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Generalized Tenney norms and Tp interval space: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 11:12:45 UTC</tt>.<br>
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[[math]]
[[math]]


In this scheme the ordinary Tenney norm now becomes the **T1 norm**, and in general we call an interval space that's been given a Tp norm **Tp interval space**. We may sometimes notate this as **Tp&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;**, where **G** is the associated group the interval space is built around.
In this scheme the ordinary Tenney norm now becomes the **T1 norm**, and in general we call an interval space that's been given a Tp norm **Tp interval space**. We may sometimes notate this as **Tp&lt;span style="font-size: 80%; vertical-align: super;"&gt;G&lt;/span&gt;**, where **G** is the associated group the interval space is built around.


Note that the || · ||**&lt;span style="font-size: 80%; vertical-align: sub;"&gt;Tp&lt;/span&gt;** norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · ||**&lt;span style="font-size: 80%; vertical-align: sub;"&gt;p&lt;/span&gt;** norm on the right side of the equation now has a subscript of p rather than 1. The Generalized Tenney Norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary Lp norm rather than restricting our consideration to the L1 norm.
Note that the || · ||**&lt;span style="font-size: 80%; vertical-align: sub;"&gt;Tp&lt;/span&gt;** norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · ||**&lt;span style="font-size: 80%; vertical-align: sub;"&gt;p&lt;/span&gt;** norm on the right side of the equation now has a subscript of p rather than 1. The Generalized Tenney Norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary Lp norm rather than restricting our consideration to the L1 norm.
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  --&gt;&lt;script type="math/tex"&gt;\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{p}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
  --&gt;&lt;script type="math/tex"&gt;\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{p}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In this scheme the ordinary Tenney norm now becomes the &lt;strong&gt;T1 norm&lt;/strong&gt;, and in general we call an interval space that's been given a Tp norm &lt;strong&gt;Tp interval space&lt;/strong&gt;. We may sometimes notate this as &lt;strong&gt;Tp&lt;span style="font-size: 80%; vertical-align: sub;"&gt;G&lt;/span&gt;&lt;/strong&gt;, where &lt;strong&gt;G&lt;/strong&gt; is the associated group the interval space is built around.&lt;br /&gt;
In this scheme the ordinary Tenney norm now becomes the &lt;strong&gt;T1 norm&lt;/strong&gt;, and in general we call an interval space that's been given a Tp norm &lt;strong&gt;Tp interval space&lt;/strong&gt;. We may sometimes notate this as &lt;strong&gt;Tp&lt;span style="font-size: 80%; vertical-align: super;"&gt;G&lt;/span&gt;&lt;/strong&gt;, where &lt;strong&gt;G&lt;/strong&gt; is the associated group the interval space is built around.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Note that the || · ||&lt;strong&gt;&lt;span style="font-size: 80%; vertical-align: sub;"&gt;Tp&lt;/span&gt;&lt;/strong&gt; norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · ||&lt;strong&gt;&lt;span style="font-size: 80%; vertical-align: sub;"&gt;p&lt;/span&gt;&lt;/strong&gt; norm on the right side of the equation now has a subscript of p rather than 1. The Generalized Tenney Norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary Lp norm rather than restricting our consideration to the L1 norm.&lt;br /&gt;
Note that the || · ||&lt;strong&gt;&lt;span style="font-size: 80%; vertical-align: sub;"&gt;Tp&lt;/span&gt;&lt;/strong&gt; norm on the left side of the equation now has a subscript of Tp rather than T1, and that the || · ||&lt;strong&gt;&lt;span style="font-size: 80%; vertical-align: sub;"&gt;p&lt;/span&gt;&lt;/strong&gt; norm on the right side of the equation now has a subscript of p rather than 1. The Generalized Tenney Norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary Lp norm rather than restricting our consideration to the L1 norm.&lt;br /&gt;
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