Group Theory Research Articles (Page 1)

Lie symmetry analysis is a powerful tool for solving ordinary differential equations whose results have been a huge driving force in the history of mathematics. Lie group theory is a very practical and handy mathematical approach and it can be used to obtain solutions to diverse problems in applied mathematics. Symmetry is an operation which leaves invariant that upon which it operates. A lie group or symmetry group is a group of transformations which maps any solution of the system to another solution of the same system. The symmetries of a given system of ordinary differential equations inform a lot about the closed form or the ability of the differential equation to solve. In this study, we use infinitesimal generators to obtain a sixth parameter symmetry for an ordinary differential equation. Symmetry properties and reduction of order differential equations yield solutions that are important in the field of science.

To address the urgent demand for high-precision positioning in power industry operations within sparse reference station areas, this paper proposes a real-time kinematic positioning method integrating BeiDou multi-antenna Precise Point Positioning–Real-Time Kinematic (PPP-RTK) with inertial measurement unit (IMU) assistance. By combining the strengths of Precise Point Positioning (PPP) and Real-Time Kinematic (RTK) technologies, we establish a multi-antenna observation model based on State Space Representation (SSR), incorporating satellite-based augmentation signals and atmospheric correction information from sparse reference station networks. Lie group theory is employed to enhance the Extended Kalman Filter (EKF) for simultaneous estimation of position, attitude, and ambiguity parameters. The integration of IMU measurements significantly improves robustness against environmental interference in dynamic scenarios. Experimental results demonstrate average positioning errors of 3.12 cm, 3.71 cm, and 6.23 cm in the East, North, and Up (ENU) directions, respectively, with an average convergence time of 1.62 min. Compared with non-IMU-augmented single-antenna PPP-RTK solutions, the proposed method achieves accuracy improvements up to 59.6% while maintaining stability in signal-occluded environments. This approach provides centimeter-level real-time positioning support for critical power grid operations in remote areas such as desert and Gobi regions, including infrastructure inspection and precise tower assembly, thereby significantly improving the efficiency of intelligent grid operation and maintenance.

Fuzzy group theory has evolved beyond single-valued memberships to account for dual polarity and uncertainty. Building on Dib’s fuzzy space and bipolar-valued fuzzy sets, we develop a unified algebraic theory of bipolar-valued fuzzy (BVF) subgroups, including BVF normal subgroups and BVF homomorphisms, via a BVF binary operation (BVFBO) on a BVF-space. We establish necessary and sufficient subgroup criteria, characterize normality through coset symmetry in BVF-space, and prove homomorphism properties that align BVF structures with their classical counterparts through correspondence theorems. The framework clarifies when associativity holds between subgroup elements and ambient BVF-group elements and provides constructive examples. This generalization resolves limitations tied to the absence of a bipolar fuzzy universal set and supports applications in polarity-sensitive decision systems and network analysis.

Abstract. LiDAR based simultaneous localization and mapping (SLAM) plays an important role for real-time localization and 3D mobile mapping of autonomous systems. However, the long-term scan-to-scan matching in the SLAM can introduce uncertainty into the position estimation. which results in a large drift. In this paper, we specifically focus on real-time estimation of the global positioning uncertainty of LiDAR SLAM so that it can enable the graceful weighting of LiDAR SLAM with other positioning systems in multi-sensor fusion localization. We introduce Lie group theory and multiple fault hypothesis solution separation (MHSS) method into a Kalman-filter based LiDAR SLAM framework. First, the scan-to-scan matching uncertainty is obtained by establishing fault hypothesis utilizing MHSS method. Then the global positioning uncertainty is propagated on Lie group based on the scan-to-scan matching uncertainty in terms of the relative position and rotation. The NCLT dataset is used to validate the proposed method. Experimental results show that: comparing with previous solutions that treat scan-to-scan matching uncertainty as a constant, the proposed method is more adaptive and robust. And the real-time global positioning uncertainty estimation can envelop the real SLAM absolute trajectory error (ATE) for the most of the time and can reflect the real changing tendency of ATE.

This paper investigates the periodic and weakly periodic ground states on the Cayley tree of order two and three. Utilizing techniques from statistical mechanics, algebraic graph theory, and group theory, we determine the conditions under which these ground states can exist. Our results demonstrate that the unique structure of the Cayley tree supports a variety of periodic and weakly periodic ground states, each with distinct physical properties. These findings enhance our understanding of phase transitions and critical phenomena in physical systems.

