Polynomial Equation Of Degree Research Articles

The Algebraicity of Ill-Distributed Sets

We show that every set \({S \subseteq [N]^d}\) occupying \({\ll p^{\kappa}}\) residue classes for some real number \({0 \leq \kappa < d}\) and every prime p, must essentially lie in the solution set of a polynomial equation of degree \({\ll ({\rm log} N)^C}\), for some constant C depending only on \({\kappa}\) and d. This provides the first structural result for arbitrary \({\kappa < d}\) and S.

Curva de maturação e estimativa do teor de sólidos solúveis e acidez total em função de graus-dia: uva IAC 138-22 'Máximo'

Devido à crescente importância de uvas para vinho no estado de São Paulo realizou-se a caracterização da curva de maturação da IAC 138-22 'Máximo', enxertada sobre o porta-enxerto IAC 766 Campinas e conduzida no sistema de espaldeira alta em safras de verão e de inverno. Os valores do teor de sólidos solúveis (TSS) foram superiores durante as safras de inverno (17,4 ºBrix a 17,5 ºBrix) em comparação às de verão (14,2 ºBrix a 15,6 ºBrix). A acidez titulável total (ATT) também apresentou maiores valores na safra de inverno. Verificou-se que a relação entre TSS e ATT e o acúmulo de graus-dia a partir do início de maturação (pintor) podem ser descritos com base em equações polinomiais de segunda ordem.

Drying characteristics of traditional fermented sausage Petrovská klobása: The effect of different ripening conditions and use of starter culture

A study was carried out on traditional Serbian dry-fermented sausage Petrovská klobása in order to investigate the effect of different ripening conditions and use of starter culture (SC) on drying characteristics of this typical meat product. The results related with reduction of diameter, water activity (aw) and moisture content of sausages, obtained during ripening, indicated 30 days faster drying process in controlled industrial ripening room (RR) than in traditional room (TR) (60 vs. 90 days). Along with the ambient (thermo-hygrometric) conditions, this phenomenon was also determined by pH, which was lower in RR sausages (P<0.05), especially in those made with SC. High values of coefficient of determination (r2>0.992) indicate good fit of the experimental data (average moisture content vs. drying time) using third degree polynomial equation, both for RR and TR sausages, whether the samples were produced with or without SC. Additionally, the evolution of moisture content ratio between internal and external fractions of sausages indicated more intense drying in the external ones. This was particularly pronounced in sausages made with SC after 30 days of drying. The use of SC and RR in processing of Petrovská klobása seems to be a potential technological improvement to shorten the drying period and to prolong the production season of this traditional meat product.

Análise dos Problemas propostos para o ensino de equações polinomiais do 1º grau com uma incógnita nos livros didáticos de matemática

This researcher reports an investigation that aimed to analyze the problems proposed for the tea ching of polynomial equations of the first degree with one unknown in the textbooks of Mat hematics Year 7 in Brazil, approved in the Programa Nacional do Livro Didatico (PNLD) de 2011. We classify the problems found, from the categories created by Marchand and Bednarz (1999) in their research in Canadian textbooks. As a result, highlight that Brazilians textbooks have a strong tendency to explore problems that do not favor the transition from arithmetic to algebra, called false problems, and structure problems arithmetic. Relative problems of algebraic structure, all books give preference to problems sharing amounts.

THE ROTA METHOD FOR SOLVING POLYNOMIAL EQUATIONS: A MODERN APPLICATION OF INVARIANT THEORY

Algebraic invariants are defined for the purpose of gaining insights into solving polynomial equations. Polynomial invariants are disclosed here as an alternative to and to clarify the umbral method of Gian-Carlo Rota. A process for solving cubic polynomial equations is examined and extended to quintic (or 5th degree) polynomial equations. It is proved that a general cubic polynomial is “apolar” to a quadratic polynomial. It is proved that a quadratic polynomial and cubic polynomial which are apolar either both have repeated roots or both have distinct roots. In the case of repeated roots, these roots are shared by the cubic and quadratic polynomials that are apolar. In the case, in which the derived quadratic which is apolar to a given cubic has distinct roots, it is shown the cubic polynomial px may be transformed to px = c1x − r1 + c2x − r2, where r1 and r2 are the distinct roots of the quadratic polynomial. This will allow the roots of the cubic to be found using algebraic operations. It will be shown that these methods can be extended to show that a given quintic polynomial is in general apolar to a cubic polynomial. Some remaining questions are posed at the end of the article. AMS Subject Classification: 13A50

