Related papers: Structure theorems for linear and non-linear diffe…
We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants…
Series of finite dimensional representations of the superalgebras spl(p,q) can be formulated in terms of linear differential operators acting on a suitable space of polynomials. We sketch the general ingredients necessary to construct these…
We study realizations of polynomial deformations of the sl(2,R)- Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite- and infinite-dimensional representations. The linear,…
This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision…
A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear operator, and then show that each linear operator on a finite dimensional nonzero real…
Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This…
We consider the application of Koopman theory to nonlinear partial differential equations. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear…
We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.
We apply Rossi's half-plane version of Borel's Theorem to study the zero distribution of linear combinations of $\mathcal{A}$-entire functions (Theorem 1.2). This provides a unified way to study linear $q$-difference, difference and…
Let us denote ${\cal V}$, the finite dimensional vector spaces of functions of the form $\psi(x) = p_n(x) + f(x) p_m(x)$ where $p_n(x)$ and $p_m(x)$ are arbitrary polynomials of degree at most $n$ and $m$ in the variable $x$ while $f(x)$…
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the…
Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional…
In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and…
The notion of lacunary infinite numerical sequence is introduced. It is shown that for an arbitrary linear difference operator L with coefficients belonging to the set R of infinite numerical sequences, a criterion (i.e., a necessary and…
Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in…
An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with…
Although the Hamiltonian in quantum physics has to be a linear operator, it is possible to make quantum systems behave as if their Hamiltonians contained antilinear (i.e., semilinear or conjugate-linear) terms. For any given quantum system,…
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…
We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical…