A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

This paper investigates the solvability, existence and uniqueness of solutions for a class of nonlinear fractional hybrid differential equations with Hilfer fractional derivative in a weighted normed space. The main result is proved by means of a fixed point theorem due to Dhage. An example to illustrate the results is included.

In this paper a certain class of polyhedral semigroups which has a presentation $$ is examined‎. ‎The completeness of the presentation and solvability of word problem of this class of semigroups is determined‎. ‎Moreover the combinatorial distance between two words is determined‎.

This paper is devoted to the proof of the unique solvability ofthe inverse problems for second-order differential operators withregular singularities. It is shown that the potential functioncan be determined from spectral data, also we prove a uniquenesstheorem in the inverse problem.

We consider the question of which loops are capable of expressing arbitrary Boolean functions through expressions of constants and variables. We call this property Boolean completeness. It is a generalization of functional completeness, and is intimately connected to the computational complexity of various questions about expressions, circuits, and equations defined over the loop. We say that a...

Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have shown very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus important instruments for studying limits or boundaries of solvability and unsolvability. Although ...

In this paper the output regulation of a linear distributed parameter system with a nonautonomous periodic exosystem is considered. It is shown that the solvability of the output regulation problem can be characterized by the solvability of a certain constrained infinite-dimensional Sylvester differential equation. Conditions are given for the existence of feedforward and feedback controllers s...

The paper is devoted to general elliptic problems in the Douglis-Nirenberg sense. We obtain a necessary and sufficient condition of normal solvability in the case of unbounded domains. Along with the ellipticity condition, proper ellipticity and Lopatinsky condition that determine normal solvability of elliptic problems in bounded domains, one more condition formulated in terms of limiting prob...

In this paper, finite and infinite horizon stochastic H2/H∞ control problems with all state, control input and external disturbance-dependent noise are studied. It is shown that the existence condition of finite horizon stochastic H2/H∞ control is equivalent to the solvability of four coupled matrix-valued differential equations, and that of infinite horizon stochastic H2/H∞ control is equivale...

We establish conditions that guarantee Fredholm solvability in the Banach space Lp of nonlocal boundary value problems for elliptic abstract differential equations of the second order in an interval. Moreover, in the space L2 we prove in addition the coercive solvability, and the completeness of root functions (eigenfunctions and associated functions). The obtained results are then applied to t...