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## February 5, 2010

### Sheaves Do Not Belong to Algebraic Geometry

#### Posted by Tom Leinster

…and here’s a proof.

They are, of course, very useful in algebraic geometry (as is the equals sign). Also, human beings discovered them while developing algebraic geometry, which is why many of them still make the association.

But as we’ll see, sheaves are an inevitable consequence of general ideas that have nothing to do with algebraic geometry. In fact, sheaves (and various related notions) arise automatically from two completely general categorical constructions, together with one almost imperceptibly small topological observation.

Before I give you the proof, let me make clear that it isn’t due to me. I don’t know who it is due to — I’ve never seen it in print — but I suspect it was known before I was even born. (Update: see Joachim Kock’s comment for a reference.) People who I’ve told this argument to seem to like it, so I wrote it up in a little note a few years ago; then a recent conversation reminded me of it, so I thought I’d air it here.

First categorical construction  Let $\mathbf{A}$ be a small category, $\mathbf{E}$ a category with small colimits, and $J: \mathbf{A} \to \mathbf{E}$ any functor. Then there is an induced adjunction \mathbf{Set}^{\mathbf{A}^{op}} \begin{aligned} \stackrel{\displaystyle\stackrel{\displaystyle - \otimes J}{\longrightarrow}}{\stackrel{\longleftarrow}{Hom(J, -)}} \end{aligned} \mathbf{E}. The right adjoint $Hom(J, -)$ is defined by $(Hom(J, E))(A) = Hom (J(A), E)$ ($E \in \mathbf{E}$, $A \in \mathbf{A}$). The left adjoint $- \otimes J$ is defined by the adjointness, and can be described as a certain coend or colimit.

Example: if $J: \Delta \to \mathbf{Top}$ is the standard simplex functor then $Hom(J, -)$ is the singular simplicial set functor and $- \otimes J$ is geometric realization.

Second categorical construction  Any adjunction restricts canonically to an equivalence between full subcategories.

Precisely, let \mathbf{C} \begin{aligned} \stackrel{\displaystyle \stackrel{F}{\displaystyle\longrightarrow}}{ \stackrel{\longleftarrow}{G}} \end{aligned} \mathbf{D} be an adjunction ($F$ left adjoint to $G$), with unit $\eta: 1 \to G F$ and counit $\varepsilon: F G \to 1$. Let $\bar{\mathbf{C}}$ be the full subcategory of $\mathbf{C}$ consisting of those objects $C$ for which $\eta_C: C \to G F(C)$ is an isomorphism, and dually $\bar{\mathbf{D}}$. Then the adjunction $(F, G, \eta, \varepsilon)$ restricts to an equivalence between $\bar{\mathbf{C}}$ and $\bar{\mathbf{D}}$.

Almost imperceptibly small topological observation  Any open subset of a topological space can be regarded as a space in its own right, and when one open set is contained in another, there is an induced inclusion of spaces.

Precisely, let $S$ be a topological space. Write $\mathbf{O}(S)$ for the poset of open subsets of $S$, regarded as a category (in which each hom-set has at most one element). Write $\mathbf{Top}/S$ for the category of spaces over $S$: objects are continuous maps into $S$, and maps are commutative triangles. Then there is a canonical functor $J: \mathbf{O}(S) \to \mathbf{Top}/S,$ sending an open set $U$ to the inclusion $U \hookrightarrow S$.

Punchline  Fix a topological space $S$. The category $\mathbf{Top}/S$ has small colimits, since $\mathbf{Top}$ does.

Applying the first categorical construction to the functor $J$ just defined produces an adjunction (presheaves on S) = \mathbf{Set}^{\mathbf{O}(S)^{op}} \begin{aligned} \stackrel{\displaystyle\longrightarrow}{\longleftarrow} \end{aligned} \mathbf{Top}/S = (spaces over S). The two functors here are the ones you’d guess.

Applying the second construction now gives an equivalence of categories (sheaves on S) = \mathbf{Sh}(S) \begin{aligned} \stackrel{\displaystyle\longrightarrow}{\longleftarrow} \end{aligned} \mathbf{Et}(S) = (étale spaces over S). This can be interpreted as the definition of sheaf, étale space, etc., or as a theorem, according to taste.

Going right and then left in the adjunction gives the associated sheaf, or sheafification, of a presheaf. Going left and then right gives the ‘étalification’ of a space over $S$.

Posted at February 5, 2010 9:50 PM UTC

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### Re: Sheaves Do Not Belong to Algebraic Geometry

Beautiful!

Small grammar note: ‘étale map’, but ‘étalé space’.

It is probably a sign of something that I never thought that sheaves inherently belonged to algebraic geometry. (Unlike schemes.)

Posted by: Toby Bartels on February 6, 2010 12:26 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

I wrote earlier:

It is probably a sign of something that I never thought that sheaves inherently belonged to algebraic geometry.

In contrast, I did think that stacks originally belonged to algebraic geometry. Of course, these are categorified sheaves, hence really more general. But I did think that the algebraic geometers developed them first and furthest.

I write ‘did’ only because now I expect somebody to tell me otherwise. But perhaps I was right.

Posted by: Toby Bartels on February 7, 2010 8:53 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Hi Tom,

To find these constructions together in print, one possibility
is Mac Lane / Moerdijk, Sheaves in geometry and logic, Ch. 2
(p.90–91), with reference to Chapter 1 for the first general
categorical construction.

But you put it much more forcefully! It is beautiful.

I am not sure I agree with the conclusion regarding algebraic
geometry, though :-) If a conclusion is to be drawn, I would
rather put something like “Algebraic Geometry is more than
what you think!”

Cheers,
Joachim.

Posted by: Joachim Kock on February 6, 2010 12:59 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks, Joachim. You’re right: M&M’s account of this is pretty close to what I wrote, much closer than I remembered. Pages 90-91 of my dog-eared copy have my pencilled annotations on them, so I clearly did read that part.

As for what algebraic geometry is, well, that’s a subjective matter. (But your comment does remind me a little of something that Jim Stasheff just said.) I guess I was taught that algebraic geometry was, at its heart, the study of solving polynomials. On the other hand, I never quite believed my teachers — I couldn’t believe that all that cool Grothendieck stuff (which I knew zero about at the time) could possibly be about something so unglamorous.

Posted by: Tom Leinster on February 6, 2010 1:20 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Much of “that cool Grothendieck stuff” was specifically about counting solutions of finitely many polynomial equations over a finite field when they define a smooth projective variety (glamour is clearly in the eyes of the beholder). It always seems to me to be a dreadful misunderstanding to imagine that Grothendieck was interested in building towers of abstractions just for the sake of it. My own vague impression from what I’ve read of “Récoltes et semailles” and other writings of his (e.g., the Grothendieck-Serre letters) is that he would agree with my remark, taking into account that a genius like him doesn’t have the same definition of abstract and concrete as common mortals… (For instance, it is striking how so many of the fundamental properties of étale cohomology turn out to hinge, in final analysis, on seemingly boring old facts of algebraic number theory.)

As for the history of sheaves, it might also be interesting to point out how different Leray’s original definition was: it was based on closed subsets instead of open sets (see Chern’s review of Leray’s main paper). This was re-interpreted and redefined in the late 40’s, in particular during the Cartan seminars. Cartan himself used sheaves to prove “Theorems A and B” in complex analytic geometry, but also used them (before) to re-interpret and extend a result of Oka from the same period (the coherence of the relation sheaf between finitely many holomorphic functions). Interestingly, Oka had used a rough definition of a special case of sheaves (corresponding more or less to ideal subsheaves of the structure sheaf of a complex analytic variety), which he called “idéal holomorphe de domaines indéterminés”. Before these works, very little was known concerning complex analysis (or geometry) in several variables – although there are similarities with algebraic geometry, many facts are much harder (e.g., the coherence of the ideal sheaf of a subvariety is a tautology in algebraic geometry, but is quite hard in the complex setting – this was one of Cartan’s first results beyond Oka’s).

The earliest reference I found in Numdam are here (from a lecture of April 9, 1951, where Cartan states that this is a new take on the 1948/49 definition of sheaf, which I can’t find; Cartan works in the context of a general regular topological space in this work) and here (which contains the results about coherence mentioned above).

As already mentioned by others, sheaves and (Cech, if I remember right) sheaf cohomology were present in Serre’s “Faisceaux algébriques cohérents”, which is a few years pre-schemes; so they were absolutely already at the heart of algebraic and analytic geometry when Grothendieck started working on these topics.

Posted by: Emmanuel Kowalski on February 7, 2010 8:28 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Sheaves were developed initially by Leray (although they seem to be traced by Wikipedia back further). He was interested in PDEs rather than anything algebraico-geometric. That was pre-1945. Cartan and Weil used them for a proof of de Rham’s theorem, 1947, then Serre introduced them into Algebraic Geometry only in 1954. The sheaf idea is an extension of analytic continuation. Perhaps therefore sheaf theory is a daughter of analysis, which was mistakenly disowned by its parents and taken under the wing of alg. geom.

Hopefully we can give sheaves back to PDEs and diff. geom. with all current work on approaches to smooth spaces etc. Convincing the analysts that they wrongly disinherited their daughter will be hard though!

On another point, no subject owns’ sheaf theory, and Joachim’s subversive work for the expansionist Empire of Algebraic Geometry may be a bit like the claims of one world power, A, to dominate a small country, B, essentially because one of A’s historic emperors came from a third small country, C, that conquered B. (I am reading a book on Kublai Khan at the moment if you want to unravel those slightly cryptic comments.)

Posted by: Tim Porter on February 6, 2010 7:46 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks, Tim. My history is lousy. I remembered that it was Leray who came up with sheaves, but I didn’t know or remember that it was in the context of PDEs.

Nevertheless, I’m convinced that most mathematicians (or at least, those who’ve heard of sheaves) do associate them with algebraic geometry, and that a significant number think that, in some sense, they “belong” to alg geom.

Would you say that it was algebraic geometry that made sheaves famous?

Posted by: Tom Leinster on February 7, 2010 3:37 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

The context is confused. I am only using a bit of knowledge other than web sources, such as the Wikipedia article and the St. Andrews’ History one.

http://www-history.mcs.st-andrews.ac.uk/Biographies/Leray.html

He seems to have avoided PDEs as he was in a POW camp and feared that he might be forced to do engineering or similar work for the German authorities. He used his expertise turning towards topology. I note the important quote

… algebraic topology should not only study the topology of a space, i.e. algebraic objects attached to a space, invariant under homomorphisms, but also the topology of a representation (continuous map), i.e. topological invariants of a similar nature for continuous maps.

(This is interesting as another precursor of a categorical viewpoint.)

Was his own motivation for introducing sheaves and spectral sequences topological or analytic? It is not clear.

Sheaves were used in Spanier’s Algebraic Topology book and Hirzebruck’s work, in the 1950s and 1960s, but it is almost certainly EGA and the SGAs and hence Grothendieck, who pushed them to the forefront in Algebraic Geometry, and his prestige has been transfered to them there.

I am not a great one for subdividing mathematics into somewhat arbitrary areas, and tend to emphasise links rather than differences. That is why I think category theory is at the same time very central and often politically marginalised, which route I will turn from as it goes away from the thread! The subdivisions are largely arbitrary.

I have a somewhat related point. Quillen was working in deformation theory in geometry when he started the work on Model Categories, so what area do they fit into? Nowadays people would probably say it was Algebraic Topology, or homotopy theory. The question may be a non-question in some sense.

Posted by: Tim Porter on February 7, 2010 7:36 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Tim Porter makes some nice observations and some remarks I find myself nodding along to, but unable to add much coherent to. I thought I’d just point out, for those who are interested in some of the history behind Leray’s work during and just after the war, that Haynes Miller has written an interesting account of this:

Leray in Oflag XVIIA: The origins of sheaf theory, sheaf cohomology, and spectral sequences

There’s a copy which can be found on his website.

