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From my limited perspective, it appears that the understanding of a mathematical phenomenon has usually been achieved, historically, in a continuous setting before it was fully explored in a discrete setting. An example I have in mind is the relatively recent activity in discrete differential geometry, and discrete minimal surfaces in particular:


      Schwartz-P
      Image: A discrete Schwarz-P surface. Bobenko, Hoffmann, Springborn: 2004 arXiv abstract.


I would be interested in examples where the reverse has happened: a topic was first substantively explored in a discrete setting, and only later extended to a continuous setting. One example—perhaps not the best example—is the contrast between $n$-body dynamics and galactic dynamics. Of course the former is hardly "discrete," but rather is more discrete than the study of galaxy dynamics leading to the conclusion that there must exist vast quantities of dark matter.

Perhaps there are clearer examples where discrete understanding preceded continuous exploration?

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    $\begingroup$ Fluid flow, and statistical mechanics more generally? (Your post suggests you're OK with physical examples.) $\endgroup$ – LSpice May 29 '17 at 0:43
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    $\begingroup$ Zeno's paradoxes erroneously use discrete arguments to make conclusions about the continuous (change and motion are illusions)? $\endgroup$ – Tom Copeland May 29 '17 at 19:13
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    $\begingroup$ Extension of the factorials to the gamma function and, in parallel, the extension of the usual calculus of integral powers of an operator to fractional powers. Related also to generalizing binomial expansions for integral powers to real powers. $\endgroup$ – Tom Copeland May 29 '17 at 20:27
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    $\begingroup$ A similar question on Maths SE: (math.stackexchange.com/q/1774670) that I asked but had received no answers... $\endgroup$ – Jean Marie Becker Jun 1 '17 at 5:26
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    $\begingroup$ Related: Computing the Continuous Discretely by Beck and Robins. $\endgroup$ – Tom Copeland Jun 20 '17 at 22:42

21 Answers 21

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I would say that a lot of topology was discrete before it was continuous. The Euler characteristic was first observed (in 1752) as an invariant of polyhedra. Around 1900 Poincaré first calculated Betti numbers, and generalized the Euler characteristic, in a polyhedral setting. The first general treatment of topology, by Dehn and Heegaard in their Enzyklopädie article of 1907, was also in a polyhedral setting.

It was only after 1910, with simplicial approximation methods introduced by Brouwer, that the topological invariance of various combinatorial invariants was proved. For example, Alexander proved the topological invariance of the Betti numbers in 1915.

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    $\begingroup$ Excellent example---Thanks! $\endgroup$ – Joseph O'Rourke May 29 '17 at 0:23
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    $\begingroup$ And of course the use of geometric flows such as Ricci flow as a continuous method for understanding topology came much later than all of this... $\endgroup$ – Terry Tao May 29 '17 at 0:50
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Probability and stochastic processes

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    $\begingroup$ With one recent expansion of the concept of probability being the jump from the Heisenberg/Bohr matrix formulation of quantum mechanics of transitions among discrete states to the Schrodinger rep.in terms of continuous probability amplitudes defined by a differential eqn. $\endgroup$ – Tom Copeland May 29 '17 at 19:56
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    $\begingroup$ @TomCopeland, the matrix formulation of quantum mechanics was not Bohr's, and Schrodinger and Heisenberg made their contributions to quantum mechanics independently, so I don't think this is a good example. $\endgroup$ – Matt F. Jun 2 '17 at 14:02
  • $\begingroup$ @Mattf, Planck, Einstein, and Bohr all devloped discrete quantum models preceding and motivating the work of Heisenberg, Born, Kramers, Jordan, and Schrodinger; ergo, the later work can't be considered independent. The devlopment proceeded from models of discrete quantum observables (energy, momentum) to a directly unobservable continuous wave model of probability amplitude. $\endgroup$ – Tom Copeland Jun 2 '17 at 18:34
  • $\begingroup$ Oops, left out de Broglie (1924), rather important. $\endgroup$ – Tom Copeland Jun 2 '17 at 18:39
  • $\begingroup$ @TomCopeland, I agree that the discrete physical understanding preceded Schrodinger. If the first comment omitted "Heisenberg/" and "matrix" I would agree with that too. $\endgroup$ – Matt F. Jun 2 '17 at 18:53
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Group theory might be a good example. The first examples of groups were in a discrete setting, namely Galois groups of number fields, which were first understood as permutation groups of the finite set of roots of a polynomial. The emergence of Lie's theory of continuous groups and Klein's Erlangen program came somewhat later (although, in writing this question, I discovered that it was closer than I expected).

