Additional Polynomial Research Articles

Simpler characterizations of total orderization invariant maps

Given a finite subset [Formula: see text] of a distributive lattice, its total orderization [Formula: see text] is a natural transformation of [Formula: see text] into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms.

Exploring the use of extended multiplicative scattering correction for near infrared spectra of wood with fungal decay

Extended Multiplicative Signal Correction (EMSC) is a multivariate linear modelling technique for multi-channel measurements that can identify and correct for different types of systematic variation patterns, known or unknown. It is typically used for pre-processing to separate light absorbance spectra, obtained by diffuse reflectance of intact samples, into three main sources of variation: additive variations due to chemical composition (≈Beer's law), mixed multiplicative and additive variations due to physical light scattering (≈Lambert's law) and more or less random measurement noise. The present work evaluates the use of EMSC to pre-process near infrared spectra obtained by hyperspectral imaging of Scots pine sapwood, inoculated with two different basidiomycete fungi and at various degradation stages. The spectral changes due to fungal decay and resulting mass loss are assessed by interpretation of the EMSC parameters and the partial least squares regression (PLSR) results. Including a cellulose (analyte) or bound water (interferent) spectral profile in the EMSC pre-processing model generally improves the predictive performance of the PLS modelling, but it can also make it worse. The inclusion of the additional polynomial baselines does not necessarily lead to a better separation of the physical and chemical effects present in the spectra. The estimated EMSC parameters provide insight into the differences in decay mechanisms. A detailed analysis of the EMSC results highlights advantages and disadvantages of using a complex pre-processing model.

Local Galois representations associated to additive polynomials

For an additive polynomial and a positive integer, we define an irreducible smooth representation of a Weil group of a non-archimedean local field. We study several invariants of this representation. We obtain a necessary and sufficient condition for it to be primitive.

Extremal structure of cones of positive homogeneous polynomials

It was proved by Anthony Wickstead that the cone of positive linear operators between Banach lattices E and F coincides with the strongly closed convex hull of the set of lattice homomorphisms from E to F if and only if the cone of positive elements on E is the weakly closed convex hull of the union of extremal rays of the cone of positive liner functionals on E. This note aims to show that this result extends to a wider class of operators, namely, orthogonally additive homogeneous polynomials acting between vector lattices.

On the geometric and analytical properties of the anharmonic oscillator

Here we consider the anharmonic oscillator that is a dynamical system given by yxx+δyn=0. We demonstrate that to this equation corresponds a new example of a superintegrable two-dimensional metric with a linear and a transcendental first integrals. Moreover, we show that for particular values of n the transcendental first integral degenerates into a polynomial one, which provides an example of a superintegrable metric with additional polynomial first integral of an arbitrary even degree. We also discuss a general procedure of how to construct a superintegrable metric with one linear first integral from an autonomous nonlinear oscillator that is cubic with respect to the first derivative. We classify all cubic oscillators that can be used in this construction. Furthermore, we study the Liénard equations that are equivalent to the anharmonic oscillator with respect to the point transformations. We show that there are nontrivial examples of the Liénard equations that belong to this equivalence class, like the generalized Duffing oscillator or the generalized Duffing–Van der Pol oscillator.

Galois groups of random additive polynomials

We study the distribution of the Galois group of a random q q -additive polynomial over a rational function field: For q q a power of a prime p p , let f = X q n + a n − 1 X q n − 1 + … + a 1 X q + a 0 X f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots +a_1X^q+a_0X be a random polynomial chosen uniformly from the set of q q -additive polynomials of degree n n and height d d , that is, the coefficients are independent uniform polynomials of degree deg ⁡ a i ≤ d \deg a_i\leq d . The Galois group G f G_f is a random subgroup of GL n ⁡ ( q ) \operatorname {GL}_n(q) . Our main result shows that G f G_f is almost surely large as d , q d,q are fixed and n → ∞ n\to \infty . For example, we give necessary and sufficient conditions so that SL n ⁡ ( q ) ≤ G f \operatorname {SL}_n(q)\leq G_f asymptotically almost surely. Our proof uses the classification of maximal subgroups of GL n ⁡ ( q ) \operatorname {GL}_n(q) . We also consider the limits: q , n q,n fixed, d → ∞ d\to \infty and d , n d,n fixed, q → ∞ q\to \infty , which are more elementary.

