Numerator Polynomials Research Articles

On Anderson-Badawi conjectures

Let R be a commutative ring with identity. Badawi (Bull Aust Math Soc 75(3), 417–429, 2007) introduced a generalization of prime ideals called 2-absorbing ideals, and this idea is further generalized in a paper by Anderson and Badawi (Commun Algebra 39(5), 1646–1672, 2011) to a concept called n-absorbing ideals. A proper ideal I of R is said to be an n-absorbing ideal if whenever x_1ldots x_{n+1}in I for x_1,ldots ,x_{n+1}in R then there are n of the x_i’s whose product is in I. It was conjectured by Anderson and Badawi (Commun Algebra 39(5), 1646–1672, 2011) that if I is an n-absorbing ideal of R then I is strongly n-absorbing (Conjecture 1) and Rad(I)^nsubseteq I (Conjecture 2). In Cahen et al. (in: Fontana et al., Commutative rings. Integer-valued polynomials, and polynomial function, Springer, New York, 2014, Problem 30c), it was conjectured also that I[X] is an n-absorbing ideal of the polynomial ring R[X] for each n-absorbing ideal of the ring R (Conjecture 3). In this paper we give an answer to (Conjecture 2) for n=3, n=4 and n=5 and we prove that (Conjecture 1) and (Conjecture 3) hold in various classes of rings.

ON THE DIVISOR-CLASS GROUP OF MONADIC SUBMONOIDS OF RINGS OF INTEGER-VALUED POLYNOMIALS

Let $R$ be a factorial domain. In this work we investigate the connections between the arithmetic of ${\rm Int}(R)$ (i.e., the ring of integer-valued polynomials over $R$) and its monadic submonoids (i.e., monoids of the form $\{g\in {\rm Int}(R)\mid g\mid_{{\rm Int}(R)} f^k$ for some $k\in\mathbb{N}_0\}$ for some nonzero $f\in {\rm Int}(R)$). Since every monadic submonoid of ${\rm Int}(R)$ is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between ${\rm Int}(R)$ and its monadic submonoids. If $R=\mathbb{Z}$ or more generally if $R$ has sufficiently many nice atoms, then we prove that the infinitude of the elasticity and the tame degree of ${\rm Int}(R)$ can be explained by using the structure of monadic submonoids of ${\rm Int}(R)$.

Pole–Zero Assignment of All-Pass-Based Notch Filters

In this brief, a new technique for designing the all-pass-based notch filter is proposed. An all-pass-based notch filter consists of an all-pass filter, which has the mirror-image symmetry relation between the numerator and denominator polynomials. With this property, a relationship between the pole angel and the zero angle of an all-pass-based notch filter is derived. Knowing this relationship, we can design an all-pass-based notch filter by assigning the pole position according to the zero position. The process of the new technique is illustrated in this brief. Moreover, experiments are presented to examine a track of the pole position at the specific frequency and to demonstrate a design example of an all-pass-based notch filter.

Extraction of orthogonal polynomials from generating function for reciprocal of odd numbers

In the present paper the orthogonality relations, exhibited by both numerator and denominator polynomials of both even and odd order convergents of a regular C-fraction of a power series with coefficients as reciprocal of odd numbers are described. The two sequences of convergents are nothing but diagonal and upper diagonal Pade approximants for the power series. The two orthogonal polynomials extracted from denominators are shown to be classical orthogonal polynomials and two orthogonal polynomials extracted from numerators are shown to be non-classical orthogonal polynomials..

Pólya groups of the imaginary bicyclic biquadratic number fields

Let OK and CK be respectively the ring of integers and the class group of a number field K. For each integer q≥2, denote by ∏q(K) the product of all the maximal ideals of OK with norm q, if these ideals do not exist we set ∏q(K)=OK. The Pólya group of K is the subgroup of CK generated by the classes of the ideals ∏q(K), and K is called a Pólya field if the module of integer-valued polynomials over OK has a regular basis. In this paper, we determine Pólya group of any imaginary bicyclic biquadratic number field, and thus we deduce all the imaginary bicyclic biquadratic Pólya fields.

On the Real-Rootedness of the Veronese Construction for Rational Formal Power Series

We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the numerator polynomial of the subsequence $\{ a_{rn+i}\}_{n\in \mathbb{N}}$, $0\leq i<r$, has only nonpositive, real roots for all $r\geq s-i$. We apply our results to combinatorially positive valuations on polytopes and to Hilbert functions of Veronese submodules of graded Cohen-Macaulay algebras. In particular, we prove that the Ehrhart $h^\ast$-polynomial of the $r$-th dilate of a $d$-dimensional polytope has only distinct, negative, real roots if $r\geq \min \{s+1,d\}$. This proves a conjecture of Beck and Stapledon (2010).