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Pretty sure summation of finite discrete series was well-understood, conceptually, long before integration. And I would not be surprised to see similarly for many other ideas of the differential/integral calculus (though not all…).

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    $\begingroup$ Expanding on this, surely the principle "From a discrete series, we can take its differences, and conversely, from any series of putative differences, we may through summation reconstruct the series from which it arises, uniquely up to an additive constant" was understood before the Fundamental Theorem(s) of Calculus. $\endgroup$ – Sridhar Ramesh May 29 '17 at 0:42
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    $\begingroup$ Do you have any historical references for such claims, or are you just guessing? Apparently, the summation symbol was introduced later (by Euler) than the integration symbol (Leibniz). $\endgroup$ – Michael Bächtold May 29 '17 at 12:24
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    $\begingroup$ Although, Leibniz apparently used $\int$ for both integrals and discrete sums. See Cajori, A history of mathematical notations. $\endgroup$ – Michael Bächtold May 29 '17 at 13:44
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    $\begingroup$ Nothing I mean depends on the sigma or long S symbols for summation; just the concept of addition, "5 + 3 + 8 + 2 + 4 = 22" and so on. $\endgroup$ – Sridhar Ramesh May 29 '17 at 19:48
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    $\begingroup$ For what it's worth, Archimedes apparently found the area under a parabola. 2300 years ago. I don't know if you call that integration though. $\endgroup$ – Mehrdad May 30 '17 at 2:42
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Theoretical computer science offers lots of examples of what you want, since people almost always think first about the discrete setting, even if the continuous setting also turns out to be important. Two examples off the top of my head:

Grover's search algorithm, one of the central quantum computing algorithms, was first understood in the discrete-time setting and only later in the continuous-time one. (In an amusing turnabout, the Farhi-Goldstone-Gutmann algorithm, which generalizes Grover's algorithm to games of alternation like chess and Go, was first understood in a continuous-time setting and only later in a discrete-time one.)

The number of samples needed to learn a hypothesis was first understood in the discrete case (through the concept of VC-dimension), and only later generalized to the continuous case (through the concept of fat-shattering dimension).

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Integers to rationals to reals.

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  • $\begingroup$ In thought perhaps, but 19th century formalisation went in the opposite direction $\endgroup$ – Henry Jun 1 '17 at 21:29
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    $\begingroup$ There are still tribes whose languages have words for one, two, three, and then only more than three. I'm pretty sure that reflects the natural prehistorical and historical development of thoughts on number from the simple to the complex. The Pythagorean cult was pretty upset with the proof of the existence of irrationals--undermined the authority of their monolithic theory of harmony. The reals, of course, are a continuum of the rationals and irrationals, addressed in decimal form. Don't really give a care about when/how academics put a formal gloss on these ideas--that's not the question. $\endgroup$ – Tom Copeland Jun 1 '17 at 22:46
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A recent example would be the generalization of graphs to continuous objects graphons, symmetric measurable functions on a square. Topological properties of spaces of these objects (e.g., compactness) have been shown to yield known properties in the discrete setting (existence of Szemeredi partitions). Also the continuous setting allows graphons to be interpreted as probability distributions which form a very general model for random graphs. Such things are detailed in the book of Lovasz, "Large networks and graph limits".

