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In each equation u is the gray level at a particular point, D[u, t] is the rate of change (derivative) of the gray level with time, and D[u,t,t] is the rate of change of that rate of change (second derivative).
Systematic methods now exist to find conserved quantities that are given by integrals of polynomials of any given degree in the dependent variables and their derivatives.
Whether one looks at stocks, bonds, commodities, currencies, derivatives or essentially any other kind of financial instrument, the sequences of prices that one sees at successive times show some overall trends, but also exhibit varying amounts of apparent randomness.
, 2] and 2n - Log[2, Denominator[Derivative[n][(1 - #) -1/2 &][0]/n!]]
And these kinds of expressions often arise in Mathematica when one manipulates functions as a whole before applying them to arguments. ( ∂ xx f[x] for example gives f''[x] which is Derivative[2][f][x] .)
Beyond a few thousand cells, however, individual cells seem to be less relevant, and instead what appears to happen is that chemicals such as retinoic acid (a derivative of vitamin A) produced by particular cells diffuse to affect all cells in a region a tenth of a millimeter or so across.
Gauge invariance It is often convenient to define quantities for which only differences or derivatives matter.
In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.)
Since the late 1970s, it has been common to assume that the response of a cell can be modelled by derivatives of Gaussians such as those shown below, or perhaps by Gabor functions given by products of trigonometric functions and Gaussians.
(It seems possible however to construct functions computable in this sense whose derivatives are not computable.)
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