[HN Gopher] The derivative of a number (2014)
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The derivative of a number (2014)
Author : tempodox
Score : 111 points
Date : 2024-05-11 06:03 UTC (16 hours ago)
(HTM) web link (rjlipton.com)
(TXT) w3m dump (rjlipton.com)
| Almondsetat wrote:
| I have found the Wikipedia entry to be much clearer and
| interestig than this article
|
| https://en.m.wikipedia.org/wiki/Arithmetic_derivative
| tromp wrote:
| Non-mobile link:
| https://en.wikipedia.org/wiki/Arithmetic_derivative
| keepamovin wrote:
| Yeh the link is good, and I think the article is good too,
| discussing some limitations / gotchas / cons. Gives me the
| impression maybe a more useful definition could be crafted, but
| also that this D function could be unexplored and yield some
| cool stuff :) haha!
| yau8edq12i wrote:
| The article also contradicts Wikipedia, e.g., the article
| claims that Barbeau introduced the derivative in 1961, but
| Wikipedia gives several references to Shelly's definition of
| the derivative in 1911. You'd expect an emeritus professor to
| have enough time on their hands to check their sources...
| playingalong wrote:
| They likely weren't emeritus in 1961.
| lupire wrote:
| The article is 2014, when Lipton was 64. So maybe emeritus,
| maybe not.
| yau8edq12i wrote:
| I'm talking about the author of the article, Lipton. Not
| Barbeau who is the subject of the article itself, not its
| author.
| samatman wrote:
| One of the nice things about Wikipedia is that you can use it
| as a time machine, instead of just blindly throwing about
| accusations.
|
| If you look at the first page in 2014, the time of the
| article, you'll find this: https://en.wikipedia.org/w/index.p
| hp?title=Arithmetic_deriva...
|
| > _E. J. Barbeau was most likely the first person to
| formalize this definition._
|
| This suggests to me that Shelly's earlier definition in 1911
| was not generally known. Indeed, Wikipedia has been a
| significant driver in rescuing earlier results in mathematics
| from obscurity.
|
| But you can't cite what you don't know. So I wouldn't expect
| an emeritus professor, or anyone, to have either a time
| machine or a crystal ball.
| bjornsing wrote:
| This function D on the integers might be interesting, but why
| call it a derivative? I can't see much conceptual similarity with
| calculus derivatives.
| keepamovin wrote:
| I'm just reading about it now, but I think because the
| derivative power rule holds for it, and you can do partial
| derivatives based on primes. The linked wikipedia article
| covers it well in Elementary properties.
| tromp wrote:
| As Wikipedia puts it, the
|
| > number derivative is a function defined for integers, based
| on prime factorization, by analogy with the product rule for
| the derivative of a function that is used in mathematical
| analysis.
| bandrami wrote:
| I'm now curious if this can be defined on Gaussian
| integers...
| lupire wrote:
| It's already defined, as given, for all unique (prime)
| factorization domains, including Gaussian integers.
| constantcrying wrote:
| Because it obeys a similar rule to derivatives on functions.
| The pattern of taking a concept and extending it to other
| things by looking only at what happens under one rule, is
| extremely common in mathematics. That way you can e.g. extend
| the definition of a derivative to non-continous functions.
| im3w1l wrote:
| > The arithmetic derivative can also be extended to any unique
| factorization domain (UFD),[6] such as the Gaussian integers
| and the Eisenstein integers, and its associated field of
| fractions. If the UFD is a polynomial ring, then the arithmetic
| derivative is the same as the derivation over said polynomial
| ring
|
| I was just skimming the wikipedia article but this seems like a
| good argument.
| gizmo686 wrote:
| It is pretty common to talk about derivatives in abstract
| algebra in contexts where the calculus definition makes no
| sense.
|
| Broadly, there is a class of functions refered to as
| "derivations" that can be viewed as a generalization of the
| derivative. In particular, a derivation satisfies 2 properties:
|
| 1) It is linear
|
| 2) It satisfies the product rule.
|
| Any function that satisfies these rules is often called a
| "derivative".
|
| Notably, the function discussed in the article fails the
| linearity test, which is a pretty big problem for calling it a
| derivative.
