For example, the tree-height function h(age) = age×20 makes no sense for an age less than zero. ... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things. So we need something more powerful, and that is where sets come in:A...
This can be used to imply that the convolution action of every minimal idempotent either preserves or annihilates it, which makes itself an idempotent, which makes normal. Read the rest of this entry » Quasirandom groups and a cheap version of the Brauer-Fowler theorem 2 May, 2013 in ...
As with objects in many other algebraic categories, a basic way to understand a Lie algebra is to factor it into two simpler algebras via a short exact sequence thus one has an injective homomorphism from to and a surjective homomorphism from to such that the image of the former homomorphism...
However, Python’s built-in sorting functions work in a different way. Instead of taking a comparison function, they take a key function, which is a function of a single argument that may return anything. This function is used to generate a “key” for each object in the list to be sor...
We start by considering a finite-dimensional, boundaryless submanifold G of G˜ that is rich enough to allow us to perturb each payoff function in each of the directions xim, xjn and ximxjm independently and obtain a new pair of payoff functions in G. To formalize this idea, let p q ...
If one makes the ansatz for some smooth and , some calculation shows that the system now reduces to a system purely on : The metric is hidden in this system through the covariant derivative . To eliminate the metric, we can lower indices to write Here the divergence is relative to ...
This makes it somewhat difficult to directly interpret the situation in terms of group cohomology. However, thanks to tools such as the Balog-Szemerédi-Gowers lemma, one can upgrade this sort of -structure to -structure – at the cost of restricting the domain to a smaller set. Here I ...
In the traditional foundations of probability theory, one selects a probability space , and makes a distinction between deterministic mathematical objects, which do not depend on the sampled state , and stochastic (or random) mathematical objects, which do depend (but in a measurable fashion) on ...
Now for a key observation: whereas the left-hand side of (6) only makes sense when is a natural number, the right-hand side is meaningful when takes a fractional value (or even when it takes negative or complex values!), interpreting the binomial coefficient as a polynomial in . As such...
Note that a connection can be determined by the group elements it assigns to each -simplex . (I have written the simplex from right to left, as this makes the composition law cleaner.) So far, only the -skeleton (i.e. the simplices of dimension at most ) of the complex have been ...