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Sperner's lemma by

Universal instantiation by

Universal instantiation is a rule of inference in formal logic that allows one to derive a specific instance from a universally quantified statement. In simple terms, if something is true for all members of a certain set (as stated by a universal quantifier), one can conclude that it is true for any particular member of that set.

Random permutation statistics by

Newton's law of universal gravitation by

Quasar by

Star by

Bacterial cellular morphogology by

Ciro Santilli 37 Updated 2025-07-16

In vivo by

Ciro Santilli 37 Updated 2025-07-16

Pharmacy by

Ciro Santilli 37 Updated 2025-07-16

Potentiometer by

Ciro Santilli 37 Updated 2025-08-08

Variable resistance element.

Canadian company by

Ciro Santilli 37 Updated 2025-07-16

Fast neutron by

Ciro Santilli 37 Updated 2025-07-16

First-order logic by

Ciro Santilli 37 Updated 2025-07-16

Builds on top of propositional logic, adding notably existential quantification.

Alpha decay by

Ciro Santilli 37 Updated 2025-07-16

Most of the helium in the Earth's atmosphere comes from alpha decay, since helium is lighter than air and naturally escapes out out of the atmosphere.

Wiki mentions that alpha decay is well modelled as a quantum tunnelling event, see also Geiger-Nuttall law.

As a result of that law, alpha particles have relatively little energy variation around 5 MeV or a speed of about 5% of the speed of light for any element, because the energy is inversely exponentially proportional to half-life. This is because:

if the energy is much larger, decay is very fast and we don't have time to study the isotope
if the energy is much smaller, decay is very rare and we don't have enough events to observe at all

Video 1.

Quantum tunnelling and the Alpha particle Paradox by Physics Explained (2022)

Source.

youtu.be/_f8zeEI0oys?t=796 George Gamow and Edward Condon proposed the quantum tunnelling explanation
youtu.be/_f8zeEI0oys?t=1725 worked out example that predicts the half-life of polonium-210 based on its emission energy

Kramer graph by

The Kramer graph, often referred to as the "K4,4" graph or "Kramer graph," is a specific type of graph used in combinatorial design and graph theory. Here are some key points about the Kramer graph: 1. **Properties**: - The Kramer graph is a bipartite graph, meaning it can be divided into two distinct sets of vertices such that no two vertices within the same set are adjacent.

Latin alphabet by

Ciro Santilli 37 Updated 2025-07-16

W boson by

Ciro Santilli 37 Updated 2025-07-16

No-Nonsense Quantum Field Theory by Jakob Schwichtenberg (2020) by

Ciro Santilli 37 Updated 2025-07-16

www.amazon.com/No-Nonsense-Quantum-Field-Theory-Student-Friendly/dp/3948763011

This book really tries to recall basic things to ensure that the reader will be able to understand the more advanced ones.

Sometimes it goes a little bit overboard, like defining what a function does several times.

But Ciro Santilli really prefers it when authors error on the side of obvious.

Lagrangian density by

Ciro Santilli 37 Updated 2025-07-16

When we particles particles, the action is obtained by integrating the Lagrangian over time:

A c t i o n = \int_{t_{0}}^{t} L (x (t), \frac{\partial x ( t )}{\partial t}, t) d t

(1)

In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.

Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.

E.g. for a 2-dimensional field

f (x, y, t)

A c t i o n = \int_{t_{0}}^{t} L (f (x, y, t), \frac{\partial f ( x , y , t )}{\partial x}, \frac{\partial f ( x , y , t )}{\partial y}, \frac{\partial f ( x , y , t )}{\partial t}, x, y, t) d x d y d t

(2)

Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with

x \in R^{2}

A c t i o n = \int_{t_{0}}^{t} L (f (x, t), \nabla f (x, t), x, t) d x d t

(3)

And in the context of special relativity, people condense that even further by adding

t

to the spacetime Four-vector as well, so you don't even need to write that separate pesky

t

The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.