Search Results (22)

Community detection in graphs can be viewed as the estimation of a partition map that remains stable under admissible perturbations of graph topology and node attributes. While modern graph neural networks (GNNs) achieve strong empirical accuracy, they often exhibit severe assignment drift under minor perturbations, leading to illusory community structures. In this work, we propose DISPEL-GNN, a stability-aware graph learning framework that integrates spectral operator regularization, Bayesian uncertainty modeling, and risk-aware dynamic attention for perturbation-bounded community detection. The model explicitly constrains graph operators through uniform spectral norm bounds, high-frequency energy suppression, and commutator alignment while dynamically modulating message passing based on node-level spectral risk and epistemic uncertainty. We further formalize instability via assignment of drift functional and establish perturbation bounds linking drift to operator norms and spectral gaps, complemented by a PAC-Bayesian generalization guarantee. Extensive experiments on real-world benchmarks including Cora, Citeseer, Pubmed, Cora-Full, and DBLP demonstrate that DISPEL-GNN consistently reduces assignment drift by 18–35% under feature noise and edge perturbations while improving clustering quality with up to +3.0 NMI and +0.04 ARI compared to strong baselines such as GAT and Bayesian GNNs. The normalized mutual information (NMI), adjusted Rand index (ARI), and PAC-Bayesian (PAC) constraints serve as evaluative and theoretical instruments in this study. Additional studies on synthetic graphs with controlled spectral gaps confirm that the proposed method maintains stable community assignments in low-gap regimes where classical spectral and GNN-based methods degrade sharply. These results establish DISPEL-GNN as a mathematically grounded and practically effective framework for robust and interpretable community detection. A metric-wise dominance analysis shows that DISPEL-GNN achieves metric-wise dominance across most accuracy and robustness criteria, with minor tradeoffs in modularity on selected datasets. These results indicate that explicitly modeling stability and uncertainty provides a principled pathway toward reliable and interpretable community detection in noisy graph environments. Full article

This paper addresses the solvability and controllability of fractional delay differential Sylvester matrix equations with non-permutable coefficient matrices. By applying a vectorization approach and Kronecker product algebra, we transform the matrix-valued problem into an equivalent vector system, enabling the derivation of explicit solution representations using a delayed perturbation of two-parameter Mittag-Leffler-type matrix functions. We establish necessary and sufficient conditions for controllability via a fractional delay Gramian matrix, providing a computationally verifiable criterion that requires no commutativity assumptions. The theoretical results are validated through numerical examples, demonstrating effectiveness in noncommutative scenarios where classical methods fail. Full article

We study compactness for the complex Green operator ${\mathsf{G}}_q$ associated with the Kohn Laplacian ${\square }_b$ on boundaries of pseudoconvex domains in Stein manifolds. Let $\Omega ⋐X$ be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n with $C^1$ boundary. For $1\le q\le n-2$, we first prove a compactness theorem under weak potential-theoretic hypotheses: if $b\Omega$ satisfies weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$, then ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$ are compact on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$. This extends known $C^{\infty }$ results in ${\mathbb{C}}^n$ to the minimal regularity $C^1$ and to the Stein setting. On locally convexifiable $C^1$ boundaries, we obtain a full characterization: compactness of ${\mathsf{G}}_q$ is equivalent to simultaneous compactness of ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$, to compactness of the $\overline{\partial }$-Neumann operators ${\mathsf{N}}_q$ and ${\mathsf{N}}_{n-1-q}$ in the interior, to weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$, and to the absence of (germs of) complex varieties of dimensions q and $n-1-q$ on $b\Omega$. A key ingredient is an annulus compactness transfer on ${\Omega }_{+}={\Omega }_2\setminus \overline{{\Omega }_1}$, which yields compactness of ${\mathsf{N}}_q^{{\Omega }_{+}}$ from weak $\left(\mathcal{P}\right)$ near each boundary component and allows us to build compact ${\overline{\partial }}_b$-solution operators via jump formulas. Consequences include the following: compact canonical solution operators for ${\overline{\partial }}_b$, compact resolvent for ${\square }_b$ on the orthogonal complement of its harmonic space (hence discrete spectrum and finite-dimensional harmonic forms), equivalence between compactness and standard compactness estimates, closed range and ${\mathsf{L}}^2$ Hodge decompositions, trace-class heat flow, stability under $C^1$ boundary perturbations, vanishing essential norms, Sobolev mapping (and gains under subellipticity), and compactness of Bergman-type commutators when $q=1$. Full article

