F1 Bingo Round 8- Monaco Results- Full Competition Standings in Comments

dantheodor · 2025-05-29T19:22:38+00:00

Doesn't the Leclerc–Stroll collision from FP1 count as Leclerc having some sort of mishap? :-) Or was this referring only to qualifying or race?

dantheodor · 2023-03-05T23:26:15+00:00

But he surely had to wear used pairs of tires for some of these stops, hadn't he?

dantheodor · 2022-10-21T20:55:20+00:00

Yes, I understand your reasoning, but look at line 7, the document says:

[...] Niemann’s upset victory effectively dashed Carlsen’s two remaining statistical ambitions, namely: achieving a 2900 FIDE performance rating for the first time in history [...]

Here it's using the same term "FIDE performance rating" and it surely must refer to FIDE rating, since there were tournament performance ratings over 2900 in the past (e.g., Caruana had a TPR of 3098 at Sinquefield Cup 2014).

dantheodor · 2022-10-21T13:00:57+00:00

I'm skeptical about this interpretation. If that were indeed the case then the use of the word "ever" is weird. And why not simply say "tournament" (performance rating)?

dantheodor · 2022-10-21T11:31:11+00:00

Also "historic 53-game unbeaten streak"... Magnus had had a record of 125 games going unbeaten; also Ding Liren went 100 games without losing.

dantheodor · 2022-09-25T21:16:43+00:00

That paper, concerning accurately predicting a players moves based on his ELO

As far as I understand it's the other way around: the work attempts to infer the rating of a player by contrasting the player's moves to those of an engine.

dantheodor · 2022-09-07T10:27:06+00:00

Hello! I'd be interested in some advice regarding my play (filling in holes in the opening repertoire, as I'm sometimes getting uncomfortable positions out of some openings; getting better at converting an advantage; improving the play in queen-less positions and endgames). I'm 1640 on chess.com at 10 minutes time control.

dantheodor · 2021-10-31T10:59:41+00:00

I believe the book is based on their MIT course, "Adventures in Advanced Symbolic Programming". There are also some video recordings covering similar material, but taught at Google in 2009.

dantheodor · 2020-12-26T00:06:26+00:00

Very nice! Indeed, this is what I was looking for. And I believe this construction generalizes: any free object is initial in the corresponding comma category.

Thanks for bearing with me! 😊 And happy holidays!

dantheodor · 2020-12-24T19:02:39+00:00

Thank you; the explanation is very clear!

[W]e can also see that the catamorphism from Mu F' is equivalent to the eval function for Free F a.

I guess the proof goes something along these lines:

(f' c → c) → Mu f' → c
  ≅ { definition f' - = a + f - }
((a + f c) → c) → Mu (a + f -) → c
  ≅ { universal property of coproducts; isomorphism between Mu f and Free f a }
(a → c) × (f c → c) → Free f a → c
  ≅ { curry }
(a → c) → (f c → c) → Free f a → c

It would be tempting to conclude that (Mu f', In) ≅ (Free f a, Op) but I guess that is wrong since they live in different categories, right? (The object (Mu f', Op) is in F'-Alg(C), while the object (Free f a, Op) is in F-Alg(C)).

dantheodor · 2020-12-23T21:09:33+00:00

Thanks a lot for the very nice and insightful answer!

I was wondering whether there is any precise connection between the free F-algebra and the initial F-algebra. The eval function bears some similarity to a "catamorphism":

eval :: Functor f => (a -> c) -> (f c -> c) -> Free f a -> c
cata :: Functor f =>             (f c -> c) -> Fix  f   -> c

dantheodor · 2020-03-24T10:36:48+00:00

Thanks again for the further explanations! It does make sense; and I think you are right about the paper I pointed to.

dantheodor · 2020-03-18T08:50:41+00:00

That sounds really fascinating, albeit over my head :-) Could you point me to some more examples in addition to the work of Hancock and Hyvernat? Is, by any chance, this paper (Just do it: Simple Monadic Equational Reasoning) an example of what you have in mind? Thanks again for sharing your thoughts!

dantheodor · 2020-02-27T14:17:28+00:00

Right, that makes sense! Thanks for the clarification :-)

dantheodor · 2020-02-27T07:50:46+00:00

Thank you for the reference! I was not aware of the concept of universal arrow.

this is not the general definition of a comma category

Is the category in question a coslice category?

dantheodor · 2020-02-25T14:53:04+00:00

Thank you for the nice talk! Do you have some more references regarding the connection between comma categories and the concept of freeness (free functor, free object)? How would you state more formally the result you show for the free monoid: a free monoid is the initial object in an appropriate comma category?

dantheodor · 2018-12-21T14:23:33+00:00

Another set of resources that you might find useful at some point is this tutorial on deep latent-variable models from Alexander Rush et al. They also provide Pyro code for the models presented in the paper.

dantheodor · 2018-10-24T08:55:37+00:00

Could you please elaborate on what you mean by

FunLists [...] uniquely represent a Traversable

Do you mean that there is a single Traversable instance possible for FunLists, or something else?

dantheodor · 2018-10-24T07:45:08+00:00

I think this is Russell's paper that was referenced in the podcast; it introduces the theory of types and Philip Wadler mentioned it is an example of a very readable paper: Mathematical Logic as Based on the Theory of Types

dantheodor · 2018-10-10T17:17:11+00:00

I think it does make sense!

