Search Results (409)

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 $A*B=AB+\epsilon \omega (A,B)$, 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

We propose a unified mathematical framework showing that the representational universality of modern foundational models arises from a shared finite latent domain. Building on the Finite Ring Continuum (FRC) framework, we model all modalities as epistemic projections of a common latent set $\mathcal{Z}\subset {\mathcal{U}}_t$, where ${\mathcal{U}}_t$ is a symmetry-complete finite-field shell. Using the uniqueness of minimal adequate representations, we prove the Universal Subspace Theorem, establishing that independently trained embeddings coincide, up to bijection, as coordinate charts on the same latent structure. This result explains cross-modal alignment, transferability, and semantic coherence as consequences of finite relational geometry rather than architectural similarity. The framework links representation learning, sufficiency theory, and FRC algebra, providing a principled foundation for universal latent structure in multimodal models. 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

We construct a finite-dimensional Z₃-graded Lie superalgebra of dimensions (12,4,3), featuring a grade-2 sector that obeys a cubic bracket relation with the fermionic sector. This induces an emergent triality symmetry cycling the three components. The full set of graded Jacobi identities is verified analytically in low dimensions and numerically in a faithful 19-dimensional matrix representation, with residuals $\le 8\times {10}^{-13}$ over ${10}^7$ random tests. Explicit quadratic and cubic Casimir operators are computed, with proofs of centrality, and the adjoint representation is shown to be anomaly-free. The algebra provides a minimal, closed extension beyond conventional ${\mathrm{Z}}_2$ supersymmetry and may offer an algebraic laboratory for models with ternary symmetries. Full article

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra $GL\left(V\right)$ built from two commuting inner automorphisms $\alpha =A{d}_{\psi }$ and $\beta =A{d}_{\varphi }$, with $\psi ,\varphi \in GL\left(V\right)$ and integers $i,j$. We prove that $(GL\left(V\right),{[·,·]}_{(\psi ,\varphi )}^{(i,j)},\alpha ,\beta )$ is a BiHom–Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $\varphi =\psi$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\tilde{L}={\psi }^{-1}L\psi$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times 2$ blocks, we derive explicit trigonometric and hyperbolic formulae that make symmetry constraints (e.g., tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times 2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values. Full article

We propose an approach to extend the concept of a Lie algebra to ternary structures based on $\omega$-symmetry, where $\omega$ is a primitive cube root of unity. We give a definition of a corresponding structure, called a ternary Lie algebra at cube roots of unity, or a ternary $\omega$-Lie algebra. A method for constructing ternary associative algebras has been developed. For ternary algebras, the notions of the ternary $\omega$-associator and the ternary $\omega$-commutator are introduced. It is shown that if a ternary algebra possesses the property of associativity of the first or second kind, then the ternary $\omega$-commutator on this algebra determines the structure of a ternary $\omega$-Lie algebra. Ternary algebras of cubic matrices with associative ternary multiplication of the second kind are considered. The structure of the 8-dimensional ternary $\omega$-Lie algebra of cubic matrices of the second order is studied, and all its subalgebras of dimensions 2 and 3 are determined. Full article

Symmetric structures are key in non-associative algebras. A Mock-Lie algebra, defined by commutativity and the Jacobi identity, shows strong algebraic symmetry. This paper studies the quasi-centroid, which captures the symmetry of linear operators commuting with the algebra’s product. We define the quasi-centroid and set its condition for linear endomorphisms under the bracket operation. We classify matrix representations of quasi-centroids for all Mock-Lie algebras of dimensions 2 to 5 by computing matrices and analyzing coefficient relations. These results provide a foundation for further structural study. We also show that in each case, the centroid is strictly contained in the quasi-centroid, confirming proper containment for all these algebras. 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

