Nosimpler

In physics, the term general covariance is meant to indicate the property of a physical system or model (in theoretical physics) whose configurations, action functional and equations of motion are all equivariant under the action of the diffeomorphism group on the smooth manifold underlying the spacetime or the worldvolume of the system. The archetypical example of a generally covariant system is of course Einstein-gravity / “general relativity”. I indicate here how general covariance has a natural formalization in homotopy type theory, hence internal to any ∞-topos. For background and all details see at general covariance on the nLab, and the links given there. Of course a basic idea of traditional dependent type theory is that types A may appear in context of other types Γ. The traditional interpretation of such a dependent type x:Γ⊢A(x):Type is that of a Γ-parameterized family or bundle of types, whose fiber over x∈Γ is A(x). But in homotopy type theory, types are homotopy types, of course, and, hence so are the contexts. A type in context is now in general something more refined than just a family of types. It is really a family of types equipped with equivariance structure with respect to the homotopy groups of the context type. Specifically, if the context type is connected and pointed, then it is equivalent to the delooping Γ≃BG of an ∞-group G. One finds (this is discussed here) – that the context defined by the type BG is that of G-equivariance: every type in the context is a type in the original context, but now equipped with a G-∞-action. In a precise sense, the homotopy type theory of G-∞-actions is equivalent to plain homotopy type theory in context BG. Consider this for the case that G is an automorphism ∞-group of a type Σ which is regarded as representing spacetime or a worldvolume. We show that in this context the rules of homotopy type theory automatically induce the principle of general covariance and naturally produce configurations spaces of generally covariant field theories: for Fields a moduli object for the given fields, so that a field configuration is a function ϕ:Σ→Fields, the configuration space of covariant fields is the function type (Σ→Fields) but formed in the “general covariance context” BAut(Σ). When interpreted in smooth models, Conf is the smooth groupoid of field configurations and diffeomorphism gauge transformations acting on them, the Lie integration of the BRST complex whose degree-1 elements are accordingly called the diffeomorphism ghosts. More precisely, I show the following (and thanks again to Mike, for discussion of this here). Definition. Consider in homotopy type theory two types ⊢Σ,Fields:Type, to be called spacetime and field moduli. Let ⊢BAut(Σ):Type be the image of the name of Σ, with essentially unique term ⊢Σ:BAut(Σ). Perform the canonical context extension of Σ and trivial context extension of Fields to get types in context Σ:BAut(Σ)⊢Σ:Type and Σ:BAut(Σ)⊢Fields:Type. Form then the type of field moduli “Conf≔(Σ→Fields)” in this context: Conf≔Σ:BAut(Σ)⊢(Σ→Fields):Type. Proposition. The categorical semantics of Conf in the model of smooth cohesion, and for Σ a smooth manifold, is given by the diffeological groupoid Conf=[Σ,Fields]⫽Diff(Σ) whose objects are field configurations on Σ and whose morphisms are diffeomorphism gauge transformations between them. In particular the corresponding Lie algebroid is dual to the (off-shell) BRST complex of fields with diffeomorphism ghosts in degree 1.

This fractal consists in a combination (using complex division) of two Julia II fractals.

The following is a paraphrase of a comment at the end of the Suggested Readings section of Spivak's calculus book:

Abel remarked that he attributed his profound knowledge of mathematics to the fact that he read the masters, rather than the pupils.

Are you better off reading the Masters or the pupils?  This of course depends on the masters and the pupil and other factors.

1. I have heard that Godel's original papers (even when translated) are well written and show a profound understanding of the subject and why its important.
   
2. However, we now have a better understanding of what Godel did and better ways to express it.
   
3. The Masters may include the motivation which may be lost in later papers.
   
4. Often the first proof of anything is ugly or odd and later proofs really clean it up.
   
5. Often the first proof of anything uses only basic concept- later abstractions may hide the heart of the proof.
   
6. As a practical matter sometimes the early papers are not available (thanks to paywalls or obscurity) or in a language you do not read.
   
7. If Lance and I ever do a book-of-blog-posts I will clean up some of the spelling, make some of the arguments more clear (perhaps indicate where I am being sarcastic in cases where it was not understood), improve the writing. This will make it better than the blog but less authentic.

