Polynomial Type Research Articles

Some K-Banhatti Polynomials of First Dominating David Derived Networks

Chemical compounds, characteristics, and molecular structures are inevitably connected. Topological indices are numerical values connected with chemical molecular graphs that contribute to understanding a chemical compounds physical qualities, chemical reactivity, and biological activity. In this study, we have obtained some topological properties of the first dominating David derived (DDD) networks and computed several K-Banhatti polynomials of the first type of DDD.

A Note on q‐Analogues of Degenerate Catalan‐Daehee Numbers and Polynomials

Recently, Yuankui et al. (Filomat J. 35 (5):17, 2022) studied q‐analogues of Catalan‐Daehee numbers and polynomials by making use of p‐adic q‐integrals on ℤp. Motivated by this study, we consider q‐analogues of degenerate Catalan‐Daehee numbers and polynomials with the help of p‐adic q‐integrals on ℤp. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we also derive some new identities and properties of this type of polynomials and numbers.

Asymptotically polynomial type solutions for 2-dimensional coupled non-linear ODEs with derivative terms

Asymptotically polynomial type solutions for 2-dimensional coupled non-linear ODEs with derivative terms

Joint Detection and Localization of Stealth False Data Injection Attacks in Smart Grids Using Graph Neural Networks

False data injection attacks (FDIA) are a main category of cyber-attacks threatening the security of power systems. Contrary to the detection of these attacks, less attention has been paid to identifying the attacked units of the grid. To this end, this work jointly studies detecting and localizing the stealth FDIA in power grids. Exploiting the inherent graph topology of power systems as well as the spatial correlations of measurement data, this paper proposes an approach based on the graph neural network (GNN) to identify the presence and location of the FDIA. The proposed approach leverages the auto-regressive moving average (ARMA) type graph filters (GFs) which can better adapt to sharp changes in the spectral domain due to their rational type filter composition compared to the polynomial type GFs such as Chebyshev. To the best of our knowledge, this is the first work based on GNN that automatically detects and localizes FDIA in power systems. Extensive simulations and visualizations show that the proposed approach outperforms the available methods in both detection and localization of FDIA for different IEEE test systems. Thus, the targeted areas can be identified and preventive actions can be taken before the attack impacts the grid.

Asymptotically polynomial type solutions for 2-dimensional coupled non-linear ODEs with derivative terms

Asymptotically polynomial type solutions for 2-dimensional coupled non-linear ODEs with derivative terms

On Degenerate Array Type Polynomials

In the paper, with the help of the Faá di Bruno formula and an identity of the Bell polynomials of the second kind, the authors define degenerate λ-array type polynomials, establish two explicit formulas, and present several recurrence relations of degenerate λ-array type polynomials and numbers.

Lacunary Generating Functions of Hybrid Type Polynomials in Viewpoint of Symbolic Approach

In this paper, we introduce mon-symbolic method to obtain the generating functions of the hybrid class of Hermite-associated Laguerre and its associated polynomials. We obtain the series definitions of these hybrid special polynomials. Also, we derive the double lacunary generating functions of the Hermite-Laguerre polynomials and the Hermite-Laguerre-Wright polynomials. Further, we find multiplicative and derivative operators for the Hermite-Laguerre-Wright polynomials which helps to find the symbolic differential equation of the Hermite-Laguerre-Wright polynomials. Some concluding remarks are also given.

A generalization of the array type polynomials

We introduce a generalization of the array type polynomials by using two specific generating functions and investigate some of its basic properties in the sequel. A recurrence relation and two summation formulas involving these polynomials are also given.

Speed of Sound, Density, Viscosity and Related Thermodynamic Excess Properties of Binary Liquid Mixtures Containing Butyl Carbitol with 1-Butanol, Toluene and Diethylmalonate at 308.15 and 318.15 K

In present communication, measurements of experimental density, viscosity and ultrasonic velocity of binary liquid mixtures of butyl carbitol with 1-butanol, toluene and diethylmalonate have been carried out at 308.15 and 318.15 K at atmospheric pressure over the entire composition range of mole fractions. To study the intermolecular interactions present in these liquid mixtures, thermodynamic parameters like VE, ΔKS, Δη, ΔLf, ΔVf and ΔZ have been calculated. The excess functions and their deviations from ideality have also been correlated using Redlich-Kister type polynomial equation by the method of least-squares. The variations of thermodynamic parameters with composition and temperature have been discussed in terms of molecular interaction in these liquid mixtures. Further, IR spectra of these liquid mixtures have also been recorded and the obtained results have been used to analyze the mixing behaviour of the components.

A Determinantal Approach to Post Quantum Analogue of Appell Type Polynomials

A Determinantal Approach to Post Quantum Analogue of Appell Type Polynomials

Formulae to Fubini Type Numbers emerge from Application of p-adic Integrals

The aim of this manuscript is to examine and survey various formulae for Fubini type numbers and polynomials with application of the p-adic integrals to some special polynomials. Relations and formulae related to the Fubini type numbers and polynomials, the Bernoulli numbers, the Euler numbers, Stirling type numbers, and combinatorial numbers are given. Moreover, by using generating functions with their functional equations, some new formulae including the Hermite polynomials, the Fubini type polynomials, and the Lah numbers are given. Finally, remarks on the results of this manuscript are presented.

