Fourth-order Polynomial Equation Research Articles

Mathematical Description of Harmful Industrial Factors for Assessing the Quarry Workers' Occupational Risk

Mining industry is highly important in the economy of each country. At the same time, the mining industry is an industry with a high level of morbidity among workers. Quarry workers are often exposed to hazardous environments, while at the same time a low level of process safety organization. This study aims to develop a mathematical description of modelling the values of harmful factors in the production environment at the workplaces of quarry workers, which will help ensure high-quality engineering control during the work shift. The initial statistical data for the development of mathematical models were obtained directly by the authors through personal observations of the technological process and measurements of the level of harmful factors in the working environment. Measurements were taken using portable digital devices. To develop mathematical description based on accumulated statistical data, polynomial regression was used. The construction of models using polynomial regression is based on the property of functional dependencies, which can be formulated as follows: with a gradual increase in the complexity of the model (polynomial degree), the magnitude of errors (approximation error, forecast error) monotonically decreases. As a result, fourth-order polynomial equations were obtained to model the values of dust concentration and noise level at certain time intervals of the work shift. The correlation coefficient of these models is R = 0.98 and R = 0.99, and the relative error is 9.76% and 6.11% of dust concentration and noise level respectively, which indicates sufficient accuracy of the model. It was established that the studied harmful factors during a normal production process, have a level of impact at which early forms of occupational diseases develop. The proposed mathematical description of modelling the values of these factors makes it possible for high-quality engineering control over current technology processes in order to avoid abnormal excesses of the values. Further research, according to the authors, should be aimed at identifying and mathematically describing other potentially harmful and dangerous factors in the working environment. The originality of the current study lies in the fact that an attempt was made to propose a non-classical approach to quickly identify deteriorating working conditions directly in the work process using mathematical modelling.

Threshold amounts of nutrients and the relationship with chlorophyll a during eutrophication phenomenon in small-scale artificial reservoirs

This study aimed at determining the relationship of chlorophyll a (Chl a) and the nutrients in two reservoirs of different sizes, namely Bueng Si Than and Bueng Thung Sang, which are located in Khon Kaen Province in Thailand. The evaluation of the trophic state index and the estimated threshold amounts of nutrients was consequently considered. Water samples were collected during both the dry and rainy seasons with twenty-seven sampling points in each reservoir. Parameters indicating water quality and nutrients were based on the APHA method. The results showed a significant difference in the average physicochemical properties and nutritional parameters of the water in a smaller proportion, with Bueng Si Than having more than Bueng Thung Sang (p < 0.05). Nutrient factors (TP, TKN) showed the impact on the occurrence of eutrophication during both the dry season and the rainy season. Average ratios of nutrients and Chl a together with the trophic state index revealed the occurrence of hypertrophic conditions in both reservoirs. The high value of Chl a:TP signaled that the algal growth in these reservoirs could be controlled by TP. The Pearson correlation analysis indicated a strong positive relationship of Chl a and TP in Bueng Si Than. Moreover, the obtained results from the simulation were achieved using MATLAB R2022a, and were performed with the fourth order polynomial equation, which indicated the boundary values of 28.25–29.42 μg/L and 4.15-4.19 mg/L for the threshold amounts of TP and TKN prior to Eutrophic occurrence, ascertain further eutrophication management in small-scale artificial reservoirs.

Flujo de calor y capacidades calorífica específica en la etapa de deshidratación de la pirólisis de biomasas por análisis térmicos

This study aims to investigate the influence of the moisture of energy cane and coconut fiber on heat flow and specific heat capacity in the dehydration stage from the pyrolysis process. The experiments were carried out in a simultaneous thermogravimetry and differential scanning calorimetry analyzer using a heating rate of 20 K/min in an inert atmosphereThree decomposition stages were identified: dehydration (marked by an expressive endothermic peak), pyrolysis, and carbonization). From the analyses of the water contributions, it was observed that the heat flow from the heat capacity of remaining water (Qwc) is negligible compared to the heat flow from the water evaporation (Qwe), for both biomasses. Also, we calculated the heat flow from the heat capacity (Qb) and the experimental specific heat capacity (cp,b) of biomasses such as 686-2371 J/kg K and 1076-2113 J/kg K, respectively. Then, for the dehydration stage, third- and fourth-order theoretical polynomial equations have been proposed to predict the heat required for the biomass heating.

