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# A little question: is x-boson included in su(5)?

1. Oct 2, 2006

### francesco_barbero

A little question:

is x-boson included in su(5)? (or in some generalization of it) :uhh:

2. Oct 2, 2006

### garrett

Yes.

The su(5) Lie algebra has 24 dimensions -- so 24 gauge bosons. In the su(5) GUT, 4 of these get identified with the electroweak, W and Z, bosons and 8 get identified with the gluons, G. This leaves 12 gauge bosons -- the X -- which are given large masses to explain why we haven't seen them.

3. Oct 2, 2006

Thank you!


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## July 11, 2011

### Doctrinal and Tannakian Reconstruction

#### Posted by David Corfield

Interest in doctrines at the Café goes right back to one of its earliest posts nearly five years ago, and even to one a few days earlier. We’ve begun to record some material at the doctrine page in nLab, but I’m sure there’s more wisdom from the blog to be extracted. Looking over the material again raised a question or two in my mind, which I’d like to pose now.

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the recovery of a theory from the category of models of that theory.

There are a range of similar dualities:

1. Adámek, Lawvere and Rosicky established a similar biequivalence between the 2-category of small Cauchy complete categories with finite products, finite product preserving functors and natural transformations, and the 2-category of multisorted finitary varieties, algebraically exact functors and natural transformations in On the duality between varieties and algebraic theories.

2. Todd tells us here of the duality between the 2-category of Cauchy complete categories on one side, and the 2-category of presheaf toposes and bicontinuous functors on the other.

3. I think I have it right that there’s a biequivalence between the 2-category of finite product categories and the 2-category of categories with all limits and colimits and with functors which preserve limits, filtered colimits, and regular epimorphisms. [EDIT: this needs modifying.]

This talk of recovering something from its category of models put me in mind of Tannakian reconstruction of algebraic entities from their category of modules or comodules or whatever. So I was wondering where there is a way to see doctrinal, Gabriel-Ulmer type, duality and Tannaka duality as instances of something larger. The only reference I have found is Brian Day in Enriched Tannaka reconstruction writing

Our approach provides a synthesis of Tannaka reconstruction and Gabriel-Ulmer duality.

Unfortunately, I find that hard to understand.

John seems to see a link. In TWF200 he writes

fans of the Tannaka-Krein reconstruction theorem for quantum groups will recognize this duality between “theories and their category of models” as just another face of the duality between “algebras and their category of representations” - the classic example being the Fourier transform and inverse Fourier transform!

Daniel Schäppi discusses the semantics-structure adjunction in Tannaka duality for comonoids in cosmoi

The functor which sends a comonad to its category of comodules is traditionally known as the semantics functor. It has a partial left adjoint, which is called the structure functor (cf. [Dub70], [Str72]). The partial here means that the left adjoint of the semantics functor is only defined on a certain subcategory. These names go back to Lawvere (see [Law04, p. 77]). A monad can be viewed as a sort of logical theory, and from this viewpoint the semantics functor sends it to its category of models; the study of the models of a logical theory is generally called its semantics.

So, in the case of a fiber functor, say the functor from the category of representations of a group to their underlying vector spaces, Tannaka reconstruction allows us to recover the group from the enriched endomorphisms of the fiber functor. Now, is it possible that this could be described as in the case of doctrinal duality?

I see the Urs of five years ago wrote

So instead of admitting that I am talking about a linear representation of some group $G$, I could equivalently say that I am talking about a model in $Vect$ of the theory $\Sigma(G)$ with respect to the doctrine $Cat$ of no structure.

If $Rep_{C}(G) = Hom_{Cat}(\Sigma (G), C)$ is the category of representations of $G$ in $C$, is there a $C$ and a doctrine, $\mathcal{D}$, for which $Hom_{\mathcal{D}}(Rep_{C}(G), C)$ is equivalent to the category with single object the fiber functor and its invertible endomorphisms, or $\Sigma{G}$?

Mike said back here that $Cat^{op}$ should be something like

“cocomplete categories equipped with a strongly-generating set of tiny objects, and functors having both left and right adjoints (equivalently, by the adjoint functor theorem, preserving small limits and colimits) whose left adjoint preserves the chosen generators”. That’s obviously similar to “complete atomic Boolean algebras” except that we have to consider the “atoms” as structure, rather than a property. If you make it into a property and consider instead “cocomplete categories for which there exists a strongly-generating set of tiny objects, and functors having both left and right adjoints” then you get the subcategory $Cat_{cc}^{op}$ of Cauchy-complete categories.

Is it possible that this is the doctrine $\mathcal{D}$ I’m after?

A couple more questions then occur

1. Given that the groupoidification program seeks to avoid linearisation of representations via vector spaces in favour of groupoids, how do groupoidification and Tannakian reconstruction fit together?

2. I first met fiber functors when reading about reconstructing the fundamental groupoid of a space from its category (topos) of sheaves. Is there a doctrinal dual to Toposes which would allow for a duality for this reconstruction?

You might then think about collecting doctrines together in a 3-category. In the doctrine page of John’s web, it says

Categorified Gabriel-Ulmer duality is all about recovering the syntactic doctrine from the semantic doctrine,

with a view to taking models of the syntactic doctrine in the 2-category of groupoids. But few details are given there.

I wonder if there is something like the Pontryagin duality for locally compact Hausdorff abelian groups, with its induced duality between properties such as discrete and compact or Lie and finite rank? Is there a duality for doctrines, which restricts to one between limit doctrines and some other kind, or to one between syntactic and semantic doctrines?

Posted at July 11, 2011 11:01 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2417

### Re: Doctrinal and Tannakian Reconstruction

I see Jim Dolan uttering ‘doctrine’ and ‘Tannaka-Krein’ in the same breath, or at least same comment.

Hmm,

the finite-dimensional representations of gl(n) form the free symmetric monoidal finitely cocomplete algebroid on one “n-dimensional” object.

And he talks earlier of

the “doctrine” of symmetric monoidal finitely cocomplete algebroids.

Why the scare quotes?

Posted by: David Corfield on July 11, 2011 12:27 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

David wrote:

Why the scare quotes?

Jim Dolan really hates it when I speak for him. But here I go again:

Maybe the square quotes are there because Jim hadn’t in that comment given his favorite definition of ‘doctrine’, which differs from Lawvere’s original definition of ‘doctrine’ as a monad on the category Cat, and also differs from the later definition of ‘doctrine’ as a pseudomonad on the 2-category Cat. I imagine he was wanting to warn us that the best definition is somewhat up for grabs.

Posted by: John Baez on July 15, 2011 7:10 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

So many paths in this post lead to Jim. There are bits and pieces spread over various webs. Besides your doctrines, there are Jim’s Algebraic Geometry, Todd’s Notes on conversations with James Dolan in Todd Trimble, and Alex’s Doctrines in Alex Hoffnung. I see also Jeffrey Morton comments on groupoidification and Tannaka reconstruction.

Posted by: David Corfield on July 15, 2011 10:37 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Geoffrey Morton replies to a question of mine on groupoidification and reconstruction.

Posted by: David Corfield on July 20, 2011 1:51 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Hi David,

by what I said in the other reply, all that is necessary to make bare Tannaka reconstruction work is the Yoneda lemma.

So if you want Tannaka reconstruction for representations in $Span(Grpd)$, what you need is a Yoneda-lemma in $Span(Grpd)$-enriched (higher) category theory.

That shouldn’t be a problem.

Posted by: Urs Schreiber on July 20, 2011 8:50 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I’d forgotten about the categorified universal properties of HDA II. E.g.,

$Rep(U(n))$ is the free connected symmetric 2-$H^{\ast}$-algebra on an even object $x$ of dimension $n$.

Posted by: David Corfield on July 19, 2011 10:13 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Here’s something I’d like to understand. Given a finite limit category $A$, we get a new category $FinLim(A, Set)$. Gabriel-Ulmer duality says (among other things) that we can recover $A$ up to equivalence from $FinLim(A, Set)$: it’s the full subcategory of finitely presentable objects, right?

Now take a finite product category $A$. We get a new category $FinProd(A, Set)$. Can we recover $A$ up to equivalence from $FinProd(A, Set)$?

A closely related question: can we recover a Lawvere theory from its category of models (in $Set$)? Of course we can do so if we’re also given the forgetful functor from the category of models to $Set$, but that’s not what I’m asking.

Or to put it another way: can different Lawvere theories have the same category of models?

Posted by: Tom Leinster on July 11, 2011 3:00 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Can we recover $A$ up to equivalence from $FinProd(A,Set)$?

I’ll wait for the experts, but isn’t that what comes about with my example 3, i.e, if you take morphisms in the 2-category of categories with all limits and colimits and with functors which preserve limits, filtered colimits, and regular epimorphisms? That’s what I gleaned.

There was also some interesting material about fixing a concept and looking at how the theory changes in different doctrines beginning with Todd here.

Posted by: David Corfield on July 11, 2011 3:16 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I have an answer to that. Yes! The Lawvere theory given by powers of a 2-element set (as a subcategory of $Fin$) has the same models as the Lawvere theory of powers of a 3-element set. For they both have the same Cauchy completion (cf. the discussion here).

Of course, finite limit theories escape this fate because finitely complete categories are Cauchy complete.

