In intuitive terms it consists of all integer functions, possibly with multiple input arguments, that can be written only with a sequence of:and such that n does not change inside the loop body, i.e. no while loops with arbitrary conditions.

n does not have to be a constant, it may come from previous calculations. But it must not change inside the loop body.

Primitive recursive functions basically include every integer function that comes up in practice. Primitive recursive functions can have huge complexity, and it strictly contains EXPTIME. As such, they mostly only come up in foundation of mathematics contexts.

The cool thing about primitive recursive functions is that the number of iterations is always bound, so we are certain that they terminate and are therefore computable.

This also means that there are necessarily functions which are not primitive recursive, as we know that there must exist uncomputable functions, e.g. the busy beaver function.

Adding unbounded while loops of course enables us to simulate arbitrary Turing machines, and therefore increases the complexity class.

More finely, there are non-primitive total recursive functions, e.g. most famously the Ackermann function.

When debugging complex software, make sure to keep notes of every interesting find you make in a note file, as you extract it from the integrated development environment or debugger.

Especially if your memory sucks like Ciro's.

This is incredibly helpful in fully understanding and then solving complex bugs.

E.g. showing live data from a scientific instrument! TODO:

Inward Bound by Abraham Pais (1988) page 282 shows how this can be generalized from the Maxwell-Boltzmann distribution

Intuition, please? Example? mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to Hamiltonian mechanics. The two arguments of the bilinear form correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.

Seems to be set of matrices that preserve a skew-symmetric bilinear form, which is comparable to the orthogonal group, which preserves a symmetric bilinear form. More precisely, the orthogonal group has:and its generalization the indefinite orthogonal group has:where S is symmetric. So for the symplectic group we have matrices Y such as:where A is antisymmetric. This is explained at: www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous orthogonal group, that definition ends up excluding determinant -1 automatically.

Therefore, just like the special orthogonal group, the symplectic group is also a subgroup of the special linear group.