for each $t>0$ and $ \\varphi|_{t=0} = \\varphi_0$ and $\\psi|_{t=0} = \\psi_0$. Moreover, $ (\\phi, \\psi)$ satisfies the following inequality: $$\\begin{aligned}\n\\label{energyineq} \\sup_{t\\in [0, \\infty)}\\Big(\\| \\varphi\\|^2_{H^3_w} + \\| \\psi\\|^2_{H^3} + \\| {\\nabla}\\psi \\|^2_{H^2_w}\\Big) \n+\\int_0^{\\infty} \\Big( \\| {\\nabla}{\\varphi}\\|_{H^3_w}^2\n+ \\| {\\nabla}{\\psi}\\|_{H^2_w}^2\n\\Big)dt\n \\le C_0M_0.",
"\n \\end{aligned}$$\n\n1. ",
" The condition $ u= {{\\nabla}\\cdot}{\\varphi}$ implies the zero integral condition on $u$, $\\int u dzdy = 0$. Equivalently, $n$ and $N$ satisfy $$\\int n(x, y, t) - N(x-st) dxdy = 0,$$ which is preserved over time, provided it holds at $t=0$ by the mass conservation property of $n$ equation.",
"\n\n2. ",
" The weight function $w$ is of a same order as $1/N$ in $z>0$ direction. ",
"More precisely, for a fixed traveling wave, there exists a constant $M$ satisfying $$\\frac{w}{M}\\leq\\frac{1}{N}\\leq Mw,$$ which is proved in Section $2.4$.\n\n3. ",
" [ Theorem \\[theoremnc\\] implies the asymptotic convergence of the primary perturbation of $(n, c)$ to $(N, C)$. From $$\\int_0^{\\infty} \\Big( \\| {\\nabla}{\\varphi}\\|_{H^3_w}^2\n + \\| {\\nabla}{\\psi}\\|_{H^2_w}^2\n \\Big)dt\n \\le C_0M_0,$$ we get a weak asymptotical estimate $$\\liminf_{t \\rightarrow \\infty} \\Big( \\| {\\nabla}{\\varphi}(t)\\|_{H^3_w}^2\n + \\| {\\nabla}{\\psi}(t)\\|_{H^2_w}^2\n \\Big) \n = 0.$$ In terms of $n(\\cdot_z +st )-N={{\\nabla}\\cdot}{\\varphi}$ and $c(\\cdot_z +st)/C=e^{-\\psi}$, we have $$\\liminf_{t \\rightarrow \\infty} \\Big( \\|n(\\cdot_z +st,\\cdot_y,t)-N(\\cdot_z) \\|_{H^3_w}^2\n + \\| \\nabla\\big( \n \\log c (\\cdot_z +st,\\cdot_y,t) - \\log C(\\cdot_z)\n \\big) \\|_{H^2_w}^2\n \\Big)\n = 0.$$]{}\n\n4. ",
" The one-sided decay appears naturally with respect to the solvability of ${{\\nabla}\\cdot}{\\varphi}= u$ in the infinite strip domain $\\Omega$ [@Sol]. [",
"^2] Indeed, if there is a vector function ${\\varphi}\\in L^2({\\Omega})$ satisfying the divergence equation, assuming Dirichlet or the periodic boundary condition on the lateral boundary, we have $$\\int_{z> x} \\int u(z, y) dz dy= \\int_{z> x} \\int_{[0, {\\lambda}]} {{\\nabla}\\cdot}{\\varphi}dz dy = -\\int_{[0, {\\lambda}]} {\\varphi}^1( x, y) dy.$$ Define $ {\\mathcal U}(x) = \\int_{z> x}\\int u dy dz $, then ${\\mathcal U}(x)$ is found in $L^2([0, \\infty))$ because $$\\int_0^{\\infty} {\\mathcal U}^2(x) dx = \\int_0^{\\infty} {\\left( \\int _{[0, {\\lambda}]} {\\varphi}^1 (x, y) dy \\right)}^2 dx \\le\n \\int _0 ^{\\infty} {\\lambda}\\int _{[0, {\\lambda}]} |{\\varphi}^1|^2(x, y) dy dx = {\\lambda}\\| {\\varphi}^1\\|_{L^2}^2.$$ In this paper, we do not pursuit solving ${{\\nabla}\\cdot}{\\varphi}= u$ for a given perturbation $u$. Instead we are given ${\\varphi}\\in H^3_w$. Note that the condition $ \\| {\\varphi}\\|_{H^3_w} < \\infty$ implies $ wu \\in L^{\\infty}$, which is enough for $u$ to satisfy $\\int_0^{\\infty} {\\mathcal U}^2 dx < \\infty$ and to be consistent with the necessary condtion for solvability. ",
"We refer to [@Sol] for the solvability of the divergence equation on the more general noncompact domain.",
"\n\nWhen the chemical diffusion is present $({\\epsilon}>0)$, the above result of Theorem \\[theoremnc\\] is hard to obtain for technical reasons. ",
"Instead, we obtain a linear stability of the planar traveling wave solution $(N, C)$ of the system when ${\\epsilon}>0$ for perturbations in a smaller class(mean-zero in $y$). ",
"Indeed, as will be derived in Section \\[hopf\\], the pertubation $(\\varphi, \\psi)$ satisfies $$\\begin{aligned}\n\\label{eq0:nonlinear}\\begin{aligned}\n\\varphi_t - s \\varphi_z - {\\Delta}\\varphi\n& = N {\\nabla}\\psi + P{\\nabla}\\cdot \\varphi + {\\nabla}\\cdot \\varphi {\\nabla}\\psi, \\\\\n{\\psi}_t - s {\\psi}_z -{\\epsilon}{\\Delta}{\\psi}&= -2{\\epsilon}P \\cdot{\\nabla}{\\psi}-{\\epsilon}|{\\nabla}{\\psi}|^2 + {{\\nabla}\\cdot}\\varphi\n\\end{aligned}\\end{aligned}$$ [where $P :=-(C'/C, 0)$]{}. ",
"Quadratic terms dropped out, the linearized system of $({\\varphi}, {\\psi})$ is as follows: $$\\begin{aligned}\n\\label{eq0:linear} \\begin{aligned}\n\\varphi_t - s \\varphi_z - {\\Delta}\\varphi\n& = N {\\nabla}\\psi + P{\\nabla}\\cdot \\varphi, \\\\\n{\\psi}_t - s {\\psi}_z -{\\epsilon}{\\Delta}{\\psi}&= -2{\\epsilon}P \\cdot{\\nabla}{\\psi}+ {{\\nabla}\\cdot}\\varphi.",
"\n\\end{aligned}\\end{aligned}$$ What it follows we denote $\\bar \\phi (z) = \\int_0^{{\\lambda}} \\phi(z, y) dy$.\n\n\\[theoremli\\] For sufficiently small ${\\epsilon}>0$ and for any given planar traveling wave solution $(N,C)$ of the system with , and , the wave $(N,C)$ is linearly stable in the following sense:\\\nThere exist constants $ C_0>0,$ and ${\\lambda}_0>0$ such that if $0<{\\lambda}\\le {\\lambda}_0$, then for any initial data $({\\varphi}_0, {\\psi}_0)$ which is ${\\lambda}-$periodic in $y$ with $$\\bar{\\varphi}_0= \\bar {\\psi}_0 =0 \\mbox{ and } M_0:=\\| {\\varphi}_0\\|^2_{H^3_w} + \\| {\\psi}_0\\|^2_{H^3} + \\|\\nabla\\psi_0\\|^2_{H^2_w} \\color{black}<\\infty,$$ there exists a unique global classical solution $({\\varphi}, {\\psi})$ of which is ${\\lambda}-$periodic in $y$ and $\\bar{\\varphi}= \\bar {\\psi}=0$ for each $t>0$ and which satisfies the inequality: $$\\begin{aligned}\n \\label{energyineqwith} \\sup_{t\\in [0, \\infty)}\\Big(\\| \\varphi\\|^2_{H^3_w} + \\| \\psi\\|^2_{H^3} + \\| {\\nabla}\\psi \\|^2_{H^2_w}\\Big) \n+\\int_0^{\\infty} \\Big( \\| {\\nabla}{\\varphi}\\|_{H^3_w}^2\n+ \\| {\\nabla}{\\psi}\\|_{H^2_w}^2\n{+{\\epsilon}\\| {\\nabla}^{4} {\\psi}\\|_{H^0_w}^2} \\Big)dt\n \\le C_0M_0.\\end{aligned}$$\n\nThe next theorem gives a hint why studying planar traveling waves is natural instead of studying general traveling waves in ${\\Omega}$. Roughly speaking, any smooth bounded solution with our front boundary conditions , becomes eventually planar under some strong regularity conditions when ${\\epsilon}>0$. This kind of nonlinear stability can be found in [@Const] on the stability of a reactive Boussinesq system in an infinite vertical strip. ",
"It also hints the asymptotic stability as in Theorem \\[theoremnc\\] might hold as well in a nonzero diffusion case. ",
"We state this theorem in terms of $(n, q)$ in which will be obtained in the subsection \\[hopf\\] by applying the Cole-Hopf transformation $q:=-\\frac{\\nabla c}{c}$ to .",
"\n\n\\[thm\\_eventual\\] Let ${\\epsilon}>0$ and $(n, q)$ be a global smooth solution of\n\n[ It implies $ \n\\lim_{t \\rightarrow \\infty} \\Big( \\| n_y\\|_{L^2} + \\| q_y \\|_{L^2}\n\\Big) \n= 0.$]{}\n\nThe remaining parts of the paper are organized as follows. ",
"In Section \\[backg\\], we introduce the backgroud materials including the existence of traveling wave solutions and the Cole-Hopf trasformation. ",
"In Section \\[sectionuniform\\], we prove Proposition \\[zerolocal\\] and \\[zeroglobal\\], then establish Theorem \\[theoremnc\\]. ",
"In Section \\[sectionuniformwith\\], we prove the energy inequality in Theorem \\[theoremli\\]. ",
"With the existence part of Theorem \\[theoremli\\] follows the same argument that works for the ${\\epsilon}=0$ case, therefore we omitted. ",
"In the last section, we proves Theorem \\[thm\\_eventual\\] which points out a nonlinear stability on a thin cylindrical domain for ${\\epsilon}>0$ case would be expected as well.\\\nThere are many prior results on the existence of traveling wave solution and its stability. ",
"Among earlier analytic works is [@NaI], where they proved the existence and linear instability of traveling wave solution of the one dimensional system when the bacterial consumption rate is constant and $\\chi(c) = 1/c$. The existence of the traveling wave solution with the front conditions and was shown in [@WaHi] when ${\\epsilon}=0$, and [@LiWa; @Wa] when ${\\epsilon}>0$. In the one dimensional system, the nonlinear asymptotic stability results were shown under the aforementioned restrictions on $\\chi(c)$ and $m$ in [@JinLiWa] when ${\\epsilon}=0$, and [@LiWa] when ${\\epsilon}>0$ is small. ",
"To our knowledge, our Theorem \\[theoremnc\\] is the first result on a nonlinear stability of traveling wave solutions in a higher dimension.\\\nFor the results on the Cauchy problem of , see [@CPZ1; @CPZ2; @FoFr; @FrTe; @LiLi], where [@CPZ1; @CPZ2] prove the existence of a global weak solution, and [@LiLi] proves that of a global classical solution. ",
"Both results consider the zero chemical diffusion case in a multi-dimension.\\\n\nBackground {#backg}\n==========\n\n[Existence of traveling wave solutions]{}\\[existence\\_traveling\\] The traveling wave solution $(N, C)$ in solves the following ODE system, $$\\begin{aligned}\n \\begin{aligned} \\label{NC}\n-sN' - N'' = - ( \\chi(C) C' N)',\\\\\n-s C' - {\\epsilon}C'' = - C^m N.\n\\end{aligned} \\end{aligned}$$ It is easy to see the above system integrable. ",
"We briefly discuss its solvability and properties of the solution for reader’s convenience. ",
"What it follows in this subsection has been already well known (for instance in [@LiLiWa; @Wa]).",
"\n\nLet $H'(\\cdot) = \\chi(\\cdot)$. We first solve $N$ by $$N(z) = N_0 e^{-sz} e^{H(C)}$$ for some positive constant $N_0$, [which means a translation by $z_0$ when $N_0=e^{s z_0}$.]{} The system with the front conditions and is translation invariant. ",
"It causes the undetermined constant $N_0$.\n\nTo satisfy the front condition , we impose $e^{H(C)} \\to 0$ as $z \\to -\\infty$, which implies $H(\\cdot)$ is singular at zero such as $\\lim_{C \\to 0} H(C) = -\\infty$. Substituting $N$ to the C equation, we have $$sC' + {\\epsilon}C'' = C^m e^{-sz} e^{H(C)} {N_0}.$$ What it follows we let $$H(C) = \\ln C, \\quad m=1.$$ We result in $$\\begin{aligned}\n\\label{NC_copy}\nN= e^{-sz} C {N_0}, \\quad sC' + {\\epsilon}C'' = e^{-sz}C^2 {N_0}.\\end{aligned}$$ By introducing $$W = e^{-sz}C$$ we have a KPP-type equation for $W$: $$\\label{W}\n {\\epsilon}W'' + (s + 2s{\\epsilon}) W' = - ({\\epsilon}+1)s^2 W +{N_0}W^2: = -f(W).$$ Since $ N= WN_0$, front conditions , force $W$ to have $$\\lim_{z \\to -\\infty} W (z) = \\frac{1}{N_0}n_- \\quad \\lim_{z \\to \\infty} W (z) = 0.$$ It is well known that the KPP- fisher equation has a nonnegative solution if and only if $(s+2{\\epsilon}s) \\ge 2\\sqrt{{\\epsilon}({\\epsilon}+1)}s$, which automatically holds in this case. ",
"More precisely we refer to the following lemma in [@Wa], which is based on the standard argument for the Fisher equation in [@LeNe; @Mu] etc.",
"\n\n\\[KPPtheorem\\]\\[Wang, Lemma 3.2.\\] A nonnegative traveling wave solution $W$ of exists. ",
"It satisfies $W' <0,$ $$W_- = \\frac{(1+{\\epsilon})s^2}{N_0}, \\mbox{ and } \\quad W_+ = 0.$$ Moreover it is unique up to a translation in $x$, or $t$ and has the following asymptotic behavior: $$W(x) - \\frac{({\\epsilon}+1)s^2}{N_0} \\sim Ce^ {{\\lambda}x} \\mbox{ as } x \\to -\\infty \\mbox{ and } \\quad \nW(x) \\sim (e^{-sx}) \\mbox{ as } x \\to \\infty$$ with $ {\\lambda}=\\frac{-s(1+ 2{\\epsilon}) + s \\sqrt{ (1+2{\\epsilon})^2 + 4{\\epsilon}(1+{\\epsilon})}}{2{\\epsilon}}$ when ${\\epsilon}>0$ and ${\\lambda}= -s$ when ${\\epsilon}=0$.\n\nTo be consistent with Lemma \\[KPPtheorem\\], the wave speed $s$ is uniquely determined by $n_-$, $$s^2 = \\frac {n_-}{1+{\\epsilon}}.$$ Also from $(C'-sC) = e^{sz} W' < 0$ and the boundary condition $C(-\\infty)= C'(-\\infty)=0$, we have $$W' (-\\infty) =0.$$ We convert back $N$, $C$ satisfying from $W$ with the required front condition , and . ",
"The monotonicity of $N, C$ follows from $W' < 0$; the profile of $N$ is decreasing, and $C$ is increasing left to rightward.",
"\n\n\n\nIn the zero diffusion (${\\epsilon}=0$) we can explicitly solve $$sC' = e^{-sz} C^2 N_0$$ such that $$C = \\frac{c_+}{ c_+ \\frac{N_0}{s^2} e^{-sz} +1}.$$ Then $$N = \\frac{c_+ N_0 e^{-sz}}{c_+ \\frac{N_0}{s^2} e^{-sz} +1}.$$ We have the relation $ n_- = s^2$ and the monotone profile of $(N, C)$.\n\n[Cole-Hopf transformation]{}\\[hopf\\]\n\nBy the Cole-Hopf transformation $$q = - {\\nabla}\\ln c = -\\frac{{\\nabla}c}{c}$$ We translate into the divergence type equations of $(n, q)$ $$\\begin{aligned}\n\\label{nq} \\begin{aligned}\n{\\partial}_t n - {\\Delta}n& = {\\nabla}\\cdot (nq), \\\\\n{\\partial}_t q - {\\epsilon}{\\Delta}q & = -2{\\epsilon}(q \\cdot {\\nabla}) q + {\\nabla}n,\n\\end{aligned} \\end{aligned}$$ with $ ((q\\cdot {\\nabla}) q )_i = \\sum_{k=1,2} q_k {\\partial}_k q_i$. This works only for $m=1$ in . ",
"The Cole-Hopf transformation is initiated in [@LiWa] for the stability problem of the traveling wave equation in the one-dimensional angiogenesis. ",
"It appears in [@LiLi] studying the multi-dimensional angiogenesis equation for ${\\epsilon}=0$ with vanishing boundary conditions, too.\\\nWe shall add the condition $$\\label{curl} {\\nabla}\\times q = {\\partial}_2q_1 - {\\partial}_1 q_2 = 0.$$ This curl free condition on $q$ is preserved in time until the classical local solution $(n, q)$ exists. ",
"It holds that $$\\label{na}\n q \\cdot {\\nabla}q = \\frac 12 {\\nabla}|q|^2$$ with . ",
"We denote $$\\label{CP} P=({\\mathcal P},0)= -(C'/C, 0)$$ then the perturbation around $(N, C)$ is written by $$\\begin{aligned}\n\\label{up} \nn(x, y, t) = N(x- st) + u (x-st, y, t), \n\\quad \nq(x, y, t) = P(x- st) + {\\nabla}\\psi (x-st, y, t).\\end{aligned}$$ Note that $N, P(:= ({\\mathcal P}, 0)) $ solves $$\\begin{aligned}\n\\label{np} \\begin{aligned}\n- sN'- N''& = (N{\\mathcal P})', \\\\\n-s{\\mathcal P}' - {\\epsilon}{\\mathcal P}''& = -2{\\epsilon}{\\mathcal P}{\\mathcal P}'+ N'\n\\end{aligned} \\end{aligned}$$ The boundary conditions of $(N, P)$ are inherited from those of $(N, C)$: $$\\begin{aligned}\n\\label{np_copy_bd} \\begin{aligned}\n & N(-\\infty) = (1+{\\epsilon}) s^2, N(\\infty) =0, \\quad {\\mathcal P}(-\\infty) = -s , {\\mathcal P}(\\infty)=0,\\\\\n & N'( \\pm \\infty) = 0, \\quad P'(\\pm \\infty) =0.",
"\n \\end{aligned}\\end{aligned}$$ For a while we denote $${\\nabla}\\psi = p.$$ So the curl-free condition is endowed to $p$. The perturbation $(u, p)$ satisfies $$\\begin{aligned}\n\\label{perturb} \\begin{aligned}\n&u_t - s u_z - {\\Delta}u\n= {\\nabla}\\cdot ( Np + Pu+ up ),\\\\\n&p_t - s p_z - {\\epsilon}{\\Delta}p = -2{\\epsilon}\\left( ( (P+p )\\cdot {\\nabla}) (P+p) - (P\\cdot {\\nabla}) P \\right) + {\\nabla}u.",
"\n\\end{aligned}\\end{aligned}$$\n\nIf we let $$\\begin{aligned}\n u = {\\nabla}\\cdot \\varphi, \\quad p = {\\nabla}\\psi,\n \\end{aligned}$$ the equation is put into the equation on $(\\varphi, \\psi)$ $$\\begin{aligned}\n\\label{eq0:phipsi_with_ep} \\begin{aligned}\n\\varphi_t - s \\varphi_z - {\\Delta}\\varphi\n& = N {\\nabla}\\psi + P{\\nabla}\\cdot \\varphi + {\\nabla}\\cdot \\varphi {\\nabla}\\psi, \\\\\n{\\psi}_t - s {\\psi}_z -{\\epsilon}{\\Delta}{\\psi}&= -2{\\epsilon}P \\cdot{\\nabla}{\\psi}-{\\epsilon}|{\\nabla}{\\psi}|^2 + {{\\nabla}\\cdot}\\varphi.",
"\n\\end{aligned}\\end{aligned}$$\n\nMain proposition for Theorem \\[theoremnc\\]\n------------------------------------------\n\nFrom now on, we fix a planar traveling wave solution $(N,C)$ of the system for ${\\epsilon}=0$ with , and , obtained in Section \\[existence\\_traveling\\]. ",
"The pertubation $({\\varphi}, {\\psi})$ satisfies the system $$\\begin{aligned}\n\\label{eq0:phipsi} \\begin{aligned}\n\\varphi_t - s \\varphi_z - {\\Delta}\\varphi\n& = N {\\nabla}\\psi + P{\\nabla}\\cdot \\varphi + {\\nabla}\\cdot \\varphi {\\nabla}\\psi, \\\\\n{\\psi}_t - s {\\psi}_z &= {{\\nabla}\\cdot}\\varphi\n\\end{aligned}\\end{aligned}$$ for $ (t, z, y) \\in [0, \\infty)\\times {\\mathbb R}\\times [0, {\\lambda}]$ where $P=-(C'/C,0)$ as in . ",
"In addition, recall the function spaces and the periodic condition (in $y$-direction) for $({\\varphi},{\\psi})$ which are discussed prior to the main Theorem \\[theoremnc\\].\\\nFirst we state a result on the local existence of solutions for the system which can be proved in the standard manner. ",
"The proof is presented at the end of the section 3 (Subsection \\[pf\\_local\\_exist\\]).",
"\n\n\\[zerolocal\\] For ${\\lambda}>0$ and $M>0$, there exists $T_0>0$ such that for any initial data $ (\\varphi_0, \\psi_0)$ which is ${\\lambda}-$periodic in $y$ with $ \\| \\varphi_0\\|^2_{H^3_w} + \\| \\psi_0\\|^2_{H^3} + \\|{\\nabla}\\psi_0\\|^2_{H^2_w} < M $, the system has a unique solution $({\\varphi}, \\psi)$ on $[0, T_0]$ which is ${\\lambda}-$periodic in $y$ for each $t\\in[0,T_0]$ with $ \\varphi|_{t=0} = \\varphi_0$ and $\\psi|_{t=0} = \\psi_0$ and $$\\varphi \\in L^{\\infty}(0, T_0; H^3_w), \\quad \\psi \\in L^{\\infty}(0, T_0; H^3), \\quad \\nabla\\psi \\in L^{\\infty}(0, T_0; H^2_w) \\color{black} .$$ Moreover it holds the following inequality: $$\\label{double}\n \\sup_{t\\in [0, T_0]}\\Big( \\| \\varphi\\|^2_{H^3_w}+ \\| \\psi \\|^2_{H^3 }+\\| {\\nabla}\\psi\\|^2_{H^2_w}\\Big) \\le 2 M.$$\n\nLet us state the result on the existence of global classical solution to the system . ",
"Due to our derivation of from , the main Theorem \\[theoremnc\\] is the direct consequence of Proposition \\[zeroglobal\\], which will be proved in Section $3$.\n\n\\[zeroglobal\\] There exist constants $m_0>0,\\, C_0>0$, and ${\\lambda}_0>0$ such that for $0<{\\lambda}\\le {\\lambda}_0$ and for any initial data $({\\varphi}_0, {\\psi}_0)$ which is ${\\lambda}-$periodic in $y$ with $M_0:=\\| {\\varphi}_0\\|^2_{H^3_w} + \\| {\\psi}_0\\|^2_{H^3} + \\|\\nabla\\psi_0\\|^2_{H^2_w} \\color{black}\\le m_0$, there exists a unique global classical solution $({\\varphi},{\\psi})$ of which is ${\\lambda}-$periodic in $y$ for each $t>0$ with $ \\varphi|_{t=0} = \\varphi_0$ and $\\psi|_{t=0} = \\psi_0$. Moreover, $ (\\phi, \\psi)$ satisfies the inequality $$\\sup_{t\\in [0, \\infty)}\\Big(\\| \\varphi\\|^2_{H^3_w} + \\| \\psi\\|^2_{H^3} + \\| {\\nabla}\\psi \\|^2_{H^2_w}\\Big) \n+\\int_0^{\\infty} \\Big( \\| {\\nabla}{\\varphi}\\|_{H^3_w}^2\n+ \\| {\\nabla}{\\psi}\\|_{H^2_w}^2\n \\Big)dt\n \\le C_0M_0.$$\n\nIn the below we summarize the notations used in the paper. ",
"$$w(z) : = 1 + e^{sz},$$ $$M(t):= \\sup_{ s\\in [0, t]}( \\| \\varphi(s)\\|^2_{H^3_w} \n+ \\|\\psi(s)\\|^2_{H^3} +\\|{\\nabla}\\psi(s)\\|^2_{H^2_w}),$$ $$M_0:= ( \\| \\varphi_0\\|^2_{H^3_w} + \\|\\psi_0\\|^2_{H^3}+\\|{\\nabla}\\psi_0\\|^2_{H^2_w}),$$ $$\\| f\\|:=\\| f\\|_{L^2({\\Omega})},$$ $$\\| f\\|^2_{k}:=\\| f\\|^2_{H^k}=\\sum_{|\\alpha|=0}^k\\int_\\Omega|D^\\alpha f|^2\\,dzdy,$$ $$\\| f\\|^2_{k,w}:=\\| f\\|^2_{H^k_w}=\\sum_{|\\alpha|=0}^k\\int_\\Omega|D^\\alpha f(z,y)|^2\\,w(z)dzdy,$$ $$\\int f:=\\int_\\Omega f(z,y) dzdy,$$ $$\\int_0^t g:=\\int_0^t g(s) ds.$$\n\nUniform in time estimate when ${\\epsilon}=0$ {#sectionuniform}\n============================================\n\nIn this section we introduce the main proposition \\[uniform\\_\\] below which implies Proposition \\[zeroglobal\\].",
"\n\n\\[uniform\\_\\] There exist constants $\\delta_0>0,\\, C_0$, and ${\\lambda}_0>0$ such that for ${\\lambda}\\in(0,{\\lambda}_0]$, we have the following:\\\nIf $(\\varphi, \\psi)$ be a local solution of on $[0,T]$ for some $T>0$ with $$M(T)\\leq\\delta\\quad \\mbox{(the bootstrap assumption)}$$ [for some initial data $(\\varphi_0, \\psi_0)$ which is ${\\lambda}$-periodic in $y$]{}, then we have $$\\label{improve}\n M(T) + \\int_0^{T} \\sum_{l = 1}^4 \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^{T} \\sum_{l = 1}^3\\| {\\nabla}^l {\\psi}\\|_w^2 \\le C_0M_0.$$\n\nThis proposition will be proved in Subsection \\[mainproof\\].",
"\n\nNote that the constant $C_0$ in the above proposition does not depend on the size of $T>0$. This fact immediately improves the bootstrap assumption if we take a sufficiently small initial data, for instance, satisfying $C_0M_0<\\delta_0/2$. Once we prove the proposition, then the global-existence statement in Proposition \\[zeroglobal\\] follows the standard continuation argument as shown below:\n\nLet $M:=\\delta_0/2, \\, m_0:=M/C_0$ where $\\delta_0$ and $C_0$ are the constants in Proposition \\[uniform\\_\\]. ",
"We may assume $C_0\\ge 1$. Consider the initial data $({\\varphi}_0, {\\psi}_0)$ with $M(0)=M_0\\leq m_0\\leq M$. By using this constant $M$ to the local-existence result (Proposition \\[zerolocal\\]), there exist $T_0>0$ and the unique local solution $({\\varphi}, {\\psi})$ on $[0,T_0]$ with $M(T_0)\\leq 2M$. Due to $M(T_0)\\leq2M\\leq \\delta_0$, we can use the result of Proposition \\[uniform\\_\\] to obtain $M(T_0)\\leq C_0 M_0$, which implies $M(T_0)\\leq C_0 m_0\\leq M$. Hence we can extend the solution up to the time $2T_0$ by Proposition \\[zerolocal\\] and we obtain $M(2T_0) \\leq 2M\\leq \\delta_0$. Due to Proposition \\[uniform\\_\\], it implies $M(2T_0)\\leq C_0M_0\\leq M$. Thus we can repeat this process of extension to get $M(kT_0)\\leq C_0M_0$ for any $k\\in\\mathbb{N}$.\n\nFirst we summarize some properties of traveling waves $(N, \\mathcal P)$ for later analysis.",
"\n\n[Properties on $(N, \\mathcal P)$]{} From the equations: $$\\begin{aligned}\n\\label{np_copy} \\begin{aligned}\n- sN'- N''& = (N{\\mathcal P})', \\\\\n-s{\\mathcal P}' & = N',\n\\end{aligned} \\end{aligned}$$ we observe the relations $$\\label{relation}\n {\\frac{ {\\mathcal P}}{N}} = -\\frac{1}{s} \\quad \\mbox{and } \\left( {\\frac{ 1}{N}} \\right)^{'' }= s\\left( {\\frac{ 1}{N}} \\right)^{'}$$ from where $s>0$, which is fixed throughout this paper, is the speed of our planar traveling wave $(N,C)$. In fact, $(N, P)$ has the explicit formula: $$\\label{formula:NP} N = \\frac{c_+ N_0 e^{-sz}}{c_+ \\frac{N_0}{s^2} e^{-sz} +1}, \\quad P = - \\left ( \\frac{c_+ N_0e^{-sz}}{ c_+ \\frac{N_0}{s} e^{-sz} +s}, 0 \\right).$$ Note that $N$ and ${\\mathcal P}$ are monotone with the boundary condition . ",
"In the following lemma, we summarize some properties of $(N,{\\mathcal P})$.\n\n\\[wave\\_prop\\] There exists a constant $M>0$ such that we have:\\\n$$\\frac{w}{M}\\leq\\frac{1}{N }\\leq Mw,$$ $$|{N^{ (k)}}| \\leq M, \\quad |{{\\mathcal P}^{(k)}}| \\leq M,\\quad \\mbox{ for } 0\\leq k\\le 2,\\quad \\mbox{ and }$$ $$\\frac{N'}{N}=-({\\mathcal P}+s), \\quad \n\\Big|(\\frac{1}{N})'\\Big|+\\Big|(\\frac{1}{N})''\n\\Big|\\leq \\frac{M}{N},\\quad \n\\Big|(\\frac{1}{\\sqrt {N}})'\\Big|\\leq \\frac{M}{\\sqrt {N}}.$$ ($N^{(k)}$ is any $k-$th derivative of $N$)\n\nThe first inequality follows from Lemma \\[KPPtheorem\\]. ",
"The others are easily obtained from the equation and .",
"\n\n[Proof of Proposition \\[uniform\\_\\]]{}\\[mainproof\\]\n\n[ First assume $0<\\delta_0\\leq\n 1$. During the proof, we will take $\\delta_0$ as small as needed to have the weighted energy estimate (e.g. Lemma \\[lemma1\\_2\\]).]{} ",
"Let us remind the system $$\\begin{aligned}\n\\label{eq:phipsi} \\begin{aligned}\n\\varphi_t - s \\varphi_z - {\\Delta}\\varphi\n &= N\n {\\nabla}\\psi + P\n {\\nabla}\\cdot \\varphi + {\\nabla}\\cdot \\varphi {\\nabla}\\psi, \\\\\n{\\psi}_t - s {\\psi}_z &= {{\\nabla}\\cdot}\\varphi \\\\ \n\\end{aligned}\\end{aligned}$$ for $ (t, z, y) \\in [0,T]\\times {\\mathbb R}\\times [0, {\\lambda}]$. We will collect a few lemmas on energy estimates before proving Proposition \\[uniform\\_\\]. ",
"Here the constants $C$ which will appear in the following lemmas are independent of $T>0$.\n\n\\[lemma0\\_\\] If ${\\lambda}>0$ is sufficiently small, we have the following: $$\\label{eq_lemma0_}\n\\| \\psi \\|^2 + \\| \\varphi\\|_w^2 + \\int_0^t \\| {\\nabla}\\varphi\\|_w^2\n\\le C ( \\| \\psi_0\\|^2 + \\| \\varphi_0\\|_w^2 )+ C {{M(t)}} \\int_0^t \n \\|{\\nabla}\\psi\\|_w^2.",
"\n$$\n\nThis estimate is not closed due to the second term in the right-hand side. ",
"This obstacle will be studied sequentially in the next lemmas.",
"\n\nMultiply ${\\frac{ \\varphi}{N}}$ to the $\\varphi$ equation and $\\psi$ to the $\\psi$ equation. ",
"Integrating by parts, we have $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}\\left( \\int {\\frac{ |\\varphi|^2}{N}} +\\int |\\psi|^2 \\right) \n+ \\int{\\frac{ \\sum_{i} |{\\nabla}\\varphi^i|^2}{N}} + \n\\frac s 2 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'\\\\\n&= \\frac 12 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)''+\n\\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1 {\\nabla}\\cdot \\varphi \n+ \\int {\\frac{ \\varphi \\cdot {\\nabla}\\psi}{N}} {\\nabla}\\cdot \\varphi.",
"\n$$ We estimate the cubic term first: $$\\begin{aligned}\n \\int {\\frac{ \\varphi \\cdot {\\nabla}\\psi}{N}} {\\nabla}\\cdot \\varphi & \\le { C\\int {\\frac{ \\| {\\varphi}\\|_{L^{\\infty}}^2| {\\nabla}\\psi|^2}{N}} \n + \\frac{1}{4} \\int {\\frac{ |{\\nabla}{\\varphi}|^2}{N}} \\le C {M(t)} \\int {\\frac{ | {\\nabla}\\psi|^2}{N}} + \\frac{1}{4}\\int {\\frac{ |{\\nabla}{\\varphi}|^2}{N}}\n }\\end{aligned}$$ where we used $\\|{\\varphi}\\|_{L^\\infty}\\leq C \\|{\\varphi}\\|_{H^2}\\leq C \\sqrt{M(t)}$ by the Sobolev embedding.",
"\n\nThanks to the relations , the quadratic term becomes $$\\begin{aligned}\n\\label{cancelation}\n \\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1{\\nabla}\\cdot \\varphi & =\n-\\frac 1s \\int \\varphi^1({\\partial}_z{\\varphi}^1+{\\partial}_y{\\varphi}^2)\n=-\\frac 1s \\int ({\\varphi}^1- \\overline{{\\varphi}}^1){\\partial}_y {\\varphi}^2 \n \\end{aligned}$$ where $\\overline{{\\varphi}}(z)=(1/{\\lambda})\\int_{[0,{\\lambda}]}{\\varphi}(z,y)dy$ is the average in $y$ of ${\\varphi}$ for each $z$. Note that we used the periodic condition for ${\\varphi}^2$. As a result, we get $$\\begin{aligned}\n \\Big|\\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1{\\nabla}\\cdot \\varphi\\Big| & \n \\le C\\frac {{\\lambda}}{s} \\| {\\partial}_y{\\varphi}^1\\|_2 \\| {\\partial}_y{\\varphi}^2\\|_2 \\le C {\\lambda}\\| {\\nabla}{\\varphi}\\|^2_w\\\\\n \\end{aligned}$$ where we used the Poincaré inequality $$\\label{ineq:poincare2_periodic} \n\\| {\\varphi}(z,\\cdot_y)-\\overline{{\\varphi}}(z) \\|_{L^2_y([0,{\\lambda}])} \\le C{\\lambda}\\| {\\partial}_{y} {\\varphi}(z,\\cdot_y)\\|_{L^2_y([0,{\\lambda}])} \\quad\\mbox{ for } z\\in\\mathbb{R}.$$ Also we have $${\\frac s 2 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'- \\frac 12 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)''} { = 0}$$ thanks to the relations . ",
"By assuming smallness of ${\\lambda}$, we get the following: $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}\\left( \\int {\\frac{ |\\varphi|^2}{N}} +\\int |\\psi|^2 \\right) +\\frac 18 \\int{\\frac{ |{\\nabla}\\varphi|^2}{N}} \n \\leq C {{M(t)}} \\int {\\frac{ | {\\nabla}\\psi|^2}{N}}.\\end{aligned}$$ It proves Lemma \\[lemma0\\_\\].\\\n\nFrom now on we study the diffusion case(${\\epsilon}>0$). ",
"Note that we do not have the simple relation . ",
"Instead, we estimate the quadratic term $\\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1{\\nabla}\\cdot \\varphi$ by $$\\begin{aligned}\n \\Big|\\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1{\\nabla}\\cdot \\varphi\\Big| & \\le \\| {\\varphi}^1/\\sqrt{N}\\| \\| {\\nabla}{\\varphi}/\\sqrt{N}\\|\n \\le C {\\lambda}\\| {\\nabla}{\\varphi}\\|^2_w,\\\\\\end{aligned}$$ where we used $|{\\mathcal P}|\\leq C\n$ (lemma \\[wave\\_prop\\]) with the Dirichlet condition for ${\\varphi}$ (${\\varphi}(z,y)=0$ when $y=0$ or $y={\\lambda}$) in order to use the Poincaré inequality $$\\label{ineq:poincare2} \n\\| {\\varphi}(z,\\cdot_y) \\|_{L^2_y([0,{\\lambda}])} \\le C{\\lambda}\\| {\\partial}_{y} {\\varphi}(z,\\cdot_y)\\|_{L^2_y([0,{\\lambda}])} \\quad\\mbox{ for } z\\in\\mathbb{R}.$$ Also we estimate $$\\begin{aligned}\n&\\frac 12 \\int |{\\varphi}|^2 {\\left( {\\frac{ 1}{N}} \\right)}''-\\frac s 2 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'\n\\le -\\int {\\varphi}\\cdot{\\varphi}_z \\left( {\\frac{ 1}{N}}\\right)'+\\frac s 2 \\int |\\varphi|^2 \\left|\\left( {\\frac{ 1}{N}}\\right)'\\right|\\\\\n&\\le C\\int |{\\varphi}||{\\varphi}_z | \\left( {\\frac{ 1}{N}}\\right) +\\frac {Cs}{2} \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right) \n\\le \\frac{1}{4}\\int |{\\varphi}_z |^2\\left( {\\frac{ 1}{N}}\\right)+(C+\\frac {Cs}{2} ) \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)\\\\\n&\\le \\frac{1}{4}\\int |{\\varphi}_z |^2\\left( {\\frac{ 1}{N}}\\right)+C{\\lambda}^2(C+\\frac {Cs}{2} ) \\int |\\varphi_y|^2 \\left( {\\frac{ 1}{N}}\\right)\\le \n (\\frac{1}{4}+C{\\lambda}^2) \\int \\frac{|\\nabla\\varphi|^2 }{N}\n \\end{aligned}$$ where we used the estimate $|(1/N)'|\\leq C/N$ in Lemma \\[wave\\_prop\\] and the Poincaré inequality .",
"\n\nFor the ${\\epsilon}$-terms, we estimate $$\\begin{aligned}\n - {\\epsilon}\\int |{\\nabla}{\\psi}|^2 {\\psi}&\\leq {\\epsilon}C \\sqrt{M(t)} \\|{\\nabla}\\psi\\|^2, \\\\\n -2{\\epsilon}\\int {\\mathcal P}{\\psi}_z \\psi &\\leq {\\epsilon}C \\|\\psi_z\\| \\|\\psi\\| \n \\le {\\epsilon}C {\\lambda}\\|\\psi_z\\| \\|\\psi_y\\|\\le {\\epsilon}C {\\lambda}\\|{\\nabla}\\psi\\|^2.\\end{aligned}$$ For the first estimate, we used $\\|\\psi\\|_{L^\\infty}\\le C\\|\\psi\\|_{H^2}\\leq C\\sqrt{ M(t)}$. For the second estimate, we used the Dirichlet condition for $\\psi$ ($\\psi(z,y)=0$ when $y=0$ or y=${\\lambda}$) in order to use the Poincaré inequality $$\\label{ineq:poincare3} \n\\| {\\psi}(z,\\cdot_y) \\|_{L^2_y([0,{\\lambda}])} \\le C{\\lambda}\\| {\\partial}_{y} {\\psi}(z,\\cdot_y)\\|_{L^2_y([0,{\\lambda}])} \\quad\\mbox{ for } z\\in\\mathbb{R}.$$\n\nBy assuming smallness of $M(t)$ and ${\\lambda}$, we arrive at $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}\\left( \\int {\\frac{ |\\varphi|^2}{N}} +\\int |\\psi|^2 \\right) \n+\\frac 18 \\int{\\frac{ |{\\nabla}\\varphi|^2}{N}} + \\frac{{\\epsilon}}{2}\\int |{\\nabla}\\psi|^2 \\leq \nC \\sqrt{M(t)} \\int {\\frac{ | {\\nabla}\\psi|^2}{N}}\\end{aligned}$$ which proves Lemma \\[lemma0\\_\\] for ${\\epsilon}>0$. Note that we used the smallness of ${\\epsilon}$ to use Lemma \\[wave\\_prop\\].",
"\n\nLet’s study the first order estimate:\n\n\\[lemma1\\_0\\] If ${\\lambda}>0$ is small enough, then we have $$\\begin{split}\\label{ineq:lem1_}\n&\\| {\\nabla}{\\varphi}\\|_w^2 + \\| {\\nabla}{\\psi}\\|^2+ \\int_0^t \\| {\\nabla}^2 {\\varphi}\\|_w^2 \\\\\n&\\le C ( \\| {\\nabla}{\\varphi}_0\\|_w^2 + \\| {\\nabla}{\\psi}_0\\|^2+ \\| \\psi_0\\|^2 + \\| \\varphi_0\\|_w^2 ) \n+ C \\int_0^t \\int {N} |{\\nabla}{\\psi}|^2 + C {{M(t)} }\\int_0^t \\int {\\frac{ {\\vert{\\nabla}{\\psi}\\vert^2}}{N}}.",
"\n\\end{split}$$\n\nThis estimate is not closed yet due to the last two terms. ",
"Later in Lemma \\[lemma01\\_\\] (see the estimate ), it will be closed by Lemma \\[lemma1\\_1\\] and Lemma \\[lemma1\\_2\\]. ",
"Indeed, the first one $\\int_0^t \\int {N} |{\\nabla}{\\psi}|^2 $ among these two will be covered by the second one $\\sqrt{M(t)} \\int_0^t \\int {\\frac{ {\\vert{\\nabla}{\\psi}\\vert^2}}{N}}$ (see Lemma \\[lemma1\\_1\\]) while the second term will be absorbed into the dissipation term $\\int_0^t \\| {\\nabla}^2 {\\varphi}\\|_w^2 $ on the left-hand side of (see Lemma \\[lemma1\\_2\\]).",
"\n\nDifferentiating in $z$, we have $$\\begin{aligned}\n{\\varphi}_{tz} - s{\\varphi}_{zz} - {\\Delta}{\\varphi}_z & = N' {\\nabla}\\psi + N {\\nabla}\\psi_z + P' {{\\nabla}\\cdot}{\\varphi}+ P {{\\nabla}\\cdot}{\\varphi}_z + {\\nabla}\\psi_z {{\\nabla}\\cdot}{\\varphi}+ {\\nabla}\\psi {{\\nabla}\\cdot}{\\varphi}_z,\\\\\n\\psi_{tz} -s\\psi_{zz} \n& =\n {{\\nabla}\\cdot}{\\varphi}_z.",
"\n$$ Multiply ${\\frac{ \\varphi_z}{N}}$ and $\\psi_z$ to the above equation. ",
"Integrating by parts, we have $$\\begin{aligned}\n \\frac 12 {\\frac{d}{dt}}{\\left( \\int {\\frac{ {\\vert{\\varphi}_z\\vert^2} }{N}} + {\\vert{\\psi}_z\\vert^2} \\right)} + \\int \\sum_{ij}{\\frac{ {\\vert{\\partial}_j {\\varphi}^i_{z}\\vert^2}}{N}} & = { \\frac 12 \\int {\\vert{\\varphi}_z\\vert^2} \\left( {\\frac{ 1}{N}} \\right)''- \\frac s2 \\int {\\vert{\\varphi}_z\\vert^2} {\\left({\\frac{ 1}{N}} \\right)}' }\n \\\\\n & +\\int {\\frac{ N'}{N}} {\\nabla}{\\psi}{\\varphi}_z +\\int {\\frac{ P'}{N}} {{\\nabla}\\cdot}{\\varphi}{\\varphi}_z + \\int {\\frac{ P}{N}} {{\\nabla}\\cdot}{\\varphi}_z {\\varphi}_z\\\\\n &+ \\int {\\nabla}{\\psi}_z {{\\nabla}\\cdot}{\\varphi}{\\frac{ {\\varphi}_z}{N}} + \\int {\\nabla}{\\psi}{{\\nabla}\\cdot}{\\varphi}_z {\\frac{ {\\varphi}_z}{N}}.\\\\\n$$ Similarly, we get $$\\begin{aligned}\n\\frac 12 {\\frac{d}{dt}}{\\left( \\int {\\frac{ {\\vert{\\varphi}_y\\vert^2} }{N}} + {\\vert{\\psi}_y\\vert^2} \\right)} +\\int \\sum_{ij}{\\frac{ {\\vert{\\partial}_j {\\varphi}^i_{y}\\vert^2}}{N}} \n &= \n { \\frac 12 \\int {\\vert{\\varphi}_y\\vert^2} \\left( {\\frac{ 1}{N}} \\right)'' - \\frac s2 \\int {\\vert{\\varphi}_y\\vert^2} {\\left({\\frac{ 1}{N}} \\right)}' }\n \\\\\n &+\\int {\\frac{ P}{N}} {{\\nabla}\\cdot}{\\varphi}_y {\\varphi}_y \\\\\n &+ \\int {\\nabla}{\\psi}_y {{\\nabla}\\cdot}{\\varphi}{\\frac{ {\\varphi}_y}{N}} + \\int {\\nabla}{\\psi}{{\\nabla}\\cdot}{\\varphi}_y {\\frac{ {\\varphi}_y}{N}}.\\\\\n$$\n\nFirst, we estimate by using Lemma \\[wave\\_prop\\] $$\\begin{aligned}\n&\\frac 12 \\int |{\\nabla}{\\varphi}|^2 {\\left( {\\frac{ 1}{N}} \\right)}''-\\frac s 2 \\int |{\\nabla}\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'\n\\le -\\int {\\nabla}{\\varphi}\\cdot{\\nabla}{\\varphi}_z \\left( {\\frac{ 1}{N}}\\right)'+\\frac s 2 \\int |{\\nabla}\\varphi|^2 \\left|\\left( {\\frac{ 1}{N}}\\right)'\\right|\\\\\n&\\le C\\int |{\\nabla}{\\varphi}||{\\nabla}{\\varphi}_z | \\left( {\\frac{ 1}{N}}\\right) +\\frac {Cs}{2} \\int |{\\nabla}\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right) \n\\le \\frac{1}{4}\\int |{\\nabla}{\\varphi}_z |^2\\left( {\\frac{ 1}{N}}\\right)+(C+\\frac {Cs}{2} ) \\int |{\\nabla}\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right).",
"\n \\end{aligned}$$ We estimate the quadratic terms as follows: $$\\begin{aligned}\n\\int {\\frac{ P}{N}} {{\\nabla}\\cdot}{\\varphi}_z {\\varphi}_z+\\int {\\frac{ P}{N}} {{\\nabla}\\cdot}{\\varphi}_y {\\varphi}_y \\le\n \\| P\\|_{L^{\\infty}} (\\| \\frac{{\\nabla}{\\varphi}_z}{ \\sqrt{N}} \\|\\| \\frac{{\\varphi}_z}{\\sqrt{N}}\\| \n+\\| \\frac{{\\nabla}{\\varphi}_y}{ \\sqrt{N}} \\|\\| \\frac{{\\varphi}_y}{\\sqrt{N}}\\| )\n \\le\n\\frac{1}{4}\\| \\frac{{\\nabla}^2{\\varphi}}{ \\sqrt{N}} \\|^2+C\\| \\frac{{\\nabla}{\\varphi}}{\\sqrt{N}}\\|^2,\\end{aligned}$$ $$\\begin{aligned}\n\\int {\\frac{ N'}{N}} {\\nabla}{\\psi}{\\varphi}_z & \\le \\| {\\frac{ N'}{N}}\\|_{L^{\\infty}}\\| {\\sqrt N}{{\\nabla}{\\psi}}\\| \\| \\frac{{\\varphi}_z}{ \\sqrt{N}} \\|\n\\le C\\| {\\sqrt N}{{\\nabla}{\\psi}}\\|^2+ C \\| \\frac{{\\nabla}{\\varphi}}{ \\sqrt{N}} \\|^2,\\\\\n\\int {\\frac{ P'}{N}} {{\\nabla}\\cdot}{\\varphi}{\\varphi}_z &\\le \n\\| P'\\|_{L^{\\infty}} \\| \\frac{{\\nabla}{\\varphi}}{\\sqrt N} \\|^2\n\\le \nC\\|\\frac{{\\nabla}{\\varphi}}{\\sqrt N} \\|^2.\\end{aligned}$$\n\nThe cubic terms are bounded by $$\\begin{aligned}\n\\int | {\\nabla}{\\psi}|| {\\nabla}{\\varphi}| {\\frac{ |{\\nabla}^2{\\varphi}|}{N}} \n+\\int | {\\nabla}{\\psi}|| {\\nabla}{\\varphi}|^2 \\underbrace{|({1}/{N})'|}_{\\leq C/{N}}\n& {\\le \\frac{1}{4} \\| {\\nabla}^2 {\\varphi}/\\sqrt{N} \\|^2 + C(M(t)+\\sqrt{M(t)})\\| {\\nabla}{\\varphi}/\\sqrt{N}\\|^2}\\\\\n& { \\le \\frac{1}{4} \\| {\\nabla}^2 {\\varphi}/\\sqrt{N} \\|^2 + C\\| {\\nabla}{\\varphi}/\\sqrt{N}\\|^2}\\end{aligned}$$ by $ \\| {\\nabla}{\\psi}\\|_{L^{\\infty}} \\le C\\sqrt{M(t)}$ and [by the assumption $M(t)\\leq M(T)\\leq 1$.]{}\\\nFor the ${\\epsilon}$ terms, we estimate $$\\begin{aligned}\n&-2{\\epsilon}\\int ({\\mathcal P}{\\psi}_z)_z \\psi_z - 2{\\epsilon}\\int {\\mathcal P}{\\psi}_{zy} \\psi_y=\n -2{\\epsilon}\\Big(\\int {\\mathcal P}' {\\psi}_z \\psi_z\n-2{\\epsilon}\\int {\\mathcal P}{\\psi}_{zz} \\psi_z \n - 2{\\epsilon}\\int {\\mathcal P}{\\psi}_{zy} \\psi_y \\Big)\\\\\n &\\leq C{\\epsilon}\\|P\\|_{L^\\infty}\\|{\\nabla}{\\psi}\\|\\|{\\nabla}^2{\\psi}\\|+C{\\epsilon}\\|P'\\|_{L^\\infty}\\|{\\nabla}{\\psi}\\|^2 \\\\\n &\\leq C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+\\frac{{\\epsilon}}{4}\\|{\\nabla}^2{\\psi}\\|^2 \\end{aligned}$$ and\n\n$$\\begin{aligned}\n&- {\\epsilon}\\int (|{\\nabla}{\\psi}|^2)_z {\\psi}_z \n- {\\epsilon}\\int (|{\\nabla}{\\psi}|^2)_y {\\psi}_y=\n- {\\epsilon}\\int (|{\\nabla}{\\psi}|^2) {\\psi}_{zz} \n- {\\epsilon}\\int (|{\\nabla}{\\psi}|^2) {\\psi}_{yy}\\\\\n&\\leq C{\\epsilon}M(t) \\int |{\\nabla}{\\psi}||{\\nabla}^2 {\\psi}|\n\\leq C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+\\frac{{\\epsilon}}{4}\\|{\\nabla}^2 {\\psi}\\|^2.\\end{aligned}$$\n\nAdding up all the estimates above, we have $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}{\\left( \\int {\\frac{ {\\vert{\\nabla}{\\varphi}\\vert^2}}{N}} + \\int {\\vert{\\nabla}{\\psi}\\vert^2} \\right)} + \\frac 14\\int {\\frac{ {\\vert {\\nabla}^2{\\varphi}\\vert^2} }{N}}\n\\le \nC\\|{\\sqrt{N}} {{\\nabla}{\\psi}}\\|^2 + C \\| \\frac{{\\nabla}{\\varphi}}{\\sqrt{N}}\\|^2.",
"\n$$ After integration in time, thanks to Lemma \\[lemma0\\_\\], we can control the last term above so that we arrive at .",
"\n\n\\[lemma1\\_1\\] If ${\\lambda}>0$ is small enough, then we have $$\\begin{aligned}\n \\label{claim1_lemma1_}\n& \\int_0^t \\int N |{\\nabla}{\\psi}|^2 \n\\le C {\\left( \\| {\\nabla}{\\psi}_0 \\|^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_w^2 \\right)} \n+ C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}}.\\end{aligned}$$\n\n1. ",
" Roughly speaking, this estimate moves $N(z)$ to the denominator, while the integral multiplied by $\\sqrt{M(t)}$ which can be assumed small enough.",
"\n\n2. ",
" This estimate for $ \\int_0^t \\int N |{\\nabla}{\\psi}|^2$ is due to the special structure . ",
"Note that the ${\\varphi}$-equation has the term $N{\\nabla}{\\psi}$ on its right hand.",
"\n\nMultiplying ${\\nabla}{\\psi}$ to the ${\\varphi}$-equation, we have $$\\label{eq_lemma0__first_claim_intermediate}\n N |{\\nabla}{\\psi}|^2 = \\underbrace{{\\varphi}_t \\cdot {\\nabla}{\\psi}}_{(A)} - s{\\varphi}_z \\cdot{\\nabla}{\\psi}-\\underbrace{{\\Delta}{\\varphi}\\cdot{\\nabla}{\\psi}}_{(B)} \n-({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}- {{\\nabla}\\cdot}{\\varphi}| {\\nabla}{\\psi}|^2.$$ Let us rewrite the terms $(A)$ and $(B)$. $$\\begin{aligned}\n(A)& = ({\\varphi}\\cdot{\\nabla}{\\psi})_t - {\\varphi}\\cdot{\\nabla}{\\psi}_t \\\\&=\n ({\\varphi}\\cdot{\\nabla}{\\psi})_t - {\\varphi}\\cdot(s {\\nabla}{\\psi}_z + {\\nabla}({{\\nabla}\\cdot}{\\varphi})\n $$ and $$(B) \n = ({{\\nabla}\\cdot}{\\varphi})_z {\\psi}_z + ( {\\varphi}^1_{yy} - {\\varphi}^2_{zy}) {\\psi}_z + ( {{\\nabla}\\cdot}{\\varphi})_y {\\psi}_y \n+ ({\\varphi}^2_{zz} - {\\varphi}^1_{zy}) {\\psi}_y.$$ Integration by parts gives $$\\int(B)\n = \\int ({{\\nabla}\\cdot}{\\varphi})_z {\\psi}_z + ( {{\\nabla}\\cdot}{\\varphi})_y {\\psi}_y\n = \\int {\\nabla}({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}{\\psi}.$$ By using the ${\\psi}$-equation which has ${\\nabla}\\cdot{\\varphi}$ on its right side, we get $$\\begin{aligned}\n \\int {\\nabla}({{\\nabla}\\cdot}{\\varphi})\\cdot{\\nabla}{\\psi}&= \\int {\\nabla}({\\psi}_{t} - s {\\psi}_{z}\n)\\cdot {\\nabla}{\\psi}= \\frac 12 {\\frac{d}{dt}}\\int |{\\nabla}{\\psi}|^2.",
"\n$$ Integration on gives us $$\\begin{aligned}\n\\int N |{\\nabla}{\\psi}|^2 =& \n{\\frac{d}{dt}}\\int {\\varphi}\\cdot{\\nabla}{\\psi}- \\int s {\\varphi}\\cdot {\\nabla}{\\psi}_z -\\int{\\varphi}\\cdot ({\\nabla}({{\\nabla}\\cdot}{\\varphi})\n)\\\\ \\nonumber\n&- \\int s {\\varphi}_z\\cdot {\\nabla}{\\psi}\\\\ \\nonumber\n&- \\frac 12 {\\frac{d}{dt}}\\int( {\\psi}_z^2 + {\\psi}_y^2)\\\\ \\nonumber\n&- \\int ({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}- \\int {{\\nabla}\\cdot}{\\varphi}| {\\nabla}{\\psi}|^2. ",
"\n \\end{aligned}$$ We rearrange the above to get $$\\begin{aligned}\n\\label{ineq_claim1_lemma1}\n\\int N |{\\nabla}{\\psi}|^2 \n=& {\\frac{d}{dt}}\\int {\\varphi}\\cdot{\\nabla}{\\psi}- \\frac 12 {\\frac{d}{dt}}\\int |{\\nabla}{\\psi}|^2 \\\\& \\nonumber + \\int | {{\\nabla}\\cdot}{\\varphi}|^2 - \\int ({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}- \\int {{\\nabla}\\cdot}{\\varphi}| {\\nabla}{\\psi}|^2\\\\ \\nonumber\n=(I)+(II). ",
"\\end{aligned}$$ Note that $$\\begin{aligned}\n {\\int_0^t(I)}=\\int_0^t{\\left( {\\frac{d}{dt}}\\int {\\varphi}\\cdot{\\nabla}{\\psi}- \\frac 12 {\\frac{d}{dt}}\\int |{\\nabla}{\\psi}|^2 \\right)}\\leq {C(\\|{\\varphi}(t)\\|^2 + \\|{\\nabla}{\\psi}_0\\|^2+\\|{\\varphi}_0\\|^2)}&\n \\end{aligned}$$ by $ \\int {\\varphi}{\\nabla}{\\psi}\\le C\\| {\\varphi}\\| ^2 + \\frac 12 \\| {\\nabla}{\\psi}\\|^2$.\n\nFor the second term $(II)$, we estimate $$\\begin{aligned}\n \\int | {{\\nabla}\\cdot}{\\varphi}|^2-\\int ({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}& \\le C\\int {\\frac{ | {\\nabla}{\\varphi}|^2}{N}} + \\frac 14 \\int N |{\\nabla}{\\psi}|^2, \\\\\n \\int |{{\\nabla}\\cdot}{\\varphi}| | {\\nabla}{\\psi}|^2 & \\le C\\sqrt{M(t)} \\int {| {\\nabla}{\\psi}|^2} \\le C\\sqrt{M(t)} \\int {\\frac{ | {\\nabla}{\\psi}|^2}{N}}\n \\end{aligned}$$ by bounding $\\| {{\\nabla}\\cdot}{\\varphi}\\|_{L^{\\infty}} \\le C \\| {{\\nabla}\\cdot}{\\varphi}\\|_{H^2}\\le C \\sqrt{M(t)}$.\n\nFor the ${\\epsilon}$-term $(III)$, we estimate $$\\begin{aligned}\n {\\epsilon}\\int {\\nabla}( {\\Delta}{\\psi}-2 P \\cdot{\\nabla}{\\psi}- |{\\nabla}{\\psi}|^2)\\cdot{\\nabla}{\\psi}&=\n-{\\epsilon}\\int |{\\nabla}^2{\\psi}|^2\n-{\\epsilon}\\int {\\nabla}( 2 P \\cdot{\\nabla}{\\psi}+ |{\\nabla}{\\psi}|^2)\\cdot{\\nabla}{\\psi}\\\\\n&\\leq-{\\epsilon}\\|{\\nabla}^2{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+\\frac{{\\epsilon}}{4}\\|{\\nabla}^2{\\psi}\\|^2\\\\\n$$ and $$\\begin{aligned}\n - {\\epsilon}\\int{\\varphi}\\cdot( {\\Delta}{\\nabla}{\\psi}-2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) - {\\nabla}(|{\\nabla}{\\psi}|^2) ) \n&\\leq C{\\epsilon}\\int|{\\nabla}{\\varphi}||{\\nabla}^2{\\psi}| + |{\\nabla}{\\varphi}||{\\nabla}{\\psi}| +|{\\nabla}{\\varphi}||{\\nabla}{\\psi}|^2\\\\\n&\\leq \\frac{\\epsilon}4\\|{\\nabla}^2{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}{\\varphi}\\|^2+C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+C{\\epsilon}M(t) \\|{\\nabla}{\\psi}\\|^2\\\\\n&\\leq \\frac{\\epsilon}4\\|{\\nabla}^2{\\psi}\\|^2 +C \\|{\\nabla}{\\varphi}\\|_w^2+C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2.\\end{aligned}$$\n\nNow integrating in time, we get $$\\begin{aligned}\n\\int_0^t\\int& N |{\\nabla}{\\psi}|^2 \n\\leq \nC(\\|{\\varphi}(t)\\|^2 + \\|{\\nabla}{\\psi}_0\\|^2+\\|{\\varphi}_0\\|^2)\\\\\n& +\\int_0^t\\Big(C \\|{\\nabla}{\\varphi}\\|_w^2+ \\frac 14 \\int N |{\\nabla}{\\psi}|^2 +C\\sqrt{M(t)} \\int {\\frac{ | {\\nabla}{\\psi}|^2}{N}} \\Big).",
"\n \\end{aligned}$$\n\nBy Lemma \\[lemma0\\_\\] and [by $M(t) \\leq \\sqrt{M(t)}$,]{} we have .",
"\n\nWhat remains to close is to control the term $\\sqrt{M(t)}\\int _0^ t \\int {\\frac{ |{\\nabla}{\\psi}|^2 }{N}}$. Note that $\\psi$-equation lacks diffusion of closing the estimates without weight. ",
"However, our choice of the one-sided exponential weight function $w(\\cdot)$ gives a one-sided dissipation $\\int_0^t\\int_{z>0}\\frac{|\\nabla \\psi|^2}{N}$ as shown below.",
"\n\n\\[lemma1\\_2\\] If $M(T)>0$ and ${\\lambda}>0$ are small enough, then we have $$\\begin{aligned}\n& \\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}} + \\int _0^ t \\int {\\frac{ |{\\nabla}{\\psi}|^2 }{N}}\n\\le C( \\| {\\nabla}{\\psi}_0 \\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{w}^2) + C\\int_0^t \\int {\\frac{ |{\\nabla}^2{\\varphi}|^2}{N}}. ",
"\\label{claim2_lemma1_}\\end{aligned}$$\n\nFirst we take ${\\nabla}$ to the ${\\psi}$-equation then multiply by $ w {\\nabla}{\\psi}$ to get $$\\begin{aligned}\n&\\frac 12 (w |{\\nabla}{\\psi}|^2)_t - \\frac s2 ( w |{\\nabla}{\\psi}|^2)_z + \\frac s2 w' |{\\nabla}{\\psi}|^2 \n = w {\\nabla}({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}{\\psi}. ",
"\n$$\n\nNote that for some $c>0$ $$\\begin{aligned}\nw' = se^{sz} > c w & \\quad \\mbox{for } z > 0,\\label{positive_comp} \\\\\nw = 1+e^{sz} \\le 2\n< \\frac{1}{c}N \n & \\quad \\mbox{for } z\\le0.\\label{negative_comp}\\end{aligned}$$ Integrating on each half strip (notation : $\\int_{z>0}f:=\\int_0^\\infty\\int_{[0,{\\lambda}]} f (z,y,t)dy dz$) and in time, we get $$\\begin{aligned}\n\\int _{z>0} w |{\\nabla}{\\psi}|^2 \n \\le & \\int_{z>0} w |{\\nabla}{\\psi}_0|^2 + \\int_0^t \\int_{z>0} w {\\nabla}({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}{\\psi}-\nc\\int_0^t \\int _{z>0} w |{\\nabla}{\\psi}|^2 \\\\& -\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}{\\psi}|^2(0, y) dy \\quad (\\mbox{by } \\eqref{positive_comp}) \\\\\\le & \\int _{z>0} w |{\\nabla}{\\psi}_0|^2 -\\frac c2 \\int_0^t \\int_{z>0} w |{\\nabla}{\\psi}|^2 +\n C \\int_0^t \\int_{z>0} w |{\\nabla}({\\nabla}\\cdot {\\varphi})|^2 \\\\& -\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}{\\psi}|^2(0, y) dy\n \\end{aligned}$$ and $$\\begin{aligned}\n \\int _{z<0} w |{\\nabla}{\\psi}|^2 \\le & \\int_{z<0} w |{\\nabla}{\\psi}_0|^2 + \\int_0^t \\int_{z<0} w {\\nabla}({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}{\\psi}\\\\& +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}{\\psi}|^2(0, y) dy\\\\\\le & \\int_{z<0} w |{\\nabla}{\\psi}_0|^2 + 2\\int_0^t \\int_{z<0} |{\\nabla}({{\\nabla}\\cdot}{\\varphi})|| {\\nabla}{\\psi}|\n \\\\& +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}{\\psi}|^2(0, y) dy \\\\ \\le & \\int_{z<0} w |{\\nabla}{\\psi}_0|^2 + C\\int_0^t \\int_{z<0} {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}}+ \\int_0^t \\int_{z<0} N |{\\nabla}{\\psi}|^2 \n \\\\& +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}{\\psi}|^2(0, y) dy. ",
" $$ Adding the above two then adding $\\frac c2 \\int_0^t \\int_{z<0} w |{\\nabla}{\\psi}|^2 $ to the both sides, we get $$\\begin{aligned}\n\\int w |{\\nabla}{\\psi}|^2 + \\frac c2 \\int_0^t \\int w |{\\nabla}{\\psi}|^2 \\le & \\int w |{\\nabla}{\\psi}_0|^2 + \\int_0^t \\int N|{\\nabla}{\\psi}|^2 \\\\\n& + \nC\\int_0^t \\int w |{\\nabla}^2 {\\varphi}|^2+\\frac c2 \\int_0^t \\int_{z<0} w |{\\nabla}{\\psi}|^2 \\\\\\le & \\int w |{\\nabla}{\\psi}_0|^2 + C\\int_0^t \\int N|{\\nabla}{\\psi}|^2 + \nC\\int_0^t \\int w |{\\nabla}^2 {\\varphi}|^2 \\quad (\\mbox{by } \\eqref{negative_comp}) \\\\\n\\le & C ( \\| {\\nabla}{\\psi}_0\\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_w^2)\\\\ & + C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}}\n+ C \\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}} $$ where we used the previous estimate for the last inequality. ",
"For the ${\\epsilon}$[-terms]{}, we estimate\n\n$$\\begin{aligned}\n&\\int {{\\epsilon}\\mbox{-terms}}={\\epsilon}\\int { w {\\nabla}{\\psi}\\cdot{\\left( {\\Delta}{\\nabla}{\\psi}-2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) - {\\nabla}(|{\\nabla}{\\psi}|^2) \\right)}}\\\\\n&={\\epsilon}\\int \\Big({ -w|{\\nabla}^2{\\psi}|^2 -w' {\\nabla}{\\psi}\\cdot {\\nabla}{\\psi}_z\n- w {\\nabla}{\\psi}\\cdot{\\left( 2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) \\right)} \n+ (w'\\psi_z+w{\\Delta}{\\psi})\n|{\\nabla}{\\psi}|^2}\\Big)\\\\\n&\\leq-{\\epsilon}\\int { w|{\\nabla}^2{\\psi}|^2 +C{\\epsilon}\\int \\Big( w |{\\nabla}{\\psi}|| {\\nabla}^2{\\psi}| + w| {\\nabla}{\\psi}|( |{\\nabla}^2{\\psi}|+|{\\nabla}{\\psi}| )\n+w |{\\nabla}{\\psi}| |{\\nabla}{\\psi}|^2}+w|{\\nabla}^2{\\psi}||{\\nabla}{\\psi}|^2\\Big)\\\\\n&\\leq-\\frac {\\epsilon}4\\int w|{\\nabla}^2{\\psi}|^2 +C{\\epsilon}\\int w |{\\nabla}{\\psi}|^2 \n +C{\\epsilon}\\sqrt{M(t)}\\int w |{\\nabla}{\\psi}|^2\n\\leq-\\frac {\\epsilon}4\\int w|{\\nabla}^2{\\psi}|^2 +C{\\epsilon}\\int w |{\\nabla}{\\psi}|^2 \n$$\n\nwhere we used the estimate $|w'|\\leq C|w|$ and for the last inequality, we assumed $M(t)$ small enough.",
"\n\nThus we have $$\\begin{aligned}\n&\\int w |{\\nabla}{\\psi}|^2 + (\\frac{c}{2}-C\n\\sqrt{M(t)})\n\\int_0^t \\int w |{\\nabla}{\\psi}|^2\\\\&\\le C ( \\| {\\nabla}{\\psi}_0\\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_w^2) + C \\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}}. ",
" \\end{aligned}$$ Then, by making $M(t)$ small enough, it proves the estimate .",
"\n\nWe are ready to obtain a closed energy estimate up to the first order derivatives:\n\n\\[lemma01\\_\\] If $M(T)>0$ and ${\\lambda}>0$ are small enough, then we have $$\\begin{split}\\label{eq_lemma01_}\n\\| {\\varphi}\\|_{1,w}^2 + \\| {\\psi}\\|^2 + \\| {\\nabla}{\\psi}\\|_w^2+ \\int_0^t \\sum_{l = 1,2} \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^t \\| {\\nabla}{\\psi}\\|_w^2\n \\le\n C ( \\| {\\varphi}_0\\|_{1,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_w^2+ \\| {\\psi}_0\\|^2). ",
"\n\\end{split}$$\n\nPlugging the estimates and into , we have $$\\begin{aligned}\n &\\| {\\nabla}{\\varphi}\\|_w^2 + \\| {\\nabla}{\\psi}\\|^2 + \\int_0^t \\| {\\nabla}^2 {\\varphi}\\|_w^2 \\\\ &\\le C ( \\| {\\nabla}{\\psi}_0\\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{1,w}^2) + C\\sqrt{{M(t)}} \\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}}\\end{aligned}$$ which gives us $$\\begin{aligned}\n\\label{ineq:lem1_closed}\n &\\| {\\nabla}{\\varphi}\\|_w^2 + \\| {\\nabla}{\\psi}\\|^2 + \\int_0^t \\| {\\nabla}^2 {\\varphi}\\|_w^2 \\le C ( \\| {\\nabla}{\\psi}_0\\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{1,w}^2) \\end{aligned}$$ if we assume $M(t)$ small enough. ",
"In addition, from the estimate with the above estimate , we get $$\\begin{aligned}\n\\label{claims_lemma1_}\n\\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}} + \\int_0^t \\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}} \\le C ( \\| {\\varphi}_0\\|_{1,w}^2 + \\| {\\nabla}{\\psi}_0\\|_w^2\n+ \\| {\\psi}_0\\|^2).\\end{aligned}$$ Adding Lemma \\[lemma0\\_\\] and the above and and applying the estimate into the sum, we have . ",
"Note that we have used $M(t)$ for $ \\| {\\varphi}\\|^2_{L^{\\infty}} \\le M(t)$, $ \\| {{\\nabla}\\cdot}{\\varphi}\\|_{L^{\\infty}}^2 \\le M(t)$, and $ \\|{\\nabla}{\\psi}\\|^2_{L^{\\infty}} <M(t)$.\n\nFrom now on, we will repeat the preceding energy estimates up to the third order derivatives to have the weighted energy estimate . [",
"Since the key ideas were already presented in detail on the previous lemmas, we do not split the higher order estimates into several lemmas as in the first order case (Lemma \\[lemma0\\_\\], \\[lemma1\\_0\\], \\[lemma1\\_1\\], \\[lemma1\\_2\\] and \\[lemma01\\_\\]) but state them in the following single lemma, and we do not write its proof in detail.]{}",
"\n\n\\[lemma23\\_\\]If $M(T)>0$ and ${\\lambda}>0$ are small enough, then for $k=2, 3$ we get $$\\begin{aligned}\n&\\| {\\nabla}^k {\\varphi}\\|^2_w + \\| {\\nabla}^k {\\psi}\\|^2_w + \\int_0^t { \\|{\\nabla}^{k+1}{\\varphi}\\|_w^2}\n+ \\int_0^t { \\|{\\nabla}^{k}{\\varphi}\\|_w^2}\n \\le C ( \\| {\\varphi}_0\\|_{k,w}^2+\n \\| {\\nabla}{\\psi}_0\\|_{k-1,w}^2+ \\| {\\psi}_0\\|^2).\\end{aligned}$$\n\n1. ",
" The proof for $k=2$ is similar with that of the first order estimate except and .",
"\n\n2. ",
" For $k=3$, we use $$\\|({\\nabla}^2 {\\varphi})/\\sqrt{N}\\|_{L^4}\\leq \\|{\\varphi}\\|_{3,w}\\leq \\sqrt{M(t)}$$ (see and ) as well as $\\|{\\nabla}{\\varphi}\\|_{L^\\infty}\\leq \\sqrt{M(t)}$.\n\nDifferentiating the ${\\varphi},{\\psi}$ equations $i+j$ times in $y$ or $z$, we have $$\\begin{aligned}\n&{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}_t - s {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}_z - {\\Delta}{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}\\\\\n&\\phantom{{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}_t - s {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}_z} =\n [ {{\\partial}_y^{i}}{{\\partial}_z^{j}}, N ] {\\nabla}{\\psi}+ N {\\nabla}{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}+ {{\\partial}_y^{i}}{{\\partial}_z^{j}}( P {{\\nabla}\\cdot}{\\varphi}) + {{\\partial}_y^{i}}{{\\partial}_z^{j}}( {{\\nabla}\\cdot}{\\varphi}{\\nabla}{\\psi}),\\\\\n&{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}_t - s {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}_z\n = {{\\nabla}\\cdot}{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}.\\end{aligned}$$ Thus we get $$\\begin{aligned}\n \\label{higher_}\n& \\frac 12 {\\frac{d}{dt}}\\int {\\left( {\\frac{ | {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}|^2}{N}} + | {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}|^2 \\right)} \n+ \\int {\\frac{ | {\\nabla}{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}|^2}{N}}\n\\\\\n &=\n { \\frac 12 \\int {\\vert{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}\\vert^2} \\left( {\\frac{ 1}{N}} \\right)''- \\frac s2 \\int {\\vert{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}\\vert^2} {\\left({\\frac{ 1}{N}} \\right)}'}\n \\nonumber\\\\\n &\\underbrace{+\\int ( {{\\partial}_y^{i}}{{\\partial}_z^{j}}(N {\\nabla}{\\psi}) - N {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\nabla}{\\psi})\\cdot {\\frac{ {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}}{N}} + {{\\partial}_y^{i}}{{\\partial}_z^{j}}(P {{\\nabla}\\cdot}{\\varphi}) \\cdot{\\frac{ {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}}{N}} }_{\\mbox{ Quadratic term }}\\nonumber\\\\\n& \\underbrace{+ {{\\partial}_y^{i}}{{\\partial}_z^{j}}( {{\\nabla}\\cdot}{\\varphi}{\\nabla}{\\psi}) {\\frac{ {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}}{N}}}_{\\mbox{ Cubic term }}. ",
"\\nonumber\\\\\n$$\n\n$\\bullet$ Case $ k= i+j =2$\\\nFirst, we estimate $$\\begin{aligned}\n&\\frac 12 \\int |{\\nabla}^2{\\varphi}|^2 {\\left( {\\frac{ 1}{N}} \\right)}''-\\frac s 2 \\int |{\\nabla}^2\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'\n\\le -\\int {\\nabla}^2{\\varphi}\\cdot{\\nabla}^2{\\varphi}_z \\left( {\\frac{ 1}{N}}\\right)'+\\frac s 2 \\int |{\\nabla}^2\\varphi|^2 \\left|\\left( {\\frac{ 1}{N}}\\right)'\\right|\\\\\n&\n\\le C\\int |{\\nabla}^2{\\varphi}||{\\nabla}^2{\\varphi}_z | \\left( {\\frac{ 1}{N}}\\right) +\\frac {Cs}{2} \\int |{\\nabla}^2\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right) \n\\le \\frac{1}{4}\\int |{\\nabla}^3{\\varphi}|^2\\left( {\\frac{ 1}{N}}\\right)+C \\int |{\\nabla}^2\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right).",
"\n \\end{aligned}$$\n\nWhat it follows we do not distinguish ${\\partial}_y$ and $ {\\partial}_z$ derivatives.\\\nThe quadratic terms are symbolically $$\\begin{aligned}\n {\\frac{ N''}{N}} {\\nabla}{\\psi}{\\nabla}^2 {\\varphi}, \\quad {\\frac{ N'}{N}} {\\nabla}^2 {\\psi}{\\nabla}^2 {\\varphi}, {\\frac{ P''}{N}} {\\nabla}{\\varphi}{\\nabla}^2 {\\varphi}, \\quad {\\frac{ P'}{N}} {\\nabla}^2 {\\varphi}{\\nabla}^2 {\\varphi}\\quad \\mbox{ and } \\quad {\\frac{ P}{N}} {\\nabla}^3 {\\varphi}{\\nabla}^2 {\\varphi}.\\\\\n $$ By Lemma \\[wave\\_prop\\], these terms are estimated by $$\\begin{aligned}\n& C\\int \\Big|{\\frac{ N''}{N}}\\Big|| {\\nabla}{\\psi}|| {\\nabla}^2 {\\varphi}| +\n \\Big|{\\frac{ P''}{N}} \\Big||{\\nabla}{\\varphi}||{\\nabla}^2 {\\varphi}|+ \\Big|{\\frac{ P'}{N}} \\Big||{\\nabla}^2 {\\varphi}||{\\nabla}^2 {\\varphi}|\\\\\n &\\leq C {\\left( \\| \\frac{{\\nabla}{\\psi}}{\\sqrt{N}}\\|^2\n +\\| \\frac{{\\nabla}{\\varphi}}{\\sqrt{N}}\\|^2+\\| \\frac{{\\nabla}^2{\\varphi}}{\\sqrt{N}}\\|^2 \\right)},\n $$ $$\\begin{aligned}\nC\\int \\Big|\\frac{P}{N}\\Big| |{\\nabla}^3 {\\varphi}| |{\\nabla}^2{\\varphi}| \\le\n C \\| \\frac{{\\nabla}^3{\\varphi}}{ \\sqrt{N}} \\|\\| \\frac{{\\nabla}^2{\\varphi}}{\\sqrt{N}}\\| \\le\n \\frac{1}{8} \\| \\frac{{\\nabla}^3{\\varphi}}{ \\sqrt{N}} \\|^2 +C\\| \\frac{{\\nabla}^2{\\varphi}}{\\sqrt{N}}\\|^2\\end{aligned}$$ and $$\\begin{aligned}\n& C\\Big|\\int {\\frac{ N'}{N}} {\\nabla}^2 {\\psi}{\\nabla}^2 {\\varphi}\\Big|\\leq C \\int | {\\nabla}{\\psi}| |{\\nabla}^2 {\\varphi}|\n+C\\int |{\\nabla}{\\psi}||{\\nabla}^3 {\\varphi}|\\\\\n &\\leq C \\| {{\\nabla}{\\psi}} \\|{\\left( \\| {{\\nabla}^2{\\varphi}} \\|+\\| {{\\nabla}^3{\\varphi}} \\| \\right)}\\leq \n C \\| {{\\nabla}{\\psi}} \\|^2+ C\\| {{\\nabla}^2{\\varphi}} \\|^2+\\frac 1 8 \\| {{\\nabla}^3{\\varphi}} \\|^2\\end{aligned}$$ where we used integration by parts for the last estimate.",
"\n\nThe cubic terms are symbolically written as $ {\\nabla}^2 ({\\nabla}{\\varphi}{\\nabla}{\\psi}) {\\frac{ {\\nabla}^2{\\varphi}}{N}}$. By using integration by parts once, it can be written as $$\\begin{aligned}\n& {\\nabla}({\\nabla}{\\varphi}{\\nabla}{\\psi}) {\\frac{ {\\nabla}^3{\\varphi}}{N}} \\quad \\mbox{and } \\quad {\\nabla}({\\nabla}{\\varphi}{\\nabla}{\\psi}) {{\\nabla}^2{\\varphi}}{(\\frac{1}{N})'} .\\end{aligned}$$ So by assuming smallness of $M(t)$, these terms are estimated by $$\\begin{aligned}\n& C\\int (|{\\nabla}^2 {\\varphi}|| {\\nabla}{\\psi}|+|{\\nabla}{\\varphi}||{\\nabla}^2 {\\psi}|) {\\frac{ |{\\nabla}^3{\\varphi}|}{N}} \\leq\nC\\sqrt{M(t)}\\int (|{\\nabla}^2 {\\varphi}| + |{\\nabla}^2 {\\psi}|) {\\frac{ |{\\nabla}^3{\\varphi}|}{N}}\\\\\n&\\leq C \\|\\frac{{\\nabla}^2 {\\varphi}}{\\sqrt{N}}\\|^2+ C \\sqrt{M(t)}\\|\\frac{{\\nabla}^2 {\\psi}}{\\sqrt{N}}\\|^2 +\\frac{1}{8}\\|\\frac{{\\nabla}^3{\\varphi}}{\\sqrt{N}}\\|^2\\end{aligned}$$ and\n\n$$\\begin{aligned}\n& C\\int(|{\\nabla}^2 {\\varphi}||{\\nabla}{\\psi}|\n+|{\\nabla}{\\varphi}|| {\\nabla}^2 {\\psi}|){{\\frac{ |{\\nabla}^2{\\varphi}|}{N}}}\\leq C \\|\\frac{{\\nabla}^2 {\\varphi}}{\\sqrt{N}}\\|^2+ C \\sqrt{M(t)}\\|\\frac{{\\nabla}^2 {\\psi}}{\\sqrt{N}}\\|^2.\\end{aligned}$$\n\nNote that we used $M(t)$ for $ \\| {\\nabla}{\\varphi}\\|^2_{L^{\\infty}} \\le M(t)$ and $ \\|{\\nabla}{\\psi}\\|^2_{L^{\\infty}} <M(t)$.\n\nFor the ${\\epsilon}$ terms, we can write them symbolically: $${\\epsilon}\\int {\\nabla}^2({\\mathcal P}{\\psi}_z) {\\nabla}^2\\psi \\quad \\mbox{and } \\quad\n {\\epsilon}\\int\\nabla^2(|{\\nabla}{\\psi}|^2) \\nabla^2{\\psi}.$$ After intergration by parts, we can estimate them by $$\\begin{aligned}\n & C{\\epsilon}\\int |{\\nabla}({\\mathcal P}{\\psi}_z)| |{\\nabla}^3\\psi|\\leq\n C{\\epsilon}\\int (|{\\nabla}{\\psi}| +|{\\nabla}^2 {\\psi}|) |{\\nabla}^3\\psi|\n \\leq C{\\epsilon}( \\|\\nabla{\\psi}\\|^2+ \\|\\nabla^2{\\psi}\\|^2) +\\frac{{\\epsilon}}{4}\\|\\nabla^3{\\psi}\\|^2\\\\\\end{aligned}$$ and $$\\begin{aligned}\n&C{\\epsilon}\\int|\\nabla(|{\\nabla}{\\psi}|^2)| |\\nabla^3{\\psi}|\\leq C{\\epsilon}\\int|\\nabla{\\psi}||\\nabla^2{\\psi}| |\\nabla^3{\\psi}|\\\\\n&\\leq C{\\epsilon}\\sqrt{M(t)}\\int |\\nabla^2{\\psi}| |\\nabla^3{\\psi}|\\leq C{\\epsilon}M(t) \\|\\nabla^2{\\psi}\\|^2 +\\frac{{\\epsilon}}{4}\\|\\nabla^3{\\psi}\\|^2.\\end{aligned}$$ Up to now, by Lemma \\[lemma01\\_\\], we estimate by $$\\begin{aligned}\n\\label{beforeclaims_lemma23_} &\\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}} +\\int |{\\nabla}^2 {\\psi}|^2 + \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} \\\\\n &\\le C( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2) \n +C \\sqrt{M(t)}\\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2}{N}}. ",
" \\nonumber \n \\end{aligned}$$ This estimate is the second order version of Lemma \\[lemma1\\_0\\].\\\nNow we claim the following two estimates which are the second order versions of Lemma \\[lemma1\\_1\\] and \\[lemma1\\_2\\]:\n\n$$\\begin{aligned}\n \\label{claim1_lemma23_}\n& \\int_0^t \\int N |{\\nabla}^2 {\\psi}|^2 \\le C {\\left( \\| {\\varphi}_0\\|_{1,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_w^2+ \\| {\\psi}_0\\|^2+\\|{\\Delta}{\\psi}_0\\|^2 \\right)} + C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2}{N}}, \\\\\n& \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2}{N}} + \\int _0^ t \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2 }{N}} \n\\le C ( \\| {\\nabla}{\\psi}_0 \\|_{1,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{1,w}^2) + C\\int_0^t \\int {\\frac{ |{\\nabla}^3{\\varphi}|^2}{N}}. ",
"\\label{claim2_lemma23_}\\end{aligned}$$\n\nAs we did in Lemma \\[lemma1\\_1\\] and \\[lemma1\\_2\\], our plan is to prove first and to use the result in order to get . ",
"Then we will close the estimate by using them.\\\n$\\bullet$ Proof of\\\nTaking $ {{\\nabla}\\cdot}$ to ${\\varphi}$ equation, we have $$\\begin{aligned}\n&{{\\nabla}\\cdot}{\\varphi}_t - s {{\\nabla}\\cdot}{\\varphi}_z - {\\Delta}{{\\nabla}\\cdot}{\\varphi}\\\\\n& = \\quad N {\\Delta}{\\psi}+ \\underbrace{ {\\nabla}N \\cdot {\\nabla}{\\psi}+ {\\mathcal P}' {{\\nabla}\\cdot}{\\varphi}+ {\\mathcal P}( {{\\nabla}\\cdot}{\\varphi})_z \n + {{\\nabla}\\cdot}({{\\nabla}\\cdot}{\\varphi}{\\nabla}{\\psi})}_{R_1 }.\\end{aligned}$$ We multiply ${\\Delta}{\\psi}$ on the both sides and plug it in the equation $${\\Delta}{\\psi}_t - s{\\Delta}{\\psi}_z = {\\Delta}({{\\nabla}\\cdot}{\\varphi})$$ in order to get $$\\begin{aligned}\nN |{\\Delta}{\\psi}|^2 &= ( {{\\nabla}\\cdot}{\\varphi}_t - s{{\\nabla}\\cdot}{\\varphi}_z - {\\Delta}{{\\nabla}\\cdot}{\\varphi}) {\\Delta}{\\psi}- \\mbox{ R}_1{\\Delta}{\\psi}\\\\\n&= ({{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi})_t - {{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi}_t - s {{\\nabla}\\cdot}{\\varphi}_z {\\Delta}{\\psi}-\\underbrace{ {\\Delta}{{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi}}_{(C)} -\\mbox{ R}_1{\\Delta}{\\psi}\\\\\n&= ({{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi})_t - {{\\nabla}\\cdot}{\\varphi}( s {\\Delta}{\\psi}_z + {\\Delta}{{\\nabla}\\cdot}{\\varphi}) - s {{\\nabla}\\cdot}{\\varphi}_z {\\Delta}{\\psi}-(C)\n-\\mbox{ R}_1{\\Delta}{\\psi}.",
"\n$$\n\nFor $(C)$, we get $$\\begin{aligned}\n \\int(C)=&\\int {\\Delta}({{\\nabla}\\cdot}{\\varphi}) {\\Delta}{\\psi}=\\int {\\Delta}({\\psi}_t - s {\\psi}_z ) {\\Delta}{\\psi}\\\\\n & =\\frac 12 {\\frac{d}{dt}}\\int |{\\Delta}{\\psi}|^2. ",
"\n \\end{aligned}$$ So, integrating on the strip, we have $$\\begin{aligned}\n \\int N |{\\Delta}{\\psi}|^2 &= {\\frac{d}{dt}}\\int {{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi}+ \\int |{\\nabla}{{\\nabla}\\cdot}{\\varphi}|^2 - \\frac 12 {\\frac{d}{dt}}\\int |{\\Delta}{\\psi}|^2 - \\int \\mbox{ R}_1{\\Delta}{\\psi}. ",
" \\end{aligned}$$\n\nNote that $$\\begin{aligned}\n \\int_0^t{\\left( {\\frac{d}{dt}}\\int {{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi}- \\frac 12 {\\frac{d}{dt}}\\int |{\\Delta}{\\psi}|^2 \\right)}\\leq {C(\\|{\\nabla}{\\varphi}(t)\\|^2 \n +\\|{\\Delta}{\\psi}_0\\|^2+\\|{\\nabla}{\\varphi}_0\\|^2)}&\n \\end{aligned}$$ by $ \\int |{{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi}| \\le C\\| {\\nabla}{\\varphi}\\| ^2 + \\frac 12 \\| {\\Delta}{\\psi}\\|^2$. The terms in $R_1$ are estimated as follows; $$\\begin{aligned}\n& \\int ( {\\nabla}N \\cdot {\\nabla}{\\psi}) {\\Delta}{\\psi}\\le C\\int {\\frac{ |{\\nabla}{\\psi}|^2 }{N}} + \\frac 14 \\int N |{\\Delta}{\\psi}|^2, \\\\\n& \\int {\\mathcal P}' {{\\nabla}\\cdot}{\\varphi}{\\Delta}{\\psi}+ {\\mathcal P}( {{\\nabla}\\cdot}{\\varphi}_z) {\\Delta}{\\psi}\\le \nC {\\left( \\int {\\frac{ |{\\nabla}{\\varphi}|^2}{N}} + \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}} \\right)} + \\frac 18 \\int N |{\\Delta}{\\psi}|^2,\\\\\n&\\int {{\\nabla}\\cdot}({{\\nabla}\\cdot}{\\varphi}{\\nabla}{\\psi}) {\\Delta}{\\psi}\\le C ( \\| {\\nabla}{\\psi}\\|_{L^{\\infty}} + \\| {\\nabla}{\\varphi}\\|_{L^{\\infty}})\n {\\left( C \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}} + C\\int {\\frac{ |{\\Delta}{\\psi}|^2}{N}} + \\frac 14 \\int N |{\\Delta}{\\psi}|^2 \\right)}\\\\\n&\\quad \\qquad\\qquad\\qquad\\qquad\\leq C \n \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}} + C\\sqrt{M(t)}\\int {\\frac{ |{\\Delta}{\\psi}|^2}{N}} + \\frac 14 \\int N |{\\Delta}{\\psi}|^2\n \\end{aligned}$$ by assuming smallness of $M(t)$.\n\nFor the ${\\epsilon}$ terms, we estimate them by $$\\begin{aligned}\n & {\\epsilon}\\int {\\left( {\\Delta}( \\mbox{R}_2 ) {\\Delta}{\\psi}- {{\\nabla}\\cdot}{\\varphi}( \n {\\Delta}\\mbox{R}_2) \\right)} \\leq -{\\epsilon}\\int|{\\nabla}{\\Delta}{\\psi}|^2\\\\\n &\\quad\\qquad\\qquad + {\\epsilon}\\int {\\left( {\\Delta}( -2 P \\cdot{\\nabla}{\\psi}- |{\\nabla}{\\psi}|^2 ) {\\Delta}{\\psi}- {{\\nabla}\\cdot}{\\varphi}( \n {\\Delta}{\\left( {\\Delta}{\\psi}-2 P \\cdot{\\nabla}{\\psi}- |{\\nabla}{\\psi}|^2 \\right)}) \\right)}. ",
" \\end{aligned}$$ For the second term and the third term, we estimate $$\\begin{aligned}\n &{\\epsilon}\\int -{\\Delta}( 2 P \\cdot{\\nabla}{\\psi}+ |{\\nabla}{\\psi}|^2){\\Delta}{\\psi}={\\epsilon}\\int {\\nabla}( 2 P \\cdot{\\nabla}{\\psi}+ |{\\nabla}{\\psi}|^2){\\nabla}{\\Delta}{\\psi}\\\\\n&\\leq C{\\epsilon}{\\left( \\|{\\nabla}{\\psi}\\|^2+\\|{\\nabla}^2{\\psi}\\|^2 \\right)}+\\frac{{\\epsilon}}{4}\\|{\\nabla}{\\Delta}{\\psi}\\|^2\\\\\n$$ and $$\\begin{aligned}\n&- {\\epsilon}\\int{{\\nabla}\\cdot}{\\varphi}( {\\Delta}{\\left( {\\Delta}{\\psi}-2 ( P \\cdot{\\nabla}{\\psi}) - |{\\nabla}{\\psi}|^2 \\right)})\n={\\epsilon}\\int{\\nabla}{{\\nabla}\\cdot}{\\varphi}( {\\nabla}{\\left( {\\Delta}{\\psi}-2 ( P \\cdot{\\nabla}{\\psi}) - |{\\nabla}{\\psi}|^2 \\right)})\\\\\n&\\leq C{\\epsilon}\\int|{\\nabla}^2{\\varphi}||{\\nabla}{\\Delta}{\\psi}| + |{\\nabla}^2{\\varphi}||{\\nabla}{\\psi}| +|{\\nabla}^2{\\varphi}||{\\nabla}^2 {\\psi}|+|{\\nabla}^2{\\varphi}||{\\nabla}^2 {\\psi}||{\\nabla}{\\psi}|\\\\\n&\\leq \\frac{\\epsilon}4\\|{\\nabla}{\\Delta}{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}^2{\\varphi}\\|^2+C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+C{\\epsilon}\\|{\\nabla}^2{\\psi}\\|^2.\\\\\n$$ As a result, we get$$\\begin{aligned}\n &{\\epsilon}\\int {\\left( {\\Delta}( \\mbox{R}_2 ) {\\Delta}{\\psi}- {{\\nabla}\\cdot}{\\varphi}( \n {\\Delta}\\mbox{R}_2) \\right)} \\leq -\\frac {\\epsilon}2 \\|{\\nabla}{\\Delta}{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}^2{\\varphi}\\|^2+C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+C{\\epsilon}\\|{\\nabla}^2{\\psi}\\|^2.",
"\n \\end{aligned}$$ Collecting the above estimates and using Lemma \\[lemma01\\_\\], we have $$\\begin{aligned}\n&\\int_0^t \\int N |{\\Delta}{\\psi}|^2\n \\le C ( \\| {\\varphi}_0\\|_{1,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_w^2+ \\| {\\psi}_0\\|^2+\\|{\\Delta}{\\psi}_0\\|^2) \n + C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\Delta}{\\psi}|^2}{N}}.\\end{aligned}$$\n\nTo get from the above estimate, we have to control $ \\int N |{\\nabla}^2{\\psi}|^2$ by $\\int N |{\\Delta}{\\psi}|^2$ with lower order estimates. ",
"Observe $$\\begin{aligned}\n&\\int N |{\\Delta}{\\psi}|^2 = \\int N( ({\\partial}_{zz}{\\psi})^2 + ({\\partial}_{yy}{\\psi})^2+ 2{\\partial}_{zz}{\\psi}{\\partial}_{yy}{\\psi})\\\\\n&= \\underbrace{\\int N( ({\\partial}_{zz}{\\psi})^2 + ({\\partial}_{yy}{\\psi})^2+ 2({\\partial}_{zy}{\\psi})^2)}_{\\int N |{\\nabla}^2{\\psi}|^2}\n-2 \\int N' {\\partial}_z{\\psi}{\\partial}_{yy} {\\psi}\\end{aligned}$$\n\nand $$\\label{la_0}\\Big| \\int N' {\\partial}_1 {\\psi}{\\partial}_{22} {\\psi}\\Big|= \\Big|-\\int(P+s) N {\\partial}_1{\\psi}{\\partial}_{22}{\\psi}\\Big|\n\\le \\frac 14 \\int N( {\\partial}_{22}{\\psi})^2\n+ C\\int N |{\\nabla}{\\psi}|^2$$ by $ N'=-(P+s)N$ from . ",
"Thus we get $$\\begin{aligned}\n \\label{la_1} \\int N |{\\nabla}^2{\\psi}|^2\n&\\leq 2 \\int N( ({\\partial}_{11}{\\psi})^2 + \\frac 12 ({\\partial}_{22}{\\psi})^2+ 2({\\partial}_{12}{\\psi})^2)\\\\\n&= 2\\int N |{\\Delta}{\\psi}|^2+4\\int N' {\\partial}_1 {\\psi}{\\partial}_{22} {\\psi}- \\int N( {\\partial}_{22}{\\psi})^2 \\nonumber\\\\\n&\\leq 2\\int N |{\\Delta}{\\psi}|^2 + \\int N( {\\partial}_{22}{\\psi})^2\n+ C\\int N |{\\nabla}{\\psi}|^2 - \\int N( {\\partial}_{22}{\\psi})^2\\nonumber \\\\\n&\\leq 2\\int N |{\\Delta}{\\psi}|^2 \n+ C\\int N |{\\nabla}{\\psi}|^2.\\nonumber \n \\end{aligned}$$ So we have $$\\int_0^t\\int N |{\\nabla}^2{\\psi}|^2 \\le \n C ( \\| {\\varphi}_0\\|_{1,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_w^2+ \\| {\\psi}_0\\|^2)+ C\\int_0^t\\int N |{\\Delta}{\\psi}|^2$$ by Lemma \\[lemma01\\_\\]. ",
"Thus we proved .\\\n$\\bullet$ Proof of\\\nMultiplying $ w {\\nabla}^2 {\\psi}$ to the equation $${\\nabla}^2 {\\psi}_t - s {\\nabla}^2{\\psi}_z\n = {\\nabla}^2 ({{\\nabla}\\cdot}{\\varphi}),\n $$ we have $$\\begin{aligned}\n&\\frac 12 (w |{\\nabla}^2 {\\psi}|^2)_t - \\frac s2 ( w |{\\nabla}^2 {\\psi}|^2)_z + \\frac s2 w' |{\\nabla}^2 {\\psi}|^2 \n= w {\\nabla}^2 ({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}^2 {\\psi}. ",
"\n$$\n\nRecall that for some $c>0$, we have $$\\begin{aligned}\nw' = se^{sz} > c w & \\quad \\mbox{for } z > 0, \\\\\nw = 1+e^{sz} \\le 2\n< \\frac{1}{c}N \n & \\quad \\mbox{for } z\\le0.\\end{aligned}$$\n\nIntegrating on each half strip and in time, we get $$\\begin{aligned}\n\\int _{z>0} w |{\\nabla}^2 {\\psi}|^2 &\n \\le \\int_{z>0} w |{\\nabla}^2 {\\psi}_0|^2 + \\int_0^t \\int_{z>0} w {\\nabla}^2 ({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}^2 {\\psi}-\nc\\int_0^t \\int _{z>0} w |{\\nabla}^2 {\\psi}|^2 \\\\&\\quad -\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^2 {\\psi}|^2(0, y) dy \\\\\n& \\le \\int _{z>0} w |{\\nabla}^2 {\\psi}_0|^2 -\\frac c2 \\int_0^t \\int_{z>0} w |{\\nabla}^2 {\\psi}|^2 +\n C \\int_0^t \\int_{z>0} w |{\\nabla}^2({\\nabla}\\cdot {\\varphi})|^2 \\\\&\\quad -\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^2 {\\psi}|^2(0, y) dy,\n \\end{aligned}$$ $$\\begin{aligned}\n \\int _{z<0} w |{\\nabla}^2 {\\psi}|^2 \\le & \\int_{z<0} w |{\\nabla}^2 {\\psi}_0|^2 + \\int_0^t \\int_{z<0} w {\\nabla}^2 ({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}^2 {\\psi}\\\\&\\quad\n +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^2 {\\psi}|^2(0, y) dy\n \\\\\n&\\le \\int_{z<0} w |{\\nabla}^2 {\\psi}_0|^2 + C\\int_0^t \\int_{z<0} | {\\nabla}^3 {\\varphi}|||{\\nabla}^2 {\\psi}| \\\\&\\quad\n +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^2 {\\psi}|^2(0, y) dy\n \\\\\n &\\le \\int_{z<0} w |{\\nabla}^2 {\\psi}_0|^2 + C\\int_0^t \\int_{z<0} N|{\\nabla}^2 {\\psi}|^2 \n + C\\int_0^t \\int_{z<0} { {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}}} \\\\&\\quad\n +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^2 {\\psi}|^2(0, y) dy.",
"\n $$ Adding those, we get $$\\begin{aligned}\n&\\int w |{\\nabla}^2 {\\psi}|^2 + \\frac c2 \\int_0^t \\int w |{\\nabla}^2 {\\psi}|^2 \\le \\int w |{\\nabla}^2 {\\psi}_0|^2\n+ C\\int_0^t \\int \\Big(N |{\\nabla}^2{\\psi}|^2 + \n w |{\\nabla}^3 {\\varphi}|^2 \\Big).",
"\n$$\n\nFor the ${\\epsilon}$[-terms]{}, as before, we estimate\n\n$$\\begin{aligned}\n&\\int {{\\epsilon}\\mbox{-terms}}={\\epsilon}\\int { w {\\nabla}^2 {\\psi}\\cdot{\\left( {\\Delta}{\\nabla}^2{\\psi}-2 {\\nabla}^2( P \\cdot{\\nabla}{\\psi}) - {\\nabla}^2(|{\\nabla}{\\psi}|^2) \\right)}}\\\\\n&={\\epsilon}\\int { \\Big(-w|{\\nabla}^3{\\psi}|^2 -w' {\\nabla}^2 {\\psi}\\cdot {\\nabla}^2{\\psi}_z\n- w {\\nabla}^2 {\\psi}\\cdot{\\left( 2 {\\nabla}^2( P \\cdot{\\nabla}{\\psi}) \\right)} \n+ (w'{\\nabla}\\psi_z+w{\\Delta}{\\nabla}{\\psi})\n{\\nabla}(|{\\nabla}{\\psi}|^2)}\\Big)\\\\\n&\\leq-{\\epsilon}\\int w|{\\nabla}^3{\\psi}|^2 \\\\&\\,\\, +C{\\epsilon}\\int \\Big(w |{\\nabla}^2 {\\psi}|| {\\nabla}^3{\\psi}| + w| {\\nabla}^2 {\\psi}|( |{\\nabla}^3{\\psi}|+|{\\nabla}^2{\\psi}|+|{\\nabla}{\\psi}| ) \n+w|{\\nabla}{\\psi}|( |{\\nabla}^2 {\\psi}| |{\\nabla}^2 {\\psi}| +|{\\nabla}^3{\\psi}| |{\\nabla}^2 {\\psi}|)\\Big)\\\\\n&\\leq-\\frac {\\epsilon}4\\int w|{\\nabla}^3{\\psi}|^2 +C{\\epsilon}\\int w |{\\nabla}^2 {\\psi}|^2 +C{\\epsilon}\\int w |{\\nabla}{\\psi}|^2 \n $$\n\nwhere we used the estimate $|w'|\\leq C|w|$ and for the last inequality, we assumed $M(t)$ small enough.\\\nCollecting the above estimates, and using Lemma \\[lemma01\\_\\] and the previous claim , we have $$\\begin{aligned}\n&\\int w |{\\nabla}^2 {\\psi}|^2 + (\\frac{c}{2}-C\n\\sqrt{M(t)})\n \\int_0^t \\int w |{\\nabla}^2 {\\psi}|^2 \n \\\\\n&\\le C ( \\| {\\nabla}{\\psi}_0\\|_{1,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{1,w}^2) \n+ C \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} . ",
"\\end{aligned}$$\n\nThen, by making $M(t)$ small enough, it proves the claim .\\\nNow we are ready to finish this proof for Lemma \\[lemma23\\_\\] for the second order. ",
"Plugging into with $M(t)$ small, we have $$\\begin{aligned}\n &\\| {\\nabla}^2{\\varphi}\\|_w^2 + \\| {\\nabla}^2{\\psi}\\|^2 + \\int_0^t \\| {\\nabla}^3 {\\varphi}\\|_w^2 \n \\le C ( \\| {\\nabla}{\\psi}_0\\|_{1,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{2,w}^2). ",
"$$ In turns, we have $$\\begin{aligned}\n\\int {\\frac{ |{\\nabla}^2 {\\psi}|^2}{N}} + \\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2}{N}}\n \\le C ( \\| {\\varphi}_0\\|_{2,w}^2 + \\| {\\nabla}{\\psi}_0\\|_{1,w}^2\n+ \\| {\\psi}_0\\|^2).\\end{aligned}$$ This proves Lemma \\[lemma23\\_\\] for case $k=i+j=2$.\n\nTogether with \\[lemma01\\_\\], we have proved $$\\begin{aligned}\n\\label{lemma2_} &\n\\| {\\varphi}\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}\\|_{1,w}^2+ \\| {\\psi}\\|^2\n+ \\int_0^t \\sum_{l = 1,2,3} \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^t \\sum_{l = 1,2}\\| {\\nabla}^l {\\psi}\\|_w^2\n\\\\\n &\\le C( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2). ",
"\\nonumber\\end{aligned}$$\n\n \\\n$\\bullet$ Case $ k=i + j =3$\\\n \\\nWe repeat the previous argument for $k=3$.\n\nFirst, we recall the equation . ",
"As before, we estimate $$\\begin{aligned}\n&\\frac 12 \\int |{\\nabla}^3{\\varphi}|^2 {\\left( {\\frac{ 1}{N}} \\right)}''-\\frac s 2 \\int |{\\nabla}^3\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'\n\\le \\frac{1}{8}\\int |{\\nabla}^4{\\varphi}|^2\\left( {\\frac{ 1}{N}}\\right)+C \\int |{\\nabla}^3\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right). ",
" \\end{aligned}$$ The quadratic terms are symbolically $$\\begin{aligned}\n {\\frac{ N^{(k)}}{N}} {\\nabla}{\\nabla}^{3-k} {\\psi}{\\nabla}^3 {\\varphi}\\quad \\mbox{ for } k=1,2,3 \\quad \\mbox{ and }\\\\\n {\\frac{ P^{(k)}}{N}} {{\\nabla}\\cdot}({\\nabla}^{3-k} {\\varphi}) {\\nabla}^3 {\\varphi}\\quad \\mbox{ for } k=0,1,2,3.\\\\\n $$ By Lemma \\[wave\\_prop\\], these terms are estimated by $$\\begin{aligned}\n& \\int \n \\Big|{\\frac{ N'''}{N}}\\Big|\n| {\\nabla}{\\psi}|| {\\nabla}^3 {\\varphi}| +\\Big|{\\frac{ N''}{N}}\\Big|\n| {\\nabla}^2 {\\psi}|| {\\nabla}^3 {\\varphi}|+\\Big|{\\frac{ P'''}{N}} \\Big||{\\nabla}{\\varphi}||{\\nabla}^3 {\\varphi}|+\n \\Big|{\\frac{ P''}{N}} \\Big||{\\nabla}^2 {\\varphi}||{\\nabla}^3 {\\varphi}|\n +\\Big|{\\frac{ P'}{N}} \\Big||{\\nabla}^3 {\\varphi}||{\\nabla}^3 {\\varphi}|\\\\\n &\\leq C {\\left( \\| \\frac{{\\nabla}{\\psi}}{\\sqrt{N}}\\|^2+\\| \\frac{{\\nabla}^2{\\psi}}{\\sqrt{N}}\\|^2\n +\\| \\frac{{\\nabla}{\\varphi}}{\\sqrt{N}}\\|^2+\\| \\frac{{\\nabla}^2{\\varphi}}{\\sqrt{N}}\\|^2 +\\| \\frac{{\\nabla}^3{\\varphi}}{\\sqrt{N}}\\|^2 \\right)},\n $$\n\n$$\\begin{aligned}\nC\\int \\Big|\\frac{P}{N}\\Big| |{\\nabla}^4 {\\varphi}| |{\\nabla}^3{\\varphi}| \n \\le\n \\frac{1}{8} \\| \\frac{{\\nabla}^4{\\varphi}}{ \\sqrt{N}} \\|^2 +C\\| \\frac{{\\nabla}^3{\\varphi}}{\\sqrt{N}}\\|^2\\end{aligned}$$\n\nand $$\\begin{aligned}\n& C\\Big|\\int {\\frac{ N'}{N}} {\\nabla}^3 {\\psi}{\\nabla}^3 {\\varphi}\\Big|\\leq C \\int | {\\nabla}^2 {\\psi}| |{\\nabla}^3 {\\varphi}|\n+C\\int |{\\nabla}^2 {\\psi}||{\\nabla}^4 {\\varphi}|\\\\\n &\\leq C \\| {{\\nabla}^2{\\psi}} \\|{\\left( \\| {{\\nabla}^3{\\varphi}} \\|+\\| {{\\nabla}^4{\\varphi}} \\| \\right)}\\leq \n C \\| {{\\nabla}^2{\\psi}} \\|^2+ C\\| {{\\nabla}^3{\\varphi}} \\|^2+\\frac 1 8 \\| {{\\nabla}^4{\\varphi}} \\|^2\\end{aligned}$$ where we used integration by parts for the last estimate.",
"\n\nThen by a similar argument, we have $$\\begin{aligned}\n &\\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} +\\int |{\\nabla}^3 {\\psi}|^2 + \\int_0^t \\int {\\frac{ |{\\nabla}^4 {\\varphi}|^2}{N}} \n \\\\\n &\\le C( \\| {\\varphi}_0\\|_{3,w}^2 + \\| {\\nabla}^3 {\\psi}_0\\|^2+\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2) +\\mbox{the cubic terms }. ",
"\\end{aligned}$$\n\nThe cubic terms are estimated by $$\\begin{aligned}\n C\\left|\\int {\\nabla}^3 ({\\nabla}{\\varphi}{\\nabla}{\\psi}) {\\frac{ {\\nabla}^3 {\\varphi}}{N}} \\right|\n&\\le C\\left| \\int {\\nabla}^2 ( {\\nabla}{\\varphi}{\\nabla}{\\psi}) {\\left( {\\frac{ {\\nabla}^4 {\\varphi}}{N}} + {{\\nabla}^3 {\\varphi}}(\\frac{1}{N})' \\right)} \\right |\\\\\n&\\le C\\left| \\int {\\nabla}^2 ( {\\nabla}{\\varphi}{\\nabla}{\\psi}) {\\frac{ {\\nabla}^4 {\\varphi}}{N}} \\right |\n+C\\left| \\int {\\nabla}^2 ( {\\nabla}{\\varphi}{\\nabla}{\\psi}) {{\\nabla}^3 {\\varphi}}(\\frac{1}{N})' \\right |.",
"\n \\end{aligned}$$ From $\\left|(\\frac{1}{N})'\\right|+\\left|(\\frac{1}{N})''\\right|\\leq \\frac{C}{N}$ (Lemma \\[wave\\_prop\\]), we can estimate each term. ",
"Indeed, recall $$\\|{\\nabla}{\\psi}\\|_{L^\\infty}+\n\\|{\\nabla}{\\varphi}\\|_{L^\\infty}\\leq C\\sqrt{M(t)}.$$ So we get $$\\begin{aligned}\n&C\\left| \\int {\\nabla}^3 {\\varphi}{\\nabla}{\\psi}{\\frac{ {\\nabla}^4{\\varphi}}{N}}\\right| + C\\left|\\int {\\nabla}{\\varphi}{\\nabla}^3 {\\psi}{\\frac{ {\\nabla}^4 {\\varphi}}{N}}\\right|\n\\le C\\sqrt{M(t)} {\\left( \\int {\\frac{ |{\\nabla}^3{\\varphi}|^2}{N}} + {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}} \\right)}+\\frac{1}{8}\\int {\\frac{ |{\\nabla}^4 {\\varphi}|^2}{N}} ,\\end{aligned}$$ $$\\begin{aligned}\nC\\left|\\int {\\nabla}^3 {\\varphi}{\\nabla}{\\psi}{ {\\nabla}^3{\\varphi}} (\\frac{1}{N})' \\right| + \\left|\\int{\\nabla}{\\varphi}{\\nabla}^3 {\\psi}{ {\\nabla}^3 {\\varphi}}(\\frac{1}{N})' \\right|\n&\\le C \\int |{\\nabla}^3 {\\varphi}||{\\nabla}{\\psi}|{\\frac{ | {\\nabla}^3{\\varphi}|}{N}} + |{\\nabla}{\\varphi}|| {\\nabla}^3 {\\psi}| {\\frac{ |{\\nabla}^3 {\\varphi}|}{N}} \\\\\n&\\le C\\sqrt{M(t)} {\\left( \\int {\\frac{ |{\\nabla}^3{\\varphi}|^2}{N}} + {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}} \\right)},\\end{aligned}$$ $$\\begin{aligned}\n& C\\left|\\int {\\nabla}^2 {\\varphi}{\\nabla}^2 {\\psi}{{\\nabla}^3 {\\varphi}} (\\frac{1}{N})' \\right|\\\\&\\,\\,\n\\leq C\\left|\\int {\\nabla}^3 {\\varphi}{\\nabla}{\\psi}{{\\nabla}^3 {\\varphi}} (\\frac{1}{N})'\\right| +C\\left|\n\\int {\\nabla}^2 {\\varphi}{\\nabla}{\\psi}{{\\nabla}^4 {\\varphi}} (\\frac{1}{N})' \\right|\n+C\\left|\\int {\\nabla}^2 {\\varphi}{\\nabla}{\\psi}{{\\nabla}^4 {\\varphi}} (\\frac{1}{N})''\\right| \\\\\n&\\,\\,\\le C \\int |{\\nabla}^3 {\\varphi}|| {\\nabla}{\\psi}|{\\frac{ |{\\nabla}^3{\\varphi}|}{N}} + |{\\nabla}^2 {\\varphi}||{\\nabla}{\\psi}| {\\frac{ |{\\nabla}^4 {\\varphi}|}{N}} \n\\le C \\sqrt{M(t)} {\\left( \n \\int {\\frac{ |{\\nabla}^3{\\varphi}|^2}{N}} + {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}}\n \\right)} + \\frac{1}{8} \\int {\\frac{ |{\\nabla}^4{\\varphi}|^2}{N}}.\\end{aligned}$$ Then the term $ \\int {\\nabla}^2{\\varphi}{\\nabla}^2 {\\psi}{\\frac{ {\\nabla}^4 {\\varphi}}{N}}$ remains. ",
"All cubic terms can be estimated very similarly except $ \\int {\\nabla}^2{\\varphi}{\\nabla}^2 {\\psi}{\\frac{ {\\nabla}^4 {\\varphi}}{N}}$. Note that by the Sobolev embedding, $$\\|f\\|_{L^4}\\leq C(\\|f\\|_{L^2}+\\|{\\nabla}f\\|_{L^2}).$$ So we estimate $$\\begin{aligned}\n\\label{k3e1}\n&C\\Big|\\int {\\nabla}^2{\\varphi}{\\nabla}^2 {\\psi}{\\frac{ {\\nabla}^4 {\\varphi}}{N}}\\Big| \n\\le C \\| {{\\nabla}^2 {\\psi}} \\|_{L^4}\n\\| \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\|_{L^4}\\| \\| \\frac {{\\nabla}^4 {\\varphi}}{\\sqrt N}\\|_{L^2} \\\\\n& \\le \\frac 18 \\| \\frac {{\\nabla}^4 {\\varphi}}{\\sqrt N}\\|^2_{L^2} + \nC {\\left( \\| {\\nabla}\\left( \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\right)\\|_{L^2} +\\| \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\|_{L^2} \\right)}^2 \n\\cdot {\\left( \\| {{\\nabla}^3 {\\psi}} \\|_{L^2} +\\| {{\\nabla}^2 {\\psi}} \\|_{L^2} \\right)}^2 \\nonumber\\\\\n& \\le \\frac 18 \\| \\frac {{\\nabla}^4 {\\varphi}}{\\sqrt N}\\|^2_{L^2} + \nC {\\left( \\| \\frac {|{\\nabla}^3 {\\varphi}|}{\\sqrt N}+|{\\nabla}^2{\\varphi}|\n{|(\\frac{1}{\\sqrt{N}})'|}\\|_{L^2} +C\\| \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\|_{L^2} \\right)}^2 \n\\cdot {\\left( \\| {{\\nabla}^3 {\\psi}} \\|_{L^2} +{\\| {{\\nabla}^2 {\\psi}} \\|_{L^2}} \\right)}^2. ",
"\\nonumber\\end{aligned}$$ Using ${|(\\frac{1}{\\sqrt{N}})'|}{\\leq \\frac{C}{\\sqrt{N}}}$ (Lemma \\[wave\\_prop\\]), we get $$\\begin{aligned}\n\\label{k3e2}& {\\left( \\| \\frac {|{\\nabla}^3 {\\varphi}|}{\\sqrt N}+|{\\nabla}^2{\\varphi}|\n {|(\\frac{1}{\\sqrt{N}})'|} \\|_{L^2}\n+C\\| \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\|_{L^2} \\right)}^2 \n\\leq \nC {\\left( \\| \\frac {{\\nabla}^3 {\\varphi}}{\\sqrt N} \\|^2_{L^2} +\\| \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\|_{L^2}^2 \\right)}\n\\leq CM(t).\\end{aligned}$$\n\nFor the ${\\epsilon}$-terms, we can write them symbolically: $${\\epsilon}\\int {\\nabla}^3({\\mathcal P}{\\psi}_z) {\\nabla}^3\\psi \\quad\\mbox{ and }\\quad\n {\\epsilon}\\int\\nabla^3(|{\\nabla}{\\psi}|^2) \\nabla^3{\\psi}.$$ After integration by parts, we can estimate these terms by $$\\begin{aligned}\n & C{\\epsilon}\\int |{\\nabla}^2({\\mathcal P}{\\psi}_z)| |{\\nabla}^4\\psi|\\leq\n C{\\epsilon}\\int (|{\\nabla}{\\psi}| +|{\\nabla}^2 {\\psi}|+|{\\nabla}^3 {\\psi}|) |{\\nabla}^4\\psi|\\\\\n &\\leq C{\\epsilon}( \\|\\nabla{\\psi}\\|^2+ \\|\\nabla^2{\\psi}\\|^2+\\|\\nabla^3{\\psi}\\|^2) +\\frac{{\\epsilon}}{4}\\|\\nabla^4{\\psi}\\|^2\\\\\\end{aligned}$$ and $$\\begin{aligned}\n&C{\\epsilon}\\int|{\\nabla}^2(|{\\nabla}{\\psi}|^2)| |{\\nabla}^4{\\psi}|\n\\leq C{\\epsilon}\\int(|\\nabla{\\psi}||\\nabla^3{\\psi}|+|\\nabla^2{\\psi}||\\nabla^2{\\psi}|) |\\nabla^4{\\psi}|\\\\\n&\\leq C{\\epsilon}\\sqrt{M(t)}\\int \\Big(|\\nabla^3{\\psi}| |\\nabla^4{\\psi}|+\\|{\\nabla}^2{\\psi}\\|^2_{L^4}\\cdot\\|{\\nabla}^4{\\psi}\\|_{L^2}\\Big)\\\\\n&\\leq C{\\epsilon}\\sqrt{M(t)} \\|\\nabla^3{\\psi}\\|^2 +\\frac{{\\epsilon}}{8}\\|\\nabla^4{\\psi}\\|^2+C{\\epsilon}\\underbrace{(\\|{\\nabla}^3{\\psi}\\|^2_{L^2}\n+\\|{\\nabla}^2{\\psi}\\|^2_{L^2})}_{\\leq C M(t)}\\cdot\\|{\\nabla}^4{\\psi}\\|_{L^2}\\\\\n&\\leq C{\\epsilon}\\sqrt{M(t)} \\|\\nabla^3{\\psi}\\|^2 +\\frac{{\\epsilon}}{4}\\|\\nabla^4{\\psi}\\|^2.\\end{aligned}$$\n\nCollecting the above estimates and using , we get the third order version of : $$\\begin{aligned}\n\\label{beforeclaims34_lemma23_} &\\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} +\\int |{\\nabla}^3 {\\psi}|^2 + \\int_0^t \\int {\\frac{ |{\\nabla}^4 {\\varphi}|^2}{N}} \\\\\n &\\le C( \\| {\\varphi}_0\\|_{3,w}^2 + \\| {\\nabla}^3 {\\psi}_0\\|^2+\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2) \n +C\\sqrt{M(t)}\\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}}. ",
" \\nonumber\n \\end{aligned}$$\n\nAs before, we claim the following two estimates for the third order derivatives: $$\\begin{aligned}\n \\label{claim3_lemma23_}\n&\\int_0^t \\int N | {\\nabla}^3 {\\psi}|^2 \n \\le C ( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2+\\|{\\nabla}{\\Delta}{\\psi}_0\\|^2) \n+ C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}}, \\\\\n& \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}} + \\int _0^ t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2 }{N}} \n\\le C ( \\| {\\nabla}{\\psi}_0 \\|_{2,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{2,w}^2)\n+ C\\int_0^t \\int {\\frac{ |{\\nabla}^4{\\varphi}|^2}{N}}. ",
"\\label{claim4_lemma23_}\\end{aligned}$$\n\n[These two claims can be proved similarly so we skip the proofs.]{} ",
"We will prove below and we will use the result in order to get . ",
"Then we will apply to close .\\\n$\\bullet$ Proof of\\\nTaking $ D{{\\nabla}\\cdot}$ to ${\\varphi}$ equation where $D$ is either $\\partial_z$ or $\\partial_y$, we have $$\\begin{aligned}\n&D{{\\nabla}\\cdot}{\\varphi}_t - s D{{\\nabla}\\cdot}{\\varphi}_z - {\\Delta}D{{\\nabla}\\cdot}{\\varphi}\\\\\n& = \\quad N D{\\Delta}{\\psi}+ \\underbrace{D N{\\Delta}{\\psi}+D\\Big({\\nabla}N \\cdot{\\nabla}{\\psi}+ {{\\nabla}\\cdot}P {{\\nabla}\\cdot}{\\varphi}+ P\\cdot{\\nabla}( {{\\nabla}\\cdot}{\\varphi}) + {{\\nabla}\\cdot}( {{\\nabla}\\cdot}{\\varphi}{\\nabla}{\\psi})\\Big)}_{R_1 }.\\end{aligned}$$\n\nWe multiply $D{\\Delta}{\\psi}$ on both sides and plug in the equation $$D{\\Delta}{\\psi}_t - sD{\\Delta}{\\psi}_z \n = D{\\Delta}({{\\nabla}\\cdot}{\\varphi})$$ so that $$\\begin{aligned}\n&N |D{\\Delta}{\\psi}|^2 = D( {{\\nabla}\\cdot}{\\varphi}_t - s{{\\nabla}\\cdot}{\\varphi}_z - {\\Delta}{{\\nabla}\\cdot}{\\varphi}) D{\\Delta}{\\psi}- \\mbox{ R}_1D{\\Delta}{\\psi}\\\\\n&= (D{{\\nabla}\\cdot}{\\varphi}D{\\Delta}{\\psi})_t - D{{\\nabla}\\cdot}{\\varphi}D{\\Delta}{\\psi}_t - s D{{\\nabla}\\cdot}{\\varphi}_z D{\\Delta}{\\psi}-\\underbrace{ D{\\Delta}{{\\nabla}\\cdot}{\\varphi}D{\\Delta}{\\psi}}_{(C)} -\\mbox{ R}_1D{\\Delta}{\\psi}\\\\\n&= (D{{\\nabla}\\cdot}{\\varphi}D{\\Delta}{\\psi})_t - D{{\\nabla}\\cdot}{\\varphi}( s D{\\Delta}{\\psi}_z + D{\\Delta}{{\\nabla}\\cdot}{\\varphi}) - s D{{\\nabla}\\cdot}{\\varphi}_z D{\\Delta}{\\psi}- (C)-\\mbox{ R}_1D{\\Delta}{\\psi}. ",
"$$ For $(C)$, we note $$\\begin{aligned}\n \\int(C)&=\\int D{\\Delta}({{\\nabla}\\cdot}{\\varphi}) D{\\Delta}{\\psi}=\\int D{\\Delta}({\\psi}_t - s {\\psi}_z ) D{\\Delta}{\\psi}=\\frac 12 {\\frac{d}{dt}}\\int |D{\\Delta}{\\psi}|^2 \\end{aligned}$$\n\nSo, integrating on the strip, we have $$\\begin{aligned}\n \\int N |D{\\Delta}{\\psi}|^2 =& {\\frac{d}{dt}}\\int D{{\\nabla}\\cdot}{\\varphi}D{\\Delta}{\\psi}+ \\int |{\\nabla}D {{\\nabla}\\cdot}{\\varphi}|^2 - \\frac 12 {\\frac{d}{dt}}\\int |D{\\Delta}{\\psi}|^2 - \\int \\mbox{ R}_1D{\\Delta}{\\psi}. ",
" \\end{aligned}$$\n\nNote that $$\\begin{aligned}\n \\int_0^t{\\left( {\\frac{d}{dt}}\\int D{{\\nabla}\\cdot}{\\varphi}D{\\Delta}{\\psi}- \\frac 12 {\\frac{d}{dt}}\\int |D{\\Delta}{\\psi}|^2 \\right)}\\leq { C(\\|{\\nabla}^2{\\varphi}(t)\\|^2 \n +\\|D{\\Delta}{\\psi}_0\\|^2+\\|{\\nabla}^2{\\varphi}_0\\|^2)}&\n \\end{aligned}$$ by $ \\int |D{{\\nabla}\\cdot}{\\varphi}D{\\Delta}{\\psi}| \\le C\\| {\\nabla}^2{\\varphi}\\| ^2 + \\frac 12 \\| D{\\Delta}{\\psi}\\|^2$.\\\nThe integral containing $R_1$ is estimated as follows; the quadratic terms: $$\\begin{aligned}\n &C\\Big|\\int (DN)({\\Delta}{\\psi})(D{\\Delta}{\\psi})\\Big|\\leq C \\int |{\\Delta}{\\psi}|^2 + \\frac 18 \\int N |D{\\Delta}{\\psi}|^2,\\\\\n &C\\Big|\\int D( {\\nabla}N \\cdot {\\nabla}{\\psi}) D{\\Delta}{\\psi}\\Big| \\le C\\int {\\frac{ |{\\nabla}{\\psi}|^2 +|{\\nabla}^2{\\psi}|^2}{N}} + \\frac 18 \\int N |D{\\Delta}{\\psi}|^2, \\\\\n&C \\Big|\\int D({\\mathcal P}' {{\\nabla}\\cdot}{\\varphi}) D{\\Delta}{\\psi}+ D({\\mathcal P}( {{\\nabla}\\cdot}{\\varphi}_z))D {\\Delta}{\\psi}\\Big| \\le \nC {\\left( \\int {\\frac{ |{\\nabla}{\\varphi}|^2+|{\\nabla}^2 {\\varphi}|^2+|{\\nabla}^3 {\\varphi}|^2}{N}} \\right)} + \\frac 18 \\int N |D{\\Delta}{\\psi}|^2\\\\\n& \\mbox{ and the cubic term: } C\\Big|\\int D({{\\nabla}\\cdot}({{\\nabla}\\cdot}{\\varphi}{\\nabla}{\\psi}) )D{\\Delta}{\\psi}\\Big|\\\\& \\le C ( \\| {\\nabla}{\\psi}\\|_{L^{\\infty}} + \\| {\\nabla}{\\varphi}\\|_{L^{\\infty}})\n {\\left( C \\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} + C\\int {\\frac{ |D{\\Delta}{\\psi}|^2}{N}} + \\frac 14 \\int N |{\\Delta}{\\psi}|^2 \\right)}\n + C\\int |{\\nabla}^2{\\varphi}||{\\nabla}^2{\\psi}||D{\\Delta}{\\psi}|\\\\\n& \\leq C \n \\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} + C\\sqrt{M(t)}\\int {\\frac{ |D{\\Delta}{\\psi}|^2}{N}} + \\frac 14 \\int N |D{\\Delta}{\\psi}|^2\n + C\\int |{\\nabla}^2{\\varphi}||{\\nabla}^2{\\psi}||D{\\Delta}{\\psi}| \n \\end{aligned}$$ for small $M(t)$. The last term in the above can be estimated:\n\n$$\\begin{aligned}\n&C\\int |{\\nabla}^2{\\varphi}||{\\nabla}^2{\\psi}||D{\\Delta}{\\psi}|=C\\int \\frac{|{\\nabla}^2{\\varphi}|}{\\sqrt{N}}{|{\\nabla}^2{\\psi}|}\\sqrt{N}|D{\\Delta}{\\psi}|\n\\le C \\| {{\\nabla}^2 {\\psi}} \\|_{L^4}\n\\| \\frac{{\\nabla}^2 {\\varphi}}{\\sqrt{N}}\\|_{L^4}\\| \\| \\sqrt{N} D{\\Delta}{\\psi}\\|_{L^2} \\\\\n& \\le \\frac 18 \\| \\sqrt{N} D{\\Delta}{\\psi}\\|^2_{L^2} + \nC {\\left( \\| {\\nabla}\\left( \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\right)\\|_{L^2} +\\| \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\|_{L^2} \\right)}^2 \n\\cdot {\\left( \\| {{\\nabla}^3 {\\psi}} \\|_{L^2} +\\| {{\\nabla}^2 {\\psi}} \\|_{L^2} \\right)}^2\\\\\n& \\le \\frac 18 \\| \\sqrt{N} D{\\Delta}{\\psi}\\|^2_{L^2} + \nC \\underbrace{{\\left( \\| \\frac {|{\\nabla}^3 {\\varphi}|}{\\sqrt N}+|{\\nabla}^2{\\varphi}|\n{|(\\frac{1}{\\sqrt{N}})'|}\\|_{L^2} +C\\| \\frac {{\\nabla}^2 {\\varphi}}{\\sqrt N}\\|_{L^2} \\right)}^2}_{\\leq CM(t)}\n\\cdot {\\left( \\| {{\\nabla}^3 {\\psi}} \\|_{L^2} +{\\| {{\\nabla}^2 {\\psi}} \\|_{L^2}} \\right)}^2.\\\\\\end{aligned}$$\n\nUp to now, using , we have $$\\begin{aligned}\n&\\int_0^t \\int N |D{\\Delta}{\\psi}|^2 \\le C ( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2+\\|D{\\Delta}{\\psi}_0\\|^2) \n+ C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}}. ",
" \\end{aligned}$$ For ${\\epsilon}$-terms, we estimate them by $$\\begin{aligned}\n &{\\epsilon}\\int {\\left( D{\\Delta}( \\mbox{R}_2 ) D{\\Delta}{\\psi}- D{{\\nabla}\\cdot}{\\varphi}( \n D{\\Delta}\\mbox{R}_2) \\right)} \\leq -{\\epsilon}\\int|{\\nabla}D{\\Delta}{\\psi}|^2\\\\\n &+ {\\epsilon}\\int {\\left( D{\\Delta}( -2 P \\cdot{\\nabla}{\\psi}- |{\\nabla}{\\psi}|^2 ) D{\\Delta}{\\psi}- D{{\\nabla}\\cdot}{\\varphi}( \n D{\\Delta}{\\left( {\\Delta}{\\psi}-2 P \\cdot{\\nabla}{\\psi}- |{\\nabla}{\\psi}|^2 \\right)}) \\right)}. ",
" \\end{aligned}$$ For the last two terms, we estimate $$\\begin{aligned}\n &{\\epsilon}\\int -{\\Delta}D( 2 P \\cdot{\\nabla}{\\psi}+ |{\\nabla}{\\psi}|^2)D{\\Delta}{\\psi}={\\epsilon}\\int {\\nabla}D( 2 P \\cdot{\\nabla}{\\psi}+ |{\\nabla}{\\psi}|^2){\\nabla}D{\\Delta}{\\psi}\\\\\n&\\leq C{\\epsilon}{\\left( \\|{\\nabla}{\\psi}\\|^2+\\|{\\nabla}^2{\\psi}\\|^2+\\|{\\nabla}^3{\\psi}\\|^2 \\right)}+\\frac{{\\epsilon}}{4}\\|{\\nabla}D{\\Delta}{\\psi}\\|^2\\\\\n$$ and\n\n$$\\begin{aligned}\n&- {\\epsilon}\\int D{{\\nabla}\\cdot}{\\varphi}( {\\Delta}D{\\left( {\\Delta}{\\psi}-2 ( P \\cdot{\\nabla}{\\psi}) - |{\\nabla}{\\psi}|^2 \\right)})\n={\\epsilon}\\int{\\nabla}D{{\\nabla}\\cdot}{\\varphi}( {\\nabla}D{\\left( {\\Delta}{\\psi}-2 ( P \\cdot{\\nabla}{\\psi}) - |{\\nabla}{\\psi}|^2 \\right)})\\\\\n&\\leq C{\\epsilon}\\int|{\\nabla}^3{\\varphi}|{\\left( |{\\nabla}D{\\Delta}{\\psi}| + |{\\nabla}{\\psi}| + |{\\nabla}^2 {\\psi}|+ |{\\nabla}^3 {\\psi}|+ |{\\nabla}^3 {\\psi}||{\\nabla}{\\psi}|+ |{\\nabla}^2{\\psi}|^2 \\right)}\\\\\n&\\leq \\frac{\\epsilon}4\\|{\\nabla}D{\\Delta}{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}^3{\\varphi}\\|^2+C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+C{\\epsilon}\\|{\\nabla}^2{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}^3{\\psi}\\|^2\n+C{\\epsilon}{ \\|{\\nabla}^2{\\psi}\\|_{L^4}^2}. ",
"$$\n\nNote that the last term $C{\\epsilon}\\|{\\nabla}^2{\\psi}\\|_{L^4}^2$ is controlled by $ C{\\epsilon}(\\|{\\nabla}^2{\\psi}\\|^2 + \\|{\\nabla}^3{\\psi}\\|^2)\\leq C{\\epsilon}M(t)$. As a result, we get $$\\begin{aligned}\n {\\epsilon}\\int {\\left( D{\\Delta}( \\mbox{R}_2 ) D{\\Delta}{\\psi}- D{{\\nabla}\\cdot}{\\varphi}( \n D{\\Delta}\\mbox{R}_2) \\right)} \\leq & -\\frac {\\epsilon}2 \\|{\\nabla}D{\\Delta}{\\psi}\\|^2 \\\\&+C{\\epsilon}{\\left( \\|{\\nabla}^3{\\varphi}\\|^2+ \\|{\\nabla}{\\psi}\\|^2\n + \\|{\\nabla}^2{\\psi}\\|^2+ \\|{\\nabla}^3{\\psi}\\|^2 \\right)}.\\\\\n \\end{aligned}$$\n\nWe use again to get $$\\begin{aligned}\n&\\int_0^t \\int N |{\\Delta}D {\\psi}|^2 +{\\epsilon}\\int_0^t \\int |{\\nabla}D{\\Delta}{\\psi}|^2 \\\\&\\le C ( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2+\\|D{\\Delta}{\\psi}_0\\|^2) \n+ C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}}.\\end{aligned}$$\n\nWe replace $D$ with $\\partial_z$ and $\\partial_y$ and add these two estimates to get $$\\begin{aligned}\n\\label{claim3_middle}&\\int_0^t \\int N |{\\Delta}{\\nabla}{\\psi}|^2 \n \\le C ( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2+\\|{\\nabla}{\\Delta}{\\psi}_0\\|^2) \n+ C\\sqrt{M(t)} \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}}. ",
"\\end{aligned}$$ To get from the above estimate, we have to estimate $ \\int N |{\\nabla}^3{\\psi}|^2=\\int N |{\\nabla}^2 ({\\nabla}{\\psi})|^2$ from $\\int N |{\\Delta}({\\nabla}{\\psi})|^2$ with lower order estimates. ",
"We apply the estimate by replacing ${\\psi}$ with ${\\nabla}{\\psi}$ then we get $$\\begin{aligned}\n \\int N |{\\nabla}^2({\\nabla}{\\psi})|^2\n&\\leq 2\\int N |{\\Delta}({\\nabla}{\\psi})|^2 \n+ C\\int N |{\\nabla}({\\nabla}{\\psi})|^2 \\\\\n \\end{aligned}$$\n\nSo $$\\int_0^t\\int N |{\\nabla}^3{\\psi}|^2 \\le \n C\\int_0^t\\int N |{\\Delta}{\\nabla}{\\psi}|^2 + C ( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2)$$ by . ",
"Together with , we proved .\\\n$\\bullet$ Proof of\\\nMultiplying $ w {\\nabla}^3{\\psi}$ to the equation $${\\nabla}^3 {\\psi}_t - s {\\nabla}^3{\\psi}_z \n= {\\nabla}^3 ({{\\nabla}\\cdot}{\\varphi}),\n$$ we have $$\\begin{aligned}\n&\\frac 12 (w |{\\nabla}^3 {\\psi}|^2)_t - \\frac s2 ( w |{\\nabla}^3 {\\psi}|^2)_z + \\frac s2 w' |{\\nabla}^3 {\\psi}|^2 = w {\\nabla}^3 ({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}^3 {\\psi}.",
"\n$$\n\nAs before, we will use $$\\begin{aligned}\nw' = se^{sz} > c w & \\quad \\mbox{for } z > 0, \\\\\nw = 1+e^{sz} \\le 2\n< \\frac{1}{c}N \n & \\quad \\mbox{for } z\\le0\\end{aligned}$$ where $c>0$ is a constant depending only on $s>0$. Integrating on each half strip and in time, we get $$\\begin{aligned}\n\\int _{z>0} w |{\\nabla}^3 {\\psi}|^2 &\n \\le \\int_{z>0} w |{\\nabla}^3 {\\psi}_0|^2 + \\int_0^t \\int_{z>0} w {\\nabla}^3 ({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}^3 {\\psi}-\n\\frac{c}{2}\\int_0^t \\int _{z>0} w |{\\nabla}^3 {\\psi}|^2 \\\\&\\quad -\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^3 {\\psi}|^2(0, y) dy\n\\\\\n& \\le \\int _{z>0} w |{\\nabla}^3 {\\psi}_0|^2 -\\frac c2 \\int_0^t \\int_{z>0} w |{\\nabla}^3 {\\psi}|^2 +\n C \\int_0^t \\int_{z>0} w |{\\nabla}^3({\\nabla}\\cdot {\\varphi})|^2 \\\\&\\quad -\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^3 {\\psi}|^2(0, y) dy,\n \\end{aligned}$$ $$\\begin{aligned}\n \\int _{z<0} w |{\\nabla}^3 {\\psi}|^2 \\le & \\int_{z<0} w |{\\nabla}^3 {\\psi}_0|^2 + \\int_0^t \\int_{z<0} w {\\nabla}^3 ({{\\nabla}\\cdot}{\\varphi}) \n \\cdot {\\nabla}^3 {\\psi}\\\\&\\quad \n +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^3 {\\psi}|^2(0, y) dy\n \\\\\n&\\le \\int_{z<0} w |{\\nabla}^3 {\\psi}_0|^2 + C\\int_0^t \\int_{z<0} | {\\nabla}^4 {\\varphi}|||{\\nabla}^3 {\\psi}| \n \\\\&\\quad +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^3 {\\psi}|^2(0, y) dy\n \\\\\n &\\le \\int_{z<0} w |{\\nabla}^3 {\\psi}_0|^2 + C\\int_0^t \\int_{z<0} N|{\\nabla}^3 {\\psi}|^2 \n + C\\int_0^t \\int_{z<0} { {\\frac{ |{\\nabla}^4 {\\varphi}|^2}{N}}}\n \\\\&\\quad +\\frac s2\\int_0^t \\int_{[0,{\\lambda}]} w |{\\nabla}^3 {\\psi}|^2(0, y) dy.",
"\n $$\n\nAdding the above two estimates, we get $$\\begin{aligned}\n&\\int w |{\\nabla}^3 {\\psi}|^2 + \\frac c2 \\int_0^t \\int w |{\\nabla}^3 {\\psi}|^2 \\le \\int w |{\\nabla}^3 {\\psi}_0|^2\n+ C\\int_0^t \\int \\Big(N |{\\nabla}^3{\\psi}|^2 + \n w |{\\nabla}^4 {\\varphi}|^2 \\Big ).",
"\n$$\n\nFor ${\\epsilon}$[-terms]{}, as before, we estimate\n\n$$\\begin{aligned}\n&\\int {{\\epsilon}\\mbox{-terms}}={\\epsilon}\\int { w {\\nabla}^3 {\\psi}\\cdot{\\left( {\\Delta}{\\nabla}^3{\\psi}-2 {\\nabla}^3( P \\cdot{\\nabla}{\\psi}) - {\\nabla}^3(|{\\nabla}{\\psi}|^2) \\right)}}\\\\\n&={\\epsilon}\\int \\Big({ -w|{\\nabla}^4{\\psi}|^2 -w' {\\nabla}^3 {\\psi}\\cdot {\\nabla}^3{\\psi}_z\n- w {\\nabla}^3 {\\psi}\\cdot{\\left( 2 {\\nabla}^3( P \\cdot{\\nabla}{\\psi}) \\right)} \n+ (w'{\\nabla}^2\\psi_z+w{\\Delta}{\\nabla}^2{\\psi})\n{\\nabla}^2(|{\\nabla}{\\psi}|^2)}\\Big)\\\\\n&\\leq-{\\epsilon}\\int w|{\\nabla}^4{\\psi}|^2 +C{\\epsilon}\\int \\Big(w |{\\nabla}^3 {\\psi}|| {\\nabla}^4{\\psi}| + w| {\\nabla}^3 {\\psi}|( |{\\nabla}^4{\\psi}|+|{\\nabla}^3{\\psi}|+|{\\nabla}^2{\\psi}|+|{\\nabla}{\\psi}| )\\\\ &\\quad\n+w |{\\nabla}^3 {\\psi}| {\\left( |{\\nabla}^3 {\\psi}||{\\nabla}{\\psi}|+ |{\\nabla}^2 {\\psi}||{\\nabla}^2{\\psi}| \\right)}+w|{\\nabla}^4{\\psi}| {\\left( |{\\nabla}^3 {\\psi}||{\\nabla}{\\psi}|+ |{\\nabla}^2 {\\psi}||{\\nabla}^2{\\psi}| \\right)}\\Big)\\\\\n&\\leq-\\frac {\\epsilon}4\\int w|{\\nabla}^4{\\psi}|^2 +C{\\epsilon}\\int w \\sum_{k=1}^3 |{\\nabla}^k {\\psi}|^2\n+C{\\epsilon}\\int \\Big(w |{\\nabla}^3 {\\psi}| |{\\nabla}^2 {\\psi}|^2+w|{\\nabla}^4{\\psi}| |{\\nabla}^2 {\\psi}|^2\\Big).\\\\\n $$\n\nwhere we used the estimate $|w'|\\leq C|w|$. For the last term(the cubic term), we estimate $$\\begin{aligned}\n&C{\\epsilon}\\int \\Big(w |{\\nabla}^3 {\\psi}| |{\\nabla}^2 {\\psi}|^2+w|{\\nabla}^4{\\psi}| |{\\nabla}^2 {\\psi}|^2\\Big)\n\\le C{\\epsilon}{\\left( \\|\\sqrt{w}{\\nabla}^3{\\psi}\\|+\\|\\sqrt{w}{\\nabla}^4{\\psi}\\| \\right)}\\|\\sqrt{w}{\\nabla}^2{\\psi}\\|_{L^4}\\| {\\nabla}^2{\\psi}\\|_{L^4}\\\\\n&\\le C{\\epsilon}{\\left( \\|\\sqrt{w}{\\nabla}^3{\\psi}\\|+\\|\\sqrt{w}{\\nabla}^4{\\psi}\\| \\right)}{\\left( \\||\\sqrt{w}{\\nabla}^3{\\psi}|+\\underbrace{|(\\sqrt{w})'|}_{\n\\leq C\\sqrt{w}}\n|{\\nabla}^2{\\psi}|\\|+\\|\\sqrt{w}{\\nabla}^2{\\psi}\\| \\right)}(\\| {\\nabla}^3{\\psi}\\|+\\| {\\nabla}^2{\\psi}\\|)\\\\\n&\\le C {\\epsilon}{\\left( \\|\\sqrt{w}{\\nabla}^3{\\psi}\\|+\\|\\sqrt{w}{\\nabla}^4{\\psi}\\| \\right)}\\cdot\\sqrt{ M(t)}\\cdot\n(\\| {\\nabla}^3{\\psi}\\|+\\| {\\nabla}^2{\\psi}\\|)\\\\\n& \\le \\frac {\\epsilon}8\\int w|{\\nabla}^4{\\psi}|^2 +C{\\epsilon}\\int w\\sum_{k=2}^3 |{\\nabla}^k {\\psi}|^2\\end{aligned}$$ for small $M(t)$.\\\n\nUsing the estimate and the previous claim , we obtain$$\\begin{aligned}\n&\\int w |{\\nabla}^3 {\\psi}|^2 + (\\frac{c}{2}-C\n\\sqrt{M(t)})\n \\int_0^t \\int w |{\\nabla}^3 {\\psi}|^2 \\\\\n&\\le C (\\| {\\nabla}{\\psi}_0\\|_{2,w}^2+\n \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{2,w}^2\n)+ C \\int_0^t \\int {\\frac{ |{\\nabla}^4 {\\varphi}|^2}{N}}. ",
"\\end{aligned}$$ By making $M(t)$ small enough, we proved the claim .\\\n\nNow we can finish the proof of Lemma \\[lemma23\\_\\] fully. ",
"Indeed plugging into with $M(t)$ small, we have $$\\begin{aligned}\n &\\| {\\nabla}^3{\\varphi}\\|_w^2 + \\| {\\nabla}^3{\\psi}\\|^2 + \\int_0^t \\| {\\nabla}^4 {\\varphi}\\|_w^2\n \\le C ( \\| {\\nabla}{\\psi}_0\\|_{2,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{3,w}^2). ",
"$$ So from , we have $$\\begin{aligned}\n\\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}} + \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}} \n \\le C ( \\| {\\varphi}_0\\|_{3,w}^2 + \\| {\\nabla}{\\psi}_0\\|_{2,w}^2\n+ \\| {\\psi}_0\\|^2).\\end{aligned}$$ This proves Lemma \\[lemma23\\_\\] for case $k=i+j=3$.\n\nFinally, we obtain Proposition \\[uniform\\_\\]:\n\nAdding Lemma \\[lemma01\\_\\] and Lemma \\[lemma23\\_\\] for $k=2,3$, we have\\\n$$\\begin{split}\\label{lemma3_}\n& {\\sup_{s\\in[0,t]}\\| {\\varphi}(s)\\|_{3,w}^2 + \\| {\\nabla}{\\psi}(s)\\|_{2,w}^2 + \\| {\\psi}(s)\\|^2} + \\int_0^t \\sum_{l = 1}^4 \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^t \\sum_{l = 1}^3\\| {\\nabla}^l {\\psi}\\|_w^2\n\\\\\n& \\le\n C { ( \\| {\\varphi}_0\\|_{3,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{2,w}^2+ \\| {\\psi}_0\\|^2)}.",
"\n\\end{split}$$ This proves Proposition \\[uniform\\_\\].",
"\n\nNote that the above estimate does not contain any control of $\\|{\\psi}\\|_w$. This is not a problem when ${\\epsilon}>0$. Indeed, for ${\\epsilon}>0$, thanks to the Dirichlet condition (in $y$), we can use the Poincaré inequality (see ) to get\n\n$$\\| {\\psi}\\|^2_w=\\int \\frac{|{\\psi}|^2}{N}\\leq C{\\lambda}^2\\int \\frac{|{\\psi}_y|^2}{N}\\leq \\| {\\nabla}{\\psi}\\|_{w}^2\\leq \nC { ( \\| {\\varphi}_0\\|_{3,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{2,w}^2+ \\| {\\psi}_0\\|^2)}.$$ However, when ${\\epsilon}=0$, the periodic condition (in $y$) is not enough to get such an estimate.",
"\n\n \\\n\nProof of Proposition \\[zerolocal\\] {#pf_local_exist}\n----------------------------------\n\nLet us remind the equation : $$\\begin{aligned}\n\\label{eq00:phipsi} \\begin{aligned}\n\\varphi_t - s \\varphi_z - \\Delta \\varphi\n& = N {\\nabla}\\psi + P{\\nabla}\\cdot \\varphi + {\\nabla}\\cdot \\varphi {\\nabla}\\psi, \\\\\n\\psi_t - s \\psi_z \n &= \n {{\\nabla}\\cdot}\\varphi\n\\end{aligned}\\end{aligned}$$ with data $(\\varphi_0, \\psi_0) $ satisfying $ \\| \\varphi_0\\|^2_{H^3_{w,p}} + \\| \\psi_0\\|^2_{H^3_p} + \\|{\\nabla}\\psi_0\\|^2_{H^2_{w,p}} < M $. ",
"First we shall find the local solution $\\Phi = (\\varphi, {\\psi})$ of in $$X_T = \\{ \\Phi \\,|\\, {\\varphi}\\in L^{\\infty}((0, T): H^3_p) \\cap L^2((0, T): \\dot{H}_p^4), \\psi \\in L^{\\infty}((0, T): H^3_p)\\}$$ $$\\| \\Phi\\|_{X_T} = \\sup_{ 0<t <T} \\| \\Phi\\|_{H^3_p} + \\int_0^T \\| {\\nabla}{\\varphi}\\|_{H^3_p} ds$$ for a sufficiently small $T$.\n\nStep 1. ",
"We define the linear map $\\mathcal F:X_T \\to X_T $ mapping ${\\tilde\\Phi:=}\\color{black}( \\tilde \\varphi, \\tilde {\\psi})$ to ${{\\Phi}:=}\\color{black}(\\varphi, {\\psi})$. $$\\begin{aligned}\n\\label{eq:linear}\\begin{aligned}\n\\varphi_t - s \\varphi_z - \\Delta \\varphi\n& = N {\\nabla}\\psi + P{\\nabla}\\cdot \\varphi + {\\nabla}\\cdot \\tilde \\varphi {\\nabla}\\tilde \\psi, \\\\\n\\psi_t - s \\psi_z\n &= {{\\nabla}\\cdot}\\varphi,\\\\\n(\\varphi, \\psi)|_{t=0}& = (\\varphi_0, \\psi_0).",
"\n\\end{aligned}\\end{aligned}$$ We approximate the above linear system by Galerkin method. ",
"Since $H^3_p (\\Omega)$ is a separable Hilbert space, there exists an orthonormal basis $ {\\omega}_j, j=1, 2, 3 \\dots$ of $H^3_p$. We define $\\Phi^k = ( \\varphi^{k1}, \\varphi^{k2}, \\psi^k)^t $ by $$( \\varphi^{k1},\\varphi^{k2}, \\psi^k )^t= \\sum_{j=1}^k {\\omega}_j g^{jk}(t),$$ where $g^{jk} = (g^{jk}_1, g^{jk}_2, g^{jk}_3)^t $ solves the following ode system $$\\label{eq:ode}\n {\\frac{d}{dt}}{\\langle\\!\\langle}\\Phi^k, {\\omega}_j{\\rangle\\!\\rangle}+ {\\langle\\!\\langle}B\\Phi^k, {\\omega}_j {\\rangle\\!\\rangle}= \n {\\langle\\!\\langle}f, {\\omega}_j {\\rangle\\!\\rangle}$$ with $$B = \\begin{pmatrix}\n-{\\Delta}-s\\partial_z +{\\mathcal P}{\\partial}_z & 0 & -N{\\partial}_z \\\\ 0& -{\\Delta}-s\\partial_z & -N{\\partial}_y \\\\\n-{\\partial}_z &-{\\partial}_y& -s\\partial_z\n\\end{pmatrix}, \\quad f = \\begin{pmatrix} {{\\nabla}\\cdot}\\tilde {\\varphi}{\\partial}_z \\tilde {\\psi}\\\\\n {{\\nabla}\\cdot}\\tilde {\\varphi}{\\partial}_y \\tilde {\\psi}\\\\ 0 \\end{pmatrix}.$$ Here, ${\\langle\\!\\langle}\\Phi, w {\\rangle\\!\\rangle}$ is the vector valued function defined by $${\\langle\\!\\langle}\\Phi, w {\\rangle\\!\\rangle}=\n ( \\langle \\Phi_1, w\\rangle, \\langle \\Phi_2, w\\rangle, \\langle \\Phi_3, w\\rangle)^t,$$ where $ \\langle \\cdot, \\cdot \\rangle$ is the $H^3$ inner product of scalar valued functions, that is $$\\langle f, g \\rangle = \\sum_{i+j \\le 3}\\int {\\partial}_y^i {\\partial}_z^j f \\overline{{\\partial}_y^i {\\partial}_z^j g} dydz.$$ The initial data $g^{jk}(0)$ is determined by $\\Phi_0 = \\sum_{j=1}^{\\infty} c^{j}w_j$ such that $$g^{jk}(0) = c^j \\quad 0\\le j\\le k \\mbox{ for any } k\\ge 0.$$ The determines the $ 3k \\times 3k $ system of ordinary differential equations with respect to $ g_l^{jk}, 1\\le j\\le k, l= 1, 2, 3$, hence $g_l^{jk}$ exists globally. ",
"By multiplying $(g^{jk})^t$ on the left to and summing over $j$, we have also $$\\label{eq:kthinner}\n\\frac{1}{2} {\\frac{d}{dt}}\\langle \\Phi^k , \\Phi^k \\rangle + \\langle B\\Phi^k, \\Phi^k \\rangle =\n \\langle f, \\Phi^k \\rangle .$$ (The inner product of vectors are defined componentwise.)\\\nStep 2. ",
"We apply the integration by parts to the term $\\langle B\\Phi^k, \\Phi^k \\rangle$, and by usual energy estimates we arrive at $$\\| \\Phi^k\\|^2_{H^3_p} + \\int_0^ t \\| {\\nabla}\\varphi^k \\|^2_{H^3_p} ds \\le C \\| \\Phi_0\\|^2_{H^3_p} +\nC \\int_0^t (\\|\\Phi^k\\|_{H^3_p} + \\| {\\nabla}\\tilde {\\varphi}\\|_{H^3_p}^2 + \\| \\tilde \\Phi\\|_{H^3_p}^2 )ds$$ with $C$ depending on $\\| ({\\mathcal P}, N)\\|_{L^{\\infty}}$. Finally by Gronwall’s inequality, we have the uniform estimate of $\\Phi^k$ in $X_T$ for a small $T$ depending only on $\\Phi_0$ and $\\|\\Phi\\|_{X_T}$. Then $\\Phi^k$ converges weakly to $\\Phi\\in X_T$, which is a distribution solution of . ",
"Note that is linear, so the weak convergence is sufficient for passing to the limit. ",
"Moreover $\\Phi$ satisfies $${\\frac{d}{dt}}\\Phi + B\\Phi = f \\,\\, a.e \\,\\mbox{ on } (0, T), \\quad \\Phi(0)= \\Phi_0.$$ for which we refer to Theorem $1.1$ in Chapter $3$ of [@Te]. ",
"From the equation above, we have also $\\Phi = ({\\varphi}, {\\psi}) \\in \nW^{1, \\infty}((0, T): H^1_p\\times H^1_p)$. In particular, $\\Phi$ is in $C([0, T): H^3_p\\times H^3_p)$. Step 3. ",
"We can write into the following Duhamel’s formula. ",
"$$\\Phi = S(t) \\Phi_0 + \\int_0^t S(t-s)( \\mbox{ RHS of }\\eqref{eq:linear}) (s) ds,$$ where $S(t)\\Phi_0 = ( e^{-t{\\Delta}}\\varphi_0, e^{-{\\epsilon}t{\\Delta}} {\\psi}_0)^t$. The injectivity of the map $\\mathcal F$ is clear from the formula above. ",
"We proceed as usual. ",
"Let $\\| \\Phi_0\\|_{H^3_p \\times H^3_p} \\le M/2$ and $\\| \\tilde \\Phi\\|_{X_T} \\le M$. We can show by heat kernel estimates $$\\| \\Phi\\|_{X_T} \\le M, \\quad \n\\| \\Phi - \\Psi\\|_{X_T} \\le \\frac 12 \\| \\tilde \\Phi - \\tilde \\Psi\\|_{X_T}$$ if $T$ is sufficiently small. ",
"By the fixed point theorem we obtain the solution of in $X_T$. Then the similar energy estimates as presented in Section 3 show that the solution $(\\varphi, \\psi)$ satisfies $ \\varphi \\in H^3_{w,p}, {\\nabla}{\\psi}\\in H^3_{w, p}, \\psi \\in H^3_p, {\\nabla}\\psi \\in H^2_{w,p}$ and holds, for which we omit details.",
"\n\n \\\n\nProof of Theorem \\[theoremli\\] {#sectionuniformwith}\n==============================\n\nIn this section we introduce the proposition \\[uniform\\_with\\] below which implies Theorem \\[theoremli\\].",
"\n\n\\[uniform\\_with\\] There exist constants ${\\epsilon}_0>0,\\,\n C_0$, and ${\\lambda}_0>0$ such that for $ 0< {\\epsilon}\\leq {\\epsilon}_0$, and ${\\lambda}\\in(0,{\\lambda}_0]$, we have the following:\\\nIf $(\\varphi, \\psi)$ be a local solution of on $[0,T]$ for some $T>0$ with $M(T)<\\infty$ [for some initial data $(\\varphi_0, \\psi_0)$ which is ${\\lambda}$-periodic in $y$ and which has the zero average in $y$ ($\\bar{\\varphi_0}=\\bar{\\psi_0}=0$)]{}, then we have $$M(T) + \\int_0^{T} \\sum_{l = 1}^4 \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^{T} \\sum_{l = 1}^3\\| {\\nabla}^l {\\psi}\\|_w^2 + {\\epsilon}\\int_0^{T} \\| {\\nabla}^4 {\\psi}\\|_w^2 \\le C_0M(0).$$\n\n1. ",
" [For this linear case, we do not need any smallness of $M(T)$ since such a condition can pop up only from the nonlinear term estimate which is not present in this linear case .]{}",
"\n\n2. ",
" As in Proposition \\[uniform\\_\\], $C_0$ does not depend on the size of $T>0$.\n\nFor the proof we shall proceed the similar line of arguments as for Propotision \\[uniform\\_\\] with a minimal modification. ",
"First we recall that we assumed $\\bar {\\varphi}_0=0$ and $ \\bar {\\psi}_0=0$. Since the coefficient functions in the system depend on $z$ only, the zero average conditions are preserved over time. ",
"It enables us to use the Poincaré inequality to $({\\varphi}, {\\psi})$ with respect to $y-$variable: $$\\| {\\varphi}(z,\\cdot_y) \\|_{L^2_y([0,{\\lambda}])} \\le C{\\lambda}\\| {\\partial}_{y} {\\varphi}(z,\\cdot_y)\\|_{L^2_y([0,{\\lambda}])}, \\quad\n\\| {\\psi}(z,\\cdot_y) \\|_{L^2_y([0,{\\lambda}])} \\le C{\\lambda}\\| {\\partial}_{y} {\\psi}(z,\\cdot_y)\\|_{L^2_y([0,{\\lambda}])},\n\\, z\\in\\mathbb{R}.$$ Second we collect some properties of traveling waves $(N, \\mathcal P)$. Note that ${\\epsilon}>0$ and $N, P= ({\\mathcal P}, 0) $ solves $$\\begin{aligned}\n\\label{npp}\\begin{aligned} \n- sN'- N''& = (N{\\mathcal P})', \\\\\n-s{\\mathcal P}' - {\\epsilon}{\\mathcal P}''& = -2{\\epsilon}{\\mathcal P}{\\mathcal P}'+ N'\n\\end{aligned}\\end{aligned}$$ with the boundary condition . ",
"All the properties in Lemma \\[wave\\_prop\\] hold still true when ${\\epsilon}>0$.\n\n\\[wave\\_propwith\\] For ${\\epsilon}_0>0$, there exists a constant $M>0$ such that for any $0<{\\epsilon}\\leq{\\epsilon}_0$, we have:\\\n$$\\frac{w}{M}\\leq\\frac{1}{N }\\leq Mw,$$ $$|{N^{ (k)}}| \\leq M, \\quad |{{\\mathcal P}^{(k)}}| \\leq M,\\quad \\mbox{ for } 0\\leq k\\le 2,\\quad \\mbox{ and }$$ $$\\frac{N'}{N}=-({\\mathcal P}+s), \\quad \n\\Big|(\\frac{1}{N})'\\Big|+\\Big|(\\frac{1}{N})''\n\\Big|\\leq \\frac{M}{N},\\quad \n\\Big|(\\frac{1}{\\sqrt {N}})'\\Big|\\leq \\frac{M}{\\sqrt {N}}.$$ ($N^{(k)}$ is any $k-$th derivative of $N$)\n\nThe first inequality follows Lemma \\[KPPtheorem\\]. ",
"For the second line, it is obvious that $k=0$ case holds. ",
"If $|\\mathcal P^{(k)}|<M$ $(k=1,2)$ are shown, the other items follow the equation . ",
"In the below we present the proof of $|\\mathcal P^{(k)}|<M$ $(k=1,2)$.\\\n$\\bullet$ $ k=1$\\\nThis case $|\\mathcal P'|<C$ was proved in [@LiLiWa]. ",
"We sketch its proof for readers convenience.\\\nMultiplying ${{\\mathcal P}''}$ to the second equation of and integraing over $(-\\infty, z)$, we have $$\\begin{aligned}\n \\frac s2 {{\\mathcal P}'}^2 + {\\epsilon}{\\int_{-\\infty}^z{{{\\mathcal P}''}^2} dz}& \\le {\\int_{-\\infty}^z{( {\\epsilon}{{\\mathcal P}}^2-N)'{{\\mathcal P}''}} dz} \\\\\n & ={\\epsilon}{\\int_{-\\infty}^z{2 {{\\mathcal P}}{{\\mathcal P}'}{{\\mathcal P}''}} dz} - {\\int_{-\\infty}^z{N'{{\\mathcal P}''}} dz}\\\\\n &= {\\epsilon}{\\int_{-\\infty}^z{{{\\mathcal P}}(({{\\mathcal P}'})^2)'} dz} + {\\int_{-\\infty}^z{ (-sN' - (N{{\\mathcal P}})'){{\\mathcal P}'}} dz} - N'{{\\mathcal P}'}\\\\\n & = {\\epsilon}{{\\mathcal P}}{{\\mathcal P}'}^2 - {\\epsilon}{\\int_{-\\infty}^z{{{\\mathcal P}'}^3} dz} - s {\\int_{-\\infty}^z{N'{{\\mathcal P}'}} dz} - {\\int_{-\\infty}^z{(N{{\\mathcal P}'}^2+N'{{\\mathcal P}}{{\\mathcal P}'})} dz}- N'{{\\mathcal P}'}\\\\\n &\\le -s {\\int_{-\\infty}^z{N'{{\\mathcal P}'}} dz} + \\frac s4 {{\\mathcal P}'}^2 + \\frac{N'^2}{s}\n \\end{aligned}$$ by using monotonicity of $N, {{\\mathcal P}}$ and ${{\\mathcal P}}<0$. Next multiplying ${{\\mathcal P}'}$ to the same equation, we can show $${\\int_{-\\infty}^z{ {{\\mathcal P}'}^2} dz} \\le \\frac{1}{s^2} {\\int_{-\\infty}^z{{N'}^2} dz}.$$ On the other hand, from the first equation ${N'}= -sN - N{{\\mathcal P}}$, we find $${\\int_{-\\infty}^z{ {N'}^2} dz} \\le - \\| {N'}\\|_{L^{\\infty}}{\\int_{-\\infty}^z{{N'}} dz} \n \\le C.$$ Thus we have $$\\begin{aligned}\n\\label{Ncp}\n {\\int_{-\\infty}^z{{{\\mathcal P}'}^2} dz} \\le C, \\quad {\\int_{-\\infty}^z{{N'}^2} dz} \\le C.\n \\end{aligned}$$ Also note that ${N'}^2 \\le C$. ",
"Plugging the bounds in the inequality $$\\frac s2 {{\\mathcal P}'}^2 + {\\epsilon}{\\int_{-\\infty}^z{{{\\mathcal P}''}^2} dz} \\le -s {\\int_{-\\infty}^z{N'{{\\mathcal P}'}} dz} + \\frac s4 {{\\mathcal P}'}^2 + \\frac{N'^2}{s},$$ we conclude $ |{{\\mathcal P}'}| < C$.\\\n$\\bullet$ $k=2$\\\nFirst we observe ${{\\mathcal P}}'' (-\\infty)= N''(-\\infty)=0$ from the equations for any ${\\epsilon}>0$. Multiplying ${{\\mathcal P}}'''$ to the equation $-s {{\\mathcal P}''}- {\\epsilon}{{\\mathcal P}}''' = -{\\epsilon}({{\\mathcal P}}^2)'' + N''$, and integrating over $(-\\infty, z)$, we have $$\\begin{aligned}\n\\label{cpq} \\begin{aligned}\n\\frac s2 {{\\mathcal P}}''^2 + {\\epsilon}{\\int_{-\\infty}^z{ {{\\mathcal P}}'''^2} dz} & = {\\int_{-\\infty}^z{ ( {\\epsilon}{{\\mathcal P}}^2 -N)'' {{\\mathcal P}}'''} dz}\\\\\n& = 2{\\epsilon}{\\int_{-\\infty}^z{ {{\\mathcal P}'}^2 {{\\mathcal P}}''' } dz}+ {\\epsilon}{{\\mathcal P}}{{\\mathcal P}''}^2 - {\\epsilon}{\\int_{-\\infty}^z{ {{\\mathcal P}'}{{\\mathcal P}''}^2} dz}\n- {\\int_{-\\infty}^z{N'' {{\\mathcal P}}''' } dz}\\\\\n&\\le 2{\\epsilon}{\\int_{-\\infty}^z{{{\\mathcal P}'}^2 {{\\mathcal P}}'''} dz}- {\\int_{-\\infty}^z{N'' {{\\mathcal P}}'''} dz}\\\\\n& = 2 {\\int_{-\\infty}^z{ {{\\mathcal P}'}^2 ( -s{{\\mathcal P}''}+ ({\\epsilon}{{\\mathcal P}}^2 - N)'')} dz} + {\\int_{-\\infty}^z{ N''' {{\\mathcal P}''}} dz} - N''{{\\mathcal P}''}.",
"\n\\end{aligned}\\end{aligned}$$ We use the monotonicity of $N, {{\\mathcal P}}$ and ${{\\mathcal P}}<0$ in the third line. ",
"We find $$\\begin{aligned}\n{\\int_{-\\infty}^z{ {{\\mathcal P}'}^2 ( -s{{\\mathcal P}''}+ ({\\epsilon}{{\\mathcal P}}^2 - N)'')} dz} & = -\\frac{s}{3} { {{\\mathcal P}'}^3} + {\\int_{-\\infty}^z{{{\\mathcal P}'}^2 ({\\epsilon}{{\\mathcal P}}^2 - N)''} dz}\\\\\n& \\le {\\int_{-\\infty}^z{{{\\mathcal P}'}^2 ({\\epsilon}{{\\mathcal P}}^2 - N)''} dz}\\end{aligned}$$ which can be shown bounded by substituting $N''$, ${\\epsilon}{{\\mathcal P}''}$ with lower order terms following the equations and using $ N^{(k)} <C$ , $|{{\\mathcal P}}^{(k)}|<C$ for $k=0, 1$ and . ",
"On the other hand, we find $$\\begin{aligned}\n{\\int_{-\\infty}^z{ N''' {{\\mathcal P}''}} dz}& = {\\int_{-\\infty}^z{( -s N' - (N{{\\mathcal P}})')' {{\\mathcal P}''}} dz} \\\\\n& = {\\int_{-\\infty}^z{ -s N'' {{\\mathcal P}''}} dz} - {\\int_{-\\infty}^z{ (N{{\\mathcal P}})'' {{\\mathcal P}''}} dz} ,\\end{aligned}$$ $$\\begin{aligned}\n-{\\int_{-\\infty}^z{ (N{{\\mathcal P}})'' {{\\mathcal P}''}} dz}& = -{\\int_{-\\infty}^z{ N {{\\mathcal P}''}^2} dz} - \\frac 12{\\int_{-\\infty}^z{ N' ({{\\mathcal P}'}^2)'} dz} -{\\int_{-\\infty}^z{{{\\mathcal P}}N'' {{\\mathcal P}''}} dz}\\\\\n&\\le \\frac 12 {\\int_{-\\infty}^z{ ( -sN' - (N{{\\mathcal P}})'){{\\mathcal P}'}^2} dz} - \\frac 12 N' {{\\mathcal P}'}^2 - {\\int_{-\\infty}^z{{{\\mathcal P}}N'' {{\\mathcal P}''}} dz}\\end{aligned}$$ The first integral and the second term are bounded. ",
"It remains to estimate ${\\int_{-\\infty}^z{N''{{\\mathcal P}''}} dz} $ and $N''{{\\mathcal P}''}$ by the Cauchy-Schwartz inequality. ",
"From the first equation, it is easy to see ${\\int_{-\\infty}^z{ N''^2} dz} < C $. ",
"By multiplying ${{\\mathcal P}''}$ to the equation $-s {{\\mathcal P}''}- {\\epsilon}{{\\mathcal P}}''' = -{\\epsilon}({{\\mathcal P}}^2)'' + N''$ and integrating over $(-\\infty, z)$, we have $$\\begin{aligned}\n \\frac s2 {\\int_{-\\infty}^z{{{\\mathcal P}''}^2} dz} + \\frac{{\\epsilon}}{2} {{\\mathcal P}''}^2 \n & \\leq {\\epsilon}{\\int_{-\\infty}^z{ ({{\\mathcal P}}^2)'' {{\\mathcal P}''}} dz}+ \\frac{1}{2s} {\\int_{-\\infty}^z{(N'')^2} dz}\\\\\n & = 2{\\epsilon}{\\int_{-\\infty}^z{ {{\\mathcal P}''}^2 {{\\mathcal P}}} dz} + \\frac{2{\\epsilon}}{3} ({{\\mathcal P}'})^3 + \\frac{1}{2s} {\\int_{-\\infty}^z{(N'')^2} dz}\\\\\n & \\leq \\frac{2{\\epsilon}}{3} ({{\\mathcal P}'})^3 + \\frac{1}{2s} {\\int_{-\\infty}^z{(N'')^2} dz}\\leq C.\n \\end{aligned}$$ So we have ${\\int_{-\\infty}^z{{{\\mathcal P}''}^2} dz} \\le C$. ",
"Finally we estimate $$\\begin{aligned}\n |N''{{\\mathcal P}''}| \\le \\frac s 4 {{\\mathcal P}''}^2 + \\frac{N''^2}{s} \\le \\frac s4 {{\\mathcal P}''}^2 + \\frac{1}{s} ( sN' + (N{{\\mathcal P}})')^2\\leq \\frac s4 {{\\mathcal P}''}^2 + C.\n \\end{aligned}$$ Plugging the estimates above into , we obtain $|{{\\mathcal P}''}|<C$.\n\nProposition \\[uniform\\_with\\] is proved by the lemmas below which are parallel to Lemma \\[lemma0\\_\\], \\[lemma1\\_0\\], \\[lemma1\\_1\\], \\[lemma1\\_2\\], \\[lemma01\\_\\], \\[lemma23\\_\\]. ",
"Here we present in detail the zeroth order estimate of ${\\epsilon}>0$ case, where the zero average condition, hence the Poincaré inequality is essentially used. [",
"For the higher order estimates, we sketch their proof. ]{}",
"\n\n\\[lemma0\\_e\\] If ${\\lambda}>0$ is sufficiently small, we have the following: $$\\label{eq_lemma0_e}\n\\| \\psi \\|^2 + \\| \\varphi\\|_w^2 + \\int_0^t \\| {\\nabla}\\varphi\\|_w^2\n+{{\\epsilon}}\\int_0^t \\|{\\nabla}\\psi\\|^2\n\\le C ( \\| \\psi_0\\|^2 + \\| \\varphi_0\\|_w^2 ).$$\n\nRecall the equations . ",
"Multiply ${\\frac{ \\varphi}{N}}$ to the $\\varphi$ equation and $\\psi$ to the $\\psi$ equation. ",
"Integrating by parts, we have $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}\\left( \\int {\\frac{ |\\varphi|^2}{N}} +\\int |\\psi|^2 \\right) \n+ \\int{\\frac{ \\sum_{i} |{\\nabla}\\varphi^i|^2}{N}} + {\\epsilon}\\int |{\\nabla}\\psi|^2+ \n\\frac s 2 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'\\\\\n&= \\frac 12 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)''+\n\\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1 {\\nabla}\\cdot \\varphi \n- 2{\\epsilon}\\int {\\mathcal P}{\\psi}_z \\psi\n $$\n\nNote that does not hold for ${\\epsilon}>0$. Instead, we estimate the quadratic term $\\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1{\\nabla}\\cdot \\varphi$ by $$\\begin{aligned}\n \\Big|\\int {\\frac{ {\\mathcal P}}{N}} \\varphi^1{\\nabla}\\cdot \\varphi\\Big| & \\le \\| {\\varphi}^1/\\sqrt{N}\\| \\| {\\nabla}{\\varphi}/\\sqrt{N}\\|\n \\le C {\\lambda}\\| {\\nabla}{\\varphi}\\|^2_w,\\\\\\end{aligned}$$ where we used $|{\\mathcal P}|\\leq C\n$ (Lemma \\[wave\\_propwith\\]) with the mean-zero condition for ${\\varphi}_0$ ($\\bar{{\\varphi}_0}(z)=0$), which is preserved in time, in order to use the Poincaré inequality $$\\label{ineq:poincare2} \n\\| {\\varphi}(z,\\cdot_y) \\|_{L^2_y([0,{\\lambda}])} \\le C{\\lambda}\\| {\\partial}_{y} {\\varphi}(z,\\cdot_y)\\|_{L^2_y([0,{\\lambda}])} \\quad\\mbox{ for } z\\in\\mathbb{R}.$$ Also we estimate $$\\begin{aligned}\n&\\frac 12 \\int |{\\varphi}|^2 {\\left( {\\frac{ 1}{N}} \\right)}''-\\frac s 2 \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)'\n\\le -\\int {\\varphi}\\cdot{\\varphi}_z \\left( {\\frac{ 1}{N}}\\right)'+\\frac s 2 \\int |\\varphi|^2 \\left|\\left( {\\frac{ 1}{N}}\\right)'\\right|\\\\\n&\\le C\\int |{\\varphi}||{\\varphi}_z | \\left( {\\frac{ 1}{N}}\\right) +\\frac {Cs}{2} \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right) \n\\le \\frac{1}{4}\\int |{\\varphi}_z |^2\\left( {\\frac{ 1}{N}}\\right)+(C+\\frac {Cs}{2} ) \\int |\\varphi|^2 \\left( {\\frac{ 1}{N}}\\right)\\\\\n&\\le \\frac{1}{4}\\int |{\\varphi}_z |^2\\left( {\\frac{ 1}{N}}\\right)+C{\\lambda}^2(C+\\frac {Cs}{2} ) \\int |\\varphi_y|^2 \\left( {\\frac{ 1}{N}}\\right)\\le \n (\\frac{1}{4}+C{\\lambda}^2) \\int \\frac{|\\nabla\\varphi|^2 }{N}\n \\end{aligned}$$ where we used the estimate $|(1/N)'|\\leq C/N$ in Lemma \\[wave\\_propwith\\] and the Poincaré inequality .",
"\n\nFor the ${\\epsilon}$-term, we estimate $$\\begin{aligned}\n -2{\\epsilon}\\int {\\mathcal P}{\\psi}_z \\psi &\\leq {\\epsilon}C \\|\\psi_z\\| \\|\\psi\\| \n \\le {\\epsilon}C {\\lambda}\\|\\psi_z\\| \\|\\psi_y\\|\\le {\\epsilon}C {\\lambda}\\|{\\nabla}\\psi\\|^2$$ where we used the mean-zero condition for $\\psi$ to use the Poincaré inequality for $\\psi$: $$\\label{ineq:poincare3} \n\\| {\\psi}(z,\\cdot_y) \\|_{L^2_y([0,{\\lambda}])} \\le C{\\lambda}\\| {\\partial}_{y} {\\psi}(z,\\cdot_y)\\|_{L^2_y([0,{\\lambda}])} \\quad\\mbox{ for } z\\in\\mathbb{R}.$$\n\nBy assuming ${\\lambda}>0$ small enough, we arrive at $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}\\left( \\int {\\frac{ |\\varphi|^2}{N}} +\\int |\\psi|^2 \\right) \n+\\frac 18 \\int{\\frac{ |{\\nabla}\\varphi|^2}{N}} + \\frac{{\\epsilon}}{2}\\int |{\\nabla}\\psi|^2 \\leq \n0\\end{aligned}$$ which proves Lemma \\[lemma0\\_e\\].",
