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A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these…
We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The operators should mirror the standard Sobolev mapping property in the holomorphic analogues. The setting is a closed real-analytic Riemannian manifold, or…
We present an algorithm to compute $\mathrm{LCLM}$-decompositions for linear differentials operators with coefficients in the rational function field of characteristic $p$, $\mathbb{F}_{p^n}(t)$. We show that for an operator $L$ of order…
This paper investigates the eigenvalue problem of integral operators whose kernels can be expressed as a finite sum of pairwise products of single-variable functions, making them separable. By consdiering the matrix form of the separable…
Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional…
In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes…
Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we…
Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO),…
Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) are fundamental for modeling stochastic dynamics across the natural sciences and modern machine learning. Learning their solution operators with…
We give an algorithm to write down all conformally invariant differential operators acting between scalar functions on Minkowski space. All operators of order k are nonlinear and are functions on a finite family of functionally independent…
The central role played by pseudodifferential operators in relativistic dynamics is very well know. In this work, operators as the Schrodinger one (e.g: square root) are treated from the point of view of the non-local pseudodifferential…
The comprehensive generalization of summation-by-parts of Del Rey Fern\'andez et al.\ (J. Comput. Phys., 266, 2014) is extended to approximations of second derivatives with variable coefficients. This enables the construction of…
There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. The third problem, however, should be added to the list: separation…
A general expression for a relative invariant of a linear ordinary differential equations is given in terms of the fundamental semi-invariant and an absolute invariant. This result is used to established a number of properties of relative…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
The Partial Integral Equation (PIE) framework was developed to computationally analyze linear Partial Differential Equations (PDEs) where the PDE is first converted to a PIE and then the analysis problem is solved by solving operator-valued…
The scalar auxiliary variable (SAV) method was introduced by Shen et al. and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into…
We find shift operators for the Dotsenko-Fateev equation, which is a differential equation of order 3, and for the three Fuchsian differential equations of order 4, 5 and 6, respectively, which are connected with the Dotsenko-Fateev…
We analyze the concepts of analytically weak solutions of stochastic differential equations (SDEs) in Hilbert spaces with time-dependent unbounded operators and give conditions for existence and uniqueness of such solutions. Our studies are…
Divided difference operators are degree-reducing operators on the cohomology of flag varieties that are used to compute algebraic invariants of the ring (for instance, structure constants). We identify divided difference operators on the…