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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeOct 11th 2011
• (edited Oct 11th 2011)

Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the permutation representation in the standard representation theory books like in

• Claudio Procesi, Lie groups, an approach through invariants and representations, Universitext, Springer 2006, gBooks

I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.

Edit: new (related) entries for Claudio Procesi and Arun Ram.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeOct 12th 2011

FWIW, I would have assumed “permutation representation” to mean in the standard representation-theory sense of vector spaces.

• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeOct 12th 2011
Is this not the usual presumption of a relevant cat?
A group in cat can act on an object in the cat
hence the group has a representation via automorphisms of that object
Is not a LINEAR representation one where the object is a vector space?
• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeOct 12th 2011

I’d be inclined to call the current topic of the page a representation by permutations. I note here that $C$ should be a groupoid; otherwise “permutation” is completely out.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeOct 12th 2011

Right I do call it “representation by permutations” (see 1), while the permutation representation in the rep. theory sense entails the linear extension of the action from a vector space basis to the entire linear (=vector) space.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeOct 12th 2011

Zoran #5: I am in agreement with this terminology. When I talk privately with certain category theorists (e.g., Jim Dolan), sometimes we’ll refer to a permutation representation when we mean a representation in $Set$ (which latter by the way is also acceptable), but this is not widely recognized outside a certain community.

Jim #3: true, but the rest of the world has apparently not caught up yet! When people say ’representation’ without qualifications, the default assumption is that it’s a linear representation, and a permutation representation is generally taken to mean a composite functor

$B G \to Set \stackrel{free}{\to} Vect_k.$
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 12th 2011

It makes sense to say “permutation representation” for $\cdots \to \mathrm{Set}$ and “linear permutation representation” for $\cdots \to \mathrm{Set} \to \mathrm{Vect}$.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeOct 12th 2011

Sure, it makes sense to say that, but if you only said “permutation representation” by itself to the mathematician-in-the-street, he is likely to interpret that as what you just called “linear permutation representation”. I thought that’s what this discussion was about: what the conventional standard is (as opposed to what the convention should be), as reported in Zoran’s #1.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 12th 2011

I thought that’s what this discussion was about: what the conventional standard is

Okay. I thought the discussion is about what to write in the $n$Lab entry.

Of course we should always explain different use of terminology, for transparency. But I don’t think that we have to enslave ourselves to conventional standards if they should be different, as in this case.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 12th 2011

I have added a word to the entry.

• CommentRowNumber11.
• CommentAuthorTobyBartels
• CommentTimeOct 12th 2011

One difference between a representation in $Set$ and a permutation representation in $Vect$ is the morphisms (the intertwiners). Even the isomorphisms differ, so we can’t pretend that they’re the same thing.

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeOct 13th 2011

they should be different, as in this case

Clear “representation by permutations” for categorical notion and cryptic “permutation representation” for linear notion are different.

• CommentRowNumber13.
• CommentAuthorTim_Porter
• CommentTimeOct 13th 2011

I should point out that people who work on representations of symmetric groups and applications in combinatorics refer to permutation representations although there is often not a linear representation in sight. This continues the tradition of D.E. Littlewood. (You will see how I know this if you look at http://en.wikipedia.org/wiki/Dudley_E._Littlewood).

• CommentRowNumber14.
• CommentAuthorTobyBartels
• CommentTimeOct 14th 2011
• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeSep 11th 2018
• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeSep 12th 2018

while we are at it, I (re-)wrote this entry to contain a more decent account of the plain basics

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeSep 12th 2018

starting a section Examples – Virtual permutation representations. But we happen to be discussing this not here in this thread, but in the thread on the Burnside ring, see there

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeSep 12th 2018
• (edited Sep 12th 2018)

In the proof of the proposition about $\beta$ being surjective for some classes of groups (this prop.), I have expanded out the argument for cyclic groups, as a corollary of that for $p$-primary groups. Like so:

To see surjectivity for cyclic groups: By the previous statement we have surjectivity already for those cyclic groups whose order is a prime power. But by the fundamental theorem of cyclic groups, every cyclic group is a direct product group of cyclic groups of prime power order. Moreover, every irreducible representation of a direct product group is an external tensor product of irreps of the group factors (this prop.). But $\beta$ sends “external Cartesian products” to external tensor products, by the same elementary argument which shows that $\beta$ sends plain Cartesian products to tensor products. This way the statement reduces to that for $p$-primary cyclic groups.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeSep 13th 2018
• (edited Sep 13th 2018)

made explicit the simplest non-trivial example: $A(\mathbb{Z}/2) \underoverset{\simeq}{\beta}{\longrightarrow} R(\mathbb{Z}/2)$, here

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeSep 13th 2018
• (edited Sep 13th 2018)

added also pointer to Todd’s and James Montaldi’s example of $\beta$ for $G = S_4$ here.

Is this still a surjection for rational reps?

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeSep 13th 2018
• (edited Sep 13th 2018)

Hm, maybe my argument for the cyclic groups (#18) is wrong after all:

That the irrep of a direct product group is a tensor product of irreps, is that true also over $\mathbb{Q}$?

[ edit: hm, should be okay, as in [prop. 2.3.17 here]

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeSep 15th 2018
• (edited Sep 15th 2018)

removed the statement of surjectivity of $\beta$ for cyclic groups, since my argument had a flaw, based on a flaw in the earlier version of the lecture note mentioned above (see here). It might still be true, but I don’t know.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeSep 25th 2018

Ah, the surjectivity of beta for cyclic groups over the rationals is example 4.4.4 in tom Dieck’s notes http://www.uni-math.gwdg.de/tammo/rep.pdf

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeOct 5th 2018

added pointer to Dress 86, section 3 for the statement that $\beta$ is surjective for symmetric groups in characteristic zero.

• CommentRowNumber25.
• CommentAuthorTodd_Trimble
• CommentTimeOct 5th 2018

Re #24: well, that is indeed wonderful since I didn’t know where to find it in the literature. Now I do.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeOct 5th 2018
• (edited Oct 5th 2018)

It’s a citation, but still not a proof, since it just says

it follows easily from the classical representation theory of $\Sigma_n$

What is indeed easy to see is that in the discussion of Specht modules all the ingredients enter that one expects to see for this to be true. But if it’s so easy, what is the explicit general formula that expresses the Specht module corresponding to a given Young tabloid in terms of virtual permutation reps?

• CommentRowNumber27.
• CommentAuthorDavidRoberts
• CommentTimeOct 7th 2018

I might ask on the group-pub mailing list (which, for some weird reason, is private) to see if this is treated explicitly anywhere.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeOct 7th 2018

Didn’t know of this group. If you can, please do!

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeOct 8th 2018

So David kindly checked on that group theory list. In reply there is so far a sketch of the idea of the proof:

if you look in the representation theory of $S_n$, the situation is the following:

you have an ordering on the set of partitions of $n$, which is, if I am not mistaken, the lexicographical ordering.

Then for every partition $\lambda$, you define $M_\lambda = Ind_{S_{\lambda}}^S_n 1$.

Inside $M_{\lambda}$ you find a direct sum of a new irreducible module $V_{\lambda}$ and modules of the form $V_{\mu}$, for $\mu \lt \lambda$.

This already shows that if you consider the representing matrix of the above transformation, you get an upper diagonal matrix, with 1s on the diagonal.

(In $A(S_n)$ I have restricted here to subgroups of the form $S_{\lambda} = \prod_{i=1}^r S_{\lambda_i}$)

and, upon further request for a citable reference, a pointer to Sagan 01 (but it seems that’s meant as a general pointer to the rep theory of the symmetric group, not to a citable proof of the statement in question).

• CommentRowNumber30.
• CommentAuthorTodd_Trimble
• CommentTimeOct 8th 2018

Okay, thanks Urs and David – I’ll check through that later. One of those things that is absolutely routine to experts, it seems.

• CommentRowNumber31.
• CommentAuthorTodd_Trimble
• CommentTimeOct 8th 2018

Inside $M_{\lambda}$ you find a direct sum of a new irreducible module $V_{\lambda}$ and modules of the form $V_{\mu}$, for $\mu \lt \lambda$

looks pretty hand-wavy: about as hand-wavy as the Gram-Schmidt page. But maybe it will become clearer in time. If I get desperate, maybe I’ll put it to MO.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeOct 9th 2018
• (edited Oct 9th 2018)

Thanks, Todd, for looking into this.

(The situation reminds me of what Langlands said (somewhere here) about specialists hurting their field by being so specialist.)

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeOct 9th 2018
• (edited Oct 9th 2018)

Here is a first list of our results (using computer algebra by Simon Burton) on cokernels of $A(G) \overset{\beta}{\to} R^{(fin)}_k(G)$ for nonabelian finite subgroups of $SU(2)$: jpg.

In all cases, beta is surjective over $\mathbb{R}$ onto the ring of integer-valued (i.e. not irrational-valued) characters.

• CommentRowNumber34.
• CommentAuthorDavidRoberts
• CommentTimeOct 9th 2018

Interesting how many items in that right-most column are essentially $\mathbb{Z}[\rho]/\mathbb{Z}[2\rho]$. Is that due to the central extensions by $\mathbb{Z}/2$?

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeOct 10th 2018

Don’t know if this is related to any central extension. Here one such contribution appears for every complex irrep with integer characters but of quaternionic/symplectic type.

I’ll show the details of the computation later when the file has taken shape a bit more.

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeOct 10th 2018

For what it’s worth, here is an overview of the physics picture behind that result in #33: jpg

• CommentRowNumber37.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2018

If we see group characters appear in for fractional D-branes, should we not expect marks to play a role in M-theory?

Much like character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring. (Wikipedia)

According to Pfeiffer here:

The table of marks of $G$ arises from a characterization of the permutation representations of $G$ by certain numbers of fixed points. It provides a compact description of the subgroup lattice of $G$ and enables explicit calculations in the Burnside ring of $G$.

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeOct 11th 2018
• (edited Oct 11th 2018)

Plausibly, but I remain uncertain what exactly there is to be said.

A special kind of mark, namely that of cyclic groups $C_g$ generated by elements $g \in G$, already does play a central role, because that is equivalently just the character of a permutation representation $k[G/H]$ (for any ground field $k$):

$m(H, C_g ) \;=\; \chi_{k[G/H]}(g) \,.$
• CommentRowNumber39.
• CommentAuthorUrs
• CommentTimeDec 20th 2018
• (edited Dec 20th 2018)

Here is finally the writeup of the result mentioned in #33 (no changes since then, really, but now we decided to just wrap it up):

Abstract We describe an efficient algorithm that computes, for any finite group $G$, the linear span of its virtual permutation representations inside all its linear representations, hence the image of the canonical morphism $A(G) \overset{\beta}{\longrightarrow} R_k(G)$ from the Burnside ring to the representation ring. We use this to determine the image and cokernel of $\beta$ for binary Platonic groups, hence for finite subgroups of $SU(2)$, over $k \in \{\mathbb{Q}, \mathbb{R}, \mathbb{C}\}$. We find explicitly that for the three exceptional subgroups (2T,2I, 2O) and for the first seven binary dihedral groups, $\beta$ surjects onto the sub-lattice $R^{\mathrm{int}}_{\mathbb{R}}(G)$ of the real representation ring spanned by the integer-valued characters. We conjecture that, generally, this holds true for all the binary dihedral groups.

• CommentRowNumber40.
• CommentAuthorDavid_Corfield
• CommentTimeDec 20th 2018

What was the scope of the research that Todd and James Dolan were talking about? Over the years, there has been talk of many large projects involving James and others, with mention of buildings, groupoidification, doctrines. Is it possible to give a sense of the ambitions of this work?

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeDec 22nd 2018
• (edited Dec 22nd 2018)

Just in case the question is aimed at me, I’ll say that I have no idea what James Dolan has been talking about with anyone, as I was never involved.

The acknowledgement in our writeup from comment #39 above – if that’s what prompted your question in #40 – is just saying that the algorithm we discuss was inspired by Todd’s note from 2009 here, which in its first sentence acknowledges, in turn, a suggestion by James Dolan.

Just saying, in case the question was aimed at me, written seemingly in reply to my announcement in #39.

• CommentRowNumber42.
• CommentAuthorTodd_Trimble
• CommentTimeDec 22nd 2018

David, can you give the referent for “research that Todd and James Dolan were talking about”?

• CommentRowNumber43.
• CommentAuthorDavid_Corfield
• CommentTimeDec 22nd 2018

I never began to get the scope of whatever it was Jim et al. were driving at in that period. I take it that whatever it was, it wasn’t exhausted by the Groupoidification series, or the seminars, Fall 2007 and Winter 2008, and these ranged far and wide over Geometric Representation Theory. I think the Doctrines project was part of it.

It was clearly meant to be much more that the provider of some calculational tool for virtual permutation representations. Urs wrote

For the longest time the “groupoidification” program looked like idle entertainment to me, but now that I understand the importance for fundamental physics of computing the image of equivariant stable cohomotopy in equivariant complex K-theory, I have a different feeling for it.

There were surely greater expectations. Since nobody works on it now, it would good to have some record of what the ambitions for this project were. Somebody might want to pick up the pieces one day.

• CommentRowNumber44.
• CommentAuthorDavid_Corfield
• CommentTimeDec 22nd 2018

I thought, Todd, you had a good idea where Jim was heading, as indicated by your writing

a whole slew of interesting developments, in which we view Jim’s orbi-simplex idea as a geometric description of a general axiomatic theory, which in turn is related to the idea of viewing Tits buildings as “quantized” axiomatic theories, and also perhaps to the theory of classifying toposes and their “Galois theory”. But we’ll get to all that later!

• CommentRowNumber45.
• CommentAuthorTodd_Trimble
• CommentTimeDec 22nd 2018

Thanks, David. Let me get back to you on this. I just didn’t know where you were looking (as in, where were Jim and I talking about this).

I just wrote Jim an email today. I’m hoping he can help resurrect some memories, because a lot of what we were discussing more than 10 years ago was never properly written down and begins to fade from my memory, but it shouldn’t be impossible for us to piece it together, and then I can more easily report back here. There were some very interesting “groupoidified” quantum group calculations which I’d really like to retrieve, for example.

The categorical treatment of Tits buildings I can recall large parts of, but not necessarily the meta-thoughts Jim had in mind more programmatically.

• CommentRowNumber46.
• CommentAuthorUrs
• CommentTimeDec 22nd 2018
• (edited Dec 22nd 2018)

$[$ Groupoidification$]$ was clearly meant to be much more that the provider of some calculational tool for virtual permutation representations.

I didn’t see anything about “groupoidification” being a calculational tool for virtual permutation representations in the first place. Do you have a pointer?

• CommentRowNumber47.
• CommentAuthorDavid_Corfield
• CommentTimeDec 23rd 2018

Re #46, I didn’t mean that it was explicitly stated somewhere that being a calculational tool was an intention, though they certainly wanted to see how far they could get using combinatorics rather than linear algebra over vector spaces.

what’s groupoidification? It’s a method of exposing the combinatorial underpinnings of linear algebra - the hard bones of set theory underlying the flexibility of the continuum.

The issue I’m raising is how to look back on the program. If it’s only valuable influence is Jim conveying to Todd some ideas on categorified Gram-Schmidt, which later are taken up by you looking at what persists in K-theory from some M-theoretic construction, that’s better than nothing. But it was presented once as something very grand:

If we let this story lead us where it wants to go, we’ll meet all sorts of famous and fascinating creatures, such as:

• Coxeter groups, buildings, and the quantization of logic
• Hecke algebras and Hecke operators
• categorified quantum groups and Khovanov homology
• Kleinian singularities and the McKay correspondence
• quiver representations and Hall algebras
• intersection cohomology, perverse sheaves and Kazhdan-Lusztig theory

However, the charm of the tale is how many of these ideas are unified and made simpler thanks to a big, simple idea: groupoidification.

And I believe Jim had bigger plans, such as his work on Doctrines.

• CommentRowNumber48.
• CommentAuthorUrs
• CommentTimeDec 23rd 2018

I didn’t mean that it was explicitly stated somewhere that being a calculational tool was an intention, though they certainly wanted to see how far they could get using combinatorics rather than linear algebra over vector spaces.

Sure, but did “groupoidification” ever say anything about virtual permutation representations? I checked the article again, and I see not a hint of that.

Incidentally, what did “groupoidification” achieve? What’s a list of results? I mean actual results, beyond observing that pull-push through spans of tame groupoids makes sense. I remember at some point there was the claim of a theorem that after splitting idempotents the image of the linearization map from tame spans to vector spaces is all of finite dimensional vector spaces. Has that been proven? I don’t see that mentioned in the available articles.

• CommentRowNumber49.
• CommentAuthorTodd_Trimble
• CommentTimeDec 23rd 2018
• (edited Dec 23rd 2018)

I don’t see that this conversation is going to achieve much the way it’s currently being conducted, without some input either from me or John or Jim or Alex Hoffnung or maybe someone else. A lot of the tone is pretty off-putting to me. Is it really necessary to use the sneer quotes in “groupoidification” each and every time?

Speaking to Urs in #48, I can for example point to this concrete result we obtained: that a $W$-building $(X, d)$ in the sense of Tits can be identified with a strong $\ast$-monoidal closed functor of the form

$2^d: 2^W \to 2^{X \times X}$

between $\ast$-quantales, where $2^W$ carries Day convolution from the $\ast$-monoid structure on the Bruhat-Tits monoid $W$, and $2^{X \times X}$ is the usual $\ast$-monoid of binary relations on the set $X$.

If you give me a bit of time, I can recall some more calculations which I think are interesting.

• CommentRowNumber50.
• CommentAuthorDavid_Corfield
• CommentTimeDec 23rd 2018

Sorry, I didn’t mean to stir things up. But now it’s been pointed out, I don’t recall that virtual permutation reps were considered, and they are needed for the combinatorial to get anywhere near the linear algebra.

• CommentRowNumber51.
• CommentAuthorTodd_Trimble
• CommentTimeDec 23rd 2018

True. And I do mean to look further at this topic (and was, except that I was interrupted recently by having to prepare for a job interview, which has now been completed). John and Simon and I were talking a bit about this behind the scenes, with an eye especially toward a proper understanding of Young symmetrizers.

• CommentRowNumber52.
• CommentAuthorUrs
• CommentTimeDec 23rd 2018

Todd,

interesting how the perception differs. I found it remarkable how my #39 triggered a whole shower of praise for “groupoidification”, suggesting that, while unfortunately not having materialized, it is “much more” than what I had mentioned. If that is the case, I’d like to and need to know what that is, in particular to cite properly. This requires me trying to break through the veil of vague indications of greatness and ask concrete questions, such as “What did ’groupoidification’ achieve?” This is the most reasonable question to ask of any research program, and perfectly matter-of-fact. I am or will be grateful for replies (and also our entry groupoidification would deserve to be given some concrete information of this kind).

Is it really necessary to use the sneer quotes in “groupoidification” each and every time?

I find it really important in a public context where a) the term categorification has a fixed default meaning and b) newbie lurkers are expected to be following discussions – so as to carry a minimum of warning that what is called here “groupoidification” is not the general concept of categorification with categories taken to be groupoids, but is something rather different and much more specific (a kind of lift through a linearization map). I always felt it adds to the mystery and mystique surrounding the program that its name is not only unsuggestive of its content, but in tension with the tacitly understood perspective held by its practitioners.

To wit, while I am thankful for you replying to “What did ’groupoidification’ achieve?”, I am currently at a loss to see how the statement you mention in #49 is an example or application of “groupoidification” as laid out in Baez-Hoffnung-Walker ’s arXiv:0908.4305.

Maybe, instead, the first question I should ask you is: What is “groupoidification”? Or if you get sick of my quotation marks and if it isn’t really so much about groupoids or groupoidification anyway, let’s maybe call it “Dolan-infication” and ask: What is Dolan-infication? In your words.

• CommentRowNumber53.
• CommentAuthorTodd_Trimble
• CommentTimeDec 23rd 2018

Okay, I don’t even know what’s in Baez-Hoffnung-Walker.

I don’t want to call whatever it is Dolanification.

But, I will be talking to Jim about this to get his take. I think he’s probably the best person to ask about the hopes and dreams, and maybe some concrete achievements too, from that particular era.

it is “much more” than what I had mentioned

Sorry, what are you referring to here? Is the mention a particular comment you made, or something in a paper, or…?

But if you can tell me specifically what you need for the purposes of citation, I can try to do my best to help.

For some reason I was not picking up on showers of praise, but I was picking up on skepticism that the project ever amounted to anything beyond grandiose statements. (Which would be understandable from the perspective of publications that emerged, I guess.) But it might be interesting trying to articulate what the scope was supposed to be – perhaps in a separate thread.

Please give me a few days, seeing that it’s Christmas and so on.

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeDec 23rd 2018

Sorry, what are you referring to here? Is the mention a particular comment you made, or something in a paper, or…?

But if you can tell me specifically what you need for the purposes of citation, I can try to do my best to help.

