Ciro Santilli @cirosantilli 35

The 3D regular convex polyhedrons are super famous, have the name: Platonic solid, and have been known since antiquity. In particular, there are only 5 of them.

The counts per dimension are:

The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.

The usual definition of a polynomial is over a field as shown at polynomial over a field.

However, there is nothing in the immediate definition that prevents us from having a ring instead, i.e. a field but without the commutative property and inverse elements.

The only thing is that then we would need to differentiate between different orderings of the terms of multivariate polynomial, e.g. the following would all be potentially different terms:while for a field they would all go into a single term:so when considering a polynomial over a ring we end up with a lot more more possible terms.

If the ring is a commutative ring however, polynomials do look like proper polynomials: Section "Polynomial over a commutative ring".

where:

Note that this is the sum of the:Note that the relationship between $\psi$ and $F$ is not explicit. However, if we knew what type of particle we were talking about, e.g. electron, then the knowledge of psi would also give the charge distribution and therefore $F$

As mentioned at the beginning of Quantum Field Theory lecture notes by David Tong (2007):

First sequenced variant: www.ncbi.nlm.nih.gov/genome/?term=86693

Genes at: www.ncbi.nlm.nih.gov/nuccore/MN908947.3 TODO protein list on a database?

30kbp, 10 genes, 29 proteins: cen.acs.org/biological-chemistry/infectious-disease/know-novel-coronaviruss-29-proteins/98/web/2020/04

50-200 nanometers in diameter.

Gene overview:

2023 full connectome published on Science by Marta Zlatic's lab at the MRC:

The hard part then is how to make any predictions from it:

2024: www.nature.com/articles/d41586-024-03190-y Largest brain map ever reveals fruit fly's neurons in exquisite detail

As of 2022, it had been almost fully decoded by post mortem connectome extraction with microtome!!! 135k neurons.

That article mentions the humongous paper elifesciences.org/articles/66039 elifesciences.org/articles/66039 "A connectome of the Drosophila central complex reveals network motifs suitable for flexible navigation and context-dependent action selection" by a group from Janelia Research Campus. THe paper is so large that it makes eLife hang.

Observation that all solids appear to have the same constant heat capacity per mole.

It can be seen as the limit case of an Einstein solid at high temperatures. At lower temperatures, the heat capacity depends on temperature.

Step of electronic design automation that maps the register transfer level input (e.g. Verilog) to a standard cell library.

The output of this step is another Verilog file, but one that exclusively uses interlinked cell library components.

In the context of Maxwell's equations, it is vector field that is one of the inputs of the equation.

Section "Maxwell's equations with pointlike particles" asks if the theory would work for pointlike particles in order to predict the evolution of this field as part of the equations themselves rather than as an external element.

Measured in amperes in the International System of Units.

This section is about the definition of the dot product over ${\mathbb{C}}^n$, which extends the definition of the dot product over ${\mathbb{R}}^n$.

Some motivation is discussed at: math.stackexchange.com/questions/2459814/what-is-the-dot-product-of-complex-vectors/4300169#4300169

The complex dot product is defined as:

E.g. in ${\mathbb{C}}^1$:

We can see therefore that this is a form, and a positive definite because:

Just like the usual dot product, this will be a positive definite symmetric bilinear form by definition.