Geometric Invariants Research Articles

A novel class of zipper fractal Bézier curves and its graphics applications

In this article, we present a novel generalization of a Bézier curve that can be non-smooth. We name it the zipper fractal Bézier curve. This new curve is constructed using a class of zipper α-fractal polynomials corresponding to Bernstein basis polynomials. We establish sufficient conditions on the parameters to ensure these zipper α-fractal polynomials exhibit properties such as non-negativity, partition of unity, linear precision, and symmetry. Utilizing these properties, we show that the zipper fractal Bézier curve can achieve endpoint interpolation, symmetry, the convex-hull property, and geometric invariance. Using the zipper fractal Bézier curves, we create attractive deltoid-like, astroid-like, exoid-like, and six-leaf flower designs. Our findings have applications in various fields, including approximation theory, CAGD, computer graphics, art, and design.

Geometrical deformation of brane matter field within f(R, Q, P) gravity

In the context of braneworld scenarios, the real scalar field plays a crucial role by providing thickness for brane. In this work, we investigate a codimension-one thick brane within the framework of f(R, Q, P) gravity, where R represents the curvature scalar, while Q=RμνRμν and P=RμναβRμναβ are quadratic geometric invariants. Our interest is to investigate the influence of these invariants on the scalar field solution and the energy density of the brane. Furthermore, we analyze the localization of spin 1/2 fermions and the gravitino by employing a Yukawa-like coupling with the scalar field background. Such a coupling is able to produce a normalizable zero-mode for both fields.

Revisiting non-learned operators based deep learning for image classification: a lightweight directional-aware network

Due to the stable feature representation capability provided by non-learned operators, the integration with deep learning models, i.e., non-learned operator based deep learning models, has become a paradigm, however, performance-wise, it is still not promising. In this paper, by revisiting non-learned operator based deep learning models, we reveal the reasons for their underperformance: lack of geometric invariance, insufficient sparsity, and neglect of directional importance. In response, we present a Lightweight Directional-Aware Network (LDAN) for image classification. Specifically, to generate sparse geometric-invariant features, we propose a ShearletNet to capture multi-directional features in three different levels. Then, a Directional-Aware module is designed to highlight the discriminative multi-directional features and generate multi-scale features. Finally, a Pointwise Convolution module is used to integrate the multi-directional features with the multi-scale ones for reducing the computational resources. Experiments on the commonly used CIFAR10, CIFAR100, Self-Taught Learning 10 (STL10), and Tiny ImageNet datasets demonstrate the efficiency and effectiveness of the proposed LDAN. Compared to the existing non-learned operator based models, LDAN reduces the parameter count by 80.83% while achieving a 6.32% increase in accuracy.

Unsupervised Part Discovery via Dual Representation Alignment.

Object parts serve as crucial intermediate representations in various downstream tasks, but part-level representation learning still has not received as much attention as other vision tasks. Previous research has established that Vision Transformer can learn instance-level attention without labels, extracting high-quality instance-level representations for boosting downstream tasks. In this paper, we achieve unsupervised part-specific attention learning using a novel paradigm and further employ the part representations to improve part discovery performance. Specifically, paired images are generated from the same image with different geometric transformations, and multiple part representations are extracted from these paired images using a novel module, named PartFormer. These part representations from the paired images are then exchanged to improve geometric transformation invariance. Subsequently, the part representations are aligned with the feature map extracted by a feature map encoder, achieving high similarity with the pixel representations of the corresponding part regions and low similarity in irrelevant regions. Finally, the geometric and semantic constraints are applied to the part representations through the intermediate results in alignment for part-specific attention learning, encouraging the PartFormer to focus locally and the part representations to explicitly include the information of the corresponding parts. Moreover, the aligned part representations can further serve as a series of reliable detectors in the testing phase, predicting pixel masks for part discovery. Extensive experiments are carried out on four widely used datasets, and our results demonstrate that the proposed method achieves competitive performance and robustness due to its part-specific attention.

