McCoy property of Hurwitz series rings

Based on a theorem of McCoy on commutative rings, Nielsen called a ring [Formula: see text] right McCoy if for any nonzero polynomials [Formula: see text] over [Formula: see text], [Formula: see text] implies [Formula: see text] for some [Formula: see text]. In this note, we introduce and investigate McCoy and [Formula: see text]-properties of Hurwitz series ring [Formula: see text] and its Hurwitz polynomial subring [Formula: see text]. We show that when [Formula: see text] is a reversible or duo ring and [Formula: see text] then the Hurwitz polynomial ring [Formula: see text] is McCoy.

Composite Hurwitz Rings as PF-Rings and PP-Rings

Let R ⊆ T be an extension of commutative rings with identity and H ( R , T ) (respectively, h ( R , T ) ) the composite Hurwitz series ring (respectively, composite Hurwitz polynomial ring). In this article, we study equivalent conditions for the rings H ( R , T ) and h ( R , T ) to be PF-rings and PP-rings. We also give some examples of PP-rings and PF-rings via the rings H ( R , T ) and h ( R , T ) .

Singular ideals of skew Hurwitz polynomial rings

For a ring R and an endomorphism $$\alpha $$ α of R, we provide a full description of left and right singular ideals of the skew Hurwitz polynomial ring $$(hR,\alpha )$$ ( h R , α ) . We obtain that if $$\alpha $$ α is an automorphism of R, then R is right (resp., left) nonsingular if and only if $$ (hR,\alpha ) $$ ( h R , α ) is right (resp., left) nonsingular. We give an example of a ring R and an endomorphism $$\alpha $$ α of R such that the skew Hurwitz polynomial ring $$(hR,\alpha )$$ ( h R , α ) is left nonsingular, but not right nonsingular.

Chain conditions on composite Hurwitz series rings

In this paper, we study chain conditions on composite Hurwitz series rings and composite Hurwitz polynomial rings. More precisely, we characterize when composite Hurwitz series rings and composite Hurwitz polynomial rings are Noetherian, S-Noetherian or satisfy the ascending chain condition on principal ideals.

Stability and Multiscroll Attractors of Control Systems via the Abscissa

We present an approach to generate multiscroll attractors via destabilization of piecewise linear systems based on Hurwitz matrix in this paper. First we present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, we consider Hurwitz polynomials. The starting point is the Gauss–Lucas theorem, we provide lower bounds for Hurwitz polynomials, and by successively decreasing the order of the derivative of the Hurwitz polynomial one obtains a sequence of lower bounds. The results are extended in a straightforward way to interval polynomials; then we apply the abscissa as a measure to destabilize Hurwitz polynomial for the generation of a family of multiscroll attractors based on a class of unstable dissipative systems (UDS) of affine linear type.

Properties of the Set of Hadamardized Hurwitz Polynomials

We say that a Hurwitz polynomialptis a Hadamardized polynomial if there are two Hurwitz polynomialsftandgtsuch thatf∗g=p, wheref∗gis the Hadamard product offandg. In this paper, we prove that the set of all Hadamardized Hurwitz polynomials is an open, unbounded, nonconvex, and arc-connected set. Furthermore, we give a result so that a fourth-degree Hurwitz interval polynomial is a Hadamardized polynomial family and we discuss an approach of differential topology in the study of the set of Hadamardized Hurwitz polynomials.

A Test for the Stability of Networks

Summary A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].