Posted by: Todd Trimble on July 11, 2011 3:20 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

And that Adamek et al. paper says

If a variety $\mathcal{V}$ is viewed as an abstract category, i.e., without reference to underlying sets and mappings, algebraic theories turn out not to be entirely satisfactory: every non-trivial variety has many (non-equivalent) algebraic theories. For example the variety $Set$ of algebras with no operations is equivalent to $Mod \mathcal{T}$ where $\mathcal{T}$ is the full subcategory of $Set^{op}$ whose objects are all natural numbers. But it is also equivalent to $Mod \mathcal{T}'$ where $\mathcal{T}'$ is the full subcategory of $Set^{op}$ whose objects are just the even numbers.

The paper proposes a “remedy”.

Posted by: David Corfield on July 11, 2011 3:25 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Just to be clear, my “yes!” was in reply to Tom’s final question. It implies a “no!” to the other questions: Tannaka reconstruction doesn’t generally hold for finite product theories.

Posted by: Todd Trimble on July 11, 2011 3:27 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

But it’s right to say, isn’t it Todd, that a finite product theory can be recovered from the category of its models in sets, by taking set-valued functors that preserve all limits, filtered colimits, and regular epimorphisms?

Posted by: David Corfield on July 11, 2011 3:37 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Oh, I think I misread Forssell’s thesis and this only holds for equational (or algebraic) theories, Section 1.2.2.

Posted by: David Corfield on July 11, 2011 3:52 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

David, I’m getting a 404 error when I click on that link. But even for algebraic theories, something seems wrong to me here.

A functor $Alg \to Set$ that preserves limits is representable, a representable that preserves filtered colimits is represented by a finitely presented algebra $A$, and on top of all this, if $\hom(A, -): Alg \to Set$ preserves regular epimorphisms, then $A$ is what I would call a finitely presented projective algebra. The category opposite to finitely presented projective algebras ought to be the Cauchy completion of the category opposite to the category of finitely generated free algebras, i.e., the Cauchy completion of a Lawvere theory. But f.p. projective algebras need not be the same as f.g. free algebras, as I was trying to indicate in my example.

Let me be more concrete about this. Take the algebraic theory of Boolean algebras. I claim that any finite (nonterminal) Boolean algebra $B$, not necessarily a free one (with respect to the usual underlying functor $U: Bool \to Set$), is a projective algebra, i.e., that

$Bool(B, -): Bool \to Set$

preserves regular epimorphisms. To prove this, we only need to show $B$ is a retract of a free finite Boolean algebra, since free Boolean algebras are certainly projective.

In the dual Stone space picture, this is easy to see: it’s the fact that any finite nonempty set $S$ with the discrete topology is a retract of the Stone space of a free Boolean algebra, e.g., the power set $2^S$.

Posted by: Todd Trimble on July 11, 2011 4:46 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I wrote

A functor $Alg \to Set$ that preserves limits is representable

but that’s a bunch of baloney. André Joyal posted a counterexample on the categories mailing list recently, which came to my attention via Mike Shulman in a recent answer on Math Overflow.

I’ve made this assertion not once but several times recently; hopefully it won’t pass my lips again. However, a functor $Alg \to Set$ that preserves limits and is accessible will be representable. (And if $Alg \to Set$ preserves filtered colimits, we do get accessibility of course.)

Posted by: Todd Trimble on July 13, 2011 11:37 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Fixed the link. It’s the material leading up to p. 14. This is just the right kind of muddle where, if I can sort it out in my mind, things become clearer for me.

Posted by: David Corfield on July 11, 2011 5:05 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Thanks for the link, David. I saw what you were referring to on page 14, and it seems to me he’s simply made a mistake. (I don’t see a proof given for the assertion.)

From the standpoint of models, since a finite product theory has the same models as its Cauchy completion, maybe Cauchy complete finite product theories are the ‘right’ thing one ought to consider for this sort of reconstruction theorem. Certainly one can repair his assertion by adding the modifier ‘Cauchy complete’.

Posted by: Todd Trimble on July 12, 2011 10:11 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

But Todd’s example is a pair of algebraic theories (single-sorted, even). So something seems to be amiss.

Posted by: Tom Leinster on July 11, 2011 4:35 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Nice example, Todd! This circle of ideas about Boolean algebras gets more and more interesting.

Of course, finite limit theories escape this fate because finitely complete categories are Cauchy complete.

Right, good point. So I was overlooking an obvious source of examples. Let $A$ be a finite product category that isn’t Cauchy complete. Let $\overline{A}$ be its Cauchy completion, which also has finite products. Then

$FinProd(A, Set) = FinProd(\overline{A}, Set)$

for reasons that momentarily escape me. (I’m sure it’s true, and Todd’s been mentioning it recently, again in connection with Boolean algebras; but there’s got to be a simple conceptual reason that I’m not seeing right now.)

Thus, all we have to do is find a finite product category that isn’t Cauchy complete. One way to do this is to take a nontrivial full subcategory of $Set$ whose collection of objects is closed under finite products. Todd’s examples, the finite sets of cardinality $2^n$ or $3^n$, are of this type.

Posted by: Tom Leinster on July 11, 2011 4:47 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

for reasons that momentarily escape me

I don’t think I’ve given any reasons here at the Café, but one point is:

• If $\mathcal{C}$ has finite products, then so does its Cauchy completion $\widebar{\mathcal{C}}$

because the cartesian product of two retracts of objects $A, B$ of $\mathcal{C}$ is a retract of $A \times B$ (just take the cartesian product of the two retraction pairs).

The second point is that any functor $F: \mathcal{C} \to Set$ has, up to isomorphism, a unique extension to a functor $\widebar{\mathcal{C}} \to Set$, and if $F$ preserves products, so does the extension. This should follow straight from the description of cartesian products in $\widebar{\mathcal{C}}$ given in the paragraph above.

Posted by: Todd Trimble on July 11, 2011 5:29 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Thanks, Todd. Am I right in thinking that the same would work for any kind of limit in place of finite products?

I’m wondering if the situation is something like the following. (I apologize for the sketchiness of what I’m about to say; I’m aware as I type that I haven’t thought this through.) The Cauchy completion of a category is its (free) cocompletion under absolute colimits. Tautologously, finite products commute with absolute colimits. I’m wondering if that can be seen as the reason for which

$FinProd(A, Set) = FinProd(\overline{A}, Set)$

for all finite product categories $A$ (or indeed the same for any other finite product category in place of $Set$).

To test out the validity of this idea, I should try it on some less trivial case. (I say “trivial” because all limits commute with absolute colimits.) For example, if what I have in mind is correct then it should work for finite limits vs. filtered colimits. The filtered cocompletion of a category $A$ is $Flat(A^{op}, Set)$, so if my idea is right then, for starters, $Flat(A^{op}, Set)$ should have finite limits whenever $A$ does. That’s not obvious to me, but perhaps it’s true. I should think about this more carefully.

Posted by: Tom Leinster on July 12, 2011 11:48 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Tom wrote in part

(or indeed the same for any other finite product category in place of $Set$)

You’ll also want the receiving category $C$ in place of $Set$ to be Cauchy complete; otherwise you won’t be able to extend to a functor $\widebar{A} \to C$.

Your general surmise reminds me of distributive laws. If $F$ is a free cocompletion monad of a certain type (e.g., free cocompletion w.r.t. absolute colimits, filtered colimits, etc.) and $G$ is a free completion monad, then it looks like we are considering a distributive law of type

$\theta: G F \to F G$

If $A$ has a $G$-algebra (or pseudo $G$-algebra) structure $\alpha: G A \to A$, then $F A$ acquires a $G$-algebra structure via the composition

$G F A \stackrel{\theta A}{\to} F G A \stackrel{F \alpha}{\to} F A$

Here we could take for instance $F$ to be the filtered colimit cocompletion, $G$ to be the finite limit completion, and I’m imagining that the distributive law $\theta$ here is actually an isomorphism, although I haven’t checked that thoroughly. Similarly if $F$ is the absolute colimit cocompletion.

An example of this where I don’t think $\theta$ is an isomorphism is where $F$ is the cocompletion w.r.t. all small colimits and $G$ is the finite limit completion. (In this case, I’m thinking the 2-monad $F G$ “is morally” the free topos monad. Here I’m thinking toposes and left exact left adjoints as 2-monadic over $Cat$, assuming that naive algebraic intuition pans out correctly w.r.t. size issues.) Even though $\theta$ is not an isomorphism, it’s still true that the free cocompletion is finitely complete. Similarly, I think, taking $G$ to be the small limit completion.

Posted by: Todd Trimble on July 13, 2011 1:52 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Panagis Karazeris wrote to me shortly after I posted my question. That was 15 July, but I was away then for CT11 in Vancouver, so my efficiency plummeted and I’ve only just got round to it.

Anyway, here’s what Panagis told me. I wrote:

if my idea is right then, for starters, $Flat(A^{op}, Set)$ should have finite limits whenever $A$ does. That’s not obvious to me, but perhaps it’s true.

Panagis teased me gently for not remembering the contents of his talk in Barcelona in 2007, when he covered exactly such questions. My one-sentence note from that talk was:

Existence of limits in a cocompletion (w.r.t. some class of colimits) is related to some kind of weak limit property in the original category.

The precise statement that Panagis just told me was:

$Flat(A^{op}, Set)$ has finite limits iff for every finite diagram $D$ in $A$, the category of cones on $D$ is filtered.

A category with a terminal object is certainly filtered, so if $A$ has finite limits then $Flat(A^{op}, Set)$ does indeed have finite limits, as I’d hoped.

The result that Panagis mentioned is here:

Panagis Karazeris, Jiří Velebil, Representability relative to a doctrine, Cahiers de Topologie et Géometrie Différentielle Catégoriques 50 (2009), 3–22.