We develop detailed quaternionic and octonionic frameworks for quantum computation grounded on normed division algebras. Our central result is to prove the polynomial computational equivalence of quaternionic and complex quantum models: Computation over $\mathbb{H}$ is polynomially equivalent to the standard complex quantum circuit model and hence captures the same complexity class BQP up to polynomial reductions. Over $\mathbb{H}$, we construct a complete model—quaternionic qubits on right $\mathbb{H}$-modules with quaternion-valued inner products, unitary dynamics, associative tensor products, and universal gate sets—and establish polynomial equivalence with the standard complex model; routes for implementation at fidelities exceeding $99\%$ via pulse-level synthesis on current hardware are discussed. Over $\mathbb{O}$, non-associativity yields path-dependent evolution, ambiguous adjoints/inner products, non-associative tensor products, and possible failure of energy conservation outside associative sectors. We formalize these obstructions and systematize four mitigation strategies: Confinement to associative subalgebras, $G_2$-invariant codes, dynamical decoupling of associator terms, and a seven-factor algebraic decomposition for gate synthesis. The results delineate the feasible quaternionic regime from the constrained octonionic landscape and point to applications in symmetry-protected architectures, algebra-aware simulation, and hypercomplex learning. Full article

Modeling of dynamic systems with nonholonomic constraints usually involves constraint multipliers. Consequently, the dynamic equations in the laboratory coordinate system have a complex form, and as a result, the corresponding numerical algorithms need to be improved in terms of both efficiency and accuracy. This paper addresses establishing the mathematical model of the hydrodynamic sleigh in the Hamel framework. Firstly, the Lie symmetry and the Noether theorem conserved quantities of classic Chaplygin sleigh in which the inertial frame is reviewed. Based on the symmetries and the nonholonomic constraints, the frame of the sleigh can be directly realized in the algebraic space. Based on the mutual coupling mechanism between the fluid and the sleigh in a potential flow environment, the reduced equations in the moving frame are proposed in nonintegrable constraint distributions. The corresponding Hamel integrator is constructed based on the discrete variational principle. For the sleigh model in potential flow, the Hamel integrator is used to verify the feasibility of parameter control based on rotation angles and mass distribution, and to obtain the dynamic characteristics of the sleigh blade with both a rotational offset and translational offset. Numerical results indicate that the modeling method in the Hamel framework provides a more concise and efficient approach for exploring the dynamic behavior of the hydrodynamic sleigh. Full article

This manuscript presents a comprehensive Lie symmetry analysis of the KdV-Burgers equation, a prototypical model for nonlinear wave dynamics incorporating dissipation and dispersion. We systematically derive its six-dimensional Lie algebra and construct an optimal system of one-dimensional subalgebras. This framework is used to perform a symmetry reduction, transforming the governing partial differential equation into a set of ordinary differential equations. A key contribution of this work is the identification and analysis of several non-trivial invariant solutions, including a new Galilean-boost-invariant solution related to an accelerating reference frame, which extends beyond standard traveling waves. Through a detailed physical interpretation supported by phase plane analysis and asymptotic methods, we elucidate how the mathematical symmetries directly manifest as fundamental physical behaviors. This reveals a clear classification of distinct wave regimes—from monotonic and oscillatory shocks to solitary wave trains governed by the interplay between nonlinearity, dissipation and dispersion. The numerical validation verify the accuracy and physical relevance of the derived invariant solutions, with errors less than $0.5\%$ in the Burgers limit and $3.2\%$ in the weak dissipation regime. Our work establishes a direct link between the model’s symmetry structure and its observable dynamics, providing a unified framework validated both analytically and through the examination of universal scaling laws. The results offer profound insights applicable to fields ranging from plasma physics and hydrodynamics to nonlinear acoustics. Full article

This paper presents the novel framework of neutrosophic modules, an algebraic structure that arises by superimposing neutrosophic sets on classical module theory. The core of this study lies in the investigation of the structural symmetry between the axioms of a module and the trivalent nature of the neutrosophic set. We define a new class of modules based on the neutrosophic set. In addition, this study establishes and examines the categorical structure corresponding to neutrosophic $\mathfrak{R}$-modules. Furthermore, it presents the procedures by which finitely generated variants of these modules can be formulated and studied. Partial characterizations are given for a particular type in which the distribution of neutrosophic values remains within a finite set. Full article

