Representing the Boolean OR Function by Quadratic Polynomials Modulo 6

2007 ◽  
pp. 322-327
Author(s):  
Gyula Győr
Keyword(s):  
Quadratic Polynomials

scholarly journals Distribution of polynomials with cycles of a given multiplier

2011 ◽  
Vol 201 ◽  
pp. 23-43 ◽  
Author(s):  
Giovanni Bassanelli ◽  
François Berteloot
Keyword(s):  
Quadratic Polynomials ◽  
Bifurcation Locus

AbstractIn the space of degreedpolynomials, the hypersurfaces defined by the existence of a cycle of periodnand multipliereiθare known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

2008 ◽  
Vol 15 (1) ◽  
pp. 189-194
Author(s):  
Ahmad Zireh
Keyword(s):  
Mandelbrot Set ◽  
Complex Numbers ◽  
Quadratic Polynomials ◽  
Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

Quadratic Polynomials

1993 ◽  
pp. 123-159
Author(s):  
Lennart Carleson ◽  
Theodore W. Gamelin
Keyword(s):  
Quadratic Polynomials

LIMITING DYNAMICS FOR THE COMPLEX STANDARD FAMILY

1995 ◽  
Vol 05 (03) ◽  
pp. 673-699 ◽  
Author(s):  
NÚRIA FAGELLA
Keyword(s):  
Cross Sections ◽  
Julia Set ◽  
Three Dimensional ◽  
Mandelbrot Set ◽  
Dimensional Parameter ◽  
Quadratic Polynomials ◽  
New Family ◽  
Standard Family ◽  
The Family ◽  
Characteristic Features

The complexification of the standard family of circle maps Fαβ(θ)=θ+α+β+β sin(θ) mod (2π) is given by Fαβ(ω)=ωeiαe(β/2)(ω−1/ω) and its lift fαβ(z)=z+a+β sin(z). We investigate the three-dimensional parameter space for Fαβ that results from considering a complex and β real. In particular, we study the two-dimensional cross-sections β=constant as β tends to zero. As the functions tend to the rigid rotation Fα,0, their dynamics tend to the dynamics of the family Gλ(z)=λzez where λ=e−iα. This new family exhibits behavior typical of the exponential family together with characteristic features of quadratic polynomials. For example, we show that the λ-plane contains infinitely many curves for which the Julia set of the corresponding maps is the whole plane. We also prove the existence of infinitely many sets of λ values homeomorphic to the Mandelbrot set.

APPROXIMATING THE POTASSIUM FERTILIZATION REQUIREMENT OF COTTON ON SOME REPRESENTATIVE SEMIARID TROPICAL SAVANNAH SOILS OF NIGERIA

1981 ◽  
Vol 61 (3) ◽  
pp. 507-516
Author(s):  
GANYIR LOMBIN
Keyword(s):  
Cotton Yield ◽  
Seed Cotton ◽  
Potassium Fertilizer ◽  
Quadratic Polynomials ◽  
Independent Variables ◽  
Regression Procedure ◽  
Fertilizer Requirement ◽  
Relevant Variables ◽  
Multiple Determination

Potassium fertilizer requirement of rain-fed cotton was evaluated in a 3-yr field study conducted at three locations using four rates (0, 25, 50 and 75 kg∙ha−1)of K. Significant response was not obtained above 25 kg∙ha−1 applied K. Quadratic polynomials, using leaf K, exch. K, exch. (Ca + Mg)/K and applied K as independent variables, were fitted to the seed-cotton yield. As a single parameter, leaf K emerged the best predictor of yield with a coefficient of multiple determination (R2) of 86% and a corresponding regression equation of: Y (yield) = 3099.2 + 6031.6%K − 1643.3 (%K)2, followed by fertilizer K with a coefficient of multiple determination of 81% and a yield equation of: Y = 1302.3 + 53.96 app. K − 0.54 (app. K)2. Soil exch. K and (Ca + Mg)/K ratio were slightly less efficient in predicting yield giving R2 values of 0.62 and 0.76, respectively, when both the linear and quadratic terms were entered into their respective yield equations. When all the 12 possible entries (linear, and second-order terms and their square root transformations) were fed into the computer and regressed over cotton yield using a step-wise regression procedure, only two variables, leaf %K and (Ca + Mg)/K, significantly fitted the yield equation giving a predictive value of 87%. But the improvement in the precision of yield predictability as measured by the R2 value was only marginal and would not justify recommending the equation considering the extra laboratory work that will be needed to obtain the relevant variables. Critical values of 1.84% in index leaf, 0.19 meq/100 g exch. K and 50 kg∙ha−1 applied K were approximated for a maximum predicted yield range of 2440–2700 kg∙ha−1.

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