| codeflo wrote:
| Also, the abstract algebra definition is an actual
| generalization, because the (analysis) derivative acts as a
| linear operator in the vector space of (suitably smooth)
| funtions. It blew my mind when I first encountered this way
| of looking at derivatives. And from there, it makes complete
| sense to look at operators with similar properties in other
| vector spaces.
|
| The definition presented here is a loose analogy to
| derivatives rather than an actual generalization, which
| doesn't fully justify using the name IMO.
| JadeNB wrote:
| > And from there, it makes complete sense to look at
| operators with similar properties in other vector spaces.
|
| It does, but this one's on a rig (= ring - negatives), not
| a vector space over any field.
| codeflo wrote:
| I don't see how you would define linearity in a ring (I
| don't mean a module, just a ring). I.e. D(af) = aD(f)
| doesn't make sense if you don't have scalar
| multiplication.
| gizmo686 wrote:
| The minimum would be to ask for D(a+b) = D(a) + D(b).
|
| EDIT: Actually, according to wikipedia, that is exactly
| what is done in differential algebra
|
| https://en.wikipedia.org/wiki/Differential_algebra
| magicalhippo wrote:
| > It satisfies the product rule.
|
| It's been many years since I had calculus at uni, and never
| any abstract math. Why is the product rule picked as the
| "interesting" attribute of derivatives, ie to serve as the
| basis for the generalization?
|
| Is there some deeper connection of the product rule in
| ordinary derivatives that singles out the product rule over
| the other properties a derivative has?
|
| For me, a key aspect of derivatives is that it allows for
| something like Taylor expansion or integrals to exist. Are
| there any equivalent things to these product-rule-generalized
| derivatives?
| gizmo686 wrote:
| I actually think it is better to not think of derivations
| as a generalization of derivatives; but to think of
| derivatives as just one of many examples of a derivation.
| That the name 'derivative' comes from calculus and became
| the name for this abstraction is just a quirk of history.
| You could also consider the notion of a linear function to
| be a generalization of derivatives. Or even the notion of
| homomorphisms in general.
|
| Having said that, there are many usages of other
| derivations where the calculus inspiration is clear, even
| if the geometric meaning that motivated the calculus is
| lost.
|
| For instance, we often talk about polynomials over
| arbitrary fields. In general, there is no way to graph such
| polynomials. There is no notion of tangent lines, slope,
| "continuous", or even "less than". There is, however, still
| the notion of roots and the multiplicity of roots. These
| notions turn out to be quite important.
|
| When working with any polynomial, you can define the
| "formal derivative" as a derivation that also satisfies
| D(x) = 1, D(a) = 0 (where a is an element of the underlying
| field). This operator behaves as you would naively expect a
| derivative to behave over polynomials. In Galois theory, it
| is important to distinguish between polynomials that have
| repeated roots, and those that do not. If you have a
| polynomial f(x), you can determine this by taking its
| formal derivative f'(x). Then, you can easily compute their
| greatest common divisor [0]. If this is a constant, then
| you know f has no repeated roots.
|
| [0] Using Euclid's algorithm, this is a purely mechanical
| process that can be done without needing to factor either f
| or f'. A similar trick has actually been used to attack
| real word cryptography. If there are secret primes p and q,
| and a public number pq, many cryptosystems assume that it
| is infeasible to determine what p and q are. However, if
| there is a bad random number generator, you might get 2
| different keys that share a prime, so have the public
| numbers pq and ps, then you can easily determine that p is
| a common factor, from which you can easily recover q and s.
| This means that you can look for a large collection of
| public keys and try this attack on possible pair of them.
| magicalhippo wrote:
| Thanks, that makes more sense. Most of the time these
| generalizations make sense to me, sometimes it can be
| really obscure. Flipping the viewpoint makes it more
| clear in this case.
| poizan42 wrote:
| The product rule is really the simplest rule for
| derivatives that apply to operations on numbers that gives
| something interesting.
|
| Other basic rules would be addition:
|
| (f(x) + g(x))' = f'(x) + g'(x)
|
| This is just linearity (together with constant
| multiplication)
|
| And function composition (the chain rule):
|
| (f [?] g)' = (f' [?] g)[?]g'
|
| We would need to somehow figure out what should correspond
| to function composition.
|
| So if we want something that captures some important
| algebraic properties of derivatives the product rule would
| be a good place to look.
| bjornsing wrote:
| > 1) It is linear
|
| This "derivative" is not linear though, and that was sort of
| what motivated my question.
