Polynomial Type Research Articles

Ultracool Dwarf Absolute Magnitude versus Spectral Type Relations for JWST NIRCam Filters

We synthesize JWST NIRCam photometry for a range of narrow, medium, and wide filters using SpeX prism spectra and parallaxes of 688 field-age and 151 young (≲200 Myr) ultracool dwarfs (spectral types M6–T9). We derive absolute magnitude-spectral type polynomial relations for the F164N, F187N, and F212N narrow filters; the F140M, F162M, F182M, and F210M medium filters; and the F115W, F150W, and F200W wide filters. Our relations enable the calculation of photometric distances for ultracool dwarfs observed with JWST in the absence of parallax measurements. Additionally, using the synthesized photometry to generate color–color figures can help distinguish high-redshift galaxies from brown dwarf interlopers.

A Hybrid ROS-SVM Model for Detecting Target Multiple Drug Types

Misleading in determining the decision to use the target drug will be fatal, even to death. This study examines five pharmacological targets designated as types A, B, C, X, and Y. Early detection of misleading drug targeting will reduce the risk of death. This study aims to develop hybrid random oversampling techniques (ROS) and support vector machine (SVM) methods. The use of the oversampling technique in this study aims to balance classes in the dataset; due to the data collection in each class, there is a relatively large gap. This study applies five schemes to see which combination of models produces the highest accuracy. This study also uses five types of SVM kernels, linear, polynomial, gaussian, RBF, and sigmoid, combined with the ROS oversampling technique. Our proposed model combines the ROS oversampling technique with a linear SVM kernel. We evaluated the proposed model and resulted in an accuracy of 97% and compared it with several experiments, including the ROS technique with a sigmoid kernel which only resulted in 50% accuracy. It can be seen from the results obtained that the linear kernel is very adaptive to data types in the form of numeric and nominal compared to other kernels. The method proposed in this study can be applied to other medical problems. Future research can be carried out using a combination of other sampling techniques with deep learning-based methods on this issue.

The Newton polytope of the Morse discriminant of a univariate polynomial

In this paper we compute the Newton polytope MA of the Morse discriminant in the space of univariate polynomials with a given support set A. Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of MA.

Parametric Dependence of Topside Ionospheric Empirical Scale Height and Electron Density Profiles in NeQuick2 Model Over the Equatorial and Low Latitudes and Its Consequences on the Estimation of TEC

AbstractConstructing realistic vertical electron density profiles is a crucial step in accurately estimating the ionospheric total electron content (TEC) in empirical models. Inadequate representation of electron density profiles leads to under‐ or over‐estimation of TEC in model outputs. One of the widely used ionospheric empirical model, the NeQuick2 exhibits large uncertainties over equatorial and low latitudes, particularly in the topside ionosphere. The NeQuick2 model employs semi Epstein type of polynomials to characterize the topside ionospheric structure, wherein, the scale height (H) is the key parameter. NeQuick2 model estimates H value using empirical formulations, which contain three major parameters namely, Ho—the scale height at the F‐layer peak, g—height gradient in H, and r—which controls the increase in H at higher altitudes. In this study, a systematic analysis has been carried out with particular focus on the equatorial and low latitudes to explore the variability of topside scale height and electron density profiles on the above three empirical parameters using COSMIC observations over Trivandrum and Ahmedabad, in India. Model uncertainties in estimating Ho and their impacts on the estimation of TEC are investigated. Data assimilation analysis has been carried out to obtain a quantitative understanding on the effect of deviations in topside scale height on the uncertainties in accurate estimation of TEC over equatorial and low latitudes. This study reveals the impact of scale height uncertainties on errors in modeled TEC and demonstrates the need for improvement in empirical formulations for improving accuracies in ionospheric modeling over equatorial and low latitude sectors.

