CINXE.COM

<!doctype html> <html lang="en" class="calculusandanalysis mathworldcontributors"> <head> <title>Wiener Algebra -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Wiener Algebra" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="Suppose W is the set of all complex-valued functions f on the interval [0,2pi] of the form f(t)=sum_(k=-infty)^inftyalpha_ke^(ikt) (1) for t in [0,2pi], where the alpha_k in C and sum_(k=-infty)^(infty)|alpha_k|<infty. The set W with the usual pointwise operations and with the norm ||f||=sum_(k=-infty)^infty|alpha_k| (2) is a commutative Banach algebra and is called the Wiener algebra. There is an isometric isomorphism between l^1(Z) and W given by f->f^~, where ..." /> <meta name="description" content="Suppose W is the set of all complex-valued functions f on the interval [0,2pi] of the form f(t)=sum_(k=-infty)^inftyalpha_ke^(ikt) (1) for t in [0,2pi], where the alpha_k in C and sum_(k=-infty)^(infty)|alpha_k|<infty. The set W with the usual pointwise operations and with the norm ||f||=sum_(k=-infty)^infty|alpha_k| (2) is a commutative Banach algebra and is called the Wiener algebra. There is an isometric isomorphism between l^1(Z) and W given by f->f^~, where ..." /> <meta name="DC.Date.Created" scheme="W3CDTF" content="2004-03-31" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Calculus and Analysis:Functional Analysis" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Moslehian" /> <meta name="DC.Subject" scheme="MSC_2000" content="46" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement." /> <meta name="DC.Format" scheme="IMT" content="text/html" /> <meta name="DC.Identifier" scheme="URI" content="https://mathworld.wolfram.com/WienerAlgebra.html" /> <meta name="DC.Language" scheme="RFC3066" content="en" /> <meta name="DC.Publisher" content="Wolfram Research, Inc." /> <meta name="DC.Relation.IsPartOf" scheme="URI" content="https://mathworld.wolfram.com/" /> <meta name="DC.Type" scheme="DCMIType" content="Text" /> <meta name="Last-Modified" content="2004-03-31" /> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_WienerAlgebra.png"> <meta property="og:url" content="https://mathworld.wolfram.com/WienerAlgebra.html"> <meta property="og:type" content="website"> <meta property="og:title" content="Wiener Algebra -- from Wolfram MathWorld"> <meta property="og:description" content="Suppose W is the set of all complex-valued functions f on the interval [0,2pi] of the form f(t)=sum_(k=-infty)^inftyalpha_ke^(ikt) (1) for t in [0,2pi], where the alpha_k in C and sum_(k=-infty)^(infty)|alpha_k|<infty. The set W with the usual pointwise operations and with the norm ||f||=sum_(k=-infty)^infty|alpha_k| (2) is a commutative Banach algebra and is called the Wiener algebra. There is an isometric isomorphism between l^1(Z) and W given by f->f^~, where ..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Wiener Algebra -- from Wolfram MathWorld"> <meta name="twitter:description" content="Suppose W is the set of all complex-valued functions f on the interval [0,2pi] of the form f(t)=sum_(k=-infty)^inftyalpha_ke^(ikt) (1) for t in [0,2pi], where the alpha_k in C and sum_(k=-infty)^(infty)|alpha_k|<infty. The set W with the usual pointwise operations and with the norm ||f||=sum_(k=-infty)^infty|alpha_k| (2) is a commutative Banach algebra and is called the Wiener algebra. There is an isometric isomorphism between l^1(Z) and W given by f->f^~, where ..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_WienerAlgebra.png"> <link rel="canonical" href="https://mathworld.wolfram.com/WienerAlgebra.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"></noscript> </head> <body id="topics"> <main id="entry"> <div class="wrapper"> <section id="container"> <header class="text-align-c"> <div id="header-dropdown-menu"> <img src="/images/header/menu-icon.png" width="18" height="12" id="menu-icon"> <span class="display-n__600">TOPICS</span> </div> <svg version="1.1" id="logo" width="490" height="30" class="hide__600" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 480.7 30" style="enable-background:new 0 0 480.7 30;" xml:space="preserve"> <g> <g> <g> <polygon fill="#0095AA" points="144.2,17.1 137.8,4 133.2,4 133.2,28.5 136.5,28.5 136.5,8.6 143,21.8 143,21.9 145.2,21.9 151.7,8.6 151.7,28.5 155,28.5 155,4 150.6,4"/> <path fill="#0095AA" d="M170.8,10.8c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.3,0.8c-1.2,0.6-2.1,1.4-2.6,2.4l0,0.1l2.6,1.6l0-0.1 c0.3-0.5,0.8-1,1.4-1.4c0.7-0.4,1.6-0.6,2.8-0.6c1.4,0,2.5,0.3,3.2,0.8c0.8,0.5,1.1,1.4,1.1,2.8v0.8l-4.5,0.4 c-1.8,0.2-3.2,0.6-4.1,1.1c-1,0.5-1.7,1.2-2.2,2c-0.5,0.8-0.8,1.9-0.8,3.2c0,1.7,0.5,3.1,1.5,4.2c1,1.1,2.4,1.6,4.1,1.6 c1.2,0,2.3-0.3,3.3-0.8c0.9-0.5,1.8-1.2,2.7-1.9v2.2h3.2V15.9C172.6,13.5,172,11.8,170.8,10.8z M169.4,19.2v4.4 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6c-0.4,0.2-0.9,0.2-1.3,0.2c-1.1,0-1.9-0.2-2.4-0.7 c-0.5-0.5-0.8-1.2-0.8-2.3c0-1.1,0.4-2,1.2-2.6c0.8-0.6,2.2-1,4.2-1.2L169.4,19.2z"/> <path fill="#0095AA" d="M184.9,25.9c-0.7,0.3-1.6,0.5-2.4,0.5c-1,0-1.6-0.3-1.9-0.9c-0.3-0.6-0.4-1.7-0.4-3.2v-9.7h4.7V9.9h-4.7V4.7 h-3.2v5.1h-2.6v2.7h2.6v10.3c0,2.1,0.4,3.7,1.2,4.7c0.8,1,2.2,1.5,4.1,1.5c1.2,0,2.3-0.2,3.5-0.6l0.1,0L184.9,25.9L184.9,25.9z" /> <path fill="#0095AA" d="M198.1,9.3c-1.2,0-2.3,0.3-3.5,1c-1.1,0.6-2,1.3-2.9,2.1V2.1h-3.2v26.4h3.2V15.4c0.9-0.8,1.9-1.5,2.9-2.1 c1-0.6,1.9-0.9,2.7-0.9c0.8,0,1.3,0.1,1.7,0.4c0.4,0.3,0.7,0.7,0.8,1.2c0.2,0.6,0.3,1.7,0.3,3.5v11h3.2V16.1 c0-2.4-0.4-4.2-1.3-5.2C201.2,9.9,199.9,9.3,198.1,9.3z"/> <polygon fill="#0095AA" points="225.2,23.5 220.6,4 220.6,4 