| GrantMoyer wrote:
| D(n) is equal to f'(0) where f(x) = P_(p[?]P(n)) x + p and P(n)
| is the set of prime factors of n.
| rm445 wrote:
| It seems pretty confusing given that operator D already exists
| as notation for the usual differential calculus. (Not the most
| common notation, but useful sometimes). It may not be very
| exciting that D{n} = 0 for all numbers, but that's the actual
| derivative of a number.
| Someone wrote:
| > It may not be very exciting that D{n} = 0 for all numbers,
| but that's the actual derivative of a number.
|
| No, that's the derivative of the function that return _n_
| whatever its argument, which also can be written as _lx.n_ or
| in a zillion different ways.
|
| That can be written as _n_ , but is different from the number
| _n_.
|
| Also, one man's "pretty confusingly is another man's "similar
| things should have similar names". There's a rich history in
| mathematics of overloading the meaning of terms and symbols
| as long as there's some similarity between them, for example
| when using x for both the multiplication of numbers and of
| matrices (where the former is commutative, but the latter
| isn't, barring some exceptions such as 1 x 1 matrices).
|
| (See also the comment elsewhere in this thread which says
| _"Mathematicans like to call two things with the same
| "structure" by the same name, even if it's not obvious how
| they're otherwise related"_
| (https://news.ycombinator.com/item?id=40327885)
| lordfrito wrote:
| I my naive mind, if you can differentiate a number than you
| should be able to integrate it. So what then is the integral of a
| number?
|
| if I(n) is the integral of n, then shouldn't I(D(n)) == n?
|
| But if D(prime) = 1, then what prime is the answer to I(1) ??
|
| If this can't be done, then what were doing here isn't
| differentiation as I understand it. So why call this the
| derivative of a number? Why not call it something else?
| yau8edq12i wrote:
| It is, indeed, not differentiation in the sense you're used to.
| Here it just means a map that satisfies the Leibniz rule, (fg)'
| = f' g + f g' or D(fg) = D(f) g + f D(g). Maps with this
| property are usually called "derivations".
|
| Although you might also want to consider that "integration"
| (really, indefinite integrals, AKA antiderivative) is only
| defined up to a constant. So why couldn't it be the same for
| this "number derivative"? Perhaps the "antiderivative" is only
| defined up to something. It'd be a fun exercise, if you're
| interested. Can you figure out under what conditions do you get
| D(a) = D(b)? Put differently, given an integer c, what are the
| solutions to D(x) = c?
| plank wrote:
| Solutions to D(x)=C is probably something like p_a^a * p_b^b
| * ... * p_n^n with a+b+d+...+n (yes, I am skipping c) equal
| to your constant C, and p_a, p_b etc all prime.
| yau8edq12i wrote:
| No, that looks wrong. Take c = 2, for example. You seem to
| be saying that p^2 ought to be a solution. But D(p^2) = 2p
| (because D(p^2) = D(p) p + p D(p) = p + p = 2p).
| GrantMoyer wrote:
| Starting from my discussion in [1], you can find a sort of
| antiderivative of natural numbers by finding which natural
| number (if any) [?]F_n maps to. Note however that D is not
| injective and has no inverse.
|
| [1]: https://news.ycombinator.com/item?id=40327885
| rm445 wrote:
| Heh, the "integral" I(1) = 2 + C, where C is a constant from
| the set {0, 1, 3, 5, 9,...} (has to add up to another prime
| number).
| xelxebar wrote:
| Comments on the OEIS sequence D(n) are insteresting:
| https://oeis.org/A003415. One in particular points out that if we
| take the prime factorization of n = p0*p1*...*pk, where the pi
| aren't necessarily distinct, then over the reals
| lim (p0+h)(p1+h)...(pk+h) - p0*p1*...*pk D(n) = h->0
| ------------------------------------ .
| h
|
| This motivates the D(p)=1 definition, meaning we can take D to be
| defined just by the Leibniz rule. (Note that D(1) = D(1*1) =
| 1*D(1) + D(1)*1 = 2*D(1), implying that D(1)=0, so that part of
| the definition is redundant.)
|
| Others have pointed out that the definition also works for any
| unique factorization domain, and in the case of polynomials, the
| Leibniz rule guarantees D agrees with the standard derivative,
| which is also a nice sanity check.
