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Dan Piponi
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Apparently some people are aphantastic and can't visualise images: http://www.bbc.com/news/health-34039054

I don't know about that. All I've discovered is that I'm apsychic. As I can't read your mind, I've no idea what happens when you visualise something, and I have no idea how to compare it with my imagination. And that means I have no idea to answer questions about how vivid the colours I imagine are. The only reasonable answers I can think of are: (1) they're not vivid at all as imaginary colours aren't colours and it's a category error to think they are or (2) they're precisely as vivid as I imagine them to be as that's the nature of imagination.
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I can't say I enjoy it; my inner pictures are not at all pretty, and have no color.
But I find it quite useful. When I hear, say, an age or a time of day I see their position in my picture. And when I later try to remember them I might not remember the exact numbers, but I do remember roughly where they are in my picture, so I know an approximate answer. 
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Apple Music is an amazing service. But it's pretty complicated. I thought I'd try to understand the different states a single track could be in with respect to iTunes, Apple Music's host application.

If you just have iTunes I think there's:

(A) Tracks that have nothing to do with iTunes, for example mp3 files you shamelessly stole using Bit Torrent and store outside of iTunes.
(B) Tracks that came from somewhere unconnected to iTunes but which you imported into iTunes.
(C) DRM-free tracks you bought from iTunes. They're probably very similar to class (B). But they have personal info embedded in them so legally they're in a different class to (B).
(D) DRMed tracks from iTunes. These won't function outside of the Apple ecosystem.
(E) Tracks on the iTunes store you haven't bought yet :-)

If you subscribe to iTunes Match it gets a bit more complex:

(F) Tracks you obtained from elsewhere but which Apple has a version of. Your file will be copied into the cloud but the version you stream from elsewhere will be Apple's version which won't be bit-for-bit identical and may have a different bit-rate. The metadata may also be rewritten.
(G) Tracks you obtained from elsewhere but which Apple doesn't have a version of. Your version will be copied into the cloud and accessible from other instances of iTunes.

If you subscribe to Apple Music it gets even more complicated:

(H) Tracks you stream but otherwise are not part of "your" music. (You may have a local copy of these cached.)
(I) Music added to your collection from Apple Music. These tracks look like part of your music collection but they're really just streamable tracks.
(J) Streamable music you've explicitly asked to make a local copy of. I'm not sure what the relationship with class (I) is but I don't think they're the same thing.

It can be pretty important to know which class your tracks are in. You may find vast swathes of what was apparently "your" music disappearing if you unsubscribe from one or other service. You need to know which files you can legally give to someone else. You need to know which music requires an Internet connection. The UI doesn't always make clear which class your tracks fall into, especially on iOS. But as long as you keep forking over the money, some of the distinctions between these classes matter less.

There are lots of things I don't know. For example if you have a track that gets uploaded to iTunes Match but isn't recognized, and you download that track in another instance of iTunes, will the second copy go away if you unsubscribe from iTunes Match? And what happens to metadata for tracks that clash with tracks Match or Music recognise?
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I am afraid to subscribe to it and to insatll a new itunes ...I've not done yet :)
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I took my old formal power series library for combinatorics [2] (see also [3]) and tweaked it to work when the base ring isn't commutative. I can now use Haskell code to manipulate infinite series of powers of (one pair of) creation and annihilation operators.

I put the code at [1]. There are countless applications. Think of the things you can can enumerate with commutative generating functions, and now allow the possibility of connecting those objects with wires.

I included toy examples for:

0. basic stuff like counting non-capturing rook placements
1. using the Euler-Maclaurin formula to convert an integral to a sum
2. some polynomial manipulations related to umbral calculus
3. some quantum optics calculations
4. some "experimental" mathematics. See [4] for explanations.

Caveat: it's experimental incomplete code. I just wanted to see if the idea worked out.

[1] https://github.com/dpiponi/formal-weyl
[2] http://blog.sigfpe.com/2007/11/small-combinatorial-library.html
[3] https://hackage.haskell.org/package/species
[4] http://arxiv.org/abs/1010.0354
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+Dan Piponi: there's been a fair amount of work on categorifying the Weyl algebra that's relevant here. See, for example, http://arxiv.org/abs/math/9906166. 
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In quantum field theory (QFT), fields behave much like random variables and we do things like ask about the expected value of a field φ at a point x, commonly written as <φ(x)>.

