Search Results (544)

The description of the Earth’s gravitational field, governed by the fundamental potential equation (the Laplace equation), is conventionally expressed using spherical harmonics, yet the Cartesian formulation, using a Taylor series representation, offers significant algebraic advantages. This paper proposes a novel Direct Cartesian Method for generating spherical basis functions and coefficients directly within the Cartesian coordinate system, utilising the partial derivatives of the inverse distance ($1/R$) function. The present study investigates the structural correspondence between the Cartesian form of spherical basis functions and the high-order partial derivatives of $1/R$. The study reveals that spherical basis functions can be categorised into four distinct groups based on the parity of the degree n and order m. It is demonstrated that each spherical basis function is equivalent to a weighted summation of the partial derivatives of the inverse distance ($1/R$) with respect to Cartesian coordinates. Specifically, the basis functions are combined with those derivatives that share the same order of Z-differentiation and possess matching parities in their orders of differentiation with respect to X and Y. In order to facilitate the practical calculation of these high-degree derivatives, a recursive numerical algorithm has been developed. The method generates the polynomial coefficients for the numerator of the $1/R$ derivatives. A pivotal innovation is the implementation of a step-wise normalization scheme within the recursive relations. The integration of the recursive ratios of global normalization factors (including full Schmidt normalization) into each step of the algorithm effectively neutralises factorial growth, rendering the process immune to numerical overflow. The validity and numerical stability of the proposed method are demonstrated through a detailed step-by-step derivation of a sectorial basis function ($n=8,m=2$). Full article

Quantum anomalies are traditionally understood as classical symmetries that fail to survive quantization, while experimental “anomalies” denote deviations between theoretical predictions and measured values. In this work, we develop a unified framework in which both phenomena can be interpreted through the lens of algebraic quantum field theory (AQFT). Building on the renormalization group viewed as an extension problem, we show that renormalization ambiguities correspond to nontrivial elements of Hochschild cohomology, giving rise to a deformation of the observable algebra $A\ast B=AB+\epsilon \omega (A,B)$, where $\omega$ is a Hochschild 2-cocycle. We interpret $\omega$ as an intrinsic algebraic curvature of the net of local algebras, namely the (local) Hochschild class that measures the obstruction to trivializing infinitesimal scheme changes by inner redefinitions under locality and covariance constraints. The transported product is associative; its first-order expansion is associative up to $\mathcal{O}\left({\epsilon }^2\right)$ while preserving the ∗-structure and Ward identities to the first order. We prove the existence of nontrivial cocycles in the perturbative AQFT setting, derive the conditions under which the deformed product respects positivity and locality, and establish the compatibility with current conservation. The construction provides a direct algebraic bridge to standard cohomological anomalies (chiral, trace, and gravitational) and yields correlated deformations of physical amplitudes. Fixing the small deformation parameter $\epsilon$ from the muon $(g-2)$ discrepancy, we propagate the framework to predictions for the electron $(g-2)$, charged lepton EDMs, and other low-energy observables. This approach reduces reliance on ad hoc form-factor parametrizations by organizing first-order scheme-induced deformations into correlation laws among low-energy observables. We argue that interpreting quantum anomalies as manifestations of algebraic curvature opens a pathway to a unified, testable account of renormalization ambiguities and their phenomenological consequences. We emphasize that the framework does not eliminate renormalization or quantum anomalies; rather, it repackages the finite renormalization freedom of pAQFT into cohomological data and relates it functorially to standard anomaly classes. Full article

This paper introduces and systematically develops the theory of Zappa–Szép skew braces, a novel algebraic structure that provides a unified framework for bidirectional group interactions, thereby generalizing the classical constructions of semidirect skew braces and matched-pair factorizations (ZS1–ZS4, BC1–BC2). We establish the complete axiomatic foundation for these objects, characterizing them through necessary and sufficient compatibility conditions that encode mutual actions between two digroups. Central results include a semidirect embedding theorem, explicit constructions of nontrivial examples—notably a fully mutual brace of order 12 built from $V_4$ and $C_3$—and a detailed analysis of key structural invariants such as the socle, center, and automorphism groups. The framework is further elucidated via universal properties and categorical adjunctions, positioning Zappa–Szép skew braces as fundamental objects within noncommutative algebra. Applications to representation theory, cohomology, and the construction of set-theoretic solutions to the Yang–Baxter equation are derived, demonstrating both the generality and utility of the theory. Full article

