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Only a monster would have made such a monster.
DennisN
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Do you have a quote? I know Cayley as ##G \leq \operatorname{Sym}_{|G|}## in the finite case.nrqed said:Cayley's theorem, which says that any group is isomorphic to a certain subgroup of that group's group of isomorphisms.
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https://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Monoidsfresh_42 said:Do you have a quote? I know Cayley as ##G \leq \operatorname{Sym}_{|G|}## in the finite case.
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Thanks, and yes, that's the proof, but I don't see that ##f_a## are isomorphisms, only bijections! E.g. ##f_a(1)=a \stackrel{i.g.}{\neq}1##. That was my problem: permutations are no homomorphisms. We have ##G \longrightarrow \operatorname{Sym}(G)## not ##\operatorname{Aut}(G)##. If we restricted ourselves to conjugations, we would get all central elements in the kernel, so that doesn't work. Where are the automorphisms?nrqed said:
All we have are automorphisms in the category ##\mathcal{Set}## but this is a very misleading wording in this context, and to say "its isomorphisms" is wrong - at least as far as I know.
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DennisN
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That explains global warming, too!
Anthropomorphic global warming.
Anthropomorphic global warming.
DennisN
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Don't forget the paper! We were supposed to save paper!DennisN said:
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Actually, I think that we have reached a point of diminishing paper. I basically no longer get any paper invoices and I read the newspapers online. Looking at the students, more and more are buying the textbooks as e-books and many assignments go directly from their computer to mine with no wood-derived product in-between.fresh_42 said:Don't forget the paper! We were supposed to save paper!
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All diagrams I can find on the internet are ##O(t)## since 1950. I haven't found a worldwide chart and the ones I've found ended in 2005 or so, which is why I don't post them. Yes, some paper traffic has gone electric, but the vast majority of papers are the n-th copy of a meeting protocol, the 10th-copy of a joke going around in emails, or the private copies of anything. Paper is still cheap and everybody is a Gutenberg. A few online invoices don't change the trend.DrClaude said:Actually, I think that we have reached a point of diminishing paper.
Keith_McClary
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https://www.statista.com/statistics/270317/production-volume-of-paper-by-type/fresh_42 said:I haven't found a worldwide chart and the ones I've found ended in 2005
Definition:
Graphic papers products are all around you and can be found in almost any home, office or business. These include a very wide variety of paper products for uses such as newspapers, inserts, supplements, A4 copy paper and printer paper, book paper, envelopes, forms, writing pads, high-quality magazines and brochures, company magazines, catalogues and annual reports.
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DennisN
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@Greg Bernhardt , when will PF/Latex support hieroglyphs?fresh_42 said:
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Yeah!
And Mayan codex too!
And Mayan codex too!
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Haven't you noticed? It already does:DennisN said:@Greg Bernhardt , when will PF/Latex support hieroglyphs?
I think it says: 'Ptolemy was here', but Ptah only knows.DarMM said:$$H(g,\kappa) = H_{\phi,\kappa} + H_{\psi,\kappa} + g\colon\bar{\psi}\phi\psi\colon - \delta m^2(g,\kappa) - E(g,\kappa)$$
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DennisN said:@Greg Bernhardt , when will PF/Latex support hieroglyphs?
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DennisN
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BillTre said:Yeah!
And Mayan codex too!
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Keith_McClary
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256bits
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That's funny.
He better be back before his supper gets cold, or if he's lucky maybe she will keep it warm in the oven.
He better be back before his supper gets cold, or if he's lucky maybe she will keep it warm in the oven.
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Oh, humour. That elusive beast.
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Keith_McClary
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In case this is not in your Physics curriculum:Keith_McClary said:I’m Going on a Random Walk
Section 3.1 (and the referenced Chapters 1-2).Once again, however, our focus is on simple diffusion or the nearest-neighbor random walk. For these processes, the possibility of a diffusing particle making arbitrarily large excursions before certain trapping takes place leads to an infinite mean lifetime. On the other hand, the recurrence of diffusion in one dimension means that the particle must eventually return to its starting point. This dichotomy between infinite lifetime and certain trapping leads to a variety of extremely surprising first-passage-related properties both for the semi-infinite interval and the infinite system.
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