We consider methods of reduction of differential operator equations with the Fredholm operator in the main expression to regular problems. Relation between the initial conditions choice problem and the Jordan structure of operator coefficients of equations is shown. The theorem of existence and uniqueness of the initial problem is proved. The method of fundamental operators is used for construc...

Keywords: Approximate solutions to integral equations Radial and kernel-based networks Gaussian kernels Model complexity Analysis of algorithms a b s t r a c t Approximate solutions to inhomogeneous Fredholm integral equations of the second kind by radial and kernel networks are investigated. Upper bounds are derived on errors in approximation of solutions of these equations by networks with in...

It is recently shown that a split-step quantum walk possesses chiral symmetry, and certain well-defined index can be naturally assigned to it. The Fredholm if only the associated unitary time-evolution operator has spectral gaps at both $+1$ $-1.$ In this paper we extend existing formula for case encompass non-Fredholm (i.e., gapless case). We make use of natural extension case, known as Witten...

‎in this paper we try to extend geometric concepts in the context of operator valued tensors‎. ‎to this end‎, ‎we aim to replace the field of scalars $ mathbb{r} $ by self-adjoint elements of a commutative $ c^star $-algebra‎, ‎and reach an appropriate generalization of geometrical concepts on manifolds‎. ‎first‎, ‎we put forward the concept of operator-valued tensors and extend semi-riemannian...

Schwinger terms of current algebra can be identiied with nontrivial cyclic cocycles of a Fredholm module. We discuss its temperature dependence. Similar anomalies may occur also in spin systems. In simple examples already an operator{valued cocycle shows up.

“Weyl’s theorem holds” for an operator T on a Banach space X when the complement in the spectrum of the “Weyl spectrum” coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the “Browder spectrum”, which in turn ought to, but does not, guarantee the spectral mapping theorem...