You can also read some talk slides on the sifted case (i.e. finite products rather than finite limits).

Posted by: Tom Leinster on August 1, 2011 3:33 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

The precise statement that Panagis just told me was:

Best to record such facts on the $n$Lab, where they can be found.

Posted by: Urs Schreiber on August 1, 2011 12:57 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Posted by: Tom Leinster on August 1, 2011 1:24 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Incidentally, I think these questions about calculating the Cauchy completion of finite product theories, even of algebraic theories, are quite interesting and often pretty non-trivial.

I think I managed to convince myself once that the Lawvere theory of groups is Cauchy complete (and therefore has a Tannaka reconstruction along the lines David gave). You can see in this Math Overflow answer a clever and concerted response that implicitly shows (if I’m not mistaken) that the Lawvere theory of commutative rings is also Cauchy complete.

It would be nice to have more examples and counterexamples! Computing the class of idempotent algebra maps is all by itself an interesting and often nontrivial problem.

Posted by: Todd Trimble on July 11, 2011 5:44 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Oh dear, my memory here was very wrong. The Lawvere theory of commutative rings is not Cauchy complete. See the very first page of this article, particularly the reference to D.L. Costa’s paper.

Posted by: Todd Trimble on July 11, 2011 5:55 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

So, I’d still like to know about my central question, which in terms of groups could be posed as

Is there a common framework which includes the (doctrinal) reconstruction of the theory of groups from its category of models and the (Tannakian) reconstruction of a specific group from its category of representations?

Looking at Isbell duality I guess the answer is ‘Yes’. Is there any duality that isn’t covered by it?

Posted by: David Corfield on July 13, 2011 12:07 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I can offer some very tautological ways in which the answer is ‘yes’.

In Tannaka reconstruction, we have not just a category of representations but also an underlying or forgetful functor to work with. In the most tautological formulations, we work with the category of all representations and its underlying-set functor

$U: Set^G \to Set.$

This is representable: $U = Set^G(G, -)$. A Yoneda lemma argument gives that the monoid of endomorphisms $Endo(U)$ is isomorphic to the monoid of $G$-set maps $G \to G$, and this monoid is isomorphic to $G$ as a group. In this way we recover $G$ from $U$.

For the category of models of a Lawvere theory $T$, we again have a forgetful functor

$U: Mod(T) \to Set$

which is again representable by $F[1]$, the free $T$-algebra on one generator. This time we don’t form the endomorphism monoid, but an “endomorphism theory” on $U$ whose $n$-ary operations are

$Endo(U)[n] = \hom(U^n, U)$

and here too $T$ is isomorphic (as a Lawvere theory) to $Endo(U)$. Again, this is by a Yoneda lemma argument (e.g., $U^n$ is represented by the free algebra $F[n]$).

There has to be a very general sort of doctrinal story here. Thinking of the objects of a doctrine (in Jim’s sense) as ‘theories’, in one case the ‘theories’ are groups or monoids, in another they are Lawvere theories. In these cases the theory is recovered from an “endomorphism theory” of an underlying functor on its category of models.

It might be thought these cases are slight cheats, because we’ve been concentrating on “single-sorted theories”, where we deal with just one underlying functor $U$ which plays the role of a “universal sort” which allows us a toehold to get at the theory. But there’s a trick which allows us to deal with say multi-sorted Lawvere theories, where we can package multiple universal sorts $U_s: Mod(T) \to Set$ (ranging over a set of sorts $S$) as a single functor $U: Mod(T) \to Set/S$. (One role of $Set/S$ here is that it is the free category with small products generated by a discrete category $S$ – it’s a good environment for not just finitary algebraic theories, but various infinitary theories.)

I have to leave soon, but I’d like to come back to Gabriel-Ulmer duality (and beyond).

Posted by: Todd Trimble on July 15, 2011 2:57 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Thanks for this Todd. That’s helpful.

Your ‘and beyond’ sounds intriguing. Presumably we ought to be looking for 2-theories and their models. Like you showed us that the group concept resides in different doctrines, I take it that there will be a Lex concept which will reside as a theory in different 2-doctrines, models from each into $Cat$ giving $Lex$.

Posted by: David Corfield on July 15, 2011 3:56 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Note: if we ever do follow up categorified Gabriel-Ulmer duality, check out Makkai’s talk 2-dimensional Gabriel-Ulmer duality.

Posted by: David Corfield on July 19, 2011 1:51 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Just to amplify on what Todd just said about how Tannaka reconstruction is essentially just the Yoneda lemma argument:

the details of the Yoneda-aspect are at the beginning of the entry Tannaka reconstruction . We once wrote this out when I was thinking about $(\infty,1)$-Tannaka duality. Simple and useful as the argument is, I am not aware that it is stated explicitly anywhere except in this $n$Lab entry. (If you know a reference, please let me know.)

But notice that most of what is called Tannaka duality in the literature does contain a second step, which is not just general abstract:

There is a general-abstract and a concrete aspect to Tannaka duality. The general abstract one says that an algebra/group $A$ is reconstructible from the fiber functor on the category of all its modules/representations. This is a pure general abstract Yoneda argument. The concrete one says that in nice cases it is reconstructible from the category of dualizable (finite dimensional) modules, even if it is itself not finite dimensional.

More precisely…

(see the entry for the details).

But this means, conversely, that if we are content with looking at a kind of Tannaka duality for the collection of all representations, hence just at the Yoneda aspect of it, the argument is of vast generality. In particular it immediately generalizes to ∞-algebras over algebraic ∞-theories. At Tannaka duality in higher category theory two applications are mentioned: Tannaka duality for $(\infty,1)$-permutation representations, and their application to Galois theory in cohesive ∞-toposes .

Posted by: Urs Schreiber on July 19, 2011 8:41 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I first learned that Tannaka duality reduces to abstract nonsense when you look at all modules/representations from Daniel Schäppi’s paper Tannaka duality for comonoids in cosmoi, which is linked from the nLab page. He phrases it in different terminology (the comonadicity theorem for cocontinuous comonads), but all abstract nonsense is interconvertible to any other abstract nonsense, right?

Posted by: Mike Shulman on July 20, 2011 3:13 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

…all abstract nonsense is interconvertible to any other abstract nonsense.

An interesting thesis, in accord with Mac Lane’s

The notion of Kan extensions subsumes all the other fundamental concepts of category theory.

Posted by: David Corfield on July 20, 2011 10:00 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

right?

Right, I should have mentioned this. In fact, I was scanning the references of the entry yesterday trying to remember which one that was, since we had talked about it at length back then. But then I couldn’t find it before I ran out of time.

It still seems to me that the Yoneda argument is noteworthy for its sheer simplicity. As far as I remember, Schäppi’s presentation does not quite convey the striking fact that one can teach half of Tannaka duality as a 1-line excercise proof right after introducing the Yoneda lemma by simply saying: “hint: apply it four times in a row”.

I think this is what I am wondering if nobody noticed it in the literature before.

Posted by: Urs Schreiber on July 20, 2011 10:38 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Re Tannaka duality for $(\infty,1)$-categories - James Wallbridge (a student of Toen, Murray and Varghese) did this in his thesis, which he recently defended. AFAIK he doesn’t have an online presence.

Posted by: David Roberts on July 20, 2011 7:51 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

James Wallbridge (a student of Toën, Murray and Varghese) did this in his thesis

I didn’t know this. Hopefully I’ll be able to get hold of this thesis.

AFAIK he doesn’t have an online presence.

Hm, I see a homepage of a James Wallbridge in Adelaide here. While this mentions a seminar titled Higher stacks and homotopy theory II: the motivic context it also speaks of a PhD thesis on Noncommutative geometry in string and M theory . Is that him, or not? Did he write two theses?

Indeed, here I see an announcement of a talk by him on Yoneda in homotopical algebraic geometry .

Posted by: Urs Schreiber on July 20, 2011 10:28 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

No, not two theses; the NC-geometry title was a provisional one from when he started his candidature, the higher category one, “Higher Tannaka duality”, is the real one. The talk about higher stacks was also him. I have handwritten notes, but there was a lot of background in the talk, so no new stuff. He gave at the end as an example the Morel-Voevodsky stable motivic (oo,1)-category.

Posted by: David Roberts on July 20, 2011 10:50 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

the NC-geometry title was a provisional one from when he started his candidature, the higher category one, “Higher Tannaka duality”, is the real one.

I see, thanks. Okay, so I have recorded what we have so far here. Let’s update this as soon as we know more.

Posted by: Urs Schreiber on July 20, 2011 10:59 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

So what was in Toën’s MSRI talk Higher Tannaka duality?

Posted by: David Corfield on July 20, 2011 11:55 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

There is a general-abstract and a concrete aspect to Tannaka duality. The general abstract one says that an algebra/group $A$ is reconstructible from the fiber functor on the category of all its modules/representations.

Does this suggest that there are (at least) two concrete aspects:

1. Reconstruction from finite dimensional modules.

2. Reconstruction from all modules but without a specified fiber functor.

Posted by: David Corfield on July 20, 2011 8:30 AM | Permalink | Reply to this

format: plain


text:
Dataset Open Access

# Galactic dust absorption maps

Leike, Reimar H.; Enßlin, Torsten A.

### Dublin Core Export

<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Leike, Reimar H.</dc:creator>
<dc:creator>Enßlin, Torsten A.</dc:creator>
<dc:date>2019-01-21</dc:date>
<dc:description>This dataset contains the 3D structure of interstelllar dust in the vicinity of the Sun, as well as its logarithmic spatial correlation power spectrum. It was computed from Gaia absorption data using Gaussian process regression.

The simple_cube contains only mean and standard deviation while the full_cube contains 50 approximate posterior samples.

The corresponding paper is available on arxiv:1901.05971

In this version the documentation was corrected. The coordinate system does not have the half-pixel shift that is described in the previous data set versions, leading to an exactly centered box. Additionally, the description of the relation of power spectrum and dust realization was extended to include the a-priori mean of 0.001 e-folds of absorption per parsec.
The numerical content of the data set is identical to version 1.1</dc:description>
<dc:identifier>https://zenodo.org/record/2577337</dc:identifier>
<dc:identifier>10.5281/zenodo.2577337</dc:identifier>
<dc:identifier>oai:zenodo.org:2577337</dc:identifier>
<dc:language>eng</dc:language>
<dc:relation>doi:10.5281/zenodo.2542807</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:title>Galactic dust absorption maps</dc:title>
<dc:type>info:eu-repo/semantics/other</dc:type>
<dc:type>dataset</dc:type>
</oai_dc:dc>

376
176
views


text:
My Math Forum no. of ordered pair

 Algebra Pre-Algebra and Basic Algebra Math Forum

 March 28th, 2011, 06:02 PM #1 Member   Joined: Jan 2011 Posts: 40 Thanks: 0 no. of ordered pair The no. of ordered pair of the equation $\frac{sin^{-1}x}{x}+\frac{sin^{-1}y}{y}= 2$
 March 28th, 2011, 09:09 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,930 Thanks: 1124 Math Focus: Elementary mathematics and beyond Re: no. of ordered pair My guess: 1. I've spent several minutes working on this and intend to return in the morning. (hours, minutes, whatever . . .)
 March 29th, 2011, 01:03 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,930 Thanks: 1124 Math Focus: Elementary mathematics and beyond Re: no. of ordered pair The range of $\frac{\arcsin(x)}{x}$ is $\(1,\,\frac{\pi}{2}\]$ so there are no ordered pairs $(x,\,y)$ satisfying $\frac{\arcsin(x)}{x}\,+\,\frac{\arcsin(y)}{y}\,=\, 2$

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text:
# Ruby for Newbies: Operators and their Methods

This post is part of a series called Ruby for Newbies.
Ruby for Newbies: Iterators and Blocks
Ruby for Newbies: Working with Directories and Files

Ruby is a one of the most popular languages used on the web. We've started a new screencast series here on Nettuts+ that will introduce you to Ruby, as well as the great frameworks and tools that go along with Ruby development. In this lesson, we’ll be taking a deeper look at operators in Ruby, and why they are different from anything you’ve ever seen before.

## Operators

You’ve familiar with operators.

 1 1 + 2 # 3  2   3 person[:name] = "Joe"

Operators are things like the plus sign (one of the arithmetic operators), or the equal sign (the assignment operator). These things don’t look to much different from the ones you use in JavaScript, PHP, or any other language. But—like most of Ruby—there’s a lot more than meets the eye going on here.

Here’s the secret: operators in Ruby are really method calls. Try this:

 1 1.+(2) # 3

Here, we’re calling the + operator on the object 1, passing in the object 2 as a parameter. We get back the object 3. We can do this with strings too:

 1 name = "Joe"  2   3 name.+(" Smith") # "Joe Smith", but name is still "Joe"  4   5 name += " Smith" # name is now "Joe Smith"

As you can see, we can do string concatenation with the + method. As a bonus here, ruby defines the += operator based on the + operator (note: you can’t use += as a method).

As you might realize, this gives us incredible power. We can customize the meaning of adding, subtracting, and assigning objects in our custom classes. We saw how this works with properties on objects in our lesson on classes (we defined a property and property= method in the class, and got the expected syntax sugar for using them). What we’re looking at here is taking that a step further.

## Building our own Operator Methods

Let’s try to create one of these methods ourselves. For this example, let’s create a refrigerator object, that we can add things to via the + operator and take things out of via the - operator.

Here’s the start of our class:

 1 class Fridge  2  def initialize (beverages=[], foods=[])  3  @beverages = beverages  4  @foods = foods  5  end  6   7  def + (item)  8   9  end  10   11  def - (item)  12   13  end  14 end

Our initialize function is pretty simple: we take two parameters (that fall back to empty arrays if nothing is given), and assign them to instance variables. Now, let’s build those two functions:

 1 def + (item)  2  if item.is_a? Beverage  3  @beverages.push item  4  else  5  @foods.push item  6  end  7 end

This is pretty simple. Every object has an is_a? method that takes a single parameter: a class. If the object is an instance of that class, it will return true; otherwise, it will return false. So, this says that if the item we’re adding to the fridge is a Beverage, we’ll add it to the @beverages array. Otherwise, we’ll add it to the @food array.

That’s good; now, how about taking things out of the fridge? (Note: this method is different from the one shown in the video; this shows you that these operator method give us a great deal of flexibility; they are really just normal methods that you can do anything with. Also, I think this is a better version of the method; however, it’s more complex.)

 1 def - (item)  2  ret = @beverages.find do |beverage|  3  beverage.name.downcase == item.downcase  4  end  5   6  return @beverages.delete ret unless ret.nil?  7   8  ret = @foods.find do |food|  9  food.name.downcase == item.downcase  10  end  11   12  @foods.delete ret  13 end

Here’s what’s going on when we use the minus operator. The parameter that it takes is a string, with the name of the item we’re looking for (By the way, we’ll create the Beverage and Food classes soon). We start by using the find method that arrays have. There are a few ways to use this method; we’re passing it a block; this block says that we’re trying to find the item in the array which has a name property that’s the same as the string we passed in; note that we’re converting both strings to lowercase, to be safe.

If there’s an item that matches in the array, that will be stored in ret; otherwise, ret will be nil. Next, we’ll return the result of @beverage.delete ret, which removes the item from the array and returns it. Notice we’re using a statement modifier at the end of that line: we do this unless ret is nil.

You might wonder why we’re using the keyword return here, since it’s not required in Ruby. If we didn’t use it here, the function wouldn’t return yet, since there’s more code to the function. Using return here allows us to return a value from a place the function wouldn’t normally return.

If we don’t return, that means the item wasn’t found in @beverages. Therefore, we’ll assume it’s in @foods. We’ll do the same thing to find the item in @foods and then return it.

Before testing this out, we’ll need our Food and Beverages classes:

 1 class Beverage  2  attr_accessor :name  3   4  def initialize name  5  @name = name  6  @time = Time.now  7  end  8 end  9 class Food  10  attr_accessor :name  11   12  def initialize name  13  @name = name  14  @time = Time.now  15  end  16 end

Note that in the video, I didn’t make @name accessible from outside the object. Here, I’m doing that with attr_accessor :name, so that we can check the name of these object when they’re inside a fridge.

So, let’s test it out in irb; we’ll start by requiring the file that holds the code; then, give the classes a try; note that I’ve added line breaks to the output for easier reading.

 1 > require './lesson_6'  2 => true  3   4 > f = Fridge.new  5 => #  6   7 > f + Beverage.new("water")  8 => [#]  9   10 > f + Food.new("bread")  11 => [#]  12   13 > f + Food.new("eggs")  14 => [  15  #,  16  #  17  ]  18   19 > f + Beverage.new("orange juice")  20 => [  21  #,  22  #  23  ]  24   25 > f  26 => #,  29  # ],  30  foods[  31  #,  32  # ]  33 > f - "bread"  34 => #  35 > f  36 => #,  39  #],  40  foods[#]

As we go along, you can see things being added to the @beverages and @foods arrays, and then subsequently removed.

## Get and Set Operators

Now let’s write methods for the get and set operators used with hashes. You’ve seen this before:

 1 person = {}  2   3 person[:name] = "Joe"

But, since these operators are methods, we can do it this way:

 1 person.[]=(:age, 35) # to set  2   3 person.[](:name) # to get

That’s right; these are normal methods, with special sugar for your use.

Let’s give this a try; we’ll make a Club class. Our club with have members with different roles. However, we may want to have more than one member with a given role. So, our Club instance will keep track of members and their roles with a hash. If we try to assign a second member to a role, instead of overwriting the first one, we’ll add it.

 1 class Club  2  def initialize  3  @members = {}  4  end  5   6  def [] (role)  7  @members[role]  8  end  9   10  def []= (role, member)  11   12  end  13 end

The get version is pretty simple; we just forward it to the @members array. But set is a little more complicated:

 1 def []== (role, member)  2  if @members[role].nil?  3  @members[role] = member  4  elsif @members[role].is_a? String  5  @members[role] = [ @members[role], member ]  6  else  7  @members[role].push member  8  end  9 end

If that role has not been set, we’ll just set the value of that key to our member hash. If it has been set as a string, we want to convert that to an array, and put the original member and the new member in that array. Finally, if neither of those options are true, it’s already an array, and so we just push the member into the array. We can test this class this way:

 1 c = Club.new  2   3 c[:chair] = "Joe"  4   5 c[:engineer] = "John"  6   7 c[:engineer] = "Sue"  8   9 c[:chair] # "Joe"  10   11 c[:engingeer] # [ "John", "Sue" ]

There you go!

