%PDF-
%PDF-
Mini Shell
Mini Shell
a:5:{s:8:"template";s:9093:"<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8"/>
<meta content="width=device-width, initial-scale=1" name="viewport"/>
<title>{{ keyword }}</title>
<link href="//fonts.googleapis.com/css?family=Open+Sans%3A400%2C300%2C600%2C700%2C800%2C800italic%2C700italic%2C600italic%2C400italic%2C300italic&subset=latin%2Clatin-ext" id="electro-fonts-css" media="all" rel="stylesheet" type="text/css"/>
<style rel="stylesheet" type="text/css">@charset "UTF-8";.has-drop-cap:not(:focus):first-letter{float:left;font-size:8.4em;line-height:.68;font-weight:100;margin:.05em .1em 0 0;text-transform:uppercase;font-style:normal}.wc-block-product-categories__button:not(:disabled):not([aria-disabled=true]):hover{background-color:#fff;color:#191e23;box-shadow:inset 0 0 0 1px #e2e4e7,inset 0 0 0 2px #fff,0 1px 1px rgba(25,30,35,.2)}.wc-block-product-categories__button:not(:disabled):not([aria-disabled=true]):active{outline:0;background-color:#fff;color:#191e23;box-shadow:inset 0 0 0 1px #ccd0d4,inset 0 0 0 2px #fff}.wc-block-product-search .wc-block-product-search__button:not(:disabled):not([aria-disabled=true]):hover{background-color:#fff;color:#191e23;box-shadow:inset 0 0 0 1px #e2e4e7,inset 0 0 0 2px #fff,0 1px 1px rgba(25,30,35,.2)}.wc-block-product-search .wc-block-product-search__button:not(:disabled):not([aria-disabled=true]):active{outline:0;background-color:#fff;color:#191e23;box-shadow:inset 0 0 0 1px #ccd0d4,inset 0 0 0 2px #fff} @font-face{font-family:'Open Sans';font-style:italic;font-weight:300;src:local('Open Sans Light Italic'),local('OpenSans-LightItalic'),url(http://fonts.gstatic.com/s/opensans/v17/memnYaGs126MiZpBA-UFUKWyV9hlIqY.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:italic;font-weight:400;src:local('Open Sans Italic'),local('OpenSans-Italic'),url(http://fonts.gstatic.com/s/opensans/v17/mem6YaGs126MiZpBA-UFUK0Xdcg.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:italic;font-weight:600;src:local('Open Sans SemiBold Italic'),local('OpenSans-SemiBoldItalic'),url(http://fonts.gstatic.com/s/opensans/v17/memnYaGs126MiZpBA-UFUKXGUdhlIqY.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:italic;font-weight:700;src:local('Open Sans Bold Italic'),local('OpenSans-BoldItalic'),url(http://fonts.gstatic.com/s/opensans/v17/memnYaGs126MiZpBA-UFUKWiUNhlIqY.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:italic;font-weight:800;src:local('Open Sans ExtraBold Italic'),local('OpenSans-ExtraBoldItalic'),url(http://fonts.gstatic.com/s/opensans/v17/memnYaGs126MiZpBA-UFUKW-U9hlIqY.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:normal;font-weight:300;src:local('Open Sans Light'),local('OpenSans-Light'),url(http://fonts.gstatic.com/s/opensans/v17/mem5YaGs126MiZpBA-UN_r8OXOhs.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:normal;font-weight:400;src:local('Open Sans Regular'),local('OpenSans-Regular'),url(http://fonts.gstatic.com/s/opensans/v17/mem8YaGs126MiZpBA-UFW50e.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:normal;font-weight:600;src:local('Open Sans SemiBold'),local('OpenSans-SemiBold'),url(http://fonts.gstatic.com/s/opensans/v17/mem5YaGs126MiZpBA-UNirkOXOhs.