By Yorozu S.

We supply a generalization of the end result acquired via C. Currais-Bosch. Weconsider the -operator linked to a transverse Killing box v on acomplete foliated Riemannian manifold . less than a undeniable assumption,we end up that, for every belongs to the Lie algebra of the linearholonomy staff . a distinct case of our end result, the model of the foliationby issues, implies the consequences given by way of B. Kostant (compact case) andC. Currfis-Bosch (non-compact case).

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CityData "Magenta" Magenta, Lombardy, Italy , Magenta, ChampagneArdenne, France Mathematica can provide web links to maps for these two cities. 97&z 12&t h 22 Chapter 0 A Brief Introduction There is also much useful data in the standard packages. Get "PhysicalConstants`" ? 6704 10 8 N Watt Kelvin4 Meter2 Get "Units`" If one wants to know how much water a 1000 cubic-foot-per-second stream will generate in a year, in gallons, that is simply done as follows. 1000. 35906 10 , Gallon Year Gallon Year And of course there is much mathematical information in these data bases.

It is not immediately obvious why the speeds 1, 7, and 17 should lead to such symmetry; Farris provides an explanation in his paper. Next we define the function and use Manipulate to study the effect of changing the parameters. f1 t_, s1_ : Cos s1 t , Sin s1 t ; f2 t_, r2_, s2_ : r2 Cos s2 t , Sin s2 t ; f3 t_, r3_, s3_, offset_ : r3 Cos s3 t offset , Sin s3 t offset ; epi r2_, r3_, s1_, s2_, s3_, offset_ t_ : f1 t, s1 f2 t, r2, s2 f3 t, r3, s3, offset ; We will let the reader experiment with the output of the following manipulation.

Here's an example. RSolve F n F n 1 aF n 2 ,F 0 0, F 1 1 ,F n ,n 40 Chapter 1 Plotting 2 n 1 1 4a n 1 1 4a n F n 1 Setting a 4a 2 shows a very simple form for that Fibonacci generalization. But we digress. Let's return to ListPlot and its use. Here is an interesting little puzzle: what is the rightmost nonzero digit of n ? Here we will examine only a few modest values of this function. The IntegerDigits function gives us the list of digits. ] 1, 5, 5, 1, 1, 2, 1, 0, 0, 4, 3, 3, 3, 0, 9, 8, 5, 9, 8, 4, 0, 0, 0, 0, 0, 0 DeleteCases is a handy way to erase what we don't want.