We consider the nonlinear reaction–diffusion equation on the unit sphere $u_t={\mathrm{\Delta }}_{{\mathbb{S}}^2}u+f\left(u\right)$, $f_{uu}\ne 0$, and carry out a complete Lie point symmetry analysis. Solving the associated determining system yields a rigidity theorem: for every genuinely nonlinear $f\left(u\right)$, the admitted symmetry algebra is $\mathfrak{so}\left(3\right)\oplus ⟨{\partial }_t⟩$, generated by the rotational Killing fields and time translation. We further show through a group classification that the source families that enlarge symmetries in Euclidean space do not produce any additional point symmetries on ${\mathbb{S}}^2$. From an optimal system of subalgebras, we derive curvature-adapted reductions in which the Laplace–Beltrami operator becomes a Legendre-type operator in intrinsic invariants. For the specific nonlinear source $f\left(u\right)=-e^u-2$, specific reduced ODEs admit a hidden one-parameter symmetry, yielding a first integral and explicit steady states on ${\mathbb{S}}^2$. Full article

The past decade has seen growing applications of the information flow-based causality analysis, particularly with the concise formula of its maximum likelihood estimator. At present, the algorithm for its estimation is based on differential dynamical systems, which, however, may raise an issue for coarsely sampled time series. Here, we show that, for linear systems, this is suitable at least qualitatively, but, for highly nonlinear systems, the bias increases significantly as the sampling frequency is reduced. This study provides a partial solution to this problem, showing how causality analysis can be made faithful with coarsely sampled series, provided that the statistics are sufficient. The key point here is that, instead of working with a Lie algebra, we turn to work with its corresponding Lie group. An explicit and concise formula is obtained, with only sample covariances involved. It is successfully applied to a system comprising a pair of coupled Rössler oscillators. Particularly remarkable is the success when the two oscillators are nearly synchronized. As more often than not observations may be scarce, this solution, albeit partial, is very timely. Full article

This paper presents new aggregation operators for $p,q,r$-fractional fuzzy sets based on the Frank t-norm and t-conorm. We introduce the $p,q,r$-fractional fuzzy Frank weighted average and $p,q,r$-fractional fuzzy Frank weighted geometric operators and discuss their algebraic properties, including closure, boundedness, idempotency, and monotonicity. Based on new operations, we develop a multi-criteria group decision-making framework that integrates the evaluations of multiple experts via the proposed Frank operators and ranks the alternatives under $p,q,r$-fractional fuzzy information. The model is applied to a cryptocurrency stability assessment problem, where four coins are evaluated with respect to six criteria. The results show that both aggregation operators yield consistent rankings with good discriminatory power among the alternatives. A sensitivity analysis is conducted to check the stability of the model under parameter variations. A comparative study further demonstrates the compatibility and advantages of the proposed method over several existing decision-making approaches. The proposed framework is well suited to decision-making scenarios in which multiple experts’ opinions must be integrated within a complex fuzzy information environment. Full article

In response to the growing complexity and uncertainty in real-world decision-making, this study introduces a novel framework based on the polytopic fuzzy tensor (PFT) model, which unifies the geometric structure of polytopes with the representational power of fuzzy tensors. The PFT framework is specifically designed to handle high-dimensional, imprecise, and ambiguous information commonly encountered in multi-criteria group decision-making scenarios. To support this framework, we define a suite of algebraic operations, aggregation mechanisms, and theoretical properties tailored to the PFT environment, with comprehensive mathematical formulations and illustrative validations. The effectiveness of the proposed method is demonstrated through a real-world application involving the evaluation of six pancreatic cancer treatment strategies. These alternatives are assessed against five key criteria: quality of life, side effects, treatment accessibility, cost, and duration. Our results reveal that the PFT-based approach outperforms traditional fuzzy decision-making techniques by delivering more consistent, interpretable, and reliable outcomes under uncertainty. Moreover, comparative analysis confirms the model’s superior ability to handle multidimensional expert evaluations and integrate conflicting information. This research contributes a significant advancement in the field of fuzzy decision science by offering a flexible, theoretically sound, and practically applicable tool for complex decision problems. Future work will focus on improving computational performance, adapting the model for real-time data, and exploring broader interdisciplinary applications. Full article

