Mathematician Measures the Repulsive Force Within Polynomials
quantamagazine.org
quantamagazine.org
Random matrices: tail bounds for gaps between eigenvalues
Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications."
https://arxiv.org/abs/1504.00396
Real roots of random polynomials: expectation and repulsion
https://arxiv.org/abs/1409.4128
Zero repulsion in families of elliptic curve L-functions and an observation of Miller
https://academic.oup.com/blms/article-abstract/45/1/80/29767...
Integral Points on Elliptic Curves and the Bombieri-Pila Bounds
Let C be an affine, plane, algebraic curve of degree d with integer coefficients. In 1989, Bombieri and Pila showed that if one takes a box with sides of length N then C can obtain no more than O_{d,\epsilon}(N^{1/d+\epsilon}) integer points within the box. Importantly, the implied constant makes no reference to the coefficients of the curve. Examples of certain rational curves show that this bound is tight but it has long been thought that when restricted to non-rational curves an improvement should be possible whilst maintaining the uniformity of the bound. In this paper we consider this problem restricted to elliptic curves and show that for a large family of these curves the Bombieri-Pila bounds can be improved. The techniques involved include repulsion of integer points, the theory of heights and the large sieve. As an application we prove a uniform bound for the number of rational points of bounded height on a general del Pezzo surface of degree 1.
To add to your list (for less number-theory specific topics):
- Lyapunov functions? Energy in physics (just a mathematical surrogate)
- Exponential families? Canonical ensemble in physics
- Convex duality? Lagrange duality in classical mechanics
and the list continues.
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[0] And, admittedly, a member of a physics lab, even though I don't really do any physics.
https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneadi...
EDIT: I also really like the fact that even the original paper mentioned in the quanta article references other (quite pictorially fun) things, such as the aptly named Hedgehog spaces [1] (see Theorem 3, for example).
Perhaps it's violating the prime directive to be mentioning this, but it's a little bit of a shame to see what I think is a good article so quickly dismissed by what is essentially just bikeshedding (that isn't even justified!).
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[0] https://link.springer.com/article/10.1007/s10955-011-0284-x
Again, big props to Hartnett (and Dimitrov, of course)!
> plot(polyroot(c(1,-1,1,-1,1,-1)))
Mathematicians borrow the terminology that is most helpful for explaining an idea. such a thing cannot be seen as harmful. Readers are not infants, and should not be treated like infants. They are responsible for the conclusions they come to, especially in mathematics.
What do you mean?
In any science, engineering, etc. field you are responsible for the conclusions you come in your papers/projects/etc.
Even in sub-fields of those with an empirical component (if that is your angle) there are standards you have to reach to claim a discovery/success.
Also, there is nothing wrong with appropriating terminology -- when it is appropriate (no pun intended). But, to quote Tom Stoppard, "If there is any point to using language at all, it is that a word is taken to stand for a particular fact or idea and not for other facts or ideas." The phrase "measure a repulsive force" has an established meaning in English, and that meaning is intimately bound to the physical world. There is nothing wrong with using that physical phenomenon as a metaphor, but that's not what the headline does. The headline says "Mathematician Measures the Repulsive Force Within Polynomials" and that is simply false under the well-established English semantics of the phrase "measure a ... force." No mathematician has ever measured a force, at least not in the course of conducting the business of being a mathematician.
Read the histories of Liebnitz and Newton. You are grossly misinformed about both mathematics and physics.
What instrument does one use to "measure the repulsive force within polynomials"? In what units are the results expressed? Can this force be used to do useful work? Does this force obey Newton's first law?
Which gave them the requisite experience to invent calculus to more accurately describe those physical forces. Which ends up being wrong because the universe isn't diffenentiable, after all. Did their imperfect analogies between differentiable functions and physical forces do HARM or did they advance the fields of math and physics by leaps and bounds?
In short, your hard delineation between math and physics is ahistorical at best and doesn't even model present-day research in either field.
(There is an additional problem, which is that the analogy being used is a bad analogy. Forces cause things to move in accordance with Newton's first law. But these alleged "repulsive forces between polynomials" don't cause anything to move. Indeed, the whole concept of a polynomial being "moved" by a "repulsive force" is non-sensical. Only things that occupy locations in physical space can be moved by forces.)
Me thinks thou doth protest too much
https://en.m.wikipedia.org/wiki/Virtual_work#Principle_of_vi...
Me also thinks you have no clue what you're talking about
When you say "actively harmful" -- do you mean that it's doing physical harm to something, or are you using an incomplete metaphor of the kind you're railing against? Because I'm speaking as somebody with some training in number theory and teaching advanced math and I think you're way off-base.
How is it harmful?
The problem here is not that the article uses an analogy, the problem is that it doesn't present it as an analogy, it presents it as literal. The headline says that mathematicians have measured a force. That's nonsense. They did no such thing. But in order to realize that they did no such thing and that the "repulsive force" idea is an analogy you have to already know what is going on.
Here is another example from the body of the article:
"“You can think of the roots of a polynomial as negatively charged particles that repel each other with a force that decays when the distance increases,” Breuillard said.
No, you can't. If you could, then the "force" that these "particles" exerted on each other would cause them to move in accordance with Newton's first law. So not only is it an analogy presented as literal, it's actually a bad analogy.
> No, you can't.
Well, yes, you can, and a professional mathematician just said so, yet it seems you insist that mathematics is all about dotting the i's and crossing the t's. Thinking of the roots in the way described is exactly what Dmitrov did: he explored this analogy, worked with a formalized version of it, and used it to prove a significant theorem.
I agree that the article writer's intro takes the analogy much too far, to the point of being just false — mathematicians don't look at the number line and see "forces" at work maintaining the distances between the integers.
Let them. It is their fundamental right to be wrong.
ironically, their misuse of the term harmful can also lead to confusion or be misinterpreted
all in all, we should use more precise words (and thinking), but this requires more effort from both the writer and the reader
Instead it gets people to overlook it as more new age nonsense woo that abuses appropeiated quantum physics terminology.
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Schinzel and Zassenhaus predicted that every non-cyclotomic polynomial must have at least one root that’s outside the unit circle and at least some minimum distance away.
Or, to put the Schinzel-Zassenhaus conjecture in terms of repulsion, it predicted that the smallest roots of a non-cyclotomic polynomial — which might fall within the unit circle — effectively push other roots outside the unit circle, like magnets pushing each other away.
You can think of the roots of a polynomial as negatively charged particles that repel each other with a force that decays when the distance increases,
"""
So if they can write a formula for distance between these roots, and it's always positive and has a similar dynamic to Newton's (or Maxwell's, or Ampere's) force, then why not call it repulsive force?
I would certainly not only specifically not say this is borrowed from quantum physics (perhaps classical EM?), but in fact the whole point of the theorem is to show that such roots do interact in a specific way. From Theorem 1 of the paper:
> Let P ∈ Z[X] be a monic integer irreducible polynomial of degree n > 1. If P is not cyclotomic, then [the largest root of P is bounded away from 0 with norm at least 2^(1/4n)].
In particular, it is a statement about a specific root (the largest one) lying outside of the unit disk, whereas the remaining roots could potentially lie within it (and specific examples of this 'tightness' condition are given in applications in section 5).