Polynomial Equation Research Articles

Integral Solutions for the Diophantine Equation of Higher Degree with Six Unknowns x⁶ − y⁶ − 3456z³ = 800(p² − q²)R⁸

Our focus in this paper has been on solving high-power Diophantine equations - those with variables raised to a high degree. These types of equations can be particularly challenging as they involve finding integer solutions that satisfy the given polynomial equation. We have utilized various techniques such as brute force methods and substitution strategies to solve these high power Diophantine equations, successfully deriving their solutions. Furthermore, our investigation has led us to uncover intriguing relationships among these solutions, which manifest in four distinct patterns. The equation x⁶ − y⁶ − 3456z³ = 800(p² − q²)R⁸ is analysed with properties. Some of the special numbers are discussed in properties. Special numbers are unique and have special qualities that set them apart from other numbers. Learning about these special qualities helps us understand how numbers work and their significance in different areas of math and science.

Solvable non-Hermitian skin effects and real-space exceptional points: non-Hermitian generalized Bloch theorem

Non-Hermitian systems can exhibit extraordinary boundary behaviors, known as the non-Hermitian skin effects, where all the eigenstates are localized exponentially at one side of lattice model. To give a full understanding and control of non-Hermitian skin effects, we have developed the non-Hermitian generalized Bloch theorem to provide the analytical expression for all solvable eigenvalues and eigenstates, in which translation symmetry is broken due to the open boundary condition. By introducing the Vieta’s theorem for any polynomial equation with arbitrary degree, our approach is widely applicable for one-dimensional non-Hermitian tight-binding models. With the non-Hermitian generalized Bloch theorem, we can analyze the condition of existence or non-existence of the non-Hermitian skin effects at a mathematically rigorous level. Additionally, the non-Hermitian generalized Bloch theorem allows us to explore the real-space exceptional points. We also establish the connection between our approach and the generalized Brillouin zone method. To illustrate our main results, we examine two concrete examples including the Su–Schrieffer–Heeger chain model with long-range couplings, and the ladder model with non-reciprocal interaction. Our non-Hermitian generalized Bloch theorem provides an efficient way to analytically study various non-Hermitian phenomena in more general cases.

Bridging theory and application: An in-depth study of the greatest common divisor and least common multiple in mathematics and real-world settings

The study of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) explores the fundamental field of number theory. In addition to being essential to theoretical number theory, these mathematical ideas have extensive applications in computer science, algebra, and real-world settings like network routing and scheduling. Using techniques like the Euclidean algorithm for GCD calculation and prime factorization for LCM, the study starts by examining the fundamental concepts and notations of GCD and LCM. Then the author goes further into the detail about how these ideas are used to polynomial equations, highlighting how they may be used to solve challenging issues and simplify algebraic expressions. The studys practical applications, which demonstrate how GCD and LCM optimize scheduling tasks and boost network routing efficiency, are presented in the conclusion. The results of this study highlight the broad impact and adaptability of LCM and GCD in theoretical and applied mathematics, providing valuable insights for further research in these domains.

A quadratic conservation algorithm for non‐hydrostatic models

AbstractWe propose a quadratic conservation algorithm for non‐hydrostatic models, focusing on energy conservation and consistency. To ensure that the discrete potential and continuity equations are equivalent and to achieve conservation of mass and momentum transport, we introduce a potential reference height into the discrete potential equation. The time integration scheme is based on an explicit fourth‐order Runge–Kutta method with varying time steps, which is specially designed to maintain the quadratic conservation property. To suppress numerical noise, we suggest an adaptive diffusion method that does not lose energy. Numerical noise is estimated by higher order terms of polynomial equations, and diffusion coefficients are determined using a least‐squares method. Numerical tests demonstrate that the quadratic conservation algorithm accurately maintains total energy conservation and produces solutions of comparable quality to those reported in existing literature. Furthermore, it can resolve problems with small‐scale features.