Dynamic multi-criteria group decision-making (MCGDM) is widely applied in complex real-world settings where multiple experts evaluate alternatives across diverse criteria under uncertain and evolving conditions. This study proposes a transparent and interpretable linguistic (L-) framework for dynamic MCGDM grounded in hedge algebras (HA), a mathematical formalism that provides explicit algebraic and semantic structures for L-domains. A novel binary L-aggregation operator is developed using the 4-tuple semantic representation of HA, ensuring closure, commutativity, monotonicity, partial associativity, the existence of an identity element, and semantic consistency throughout the aggregation process. Using this operator, a two-stage dynamic decision-making model is developed—(i) L-FAHP for determining the criterion weights in dynamic environments, and (ii) L-FTOPSIS for ranking alternatives—where both methods are formulated on HA L-scales. To address temporal dynamics, a dynamic L-aggregation mechanism is further proposed to integrate current expert judgments with historical evaluations through a semantic decay factor, enabling the controlled attenuation of outdated information. A case study on enterprise digital transformation readiness illustrates that the proposed framework enhances semantic interpretability, maintains stability under uncertainty, and more accurately captures the temporal evolution of expert assessments. These results underscore the practical value and applicability of the HA-based dynamic L-approach in complex decision environments where expert knowledge and temporal variability are critical. Full article

In this paper we study the existence of Killing vector fields for right-invariant metrics on five-dimensional Lie groups. We begin by providing some explanation of the classification lists of the low-dimensional Lie algebras. Then we review some of the known results about Killing vector fields on Lie groups. We take as our invariant metric the sum of the squares of the right-invariant Maurer–Cartan one-forms, starting from a coordinate representation. A number of such metrics are uncovered that have one or more extra Killing vector fields, besides the left-invariant vector fields that are automatically Killing for a right-invariant metric. In each case the corresponding Lie algebra of Killing vector fields is found and identified to the extent possible on a standard list. The computations are facilitated by use of the symbolic manipulation package MAPLE. Full article

This paper establishes a comprehensive framework for the hereditary transfer of categorical completeness and cocompleteness to categories of internal group objects in D-modules. We prove that while completeness of $\mathbf{Grp}\left(\mathbf{D}\text{-}\mathbf{Mod}\right)$ follows unconditionally from the completeness of the base category $\mathbf{D}\text{-}\mathbf{Mod}$, cocompleteness requires $\mathbf{D}\text{-}\mathbf{Mod}$ to be regular, cocomplete, and admit a free group functor left adjoint to the forgetful functor. Explicit constructions are provided for limits via componentwise operations and for colimits through coequalizers of relations induced by group axioms over free group objects. The theory reveals fundamental geometric obstructions: differentially constrained subcategories such as holonomic D-modules fail to be cocomplete due to characteristic variety constraints that prevent free group constructions. Applications demonstrate cocompleteness in topological D-module groups and D-module sheaves, while counterexamples in differential geometric groups exhibit necessary analytic constraints. Additional results include regularity inheritance under product-preserving free group functors, internal hom-object constructions in locally Cartesian closed settings yielding Tannaka-type dualities, and monadicity criteria for locally presentable base categories. This work unifies categorical algebra with differential geometric obstruction theory, resolving fundamental questions on exactness transfer while enabling new constructions in homotopical algebra and internal representation theory for D-modules. Full article

A unified geometric framework for the full Lorentz group is introduced, based on a bounded angle parametrization of spacetime transformations within a hyper-spherical geometry. By mapping the unbounded hyperbolic angle $\phi$ to a bounded angle $\beta$ using the Gudermannian function, hyperbolic, causal, and Euclidean three-spheres are brought together into a single structure: hyper-spacetime. This structure unifies the Euclidean ${\mathbb{R}}^4$ and Minkowski ${\mathbb{R}}^{\mathrm{1,3}}$ domains, incorporates discrete symmetries in a continuous way, and removes discontinuities at the lightlike boundary. Each three-sphere carries a natural spinor set, encoding symmetry, and acting as eigen-spinors of corresponding observables. These reproduce the Dirac spectrum while confining singular behavior to a scalar factor. The bounded angle parametrization therefore provides a continuous, closed representation of the full Lorentz group and a transparent geometric basis for spacetime symmetry. Full article

