advanced-numerical-methods-in-computational-mathematics

Statistics isn't a supporting subject in data science, it's the thing that makes the "science" part of data science actually mean something. Code Gives You Numbers. Statistics Tells You What They Mean. Anyone can write a few lines of code to calculate an average, plot a chart, or run a model. What statistics adds is the ability to know whether any of that actually means anything. Is a 3% increase…

https://github.com/black-210/neva I started working on a small project called Neva because I wanted to experiment with lattice mathematics without starting with a huge framework. The original idea was pretty simple: Take a lattice, enter its basis vectors, and let the program show me what is happening mathematically. I used Python and NumPy for the first version. What Neva Does Neva currently let…

If the domains of functions is R^N then embedding theorem follows even if on bounded domains without extension operators.

I needed to figure out what the Moore-Penrose inverse is, so I put together these basic notes for myself. Thought I’d share them here in case they’re helpful to anyone else.

The definition of a continuous function at a point ##x_0## is: $$\forall \epsilon >0 \exists \delta >0 \forall x(|x-x_0|<\delta \rightarrow |f(x)-f(x_0)|<\epsilon)$$ What would happen if we change it to another defintion as: ##\exists \epsilon >0 \forall \delta>0 \forall x(|x-x_0|<\delta... Read more

The definition of a continuous function at a point ##x_0## is: $$\forall \epsilon >0 \exists \delta >0 \forall x(|x-x_0|<\delta \rightarrow |f(x)-f(x_0)|<\epsilon)$$ What would happen if we change it to another defintion as: ##\exists \epsilon >0 \forall \delta>0 \forall x(|x-x_0|<\delta... Read more

The definition of a continuous function at a point ##x_0## is: $$\forall \epsilon >0 \exists \delta >0 \forall x(|x-x_0|<\delta \rightarrow |f(x)-f(x_0)|<\epsilon)$$ What would happen if we change it to another defintion as: ##\exists \epsilon >0 \forall \delta>0 \forall x(|x-x_0|<\delta... Read more

The Nitescence Theorem constructs a canonical mechanism for navigating Gödelian incompleteness through structural ascension. Given a consistent, Σ1-sound effective theory S, the nitescence operator N(S) = S + Con*(S)* resolves an obstruction determined by S itself rather than arbitrarily chosen among its undecidable propositions. The construction is classical—the Turing–Feferman consistency progr…

This is an application of functional analysis to the existence and smoothness of the Navier–Stokes equations using elementary weak solutions in Sobolev spaces. We solve the problem in mathematics. The problems are not in physics, so we do not use any physics or assumptions-falsified mathematics. We use mathematics only. We can solve the problem by using an exactly and completely FALSIFIED resolut…

There are several contexts in which it is of relevance that a certain property of a morphism is preserved (or stable) under pullback, i.e. also shared by the the morphism for any pullback diagram Geometers prefer to say “stable under base change”. Monomorphisms are always stable under pullback; that is, if is a monomorphism, then so is . Epimorphisms are not necessarily stable under pullback. In …

The Jordan-Hölder theorem says that every composition series of a given group, and every Jordan-Hölder sequence on a given object in an abelian category, has the same length, and the same simple factors, up to permutation. In particular says that the length of an object in an abelian category is well defined. More generally, a form of the theorem holds in any homological category. This is a proof…

This week we’re talking about Mark Zuckerberg’s latest essay, “The Future Is for Everyone,” which outlines his positive new vision about the potential of A.I. But do we think it’s credible? Then, Pangram’s chief executive, Max Spero, joins us to talk about the breakout success of his A.I. slop detector. And finally, it’s time for our new segment all about math — we’re Running the Numbers.

For the first time in 30 years, computer scientists have found a better way to allocate objects evenly between two groups. The post ‘Huge Breakthrough’ in the Math of Imbalance first appeared on Quanta Magazine