scholarly journals Cayley Forms and Self-Dual Varieties

2013 ◽  
Vol 57 (1) ◽  
pp. 89-109 ◽  
Author(s):  
F. Catanese
Keyword(s):  
Quadratic Equation ◽  
Space Curve ◽  
Zero Set ◽  
Dual Variety ◽  
Quadratic Equations ◽  
Dual Varieties ◽  
Chow Forms ◽  
The Ideal

AbstractGeneralized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1, 3). Our main result is that F is a Cayley form if and only if Z = G(1, 3) ∩ {F = 0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quadratic equations, since there is a unique representative F0 + QF1 of F, with F0, F1 harmonic, such that the harmonic projection of the Cayley equation is identically 0. We also give new equations for honest Cayley forms, but show, with some calculations, that the variety of honest Cayley forms does not seem to be defined by quadratic and cubic equations.

Small Prime Solutions of Quadratic Equations

2002 ◽  
Vol 54 (1) ◽  
pp. 71-91 ◽  
Author(s):  
Kwok-Kwong Stephen Choi ◽  
Jianya Liu
Keyword(s):  
Quadratic Equation ◽  
List Type ◽  
Quadratic Equations

AbstractLet b1,…,b5 be non-zero integers and n any integer. Suppose that b1 + … + b5 ≡ n (mod 24) and (bi, bj) = 1 for 1 ≤ i < j ≤ 5. In this paper we prove that(i)if all bj are positive and , then the quadratic equation is soluble in primes pj, and(ii)if bj are not all of the same sign, then the above quadratic equation has prime solutions satisfying .

scholarly journals Quadratic Equations in a Cyclic Number System

1931 ◽  
Vol 2 (3) ◽  
pp. 151-157
Author(s):  
R. Wilson
Keyword(s):  
Division Algebra ◽  
Number System ◽  
Quadratic Equation ◽  
The Real ◽  
Quadratic Equations ◽  
Complex Quaternion ◽  
The Common ◽  
Real Quaternions

Mr D. E. Littlewood has recently discussed the properties of the quadratic equation over the real quaternions and shown that the solutions correspond to the common intersections of four quadrics in four-space. Although complex quaternion solutions may arise, the system of real quaternions to which the coefficients belong is a division algebra. It is of interest, therefore, to discuss the solution of the quadratic when the coefficients are drawn from a system containing divisors of zero.

Sharing Teaching Ideas: A Squeeze Play on Quadratic Equations

1982 ◽  
Vol 75 (2) ◽  
pp. 132-136
Keyword(s):  
High School ◽  
Mathematics Teacher ◽  
Quadratic Equation ◽  
Senior Year ◽  
Quadratic Formula ◽  
Mathematics Courses ◽  
Quadratic Equations ◽  
Teaching Ideas

As a mathematics teacher whose present assignment is to teach science, I was somewhat dismayed when my physics class wa unable to solve a nontrivial quadratic equation. These students are all enrolled in senior-year mathematics and had taken all lower level mathematics courses available in our small Western Kansas high school. They charged this inability to having forgotten the quadratic formula. To the e students the quadratic formula is a magic passkey to solving “unfactorable” quadratic equations. On further di scussion, l discovered that they vaguely remembered having heard of the method of completing the square, but they saw no connection between the quadratic formula and that method of solving a quadratic equation. They could solve simple quadratics by hit-and-miss factoring, but that was their only tool with which to attack this problem.

A DESCRIPTION OF SOLUTIONS OF QUADRATIC EQUATIONS IN HYPERBOLIC GROUPS

1992 ◽  
Vol 02 (03) ◽  
pp. 237-274 ◽  
Author(s):  
R.I. GRIGORCHUK ◽  
I.G. LYSIONOK
Keyword(s):  
Class Group ◽  
Hyperbolic Group ◽  
Quadratic Equation ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Parametric Solutions ◽  
Quadratic Equations ◽  
All Solutions ◽  
Finite Set ◽  
Special Mapping

A description is given for the set of solutions of a quadratic equation in a hyperbolic group. It consists of a finite set of parametric solutions of the equation which generates all solutions by the action of a group which may be interpreted as a special mapping class group of a compact surface.

scholarly journals Modification Model of Soil Fertility Evaluation FAO-UNESCO on the Slopes of Merapi Volcano, Indonesia

2021 ◽  
Vol 1 (1) ◽  
pp. 494-499
Author(s):  
Eko Adi Julianto ◽  
Partoyo Partoyo ◽  
Sri Suharsih
Keyword(s):  
Soil Fertility ◽  
Goodness Of Fit ◽  
Linear Equations ◽  
Quadratic Equation ◽  
Merapi Volcano ◽  
Volcanic Material ◽  
Quadratic Equations ◽  
Natural Logarithm ◽  
Schwarz Criterion

One of the mountains known as active volcanoes in the world was Merapi volcano. From the very active impact of Merapi activity, there was a continuous addition of volcanic material associated with soil fertility, which can be evaluated using the Soil Fertility Evaluation (SFE) system. This study aims to obtain a more adaptive SFE system to the southern slopes of Merapi volcano by modifying the FAO- UNESCO version of SFE system that still uses linear equations. In this research used system of quadratic equation, use of natural logarithm (ln), and modification of new parameter. From the evaluation of soil fertility is then connected with the production component of paddy rice (dry grains crop). There were several indicators that used to see the quality of the model or test the goodness of fit of the model we make, for example from its R2. In this study the quality of a model was seen from: Akaike Info Criterion (AIC) and Schwarz Criterion (SC), and the data was done by using EViews 9. The results showed the parameters that influence big in the model can be seen from the correlation and influence the parameters in single. Quadratic equations can improve the quality of a model over a linear equation. The standard SFE model which is modified by using the nat

scholarly journals Optimization of crude protein in diets for Nile tilapia reared in net pens: performance, hematology, and water quality

