Formulas For Polynomials Research Articles

Analysis of Numerical Differential Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer

The paper considers numerical differentiation of functions with large gradients in the region of an exponential boundary layer. This topic is important, since the application of classical polynomial difference formulas for derivatives to such functions in the case of a uniform mesh leads to unacceptable errors if the perturbation parameter is comparable with the mesh size. The numerical differentiation formula with a given number of nodes in the difference stencil is built on subintervals covering the original interval. The accuracy of numerical differentiation formulas on a Bakhvalov mesh, which is widely used in the construction of difference schemes for singularly perturbed problems, is analyzed. For the original function of one variable, a representation in the form of a sum of regular and boundary-layer components, based on the Shishkin decomposition, is used to solve a singularly perturbed problem. Previously, such a decomposition was used to prove the convergence of difference schemes. An estimate of the error of classical polynomial formulas for numerical differentiation on a Bakhvalov mesh is obtained. The error estimate on a Bakhvalov mesh is obtained in the general case, when a derivative of an arbitrarily given order is calculated and the difference stencil for this derivative contains a given number of nodes. The error estimate depends on the order of the calculated derivative and the number of nodes in difference stencil and takes into account the uniformity in the parameter . The results of numerical experiments are presented, which are consistent with the error estimates obtained.

Closed-Form Formulas for the nth Derivative of the Power-Exponential Function xx

In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function xx, establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function xx and for related functions and integer sequences.

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring’s problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz–Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.

A collisional global sheath – Bulk model of argon plasma for semiconductor scale manufacturing

Plasma processes enhance the high-quality, semiconductor fabrication employed in large-scale industries. A crucial element in this process are the many gases that generate different reactive species at low temperatures. Capacitive plasma discharge radio frequency CCPs play a major role in semiconductor scale fabrication. The analysis of nonlinear dynamics in CCPs are complicated even under simple approaches. Therefore, we study self consistent collision fluid model, particle in cell simulation PIC, step approximation and finally the global model to find the accurate solution of merging plasma sheath -bulk region. For this purpose we study the self consistent collisional fluid model to find the accurate charge–voltage distribution V(Q) which controls the nonlinear dynamics of CCPs. The results against the PIC simulations and the more elaborate model step approximation is accomplished with higher accuracy. Consequently, the cubic polynomial formula V(Q) are applied in the global model. Accurate analytical global sheath-bulk calculations of the higher harmonic effect-exhibiting average power distributions, bulk potentials, and currents are obtained. This study provides an effective solution of collision regime for any arbitrary plasma reactor designs, including single-frequency and double frequency CCPs. Moreover, we used the argon gas as a good example in our study for a good ion etching. Finally, we can assist in achieving effective results that are consistent with experimental data and in large-scale applications in the semiconductor sector.

Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.

Partial-fraction decomposition of a rational function and its application

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function frac{x^{M}}{(x+1)^{lambda}_{n}} (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu’ results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind.

The characteristic polynomials of semigeneric threshold arrangements

<abstract><p>The semigeneric threshold arrangement is the hyperplane arrangement defined by $ x_i+x_j = a_{i} $, where $ 1 \leq i < j \leq n $ and $ a_i\text{'s} $ are generic elements. In this paper, we obtain a necessary and sufficient condition for subarrangements of the semigeneric threshold arrangement to be central from the perspective of simple graphs. Combining it with Whitney's theorem, we provide a formula for the characteristic polynomials of the semigeneric threshold arrangement and its subarrangements.</p></abstract>

SHORTER PROOF OF THE RIESZ INTERPOLATION FORMULA FOR TRIGONOMETRIC POLYNOMIALS

By means of partial fraction decompositions, a shorter proof is presented for the important interpolation formula of trigonometric polynomials discovered by Riesz (1914).

Nominal tensile strength reduction and its mechanism of ultra-high performance concrete with steel bar reinforcements

A comprehensive experimental study was designed to investigate the tensile response of four series of ultra-high performance concrete (UHPC) with different steel bars. The changing rule of the nominal tensile strength of UHPC in reinforced UHPC with respect to reinforcement details was revealed and the relative mechanism was clarified to both make adequate capacity calculation for structural design and develop the refined material model for finite-element analysis. Polynomial fitting formulas to account for the nominal tensile strength reduction of UHPC with steel bar ratios of 1.57–6.28% were proposed. UHPC nominal elastic tensile strength reduction in reinforced UHPC was found linearly dependent on the restrained degree of steel bars on UHPC shrinkage based on theoretical analysis. The further nominal ultimate tensile strength reduction of strain hardening UHPC in reinforced UHPC was verified to be attributed to the localized damage aggravation around the ribs of the deformed steel bar based on acoustic emission parametric analysis.

On atomic density of numerical semigroup algebras

A numerical semigroup S is a cofinite, additively closed subset of the nonnegative integers that contains 0. We initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring 𝔽q[x] is zero for any finite field 𝔽q; we prove that the numerical semigroup algebra 𝔽q[S] also has atomic density zero for any numerical semigroup S. We also examine the particular algebra 𝔽2[x2,x3] in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field 𝔽q.

Matiyasevich Formula for Chromatic and Flow Polynomials and Feynman Amplitudes

Matiyasevich Formula for Chromatic and Flow Polynomials and Feynman Amplitudes

Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and 1432-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier.We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015).

Recurrence relation for the Appell sequences

In this paper, we present several explicit formulas for Appell polynomials in terms of the weighted Stirling numbers of the second kind. We also provide a unified approach to obtain a three-term recurrence formula for the computation of Appell polynomials. Several examples are given to illustrate our results. As an application, we define and study a new class of polynomials that we call SPI polynomials. They are related to sums of powers of integers.

Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi

Abstract In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function.

Characterization of metastable high pressure phase transition positions and its influence on the behavior of microbial destruction

Effect of perturbation factors on phase transition metastable positions of whole milk (4% fat content) and their influence on microbial destruction characteristics of non-pathogenic Escherichia coli inoculated in milk subjected to high pressure low temperature treatment were evaluated using a specially developed high pressure (HP) cooling system. Initially, the phase transition data of milk transitioning through the metastable phases were obtained and fitted successfully using Simon-like models as done in previous studies and polynomial formulas with R2 of 0.997 & 0.996 for ice I, and 0.989 & 0.989 for ice III, respectively. The phase transition position of milk was explored with 5% and 10% sodium chloride solution as perturbation sources, respectively. Results showed that the 5% sodium chloride solution can reduce the transition pressure of milk by 43 MPa and increase the transition temperature by 4.1 °C, so that the milk can achieve phase transition at lower pressure and higher temperature. Phase transition microbial destruction was characterized by discontinuity, mutation and segmentation when the phase transition pressure interval 250– 300 MPa was carefully refined. The inactivation amount of E. coli before the phase transition (250 MPa) was 1.11 log and the phase transition process itself brought an additional 1.26 log destruction of E. coli population in milk. Industrial relevanceHigh pressure low temperature (HPLT) phase change kinetics were employed to enhance microbial destruction. HPLT was established based on a self-cooling unit positioned inside conventional HP chamber offering opportunities for scale up and commercialization. The effectiveness of HPLT phase transition for Escherichia coli destruction was demonstrated. The related research in metastable state provides a reference point for commercial application of high-pressure-low-temperature technology for microbial destruction and quality enhancement.

Quiver Hall–Littlewood functions and Kostka–Shoji polynomials

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $\mu$, we define a quiver current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},\mu(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ as the result of acting on $1\in\Lambda^Q$ by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle on Lusztig's convolution diagram determined by the sequences $\mathbf{i},\mathbf{a}$. For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to $K$-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type $\mathfrak{sl}_r$.

Specializing Koornwinder polynomials to Macdonald polynomials of type B, C, D and BC

We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type \((C_n^\vee ,C_n)\) in the sense of Macdonald (Affine Hecke algebras and orthogonal polynomials, Cambridge tracts in mathematics, Cambridge Univ Press, 2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type B, C and D via Ram–Yip type formulas of non-symmetric Koornwinder and Macdonald polynomials.

Closing the Feedback of Evapotranspiration on the Atmospheric Evaporation Demand Based on a Complementary Relationship

Evapotranspiration is the important feedback of the catchment into the atmosphere. However, in catchment hydrological modeling, the feedback of evaporation into the atmosphere is not closed and potential evaporation is always a meteorological forcing which is not dependent on the actual evaporation. A modeling framework to close the feedback of evapotranspiration into the atmosphere (FCEA) based on the evapotranspiration complementary relationship was proposed in the catchment hydrological modeling, and the effect of land-use changes on the runoff and evapotranspiration in the upper reach of Han River of China was investigated in the FCEA. Brutsaert uses the boundary condition analysis method to propose a nonlinear complementary relationship based on polynomial formula (B2015 function), which was applied in the study area, and the parameters were calibrated based on the catchment water balance of 1972–1990 and validated in 1991–2017. The actual evapotranspiration (AET) in the study area was estimated based on the complementary model in the upper reach of Han River. The SWAT model was used to simulate the catchment hydrological processes in the study area from 1972 to 2017. The evapotranspiration in the upper reach of Han River was studied in four scenarios to realize the feedback of evapotranspiration to the atmosphere and analyze the impact of the evapotranspiration feedback to the change of runoff in the basin. The results showed that the annual runoff in the upper reach of the Han River will increase, and the annual actual evapotranspiration will decrease in the long-term simulations in Scenarios 1 and 4. In Scenarios 2 and 3, with the increase of woodland, the annual runoff will decrease due to the feedback to the atmosphere, and annual actual evapotranspiration will increase, which is related to the increase in ecological water demand caused by the increase in woodland. Converting grassland into farmland will increase the runoff of the watershed. It is important to improve the land-use planning policy in the Han River Basin in order to realize the sustainable development of the river basin.

Variable Time Magnetization Current Trajectory Control Method for Variable Flux Memory Machines

Magnetization state (MS) manipulation is crucial to the comprehensive performance improvement of variable flux memory machine (VFMM) drive system. This article proposes a novel MS manipulation control method wherein variable time magnetization current trajectories can be adaptively controlled to reduce the MS manipulation losses. A fuzzy proportional integral (PI) feedforward current controller is newly structured to track the trajectories, improving the current tracking ability. First, dynamic MS manipulation processes are simulated and analyzed deeply, in which the variable machine parameters are calculated by finite element method (FEM) and fitted with polynomial formulas. Based on these, the maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">di</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dt</i> of magnetization current pulse trajectory, namely the shortest MS manipulation time, can be adaptively acquired to reduce the MS manipulation losses at different running conditions. Then, a fuzzy PI feedforward current controller is developed by combining the feedforward and fuzzy logic controls, ensuring the real current tracks its reference well at different speed and load conditions, particularly in the MS manipulation process. Finally, the effectiveness and feasibility of the proposed MS manipulation method are verified by experimental results on a hybrid-magnetic-circuit VFMM (HMC-VFMM) prototype.

Tropical Newton-Puiseux polynomials II

Tropical Newton-Puiseux polynomials, defined as piece-wise linear functions with rational coefficients of the variables, play a role as tropical algebraic functions. We provide explicit formulas for tropical Newton-Puiseux polynomials being the tropical zeroes of a univariate tropical polynomial with parametric coefficients.