Related papers: Maxwell operator in a cylinder. Separation of vari…
Under certain assumptions (including $d\ge 2)$ we prove that the spectrum of a scalar operator in $\mathscr{L}^2(\mathbb{R}^d)$ \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} covers interval…
We investigate a linear operator associated with a functional equation that arises from studying some class of invariant measures under multidimensional transformations. By examining its iterates, we derive an explicit solution formula for…
In this paper, the linear differential expression of order $n \ge 2$ with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we…
We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces.…
In this work, we analyze the combinatorial properties of cylinders and subdivisions of augmented semi-simplicial sets. These constructions are obtained as particular cases of a certain action from a co-semi-simplicial set on an augmented…
This article deals with the study of electromagnetic waves equations and the Lorentz condition, as emergent properties of Maxwell's system in the context of systems theory. To do this, the wave equations and the Helmholtz equation are first…
In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three…
Maxwell equation in geometric algebra formalism with equally weighted basic solutions is subjected to continuously acting Clifford translation. The received states, operators acting on observables, are analyzed with different values of the…
We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k…
In this paper we prove that a class of non self-adjoint second order differential operators acting in cylinders $\Omega\times\mathbb R\subseteq\mathbb R^{d+1}$ have only real discrete spectrum located to the right of the right most point of…
Closed form expressions for a logarithm of general multivector (MV) in base-free form in real geometric algebras (GAs) Cl(p,q) are presented for all n=p+q=3. In contrast to logarithm of complex numbers (isomorphic to Cl(0,1), 3D logarithmic…
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…
For a discrete dynamical system on $\R$ generated by a quadratic function, we show, using elementary computations, that the existence, number, and stability of 3-cycles are determined by a single parameter depending on the coefficients of…
Spectrum of the volume integral operator of the three-dimensional electromagnetic scattering is analyzed. The operator has both continuous essential spectrum, which dominates at lower frequencies, and discrete eigenvalues, which spread out…
We begin by reviewing the derivation of generalized Maxwell equations from an operational definition of the electromagnetic field and the most basic notions of what constitutes a dynamical field theory. These equations encompass the…
We consider a polyharmonic operator $H=(-\Delta)^l+V(\x)$ in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential $V(\x)$. We prove that the spectrum of $H$ contains a semiaxis and there is a family of…
Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We…
We consider the maximal operator with respect to uncentered cubes on Euclidean space with arbitrary dimension. We prove that for any function with bounded variation, the variation of its maximal function is bounded by the variation of the…
Let X be a finite set of complex numbers and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e_1,e_2, ... for H, A gives rise to a matrix whose diagonal is a…
We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to…