"\n\n\\[lemma1\\_0\\_e\\] If ${\\lambda}>0$ is small enough, then we have $$\\begin{split}\\label{ineq:lem1__e}\n&\\| {\\nabla}{\\varphi}\\|_w^2 + \\| {\\nabla}{\\psi}\\|^2+ \\int_0^t \\| {\\nabla}^2 {\\varphi}\\|_w^2 \n+ {\\epsilon}\\int_0^t \\|{\\nabla}^2 \\psi\\|^2\\\\\n&\\le C ( \\| {\\nabla}{\\varphi}_0\\|_w^2 + \\| {\\nabla}{\\psi}_0\\|^2+ \\| \\psi_0\\|^2 + \\| \\varphi_0\\|_w^2 ) \n+ C \\int_0^t \\int {N} |{\\nabla}{\\psi}|^2.\\end{split}$$\n\nDifferentiating in $z$, we have $$\\begin{aligned}\n{\\varphi}_{tz} - s{\\varphi}_{zz} - {\\Delta}{\\varphi}_z & = N' {\\nabla}\\psi + N {\\nabla}\\psi_z + P' {{\\nabla}\\cdot}{\\varphi}+ P {{\\nabla}\\cdot}{\\varphi}_z,\\\\\\psi_{tz} -s\\psi_{zz} -{\\epsilon}{\\Delta}{\\psi}_z & = -2{\\epsilon}(P \\cdot{\\nabla}{\\psi})_z \n+ {{\\nabla}\\cdot}{\\varphi}_z.",
"\n \\end{aligned}$$ Multiply ${\\frac{ \\varphi_z}{N}}$ and $\\psi_z$ to the above equation. ",
"Integrating by parts, we have $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}{\\left( \\int {\\frac{ {\\vert{\\varphi}_z\\vert^2} }{N}} + {\\vert{\\psi}_z\\vert^2} \\right)} + \\int \\sum_{ij}{\\frac{ {\\vert{\\partial}_j {\\varphi}^i_{z}\\vert^2}}{N}}\n+ {\\epsilon}\\int |{\\nabla}\\psi_z|^2\\\\\n & = { \\frac 12 \\int {\\vert{\\varphi}_z\\vert^2} \\left( {\\frac{ 1}{N}} \\right)''- \\frac s2 \\int {\\vert{\\varphi}_z\\vert^2} {\\left({\\frac{ 1}{N}} \\right)}' }\n \\\\\n & +\\int {\\frac{ N'}{N}} {\\nabla}{\\psi}{\\varphi}_z +\\int {\\frac{ P'}{N}} {{\\nabla}\\cdot}{\\varphi}{\\varphi}_z + \\int {\\frac{ P}{N}} {{\\nabla}\\cdot}{\\varphi}_z {\\varphi}_z\\\\\n& -\\underbrace{ 2{\\epsilon}\\int (P\\cdot {\\nabla}{\\psi})_z \\psi_z }_\n{= 2{\\epsilon}\\int ({\\mathcal P}{\\psi}_z)_z \\psi_z}.",
"\n$$ Similarly, we get $$\\begin{aligned}\n&\\frac 12 {\\frac{d}{dt}}{\\left( \\int {\\frac{ {\\vert{\\varphi}_y\\vert^2} }{N}} + {\\vert{\\psi}_y\\vert^2} \\right)} +\\int \\sum_{ij}{\\frac{ {\\vert{\\partial}_j {\\varphi}^i_{y}\\vert^2}}{N}} \n+ {\\epsilon}\\int |{\\nabla}\\psi_y|^2\\\\\n &= \n { \\frac 12 \\int {\\vert{\\varphi}_y\\vert^2} \\left( {\\frac{ 1}{N}} \\right)'' - \\frac s2 \\int {\\vert{\\varphi}_y\\vert^2} {\\left({\\frac{ 1}{N}} \\right)}' }\n \\\\\n &+\\int {\\frac{ P}{N}} {{\\nabla}\\cdot}{\\varphi}_y {\\varphi}_y \\\\\n & -\\underbrace{ 2{\\epsilon}\\int (P\\cdot {\\nabla}{\\psi})_y \\psi_y }_\n{= 2{\\epsilon}\\int {\\mathcal P}{\\psi}_{zy} \\psi_y}.",
"\n$$\n\nWe repeat the process in the proof of Lemma \\[lemma1\\_0\\]. ",
"For the ${\\epsilon}$ term, we estimate $$\\begin{aligned}\n&-2{\\epsilon}\\int ({\\mathcal P}{\\psi}_z)_z \\psi_z - 2{\\epsilon}\\int {\\mathcal P}{\\psi}_{zy} \\psi_y=\n -2{\\epsilon}\\Big(\\int {\\mathcal P}' {\\psi}_z \\psi_z\n-2{\\epsilon}\\int {\\mathcal P}{\\psi}_{zz} \\psi_z \n - 2{\\epsilon}\\int {\\mathcal P}{\\psi}_{zy} \\psi_y \\Big)\\\\\n &\\leq C{\\epsilon}\\|P\\|_{L^\\infty}\\|{\\nabla}{\\psi}\\|\\|{\\nabla}^2{\\psi}\\|+C{\\epsilon}\\|P'\\|_{L^\\infty}\\|{\\nabla}{\\psi}\\|^2 \n \\leq C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+\\frac{{\\epsilon}}{4}\\|{\\nabla}^2{\\psi}\\|^2 \\end{aligned}$$ The last term is absorbed into the chemical diffusion term. ",
"Then we have $$\\begin{aligned}\n& {\\frac{d}{dt}}{\\left( \\int {\\frac{ {\\vert{\\nabla}{\\varphi}\\vert^2}}{N}} + \\int {\\vert{\\nabla}{\\psi}\\vert^2} \\right)} + \\int {\\frac{ {\\vert {\\nabla}^2{\\varphi}\\vert^2} }{N}}\n+ {{\\epsilon}} \\int |{\\nabla}^2 \\psi|^2\n\\le \nC\\|{\\sqrt{N}} {{\\nabla}{\\psi}}\\|^2 + C \\| \\frac{{\\nabla}{\\varphi}}{\\sqrt{N}}\\|^2 \n +C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2.\\end{aligned}$$ For the last two terms above we use Lemma \\[lemma0\\_e\\].",
"\n\n\\[lemma1\\_1\\_e\\] If ${\\lambda}>0$ is small enough, then we have $$\\begin{aligned}\n \\label{claim1_lemma1__e}\n& \\int_0^t \\int N |{\\nabla}{\\psi}|^2 +{\\epsilon}\\int_0^t \\int |{\\nabla}^2 {\\psi}|^2 \\le C {\\left( \\| {\\nabla}{\\psi}_0 \\|^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_w^2 \\right)}. ",
"\n$$\n\nAs in the proof of Lemma \\[lemma1\\_1\\], we get\n\nMultiplying ${\\nabla}{\\psi}$ to the ${\\varphi}$-equation, we have $$\\label{eq_lemma0__first_claim_intermediate_e}\n N |{\\nabla}{\\psi}|^2 = \\underbrace{{\\varphi}_t \\cdot {\\nabla}{\\psi}}_{(A)} - s{\\varphi}_z \\cdot{\\nabla}{\\psi}-\\underbrace{{\\Delta}{\\varphi}\\cdot{\\nabla}{\\psi}}_{(C)} \n-({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}.",
"\n$$ Let us rewrite the terms $(A)$ and $(C)$. $$\\begin{aligned}\n(A)& = ({\\varphi}\\cdot{\\nabla}{\\psi})_t - {\\varphi}\\cdot{\\nabla}{\\psi}_t \\\\&=\n ({\\varphi}\\cdot{\\nabla}{\\psi})_t - {\\varphi}\\cdot(s {\\nabla}{\\psi}_z + {\\nabla}({{\\nabla}\\cdot}{\\varphi})\n +{\\epsilon}{\\Delta}{\\nabla}{\\psi}-2{\\epsilon}{\\nabla}( P \\cdot{\\nabla}{\\psi}) )\\end{aligned}$$ and $$(C) \n = ({{\\nabla}\\cdot}{\\varphi})_z {\\psi}_z + ( {\\varphi}^1_{yy} - {\\varphi}^2_{zy}) {\\psi}_z + ( {{\\nabla}\\cdot}{\\varphi})_y {\\psi}_y \n+ ({\\varphi}^2_{zz} - {\\varphi}^1_{zy}) {\\psi}_y.$$ Integration by parts gives $$\\int(C)\n = \\int ({{\\nabla}\\cdot}{\\varphi})_z {\\psi}_z + ( {{\\nabla}\\cdot}{\\varphi})_y {\\psi}_y\n = \\int {\\nabla}({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}{\\psi}.$$ By using the ${\\psi}$-equation which have ${\\nabla}\\cdot{\\varphi}$ on its right side, we get $$\\begin{aligned}\n \\int {\\nabla}({{\\nabla}\\cdot}{\\varphi})\\cdot{\\nabla}{\\psi}&= \\int {\\nabla}({\\psi}_{t} - s {\\psi}_{z}\n -{\\epsilon}{\\Delta}{\\psi}+2{\\epsilon}P \\cdot{\\nabla}{\\psi})\\cdot {\\nabla}{\\psi}\\\\\n&= \\frac 12 {\\frac{d}{dt}}\\int |{\\nabla}{\\psi}|^2\n+\\int {\\nabla}( \n -{\\epsilon}{\\Delta}{\\psi}+2{\\epsilon}P \\cdot{\\nabla}{\\psi})\\cdot{\\nabla}{\\psi}.\\end{aligned}$$ Integration on gives us $$\\begin{aligned}\n\\int N |{\\nabla}{\\psi}|^2 =& \n{\\frac{d}{dt}}\\int {\\varphi}\\cdot{\\nabla}{\\psi}- \\int s {\\varphi}\\cdot {\\nabla}{\\psi}_z -\\int{\\varphi}\\cdot ({\\nabla}({{\\nabla}\\cdot}{\\varphi})\n+ {\\epsilon}{\\Delta}{\\nabla}{\\psi}-2{\\epsilon}{\\nabla}( P \\cdot{\\nabla}{\\psi}) )\\\\ \\nonumber\n&- \\int s {\\varphi}_z\\cdot {\\nabla}{\\psi}\\\\ \\nonumber\n&- \\frac 12 {\\frac{d}{dt}}\\int( {\\psi}_z^2 + {\\psi}_y^2)\n+\\int {\\nabla}( \n {\\epsilon}{\\Delta}{\\psi}-2{\\epsilon}P \\cdot{\\nabla}{\\psi})\\cdot{\\nabla}{\\psi}\\\\ \\nonumber\n&- \\int ({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}- \\int {{\\nabla}\\cdot}{\\varphi}| {\\nabla}{\\psi}|^2. ",
"\n \\end{aligned}$$ We rearrange the above to get $$\\begin{aligned}\n\\label{ineq_claim1_lemma1_e}\n\\int N |{\\nabla}{\\psi}|^2 \n=& {\\frac{d}{dt}}\\int {\\varphi}\\cdot{\\nabla}{\\psi}- \\frac 12 {\\frac{d}{dt}}\\int |{\\nabla}{\\psi}|^2 \\\\& \\nonumber + \\int | {{\\nabla}\\cdot}{\\varphi}|^2 - \\int ({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}\\\\ \\nonumber\n&+{\\epsilon}\\int {\\nabla}( \n {\\Delta}{\\psi}-2 P \\cdot{\\nabla}{\\psi})\\cdot{\\nabla}{\\psi}- {\\epsilon}\\int{\\varphi}\\cdot( {\\Delta}{\\nabla}{\\psi}-2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) )\\\\& \\nonumber=(I)+(II)+(III).",
"\n \\end{aligned}$$ For $(I)$ and $(II)$, we refer to the proof of Lemma \\[lemma1\\_1\\]. ",
"Note that $$\\begin{aligned}\n {\\int_0^t(I)}=\\int_0^t{\\left( {\\frac{d}{dt}}\\int {\\varphi}\\cdot{\\nabla}{\\psi}- \\frac 12 {\\frac{d}{dt}}\\int |{\\nabla}{\\psi}|^2 \\right)}\\leq {C(\\|{\\varphi}(t)\\|^2 + \\|{\\nabla}{\\psi}_0\\|^2+\\|{\\varphi}_0\\|^2)}&\n \\end{aligned}$$ by $ \\int {\\varphi}{\\nabla}{\\psi}\\le C\\| {\\varphi}\\| ^2 + \\frac 12 \\| {\\nabla}{\\psi}\\|^2$.\n\nFor the second term $(II)$, we estimate $$\\begin{aligned}\n \\int | {{\\nabla}\\cdot}{\\varphi}|^2-\\int ({{\\nabla}\\cdot}{\\varphi})P\\cdot{\\nabla}{\\psi}& \\le C\\int {\\frac{ | {\\nabla}{\\varphi}|^2}{N}} + \\frac 14 \\int N |{\\nabla}{\\psi}|^2 \n \\end{aligned}$$\n\nFor the ${\\epsilon}$-term $(III)$, we estimate $$\\begin{aligned}\n {\\epsilon}\\int {\\nabla}( {\\Delta}{\\psi}-2 P \\cdot{\\nabla}{\\psi})\\cdot{\\nabla}{\\psi}&=\n-{\\epsilon}\\int |{\\nabla}^2{\\psi}|^2\n-{\\epsilon}\\int {\\nabla}( 2 P \\cdot{\\nabla}{\\psi})\\cdot{\\nabla}{\\psi}\\\\\n&\\leq-{\\epsilon}\\|{\\nabla}^2{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2+\\frac{{\\epsilon}}{4}\\|{\\nabla}^2{\\psi}\\|^2\\\\\n$$ and $$\\begin{aligned}\n - {\\epsilon}\\int{\\varphi}\\cdot( {\\Delta}{\\nabla}{\\psi}-2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) \n ) \n&\\leq C{\\epsilon}\\int|{\\nabla}{\\varphi}||{\\nabla}^2{\\psi}| + |{\\nabla}{\\varphi}||{\\nabla}{\\psi}|\n \\\\\n&\\leq \\frac{\\epsilon}4\\|{\\nabla}^2{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}{\\varphi}\\|^2+C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2.",
"\n$$\n\nIntegrating in time, we get $$\\begin{aligned}\n\\int_0^t\\int& N |{\\nabla}{\\psi}|^2 \n\\leq \nC(\\|{\\varphi}(t)\\|^2 +\\|{\\nabla}{\\psi}_0\\|^2+\\|{\\varphi}_0\\|^2)\\\\\n& +\\int_0^t\\Big(C \\|{\\nabla}{\\varphi}\\|_w^2+ \\frac 14 \\int N |{\\nabla}{\\psi}|^2 \n- \\frac{\\epsilon}2\\|{\\nabla}^2{\\psi}\\|^2 +C{\\epsilon}\\|{\\nabla}{\\psi}\\|^2\\Big).",
"\n \\end{aligned}$$\n\nBy Lemma \\[lemma0\\_e\\], we have .",
"\n\n\\[lemma1\\_2\\_e\\] If ${\\lambda}>0$ and ${\\epsilon}>0$ are small enough, then we have $$\\begin{aligned}\n& \\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}} + \\int _0^ t \\int {\\frac{ |{\\nabla}{\\psi}|^2 }{N}}\n+ {\\epsilon}\\int_0^t\\int {\\frac{ |{\\nabla}^2{\\psi}|^2}{N}}\n\\le C( \\| {\\nabla}{\\psi}_0 \\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{w}^2) + C\\int_0^t \\int {\\frac{ |{\\nabla}^2{\\varphi}|^2}{N}}. ",
"\\label{claim2_lemma1__e}\\end{aligned}$$\n\nHere we used the smallness condition on ${\\epsilon}$ first in order to get the weighted $L^2$-estimate for $\\nabla \\psi$ above as in Lemma \\[lemma1\\_2\\].",
"\n\nFirst we take ${\\nabla}$ to the ${\\psi}$-equation then multiply by $ w {\\nabla}{\\psi}$ to get $$\\begin{aligned}\n&\\frac 12 (w |{\\nabla}{\\psi}|^2)_t - \\frac s2 ( w |{\\nabla}{\\psi}|^2)_z + \\frac s2 w' |{\\nabla}{\\psi}|^2 \\\\\n&= w {\\nabla}({{\\nabla}\\cdot}{\\varphi}) \\cdot {\\nabla}{\\psi}+\\underbrace{{\\epsilon}w {\\nabla}{\\psi}\\cdot{\\left( {\\Delta}{\\nabla}{\\psi}-2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) \\right)}}_{{\\epsilon}\\mbox{-terms}}.\\end{aligned}$$ Note that for some $c>0$ $$\\begin{aligned}\nw' = se^{sz} > c w & \\quad \\mbox{for } z > 0, \\\\\nw = 1+e^{sz} \\le 2\n< \\frac{1}{c}N \n & \\quad \\mbox{for } z\\le0.\\end{aligned}$$ Integrating on each half strip and in time then summing these two,\n\nAs in the proof of Lemma \\[lemma1\\_2\\], we get $$\\begin{aligned}\n&\\int w |{\\nabla}{\\psi}|^2 + \\frac c2 \\int_0^t \\int w |{\\nabla}{\\psi}|^2 \\le \\int w |{\\nabla}{\\psi}_0|^2 + \\int_0^t \\int N|{\\nabla}{\\psi}|^2 + \nC\\int_0^t \\int w |{\\nabla}^2 {\\varphi}|^2 +\\int_0^t \\int {{\\epsilon}\\mbox{-terms}}\\\\\n&\\le C ( \\| {\\nabla}{\\psi}_0\\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_w^2)\n+ C \\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}}+\\int_0^t \\int {{\\epsilon}\\mbox{-terms}}, \\end{aligned}$$ where we used the previous estimate .",
"\n\nFor the ${\\epsilon}$[-terms]{}, we estimate $$\\begin{aligned}\n&\\int {{\\epsilon}\\mbox{-terms}}={\\epsilon}\\int { w {\\nabla}{\\psi}\\cdot{\\left( {\\Delta}{\\nabla}{\\psi}-2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) \\right)}}\\\\\n&={\\epsilon}\\int \\Big({ -w|{\\nabla}^2{\\psi}|^2 -w' {\\nabla}{\\psi}\\cdot {\\nabla}{\\psi}_z\n- w {\\nabla}{\\psi}\\cdot{\\left( 2 {\\nabla}( P \\cdot{\\nabla}{\\psi}) \\right)} \n}\\Big)\\\\\n&\\leq-{\\epsilon}\\int { w|{\\nabla}^2{\\psi}|^2 +C{\\epsilon}\\int \\Big( w |{\\nabla}{\\psi}|| {\\nabla}^2{\\psi}| + w| {\\nabla}{\\psi}|( |{\\nabla}^2{\\psi}|+|{\\nabla}{\\psi}| )\n } \\Big)\\\\\n&\\leq-\\frac {\\epsilon}4\\int w|{\\nabla}^2{\\psi}|^2 +C{\\epsilon}\\int w |{\\nabla}{\\psi}|^2 \n $$ where we used the estimate $|w'|\\leq C|w|$. Plugging this estimate, we have $$\\begin{aligned}\n&\\int w |{\\nabla}{\\psi}|^2 + (\\frac{c}{2}-C{\\epsilon}) \\int_0^t \\int w |{\\nabla}{\\psi}|^2 +\\frac{ {\\epsilon}}{4} \\int_0^t\\int w|{\\nabla}^2{\\psi}|^2 \\\\\n&\\le C ( \\| {\\nabla}{\\psi}_0\\|_w^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_w^2) + C \\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}}. ",
" \\end{aligned}$$ Then, by making ${\\epsilon}$ small enough, it proves the estimate .",
"\n\nBy adding all the above lemmas, we get\n\n\\[lemma01\\_\\_e\\] If ${\\lambda}>0$ and ${\\epsilon}>0$ are small enough, then we have $$\\begin{split}\\label{eq_lemma01__e}\n\\| {\\varphi}\\|_{1,w}^2 + \\| {\\psi}\\|^2 + \\| {\\nabla}{\\psi}\\|_w^2+ \\int_0^t \\sum_{l = 1,2} \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^t \\| {\\nabla}{\\psi}\\|_w^2\n+{\\epsilon}\\int_0^t \\| {\\nabla}^{2} {\\psi}\\|_w^2\n\\\\\n\\quad \\le\n C ( \\| {\\varphi}_0\\|_{1,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_w^2+ \\| {\\psi}_0\\|^2). ",
"\n\\end{split}$$\n\nPlugging the estimates into , we have\n\n$$\\begin{aligned}\n\\label{ineq:lem1_closed_e}\n &\\| {\\nabla}{\\varphi}\\|_w^2 + \\| {\\nabla}{\\psi}\\|^2 + \\int_0^t \\| {\\nabla}^2 {\\varphi}\\|_w^2 + {\\epsilon}\\int_0^t\\|{\\nabla}^2{\\psi}\\|^2\n\\le C ( \\| {\\nabla}{\\psi}_0\\|^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{1,w}^2) \\end{aligned}$$\n\nIn addition, from the estimate with the above estimate , we get $$\\begin{aligned}\n\\label{claims_lemma1__e}\n\\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}} + \\int_0^t \\int {\\frac{ |{\\nabla}{\\psi}|^2}{N}} + \n {\\epsilon}\\int_0^t \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2}{N}} \\le C ( \\| {\\varphi}_0\\|_{1,w}^2 + \\| {\\nabla}{\\psi}_0\\|_w^2\n+ \\| {\\psi}_0\\|^2).\\end{aligned}$$ Adding Lemma \\[lemma0\\_e\\] and the above and , we have .",
"\n\nThen we can repeat this process up to the highest order:\n\n\\[lemma23\\_e\\]If ${\\lambda}>0$ and ${\\epsilon}> 0$ are small enough, then for $k=2, 3$ we get $$\\begin{aligned}\n&\\| {\\nabla}^k {\\varphi}\\|^2_w + \\| {\\nabla}^k {\\psi}\\|^2_w + \\int_0^t { \\|{\\nabla}^{k+1}{\\varphi}\\|_w^2}\n+ \\int_0^t { \\|{\\nabla}^{k}{\\varphi}\\|_w^2}\n+ {\\epsilon}\\int_0^t { \\|{\\nabla}^{k+1}{\\psi}\\|_w^2}\\\\\n&\\le C ( \\| {\\varphi}_0\\|_{k,w}^2+\n \\| {\\nabla}{\\psi}_0\\|_{k-1,w}^2+ \\| {\\psi}_0\\|^2).\\end{aligned}$$\n\nWe skip the proof(refer to that of Lemma \\[lemma23\\_\\]).",
"\n\n$$\\begin{aligned}\n \\label{higher__e}\n& \\frac 12 {\\frac{d}{dt}}\\int {\\left( {\\frac{ | {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}|^2}{N}} + | {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}|^2 \\right)} \n+ \\int {\\frac{ | {\\nabla}{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}|^2}{N}}\n+ {\\epsilon}\\int { | {\\nabla}{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\psi}|^2}\\\\\n &=\n { \\frac 12 \\int {\\vert{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}\\vert^2} \\left( {\\frac{ 1}{N}} \\right)''- \\frac s2 \\int {\\vert{{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}\\vert^2} {\\left({\\frac{ 1}{N}} \\right)}'}\n \\nonumber\\\\\n &\\underbrace{+\\int ( {{\\partial}_y^{i}}{{\\partial}_z^{j}}(N {\\nabla}{\\psi}) - N {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\nabla}{\\psi})\\cdot {\\frac{ {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}}{N}} + {{\\partial}_y^{i}}{{\\partial}_z^{j}}(P {{\\nabla}\\cdot}{\\varphi}) \\cdot{\\frac{ {{\\partial}_y^{i}}{{\\partial}_z^{j}}{\\varphi}}{N}} }_{\\mbox{ Quadratic term }}\\nonumber\\\\\n& \\underbrace{ -2{\\epsilon}\\int {{\\partial}_y^{i}}{{\\partial}_z^{j}}({\\mathcal P}{\\psi}_z) {{\\partial}_y^{i}}{{\\partial}_z^{j}}\\psi\n}_{{\\epsilon}-\\mbox{term}}. ",
"\\nonumber\\end{aligned}$$\n\n$\\bullet$ Case $ k= i+j =2$\\\nFor the ${\\epsilon}$ terms, after intergration by parts, we can estimate them by $$\\begin{aligned}\n & C{\\epsilon}\\int |{\\nabla}({\\mathcal P}{\\psi}_z)| |{\\nabla}^3\\psi|\\leq\n C{\\epsilon}\\int (|{\\nabla}{\\psi}| +|{\\nabla}^2 {\\psi}|) |{\\nabla}^3\\psi|\n \\leq C{\\epsilon}( \\|\\nabla{\\psi}\\|^2+ \\|\\nabla^2{\\psi}\\|^2) +\\frac{{\\epsilon}}{4}\\|\\nabla^3{\\psi}\\|^2\\\\\\end{aligned}$$\n\nUp to now, by Lemma \\[lemma01\\_\\_e\\], we estimate by $$\\begin{aligned}\n\\label{beforeclaims_lemma23__e} &\\int {\\frac{ |{\\nabla}^2 {\\varphi}|^2}{N}} +\\int |{\\nabla}^2 {\\psi}|^2 + \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} +\n {\\epsilon}\\int_0^t \\int{ |{\\nabla}^3 {\\psi}|^2}\\\\\n &\\le C( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2). ",
" \\nonumber \n \\end{aligned}$$ This estimate is the second order version of Lemma \\[lemma1\\_0\\_e\\].\\\nNow we claim the following two estimates which are the second order versions of Lemma \\[lemma1\\_1\\_e\\] and \\[lemma1\\_2\\_e\\]:\n\n$$\\begin{aligned}\n \\label{claim1_lemma23__e}\n& \\int_0^t \\int N |{\\nabla}^2 {\\psi}|^2 + {\\epsilon}\\int _0^ t \\int {|{\\nabla}{\\Delta}{\\psi}|^2 } \\\\&\\qquad\\qquad\\qquad \\qquad \\le C {\\left( \\| {\\varphi}_0\\|_{1,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_w^2+ \\| {\\psi}_0\\|^2+\\|{\\Delta}{\\psi}_0\\|^2 \\right)} \\nonumber, \\\\\n& \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2}{N}} + \\int _0^ t \\int {\\frac{ |{\\nabla}^2 {\\psi}|^2 }{N}} \n+ {\\epsilon}\\int _0^ t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2 }{N}} \\le C ( \\| {\\nabla}{\\psi}_0 \\|_{1,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{1,w}^2) + C\\int_0^t \\int {\\frac{ |{\\nabla}^3{\\varphi}|^2}{N}}. ",
"\\label{claim2_lemma23__e}\\end{aligned}$$\n\n$\\bullet$ Proof of\\\nAs we did in the proof of (and Lemma \\[lemma1\\_1\\_e\\] for ${\\epsilon}$-control), we can get $$\\begin{aligned}\n\\int_0^t \\int N |{\\Delta}{\\psi}|^2 +{\\epsilon}\\int_0^t \\int |{\\nabla}{\\Delta}{\\psi}|^2 \\le & C ( \\| {\\varphi}_0\\|_{1,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_w^2+ \\| {\\psi}_0\\|^2+\\|{\\Delta}{\\psi}_0\\|^2).\\end{aligned}$$ Then by , we get .\\\n$\\bullet$ Proof of\\\nAs we did in the proof of (and Lemma \\[lemma1\\_2\\_e\\] for ${\\epsilon}$-control), we can get $$\\begin{aligned}\n&\\int w |{\\nabla}^2 {\\psi}|^2 + (\\frac{c}{2}-C {\\epsilon}) \\int_0^t \\int w |{\\nabla}^2 {\\psi}|^2 + \\frac{\\epsilon}4 \\int_0^t\\int w|{\\nabla}^3{\\psi}|^2 \\\\\n&\\le C ( \\| {\\nabla}{\\psi}_0\\|_{1,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{1,w}^2) \n+ C \\int_0^t \\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} . ",
"\\end{aligned}$$\n\nThen, by making ${\\epsilon}$, it proves the claim .\\\nUsing and with , we have Lemma \\[lemma23\\_e\\] for case $k=i+j=2$. Indeed together with Lemma \\[lemma01\\_\\_e\\], we have proved $$\\begin{aligned}\n\\label{lemma2__e} &\n\\| {\\varphi}\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}\\|_{1,w}^2+ \\| {\\psi}\\|^2\n+ \\int_0^t \\sum_{l = 1,2,3} \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^t \\sum_{l = 1,2}\\| {\\nabla}^l {\\psi}\\|_w^2\n+{\\epsilon}\\int_0^t \\sum_{l = 1,2,3} \\| {\\nabla}^{l} {\\psi}\\|_w^2\\\\\n &\\le C( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2). ",
"\\nonumber\\end{aligned}$$\n\n \\\n$\\bullet$ Case $ k=i + j =3$\\\n \\\nWe repeat the previous argument for $k=3$.\n\nCollecting the above estiamtes and using , we get the third order version of : $$\\begin{aligned}\n\\label{beforeclaims34_lemma23__e} &\\int {\\frac{ |{\\nabla}^3 {\\varphi}|^2}{N}} +\\int |{\\nabla}^3 {\\psi}|^2 + \\int_0^t \\int {\\frac{ |{\\nabla}^4 {\\varphi}|^2}{N}} +\n {\\epsilon}\\int_0^t \\int{ |{\\nabla}^4 {\\psi}|^2}\\\\\n &\\le C( \\| {\\varphi}_0\\|_{3,w}^2 + \\| {\\nabla}^3 {\\psi}_0\\|^2+\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2) \n \\nonumber\n \\end{aligned}$$\n\nAs before, we can show the following two estimates for the third order derivatives: $$\\begin{aligned}\n \\label{claim3_lemma23__e}\n&\\int_0^t \\int N | {\\nabla}^3 {\\psi}|^2 +{\\epsilon}\\int_0^t \\int |{\\nabla}^2{\\Delta}{\\psi}|^2 \\le C ( \\| {\\varphi}_0\\|_{2,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{1,w}^2+ \\| {\\psi}_0\\|^2+\\|{\\nabla}{\\Delta}{\\psi}_0\\|^2),\n\\\\\n& \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2}{N}} + \\int _0^ t \\int {\\frac{ |{\\nabla}^3 {\\psi}|^2 }{N}} \n+ {\\epsilon}\\int _0^ t \\int {\\frac{ |{\\nabla}^4 {\\psi}|^2 }{N}} \\le C ( \\| {\\nabla}{\\psi}_0 \\|_{2,w}^2 + \\| {\\psi}_0\\|^2 + \\| {\\varphi}_0\\|_{2,w}^2)\n+ C\\int_0^t \\int {\\frac{ |{\\nabla}^4{\\varphi}|^2}{N}}. ",
"\\label{claim4_lemma23__e}\\end{aligned}$$\n\nBy using and with , we can prove Lemma \\[lemma23\\_e\\] for case $k=i+j=3$.\n\n \\\nFinally, we obtain Proposition \\[uniform\\_with\\]:\n\nAdding all the lemmas in this section, we have\\\n$$\\begin{split}\\label{lemma3_e}\n& {\\sup_{s\\in[0,t]}\\Big(\\| {\\varphi}(s)\\|_{3,w}^2 + \\| {\\nabla}{\\psi}(s)\\|_{2,w}^2 + \\| {\\psi}(s)\\|^2}\\Big) + \\int_0^t \\sum_{l = 1}^4 \\| {\\nabla}^{l} {\\varphi}\\|_w^2\n+\\int_0^t \\sum_{l = 1}^3\\| {\\nabla}^l {\\psi}\\|_w^2\n+{\\epsilon}\\int_0^t \\| {\\nabla}^{4} {\\psi}\\|_w^2\n\\\\\n& \\le\n C { ( \\| {\\varphi}_0\\|_{3,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{2,w}^2+ \\| {\\psi}_0\\|^2)}.",
"\n\\end{split}$$\n\nNote that the above estimate does not contain any control of $\\|{\\psi}\\|_w$. This is not a problem when ${\\epsilon}>0$. Indeed, for ${\\epsilon}>0$, thanks to the mean-zero assumption (in $y$), we can use the Poincaré inequality (see ) to get\n\n$$\\| {\\psi}\\|^2_w=\\int \\frac{|{\\psi}|^2}{N}\\leq C{\\lambda}^2\\int \\frac{|{\\psi}_y|^2}{N}\\leq \\| {\\nabla}{\\psi}\\|_{w}^2\\leq \nC { ( \\| {\\varphi}_0\\|_{3,w}^2 +\n \\| {\\nabla}{\\psi}_0\\|_{2,w}^2+ \\| {\\psi}_0\\|^2)}.$$ However, when ${\\epsilon}=0$, just having the periodic condition (in $y$) is not enough to get such an estimate.",
"\n\nProof of Theorem \\[thm\\_eventual\\]\n==================================\n\nTaking a $y$-derivative in we have $$\\begin{aligned}\n{\\partial}_t n_y - {\\Delta}n_y & = {{\\nabla}\\cdot}(nq)_y \\\\\n{\\partial}_t q_y - {\\epsilon}{\\Delta}q_y & = -2{\\epsilon}( (q \\cdot {\\nabla}) q )_y + {\\nabla}n_y.\\end{aligned}$$ Integration by parts gives $$\\begin{aligned}\n\\frac 12 {\\frac{d}{dt}}\\| n_y\\|^2 + \\|{\\nabla}n_y\\|^2 & \\le \\| q\\|_{L^{\\infty}} ( {\\delta}^{-1}\\| n_y\\|^2 + {\\delta}\\| {\\nabla}n_y\\|^2)+ \\|n\\|_{L^{\\infty}} ( {\\delta}^{-1} \\| q_y\\|^2 + {\\delta}\\|{\\nabla}n_y\\|^2) \\label{ny}\\\\\n\\frac 12 {\\frac{d}{dt}}\\|q_y\\|^2 + {\\epsilon}\\|{\\nabla}q_y\\|^2 &\\le -2{\\epsilon}\\int ( (q\\cdot {\\nabla})q)_y \\cdot q_y + \\int {\\nabla}n_y \\cdot q_y\n\\label{qy}\\end{aligned}$$ where ${\\delta}>0$ is small parameter to be chosen. ",
"The right hand side of the second inequality is bounded by $${\\epsilon}\\| {\\nabla}q\\|_{L^{\\infty}}\\| q_y\\|^2 \n+ {\\epsilon}\\|q\\|_{L^{\\infty}}( {\\delta}\\| {\\nabla}q_y\\|^2 +{\\delta}^{-1} \\| q_y\\|^2) + {\\delta}\\|{\\nabla}n_y\\|^2 + {\\delta}^{-1}\\|q_y\\|^2.$$ We add and . ",
"For sufficiently small ${\\delta}>0$, we make $\\| {\\nabla}n_y\\|^2$ and $\\epsilon\\| {\\nabla}q_y\\|^2$ terms of the right hand side smaller than $ \\frac 12\\|{\\nabla}n_y\\|^2 + \\frac {{\\epsilon}}{2} \\|{\\nabla}q_y\\|^2 $. ",
"By the Poincaré inequality (in $y$) we have $$\\frac 12\\|{\\nabla}n_y\\|^2 \\ge \\frac{\\| n_y\\|^2}{C{\\lambda}^2}, \\quad \\frac 12 \\|{\\nabla}q_y\\|^2 \\ge \\frac{\\| q_y\\|^2}{C{\\lambda}^2}.$$ Then we have $$\\frac 12{\\frac{d}{dt}}( \\|n_y\\|^2 +\\| q_y\\|^2) + \\frac{ \\|n_y\\|^2 +{\\epsilon}\\| q_y\\|^2}{C{\\lambda}^2}$$ $$\\le \nC_1 {\\delta}^{-1} (\\|n_y\\|^2 +\\| q_y\\|^2) + {\\epsilon}C_1 (1 +{\\delta}^{-1})\\|q_y\\|^2 + {\\delta}^{-1} \\|q_y\\|^2.$$ Thanks to $\\epsilon>0$, we decrease ${\\lambda}>0$ to get $${\\frac{d}{dt}}( \\|n_y\\|^2 +\\| q_y\\|^2) + \\frac{ \\|n_y\\|^2 +{\\epsilon}\\| q_y\\|^2}{C{\\lambda}^2} \\le 0.$$ Thus, there exists a constant $c=c({\\epsilon},{\\lambda})>0$ such that $${\\frac{d}{dt}}( \\|n_y\\|^2 +\\| q_y\\|^2) \\leq -c ({ \\|n_y\\|^2 +\\| q_y\\|^2}).$$\n\nacknowledgement {#acknowledgement .unnumbered}\n===============\n\nMC’s work was supported by NRF-2015R1C1A2A01054919. ",
"KC’s work was supported by the 2015 Research Fund(1.150064.01) of UNIST(Ulsan National Institute of Science and Technology) and by NRF-2015R1D1A1A01058614. ",
"KK’s work was supported by NRF-2014R1A2A1A11051161 and NRF-20151009350. ",
"JL’s work was supported by NRF-2009-0083521.",
"\n\n[00]{} R. A. Anderson and M.A,J. Chaplain, [*Modelling the growth and form of capillary networks in On Growth and Form: Spatio-Tempotal Pattern formation in Biology*]{}, Wiley, 225–249, 1999. ",
"P.K. Brazhnik, J. J. Tyson, [*On traveling wave solutions of fisher’s equation in two spatial dimensions*]{}, SIAM J. Appl. ",
"Math., ",
"Vol. ",
"60, no 2, 371–391, 2000. ",
"P. Constantin, A. Kiselev, and L. Ryzhik, [*Fronts in reactive convection: bounds, stability and instability*]{}, Comm. ",
"Pure and Appl. ",
"Math., ",
"56, 1781–1803, 2003. ",
"L. Corrias, B. Perthame, H. Zaag, [*A chemotaxis model motivated by angiogenesis*]{}, C. R. Math. ",
"Acad.",
"Sci. ",
"Paris, 336(2); 141–146, 2003. ",
"L. Corrias, B. Perthame, H. Zaag, [*Global solutions of some chemotaxis and angiogenesis systems in high space dimensions*]{}, Milan J. Math., ",
"72(1); 1–28, 2004. ",
"Bruce Alberts et al., *",
"Molecular biology of the cell*, 4th edition, Garland Science, New York, 2002. ",
"M. A. Fontelos, A. Friedman and B. Hu, *Mathematical analysis of a model for the initiation of angiogenesis,* SIAM. ",
"J. Math. ",
"Anal., ",
"Vol. ",
"33, 1330–1355, 2002. ",
"A. Friedman and J.I.Tello, *Stability of solutions of chemotaxis equations in reinforced random walks*, J. Math. ",
"Anal. ",
"Appl., ",
"Vol. ",
"272, 136–163, 2002. ",
"H-Y. Jin, J. Li and Z.A. Wang, *Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity*, J. Differential equations, Vol. ",
"225, 193–210, 2013. ",
"J.A.Leach and D.J.Needham. *",
"Matched asymptotic expansions in reaction-diffusion theory*, Springer monograghs in mathematics, 2004. ",
"H.A.Levine, B. D. Sleeman and M. Nilson-Hamilton. *",
"A mathemacial model for the roles of pericytes and macrophages in the onset of angiogenesis: I. the role of protease inhibitors in preventing angiogenesis.* ",
"Math. ",
"Biosci., ",
"168, 77–115, 2000. ",
"D. Li and T. Li, [*On a hyperbolic-parabolic system modeling chemotaxis*]{}, Math. ",
"Models Methods Appl. ",
"Sci. ",
"21, No. ",
"8,1631–1650, 2011. ",
"J. Li, T. Li and Z.A. Wang, *Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity*, Math. ",
"Models and Methods in Appl. ",
"Sci., ",
"Vol 24, no 14, 2819–2849, 2014. ",
"T. Li and Z. A. Wang, [*Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis*]{}, J. Differential equations, Vol. ",
"250, 1310–1333, 2011. ",
"D. Horstmann. *",
"From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I*, Jahresber. ",
"Deutsch. ",
"Math.-Verein., ",
"105, 103–-165, 2003. ",
"D. Horstmann and A. Stevens, *A constructive approch to traveling waves in chemotaxis*, J. Nonlinear Sci. ",
"Vol. ",
"14, 1–25, 2004. ",
"E. F. Keller and L. A. Segel, *Traveling bands of chemotactic bacteria: A Theoretical Analysis*, J. theor. ",
"Biol. ",
"30, 235-–248, 1971. ",
"J. D. Murray Mathematical Biology I: An Introduction. ",
"Springer, Berlin, 3rd edition, 2002. ",
"T. Nagai and T. Ikeda, *Traveling waves in a chemotactic model*, J. Math. ",
"Biol. ",
"30, 169–184, 1981. ",
"B. Perthame. ",
"PDE models for chemotactic movement: parabolic, hyperbolic and kinetic. *",
"Applications of Mathematics.*, ",
"49, 539–564, 2004. ",
"H. Schwetlick, Travelling waves for chemotaxis systems. ",
"Proc. ",
"Appl. ",
"Math. ",
"Mech., ",
"2: 476–478, 2003. ",
"J. A. Shettatt, Travelling Wave solutions of a mathematical model for tumor encapulation, Siam J. Appl. ",
"Math. ",
"Vol. ",
"60, no. ",
"2, 392–407, 1999. ",
"V. A. Solonnikov, *On the solvability of boundary and initial boundary value problems fir the Navier-Stokes system in domains with noncompact boundaries*, Pacific J. Math., ",
"Vol. ",
"93, no. ",
"2, 443–458, 1981. ",
"R. Temam, *Navier-Stokes equations*, AMS Chelsea publishing, 2001. ",
"Z. A. Wang, *Wavefront of an angiogenesis model*, DCDS-B, Vol. ",
"17, no. ",
"8, 2849–2860, 2012. ",
"Z.A.Wang and T. Hillen, *Shock formation in a chemotaxis model*, Math. ",
"Methods, Appl. ",
"Sci., ",
"Vol. ",
"31, 45–70, 2008.",
"\n\n[^1]: [In Bruce Alberts et al [@Ce], it is summarized as follows: “Cells that are short of oxygen increase their concentration of certain protein (HIF-1), which stimulates the production of vascular endothelial growth factor (VEGF). ",
"VEGF acts on endothelial cells, causing them to proliferate and invade the hypoxic tissue to supply it with new blood vessels.\"]{}",
"\n\n[^2]: In the infinite strip domain $\\Omega$, it is shown that for any $u$ satisfying $ \\| u \\|_{L^2(\\Omega)}^2 \n + \\int_0^{\\infty} \\mathcal U^2 (x) dx <\\infty$ there exist a vector function $\\varphi$ in $H_0^1(\\Omega)$ such that ${{\\nabla}\\cdot}{\\varphi}= u$ in [@Sol].",
"\n"
] | {
"pile_set_name": "ArXiv"
} | [
0.03125,
0.01818181818181818,
0,
0.005681818181818182,
0,
0,
0,
0.013368983957219251,
0.037037037037037035,
0.004750593824228029,
0.006968641114982578,
0,
0,
0,
0,
0.002229654403567447,
0.00931098696461825,
0.004703668861712135,
0,
0.009242144177449169,
0,
0,
0,
0.00398406374501992,
0.005311077389984826,
0,
0.003436426116838488,
0,
0,
0,
0.0026845637583892616,
0.006622516556291391,
0.0054694621695533276,
0.009615384615384616,
0.007042253521126761,
0,
0,
0,
0.0043023970497848806,
0,
0,
0.008130081300813009,
0,
0.008064516129032258,
0,
0,
0.0037174721189591076,
0.010050251256281407,
0.02865329512893983,
0.006741573033707865,
0,
0.020833333333333332,
0.004016064257028112,
0.004048582995951417,
0.028368794326241134,
0.011111111111111112,
0.002304147465437788,
0.008064516129032258,
0,
0.0011890606420927466,
0.006802721088435374,
0.005813953488372093,
0.012345679012345678,
0.010113780025284451,
0.005012531328320802,
0,
0.0036900369003690036,
0.002364066193853428,
0.003424657534246575,
0,
0.004689331770222743,
0.0049800796812749,
0.002677376171352075,
0.0033500837520938024,
0.003929273084479371,
0.008168028004667444,
0.002577319587628866,
0.003472222222222222,
0,
0.004545454545454545,
0.0043859649122807015,
0.005780346820809248,
0,
0,
0,
0,
0.0081799591002045,
0.004071661237785016,
0.002652519893899204,
0,
0.003109452736318408,
0.005627009646302251,
0.0022371364653243847,
0,
0.02586206896551724,
0.00546448087431694,
0,
0,
0.0019940179461615153,
0.006075768406004289,
0.00847457627118644,
0,
0,
0,
0,
0,
0.002304147465437788,
0.0021551724137931034,
0,
0.001358695652173913,
0.011627906976744186,
0,
0,
0,
0,
0.002489110143123833,
0.00620347394540943,
0.0009407337723424271,
0.007633587786259542,
0.01282051282051282,
0,
0.0048,
0.005194805194805195,
0.006309148264984227,
0.0058823529411764705,
0,
0,
0,
0.0009610764055742432,
0.001455604075691412,
0.007488479262672811,
0.004700352526439483,
0.01084010840108401,
0,
0.0007855459544383347,
0.004651162790697674,
0,
0.00315955766192733,
0.0035511363636363635,
0.002105263157894737,
0.0016181229773462784,
0.003989361702127659,
0.007731958762886598,
0.009721322099805573,
0,
0.008287292817679558,
0.012422360248447204,
0.0040650406504065045,
0.0046439628482972135,
0.014492753623188406,
0.0031645569620253164,
0.005827505827505828,
0,
0.0017793594306049821,
0.006711409395973154,
0.005835543766578249,
0.008375209380234505,
0.0031832651205093224,
0.011041009463722398,
0,
0,
0.0007407407407407407,
0.001949317738791423,
0.006144890038809832,
0.0020161290322580645,
0.00505902192242833,
0.004035512510088781,
0.004784688995215311,
0.00234192037470726,
0.007575757575757576,
0.0031426775612822125,
0.0037313432835820895,
0.004786597526924611,
0.015503875968992248,
0.003968253968253968,
0.006631299734748011,
0,
0.0017985611510791368,
0.0018975332068311196,
0.005847953216374269,
0.002188183807439825,
0.011235955056179775,
0.007549361207897794,
0,
0.00315955766192733,
0,
0.017045454545454544,
0,
0,
0,
0,
0.011673151750972763,
0.0032258064516129032,
0.00510204081632653,
0.004601226993865031,
0,
0,
0,
0,
0.00267379679144385,
0.0031201248049922,
0,
0.011764705882352941,
0.006993006993006993,
0.005050505050505051,
0.002277904328018223,
0,
0.0018552875695732839,
0.0050568900126422255,
0.007692307692307693,
0,
0,
0.008064516129032258,
0,
0,
0.0070921985815602835,
0.010752688172043012,
0.0028222013170272815,
0.0036188178528347406,
0.005479452054794521,
0,
0.0027624309392265192,
0.004807692307692308,
0.015625,
0.0049261083743842365,
0.0022271714922048997,
0,
0.0026109660574412533,
0.0016339869281045752,
0,
0.011627906976744186,
0,
0.0030959752321981426,
0,
0.0025380710659898475,
0,
0.0016542597187758478,
0.0028462998102466793,
0,
0,
0.0013386880856760374,
0.001838235294117647,
0.0008771929824561404,
0.0012578616352201257,
0.010625737898465172,
0.009535160905840286,
0.005199306759098787,
0.005714285714285714,
0.004784688995215311,
0.0017152658662092624,
0.0037593984962406013,
0,
0,
0.00351288056206089,
0.019230769230769232,
0.027777777777777776,
0.045454545454545456,
0.02577319587628866,
0.016129032258064516,
0,
0,
0,
0.025,
0,
0,
0,
0.04081632653061224,
0,
0,
0,
0.027972027972027972,
0,
0.043478260869565216,
0.01282051282051282,
0.02586206896551724,
0.1111111111111111,
0,
0,
0,
0.017699115044247787,
0,
0,
0,
0,
0.025974025974025976,
0.05,
0,
0,
0.0392156862745098,
0.006369426751592357,
0,
0,
0,
0.024096385542168676,
0,
0,
0,
0,
0.02459016393442623,
0,
0,
0,
0.019230769230769232,
0,
0.06666666666666667,
0.02040816326530612,
0,
0,
0,
0.02830188679245283,
0,
0,
0.028037383177570093,
0,
0,
0.018518518518518517,
0.02702702702702703,
0.02702702702702703,
0,
0,
0.07692307692307693,
0.0136986301369863,
0,
0,
0.017857142857142856,
0,
0,
0,
0,
0,
0.028846153846153848,
0,
0,
0,
0,
0.011560693641618497,
0,
0,
0,
0.029850746268656716,
0.015873015873015872,
0,
0,
0.014084507042253521,
0,
0,
0,
0,
0.00851063829787234,
0.007692307692307693,
0.007246376811594203,
0
] | 0.006096 | 5 |
[
"Waqar Malik\n\nWaqar Malik (born 15 February 1995) is a Pakistani cricketer. ",
"He made his first-class debut for Water and Power Development Authority in the 2018–19 Quaid-e-Azam Trophy on 8 September 2018.",
"\n\nReferences\n\nExternal links\n \n\nCategory:1995 births\nCategory:Living people\nCategory:Pakistani cricketers\nCategory:Water and Power Development Authority cricketers\nCategory:Place of birth missing (living people)"
] | {
"pile_set_name": "Wikipedia (en)"
} | [
0.02666666666666667,
0.007874015748031496,
0
] | 0.011514 | 5 |
[
"The Ocean Foundation and University of Havana’s Marine Research Center: 21 Years of Science, Discovery, and Friendship\n\nThis week The Ocean Foundation attended the 50th Anniversary Celebration of the University of Havana’s Centro de Investigaciones Marinas (CIM, Marine Research Center) where TOF was recognized for its 21 years of collaboration with CIM on marine science in Cuba. ",
"TOF’s work with CIM began in 1999 when TOF’s Fernando Bretos met the CIM Director at the time Doctor Maria Elena Ibarra. ",
"Dr. Ibarra’s passion for marine conservation and in partnering with international groups was the driving force behind TOF’s first collaborations with CIM.",
"\n\nThe first TOF-CIM collaborative project involved analyses of CIM’s taxonomic collection in 1999. ",
"Since then, TOF-CIM collaborations have grown to include sea turtle conservation in Cuba’s Guanahacabibes National Park, research cruises along nearly the entirety of the Cuban coastline, international fisheries learning exchanges, expeditions to monitor coral spawning, and most recently a project to study and protect sawfish in Cuba. ",
"These collaborations have led to important conservation outcomes and form the basis for over 30 doctoral and master’s dissertations for CIM students. ",
"CIM has also been long-time partners in TOF’s Trinational Initiative for Marine Science and Conservation in the Gulf of Mexico & Western Caribbean.",
"\n\nKatie Thompson (left) and CIM Director, Patricia González\n\nTOF’s Alejandra Navarrete and Katie Thompson attended this week’s celebration. ",
"Mrs. Navarrete received an award from CIM for TOF’s decades of collaboration with and support of CIM. ",
"Ms. Thompson gave the presentation “The Ocean Foundation and CIM: 21 years of science, discovery, and friendship” on the panel “International Scientific Relations and Capacity Building” moderated by CIM Director Patricia González. ",
"TOF is excited to continue collaborating with CIM for many more years on marine science and conservation in Cuba and the Wider Caribbean Region.",
"\n\nRead More\n\nKeynoteWednesday, 9 October 2019 Honorable Senators and distinguished guests.",
"My name is Mark Spalding, and I am the President of The Ocean Foundation, and of the A.C. Fundación Mexicana para el Océano This is my 30th year of …\n\nConference of Parties for the Cartagena Convention will meet in Roatan, Honduras to address marine environmental issues Regional Experts look forward to finding solutions for common challenges in the Wider Caribbean …"
] | {
"pile_set_name": "Pile-CC"
} | [
0.015706806282722512,
0.01652892561983471,
0.006493506493506494,
0,
0.002967359050445104,
0,
0.013605442176870748,
0.02857142857142857,
0.00980392156862745,
0.008658008658008658,
0.006944444444444444,
0,
0.010869565217391304
] | 0.009242 | 5 |
[
"EDIT: Apparently this gif I made out of boredom is in some front page (My DA setting is such that I only see the newest stuff lol...). ",
"I thank you all for letting this work be up there but as I said in this comment thread, I was not quite aware about this big deal of a Front Page until recently. ",
"So yea, that was not intentional on my part...the DA community - or a good fraction of it - likes it enough to be given that recognition. ",
"So if you feel like this is injustice, I suggest you flame or post a forum thread about it elsewhere."
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0.006172839506172839,
0,
0
] | 0.001543 | 5 |
[
"1. ",
"Field of the Invention\nThe present invention relates to controllers for vending machines and, more particularly, to a controller for a vending machine, wherein a mode for enabling selling of commodities can be selected even when change lacks.",
"\n2. ",
"Description of the Related Art\nConventionally, a vending machine has an empty detector which detects as lack of change when the coins stored in change coin tubes arrives at a minimum number necessary for paying out the change. ",
"It is arranged such that when the empty detector detects the lack of change, all the selling operations involving payment of change are inhibited. ",
"Otherwise, payment of change is not ensured, so that a user would suffer from unexpected loss.",
"\nHowever, there may be cases where a person who uses vending machines wishes to buy commodities even if he cannot receive change: for example, in a case where there is no way to buy commodities except from a nearby vending machine; in a case where a nearby vending machine from which he can buy commodities is not in service because of time; and in a case where he somehow wants to get commodities at once even if he may lose small change, etc. ",
"Even such cases, the selling operation involving payment of change is completely inhibited when there is no change stored in a conventional vending machine. ",
"Therefore, it is impossible for a person to get desired commodities from a conventional vending machine which sells those commodities even though the vending machine is located near him, if he does not have kinds of denominations which do not require change."
] | {
"pile_set_name": "USPTO Backgrounds"
} | [
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0 | 5 |
[
"Q:\n\nWhat is the equivalent of \"android:fontFamily=\"sans-serif-light\" in Java code?",
"\n\nMy question is quite simple: \nIn every of my textview, I am currently using the attribute \nandroid:fontFamily=\"sans-serif-light\"\n\nto provide a gorgeous look on post HC devices.",
"\nUnfortunately, this doesn't work with every widget and for my Spinners, I need to overwrite the Adapter.",
"\n@Override\npublic View getView(int position, View convertView, ViewGroup parent) {\n //You can use the new tf here.",
"\n\n if(convertView == null || convertView.getTag() == null) {\n // new view - populate \n convertView = inflater.inflate(android.",
"R.layout.simple_spinner_dropdown_item, parent, false);\n convertView.setTag(new Object());\n }\n\n CheckedTextView spinner_text=(CheckedTextView) convertView.findViewById(android.",
"R.id.text1);\n //Typeface should be set here...\n return spinner_text;\n }\n}\n\nSo, is there a way to get exactly the same result by code?",
"\nPS: No, I don't want to put a typeface in asset folder, I just want to use system one.",
"\n\nA:\n\nIt should be possible with setTypeface() and Typeface.create():\nconvertView.setTypeface(Typeface.create(\"sans-serif-light\", Typeface.",
"NORMAL));\n\nSee Docs:\n\nCreate a typeface object given a family name, and option style\n information. ",
"If null is passed for the name, then the \"default\" font\n will be chosen. ",
"The resulting typeface object can be queried\n (getStyle()) to discover what its \"real\" style characteristics are.",
"\n\nNote that excessively using Typeface.create() is bad for your memory, as stated in this comment. ",
"The suggested Hashtable is a good solution, but you have to modify it a little since you don't create your typeface from an asset.",
"\n\nA:\n\nAndroid 4.1 (API Level 16) and Support Library 26 and higher\n\nIf you are using res -> font folder, you can use like this \n val typeface = ResourcesCompat.getFont(Context, R.font.",
"YOUR_FONT)\n TextView.setTypeface(typeface)\n\nA:\n\nIn my opinion there is still a way to apply system fonts programatically on TextView without having any memory issue and that is using textview.setTextAppearance method :\n<style name=\"styleA\">\n <item name=\"android:fontFamily\">sans-serif</item>\n <item name=\"android:textStyle\">bold</item>\n <item name=\"android:textColor\">?android:attr/textColorPrimary</item>\n</style>\n<style name=\"styleB\">\n <item name=\"android:fontFamily\">sans-serif-light</item>\n <item name=\"android:textStyle\">normal</item>\n <item name=\"android:textColor\">?android:attr/textColorTertiary</item>\n</style>\n\nif(condition){\n textView.setTextAppearance(context,R.style.styleA);\n}else{\n textView.setTextAppearance(context,R.style.styleB);\n}\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.012195121951219513,
0,
0,
0.017094017094017096,
0.006993006993006993,
0.015957446808510637,
0,
0,
0.014388489208633094,
0,
0,
0,
0,
0,
0.005405405405405406,
0
] | 0.004502 | 5 |
[
"Story highlights Republican Sen. Jeff Flake has broken from Trump in the past\n\nTrump is trying to find someone to run against him in his Senate primary\n\nWashington (CNN) Republican Sen. Jeff Flake insists that he wants to \"work with\" President Donald Trump -- despite Trump's attempts to find a challenger to replace the sitting senator in his upcoming election.",
"\n\nThe frequent Trump critic refrained from adding fuel to the fire Thursday morning on Fox News after Trump's rally in Arizona on Tuesday -- at which the President, without using his name, slammed Flake as \"weak\" in front of his home-state crowd.",
"\n\nWhen asked why Trump is not his biggest fan, Flake told Fox News, \"I don't know. ",
"I want to work with the President when I think that he's right, and I'll challenge him when I think that he's wrong.\"",
"\n\nAfter Tuesday's rally, Trump tweeted a day later: \"Phoenix crowd last night was amazing -- a packed house. ",
"I love the Great State of Arizona. ",
"Not a fan of Jeff Flake, weak on crime & border!\"",
"\n\nPhoenix crowd last night was amazing - a packed house. ",
"I love the Great State of Arizona. ",
"Not a fan of Jeff Flake, weak on crime & border! — ",
"Donald J. Trump (@realDonaldTrump) August 23, 2017\n\nBefore taking the stage in Phoenix, Trump huddled privately with potential Republican primary challengers to Flake.",
"\n\nRead More"
] | {
"pile_set_name": "OpenWebText2"
} | [
0.019337016574585635,
0.012195121951219513,
0.024096385542168676,
0,
0.009174311926605505,
0,
0.04081632653061224,
0,
0,
0.0392156862745098,
0.017964071856287425,
0
] | 0.013567 | 5 |
[
"Q:\n\nFinding a direct basis for tangent space of piece with boundary of an oriented manifold.",
"\n\nI have the following definition (from Hubbard's vector calculus book) for an oriented boundary of piece with boundary of an oriented manifold:\n\nLet $M$ be a $k$ dimensional manifold oriented by $\\Omega$ and $P$ a piece with boundary of $M$. Let $x$ be a point of the smooth boundary $\\partial^{ \\ S}_MP$ and let $\\vec{V}_{\\text{out}}\\in T_xM$ be an outward pointing bector. ",
"Then the function $\\Omega^\\partial : \\mathcal{B}(T_x\\partial P)\\to\\left\\{+1,-1\\right\\}$ given by\n $$\n\\Omega_x^\\partial(\\vec{v}_1,...,\\vec{v}_{k-1}) = \\Omega_x(\\vec{V}_{\\text{out}},\\vec{v}_1,...,\\vec{v}_{k-1})\n$$\n defines an orientation on the smooth boundary $\\partial_M^{ \\ S}P,$ where $\\vec{v}_1,...,\\vec{v}_{k-1}$ is an ordered basis of $T_x\\partial_M^{ \\ S}P$.\n\nI'm working on a problem that asks me to find a basis for the $T_x\\partial P$ that is direct to a certain orientation (given by an elementary 3-form). ",
"My question is this:\nWhen I choose a basis for $T_x\\partial P$, does this basis also need to lie in $T_xM$? ",
"Also, are there restrictions to how I should choose $\\vec{V}_{\\text{out}}$? ",
"In other words, does $\\vec{V}_{\\text{out}}$ need only lie in $T_xM$ and not in $T_x\\partial P$?",
"\n\nA:\n\nYes, $T_x\\partial P$ is a hyperplane in $T_xM$. $\\vec V_{\\text{out}}$ is by definition the outward-pointing normal to $\\partial P$. This means that if $\\vec v_1,\\dots,\\vec v_{k-1}$ are chosen as a basis for $T_x\\partial P$, then $\\vec V_{\\text{out}},\\vec v_1,\\dots,\\vec v_{k-1}$ will give you a basis for $T_xM$. The whole point of this orientation stuff is that when you pick $\\vec v_1,\\dots,\\vec v_{k-1}$ so that $\\vec V_{\\text{out}},\\vec v_1,\\dots,\\vec v_{k-1}$ gives you a positively-oriented basis for $T_xM$, then you win: You have achieved a positively-oriented basis for $T_x\\partial P$.\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0,
0,
0.0038535645472061657,
0,
0,
0,
0
] | 0.000551 | 5 |
[
"Neurogenins in brain development and disease: an overview.",
"\nThe production of neurons, astrocytes and oligodendrocytes is regulated by a group of transcription factors, which determine cell fates and specify subtype identities in the nervous system. ",
"Here we focus on profiling the distinct roles of Neurogenin (Ngn or Neurog) family members during the neuronal development. ",
"Ngn proteins are tightly regulated to be expressed at defined times and positions of different progenitor cell pools. ",
"In addition to their well-elucidated proneural function, Ngn proteins play various critical roles to specify or maintain cell fate and regulate neurite outgrowth and targeting in the central nervous system. ",
"Finally, Ngns have been associated with neuronal disorders. ",
"Therefore understanding the function and regulation of Ngns will not only improve the understanding of the molecular mechanism underlying the development of nervous system, but may also provide insight into neuronal disease."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0.008064516129032258,
0.00847457627118644,
0.004830917874396135,
0.016666666666666666,
0.004464285714285714
] | 0.006072 | 5 |
[
"Treatment of the agitation of late-life psychosis and Alzheimer's disease.",
"\nIn older as well as younger people, antipsychotic medication is commonly used to treat psychoses. ",
"In clinical practice, antipsychotic medication is also used to control severe behavioral disturbances such as agitation, wandering, self-mutilation, as well as assaultiveness. ",
"Neuroleptic and non-neuroleptic drug treatments are used to control severe agitation and disruptive behavior. ",
"Among typical neuroleptics, very low doses (e.g., 0.25 mg of haloperidol 1-4 times per day) may be effective and limit the development of severe extrapyramidal reactions. ",
"Recent experience suggests that the atypical neuroleptics, olanzaine, risperidone, and quetiapine, are also useful for controlling severe agitation in elderly demented nursing home residents. ",
"The benzamides are also known for the treatment of behavioral disturbances in Europe, but there is little experience in the U.S.A. Although research studies in this area are virtually nonexistent, growing clinical experience suggests that the following may be quite useful: 1) trazodone; 2) buspirone (Buspar); 3) anticonvulsants (e.g., valproate); and 4) beta-blockers."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0,
0,
0,
0,
0.002702702702702703
] | 0.000386 | 5 |
[
"Isabel Sandoval’s “Lingua Franca,” which tells the story of a Filipino transgender immigrant in New York, will be one of 11 films that will compete—and have its world premiere—at the 2019 Venice Days (or Giornate degli Autori), the independent section that runs alongside the Venice Film Festival in Italy, its producer Darlene Malimas told the Inquirer.",
"\n\nMalimas explained that she and fellow producers, Sandoval, Jhett Tolentino and Carlo Velayo, have aimed for “Lingua Franca” to premiere at the Venice Days because “it is one of the world’s most prestigious festivals. ",
"Plus, our past films were able to compete in Locarno (“Senorita,” 2010) and Busan (“Aparisyon,” 2012); and so to have another prestigious fest to debut our third collaboration is surreal.”",
"\n\nADVERTISEMENT\n\nMalimas added that they hope to “attract distributors and engaging reviews … [since] all eyes will be on those films that show promise.”",
"\n\nThe film, which also features Sandoval as a transgender immigrant named Olivia, is set in Brighton Beach, New York. ",
"Desperate to avoid deportation, Olivia gets involved with a Russian slaughterhouse worker who is unaware that she is trans.",
"\n\n“Lingua Franca” was chosen out of 900 submissions, Malimas pointed out. “",
"This film is very personal to [Sandoval]. ",
"Also, it will go down in history as the first film written and directed by a transwoman of color,” she added.",
"\n\nAsked to explain how she came up with the story, the New York-based Sandoval said: “We have always been fascinated by the female gaze and female character-driven pieces. ",
"Olivia was no different. ",
"We wanted to tell a story that touches on the hot-button issues of Trump’s America such as LGBTQIA+, immigration and underrepresentation, but is intimate enough and accessible. ",
"Hence, ‘Lingua Franca’ became a complex love story of two immigrants from very disparate backgrounds.”",
"\n\nSandoval also told the Inquirer: “It isn’t often that minority filmmakers, like me, are given the platform to tell stories about my own community, which is why I feel an enormous responsibility to tell the story with candor and compassion.”",
"\n\nSandoval pointed out that, drawing from personal experiences, she tried her best to make Olivia “as layered and nuanced as I could. ",
"I’d like to think that my first-person perspective gives ‘Lingua Franca’ a lived-in authenticity.”",
"\n\nThe 10 other competing films are Dominik Moll’s “Only Beasts,” Amjad Abu Alala’s “You Will Die at 20,” Igort’s “5 is the Perfect Number,” Fabienne Berthaud’s “A Bigger World,” Javro Bustamante’s “The Weeping Woman,” Dag Johan Haugerud’s “Beware of Children,” Jan Komasa’s “Corpus Christi,” Mattie Do’s “The Long Walk,” Manele Labidi Labbe’s “Arab Blue” and Jo Odagiri’s “They Say Nothing Stays the Same.”",
"\n\nVenice Days will take place between Aug. 28 to Sept. 7. ",
"The winner will receive a $22,000 grand prize. —",
"MARINEL R. CRUZ\n\nADVERTISEMENT\n\nRead Next\n\nEDITORS' PICK\n\nMOST READ"
] | {
"pile_set_name": "OpenWebText2"
} | [
0.005649717514124294,
0.0182648401826484,
0.005319148936170213,
0.006535947712418301,
0.00847457627118644,
0,
0.013333333333333334,
0,
0,
0.005813953488372093,
0,
0.005649717514124294,
0,
0.004132231404958678,
0,
0,
0.019704433497536946,
0,
0,
0.014925373134328358
] | 0.00539 | 5 |
[
"Q:\n\nCan I output to a VM running SQL Server from Azure Analytics job?",
"\n\nGenerating output to a Azure SQL database is supported, but I was shocked when I found that the portal does not allow to specify a SQL Server database running on a VM. ",
"Is not this supported?",
"\nWe need to store lots of data coming through the ASA jobs, and use SQL Jobs, that's why we were planning to use a SQL Server VM.",
"\n Thanks!",
"\n\nA:\n\nYou cannot configure the SQL Database running on VM as an output to the ASA job.",
"\nHowever, Azure provides SQL services with 2 variants\nMicrosoft Azure SQL Database (Azure SQL Database) as PaaS\nwhere lower stack is managed by Microsoft Azure and billed as pay-as-you-go model.",
"\nand\nSQL Server in Azure Virtual Machine (VM) as IaaS where user owns the VM and make any changes, including licences for the SQL database.",
"\n\nthe Microsoft Azure SQL Database provided as PaaS is configurable as\nASA output.",
"\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.028985507246376812,
0.0058823529411764705,
0,
0.023255813953488372,
0,
0,
0.015384615384615385,
0.014285714285714285,
0.012195121951219513,
0
] | 0.009999 | 5 |
[
"Q:\n\nSelectList wont accept my SelectListItem object\n\nIm trying to populate my Html.",
"Listbox with all the users of my ASP.NET user database.",
"\nI have created a ViewModel that looks like this:\nusing System;\nusing System.",
"Collections.",
"Generic;\nusing System.",
"ComponentModel.",
"DataAnnotations;\nusing System.",
"Globalization;\nusing System.",
"Web.",
"Mvc;\nusing System.",
"Web.",
"Security;\n\nnamespace SkyLearn.",
"Areas.",
"Users.",
"Models\n{\npublic class UsersViewModel\n {\n MembershipUserCollection membershipuserscollection { get; set; }\n public List<MembershipUser> membershipuserslist { get; set; }\n public List<SelectListItem> AvailableUsers { get; set; }\n\n public UsersViewModel()\n {\n availableUsers = new List<SelectListItem>();\n } \n }\n}\n\nAnd a Actionresult that looks like this:\n public ActionResult Index()\n {\n var users = Membership.",
"GetAllUsers().Cast<MembershipUser>();\n\n UsersViewModel model = new UsersViewModel();\n\n foreach (MembershipUser user in users)\n {\n SelectListItem selectuser = new SelectListItem();\n\n selectuser.",
"Text = user.",
"UserName;\n selectuser.",
"Value = user.",
"UserName;\n\n model.",
"AvailableUsers.",
"Add(selectuser);\n }\n\n return View(model);\n }\n\nOn my view i try to populate the list like this:\n<%@ Page Title=\"\" Language=\"C#\" MasterPageFile=\"~/Views/Shared/Site.",
"Master\" Inherits=\"System.",
"Web.",
"Mvc.",
"ViewPage<SkyLearn.",
"Areas.",
"Users.",
"Models.",
"UsersViewModel>\" %>\n\n<%= Html.",
"ListBox(\"AvailableUsers\", Model.",
"AvailableUsers)%>\n\nWhen i run the code i get the follow error:\nCS1579: foreach statement cannot operate on variables of type 'SkyLearn.",
"Areas.",
"Users.",
"Models.",
"UsersViewModel' because 'SkyLearn.",
"Areas.",
"Users.",
"Models.",
"UsersViewModel' does not contain a public definition for 'GetEnumerator'\n\nA:\n\nAvailableUsers is not initialized, you can do this in the constructor of your ViewModel, for example.",
"\npublic class UsersViewModel\n {\n public UsersViewModel {\n AvailableUsers = new List<SelectListItem>();\n }\n MembershipUserCollection membershipuserscollection { get; set; }\n public List<MembershipUser> membershipuserslist { get; set; }\n public List<SelectListItem> AvailableUsers { get; private set; }\n}\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.012048192771084338,
0,
0.025974025974025976,
0,
0.09090909090909091,
0.06666666666666667,
0.06666666666666667,
0.07142857142857142,
0,
0.1111111111111111,
0,
0.03333333333333333,
0,
0,
0.00411522633744856,
0,
0,
0.030303030303030304,
0,
0.034482758620689655,
0.06666666666666667,
0.005555555555555556,
0,
0,
0,
0.05555555555555555,
0,
0,
0,
0,
0.03125,
0.014814814814814815,
0,
0,
0,
0.029411764705882353,
0,
0,
0,
0.0111731843575419,
0.002976190476190476
] | 0.018645 | 5 |
[
"It’s certainly not unusual for conflicts between business and labor to be played out in the Capitol.",
"\n\nIt is, however, very unusual for conflicts between two labor unions to reach the Capitol. ",
"And one such duel now presents Gov. Jerry Brown with a dilemma.",
"\n\nThe state’s building trades unions – carpenters, plumbers, electricians, etc. – ",
"sponsored legislation that would require construction work on oil refineries to be done by a “skilled and trained workforce.”",
"\n\nComing in the wake of a disastrous fire at Chevron’s Richmond refinery, that requirement doesn’t sound unreasonable from the standpoint of public safety. ",
"Public safety, in fact, is the official rationale for Senate Bill 54 from its author, Sen. Loni Hancock, D-Berkeley, and its sponsors.",
"\n\nBut, as with all such bills, the devil is in the details, and in this case, the definition of a “skilled and trained workforce” involves very detailed specifications of apprenticeship programs. ",
"And those specifications, not surprisingly, exactly mirror the programs operated by State Building and Construction Trades Council’s member unions.",
"\n\nThe effect of the legislation, which gained heavy support from the Legislature’s majority Democrats, apparently would be to exclude members of the United Steelworkers Union, which represents 5,000 refinery workers in the state. ",
"The USW has, therefore, mounted a campaign to persuade Brown to veto the bill.",
"\n\n“Instead of raising the bar for training,” United Steelworkers says in a letter to Brown, “SB54 lowers it by excluding our current trained and experienced workforce.”",
"\n\nThe union calls it “a jurisdictional dispute” that should be resolved “within the house of labor,” rather than “legislation that will devastate the lives of our members and their families.”",
"\n\nUnions aren’t the only ones divided by the bill. ",
"Contractors who employ building trades union members support it because it would, in effect, compel Chevron and other refineries to hire them to do construction work. ",
"But other contractors, along with the oil industry and other business groups, oppose it.",
"\n\nThe state’s largest array of refineries is to be found along San Francisco Bay and its estuaries in Contra Costa County, so the bill has received special attention in that area.",
"\n\nThe Contra Costa Times newspaper editorialized that “the Legislature has recklessly voted to place the entire Bay Area at risk” with the bill because it would undermine Contra Costa County’s industrial safety ordinance.",
"\n\nWe’d all like to think that the Legislature acts only after objectively seeing a problem, weighing pro and con arguments and informing itself of the facts.",
"\n\nSB54 is, unfortunately, further proof that most lawmakers’ votes are driven only by which special interest groups support and oppose a bill.",
"\n\nTrump says any NFL player who sits during anthem is 'son of a bitch' and should be fired\n\nShanahan: 'We just need to get better'\n\nBay Area toll bridge robbers caught on camera\n\nDan Walters: Big election, but most people 'could care less'\n\nThe Sacramento Bee's Dan Walters is excited that for the first time in a long time Californians were actually treated to a full blown presidential campaign, with all nominees from both parties campaigning up and down the state. ",
"Despite California voters actually being treated like they mattered, it still seems as if most people 'could care less.'"
