advanced-numerical-methods-in-computational-mathematics

Ho(Cat) is a name for the homotopy category of Cat. That is, is the category This is an instance of a general construction which, given a 2-category, or more generally an n-category, produces a 1-category with the same objects and whose morphisms are equivalence classes of 1-morphisms in the original -category. Sometimes this is called the 1-truncation and denoted . It can also be viewed as an in…

Consider the following predator-prey system: $$ \begin{cases} \dfrac{dN}{dT} &= rN\left(1-\dfrac{N}{K(T)}\right) -aN^\alpha P -\dfrac{qEN}{mE+nN},\\ \dfrac{dP}{dT} &=-d_1P+aeN^\alpha P. ...

The product of four consecutive Fibonacci numbers equals the product of two consecutive integers. For example, 3 × 5 × 8 × 13 = 39 × 40. I ran across this theorem in a note [1] that says “The product of any four consecutive Fibonacci numbers is twice a triangular number.” Since triangular numbers have […] The post Fibonacci product first appeared on John D. Cook .

The other day, I found myself wondering how big 52! (52 factorial) is, and that led me to ponder how these could be estimated without a calculator or a computer. It turns out there’s some fairly interesting math behind being able to estimate the size (number of digits) of a factorial reasonably accurately. This post will start by stating how to do the estimate, and if you’re curious you can read …

algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on operads model structure on algebras over an operad A Jordan algebra is an algebra that…

Moser’s circle problem can be used to divide a pizza into one, two, four, eight, 16 or 31 slices. It’s a perfect example of how the numbers in a sequence aren’t as clear-cut as they may first appear

After an OpenAI model presented a possible solution to the Navier-Stokes problem, mathematicians are speculating about which major unsolved question could be resolved next

Hi everyone, I’ve been thinking about how beginners learn algebra and wanted to get some opinions. When solving a basic equation like: 2x + 6 = 14 it’s easy to learn the steps and arrive at x = 4. But is it more important for a beginner to understand why we can subtract 6 from both... Read more

The Origin of Probability: A Complete Derivation in Three Parts is the collected edition of a three-part theoretical investigation into the structural origin of probability within a deterministic closed-description framework. The work asks a deliberately narrow but foundational question: under what conditions can a system whose underlying description remains deterministic acquire a lawful probabi…