The concept of symmetry is fundamental to the study of algebra; it serves as the basis for a branch of group theory that is essential to abstract algebra. A semigroup is a structure that builds upon the concept of a group, similarly extending the idea of symmetry found within groups. In this study, we specifically focus on semigroups. The main objective of this research is to apply the notion of m-polar picture fuzzy sets (m-PPFSs), with m being a natural number, in investigations into semigroups, as this concept generalizes m-polar fuzzy sets (m-PFSs) and picture fuzzy sets (PFSs). This research introduces the concepts of m-polar picture fuzzy left ideals (m-PPFLs), m-polar picture fuzzy right ideals (m-PPFRs), m-polar picture fuzzy ideals (m-PPFIs), m-polar picture fuzzy bi-ideals (m-PPFBs), and m-polar picture fuzzy generalized bi-ideals (m-PPFGBs) in semigroups. This study examines the relationships between these concepts, showing that every m-PPFL (m-PPFR) in the semigroups is also an m-PPFB, and that every m-PPFB in the semigroups is an m-PPFGB. However, the opposite is not true. Additionally, we provide the characteristics of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in semigroups. We further discuss the connections between the m-PPFLs (m-PPFIs) and the m-PPFBs within the framework of regular semigroups, and most importantly, we show that, if the semigroup is regular, then the m-PPFBs and m-PPFGBs are equal. Finally, we utilize the properties of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs within semigroups to explore the classifications of regular semigroups. Full article

This paper focuses on the invariant properties of fractal operators and aims to achieve the axiomatization of the theory of fractal operators. Building upon the derivative and integral theorems of the Laplace transform, we redefine the time differential operator and demonstrate that the newly defined operator exhibits form invariance under the Laplace transform. This property is further generalized to encompass broader classes of operators, including non-rational and fractional fractal operators. Inspired by Klein’s concept of “invariance under transformation groups”, we propose a postulate asserting the “form invariance of operators under the Laplace transform group”. Based on this postulate, we clarified the algebraic operational rules of fractal operators and constructed a rigorously axiomatized theory of fractal operators. Full article

Many of the most fundamental observables—position, momentum, phase point, and spin direction—cannot be measured by an instrument that obeys the orthogonal projection postulate. Continuous-in-time measurements provide the missing theoretical framework to make physical sense of such observables. The elements of the time-dependent instrument define a group called the instrumental group (IG). Relative to the IG, all of the time dependence is contained in a certain function called the Kraus-operator density (KOD), which evolves according to a classical Kolmogorov equation. Unlike the Lindblad master equation, the KOD Kolmogorov equation is a direct expression of how the elements of the instrument (not just the total quantum channel) evolve. Shifting from continuous measurements to sequential measurements more generally, the structure of combining instruments in sequence is shown to correspond to the convolution of their KODs. This convolution promotes the IG to an involutive Banach algebra (a structure that goes all the way back to the origins of POVM and C*-algebra theory), which will be called the instrumental group algebra (IGA). The IGA is the true home of the KOD, similar to how the dual of a von Neumann algebra is the true home of the density operator. Operators on the IGA, which play the analogous role for KODs as superoperators play for density operators, are called ultraoperators and various important examples are discussed. Certain ultraoperator–superoperator intertwining relationships are also considered throughout, including the relationship between the KOD Kolmogorov equation and the Lindblad master equation. The IGA is also shown to have actually two distinct involutions: one respected by the convolution ultraoperators and the other by the quantum channel superoperators. Finally, the KOD Kolmogorov generators are derived for jump processes and more general diffusive processes. Full article

We study the controllability of right-invariant bilinear systems on the complex and quaternionic special linear groups $\mathrm{Sl}(n,\mathbb{C})$ and $\mathrm{Sl}(n,\mathbb{H})$. The analysis relies on the Lie saturate$\mathrm{LS}(\Gamma )$, which characterizes controllability through convexity and closure properties of attainable sets, avoiding explicit Lie algebra computations. For $\mathrm{Sl}(n,\mathbb{C})$ with a strongly regular diagonal control matrix, we show that controllability is equivalent to the irreducibility of the drift matrix A, a property verified by the strong connectivity of its associated directed graph. For $\mathrm{Sl}(n,\mathbb{H})$, we derive controllability criteria based on quaternionic entries and the convexity of ${\mathbb{T}}^2$-orbits, which provide efficient sufficient conditions for general n and exact ones in the $2\times 2$ case. These results link algebraic and geometric viewpoints within a unified framework and connect to recent graph-theoretic controllability analyses for bilinear systems on Lie groups. The proposed approach yields constructive and scalable controllability tests for complex and quaternionic systems. Full article