Drazin-Star and Star-Drazin Inverses of Bounded Finite Potent Operators on Hilbert Spaces

The aim of this work is to extend to bounded finite potent endomorphisms on arbitrary Hilbert spaces the notions of the Drazin-Star and the Star-Drazin of matrices that have been recently introduced by D. Mosić. The existence, structure and main properties of these operators are given. In particular, we obtain new properties of the Drazin-Star and the Star-Drazin of a finite complex matrix. Moreover, the explicit solutions of some infinite linear systems on Hilbert spaces from the Drazin-Star inverse of a bounded finite potent endomorphism are studied.

Locally finite complexes, modules and generalized information systems

Simplicial complexes (here briefly complexes) are set systems on an arbitrary set which are object of study in many areas of both mathematics and theoretical computer science. Usually, they are investigated over finite sets. However, in general, when we consider an arbitrary set [Formula: see text] (not necessarily finite) and a complex [Formula: see text] on [Formula: see text], the most natural property related to finiteness is the following: for any subset [Formula: see text] of [Formula: see text], if [Formula: see text] for all finite subsets [Formula: see text] of [Formula: see text], then [Formula: see text]. We call locally finite any complex [Formula: see text] having such a property. Bearing in mind some motivations and constructions derived from the analysis of information systems in rough set theory, in this paper we associate with any locally finite complex [Formula: see text] a corresponding pre-closure operator [Formula: see text] and, through it, we establish several properties of [Formula: see text]. Next, we investigate the main features of the specific sub-class of locally finite complexes [Formula: see text] for which [Formula: see text] is a closure operator. We call these complexes closable and exhibit a particular family of closable locally finite complexes using left-modules on rings with identity. Finally, we establish a representation result according to which we can associate a pairing structure with any closable locally finite complex.

Some unity results on entire functions and their difference operators related to 4 CM theorem

This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let $f(z)$ f ( z ) be a nonconstant entire function of finite order, and let $a_{1}$ a 1 , $a_{2}$ a 2 be two distinct finite complex constants. If $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 “CM”, then $f(z)\equiv \Delta _{\eta }^{n} f(z)$ f ( z ) ≡ Δ η n f ( z ) , and hence $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 CM.

On the Regular Integral Solutions of a Generalized Bessel Differential Equation

The original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves. The solutions of the generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and degree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the whole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation. The regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions; these reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree.

Two remarks on Wall's D2 problem

If a finite group G is isomorphic to a subgroup of SO(3), then G has the D2-property. Let X be a finite complex satisfying Wall's D2-conditions. If π1(X) = G is finite, and χ(X) ≥ 1 − def(G), then X ∨ S2 is simple homotopy equivalent to a finite 2-complex, whose simple homotopy type depends only on G and χ(X).

Three-dimensional representations of braid groups associated with some finite complex reflection groups

We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in [Formula: see text]. In many cases, we show the rigidity. For rigid representations, we give explicit forms of the representations, which turns out to be the monodromy representations of uniformization equations of Saito–Kato–Sekiguchi [Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three, Kyushu J. Math. 68 (2014) 181–221; On the uniformization of complements of discriminant loci, RIMS Kokyuroku 287 (1977) 117–137]. Invariant Hermitian forms are also studied.

ON UNICITY OF MEROMORPHIC SOLUTIONS TO DIFFERENCE EQUATIONS OF MALMQUIST TYPE

In this note, we prove a uniqueness theorem for finite-order meromorphic solutions to a class of difference equations of Malmquist type. Such solutions $f$ are uniquely determined by their poles and the zeros of $f-e_{j}$ (counting multiplicities) for two finite complex numbers $e_{1}\neq e_{2}$.