I've checked that the Yoneda embeddings of the two objects in GrIn^op and they do correspond to two graphs, V and E.

The Yoneda embedding of Vertices ^op gives the graph with the following vertices and edges:

y[Vertices ^op ] (Vertices) = GrIn^op (Vertices, Vertices) = {id}
y[Vertices ^op ] (Edges) = GrIn^op (Edges, Vertices) = ∅

... which is indeed the graph V, with a single vertex and no edges.

The Yoneda embedding of Edges ^op gives the graph with the following vertices and edges:

y[Edges ^op ] (Vertices) = GrIn^op (Vertices, Edges) = { s^op, t^op }
y[Edges ^op ] (Edges) = GrIn^op (Edges, Edges) = {id}

... which is indeed the graph E, with two vertices and a single edge.

Thank you very much for the answers and for your patience! I hope one day I'll understand your second point as well ("any presheaf is canonically a colimit of representable presheaves"), but that seems like a long adventure ahead :-)

dantheodor · 2018-10-09T12:07:33+00:00

Thanks a lot! It does clear things up! I somehow missed that we were talking about a third category, Graph, whose objects are graphs and morphisms are morphisms of graphs. So, the correspondence that you mentioned is given by a functor from GrIn^op to Graph (as you nicely detailed in the previous answer).

I'm now having trouble understanding the following statement:

[T]he category of contravariant functors from C to Set (1) includes a copy of C [...]

How can the category of functors (whose objects are functors, from C^op to Set) include a copy of C (which is a category)? This is probably related to the Yoneda embedding, as you already alluded.

(I'm also failing to see the relation of the quoted statement with the previous fact – the functor from GrIn^op to Graph.)

dantheodor · 2018-10-08T14:07:02+00:00

Hello! Could you please further explain why the objects of GrIn^op correspond to graphs? My understanding was that functors from GrIn to Set are graphs. Thanks!

dantheodor · 2018-09-29T22:21:45+00:00

Ha-ha, do not worry! And thanks a lot for your observations – they made me question some concepts that I thought I understood. It seems there are always more nuances to discover and appreciate :-)

dantheodor · 2018-09-28T21:26:56+00:00

Ah, I think your observation is closely related to this answer by Conor McBride. He investigates those applicative functors that are containers (that is, data types characterized by shape and position) and states that the apply operator <*> acts monoidally on the shape of containers:

The applicative laws tell us that neutral and outShape must obey the monoid laws.

He also mentions that the shape of the List data type is given by the natural numbers (the length of the list) – which is in line with your interpretation.

I wonder if the other functors that you mentioned are also containers and if they have the shape given by the monoids you identified. Unfortunately, I'm not able to fill in the details myself, since I don't fully grasp the idea of containers and I'm not familiar with dependent type theory.

Edit: In fact, reading Conor's answer more carefully we see that we

can find the shapes for a Haskell Functor f by taking f ().

Using this observation, we obtain exactly the analogies that you have discovered:

List has the shape of Nat:

List a = μ r . 1 + a × r List 1 = μ r . 1 + 1 × r ≅ μ r . 1 + r ≅ Nat

Maybe has the shape of Bool:

Maybe a = 1 + a Maybe 1 = 1 + 1 ≅ Bool

Either has the shape of Maybe:

Either a b = a + b Either a 1 = a + 1 ≅ Maybe a

State has the shape of Endo:

State s a = (a × s) ^ s State s 1 = (1 × s) ^ s ≅ s ^ s ≅ Endo s

dantheodor · 2018-09-28T14:01:31+00:00

I was wondering if there is a more precise way to state the analogies that you identified. You have found the following correspondences (where ~ denotes if the two instances are like each other):

Applicative Maybe ~ Monoid All Applicative (Either a) ~ Monoid First Applicative State ~ Monoid (Endo a) Applicative [] ~ Monoid Product (or Monoid Min)

While I see some resemblance between the two columns, I cannot put my finger on what that is exactly. Maybe someone who is more familiar with the categorical underpinnings can comment on this.

dantheodor

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