By the way: I was under the impression, albeit based on little except some browsing in libraries while looking for other things, that there was a period when analysts working in several complex variables took sheaf cohomology (Theorems A, B and all that jazz) pretty seriously and used it.

While this reminds me: has anyone here had cause to look at Finnur Larusson’s recent work advocating a model-categoric/simplicial viewpoint on various parts of holomorphic geometry in higher dimensions? See e.g. this arXiv preprint and various of his sequels.

Posted by: Yemon Choi on February 7, 2010 9:12 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks for that reference to Haynes Miller’s article. I will have a look at it as that period has always interested me for its influence on the maths I grew up with as a postgrad.

I should have mentioned
The Theory of Sheaves. By RICHARD G. SWAN. Chicago lectures in mathematics. (University of Chicago Press), mid 1960s.

They were notes of some of his lectures. (Someone well known to me took the notes so I should not have forgotten them.)

I should also mention Grothenideick’s notes from Kansas. Can someone provide a full title (and link to a scanned copy???)

Posted by: Tim Porter on February 7, 2010 9:34 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Do you mean these lectures?

Posted by: Thomas on February 7, 2010 10:07 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Yes, exactly, that is useful. I have an original hard copy but do love clogging up my hard disc with electronic copies of things!

Posted by: Tim Porter on February 7, 2010 10:15 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Hold on a sec. Have you pulled a fast one on us again? Your definition of sheaf is not the usual one. Are they obviously equivalent? If so, can you do this for any Grothendieck topology? I had always thought that abstract sheaf theory had exactly one theorem—that the sheafification functor exists. I’d be impressed if you could get around that!

Posted by: James on February 6, 2010 9:31 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Ha! You haven’t forgiven me for this. I don’t think I’m pulling a fast one. Like I say, I believe this is something old. And like Joachim and Urs say, it’s in Mac Lane and Moerdijk.

Here you say:

I would expect that you’ve just pushed the hard work to proving that this definition of sheaf agrees with the usual one, i.e. a presheaf which behaves in the usual way on covers.

In a sense I agree. If you want to develop the notion of sheaf, starting from the beginning and arriving at the set of basic facts about sheaves that you’re used to, then this won’t save you any work. But I also like the GPS analogy that Urs gives in reply. (Really GPS stands for Gordon, Power and Street, but I also understand this new-fangled usage.) And whenever a definition has several different formulations, people will have subjective feelings about which is the best one.

Posted by: Tom Leinster on February 6, 2010 11:09 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

OK, good. I just wanted to make sure I wasn’t missing a royal road to sheaf theory!

Posted by: James on February 7, 2010 12:09 AM | Permalink | Reply to this

### The abstract definition of sheaves

It’s more general abstract than that. You don’t need to restrict to sheaves on topological spaces.

In full generality:

For $C$ any small category, Grothendieck topologies and categories of sheaves $Sh(C)$ on $C$ are equivalent to fully faithful functors

$Sh(C) \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh(C)$

into the category of presheaves $PSh(C) = Func(C^{op},Set)$, that have a left exact left adjoint (a left adjoint that preserves finite limits).

Such a morphism is also called

This definition of sheaves is the general abstract one, that allows to define the notion of sheaves in higher category theory, or whenever the notions

• presheaf, adjoint functor, fully faithful functor, left exact functor

are available.

Notably $(\infty,1)$-sheaves (aka $\infty$-stacks) are defined this way, as left exact localizations of presheaf $(\infty,1)$-toposes

$Sh_{(\infty,1)}(C) \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.$

An exposition of this central fact of higher topos theory is at models for $\infty$-stack $(\infty,1)$-toposes.

Posted by: Urs Schreiber on February 6, 2010 11:56 AM | Permalink | Reply to this

### Re: The abstract definition of sheaves

OK, but what I intended to ask was whether this formal approach allows you to bypass the hard work of constructing the sheafification functor in the usual approach. I would expect that you’ve just pushed the hard work to proving that this definition of sheaf agrees with the usual one, i.e. a presheaf which behaves in the usual way on covers.

Posted by: James on February 6, 2010 12:50 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

whether this formal approach allows you to bypass the hard work

This formal definition allows you to bypass any confusion as to what the notion of sheaf is on an abstract level. This helps you to understand what it is that you need to do in any given situation. Then actually doing it may be hard work, but at least you are doing it efficiently then.

Abstract theory is like a GPS sensor: there is a jungle that you have to cross by foot. It will be hard work to make yourself a way through there. But the general abstract GPS sensor that you carry with you will at least ensure that you go on the straight and direct way through. So as to prevent that on top of all the perils that lurk anyway, you’ll not also lose your way and get lost in the thicket.

constructing the sheafification functor in the usual approach

There is no usual and unusual approach here. All there is is the insight that not only does the sheafification functor turn out to be left exact and left adjoint to the inclusion functor, but that conversely it is entirely characterized by this property.

This is not a new insight, mind you. This is the way the standard textbook on abstract sheaf theory discusses it: Sheaves in geometry and logic.

Posted by: Urs Schreiber on February 6, 2010 1:02 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

This formal definition allows you to bypass any confusion as to what the notion of sheaf is on an abstract level.

I think that’s an overly broad statement. I feel that “the notion of sheaf” does not have a unique meaning, and we shouldn’t assert that any one way of thinking about it is the One True Way. This formal definition is one very nice way of thinking about it, which is quick and convenient and gives some good intuition. But the traditional definitions are valid and useful as well, and there are also other valid and useful intuitions.

Posted by: Mike Shulman on February 6, 2010 11:26 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

This formal definition is one very nice way of thinking about it, which is quick and convenient and gives some good intuition. But the traditional definitions are valid and useful as well

Sure, because these traditional definitions are a consequence of this formal definition.

Posted by: Urs Schreiber on February 7, 2010 11:01 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

As I said above, I just wanted to be sure that this point of view doesn’t really cut down on the work you have to do.

That said, I do like this point of view a lot. Somehow I had never appreciated it before. To me, though, it’s particularly worth pointing out that the list of things you need (presheaf, adjoint functor, fully faithful functor, left exact functor) does not include anything like a Grothendieck topology.

Is it true that the point is not so much “sheaves do not belong to algebraic geometry” as much as “sheaves can be defined without sites”?

Posted by: James on February 7, 2010 12:18 AM | Permalink | Reply to this

### Re: The abstract definition of sheaves

You don’t need to restrict to sheaves on topological spaces.

What you wrote is of course correct and interesting, but it doesn’t look to me like a generalization of what Tom wrote. I’m not sure if you were saying that it is, but for sheaves on a general site, there is no analogue of the category $\mathbf{Top}/S$, as far as I know.

Posted by: Mike Shulman on February 6, 2010 11:28 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

but it doesn’t look to me like a generalization of what Tom wrote. I’m not sure if you were saying that it is,

I read Tom’s post as saying: even though sheaves appeared in algebraic geometry, they can be understood as just being structures in plain category theory.

To which I meant to remark: sure, and that’s called topos theory. Where categories of sheaves are precisely characterized as the geometric subtoposes of presheaf toposes.

Posted by: Urs Schreiber on February 7, 2010 11:05 PM | Permalink | Reply to this

### Where sheaves belong to

The most clear-sighted account of what geometry is in full generality and how the notion interacts with the notion of sheaves in full generality is described at notions of space.

As you choose different sites and pregeometries (as described there) you get different notions of geometry.

Posted by: Urs Schreiber on February 6, 2010 12:09 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Sheaf theory (higher sheaf theory [higher stack theory]) is perhaps one of my favorite subjects. It is a beautiful theory with connections to every part of mathematics.

Posted by: Harry Gindi on February 7, 2010 6:04 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

I agree with Mike that the statement quoted by Urs is not a generalization of the statement Tom phrased in the introduction. Tom looks at the adjunction between presheaves on X and all spaces over X. Now this adjunction RESTRICTS to an adjunction between a subcategory which can be called sheaves at either side, but Urs’s statement chooses topologically less intuitive side of the “sheaves as presheaves”. By this stage both the info on bigger category Top/X and the geometric look of its subcategory espaces étalé over X is lost. Now Urs explains the generalization of this void-of-usual-spaces-setup corollary (this does not matter at some different levels of questions, but in elementary intuitive foundations it does matter). Now there is a warm up question for Urs: what is REAL higher categorical generalization of étalé spaces (no equivalent categories, description which is analoguous to them, I mean a sort of categorified generalized covering spaces). This is not that simple task. Tim Porter will remember recent thesis at Bangor treating n=2 as one of the issues.

Now, on history. I am so much surprised with the attitude that “sheaves are algebraic geometry”. I was quite knowledgeable about sheaves as undergraduate and studied related algebaric topology a lot then and did not have ANY idea at the time what an algebraic variety is. In our topology seminar which I attended then we used old definitions from 1956 Godement’s excellent book for whom the sheaf is what is here called étalé space and in the Russian edition its total space is called nakryvajushchee prostranstvo, that is covering space; the presheaf point of view is compared of course there in detail (this equivalence of categories proved but not more general adjunction above). Leray, Cartan etc. at the time were interested in analytic continuation like Oka’s problem, Cousin’s problem in cohomology etc. This resulted in EARLY 1950s, that is before Spanier and similar resources irrelevant for early history to lots of results in analytic geometry, Grauert and Remmert taking place on German side and Henri Cartan on French, and of course many others; Cartan’s theorems A and B being some of the famous results. Serre was in the same milleu and transferred in his famous FAC article some of the Cartan seminaire methods to algebraic geometry. Grothendieck was doing tensor products of topological vector spaces at the time. Then he went to Kansas state to do some topology, and there were the breakthrough in sheaf theory, NOT in algebraic geometry. I mean his Kansas report (there are two, the booklet on sheaf theory, and also the official report to the funding agency, where he claims his failure to achieve the goals and then goes on with big discoveries), and his seminar in Spring 1955 where he obtained most of the Tohoku results which were published only 2 years later. Tohoku is still written with main view toward the cohomology of sheaves in topological and equivariant setup. Later AG claimed that the intuition on abelian categories was to isolate the properties he thought of sheaf categories, and that topos theory later was more or less just a later variant of the same initial idea. I mean the characterization of Grothendieck topoi like in Giraud’s theorem is according to him, one of the principal reasons to find the axiomatics of the abelian categories as he phrased it in Tohoku.

In complex analytic geometry according to an intro to a book of Walter Rudin (the one on the complex functions in the unit ball) the sheaves went into second plane only with the discovery of new, quite different, methods of 1970s, which were more effective to then actual questions.

John Baez has recently anounced at the cafe some very recent discovery of Freyd (?) on comparison between abelian and topos case finding a new formulation with closer similarity. I would like to hear a more precise statement and update on this exciting story.

Posted by: Zoran Skoda on February 7, 2010 7:57 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

what is REAL higher categorical generalization of étalé spaces (no equivalent categories, description which is analoguous to them, I mean a sort of categorified generalized covering spaces). This is not that simple task. Tim Porter will remember recent thesis at Bangor treating n=2 as one of the issues.

I would posit that for n=2 the answer is: 2-covering spaces, as defined in my thesis (see chapter 4). The thesis you refer to, by Richard Lewis, takes a more algebro-combinatorial approach, with crossed modules acting on simplicial sets.

Posted by: David Roberts on February 8, 2010 12:04 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks Zoran and David.

Yes. I felt from way back when I started working on 2-stack type things (actually before we at Bangor got the start of the Grothendieck correspondence, (in the sense of letters not a bijection!!!!), but after, by several years, AG’s letter to Breen (note not the later letter to Quillen)), that the 2-type of a space classified its 2-covering spaces, that that was very important, and that if I was to attempt to convince analysts, geometers and others that this was important, there had to be fairly concrete models of the covering spaces. The solution using abstract sheaf theoretic models should be doable (and probably is essentially there in places that I should have read!!!) such as David’s thesis, which I have only browsed, but what about Galois theory, Riemann surface theory, ramifications, complex analytic continuation, etc. How were generalisations of those to be found and then sold’ to the non-categorical mathematical audience? That needed something that I could not visualise. Something more concrete than just an abstract construction in a Quillen model category, however pretty, is needed. I had seen algebraic topologists in seminars ‘turn off’ if you mentioned anything non-stable, yet what I wanted was very basic, some sort of elementary approach to those ideas.

