Vladimir Sotirov

@MarianoSuárez-Álvarez distinct maximal atlases may determine diffeomorphic manifolds (see math.stackexchange.com/q/3653191 ), which seems to contradict your suggestion of checking manifolds are the same by comparing maximal atlases.

@ArturoMagidin you could give an answer explaining your position (which I could then downvote) instead of leaving condescending comments...

thanks! I got bit by this while doing some of the phase I problems :-)

@Adám I would have expected to receive an empty array whenever ⍺>≢⍵

@Adám is there a good reason why {⍺ ⍺⍺/ ⍵} gives a length error if ⍺>1+≢⍵?

@Jarmex I didn't really wrap my head aroung trains until I watched Tacit Techniques with Dyalog version 18.0 Operators available at dyalog.tv/Dyalog19/?v=czWC4tjwzOQ ; it might be helpful if you haven't seen this talk

anyway, thank you all for the discussion!

it might be possible to justy jot the weights to ⌹ directly rather than going through ⍣¯1 on the modified inner product; that's just a matter of me working out the linear algebra though.

what I have in mind is this: if do a coordinate transformation where you scale each axis by L, then you have a weighted dot product {⍺+.×L×L×⍵}. I am curious to what extend ⍣¯1 of that will end up behaving like ⌹ in terms of giving closest points relative to this new dot product

ok. so second/last question: if I want to use an inner product weighted by a list of weights W, as in I think +.×∘(W∘×), would +.×∘(W∘×)⍣¯1⍨ give me weighted matrix divide (for vectors at least)?

is (v⌹w)≡v+.×⍣¯1⊢w

2 questions: is +.×⍣¯1 an idiom for ⌹?

cool; thank you! this was very illuminating.

so 12 15 1.1⌹E⍪1 1 gives x y such that (E⍪1 1)+.× x y is closest to 12 15 1.1?

@JeffZeitlin whereas multilpying real numbers together (or real number by vector) corresponds to sclaing, multiplying complex numbers together in general corresponds to scaling by the magnitude and rotating by the angle from the x-axis

@JeffZeitlin if you identify complex numbers with vectors, 2j7÷3j1 is the complex number which when scaled up by the norm of3j1 and rotated by the angle 3j1 makes with the x-axis lands you on 2j7

ok, cool, thanks! so then the coordinates of the intersection point are (2 7⌹3 1)×3 1, so the "length" in the 3 1-direction is what scalar multiple of 3 1 gives those coordinates?

wait, so (⌹v)≡ v÷+.×⍨v for vectors?

@JamesHeslip draw the perpendicular segment from the point 2 7 to the line through 0 0 and 3 1. The intersection point has coordinates (2 7⌹3 1)×⌹3 1: ⌹3 1 is the unit vector in the same direction as 3 1; the distance from 0 0 to the point is |2 7⌹3 1.

ooh, so 2 7⌹3 1 is the "length" of the component of 2 7 in the 3 1-direction?

thank you for the prompt response!

@Adám dyalog.com/student-competition.htm indicates this year's compeition is live at dyalogaplcompetition.com but the latter is still hosting last year's.

thanks!

Cool! Does that mean Dyalog v18.0 will also be out tomorrow? :-)

@Adám will there be an APL problem solving competition this year?

@user251222 you seem to be confusing tensor products of $R$-algebras with pushouts in the category of $R$-algebras: what you take the tensor product over depends on the corner of the diagram, not which category of algebras the diagram is in. You're also confusing colimits of algebras with colimits of schemes: colimits of algebras correspond to limits of schemes.