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This question was cross-posted on Math Stack Exchange. Here is a copy of my answer for it there.
This is it. The perfectly centered billiards break. Behold.
Setup
This break was computed in Mathematica using a numerical differential equations model. Here are a few details of the model:
All balls are assumed to be perfectly elastic and ...

74

Let's give it a try. Of course, the precise mathematical meaning is perhaps absent, so the answers are sort of heuristic. But if I understand correctly, you want to gain intuition ;)
The first observation is that Planck's constant has units, it is not a numerical constant but carries physical dimension. In some sense, this makes all the difference: in your ...

46

Alon Amit: "There are some things that break my heart more thoroughly than reading nonsensical conclusions from Gödel's Theorems to the limitations of physics published by eminent scientists, but they are few."
Johannes Koelman: "Stating that Gödel (or Turing, or gravity) implies the logical impossibility of a TOE, is the same as stating that because of the ...

answered Mar 14 '18 at 10:43

Carlo Beenakker

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43

First let us recall why string theory is attractive. As of now, we have two experimentally verified, but mutually incompatible theories describing fundamental physical phenomena. The standard model of particle physics is a quantum field theory describing all the elementary particle interactions except for gravitation. General relativity is a classical theory ...

39

If you allow such a comprehensive reference to re-introduce basic mathematics, then either as a layman or a working mathematician your prayers are answered by the following (he even prefaces by saying that his intended layman-audience must have some mathematical sophistication):
The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger ...

33

To build intuition for the Planck constant $\hbar$, which I understand is the purpose of the OP, I would start by noting that $\hbar$ is not a dimensionless number: it has dimensions of energy $\times$ time, or of momentum $\times$ position, meaning that it represents an action. (Equivalently, it could represent an angular momentum, I will get back to that ...

answered Sep 11 '19 at 9:46

Carlo Beenakker

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The problem with this question, for mathematicians, and actually for anyone, is that the term "string theory" is not well-defined, making the question of falsifiability much more complicated.
The most well-defined interpretation of "string theory" would be the superstring in 10 flat space-time dimensions, which is defined by a series expansion. The details ...

28

You can understand this philosophy as a generalization of Noether's theorem. Let me only state Noether's theorem in the quantum case because it's actually easier to understand there than in the classical case (for me, anyway).
As setup, we have a Hilbert space of states $V$ and a state vector $\psi \in V$ which evolves according to a Hamiltonian $H$, ...

answered Mar 16 '17 at 19:22

Qiaochu Yuan

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The article by J.-L. Dorier in On the Teaching of Linear Algebra suggests the answer to your question will be different for the UK and for continental Europe:
In an attempt to answer your question more directly, I have searched for early University text books that introduce matrix multiplication. It was introduced in the context of the theory of ...

answered Nov 2 '14 at 8:11

Carlo Beenakker

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22

The difficulty is not in the language that Newton uses, but in the incredibly original viewpoint that he takes. Get a good translation, by all means, but you'll also need the help of a good guide.
One of the best such guides I ever found was S. Chandrasekhar's fantastic "Newton's Principia for the Common Reader", Oxford University Press, 1995. (Don't be ...

answered Mar 28 '18 at 18:52

Robert Bryant

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This is more of a comment inspired by Jim Belk's answer than an answer to the question in itself. It is also more physics than mathematics. However, I hope I can help readers see that this system is related to some very interesting science.
I wanted to say a little bit about the dynamics of the energy transfer in the billiard break. Very naïvely, you might ...

18

The answers so far (by Carlo Beenakker and p6majo) do not directly address Hawking's main argument, which is presented near the end of his lecture.
Hawking first presents a heuristic, motivational argument based on Gödel's theorem.
In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ...

answered Mar 18 '18 at 20:59

Timothy Chow

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17

A really "real" example is the relation between integral quadratic forms (i.e. integral lattices) and conformal field theories of free compact bosons. The latter is basically defined by an integral lattice, so the connection to integral quadratic forms is natural. They describe the excitations on the boundary of a novel quantum state of matter, the ...

17

The constant is $2$. Let $n=\lfloor d_2/d_1 \rfloor \geq 1$, and let $t_k$ be the time which
the long distance runner takes to arrive at the distance $kd_1$ from the origin,
$1\leq k\leq n$.
Proving by contradiction,
suppose that on every interval $[(j-1)d_1,jd_1], j=1,...,k$ the average speed of the long distance runner is less than
$v_2/2$. Then $t_n>...

answered Dec 5 '18 at 16:47

Alexandre Eremenko

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It may take a bit of extraction, but positive answers to both of your questions follow from my results joint with Gaiotto in Condensations in higher categories (arXiv:1905.09566). In that paper we build a $\mathbb{C}$-linear $(d+1)$-category that we call $\Sigma^d\mathrm{Vect}_{\mathbb{C}}$. ($d$ is arbitrary, and so is the ground field, but $\mathbb{C}$ is ...

answered Jun 11 '19 at 22:37

Theo Johnson-Freyd

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16

You can find a huge collection of examples in this website:
Number Theory and Physics Archive
As for a concrete example, the critical temperature of the Bose-Einstein condensate is
$$T=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi \hbar^2}{mk_B}$$
where $\zeta(s)$ is the Riemann zeta function.
Particularly after the work of Ketterle, this seems as "...

