westurner 4 years ago

awesome-fluid-dynamics https://github.com/lento234/awesome-fluid-dynamics :

- "12 Steps to Navier-Stokes"

- Differential programming https://en.wikipedia.org/wiki/Differentiable_programming : gradients; sometimes gradient descent

- Neural Networks for PDE

  • westurner 4 years ago

    "Differentiable function" https://en.wikipedia.org/wiki/Differentiable_function

    > In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain

    If there is a ZeroDivisionError because a denominator is zero, is that actually a differentiable function?

      -x**-1
  -1/x

    There's a symbolic result at or approaching that limit. IDK how exactly that applies to quantum-scale fluid effects at room temperature? Shouldn't there be a triality in terms of relativity, post-relativity QM with hair, and fluids converging or diverging to infinity?

    Divergence theorem ... Fluid diverconvergences https://en.wikipedia.org/wiki/Divergence_theorem :

    > The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem. [...] Explanation using liquid flow [...]

    "Logarithms yearning to be free" re: symbolic limits and currently non-axiomatic Infinity https://news.ycombinator.com/item?id=31015139

    ... Quantum thermodynamics, fluids, and chaotic divergence

    • adrian_b 4 years ago

      The "derivative" at a certain point is defined as the limit of the ratio between function differences and argument differences.

      In the context of derivatives, "exists" means that this limit is a finite number.

      If the derivative function has an expression which is a fraction whose denominator becomes null at a point, the derivative may exist or not exist at that point.

      If the corresponding numerator at that point is non-null (i.e. the derivative is infinite), then the derivative does not exist and the original function is not differentiable at that point.

      If the numerator is also null, the derivative may exist or not at that point, depending on whether the limit of the derivative exists or not.

      • westurner 4 years ago

        Are there no symbolic results from derivatives of any order?

        Is it more correct to say that, if the denominator is zero at that point, the derivative is just non-Real because it's e.g. `n*x*oo`?

        Practically, do we just say that such actually discontinuous functions are still mostly differentiable but the derivative does not exist in non-symbolic space?

    • westurner 4 years ago

      > ... Quantum thermodynamics, fluids, and chaotic divergence

      FWIW, does [quantum] thermodynamics predict emergent behaviors amongst self-organizing systems apparently at least temporarily contradicting a tendency to entropic decay?

      My understanding is that no: fluid dynamics, quantum fluid dynamics, and quantum chemistry are not sufficient to describe and thus cannot predict emergent behaviors in complex nonlinear - possibly emergently adaptive - complex systems.

      Emergence occurs in/of/by/within/betwixt/between systems; in application emergent programs require human-level intelligence ethical filters: https://en.wikipedia.org/wiki/Emergence

      Perhaps before describing physical systems with current best known descriptions as multi-field (QFT,QQ,) wave-particle[-fluid] interactions with convergent and divergent e.g convection, it's appropriate to compare the difference between Classical and Quantum Wave Interference: https://en.wikipedia.org/wiki/Wave_interference#Quantum_inte...

      > some of the differences between classical wave interference and quantum interference: (a) In classical interference, two different waves interfere; In quantum interference, the wavefunction interferes with itself. (b) Classical interference is obtained simply by adding the displacements from equilibrium (or amplitudes) of the two waves; In quantum interference, the effect occurs for the probability function associated with the wavefunction and therefore the absolute value of the wavefunction squared. (c) The interference involves different types of mathematical functions: A classical wave is a real function representing the displacement from an equilibrium position; a quantum wavefunction is a complex function. A classical wave at any point can be positive or negative; the quantum probability function is non-negative.

      Thus our best descriptions of emergent behavior in fluids (and chemicals and fields) must presumably be composed at least in part from quantum wave functions that e.g. Navier-Stokes also fit for; with a fitness function.