THE CONNECTEDNESS LOCUS OF A FAMILY OF REAL BIQUADRATIC POLYNOMIALS

2007 ◽  
Vol 17 (11) ◽  
pp. 4219-4222 ◽  
Author(s):  
YESHUN SUN ◽  
YONGCHENG YIN
Keyword(s):  
Real Numbers ◽  
Precise Description ◽  
The Family ◽  
Connectedness Locus

In this paper we present a precise description of the connectedness locus of the family of polynomials (z2 + x)2 + y, where x, y are real numbers.

BIFURCATION CURVES OF A FAMILY OF ONE-DIMENSIONAL BIQUADRATIC POLYNOMIAL MAPS, DEPENDING ON THE COMPLEX VARIABLE, AND TWO REAL PARAMETERS

2010 ◽  
Vol 20 (12) ◽  
pp. 4119-4125
Author(s):  
HISASHI ISHIDA ◽  
TSUYOSHI ITOH
Keyword(s):  
Fixed Points ◽  
Complex Variable ◽  
Elementary Proof ◽  
Precise Description ◽  
One Dimensional ◽  
Polynomial Maps ◽  
The Family ◽  
Connectedness Locus

Sun and Yin [2007] had presented a precise description of the connectedness locus of the family of real biquadratic polynomials {pa,b(z) = (z2 + a)2 + b}. We shall first give an elementary proof of their result. Second, we shall give a precise description of the sets of parameters (a, b) such that the family {pa,b} has attracting fixed points.

scholarly journals A generalized family of multidimensional continued fractions: TRIP Maps

2014 ◽  
Vol 10 (08) ◽  
pp. 2151-2186 ◽  
Author(s):  
Krishna Dasaratha ◽  
Laure Flapan ◽  
Thomas Garrity ◽  
Chansoo Lee ◽  
Cornelia Mihaila ◽  
...  
Keyword(s):  
Number Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Multidimensional Continued Fractions ◽  
The Family ◽  
Before And After ◽  
Multidimensional Continued Fraction

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

scholarly journals Dynamics of a Higher-Order System of Difference Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Qi Wang ◽  
Qinqin Zhang ◽  
Qirui Li
Keyword(s):  
Positive Integer ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Order System ◽  
Real Numbers ◽  
Precise Description ◽  
System Of Difference Equations ◽  
Positive Real

Consider the following system of difference equations:xn+1(i)=xn-m+1(i)/Ai∏j=0m-1xn-j(i+j+1)+αi,xn+1(i+m)=xn+1(i),x1-l(i+l)=ai,l,Ai+m=Ai,αi+m=αi,i,l=1,2,…,m;n=0,1,2,…,wheremis a positive integer,Ai,αi,i=1,2,…,m, and the initial conditionsai,l,i,l=1,2,…,m,are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.

PRESERVATION OF LOG-CONCAVITY AND LOG-CONVEXITY UNDER OPERATORS

2020 ◽  
pp. 1-14
Author(s):  
Wanwan Xia ◽  
Tiantian Mao ◽  
Taizhong Hu
Keyword(s):  
Distribution Function ◽  
Operations Research ◽  
Random Variable ◽  
Group Property ◽  
Semi Group ◽  
Real Numbers ◽  
Bernstein Type ◽  
Stochastic Comparisons ◽  
Weighted Distributions ◽  
The Family

Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$ , θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

scholarly journals The uniform limit of Lipschitz functions on a Banach space

1973 ◽  
Vol 15 (3) ◽  
pp. 325-331
Author(s):  
J. V. Herod
Keyword(s):  
Banach Space ◽  
Lipschitz Functions ◽  
Bounded Subset ◽  
Real Numbers ◽  
Uniform Limit ◽  
The Family ◽  
Class Of Functions

With R the set of real numbers and S a Banach space, let ℒ be the class of functions A from R × S to S which have following properties: (1) if B is a bounded subset of S then the family {A(·, P): P is in B} is equicontinuous; i.e., if t is a number and ε > 0 then there is a positive number δ such that if |s – t | < δ and P is in B then | A(s, P) – A(t, P)| < ε.

scholarly journals Automorphisms of the semigroup of all differentiable functions

1967 ◽  
Vol 8 (1) ◽  
pp. 63-66 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Unit Interval ◽  
Real Numbers ◽  
Identity Function ◽  
Differentiable Functions ◽  
The Family

Let R denote the space of real numbers and let D(R) denote the family of all functions mapping R into R that are (finitely) differentiable at each point of R. Since the composition f o g of two differentiable functions is also differentiable and since the composition operation is associative, it follows that D(R) is a semigroup with this operation. Such semigroups have been studied previously. Nadler, in [4], has shown that the semigroup of al differentiable functions mapping the closed unit interval into itself has no idempotent elements other than the identity function and the constant functions. The proof of that result carries over easily to the semigroup D(R).

scholarly journals Mating quadratic maps with the modular group II

2019 ◽  
Vol 220 (1) ◽  
pp. 185-210
Author(s):  
Shaun Bullett ◽  
Luna Lomonaco
Keyword(s):  
Modular Group ◽  
Parameter Family ◽  
Quadratic Polynomial ◽  
Complex Parameter ◽  
Rational Maps ◽  
Quadratic Maps ◽  
The Family ◽  
The One ◽  
Holomorphic Correspondences ◽  
Connectedness Locus

Abstract In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences $$\mathcal {F}_a$$Fa: $$\begin{aligned} \left( \frac{aw-1}{w-1}\right) ^2+\left( \frac{aw-1}{w-1}\right) \left( \frac{az+1}{z+1}\right) +\left( \frac{az+1}{z+1}\right) ^2=3 \end{aligned}$$aw-1w-12+aw-1w-1az+1z+1+az+1z+12=3and proved that for every value of $$a \in [4,7] \subset \mathbb {R}$$a∈[4,7]⊂R the correspondence $$\mathcal {F}_a$$Fa is a mating between a quadratic polynomial $$Q_c(z)=z^2+c,\,\,c \in \mathbb {R}$$Qc(z)=z2+c,c∈R, and the modular group $$\varGamma =PSL(2,\mathbb {Z})$$Γ=PSL(2,Z). They conjectured that this is the case for every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus. We show here that matings between the modular group and rational maps in the parabolic quadratic family $$Per_1(1)$$Per1(1) provide a better model: we prove that every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus is such a mating.

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