I felt those questions were important then and still do now. Some of John’s ideas and a lot of the stuff here in the café and in the Lab go a long way towards answering them.

Grothendieck’s letters to Ronnie, and their transformation into Pursuing Stacks made me very pleased!!! But then I had to start learning all the bits of theory, that would be needed to do put his ideas into practice. I am still learning!

Posted by: Tim Porter on February 8, 2010 7:02 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Hi Zoran,

you write:

what is REAL higher categorical generalization of étalé spaces (no equivalent categories, description which is analoguous to them, I mean a sort of categorified generalized covering spaces).

I would think this is the topic of section 7.1 in HTT, leading up to theorem 7.1.4.4 on page 555:

For $X$ a topological space, sufficiently nice, there is an adjoint equivalence of $(\infty,1)$-categories

$Top/X \stackrel{\leftarrow}{\to} Sh_{(\infty,1)}(X)$

where left-to-right is taking sections and right-to-left is the analog of forming étale spaces.

Indeed, the +-construction for $\infty$-stacks on $X$ is the immediate generalization of the 1-categorical familiar “form étale space and then take sections”. This is theorem 7.1.4.3 on p. 554.

Posted by: Urs Schreiber on February 8, 2010 7:10 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Here’s another interesting aspect of this. It’s certainly true that any adjunction restricts to an equivalence between the full subcategories consisting of the objects for which the unit or counit (as appropriate) is an isomorphism. However, for many adjunctions, these full subcategories can be very small. For instance, for the free-forgetful adjunction of any algebraic theory (monoids, groups, rings, etc.), these full subcategories are empty.

But the adjunction in question here has a special property: it’s idempotent. That means that the unit $\eta_C \colon C\to G F (C)$ is an isomorphism whenever $C$ is of the form $G(D)$, and (equivalently!) that the counit $\varepsilon_D\colon F D(D) \to D$ is an isomorphism whenever $D$ is of the form $F(C)$.

In general, you could think of starting with an object $C$ and trying to get it into one of the full subcategories that are equivalent by successively applying the functors: $C, F(C), G F(C), F G F(C), \dots$. In general, you’ll never get there, while the adjunction being idempotent means that you get there after only one step. (Off the top of my head, I don’t know any examples where you get there in some finite number of steps $\gt 1$.)

Posted by: Mike Shulman on February 8, 2010 7:30 PM | Permalink | PGP Sig | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Yes — and it’s important that it’s idempotent. The question is, is there a really slick way to make idempotency pop out of this derivation of the adjunction?

Posted by: Tom Leinster on February 8, 2010 11:59 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Good question. I don’t immediately see any such slick argument.

Posted by: Mike Shulman on February 9, 2010 5:25 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Hi I know its an awfully long time since this was posted but i just read it and couldn’t help thinking about this. The question being how does one show “slickly” that the adjunction derived in the first place is idempotent? For this i propose the following Lemma:

Proposed Lemma: Let $\mathbb{C}$ be small, $\mathbb{E}$ co-complete, $z \;:\; \mathbb{C} \rightarrow \mathbb{E}$, and $F \;:\; \hat{\mathbb{C}} \rightharpoonup \mathbb{E} \;:\; G$ the standard adjunction arising from this situation; Then it is idempotent provided that $z$ is full.

If this lemma is correct (i think the proof is below), then the situation with sheaves is a corollary since in that case $z : \mathcal{O}(X) \rightarrow \text{Top}/X$ is clearly full since the only arrows of the form $z(U) \rightarrow z(V)$ in $\text{Top}/X$ must be inclusions.

Here is a proposed proof: Let $\eta$ be the unit of the adjunction. $\eta G$ is trivially a split monic from the triangle identities, so we are only concerned with showing it is epi, for which it suffices to show that each of the embedded components, of the following form, is a surjection of sets:

$\qquad \qquad (\eta_{G D})_C \;:\; \text{Hom}_{\mathbb{C}}(z C,D) \rightarrow \text{Hom}_{\mathbb{C}}(G z C,G D)$

… which function acts by mapping the arrows through $G$, so an arrow $h \;:\; z C \rightarrow D$ on the left becomes $h_{*}$ on the right.

Conversely, consider any arrow $\varphi \;:\; G z C \rightarrow G D$ on the right. Written out in full, this is a natural transformation $\varphi \;:\; \text{Hom}_{\mathbb{E}}(z -,z C) \rightarrow \text{Hom}_{\mathbb{E}}(z -,D)$. Draw a naturality square for this with respect to an arbitrary arrow $g \;:\; C' \rightarrow C$ and chase the unit $\text{id}_{z C}$ around the diagram to learn that $\varphi_{C'} \;:\; (z g) \mapsto \varphi_{C}(\text{id}_C) \cdot (z g)$. Provided that $z$ is full, this allows us to conclude that $\varphi$ is of the form $h_*$ where $h = \varphi_C(\text{id}_C)$ in all cases, i.e. that $(\eta_{G D})_C$ is surjective as required, and the adjunction is idempotent.

Posted by: hypnocat on February 19, 2015 2:23 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

I’m afraid I’ve only just sat down to work through this.

I’d love to see a simple criterion for when the adjunction $\widehat{\mathbb{C}} \stackrel{\leftarrow}{\rightarrow} \mathbb{E}$ induced by a functor $\mathbb{C} \to \mathbb{E}$ is idempotent, but I don’t think this lemma can be right.

Here’s what appears to be a counterexample. First recall that an adjunction is idempotent if and only if the comonad it induces is idempotent (that is, the multiplication part of the comonad is an isomorphism).

Now consider the inclusion $FinSet \to Set$ of the full subcategory of finite sets, and form its opposite $FinSet^{op} \to Set^{op}$, which is certainly full (and faithful). The adjunction induced by this latter functor is an adjunction between $Set^{op}$ and $Set^{FinSet}$. It induces a comonad on $Set^{op}$, which of course can be thought of as a monad on $Set$.

As originally shown by Kennison and Gildenhuys, and recounted more recently in Section 2 here, that monad is the ultrafilter monad.

However, the ultrafilter monad $U$ is not idempotent (at least, assuming that $Set$ satisfies the axiom of choice). For if $U$ is idempotent then for each set $X$, the multiplication map $U U X \to U X$ is a bijection, or equivalently the unit map $\eta_{U X} : U X \to U U X$ is a bijection, and therefore every ultrafilter on $U X$ is principal. But if we take $X$ to be infinite, then $U X$ is also infinite, so (by choice) $U X$ admits a non-principal ultrafilter — a contradiction.

What’s wrong with hypnocat’s proof? Well, it seems to me that the codomain of $(\eta_{G D})_C$ (shown in the displayed equation) is wrong. (Here $C \in \mathbb{C}$ and $D \in \mathbb{E}$.) I think that display should read

$(\eta_{G D})_C \colon Hom_{\mathbb{E}}(z C, D) \to Hom_{\mathbb{E}}(z C, F G D).$

If $F$ were right adjoint to $G$ then the codomain could be rewritten as $Hom_{\widehat{\mathbb{C}}}(G z C, G D)$, which is (almost) what hypnocat wrote. But it’s left adjoint.

Posted by: Tom Leinster on August 31, 2015 7:48 PM | Permalink | Reply to this

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text:
# 9.4: PID tuning via Frequency Responses with Bode Plots

## Description of Amplitude Ratio, Phase Shift and Frequency Response

A good way to understand the physical meaning of amplitude ratio and phase shift and how they change as a function of the frequency of the input stream is through the analysis of a simple system. A well-insulated continuously-stirred tank reactor (CSTR) with an input flow equal to its output flow is a good example (Figure $$\PageIndex{1}$$). For this description we consider inlet and outlet temperature, but inlet concentration or feed rate could also be used.

If the CSTR is initially at 55˚C, and the inlet flow has a temperature that fluctuates sinusoidally between 50˚C and 60˚C, the outlet temperature will also vary sinusoidally. If the fluid has a large residence time in the tank, and the frequency of the temperature variation of the input fluid is high relative to the residence time. Then the outlet temperature will also vary quickly in a sinusoidal fashion, but with a significantly smaller amplitude than that of the inlet. The large hold up in the well mixed tank dampens the fluctuations in the inlet temperature. In contrast, if the the frequency of the inlet temperature sine wave is low relative to the fluid residence time, the output will also be a sine wave with temperatures ranging from 50˚C to 60˚C.

In both cases described above there will be a lag or phase shift between the input and the output of the system (see the figure immediately below). In our example, the phase shift is controlled by the average residence time of fluid in the tank. The time constant of this system is the average residence time of the tank. A system with a small residence time will respond very quickly to changes in input flow rate, and large residence times will result in sizable lag time.

As frequency of the temperature variation increases, the period of the input decreases and the phase shift becomes larger. This can be seen in Figure $$\PageIndex{2}$$:

Imagine that we have a sensitive reaction that must be maintained with 3˚C of 55˚C, it now becomes of utmost importance that we understand the frequency response of this CSTR system. Here in lies the utility of Bode plots; they depict a range of different frequency responses in two plots. This allows a relatively rapid determination of system robustness. In addition, frequency response and Bode plot analysis can be used to tune PID control systems.

## Frequency Response

For a given process described by $$Y(t)=\hat{G} X(t)$$, one considers a sinusoidal input

$X(t)=\sin (\omega t)$

After a long time, the solution will also be sinusoidal, with

$Y(t) \sim A R \sin (\omega t+\phi)$

where $$AR$$ is the amplitude ratio (must be a positive number) and $$\phi$$is the phase lag ($$\phi < 0$$) or phase lead ($$\phi > 0$$).

Sometimes, to solve for $$Y(t)$$ using an ODE solver, it is useful to have the process equation in the form $$X(t)=\hat{G}^{-1} Y(t)$$ and define $$\hat{G}^{-1}$$ instead of $$\hat{G}$$.

### Expressions for AR and $$\phi$$

First-Order System

$A R=\frac{1}{\sqrt{1+\omega^{2} \tau_{p}^{2}}}$

$\phi=\tan ^{-1}\left(-\omega \tau_{p}\right)$

Second-Order System

$A R=\frac{1}{\sqrt{\left(1-\omega^{2} \tau_{p}^{2}\right)^{2}+\left(2 \zeta \omega \tau_{p}\right)^{2}}}$

$\phi=\tan ^{-1}\left(\frac{-2 \zeta_{p} \omega \tau_{p}}{1-\omega^{2} \tau_{p}^{2}}\right)$

$A R=1] \[\phi=-\omega \tau_{\text {dead}}$

Systems in series

$A R=A R_{1} \cdot A R_{2} \cdot \ldots$

$\log A R=\log A R_{1}+\log A R_{2}+\ldots$

$\phi=\phi_{1}+\phi_{2}+\ldots$

P Controller

$\hat{G}_{c}=K_{c}$

$A R=K_{c}$

$\phi=0$

PI Controller

$\hat{G}_{c}=K_{c}\left(1+\frac{1}{\tau_{I}} \int_{0}^{t} d t^{\prime}\right)$

$\phi=\tan ^{-1}\left(-\frac{1}{\omega \tau_{I}}\right)$

PD Controller

$A R=K_{c} \sqrt{1+\omega^{2} \tau_{D}^{2}}$

$\phi=\tan ^{-1}\left(\omega \tau_{D}\right)$

PID Controller

$\phi=\tan ^{-1}\left(\omega \tau_{D}-\frac{1}{\omega \tau_{I}}\right)$

## Introduction to Bode Plots

Because it’s impossible to perfectly model a real chemical process, control engineers are interested in characterizing the robustness of the system – that is, they want to tune controllers to withstand a reasonable range of change in process parameters while maintaining a stable feedback system. Bode (pronounced boh-dee) plots provide an effective means to quantify the system’s stability. Bode plots describe an open or closed-loop system as a function of input frequency and give a picture of the system’s stability. If all inputs into a system were constant, it would be a relatively simple task to control the system and its output. However, if an input value, such as temperature, varies sinusoidally, then the output should also be describable as a sinusoidal function. In this case Bode plots are a useful tool for predicting the response of the system. There are two Bode plots used to describe a system. The first shows amplitude ratio as a function of frequency. The amplitude ratio is the amplitude of the output sine wave divided by the input sine wave. The second plot graphs phase shift as a function of frequency, where phase shift is the time lag between the output and the input sine curves divided by the period of either of the sine curves and then multiplied by -360° (or $$-2π$$ radians) to express this shift as an angle.

