Table-Based Methods

Many algebraic transformations of the hypergeometric equation σ(x)z"(x) + τ(x)z'(x) + lz(x) = 0, where σ, τ, l are polynomial functions of degrees 2 (at most), 1, 0, respectively, are well known. Some of them involve x = x(t), a polynomial of degree r, in order to recover the Heun equation, extension of the hypergeometric equation by one more singularity. The case r = 2 was investigated by K. Kuiken (see 1979 SIAM J. Math. Anal. 10 (3) 655-657) and extended to r = 3,4, 5 by R. S. Maier (see 2005 J. Differ. Equat. 213 171 - 203). The transformations engendered by the function y(x) = A(x)z(x), also very popular in mathematics and physics, are used to get from the hypergeometric equation, for instance, the Schroedinger equation with appropriate potentials, as well as Heun and confluent Heun equations. This work addresses a generalization of Kimura's approach proposed in 1971, based on differential transformations of the hypergeometric equations involving y(x) = A(x)z(x) + B(x)z'(x). Appropriate choices of A(x) and B(x) permit to retrieve the Heun equations as well as equations for some exceptional polynomials. New relations are obtained for Laguerre and Hermite polynomials.

The study was conducted at the experimental base of the Menkovsky branch of the Agrophysical Research Institute in the Leningrad Region in 2015–2021. The purpose of the experiment is to develop a model of the seasonal dynamics of the NDVI index of spring barley crops. Comparing the current NDVI measurements during the growing season with the model data will allow monitoring and, if necessary, adjusting the condition of crops using agrotechnological techniques. Measurements were carried out once a week at permanent accounting sites in two field experiments. The NDVI dynamics for experiments with the cultivation of spring barley mixed with perennial grasses and in pure form were different. In the first case, the maximum value of the index occurred on the 28th week from the beginning of the year, in the second – on the 25th week. The indicated difference was due to the influence of perennial grasses, due to the increase in the above-ground mass of which the effects of herbicide treatment were leveled. Models of seasonal dynamics of NDVI, built on average long-term data, turned out to be more reliable for general analysis than one-year ones, since they reduced the influence of random factors. In both cases of modeling the variability of the seasonal course of the index, as applied to two experiments with the cultivation of spring barley, good results were obtained using a polynomial function of the 2nd degree (R = 0,97 and 0,80, p = 0,00003 and 0,008). At the same time, complication of the model to a polynomial of the 3rd degree led to an increase in the accuracy (R = 0,92) and reliability of the forecast (p = 0,004) only for the second experiment. In the models describing the seasonal dynamics of NDVI in individual years, higher accuracy corresponded to a polynomial of the 2nd degree.

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

The most turbulent vortices that populate the forest-atmosphere interface have canopy height length scales. These vortices are mainly responsible for turbulent exchanges between inside and above canopy region. Thus, we used the vertical wind profiles obtained by 10 anemometers installed inside and above the forest canopy of the Rebio-Jarú experimental site, in the Amazon Rainforest. A third degree polynomial function was developed to better fit the wind profile and therefore estimate the inflection point height of the vertical wind profile (zi) a length scale associated with wind shear (Ls), and the wind speed at height zi. These length and velocity scales were used to obtain better fits for the dimensional wind profiles and turbulence statistical moments. Three dimensionless profile models were compared using friction velocity, wind velocity in zi and wind velocity at canopy height. It was observed that the dimensionless profiles using the velocity and shear calculated at zi provided support for the elaboration of more realistic parameterization of the turbulent exchange processes that occur both at the forest-atmosphere interface and inside the canopy.

Identification and dose assessment of irradiated cardamom and cloves by EPR spectrometry

The purpose of this paper is to analyze a mixed method for the linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise (<i>k</i> + 1), <i>k</i>, and (<i>k</i> + 1) th degree polynomial functions (<i>k</i> ≥ 1), respectively. The numerical eigenfunction of stress is symmetric. By the discrete <i>H</i><sup style="">1</sup>-stability of numerical displacement, we prove an <i>O </i>(<i>h^k</i>⁺²) approximation to the <i>L</i><sup style="">2</sup>-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem with a proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to the Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an <i>O </i>(<i>h</i>²) initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete <i>H</i>¹-stability of numerical displacement, while only an <i>O </i>(<i>h</i>) approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments. © 2023 The Author(s).

With the growing importance of Human-Robot Interaction (HRI), the movement of the robots requires more and more attention to address the issues related to safety, efficiency and ergonomics. Trajectories are excellent candidates in the making of desirable motions designed for collaborative robots, because they allow to simply and precisely describe the motions. Despite the large number of works available for Online Trajectory Generation (OTG), there was, to our knowledge, no complete solution capable to simultaneously meet all the requirements of these new applications. In this paper we present the first complete trajectory generation algorithm that build trajectories from arbitrary initial and final conditions, subject to general asymmetric bounds on jerk, acceleration and velocity. A review of the state of the art exposes the limits of the previous OTG works and reveals the difficult problem of non-linearity related with short motions. We explain how these non-linearities introduce discontinuities and we propose a solution based on sequences of segment of third degree polynomial functions.

Mechanistic force modeling of single point cutting tool in terms of grinding angles

Electrocoagulation makes an alternative method to chemical coagulation. This paper presents the results obtained during the electrocoagulation of the model wastewater using aluminum electrodes. The wastewater was treated by means of chronopotentiometric electrocoagulation process in a static system, at the constant current I = 0.3 A; therefore higher doses of electrocoagulant required longer electrocoagulation time. Changes in zeta potential, pH, turbidity, chemical oxygen demand (COD), suspended solids and total phosphorus concentrations in the treated wastewater were determined. A new method for determining the optimal dosage of the aluminum electrocoagulant was proposed through application of the third degree polynomial function rather than the parabolic equation. An increase in the electrocoagulant dose raised the share of sweep fl occulation in the studied treatment process, resulting in the effective removal over 90% of phosphorus compounds from the system.