15

Too long for a comment, but this premise of the question:
I know that matrix multiplication was introduced by Cayley (correct me if I am wrong)
is indeed wrong. Gauss in Disquisitiones Arithmeticae (1801) has something called not matrix multiplication but combination of substitutions — e.g. he writes near the end of §294:
$(S)=\left\{\!\!\begin{...

answered Dec 19 '19 at 10:18

Francois Ziegler

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To supplement the excellent answers previously given, some more remarks situated in the canonical quantization formalism, which is equivalent (at physicist level) to the path integral formalism referred to by Carlo Beenakker:
When we open our eyes, we see that we're given a playground in which we can organize what we see according to position $x$, time $t$, ...

answered Sep 11 '19 at 17:39

Michael Engelhardt

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Consider what path is traced out by the projectile in the 2d velocity space (horizontal velocity x-axis; horizontal is "after rotating up so the ground is flat, gravity no longer vertical"). It starts somewhere on a circular arc, and thereafter follows a path 'down and to the right' at an angle $\phi$ to the vertical, at constant speed (corresponding to ...

13

The nLab has a page devoted to this question:
ncatlab.org/nlab/show/string+theory+FAQ
I'll be glad to further expand this as need be.

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This is really just a handwavy but perhaps more "visual" description of Pontryagin's result as cited by solbap in the comments above. Though I've written a huge block of text, there are some reasonably concrete three-dimensional pictures that you can build up in your head in this case, but it does take quite a bit of practice.
First, I assume that ...

12

Probably, the best-known application of ideas from physics to computational number theory is Shor's algorithm.
You may also be interested in other examples from “unusual and physical methods for finding prime numbers”.

12

Birkhoff and Mac Lane's 1941 "Survey of Modern Algebra" had a chapter on "The Algebra of Matrices". That book was influential in US curricula, and it would be a good place to look.
That answer is consistent with the influence of quantum mechanics, since quantum mechanics at Gottingen (as with Heisenberg) influenced algebra at Gottingen (as with Noether and ...

12

The Algebra of Grand Unified Theories, by John Baez and John Huerta may well be to your liking:
A full-fledged treatment of particle physics requires quantum field
theory, which uses representations of a noncompact Lie group called
the Poincaré group on infinite-dimensional Hilbert spaces. This brings
in a lot of analytical subtleties, which make it ...

answered Mar 16 '17 at 18:55

Carlo Beenakker

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About your Q1: I think that the simplest—and most obvious—way to think mathematically about the physical meaning of Planck's constant $h$ is that it is a kind of quantitative measure of the departure from commutativity:
Although the word "quantization" does not have a uniquely defined mathematical meaning, it is almost always connected with the passage of ...

answered Sep 11 '19 at 22:54

Konstantinos Kanakoglou

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11

This calculation of $\pi_4(S^3)$ is also obtained in the paper R. Brown and J.-L. Loday, Topology, 26 (1987) 311-334, and also available here. In that paper, $S^3$ is regarded as the double suspension $SS$ of the circle $S^1$, which is itself seen as an Eilenberg-Mac Lane space $K(\mathbb Z,1)$. We obtain in Proposition 4.10 a determination of $\pi_4$ of the ...

11

Well, there is always the trivially enforced solution
$$\tag{1} S[x,\lambda]~=~\int\! dt \sum_{i=1}^3\lambda_i(t) \left(\ddot{x}^i(t)+\alpha \dot{x}^i(t) \right),$$
where $\lambda_i(t)$ are three Lagrange multiplier variables. From now on we assume that we are not allowed to use other variables than $x^i(t)$.
Whether the 3D ODE $\ddot{\bf x}+\alpha\dot{\bf ...

11

This is an interesting line of research in modern physics, whether Nature at its most fundamental level is described by real or by complex numbers. Volovik and Zubkov have written about this in Emergent Weyl fermions and the origin of $i=\sqrt{-1}$ in quantum mechanics:
Conventional quantum mechanics is described in terms of complex
numbers. However, ...

answered Mar 23 '18 at 11:03

Carlo Beenakker

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11

I would recommend this 1999 version by Cohen & Whitman:
The Principia: The Authoritative Translation and Guide: Mathematical Principles of Natural Philosophy
This edition has comments that help you understand the original. It is also quite cheap.

answered Mar 28 '18 at 17:31

Piotr Hajlasz

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11

Perhaps to resolve this issue it helps to work out a simple example.
Take a region $D$ consisting of the strip $|x|<1$, $0<y<1$, and a $y$-independent conductivity profile
$$\sigma(x)=\begin{cases}
1 &\text{for} -1<x<0,\\
2& \text{for}\;\;\;\; 0<x<1.
\end{cases}
$$
The solution of the Poisson equation $\nabla \cdot (\sigma\nabla ...

answered Nov 1 '18 at 19:07

Carlo Beenakker

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