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gave wavelength a bare minimum of content
needed to be able to point to wave vector, so I created a bare minimum entry
needed to be able to point to plane wave, so I created a bare minimum entry
the entry valuation would deserve more clarification on that issue alluded to under “Sometimes one also…” and where the min-style definition appears the max-style definition should also appear.
The entry should say that at least with some qualification added, then a valued field is a normed field with multiplicative norm. – Or should it be semi-normed?
I could fiddle with it, but I feel I don’t quite get why the terminology here is so non-uniform that I am afraid I am missing something and maybe a more expert person should help.
In Scholze 11, remark 2.3 is a useful comment:
The term valuation is somewhat unfortunate: If Γ=ℝ≥0, then this would usually be called a seminorm, and the term valuation would be used for (a constant multiple of) the map x↦−log|x|. On the other hand, the term higher-rank norm is much less commonly used than the term higher-rank valuation.
gave rapidly decreasing function its own entry, for completeness
I gave Parseval”s theorem a stub entry (in the course of bringing in substance into the entry Fourier transform)
I gave Fourier inversion theorem a stub entry (in the course of bringing in substance into the entry Fourier transform)
Intending to write something on Fourier transforms of products of functions, I realized that “convolution product” used to redirect to group algebra, where only the discrete version of the concept was mentioned, and only hidden in a remark somewhere.
As a first step, I have removed the redirect now and created with convolution product a disambiguation entry that indicates the general idea and mentions some classes of examples.
Could be expanded much further.
since in other entries I keep having the need to directly point to it, I created a brief entry non-singular distribution.
The only proposition there currently is (here) the density of the embedding C∞cp(ℝn)↪𝒟′(ℝn). I should find a canonical citation that also C∞(ℝn)↪ℰ′(ℝn) is dense.
Just a brief definition to satisfy links: meromorphic function.
All our entries which wanted to point to something like dualizing module (such as at Verdier duality) or the more general concept in a closed monoidal category (such as at star-autonomous category) used to point to the entry dualizable object, which however did not really discuss this specific concept of “dualizable object”.
Therefore I have now created dualizing object in a closed category and made these entries point to that, instead.
Mentioned that in homological algebra/stable homotopy theory one usually puts additional finiteness conditions on the would-be dualizing object and added a brief remark on Anderson duality as a fundamental example.
I’m a little mystified as to why Chenchang Zhu is putting up her course Winter 2017-2018 seminar on higher structures (Higher Lie Groupoids) on nLab. There isn’t any linkage to the standard entries. A previous version is almost empty Winter 2016-2017 seminar on higher structures, likewise seminar on Grothendieck-Teichmuller theory. lectures on higher bundle theory has a list of links at least.
I have added to the entries
warning that there is also the respective other author, in the same field.
We had attributions mixed up in at least one case, namely Whitehead theorem used to be listed at George Whitehead instead of at J. H. C. Whitehead. (May fault, probably. I have fixed it now.) Somebody might want to check for further inconsistencies.
Also I made John Whitehead another redirect to J. H. C. Whitehead.
I have added to Saunders MacLane and to classical mechanics pointer to
This made me go and harmonize the formatting at Saunders MacLane – Writings a little.
I note that there is a page on codiscrete groupoid and a page on indiscrete category. As they were they did not link to each other. I have added some links, but perhaps some additional remarks on the terminology would be useful.
I started a page on a recent preprint, categorified Dold-Kan correspondence.
while I was adding more references and pointers to KR-theory I have created a brief stub for real algebraic K-theory, just to record the (still unpublished…?) references
gave Schwartz kernel theorem some minimal content
I have split off a simple entry sphere bundle from spherical fibration in order to allow more precise linking (for instance from Thom space and from wave front set): “sphere bundle” is about fiber bundles whose fibers are isomorphic to spheres, while “spherical fibrations” is about those whose fibers have the (stable) homotopy type of spheres.
Often when typing “sub-anything” into some nLab entry, I hesitate, wondering if this should come with a hyperlink. Maybe in general this is overkill, but right now, after creating unit sphere bundle, I felt like creating a simple entry sub-bundle, just for completeness.
