Because of phase coherence across the A-B interface, the hypercooled phase transition is not an onset phenomenon in this system, but rather the limiting behavior of very large interface velocities. At lower velocities, the chemical potential is continuous across the interface and the growing B phase is colder than the receding A phase.

In a linear world, averages make perfect sense. Something too big is compensated by something too small. We show, however that the underlying differential equations (e.g. unlimited growth) rather than the equations themselves (e.g. exponential growth) need to be linear. Especially in finance and economics non-linear differential equations are used although the input parameters are average quantities (e.g. average spending). It leads to the sad conclusion that almost all results are at least doubtful. Within one model (diffusion model of marketing) we show that the error is tremendous. We also compare chaotic results to random ones. Though these data are hardly distinguishable, certain limits prove to be very different. Implications for finance can be important because e.g. stock prices vary generally, chaotically, though the evaluation assumes quite often randomness.

In financial markets the demand curve is positively sloped in most cases. We give a rigorous mathematical prove that this leads to an instable equilibrium price. Therefore stock prices may fluctuate chaotically, making them unpredictable in many cases. Financial investments have therefore lots in common with gambling. In order to take the analogy further, we suggest a special gambling strategy (betting on a color in roulette). In doing so we have a model which may create a substantial amount of cash each year until it crashes after many years. Both gambling and financial speculation will never create money in the very long run. Because our gambling model is at least statistically predictable, it is "better" than speculative investment.

Diseases are spread due to the behavior of people. One may have millions of differently behaving people. For simplicity, humankind is only considering the average behavior. However, this simplification can lead to results tremendously different from the exact solution in at least some applications when non-linear differential equations are used. In this letter, we prove that the mistake is very big for the spread of infections like Covid-19. Ten percent ignoring the rules can almost ruin the extremely careful behavior of the remaining ninety percent. This is totally different to most business situations where considering eighty percent is sufficient ("80-20-rule"). This may explain why the spread of the Covid-19 pandemic and the impact of measures against it are hard to predict for decision makers.

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