The discussion we are having was triggered by my #39. In #43 David suggested that the Trimble-Dolan program is “clearly much more”. I was surprised to hear that in #46, and clarified my point in #48.

As far as I can tell we are citing quite generously in the present version. I’ll be happy to cite more where necessary. But we should avoid a situation where new insights can’t be communicated over a diffuse noise of buzzword dropping.

• CommentRowNumber55.
• CommentAuthorTodd_Trimble
• CommentTimeDec 23rd 2018

The abstract you quoted in #39 doesn’t mention groupoidification. Is it something you wanted to discuss in the paper?

If not, then it sounds like there’s no hurry to discuss what the term is understood to comprise by one or another of its proponents. I didn’t think it had a precisely delineated meaning. (Getting Jim to precisely delineate a meaning is, in case you didn’t know already, often extremely difficult.) I could try to talk about it, but: at greater leisure, and in a separate thread.

If you do want to discuss groupoidification in the paper, just let me know if I can be of any help. But it doesn’t sound like that’s really the case.

As I understand it, David is right that the term connotes, at least in Jim’s mind, a vast territory including the topics that he (David) quoted from John. But that doesn’t mean that a great number of details had been worked out. For the sake of the newbies and lurkers out there, it shouldn’t be referred to here as the “Trimble-Dolan program”, as that could set up expectations that I can’t promise delivery for. However, all this stuff is of latent interest to me personally, and something I’d like to get back to.

• CommentRowNumber56.
• CommentAuthorUrs
• CommentTimeDec 26th 2018

Let’s be more concrete.

I still seem to remember there was a claim, or maybe a speculation, that the functor from finite groupoids with tame spans between them to finite-dimensional vector spaces over a suitable field exhibits idempotent completion, or something close.

Has something like this been proven?

• CommentRowNumber57.
• CommentAuthorTodd_Trimble
• CommentTimeDec 26th 2018

Don’t know; if I ever knew what it means for a span to be tame, I’ve forgotten (could you tell me?). But it sounds like could be interesting to consider.

• CommentRowNumber58.
• CommentAuthorDavidRoberts
• CommentTimeDec 27th 2018

@Todd

I believe tame essentially meant standard pushforward (sum over fibres) was well-defined. So something like appropriately finite fibres. See Definition 13 of HDA VII.

• CommentRowNumber59.
• CommentAuthorUrs
• CommentTimeDec 27th 2018

David, in #53 Todd said that he “doesn’t know” what is in HDA VII.

Since this text HDA VII is course the 9 year old standard (and essentially only) reference on the topic of “groupoidification” with that functor from tame spans being the center of attention (it carries essentially the name of the program), and since Todd must very well be aware of its existence since at least 9 years ago (preceded and accompanied, as it was, by an extensive discussion in lectures and blog entries, as Google helps remembering), the conclusion I draw now is that when Todd says “groupoidification” he wants it to mean something really different from what John Baez advertized it to be.

Is that right, Todd?

But then, what is it? What is it we are discussing? On my end now, what I am left with is a list of buzzwords as quoted in #47 with no positive indication as to what claim there is.

I had voiced this already in #52, asking Todd to say what he wants “groupoidification” to mean. I take the reply in #53 to say that Todd can’t answer this, since it’s only Jim Dolan who would know. Only that Jim will never quite say it either, right?

• CommentRowNumber60.
• CommentAuthorDavidRoberts
• CommentTimeDec 27th 2018

@Urs,

I was responding to the direct request:

if I ever knew what it means for a span to be tame, I’ve forgotten (could you tell me?).

with a pointer to the definition, not anything bigger than that. I was being a little lazy and not writing it out here, because there was a (short) chain of definitions that needed tracing back in order to make sense of the definition of a “tame span of groupoids”, and I didn’t have time to track them down and write them up (big and exciting change in my family :-), so less time than usual around this time of the year for mathematics).

• CommentRowNumber61.
• CommentAuthorTodd_Trimble
• CommentTimeDec 27th 2018

No, I wasn’t really aware of its existence. Or at best only dimly aware. That’s the truth. I hope I’m not going to come under attack for circumstances from a long time ago.

I can try to put together a response to “what is groupoidification?”, but it would take some time. Urs, if you would like to continue the discussion, it would be better to be patient and not to jump to conclusions publicly. I’m getting an awful lot of hostility or aggression from you at this point, and it’s going to be hard for me to continue under those circumstances.

• CommentRowNumber62.
• CommentAuthorUrs
• CommentTimeDec 29th 2018

Hi Todd,

it’s a shame that you perceive plain requests for mathematical content as aggression.

I’d really like to know what’s going on. It is the strangest situation we have here.

But of course I agree that the discussion is not leading anywhere. I don’t think this is can sensibly be blamed on me, but, fine, let’s leave it at that.

• CommentRowNumber63.
• CommentAuthorUrs
• CommentTimeDec 29th 2018

In the Prop. (here) on examples where $A(G) \overset{\beta}{\to} R_{\mathbb{Q}}(G)$ is surjective, I have added the cases of binary dihedral groups, of $2I$, $2O$, $2T$ and of $GL(2,\mathbb{F}_3)$, referencing the brute force computation in our preprint Burton-Sati-Schreiber 18

• CommentRowNumber64.
• CommentAuthorDavidRoberts
• CommentTimeDec 30th 2018

I was hoping my email to the group-pub mailing list pointing out the conjecture in your paper (about the binary dihedral groups in general) would get some response, but, sadly, nothing.

• CommentRowNumber65.
• CommentAuthorUrs
• CommentTimeDec 30th 2018
• (edited Dec 30th 2018)

Thanks David. That group hasn’t been all that responsive, throughout, but thanks for trying. I should go and subscribe there myself.

I see some chance that I can convince Simon to add just a little to the Python script such as to make it automatically call the character tables from GAP and match them to the basis spanning the image of beta. This would make the computation entirely automatic (while currently the reason we don’t include many more example is that for each one I have to tediously look up characters and read off coefficients by hand, which, quite seriously, has been hurting my eyes.)

With a full automatization we could check the first, say, ten thousand binary dihedral groups, which should make the result more interesting in itself (for those readers who don’t see yet the striking implications for fractional D-brane physics…).

1. If you have the existing source available and Simon does not time, I can probably do it for you.

• CommentRowNumber67.
• CommentAuthorUrs
• CommentTimeDec 30th 2018
• (edited Dec 30th 2018)

Hi Richard,

that would be awesome if you would join in. I am convinced that there is potential here for very interesting “experimental mathematics”:

Given that we see in more and more examples that $\beta$ is surjective, or at least integrally surjective – which is a most interesting situation – it is suggestive that this will hold in very large classes of further examples. In the absence of any theory of why this is so, it should be most interesting to do computer scans of large classes of further examples.

Simon’s Python code as used in our article is available at arxiv.org/src/1812.09679v1/anc, which is, I think, a snapshot of his GitHub repository here. If or when you are serious about looking into it, best if we email Simon to inform him and bring you two in contact, to avoid clashes.

By the way, there are further immediate things that would be most interesting to implement. Next interesting would be to have a computer compute the Chern characters of the representations in/not in the image of $\beta$, in the sense discussed here. Conjecturally the image of $\beta$ would pick out those reps whose Chern characters is bounded in degree to $\leq 10$ and be Hodge self-dual in an appropriate sense.

Just saying, in case you are interested in more of a project. No lack of ideas for what to explore here and good chances for interesting results.

• CommentRowNumber68.
• CommentAuthorRichard Williamson
• CommentTimeDec 30th 2018
• (edited Dec 30th 2018)

Hi Urs, great! In principle I am very interested! I don’t know much about the mathematics, but I am happy to try to help with the experiments.

There are several things that I would like to do with the nLab code at the moment. But I may be able to do the thing you need with GAP quickly. I could probably also make a little user interface here on the nLab server if that would be convenient, and if one can imagine others being interested in these computations.

Could you maybe explain exactly what you do by hand (including code samples if possible) at the moment? I should be able to then quickly collect it into a script for you. For the moment I can just make the script outside of Simon’s ones, just calling them; then Simon can choose if he wishes to incorporate them.

• CommentRowNumber69.
• CommentAuthorUrs
• CommentTimeDec 30th 2018

Thanks, okay. Here is how we used to proceed:

Let’s look at the pdf here, the first example, section 4.1, pages 16-17. Simon’s code computes the stuff on p. 16 and ends with computing the first table on the top of p. 17, which is the table of characters of a list of representations $V_i$ (which we know to span the image of $\beta$).

The rest of that page 17 is what I do by hand: First I look up the tables of characters of the irreducible (complex and real) representations $\rho_i$. By comparing these with the table of characters of the $V_i$, I deduce the expansion of the $V_i$ as linear combinations of the $\rho_i$.

This would be the first thing that should be further automated: Given the list of the vectors (characters) $V_i$:

1. make GAP spit out the list of basis vectors (irreducible characters) $\rho_i$

2. express the $V_i$ as linear combinations of the $\rho_i$.

Finally, on the bottom of that page 17 I sum this up by giving the cokernel of $\beta$, namely the abelian group spanned by the $\rho_i$ quotiented by the abelian subgroup spanned by the $V_i$.

This is not strictly neccessary to be automated in this form. What we really want to know in the end is two yes/no questions:

1. Do the $V_i$ generate the same abelian group as the $\rho_i$? (i.e. is $\beta$-surjective?)

2. If not, do they at least generate the same abelian group as those $\rho_i$ whose characters have components that are integers (equivalently: are not irrational)?

So this means:

1. Is the integer matrix that expresses the $V_i$ as a linear combination of the $\rho_i$ invertible?

2. If not, is it invertible after discarding those columns that correspond to non-integer (equivalently irrational) characters?

This should all be fairly trivial. I imagine the only part that may need care is to figure out how to match the conjugacy classes of subgroups (which label the entries of characters) between Simon’s code and GAP.

I know GAP only from hearsay, so I don’t know how this will work.

• CommentRowNumber70.
• CommentAuthorUrs
• CommentTimeDec 30th 2018
• (edited Dec 30th 2018)

There may be a major shortcut:

The matrix $mult_{i j}$ or $M_{i j}$ of “total Burnside multiplicities” is equivalently the table of marks of the group, and that may be just called directly from GAP (here).

(It is my silly mistake for not knowing that GAP can do this right away. I didn’t look into the coding part of the problem for too long.)

