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Such a spectrum implies the presence of large low-frequency fluctuations, and indeed fluctuations are often seen over timescales of many minutes or even hours.
Current thinking on the Second Law
The vast majority of current physics textbooks imply that the Second Law is well established, though with surprising regularity they say that detailed arguments for it are beyond their scope.
The discoveries in this book imply however that there are much simpler components that can also be used to set up systems that have behavior with essentially any degree of sophistication.
Central to computational complexity theory are a collection of hypotheses that imply that NC, P, NP and PSPACE form a strict hierarchy.
(A corollary to Gödel's proof had in fact already supplied another explicit undecidable problem by implying that no finite procedure based on recursive functions could decide whether a given primitive recursive function is identically 0.)
Quantitative comparisons of pure power laws implied by the simplest fractals with observations of natural systems have had somewhat mixed success, leading to the introduction of multifractals with more parameters, but Mandelbrot's general idea of the importance of fractals is now well established in both science and mathematics.
Most games have rules which imply that if certain states are reached one player can be forced in the end to lose, regardless of what specific moves they make.
The shape of the front is presumably stabilized by the Gibbs–Thomson effect, which implies that the freezing temperature is lower when the front is more curved.
(Precise definitions of dimension based for example on the maximum ever achieved by d will often in general imply formally non-computable values, as in the discussion of page 1138 .)
But as the pictures below illustrate, it implies that light cones tip as v increases, so that the apparent speed of light changes, and for example Maxwell's equations must change their form.