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0,
0.015873015873015872,
0,
0,
0.00641025641025641,
0.022388059701492536,
0,
0.006802721088435374,
0.008695652173913044,
0.02564102564102564,
0.011904761904761904,
0,
0,
0.005988023952095809,
0,
0.00558659217877095,
0.013574660633484163,
0.006369426751592357,
0,
0.008528784648187633,
0
] | 0.006262 | 5 |
[
"Q:\n\nSwift Delegate Not Being Called\n\nI have a view with a delegate that I want to have call numpadView(numpadView:) in GameController upon button press, however I can't get it to work. ",
" The overload of touchesBegan() works fine, it's really the line with pressDelegate?.numpadView(self) which doesn't call the delegate function in the GameController class. ",
" I'm stumped as to what's missing to get it to work?",
"\nI cleaned the code to leave only the essential related to the problem for simplicity.",
"\nNumpadView.swift\nprotocol NumpadPressDelegateProtocol {\n func numpadView(numpadView: NumpadView)\n}\n\nclass NumpadView: UIButton{\n var pressDelegate: NumpadPressDelegateProtocol?",
"\n\n init(char: Character) {\n let frame = CGRectMake(160, 100, 50, 50)\n super.init(frame:frame)\n\n self.setTitle(\"\\(char)\", forState: UIControlState.",
"Normal)\n self.userInteractionEnabled = true\n }\n\n override func touchesBegan(touches: NSSet, withEvent event: UIEvent) {\n pressDelegate?.numpadView(self)\n } \n}\n\nGameController.swift\nclass GameController: NumpadPressDelegateProtocol {\n\n func numpadView(numpadView: NumpadView) {\n //do something\n }\n\n}\n\nA:\n\nDeclaring GameController as NumpadPressDelegateProtocol is not enough for you to get a callback. ",
"You also need to set the pressDelegate of the NumpadView instance inside the Gamecontroller. ",
"Assuming numpadView is the instance variable be it IBOutlet or not, you have to set the delegate like\nInside GameController init\n numpadView.pressDelegate = self\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.005405405405405406,
0.011627906976744186,
0,
0,
0.0111731843575419,
0.00641025641025641,
0.00963855421686747,
0,
0.012269938650306749
] | 0.006281 | 5 |
[
"Q:\n\nstr::contains doesn't work when the supplied input is a string reference\n\nuse std::collections::HashSet;\n\nfn character_count(needles: &HashSet<char>, needle_type: &str, input: &str) -> i32 {\n for needle in needles {\n let mut needle_custom;\n\n if needle_type == \"double\" {\n needle_custom = needle.to_string() + &needle.to_string();\n } else {\n needle_custom = needle.to_string() + &needle.to_string() + &needle.to_string();\n }\n\n if input.contains(needle_custom) {\n println!(\"found needle {:?}\", ",
"needle_custom);\n }\n }\n\n return 1;\n}\n\nerror[E0277]: expected a `std::ops::FnMut<(char,)>` closure, found `std::string::String`\n --> src/lib.rs:13:18\n |\n13 | if input.contains(needle_custom) {\n | ^^^^^^^^ expected an `FnMut<(char,)>` closure, found `std::string::String`\n |\n = help: the trait `std::ops::FnMut<(char,)>` is not implemented for `std::string::String`\n = note: required because of the requirements on the impl of `std::str::pattern::Pattern<'_>` for `std::string::String`\n\nThe code works if I replace needle_custom with \"test\".",
"\n\nA:\n\nThe contains method will accept an &str or a char but not a String.",
"\nThe contains method is declared as \npub fn contains<'a, P>(&'a self, pat: P) -> bool \nwhere\n P: Pattern<'a>, \n\nAnd if you look at the implementors of Pattern, you'll see its implemented for char and for &str.",
"\nThis means that you need to pass a &str to contains instead of your owned String. ",
"Because &String coerces to &str, this is an easy change:\n-- if input.contains(needle_custom) {\n\n++ if input.contains(&needle_custom) {\n\nHere is your code with this small change working in the Playground.",
"\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.001763668430335097,
0.0034071550255536627,
0,
0.0047169811320754715,
0,
0.0049261083743842365,
0
] | 0.002116 | 5 |
[
"Bahrin\n\nBahrin (, also Romanized as Baḥrīn) is a village in Nehzatabad Rural District, in the Central District of Rudbar-e Jonubi County, Kerman Province, Iran. ",
"At the 2006 census, its population was 90, in 17 families.",
"\n\nReferences \n\nCategory:Populated places in Rudbar-e Jonubi County"
] | {
"pile_set_name": "Wikipedia (en)"
} | [
0.006211180124223602,
0,
0
] | 0.00207 | 5 |
[
"Enhanced recovery after liver surgery.",
"\nIn a majority of cases, enhanced recovery after surgery program (ERP) leads to a reduced rate of postoperative complications and shortened hospital stays following digestive surgery. ",
"The program's preoperative, perioperative and postoperative measures are implemented by the members of a motivated multidisciplinary team. ",
"Having shown its merits in digestive surgery, ERP would be particularly useful in liver surgery due to the elevated rates of morbidity and mortality this type of operation continues to entail. ",
"The objective of this review was to evaluate the efficacy of ERP in liver surgery. ",
"This is a systematic narrative review of the literature on the efficacy of ERP in liver surgery by laparotomy or laparoscopy. ",
"Notwithstanding a number of studies (n=30: 5 randomized trials, 14 cohort studies and 11 meta-analyses) less sizable than with regard to digestive surgery in general and colorectal surgery in particular, analysis of the literature confirms that in liver surgery, ERP is associated with an overall decrease in complications by 30 to 60%, but without improvement in the rates of hospital readmission and postoperative mortality. ",
"All of the studies report a reduction in average length of stay (ALOS) by 2.3 days and in functional recovery, a more objective indicator than ALOS, by 2.5 days. ",
"As of now, the economic impact of the ERP programs in liver surgery is neither positive nor negative, the above-mentioned savings being counterbalanced by heightened costs for material and equipment. ",
"Laparoscopic surgery is independently associated with better outcomes in terms of complications, functional recovery and ALOS; that is why it is important to incorporate this surgical approach in ERP as often as possible. ",
"Given a lack of robust evidence, Prehabilitation, which is a preoperative optimization process leading to improved functional reserve, has yet to be assigned a place in ERP programs pertaining to liver surgery. ",
"Possible roadblocks to application of an ERP program can be overcome through coordination by a team leader, a motivated multidisciplinary team, training courses and dedicated teaching sessions. ",
"ERP is a care improvement process that has a major play to play in organization of liver surgery, and its large-scale application is to be recommended."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0.005434782608695652,
0,
0.0051813471502590676,
0.012048192771084338,
0.007936507936507936,
0.00468384074941452,
0.012345679012345678,
0.005,
0.009009009009009009,
0.009478672985781991,
0.005154639175257732,
0.006622516556291391
] | 0.006377 | 5 |
[
"Forlaget: «Dette er sterke uttrykk som foreldre, skole og lærere kan synes er uheldig»\n\nFY SØREN, SÅ ELENDIG: Først trodde han det var fleip, så en glipp, men til slutt forstod han det slik at Cappelen Damm vil skrive om ordene «faen» til «søren» og «ræva» til «elendig» i frykt for at noen lærere skal reagere.– ",
"Det er jo ikke akkurat i tråd med norske forlagstradisjoner, sier Langeland. ",
"Foto: Håkon Eikesdal\n\nForlag ba forfatter endre «fy faen» til «fy søren» og «ræva» til «elendig»\n\nHenrik Langeland (43) reagerer sterkt på at Cappelen Damm ville sensurere setningen «Fy faen jeg visste ikke at du var så ræva til å padle kajakk!» - ",
"fordi den kunne være for sterk for norske 10. ",
"klassinger.",
"\n\nOppdatert 3. ",
"mai 2016\n\nArtikkelen er over fire år gammel\n\nCappelen Damm undervisning skulle bruke et utdrag fra Henrik Langelands roman «Verdensmestrene» i leseprøver for norske 10. ",
"klassinger. ",
"De ville endre nevnte setning til: «Fy søren jeg visste ikke at du var så elendig til å padle kajakk!»",
"\n\n– Det høres jo ut som en replikk fra Olsenbanden jr., ",
"ler Henrik Langeland - men blir raskt mer alvorlig:\n\n– Dette er ikke bare latterlig, det er rett og slett betenkelig hvis denne formen for fryktdrevet selvsensur er vanlig praksis i Cappelen Damm. ",
"Som forfatter har jeg jo ment noe med ordvalget mitt. ",
"Dersom de hadde ønsket seg en bedehustekst, da er ikke mitt forfatterskap stedet å lete. ",
"Men jeg tror ærlig talt at norske 15-åringer tåler å lese ordene ræva og faen, sier Henrik Langeland som blant annet er forfatteren bak bestselgeren «Wonderboy» - og som fikk terningkast 5 for sin forrige roman «Fyrsten».",
"\n\nOppdatering: Cappelen Damm-sjef beklager\n\nDet var Langelands eget forlag, Tiden, som i første omgang reagerte på mailen som Cappelen Damm sendte Langeland. ",
"I mailen står det blant annet: «Ettersom dette er en prøve som skal brukes i grunnskolen kreves det en omskriving av bannord og «sterke uttrykk». ",
"Det vil si at det er nødvendig å oversette «faen» til «søren» og ræva til «elendig». ",
"Hva tenker du om det?»",
"\n\nJo, Henrik Langeland tenker følgende:\n\n– Først trodde jeg det var en fleip. ",
"sier Langeland som ringte redaktøren og spurte om de virkelig mente dette.",
"\n\nBokinteressert? ",
"Les ferske anmeldelser og boktips her!",
"\n\n– Jeg antok at de var underlagt noen statlige retningslinjer av noe slag, men nei, jeg forstod det slik at det bare er Cappelen Damm selv som vil sensurere i frykt for at noen lærere skal reagere. ",
"Det er jo ikke akkurat i tråd med norske forlagstradisjoner, sier Langeland.",
"\n\nTrodde det var en glipp\n\n– Jeg tenkte at dette må være en glipp av en ung og uerfaren redaktør, men så ringte det en redaksjonssjef som har vært i bransjen i mange år. ",
"Hun insisterte på å gjennomføre endringene.",
"\n\nLATTERLIG: Henrik Langeland synes ikke bare det er latterlig at Cappelen Damm undervisning vil skrive om et utdrag fra hans roman «Verdensmestrene», han mener også et det er betenkelig at denne formen for selvselsur er praksis i Cappelen Damm. ",
"Foto ble tatt i 2010 - i forbindelse med lanseringen av nettopp «Verdensmestrene». ",
"Foto: Robert S. Eik VG\n\nDa Henrik Langeland sa at han ikke ville forandre ordene i sin skjønnlitterære tekst, reagerte redaksjonsssjefen med sinne og slengte på telefonrøret, ifølge forfatteren. ",
"Forlaget har nå gitt ham beskjed om at de ikke kommer til å trykke teksten - med mindre han går med på å bruke «søren» og «elendig» i stedenfor «faen» og «ræva».",
"\n\nBokpodden Pocket: Den erotiske thrilleren «Maestra» - og Downton Abbey-skaperens nye herskapsserie\n\n– Skal jeg som forfatter akseptere språklige endringer i en skjønnlitterær tekst fordi forlaget er redd for represalier? ",
"Det høres ut som praksis i land vi ikke liker å sammenlikne oss med, sier Langeland som nå kan se langt etter honoraret han ville ha fått for bruken av romanutdraget.",
"\n\n– Jeg selger selvfølgelig ikke forfatterintegriteten min for noen tusenlapper. ",
"Jeg tar jo ikke hensyn til at noen kanskje reagerer på språkbruken når jeg skriver romaner, sier Langeland som understreker at han i utgangspunktet setter pris på at de ønsker å bruke utdrag fra hans bok.",
"\n\nLes også: Forlag kjøper seg gode anmeldelser\n\nForlaget: – Tekstene skal ikke støte noen\n\nForlagssjef i Cappelen Damm grunnskole, Espen Skovdahl, mener begrepet sensur i denne sammenhengen blir feil.",
"\n\n– Det er alltid et redaksjonelt arbeid som ligger bak de tekstene vi velger å bruke i våre lærebøker eller i norskprøver. ",
"Det er veldig veldig mange tekster som det ikke passer å bruke av ulike grunner, det kan være fordi de er politisk betent, uheldig i forhold til religion, diskriminering og lignende. ",
"Tekstene skal ikke støte noen, sier Skovdahl og bekrefter at dette er vanlig praksis.",
"\n\n– Men her sier dere at det kreves en omskriving av ordene faen og ræva - det er vel det samme som å sensurere?",
"\n\n– Vi mener ikke dette er å sensurere, det er heller en vurdering av hvordan vi formulerer oss. ",
"Vi søker å treffe bredest mulig og å ikke provosere noen. ",
"Dette er sterke uttrykk som foreldre, skole og lærere kan synes er uheldig.",
"\n\n– Langeland mener dette nettopp viser at dere driver selvpålagt sensur fordi noen kan komme til å reagere?",
"\n\n– I denne sammenhengen synes vi det var mulig å endre litt på ordlyden. ",
"Vi kan forstå at forfatteren reagerer, men det handler ikke om at vi er redd for å få klager. ",
"Vi ønsker bare ikke at noen skal bli støtt.",
"\n\n– Det er vel det samme som å si at dere sensurerer fordi dere er redd for at noen reagerer?",
"\n\n– Hvis ikke forfatteren synes det er greit, så bruker vi en annen tekst. ",
"Og det gjør vi nå i dette tilfellet. ",
"Det handler jo ikke om å utgi en bok eller ikke her, det er bare snakk om å bruke en tekst i en leseprøve - og vi mener at vi ikke er tjent med å bruke slike kraftuttrykk her, sier Skovdahl.",
"\n\n– Så dere tenker at «faen» og «ræva» er for kraftfulle uttrykk for en 10. ",
"klassing?",
"\n\n– Nei, selvfølgelig ikke. ",
"Men vi er jo alle kjent med at lærere ikke ønsker banning i klasserommet og på skolen generelt - og at de daglig jobber med å motvirke dette. ",
"Hvis man da stadig møter sånn ordbruk i offentlige sammenhenger, så vet vi at det er vanskeligere å motkjempe det i klasserommet.",
"\n\n– Og det er deres oppgave?",
"\n\n– Det er ikke vår oppgave å si hva som er feil eller ikke, men når vi velger en tekst der hensynet ikke er å provosere, så trenger vi jo ikke å provosere unødvendig, sier Skovdahl som kan forstå at Langeland ikke vil endre på en romantekst.",
"\n\n– Forfatteren har svart nei på vår forespørsel, noe vi respekterer fullt ut, sier Skovdahl og beklager samtidig at en redaksjonssjef brøt telefonsamtalen med Langeland.",
"\n\nHar du lest? ",
"VG slakter boken forleggere får bloggere til å «elske»\n\nPublisert: 03.05.16 kl. ",
"10:15 Oppdatert: 03.05.16 kl. ",
"14:53\n\nLes også\n\nMer om Litteratur"
] | {
"pile_set_name": "OpenWebText2"
} | [
0.009584664536741214,
0.025974025974025976,
0.012096774193548387,
0.021739130434782608,
0,
0,
0.005917159763313609,
0,
0.0196078431372549,
0.017857142857142856,
0.025380710659898477,
0,
0.011235955056179775,
0.004524886877828055,
0.03164556962025317,
0,
0.011764705882352941,
0,
0.038461538461538464,
0.013513513513513514,
0,
0,
0.01507537688442211,
0.02631578947368421,
0.01764705882352941,
0,
0.012195121951219513,
0,
0.015384615384615385,
0.012422360248447204,
0.004484304932735426,
0.018072289156626505,
0.012345679012345678,
0.014705882352941176,
0.01,
0.008064516129032258,
0.01092896174863388,
0,
0.008928571428571428,
0,
0,
0,
0.027777777777777776,
0,
0,
0,
0.021505376344086023,
0,
0,
0.005263157894736842,
0,
0,
0,
0,
0.007751937984496124,
0,
0.008264462809917356,
0.01764705882352941,
0,
0,
0,
0
] | 0.008453 | 5 |
[
"Q:\n\nWhat constitutes a \"musical\" EQ?",
"\n\nIn reading product advertisements and gear reviews in magazines like Sound on Sound, I've come across the term \"musical\" a few times referring to an apparently positive quality of an equalizer device or plugin.",
"\nWhat does it mean for an EQ to be musical? ",
" Is there a generally-agreed-upon definition for this term, in this context? ",
" Is there such a thing as a deliberately \"un-musical\" EQ? ",
" \n\nA:\n\nI think this is largely marketingspeak. ",
"And they stole it from engineers who were trying to convey that some EQ voicings are just more right than others when you're working with EQs that have limited controls.",
"\nIn particular, when you've got an EQ that's not fully parametric, or configured with enough bands to cover the entire audible frequency spectrum, you're at the mercy of what some electrical engineer thought was the best parameters for Q and taper and whatnot on the EQ.",
"\nThis is usually the case when you're working with channel strips on analog mixing consoles: each strip has an EQ section, but it's simple in what you can do because there's limited room on the physical strip for potentiometers to play with. ",
"As such, the channel strip designer has to make some tough decisions about how that EQ can be controlled and how each dial changes the EQ voicing. ",
"A \"musical\" EQ in this context would be one where, when you move a dial on the EQ, gives you pleasant results -- the Q and taper and center frequencies of the bands in the EQ all change in a way that makes working with the EQ easy.",
"\n\nA:\n\nKeep in mind that even EQs with fully customizable band parameters will not all sound the same when set to identical settings. ",
" The type of electrical circuit and quality of components (or the type of digital algorithm) used to achieve the equalization will be different for each EQ model, and these differences have very real effects on how the signal is processed. ",
" A lot of these show up as phase shifts or effects on harmonically-related frequencies that aren't actually in the intended band.",
"\nThese differences are what make some EQs better than others for certain applications, and could be part of what the \"musical EQ\" is referring to.",
"\nThough the point Ian C. made is probably what is most often meant.",
"\nI just posted this addition as something to keep in mind.",
"\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.027777777777777776,
0,
0.022727272727272728,
0,
0.017241379310344827,
0,
0.011834319526627219,
0.007407407407407408,
0.004132231404958678,
0.006802721088435374,
0.017316017316017316,
0.007518796992481203,
0.004166666666666667,
0,
0.0136986301369863,
0.014925373134328358,
0,
0
] | 0.008642 | 5 |
[
"Effects of neonatal and adolescent neuroactive steroid manipulation on locomotor activity induced by ethanol in male wistar rats.",
"\nNeonatal neuroactive steroids levels are crucial for brain development. ",
"Alterations of neonatal neuroactive steroids levels induce anxiolytic-like effects and improve exploration in novel environments in adulthood. ",
"These behavioural traits, i.e. sensation/novelty seeking, anxiety or impulsivity, are associated with vulnerability to drug use and abuse. ",
"Adolescence is also recognized as a particularly critical developmental phase to contribute to vulnerable phenotype. ",
"However, the influence of neuroactive steroids during development in the vulnerability to drug addiction has been poorly studied. ",
"The aim of the present experiment is to study the effect of early neonatal and adolescent manipulations of neuroactive steroids on the sensitivity to the stimulant effects of ethanol in adult male rats. ",
"Therefore, allopregnanolone or finasteride, an allopregnanolone synthesis inhibitor, were injected from postnatal day 5-9. ",
"In early adolescence, half of the subjects were injected with progesterone, the main allopregnanolone precursor, and the elevated plus-maze anxiety test was performed. ",
"Results indicated that early adolescent progesterone induced anxiolytic-like effects (increase in the percentage of entries and time in open arms). ",
"Neonatal finasteride administration decreased locomotor activity induced by ethanol in adolescent vehicle subjects. ",
"Interestingly, differences induced by neonatal treatments were not present in the animals that received progesterone in the early adolescence. ",
"In conclusion, neuroactive steroid manipulations in crucial stages of development could be playing an important role in behavioural effects of alcohol such as the sensitivity to locomotor stimulation."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0 | 5 |
[
"The election was a crushing defeat – but not for either of the major parties. ",
"The faction that now retreats in utter disarray wasn’t technically standing, though in the past it has arguably wielded more power than the formal contestants. ",
"I’m talking about the media.",
"\n\nThe rightwing press threw everything it had at Jeremy Corbyn, and failed to knock him over. ",
"In doing so, it broke its own power. ",
"Its wild claims succeeded in destroying not Corbyn’s credibility but its own. ",
"However the problem is by no means confined to the corporate media. ",
"The failure also belongs to media outside the grasp of billionaires. ",
"It is one from which some platforms may struggle to recover.",
"\n\nThere is no point in trying to hide or minimise this: the election has been a disaster for mainstream outlets. ",
"They missed the moment because they were constitutionally destined to do so. ",
"The issue that caused this disaster is the one that eventually fells all forms of power: the media has created a hall of mirrors, in which like-minded people reflect and reproduce each other’s opinions.",
"\n\nThe broadcasters echo what the papers say, the papers pick up what the broadcasters say. ",
"A narrow group of favoured pundits appear on the news programmes again and again. ",
"Press prizes are awarded to those who reflect the consensus, and denied to those who think differently. ",
"People won’t step outside the circle for fear of ridicule and exclusion.",
"\n\nA study at Cardiff University shows how the broadcasters allow themselves to be led by the newspapers, despite the massive bias of the printed press. ",
"For example, during the 2015 election campaign opinion polls revealed that the NHS came top of the list of voters’ concerns while the economy came third. ",
"But the economy, on which the Conservatives were perceived to be strongest, received four times as much coverage on TV news as the NHS, which was seen as Labour’s strongest suit. ",
"This appeared to reflect the weight given to these issues in the papers, most of which sought a Conservative victory.",
"\n\nAn analysis by the Media Reform Coalition and Birkbeck College found that, despite the rules on impartiality and balance, when Corbyn’s leadership was being challenged last summer, the BBC’s evening news bulletins gave almost twice as much airtime to his critics as they gave to his supporters. ",
"They often ascribed militancy and aggression to him and his supporters, but never to his challengers. ",
"One report on the BBC News at Six finished with the words “This is a fight only one side can win. ",
"The others are being carted off to irrelevance. ",
"The place for political losers.” ",
"The accompanying shot showed a dustbin lorry setting off, painted with the word Corbyn.",
"\n\nThis problem also affects the Guardian. ",
"According to a study by the London School of Economics of the representation of Corbyn by newspapers in his first two months as Labour leader (in the autumn of 2015), around a fifth of news articles in this paper lacked balance. ",
"Overall, roughly 18% of its coverage was judged supportive of Corbyn, 53% was neutral, and 29% was negative. ",
"This is a better balance than in the other liberal papers. ",
"Indeed, the Guardian had more diverse and more pro-Corbyn voices than any other mainstream outlet. ",
"Only the Guardian and the Mirror enthusiastically supported both Labour and Corbyn in election editorials.",
"\n\nBut the scales still didn’t balance. ",
"Even I, who was supportive at the beginning of Corbyn’s candidacy and during his election campaign, fell into despair about his leadership during the winter after a series of fiascos in parliament, and tweeted: “I have now lost all faith.”",
"\n\nThe net result is that the most dynamic political force this nation has seen for decades feels alienated by the media, and not just by the Tory press. ",
"Those who have thrown so much energy into the great political revival, many of whom are young, have been almost unrepresented; their concerns and passions have been unheeded, misunderstood or reviled. ",
"When they have raised complaints, journalists have often reacted angrily, writing off movements that have gathered in hope as a rabble of Trots and wreckers.",
"\n\nThis response has been catastrophic in the age of social media. ",
"What many people in this movement now perceive is a solid bloc of affluent, middle-aged journalists instructing young people mired in rent and debt to abandon their hopes of a better world. ",
"Why has it come to this, even in the media that’s not owned by billionaires?",
"\n\nIt is partly because this industry, in which people without a degree could once work their way up from the floor, now tends to select its entrants from a small, highly educated pool. ",
"The use of internships narrows the selection further. ",
"Wherever they come from, journalists, on average, end up better paid than most people. ",
"Whatever their professed beliefs, they tend to be drawn towards their class interests.",
"\n\nBut the biggest problem, I believe, is that we spend too much time in each other’s company, a tendency that is fatal in an industry that is meant to reflect the world. ",
"There has been a major effort by some media, including the Guardian, to get away from Westminster and to hear the voices of the rest of the country. ",
"This is good, but it is not enough. ",
"What counts is not only the new people and new ideas you encounter, but also the old ones you leave behind. ",
"The first ambition of a journalist should be to know as few journalists as possible – to escape the hall of mirrors.",
"\n\nThe media as a whole has succumbed to a new treason of the intellectuals, first absorbing dominant ideologies, then persuading each other that these are the only views worth holding. ",
"If we are to reclaim some relevance in these times of flux and crisis, we urgently need to broaden the pool of contributors and perspectives.",
"\n\nWe should actively recruit people from poorer backgrounds, and diversify our expertise. ",
"Newsrooms tend to be largely peopled by humanities graduates. ",
"Over the years, I have found myself explaining to other journalists how to calculate percentages, that two orders of magnitude greater does not mean double, and that animals and mammals are not synonyms. ",
"There is a lack of contact not only with most of the population, but also with the material world and its physical parameters.",
"\n\nWe need to interrogate every item of the news agenda and the way in which it is framed. ",
"We should ask ourselves where our own interests lie – and how we might avoid reproducing them. ",
"And, to the greatest extent possible, we should avoid each other like the plague."
] | {
"pile_set_name": "OpenWebText2"
} | [
0,
0,
0,
0.010638297872340425,
0,
0.01282051282051282,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.006578947368421052,
0.006493506493506494,
0.0111731843575419,
0,
0.010101010101010102,
0,
0.01020408163265306,
0,
0,
0.011494252873563218,
0.023809523809523808,
0.008733624454148471,
0.009174311926605505,
0,
0.020202020202020204,
0.03773584905660377,
0,
0.0041841004184100415,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.006711409395973154,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0.003277 | 5 |
[
"Q:\n\nProving linear independence\n\nLet $A$ be an $n \\times n$ matrix and suppose $v_1, v_2, v_3 \\in \\mathbb{R}^n$ are nonzero vectors that satisfy:\n$$\nAv_1 = v_1 \\\\\nAv_2 = 2v_2 \\\\\nAv_3 = 3v_3 $$\nProve that $\\{v_1, v_2, v_3\\}$ is linearly independent. (",
"Hint: Start by showing that $\\{v_1, v_2\\}$ must be linearly independent.)",
"\nI know if $A$ is non-singular,it is easy. ",
"But if $A$ is singular, I have no idea to get that.",
"\n\nA:\n\nHINTS\n\nSuppose $v_1$ and $v_2$ are linearly dependent, so that $v_1 = cv_2$.\nWhat is $Av_1$? ",
"What about $A cv_2$? ",
"What about $cAv_2$?",
"\nDo the same idea for $v_3$ independence.",
"\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.00398406374501992,
0.0136986301369863,
0,
0,
0.010101010101010102,
0,
0,
0,
0
] | 0.003087 | 5 |
[
"Grateful Dead\n\nOn Wisconsin Vine\n\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\n7\n\n8\n\n9\n\n10\n\n11\n\n12\n\n13\n\n14\n\n15\n\n16\n\n17\n\n18\n\n19\n\n20\n\n21\n\n22\n\n23\n\n24\n\n25\n\n26\n\n27\n\n28\n\n29\n\n30\n\nWe're on a mission to make a miracle every day. ",
"As a token of our appreciation for making 2010 an epic year, we're giving away a high-quality 320Kbps download every day this November. ",
"That's 30 days of Dead! ",
"Intrigued? ",
"We're also going to put your knowledge to the test and give you the chance to win some great music from the Dead.+ More InfoYou know your Ables from your Bakers from your C's, but can your finely tuned ears differentiate the cosmic \"comeback\" tour from a spacey 70's show? ",
"Each day we'll post a free download from one of the Dead's coveted shows. ",
"Will it be from that magical night at Madison Square Garden in '93 or from way back when they were just starting to warm it up at Winterland? ",
"Is that Pigpen's harmonica we hear? ",
"Brent on keys? ",
"Step right up and try your hand.",
"\n\nEach day, the first person to guess the venue & date correctly will get a Road Trips of their choice. ",
"If you’re not first but you've answered correctly, you will also be automatically entered for a weekly prize of THE WARNER BROTHERS STUDIO ALBUMS VINYL BOX (EXCLUSIVE EDITION) or a FORMERLY THE WARLOCKS boxed set. ",
"Log in now and enter to win!",
"\n- Less Info\n\nWe haven't selected a daily winner yet. ",
"The first person to answer correctly will get to pick a Road Trips of their choice. ",
"To get started, login or register using the form above.",
"\n\n....never heard this \"dark hollow\" verse before ~ reminiscent of the \"muddy water\" verse in Rider.",
"\nFunny, the recent May '70 road trips was the first live version I've heard where Bob actually sings the \"don't make it hard to believe in you\" verse (does anyone else get the impression he wrote that one himself? ",
"It's very \"ring around the sun\"-ish for an old swamp tune).",
"\n\nI'll definitely take a stab at this one, though I've never heard it before today.",
"\n\nI really don't understand how people are so confident that they can guess when and where each song was performed. ",
"I have been a dead head for many years and I own most of their live shows that were officially released on CD. ",
"I also have a few old bootleg cassette tapes that I acquired from several friends. ",
"Even though I usually can tell by listening to the song a rough estimate of what year it was but to actually know what city the show was actually recorded at is amazing. ",
"I even have the dead base book XI. ",
"so I have used it to look up the information on shows and I admit I came fairly close on a few of my guesses but I have been wrong every time so far. ",
"My guess is that people out there who are actually winning this contest are way more dedicated to the Grateful Dead then I am. ",
"I must give these people a real patt on the back for being able to guess correctly on every song. ",
"If anybody out there wants to give me some tips or hints on how better to correctly guess these songs I would appreciate it. ",
"Anyways, I am just happy that I am able to add new songs to my collection. ",
"Also I would like to thank the people who have posted the links to past songs. ",
"There were several days that I was unable to download the song or even get on to the site. ",
"If it wasn't for those people I would not have the complete list including the two songs with the \"B\" versions.",
"\n\nThe sun will shine at my back door some day, March winds will blow all my troubles away.",
"\n\nYeah definitely agree with Jace and others, this has to be from '71, such a good year, not so early as the Port Chester run though I would dare venture. ",
"Just love that '71 sound!",
"\nThanks to the band and all at Dead.net for this one, Truckin' has to be one of the two or three master all time essential Dead numbers! ",
"Yeah:-)\n\nGetting to \"about when\" isn't too hard - who's there and who's not - but I don't have access to all the shows in that time period so I'mma rollin' the dice here... Thinkin' if you don't already have this show you might have a harder time than usual getting it.",
"\n\nConversation is always more interesting than recitation, so speak your mind and not someone else's.",
"\n\nwhat happened to all the links in the comments for downloading past days songs, I missed a couple of days. ",
"It would be great if someone could point me to a place where they are available.",
"\nUntil then I guess I'll keep Truckin'\nThanks Folks,\n\nWHAT is with the ends of these songs.? ",
"It seems like every download, I wind up re downloading, just to make sure that the track they are giving us really ends that way. ",
"really.? ",
"is this the way Truckin ends.? ",
"I get 8:27 and 19.38 MBs. ",
"Anyone get any different.? ",
"Its nice music. ",
"I greatly appreciate having a new song to be my song of the \"day\" but do something with the dangling ends. ",
"HUH.? ",
"Enjoy your days. !",
"\n\nThat's my best guess at this point. ",
"Cannot hear Pigpen's voice, but can hear his Hammond at points. ",
"Nice and raucous performance, even a little raw, and so perhaps somewhere in the February Port Chester shows, Quick guess though, and I could be off. ",
"Cannot pick a specific date at the moment. ",
"So many of these splendid 1971 Truckins. ",
"There's sort of a focused earnestness to the vocals of those February Port Chesters, which is why they come to mind. ",
"Monstrously good guitar work from Bobby and Jerry in the last third of the song, starting at 5:31. ",
"Lingering drums at the end, and so perhaps one of the many truckin-drums-other sequences. ",
"Could also be one of the as yet unreleased Houseboat Tapes. ",
"That leaves plenty to choose from ... (smile)\n\n...so good, in fact, that it seems a shame to waste all that server work on a contest that will end in 17 days. ",
"I propose A Jerry Little Christmas: twelve days of Garcia downloads followed by a Christmas release of a new Pure Jerry set.",
"\n\nSomebody call up the Garcia girls, let's get this in motion ;)\n(Love ya, Sunshine!)",
"\n\nWho's Online\n\nOnline users\n\nNewsletter\n\nGet the latest news on Grateful Dead releases and more straight to your inbox.",
"\n\nBy submitting your email address you acknowledge and agree to our Privacy Policy and Terms of Use and are okay with receiving news, updates, special offers and occasional marketing messages from us and our affiliates.",
"\n\nNewsletter\n\nGet the latest news on Grateful Dead releases and more straight to your inbox.",
"\n\nSignup!",
"\n\nBy submitting your email address you acknowledge and agree to our Privacy Policy and Terms of Use and are okay with receiving news, updates, special offers and occasional marketing messages from us and our affiliates."
] | {
"pile_set_name": "Pile-CC"
} | [
0.0051813471502590676,
0,
0,
0,
0,
0.013513513513513514,
0,
0,
0,
0,
0.009615384615384616,
0.009345794392523364,
0,
0,
0,
0,
0.01,
0.004672897196261682,
0,
0,
0,
0,
0,
0,
0,
0,
0.007874015748031496,
0,
0,
0,
0,
0,
0,
0,
0.0064516129032258064,
0,
0.014598540145985401,
0.007434944237918215,
0,
0,
0,
0.010752688172043012,
0,
0,
0.03225806451612903,
0,
0,
0,
0,
0,
0,
0,
0.015625,
0.013333333333333334,
0,
0,
0,
0.020202020202020204,
0,
0.016666666666666666,
0,
0.008064516129032258,
0.011764705882352941,
0,
0,
0,
0,
0
] | 0.003196 | 5 |
[
"THURSDAY, DECEMBER 15, 2011\n\nIMAGE COMICS FOR MARCH\n\nSAGA #1\n\nWritten by BRIAN K. VAUGHAN\nart / cover FIONA STAPLES\n\nY: THE LAST MAN writer BRIAN K. VAUGHAN returns to comics with red-hot artist FIONA STAPLES for an all-new ONGOING SERIES! ",
"Star Wars-style action collides with Game of Thrones-esque drama in this original sci-fi/fantasy epic for mature readers, as new parents Marko and Alana risk everything to raise their child amidst a never-ending galactic war. ",
"The adventure begins in a spectacular DOUBLE-SIZED FIRST ISSUE, with forty-four pages of STORY with no ads for the regular price of just $2.99!",
"\n\n40 pages, $2.99, in stores on March 14.",
"\n\nTHE MANHATTAN PROJECTS #1\n\nWritten by JONATHAN HICKMAN\nart NICK PITARRA / CRIS PETER\n\n\"INFINITE OPPENHEIMERS\"\nWhat if the research and development department created to produce the first atomic bomb was a front for a series of other, more unusual, programs? ",
"What if the union of a generation's brightest minds was not a signal for optimism, but foreboding? ",
"What if everything… went wrong?",
"\n\nExtreme's zombie-black op-superheroes return! ",
"Cabbot Stone is the last unliving Bloodstrike agent, an unstoppable weapon in America's war on terror! ",
"But being dead is a hell of way to make a living, and Cabbot is losing his faith. ",
"Will his latest mission be his last?",
"\n\n32 pages, $2.99, in stores on March 28.",
"\n\nHELL YEAH #1\n\nWritten by JOE KEATINGE\nart & cover ANDRE SZYMANOWICZ\n\nTHE LAST GENERATION OF HEROES IS HERE!",
"\nTwenty years ago, the first-ever superheroes debuted without warning and forever altered our global culture! ",
"Now, the generation born in their wake fight to claim their place in a world evolved beyond them! ",
"Written by Eisner & Harvey award-winner JOE KEATINGE and illustrated by ELEPHANTMEN and POPGUN's ANDRE SZYMANOWICZ, HELL YEAH combines the over-the-top excitement of the original Image Comics launch with modern, innovative storytelling. ",
"It all begins with a giant-sized first issue containing a full thirty-two pages of story with no ads for the regular price of just $2.99!",
"\n\n\"GETTING AWAY WITH MURDER\"\nCryptids. ",
"Aliens. ",
"Monsters. ",
"All the world's bizarre secrets--what if they were real? ",
"Their existence would be debunked by a reality TV show! ",
"HOAX HUNTERS is that show, publicly disproving all variety of lore. ",
"But their real goal is the opposite: as the world's dark corners surface, the HOAX HUNTERS cover them up.",
"\nWhen a mysterious astronaut made of crows appears in Russia, the HOAX HUNTERS are on the case. ",
"From the pages of HACK/SLASH!",
"\n\n32 pages, $2.99, in stores on March 21.",
"\n\nREBEL BLOOD #1\n\nWritten by ALEX LINK & RILEY ROSSMO\nart / cover RILEY ROSSMO\n\nFROM THE DESK OF OFFICER RED PROSPERO; WHILE YOU WERE OUT:\n\"The world went to hell in a hand basket. ",
"A highly infectious disease has turned every living thing into zombies. ",
"Your neighbors will rend your flesh with their bare teeth. ",
"So will the rats. ",
"And if the rats don't, the squirrels will. ",
"Save yourself. ",
"Stay in the woods. ",
"Pray it hasn't reached the wolves…and keep your gun loaded.\"",
"\n\nNEW CREATIVE TEAM! ",
"BOLD NEW DIRECTION!",
"\n\"THE CRACK IN EVERYTHING\", PART 1\nHot on the heels of THE DARKNESS II video game release, Top Cow proudly welcomes the new creative team of DAVID HINE (THE DARKNESS: FOUR HORSEMEN, THE BULLETPROOF COFFIN) and JEREMY HAUN (ARTIFACTS, BATTLE HYMN, Detective Comics)! ",
"On the surface, Jackie Estacado has everything he ever wanted – control of the Darkness, a successful career, and happy family life. ",
"Only his desire to make his life truly perfect will be his undoing.",
"\n\n32 pages, $2.99, in stores on March 21.",
"\n\n'68, VOL. ",
"1: BETTER RUN THROUGH THE JUNGLE TP\n\nWritten by MARK KIDWELL\nart / cover NAT JONES & JAY FOTOS\n\nThere are zombies in the razorwire. ",
"Welcome to 1968…and the end of the world. ",
"From the steaming jungles of Viet Nam to the brightly lit campus of demonstration-torn Berkeley, California, ravenous hordes of unstoppable ghouls are changing the face of the Age of Aquarius. ",
"Collected for the first time, this 178-page trade paperback re-presents the first four-issue story arc from the '68 ongoing series, along with the re-colored and re-lettered original one-shot from 2006! ",
"Plus, creators MARK KIDWELL, NAT JONES and JAY FOTOS have included tons of behind-the-scenes extras to make this a must-have for zombie and horror fans everywhere! ",
"Collects '68 #1-4, plus the 2006 '68 one-shot.",
"\n\nPHIL HESTER'S EPIC RUN CONCLUDES!",
"\nJackie Estacado has been pushed to the limit and beyond. ",
"He finally has something worth losing in his life and there is only one true threat to his happiness – the Darkness, itself. ",
"Utilizing the Sun Dagger, Estacado will die and plunge himself into the Darkness dimension itself to confront the very source of his gift and his curse.",
"\nLongtime writer PHIL HESTER (Wonder Woman, Bionic Man) concludes his critically acclaimed run on one of Top Cow's flagship franchises with art by ROMANO MOLENAAR (Within Temptation's the Unforgiving) and LEANDRO OLIVEIRA (Superman) with \"Total Darkness.\" ",
"The perfect way to catch up before starting DAVID HINE and JEREMY HAUN's new run! ",
"Collects THE DARKNESS #96-100.",
"\n\n160 pages, $19.99, in stores on March 28.",
"\n\nTHE LAST OF THE GREATS, VOL. ",
"1 TP\n\nWritten by JOSHUA HALE FIALKOV\nart / cover BRENT PEEPLES\n\nMEET THE LAST - He's mankind's last hope and he f#@&ing HATES us.",
"\nAs the Earth stands on the brink of annihilation we must beg the last survivor of a family of superheroes to put aside his hatred and save our asses. ",
"From the writer of ECHOES, Tumor and I, Vampire comes the complete first volume. ",
"Collects LAST OF THE GREATS #1-5\n\nMAGDALENA RETURNS! ",
"GUEST STARRING WITCHBLADE!",
"\nPatience, the current Magdalena, has reconciled with the Catholic Church and agreed to work with her former masters. ",
"But there are still forces within the Church that feel another might serve more willingly as the bearer of the Spear of Destiny. ",
"Patience must fight for her place and then with her partner and mentor, Kristof, recover a piece of the True Cross from mercenaries.",
"\nWritten by Top Cow Universe architect RON MARZ (ARTIFACTS, WITCHBLADE) and featuring art from NELSON BLAKE II (PILOT SEASON: MURDERER), DAVID MARQUEZ (Fantastic Four: Season One, Days Missing), and KEU CHA (WITCHBLADE, RISING STARS) this collections collects the \"Sisters\" and \"The One, True Cross\" story arcs! ",
"Collects MAGDALENA #7-12.",
"\n\nSixty years ago the oil ran out and debts were called in. ",
"Civil war followed that splintered America into warring fiefdoms. ",
"New San Diego is a technocratic utopia that offers the last bastion of peace and prosperity, provided you live within its walls. ",
"Drake McCoy is its best protector. ",
"One of a select group of Marksmen, descended from the Navy Seals, McCoy defends the city from the numerous threats in the wasteland outside the walls. ",
"But when the oil rich Lone Star state sends a powerful army to steal New San Diego's energy technology, even the Marksmen's skill may not be enough to fend off the siege. ",
"Collects MARKSMEN #1-6.",
"\n\nIn the wake of THE DARK CHAMBER, Professor Moriarty finds himself afflicted with unsettling visions. ",
"He sets sail for Burma in search of an old ally, but finds himself caught up in a city on the brink of revolt, a troublesome Imperial Policeman, and a secret that holds the key to his destiny. ",
"Collects MORIARTY #5-9.",
"\n\n136 pages, $14.99, in stores on March 28.",
"\n\nSAVAGE DRAGON ULTIMATE COLLECTION, VOL. ",
"1 HC\n\nWritten by / art / cover ERIK LARSEN\n\nThe earliest adventures of fandom's favorite fin-head are collected for the first time in an oversized volume in full color! ",
"It's all here! ",
"Overlord! ",
"SuperPatriot! ",
"Mako! ",
"Arachnid! ",
"Freak Force! ",
"It's the beginning of the greatest comic book in the history of comic books! ",
"Savage Dragon defends Chicago from Overlord and the Vicious Circle and the stakes--and body count--couldn't be higher! ",
"It's the definitive Savage Dragon collection! ",
"Guest-starring Rob Liefeld's Badrock and the Teenage Mutant Ninja Turtles. ",
"Comes with our highest possible recommendation! ",
"Collects the expanded five-issue miniseries and Savage Dragon #1-7, plus covers, sketches and behind the scenes material.",
"\n\nRelive the excitement of one of the best-selling independent comics of all time and grow your SPAWN: ORIGINS collection! ",
"In this volume: The ghost of slain child killer Billy Kincaid returns to taunt Spawn! ",
"Possessing human souls, Kincaid carries out his dirty work vicariously through innocents. ",
"Sam and Twitch find themselves knee deep in it when current police officers become victims to Kincaid's torment. ",
"Can they help Spawn save the people without killing all of them too!? ",
"Collects SPAWN #81-86.",
"\n\nWritten by & art ROBERT KIRKMAN\ncover by CHARLIE ADLARD & CLIFF RATHBURN\n\nFor the first time ever: an unprecedented, behind-the-scenes look at ROBERT KIRKMAN's original, hand-written plot lines for the early issues of the Eisner-award winning series! ",
"See what plot lines were left on the cutting room floor and get an in-depth look at how the series came together. ",
"This collection includes commentaries on KIRKMAN's original plots with never before seen material. ",
"This is the WALKING DEAD collector's item of the year!",
"\nCollects NEVER BEFORE SEEN WALKING DEAD PLOTS!",
"\n\n112 pages, $19.99, in stores on March 14.",
"\n\nTHE ACTIVITY #4\n\nWritten by NATHAN EDMONDSON\nart & cover MITCH GERADS\n\n\"WHAT GOES UP MUST COME DOWN\"\nFort Bragg is calling: DELTA has a problem with an out of control Columbian drug lord, and it's up to Danny Locke's Team Omaha to find a solution. ",
"THE ACTIVITY #4 is a great jumping-on point as the series continues to grind high-octane realism at all latitudes.",
"\n\nThe action-packed second issue opens with our heroine getting her first taste of the new apocalyptic world. ",
"On the run from rabid women and mysterious men in black rubber suits, Judith flees to the suburbs where she meets Frank, a sex starved teenager with a horrible secret. ",
"Joining forces, the two teens set out to evade evisceration and find Judith's brother!",
"\n\n32 pages, $2.99, in stores on March 14.",
"\n\nARTIFACTS #17\n\nWritten by RON MARZ\nart / cover STJEPAN SEJIC\n\n\"SEARCH FOR THE ANGELUS,\" Part Four\nTom Judge and Tilly Grimes were prepared that not all the bearers of the 13 Artifacts would be receptive to their call to action. ",
"They even anticipated that it would shake some to the core. ",
"But they never expected what came next. ",
"RON MARZ (MAGDALENA, SHINKU) and STJEPAN SEJIC (ANGELUS) continue the story of the Top Cow Universe in directions that will top everything that has come before!",
"\n\n\"THE HUNT,\" Part Three\nGuest starring the SEVEN DEADLY SINS. ",
"The manhunt for Father Michael continues, as Avengelyne and Passover search for Priest. ",
"Meanwhile, the Red Dragon chooses who wields the Luci-Fire weapon. ",
"Plus, an Extreme Origins back-up written by Mark Poulton with art by a guest artist!",
"\n\n\"DEAD HEAT\"\nWith just three episodes left in the season, the race for revenge is on and the fight for survival comes down to a photo finish! ",
"See who goes to the mattresses, who sleeps with the fishes and who just stays groovy as Blue Estate scores another bull's-eye with part ten of the fast, funny, 100% cool series.",
"\n\n32 pages, $2.99, in stores on March 7.",
"\n\nTHE BULLETPROOF COFFIN: DISINTERRED #3\n\nWritten by DAVID HINE\nart & cover SHAKY KANE\n\n\"AS FLIES TO WANTON BOYS…\"\nCoffin Fly rises again to join The Unforgiving Eye in a race against time as The Red Menace launches an attack on planet Earth. ",
"But who is really behind this nefarious plan for world domination? ",
"Could it be that we are all no more than playthings in the hands of a 10-year-old sociopath?",
"\n\n32 pages, $3.99, in stores on March 28.",
"\n\nELEPHANTMEN #41\n\nWritten by / art DAVID HINE\ncover BOO COOK\n\n\"A LONG AND GHASTLY KITCHEN\"\nDAVID HINE, co-creator of THE BULLETPROOF COFFIN, new writer of THE DARKNESS and creator of STRANGE EMBRACE, tells the story of the last days of MAPPO geneticist Javier Kubec...\n\nWhat will Detective Walt Booker sacrifice, to what gods? ",
"And is he\nlooking for redemption or revenge? ",
"In part three of BRUBAKER and\nPHILLIPS' new horror/noir masterpiece, the threads of the first arc twist together... into a noose around the neck of a corrupt cop in late ‘50s San Francisco.",
"\n\n32 pages, $3.50, in stores on March 7.",
"\n\nGLORY #24\n\nWritten by JOE KEATINGE\nart & cover ROSS CAMPBELL\n\n\"ONCE AND FUTURE,\" Part Two\nGlory discovers some embers of war can never be extinguished as secrets of her past threaten to devastate the present. ",
"The bold new direction written by JOE KEATINGE and illustrated by Wet Moon's ROSS CAMPBELL continues with one of the most brutal opening sequences in Extreme history.",
"\n\n\"MONSTER BAITING,\" Part Three\nCat Curio has been kidnapped by the Gorillaconda! ",
"Cassie is prey for a demented hunter slasher! ",
"Samhain has to battle his way through Psychosaurs, and make the ultimate decision! ",
"Will he become a slasher again to save Cassie and Cat?",
"\n\n32 pages, $3.50, in stores on March 21.",
"\n\nHAUNT #24\n\nWritten by JOE CASEY\nart / cover NATHAN FOX\n\nIt's mayhem on the streets of Manhattan! ",
"A relentless creature of fury andfire running amok! ",
"Only Haunt can save the city from an apocalyptic end! ",
"But at what cost? ",
"The new nightmare rolls on in 2012!",
"\n\n32 pages, $2.99, in stores on March 28.",
"\n\nTHE INFINITE #7\n\nWritten by ROBERT KIRKMAN\nart / cover ROB LIEFELD\n\nBowen's plan is falling apart. ",
"He's on the run, his team is in disarray--he only has one place to go if he's going to turn things around. ",
"In order to defeat The Infinite and save the world--there's only one man he can rely on... his father.",
"\n\n32 pages, $2.99, in stores on March 7.",
"\n\nINVINCIBLE #91\n\nWritten by ROBERT KIRKMAN\nart & cover RYAN OTTLEY & JOHN RAUCH\n\nThe new Invincible will not back down, will not give in, and he is not going to play nice. ",
"What happens when Mark Grayson isn't happy with his replacement? ",
"All this and… What exactly happened between Robot and Monster Girl in the Flaxan Dimension?",
"\n\n32 pages, $2.99, in stores on March 14.",
"\n\nTHE LI'L DEPRESSED BOY #12\n\nWritten by S. STEVEN STRUBLE\nart / cover SINA GRACE\n\n\"BRAND NEW SHOES\"\nThe Li'l Depressed Boy may have met a new crush... but does he have the guts to ask if the spark is mutual? ",
"It all begins with a question."
] | {
"pile_set_name": "Pile-CC"
} | [
0.016666666666666666,
0.004424778761061947,
0.006993006993006993,
0,
0.015384615384615385,
0,
0,
0,
0.019417475728155338,
0.012195121951219513,
0,
0,
0.009174311926605505,
0,
0,
0.016877637130801686,
0,
0,
0,
0,
0,
0,
0,
0.009523809523809525,
0.010416666666666666,
0.034482758620689655,
0,
0.0055248618784530384,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.011278195488721804,
0.007518796992481203,
0,
0,
0,
0.022727272727272728,
0,
0,
0,
0.018292682926829267,
0,
0.02857142857142857,
0,
0,
0.013157894736842105,
0.015625,
0.024390243902439025,
0,
0,
0,
0.007751937984496124,
0,
0.024691358024691357,
0,
0,
0.01694915254237288,
0.015503875968992248,
0.015151515151515152,
0.009615384615384616,
0,
0,
0,
0.007751937984496124,
0,
0.013245033112582781,
0.005847953216374269,
0,
0.009708737864077669,
0.0051813471502590676,
0,
0,
0,
0.005917159763313609,
0,
0,
0,
0.16666666666666666,
0,
0.07692307692307693,
0,
0,
0,
0.04,
0,
0,
0,
0.011627906976744186,
0.011111111111111112,
0.02654867256637168,
0,
0,
0.015810276679841896,
0,
0.010101010101010102,
0,
0,
0,
0.016,
0,
0,
0.011904761904761904,
0.011627906976744186,
0,
0.017391304347826087,
0,
0,
0.01875,
0,
0.022727272727272728,
0,
0.011904761904761904,
0,
0.005649717514124294,
0,
0.00823045267489712,
0,
0,
0,
0.021341463414634148,
0,
0.005291005291005291,
0,
0.009478672985781991,
0.018072289156626505,
0.012195121951219513,
0,
0,
0.037037037037037035,
0,
0.020202020202020204,
0,
0,
0,
0,
0,
0.0297029702970297,
0,
0.00980392156862745,
0,
0.011560693641618497,
0.015384615384615385,
0.01098901098901099,
0,
0.014354066985645933,
0
] | 0.007489 | 5 |
[
"America's Top 120\n\nGet our great core programming with over 120 of your favorite channels, including CNN, ESPN, Nickelodeon and Discovery Channel. ",
"From sports to movies to the best TV programs, we've got something for everyone--and all at a great price.",
"\n\nChannel\n\nName\n\nDescription\n\n180\n\nABCF\n\nABC Family (ABCF)\n\nABC family features programming reflecting today\\'s families, entertaining and connecting with adults through relevant stories about today\\'s relationships.",
"\n\n167\n\nAXS\n\nAXS TV (AXS)\n\nAXS TV is your LIVE source for anything and everything happening now in pop culture, music, fashion and more.",
"The future of TV is AXS TV, providing you LIVE updates wherever unique happens.",
"\n\n4818\n\nALMA\n\nAlma Vision Hispanic Network (ALMA)\n\nAlmavisión is a Christian television network that teaches family values with biblical principals through the five full ministries: apostle, prophet, evangelist, pastor and teacher, and by establishing churches for the community and helping families in need.",
"\n\n219\n\nALIVE\n\nAmerica Live (ALIVE)\n\nAmerica\\'s Auction Network\n\n131\n\nAMC\n\nAmerican Movie Classics (AMC)\n\nAMC boasts a comprehensive library of the most entertaining movies of all time, as well as award-winning dramas and non-scripted programming.",
"\n\nA&E HD is television that you can\\'t turn away from. ",
"A&E offers a diverse mix of high quality entertainment ranging from the network\\'s original scripted series to signature non-fiction franchises.",
"\n\n4369\n\nBYUTV\n\nBYU Television (BYUTV)\n\nBYU Television is dedicated to programming that reflects the values and mission of Brigham Young University. ",
"Orchestral, choral and jazz performances originating directly from the BYU campus reflect those same values. ",
"We also feature the best of collegiate sports, including football, men\\'s and women...\n\n823\n\nBABY1\n\nBaby First (BABY1)\n\nBabyFirstTV is the nation\\'s only TV network made just for babies, toddlers and parents. ",
"Developed by educational experts, BabyFirstTV programming offers an inspiring journey filled with numbers, animals, colors, music and more. ",
"Plus, it\\'s completely commercial-free!",
"\n\n375\n\nBBHME\n\nBlockbuster @ Home (BBHME)\n\n102\n\nBLOCK\n\nBlockbuster Studio (BLOCK)\n\n102\n\nBLOCK\n\nBlockbuster Studio (BLOCK) HD\n\n210\n\nCSPAN\n\nC-SPAN (CSPAN)\n\nC-SPAN offers unique news and information programming, including 24-hour coverage of important political events from Washington, D.C. and around the nation including live debate from the floor of the U.S. House.",
"\n\n211\n\nCSPN2\n\nC-SPAN2 (CSPN2)\n\nC-SPAN2 offers24-hour coverage of important political events from Washington, D.C. and around the nation including live debate from the U.S. Senate.",
"\n\n279\n\nCCNEW\n\nCCTV News (CCNEW)\n\nCCTV-News, an English-language Chinese channel, offers the country the best window into China and its culture. ",
"CCTV-News\\' programming includes news and cultural and entertainment programs.",
"\n\n884\n\nCCTVE\n\nCCTV-E (CCTVE)\n\nCCTV-E broadcasts programs in Spanish including news, features, education and entertainment programs in a timely, narrative and culturally flavored way.",
"\n\n208\n\nCNBC\n\nCNBC (CNBC)\n\nCNBC is the recognized global leader in business news, providing real-time financial market coverage and business information.",
"\n\n200\n\nCNN\n\nCNN (CNN)\n\nEvery day, every night, everywhere, viewers depend on CNN - for its unparalleled coverage of breaking news and dynamic blend of debate and in-depth analysis.",
"\n\n176\n\nTOON\n\nCartoon Network (TOON)\n\nCartoon Network is an outrageous environment that celebrates all kinds of youth-targeted entertainment, from original animation and reality series to the best in action-adventure and animé.",
"\n\n177\n\nTOONW\n\nCartoon Network - West (TOONW)\n\nCartoon Network is an outrageous environment that celebrates all kinds of youth-targeted entertainment, from original animation and reality series to the best in action-adventure and animé.",
"\n\nThe Church Channel, a digital network from TBN, reflects the diversity and popularity of Christian faith across America. ",
"Featuring programs from Protestant, Catholic and Messianic Jewish leaders 24 hours a day, The Church Channel was create...\n\n4563\n\nARTS\n\nClassic Arts Showcase (ARTS)\n\nClassic Arts Showcase is a 24-hour cable channel that plays contemporary, vintage and rare video and film clips of the arts.",
"\n\nAvailable at no additional cost to Dish customers. ",
"May require additional dish antenna to view this programming. ",
"Additional fees may apply.",
"\n\n107\n\nCMDY\n\nComedy Central (CMDY)\n\nCOMEDY CENTRAL is comedy.",
"In all its forms.",
"Hands-down, the #1 provider of surprising, cutting-edge, laugh-out-loud programming from today’s hottest comic talent.",
"\n\n166\n\nCMT\n\nCountry Music Television (CMT)\n\nCMT is the #1 destination for contemporary country music.",
"CMT\\'s programming mix blends hit music franchises with family, adventure and comedy-oriented series that reach the heart of the country audience.",
"\n\nThe Pentagon Channel broadcasts military news and information for the 2.6 million members of the U.S. Armed Forces.",
"\n\n263\n\nDYSTR\n\nDaystar (DYSTR)\n\nWith an extensive blend of interdenominational and multi-cultural programming, Daystar is committed to providing quality television that will reach our viewers, refresh their lives, and renew their hearts.",
"\n\n182\n\nDISC\n\nDiscovery (DISC)\n\nIllustrating the world in all its wonder, diversity and amazement, Discovery Channel creates content that ignites viewers curiosity, immerses them in the how and why and satisfies them with information.",
"\n\n969\n\nDCD_Aura\n\nDish CD - Acoustic Crossroads (CD 20)\n\nContemporary Folk: The acoustic roots of American music from a contemporary perspective\n\n965\n\nDCD_Aura\n\nDish CD - Ambrosia (CD 16)\n\nA celebration of the classic soft rock and pop sounds that defined the easy rock explosion of the \\'70s and \\'80s\n\n977\n\nDCD_Aura\n\nDish CD - Aura (CD 28)\n\nNew Age: Inspiration in the form of a rich and diverse mix featuring contemporary new age artists\n\n980\n\nDCD_Aura\n\nDish CD - BYU Radio Network (CD 31)\n\nBYU Radio Network features a mix of instrumental and vocal inspirational music, as well as selected speeches, BYU General Conference talks, and programs with gospel themes.",
"\n\n957\n\nDCD_Aura\n\nDish CD - Backpages (CD 8)\n\nAdult Alternative: a diverse an intelligent mix of the latest generation of adult alternative music\n\nDisney Channel has something for everyone! ",
"Preschoolers play and learn with Playhouse Disney every morning.",
"In the afternoons, Disney Channel offers original series and movies perfect for older kids and families.",
"\n\n173\n\nDISW\n\nDisney Channel - West (DISW)\n\nDisney Channel has something for everyone! ",
"Preschoolers play and learn with Playhouse Disney every morning.",
"In the afternoons, Disney Channel offers original series and movies perfect for older kids and families.",
"\n\n114\n\nE!",
"\n\nE! ",
"Entertainment Television (E!)",
"\n\nE! ",
"inspires and entertains the Pop Culture fan in all of us. ",
"We never let you go a day, an hour, a minute without your Pop of Culture.",
"\n\n140\n\nESPN\n\nESPN (ESPN)\n\nESPN offers sports fans live major events and original studio programming with the authority that they trust and personalities that they love.",
"\n\n145\n\nESPNA\n\nESPN Alternate (ESPNA)\n\nESPN offers the widest and most diverse schedule of sports programming available. ",
"ESPN sports coverage includes Major League Baseball, Sunday Night Football and NCAA Basketball.",
"\n\n144\n\nESPN2\n\nESPN2 (ESPN2)\n\nESPN2 offers sports fans a wider variety of major live events and studio programming with the authority that they trust and the personalities that they love.",
"\n\n146\n\nESP2A\n\nESPN2 Alternate (ESP2A)\n\nESPN2 is a differentiated 24-hour sports network that features more than 2,500 hours a year of original sports programming targeting young and light to moderate sports viewers, yet appeals to all sports fans. ",
"ESPN2\\'s event programming inclu...\n\n142\n\nESNWS\n\nESPNEWS (ESNWS)\n\nESPNEWS, featuring up-to-the-minute scores through a continuous BottomLine, is the definitive and only 24-hour sports news television network in America.",
"\n\n141\n\nESPNU\n\nESPNU (ESPNU)\n\nESPNU delivers live game action from 27 Division I Conferences including ACC, Big East, Big Ten, Big 12, Pac-10 and SEC, as well as enhanced coverage of 22 NCAA Championships.",
"\n\n4870\n\nENLC\n\nEnlace (ENLC)\n\nEnlace allows you to watch recognized lecturers and panelists while they share values, and lessons to help you reach integral success. ",
"Merengue, Salsa, Bachata, Rock and Gospel, have a special place in Enlace, since we feature the best videos and concerts to inspire Children, youth and adults as we...\n\nFOX News Channel is the news network for the next generation of news viewers. ",
"We offer viewers what they want from the news today: more information presented in an objective format.",
"\n\n136\n\nFX\n\nFX (FX)\n\nFX is reinventing the programming landscape and amassing critical acclaim with original scripted series, and a growing roster of blockbuster movies and acquired hit series.",
"\n\n110\n\nFOOD\n\nFood Network (FOOD)\n\nFood Network is a unique lifestyle network that strives to be way more than cooking, exploring new and different ways to approach food - through pop culture, competition, adventure, and travel.",
"\n\n103\n\nPREVW\n\nFree Preview Guide (PREVW)\n\n4379\n\nFSTV\n\nFree Speech TV (FSTV)\n\nWorking for progressive social change\n\n164\n\nFUSE\n\nFuse (FUSE)\n\nFuse, the national television network dedicated exclusively to music, brings viewers closer to their favorite artists and bands by featuring original series and specials, exclusive interviews, live concerts and video blocks – all rooted in ...\n\n229\n\nGEMS\n\nGems and Jewelry (GEMS)\n\nGem Shopping Network-Live Shopping, Infomercial\n\n843\n\nHITN\n\nHITN (HITN)\n\nHITN is the first Hispanic Network of Public Television. ",
"HITN is dedicated to the issues of Hispanic immigrants and offers educational programming to better the Hispanic experience in the US.",
"\n\n202\n\nHLN\n\nHeadline News Network (HLN)\n\nHLN updates viewers constantly, 24 hours each day, with complete and concise national newscasts throughout the day and a compelling slate of legal, entertainment and talk programs in primetime\n\n120\n\nHIST\n\nHistory (HIST)\n\nHistory is a 24-hour programming service devoted to fulfilling America\\'s passion for history. ",
"The network features original and exclusive historical documentaries, movies, and mini-series.",
"\n\n222\n\nHSN\n\nHome Shopping Network (HSN)\n\nHSN features programs geared to a variety of lifestyles and special interest categories. ",
"Great values and quality brand merchandise make HSN a unique and entertaining interactive shopping experience.",
"\n\n226\n\nHSN2\n\nHome Shopping Network 2 (HSN2)\n\nHSN 2 features programs geared to a variety of lifestyles and special interest categories. ",
"Great values and quality brand merchandise make HSN 2 a unique and entertaining interactive shopping experience.",
"\n\n112\n\nHGTV\n\nHome and Garden Television (HGTV)\n\nHGTV presents some of America\\'s best home builders, decorators, gardeners and craft experts as they provide practical information to help people make the most of their lives at home.",
"\n\n398\n\nHRTV\n\nHorseracing Television (HRTV)\n\nHRTV, The Network for Horse Sports, is dedicated to equine sports including Thoroughbred horseracing from top U.S. and international racetrack, a wide range of world-class English and Western horse competitions, and a variety of equestrian ...\n\n216\n\nION\n\nION (ION)\n\nION offers timeless, diverse entertainment with a mix of television series, classic TV favorites, movies, specials and sports the whole family can enjoy.",
"\n\nThe Independent Film Channel is an American TV network that airs independent film and related programming.",
"\n\n259\n\nINSP\n\nInspiration Network (INSP)\n\nAmerica’s fastest-growing network for faith, INSP is where America goes for inspiration! ",
"INSP features a diverse lineup with programs from more than 70 ministries and content for the entire family, including specials, original kids shows, and action sports and music programs for teens.",
"\n\n227\n\nJTV\n\nJewelry Television (JTV)\n\nJewelry Television specializes in offering a vast selection of jewelry and gemstones, live - 24 hours-a-day, 7 days-a-week with a commitment to customer satisfaction.",
"\n\n264\n\nKTV\n\nKids and Teens Television (KTV)\n\nHigh-quality programming for youth, from toddlers to teens. ",
"Animated shows, talk, music videos, dramas, and educational, health and fitness programs.",
"\n\n183\n\nTLC\n\nLearning Channel (TLC)\n\nTLC engages and entertains with insightful programming that transports viewers into the authentic and revealing lives of real-life extraordinary characters you can learn from....and always does it with heart.",
"\n\n108\n\nLIFE\n\nLifetime (LIFE)\n\nLifetime keeps women entertained, informed and in touch with quality series, original world premiere movies and exciting original specials, as well as public outreach campaigns.",
"\n\n4383\n\nLINK\n\nLink Television (LINK)\n\nLink TV is linking Americans to the world, providing audiences with a global perspective on cultural, political and humanitarian issues, which impact every American\\'s life.",
"\n\nA 24-hour music video channel, MTV2 is custom-built for music enthusiasts, blending a diverse mix of music with the power of interactive technology.",
"\n\n470\n\nNBA\n\nNBA League Pass (NBA)\n\nNBA LEAGUE PASS takes you courtside every week with up to 40 live games from around the league. ",
"Follow your favorite team, player or match-up from outside your local viewing area. ",
"NBA LEAGUE PASS is your ticket to the most NBA action.",
"\n\n561\n\nNBA\n\nNBA League Pass (NBA) HD\n\nNBA LEAGUE PASS takes you courtside every week with up to 40 live games from around the league. ",
"Follow your favorite team, player or match-up from outside your local viewing area. ",
"NBA LEAGUE PASS is your ticket to the most NBA action.",
"\n\n475\n\nNHL\n\nNHL Center Ice\n\nFollow your favorite teams and players all season long. ",
"With up to 40 out-of-market games a week, keeping up with all the action from around the NHL® has never been easier. ",
"Don’t miss your chance to make every game a home game.",
"\n\n574\n\nNHL\n\nNHL Center Ice (NHLHD) HD\n\nFollow your favorite teams and players all season long. ",
"With up to 40 out-of-market games a week, keeping up with all the action from around the NHL® has never been easier. ",
"Don’t miss your chance to make every game a home game.",
"\n\nNickelodeon offers everything from live action to comedies to animation. ",
"In the evening, the network delivers family fun as it switches into Nick at Nite, featuring the best loved and time-tested shows.",
"\n\n149\n\nNOTAX\n\nNo Tax (NOTAX)\n\nNo Tax (NOTAX)\n\n393\n\nPRST\n\nPRST (PRST)\n\nCreated for the outdoors by outdoorsmen, with programming that covers fishing, hunting, archery, camping and other outdoor activities. ",
"With news, competition coverage and family-friendly programming that offers exciting adventures and fascinating explorations.",
"\n\n197\n\nPIVOT\n\nPivot (PIVOT)\n\npivot focuses on entertainment that sparks conversation and inspires change, serving passionate Millennials the type of socially relevant programming they crave through a diverse slate of talent and a mix of original and acquired series, fi...\n\n256\n\nPRAYR\n\nPrayer (PRAYR)\n\nPRAYER is a new religious channel featuring Shepherd’s Chapel with Pastor Arnold Murray. ",
"Pastor Murray has been broadcasting his daily scripture readings part-time on DISH and will now be expanding his coverage to 24 hours a day, Monday thr...\n\n137\n\nQVC\n\nQVC (QVC)\n\nQVC is a home shopping network featuring name brand products at super savings. ",
"Fashion, fitness, electronics, home furnishings, tools, jewelry and gift merchandise are highlighted.",
"\n\n299\n\nREELZ\n\nReelz (REELZ)\n\nReelzChannel is the network for movie fans. ",
"We bring viewers the latest movie news and info, interviews with Hollywood\\'s biggest names, and behind-the-scenes footage.",
"\n\nRT News Channel broadcasts from its studios in Moscow and Washington, DC. ",
"Tune in to watch news reports, features and talk shows with a totally different perspective. ",
"We report on the other side of the story, not making any conclusions, but raising the unanswered questions.",
"\n\n225\n\nSALE\n\nSale (SALE)\n\nSale brings you Bargin left and right on some of your favorite Collectables!",
"\n\n104\n\nIDEA\n\nSatellite Response Interactive (IDEA)\n\nSatellite Response Interactive brings you the latest products in fitness, house wares, health and beauty, business opportunities and more! ",
"Convenient shopping via 800 numbers and websites affords the ability to shop at leisure and obtain g...\n\nThe Savoy Express with Big Bands, classic vocal stars and hits from the \\'40s and beyond!",
"\n\n6005\n\nSXM05\n\nSiriusXM - \\'50s on 5 (SXM05)\n\nThe roots of rock \\'n\\' roll.",
"\n\n6006\n\nSXM06\n\nSiriusXM - \\'60s on 6 (SXM06)\n\nRelive the musical revolution of the \\'60s.",
"\n\n6007\n\nSXM07\n\nSiriusXM - \\'70s on 7 (SXM07)\n\nThe most popular music from the \\'70s.",
"\n\n6008\n\nSXM08\n\nSiriusXM - \\'80s on 8 (SXM08)\n\nThe defining hit music of the \\'80s. ",
"Hosted by the original MTV VJs.",
"\n\n6009\n\nSXM09\n\nSiriusXM - \\'90s on 9 (SXM09)\n\nFrom the emergence of grunge music to the resurgence of pop music.",
"\n\n6065\n\nSXM65\n\nSiriusXM - enLighten (SXM65)\n\nNorth America\\'s southern gospel music station.",
"\n\n257\n\nSBN\n\nSonlife Broadcasting Network (SBN)\n\nSBN SonLife Broadcasting Network, an outreach of Jimmy Swaggart Ministries, is a Christian multi-media network. ",
"This network offers a variety of live and prerecorded programs, specializing in music and teaching, that appeal to audiences of ...\n\n241\n\nSPIKE\n\nSpike Television (SPIKE)\n\nSpike knows what guys like, speaking to the bold, adventuresome side of men with action-packed entertainment, including a mix of comedy, blockbuster movies, sports, innovative originals and live events.",
"\n\n122\n\nSYFY\n\nSyfy Channel (SYFY)\n\nSyfy is a television network where \"what if\" is what\\'s on. ",
"Syfy fuels the imagination of viewers with original series and events, blockbuster movies and classic science fiction and fantasy programming.",
"\n\n399\n\nTVGAM\n\nTV Games Network (TVGAM)\n\nTV Games is the interactive horseracing network that combines live, televised coverage from over 45 of America\\'s greatest tracks with the convenience of wagering from home - either online or by phone!",
"\n\n117\n\nTVGN\n\nTV Guide Network (TVGN)\n\nTV Guide Network is the true entertainment insider and the only network that celebrates Hollywood, its stars and the shows that fans adore.",
"\n\n106\n\nTVLND\n\nTV Land (TVLND)\n\nTV Land showcases the best in original programming, acquired classic shows and hit movies from the 1950s through the 1980s.",
"\n\n9393\n\n3ABN\n\nThree Angels Broadcasting Network (3ABN)\n\nThree Angels Broadcasting Network.",
"\n\n196\n\nTRV\n\nTravel Channel (TRV)\n\nThe place to satisfy your urge to experience new things, meet amazing people, and see things you\\'ve never seen before. ",
"Travel Channel connects subscribers to the world\\'s excitement... CATCH IT!",
"\n\n260\n\nTBN\n\nTrinity Broadcasting Network (TBN)\n\nTBN features the best in faith and inspirational programs including a variety of denominations. ",
"Its program formats include worship, variety, music, children\\'s, news, sports, talk and live anointed programs.",
"\n\n139\n\nTBS\n\nTurner Broadcasting System (TBS)\n\nTBS is home to original comedy series, original late-night series, hot contemporary comedies, special events, blockbuster movies; and hosted movie showcases.",
"\n\n138\n\nTNT\n\nTurner Network Television (TNT)\n\nAs the first and only network dedicated to drama, TNT transports viewers from their daily lives onto an emotional and engaging ride filled with riveting stories and unpredictable plots.",
"\n\n105\n\nUSA\n\nUSA Network (USA)\n\nUSA Network is cable television’s leading provider of original series and feature movies, sports and entertainment events, off-net television shows and blockbuster theatrical films.",
"\n\n846\n\nV-ME\n\nV-ME (V-ME)\n\nThe first national Spanish-language public interest channel. ",
"V-ME is intelligent entertainment for Latino families in the U.S. With the best children\\'s programming, history, science, technology, news, film, religion, lifestyle and more, V-ME promises to entertain, empower, inspire and educate.",
"\n\n162\n\nVH1\n\nVH1 (VH1)\n\nVH1 connects viewers to the music, artists, pop culture and nostalgia they love.",
"\n\n875\n\nVALU\n\nValue TV (VALU)\n\nValu has the best products offered on TV.",
"\n\n214\n\nWEATH\n\nWeather Channel (WEATH)\n\nThe Weather Channel offers live 24-hour-a-day programming, which provides local forecasts to over 3,000 weather zones with regional and national weather conditions.",
"\n\n128\n\nWE\n\nWomen\\'s Entertainment (WE)\n\nWEtv is an American television channel that is marketed towards women. ",
"It features some original programming, such as non-fiction talk shows, as well as made-for-TV movies, older feature films, and reruns of TV shows, it also shows art and ...\n\nIV SUPPORT HOLDINGS LLC is an authorized contractor retailer of DISH Network L.L.C. DISH, DISH Network and DISH Network logos are registered trademarks and/or\nservice marks of DISH Network L.L.C. and/or its affiliate(s). ",
"The DISH Network trademarks and/or service marks are used by authority of DISH Network L.L.C. and/or its\napplicable affiliate(s).",
"\n\nIV SUPPORT HOLDINGS LLC in Frisco, CO is your local trusted authorized retailer for DISH Network. ",
"We offer the best value and the best deals\nin programming and services for satellite television. ",
"Hopper® technology from\nDISH delivers the shows you love to all your mobile devices wherever you go, all from IV SUPPORT HOLDINGS LLC in Frisco, CO.",
"\n\nIV SUPPORT HOLDINGS LLC is an authorized contractor retailer of DISH Network L.L.C. DISH, DISH Network and DISH Network logos are registered trademarks and/or\nservice marks of DISH Network L.L.C. and/or its affiliate(s). ",
"The DISH Network trademarks and/or service marks are used by authority of DISH Network L.L.C. and/or its\napplicable affiliate(s).",
"\n\nIV SUPPORT HOLDINGS LLC is an authorized contractor retailer of DISH Network L.L.C. DISH, DISH Network and DISH Network logos are registered trademarks and/or\nservice marks of DISH Network L.L.C. and/or its affiliate(s). ",
"The DISH Network trademarks and/or service marks are used by authority of DISH Network L.L.C. and/or its\napplicable affiliate(s).",
"\n\nIV SUPPORT HOLDINGS LLC in Frisco, CO is your local trusted authorized retailer for DISH Network. ",
"We offer the best value and the best deals\nin programming and services for satellite television. ",
"Hopper® technology from DISH delivers the shows you love to all your mobile devices wherever you go, all\nfrom IV SUPPORT HOLDINGS LLC in Frisco, CO."