Made a start at coordination. I’m unsure whether it’s worth spending too long on the intricate accounts of Schlick and Reichenbach, and then of whose makes best sense of Einstein’s proposals. Then there’s plenty of recent literature on the subject.
For me, it would probably only be worth expanding if we could thrash out an account of what the nPOV has to say on the subject. Urs has suggested we look at Bohrification. That sounds like the best lead. Reading through the Bohr topos entry, however, it seemed to me that little is said there about how to integrate that with other parts of the synthetic QFT story. There’s the idea of the ’fifth axiom’, but shouldn’t we expect these quantum phase spaces to have appeared earlier as part of the quantization process. Or do we see it merely as way to interpret our way back from the weird quantum world to something as classical as possible so as to be able to relate theory to the recordings of our classical instruments?
Created linear bicategory.
Created local colimit.
I have started an entry on shuffles. It is meant to be an ’elementary introduction’ so there will be room for deeper exploration of them in follow-on entries.
I wrote the article distribution. I'm by no means an expert though. I left open a section "Applications" in case someone would like to add some, or if not I'll try to fill this in soon.
Created a stub tangent bundle categories as a link target to be disambiguated from tangent categories (with a hatnote at the latter). What I’m calling “tangent bundle categories” here are usually called just “tangent categories”, but that clashes with our page tangent category, so I invented a variation. Better suggestions are welcome.
New stub order category (redirecting also the more general case of preorder category).
just for completeness, some terminology explanation at auxiliary field.
Started bornological set. Some people call it a bornological space, but that conflicts with the terminology in functional analysis which refers to a locally convex TVS with a suitable “bounded = continuous” property. I quickly wrote that uniformly continuous maps between metric spaces induce bounded maps, but I’ll recheck when I have a free moment.
Gave Nakanishi-Lautrup field a brief entry, and touched the corresponding entry antighost field.
Presently the actual technical content that should go into this entry is this example at A first idea of quantum field theory. But before I paste that into the entry, I will expand and polish it a bit more tomorrow, when I am more awake.
Needed to be able to point to contractible chain complex and discovered that we didn’t have an entry for that, so I quickly created one.
I gave Koszul complex and Idea-section and stated two key Properties in citable form (but without proof), one of them the statement that a sufficient condition for the Koszul complex to be a resolution of R/(x1,⋯,xn) is that R is Noetherian, the xi are in the Jacobson radical, and the cohomology in degree -1 vanishes.
Finally I stated the special case of this (here) where R is a formal power series algebra over a field and the elements xi are formal power series with vanishing constant term.
(I have added the relevant facts as citable numbered examples at Noetherian ring and at Jacobson radical.)
This happens to be the case that one need in BV-formalism in field theory. I am writing this out now at A first idea of quantum field theory (here).
balanced monoidal category used to redirect to twist, which started off with a length disambiguation list before moving on to talk about balanced monoidal categories. I thought this was ugly and unhelpful, so I renamed this page to balanced monoidal category and made twist a pure disambiguation page.
I tried to impose some uniformity on the pages about various different kinds of monoidal 2-categories, listing them all at the top of monoidal 2-category and making their Idea sections all parallel. I also added some discussion of the microcosm principle (defining pseudomonoids in monoidal 2-categories of various sorts) to monoidal 2-category as well as compact closed 2-category and microcosm principle.
I tried over category and found the arrows (and a lot else) were not showing up. (On Firefox 56.0, and on Chrome and Safari)
This is after discovering that the accents on Myles Tierney were not appearing. Strangely they do appear at Alain Prouté, (perhaps they are a different font????)
I had occation to link to “formally integrable PDE” from somewhere, and so I created a stub entry just so that the link works. Also created a stub for integrable PDE and cross-linked with integrable system.
Arnold Neumaier contacted me about the previous stubbiness of the entry resurgence and pointed me to his PO comment on the topic (here). I have put that into the Idea-section of the entry, equipped with some links.
Started something at local BRST complex.
on my personal web I am starting a page derived critical locus (schreiber) with some notes.
I think so far I can convince myself of the claim that the page currently ends with (without proof). My next goal is to show that the homotopy fibers discussed there are given by BV-BRST complexes. But I have to interrupt now.
