A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.

Dihedral group and classification of G-circuits of length 10

In this paper, we classify G-circuits of length 10 with the help of the location of the reduced numbers lying on G-circuit. The reduced numbers play an important role in the study of modular group action on P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -subset of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ . For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of real quadratic fields. In particular, we classify P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ = ⋃ k ∈ N Q * k 2 m $={\bigcup }_{k\in N}{Q}^{{\ast}}\left(\sqrt{{k}^{2}m}\right)$ containing G-circuits of length 10 and determine that the number of equivalence classes of G-circuits of length 10 is 41 in number. We also use dihedral group to explore cyclically equivalence classes of circuits and use cyclic group to explore similar G-circuits of length 10 corresponding to 10 of these circuits. By using cyclically equivalent classes of circuits and similar circuits, we obtain the exact number of G-orbits and the structure of G-circuits corresponding to cyclically equivalent classes. This study also helps us in classifying the reduced numbers lying in the P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits.

Wave Transport and Localization in Prime Number Landscapes

In this paper, we study the wave transport and localization properties of novel aperiodic structures that manifest the intrinsic complexity of prime number distributions in imaginary quadratic fields. In particular, we address structure-property relationships and wave scattering through the prime elements of the nine imaginary quadratic fields (i.e., of their associated rings of integers) with class number one, which are unique factorization domains (UFDs). Our theoretical analysis combines the rigorous Green’s matrix solution of the multiple scattering problem with the interdisciplinary methods of spatial statistics and graph theory analysis of point patterns to unveil the relevant structural properties that produce wave localization effects. The onset of a Delocalization-Localization Transition (DLT) is demonstrated by a comprehensive study of the spectral properties of the Green’s matrix and the Thouless number as a function of their optical density. Furthermore, we employ Multifractal Detrended Fluctuation Analysis (MDFA) to establish the multifractal scaling of the local density of states in these complex structures and we discover a direct connection between localization, multifractality, and graph connectivity properties. Finally, we use a semi-classical approach to demonstrate and characterize the strong coupling regime of quantum emitters embedded in these novel aperiodic environments. Our study provides access to engineering design rules for the fabrication of novel and more efficient classical and quantum sources as well as photonic devices with enhanced light-matter interaction based on the intrinsic structural complexity of prime numbers in algebraic fields.