] | {
"pile_set_name": "Pile-CC"
} | [
0.027210884353741496,
0,
0.004629629629629629,
0.02962962962962963,
0.012658227848101266,
0.006493506493506494,
0.02032520325203252,
0,
0.013888888888888888,
0.013513513513513514,
0,
0.009569377990430622,
0,
0.02564102564102564,
0.016483516483516484,
0.00558659217877095,
0.020833333333333332,
0.01282051282051282,
0,
0.02631578947368421,
0.022222222222222223,
0.004424778761061947,
0.01276595744680851,
0.016260162601626018,
0.006896551724137931,
0,
0,
0,
0.03278688524590164,
0,
0,
0.019801980198019802,
0.00684931506849315,
0.008547008547008548,
0.00423728813559322,
0,
0.007518796992481203,
0.005291005291005291,
0.015625,
0.009615384615384616,
0.023255813953488372,
0.015625,
0.009615384615384616,
0,
0,
0.034482758620689655,
0,
0,
0,
0.023809523809523808,
0.016666666666666666,
0.021052631578947368,
0.016129032258064516,
0.004032258064516129,
0,
0.0196078431372549,
0.006097560975609756,
0.008097165991902834,
0,
0,
0,
0.014519056261343012,
0.007462686567164179,
0.008403361344537815,
0,
0.015384615384615385,
0,
0,
0,
0.012987012987012988,
0.0021598272138228943,
0,
0.007692307692307693,
0,
0.00980392156862745,
0,
0,
0,
0,
0,
0,
0.007633587786259542,
0,
0.037037037037037035,
0.007462686567164179,
0,
0.037037037037037035,
0.023809523809523808,
0.008547008547008548,
0,
0.021052631578947368,
0.008547008547008548,
0,
0,
0.007751937984496124,
0.014634146341463415,
0,
0.0025575447570332483,
0.015625,
0,
0.0273972602739726,
0.008130081300813009,
0.013157894736842105,
0,
0,
0.00980392156862745,
0.005235602094240838,
0.010309278350515464,
0,
0,
0.011904761904761904,
0,
0.03225806451612903,
0,
0,
0.00625,
0,
0.031914893617021274,
0,
0.008298755186721992,
0,
0,
0.022222222222222223,
0.006493506493506494,
0,
0.013888888888888888,
0.008928571428571428,
0.0049261083743842365,
0.017391304347826087,
0.0047169811320754715,
0,
0.004273504273504274,
0.009708737864077669,
0,
0,
0,
0.010126582278481013,
0.007751937984496124,
0.02,
0,
0.013513513513513514,
0.017937219730941704,
0.007751937984496124,
0.017937219730941704,
0.007751937984496124,
0.02,
0,
0.02027027027027027
] | 0.008698 | 5 |
[
"Bishops of Rome under Constantine the Great\n\nConstantine the Great's (272–337) relationship with the four Bishops of Rome during his reign is an important component of the history of the Papacy, and more generally the history of the Catholic Church.",
"\n\nThe legend surrounding Constantine I's victory in the Battle of the Milvian Bridge (312) relates his vision of the Chi Rho (☧) and the text in hoc signo vinces in the sky and his reproducing this symbol on the shields of his troops. ",
"The following year Constantine and Licinius proclaimed the toleration of Christianity with the Edict of Milan, and in 325 Constantine convened and presided over the First Council of Nicaea, the first ecumenical council. ",
"None of this, however, has particularly much to do with the popes, who did not even attend the Council; in fact, the first bishop of Rome to be contemporaneously referred to as \"Pope\" (πάππας, or pappas) is Damasus I (366-384). ",
"Moreover, between 324 and 330, he built Constantinople as a new capital for the empire, and—with no apologies to the Roman community of Christians—relocated key Roman families and translated many Christian relics to the new churches.",
"\n\nThe Donation of Constantine, an 8th-century forgery used to enhance the prestige and authority of popes, places the pope more centrally in the narrative of Constantinian Christianity. ",
"The legend of the Donation claims that Constantine offered his crown to Sylvester I (314-335), and even that Sylvester baptized Constantine. ",
"In reality, Constantine was baptized (nearing his death in May 337) by Eusebius of Nicomedia, who, unlike the pope, was an Arian bishop of Constantinople. ",
"Sylvester was succeeded by Mark (336) and Julius I (337-352) during the life of Constantine.",
"\n\nAlthough the \"Donation\" never occurred, Constantine did hand over the Lateran Palace to the bishop of Rome, and begin the construction of Old Saint Peter's Basilica (the \"Constantinian Basilica\"). ",
"The gift of the Lateran probably occurred during the reign of Miltiades (311-314), Sylvester I's predecessor, who began using it as his residence. ",
"Old St. Peter's was begun between 326 and 330 and would have taken three decades to complete, long after the death of Constantine. ",
"Constantine's legalization of Christianity, combined with the donation of these properties, gave the bishop of Rome an unprecedented level of temporal power, for the first time creating an incentive for secular leaders to interfere with papal succession.",
"\n\nBackground\n\nIn spite of the Diocletian Persecution, Christians constituted approximately one-tenth of the population of the Roman Empire at the time of Constantine's rise to power. ",
"Christianity was legalized by Galerius, who was the first emperor to issue an edict of toleration for all religious creeds including Christianity in April 311. ",
"Eamon Duffy characterizes the church in Rome before Constantine as \"not one congregation, but a loose constellation of churches based in private houses or, as time went on and the community grew, meeting in rented halls in markets and public baths. ",
"It was without any single dominant ruling officer, its elders or leaders sharing responsibility, but distributing tasks, like that of foreign correspondent. ",
"By the eve of the conversion of Constantine, there were more than two dozen of these religious community-centers or tituli\".",
"\n\nThe Roman church was a small community, and its bishop exercised little influence outside its members in the time of Constantine. ",
"Constantine was the first Roman Emperor to embrace Christianity, although he likely continued in his pre-Christian beliefs. ",
"He and co-Emperor Licinius bestowed imperial favor on Christianity through the Edict of Milan promulgated in 313. ",
"After the Edict of Milan, the church adopted the same governmental structure as the Empire: geographical provinces ruled by bishops. ",
"These bishops of important cities (Metropolitan bishops) therefore rose in power over the bishops of lesser cities (later called Suffragan bishops).",
"\n\nWhatever his personal beliefs, Constantine's political interest in Christianity was as a unifying force and his policy of \"the imposition of unity on the churches at all costs\" soon set him on a \"collision course with the popes.\"",
"\n\nPopes under Constantine\n\nMiltiades\n\nMiltiades (311–314) was pope at the time of Constantine's victory, and Constantine gifted to Miltiades the Lateran Palace, where he relocated, holding a synod in 313. ",
"Constantine designated Miltiades as one of four bishops to adjudicate the case of the Donatists, but he had no authority to decide the case or publish the result without the approval of the emperor himself. ",
"Customarily, the African bishops may have gone to the bishop of Rome as a respected, neutral figure, but it was well known that Miltiades would not agree with the Donatist position that ordination by a \"traitor\" bishop would invalidate the sacrament.",
"\n\nTurning to Constantine was a strange move because he had not yet been baptized, and word of his budding conversion may not yet have reached Alexandria. ",
"Constantine therefore referred the matter to Miltiades, requiring him to collaborate with three bishops from Gaul. ",
"Eamon Duffy calls this the \"first direct intervention by an emperor in the affairs of the church.\" ",
"When Miltiades invited fifteen additional Italian bishops to participate in the synod and ruled against the Donatists, they appealed to Constantine again, who called for a new synod in Arles, this time headed by the bishops of Arles and Syracuse.",
"\n\nSylvester I\n\nMiltiades died, and his successor, Sylvester I (313–335), did not travel to Arles. ",
"The Arles synod gave Silvester I somewhat of a nod by asking him to circulate their decisions to the other bishops, although he had no part in the process. ",
"During Silvester I's reign, construction began on the Lateran Basilica, Santa Croce in Gerusalemme, and St. Peter's. ",
"Silvester did not attend the first ecumenical council, the First Council of Nicaea (325), but sent two priests as his representatives; the Western bishops of Carthage and Milan were also in attendance.",
"\n\nSilvester would have viewed Arianism as a heresy; Constantine himself probably did not understand the complex theological issues in dispute, although he had surrounded himself with many followers of Arius, including Bishop Eusebius of Nicomedia, his eventual baptizer. ",
"Arius's followers did poorly at Nicaea, and the Nicene Creed that was adopted was squarely against their Christological position. ",
"The Arians were \"silenced, not persuaded\" by the Council, and the controversy in the larger Christian community was not resolved. ",
"The Arian position would persist in the East for three generations, and even eventually be adopted by Constantine's son, Constantius II. ",
"Constantine himself supported the Nicene position mainly because it was \"his Council\" sought a compromise text that would \"paper over the differences between the two sides.\" ",
"Eusebius remained an Arian, although he assured Constantine his views were compatible with his interpretation of the Nicene Creed, and baptized Constantine in 337.",
"\n\nThe chief supporter of the Nicene Creed was Bishop Athanasius of Alexandria, but in 335 he was deposed and exiled to Gaul over unrelated fears about the Egyptian corn supply to Constantinople. ",
"His followers were rounded up and picked off. ",
"The key debates in Nicaea had been conducted in the Greek language, with the subtleties of particular words inciting great controversy; the Latin language of the West simply \"did not yet even possess adequate terminology to handle the debate properly.\" ",
"The pope's legates signed onto the findings of the Council, and it continued to be supported by the next two popes, just as followers of Athanasius remained welcome in Rome, but the theological issues were little examined in the West.",
"\n\nMark\nMark (336) was the first to designate the bishop of Ostia as the first among the consecrators of the new bishop of Rome (the bishop of Ostia is currently the Dean of the College of Cardinals).",
"\n\nJulius I\nThe influence of Constantine would help solidify a strong role for the Roman emperor in the selection process: Constantine chose Julius I (337–352) for all intents and purposes, and his son Constantius II exiled Liberius and installed Felix II (an Arian) as his successor.",
"\n\nJulius I received Athanasius, and invited the Arian Eastern bishops then in Antioch to join him in Rome. ",
"The Eastern bishops would have regarded Julius I as having an equal dignity to their own episcopate, but were not pleased that he took into his communion a bishop condemned by an Eastern synod.",
"\n\nAftermath\nUnder Pope Liberius (352-366), the Arian conflict between the emperor and bishop of Rome culminated in the Synod of Arles (353), convened by Constantius II. ",
"Therein, Liberius's legates signed a declaration condemning the Council of Nicaea. ",
"When Liberius himself refused to cooperate, he was exiled. ",
"Pope Damasus, (366-384) was able to largely suppress the Arians with the help of Emperor Theodosius, Gregory of Nazianzus, Gregory of Nyssa, and Ambrose of Milan.",
"\n\nHistoriography\n\n \n\nThe Donation of Constantine purported to memorialize the transfer to Sylvester I and his successors of dominion over the entire Western Roman Empire for the consideration of Sylvester I's instruction of Constantine in Christianity, baptism of Constantine, and curing Constantine of leprosy. ",
"Constantine allegedly kept for himself only the Eastern Roman Empire. ",
"The forgery was probably constructed during the Frankish Papacy, when Pope Stephen II became the first pope to cross the Alps to crown Pepin the Short, who issued the Donation of Pepin (a non-forgery), granting the pope control of the lands of the Lombards, which coalesced into the first fragments of the Papal States.",
"\n\nIt was not long before the document was denounced as a forgery, notably by Otto III, Holy Roman Emperor (r. 983-1002). ",
"By the mid 15th century, not even the popes themselves regarded the document as genuine. ",
"Italian humanist Lorenzo Valla further proved its falsity in 1440 by showing that its Latin language did not correspond to that of the 4th century. ",
"The \"Donation\" purports to acknowledge the primacy of Rome over Antioch, Jerusalem, Alexandria, and Constantinople, even though the last of these had not even been founded at the time of the claimed Donation.",
"\n\nThe myth of the \"Donation of Constantine\" is embellished further in a 5th-century hagiographic text Vita S Silvestri (or Actus S Silvestri). ",
"Among other things, the document claims that Silvester I slew a dragon that had been threatening Rome. ",
"The text says that all of these events occurred right after Constantine's entry into the city following the Battle of the Milvian Bridge, despite the fact that Silvester I did not become bishop until 314. ",
"These events were incorporated uncritically into the Liber pontificalis c. 530. ",
"John Malalas further embellished the story in his Chronicon, which claimed that Silvester I baptized not only Constantine, but his mother Helena, and—for good measure—a large group of his relatives and Roman bystanders. ",
"Theophanes the Confessor in his Chronicle c. 815-820 adds Constantine's son Crispus to the list and viciously attacks contrary accounts as Arian lies; Theophanes refers to the Lateran Baptistery as the \"Baptistry of Constantine.\"",
"\n\nPope Pius V's Breviarum Romanum (1568) and Pope Gregory XIII's Martyrologium Romanum (1584) also asserted that Silvester I was the baptizer of Constantine. ",
"When Pope Sixtus V erected the Egyptian obelisk to the north of the Lateran Basilica in 1588, he added to the base the inscription \"Constantine was baptized here.\" ",
"Cardinal Cesare Baroni continued to claim this in his Annales Ecclesiastici (1592) and a French author has claimed this as recently as 1906.",
"\n\nThis falsified version of Constantine's baptism has found its way into a great deal of ecclesiastical art. ",
"Depictions include the Stavelot Triptych (c. 1165), frescoes in I Santi Quattro Coronati, stained glass in the St Michael and All Angels' Church, Ashton-under-Lyne, and—most famously—Raphael's The Baptism of Constantine in the Raphael Rooms of the Apostolic Palace.",
"\n\nNotes\n\nReferences\nBaumgartner, Frederic J. 2003. ",
"Behind Locked Doors: A History of the Papal Elections. ",
"Palgrave Macmillan. .",
"\n \nHurst, John Fletcher. ",
"1897. ",
"History of the Christian church, Volume 1.",
"\nPohlsander, Hans A. 2005 (ed. ",
"2). ",
"The Emperor Constantine.",
"\n\nCategory:History of the papacy\nCategory:Constantine the Great and Christianity"
] | {
"pile_set_name": "Wikipedia (en)"
} | [
0.008032128514056224,
0.00851063829787234,
0.01818181818181818,
0.0043859649122807015,
0,
0,
0.02127659574468085,
0.012903225806451613,
0.021739130434782608,
0,
0.013605442176870748,
0.007633587786259542,
0.003937007874015748,
0.01092896174863388,
0.00625,
0.004016064257028112,
0,
0.008064516129032258,
0,
0.024193548387096774,
0.008771929824561403,
0.007518796992481203,
0,
0.004329004329004329,
0.014634146341463415,
0.00966183574879227,
0.008,
0.006493506493506494,
0.008695652173913044,
0.010101010101010102,
0.016260162601626018,
0.01020408163265306,
0.00641025641025641,
0.008547008547008548,
0.004975124378109453,
0.007380073800738007,
0.023076923076923078,
0.007692307692307693,
0.014598540145985401,
0.017241379310344827,
0.018404907975460124,
0.015384615384615385,
0,
0,
0.008547008547008548,
0.010050251256281407,
0.01060070671378092,
0.009345794392523364,
0.010362694300518135,
0.005917159763313609,
0.024096385542168676,
0,
0.012345679012345678,
0.00641025641025641,
0.02857142857142857,
0.006269592476489028,
0.01652892561983471,
0,
0.006756756756756757,
0.004807692307692308,
0,
0,
0.004878048780487805,
0.025,
0.01818181818181818,
0.013100436681222707,
0.02531645569620253,
0.006097560975609756,
0.007142857142857143,
0.009174311926605505,
0.007547169811320755,
0.0196078431372549,
0,
0.047619047619047616,
0.08,
0,
0,
0.03225806451612903,
0,
0.041666666666666664,
0
] | 0.011114 | 5 |
[
"// (C) Copyright 2008 CodeRage, LLC (turkanis at coderage dot com)\r\n// (C) Copyright 2004-2007 Jonathan Turkanis\r\n// Distributed under the Boost Software License, Version 1.0. (",
"See accompanying\r\n// file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt.)",
"\r\n\r\n// See http://www.boost.org/libs/iostreams for documentation.",
"\r\n\r\n// Contains the definition of the class template \r\n// boost::iostreams::detail::double_object, which is similar to compressed pair\r\n// except that both members of the pair have the same type, and \r\n// compression occurs only if requested using a boolean template\r\n// parameter.",
"\r\n\r\n#ifndef BOOST_IOSTREAMS_DETAIL_DOUBLE_OBJECT_HPP_INCLUDED\r\n#define BOOST_IOSTREAMS_DETAIL_DOUBLE_OBJECT_HPP_INCLUDED\r\n\r\n#if defined(_MSC_VER)\r\n# pragma once\r\n#endif \r\n\r\n#include <algorithm> // swap.",
"\r\n#include <boost/detail/workaround.hpp>\r\n#include <boost/mpl/if.hpp>\r\n#if BOOST_WORKAROUND(__MWERKS__, > 0x3003)\r\n# include <msl_utility>\r\n#else\r\n# include <boost/call_traits.hpp>\r\n#endif\r\n\r\nnamespace boost { namespace iostreams { namespace detail {\r\n\r\ntemplate<typename T>\r\nclass single_object_holder {\r\npublic:\r\n#if BOOST_WORKAROUND(__MWERKS__, > 0x3003)\r\n typedef Metrowerks::call_traits<T> traits_type;\r\n#else\r\n typedef boost::call_traits<T> traits_type;\r\n#endif\r\n typedef typename traits_type::param_type param_type;\r\n typedef typename traits_type::reference reference;\r\n typedef typename traits_type::const_reference const_reference;\r\n single_object_holder() { }\r\n single_object_holder(param_type t) : first_(t) { }\r\n reference first() { return first_; }\r\n const_reference first() const { return first_; }\r\n reference second() { return first_; }\r\n const_reference second() const { return first_; }\r\n void swap(single_object_holder& o)\r\n { std::swap(first_, o.first_); }\r\nprivate:\r\n T first_;\r\n};\r\n\r\ntemplate<typename T>\r\nstruct double_object_holder {\r\npublic:\r\n#if BOOST_WORKAROUND(__MWERKS__, > 0x3003)\r\n typedef Metrowerks::call_traits<T> traits_type;\r\n#else\r\n typedef boost::call_traits<T> traits_type;\r\n#endif\r\n typedef typename traits_type::param_type param_type;\r\n typedef typename traits_type::reference reference;\r\n typedef typename traits_type::const_reference const_reference;\r\n double_object_holder() { }\r\n double_object_holder(param_type t1, param_type t2)\r\n : first_(t1), second_(t2) { }\r\n reference first() { return first_; }\r\n const_reference first() const { return first_; }\r\n reference second() { return second_; }\r\n const_reference second() const { return second_; }\r\n void swap(double_object_holder& d)\r\n { \r\n std::swap(first_, d.first_); \r\n std::swap(second_, d.second_); \r\n }\r\nprivate:\r\n T first_, second_;\r\n};\r\n\r\ntemplate<typename T, typename IsDouble>\r\nclass double_object \r\n : public mpl::if_<\r\n IsDouble, \r\n double_object_holder<T>, \r\n single_object_holder<T>\r\n >::type\r\n{\r\nprivate:\r\n typedef typename \r\n mpl::if_<\r\n IsDouble, \r\n double_object_holder<T>, \r\n single_object_holder<T>\r\n >::type base_type;\r\npublic:\r\n#if BOOST_WORKAROUND(__MWERKS__, > 0x3003)\r\n typedef Metrowerks::call_traits<T> traits_type;\r\n#else\r\n typedef boost::call_traits<T> traits_type;\r\n#endif\r\n typedef typename traits_type::param_type param_type;\r\n typedef typename traits_type::reference reference;\r\n typedef typename traits_type::const_reference const_reference;\r\n double_object() : base_type() {}\r\n double_object(param_type t1, param_type t2)\r\n : base_type(t1, t2) { }\r\n bool is_double() const { return IsDouble::value; }\r\n};\r\n\r\n} } } // End namespaces detail, iostreams, boost.",
"\r\n\r\n#endif // #ifndef BOOST_IOSTREAMS_DETAIL_DOUBLE_OBJECT_HPP_INCLUDED\r\n"
] | {
"pile_set_name": "Github"
} | [
0.01694915254237288,
0.01098901098901099,
0.015384615384615385,
0.0035587188612099642,
0.013157894736842105,
0.0032061558191728116,
0.0136986301369863
] | 0.010992 | 5 |
[
"American Independent Party\n\nThe American Independent Party (AIP) is a far-right political party in the United States that was established in 1967. ",
" The AIP is best known for its nomination of former Governor George Wallace of Alabama, who carried five states in the 1968 presidential election running on a segregationist “law and order” platform against Richard M. Nixon and Hubert H. Humphrey. ",
" The party split in 1976 into the modern American Independent Party and the American Party. ",
"From 1992 until 2008, the party was the California affiliate of the national Constitution Party. ",
"Its exit from the Constitution Party led to a leadership dispute during the 2016 election.",
"\n\nBackground Information\n\nWallace campaign and early history\n\nIn 1968, the AIP was founded by Bill Shearer and his wife, Eileen Knowland Shearer. ",
"It nominated George C. Wallace (Democrat) as its presidential candidate and retired U.S. Air Force General Curtis E. LeMay as the vice-presidential candidate. ",
"Wallace ran on every state ballot in the election, though he did not represent the American Independent Party in all fifty states: in Connecticut, for instance, he was listed on the ballot as the nominee of the \"George Wallace Party.\" ",
"The Wallace/LeMay ticket received 13.5 percent of the popular vote and 46 electoral votes from the states of Arkansas, Louisiana, Mississippi, Georgia, and Alabama. ",
"No third-party candidate has won more than one electoral vote since the 1968 election.",
"\n\nIn 1969, representatives from forty states established the American Party as the successor to the American Independent Party. ",
"In some places, such as Connecticut, the American Party was constituted as the American Conservative Party. (",
"The modern American Conservative Party, founded in 2008, is unrelated to the Wallace-era party.) ",
"In March 1969, the party ran a candidate in a special election in Tennessee's 8th congressional district in northwestern Tennessee, where Wallace had done well the previous November, to replace Congressman Robert \"Fats\" Everett, who had died in office. ",
"Their candidate, William J. Davis, out-polled Republican Leonard Dunavant, with 16,375 votes to Dunavant's 15,773; but the race was carried by moderate Democrat Ed Jones, with 33,028 votes (47% of the vote).",
"\n\nThe party flag, adopted on August 30, 1970, depicts an eagle holding a group of arrows in its left talons, over a compass rose, with a banner which reads \"The American Independent Party\" at the eagle's base.",
"\n\nThe American Party, as it was commonly called and legally styled in several states, ran occasional congressional and gubernatorial candidates, but few made any real impact. ",
"In 1970, the AIP fielded a candidate for governor of South Carolina, Alfred W. Bethea, a former Democratic member of the South Carolina House of Representatives from Dillon County. ",
" Democrat John C. West defeated the Republican nominee, Albert Watson, an outgoing member of the United States House of Representatives. ",
"Bethea finished with only 2 percent of the votes cast. ",
"In another 1970 gubernatorial race, the Arkansas American Party ran Walter L. Carruth (1931–2008), a justice of the peace from Phillips County in eastern Arkansas, against Republican Winthrop Rockefeller and Democrat Dale Bumpers. ",
"Carruth received 36,132 votes (5.9 percent), not enough to affect the outcome in which Bumpers handily unseated Rockefeller. ",
"The American Party had gained ballot access in Tennessee in 1970 as the result of George Wallace's strong (second-place) showing in the state in 1968, easily crossing the 5 percent threshold required, and held a primary election which nominated a slate of candidates including businessman Douglas Heinsohn for governor. ",
" However, neither Heinsohn nor any other candidate running on the American Party line achieved the 5 percent threshold in the 1970 Tennessee election, and it likewise failed to do so in 1972, meaning that the party lost its newfound ballot access, which as of 2017 it has never regained.",
"\n\nIn 1972, the American Party nominated Republican Congressman John G. Schmitz of California for president and Tennessee author Thomas Jefferson Anderson, both members of the John Birch Society, for vice president (they received well over a million votes). ",
"In that election, Hall Lyons, an oilman from Lafayette, Louisiana, and a former Republican, ran as the AP U.S. Senate nominee but finished last in a four-way race dominated by the Democratic nominee, J. Bennett Johnston, Jr.\n\nAfter the 1976 split\nIn 1976, the American Independent Party split into the more moderate American Party, which included more northern conservatives and Schmitz supporters, and the American Independent Party, which focused on the Deep South. ",
"Both parties have nominated candidates for the presidency and other offices. ",
"Neither the American Party nor the American Independent Party has had national success, and the American Party has not achieved ballot status in any state since 1996.",
"\n\nIn the early 1980s, Bill Shearer led the American Independent Party into the Populist Party. ",
"From 1992 to 2008, the American Independent Party was the California affiliate of the national Constitution Party, formerly the U.S. Taxpayers Party, whose founders included the late Howard Phillips.",
"\n\n2007 leadership dispute\nA split in the American Independent Party occurred during the 2008 presidential campaign, one faction recognizing Jim King as chairman of the AIP with the other recognizing Ed Noonan as chairman. ",
"Noonan's faction claims the old AIP main website while the King organization claims the AIP's blog. ",
"King's group met in Los Angeles on June 28–29, elected King to state chair. ",
"Ed Noonan's faction, which included 8 of the 17 AIP officers, held a convention in Sacramento on July 5, 2008. ",
"Issues in the split were U.S. foreign policy and the influence of Constitution Party founder Howard Phillips on the state party.",
"\n\nThe King group elected to stay in the Constitution Party and supported its presidential candidate, Chuck Baldwin. ",
"It was not listed as the \"Qualified Political Party\" by the California Secretary of State and Baldwin's name was not printed in the state's ballots. ",
"King's group sued for ballot access\n and their case was dismissed without prejudice.",
"\n\nThe Noonan group voted to pull out of the Constitution Party and join a new party called America's Party, put together by perennial candidate and former United Nations Ambassador Alan Keyes as a vehicle for his own presidential campaign. ",
"Since Noonan was on record with the California Secretary of State as (outgoing) party chairman, Keyes was added to the state ballots as the AIP candidate. ",
"This group elected Markham Robinson as its new chair at the convention.",
"\n\nPresidential tickets\n\nSince the fracture of the American Independent Party between the King and Noonan factions, control of the State Party, and thus the ballot line, has been in the hands of the Noonan faction. ",
"Attempts to nominate Chuck Baldwin (the 2008 Constitution nominee) or Virgil Goode (the 2012 Constitution nominee) were unsuccessful, as were their independent efforts to make it onto the California presidential ballot.",
"\n\nNominee\n\nThe King faction website is now billed as the Constitution Party of California, stating \"Vote Constitution Party--the only party fighting for a return to Constitutional government!\" ",
"Its blog provides a list of \"2010 California Candidates: We endorse the following candidates who are running under the \"American Independent\" banner in California for 2010!\"",
"\n\nCalifornia gubernatorial candidates\n\nChairmen/Vice-Chairmen\nBill Shearer: 1967–1999\nNathan Johnson: 1999–2002\nJim King/Reed R. Heustis: 2002–2004\nNancy Spirkoff: 2004–2006\nEdward C. Noonan/Mark Seidenberg: 2006–2008\n Disputed: Jim King and Markham Robinson claim chairmanship: 2008–present\n\nMembership and accidental-membership phenomenon\nAs of 2016, about 3% of California's 17.2 million voters are registered with the AIP, making the party the third-largest of California's political parties, although it is far behind the numbers registered with the Democrats (43%), Republicans (28%) and those stating \"no party preference\" (24%).",
"\n\nHowever, it has long been thought by political analysts that the party, which has received very few votes in recent California elections, maintains its state ballot status because people join the American Independent Party mistakenly believing that they are registering as \"independent\" voters. ",
"This was confirmed in a Los Angeles Times investigation in 2016, which found \"overwhelming\" and \"indisputable\" evidence that thousands of California voters who are registered as affiliated with the American Independent Party on voter forms in fact intended to be registered as \"no party preference\" (i.e., as independent voters). ",
"A 2016 poll conducted of California voters registered with the AIP showed that 73% identify themselves as \"no affiliation\" and 3% identify themselves as \"undecided.\" ",
"Upon learning the AIP platform, 50% of registered AIP voters wanted to leave the AIP. ",
"A Times review of voting records revealed a wide array of Californians have fallen victim to this error, including celebrities such as Sugar Ray Leonard, Demi Moore and Emma Stone, and Kaley Cuoco. ",
" Similarly, in 2008, Jennifer Siebel, fiancée of San Francisco's liberal Democratic mayor Gavin Newsom, attempted to change her party affiliation from Republican to unaffiliated, but \"checked the American Independent box thinking that was what independent voters were supposed to do.\"",
"\n\nThis confusion results in accidentally registered AIP members being unable to vote in presidential primary elections and, in prior years, in all partisan primary elections other than those of the AIP. ",
"A number of California registrars of voters had expressed concern over the confusion that the party's name causes. ",
"Kim Alexander, president of the nonpartisan California Voter Foundation, said that the California voter form was \"confusing and somewhat misleading.\" ",
"However, since the advent of the \"top-two\" blanket primary in California in 2012, all voters may participate in non-presidential primary elections where nominations for public office are to be made. ",
"Presidential nominations and elections of members of party county central committees are still restricted to voters registered in the party where such contests are held, but a party may choose to allow voters with No Party Preference to vote in their presidential primary.",
"\n\nReferences\n\nExternal links\nAmerican Independent Party of California\nAmerican Independent Party at JoinCalifornia\n\nCategory:Political parties established in 1967\n \nCategory:Far-right political parties in the United States\nCategory:Paleoconservative parties in the United States\nCategory:American nationalist parties\nCategory:Organizations that oppose same-sex marriage"
] | {
"pile_set_name": "Wikipedia (en)"
} | [
0.02040816326530612,
0.016129032258064516,
0.021739130434782608,
0.010309278350515464,
0.011111111111111112,
0.02054794520547945,
0.018867924528301886,
0.00851063829787234,
0,
0,
0.015625,
0.01834862385321101,
0.010309278350515464,
0.007905138339920948,
0.01932367149758454,
0.009569377990430622,
0.005714285714285714,
0.011049723756906077,
0.029197080291970802,
0.01818181818181818,
0.017316017316017316,
0.016,
0.009375,
0.006968641114982578,
0.01556420233463035,
0.019230769230769232,
0,
0.018072289156626505,
0.031578947368421054,
0.020100502512562814,
0.018018018018018018,
0.04,
0.013157894736842105,
0.018018018018018018,
0.015625,
0.02586206896551724,
0.013422818791946308,
0.011904761904761904,
0.020833333333333332,
0.025806451612903226,
0.014084507042253521,
0.02336448598130841,
0.0091324200913242,
0.0051813471502590676,
0.005780346820809248,
0.012578616352201259,
0.003367003367003367,
0.006060606060606061,
0.006024096385542169,
0.03488372093023256,
0.025252525252525252,
0.007042253521126761,
0.009852216748768473,
0,
0.013333333333333334,
0,
0.003676470588235294,
0.008130081300813009
] | 0.014094 | 5 |
[
"Gurmeet Singh\n\nGurmeet Singh is founder and CEO of DEEP AERO, which helps individuals and organisations to understand the revolutionary benefits of drone technology, and gain the skills and connections to implement drone technology in their workplaces.",
"\n\nDrones have been used for a variety of purposes across the world – from transporting medical facilities in Africa to mapping agricultural fields and wildlife.",
"\n\nThe robots are coming! ",
"Advances in deep technology, machine learning and automation are ushering a new era of digital workers. ",
"In the near future, drones, artificial intelligence and driverless cars will seamlessly coordinate and transport goods and people across the globe at rather smaller cost. ",
"In fact, drones in particular have caught the interest of several bodies and policymakers across the globe. ",
"Countries across the world are exploring the possibilities of drones and their extent of usage in different scenarios.",
"\n\nFrom delivering online grocery orders at the doorstep, to providing emergency medical supplies to remote areas, or facilitating unmanned surveillance in dangerous warzones, there are many more ways in which Unmanned Aerial Vehicles (UAVs) or drones are changing the commerce landscape as well as our lives. ",
"This change is much faster than we could have thought 4-5 years ago. ",
"Autonomous operations of drones or Unmanned Aerial Vehicles (UAVs) through AI (artificial intelligence) and Blockchain may sound like science fantasy, however it is a reality that’s being explored actively today. ",
"Let us look at how the usage of AI and blockchain can pave a new route map for drones and consequently the world transport and logistic economy at large.",
"\n\nAccording to a new study for Mercatus Center at George Mason University, scholars Adam Thierer and Ryan Hagemann found that driver errors in accidents cost $300 billion in USA alone. ",
"Additionally, in 2011, congestion caused drivers to spend an extra 5.5 billion hours on the road while having to purchase 2.9 billion gallons of fuel at a cost of $121 billion. ",
"Adoption of autonomous and intelligent vehicles like drones will not only reduce congestion and human-initiated driving errors but also drastically cut costs in fuel, accident insurance premiums and provide more safety for both passengers and cargo.",
"\n\nDrones have been used for a variety of purposes across the world – from transporting medical facilities in Africa to mapping agricultural fields and wildlife. ",
"Merely flying at a low altitude is not what makes drones special. ",
"It is their capability to capture remote data, surveillance videos and also analyse it for various objectives without much human intervention is what makes drones so lucrative as dynamic sensors. ",
"The integration of AI and deep learning allows these autonomous flying objects to understand and analyse the surroundings. ",
"For instance, without AI, a team of humans cannot keep an eye on video footage received from hundreds of drones in real time to find a missing vehicle. ",
"These videos can only be created and archived for future reference. ",
"However, with AI, this is not the case. ",
"The smart algorithm can analyse the footage received from different cameras and identify the searched object in real time. ",
"In another use case, the AI powered drones can have predictive capabilities by analysing the past data. ",
"For example, a month long footage captured from a construction site can be used to project how the site may look like in the next week or so.",
"\n\nHowever, the biggest challenge for the commercial drone industry is air traffic management in the shared airspace. ",
"With exponentially increasing number of drone fights, it is not possible to manage the aerial traffic using conventional air traffic management. ",
"While these methods may have done well with manned flights, they are not sufficient to handle autonomous drone flights. ",
"This is where AI and blockchain come into the picture."
] | {
"pile_set_name": "Pile-CC"
} | [
0.007936507936507936,
0,
0,
0,
0,
0,
0,
0.003236245954692557,
0,
0.004694835680751174,
0.006535947712418301,
0.021621621621621623,
0,
0,
0,
0,
0,
0,
0.006578947368421052,
0,
0.025,
0,
0,
0,
0,
0,
0,
0
] | 0.0027 | 5 |
[
"A randomized controlled trial of prolonged exposure therapy versus relaxation training for older veterans with military-related PTSD.",
"\nAlthough prolonged exposure (PE) has strong support for treating posttraumatic stress disorder (PTSD), there is little research on PE for older adults. ",
"Likewise, Relaxation Training (RT) has shown some benefit for PTSD, but has not been adequately tested in this population. ",
"This study represents the first randomized controlled trial of two active psychotherapies for PTSD among older adults. ",
"Male combat veterans (N = 87; mean age = 65 years) were randomly assigned to 12 sessions of PE (n = 41) or RT (n = 46). ",
"Clinician-administered and self-report assessments were conducted at pre-treatment, post-treatment, and six-month follow-up; self-reported symptoms were also measured at each treatment session. ",
"Multi-level modeling indicated that Clinician-Administered PTSD Scale scores significantly decreased from pre-treatment to follow-up, but the time by treatment condition interaction was not significant. ",
"Pre- to post-treatment change was large in PE and moderate in RT, but many gains were lost at follow-up. ",
"For self-reported PTSD symptoms, a significant time by treatment condition interaction emerged, suggesting that participants who received PE had both greater decreases in symptoms and a greater rebound in self-reported PTSD symptoms than those who received RT. ",
"Unlike PTSD symptoms, depression symptoms neither changed nor were moderated by treatment condition from pre-treatment to follow-up. ",
"For self-reported PTSD and depression symptoms assessed at each session, time significantly predicted symptom reductions across psychotherapy sessions. ",
"PE and RT are well-tolerated, feasible, and effective for older adults, though treatment gains were not maintained at follow-up. ",
"clinicaltrials.gov Identifier: NCT00539279."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0,
0,
0.008333333333333333,
0,
0.0049261083743842365,
0.009523809523809525,
0.0038314176245210726,
0,
0,
0.023255813953488372,
0
] | 0.003836 | 5 |
[
"Thursday, December 25, 2008\n\nTuesday, December 23, 2008\n\nHitting level 76 has turned the old level 70 daily quests green, which reduces the XP by 14,000 each day (including the Shattrath cooking daily, which I can't always do anyway). ",
"I've done all the Sholazar Basin quests that I can do, and the dailies are still not available. ",
"El of El's Anglin' left a comment on my previous post that some new pacifistic dailies should open up at level 77. ",
"Until then, I think I'll try to maximize my XP by skipping the fishing daily (0 XP) and re-doing the Troll Patrol daily to do the Alchemist's Apprentice four extra times (five if I can't do the Shattrath cooking daily that day). ",
"That will keep my total number of dailies to 25 per day (in case the Troll Patrol trick is considered an exploit) and get me the most XP each day. ",
"Of course, if I can't do all the Troll Patrol subquests, I'll just substitute six more Alchemist's Apprentice quests.",
"\n\nMonday, December 22, 2008\n\nMade it to level 76 by fishing up my hundredth Strange Engine Part and doing the Alchemist's Apprentice over & over because I couldn't do all of today's Troll Patrol subquests. ",
"I also did Digging Through Bones (which I had tried once before, long ago) and I didn't have to do anything to help. ",
"Unfortunately, the quest was gray so I only got 1100 XP. ",
"I've also done over 500 daily quests now.",
"\n\nWowhead says there are two new daily quests I can do at level 76, but I think they're behind some level 77 prerequisites, since the questgivers have no \"!\"...",
"\n\nMonday, December 15, 2008\n\nOK, I managed to do the first timed Troll Patrol (all the quests were non-violent ones) and did it just in time to get the Congratulations! ",
"bonus, which gave me even more XP than wowhead claims (31,750; may vary by level). ",
"The next day I couldn't do two of the four subquests, but after doing all the other dailies I went back and discovered I could drop & re-get Troll Patrol and do the dailies again! ",
"The subquests don't change, but I can always do at least two of the four subquests, each of which is worth 5300 XP. ",
"I mostly did the Alchemist's Apprentice to practice it.",
"\n\nI might skip the fishing daily, because it's worth 0 XP, and I can substitute a Troll Patrol subquest. ",
"The subquests are worth the least of my dailies, so I won't drop any others yet. ",
"I'll do the fishing daily again when I hit 80.",
"\n\nI'll try to do Troll Patrol last, so I know exactly how many dailies I have left (sometimes I can't do the Shattrath cooking daily, and it's possible I won't be able to get Venture Bay to be Alliance so I can do Riding the Red Rocket). ",
"I can discover if I can do all four subquests (a 44.4% chance), and do the subquests until I have 6 dailies left, which should be just enough to do all of Troll Patrol and Congratulations (assuming I can do them fast enough). ",
"Of course, if I can't do all four subquests, I can just do some of them over & over until I hit 25 dailies done.",
"\n\nWith this additional XP, I should be able to level up faster than one level per week.",
"\n\nUPDATE: It's even better than that, though this might be considered an exploit...\n\nIf you abandon Troll Patrol after having done some dailies, those dailies DON'T COUNT once you log off & on. ",
"While logged in, they count towards your maximum of 25 per day, but logging recalculates the totals in some way. ",
"The upshot is, I can do Troll Patrol subquests as many times as I want. ",
"However, each subquest is only worth about 1/15 of an XP bar, so I will probably only do it if I'm close to a new level. ",
"Addendum: it seems it's the \"duplicate\" dailies that don't count, i.e. if you do Alchemist's Apprentice the first time, it counts as a daily, but if you do it a second time, that one won't count if you log out and back in.",
"\n\nUPDATE TWO: Keep in mind that if you can't do the subquests fast enough to get the Congratulations! ",
"quest, just DON'T turn in \"Troll Patrol\" - drop it, (log off & on if you have fewer than 6 daily quest slots remaining), and take Troll Patrol again until you do them within the 20 minute limit.",
"\n\nUPDATE THREE: It appears the random subquests aren't independent; you get one of three random 'sets' of subquests, and luckily, the two quests I want to avoid are in the same set, so I have a 2/3 chance to do all the subquests instead of 4/9...\n\nSaturday, December 13, 2008\n\nNoor made it to level 74, which opens up the Troll Patrol dailies (luckily, I didn't have to do the quest to kill 10 scourge to open it, just the parachute one). ",
"I won't be able to do Troll Patrol every day, since one of the random quests is to kill some of the swamp elementals, and I'm not sure about Throwing Down, where you blow up spider tunnels (because a bunch of spiders come out and die after you do it). ",
"I did Throwing Down once as a non-daily to open up the daily quests themselves.",
"\n\nIf I can do all the dailies (including the Shattrath cooking daily) and do Troll Patrol fast enough to get the Congratulations! ",
"quest, I can get 318,550 XP each day (4 bars of XP at level 74). ",
"Without the cooking daily and only doing 2 of the Troll Patrol quests, I can get 247,650 XP (3.1 bars).",
"\n\nMonday, December 8, 2008\n\nWell, after trying to skill up skinning on Saturday morning/afternoon by only skinning found corpses for eight hours and getting to a skill level of just 77, I decided I had to get out my Deathknight to create some corpses (Maiev is still a pretty low population server). ",
"I only cheated in the old world until Noor was at 300 skinning. ",
"She needed to be at least 365 skinning (plus her +5 skinning enchant) to skin the Jormungar Tunnelers, which are level 73 and 74. ",
"She got from 300 to 365 in Outland without more cheating, as there were a lot of DKs out killing beasts and my fast roflcopter made it fairly easy.",
"\n\nSince I still can't take a grandmaster gathering skill without losing the still-buggy Gaining the Advantage daily, I decided to drop skinning and take up herbalism, and got that up high enough not to miss a daily.",
"\n\nTuesday, December 2, 2008\n\nLevel 72 opened up some quests that I could do in rather creative ways...\n\nNo Place to Run, while not a very pacifistic quest (since you're helping Serinar set up a trap to slaughter all the cultists), I was able to do with more help from Coopthar the paladin, healing the crystal until it was fully charged. ",
"The crystal aggros mobs, so my earlier tries using a target dummy wouldn't distract them long enough. ",
"But the joke's on Serinar -- cultists can still chase me beyond the limit where the destructive ward is supposed to stop them.",
"\n\nA similar quest, Mystery of the Infinite, has me and my future level 80 self defend an Hourglass of Eternity from dragon mobs. ",
"I wielded my Big Iron Fishing Pole and my future self (who apparently has given up pacifism) was able to take on all the mobs long enough to finish the quest, while I just healed and blinded.",
"\n\nFinally, after thinking I had done all the quests I could get from the goblins in north-central Dragonblight, I found out that I could do Shaved Ice entirely by mining Crystalline Ice Elemental corpses that other players left (in my case, a Blood Elf death knight)... and this brings me to my next weekend project.",
"\n\nIn order to continue to do Gaining the Advantage, I'll need to drop mining and get my skinning up high enough to avoid missing a day. ",
"I think on Saturday morning after doing Gaining the Advantage with mining, I'll switch to skinning. ",
"This will give me nearly two days to get my skinning up to at least 285 so I can skin the lowest-level (58) Hellfire Peninsula mobs using +5 skinning gloves by Sunday night. ",
"Then I can get my skinning up until I can do Soft Packaging, which I can do with just Master skinning in case Gaining the Advantage is still bugged and won't give the quest to Grandmasters.",
"\n\nAfter that, I think I'll just go right to Herbalism. ",
"Death Knights made releveling mining difficult since many of them seem to have blacksmithing/mining, so Herbalism (with +10 gloves) should be faster to level the next weekend.",
"\n\nI'll try to skill up skinning legitimately (by only skinning found corpses), but if I really have to, I can get out my death knight and create some..."
] | {
"pile_set_name": "Pile-CC"
} | [
0.00425531914893617,
0,
0.017391304347826087,
0.004366812227074236,
0.006802721088435374,
0.008547008547008548,
0.0048543689320388345,
0,
0,
0,
0,
0.011834319526627219,
0.012048192771084338,
0.005555555555555556,
0.008620689655172414,
0,
0.009523809523809525,
0,
0,
0.008403361344537815,
0.004424778761061947,
0,
0.011494252873563218,
0.005154639175257732,
0,
0,
0.008264462809917356,
0,
0.00980392156862745,
0,
0,
0.007936507936507936,
0,
0.007692307692307693,
0.015384615384615385,
0,
0,
0,
0.007692307692307693,
0,
0,
0,
0,
0,
0.007751937984496124,
0,
0.0031645569620253164,
0,
0,
0.005747126436781609,
0.005291005291005291,
0,
0,
0
] | 0.003741 | 5 |
[
"Pseudonocardia tropica\n\n{{Taxobox\n| name = Pseudonocardia tropica\n| domain = Bacteria\n| phylum = \"Actinobacteria\"\n| classis = Actinobacteria\n| subclassis = Actinobacteridae\n| ordo = Actinomycetales\n| subordo = Pseudonocardineae\n| familia = Pseudonocardiaceae\n| genus = Pseudonocardia\n| binomial = Pseudonocardia tropica| species = P. tropica| binomial_authority = Qin et al. ",
"2010\n| type_strain = CCTCC AA 208018, DSM 45199, YIM 61452\n| synonyms = \n}}Pseudonocardia tropica is a bacterium from the genus of Pseudonocardia which has been isolated from the stem of the tree Maytenus austroyunnanensis'' in Xishuangbanna in China.",
"\n\nReferences\n\nCategory:Pseudonocardineae\nCategory:Bacteria described in 2010"
] | {
"pile_set_name": "Wikipedia (en)"
} | [
0.013333333333333334,
0.00398406374501992,
0.013157894736842105
] | 0.010158 | 5 |
[
"Ride Sunday 2018\n\nWhen\n\nWhere\n\nOn Sunday, 3rd June 2018, tens-of-thousands of two-wheeled enthusiasts around the world will lace up their boots, strap on their helmets and ride together in aid of various charities that together, work tirelessly to improve the lives of so many.",
"\n\nBy joining Ride Sunday and selecting Prostate Cancer Foundation of Australia (PCFA) as your charity of choice, you can help fund life-saving prostate cancer research and provide vital support to the many men and their families affected by this devastating disease, before it's too late.",
"\n\nRide Sunday is open to all motorcyclists, whether you intend on riding solo, or in a group. ",
"It's free to sign up and you can choose to join an existing ride, or create your own.",
"\n\nNo matter who you are, or where you are, you can Ride Sunday in support of the many men and their loved ones affected by prostate cancer."
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0.006944444444444444,
0,
0,
0
] | 0.001389 | 5 |
[
"Mexico travel warning: 5 Mexican states get highest “do not travel” warning under State Department system\n\nMEXICO CITY — Five states in Mexico now have the sternest “do not travel” advisories under a revamped U.S. State Department system unveiled Wednesday, putting them on the same level as war-torn countries like Syria, Afghanistan, Yemen and Somalia. ",
"The five states are Tamaulipas on the U.S. border and Sinaloa, Colima, Michoacan and Guerrero on the Pacific coast. ",
"All the states are hotspots of drug cartel activity, either hosting trafficking routes or extensive drug-crop cultivation.",
"\n\nThe State Department had previously discouraged travel to all or part of those states but the new warnings are sterner, placing them on a level 4 warning, the highest level of potential danger.",
"\n\nMexico as a whole has a level 2 rating, meaning Americans should “exercise increased caution” because of concerns about crime. “",
"Violent crime, such as homicide, kidnapping, carjacking, and robbery, is widespread,” a travel advisory dated January 10 says.",
"\n\nBut an additional 11 Mexican states got a level 3 warning Wednesday, which urges people to “reconsider travel” there. ",
"Mexico has 31 states, half of which are now under level 3 or 4 warnings.",
"\n\nThose states where Americans are urged to reconsider travel include the State of Mexico — Mexico’s most populous state, which includes most suburbs of Mexico City — and Jalisco, home to the city of Guadalajara, the Puerto Vallarta resorts and the lakeside expat community of Chapala and Ajijic. ",
"But the travel advisory said there are “no restrictions on U.S. government employees for stays in … Guadalajara, Puerto Vallarta, Chapala, and Ajijic.”",
"\n\nMost of northern Mexico, including the border states of Chihuahua, Coahuila, Nuevo Leon and Sonora as well as Durango, Zacatecas and San Luis Potosi, are under level 3 warnings.",
"\n\nMexico’s federal tourism department was not immediately available to comment on the new warnings. ",
"But the government’s Mexico Tourism Board said in a statement that “Mexico’s major international tourism destinations have been explicitly listed as having no travel restrictions,” apparently a reference to major resorts like Cancun, Puerto Vallarta and Huatulco.",
"\n\nHowever, at least two Mexican resorts — Ixtapa-Zihuatanejo and Acapulco — are in a do-not-travel state, Guerrero, and last year, the State Department extended a total ban on personal travel by U.S. government personnel there.",
"\n\nU.S. personnel had previously been allowed to fly to Ixtapa, the last place in Guerrero where they had been allowed to go. ",
"Personal travel by land and to the resort city of Acapulco had already been prohibited.",
"\n\nThe no-travel states had mostly already lost much foreign tourism. ",
"Tamaulipas has long been riven by turf wars between rival drug cartels, and Sinaloa is home to the cartel of the same name. ",
"Michoacan was so dominated by a drug cartel that vigilantes took up arms in 2013 to drive them out.",
"\n\nPreliminary figures suggest Mexico saw a record number of murders last year, the BBC reports. ",
"The year that previously had the most homicides was 2011, when over 27,000 people were killed, according to official figures.",
"\n\nColima has seen homicides skyrocket in recent years due to the growth of the Jalisco New Generation drug cartel, and the state now has Mexico’s highest homicide rate, with 83.3 killings per 100,000 residents, according to figures for the first 11 months of 2017.",
"\n\nThe U.S. government did note its employees are allowed to go to the seaside city of Manzanillo, Colima.",
"\n\nThe state with second-highest homicide rate — 61.6 per 100,000 — was Baja California Sur, home to the twin resorts of Los Cabos. ",
"The state conserved its level 2 advisory, “exercise increased caution,” despite a series of shootouts and killings in recent months.",
"\n\nRising levels of violence have not so far affected Los Cabos, which saw a 16 percent increase in tourism arrivals and an 18 percent rise in hotel occupancy in 2017, said Rodrigo Esponda, managing director of the Los Cabos Tourism Board.",
"\n\nEsponda said local officials and tourism operators are investing in increased security, including camera systems and the construction of a new marine base.",
"\n\n“We are going to keep working very hard in 2018 to make sure that Los Cabos continues as a safe destination,” Esponda said.",
"\n\nSpeaking to local media earlier this week, Tourism Secretary Enrique De la Madrid said, “In my opinion, the most important challenge we have in the tourism sector are crime events occurring where they didn’t before, for example in Cancun, la Paz and Los Cabos.”"
] | {
"pile_set_name": "Pile-CC"
} | [
0.005633802816901409,
0.02586206896551724,
0,
0.005128205128205128,
0,
0,
0,
0,
0.010101010101010102,
0.006622516556291391,
0.0335195530726257,
0,
0.0076045627376425855,
0.013215859030837005,
0.016,
0,
0,
0.008064516129032258,
0,
0.010416666666666666,
0,
0.007575757575757576,
0,
0.015267175572519083,
0,
0.012605042016806723,
0,
0.016,
0.011406844106463879
] | 0.00707 | 5 |
[
"Src-like adaptor protein (Slap) is a negative regulator of mitogenesis.",
"\nThe Src-like adaptor protein (Slap) is a recently identified adaptor protein containing Src homology 3 (SH3) and SH2 domains. ",
"Slap is found in a wide range of cell types and was shown to interact with the Eck receptor tyrosine kinase in a yeast two-hybrid interaction screen [1]. ",
"Here, we found that Slap is expressed in NIH3T3 cells and could associate with the activated platelet-derived growth factor (PDGF) receptor. ",
"Using mutated versions of the PDGF receptor and phosphopeptide competition experiments, we determined that Slap has the highest affinity for the Src-binding site of the PDGF receptor. ",
"Our inability to produce cell lines that stably expressed Slap suggested that Slap inhibited cell growth. ",
"We further investigated this issue by transiently expressing Slap by microinjection. ",
"Overexpression of Slap by this method inhibited DNA synthesis induced by PDGF and serum, whereas overexpression of the adaptor proteins Grb2 and Shc did not. ",
"Finally, microinjection of a Slap antibody into NIH3T3 cells that had been stimulated with suboptimal doses of growth factors potentiated the effects of the growth factors. ",
"These data suggest that, unlike other adaptor proteins, Slap is a negative regulator of signalling initiated by growth factors."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0.006493506493506494,
0.02127659574468085,
0.016304347826086956,
0.018867924528301886,
0.011764705882352941,
0.012658227848101266,
0.011560693641618497,
0.007874015748031496
] | 0.01068 | 5 |
[
"ECE Professor Ryan Zurakowski & Fabian Cardozo\n\nElectrical & Computer Engineering professor Ryan Zurakowski along with student Fabian Cardozo are studying a more effective way to judge the effectiveness of drugs on AIDS patients by testing the “markers” in their blood."
] | {
"pile_set_name": "Pile-CC"
} | [
0.01858736059479554
] | 0.018587 | 5 |
[
"[Regulation of ursolic acid on TNF-alpha and collagen in silicotic rats].",
"\nTo explore the regulation of ursolic acid on the expressions of TNF-alpha and collagen on Silicotic Rats. ",
"Seventy-five Wistar rats (class SPF) were divided into there groups according to the randomized block design, namely, control, model, ursolic acid groups with twenty-five rats in each group. ",
"SiO2 powders (250 mg/kg) were douched in the trachea of rat to make the silicotic model in model and ursolic acid groups. ",
"Ursolic acid (40 mg/kg) was injected into stomach cavity in ursolic acid group from the second day of SiO2, while the rats in control group were given sodium chloride in the same condition for 28 consecutive days. ",
"All rats were put to death on the 7th,14th and 28th day. ",
"TNF-alpha contents in serum were detected by ELISA. ",
"Total collagen contents in lung tissue were determined by hydroxyproline kits. ",
"Collagen I and III in lung tissue were investigated by western blot technique. ",
"After four weeks of intervention, the contents of TNF-alpha in serum of the model group had rised, showing statistically significant difference in each time compared to those of the control group (P < 0.05). ",
"Ursolic acid had depressant effect on the contents of TNF-alpha (P < 0.05). ",
"Compared with control group, the content of total collagen was significantly improved in model group (P < 0.05). ",
"However, ursolic acid depressed the expression of total collagen protein compared to those of the model group (P < 0.05). ",
"The expression levels of collagen I and III in ursolic acid and model groups were similar with the total collagen. ",
"Ursolic acid could decrease the expressions of TNF-alpha and collagen in the process of silicosis fibrosis."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0.005235602094240838,
0.00819672131147541,
0,
0,
0.019230769230769232,
0,
0.012658227848101266,
0,
0,
0,
0,
0.008695652173913044,
0
] | 0.003601 | 5 |
[
"Treatment outcomes among women and men methamphetamine abusers in California.",
"\nThis prospective longitudinal study examined treatment outcomes among 1,073 methamphetamine-abusing patients (567 women, 506 men) from 32 community-based outpatient and residential programs in 13 California counties. ",
"Data were collected at intake and at 3 months and 9 months after admission. ",
"With one exception, improvements from baseline to follow-up were observed in all areas measured by the Addiction Severity Index for both women and men in either modality. ",
"Compared to men, women demonstrated greater improvement in family relationships and medical problems, and similar improvement in all other areas, despite the fact that more women were unemployed, had childcare responsibilities, were living with someone who also used alcohol or drugs, had been physically or sexually abused, and reported more psychiatric symptoms. ",
"Implications for service improvement are discussed."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0,
0.005847953216374269,
0,
0
] | 0.000975 | 5 |
[
"Wildlife\n\nYour childhood science textbooks lied to you — finches and giant tortoise aren’t the coolest species in the Galapagos Islands.",
"\n\nNow that’s not to say that these classic Galapagos species aren’t cool — both groups are undeniably awesome. ",
"But this archipelago has a lot more to offer a naturalist than just these often-hyped examples. ",
"Try these: Pink iguanas, giant daisy trees, lava-loving cacti, and painted locusts, anyone?",
"\n\nHere are our picks for awesome non-finch, non-tortoise species from The Enchanted Isles."
] | {
"pile_set_name": "OpenWebText2"
} | [
0,
0,
0,
0,
0
] | 0 | 5 |
[
"Q:\n\nServer software - Which to run as root?",
"\n\nHere is a list of my stack:\n\nNginx UWSGI Tomcat Solr Virtualenv Supervisor\n\nI'm trying to set up my server correctly and I am wondering which of these should be run as root and which should have their own user accounts. ",
"If they shouldnt run as user accounts, should every one have its own account, or should programs like Nginx and Uwsgi be grouped under one account called \"web\" for example.",
"\nAny feedback on this would be much appreciated!",
"\n\nA:\n\nI would not run any of those services as root. ",
"I personally use each of them under its own user account.",
"\nUse root only to perform installations or other maintenance tasks. ",
"But not to run any services or user programs.",
"\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0,
0.0045045045045045045,
0.011627906976744186,
0,
0,
0,
0,
0,
0
] | 0.001792 | 5 |
[
"Credit\n\nRestrictions:\n\nThis image may not be used to state or imply ROE endorsement of any company or product.",
"\n\nCaption\n\nStarfield with Antares (alpha Scorpii), a red supergiant star 500 light years from the Earth. ",
"The red supergiant phase is one of the final stages in the life of a normal star, when almost all its hydrogen content is burnt. ",
"In this phase the outer layers of the star are greatly inflated by a thin shell of hydrogen burning around a central core of helium. ",
"Antares has a diameter of roughly 1 billion km, a surface temperature of 3100K and a mass 10-15 times the solar mass. ",
"The globular star cluster NGC 6144 is at top right."
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0.02857142857142857,
0,
0,
0,
0
] | 0.004762 | 5 |
[
"Treating AIDS: dilemmas of unequal access in Uganda.",
"\nThe price of antiretroviral (ARV) medicines in Uganda has fallen dramatically in recent years and more people are under treatment. ",
"By mid-2003 it was estimated that 10 000 people were taking ARVs. ",
"Drawing on participant observation, qualitative interviews, work with key informants and document reviews, we seek to map out the channels through which ARVs are being made available to people and to describe and assess the social implications of the present system of distribution. ",
"Four channels of access to ARV medicines were common in mid-2003: (i) Medicines were provided free in structured research and treatment programmes funded by donors, but only to those who lived in a defined catchment area and met inclusion criteria. (",
"ii) Gazetted treatment centres provided drugs on a fee-for-service basis; these urban-based institutions account for the largest number of drugs dispensed. (",
"iii) Private practitioners, mainly based in Kampala, provided discrete treatment for those who could afford it. (",
"iv) Finally, medicines were 'facilitated' along informal networks, supplying friends and relatives on a less regular basis, sometimes for free, sometimes for cash. ",
"However, access to ARVs remains highly uneven. ",
"We argue that cheaper drugs make possible different kinds of access, different qualities of care, and a growing awareness of inequity. ",
"Because the price of drugs has fallen drastically, middle-class families now have the possibility of buying them. ",
"But this requires tough prioritising and many cannot follow the regimen regularly. ",
"Health workers must consider whether patients will be able to purchase the drugs or not. ",
"In a kind of popular social pharmacy, people assess who can and should and does get access to ARVs. ",
"Further research should examine the whole range of ARV access channels in different countries and the associated patterns of social differentiation and exclusions."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0.007575757575757576,
0,
0,
0.004,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.006134969325153374
] | 0.001181 | 5 |
[
"Keith Meyer\n\nKeith Meyer (born June 20, 1938) is an American speed skater. ",
"He competed in the men's 1500 metres event at the 1960 Winter Olympics.",
"\n\nReferences\n\nCategory:1938 births\nCategory:Living people\nCategory:American male speed skaters\nCategory:Olympic speed skaters of the United States\nCategory:Speed skaters at the 1960 Winter Olympics\nCategory:People from Geneva, Illinois"
] | {
"pile_set_name": "Wikipedia (en)"
} | [
0.02666666666666667,
0,
0
] | 0.008889 | 5 |
[
"The tax hikes endorsed by many Democratic presidential candidates will eliminate up to 413,000 jobs, according to a new analysis.",
"\n\nFormer vice president Joe Biden, Sen. Elizabeth Warren (D., Mass.), and other 2020 Democratic hopefuls have pledged to reverse President Donald Trump's 2017 tax cuts and hike corporate tax rates to as high as 35 percent to fund expensive government programs. ",
"Such policies will discourage corporate investment, thereby limiting growth in wages, employment, and productivity, according to an analysis from the Tax Foundation.",
"\n\n\"The corporate income tax is the most harmful tax to economic growth, so it shouldn't be seen as a good option for funding various policy proposals,\" Erica York, the economist who compiled the study, told the Washington Free Beacon.",
"\n\nThe 2017 tax cuts reduced corporate tax rates from 35 to 21 percent, helping spur a modest economic uptick. ",
"That policy has been a top target for Democrats seeking to run against Trump in 2020. ",
"Biden has suggested raising corporate tax rates to 28 percent, while Warren, Sen. Bernie Sanders (I., Vt.), ",
"and South Bend mayor Pete Buttigieg have advocated for returning the tax rate to 35 percent.",
"\n\nAll of those proposals would eliminate jobs and hurt the broader economy, according to the study.",
"\n\nBiden's corporate tax plan will cut the GDP by 1 percent and shave off 187,000 jobs from the U.S. economy. ",
"Meanwhile, the tax plan backed by Warren, Sanders, and Buttigieg would leave 413,000 people without work.",
"\n\nThe Tax Foundation analysis studies the negative economic effect of corporate tax hikes in isolation from how the additional government spending fueled by the higher tax revenue may affect the economy. ",
"York said that returning to pre-2017 tax levels will restore corporate tax burdens to exceptionally high levels compared with other advanced industrial countries.",
"\n\n\"When we had a 35 percent rate, we were an outlier amongst most of our competitors,\" she said. \"",
"We had one of the highest corporate tax rates in the industrialized world. ",
"Lowering it made us more competitive, raising it back up would be a step in the wrong direction.\"",
"\n\nDemocratic hopefuls have characterized the 2017 tax reform as a handout to the rich, despite the fact that four in five American taxpayers have received smaller tax bills since it passed. ",
"Warren has gone even further and called for an additional 7 percent tax on every dollar of profit above $100 million.",
"\n\nYork said such hikes would reverberate through the wider economy, because increasing corporate taxes would leave corporations less inclined to invest in the workforce or in future development.",
"\n\n\"Higher corporate income tax rates will mean some investments that would have been worthwhile under the old tax codes will no longer be worthwhile under the higher tax rate,\" York said."
] | {
"pile_set_name": "OpenWebText2"
} | [
0,
0.011494252873563218,
0.006060606060606061,
0.008547008547008548,
0,
0.011627906976744186,
0.037037037037037035,
0.010869565217391304,
0,
0.009174311926605505,
0.02857142857142857,
0,
0.006172839506172839,
0,
0,
0,
0,
0,
0,
0.0053475935828877
] | 0.006745 | 5 |
[
"1. ",
"Field of the Invention\nThe present invention relates to a digital signal processing for such displays as a liquid crystal display and plasma display, and more particularly to a y correction circuit and a xcex3 correction method where the xcex3 correction of digital video signals is executed by line graph approximation.",
"\n2. ",
"Description of the Related Art\nThe relationship of the luminance value (X) of the video signal to be input to the display to the output luminance (Z) of the display is called the xe2x80x9cxcex3 characteristicxe2x80x9d, and the characteristic of a cathode ray tube display, which now dominates the market, is approximated as Z=kxc2x7X{circumflex over ( )}xcex3 (xcex3th power of X). ",
"Therefore in NTSC type television broadcasting, the xcex3 characteristic of a cathode ray tube display as an image receiver is considered and video signals are transmitted after executing reverse xcex3 correction at xcex3=2.2.",
"\nIn the case of a liquid crystal display, on the other hand, the xcex3 characteristic is different from that of a cathode ray tube display, so if the television images and computer images created by a cathode ray tube display are displayed on a liquid crystal display, luminance reproduction will be distorted. ",
"To control the distortion of luminance, it is necessary to include a xcex3 correction circuit in the display so as to execute xcex3 correction on the input video signals at an optimum xcex3 value before displaying. ",
"However, a xcex3 value optimum for video signals is generally not always the same where xcex3=2.2 is used in the case of a television, and xcex3=1.0 is used to expressed the fine difference of gradation and colors when images are created by a personal computer. ",
"Also the xcex3 value is different depending on the cathode ray tube display, so in order to reproduce ideal luminance for computer images, it is necessary to execute xcex3 correction based on the xcex3 characteristic of the display used for image creation.",
"\nTherefore in order to display these input images with an optimum xcex3 characteristic, a line graph xcex3 correction circuit, which approximates the xcex3 characteristic curve with a plurality of straight lines and executes non-linear processing on digital video signals, is used, as stated in the Japanese Patent Laid-Open No. ",
"8-18826.",
"\nAlso as a xcex3 correction circuit control method, a display stated in Japanese Patent Laid-Open No. ",
"7-152347 exists, where a plurality of sets of xcex3 correction data, such as slopes and intercepts, are prepared and a switching operation is executed by input signals.",
"\nAlso as Japanese Patent Laid-Open No. ",
"10-145806 states, there is a display which holds the plurality of xcex3 correction data separately for each R, G and B, and white balance adjustment is executed by the xcex3 correction circuit at the same time as xcex3 correction.",
"\nA conventional type xcex3 correction method will now be explained with reference to FIG. ",
"2. ",
"In FIG. ",
"2, 220 is a storage device for storing xcex3 correction data corresponding to n number of xcex3 values, such as a first xcex3 correction data 221, a second xcex3 correction data 222 and so on until the nth xcex3 correction data 223. ",
"230 is a line graph xcex3 correction circuit. ",
"In the line graph xcex3 correction circuit 230, 231 is a decoder which outputs a signal to the line graph block for the input video signal data. ",
"232 is a slope data selector which outputs a slope data, 233 is an intercept data selector which outputs an intercept data, 234 is a multiplier, 235 is an adder and 236 is a limiter which executes limiter processing on the video data.",
"\n210 is a xcex3 conversion section. ",
"A xcex3 correction selection section 211 reads one data from the n number of re-xcex3 corrected data in the storage device 220 based on the xcex31 value which was input. ",
"The calculation section 212 calculates the slope and intercept of each straight line of the line graph in FIG. ",
"3 using the re-xcex3 corrected data which was read.",
"\nBy storing xcex3 correction data having a target xcex3 characteristic in the storage device 220 in advance, optimum xcex3 correction is possible for a video signal having a plurality of xcex3 characteristics. ",
"White balance can also be adjusted by preparing the configuration in FIG. ",
"2 for three colors: R, G and B.\nWith the conventional method, however, the correction data must be stored in the storage device for the assumed number of xcex3 characteristics and white balances, so an increase in the capacity of the storage device is indispensable to support the subtle changes of xcex3 values and white balance.",
"\nWith the foregoing in view, it is an object of the present invention to provide a display which implements subtle xcex3 correction, and digital white balance adjustment, as well as black level and contrast adjustment by digital signal processing using the above line graph xcex3 correction circuit and a small capacity storage device.",
"\nTo solve the above problems, the display of the present invention has a xcex3 correction circuit for correcting digital video signals by line graph approximation, where the storage device holds a xcex3 characteristics of the display device, and video signals being input are displayed at the desired xcex3 characteristics, black level, contrast and white balance using a micro-controller, which calculates the slope and intercept of the line graph by the xcex3 characteristic of the display device, data which was input by the xcex3 value input means, white balance input means, black level input means, and contrast value input means, and the ideal xcex3 curve, and sets the slope and intercept of the line graph to the xcex3 correction circuit.",
"\nA first aspect of the present invention is a xcex3 correction circuit for executing reverse xcex31 correction on pre-xcex31 corrected input video data and then executing xcex32 correction, comprising: means for setting a representative luminance value; primary conversion means for executing reverse xcex31 correction on the representative luminance value so as to generate a primary conversion value; secondary conversion means for executing xcex32 correction on the primary conversion value so as to generate a secondary conversion value; means for generating a slope and intercept of each straight line of a line graph which pole is the secondary conversion value; and line graph xcex3 correction means for executing xcex3 correction on the input video data by the line graph.",
"\nA second aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned primary conversion means receives a xcex31 value.",
"\nA third aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned primary conversion means receives a xcex31 value and a black level value.",
"\nA fourth aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned primary conversion means receives a xcex31 value and a contrast value.",
"\nA fifth aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned primary conversion means receives a xcex31 value, a red adjustment value, a green adjustment value and a blue adjustment value.",
"\nA sixth aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned secondary conversion means has a table denoting a relationship between pre- xcex32 correction and post xcex32 correction.",
"\nA seventh aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned y1 value is a xcex3 correction value for a cathode ray tube display.",
"\nAn eighth aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned xcex32 correction is a xcex3 correction for a liquid crystal display.",
"\nA ninth aspect is a xcex3 correction circuit according to the first aspect, wherein the above mentioned xcex32 correction is a xcex3 correction for a plasma display.",
"\nA tenth aspect is a xcex3 correction method for executing reverse xcex31 correction on pre-xcex31 corrected input video data and then executing xcex32 correction comprising steps of: setting a representative luminance value; executing a reverse xcex31 correction on the representative luminance value so as to generate a primary conversion value; executing xcex32 correction on the primary conversion value so as to generate a secondary conversion value; generating a slope and intercept of each straight line of a line graph which pole is the secondary conversion value; and executing xcex3 correction on the input video data by the line graph."
] | {
"pile_set_name": "USPTO Backgrounds"
} | [
0,
0,
0,
0.002617801047120419,
0.004424778761061947,
0,
0,
0.003816793893129771,
0,
0,
0,
0,
0,
0,
0,
0.011111111111111112,
0,
0.125,
0,
0,
0,
0,
0,
0.0058823529411764705,
0.009009009009009009,
0,
0,
0.013513513513513514,
0,
0,
0,
0.002564102564102564,
0.0064516129032258064,
0.0056179775280898875,
0.005681818181818182,
0.004310344827586207,
0.008849557522123894,
0,
0.005681818181818182,
0.006024096385542169,
0.0030959752321981426
] | 0.005455 | 5 |
[
"Rosai-Dorfman disease mimicking parasagittal meningioma: case presentation and review of literature.",
"\nSinus histiocytosis with massive lymphadenopathy was originally described by Rosai and Dorfman in 1969. ",
"It usually presents with bilateral painless cervical lymphadenopathy. ",
"In extremely rare circumstances, the CNS can be affected. ",
"Only 21 prior cases of intracranial involvement have been reported. ",
"A 33-year-old white male presented with a 2-week history of progressive cephalgia. ",
"The patient underwent MRI testing that revealed an enhancing mass in the right parasagittal region with associated edema. ",
"Preoperative diagnosis was right parasagittal meningioma. ",
"The patient underwent craniotomy with complete resection of the mass. ",
"Histopathology was compatible with Rosai-Dorfman disease (RDD). ",
"Rosai-Dorfman disease is rarely found intracranially; however, its ability to mimic meningioma as well as other pathologies underlines its importance. ",
"With so few reported cases of intracranial involvement, more experience will be necessary before this clinical presentation and prognosis can be clearly outlined."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0.01904761904761905,
0,
0.017241379310344827,
0,
0,
0,
0,
0,
0.015625,
0,
0
] | 0.004326 | 5 |
[
"Q:\n\nReact-Redux Uncaught Error: Expected the reducer to be a function\n\nThe github repo for this project: https://github.com/leongaban/react_starwars\nExpected\nMy basic app runs without errors, and in the chrome devtools Redux tab, I see a list of star wars characters\nResults\n\nThe main index.js\nimport React from 'react'\nimport ReactDOM from 'react-dom'\nimport App from './components/App'\nimport { createStore, applyMiddleware, compose } from 'redux'\nimport { Provider } from 'react-redux'\nimport thunk from 'redux-thunk'\nimport reducer from './reducer'\nimport { getCharacters } from './reducer/characters/actions'\n\nconst store = createStore(reducer, compose(\n applyMiddleware(thunk),\n window.devToolsExtension ? ",
"window.devToolsExtension : f => f\n));\n\nstore.dispatch(getCharacters())\n\nconst container = document.getElementById('app-container');\n\nReactDOM.render(\n <Provider store={store}>\n <App />\n </Provider>\n, container);\n\nMy src > reducer > characters > actions.js\nimport { API_URL } from '../../constants'\n\nexport const SET_CHARACTERS = 'SET_CHARACTERS'\n\nexport function getCharacters() {\n return dispatch => {\n fetch(`${API_URL}/people`)\n .then(res => res.json())\n .then(res => res.results)\n .then(characters => \n dispatch(setCharacters(characters))\n )\n }\n}\n\nexport function setCharacters(characters) {\n return {\n type: SET_CHARACTERS,\n characters\n }\n}\n\nreducer > characters > index.js\nimport { SET_CHARACTERS } from './actions'\n\nconst initialState = [];\n\nexport default (state = initialState, action) => {\n switch(action.type) {\n case SET_CHARACTERS:\n return action.characters;\n default:\n return state;\n }\n}\n\nreducer > index.js\nimport { combineReducers } from 'redux'\nimport characters from './characters'\n\nexport default combineReducers({\n characters\n})\n\nA:\n\ntry\nconst createFinalStore = compose(\n applyMiddleware(thunk)\n)(createStore)\n\nconst store = createFinalStore(reducer, initialState /* or {} */);\n\ni don't think the createStore redux api wants you to pass the middleware as the second argument. ",
"that should be the initial/preloaded state \nhttp://redux.js.org/docs/api/createStore.html\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.002785515320334262,
0.0020718232044198894,
0.01098901098901099
] | 0.005282 | 5 |
[
"Q:\n\nError Undefined offset\n\nI want insert following checkbox in a row from database table, but have error, how is fix it? (",
"I can not change 5 in name checkbox)\n<input type=\"checkbox\" name=\"checkbox_units_a[5][]\" value=\"tv\">\n<input type=\"checkbox\" name=\"checkbox_units_a[5][]\" value=\"radio\">\n\n$name_un = $this -> input -> post('name_units_a');\n$service_un = $this -> input -> post('checkbox_units_a');\n$data3 = array();\nforeach($name_un as $idx => $name) {\n $data3[] = array(\n 'name_un' => $name_un[$idx],\n 'service_un' => json_encode($service_un[$idx]), ); //This is line 210\n};\n$this -> db -> insert_batch('hotel_units', $data3);\n\nError:\n\nA PHP Error was encountered Severity: Notice Message:\n Undefined offset: 0 Filename: residence.php Line Number:\n 210\n\nMy output in var_dump from $name_un:\narray(1) {\n [0] = > string(6)\"accessories\"\n}\n\nMy output in var_dump from $service_un:\narray(1) {\n [5] = > array(2) {\n [0] = > string(15)\"tv\" [1] = > string(12)\"radio\"\n }\n}\n\nA:\n\nError says your offset is equal 0 propably your $idx == 0 and $service_un[$idx] == null ;)\nEDIT\nAs I thought $idx == 0 is causing undefined offset 0 in json_encode try to reindex your $service_un array (if you cannot change the 5 in HTML) to make it start with 0 example:\n$service_un = array_values ( $service_un ); // reindexing array\n\nit will look like:\narray(1) {\n [0] => array(2) {\n [0] => string(2) \"tv\"\n [1] => string(5) \"radio\"\n }\n}\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0,
0.0007547169811320754
] | 0.000377 | 5 |
[
"Metformin-inclusive therapy reduces the risk of stroke in patients with diabetes: a 4-year follow-up study.",
"\nMetformin is a kind of oral hypoglycemic agents commonly prescribed to patients with diabetes mellitus. ",
"Although past studies had proven its protective effect on cardiovascular risk and related mortality, the evidence of metformin on stroke prevention was still insufficient and conflicting. ",
"Our study randomly selected 14,856 patients with diabetes from the database provided by the Taiwan National Health Research Institute, and 2 cohorts were formulated according to whether metformin was in the prescription record. ",
"All cases were followed up for 4 years to track their stroke incidence. ",
"As a result, 701 (17.5%) of 3999 diabetic patients had stroke without metformin use, whereas 994 (9.2%) of 10,857 patients had stroke with metformin use. ",
"Cox proportional hazard regressions showed that the stroke hazard ratio (HR) of metformin was .383. ",
"After adjustment for the patients' age, gender, hypertension, atrial fibrillation, hyperlipidemia, coronary artery disease, and medications including antiplatelets, coumadin, statin, and estrogen use, the HR was still .468. ",
"Further stratified analysis revealed that metformin had more protective effect in the patients with higher risk of stroke. ",
"Therefore, metformin should be placed in priority when prescribing oral hypoglycemic agents for diabetic patients when considering stroke prevention according to our study."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0.009523809523809525,
0,
0.0043859649122807015,
0,
0,
0,
0.004464285714285714,
0,
0
] | 0.001837 | 5 |
[
"1966 Batman 1/6 Scale Figure by Hot Toys\n\nHot Toys brings us Batman based on the 1960's TV Series portrayed by Adam West.",
"\n\nPackaging : The outer box has a great pop art design. ",
"When you remove the slipcase, the inner box reveals a plastic tray that secures the figure and accessories. ",
"The part that the tray lays in also doubles as a two sided window backdrop display.",
"\n\nSculpt:The likeness of Adam West is near perfect. ",
"The ears on the cowl seem to point out a bit much, but that's my only complaint. ",
"The texture on the boots and gloves are really well done.",
"Paint : The paint application on the lower face plates are incredible. ",
"Clothing : The cape and bodysuit closely resemble the costume Adam West wore on the show. ",
"There are snaps on the back to position the cape differently.",
"Articulation : There are numerous points of articulation on this figure.",
"\n\nAccessories : .There are a huge amount of accessories included with this figure. ",
"There are thirteen different hands, three removable face plates, utility belt, shark repellent, bat radio, big bomb, figure stand and a batarang with rope.",
"\n\nPrice : This figure can be obtained for $204.99 from Sideshow . ",
"Payment can be made in two installments as well.",
"\n\nThe Verdict: This is the definitive Adam West Batman toy. ",
"The sculpt, costume, accessories and even the box are all incredible. ",
"The expressions on the face plates and the accessories included were perfect choices."
] | {
"pile_set_name": "Pile-CC"
} | [
0.024793388429752067,
0,
0,
0,
0.019230769230769232,
0,
0,
0,
0.022222222222222223,
0,
0,
0,
0,
0,
0,
0.03333333333333333,
0,
0
] | 0.005532 | 5 |
[
"MANILA (Reuters) - The Philippines said on Saturday it was planning to change the name of a stretch of water east of the country in a bid to highlight its sovereignty over the area, which was surveyed recently by a Chinese vessel.",
"\n\nA Chinese survey ship was tracked for several months late last year moving around Benham Rise -- declared part of the Philippines’ continental shelf in 2012 by the United Nations, stirring concern in Manila about Beijing’s possible intentions.",
"\n\nChina says the ship was simply passing through the area and was not engaged in any other activity, and the country’s foreign minister said last week China fully respects the Philippines’ maritime area rights over Benham Rise.",
"\n\nManila said on Saturday it wanted to rename the area, which is roughly the size of Greece and believed by some scientists to be rich in biodiversity and tuna, “Philippine Rise”.",
"\n\n“A motion has been made subject to the conduct of the requisite legal and logistical study to effect the change,” presidential spokesman Ernesto Abella said in a statement.",
"\n\nThe president ordered the foreign ministry and executive secretary’s offices to study changing the territory’s name to emphasize Philippine sovereign rights, he said.",
"\n\nTerritorial rows with China have usually centered on the South China Sea, west of the Philippines, a conduit for about $5 trillion of shipped goods annually. ",
"China lays claim to almost the entire South China Sea.",
"\n\nPresident Rodrigo Duterte, who has sought a warm relationship with Beijing following years of territorial spats under the previous president, ordered the navy to put up “structures” in Benham Rise to assert sovereignty over the area.",
"\n\nHe has come under pressure of late for what critics say is a defeatist stance, accusing him of turning a blind eye to China’s island-building in the exclusive economic zone of the Philippines and its activities at Benham Rise.",
"\n\nA Senate hearing this week revealed China sought permission two years ago to survey Benham Rise, but the request was rejected because it would not let Philippine experts take part."
