Polynomials Of Degree Research Articles

The Activation Energy Temperature Dependence for Viscous Flow of Chalcogenides

For some chalcogenide glasses, the temperature dependence of the activation energy E(T) of viscous flow in the glass transition region was calculated using the Williams–Landel–Ferry (WLF) equation. A method for determining the activation energy of viscous flow as a function of temperature is proposed using the Taylor expansion of the function E(T) using the example of chalcogenide glasses As-Se, Ge-Se, Sb-Ge-Se, P-Se, and AsSe-TlSe. The calculation results showed that the temperature dependence of the activation energy for the Ge-Se, As-Se, P-Se, AsSe-TlSe, and AsSe systems is satisfactorily described by a polynomial of the second degree, and for Sb-Ge-Se glass by a polynomial of the third degree. The purpose of this work is to compare the values of the coefficients obtained from the Taylor series expansion of E(T) with the characteristics of the E(T) versus (T − Tg) curves obtained directly from the experimental temperature dependence of viscosity. The nature of the dependence E(T) is briefly discussed.

Modeling of energy recovery processes in railway traction power supply systems

Regenerative braking of trains can provide significant energy saving and a sizeable reduction in greenhouse gas emissions. The efficiency of energy recovery can be defined using an efficiency indicator equal to the product of the free-fall acceleration and the fraction of potential energy used in the recovery. Its slope dependence is well approximated by a polynomial of the third degree. The operating conditions of the traction power supply system of the mountain pass section of the main railway were modeled. The results obtained allowed the following conclusions to arrive at: regenerative braking is accompanied by circulating currents in adjacent inter-substation areas and leads to an absolute and relative increase in electrical energy losses in the traction network; the share of active energy losses rises by several times; the energy recovery causes a change in the direction of active energy flows at traction substations with a steep increase in reactive power consumption by trains; the energy recovery produces a significant decrease in voltage on the current collectors of electric locomotives; the voltage on the section under consideration, with three trains weighing tons decreased to 23 kV due to the large reactive energy consumption. © 2024 The Authors. Published by Elsevier Ltd.

Some new results on the negative polynomial Pell’s equation

We consider the negative polynomial Pell's equation $P^2(X)-D(X)Q^2(X)=-1$, where $D(X)\in \mathbb{Z}[X]$ be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions $P(X), Q(X)$ with integer coefficients.

Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree

Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree

Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

AbstractLet be a random matrix distributed according to uniform probability measure on the finite general linear group . We show that equidistributes on as as long as and that this range is sharp. We also show that nontrivial linear combinations of equidistribute as long as and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for , where depends on , due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of , we end up showing that certain explicit character sums modulo exhibit cancellation when averaged over monic polynomials of degree in as long as . This goes far beyond the classical range due to Montgomery and Vaughan. To study these sums, we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

A revisit on the hydrogen atom induced by a uniform static electric field

In this paper, we revisit the Stark effect of the hydrogen atom induced by a uniform static electric field. In particular, a general formula for the integral of associated Laguerre polynomials was derived by applying the method for Hermite polynomials of degree n proposed in the work (Anh-Tai T.D. et al., 2021 AIP Advances 11 085310). The quadratic Stark effect is obtained by applying this formula and the time-independent non-degenerate perturbation theory to hydrogen. Using the Siegert state method, numerical calculations are performed and serve as data for benchmarking. The comparisons are then illustrated for the ground and some highly excited states to provide an insightful look at the applicable limit and precision of the quadratic Stark effect formula for other atoms with comparable properties.

Fractional parts of generalized polynomials at prime arguments

Let f(X) be a monic polynomial of degree d≥2 with real coefficients. In this paper, we obtain an upper bound for the discrepancy of the sequence ([f(p)α]β) generated by the generalized polynomial [f(X)α]β, where p runs through the set of all primes and α, β are irrational numbers satisfying certain conditions. We also study the small fractional parts of the sequence ([f(p)α]β) and prove a result analogous to that of Madritsch and Tichy (Theorem 2.3, [13]).

Hausdorff limits of external rays: The topological picture

AbstractWe study Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials of degree with connected Julia sets.