216.8,4 212.2,23.3 207.8,4 207.8,4 204.3,4 210.1,28.4 210.1,28.5 214.1,28.5 218.6,9.3 223.1,28.4 223.2,28.5 227.2,28.5 233.1,4 229.7,4"/> <path fill="#0095AA" d="M264.2,9.6c-1.2,0-2.2,0.3-3.2,1c-0.9,0.6-1.7,1.3-2.5,2.1V9.9h-3.2v18.7h3.2V15.7c1.1-1.1,2.1-1.8,2.9-2.3 c0.8-0.4,1.7-0.6,2.6-0.6c0.6,0,1.1,0.1,1.4,0.2l0.1,0l0.7-3.1l-0.1,0C265.6,9.7,264.9,9.6,264.2,9.6z"/> <rect x="269" y="2.1" fill="#0095AA" width="3.2" height="26.4"/> <path fill="#0095AA" d="M287.6,2.1v9.2c-1.5-1.3-3.2-1.9-5-1.9c-2.3,0-4.2,0.9-5.5,2.7c-1.3,1.8-2,4.3-2,7.3c0,3.1,0.6,5.5,1.8,7.2 c1.2,1.7,3,2.5,5.2,2.5c1.1,0,2.2-0.2,3.2-0.7c0.9-0.4,1.7-0.9,2.4-1.6v1.7h3.2V2.1H287.6z M287.6,14.2v9.7 c-0.6,0.6-1.4,1.1-2.2,1.6c-0.9,0.5-1.8,0.7-2.7,0.7c-2.9,0-4.3-2.3-4.3-6.9c0-2.4,0.4-4.2,1.3-5.4c0.9-1.1,2-1.7,3.3-1.7 C284.5,12.2,286,12.9,287.6,14.2z"/> </g> <g> <polygon fill="#666666" points="29,3.3 25.4,3.3 21,22.5 16.5,3.4 16.5,3.3 12.5,3.3 8,22.4 3.7,3.3 0,3.3 5.8,28.1 5.8,28.2 10,28.2 14.4,9.3 18.8,28.1 18.8,28.2 23.1,28.2 29,3.5"/> <path fill="#666666" d="M37,8.7c-2.7,0-4.8,0.8-6.2,2.5c-1.4,1.7-2.1,4.2-2.1,7.6c0,3.3,0.7,5.8,2.1,7.5c1.4,1.7,3.5,2.5,6.2,2.5 c2.7,0,4.8-0.9,6.2-2.6c1.4-1.7,2.1-4.2,2.1-7.5c0-3.3-0.7-5.8-2.1-7.5C41.8,9.5,39.7,8.7,37,8.7z M40.5,24 c-0.8,1.1-1.9,1.7-3.6,1.7c-1.7,0-2.9-0.5-3.7-1.7c-0.8-1.1-1.2-2.9-1.2-5.3c0-2.5,0.4-4.3,1.2-5.4c0.8-1.1,2-1.6,3.6-1.6 c1.6,0,2.8,0.5,3.5,1.6c0.8,1.1,1.2,2.9,1.2,5.4C41.7,21.2,41.3,22.9,40.5,24z"/> <rect x="48.7" y="1.4" fill="#666666" width="3.4" height="26.8"/> <path fill="#666666" d="M67,1.6c-1.1-0.5-2.3-0.7-3.6-0.7c-2.2,0-3.7,0.6-4.6,1.7c-0.9,1.1-1.3,2.8-1.3,5.1v1.6h-2.7v3h2.7v16.1h3.4 V12.2h4.1v-3h-4.1V7.6c0-1.5,0.2-2.5,0.6-3c0.4-0.5,1-0.7,2-0.7c0.4,0,0.8,0.1,1.3,0.2c0.5,0.1,0.9,0.3,1.2,0.4l0.2,0.1l1-3 L67,1.6z"/> <path fill="#666666" d="M76.3,8.9c-1.2,0-2.3,0.4-3.2,1c-0.8,0.6-1.6,1.2-2.3,2V9.2h-3.4v19h3.4v-13c1.1-1.1,2-1.8,2.8-2.2 c0.8-0.4,1.7-0.6,2.5-0.6c0.6,0,1,0.1,1.3,0.2l0.2,0.1l0.7-3.3l-0.1-0.1C77.7,9.1,77.1,8.9,76.3,8.9z"/> <path fill="#666666" d="M92.3,10.2c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.4,0.8c-1.2,0.6-2.1,1.4-2.7,2.5l-0.1,0.2l2.8,1.7 l0.1-0.2c0.3-0.5,0.7-0.9,1.4-1.3c0.6-0.4,1.6-0.6,2.8-0.6c1.3,0,2.4,0.3,3.2,0.7c0.7,0.5,1.1,1.4,1.1,2.8v0.7l-4.4,0.4 c-1.8,0.2-3.2,0.6-4.2,1.1c-1,0.5-1.7,1.2-2.2,2.1c-0.5,0.9-0.8,2-0.8,3.3c0,1.7,0.5,3.2,1.5,4.3c1,1.1,2.4,1.7,4.2,1.7 c1.2,0,2.3-0.3,3.3-0.8c0.8-0.5,1.7-1.1,2.5-1.8v2.1h3.4V15.4C94.1,12.9,93.5,11.2,92.3,10.2z M85.8,25.6c-1,0-1.8-0.2-2.3-0.7 c-0.5-0.4-0.8-1.2-0.8-2.2c0-1.1,0.4-1.9,1.1-2.5c0.8-0.6,2.2-1,4.1-1.1l2.7-0.3V23c-0.4,0.3-0.7,0.6-1.1,0.9 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6C86.7,25.5,86.2,25.6,85.8,25.6z"/> <path fill="#666666" d="M122,11.6c-0.3-0.9-0.9-1.7-1.6-2.2c-0.7-0.5-1.7-0.8-2.8-0.8c-1.3,0-2.4,0.4-3.4,1.1 c-0.9,0.6-1.8,1.4-2.7,2.3c-0.3-1.1-0.8-2-1.5-2.5c-0.8-0.6-1.8-0.9-3.1-0.9c-1.2,0-2.3,0.3-3.3,1c-0.8,0.6-1.6,1.2-2.4,2V9.2 h-3.4v19h3.4V15c0.9-0.9,1.8-1.7,2.7-2.2c0.8-0.5,1.6-0.8,2.2-0.8c0.7,0,1.2,0.1,1.5,0.3c0.3,0.2,0.5,0.6,0.6,1.2 c0.1,0.6,0.2,1.8,0.2,3.5v11.2h3.4V15c1.2-1.1,2.2-1.9,2.9-2.3c0.7-0.4,1.4-0.6,2-0.6c0.7,0,1.2,0.1,1.5,0.3 c0.3,0.2,0.5,0.6,0.7,1.1c0.1,0.6,0.2,1.8,0.2,3.6v11.2h3.4V15.6C122.5,13.9,122.3,12.5,122,11.6z"/> </g> <path fill="#0095AA" d="M242.6,8.3c-5.8,0-10.5,4.7-10.5,10.5s4.7,10.5,10.5,10.5c5.8,0,10.5-4.7,10.5-10.5S248.4,8.3,242.6,8.3z M235.7,23.9c0-0.5,0.1-1,0.2-1.6c1.1,0.9,2.3,1.7,3.7,2.4l-0.6,1.4C237.8,25.5,236.6,24.7,235.7,23.9z M238.8,26.5l-0.2,0.5 c-0.9-0.4-1.8-1-2.5-1.6c-0.2-0.2-0.3-0.5-0.3-0.8C236.6,25.4,237.7,26,238.8,26.5z M235.1,23.4c-0.8-0.8-1.4-1.6-1.6-2.4 c-0.1-0.5-0.1-1.1-0.1-1.6c0.4,0.8,1.1,1.7,2,2.5C235.3,22.4,235.2,23,235.1,23.4z M235.2,24.6L235.2,24.6 c-0.3-0.4-0.6-0.8-0.8-1.2c0.2,0.2,0.5,0.5,0.7,0.7C235.2,24.3,235.2,24.4,235.2,24.6z M233.4,17.5c0-0.1,0-0.1,0-0.2 c0.1-0.7,0.4-1.5,0.7-2.2c0-0.1,0-0.1,0.1-0.2c0,0.3,0.1,0.6,0.2,0.9c-0.3,0.4-0.5,0.8-0.7,1.2C233.5,17.1,233.5,17.3,233.4,17.5z M233.6,18.5c0.1-0.4,0.2-0.8,0.4-1.2c0.1-0.3,0.3-0.6,0.5-0.9c0.4,0.9,1.1,1.9,2.1,2.8c-0.2,0.4-0.4,0.8-0.6,1.2 c-0.1,0.3-0.3,0.6-0.4,0.9C234.6,20.4,234,19.4,233.6,18.5z M250.7,15.2c-0.1,0.4-0.3,0.8-0.8,1c-0.3-0.8-0.8-1.6-1.3-2.4 c0.2-0.1,0.3-0.3,0.4-0.5C249.7,14,250.3,14.6,250.7,15.2z M249.2,12.4c0.4,0.2,0.7,0.5,1,0.8c0.3,0.4,0.4,0.7,0.5,1 c0,0,0,0.1,0,0.1c-0.4-0.5-0.9-1-1.4-1.5C249.2,12.7,249.2,12.6,249.2,12.4C249.2,12.4,249.2,12.4,249.2,12.4z M251.8,17.8 c-0.1,0.6-0.5,1.1-1.1,1.5c-0.1-0.8-0.3-1.7-0.6-2.5c0.4-0.2,0.7-0.6,0.9-0.9C251.4,16.5,251.7,17.1,251.8,17.8z M251.3,15.2 C251.3,15.2,251.3,15.2,251.3,15.2C251.3,15.2,251.3,15.2,251.3,15.2C251.3,15.2,251.3,15.2,251.3,15.2z M239.2,16.5 c0.6-0.7,1.2-1.3,1.8-1.9c0.7,0.6,1.5,1.1,2.5,1.6l-1.1,2.4C241.2,18,240.1,17.3,239.2,16.5z M242.2,19.1l-1.1,2.4 c-1.4-0.7-2.7-1.5-3.7-2.4c0.4-0.7,0.9-1.4,1.5-2.1C239.8,17.8,241,18.5,242.2,19.1z M244,16.4c0.8,0.3,1.5,0.6,2.3,0.7 c0.2,0,0.4,0.1,0.6,0.1c0,0.9-0.1,1.8-0.2,2.6c-0.3,0-0.6-0.1-0.9-0.1c-1-0.2-2-0.5-3-0.9C243.3,18,243.7,17.2,244,16.4z M246.7,14.2c0.1,0.8,0.2,1.6,0.3,2.4c-0.2,0-0.4-0.1-0.5-0.1c-0.7-0.1-1.4-0.4-2.1-0.7c0.3-0.7,0.6-1.5,0.9-2.1 c0.4,0.2,0.8,0.3,1.2,0.4C246.5,14.2,246.6,14.2,246.7,14.2z M246.6,12.2c0.5,0.4,0.9,0.9,1.3,1.4c-0.2,0.1-0.5,0.1-0.8,0.1 C247,13.1,246.8,12.6,246.6,12.2z M245,13.1c-0.3-0.2-0.7-0.4-1-0.6c0.5-0.3,1.1-0.6,1.6-0.7L245,13.1z M244.7,13.6l-1,2.1 c-0.8-0.4-1.6-0.9-2.3-1.4c0.7-0.6,1.3-1,2-1.5C243.9,13.1,244.3,13.4,244.7,13.6z M241.1,13.9c-0.6-0.6-1-1.1-1.2-1.7 c0.8-0.3,1.7-0.5,2.6-0.7c0.1,0.3,0.3,0.6,0.6,0.9C242.5,12.8,241.8,13.3,241.1,13.9z M242.3,11c-0.9,0.1-1.8,0.3-2.6,0.6 c-0.1-0.5,0-0.9,0.2-1.3c0.8-0.1,1.6-0.1,2.4,0C242.3,10.6,242.3,10.8,242.3,11z