Numerator Polynomials Research Articles

On rings of integer-valued rational functions

Let D ⊆ B be an extension of integral domains and E a subset of the quotient field of D. We introduce the ring of D -valued B-rational functions on E, denoted by Int B R ( E , D ) , which naturally extends the concept of integer-valued polynomials, defined as Int B R ( E , D ) : = { f ∈ B ( X ) ; f ( E ) ⊆ D } . The notion of Int B R ( E , D ) boils down to the usual notion of integer-valued rational functions when the subset E is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from Int R ( E , D ) to D.

Limiting Properties of Linear Control Systems, Synthesis by LQR Methods According to Quadratic Criteria with One Variable Coefficient

The work provides a comparative analysis of the properties of optimal control systems for linear stationary single-channel objects, the distinctive feature of which is that the numerator polynomial of the object’s transfer function is Hurwitz. Systems are synthesized by two main methods of the theory of analytical design of optimal regulators (ADOR) — methods the Letov-Kalman and A. A. Krasovsky, which use quality functionals based on an integral criterion containing only two terms: the square of the object’s control signal and the square of its output coordinate with a weighting coefficient q . The limiting properties of the synthesized systems are studied at q q → ∞ . The common known property of these systems, for brevity called the Letov-Kalman and Krasovsky systems, is the property of their stability at the limiting value of the weight coefficient and, accordingly, an infinitely large increase in the total gain of these systems, which for them ensures obtaining a given value static control error. Other analyzed properties of the systems turned out to be significantly different and even mutually opposite. For example, at limiting values q q → ∞ , the coefficients of the optimal Letov-Kalman controller do not depend on the parameters of the object, while the coefficients of the Krasovsky controller, on the contrary, are determined exclusively by the parameters of the object. We also note that, unlike Letov-Kalman control systems, the time of transient processes of Krasovsky systems at q q → ∞ cannot be reduced less than a certain finite value, despite the absence of restrictions on the magnitude of the control signal. In this work, with the aim of combining in one control system the indicated positive, but contradictory properties of the analyzed systems, the so-called combined ADOR method is proposed. Its main idea is to represent the control signal of an object by two terms, which are then sequentially determined by using two main methods of the ADOR theory, and at the first stage of synthesis there is control using the Letov-Kalman method, which ensures the direct application of the combined synthesis method to unstable objects. This synthesis method, when using a quality functional with one variable weight coefficient, allows for the considered control objects of order n m 5 to design systems with given values of the static error and control time, for which overregulation does not exceed 5.2 %.

Waring’s problem for locally nilpotent groups: The case of discrete Heisenberg groups

Kamke (Kamke, 1921) solved an analog of Waring’s problem with nth power monomials replaced by integer-valued polynomials. Larsen and Nguyen (Larsen and Nguyen, 2019) explored the view of algebraic groups as a natural setting for Waring’s problem. This paper applies the theory of polynomial maps and polynomial sequences in locally nilpotent groups developed in a previous work (Hu, 2024) to solve an analog of Waring’s problem for the general discrete Heisenberg group H2n+1(Z) for each integer n≥1.

Factorization in rings of integer-valued rational functions

For a domain D, the ring Int ( D ) of integer-valued polynomials over D is atomic if D satisfies the ascending chain condition on principal ideals. However, even for a discrete valuation domain V, the ring Int R ( V ) of integer-valued rational functions over V is antimatter. We introduce a family of atomic rings of integer-valued rational functions and study various factorization properties in these rings.

Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Let D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and 1<n1≤⋯≤nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < n_1 \\le \\cdots \\le n_k$$\\end{document}, there exists an integer-valued polynomial on D, that is, an element of Int(D)={f∈K[X]∣f(D)⊆D}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D) = \\{ f \\in K[X] \\mid f(D) \\subseteq D \\}$$\\end{document}, which has precisely k essentially different factorizations into irreducible elements of Int(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D)$$\\end{document} whose lengths are exactly n1,…,nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1, \\ldots , n_k$$\\end{document}. Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.

Travelling Ionospheric Disturbances detection: a statistical study of detrending techniques, induced period error and near real-time observables

