Abstract Inspired by the Kadanoff transformation in the standard renormalization group theory, we propose a temporal renormalization scheme. A Boltzmann factor that explicitly depends on the renormalized timescale is constructed, permitting thermodynamic quantities to be evaluated self-consistently across different timescales. By applying the scheme to the long-time dynamics of supercooled liquids, we uncover critical-like behaviors of supercooled liquid with three characteristic renormalization timescales: At the first timescale s α , the system appears to be “thermodynamically frozen”, i.e., the energy fluctuation becomes temperature-independent throughout the supercooled regime. At the second timescale s β , the third-order moment of energy distribution reaches a maximum, and s β is nearly temperature-independent. At the third timescale s γ , the third-order moment of energy distribution passes through a minimum, and s γ diverges as a power law s γ ~| T - T c | -γ . The scaling relations may reveal an intrinsic behavior in supercooled liquids, highlighting their unique feature. The current findings also demonstrate that temporal renormalization provides a powerful lens for investigating the timescale-specific dynamics.

We extend our theoretical description of optical excitations in binary plasmonic supercrystals made of two types of metal nanoparticles to three common cubic structures: sodium chloride (NaCl), zinc blende (ZnS), and auricupride ( Cu 3 Au ). Our microscopic model of plasmon polaritons in supercrystals incorporates dipole and quadrupole nanoparticle excitations, their interactions, and their coupling with photons. We calculate the plasmon-polariton dispersion for the three cubic structures starting from the dipole and quadrupole energies obtained in the quasistatic approximation. The bands depend on the properties of the nanoparticles as well as the symmetry and the lattice parameters of the supercrystals. To gain further insight into the coupling of different excitations and develop a set of symmetry selection rules for each structure, we complement our calculations with an extensive group theory analysis. Comparing the results across various structures makes our symmetry analysis more general, enabling us to observe selection rules that apply to multiple structures.

Right groups form an important bridge between group theory and semigroup theory, combining the algebraic symmetry of groups with the one-sided structure of right zero semigroups. This paper develops a complete structural and enumerative theory of right groups. We describe all sub-right groups, Green’s equivalences, Rees’ congruences, and quotients in explicit algebraic terms. Closed formulas and asymptotic estimates are obtained for right groups, sub-right groups, subsemigroups, normal sub-right groups, Rees congruences, and quotient structures, together with a precise comparison between projection and Rees’ quotients. The results unify the structure, combinatorics, and quotient theory of right groups and provide a framework for further investigations into categorical and varietal aspects of right groups.

The power graph P(G) is a simple graph associated with a group G that represents power relations among its elements. Although power graphs have been widely studied in connection with domination and connectivity, the effect of removing dominating sets, particularly those excluding the identity element, on graph connectivity has not been examined in detail. This study aims to characterize dominating sets in power graphs of finite groups and to investigate whether connectivity is preserved after their removal, with emphasis on symmetric groups and cyclic groups. This research employs a theoretical and analytical approach based on group theory and algebraic graph theory. The results show that, for symmetric groups Sn, there exists a dominating set excluding the identity element such that the power graph remains connected after its removal. Furthermore, for cyclic groups Cn, any generator forms a minimum dominating set, and the power graph remains connected after its removal.

We prove that a pseudoconcave complex homogeneous space of a connected solvable linear algebraic group is necessarily compact. This resolves a central conjecture in the theory, showing that pseudoconcavity characterizes compactness for this large class of solvmanifolds. The proof combines new pluriharmonic obstructions for C∗-bundles with the structure theory of solvable groups, demonstrating that any noncompact solvmanifold admits a nonconstant pluriharmonic function, contradicting pseudoconcavity. Our result unifies and extends all known partial classifications for this class of solvable linear algebraic groups, establishing pseudoconcavity as a definitive geometric property that forces compactness in the solvable setting.