Exact Solutions for Vector Phase-Matching Conditions in Nonlinear Uniaxial Crystals

The transcendental equations of vector phase matching are transformed into a fourth-order polynomial equation that admits an analytical solution. The real roots of this equation provide the optical axis orientations that are useful for efficient down-conversion in nonlinear uniaxial crystals. The production of entangled photon pairs is discussed in both collinear and non-collinear configurations of the spontaneous parametric down-conversion (SPDC) process. Degenerate and non-degenerate cases are also distinguished. As a practical example, SPDC processes of type-I and type-II are studied for beta-barium borate (BBO) crystals. The predictions are in very good agreement with experimental measurements already reported in the literature and include theoretical results of other authors as particular cases. Some properties that seem to be exclusive to BBO crystals are reported; the experimental verification of the latter would allow a better characterization of these crystals.

Slowness vector versus ray direction in polar anisotropic media

SUMMARY The inverse problem of finding the slowness vector from a known ray direction in general anisotropic elastic media is a challenging task, needed in many wave/ray-based methods, in particular, solving two-point ray bending problems. The conventional resolving equation set for general (triclinic) anisotropy consists of two fifth-degree polynomials and a sixth-degree polynomial, resulting in a single physical solution for quasi-compressional (qP) waves and up to 18 physical solutions for quasi-shear waves (qS). For polar anisotropy (transverse isotropy with a tilted symmetry axis), the resolving equations are formulated for the slowness vectors of the coupled qP and qSV waves (quasi-shear waves polarized in the axial symmetry plane), and independently for the decoupled pure shear waves polarized in the normal (to the axis) isotropic plane (SH). The novelty of our approach is the introduction of the geometric constraint that holds for any wave mode in polar anisotropic media: The three vectors—the slowness, ray velocity and medium symmetry axis—are coplanar. Thus, the slowness vector (to be found) can be presented as a linear combination of two unit-length vectors: the polar axis and the ray velocity directions, with two unknown scalar coefficients. The axial energy propagation is considered as a limit case. The problem is formulated as a set of two polynomial equations describing: (i) the collinearity of the slowness-related Hamiltonian gradient and the ray velocity direction (third-order polynomial equation) and (ii) the vanishing Hamiltonian (fourth-order polynomial equation). Such a system has up to twelve real and complex-conjugate solutions, which appear in pairs of the opposite slowness directions. The common additional constraint, that the angle between the slowness and ray directions does not exceed ${90^{\rm{o}}}$, cuts off one half of the solutions. We rearrange the two bivariate polynomial equations and the above-mentioned constraint as a single univariate polynomial equation of degree six for qP and qSV waves, where the unknown parameter is the phase angle between the slowness vector and the medium symmetry axis. The slowness magnitude is then computed from the quadratic Christoffel equation, with a clear separation of compressional and shear roots. The final set of slowness solutions consists of a unique real solution for qP wave and one or three real solutions for qSV (due to possible triplications). The indication for a qSV triplication is a negative discriminant of the sixth-order polynomial equation, and this discriminant is computed and analysed directly in the ray-angle domain. The roots of the governing univariate sixth-order polynomial are computed as eigenvalues of its companion matrix. The slowness of the SH wave is obtained from a separate equation with a unique analytic solution. We first present the resolving equation using the stiffness components, and then show its equivalent forms with the well-known parametrizations: Thomsen, Alkhalifah and ‘weak-anisotropy’. For the Thomsen and Alkhalifah forms, we also consider the (essentially simplified) acoustic approximation for qP waves governed by the quartic polynomials. The proposed method is coordinate-free and can be applied directly in the global Cartesian frame. Numerical examples demonstrate the advantages of the method.