Here are examples where the Masters papers may not be worth reading:

1. Recursion theory in the early 1960's had several infinite injury arguments. I have heard that they were known to work only because the lemmas and proofs worked out.  Only after Bob Soare's excellent article on the topic were they really understood. For 0''' priority arguments it is also true that the early papers are not the ones to read.
   
2. Example (and the real motivation for this post). I have tried to read Ramsey's original article. I knew that his goal was a problem in logic, and I wanted to know what that problem was. I had a hard time reading the paper.  (I did  my own writeup.) Why was his version so hard to read?  (1) He never uses the words coloring or graph or hypergraph. He doesn't mention that if you have six people at a party either three of them know each other or three of them don't know each other. Perhaps he didn't go to many parties.  (2) He uses odd terms at time.  (3) His paper is rather abstract. If he had just proven a simple case then it would be obvious how to proceed to his abstract case.  This is true for both his combinatorial theorem--- he only proves (what we would call) the hypergraph version, and also the Logic theorem.
   
3. The Cliff notes for Atlas Shrugged are far better than the book. Shorter too.  They are online for free here which makes sense since Ayn Rand was known for her altruism.

SO- what do you think? Examples of cases where the Master is better to read?
Examples of cases where the Pupil (or more generally later summaries, surveys, expositions) is better to read?

It’s the day before the Kalends of September and you know what that means: it’s the birthday of Gaius Julius Caesar Augustus Germanicus, aka the emperor Caligula... His father dressed little Gaius in a miniature army outfit, including child-sized versions of caligae, the hobnailed sandal boots worn by common soldiers. That’s how he got his nickname, Caligula, meaning little caliga. 

His ascent to power was fueled by cutting taxes, which unfortunately led to the bankrupting of the country, to which he responded by starting a brothel staffed by the senators' wives and daughters.

More details at The History Blog.  Photo credit Munich Archaeological Museum.

In February, 1971, Apollo 14 astronaut Edgar Mitchell experienced the little understood phenomenon sometimes called the “Overview Effect”. He describes being completely engulfed by a profound sense of universal connectedness. Without warning, he says, a feeing of bliss, timelessness, and connectedness began to overwhelm him. He describes becoming instantly and profoundly aware that each of his constituent atoms were connected to the fragile planet he saw in the window and to every other atom in the Universe. He described experiencing an intense awareness that Earth, with its humans, other animal species, and systems were all one synergistic whole. He says the feeling that rushed over him was a sense of interconnected euphoria. He was not the first—nor the last—to experience this strange “cosmic connection”...

Their experiences, along with dozens of other similar experiences described by other astronauts, intrigue scientists who study the brain. This “Overview Effect”, or acute awareness of all matter as synergistically connected, sounds somewhat similar to certain religious experiences described by Buddhist monks, for example. Where does it come from and why? ...

Mitchell believes that perhaps both the theologians and scientists have missed the mark.

“All I can suggest to the mystic and the theologian is that our gods have been too small; they fill the universe. And to the scientist all I can say is that the gods do exist; they are the eternal, connected, and aware Self experienced by all intelligent beings."

Text from The Daily Galaxy.  Photo: The Sombrero Galaxy (M104), from The HubbleSite, via Conservation Report.

Theriogenology. 2012 Aug 25; Gatti M, Ungerfeld RVaginal secretions are an important source of chemical signals, which affect ewes' attractiveness. Moreover, alterations of vaginal flora reduce sexual attractiveness of estrous ewes. As intravaginal sponges containing progestagens (widely used for estrous synchronization) affect vaginal flora, our aims were to determine if estrous ewes pretreated with intravaginal sponges were less attractive than ewes displaying spontaneous estrus, and if the addition of antibiotic to the sponge mitigated the decreased sexual attractiveness. Seventy-two estrous ewes were used in experiment 1: in 36, estrus was synchronized with commercial intravaginal sponges (50 mg medroxyprogesterone acetate for 14 days, group MAP1), whereas the other 36 were given a PGF2α analogue 19 to 20 days earlier and displayed spontaneous estrus (group C1). In experiment 2, 72 ewes were treated with intravaginal sponges for 14 days; for 36 ewes, the sponges contained 0.02 mg oxytetracycline (group Ox), whereas there was no antibiotic in the sponges for the remaining 36 ewes (group MAP2). In both experiments, sexual attractiveness was determined in 12 groups of six estrous ewes (three MAP1 vs. three C1, and three MAP2 vs. three Ox for Experiments 1 and 2, respectively) located in a 4 × 4 m pen. Courting and mating time that each ram spent with each ewe was recorded. After 5 min, the ewe with which the ram spent more time (most attractive ewe, ranked one, scale one to six) was taken out from the pen. The procedure was repeated until the ram ranked all six ewes, and repeated in the 12 groups in both experiments. In experiment 1, C1 ewes were more attractive than MAP1 ewes (ranks: 2.9 ± 0.3 vs. 4.1 ± 0.3, mean ± SEM, respectively; P