Abstract Recently, the author published a book [A. A. Ivanov, Ever-Evolving Groups—an Introduction to Modern Finite Group Theory , Algebr. Appl. 32, Springer, Cham, 2025] where he summarised the recent progress in the geometric theory of sporadic groups and outlined some geometries which require further investigation. Among them was a geometry for the smallest Conway sporadic simple group Co 3 \mathrm{Co}_{3} , with diagram originally introduced in [M. A. Ronan and G. Stroth, Minimal parabolic geometries for the sporadic groups, European J. Combin. 5 (1984), 1, 59–91] (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs , London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]), which we denote by G ⁢ ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) . The 2-local geometries for the other Conway groups Co 1 \mathrm{Co}_{1} and Co 2 \mathrm{Co}_{2} are the tilde and Petersen geometries which have been intensively studied (cf. [A. A. Ivanov and S. V. Shpectorov, The flag-transitive tilde and Petersen-type geometries are all known, Bull. Amer. Math. Soc. (N. S.) 31 (1994), 2, 173–184]). However, G ⁢ ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) seems to be studied less. There is another geometry associated with Co 3 \mathrm{Co}_{3} (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs , London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]) with diagram The elements of type 1, 2, 3, and 4 in the geometry G 276 \mathcal{G}_{276} correspond to cliques of size 1, 2, 3, and 6, respectively, in a double cover of the complete graph on 276 vertices. The group Co 3 \mathrm{Co}_{3} acts doubly transitively on the vertex set of the complete graph and flag-transitively on G 276 \mathcal{G}_{276} . This double cover is naturally associated with the well-known 2-graph of Co 3 \mathrm{Co}_{3} . The geometry G 276 \mathcal{G}_{276} is simply connected, as established by [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs , London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Proposition 6]. In the present paper, we recover the distance 2 graph of G 276 \mathcal{G}_{276} from the universal cover of G ⁢ ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) , in particular re-proving the simple connectedness of the latter geometry originally established in [A. Chermak, B. Oliver and S. Shpectorov, The linking systems of the Solomon 2-local finite groups are simply connected, Proc. Lond. Math. Soc. (3) 97 (2008), 1, 209–238]. A key step in our proof makes use of the simple connectedness of the Petersen-type geometry of the McLaughlin group, as shown in [B. Baumeister, A. A. Ivanov and D. V. Pasechnik, A characterization of the Petersen-type geometry of the McLaughlin group, Math. Proc. Cambridge Philos. Soc. 128 (2000), 1, 21–44].

Abstract We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact‐by‐discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible nongeometric 3‐manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one‐ended virtually torsion‐free graphically discrete groups are action rigid within the class of virtually torsion‐free groups. We also prove quasi‐isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are nonelementary quasi‐convex subgroups. This includes the case of two hyperbolic 3‐manifold groups amalgamated along a quasi‐convex malnormal non‐abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.

Abstract In this present paper we apply the Lie group theory associated fractional calculus to obtain the symmetries of the $$\zeta (t)$$ ζ ( t ) -KdV fractional partial differential equation, which is given in terms of the $$\zeta (t)$$ ζ ( t ) -Riemann-Liouville time fractional partial derivative, in which a particular case of the $$\zeta (t)$$ ζ ( t ) -Hilfer fractional partial derivative is obtained. The calculus of symmetries consider the explicit formula of this infinitesimal extension of the fractional operator. The fractional partial equation is reduced to a fractional differential ordinary equation, and an analytical solution is proposed in the form of a power series. We obtain a nonlinear recurrence for the coefficients of the series. We discuss the linearized case for the fractional KdV equation, obtaining the Mainardi function as the solution, and the results are interpreted graphically. We then present the conservation law theorem for fractional $$\zeta (t)$$ ζ ( t ) operators, and we applied the law to find the law associated with each symmetry.

The q-Heisenberg algebra hn(q) is a significant class of solvable polynomial algebras, and it unifies the canonical commutation relations of Heisenberg algebras and the deformation theory of quantum groups. In this paper, we employ Gröbner-Shirshov basis theory and PBW (Poincare´-Birkhoff-Witt) basis techniques to systematically investigate hn(q). Our main results establish that: hn(q) possesses an iterated skew-polynomial algebra structure, and it satisfies the important homological regularity properties of being Auslander regular, Artin-Schelter regular, and Cohen-Macaulay. These findings provide deep insights into the algebraic structure of hn(q), while simultaneously bridging the gap between noncommutative algebra and quantum representation theory. Furthermore, our constructive approach yields computable methods for studying modules over hn(q), opening new avenues for further research in deformation quantization and quantum algebra.

For a function f \colon \{0,1\}^{n} \to \mathbb{R} with Fourier expansion f=\sum_{S \subset \{1,\ldots,n\}}\hat f(S)\chi_{S} , the hypercontractive inequality for the noise operator allows bounding norms of T_{\rho} f = \sum_{S} \rho^{|S|}\hat f(S)\chi_{S} in terms of norms of f . If f is Boolean-valued, the level- d inequality allows bounding the norm of f^{=d}=\sum_{|S|=d}\hat f(S)\chi_{S} in terms of \mathbb{E}[f] . These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube \{0,1\}^{n} is endowed with the uniform measure, it is easy to show that they do not hold for more general discrete product spaces, and finding a ‘natural’ generalization was a long-standing open problem. Keevash, Lifshitz, Long, and Minzer [J. Amer. Math. Soc. 37, 245–279 (2024)] obtained a hypercontractive inequality for general discrete product spaces, that holds for functions which are ‘global’ – namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group S_{n} and to other settings where sharpness of the bound is crucial. Also, no sharp level- d inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level- d inequality for global functions over discrete product spaces. Our inequalities open the way for diverse applications to extremal set theory, group theory, theoretical computer science, and number theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications that were obtained using our results. Those contain applications to the study of functions over the symmetric group S_{n} – including hypercontractivity and level- d inequalities, character bounds, variants of Roth’s theorem and of Bogolyubov’s lemma, and diameter bounds, as well as an application to the Furstenberg–Sárközy problem on the maximal size of a subset of \{1,\ldots,n\} which does not contain two elements that differ by a perfect square.

