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The zeros of polynomials whose coefficients exhibit a +/-/+ sign pattern are sometimes excluded from sectors in the complex plane. It is shown how to obtain such sectors, as well as the number of zeros that lie between the sectors, along with bounds on their magnitudes. The results are illustrated with an application to polynomial eigenvalue problems. [ABSTRACT FROM AUTHOR]

A well-known theorem due to Ankeny and Rivlin states that if p(z) is a polynomial of degree n such that p(z) has no zero in | z | < 1 , then max | z | = R ≥ 1 | p (z) | ≤ R n + 1 2 max | z | = 1 | p (z) |. This research investigates the polynomial p(z) ensuring it possesses no zero in | z | < k where k ≥ 1 . Simultaneously, we explore the s th derivative, where 0 ≤ s < n , of this polynomial. In an attempt to obtain an integral setting of the inequalities concerning derivatives of this class of polynomials p(z), we have been able to establish extensions and generalizations of the above Ankeny and Rivlin's inequality to integral analogs. As a consequence of our result, we obtained an improvement of the result due to Jain [Turk. J. Math., 31 (2007), 89 - 94 ] which we have also compared by considering an example. Further, it is worth to note that we have compared our results by means of this numerical example where the bounds that are in terms of integral means are estimated numerically by numerical integration using Simpson's 1 3 rd rule and illustrate graphically the obtained inequalities as regards sharpness. [ABSTRACT FROM AUTHOR]

Let τ be a topology on the finite set X n . We consider the open-set polynomial associated with the topology τ. Its coefficients are the cardinalities of sets U j = U j (τ) of open sets of size j = 0 , ... , n. We prove that this polynomial has only real zeros only in the trivial case where τ is the discrete topology. Hence, we answer a question raised by J. Brown. We give a partial answer to the question: for which topology is this polynomial log-concave, or at least unimodal? More specifically, we prove that if the topology has a large number of open sets, its open polynomial is unimodal. The idea of degree of log-concavity is introduced and it is shown to be limited for polynomials of non-trivial topologies. Furthermore, the maximum-sized topologies that omit open sets of given sizes are derived. Moreover, all topologies over n points with at least (3 / 8) 2 n open sets are proved to be unimodal, completing previous results. [ABSTRACT FROM AUTHOR]

Background/Objectives: The development of dental arches is a complex adaptive system with interactions between genetic and environmental factors. At different developmental stages, the relative contribution of these factors varies. The aims of this project were to identify the longitudinal changes of dental arches in the primary, mixed and permanent dentition stages, using curve fitting methods on serial dental casts, and to investigate the contribution of the genotype to dental arch development. Methods: Longitudinal dental records from 125 monozygotic same-sex twin pairs, 89 dizygotic same-sex twin pairs, and 49 opposite-sex dizygotic twin pairs were used. Standardized model photographs were collected, and key landmarks were digitized. Fourth-order orthogonal polynomials were applied to the Cartesian data. Descriptive statistics were calculated, and structural equation models were developed to analyze the individual polynomial coefficients. The final models employed a genetic simplex framework, enabling the evaluation of how genetic and environmental influences changed over time. These changes were examined both quantitatively (e.g., variations in heritability) and qualitatively (e.g., the influence of different genes at various stages). Results: In the primary dentition, arches were typically parabolic, while in the permanent dentition, they tended to be more square-shaped. Asymmetry made a minor contribution to variation across all stages of development. Genetic analysis revealed that a core group of genes influenced arch shape over time, though their impact varied. Additionally, some genes were specific to certain developmental stages, with their relative contributions differing significantly. Notably, there was evidence of sexual heterogeneity in arch shape, particularly in the permanent dentition. Heritability was consistently high, both at individual developmental stages and throughout the overall developmental process. Conclusions: The degree of genetic influence at each developmental stage was substantial but it fluctuated between the primary, mixed, and permanent dentition stages. [ABSTRACT FROM AUTHOR]

In this study, a broader class of rational functions r(u(z)) of degree mn, where u(z) is a polynomial of degree m is taken into consideration and obtain certain sharp compact generalization of well-known inequalities for rational functions. [ABSTRACT FROM AUTHOR]

Different public key exchange protocols can be employed to design a cryptosystem. The most famous example is the ElGamal cryptosystem which is based on the Diffie–Hellman key establishment to do encryption and decryption. In the same fashion, we propose a public key cryptosystem using a public key exchange protocol. This variant uses the polynomials over noncommutative matrix ring as underlying work structure. The useful feature of the proposed cryptosystem is that it provides favorable security because of use of the generalized decomposition problem over the noncommutative ring of matrices. The issues regarding the choice of parameters and platform are discussed. Further, a brief note on security analysis of the proposed cryptosystem is also presented. [ABSTRACT FROM AUTHOR]

This paper mainly studies the well-known Turán type inequalities that relate the uniform norm of a polynomial to its derivative on the unit disk in the plane. We derive some new inequalities that relate the uniform norm of the polar derivative to the underlying polynomial when the zeros are within a closed disk. The obtained results produce various inequalities that are sharper than those available in a very rich literature on the subject. [ABSTRACT FROM AUTHOR]

In this paper, we present for a polynomial p(z) of degree n, the sth derivative (0 ≤ s < n) concept on a result due to Govil et al. [Illinois J. Math., 23 (1979), 319--329]. As an application of this result, we obtain improved generalizations of the well-known theorem due to Ankeny and Rivlin which states that if p(z) is a polynomial of degree n such that p(z) has no zero in [z] < 1, then ... Moreover, these achievements lead to enhancements of a previous result attributed to Jain [Turk. J. Math., 31 (2007), 89--94] which we have compared by considering a concrete numerical example and analyzed graphically to illustrate their sharpness. [ABSTRACT FROM AUTHOR]

Hayes equivalence is defined on monic polynomials over a finite field F q in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial Q. We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes equivalence class. It is well known that the number of distinct zeros of a random polynomial over F q is asymptotically Poisson with mean 1. We show that this is also true for random polynomials in any given Hayes equivalence class. Asymptotic formulas are also given for the number of such polynomials when the degree of such polynomials is proportional to q and the degree of Q and the number of prescribed leading coefficients are bounded by q. When Q = 1 , the problem is equivalent to the study of the distance distribution in Reed-Solomon codes. Our asymptotic formulas extend some earlier results and imply that all words for a large family of Reed-Solomon codes are ordinary, which further supports the well-known Deep-Hole Conjecture. [ABSTRACT FROM AUTHOR]

R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials over F q 3 in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed permutation polynomials and provide their compositional inverses. [ABSTRACT FROM AUTHOR]