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reference request – Applications of quantity principle in dynamical techniques

From Wikipedia: “Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers.”

First behold examples within the MO-Q “Where is number theory used in the rest of mathematics?”

I commented there, “Goldman in his book The Queen of Mathematics alludes to an article by Weinberg where the partition function of number theory is related to the states of a vibrating string.”

Edit Jan, 10, 2021: (Start)

Thanks to a observation by Terry Tao on the adeles and the purposeful balance equation for the Riemann zeta duty, I create these fascinating succession of papers

1. “Linear Fractional p-Adic and Adelic Dynamical Systems” by Dragovich, Khrennikov, and Mihajlovic

2. “Adeles in mathematical physics” by Dragovich, by which the writer states “It is remarkable that such a simple physical system as a harmonic oscillator is related to so significant a mathematical object as the Riemann zeta function.”

This is simply stunning if one focuses solely on the benchmark quantity theorist’s narrative on the algebra and calculus and doesn’t point to that, by way of the Mellin remodel, the e.g.f. of the Bernoulli polynomials underpin the Riemann, Hurwitz, and Lerch zeta features and that of the Hermite numbers underlie the corresponding Landau-Riemann xi duty through the related Jacobi theta features. Both the Bernoulli and the (arrogate) Hermite polynomials are Appell sequences with reducing/destruction/annihilation and elevating/creation ops, or ladder ops. A household of Hermite polynomials pops up in eigenfunctions for quantum harmonic oscillators and due to this fact in QFT; as early as Schwinger, the Hurwitz zeta duty was associated to a specific mannequin in QED; and later zeta duty regularization was launched into physics by Hawking. This isn’t a surprise for the reason that Sheffer Appell polynomial formalism parallels that of symmetric polynomials/features with related vestige formulation (eigenvalue/zero/pole sums), volumes related to determinants, log/exp mappings (Newton identities), and so on. (Langlands-program, GUE and random matrix integration, and all that stuff). What would breathe stunning is that if there have been no connections.

1. “Adelic harmonic oscillator” by Dragovich, by which he states, “The Mellin remodel of a easiest vacuum condition results in the well-known purposeful relation for the Riemann zeta duty. “

2. “From p-adic to zeta strings” by Dragovich, by which he “briefly discusses some properties of … Lagrangians, related potentials, equations of motion, mass spectra and possible connections with ordinary strings. This is a review of published papers with some new views.”

3. “p-Adic Mathematical Physics: The First 50 Years” by Dragovich, Khrennikov, Kozyrev, Volovich, and Zelenov–a barely earlier survey article to the earlier one, which relates p-adics to various fields of analysis.

An article with an identical taste is “A Correspondence Principle” by. Hughe and Ninham, the place parallels are drawn among the many Poisson summation components, Dirac combs, and purposeful balance equations, and these are related to the properties of options of assorted differential equations of mathematical physics.

(End)

The Dedekind eta duty enters into the dynamics of modular flows and the Lorenz equations by way of knot principle. (It too has purposes in statistical mechanics and string principle.) See refs to Ghys labor on knots and dynamics in MO-Q “The Dedekind eta function in physics” and solutions to the duplicate query on PhysicsSwarm.

With a extra combinatorial taste:

Solutions to the inviscid Burgers’ the KdV, and the KP equations of hydrodynamics and to the common evolution equations for current fields generated by tangent vectors are associated to basic integer arrays. (Not so stunning for the reason that iterated operators $$(x^{m+1}d/dx)^n$$ are associated to basic integer arrays and combinatorics.)

The integers narrate to options of the Burgers’ equation by way of the combinatorics of the associahedra and its relations to compositional inversion by way of OEIS A133437 as sketched within the respond to MO-Q “Why is there a connection between enumerative geometry and nonlinear waves?”

A bivariate e.g.f. for the Eulerian numbers (A008292/A123125) with its related quadratic ($$sl_2$$) infinigen gives a soliton resolution of the 1-D KdV equation. (The Eulerians are rife with ($$A_n$$ and $$B_n$$) connnections to enumerative algebraic geometry, as mentioned by Hirzebruch, Losev and Manin, Batryev and Blume, Cohen, et al.)

Lauren Williams in “Enumeration of totally positive Grassmann cells” develops a polynomial producing duty $$A_{ok,n}(q)$$ whose $$q^d$$ coefficient is the quantity A046802 of completely optimistic cells in $$G^+(ok,n)$$ which have dimension $$d$$ and goes on to display that for the binomial remodel $$hat{E}_{ok,n}(q)=q^{k-n}sum^n_{i=0} (-1)^i binom{n}{i} A_{ok,n-i}(q)$$ that $$hat{E}_{ok,n(}(1)=E_{ok,n}$$, the Eulerian numbers A008292, and $$hat{E}_{ok,n}(0)=N_{ok,n}$$, the Narayana numbers A001263. She reiterates this in her presentation “The Positive Grassmannian (a mathematician’s perspective)” and relates $$G^+$$ to soliton shallow-water-wave options of a KP equation, noting too the roles of $$G^+$$ in computing scattering amplitudes in string principle, a relation to free likelihood, and the happening of the Eulerians and Narayanaians within the BCFW recurrence and twistor string principle. (See hyperlinks to Williams’ papers in A046802.)

The refined Eulerian numbers A145271 come up within the succession growth for current fields generated by exponentiation of tangent vectors.

The conservation equations related to the Burgers’ and KdV equations too have connections to basic integer sequences. See too “Set partitions and integrable hierarchies” by V.E. Adler.

See too some refs and feedback on the relation of the cycle index polynomials for the symmetric group $$S_n$$ A036039 (refined Stirling polynomials of the primary kindly, associated to the elementary Schur polynomials and Faber polynomials) to tau features and integrable hierarchies (and zeta features). The Faber polynomials A263916 are too associated to integrable techniques, evolution equations, and quantity theoretic relations.

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