The Problem Parameters Effects on Transient Behavior of MHD Duct Flow

The present study focuses the effects of Reynolds number Re and magnetic Reynolds number Rm on the transient behavior of the MHD flow. The incompressible, electrically conducting and viscous fluid flows through a long pipe subjected to magnetic field B0(t)=B0f(t). B0 is the intensity and f(t) is the time varying function of the magnetic field which is chosen as polynomial, trigonometric, exponential and logarithmic function to illustrate the problem parameters effects. The Re and Rm effects on the behavior of the flow at transient levels are studied with these functions by taking Hartmann number Ha value as 20. The unsteady MHD equations in coupled form are treated by using the dual reciprocity boundary element method (DRBEM). The study reveals that, when Re or Rm increases the time level where the flow elongates is postponed to a further time level. Moreover, the increase in Re flattens the flow as in the increase of Hartmann number. However, the increase in Rm increases the flow magnitude. The transient flow and induced current contours are demonstrated for several Re and Rm values. After the flow elongates, the flow and induced current lines preserve the behavior for polynomial, exponential and logarithmic type f(t) while trigonometric type f(t) causes the flow to show periodic behavior.

On the use of L-functionals in regression models

Abstract In this article, we survey and unify a large class or L L -functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L L -moments (G. Sillito, Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample, Biometrika 56 (1969), no. 3, 641–650.), L L -skewness, and L L -kurtosis (J. R. M. Hosking, L-moments: analysis and estimation of distributions using linear combinations or order statistics, J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990), no. 1, 105–124.) and introduce order numbers for a large class of L L -functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2), respectively, and describe how a family of L L -functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre, or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follows a transformed linear model, with a link function that determines the appropriate class of L L -functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a framework for asymptotic theory of estimates of L L -functionals and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context, a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.

Determination of Interaction Parameters between Mn and Al and the Influence of Mn on Al2O3 Inclusions Formation in High Mn and Al Content Fe-Mn-Al-O Melts at 1873 K

For the purpose of determining the interaction parameters between Mn and Al, and the influence of Mn on Al2O3 inclusions formation in the Fe-Mn-Al-O melts with high Mn and Al contents, three groups of Fe-Mn-Al-O melts with the initial Al content of 3, 5, and 7 mass% and different Mn contents were equilibrated with pure solid Al2O3 in an Al2O3 crucible at 1873 K and Ar-H2 atmosphere. Then, the interaction parameters between Mn and Al were deduced using the WIPF (Wagner’s Interaction Parameter Formalism) and the R-K polynomial (Redlich-Kister type polynomial), respectively. From the WIPF, the first- and second-order interaction parameters, eAlMn and rAlMn, were determined to be 0.0292 and −0.00016, respectively. From the R-K polynomial, the binary interaction parameters, ΩMn-Al0 and ΩMn-Al1, were determined to be 73,439 J/mol and −34,919 J/mol, respectively. The applicability of the WIPF to high Mn and Al content Fe-Mn-Al-O melts was investigated by comparing the Al activity calculated by the WIPF and the R-K polynomial using the obtained data. The results showed that WIPF can be used in high Mn and Al content melts in the current concentration range. Further from the iso-activity contours of Al, the activity of Al increases with increasing Al or Mn content. Finally, the thermodynamic calculations show that the addition of Mn decreases the equilibrium O content at the same Al content, making the formation of Al2O3 inclusions easier.

Certain Properties and Characterizations of Multivariable Hermite-Based Appell Polynomials via Factorization Method

This paper introduces a new type of polynomials generated through the convolution of generalized multivariable Hermite polynomials and Appell polynomials. The paper explores several properties of these polynomials, including recurrence relations, explicit formulas using shift operators, and differential equations. Further, integrodifferential and partial differential equations for these polynomials are also derived. Additionally, the study showcases the practical applications of these findings by applying them to well-known polynomials, such as generalized multivariable Hermite-based Bernoulli and Euler polynomials. Thus, this research contributes to advancing the understanding and utilization of these hybrid polynomials in various mathematical contexts.

Analysis of the numerical scheme of the one-dimensional fractional Rayleigh–Stokes model arising in a heated generalized problem

In this paper, we present a well-organized method to estimate the one-dimensional fractional Rayleigh–Stokes model using the construction of orthogonal Gegenbauer polynomials (GBPs) and Lagrange square interpolation to estimate the time derivative. Therefore, we design an authentic and fast numerical calculation approach based on the elaborated convergence rate recovery method. The matrix of the derivative operation of an orthogonal GBP is gained by employing the characteristic of this type of polynomial. The privilege of the numerical method is the orthogonality of the GBP and operational matrices, which reduces time computation and increases speed. Eventually, we propose three numerical examples to check the validity and numerical studies to illustrate the precision and efficiency of the new approach.