2017 ◽  
Vol 47 (3) ◽  
Author(s):  
Débora Del Puppo ◽  
◽  
Douglas Haese ◽  
Levy de Carvalho Gomes ◽  
João Luís Kill ◽  
...  
Keyword(s):  
Nile Tilapia ◽  
Crude Protein ◽  
Hematological Parameters ◽  
Ammonia Excretion ◽  
Optimal Level ◽  
Quadratic Equation ◽  
Experimental Unit ◽  
Feed Conversion ◽  
Ideal Protein ◽  
The Ideal

ABSTRACT: Two experiments were conducted to evaluate the effects of reducing dietary crude protein (CP), based on the ideal protein concept for Nile tilapia reared in net pens. The experimental (isocaloric, isocalcium, and isophosphoric) diets were formulated to contain 270, 300, 330, and 360g kg-1 CP. In experiment 1, 4320 Nile tilapia (13.5±0.82g) were used to evaluate the performance and hematological parameters. The experimental design was completely randomized and the fish were distributed in 24 net pens (1.0m3), with four diets, six replicates, and 180 fishes per experimental unit. In experiment 2, 40 Nile tilapia (22.5±0.56g) were used to evaluate the ammonia excretion. Fish were distributed in 40 aquaria (3.0L), with one fish per aquarium (n=10). No protein reduction effect was observed in feed intake and the hematocrit and hemoglobin values. Regarding the statistic models used in the present study, difference was observed between CP values. The optimal level estimated by the quadratic equation first interception with the linear response plateau (LRP) as a response to CP changes in the diet was determined for weight gain (324.3g kg-1) and feed conversion (317.8g kg-1). After reduction in the CP levels, a linear reduction was observed in the ammonia excretion in water. Based on the ideal protein concept for Nile tilapia reared in net pens, reducing the CP levels in the diets is possible, without change in the performance and hematological parameters, and with a reduction in the levels of ammonia excretion in water, since amino acids are supplemented.

scholarly journals ANALYSIS OF MATHEMATICS PROBLEMS IN THE 2013 CURRICULUM AND CAMBRIDGE CURRICULUM MATHEMATICS TEXTBOOKS

2021 ◽  
Vol 6 (1) ◽  
pp. 99-110
Author(s):  
Ratu Sarah Fauziah Iskandar ◽  
Aji Raditya ◽  
Trisna Roy Pradipta
Keyword(s):  
School Quality ◽  
Quadratic Equation ◽  
Mathematics Textbooks ◽  
Analysis Method ◽  
Mathematics Textbook ◽  
Quadratic Equations ◽  
Mathematics Problems ◽  
Data Collection Technique ◽  
Dimensional Analysis Method

Several factors influence the success of learning; one of them is the quality of textbooks. Textbooks have a pivotal role in learning, namely, representing the teacher's explanation in front of the class. Curricula have continuously changed because they are far from the expectations. In Indonesia, many schools have implemented an international curriculum to improve school quality. One of the curricula used is the Cambridge curriculum. This study analyzed the types of problems in the Cambridge and 2013 curriculum mathematics textbooks, especially on quadratic equations. This research utilized a six-dimensional analysis method which consists of mathematical activities, complexity level, answer form, contextual features, response types, and mathematical features. Furthermore, the data collection technique was carried out by analyzing and describing the types of questions in the 2013 curriculum and the Cambridge curriculum mathematics textbooks. The analysis focused on the quadratic equation topic in the 2013 curriculum and the Cambridge curriculum mathematics textbooks. The results shows that there is no difference between the types of problems in the 2013 curriculum and the Cambridge curriculum mathematics textbooks for quadratic equation topics. The framework of this study could be a reference for further research and used by mathematics textbook writers to create more diverse types of questions.              

scholarly journals A new method for solving interval and fuzzy quadratic equations of dual form

2021 ◽  
Vol 5 (2) ◽  
pp. 81-89
Author(s):  
Kamal Mamehrashi
Keyword(s):  
Numerical Method ◽  
Fuzzy Number ◽  
Quadratic Equation ◽  
New Method ◽  
Numerical Examples ◽  
Quadratic Equations ◽  
Dual Form ◽  
Interval Extension

In this paper, we present a numerical method for solving a quadratic interval equation in its dual form. The method is based on the generalized procedure of interval extension called” interval extended zero” method. It is shown that the solution of interval quadratic equation based on the proposed method may be naturally treated as a fuzzy number. An important advantage of the proposed method is that it substantially decreases the excess width defect. Several numerical examples are included to demonstrate the applicability and validity of the proposed method.

Approaching quadratic equations from a right angle

2017 ◽  
Vol 101 (552) ◽  
pp. 424-438
Author(s):  
King-Shun Leung
Keyword(s):  
Secondary School ◽  
Mathematics Curriculum ◽  
Quadratic Equation ◽  
Bisection Method ◽  
Quadratic Formula ◽  
Secondary School Mathematics ◽  
Quadratic Equations ◽  
Newton Raphson ◽  
Raphson Method ◽  
Comprehensive Discussion

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two).

scholarly journals Quaternion quadratic equations in characteristic 2

2014 ◽  
Vol 14 (03) ◽  
pp. 1550033 ◽  
Author(s):  
Adam Chapman
Keyword(s):  
Division Algebra ◽  
Quadratic Equation ◽  
Quadratic Equations ◽  
Characteristic 2 ◽  
Quaternion Division Algebra

In this paper, we present a solution for any standard quaternion quadratic equation, i.e. an equation of the form z2 + μz + ν = 0 where μ and ν belong to some quaternion division algebra Q over some field F, assuming the characteristic of F is 2.

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