Fluid flow in three-dimensional porous systems shows power law scaling with Minkowski functionals

Integral geometry uses four geometric invariants—the Minkowski functionals—to characterize certain subsets of three-dimensional (3D) space. The question was, how is the fluid flow in a 3D porous system related to these invariants? In this work, we systematically study the dependency of permeability on the geometrical characteristics of two categories of 3D porous systems generated: (i) stochastic and (ii) deterministic. For the stochastic systems, we investigated both normal and lognormal size distribution of grains. For the deterministic porous systems, we checked for a cubic arrangement and a hexagonal arrangement of grains of equal size. Our studies reveal that for any three-dimensional porous system, ordered or disordered, permeability k follows a unique scaling relation with the Minkowski functionals: (a) volume of the pore space, (b) integral mean curvature, (c) Euler characteristic, and (d) critical cross-sectional area of the pore space. The cubic and the hexagonal symmetrical systems formed the upper and lower bounds of the scaling relations, respectively. The disordered systems lay between these bounds. Moreover, we propose a combinatoric F that weaves together the four Minkowski functionals and follows a power-law scaling with permeability. The scaling exponent is independent of particle size and distribution and has a universal value of 0.428 for 3D porous systems built of spherical grains.

Singularities of worldsheets along framed particle trajectories in Minkowski three-space

Modeling the worldsheets along the trajectory (a string) of a point particle (a 0-brane) as two timelike surfaces in 3-subspace of Minkowskian spacetime, respectively, the work aims at analyzing the topological structures of these two worldsheets along the trajectory of the point particle from the view point of singularity theory. Different from the regular curves, the traveling trajectory (modeled as framed timelike curve) of the particle is allowed to be singular, two worldsheets are generated by the traveling trajectories of the particle. As applications of singularity theory, we classify the singularities of these two worldsheets along the traveling trajectory of the particle. Using the approach of the unfolding theory in singularity theory, we find two new geometric invariants which are useful for characterizing the local topological structures of singularities of these two worldsheets along the particle. It is revealed that there exist cuspidal edge type and swallowtail type of singularities for these two worldsheets under the appropriate conditions of geometric invariants. Meanwhile, it is also pointed out that the types of singularities of these two worldsheets have a close relationships to the order of contacts between these two worldsheets and two timelike planes, respectively. Finally, some examples are presented to interpret our theoretical results.

Comparing inflationary models in extended Metric-Affine theories of gravity

We study slow-roll inflation as a common feature in Metric-affine Theories of Gravity. In particular, we take into account extended metric, teleparallel and symmetric-teleparallel theories of gravity, based on different geometric invariants, discussing analogies and differences. The analysis for each model is performed in two approaches. First, we focus on the potential-slow-roll approach by studying the reconstructed potentials for different forms of the extended models related to the considered gravitational theory in the Einstein frame. Secondly, we investigate the Hubble-slow-roll approach for some conventional inflationary potentials related to the specific extended model in the Jordan frame. We compare all results with cosmic microwave background anisotropy observations coming from Planck 2018 and BICEP2/Keck array satellites in order to find the observational constraints on the parameters space of the models as well as their prediction from the spectral parameters. Eventually, we attempt to present a qualitative comparison between three classes of considered modified gravities using the obtained inflationary results. The aim is to select, in principle, cosmological signatures capable of discriminating among concurrent models in view to point out the representation of gravity, according with geometric invariants, which better addresses the early universe dynamics.

Desynchronization Attacks Resistant Watermarking for Remote Sensing Images Based on DWT‐SVD and Normalized Feature Domain

ABSTRACTMost existing remote sensing image watermarking algorithms concentrate on excavation of particular embedding templates, image features, or geometric invariant domains, which present challenges in terms of resistance to desynchronization attacks, embedding domain repetition, and insufficient algorithm versatility. To address these issues, this study proposes a watermarking algorithm that is robust to desynchronization attacks and can adapt to different types of remote sensing images using the geometric invariant domain and hybrid frequency domain. The algorithm uses the multi‐scale SIFT to identify feature points in remote sensing images, then creates a Delaunay triangulation network (DTN) based on these feature points, extracts the tangent circles of triangles, and normalizes these tangent circles using image moment and affine transformation, and the feature domains with geometric invariance are constructed. On this basis, the discrete wavelet transform (DWT) transforms the feature domain to the frequency decomposition state, and the singular value decomposition (SVD) further mines the watermark embedding domain, ensuring the stability of the watermark transforming back and forth in the embedding domain and improving the overall invisibility of the watermarking algorithm. The experimental results indicate that, compared to related algorithms, the proposed watermarking algorithm not only adapts better to remote sensing images with different bands and bit depths but also provides superior invisibility and demonstrates strong robustness against various desynchronization attacks such as splicing, panning, rotating, as well as image processing like noise addition, filtering, and compression.

