Related papers: Separation of variables via integral transformatio…
A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a…
The distribution of the non-trivial zeroes of the Riemann zeta function, according to the Riemann hypothesis, is tantalisingly similar to the zeroes of the partition functions (Fisher and Yang-Lee zeroes) of statistical mechanical models…
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the…
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with non integer exponents. We apply the method to equations resembling generalizations…
In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…
This article gives a fundamental discussion on variable coefficients, self-adjoint, formally partially hypoelliptic differential operators. A generalization of the results to pseudo differential operators, is given in a following article in…
The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme}…
We present a method for calculating the results of operation of differential operators operating on components of vector in generalized coordinates not restricted to orthogonal one. For this we use the relationships between covariant,…
The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special…
We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The…
In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a…
We introduce a symmetric operad whose algebras are the Operator Product Expansion (OPE) Algebras of quantum fields. There is a natural classical limit for the algebras over this operad and they are commutative associative algebras with…
Solving partial differential equations (PDEs) efficiently is essential for analyzing complex physical systems. Recent advancements in leveraging deep learning for solving PDE have shown significant promise. However, machine learning…
A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit…
Additive processes are obtained from L\'{e}vy ones by relaxing the condition of stationary increments, hence they are spatially (but not temporally) homogeneous. By analogy with the case of time-homogeneous Markov processes, one can define…
Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high-fidelity data from numerical solvers, limiting applications in complex systems. We…
An explicit formula for the wave operators associated with Schroedinger operators on the discrete half-line is deduced from their stationary expressions. The formula enables us to understand the wave operators as one dimensional…
Consider the Plancherel decomposition of the tensor product of a highest weight and a lowest weight unitary representations of $SL_2$. We construct explicitly the action of the Lie algebra $sl_2 + sl_2$ in the direct integral of Hilbert…
For a quasi-split Satake diagram, we define a modified $q$-Weyl algebra, and show that there is an algebra homomorphism between it and the corresponding $\imath$quantum group. In other words, we provide a differential operator approach to…
There being no precise definition of the quantum integrability, the separability of variables can serve as its practical substitute. For any quantum integrable model generated by the Yangian Y[sl(3)] the canonical coordinates and the…