Search Results (77)

The increasing integration of wind power reduces the equivalent inertia of power systems, leading to lower frequency nadirs and higher rate of change of frequency following disturbances. In Type-4 wind turbine systems, conventional maximum power point tracking (MPPT) may counteract the additional inertial power command during frequency support and cause secondary frequency dips during rotor-speed recovery. To address these issues, this paper proposes a virtual-inertia rate-of-change-of-frequency (VI-RoCoF) frequency-modulated Andronov-Hopf oscillator (AHO)-based adaptive inertia enhancement method together with an adaptive MPPT coordination strategy. The proposed method constructs a frequency-support demand from frequency deviation and VI-filtered RoCoF and embeds it into the instantaneous angular-frequency evolution of the AHO. Different from a conventional linear virtual-inertia controller that directly converts frequency-deviation and RoCoF signals into an algebraic power command, the proposed method realizes the additional support through a bounded limit-cycle frequency-forming process, thereby preserving phase continuity and nonlinear amplitude self-regulation during frequency modulation. Meanwhile, the adaptive MPPT strategy adjusts the power reference in stages to suppress the counteractive effect of conventional MPPT on inertial support and to ensure a smooth transition back to maximum power point tracking. Theoretical analysis shows that the proposed modulation maintains the limit-cycle stability of the AHO under bounded control constraints while improving the equivalent inertia and damping characteristics of the system. Simulation results, including both averaged-model and switching-level SPS simulations, demonstrate that, compared with conventional AHO-based, fixed-inertia AHO-based, and linear VI-RoCoF benchmark schemes without AHO dynamics, the proposed AHO-MPPT coordinated control strategy increases the frequency nadir, reduces the peak RoCoF, improves recovery-stage frequency dynamics, mitigates secondary frequency dips, maintains bounded AHO internal variables, and preserves DC-link voltage stability. Full article

We develop a comparative synthesis of quantum-information geometry beyond complex numbers, with emphasis on what different algebraic frameworks contribute to information-processing structure rather than on their formal novelty alone. The organizing idea is a layer-by-layer test of the standard complex Hilbert-space formalism: each non-complex or deformed framework modifies the scalar field, phase group, projective state space, Born-probability semantics, composition rule, measurement geometry, symmetry algebra or representation category. The central thesis is that such frameworks are physically meaningful when they identify which assumptions make complex quantum mechanics operationally stable: positive probabilities, associative multipartite composition, reversible dynamics, experimentally testable phases, locality constraints, informationally complete measurements, error bases and clear operational semantics. Real quantum theory probes the necessity of complex phases and local tomography; quaternionic quantum mechanics probes non-Abelian phase while retaining associativity and admitting complex embeddings; octonionic proposals probe the boundary where exceptional geometry survives but generic circuit composition is obstructed by non-associativity; Jordan algebras test ordered probabilistic state spaces; Clifford algebras and Bott periodicity provide the spinorial and topological grammar connecting gates, Hopf maps and periodic dimensions; and quantum-group or q-deformed constructions probe coproducts, braiding and representation categories rather than scalar amplitudes. We distinguish three roles that are often conflated: genuine hypercomplex kinematics, Hopf-fibration coordinates for ordinary complex multipartite entanglement, and deformed algebraic or categorical structures. The resulting map separates established equivalence and experimental-constraint results from useful representation tools and speculative programs, while identifying concrete open problems for non-complex quantum information. Full article

In this paper, we mainly introduce the notion of a braided algebraic group, which unifies the notions of a braided Hopf algebra with an integral, a Hopf group-coalgebra with a group-integral and an algebraic quantum group. For this, we introduce the notion of braided multiplier rings and study their properties. Then we study some structural properties of a braided multiplier algebra with a nontrivial example. Full article

In this note, we study the Hopf-algebra-based (HAB) formula of Yang–Mills-Scalar (YMS) amplitudes, which expands a YMS amplitude with massive scalars as a combination of propagator matrices that mix massless scalars corresponding to gluons with the original massive scalars. We propose a recursive formula that conveniently expresses the HAB formula. In this formula, gluons are converted into massless scalars. Thus it expresses a YMS amplitude with massive scalars by amplitudes with fewer gluons, massive scalars and massless scalars. We verify this formula by using the soft behavior of amplitudes. We further show the equivalence between the massless limit of the HAB formula and an earlier proposed recursive expansion formula through explicit calculations on amplitudes with one and two gluons. Full article

Firstly, we define and study the notions of a smash product for actions of multiplier left Hopf algebras on algebras and of an integral on such smash products. Then we construct an analogue of Radford’s biproduct in the framework of multiplier left Hopf algebras under assumption of a multiplier left Hopf algebra having an anti-bialgebra homomorphic left antipode. Finally, we study a duality theorem for smash products of a left Hopf algebra of dimension n which is a special multiplier left Hopf algebra. Full article