M240.7,14.3c-0.7,0.6-1.3,1.2-1.9,1.9 c-0.8-0.8-1.4-1.6-1.7-2.4c0.7-0.5,1.5-0.9,2.3-1.3C239.6,13,240.1,13.7,240.7,14.3z M239.2,11.9c-0.8,0.3-1.6,0.8-2.3,1.2 c-0.2-0.7,0-1.4,0.4-1.9c0.6-0.3,1.3-0.6,2-0.8C239.1,10.9,239.1,11.4,239.2,11.9z M238.4,16.6c-0.6,0.7-1.1,1.4-1.5,2.1 c-1-0.9-1.7-1.9-2-2.9c0.5-0.6,1-1.2,1.7-1.8C236.9,14.9,237.6,15.8,238.4,16.6z M237,19.6c1.1,0.9,2.4,1.8,3.8,2.4l-1,2.1 c-1.4-0.7-2.7-1.5-3.8-2.4c0.1-0.3,0.3-0.7,0.4-1.1C236.7,20.3,236.8,20,237,19.6z M240.2,24.9c1.2,0.5,2.4,0.9,3.5,1.1 c0.3,0.1,0.6,0.1,0.9,0.1c-0.3,0.5-0.7,0.9-1.1,1.2c-0.3,0-0.7-0.1-1-0.1c-1-0.2-2-0.5-3-0.9C239.7,25.9,239.9,25.5,240.2,24.9z M243.7,27.9c0.3,0,0.6,0,0.9,0c-0.4,0.1-0.9,0.2-1.3,0.2C243.5,28.1,243.6,28,243.7,27.9z M244.3,27.4c0.3-0.3,0.7-0.7,1-1.2 c1.1,0.1,2.2,0,3-0.2c-0.4,0.4-0.8,0.7-1.2,1C246.3,27.3,245.4,27.5,244.3,27.4z M244.9,25.7c-0.4,0-0.7-0.1-1.1-0.2 c-1.1-0.2-2.3-0.6-3.4-1.1c0.1-0.3,0.3-0.6,0.5-1l0.5-1.1c1.2,0.5,2.4,0.9,3.5,1.1c0.3,0.1,0.7,0.1,1,0.2 c-0.1,0.4-0.3,0.7-0.4,1.1C245.3,25,245.1,25.3,244.9,25.7z M245.1,22.8c-1.1-0.2-2.3-0.6-3.4-1.1c0.4-0.8,0.7-1.6,1.1-2.4 c1,0.4,2.1,0.8,3.1,1c0.3,0.1,0.6,0.1,0.9,0.1c-0.1,0.9-0.3,1.7-0.6,2.5C245.8,22.9,245.4,22.9,245.1,22.8z M247.3,20.5 c1.1,0.1,2.2,0,3-0.4c0,0.8,0,1.7-0.2,2.4c-0.9,0.4-2,0.5-3.4,0.5C247,22.2,247.1,21.4,247.3,20.5z M247.4,19.9 c0.1-0.9,0.2-1.8,0.1-2.6c0.8,0,1.6,0,2.2-0.3c0.3,0.8,0.5,1.7,0.5,2.5C249.5,19.9,248.5,20,247.4,19.9z M249.5,16.5 c-0.5,0.2-1.2,0.3-2,0.2c0-0.9-0.1-1.7-0.3-2.5c0.4,0,0.7,0,1-0.1C248.8,14.8,249.2,15.6,249.5,16.5z M248.4,13.4 c-0.3-0.4-0.7-0.8-1.1-1.2c0.5,0.3,0.9,0.5,1.3,0.9C248.6,13.2,248.5,13.3,248.4,13.4z M247.4,11.7c0.4,0.1,0.7,0.2,1,0.3 c0.1,0.2,0.2,0.3,0.2,0.5C248.2,12.2,247.8,11.9,247.4,11.7z M246.7,11.1c0.2-0.1,0.4-0.2,0.5-0.2c0.2,0.1,0.3,0.2,0.5,0.4 C247.4,11.2,247.1,11.2,246.7,11.1z M246.2,10.9c0-0.2-0.1-0.5-0.1-0.7c0.2,0.1,0.5,0.2,0.7,0.3C246.6,10.7,246.4,10.8,246.2,10.9 z M246.6,13.6c0,0-0.1,0-0.1,0c-0.3-0.1-0.7-0.2-1-0.3c0.2-0.5,0.4-1,0.6-1.4C246.2,12.4,246.4,13,246.6,13.6z M244.8,10 c0.1,0,0.2,0,0.3,0.1c0.1,0,0.2,0,0.3,0.1c0.1,0.1,0.2,0.3,0.2,0.6C245.4,10.4,245.1,10.2,244.8,10z M245,11.4 c-0.5,0.2-0.9,0.4-1.4,0.7c-0.2-0.2-0.4-0.5-0.5-0.7C243.8,11.4,244.4,11.4,245,11.4z M244.6,11c-0.6,0-1.1,0-1.7,0 c0-0.2,0-0.3,0.1-0.5C243.5,10.6,244.1,10.8,244.6,11z M243.4,10.1c0.1-0.1,0.3-0.1,0.4-0.1c0.3,0.1,0.6,0.3,0.9,0.5 C244.3,10.4,243.9,10.2,243.4,10.1z M242.7,10c-0.5-0.1-1-0.1-1.5-0.1c-0.2,0-0.4,0-0.5,0c0.3-0.2,0.7-0.3,1.1-0.4 c0.5,0,0.9,0,1.4,0.2C243,9.8,242.8,9.9,242.7,10z M236.4,13.5c-0.6,0.5-1.2,1.1-1.7,1.7c-0.1-0.6-0.1-1.1,0.1-1.5 c0.4-0.7,1-1.3,1.6-1.8C236.2,12.4,236.2,12.9,236.4,13.5z M239.3,26.8c1,0.4,2,0.8,3.1,0.9c0.2,0,0.3,0.1,0.5,0.1 c-0.3,0.1-0.5,0.2-0.8,0.2c-0.3,0-0.5-0.1-0.8-0.1c-0.7-0.1-1.5-0.4-2.2-0.7C239.2,27.1,239.2,27,239.3,26.8z M245.6,25.7 c0.2-0.3,0.3-0.6,0.5-0.9c0.2-0.4,0.3-0.8,0.5-1.2c1.3,0.1,2.4,0,3.4-0.4c-0.1,0.3-0.2,0.6-0.3,0.9c-0.2,0.4-0.4,0.8-0.6,1.1 C248,25.6,246.9,25.8,245.6,25.7z M250,24.3c0.2-0.4,0.3-0.8,0.5-1.3c0.2-0.1,0.5-0.3,0.7-0.4c-0.3,0.7-0.7,1.3-1.1,1.9 c-0.1,0.1-0.1,0.1-0.2,0.2C249.9,24.5,250,24.4,250,24.3z M250.7,22.3c0.1-0.8,0.2-1.5,0.1-2.3c0.5-0.3,0.9-0.6,1.2-1 c0,0.1,0,0.2,0,0.3c0,0.6-0.1,1.3-0.3,1.8C251.5,21.5,251.2,21.9,250.7,22.3z"/> </g> <g> <path fill="#0095AA" d="M297.9,7.6h4.5v0.8h-3.5V11h2.9v0.8h-2.9v3.5h-1V7.6z"/> <path fill="#0095AA" d="M303.7,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4H308l-1.9-3.3h-1.4v3.3h-1V7.6z M306,11.2 c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H306z"/> <path fill="#0095AA" d="M309.9,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C311.2,15.5,309.9,13.9,309.9,11.4z M315.5,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C314.6,14.6,315.5,13.3,315.5,11.4z"/> <path fill="#0095AA" d="M318.1,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4H321l-1.4-4L319,8.7h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M330.7,8.4h-2.3V7.6h5.7v0.8h-2.3v6.9h-1V8.4z"/> <path fill="#0095AA" d="M335.4,7.6h1v3.2h3.6V7.6h1v7.7h-1v-3.6h-3.6v3.6h-1V7.6z"/> <path fill="#0095AA" d="M343.1,7.6h4.5v0.8h-3.5v2.4h2.9v0.8h-2.9v2.8h3.6v0.8h-4.6V7.6z"/> <path fill="#0095AA" d="M351.7,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4h-0.7l-1.4-4l-0.6-1.8h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M361.8,7.6h1.1l2.6,7.7h-1.1l-0.7-2.3H361l-0.7,2.3h-1L361.8,7.6z M361.2,12.2h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L361.2,12.2z"/> <path fill="#0095AA" d="M366.7,7.6h1v3.9h0l3.2-3.9h1.1l-2.4,2.9l2.8,4.8h-1.1l-2.3-4l-1.3,1.6v2.4h-1V7.6z"/> <path fill="#0095AA" d="M373,7.6h4.5v0.8H374v2.4h2.9v0.8H374v2.8h3.6v0.8H373V7.6z"/> <path fill="#0095AA" d="M379.2,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4h-1.1l-1.9-3.3h-1.4v3.3h-1V7.6z M381.5,11.2c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H381.5z"/> <path fill="#0095AA" d="M385.4,14.3l0.6-0.6c0.6,0.6,1.4,0.9,2.2,0.9c1,0,1.6-0.5,1.6-1.3c0-0.8-0.6-1-1.3-1.4l-1.1-0.5 c-0.7-0.3-1.6-0.8-1.6-2c0-1.2,1-2.1,2.4-2.1c0.9,0,1.7,0.4,2.3,0.9L390,9c-0.5-0.4-1.1-0.7-1.7-0.7c-0.9,0-1.4,0.4-1.4,1.1 c0,0.7,0.7,1,1.3,1.3l1.1,0.5c0.9,0.4,1.6,0.9,1.6,2.1c0,1.2-1,2.2-2.6,2.2C387,15.5,386.1,15,385.4,14.3z"/> <path fill="#0095AA" d="M393.7,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C395.1,15.5,393.7,13.9,393.7,11.4z M399.3,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C398.4,14.6,399.3,13.3,399.3,11.4z"/> <path fill="#0095AA" d="M402,7.6h4.5v0.8H403V11h2.9v0.8H403v3.5h-1V7.6z"/> <path fill="#0095AA" d="M410.1,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4H413l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M420.2,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L420.2,7.6z