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I looked at real number and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:
A real number is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a number field, denoted ℝ. The underlying set is the completion of the ordered field ℚ of rational numbers: the result of adjoining to ℚ suprema for every bounded subset with respect to the natural ordering of rational numbers.
The set of real numbers also carries naturally the structure of a topological space and as such ℝ is called the real line also known as the continuum. Equipped with both the topology and the field structure, ℝ is a topological field and as such is the uniform completion of ℚ equipped with the absolute value metric.
Together with its cartesian products – the Cartesian spaces ℝn for natural numbers n∈ℕ – the real line ℝ is a standard formalization of the idea of continuous space. The more general concept of (smooth) manifold is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see spacetime), or at least in classical physics. See at geometry of physics for more on this.
category: people page for the reference
Anonymouse
added hyperlinks to the text at induced representation. Made sure that it is cross-linked with Frobenius reciprocity.
Stub Frobenius reciprocity.
stub for confinement, but nothing much there yet. Just wanted to record the last references there somewhere.
I added the reference
a stub entry, for the time being just to satisfy a link that has long been requested at Quillen Q-construction
added pointer to:
created black holes in string theory, since somebody asked me: a brief paragraph explaining how the entropy-counting works and some references.
have created a stub for supersymmetric quantum mechanics
Zoran, I see that you once dropped a big query box at quantum mechanics with a complaint. I disagree with the point you make there: we have fundamental definitions of quantum field theory and restricting them to 1 dimension gives quantum mechanics. If you want to turn this around and understand all QFTs as infinite-dimensional quantum mechanics (which, yes, one can do) you are discarding the nice conceptual models and kill the concept of extended QFT.
In any case, I think remarks like this (in the style of “we can also regard this the other way round like this”) are better added into an entry as what they are – remarks – than as query boxes that give the impression that there is something fishy about the rest of the entry.
I came across this tiny page which is called by ’extension’ from the page central extension.
But what’s this page trying to be? Merely about a certain kind of field extension?
Created:
[…]
Introduced by
Survey:
starting something, for the moment just to record the basic characterization by Fadell & VanBuskirk 1961
Have added to HowTo a description for how to label equations
In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.
stub for jet bundle
was initially just looking for a page to host this reference:
but now I wrote a little bit of an Idea-section, too. Leaving much room to be further expanded, of course.
made some cosmetic adjustments to the entry,
and slightly expanded the last remark (which I made a Remark) relating to order theory (prodded by this comment)
Added a budget of links from the FAQ to the Eunuch-Code Data-Bank.
For some odd reason, the TOC generator is not picking up the first heading — ???
I added a synthetic definition of open subspace due to Penon.
Created:
An intrinsic notion of an open subobject in an elementary topos.
A monomorphism U→X in an elementary topos E is a Penon open if the following statement holds in the internal logic of E:
∀x∈X∀y∈X(x∈U)→(¬(y=x)∨y∈U).If U→X is a Penon open, then
∀x∈U({y∈X∣¬¬(y=x)}⊂U).Jacques Penon, De l’infinitésimal au local (Thèse de Doctorat d’État), Diagrammes S13 (1985), 1-191. numdam.
Eduardo J. Dubuc, Jacques Penon, Objets compacts dans les topos, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40:2 (1986), 203-217. doi.
Jacques Penon, Infinitésimaux et intuitionnisme, Cahiers de topologie et géométrie différentielle 22:1 (1981), 67-72. numdam.
Oscar P. Bruno, Logical opens of exponential objects, Cahiers de Topologie et Géométrie Différentielle Catégoriques 26:3 (1985), 311-323.
Marta C. Bunge, Felipe Gago, Ana María San Luis, Synthetic Differential Topology, Cambridge University Press, 2018. ISBN: 9781108553490, DOI.
Numerous bibliographic additions to linguistics and new stub mathematical linguistics.
added pointer to:
came to fix the typesetting glitches of revision 2, ended up re-typing the paragraph from scratch.
created a brief entry cohomological field theory and cross-linked a good bit.
added these pointers:
Dam Thanh Son, Mikhail Stephanov: Relativistic Guiding-Center Motion: Action Principle, Kinetic Theory, and Hydrodynamics, Phys. Rev. Lett. 133 (2024) 145201 [arXiv:2405.08073, doi:10.1103/PhysRevLett.133.145201]
Dam Thanh Son: Lorentz-covariant description of relativistic guiding-center motion, talk at Simons Center Physics Seminar (2025-02-12) [webpage]
(which may ultimately want to go to a more specialized entry, but for the moment I see no better place than to have them here)
a bare minimum, for the moment just to make links work (such as at probability amplitude and at Wigner’s theorem)
this is a bare list of references, meant to be !include-ed into the relevant References-sections at black hole information paradox and Bekenstein-Hawkind entropy
Zoran created monadic descent
I added to higher homotopy van Kampen theorem the statement of the theorem by Jacob Lurie.
Created.
Removed a query:
+– {: .query} Bruce: I’m shooting in the dark here with this ω-groupoid sentence above. Am I right? What does that boil down to concretely? =–
+– {: .query}
Bruce: What’s a geometric stack?
Chris: A stack is geometric if it is quasi-compact (any open cover has a finite sub-cover) and the diagonal morphism is representable and affine, though that probably doesn’t help much. I don’t know much about stacks yet, but maybe someone else can explain this. I think the point is that one needs some hypotheses to actually prove stuff for stacks.
=–
Created homotopy equivalence of toposes.
At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group π2(X,A) together with the bundary map δ:π2(X,A)→π1(A) and the π1(A)-action on π2(X,A). As someone confirms this example is correct I’ll add it to crossed module.
Is this page a duplicate of Jiří Vanžura?