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In this paper, we investigate the average geodesic distance on Sierpiński torus networks. We construct Sierpiński torus networks based on the classic Sierpiński carpet in an iterative way. By applying finite patterns on integrals, we deduce the exact value of the average geodesic distance of the Sierpiński torus. Furthermore, the asymptotic formula for the average geodesic distance of the corresponding networks can be obtained.

The Local Multi-Scale Discriminant Analysis (LMSDA) is a novel dimensionality reduction technique that proficiently maps high-dimensional data into a low-dimensional space. It seamlessly integrates the collaborative representation framework with dual constraint optimization in intrinsic and penalty graphs. The intrinsic graph captures intricate local manifold structures by leveraging multi-scale reconstruction relationships, ensuring class cohesion during reduction. Meanwhile, the penalty graph sharpens discriminability by maximizing inter-class disparities, fostering clearer class boundaries. This dual strategy not only preserves the essence of intra-class data but also bolsters its discriminative powers. Extensive experiments on AR, CMU_PIE, and Yale B datasets affirm LMSDA’s superiority over contemporary methods, underscoring its potential for diverse applications.

Similarity measurement is crucial for classification. Based on the manifold assumption, many graph-based algorithms were developed. Almost all methods follow the k-rule or ε-rule to construct a graph, and then focus on the algorithms based on the graph. However, the graph may not represent the local structure well, and it does not fully utilize the label information yet. The local structure can be presented by the local density and the distance between the samples and their neighbors. And the graph constructed by the guidance of label information will be better approximate of the relationship of the input data. In this paper, we propose an adaptive semi-supervised graph constructing method. The similarity is learned when constructing the graph. The advantages of the similarity learned by our method include: (1) The similarity is measured along the manifold by constructing a graph; (2) nearby points and points in the same cluster share high similarity; (3) samples from the same class have higher similarity than samples from different classes. Experimental results show that using the proposed similarity for classification task could get better recognition accuracy.

The manifold learning technique is a class of machine learning techniques that converts the intrinsic geometry of the data from higher to lower dimensional representation. The manifold learning technique in image analysis is used to view as a single point in a very high-dimensional space, a set of such points for population of images that may be well represented by a sub-manifold of the space that is likely to be nonlinear and of significantly lower dimensions. In this paper, we presented a new method to generate optimal random fields for image segmentation using local linear embedding manifold learning technique. This method gives better segmentation results compared to entropy classification and occlusion reasoning techniques.

We present a new continuation method for computing implicitly defined manifolds. The manifold is represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a boundary point. The boundary point is found using an expression for the boundary in terms of the vertices of a set of finite, convex polyhedra. The resulting algorithm is quite simple, allows adaptive spacing of the computed points, and deals with the problems of local and global overlap in a natural way. The algorithm is robust (the new points need only be near the boundary), and is well suited to problems with large embedding dimension, and small to moderate dimension.

This paper deals with two types of bifurcation behaviors of charged particles moving on rough surface under different damping effects. Based on the derived models, the stability of the particle system is judged by eigenvalue analysis and then the eigenvalue movement of the Jacobian matrix is analyzed to reveal the underlying mechanism of the dynamical evolution. It is shown that in the particle system with constant damping force, the system loses stability via Neimark–Sacker bifurcation, whereas in the system with time-dependent damping force, the stability is lost by way of period-doubling bifurcation. In addition, a powerful tool called manifold is employed to meticulously characterize the phase space so as to clearly describe the process of energy evolution, which leads to the inherent understanding of the complex behaviors and particularly the global dynamical properties in the particle system. Finally, some bifurcation diagrams are obtained to give a more evident explanation of complex behaviors. These results are very useful for the entire transport knowledge of charged particle system.

We focus on the two-dimensional stable manifold of Chen system. Based on the Chen system, a new system with nonlinear control is put forward, which exhibits brand new dynamical phenomena such as the coexistence of heteroclinic orbits with periodic solutions, or two new attractors. The emergence of manifold surface is found by tangential self-intersection. The two-dimensional unstable manifold of saddle-focus tends to another equilibrium solution asymptotically, the underlying heteroclinic bifurcation and the “twins” unstable manifolds are bulit from two saddle-focus respectively.

Two-dimensional unstable manifolds of the modified Chen system are constructed at equilibrium solution by “expanding up” along the unstable eigen-direction, hence it is tangent to the unstable eigenspace. In general, unstable manifold expands to the attraction basin of the corresponding limit cycle or attractor. With the introduction of time delay, the two-dimensional unstable manifold of an unstable focus is simulated via expanding solution orbits with restriction condition on the associated foliations. The simulated unstable manifold coincides with the attraction basin of the limit cycle of the delay differential equations.

Given a cycle of length k on a triangulated 2-manifold, we determine if it is null-homologous (bounds a surface) in O(n+k) optimal time and space where n is the size of the triangulation. Further, with a preprocessing step of O(n) time we answer the same query for any cycle of length k in O(g+k) time, g the genus of the 2-manifold. This is optimal for k < g.

We present a novel way to efficiently compute Riemannian geodesic distance over a two- or three-dimensional domain. It is based on a previously presented method for computation of geodesic distances on surface meshes. Our method is adapted for rectangular grids, equipped with a variable anisotropic metric tensor. Processing and visualization of such tensor fields is common in certain applications, for instance structure tensor fields in image analysis and diffusion tensor fields in medical imaging.

