Related papers: Stokes' formulae on classical symbol valued forms …
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the…
We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately…
We study Stokes phenomena of the k \times k isomonodromy systems with an arbitrary Poincar\'e index r, especially which correspond to the fractional-superstring (or parafermionic-string) multi-critical points (\hat p,\hat q)=(1,r-1) in the…
We generalize the linear algebra setting of Tate's central extension to arbitrary dimension. In general, one obtains a Lie (n+1)-cocycle. We compute it explicitly. The construction is based on a Lie algebra variant of Beilinson's adelic…
We extend the Cauchy residue theorem to a large class of domains including differential chains that represent, via canonical embedding into a space of currents, divergence free vector fields and non-Lipschitz curves. That is, while the…
We form real-analytic Eisenstein series twisted by Manin's noncommutative modular symbols. After developing their basic properties, these series are shown to have meromorphic continuations to the entire complex plane and satisfy functional…
We extend formulae which measure discrepancies for regularized traces on classical pseudodifferential operators to regularized trace cochains, regularized traces corresponding to 0-regularized trace cochains. This extension from 0-cochains…
In a previous work [arXiv:2211.02945] we made an attempt to set up a discrete octonionic Stokes' formula. Due to an algebraic property that we have not considered in that attempt, the formula however turned out to involve an associator term…
In this paper, we establish the existence of Stokes waves with piecewise smooth vorticity in a two-dimensional, infinitely deep fluid domain. These waves represent traveling water waves propagating over sheared currents in a semi-infinite…
This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of `maximal formula', `segment' and `maximal segment' suitable to the system, and gives…
Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis and Guillemin. In this article we give a definition of the Wodzicki residue, using the langage of r-fibered distributions in the context of filtered…
We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from a natural number $m\in\mathbb N$ to the falling factorials…
Given a holomorphic regularisation procedure (e.g. Riesz or dimensional regularisation) on classical symbols, we define renormalised multiple integrals of radial classical symbols with linear constraints. To do so, we first prove the…
While detailed information about the semiclassics for single-particle systems is available, much less is known about the connection between quantum and classical dynamics for many-body systems. As an example, we focus on spin chains which…
The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules…
We give a topological description of the behaviour of Stokes matrices under the Fourier transform from infinity to infinity in a large number of cases of one level. This explicit, algorithmic statement is obtained by building on a recent…
We define an integral version of Sczech's Eisenstein cocycle on GLn by smoothing at a prime ell. As a result we obtain a new proof of the integrality of the values at nonpositive integers of the smoothed partial zeta functions associated to…
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We…
The commutator anomalies (Schwinger terms) of current algebras in $3+1$ dimensions are computed in terms of the Wodzicki residue of pseudodifferential operators; the result can be written as a (twisted) Radul 2-cocycle for the Lie algebra…