scholarly journals Entropy Monotonicity and Superstable Cycles for the Quadratic Family Revisited

Entropy ◽  
2020 ◽  
Vol 22 (10) ◽  
pp. 1136
Author(s):  
José M. Amigó ◽  
Ángel Giménez

The main result of this paper is a proof using real analysis of the monotonicity of the topological entropy for the family of quadratic maps, sometimes called Milnor’s Monotonicity Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis. Our proof is based on tools and algorithms previously developed by the authors and collaborators to compute the topological entropy of multimodal maps. Specifically, we use the number of transverse intersections of the map iterations with the so-called critical line. The approach is technically simple and geometrical. The same approach is also used to briefly revisit the superstable cycles of the quadratic maps, since both topics are closely related.


1986 ◽  
Vol 6 (2) ◽  
pp. 295-309 ◽  
Author(s):  
Mariusz Urbański

AbstractGiven an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).



2015 ◽  
Vol 11 (4) ◽  
pp. 44-54 ◽  
Author(s):  
N.V. Nosikova

This paper aims to carry out a structural analysis of the psychosemantic system of goal-directedness in families. The hypothesis suggests that family semantic criterion makes it possible to differentiate the structure/level organization of meaning elements and their associative connections and to reveal the content of the object. The research involved 135 young women and 134 young men (aged 15—18); 150 women (aged 21—64) and 38 men (aged 20—55), married, with children. A modified version of I.L. Solomin’s technique of semantic differential was used. A comparative analysis of structural and hierarchical levels of psychosemantics of social psychological goal-directedness of families was carried out basing on the family semantic criterion. The general system level consists of structures representing two generations of a fam¬ily – parents and modern generation. The structure of the subsystem level is common for all groups of respondents and consists of five sublevels with different functional tasks: “My parental family”, “My father”, “My future family”/ “My family”, “My husband”/ “My wife”, and “Birth of a child”. Concepts that define members of the family and family groups, ideal representations of them, events and types of activity related to family life, constitute the component level. The concept of divorce due to its semantics does not belong to the family psychosemantic system. The component content of the subsystem level differs according to age, sex and marital status, which highlights the necessity of further functional research and complex analysis of the psychosemantic system relevant for practical family psychology.



2012 ◽  
Vol 22 (06) ◽  
pp. 1250150 ◽  
Author(s):  
ALEXANDRA NASCIMENTO BAPTISTA ◽  
CARLOS CORREIA RAMOS ◽  
NUNO MARTINS

We study the iteration of a quadratic family in the algebra of 2 × 2 real matrices, parameterized by a matrix C. We analyze and classify the existing cycles (periodic orbits) and their dependence on the parameter matrix. We discuss how new dynamical phenomena occur as a consequence of the noncommutativity of the matrix product. In particular, we show that the commutator of the initial condition with parameter matrix C has a decisive role in the overall dynamics.



2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Syed Ghoos Ali Shah ◽  
Saqib Hussain ◽  
Akhter Rasheed ◽  
Zahid Shareef ◽  
Maslina Darus

Inequalities play a fundamental role in many branches of mathematics and particularly in real analysis. By using inequalities, we can find extrema, point of inflection, and monotonic behavior of real functions. Subordination and quasisubordination are important tools used in complex analysis as an alternate of inequalities. In this article, we introduce and systematically study certain new classes of meromorphic functions using quasisubordination and Bessel function. We explore various inequalities related with the famous Fekete-Szego inequality. We also point out a number of important corollaries.



2019 ◽  
Vol 220 (1) ◽  
pp. 185-210
Author(s):  
Shaun Bullett ◽  
Luna Lomonaco

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.



2021 ◽  
pp. 47-51
Author(s):  
T.I. Grabelnykh ◽  
◽  
N.A. Sablina

This study examined the transformation of the family’s resource supply system, taking into account the educational strategies of its members. It is substantiated that the development of the family as a social institution becomes possible only through the creation of an open system of its resource supply in compliance with the principles of joint activities and gender equality, which ensures the integration of individual functions of the family and higher education. On the issue of women’s access to available resources, including educational ones, the work revealed a social contradiction, when, on the one hand, women retain an active position in the provision of resources to the family, there is equal access with men to information and intellectual resources, on the other hand, there is a limited access of women to power, material and financial resources. In the field of complex analysis and assessment of the family's resource supply, the authors have proposed new indicators of information-technological and educational growth of a social institution: technologies of housekeeping; use of information technologies in the family resource supply system. The conclusion is made about the growing role of modern technologies of public participation, contributing to the improvement of the status of women and strengthening their role in the resource provision of the family by increasing the educational level and status.



