Subgroup Structure Research Articles

2-Adic Galois images of isogeny-torsion graphs over ℚ with CM

Let [Formula: see text] be a [Formula: see text]-isogeny class of elliptic curves defined over [Formula: see text]. The isogeny graph associated to [Formula: see text] is a graph which has a vertex for each elliptic curve over [Formula: see text] of [Formula: see text] and an edge for each [Formula: see text]-isogeny of prime degree that maps one elliptic curve in [Formula: see text] to another elliptic curve in [Formula: see text], with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to [Formula: see text] is the isogeny graph associated to [Formula: see text] where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over [Formula: see text] of the corresponding elliptic curve. The main result of the article is a classification of the [Formula: see text]-adic Galois image at each vertex of the isogeny-torsion graphs whose associated [Formula: see text]-isogeny class consists of elliptic curves over [Formula: see text] with complex multiplication.

DT-k-group structures on digital objects and an answer to an open problem

The paper examines various properties of [Formula: see text]-[Formula: see text]-subgroup structures and addresses an open problem on the existence of topological group structures on the [Formula: see text]-dimensional Khalimsky ([Formula: see text]-, for brevity) topological space and Marcus–Wyse ([Formula: see text]-, for short) topological plane. In particular, we obtain many types of totally [Formula: see text]-disconnected or [Formula: see text]-connected subgroups of a [Formula: see text]-connected [Formula: see text]-[Formula: see text]-group. Besides, we prove that each of the [Formula: see text]-dimensional [Formula: see text]-topological space and the [Formula: see text]-topological plane cannot be a typical topological group. Unlike an existence of a [Formula: see text]-[Formula: see text]-group structure of [Formula: see text] (see Proposition 4.7), we prove that neither of [Formula: see text] and [Formula: see text] is a topological group, where [Formula: see text] (respectively, [Formula: see text]) is a simple closed [Formula: see text]- (respectively, [Formula: see text]-) topological curve with [Formula: see text] elements in [Formula: see text] (respectively, [Formula: see text]) and the operation “∗” is a special kind of binary operation for establishing a group structure of each of [Formula: see text] and [Formula: see text]. Finally, given a [Formula: see text]-[Formula: see text]-group structure of [Formula: see text], we find several types of [Formula: see text]-[Formula: see text]-subgroup structures of it (see Theorem 5.7).

Minimal CSS-supplemented subgroups of finite groups

Over the years, group theorists have evaluated the impact of studying supplemented subgroups on the theory of finite groups. As a result, they have described dozens of criteria for nilpotency and solubility that depend on various supplemented embedded classes of subgroups. In keeping with this approach, we have also established a new concept of CSS-supplemented subgroups. A subgroup H of a group G is described as a CSS-supplemented subgroup in G if T ⊵ G such that G = HT and H ∩ T is SS-supplemented in G. A subgroup H of G is known as an SS-supplemented subgroup in G if there is a subgroup T of G such that HT = G and H ∩ T is SS-quasinormal in T. The SS-supplemented subgroup is expanded based on this agreement by the CSS-supplemented subgroup. By describing new criteria involving formation, we have examined the minimal CSS-subgroup of G based on the structure of the SS-supplemented subgroup that affected p-solubility and p-nilpotency.

HOMOMORPHISM OF IMAF NORMAL SUBGROUP

ABSTRACT In this article an attempt has been made to study some new algebraic structures of intuitionistic multi-anti fuzzy normal subgroups under homomorphism are discussed. Keywords Intuitionistic multi-fuzzy set (IMFS), Intuitionistic multi-anti fuzzy subgroup (IMAFSG), Intuitionistic multi-anti fuzzy normal subgroup (IMAFNSG). Mathematics Subject Classification 20N25, 03E72, 08A72, 03F55, 06F35, 03G25, 08A05

Evolutionary dynamics in the cyclic competition system of seven species: Common cascading dynamics in biodiversity

Complex systems in ecological science can be generally defined by either the number of different species or the structure among species having many relations, and understanding the given interaction structure is essential to predict the evolution of ecosystems. In this paper, we propose a multi-species system whose competition can occur cyclically. By exploiting the generalized system of cyclic competition among seven species, we explore how species biodiversity can appear when the generalized system is established by possessing the underlying mechanism of rock–paper–scissors (RPS) and rock–paper–scissors–lizard–spock (RPSLS) games. Through Monte-Carlo simulations, similar to the RPSLS system having the phase transition in biodiversity from five to one containing the three species survival in the middle, the model for seven species also exhibits similar cascading features in the biodiversity as mobility increases, validated by measuring the survival probability. We also found that not every cyclic structured system among seven species exhibits a common cascading feature in the transition in biodiversity. It is revealed that such a characteristic may require sufficient structures of RPS-like subgroups. Our findings may provide insights into the biodiversity of cyclically competing species and the link to predict biodiversity associated with the interaction structure in the microscopic framework.