## Other Operators

These aren’t the only operators that we can do this with, of course. Here’s the whole list:

• Arithmetic Operators: + - * \
• Get and Set Operators: [] []=
• Shovel Operator: <<
• Comparison Operators: == < > <= >=
• Case equality Operator: ===
• Bit-wise Operator: | & ^


text:
# Partial fraction integration with unclear roots

Let's look at a simple example like $\frac{1}{x^3+2x+1}$ here. We know that the denominator has a real root between $0$ and $-1$ (could go closer, but that's not the point). By the concept of slope of a cubic being always positive or always negative, now I need to turn it into ($ax+b/$quadratic$)+(1/$linear form$)$. For that, I'm going to need that root. So, in general, like the case of cubic where the roots are hard to calculate (with non real ones being in pairs as per theory), is there another way to factorise the denominator into linear and quadratics, or do I need to find suitable substitutions to find the actual root of equation like in quadratic which is near about impossible during examination time? (I can take care of repeated roots and other fundamental things related to partial fractions.)

• If you can factorize the denominator into the product of a linear factor $x - a$ and a quadratic, then you've found a root $a$, and vice versa, no? – Travis Willse Jan 21 '16 at 22:22
• I think one needs the factors just to set up partial fractions ... – coffeemath Jan 21 '16 at 22:22

It's not a trivial amount of work but at least this is a depressed cubic already, and you can use something like Vieta's substitution. Let $x = w - \frac{2}{3w}$ to get:

$$x^3 + 2x + 1 = w^3 + 1 - \frac{8}{27w^3} = 0.$$

Multiply by $w^3$:

$$w^6 + w^3 - \frac{8}{27} = 0,$$

with a larger root

$$w^3 = \frac{-1 + \sqrt{1 + \frac{32}{27}}}{2}\equiv C.$$

Then the three cube roots of $w^3$ are $\sqrt[3]{C}, (\pm \frac{\sqrt{3}i}{2}-\frac{1}{2})\sqrt[3]{C},$ from which you can get the corresponding $x$'s.

• its quite easy to eliminate the x^2 term, so didnt bother with it, but what about higher polynomials like x^5 in place of x^3 in the question , how would we factorise it? – Mrigank Jan 21 '16 at 23:00
• It gets down to your bag of tricks for factorizing. In general there isn't an algorithm that allows you to factorize a quintic. There may be for your specific quintic, but not always. – John Jan 25 '16 at 20:16


text:
LCD won't play right

MrDEB

Well-Known Member
seems I have been here before. I ordered some crystals but what gets me is I had this operational before.
Same circuit but uing a 4520. Now have a 18f4321. Have two LCDs that only show white squares but different patterns.
I think it's a timing issue but?
here is my code (sorry forgot how to correctly post code)

Device=18f4321
Clock = 8 '8MHz
Config OSC = INTIO2
#option LCD_DATA = PORTD.4
#option LCD_RS = PORTD.1
#option LCD_EN = PORTD.0
#option SWORDFISH_SE = true
Include "IntOSC.bas"
Include "SetDigitalIO.bas"
Include "Utils.bas"
Include "LCD.bas"

// some LCD options...

// set this option true if building with the SE compiler

// program start...
While true
SetAllDigital
Cls
WriteAt(1,1,"Hello World")
Wend

Member
Disclaimer - I know nothing about Swordfish, I don't own a PIC18F4321 and I have no idea what LCD display you are using...

This line bothers me...
Code:
#option LCD_DATA = PORTD.4
In my experience, LCD data port is 4 bits wide, or 8 bits.
PORTD.4 implies a single bit definition in my head
Should this be...
Code:
#option LCD_DATA = PORTD
Just guessing...

MrDEB

Well-Known Member
the LCD can be run in 4 or 8 bit mode. If using 4 bit the portD.4 inducates that the last port bits are used for data.
this is a Hitachi HD44780

Dan Soze

Member
Disclaimer - I know nothing about Swordfish, I don't own a PIC18F4321 and I have no idea what LCD display you are using...
From page 46 of Swordfish Compiler Language Reference Guide:
============================================================
Code:
#option
#option identifier [= expression]
Creates an option identifier. An option identifier is global and identical to the #define directive in every way, except for two main differences. Firstly, an option identifier can be seen and used in the declaration block of a user program. For example,
Code:
#option BUFFER_SIZE = 64
const MyBufferSize = BUFFER_SIZE
In the above example, the program constant MyBufferSize becomes equal to 64, as this is the value assigned to the option identifier BUFFER_SIZE. It is important to note that a user program identifier will always take precedence over an option directive. For example,
Code:
#option BUFFER_SIZE = 64
const BUFFER_SIZE = 16
const MyBufferSize = BUFFER_SIZE
In this example, MyBufferSizebecomes equal to 16. The other difference between option and define directives is that re-declaring an option twice, using the same identifie,r will not cause an error. For example,
Code:
#option BUFFER_SIZE = 64
#option BUFFER_SIZE = 16
In this example, BUFFER_SIZE will become equal to 64, which is the first option found by the preprocessor. The second declaration of BUFFER_SIZE is ignored.

This capability may appear a little redundant at first, but it's an extremely useful technique to enable users to configure a module from their main program code.

============================================================

In my opinion the Swordfish Compiler directive "#option" seems to be a method to inject syntax bugs in a way the compiler will not find.

This issue seems to concern the developers as the next directive described on page 47 is "IsOption(Identifier)".

This directive is used to validate that a symbol declared using the #option directive is a member of a valid list of options.

The syntax of the example in the Reference Guide on page 47 is wrong, it is missing the #else part.

The correct example should be something like:
Code:
#if IsOption(USART_LS) and not(USART_LS in(true, false))
#error USART_LS, "Invalid option. LS must be TRUE or FALSE."
#else
#option USART_LS = false
#endif

Note that threads involving MrDEB can run a bit long, see:

Last edited:

Pommie

Well-Known Member
Note that threads involving MrDEB can run a bit long, see:
Stop it. That pain had diminished.

Mike.

Dan Soze

Member
Stop it. That pain had diminished.
hexreader should be forewarned wherein he treads.

Where is your compassion?

If we try maybe we can actually help MrDEB to a better place.

If we fail how about a thread that runs more than 400 posts?

Macbeth, Act 5, Scene5, Line 25.

be80be

Well-Known Member
What do you think happens here
Code:
While true
SetAllDigital
Cls
WriteAt(1,1,"Hello World")
Wend
your code just keeps clearing the screen as fast as it writes it.

Member
hexreader should be forewarned wherein he treads.
Many thanks - forewarned is fore-armed.
Much appreciated.

This thread motivated me to download Swordfish BASIC and the manual - now that I have tried it, I hate it - ROFL

be80be

Well-Known Member
Code:
delayms(1000)
like
Code:
SetAllDigital
While true
Cls
WriteAt(1,1,"Hello World")
delayms(1000)
Wend

MrDEB

Well-Known Member
will give it a go Be80using the example from the samples included w/ swordfish but will try the delay.
Jonsea suggested along time ago to include the use of a crystal. Waiting on parts I ordered from Sparkfun.
This thread will be historic as it won't go past 2 pages

Ian Rogers

User Extraordinaire
Forum Supporter
This thread will be historic as it won't go past 2 pages
So what if it does.... You take your time.... Members don't have to read it.... If you want to break a record, lets go for it!!

be80be

Well-Known Member
I just reposted what you posted MrDeb
The CLS clears the screen you wrapped it inside a loop while wend
So this is what happens the loop goes as fast as it can setting and clearing the screen faster then you can see it as a matter of
fact faster then it can more then likely write it.

Pommie

Well-Known Member
So what if it does.... You take your time.... Members don't have to read it.... If you want to break a record, lets go for it!!
I'm kinda looking forward to it in a nostalgic kinda way.

Mike.

be80be

Well-Known Member
Whats funny is ok 18f4520. cost 6 bucks MrBed you really need a newer basic that can use the newer 16fxxxx chip's

Pommie

Well-Known Member
Whats funny is ok 18f4520. cost 6 bucks MrBed you really need a newer basic that can use the newer 16fxxxx chip's
Or just learn C. Oh wait!

Mike.

MrDEB

Well-Known Member
been looking at C but at my age I feel it would be three steps backward.
have chips on order

Pommie

Well-Known Member
I'm 61 and C works for me.

Mike.

MrDEB

Well-Known Member
but to learn something new.
After a bypass surgery a year ago my short term memory is not real sharp.
for the small projects I am playing with Swordfish seems fine but maybe need to rethink what pic I am using.

MrDEB

Well-Known Member
I moved things around and still get only white squares.
checked and rechecked wiring with ohm meter etc.
thinking maybe because I have the Vcc and VSS daisy chaned on a perf board? could also be in need of a crystal but have operated with good results before.
going to rewire but that dosen't explain why SAME board worked before. BAD LCD? been trying using three different LCD modules.

MrDEB

Well-Known Member
Houston we have landed!!
used a different perf without daisy chaining the VCC or VSS and a different code (one I used for reading a keypad)
The LCD displays text (turns out one of the LCD modules was bad, circular filed it) now I can progress on this project.
Waiting for tactle switches w/LEDs


text:
Browse Questions
Home  >>  AIMS  >>  Class11  >>  Math  >>  Sets

# From the figure (A $\cap$ B)’ =

{1, 2, 4, 5, 6, 7, 8}
Hence (C) is the correct answer.


text: Just Accepted This paper has been accepted to be published in the JGAA Special issue on Selected papers from the Twenty-seventh International Symposium on Graph Drawing and Network Visualization, GD 2019. The paper will receive a volume, an issue number, and page numbers when the whole special issue will be published. Parameterized Algorithms for Book Embedding Problems Vol. 0, no. 0, pp. 0-0, 0. Regular paper. Abstract A $k$-page book embedding of a graph $G$ draws the vertices of $G$ on a line and the edges on $k$ half-planes (called pages) bounded by this line, such that no two edges on the same page cross. We study the problem of determining whether $G$ admits a $k$-page book embedding both when the linear order of the vertices is fixed, called ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$, or not fixed, called ${\rm B{\small OOK}~T{\small HICKNESS}}$. Both problems are known to be ${\sf NP}$-complete in general. We show that ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$ and ${\rm B{\small OOK}~T{\small HICKNESS}}$ are fixed-parameter tractable parameterized by the vertex cover number of the graph and that ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$ is fixed-parameter tractable parameterized by the pathwidth of the vertex order. Submitted: October 2019. Reviewed: January 2020. Revised: March 2020. Accepted: April 2020. Final: April 2020. Appeared on-line: April 2020. Communicated by Daniel Archambault and Csaba D. Tóth article (PDF) BibTeX


text:
# Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x^3 and y = x.?