Some Explicit Formula of Hurwitz Lerch type Poly-Cauchy Polynomials and Poly-Bernoulli Polynomials

In this paper, the Hurwitz-Lerch poly-Cauchy and poly-Bernoulli polynomials are defined using polylogarithm factorial function. Some properties of these types of polynomials were also established. Specifically, two different forms of explicit formula of Hurwitz-Lerch type poly-Cauchy polynomials were obtained using Stirling numbers of the first and second kinds and an explicit formula of Hurwitz-Lerch type poly-Bernoulli polynomials was established using the Stirling numbers of the first kind.

Some results related to Hurwitz stability of combinatorial polynomials

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to observe some results and applications for the Hurwitz stability of polynomials in combinatorics and study other related problems.We first present a criterion for the Hurwitz stability of the Turán expressions of recursive polynomials. In particular, it implies the q-log-convexity or q-log-concavity of the original polynomials. We also give a criterion for the Hurwitz stability of recursive polynomials and prove that the Hurwitz stability of any palindromic polynomial implies its semi-γ-positivity, which illustrates that the original polynomial with odd degree is unimodal. In particular, we get that the semi-γ-positivity of polynomials implies their parity-unimodality and the Hurwitz stability of polynomials implies their parity-log-concavity. Those results generalize the connections between real-rootedness, γ-positivity, log-concavity and unimodality to Hurwitz stability, semi-γ-positivity, parity-log-concavity and parity-unimodality (unimodality). As applications of these criteria, we derive some Hurwitz stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating run polynomials of types A and B and the Hurwitz stability for alternating run polynomials defined on a dual set of Stirling permutations.Finally, we study a class of recursive palindromic polynomials and derive many nice properties including Hurwitz stability, semi-γ-positivity, non-γ-positivity, unimodality, strong q-log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be implied in a unified approach.

ON CAUCHY-TYPE BOUNDS FOR THE EIGENVALUES OF A SPECIAL CLASS OF MATRIX POLYNOMIALS

Let \(\mathbb{C}^{m\times m}\) be the set of all \(m\times m\) matrices whose entries are in \(\mathbb{C},\) the set of complex numbers. Then \(P(z):=\sum\limits_{j=0}^nA_jz^j,\) \(A_j\in \mathbb{C}^{m\times m},\) \(0\leq j\leq n\) is called a matrix polynomial. If \(A_{n}\neq 0\), then \(P(z)\) is said to be a matrix polynomial of degree \(n\). In this paper we prove some results for the bound estimates of the eigenvalues of some lacunary type of matrix polynomials.

Design of progressive fuzzy polynomial neural networks through gated recurrent unit structure and correlation/probabilistic selection strategies

This study focuses on two critical design aspects of a progressive fuzzy polynomial neural network (PFPNN): the influence of the gated recurrent unit (GRU) structure and the implementation of fitness-based candidate neuron selection (FCNS) through two probabilistic strategies. The primary objectives are to enhance modeling accuracy and to reduce the computational load associated with nonlinear regression tasks. Compared with the existing fuzzy rule-based modeling architecture, the proposed dynamic model consists of the GRU structure and the hybrid fuzzy polynomial architecture. In the initial two layers of the PFPNN, we introduce three types of polynomial and fuzzy rules into the GRU neurons (GNs) and fuzzy polynomial neurons (FPNs), which can effectively reveal potential complex relationships in the data space. The synergy of the FCNS strategies and the ℓ2 regularization learning method is to design a progressive regression model adept at melding the GRU structure with a self-organizing architecture. The proposed GRU structure and polynomial-based neurons significantly improve the modeling accuracy for time-series datasets. The rational utilization of FCNS strategies can reinforce the network structure and discover the potential performance of neurons of the network. Furthermore, the inclusion of ℓ2 norm regularization provides additional stability to the proposed model and mitigates the overfitting issue commonly encountered in many existing learning methods. We validated the proposed neural networks using six time-series, four machine learning, and two real-world datasets. The PFPNN outperformed other models in the comparison studies in 83.3% of the datasets, emphasizing its superiority in terms of developing a stable deep structure from diverse candidate neurons and reducing computational overhead.

Twists by Dirichlet characters and polynomial Euler products of L-functions

We prove that suitable properties of the twists by Dirichlet characters of an L-function of degree 2 imply that its Euler product is of polynomial type.