M420,12.2h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L420,12.2z"/> <path fill="#0095AA" d="M425.6,8.8h-2.2V7.6h5.8v1.2H427v6.5h-1.4V8.8z"/> <path fill="#0095AA" d="M430,7.6h1.4v3.1h3.1V7.6h1.4v7.7h-1.4v-3.4h-3.1v3.4H430V7.6z"/> <path fill="#0095AA" d="M437.4,7.6h4.7v1.2h-3.3v1.9h2.8v1.2h-2.8v2.2h3.4v1.2h-4.8V7.6z"/> <path fill="#0095AA" d="M443.3,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4h-0.9l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M453.3,7.6h1.7l2.5,7.7H456l-0.6-2.1h-2.6l-0.6,2.1h-1.4L453.3,7.6z M453.2,12.2h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L453.2,12.2z"/> <path fill="#0095AA" d="M459,8.8h-2.2V7.6h5.8v1.2h-2.2v6.5H459V8.8z"/> <path fill="#0095AA" d="M463.7,7.6h1.4v7.7h-1.4V7.6z"/> <path fill="#0095AA" d="M466.4,11.5c0-2.5,1.6-4,3.5-4c1,0,1.7,0.4,2.2,1l-0.7,0.9c-0.4-0.4-0.9-0.6-1.5-0.6c-1.2,0-2.1,1-2.1,2.8 c0,1.7,0.8,2.8,2.1,2.8c0.7,0,1.2-0.3,1.6-0.7l0.8,0.8c-0.6,0.7-1.5,1.1-2.4,1.1C467.9,15.5,466.4,14,466.4,11.5z"/> <path fill="#0095AA" d="M474.8,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L474.8,7.6z M474.7,12.2h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L474.7,12.2z"/> <path fill="#0095AA" d="M299.5,20.7h1.1l2.6,7.7h-1.1l-0.7-2.3h-2.8l-0.7,2.3h-1L299.5,20.7z M298.9,25.3h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L298.9,25.3z"/> <path fill="#0095AA" d="M304.1,20.7h1.1l2.8,4.8c0.3,0.5,0.6,1.1,0.8,1.7h0c-0.1-0.8-0.1-1.7-0.1-2.5v-4h0.9v7.7h-1.1l-2.8-4.8 c-0.3-0.5-0.6-1.1-0.8-1.7h0c0.1,0.8,0.1,1.6,0.1,2.4v4.1h-0.9V20.7z"/> <path fill="#0095AA" d="M311.4,20.7h1.9c2.4,0,3.7,1.4,3.7,3.8c0,2.5-1.3,3.9-3.6,3.9h-2V20.7z M313.3,27.6c1.8,0,2.7-1.1,2.7-3.1 c0-1.9-0.9-3-2.7-3h-0.9v6.1H313.3z"/> <path fill="#0095AA" d="M320.4,20.6l1.4,0l0.6,3.9c0.1,0.8,0.2,1.6,0.3,2.5l0,0c0.2-0.9,0.3-1.7,0.5-2.5l1-3.9l1.3,0l0.9,3.9 c0.2,0.8,0.3,1.6,0.5,2.5l0,0c0.1-0.9,0.2-1.7,0.4-2.5l0.7-3.9l1.3,0l-1.6,7.7l-1.8,0l-0.8-4.1c-0.1-0.6-0.2-1.2-0.3-1.9l0,0 c-0.1,0.7-0.2,1.2-0.3,1.9l-0.9,4l-1.8,0L320.4,20.6z"/> <path fill="#0095AA" d="M329.8,24.5c0-2.5,1.5-3.9,3.5-3.9c2,0,3.4,1.5,3.3,4c0,2.5-1.5,4-3.5,4C331.2,28.5,329.8,27,329.8,24.5z M335.2,24.6c0-1.7-0.7-2.7-1.9-2.8c-1.2,0-2,1-2,2.7c0,1.7,0.7,2.8,1.9,2.9C334.4,27.4,335.2,26.3,335.2,24.6z"/> <path fill="#0095AA" d="M337.8,20.7l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L337.8,20.7z"/> <path fill="#0095AA" d="M343.4,20.7l4.7,0.1l0,1.2l-3.3-0.1l0,2.2l2.8,0l0,1.2l-2.8,0l-0.1,3.2l-1.4,0L343.4,20.7z"/> <path fill="#0095AA" d="M349,20.7l2.6,0c1.6,0,2.8,0.6,2.8,2.3c0,1.2-0.6,1.9-1.6,2.2l1.8,3.2l-1.6,0l-1.6-3l-1.1,0l-0.1,3l-1.4,0 L349,20.7z M351.4,24.3c1,0,1.6-0.4,1.6-1.3c0-0.9-0.5-1.2-1.6-1.2l-1.1,0l0,2.5L351.4,24.3z"/> <path fill="#0095AA" d="M357.4,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L357.4,20.7z M357.2,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L357.2,25.2z"/> <path fill="#0095AA" d="M362.5,20.6l1.7,0l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5l0,0c0.2-0.6,0.3-1,0.5-1.5l1.4-3.8l1.7,0l-0.1,7.7l-1.3,0 l0.1-3.5c0-0.7,0.1-1.8,0.3-2.5l0,0l-0.6,1.9l-1.3,3.4l-0.9,0l-1.2-3.5l-0.6-1.9l0,0c0.1,0.7,0.2,1.8,0.2,2.5l-0.1,3.5l-1.3,0 L362.5,20.6z"/> <path fill="#0095AA" d="M370.8,19.1l0.9,0L371.5,30l-0.9,0L370.8,19.1z"/> <path fill="#0095AA" d="M375.1,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L375.1,20.7z M374.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L374.8,25.2z"/> <path fill="#0095AA" d="M379.9,20.7l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L379.9,20.7z"/> <path fill="#0095AA" d="M385.6,20.7l2.4,0c1.7,0,3,0.7,3,2.4c0,1.7-1.3,2.5-3,2.4l-1.1,0l0,2.8l-1.4,0L385.6,20.7z M387.9,24.5 c1.1,0,1.7-0.4,1.7-1.4c0-0.9-0.6-1.3-1.7-1.3l-0.9,0l0,2.7L387.9,24.5z"/> <path fill="#0095AA" d="M392,20.6l1.4,0l-0.1,3.1l3.1,0.1l0.1-3.1l1.4,0l-0.1,7.7l-1.4,0l0.1-3.4l-3.1-0.1l-0.1,3.4l-1.4,0L392,20.6z "/> <path fill="#0095AA" d="M401,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L401,20.7z M400.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L400.8,25.2z"/> </g> </g> <a href="https://www.wolfram.com/mathematica/"><rect x="409" y="5.9" style="fill:#ffffff00;" width="70" height="11.6"/></a> <a href="https://wolframalpha.com/"><rect x="319.6" y="18.2" style="fill:#ffffff00;" width="86.9" height="11.6"/></a> <a href="/"><rect y="0.1" style="fill:#ffffff00;" width="292.4" height="29.9"/></a> </svg> <svg version="1.1" id="logo-600" class="hide show__600" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 480.7 30" style="enable-background:new 0 0 480.7 30;" xml:space="preserve"> <g> <g> <g> <polygon fill="#0095AA" points="145.2,17.1 138.8,3.9 134.2,3.9 134.2,28.5 137.5,28.5 137.5,8.6 144,21.8 144,21.9 146.2,21.9 152.7,8.6 152.7,28.5 156,28.5 156,3.9 151.6,3.9"/> <path fill="#0095AA" d="M171.8,10.8c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.3,0.8c-1.2,0.6-2.1,1.4-2.6,2.4l0,0.1l2.6,1.6l0-0.1 c0.3-0.5,0.8-1,1.4-1.4c0.7-0.4,1.6-0.6,2.8-0.6c1.4,0,2.5,0.3,3.2,0.8c0.8,0.5,1.1,1.4,1.1,2.8v0.8l-4.5,0.4 c-1.8,0.2-3.2,0.6-4.1,1.1c-1,0.5-1.7,1.2-2.2,2c-0.5,0.8-0.8,1.9-0.8,3.2c0,1.7,0.5,3.1,1.5,4.2c1,1.1,2.4,1.6,4.1,1.6 c1.2,0,2.3-0.3,3.3-0.8c0.9-0.5,1.8-1.2,2.7-1.9v2.2h3.2V15.9C173.6,13.5,173,11.8,171.8,10.8z M170.4,19.1v4.4 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6c-0.4,0.2-0.9,0.2-1.3,0.2c-1.1,0-1.9-0.2-2.4-0.7 c-0.5-0.5-0.8-1.2-0.8-2.3c0-1.1,0.4-2,1.2-2.6c0.8-0.6,2.2-1,4.2-1.2L170.4,19.1z"/> <path fill="#0095AA" d="M185.9,25.8c-0.7,0.3-1.6,0.5-2.4,0.5c-1,0-1.6-0.3-1.9-0.9c-0.3-0.6-0.4-1.7-0.4-3.2v-9.7h4.7V9.8h-4.7V4.7 