Coordinate Transformation Using the Author’s Software in GIS Class System – Case Study

In order to implement the provisions of the INSPIRE Directive, it is necessary for the Member States of the European Union to take appropriate measures to enable combining in a uniform manner spatial data deriving from different sources and sharing use of them by many users and many applications. Spatial data regarding underground hard coal mining in Poland should also be available in the national spatial reference system. Mining enterprises run a cartographic resource in the different rectangular flat coordinate systems. The standard transformation procedure does not provide the required accuracy because these are areas affected by mining activity, and the stability of points is limited, hence, studies were undertaken. The result is the development of software that can be used in Geographic Information Systems to transform spatial data from a system used in mine to the national system. The article described shortly a chosen coordinates systems used in Polish underground mines, elaborated procedure for selection of the degree and the type of a transformation polynomial in the transformation task. It presents its practical application of procedure for the area of one of hard coal mines using the author’s software elaborated in the results of above-mentioned research.

Solution for elliptic inclusion in an infinite plate with remote loading and Eshelby’s eigenstrain of polynomial type

In this paper, a particular inhomogeneous inclusion problem is studied. In the problem, Eshelby’s eigenstrain takes the type , where m+n = 2, and the remote loadings , are applied. In the solution, the complex variable method is used. The continuity conditions along the interface of the matrix and the inclusion are formulated exactly. Because the stress field is no longer uniform in inclusion in this case, the studied problem has an inherent difficulty. After some manipulation, the final result for stress components , and in inclusion are obtainable. In the present study, , and are no longer uniform.

The polynomial Hermite-Padé -system for meromorphic functions on a compact Riemann surface

Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface . We show that if for some meromorphic function on , then with the help of the ratios of polynomials of the Hermite-Padé -system we recover the values of on all sheets of the Nuttall partition of , apart from the last sheet. Bibliography: 18 titles.

A New Modified Algorithm Based Upon Exponential and Polynomial Functions for Solving Initial Value Problems

The main focus of this paper is to propose a fourth-order convergent modified numerical algorithm by using an exponential and polynomial type function for getting the approximate solution of initial value problems (IVPs) in ordinary differential equations (ODEs). The proposed algorithm has been tested and analyzed for the convergence, stability, consistency, of several numerical experiments that proved the superiority of the proposed algorithm against algorithms including the fourth-order arithmetic mean (AM), the fourth-order harmonic mean (HM), and the fourth-order Contra harmonic mean( ). Keywords: Modified numerical algorithm, Ordinary differential equation, Order of convergence, Initial value problems, Stability, and Consistency. DOI: 10.7176/MTM/11-6-01 Publication date: December 31 st 2021

Polynomials associated to non-convex bodies

Polynomial spaces associated to a convex body C in \((\mathbb{R}^{+})^d\) have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex C. We develop some basic pluripotential theory including notions of C−extremal plurisubharmonic functions VC,K for \(K\subset \mathbb{C}^d\) compact. Using this, we discuss Bernstein−Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.

Stochastic matrices realising the boundary of the Karpelevič region

A celebrated result of Karpelevič describes Θn, the collection of all eigenvalues arising from the stochastic matrices of order n. The boundary of Θn consists of roots of certain one-parameter families of polynomials, and those polynomials are naturally associated with the so-called reduced Ito polynomials of Types 0, I, II and III.In this paper we explicitly characterise all n×n stochastic matrices whose characteristic polynomials are of Type 0 or Type I, and all sparsest stochastic matrices of order n whose characteristic polynomials are of Type II or Type III. The results provide insights into the structure of stochastic matrices having extreme eigenvalues.

Thermal management and performance improvement by using coupled effects of magnetic field and phase change material for hybrid nanoliquid convection through a 3D vented cylindrical cavity

In this study, effects of using magnetic field and packed bed phase change material (PCM) system in a 3D cavity having ventilation ports on the performance improvements are analyzed during hybrid nanoliquid convection. Two different locations of inlet port is considered while the numerical study is conduced for various values (Reynolds number (Re, between 250 and 750), Hartmann number (Ha, between 0 and 100), size of the inlet (wd, between 0.15H and 0.85H) and nanoparticle loading amount (between 0.02% and 0.1%). When PCM is used in the vented cavity, 13% and 16.5% enhancements of average Nusselt (Nu) number are obtained as compared to no-PCM case at Ha=0 and Ha=100. A critical Ha is obtained beyond which the phase transition time (tc) is reduced and the value depends upon the inlet port location. The location of the inlet port has significant impacts on phase change dynamics and transition time. When it is closer to the wall (case-C2), tc is reduced. 65% and 80% of reductions in the tc are observed at the highest Re for configurations C1 and C2. The higher port size resulted in fast phase transition while reduction of 89% in tc is obtained at Re=750 for case C1. Nanoparticle loading accelerates the phase transition and tc is reduced by 10.4% and 9% for cases C1 and C2. However, the average Nu variation with PCM shows different behavior for cases C1 and C2. At the highest particle loading, 11% (t=52 min) and 13% (t=250 min) increments in the average Nu are achieved C1 and C2. A polynomial type correlation for tc is obtained in terms of Ha and nanoparticle amount in the heat transfer fluid.

Solvability of a New q-Differential Equation Related to q-Differential Inequality of a Special Type of Analytic Functions

The current study acts on the notion of quantum calculus together with a symmetric differential operator joining a special class of meromorphic multivalent functions in the puncher unit disk. We formulate a quantum symmetric differential operator and employ it to investigate the geometric properties of a class of meromorphic multivalent functions. We illustrate a set of differential inequalities based on the theory of subordination and superordination. In this real case study, we found the analytic solutions of q-differential equations. We indicate that the solutions are given in terms of confluent hypergeometric function of the second type and Laguerre polynomial.

Alternating Eulerian polynomials and left peak polynomials

We first present grammatical interpretations for the alternating Eulerian polynomials of types A and B. As applications, we then derive several properties of the type B alternating Eulerian polynomials, including recurrence relations, generating function and unimodality. And then, we establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials, which implies that the type B alternating Eulerian polynomials have gamma-vectors that alternate in sign.