| xpe wrote:
| People may disagree about the use of "derivative" as used in the
| article. May the bludgeoning-by-definition continue! Alas, if one
| day we tire of definitional-skull-splitting...
|
| ... for some given function, we can simply recognize the
| difference between: (a) the function definition; (b) properties
| of the function.
|
| For the (calculus) derivative: (a) means "rate of change"; (b)
| means the usual derivative properties e.g. the product rule and
| chain rule
|
| For the arithmetic derivative [1] (or number derivative): (a)
| means "1 for any prime; everything else calculated via the
| product rule" [2]; (b) means the same as above
|
| There are other examples of the above a/b split in mathematics.
| Finding examples is left as an exercise for the reader.
|
| [1] https://oeis.org/wiki/Arithmetic_derivative
|
| [2] Yes, the definition of (a) makes (b) obvious.
| nobodyandproud wrote:
| The skull splitting is a natural reaction because this has
| nothing to do with differentiation.
|
| Throw in Leibniz's rule and the reader is reduced to reading
| the fine print to understand.
|
| The articles (Wikipedia included) are as guilty of this as
| whoever chose this name.
| eigenket wrote:
| This definition does actually have a surprising amount to do
| with differentiation. The definition works for any unique
| factorization domain and in particular for polynomials.
|
| It turns out that the definition here exactly matches the
| usual derivative for polynomials.
| xpe wrote:
| Are you saying that for arithmetic derivatives, the
| definition (part "a" above) "1 for any prime; everything
| else calculated via the product rule" has a surprising
| amount to do with differentiation?
|
| If so, can you connect the dots?
|
| Or did you mean the _properties_ (part "b" above)?
| eigenket wrote:
| Yes,
|
| > 1 for any prime; everything else calculated via the
| product rule
|
| does indeed have a surprising amount to do with
| differentiation!
|
| If you take the usual polynomial functions in one
| variable (lets say x is the variable and all our things
| are complex numbers) then these can be factored: e.g. x^2
| + 3x + 2 = (x+1)(x+2). They form a (so called) unique
| factorization domain, which essentially means that
| factorization into "primes" works exactly the same as it
| does for integers. In the example above (x+1) and (x+2)
| are examples of prime factors which can't be factored any
| further.
|
| If you take the definition "1 for any prime; everything
| else calculated via the product rule" and apply it to
| this system where our "numbers" are polynomials and our
| "primes" are the polynomials we can't factor any further
| you get a definition of an "arithmetic derivative" for
| polynomials.
|
| The fun fact then is that this arithmetic derivative we
| just defined is _exactly_ the same as the usual
| definition of the derivative from calculus:
|
| D[(x+1)(x+2)] = (x+1)D[(x+2)] + (x+2)D[(x+1)] = (x+1) +
| (x+2) = 2x+3
|
| whereas
|
| d/dx (x^2 + 3x + 2) = 2x + 3
| gjm11 wrote:
| More the first than the second.
|
| There are things other than the integers for which it
| makes sense to talk about "the primes". One example is:
| polynomials (with coefficients in, let's say, the complex
| numbers). In this case it turns out that the "primes" are
| exactly the _linear polynomials_ (ax+b) where a is
| nonzero.
|
| There's a bit of ambiguity there, just as there is in the
| integers; 7 and -7 are "the same prime number", and x+3
| and 5x+15 are "the same prime polynomial"; if we're going
| to say D(p)=1 then we need to pick which "version" of p
| has this property, and the obvious choice is the one of
| the form (x+a).
|
| So, now, if we apply _the same definition as for
| integers_ to polynomials with these conventions, it says:
| (1) D(x+a) = 1 and (2) D(fg) = fD(g) + D(f)g when f,g are
| polynomials. And that turns out to give the exact same
| result as the "ordinary" derivative for polynomials.
|
| Whether "exactly identical to" implies "a surprising
| amount to do with" depends on how easily surprised you
| are, I guess.
|
| ... I glossed over the sense in which the "primes" are
| precisely the linear polynomials, so here are a few words
| about that for anyone who's curious.
