Abstract. The Balescu-Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau’s fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing ...

Given a commuting d-tuple T̄ = (T1, . . . , Td) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator DT̄ . Significant attributes of the d-tuple are best expressed in terms of DT̄ , including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T̄ derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimen...

MAI2TIN MATHILU This article summarizes a series of lectures clelivered at the Matherna.tics Departrnent of the University of Leipzig, Cerinany in April 1991, which were te overview teclrniques for solving operator ecluations en C*-algebras connected with methocis developed in a Spanish-Gerrnan research project en "Structure ancí Applications of C*-Algebras oí' Quotients" (SACA) . One of the re...

We provide a new analytical approach to operator splitting for equations of the type ut = Au + B(u) where A is a linear operator and B is quadratic. A particular example is the Korteweg–de Vries (KdV) equation ut−uux +uxxx = 0. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.

In the paper the equation of the form A+B = P , w h e r e A and B are binary aggregation operators, is investigated. A necessary and suu-cient condition for binary aggregation operators to be solutions of this equation, is proved. Already known solutions in some special classes of aggrega-tion operators, e.g., in the class of triangular norms and conorms, in the class of quasi{arithmetic means ...

Let H be a separable Hilbert space with an orthonormal basis {en/n ∈ N}, T be a bounded tridiagonal operator on H and Tn be its truncation on span ({e1, e2, . . . , en}). We study the operator equation T x = y through its finite dimensional truncations Tnx = yn. It is shown that if {‖T−1 n en‖} and {‖T ∗−1 n en‖} are bounded, then T is invertible and the solution of T x = y can be obtained as a...

Modeling of wave propagation in a heterogeneous medium requires input data that varies on many different spatial and temporal scales. Operator-based upscaling allows us to capture the effect of the fine scales on a coarser domain without solving the full fine-scale problem. The method applied to the constant density, variable sound velocity acoustic wave equation consists of two stages. First, ...

(Abstract) The ordinary Schrödinger equation with minimal coupling for a nonrelativistic electron interacting with a single-mode photon field is not satisfied by the nonrelativistic limit of the exact solutions to the corresponding Dirac equation. A Schrödinger-like equation valid for arbitrary photon intensity is derived from the Dirac equation without the weak-field assumption. The ”eigenvalu...

this paper deals with the boundary value problem involving the differential equationbegin{equation*}    ell y:=-y''+qy=lambda y, end{equation*} subject to the standard boundary conditions along with the following discontinuity  conditions at a point $ain (0,pi)$ begin{equation*}    y(a+0)=a_1 y(a-0),quad y'(a+0)=a_1^{-1}y'(a-0)+a_2 y(a-0),end{equation*}where $q(x),  a_1 , a_2$ are  real, $qin l...

Consider an operator equation F(u) = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. If F is monotone C2 loc(H) operator, then we construct a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends t...