Evaluation of Piecewise Polynomial Equations for Two Types of Thermocouples

Thermocouples are the most frequently used sensors for temperature measurement because of their wide applicability, long-term stability and high reliability. However, one of the major utilization problems is the linearization of the transfer relation between temperature and output voltage of thermocouples. The linear calibration equation and its modules could be improved by using regression analysis to help solve this problem. In this study, two types of thermocouple and five temperature ranges were selected to evaluate the fitting agreement of different-order polynomial equations. Two quantitative criteria, the average of the absolute error values |e|ave and the standard deviation of calibration equation estd, were used to evaluate the accuracy and precision of these calibrations equations. The optimal order of polynomial equations differed with the temperature range. The accuracy and precision of the calibration equation could be improved significantly with an adequate higher degree polynomial equation. The technique could be applied with hardware modules to serve as an intelligent sensor for temperature measurement.

Drainage Basin Delineation and Quantitative Analysis of Panamaram Watershed of Kabani River Basin, Kerala Using Remote Sensing and GIS

Abstract The morphometric analysis was carried out to determine the drainage characteristics of Panamaram watershed (PW) of Kabani river basin, Kerala, India with emphasis on fourth order sub-basins (FOSBs) using Geocoded imageries of IRS-IC LISS III FCCs (1997) of 1:50,000 scale, aerial photographs (1990) of 1:15,000 scale and Survey of India toposheets (1968) on 1: 50,000 scale. The main idea was to examine the stream properties based on the measurement of various stream attributes. The different drainage parameters studied and the measurements related to perimeter, area of sub-basins, basin length and number of rivers were determined by using Map Info 6.5 techniques. The drainage pattern of the PW is dendritic, a sixth order stream. There are 587, 135, 36 and 12 first, second, third and fourth order Hortonian streams, respectively in PW. The mean bifurcation ratio indicates that the drainage pattern is not much influenced by geological structures. The shape parameters reveal the elongation of the basin and sub-basins. The applicability of Horton’s Laws on Stream numbers, Stream lengths and Stream areas is tested by using the theory of regression by estimating theoretically bifurcation ratio, length ratio and area ratio. For the fourth order sub-basins of 1, 2, 4, 5 and 8 of PW, a second degree polynomial equation seems to be a better model than the Hortonian model.

Mean-square filter design for stochastic polynomial systems with Gaussian and Poisson noises

This paper addresses the mean-square finite-dimensional filtering problem for polynomial system states with both, Gaussian and Poisson, white noises over linear observations. A constructive procedure is established to design the mean-square filtering equations for system states described by polynomial equations of an arbitrary finite degree. An explicit closed form of the designed filter is obtained in case of a third-order polynomial system. The theoretical result is complemented with an illustrative example verifying performance of the designed filter.

ESTABLISHING LEAF NUTRIENT NORMS FOR ARECANUT BY BOUNDARY LINE APPROACH

Critical and optimum leaf nutrient standards for arecanut are unknown. As nutrients are becoming yield limiting factors in arecanut, corrective measures require development of leaf nutritional standards. The objective of this paper was to determine optimum concentration and range of all nutrients through boundary line approach. Optimum foliar concentrations for nitrogen (N), phosphorus (P), potassium (K), calcium (Ca) and magnesium (Mg) were established as 2.70, 0.23, 1.12, 0.61, and 0.20%, respectively, using leaf composition data from 12-year old arecanut plantation. Optima of major nutrient concentrations were nearly identical between boundary line approach and mean of high yielding population. Optimum micronutrient concentrations (mg kg−1) were estimated at 146 for iron (Fe), 56.5 for manganese (Mn), 2.6 for copper (Cu), 45.8 for zinc (Zn), 39.5 for boron (B), 432 for aluminum (Al) and 63 for sodium (Na). Optimum ranges and ratios of different nutrients were found out from second degree polynomial equations.

