Duality in elliptic Ruijsenaars system and elliptic symmetric functions

We demonstrate that the symmetric elliptic polynomials $$E_\lambda (x)$$ E λ ( x ) originally discovered in the study of generalized Noumi–Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars–Schneider (eRS) Hamiltonians that act on the mother function variable $$y_i$$ y i (substitute of the Young-diagram variable $$\lambda $$ λ ). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $$P_\lambda (x)$$ P λ ( x ) are eigenfunctions of the elliptic reduction of the Koroteev–Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS Hamiltonians by a somewhat mysterious orthogonality transformation, which is well defined only on the full space of time variables, while the coordinates $$x_i$$ x i appear only after the Miwa transform. This observation explains the difficulties with getting the apparently self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.

An abstract form of the first epsilon theorem

We present a new method of computing Herbrand disjunctions. The up-to-date most direct approach to calculate Herbrand disjunctions is based on Hilbert’s epsilon formalism (which is in fact also the oldest framework for proof theory). The algorithm to calculate Herbrand disjunctions is an integral part of the proof of the extended first epsilon theorem. This paper introduces a more abstract form of epsilon proofs, the function variable proofs. This leads to a computational improved version of the extended first epsilon theorem, which allows a nonelementary speed up of the computation of Herbrand disjunctions. As an application, sequent calculus proofs are translated into function variable proofs and a variant of the axiom of global choice is shown to be removable from proofs in Neumann–Bernays–Gödel set theory.

Lexicalizing disjunction scope

Tiwa (Tibeto-Burman; India) has two disjunctive particles, which give rise to different interpretations in sentences with other operators. I argue that this semantic distinction is not one of inclusive vs. exclusive disjunction, but one of scope. I provide an analysis that captures this scopal distinction: one particle lexicalizes a choice function variable (subject to existential closure high in the structure), and the other lexicalizes an alternative-set former which interacts with higher operators. I also show that wide scope disjunction in Tiwa behaves differently from wide scope readings of English "or", and suggest that they warrant different analyses.

UNIFORMLY BOUNDED COMPOSITION OPERATORS

We prove that if a uniformly bounded (or equidistantly uniformly bounded) Nemytskij operator maps the space of functions of bounded ${\it\varphi}$-variation with weight function in the sense of Riesz into another space of that type (with the same weight function) and its generator is continuous with respect to the second variable, then this generator is affine in the function variable (traditionally, in the second variable).

Solution of the Parallel sheaR Layer Green's Function Using Conservation Equations

A parallel shear flow representation of a jet is a standard way to solve for the wave propagation terms in jet noise modeling using the acoustic analogy. In this paper we show by introducing a new primary Green's function variable, proportional to the convective derivative of the pressure-like Green's function, the wave propagation equations reduce to an exact conservation form that does not include any derivatives of the mean flow. We analyze this Green's function variable numerically and show its utility when the mean flow is defined by a CFD solution and known only at a discrete set of points.