We investigate the three-dimensional incompressible fractional Navier–Stokes system with directional fractional diffusion: a vertical dissipative operator of order $2\alpha \in (0,2]$ acting on the full velocity field together with a horizontal fractional operator of order $2\beta \in (0,2]$ applied to the vertical average of one horizontal component. This anisotropic, nonlocal structure captures media in which smoothing acts with unequal strength by direction. For small, divergence-free initial data in an anisotropic energy class, we establish global well-posedness and stability of the zero state, including uniqueness and continuous dependence on the data. The analysis crucially relies on an average oscillation decomposition in the vertical variable, a fractional Poincaré inequality aligned with the vertical direction, and sharp product/commutator bounds compatible with the anisotropic splitting. We provide explicit estimates for direction-dependent smoothing and algebraic decay governed by $(\alpha ,\beta )$, and we show that the conclusions persist under small perturbation of the dissipation parameters. Full article

The increasing integration of artificial intelligence agents (AIAs) based on large language models (LLMs) is transforming many spheres of society. These agents act as human assistants, forming Distributed Intelligent Systems (DISs) and engaging in opinion formation, consensus-building, and collective decision-making. However, complex DIS network topologies introduce significant uncertainty into these processes. We propose a quantum-inspired graph signal processing framework to model collective behavior in a DIS interacting with an external environment represented by an influence matrix (IM). System topology is captured using scale-free and Watts–Strogatz graphs. Two contrasting interaction regimes are considered. In the first case, the internal structure fully aligns with the external influence, as expressed by the commutativity between the adjacency matrix and the IM. Here, a renormalization-group-based scaling approach reveals minimal reservoir influence, characterized by full phase synchronization and coherent dynamics. In the second case, the IM includes heterogeneous negative (antagonistic) couplings that do not commute with the network, producing partial or complete spectral disorder. This disrupts phase coherence and may fragment opinions, except for the dominant collective (Perron) mode, which remains robust. Spectral entropy quantifies disorder and external influence. The proposed framework offers insights into designing LLM-participated DISs that can maintain coherence under environmental perturbations. Full article

We present a fully discrete splitting-Galerkin scheme for second-order, non-autonomous abstract Cauchy problems with time-dependent perturbations. By reformulating the second-order equation as a first-order system in the product space, we apply a Galerkin semi-discretization in space of order $O\left(h^k\right)$ and a Strang splitting in time of order $O(\Delta t^2)$. An embedded Runge–Kutta controller provides adaptive time-stepping to handle rapid temporal variations in the perturbation operator $B\left(t\right)$. Under standard regularity and commutator assumptions on $A\left(t\right)$ and $B\left(t\right)$, we establish a priori error estimates $\underset{0\le t_n\le T}{max}{\parallel u\left(t_n\right)-u_n\parallel }_Z=O(h^k+\Delta t^2).$ Numerical experiments for a 1D perturbed wave equation confirm the theoretical convergence rates, illustrate stability thresholds in the unstable regime, and demonstrate up to 40% savings in computational cost via adaptivity. Full article

We develop a general framework for response theory in Markovian diffusion processes governed by Fokker–Planck equations. Our formalism, based on the notion of the dissipation function, is capable of handling dynamics that are driven by an external perturbation arbitrarily far from a given reference process. Using the analytically solvable Brownian oscillator model, we derive exact response formulae for both overdamped and underdamped dynamics of harmonically bound Brownian particles. We also demonstrate that for certain observables and under suitable time scaling, the operations of model reduction and response formula extraction commute, which highlights a relevant symmetry of the adopted mathematical formalism. Notably, the time reversal invariance symmetry, which is manifested as detailed balance in stochastic processes and is often required in statistical mechanics, is not necessary in our response framework. Full article