A weighted combination of reproducing kernel particle shape functions with cardinal functions of scalable polyharmonic spline radial kernel utilized in Galerkin weak form of a mathematical model related to anti-angiogenic therapy

In this manuscript, a new localized meshfree (meshless) approximation, i.e., a combination of the reproducing kernel particle (RKP) shape functions with the cardinal functions based on the scalable polyharmonic spline (PHS) radial function is introduced. It is called the RKP+PHS+poly approximation, and its convergence rate is of order O(hm+1), where m is the total degree of polynomials. We apply this method to construct the spaces of trial and test functions in a Galerkin scheme of an extended version of the biological mathematical model in two dimensions describing the interactions between endothelial cells, fibronectin, angiogenic growth factors, and fasentin concentrations. By considering the row-sum method, the eigenvalue stability is also numerically carried out for the discrete equations corresponding to the obtained weak formulation. Accordingly, a semi-implicit form of the backward difference method of order 1 (SBDF1) has been utilized to approximate the weak form in time. We complete our numerical algorithm by solving the obtained full-discretized problem overtime via the biconjugate gradient stabilized (BiCGSTAB) solver with a proper preconditioner. Some simulation results are investigated by estimating the maximum velocity parameter of an enzymatic reaction from the experimental dose–response curve of fasentin to demonstrate the effect of using fasentin drug.

Mutually unbiased bases: polynomial optimization and symmetry

A set of k orthonormal bases of Cd is called mutually unbiased if |⟨e,f⟩|2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d. The (well-known) upper bound k≤d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk.We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk-module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12.

A nodally bound-preserving finite element method for reaction–convection–diffusion equations

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of [Formula: see text] in the energy norm, where [Formula: see text] represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach.

Experimental studies of a remote device for biometric identification of the pre-mastitis state of the udder of dairy cows

Analysis of development of dairy cattle breeding in the Republic of Belarus is presented. The problems of reducing milk productivity and the period of economic use of dairy cows as a result of mastitis disease are noted. The importance of timely detection of sick animals, as well as treatment of the inflammatory process of the udder in the early stages is noted. The results of theoretical and experimental studies of a mock-up sample of a biometric identification device for the pre-mastitis state of udder of dairy cows are presented. In the course of experimental studies, a significant influence of ambient temperature on the thermal imaging picture was revealed, based on which the values of the radiation coefficient for cows were determined. The final adjustment of the previously established temperature ranges was made when determining one or another form of mastitis, namely, within 32.0−36.3 ℃ – the range of normal temperatures, 36.4−37.7 ℃ – “Subclinical mastitis”, 37.8−39.0 ℃ – “Clinical mastitis”. The most significant factors and optimization parameter have been determined. The percentage of the useful area of the studied object is selected as the optimization parameter. Based on the processing of experimental data, a mathematical model is obtained, described by a regression equation in the form of a polynomial of the second degree. The rational parameters of the device for biometric identification of the pre-mastitis state of the udder of dairy cows were obtained, namely: the angle of inclination of the thermal imaging module αо – 2,7o, the focal length to the object l – 794 mm, installation height of the thermal imaging module h – 495 mm. Also, based on the conducted research, priority areas and opportunities for development of the domestic agro-industrial complex through creation of integrated systems for monitoring the physiological state of animals within the framework of development of digital automated technologies of a “smart” farm have been identified.

‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of 0 , 1 , … , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

Assessing reproductive performance and predictive models for litter size in Landrace sows under tropical conditions.