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Discrete (more specifically, finite field version) Kakeya conjecture is solved by Zeev Dvir using polynomial method, while original continuous problem is wide open.

If you let me speculate on the reasons, I would say that though we have some ideas what are 'continuous polynomials' (say, Fourier integrals are analogues of trigonometric polynomials), we do not understand what is a right substitute for degree of a polynomial.

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  • $\begingroup$ Re: degree of a polynomial—at least in the complex case, what about the winding number? I guess this privileges certain points above others. $\endgroup$ – LSpice May 29 '17 at 4:15
  • $\begingroup$ @LSpice winding is the number of intersections of a curve and a line (well, sign respected). So, it looks to be an invariant in the spirit of "number of roots", in general not well defined for a function which is not a polynomial. Maybe I am too pessimistic. $\endgroup$ – Fedor Petrov May 29 '17 at 7:19
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Another example is coin tossing and other probabilistic models. Coin tossing was studied long before Brownian motion. Discrete probability precedes continuous probability.

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In general you seem to be right. Since the invention of Calculus, "continuous" became easier, and usually is investigated before the discrete. One example of the opposite is the differential equations vs difference equations. As I understand the first difference equation studied was the one that defined Fibonacci sequence. But this was long before Calculus.

EDIT. As I recently learned, Fibonacci discussed this sequence but he did not derive the formula for the general term, did not "solve" the difference equation. It was solved by D. Bernoulli AFTER the invention of calculus. The solution is called Binet formula.

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    $\begingroup$ Calculus itself was built out of earlier discrete ideas, since Newton knew the calculus of finite differences very well, and that informed his methods of interpolation leading him to power series and the rest of calculus as he knew it. $\endgroup$ – KConrad Dec 5 '17 at 22:55
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I can think of two good examples. The first is rather straight forward. The hyper-operators. Namely,

$$a \uparrow^n b : \mathbb{N}^3 \to \mathbb{N}$$ $$a \uparrow^0 b = a \cdot b$$ $$a \uparrow^n 1 = a$$ $$a \uparrow^{n} (a \uparrow^{n+1} b) = a \uparrow^{n+1} (b+1)$$ $$a \uparrow^{n+1} b = a \uparrow^n a \uparrow^n ...(b\,times)... \uparrow^n a$$

It is a rather interesting open problem to construct the same object in analysis. Namely, replace the $\mathbb{N}$'s with some domain in $\mathbb{C}$. Pivotally, tetration, or $e \uparrow^2 z$, is somewhat solved (though controversy exists as to which solution is the right solution). You can perceive the problem better with tetration. It's very easy to get

$$e \uparrow^2 k = e^{e^{...k\,times...^e}}$$

which satisfies

$$e^{e \uparrow^2 k} = e \uparrow^2 (k+1)$$

but how do we get holomorphic $e \uparrow^2 z$? Surely in this scenario the discrete instance inspired the continuous instance.

Another one would be indefinite summation. Everyone knows that

$$\sum_{j=1}^n f(j) = a(n)$$

is well defined, and satisfies $a(n) + f(n+1) = a(n+1)$, but how do we get holomorphic

$$\sum_{j=1}^z f(j) = a(z)$$

where $a(z) + f(z+1) = a(z+1)$. Surely the discrete instance inspired the continuous instance, again.

In fact, the whole field of study dedicating itself to extending recursive relationships defined on the naturals to the complex plane fits the bill of discrete before continuous very well.