2018 ◽  
pp. 130-143
Author(s):  
Yaroslav Volodymyrovych Yuvsechko

The article analyzes the beliefs and practical activities of synthetic neo-religions on issues of family, marriage, marital life, children’s education, attitude to parents, etc. In particular, the position of Baha'i Faith, Unification Church and Church of Scientology is considered. The peculiarity of this research is the complex analysis of the doctrine and practice of these neo-religious movements and finding of common aspects in their views on family values, both among themselves and with traditional religions. It emphasizes their syncretism  and refute the available warning in society about the destructive influence of neo-religions’ beliefs on established family values. In the teaching of the Unification Church, the issue of the family, marital relations, holiness and purity of marital ties, the inadmissibility of premarital and extra-marital relations occupy one of the central places. In the doctrine of the Baha'i Faith, the vital importance is given to the institution of the family. It emphasizes the sanctity of marriage, the equality of men and women in their rights, privileges, upbringing and social status. The Baha'i recognize the principle of equal rights, opportunities and privileges for men and women, the requirement of monogamy and marital fidelity. In the teaching of the Church of Scientology, the family is regarded as an important bricks of society: the biological model of family relationships and the development of an organism is that ensures the continuation of human existence. Marriage is the basis of a family. The family is the closest union in a society, which provides itself for the continuation of own existence and own protection. The family is also necessary for the society by an economic point of view. According to Scientologists, the whole culture will perish if its foundation - the family - will cease to exist. Thus, in their opinion, there is no doubt that the one who destroys the marriage union also destroys civilization. It is emphasized that despite the claims of these religious organizations to the exclusivity and authority of their own religious sources, their positions on family values ​​are quite similar to each other. Also they often overlap with the principles of Christianity and other world religions. The author draws attention to the lack of awareness of the general public with the basics of dogma of the Baha'i Faith, the Unification Church and the Church of Scientology. As a result, there is a fear in society about the spread of doctrines of synthetic neo-religions, despite the fact that their positions on family values ​​do not contradict the generally accepted norms of social morality and mostly accord with them.



2021 ◽  
Author(s):  
Almouid Mohammed Hasibul Haque

In this paper, I attempt to solve one of the most difficult problems in modern mathematics-'The Riemann Hypothesis'. I redefine the gamma function and use that modified form along with some identities from Fourier analysis and concepts from complex analysis to show that all the non-trivial zeros of the Riemann zeta function must lie on the critical line and then by recalling Hardy's theorem I prove the Riemann hypothesis.



1992 ◽  
Vol 02 (02) ◽  
pp. 241-250 ◽  
Author(s):  
W. METZLER ◽  
A. BRELLE ◽  
K.-D. SCHMIDT

From experimental work, nonanalytic quadratic maps are known which generate the Mandelbrot set as well as Julia sets not different from those obtained by the complex analytic quadratic family z↦z2+c, c∈C. This paper presents a tutorial on how to generate analytically the complete class of real parameter-dependent quadratic maps of the plane, all having, for instance, the Mandelbrot set in common with the quadratic family. In the analytical case, the transformations involved are rotations or reflections.



2016 ◽  
Vol 37 (5) ◽  
pp. 1389-1412 ◽  
Author(s):  
ERIC BEDFORD ◽  
JOHN SMILLIE

We consider the family of quadratic Hénon diffeomorphisms of the plane $\mathbb{R}^{2}$. A map will be said to be a ‘horseshoe’ if its restriction to the non-wandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for being a horseshoe based on an auxiliary coding which describes positions of points relative to the stable manifold of one of the fixed points. In addition we describe the topological conjugacy type of maps on the boundary of the horseshoe locus. We use complex techniques and we work with maps in a parameter region which is a two-dimensional analog of the familiar ‘$1/2$-wake’ for the quadratic family $p_{c}(z)=z^{2}$.



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