Litter size and piglet loss at birth significantly impact piglet production and are closely associated with sow parity. Understanding how these traits vary across different parities is crucial for effective herd management. This study investigates the patterns of the number of born alive piglets (NBA), number of piglet losses (NPL), and the proportion of piglet losses (PPL) at birth in Landrace sows under tropical conditions. Additionally, it aims to identify the most suitable model for describing these patterns. A dataset comprising 2,322 consecutive reproductive records from 258 Landrace sows, spanning parities from 1 to 9, was analyzed. Modeling approaches including 2nd and 3rd degree polynomial models, the Wood gamma function, and a longitudinal model were applied at the individual level to predict NBA, NPL, and PPL. The choice of the best-fitting model was determined based on the lowest mean and standard deviation of the difference between predicted and actual values, Akaike information criterion (AIC), and Bayesian information criterion (BIC). Sow parity significantly influenced NBA, NPL, and PPL (p<0.0001). NBA increased until the 4th parity and then declined. In contrast, NPL and PPL decreased until the 2nd parity and then steadily increased until the 8th parity. The 2nd and 3rd degree polynomials, and longitudinal models showed no significant differences in predicting NBA, NPL, and PPL (p>0.05). The 3rd degree polynomial model had the lowest prediction standard deviation and yielded the smallest AIC and BIC. The 3rd degree polynomial model offers the most suitable description of NBA, NPL, and PPL patterns. It holds promise for applications in genetic evaluations to enhance litter size and reduce piglet loss at birth in sows. These findings highlight the importance of accounting for sow parity effects in swine breeding programs, particularly in tropical conditions, to optimize piglet production and sow performance.

Minimum Degree Energy of Graphs and Pebbling Graphs

Objectives: The minimum degree energy concept is applied to pebbling graphs. This study establishes a connection between the minimum degree energy of graphs and the minimum degree energy of pebbling graphs by applying the lowest degree energy of pebbling notion to twenty standard graphs. Methods: The minimum degree energy of pebbling graph with the matrix whose was calculated using . The characteristic polynomial of the minimum degree, is must be found from the matrix. Next, the eigen values of the matrix were calculated using and the sum of all the eigen values gives the minimum degree energy. The lower and upper bounds for the minimum degree energy of graphs are established along with the algorithm for computing the minimum degree energy of graphs. Findings: The lower and upper bounds were found for the minimum degree energy of pebbling graphs. For twenty standard graphs and pebbling graphs, the minimum degree energy values were calculated, and their relation was tabulated. Novelty: Pebbling graphs were subjected to the minimal energy idea and a relationship was found between the minimum energy of pebbling graphs and the minimum energy of graphs in general. Keywords: Energy, Minimum degree energy, Grotzsch graph, Pebbling graph, Data mining

Sharpening of Tur´an-type inequality for polynomials

For the polynomial P(z) = n \sum j=0 cjzj of degree n having all its zeros in | z| \leq k, k \geq 1, V. Jain in “On the derivative of a polynomial”, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that max | z| =1 | P \prime (z)| \geq n \biggl( | c0| + | cn| kn+1 | c0| (1 + kn+1) + | cn| (kn+1 + k2n) \biggr) max | z| =1 | P(z)| . In this paper we strengthen the above inequality and other related results for the polynomials of degree n \geq 2.

Some Lp norm involving the polar derivative of a polynomial

Let [Formula: see text], [Formula: see text], is a polynomial of degree [Formula: see text] having all zeros in [Formula: see text], [Formula: see text], then for [Formula: see text] with [Formula: see text], [Formula: see text] and for each [Formula: see text] with [Formula: see text]; it was proved by (N. A. Rather et al., Some [Formula: see text] inequalities involving the polar derivative of a polynomial, Appl. Math. E-Notes 14 (2014) 116–122) [Formula: see text] In this paper, we establish some refinements and generalizations of above and some known polynomial inequalities concerning the polar derivative of a polynomial.

On stability behaviors of 5D M-theory black objects

Using N = 2 supergravity formalism, we investigate certain behaviors of five-dimensional black objects from the compactification of M-theory on a Calabi–Yau three-fold. The manifold has been constructed as the intersection of two homogeneous polynomials of degrees (ω + 2, 1) and (2, 1) in a product of two weighted projective spaces given by WP4(ω,1,1,1,1)×P1 . First, we determine the allowed electric charge regions of the BPS and non BPS black holes obtained by wrapping M2-branes on appropriate two cycles in such a proposed Calabi–Yau three-fold. After that, we calculate the entropy of these solutions which takes a maximal value corresponding to ω = 1 defining the ordinary projective space P4 . For generic values of ω, we show that the non BPS states are unstable. Then, we conduct a similar study of five-dimensional black strings. Concerning the allowed magnetic charge regions of the BPS and non BPS black stringy solutions derived from M5-branes on dual divisors, we calculate the tension taking a minimal value for P4 . By determining the recombination factor, we show that the non-BPS black string states are stable in the allowed regions in the magnetic charge space.

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.