The purpose of study was to assess the rate of water loss from the strings of vines in relation to internodes position on vine chords and degree of wood maturity. The biological material was represented by Burgund vine cultivar, to which were analyzed individually all internodes distributed on the chord length (internodes number IN2-19). The rate of water loss and associated parameters (maximum rate of water loss - RWLMax, total drying time - T, time to achieve RWLMax - tRWLMax) were determined from the Burgund vine cultivar, in controlled condition. Parameters studied were associated with dry matter content (refractometric method), degree of maturity of the wood, and internodes positions on chord. RWL Max had higher values in the basal internodes (IN2; RWL Max = 0.252±0.005 g/min) and lower in the apical internodes (IN17-19; RWLMax = 0.202±0.011 g/min). RWLMax distributions values, according to the position of internodes on chord was described by a third degree polynomial function, statistical safety (R2 = 0.949, p&lt;0.01). Dry substance content in internodes (sugar) was correlated with the total time of water loss (T; R2 = 0.945), the time to reach the maximum rate of water loss (tRWLMax; R2 = 0.855), and maximum rate of water loss (RWLMax; R2 = 0.984).

Given a unicursal quartic in a plane and four bi-tangential straight lines, there exist a pair of conics four times tangential to it. Choosing an arbitrary point K0 outside the plane, the lines and either one of the conics determine four planes and a cone, and a quartic surface can be found that stays tangential to them and presents 15 conic point singularities, including K0. We introduce a one-parameter family of symmetric tensors which are polynomial functions of degree 7 of the Cartesian coordinates. Each of them determines two families of curves on the surface, which are the null curves of the associated metric and are the integral curves of a differential system which turns out to possess the Painlevé property and to be integrable by quadratures. Differential systems of that form have been found (Gaffet 2006 J. Phys. A: Math. Gen. 39 99) to represent the rotating motion with precession of gas clouds expanding into a vacuum.

A simple gas-blow switch, consisting of an axisymmetric tulip/pin contact and a cylindrical polytetrafluoroethylene (PTFE) nozzle, has been used for empirical studies of medium-voltage load current interruption in air. Only the thermal phase of the interruption is considered. Based on the results of more than 3000 interruption tests, a logistic regression model has been developed to describe the relationships between the switch design and test circuit parameters, and the current interruption capability. The most important design recommendation is that the inner diameter of the nozzle should be only slightly larger than the pin contact. The upstream overpressure needed for successful interruption increases as a polynomial function of degree 0.5–1 with the rate of rise of recovery voltage and of degree 2-3 with increasing current.

The effect of gamma irradiation on the radiofrequency dielectric properties of bovine kidney tissue has been investigated using gamma irradiator (Gs 1000), booton (7200) capacitance meter, signal generator (Lodstars SG-416013) and Dielectric cells Mathematic models of the following dielectric structural parameter, Dielectric spread parameter (), dielectric decrement () and Dielectric relaxation time (), were also developed. The mathematics models were found to be polynomial functions of degree 4,4, and 5 respectively. The coefficient of fit for dielectric decrement, relaxation time and spread parameter (), were found to be 98.2, 83.9 and 99.5, respectively. This shows that, the mathematical model can be effectively used to generate dielectric data that will facilitate the prediction of the extent of mammalian kidney tissue damage in gamma irradiation doses regime usually encountered in diagnostic and therapeutic radiology. &nbsp; Key words: Gamma irradiation, spread parameter (), dielectric decrement (), dielectric relaxation time (), dielectric permittivity (). &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp

The presence of non-conforming components instead of, or in addition to, the usual assembly errors results in N- or W-shaped hazard rate (HR) functions rather than the usual bathtub (i.e. U-shaped) ones. Although there have been numerous models for bathtub-shaped HR functions, N- and W-shaped HR functions are usually modelled using mixtures of two or more distributions. While this approach does sometimes lead to tidy interpretation, there can be a degree of overparameterization, with consequent problems in stability and fitting. For this reason, the present paper revisits the natural approach of modelling N- and W-shaped HR functions using polynomial functions of degree three or four. Although the non-negativity of the hazard rate function becomes non-trivial, this ensures a minimal number of parameters. The polynomial approach also allows the use of a parametric model without imposing a particular shape of hazard rate function on the data, which usually requires a non-parametric approach. The possible hazard rate shapes obtainable are characterized, and detailed formulae for local minima and maxima of the functions provided. The performance of the models is compared to that of several generalizations of the Weibull distribution, with promising results.

Let \({\mathbb {N}}\) denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid \(S\subseteq {\mathbb {N}}\). For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid \(S\subseteq {\mathbb {N}}^d\). The cardinality of \({\mathbb {N}}^d \setminus S\) is called the genus. We describe a family of algorithms, parameterized by (relaxed) monomial orders, that can be used to generate trees of semigroups with each GNS appearing exactly once. Let \(N_{g,d}\) denote the number of generalized numerical semigroups \(S\subseteq {\mathbb {N}}^d\) of genus \(g\). We compute \(N_{g,d}\) for small values of \(g,d\) and provide coarse asymptotic bounds on \(N_{g,d}\) for large values of \(g,d\). For a fixed \(g\), we show that \(F_g(d)=N_{g,d}\) is a polynomial function of degree \(g\). We close with several open problems/conjectures related to the asymptotic growth of \(N_{g,d}\) and with suggestions for further avenues of research.