I know how to use the midpoint rule to get the area under a curve but I'm confused on how to get the area between the two curves. Do I subtract them somehow? How do I set it up?

1. 👍
2. 👎
3. 👁
4. ℹ
5. 🚩


text:
arXiv:1912.05640
IPPP/19/92
MCnet-19-26
Frascati Phys.Ser. 70 (2019)

by: Ferrario Ravasio, Silvia (Durham U., IPPP) et al.

Abstract:
In these proceedings we present three possible interpretations of the ordering variable implemented in the Herwig7 angular-ordered parton shower. Each interpretation determines a different recoil-scheme prescription and we show how it can impact the logarithmic accuracy of the algorithm. We also present comparisons with LEP data.


text:
• ### High temperature antiferromagnetism proximate to a Kondo insulator in $\alpha$-YbAl$_{1-x}$Mn$_{x}$B$_{4}$(1803.09366)

March 25, 2018 cond-mat.str-el
Conventionally, the parallelism between the physical properties of Ce and Yb based magnets and heavy fermions has been considered to arise as a consequence of the electron-hole symmetry known for the band theory in condensed matter physics. Namely, the trivalent state of Ce has $4f^{1}$ configuration, while $4f^{13}$ configuration for Yb$^{3+}$ can be viewed as a state carrying one 4$f$ hole. However, in the light of this parallelism, the magnetism of Yb based intermetallic compounds has been a mystery over decades; the transition temperature of the Yb based compounds is normally very small, as low as $\sim$ 1 K or even lower, while Ce counterparts may often have the transition temperature well exceeding 10 K. Here, we report our experimental discovery of the transition temperature reaching 20 K for the first time in an Yb based compound. Our single crystal study using transport, thermomagnetic and photoemission spectroscopy measurements reveals that the Mn substitution at the Al site in an intermediate valence state of $\alpha$-YbAlB$_{4}$ not only induces antiferromagnetic transition at a record high temperature of 20 K but also transform the heavy fermion liquid state in $\alpha$-YbAlB$_{4}$ into a highly resistive metallic state proximate to a Kondo insulator. The high temperature antiferromagnetism in a system close to a Kondo insulator is unusual and suggest a novel type of itinerant magnetism in the Yb based heavy fermion systems.
• ### Observation of topological superconductivity on the surface of an iron-based superconductor(1706.05163)

Topological superconductors, whose edge hosts Majorana bound states or Majorana fermions that obey non-Abelian statistics, can be used for low-decoherence quantum computations. Most of the proposed topological superconductors are realized with spin-helical states through proximity effect to BCS superconductors. However, such approaches are difficult for further studies and applications because of the low transition temperatures and complicated hetero-structures. Here by using high-resolution spin-resolved and angle-resolved photoelectron spectroscopy, we discover that the iron-based superconductor FeTe1-xSex (x = 0.45, Tc = 14.5 K) hosts Dirac-cone type spin-helical surface states at Fermi level, which open an s-wave SC gap below Tc. Our study proves that the surface states of FeTe0.55Se0.45 are 2D topologically superconducting, and thus provides a simple and possibly high-Tc platform for realizing Majorana fermions.
• ### Antiphase Fermi-surface modulations accompanying displacement excitation in a parent compound of iron-based superconductors(1803.02599)

We investigate the transient electronic structure of BaFe2As2, a parent compound of iron-based superconductors, by time- and angle-resolved photoemission spectroscopy. In order to probe the entire Brillouin zone, we utilize extreme ultraviolet photons and observe photoemission intensity oscillation with the frequency of the A1g phonon which is antiphase between the zone-centered hole Fermi surfaces (FSs) and zone-cornered electron FSs. We attribute the antiphase behavior to the warping in one of the zone-centered hole FSs accompanying the displacement of the pnictogen height, and find that this displacement is the same direction as that induced by substitution of P for As, where superconductivity is induced by a structural modification without carrier doping in this system.
• ### Topological Dirac semimetal phase in the iron-based superconductor Fe(Te,Se)(1803.00845)

Topological Dirac semimetals (TDSs) exhibit bulk Dirac cones protected by time reversal and crystal symmetry, as well as surface states originating from non-trivial topology. While there is a manifold possible onset of superconducting order in such systems, few observations of intrinsic superconductivity have so far been reported for TDSs. We observe evidence for a TDS phase in FeTe$_{1-x}$Se$_x$ ($x$ = 0.45), one of the high transition temperature ($T_c$) iron-based superconductors. In angle-resolved photoelectron spectroscopy (ARPES) and transport experiments, we find spin-polarized states overlapping with the bulk states on the (001) surface, and linear magnetoresistance (MR) starting from 6 T. Combined, this strongly suggests the existence of a TDS phase, which is confirmed by theoretical calculations. In total, the topological electronic states in Fe(Te,Se) provide a promising high $T_c$ platform to realize multiple topological superconducting phases.
• ### Direct observation of multiple topological phases in the iron-based superconductor Li(Fe,Co)As(1803.00846)

Topological insulators/semimetals and unconventional iron-based superconductors have attracted major attentions in condensed matter physics in the past 10 years. However, there is little overlap between these two fields, although the combination of topological states and superconducting states will produce more exotic topologically superconducting states and Majorana bound states (MBS), a promising candidate for realizing topological quantum computations. With the progress in laser-based spin-resolved and angle-resolved photoemission spectroscopy (ARPES) with very high energy- and momentum-resolution, we directly resolved the topological insulator (TI) phase and topological Dirac semimetal (TDS) phase near Fermi level ($E_F$) in the iron-based superconductor Li(Fe,Co)As. The TI and TDS phases can be separately tuned to $E_F$ by Co doping, allowing a detailed study of different superconducting topological states in the same material. Together with the topological states in Fe(Te,Se), our study shows the ubiquitous coexistence of superconductivity and multiple topological phases in iron-based superconductors, and opens a new age for the study of high-Tc iron-based superconductors and topological superconductivity.
• ### Capturing ultrafast magnetic dynamics by time-resolved soft x-ray magnetic circular dichroism(1701.03156)

Experiments of time-resolved x-ray magnetic circular dichroism (Tr-XMCD) and resonant x-ray scattering at a beamline BL07LSU in SPring-8 with a time-resolution of under 50 ps are presented. A micro-channel plate is utilized for the Tr-XMCD measurements at nearly normal incidence both in the partial electron and total fluorescence yield (PEY and TFY) modes at the L2,3 absorption edges of the 3d transition-metals in the soft x-ray region. The ultrafast photo-induced demagnetization within 50 ps is observed on the dynamics of a magnetic material of FePt thin film, having a distinct threshold of the photon density. The spectrum in the PEY mode is less-distorted both at the L2,3 edges compared with that in the TFY mode and has the potential to apply the sum rule analysis for XMCD spectra in pump-probed experiments.
• ### Unconventional superconductivity in the BiS$_2$-based layered superconductor NdO$_{0.71}$F$_{0.29}$BiS$_2$(1703.08379)

We investigate the superconducting-gap anisotropy in one of the recently discovered BiS$_2$-based superconductors, NdO$_{0.71}$F$_{0.29}$BiS$_2$ ($T_c$ $\sim$ 5 K), using laser-based angle-resolved photoemission spectroscopy. Whereas the previously discovered high-$T_c$ superconductors such as copper oxides and iron-based superconductors, which are believed to have unconventional superconducting mechanisms, have $3d$ electrons in their conduction bands, the conduction band of BiS$_2$-based superconductors mainly consists of Bi 6$p$ electrons, and hence the conventional superconducting mechanism might be expected. Contrary to this expectation, we observe a strongly anisotropic superconducting gap. This result strongly suggests that the pairing mechanism for NdO$_{0.71}$F$_{0.29}$BiS$_2$ is unconventional one and we attribute the observed anisotropy to competitive or cooperative multiple paring interactions.
• ### Topological Weyl phase transition in Mo$_x$W$_{1-x}$Te$_2$(1612.07793)

Dec. 22, 2016 cond-mat.mtrl-sci
Topological phases of matter exhibit phase transitions between distinct topological classes. These phase transitions are exotic in that they do not fall within the traditional Ginzburg-Landau paradigm but are instead associated with changes in bulk topological invariants and associated topological surface states. In the case of a Weyl semimetal this phase transition is particularly unusual because it involves the creation of bulk chiral charges and the nucleation of topological Fermi arcs. Here we image a topological phase transition to a Weyl semimetal in Mo$_x$W$_{1-x}$Te$_2$ with changing composition $x$. Using pump-probe ultrafast angle-resolved photoemission spectroscopy (pump-probe ARPES), we directly observe the nucleation of a topological Fermi arc at $x_c \sim 7\%$, showing the critical point of a topological Weyl phase transition. For Mo dopings $x < x_c$, we observe no Fermi arc, while for $x > x_c$, the Fermi arc gradually extends as the bulk Weyl points separate. Our results demonstrate for the first time the creation of magnetic monopoles in momentum space. Our work opens the way to manipulating chiral charge and topological Fermi arcs in Weyl semimetals for transport experiments and device applications.
• ### Discovery of a new type of topological Weyl fermion semimetal state in Mo$_x$W$_{1-x}$Te$_2$(1612.05990)

