My Science & Epistemology Researches
Posts related to mathematical topics.
I had second thoughts about the draft proof already posted here.
Using a couple of theorems (called 1.1.3 and 1.1.8, attached), the proof becomes much simpler and more straightforward.
But I think I am justified in my oversight, as I studied these theorems so long ago that Tim Berners Lee had yet to invent the www ; )
Brussi 2026_Theorem_1.1.3 and 1.1.8
Brussi 2026_Dedekind theorem 2
The chain rule in differential equations is a small but powerful trick often used to derive composite functions by finding the derivative of the outer function while keeping the inner function unchanged.
For example, the expression dy/dx can be rewritten as:
[1] dy/dx = dy/du * du/dx .
In this way, we can obtain a derivative with respect to a variable that is more useful, for example because we know its value (possibly in a simpler formal expression, or already known), or is easier to integrate.
In simple terms, this means transforming the rate of change of x with respect to y into the product of the rate of change of u with respect to y and the rate of change of x with respect to u .
Regarding rules for deriving functions, the chain rule leads to the proof that if F(x) = f(g(x)) , its derivative is:
[2] F'(x) = f'(g(x) * g'(x) .
But while this rule for deriving functions is used every day without thinking about it, the chain rule can be a useful trick in solving many physics problems when we don’t know the direct rate of change between two variables.
A simple example of its use in astrophysics is in the study of star formation and Jeans’ Mass (the critical threshold required for a cloud of interstellar gas and dust to collapse under its own gravity to form stars). The chain rule is used to convert acceleration into a form that depends on position rather than time, making the differential equation integrable. In formal terms, as in [1] , we can write:
[3] d²R/dt² = dv/dt = dv/dR * dR/dt = v * dv/dR ,
where v = dR/dt .
In this case, this substitution allows us to integrate the equation of motion with respect to radius R to determine the free-fall time of a collapsing gas cloud (off-topic calculation omitted).
Collapsing gas cloud(source accademiadellestelle.org)
Some considerations for true beginners.
The equal sign “=” is a mathematical symbol (coined by Welsh mathematician Robert Recorde in 1557) of two parallel horizontal lines used to show that two expressions have the same value or are identical.
It is taught very early on in elementary school, but I don’t think the power of its deep meaning is explained well. At least, that’s what we can see from the difficulties experienced by slightly older students.
Equal means exactly equal. What is on the left of the “=” sign has the same value as what is on the right. So if there is a symbol on the left and a number on the right, in that context that symbol is worth exactly the amount indicated. And if two symbols are equal, they can be used interchangeably. Trivial, but a source of great doubt for those who have not fully grasped the basic concept.
From a formal point of view, the equal sign satisfies the following conditions:
– Reflexive Property: any value is equal to itself, a = a
– Symmetric property: if a = b then b = a (the order of the expressions can be swapped without changing the truth)
– Transitive Property: if a = b and b = c then a = c
– Substitution Property: informally, this just means that if a = b , then a can replace b in any mathematical expression or formula without changing its meaning; formally, for every a and b , and any formula ϕ(x) with a free variable x , if a = b , then ϕ(a) implies ϕ(b) ; we can call this a function application.
Even without going into further detail, these simple properties allow us, for example, to solve first-degree equations directly. They allow us to invert formulas simply by adding or multiplying the same quantity on both sides of the equation. OK, taking care not to divide by zero.
If the unknown quantities (the so-called variables) are of a higher degree, things get more complicated, but the effectiveness of “=” remains the same.
A powerful little trick based on this is to use new variables to replace more complicated expressions or those with higher powers.
A draft written in Italian proves the existence and uniqueness of an element that separates two non-empty sets representing the field of real numbers.
This allows us to prove that the real numbers are a complete space (i.e., that every Cauchy sequence is convergent).
Note: in the case of irrational numbers instead of rational numbers, it does not work because they are not complete, i.e., it is not possible to define an element that is an extreme.
In layman’s terms, one could say that if rational numbers are removed from the real numbers to obtain irrational numbers, then ‘gaps remain’ and completeness no longer exists.
upgrade: a reconsideration of the demonstration
A little reminder about a useful function used in Astronomy: the Abel integral equation (source: Google AI)
At school, 50 years ago, they taught me a proof of the Pythagorean theorem.
I didn’t remember it and tried to make my own.
Among the dozen I’ve made, I’ll list a couple of demonstration drafts.
Brussi 2026_Pythagorean theorem 2
Brussi 2026_Pythagorean theorem 9
A very thorough article with examples of the Feynman trick for integration (differentiation under the integral sign). But even in the simplest cases it is a precious tool.
Zaharia Burghelea “Feynman’s Trick”
After a hiatus from 2021, I’m resuming my online presence with this small collection of suggestions, seeking intellectual dialogue. Let me know what you think in the comments or via email.
Thanks
p.s. I posted with the dates of the original documents
We derive the II Jeans equation (the Jeans equations are a set of partial differential equations that describe the motion of a collection of stars in a gravitational field) from the collisionless Boltzman equation (CBE) in cylindrical coordinates.
Brussi 2024_Jeans equation in cylindrical coordinates
An in-depth analysis that derives the photon distribution function, known as Planck’s distribution or black body law, in two different ways.
Written in italian
Brussi 2023_Funzione di distribuzione dei fotoni