Real-time slope stability quantification: Emergent framework coupling landslide displacement with damage mechanics and renormalization group theory

Quantum chemical calculations of ultrathin nanorods cut from two bulk selenium phases were performed. Two sets of nanorods with trigonal and hexagonal geometric shapes described by the rod symmetry groups p31 and p3121, respectively, were constructed from the most stable Se-I (P3121) phase. The ultrathin nanorods generated by the Se-I phase were found to be unstable with respect to spontaneous torsion deformations, which slightly shift the helical axis order away from its crystallographic integer value of 3. In order to describe their correct atomic structure, one should use the line symmetry groups and determine the exact order of the helical axis for each nanorod. As the nanorod thickness increases, the true order quickly approaches the crystallographic value, but is never equal to it. Nanorods with a square geometric shape were constructed from the Se-II' (I41/acd) phase. Depending on their thickness, these nanorods are classified as either chiral or achiral, exhibiting p4122 or p4c2 symmetries, respectively. It was shown that square nanorods represent a unique class of nanostructures that alternately exhibit chiral and achiral properties as their thickness increases. Chiral square nanorods are unstable with respect to spontaneous torsion deformations, which shift the helical axis order from the crystallographic integer value of 4 (similar to nanorods cut from the Se-I phase). At the same time, achiral square nanorods are stable with respect to spontaneous torsion deformations.

ABSTRACT Background Individuals from specific ethnic populations are at increased risk of developing psychosis. Although socio-environmental adversity is cited as the main cause, current theoretical perspectives remain limited. This study adds to the debate about psychosis excess, by examining the qualitative accounts and perspectives of racialised patients. Method Drawing on a phenomenological and life narratives approach, 13 semi-structured interviews were conducted with Black/Black-British Caribbean patients recovering from their first episode of psychosis. Interviews elicited key aspects of the patients’ lives and obtained their reflections on the perceived cause of their first episode. Interviews were analysed thematically. Results Seven themes captured the life events of the patients. These were: Interpersonal Relationships and Social Interactions; Employment; Hobbies and Personal Interests; Educational Attainment; Cannabis use; Migrating to England and Religiousness. Being mistreated by others, cannabis use, and the cumulative effect of all of life’s negative circumstances were cited as causes of their first episode. Discussion Our findings are discussed in the context of interpersonal trauma, cultural differences in the response to mental distress and social group theory. We also consider the role of cannabis use as a social process and the cumulative effect of negative life events. The limitations of this work are discussed along with recommendations for further research.

Symmetry breaking in van der Waals materials enables the realization of quantum states and advanced device functionalities. Janus transition-metal dichalcogenides (TMDs) exhibit distinctive nonlinear optical properties due to their broken out-of-plane mirror symmetry. However, the dynamic control of second harmonic generation (SHG) anisotropy and resonance behavior via optical excitation remains elusive. In this work, we investigate the SHG response of Janus MoSSe/MoS2 heterostructures with 2H and 3R stackings. We can tune the SHG response by varying the incident photon wavelength from 800 to 1000 nm, which shows a resonance-dependent enhancement in intensity and a deviation from 6-fold symmetry, indicating wavelength-dependent anisotropy. The ratio between maximum and minimum intensity in the armchair directions, associated with the SHG anisotropy, reaches a value of 1.73 at the excitation wavelength of 1000 nm. Group theory analysis and first-principles calculations reveal that the observed anisotropy arises from optically induced strain. Our findings highlight the role of symmetry breaking and optical resonance contributing to the optomechanical tuning of SHG anisotropy, offering opportunities for developing Janus TMD-based photonic devices for frequency conversion, light generation, and optical switching.

