Discrete-Continuous-Discrete

  • I see no hope for the future of our people if they are dependent on the frivolous youth of today,for certainly all youth are reckless beyond words. When I was a boy, we were taught to be discrete and respectful of elders, but the present youth are exceedingly wise and impatient of restraint. (Hesiod)
  • Life defies our phrases, it is infinitely continuous and subtle and shaded, whilst our verbal terms are discrete, rude and few. (William James)
  • And the continuity of our science has not been affected by all these turbulent happenings, as the older theories have always been included as limiting cases in the new ones. (Max Born)
  • At the point when continuity was interrupted by the first nuclear explosion, it would have been too easy torecover the formal sediment which linked us with an age of poetic decorum, of a preoccupation with poetic sounds. (Salvatore Quasimodo)

Life at 24 Frames/Second

You will find that computational simulations are performed by computing a sequence of frames or pictureswhich make up a film as a sequence of frames following upon each other with a certain time step

Each frame is computed by updating a finite number of variables, and the simulation is thus performed by using a model which is discrete in both time and space, that is, the model involves the values of a finite number of variables representing values at certain points in space at a discrete sequence time instants.

You will first meet such discrete models in the form of mass-spring systems modeling the motion of elastic bodies.

By increasing the number of discrete points in time and space, that is increasing the resolution in space (increasing the number of pixels of a picture) and decreasing the time step, we will be led to emph{continuous models} in the form of differential equations.

For elastic solids, we thus start with discrete models and arrive at continuous models as idealizations withinfinitely fine resolution in time and space.

For fluids it is more rational to start with continuous models and then perform the discretization into discrete models by the Finite Element Method FEM based on

  • variational formulation or Galerkin’s method,
  • piecewise polynomial approximation.

We will recover certain discrete models for elastic solids discretizing continuous models by FEM and we will thus become familiar with the full circle discrete-continuous-discrete, where the computer only accepts discrete models and continuous models are useful to prepare for discretization by FEM.

The main tools for formulating continuous models as differential equations is

  • Calculus: functions, derivatives and integrals.

The main tool for discretizing by FEM and solving discrete systems by computers, is

  • Linear Algebra: vectors, matrices, linear transformations.

You collect experience of the interplay between discrete and continuous and understand that the continuous models are fictional in the sense that their solutions are “untouchable” or “unknowable” in complete detail.

Nevertheless the continuos models are useful by their extreme economy of expression, which is helpful forboth computation and imagination through the ingenious Calculus by Leibniz.

 

 

 

 

 

 

To Watch:

The Illusionist

We shall discover that analytical mathematics is a form of illusion playing with symbols like sqrt{2}, pi, sin(1), and exp(1) , which represent numbers with neverending non-repeating decimals expansions, which can only be specified or made known to a finite number of decimals. Analytical mathematics is thus similar to a novel as a play with words the exact meaning of which cannot be specified.

Computational mathematics on the other hand plays directly with digits and decimals and in this sense is more concrete and knowable. On the other hand it is impossible to follow all the steps of a long digital computation performed by a computer. We can thus inspect in clear light the output of a long computation, but not follow the individual steps leading to the result.

On the other hand, in an analytical argument, or derivation of an analytical formula, we are supposed to follow all the steps, but the precise nature of the result remains hidden to us.

We thus can chose bewteen the following possible ways of using mathematics:

  • analytical: following the steps in detail to a partly hidden result,
  • computational: not following the steps in detail to a fully visible result.

But we cannot, except in simple cases, follow all the steps in complete detail to a fully visible result.

Watch Illusionist.

The Genetic Code and Emergence

The genetic code is the set of rules by which information encoded in genetic material or genome (DNA or mRNA sequences) is translated into proteins (amino acid sequences) by living cells.

The human genome is the genome of Homo sapiens, which is stored on 23 chromosome pairsof 3 billion DNA base pairs, contains ca. 23,000 protein-coding genes (about 1.5 percentof the genome) while the rest consists of non-coding RNA genes, regulatory sequences, introns, and (controversially named) “junk” DNA.

The genom can be seen as a computer code which generates the life of an individual upon executionin interaction with the environment. The human genome project has listed the code, but the task to understand the code remains.

Life is an example of emergence with complex systems and patterns arising from repeated simple interactions: A short computer code, like the genome, which generates complex output, is an example of emergence: The code itself does not display the complexity which comes ou upon execution.

Fractal images are complex emergent patterns generated by one-line code loops.Turbulence is a prime non-organic example of emergence, which you will meet below.

Fractal image of broccoli generated by one-line code.

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