The Suggested Reciprocal Relationship between Maximum, Minimum and Optimum Usable Frequency Parameters Over Iraqi Zone

In this work, the relationship between the ionospheric parameters (Maximum Usable Frequency (MUF), Lowest Usable Frequency (LUF) and Optimum working Frequency (OWF)) has been studied for the ionosphere layer over the Iraqi zone. The capital Baghdad (44.42oE, 33.32oN) has been selected to represent the transmitter station and many other cities that spread over Iraqi region have represented as receiver stations. The REC533 communication model considered as one of the modern radio broadcasting version of ITU has been used to calculate the LUF parameter, while the MUF and OWF ionospheric parameters have been generated using ASAPS international communication model which represents one of the most advanced  and accurate HF sky wave propagation models. The study has been conducted for the annual and seasonal time periods of the years (2009 and 2014) of the solar cycle 24. The results of the seasonal and annual tests have indicated that the interrelationship between the MUF and OWF with LUF was a fourth order polynomial equation, while the reciprocal relationship between the MUF and OWF was a simple relationship that could be represented by a linear regression equation. The reciprocal relationships between MUF, LUF and OWF parameters (present values) have shown a good fitting with the data generated using the international models (predicted values) and theoretical values calculated from the criterion equation.

Density, speed of sound, viscosity, refractive index, surface tension and solubility of С60[C(COOH)2]3

In this paper, experimental data on different physicochemical properties of С60[C(COOH)2]3 aqueous solutions are presented. Concentration and temperature dependences on density (ρ), viscosity (η), refractive index (nD), speed of sound (u), surface tension (γ), as well as nanoparticle size dimensions (δi) of water solutions were obtained and characterized. The obtained T-x dependences of density, viscosity, refraction index and speed of sound were correlated using fourth-order polynomial equation. Moreover, the solubility diagram of carboxylated fullerene derivative in ternary system С60[C(COOH)2]3–NaCl–H2O at T = 298.15 K was investigated.

Comment on “Measurement and Correlation of Solubility of Two Isomers of Cyanopyridine in Eight Pure Solvents from 268.15 to 318.15 K”

Several mathematical errors in the published paper by Zhang and co-workers are identified. The errors pertain to the published equation coefficients for the modified Apelblat, for the empirical fourth-order polynomial equation, and for the Buchlowski–Ksiazcyak λh. The published curve-fit equation coefficients do not yield the back-calculated mole fraction solubilities that are reported in the published paper.

Analysis and validation of a new extended method for estimating plasma free cortisol including neutrophil elastase and competition from other steroids

A non-linear mechanistic model for the distribution of cortisol in plasma on free and bound forms is proposed. The influence of progesterone, testosterone and neutrophil elastase on the cortisol distribution in the blood is investigated.The activity of neutrophil elastase is directly included in the model with the concentration of elastase and the kinetic constants describing the activity of elastase collected in one single input variable. The model is very sensitive towards this input variable and fits data excellently, when it is allowed to be subject specific.The analysis shows that steroids such as testosterone with low affinity for corticosteroid-binding globulin (CBG) do not significantly influence the concentration of free cortisol. Progesterone has a high affinity for CBG, but low plasma concentrations compared to cortisol. Contrary to expectations, progesterone is shown to impact the distribution of cortisol in plasma both under circumstances with high levels as seen in pregnancy and during the normal menstrual cycle of women.Comparing the predictions of our model with predictions made with the equilibrium models by Coolens et al. [1], Dorin et al. [2] and Nguyen et al. [3] shows that the models differ considerably not only in their predictions for free cortisol, but also for cortisol on bound forms; i.e. bound to albumin, intact CBG and elastase-cleaved CBG.Disregarding some of the smallest terms of the model equations a reduced version of the model in form of a fourth order polynomial equation is obtained. The reduced version of the model performs almost identically to the full version and serves as a new formula for calculating the plasma free cortisol concentration.

Human evaluation in association to the mathematical analysis of arch forms: Two-dimensional study

To evaluate the relationship between human evaluation of the dental-arch form, to complete a mathematical analysis via two different methods in quantifying the arch form, and to establish agreement with the fourth-order polynomial equation. This study included 64 sets of digitised maxilla and mandible dental casts obtained from a sample of dental arch with normal occlusion. For human evaluation, a convenient sample of orthodontic practitioners ranked the photo images of dental cast from the most tapered to the less tapered (square). In the mathematical analysis, dental arches were interpolated using the fourth-order polynomial equation with millimetric acetate paper and AutoCAD software. Finally, the relations between human evaluation and mathematical objective analyses were evaluated. Human evaluations were found to be generally in agreement, but only at the extremes of tapered and square arch forms; this indicated general human error and observer bias. The two methods used to plot the arch form were comparable. The use of fourth-order polynomial equation may be facilitative in obtaining a smooth curve, which can produce a template for individual arch that represents all potential tooth positions for the dental arch.