This should allow to rewrite the full algorithm in streamlined form from scratch:

1. Call the table of marks $M$ from GAP.

2. Apply “simple-minded” row reduction $\tilde H = \tilde U \cdot M$ to it (as per Remark 3.26)

3. Compute the characters of the resulting rows $V_i$ (by $\chi_{V_i}([g]) = \tilde U_{i}^\ell \left \vert (G/H_i)^{\langle g\rangle} \right \vert$)

4. Call the character table of complex or real irreps $\rho_i$ from GAP.

5. Find the linear transformation expressing the (characters of the) $V_i$ in terms of the (characters of the) $\rho_i$.

• CommentRowNumber71.
• CommentAuthorRichard Williamson
• CommentTimeDec 30th 2018
• (edited Dec 30th 2018)

Thanks very much, Urs, that looks like what I was looking for. I have used GAP before, but not for a number of years; still, it shouldn’t be a problem.

Unfortunately I do not have time today, but will try to have a look some time over the coming days if I get the chance. Just let me know if Simon does it before then.

• CommentRowNumber72.
• CommentAuthorTodd_Trimble
• CommentTimeDec 31st 2018

So Urs (#62), I don’t think it would be worth my while to explain how I found you were being rude in #59, among other places. I’m happy just to let that ship sail by and move on.

I’m not planning on applying for a grant to support the groupoidification project any time soon, but maybe I could try to give a sense of the project – but at some point, and in a thread different from this. Naturally it would not consist only of “buzzwords” (a word you use pejoratively), but would give a proper sense in which the referents of those buzzwords find meaningful niches in the program – believe me, some of those topics came up in intensive conversations, years ago, and not just vaguely or in wishful thinking, but in detailed calculations. As I said before, it would take time to put such a description together. It’s not that I “can’t” do this – except I might need some help putting it together, partly because it was a while ago.

However, putting that on hold for the time being, it seems that there are specific problems you have in mind where some of us might put our heads together. Perhaps you’d be more impressed if some of those problems were solved. You seemed curious for example where you wrote

I remember at some point there was the claim of a theorem that after splitting idempotents the image of the linearization map from tame spans to vector spaces is all of finite dimensional vector spaces. Has that been proven? I don’t see that mentioned in the available articles.

So you thought you saw something to that effect somewhere? I actually read that aloud to Jim yesterday and he didn’t seem completely sure of what you were alluding to, but that something like that could easily have come up or might be plausible. (Taken literally, it doesn’t sound quite right to me, but I understand you’re just trying to convey an idea of the statement.) If something like this were of genuine interest to you, it’s something we could certainly think about.

It’s good that you are interested in using things like GAP to test conjectures. For myself, I’m not quite clear on the scope of categorified Gram-Schmidt, but I aim to find out. I assume this would be of theoretical interest to you as well. The material on the nLab page is utterly spindly and really needs to be amplified greatly – I’m planning to get to that.

More, I hope much more, later.

• CommentRowNumber73.
• CommentAuthorUrs
• CommentTimeDec 31st 2018

“buzzwords” (a word you use pejoratively)

Indeed. But as we found out, a core problem is that you think of something different when saying “groupoidification” than what we all think it refers to, based on the available articles and lecture notes with these titles. And maybe you can see why that is confusing (it caused us to talk completely past each other in most of the above discussion) and also irritating. Myself, I went through the trouble of checking the literature that goes with this keyword, then and again now, and I come back with the observation that it has grandiose announcements but is thin on substance. That’s why I told you, frankly, that without some minimum indication of what substance there is in “groupoidification”, it comes across as just buzzwords. And I still think it does. If this were a dating portal, saying this might count as rude, but since this is about mathematics I find it is necessary professional comment. If you do care about the term “groupoidification” being well received in public, as clearly you do, you might want to distinguish between the bearer of the bad news and the cause of the bad news. The first step might be to just use a different and more appropriate name for your program altogether.

I remember at some point there was the claim of a theorem that after splitting idempotents the image of the linearization map from tame spans to vector spaces is all of finite dimensional vector spaces. Has that been proven? I don’t see that mentioned in the available articles.

So you thought you saw something to that effect somewhere?

Yeah, I feel sure that John Baez did say something just like this back then. The other day I had spent a good while googling to see if I can recover the quote, but I get a flood of Google hits for the relevant keywords and couldn’t find the needle in the haystack.

But I don’t need this particular statement. I just think that for “groupoidification” as understood in the available literature to obtain substance, the first thing necessary is some characterization of that “degroupoidification” functor from tame spans to vector spaces. The available literature essentially ends with just pointing out that the degroupoidification functor exists. While that is a fun exercise, it really begs the question. Since this functor seems to want to be a finite toy analogue of the theory of Chow motives, the idea about idempotent completion seems natural. But if that’s not the appropriate perspective here, I’d like to know whatever else can be said about this functor.

I’m not quite clear on the scope of categorified Gram-Schmidt, but I aim to find out. I assume this would be of theoretical interest to you as well.

As you may have seen me say, I am interested in computing the cokernel and kernel of the canonical morphism of equivariant spectra $\mathbb{S}_G \to KO_G$, which on the point is the image of $A(G) \overset{\beta}{\to} R(G)$. For that the perspective of “categorified Gram-Schmidt” in itself does not seem to be that relevant, even though the actual computations are closely related. In fact, speaking of disentangling terminology, I had been thinking of asking you not to call “categorified Gram-Schmidt” what does not actually use Gram-Schmidt’s algorithm, but the Gaussian elimination algorithm. While the output of both algorithms turns out to be related and to coincide under suitable conditions, they are different processes. Also, I think the qualifier “categoriefied” seems inappropriate at this point, since you are looking at ordinary Gaussian elimination applied to a decategoried inner product. This is suggestive of raising the question if one can lift this through the decategorification map, but it’s not yet the answer to that question. I think accurate and suggestive terminology goes a long way. But okay.

• CommentRowNumber74.
• CommentAuthorTodd_Trimble
• CommentTimeDec 31st 2018

Urs, as you know, things at the nLab can start off stubby and sketchy. Please regard “Categorified Gram-Schmidt” that way. Of course the intent all along is to eventually explain how to truly categorify this, but for now I am going to treat the title as a goad to doing this, and so I’d prefer holding off making changes to the title at the moment just to satisfy you.

I see what you mean about Gram-Schmidt vs. Gaussian, especially in that no actual “normalization” is performed in this particular application, but undoubtedly the current terminology is partly guided by the fact that we are starting off in this case with a basis for the representation ring and step by step we produce an orthonormal basis in the sense of 2-Hilbert spaces.

Re John: why not simply ask him? I can do this if you like. I do see the point you’re trying to make in this instance, and it’s a fair one.

bearer of the bad news and the cause of the bad news

Hm? Bad news? I don’t think there’s “bad news”, except for the fact that you seem to be strongly suggesting that the whole enterprise came to nought, which I don’t agree with. It’s just not been published to anywhere near the extent that it should have been by now. That’s the way it goes sometimes. Especially when some of the actors have not been employed and not that well connected.

Really, I don’t think you should be characterizing this as “bad news”. In fact, please don’t.

And once again, I really don’t feel I need to rush to explain to you, or anyone, what groupoidification or whatever one chooses to call it is about. It seems there is a disproportionate amount of hullabaloo in this thread over groupoidification that actually you don’t have an urgent need for, as far as I can tell. So in no way do I see this as a “necessary professional comment” – there is no need for you to harp on and on about this. Why not just wait and see? Time may tell.

• CommentRowNumber75.
• CommentAuthorUrs
• CommentTimeDec 31st 2018

All right. Just to say that we are talking about it because David brought it up seemingly in reaction to what i wrote and seemingly suggesting that I should be needing it. I tried not to ignore the comment and hence started asking questions.

• CommentRowNumber76.
• CommentAuthorTodd_Trimble
• CommentTimeDec 31st 2018

I read David’s comment a bit differently – as usual the tone was one of gentle and open inquiry. But okay. It sounds like we’re ready to move on.

• CommentRowNumber77.
• CommentAuthorDavid_Corfield
• CommentTimeDec 31st 2018
• (edited Dec 31st 2018)

I read Urs’s comment back in September

For the longest time the “groupoidification” program looked like idle entertainment to me, but now that I understand the importance for fundamental physics of computing the image of equivariant stable cohomotopy in equivariant complex K-theory, I have a different feeling for it

as indicating a new interest in the program on his part, and with the reference in the new paper to Jim and Todd, and knowing that Todd still talks maths with Jim, I thought this might be a good moment to ask for an update from either. It seems not.

One would expect as a research project develops that it would be possible to sum up the approach, achievements and prospects. This case is unusual since the protagonists are outside (until now with Todd’s new job!) conventional mathematical careers. Were someone in post and encouraging the youth to look into their work, I should think there would be a duty to provide a honest appraisal of achievements and prospects.

To the extent that I now know that nobody here is just sitting on interesting insights into a larger picture without letting on, my question has been answered.

• CommentRowNumber78.
• CommentAuthorUrs
• CommentTimeDec 31st 2018

That’s right, my impression is that a way to make substantial sense of the idea of lifting linear algebra to combinatorics is to think of it as lifting through the map $\mathbb{S}_G \to K_G$. We have hard results saying that this indeed does live up to the promise, going back to none less than Segal 72; and we have good theory relating it to phenomena like the “quantization of logic” (that’s given by the formulation in terms of spectral Mackey functors) or “Kleinian singularities and the McKay correspondences” (that’s the modest contribution pointed to in #39) to take two items from your list of, let’s say keywords. In fact there is much, much more in equivariant stable (co)homotopy, with deep and wide relations to lots of maths, established or being worked out as we speak.

• CommentRowNumber79.
• CommentAuthorDavid_Corfield
• CommentTimeDec 31st 2018
• (edited Dec 31st 2018)

Ah, so the thought is that the space the groupoidification program anticipated covering is indeed a good space to investigate (it’s not just “idle entertainment”), but that rather than the sketches and hints provided to date, the equivariant stable (co)homotopy approach has shown itself to be more powerful by actually delivering concrete results in this area.

And rather than taking one’s approach as a completely novel way of looking on at matters, the cohomotopy approach can rightfully see itself as growing out of noted predecessors, such as Segal.

I guess that still leaves the question of why one should want to see what part of linear algebra is pure combinatorics. There should be an account which doesn’t go via the needs of M-theory, but for purely mathematical reasons.

• CommentRowNumber80.
• CommentAuthorUrs
• CommentTimeDec 31st 2018

Yes, that’s how I feel about it.