Yesterday I finished reading Chris Hayes’s book “Twilight of the Elites,” and although I enjoyed it, I have to say it was more about the elites than about their twilight.

He focused on the enormous distance between people in society, how the myth of meritocracy is widening that gap (with healthy references to Karen Ho’s book Liquidated, which I blogged about here), and how, as the entrenched elite get more and more entrenched, they get less and less competent.

But Hayes didn’t really paint a picture of how things would end, although he mentioned the Tea Party and Occupy as possible important sources of resistance, not unlike Barofsky’s recent book Bailout (which I blogged about here), in which Barofsky appealed to the righteous anger of the people to whom government is no longer accountable.

Well, I guess Hayes did add one wrinkle which surprised me. He said it would be the upper middle class, educated class that actually foments the coming revolution. Oh, and the bloggers (because the mainstream media is so captured they’re useless). So me and my friends.

His argument is that we are the ones sufficiently educated and sufficiently insiderish that we will be at the window, with our faces pressed against the glass, looking in at the true insider elites, and seeing how stupid and incompetent those guys are, and how they are rigging the system against the rest of us, and we’ll eventually explode with disgust and righteous anger and that will signal the end.

Kind of feels like that’s already happened, but maybe I’m being impatient.

Two things I really enjoyed about his book:

First, the fact that practically everyone thinks they’re an underdog and has fought tooth and nail to succeed in this world. Absolutely true, including the guys I worked with in finance. I think the phrase he used is “people born on third base think they hit a triple”.

Second, he does a really good job describing the never-can-be-too-rich culture of our country; his example of going to Davos is an excellent one and brings that concept to life perfectly.

It’s enough to get you kind of depressed overall, though. If we are to believe this book’s thesis, our entrenched elite and dysfunctional political structure and economic system are doomed to fail at some future moment, and the best we can hope for is a moment where the hypocrisy collapses in on itself. What is there to look forward to exactly?

I asked that of a friend of mine, and how it was getting me down. His advice to me was to own it more. To make the coming apocalypse an event, kind of like the 4th of July or a vacation, that you plan for and enjoy thinking about.

He said plenty of people do this, it’s in fact a huge industry of doom and gloom. The country is going to hell, whaddya gonna do, he said, might as well have some fun with it.

What? Who are these doom and gloom people? Start here, where Dmitry Orlov compares the preparedness of the US to the former USSR for the coming inevitable apocalypse. He calls this the “Collapse Gap”.

It’s got some great points (although he can’t both say that lawlessness ensues and people take what they want, and also say that people behind in their mortgages will be homeless) and it’s really funny as well, in a completely cynical, Russian way of course. My favorite lines:

One area in which I cannot discern any Collapse Gap is national politics. The ideologies may be different, but the blind adherence to them couldn’t be more similar.

It is certainly more fun to watch two Capitalist parties go at each other than just having the one Communist party to vote for. The things they fight over in public are generally symbolic little tokens of social policy, chosen for ease of public posturing. The Communist party offered just one bitter pill. The two Capitalist parties offer a choice of two placebos. The latest innovation is the photo finish election, where each party buys 50% of the vote, and the result is pulled out of statistical noise, like a rabbit out of a hat.

Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Aug; 84(2 Pt 1): 021121
Tempesta P

The notion of group entropy is proposed. It enables the unification and generaliztion of many different definitions of entropy known in the literature, such as those of Boltzmann-Gibbs, Tsallis, Abe, and Kaniadakis. Other entropic functionals are introduced, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.