Hydrodynamic Performance Analysis of Camber Ratio Variations on B-series Propeller Types

The hydrodynamic performance of the B-series propeller can be determined by calculating the polynomial equation published by MARIN. Furthermore, analysis and evaluation of B-series propeller modifications can be done by varying the camber ratio of the foil. The camber ratio affects the lift force on the propeller foil, directly affecting the propeller thrust and torque. Numerical calculations were carried out using Computational Fluids Dynamics (CFD), based on Reynolds Averaged Navier Stokes Equations (RANSE) and turbulence model in the form of explicit algebraic stress models (EASM). The overall results of this study show an increase in efficiency of 4.182 on the foil with a camber ratio of 2.2% when compared to the foil camber ratio of 0%.

The One-Cushion Escape from Snooker in a Circular Table

Abstract The one-cushion escape from snooker in a circular table can be viewed as a ge-ometric problem involving the reflection of a light ray in a circular mirror. There are at most four escapes in any given configuration. We will obtain specific configurations in which there are exactly 0, 1, 2, 3 or 4 escapes. The details consist in determining the number of real solutions in the interval (−1; 1) of certain polynomial equations, of degrees two, three or four.

ALGEBRAIC STRUCTURES ON A SET OF DISCRETE DYNAMICAL SYSTEM AND A SET OF PROFILE

A discrete dynamical system is represented as a directed graph with graph nodes called states that can be seen on the dynamical map. This discrete dynamical system is symbolized by , where  is a finite set of states and the function g  is a function from  to . In the dynamical map, the discrete dynamical system has a height where the number of states in each height is called a profile. The set of discrete dynamical systems has an addition operation defined as a disjoint union on the graph and a multiplication operation defined as a tensor product on the graph. The set of discrete dynamical systems and the set of profiles are very interesting to observe from the algebraic point of view. Considering operation on the set of discrete dynamical systems and the set of profiles, we can see their algebraic structure. By recognizing the algebraic structure, it will be easy to solve the polynomial equation in the discrete dynamical system and in the profile. In this research, we will investigate the algebraic structure of discrete dynamical systems and the set of profiles. This research shows that the set of discrete dynamical system has an algebraic structure, which is a commutative semiring and the set of profiles has an algebraic structure, which is a commutative semiring and -semimodule. Moreover, both sets have the same property, which is isomorphic to the set of non-negative integers.

Anisotropic media with singular slowness surfaces

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

Development of a novel fluorimetric sensor based on polypyrrole doped with 4-(methylsulfonyl)-2-nitrobenzoic acid (MSNBA) for sensitive detection of copper (Cu2+)

In this study, we successfully electrochemically synthesized a polypyrrole film doped with 4-methylsulfonyl-2-nitrobenzoic acid (MSNBA) within an aqueous medium containing 0.01 M MSNBA, 0.003 M NaOH, and 0.01 M pyrrole. Our investigation encompassed a thorough exploration of both electrochemical and spectroscopic properties, confirming the effective incorporation of MSNBA as a dopant within the electroactive PPy chain. The morphological characterization of the film was undertaken using scanning electron microscopy (SEM). The optical properties, particularly UV–visible absorption, and fluorescence spectroscopy, were scrutinized in a dilute DMF solution. The UV–visible absorption spectra revealed a broad absorption band centered around 292 nm, attributed to the π→π* 1B - 1La/1Lb transitions of the oligo-benzenes in the film. In parallel, fluorescence spectra exhibited remarkable consistency between the solid state and solution, featuring an excitation peak at 345 nm (346 nm in solution) and an emission peak around 401 nm (403 nm in solution). Exploring the impact of copper (II) ions on absorption and fluorescence spectra unveiled a notable quenching effect along two distinct linearity domains. Various mathematical treatments, including the Stern-Volmer equation (R1 = 0.98, Ksv1 = 5.104 and R2 = 0.99, Ksv2 = 1.54.106), the Perrin equation, and the polynomial equation, were applied to elucidate the nature of this fluorescence quenching. Our findings indicated that the fluorescence extinction, as a function of Cu2+ ion concentration, aligns well with the polynomial equation, signifying a combination of dynamic and static quenching (Io/I = 1.08(±0.04) + 1.46(±0.44)105[Q]- 1.15(±0.75)1010 [Q]2). Moreover, we explored the potential application of this method for the determination of Cu2+. Calibration graphs demonstrated excellent linearity between fluorescence intensity and Cu2+ concentration in the range of 0–5.4 μM. Notably, the detection limit of 1.2 μM surpassed the LOD for Cu2+ (20 μM) in drinking water set by the EPA, suggesting the viability of this approach for sensitive detection in environmental monitoring.

