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Counting Rotational Sets for Laminations of the Unit Disk from First Principles
Full text linkBy studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, d, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of rotational sets\u27\u27, which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure of these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials.
These rotational sets are created by uniting rotational orbits, as we define in this paper. The number of such sets for a given degree d, rotation number p/q, and cardinality k can be determined by analyzing the potential placements of pre-images of zero on the unit circle with respect to the rotational set under the d-tupling map. We obtain a closed-form formula for the count. Though this count is already known based upon some sophisticated results, our count is based upon elementary geometric and combinatorial principles, and provides an intuitive explanation
The Intricacies of Pairwise Modular Multiplicative Inverse in Lucas Numbers
Full text linkLet (p,q) be a pair of relatively prime integers greater than 1. The pairwise modular multiplicative inverse (PMMI) of (p,q) is defined as the unique pair of positive integers (p′, q′) such that p p′ ≡ 1 (mod q), p′ \u3c q, qq′ ≡ 1 (mod p), q′ \u3c p. In this paper, we determine all pairs of Lucas numbers such that their PMMIs are pairs of Lucas numbers
How Many Symmetries of the Regular n-gon Are Even?
Full text linkOne of the simplest classes of finite groups used as a source of counterexamples in a first course of modern algebra is the class of finite dihedral groups. Among the subgroups of dihedral group, finding subgroups of index 2 is of interest in part because these subgroups are normal subgroups. In this article, we use the representations of the symmetries of the dihedral groups as permutations of the vertices and determine concretely all its subgroups of index 2. Under this representation or embedding, the article determines the intersection of the dihedral group with the corresponding alternating groups when they are naturally viewed as subgroup of the symmetry group on the set of vertices of the regular n-gon. In its second part, the article considers the question of embedding an arbitrary finite group into a symmetry group using the well-known Cayley embedding. In this more general context where every element of the group is viewed as a permutation, one counts the even elements of the group
An elementary approach to the probability distribution of the product of multiple random variables
Full text linkAbstract There arenindependent identically distributed (i.i.d.) uniform random variables defined as ξ1,ξ2,…,ξn, valued in interval [0,k](k\u3e0), and there is a constantmvalued in interval (0,kn). We study the distribution of the product of these random numbers and prove a formula calculating Pr(∏i=1nξi≤m). Interestingly, we find that the result is exactly the sum of the firstnterms in the Taylor series expansion of the function exp(x) with x=nlnk-lnm. Through considering the corresponding probability density function, we make an extension of the formula calculating Pr(∏i=1nξi≤m) to any positive realn, and the extended formula can be written in a concise form of regularized gamma function. Then we discuss the probability in the limit n→∞ with m=1 and prove that it tends to 1,1/2 and 0 when k∈(0,e), k=e and k∈(e,∞), respectively
Fixing the Potholes on the Road to Academic Success: A Curriculum for Engineering Educators to Create and Sustain Meaningful Change
No full textCounting Rotational Subsets of the Circle under the Angle-Multiplying Map
Full text linkA rotational set is a finite subset of the unit circle such that the angle-multiplying map maps onto itself by a cyclic permutation of its elements. Each rotational set has a geometric rotation number . Lisa Goldberg introduced these sets to study the dynamics of complex polynomial maps. In this paper, we provide a necessary and sufficient condition for a set to be -rotational with rotation number . As applications of our condition, we recover two classical results and enumerate -rotational sets with rotation number that consist of a given number of orbits
The Frequency of Elliptic Curves Over with Fixed Torsion
Full text linkMazur\textsc{\char13}s Theorem states that there are precisely 15 possibilities for the torsion subgroup of an elliptic curve defined over the rational numbers. It was previously shown by Harron and Snowden that the number of isomorphism classes of elliptic curves of height up to that have a specific torsion subgroup is on the order of , for some positive depending on . We compute for these groups over \Qi. Furthermore, in a collection of recent papers it was proven that there are 9 more possibilities for the torsion subgroup in the base field \Qi. We compute the value of for these new groups