Related papers: Riemann-Hilbert factorization of matrices invarian…
This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and…
Some preliminaries and basic facts regarding unbounded Wiener-Hopf operators (WH) are provided. WH with rational symbols are studied in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency…
In this paper, we investigate the inverse scattering transform(IST) for the focusing and defocusing mKdV equation with fully asymmetric nonzero boundary conditions. Our analysis focuses on the properties of the Jost function, allowing us to…
We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a…
Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear…
We consider the inverse problems of for the fractional Schr\"odinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal…
These lectures introduce the method of nonlinear steepest descent for Riemann-Hilbert problems. This method finds use in studying asymptotics associated to a variety of special functions such as the Painlev\'{e} equations and orthogonal…
In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective…
An upper bound of the variation of argument of a holomorphic function along a curve on a Riemann surface is given. This bound is expressed through the Bernstein index of the function multiplied by a geometric constant. The Bernstein index…
We address the question whether the condition on a fusion category being solvable or not is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of…
Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We…
A factorization formula for wave functions, which is basic in the inverse spectral transform approach to initial-boundary value problems, is proved in greater generality than before. Applications follow. Related compatibility questions for…
Extending earlier work by Rogers, Wiener-Hopf factorisation is studied for a class of functions closely related to Nevanlinna-Pick functions and complete Bernstein functions. The name 'Rogers functions' is proposed for this class. Under…
In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with quasi-permutation monodromy groups which correspond to non-singular branched coverings of $\CP1$. The solution is given in terms of Szeg\"o kernel on the…
We study holomorphic vector fields whose singular locus contains a local complete intersection smooth positive-dimensional component. We prove global and local formulas expressing the limiting Milnor/Poincare-Hopf contribution along such a…
The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions $\mathcal{M}$ can be non-compact due to the appearance…
We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a…
We use inverse scattering methods, generalized for a specific class of complex potentials, to construct a one parameter family of complex potentials V(s, r) which have the property that the zero energy s-wave Jost function, as a function of…
We present in this report 1+1 dimensional nonlinear partial differential equation integrable through inverse scattering transform. The integrable system under consideration is a pseudo-Hermitian reduction of a matrix generalization of…
The focusing Kundu-Eckhaus (KE) equation with non-zero boundary conditions at infinity, under two cases: simple zeros and double zeros, is investigated systematically via Riemann-Hilbert (RH) problem. We derive some new results for the…