Sometimes we need to know the expected value of the square of the field <φ(x)²>. As so often happens in QFT, when you try to calculate this you end up with something that diverges.

We can sometimes try to get this divergence under control by considering the limit as y→x of <φ(x)φ(y)>. Quite often this quantity is a straightforward pole, for example it might take the form <φ(x)φ(y)>=1/(x-y)² + something finite. We could just try subtracting off the pole and see what happens. So we could try replacing <φ(x)²> with the limit as y→x of <φ(x)φ(y)>-1/(x-y)².

Sometimes our physical system has a symmetry. For example, if you're doing physics in 2D (one space+one time) it's not unusual to have conformal invariance, ie. invariance under transformations of spacetime that preserve angle. So if we think of our physics taking place in the complex plane, this means that if φ is a good solution to the equations of motion in some region, so should φ○f be for any analytic bijection f, because analytic functions preserve angles. (Sometimes it's a little more complex than this because we might have some kind of covariance instead of invariance.)

But if we transform our underlying space using f, the pole we subtracted off will get replaced by 1/(f(x)-f(y))² instead of 1/(x-y)². In partciular this means that if we change coordinates using an analytic function, which should have no effect, our kludge of subtracting off the pole subtracts something different off. This is known as the conformal anomaly. It's the extra term that pops up when you transform your spacetime with f and it's the thing that makes string theories only work in certain dimensions like 10 or 26.

The exact details depend on the type of field, but for many physical systems the anomaly is proportional to the so-called Schwarzian derivative of f. Check out the wikipedia page at https://en.wikipedia.org/wiki/Schwarzian_derivative to see how it is defined and why it pops up when you apply holomorphic functions to warp your spacetime in the vicinity of a pole (e.g. see the bit starting "Introducing the function of two complex variables"). Note also the surprising "chain" rule.

The ordinary second derivative of a function f is a measure of how much it fails to take the form f(z)=az+b. The Schwarzian derivative measures how much f fails to fit the form f(z)=(az+b)/(cz+d).

The Schwarzian derivative pops up all over the place in mathematics from chaos theory and string theory to projective geometry and the theory of differential equations.

Anyway, the above is some of what I wrote about in my PhD thesis many years ago. I always found the Schwarzian derivative very weird and never really got to grips with it. I think if I'd carried on in academia I would have looked into the Schwarzian derivative some more.

This is the nearest thing I know of to a pop math article on this "derivative": http://math.univ-lyon1.fr/~ovsienko/Publis/what-is.pdf.
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+Dan Piponi"Creation/Annihilation" operators? Have you found a mathematical imperative that overrides 'constancy?' (by that I mean a 'QF/T' assumption of a given energy potential assigned to matter)
Or did I misread something? 
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A few times now I've seen people take for granted that the set of true mathematical propositions exists. After all, in Set Theory the set of all propositions exists (when propositions are suitably encoded as sets) and for the set of true propositions and we merely want a subset.

If we take true mathematical propositions to be true propositions of ZF, and work in ZF, I don't think it makes any sense to talk of the set of true propositions. The obvious construction of this putative set  is to use the axiom of separation to form the subset of true propositions from the set of all propositions. But Tarsi showed us we don't have a truth predicate and so we can't follow this path. I don't think we can even formalise what it means for this set to exist.

But I'm pretty sure a lot of people think it's completely obvious this set exists.

You can construct this set it if you're prepared to use a stronger axiom system than ZF to talk about the true statements of ZF. But as soon as you do that you're playing a different game and I expect that the set (and whether or not it even exists) will depend on the choice of stronger system you use.

I think some people are using the intuitive argument that we don't need to know precisely what the set of true propositions is. The power set of the set of all propositions exists, and so do all of its elements, and therefore the set of true propositions exists. But I don't
find this compelling.

Am I crazy for thinking people shouldn't be asserting the existence of this set without careful qualification? Or is everyone else crazy?
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I wish I knew this topic well enough to collect up what has been said into a coherent essay on this topic. Maybe one day someone qualified will write one. I can guarantee them one reader if it's not pitched at too high a level.
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One of the pillars of quantum mechanics is the photoelectric effect. Light below a certain frequency is unable to dislodge electrons from a material even when provided by a high power beam. The argument in almost every textbook, due to Einstein, says that the energy must be arriving in discrete packets, photons, and that you need a single packet on its own to have enough energy to kick out an electron.