A two-stage Polynomial Chaos Expansion application for bound estimation of uncertain FRFs

Polynomial Chaos Expansion (PCE) is a method for analysing uncertain vibratory structures with lower computational effort. It may simply be described as a curve fitting method with orthogonal basis terms, where the polynomial type, dimension and order are predefined for the uncertain responses. However, the polynomial order in PCE must be very high to accurately estimate statistical moments of the frequency response function in resonance regions of lightly damped and uncertain structures. To solve this issue different transformation techniques are reported in the literature, where implementations of PCE produce higher accuracy with a lower order polynomial. However, these transformations lose the attraction for using PCE, since they require some additional mathematical operations and, mostly, they present high accuracy if the higher orders of polynomials are again of interest. In this study, an efficient approach is presented for the upper bound estimation of the uncertain frequency response functions (FRFs) via PCE with lower order terms without performing any transformation. Rather than one-stage application of PCE for the desired response of an uncertain problem, the approach comprises a two-stage application of the classical PCE, i.e. first for the natural frequencies and then for the FRF calculations. As an example application of the approach, a thin beam for two different uncertainty cases is considered, namely local and global uncertainty. The local and global input uncertainties are generated by variability of lumped masses added at the boundary and Young's modulus, respectively. The FRF bounds are compared with extensive experimental and numerical Monte Carlo simulations, showing that low order polynomials are sufficient to calculate the bounds accurately with the technique described.

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained.

A Numerical Evaluation of Forming Failure of an Aluminum Sheet due to Splitting Damage in Hole Expansion Process

Abstract The reduction of car body weight is the principal issue of car manufacturers for reducing fuel consumption. Aluminum alloys are attractive materials for the automotive industry because they have low density and adequate strength, but they may exhibit crack formation during manufacturing processes. Generally, crack formations emerge because of tool geometry and material anisotropy. Accordingly, determination of the forming limits of aluminum alloys is essential. The hole expansion test (HET) is a significant formability process used in the automotive industry because it gives information about the stretch-flangeability limits of the material. Edge splitting (edge fracture) is a failure type seen in HET, and it limits the stretch-flangeability of the material. Therefore, the prediction of edge splitting is an essential issue for engineers in the automotive industry. In this work, HET of AA6016-O aluminum alloy was simulated with the finite element (FE) method to assess the influence of yield functions on failure prediction in HET. To this end, Hill48, Yld91, and a homogeneous fourth-order polynomial type yield criteria (HomPol4) were selected to identify the anisotropic behavior of the sheet. Analyses were carried out in Marc commercial FE software, and the Hypela2 user subroutine was incorporated into FE code. Thickness distributions in the rolling direction (RD), diagonal direction, and transverse direction (TD) of the part and around the hole edge were also predicted, and it was observed that only the HomPol4 criterion predicted excessive thinning at two locations near the RD and TD, separately. On the other hand, Yld91 and Hill48 predicted lower strain levels when compared with HomPol4. Finally, plastic work distribution around the hole edge was considered, and the predictions were compared with the experimental damaged sample. This comparison showed that HomPol4 predicted a robust plastic work localization in RD, which is consistent with the damaged sample.

Excess Volume for Acetylene Tetrachloride with Pyridine, Methoxybenzene, 2-Butanone and 1,4-Dioxane at 298.15K

A dilatometric measurement have been carried out for getting excess volumes, VE, in the liquid state for mixtures of acetylene tetrachloride (CHCl2CHCl2), with pyridine (C5H5N), methoxybenzene (C6H5O CH3), 2-butanone (CH3COC2H5) and 1,4-dioxane (1,4-C4H8O2) at 298.15K and under atmospheric pressure. The values of excess molar volume, VE, are negative in sign for all the mixtures over whole composition range. Obtained data have been fitted by Redlich-Kister type polynomial smoothing equation. The results are discussed in the light of molecular interactions occurred between components in the liquid state.

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.

Infinite Horizon Optimal Control Problems with Discount Factor on the State, Part I: Analysis of the Controlled State Equation

This is the first part of an investigation of infinite horizon optimal control problems subject to semilinear parabolic equations. A discount factor on the state variable is introduced in the cost. This allows for the treatment of infinite horizon problems without stabilizability assumptions. The nonlinearities can be of polynomial type, thus covering reaction-diffusion equations which are important for applications. The control-to-state mapping and its regularity are analyzed in detail. This involves the relation between the type of the nonlinearity and the discount factor.