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation (slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.

Trace formulae for actions of finite unitary groups on cohomology of Artin–Schreier varieties

Associated to a certain additive polynomial, we introduce an Artin–Schreier variety admitting an action of a finite unitary group. We calculate the character of the cohomology as a representation of a finite unitary group. One of our main ingredients is explicit character formulae for Weil representations of unitary groups due to Gérardin. We give another trace formula for a projective hypersurface admitting an action of a finite unitary group.

Impact of geographical, meteorological, demographic, and economic indicators on the trend of COVID-19: A global evidence from 202 affected countries

Research problemPublic health and the economy face immense problems because of pathogens in history globally. The outbreak of novel SARS-CoV-2 emerged in the form of coronavirus (COVID-19), which affected global health and the economy in almost all countries of the world. Study designThe objective of this research is to examine the trend of COVID-19, deaths, and transmission rates in 202 affected countries. The virus-affected countries were grouped according to their continent, meteorological indicators, demography, and income. This is quantitative research in which we have applied the Poisson regression method to assess how temperature, precipitation, population density, and income level impact COVID-19 cases and fatalities. This has been done by using a semi-parametric and additive polynomial model. FindingsThe trend analysis depicts that COVID-19 cases per million were comparatively higher for two groups of countries i.e., (a) average temperature below 7.5 °C and (b) average temperature between 7.5 °C and 15 °C, up to the 729th day of the outbreak. However, COVID-19 cases per million were comparatively low in the countries having an average temperature between 22.5 °C and 30 °C. The day-wise trend was comparatively higher for the countries having average precipitation between (a) 1 mm and 750 mm and (b) 750 mm and 1500 mm up to the 729th day of the outbreak. The day-wise trend was comparatively higher for the countries having more than 1000 people per sq. km. Discussing the COVID-19 cases per million, the day-wise trend was higher for the HICs, followed by UMICs, LMICs, and LIC. ConclusionThe study highlights the need for targeted interventions and responses based on the specific circumstances and factors affecting each country, including their geographical location, temperature, precipitation levels, population density, and per capita income.

Additive polynomial superfunctors and cohomology

We prove a classification of additive polynomial superfunctors, which allows us to compute some extensions of a superfunctor of the form F∘A where F is a classical polynomial functor and A is additive. We get a formula which relates these extensions to the classical ones of F. A possible generalisation is conjectured at the end.

Degenerate Apostol-Frobenius Type Poly-Genocchi Polynomials of Higher Order with Parameters a and b

This paper introduces another variation of poly-Genocchi polynomials by mixing the concept of modified degenerate polyexponential function, Apostol-Genocchi polynomials and Frobenius polynomials. These polynomials are called the degenerate Apostol-Frobenius-type poly-Genocchi polynomials with parameters a and b. Several identities and formulas are derived including recurrence relations, explicit formulas and certain differential identity. Moreover, some relations are established connecting these polynomials to degenerate Stirling numbers of the first and second kind, higher order degenerate Bernoulli polynomials, and higher order degenerate Frobenius-Euler polynomials.

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

Effects of temperature, relative humidity, and illumination on the entomological parameters of Aedes albopictus: an experimental study.

Aedes albopictus (Diptera: Culicidae) is a mosquito from Asia that can transmit a variety of diseases. This paper aimed to explore the effects of temperature, relative humidity, and illumination on the entomological parameters related to the population growth of Aedes albopictus, and provide specific parameters for developing dynamic models of mosquito-borne infectious disease. We used artificial simulation lab experiments, and set 27 different meteorological conditions to observe and record mosquito's hatching time, emergence time, longevity of adult females, and oviposition amount. We then applied generalized additive model (GAM) and polynomial regression to formulate the effects of temperature, relative humidity, and illumination on the biological characteristics of Aedes albopictus. Our results showed that hatchability closely related to temperature and illumination. The immature stage and the survival time of adult female mosquitoes were associated with temperature and relative humidity. The oviposition rate related to temperature, relative humidity, and illumination. Under the control of relative humidity and illumination, ecological characteristics of mosquitoes such as hatching rate, transition rate, longevity, and oviposition rate had an inverted J shape with temperature, and the thresholds were 31.2°C, 32.1°C, 17.7°C, and 25.7°C, respectively. The parameter expressions of Aedes albopictus using meteorological factors as predictors under different stages were established. Meteorological factors especially temperature significantly influence the development of Aedes albopictus under different physiological stages. The established formulas of ecological parameters can provide important information for modeling mosquito-borne infectious diseases.