## Constructing Bode Plots

Bode plots concisely display all relevant frequency input and output information on two plots: amplitude ratio as a functions of frequency and phase shift as a function of frequency. The amplitude ratio plot is a log-log plot while the phase angle plot is a semilog (or log-linear) plot.

To construct a Bode plot, an engineer would have empirical data showing input and output values that vary as sinusoidal functions of time. For instance, there might be inlet temperature data that varies sinusoidally and the outlet temperature data that also varies sinusoidally. For example, given a heat exchanger that is used to cool a hot stream, we can vary the flow rate of the hot stream and model it sinusodially. (This can be done using ODEs & an Excel model of a heat exchanger.)

Figure $$\PageIndex{3}$$ includes inlet flowrate (process fluid stream) varying as a sinusoidal function and the hot temperature exiting the heat exchanger likewise varying as a sinusoidal function. Portions of this graph are not yet at steady-state.

Bode plots are constructed from steady-state data. Figure $$\PageIndex{4}$$ shows part of the steady-state region of the same data used for Figure $$\PageIndex{3}$$.

To collect a single data point for a Bode plot we will use the information from a single period of the inlet flow rate and the corresponding temperature from the hot exiting stream.

The amplitude ratio, $$AR$$, is the ratio of the amplitude of the output sinusoidal curve divided by the amplitude of the input sinusoidal curve.

$A R=\dfrac{\text { output amplitude }}{\text {input amplitude}} \nonumber$

The value of the amplitude ratio should be unitless so if the units of the input frequency and the units of the output frequency are not the same, the frequency data should first be normalized. For example, if the input frequency is in ˚C/min and the output frequency is also in ˚C/min, then AR is unitless and does not need to be normalized. If, instead, the input frequency is in L/min and the output frequency is in ˚C/min, then AR = ˚C/liters. In this case the inlet and outlet frequencies need to be normalized because the ratio AR = ˚C/L doesn’t say anything about the physical meaning of the system. The value of AR would be completely different if the units of AR were Kelvin/gal.

To find the phase shift, the periods of the input and output sine curves need to be found. Recall that the period, $$P$$, is the length of time from one peak to the next.

$P=\frac{1}{f}=\frac{2 \pi}{\omega}$

with $$f$$ is the frequency and $$\omega$$ is the angular frequency (rad/sec)

The phase shift is then found by

$\phi=-2 \pi \left(\dfrac{\Delta P}{P}\right)$

Using these values found from multiple perturbations in feed flow rate it is possible to construct the following Bode plots:

### Rules of Thumb when analyzing Bode plots

Generally speaking, a gain change shifts the amplitude ratio up or down, but does not affect the phase angle. A change in the time delay affects the phase angle, but not the amplitude ratio. For example, an increase in the time delay makes the phase shift more negative for any given frequency. A change in the time constant changes both the amplitude ratio and the phase angle. For example, an increase in the time constant will decrease the amplitude ratio and make the phase lag more negative at any given frequency.

Table $$\PageIndex{1}$$: Rules of Thumb

Change

Effect

Gain Change

Amplitude ratio is shifted up or down; no change in phase angle

Time Delay

Phase angle changes, but no change in amplitude ratio

Historical Note

Prior to the advent of powerful computer modeling tools, controls engineers would model systems using transfer functions. Readers interested in learning more about how these were used to construct Bode plots should refer to Bequette's Process Control: Modeling, Design, and Simulation (see References). This wiki assumes that the engineer already has data in Excel, etc, that shows the sinusoidal behavior of input and outputs.

## Bode Stability Criterion

A control system is unstable if the open-loop frequency response exhibits an AR exceeding unity at the frequency for which the phase lag is 180 degrees. This frequency is called the crossover frequency. Adjusting the gain so that the AR < 1 for - phi < (180 minus phase margin) results in acceptable stability if the phase margin is greater than 30 degrees.

As previously mentioned, the controls engineer uses Bode plots to characterize the stability of the system. If you were to find that the amplitude ratio is greater than 1 the system would be unstable at that frequency. An important stability criterion is that AR < 1 when phase shift = -180 degrees (or -pi radians). This is called the Bode stability criterion and if it holds for a closed-loop system, then that system will be stable. If this criterion is not satisfied and you were to put a feedback controller on this process, as done in PID tuning via optimization [PID_HeatExchange.xls – Note there are still errors in this file and a link will be provided once the file is operational.], the process would explode due to the response lag. The reason for this is that the system might think temperature is too low when it's actually to high, so it would decrease cooling water rather than increase it. Also, with an AR < 1, an error in input is amplified in the output, so the system will lose control due to amplification of error. This could obviously be a dangerous situation, for example in an exothermic CSTR, which could explode. This can also be seen by following the procedure above for the PID_HeatExchange.xls file.

## Frequency Response in Mathematica

Bode plot for PD controller, ΤD = 1:

AR = Sqrt[1 + omega^2]

phi = 180/Pi*ArcTan[omega]

LogLogPlot[AR,{omega,0.01,100}]

LogLinearPlot[phi,{omega,0.01,100}]

Bode plot for PID controller, using the \Manipulate" function to watch the effect of varying the values of both Tau_I and Tau_D:

Manipulate[LogLogPlot[Sqrt[1 + (omega*tauD - 1/(tauI*omega))^2],{omega,0.01,100}],{tauI,0.1,10},{tauD,0,10}]

Manipulate[LogLinearPlot[180/Pi*ArcTan[omega*tauD - 1/(tauI*omega)],{omega,0.01,100}],{tauI,0.1,10},{tauD,0,10}]

Bode plot for open-loop operation of third-order system with a P controller:

AR = Sqrt[1/(1+omega^2)]*Sqrt[1/(1+omega^2)]*Sqrt[1/(1+25*omega ^ 2)] phi = 180/Pi*(ArcTan[-omega] + ArcTan[-omega] + ArcTan[-5*omega]) LogLogPlot[AR,{omega,0.01,100}] LogLinearPlot[phi,{omega,0.01,100}]

Find crossover frequency:

FindRoot[phi==-180,{omega,1}]

This yields omega = 1.18322 which is the crossover frequency, and can be used to determine the amplitude ratio (ask mathematica for AR) the gain (Kc=1/AR)

To determine what the gain should be set to at a phase margin of 30 degrees (safe operating conditions) use the following in mathematica:

FindRoot[phi==-150,{omega,1}]

The phase margin of Cohen-Coon and other frequency analysis (such as Zielger-Nicholas) can be calculated by using the gain (Kc) value gotten from the method a typing the following into mathematica:

NSolve[Kc*AR==1,omega]

## Derivations

Assume a first-order process with input $$X(t)$$, output $$Y(t)$$, and operator

$\hat{G}^{-1}=\left(\tau \frac{\delta}{\delta t}+1\right)$

where

$X(t)=\hat{G}^{-1} Y(t)=\left(\tau \frac{\delta}{\delta t}+1\right) Y(t)=\tau Y^{\prime}(t)+Y(t) \label{1}$

For Bode Plot Analysis, we assume a periodic input to the system:

$X(t) = \sin(ωt)$

Based on this input, we can hypothesize that:

$Y(t) = AR * \sin(ωt + φ)$

From this hypothesis, we can replace Equation \ref{1} with our functions for $$X(t)$$ and $$Y(t)$$. This is shown below.

$X(t) = \tau Y'(t) + Y(t) \label{1a}$

\begin{align} \sin (\omega t) &=\tau \frac{\delta}{\delta t}(A R \sin (\omega t+\phi))+A R \sin (\omega t+\phi) \label{2} \\[4pt] &= AR(τω \cos(ωt + φ)) + \sin(ωt + φ) \label{3} \end{align}

Using trigonometric identities, we can replace $$\cos(ωt + φ)$$ with $$\cos(ωt)cos(φ) − \sin(ωt)sin(φ)$$ and $$\sin(ωt + φ)$$ with $$\sin(ωt)cos(φ) + \cos(ωt)sin(φ)$$ to get:

$\sin (\omega t)=A R(\omega T(\cos (\omega t) \cos (\varphi)-\sin (\omega t) \sin (\varphi))+\sin (\omega t) \cos (\varphi)+\cos (\omega t) \sin (\varphi)) \label{4a}$

To solve for $$AR$$ and $$φ$$, we can match the coefficients for the $$\sin(ωt)$$ term and $$\cos(ωt)$$ term on the left and right side of Equation \ref{4a}.

Coefficient of $$\cos(ωt)$$:

$0 = AR(ωτ\cos(φ) + \sin(φ)) \label{5}$

Coefficient of $$\sin(ωt)$$:

$1 = AR( − ωτ\sin(φ) + \cos(φ)) \label{6}$

From Equation \ref{5}, we can divide both sides by $$AR\cos(φ)$$ and rearranging to get:

$\tan (\phi)=-\omega \tau \label{7a}$

or

$\phi=\tan^{-1}(-\omega \tau) \label{7b}$

Based on this relationship, we can draw a triangle with the angle $$φ$$ and the relationship shown in Equation \ref{7b}. A picture of the triangle is shown below.

From Equation \ref{6} and the triangle above, we can solve for AR:

\begin{align} AR &=\frac{1}{-\omega \tau(\sin (\phi))+\cos (\phi)} \\[4pt] &=\frac{1}{-\omega \tau\left(\frac{-\omega \tau}{\sqrt{1+\omega^{2} \tau^{2}}}\right)+\frac{1}{\sqrt{1+\omega^{2} \tau^{2}}}} \\[4pt] &=\frac{1}{\sqrt{1+\omega^{2} \tau^{2}}} \label{8} \end{align}

## Determining Stable Controller Gain and Phase Margin

Tuning via phase margin is a more precise method than tuning through Cohen-Coon Rule and Ziegler-Nichols Tuning Rule. We can find the gain and phase margin by using bode plots or by an analytical method.

Graphical Method

To find a safe controller gain, use the following steps:

1. Construct Bode Plot of $$\log AR$$ versus $$\log ωτ$$ and $$φ$$ versus $$\log ωτ$$
2. Use the above plot to find $$ωτ$$ such that $$φ = -180^o + φ_{PM}$$.
3. Use $$ωτ$$ found from above step to find $$AR$$ in $$\log AR$$ versus $$\log ωτ$$ plot.
4. $$K_{C s a f e}=\frac{1}{A R}$$

To find the phase margin, use the following steps:

1. Construct Bode Plot of LogAR versus Log ωτ and φ versus Log ωτ
2. Use the above plot to find ωτ such that AR=1.
3. Use ωτ found from above step to find φ in φ versus Log ωτ plot.
4. φPM = 180 − | φAR = 1 |

Analytical Method

Finding the gain with a phase margin:

1. Use your expression for Φ, set Φ to the desired value and find the root for ω.
2. Input the value obtained for ω in your expression for AR.
3. $$K_{C}=\frac{1}{A R}$$

Finding the phase margin with a gain:

1. Use your expression for AR, set and solve for ω.
2. Input the value obtained for ω in your expression for Φ.

## Determining the Slope of a Bode Plot

Sometimes it is necessary to determine the relative shape of a Bode Plot without the use of a computer or ODE equation solver. In sketching the relative shape, there are three main parts of a basic Bode Plot that must be considered.