A P3P problem solver representing all parameters as a linear combination

We propose a novel strategy for the Perspective-Three-Point (P3P) problem that determines the position and orientation of a calibrated camera from three known point pairs of 2D-3D correspondences. Starting from three similarity transformation equations that relate the global and the camera-oriented coordinates, our method treats all the extrinsic camera parameters as a linear combination of known vectors with unknown coefficients. By reducing the number of unknowns and using a Gröbner basis, the problem is converted into a fourth-order polynomial equation with a single unknown parameter. Experimental results show that our method is highly practical and precise. Moreover, the performance of our method and its robustness to image noise are similar to those of a state-of-the-art method. In addition, our method exhibits greater computational efficiency than other methods.

Optimization Technique to Determine the p-y Curves of Laterally Loaded Stiff Piles in Dense Sand

Abstract Lateral loading is often the governing design criteria for piles supporting offshore wind turbines and with the recent growth of this sector, the reliability of traditional design approaches is receiving renewed interest. To accurately assess the behavior of a laterally loaded pile requires a detailed understanding of the soil reaction that is mobilized as the lateral deflection of the pile occurs. Currently, the p-y curve method is widely adopted to model the response of laterally loaded piles. The limitations of existing p-y formulations are widely known, and there is acceptance that load tests on large-diameter stiff monopiles are urgently required to formulate appropriate design methods. However, interpretation of the data from instrumentation placed on stiff monopiles is not straightforward. This paper proposes an optimization technique to derive the soil reaction profile along the shaft of instrumented piles, from which the correlated p-y curves can then be obtained. The method considers force equilibrium, pile deflection, and additional boundary conditions. A set of fourth-order polynomial equations are assumed to model the soil reaction profile under each load step during a monotonic load test. By minimizing the difference between the measured and calculated bending moment and considering equilibrium of the shear forces acting on the pile, the soil reaction profile and the concentrated tip resistance can be obtained simultaneously. A stiff instrumented test pile installed in over-consolidated sand was load tested and the results were used to test the performance of the proposed method. The results are compared with other methods used in literature and practice. The method provides a consistent framework to derive p-y curves from measured strain data. The results of the field test and derived p-y curves confirmed that existing design methods do not accurately capture the lateral loading response of piles in dense sand.

Recognition of silver nanoparticles surface-adsorbed citrate anions by macrocyclic polyammonium cations: a spectrophotometric approach to study aggregation kinetics and evaluation of association constant.

In this report, we have studied the recognition of citrate anions adsorbed on the surface of silver nanoparticles (cit-Ag-NPs), by macrocyclic polyammonium cations (MCPACs): Me6 [14]ane-N4 H8 (4+) (Tet-A/Tet-B cations) and [32]ane-N8 H16 (8+) , which are well reputed anion recognizers and are treated as to mimic of biological polyamines. The study was monitored on ultraviolet-visible spectroscopy by performing a titration of the aqueous dispersion of the cit-Ag-NPs by the aqueous solution of MCPACs. The ultraviolet-visible time-scan plots over the reduction of the absorption band of surface plasmon resonance of cit-Ag-NPs at 390 nm are well fitted with fourth-order polynomial equation and are employed to determine the initial aggregation rate constants. It has been stated that the aggregation is the result in electrostatic attraction followed by H-bond formation between the surface-adsorbed citrate anions and added MCPACs. The atomic force microscopy results have evidenced aggregation of cit-Ag-NPs in presence of MCPACs. The evaluated H-bonded association constant (Kasso ) using Benesi-Hildebrand method reveals that [32]ane-N8 H16 (8+) cations form stronger association complex, as expected, with the citrate anions than the Me6 [14]ane-N4 H8 (4+) cations. The study would thus provide the insight of molecular interactions involved in nanoparticle surface-adsorbed anions with biological polyamines. Copyright © 2016 John Wiley & Sons, Ltd.