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  • $\begingroup$ Thanks for bringing in tetration, about which I knew little. $\endgroup$ – Joseph O'Rourke May 30 '17 at 1:57
  • $\begingroup$ It's a great subject, but the literature is sparse to almost non-existent. $\endgroup$ – user78249 May 30 '17 at 2:14
  • $\begingroup$ I went looking for indefinite summation a while ago, spurred by what I don't know, and had a hard time finding anything non-handwavy. Are there any good expository or introductory papers on it? $\endgroup$ – LSpice Dec 1 '17 at 20:59
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    $\begingroup$ You might have more luck looking at books on finite differences, namely solving the equation $\Delta y = g(z)$. Everything is phrased in a different language, but the problem is the same. The most researched version is probably due to Ramanujan, who solves the case when $|g(z)| < Ce^{\tau|\Im(z)| + \rho|\Re(z)|}$ with $\tau < \pi$. In this case it's called Ramanujan Summation. Ramanujan's notes themselves are handwavy, but enough rigor has been added over the years. Sadly I can't think of any articles off the top of my head. But the keywords Ramanujan Summation is a good starting point. $\endgroup$ – user78249 Dec 4 '17 at 13:47
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To add to the theoretical computer science angle: the study of graph algorithms has traditionally been done in a combinatorial setting, because that is way more natural at first of course. But recently our understanding has been greatly aided by methods from linear algebra and continuous optimization, especially in the context of the central to the field maximum flow problem. These insights have in particular lead to the fastest known graph algorithms for certain problems. For some overview of this, see these recent slides by Madry.

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Investigations of harmonic motion and heat flow (by Huygens, the Bernoullis, Euler, Fourier, et al.) as discussed in Sections 1.2, 3, and 5 of "The acoustic origins of harmonic analysis" by Olivier Darrigol involved using discrete models as a stepping stone to the continuous.

(This research motivated much work devoted to establishing rigourous foundations of mathematical analysis, such as the definition of a function.)

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    $\begingroup$ More precisely, Lagrange obtained for the first time eigenvalues and eigenfunctions of a homogeneous string by approximating it by a weightless srting with equally spaced beads of equal mass. $\endgroup$ – Alexandre Eremenko May 29 '17 at 21:10
  • $\begingroup$ @AlexandreEremenko, I would say "In particular ... ." There is much earlier and later work than Lagrange's that is important. $\endgroup$ – Tom Copeland May 29 '17 at 21:36
  • $\begingroup$ @Tom Copeland: Who rigorously derived the eigenvalues of a homogeneous string before Lagrange? $\endgroup$ – Alexandre Eremenko May 30 '17 at 20:14
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    $\begingroup$ @AlexandreEremenko, it's just the old forest versus the trees scenario. My answer(s) addresses the question of the discrete scenario informing the continuous not necessarily completely precisely defining it in some selected sense in some limit. No reason to emphasize precise results on spectra here, and, indeed, Lagrange wasn't uninfluenced by his predecessors and contemporaries--he didn't stand alone (he even pays tribute to D Bernoulli). The point is the use of discrete models to draw conclusions, whether heuristic or precise, about associated continuous ones not the exact rigourous results. $\endgroup$ – Tom Copeland May 30 '17 at 21:13
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The main line of inquiry in information theory concerns itself with theoretical limits on data compression for communication and storage, as well as theoretical limits on transmission rates for noise-resilient communication (or fault-tolerant computing, etc.). Claude Shannon's famous 1948 paper, which lays the foundation for the field rather comprehensively, concerns itself entirely with discrete information sources (that is discrete in both time and state space), modeled as stochastic processes. Later treatments (even until now) remain much more concerned with discrete information sources. As far as I know, there is substantial treatment of continuous sources (especially continuous state-space), but it was not pursued first and it remains secondary.

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Fredholm integral equations of the first kind can be viewed as continuous analogues of matrix multiplication of a vector. The use of the terminology for matrices, such as the trace and determinant, for integral kernels are vestiges of the development from the discrete to the continuous as explained in "On the origin and early history of functional analysis" by Lindstrom (see in particular Section 4).