We present new additive results for the group inverse in a Banach algebra under certain perturbations. The upper bound of ${\parallel (a+b)}^{\#}-a^d\parallel$ is thereby given. These results extend the main results presented by Liu, Qin, and Wei. We then apply them to establish the representation of the group inverse of $a+b$ under a kind of commutative perturbation condition. Finally, some numerical examples are given to demonstrate the main results. Full article

In this work we study the spectral dimensionality of spacetime around a radiating Schwarzschild black hole using a recently introduced formalism of quantum gravity, where the alterations of the gravitational field produced by the radiation are represented on an extended manifold, and describe a non-commutative and nonlinear quantum algebra. The relation between classical and quantum perturbations of spacetime can be measured by the parameter $z\ge 0$. In this work we have found that when $z=(1+\sqrt{3})/2\simeq 1.3660$, a relativistic observer approaching the Schwarzschild horizon perceives a spectral dimension $N\left(z\right)=4\left[\theta \left(z\right)-1\right]\simeq 2.8849$, which is related to quantum gravitational interference effects in the environment of the black hole. Under these conditions, all studied Schwarzschild black holes with masses ranging from the Planck mass to ${10}^{46}$ times the Planck mass present the same stability configuration, which suggests the existence of a universal property of these objects under those particular conditions. The difference from the spectral dimension previously obtained at cosmological scales leads to the conclusion that the spacetime dimensionality is scale-dependent. Another important result presented here is the fundamental alteration of the effective gravitational potential near the horizon due to Hawking radiation. This quantum phenomenon prevents the potential from diverging to negative infinity as the observer approaches the Schwarzschild horizon. Full article

We present a study of relic gravitational waves based on a foliated gauge field theory defined over a spacetime endowed with a noncommutative algebraic–geometric structure. As an ontological extension of general relativity—concerning manifolds, metrics, and fiber bundles—the conventional space and time coordinates, typically treated as classical numbers, are replaced by complementary quantum dual fields. Within this framework, consistent with the Bekenstein criterion and the Hawking–Hertog multiverse conception, singularities merge into a helix-like cosmic scale factor that encodes the topological transition between the contraction and expansion phases of the universe analytically continued into the complex plane. This scale factor captures the essence of an intricate topological quantum-leap transition between two phases of the branching universe: a contraction phase preceding the now-surpassed conventional concept of a primordial singularity and a subsequent expansion phase, whose transition region is characterized by a Riemannian topological foliated structure. The present linearized formulation, based on a slight gravitational field perturbation, also reveals a high sensitivity of relic gravitational wave amplitudes to the primordial matter and energy content during the universe’s phase transition. It further predicts stochastic homogeneous distributions of gravitational wave intensities arising from the interplay of short- and long-spacetime effects within the non-commutative algebraic framework. These results align with the anticipated future observations of relic gravitational waves, expected to pervade the universe as a stochastic, homogeneous background. Full article

With the large-scale deployment of renewable energy, transmission systems for new energy sources are increasingly exposed to transient overvoltage issues induced by DC blockages and commutation failures. To address the challenges of an imprecise response to multiple fault scenarios and the inefficiency of independent device actions in existing dynamic reactive power control schemes, this paper proposes a coordinated optimization control strategy integrating multiple dynamic reactive power sources tailored to different fault characteristics. An equivalent model of the renewable energy DC transmission system is established to elucidate the underlying mechanisms of transient overvoltage formation under various fault conditions. By employing trajectory sensitivity analysis and parameter perturbation methods, the influence patterns of control parameters on transient overvoltage behaviors across different fault scenarios are quantitatively assessed, thereby overcoming the limitations of traditional empirical parameter tuning approaches. Subsequently, a multi-source coordinated optimization model is developed with the objective of minimizing transient overvoltages under simultaneous dual-fault conditions. A multi-objective particle swarm optimization algorithm, incorporating comprehensive trajectory sensitivity and dynamically adaptive inertia weights, is introduced, alongside Pareto front theory, to achieve rapid and balanced optimization among competing objectives. Simulation results validate that the proposed strategy significantly enhances transient overvoltage suppression across diverse fault conditions. The findings provide robust theoretical foundations and practical guidance for the refined parameter tuning and high-efficiency coordinated control of dynamic reactive power sources in renewable energy transmission systems. Full article