Stability Characterization of Three Porous Layers Model in the Presence of Transverse Magnetic Field

The current study concerns, the effect of a horizontal magnetic field on the stability of three horizontal finite layers of immiscible fluids in porous media. The problem examines few representatives of porous media, in which the porous media are assumed to be uniform, homogeneous and isotropic. The dispersion relations are derived using suitable boundary and surface conditions in the form of two simultaneous Mathieu equations of damping terms having complex coefficients. The stability conditions of the perturbed system of linear evolution equations  are investigated  both analytically and numerically and stability diagrams are obtained. The stability diagrams are discussed in detail in terms of various parameters governing the flow on the stability behavior of the system such as the streaming velocity, permeability of the porous medium and the magnetic properties. In the special case of uniform velocity, the fluid motion has been displayed in terms of streamlines concept, in which the streamlines contours are plotted. In the  uniform velocity motion, a fourth order polynomial equation with complex coefficients is obtained. According to the complexity of the mathematical treatments, when the periodicity of the velocity is taken into account, the method of multiple scales is applied to obtain stability solution for the considered system.&lt;br /&gt;It is found that a stability effect is found for increasing, the magnetic permeability ratio, the magnetic field, and  the permeability parameter  while the opposite influence is observed for increasing the upper layer velocity.

Salinity and turbidity distributions in the Brisbane River estuary, Australia

Summary The Brisbane River estuary (BRE) in Australia not only plays a vital role in ecosystem health, but is also of importance for people who live nearby. Comprehensive investigations, both in the short- and long-term, into the salinity and turbidity distributions in the BRE were conducted. Firstly, the analysis of numerical results revealed that the longitudinal salinity varied at approximately 0.45 and 0.61 psu/h during neap and spring tides, respectively. The turbidity stayed at a higher level and was less impacted by tide in the upper estuary, however, the water cleared up while the tide changed from flood to ebb in the mid and lower estuary. The second investigation into the seasonal variations of salinity and turbidity in the BRE was conducted, using ten-year field measurement data. A fourth-order polynomial equation was proposed, describing the longitudinal variation in salinity dilution changes as the upstream distance in the BRE during the wet and dry seasons. From the observation, the mid and upper estuaries were vertically well-mixed during both seasons, but the lower BRE was stratified, particularly during the wet season. The estuary turbidity maximum (ETM) zone was about 10 km longer during the wet season than the dry season. Particular emphasis was given to the third investigation into the use of satellite remote sensing techniques for estimation of the turbidity level in the BRE. A linear relationship between satellite observed water reflectance and surface turbidity level in the BRE was validated with an R 2 of 0.75. The application of satellite-observed water reflectance therefore provided a practical solution for estimating surface turbidity levels of estuarine rivers not only under normal weather conditions, but also during flood events. The results acquired from this study are valuable for further hydrological research in the BRE and particularly prominent for immediate assessment of flood impacts.

Liquid–Liquid Equilibrium of Primary and Secondary Aqueous Two-Phase Systems Composed of Sucrose + Triton X-114 + Water at Different Temperatures

Liquid–liquid equilibrium (LLE) data for aqueous two-phase systems (ATPSs) composed of sucrose, Triton X-114, and water were determined at 288 K, 293 K, and 298 K. At 293 K and 298 K, the top and bottom phases of the ATPS were enriched with sucrose and Triton X-114, respectively. However, phase inversion was observed in the ATPS formed at a lower temperature (288 K). When the temperature increased, the area of the biphasic region became larger, whereas the slopes and the lengths of the tie lines decreased. Binodal data obtained at different temperatures were correlated using a fourth-order polynomial equation, while tie-line data were correlated using the Othmer–Tobias and Bancroft equations. Moreover, the Setschenow equation was used to investigate the sugaring-out effect of sucrose on the formation of the ATPS. The correlated results and fitting parameters from these empirical equations confirmed the reliability and reproducibility of the experimental results. Furthermore, both top- and bottom-phase solutions can be heated to 303 K to form the secondary ATPSs composed of sucrose and Triton X-114, respectively. The LLE data for secondary ATPSs are presented in this paper.

Complete solutions of zoom curves of three-component zoom lenses with the second component fixed.