Inspired by noncommutative geometry in string theory, we propose extended derivatives in black hole physics by incorporating a real antisymmetric tensor of rank 2 carrying similarities of certain stringy fields. Using gauge theory formulation of gravity via de Sitter group theory, we first find the associated black hole solutions by solving the Einstein field equations. Then, we study the thermodynamic properties by approaching the stability analysis, the criticality and the phase transitions. Concretely, we investigate the P-V criticality behavior of the obtained solution. We compute and examine the Gibbs free energy revealing comparable attitudes with the Van der Waals phase transitions. Combining such results, we provide constraints on the deformed parameter B and the charge Q with the help of CUDA numerical methods exploited in machine learning computations. Precisely, we show that there are suitable ranges for such parameters where the obtained black holes behave like the Van der Waals fluid systems.

The Shortest Vector Problem (SVP) is a fundamental challenge in computational number theory and lattice-based cryptography, with critical implications for cryptanalysis, coding theory, and computational geometry. Classical algorithms such as enumeration, sieving, and basis reduction techniques either suffer from exponential time complexity or prohibitive memory demands. This work presents a novel SVP-solving framework rooted in sheaf theory and the geometry of Etalé spaces, providing a categorical and topological interpretation of vector localization, difference computation, and sieving procedures. By embedding the lattice problem into a manifold structure and defining fractional coordinate spaces via exact sequences, we develop a generalized coboundary map formulation that subsumes classical vector-difference methods used in sieving. Orthogonalization through algorithms such as LLL reduces search complexity, while the Etalé-space-based sieving allows for localized, memory-aware updates. Complexity analysis shows that while the algorithm retains exponential dependence on lattice dimension in the worst case, careful selection of subset coverings and coarse-graining strategies can reduce memory requirements from exponential to polynomial scaling. The proposed approach bridges the gap between geometric group theory, algebraic topology, and algorithmic lattice reduction, offering a mathematically rigorous and computationally viable method for SVP in moderate dimensions.

There are various challenges to foreign policy execution in South Africa. Disagreements over policy direction among political parties, factions, and interest groups may be the most significant of them. As a result, South Africa is at a loss about which partners to choose between traditional Western allies or eastern partners like China and other BRICS partners. This chapter explores how the discord in policy positions and consequent geopolitical implications affect South Africa's international development cooperation agenda. The study adopts a qualitative literature review to draw its conclusion. Theoretically, the chapter leans on Group Theory to guide its research and conclusion. The study shows significant disparities in foreign policy adoption, leading to significant geopolitical and domestic problems. Political parties, foreign actors and elements like NGOs and other interest groups all have disagreements. The chapter makes the argument that these disagreements and consequent delays cause major challenges for South Africa’s development cooperation agenda. In this regard, the chapter makes various recommendations on how South Africa can navigate this conundrum while balancing relations between both Eastern and Western partners for a sustainable development path. Furthermore, it is also recommended that South Africa set a foreign policy agenda that surpasses partisan and party-political interests to minimise these effects. This chapter seeks to contribute to wider discussions on international development cooperation in the changing geopolitical and North-South relations.

Civil Society Organizations are voluntary associations such as the NGOs, community-based organizations, professional associations, faith-based organizations, labour unions, youth groups, and advocacy networks that work to promote social, economic, and political development. Corroboratively, the study sought to ascertain whether the use of voter’s education and election monitoring by civil society organizations (CSOs) have helped in mitigating election violence in Sub-Saharan Africa with particular reference to Nigeria. The study’s theoretical thrust is anchored on the Group Theory of State. The sample size for the study consists of 292 registered members of the Transition Monitoring Group (TMG) and the Centre for Democracy and Development (CDD) selected purposively using Taro Yamane Formula, whereas, the Simple Linear Regression (SLR) was used to test the null hypotheses at 5% error limit. The results revealed a concurrent significant influence of voters’ education and election monitoring by civil society organizations on election violence in Sub-Saharan Africa and particularly Nigeria. Based on the finding, it was recommended that civil society organizations should continuously train and mobile election observers prior to the election periods, as this will help to decentralize its operational scope to cover the rural areas; that there should be increased security and welfare for election monitors/observers in the field in order to guarantee their safety; and Civil society organizations should continuously engage in voters’ education especially by laying emphasis on fairness, peace, and equity as well as respect for election adjudication processes (electoral tribunals).