The mathematical universe of the quantum topos, which is formulated on the basis of classical Boolean snapshots, delivers a neo-realist description of quantum mechanics that preserves realism. The main contribution of this article is developing formal objectivity in physical theories beyond quantum mechanics in the topos-theory approach. It will be shown that neo-realist responses to non-perturbative structures of quantum field theory do not preserve realism. In this regard, the method of Feynman graphons is applied to reframe the task of describing objectivity in quantum field theory in terms of replacing the standard Hilbert-space/operator-algebra ontology with a new context category built from a certain family of topological Hopf subalgebras of the topological Hopf algebra of renormalization as algebraic/combinatorial data tied to non-perturbative structures. This topological-Hopf-algebra ontology, which is independent of instrumentalist probabilities, enables us to reconstruct gauge field theories on the basis of the mathematical universe of the non-perturbative topos. The non-Boolean logic of the non-perturbative topos cannot be recovered by classical Boolean snapshots, which is in contrast to the quantum-topos reformulation of quantum mechanics. The article formulates a universal version of the non-perturbative topos to show that quantum field theory is a globally and locally neo-realist theory which can be reconstructed independent of the standard Hilbert-space/operator-algebra ontology. Formal objectivity of the universal non-perturbative topos offers a new route to build objective semantics for non-perturbative structures. Full article

This paper introduces and investigates Ore extensions in the context of multiplier Hopf coquasigroups, a structure that generalizes both multiplier Hopf algebras and Hopf coquasigroups. We establish necessary and sufficient conditions under which an Ore extension of a regular multiplier Hopf coquasigroup itself forms a regular multiplier Hopf coquasigroup. Furthermore, we explore the isomorphism problem for such Ore extensions, providing criteria for the equivalence of two extensions. The case of multiplier Hopf coquasigroups is also analyzed, with conditions derived for the Ore extension to inherit the structure. Our results unify and extend prior work on Ore extensions in the settings of Hopf algebras, multiplier Hopf algebras, and Hopf coquasigroups. Full article

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that $di{m}_K\left(L\right)=\mid L/K\mid <\infty$, the transcendence degree $trd(L/K)<\infty$ and $trd(L/K)=\infty$ with $difftrd(L/K)<\infty$, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension $L=K\left(G\right)$; then, there is a Galois correspondence between the intermediate fields $K\subset K^{\prime }\subset L$ and the subgroups $e\subset G^{\prime }\subset G$, provided that $K^{\prime }$ is stable under a Lie algebra $\Delta$ of invariant derivations of $L/K$. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations. Full article

In this paper, we investigate the module algebra structures of $X_q\left(A_1\right)$ on the quantum polynomial algebra ${\mathbb{C}}_q[x,y,z]$. The automorphism group of ${\mathbb{C}}_q[x,y,z]$ is denoted by $\mathrm{Aut}\left({\mathbb{C}}_q[x,y,z]\right)$, and we give a complete classification of $X_q\left(A_1\right)$-module algebra structures on ${\mathbb{C}}_q[x,y,z]$, where $K_1,K_2\in \mathrm{Aut}\left({\mathbb{C}}_q[x,y,z]\right)$. Full article

Research at the intersection of quantum gravity and quantum information theory has seen significant success in describing the emergence of spacetime and gravity from quantum states whose entanglement entropy approximately obeys an area law. In a different direction, the Kaluza–Klein proposal aims to recover gauge symmetries by means of dimensional reduction in higher-dimensional gravitational theories. Integrating both of these, gravitational and gauge degrees of freedom in $3+1$ dimensions may be obtained upon dimensional reduction in higher-dimensional emergent gravity. To this end, we show that entangled systems of two and three qubits can be associated with $5+1$- and $9+1$-dimensional spacetimes, respectively, which are reduced to $3+1$ dimensions upon singling out a preferred complex direction. Depending on the interpretation of the residual symmetry, either the Standard Model gauge group, $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)/{\mathbb{Z}}_6$, or the symmetry of Minkowski spacetime together with the gauge symmetry of a right-handed ‘half-generation’ of fermions can be recovered. Thus, there seems to be a natural way to accommodate the chirality of the weak force in the given construction. This motivates a picture in which spacetime emerges from the area law contribution to the entanglement entropy, while gauge and matter degrees of freedom are obtained due to area-law-violating terms. Furthermore, we highlight the possibility of using this construction in quantum simulations of Standard Model fields. Full article

Bi-Frobenius algebras are a class of Frobenius algebras and Frobenius coalgebras with some compatible conditions. In this paper, we construct a class of bi-Frobenius algebras generated by n elements on graded algebra $A.$ The comultiplication and counit are defined via a permutation $\pi$ on $A,$ such that A becomes a bi-Frobenius algebra. For any n, these bi-Frobenius algebras are neither Hopf algebras nor S-type bi-Frobenius algebras. Full article

In this paper, we introduce and study the notion of a multiplier left Hopf algebra, which can be seen as an extension of the Van Daele’s multiplier Hopf algebras and the Green–Nichols–Taft’s left Hopf algebras. In particular, we investigate the relation between the notion of the Van Daele’s left multiplier Hopf algebras and the one of our multiplier left Hopf algebras. Finally, we determine the case when a multiplier left Hopf algebra becomes a multiplier Hopf algebra. Full article

There has been strong interest in the fate of relativistic symmetries in some quantum spacetimes, partly because of its possible relevance for high-precision experimental tests of relativistic properties. However, the main technical results obtained so far concern the description of suitably deformed relativistic symmetry transformation rules, whereas the properties of the associated Noether charges, which are crucial for the phenomenology, are still poorly understood. Here, we tackle this problem focusing on first-quantized particles described within a Hamiltonian framework and using as a toy model the so-called “spatial kappa-Minkowski noncommutative spacetime”, where all the relevant conceptual challenges are present but, as here shown, in technically manageable fashion. We derive the Noether charges, including the much-debated total momentum charges, and we reveal a strong link between the properties of these Noether charges and the structure of the laws of interaction among particles. Full article