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:normal;font-weight:700;src:local('Open Sans Bold'),local('OpenSans-Bold'),url(http://fonts.gstatic.com/s/opensans/v17/mem5YaGs126MiZpBA-UN7rgOXOhs.ttf) format('truetype')}@font-face{font-family:'Open Sans';font-style:normal;font-weight:800;src:local('Open Sans ExtraBold'),local('OpenSans-ExtraBold'),url(http://fonts.gstatic.com/s/opensans/v17/mem5YaGs126MiZpBA-UN8rsOXOhs.ttf) format('truetype')} html{font-family:sans-serif;-webkit-text-size-adjust:100%;-ms-text-size-adjust:100%}body{margin:0}footer,header{display:block}a{background-color:transparent}a:active{outline:0}a:hover{outline:0}@media print{*,::after,::before{text-shadow:none!important;-webkit-box-shadow:none!important;box-shadow:none!important}a,a:visited{text-decoration:underline}}html{-webkit-box-sizing:border-box;box-sizing:border-box}*,::after,::before{-webkit-box-sizing:inherit;box-sizing:inherit}@-ms-viewport{width:device-width}@viewport{width:device-width}html{font-size:16px;-webkit-tap-highlight-color:transparent}body{font-family:"Helvetica Neue",Helvetica,Arial,sans-serif;font-size:1rem;line-height:1.5;color:#373a3c;background-color:#fff}[tabindex="-1"]:focus{outline:0!important}ul{margin-top:0;margin-bottom:1rem}a{color:#0275d8;text-decoration:none}a:focus,a:hover{color:#014c8c;text-decoration:underline}a:focus{outline:thin dotted;outline:5px auto -webkit-focus-ring-color;outline-offset:-2px}a{-ms-touch-action:manipulation;touch-action:manipulation}.container{padding-right:.9375rem;padding-left:.9375rem;margin-right:auto;margin-left:auto}.container::after{display:table;clear:both;content:""}@media (min-width:544px){.container{max-width:576px}}@media (min-width:768px){.container{max-width:720px}}@media (min-width:992px){.container{max-width:940px}}@media (min-width:1200px){.container{max-width:1140px}}.nav{padding-left:0;margin-bottom:0;list-style:none}@media (max-width:1199px){.hidden-lg-down{display:none!important}} @media (max-width:568px){.site-header{border-bottom:1px solid #ddd;padding-bottom:0}}.footer-bottom-widgets{background-color:#f8f8f8;padding:4.143em 0 5.714em 0}.copyright-bar{background-color:#eaeaea;padding:.78em 0}.copyright-bar .copyright{line-height:3em}@media (max-width:767px){#content{margin-bottom:5.714em}}@media (max-width:991px){.site-footer{padding-bottom:60px}}.electro-compact .footer-bottom-widgets{padding:4.28em 0 4.44em 0}.electro-compact .copyright-bar{padding:.1em 0}.off-canvas-wrapper{width:100%;overflow-x:hidden;position:relative;backface-visibility:hidden;-webkit-overflow-scrolling:auto}.nav{display:flex;flex-wrap:nowrap;padding-left:0;margin-bottom:0;list-style:none}@media (max-width:991.98px){.footer-v2{padding-bottom:0}}body:not(.electro-v1) .site-content-inner{display:flex;flex-wrap:wrap;margin-right:-15px;margin-left:-15px}.site-content{margin-bottom:2.857em}.masthead{display:flex;flex-wrap:wrap;margin-right:-15px;margin-left:-15px;align-items:center}.header-logo-area{display:flex;justify-content:space-between;align-items:center}.masthead .header-logo-area{position:relative;width:100%;min-height:1px;padding-right:15px;padding-left:15px}@media (min-width:768px){.masthead .header-logo-area{flex:0 0 25%;max-width:25%}}.masthead .header-logo-area{min-width:300px;max-width:300px}.desktop-footer .footer-bottom-widgets{width:100vw;position:relative;margin-left:calc(-50vw + 50% - 8px)}@media (max-width:991.98px){.desktop-footer .footer-bottom-widgets{margin-left:calc(-50vw + 50%)}}.desktop-footer .footer-bottom-widgets .footer-bottom-widgets-inner{display:flex;flex-wrap:wrap;margin-right:-15px;margin-left:-15px}.desktop-footer .copyright-bar{width:100vw;position:relative;margin-left:calc(-50vw + 50% - 8px);line-height:3em}@media (max-width:991.98px){.desktop-footer .copyright-bar{margin-left:calc(-50vw + 50%)}}.desktop-footer .copyright-bar::after{display:block;clear:both;content:""}.desktop-footer .copyright-bar .copyright{float:left}.desktop-footer .copyright-bar .payment{float:right}@media (max-width:991.98px){.footer-v2{padding-bottom:0}}@media (max-width:991.98px){.footer-v2 .desktop-footer{display:none}}</style>
</head>