A Classic New Method to Solve Quartic Equations

Polynomials of high degrees often appear in many problems such as optimization problems. Equations of the fourth degree or so called quartics are one type of these polynomials. In this paper we give a new Classic method for solving a fourth degree polynomial equation (Quartic). We will show how the quartic formula can be presented easily at the precalculus level.

Tensiometric determination of Gibbs surface excess and micelle point: A critical revisit

Amphiphile adsorption at the air/water interface lowers the surface tension (γ) of the solution. After a critical surfactant concentration (C), γ becomes constant (with a break in the γ–logC plot), which is considered the critical micelle concentration (CMC). At very low amphiphile concentration, γ decreases slowly, forming a plateau, then decreases sharply and often nonlinearly by a co-operative adsorption process till the second plateau is reached at CMC. To get the Gibbs surface excess (Γ) of the amphiphile relative to water, a polynomial equation of appropriate degree needs to be used, since the drop in γ progresses with continuous changing slope, which maximizes at CMC and becomes zero afterward. Recent research has evidenced that a complete saturated Gibbs monolayer may not always form at CMC; there may be formation of multilayer of micelles below the Gibbs monolayer, which cannot be assessed by ST measurements. A method like neutron reflectometry (NR) can evaluate the Γ beyond CMC. A procedure for determining Γmax from tensiometric results is herein proposed. Amphiphiles do sometimes show a linear decline in γ with logC followed by a break with a plateau at CMC. There, a single slope leading to a single surface excess quantity is obtained for the Gibbs equation at all concentrations up to CMC. Possible reasons for such results are given. Current conflicting ideas and criticisms on the issue of Gibbs equation and determination of Γ and Γmax have been addressed.

A simple proof of the solutions of the Helmert- and the overdetermined nonlinear 7-parameter datum transformation

The laws of nature in general, and the relations and laws in geodesy in particular can be expressed in most cases by nonlinear equations which are in general solved by transforming them to linear form and applying iteration. The process of bringing the equations to linear form implies neglections and approximation. In certain cases it is possible to obtain exact, correct solutions for nonlinear problems. In the present work we introduce parameters into the rotation matrix, and using this we derive solutions for the 2D and 3D similarity transformations. This method involves no iteration, and it does not require the transformation of the equations to linear form. The scale parameter is determined in both cases by solving a polynomial equation of second degree. This solution is already known, but our derivation is worth consideration because of its simple nature.

The inverse sieve problem in high dimensions

We show that if a big set of integer points S⊆[0,N]d, d>1, occupies few residue classes mod p for many primes p, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of Helfgott and Venkatesh.

Assessing soil fertility of a laterite soil in relation to yield of arecanut (Areca catechu L.) in humid tropics of India

The purpose of the study was to determine optimum nutrient norms for laterite soils (Ultisols) and to assess quantitative relationship of soil fertility, leaf nutrients and nutrient uptake with yield of arecanut. At 0–30cm soil depth, optimum nutrient concentration for P, K, Ca, Mg, Fe, Mn, Cu, Zn and B was established as 15, 192, 925, 179, 37, 88, 26, 5.5 and 1.4mgkg−1, respectively. Optimum ranges for different nutrients were found out from second degree polynomial equations. Optimum nutrient values were higher for laterite soils in arecanut tract than generalized guidelines for interpretation of soil analysis data. Multivariate analysis of nutrient data indicated that nutrient uptake/removal would explain the variability in yield more than actual nutrient concentrations in soil and leaf with high degree of relation (R2=0.92–0.98). Stepwise regression analysis identified the soil and leaf nutrient variables with a significant influence on the kernel yield of arecanut. The loadings of principal component analysis were in confirmation with the correlations among the nutrient variables and yield. The study clearly establishes the direct and significant relationship between soil nutrient status/nutrient uptake and yield. The findings of the study give scope for precision agriculture and judicious use of inputs.