Robust inference for subgroup analysis with general transformation models

A crucial step in developing personalized strategies in precision medicine or precision marketing is to identify the latent subgroups of patients or customers of a heterogeneous population. In this article, we consider a general class of heterogeneous transformation models for subgroup identification, under which an unknown monotonic transformation of the response is linearly related to the covariates via subject-specific regression coefficients with unknown error distribution. This class of models is broad enough to cover many popular models, including the heterogeneous linear model, the heterogeneous Cox’s proportional hazard model and the heterogeneous proportional odds model. Without any priori grouping information, we propose a robust method based on the maximum rank correlation and a concave fusion to automatically determine the number of subgroups, identify the latent subgroup structure, and estimate the subgroup-specific covariate effects simultaneously. We establish the theoretical properties of our proposed estimate under regularity conditions. A random weighting resampling scheme is used for variance estimation. The proposed procedure can be easily extended to handle censored data. Numerical studies including simulations and two real data analysis demonstrate that the proposed method performs reasonably well in practical situations.

Characterization of the Structural Changes of the Genomic DNA of Staphylococcus aureus Due to Femtosecond Laser Irradiation by Fourier Transform Infrared (FT-IR) Spectroscopy

Medical application of femtosecond laser irradiation for possible cancer treatment is a primary interest for this study. In this study the effects of femtosecond laser irradiation time, treatment wavelength and strain type upon the structure of DNA from methicillin-resistant Staphylococcus aureus (strain I) and methicillin-susceptible Staphylococcus aureus (strain II) is presented using Fourier transform infrared (FT-IR) spectroscopy. Six sets of irradiation parameters are considered for the bacterial strains. Fourier transform infrared spectra of genomic nucleic acids extracted from the two Staphylococcus aureus bacterial strains were characterized for control and irradiated samples. FT-IR assignments and structure of nucleic acids subgroups were analyzed from 800 to 1800 cm−1. In particular, the bands at 1074 cm−1 [PO2 − symmetric stretching (υsPO2 −)], 1233 cm−1 (antisymmetric stretching υaPO2 −, possible A-form), 1330 cm−1 [deoxyadenosine (dA)], 1400 cm−1 (C3'-endo/anti, A-form), 1540 cm−1 [deoxycytidine (dC)] and 1650 cm−1 [deoxycytidine (dC) (C2 = O2)] were considered to evaluate the structural changes of DNA functional groups following irradiation.

Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles

We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the symmetry transformations and the corresponding generators. The constructed loss functions ensure that the applied transformations are symmetries and the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry. In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups SO(2), SO(3), and SO(4), and of the Lorentz group . Other examples include squeeze mapping, piecewise discontinuous labels, and SO(10), demonstrating that our method is completely general, with many possible applications in physics and data science. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.

Algebraic Morphology of DNA–RNA Transcription and Regulation

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π, we compute the SL(2,C) character variety, where SL(2,C) is simultaneously a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks similar to a free group Fr (r=1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (such as the well known Cayley cubic) within G is a signature of a potential disease even when π looks similar to a free group Fr in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in {A,T/U,G,C}. We produce a few tables of human TFs and miRNAs showing that a disease may occur when either π is away from a free group or G contains surfaces with isolated singularities.

Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3

We introduce an algorithm to compute the structure of the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points over finite fields using the Kummer variety and to check whether they are rational using heights. Both have been made explicit for Jacobians of hyperelliptic curves of genus 3 by Stoll. This article is partially based on the second-named author’s Master thesis.

Crowdsourcing Utilizing Subgroup Structure of Latent Factor Modeling

Crowdsourcing has emerged as an alternative solution for collecting large scale labels. However, the majority of recruited workers are not domain experts, so their contributed labels could be noisy. In this article, we propose a two-stage model to predict the true labels for multicategory classification tasks in crowdsourcing. In the first stage, we fit the observed labels with a latent factor model and incorporate subgroup structures for both tasks and workers through a multi-centroid grouping penalty. Group-specific rotations are introduced to align workers with different task categories to solve multicategory crowdsourcing tasks. In the second stage, we propose a concordance-based approach to identify high-quality worker subgroups who are relied upon to assign labels to tasks. In theory, we show the estimation consistency of the latent factors and the prediction consistency of the proposed method. The simulation studies show that the proposed method outperforms the existing competitive methods, assuming the subgroup structures within tasks and workers. We also demonstrate the application of the proposed method to real world problems and show its superiority. Supplementary materials for this article are available online.