In this paper, I propose integrating group theory into relational psychoanalytic training to create structural change within psychoanalytic institutes. I argue that candidates should undergo mandatory experiential participation in small, medium, and large group settings to enhance clinical competence and institutional awareness. Drawing from my experience directing the Institute for Relational and Group Psychotherapy, in Athens, Greece, I highlight how group dynamics—particularly within large groups—can deconstruct hierarchical idealizations and foster more authentic, horizontal relationships among trainees and faculty. I suggest rotating group supervision models and shared leadership structures as institutional mechanisms to promote democratic participation and reduce abuses of authority, so that candidates are equipped to recognize and work with both sibling and institutional transferences. By embedding group theory into the heart of psychoanalytic pedagogy, I contend we can create more ethically responsive, inclusive, and resilient training environments.

Abstract In a short paper that appeared in the Journal of the London Mathematical Society in 1934, H. S. M. Coxeter completed the classification of finite Coxeter groups. In this survey, we describe what Coxeter did in this paper and examine an assortment of topics that illustrate the broad and enduring influence of Coxeter's paper on developments in algebra, group theory, and geometry.

The key property of NVIS radio communication is the use of reflective properties of the lower ionospheric boundary to ensure radio contact with the receiver at distances beyond the direct visibility of "classical" radio communications. The task of NVIS radio communication with moving objects (or with consideration of the mutual kinematics of source-receiver), taking into account reflection from an inhomogeneous ionosphere, is a separate problem and at the mathematical level requires the implementation of real-time signal processing algorithms, as well as taking into account the influence of the Doppler effect. In this paper, the structure of the reflected from the ionosphere wave packet is briefly considered, while the kinematics of the receiver is taken into account using a group theory. In this paper, a dispersion relation is obtained, the fulfillment of which gives limitations on the use of the proposed approach.

Cyclotomic classes of finite abelian groups have been extensively investigated for many decades, largely because of their nice algebraic structure and the breadth of their theoretical and practical applications. They naturally arise in diverse areas of mathematics, ranging from number theory and polynomial factorization to the decomposition of group algebras, and they play a particularly important role in the study of algebraic coding theory, where they provide fundamental tools for analyzing cyclic and abelian codes as well as their generalizations. Motivated by these connections, the present work concentrates on certain special types of q 2 ‐cyclotomic classes arising in a direct product of finite abelian groups, with q denoting a prime power. The main contributions of this work include a detailed characterization and enumeration of these cyclotomic classes in terms of the corresponding classes in their component groups. Several concrete examples and explicit computations are provided to illustrate the general results and to demonstrate how the cyclotomic structure behaves under group product decompositions. As an application, the paper revisits the enumeration of Hermitian complementary dual and Hermitian self‐dual abelian codes defined over group algebras, presenting refined counting formulas derived from the structure of q 2 ‐cyclotomic classes. These results not only deepen our understanding of the interplay between group theory and coding theory but also contribute to the ongoing development of algebraic coding techniques grounded in group‐theoretic methods.

The complex intuitionistic fuzzy (CIF) set is an advanced version of the regular intuitionistic fuzzy set. It is made to better show the uncertainty and complexity that arise in real‐life problems. The grading and nongrading degrees in the CIF set are shown by complex‐valued functions that are defined on the unit disc of the complex plane. A CIF set combines both magnitude and phase terms to establish a more robust mathematical framework to understand and address, such as decision‐making and pattern detection, where conventional intuitionistic fuzzy sets may be insufficient. The £‐CIF set is the generalization of the CIF set. The extension of the traditional Sylow theorems in the context of £‐CIF brings in the idea of the £‐CIF conjugate element within the £‐CIF subgroup of a group. In this paper, we introduce the paradigmatic idea of £‐CIF subgroups of a group, and various features of this concept are demonstrated. Furthermore, the construction of the £‐CIF version of the Cauchy theorem is derived. Finally, we look into the £‐CIF Sylow p subgroup for a finite group and show how Sylow’s theorems can be extended in a £‐CIF environment.

Ytterbium Surprise – the meeting of a rare earth element with symmetry, group theory and culture