Search Results (2,450)

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples. Full article

A real-space renormalization group (RG) framework is formulated for classical SO(n) spin models defined on d-dimensional crystal lattices composed of corner-sharing hyper-tetrahedra, a class of geometrically frustrated crystal structures. This includes, as specific instances, the classical Heisenberg model on the kagome and pyrochlore crystals. The approach involves computing the partition function and corresponding order parameters for spin clusters embedded in the crystal, to leading order in symmetry-breaking fields generated by surrounding spins. The crystal geometry plays a central role in determining the scaling relations and the associated critical behavior. To illustrate the efficacy of the method, a reduced manifold of symmetry-allowed ordered states for isotropic nearest-neighbor interactions is analyzed. The RG flow systematically excludes the emergence of a $\overrightarrow{q}=\mathbf{0}$ ordered phase within the antiferromagnetic sector, independently of both the spatial dimensionality of the crystal and the number of spin components. Extensions to incorporate more elaborate crystal-symmetry-induced ordering patterns and fluctuation-driven phenomena—such as order-by-disorder—are also discussed. Full article

This article presents a novel mathematical formalism for advanced manifold–metric pairs, enhancing the frameworks of geometry and topology. We construct various D-dimensional manifolds and their associated metric spaces using functional methods, with a focus on integrating concepts from mathematical physics, field theory, topology, algebra, probability, and statistics. Our methodology employs rigorous mathematical construction proofs and logical foundations to develop generalized manifold–metric pairs, including homogeneous and isotropic expanding manifolds, as well as probabilistic and entropic variants. Key results include the establishment of metrizability for topological manifolds via the Urysohn Metrization Theorem, the formulation of higher-rank tensor metrics, and the exploration of complex and quaternionic codomains with applications to cosmological models like the expanding spacetime. By combining spacetime generalized sets with information-theoretic and probabilistic approaches, we achieve a unified framework that advances the understanding of manifold–metric interactions and their physical implications. Full article

Large-scale wind power grid-connected systems can trigger the risk of power system instability. In order to enhance the stability margin of grid-connected systems, this paper accurately characterizes the topology of the global boundary of stability domain (BSD) of the grid-connected system based on BSD theory, using the method of combining the manifold topologies and singularities at infinity. On this basis, the effect of large-scale doubly fed induction generators (DFIGs) replacing synchronous units on the BSD of the system is analyzed. Simulation results based on the IEEE 39-bus system indicate that the negative impedance characteristics and low inertia of DFIGs lead to a contraction of the stability domain. The principle of singularity invariance (PSI) proposed in this paper can effectively expand the BSD by adjusting the inertia and damping, thereby increasing the critical clearing time by about 5.16% and decreasing the dynamic response time by about 6.22% (inertia increases by about 5.56%). PSI is superior and applicable compared to traditional energy functions, and can be used to study the power angle stability of power systems with a high proportion of renewable energy. Full article

In this study, theorems and proofs related to spherical and focal curves are presented in the BCV-Sasakian space. An approximate solution to the differential equation characterizing spherical curves in the BCV-Sasakian manifold ${\mathfrak{M}}^3$ is obtained using the Taylor matrix collocation method. The general equations of canal and tubular surfaces are provided within this geometric framework. Additionally, the curvature properties of the tubular surface constructed around a non-vertex focal curve are computed and analyzed. All of these results are presented for the first time in the literature within the context of the BCV-Sasakian geometry. Thus, this study makes a substantial contribution to the differential geometry of contact metric manifolds by extending classical concepts into a more generalized and complex geometric structure. Full article

The purpose of this paper is to use an informatics-based analysis to develop a rational design approach to the accelerated screening of nano-composite materials. Using existing nano-composite data, we develop a quantitative structure–activity relationship (QSAR) as a function of polymer matrix chemistry and nano-additive volume, with the property predicted being electrical conductivity. The development of a QSAR for the electrical conductivity of nano-composites presents challenges in representing the polymer matrix chemistry and backbone structure, the additive content, and the interactions between the components while capturing the non-linearity of electrical conductivity with changing nano-additive volume. An important aspect of this work is designing chemistries with small training data sizes, as the uncertainty in modeling is high, and potentially the representated physics may be minimal. In this work, we explore two important components of this aspect. First, an assessment via Uniform Manifold Approximation and Projection (UMAP) is used to assess the variability provided by new data points and how much information is contributed by data, which is significantly more important than the actual data size (i.e., how much new information is provided by each data point?). The second component involves assessing multiple training/testing splits to ensure that any results are not due to a specific case but rather that the results are statistically meaningful. This work will accelerate the rational design of polymer nano-composites by fully considering the large array of possible variables while providing a high-speed screening of polymer chemistries. Full article