] | {
"pile_set_name": "OpenWebText2"
} | [
0.004347826086956522,
0.00816326530612245,
0.004405286343612335,
0,
0.005747126436781609,
0.005952380952380952,
0,
0,
0.00851063829787234,
0.0043859649122807015,
0.01098901098901099
] | 0.004773 | 5 |
[
"Engineering\n\nDevelopment\n\nID:\n\n14502\n\nComments:\n\n637\n\nDate:\n\nFebruary 14th 2015\n\nDesign Notes: Rental Equipment Credits\n\nDesign: Rental Equipment Credits\n\nNote from the Chairman\n\nGreetings Citizens,\n\nLast week, Kotaku ran an article showcasing Arena Commander’s combat. ",
"While the focus was that dogfighting looks great in action, one other thing immediately stood out: the video was an Aurora pilot flying with a HOTAS and TrackIR setup… and he was schooling Hornet after Hornet! ",
"The video, embedded below, tells me that we’re on the right track in creating a skill based game where battles are decided by the abilities of the pilots just as much as the capabilities of their equipment.",
"\n\nOne of the design tenets of Star Citizen is that it needs to be a complete experience regardless of what you have purchased. ",
"Someone buying a starter package needs to have exactly as much potential as someone supporting development by pledging for a new ship or a purchasing a new weapon. ",
"I do not want to make a game where you feel compelled to spend anything but time to continue playing.",
"\n\nIn the Persistent Universe, that will be easy: you’ll earn credits flying missions, locating star systems, hauling goods, hunting bounties and through a dozen other activities. ",
"You’ll rapidly explore and expand your universe. ",
"But the pilots helping us develop our spaceflight in Arena Commander don’t have that ability… yet!",
"\n\nWith Arena Commander 1.1, we intend to roll out a system called REC, or Rental Equipment Credits. ",
"In essence, you will be able to earn credits to rent equipment to use in Arena Commander by playing Arena Commander’s ranked modes. ",
"We want to reward backers for their time spent with the early game with a more expansive experience, not prevent them from accessing things that will be readily available in the Persistent Universe. ",
"You’ll still be able to support the development of the game by pledging for ships and weapons, and these will carry across into the Persistent Universe but you will no longer have to contribute additional money to fly a different ship or use a new weapon; You’ll have a choice. ",
"Spend time playing the game, which will earn REC to be spent on ships and weapons for use in Arena Commander or pledge for these items, supporting the ongoing development and running costs of the game. ",
"It’s the classic time vs money proposition and either way you’ll be helping making Arena Commander and Star Citizen better! ",
"This is always how the final game was intended to work and we’ve decided that we need to bring this dynamic into play sooner than later!",
"\n\nStar Citizen designer Calix Reneau details the system below, which we believe is a winner. ",
"Now, we’d like your feedback: is this the right direction for the system? ",
"Where would you like to see it go, in the near term and the long? ",
"Letting players earn credits in-game is a no-brainer, but working out the specifics is going to be an ongoing process no different from how we are continuing to work out flight controls and weapon balance!",
"\n\nFrom my perspective, Arena Commander is shaping up to be a fantastic standalone experience, something backers will want to continue to play in their new sim pods even after the Persistent Universe launches. ",
"I’m grateful for the testing and support the community has provided to get us to this point. ",
"It’s something every one of you should be proud of… and now let’s make it even better!",
"\n\n— Chris Roberts\n\nDesign: Rental Equipment Credits\n\nDesign: Rental Equipment Credits\n\nWe’ve all been there: a budding young Aurora pilot loads up Arena Commander to play for the first time. ",
"That first battle isn’t a great success, but between the ‘First!’ ",
"medal bonus and a lucky missile, enough REC is earned that an F7C Hornet doesn’t seem that impossible to earn. ",
"A few matches later, the mechanics of space flight are starting to be understood, so the REC flows a bit easier and it isn’t much longer before even the Super Hornet is within reach. ",
"Diving all in for the big guns, our plucky hero has become death, destroyer of worlds… and it feels good! ",
"But as often happens, life gets in the way, and it’s over a week later before the Star Citizen logo comes back on screen. ",
"Imagine the surprise to find that Super Hornet still selectable in the Arena Commander start menu! ",
"A few matches later the rental use runs out, but it’s enough time to earn up the REC to take advantage that sweet renewal discount. ",
"Dogfighting is a great thrill of course, but eventually curiosity takes hold and the next rental is an M50. ",
"What a difference! ",
"Racing seemed so trivial before, but this thing can really fly. ",
"At this point, any new arrangement of ships or equipment is always attainable with just a little bit of focused play. ",
"Still, there was something special about that first Super Hornet, so maybe it’s worth keeping around…\n\nWhat are REC?",
"\n\nREC, or Rental Equipment Credits, is a virtual currency that can be spent in Voyager Direct or the mobiGlas virtual store to rent software licenses on any equipment that can be used in the Sim-Pod (future iterations of Arena Commander will be accessed through a pod added to your Hangar.) ",
"Players earn REC through success in Sim-Pod activities – Arena Commander, the upcoming FPS Module, etc – with additional medals providing bonus REC and recurring incentives for progression.",
"\n\nHow do I Earn REC?",
"\n\nREC is earned by playing competitive (ranked) multiplayer matches of Arena Commander, the FPS Module or any other Simulated Experience in Star Citizen. ",
"Each match awards REC from a common pot, awarding the greater share to the winners, which is then split among the members of the team based on performance. ",
"For example, in Arena Commander:\n\nSquadron Battle: – REC is earned based on the number of kills plus the k:d if positive, weighted against the ticket difference.",
"\n\nMurray Cup: – REC is earned based on the final position multiplied by the proximity to World Record (by ship), weighted against Personal Best (by ship)\n\nCapture the Core: – REC is earned based on personal score weighted against the score difference.",
"\n\nPlayers can also earn bonus REC from medals, which are awarded for various milestones – umber of kills earned, number of missiles dodged, total matches, etc. ",
"These medals track over multiple matches and are recurring, so players can always make some progression even if they only play a match or two here and there.",
"\n\nTo prevent exploitation, this pot is associated with a total score expected from all participants, and scales REC yield based on the total score actually achieved. ",
"It also scales based on the number of participants. ",
"For example, a REC pot of 12,000 with a total score of 100,000 will only split 9600 REC if the players only collectively manage to score a total of 80,000 points. ",
"That same game will only split 4800 REC if the 10 player match only gets 5 participants.",
"\n\nHow does the rental system work?",
"\n\nAll rented equipment is rented for one week of real time use – guaranteeing that you always get at least a little bit of time with the ship. ",
"Ships and weapons cost 10% of their UEC price, rounded up. ",
"Once an item has been rented, it can be renewed for another week at a 20% discount. ",
"Renewals can only be bought a week in advance. (",
"Remaining rental time is visible in your RSI account and within the Sim-Pod HoloTable)\n\nExamples:\n\n9-series Longsword: 4,000 UEC; 400 REC. (",
"320 REC renewal)\n\nM4A Laser Cannon: 8,000 UEC; 800 REC. (",
"640 REC renewal)\n\nHornet F7C: 140,000 UEC; 14,000 REC. (",
"11,200 REC renewal)\n\nRedeemer: 250,000 UEC; 25,000 REC. (",
"20,000 REC renewal)\n\nThere are also broader goals available for the REC endgame – for example, we envision building a Community Chest into which players can donate REC that builds towards permanent prizes and other gameplay experiences. ",
"Imagine having the option between renting new equipment or helping the community earn something together… it should make for an interesting challenge!",
"\n\nCurrent Balance Plan\n\nAveraged at 1800 REC per hour. ",
"Please note that this currently represents ‘back of the envelope’ thinking; actual balance will likely alter significantly after the system launches and we determine how players use the system. ",
"But the goal remains to create a system that makes renting additional equipment a fun challenge rather than an impossible chore.",
"\n\nShip Cost: at 10% UEC, F7C rents for 14,000 REC. (",
"7.7hr)\n\nWeapon Cost: at 10% UEC, M4A rents for 800 REC. (",
"26.6min)\n\nRenewal Discount: at 20%, F7C renews for 11200 REC. (",
"6.2hr)\n\nReal Time Rented: 1 week.",
"\n\nPayout Pot (Per Game Mode): 10k for full Squad Battle, split 70-30.",
"\n\nGame Modes that Yield REC\n\nSquadron Battle\n\nBattle Royale\n\nNew Horizons Raceway\n\nCapture the Core\n\nCan I use rented ships and equipment in non-ranked modes?",
"\n\nYes, and there will be no limit on such uses.",
"\n\nHow do I spend REC?",
"\n\nREC is spent on ship and equipment rentals or Sim Pod unlocks. ",
"This can be done from Voyager Direct, the mobiGlas, or the Sim Pod. ",
"Once a ship/item is rented, it can be renewed for 1 additional week at a discount.",
"\n\nAs the system is developed, REC can also be spent to unlock additions to your Sim Pod, such as aesthetic changes like different background images/theme in the menus, or gameplay changes like opt-in handicaps which can be taken to boost REC yield at the cost of additional gameplay restrictions. ",
"Finally, REC can also (eventually) be donated to the Community Chest, a mystery box that results in Community Events when enough total REC is accumulated. ",
"Community Events benefit all players who log on while they are in effect.",
"\n\nWill my equipment carry over into the Persistent Universe?",
"\n\nNo, equipment rented in Arena Commander will be available only in your sim pod. ",
"When the Persistent Universe launches, you will have access only to upgrades purchased with UEC (or provided as gifts during the pledge campaign.) ",
"But we anticipate Arena Commander continuing to exist in the Star Citizen universe; it is, after all, a game played in SC’s future!",
"\n\nWhy have rental items instead of permanent ones?",
"\n\nWe know that choosing rental equipment instead of permanent AC-only unlocks will likely be the most controversial part of this plan. ",
"We’ve run through countless scenarios offering one, the other or a mix of both… and what we’ve decided is that at this time, rental is the way to go. ",
"What follows is the five key points we used to make this decision:\n\nArena Commander progression should NOT equal Persistent Universe progression. ",
"We believe strongly that players need to go out into the world to truly progress; a system that permanently rewards you for staying home and playing the simulator will reduce that experience in the long run.",
"\n\nIt is a buffer against exploits. ",
"We fully expect players to discover bugs and loopholes in the REC system… but with a one-week rental deadline, the most damage they can do to the game is unbalancing it for one week.",
"\n\nIt’s a first blush at future balance goals. ",
"The goal has always been to balance effort and convenience – whether you have time or money, you should be able to have a good experience. ",
"That’s going to take a lot of work to get right, so this is the earliest possible version of it to help us start to understand how this will work out in the wild.",
"\n\nIt allows for a tool for ‘directing’ testing. ",
"If we are going to be working on a particular ship or item, we can put it on sale – a tactic that wouldn’t affect players who had already unlocked it.",
"\n\nIt prevents stagnation during alpha. ",
"It should not ever be possible to “complete” (or win) this pre-alpha game.",
"\n\nWhat’s Next?",
"\n\nWe envision enhancing the REC system alongside everything else in Arena Commander going forward! ",
"In addition to building the system out based on player feedback, we have several systems we hope to deploy in future patches:\n\nDaily Challenges\n\nREC can be earned by completing personal and community daily challenges. ",
"Community challenges pay out at the end of the day if they have been fulfilled, and reward all players who have contributed to the challenge at all. ",
"These also have the benefit of allowing new gameplay features to be highlighted.",
"\n\nPersonal Challenge Examples\n\nGet 10 kills at 50% throttle\n\nDodge 10 missiles without using countermeasures.",
"\n\nCapture the Core 10 times.",
"\n\nCommunity Challenge Examples\n\nKill 3000 vanduul in Coop Swarm\n\nComplete 500 laps in Old Vanderval\n\n5000 total kills.",
"\n\nCommunity Chest\n\nThe Community Chest accumulates with REC donated to it until it reaches 100%, when it activates a surprise Community Event that is available to all players during its duration (which can be set individually for each event, but which would generally last 2 weeks). ",
"When it reaches 100%, a Community Event begins and is available to all players during its duration.",
"\n\nDiscounts / Bonus Yield\n\nREC discounts on specific ships and equipment can be used to encourage users towards their use. ",
"Ships can also give bonuses to REC as additional incentive.",
"\n\nWe are hoping to launch the REC system with Arena Commander’s next major patch, currently scheduled to be AC 1.1. ",
"This will allow backers to earn REC and spend them on store items. ",
"Additional functionality will be added as we go, just like the rest of the AC development process! ",
"Until then, please provide your feedback below. ",
"You’re helping us build Star Citizen AND Arena Commander, and we want to know what you think!",
"\n\nDISCLAIMER\n\nAs with any design post, this article represents our current thinking for the systems covered. ",
"All content is subject to change as a result of testing. ",
"The numbers provided here represent data that is current as of publication, but which will likely be altered in a number of ways before launch to create the best possible Star Citizen experience."
] | {
"pile_set_name": "Pile-CC"
} | [
0.007407407407407408,
0.004761904761904762,
0,
0.007874015748031496,
0,
0,
0,
0,
0,
0.01,
0,
0,
0,
0,
0.008064516129032258,
0,
0.010752688172043012,
0,
0,
0,
0,
0,
0,
0.010471204188481676,
0,
0,
0,
0,
0,
0.010101010101010102,
0,
0,
0,
0,
0,
0.008620689655172414,
0.006872852233676976,
0,
0,
0,
0,
0,
0.00398406374501992,
0,
0,
0,
0,
0,
0,
0,
0,
0.03389830508474576,
0,
0,
0.02142857142857143,
0.017543859649122806,
0.017857142857142856,
0.017543859649122806,
0.004219409282700422,
0,
0,
0,
0,
0.019230769230769232,
0.017543859649122806,
0,
0,
0.014492753623188406,
0,
0,
0,
0.015384615384615385,
0,
0,
0.003367003367003367,
0.012903225806451613,
0,
0,
0,
0.006802721088435374,
0.022900763358778626,
0,
0.007407407407407408,
0,
0.00684931506849315,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.006711409395973154,
0,
0,
0,
0.00847457627118644,
0,
0,
0,
0.01694915254237288,
0.008620689655172414,
0,
0.010101010101010102,
0,
0.010752688172043012,
0,
0,
0.005128205128205128
] | 0.003435 | 5 |
[
"I’ve never been to Bear Creek Park – it’s one of those locations that photographer’s refer to, but I’d yet to have a chance to visit. ",
"The park, while beautiful holds nothing on this stunning family. ",
"I went to college with the mom –...\n\nI enjoyed capturing this sweet growing family this past fall, before the leaves turned colors and still in that sweet spot of Texas weather where it might be hot or cool, dry or humid. ",
"We visited Shady Oaks Park in Azle and captured their most current milestone: a new...\n\nBefore I ever became a professional, I shot here and there with my little Rebel. ",
"I say that because everyone does start somewhere. ",
"I wasn’t paid (because I wasn’t a legitimate tax-paying business!) ",
"but I captured my cousin’s wedding. ",
"He was the first...\n\nThis sweet momma drove all the way from Brownwood with her grandmother, husband and her son. ",
"We had a few moments with this precious little girl and captured just a few lovely images in the home studio. ",
"The moments were quick as there was a clock on the time we could...\n\nThis is quite possibly the sweetest love story I know. ",
"It’s a true romance story. ",
"They met on a train. ",
"In Europe. ",
"They both spoke little to none of the other’s language and they both drew on paper to communicate. ",
"They kept in contact. ",
"They dated across the...\n\nThis girl has a ring of love surrounding her – she has been prayed for and was the answer to years of prayers. ",
"She may be small, but she will be mighty. ",
"It was a true honor to capture this newest addition to Ryan and Nicole’s family. ",
"We captured her six...\n\nThis young man I saw grow up a little bit as a teacher – he was always the most wonderful young man in the hallways and worked hard. ",
"He’s off to Texas Tech in the fall to adventure in the same school his mom and dad met. ",
"It was a slightly warm day, but he...\n\nMr. Devon grew up in Fort Worth, played baseball and is a proud graduate of the class of 2016. ",
"Our session was split between two graduating young men who happen to also be friends. ",
"We captured him off of Lancaster near the T&P Train Station as well as at Tandy...\n\nEach summer, no matter the age of your children, there is a lull, a downtime from the daily educational endeavors grind that requires filling time and still engaging in learning. ",
"What’s great about summer, is it gives kids the opportunity to study one thing for...\n\nIn an age where everyone can do anything they want thanks to owning the tool and then watching 10 minutes of Youtube videos, we have people who believe they are makeup artists, but don’t wash their brushes cashing catastrophe, cake artists who still...\n\nCategories\n\nRecent Posts\n\nChristine Anne Peirce Coleman is a Dallas/Fort Worth family lifestyle photographer offering birth, celebration, wedding and portrait photography in greater Fort Worth including Weatherford, Aledo, Grapevine, Keller, Arlington, Burleson and travels as well to Katy, College Station and Austin."
] | {
"pile_set_name": "Pile-CC"
} | [
0.007462686567164179,
0,
0,
0.005917159763313609,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.024691358024691357,
0,
0.011363636363636364,
0.007462686567164179,
0,
0.0076045627376425855,
0.004552352048558422
] | 0.002877 | 5 |
[
"Thank you for your request.",
"\n\nYou will be notified via eMail when your request has been processed. ",
"You can \ncheck the progress of your request by doing one of the following:\nIn the SRRS application, click on Requests I Have Made. ",
"A listing of recent \nrequests will be appear sorted by resource name.",
"\nClick on the Request Link below. ",
"This is a link to information regarding your \nrequest.",
"\n\nBelow is a summary of your request.",
"\n\nRequested For: Phillip M Love/HOU/ECT\nRequest Date: 5/9/2000 9:09:07 AM\nRequest Type: Update Access\nRequest Link: \n\nRequested Resources\nO\n/ECT_Trading\n\n\nIf you have any questions, please contact Information Risk Management at \n35536"
] | {
"pile_set_name": "Enron Emails"
} | [
0,
0,
0,
0,
0,
0,
0,
0.008403361344537815
] | 0.00105 | 5 |
[
"Saturday, July 19, 2014\n\nBirthday Party Shopping at Smart and Final Grand Opening in Carmichael CA\n\nThis shop has been compensated by Collective Bias, Inc. and its advertiser. ",
"All opinions are mine alone.",
"\n\nMy family often shops at Smart and Final to buy warehouse size items like paper towels, snacks, household cleaning items and meat. ",
"We can get many club size items there without the membership fees of other places. ",
"The reason we like Smart and Final is we can shop for the bulk stuff but also smaller items like fresh produce and bakery goods. ",
"We are a family of 3, we do not need everything we buy to be in club store sizes so it is a nice mix of items. ",
"I am celebrating my birthday next week and needed to do some birthday party shopping. ",
"A new Smart and Final store opened near my mother in laws house so we decided to stop by the celebration and shop before heading to her home.",
"\n\nThe first thing we see when we enter the store is the produce department filled with lively colors. ",
"The prices I saw next and gasped! ",
"Only $1 each for my 3 favorite berries, in that moment we changed dessert plans from ice cream to mixed berries with cream. ",
"I always find good deals at Smart and Final but for the Grand Opening celebration in Carmichael CA we found super savings to celebrate.",
"\n\nEach birthday party in my family I am always the one who buys and puts up the party decorations so naturally I pick out my own. ",
"I think I would be shocked if I ever woke up and the house was already decorated for my birthday (hint hint to my husband). ",
"I like that I can find all the many types of products I need in one store, bulk or single items. ",
"My son spotted a giant bag of balloons for making balloon animals, being the fun Mom I am I got them and we plan to google how to make the animals for my birthday party.",
"\n\nAmong all the great deals we found there were little sample stations, coupon booklets being handed out and friendly staff restocking the shelves. ",
"It was crowded in the store but the lines were short and the deals were great. ",
"The location of the new Smart and Final we visited is 7223 Fair Oaks Blvd., ",
"Carmichael, CA 95608. ",
"There will be Grand Opening celebrations all weekend at this location until Sunday July 20th.",
"\n\nI enjoy shopping the bakery section at Smart and Final when we need a quick treat for a party or family gathering. ",
"The store carries a delicious lemon pound cake that has been the base for many desserts that my family has enjoyed. ",
"This trip we decided to get mini cupcakes for my birthday. ",
"This year I am wanting to keep my birthday simple and inexpensive. ",
"I think I have a smart start by the money I saved at the Carmichael Smart and Final Grand Opening celebration. #",
"ChooseSmart #CollectiveBias\n\nMy son and I had to \"test\" some of the cupcakes before we put away all the party decorations and food for later in the week. ",
"I liked that we were able to find the party decor, the meat for BBQ, the produce and drinks for the birthday party all in one place without the overwhelming size of a warehouse store.",
"\n\nIf you would like some recipe ideas for making some great desserts for your next birthday party you can find them at the Smart and Final community. ",
"To connect with Smart and Final you can follow them on twitter and facebook."
] | {
"pile_set_name": "Pile-CC"
} | [
0.005681818181818182,
0,
0.007518796992481203,
0,
0.007751937984496124,
0,
0,
0.0070921985815602835,
0,
0,
0,
0.007407407407407408,
0,
0,
0,
0,
0,
0,
0.013157894736842105,
0,
0.010752688172043012,
0.008547008547008548,
0,
0,
0,
0,
0.006493506493506494,
0.00546448087431694,
0.006666666666666667,
0.013157894736842105
] | 0.003323 | 5 |
[
"PLEASE INCLUDE NAME, EMAIL AND PHONE FOR INFORMATIONLooking for a great new home At The Lakes of Holland, we have beautifully renovated 1017 sq. ",
"ft. ",
"2 bedroom, 1.5 bathroom apartment that is just waiting for you. ",
"Enjoy the convenience of having your own private entry with a patio. ",
"This home comes complete with every major kitchen appliance, granite-like countertops, and modern wood vinyl floori...\n\nCheck out all this gym equipment that is available weights inversion table and everything you need for a home gym. ",
"$2,000 $100 $200 Nautilus Home Gym, treadmill power blocks weight system. ",
"Call or text .",
"\n\nIf you are looking to build a garage style functional fitness gym we have everything you would need We have new equipment from several top of the line manufacturers in stock and ready for pickup. ",
"From a personal garage gym to outfitting an entire facility we are here to help We are located just 15 minutes outside of historic downtown Charleston. ",
"Feel free to come by and check us out.",
"Please feel...\n\nIf you are looking to build a garage style functional fitness gym we have everything you would need We have new equipment from several top of the line manufacturers in stock and ready for pickup. ",
"From a personal garage gym to outfitting an entire facility we are here to help We are located just 15 minutes outside of historic downtown Charleston. ",
"Feel free to come by and check us out.",
"Please feel..."
] | {
"pile_set_name": "Pile-CC"
} | [
0.006896551724137931,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0.000493 | 5 |
[
"NC State had by far its best performance of the season in a 37-13 win over No. ",
"7 Clemson Saturday afternoon at Carter-Finley Stadium in Raleigh.",
"\n\nThe Wolfpack improved to 6-5 overall and have to defeat Clemson and Maryland next Saturday to become bowl eligible.",
"\n\nHere are the Keys To The Game - Revisited:\n\nSlow down the pace\n\nWhat we wrote: Clemson's offense will try and make the game a track meet, which won't work for NC State. ",
"The Tigers have scored at least 23 points in every game but one, which was the 31-17 loss to Georgia Tech. ",
"The Yellow Jackets jumped the Tigers early and never looked back in an emotional victory. ",
"NC State will need to follow the same script.",
"\n\nClemson has scored 35 or more points in eight of its nine wins, and NC State just doesn't have the big-play potential to get into a shootout.",
"\n\nWhat happened: NC State not only did that, they were able to keep Clemson's off the field thanks to a stout defensive performance and a time-consuming offense. ",
"The Pack had the ball for 36:29. ",
"That was the most State has had possession this year and conversely the second fewest Clemson had behind 21:00 against the Yellow Jackets.",
"\n\nExpose Clemson's rush defense\n\nWhat we wrote: Wake Forest rushed for 145 yards on 35 carries last Saturday thanks to the effort of running back Brandon Pendergrass, who tallied 134 yards and two touchdowns on 20 carries.",
"\n\nGeorgia Tech rushed for 383 yards on 67 carries in its victory over Clemson. ",
"UNC had to abandon the running game, but its one-two punch of Giovani Bernard and Ryan Houston combined for 107 yards and two touchdowns on 23 carries.",
"\n\nMaryland quarterback C.J. Brown had 162 yards and a touchdown on 22 carries, and Boston College's top two backs had 138 yards and a score on 30 carries.",
"\n\nNC State might not have a running quarterback like the Terrapins, or run the option like the Yellow Jackets, but it has been proven that teams can run on the Tigers this season.",
"\n\nWhat happened: The Pack had a solid effort rushing thanks to a committed rushing attack. ",
"The duo of junior James Washington and redshirt freshman Tony Creecy ran a combined 37 times for 151 yards and a score. ",
"State finished the game with 145 yards rushing as a team when sacks and kneel downs were included, but that still represented the Pack's fourth best rushing performance of the year.",
"\n\nGet a few lucky bounces\n\nWhat we wrote: The Wolfpack passing game is due for a big play or two Saturday, and the defense could use some turnovers in Tigers' territory.",
"\n\nRedshirt junior quarterback Mike Glennon has a big arm and is adept at rolling out and throwing downfield on the run. ",
"He'll need to take some chances to keep pace with Clemson's offense.",
"\n\nNC State sophomore cornerback David Amerson has 10 interceptions, and is one away from tying the ACC record, held by former UNC star cornerback Dre Bly. ",
"Whether it's Amerson or another NC State defender, the Wolfpack will need to create some turnovers and turn them into points.",
"\n\nWhat happened: Whether or not they were lucky bounces, State forced four turnovers and that was one of the key stats from the game. ",
"The Pack picked off a pair of passes, one of them by Amerson, and also picked up two fumbles. ",
"One was a lucky bounce. ",
"With State up 14-3, Clemson tried a reverse to freshman running back Mike Bellamy at its own 19, but the exchange was botched and redshirt junior linebacker Terrell Manning picked it up at the Clemson 18.",
"\n\nGive Glennon some time\n\nWhat we wrote: NC State has allowed 25 sacks this season, including three in the loss to Boston College last week. ",
"Glennon has also had several passes batted down at the line of scrimmage this week.",
"\n\nThe Wolfpack offense will need to protect Glennon against the Tigers fierce pass rush. ",
"The onus isn't just on the offensive line. ",
"Tight ends, running backs or fullbacks who are picking up the blitz, trying to knock down defenders or chipping away at them, will also need to come through and complete the block.",
"\n\nWhat happened: Glennon was sacked twice, but as the game developed, he began to get more time to throw the football and started playing more like the Glennon that State fans saw earlier this season.",
"\n\nMatchups to watch\n\nWhat we wrote: Clemson senior defensive end Andre Branch vs. NC State redshirt junior left tackle R.J. Mattes: This is the kind of matchup NFL scouts will love to watch. ",
"Branch is second on the Tigers with 60 tackles and leads the ACC with 8.5 sacks. ",
"He also has 12 quarterback pressures and one forced fumble. ",
"Mattes is coming off a foot injury and has allowed four sacks this season, one of which came when he was hobbled vs. UNC.",
"\n\nWhat happened: Branch had a solid afternoon, making nine tackles, including 1.5 for loss, and pressuring the quarterback twice, but Branch was not a thorn in State's side for the afternoon either.",
"\n\nWhat we wrote: NC State cornerbacks Amerson and Rashard Smith vs. Clemson's wide receivers: The two NC State cornerbacks will be running all over the field against Clemson's passing attack, which threw the ball 44 times last week against Wake Forest. ",
"Amerson and Smith will get matched up against sophomore wide receiver DeAndre Hopkins, junior wide receiver Jaron Brown, and if he plays, sensational freshman Sammy Watkins, who is listed as \"questionable\" with a shoulder injury.",
"\n\nWhat happened: Redshirt junior corner C.J. Wilson actually got the start over Smith, but Wilson gave up a long 43-yard completion to Hopkins on one play. ",
"Overall though State held its own, limiting Clemson's big plays. ",
"It helped that Watkins did not play.",
"\n\nStats to watch\n\nWhat we wrote: 5: Number of first quarters this season where the Wolfpack haven't scored a point.",
"\n\nWhat happened: Make it six.",
"\n\nWhat we wrote: 92: Difference in rushing yards for NC State when the Wolfpack play at home compared to away games. ",
"NCSU averages 148 rushing yards per game at home, but 56 in road contests.",
"\n\nWhat happened: Par for the course at home, NC State ran for 145 yards, 89 more their road average.",
"\n\nPlay of the game: State redshirt freshman defensive end Art Norman nailed Boyd from behind at the Clemson 10, forcing a fumble that was picked up by Manning and returned to the Clemson 6, setting up State's second touchdown of the game and giving the Pack a surprising 14-3 second quarter lead with 10:33 left before halftime.",
"\n\nRun of the game: It came on a pass, but it deserves kudos for the run after the catch. ",
"Palmer caught a flair pass from Glennon at the 43, sprinted down the line and then escaped about six potential tackles on his way for a 43-yard touchdown with 5:41 left in the third quarter to give the Pack a 37-6 lead.",
"\n\nPass of the game: NC State took a 7-3 lead with 11:35 left in the first half when Glennon whizzed a pass over a Clemson defender's hand that fifth-year senior tight end George Bryan caught and then tight-roped the sideline for an 11-yard score.",
"\n\nHit of the game: Norman's sack was a nice hit as well.",
"\n\nSpecial teams play of the game: Graham nearly returned a punt for a touchdown, returning it 34 yards to the Clemson 6. ",
"It would have been Graham's fifth career return for a score if he had kept his balance along the sideline.",
"\n\nOffensive player of the game: Glennon did not play a perfect game, but he completed 19 of 29 passes for 253 yards and three scores without an interception and managed the game fairly well.",
"\n\nDefensive player of the game: Manning was all over the place for NC State. ",
"He was credited with eight tackles, including 3.5 for loss and assisting on a sack, forced a fumble, recovered a pair and also broke up a pass.",
"\n\nSpecial teams player of the game: Freshman kicker Niklas Sade had his best game kicking. ",
"He made all three field goal attempts, the longest being from 32 yards."
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0.015384615384615385,
0.008547008547008548,
0.005847953216374269,
0.018691588785046728,
0.011111111111111112,
0,
0,
0.006172839506172839,
0,
0.021739130434782608,
0.013513513513513514,
0.012658227848101266,
0.019867549668874173,
0.012987012987012988,
0.01675977653631285,
0,
0.016666666666666666,
0.0055248618784530384,
0.005917159763313609,
0.008333333333333333,
0.014705882352941176,
0.01935483870967742,
0.016,
0.007462686567164179,
0.010638297872340425,
0,
0.014705882352941176,
0.014184397163120567,
0,
0.02247191011235955,
0,
0,
0.005,
0.015706806282722512,
0.024691358024691357,
0,
0.008264462809917356,
0.015151515151515152,
0.019762845849802372,
0.017467248908296942,
0.01282051282051282,
0.03076923076923077,
0.027777777777777776,
0.008695652173913044,
0,
0.008547008547008548,
0,
0,
0.009146341463414634,
0,
0,
0.008130081300813009,
0.017857142857142856,
0.008264462809917356,
0.009433962264150943,
0,
0,
0,
0.01098901098901099,
0
] | 0.009471 | 5 |
[
"Q:\n\nCan't login on SO/SU/SF/... with stackexchange openID\n\nI have a stackexchange openID account and a myOpenID account, as well. ",
"I never managed it to login with the stackexchange, while myOpenID works.",
"\nI go to https://serverfault.com/users/login and click on the stackexchange picture and then nothing happens. ",
"There are the three pulsing dots, which just pulse forever.",
"\nI am using Chrome on Windows 7 (different computers). ",
"When I was using internet explorer just to give it a try there, I got a certificate error (\"Internet Explorer blocked this webpage from displaying content with security certificate errors.\"). ",
"When I click on \"Show content\" it works. ",
"So it seems that Chrome hides this error from me.",
"\nI am wondering, because Chrome is a widely used browser, but I was not able to find any related questions here on Meta, nor any advice how to deal with the problem.",
"\nI assume I could try to import the certificate from stackexchange.com, but is this the preferred way? ",
" \n\nA:\n\nI haven't been able to reproduce this using Internet Explorer, nor Firefox or Chrome.",
"\nThe *.stackexchange.com certificate appears as so to me:\n\nHave your trusted root certificates been muddled with perhaps?",
"\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0,
0,
0.00909090909090909,
0,
0,
0,
0,
0.02040816326530612,
0.012121212121212121,
0,
0.021739130434782608,
0,
0
] | 0.004874 | 5 |
[
"A systematic review of ETEC epidemiology focusing on colonization factor and toxin expression.",
"\nVaccine development for enterotoxigenic Escherichia coli (ETEC) is dependent on in-depth understanding of toxin and colonization factor (CF) distribution. ",
"We sought to describe ETEC epidemiology across regions and populations, focusing on CF and toxin prevalence. ",
"We conducted a systematic review of the published literature, including studies reporting data on ETEC CF and toxin distributions among those with ETEC infection. ",
"Point estimates and confidence intervals were calculated using random effects models. ",
"Data on 17,205 ETEC isolates were abstracted from 136 included studies. ",
"Approximately half of the studies (49%) involved endemic populations, and an additional 17% involved only travel populations. ",
"Globally, 60% of isolates expressed LT either alone (27%) or in combination with ST (33%). ",
"CFA/I-expressing strains were common in all regions (17%), as were ETEC expressing CFA/II (9%) and IV (18%). ",
"Marked variation in toxins and CFs across regions and populations was observed. ",
"These results demonstrate the relative importance of specific CFs in achieving target product profiles for a future ETEC vaccine. ",
"However, heterogeneity across time, population, and region, confounded by variability in CF and toxin detection methodologies, obfuscates rational estimates for valency requirements."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0.010638297872340425,
0.019230769230769232,
0.009174311926605505,
0.006134969325153374,
0,
0,
0,
0,
0.027522935779816515,
0,
0.007692307692307693,
0
] | 0.006699 | 5 |
[
"Gentlemen, Yes that means you, Eric. ",
" Please review the attached and give me \nany comments before I distribute to the group."
] | {
"pile_set_name": "Enron Emails"
} | [
0.02702702702702703,
0
] | 0.013514 | 5 |
[
"Mobile Software\n\nFrom apps for iPhone and Android to mobile websites, we'll help you reach your customers on any device.",
"\n\nYour brand’s presence on any device from any location\n\nBeyond robust functionality, we also understand how users think. ",
"Incorporating the latest research in digital psychology with superior architecture, we build mobile-first web applications with both power and usability in mind.",
"\n\nWith mobile digital time at a staggering 51% and desktop usage down to 42% of all users, the implications are clear - if you're not providing a satisfactory mobile experience you'll lose business to competitors that are.",
"\n\nOur Approach\n\nWe know how to think \"mobile friendly\" and how to translate that into reality. ",
"The Discovery and Architecture phase will involve deliberate discussion around how your users will interact with your site, products, and services on smaller devices. ",
"For some of our clients, this has meant putting on our \"Agile Development\" hat and launching progressive iterations of a fully featured iPhone app. ",
"For others, this has meant converting \"what works\" on a desktop to a variable method of feature interaction in a responsive version of their site.",
"\n\nOur Expertise Includes\n\nFully response and intuitive custom designs\n\niPhone and Android app development\n\nIntelligent Information Architecture\n\nE-Commerce shopping from any web-capable device\n\nContent and User migrations\n\nSocial Media integration\n\nand much more\n\nWhy Us?",
"\n\nThere are some things only a team of industry experts can do. ",
"We work hard to truly understand the intricacies of your business and how you operate. ",
"Specifically, we will look deeply into your mobile users and what they expect to see on your mobile system. ",
"This rich understanding allows us to develop a product with you that caters to your specific needs and goals to help you get ahead in the market.",
"\n\nHere’s some of what makes us stand out:\n\nPioneering iPhone and Android development using Drupal (the world's most powerful open-source CMS) as a framework.",
"\n\nCreated native iOS app to match and interact with an online commerce B2C marketing platform."
] | {
"pile_set_name": "Pile-CC"
} | [
0.016666666666666666,
0,
0,
0,
0,
0,
0,
0,
0.007380073800738007,
0,
0,
0,
0,
0.01910828025477707,
0
] | 0.002877 | 5 |
[
"---\ntitle: Search\ntaxonomy:\n category: docs\nprocess:\n twig: true\nnever_cache_twig: true\n---\n\nA search box is added to the top of the dropdown automatically for select boxes where only a single option can be selected. ",
"The behavior and appearance of the search box can be easily customized with Select2.",
"\n\n## Customizing how results are matched\n\nWhen users filter down the results by entering search terms into the search box, Select2 uses an internal \"matcher\" to match search terms to results. ",
"You may customize the way that Select2 matches search terms by specifying a callback for the `matcher` configuration option.",
"\n\nSelect2 will pass each option as represented by its [internal representation](/options) into this callback to determine if it should be displayed:\n\n```\nfunction matchCustom(params, data) {\n // If there are no search terms, return all of the data\n if ($.trim(params.term) === '') {\n return data;\n }\n\n // Do not display the item if there is no 'text' property\n if (typeof data.text === 'undefined') {\n return null;\n }\n\n // `params.term` should be the term that is used for searching\n // `data.text` is the text that is displayed for the data object\n if (data.text.indexOf(params.term) > -1) {\n var modifiedData = $.extend({}, data, true);\n modifiedData.text += ' (matched)';\n \n // You can return modified objects from here\n // This includes matching the `children` how you want in nested data sets\n return modifiedData;\n }\n \n // Return `null` if the term should not be displayed\n return null;\n}\n \n$(\".js-example-matcher\").select2({\n matcher: matchCustom\n});\n```\n\n>>>> `matcher` only works with **locally supplied data** (e.g., via an [array](/data-sources/arrays)! ",
" When a remote data set is used, Select2 expects that the returned results have already been filtered on the server side.",
"\n\n### Matching grouped options\n\nOnly first-level objects will be passed in to the `matcher` callback. ",
" If you are working with nested data, you must iterate through the `children` array and match them individually. ",
" This allows for more advanced matching when working with nested objects, allowing you to handle them however you want.",
"\n\nThis example matches results only if the term appears in the beginning of the string:\n\n<div class=\"s2-example\">\n <select class=\"js-example-matcher-start js-states form-control\"></select>\n</div>\n\n<pre data-fill-from=\".js-code-example-matcher\"></pre>\n\n<script type=\"text/javascript\" class=\"js-code-example-matcher\">\n\nfunction matchStart(params, data) {\n // If there are no search terms, return all of the data\n if ($.trim(params.term) === '') {\n return data;\n }\n\n // Skip if there is no 'children' property\n if (typeof data.children === 'undefined') {\n return null;\n }\n\n // `data.children` contains the actual options that we are matching against\n var filteredChildren = [];\n $.each(data.children, function (idx, child) {\n if (child.text.toUpperCase().indexOf(params.term.toUpperCase()) == 0) {\n filteredChildren.push(child);\n }\n });\n\n // If we matched any of the timezone group's children, then set the matched children on the group\n // and return the group object\n if (filteredChildren.length) {\n var modifiedData = $.extend({}, data, true);\n modifiedData.children = filteredChildren;\n\n // You can return modified objects from here\n // This includes matching the `children` how you want in nested data sets\n return modifiedData;\n }\n\n // Return `null` if the term should not be displayed\n return null;\n}\n\n$(\".js-example-matcher-start\").select2({\n matcher: matchStart\n});\n\n</script>\n\n>>> A [compatibility module](/upgrading/migrating-from-35#wrapper-for-old-style-matcher-callbacks) exists for using v3-style matcher callbacks.",
"\n\n## Minimum search term length\n\nSometimes when working with large data sets, it is more efficient to start filtering the results only when the search term is a certain length. ",
"This is very common when working with remote data sets, as it allows for only significant search terms to be used.",
"\n\nYou may set a minimum search term length by using the `minimumInputLength` option:\n\n```\n$('select').select2({\n minimumInputLength: 3 // only start searching when the user has input 3 or more characters\n});\n```\n\n## Maximum search term length\n\nIn some cases, search terms need to be limited to a certain range. ",
"Select2 allows you to limit the length of the search term such that it does not exceed a certain length.",
"\n\nYou may limit the maximum length of search terms by using the `maximumInputLength` option:\n\n```\n$('select').select2({\n maximumInputLength: 20 // only allow terms up to 20 characters long\n});\n```\n\n## Limiting display of the search box to large result sets\n\nThe `minimumResultsForSearch` option determines the minimum number of results required in the initial population of the dropdown to display the search box.",
"\n\nThis option is useful for cases where local data is used with a small result set, and the search box would simply be a waste of screen real estate. ",
"Set this value to -1 to permanently hide the search box.",
"\n\n```\n$('select').select2({\n minimumResultsForSearch: 20 // at least 20 results must be displayed\n});\n```\n\n## Hiding the search box\n\n### Single select\n\nFor single selects, Select2 allows you to hide the search box using the `minimumResultsForSearch` configuration option. ",
"In this example, we use the value `Infinity` to tell Select2 to never display the search box.",
"\n\n<div class=\"s2-example\">\n <select id=\"js-example-basic-hide-search\" class=\"js-states form-control\"></select>\n</div>\n\n<pre data-fill-from=\"#js-code-example-basic-hide-search\"></pre>\n\n<script type=\"text/javascript\" id=\"js-code-example-basic-hide-search\">\n\n$(\"#js-example-basic-hide-search\").select2({\n minimumResultsForSearch: Infinity\n});\n\n</script>\n\n### Multi-select\n\nFor multi-select boxes, there is no distinct search control. ",
"So, to disable search for multi-select boxes, you will need to set the `disabled` property to true whenever the dropdown is opened or closed:\n\n<div class=\"s2-example\">\n <select id=\"js-example-basic-hide-search-multi\" class=\"js-states form-control\" multiple=\"multiple\"></select>\n</div>\n\n<pre data-fill-from=\"#js-code-example-basic-hide-search-multi\"></pre>\n\n<script type=\"text/javascript\" id=\"js-code-example-basic-hide-search-multi\">\n\n$('#js-example-basic-hide-search-multi').select2();\n\n$('#js-example-basic-hide-search-multi').on('select2:opening select2:closing', function( event ) {\n var $searchfield = $(this).parent().find('.select2-search__field');\n $searchfield.prop('disabled', true);\n});\n</script>\n\nSee [this issue](https://github.com/select2/select2/issues/4797) for the source of this solution.",
"\n"
] | {
"pile_set_name": "Github"
} | [
0,
0.011904761904761904,
0.005208333333333333,
0.008064516129032258,
0.004370629370629371,
0.008264462809917356,
0,
0,
0,
0.00253324889170361,
0,
0,
0,
0.009615384615384616,
0,
0,
0.017857142857142856,
0.0036363636363636364,
0.010752688172043012,
0.002288329519450801,
0.00245398773006135,
0
] | 0.003952 | 5 |
[
"Media playback is unsupported on your device Media caption Donald Trump was booed by protesters while playing golf at Turnberry\n\nThousands took to the streets of Edinburgh in a third day of protests against Donald Trump's UK visit.",
"\n\nThey demonstrated as the US president played golf during a visit to his Turnberry resort on the Ayrshire coast.",
"\n\nMr Trump is in Scotland on a private visit with his family ahead of a summit with Vladimir Putin in Helsinki.",
"\n\nNo protesters were arrested in Scotland, but the Met Police said 12 people were detained following pro-Trump processions in London.",
"\n\nDuring his Saturday afternoon golf game he was booed by demonstrators gathered at the perimeter of the Turnberry complex.",
"\n\nEarlier, Police Scotland confirmed that they were investigating how a paraglider was able to fly over the resort hotel with a banner criticising Mr Trump.",
"\n\nThe president's visit to Scotland has been accompanied by a major security operation.",
"\n\nPolice snipers are positioned on tiers of temporary scaffolding overlooking the golf course, with a large number of other officers patrolling the grounds and surrounding area.",
"\n\nImage copyright Getty Images Image caption The march and protest in Edinburgh began at midday on Saturday\n\nImage caption The police estimated that 9,000 people joined the Edinburgh demonstration. ",
"Organisers said they believed some 60,000 had taken to the streets\n\nMr Trump travelled north of the border for the weekend after meeting Prime Minister Theresa May at Chequers and The Queen at Windsor on Friday.",
"\n\nSpeaking after talks with Mrs May, the president said a US-UK trade deal \"will absolutely be possible\". ",
"That came just hours after he told The Sun Theresa May's Brexit plan could kill an agreement.",
"\n\nMr Trump has often spoken of his love for Scotland and its people and had been a regular visitor to the country for many years, but this is his first trip since becoming the US leader.",
"\n\nSecretary of State for Scotland David Mundell greeted Mr Trump after his Air Force One jet landed at Prestwick Airport in Scotland on Friday evening.",
"\n\nThe American president did not meet Scotland's First Minister Nicola Sturgeon, however, she denied she had refused to see him.",
"\n\nAt a gay pride event in Glasgow on Saturday, she told the BBC: \"If the opportunity arises to meet the president I will do that and I'm sure if the opportunity arises in the future we will have lots to talk about, including the close and very important links between our two countries.",
"\n\n\"In democracies, it's also important to be able to focus also on where we perhaps disagree, and lots of people disagree with the policies of the Trump administration.\"",
"\n\nImage copyright Getty Images Image caption Donald Trump spent most of Saturday afternoon playing a game of golf at Turnberry, the course he bought in 2014\n\nImage copyright Getty Images Image caption Snipers are positioned around the resort, with a heavy police presence\n\nWhile Mr Trump was at Turnberry, police estimated that about 9,000 protesters walked from the Scottish Parliament in Edinburgh to the Meadows for a \"carnival of resistance\".",
"\n\nThe event featured the giant Trump Baby balloon, which was banned from both Turnberry and Holyrood.",
"\n\nTens of thousands of people - organisers claim up to 250,000 - protested in London on Friday against Mr Trump's UK visit.",
"\n\nDemonstrators also filled George Square in Glasgow ahead of his arrival in Scotland.",
"\n\nHowever, the UK government's International Trade Secretary Liam Fox told the BBC that the anti-Trump campaigners were \"an embarrassment to themselves\".",
"\n\nHe believed that the large scale protests in London and other cities \"did not reflect the genuine good manners and hospitality of the British people\".",
"\n\nAs well as the protests in Edinburgh and at Turnberry, there were supporters of Mr Trump at London's \"Welcome Trump\" and \"Free Tommy Robinson\" processions.",
"\n\nEnglish right-wing activist Robison is currently serving a 13 month jail sentence for potentially prejudicing a court case.",
"\n\nThe Met Police said that there were clashes during those processions and 12 people were arrested for a range of alleged crimes, including violent disorder, assault and and public order offences.",
"\n\nMedia playback is unsupported on your device Media caption Liam Fox: 'Trump protesters an embarrassment to themselves'\n\nMr Trump - whose mother was Scottish - and members of his family are spending the weekend at the hotel he bought in 2014 before departing on Sunday for the meeting with Russian leader Mr Putin in Finland the following day.",
"\n\nThe paraglider stunt, which happened shortly after Mr Trump and his entourage arrived at Turnberry, was reportedly staged by Greenpeace.",
"\n\nBen Stewart, a spokesman for the organisation, told the BBC: \"It wasn't dangerous at all. ",
"We let the police know about 10 or 15 minutes before that we were coming in.",
"\n\n\"We phoned them, we had someone on the police line who informed them. ",
"We thought it was important that the president actually saw a real-life protester.",
"\n\n\"There's tens, maybe hundreds of thousands of people on the streets around the UK.\"",
"\n\nThe police said no arrest had been made.",
"\n\nMedia playback is unsupported on your device Media caption Glider protester at Trump's resort\n\nPolice insisted they wanted to strike a balance between \"protection and public safety and the public's right to peacefully protest\".",
"\n\nHowever, Assistant Chief Constable Mark Williams said officers were committed to tracing the person who flew the aircraft.",
"\n\nHe added: \"There are armed assets protecting the president - both from the US Secret Service and ourselves and the Met Police, who offer a close protection function as well.",
"\n\n\"And there's no doubt anybody who breaches security around him puts themselves in grave danger.",
"\n\n\"On this occasion we could assess the situation and we realised there was no direct threat to the president however it's absolutely something that is very serious.\"",
"\n\nAs well as playing host to one of the world's most powerful men, Turnberry was the venue for a wedding on Saturday afternoon.",
"\n\nIt is understood that the bride and groom - as well as their guests and the humanist celebrant who conducted the service - had to be security checked ahead of their big day.",
"\n\nImage copyright EPA Image caption Supporters of Donald Trump gathered at the US Embassy in London on Saturday afternoon"
] | {
"pile_set_name": "OpenWebText2"
} | [
0.012987012987012988,
0.008849557522123894,
0.018018018018018018,
0.007518796992481203,
0.008130081300813009,
0,
0,
0,
0.005050505050505051,
0,
0.009433962264150943,
0.021505376344086023,
0.005376344086021506,
0.019867549668874173,
0.0078125,
0.0034965034965034965,
0.005917159763313609,
0.01569506726457399,
0.009900990099009901,
0.008130081300813009,
0.023255813953488372,
0.0196078431372549,
0,
0.006369426751592357,
0.008,
0.00510204081632653,
0.005813953488372093,
0.014492753623188406,
0.03260869565217391,
0,
0,
0,
0,
0,
0.004366812227074236,
0.008064516129032258,
0.011428571428571429,
0,
0,
0.007874015748031496,
0,
0.024793388429752067
] | 0.008083 | 5 |
[
"/*!",
"\n * define-property <https://github.com/jonschlinkert/define-property>\n *\n * Copyright (c) 2015-2018, Jon Schlinkert.",
"\n * Released under the MIT License.",
"\n */\n\n'use strict';\n\nvar isobject = require('isobject');\nvar isDescriptor = require('is-descriptor');\nvar define = (typeof Reflect !",
"== 'undefined' && Reflect.defineProperty)\n ? ",
"Reflect.defineProperty\n : Object.defineProperty;\n\nmodule.exports = function defineProperty(obj, key, val) {\n if (!",
"isobject(obj) && typeof obj !",
"== 'function' && !",
"Array.isArray(obj)) {\n throw new TypeError('expected an object, function, or array');\n }\n\n if (typeof key !",
"== 'string') {\n throw new TypeError('expected \"key\" to be a string');\n }\n\n if (isDescriptor(val)) {\n define(obj, key, val);\n return obj;\n }\n\n define(obj, key, {\n configurable: true,\n enumerable: false,\n writable: true,\n value: val\n });\n\n return obj;\n};\n"
] | {
"pile_set_name": "Github"
} | [
0,
0.017094017094017096,
0,
0,
0.021739130434782608,
0,
0.034482758620689655,
0,
0,
0
] | 0.007332 | 5 |
[
"Rep. Matt Gaetz Matthew (Matt) GaetzTrump faces tricky choice on Supreme Court pick Florida attorney general scrutinizing Bloomberg paying fines for felons to vote Lara Trump campaigns with far-right activist candidate Laura Loomer in Florida MORE (R-Fla.) said he and a group of lawmakers reviewed and spoke with President Trump Donald John TrumpFederal prosecutor speaks out, says Barr 'has brought shame' on Justice Dept. ",
"Former Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. ",
"1 Supreme Court pick MORE about the memo of his call with Ukrainian President Volodymyr Zelensky at the White House Wednesday.",
"\n\n\"I've had the chance to review the transcripts and chat with the president about them,\" Gaetz told MSNBC's Hallie Jackson.",
"\n\nGaetz and other Republicans got early peek at the WH call memo this morning with Trump pic.twitter.com/evMGaZXpCH — TPM Livewire (@TPMLiveWire) September 25, 2019\n\nWhen pressed by Jackson about who attended the White House meeting and whether the group included Republicans or Democrats, Florida representative said it was \"a group of legislators.\"",
"\n\n\"I didn't check anybody's voter registration card,\" Gaetz said.",
"\n\nThe representative defended the president saying no wrongdoing was committed in the phone call, in which he said the presidents commended each other for fighting corruption.",
"\n\n\"The bottom line here is: In this transcript there's no quid pro quo, there's no improper leverage and the overall tone of this transcript is that it's mutually laudatory,\" he said.",
"\n\nADVERTISEMENT\n\nThe White House released a memorandum of the phone conversation between Trump and Zelensky Wednesday after mounting pressure from Democrats.",
"\n\nThe release confirmed previous reports of the call that said the president asked Zelensky to \"look into\" former Vice President Joe Biden Joe BidenFormer Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. ",
"1 Supreme Court pick Bloomberg rolls out M ad buy to boost Biden in Florida MORE after withholding military aid from the country days earlier. ",
"This exchange has provoked debates over whether Trump offered a quid pro quo to the Ukrainian president.",
"\n\nThe call came to Congress's attention after the Trump administration refused to provide the House Intelligence Committee with a whistleblower report on the conversation that was subpoenaed. ",
"Media reports soon revealed the whistleblower report referred to this conversation.",
"\n\nSeveral Democratic House members have voiced their support for an impeachment inquiry following news of the conversation, which House Speaker Nancy Pelosi Nancy PelosiDemocratic senator to party: 'A little message discipline wouldn't kill us' Overnight Health Care: New wave of COVID-19 cases builds in US | Florida to lift all coronavirus restrictions on restaurants, bars | Trump stirs questions with 0 drug coupon plan Overnight Defense: Appeals court revives House lawsuit against military funding for border wall | Dems push for limits on transferring military gear to police | Lawmakers ask for IG probe into Pentagon's use of COVID-19 funds MORE (D-Calif.) announced Tuesday.",
"\n\n-- Updated at 2:38 p.m."
] | {
"pile_set_name": "OpenWebText2"
} | [
0.021176470588235293,
0.030534351145038167,
0.015873015873015872,
0.016129032258064516,
0.017142857142857144,
0,
0,
0,
0.01910828025477707,
0.021505376344086023,
0.02097902097902098,
0.009615384615384616,
0.015625,
0,
0.008771929824561403,
0
] | 0.012279 | 5 |
[
"Manage and secure multiple relational database platforms from a single user interface\n\nManage multiple platforms (Oracle, SQL Server, Db2, Sybase) from one UI\n\nProactively manage space, data and performance with built-in analytics\n\nEasily find and fix performance issues using intelligent diagnostics\n\nMove schema and table data across different platforms with ease\n\nProtect data security with view, grant and revoke of permissions\n\nManage Multiple Platforms with One Interface\n\nProactively manage all major DBMSs (Oracle, SQL Server, Db2 and Sybase) from a single common interface. ",
"Reduce training requirements and streamline collaboration among different teams across the organization. ",
"Manage multiple Oracle-specific schema object types and SQL Server advanced object properties such as temporal tables, in-memory tables, and natively compiled triggers, procedures and functions.",
"\n\nManage Space, Data and Performance Analytics\n\nComprehensive tools help you manage space, data and performance to keep your databases optimized and available. ",
"Manage the performance of your databases with a built-in process monitor that helps you understand who is connected to your database along with each user's current activity and session-related data. ",
"Sophisticated diagnostics help you pinpoint performance inefficiencies that result in poor space management, tracking key database metadata and performance metrics over time.",
"\n\nIdentify and Resolve Performance Issues\n\nView execution plans using a graphical and user-customizable display of the database’s internal processing order and priorities for queries. ",
"Probe into the performance of SQL statements by providing visibility and insight as to how syntax, cardinality, index usage and data skew affect and vary performance over time in a variety of situations.",
"\n\nSimplify Schema and Data Migrations\n\nAdvanced data management tools help to move schema and corresponding table data across the same or different platforms. ",
"The Migration Wizard guides administrators through a multi-step process that lets you select source and target data sources, select object types and specific objects to be migrated, specify a flexible set of migration options, and view progress of the operation.",
"\n\nManage Database Security Access and Permissions\n\nEstablish and maintain database security whether you are managing an existing production database or setting up a new environment. ",
"You'll find consistent support across all of the supported platforms as DBArtisan lets you view, grant, and revoke permissions or privileges. ",
"When you open a security-related object, such as user or role, the permissions tab lets you assign permissions on an object-by-object basis."
] | {
"pile_set_name": "OpenWebText2"
} | [
0.00686106346483705,
0,
0.005154639175257732,
0.00625,
0,
0,
0,
0.0049261083743842365,
0.006289308176100629,
0,
0,
0,
0
] | 0.002268 | 5 |
[
"Effect of some organic acids in the incubation step on plasma renin activity.",
"\nThe organic acids potassium hydrogen phthalate (KHphthalate), maleic, critic, acetic, formic, and oxalic were used to adjust the pH of plasma to 5.6 +/- 0.2 in the incubation step of the procedure for the determination of plasma renin activity (PRA). ",
"When compared to the PRA determined after pH adjustment with hydrochloric acid (HCl), it was found that KHphthalate and maleic acid produced a statistically significant increase in PRA. ",
"In comparison to HCl, KHphthalate and maleic acid showed a mean percent difference of 61% and 27% respectively for a one hour incubation; and, 57% and 34% respectively for a three hour incubation."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0.007936507936507936,
0.010752688172043012,
0.00510204081632653
] | 0.005948 | 5 |
[
"<!",
"DOCTYPE html>\n<html>\n<head>\n<title>NCD in Javascript</title>\n<meta http-equiv=\"Content-Type\" content=\"text/html; charset=UTF-8\" />\n</head>\n<body>\n\n<script src=\"emncd.js\"></script>\n\n<script>\n\nvar cfunc_emncd_start = Module.cwrap('emncd_start', 'null', ['string']);\nvar cfunc_emncd_stop = Module.cwrap('emncd_stop', 'null', []);\n\nfunction on_start ()\n{\n var code = document.getElementById('codetext').value;\n \n var loglevel = 0;\n for (i = 0; i <= 5; i++) {\n if (document.getElementById('loglevel' + i).checked) {\n loglevel = i;\n }\n }\n\n cfunc_emncd_start(code, loglevel);\n}\n\nfunction on_stop ()\n{\n cfunc_emncd_stop();\n}\n\nfunction load_example (num)\n{\n document.getElementById('codetext').value = document.getElementById('example' + num).value;\n}\n\n</script>\n\nThis is a quick port of my <a href=\"http://code.google.com/p/badvpn/wiki/NCD\">NCD programming language</a>\nto Javascript using the <a href=\"https://github.com/kripken/emscripten\">Emscripten</a> compiler.",
"\n<br />\n\nPlease open the Javascript console so you can see the output (Chrome: CTRL+Shift+J. Firefox: CTRL+Shift+K).",
"\n<br />\n\n<textarea id=\"example1\" style=\"display:none;\">\nprocess hello {\n println(\"hello, world\");\n exit(\"0\");\n}\n</textarea>\n\n<textarea id=\"example2\" style=\"display:none;\">\nprocess foo {\n println(\"Starting up, please wait...\");\n rprintln(\"Goodbye World!\");",
"\n\n sleep(\"500\", \"300\"); # sleeps 500ms on init and 300ms on deinit\n\n println(\"Hello World!\");",
"\n rprintln(\"Shutting down, please wait...\");\n}\n</textarea>\n\n<textarea id=\"example3\" style=\"display:none;\">\nprocess hello {\n var(\"0\") ctr;\n\n blocker() blk;\n blk->up();\n blk->use();\n\n num_modulo(ctr, \"3\") m_three;\n num_modulo(ctr, \"5\") m_five;\n num_equal(m_three, \"0\") d_three;\n num_equal(m_five, \"0\") d_five;\n and(d_three, d_five) d_three_five;\n\n If (d_three_five) {\n var(\"fizzbuzz\") text;\n } Elif (d_three) {\n var(\"fizz\") text;\n } Elif (d_five) {\n var(\"buzz\") text;\n } else {\n var(ctr) text;\n } branch;\n\n println(branch.text);\n\n num_add(ctr, \"1\") i;\n num_modulo(i, \"20\") i;\n ctr->set(i);\n\n sleep(\"100\", \"0\");\n blk->downup();\n\n}\n</textarea>\n\n<textarea id=\"example4\" style=\"display:none;\">\nprocess main {\n var({\"0\", \"1\", \"3\", \"2\", \"2\", \"3\", \"1\", \"1\", \"6\", \"end\"}) list;\n value([\"1\":\"one\", \"2\":\"two\", \"3\":\"three\"]) map;\n\n call(\"replace\", {\"_caller.list\", \"_caller.map\"}) replace;\n\n to_string(replace.result) str;\n println(str);\n exit(\"0\");\n}\n\ntemplate replace {\n alias(_arg0) list;\n alias(_arg1) map;\n\n value({}) new_list;\n Foreach (list As elem) {\n map->try_get(elem) y;\n If (y.exists) {\n new_list->insert(new_list.length, y);\n } Else {\n new_list->insert(new_list.length, elem);\n };\n };\n\n alias(\"new_list\") result;\n}\n</textarea>\n\n<textarea id=\"example5\" style=\"display:none;\">\nprocess hello {\n println(\"Hello, NCD in Javascript!\");",
"\n println(\"Will now wait 2 seconds.\");",
"\n rprintln(\"Goodbye.\");",
"\n\n sleep(\"2000\", \"3000\");\n rprintln(\"Waiting 3 seconds before dying.\");",
"\n\n call(\"func\", {\"FirstArg\", \"SecondArg\"});\n \n var({\"one\", \"two\", \"three\"}) list;\n to_string(list) str;\n println(str);\n \n Foreach (list As elem) {\n println(\"start \", elem);\n rprintln(\"stop \", elem);\n };\n\n println(\"We're finished. ",
"Press \\\"Request termination\\\" to unwind us.\");",
"\n rprintln(\"Terminating...\");\n}\n\ntemplate func {\n println(\"func here, my args are: \", _arg0, \" \", _arg1);\n rprintln(\"func going away\");\n}\n</textarea>\n\n<textarea id=\"example6\" style=\"display:none;\">\nprocess main {\n # Turing machine specification.",
"\n var(\"B\") blank;\n var([\n {\"0\", \"0\"}:{\"0\", \"0\", \"right\"},\n {\"0\", \"1\"}:{\"1\", \"x\", \"right\"},\n {\"1\", \"1\"}:{\"1\", \"1\", \"right\"},\n {\"1\", \"0\"}:{\"2\", \"0\", \"right\"},\n {\"2\", \"0\"}:{\"2\", \"0\", \"right\"},\n {\"2\", \"1\"}:{\"3\", \"1\", \"right\"},\n {\"3\", \"1\"}:{\"3\", \"1\", \"right\"},\n {\"3\", \"0\"}:{\"4\", \"1\", \"left\"},\n {\"3\", \"B\"}:{\"4\", \"1\", \"left\"},\n {\"4\", \"1\"}:{\"4\", \"1\", \"left\"},\n {\"4\", \"0\"}:{\"5\", \"0\", \"left\"},\n {\"5\", \"0\"}:{\"5\", \"0\", \"left\"},\n {\"5\", \"1\"}:{\"6\", \"1\", \"left\"},\n {\"5\", \"x\"}:{\"h\", \"x\", \"stay\"},\n {\"6\", \"1\"}:{\"6\", \"1\", \"left\"},\n {\"6\", \"x\"}:{\"0\", \"x\", \"right\"},\n {\"6\", \"0\"}:{\"0\", \"0\", \"right\"}\n ]) rules;\n var(\"0\") initial_state;\n var({}) initial_tape_left;\n var({\n \"1\", \"1\", \"1\", \"1\", \"0\", \"0\", \"1\", \"1\", \"1\", \"1\"\n }) initial_tape_right;\n\n # Perform the computation, stopping when no rule matches.",
"\n call(\"turing\", {blank, rules, initial_state, initial_tape_left, initial_tape_right}) results;\n\n # Print results.",
"\n to_string(results.tape_left) tape_left;\n to_string(results.tape_right) tape_right;\n to_string({results.side, results.pos}) head_pos;\n to_string(results.state) head_state;\n println(\"Tape L: \", tape_left);\n println(\"Tape R: \", tape_right);\n println(\"Head position: \", head_pos);\n println(\"Head state: \", head_state);\n\n exit(\"0\");\n}\n\ntemplate turing {\n alias(\"_arg0\") blank;\n value(_arg1) rules;\n alias(\"_arg2\") initial_state;\n alias(\"_arg3\") initial_tape_left;\n alias(\"_arg4\") initial_tape_right;\n\n # Head state.",
"\n var(initial_state) state;\n\n # Tape. ",
"Positions go like this: ... L2 L1 L0 R0 R1 R2 ... \n value(initial_tape_left) tape_left;\n value(initial_tape_right) tape_right;\n\n # Make sure each side of the tape has at least one symbol so we can flip easily.",
"\n tape_left->insert(tape_left.length, blank);\n tape_right->insert(tape_right.length, blank);\n\n # Head position.",
"\n var(\"right\") side;\n var(\"0\") pos;\n\n # Enter loop.",
"\n blocker() loop_blk;\n loop_blk->up();\n loop_blk->use();\n\n # Get symbol under head.",
"\n concat(\"tape_\", side) tape_name;\n alias(tape_name) cur_tape;\n cur_tape->get(pos) symbol;\n\n # Look for a matching rule.",
"\n rules->try_get({state, symbol}) rule;\n\n If (rule.exists) {\n # Extract directions from rule.",
"\n rule->get(\"0\") new_state;\n rule->get(\"1\") new_symbol;\n rule->get(\"2\") move;\n\n # Change head state.",
"\n state->set(new_state);\n\n # Replace symbol under head.",
"\n cur_tape->remove(pos);\n cur_tape->insert(pos, new_symbol);\n\n # Branch based on how we move.",
"\n strcmp(move, side) is_outside;\n strcmp(move, \"stay\") is_stay;\n strcmp(pos, \"0\") is_zero;\n If (is_outside) {\n # Increment position.",
"\n num_add(pos, \"1\") new_pos;\n pos->set(new_pos);\n\n # If the new position is out of range, extend tape.",
"\n strcmp(pos, cur_tape.length) need_extend;\n If (need_extend) {\n cur_tape->insert(pos, blank);\n };\n } elif (is_stay) {\n # Nop.",
"\n getargs();\n } elif (is_zero) {\n # Flip side, leave pos at zero.",
"\n side->set(move);\n } else {\n # Decrement position.",
"\n num_subtract(pos, \"1\") new_pos;\n pos->set(new_pos);\n };\n\n # Continue loop.",
"\n loop_blk->downup();\n };\n}\n</textarea>\n\nExamples:\n<button onclick=\"load_example(1)\">Hello World</button>\n<button onclick=\"load_example(2)\">Sleep</button>\n<button onclick=\"load_example(3)\">FizzBuzz</button>\n<button onclick=\"load_example(4)\">Replace</button>\n<button onclick=\"load_example(5)\">Foreach, call</button>\n<button onclick=\"load_example(6)\">Turing Machine</button>\n<br />\n\n<textarea rows=30 cols=80 id=\"codetext\">\nprocess hello {\n println(\"hello, world\");\n exit(\"0\");\n}\n</textarea>\n<br />\n\nLoglevel: \n<input type=\"radio\" name=\"loglevel\" id=\"loglevel0\"> None\n<input type=\"radio\" name=\"loglevel\" id=\"loglevel1\"> Error\n<input type=\"radio\" name=\"loglevel\" id=\"loglevel2\" checked> Warning\n<input type=\"radio\" name=\"loglevel\" id=\"loglevel3\"> Notice\n<input type=\"radio\" name=\"loglevel\" id=\"loglevel4\"> Info\n<input type=\"radio\" name=\"loglevel\" id=\"loglevel5\"> Debug <br />\n\n<button onclick=\"on_start()\">Start NCD</button>\n<button onclick=\"on_stop()\">Request termination</button>\n<br />\nNote: if you get the interpreter stuck and unable to terminate, just refresh the page\n\n</body>\n</html>\n"
] | {
"pile_set_name": "Github"
} | [
0,
0.004965243296921549,
0.008620689655172414,
0.003745318352059925,
0.020202020202020204,
0.004638833664678595,
0,
0,
0,
0.0036900369003690036,
0,
0,
0.0010741138560687433,
0,
0.005385996409335727,
0,
0,
0,
0,
0.010526315789473684,
0.015151515151515152,
0,
0,
0,
0.008771929824561403,
0.005847953216374269,
0.007407407407407408,
0,
0.010638297872340425,
0,
0,
0.0018099547511312218
] | 0.003515 | 5 |
[
"Q:\n\nКак изменить значение background-size с помощью jquery?",
"\n\nЕсть такой код:\n$('.header-x').css({'background-size': screenWidth + ' auto'});\n\nНо он не работает.",
"\nМожет кто поможет с правильным решением?",
"\n\nA:\n\nsetTimeout(() => {\r\n\r\n screenWidth = 50;\r\n document.querySelector('.header-x').style.backgroundSize = screenWidth+'px auto'\r\n\r\n}, 666);\n.header-x{\r\n width: 100px;\r\n height: 100px;\r\n border: 1px solid black;\r\n background-image: url(https://cdn2.iconfinder.com/data/icons/travel-roundline/512/wall-512.png);\r\n \r\n // Ваш косяк в том, что вы забыли единицы измерения.",
"\r\n // Так это не работает:\r\n background-size: 50 auto; \r\n}\n<div class='header-x'></div>\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0,
0,
0,
0.007957559681697613,
0
] | 0.001592 | 5 |
[
"Establishment of a basic method for manipulating preantral follicles: effects of retrieval method on in vitro growth of preantral follicles and intrafollicular oocytes.",
"\nThe aim of this study was to establish a basic manipulation protocol of preantral follicles for deriving developmentally competent oocytes. ",
"Primary, early and late secondary follicles retrieved from the ovaries of 14-day-old F1 (C57BL/6 x DBA2) female mice mechanically or enzymatically were cultured singly and in vitro growth of the follicles and maturation of intrafollicular oocytes were subsequently monitored. ",
"A mechanical method retrieved more (p < 0.0001) follicles (339 +/- 48 vs. 202 +/- 28) than an enzymatic method. ",
"However, the enzymatic method collected more singly isolated follicles that could be provided for subsequent culture (102 +/- 26 vs. 202 +/- 28). ",
"When an enzymatic method was employed, early and late secondary follicles required 9 and 6 days for reaching the maximal incidence of the pseudoantral stage. ",
"However, primary follicles were not possible to develop into the pseudoantral stage. ",
"The optimal duration of oocyte maturation from the onset of follicle culture was 7 days and 5-7 days for early and late secondary follicles, respectively. ",
"A general decrease in oocyte diameter (65.2-65.53 microm vs. 75 microm) and zona thickness (5.41-5.74 microm vs. 7.76 microm) was detected in in vitro-derived compared with in vivo-derived matured oocytes. ",
"Pronuclear formation was detected in 86-94% of mature oocytes after parthenogenetic activation and no significant difference was detected among groups. ",
"These results showed that preantral follicles retrieved by an enzymatic method underwent step-by-step growth in vitro, which could yield mature oocytes."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0.0036231884057971015,
0,
0,
0,
0,
0,
0.0048543689320388345,
0,
0
] | 0.000771 | 5 |
[
"Chlamydia pneumoniae infection and inflammation in adults with asthma.",
"\nChlamydia pneumoniae infection and immune response to the C. pneumoniae heat shock protein 60 (CpHsp60) have been suggested to be associated with asthma. ",
"To study whether a slightly elevated C-reactive protein (CRP) level as a marker of low-grade systemic inflammation has a role in this association, we collected serum and sputum samples from 103 asthma patients with disease severity ranging from mild to moderate and from 30 healthy volunteers. ",
"IgA and IgG antibodies to C. pneumoniae elementary bodies (CpEB) and CpHsp60 were measured by enzyme immunoassay. ",
"Serum CRP levels were measured with a rapid two-site ultra-sensitive assay based on time-resolved immunofluorometry. ",
"The asthma patients, especially those with moderate asthma, had higher serum IgA antibody levels to CpHsp60 than the healthy controls (test for trend, p = 0.05), whereas antibody levels to CpEB antigen did not differ between the study groups. ",
"CRP levels were higher in both asthma groups compared to the control group and moreover, the patients with moderate asthma had higher CRP levels than those with mild asthma (test for trend, p < 0.01). ",
"The subjects with a slightly elevated CRP level, defined as > or =1.8 mg/l, had higher CpEB IgA (p = 0.001), CpEB IgG (p = 0.008) and CpHsp60 IgA (p = 0.023) antibody levels in serum compared to the subjects with lower CRP levels. ",
"Slightly elevated CRP levels as a marker of low-grade systemic inflammation may be associated with C. pneumoniae infection in asthma patients."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0.003401360544217687,
0.017543859649122806,
0.008547008547008548,
0.00411522633744856,
0.009950248756218905,
0.021645021645021644,
0.007042253521126761
] | 0.008027 | 5 |
[
"To prepare, the Cavs, Heat and NBA provided extra security. ",
"Fans booed, yelled obscenities and some threw things, including one battery and cups filled with drinks. ",
"They held their signs: 'Quitness.' '",
"11-8? ",
"Looks like you left your talents in South…\n\nThe Pro Football Hall of Fame unveiled its 25 modern-era semifinalists for the Class of 2019 on Tuesday. ",
"The list includes three first-time candidates: cornerback Champ Bailey, tight end Tony Gonzalez and safety Ed Reed. ",
"The semifinalist group also…"
] | {
"pile_set_name": "Pile-CC"
} | [
0.03333333333333333,
0,
0,
0,
0.006711409395973154,
0.02586206896551724,
0
] | 0.009415 | 5 |
[
"Jean,Pretty unusual, but I guess it serves it's purpose when you are swooshing through the mud in search of food.",
"\n\nAunty,Actually, I am not much of a slaw fan. ",
"When I do eat it, I like it coarse, peppery, and not gooped up with mayo. ",
"Junior is totally addicted to a series of books by George R. Martin ... A Clash Of Kings, A Game of Thrones.",
"He is set to go as a character called Jon Longclaw. ",
"Shield is almost done, tonight I will put the straps on the backside.",
"\n\n\"Avocet is the Anglicized form of the Italian \"avocetta\" and \"avosetta,\" the origins of which are unknown. ",
"Newton, with great hesitation, suggests that the Italian is derived from the Latin \"avis,\" to which a diminutive suffix indicating gracefulness is added. ",
"This charming idea is not, however, generally accepted.\""
] | {
"pile_set_name": "Pile-CC"
} | [
0.008849557522123894,
0,
0.013513513513513514,
0.009259259259259259,
0.019230769230769232,
0.014492753623188406,
0.009174311926605505,
0,
0
] | 0.00828 | 5 |
[
"Blue-Eyes White Dragon – perhaps the most well-known cards in the game. ",
"With such power, it’s almost guaranteed to get support every few years. ",
"With The King of D. and Dragon Revival Rhapsody coming out on March 9th, it’s a perfect time to analyze the place of Blue-Eyes in the current climate.",
"\n\nWhat are the Blue-Eyes variants?",
"\n\nSynchro\n\nThe “classic” build, most similar to its heyday. ",
"It focuses on Blue-Eyes Spirit Dragon‘s ability to tag into Azure-Eyes Silver Dragon, protecting your high-damage boards. ",
"In conjunction with Azure-Eyes, Synchro variants can oppress Pendulum and Paleo match-ups. ",
"Synchro variants have ample amounts of grind game, with tons of revival spells and The White Stone of Ancient’s GY effect. ",
"Return of the Dragon Lords‘ GY effect can even cover Azure’s weakness to being run over by cards like Number S39: Utopia the Lightning – it doesn’t start a chain!",
"\n\nOther than Azure-Eyes, Blue-Eyes Spirit Dragon can tag into Ancient Fairy Dragon to Special Summon another Level 1 Tuner to Synchro for Crystal Wing Synchro Dragon. ",
"Even if Azure-Eyes is run over, you can Synchro Summon Cloudcastle to revive it, and XYZ Summon True King of All Calamities!",
"\n\nXYZ\n\nXYZ variants are explosive and dominating. ",
"By using Number 95: Galaxy-Eyes Dark Matter Dragon‘s mill effect, it can send Amorphage Goliath and Arkbrave Dragon, XYZ variants can steal games where the opponent does not draw removal. ",
"On your opponent’s Standby Phase, Arkbrave Dragon will revive Amorphage Goliath, meaning your opponent cannot summon monsters from the Extra Deck. ",
"Return of the Dragon Lords benefits from this variant as well, protecting Goliath from mass removal.",
"\n\nOther than Dark Matter, the Deck also has an easier time summoning big boss monsters like Number 38: Hope Harbringer Dragon Titanic Galaxy. ",
"It also has significantly less dead draws due to having a smaller amount of Normal Summons compared to the tuner-centric Synchro variant.",
"\n\nChaos MAX\n\nThe most offensive variant, Blue-Eyes Chaos MAX Dragon serves as both a pressure tool and a self-protecting boss monster. ",
"Unlike the other variant’s boss monsters, Chaos MAX does not occupy the Extra Monster Zone. ",
"In a high roll situation, Chaos MAX can be accompanied by Azure-Eyes or Number 38: Hope Harbinger Dragon Titanic Galaxy.",
"\n\nThe ease of summoning Chaos MAX is not to be understated. ",
"Most builds run four to five Ritual Spells, Manju of the Ten Thousand Hands, and can search Chaos MAX with The Melody of Awakening Dragon. ",
"However, this consistency is the most easily disrupted by Ash Blossom & Joyous Spring, as opposed to Synchro or XYZ variants. ",
"The latter is harder to stop with hand traps, but their combo pieces – Tuners and cards like Ancient Rules respectively – have to be drawn.",
"\n\nWhat are the Differences in Non-engine Cards?",
"\n\nWhile each Blue-Eyes variant has similar engine cards, they all have entirely different goals. ",
"Choose tech and removal with this in mind.",
"\n\nThe greatest boon of Synchro variants is its ability to hold on to boards, and continuously summoning Dragon Spirit of White. ",
"Thus, cards that may revive either card would benefit this game plan, such as Back to the Front. ",
"Cards that may send Ancients to the GY apply pressure and accelerate Dragon Spirit’s summon. ",
"On your turn, Dragon Shrine can achieve this. ",
"Phoenix Wing Wind Blast can send Ancients on the opponent’s turn, and can blow out Pendulum match-ups. ",
"This also solves having too many cards in hand that require a Normal Summon.",
"\n\nSpeaking of which, the Normal Summon in Blue-Eyes variants are greatly important. ",
"Clearly, in a Synchro variant, tuners are the required Normal Summon. ",
"This is important when considering support such as The King of D.\n\nXYZ variants rely heavily on dropping Goliath, but it needs to be able to protect it going first or second. ",
"Interrupted Kaiju Slumber is the perfect out to wide boards, and gives you a Level 8 to XYZ Summon with. ",
"Furthermore, the level 8 Kaijus can be discarded with Trade-In or The Melody of Awakening Dragon.",
"\n\nFoolish Burial Goods is an unorthodox choice, but it makes a package of advantage with The King of D. Both cards can use Metalfoes Fusion as a pseudo-Upstart Goblin. ",
"The King of D. fits snugly into this variant, replacing Tuners since the goal of the deck is to summon Dark Matter. ",
"The King of D. is especially great for the XYZ variant because it can search for The Flue of Summoning D. to complete a combo and unbrick an otherwise unsalvageable hand, or fetch combo pieces with The Melody of Awakening Dragon.",
"\n\nChaos MAX variants crave the ability to steal games. ",
"It can do so against Trickstar and Paleozoic with ample S/T removal. ",
"Specifically, the ability to bait out Scapegoat and Paleozoic summons is critical for the Deck, so cards like Twin Twisters are critical. ",
"It also wants to play mass “removal” like Swords of Concealing Light, which double as OTK tools in conjunction with Chaos MAX’s double piercing damage. ",
"Unfortunately, Manju of the Ten Thousand Hands is necessary for consistency in drawing Advanced Ritual Art, so The King of D. should not be played.",
"\n\nAn important fundamental in playing this deck is to be able to bait Ash Blossom & Joyous Spring. ",
"It can negate Advanced Ritual Art, leaving you defenseless, so it’s more optimal to let any other search card be negated before that happens. ",
"Unfortunately, Spell-wise, Pot of Desires is the best bait, being a high-priority target that doesn’t make you discard a card. ",
"Manju can also work if you already have a Ritual Spell and Chaos MAX in hand.",
"\n\nPSY-Framegear Gamma could be considered to negate hand traps. ",
"However, the consistency of Chaos MAX decks should not be overestimated; the deck has enough to consistently summon MAX, but just enough. ",
"Any more dead draws that don’t have synergy with Trade-In risks losing before the game begins.",
"\n\nEach Blue-Eyes variant has solid match-ups against the meta of today. ",
"Synchro variants can bully Pendulum decks by banishing scales and Azure-Eyes protecting the board. ",
"XYZ variants oppress SPYRAL and Trickstar by limiting their Extra Deck. ",
"Chaos MAX can unexpectedly swipe games from Trickstar and rogue decks. ",
"Right now, it would seem XYZ variants have the upper hand, but with a tri-deck format like this, the most popular deck changes frequently! ",
"The key to playing Blue-Eyes is to understand the purpose of each variant as we have here, and apply that logic to the current meta."
