The depth of continued fraction expansion for some classes of rational functions

For nonzero polynomials [Formula: see text] and [Formula: see text] over a field [Formula: see text], let [Formula: see text] be the depth (length) of the continued fraction expansion for [Formula: see text]. An upper bound on [Formula: see text], for nonzero polynomial [Formula: see text] and rational function [Formula: see text] is obtained. Applying this result, an upper bound on the depth of a linear fractional transformation is also established.

Everywhere vanishing polynomial functions

If [Formula: see text] is a finite commutative ring, it is well known that there exists a nonzero polynomial in [Formula: see text] which is satisfied by every element of [Formula: see text]. In this paper, we classify all commutative rings [Formula: see text] such that every element of [Formula: see text] satisfies a particular monic polynomial. If the polynomial, satisfied by the elements of [Formula: see text], is not required to be monic, then we can give a classification only for Noetherian rings, giving examples to show that the characterization does not extend to arbitrary commutative rings.

Remarks on the Number of Rational Points on a Class of Hypersurfaces over Finite Fields

Let 𝔽q be the finite field of q elements and f be a nonzero polynomial over 𝔽q. For each b ϵ 𝔽q, let Nq(f = b) denote the number of 𝔽q-rational points on the affine hypersurface f = b. We obtain the formula of Nq(f = b) for a class of hypersurfaces over 𝔽q by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.

Meromorphic functions whose certain differential polynomial share a polynomial

In this paper, we use the idea of normal family to investigate the uniquenessproblems of meromorphic functions when certain non-linear dierential polynomial sharinga nonzero polynomial with certain degree. We obtain some results which will not only rectifythe recent results of P. Sahoo and H. Karmakar [9] but also improve and generalize somerecent results of L. Liu [7], H. Y. Xu, T. B. Cao and S. Liu [12] and P. Sahoo and H.Karmakar [9] in a large extent.

On a conjecture of Y. H. Cao and X. B. Zhang

In this paper we investigate the possible relation between two meromorphic functions f n f (k) and g n g ( k ) that share a nonzero polynomial and obtain two results which are related to a conjecture of X. Y. Cao and B. X. Zhang [Uniqueness of meromorphic functions sharing two values, J. Ineq. Appl., 1(100) (2012)]

On the uniqueness of certain type of differential-difference polynomials sharing a polynomial

The purpose of the paper is to study uniqueness problems of certain types of differential-difference polynomials sharing a nonzero polynomial of certain degree under relaxed sharing hypotheses. We not only point out some gaps in the proof of the main results in [

On Modelling AM/AM and AM/PM Conversions via Volterra Series

In this paper, we present the expressions, not published up to now, that describe the AM/AM and AM/PM conversions of communication power amplifiers (PAs) via the Volterra series based nonlinear transfer functions. Furthermore, we present a necessary and sufficient condition of occurrence of the nonzero values of AM/PM conversion in PAs. Moreover, it has been shown that Saleh’s approach and related ones, which foresee nonzero level of AM/PM conversion, are not models without memory. It has been also shown that using a polynomial description of a PA does not lead to a nonzero AM/PM conversion. Moreover, a necessary condition of occurrence of an AM/AM conversion in this kind of modelling is existence of at least one nonzero polynomial coefficient associated with its odd terms of degree greater than one.

On an Open Problem of Xiao-Bin Zhang and Jun-Feng Xu

The purpose of the paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial. The results of the paper improve and generalize the recent results due to X. B. Zhang and J. F. Xu [19]. We also solve an open problem as posed in the last section of [19].