h-3.2v5.1h-2.6v2.7h2.6v10.3c0,2.1,0.4,3.7,1.2,4.7c0.8,1,2.2,1.5,4.1,1.5c1.2,0,2.3-0.2,3.5-0.6l0.1,0L185.9,25.8L185.9,25.8z" /> <path fill="#0095AA" d="M199.1,9.3c-1.2,0-2.3,0.3-3.5,1c-1.1,0.6-2,1.3-2.9,2.1V2.1h-3.2v26.4h3.2V15.4c0.9-0.8,1.9-1.5,2.9-2.1 c1-0.6,1.9-0.9,2.7-0.9c0.8,0,1.3,0.1,1.7,0.4c0.4,0.3,0.7,0.7,0.8,1.2c0.2,0.6,0.3,1.7,0.3,3.5v11h3.2V16.1 c0-2.4-0.4-4.2-1.3-5.2C202.2,9.8,200.9,9.3,199.1,9.3z"/> <polygon fill="#0095AA" points="226.2,23.5 221.6,4 221.6,3.9 217.8,3.9 213.2,23.3 208.8,4 208.8,3.9 205.3,3.9 211.1,28.4 211.1,28.5 215.1,28.5 219.6,9.3 224.1,28.4 224.1,28.5 228.2,28.5 234.1,3.9 230.7,3.9 "/> <path fill="#0095AA" d="M265.2,9.6c-1.2,0-2.2,0.3-3.2,1c-0.9,0.6-1.7,1.3-2.5,2.1V9.8h-3.2v18.7h3.2V15.7c1.1-1.1,2.1-1.8,2.9-2.3 c0.8-0.4,1.7-0.6,2.6-0.6c0.6,0,1.1,0.1,1.4,0.2l0.1,0l0.7-3.1l-0.1,0C266.6,9.7,265.9,9.6,265.2,9.6z"/> <rect x="270" y="2.1" fill="#0095AA" width="3.2" height="26.4"/> <path fill="#0095AA" d="M288.6,2.1v9.2c-1.5-1.3-3.2-1.9-5-1.9c-2.3,0-4.2,0.9-5.5,2.7c-1.3,1.8-2,4.3-2,7.3c0,3.1,0.6,5.5,1.8,7.2 c1.2,1.7,3,2.5,5.2,2.5c1.1,0,2.2-0.2,3.2-0.7c0.9-0.4,1.7-0.9,2.4-1.6v1.7h3.2V2.1H288.6z M288.6,14.1v9.7 c-0.6,0.6-1.4,1.1-2.2,1.6c-0.9,0.5-1.8,0.7-2.7,0.7c-2.9,0-4.3-2.3-4.3-6.9c0-2.4,0.4-4.2,1.3-5.4c0.9-1.1,2-1.7,3.3-1.7 C285.5,12.2,287,12.8,288.6,14.1z"/> </g> <g> <polygon fill="#666666" points="30,3.2 26.4,3.2 22,22.5 17.5,3.4 17.5,3.2 13.5,3.2 9,22.4 4.7,3.2 1,3.2 6.8,28 6.8,28.2 11,28.2 15.4,9.2 19.8,28 19.8,28.2 24,28.2 30,3.5"/> <path fill="#666666" d="M38,8.7c-2.7,0-4.8,0.8-6.2,2.5c-1.4,1.7-2.1,4.2-2.1,7.6c0,3.3,0.7,5.8,2.1,7.5c1.4,1.7,3.5,2.5,6.2,2.5 c2.7,0,4.8-0.9,6.2-2.6c1.4-1.7,2.1-4.2,2.1-7.5c0-3.3-0.7-5.8-2.1-7.5C42.8,9.5,40.7,8.7,38,8.7z M41.5,24 c-0.8,1.1-1.9,1.7-3.6,1.7c-1.7,0-2.9-0.5-3.7-1.7c-0.8-1.1-1.2-2.9-1.2-5.3c0-2.5,0.4-4.3,1.2-5.4c0.8-1.1,2-1.6,3.6-1.6 c1.6,0,2.8,0.5,3.5,1.6c0.8,1.1,1.2,2.9,1.2,5.4C42.7,21.1,42.3,22.9,41.5,24z"/> <rect x="49.7" y="1.4" fill="#666666" width="3.4" height="26.8"/> <path fill="#666666" d="M67.9,1.5c-1.1-0.5-2.3-0.7-3.6-0.7c-2.2,0-3.7,0.6-4.6,1.7c-0.9,1.1-1.3,2.8-1.3,5.1v1.6h-2.7v3h2.7v16.1 h3.4V12.2h4.1v-3h-4.1V7.6c0-1.5,0.2-2.5,0.6-3c0.4-0.5,1-0.7,2-0.7c0.4,0,0.8,0.1,1.3,0.2c0.5,0.1,0.9,0.3,1.2,0.4l0.2,0.1l1-3 L67.9,1.5z"/> <path fill="#666666" d="M77.3,8.9c-1.2,0-2.3,0.4-3.2,1c-0.8,0.6-1.6,1.2-2.3,2V9.2h-3.4v19h3.4v-13c1.1-1.1,2-1.8,2.8-2.2 c0.8-0.4,1.7-0.6,2.5-0.6c0.6,0,1,0.1,1.3,0.2l0.2,0.1l0.7-3.3l-0.1-0.1C78.7,9,78.1,8.9,77.3,8.9z"/> <path fill="#666666" d="M93.3,10.2c-1.2-1-3.1-1.5-5.8-1.5c-1.7,0-3.2,0.3-4.4,0.8c-1.2,0.6-2.1,1.4-2.7,2.5l-0.1,0.2l2.8,1.7 l0.1-0.2c0.3-0.5,0.7-0.9,1.4-1.3c0.6-0.4,1.6-0.6,2.8-0.6c1.3,0,2.4,0.3,3.2,0.7c0.7,0.5,1.1,1.4,1.1,2.8v0.7l-4.4,0.4 c-1.8,0.2-3.2,0.6-4.2,1.1c-1,0.5-1.7,1.2-2.2,2.1c-0.5,0.9-0.8,2-0.8,3.3c0,1.7,0.5,3.2,1.5,4.3c1,1.1,2.4,1.7,4.2,1.7 c1.2,0,2.3-0.3,3.3-0.8c0.8-0.5,1.7-1.1,2.5-1.8v2.1h3.4V15.4C95.1,12.9,94.5,11.2,93.3,10.2z M86.8,25.6c-1,0-1.8-0.2-2.3-0.7 c-0.5-0.4-0.8-1.2-0.8-2.2c0-1.1,0.4-1.9,1.1-2.5c0.8-0.6,2.2-1,4.1-1.1l2.7-0.3V23c-0.4,0.3-0.7,0.6-1.1,0.9 c-0.4,0.3-0.8,0.6-1.2,0.9c-0.4,0.2-0.9,0.5-1.3,0.6C87.7,25.5,87.2,25.6,86.8,25.6z"/> <path fill="#666666" d="M123,11.6c-0.3-0.9-0.9-1.7-1.6-2.2c-0.7-0.5-1.7-0.8-2.8-0.8c-1.3,0-2.4,0.4-3.4,1.1 c-0.9,0.6-1.8,1.4-2.7,2.3c-0.3-1.1-0.8-2-1.5-2.5c-0.8-0.6-1.8-0.9-3.1-0.9c-1.2,0-2.3,0.3-3.3,1c-0.8,0.6-1.6,1.2-2.4,2V9.2 h-3.4v19h3.4V14.9c0.9-0.9,1.8-1.7,2.7-2.2c0.8-0.5,1.6-0.8,2.2-0.8c0.7,0,1.2,0.1,1.5,0.3c0.3,0.2,0.5,0.6,0.6,1.2 c0.1,0.6,0.2,1.8,0.2,3.5v11.2h3.4V14.9c1.2-1.1,2.2-1.9,2.9-2.3c0.7-0.4,1.4-0.6,2-0.6c0.7,0,1.2,0.1,1.5,0.3 c0.3,0.2,0.5,0.6,0.7,1.1c0.1,0.6,0.2,1.8,0.2,3.6v11.2h3.4V15.6C123.5,13.9,123.3,12.5,123,11.6z"/> </g> <path fill="#0095AA" d="M243.6,8.3c-5.8,0-10.5,4.7-10.5,10.5s4.7,10.5,10.5,10.5c5.8,0,10.5-4.7,10.5-10.5S249.4,8.3,243.6,8.3z M236.7,23.9c0-0.5,0.1-1,0.2-1.6c1.1,0.9,2.3,1.7,3.7,2.4L240,26C238.8,25.5,237.6,24.7,236.7,23.9z M239.8,26.5l-0.2,0.5 c-0.9-0.4-1.8-1-2.5-1.6c-0.2-0.2-0.3-0.5-0.3-0.8C237.6,25.4,238.7,26,239.8,26.5z M236.1,23.4c-0.8-0.8-1.4-1.6-1.6-2.4 c-0.1-0.5-0.1-1.1-0.1-1.6c0.4,0.8,1.1,1.7,2,2.5C236.3,22.4,236.2,22.9,236.1,23.4z M236.2,24.5L236.2,24.5 c-0.3-0.4-0.6-0.8-0.8-1.2c0.2,0.2,0.5,0.5,0.7,0.7C236.2,24.3,236.2,24.4,236.2,24.5z M234.4,17.4c0-0.1,0-0.1,0-0.2 c0.1-0.7,0.4-1.5,0.7-2.2c0-0.1,0-0.1,0.1-0.2c0,0.3,0.1,0.6,0.2,0.9c-0.3,0.4-0.5,0.8-0.7,1.2C234.5,17.1,234.5,17.3,234.4,17.4z M234.6,18.4c0.1-0.4,0.2-0.8,0.4-1.2c0.1-0.3,0.3-0.6,0.5-0.9c0.4,0.9,1.1,1.9,2.1,2.8c-0.2,0.4-0.4,0.8-0.6,1.2 c-0.1,0.3-0.3,0.6-0.4,0.9C235.6,20.4,235,19.4,234.6,18.4z M251.7,15.2c-0.1,0.4-0.3,0.8-0.8,1c-0.3-0.8-0.8-1.6-1.3-2.4 c0.2-0.1,0.3-0.3,0.4-0.5C250.7,13.9,251.3,14.5,251.7,15.2z M250.2,12.3c0.4,0.2,0.7,0.5,1,0.8c0.3,0.4,0.4,0.7,0.5,1 c0,0,0,0.1,0,0.1c-0.4-0.5-0.9-1-1.4-1.5C250.2,12.7,250.2,12.5,250.2,12.3C250.2,12.4,250.2,12.3,250.2,12.3z M252.8,17.8 c-0.1,0.6-0.5,1.1-1.1,1.5c-0.1-0.8-0.3-1.7-0.6-2.5c0.4-0.2,0.7-0.6,0.9-0.9C252.4,16.5,252.7,17.1,252.8,17.8z M252.3,15.2 C252.3,15.2,252.3,15.2,252.3,15.2C252.3,15.2,252.3,15.2,252.3,15.2C252.3,15.2,252.3,15.2,252.3,15.2z M240.2,16.5 c0.6-0.7,1.2-1.3,1.8-1.9c0.7,0.6,1.5,1.1,2.5,1.6l-1.1,2.4C242.2,18,241.1,17.3,240.2,16.5z M243.2,19.1l-1.1,2.4 c-1.4-0.7-2.7-1.5-3.7-2.4c0.4-0.7,0.9-1.4,1.5-2.1C240.8,17.8,242,18.5,243.2,19.1z M245,16.4c0.8,0.3,1.5,0.6,2.3,0.7 c0.2,0,0.4,0.1,0.6,0.1c0,0.9-0.1,1.8-0.2,2.6c-0.3,0-0.6-0.1-0.9-0.1c-1-0.2-2-0.5-3-0.9C244.3,18,244.7,17.2,245,16.4z M247.7,14.2c0.1,0.8,0.2,1.6,0.3,2.4c-0.2,0-0.4-0.1-0.5-0.1c-0.7-0.1-1.4-0.4-2.1-0.7c0.3-0.7,0.6-1.5,0.9-2.1 c0.4,0.2,0.8,0.3,1.2,0.4C247.5,14.1,247.6,14.2,247.7,14.2z