1. Crossing point
2. Starting value
3. Slope

### Crossing Point

The crossing point of the Bode Plot is when the frequency, ω, is equal to 1 / τ . If the x-axis of the Bode Plot is ωτ, the crossing point will be at x-axis = 1. If there are multiple values of τ, there may be multiple crossing points, each at the frequency where ω is equal to 1 / τ . If there are multiple τ values, and the x-axis is ωτ, it must be specified which τ the x-axis is referring to.

### Starting Value

The starting value depends on the $$K_C$$ and $$K_P$$ values contributing to the process.

### Slope

The slope for an uncontrolled process is equal to the negative of the order of the process. For example, if there is a first-order process with no controller the slope of the Bode Plot would be -1 (-1 for first order), after the crossing point has been reached. A picture of this Bode Plot is shown below.

For a process controlled by a PD controller, the AR is essentially the inverse of the first-order system AR, meaning that the slope addition from a PD controller is a +1, instead of a -1 as in a first-order process. For a second-order system with a PD controller the final slope will be -1 (-2 from the second order process, +1 from PD). However, there may be multiple crossing points, whose location depends on the value of τP and τD. Remember that the crossing point is where ωτ equals one. Since the slope contribution only comes after the crossing point, a Bode Plot for τD less than τP would look like the picture below.

• Crossing point 1 is where ωτP is 1 and crossing point 2 is where ωτD is 1.
• Similarly, if τD is greater than τ, the slope would first go to +1, then to -1.

For a PI controller, the slope contribution from the controller comes before the crossing point, and then goes to zero after the crossing point. A PID controller would therefore look like the picture shown below, assuming τD = τI .

Example $$\PageIndex{1}$$: Determine the Amplitude Ratio

You are a controls engineer and wish to characterize a heat exchanger used in a chemical process. One of the many things you are interested in knowing about the system is how the hot outlet temperature responds to fluctuations in the inlet flow rate. Using data for a particular inlet flow rate, you graphed normalized (why?) flow rate and normalized hot outlet temperature vs. frequency (rad/min). Use this graph to determine the amplitude ratio.

Solution

Because flow rate units and temperature do not have the same units, these values needed to be normalized before calculating the amplitude ratio. To normalize flow, use the following equation:

$\text {F}_\text{norm}=\frac{F-\text {Fmin}}{\text {Fmax}-\text {Fmin}} \nonumber$

where $$F$$ is the flow rate of a particular data point.

In this problem temperature, $$T$$, is in ˚C, so

$\text {Tnorm}=\frac{T-273.15}{100} \nonumber$

To find the amplitude of both wave functions, first recall that the amplitude of a wave is the maximum height of a wave crest (or depth of a trough). For one steady-state wave produced from a column of values in Excel, you could quickly calculate the amplitude by using the max( ) and min( ) Excel functions. [This can be found using Excel help.] If you opted to find the amplitude this way, then the amplitude for a single wave function would be

$\text {Amplitude}=\frac{[\max ()-\min ()]}{2} \nonumber$

Note that this is just one way to find the amplitude of a wave. It could also be found simply by reading the values off the graph of a wave.

Once the amplitudes of the inlet and outlet waves are found, use the values to find the amplitude ratio, AR, as follows:

$AR=\frac{\text { outlet streams amplitude }}{\text {inlet streams amplitude}}=\frac{0.0486}{0.499}=0.0974$

Example $$\PageIndex{1}$$: Determine the Phase Shift

The following graph shows how inlet flow and both the hot and cold outlet temperatures vary as sinusoidal functions of time. This graph was generated using the same data for the heat exchanger of Example 1. Use this graph to find the phase shift between the inlet flow and the hot outlet stream.

Solution

Determine the period (P) – This can be done by finding the time from one peak to the next of a given wave. In this case, we want to know the period of the inlet flow rate, so $$P = 1.14\,s$$.

Determine the lag (delta P) – This can be done by finding the time difference between the peak of the inlet flow rate and the peak of the hot outlet stream. (Remember that the hot outlet wave lags the wave of the inlet flowrate).

$$\delta P = 0.87s$$

$\text {Phaseshift}=\frac{0.87\, s}{1.14\, s} *-2 \pi=-4.80$

Since we're only concerned with time values for finding the phase shift, the data doesn't need to be normalized.

## References

• Bequette, B. Wayne. Process Control: Modeling, Design, and Simulation. pp 172-178, 214-235. Upper Saddle River, N.J. Prentice Hall, PTR 2003.
• Liptak, Bela G. Process Control and Optimization. Volume 2. Instrument Engineers Handbook. Fourth Edition. CRC Press, Boca Raton Fl. 2003. Insertformulahere

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# All Activity

1. Past hour
2. ## Complex spaces

I have limited knowledge of this but was ”complex spaces” not possible before? I think have come across the term earlier, maybe more definitions are needed? What complex spaces are we discussing? what is GeneroiKantaolioidenTulot? Generates Constants Revenue? Lack of comments combined with usage of non-english definitions makes the program hard to read.
3. ## Complex spaces

What is this program supposed to do?
4. Today

6. ## Einstein's first Law

I was thinking of the FOR the physical experiment was carried out in.(if that helps)
7. ## Einstein's first Law

It says the same laws of physics apply. There is no experiment that shows that one frame must be at rest, while another must be in motion. Energy is a relative value so in general different frames will measure a different energy.
8. ## Is there an experiment that verifies what happens when two unrelated matter waves collide?

You will see wave behavior if your experiment is set up to see that, and particle behavior in an experiment designed to see particle behavior. It’s not like a quantum particle acting as a wave can never exhibit particle behavior in some other interaction. Observing isn’t an issue - this isn’t related to wave function collapse
9. ## Pascal's wager and atheism (split from What made you stop believing in God?)

Very true, it needs physical evidence. For me space and time itself, the atoms, electrons, photons, waves are sign of God. For me God=Nature I rather believe, (gives power) than not. I think Pascal sensed that. I need the best physical evidence and I got Nature. I understand how small and simple I am. I understand that I am still Valuable. I have Energy, Matter and Information in Space(Time). Of course they are, it's right there in pascals wager What atheists betting on? It is a bit unclear even I read this forum since a while. As far as I understand they are betting on Nothing when it comes to God. The crazy thing is that the only supernatural I can sense is the sense of Nothing. Space, time, energy and matter free state. 0. God = 0 0,0 0,00. 0.00....0=time
10. ## Why do humans walk upright?

! Moderator Note This is a discussion forum, not your blog or a place to publish scientific papers. I left the abstract. If you want to discuss your idea you are free to do that. If you want your paper published then submit it to a journal. ! Moderator Note Moved to Speculations. Please read the special rules for this section of the forum.
11. ## The solution of the Cosmological constant problem ?

a would be the differential between your H/H_0 recall that a as the scale factor is just a dimensionless ratio parameter. You set scale factor at cosmological time now at 1. a would be the differential between your H/H_0 recall that a as the scale factor is just a dimensionless ratio parameter. You set scale factor at cosmological time now at 1. Here this will help ${\small\begin{array}{|c|c|c|c|c|c|}\hline T_{Ho} (Gy) & T_{H\infty} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$ ${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{gen}/c&H/Ho \\ \hline 0.001&1090.000&0.000373&0.000628&45.331596&0.041589&0.056714&21.023&22915.263\\ \hline 0.001&863.334&0.000551&0.000915&45.159913&0.052309&0.071406&18.232&15740.128\\ \hline 0.001&683.804&0.000810&0.001326&44.962398&0.065753&0.089864&15.881&10859.192\\ \hline 0.002&541.606&0.001183&0.001915&44.736079&0.082599&0.113040&13.885&7520.218\\ \hline 0.002&428.979&0.001722&0.002756&44.477683&0.103683&0.142116&12.180&5224.758\\ \hline 0.003&339.773&0.002496&0.003956&44.183524&0.130038&0.178562&10.712&3639.803\\ \hline 0.004&269.117&0.003606&0.005666&43.849475&0.162938&0.224202&9.443&2541.361\\ \hline 0.005&213.154&0.005194&0.008100&43.470902&0.203941&0.281289&8.340&1777.702\\ \hline 0.006&168.829&0.007463&0.011563&43.042568&0.254948&0.352603&7.377&1245.393\\ \hline 0.007&133.721&0.010698&0.016484&42.558633&0.318265&0.441559&6.533&873.554\\ \hline 0.009&105.913&0.015309&0.023478&42.012463&0.396668&0.552333&5.791&613.344\\ \hline 0.012&83.889&0.021873&0.033412&41.396601&0.493471&0.690005&5.138&430.988\\ \hline 0.015&66.444&0.031211&0.047518&40.702622&0.612585&0.860719&4.561&303.042\\ \hline 0.019&52.627&0.044487&0.067545&39.921133&0.758568&1.071848&4.051&213.190\\ \hline 0.024&41.683&0.063355&0.095974&39.041469&0.936624&1.332155&3.600&150.041\\ \hline 0.030&33.015&0.090158&0.136321&38.051665&1.152552&1.651928&3.200&105.633\\ \hline 0.038&26.150&0.128224&0.193578&36.938267&1.412573&2.043059&2.845&74.389\\ \hline 0.048&20.712&0.182274&0.274818&35.686105&1.722983&2.519001&2.530&52.398\\ \hline 0.061&16.405&0.258995&0.390062&34.278330&2.089532&3.094542&2.250&36.917\\ \hline 0.077&12.993&0.367873&0.553490&32.695921&2.516347&3.785220&2.002&26.017\\ \hline 0.097&10.291&0.522342&0.785104&30.917756&3.004225&4.606237&1.782&18.342\\ \hline 0.123&8.151&0.741396&1.112970&28.920472&3.547949&5.570564&1.587&12.938\\ \hline 0.155&6.456&1.051751&1.575989&26.679131&4.132295&6.685941&1.415&9.137\\ \hline 0.196&5.114&1.490772&2.226851&24.167785&4.726112&7.950210&1.265&6.467\\ \hline 0.247&4.050&2.109877&3.133394&21.362526&5.274330&9.344906&1.135&4.596\\ \hline 0.312&3.208&2.977691&4.373615&18.247534&5.688090&10.827382&1.026&3.292\\ \hline 0.394&2.541&4.180384&6.013592&14.827243&5.835394&12.323993&0.942&2.395\\ \hline 0.497&2.013&5.813076&8.053192&11.147771&5.539179&13.731340&0.888&1.788\\ \hline 0.627&1.594&7.955449&10.346218&7.320583&4.592515&14.935503&0.873&1.392\\ \hline 0.792&1.263&10.632280&12.576261&3.528946&2.795101&15.853609&0.907&1.145\\ \hline 1.000&1.000&13.787206&14.399932&0.000000&0.000000&16.472274&1.000&1.000\\ \hline 1.263&0.792&17.302122&15.660852&3.144839&3.970507&16.845080&1.161&0.919\\ \hline 1.571&0.637&20.811932&16.388509&5.639833&8.860361&17.041221&1.380&0.879\\ \hline 1.955&0.512&24.444731&16.808523&7.715938&15.083827&17.146441&1.675&0.857\\ \hline 2.433&0.411&28.147021&17.039505&9.416655&22.906412&17.198866&2.056&0.845\\ \hline 3.027&0.330&31.886775&17.163312&10.797410&32.682676&17.221760&2.540&0.839\\ \hline 3.766&0.265&35.646534&17.228668&11.913050&44.870264&17.228668&3.148&0.836\\ \hline 4.687&0.213&39.416981&17.262737&12.812202&60.047793&17.262737&3.910&0.834\\ \hline 5.832&0.171&43.192748&17.280614&13.535828&78.939747&17.280614&4.860&0.833\\ \hline 7.257&0.138&46.971377&17.289940&14.117812&102.450932&17.289940&6.044&0.833\\ \hline 9.030&0.111&50.751666&17.294639&14.585725&131.708610&17.294639&7.519&0.833\\ \hline 11.236&0.089&54.532554&17.297194&14.961819&168.115662&17.297194&9.354&0.833\\ \hline 13.982&0.072&58.313845&17.298546&15.264097&213.418728&17.298546&11.639&0.832\\ \hline 17.398&0.057&62.095516&17.299100&15.507045&269.791380&17.299100&14.482&0.832\\ \hline 21.649&0.046&65.877123&17.299503&15.702284&339.937814&17.299503&18.020&0.832\\ \hline 26.939&0.037&69.658785&17.299738&15.859189&427.223615&17.299738&22.423&0.832\\ \hline 33.521&0.030&73.440477&17.299885&15.985286&535.836513&17.299885&27.902&0.832\\ \hline 41.711&0.024&77.222357&17.299813&16.086628&670.987643&17.299813&34.719&0.832\\ \hline 51.902&0.019&81.004072&17.299891&16.168066&839.160848&17.299891&43.202&0.832\\ \hline 64.584&0.015&84.785791&17.299957&16.233514&1048.424819&17.299957&53.758&0.832\\ \hline 80.364&0.012&88.567685&17.299844&16.286113&1308.819673&17.299844&66.893&0.832\\ \hline 100.000&0.010&92.349407&17.299900&16.328381&1632.838131&17.299900&83.237&0.832\\ \hline \end{array}}$ Row 1.00 is today after row 1.00 is future. Previous is past.
12. ## Pascal's wager and atheism (split from What made you stop believing in God?)

