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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{...


11

In the form (1), if you compute the variation $\delta S / \delta x(t) = E(t)$, you find that $E(t) = E(x(t),\dot{x}(t), \ddot{x}(t) ,t)$ is a local/differential expression (the value of $E(t)$ does not depend on $x(t')$ or its derivatives at other times $t'\ne t$). This is no longer true if you use $\exp(S)$ instead of $S$. There is no dispute that $S$ and $\...


8

The previous answers all give good discussions of the physical significance of $\hbar$ and the classical limit "$\hbar \to 0$" (in large quotation marks), but few of them discuss your "motivation" comment that $\hbar$ rarely appears in the study of quantum computing and information. David Mermin dedicates an entire section of his great ...


6

Based on some of the comments I have seen, some commentators are finding Hawking's arguments vague or unconvincing. These are legitimate criticisms. But I think he has been partially, if not completely vindicated, by two recent advances that were not fully developed at the time the paper was published. The undecidability of the spectral gap problem. ...


5

I interpret a "gadget" as a physical device that operates in an analog, rather than a digital way (to exclude a computer). The OP asks for "primality tests", but if I may broaden the question to include "prime number generators", there is a variety of such gadgets, collected at unusual and physical methods for finding prime numbers. The gadgets use effects ...


4

In a sense, all the Lagrangians giving the same Euler-Lagrange equations are exhausted by transformations of your type (b), which adds a total derivative/total divergence/boundary term/... Transformations of your type (a) can alter the Euler-Lagrange equations, for instance if $a\ne 1$, then the EL equations get rescaled by the same constant $a$. Perhaps ...


4

Not an answer, only an attempt to clarify. I'm guessing this is what is meant by "generalized geometry": "Generalized geometry is based on two premises – the first is to replace the tangent bundle $T$ of a manifold $M$ by $T \oplus T^*$, and the second to replace the Lie bracket on sections of $T$ by the Courant bracket. The idea then is to use one’s ...


3

We have a paper that contains lists of simple fermionic topological orders in 2+1D: https://arxiv.org/abs/1507.04673 . For fermionic topological without symmetry, there is no filling fraction. So our lists are based on the number of anyon types, together with their quantum dimensions and topological spins. Each entry in the table corresponds to a sequence of ...


2

This problem is discussed in Bryant, Griffiths, Hsu, Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, for Lagrangians for scalar fields.


2

Okay, so I think I may have found the answer myself. So, really, the absolute value symbol is a trick. You can get rid of it by pulling out an $i$, and then you have $$\mathcal{I}:=\frac{1}{2\pi i}\int_a^b dx \frac{\log(x e^{-t_1})}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{(x-a)(x-b)}}$$ What you have to do is take a dumbbell contour around taking a clockwise &...


2

I suspect not, at least for regular polygons. (For nonregular polygons, there are orientation issues which I believe won't be solved by rubber bands.) Let me illustrate with squares. Consider 8 squares in a three by three arrangement with a central hole. This configuration by itself does not tile when you apply a rubber band, but if you have a larger ...


2

There are already good answers for Q1-Q3, so I'll clarify Q4. In physics, quantities typically have a dimension, such as length, mass, time etc., and a unit, such as metre, kilogram, second, etc. Pure mathematical numbers have neither a dimension nor a unit. Let's consider time. A possible mathematical model for time is to use a one-dimensional oriented ...


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