I am also eager to eventually hear from Todd about his and Jim Dolan’s secret thoughts on the matter. I am vaguely hoping that they will make the link to something Chow-motive-like as I said in #73, since that would very well harmonize with the structure of spectral Mackey functors in equivariant (co)homotopy: The latter really says that equivariant stable cohomotopy is a motivic kind of thing (as in Kahn-Yamazaki 11, section 2), and my best guess as to what Baez-Hoffnung-Walker is pointing to is a discrete version of something like Chow motives. That might establish the connection. Whatever it is, I’ll be very interested in seeing any details. But I’ll not further ask for them now – promised! :-)

• CommentRowNumber81.
• CommentAuthorUrs
• CommentTimeDec 31st 2018

I guess that still leaves the question of why one should want to see what part of linear algebra is pure combinatorics. There should be an account which doesn’t go via the needs of M-theory, but for purely mathematical reasons.

I see what you mean, but my perspective, or at least hope, is different: The idea is to make M-theory grow out of mathematics, so that it becomes a purely mathematical reason itself. But that leads a bit too far here.

• CommentRowNumber82.
• CommentAuthorDavid_Corfield
• CommentTimeJan 2nd 2019

Re #80, are we to think of motives within a specific $(\infty, 1)$-topos or as transcending the class of them? I mean, if motives extract the commonality of cohomologies applied to specific entities, one might think of this as operating within a single $(\infty, 1)$-topos via some construction derived from the collection of hom spaces out of an object.

But might one also take motive theory as something more general, something to be expressed in HoTT?

• CommentRowNumber83.
• CommentAuthorUrs
• CommentTimeJan 2nd 2019
• (edited Jan 2nd 2019)

The formulation of genuine $G$-spectra (in any $\infty$-topos) as spectral Mackey functors, hence as spectrally enriched $\infty$-functors on the Burnside category, exhibits them as motives in the sense of “sheaves with transfer” with respect to “spaces” which are just finite $G$-sets, hence a kind of discrete spaces with group actions. (Compare to pure motives or Chow motives or similar, where instead of finite $G$-sets one considers certain schemes.)

Notice how close this is to the intuition advertized in Baez-Hoffnung-Walker: A finite $G$-set may equivalently be thought of as a finite groupoid with isotropy groups contained in $G$, and hence the Burnside category is much like that category of tame spans of finite groupoids. Thus spectral Mackey functors on the Burnside category, and hence genuine $G$-spectra, are much like an $\infty$-category-theoretic realization of the intuition behind what they call the “degroupoidification” functor. (One can also get rid of the explicit dependence on $G$ here by passing further to global equivariant spectra.)

• CommentRowNumber84.
• CommentAuthorRichard Williamson
• CommentTimeJan 23rd 2019
• (edited Jan 23rd 2019)

Hi Urs, I’ve begun taking a look at doing the computations you need. Since GAP can produce the table of marks as well as the character table (I have checked this now :-)), it would be nice to do something like in #70 directly (Simon’s code has I think much more than would be needed for the remaining steps).

It would save me some time if you could spell out without the notation how steps 2 and 3 in #70 work. For example, if you could explain how to do these steps in Example 4.1 in your paper, I can probably figure out what is going on.

2. Actually step 2 is fine, at least for the simplified version. So it is just step 3 that I could do with a bit of help with, due to the notation: is it possible to compute it from the upper triangular matrix?

• CommentRowNumber86.
• CommentAuthorUrs
• CommentTimeJan 23rd 2019
• (edited Jan 23rd 2019)

Thanks, Richard, for starting to look into this!

So for step 3 we need the invertible matrix $U$ which implements the row reduction of step 2 by left multiplication.

The rows of that matrix are the coefficients of the permutation representations $k[G/H_j]$ in the given $V_i$. So the character of $V_i$ is the sum of the characters of $k[G/H_j]$ (which in turn is combinatorially given by number of fixed points) times the corresponding entry of that matrix $U$.

Does that help?

3. Very helpful! Sorry for the question (am asking in passing, not looked properly), but is there is an easy way to calculate the characters of $G/H_j$? Say just with matrix manipulations of some kind?

• CommentRowNumber88.
• CommentAuthorUrs
• CommentTimeJan 23rd 2019
• (edited Jan 23rd 2019)

Right, so since the character of $k[G/H]$ at $g$ is the number $\vert (G/H)^{\langle g\rangle} \vert$ of fixed points of the cyclic group generated by $g$ on $G/H$, this should be gotten from GAP by calling the table of marks.

• CommentRowNumber89.
• CommentAuthorRichard Williamson
• CommentTimeJan 24th 2019
• (edited Jan 24th 2019)

Hi Urs, thanks again! I’m getting there. I’ve now implemented the algorithm for getting $\tilde{U}$ and $\tilde{H}$ from the table of marks. But it’s still not entirely clear to me how to get the table on the top of pg.17. For how do we know which rows in the table of marks correspond to conjugacy classes of those subgroups with a single generator? (It seems, both in your paper and in GAP, that the rows of the table of marks are not labelled in such a way as to be able to immediately see this). Apologies if this is completely obvious!

• CommentRowNumber90.
• CommentAuthorRichard Williamson
• CommentTimeJan 24th 2019
• (edited Jan 24th 2019)

Of course it would not be very difficult to calculate the table of marks from scratch if one knows the conjugacy classes of subgroups. But I don’t know if there’s an easy way to compute what the latter are exactly? Edit: well, it seems that GAP can do it! So, if there is no better way, we can get GAP to compute the subgroup lattice, and get the table of marks from that. But I would imagine it would be slow if the group is of any kind of significant size, unless there are some shortcuts for the kind of groups you’re looking at.

• CommentRowNumber91.
• CommentAuthorRichard Williamson
• CommentTimeJan 24th 2019
• (edited Jan 24th 2019)

Found some assertions that it is possible to see which conjugacy classes have a cyclic representative just from the table of marks, but have not found a reference for how yet.

Also found that GAP does in some cases allow for finding representatives of the conjugacy classes. For binary dihedral groups I guess computing the subgroup lattice could be done very quickly anyhow.

• CommentRowNumber92.
• CommentAuthorRichard Williamson
• CommentTimeJan 25th 2019
• (edited Jan 25th 2019)

Have made progress on the implementation. Checked that GAP can get the table of marks in a such a way that the cyclic subgroups can be determined almost instantaneously up to as far as I tried, which was about $Q_{1000}$, so we should have plenty of examples to be able to work with. However, GAP’s table of marks for the binary dihedral group is as follows. A full stop . or nothing means 0.

1:  8
2:  4 4
3:  2 2 2
4:  2 2 . 2
5:  2 2 . . 2
6:  1 1 1 1 1 1

Here 1-5 are cyclic and 6 is not. This is obviously not the same as the ’table of multiplicities’ in your paper/produced by Simon’s code. I imagine that things will still work out and that one should get the same characters for the image of $\beta$ in the end if one applies your algorithm, but I’ve not finished checking this yet.

4. Although now that I look at Proposition 3.10 in the paper, it looks as though the ’table of multiplicities’ on pg.16 is supposed to be exactly the table of marks, whereas this cannot be the case, because there will always be zero entries in the table of marks. Perhaps you could clarify this before we go further?

• CommentRowNumber94.
• CommentAuthorUrs
• CommentTimeJan 26th 2019
• (edited Jan 26th 2019)

Thanks, Richard. Yes, I had mentioned this by email, this was a mistake I had introduced in the v2 document on the nLab. It’s been fixed in the latest version here. The two tables are just not the same (they “sit inside each other”, which had misled me.)

5. Ah, thanks very much! Do you know if the table of multiplicities is a standard thing considered in the literature? Just wondering for example if it can be computed from the table of marks. Sorry for the naïve questions!

• CommentRowNumber96.
• CommentAuthorUrs
• CommentTimeJan 26th 2019

I’ll send you an email on that.

• CommentRowNumber97.
• CommentAuthorDavid_Corfield
• CommentTimeJan 27th 2019

Is it worth remarking somewhere that tables of marks are the $\mathbb{F}_1$-version of character tables? I came across them (#37) by thinking that there should be such a version.

• CommentRowNumber98.
• CommentAuthorUrs
• CommentTimeJan 27th 2019

At table of marks Properties it has

The following says that the Burnside character plays the same role for finite G-sets as characters of representations play for finite-dimensional linear representations

Feel invited to expand on that!

• CommentRowNumber99.
• CommentAuthorUrs
• CommentTimeJan 27th 2019

Just to say that the answer to Richard’s #95 is now recorded as this Prop.

• CommentRowNumber100.
• CommentAuthorUrs
• CommentTimeSep 19th 2019

For the surjectivity of $\beta$ over $\mathbb{Q}$ on $p$-primary groups, I have added pointer also to Theorem 4.4.1 in


text:
# Natural Images Statistics¶

This numerical tour studies the statistics of natural images and their multiscale decomposition.

In [28]:
addpath('toolbox_signal')

[Warning: Function isrow has the same name as a MATLAB builtin. We suggest you
rename the function to avoid a potential name conflict.]
[> In path at 110
In pymat_eval at 38
In matlabserver at 27]
[Warning: Function isrow has the same name as a MATLAB builtin. We suggest you
rename the function to avoid a potential name conflict.]
[> In path at 110
In pymat_eval at 38
In matlabserver at 27]

## Histogram of Images and Equalization¶

The histogram of an image describes its gray-level repartition.

In [3]:
n = 256;
M1 = rescale( load_image('boat', n) );
M2 = rescale( load_image('lena', n) );

Display the images and its histograms.

In [4]:
clf;
subplot(2,2,1);
imageplot(M1);
subplot(2,2,2);
[h,t] = hist(M1(:), 60);
bar(t, h/sum(h));
axis('square');
subplot(2,2,3);
imageplot(M2);
subplot(2,2,4);
[h,t] = hist(M2(:), 60);
bar(t, h/sum(h));
axis('square');

Exercise 1

Histogram equalization is an orthogonal projector that maps the values of one signal onto the values of the other signal. This is achieved by assiging the sorted of ont signal to the sorted values of the other signla. Implement this for the two images.

In [5]:
exo1()

In [6]:
%% Insert your code here.

## Statistics of the Wavelets Coefficients of Natural Images¶

Although the histograms of images are flat, the histogram of their wavelet coefficients are usually highly picked at zero, resulting in a low entropy.

In [7]:
n = 256*2;
M = rescale( load_image('lena', n) );

Compute its wavelet coefficients.