We present a concise and self-contained extension of the Finite Ring Continuum (FRC) program, showing that symmetry-complete prime shells ${\mathbb{F}}_{\mathsf{p}}$ with $\mathsf{p}=4\mathsf{t}+1$ exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot be realized within a single space-like prime shell ${\mathbb{F}}_{\mathsf{p}}$, since to split time from space one requires a time coefficient $c^2$ in the nonsquare class of ${\mathbb{F}}_{\mathsf{p}}^{\times }$, but then $c\notin {\mathbb{F}}_{\mathsf{p}}$. An explicit finite-field Lorentz transformation is subsequently derived that preserves the Minkowski form and generates a finite orthogonal group $O(Q_{\nu },{\mathbb{F}}_{p^2})$ of split type (Witt index 1). These results demonstrate that the essential algebraic features of special relativity—the invariant interval and Lorentz symmetry—emerge naturally within finite-field arithmetic, thereby establishing an intrinsic relativistic algebra within FRC. Finally, this dichotomy implies the algebraic origin of causality: Euclidean invariants reside within a space-like shell ${\mathbb{F}}_{\mathsf{p}}$, while Lorentzian structure and causal separation arise in its quadratic spacetime extension ${\mathbb{F}}_{{\mathsf{p}}^2}$. Full article

The pure states of two-qubit quantum systems are described by a four-dimensional vector of complex numbers, and unitary operators transferring a two-qubit quantum system from one state to another have the form of a $4\times 4$ matrix with complex elements. This fact brings to mind the idea of studying the spaces of four-dimensional numbers with complex components. Moreover, the results obtained by the authors for four-dimensional numbers with real components inspire some optimism. In this paper we construct four-dimensional spaces of complex numbers by analogy with four-dimensional spaces of real numbers. Each four-dimensional number is mapped to a matrix formed from its components and it is proved that the constructed mapping is a bijection and a homomorphism. In the space of four-dimensional numbers of the eight basis elements, half are real and half are imaginary. The presence of such symmetry distinguishes these spaces from the space of quaternions, in which one basis element is real and the rest are imaginary. The symmetry of the basis numbers makes these spaces a natural generalization of one-dimensional and two-dimensional (complex) algebra. The conditions under which the corresponding matrices are gates for two-qubit quantum systems are defined. The notion of a unitary state of a two-qubit quantum system is introduced, to which various gates from commutative groups of gates correspond. It is shown that any gate of a unitary state transforms a unitary state into a unitary state and a non-unitary state into a non-unitary state. Almost all gates used in the construction of quantum circuits, in particular H, SWAP, CX, CY, and CZ, have the same properties. The problem of searching for a gate that transfers a quantum system from one unitary state to another unitary state has been solved. Thus, with the help of four-dimensional spaces of complex numbers it was possible to construct whole classes of two-qubit gates, which opens new possibilities for the construction of quantum algorithms. The results obtained have important theoretical and practical implications for quantum computing. Full article

It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. This paper introduces a novel methodological framework based on the SO(3) Lie method to estimate time-dependent correlation matrices (correlation flows) among three variables that have chaotic, entropy, and fractal characteristics, from 11 April 2011 to 31 December 2024 for daily data; from 10 April 2011 to 29 December 2024 for weekly data; and from April 2011 to December 2024 for monthly data. So, it develops the stochastic SO(2) Lie method into the SO(3) Lie method that aims to obtain the correlation flow for three variables with chaotic, entropy, and fractal structure. The results were obtained at three stages. Firstly, we applied entropy (Shannon, Rényi, Tsallis, Higuchi) measures, Kolmogorov–Sinai complexity, Hurst exponents, rescaled range tests, and Lyapunov exponent methods. The results of the Lyapunov exponents (Wolf, Rosenstein’s Method, Kantz’s Method) and entropy methods, and KSC found evidence of chaos, entropy, and complexity. Secondly, the stochastic differential equations which depend on S² (SO(3) Lie group) and Lie algebra to obtain the correlation flows are explained. The resulting equation was numerically solved. The correlation flows were obtained by using the defined covariance flow transformation. Finally, we ran the robustness check. Accordingly, our robustness check results showed the SO(3) Lie method produced more effective results than the standard and Spearman correlation and covariance matrix. And, this method found lower RMSE and MAPE values, greater stability, and better forecast accuracy. For daily data, the Lie method found RMSE = 0.63, MAE = 0.43, and MAPE = 5.04, RMSE = 0.78, MAE = 0.56, and MAPE = 70.28 for weekly data, and RMSE = 0.081, MAE = 0.06, and MAPE = 7.39 for monthly data. These findings indicate that the SO(3) framework provides greater robustness, lower errors, and improved forecasting performance, as well as higher sensitivity to nonlinear transitions compared to standard correlation measures. By embedding time-dependent correlation matrix into a Lie group framework inspired by physics, this paper highlights the deep structural parallels between financial markets and complex physical systems. Full article