Purpose Guided by the source credibility model and reference group theory, this study proposed a theoretical framework to uncover the mechanisms of effective social media influencer marketing. Specifically, the research investigated the impact of virtual social media influencers’ (VSMIs) source credibility and self-congruity on brand image and trust associated with the VSMIs’ endorsement, subsequently influencing purchase intention toward the brand. Design/methodology/approach The present study utilized a between-subjects experimental design. Two distinct Instagram ads were created to represent VSMIs’ endorsements for fast fashion and luxury fashion brands. Female participants aged 18–25 were randomly assigned to view one of the two Instagram ads. Data were collected through an online survey administered via Qualtrics software, with 166 participants completing the survey. Findings The results of structural equation modeling revealed that source credibility significantly impacts brand image and trust associated with the VSMIs’ endorsement. Conversely, self-congruity notably enhances brand image but does not directly influence trust in a fashion brand endorsed by the influencer. Brand image significantly affects trust in a fashion brand endorsed by the influencer, which subsequently influences purchase intention toward the fashion brand. The effects of VSMI characteristics remain consistent across both fast fashion and luxury fashion brands. Both brand image and trust mediate the relationship between the attributes of VSMIs and purchase intention toward the fashion brand endorsed by the influencer. Originality/value This study contributes to the influencer marketing literature by identifying VSMI attributes that shape source credibility and self-congruity, which are vital in influencer marketing. The findings offer theoretical and practical implications, particularly for fashion brand managers. The study provides valuable insights into curating social media content, emphasizing the strategic advantages of partnering with VSMIs and avoiding misalignments that could damage brand image or trust.

This paper investigates bijections on a group G that arise from products of the form f(x)g(x), a problem that is centrally connected to the concept of a complete mapping. We introduce the notion of a chain-k mapping to analyze the structure of certain full-cycle permutations and explore the relationship between complete mappings and the spectral properties of their associated permutation matrices. Key results include a proof that the cyclic group Z/nZ admits a complete mapping if and only if n is odd, and a characterization of the cycle structures of permutations for which the maps id+ and id are automorphisms of a specific vector subspace. This work establishes a connection between combinatorial group theory and the eigenvalue theory of permutation matrices, viewed through the novel lens of (-1)-elliptic elements. We will begin with the motivation that led to this study, and look at problems including elliptic elements and complete mappings over finite fields.

This paper examines the factor that has remained relatively unstudied in the literature, namely the influence of non-governmental organizations (NGOs) in informing legislative developments in the Kurdistan region of Iraq (KRI) with emphasis on the Civil Society Law of 2011. Ultimately, filling an important gap in the literature, the study sought to explore the tactics that NGOs use towards the influencing of legislation especially within inter politic systems. Still, more work is required in clarifying the exact ways and tactics through which NGOs influenced this body of legislation, which has, certainly, affected civil society organizations. To collect desired data, this paper adopts a qualitative case study method with both interviews and documentary analysis to capture the strategies and experiences of the NGOs as well as the effects of the NGO engagements on policy making process from the perspective of the target beneficiaries from the Kurdistan parliamentary and non-governmental organizations. The paper is found that NGOs used lobbying, advocacy and alliances in the management of the civil society and the shaping of the Civil Society law. By thus applying the Interest Group Theory to the semi-autonomous regions of a developing country, this study enriches the approaches to analyzing civil society and governance. Thus, it has serious policy implications for countries with opaque governance structures for the need-to-know policymakers, NGOs, and international donors in the pursuit of more effective civil society participation in legal change.

ABSTRACT This paper systematically examines the historical evolution of China’s permanent employment system (Gudinggong) and subsequent reforms, including the establishment of the labor contract system (Hetongzhi), implementation of the personnel agency system (Renshi dailizhi), and transfer of welfare responsibilities from work unit (danwei) to society. This study explores the strategies employed in implementing these reforms, the subsequent variations resulting from them, and the anomalous development of the labor dispatch system (Paiqianzhi), and the resulting identity segregation and institutionalization within danwei. The authors argue that China’s danwei system, though a product of Marxist ideology, bears certain resemblances to Émile Durkheim’s ideal of modern professional groups. Since the 1980s, labor and personnel reforms have addressed the old danwei system’s shortcomings – such as poor labor mobility and excessive egalitarianism – while simultaneously dismantling the organic solidarity of these quasi-professional groups. Drawing on Durkheim’s theory of professional groups, the authors highlight that the absence of transparent recruitment norms, coupled with a lack of value-rational management, ultimately led to a decline in individual morality and broader societal ethics. They contend China’s post-1980s problems stem less from the rise of capitalism than from the erosion of social morality due to the dissolution of these quasi-professional groups.

Group theory is a very important concept in mathematics with many interesting theories that have been widely applied in other areas of mathematics. As one of the fundamental tools in abstract algebra, it provides a unifying language for studying symmetries, structures, and transformations, making it central to both theoretical and applied mathematics. This paper proves the orbit stability theorem based on the theory of group actions. Then, this paper introduces the application of the orbit stabilizer in other parts of mathematics and its full proof. Among these theorems, compared with other proof methods, the orbit stabilizer theorem is more concise and easier to understand. These examples show the wide application of the orbit stability theorem in mathematics, proving its practicality. Furthermore, the theorem serves as a foundation for exploring topics such as combinatorics, number theory, and geometry, where orbit-stabilizer arguments simplify otherwise complex counting and classification problems. In this way, the study highlights how group theory not only develops its own framework but also contributes essential insights to broader mathematical investigations.

