Genady

I suppose it was not serious.

It is often mentioned that the number theory has applications in computer science, esp. in cryptography. It seems to me that these applications are very few compared to the size of the number theory itself, aren't they? More generally, are there any applications of the number theory that are not in computer science?

Here are two of them mating, but the picture didn't turn out clear.

Yeah. And I can quote Mao Zedong stuff and compare it to quantum machines. Still BS.

BS

Check this exercise: I think, the text is mistaken. In the category of sets, a direct product of A and B is their Cartesian product C, and the morphisms are maps from C to A and from C to B, which are not injective, i.e., monomorphisms, but rather surjective, i.e., epimorphisms. OTOH, a direct sum of sets A and B is their disjoint union C, and the morphisms are maps from A to C and from B to C, which are not surjective (epimorphisms), but rather injective (monomorphisms). Do you agree?

Today I've found a simple example of a morphism which is monomorphism, i.e., one-to-one, and epimorphism, i.e., onto, but is not an isomorphism, i.e., does not have an inverse.

Here it is: Mapping [math]f: A \to B[/math] is onto if for any X and any mappings [math]p: B \to X[/math] and [math]q: B \to X[/math], [math]p \circ f = q \circ f \Rightarrow p=q[/math].

TIL that "one-to-one" mapping between two sets can be defined as an external property of the mapping, i.e., without any reference to elements of the sets and to what happens to them under the mapping. Here we go: The mapping [math]f: A \to B[/math] is one-to-one if for any X and any mappings [math]p: X \to A[/math] and [math]q: X \to A[/math], [math]f \circ p = f \circ q \Rightarrow p=q[/math]. Can you come up with a similarly external definition of "onto" mapping?

It's from fb: https://www.facebook.com/permalink.php?story_fbid=pfbid028dPTEhRvxf6j3HMNy52Ymv742GMuQXekebvDo8DzjkAU9EuiENkDdM1Kt2TAUM8hl&id=100090706310719&__cft__[0]=AZZHIbs76gT7OBtsHj6mMXIx5moXtwff6b04aSxChSwB5lEXYpPMSGOGKRrOS9uci6M4uNynPdHRcXUqgAn7915tu-8VZC_TjOrw2r4iP4O92OMOlQ6FBa5PGInZyi69m869_BZmfljXXH2ce7DMOVWjbE4olX7ZzZg8C7glt3qkEg&__tn__=%2CO%2CP-R

These are not words in English.

Yes, it does. It is an effective manipulation technique.

Because it works.

I remember getting a question on my US citizenship exam (many-many years ago), "What is Constitution?" with one line for an answer. I've answered, correctly, "Constitution is the supreme law of the land."

This notation, (rather than, e.g., S or s) is new to me. Only two days ago it was new to me, and it is already in my next book:

I've read it only last month and they've made a movie already 🙂

A good math joke, albeit 30 years old:

Right. And my book says, but still, since all the numbers here are integer, the definitions [math]|\lambda(m+n)-(\lambda(m)+\lambda(n))| < M_{\lambda}[/math] and [math]\left\{ \lambda(m+n)-(\lambda(m)+\lambda(n)) \right\} \,\text{is finite}[/math] are equivalent.

I still don't see a difference between the two constructions mentioned above. The first says, The second, Does anybody see how they are different?

Whoever it was that downvoted you, I've balanced it by an upvote, just as you've done for me earlier.

Yes, I'm studying this book: This notation, (rather than, e.g., S or s) is new to me.

This note of yours has a direct relevance to this "Quiz" of mine here: https://www.scienceforums.net/topic/140398-from-naturals-to-integers-quiz/ 🙂

The construction that I've learned recently follows closely the "2.12. Schanuel (et al.)’s construction using approximate endomorphisms of Z ([2, 11, 16, 29, 30, 1985])" in your first linked paper. Interestingly, my book cites rather "Norbert A’Campo, A natural construction for the real numbers, Elemente der Mathematik, vol. 76 (2021)." P.S. Ah, I see that A'Campo's is your second linked paper. Perhaps, there is some difference that I didn't see yet.

Thank you! I didn't know about 10 different ones, only about three, I think. And they all constructed rational numbers before constructing reals. So, a direct route from Z to R without Q was interesting.