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    $\begingroup$ +1 even though I hate using the matrix representation, and hate when one has to digest a proof in terms of matrices. $\endgroup$ – user78249 May 29 '17 at 18:25
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    $\begingroup$ @james.nixon, I think it is important for students wishing to become innovators (in any field) to appreciate the struggles/messiness of creation not just the beauty of the final product. $\endgroup$ – Tom Copeland May 29 '17 at 18:57
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    $\begingroup$ I'm all for that... except when it involves matrices ;) $\endgroup$ – user78249 May 29 '17 at 18:58
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    $\begingroup$ @james.nixon, I'm reminded, on the other hand, of some anecdote (maybe by Rota or Arnold) of a researcher who dismissively said, "I do discrete not continuous." $\endgroup$ – Tom Copeland May 29 '17 at 19:03
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In general you seem to be right. Since the invention of Calculus, "continuous" became easier, and usually is investigated before the discrete. One example of the opposite is the differential equations vs difference equations. As I understand the first difference equation studied was the one that defined Fibonacci sequence. But this was long before Calculus.

On the contrary, I would argue this is a wholly 'modern' 20th century perspective. Pretty much all the great mathematical-physicists that introduced major classes of new mathematical tools, from Newton to Maxwell to Feynman, were very much constructive mathematicians, in their thinking (Einstein comes to mind as a counterexample; sure there are more).

Newton pretty much hated infinitesimal calculus, and wrote all his original proof in terms of finite geometrical constructions; that the differential algebraic method was a more convenient method of obtaining the same results is definitely the opinion of people that came after him, not of the guy that came up with the original logic of it.

Maxwell preferred to think about vector calculus in terms of finite elements / mechanical gears; and if I had to teach someone electrodynamics today, I would most certainly prefer to do it using discrete exterior calculus rather than using vector calculus (and similarly, einstein notwithstanding, I think Regge calculus is a lot more instructive than differential tensor equations).

The same applies to Feynman. His methods were developed using discrete logic; and he was highly critical of the logical acrobatics required to make sense of the UV divergences as you take the limit to continuous space.

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    $\begingroup$ "Newton pretty much hated infinitesimal calculus, and wrote all his original proof in terms of finite geometrical constructions" could you provide some references for this? $\endgroup$ – Michael Bächtold May 29 '17 at 12:53
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    $\begingroup$ It seems that Newton preferred "geometry" over "algebra". Which --to my mind-- are different opposite than "discrete" vs. "continuous". At least that's how I interpret the review you linked to. $\endgroup$ – Michael Bächtold May 29 '17 at 15:25
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    $\begingroup$ Do you have a reference for Maxwell's or any subsequent development of electrodynamics using finite elements/mechanical gears? That would be very interesting. Any treatment would be worth looking at). $\endgroup$ – Francis Davey May 30 '17 at 12:41
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    $\begingroup$ Newton said a lot of things. Do you really believe the story of the falling apple? For a guy who discovered so much of analytic importance, I can't seriously take any claim that he was truly averse to analytic methods as opposed to geometric even if he indeed claimed that. I do think he became weary of concentrating on math in his later years. See the intro to Needham's Visual Complex Analysis on some related thoughts. $\endgroup$ – Tom Copeland May 31 '17 at 5:17
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    $\begingroup$ @MichaelBächtold, from Wikipedia: "Chandrasekhar worked on a project devoted to explaining the detailed geometric arguments in ... Newton's ... Principia Mathematica using the language and methods of ordinary calculus. The effort resulted in the book Newton's Principia for the Common Reader, published in 1995." // Geometric proofs were de rigeur in Newton's time and he had some difficulty getting his work on calculus published. $\endgroup$ – Tom Copeland May 31 '17 at 14:10
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Quantum field theory is a very cogent example of contemporary interest: Path integrals and quantum fields on a lattice are easy to define (and lattice quantum chromodynamics is a big industry nowadays), but establishing even a good definition for the continuum analogs has proved hitherto elusive. You can win a million dollars for proving that such a continuum field theory exists.

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Pretty much every Analytic Continuation from the integer, or other discrete, into real (or other continuous) domain is an example of this.

The Gamma function extending the factorial into real domain is a classic example of analytic continuation, but there are many more.