] | {
"pile_set_name": "OpenWebText2"
} | [
0.013888888888888888,
0,
0.006666666666666667,
0.029411764705882353,
0,
0.01639344262295082,
0.02197802197802198,
0.016260162601626018,
0.006172839506172839,
0.017964071856287425,
0.024193548387096774,
0,
0.010638297872340425,
0.02040816326530612,
0.01,
0.02112676056338028,
0.0072992700729927005,
0.014814814814814815,
0.010869565217391304,
0.016666666666666666,
0.016666666666666666,
0.014388489208633094,
0.007936507936507936,
0.014388489208633094,
0.02127659574468085,
0.010309278350515464,
0,
0,
0.010309278350515464,
0.021505376344086023,
0.021739130434782608,
0.009708737864077669,
0.013157894736842105,
0.011904761904761904,
0.014285714285714285,
0.005714285714285714,
0.009523809523809525,
0,
0,
0.017241379310344827,
0.004366812227074236,
0.01818181818181818,
0.028985507246376812,
0.028985507246376812,
0.006578947368421052,
0.006802721088435374,
0.010101010101010102,
0,
0,
0,
0,
0.007246376811594203,
0.010638297872340425,
0,
0.010101010101010102,
0.041666666666666664,
0.014084507042253521,
0,
0.007575757575757576
] | 0.011528 | 5 |
[
"1. ",
"Introduction {#sec1}\n===============\n\nAcute myocardial infarction (AMI) is one of the main causes of morbidity and mortality in coronary heart disease. ",
"Timely restoration of the blood flow (reperfusion) remains the mainstay of all current therapeutic approaches to rescue the ischemic myocardium. ",
"However, reperfusion may paradoxically exacerbate tissue injury, and this additional damage is called ischemia/reperfusion (IR) injury \\[[@B1]\\].",
"\n\nSignal transducer and activator of transcription (STAT) 3, a transcription factor that belongs to the STAT family, participates in a wide variety of physiological processes (e.g., proliferation and apoptosis) and protects the hearts against myocardial hypertrophy and myocardial IR injury \\[[@B2]--[@B4]\\]. ",
"STAT3 can be activated through phosphorylation at two residues: serine (Ser) 727 and tyrosine (Tyr) 705. ",
"When phosphorylated at Ser727, STAT3 translocates into the mitochondria and regulates mitochondrial integrity and biogenesis, resulting in the reduction of reactive oxygen species (ROS) production \\[[@B5]\\]. ",
"However, when phosphorylated at Tyr705, STAT3 translocates into the nucleus where it promotes transcription of cardioprotective genes and improves cellular antioxidant property \\[[@B6]\\]. ",
"Cardiac-specific STAT3 knockout mice demonstrated increased postischemic mortality and cardiac injury following myocardial IR \\[[@B7]\\]. ",
"Moreover, cardiomyocyte-restricted STAT3 deletion rendered the hearts more sensitive to lipopolysaccharide-induced inflammatory damage \\[[@B8]\\]. ",
"All these indicate an important role of STAT3 activation in myocardial protection. ",
"Thus, effective means that can activate STAT3 may attenuate myocardial IR injury through concomitantly reducing oxidative stress by improving mitochondrial biogenesis \\[[@B9]\\] and increasing antioxidant capacity by promoting the expression of nuclear antioxidant genes.",
"\n\nIschemic postconditioning (IPo), a phenomenon that brief repetitive episodes of ischemia and reperfusion applied at the immediate onset of reperfusion, has been proven to be an effective cardioprotective strategy that protects the hearts against myocardial IR injury \\[[@B10]\\]. ",
"In mice with cardiomyocyte-restricted deletion of STAT3, the threshold of IPo was increased (more cycles of short periods of IR was required for IPo to confer cardioprotection), indicating that the cardioprotective effects of IPo were reduced \\[[@B11]\\]. ",
"Pharmacological inhibition of STAT3 abolished IPo-mediated cardioprotection in pigs with regional myocardial IR \\[[@B12]\\], while strategies that can enhance STAT3 activation confer cardioprotection both in animal models of myocardial IR injury \\[[@B13], [@B14]\\] and in humans \\[[@B4]\\]. ",
"These findings collectively indicate that STAT3 activation plays an essential role in IPo cardioprotection against myocardial IR injury. ",
"However, how IPo activates STAT3 to confer cardioprotection remains unclear.",
"\n\nWe previously showed that adiponectin (APN), a protein secreted from adipocytes and cardiomyocytes, is essential in IPo-mediated mitochondrial STAT3 activation and the subsequent protection against myocardial IR injury \\[[@B3]\\]. ",
"APN has been shown to reduce liver fibrosis by modulating the Jak-STAT3 signaling pathway which required AMP-activated protein kinase (AMPK) in the liver \\[[@B15]\\]. ",
"Given that AMPK is well known as the main downstream effector in the APN signaling pathway \\[[@B16]\\], it is possible that AMPK may play a role in APN-mediated STAT3 activation in IPo cardioprotection. ",
"Interestingly, a recent study showed that although AMPK activation was increased in IPo-induced cardioprotection, inhibition of AMPK had no impact on IPo-induced cardioprotection \\[[@B17]\\], suggesting that AMPK may not be the major cellular mediator in IPo cardioprotection. ",
"Along with the most recent finding showing that APN-mediated cardioprotection can be AMPK-dependent and AMPK-independent \\[[@B18]\\], it can be speculated that IPo may activate STAT3 through APN in an AMPK-independent manner. ",
"Therefore, we hypothesized that APN may facilitate IPo-mediated AMPK-independent activation of mitochondrial STAT3 and that APN supplementation may provide additional/synergistic effects on IPo cardioprotection by activating nuclear STAT3 in an AMPK-dependent manner.",
"\n\n2. ",
"Materials and Methods {#sec2}\n========================\n\n2.1. ",
"Animals {#sec2.1}\n------------\n\nMale Sprague-Dawley rats (250 ± 10 g, 6-8 weeks) supplied by the laboratory Animal Service Center (The University of Hong Kong) were used. ",
"All rats were housed and given free access to standard rat chow and water in accordance with the principles of Animal Care of the University of Hong Kong. ",
"The investigation conformed to the procedures described in the Guide for the Care and Use of Laboratory Animals published by the United States National Institutes of Health (NIH Publication number 85-23, revised 1996). ",
"The experimental protocol used in this study was approved by the Wenzhou Medical University Laboratory Animal Ethics Committee and the Committee for Use of Live Animals in Teaching and Research (CULATR) of the University of Hong Kong. ",
"Animal experiments were conducted in the Wenzhou Medical University and the University of Hong Kong.",
"\n\n2.2. ",
"Animal Experimental Protocol {#sec2.2}\n---------------------------------\n\nRats were randomly divided into five groups: (1) sham-operated group (group sham); (2) rats subjected to myocardial IR (group IR); (3) rats injected with one dose of recombinant APN adenovirus (1 × 10^9^ pfu, tail vein injection) 7 days before being subjected to myocardial IR (group IR+APN), as previously described \\[[@B19]\\]; (4) rats subjected to myocardial IR with IPo achieved by inducing three cycles of 10 seconds of reocclusion and 10 seconds of reperfusion at the onset of reperfusion (group IPo); and (5) rats injected with recombinant APN adenovirus 7 days before myocardial IR and then received IPo (group IPo+APN).",
"\n\n2.3. *",
"In Vivo* Model of Myocardial Ischemic Reperfusion and Ischemic Postconditioning {#sec2.3}\n-------------------------------------------------------------------------------------\n\nAfter anesthesia by intraperitoneal injection of sodium pentobarbital (65 mg/kg), animals were randomized to receive sham operation, myocardial IR, or IPo. ",
"Myocardial IR was induced by temporarily exteriorizing the heart via a left thoracic incision and occluding the left anterior descending (LAD) coronary artery 30 minutes (min) followed by reperfusion for 2 hours \\[[@B3], [@B20]\\]. ",
"IPo was produced by three cycles of 10 seconds of reocclusion and 10 seconds of reperfusion immediately after the completion of ischemia \\[[@B3]\\]. ",
"At the completion of 2 hours of reperfusion, left ventricular tissue was harvested and immediately frozen in liquid nitrogen until analyzed.",
"\n\n2.4. ",
"Measurement of Left Ventricular Function {#sec2.4}\n---------------------------------------------\n\nHemodynamics were continuously monitored using subcutaneous stainless-steel electrodes that were connected via a cable to a PowerLab monitoring system (ML750 PowerLab/4sp with MLT380 Reusable BP Transducer; AD Instruments, CO Springs, CO). ",
"Heart rate, mean arterial pressure (MAP), and the rate pressure product (RPP, the product of heart rate and systolic pressure) were recorded. ",
"The RPP was calculated to evaluate the myocardial oxygen demand of the animals \\[[@B21]\\].",
"\n\n2.5. ",
"Determination of Myocardial Infarct Size {#sec2.5}\n---------------------------------------------\n\nAt the end of reperfusion, myocardial infarct size was measured using TTC (1% 2,3,5-triphenyltetrazolium chloride) staining as described \\[[@B21]\\] and the operators were initially blinded to the information of study design and intervention. ",
"Briefly, the LAD was reoccluded and cannulated just distal to the occlusion site. ",
"One milliliter (mL) of 5% Evans Blue was intravenously administrated. ",
"The stained region of the heart was identified as the normal region without ischemia-reperfusion episodes. ",
"The unstained region was considered as the area-at-risk (AAR). ",
"The heart was immediately fibrillated, removed, and sliced into serial transverse sections 6 to 7 mm in width and incubated in 1% TTC buffer for 30 min at room temperature. ",
"The heart slices were then fixed with 10% fresh formalin for 24 hours. ",
"The area unstained by TTC was identified as the infarcted tissue. ",
"Myocardial infarct size was expressed as a percentage of the AAR.",
"\n\n2.6. ",
"Measurement of Plasma Creatinine Kinase-MB Levels {#sec2.6}\n------------------------------------------------------\n\nCreatinine kinase-MB (CK-MB) isoenzyme is a major biomarker for myocardial cellular injury. ",
"After 2 hours of reperfusion, blood samples were collected for measurement of CK-MB by enzyme immunoassay using a commercial kit (Cloud-Clone Corp, Houston, TX) as described \\[[@B21]\\].",
"\n\n2.7. ",
"Apoptotic Cell Death Detection Using Terminal Deoxynucleotidyl Transferase dUTP Nick-End Labeling {#sec2.7}\n------------------------------------------------------------------------------------------------------\n\nTerminal deoxynucleotidyl transferase dUTP nick-end labeling (TUNEL) reaction was performed using an *in situ* cell death detection kit (Roche Supplementary Diagnostics GmbH, Mannheim, Germany) as previously described \\[[@B3]\\]. ",
"The sections were observed in the fluorescent microscope by an investigator, who was initially blinded to treatment groups, and five randomly selected fields of each slide were analyzed, and the apoptotic index was calculated as a percentage of apoptotic nuclei to total nuclei.",
"\n\n2.8. ",
"Adult Rat Ventricular Cardiomyocyte Isolation and Hypoxia/Reoxygenation {#sec2.8}\n----------------------------------------------------------------------------\n\nCalcium-tolerant cardiomyocytes were isolated from rat ventricles via a modified method as previously described \\[[@B22]\\]. ",
"Rats were sacrificed with an intraperitoneal injection of overdose sodium pentobarbital (220 mg/kg) and heparinized. ",
"The hearts were rapidly removed and mounted on a Langendorff perfusion apparatus and proceeding to cardiomyocyte isolation as we described \\[[@B3]\\]. ",
"Cells isolated from a single rat heart were plated onto Matrigel-coated culture dishes and allowed to recover for 3 hours. ",
"Cultured ventricular cardiomyocytes were incubated in Medium 199 (Gibco, Grand Island, NY) at 37°C.",
"\n\nCardiomyocytes were, respectively, treated with stattic (a specific STAT3 inhibitor; 100 mmol/L, 10 min), Dorsomorphin (also named compound C, CC, a specific AMPK inhibitor; 5 *μ*mol/L, 1 hour), or recombinant globular APN (gAd) (2 *μ*mol/L, 24 hours) \\[[@B3]\\] before being subjected to hypoxia/reoxygenation (HR) and hypoxic postconditioning (HPo). ",
"Stattic, the first nonpeptidic small molecule, potently inhibits STAT3 activation and nuclear translocation with IC50 of 5.1 *μ*M in cell-free assays. ",
"Dorsomorphin (compound C) is a potent, reversible, and selective AMPK inhibitor with Ki of 109 nM in cell-free assays, which exhibits no significant inhibition for several structurally related kinases including ZAPK, SYK, PKA, and JAK3.",
"\n\nHR was achieved by subjecting the cells to hypoxia for 45 min followed by 2 hours of reoxygenation. ",
"Hypoxia conditions were obtained by equilibrating a humidified Plexiglas chamber containing myocytes with 95% N~2~ and 5% CO~2~ and confirmed by measuring the chamber O~2~ concentration falling to 0.1%. ",
"Reoxygenation was achieved by exposing cells to room air. ",
"HPo was achieved by three cycles of 5 min of reoxygenation and 5 min of hypoxia before prolonged reoxygenation \\[[@B3]\\]. ",
"At the end of treatments, cells were fixed for immunofluorescence staining or collected and snap-frozen in liquid nitrogen for future analysis, as described below. ",
"Experiments were repeated three times and each in duplicate.",
"\n\n2.9. ",
"Measurement of Cellular ROS in Cultured Cardiomyocyte {#sec2.9}\n----------------------------------------------------------\n\nSuperoxide generation in cultured cardiomyocytes was estimated by dihydroethidium (DHE) staining as previously described \\[[@B23], [@B24]\\]. ",
"Briefly, cardiomyocytes were loaded with DHE at a concentration of 10 *μ*M for 30 min at 37°C. ",
"The DHE fluorescence of DHE-labeled positive nuclei was calculated in each of five randomly selected fields and was expressed as a percentage of the DHE-stained positive myocyte nuclei compared with control by a quantitative morphometric method.",
"\n\n2.10. ",
"Determination of Cellular Injury {#sec2.10}\n--------------------------------------\n\nCell apoptosis was detected by double immunofluorescence staining of TUNEL, using the *In Situ* Cell Death Detection Kit (Roche, Indianapolis, IN, USA), and cell lactate dehydrogenase (LDH) content was measured with an LDH Cytotoxicity Assay Kit (Roche, Indianapolis, IN, USA) as described \\[[@B25]\\].",
"\n\n2.11. ",
"Separation of Myocardial Cytosolic and Nuclear Fractions and Isolation of Mitochondria {#sec2.11}\n--------------------------------------------------------------------------------------------\n\nAt the end of 2 hours of postischemic reperfusion, heart tissues were immediately collected for separation of cytosolic and nuclear fractions and for isolation of mitochondria. ",
"Hearts were cleared of blood by washing thoroughly in Tyrode buffer and aortic and atrial sections removed from the ventricles. ",
"Ventricular tissue was freeze-clamped in liquid nitrogen and stored until fractionated to isolate cytosolic and nuclear fractions (Nuclear and Cytoplasmic Extraction Kit, Thermo, Chicago, IL) and mitochondria (Mitochondria Extraction Kit, Thermo, Chicago, IL) according to the manufacturer\\'s protocol as described \\[[@B3]\\].",
"\n\n2.12. ",
"Extraction of Total RNA and Quantitative Real-Time Polymerase Chain Reaction Analysis {#sec2.12}\n-------------------------------------------------------------------------------------------\n\nTotal RNA was extracted using Trizol (Invitrogen Life Technologies, Carlsbad, CA), equal amounts of RNA were reverse-transcribed and processed using the PrimeScript RT Master Mix Kit, and quantitative real-time polymerase chain reaction (PCR) was performed with an SYBR green PCR master mix (Takara) on a 7300 ABI-Prizm Sequence Detector (Applied Biosystems) as described \\[[@B19]\\], for measurement of superoxide dismutase 1 (SOD1, MnSOD), nuclear factor erythroid 2-related factor (Nrf) 2, heme oxygenase 1 (HO-1), Nrf1, and peroxisome proliferator-activated receptor-*γ* coactivator-1*α* (PGC-1*α*) genes with *β*-Actin as reference. ",
"The conditions for amplification were 30 seconds at 95°C for denaturation, followed by 40 cycles of 5 seconds at 95°C and 30 seconds at 60°C. ",
"Gene-specific primers were as follows: SOD1 forward, 5′-AACCAGTTGTGGTGTCAGGA-3′, reverse, 5′-CTCCTGAGAGTGAGATCACA-3′; catalase forward, 5′-TTCTACACTGAAGATGGTAACTG-3′, reverse, 5′-GAAAGTAACCT GATGGAGAGAC-3′; Nrf2 forward, 5′-GAATAAAGTTGCCGCTCAGAA-3′, reverse, 5′-AA GGTTTCCCATCCTCATCAC-3′; HO-1 forward, 5′-TGCTCGCATGAACACTCTG-3′, reverse, 5′-TCCTCTGTCA GCAGTGCCT-3′; Nrf1 forward, 5′-GATGCTTCAGAACTGCCAACCA-3′, reverse, 5′-GGTC ATTTCACCGCCCTGTAAC-3′; PGC-1*α* forward, 5′-CACTGACAGATGGA GCCGTGA-3′, reverse, 5′-TGTTGGCTGGTGCCAGTAAGAG-3′; and *β*-Actin, served as an endogenous control, forward, 5′-AGGCCAACCGTGAAAAGATG-3′, reverse, 5′-ACCAGAGG CATACAGGGACA A-3′. The level of target gene expression was analyzed using the 2(-*Δ*ΔCT) method and normalized against the *β*-actin gene.",
"\n\n2.13. ",
"Western Blot Analysis for the Protein Expression and Signaling Proteins {#sec2.13}\n-----------------------------------------------------------------------------\n\nEqual protein amounts from isolated cardiomyocytes, rat heart, and isolated mitochondria were resolved by 8-12% SDS-PAGE and transferred to polyvinylidene fluoride membrane for immunoblot analysis, as previously described \\[[@B26]\\]. ",
"Membranes were blocked with 5% nonfat milk in Tris-Buffered Saline- (TBS-) Tween and were incubated with primary antibodies overnight at 4°C at the following dilutions: AMPK, phosphorylated AMPK (Thr172), STAT3, phosphorylated STAT3 at Tyr705 (p-STAT3 Tyr705), Bax, Bcl2, caspase-3, and cleaved caspase-3 (Cell Signaling Technology, Beverly, MA) 1 : 1000; phosphorylated STAT3 at Ser727 (p-STAT3 Ser727) (Cell Signaling Technology, Beverly, MA) 1 : 500. ",
"After washing with TBS-Tween, immunoreactive bands were visualized by an enzymatic chemiluminescence method and quantified with Quantity One image software.",
"\n\n2.14. ",
"Statistical Analysis {#sec2.14}\n--------------------------\n\nAll the values were expressed as mean ± standard error of the mean (SEM). ",
"One-way analysis of variance (ANOVA) was used for statistical analyses (GraphPad Prism, USA) of data obtained within the same group and between groups, respectively, followed by Tukey\\'s test for multiple comparisons of group means. ",
"A *p* \\< 0.05 was considered statistically significant.",
"\n\n3. ",
"Results {#sec3}\n==========\n\n3.1. ",
"Adiponectin-Activated Posthypoxic Cardiomyocyte Nuclear STAT3 and Mitochondrial STAT3, Respectively, via AMPK-Dependent and AMPK-Independent Pathways {#sec3.1}\n----------------------------------------------------------------------------------------------------------------------------------------------------------\n\nAs shown in [Figure 1](#fig1){ref-type=\"fig\"}, in cultured isolated cardiomyocytes, posthypoxic cell death was significantly increased, manifested as elevated LDH release, enhanced protein expression of cleaved caspase-3 (marker of cell apoptosis), and increased TUNEL-positive cells (Figures [1(a)](#fig1){ref-type=\"fig\"}--[1(e)](#fig1){ref-type=\"fig\"}), which were associated with enhanced ROS evidenced by increased number of DHE-labeled positive nuclei (Figures [1(c)](#fig1){ref-type=\"fig\"} and [1(e)](#fig1){ref-type=\"fig\"}). ",
"All these changes were attenuated or prevented by globular APN (gAd) supplementation (Figures [1(a)](#fig1){ref-type=\"fig\"}--[1(e)](#fig1){ref-type=\"fig\"}). ",
"However, these protective effects of gAd were cancelled by either AMPK inhibition with the specific inhibitor compound C (CC) or by STAT3 inhibition with the specific inhibitor stattic (Figures [1(a)](#fig1){ref-type=\"fig\"}--[1(e)](#fig1){ref-type=\"fig\"}). ",
"HR induced a significant increase of AMPK phosphorylation/activation (p-AMPK), which was concomitant with significant reductions of STAT3 phosphorylation at both Ser727 and Tyr705, while gAd treatment further enhanced p-AMPK and reversed HR-induced reductions of STAT3 phosphorylation at both Ser727 and Tyr705 (Figures [1(f)](#fig1){ref-type=\"fig\"}--[1(i)](#fig1){ref-type=\"fig\"}). ",
"Interestingly, AMPK inhibition cancelled gAd-induced activation of STAT3 at Tyr705 but not at Ser727 (Figures [1(h)](#fig1){ref-type=\"fig\"} and [1(i)](#fig1){ref-type=\"fig\"}), while STAT3 inhibition had no effect on gAd-induced AMPK activation ([Figure 1(g)](#fig1){ref-type=\"fig\"}), suggesting that gAd-induced STAT3 activation at Tyr705 is AMPK-dependent while gAd-induced mitochondrial STAT3 activation at Ser727 is AMPK-independent.",
"\n\n3.2. ",
"Hypoxic Postconditioning (HPo) Enhanced Mitochondrial STAT3 (Ser727) Activation but Not Nuclear STAT3 (Tyr705) Activation Independent of AMPK Activation {#sec3.2}\n-------------------------------------------------------------------------------------------------------------------------------------------------------------\n\nIn parallel to HR-induced significant cell injury (Figures [2(a)](#fig2){ref-type=\"fig\"}--[2(e)](#fig2){ref-type=\"fig\"}), the phosphorylation of STAT3 at both Tyr705 and Ser727 (Figures [2(f)](#fig2){ref-type=\"fig\"}, [2(h)](#fig2){ref-type=\"fig\"}, and [2(i)](#fig2){ref-type=\"fig\"}) was significantly reduced despite a significant increase of AMPK activation ([Figure 2(g)](#fig2){ref-type=\"fig\"}). ",
"This HR-induced cellular injury was attenuated by HPo (Figures [2(a)](#fig2){ref-type=\"fig\"}--[2(e)](#fig2){ref-type=\"fig\"}) with concomitant enhancement of STAT3 activation at Ser727 ([Figure 2(h)](#fig2){ref-type=\"fig\"}) but not at Tyr705 ([Figure 2(i)](#fig2){ref-type=\"fig\"}). ",
"STAT3 inhibition reverted HPo-mediated attenuation of cellular LDH release and cell apoptosis and the HPo-mediated attenuation of oxidative stress (Figures [2(a)](#fig2){ref-type=\"fig\"}--[2(e)](#fig2){ref-type=\"fig\"}), while AMPK inhibition had no impact on HPo-mediated reduction of apoptosis but cancelled HPo-mediated reduction of oxidative stress (Figures [2(c)](#fig2){ref-type=\"fig\"} and [2(e)](#fig2){ref-type=\"fig\"}). ",
"Posthypoxic AMPK activation was moderately but significantly increased that was concomitant with reduced STAT3 activation at both phosphorylation residues (Figures [2(f)](#fig2){ref-type=\"fig\"}--[2(i)](#fig2){ref-type=\"fig\"}). ",
"HPo treatment further significantly enhanced posthypoxic AMPK activation and increased posthypoxic STAT3 activation at Ser727 but not at Tyr705 ([Figure 2(i)](#fig2){ref-type=\"fig\"}). ",
"AMPK inhibition had no impact on HPo-induced STAT3 activation (Figures [2(h)](#fig2){ref-type=\"fig\"} and [2(i)](#fig2){ref-type=\"fig\"}). ",
"Similarly, STAT3 inhibition had no significant effect on HPo-induced AMPK activation ([Figure 2(g)](#fig2){ref-type=\"fig\"}). ",
"This suggests that HPo-induced mitochondrial STAT3 (Ser727) activation is AMPK-independent.",
"\n\n3.3. ",
"Adiponectin Provided Additive Cardioprotective Effects to Ischemic Postconditioning (IPo) *In Vivo* {#sec3.3}\n--------------------------------------------------------------------------------------------------------\n\nAdiponectin supplementation at the dose used did not cause animal death. ",
"As shown in [Table 1](#tab1){ref-type=\"table\"} and [Figure 3](#fig3){ref-type=\"fig\"}, in rats subjected to *in vivo* myocardial IR, postischemic cardiac function was compromised, manifested as reduced heart rate, MAP, and RPP that was associated with myocardial injury evidenced by increased postischemic infarct size, elevated plasma CK-MB release, and enhanced myocardial cell apoptosis manifested as increased TUNEL-positive cells (Figures [3(a)](#fig3){ref-type=\"fig\"}--[3(e)](#fig3){ref-type=\"fig\"}). ",
"Either APN or IPo treatment significantly attenuated the impairment in postischemic cardiac function and reduced myocardial injuries, while APN in combination with IPo conferred superior protective effects than either APN or IPo alone manifested as more significant reductions in postischemic infarct size, plasma CK-MB, and apoptosis (Figures [3(a)](#fig3){ref-type=\"fig\"}--[3(e)](#fig3){ref-type=\"fig\"}). ",
"Myocardial IR significantly reduced SOD1 (Cu-Zn-SOD) ([Figure 3(f)](#fig3){ref-type=\"fig\"}) but increased Nrf2 ([Figure 3(g)](#fig3){ref-type=\"fig\"}) and catalase ([Figure 3(i)](#fig3){ref-type=\"fig\"}) mRNA expression in the nuclear that were associated with decreased PGC-1*α* and Nrf1 mRNA expression in the mitochondria (Figures [3(j)](#fig3){ref-type=\"fig\"} and [3(k)](#fig3){ref-type=\"fig\"}), while nuclear HO-1 mRNA expression remained unchanged ([Figure 3(h)](#fig3){ref-type=\"fig\"}). ",
"Either APN or IPo significantly enhanced SOD1, Nrf2, HO-1, catalase, PGC-1*α*, and Nrf1 mRNA expression (Figures [3(f)](#fig3){ref-type=\"fig\"}--[3(k)](#fig3){ref-type=\"fig\"}), while APN further significantly potentiated IPo-induced enhancements of SOD1 and Nrf2 mRNA expression.",
"\n\n3.4. ",
"Adiponectin and Ischemic Postconditioning Exerted Distinct Effects on Cardiac AMPK and STAT3 Activation *In Vivo* {#sec3.4}\n----------------------------------------------------------------------------------------------------------------------\n\nAs shown in [Figure 4](#fig4){ref-type=\"fig\"}, postischemic myocardial STAT3 activation (at Tyr705 and Ser727) was slightly increased but this increase did not reach statistical significance (Figures [4(a)](#fig4){ref-type=\"fig\"} and [4(b)](#fig4){ref-type=\"fig\"}). ",
"APN significantly increased the activation of posthypoxic cardiac STAT3 at both residues (Ser727 and Tyr705) and the activation of AMPK (Figures [4(a)](#fig4){ref-type=\"fig\"}--[4(d)](#fig4){ref-type=\"fig\"}) either in the absence or presence of IPo. ",
"IPo alone significantly enhanced STAT3 activation at Ser727 but had no impact on Tyr705 nor did it affect AMPK activation (Figures [4(a)](#fig4){ref-type=\"fig\"}--[4(d)](#fig4){ref-type=\"fig\"}). ",
"APN and IPo combination resulted in a further increase in STAT3 activation at Ser727 when compared to either APN or IPo alone ([Figure 4(b)](#fig4){ref-type=\"fig\"}).",
"\n\n3.5. ",
"STAT3 Inhibition Abolished While AMPK Inhibition Compromised the Cardioprotection Conferred by Adiponectin in Combination with Hypoxic Postconditioning {#sec3.5}\n------------------------------------------------------------------------------------------------------------------------------------------------------------\n\nIn order to determine the role of AMPK and STAT3 in the cardioprotection of APN in combination with IPo, primarily isolated cardiomyocytes were subjected to HR and treated with APN and HPo in the absence or presence of the AMPK inhibitor CC or the STAT3 inhibitor stattic. ",
"As shown in [Figure 5](#fig5){ref-type=\"fig\"}, in cultured cardiomyocytes, the posthypoxic cellular injuries were significantly attenuated by either APN or HPo treatment while APN in combination with HPo conferred superior protective effects to either APN or HPo alone (Figures [5(a)](#fig5){ref-type=\"fig\"}--[5(f)](#fig5){ref-type=\"fig\"}). ",
"The protective effects of APN in combination with HPo in reducing posthypoxic LDH release and ROS production were reverted by either AMPK inhibition or STAT3 inhibition (Figures [5(a)](#fig5){ref-type=\"fig\"}--[5(f)](#fig5){ref-type=\"fig\"}), while STAT3 inhibition but not AMPK inhibition significantly compromised APN in combination with HPo-mediated attenuation of posthypoxic apoptotic cell death ([Figure 5(d)](#fig5){ref-type=\"fig\"}). ",
"Posthypoxic AMPK activation was significantly increased while STAT3 activation was reduced at both Ser727 and Tyr705 (all *p* \\< 0.05, group HR *vs.* group c, Figures [5(g)](#fig5){ref-type=\"fig\"}--[5(i)](#fig5){ref-type=\"fig\"}). ",
"APN alone or its combination with HPo significantly increased the activation of AMPK and the activation of STAT3 at both phosphorylation residues (Figures [5(g)](#fig5){ref-type=\"fig\"}--[5(i)](#fig5){ref-type=\"fig\"}), and APN in combination with HPo induced more significant increase of STAT3 activation at Ser727 than APN alone (*p* \\< 0.05, group APN+HPo *vs.* group APN, [Figure 5(h)](#fig5){ref-type=\"fig\"}), while HPo alone enhanced AMPK activation and STAT3 activation at Ser727 but had no impact on STAT3 activation at Tyr705 (Figures [5(g)](#fig5){ref-type=\"fig\"} and [5(i)](#fig5){ref-type=\"fig\"}). ",
"AMPK inhibition cancelled the AMPK activation and the activation of STAT3 at Tyr705 induced by combinational treatment with APN and HPo (Figures [5(g)](#fig5){ref-type=\"fig\"}--[5(i)](#fig5){ref-type=\"fig\"}), while STAT3 inhibition cancelled APN and HPo-induced STAT3 activation but had no effect on AMPK activation induced by combinational treatment with APN and HPo (Figures [5(g)](#fig5){ref-type=\"fig\"}--[5(i)](#fig5){ref-type=\"fig\"}).",
"\n\n4. ",
"Discussion {#sec4}\n=============\n\nFindings from our *in vitro* and *in vivo* studies provided strongly suggest that APN confers cardioprotection through AMPK-dependent nuclear STAT3 activation and AMPK-independent mitochondrial STAT3 activation, while IPo confers cardioprotection mainly through mitochondrial STAT3 activation which is AMPK-independent. ",
"APN and IPo synergistically attenuated myocardial IR injury by activating STAT3 via distinct signaling pathways despite that they also share common mechanisms such as both APN and IPo can activate p-STAT3 at Ser727. ",
"In isolated primary cardiomyocytes, supplementation of APN attenuated HR-induced oxidative stress and cellular injury, which were reduced by AMPK inhibition or abolished by STAT3 inhibition. ",
"APN supplementation increased the activation of AMPK and the activation of STAT3 at both phosphorylation residues, but AMPK inhibition cancelled APN-induced STAT3 activation at Tyr705 but not Ser727, while STAT3 inhibition had no effect on APN-induced AMPK activation, suggesting that APN conferred cardioprotection through AMPK-dependent STAT3 activation at Tyr705 and AMPK-independent STAT3 at Ser727. ",
"HPo increased p-AMPK and activated STAT3 at Ser727 but not Tyr705, while AMPK inhibitor CC, at the concentration used, had no effect on HPo-induced STAT3 activation, indicating that HPo conferred cardioprotection mainly through activating STAT3 at Ser727 which is AMPK-independent. ",
"Collectively, we conclude that APN and IPo attenuate myocardial IR by activating STAT3 via distinct signaling pathways (i.e., AMPK-dependent and AMPK-independent).",
"\n\nOxidative stress resulting from the excessive production of ROS and/or inadequate antioxidant defense is the main cause of myocardial IR injury \\[[@B27]--[@B30]\\]. ",
"In the current study, we showed that either APN, IPo, or their combination reduced post-ischemic oxidative stress and cardiac injury, which were abolished by STAT3 inhibition, pointing out the importance of STAT3 in APN and IPo-mediated reduction of oxidative stress to confer their cardioprotection. ",
"STAT3 is activated via the tyrosine phosphorylation (Tyr705) and serine phosphorylation (Ser727) cascades \\[[@B6]\\]. ",
"When being phosphorylated at Tyr705, STAT3 translocates into the nucleus where STAT3 promotes transcription of antioxidant genes (i.e., Mn-SOD, Nrf2, and HO-1) and exerts its antioxidant effects \\[[@B31]\\]. ",
"When being phosphorylated at Ser727, STAT3 shuttles into the mitochondria primarily where it interacts with Complex I and maintains mitochondrial biogenesis thereby reduces ROS production \\[[@B5]\\]. ",
"In the current study, APN alone increased STAT3 activation at both phosphorylation residues (Ser727 and Tyr705) thereby activated the downstream effectors that target antioxidant molecules in the nucleus (i.e., SOD1, Nrf2, HO-1, and catalase) and in the mitochondria (Nrf1 and PGC-1*α*), while IPo alone enhanced STAT3 activation only at Ser727 and activated Nrf2, HO-1, and catalase in the nucleus and Nrf1 and PGC-1*α* in the mitochondrial but had no effect on SOD1. ",
"These findings not only suggest that different regulatory patterns of STAT3 activation in APN and IPo cardioprotection but may also provide a clue that APN through activating STAT3 at Tyr705 facilitates IPo cardioprotection against myocardial IR injury. ",
"Indeed, our study demonstrated that combined use of APN and IPo led to further increase of STAT3 activation at both residues and further increase of Nrf2 in the nucleus and conferred superior cardioprotection to either APN or IPo alone. ",
"This suggests that maximal activation of STAT3 (phosphorylation at two residues, Ser727 and Tyr705) credits superior cardioprotection by integrating the two distinct pathways, i.e., by promoting nuclear antioxidant genes transcription and by reducing mitochondrial ROS production, while partial activation of STAT3 (Ser727 or Tyr705) favors stimulation of either of the above-mentioned pathways.",
"\n\nDespite the proven importance of STAT3 activation in cardioprotective interventions (e.g., APN and IPo), it is still unclear how APN and IPo activate STAT3 to confer cardioprotection. ",
"AMPK is well known as the downstream effector of APN biological function \\[[@B32]--[@B34]\\], and a tight regulatory effect of AMPK on STAT3 activation was evidenced by the findings that up-regulation of AMPK by APN or by specific agonist-attenuated fibrosis and inflammatory response in the liver through inhibition of STAT3 activation \\[[@B15], [@B35]\\]. ",
"Furthermore, pharmacological activation of AMPK can inhibit STAT3 activation and consequently inhibit monocyte-to-macrophage differentiation and attenuate angiotensin-II-induced atheromatous plaque formation \\[[@B36]\\]. ",
"However, the direct regulatory effect of AMPK on STAT3 in the heart, particularly in the setting of APN and IPo, has not been reported. ",
"We found that, in cultured cardiomyocytes, AMPK inhibition cancelled APN-induced STAT3 activation at Tyr705 but not at Ser727 and abolished APN-mediated protection against post-hypoxic cellular injury, indicating that AMPK is involved in the APN-mediated Tyr705 STAT3 activation while APN-mediated Ser727 STAT3 activation is AMPK-independent. ",
"Our results are consistent with the findings of previous studies demonstrating that APN cardioprotection is partially but not completely mediated by AMPK \\[[@B33], [@B37]\\], which involved the AMPK-independent pathway \\[[@B18]\\]. ",
"This was further confirmed by our current results, showing that inhibition of STAT3 activation at both residues led to a further exacerbation of myocardial injury than that of AMPK inhibition in APN-mediated cardioprotection. ",
"The findings indicate that an AMPK-independent pathway exists in APN-induced STAT3 activation and support the notion that STAT3 activation may work as the end-effector in APN cardioprotection \\[[@B6]\\].",
"\n\nOxidative stress-induced cardiomyocyte apoptotic cell death is the main cause of post-ischemic cardiac injury. ",
"In the current study, we found that either APN or IPo or their combination reduced post-ischemic cell apoptosis *in vivo* in the heart and *in vitro* in cultured cardiomyocytes, while these anti-apoptotic effects of APN or IPo or their combination were abolished by STAT3 inhibition, suggesting that STAT3 activation plays an essential role in their anti-apoptotic effects. ",
"Interestingly, in IPo cardioprotection, we found that AMPK was activated by IPo while AMPK inhibition cancelled IPo-induced reduction of oxidative stress but had no impact on IPo-mediated STAT3 activation and the subsequent reduction of cell apoptosis. ",
"This finding may be partially explainable by the notion that there exists reactive oxygen species-independent apoptosis in reperfusion injury \\[[@B38]\\]. ",
"Alternatively, differences might exist in the extent of ROS decrease and/or the decrease might occur in different compartments and some of which could be more relevant for causing cell injury. ",
"Further, this finding not only suggests that AMPK-mediated reduction of oxidative stress in IPo may not be efficient to confer anti-apoptotic effects but also indicates that AMPK may not be the principal mechanism in IPo cardioprotection. ",
"This was in contrast to the recent finding by Hu et al. ",
"\\[[@B39]\\] who demonstrated that AMPK inhibition cancelled IPo-mediated reduction of oxidative stress and attenuation of myocardial injury in neonatal rat cardiomyocytes subjected to HR. ",
"The inconsistencies of these results may be due to (1) difference of cell models (cardiomyocytes isolated from adult rats in our study *vs.* cardiomyocytes isolated from neonatal rats in Hu\\'s study), (2) variation of HR models (45 min hypoxia with 2 hours reoxygenation in our study *vs.* 3 hours hypoxia with 6 hours reoxygenation in Hu\\'s study), and (3) variation in AMPK inhibition by compound C (1 hour before hypoxia in our study *vs.* 6 hours after hypoxia in Hu\\'s study). ",
"However, these findings together with our and others\\' previous finding that APN-mediated STAT3 activation is essential in IPo cardioprotection \\[[@B3]\\] and that APN may exert its biological effects through the AMPK-independent pathway \\[[@B33]\\] collectively suggest that APN regulates STAT3 activation through AMPK-independent pathway in IPo cardioprotection. ",
"More importantly, this provides the possibility that APN may act through AMPK-dependent STAT3 activation to facilitate IPo cardioprotection. ",
"Thus, effective means that can activate APN signaling pathways (such as the newly identified APN receptor agonist \\[[@B40]\\] and N-Acetylcysteine \\[[@B21]\\]) may provide additional cardioprotection on top of IPo or rescue IPo cardioprotection in combating myocardial IR injury, in particular, under conditions that IPo cardioprotection is diminished or abolished such as in hearts from subjects with diabetes or the elderly \\[[@B3]\\]. ",
"It should be noted that despite of the obvious synergistic cardioprotective effects observed between APN and IPo in an *in vivo* model of myocardial IR, mechanistic explorations in our current study were mainly conducted in isolated cultured cardiomyocytes that have excluded the potential impacts of systemic factors such as hormonal changes that may be induced by APN and/or IPo *in vivo* and the potential interactions of different type of cells such as cardiomyocytes and vascular endothelial cells. ",
"Nevertheless, current studies shall stimulate further studies to be conducted both *in vivo* and *ex vivo* to confirm and to explore the underlying mechanisms, such as why cardiomyocyte hypoxia/reoxygenation leads to increased AMPK activation with concomitant reductions in STAT3 activation but a further increase in AMPK activation induced by HPo and/or by APN-increased STAT3 activation \\[[@B41]\\].",
"\n\n5. ",
"Conclusions {#sec5}\n==============\n\nIn summary, as illustrated in [Figure 6](#fig6){ref-type=\"fig\"}, our study demonstrated that APN should have conferred cardioprotection by enhancing antioxidant capacity through activating STAT3 at Tyr705 which is AMPK-dependent and by reducing ROS production through activating STAT3 at Ser727 which is AMPK-independent. ",
"By contrast, IPo conferred cardioprotection by reducing ROS production through AMPK-independent activation of STAT3 at Ser727. ",
"Combinational use of APN and IPo conferred synergistic cardioprotective effects against myocardial IR injury. ",
"However, caution should be exercised when choosing the dose and regimen of exogenous APN to avoid the occurrence of hyperadiponectinemia and the subsequent potential adverse effects, as IPo per se could also enhance endogenous adiponectin. ",
"Also, it should be noted that adenoviral-mediated APN expression may not be an ideal model of physiological relevance, but the insight gained from this mechanistic exploratory study may serve to stimulate the use of drugs that can enhance endogenous APN secretion as an adjuvant therapy to increase the effectiveness of ischemic postconditioning in clinical settings.",
"\n\nThe authors thank Dr. Ruby Lai-chiong Hoo (Department of Medicine, University of Hong Kong, Hong Kong SAR, China) for excellent technical assistance in generating adiponectin adenovirus. ",
"Part of these data were presented at the Experimental Biology 2016, San Diego, CA, April 2-6, 2016, and published as an abstract in FASEB J 30:lb526. ",
"This study was supported in part by the Hong Kong RGC/GRF grants (17124614M and 17123915M), in part by the Zhejiang Provincial Top priority first-level discipline grants (to Ren-shan Ge and Zhengyuan Xia), the Science and Technology Project of Guangdong Province (2017A030313467 to Haobo Li), the National Natural Science Foundation of China (81601722, 81601729 and 81670770), and the Wenzhou Municipal Science and Technology Bureau (Y20160111 and Y20160133).",
"\n\nData Availability\n=================\n\nThe data used to support the findings of this study are included in the article.",
"\n\nConflicts of Interest\n=====================\n\nThe authors confirm that there are no conflicts of interest.",
"\n\nAuthors\\' Contributions\n=======================\n\nHaobo Li and Zhengyuan Xia designed the experiments; Qiqi Zhu, Haobo Li, Xiang Xie, and Xiaozhen Chen performed experiments and analyzed data; Qiqi Zhu, Haobo Li, Ramoji Kosuru, Sisi Li, Qingquan Lian, Chi Wai Cheung, Michael G. Irwin, and Zhengyuan Xia interpreted the results of experiments; Qiqi Zhu and Haobo Li drafted the manuscript; Ren-shan Ge, Michael G. Irwin, and Zhengyuan Xia edited and revised the manuscript; Zhengyuan Xia approved the final version of the manuscript; Zhengyuan Xia has full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis; and all authors reviewed the manuscript. ",
"Qiqi Zhu and Haobo Li contributed equally to this work.",
"\n\n{#fig1}\n\n{#fig2}\n\n{#fig3}\n\n{#fig4}\n\n{#fig5}\n\n{#fig6}\n\n###### \n\nHemodynamic measurements at baseline and 2 hours of reperfusion in rats with sham and myocardial ischemia-reperfusion with or without adiponectin and ischemic postconditioning.",
"\n\n Sham IR IR+APN IR+IPo IR+APN+IPo\n -------------------------- ------------- ------------ -------------- -------------- --------------\n Baseline \n Heart rate (min^−1^) 280 ± 10 279 ± 7 276 ± 10 281 ± 10 277 ± 9\n MAP (mmHg) 115 ± 3 113 ± 5 116 ± 5 112 ± 5 110 ± 6\n RPP (mmHg·min^−1^/10000) 33 ± 4 36 ± 3 34 ± 4 37 ± 3 38 ± 2\n \n 2 hours of reperfusion \n Heart rate (min^−1^) 287 ± 9^\\#^ 155 ± 8^∗^ 203 ± 6^∗\\#^ 197 ± 7^∗\\#^ 197 ± 9^∗\\#^\n MAP (mmHg) 113 ± 4^\\#^ 62 ± 8^∗^ 88 ± 4^∗\\#^ 88 ± 4^∗\\#^ 91 ± 3^∗\\#^\n RPP (mmHg·min^−1^/10000) 36 ± 2^\\#^ 18 ± 1^∗^ 26 ± 1^∗\\#^ 26 ± 2^∗\\#^ 27 ± 1^∗\\#^\n\nRats were subjected to 30 min of coronary occlusion and 2 hours of reperfusion (IR) with or without adiponectin (APN, tail vein-injected 7 days before inducing IR) or ischemic postconditioning (IPo) or their combination (APN+IPo). ",
"All values are mean ± SEM. *",
"n* = 8 animal *per* group. ",
"Heart rate, mean arterial pressure (MAP), and rate pressure product (RPP) were measured at baseline and 2 hours of reperfusion. ",
"^∗^*p* \\< 0.05 or *p* \\< 0.01*vs.* ",
"their corresponding baseline, ^\\#^*p* \\< 0.05 or *p* \\< 0.01*vs.* ",
"their corresponding IR.",
"\n\n[^1]: Academic Editor: Neelam Khaper\n"
] | {
"pile_set_name": "PubMed Central"
} | [
0,
0.006578947368421052,
0,
0.006896551724137931,
0.012944983818770227,
0.009523809523809525,
0.014423076923076924,
0.015957446808510637,
0.021897810218978103,
0.0136986301369863,
0.012048192771084338,
0.007407407407407408,
0.0035587188612099642,
0.00784313725490196,
0.020761245674740483,
0.0072992700729927005,
0.013157894736842105,
0.017241379310344827,
0.018072289156626505,
0.0297029702970297,
0.014492753623188406,
0.03111111111111111,
0.02247191011235955,
0,
0,
0.011695906432748537,
0.0064516129032258064,
0.0091324200913242,
0.01276595744680851,
0.02,
0,
0.008547008547008548,
0,
0.003003003003003003,
0.008658008658008658,
0.006756756756756757,
0,
0,
0.008875739644970414,
0.014084507042253521,
0.022222222222222223,
0,
0.0058823529411764705,
0,
0,
0,
0.015873015873015872,
0,
0,
0,
0.015384615384615385,
0,
0.009615384615384616,
0.021621621621621623,
0,
0.006802721088435374,
0,
0,
0.01056338028169014,
0,
0.013333333333333334,
0,
0.020202020202020204,
0.025495750708215296,
0.013245033112582781,
0.0211864406779661,
0,
0.009852216748768473,
0,
0.00819672131147541,
0,
0,
0,
0.007547169811320755,
0.010526315789473684,
0.00816326530612245,
0,
0.012987012987012988,
0,
0,
0.015625,
0.009230769230769232,
0,
0.012091898428053204,
0,
0.0012787723785166241,
0,
0.005050505050505051,
0.024229074889867842,
0.01282051282051282,
0,
0,
0.008583690987124463,
0,
0,
0,
0.00589622641509434,
0.006369426751592357,
0.007782101167315175,
0.018276762402088774,
0.022935779816513763,
0,
0.008321775312066574,
0.0071174377224199285,
0.004694835680751174,
0.004405286343612335,
0.016304347826086956,
0.014598540145985401,
0.016,
0.02197802197802198,
0,
0.006920415224913495,
0.007905138339920948,
0.009828009828009828,
0.0020325203252032522,
0.01079136690647482,
0,
0.0058823529411764705,
0.012048192771084338,
0.010309278350515464,
0.01818181818181818,
0,
0.011804384485666104,
0.008797653958944282,
0.015945330296127564,
0.013043478260869565,
0.018092105263157895,
0.0228310502283105,
0,
0.01977401129943503,
0.018518518518518517,
0.015706806282722512,
0.04207920792079208,
0.02127659574468085,
0.024539877300613498,
0.012048192771084338,
0.013289036544850499,
0.02564102564102564,
0.028985507246376812,
0.010050251256281407,
0.01279317697228145,
0.01968503937007874,
0.012658227848101266,
0.012658227848101266,
0.021505376344086023,
0.03089887640449438,
0.013636363636363636,
0.022058823529411766,
0.03206997084548105,
0.02608695652173913,
0.01327433628318584,
0.0297029702970297,
0,
0.0106951871657754,
0.011857707509881422,
0.006493506493506494,
0,
0.008368200836820083,
0.017857142857142856,
0.0106951871657754,
0.002074688796680498,
0.024793388429752067,
0.02127659574468085,
0.011494252873563218,
0.005952380952380952,
0.015,
0,
0.01675977653631285,
0.015748031496062992,
0.00909090909090909,
0.004166666666666667,
0.005449591280653951,
0.010582010582010581,
0,
0.015250544662309368,
0,
0,
0.02880658436213992,
0.03636363636363636,
0,
0.033112582781456956,
0.01639344262295082,
0,
0,
0,
0,
0.013513513513513514,
0,
0.042105263157894736,
0.013333333333333334,
0.03787878787878788,
0.010256410256410256,
0,
0,
0,
0,
0.013513513513513514,
0,
0.042105263157894736,
0.013333333333333334,
0.011764705882352941,
0.005649717514124294,
0.006896551724137931,
0,
0,
0.0136986301369863,
0.016129032258064516,
0.02608695652173913,
0.0111731843575419,
0.012658227848101266,
0.011904761904761904,
0.017543859649122806,
0,
0.008928571428571428,
0.013071895424836602,
0.012552301255230125,
0,
0,
0,
0,
0.04081632653061224,
0.013333333333333334,
0.014285714285714285,
0.034782608695652174,
0.01,
0.015957446808510637,
0.006535947712418301,
0.009708737864077669,
0.006535947712418301,
0.0031520882584712374,
0,
0,
0.0078125,
0.02857142857142857,
0.015151515151515152,
0.043478260869565216,
0.02564102564102564
] | 0.010721 | 5 |
[
"---\nabstract: 'Let $\\,M=G/K \\,$ be a Riemannian homogeneous manifold with $\\,dim_\\C G^\\C = dim_\\R G\\,$, where $\\,G^\\C\\,$ denotes the universal complexification of $\\,G$. Under certain extensibility assumptions on the geodesic flow of $\\,M$, we give a characterization of the maximal domain of definition in $\\,TM\\,$ for the adapted complex structure and show that it is unique. ",
"For instance, this can be done for generalized Heisenberg groups and naturally reductive Riemannian homogeneous spaces. ",
"As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definitions for the adapted complex structure which are neither holomorphically separable, nor holomorphically convex.'",
"\nauthor:\n- 'S. Halverscheid'\n- 'A. Iannuzzi\\*'\ntitle: Maximal Complexifications of certain Riemannian Homogeneous Manifolds\n---\n\n[^1] [^2] [^3]\n\nIntroduction\n============\n\nIt is well known that complexifications of a real-analytic manifold $\\,M\\,$ exist and are equivalent near $\\,M,$ but differ usually very much in nature. ",
"If a complete real-analytic metric on $\\,M\\,$ is given, one can construct canonical complexifications of $\\,M\\,$ compatible with the metric by defining an [*adapted*]{} complex structure on a domain $\\,\\Omega\\,$ of the tangent bundle $\\,TM$ (see [@GS] and [@LS]). ",
"This can be characterized by the condition that the “complexification\" $\\,(x + iy) \\mapsto y \\, \\gamma'(x) \\in\n\\Omega\\,$ of any geodesic $\\,x \\mapsto \\gamma(x)\\,$ of $\\,M\\,$ be a complex submanifold near the zero section. ",
"By the results of Guillemin-Stenzel and Lempert-Sz[ő]{}ke cited above, the adapted complex structure exists and is unique on a sufficiently small neighborhood of $\\,M\\,$. Here $\\,M\\,$ is identified with its zero section in $\\,TM\\,$.\n\nIn particular it is natural to ask for maximal domains around $\\,M\\,$ on which it exists which, by functoriality of the definition, may be regarded as invariants of the metric, i.e., isometric manifolds have biholomorphic maximal domains. ",
"For instance examples are known for symmetric spaces of non-compact type ([@BHH]), compact normal Riemannian Homogeneous spaces ([@Sz2]), compact symmetric spaces ([@Sz1]) and spaces obtained by Kählerian reduction of these ([@A]). ",
"Note that in the mentioned cases maximal domains turn out to be Stein.",
"\n\nThe aim of this work is to characterize maximal domains for the adapted complex structure for a class of Riemannian homogeneous spaces with “big” isometry group. ",
"Let $\\,M=G/K\\,$, with $\\,G\\,$ a Lie group of isometries and $\\,K\\,$ compact, and assume that $\\,dim_\\C G^\\C = dim_\\R G\\,$, where $\\,G^\\C\\,$ is the universal complexification of $\\,G\\,$. Then $\\,K^\\C\\,$ acts on $\\,G^\\C\\,$, the left action on $\\,M\\,$ induces a natural $\\,G\\,$-action on $\\,TM\\,$ and under certain extensibility assumptions on the geodesic flow of $\\,M\\,$ one obtains a real-analyti and $\\,G$-equivariant map $\\,P:TM \\to G^\\C / K^\\C\\,$ such that (see Theorem \\[sliceflow\\] for the precise statement)\n\nThis applies to the case of naturally reductive Riemannian homogeneous spaces (corollary \\[reductive\\]) and of generalized Heisenberg groups (see Sect.$\\,$4 and 5).",
"\n\nAs an application it is shown that for all generalized Heisenberg groups such maximal domain is neither holomorphically separable, nor holomorphically convex (Proposition \\[generalized\\]). ",
"We are not aware of previous non-Stein examples. ",
"In the case of the $3$-dimensional Heisenberg group we determine its envelope of holomorphy as well as a certain maximal Stein subdomain (Proposition \\[notstein\\]).",
"\n\nPreliminaries\n=============\n\nHere we introduce notations and briefly recall basic results we will need in the present paper. ",
"Let $\\,M\\,$ be a complete real-analytic Riemannian manifold which will be often identified with the zero section in its tangent bundle $\\,TM\\,$. Following [@LS] (see also [@GS] for an equivalent characterization) we say that a real-analytic complex structure defined on a domain $\\,\\Omega\\,$ of $\\,TM\\,$ is adapted if all complex leaves of the Riemannian foliation are submanifolds with their natural complex structure, i. e., for any geodesic $\\,\\gamma:\\R \\to \\C \\,$ the induced map $\\,f: \\C \\to TM\\,$ defined by $\\,(x + iy) \\mapsto y \\, \\gamma'(x)\\,$ is holomorphic on $\\,f^{-1}(\\Omega)\\,$ with respect to the adapted complex structure. ",
"Here $y \\, \\gamma^{\\prime}(t) \\in T_{\\gamma(t)} M$ is the scalar multiplication in the vector space $T_{\\gamma(t)} M$.\n\nThe adapted complex structure exists and is unique on a sufficiently small neighborhood of $\\,M\\,$ and if $\\,\\Omega\\,$ is a domain around $\\,M\\,$ in $\\,TM\\,$ on which it is defined, we refer to it as an [*adapted complexification*]{}. ",
"Sometimes these are called Grauert tubes. ",
"For later use we need the following\n\n\\[holomorphicity\\] Let $\\,F:TM \\to \\C\\,$ be a real-analytic map which is holomorphic on any complex leaf of $\\,TM\\,$ in a neighbourhood of $\\,M\\,$. Then $\\,F\\,$ is holomorphic on every adapted complexification.",
"\n\nFollowing the proof of $\\,$\\[Sz1, Prop.3.2 p.416\\]$\\,$ one checks that the restriction of $\\,F\\,$ to $\\,M\\,$ extends to a holomorphic map $\\,\\hat F\\,$ in a neighborhood $\\,U\\,$ of $\\,M \\subset TM\\,$ where the adapted complex structure $\\,J_0\\,$ exists and, in order to have connected leaves, $\\,U\\,$ may be assumed to be starshaped. ",
"By assumption we can also assume that for any geodesic $\\,\\gamma:\\R \\to M\\,$ the map $\\,x+iy \\,\\mapsto \\, F(y \\cdot \\gamma'(x))\\,$ is holomorphic for all $\\,x +iy\\,$ such that $\\,y \\, \\gamma'(x)\\in U$. Now $\\,F= \\hat F\\,$ on $\\,\\gamma(\\R)\\subset M\\subset TM\\,$, therefore $\\,F= \\hat F\\,$ on every complex leaf, i.e., on $\\,U\\,$. In particular $\\,DF \\circ J_0 = i \\,DF\\,$ on $\\,U\\,$ and since all maps are real-analytic the statement follows from the identity principle.",
"\n\nA real Lie group$\\,G\\,$ acts on a complex manifold $\\,X\\,$, i.e., $\\,X\\,$ is a $\\,G$-manifold, if there exists a real-analytic surjective map $\\,G \\times X \\to X\\,$ given by $\\,(g,x) \\mapsto g\\cdot x\\,$ such that for fixed $\\,g\\in G\\,$ the map $\\,x\\mapsto g\\cdot x\\,$ is holomorphic and $\\ (gh)\\cdot x=g\\cdot(h \\cdot x) \\ $ for all $\\,h,g \\in G\\,$ and $\\,x\\in X$. Furthermore if $\\,\\dim_\\R \\,G = \\dim_\\C \\,G^\\C\\,$, where $\\,\\iota:G \\to G^\\C\\,$ is the universal complexification of $\\,G\\,\\,$ (see e.g. [@Ho]),$\\,$ then $\\,Lie(G^\\C)= \\g^\\C\\,$ and one obtains an induced local holomorphic $\\,G^\\C$-action by integrating the holomorphic vector fields given by the $\\,G$-action. ",
"Here $\\, \\g\\,$ denotes the Lie algebra of $\\,G$.\n\nLet $\\,M=G/K\\,$ be a Riemannian homogeneous manifold with $\\,G\\,$ a connected Lie group of isometries and $\\,K\\,$ compact, and consider the induced $\\,G$-action on $\\,TM\\,$ defined by $\\,g \\cdot w :=g_*w\\,$ for all $\\,g \\in G\\,$ and $\\,w \\in TM\\,$. Then if $\\,\\Omega \\,$ is a $\\,G$-invariant adapted complexification, as an easy consequence of the definitions $\\,g_*\\,$ is a biholomorphic extension of the isometry $\\,g\\,$, i.e., $\\,G \\subset Aut(\\Omega)\\,$.\n\nIf one assumes that $\\,\\dim_\\R \\,G = \\dim_\\C \\,G^\\C\\,$, then the natural map $\\,\\iota:G \\to G^\\C\\,$ is an immersion and from the universality property of the universal complexification $\\,K^\\C\\,$ of $\\,K\\,$ it follows that the restriction $\\,\\iota |_K\\,$ of $\\,\\iota\\,$ to $\\,K\\,$ extends to an immersion $\\,\\iota^{\\C} :K^{\\C} \\to\nG^\\C\\,$. Moreover the subgroup $\\,\\iota^\\C (K^\\C)\\,$ acts by right multiplication on $\\,G^\\C\\,$ and one has a commutative diagram $$\\begin{matrix} G \\ & \\buildover{\\iota} \\to \\ & \\ \\ G^\\C \\cr\n & & \\cr \n \\downarrow\\ & & \\ \\downarrow \\cr\n & & \\cr\n G/K \\ & \\to \\ & \\ \\ \\ \\ G^\\C/\\iota^\\C(K^\\C) \\,.\\cr \n\\end{matrix}$$\n\nAlso notice the $\\,G$-action on $\\,G^\\C/\\iota^\\C(K^\\C)\\,$ defined by $\\,g \\cdot h\\,\\iota^\\C(K^\\C) := \\iota(g)\\, h\\,\\iota^\\C(K^\\C)\\,$ for all $\\,g \\in G\\,$ and $\\,h \\in G^\\C\\,$.\n\n\\[goodquotient\\] Let $\\,G\\,$ be a connected Lie group, $\\,K\\,$ a compact subgroup and assume that $\\dim_\\C G^\\C = \\dim_\\R G$. Then $G^\\C/\\iota^\\C(K^\\C)\\,$ is a complex $\\,G$-manifold and $\\,\\dim_\\C \\,G^\\C/\\iota^\\C(K^\\C) = \\dim_\\R\\, G/K.$\n\nOne needs to show that $\\,\\iota^\\C(K^\\C)\\,$ is closed in $\\,G^\\C\\,$. Note that $\\,G^\\C/\\iota^\\C(K^\\C)\\,$ is the orbit space with respect to the $\\,K^{\\mathbb{C}}$-action on $G^{\\mathbb{C}}\\,$ defined by $\\,k \\cdot h := h\\,\\iota(k^{-1})\\,$ for all $\\,k \\in K^\\C\\,$ and $\\,h \\in G^\\C\\,$. Since $\\,G^\\C\\,$ is Stein ([@He]) and $\\,K^\\C\\,$ is reductive it follows that every fiber of the categorical quotient $\\,G^{\\mathbb{C}} \\to G^{\\mathbb{C}} \\qq K^{\\mathbb{C}}\\,$ is equivariantly biholomorphic to an affine algebraic variety on which $\\,K^{\\mathbb{C}}\\,$ acts algebraically ([@Sn]). ",
"In particular there exists at least one closed $\\,K^\\C$-orbit and consequently $\\,\\iota^\\C (K^{\\mathbb{C}})\\,$ is closed in $\\,G^{\\mathbb{C}}\\,$. Thus $\\,G^{\\mathbb{C}}/\\iota^\\C(K^{\\mathbb{C}})\\,$ is a complex $\\,G$-manifold and by construction its complex dimension is $\\,\\dim_\\R\\,G/K \\,$.\n\nA characterization of maximal adapted complexifications\n=======================================================\n\nIf $\\,M=G/K\\,$ is a symmetric space of the non-compact type, then $\\,G^\\C/K^\\C\\,$ is a natural candidate for a complexification of $\\,M\\,$ and there exists a $\\,G$-equivariant map $\\,P:TM \\to \\,G^\\C/K^\\C\\,$ embedding holomorphically a maximal adapted complexification of $\\,M\\,$ (see [@BHH], [@Ha], [@AG]). ",
"As a matter of fact one may show that $\\,DP\\,$ is singular on the border $\\,\\partial \\Omega\\,$ of $\\,\\Omega$.\n\nHere we consider a Riemannian homogeneous manifold $\\,M=G/K\\,$ endowed with the additional data of a certain real-analytic $\\,G$-equivariant map $\\,P\\,$ from $\\,TM\\,$ to a suitable complex $\\,G$-manifold, characterizing a maximal adapted complexification $\\,\\Omega_M\\,$ as the connected component of $\\,\\{\\,DP$ not singular $\\}\\,$ containing $\\,M$. Unicity of $\\,\\Omega_M\\,$ follows.",
"\n\nThe existence of such a data is proved when $\\dim_\\C G^\\C = \\dim_\\R G$ and the geodesic flow “extends” holomorphically on $G^\\C/\\iota^\\C(K^\\C)\\,$ (cf. ",
"Lemma\\[goodquotient\\]). ",
"As a consequence the characterization applies to the case of naturally reductive Riemannian homogeneous spaces and of generalized Heisenberg groups.",
"\n\n\\[equivariant-theorem\\] Let $\\,M=G/K\\,$ be an $n$-dimensional Riemannian homogeneous space and $\\,X\\,$ a $\\,G$-complex manifold of complex dimension $n$ such that the induced local $\\,G^\\C\\,$-action is locally transitive. ",
"Assume there exists a real-analytic map $\\,P:TM \\to X\\,$ which is $$\\begin{aligned}\n \\ \\ & i) \\ \\ \\ {G-equivariant \\ and} & \\\\\n \\ \\ & ii) \\ \\ \n{holomorphic \\ on\\ every \\ complex \\ leaf \\ of \\ TM.} ",
" & \n\\quad \\quad \\quad \\quad \\quad \\quad \\quad\n\\quad \\quad \\quad \\quad \\end{aligned}$$\n\nThen the connected component $\\,\\Omega_M\\,$ of $\\, \\{ \\,p \\in TM \\, : \\, DP_p {\\rm \\ \\ not\\ singular\\ }\\,\\}\\,$ containing $\\,M\\,$ is the unique maximal adapted complexification and $\\,P|_{\\Omega_M}\\,$ is locally biholomorphic.",
"\n\nFirst we show that $\\,\\Omega_M\\,$ is well defined, i.e., $\\,DP\\,$ has maximal rank along $\\,M\\,$. Since from Lemma \\[holomorphicity\\] it follows that $\\,P\\,$ is holomorphic on $\\,M\\,$ with respect to the adapted complex structure, this is a consequence of the following\n\nAssume that $\\,P\\,$ is holomorphic in $\\,p \\in TM$. Then $\\,DP_p\\,$ has maximal rank.",
"\n\n$\\ \\ $ Since $\\,G^\\C\\,$ acts locally transitively on $\\,X\\,$, there exist elements $\\,\\xi_1, \\cdots, \\xi_n\\,$ of $\\, \\mathfrak g \\,$ such that the induced vector fields $\\,\\xi_{X,1}, \\cdots,\n\\xi_{X,n}\\,$ on $\\,X\\,$ span a totally real and maximal dimensional subspace $\\,V_{P(p)} \\,$ of $\\,T_{P(p)}X\\,$, where $$\\,\\xi_{X,j}(x):=\\frac{d}{dt}\\biggr|_0 \\exp_{G^\\C}(t\\xi_j) \\cdot x\\,$$ for $\\,j=1, \\cdots, n\\,$ and all $x \\in X$. By equivariance it follows that $\\,DP_p(V_p) =V_{P(p)}\\,$, where $\\,V_p\\,$ is the subspace of $\\,T_pTM\\,$ spanned by $\\,\\xi_{TM,1}, \\cdots , \\xi_{TM,n}\\,$, here $\\,\\xi_{TM,j}(q):=\\frac{d}{dt}\\biggr|_0 \\exp_{G}(t\\xi) \\cdot q\\,$ for all $\\,q \\in TM$. In particular $\\,{\\mathrm dim}_\\R V_p = n\\,$ and since $\\,P\\,$ is holomorphic in $\\,p\\,$, $\\,V_p\\,$ is totally real and $\\,DP_p\\,$ has maximal rank, proving the claim.",
"\n\nNow we see that the pulled-back complex structure $\\,J_o\\,$ on $\\,\\Omega_M\\,$ of the complex structure $\\,J\\,$ on $\\,X\\,$ is the adapted complex structure. ",
"For this consider a complex leaf $\\,f:\\C \\to TM\\,$ defined by $\\,f(x+iy):=y \\cdot \\gamma'(x)\\,$, where $\\,\\gamma\\,$ is a geodesic of $\\,M\\,$, and note that by $ii)$ $$\\,DP \\circ Df(i \\eta) =DP \\circ J_o \\circ\nDf(\\eta)\\,$$ for all $\\,\\eta \\,$ tangent in $\\,f^{-1}(\\Omega_M)\\,$. Since $\\,DP\\,$ has maximal rank on $\\,\\Omega_M\\,$, then $$\\,Df(i \\eta) = J_o \\circ Df(\\eta)\\,$$ showing that $\\,J_o\\,$ is the adapted complex structure. ",
"In particular $\\,P|_{\\Omega_M}\\,$ is locally biholomorphic.",
"\n\nIn order to prove maximality, assume that $\\,J_o\\,$ extends analytically in a neighborhood of a certain $\\,p\\in \\partial \\Omega_M\n\\subset TM\\,$. By construction $DP \\circ J_o = J \\circ DP \\,$ on $\\,\\Omega_M\\,$ and since all maps are real-analytic $\\,P\\,$ is holomorphic in $\\,p\\,$. Then the above claim shows that $\\,DP_p\\,$ has maximal rank, contradicting the definition of $\\Omega_M$.\n\nFinally we want to show that any adapted complexification $\\,\\Omega\\,$ is contained in $\\,\\Omega_M$. If this is not the case, there exists a point $\\,p\\,$ in $\\,\\Omega \\cap \\partial \\Omega_M$ and from Lemma \\[holomorphicity\\] it follows that $\\,P|_\\Omega\\,$ is holomorphic. ",
"In particular $\\,P\\,$ is holomorphic in $\\,p\\,$ and one obtains a contradiction arguing as above. ",
"Thus $\\,\\Omega_M$ is unique and this concludes the statement.",
"\n\nNow we determine a class of Riemannian homogeneous spaces to which Proposition \\[equivariant-theorem\\] may be applied in order to determine the maximal adapted complexification.",
"\n\n\\[sliceflow\\] $\\ \\ \\ \\ $ Let $\\,M =G/K\\,$ be a Riemannian homogeneous space with $\\, \\dim_\\R \\,G = \\dim_\\C \\,G^\\C\\,$ and assume there exists a map $\\, \\varphi: \\C \\times T_K M \\mapsto \\mathfrak g^\\C \\,$ real-analytic and holomophic on the first component such that $\\,\\varphi(\\R \\times T_K M) \\subset \\mathfrak g\\,$ and $\\, t \\mapsto \\exp_G \\circ \\varphi (t,v)K\\,$ is the unique geodesic tangent to $\\,v\\,$ at $\\,0\\,$ for all $\\,v \\in T_K M\\,$. Then the map $$P: TM \\mapsto G^\\C / \\iota^\\C(K^\\C)$$ defined by $$P(g_*(v)):= \\iota(g)\\,\\exp_{G^\\C}(\\varphi(i,v))\\,\n\\iota^\\C(K^\\C)$$\n\nfor all $\\, g \\in G\\,$ and $\\,v \\in T_KM$ is as in Proposition \\[equivariant-theorem\\]. ",
"In particular the connected component $\\,\\Omega_M\\,$ of $\\,\\{ \\,p \\in TM \\, : \\, DP_p {\\rm \\ \\ not\\ singular\\ }\\,\\}\\,$ containing $\\,M\\,$ is the maximal adapted complexification and $\\,P|_{\\Omega_M}\\,$ is locally biholomorphic.",
"\n\nIn order to prove that $\\,P\\,$ is well defined we need to show that if $\\,w=k_*v\\,$ for some $\\,k \\in K\\,$ and $\\,v \\in T_K M\\,$ then $\\,P(w)=P(k_*v)\\,$, i. e., $$\\label{welldefined}\n\\, \\exp_{G^\\C}(\\varphi(i,w))\\,\\iota^\\C(K^\\C) =\n\\iota(k)\\,\\exp_{G^\\C}(\\varphi(i,v))\\,\\iota^\\C(K^\\C)\\, .$$\n\nFor this note that $\\, t \\mapsto k\\,\\exp_G \\circ \\varphi (t,v)K\\,$ is the unique geodesic tangent to $\\, w \\,$ at $\\,0\\,$ in $K$, thus $$\\,\\exp_G \\circ \\varphi (t,w)K\\,=k\\,\\exp_G \\circ \\varphi (t,v)K\\,.$$\n\nThen the commutativity of the diagram $$\\begin{matrix} \\g \\ \\ \\ \\ \\ \\ & \\to \\ & \\g \\ ^\\C \\ \\ \\ \\ \\cr\n & & \\cr\n \\downarrow\\, \\exp_G & & \\ \\downarrow \\, \\exp_{G^\\C} \\cr\n & & \\cr\n G \\ \\ \\ \\ \\ \\ \\ & \\buildover{\\iota} \\to \\ & G^\\C \\ \\ \\ \\ \\ \\cr\n\\end{matrix}$$\n\nimplies that $$\\,\\exp_{G^\\C} \\circ \\varphi (t,w)\\,\\iota^\\C(K^\\C)\\,=\n\\iota(k)(\\,\\exp_{G^\\C} \\circ \\varphi (t,v)\\,) \\iota^\\C(K^\\C)\\,$$\n\nfor all $\\,t \\in \\R\\,$ and equation (\\[welldefined\\]) is a consequence of the identity principle for holomorphic maps.",
"\n\nNow define $\\,\\varPhi(z,v):=\\exp_{G^\\C} \\circ \\varphi (z,v)\\,$ for all $\\,z \\in \\C\\,$ and $\\,v \\in T_K M\\,$ and, in order to simplify notations, assume that the canonical immersion $\\,\\iota :G \\to G^\\C\\,$ is injective so that once we identify $\\, G \\,$ with $\\iota(G)$ the curve $\\, \\gamma(t) := \\varPhi(t,v)K\\,$ is the unique geodesic tangent to $\\, v\\,$ at $\\,0\\,$. In what follows it is easy to check that all arguments apply to the case where $\\,\\iota\\,$ is a non-injective immersion.",
"\n\nFix $\\,x \\in \\R\\,$, let $g:=\\varPhi(x,v)\\,$ and note that $$y \\mapsto g\\,\\varPhi(y, g_*^{-1}\\gamma'(x))K$$\n\nis the unique geodesic tangent to $\\,\\gamma'(x)\\,$ at $\\,0\\,$. Therefore one has $$\\varPhi(x,v)\\,\\varPhi(y,g_*^{-1}\\gamma'(x))K =\n\\varPhi(x+y,v)K$$\n\nfor all $\\,y\\in \\R\\,$ and by the identity principle it follows that $$\\label{flowrule}\n \\varPhi(x,v)\\,\\varPhi(z,g_*^{-1}\\gamma'(x))\\,\\iota^\\C(K^\\C) =\n\\varPhi(x+z,v)\\,\\iota^\\C(K^\\C)$$\n\nfor all $\\,v \\in T_K M, \\ x \\in \\R\\,$ and $\\, z \\in \\C\\,$. For $\\, h \\in G \\,$ and $\\,v\\in T_KM\\,$ consider the unique geodesic $\\,\\tilde \\gamma(t) :=h \\cdot \\gamma(t)\\,$ tangent to $\\,h_*(v)\\,$ at $\\,0\\,$. One has $$P(y\\tilde \\gamma'(x))=P(h_*y \\gamma'(x))=\nh\\,P(g_*g_*^{-1}y \\gamma'(x))=\nhg\\,\\varPhi(i,y \\,g_*^{-1}\\gamma'(x))\\,\\iota^\\C(K^\\C)=$$ $$h\\,\\varPhi(x,v)\\,\\varPhi(iy,\\,g_*^{-1}\\gamma'(x))\\,\\iota^\\C(K^\\C)=\nh\\,\\varPhi(x +iy,v)\\,\\iota^\\C(K^\\C),$$\n\nwhere we used (\\[flowrule\\]) and the fact that $\\, \\varPhi(z,yv)\\,\\iota^\\C(K^\\C)= \\varPhi(zy,v)\\,\\iota^\\C(K^\\C)\\,$ for all $\\,z \\in \\C\\,$, since this holds for all $\\,z \\in \\R\\,$. As a consequence the map $\\, (x+iy) \\mapsto P(y\\tilde \\gamma'(x))\\,$ is holomorphic for all geodesics $\\, \\tilde \\gamma \\,$ of $\\,M\\,$, i.e., $\\,P\\,$ is holomorphic on every complex leaf of $\\,TM$.\n\nFinally the map $\\, P \\,$ is $\\,G$-equivariant by construction and the $\\,G$-action on $G^\\C/\\iota^\\C(K^\\C)$ induces a holomorphic $G^\\C$-action which may be obtained through left multiplication on $G^\\C$. Thus it is obviously transitive and this yields the statement.",
"\n\nNow let $\\,M\\,$ be a naturally reductive Riemannian homogeneous space and $\\,M=G/K\\,$ be a natural realization of $\\,M\\,$, i.e., there exists a reductive decomposition $\\,\\g = Lie(K) \\oplus \n\\mathfrak{m}\\,$ of the Lie algebra of $\\,G\\,$ such that every geodesic in $\\,M\\,$ is the orbit of a one parameter subgroup of $\\,G\\,$ generated by an element of $\\,\\mathfrak{m}\\,$ (see e.g. [@BTV]). ",
"Consider the natural projection $\\, \\Pi: G \\to M\\, $ and note that $\\,D\\Pi_e(\\mathfrak{m}) =\nT_K\\,M$, where $\\,e\\,$ is the neutral element of $\\,G$. Denote by $\\,L: T_K M \\to \\mathfrak{m}\\,$ the inverse of the restriction of $\\,D\\Pi_e\\,$ to $\\, \\mathfrak{m}\\,$. Since $\\,L \\,$ is linear it extends $\\,\\C$-linearly from $\\,(T_K M)^\\C\\,$ to $\\, \\mathfrak{m}^\\C\\,$ and the map $\\, \\varphi: \\C \\times T_K M \\to \n\\mathfrak{g}^\\C\\,$ defined by $\\, \\varphi(z,v):=zL(v)\\,$ is as in the above Theorem. ",
"Therefore one has\n\n\\[reductive\\] Let $M =G/K$ be a natural realization of a naturally reductive Riemannian homogeneous space and assume that $\\dim_{\\mathbb{R}}G = \\dim_{\\mathbb{C}}G^{\\mathbb{C}}$. Then the map $$P: TM \\to G^{\\mathbb{C}}/\\iota(K^{\\mathbb{C}})$$ defined by $$P(g_*(v)):= \\iota(g)\\,\\exp_{G^\\C}(iL(v))\\,\\iota(K^\\C)$$\n\nfor all $\\, g \\in G\\,$ and $\\,v \\in T_KM$ meets the conditions of Proposition \\[equivariant-theorem\\]. ",
"In particular the connected component $\\,\\Omega_M\\,$ of $\\,\\{ \\,p \\in TM \\, : \\, DP_p {\\rm \\ \\ not\\ singular\\ }\\,\\}\\,$ containing $\\,M\\,$ is the maximal adapted complexification and $\\,P|_{\\Omega_M}\\,$ is locally biholomorphic.",
"\n\nThe 3-dimensional Heisenberg group\n==================================\n\nHere we apply results of the previous section in order to give a concrete description of the unique maximal adapted complexification for the 3-dimensional Heisenberg group. ",
"It turns out that such domain is neither holomorphically separable, nor holomorphically convex. ",
"We also determine its envelope of holomorphy and a particular maximal Stein subdomain. ",
"We remark that in all previous examples we are aware of, maximal adapted complexifications are Stein.",
"\n\nConsider the 3-dimensional Heisenberg group defined as a subgroup of $\\, GL_3(\\R)\\, $ by [$$\\begin{aligned}\n H := \\left \\{ \\left( \\begin{array}{ccc}\n 1 & \\alpha & \\gamma \\\\\n 0 & 1 & \\beta \\\\\n 0 & 0 & 1\n \\end{array} \\right)\\ : \\ \n\\alpha, \\beta, \\gamma \\in \\R \\right \\}, \n& \\end{aligned}$$ ]{}\n\nfix the inner product of the tangent space $\\,T_eH\\,$ in the neutral element $\\,e\\,$ for which the canonical basis determined by the global natural chart $\\,(\\alpha, \\beta, \\gamma)\\,$ is orthonormal and let $\\,(a,b,c)\\,$ be coordinates of $\\,T_eH\\,$ with respect to this basis. ",
"Endow $\\,H\\,$ with the induced $\\,H$-invariant metric $$(\\mathrm{d} \\alpha)^2 + (\\mathrm{d} \\beta)^2 + \n (\\mathrm{d} \\gamma - \\alpha \\cdot \\mathrm{d} \\beta)^2,$$\n\nlet $\\,\\h=Lie(H)\\,$ and define $\\,\\varphi:\\R \\times T_eH \\,\\to\\, \\h \\,$ by\n\n[$$\\begin{aligned}\n\\varphi(t,(a,b,c)) \\,:=\\,\n \\left (\\,a \\frac{\\sin(tc)}{c} - b \\frac{1 -\\cos(tc)}{c}\\ , \\ \n b \\frac{\\sin(tc)}{c} + a \\frac{1 -\\cos(tc)}{c}\\ , \\ \n \\left ( t +\\frac{a^2 +b^2}{2c^2} \\left ( t - \\frac{\\sin(tc)}{c}\n \\right ) \\right) c \\,\n\\right ), & \\end{aligned}$$ ]{}\n\nwhere the coordinates of $\\,\\h\\,$ are induced by those of $\\,T_eH\\,$ via the natural identification $\\,\\h \\cong T_eH$. Note that all singularities are removable and consequently $\\,\\varphi \\,$ is real-analytic. ",
"Following \\[BTV, Th.p.31\\] one checks that $\\,t \\mapsto \\exp_H \\circ\\varphi(t,(a,b,c))\\,$ is the unique geodesic tangent to $\\,(a,b,c)\\,$ at $ \\,0\\,$. Furthermore by expanding the power series it is easy to verify that $\\, \\varphi(\\, \\cdot \\, ,(a,b,c))\\, $ extends holomorphically on $\\,\\C\\,$ to $\\,(T_eH)^\\C\\,$ and by considering the polar decomposition $\\,H \\times \\h \\to H^\\C\\,$ of $\\,H^\\C\\,$ given by $\\,(g,\\xi) \\,\\mapsto \\,g\\,\\exp_{H^\\C}(i\\xi)\\,$, one obtains real-analytic functions $\\, (a,b,c) \\mapsto h_{(a,b,c)} \\in H\\,$ and $\\,(a,b,c) \\to \\xi_{(a,b,c)} \\in \\h\\,$ such that $$\\exp_{H^\\C} \\circ \\varphi(i,(a,b,c))\\,=\\,\n h_{(a,b,c)}\\, \\exp_{H^\\C} (i\\xi_{(a,b,c)}).$$\n\nDefine $\\,P:TH \\,\\to\\, H^\\C \\, \\cong \\,H \\times \\h \\,$ by $$\\,g_*(a,b,c) \\, \\mapsto \\, g\\, \\exp_{H^\\C}\\circ \\varphi(i,(a,b,c)) \\,\\cong\n\\,\\left ( \\, gh_{(a,b,c)}\\, , \\, \\xi_{(a,b,c)} \\,\\right ) .$$\n\nThen Theorem \\[sliceflow\\] implies that the connected component $\\,\\Omega_H\\,$ containing $\\,H\\,$ of $\\,\\{\\,DP$ not singular $\\}$ is the maximal adapted complexification. ",
"Note that since $\\,P\\,$ is $\\,H$-equivariant, $\\,\\Omega_H\\,$ is $\\,H$-invariant. ",
"Moreover $\\,T_eH\\, $ is a global slice for the $\\,H$-action on $\\,TH\\,$, i.e., the map $\\,H \\times T_eH \\to TH\\,$ given by $\\,(g,(a,b,c))\\to g_*(a,b,c)\\,$ is a $\\,H$-equivariant real-analytic diffeomorphism, thus $\\,\\Omega_H\\,$ is completely determined by its slice $\\,\n\\Omega_H\\cap T_eH$.\n\nFurthermore $\\,H\\,$ acts freely on the first component of $\\,H \\times \\h \\,$, then $\\,H$-equivariance of $\\, P \\,$ implies that $\\, DP_{g_*(a,b,c)} \\,$ has maximal rank if and only if $\\, D\\tilde P_{(a,b,c)} \\,$ has maximal rank, where $\\, \\tilde P\\, := \\,p_2 \\circ P|_{T_eH}:T_eH \\,\\to \\,\\h\n\\,$ is given by\n\n[$$\\begin{aligned}\n\\tilde P(a,b,c) \\,=\\, \\xi_{(a,b,c)}\\, =\\,\n\\left ( \\,a \\frac{\\sinh(c)}{c}\\ , \\ \n b \\frac{\\sinh(c)}{c} \\ , \\ \n \\left ( 1 +\\frac{a^2 +b^2}{2c^3} \\left ( c- \\sinh(c)\\,\\cosh(c)\n\\right ) \\right) c \\,\\right ).",
"\n& \\end{aligned}$$ ]{}\n\nHere $\\,p_2:H \\times \\h \\,\\to \\,\\h \\,$ is the canonical projection. ",
"It follows that $\\,\\Omega_H=H\\cdot O_0\\,$, where $\\,O_0\\,$ is the connected component of $\\{ \\,\\det(D\\tilde P)\\, \\not= \\,0 \\}$ containing $\\,0\\,$ in $\\,T_eH\\, $. ",
"Now a straightforward computation shows that [$$\\det(D\\tilde P_{(a,b,c)} )\\,=\\, \\frac{\\sinh(c)}{c} \\left (\\,\\frac{\\sinh(c)}{c}\n+ (a^2 +b^2) \\left (\\frac{\\sinh(c) -c \\,\\cosh(c)}{c^3} \\right ) \\right),$$ ]{}\n\ntherefore\n\n[$$\\begin{aligned}\nO_0= \\left \\{ \\,(a,b,c) \\in T_eH \\ : \\ a^2 + b^2 < \\frac{ c^2 \\, \n \\sinh(c)}{c\\,\\cosh(c) - \\sinh(c)} \\right \\}.",
"\n& \\end{aligned}$$ ]{}\n\nWe want to discuss injectivity of $\\,P|_{\\Omega_H}:\\Omega_H \\, \\to \\, \\, H^\\C \\,\n\\cong \\,H \\times \\h \\,$ and again this is equivalent to injectivity of $\\,\\tilde P|_{O_0}\\,$.\n\nNote that $\\,\\tilde P\\,$ is equivariant with respect to rotations around the $\\,c$-axis as well as to the reflection $\\, \\sigma \\,$ with respect to the plane $\\,\\{ \\, c=0 \\, \\}\\,$. In particular for any [$$\\begin{aligned}\n\\, (a,b,c) \\in \\left \\{ 1 +\\frac{a^2 +b^2}{2c^3}\n\\left ( c - \\sinh(c)\\,\\cosh(c)\\right ) \\,=\\,0 \\right \\}\\,\n& \\end{aligned}$$ ]{}\n\none has $\\, \\tilde P(a,b,c) =\\tilde P(a,b,-c) =\n\\left (\\,a\\,\\frac{\\sinh(c)}{c}, b\\,\\frac{\\sinh(c)}{c}, 0 \\,\\right ) \\,$. Therefore we are induced to investigate the domain\n\n[$$\\begin{aligned}\nO_1:= \\left \\{ \\,(a,b,c) \\in T_eH \\ : \\ a^2 + b^2 \n< \\frac{ 2c^3}{\\sinh(c)\\cosh(c) - c} \\, \\right \\}.",
"\n& \\end{aligned}$$]{}\n\n\\[injectivdomain\\] The domain $O_1\\,$ is the maximal $\\,\\sigma \\,$-invariant subdomain of $\\,O_0\\,$ containing $\\,0\\,$ on which $\\, \\tilde P\\,$ is injective. ",
"In particular $\\, \\tilde P|_{O_0}\\,$ is not injective.",
"\n\nLet $\\, f_j\\,$ be the real function defining $$\\,O_j\\,=\\,\\{ \\,(a,b,c) \\in T_eH \\ : \\ a^2 + b^2 \n< f_j(c)\\,\\}\\,$$ for $\\,j=0,1 \\,$. First we want to show that $\\,O_1\\,$ is a subdomain of $\\,O_0\\,$, i.e, $$f_1(c)\\,\\leq\\,f_0(c)$$\n\nfor all $\\,c \\in \\R\\,$, which is equivalent to $$2\\cosh(c) \\leq \\frac{\\sinh(c)}{c} +\\frac{\\sinh^2(c)}{c^2}\\cosh(c).$$\n\nExpanding in power series one obtains [$$\\begin{aligned}\n2 \\,+\\, c^2 &+\\frac{1}{12}c^4\\, + \\, \\cdots \\ \\,\\leq\\,\\, \n \\left( 1\\,+\\frac{1}{6}c^2\\,+\\,\\frac{1}{120}c^4\\,+\\,\\cdots \\right )\n\\,+\\,\\left ( 1\\, +\\,\\frac{5}{6}c^2\\,+\\,\\left(\\frac{5}{6}\\,+\n\\frac{1}{24}\\,+\\,\\cdots\\, \\right)c^4 \\,+ \\, \\cdots \\,\\right ). & ",
"\\end{aligned}$$]{}\n\nAll coefficients are non-negative and one easily checks that for $\\, k \\geq 2\\,$ the coefficient of $ \\,c^{2k}\\,$ in the last series on the right side is strictly greater than that in the series on the left, hence $\\,O_1\\subset O_0\\,$. Moreover $\\,\\partial O_1 \\cap \\partial O_0\\,= \\, \\{\\,\n(a,b,0) \\in T_eH \\ : \\ a^2 + b^2 =3 \\, \\}$, thus $\\,O_1\\,$ is a proper subdomain of $\\,O_0$.\n\nFurthermore by the previous remarks any $\\,\\sigma$-invariant domain containing $\\,0\\,$ on which $\\, \\tilde P\\,$ is injective is necessarily contained in $\\,O_1 \\,$.\n\nAssume that there exist $\\,(a',b',c'), \\,(a'',b'',c'') \\in O_1\\,$ such that $\\, \\tilde P(a',b',c')=\\tilde P(a'',b'',c'')=:(A,B,C).$ If $\\,C=0\\,$ then $\\,c' =c''=0\\,$ and consequently $\\,a'=a'' =A \\,$ and $\\,b=b''=B\\,$. If $\\,C\\not=0\\,$ by eventually acting with $\\,\\sigma\\,$ and a rotation around the $\\,c$-axis we may assume that $\\, a,A\\geq 0\\,$, $\\,b=B=0\\,$ and $\\, c >0\\,$ . ",