|
| If we look at polynomials with complex-number
| coefficients, a beautiful theorem says that they can all
| be written as A (x-r1) (x-r2) ... (x-rk), and then one
| polynomial divides another if and only if its set of rj
| is a subset of the other's (handling repeated roots in
| the "obvious" way). It's pretty easy to get from this
| that the linear polynomials are (1) the _irreducible_
| ones, i.e., the ones that can 't be factored into lower-
| degree polynomials, and (2) the _prime_ ones, i.e., the
| ones with the property that if p divides ab then p
| divides either a or b. (These properties are equivalent
| for the integers, as well as for polynomials with complex
| coefficients, but there are other settings in which they
| come out different, and both of them are useful, so they
| have different names.)
|
| (What happens if we use _real_ rather than _complex_
| coefficients? The Wikipedia "Arithmetic derivative" page
| claims that we still get the usual derivative, but that
| looks wrong to me, because if we work over the real
| numbers then x^2+1 is both prime and irreducible, but its
| derivative isn't 1. Maybe I'm missing something.)
| eigenket wrote:
| As far as your point in parentheses goes, I think
| wikipedia is either wrong or confusingly written
| (allowing complex factorisations of real polynomials
| makes what they're written consistent, but is a bit
| silly).
|
| See theorem (20) on page 18 of this pdf for a theorem
| along these lines
|
| https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufna
| rov...
| xpe wrote:
| This comment demonstrates my point, does it not? It looks
| like more of the same: a battle of definitions. If one's goal
| is to win a definitional battle, what do you accomplish if
| you succeed? [1]
|
| But one will not consistently win such a battle. Many people
| will resist for various reasons, whether it be "stubbornness"
| or simply feeling like the other person shows no signs of
| trying to understand what they _mean_.
|
| I propose that better goals include: (i) understanding what
| people are saying; (ii) applying the concepts to some
| productive end. By "productive" I mean some forward progress
| in an empirical or mathematical sense, whether it be
| prediction or proof.
|
| So give up the battle. Why? Not because you are wrong. [2]
| Because "being right" about a definition is rather silly.
| We're talking about concepts being communicated by language
| and symbols. The goal is shared understanding of the concepts
| (which happens _inside_ a brain), not merely enforcing a
| mapping of brain states to ink on a page (words) or
| vibrations in a physical medium (sound).
|
| [1]: Whether you win or lose, the distinction between (a:
| definition) and (b: properties) still exists.
|
| [2]: And not because you are "right" either. You can, at
| best, be consistent in your definitions and use them in
| useful ways.
| nobodyandproud wrote:
| I'm saying that if confusion and annoyance has been the
| norm, then a quick translation for the general audience is
| in order.
|
| Even better would be a differentiating name, but I realize
| that's unlikely.
| xpe wrote:
| > The skull splitting is a natural reaction because this has
| nothing to do with differentiation.
|
| I recommend rephrasing that as " _the definition_ (part "a"
| above) of arithmetic derivative is different than calculus
| definition of derivative."
|
| Do you see? Stating it this way reduces the war of words.
| Your point is made clear. [1] Then other people can say "Ok,
| sure, but don't you see how the _properties_ (part "b"
| above) are the same? And isn't that interesting?"
|
| Think of this another way. Imagine an alternate history where
| the arithmetic derivative was discovered, named, and
| socialized first. Then imagine calculus came along later. If
| so, would calculus be wrong to use the same word,
| "derivative"? ... I won't answer that question because it is
| invalid. Better to dissolve the question [2].
|
| My point? Let's try to shift away from historical battles
| over turf and terminology. Let's find ways to share insight.
|
| [1] Unless your intended point was: "how dare you use the
| word differently?"
|
| [2] https://www.lesswrong.com/posts/Mc6QcrsbH5NRXbCRX/dissolv
| ing...
| nobodyandproud wrote:
| I have zero stakes in this other than stating perhaps it's
| not the audience that's at fault, when an in-circle term
| isn't translated for a more general audience; and the
| audiences are consistently confused and annoyed.
|
| I woke up, noticed the title & blog (and references), went
| through the same tortured route and confusion, before
| stumbling on the fine print, and coming to same "oh, for
| crying out loud" reaction.
| xpe wrote:
| Is it fair to say the definitional confusion bothered you
| more than the interesting aspects (I'm presuming that
| part "b" above is more interesting) pleased you?
|
| If I were to guess... I'd say you (and many people,
| including myself, often) are weary of people redefining
| words in a way that seems wasteful, distortive (such as
| 'stealing' words that formerly had clear technical
| meanings), purely commercial, or self-promotional.