it didn't even sound clever the first time ..... good luck with your "atheism" LOL
13. ## How to make better water filter? Turn it inside out

Full story at link....https://phys.org/news/2019-08-filter.html Original story by Erica K. Brockmeier, University of Pennsylvania. a diagram of how the nanofilters are made (top panel) and their microscopic structure (bottom panel) after Abstract.....Post docs Xunda Feng, now at Donghua University, and Yizhou Zhang and graduate student Qaboos Imran, have conducted a study at the lab of Chinedum Osuji which describes a new way to create nanoscale water filters, which as well as being flexible and robust, also have anti microbial properties. When designing a nanoscale filter, engineers usually start with something that resembles a microscopic strainer or a sieve. Water travels through individual holes that are spread along the strainer and are held together by a solid material that fills the space around them. Osuji's group, experts in modifying the chemistries of block polymers, large chains of molecules with large "blocks" of repeated sequences, found something unexpected while studying a similar material. Their discovery led them to "inverting" their design strategy: turning the "holes" of the strainer into solid fibres, leaving the previously solid portions of the structure open. The material was engineered with cross links between individual fibres to add support. Also included in the polymer are chemical structures that give it antimicrobial properties, which will prevent it from being clogged by bacteria. Osuji says.. "Reverse osmosis is highly developed and very efficient at removing all but the most challenging contaminants, but there are places where it is not cost effective, such as in the treatment of brackish water, treatment of industrial wastewater before discharge, or water softening, there is a possibility to push these new membranes into those regimes".
14. ## Merge USA and Canada?

Then don't use Trump as the bar. Be specific and say the country itself sucks.

16. ## Merge USA and Canada?

When half the country follows him, that's where the bar is set. Thing is, even when Trump is gone, America will still be what it is. We used to be in the commonwealth. The queen isn't our monarch. We don't pay her anything and she doesn't decree. Trudeau Sr. patriated our constitution a half century ago. Our constitution is a modern document though, something that is not amended and absent of contentious clauses that authoritarians love to cling to, like guns.
17. ## The solution of the Cosmological constant problem ?

to make sure I'm not mistaken with the reverse : is $H/H_0=da/dt$ with $a=1$ now ? (I'm not sure of myself)
18. ## Pascal's wager and atheism (split from What made you stop believing in God?)

The irony is strong in you, my young padawan.
19. ## Pascal's wager and atheism (split from What made you stop believing in God?)

another deflection. fair enough.

LOL
21. ## Pascal's wager and atheism (split from What made you stop believing in God?)

Now that's ironic...
22. ## Pascal's wager and atheism (split from What made you stop believing in God?)

the only one who can spell out your views is you. if you want to engage in a conversation as an equal, then you would afford me that much. Fry is funny i like him, he has said some things that would put him very much in the "bad atheist" category acccording to some people.Things I would not even say. In fact, I think it was partly thanks to him that the anti blasphemy law was finally gotten rid of in Ireland. vHe was criticizing God on a television Interview and someone reported his anti God comments to the Irish Police. The police never followed it up, presumably because the will wasn't there to enforce such a draconian law. shortly afterwards we had a referrendum to get rid of the law, just a few months ago actually. Yep. 2019, not the 1600's. 2019
23. ## Complex spaces

By eliminating symbolic signs for multiplication, complex spaces are possible. I studied this problem for over a decade until it was resolved. Complex spaces fulfill all municipal conditions. Please find the code below to test the municipal rules. #include <memory.h> #include <stdlib.h> #include <stdio.h> #include <math.h> class number { public: #define DIMENSION 4 double e[DIMENSION]; number(void); ~number(void); number(char*, ...); number(int); number(int, int); number(int, int, int, int); friend void print(number); friend double abs(number); friend number operator-(number); friend number operator+(number, number); friend number operator-(number, number); friend number operator*(number, number); friend number operator/(number, number); friend number operator*(number, double); friend number operator/(number, double); friend number operator*(double, number); friend number operator/(double, number); friend number pow(number, int); }; number::number(void) { memset(this, 0, sizeof(number)); } number::~number(void) { } /********************************************************************* * Generator number (char *, ...) * * * * For example: * * * * number x("2d", 2, 3), x gets a value of 2.0000 + 3.0000i * * number y("2f", 1.2345, 6.7898), y gets a value of 1.2345 + 6.7898i * * * * The format first gives the number of chapters, for example 4. * * - 'd' means that the numbers are given as integers. * * - 'f' means that the numbers are given in floating point form. * *********************************************************************/ number::number(char *f, ...) { char format='d'; int i, kpl=atoi(f); int step=(int)sizeof(int); memset(this, 0x00, sizeof(number)); unsigned long P=(unsigned long)&f+sizeof(char*); for (i=0; f[i]; i++) { if (f[i] == 'f') { step=sizeof(double); format='f'; } } for (i=0; i<kpl; i++, P+=step) e[i] = (char)format=='d'? *(int*)P: *(double*)P; } /********************************************************************* * Somewhere around 20 elements when the parent vector ID is larger * * like 'z', the base vector ID may be a letter abnormal character. * * In fact, the calculation of the base vectors no comment is made on * * symbolic symbols. * *********************************************************************/ void print(number a) { printf("%0.9f ", a.e[0]); for (int n=1; n<DIMENSION; n++) printf("%c %0.9f%c ", a.e[n]<0? '-': '+', fabs(a.e[n]), 'h'+n); printf("\n"); } number::number(int a) { memset(this, 0, sizeof(number)); e[0]=a; } number::number(int a, int b) { memset(this, 0, sizeof(number)); e[0]=a; e[1]=b; } number::number(int a, int b, int c, int d) { memset(this, 0, sizeof(number)); e[0]=a; e[1]=b; e[2]=c; e[3]=d; } number operator-(number a) { for (int n=0; n<DIMENSION; n++) a.e[n]=-a.e[n]; return a; } number operator+(number a, number b) { for (int n=0; n<DIMENSION; n++) a.e[n]+=b.e[n]; return a; } number operator-(number a, number b) { for (int n=0; n<DIMENSION; n++) a.e[n]-=b.e[n]; return a; } number operator*(number a, number b) { int *KantaolioidenTulot(void); int *R=KantaolioidenTulot(); int i, j, n=DIMENSION*0x02; double x[DIMENSION*2]; double y[DIMENSION*2]; double t[DIMENSION*2]; for (i=j=0x00; i<DIMENSION; i++, j+=2) x[j+1]=y[j+1]=t[j]=t[j+1]=0.0, x[j]=a.e[i], y[j]=b.e[i]; for (i=0; i<n; i++) for (j=0; j<n; j++) t[R[i*n+j]]+=x[i]*y[j]; for (i=j=0x00; i<DIMENSION; i++, j+=2) t[i]=(double)(t[j] - t[j+1]); return *(number*) t; } void GeneroiKantaolioidenTulot(int *R, int n) { int I=0, J=n-1, i; int X=n, Y=J+n, j; int k=DIMENSION*2; for (i=I; i<=J; i++) for (j=I; j<=J; j++) { R[i*k+X+j]=R[i*k+j]+X; R[(X+j)*k+i]=R[i*k+j]+X; R[(Y-i)*k+Y-j]=J-R[i*k+j]; } if (n+n < DIMENSION*2) { GeneroiKantaolioidenTulot(R, n+n); } } int* KantaolioidenTulot(void) { static int R[DIMENSION*DIMENSION*4]={-1}; if (R[0] == -1) { int EnsinOliNolla=0x00; R[0]=(double)EnsinOliNolla; GeneroiKantaolioidenTulot(R, 1); } return R; } number operator/(number x, number y) { number z; for (int i, n=DIMENSION; n>=2; n/=2) { for (z=y, i=n/2; i<n; i++) z.e[i] = -z.e[i]; x=x*z; y=y*z; } return x/y.e[0]; } number operator/(number x, double k) { for (int i=0; i<DIMENSION; i++) x.e[i]/=(double)k; return x; } number operator*(double k, number x) { for (int i=0; i<DIMENSION; i++) x.e[i]*=(double)k; return x; } number operator*(number x, double k) { return k*x; } number operator/(double k, number x) { number y("1f", k); return (number)y/x; } double abs(number x) { number z; double r, c; int i, j=1, n; for (n=DIMENSION; n>=2; n/=2, j+=j) { for (z=x, i=n/2; i<n; i++) z.e[i]=-z.e[i]; x=x*z; } r=fabs(x.e[0]); c=1.0/(double)j; return pow(r, c); } number pow(number x, int n) { if (n) { number t=x; int i=n<0? -n: n; for (--i; i; i--) t=t*x; return n>0? t: number(1) / t; } else { return number(1); } } void main(void) { number x("4f", 0.12, -0.34, 0.56, -0.78); number y("4f", 0.98, -0.76, 0.54, -0.32); printf("x = "); print(x); printf("y = "); print(y); printf("\n"); printf("x*y = "); print(x*y); printf("y*x = "); print(y*x); printf("\n"); }
24. ## Pascal's wager and atheism (split from What made you stop believing in God?)

It's a shame you can't see the second video, it basically says, "you're predecessor was a raging theist and so will your progeny". Do I really have to spell it out?
25. ## Pascal's wager and atheism (split from What made you stop believing in God?)

oh, so you are still talking to me? whatever for, im not sure. I cant see the second video, it is blocked in my country. the first one is funny, so im a "bad" atheist am I? LOL I just gave my honest account. go ahead and explain why its bad, tell me the alternative (without lying about what i said or being unnecessarily patronizing) but i doubt you will at this point, that doesn't seem to be your goal.
26. ## infinity - could it lead to reincarnation?

Just so we’re clear and in case there was any confusion, I was not under the influence when typing my post. ✌
27. ## Why do humans walk upright?

I would want to participate in discussing of your findings, initially to see if your ideas and arguments are supported by mainstream science. This is a discussion forum, so your discovery will be discussed. The outcome of such a discussion will partly depend on what and how you choose to contribute. (Side note: My first language is not english, but on this forum language is seldom a problem. General attitude of most members is helpful; language issues are identified and overcome)


text:
# Is there a formal name for the “love” of arguing?

Most of people drop discussion easily. Some others don't. As if they would like to argue. They feel that they have to demonstrate something.

Is there a formal trait to describe people who enjoy arguing and the behavior of enjoying an argument?

-
I like polemical as a word, but it has little to do with cognitive science. Extending that, polemicophilia, while not actually being a word in use, might be a nice term for what you're describing (although 'polemikos' actually means 'war' in Greek). – Eoin Mar 7 '14 at 10:25
@Eoin: yes, polemic is another which I like. Are there no correlation between polemicity and any trait of personality? Or with any axis defined by the MMPI? Or with some of the myers briggs classification? – Revious Mar 7 '14 at 14:49


text:
# Binomial Series

1. Nov 20, 2004

(a) Expand

$$f(x)=\frac{x+x^2}{\left( 1-x\right) ^3}$$

as a power series.

