I was just alerted (here) that things didn’t render properly at *Science of Logic*: most hyperlinked words didn’t show up at all (not even the link text). I made a trivial edit and resubmitted, now things seem to be back to normal (also in the history, rev 261 it is back to normal, so I cannot point to a page that still has the problem).

But then I noticed that the link to *∞-representation* remains gray, while the entry *infinity-representation* does exist, and should be redirecting.

Looking at that entry, it, too had the problem that hyperlinked words didn’t display! I resaved, and it works now. But maybe this means that some/all entries need another re-rendering?

And the following issue remains: *∞-representation* still does not redirect. I copy-and-pasted it into the redirect, to be sure that there is no funny unicode ambiguity in the background, but that didn’t help here.

Plenty of people were mentioning lenses at SYCO 1. Can’t say I gained much of a feel for them. De Paiva’s Dialectica interpretation was mentioned in this respect, and apparently Mitchell Riley’s Categories of Optics is the place to go for a broader account.

Curious how many young researchers reference the nLab, but shy away from describing their work there.

]]>Corrected an arithmetic error in the last section.

]]>Now there is Sylow p-subgroup.

Is there a compilation, somewhere, of the results “the (obvious) automorphisms of a small $\mathfrak{A}$ $A$ are transitive on $A$’s maximal $\mathfrak{B}$s?” The only other example ready in my head is that the maximal tori in a compact Lie group are conjugate, but I know I’ve seen more.

]]>New article class equation, just to fill some gaps in the nLab literature. Truly elementary stuff.

]]>However diff page(https://ncatlab.org/nlab/show/diff/category+with+weak+equivalences) and history page is fine.

My browser:

User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:62.0) Gecko/20100101 Firefox/62.0 ]]>

added a subsection “Properties – As algebraic K-theory over the field with one element” (here)

]]>Created page, mainly to record a bunch of references that I am trying to collect. Additional suggested references would be welcome!

]]>maded explicit the identification of equivariant stable homotopy groups with equivariant generalized cohomology groups of the point: here

]]>starting some minimum

]]>Fixed a typo, but also I noted the last but one link is dead. Does any one know if this has moved somewhere?

(Edit: I found it. There is a link from his home page. I have updated the link.)

]]>added to *inter-universal Teichmüller theory* a pointer to the recent note

- Yamashita,
*FAQ on ‘Inter-Universality’*(pdf)

(Though after reading I am not sure if that note helps so much.)

]]>Cycling is not a sort of breakdown that the Simplex algorithm occasionally does, but the sign of the existence of extraordinary solutions that assign positive values to more than $m$ decision-variables, where $m$ is the number of our structural constraints. Simplex, by its very nature, cannot find such solutions but is able to emit a signal of their existence by cycling.

Consider the classical example given by E. M. L. Beale\footnote{The coefficient of $x_{5}$ on the objective function is altered slightly for reasons that will soon be understood.} where $n=7$ and $m=3$ [see Bazaraa et al., p.176]:

$\begin{array}{lr} max.& & & &+\frac{3}{4}x_{4} & -15x_{5} & +\frac{1}{2} x_{6} & - 6x_{7} & RHS \\ ST: &+x_{1} & & &+\frac{1}{4} x_{4} & -8x_{5} & -x_{6} & +9x_{7} & = 0 \\ & &+x_{2} & & +\frac{1}{2} x_{4} & -12x_{5} & -\frac{1}{2} x_{6} & +3x_{7} & = 0 \\ & & &+x_{3} & & & +x_{6} & & = 1 \end{array}$$\qquad \qquad$ and nonnegativity constraints.

As we all know, Simplex algorithm can assign positive values to at most three (basic) decision variables out of seven of this problem. Now let $\eta$ be a positive number. Then $x_{4}=\frac{2}{5}+\frac{112}{5}\eta$, $x_{5}=0+\eta$, $x_{6}=1$ and $x_{7}= \frac{1}{10}+\frac{4}{15} \eta$ is such an extraordinary solution that assigns positive values to four variables! The objective function accordingly assumes the unbounded value $(1+ \eta)/5$. Even if the third scarce resource is not tapped (i.e. $x_{3}=1$), it assumes the value $(0+\eta)/5$ which is again as big as we wish. Hadn’t we modified the objective function coefficient of $x_{5}$, it would come out as $-\infty$ in which case cycling must really be avoided.