If it's an advanced enough textbook then there will be a later chapter on time-dependent perturbation theory where it is shown how to calculate the rate at which electrons are kicked from one energy level to another as a function of the incoming electromagnetic wave. In almost every single textbook the argument treats the incoming wave as a classical field, demonstrating how you can in fact explain the photoelectric effect without recourse to photons.

Disappointingly these books don't simply self-annihilate in a cloud of contradiction.

(Of course the photoelectric effect is still good  evidence for QM. Just not in the way textbooks claim.)
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hmm interesting.  Your thoughts on mutual resonance increasing vibration... 
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I appear to have stepped sideways into another company. But I'm still eating at Google cafeterias.
 
3 years ago we embarked on a project to put computing inside a contact lens -- an immensely challenging technical problem with an important application to health.  While I am delighted at the progress that project has made, I could not have imagined the potential of the initiative it has grown into -- a life sciences team with the mission to develop new technologies to make healthcare more proactive.  The efforts it has spawned include  a nanodiagnostics platform, a cardiac and activity monitor, and the Baseline Study.

It’s a huge undertaking, and I am delighted to announce that the life sciences team is now ready to graduate from our X lab and become a standalone Alphabet company, with Andy Conrad as CEO.  While the reporting structure will be different, their goal remains the same. They’ll continue to work with other life sciences companies to move new technologies from early stage R&D to clinical testing—and, hopefully—transform the way we detect, prevent, and manage disease.  
   
The team is relatively new but very diverse including software engineers, oncologists, and optics experts.  This is the type of company we hope will thrive as part of Alphabet and I can’t wait to see what they do next.
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+Andrey Khalyavin Sort of. Except the main part of X left for another building leaving us Life Science people behind without our own cafe.
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From a young age you learn how to work with collections of objects, starting with collections whose sizes are small naturals. But you don't learn how to work with zero sized collections until much later. It seems like it's actually a pretty advanced subject. People with a strong scientific training, say, sometimes have a hard time with zero-sized collections even though they should be easier to work with than the size one, two and three sets they've been using since infancy. After all they have fewer elements.

For example: how many sequences of binary digits of length zero are there? It seems obvious to me that there's exactly one. But some people think this is an arbitrary stretch of the word sequence and that you could just as easily say there aren't any such sequences.

Or how many ways are there to arrange zero people in pairs? Again, I think it's obvious that there is one way to form pairs from no people. But many think it's clear that if there are no people, there's no way of forming pairs.

There are many other examples:

1. Is there a zero sized increasing subsequence of (1, 2, 3, ..., 9)?
2. Or a zero sized subset of {1,2,..100} the product of whose elements is odd?
3. Or a zero sized subset of {2,...100} of pairwise coprime elements?
4. Does the vector space with one element have a basis?

I think there's a clear answer in each case. This is, of course, an issue of semantics. But I think in each case there is a natural choice of meaning for the words that makes calculations consistent with non-zero sized sets. But maybe you disagree. And maybe this subject should be taught as part of at least one course so that everyone can agree on what exactly their words mean for zero-sized sets.

I guess it doesn't help that mathematicians don't even agree on whether zero is a natural :-)
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My thinking is heavily influenced by Haskell's types. There is one way to represent zero pairs of `a` listed:

[] :: [(a, a)]

In very specific cases I wonder if Maybe [a] is more correct than [a].
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Anyone who's studied quantum mechanics knows that the subject is largely about pairs of linear operators, a and a⁺, such that:

aa⁺ = a⁺a+1

Solving physics problems often involves rearranging expressions in a and a⁺ so that all of the a⁺ factors are on the left of each monomial and the a factors are on the right. This sort of thing:

  aa⁺aa⁺a⁺a⁺aa⁺a
= (a⁺a+1)aa⁺a⁺a⁺aa⁺a
= a⁺aaa⁺a⁺a⁺aa⁺a+aa⁺a⁺a⁺aa⁺a
= ...
= a⁺⁵a⁴+10a⁺⁴a³+23a⁺³a²+9a⁺²a

Getting to the last line takes a substantial amount of work and of course it gets worse when you have infinite sums.