M247.6,12.2c0.5,0.4,0.9,0.9,1.3,1.4c-0.2,0.1-0.5,0.1-0.8,0.1 C248,13.1,247.8,12.6,247.6,12.2z M246,13.1c-0.3-0.2-0.7-0.4-1-0.6c0.5-0.3,1.1-0.6,1.6-0.7L246,13.1z M245.7,13.6l-1,2.1 c-0.8-0.4-1.6-0.9-2.3-1.4c0.7-0.6,1.3-1,2-1.5C244.9,13.1,245.3,13.3,245.7,13.6z M242.1,13.9c-0.6-0.6-1-1.1-1.2-1.7 c0.8-0.3,1.7-0.5,2.6-0.7c0.1,0.3,0.3,0.6,0.6,0.9C243.5,12.8,242.8,13.3,242.1,13.9z M243.3,11c-0.9,0.1-1.8,0.3-2.6,0.6 c-0.1-0.5,0-0.9,0.2-1.3c0.8-0.1,1.6-0.1,2.4,0C243.3,10.6,243.3,10.8,243.3,11z M241.7,14.2c-0.7,0.6-1.3,1.2-1.9,1.9 c-0.8-0.8-1.4-1.6-1.7-2.4c0.7-0.5,1.5-0.9,2.3-1.3C240.6,13,241.1,13.6,241.7,14.2z M240.2,11.9c-0.8,0.3-1.6,0.8-2.3,1.2 c-0.2-0.7,0-1.4,0.4-1.9c0.6-0.3,1.3-0.6,2-0.8C240.1,10.9,240.1,11.4,240.2,11.9z M239.4,16.6c-0.6,0.7-1.1,1.4-1.5,2.1 c-1-0.9-1.7-1.9-2-2.9c0.5-0.6,1-1.2,1.7-1.8C237.9,14.9,238.6,15.8,239.4,16.6z M238,19.6c1.1,0.9,2.4,1.8,3.8,2.4l-1,2.1 c-1.4-0.7-2.7-1.5-3.8-2.4c0.1-0.3,0.3-0.7,0.4-1.1C237.7,20.3,237.8,20,238,19.6z M241.2,24.9c1.2,0.5,2.4,0.9,3.5,1.1 c0.3,0.1,0.6,0.1,0.9,0.1c-0.3,0.5-0.7,0.9-1.1,1.2c-0.3,0-0.7-0.1-1-0.1c-1-0.2-2-0.5-3-0.9C240.7,25.9,240.9,25.5,241.2,24.9z M244.7,27.9c0.3,0,0.6,0,0.9,0c-0.4,0.1-0.9,0.2-1.3,0.2C244.5,28,244.6,28,244.7,27.9z M245.3,27.4c0.3-0.3,0.7-0.7,1-1.2 c1.1,0.1,2.2,0,3-0.2c-0.4,0.4-0.8,0.7-1.2,1C247.3,27.3,246.4,27.4,245.3,27.4z M245.9,25.6c-0.4,0-0.7-0.1-1.1-0.2 c-1.1-0.2-2.3-0.6-3.4-1.1c0.1-0.3,0.3-0.6,0.5-1l0.5-1.1c1.2,0.5,2.4,0.9,3.5,1.1c0.3,0.1,0.7,0.1,1,0.2 c-0.1,0.4-0.3,0.7-0.4,1.1C246.3,25,246.1,25.3,245.9,25.6z M246.1,22.8c-1.1-0.2-2.3-0.6-3.4-1.1c0.4-0.8,0.7-1.6,1.1-2.4 c1,0.4,2.1,0.8,3.1,1c0.3,0.1,0.6,0.1,0.9,0.1c-0.1,0.9-0.3,1.7-0.6,2.5C246.8,22.9,246.4,22.9,246.1,22.8z M248.3,20.5 c1.1,0.1,2.2,0,3-0.4c0,0.8,0,1.7-0.2,2.4c-0.9,0.4-2,0.5-3.4,0.5C248,22.2,248.1,21.4,248.3,20.5z M248.4,19.9 c0.1-0.9,0.2-1.8,0.1-2.6c0.8,0,1.6,0,2.2-0.3c0.3,0.8,0.5,1.7,0.5,2.5C250.5,19.8,249.5,20,248.4,19.9z M250.5,16.4 c-0.5,0.2-1.2,0.3-2,0.2c0-0.9-0.1-1.7-0.3-2.5c0.4,0,0.7,0,1-0.1C249.8,14.8,250.2,15.6,250.5,16.4z M249.4,13.4 c-0.3-0.4-0.7-0.8-1.1-1.2c0.5,0.3,0.9,0.5,1.3,0.9C249.6,13.2,249.5,13.3,249.4,13.4z M248.4,11.7c0.4,0.1,0.7,0.2,1,0.3 c0.1,0.2,0.2,0.3,0.2,0.5C249.2,12.1,248.8,11.9,248.4,11.7z M247.7,11.1c0.2-0.1,0.4-0.2,0.5-0.2c0.2,0.1,0.3,0.2,0.5,0.4 C248.4,11.2,248.1,11.2,247.7,11.1z M247.2,10.9c0-0.2-0.1-0.5-0.1-0.7c0.2,0.1,0.5,0.2,0.7,0.3C247.6,10.7,247.4,10.8,247.2,10.9 z M247.6,13.6c0,0-0.1,0-0.1,0c-0.3-0.1-0.7-0.2-1-0.3c0.2-0.5,0.4-1,0.6-1.4C247.2,12.4,247.4,13,247.6,13.6z M245.8,10 c0.1,0,0.2,0,0.3,0.1c0.1,0,0.2,0,0.3,0.1c0.1,0.1,0.2,0.3,0.2,0.6C246.4,10.4,246.1,10.2,245.8,10z M246,11.4 c-0.5,0.2-0.9,0.4-1.4,0.7c-0.2-0.2-0.4-0.5-0.5-0.7C244.7,11.4,245.4,11.4,246,11.4z M245.6,10.9c-0.6,0-1.1,0-1.7,0 c0-0.2,0-0.3,0.1-0.5C244.5,10.6,245.1,10.7,245.6,10.9z M244.4,10.1c0.1-0.1,0.3-0.1,0.4-0.1c0.3,0.1,0.6,0.3,0.9,0.5 C245.3,10.3,244.9,10.2,244.4,10.1z M243.7,9.9c-0.5-0.1-1-0.1-1.5-0.1c-0.2,0-0.4,0-0.5,0c0.3-0.2,0.7-0.3,1.1-0.4 c0.5,0,0.9,0,1.4,0.2C243.9,9.7,243.8,9.8,243.7,9.9z M237.4,13.5c-0.6,0.5-1.2,1.1-1.7,1.7c-0.1-0.6-0.1-1.1,0.1-1.5 c0.4-0.7,1-1.3,1.6-1.8C237.2,12.4,237.2,12.9,237.4,13.5z M240.3,26.8c1,0.4,2,0.8,3.1,0.9c0.2,0,0.3,0.1,0.5,0.1 c-0.3,0.1-0.5,0.2-0.8,0.2c-0.3,0-0.5-0.1-0.8-0.1c-0.7-0.1-1.5-0.4-2.2-0.7C240.1,27.1,240.2,27,240.3,26.8z M246.6,25.7 c0.2-0.3,0.3-0.6,0.5-0.9c0.2-0.4,0.3-0.8,0.5-1.2c1.3,0.1,2.4,0,3.4-0.4c-0.1,0.3-0.2,0.6-0.3,0.9c-0.2,0.4-0.4,0.8-0.6,1.1 C249,25.6,247.9,25.8,246.6,25.7z M251,24.3c0.2-0.4,0.3-0.8,0.5-1.3c0.2-0.1,0.5-0.3,0.7-0.4c-0.3,0.7-0.7,1.3-1.1,1.9 c-0.1,0.1-0.1,0.1-0.2,0.2C250.9,24.5,251,24.4,251,24.3z M251.7,22.2c0.1-0.8,0.2-1.5,0.1-2.3c0.5-0.3,0.9-0.6,1.2-1 c0,0.1,0,0.2,0,0.3c0,0.6-0.1,1.3-0.3,1.8C252.5,21.5,252.2,21.9,251.7,22.2z"/> </g> <g> <path fill="#0095AA" d="M298.9,7.6h4.5v0.8h-3.5V11h2.9v0.8h-2.9v3.5h-1V7.6z"/> <path fill="#0095AA" d="M304.7,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4H309l-1.9-3.3h-1.4v3.3h-1V7.6z M307,11.2 c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H307z"/> <path fill="#0095AA" d="M310.9,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C312.2,15.4,310.9,13.9,310.9,11.4z M316.5,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C315.6,14.6,316.5,13.3,316.5,11.4z"/> <path fill="#0095AA" d="M319.1,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4H322l-1.4-4L320,8.7h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M331.7,8.4h-2.3V7.6h5.7v0.8h-2.3v6.9h-1V8.4z"/> <path fill="#0095AA" d="M336.4,7.6h1v3.2h3.6V7.6h1v7.7h-1v-3.6h-3.6v3.6h-1V7.6z"/> <path fill="#0095AA" d="M344.1,7.6h4.5v0.8h-3.5v2.4h2.9v0.8h-2.9v2.8h3.6v0.8h-4.6V7.6z"/> <path fill="#0095AA" d="M352.7,7.6h1.2l1.5,4.1c0.2,0.5,0.4,1,0.5,1.6h0c0.2-0.6,0.3-1.1,0.5-1.6l1.5-4.1h1.2v7.7h-0.9V11 c0-0.7,0.1-1.6,0.1-2.3h0l-0.6,1.8l-1.5,4h-0.7l-1.4-4l-0.6-1.8h0c0.1,0.7,0.1,1.6,0.1,2.3v4.3h-0.9V7.6z"/> <path fill="#0095AA" d="M362.8,7.6h1.1l2.6,7.7h-1.1l-0.7-2.3H362l-0.7,2.3h-1L362.8,7.6z M362.2,12.1h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L362.2,12.1z"/> <path fill="#0095AA" d="M367.7,7.6h1v3.9h0l3.2-3.9h1.1l-2.4,2.9l2.8,4.8h-1.1l-2.3-4l-1.3,1.6v2.4h-1V7.6z"/> <path fill="#0095AA" d="M374,7.6h4.5v0.8H375v2.4h2.9v0.8H375v2.8h3.6v0.8H374V7.6z"/> <path fill="#0095AA" d="M380.2,7.6h2.4c1.6,0,2.7,0.6,2.7,2.2c0,1.2-0.7,1.9-1.7,2.2l2,3.4h-1.1l-1.9-3.3h-1.4v3.3h-1V7.6z M382.5,11.2c1.2,0,1.9-0.5,1.9-1.5c0-1-0.7-1.4-1.9-1.4h-1.3v2.9H382.5z"/> <path fill="#0095AA" d="M386.4,14.3l0.6-0.6c0.6,0.6,1.4,0.9,2.2,0.9c1,0,1.6-0.5,1.6-1.3c0-0.8-0.6-1-1.3-1.4l-1.1-0.5 c-0.7-0.3-1.6-0.8-1.6-2c0-1.2,1-2.1,2.4-2.1c0.9,0,1.7,0.4,2.3,0.9L391,9c-0.5-0.4-1.1-0.7-1.7-0.7c-0.9,0-1.4,0.4-1.4,1.1 