This outcome is categorically different from the “unbounded solutions” we are familiar with. Let’s define a “sub-basis” as the set of current basis’ column vectors but the “departing” one. We already know something about such bases. For instance each row of the current basis’ multiplicative inverse (discernible as a specific part of a specific row of the current Tableau) is the outward-normal (gradient) of the subspace spanned by such a sub-basis. In every cycling example given by Gass et al., and notably in the non-trivial one due to Nering and Tucker (p.310), there is a subspace spanned by such a sub-basis and perpendicular to the RHS vector! Moreover, if such a sub-basis can be extended to a basis that spans the associated subspace positively, then, in the case of a Product Mix Problem for instance, the manufacturer has attained some self-sufficiency to produce at least one product whose production is restrained by intangible constraints only. In the Portfolio Management Problem, the investor’s “credibility”.

\textbf{References}

\renewcommand*\labelenumi{[\theenumi]}

\begin{enumerate} \item Bazaraa, M. S., Jarvis, J. J. \& Sherali, H. D., Linear Programming and Network Flows, Copyright \copyright 2010 by John Wiley \& Sons Inc. \item Beale, E. M. L., Cycling in the Dual Simplex Algorithm, Naval Research Logistics Quarterly 2(4), pp.269-276 December 1955 \item Gass, S. I. \& Vinjamuri, S., Cycling in linear programming problems, Computers \& Operations Research 31 (2004) 303-311 \end{enumerate}

]]>I am giving this generalized homology theory its own little entry, so that it becomes possible to refer to it more specifically, beyond broadly pointing to just “stable homotopy groups”.

(Curious that things are set up such that the most fundamental of homology theories is almost un-nameable, since its canonical name clashes with the name of the whole subject. Curious circularity there.

The other day I was visiting the Grand Mosque. It’s qibla wall has a huge mosaique displaying the 99 names of God in 99 flowers, plus one flower with no name it in, to represent the un-nameable (one can see it well here, only that the sheer size of it is not brought across by photographs). )

]]>Page created, but author did not leave any comments.

]]>created placeholder for Hurewicz theorem

]]>added pointer to this new textbook

- Antoine Chambert-Loir, Johannes Nicaise, JulienSebag,
*Motivic integration*, Birkhaeuser 2018

(Somebody should write a paragraph into this entry that gives an actual idea of what motivic integration is about, beyond it being an idea that Kontsevich had.)

]]>I have scratched the rough Idea-section that I had here previously, and started afresh, now with mentioning of the K-theoretic McKay correspondence. Also added references. But it’s still just a stub entry

]]>brief category:people entry for hyperlinking of references at *McKay correspondence* and *fractional D-brane*

Added the second proof of the Rezk completion.

What would the eliminator for this HIT type look like? I cannot find anything in the literature apart from the HoTT book. It seems this was originally crafted up by Mike so perhaps he has an eliminator written down somewhere?

Specifically the third and fourth constructors are giving me a headache. Because there is a higher transport involved, which even for simple spaces like a torus is a bit difficult to wrap your head around.

]]>Provided a bare minimum so as to ungrey the link. Feel free to expand/correct.

]]>trivia:

how to typeset here LaTeX’s `\not\subset`

? Instiki has `\nsubset`

, but on the nLab it comes out with a strangely vertical bar, instead of the usual slanted slash (?) while here on the nForum it comes out completely weird

Added another reference.

I was chatting with Robin Cockett yesterday at SYCO1. In a talk Robin claims to be after

The algebraic/categorical foundations for differential calculus and differential geometry.

It would be good to see how this approach compares with differential cohesive HoTT.

]]>brief category:people entry for hyperlinking references at *motivic integration*

renamed from “geometric fixed points” to “geometric fixed point spectrum”, which is clearly the better/right entry title

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