The included benchmark study shows that our method provides significantly better results in anisotropic regions in 2-D and 3-D and is faster than current stat-of-the-art solvers in 2-D grids. Additionally, our method is straightforward to code; the test implementation is less than 150 lines of C++ code. The paper is an extension of a previously presented conference paper and includes new sections on 3-D grids in particular.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

We give explicit palindrome presentations of the groups of rational knots, i.e. presentations with relators which read the same forwards and backwards. This answers a question posed by Hilden, Tejada and Toro in 2002. Using such presentations we obtain simple alternative proofs of some classical results concerning the Alexander polynomial of all rational knots and the character variety of certain rational knots. Finally, we derive a new recursive description of the SL(2, ℂ) character variety of twist knots.

The main result of this paper is the construction of two Hyperbolic manifolds, $M_1$ and $M_2$, with several remarkable properties:

• ⁽¹⁾Every closed orientable $3$-manifold is homeomorphic to the quotient space of the action of a group of order $16$ on some covering space of $M_1$ or $M_2$.
• ⁽²⁾$M_1$ and $M_2$ are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way.
• ⁽³⁾There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic $-2$ each of which is tesselated by regular right angled pentagons and embedded in $M_1$ or $M_2$. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic $-2$.

As it is well-known, all Vassiliev invariants of degree one of a knot K ⊂ ℝ³ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not ℝ³. Recently, T. Fiedler introduced such invariants of a knot in an ℝ¹-fibration over a surface F. They take values in the free ℤ-module generated by all the free homotopy classes of loops in F. Here, we generalize them to the most refined Vassiliev invariant of degree one. The ranges of values of all these invariants are explicitly described.

We also construct a similar invariant of a two-component link in an ℝ¹-fibration. It generalizes the linking number.

Generation of random fields (GRF) for image segmentation represents partitioning an image into different regions that are homogeneous or have similar facets of the image. It is one of the most challenging tasks in image processing and a very important pre-processing step in the fields of computer vision, image analysis, medical image processing, pattern recognition, remote sensing, and geographical information system. Many researchers have presented numerous image segmentation approaches, but still, there are challenges like segmentation of low contrast images, removal of shadow in the images, reduction of high dimensional images, and computational complexity of segmentation techniques. In this review paper, the authors address these issues. The experiments are conducted and tested on the Berkely dataset (BSD500), Semantic dataset, and our own dataset, and the results are shown in the form of tables and graphs.

Contour grouping is the precondition for high-level visual tasks such as region-of-interest detection and object recognition. Those methods adopting local geometric features and simple photometric attributes always lead to unreliable result or bad robustness. In this paper, we propose a grouping method using multiresolution analysis and one-dimensional manifold. By using multiresolution analysis, grouping seeds and initial clusters are detected as precondition for describing topological structure. Then one-dimensional manifold is applied to discovering intrinsic order of topological structure, which corresponds to coordinates of edges in a closed contour. These two improvements enhance our method's robustness relative to those using local features or linear combinations of them. Experiments on different classes of real images show competence of our method.

Hyperspectral image (HSI) classification draws a lot of attentions in the past decades. The classical problem of HSI classification mainly focuses on a single HSI scene. In recent years, cross-scene classification becomes a new problem, which deals with the classification models that can be applied across different but highly related HSI scenes sharing common land cover classes. This paper presents a cross-scene classification framework combining spectral–spatial feature extraction and manifold-constrained feature subspace learning. In this framework, spectral–spatial feature extraction is completed using three-dimensional (3D) wavelet transform while manifold-constrained feature subspace learning is implemented via multitask nonnegative matrix factorization (MTNMF) with manifold regularization. In 3D wavelet transform, we drop some coefficients corresponding to high frequency in order to avoid data noise. In feature subspace learning, a common dictionary (basis) matrix is shared by different scenes during the nonnegative matrix factorization, indicating that the highly related scenes should share than same low-dimensional feature subspace. Furthermore, manifold regularization is applied to force the consistency across the scenes, i.e. all pixels representing the same land cover class should be similar in the low-dimensional feature subspace, though they may be drawn from different scenes. The experimental results show that the proposed method performs well in cross-scene HSI datasets.

An intrinsic version of the integrability theorem for the classical Bäcklund theorem is presented. It is characterized by a one-form which can be put in the form of a Riccati system. It is shown how this system can be linearized. Based on this result, a procedure for generating an infinite number of conservation laws is given.

A theorem due to Lichnerowicz which establishes a lower bound on the lowest nonzero eigenvalue of the Laplacian acting on functions on a compact, closed manifold is reviewed. It is shown how this theorem can be extended to the case of a manifold with nonempty boundary. Lower bounds for different boundary conditions, analogous to the empty boundary case, are formulated and some novel proofs are presented.

The components for the frame field of a two-dimensional manifold with constant Gaussian curvature are determined for arbitrary nonzero curvature. The components of the frame fields are found from the structure equations and lead to specific nonlinear equations which pertain to surfaces with specific values of the Gaussian curvature. For negative curvature, the equation is of sine-Gordon type, and for positive curvature it is of sinh-Gordon type. The integrability and Bäcklund properties of these equations are then investigated by studying a differential ideal of two-forms which leads to the equations. As a consequence of studying the prolongation structure of each equation, a Lax pair and Bäcklund transformation are obtained.