A note on polynomial equations over algebras

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three.

Intermolecular Interactions of Binary Mixtures Comprising 2‐Benzylaminoethanol with Alcohols (C1−C3) at Different Temperatures: Experiments and Modelling

AbstractThis manuscript represents the physical quantities comprising the densities, speed of sound and refractive indices for the binary mixtures involving [2‐Benzylaminoethanol (BAE) with methanol, ethanol, 1‐propanol, or 2‐propanol] over a molar composition (xBAE=0–1). The measurements were conducted with the use of Anton Paar densimeter (DSA 5000 M) operated at different temperatures, T=(293.15–323.15) K and at atmospheric pressure. The intermolecular interactions occurring between the prepared binary mixtures were interpreted by computing the excess properties including excess molar volumes ( ), isentropic compressibilities ( ), intermolecular free length ( ), specific acoustic impedance ( ), relative association ( , relaxation strength ( ), and Rao's molar sound function ( ). The Redlich‐Kister polynomial equation was utilized to fit the computed excess properties and PC‐SAFT EoS was utilized for modeling the measured densities. In addition, the Laplace mixing rule as well as the Nomoto's relation were utilized for the modeling of refractive indices and speed of sound, respectively. The experimental data was found in good agreement with the theoretical data for the measured properties.

Airflow resistance analysis of wire screens used in waste heat recovery systems

Wire screens are widely used in electronics, refrigeration, insect screens, food processing, filtration, oil refinery, agricultural, and other applications. For instance, these wire screens are used as heat exchange materials in regenerators due to higher surface area density. Heat transmission can typically be enhanced by increasing the surface area density of wire screens, however flow obstruction is a possibility. As a result, this wire screen analysis is necessary not only to estimate the heat transfer rate but also to assess the airflow obstruction. This extensive study investigates the role of wire screens in a wind tunnel on the different velocities of air moving through them in order to compute the pressure drop. Six wire screens were examined as part of this investigation, and the experimental results were utilised to derive the coefficients of second order polynomials (a,b, and c). Permeability and inertial factor were calculated using this second order polynomial and Forchheimer’s equations. Second-order polynomial equations were found to be the best fit for the permeability versus porosity ratio and the inertial factor versus porosity ratio.

Viscosity Studies of Binary Liquid Mixture of Methyl Acrylate and Benzene and Substituted Halo Benzenes at 313.15k

Studied the binary liquid mixtures of Methyl Acrylate with Benzene (B), and Substituted Halo Benzene molecules like Chloro Benzene (CB) and Bromo Benzene (BB) at 313.15K. Initially, the density and viscosities of the mixtures are measured and this data is utilized to calculate excess properties like excess molar volume (VE), excess viscosity (ηE), and excess Gibbs free energy (ΔG*E). These excess parameter values are fitted in the Redlich–Kister Polynomial equation and the intermolecular interaction is discussed.

Temporal dynamics of volatile fatty acids profile, methane production, and prokaryotic community in an in vitro rumen fermentation system fed with maize silage.