Polynomial chaos‐based extended Padé expansion in structural dynamics

SummaryThe response of a random dynamical system is totally characterized by its probability density function (pdf). However, determining a pdf by a direct approach requires a high numerical cost; similarly, surrogate models such as direct polynomial chaos expansions are not generally efficient, especially around the eigenfrequencies of the dynamical system. In the present study, a new approach based on Padé approximants to obtain moments and pdf of the dynamic response in the frequency domain is proposed. A key difference between the direct polynomial chaos representation and the Padé representation is that the Padé approach has polynomials in both numerator and denominator. For frequency response functions, the denominator plays a vital role as it contains the information related to resonance frequencies, which are uncertain. A Galerkin approach in conjunction with polynomial chaos is proposed for the Padé approximation. Another physics‐based approach, utilizing polynomial chaos expansions of the random eigenmodes, is proposed and compared with the proposed Padé approach. It is shown that both methods give accurate results even if a very low degree of the polynomial expansion is used. The methods are demonstrated for two degree‐of‐freedom system with one and two uncertain parameters. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Numerical constructions involving Chebyshev polynomials

We propose a new algorithm for the character expansion of tensor products of finite-dimensional irreducible representations of simple Lie algebras. The algorithm produces valid results for the algebras B3, C3, and D3. We use the direct correspondence between Weyl anti-invariant functions and multivariate second-kind Chebyshev polynomials. We construct the triangular trigonometric polynomials for the algebra D3.

Novel arrangement of Routh array for order reduction of z-domain uncertain system

ABSTRACTThis paper presents a novel arrangement of Routh table array for deriving an approximate model of a higher order z-domain uncertain system. The demand for this computation is to procure a lower order model which is easy to be exercised in comparison to their original large scale systems. Additionally, the derived model should preserve fewer dynamic characteristic of the comprehensive higher order systems. The mentioned new arrangement is achieved from the arena of different combinations of numerator and denominator polynomials. The combinations are validated by their practice over the conventional example from the literature. This precise blend is then applied to a real-time system for its rational acceptability. Both the models play a significant role in establishing the algorithm. Besides this, the limitation encountered during the foundation course of the arrangement is also taken into consideration. The paper also offers a future scope for fellow researchers.

Imaginary biquadratic Polya fields of the form Q(sqrt(d), sqrt(-2))

A number field k, with ring of integers Ok, is called a Polya field if the module of integer-valued polynomials over Ok has a regular basis. In this paper, we characterize all imaginary biquadratic P\olya fields of the form ℚ(√d, √-2) where d is a square-free integer.

Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.

Model order reduction using eigen algorithm

An order reduction method has been proposed for reducing the order of the large-scale dynamic systems where denominator polynomial determined through Eigen algorithm and numerator polynomial via factor division algorithm. In Eigen algorithm, the most dominant Eigen value of both original and reduced order systems remain same. The proposed mixed method confirm stability of the reduced model if the original system is stable and has been also compared in quality with other existing model order reduction methods.Keywords: Eigen algorithm; Order reduction; Factor division algorithm; Stability; Transfer function

Some results on the integer-valued polynomials over matrix rings

ABSTRACTIn this paper, we study the ring of integer-valued polynomials over matrix rings where D is an integral domain with the field of fractions K. We introduce equalizing ideal of Mn(𝔞) in Int(Mn(D)) where 𝔞 is an ideal of D. We show that, if is finite then the set of distinct ideals of the form of Int(Mn(D)) is finite. Also, we prove that . Using this inclusion we obtain that, if D is a Noetherian local one-dimensional domain with finite residue field, which is not unibranched, then the set of ideals of the form is finite, where 𝔪 is the maximal ideal of D. We present some properties of maximal ideals of Int(Mn(D)). Also, we generalize the Skolem closure of an ideal of Int(Mn(D)). Among the other results, we present some relations between prime ideals of Mn(D) and prime ideals of Int(Mn(D)). Finally, we state a lower bound of Krull dimension of integer-valued polynomials ring Int(Mn(D)).

Quantum integer-valued polynomials

We define a $q$-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity properties: for instance, the structure constants for this ring with respect to its basis of $q$-binomial coefficient polynomials belong to $\mathbb{N}[q]$. We then classify all maps from this ring into a field, extending a known classification in the classical case where $q=1$.