Multi-channel depth segmentation network based on 3D graph convolution algorithm and its application in point cloud segmentation

At the current stage, 3D graph convolutional algorithms face the following issues: (1) The problem of determining the neighborhood in graph convolution. (2) The issue of fusing features at different depths in deep graph convolutional algorithms. (3) The challenge of feature fusion and recognition in multi-path deep graph convolutional algorithms. Based on these, the paper "Multi-Path Deep Segmentation Network Based on 3D Graph Convolutional Algorithm and Its Application in Point Cloud Segmentation" is proposed. Inspired by visual computing theory, the neighborhood for 3D graph convolution is determined based on the receptive field theory, and a 3D graph convolutional algorithm based on visual selectivity is proposed; A single-link deep feature extraction algorithm for 3D graph convolution based on visual selectivity is proposed, it achieve the fusion of features at different depths learned by the graph convolutional algorithm by a pyramid learning method; A multi-link feature extraction algorithm for 3D graph convolution based on visual selectivity is proposed, and achieve multi-link feature fusion and segmentation to utilize the RPN calculation method. compared with algorithms such as PointNet, PointNet++, KPConv deform, 3D GCN, and My Algorithm, The segmentation performance and geometric invariance of the proposed algorithm are validated on the ShapeNetPart dataset and the custom Mortise_and_Tenon_DB dataset, and Experiments demonstrate that the proposed algorithm is correct and feasible, offers superior 3D point cloud segmentation performance, and exhibits strong geometric invariance.

Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

Diffeomorphisms of the energy-momentum space: Perturbative QED

A perturbative formulation of quantum electrodynamics is given in terms of geometrical invariants of the energy-momentum space, whose geometry is taken to be one of a constant curvature. The construction is relevant for different classes of noncommutativity: the Snyder model and the so called GUP models. For the Snyder model it is shown that all the amplitudes are finite at every order of the perturbation expansion.

Two geometrical invariants for three‐dimensional systems

The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components.

Electromagnetic Plasma Reactor: Fermionic Derivation Tools

From the developed theories around of an electromagnetic plasma reactor designed and controlled through geometrical invariants has been demonstrated the existence of derived products from electromagnetic plasma as are phonons, fermions, ions, free electrons and protons, magnetic and electric drifts, which can be used as complement or development part of the electromagnetic propulsion system where the fermion flow can produce a ionic force that could be used as propulsion force considering shock waves produced with an electric field on these ions. Likewise, experimentally and considering direct current and voltage in a constant regime, has been detected as ionic wind which carries fermion products to be used too in the possible electromagnetic propulsion. These are other products proposes of previous works. Likewise, through of a caption and detection camera, is measured the electromagnetic properties of the ionic flow of the space derived from the electromagnetic plasma, and is proposed a curvature energy signal of magnetic nature to measure the force and control the electromagnetic plasma and its ionic flow.

Graph-Regularized Non-Negative Matrix Factorization for Single-Cell Clustering in scRNA-Seq Data.

The advent of single-cell RNA sequencing (scRNA-seq) has brought forth fresh perspectives on intricate biological processes, revealing the nuances and divergences present among distinct cells. Accurate single-cell analysis is a crucial prerequisite for in-depth investigation into the underlying mechanisms of heterogeneity. Due to various technical noises, like the impact of dropout values, scRNA-seq data remains challenging to interpret. In this work, we propose an unsupervised learning framework for scRNA-seq data analysis (aka Sc-GNNMF). Based on the non-negativity and sparsity of scRNA-seq data, we propose employing graph-regularized non-negative matrix factorization (GNNMF) algorithm for the analysis of scRNA-seq data, which involves estimating cell-cell sparse similarity and gene-gene sparse similarity through Laplacian kernels and p-nearest neighbor graphs ( p-NNG). By assuming intrinsic geometric local invariance, we use a weighted p-nearest known neighbors ( p-NKN) to optimize the scRNA-seq data. The optimized scRNA-seq data then participates in the matrix decomposition process, promoting the closeness of cells with similar types in cell-gene data space and determining a more suitable embedding space for clustering. Sc-GNNMF demonstrates superior performance compared to other methods and maintains satisfactory compatibility and robustness, as evidenced by experiments on 11 real scRNA-seq datasets. Furthermore, Sc-GNNMF yields excellent results in clustering tasks, extracting useful gene markers, and pseudo-temporal analysis.

Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems

Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems

Learning geometric invariants through neural networks

Learning geometric invariants through neural networks

KCC theory for a type of nonlinear damped SD oscillators

Smooth and discontinuous (SD) oscillators are the nonlinear models that will exhibit SD dynamics due to continuous variation of the smooth parameter. Kosambi–Cartan–Chern (KCC) theory is a differential geometric theory that describes the deviations from the entire trajectory of the variational equation to the nearby trajectory. This paper presents a completely new dynamical analysis of a type of SD oscillators with nonlinear damping based on the KCC theory. First, the paper gives five KCC geometrical invariants of SD oscillators in the smooth and non-smooth cases, respectively. The results show that the geometric quantities in the non-smooth case cannot be derived directly from the geometric quantities in the smooth case. Second, from these geometric quantities, KCC stability of SD oscillators trajectory at any point except [Formula: see text] is analyzed. Lastly, numerical results show that in some regions that deviate from the equilibrium point of system, the system will exhibit complex and variable dynamical behavior due to small changes in parameters. This paper shows that the KCC theory is also a useful tool in the dynamical analysis of non-smooth systems.

Jacobi Stability Analysis of Liu System: Detecting Chaos

By utilizing the Kosambi–Cartan–Chern (KCC) geometric theory, this paper is dedicated to providing novel insights into the Liu dynamical system, which stands out as one of the most distinctive and noteworthy nonlinear dynamical systems. Firstly, five important geometrical invariants of the system are obtained by associating the nonlinear connection with the Berwald connection. Secondly, in terms of the eigenvalues of the deviation curvature tensor, the Jacobi stability of the Liu dynamical system at fixed points is investigated, which indicates that three fixed points are Jacobi unstable. The Jacobi stability of the system is analyzed and compared with that of Lyapunov stability. Lastly, the dynamical behavior of components of the deviation vector is studied, which serves to geometrically delineate the chaotic behavior of the system near the origin. The onset of chaos for the Liu dynamical system is obtained. This work provides an analysis of the Jacobi stability of the Liu dynamical system, serving as a useful reference for future chaotic system research.

Horizons that gyre and gimble: a differential characterization of null hypersurfaces

Motivated by the thermodynamics of black hole solutions conformal to stationary solutions, we study the geometric invariant theory of null hypersurfaces. It is well-known that a null hypersurface in a Lorentzian manifold can be treated as a Carrollian geometry. Additional structure can be added to this geometry by choosing a connection which yields a Carrollian manifold. In the literature various authors have introduced Koszul connections to study the study the physics on these hypersurfaces. In this paper we examine the various Carrollian geometries and their relationship to null hypersurface embeddings. We specify the geometric data required to construct a rigid Carrollian geometry, and we argue that a connection with torsion is the most natural object to study Carrollian manifolds. We then use this connection to develop a hypersurface calculus suitable for a study of intrinsic and extrinsic differential invariants on embedded null hypersurfaces; motivating examples are given, including geometric invariants preserved under conformal transformations.

Repulsive gravity in regular black holes

We evaluate the effects of repulsive gravity using first order geometric invariants, i.e. the Ricci scalar and the eigenvalues of the Riemann curvature tensor, for three regular black holes, namely the Bardeen, Hayward, and Dymnikova spacetimes. To examine the repulsive effects, we calculate their respective onsets and regions of repulsive gravity. Afterwards, we compare the repulsive regions obtained from these metrics among themselves and then with the predictions got from the Reissner–Nordström and Schwarzschild–de Sitter. A notable characteristic, observed in all these metrics, is that the repulsive regions appear to be unaffected by the mass that generates the regular black hole. This property emerges due to the invariants employed in our analysis, which do not change sign through linear combinations of the mass and the free coefficients of the metrics. As a result, gravity can change sign independently of the specific values acquired by the mass. This conclusion suggests a potential incompleteness of regular solutions, particularly in terms of their repulsive effects. To further highlight this finding, we numerically compute, for the Reissner–Nordström and Schwarzschild–de Sitter solutions, the values of mass, M, that emulate the repulsive effects found in the Bardeen and Hayward spacetimes. These selected values of M provide evidence that regular black holes do not incorporate repulsive effects by means of the masses used to generate the solutions themselves. Implications and physical consequences of these results are then discussed in detail.