|
| For me, at least, the _intent_ of the redefinition
| matters. But I detect no self-interest or obvious neglect
| in the case of the arithmetic primes.
| DerekL wrote:
| Title needs (2014).
| JadeNB wrote:
| It wouldn't hurt, but why "needs?" The notion isn't changing.
| tempodox wrote:
| While we're at it, why not (1961)? That's when Edward Barbeau
| published his paper. The link to that paper in the article is
| sadly 404.
| ykonstant wrote:
| This is one of the many examples of geometric concepts being
| applied to integers (in this case, the notion of derivation,
| although a non-linear one).
|
| Another important concept is that of a curve and its ring of
| functions in algebraic geometry; for the integers, the curve is
| the prime spectrum of Z, i.e. the prime ideals generated by each
| prime number <p>. The ring of regular functions is precisely the
| ring of integers, operating as functions on prime numbers by n(p)
| = n modulo p.
|
| I wonder if D has any interpretation in terms of nonlinear
| differential operators on Spec(Z).
| bandrami wrote:
| Are you saying the derivative is a geometric concept? Tangent
| slope of a curve is simply one application of a derivative;
| it's not the derivative's identity. What the derivative _is_ is
| the inverse of an inner product.
| ykonstant wrote:
| Can you people stop with the inane pedantry? Yes, the
| derivative is a geometric concept and so is the inner
| product; they are at the core of what a Riemannian manifold
| is, they group to form the (co)tangent spaces of varieties
| and schemes and their derived structures produce the local
| geometric data of the object in question.
| lupire wrote:
| The point is that the derivative is a more general concept
| than just geometric, and is naturally defined with or
| without a geometric comtext. Of course almost anything can
| be modeled geometrically. You can draw a picture of almost
| anything. The integers themselves are obviously geometric,
| by drawing a kindergarten number line.
|
| https://en.m.wikipedia.org/wiki/Langlands_program
| ykonstant wrote:
| This is all semantic nonsense that devalues the original
| post I made about a potential connection between two
| arithmetic-geometric objects. But thanks for patronizing
| me, a research mathematician actually working on the
| arithmetic Langlands program, with a wikipedia link to
| the fucking Langlands program.
| bandrami wrote:
| Huh? Inner product only needs vectors. No geometry
| required.
| lupire wrote:
| That's one of many definitions of derivative.
| jesuslop wrote:
| Ahh, nice one dares to proffer this. I can plug my own
| wondering about if one can formulate an aritmetic derivative
| that instantiates the Kahler differential concept (but I'm
| supposed not to ask unless I already knew the answer, so
| what'd'be the point).
| GrantMoyer wrote:
| Mathematicans like to call two things with the same "structure"
| by the same name, even if it's not obvious how they're otherwise
| related. In this case, the shared structure is that both the
| derivative of a function and D(n) follow the product rule of
| derivatives. Sharing a structure means two mathematical objects
| are "morphic" (homomorphic[1], isomorphic[2], etc.) in some way,
| so in some sense they are the are same object.
|
| In this case, we can construct a morphism. Since D(n) follows the
| product rule, we only need to find a function f of x for each
| prime p which at some x has a value of p and a derivative of 1.
| Then we can compose those functions by multiplication for all
| other natural numbers. f_p(x) = x + p is one such set of
| functions, giving us the complete function F_n(x) = P_(p[?]P(n))
| x + p, where P(n) is the set of prime factors of n. Note, the
| product of the empty set is defined to be 1, so F_1(x) = 1.
|
| Finally, the homomorphism between D and the derivatives of
| functions is that D(n) = F'_n(0), so in some sense, D really is a
| derivative.
|
| [1]: https://en.wikipedia.org/wiki/Homomorphism
|
| [2]: https://en.wikipedia.org/wiki/Isomorphism
| supernikio2 wrote:
| I'm rather new to the concept of objects and morphisms myself,
| but I love how fields like category theory allow one to "zoom
| out" far enough in abstraction that two seemingly different
| concepts are actually the same applied to distinct contexts.
| BalinKing wrote:
| > so in some sense they are the are same object.