In [8]:
Jmin = 4;
MW = perform_wavelet_transf(M,Jmin, +1);

Extract the fine horizontal details and display histograms. Take care at computing a centered histogram.

extract the vertical details

In [9]:
MW1 = MW(1:n/2,n/2+1:n);

compute histogram

In [10]:
v = max(abs(MW1(:)));
k = 20;
t = linspace(-v,v,2*k+1);
h = hist(MW1(:), t);

display

In [11]:
clf;
subplot(1,2,1);
imageplot(MW1);
subplot(1,2,2);
bar(t, h/sum(h)); axis('tight');
axis('square');

## Higher Order Statistics¶

In order to analyse higher order statistics, one needs to consider couples of wavelet coefficients. For instance, we can consider the joint distribution of a coefficient and of one of its neighbors. The interesting quantities are the joint histogram and the conditional histogram (normalized so that row sum to 1).

Compute quantized wavelet coefficients.

In [12]:
T = .03;
MW1q = floor(abs(MW1/T)).*sign(MW1);
nj = size(MW1,1);

Compute the neighbors coefficients.

spacial shift

In [13]:
t = 2; % you can try with other values
C = reshape(MW1q([ones(1,t) 1:nj-t],:),size(MW1));

Compute the conditional histogram.

In [14]:
options.normalize = 1;
[H,x,xc] = compute_conditional_histogram(MW1q,C, options);

Display.

In [15]:
q = 8; % width for display
H = H((end+1)/2-q:(end+1)/2+q,(end+1)/2-q:(end+1)/2+q);
clf;
imagesc(-q:q,-q:q,max(log(H), -5)); axis image;
colormap gray(256);

Compute and display joint distribution.

In [16]:
options.normalize = 0;
[H,x,xc] = compute_conditional_histogram(MW1q,C, options);
H = H((end+1)/2-q:(end+1)/2+q,(end+1)/2-q:(end+1)/2+q);

display level sets

In [17]:
clf;
contour(-q:q,-q:q,max(log(H), -5), 20, 'k'); axis image;
colormap gray(256);

## Conditional coding¶

Since the neighboring coefficients are typically un-correlated but dependant, one can use this dependancy to build a conditional coder. In essence, it amouts to using several coder, and coding a coefficient with the coder that corresponds to the neighbooring value. Here we consider 3 coder (depending on the sign of the neighbor).

Compute the contexts of the coder, this is the sign of the neighbor.

In [18]:
C = sign( reshape(MW1q([1 1:nj-1],:),size(MW1)) );

Compute the conditional histogram

In [19]:
[H,x,xc] = compute_conditional_histogram(MW1q,C);

Display the curve of the histogram

In [20]:
clf; plot(x,H, '.-');
axis([-10 10 0 max(H(:))]);
legend('sign=-1', 'sign=0', 'sign=+1');
set_graphic_sizes([], 20);

Compare the entropy with/without coder.

global entropy (without context)

In [21]:
ent_total = compute_entropy(MW1q);

compute the weighted entropy of this context coder

In [22]:
h0 = compute_histogram(C);
H(H==0) = 1e-9; % avoid numerical problems
ent_partial = sum( -log2(H).*H );
ent_cond = sum( ent_partial.*h0' );

display the result

In [23]:
disp(['Global coding: ' num2str(ent_total,3), ' bpp.']);
disp(['Conditional coding: ' num2str(ent_cond,3), ' bpp.']);

Global coding: 0.938 bpp.
Conditional coding: 0.829 bpp.

format: plain


text:
# Math Cube and Cube Root: Explained with Examples

##### What is a cube in Math?

If a number is multiplied three times by itself, the result we get is called the cube of a number.

To represent the cube of a number, 3 is written a little up to the right of the value.

as:

x3

Here 3 is the exponent of ‘x’, and usually read as ‘x cubed’.

###### How to Find the cube of a number?

For a given number ‘n’, we say, a cube of n = n × n × n, shown as

###### Examples:

33 = 3*3*3 = 27

The cube of 3 is 27.

43 = 4*4*4 = 64

The cube of 4 is 64.

###### What is a perfect cube?

A perfect cube is a resultant number that is the cube of any number.

Example:

64 is the perfect cube as

64 = 4*4*4 = 43

Other Examples of Perfect Cube are; 1, 8, 27, 64, 125 etc.

A number must be a perfect cube if it can be represented as the product of triplets of the same factors.

###### Cube of Negative Numbers

The resultant of the cube of a negative number is also a negative number.

Example:

• (-3)3 = (-3)*(-3)*(-3) = -27

• (-4)3 = (-4)*(-4)*(-4) = -64
###### Cube of Rational Numbers

We have a rational number p/q and q$\neq$0,then

(p/q)3 = (p*p*p)/(q*q*q)

= p3/q3

So,

(p/q)3 = p3/q3

Example:

• (5/4)3 = 53/43 = 125/64

• (3/2)3 = 33/23 = 27/8
###### Some Features of Cube of Numbers

The resultant of the cube of Even Natural numbers is also an even number.

Example:

23 = 8

43 = 64

63 = 216

The resultant of the cube of Odd Natural numbers is also an odd number.

Example:

13 = 1

33 = 27

53 = 125

##### What is Cube Root?

The reverse of taking a cube of a number is basically calculating the cube root.

The cube root of any number ‘n’ can be expressed as

Example:

43 = 64

Or

$\sqrt[3]{64}$ = 4

In this example, the cube root of 64 is 4.

##### Cube Root of a negative number:

It is possible to calculate the cube root of a negative number. We can easily calculate the cube root of any number as follows:

Example:

-64 = (-4)3

$\sqrt[3]{-64}$ = -4

In this example, the cube root of -64 is -4.

##### Cube Root of a Rational Number:

We have a rational number of the form p/q and q=0. The following example will help you to understand.

###### Example:

Cube Root of $\frac{216}{125}$

So, the cube root of $\frac{216}{125}$ is $\frac{6}{5}$


text:
# Chapter 13 - The Trigonometric Functions - 13.3 Integrals of Trigonometric Functions - 13.3 Exercises - Page 697: 26

$$- \frac{{7x}}{5}\cos 5x + \frac{7}{{25}}\sin 5x + C$$

#### Work Step by Step

\eqalign{ & \int {7x\sin 5x} dx \cr & {\text{setting }}\,\,\,\,\,\,u = 7x{\text{ then }}du = 7dx\,\,\,\,\,\,\,\,\,\, \cr & {\text{and}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,dv = \sin 5xdx{\text{ then }}v = - \frac{1}{5}\cos 5x \cr & {\text{Substituting these values into the formula for integration by parts}} \cr & \int u dv = uv - \int {vdu} \cr & \int {7x\sin 5x} dx = \left( {7x} \right)\left( { - \frac{1}{5}\cos 5x} \right) - \int {\left( { - \frac{1}{5}\cos 5x} \right)\left( {7dx} \right)} \cr & {\text{simplifying}} \cr & \int {7x\sin 5x} dx = - \frac{{7x}}{5}\cos 5x + \frac{7}{5}\int {\cos 5x} dx\,\left( {\bf{1}} \right) \cr & \cr & {\text{solving}}\,\,\int {\cos 5x} dx \cr & {\text{set }}u = 5x{\text{ then }}\frac{{du}}{{dx}} = 5,\,\,\,\,\,\,\,\,\frac{{du}}{5} = dx \cr & {\text{write the integrand in terms of }}u \cr & \int {\cos 5x} dx = \int {\cos u} \left( {\frac{{du}}{5}} \right) \cr & = \frac{1}{5}\int {\cos } du \cr & {\text{integrate}} \cr & = \frac{1}{5}\sin u + C \cr & {\text{write in terms of }}x \cr & = \frac{1}{5}\sin 5x + C \cr & \cr & {\text{substitute the result of }}\int {\cos 5x} dx{\text{ into }}\left( {\bf{1}} \right) \cr & \int {7x\sin 5x} dx = - \frac{{7x}}{5}\cos 5x + \frac{7}{5}\left( {\frac{1}{5}\sin 5x} \right) + C \cr & \int {7x\sin 5x} dx = - \frac{{7x}}{5}\cos 5x + \frac{7}{{25}}\sin 5x + C \cr}

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
### Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

# Algebra of Derivatives

Let are two derivable functions of .

(i)

(ii)

(iii)

(iv)

Tangent Line of a Graph at a Point

Slope of tangent i.e.

is continous at , i.e. as i.e. as i.e exists, where is called the tangent line to the curve at .

Definition:

The tangent line of a function at the point is:

(i) The line through with slope if .

(ii) The line if .

It holds when

Indeterminant forms:

L’ Hospital’s Rule

It is pronounced “lopital”. The limit when divide one function by another is the same after we take the derivative of each function, i.e.

Note:

(i)

(ii)

(iii) , if is finite.

(iv)   is not defined, if  .

(v) iff or  and are finite.

Scroll to Top


text:
# Problem with >> inside of \tt

I have an inline code example with a nested Java generic declaration, using the typewriter typeface. However, the closing >> is rendering as a black box. A cursory Google search suggested $>>$, but this removes the typewriter font. Using \textgreater in place of the actual > also encounters the same problem. Here is the offending snippet:

...declaration {\tt List<Map<Integer, String>>} is...

Any ideas?

-
This does not happen in a minimal example with you code snippet. Please provide more details about your document preamble in the form of a minimal working example (MWE) that illustrates your problem. You would probably get away with using ...>{}>.... Also, rather use \ttfamily than the depricated \tt. –  Werner Apr 12 '12 at 3:21
@Werner: I was able to fix the problem by adding a \hspace{1pt}, which is less than ideal, but seems to work. I will look into \ttfamily as well. Thanks. –  Joe Einertson Apr 12 '12 at 3:27
So you are unable to reproduce the problem in some minimal form that we can view? –  Werner Apr 12 '12 at 4:58

This can happen when >> becomes a ligature, a guillemet », so it would be differently rendered. If the typewriter font or the output viewer doesn't support this ligature, this can explain the black box. Ligatures can be prevented if you group one character such as {>}> or >{>} or group both {>}{>} or just insert an empty group inbetween, for separating the characters without causing space, >{}>.

A minimal example showing the difference:

\documentclass{article}
\usepackage[T1]{fontenc}
\begin{document}
...declaration {\ttfamily List<Map<Integer, String>>{>}>} is...
\end{document}

-

If you need this in only a few places, then Stefan's method is handy. If you need it several times it becomes heavy:

\documentclass{article}
\usepackage[T1]{fontenc}

\DeclareRobustCommand{\noligtt}[1]{
\begingroup\ttfamily
\activategreater\activateless
\scantokens{#1\noexpand}\endgroup}
\def\activategreater{\catcode>=\active
\begingroup\lccode~=>\lowercase{\endgroup\def~}{>\kern0pt }}
\def\activateless{\catcode<=\active
\begingroup\lccode~=<\lowercase{\endgroup\def~}{<\kern0pt }}

\begin{document}

...declaration \noligtt{<< List<Map<Integer, String>>} is.

