Linear Algebra Find a basis for the solution space of the homogeneous linear system x - y + 2z = 0 2x + y = 0 x - 4y + 6z = 0 How can a basis not be orthogonal? What is a nontrivial relationship linear algebra? How do you find the basis of a polynomial vector space?
19 December, 2024 in expository, math.MG | Tags: cosmic distance ladder, quaternions, spherical trigonometry | by Terence Tao | 28 comments Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and...
De Broglie, L.: Interpretation of quantum mechanicsby the double solution theory Annales de la Fondation Louis de Broglie, Volume 12,(4) (1987) Couder, Y., Fort, E., Gautier, C.-H., Boudaoud, A.: From Bouncing to Floating: Noncoalescence of Drops on a Fluid Bath. Phys. Rev. Lett...
It seems that the first step in this is to construct a “soliton” type localized steady state solution, that is a solution , to the equation that decays in the variable; one can then hope to do a suitable stability (or instability) analysis of this soliton to perturb it to a blowup...
Vectors in Linear Algebra: In linear algebra, vectors are written as a row or column of numbers. An example of a row vector is given as follows: [123456] The number of elements of a vector indicates itsdimension. So for our example, we have a6-dimensional vector. ...
But not all matrices are diagonalizable! The solution that is usually presented in the classroom is to use the Jordan canonical form instead and to compute the matrix exponential of Jordan blocks (using that you can split a Jordan block into the sum of a diagonal matrix and a nil-potent mat...
The idea is that most of the executing code will be hitting a small set of variables often, and the rest (a much larger set of variables) infrequently. If the processor can't find the data in L1 cache, then it looks in L2 cache. If not there, then L3 cache, and if not there, ...
In detail, one can map pretty much all success-driving elements of AlphaFold2 to well-known principles of traditional Machine Learning (ML): The arrangement of amino acid sequences into data matrices that reveal evolutionary connections between proteins is nothing but “a separate pre-processing ste...
A particularly non-obvious symmetry is the pseudoconformal symmetry: if solves (1), then the pseudoconformal solution defined by for can be seen after some computation to also solve (1). (If has suitable decay at spatial infinity and one chooses a suitable branch cut for , one can extend...
24 May, 2013 in expository, math.GR, math.OA, math.RT | Tags: characters, convolution, Frobenius groups, Frobenius theorem, idempotents | by Terence Tao | 8 comments A finite group is said to be a Frobenius group if there is a non-trivial subgroup of (known as the Frobenius complemen...