c0,0.7,0.7,1,1.3,1.3l1.1,0.5c0.9,0.4,1.6,0.9,1.6,2.1c0,1.2-1,2.2-2.6,2.2C388,15.4,387.1,15,386.4,14.3z"/> <path fill="#0095AA" d="M394.7,11.4c0-2.5,1.4-4,3.3-4c1.9,0,3.3,1.5,3.3,4c0,2.5-1.4,4-3.3,4C396.1,15.4,394.7,13.9,394.7,11.4z M400.3,11.4c0-1.9-0.9-3.1-2.3-3.1c-1.4,0-2.3,1.2-2.3,3.1c0,1.9,0.9,3.2,2.3,3.2C399.4,14.6,400.3,13.3,400.3,11.4z"/> <path fill="#0095AA" d="M403,7.6h4.5v0.8H404V11h2.9v0.8H404v3.5h-1V7.6z"/> <path fill="#0095AA" d="M411.1,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4H414l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M421.2,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L421.2,7.6z M421,12.1h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L421,12.1z"/> <path fill="#0095AA" d="M426.6,8.8h-2.2V7.6h5.8v1.2H428v6.5h-1.4V8.8z"/> <path fill="#0095AA" d="M431,7.6h1.4v3.1h3.1V7.6h1.4v7.7h-1.4v-3.4h-3.1v3.4H431V7.6z"/> <path fill="#0095AA" d="M438.4,7.6h4.7v1.2h-3.3v1.9h2.8v1.2h-2.8v2.2h3.4v1.2h-4.8V7.6z"/> <path fill="#0095AA" d="M444.3,7.6h1.7l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5h0c0.1-0.6,0.3-1,0.4-1.5l1.3-3.8h1.7v7.7h-1.3v-3.5 c0-0.7,0.1-1.8,0.2-2.5h0l-0.6,1.9l-1.3,3.4h-0.9l-1.2-3.4l-0.6-1.9h0c0.1,0.7,0.2,1.8,0.2,2.5v3.5h-1.3V7.6z"/> <path fill="#0095AA" d="M454.3,7.6h1.7l2.5,7.7H457l-0.6-2.1h-2.6l-0.6,2.1h-1.4L454.3,7.6z M454.2,12.1h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L454.2,12.1z"/> <path fill="#0095AA" d="M460,8.8h-2.2V7.6h5.8v1.2h-2.2v6.5H460V8.8z"/> <path fill="#0095AA" d="M464.7,7.6h1.4v7.7h-1.4V7.6z"/> <path fill="#0095AA" d="M467.4,11.5c0-2.5,1.6-4,3.5-4c1,0,1.7,0.4,2.2,1l-0.7,0.9c-0.4-0.4-0.9-0.6-1.5-0.6c-1.2,0-2.1,1-2.1,2.8 c0,1.7,0.8,2.8,2.1,2.8c0.7,0,1.2-0.3,1.6-0.7l0.8,0.8c-0.6,0.7-1.5,1.1-2.4,1.1C468.9,15.4,467.4,14,467.4,11.5z"/> <path fill="#0095AA" d="M475.8,7.6h1.7l2.5,7.7h-1.5l-0.6-2.1h-2.6l-0.6,2.1h-1.4L475.8,7.6z M475.7,12.1h1.9l-0.3-0.9 c-0.2-0.8-0.5-1.7-0.7-2.5h0c-0.2,0.8-0.4,1.7-0.7,2.5L475.7,12.1z"/> <path fill="#0095AA" d="M300.5,20.7h1.1l2.6,7.7h-1.1l-0.7-2.3h-2.8l-0.7,2.3h-1L300.5,20.7z M299.8,25.2h2.3l-0.4-1.2 c-0.3-0.9-0.5-1.7-0.8-2.6h0c-0.2,0.9-0.5,1.7-0.8,2.6L299.8,25.2z"/> <path fill="#0095AA" d="M305.1,20.7h1.1l2.8,4.8c0.3,0.5,0.6,1.1,0.8,1.7h0c-0.1-0.8-0.1-1.7-0.1-2.5v-4h0.9v7.7h-1.1l-2.8-4.8 c-0.3-0.5-0.6-1.1-0.8-1.7h0c0.1,0.8,0.1,1.6,0.1,2.4v4.1h-0.9V20.7z"/> <path fill="#0095AA" d="M312.4,20.7h1.9c2.4,0,3.7,1.4,3.7,3.8c0,2.5-1.3,3.9-3.6,3.9h-2V20.7z M314.3,27.6c1.8,0,2.7-1.1,2.7-3.1 c0-1.9-0.9-3-2.7-3h-0.9v6.1H314.3z"/> <path fill="#0095AA" d="M321.4,20.6l1.4,0l0.6,3.9c0.1,0.8,0.2,1.6,0.3,2.5l0,0c0.2-0.9,0.3-1.7,0.5-2.5l1-3.9l1.3,0l0.9,3.9 c0.2,0.8,0.3,1.6,0.5,2.5l0,0c0.1-0.9,0.2-1.7,0.4-2.5l0.7-3.9l1.3,0l-1.6,7.7l-1.8,0l-0.8-4.1c-0.1-0.6-0.2-1.2-0.3-1.9l0,0 c-0.1,0.7-0.2,1.2-0.3,1.9l-0.9,4l-1.8,0L321.4,20.6z"/> <path fill="#0095AA" d="M330.8,24.4c0-2.5,1.5-3.9,3.5-3.9c2,0,3.4,1.5,3.3,4c0,2.5-1.5,4-3.5,4C332.2,28.5,330.8,26.9,330.8,24.4z M336.2,24.5c0-1.7-0.7-2.7-1.9-2.8c-1.2,0-2,1-2,2.7c0,1.7,0.7,2.8,1.9,2.9C335.4,27.3,336.2,26.3,336.2,24.5z"/> <path fill="#0095AA" d="M338.8,20.6l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L338.8,20.6z"/> <path fill="#0095AA" d="M344.4,20.6l4.7,0.1l0,1.2l-3.3-0.1l0,2.2l2.8,0l0,1.2l-2.8,0l-0.1,3.2l-1.4,0L344.4,20.6z"/> <path fill="#0095AA" d="M350,20.6l2.6,0c1.6,0,2.8,0.6,2.8,2.3c0,1.2-0.6,1.9-1.6,2.2l1.8,3.2l-1.6,0l-1.6-3l-1.1,0l-0.1,3l-1.4,0 L350,20.6z M352.4,24.3c1,0,1.6-0.4,1.6-1.3c0-0.9-0.5-1.2-1.6-1.2l-1.1,0l0,2.5L352.4,24.3z"/> <path fill="#0095AA" d="M358.4,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L358.4,20.7z M358.2,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L358.2,25.2z"/> <path fill="#0095AA" d="M363.5,20.6l1.7,0l1.3,3.8c0.2,0.5,0.3,0.9,0.4,1.5l0,0c0.2-0.6,0.3-1,0.5-1.5l1.4-3.8l1.7,0l-0.1,7.7l-1.3,0 l0.1-3.5c0-0.7,0.1-1.8,0.3-2.5l0,0l-0.6,1.9l-1.3,3.4l-0.9,0l-1.2-3.5l-0.6-1.9l0,0c0.1,0.7,0.2,1.8,0.2,2.5l-0.1,3.5l-1.3,0 L363.5,20.6z"/> <path fill="#0095AA" d="M371.8,19l0.9,0L372.5,30l-0.9,0L371.8,19z"/> <path fill="#0095AA" d="M376.1,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L376.1,20.7z M375.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L375.8,25.2z"/> <path fill="#0095AA" d="M380.9,20.6l1.4,0l-0.1,6.5l3.2,0.1l0,1.2l-4.6-0.1L380.9,20.6z"/> <path fill="#0095AA" d="M386.6,20.6l2.4,0c1.7,0,3,0.7,3,2.4c0,1.7-1.3,2.5-3,2.4l-1.1,0l0,2.8l-1.4,0L386.6,20.6z M388.9,24.5 c1.1,0,1.7-0.4,1.7-1.4c0-0.9-0.6-1.3-1.7-1.3l-0.9,0l0,2.7L388.9,24.5z"/> <path fill="#0095AA" d="M393,20.6l1.4,0l-0.1,3.1l3.1,0.1l0.1-3.1l1.4,0l-0.1,7.7l-1.4,0l0.1-3.4l-3.1-0.1l-0.1,3.4l-1.4,0L393,20.6z "/> <path fill="#0095AA" d="M402,20.7l1.7,0l2.4,7.8l-1.5,0l-0.6-2.1l-2.6,0l-0.6,2.1l-1.4,0L402,20.7z M401.8,25.2l1.9,0l-0.2-0.9 c-0.2-0.8-0.4-1.7-0.7-2.5l0,0c-0.2,0.8-0.5,1.7-0.7,2.5L401.8,25.2z"/> </g> </g> <a href="https://www.wolfram.com/mathematica/"><rect x="296.2" y="0.1" style="fill:#ffffff00;" width="183.8" height="16"/></a> <a href="https://wolframalpha.com/"><rect x="296.2" y="16.4" style="fill:#ffffff00;" width="123.4" height="13.6"/></a> <a href="/"><rect x="1" y="0.1" style="fill:#ffffff00;" width="292.4" height="29.9"/></a> </svg> <form method="get" action="/search/" name="search" id="search" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchField"> <img src="/images/header/search-icon.png" width="18" height="18" class="search-btn" title="Search" alt="Search"> <img src="/images/header/menu-close.png" width="16" height="16" id="search-close" class="close-btn" title="Close" alt="Close"> </form> </header> <section class="left-side"> <img