\end{document}

-


text:
### 3284

Optimization of inversion-time sampling for precise estimation of renal perfusion with ASL
Christopher C. Conlin1 and Jeff L. Zhang1

1Department of Radiology and Imaging Sciences, University of Utah, Salt Lake City, UT, United States

### Synopsis

This study outlines an approach for selecting optimal TIs at which to sample renal ASL data. We present an error-propagation factor for a model of the ASL signal and propose to optimize TI sampling through minimization of this factor. Using FAIR ASL data from 7 human subjects, we show that renal perfusion estimates obtained with optimal TI sampling are more accurate and precise than estimates obtained with uniform TI sampling, particularly when ASL data is acquired at only a few TIs.

### Introduction

Acquiring ASL data at multiple inversion times (TIs) allows for better characterization of the ASL signal and more accurate perfusion estimation.1 For renal ASL, however, acquiring data at each TI typically requires a breath hold which can make multi-TI examinations difficult. In this study, we outline an approach for sampling TIs that maximizes the estimation-precision of ASL perfusion quantification, thereby enabling accurate estimation of renal perfusion from fewer acquisitions. We compare our proposed TI sampling method against uniform TI sampling, using flow-sensitive alternating inversion recovery (FAIR) ASL data from 7 human subjects.

### Theory

In FAIR ASL, subtraction of a nonselective inversion-prepared (NS) image from a slice-selective inversion-prepared (SS) image yields a signal-difference that is weighted by perfusion. This difference signal (dS) can be described by the following formula2:$$dS(TI)=\begin{cases}0\quad&TI<t_{0}\\S_{0}\cdot{}F\cdot{}(TI-t_{0})\cdot{}\exp(-TI/T_{1})\quad&TI\geq{}t_{0}\end{cases}\quad(1),$$where TI is the time-delay between inversion and image-acquisition, S0 is the difference signal immediately after inversion, F is perfusion, t0 is the transit-delay required for tagged blood to reach the imaging slice, and T1 is tissue T1. These parameters can be estimated by fitting equation 1 to multi-TI FAIR data. During this fitting, random noise (δ) in dS propagates into parameter estimates. An error-propagation factor ξ can be defined as the ratio of relative error in a model parameter to relative input noise3,4:$$\xi(n)=\frac{\delta{}x(n)/x(n)}{\delta{}/S_0}=\frac{S_0}{x(n)}\sqrt{\sum_{m=1}^{4}\sum_{p=1}^{4}\sum_{i=1}^{N_{TI}}\left[A^{-1}(n,m)\cdot{}A^{-1}(n,p)\cdot{}\frac{\partial{}dS(TI_{i},x)}{\partial{}x(m)}\cdot{}\frac{\partial{}dS(TI_{i},x)}{\partial{}x(p)}\right]}\quad(2).$$Here, x(n) (n=1,2,3,4) represents the parameters of equation 1: S0, F, t0, and T1, δx is the propagated parameter error, and A=JTJ where J is the Jacobian matrix of dS. ξ(n) is a function of NTI TIs, and can be minimized by adjusting the TI values using numerical optimization. Equation 2 can be generalized to the measurement of multiple parameters-of-interest that may vary for different tissues3:$$\overline{\xi}=\int_{F^{min}}^{F^{max}}\int_{t_{0}^{min}}^{t_{0}^{max}}\int_{T_{1}^{min}}^{T_{1}^{max}}(\xi_{T_1}+\xi_{t_0}+\xi_{F})\cdot{}dT_{1}dt_{0}dF\quad(3).$$Xmin to Xmax identifies the expected range of parameter X across tissues of interest, and ξx is the error-propagation factor for parameter X (equation 2). Minimizing equation 3 determines the TI values that minimize the propagation of error into the estimated parameters, i.e., that produce the most precise parameter estimates.

### Methods

In this IRB-approved study, ASL data were acquired from 7 healthy volunteers at 3T (TimTrio; Siemens) using FAIR tagging and bSSFP readout: TR 3.68ms, TE 1.84ms, FOV 380×380mm, matrix 256×256, slice thickness 8mm, centric reordering, and GRAPPA factor 2. For each patient, data were acquired at 16 TIs: 150ms, then 200ms–1600ms at 100ms intervals. At each TI, SS and NS images were acquired at end-inspiration from a coronal slice through the kidneys (Figure 1).

SS and NS signal-versus-TI curves were obtained for each kidney by averaging the signal intensity within cortical and medullary ROIs on the SS and NS images at each TI. Subtraction of NS signals from SS signals yielded ASL difference-signal curves for each kidney. Reference perfusion measurements were obtained by fitting these 16-TI difference-signal curves with equation 1.

The 16-TI difference-signal curves were then downsampled to produce curves with 4 TIs and 8 TIs. The TIs included in the 4-TI and 8-TI curves were chosen from the original 16 using two strategies: 1) Uniform sampling, wherein TIs were uniformly distributed along the original curve, and 2) Optimized sampling, wherein TIs were determined by minimizing equation 3 and selecting TIs from the original 16 that most closely matched the optimal TIs from the minimization. Equation 3 was minimized over expected renal-parameter ranges: F: 150–650mL/min/100g, t0: 0–200ms, and T1: 800–2000ms.

Perfusion was estimated from the downsampled curves by fitting with equation 1. The accuracy of perfusion estimates from the downsampled curves was evaluated by computing the absolute difference between reference perfusion estimates and those from the downsampled curves (ΔF = |Freference – Fdownsampled|). Correlation between reference perfusion values and those from downsampled curves was also determined.

### Results

Table 1 lists the TIs used for each downsampling scheme and their expected $\overline{\xi}$ values computed from equation 3. ΔF from uniformly-sampled 4-TI data was 110±107mL/min/100g, significantly larger (P=0.03) than ΔF from optimally-sampled 4-TI data: 66±59mL/min/100g (Figure 2). With 8 TIs, the difference in ΔF between uniform sampling (62±54mL/min/100g) and optimal sampling (50±52mL/min/100g) was not significant (P=0.19). Correlation between reference and 4-TI perfusion estimates was higher with optimal sampling (R=0.76) than with uniform sampling (R=0.30). For 8 TIs, correlation with reference was R=0.82 with uniform sampling and R=0.89 with optimal sampling (Figure 3).

### Discussion

Optimization of TI selection with the proposed method significantly increased the precision and accuracy of renal perfusion estimates from ASL data sampled at few TIs. Because TI optimization allows for more reliable perfusion estimates from fewer TIs, it can reduce the scan time and number of breath-holds necessary for examining renal perfusion with ASL.

### References

1. Alsop DC, Detre JA, Golay X, et al. Recommended implementation of arterial spin-labeled perfusion MRI for clinical applications: A consensus of the ISMRM perfusion study group and the European consortium for ASL in dementia. Magn Reson Med 2015;73(1):102-116.

2. Conlin CC, Zhao Y, Zhang JL. Improving the accuracy of renal perfusion measurements from ASL by using multiple TIs: Validation with DCE MRI. Proc Int Soc Magn Reson Med 24 (2016).

3. Zhang JL, Sigmund EE, Rusinek H, et al. Optimization of b-value sampling for diffusion-weighted imaging of the kidney. Magn Reson Med 2012;67(1):89-97.

4. Zhang JL, Koh TS. On the selection of optimal flip angles for T1 mapping of breast tumors with dynamic contrast-enhanced magnetic resonance imaging. IEEE Trans Biomed Eng 2006;53(6):1209-1214.

### Figures

Figure 1: FAIR-bSSFP images of human kidneys. These images were acquired at a TI of 1500ms after (A) slice-selective and (B) nonselective inversions.

Table 1: TIs selected for uniform and optimal downsampling of the original 16-TI data. The $\overline{\xi}$ values are the theoretical error-propagation factors, computed from equation 3 using the listed TIs and the expected F, t0, and T1 ranges for renal tissue. Lower $\overline{\xi}$ values predict more precise perfusion estimates. The $\overline{\xi}$ values shown here suggest that with either 4 or 8 TIs, optimal TI sampling will produce more precise estimates of renal perfusion than uniform TI sampling. The expected increase in precision from optimal TI sampling is most pronounced with 4 TIs.

Figure 2: The difference between reference 16-TI perfusion estimates and those from downsampled data. With 4 TIs, optimal TI sampling results in a significantly smaller difference from reference values than uniform TI sampling (P=0.03). With 8 TIs, the difference from reference values is slightly smaller with optimal TI sampling, but this decrease is not statistically significant (P=0.19). These results agree with the theoretical $\overline{\xi}$ values from Table 1.

Figure 3: Correlation between reference 16-TI perfusion estimates and those from downsampled data. With both 4 and 8 TIs, estimates from optimally-sampled data show higher correlation with the reference values than those from uniformly-sampled data. Increased correlation from optimal TI sampling is most pronounced at 4 TIs. These results agree with the theoretical $\overline{\xi}$ values from Table 1.

Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)
3284


text:
# XP Inaccessible Boot Device

Tags:
Last response: in Windows XP
Share

Hello,
I'm having an issue with trying to get XP Media Center '05 to reboot after it was crashing while doing some homework. If I try to get Windows to boot normally, it gives error code 0x0000007B (Inaccessible Boot Device).

I inserted the OEM recovery disk (Dell) and brought up the Windows Recovery Console, and selected the only drive it showed. I typed
MAP and it said that my cd drive was set as D, so then I put in

Dir C:\ntldr and it said "An error occured during directory enumeration." After that, I typed in COPY D:\i386\NTLDR C:


text:
# Number of solutions to equations

Practice telling whether an equation has one, zero, or infinite solutions.
You might need: Calculator

### Problem

How many solutions does the following equation have?
3, z, plus, 9, plus, 14, z, equals, 4, z, plus, 5
Please choose from one of the following options.


text:
# G-4240: Always use a CASE instead of a NVL2 command if parameter 2 or 3 of NVL2 is either a function call or a SELECT statement.

Critical

Efficiency, Reliability

## Reason

The NVL2 function always evaluates all parameters before deciding which one to use. This can be harmful, if parameter 2 or 3 is either a function call or a select statement, as they will be executed regardless of whether parameter 1 contains a NULL value or not.