This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis is inadequate, as only the center manifolds are obtained. Consequently, the stability of the system is investigated through both analytical and numerical methods using Lyapunov functions and numerical simulations. The proposed model exhibits rich dynamics, including hyperchaotic behavior, which is characterized using the Lyapunov exponents, bifurcation diagrams, sensitivity analysis, attractor projections, and Poincaré map. Moreover, in this paper, we explore the model with fractional-order derivatives, demonstrating that the fractional dynamics fundamentally change the geometrical structure of the attractors and significantly change the system stability. The Grünwald–Letnikov formulation is used for modeling, while numerical integration is performed using the Caputo operator to capture the memory effects inherent in fractional models. Finally, an analog electronic circuit realization is provided to experimentally validate the theoretical and numerical findings. Full article

Background: Acute myeloid leukemia (AML) is a common and aggressive adults hematological malignancies. This study explored megakaryocyte–erythroid progenitors (MEPs) signature genes and constructed a prognostic model. Methods: Uniform manifold approximation and projection (UMAP) identified distinct cell types, with differential analysis between AML-MEP and normal MEP groups. Univariate and the least absolute shrinkage and selection operator (LASSO) Cox regression selected biomarkers to build a risk model and nomogram for 1-, 3-, and 5-year survival prediction. Results: Ten differentially expressed genes (DEGs) related to overall survival (OS), six (AHSP, MYB, VCL, PIM1, CDK6, as well as SNHG3) were retained post-LASSO. The model exhibited excellent efficiency (the area under the curve values: 0.788, 0.77, and 0.847). Pseudotime analysis of UMAP-defined subpopulations revealed that MYB and CDK6 exert stage-specific regulatory effects during MEP differentiation, with MYB involved in early commitment and CDK6 in terminal maturation. Finally, although VCL, PIM1, CDK6, and SNHG3 showed significant associations with AML survival and prognosis, they failed to exhibit pathological differential expression in quantitative real-time polymerase chain reaction (qRT-PCR) experimental validations. In contrast, the downregulation of AHSP and upregulation of MYB in AML samples were consistently validated by both qRT-PCR and Western blotting, showing the consistency between the transcriptional level changes and protein expression of these two genes (p < 0.05). Conclusions: In summary, the integration of single-cell/transcriptome analysis with targeted expression validation using clinical samples reveals that the combined AHSP-MYB signature effectively identifies high-risk MEP-AML patients, who may benefit from early intensive therapy or targeted interventions. Full article

Alzheimer’s Disease and Dementia (ADD) progresses along a continuum of cognitive decline, typically from Subjective Cognitive Impairment (SCI) to Mild Cognitive Impairment (MCI) and eventually to dementia. While many studies have focused on classifying these clinical stages, fewer have examined whether brain connectomes encode this continuum in a low-dimensional, interpretable form. Motivated by the hypothesis that structural brain connectomes undergo complex yet compact changes across cognitive decline, we propose a Graph Neural Network (GNN)-based framework that embeds these connectomes into a two-dimensional manifold to capture the evolving patterns of structural connectivity associated with cognitive deterioration. Using attention-based graph aggregation and Principal Component Analysis (PCA), we find that MCI subjects consistently occupy an intermediate position between SCI and ADD, and that the observed transitions align with known clinical biomarkers of ADD pathology. This hypothesis-driven analysis is further supported by the model’s robust separation performance, with ROC-AUC scores of 0.93 for ADD vs. SCI and 0.81 for ADD vs. MCI. These findings offer an interpretable and neurologically grounded representation of dementia progression, emphasizing structural connectome alterations as potential markers of cognitive decline. Full article