High-Degree Polynomial Integrals of a Natural System on the Two-Dimensional Torus

We study a natural mechanical system on the two-dimensional torus which admitsan additional first integral polynomial in momenta of an odd degree N andindependent of the energy integral. For N=5,7 , we obtain the estimateson the number of straight lines in the spectrum of the potential.

A Nakano carrier theorem for polynomials

We use a localisation technique to study orthogonally additive polynomials on Banach lattices. We derive alternative characterisations for orthogonal additivity of polynomials and orthosymmetry of m m -linear mappings. We prove that an orthogonally additive polynomial which is order continuous at one point is order continuous at every point and we give an example to show that this result does not extend to regular polynomials in general. Finally, we prove a Nakano Carrier theorem for orthogonally additive polynomials, generalising a result of Kusraev [Orthosymmetric bilinear operators, Vladikavkaz, preprint, 2007].

ON EFFICIENT FRACTIONAL CAPUTO-TYPE SIMULTANEOUS SCHEME FOR FINDING ALL ROOTS OF POLYNOMIAL EQUATIONS WITH BIOMEDICAL ENGINEERING APPLICATIONS

This research paper introduces a novel fractional Caputo-type simultaneous method for finding all simple and multiple roots of polynomial equations. Without any additional polynomial and derivative evaluations using suitable correction, the order of convergence of the basic Aberth–Ehrlich simultaneous method has been increased from three to [Formula: see text]. In terms of accuracy, residual graph, computational efficiency and computation CPU time, the newly proposed families of simultaneous methods outperforms existing methods in numerical applications.

H-Adaptive radial basis function finite difference method for linear elasticity problems

In this research work, the radial basis function finite difference method (RBF-FD) is further developed to solve one- and two-dimensional boundary value problems in linear elasticity. The related differentiation weights are generated by using the extended version of the RBF utilizing a polynomial basis. The type of the RBF is restricted to polyharmonic splines (PHS), i.e., a combination of the odd m-order PHS phi (r)=r^m with additional polynomials up to degree p will serve as the basis. Furthermore, a new residual-based adaptive point-cloud refinement algorithm will be presented and its numerical performance will be demonstrated. The computational efficiency of the PHS RBF-FD approach is tested by means of the relative errors measured in ell _2-norm on several representative benchmark problems with smooth and non-smooth solutions, using h-adaptive, uniform, and quasi-uniform point-cloud refinement.

Generalized Laguerre-Apostol-Frobenius-Type Poly-Genocchi Polynomials of Higher Order with Parameters a, b and c


 
 
 
 In this paper, the generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polyno- mials of higher order with parameters a, b and c are defined using the concept of polylogarithm, Laguerre, Apostol and Frobenius polynomials. These polynomials possess numerous properties including recurrence relations, explicit formulas and certain differential identity. Moreover, some connections of these higher order generalized Laguerre-Apostol-Frobenius-type poly-Genocchi poly- nomials to Stirling numbers of the second kind and different variations of higher order Euler and Bernoulli-type polynomials are obtained.
 
 
 

Sums of powers of orthogonally additive polynomials

It is proved that for a given two positive integers N,n with N<n and N orthoregular homogeneous polynomials of the same degree acting between Archimedean vector lattices and pairwise independent in some sense, the sum of these polynomials each raised to the power of n and multiplied by an orthomorphism, is orthogonally additive if and only if all these polynomials multiplied by the corresponding orthomorphisms are disjointness preserving. It is also shown that given three positive integers N,n,m, an order bounded orthogonally additive polynomial with values in a Dedekind complete vector lattice is the sum of n-th powers of N order bounded disjointness preserving m-homogeneous polynomials if and only if it is representable as a disjoint sum of N disjointness preserving mn-homogeneous polynomials.

Determining monodromy groups of abelian varieties

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford–Tate group, $$\ell $$ -adic monodromy groups, and the Sato–Tate group. Assuming the Mumford–Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree.