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The Kruskal–Szekeres coordinates also apply to space-time around a spherical object, but in that case do not give a description of space-time inside the radius of the object. Space-time in a region where a star is collapsing into a black hole is approximated by the Kruskal–Szekeres coordinates (or by the Schwarzschild coordinates). The ...
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
An R-square of 0.6 is considered the minimum acceptable level. [citation needed] An R-square of 0.8 is considered good for metric scaling and .9 is considered good for non-metric scaling. Other possible tests are Kruskal’s Stress, split data tests, data stability tests (i.e., eliminating one brand), and test-retest reliability.
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Working in a coordinate chart with coordinates (,,,) labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum.
Similarly R seems appropriate for an everywhere-spacelike coordinate. Usage in the literature seems mixed. While some use (u,v) others do use (T,R) (t,r) or similar. For example Rindler's Relativity: Special, General and Cosmological uses (t,x). This usage would be similar to the use of (T,X) for Minkowski coordinates in the Rindler coordinates ...
This is a consequence of Jacobi's two-square theorem, which follows almost immediately from the Jacobi triple product. [ 6 ] A much simpler sum appears if the sum of squares function r 2 ( n ) {\displaystyle r_{2}(n)} is defined as the number of ways of writing the number n {\displaystyle n} as the sum of two squares.
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected , it finds a minimum spanning tree . It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle . [ 2 ]