Due to advances in remote sensing of the Earth's Ionosphere through Total Electron Content (TEC) estimates by Global Navigation Satellite System (GNSS) receivers, it is possible to detect and characterize Travelling Ionospheric Disturbances (TIDs) in both post-processing and, to some extent, in near real-time (NRT). A reliable and precise TEC filtering technique must be adopted to characterize waves accurately. Specifically, TEC detrending is widely adopted to extract the amplitude and period of the detected ionospheric waves from the background ionospheric conditions. Therefore, this study aims to understand and compare how different TEC detrending techniques and their settings impact the ability to extract such parameters. We highlight that the novel Fast Iterative Filtering (FIF) and the Savitzky-Golay filter (SGOLAY) techniques are the most reliable overall compared with moving average (MA), multi-order numerical difference (DD), polynomial detrending (POLY) and Finite Impulse response (FIR) band-pass filter (BUTF). Moreover, the impact of general algorithm settings on the extracted TID period is investigated, such as the Ionospheric Piercing Point (IPP) height and elevation cut-off angle, showing that such parameters drastically impact the retrieved period, especially for slower TIDs. Finally, due to the growing interest in real-time (RT) detection and classification of TIDs, the study proposes techniques for accurately estimating the TID amplitude in an NRT scenario. Such NRT techniques are then compared with the widely used post-processing products, such as the calibrated vertical TEC (vTEC), showing a difference that is mostly lower than the typical noise level of GNSS receivers (0.05 TECu).

On the asymmetric additive energy of polynomials

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy.

Using Affine Arithmetic to Design a Controller for Large Uncertain System

This study presents a approach for creating the supervisor of expansive uncertain systems. The Controller for a particular high order system is designed using a reduced order model. The numerator and denominator polynomial in the suggested reduction approach is derived using modified polynomial differential method. A lower order model with least ISE optimization is obtained. Assuming that the initial high-order system is stable, the proposed approach guarantees the stability of the streamlined model. Through With reference to the original high-order systems, a PID controller is created for the suggested low-order model. In order to explain the method's efficiency, a few numerical examples were taken into consideration. It has been demonstrated that applying Control from the lower order model to the higher order system is improved and the controlled system's performance. Common numerical illustrations seen in the literature have been used to test the method, and the results show that it works satisfactorily.

The Skolem property in rings of integer-valued rational functions

Let D be a domain and let Int(D) and IntR(D) be the ring of integer-valued polynomials and the ring of integer-valued rational functions, respectively. Skolem proved that if I is a finitely-generated ideal of Int(Z) with all the value ideals of I not being proper, then I=Int(Z). This is known as the Skolem property, which does not hold in Z[x]. One obstruction to Int(D) having the Skolem property is the existence of unit-valued polynomials. This is no longer an obstruction when we consider the Skolem property on IntR(D). We determine that the Skolem property on IntR(D) is equivalent to the maximal spectrum being contained in the ultrafilter closure of the set of maximal pointed ideals. We generalize the Skolem property using star operations and determine an analogous equivalence under this generalized notion.

Development of a novel model matching decentralized controller design algorithm and its experimental validation through load frequency controller implementation in restructured power system using TMS320F28379D controlCARD

A three-stage decentralized controller design algorithm is developed to achieve setpoint tracking and disturbance rejection in MIMO systems with communication time delay, and nonlinearities such as saturation and dead band. The first stage involves reference model formulation. The second stage comprises equating approximate generalized time moments/approximate generalized Markov parameters of closed-loop system model with reference model at certain expansion points in s-plane to obtain synthesis-like equation, concerning products of unknown controller numerator and denominator polynomials. This process yields simultaneous linear equations, whose solution provides the coefficients of the product polynomials. In the third stage, the controller parameters are extracted from the product polynomials using exact model matching. The proposed method is illustrated by designing load frequency controller in traditional and restructured power systems under scenarios like system parameter uncertainties, random load variation, and time-varying communication delays. Simulation studies reveal the efficacy of the proposed technique over existing techniques. Furthermore, the practical implementation feasibility of the decentralized controller designed for the restructured power system is validated using the TMS320F28379D controlCARD.

Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

We study the class of univariate polynomials βk(X), introduced by Carlitz, with coefficients in the algebraic function field Fq(t) over the finite field Fq with q elements. It is implicit in the work of Carlitz that these polynomials form an Fq[t]-module basis of the ring Int(Fq[t])={f∈Fq(t)[X]|f(Fq[t])⊆Fq[t]} of integer-valued polynomials on the polynomial ring Fq[t]. This stands in close analogy to the famous fact that a Z-module basis of the ring Int(Z) is given by the binomial polynomials (Xk).We prove, for k=qs, where s is a non-negative integer, that βk is irreducible in Int(Fq[t]) and that it is even absolutely irreducible, that is, all of its powers βkm with m>0 factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that βk is not even irreducible if k is not a power of q.

Topological aspects of the ideal theory in rings of integer-valued polynomials

Let D D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of I n t ( D ) Int(D) lying over a given maximal ideal m \mathfrak m of D D . We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the m \mathfrak m -adic completion of D D . Moreover, these results are naturally extended to overrings of I n t ( D ) Int(D) . As an application we provide explicit constructions of divisorial ideals of I n t ( D ) Int(D) which are not finitely generated.