src="/images/sidebar/menu-close.png" width="16" height="16" id="dropdown-topics-menu-close" class="close-btn"> <form method="get" action="/search/" name="search" id="search-mobile" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchFieldMobile"> <img src="/images/header/search-icon.png" width="18" height="18" id="mobile-search" class="search-btn" title="Search" alt="Search" onclick="submitForm()"> </form> <nav class="topics-nav"> <a href="/topics/Algebra.html" id="sidebar-algebra"> Algebra </a> <a href="/topics/AppliedMathematics.html" id="sidebar-appliedmathematics"> Applied Mathematics </a> <a href="/topics/CalculusandAnalysis.html" id="sidebar-calculusandanalysis"> Calculus and Analysis </a> <a href="/topics/DiscreteMathematics.html" id="sidebar-discretemathematics"> Discrete Mathematics </a> <a href="/topics/FoundationsofMathematics.html" id="sidebar-foundationsofmathematics"> Foundations of Mathematics </a> <a href="/topics/Geometry.html" id="sidebar-geometry"> Geometry </a> <a href="/topics/HistoryandTerminology.html" id="sidebar-historyandterminology"> History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/CalculusandAnalysis.html">Calculus and Analysis</a> </li> <li> <a href="/topics/FunctionalAnalysis.html">Functional Analysis</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Moslehian.html">Moslehian</a> </li> </ul></nav>   <h1>Wiener Algebra</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> Suppose <img src="/images/equations/WienerAlgebra/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="W" /> is the set of all complex-valued functions <img src="/images/equations/WienerAlgebra/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="f" /> on the interval <img src="/images/equations/WienerAlgebra/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="[0,2pi]" /> of the form </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/WienerAlgebra/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="129" height="50" alt=" f(t)=sum_(k=-infty)^inftyalpha_ke^(ikt) " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/WienerAlgebra/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="77" height="21" alt="t in [0,2pi]" />, where the <img src="/images/equations/WienerAlgebra/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="alpha_k in C" /> and <img src="/images/equations/WienerAlgebra/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="108" height="23" alt="sum_(k=-infty)^(infty)|alpha_k|<infty" />. The set <img src="/images/equations/WienerAlgebra/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="W" /> with the usual pointwise operations and with the norm </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/WienerAlgebra/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="104" height="50" alt=" ||f||=sum_(k=-infty)^infty|alpha_k| " /></td><td align="right" width="3"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> is a commutative <a href="/BanachAlgebra.html">Banach algebra</a> and is called the Wiener algebra. </p> <p> There is an isometric isomorphism between <img src="/images/equations/WienerAlgebra/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="l^1(Z)" /> and <img src="/images/equations/WienerAlgebra/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="W" /> given by <img src="/images/equations/WienerAlgebra/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="46" height="23" alt="f->f^~" />, where </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/WienerAlgebra/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="145" height="50" alt=" f^~(t)=sum_(k=-infty)^inftyf(k)e^(ikt) " /></td><td align="right" width="3"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr> </table> </div> <p> with <img src="/images/equations/WienerAlgebra/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="77" height="21" alt="t in [0,2pi]" />. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">      <p class="contributor"> <i>This entry contributed by <a target="_blank" href="/topics/Moslehian.html">Mohammad Sal Moslehian</a></i> </p>   <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=2%2C5+torus+knot">2,5 torus knot</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=cross+polytope">cross polytope</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=local+extrema+sin+x%5E2">local extrema sin x^2</a></li> </ul> </div> </div>   <h2>References</h2><cite>Bonsall, F. F. and Duncan, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387063862/ref=nosim/ericstreasuretro">Complete Normed Algebras.</a></i> New York: Springer-Verlag, 1973.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/wiener_algebra/h2/vx/dl/" title="Wiener Algebra" target="_blank">Wiener Algebra</a>   <h2>Cite this as:</h2> <p> <a href="/topics/Moslehian.html">Moslehian, Mohammad Sal</a>. "Wiener Algebra." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource, created by <a href="/about/author.html">Eric W. Weisstein</a>. <a href="https://mathworld.wolfram.com/WienerAlgebra.html">https://mathworld.wolfram.com/WienerAlgebra.html</a> </p>  <h2>Subject classifications</h2><nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/CalculusandAnalysis.html">Calculus and Analysis</a> </li> <li> <a href="/topics/FunctionalAnalysis.html">Functional Analysis</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Moslehian.html">Moslehian</a> </li> </ul></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,208 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Thu Nov 21 2024</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2024 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>