|
| My understanding is that this is only the case if they're
| isomorphic--even a pair of homomorphisms between the objects is
| not itself sufficient to identify them as the same. But I also
| don't know any category theory, so I might be spouting nonsense
| :-)
| GrantMoyer wrote:
| Indeed, a homomorphism from A to B is an isomorphism from A
| to a subset of B. In other words, homomorphism is to
| injective as isomorphism is to bijective. So whether you say
| A is equivalent to B depends on how you define equivalent.
|
| For anyone curious, while Category Theory is very concerned
| with morphisms, they also come up in many other places. In
| particular, Abstract Algebra (groups and rings and such) may
| be a more approachable introduction to morphisms than
| Category Theory, and then Category Theory flows pretty
| naturally from the concepts of Abstract Algebra.
| nico wrote:
| Derivative of a binary number:
|
| Think of any binary number as a sequence of 0s and 1s in a
| certain order
|
| For example, 16 in binary is the sequence: 1000
|
| Reading the sequence from right to left, two bits at a time, for
| each one of those two bits, we can note if the value of the
| "earlier" bit in the sequence changed
|
| I can note a change as 1 and a not change as 0, then the above
| sequence becomes:
|
| 0 (0-0 no change) 0 (0-0 no change) 1 (0-1 changed)
|
| Result: 100
|
| In decimal: 8
|
| Now if I want to "integrate" that sequence, I can do the reverse,
| but now I have ambiguity, if I start with 0, the sequence would
| be the original:
|
| 0 0 (0 means no change) 0 (0 means no change) 1 (1 change)
|
| Result: 1000
|
| But if we start with 1 instead:
|
| 1 1 (0 no change) 1 (0 no change) 0 (1 change)
|
| Result: 0111
|
| Intuitively you can think of this as tracking a "discrete rate of
| change"
|
| Usually the derivative or slope of a function gives a real value,
| now imagine "zooming into" the function until you can't track a
| real value anymore, only whether what you are looking at is
| changing or not every time you look
| nico wrote:
| This can also be generalized to anything that can be expressed
| as sequence generated by a function
|
| You can always just split the thing into it's sequential
| elements, then get a pair-wise derivative in between the
| elements
|
| The sequence of pair-wise derivatives is then the equivalent of
| the derivative of the original sequence
|
| If you do this to the limit where the "space" between the
| elements is 0, then you get the continuous case
| johnthescott wrote:
| entropy of a bit string may be another way to view the bitwise
| derivative. paper here for analyzing primes:
| https://arxiv.org/abs/1305.0954
|
| [BiEntropy - The Approximate Entropy of a Finite Binary String]
| nico wrote:
| Thank you for the reference, these concepts are definitely
| closely related
|
| This is super interesting:
|
| > We successfully test the algorithm in the fields of Prime
| Number Theory (where we prove explicitly that the sequence of
| prime numbers is not periodic)
|
| What we do, and what ML algorithms try to imitate, when
| learning, is exactly that: finding loops (periodic sequences)
| within the data (or rather, fitting the data to continuous
| "loopy" representations)
| NegativeLatency wrote:
| What does D look like around the origin? I was hoping for a plot
| or something, also pretty hard to search for.
| math_dandy wrote:
| This notion of derivative of a number depends very weakly on the
| number itself. It only depends on the multiset of exponents in
| the number's factorization into a product of prime powers. In
| this way it's like the divisor counting function, the omega and
| Omega functions, the mobius function, etc. As such, its value
| distribution might be interesting. Something analytic number
| theorists might like to play around with.
| dang wrote:
| Discussed at the time:
|
| _The Derivative of a Number_ -
| https://news.ycombinator.com/item?id=8198607 - Aug 2014 (47
| comments)
| zvr wrote:
| Interesting... I'd imagine it's not common to have re-
| submissions of exact same links ten years apart, without other
| re-submissions in-between.
| pvg wrote:
| It's not that uncommon, in fact, there's another one of the
| front page right now with a bigger gap:
|
| https://news.ycombinator.com/item?id=40329173
|
| https://news.ycombinator.com/item?id=2920379
| galaxyLogic wrote:
| As an aside there also exists the notion of Derivative of Regular
| Expression which has useful applications
|
| https://jvns.ca/blog/2016/04/24/how-regular-expressions-go-f....
| sigil wrote:
| This idea was used to great effect in Matt Might's "Parsing
| with Derivatives" paper [0]! And it featured prominently in the
| Compilers class he taught at the University of Utah.
|
| [0] https://matt.might.net/papers/might2011derivatives.pdf
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