Adversarial machine learning exploits the vulnerabilities of artificial intelligence (AI) models by inducing malicious distortion in input data. Starting with the effect of adversarial methods on well-known MNIST and CIFAR-10 open datasets, this paper investigates the ability of Uniform Manifold Approximation and Projection (UMAP) in providing useful representations of both legitimate and malicious images and analyzes the attacks’ behavior under various conditions. By enabling the extraction of decision rules and the ranking of important features from classifiers such as decision trees, eXplainable AI (XAI) achieves zero false positives and negatives in detection through very simple if-then rules over UMAP variables. Several examples are reported in order to highlight attacks behaviour. The data availability statement details all code and data which is publicly available to offer support to reproducibility. Full article

Intelligent reflecting surfaces (IRSs) have attracted extensive attention in the design of future communication networks. However, their large number of reflecting elements still results in non-negligible power consumption and hardware costs. To address this issue, we previously proposed a green heterogeneous IRS (HE-IRS) consisting of both dynamically tunable elements (DTEs) and statically tunable elements (STEs). Compared to conventional IRSs with only DTEs, the unique DTE–STE integrated structure introduces new challenges in optimizing the positions and phase shifts of the two types of elements. In this paper, we investigate the element position and phase shift optimization problems in HE-IRS-assisted millimeter-wave systems. We first propose a particle swarm optimization algorithm to determine the specific positions of the DTEs and STEs. Then, by decomposing the phase shift optimization of the two types of elements into two subproblems, we utilize the manifold optimization method to optimize the phase shifts of the STEs, followed by deriving a closed-form solution for those of the DTEs. Furthermore, we propose a low-complexity phase shift optimization algorithm for both DTEs and STEs based on the Cauchy–Schwarz bound. The simulation results show that with the tailored element position and phase shift optimization algorithms, the HE-IRS can achieve a competitive performance compared to that of the conventional IRS, but with much lower power consumption. Full article

This paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants of submanifolds. The methodology involves deriving generalized Chen-type and $\delta (2,2)$ curvature inequalities using curvature tensor analysis and dual affine connections. A concrete example is provided to verify the theoretical framework. The novelty of this work lies in extending classical curvature inequalities to a newly introduced statistical structure, thereby opening new perspectives in the study of geometric inequalities in information geometry and related mathematical physics contexts. Full article

Conjecture $\mathcal{O}$ (and the Gamma Conjectures), introduced by Galkin, Golyshev, and Iritani stand as pivotal open problems in the quantum cohomology of Fano manifolds, bridging algebraic geometry, mathematical physics, and representation theory. These conjectures aim to decode the structural essence of quantum multiplication by uncovering profound connections between spectral properties of quantum cohomology operators and the underlying geometry of Fano manifolds. Conjecture $\mathcal{O}$ specifically investigates the spectral simplicity and eigenvalue distribution of the operator associated with the first Chern class $c_1$ in quantum cohomology rings, positing that its eigenvalues govern the convergence and asymptotic behavior of quantum products. Full article

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature. Full article

This study introduces a paradigm shift in permanent magnet synchronous motor (PMSM) control through the development of hybrid dual fractional-order sliding mode control (HDFOSMC) architecture integrated with evolutionary parameter learning (EPL). Conventional PMSM control frameworks face critical limitations in ultra-precision applications due to their inability to reconcile dynamic agility with steady-state precision under time-varying parameters and compound disturbances. The proposed HDFOSMC framework addresses these challenges via two synergistic innovations: (1) a dual fractional-order sliding manifold that fuses the rapid transient response of non-integer-order differentiation with the small steady-state error capability of dual-integral compensation, and (2) an EPL mechanism enabling real-time adaptation to thermal drift, load mutations, and unmodeled nonlinearities. Validation can be obtained through the comparison of the results on PMSM testbenches, which demonstrate superior performance over traditional fractional-order sliding mode control (FOSMC). By integrating fractional-order theory, sliding mode control theory, and parameter self-tuning theory, this study proposes a novel control framework for PMSM. The developed system achieves high-precision performance under extreme operational uncertainties through this innovative theoretical synthesis and comparative results. Full article