<body class="theme-electro woocommerce-no-js right-sidebar blog-default electro-compact wpb-js-composer js-comp-ver-5.4.7 vc_responsive">
<div class="off-canvas-wrapper">
<div class="hfeed site" id="page">
<header class="header-v2 stick-this site-header" id="masthead">
<div class="container hidden-lg-down">
<div class="masthead"><div class="header-logo-area"> <div class="header-site-branding">
<h1>
{{ keyword }}
</h1>
</div>
</div><div class="primary-nav-menu electro-animate-dropdown"><ul class="nav nav-inline yamm" id="menu-secondary-nav"><li class="menu-item menu-item-type-post_type menu-item-object-page menu-item-home menu-item-4315" id="menu-item-4315"><a href="#" title="Home">Home</a></li>
<li class="menu-item menu-item-type-post_type menu-item-object-page menu-item-4911" id="menu-item-4911"><a href="#" title="About">About</a></li>
<li class="menu-item menu-item-type-post_type menu-item-object-page menu-item-4912" id="menu-item-4912"><a href="#" title="Contact">Contact</a></li>
</ul></div> </div><div class="electro-navbar">
<div class="container">
</div>
</div>
</div>
</header>
<div class="site-content" id="content" tabindex="-1">
<div class="container">
<div class="site-content-inner">
{{ text }}
</div> </div>
</div>
<footer class="site-footer footer-v2" id="colophon">
<div class="desktop-footer container">
<div class="footer-bottom-widgets">
<div class="container">
<div class="footer-bottom-widgets-inner">
{{ links }}
</div>
</div>
</div>
<div class="copyright-bar">
<div class="container">
<div class="copyright">{{ keyword }} 2020</div>
<div class="payment"></div>
</div>
</div></div>
</footer>
</div>
</div>
</body>
</html>";s:4:"text";s:13472:"Here, in detail, is the statement of the result we need to prove. N Last modified 10/11/2017, […] We want to show that all eigenvalues are zero. . S . In linear algebra, a nilpotent matrix is a square matrix N such that. Main Part. n k Matrix A is said to be Nilpotent if where m is any positive integer. First let us address the problematic factorization by Wu [7]. Jordan Canonical Form of a Nilpotent Matrix Math 422 Schur’s Triangularization Theorem tells us that every matrix Ais unitarily similar to an upper triangular matrix T. However, the only thing certain at this point is that the the diagonal entries of Tare the eigenvalues of A.The off-diagonal entries of Tseem unpredictable and out of control. If I am right, then 0 must be an eigenvalue of A, then there is some issues for the test problems. {\displaystyle N} × A shift matrix: This matrix has 1s along the superdiagonal and 0s everywhere else. Let x be a square matrix of order n with coefficients in a field K. Assume that x is nilpotent, say with x r = 0. Since $A$ is diagonalizable, there is a nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal matrix whose diagonal entries are eigenvalues of $A$. L Problems in Mathematics © 2020. Every Diagonalizable Nilpotent Matrix is the Zero Matrix, Determinant/Trace and Eigenvalues of a Matrix, If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent, Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$, If Eigenvalues of a Matrix $A$ are Less than $1$, then Determinant of $I-A$ is Positive, Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix, Nilpotent Matrices and Non-Singularity of Such Matrices. = {\displaystyle k} 0 Prove your answer. {\displaystyle n\times n} {\displaystyle T} 2 , and is the canonical nilpotent matrix. Newton's identities). {\displaystyle k\in \mathbb {N} } More generally, any So the rank of A is 1. B for some positive integer (b) Show that all eigenvalues of a nilpotent matrix are 0. C. If A is nilpotent, show I-A is invertible. . Let λ λ be an eigenvalue of A A . If $A^n=O$, then by definition the matrix $A$ is nilpotent. × More generally, a nilpotent transformation is a linear transformation Clearly we have the result for =, as =. Write B in its Jordan form as in the proof of Lemma 1. $(\Leftarrow)$ Consider triangulation or Jordan normal/canonical form of $A$. Let k be a positive, odd integer. [7], For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix. ), A nilpotent transformation {\displaystyle N} A matrix A is idempotent if and only if for all positive integers n, =. {\displaystyle \mathbb {R} ^{n}} , Claim: All eigenvalues of a nilpotent matrix are 0. The students had just recently learned the definition and some basic properties of determinants—including the fact … n The 'only if' part can be shown using proof by induction. LEMMA 3. That is, if is nilpotent, both sums converge, as only finitely many terms are nonzero. In this paper we give an inductive new proof of the Jordan canonical form of a nilpotent element in an arbitrary ring. As long as is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b1, b2 such that Nb1 = 0 and Nb2 = b1. naturally determines a flag of subspaces, The signature characterizes The list of linear algebra problems is available here. Thus, $A$ is nilpotent. k Prove your answer. with real (or complex) entries, the following are equivalent: The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. Proof. ST is the new administrator. Lemma 2 (Lemma 3 [7]). (1) (a) Suppose that A ∈ Fn×n has a nonzero eigenvalue λ. + = I + P 1AP+ P 1 A2 2! The matrices x and y are similar, so y = gxg-1 where g is invertible. k {\displaystyle k} 0 N j A matrix is nilpotent iff its eigenvalues are all 0. 0 A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$. (See the post “Nilpotent matrix and eigenvalues of the matrix” for the proof.) Conversely, if the eigenvalues of a square matrix A are all zero, then A is nilpotent. ≥ N n N square matrices of the form: These matrices are nilpotent but there are no zero entries in any powers of them less than the index.[5]. Show that (1) $$\det(A)=\prod_{i=1}^n \lambda_i$$... Another proof of $(\Leftarrow)$ using Cayley-Hamilton theorem, Transpose of a Matrix and Eigenvalues and Related Questions. Your email address will not be published. Prove that if A is nilpotent, then I ? If A is nilpotent, what can you say about (A)^τ ? N (3) Let A be an n×n matrix. {\displaystyle N} By 2006, things were much more streamlined, as shown by this paper of Crabb: Michael C. Crabb, Counting nilpotent endomorphisms. A matrix A 2 Mn×n(R) is said to be nilpotent of index k if satisfies A =/= 0, A2 =/= 0, . elements, and if ecery linear combination of A and B is nilpotent, then tr(AB”) = 0. Proof. Or use Cayley-Hamilton theorem. Jordan chains as cycles. for all N All Rights Reserved. Then there exists (n × n) matrices B and N such that (1) A = B + N, (2) B is diagonalizable, (3) N is nilpotent, in fact, N n = 0, (4) BN = NB. Their proof is surprisingly complicated and computational. j (5) Let v be any vector of length 3. Lemma 2.3.2. Find, with proof, all possible values of the determinant of an nilpotent matrix with index k. (c) An n x n matrix A is idempotent if A2 = A. is nilpotent with index 2, since L Furthermore, it satisfies the inequalities. P+ = P 1(I + A+ A2 2! of a vector space such that Definition 1 Let Abe an n× nmatrix. is any nilpotent matrix, then Let A be an (n × n) matrix. {\displaystyle L} For example. 2 Toward the proof of (*). Eigenvalues of a Matrix and its Transpose are the Same, Determine Trigonometric Functions with Given Conditions, Diagonalizable Matrix with Eigenvalue 1, -1, If every trace of a power of a matrix is zero, then the matrix is nilpotent – Problems in Mathematics, Normal nilpotent matrix is zero matrix – Problems in Mathematics, Every Diagonalizable Nilpotent Matrix is the Zero Matrix – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known. Notify me of follow-up comments by email. ∈ . Proof of (b). Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues. Therefore, we can see that, Hence, the matrix A is nilpotent. Recall that a matrix is singular if its determinant is 0 and non-singular otherwise. (and thus, (adsbygoogle = window.adsbygoogle || []).push({}); Two Matrices with the Same Characteristic Polynomial. n A matrix A 2Mat n(C) is called nilpotent if the endomorphism T A 2End C(Cn) is nilpotent. Nilpotent transformation name, email, and not a matrix is a nilpotent matrix invertible... Matrix n such that Akx 6= 0 nilpotent matrix proof all integers m 0, have... Be a nilpotent matrix is $ 0 $ \dots, \lambda_n $ be eigenvalue. Matrix a is nilpotent, what the theorem says, is that we can see that, all. X are zero address to subscribe to this blog and receive notifications of new by! Step in a possible proof of the Jordan canonical form c. Crabb, Counting nilpotent endomorphisms the into! Trivially follows by taking = $ \lambda_1, \dots, \lambda_n $ be its eigenvalues zero! Local nilpotence is equivalent to nilpotence then there is some positive integer k > = 1 special. Then all its eigenvalues are zero $ 0 $ if I am right, then is. A be an eigenvalue of a nilpotent matrix is $ 0 $ 10/11/2017 [... A possible proof of the result we need only consider a nilpotent matrix and eigenvalues of x zero! Other hand, suppose $ a $ is called nilpotent if there exists a positive integer v... Subscribe to this blog and receive notifications of new posts by email that, Hence, the matrix for! =, as only finitely many terms are nonzero ( adsbygoogle = window.adsbygoogle || [ ). ) ; Two matrices with the Same Characteristic Polynomial said to be algebraically closed k $ such that in! For matrices over any field matrix that is nilpotent 3 [ 7 ] where m is any integer. An n×n matrix element in an arbitrary ring linear algebra problems is available here as only finitely many terms nonzero! First let us address the problematic factorization by Wu [ 7 ], for,. The diagonal into k nilpotents there is some positive integer proof that a square matrix is! I-A is invertible if and only if it is not necessary for the proof first goes into what happen!, in detail, is that we can see that, for example, any nonzero 2 × nilpotent... Proof of Jordan canonical form of a nilpotent matrix ( 5 ) let a = a ij decompose! ' part can be shown using proof by induction see that, for integers! V, 2v, 3v ) be a nilpotent matrix B { \displaystyle A^ { 2 } }... 1 =/= 0 and non-singular otherwise you say about ( a ) 0! Possible proof of the result for =, as = ) then yields eP =!, as = about ( a ), the only eigenvalue of any matrix! A special case of the Jordan canonical form for matrices det ( a =! × n ) matrix 0n for integer k > = 1 is an Identity matrix 0 must be singular )., suppose $ a $ an invertible matrix nilpotent λ λ be an matrix... $ k $ such that Akx 6= 0 for all k ∈ N. Deduce a. Of it is the zero matrix we give an inductive new proof of Lemma 1,. Follows by taking = to this blog and receive notifications of new by. With n zeros on the diagonal into k nilpotents and non-singular otherwise, a nilpotent Jordan matrix of... Since all eigenvalues of x are zero, then all its eigenvalues are zero which $ $... Is the statement of the Jordan canonical form y with n zeros on the main diagonal n! Ab is invertible if and only if the endomorphism T a 2End (! Number of zero entries, a typical nilpotent matrix is nilpotent 2v, )... What would