(b) Use part (a) to find the sum of the series

$$\sum _{n=0} ^{\infty} \frac{n^2}{2^n}$$

$$\hline$$

Here is what I've got:

(a)

$$f(x)=\frac{x+x^2}{\left( 1-x \right) ^3 } = \left( x + x^2 \right) \left[ 1 + (-x) \right] ^{-3} = \left( x + x^2 \right) \sum _{n=0} ^{\infty} \binom{-3}{k} (-x) ^k = \left( x + x^2 \right) \sum _{n=0} ^{\infty} (-1) ^k \binom{-3}{k} x ^k$$

$$f(x) = \left( x + x^2 \right) \left[ \binom{-3}{0} - \binom{-3}{1}x + \binom{-3}{2} x^2 - \binom{-3}{3} x^3 + \binom{-3}{4} x^4 - \cdots \right]$$

$$f(x) = \left( x + x^2 \right) \left[ 1 - \frac{(-3)}{1!}x + \frac{(-3)(-4)}{2!}x^2 - \frac{(-3)(-4)(-5)}{3!}x^3 + \frac{(-3)(-4)(-5)(-6)}{4!}x^4 -\cdots \right]$$

$$f(x) = \left( x + x^2 \right) \left[ 1 + \frac{1}{2} \sum _{n=1} ^{\infty} \frac{\left( n+2 \right)!}{n!}x^n \right] = \left( x + x^2 \right) \left[ 1 + \frac{1}{2} \sum _{n=1} ^{\infty} (n+2)(n+1)x^n \right]$$

$$f(x) = x + x^2 + \frac{1}{2} \sum _{n=1} ^{\infty} \left[ (n+2)(n+1)x^{n+1} \right] + \frac{1}{2} \sum _{n=1} ^{\infty} \left[ (n+2)(n+1)x^{n+2} \right]$$

(b)

$$\sum _{n=0} ^{\infty} \frac{n^2}{2^n} = ?$$

Thank you

Last edited: Nov 20, 2004
2. Nov 20, 2004

### vsage

Did you write the question correctly? I don't see how you can use partial fractions in the last equation you give to get the question again. I don't think you can either because for x = 2 the equivalency isn't there.

Last edited by a moderator: Nov 20, 2004
3. Nov 20, 2004

### arildno

That last equality is incorrect!

4. Nov 20, 2004

I see. You guys are right about that last equality.

If I have

$$\sum _{n=0} ^{\infty} \frac{n^2}{2^n}$$

I may need to rewrite

$$\frac{n^2}{2^n}$$

and compute it with the aid of the f(x) I found in part (a). What I can do is:

$$\frac{n^2}{2^n} = n^2 \left( \frac{1}{2} \right) ^n$$

but the connection with f(x) still is not clear.

5. Nov 20, 2004

I've just made some corrections to my 1st post.... did I get it right this time?

Thanks.

6. Nov 20, 2004

vsage,

I'm sorry. I wrote it incorrectly last time. It isn't:

$$\frac {x^2+x}{1-x^3}$$

I've just edited my 1st post to fix it. Here it is:

$$f(x)=\frac{x+x^2}{\left( 1-x\right) ^3}$$

Sorry again, and thanks for your help.

7. Nov 20, 2004

### vsage

Consider that $$\frac {1}{(1-x)^3} = \frac {d^2\frac{1}{2} \frac{1}{1-x}}{dx^2}$$

8. Nov 20, 2004

Well, I see that:

$$f(x) = \frac{x+x^2}{\left( 1-x\right) ^3} = -\frac{1}{2}\left( x + x^2 \right) \frac{d^2}{dx^2} \left[ \frac{1}{1-x} \right] = -\frac{1}{2}\left( x + x^2 \right) \frac{d^2}{dx^2} \left[ \sum _{n=0} ^{\infty} x^n \right] = -\frac{1}{2}\left( x + x^2 \right) \sum _{n=2} ^{\infty} n(n-1)x^{n-2}$$

This method is, in fact, simpler. The directions of this problem in part (a) are to expand f(x) as a power series, so it's ok to do that. But, I've got this exercise from the "Binomial Series" section in my calc book, so I tried to stick to it. That's why I used it in my 1st post.

Maybe your tip was about the connection between f(x) and the series in part (b). To be honest, I don't see how it may be related to it.

9. Nov 21, 2004

Folks, let me rephrase what I have. I was able to develop the solution quite a bit farther, but there still are some unclear areas.

Problem:

(a) Expand

$$f(x)=\frac{x+x^2}{\left( 1-x\right) ^3}$$

as a power series.

(b) Use part (a) to find the sum of the series

$$\sum _{n=1} ^{\infty} \frac{n^2}{2^n}$$

$$\hline$$

Solution:

(a)

$$f(x)=\frac{x+x^2}{\left( 1-x\right) ^3} = \left( x + x^2 \right) \left[ 1 + (-x) \right] ^{-3} = \left( x + x^2 \right) \sum _{n=0} ^{\infty} \binom{-3}{k} (-x) ^k = \left( x + x^2 \right) \sum _{n=0} ^{\infty} (-1) ^k \binom{-3}{k} x ^k$$

$$f(x) = \left( x + x^2 \right) \left[ \binom{-3}{0} - \binom{-3}{1}x + \binom{-3}{2} x^2 - \binom{-3}{3} x^3 + \binom{-3}{4} x^4 - \cdots \right]$$

$$f(x) = \left( x + x^2 \right) \left[ 1 - \frac{(-3)}{1!}x + \frac{(-3)(-4)}{2!}x^2 - \frac{(-3)(-4)(-5)}{3!}x^3 + \frac{(-3)(-4)(-5)(-6)}{4!}x^4 -\cdots \right]$$

$$f(x) = \left( x + x^2 \right) \left[ 1 + \frac{1}{2} \sum _{n=1} ^{\infty} \frac{\left( n+2 \right)!}{n!}x^n \right] = \left( x + x^2 \right) \left[ 1 + \frac{1}{2} \sum _{n=1} ^{\infty} (n+2)(n+1)x^n \right]$$

$$f(x) = x + x^2 + \frac{1}{2} \sum _{n=1} ^{\infty} \left[ (n+2)(n+1)x^{n+1} \right] + \frac{1}{2} \sum _{n=1} ^{\infty} \left[ (n+2)(n+1)x^{n+2} \right]$$

$$f(x) = x + x^2 - \frac{x\left( 3x-3x^2+x^3\right)}{\left( x-1 \right) ^3} - \frac{x^2\left( 3x-3x^2+x^3\right)}{\left( x-1 \right) ^3} \qquad \fbox{UNCLEAR 1}$$

$$f(x) = x + x^2 - \left[ x +\frac{1}{\left( x-1 \right) ^2} +\frac{1}{\left( x-1 \right) ^3} \right] - \left[ x^2 + \frac{1}{ x-1} + \frac{2}{\left( x-1 \right) ^2} + \frac{1}{\left( x-1 \right) ^3} \right]$$

$$f(x) = -\frac{1}{ x-1} -\frac{3}{\left( x-1 \right) ^2} - \frac{2}{\left( x-1 \right) ^3}$$

(b)

$$\sum _{n=1} ^{\infty} \frac{n^2}{2^n} = 6 = f\left( \frac{1}{2} \right) \qquad \fbox{UNCLEAR 2}$$

$$\fbox{UNCLEAR 1}:$$ I don't understand the transition from the series above to this result.
$$\fbox{UNCLEAR 2}:$$ I've found the sum of the series ("6") with the aid of my calculator. Later, I evaluated a guess ("1/2"), which gave the correct answer. However, it still isn't explicit the connection between

$$\sum _{n=1} ^{\infty} \frac{n^2}{2^n} \qquad \mbox{ and } \qquad f(x) = \frac{x+x^2}{\left( 1-x\right) ^3}$$

Thank you very much.

Last edited: Nov 21, 2004
10. Nov 21, 2004

Guys, I may have found the rest of the solution:

(a)

$$f(x)=\frac{x+x^2}{\left( 1-x\right) ^3} = \left( x + x^2 \right) \left[ 1 + (-x) \right] ^{-3} = \left( x + x^2 \right) \sum _{n=0} ^{\infty} \binom{-3}{n} (-x) ^n = \left( x + x^2 \right) \sum _{n=0} ^{\infty} (-1) ^n \binom{-3}{n} x ^n$$

$$f(x) = \left( x + x^2 \right) \left[ \binom{-3}{0} - \binom{-3}{1}x + \binom{-3}{2} x^2 - \binom{-3}{3} x^3 + \binom{-3}{4} x^4 - \cdots \right]$$

$$f(x) = \left( x + x^2 \right) \left[ 1 - \frac{(-3)}{1!}x + \frac{(-3)(-4)}{2!}x^2 - \frac{(-3)(-4)(-5)}{3!}x^3 + \frac{(-3)(-4)(-5)(-6)}{4!}x^4 -\cdots \right]$$

$$f(x) = \left( x + x^2 \right) \left[ 1 + \frac{1}{2} \sum _{n=1} ^{\infty} \frac{\left( n+2 \right)!}{n!}x^n \right] = \left( x + x^2 \right) \left[ 1 + \frac{1}{2} \sum _{n=1} ^{\infty} (n+2)(n+1)x^n \right]$$

$$f(x) = x + x^2 + \frac{1}{2} \sum _{n=1} ^{\infty} \left[ (n+2)(n+1)x^{n+1} \right] + \frac{1}{2} \sum _{n=1} ^{\infty} \left[ (n+2)(n+1)x^{n+2} \right]$$

$$f(x) = x + x^2 - \frac{x\left( 3x-3x^2+x^3\right)}{\left( x-1 \right) ^3} - \frac{x^2\left( 3x-3x^2+x^3\right)}{\left( x-1 \right) ^3} \qquad \fbox{UNCLEAR}$$

$$f(x) = x + x^2 - \left[ x +\frac{1}{\left( x-1 \right) ^2} +\frac{1}{\left( x-1 \right) ^3} \right] - \left[ x^2 + \frac{1}{ x-1} + \frac{2}{\left( x-1 \right) ^2} + \frac{1}{\left( x-1 \right) ^3} \right]$$

$$f(x) = -\frac{1}{ x-1} -\frac{3}{\left( x-1 \right) ^2} - \frac{2}{\left( x-1 \right) ^3} = \frac{x\left( x+1 \right)}{\left( x-1 \right) ^3} = x \frac{d}{dx} x \frac{d}{dx} \left[ \frac{1}{1-x} \right] = x \frac{d}{dx} x \frac{d}{dx} \left[ \sum _{n=0} ^{\infty} x^n \right]$$

$$f(x) = \sum _{n=1} ^{\infty} n^2 x^n$$

(b)

$$\sum _{n=1} ^{\infty} \frac{n^2}{2^n} = \sum _{n=1} ^{\infty} n^2 \left( \frac{1}{2} \right) ^n = f\left( \frac{1}{2} \right) = \frac{\frac{1}{2}+\left( \frac{1}{2} \right)^2}{\left( 1-\frac{1}{2}\right) ^3} = 6$$

$$\fbox{UNCLEAR}:$$ I don't understand the transition from the series above to this result.


text:
## Constraints on Unparticle Physics from Solar and KamLAND Neutrinos

##### Authors
Anchordoqui, Luis
Goldberg, Haim
##### Description
Interest has been directed recently towards low energy implications of a non-trivial conformal sector of an effective field theory with an IR fixed point (\Lambda), manifest in terms of unparticles'' with bizarre properties. We re-examine the implications of the limits on decay lifetimes of solar neutrinos for unparticle interactions. We study in detail the fundamental parameter space (\Lambda, M) and derive bounds on the energy scale M characterizing the new physics. We work strictly within the framework where conformal invariance holds down to low energies. We first assume that couplings of the unparticle sector to the Higgs field are suppressed and derive bounds with \Lambda in the TeV region from neutrino decay into scalar unparticles. These bounds are significant for values of the anomalous dimension of the unparticle operator 1.0 < d < 1.2. For a region of the parameter space, we show that the bounds are comparable to those arising from production rates at high energy colliders. We then relax our assumption, by considering a more natural framework which does not require a priori restrictions on couplings of Higgs-unparticle operators, and derive bounds with \Lambda in meV region from neutrino decay into vector unparticles. Such low scales for the IR fixed point are relevant in gauge theories with many flavors.
Comment: To be published in Phys. Lett. B
##### Keywords
High Energy Physics - Phenomenology


text:
# CSAPP DATALAB

### Report For Puzzles

#### Bit Manipulations

1. bitAnd:

Similar to $A \cap B=-(-A \cup -B)$, $\sim (\sim x \lvert \sim y)=x\&y$

2. getByte:

To extract byte n from x, first shift x to the right by $n«3$, thus making the wanted byte reside in the last byte in x. Then x & 0xff is the wanted byte.