In this work, we elaborate further on a (3+1)-dimensional cosmological Running-Vacuum-type-Model (RVM) of inflation based on string-inspired Chern-Simons(CS) gravity, involving axions coupled to gravitational-CS(gCS) anomalous terms. Inflation in such models is caused by primordial-gravitational-waves(GW)-induced condensation of the gCS terms, which leads to a linear-axion potential. We demonstrate that this inflationary phase may be metastable, due to the existence of imaginary parts of the gCS condensate. These are quantum effects, proportional to commutators of GW perturbations, hence vanishing in the classical theory. Their existence is quantum-ordering-scheme dependent. We argue in favor of a physical importance of such imaginary parts, which we compute to second order in the GW (tensor) perturbations in the framework of a gauge-fixed effective Lagrangian, within a (mean field) weak-quantum-gravity-path-integral approach. We thus provide estimates of the inflation lifetime. On matching our results with the inflationary phenomenology, we fix the quantum-ordering ambiguities, and obtain an order-of-magnitude constraint on the String-Mass-Scale-to-Planck-Mass ratio, consistent with previous estimates by the authors in the framework of a dynamical-system approach to linear-axion RVM inflation. Finally, we examine the role of periodic modulations in the axion potential induced by non-perturbative effects on the slow-roll inflationary parameters, and find compatibility with the cosmological data. Full article

This paper studies the stability of a real-world wind farm, Bison Wind Generation System (BWGS) in the state of North Dakota in the United States. BWGS uses an AC collector grid rated at 34.5 kV and a symmetrical bipolar high-voltage DC (HVDC) transmission grid rated at ±250 kV. The HVDC line transfers a total power of 0.5 GW, while both the HVDC rectifier and inverter substations use line-commuted converters (LCCs). The LCC-based rectifier adopts constant DC current control to regulate HVDC current, while the inverter operates in constant extinction angle control mode to maintain a fixed HVDC voltage. This paper proposes a frequency scan-based approach to obtain the d–q impedance model of (i) BWGS AC collector grids with Type 4 wind turbines that use permanent magnet synchronous generators (PMSGs) and two fully rated converters, and (ii) an LCC-HVDC system. The impedance frequency response of the BWGS is acquired by exciting the AC collector grid and LCC-HVDC with multi-sine voltage perturbations during its steady-state operation. The resulting voltage and current signals are subjected to a fast Fourier transform (FFT) to extract frequency components. By analyzing the impedance frequency response measurement of BWGS, a linear time–invariant (LTI) representation of its dynamics is obtained using the vector fitting (VF) technique. Finally, a Bode plot is applied, considering the impedance of the BWGS and grid to perform stability analyses. This study examines the influence of the short circuit ratio (SCR) of the grid and the phase lock loop (PLL) frequency bandwidth on the stability of the overall system. The findings provide valuable insights for the design and verification of an AC collector and LCC-based HVDC transmission systems. The findings suggest that the extraction of the impedance model of a real-world wind farm, achieved through frequency scanning and subsequent representation as an LTI system using VF, is regarded as a robust, suitable, and accurate methodology for investigating the dynamics, unstable operating conditions, and control interaction of the wind farm and LCC-HVDC system with the AC grid. Full article

We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (“ML spacetime”). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito’s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that—under the Robertson–Walker minimum length theory—rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents. Full article

As an option to deal with insulin-dependent disease, a recently commuted PD control strategy is designed and carefully analyzed for different clinic diabetic patients. This controller approach is mainly conceived to stabilize the glucose blood concentration in a diabetic patient around its basal value; hence, avoiding extreme situations such as hypoglycemia and hyperglycemia. This control strategy receives two inputs carefully tuned to actuate when the measured variable is out of a prescribed healthy zone. Therefore, one of these variables is invoked to decrease the glucose concentration to insulin injection, and the other is employed to increase the glucose absorption, both by using a proper PD controller. According to our numerical experiments, our controller approach performs well, even when there is an external disturbance in the controlled system. Full article