The recent discovery of a Weyl semimetal in TaAs offers the first Weyl fermion observed in nature and dramatically broadens the classification of topological phases. However, in TaAs it has proven challenging to study the rich transport phenomena arising from emergent Weyl fermions. The series Mo$_x$W$_{1-x}$Te$_2$ are inversion-breaking, layered, tunable semimetals already under study as a promising platform for new electronics and recently proposed to host Type II, or strongly Lorentz-violating, Weyl fermions. Here we report the discovery of a Weyl semimetal in Mo$_x$W$_{1-x}$Te$_2$ at $x = 25\%$. We use pump-probe angle-resolved photoemission spectroscopy (pump-probe ARPES) to directly observe a topological Fermi arc above the Fermi level, demonstrating a Weyl semimetal. The excellent agreement with calculation suggests that Mo$_x$W$_{1-x}$Te$_2$ is the first Type II Weyl semimetal. We also find that certain Weyl points are at the Fermi level, making Mo$_x$W$_{1-x}$Te$_2$ a promising platform for transport and optics experiments on Weyl semimetals.
• ### Spin Texture in Type-II Weyl Semimetal WTe2(1606.00085)

Nov. 20, 2016 cond-mat.mtrl-sci
We determine the band structure and spin texture of WTe2 by spin- and angle-resolved photoemission spectroscopy (SARPES). With the support of first-principles calculations, we reveal the existence of spin polarization of both the Fermi arc surface states and bulk Fermi pockets. Our results support WTe2 to be a type-II Weyl semimetal candidate and provide important information to understand its extremely large and nonsaturating magnetoresistance.
• ### A minimal, "hydrogen atom" version of an inversion-breaking Weyl semimetal(1610.02013)

Oct. 6, 2016 cond-mat.mes-hall
The recent explosion of research interest in Weyl semimetals has led to many proposed Weyl semimetal candidates and a few experimental observations of a Weyl semimetal in real materials. Through this experience, we have come to appreciate that typical Weyl semimetals host many Weyl points. For instance, the first Weyl semimetal observed in experiment, TaAs, hosts 24 Weyl points. Similarly, the Mo$_x$W$_{1-x}$Te$_2$ series, recently under study as the first Type II Weyl semimetal, has eight Weyl points. However, it is well-understood that for a Weyl semimetal without inversion symmetry but with time-reversal symmetry, the minimum number of Weyl points is four. Realizing such a minimal Weyl semimetal is fundamentally relevant because it would offer the simplest "hydrogen atom" example of an inversion-breaking Weyl semimetal. At the same time, transport experiments and device applications may be simpler in a system with as few Weyl points as possible. Recently, TaIrTe$_4$ has been predicted to be a minimal, inversion-breaking Weyl semimetal. However, crucially, the Weyl points and Fermi arcs live entirely above the Fermi level, making them inaccessible to conventional angle-resolved photoemission spectroscopy (ARPES). Here we use pump-probe ARPES to directly access the band structure above the Fermi level in TaIrTe$_4$. We directly observe Weyl points and topological Fermi arcs, showing that TaIrTe$_4$ is a Weyl semimetal. We find that, in total, TaIrTe$_4$ has four Weyl points, providing the first example of a minimal inversion-breaking Weyl semimetal. Our results hold promise for accessing exotic transport phenomena arising in Weyl semimetals in a real material.
• ### Measuring Chern numbers above the Fermi level in the Type II Weyl semimetal Mo$_x$W$_{1-x}$Te$_2$(1604.07079)

April 24, 2016 cond-mat.mes-hall
It has recently been proposed that electronic band structures in crystals give rise to a previously overlooked type of Weyl fermion, which violates Lorentz invariance and, consequently, is forbidden in particle physics. It was further predicted that Mo$_x$W$_{1-x}$Te$_2$ may realize such a Type II Weyl fermion. One crucial challenge is that the Weyl points in Mo$_x$W$_{1-x}$Te$_2$ are predicted to lie above the Fermi level. Here, by studying a simple model for a Type II Weyl cone, we clarify the importance of accessing the unoccupied band structure to demonstrate that Mo$_x$W$_{1-x}$Te$_2$ is a Weyl semimetal. Then, we use pump-probe angle-resolved photoemission spectroscopy (pump-probe ARPES) to directly observe the unoccupied band structure of Mo$_x$W$_{1-x}$Te$_2$. For the first time, we directly access states $> 0.2$ eV above the Fermi level. By comparing our results with $\textit{ab initio}$ calculations, we conclude that we directly observe the surface state containing the topological Fermi arc. Our work opens the way to studying the unoccupied band structure as well as the time-domain relaxation dynamics of Mo$_x$W$_{1-x}$Te$_2$ and related transition metal dichalcogenides.
• ### High-resolution three-dimensional spin- and angle-resolved photoelectron spectrometer using vacuum ultraviolet laser light(1604.06521)

April 22, 2016 cond-mat.mtrl-sci
We describe a spin- and angle-resolved photoelectron spectroscopy (SARPES) apparatus with a vacuum-ultraviolet (VUV) laser ($h\nu$= 6.994 eV) developed at the Laser and Synchrotron Research Center at the Institute for Solid State Physics, The University of Tokyo. The spectrometer consists of a hemispherical photoelectron analyzer equipped with an electron deflector function and twin very-low-energy-electron-diffraction-type spin detectors, which allows us to analyze the spin vector of a photoelectron three-dimensionally with both high energy and angular resolutions. The combination of the high-performance spectrometer and the high-photon-flux VUV laser can achieve an energy resolution of 1.7 meV for SARPES. We demonstrate that the present laser-SARPES machine realizes a quick SARPES on the spin-split band structure of a Bi(111) film even with 7 meV energy and 0.7$^\circ$ angular resolutions along the entrance-slit direction. This laser-SARPES machine is applicable to the investigation of spin-dependent electronic states on an energy scale of a few meV.
• ### Low-Temperature and High-Energy-Resolution Laser Photoemission Spectroscopy(1603.00356)

March 1, 2016 cond-mat.supr-con
We present a review on the developments in the photoemission spectrometer with a vacuum ultraviolet laser at Institute for Solid State Physics at the University of Tokyo. The advantages of high energy resolution, high cooling ability, and bulk sensitivity enable applications with a wide range of materials. We introduce some examples of fine electronic structures detected by laser photoemission spectroscopy and discuss the prospects of research on low-transition-temperature superconductors exhibiting unconventional superconductivity.
• ### Unoccupied electronic structure and signatures of topological Fermi arcs in the Weyl semimetal candidate Mo$_x$W$_{1-x}$Te$_2$(1512.09099)

Dec. 30, 2015 cond-mat.mtrl-sci
Weyl semimetals have sparked intense research interest, but experimental work has been limited to the TaAs family of compounds. Recently, a number of theoretical works have predicted that compounds in the Mo$_x$W$_{1-x}$Te$_2$ series are Weyl semimetals. Such proposals are particularly exciting because Mo$_x$W$_{1-x}$Te$_2$ has a quasi two-dimensional crystal structure well-suited to many transport experiments, while WTe$_2$ and MoTe$_2$ have already been the subject of numerous proposals for device applications. However, with available ARPES techniques it is challenging to demonstrate a Weyl semimetal in Mo$_x$W$_{1-x}$Te$_2$. According to the predictions, the Weyl points are above the Fermi level, the system approaches two critical points as a function of doping, there are many irrelevant bulk bands, the Fermi arcs are nearly degenerate with bulk bands and the bulk band gap is small. Here, we study Mo$_x$W$_{1-x}$Te$_2$ for $x = 0.07$ and 0.45 using pump-probe ARPES. The system exhibits a dramatic response to the pump laser and we successfully access states $> 0.2$eV above the Fermi level. For the first time, we observe direct, experimental signatures of Fermi arcs in Mo$_x$W$_{1-x}$Te$_2$, which agree well with theoretical calculations of the surface states. However, we caution that the interpretation of these features depends sensitively on free parameters in the surface state calculation. We comment on the prospect of conclusively demonstrating a Weyl semimetal in Mo$_x$W$_{1-x}$Te$_2$.
• ### Spin polarization and texture of the Fermi arcs in the Weyl Fermion semimetal TaAs(1510.08430)

A Weyl semimetal is a new state of matter that host Weyl fermions as quasiparticle excitations. The Weyl fermions at zero energy correspond to points of bulk band degeneracy, Weyl nodes, which are separated in momentum space and are connected only through the crystal's boundary by an exotic Fermi arc surface state. We experimentally measure the spin polarization of the Fermi arcs in the first experimentally discovered Weyl semimetal TaAs. Our spin data, for the first time, reveal that the Fermi arcs' spin polarization magnitude is as large as 80% and possesses a spin texture that is completely in-plane. Moreover, we demonstrate that the chirality of the Weyl nodes in TaAs cannot be inferred by the spin texture of the Fermi arcs. The observed non-degenerate property of the Fermi arcs is important for the establishment of its exact topological nature, which reveal that spins on the arc form a novel type of 2D matter. Additionally, the nearly full spin polarization we observed (~80%) may be useful in spintronic applications.
• ### Electronic structure and relaxation dynamics in a superconducting topological material(1511.00278)

