Mathematical reflections and “innovations” of a complete ignoramus and a dumbass in this field.

I once described the concept of an integral in the simplest and most understandable form even for philologists (See the note “Mathematics lesson by Esprit” 13 XII 2017) and also mentioned such a strange possibility – the key to everything that follows in this note:

When mathematics describes the meaning of an integral in its simplest form, namely, as an infinitely large sum of infinitely small quantities, I think it would be more accurate to say an infinitely INCREASING sum of infinitely DECREASING quantities, then there is an implicit “obvious” assumption in this reasoning: the rate of increase in the number of these decreasing quantities is EQUAL to the RATE of their decrease.

This assumption is SELF-EVIDENT.

To fill a certain area bounded on three sides by straight lines, and on the fourth side by an arbitrarily varying curve, it is necessary to thin the strips of the stepped polygon and, at the same time, increase the number of these strips-steps so that they maximally approximate the shape of the curve, having the exact shape of this curve in the limit.

And here’s the question:

And WHAT happens if these two speeds are NOT EQUAL to each other?

If the rate of “multiplication” of the stripes IS GREATER THAN the rate of their thinning, WHAT WILL HAPPEN?

There will be some kind of “compression”, violent deformation of insufficiently rapidly tapering strips, so as to REMAIN within the “allowed”, initially LIMITING area!

If the rate of thinning of the strips exceeds the rate of their numerical increase, then there will be a “vacuum” inside the specified area, a partial void filled with strips, and they will have to “inflate” themselves in order to fill this area.

“Nature does not tolerate emptiness”!

If we assume such fantastic cases, then two new types of integrals arise:

Compression integrals, that is, compressed integrals, and vacuumed integrals, that is, discharged integrals.

Perhaps this will require the introduction of additional indices of the RATE OF CHANGE of both the integral itself and the “dx” under it, or the use of double intergrals? Or maybe ACCELERATION, that is, changes in velocity, the derivative of the index over time?

(It’s funny, there will be, according to the Alexander Block, a “Breathing Integral”:

“Steel machines, where THE INTEGRAL BREATHES”)

Where can such “mutants” of the mother – mathematics find their use in practice?

In many mathematical models that describe exactly such real processes, in which the speeds of two COMPETING processes are NOT the SAME.

For example, how can an explosion or a shot be described in terms of “internal and rapid” combustion processes? Or, in general, the process of operation of an internal combustion engine, and NOT its INDIVIDUAL stages, but in a whole cycle. Here both integrals can be useful – both compressed and discharged.

To describe the processes of compression and discharge, explosion and implosion, such as in one version of the atomic bomb, processes with positive or negative feedback, gravitational collapse – the formation of “black holes” and neutron stars, the pinch effect in plasma and liquid conductors…

A demographic explosion or, conversely, the extinction of a certain population of people or any living creatures can also be mathematically well described by such integrals.

Again, these proposed two types of integrals can successfully describe any dynamic process of velocity inequality (and even it’s changes) of two mutually competing processes, because such integrals themselves arose from it!

Some MOBILE integrals will arise, combining concepts.

Throughout the woods and thickets
In front of pretty Europe
We will spread out! We’ll turn to you
With our Asian muzzles.

Come everyone, come to the Urals!
We’re clearing a battlefield there
Between steel machines breathing integrals
And the wild Tatar Horde!

The Scythians, A.Blok, 1918.

Translated by A. Wachtel, I. Kutik and M. Denner

Faciant meliora potentes.

If I’m wrong, and THIS is almost certainly applicable to THIS CASE, given the sincere subtitle, let my seniors correct me!

17 XII 2025

P.S. As an apologetic gift for my intrusion into a field in which both my knowledge and abilities are zero or even negative, I propose the candidacy of another “God of Mathematics”: HECATONHEIR, a hundred–armed titan with fifty heads, whose visual image, in my opinion, was most successfully created by the Urusov brothers artists. The only addition that seems problematic to me is the WALKING of such titanic creatures on only two legs, then it is better to stand like a tree firmly rooted in the soil.

And if you still want such three Titan brothers to MOVE fast enough in space, walking, say, on the ground, then let them use half their arms to walk like centipedes. Both fast and reliable, and the versatility of using free hands is developing!

They can swim well and even fly if they turn their palms skillfully and quickly enough. And the forces are enough if they broke off whole rocks in a fight and threw them at the enemy.

The power is there – you don’t need a mind!