Can you state what's the mechanism that makes it fast? Is this a breakthrough in rendering optimizations or does it introduce smarter data structures? Or is it an accumulation of small optimizations everywhere?
i think a better question to ask is why the others are not faster. honestly i cannot answer that question without digging into the source of each to find their bottlenecks.
allocating a ton of small objects or arrays is a very common source of slowness.
i took a raw canvas and a raw data structure of a single array per series, plus one for timestamps and made a loop to draw lines on a canvas. it turned out to be very fast. the mousemove interaction has rAF throttling applied and does a binary search over the timestamps plus some basic arithmetic. there's no "secret sauce" that makes it go fast. maybe the way it calculates scales is more efficient than the others. i honestly don't know without looking into what the others do, nor do i care enough to look into it.
haha touché! While the blog post contains few illustrations, the book that it's based on, Deep Learning Illustrated, contains north of a hundred of them.
It looks like she wants to talk about symmetric monoidal categories, not just monoidal categories, for that’s where the calculus of string diagrams is actually useful.
Tensor categories is another word for symmetric monoidal categories, indeed it’s less of a mouthful, and perhaps more “intuitive” - it’s conventional to write the symmetric monoidal structure as a tensor product.
What a confusing name in an era where homotopy-theoretic methods in arithmetic geometry are flourishing. Looks Hatcher covers non of that and use "topology" to mean "geometrically motivated".
This is a dumb question, but what's the difference between geometry on discrete sets and homotopy theory on discrete sets, eg, in the spirit of digital topology?
Every set is open, so every set is closed. So something like a closed interval, or the product of closed intervals, is just going to look like the network that defines a line segment, square, etc... right?
In that sense, it seems like the geometry of numbers and the topology of numbers are basically the same thing.
But I'm going to be honest -- I know basically nothing about arithmetic geometry.
As you observe one can't really recover any non-trivial number theoretic things by looking at integers with the discrete topology. The theory of discrete topological spaces is just the theory of sets.
Instead you can look at things like prime ideals of integers localized at some prime, and consider algebro-geometric topologies on that
I think the language and machinery of topology, even when just reconstructing the language of sets in a discrete setting, highlights interesting facets of numbers.
eg, if you look at the inverse image of various mappings, and particularly in cases where you can iterate this via a function from a set into itself, you can start building up meaningful comments on certain classes of number theory problems.
But I am curious what you mean by "prime ideals of integers localized at some prime", since I know what (prime) ideals are, but am not sure I follow what you mean by localized
The category of sets embeds into the category of topological spaces as a full subcategory, the essential image of which are the discrete spaces. Hence the equivalence I claimed is a precise statement.
> Hence the equivalence I claimed is a precise statement
They're obviously equivalent.
My point is that what's easily noticeable in one incarnation of the theory is different than what's easily noticeable in the other incarnation (or if you prefer, expressible), and switching our language for the same abstract structure can highlight different interesting features of it. And further, there's still utility to using topological perspectives and language to discuss the integers or naturals, even if it's equivalent to set theory.
I do appreciate the reference to ring localization -- will have to look at that further.
Genuinely tried reading the "examples" you wrote in this thread, can't make any sense out of it. Happy to discuss it if you clarify what you mean.
Just to address your original comment in this thread, perhaps it's relevant to note the following. Consider the homotopy theory of the category of nice topological spaces. The full subcategory of topological spaces supported on discrete topological spaces inherits a homotopy theory. This inherited homotopy theory is equivalent to the trivial homotopy theory on the category of sets: where weak equivalences are isomorphisms. This is the sense in which discrete spaces don't have an interesting homotopy theory, at least naively.
(This statement you can precise in your favorite model for the homotopy theory of spaces, via infinity categories, model categories etc.)
Some context would be nice as this (very nice library) has been around for a while: was there a new release? What are we supposed to be looking for Here?
I just felt like sharing it. I was foraging around Twitter for good links and found this. I honestly didn't know it was around for a while, but it stimulated my intellectual curiosity so I shared it.
> Our name comes from advanced mathematics and represents the numerous ways in which we simplify financial processes and products using blockchain technology.
Ugh, I'm cringing a little that they consider adjunctions 'advanced mathematics'. Adjunctions are ubiquitous even in elementary math. For example, in linear algebra whenever one writes down a matrix to represent a operator in some basis, this is using the tensor-hom adjunctions for modules.
Sorry this is just wrong. Maybe you’re trying to get at the “co/contravariant” properties of tensors, in which case your statement can be more clearly stated as, e.g., the space rank (2, 0), and rank (1,1) vectors, admit different interpretations as internal hom spaces of vector spaces. But in any interpretation of your statement the distinction is never important because all spaces distinguished by this distinction are isomorphic via cononical isomorphisms.
You might want to think that through a bit more, leaving aside the issue of needing the underlying vector space, are you sure you are comfortable with the statement that all matrices are tensors?
Had to scroll down the page for a while before understanding the point of this: drag-and-drop data transformation pipelines that comes with app integrations at both ends. It’s a great idea!
Not an answer but a general comment on learning languages: instead of setting your goal as “to learn as much of a languageas possible”, instead you can go for “I want to be able to do this class of things that will be made very easy once i know thia language”.
If you do the former it’s very easy to learn something then forget it. Furthermore, echoing other comments, you would lack clear measures of success because it almost makes no sense to ask whether you truly “know” a language.
