tag:blogger.com,1999:blog-10614348.post8432936589868179519..comments2024-01-22T11:26:37.599-08:00Comments on TGD diary: Quantum critical cosmology of TGD predicts also very fast expansionMatti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-10614348.post-87170006389841423282014-04-01T00:28:49.650-07:002014-04-01T00:28:49.650-07:00
To crow:
I already answered this message but as ...<br />To crow:<br /><br />I already answered this message but as it sometimes<br />happens the reply disappeared mysteriously although<br />it was certainly at the page (I always do the check).<br /><br />Minkowksi content is new notion to me, in no manner related to Minkowski metric. In TGD the existence of first derivatives of imbedding space coordinates is essential since otherwise induced field quantities would be ill-defined. <br /><br />One can have many-valued partial derivatives as in case of branching of space-time sheet occurring in the vertex representing particle decay. <br /><br />In any case, the geometric measures defined by the induced metric are enough. Matti Pitkanenhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-26908947473746474572014-03-31T00:19:43.406-07:002014-03-31T00:19:43.406-07:00Matti, what can be considered Minkowski content in...Matti, what can be considered Minkowski content in TGD? In the sense of<br />https://en.wikipedia.org/wiki/Minkowski_content. also offtopic , are you familiar with the urantia book?<br /><br />The Minkowski content of a set (named after Hermann Minkowski), or the boundary measure, is a basic concept in geometry and measure theory which generalizes to arbitrary measurable sets the notions of length of a smooth curve in the plane and area of a smooth surface in the space. It is typically applied to fractal boundaries of domains in the Euclidean space, but makes sense in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure.<br /><br />Definition <br /><br />Let be a metric measure space, where d is a metric on X and μ is a Borel measure. For a subset A of X and real ε > 0, let<br /><br /><br />be the ε-extension of A. The lower Minkowski content of A is given by<br /><br /><br />and the upper Minkowski content of A is<br /><br /><br />If M*(A) = M*(A), then the common value is called the Minkowski content of A associated with the measure μ, and is denoted by M(A).<br /><br />Minkowski content in Rn <br /><br />Let A be a subset of Rn. Then the m-dimensional Minkowski content of A is defined as follows. The lower content is<br /><br /><br />where αn−m is the volume of the unit (n−m)-ball and is -dimensional Lebesgue measure. The upper content is<br /><br /><br />As before, if the upper and lower m-dimensional Minkowski content of A agree, then the Minkowski content of A, Mm(A), is defined to be this common value.crowhttps://www.blogger.com/profile/14715663185910266616noreply@blogger.com