Abstract Most measurements are designed to tell you which of several alternatives have occurred, but it is also possible to make measurements that eliminate possibilities and tell you an alternative that did not occur. Measurements of this type have proven useful in quantum foundations and in quantum cryptography. Here we show how group theory can be used to design such measurements. This requires that the set of states being considered possesses a symmetry described by a group. After some general considerations, we focus on the case of measurements on two-qubit states that eliminate one state. We then move on to construct measurements that eliminate two three-qubit states and four four-qubit states. The sets of states eliminated constitute cosets of a subgroup. A condition that constrains the construction of elimination measurements is then presented. Finally, in an appendix, we briefly consider the case of elimination measurements with failure probabilities and an elimination measurement on n-qubit states.

Some Practical Applications of Group Theory Summary It is known that mathematical methods are widely used in physics, chemistry, biology, psychology, and other areas of science. This issue is addressed in the article using group theory, and specific examples are provided. The fields of application of mathematics are very vast and diverse. This results from the use of operations such as generalization, analogy, and comparison in mathematics. Group theory, created by E.Galois in the 19th century, serves to identify such analogies using binary algebraic operations defined on sets. Keywords: mathematics, algebra, group theory, method of generalization, sets

This paper explores Lagranges Theorem, a foundational result in abstract algebra that establishes a connection between the orders of a group and its subgroups. Initially introduced by Joseph Lagrange in the 18th century, the theorem asserts that the order of any subgroup divides the order of the entire group. This investigation begins with essential concepts of group theory, including cosets and bijections, leading to a rigorous proof of Lagranges Theorem. The paper also highlights significant implications of the theorem, such as its role in deriving Wilsons Theorem and Fermats Little Theorem, both of which proves pivotal in algebraic theory. Furthermore, the applications of Lagranges Theorem in modern cryptography, particularly in the RSA public-key cryptosystem, are discussed, illustrating its relevance in contemporary mathematical practices. Despite its profound impact, there is no guarantee of the existence of subgroups for every divisor by the theorem, a limitation addressed by Sylows Theorem. This paper concludes by emphasizing the enduring significance of Lagranges Theorem in linking abstract algebra to practical applications and suggests avenues for future research in Galois theory and advanced cryptographic methods.

Chronic kidney disease (CKD) inequities among midlife women in the U.S. arise from intersecting determinants across individual, interpersonal, community, and societal levels. This paper introduces the Kidney Health Inequities among Midlife Women Minority Group (KH-MWMG) Theory, a situation-specific theory (SST) developed using an integrative approach. The KH-MWMG theory illustrates how socio-ecological factors interact to influence CKD risks and outcomes over the life course. The KH-MWMG theory provides a structured framework to address CKD inequities. The SST needs to be further developed through its actual application in nursing research and practice with various minority groups of this specific population.

Iwasawa theory of fine Selmer groups over global fields

Symmetry is a fundamental principle in mathematics, physics, and biology, where it governs structure and invariance. Classical symmetry analysis focuses on exact group-theoretic descriptions, but rarely addresses how robust a symmetric configuration is to perturbations. In this work, we introduce a probabilistic framework for quantifying the stability of finite point-set symmetries under random deletions. Specifically, given a finite set of points with a prescribed nontrivial symmetry group, we define the probability PN that removing N points reduces the symmetry to the trivial group C1. The complementary quantity SN=1−PN serves as a measure of symmetry stability, providing a robustness profile of the configuration. We calculate SN explicitly for representative families of symmetric point sets, including linear arrays, polygons, polyhedra, directed necklace of points, and crystallographic unit cells. Our results demonstrate unexpected behaviors: the regular hexagon loses symmetry with a probability of 0.6 under the removal of three vertices, while cubes and tetrahedra exhibit the maximal robustness (SN=1) for all admissible N. We further introduce a Shannon entropy of symmetry stability, which quantifies the overall uncertainty of symmetry breaking across all deletion sizes. This framework extends classical symmetry studies by incorporating randomness, linking group theory with probabilistic combinatorics, and suggesting applications ranging from crystallography to defect tolerance in physical systems.