Positive existential definability of multiplication from addition and the range of a polynomial

We consider the problem of recovering multiplication in the integers from enrichments of its additive structure, in the positive existential context. We prove that if a conjecture by Caporaso–Harris–Mazur holds, then for all integer-valued polynomials F of degree at least 2, multiplication is positive-existentially definable in (Z; 0, 1,+, RF, =) where RF is the unary relation F(Z). Similar results were only known for the polynomials F(t) = t2 (under the Bombieri–Lang conjecture) and F(t) = tn (under a generalization of the abc conjecture).

Numerical polynomial homotopy continuation method to locate all the power flow solutions

The manuscript addresses the problem of finding all solutions of power flow equations or other similar non-linear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly the direct methods for transient stability analysis and voltage stability assessment. Here, the authors introduce a novel form of homotopy continuation method called the numerical polynomial homotopy continuation method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. Since finding real solutions is much more challenging, first the authors embed the real form of power flow equation in complex space, and then track the generally unphysical solutions with complex values of real and imaginary parts of the voltages. The solutions converge to physical real form in the end of the homotopy. The so-called gamma-trick mathematically rigorously ensures that all the paths are well-behaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelisable. The authors demonstrate the technique performance by solving several test cases up to the 14 buses. Finally, they discuss possible strategies for scaling the method to large size systems, and propose several applications for security assessments.

Impulse energy approximation of higher-order interval systems using Kharitonov’s polynomials

This research work proposes a method to obtain a stable reduced-order interval model from its stable higher-order interval plant. The reduced-order interval numerator and denominator polynomials are determined by using Kharitonov’s polynomials and a general form of the Routh approximation method. The proposed reduction algorithm retains stability and full impulse response energy of the higher-order interval system in its reduced-order interval model. In addition to this, the proposed method has useful features like matching of time moments and mathematical simplicity. Moreover, a few numerical examples in the literature are taken into consideration and simulated through MATLAB to illustrate the effectiveness of the proposed method.

Non-triviality conditions for integer-valued polynomial rings on algebras

Let $D$ be a commutative domain with field of fractions $K$ and let $A$ be a torsion-free $D$-algebra such that $A \cap K = D$. The ring of integer-valued polynomials on $A$ with coefficients in $K$ is ${\rm Int}_K(A) = \{f \in K[X] \mid f(A) \subseteq A\}$, which generalizes the classic ring ${\rm Int}(D) = \{f \in K[X] \mid f(D) \subseteq D\}$ of integer-valued polynomials on $D$. The condition on $A \cap K$ implies that $D[X] \subseteq {\rm Int}_K(A) \subseteq {\rm Int}(D)$, and we say that ${\rm Int}_K(A)$ is nontrivial if ${\rm Int}_K(A) \ne D[X]$. For any integral domain $D$, we prove that if $A$ is finitely generated as a $D$-module, then ${\rm Int}_K(A)$ is nontrivial if and only if ${\rm Int}(D)$ is nontrivial. When $A$ is not necessarily finitely generated but $D$ is Dedekind, we provide necessary and sufficient conditions for ${\rm Int}_K(A)$ to be nontrivial. These conditions also allow us to prove that, for $D$ Dedekind, the domain ${\rm Int}_K(A)$ has Krull dimension 2.

Multi-swarm bat algorithm for association rule mining using multiple cooperative strategies

Association Rule Mining (ARM) can be considered as a combinatorial problem with the purpose of extracting the correlations between items in sizeable datasets. The numerous polynomial exact algorithms already proposed for ARM are unadapted for large databases and especially for those existing on the web. Assuming that datasets are a large space search, intelligent algorithms was used to found high quality rules and solve ARM issue. This paper deals with a cooperative multi-swarm bat algorithm for association rule mining. It is based on the bat-inspired algorithm adapted to rule discovering problem (BAT-ARM). This latter suffers from absence of communication between bats in the population which lessen the exploration of search space. However, it has a powerful rule generation process which leads to perfect local search. Therefore, to maintain a good trade-off between diversification and intensification, in our proposed approach, we introduce cooperative strategies between the swarms that already proved their efficiency in multi-swarm optimization algorithm(Ring, Master-slave). Furthermore, we innovate a new topology called Hybrid that merges Ring strategy with Master-slave plan previously developed in our earlier work [23]. A series of experiments are carried out on nine well known datasets in ARM field and the performance of proposed approach are evaluated and compared with those of other recently published methods. The results show a clear superiority of our proposal against its similar approaches in terms of time and rule quality. The analysis also shows a competitive outcomes in terms of quality in-face-of multi-objective optimization methods.

Proof of Sun's conjectures on integer-valued polynomials

Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to prove that certain rational sums involving even powers of those polynomials are integers whenever they are evaluated at integers. This confirms two conjectures of Z.-W. Sun. We also conjecture that many of these results have neat q-analogues.