A numerical range approach to Birkhoff–James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

Integer-valued polynomials on subsets of upper triangular matrix rings

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n ( K ) ( T n ( D ) ) : = { f ∈ T n ( K ) [ x ] | f ( T n ( D ) ) ⊆ T n ( D ) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n ( R 2 ) ( Ω , T n ( R 1 ) ) : = { f ∈ T n ( R 2 ) [ x ] | f ( Ω ) ⊆ T n ( R 1 ) } for some subsets Ω ⊆ T n ( R 1 ) . We generalize Frisch’s result and show that Int T n ( R 2 ) ( T n ( R 1 ) ) : = Int T n ( R 2 ) ( T n ( R 1 ) , T n ( R 1 ) ) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings.

Entropy and Multi-Fractal Analysis in Complex Fractal Systems Using Graph Theory

In 1997, Sierpinski graphs, S(n,k), were obtained by Klavzar and Milutinovic. The graph S(1,k) represents the complete graph Kk and S(n,3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by Sie(Λ,t), already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph Sie(Λ,t). In the second step, we extract some degree-based indices of a generalized Sierpinski graph Sie(Λ,t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph Sie(Λ,t) by using the Randić index.

Dynamical Analysis of Rubella Disease Model in the Context of Fractional Piecewise Derivative: Simulations with Real Statistical Data

Rubella is a viral disease that can lead to severe health complications, especially in pregnant women and their unborn babies. Understanding the dynamics of the Rubella disease model is crucial for developing effective strategies to control its spread. This paper introduces a major innovation by employing a novel piecewise approach that incorporates two different kernels. This innovative approach significantly enhances the accuracy of modeling Rubella disease dynamics. In the first interval, the Caputo operator is employed to address initial conditions, while the Atangana–Baleanu derivative is utilized in the second interval to account for anomalous diffusion processes. A thorough theoretical analysis of the piecewise derivative for the problem is provided, discussing mathematical properties, stability, and convergence. To solve the proposed problem effectively, the piecewise numerical Newton polynomial technique is employed and the numerical scheme for both kernels is established. Through extensive numerical simulations with various fractional orders, the paper demonstrates the approach’s effectiveness and flexibility in modeling the spread of the Rubella virus. Furthermore, to validate the findings, the simulated results are compared with real data obtained from Rubella outbreaks in Uganda and Tanzania, confirming the practical relevance and accuracy of this innovative model.

Integer-valued polynomials on valuation rings of global fields with prescribed lengths of factorizations

Let V be a valuation ring of a global field K. We show that for all positive integers k and 1 < n_1 le cdots le n_k there exists an integer-valued polynomial on V, that is, an element of {{,textrm{Int},}}(V) = { f in K[X] mid f(V) subseteq V }, which has precisely k essentially different factorizations into irreducible elements of {{,textrm{Int},}}(V) whose lengths are exactly n_1,ldots ,n_k. In fact, we show more, namely that the same result holds true for every discrete valuation domain V with finite residue field such that the quotient field of V admits a valuation ring independent of V whose maximal ideal is principal or whose residue field is finite. If the quotient field of V is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.

Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains

Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one.In this paper, we study irreducible polynomials F∈Int(R) where R is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number S∈N that reduces the absolute irreducibility of F to the unique factorization of FS.To this end, we establish a connection between the factors of powers of F and the kernel of a certain linear map that we associate to F. This connection yields a characterization of absolute irreducibility in terms of this so-called fixed divisor kernel. Given a non-trivial element v of this kernel, we explicitly construct non-trivial factorizations of Fk, provided that k≥L, where L depends on F as well as the choice of v. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for k, one of which only depends on the valuation of the denominator of F and the size of the residue class field of R.

Applications of [formula omitted]-difference equation and homogeneous [formula omitted]-shift operator [formula omitted] in [formula omitted]-polynomials

In this paper, the generalized homogeneous q-shift operator is constructed. The q-difference equation is then utilized to construct numerous polynomial q-identities, such as the generating function and its extension, Rogers’ formula and its extension, and Mehler’s formula and its extension for the generalized q-hypergeometric polynomials. Also demonstrated is a transformational identity involving generating functions for the generalized q-hypergeometric polynomials.

The power-generalized k-Horadam sequences in different modules and a new cryptographic method with these sequences

In this study, unlike other power sequences, we define new ones in the integer and integer-valued polynomial module. We call these sequences the power-generalized [Formula: see text]-Horadam sequence in the integer module and the power-generalized [Formula: see text]-Horadam sequence in the integer-valued polynomial module. Using these sequences, we rebuild the ElGamal cryptosystem and then obtain a new cryptographic method by combining this and symmetric systems. We provide an encryption example where our method is applied. Finally, we see the advantages of the new cryptographic method when we compare the obtained new cryptographic method with some of the asymmetric cryptographic systems such as RSA and ElGamal. And we achieve that new cryptographic method that has more advantageous.