$\Gamma (n) = (n-1)!$

${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx} $

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  • $\begingroup$ Hi, welcome to MO. Can you add more details about your answer? $\endgroup$ – Amir Sagiv Jun 2 '17 at 13:19
  • $\begingroup$ Already mentioned in the comments to the question. For development see math.stackexchange.com/questions/1537/… $\endgroup$ – Tom Copeland Jun 2 '17 at 18:17
  • $\begingroup$ @TomCopeland: Analytic continuation? Where? Analytic continuation is a whole large class or problems which fall under the titular question, gamma is just one common example. $\endgroup$ – SF. Jun 2 '17 at 20:06
  • $\begingroup$ The extension of the Euler integral to Re(z) < 0 is an example of analytic continuation of a function analytic and therefore continuous and smooth in some domain to a larger domain. The extension of values at discrete points to a continuum is an example of interpolation, as in the MSE answer and links which is the thrust of my earlier comment. $\endgroup$ – Tom Copeland Jun 2 '17 at 20:28
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Benjamini-Schramm convergence was originally defined for graphs, later extended to metric spaces with applications especially in the setting of locally symmetric spaces. See http://annals.math.princeton.edu/2017/185-3/p01

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From SKETCHES OF KDV by E. Arbarello:

Studying a dynamical system consisting of a finite number of particles of unitary mass distributed along a line segment with forces acting on adjacent pairs, Fermi, Pasta and Ulam introduced a natural non-linear perturbation which gave rise to an unexpected, discrete version of the KdV equation. Starting from this model, ten years later, Zabusky and Kruskal [ZK65] undertook a numerical study of the KdV which exhibited, for the first time, solutions of KdV having the appearence of packets of N localized waves which collide and resurrect in the same way as Scott Russel’s waves do in a shallow canal. Because of the particle-like behaviour of these collisions, they named these solutions: N-solitons.

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The Riemann zeta function is a prime example.

Far into antiquity are investigations of partial sums of integral powers of the natural numbers, leading eventually to the Bernoulli polynomials and the discovery of their exponential generating function by Euler. It was natural to then look at partial sums of the reciprocals of the natural integers.

From Wikipedia:

The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. ... The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:

$$ \sum_{n>0} \frac{1}{n^2}$$

Excerpt from "On some historical aspects of the theory of the Riemann zeta function" by Giuseppe Iurato:

Euler's application of infinite series to different number theoretical problems was of principal importance. The study of the series $ \frac{1}{n^{2k}}$ leads to series of the form $ \frac{1}{n^s}$. .... Euler (1707-1783) was probably the first to see that these series can be applied to number theory. He was in correspondence with C. Goldbach and J.L Lagrange just on number theory questions. His proof of the existence of infinitely many primes uses the divergence of the harmonic series $\sum \frac{1}{n}$ and using the above fundamental theorem of arithmetic which says that every natural number can uniquely be written as a product of powers of primes. Afterwards, P.L.G. Dirichlet (1805-1859) systematically introduced analytical methods in number theory. Among other things, he investigated the series $ \frac{1}{n^s}$ for real $s$, while B. Riemann (1826-1866) allowed complex $s$.

There is also Euler's formula for reflection symmetry of the nascent Riemann $\xi$ function.

It is precisely the e.g.f. for the Bernoulli numbers that appears in Riemann's integral formula for $(s-1)! \zeta(s)$, as he was well aware of.

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  • $\begingroup$ "The Riemann zeta function is a prime example." Pun intended? $\endgroup$ – Gerry Myerson 2 days ago
  • $\begingroup$ @GerryMyerson, primarily. $\endgroup$ – Tom Copeland 2 days ago
  • $\begingroup$ More detail in "Euler's constant: Euler's work and modern developments" by Lagarias arxiv.org/abs/1303.1856 $\endgroup$ – Tom Copeland 2 days ago
  • $\begingroup$ I believe Dirichlet's initial should be P.G.L. rather than P.L.G. $\endgroup$ – Andreas Blass yesterday

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