Anaerobic in vitro fermentation is widely used to simulate rumen kinetics and study the microbiome and metabolite profiling in a controlled lab environment. However, a better understanding of the interplay between the temporal dynamics of fermentation kinetics, metabolic profiles, and microbial composition in in vitro rumen fermentation batch systems is required. To fill that knowledge gap, we conducted three in vitro rumen fermentations with maize silage as the substrate, monitoring total gas production (TGP), dry matter degradability (dDM), and methane (CH4) concentration at 6, 12, 24, 36, and 48 h in each fermentation. At each time point, we collected rumen fluid samples for microbiome analysis and volatile fatty acid (VFA) analysis. Amplicon sequencing of 16S rRNA genes (V4 region) was used to profile the prokaryotic community structure in the rumen during the fermentation process. As the fermentation time increased, dDM, TGP, VFA concentrations, CH4 concentration, and yield (mL CH4 per g DM at standard temperature and pressure (STP)) significantly increased. For the dependent variables, CH4 concentration and yield, as well as the independent variables TGP and dDM, polynomial equations were fitted. These equations explained over 85% of the data variability (R2 > 0.85) and suggest that TGP and dDM can be used as predictors to estimate CH4 production in rumen fermentation systems. Microbiome analysis revealed a dominance of Bacteroidota, Cyanobacteria, Desulfobacterota, Euryarchaeota, Fibrobacterota, Firmicutes, Patescibacteria, Proteobacteria, Spirochaetota, and Verrucomicrobiota. Significant temporal variations in Bacteroidota, Campylobacterota, Firmicutes, Proteobacteria, and Spirochaetota were detected. Estimates of alpha diversity based on species richness and the Shannon index showed no variation between fermentation time points. This study demonstrated that the in vitro fermentation characteristics of a given feed type (e.g., maize silage) can be predicted from a few parameters (CH4 concentration and yield, tVFA, acetic acid, and propionic acid) without running the actual in vitro trial if the rumen fluid is collected from similar donor cows. Although the dynamics of the rumen prokaryotes changed remarkably over time and in accordance with the fermentation kinetics, more time points between 0 and 24 h are required to provide more details about the microbial temporal dynamics at the onset of the fermentation.

Magnetic field alignment in low-mass molecular clouds: the role of turbulence and density of the clouds

ABSTRACT In this work, we report the role of turbulence (measured by the $^{12}\mathrm{ CO}$ linewidth, $\Delta V$) on the offset between the orientation of local magnetic field in the low-density regions of dark clouds ($\theta ^{\mathrm{ env}}_\mathrm{ B}$) and the position angle of the Galactic Plane (GP; $\theta _{\mathrm{ GP}}$) in 15 low-mass isolated molecular clouds. We find a dependency of $\Delta V$ on offset, ($\theta _{\mathrm{ off}} =|\theta ^{\mathrm{ env}}_\mathrm{ B}-\theta _{\mathrm{ GP}}|$), represented by a second-order polynomial equation, $\theta _{\mathrm{ off}}=(3.328 \pm 1.187) {\Delta V}^2 - (0.991 \pm 5.541) {\Delta V} + (4.767 \pm 5.309)$. In regions where the majority of low turbulence is observed, characterized by $\Delta V \ \lt\ 3.1 \ \mathrm{ km\,s}^{-1}$, the envelope magnetic fields closely align with the orientation of the GP ($\theta _{\mathrm{ off}}\ \lt\ 20^\circ$). Conversely, in areas exhibiting high turbulence with $\Delta V \ \gt\ 3.1 \ \mathrm{ km\,s}^{-1}$ (for clouds CB130 and CB188), the envelope magnetic field tends to misalign with the GP ($\theta _{\mathrm{ off}}\ \gt\ 50^\circ$). This suggests that turbulence could be one of the important factors influencing the alignment of the local magnetic field with the GP. We have also explored the potential link between gas density of clouds ($\rho$) and turbulence ($\Delta V$) for the first time, utilizing data from 10 isolated clouds. This analysis suggests a possible correlation, with the observed trend quantified as, $\Delta V \propto \rho ^{-0.19}$.

Creativity in High School Through Drawing with Polynomials

High school math projects center on solving real-world examples in a wide variety of situations that require a creative skill set. Creativity in high school mathematics takes place in a subtle form. In contrast, creativity presents itself conspicuously in art. Therefore, this paper will explore the utilization of polynomial equations in drawing various figures as part of algebra class projects. Additionally, these projects incorporate integrated digital tools in mathematics. The paper will analyze three algebraic projects: the first project involves drawing a bird, the second project focuses on drawing a butterfly, and the third project entails drawing a bat using polynomial equations with the technological tool Desmos. All three projects are products of student work that encompass figures requiring up to forty polynomial equations. The students' impressive work demonstrates the power of mathematics in pushing the boundaries of other subjects. As digital technology becomes increasingly integral, mathematical tools are positioned as universal tools for the future and are poised to rival artists in producing visual art. This raises the intriguing question: can a mathematician create better art with mathematical equations through digital technology than an artist?