Purely algebraic algorithms are presented for solving the zoom curves of a three-component zoom lens of which the second component is fixed on zooming. Two separated algorithms for infinite and finite conjugate imaging conditions are provided. For the infinite-conjugate condition, the transverse magnifications of the second and third components are solved to match the required system focal length, resulting in solving a quadratic equation. For the finite-conjugate condition, three nonlinear simultaneous equations regarding the system magnification, the object-to-image thickness, and the position of the second component are combined into a fourth-order polynomial equation. The roots can all be directly obtained by simple algebraic calculations. As a result, the proposed algebraic algorithms provide a more efficient and complete method than do earlier algorithms adopting scanning procedures.

Least square approximate motion generation synthesis of spherical linkages by using Chebyshev and equal spacing

In this paper, approximate motion synthesis of spherical linkages is presented. Rigid body guidance of a spherical four-bar mechanism is performed by a spherical RR open chain. In the first step, position of a point on rigid body and orientation of a rigid body on a unit sphere are described. Synthesizing function of spherical dyad is derived by means of using unit vectors that describe location of two revolute joints and tip point. Being based on the theory of function approximation and besides the linearization of nonlinear synthesis equations by using superposition method, the design procedure for real solutions of fourth order polynomial equation is developed. In the second step, approximate motion generation synthesis of spherical dyad is presented by using least-square approximation. Chebyshev spacing and equal spacing are used in the determination of poses. In the final step, two numerical examples are given to show how error graph is varied in terms of selected poses. The spherical motion generation synthesis of spherical four-bar mechanism is obtained by the combination of the two real solutions of the synthesis of two spherical dyads.

Weight Function Method With Segment-Wise Polynomial Interpolation to Calculate Stress Intensity Factors for Complicated Stress Distributions

Abstract Many solutions of the stress intensity factor have been proposed in recent years. However, most of them take only third or fourth-order polynomial stress distributions into account. For complicated stress distributions which are difficult to be represented as third or fourth-order polynomial equations over the stress distribution area such as residual stress distributions or thermal stress distributions in dissimilar materials, it is important to further improve the accuracy of the stress intensity factor. In this study, a weight function method with segment-wise polynomial interpolation is proposed to calculate solutions of the stress intensity factor for complicated stress distributions. By using this method, solutions of the stress intensity factor can be obtained without employing finite element analysis or difficult calculations. It is therefore easy to use in engineering applications. In this method, the stress distribution area is firstly divided into several segments and the stress distribution in each segment is curve fitted to segment-wise polynomial equation. The stress intensity factor is then calculated based on the weight function method and the fitted stress distribution in each segment. Some example solutions for both infinite length cracks and semi-elliptical cracks are compared with the results from finite element analysis. In conclusion, it is confirmed that this method is applicable with high accuracy to the calculation of the stress intensity factor taking actual complicated stress distributions into consideration.

Subjective classification and objective analysis of the mandibular dental-arch form of orthodontic patients

Our objective was to evaluate the relationship between subjective classification of dental-arch shape, objective analyses via arch-width measurements, and the fitting with the fourth-order polynomial equation. Twenty-seven pretreatment mandibular dental casts (from 13 males and 14 females; ages, 12-31 years) were selected. Standardized photographs of the arches were serially organized from tapered to square by 10 examiners. The mean position in the ranking of each cast was calculated as a rank of each arch form. The dental casts were analyzed with a 3-dimensional laser scanning system. Dental-arch widths at the canines and molars were measured, and then a fourth-order polynomial equation was fit to each arch. Correlations between the rank of arch shape and the objective measurements were statistically tested. The arch forms having the greatest variations among the examiners were those with an intermediate (ovoid) ranking. Statistically significant correlations were found between the ranks of arch shape, arch dimensions, and the polynomial equation analyses. Subjective clinical assessments were generally in agreement at the extremes of tapered and square arch forms; the exceptions were arches with an ovoid shape. There were statistically significant correlations between subjective dental-arch classifications and dental-arch dimensions, as well as the ratio determined from these variables and polynomial equation analyses. Therefore, fourth-order polynomial equations might be an important factor in the quantitative analysis of dental-arch form in orthodontic patients.