Since somebody put some nonsense at floor (as reported in Forum discussion), I put some content there.
in the course of writing those notes on A first idea of quantum field theory I occasionally find the need to reference more field theory jargon. In this vein I just created a simple entry “relativistic quantum field theory” just with some highlighting of terminology.
(Similarly I had also created locally variational quantum field theory a while back, but did not find the leisure yet to give it substantial content.)
Finally I gave Bogoliubov’s formula its own entry (with material taken from S-matrix)
I have seen filtered object also used in settings where there may not be an ∅ (or initial object) and where the filtration can be doubly infinite. In the current entry both seem not to be allowed. Is this deliberate with another term being used for the more general concept?
New entry (improvized, check co- etc.) resolution.
I couldn’t find an existing discussion for this page – I hope I’m not duplicating such an existing discussion.
I just added a bit to the introduction to clarify something that I was confused about – a cartesian bicategory basically abstracts the properties of V-Prof, but only for cartesian V. To me, this feels like an interesting intermediate position between abstracting the properties of V-Prof for general V, and abstracting the properties of a bicategory like Cat.
Of course, feel free to correct, rework, or roll back entirely my additions! Todd in particular has clearly put a lot of work into this page already.
stub for free loop space object
this is what should be mentioned in that section I added to smooth loop space, instead of the based loop space object notion. Maybe later.
I had need to point specifically to ideals in Lie algebras, so I gave them a little entry Lie ideal.
have created an entry for the new book: Higher Algebra
tried to polish and improve the floating TOC
The entry higher algebra itself I am not happy with (well, it’s mostly just a link list). But no time for that right now.
I have created an entry transgression of differential forms that discusses the concept using the topos of smooth sets. Apart from the traditional definition as τΣ≔∫Σev* the entry considers the formulation as
τΣ=∫Σ[Σ,−]which simply forms the internal hom into the classifying map X→Ωn of a differential form. I have spelled out the proof that the two definitions are equivalent.
Then the entry contains statement and proof of the situation of “relative” transgression over manifolds with boundary. (This is what yields, when applied to Lepage forms, Lagrangian correspondences between the phase spaces with respect to different Cauchy surfaces, which is what I currently need this material for in the exposition at A first idea of quantum fields.)
Finally there are two examples, a simplistic one and an simple but interesting one related to Chern-Simons theory. These two examples I had kept for a long time already at geometry of physics – integration in the section “Transgression”. That section I have now expanded accordingly, its content now coincides with the entry transgression of differential forms.
gave retarded product its own entry, with material split off from S-matrix.
This entry is to be expanded further. A related stub is now at quantum observable on interacting fields, also to be expanded.
I also created a table-for-inclusion-into-related-entries products in pQFT – table and included it into related entries
I have added in some additional comments plus some references. In particular there is a paper by Barbara Osofsky (The subscript of ℵn projective dimension, and the vanishing of lim(n)_, Bull. Amer. Math. Soc. Volume 80, Number 1 (1974), 8-26.) which gives a neat discussion.
I have added to causal complement the actual definition of causal complements of subsets of Lorentzian manifolds.
The entry used to contain only a more abstract concept, now kept as the second subsection of the Definition section here.
It is a classical fact that a formal deformation quantization of a Lie-Poisson structure is provided by the universal enveloping algebra of the corresponding Lie algebra. Remarkably, this statement generalizes to some extent to more general (polynomial) Poisson algebras. In particular it holds for every such up to degree three in ℏ ! This is due to Penkava-Vanhaecke 00.
I have added a quick summary of this theorem to deformation quantization in a new subsection: Existence – Deformation by universal enveloping algebras. I also gave this an entry on its own at polynomial Poisson algebra.
This is maybe remarkable, since there is possibly no physical measurement known which could detect contributions of higher than third order in ℏ. Though I’d need to check. This is subtle because order in ℏ is different from the usual loop order that is commonly stated (which is order in the coupling constant) and the relation between the two is complicated.)
Also (and that’s how I came across this article) at least in special cases this gives a way to quantize just by universal constructions on Lie algebras, hence this might potentially tell us something about the quantization of Poisson bracket Lie n-algebras (for which no analog of the corresponding Poisson algebra, i.e. with an associative product around, is known).
expansion, a stub containing just the basic definition of an expansion in shape theory.
I have started an entry on the Mittag-Leffler condition.