3. logicalShift:

To logically shift x to the right, first get $\sim ((1«31»n)«1)$ to be as the mask. Then, arithmetically shift x and x & make is the wanted result.

4. bitCount:

First, divide x into two by two. In the two bits, it’s can be easily evidenced that the sum of high order bit and low order bit equals the count of number of 1’s in the two bits. Thus, using 0x55555555 as mask, each two-digit value in (x&mask + (x»1)&mask) represents the number of 1’s in the two digits. Recursively, divide the result four b four, eight by eight and sixteen by sixteen. To eliminate the ops, in eight-digit dividing and sixteen-digit dividing, it’s reasonable to omit the mask. And at last, because the final result can’t exceed $2^5$，thus x&0x3f is the final number of 1’s in x.

5. bang:

 In bang function, only need to distinguish 0 and others. Only in 0 (~0+1), the first bit is 0. Thus, extract the first bit using x (~x+1) »31 and bitwise not it. At last, bitwise and 1 is the final result.

#### Two’s Complement Arithmetic

1. tmin:

1«31 is the TMIN

2. fitBits:

discuss in two situations using x’s sign bit. When x is positive, if all bits in x»(n+~0) equal 0, then n bits can represent x, vice versa. When x is negative, if all bits in ~ x»(n+~0) equal 0, then n bits can represent x.

n+~0=n-1

3. divpwr2:

discuss in two situations using x’s sign bit. When x is positive, x»n is the final result. When x is negative, using x+(1«n)+~0 to shift to the right for rounding toward zero.

4. negate:

~x+1=-x

5. isPositive:

!(x&(1«31)) = 1 when x is positive. Besides, taking account of zero, using the result Bitwise and !!x to get the right result.

6. isLessOrEqual:

Tacking account of overflow, discuss in two’s situation using x and y’s sign. When x’s sign and y’s sign is the same, if (y+~x+1) is positive, x is Less Or Equal y. When x’s sign and y’s sign is different, if x is negative, the x is Less Or Equal y.

7. ilog2:

First divide x into sixteen by sixteen, using a=!!(x»16) to see whether there exists 1 in the first sixteen bits. If exists, then x=x»(1«4) to shift the 1 to the right sixteen bits, vice versa. Then recursively, divide x into eight by eight to see whether the 1 reside in the second eight block from the right by b=!!(x»8). If exists, then shift x to the right by (1«3). Then, divide into four, two, one… At Last, (a«4)+(b«3)+(c«2)+(d«1)+e is the final result.

#### Floating-Point Operations

1. float_neg:

discuss in two’s situation by whether uf==NaN. When us equals NaN, return uf. Otherwise, uf ^ 0x80000000 is the final result.

using (((uf » 23) & 0xff) == 0xff) && (uf & 0x7fffff) != 0 to distinguish whether uf equals NaN

2. float_i2f:

The transformation is divided into three steps.

• use sign to store x’s sign And transform x into positive.
• shift x to left to get the highest order bit of x as the exponential
• use x & 0x01ff to distinguish whether x needs to add 1 to round and use flag to store

Then the final result is sign + exponential + x’s mantissa + flag

3. float_twice:

discuss uf into two situations by whether uf is denormalized number and NaN. First, if (uf & 0x7f800000) == 0, then shift uf’s mantissa to the left by 1. Then if uf is not denormalized and is not NaN, uf = uf + 0x00800000 represents the twice of uf. Otherwise, return uf.

### Experimental expreience

Through this experiment, I have a better command of Bit-level encoding rules of TINT, UINT numbers and float numbers. Actually, the lab is hard for me, so I struggle a lot in this lab and spend a lot of time. However, I acquire a lot from this lab in the meantime. I learn a lot about how integers and floats can be expressed in bit-level and how to accomplish various kinds of operations using just bit-level operations. It’s really worth it.


text:
# Synopsis: Tantalizing cosmic-ray electrons

A report from the PAMELA satellite says the electron flux in cosmic rays is consistent with the existence of new sources, such as dark matter particles or other astrophysical phenomena, but the results are still inconclusive.

In 2009, the satellite-borne instrument PAMELA reported that high-energy cosmic rays contained an unexpectedly large ratio of positrons to electrons. The excess flux of positrons suggested that perhaps the propagation of cosmic rays through space was different than had been assumed, or models needed to account for additional astrophysical sources, such as supernova remnants or dark matter particles—the latter being a particularly exciting possibility.

If additional sources are indeed at play, they should contribute roughly equally to positrons and to electrons, but in comparison to the larger electron flux, the contribution from extra sources would be more difficult to see than with positrons.

Now a new report from the PAMELA Collaboration appearing in Physical Review Letters contributes data on the absolute spectrum of electrons, for the first time at energies above $50\phantom{\rule{0.333em}{0ex}}\text{GeV}$. Although the new data can be fit reasonably well with a standard-model calculation of cosmic-ray propagation, a somewhat better fit can be obtained by including additional sources of electrons in the calculation. (These same additional sources provide a good fit to the excess positron fraction that PAMELA observed earlier.) The data are also consistent (within errors) with results from the Fermi Large Area Telescope, which measured the combined contribution from positrons and electrons and provided some support for the new sources hypothesis.

Although the new PAMELA results support the earlier tantalizing indications of additional sources of cosmic rays, the field is not yet at the point of confirming their existence or nature. – Stanley Brown

More Features »

### Announcements

More Announcements »

Astrophysics

## Previous Synopsis

Quantum Information

Fluid Dynamics

## Related Articles

Statistical Physics

### Viewpoint: New Clues as to Why Boyajian’s Star is Dimming

A statistical analysis links a star’s mysterious brightness fluctuations to internal nonequilibrium phenomena, rather than structures orbiting around the star. Read More »

Astrophysics

### Viewpoint: Searching for Baby Planets in a Star’s Dusty Rings

Images of gaps in the dust and gas around a young star provide the best evidence to date that these gaps host newly formed planets. Read More »

Optics

### Synopsis: Sharper Vision for Infrared Telescopes

Converting infrared light to visible light might boost the sensitivity of infrared telescope arrays. Read More »


text:
# \alpha -Helix and Mixed $3_{10}$/ \alpha -helix in Cocrystallized Conformers of Boc-Aib-Val-Aib-Aib-Val-Val-Val-Aib-Val-Aib-Ome

Karle, Isabella L and Flippen-Anderson, Judith L and Uma, Kuchibhotla and Balaram, Hemalatha and Balaram, Padmanabhan (1989) \alpha -Helix and Mixed $3_{10}$/ \alpha -helix in Cocrystallized Conformers of Boc-Aib-Val-Aib-Aib-Val-Val-Val-Aib-Val-Aib-Ome. In: Proceedings of the National Academy of Sciences of the United States of America, 86 (3). pp. 765-769.

 PDF 176(1989).pdf Restricted to Registered users only Download (854Kb) | Request a copy

## Abstract

Two molecules of Boc-Aib-Val-Aib-Aib-Val-Val-Val-Aib-Val-Aib-OMe (where Boc is t-butoxycarbonyl and Aib is \alpha-aminoisobutyryl) cocrystallize in a triclinic cell with different helical conformations. One molecule is completely \alpha-helical with seven 5 \rightarrow 1 intramolecular hydrogen bonds. It forms three head-to-tail NH··· O==C hydrogen bonds to other molecules of the same conformation. The second molecule has a mixed $3_{10}$ / \alpha -helix conformation with three 4 \rightarrow 1 hydrogen bonds and four 5 \rightarrow 1 hydrogen bonds; furthermore, there is a helix reversal at both termini. The second molecule forms only two head-to-tail hydrogen bonds with molecules of the same type, and the N(3)H group does not participate in any hydrogen bonding. The two different types of helices occur in alternate sheets in the crystal, where each sheet is composed of adjacent rods of helices formed by head-to-tail hydrogen bonding. Within each sheet, containing helices of only one type of conformation, the helices aggregate in a parallel mode. Between the sheets of different helices, the aggregation is antiparallel. The peptide, with formula $C_{51}H_{92}N_{10}O_{13}$, crystallizes in space group P1 with Z = 2 and cell parameters a = 10.047 \pm 0.002 A, b = 16.684 \pm 0.003 A, c = 19.198 \pm 0.004 A, \alpha = 80.30 deg \pm 0.01 deg, \beta = 85.74 deg \pm 0.01 deg, and \gamma = 83.03 deg \pm 0.01 deg; overall agreement factor R = 6.7% for 6053 data (\mid$F_0$\mid > 3\sigma) and 0.96- A resolution.

Item Type: Journal Article Copyright for this article belongs to National Academy of Sciences. peptide cyrstal structure;head-to-tail hydrogen bonds;helix unwinding;helix aggregation Division of Biological Sciences > Molecular Biophysics Unit 15 Feb 2005 19 Sep 2010 04:18 http://eprints.iisc.ernet.in/id/eprint/2784


text:
# TimeScales documentation¶

This is the documentarion of the TimeScales framework, which is used to run a Multiscale Model of the Medium Spiny Neuron of the Neostriatum, integrating electrical signalling with biochemical pathways.

The main script to launch the simulation is spinesIntegration.py, which accepts a certain amount of parameters.

Mattioni M, Le Novère N (2013) Integration of Biochemical and Electrical
PLoS ONE 8(7): e66811. doi:10.1371/journal.pone.0066811


text:
# Lie

(Redirected from Deceit)

A lie is what it is called when one tells the truth in a creative way. Given the usual effects of telling the truth, telling a lie is usually the only way to get people to do what you want them to do.

Professional liars study with a Zen master for ten years to receive a degree, which is called the Master of Business Administration. At the end of this period of study, they realize they could have just told people they had finished the degree without actually having done it.

## editThe science of lies

Lies have a life cycle: half-truth → tall tale → falsehood → porky → white lie → lie → dirty lie → damned lie → France → damned dirty lie → Half-Life 2: Episode 3 → political manifesto → statistics. Statistics are in turn interpreted by pundits and the media to create new lies. When dealing with statistics, one should remember that around 89% of all known statistics are made up. This has of course been proven through statistics.

The advent of quantum physics has changed the modern perception of lie formation. Albert Einstein, in his landmark book Das Kapital, promises:

"A lie is a statement based on a logical discrepancy, which draws energy from its surroundings with the relationship."

$E = (m*g) / q^2$

"Where E is the energy, m is the number of people who believed it, g is the weight of the lie, and q is the probability that you are 'not' in a movie (q = 1 - probability you are in a movie). This simple equation gives us insight to the true power of a lie."

A commonly used basic unit of measure describing the number of lies per sentence is the Goebbels or Go.


text:
Enhanced ${\cal N}=8$ Supersymmetry of three-algebra ABJM Theory
By three-algebra ABJM theory I refer to any ${\cal N} = 6$ supersymmetric Chern-Simons matter theory. In this talk I show that if a three-algebra ABJM theory has a covariantly constant unitary monopole operator carrying two symmetric three-algebra indices, then this implies that this ABJM theory has hidden ${\cal N} = 8$ supersymmetry.