Pragmatism Updated 2025-07-16
The opposite of idealism.
Mathieu group Updated 2025-07-16
Mathieu group Updated 2025-07-16
Mathieu group Updated 2025-07-16
Contains the first sporadic groups discovered by far: 11 and 12 in 1861, and 22, 23 and 24 in 1973. And therefore presumably the simplest! The next sporadic ones discovered were the Janko groups, only in 1965!
Each is a permutation group on elements. There isn't an obvious algorithmic relationship between and the actual group.
TODO initial motivation? Why did Mathieu care about k-transitive groups?
Their; k-transitive group properties seem to be the main characterization, according to Wikipedia:
Looking at the classification of k-transitive groups we see that the Mathieu groups are the only families of 4 and 5 transitive groups other than symmetric groups and alternating groups. 3-transitive is not as nice, so let's just say it is the stabilizer of and be done with it.
Mathematics and Statistics course of the University of Oxford Updated 2025-07-16
Mathematics and Philosophy masters course of the University of Oxford Updated 2025-07-16
Primitive recursive function Updated 2025-07-16
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
- variable assignments
- addition and subtraction
- integer comparisons and if/else
- for loops
for (i = 0; i < n; i++)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.
Prisoner's dilemma Updated 2025-07-16
Mathematics and Computer Science masters course of the University of Oxford Updated 2025-07-16
Corresponding undergrad: Mathematics and Computer science course of the University of Oxford
Mathematics and Computer science course of the University of Oxford Updated 2025-07-16
Public landing page: www.ox.ac.uk/admissions/undergraduate/courses/course-listing/mathematics-and-computer-science
A mixed cross department course with the Mathematical Institute of the University of Oxford.. Its corresponding masters is known as Oxford MMathCompSci. The handbook is together with the computer science one: Section "Computer science course of the University of Oxford".
Mathematical Sciences masters course of the University of Oxford Updated 2025-07-16
Mathematical formulation of quantum mechanics Updated 2025-07-16
These are the key mathematical ideas to understand!!
There are actually a few formulations out there. By far the dominant one as of 2020 has been the Schrödinger picture, which contrasts notably with the Heisenberg picture.
Another well known one is the de Broglie-Bohm theory, which is deterministic, but non-local.
Mathematical formulation of quantum field theory Updated 2025-07-16
The Dirac equation, OK, is a partial differential equation, so we can easily understand its definition with basic calculus. We may not be able to solve it efficiently, but at least we understand it.
But what the heck is the mathematical model for a quantum field theory? TODO someone was saying it is equivalent to an infinite set of PDEs somehow. Investigate. Related:
The path integral formulation might actually be the most understandable formulation, as shown at Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
Quantum electrodynamics by Lifshitz et al. 2nd edition (1982) chapter 1. "The uncertainty principle in the relativistic case" contains an interesting idea:
The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta,
polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process.
Mathematical constant Updated 2025-07-16
Mathematical analysis Updated 2025-07-16
MathDoctorBob Updated 2025-07-16
He got so old from 2012 to 2021 :-)
This dude did well. If only he had written a hyperlinked wiki rather than making videos! It would allow people to jump in at any point and just click back. It would be Godlike.
mathdoctorbob.org/About.html says:
Robert Donley received his doctorate in Mathematics from Stony Brook University and has over two decades of teaching experience at the high school, undergraduate, and graduate levels.
MathBook XML Updated 2025-07-16
Prize Updated 2025-07-16
Generally, prizes that pay big lumps of money to well established individuals are a bit useless, it would be better to pay smaller sums to struggling beginners in the field, of which there are aplenty.
The most important part about prizes should not be the money, nor the recognition, but rather explaining better what the laureates did. In this, most prizes fail. Thus Ciro Santilli's project idea: Project to explain each Nobel Prize better.
Material property Updated 2025-07-16
There are unlisted articles, also show them or only show them.