Refractive index and related properties for the mixture (1, 4-dioxane + water) at temperature T= (295.15, 298.15, 301.15, 304.15, 307.15, 310.15 and 313.15) K

ABSTRACT The refractive index n D for the mixture (1,4-dioxane (1) + water (2)) was measured over the whole composition range at different temperatures and atmospheric pressure. From measured values of the refractive indices, the molar refraction R m , reduced molar free volumes V m R m , molecular radii r and internal pressure, P i n t were calculated. On the other hand, some excess parameters were also deduced. The obtained results have been discussed and interpreted in terms of the type and nature of specific intermolecular interactions existed between the components. Furthermore, we have used nine mixing rules for refractive index proposed by different authors to calculate the refractive indices of the present mixture. A comparative study was carried out between the measured and calculated indices. The Redlich–Kister polynomial equation has been used to fit the presented excess parameters to derive the adjustable fitting coefficients and estimate the standard error values.

On Hurwitz stability for families of polynomials

AbstractThe robustness of a linear system in the view of parametric variations requires a stability analysis of a family of polynomials. If the parameters vary in a compact set , then obtaining necessary and sufficient conditions to determine stability of the family is one of the most important tasks in the field of robust control. Three interesting classes of families arise when is a diamond, a box or a ball of dimension . These families will be denoted by , , and , respectively. In this article, a study is presented to contribute to the understanding of Hurwitz stability of families of polynomials . As a result of this study and the use of classical results found in the literature, it is shown the existence of an extremal polynomial whose stability determines the stability of the entire family . In this case comes from minimizing determinants and in some cases coincides with a Kharitonov's polynomial. Thus another extremal property of Kharitonov's polynomials has been found. To illustrate our approach, it is applied to families such as , , and with . The study is also used to obtain the maximum robustness of the parameters of a polynomial. To exemplify the proposed results, first, a family is taken from the literature to compare and corroborate the effectiveness and the advantage of our perspective. Followed by two examples where the maximum robustness of the parameters of polynomials of degree 3 and 4 are obtained. Lastly, a family is proposed whose extreme polynomial is not necessarily a Kharitonov's polynomial. Finally, a family is used to exemplify that if the boundary of is given by a polynomial equation in several variables, the number of candidates to be an extremal polynomial is finite.

Instability analysis of swirling cylindrical Rivlin–Ericksen viscoelastic fluid–viscous fluid interface with heat and mass transfer

The applications of swirling interfaces with heat and mass transfer are diverse and impactful, spanning industries from energy and manufacturing to healthcare and environmental protection. This study focuses on the stability of such interfaces where a viscous fluid interacts with a Rivlin–Ericksen (RE) viscoelastic fluid, undergoing heat and mass transfer. In this paper, the fluids are enclosed between two cylinders, one stationary and the other rotating. Mathematical equations are solved using potential flow theory. The interface stability is assessed using a normal mode procedure, leading to a second-order polynomial equation. The study finds that swirling flow reduces perturbation amplification, especially when heat and mass transfer occur simultaneously. However, the viscoelastic nature of the Rivlin–Ericksen fluid destabilizes the interface. Overall, this research provides valuable insights into complex fluid behavior with applications across industries.

Determination of the solubility, dissolution enthalpy and entropy of 2,4-diaminotoluene, methyl 3-amino-4-methyl-N-phenyl carbamate and dimethyl toluene-2,4-dicarbamate in different solvents

The solubilities of 2,4-diaminotoluene (2,4-TDA), methyl 3-amino-4-methyl-N-phenyl carbamate (4-TMC) and dimethyl toluene-2,4-dicarbamate (2,4-TDC) in dimethyl carbonate (DMC) and di(2-ethylhexyl) phthalate (DOP) were determined by dynamic method at 298.15 ∼ 398.15 K under atmospheric pressure. The solubility data were correlated by the modified Apelblat equation, polynomial equation and simplified two-parameter equation. The modified Apelblat equation was more accurate than the others. In addition, the thermodynamic properties of the solution process, including the Gibbs energy, enthalpy and entropy were calculated by the modified Van't Hoff equation. The dissolution process of 2,4-TDC in DOP solvent is a non-spontaneous entropy reduction endothermic process. The dissolution processes of 2,4-TDA, 4-TMC and 2,4-TDC in DMC solvent and 2,4-TDA and 4-TMC in DOP solvent are all non-spontaneous entropy-increasing endothermic processes.