Comparative analysis of structural characteristics of China's 18 typical urban agglomerations based on flows of various elements

Flows of different elements among cities can affect urban organizational structures, leading to specific links between and roles for cities within urban agglomerations. Previous studies have mainly focused on flows produced by specific elements in a single urban agglomeration, and they cannot reveal differences in organizational structures affected by multiple elements among different urban agglomerations. Therefore we selected 18 typical urban agglomerations in China and explored their structural characteristics under comprehensive effects of four elements: economy, resources, population and technology, with gravity model and social network analysis. Guangdong-Hong Kong-Macao Urban Agglomeration had a maximum flow intensity of 5.15, this concentration range was the largest from 0.65 to 3.75, indicating that more elements flowed internally. Shandong Peninsula Urban Agglomeration had the largest network density at 0.412, the lowest efficiency at 0.567, and the lowest hierarchy at 0.000, showing that intercity relationships within SP were the tightest among the18 urban agglomerations. Geographical proximity and topological structure of subgroups proved that connections among subgroups were not complex, and cities of different provinces lacked close interactions. Furthermore, most of the key cities with high centrality indexes were municipalities directly under the Central Government or provincial capitals; these cities played important roles in networks and were distributed into “main spillover” and “bidirectional spillover” subgroups. This paper provides a unified analytical framework aimed at urban agglomerations that accounts for flows of multiple types of elements and explores urban organizational structures, and our results could provide scientific guidance for promoting the coordinated and sustainable development of urban agglomerations.

Identification of subgroups of children in the Australian Autism Biobank using latent class analysis

BackgroundThe identification of reproducible subtypes within autistic populations is a priority research area in the context of neurodevelopment, to pave the way for identification of biomarkers and targeted treatment recommendations. Few previous studies have considered medical comorbidity alongside behavioural, cognitive, and psychiatric data in subgrouping analyses. This study sought to determine whether differing behavioural, cognitive, medical, and psychiatric profiles could be used to distinguish subgroups of children on the autism spectrum in the Australian Autism Biobank (AAB).MethodsLatent profile analysis was used to identify subgroups of children on the autism spectrum within the AAB (n = 1151), utilising data on social communication profiles and restricted, repetitive, and stereotyped behaviours (RRBs), in addition to their cognitive, medical, and psychiatric profiles.ResultsOur study identified four subgroups of children on the autism spectrum with differing profiles of autism traits and associated comorbidities. Two subgroups had more severe clinical and cognitive phenotype, suggesting higher support needs. For the ‘Higher Support Needs with Prominent Language and Cognitive Challenges’ subgroup, social communication, language and cognitive challenges were prominent, with prominent sensory seeking behaviours. The ‘Higher Support Needs with Prominent Medical and Psychiatric and Comorbidity’ subgroup had the highest mean scores of challenges relating to social communication and RRBs, with the highest probability of medical and psychiatric comorbidity, and cognitive scores similar to the overall group mean. Individuals within the ‘Moderate Support Needs with Emotional Challenges’ subgroup, had moderate mean scores of core traits of autism, and the highest probability of depression and/or suicidality. A fourth subgroup contained individuals with fewer challenges across domains (the ‘Fewer Support Needs Group’).LimitationsData utilised to identify subgroups within this study was cross-sectional as longitudinal data was not available.ConclusionsOur findings support the holistic appraisal of support needs for children on the autism spectrum, with assessment of the impact of co-occurring medical and psychiatric conditions in addition to core autism traits, adaptive functioning, and cognitive functioning. Replication of our analysis in other cohorts of children on the autism spectrum is warranted, to assess whether the subgroup structure we identified is applicable in a broader context beyond our specific dataset.

Structure of low-carbon economy spatial correlation network in urban agglomeration

Currently, a low-carbon economy has become an inevitable choice for sustainable development in many countries. This research issue has been widely discussed in the previous literature. However, discussion about the spatial correlation network of low-carbon economy in urban agglomeration has been rare. Focusing on the Chengdu–Chongqing urban agglomeration, the structural characteristics of its low-carbon economy spatial correlation network are explored from three perspectives, namely, overall, individual, and subgroup, based on the low-carbon economy gravity model and social network analysis. This work finds that the spatial correlation of low-carbon economic development in the Chengdu–Chongqing urban agglomeration is increasingly positive. Spatial correlation among cities is constrained by administrative boundaries, which are slowly being eliminated. The centrality results show that the development plans of Chengdu and Chongqing largely determine the spatial orientation of the Chengdu–Chongqing urban agglomeration. Moreover, the spatial spillover and intermediation effects of Chengdu are stronger than those of Chongqing. Finally, the structure of cohesive subgroups is relatively stable, and the core–periphery hierarchy is clearly distributed. The findings contribute to the scientific and rational allocation of spatial resource elements in the Chengdu–Chongqing urban agglomeration, inspiring it to form a new paradigm of low-carbon economic development.