But now I've read http://arxiv.org/abs/0904.1506 I see that there's a much easier way of getting those coefficients: 1, 10, 23, 9.

You can translate a monomial in a and a⁺ into a path on a grid by drawing a⁺ as a horizontal line and a as a vertical line, as in the diagram. That defines a region under the path known as a Ferrers board.

1 is the number of ways of placing zero non-attacking rooks on this Ferrers board. 10 is the number of ways of placing 1 rook, 23 is the number of ways of placing 2 rooks and so on.

I can't believe I've gone all these years without coming across this simple interpretation of the coefficients before.

It's worth reading the proof in the paper. The expression aa⁺ = a⁺a+1 corresponds precisely to a single step in a recursive procedure for counting rook placements.

This is just a tiny hint of the richness of the combinatorics of a and a⁺.
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+Urs Schreiber Thanks for the clarification and multi-symplectic algebra is something I can vaguely follow.
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+Satnam Singh recently posted a view of (roughly) where he lives. This is Redwood Peak, near where I Iive.

In the areas that get coastal fog it's still green in California.
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In many programming languages, including Haskell, you can freely copy values. For example, in Haskell you can write

    let b = a in ...

Haskell is based on the internal language of Cartesian closed categories (CCCs). The feature of these that allows us to copy values is the fact that a CCC has a diagonal morphism

    Δ:A →A×A

You can imagine making the copy more explicit by thinking of the above notation as shorthand for

    let (a,b) = Δ(a) in ...

The shorthand makes clear that in some sense we're reusing the variable a to continue labelling one of the copies. But that doesn't matter here because a has the same value before and after.

If you're writing code for linear algebra it's natural to think in terms of operations in the category of vector spaces. They are equipped with a copy morphism too:

    Δ:V→V⊕V

and so a vector space language should have a let clause to allow this.

But vector spaces also come equipped with a duality operation so any linear map f:A→B gives rise to another f*:B*→A*, usually called the adjoint or transpose of f. So we'd like to have

    Δ*:V*⊕V*→V*

This map is more commonly known as addition.

As I mentioned on G+ recently [2], you can construct adjoints of computer programs by a process that involves writing your code backwards. It'd be nice to have your programming language reflect this. So what operation is dual to "let b = a"? In C it'd be written

    a += b

Because let is shorthand for reusing the variable name for one of the copies, the adjoint version also has a variable reuse. But this time a changes value so the corresponding operation is a mutation.

In CCCs you don't have a dual to Δ so it makes sense that Haskell outlaws these kinds of mutable updates. But in a language (or DSL) for linear algebra, += is as natural a statement as let.

The is a simpler formulation of something I mentioned many years ago. [3]

This also serves as a reply to Twan van Laarhoven [4]. Curiously, since then Laarhoven has played an important role in making a serviceable form of += available in Haskell [5].

[1] https://golem.ph.utexas.edu/category/2006/08/cartesian_closed_categories_an_1.html
[2] https://plus.google.com/+DanPiponi/posts/9uWDCeLPLpz
[3] http://blog.sigfpe.com/2007/01/monads-hidden-behind-every-zipper.html
[4] http://blog.sigfpe.com/2008/09/two-papers-and-presentation.html
[5] http://www.twanvl.nl/files/lenses-talk-2011-05-17.pdf
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+Bas Spitters - I don't know which * you were referring to.

The usual notation for the linear map from B* to A* coming from a linear map f from a vector space A to a vector space B is f*.   If A and B are Hilbert spaces we also have a map

f^\dagger: B -> A

but that's different.
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I enjoyed Jessica Augustsson's previous anthology so I bought this right away. Includes a story by +Lennart Augustsson so how could I refuse?
 
I'm happy and proud to announce my second anthology, Encounters, with a whole slew of new stories by talented authors. You can find it as a Kindle book or paperback from Amazon around the world! Please like and share! smile emoticon
http://www.amazon.co.uk/dp/B00ZJD1O26
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(Also, if you could perhaps consider doing a review, I'd be eternally grateful! No worries if not. Just thought I'd ask. :)
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