Topological superconductors host new states of quantum matter which show a pairing gap in the bulk and gapless surface states providing a platform to realize Majorana fermions. Recently, alkaline-earth metal Sr intercalated Bi2Se3 has been reported to show superconductivity with a Tc ~ 3 K and a large shielding fraction. Here we report systematic normal state electronic structure studies of Sr0.06Bi2Se3 (Tc ~ 2.5 K) by performing photoemission spectroscopy. Using angle-resolved photoemission spectroscopy (ARPES), we observe a quantum well confined two-dimensional (2D) state coexisting with a topological surface state in Sr0.06Bi2Se3. Furthermore, our time-resolved ARPES reveals the relaxation dynamics showing different decay mechanism between the excited topological surface states and the two-dimensional states. Our experimental observation is understood by considering the intra-band scattering for topological surface states and an additional electron phonon scattering for the 2D states, which is responsible for the superconductivity. Our first-principles calculations agree with the more effective scattering and a shorter lifetime of the 2D states. Our results will be helpful in understanding low temperature superconducting states of these topological materials.
• ### Gigantic surface life-time of an intrinsic topological insulator revealed via time-resolved (pump-probe) ARPES(1508.06000)

Aug. 25, 2015 cond-mat.mes-hall
The interaction between light and novel two-dimensional electronic states holds promise to realize new fundamental physics and optical devices. Here, we use pump-probe photoemission spectroscopy to study the optically-excited Dirac surface states in the bulk-insulating topological insulator Bi2Te2Se, and reveal optical properties that are in sharp contrast to those of bulk-metallic topological insulators. We observe a gigantic optical life-time exceeding 4 micro-sec for the surface states in Bi2Te2Se, whereas the life-time in most topological insulators such as Bi2Se3 has been limited to a few picoseconds. Moreover, we discover a surface photo-voltage in topological materials, a shift of the chemical potential of the Dirac surface states, as large as 100 mV. Our results demonstrate a rare platform to study charge excitation and relaxation in energy and momentum space in a two dimensional quantum system.
• ### Ultrafast electron dynamics at the Dirac node of the topological insulator Sb$_2$Te$_3$(1507.03475)

Topological insulators (TIs) are a new quantum state of matter. Their surfaces and interfaces act as a topological boundary to generate massless Dirac fermions with spin-helical textures. Investigation of fermion dynamics near the Dirac point is crucial for the future development of spintronic devices incorporating topological insulators. However, research so far has been unsatisfactory because of a substantial overlap with the bulk valence band and a lack of a completely unoccupied Dirac point (DP). Here, we explore the surface Dirac fermion dynamics in the TI Sb$_2$Te$_3$ by time- and angle-resolved photoemission spectroscopy (TrARPES). Sb$_2$Te$_3$ has a DP located completely above the Fermi energy ($E_F$) with an in-gap DP. The excited electrons in the upper Dirac cone stay longer than those below the Dirac point to form an inverted population. This was attributed to a reduced density of states (DOS) near the DP .
• ### Superconductivity in an electron band just above the Fermi level: possible route to BCS-BEC superconductivity(1307.7845)

Feb. 7, 2014 cond-mat.supr-con
Conventional superconductivity follows Bardeen-Cooper-Schrieffer(BCS) theory of electrons-pairing in momentum-space, while superfluidity is the Bose-Einstein condensation(BEC) of atoms paired in real-space. These properties of solid metals and ultra-cold gases, respectively, are connected by the BCS-BEC crossover. Here we investigate the band dispersions in FeTe$_{0.6}$Se$_{0.4}$($T_c$ = 14.5 K $\sim$ 1.2 meV) in an accessible range below and above the Fermi level($E_F$) using ultra-high resolution laser angle-resolved photoemission spectroscopy. We uncover an electron band lying just 0.7 meV ($\sim$ 8 K) above $E_F$ at the $\Gamma$-point, which shows a sharp superconducting coherence peak with gap formation below $T_c$. The estimated superconducting gap $\Delta$ and Fermi energy $\epsilon_F$ indicate composite superconductivity in an iron-based superconductor, consisting of strong-coupling BEC in the electron band and weak-coupling BCS-like superconductivity in the hole band. The study identifies the possible route to BCS-BEC superconductivity.
• ### Surface electronic structure of the topological Kondo insulator candidate correlated electron system SmB6(1312.1979)

Dec. 30, 2013 cond-mat.str-el
The Kondo insulator SmB6 has long been known to exhibit low temperature transport anomalies whose origin is of great interest. Here we uniquely access the surface electronic structure of the anomalous transport regime by combining state-of-the-art laser- and synchrotron-based angle-resolved photoemission techniques. We observe clear in-gap states (up to 4 meV), whose temperature dependence is contingent upon the Kondo gap formation. In addition, our observed in-gap Fermi surface oddness tied with the Kramers' points topology, their coexistence with the two-dimensional transport anomaly in the Kondo hybridization regime, as well as their robustness against thermal recycling, taken together, collectively provide by-far the strongest evidence for protected surface metallicity with a Fermi surface whose topology is consistent with the theoretically predicted topological surface Fermi surface (TSS). Our observations of systematic surface electronic structure provide the fundamental electronic parameters for the anomalous Kondo ground state of the correlated electron material SmB6.
• ### Unexpectedly robust protection from backscattering in the topological insulator Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$(1312.0073)

Nov. 30, 2013 cond-mat.mes-hall
Electron scattering in the topological surface state (TSS) of the bulk-insulating topological insulator Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$ was studied using quasiparticle interference observed by scanning tunneling microscopy. It was found that not only the 180$^{\circ}$ backscattering but also a wide range of backscattering angles of 100$^{\circ}$--180$^{\circ}$ is effectively prohibited in the TSS. This conclusion was obtained by comparing the observed scattering vectors with the diameters of the constant-energy contours of the TSS, which were measured for both occupied and unoccupied states using time- and angle-resolved photoemission spectroscopy. The unexpectedly robust protection from backscattering in the TSS is a good news for applications, but it poses a challenge to the theoretical understanding of the transport in the TSS.
• ### Two-Fermi-surface superconducting state and a nodal d-wave gap in the electron-doped Sm(1.85)Ce(0.15)CuO(4-d) cuprate superconductor(1104.2617)

We report on laser-excited angle-resolved photoemission spectroscopy (ARPES) in the electron-doped cuprate Sm(1.85)Ce(0.15)CuO(4-d). The data show the existence of a nodal hole-pocket Fermi-surface both in the normal and superconducting states. We prove that its origin is long-range antiferromagnetism by an analysis of the coherence factors in the main and folded bands. This coexistence of long-range antiferromagnetism and superconductivity implies that electron-doped cuprates are two-Fermi-surface superconductors. The measured superconducting gap in the nodal hole-pocket is compatible with a d-wave symmetry.
• ### Incommensurate Orbital Modulation in Multiferroic CuFeO2(1012.3586)

Dec. 16, 2010 cond-mat.str-el
CuFeO_2 is one of the multiferroic materials and is the first case that the electric polarization is not explained by the magnetostriction model or the spin-current model. We have studied this material using soft x-ray resonant diffraction and found that superlattice reflection 0, 1 - 2q, 0 appears in the ferroelectric and incommensurate magnetic ordered phase at the Fe L2,3 absorption edges and moreover that the rotation of the x-ray polarization such as from {\sigma} to {\pi} or from {\pi} to {\sigma} is allowed at this reflection. These findings definitely provide direct evidence that the 3d orbital state of Fe ions has a long range order in the ferroelectric state and support the spin-dependent d-p hybridization mechanism.
• ### Electronic Structure of Superconducting FeSe Studied by High-Resolution Photoemission Spectroscopy(0811.1507)

Nov. 10, 2008 cond-mat.supr-con
We have performed soft x-ray and ultrahigh-resolution laser-excited photoemission measurements on tetragonal FeSe, which was recently identified as a superconductor. Energy dependent study of valence band is compared to band structure calculations and yields a reasonable assignment of partial densities of states. However, the sharp peak near the Fermi level slightly deviates from the calculated energy position, giving rise to the necessity of self-energy correction. We have also performed ultrahigh-resolution laser photoemission experiment on FeSe and observed the suppression of intensity around the Fermi level upon cooling.


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# 4.3 Addition and Subtraction of Matrices

4.3 Addition and Subtraction of Matrices

(A) Determining whether addition or subtraction can be performed on two given matrices
1. Two matrices can be added or subtracted if both matrices have the same order.
2. The addition or subtraction of two matrices is performed by adding or subtracting the corresponding elements of the matrices.

Example 1:

(c) (p   2   4) and (2   6   q)

Solution:
(a) Cannot because the matrices are not of the same order.
(b) Can because the matrices are of the same order.
(c) Can because the matrices are of the same order.

Example 2:
Express each of the following matrices as a single matrix.

$\begin{array}{l}\left(b\right)\text{}B-A=\left(\begin{array}{l}3\\ 2\end{array}\right)-\left(\begin{array}{l}5\\ 1\end{array}\right)\\ =\left(\begin{array}{l}3-5\\ 2-1\end{array}\right)\\ =\left(\begin{array}{l}-2\\ 1\end{array}\right)\end{array}$


text:
257 Pages

This is a command for generating the symbol.

For example:

\begin{align}
\because x+3&=5\\
\therefore x&=2
\end{align}

gives:

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