On equality of certain subcentral automorphisms in finite p-groups

Let [Formula: see text] be a group and [Formula: see text] be the group of all automorphisms of [Formula: see text]. In this paper, we study the structure of certain subgroups of the automorphisms group [Formula: see text] and give necessary and sufficient conditions for equality between certain subcentral automorphisms. As consequence of these results, we state a number of interesting corollaries about central automorphisms, autocentral automorphisms and derival automorphisms for finite [Formula: see text]-groups.

Nonparametric Bayesian Two-Level Clustering for Subject-Level Single-Cell Expression Data

The advent of single-cell sequencing opens new avenues for personalized treatment. In this paper, we address a two-level clustering problem of simultaneous subject subgroup discovery (subject level) and cell type detection (cell level) for single-cell expression data from multiple subjects. However, current statistical approaches either cluster cells without considering the subject heterogeneity or group subjects without using the single-cell information. To bridge the gap between cell clustering and subject grouping, we develop a nonparametric Bayesian model, Subject and Cell clustering for Single-Cell expression data (SCSC) model, to achieve subject and cell grouping simultaneously. SCSC does not need to prespecify the subject subgroup number or the cell type number. It automatically induces subject subgroup structures and matches cell types across subjects. Moreover, it directly models the single-cell raw count data by deliberately considering the data's dropouts, library sizes, and over-dispersion. A blocked Gibbs sampler is proposed for the posterior inference. Simulation studies and the application to a multi-subject iPSC scRNA-seq dataset validate the ability of SCSC to simultaneously cluster subjects and cells.

An extension of Fermatean b-derived fuzzy soft ternary subgroup

In this paper, an extension of the fermatean uncertainty soft subgroup structures under a norm. Also, the cubic fermatean uncertainty soft ideal structures and fermatean uncertainty multigroup over multi-homomorphism are discussed in detail.

Homomorphic encoders of profinite abelian groups I

Let {Gi|i∈N} be a family of finite Abelian groups. We say that a subgroup G≤∏i∈NGi is order controllable if for every i∈N there is ni∈N such that for each c∈G, there exists a∈G satisfying that a|[1,i]=c|[1,i], supp(a)⊆[1,ni], and order(a) divides order(c|[1,ni]). In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset {gn|n∈N} that topologically generates the group and whose elements gn all have finite support. As a consequence, sufficient conditions are obtained that allow us to encode, by means of a topological group isomorphism, order controllable profinite abelian groups. Further applications of these results to group codes will appear subsequently.

Spatial heterogeneity automatic detection and estimation

Spatial regression is widely used for modeling the relationship between a dependent variable and explanatory covariates. Oftentimes, the linear relationships vary across space, such that some covariates have location-specific effects on the response. One fundamental question is how to detect the systematic variation in the model and identify which locations share common regression coefficients and which do not. Only a correct model structure can assure unbiased estimation of coefficients and valid inferences. A new procedure is proposed, called Spatial Heterogeneity Automatic Detection and Estimation (SHADE), for automatically and simultaneously subgrouping and estimating covariate effects for spatial regression models. The SHADE employs a class of spatially-weighted fusion type penalty on all pairs of observations, with location-specific weight constructed using spatial information, to cluster coefficients into subgroups. Under certain regularity conditions, the SHADE is shown to be able to identify the true model structure with probability approaching one and estimate regression coefficients consistently. An alternating direction method of multiplier algorithm (ADMM) is developed to compute the SHADE. In numerical studies, the empirical performance of the SHADE is demonstrated by using different choices of weights and comparing their accuracy. The results suggest that spatial information can enhance subgroup structure analysis in challenging situations when the spatial variation among regression coefficients is small or the number of repeated measures is small. Finally, the SHADE is applied to find the relationship between a natural resource survey and a land cover data layer to identify spatially interpretable groups.

Isotropy subgroups of some almost rigid domains

In this paper, we describe the structure of isotropy subgroups of some almost rigid domains and give necessary and sufficient conditions for an automorphism to be an element of the isotropy subgroup.