"Now one has $$\\,a'\\frac{\\sinh(c')}{c'}=a''\\frac{\\sinh(c'')}{c''}=A\\,$$ therefore $\\,(a',0,c')\\,$ and $ \\,(a'',0,c'')\\,$ lie on the same level curve $\\,\\rho_A:\\R \\to T_eH\\,$ given by $$\\rho_A(t):=\\left(\\,A \\frac{t}{\\sinh(t)},0,t\\, \\right ).$$ One has the following\n\n$\\ \\ $Let $\\, A\\geq 0\\,$ and $\\, t_0\\in\\R^{\\geq 0} \\,$ such that $\\rho_A(t_0) \\in \\overline O_0\\,$. Then $\\rho_A(t) \\in O_0\\,$ for all $\\, t >t_0\\,$.\n\n$\\ \\ $ One needs to show that $A^2\\frac{t^2}{\\sinh^2(t)} \\,< \\, f_0(t)$ for all $\\,t>t_0\\,$, that is [$$\\label{disugua}\nA^2\\,<\\,\\frac{\\sinh^2(t)}{t^2} f_0(t).$$]{}\n\nBy expanding in power series as above one has the estimate $$2t\\,\\cosh^2(t) -3\\cosh(t)\\sinh(t) + t >0,$$\n\nfor all $\\,t>0\\,$, which by a straightforward computation implies that the derivative of the function at the right hand side of (\\[disugua\\]) is positive for all $\\,t >0\\,$, proving the claim.",
"\n\nNow let $\\,t_0 := \\min(c',c'')\\,$ and note that since $\\,O_1 \\subset O_0\\,$, then as a consequence of the above claim there exists $\\,\\epsilon >0\\,$ such that $\\,\\rho_A(t) \\in O_0 \\,$ for $\\,t >t_0-\\epsilon$. In particular $\\,(a',0,c')\\,$ and $ \\,(a'',0,c'')\\,$ lie in the same connected real one dimensional submanifold $\\,N:=\\rho_A(t_0-\\epsilon, \\infty)\\,$ of $\\,O_0\\,$ and $\\,\\tilde P|_{N}:N \\,\\to\\, \\{(A,0,\\,\\cdot\\,) \\in T_eH\\} \\, \\cong \\,\\R\\,$ is locally diffeomorphic. ",
"Then a classical argument implies that $\\,\\tilde P|_{N}\\,$ is injective, thus $\\,(a',0,c')\\,= \\,(a'',0,c'')\\,$ as wished.",
"\n\nWe also want to determine the image of $\\,P|_{\\Omega_H}\\,$ in $\\,H^\\C\\,$. Note that $\\,P(\\Omega_H)\\,$ is $\\,H$-invariant and the polar decomposition implies that $\\,\\exp_{H^\\C}(i\\g)\\,$ is a global slice for the $\\,H$-action on $\\,H^\\C\\,$. Then this can be achieved by describing $\\,exp_{H^\\C}(i\\tilde P(O_0))=P(\\Omega_H)\\cap \\exp_{H^\\C}(i\\g)\\,$.\n\n\\[imagedomain\\]$\\,\\tilde P(O_0)\n\\,=\\, \\h \\setminus \\{ \\,(A,B,C) \\in \\h\n\\ :\\ A^2 +B^2 =3, \\ C=0 \\}.$\n\nLet $\\,(a,0,c) \\in \\{a^2 =f_1(c)\\} \\subset \\partial O_1\\,$ with $ \\, c > 0\\,$. From the proof of Lemma \\[injectivdomain\\] it follows that $\\,(a,0,c) \\in O_0\\,$. Since $\\,\\tilde P(a,0,c)\\,=\\,(\\,a\\frac{\\sinh(c)}{c},0,0\\,)\\,$ and $$\\sqrt{f_1(0)} = \\sqrt 3\\quad {\\rm and} \\quad \\quad \\ \\ \n\\,\\lim_{c \\to\\infty} \\sqrt{f_1(c)} \\, \\frac{\\sinh(c)}{c} = \\infty$$ it follows that $\\,(A,0,0) \\in \\tilde P(O_0)\\,$ for all $\\,A >\\sqrt 3\\,$.\n\nFor $\\,A> \\sqrt 3 \\,$ let $\\,(a,0,c) \\in O_0\\,$ such that $\\,\\tilde P(a,0,c)\\,=\\,(\\,A,0,0\\,)\\,$. By the claim in Lemma \\[injectivdomain\\] one has $\\, \\rho_A(t) \\in O_0\\,$ for all $\\, t \\geq c\\,$. Moreover one sees that\n\n$$\\label{limit}\n\\,\\tilde P(\\rho_A(t))=(A,0,C_A(t)) \\quad \\quad {\\rm with} \\quad \\quad\n\\,\\lim_{t \\to\\infty}C_A(t) = \\infty\\,.$$\n\nThen by $\\,\\sigma$-invariance of $\\,O_0 \\,$ and $\\,\\sigma$-equivariance of $\\,\\tilde P \\,$ it follows that $\\,(A,0,C) \\in\n\\tilde P(O_0)\\,$ for all $\\,C\\in \\R,$ and $\\,A >\\sqrt 3\\,$.\n\nNow note that $\\,\\tilde P(a,0,0)=(a,0,0)\\,$ and $\\,f_0(0)=3\\,$, thus $\\,(A,0,0) \\in \\tilde P(O_0)\\,$ for all $\\,A< \\sqrt 3\\,$ and arguing as above it follows that $\\,(A,0,C) \\in \\tilde P(O_0)\\,$ for all $\\,C\\in \\R\\,$ and $\\,A < \\sqrt 3\\,$.\n\nFinally $\\, \\rho_{\\sqrt 3}(0) \\in \\partial O_0\\,$, thus $\\, \\rho_{\\sqrt 3}(t) \\in O_0\\,$ for all $\\,t>0\\,$. It follows that $\\,t \\mapsto \\tilde P(\\rho_{\\sqrt 3}(t)) \\,$ is injective for $\\,t>0\\,$ and since $$\\,\\lim_{t \\to 0^+}C_{\\sqrt 3}(t) = 0\\\n\\quad {\\rm and} \\quad\n\\,\\lim_{t \\to\\infty}C_{\\sqrt 3}(t) = \\infty\\,$$\n\nthen $\\,(\\sqrt 3,0,C) \\in \\tilde P(O_0)\\,$ if and only if $\\,C\\not=0\\,$ .",
"\n\nThe statement follows from the invariance of $\\,O_0 \\,$ and the equivariance of $\\,\\tilde P \\,$ with respect to the group of rotations around the $\\,c$-axis.",
"\n\nIn the picture below one sees the border of $\\,O_0\\,$ and $\\,O_1\\,$ determined by $\\,f_0 \\,$ and $\\, f_1\\,$ respectively as well as the level curve $\\,\\rho_2\\,$ in the upper half-plane $\\,\\{\\, b =0, \\ a \\geq 0 \\}$ of $\\,T_eH$. Since $\\,O_0\\,$ and $\\,O_1\\,$ are invariant with respect to rotations around the $\\,c$-axis this completely determine their shape and, by $\\,H$-equivariance, that of $\\,\\Omega_H$.\n\n=0.5\n\n\\[notstein\\] The maximal domain $\\Omega_H\\,$ is neither holomorphically separable, nor holomorphically convex, its envelope of holomorphy is biholomorphic to $\\,\\C^3$. The maximal $\\,\\sigma$-invariant Stein subdomain of $\\,\\Omega_H\\,$ is biholomorphic to $\\,\\{\\, (z_1,z_2,z_3) \\in \n\\C^3 \\ : \\ (\\mathrm{Im}\\,z_1)^2\\, +\\, (\\mathrm{Im} z_2)^2\n < 3 \\, \\}$.\n\nFrom \\[CIT, Prop.p.543\\] it follows that any holomorphically separable Riemann $\\,H$-domain over $\\,H^\\C\\,$ is univalent. ",
"Moreover Lemma \\[injectivdomain\\] implies that $\\,P|_{\\Omega_H}:\\Omega_H \\, \\to \\, \\, H^\\C \\,$ is not injective, therefore $\\Omega_H\\,$ is not holomorphically separable.",
"\n\nBy a result of Loeb (\\[L,Th.p.186\\]), a Stein $\\,H$-invariant domain $\\,U\\,$ of $\\,H^\\C\\,$ is “geodesically\" convex, i.e., it is convex with respect to all curves of the form $\\, t \\mapsto g\\,\\exp_{H^\\C}(it \\xi)\\,$, with $\\,g \\in U\\,$ and $\\,\\xi \\in \\h\\,$. Since $\\,H^\\C\\,$ admits polar decomposition and $\\,U\\,$ is $\\,H$-invariant it is enough to consider curves of the form $\\,\\exp_{H^\\C}(i \\eta)\n\\,\\exp_{H^\\C}(it \\xi)\\,$, with $\\,\\exp_{H^\\C}(i \\eta) \\in U\\,$ and $\\,\\xi \\in \\h\\,$. Furthermore for a two steps nilpotent Lie group one has $$\\,\\exp_{H^\\C}(i\\eta)\\,\\exp_{H^\\C}(it\\xi)\\,\n=\\,\\exp_{H^\\C}( i\\eta \\,+ \\,it\\xi \\,- \\,\\frac{t}{2}[\\eta, \\xi] )\\,=$$ $$\\,\\exp_{H^\\C}( - \\,\\frac{t}{2}[\\eta, \\xi] )\\,\n\\exp_{H^\\C}( i(\\eta \\,+ \\, t\\xi)\\,)$$\n\nand using $\\,H$-invariance one more time we conclude that if $\\,U=H \\cdot \\,\n\\exp_{H^\\C}(iD)\\,$, with $\\,D \\,$ a domain in $\\,\\h\\,$, is Stein then $\\,D \\,$ is convex in the usual affine sense.",
"\n\nSince $\\,P(\\Omega_H)= H\\cdot exp_{H^\\C}(i\\tilde P(O_0))\\,$ and as a consequence of Lemma \\[imagedomain\\] the domain $\\,\\tilde P(O_0)\\,$ is not convex, then $\\,P(\\Omega_H) \\,$ is not Stein. ",
"Now $\\,H^\\C\\,$ is Stein and $\\,P|_{\\Omega_H}\\,$ is locally biholomorphic, therefore by [@R] there exists a commutative diagram $$\\begin{matrix}\n\\Omega_H & \\buildover j \\longrightarrow & \\hat \\Omega_H\\cr\n & & \\cr \n & P \\searrow &\\downarrow \\hat P \\cr\n & & \\cr \n & & H^\\C \\cr\n\\end{matrix}$$\n\nwhere $\\, \\hat \\Omega_H \\,$ is the envelope of holomorphy of $\\, \\Omega_H \\,$. Moreover $\\, H\\,$ acts on $\\, \\hat \\Omega_H \\,$ and all maps are $\\, H$-equivariant. ",
"Furthermore $\\,\\hat P \\,$ is injective by \\[CIT,Th.p.543\\] and if $\\, \\Omega_H \\,$ is holomorphically convex then $\\, j \\,$ is surjective and consequently $\\, \\hat \\Omega_H \\,$ is biholomorphic to $\\, P(\\Omega_H) \\,$, giving a contradiction. ",
"Hence $\\,\\Omega_H\\,$ is not holomorphically convex.",
"\n\nNotice that $\\,\\hat \\Omega \\cong \\hat P(\\hat \\Omega)\\,$ contains $\\, P(\\Omega_H) = H\\cdot \\exp_{H^\\C}(i\\tilde P(O_0)) \\,$ and the convex envelope of $\\, \\tilde P(O_0) \\,$ is $\\,\\h\\,$, thus by the above arguments the envelope of holomorphy $\\,\\hat \\Omega \\,$ is biholomorphic to $\\,H^\\C \\cong\n\\C^3$.\n\nFinally the maximal convex $\\,\\sigma$-invariant subdomain of $\\, \\tilde P(O_0)\\,$ is $\\,\\{ \\, (A,B,C) \\in \\h \\, : \\, A^2 +B^2 <\n3\\,\\} = \\tilde P(O_2)\\,$, where $\\,O_2:= \\{ \\, (a,b,c) \\in \\h \\, : \\,\na^2 +b^2 < 3\\frac{c^2}{\\sinh^2(c)}\\,\\}$. One checks that $\\,O_2 \\subset O_1\\,$, thus $\\,\\tilde P|_{O_2}\\,$ is injective and $\\, H \\cdot O_2\\,$ is biholomorphic to $\\,H\\cdot \\exp_{H^\\C}(i\\tilde\nP(O_2))\\,$. Moreover one has $$\\begin{aligned}\n\\exp_{H^\\C}(\\,(A',B',C') + & i(A,B,C) \\,)\\,= \\\\ \n & \\exp_H(A',B',C')\\,\n\\exp_{H^\\C}(\\,i(A,\\,B,\\,C-\\frac{1}{2}(A'B-AB')\\,)\\,).\\end{aligned}$$\n\nIt follows that $$\\,\\exp_{H^\\C}^{-1}(H\\cdot \\exp_{H^\\C}(i\\tilde P(O_2)) =\n\\,\\{\\, ({\\mathrm{Im}}\\,z_1)^2 + ({\\mathrm{Im}}\\,z_2)^2 < 3 \\,\\},$$\n\nwhere $\\,(z_1, z_2,z_3)=(A'+iA,B'+iB,C'+iC)\\,$ are natural complex coordinates of $\\,h^\\C \\cong \\C^3\\,$ and this yields the statement.",
"\n\n\\[O\\_1image\\] Since $\\,\\tilde P \\,$ is injective on $\\,O_1\\,$, then the $\\,H$-invariant domain defined by $O_1$ is holomorphically separable. ",
"As a matter of fact one may show that $\\,\\tilde P(O_1) \n\\,=\\, \\h \\setminus \\{ \\,(A,B,C) \\in \\h\n\\ :\\ A^2 +B^2 \\geq 3, \\ C=0 \\}$ and analogous arguments as above show that such $\\,H$-invariant domain is not holomorphically convex.",
"\n\nGeneralized Heisenberg groups\n=============================\n\nHere we apply results of the previous section to generalized Heisenberg groups exhibiting additional examples of non-Stein maximal domains of existence for the adapted complex structure. ",
"We refer to [@BTV] for the basic properties of generalized Heisenberg groups.",
"\n\nLet $\\,G\\,$ be a generalized Heisenberg group with Lie algebra $\\,\\g\\,$, consider its abelian subalgebra $\\,\\mathfrak{z}:= [\\mathfrak{g},\\mathfrak{g}]\\,$ and the subspace $\\,\\mathfrak{v}\\,$ orthogonal to $\\,\\mathfrak{z}\\,$ with respect to the $\\,G$-invariant metric $\\,(\\,\\cdot \\, ,\\,\\cdot \\, ) \\,$ of $\\,G\\,$. Then for all $\\,V +Y \\in \\mathfrak{v} \\oplus \\mathfrak{z} = \\g \\cong T_eG\\,$ the unique geodesic tangent to $\\,V +Y \\,$ at $\\,0\\,$ can be explicitely given by $ \\,t \\mapsto \\exp_G \\circ \\varphi_G(t, V+Y)\\,$ for a certain real-analytic map $\\,\\varphi_G: \\R \\times T_eG \\to \\g \\,$ (see \\[BTV,Th.p.31\\]).",
"\n\nA straightforward computation shows that $\\,\\varphi_G\\,$ extends holomorphically on $\\,\\C\\times T_eG\\,$ to $\\,\\g^\\C\\,$ and analogous arguments as in the previous section imply that $\\,\\Omega_G := G \\cdot O_G \\subset TG\\,$ is the maximal the adapted complexification, where $\\, O_G \\,$ is the connected component of $$\\{ \\, V+Y \\in T_eG \\ :\\ \\det(D\\tilde P_G)_{V+Y} \\not=0 \\,\\}$$\n\ncontaining $\\,0\\,$ and $\\, \\tilde P_G: T_eG \\to \\g \\,$ is given by\n\n$$\\, V+Y \\mapsto l(|Y|)V \\ + \\ (\\,1 + |V|^2m(|Y|)\\,)Y \\,.$$\n\nHere $\\,|\\, \\cdot \\,|\\,$ denotes the norm induced by $\\,(\\,\\cdot \\, ,\\,\\cdot \\, ) \\,$ and the real-analytic functions $\\,l,m: \\R \\to \\R\\,$ are defined by $$l(t) := \\frac{\\sinh(t)}{t}, \\quad \\quad \\quad m(t) :=\n \\frac{t -\\sinh(t)\\,\\cosh(t)}{2t^3}.$$\n\nNow for $\\,V+Y \\in \\mathfrak{v} \\oplus \\mathfrak{z}=T_eG\\,$ with $\\,Y \\not= 0\\,$ and $\\, U+X \\in T_{V+Y}T_eG \\cong T_eG\\,$ one has\n\n$$(D \\tilde P_G)_{V + Y}(U + X) \\,= \\,\\frac{\\partial}{\\partial t}\n\\,\\tilde P_G( (V + tU) + (Y + t X) ) \\vert_{t = 0} \\, =\n\\, l^{\\prime}(\\vert Y \\vert)\n\\frac{(Y,X)}{\\vert Y \\vert} V \\,+\\, l( \\vert Y \\vert) U$$ $$+ \\ \\left(2 (V, U) m(\\vert Y \\vert) + \\vert V \\vert^2 \n m^{\\prime}(\\vert Y \\vert) \\frac{ (Y,X) }{\\vert Y \\vert} \\right)Y\n\\ + \\ (1 + \\vert V \\vert^2 m(\\vert Y \\vert) )X.$$\n\nNote that the equation is written according to the splitting $\\, \\mathfrak{v} \\oplus \\mathfrak{z}\\,$ and since $\\,l(\\vert Y \\vert)\\,$ never vanishes, the $\\,\\mathfrak{v}\\,$-part vanishes if and only if\n\n$$U \\ = \\ \n- \\,\\frac{(Y,X) \\,l^{\\prime}(\\vert Y \\vert)}{\\vert Y \\vert\n\\, l(\\vert Y \\vert)} \\, V.$$\n\nIt follows that the central $\\mathfrak{z}$-part also vanishes if and only if $$( 1 + \\vert V \\vert^2 m(\\vert Y \\vert) ) X \\ =\\ \n \\vert V \\vert^2\\,\\frac{(X, Y) }{\\vert Y \\vert} \n \\left( 2 \\frac{ l^{\\prime}(\\vert Y \\vert)}{l(\\vert Y \\vert)}\n m(\\vert Y \\vert)\n \\ - \\ {m^{\\prime}(\\vert Y \\vert)} \\right) Y.$$\n\nIn particular $\\,Y\\,$ and $\\,X\\,$ have to be proportional. ",
"Since both sides are homogeneous of degree 1 in $\\,X\\,$, then $\\,(D \\varphi) \\vert_{(V + Y)}\\,$ is singular if and only if $$\\label{lowrank}\n1 + \\vert V \\vert^2 m(\\vert Y \\vert)\\, =\\,\n \\vert V \\vert^2 \\,\\vert Y \\vert \\, \n \\left( 2 \\frac{ l^{\\prime}(\\vert Y \\vert)}{l(\\vert Y \\vert)}\n m(\\vert Y \\vert)\\ - \\ {m^{\\prime}(\\vert Y \\vert)} \\right)$$\n\nAn analogous computation shows that $\\,(D \\tilde P_G)_{V+Y}\\,$ has maximal rank if $\\,Y=0\\,$, thus equation (\\[lowrank\\]) describes the singular locus of $\\,D \\tilde P_G \\,$. It is remarkable that this identity is independent of the fine structure of the generalized Heisenberg group, e. g. of its dimension or the dimension of its centre. ",
"In particular if $\\,H\\,$ is the 3-dimensional Heisenberg group considered in the previous section, equation (\\[lowrank\\]) determines the border of $\\,O_H = \\Omega_H \\cap T_eH\\,$. Using this fact we are now going to show that for a generalized Heisenberg group $\\,G\\,$ there exist many copies of $\\,\\Omega_H\\,$ embedded as closed submanifolds in $\\,\\Omega_G\\,$.\n\nLet $\\,G\\,$ be a generalized Heisenberg group and choose non zero elements $\\,\\bar V_1 \\in \\mathfrak v\\,$ and $\\,\\bar Y \\in \\mathfrak z\\,$. Then there exists an element $\\,\\bar V_2 \\in \\mathfrak v\\,$ such that the closed subgroup $\\,\\exp_G(\\mathrm{span}\\{\\bar V_1,\\bar V_2,\\bar Y\\})\\,$ is a totally geodesically embedded 3-dimensional Heisenberg group (see \\[BTV,p.30\\]). ",
"Denote by $\\, I:H \\to G\\,$ such an embedding and note that since $\\,\\exp_{G^\\C}: \\g^\\C \\to G^\\C \\,$ is a biholomorphism, then $\\,I\\,$ extends to a holomorphic embedding $\\,I^\\C:H^\\C\n\\to G^\\C\\,$ of the universal complexification of $\\,H\\,$ into the universal complexification of $\\,G\\,$ such that the diagram $$\\begin{matrix} \n\\h^\\C \\ \\ \\ \\ \\ \\ & \\buildover{DI^\\C} \\to & \\g \\ ^\\C \\ \\ \\ \\ \\cr\n & & \\cr\n \\downarrow\\, \\exp_{H^\\C} & & \\ \\downarrow \\, \\exp_{G^\\C} \\cr\n & & \\cr\n H ^\\C\\ \\ \\ \\ \\ \\ \\ & \\buildover{I^\\C} \\to \\ & G^\\C \\ \\ \\ \\ \\ \\cr\n\\end{matrix}$$\n\ncommutes. ",
"Now $\\,I:H \\to G\\,$ is totally geodesic, thus $\\ t \\mapsto I \\circ \\exp_H \\circ \\varphi_H(t,v)\\ $ is the unique geodesic of $\\,G\\,$ tangent to $\\,DI(v)\\,$ at $\\,0\\,$ for all $\\,v \\in \\h \\,$. Then $$I \\circ exp_H \\circ \\varphi_H(t,v) \\,=\\,\nexp_G \\circ \\varphi_G(t,DI(v))$$\n\nand by the identity principle $$\\label{diagram2}\n\\,I^\\C \\circ \\exp_{H^\\C}\\circ \\varphi_H(z, v)\n\\, =\\, \\exp_{G^\\C}\\circ \\varphi_G(z, DI(v))$$\n\nfor all $\\,z\\in \\C\\,$, since this holds for all $\\,z\\in \\R$. Commutativity of the diagram\n\n$$\\label{final}\n\\begin{matrix} \nTH \\ \\ \\ \\ & \\buildover{DI} \\to & TG\\ \\ \\ \\ \\cr\n & & \\cr\n \\downarrow\\, P_H & & \\ \\downarrow \\, P_G \\cr\n & & \\cr\n H^\\C \\ \\ \\ \\ \\ & \\buildover{I^\\C} \\to \\ & G^\\C \\ \\ \\ \\cr\n\\end{matrix}$$\n\nfollows. ",
"For this note that being $\\,I\\,$ a group homomorphism then $\\,DI:TH \\to TG\\,$ is $\\,H$-equivariant, i.e., $\\,DI(g_*w) = I(g)_*DI(w)\\,$ for all $\\,g \\in H\\,$ and $\\,w \\in TH\\,$. In particular $$P_G \\circ DI(g_*v) \\,=\\, P_G (\\, I(g)_*DI(v)\\,) =\nI(g)\\,\\exp_{G^\\C}\\circ \\varphi_G(i,DI(v))$$\n\nfor all $\\,g \\in H \\,$ and $\\,v \\in T_eH \\,$. On the other hand using equation (\\[diagram2\\]) one obtains $$I^\\C \\circ P_H (g_*v) \\,=\\, I^\\C (g\\,\n\\exp_{H^\\C} \\circ \\varphi_H(i , v)\\,)\\,=\\,\n\\, I^\\C (g)\\,I^\\C(\\,\n\\exp_{H^\\C} \\circ \\varphi_H(i , v)\\,) \\,=$$ $$I(g)\\,\\exp_{G^\\C}\\circ \\varphi_G(i , DI(v)),$$\n\nshowing that the above diagram is commutative.",
"\n\nFrom the equivariance of $\\,DI\\,$ it follows that $\\,DI(\\Omega_H) =\\,DI(H\\cdot O_H)\\,=\\,I(H) \\cdot DI(O_H)\\,$ and since $\\,DI\\,$ is isometric and the border of $\\,O_H\\,$ is defined by equation (\\[lowrank\\]) which also describes the singular locus of $\\,\\tilde P_G\\,$ one has $$DI(O_H) \\,\\subset \\, O_G \\cap \n\\mathrm{span}\\{\\bar V_1,\\bar V_2,\\bar Y\\},\n\\quad \\quad \\quad \\quad DI(\\partial O_H) \\subset \\partial O_G.$$\n\nThus $\\,DI(\\Omega_H)\n\\,\\cong I(H) \\times DI(O_H)\\,$ is closed in $\\,\\Omega_G \\cong G \\times O_G\\,$.\n\nFurthermore $\\,DI\\,$ is injective and $\\,P_H\\,$, $\\,P_G\\,$ are locally biholomorphic where the adapted complex structure is defined (cf. ",
"Theorem\\[sliceflow\\]), thus diagram (\\[final\\]) shows that $\\,DI(\\Omega_H) \\cong \\Omega_H\\,$ is a closed complex submanifold of $\\,\\Omega_G\\,$. Finally by Proposition (\\[notstein\\]) the domain $\\,\\Omega_H\\,$ is neither holomorphically separable, nor holomorphically convex, thus one has\n\n\\[generalized\\] Let $\\,G\\,$ be a generalized Heisenberg group. ",
"Then the maximal adapted complexification $\\,\\Omega_G\\,$ is neither holomorphically separable, nor holomorphically convex.",
"\n\n[99]{}\n\nAnn. ",
"Global Anal. ",
"Geom. ",
"19 (2001), no. ",
"4, 327–353\n\nMath. ",
"Ann. ",
"286 (1990), 1–12\n\nComplex analysis and geometry (Trento, 1993), 119–133, Lecture Notes in Pure and Applied Math. ",
"173\n\nPreprint, 2001, math.",
"CV/0109186\n\nLNM 1598, Springer-Verlag 1995\n\nMath. ",
"Z. 233 (2000), 535–551\n\nJ. Differential Geom. ",
"34 (1991), no. ",
"2, 561–570 (first part); J. Differential Geom. ",
"35 (1992), no. ",
"3, 627–641 (second part)\n\nBull. ",
"Soc. ",
"Math. ",
"France 121 (1993), 445–463\n\nThesis, Ruhr-Universität Bochum, 2001\n\nHolden-Day, San Francisco, 1965\n\nMath. ",
"Ann. ",
"290 (1991), 689–712\n\nSéminaire d’Analyse (P. Lelong, P. Dolbeault, H. Skoda), Lecture Notes, Springer-Verlag (1983-84)\n\nCommunications on pure and applied mathematics 16 (1963), 9–17\n\nMath. ",
"Ann. ",
"289 (1991), no. ",
"4, 355–382\n\nMath. ",
"Ann. ",
"259 (1982), 79–97\n\nMath. ",
"Ann. ",
"291 (1991), 409–428\n\nComplex analysis and applications (Warsaw, 1997) Ann. ",
"Polon. ",
"Math. ",
"70 (1998), 215–220.",
"\n\nStefan Halverscheid\n\nMath. ",
"Fakultät, Ruhr-Universität Bochum\n\nUniversitätsstr. ",
"150, D-44780 Bochum, Germany\n\n[: sth@cplx.ruhr-uni-bochum.de]{}\n\nAndrea Iannuzzi\n\nDip. ",
"di Matematica, Universit[à]{} di Bologna\n\nP.$\\,$zza di Porta S. Donato 5, I-40126 Bologna, Italy\n\n[: iannuzzi@dm.unibo.it]{}\n\n[^1]: 2000 [*Mathematics Subject Classification.*]{} ",
"Primary 32C09, 53C30, 32M05; Secondary 32Q28\n\n[^2]: [*Key words*]{}: adapted complex structure, Riemannian homogeneous spaces, Stein manifold\n\n[^3]: \\*Partially supported by University of Bologna, funds for selected research topics\n"
] | {
"pile_set_name": "ArXiv"
} | [
0.0026455026455026454,
0,
0,
0.009230769230769232,
0.007575757575757576,
0.0045045045045045045,
0.004228329809725159,
0.01293103448275862,
0.014285714285714285,
0,
0.0014727540500736377,
0,
0,
0.006097560975609756,
0,
0.006259780907668232,
0.005633802816901409,
0.023809523809523808,
0,
0,
0.0042643923240938165,
0.004437869822485207,
0.005455308434746118,
0.007022471910112359,
0.0020242914979757085,
0,
0,
0,
0.004464285714285714,
0.009708737864077669,
0.009523809523809525,
0.002793296089385475,
0.009478672985781991,
0,
0.004651162790697674,
0,
0.0015060240963855422,
0,
0,
0,
0.005970149253731343,
0.008733624454148471,
0.005037783375314861,
0.006122448979591836,
0.004527813712807244,
0.00510204081632653,
0.006072874493927126,
0.004597701149425287,
0.008771929824561403,
0,
0,
0.011494252873563218,
0,
0.009831460674157303,
0.0026490066225165563,
0.018026565464895637,
0,
0.011976047904191617,
0,
0,
0.005649717514124294,
0.01060070671378092,
0,
0,
0.004552352048558422,
0,
0.004550625711035267,
0.0041928721174004195,
0,
0.006845965770171149,
0,
0.004484304932735426,
0.011834319526627219,
0.004273504273504274,
0.015706806282722512,
0.00267379679144385,
0.012396694214876033,
0,
0.005110732538330494,
0,
0.008695652173913044,
0.004,
0.012987012987012988,
0.0016260162601626016,
0.01015228426395939,
0.004335260115606936,
0,
0.003947368421052632,
0.006423982869379015,
0.009404388714733543,
0.00303951367781155,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.02,
0.043478260869565216,
0,
0.02127659574468085,
0,
0,
0,
0,
0.009433962264150943,
0,
0.02631578947368421,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.034482758620689655,
0,
0.022988505747126436,
0.011049723756906077,
0.008547008547008548
] | 0.004945 | 5 |
[
"* Disclaimer:\nE-mail messages are received by our office manager. ",
"Please be aware that messages may not be checked on a daily basis. ",
"If you need immediate attention or would like to discuss a confidential concern, please call 707-826-3236 rather than using email, so that you can be directed to the appropriate resource. ",
"Thank you.",
"\n\nTips for Good Sleep\n\nExperiencing problems with your sleep? ",
"Here are several guidelines for getting back on track:\n\n1. ",
"Get up and go to bed at the same time every day, even on weekends. ",
"You are training your body to get into a sleep schedule. ",
"Even if you had a tough time falling to sleep the night before, it is important to get up when the alarm rings. ",
"You may be a bit tired during the day, but you will be more likely to fall asleep on schedule that night.",
"\n2. ",
"Develop sleep rituals. ",
"Give your body cues that it is time to slow down and sleep. ",
"For example, listen to relaxing music, read something soothing, have a cup of chamomile tea, meditate, do relaxation exercises.",
"\n3. ",
"Don’t try to “make” yourself sleep. ",
"If it has been over 20 minutes since you went to bed and you are still lying there awake, get up and do something boring for a short stretch of time. ",
"For example, move to a couch or chair in a dimly lit place and read something that is fairly dull. ",
"This is not the time for a new Stephen King novel. ",
"Let yourself get drowsy and then get back into bed.",
"\n4. ",
"Don’t take naps. ",
"Again, you need to establish and train your body into having a sleep schedule. ",
"Naps often interfere with your ability to go to sleep at night. ",
"If you really cannot make it through the day without a nap, make it a short one (less than 60 minutes) and early in the day (before 3 p.m.).",
"\n5. ",
"Exercise, but do it early in the day (at least 3 hours before bedtime). ",
"Exercise that gets your heart pumping (cardiovascular exercise) is most helpful. ",
"Morning/early afternoon exercise is best: 20+ minutes of cardio or 45-60 minutes of brisk walking.",
"\n6. ",
"Only use your bed for sleeping and sex. ",
"Don’t use it to watch TV, do homework, read, etc. ",
"You are training your body to associate bed with sleep.",
"\n7. ",
"Avoid caffeine, nicotine, alcohol and other drugs at least 4-6 hours before bed. ",
"Caffeine and nicotine are stimulants that interfere with your ability to fall asleep. ",
"Alcohol, while it may feel like it helps you get to sleep, will actually interfere with your ability to have a deep restful sleep (that is, you will actually wake sporadically throughout the night). ",
"If you are a caffeine drinker try limiting it to 1-2 cups in the morning; if you are still having trouble with sleep, try cutting caffeine altogether.",
"\n8. ",
"Don’t go to bed hungry or overly full. ",
"It may be helpful to have a light snack before bed. ",
"Having a glass of warm milk is actually quite helpful as this contains a natural sleep inducer (tryptophan).",
"\n9. ",
"Make sure your bed and bedroom are quiet and comfortable. ",
"If there are external noises that are bothersome, wear earplugs or use a “white noise” machine. ",
"Consider a “blackout shade” for your windows if the light bothers you in the mornings. ",
"Make sure you are comfortable in terms of temperature (that the room is neither too hot nor too cold). ",
"Typically, most people do best with the room a little on the cooler side (60s). ",
"Make sure to have adequate blankets.",
"\n10. ",
"Take a hot bath or shower 90 minutes before bedtime. ",
"There is research that shows that a drop in body temperature following a bath may leave you feeling sleepy.",
"\n11. ",
"Use sunlight to set your biological clock. ",
"As soon as you get up in the morning, go outside and turn your face to the sun for 15 minutes.",
"\n12. ",
"If you can’t sleep, try to avoid getting too upset about it. ",
"Pay attention to your thoughts and whether they are likely hindering or helping. ",
"Tell yourself that if you don’t get much sleep that night, you will likely be tired the next day, but you will be able to function adequately and your sleep is likely to be better the next night.",
"\n13. ",
"Avoid prolonged use of sleep medications. ",
"Short-term sleep aids can be helpful in getting a couple of nights of good rest during a difficult period, but long-term use of medications should be avoided. ",
"Certain medications, once stopped, can cause rebound insomnia, and medications can be habit forming. ",
"See your health care professional about the appropriate use of such medications. ",
"Overall, lifestyle changes are safer and more effective in producing quality sleep over the long-haul.",
"\n\nResources: Some of the above tips are adapted from The Anxiety & Phobia Workbook by Edmund Bourne, Ph.D.\n\nSleep Machine. ",
"Highly acclaimed sleep app with over 71,000 ratings. ",
"Provides ambient sounds which you can mix together. ",
"Current version rated at 4.5 stars. ",
"Lite version is free; full version is $1.99.",
"\n\nSleep Pillow: White Noise Machine: A variety of different nature sounds (rain, ocean, whales, etc.) ",
"to help you relax and/or sleep. ",
"Terrific reviews at 4.5 stars and almost 14,000 ratings! ",
"Lite version is free; full version is $1.99.",
"\n\nRelax Completely: hypnosis session for deep relaxation which has received rave reviews from around the world. ",
"Rated at 4.4 stars with over 1,300 ratings. ",
"Free. ",
"The paid version of this app, Sleep Deeply is also well-regarded.",
"\n\nRelax and Sleep Well: guided relaxation/self-hypnosis by Glenn Harrold. ",
"The recording is designed to help you alleviate stress and anxiety and sleep well at night. ",
"Free.",
"\n\nSleep Cycle: simple time calculator that makes it easy to manage sleep cycles so that you wake up well-rested. ",
"Free."
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0,
0.005319148936170213,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.0196078431372549,
0,
0,
0,
0,
0.015625,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.024390243902439025,
0,
0,
0.027777777777777776,
0,
0,
0,
0.017543859649122806,
0,
0,
0.022727272727272728,
0,
0.015384615384615385,
0.013513513513513514,
0.010869565217391304,
0,
0,
0
] | 0.002081 | 5 |
[
"Q:\n\nHow to get index of node in number of nodes with special attribute value\n\nxml:\n<fields>\n <field name=\"name1\" />\n <field name=\"special\" />\n <field name=\"name1\" />\n <field name=\"special\" />\n <field name=\"name1\" />\n</fields>\n\nxslt:\n<xsl:apply-temlates select=\"//field\" />\n\n...\n<xsl:template match=\"field[@name='special']\">\n <xsl:value-of select=\"_what_here?_\" /",
">\n</xsl:template>\n\nMy template must return 1 and 2 instead of 2 and 4 like position() do.",
"\nHow can i get it?",
"\n\nA:\n\nUse the preceding-sibling:: axis instead to count all relevant field elements.",
"\nStylesheet\n<?",
"xml version=\"1.0\" encoding=\"UTF-8\" ?",
">\n<xsl:transform xmlns:xsl=\"http://www.w3.org/1999/XSL/Transform\" version=\"2.0\">\n <xsl:output method=\"text\" encoding=\"UTF-8\"/>\n\n <xsl:strip-space elements=\"*\"/>\n\n <xsl:template match=\"/fields\">\n <xsl:apply-templates select=\"field\"/>\n </xsl:template>\n\n <xsl:template match=\"field[@name='special']\">\n <xsl:value-of select=\"count(preceding-sibling::field[@name='special']) + 1\" />\n </xsl:template>\n\n</xsl:transform>\n\nText Output\n12\n\nAnother solution is not to use a separate template at all, but a for-each. ",
"Inside for-each, the position() function behaves exactly as you'd like it to - it retrieves the position in the sequence of elements in for-each.",
"\n<?",
"xml version=\"1.0\" encoding=\"UTF-8\" ?",
">\n<xsl:transform xmlns:xsl=\"http://www.w3.org/1999/XSL/Transform\" version=\"2.0\">\n <xsl:output method=\"text\" encoding=\"UTF-8\"/>\n\n <xsl:strip-space elements=\"*\"/>\n\n <xsl:template match=\"/fields\">\n <xsl:for-each select=\"field[@name='special']\">\n <xsl:value-of select=\"position()\"/>\n </xsl:for-each>\n </xsl:template>\n\n</xsl:transform>\n\n"
] | {
"pile_set_name": "StackExchange"
} | [
0.010869565217391304,
0,
0,
0,
0,
0.027777777777777776,
0.005597014925373134,
0,
0,
0.027777777777777776,
0.005494505494505495
] | 0.007047 | 5 |
[
"1. ",
"Field of the Invention\nThe present invention relates to a digital X-ray detecting device, and particularly to a digital X-ray detecting panel and a method for manufacturing the same.",
"\n2. ",
"Description of the Related Art\nDigital X-ray detecting devices are widely used in a lot of fields because of nondestructive detection characteristic. ",
"A conventional digital X-ray detecting device generally includes a digital X-ray detecting panel and a driving circuit electrically connected to the digital X-ray detecting panel. ",
"The digital X-ray detecting panel is used for transforming optical signals into electric signals and sending the electric signals to the driving circuit, and thereby obtaining an image information corresponding to the optical signals. ",
"The digital X-ray detecting panel generally includes a wavelength transforming layer and a photodetector array substrate. ",
"The wavelength transforming layer (e.g. scintillator) is used for transforming X-ray into visible light. ",
"The photodetector array substrate is disposed under the wavelength transforming layer and is used for transforming the visible light into electric signals. ",
"The photodetector array substrate generally includes a substrate and a number of photodetectors disposed on the substrate and arranged in an array.",
"\nFIG. ",
"1 is a schematic view of a photodetector of a conventional digital X-ray panel. ",
"Referring to FIG. ",
"1, each photodetector 100 of a conventional X-ray panel is disposed on the substrate 105. ",
"The photodetector 100 includes an amorphous silicon thin film transistor (a-Si TFT) 110 and a photodiode 120 electrically connected to the amorphous silicon thin film transistor 110. ",
"The amorphous silicon thin film transistor 110 includes a first metal layer 111, an insulating layer 112, an amorphous silicon layer 113, an ohmic contacting layer 114 and a second metal layer 115 formed on the substrate 105 in order.",
"\nThe amorphous silicon thin film transistor 110 is electrically connected to the corresponding photodiode 120 through the second metal layer 115. ",
"However, it is well known that the amorphous silicon material has a sensitive property. ",
"After the X-ray is transformed into the visible light having a wavelength of 550 nanometers by the wavelength transforming layer 130, the visible light will cause to produce a photocurrent in the amorphous silicon layer 113 of the amorphous silicon thin film transistor 110. ",
"Thus, an electric leakage is caused in the amorphous silicon thin film transistor 110, so the digital X-ray detecting panel has a poor detecting accuracy. ",
"In order to solve this problem, a metal layer 140 is generally formed above the amorphous silicon thin film transistor 110. ",
"The metal layer 140 is used for blocking the visible light so as to prevent the visible light illuminating the amorphous silicon thin film transistor 110.",
"\nBecause of the formation of the metal layer 140, the process of manufacturing the conventional digital X-ray detecting panel becomes more complex, and thereby increasing the cost of the conventional digital X-ray detecting panel and decreasing the manufacturing efficiency of the conventional digital X-ray detecting panel.",
"\nIn addition, the amorphous silicon layer 113 of the amorphous silicon thin film transistor 110 of the conventional digital X-ray detecting panel is generally formed by a plasma enhanced chemical vapor deposition (PECVD) process. ",
"Because a processing temperature of the PECVD process is high (about 380 Celsius degrees), the high processing temperature when forming the amorphous silicon layer 113 may affect other materials of the conventional digital X-ray detecting panel, and thereby reducing the quality of the conventional digital X-ray detecting panel."
] | {
"pile_set_name": "USPTO Backgrounds"
} | [
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.05555555555555555,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.004347826086956522,
0.00303951367781155
] | 0.002518 | 5 |
[
"The Late Show Will Broadcast Live During Both Political Conventions\n\nStephen Colbert will bring viewers live political coverage this July.",
"\n\nwill also have an on-site presence at both conventions\n\nAs the Democratic and Republic political conventions approach, millions of Americans have been flooded with social media posts, hot-takes, and campaign commercials. ",
"Where's the humor and the fun?Well, fear not, concerned citizens. ",
"Stephen Colbert has got you covered. ",
"The Late Show will broadcast two weeks of live shows during the political conventions in July, covering both the Republican National Convention in Cleveland from July 18–21 and the Democratic National Convention in Philadelphia from July 25–28. ",
"Stephen will host from the Ed Sullivan Theater in New York , where he and his guests will react to the day's coverage of events, and The Late ShowBut, wait, there's more! ",
"Tickets are available for select shows , so you can be part of the fun. ",
"Hurry, they're bound to go fast!Stay tuned for more information, and be sure to watch The Late Show on CBS and CBS All Access"
] | {
"pile_set_name": "OpenWebText2"
} | [
0,
0,
0,
0.02702702702702703,
0.00816326530612245,
0.005847953216374269,
0,
0.016
] | 0.00713 | 5 |
[
"/*\n * Copyright 2017 Google, Inc.\n *\n * Licensed under the Apache License, Version 2.0 (the \"License\");\n * you may not use this file except in compliance with the License.",
"\n * You may obtain a copy of the License at\n *\n * http://www.apache.org/licenses/LICENSE-2.0\n *\n * Unless required by applicable law or agreed to in writing, software\n * distributed under the License is distributed on an \"AS IS\" BASIS,\n * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.",
"\n * See the License for the specific language governing permissions and\n * limitations under the License.",
"\n *\n */\n\npackage com.netflix.spinnaker.clouddriver.kubernetes.description.manifest;\n\nimport lombok.",
"Data;\nimport lombok.",
"EqualsAndHashCode;\n\n@EqualsAndHashCode(callSuper = true)\n@Data\npublic class KubernetesScaleManifestDescription extends KubernetesManifestOperationDescription {\n int replicas;\n}\n"
] | {
"pile_set_name": "Github"
} | [
0.011695906432748537,
0.009584664536741214,
0.009523809523809525,
0,
0.05,
0.011235955056179775
] | 0.01534 | 5 |
[
"2ND Annual DC Brunch Fest\n\nUnion Market – Dock 5\n\nFebruary 2nd, 2019\n\n11am-3pm\n\n\n\n\n\nJoin Us For an ALL-YOU-CARE-TO-TASTE Brunch Extravaganza\n\nGuests will receive a souvenir glass that will facilitate their ALL-YOU-CARE-TO-TASTE drinks.",
"\n\nSmell and taste the sausage and bacon griddles – fired up all day cooking non-stop\n\nEnjoy brunch samples from 15+ of DC’s hottest brunch spots\n\nThe Almost Famous Bloody Mary Bar will feature all of your\n\nfavorite toppings and fixin’s – build your own masterpiece!",
"\n\nThe Boozy Coffee Bar – enjoy hot and iced coffee drinks!",
"\n\nThe Fresh Mimosa Bar will feature fresh squeezed juices and lovely sparkling bubbles for your pleasure.",
"\n\nPink Playhouse featuring Rosés from around the World.",
"\n\nGreat selection of 30+ other beer, spirits and wine\n\nGet your day cranking with the hot beats from DC’s best DJ’s\n\nFun activities and games will keep you focused when you are not imbibing.",
"\n\nTasting Tickets: $70 Advance\n\nTickets are all inclusive and include a souvenir glass and unlimited alcohol samples and food samples.",
"\n\nParking / Directions\n\nUnion Market – Dock 5\n\n1309 5th St. NE\n\nWashington, DC 20002\n\nParking available at nearby lots.",
"\n\n*SALE DETAILS: Some events/tickets are not included in the sale. ",
"2 for 1 tickets are 2 tickets for the price of 1 ticket. ",
"A limited amount of these tickets will be available for purchase. ",
"This price is off of the onsite ticket price (not the discounted sale/advance price). ",
"An option will appear on the ticket page that says “2 for 1” – You must select this specific option to get the offer. ",
"Once the 2 for 1 offer is gone some tickets will be sold at a discounted rate for a limited time. ",
"Sale prices are on NEW ticket orders only. ",
"Tickets are subject to taxes and fees. ",
"All tickets are final sale.",
"\n\nGeneral Information\n\nIt’s for a good cause! ",
"A portion of event proceeds go to Spirit of Hope Children’s Foundation Visit: www.spiritofhopechildrensfoundation.org\n\nFollow us on Facebook and Twitter to see fun event photos and get information in real time @DrinkEatRelax\n\nYour tickets have been EMAILED to you, you should bring a PRINTED COPY of the ticket that you received with you to the event entrance for scanning. ",
"No phone tickets will be accepted.",
"\n\nMust have license/valid photo ID for age verfication.",
"\n\nIf you have trouble accessing your PDF tickets please email jess@triggeragency.com for assistance!",
"\n\nAsk for half pours if you are unsure about a product – you can always go back for more.",
"\n\nLooking to transfer you ticket? ",
"Absolutely, the name on the ticket is irrelevant, only the barcode is significant! ",
"Please forward your PDF e-ticket to the guest attending!",
"\n\nDon’t go hungry! ",
"Our food vendors will be sure to have a variety of delicious food, and are ready to serve you.",
"\n\nOur event staff is here to help you! ",
"Please visit the show office on the day of the event if you have any questions.",
"\n\nAlcohol Sampling. ",
"This is a sampling structured event. ",
"This means that you are given a tasting glass that will allow you to RESPONSIBLY enjoy all of the products featured at the show. ",
"These samples are all-you-care-to-taste from this sampling glass.",
"\n\nRe-Entry. ",
"You can come and go, just make sure you get your hand stamped at the exit!",
"\n\nLast Call is 20 minutes till the end of the event. ",
"We stop pouring alcohol at 15 minutes prior to the end of the event.",
"\n\nis 20 minutes till the end of the event. ",
"We stop pouring alcohol at 15 minutes prior to the end of the event.",
"\n\nLast but not least (although the most important tip on this list) Please Drink Responsibly!",
"\n\nWhat to Bring\n\nFood! ",
"You are welcome to bring your own appetizers, meals or snacks to enjoy at the event.",
"\n\nCash. ",
"Many of the vendors & food vendors will accept credit cards but to save yourself time and hassle, of having to track down an ATM if they do not, we recommend bringing cash.",
"\n\nValid ID. ",
"All guests entering will have their ID checked to verify that they are over the legal drinking age if they have purchased a tasting ticket\n\nWhat NOT to bring\n\nNo pets. ",
"Service dogs are welcome but must register in advance and onsite upon arrival with all proper documentation.",
"\n\nNo weapons. ",
"All bags are subject to search by our security team.",
"\n\nNo outside alcohol or illegal substances. ",
"All bags and coolers are subject to search by our security team.",
"\n\nIf you enjoyed this, check out more posts from the Just Grubbin team here! ",
"Also make sure to listen to The Just Grubbin podcast to hear us go into further detail about this and to hear all of our random irreverent takes on food, drink and whatever else.",
"\n\n\n\n\n\n"
] | {
"pile_set_name": "OpenWebText2"
} | [
0,
0,
0.017241379310344827,
0.009523809523809525,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.008021390374331552,
0,
0,
0.02,
0,
0,
0,
0.017857142857142856,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0.005813953488372093,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0.001427 | 5 |
[
"The problem is not limited to the judiciary. ",
"Turkey today is a far cry from the country the AKP promised 14 years ago, with the government backtracking or turning about on key matters shaping the course of the country.",
"\n\nThe first harbinger of the current turmoil was perhaps the 2010 referendum on constitutional changes that profoundly reshaped the judiciary. ",
"Erdogan drummed up popular support for the amendments, arguing they would end the supremacy of elites and make the judiciary truly independent and impartial. ",
"Soon, however, Erdogan and his aides would admit that Gulenist judges and prosecutors took advantage of the amendments to take control of the judiciary. ",
"This admission, however, happened only after massive corruption probes targeted government ministers and Erdogan’s family in December 2013, shattering the decadelong alliance between the AKP and the Gulen community.",
"\n\nThe toll of the July 15 coup attempt, blamed on followers of US-based cleric Fethullah Gulen, is staggering. ",
"Some 250 civilians were killed and around 1,500 injured. ",
"As of July 28, about 16,000 people were detained, including 9,000 in jail pending trial. ",
"Close to 50,000 public employees were dismissed, and 149 generals and admirals , roughly 40% of the total, were expelled from the military. ",
"Dozens of media outlets were shut down, and arrest warrants were issued for more than 50 journalists. ",
"The pre-charge detention period was extended from four to 30 days. ",
"The crackdown is being conducted through legislative decrees, made possible under the emergency rule declared after the coup attempt.",
"\n\nAfter 14 years of AKP rule, Turkey is in turmoil. ",
"The events that led to this state of affairs boil down to one simple reason: the AKP’s skewed understanding of democracy and popular will.",
"\n\nA July 27 cartoon by popular Turkish cartoonist Musa Kart speaks volumes. ",
"Ruminating after the failed coup attempt, President Recep Tayyip Erdogan is depicted wondering, “Somewhere I didn’t make a mistake, but where?” ",
"Indeed, the list of the Justice and Development Party’s (AKP) wrongs is so long that the rights are hardly discerned.",
"\n\nTake, for instance, emergency rule. ",
"The AKP had long boasted of ending a lengthy state of emergency in the Kurdish-majority southeast in the early 2000s, but today the whole country is under emergency rule, while dozens of cities and towns in the southeast lie in ruins after a ferocious crackdown on Kurdish militants. ",
"Similarly, the government used to boast of stamping out torture, but today Turkey is back on the watch list of leading rights groups. ",
"Even the restitution of the death penalty, removed from the books in 2004, is back on the agenda.",
"\n\nKeen to end the military’s tutelage in politics, the AKP backed sprawling trials of army officers accused of plotting to topple the government, which extended also to dissident journalists and intellectuals. ",
"The so-called Ergenekon and Sledgehammer cases led to a big purge in NATO’s second-largest standing army and upended its line of succession, which, as it turns out now, cleared the way for the ascent of the Gulenist officers accused of leading the July 15 coup attempt. ",
"Ironically, some of the officers who spent years behind bars in the sham trials were instrumental in foiling the coup, and some have been reinstated to senior positions. ",
"After the Gulenist-AKP alliance collapsed, the government grumbled how Gulenists in the police and judiciary had “framed” the military and designated the Gulen community as the “Fethullahist terrorist organization” (FETO). ",
"The coup attempt was only the culmination of the open war that had raged between the two since then.",
"\n\nOn another front, the Gulen community had long been accused of stealing the questions of public service exams to install its followers in state offices. ",
"Erdogan had defended the exams as “clean” despite ample indications to the contrary. ",
"Now, the AKP chorus says “FETO was able to infiltrate the state by stealing the exam questions.”",
"\n\nThroughout the years, the AKP was persistently cautioned by opposition parties and the media. ",
"But criticism of the Gulenists was flatly dismissed, often perceived as malicious secularist attacks on devout Muslims. ",
"AKP leaders and supporters lauded the Gulenists as an exemplary charity movement, paid visits to Gulen in Pennsylvania and praised the community’s schools, which are now being closed en masse. ",
"But not only that. ",
"The AKP saw the critics as enemies, vilified them and often portrayed them as “bootlickers” of the military. ",
"Erdogan and the AKP, after all, knew best how to run the country. ",
"Riding on a high horse, Erdogan disparaged opposition parties as “incapable of even herding a few sheep” and told critics “to keep their wisdom to themselves.”",
"\n\nThe result is a Turkey stuck in a miserable governance crisis — the product of a mindset that sees democracy as dictating one’s own will on others and rejects any compromise with opposing thinking. ",
"Lending an ear to the opposition would have probably prevented much of the current disaster. ",
"The AKP’s own backpedaling and turnabouts are arguably a proof in itself that the opposition was right on many issues of governance, not to mention foreign policy and the Syrian crisis.",
"\n\nThe price for Turkey is steep. ",
"Veering off from its European Union membership path, it is back to ground zero as a country associated with military coups, emergency rule, torture and mass detentions. ",
"The question now is whether the AKP can face up to its mistakes and move to control the damage.",
"\n\nSome positive signs have emerged to that effect. ",
"Erdogan this week convened a rare meeting with Prime Minister Binali Yildirim and the leaders of the opposition Republican People’s Party (CHP) and Nationalist Movement Party (MHP), which resulted in a deal for a constitutional amendment on judicial independence. ",
"The downside was the exclusion of the pro-Kurdish Peoples’ Democratic Party. ",
"In a further gesture, Erdogan retracted libel cases against the MHP and CHP leaders.",
"\n\nIf the AKP sticks to this course with an earnest sense of reconciliation, Turkey stands a chance to extract a silver lining from the bloody coup bid and salvage its democracy.",
"\n\nYet, one should take the gestures with a big grain of salt, given the AKP’s 14-year record and the practices unfolding under emergency rule. ",
"This is true also for this writer, who was the first in the Turkish media to use the term “parallel setup” — as uttered by intelligence chief Hakan Fidan — to describe how mighty the Gulen community had become. ",
"This was back in 2012, a whole four years before the putschists shed blood."
] | {
"pile_set_name": "OpenWebText2"
} | [
0,
0.005780346820809248,
0,
0.006329113924050633,
0.006535947712418301,
0.009302325581395349,
0,
0,
0,
0,
0,
0,
0,
0.019230769230769232,
0.007246376811594203,
0.013157894736842105,
0.006944444444444444,
0.017094017094017096,
0,
0.0035211267605633804,
0,
0,
0.004761904761904762,
0.007407407407407408,
0,
0,
0,
0,
0.011764705882352941,
0.010416666666666666,
0.010416666666666666,
0,
0.0051813471502590676,
0,
0.009174311926605505,
0.030303030303030304,
0.006289308176100629,
0,
0,
0.005405405405405406,
0,
0.005917159763313609,
0.010526315789473684,
0,
0.01893939393939394,
0,
0.03571428571428571,
0.005649717514124294,
0.006993006993006993,
0.004739336492890996,
0
] | 0.005583 | 5 |
[
"Hall was also absent from a New York Times report on the shifting late-night landscape. ",
"But on Tuesday night's show, he was trying to take things in stride. ",
"After all, he once made huge waves back in 1992 by appearing on the cover of Entertainment Weekly and declaring he was \"gonna kick Leno's ass.\" ",
"These days, he says, \"I don't want to come off as the angry black man.\"",
"\n\nHowever, he did want to let NBC anchor Brian Williams and the NBC News crew know about his displeasure, so he announced the phone number to NBC News on the air and encouraged fans to give them a ring.",
"\n\nYes, that really was the number for the \"NBC Nightly News\" and an NBC employee confirmed that they'd received several calls from Hall's fans.",
"\n\nHall may not have wanted to appear angry, but he had a friend on hand who had no trouble delivering a sinister message to Williams and the crew at NBC News. ",
"Enter, Suge Knight.",
"\n\nThe hulking music mogul, who has spent time in prison, strode out and delivered a message to Williams and \"the world of journalism.\"",
"\n\n\"Well, as far as I look at it, it's like this,\" Knight said. \"",
"You the West Coast,\" Knight indicated Hall. \"",
"So we're going to give him so many days, so many hours to get it right. ",
"And he gonna have to apply. ",
"Either he apply or he leave.\"",
"\n\nHall was quick to clarify what Knight was saying: \"Either change it or resign.\""
] | {
"pile_set_name": "Pile-CC"
} | [
0.022727272727272728,
0,
0.013888888888888888,
0,
0.019801980198019802,
0.013986013986013986,
0.012578616352201259,
0.05263157894736842,
0.007462686567164179,
0.015625,
0,
0,
0,
0,
0.024691358024691357
] | 0.012226 | 5 |
[
"Seattle, WA- “People like me are finding themselves in a tougher situation than ever,” said Seattle pizza shop employee Devin Jeran in comments to Q13 FOX News. ",
"Earlier this month, Jeran was the recipient of a raise, as Seattle’s mandated minimum wage hike began to kick in, forcing large businesses with more than 500 employees and small business franchises to first begin paying employees $11 per hour starting this month before being required to raise wages to $15 per hour by 2017. ",
"Small businesses without franchise relationships are allowed to wait until 2021 to raise their wages to $15 per hour.",
"\n\nThough Jeran will receive a bigger paycheck until August, his paychecks will stop for good after that, as the small business Z Pizza franchise at which he works is set to close under pressure from the new minimum wage law. ",
"The franchise’s owner, Ritu Shah Burnham, described the efforts she’s already taken to adapt to the phased increase to the new $11 minimum hourly wage, “I’ve let one person go since April 1, I’ve cut hours since April 1, I’ve taken them myself because I don’t pay myself.” ",
"However, the fact that her 12-employee business has a franchise relationship with Z Pizza means that, under the new minimum wage law, she is required to retool her business such that it can stay open while paying employees $15 per hour within just 2 years, rather than being able to wait until 2021 like other small businesses, a feat Burnham says she can not manage. “",
"I know that I would have stayed here if I had 7 years, just like everyone else, if I had an even playing field. ",
"The discrimination I’m feeling right now against my small business makes me not want to stay and do anything in Seattle,” she said.",
"\n\nEarlier this year, International Franchise Association representative Matthew Haller spoke out against the discriminatory nature of Seattle’s new minimum wage law in terms of how it affects small business franchises and said in comments to CNBC, “Franchisees are often small and local, and face the same challenges that other small businesses face. ",
"The franchise system is still a network of small businesses that will face irreparable harm due to these extra costs.”",
"\n\n- Newsletter -\n\nBurnham told Q13 FOX News that she worries about what her employees will do after her business closes, “I absolutely am terrified for them. ",
"I have no idea where they’re going to find jobs, because if I’m cutting hours, I imagine everyone is across the board.”"
] | {
"pile_set_name": "OpenWebText2"
} | [
0.012422360248447204,
0.003076923076923077,
0,
0.0044444444444444444,
0.003663003663003663,
0.0027100271002710027,
0,
0,
0.008547008547008548,
0,
0.006329113924050633,
0
] | 0.003433 | 5 |
[
"The devices described below provide for improved heat transfer and sealing performance for our Starion® line of thermal cautery forceps and thermal ligating shears, and in instruments such as those disclosed in Treat, et al., ",
"Electrothermal Instrument For Sealing And Joining Or Cutting Tissue, U.S. Pat. ",
"No. ",
"6,626,901 (Sep. 30, 2003) (the disclosure of which is hereby incorporated in its entirety)."
] | {
"pile_set_name": "USPTO Backgrounds"
} | [
0.004424778761061947,
0,
0,
0
] | 0.001106 | 5 |
[
"hpux-g++*:QMAKE_CXXFLAGS += -mbig-switch\n\nHEADERS += $$PWD/qparsercontext_p.h \\\n $$PWD/qmaintainingreader_p.h \\\n $$PWD/qquerytransformparser_p.h \\\n $$PWD/qtokenizer_p.h \\\n $$PWD/qtokenrevealer_p.h \\\n $$PWD/qtokensource_p.h \\\n $$PWD/qxquerytokenizer_p.h \\\n $$PWD/qxslttokenizer_p.h \\\n $$PWD/qxslttokenlookup_p.h\n\nSOURCES += $$PWD/qquerytransformparser.cpp \\\n $$PWD/qparsercontext.cpp \\\n $$PWD/qtokenrevealer.cpp \\\n $$PWD/qtokensource.cpp \\\n $$PWD/qxquerytokenizer.cpp \\\n $$PWD/qxslttokenizer.cpp \\\n $$PWD/qxslttokenlookup.cpp\n"
] | {
"pile_set_name": "Github"
} | [
0.006289308176100629
] | 0.006289 | 5 |
[
"Theresa May’s Brexit plan will be blocked by MPs even if she is able to “bounce” the Cabinet into signing it off, the Prime Minister has been warned.",
"\n\nSenior members of the Eurosceptic grouping of Tory backbenchers and the Democratic Unionist Party figures are publicly uniting to insist they will vote against Mrs May’s proposals unless she backs down.",
"\n\nTheir intervention came as senior government figures warned that the deal would still fall in Parliament even if it were forced through a reluctant Cabinet this week. ",
"A defeat for Mrs May would be likely to spark a leadership challenge.",
"\n\nThe warnings come amid opposition from across the Conservative Party to a proposed “backstop”, or insurance plan, for the UK’s relationship with the EU if no alternative deal is reached.",
"\n\nMinisters and MPs are concerned it could allow for a border between Northern Ireland and the rest of the UK, and would provide an insufficient mechanism for the country to extract itself from the EU’s structures.",
"\n\nThe plan sparked the resignation of Jo Johnson, the pro-Remain transport minister, on Friday, on the basis it amounted to “vassalage” for the UK.",
"\n\nAround 10 Cabinet ministers have rejected the idea of an exit mechanism from the backstop that depends on the EU giving its permission for the UK to leave its structures, leading to speculation some may resign. ",
"But a senior Tory claimed the ministers would ultimately “fall like paper tigers” – referring to a Chinese expression about objects that appear threatening but are ineffectual."
] | {
"pile_set_name": "OpenWebText2"
} | [
0.013422818791946308,
0.004901960784313725,
0.011834319526627219,
0,
0.010638297872340425,
0.004672897196261682,
0.006802721088435374,
0.004694835680751174,
0
] | 0.00633 | 5 |
[
"Preliminary imaging results with 18F-2-fluoro-2-deoxy-D-glucose.",
"\nTransverse section imaging with 18F-2-fluoro-2-deoxy-D-glucose (2FDG) is of considerable interest because quantitative values of regional glucose metabolism may be obtained. ",
"We present preliminary results using 2FDG produced by the Brookhaven National Laboratory and imaged with the PCII at the Massachusetts General Hospital. ",
"Three studies are reported showing the distribution in dog, monkey, and in brain of an essentially normal patient."
] | {
"pile_set_name": "PubMed Abstracts"
} | [
0,
0,
0.013071895424836602,
0
] | 0.003268 | 5 |
[
"class CGI\n extend ::EscapeUtils::HtmlSafety\n\n class << self\n alias escapeHTML _escape_html\n\n def unescapeHTML(s)\n EscapeUtils.unescape_html(s.to_s)\n end\n end\nend"
] | {
"pile_set_name": "Github"
} | [
0.011235955056179775
] | 0.011236 | 5 |
[
"Before there was Lil' Wayne or Lil' Bow Wow, there was Lil' Steven. ",
"Okay, well, actually, Little Steven aka Miami Steve aka Silvio Dante (on The Sopranos) aka second banana and onstage foil to Bruce Springsteen (especially as Clarence Clemons has gotten older and less mobile and more especially when wife Patti doesn't show up to the concert) aka the most famous bandana-wearing rocker this side of Axl Rose.",
"\n\nMost people would be thrilled to have just one nickname. ",
"When Van Zandt puts out a record, the performer is Little Steven and the producer is Miami Steve. ",
"From here on out, I'll refer to him as Miami Steve to avoid confusion.",
"\n\nMiami Steve (as I will refer to him from here on out) is my favorite of the bunch. ",
"A gifted songwriter and producer, he possesses an infectious rock 'n roll spirit that seems to make anything that he is involved in better. ",
"Though Springsteen gets the credit for advancing Southside Johnny's career because he gave him \"The Fever,\" if you check those great early records, most of the songs, including most of the title tracks, are songs written by Miami Steve. ",
"He also produces those records. ",
"Check, too, his production credits on Springsteen albums like The River. ",
"Plus, he plays the mandolin in the \"Glory Days\" video.",
"\n\nBut there were brief, shining moments, before and after Miami Steve left the E-Street band in the mid-80's and ventured off into a career fronting his own band when he put out fiery, defiant rock 'n roll with an intensity that I'm not sure even Springsteen has ever matched. ",
"This move resulted in two classic records--Little Steven and the Disciples of Soul's Men Without Women and Little Steven's Voice of America-- plus that major, multi-artist, post-\"We Are The World\" political statement, \"Sun City.\"",
"\n\nThe guy oozes street cred, which is probably how he got The Sopranos part, and how he got guys like Lou Reed and Miles Davis for the Sun City sessions.",
"\n\nLike E-Street bandmate Nils Lofgren, Keith Richards is clearly an inspiration for Van Zandt. ",
"As a guitarist, Miami Steve can churn out Stones-like riffs with the best of them, but with a different purpose and with other gifts which allow him to front a band and forge a sound in ways that the legend can't.",
"\n\nThough not a great singer, Miami Steve is a great rock 'n roll singer, by which I mean that he can sell the words and the song with pure honesty and emotion. ",
"When you listen to him, you never feel like he is going through the motions--he believes what he says, even if it's simplistic, jingoistic, or naively idealistic. ",
"And, though his voice is somewhat thin, he is lucky enough to sound quite a bit like Keith Richards, though on-key and without the jaded ennui. ",
"His lines like \"It never died inside of me\" or \"Undefeated, everybody goes home\" or \"As the things we want come closer, I feel the things we need slipping away\" would not fit in a Stones song.",
"\n\nHe can also play distinctive, idiosyncratic, searing lead guitar in a way that doesn't fit into Springsteen's sound, so when you hear his solo work, it is indeed a revelation. ",
"His high, melodic leads support the r + b based songs on Men Without Women as ably as they do the more 80's sounding, synthesizer poltical rock of Voice of America.",
"\n\nAnd he understands horns. ",
"This is no small gift. ",
"Miami Steve is, arguably, one of a handful of rock producers who know how to use horns in a song, who can write a horn chart to serve as a memorable main riff or a transition or a bridge. ",
"Horns are not window dressing to him, nor are they an experiment to try to alter a sound. ",
"The horns he heard on the Jersey shore all his life are indigenous to many of his compositions, and, coupled with his percussive rhythm guitar, the horns and guitar play off each other to push the songs to new heights. ",
"As a producer, I'm not sure he has an equal in maximizing the potential of horns and guitar interplay.",
"\n\nBut perhaps most of all, his songwriting skills set him apart. ",
"When it comes to rock 'n roll, I have an unapologetic preference for a positive, life-affirming, defiant-in-the-face-off-all-odds stance. ",
"The common strand throughout Miami Steve's beach rhythm and blues for Southside Johnny, his own solo work, and even the song, \"Sun City,\" is a never-quit, never-give-in attitude that pervades the music as much as the lyrics. ",
"The songs are a celebration of life, of country, of the power of music, even if everything doesn't work out. ",
"If you've seen him onstage with Bruce, then you know that fire hasn't gone out and he's still living that same dream.",
"\n\nAll songs above written, arranged, and produced by Steven Van Zandt. ",
"Check out his wild, neo-Yardbirdsian solo on \"Voice of America.\" ",
"Though the songs above have either been reformatted from LP or uploaded from old CD's, it would seem that amazon.com is the best source for Little Steven tunes these days. ",
"Photos #2 and #4 are courtesy of the Troutking.",
"\n\n7 comments:\n\nI still remember my original reaction to \"Sun City\": If that skeezy-lookin' dude won't play Sun City, then why the hell would anyone want to?",
"\n\nMaybe that wasn't his intent, but it worked the same nonetheless, since I've never yet played Sun City. ",
"I did just re-watch the video... damn but Pat Benatar's voice kicks booty... I didn't really know 80% of the artists on this song when it originally came out. ",
"I was such a musical infant...\n\nBut I knew Peter Wolf, and I knew he was mostly just good for dancing like a goofy white guy.",
"\n\nHere he is bringing the soul to Bruce's last show in Buffalo (about 4 mins in): http://www.youtube.com/watch?v=AaqfJznJVrY\n\nBob, you probably know that supposedly the way he joined the E Street Band was by scoring the horn lines in 10th Avenue Freezout.",
"\n\nMy two darkest admissions from the 1980's. ",
"One, I loved President Reagan. ",
"Two, I didn't like Little Steven in the Glory Days video. ",
"I thought he was too over the top. ",
"What I didn't realize was that he really believed in the transforming power and resplendent glory of rock and roll. ",
"Now I realize it's at the heart of what makes a Bruce concert the event that it is. ",
"We love you, Stevie!!!",
"\n\nI actually saw Steve with the \"Artists United Against Apartheid\" in concert. (",
"No recollection of what songs other than \"Sun City\" were played.) ",
"He was the 1st opening act for the Beach Boys. ",
"Who was the 2nd opening act, you ask? ",
"Why, Mr. John Cafferty and his band of brown beavers. ",
"Ah, the '80s. ",
"Happy times.",
"\n\nWelcome To The Bottom\n\nTwo men, shootin' the s*#t, usually over beer, with music in the background.",
"\n\nWe like to write, and we like to listen to music. ",
"As best we can, we try to coordinate those two. ",
"Sometimes the music drives the written post; sometimes the post sends us searching for the right music.",
"If you have comments, we'd appreciate hearing from you. ",
"Please drop us a note at bottomoftheglass@gmail.com.",
"\n\nBillyBob or BobBilly?",
"\n\nBob used to do a lot of things he no longer does. ",
"If he can't get back to them, he might as well write about them.",
"\n\nBilly considers the iPod his fourth child and has an unhealthy love of power pop, whiny rockers and folksy women. ",
"He is imminently grateful his daughters dislike Justin Beiber and think he's a wuss.",
"\n\nNeither is from New Orleans, though both retain a perpetual wistfulness for the city.",
"\n\nExplanation About Music Files\n\na.k.a. \"",
"Please Don't Sue Us, Pretty Record Companies\"\n\nAll songs posted on this blog are for previewing purposes only. ",
"We like these artists. ",
"Every dollar going in their direction adds to the odds they can make more songs. ",
"So please, consider Amazon.com, iTunes, or your local mom 'n' pop record shop and show your support for the songs you like in the right way.",
"\n\nWe will only keep songs active for a limited amount of time. ",
"Sooner if appropriate litigious powers that be don't approve."
] | {
"pile_set_name": "Pile-CC"
} | [
0,
0.017595307917888565,
0,
0.02040816326530612,
0,
0.011764705882352941,
0,
0.004219409282700422,
0,
0,
0,
0.0036101083032490976,
0.008733624454148471,
0.013071895424836602,
0.031578947368421054,
0,
0,
0,
0.006944444444444444,
0.005208333333333333,
0.0056179775280898875,
0.006097560975609756,
0,
0,
0,
0,
0,
0,
0,
0,
0.0044444444444444444,
0,
0,
0.014084507042253521,
0,
0.005813953488372093,
0.02127659574468085,
0,
0,
0.006289308176100629,
0.008,
0.011764705882352941,
0,
0.03225806451612903,
0.017241379310344827,
0,
0,
0,
0,
0.0125,
0,
0,
0,
0.018518518518518517,
0,
0,
0,
0,
0,
0,
0,
0.019230769230769232,
0.08695652173913043,
0.019230769230769232,
0,
0.017241379310344827,
0.011904761904761904,
0,
0,
0.009009009009009009,
0,
0,
0.014285714285714285,
0,
0.01639344262295082
] | 0.006417 | 5 |
[
"Please view the information and link below from the CDC regarding stress and anxiety in children and COVID19. ",
"Please feel free to contact any of us with questions or concenrs. ",
"https://www.cdc.gov/coronavirus/2019-ncov/prepare/managing-stress-anxiety.html\n\nWith the announcement of school closures from March 16 through at least March 27, we want to inform you of food access options throughout Wake County during these unprecedented times. ",
"A number of schools and community centers will serve as food distribution sites for Wake County families beginning Tuesday, March 17. ",
"Families may pick up food at the locations and times indicated below. ",
"Food will be distributed by volunteers at curbside to minimize safety concerns. ",
"The food will be placed in safe and sanitized food containers. ",
"ID is not required to pick up food."
] | {
"pile_set_name": "Pile-CC"
} | [
0.00909090909090909,
0,
0.003787878787878788,
0,
0,
0,
0,
0
] | 0.00161 | 5 |
[
"New World Symphony to get a New York state of mind this weekend\n\nBy David Fleshler\n\nThe contemporary music scene of New York comes to Miami Beach this weekend, as the New World Symphony adds an electric guitar to its roster for a performance of works by three of the city’s most prominent living composers.",
"\n\nContemporary classical music can be a tough sell, a reality reflected in the $15 general admission price for this concert, as opposed to the $33-$78 tickets for the following week that includes Tchaikovsky’s Symphony No. ",
"6.",
"\n\nBut the composers to be heard Saturday at the Lincoln Theatre, have all worked hard to produce works that engage audiences. ",
"They are all leaders – and two are founders – of New York’s renowned Bang on a Can collective, which puts on festivals that blend classical, rock, world music and jazz.",
"\n\nMembers of Bang on a Can – while giving away nothing in terms of musical complexity or sophistication – have tried from the start to attract a broad audience, having no interest in the sort of contempt for the public displayed by some composers of the 20th century.",
"\n\nDavid Lang\n\n“I think we went through a period of music in the 60s where the point of a piece of music was for the composer to show how much smarter he was than the audience,” said David Lang, a founder of Bang on a Can, whose Pierced will be performed Saturday. “",
"And there was a lot of good music written. ",
"But we live in a different world. ",
"This is not that kind of music.",
"\n\n“We all grew up – all three of us – playing in rock bands. ",
"We all have an incredible love and commitment to music. ",
"But we all get the view from rock bands that music should be about energy and music should be about a community of people experiencing that energy together.”",
"\n\nAll three composers will speak at the concert. ",
"Lang, who won the 2008 Pulitzer Prize for Music for his work The Little Match Girl, composed Pierced for the violin, piano and percussion ensemble Real Quiet, which will perform it Saturday. ",
"He wrote the work as a quasi-concerto, or even an anti-concerto.",
"\n\nWhere the standard 19th- entury concerto treated the soloist as a heroic protagonist in opposition to the group, his work lets the solo ensemble and the orchestra each take their own paths, although each needs the other for its harmonies to make sense. “",
"It’s pretty challenging,” Lang said. “",
"There are a lot of notes, a lot of complicated rhythms.”",
"\n\nMichael Gordon\n\nThe biggest work on the program is a 30-minute film symphony called Gotham, an exploration of New York City, by Michael Gordon, another co-founder of Bang on a Can. ",
"Gordon grew up in Miami Beach. ",
"He attended Miami Beach Senior High School and breathed the excitement of contemporary music by poking through the new music bin at Spec’s Records in Coral Gables. “",
"I really liked Led Zeppelin, but Stravinsky was a lot better,” he said. ",
"His 94-year old father still lives in Miami Beach, where Gordon said he “drives, shops and plays duplicate bridge.” ",
"But Gordon left Miami Beach at 17, now lives in the lower Manhattan neighborhood of Tribeca and composed the work to be heard Saturday for his adopted city.",
"\n\nGotham is performed along with filmed still and video images of New York by the film maker Bill Morrison, although the music was composed before the film. ",
"The three-movement work, which uses an orchestra, piano, electric guitar and electric bass, will accompany images – some vintage, some deteriorated – that show the history, energy and endless change and reinvention of the city.",
"\n\nThe opening scene shows sheep grazing in a meadow, then the camera pulls back to reveal the buildings surrounding this scene in Central Park. ",
"The middle section portrays the city’s energy. “",
"It has great sequences of fast-moving, big-city shots,” Gordon said. “",
"It has an incredible shot of an elevator at a construction site going down 80 or 90 floors.” ",
"The last movement is a canon – “almost like a tornado” – with somber images included ones of the former site of the World Trade Center.",
"\n\nThe contemporary experience of jet travel inspired the third work on the program, Evan Ziporyn’s War Chant. ",
"He put together two contrasting musical styles, the spiky, modernistic tone of the composer Iannis Xenakis and the genial, relaxed style of the Mexican composer Juan Garcia Esquivel, to bring out the menace and weirdness beneath the surface of air travel, particularly in the post-9/11 world. ",
"One style brings out the reassuring chimes, announcements and routine of the passenger cabin, while the other shows the terrifying industrial power that transports the passenger (usually) safely across the sky.",
"\n\nJeffrey Milarsky, who will conduct the performance, is a contemporary music specialist on the faculty of Juilliard and Columbia University. ",
"He described the concert as a rare opportunity to hear three prominent composers at once with full orchestra.",
"\n\nJeffrey Milarsky\n\n“You’re getting a chance to hear three sort of forces of nature and see what’s been going on in New York for years,” he said. “",
"Bang on a Can is not a specific style. ",
"They break the mold of tradition. ",
"Their quest for innovation is something you hear. ",
"It might be different listening, but it will be exciting listening. ",
"I think it’s going to be edgy, exciting and quite beautiful.”",
"\n\nAlthough the works to be performed Saturday are sophisticated, energetic and decidedly contemporary. ",
"they are also meant to be accessible, capable of engaging people who appreciate Bach, Schubert and Prokofiev. “",
"You have to make the audience realize that it’s participating in the decisions about what the future is,” Lang said. “",
"It’s not the future if people don’t like it.”",
"\n\nThe New World Symphony performs New York State of Mind: Sounds of the Times on Saturday at 7:30 p.m. at the Lincoln Theatre. ",
"Call 305-673-3331 or go to www.nws.edu."
] | {
"pile_set_name": "Pile-CC"
} | [
0.0032679738562091504,
0.004484304932735426,
0,
0.007936507936507936,
0.005952380952380952,
0,
0.007547169811320755,
0,
0,
0,
0,
0,
0,
0,
0.010471204188481676,
0,
0,
0.02631578947368421,
0,
0.01639344262295082,
0.03225806451612903,
0.012121212121212121,
0.027777777777777776,
0.008620689655172414,
0.00641025641025641,
0.012738853503184714,
0,
0,
0,
0.014285714285714285,
0,
0.007407407407407408,
0.00909090909090909,
0.006825938566552901,
0,
0.02112676056338028,
0,
0.006802721088435374,
0,
0,
0,
0,
0,
0,
0.02702702702702703,
0.00847457627118644,
0,
0.015748031496062992,
0.05128205128205128
] | 0.00715 | 5 |
[
"The implementations described herein relate generally to aircraft ram air systems, and, more specifically, to an aircraft ram system that is capable of providing cooling to multiple components of the aircraft.",
"\nAt least some known aircraft include a ram air system that provides ram air to at least one air conditioning (A/C) pack of an aircraft environmental control system (ECS). ",
"At least some ram air systems and A/C packs are both positioned within a pack bay of the aircraft, along with other aircraft components. ",
"The A/C pack generates heat during operation and discharges the heat into the pack bay and ram exhaust, while supplying cool air to the cabin. ",
"The heated air within the pack bay may cause an undesirable increase in temperature of the components within the pack bay and also the surrounding aircraft structure. ",
"For example, at least some aircraft include fuel tanks located proximate the pack bay. ",
"As such, an increase in pack bay temperature may cause an undesirable increase in fuel temperature. ",
"At least some aircraft include an insulation layer between the pack bay and the fuel tanks to reduce fuel heating. ",
"However, such insulation increases the overall weight of the aircraft and may require replacement.",
"\nThe ram air system provides cooling air to the A/C packs, and, more specifically, to a heat exchanger of the A/C pack. ",
"As such, the temperature of the exhaust flow of the ram air system is higher compared to the temperature of the inlet flow of ram air into the system and also the free stream airflow. ",
"In at least some aircraft, the exhaust from the ram air system flows along downstream skin panels of the aircraft. ",
"Continuous exposure to this high temperature ram air exhaust flow may cause unscheduled maintenance of the downstream panels or limit the service lifetime of such panels, especially in the case of carbon fiber panels. ",
"At least some known aircraft include a heat shield built into the skin panels downstream of the ram air exhaust. ",
"Such heat shields may be simply a thicker, more robust, skin panel, or may include an insulation core to insulate the skin from the hot exhaust flow. ",
"In either case, the heat shield may be quite large and relatively heavy as compared to a standard skin panel. ",
"Accordingly, such heat shields increase the overall weight of the aircraft and also the manufacturing costs."
] | {
"pile_set_name": "USPTO Backgrounds"
} | [
0,
0.005813953488372093,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | 0.000342 | 5 |