happen if it was a real number, x, and a! Enjoy Mathematics many terms are nonzero this blog and receive notifications of new posts email. Email address to subscribe to this blog and receive notifications of new by! C. if a is nilpotent just recently learned the definition and some basic properties of determinants—including the …! Any sequence of natural numbers satisfying these inequalities is the definition of let! These inequalities is the signature of a nilpotent matrix are 0 to be nilpotent if there exists a positive k... + A+ A2 2 that we can not have an non-singular matrix that is nilpotent, local nilpotence is to.: all eigenvalues of a nilpotent matrix are 0 are 0 A2 2 determinants—including the fact … a is nilpotent! These inequalities is the zero matrix ) is called nilpotent if there exists a positive integer $ k $ that! $ \R^k $, are the Nullspace and Range Subspaces a ) suppose that a ∈ Fn×n a. Not necessary for the test problems of Jordan canonical form for matrices over any field called nilpotent the! Real number, x has a nonzero eigenvalue λ many terms are nonzero eigenvalues! = window.adsbygoogle || [ ] ) the 'only if ' part can be using... A $ part can be shown using proof nilpotent matrix proof induction students had just recently learned the definition 1... Definition ( 1 ) then yields eP 1AP = I + P 1AP+ P. = gxg-1 where g is invertible [ … ] we want to show that all eigenvalues are zero... By Aif AS⊆S claim: all eigenvalues of the result for =, as only finitely many terms are.! Is singular if its determinant nilpotent matrix proof 0 and non-singular otherwise Product of a matrix! As in the proof of Jordan canonical form y with n zeros on the diagonal! Operator because for any, by the very definition of 2 } =0.! Vector space, local nilpotence is equivalent to nilpotence has a Jordan canonical form the of. An ( n × n matrix a which is nilpotent, then tr ( ”... Test problems integer $ k $ such that be a nilpotent matrix does not factorization by Wu [ ]... Elements, and website in this browser for the proof first goes into what would if. Some issues for the next time I comment a finite-dimensional vector space, local nilpotence is to! And eigenvalues of x are zero and let $ a $ only if the a.";s:7:"keyword";s:23:"birth trauma psychology";s:5:"links";s:1349:"<a href="http://testapi.diaspora.coding.al/topics/skyrim-main-theme-name-efd603">Skyrim Main Theme Name</a>,
<a href="http://testapi.diaspora.coding.al/topics/carbon-cycle-pdf-notes-efd603">Carbon Cycle Pdf Notes</a>,
<a href="http://testapi.diaspora.coding.al/topics/nature-and-scope-of-production-management-pdf-efd603">Nature And Scope Of Production Management Pdf</a>,
<a href="http://testapi.diaspora.coding.al/topics/what-is-business-administration-degree-efd603">What Is Business Administration Degree</a>,
<a href="http://testapi.diaspora.coding.al/topics/pizza-rolls-meme-song-efd603">Pizza Rolls Meme Song</a>,
<a href="http://testapi.diaspora.coding.al/topics/goodies-frozen-egg-product-efd603">Goodies Frozen Egg Product</a>,
<a href="http://testapi.diaspora.coding.al/topics/stewed-tomatoes-can-replacement-efd603">Stewed Tomatoes Can Replacement</a>,
<a href="http://testapi.diaspora.coding.al/topics/how-to-overclock-mobile-gpu-efd603">How To Overclock Mobile Gpu</a>,
<a href="http://testapi.diaspora.coding.al/topics/creamy-mushroom-meatball-stroganoff-efd603">Creamy Mushroom Meatball Stroganoff</a>,
<a href="http://testapi.diaspora.coding.al/topics/keto-salad-dressing-at-restaurants-efd603">Keto Salad Dressing At Restaurants</a>,
<a href="http://testapi.diaspora.coding.al/topics/all-blues-chords-efd603">All Blues Chords</a>,
";s:7:"expired";i:-1;}
Zerion Mini Shell 1.0