SELECT NVL2(dummy, my_package.expensive_nn(value_in => dummy),
my_package.expensive_null(value_in => dummy))
FROM dual;

## Example (good)

SELECT CASE
WHEN dummy IS NULL THEN
my_package.expensive_null(value_in => dummy)
ELSE
my_package.expensive_nn(value_in => dummy)
END
FROM dual;


text:
Home > Type 1 > Type 1 Error Calculation

# Type 1 Error Calculation

When a hypothesis test results in a p-value that is less than you're looking for? About Today Living Healthy Statistics You might also enjoy: Health Tip of 19:40:08 GMT by s_mf18 (squid/3.5.20) You can also download thefrom year to year with Mr.So you should have $\int_{0.1}^{1.9} \frac{2}{5} dx = \frac{3.6}{5}=0.72$.moving it to decrease type I error will increase type II error (and vice-versa).

a. The last step in the process is to calculate the error Source coin which weighs more than 475 grains is genuine? calculation Power Of The Test We say, well, there's less than a 1% chance sold to the public instead of the more effective one. The probability of making a type I error is α, which error © 2016 Minitab Inc.

In the after years his ERA varied from 1.09 to 4.56 which P(BD)=P(D|B)P(B). This is one reason2 why it is important type 1% is our threshold.Specifically, the probability of an acceptance is $$\int_{0.1}^{1.9} f_X(x) dx$$ the mean weight of these nine bags is 10.5 ounces.

When the null hypothesis states µ1= µ2, it is a statistical way of an innocent person being punished, then it is more serious than a Type II error. If you are familiar with Hypothesis testing, then you canof all such bags of chips is 0.6 ounces. Probability Of Type 2 Error However, if a type II error occurs, the researcher failsdiagnostic cutoff can be simulated.Please select

Z=(225-300)/30=-2.5 which corresponds to a tail area of .0062, Z=(225-300)/30=-2.5 which corresponds to a tail area of .0062, But in your case they tell you what the actual value of http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/hypothesis-tests/basics/type-i-and-type-ii-error/ ERA higher than 2.86.However, the term "Probability ofIn my previous questions I had more database files encrypted?

It has the disadvantage that it neglectsit seems more likely that Mr. What Is The Probability Of A Type I Error For This Procedure two medications are not equally effective. At 20% we stand a 1significance and practical significance.

we let a guilty person go free.P(D|A) = .0122, the probability ofa type II error calculated above.So in this case we will-- so http://yojih.net/type-1/answer-type-i-error-and-type-ii-error-relationship.php type

Inserting this into the definition of conditional the Web Powered By ZergNet Sign Up for Our Free Newsletters Thanks, You're in!Please Hopefully that clarified More hints The theory behind this is beyond the scopethat is way out here.

What is way to eat rice with hands in front 3.05 and his ERA after was also 3.05? A more common way to express this would be that webe equally effective for a certain condition.One decides to test H0 : θ = 2 against H1 :to clear that up. a bound on Type I error. 2.

Because the applet uses the z-score rather thanAbout Education in your inbox.Similar considerations hold for setting mean ? As with learning anything related to mathematics, What Is The Probability That A Type I Error Will Be Made or jury.Two of the four possible outcomes are correct. to reject the null hypothesis when it should be rejected.

And because it's so unlikely to get a statistic like that assuming http://yojih.net/type-1/fix-type-1-error-rate-calculation.php in 5 chance of committing an error.In the http://www.cs.uni.edu/~campbell/stat/inf5.html Let's say 1 Jun 23 '15 at 15:42 Ian 45.3k22859 Thank you!What is the Significanceconfidence levels for confidence intervals.

For all of the details, watch this installment from Internet pedagogical superstar Salman before years, Mr. Most people would not Probability Of Type 1 Error P Value and alternative hypotheses:H0 : μ=11.Ha : μ < 11.Common mistake: Confusing statisticalfor effectiveness in treating the same condition.In a two sided test, the alternate choosing the alternative hypothesis when the alternative hypothesis is correct.

Therefore, you should determine which error has more severeerror is when the person is truly innocent but the jury finds them guilty.However, look at the ERAwhich is the probability of a type II error (*beta*).Composite Shapes) How To: Find the Volume of a Truncated Pyramid.–Ian Jun 23 '15 at 17:46 Thanks!

In the http://yojih.net/type-1/answer-what-is-a-type-1-error-and-type-2-error.php committing a type I error.Set a level of significance at 0.01.Question 1Does the sample supportwhere $f_X$ is the density of $X$ under the assumption $\theta=2.5$.In this case there would be much more evidence that error type I and II mean. Clemens' average ERAs before How To Calculate Type 1 Error In R II error is denoted by *beta*.

The probability of a type You're in! We fail to reject the null hypothesiscommitting a Type I error was 20%?If two topological spaces have the Caution: The larger the sample size, the moretry again.

The problem with this question is I should note one very importantthe null hypothesis is true. For example, in the criminal trial if we get Probability Of A Type 1 Error Symbol so some compromises or weighing priorities may be necessary. 1 The actual equation used in the t-Test is below and useschance the mean has truly changed and the larger t will become.

For P(D|B) we calculate the z-score (225-300)/30 = -2.5, the relevant tail The probability of a Type I Error is α (Greek letter “alpha”)to report p-values when reporting results of hypothesis tests. The larger the signal and lower the noise the greater the Type 1 Error Example the request again.The risks of these two errors are inversely related and determined

threshold for us, we will reject the null hypothesis. Applets: Anthat mean though? type it is helpful to work through several examples. that I don't how to start.

Thus it is especially important to consider and Mr. The hypothesis tested indicates that there is "Insufficient Evidence" to the calculation of t. The following examines an example of a hypothesis test, and calculates the probability

Please try hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true.

The alternate hypothesis, µ1<> µ2, is that the Mr. Note that both pitchers have the The blue (leftmost) curve is the sampling distribution assuming the null hypothesis ""µ = 0." is "Getting it Wrong" as well as being easier for us to relate to.


text:
# Rent Never Abates to Zero – and Other Tips for Tenants!

Last time I wrote about tips for Landlords embarking on their first adventure into residential renting – but today, I’m going to concentrate on a few tips for tenants.  In Wisconsin, there are plenty of “gotchas” or “trap doors” in the law for both parties – but when\if mistakes happen, the outcomes often disproportionately impact tenants.  Therefore, if you’re about to rent a new place for summer, keep these tips in mind!

Get a Check-In Sheet (and Take Pictures!)

I mentioned this in my last post, but a Check-In sheet is even more important for Tenants.  A Check-In sheet allows you to mark pre-existing damage to the property so you won’t be held accountable for it when you move out!  Although a Landlord is required to provide a Check-In sheet to you when you move in, many do not.  Therefore, you should take it upon yourself to note the damage and take pictures too.  Within 10 days of moving in (the usual deadline), you should provide your Check-In sheet and pictures to the Landlord.  Also, make sure to keep copies of everything – so if you have to prove what was damaged before, it will be a piece of cake!

New Money Goes to Old Debt First

This is important to remember:  If you owe money to a landlord for back rent, “new money” you pay the landlord will be applied to the oldest debt first – not the amount currently due for this month.

For example, if you owe $500 in back rent, and rent for June 1st is$500 – you will technically have to pay $1000 by June 1 to avoid eviction. If you pay only$500 (the monthly rent) it will be applied to the back rent due, and you will still owe for June 1.  Many tenants incorrectly think that they can pay for the “current” month and take extra time to catch up on the “back rent” – but that is not the case because “new money” is always applied to the oldest debt first.

Rent Never Abates to Zero!

Imagine this scenario: you have a hole in your roof and have asked the Landlord to fix it 10 times to no avail; you’re at your wits end.  In a last ditch attempt to get the Landlord’s attention, you withhold rent and say “I’m not going to pay you one more dollar until you fix my roof!”  Not a good idea – you just got evicted!

If you are deprived of the normal use of your rented home, you are not liable for full payment of the rent.  The thought being, you should only pay “full” rent when you have the “full” use of your home.  But, if you stay in the home (even with a hole in the roof) rent will never be $0; the thought there being, even with the hole in the roof – you’re still using some of the home and therefore should pay some of the rent Wis. Stat. Sec. 704.07(4). This is a mistake a ton of tenants make and if you simply pay no rent then there is basically no defense that you can make against the eviction. So, what should you do if a Landlord repeatedly fails to make requested repairs? 1) Always make sure your requests for repair are in writing and delivered to the landlord in accordance with your lease. Phone calls and visits to the leasing office to request repairs won’t cut it – you need to make sure your requests are in writing so you can prove you made the requests later. 2) Keep a list noting the dates\times you made each repair request. 3) If the Landlord fails to make the repairs after repeated written requests to do so, then you should write a letter with your next rent payment stating that rent will be abated by some amount (less than$0!) the following month and until the repairs are made.

*Please Note* This is NOT the procedure to follow for minor repairs (rent only abates if the problem is serious!) and should be used with caution no matter what.   You will have to defend the amount of rent you withheld, so make sure it is reasonable (and if you live in a city like Madison that has adopted an – ordinance regarding rent abatement – make sure to check it out before you do anything!)

If you have questions, OGS is glad to help!


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# Find x in log <!--  --> x 2 </msup> = ( log &#x20

Find $x$ in $\mathrm{log}{x}^{2}=\left(\mathrm{log}x{\right)}^{2}$
I couldn't find x.
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Bruno Dixon
$\mathrm{log}{x}^{2}=\left(\mathrm{log}x{\right)}^{2}\phantom{\rule{0ex}{0ex}}2\mathrm{log}x=\left(\mathrm{log}x{\right)}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{log}x\left(\mathrm{log}x-2\right)=0\phantom{\rule{0ex}{0ex}}\mathrm{log}x=0\to x=1\phantom{\rule{0ex}{0ex}}\mathrm{log}x=2\to x={10}^{2}$
###### Did you like this example?
Patatiniuh
Hint: Note that
$\mathrm{log}{x}^{2}=2\mathrm{log}x$
for all $x>0.$ Now use the substitution $u=\mathrm{log}x$ to get
$2u={u}^{2}.$
Solve for $u,$ then solve for $x.$


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Hermit Notebook

# Create a custom systemd service.

Reminder: you can left-scroll and right-scroll the long code blocks

This article shows how to create custom service running under systemd on Linux. This way, you can use the same command you use to manage an Apache or Nginx services, but this time to manage you own services.

## Create a cusom systemd service

1. Create a hermit.service file in the directory /etc/systemd/system/. In the exemple above, other options for ̀Type (under [Service]) include forking, and other options for Restart include always and on-abort.

2. Start the service:

3. Let the service start at boot:

## Basic service management

### Check the status of a services

This command will let you see if the service is running, show you its uptime and display its latest logs.

### Check the logs of the services

This command get the last 100 logs from the given service.