Yeah, it's a matter of convention rather than opinion really, but among US academia the convention is to exclude 0 from the naturals. I think in France they include it.
You can imagine tracing a path along a Klein bottle to see that it only has one side. To get more precise than that requires some topological context. If you slice it down the middle it turns into two Möbius strips and an orientation of the Klein bottle would induce an orientation of the strips, which are non-orientable. Alternately it has zero top integer homology, which you can get from looking at a triangulation. The orientable double cover of a Klein bottle is a torus, which is connected (if it were orientable, the double cover would be two disconnected Klein bottles).
As long as we can put an upper bound on gayness (or more specifically on each totally ordered subset of people under the is-gayer-than relation) this follows from Zorn's lemma.
It's also true by virtue of the fact that the set of all people who will have ever lived is finite, but "the existence of a maximal element in a poset" just screams Zorn's lemma.
I mean the specific issue about the binary blobs. Something that might set off alarm bells for you or a security-focused group may not do so for some dude working on a passion project in his free time.
Software to create bootable usb drives. It's handy, you just copy ISOs into the drive and pick which one to boot into instead of overwriting the drive with a single ISO.
Dropping support for that stuff means breaking 95% of the websites people currently use. It's a non-starter, it cannot ever happen, even if you think it would be for the best.
Math builds up so much context that it's hard to avoid the use of shorthand and reused names for things. Every math book and paper will start with definitions. So it's not really on you for not recognizing it here
🍕(--, B) : C -> Set denotes the contravariant hom functor, normally written Hom(--, B).
In this case, C is a category, and B is a fixed object in that category. The -- can be replaced by either an object or morphism of C, and that defines a map from C to Set.
For any given object X in C, the hom-set Hom(X, C) is the set of morphisms X -> B in C. For a morphism f : X -> Y in C, the Set morphism Hom(f, B) : Hom(Y, B) -> Hom(X, B) is defined by sending each g : Y -> B to gf : X -> B. This is the mapping C -> Set defined by Hom(--, C), and it's a (contravariant) functor because it respects composition: if h : X -> Y and f : Y -> Z then fh : X -> Z and Hom(fh, C) = Hom(h, C)Hom(f, C) sends g : Z -> B to gfh : X -> B.
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P(n)(R) AKA RPn is the n-dimensional real projective space.
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The caveat "phi is a morphism" is probably just to clarify that we're talking about "all morphisms X -> Y [in a given category]" and not simply all functions or something.
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For more context, the derived functor of Hom(--, B) is called the Ext functor, and the exactness of that sequence (if the typo were fixed) is the statement of the universal coefficient theorem (for cohomology): https://en.wikipedia.org/wiki/Universal_coefficient_theorem The solution to this problem is the "Example: mod 2 cohomology of the real projective space" on that page. It's (Z/2Z)[x] / <x(n+1)> or 🍔[x]/<x(n+1)>, i.e. the ring of polynomials of degree n or less with coefficients in 🍔 = Z/2Z, meaning coefficients of 0 or 1.
It's not nonsense, although there is a typo that makes it technically unsolvable. If you fix the typo, it's an example calculation in the wikipedia page on the universal coefficient theorem: https://en.m.wikipedia.org/wiki/Universal_coefficient_theorem
The standard .NET C# compiler and CLI run on and build for Windows, MacOS, and Linux. You can run your ASP.NET webapps in a Linux docker container, or write console apps and run them on Linux, it doesn't matter anymore. As a .NET dev I have literally no reason to ever touch Windows, unless I'm touching legacy code from before .NET Core or building a Windows-exclusive app using a Windows app framework.
Ok, there's no such thing as native Windows apps for Linux, but there are cross platform GUI frameworks like Avalonia and Uno that can produce apps with a polished identical experience across all platforms, no electron needed
Yeah, it's a matter of convention rather than opinion really, but among US academia the convention is to exclude 0 from the naturals. I think in France they include it.