A Novel Solution to the P4P Problem for an Uncalibrated Camera

We present a novel solution for the absolute camera pose and the camera calibration (effective focal length and aspect ratio) based on perspective four point (P4P) problem. By converting perspective transformation to affine transformation and using invariance to 3D affine transformation, we explore the relationship between the dual image of the absolute conic (DIAC) and the world coordinate of camera optical center and show how the coplanar and noncoplanar cases are cast into the problems of solving a quadratic polynomial equation and an eighth degree polynomial equation in a single variable respectively using only linear algebra. In particular, geometric configurations for infinite solutions of the coplanar case are explored. We also confirm the conclusion that the upper bound of eight real solutions for noncoplanar case is attainable by an example. The performance and usefulness of our novel solution are demonstrated by thorough testing on both synthetic and real data.

Higher order transmission resonances in above-barrier reflection of ultra-cold atoms

The reflectionless transmission resonances in above-barrier reflection of Bose-Einstein condensates by the Rosen-Morse potential are considered using the mean field Gross-Pitaevskii approach. Applying an exact third order nonlinear differential equation obeyed by the condensate's density, the exact solution of the problem for the first resonance is derived. It is shown that in the nonlinear case the total transmission is possible for positive potential heights, i.e., for potential barriers. Further, it is shown that an appropriate approximation for higher-order resonances can be constructed using a limit solution of the equation for the density written as a root of a polynomial equation of the third degree. Using this limit function and the solution for the first resonance, a simple approximation for the shift of the nonlinear resonance potential's depth from the corresponding linear resonance's position is constructed for higher order resonances. The result is written as a linear function of the resonance order. This behavior notably differs from the case of the rectangular barrier for which the nonlinear shift is approximately constant for all the resonance orders.

Volumetric, viscometric and ultrasonic studies on binary mixtures of oleic acid with alkyl esters at 303.15, 308.15 and 313.15 K

Density (r), viscosity (η) and speed of sound (u) for binary mixtures of oleic acid with methyl acetate, ethyl acetate, n-propyl acetate and n-butyl acetate were measured at 303.15, 308.15 and 313.15 K over the whole composition range at atmospheric pressure. These results were used to calculate excess molar volumes (VE), deviations in viscosity (Δη), deviations in speed of sound (Δu), the excess Gibbs free energies of activation of viscous flow (G*E), deviation in adiabatic compressibility (ΔkS), acoustic impedance (Z) and deviation in acoustic impedance(ΔZ). These results were fitted to Redlich-Kister type polynomial equation of the third degree to derive the binary coefficients. The standard error values were estimated between the calculated and experimental data. Key words: Density, viscosity, ultrasonic speed, oleic acid, alkyl esters.

A proof of the set-theoretic version of the salmon conjecture

We show that the irreducible variety of 4×4×4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg–Manivel polynomials), and of degree 9 (the symmetrization conditions).

Studies on Interrelations among SO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, NO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; and PM&amp;lt;sub&amp;gt;10&amp;lt;/sub&amp;gt; Concentrations and Their Predictions in Ambient Air in Kolkata

In this paper we have first of all studied the interrelations among the concentrations of SO2, NO2 and PM10 and then predicted their future level of concentrations in the ambient air of Kolkata. The data collected from West Bengal Pollution Control Board website have been used to construct second degree, third degree and four degree polynomial equations using MATLAB software. Since a curve in a small interval can be approximated by a line segment in that small interval, we have observed that better result can be achieved if we replace the curves piece meal wise in small intervals by line segments during January-April, May-August and September-December months. The multiple regression equations among the aforesaid three parameters have been established to predict the value of each parameter in terms of the remaining two. A further improvement in terms of reducing the number of dependent variables has been made using the results of correlation coefficients. Finally, we have predicted the value of each parameter in terms of only one dependent variable.

General polynomials over division algebras and left eigenvalues

In this paper, we present an isomorphism between the ring of general polynomials over a division ring of degree $p$ over its center $F$ and the group ring of the free monoid with $p^2$ variables. Using this isomorphism, we define the characteristic polynomial of a matrix over any division algebra, i.e. a general polynomial with one variable over the algebra whose roots are precisely the left eigenvalues. Plus, we show how the left eigenvalues of a $4 \times 4$ matrices over any division algebra can be found by solving a general polynomial equation of degree 6 over that algebra.