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Fractional dissipation is a powerful tool to study non-local physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational…

In this work we study some general classes of pseudodifferential operators whose symbols are defined in terms of phase space estimates.

We consider a family of integral operators which appears when analyzing layered equilibria and front dynamics of a phase kinetics equation with a conservation law. We study the spectra of these operators in $L^2$ and derive a lower bound…

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler's identity, and Dedekind zeta functions of quadratic imaginary…

This paper is dedicated to finding the quadrature operator eigenstates and wavefunctions of the most general $f$-deformed oscillators. A definition for quadrature operator for deformed algebra is derived to obtain the quadrature operator…

In this paper, we revisit the diffusive representations of fractional integrals established in \cite{diethelm2023diffusive} to explore novel variants of such representations which provide highly efficient numerical algorithms for the…

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…

A class of non-linear eigenvalue problems defined in the form of operator polynomials is investigated. The problems are related to wave equations which appear in a relativistic quantum field theory. Spectral asymptotics for this class are…

Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in problems of mathematics, statistics and natural sciences. In this article we start considering the case of a Gauss hypergeometric…

This paper studies fractional integral operator for vector fields in weighted $L^1$. Using the estimates on fractional integral operator and Stein-Weiss inequalities, we can give a new proof for a class of Caffarelli-Kohn-Nirenberg…

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…

This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with…

We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a…

We translate Davenport's and Heilbronn's work on a quantitative version of the Oppenheim conjecture for indefinite diagonal quadratic forms in 5 variables into the setting of function fields.

We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni-…

We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm-Liouville eigentransforms and calculating Gauss-Jacobi quadrature rules. Our approach is based on the well-known fact that…

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders,…

This work further develops the properties of fractional differential forms. In particular, finite dimensional subspaces of fractional form spaces are considered. An inner product, Hodge dual, and covariant derivative are defined. Coordinate…

Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance…

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth…