Topological Planes Research Articles

DT-k-group structures on digital objects and an answer to an open problem

The paper examines various properties of [Formula: see text]-[Formula: see text]-subgroup structures and addresses an open problem on the existence of topological group structures on the [Formula: see text]-dimensional Khalimsky ([Formula: see text]-, for brevity) topological space and Marcus–Wyse ([Formula: see text]-, for short) topological plane. In particular, we obtain many types of totally [Formula: see text]-disconnected or [Formula: see text]-connected subgroups of a [Formula: see text]-connected [Formula: see text]-[Formula: see text]-group. Besides, we prove that each of the [Formula: see text]-dimensional [Formula: see text]-topological space and the [Formula: see text]-topological plane cannot be a typical topological group. Unlike an existence of a [Formula: see text]-[Formula: see text]-group structure of [Formula: see text] (see Proposition 4.7), we prove that neither of [Formula: see text] and [Formula: see text] is a topological group, where [Formula: see text] (respectively, [Formula: see text]) is a simple closed [Formula: see text]- (respectively, [Formula: see text]-) topological curve with [Formula: see text] elements in [Formula: see text] (respectively, [Formula: see text]) and the operation “∗” is a special kind of binary operation for establishing a group structure of each of [Formula: see text] and [Formula: see text]. Finally, given a [Formula: see text]-[Formula: see text]-group structure of [Formula: see text], we find several types of [Formula: see text]-[Formula: see text]-subgroup structures of it (see Theorem 5.7).

Semi-separation axioms associated with the Alexandroff compactification of the $ MW $-topological plane

<abstract><p>The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ($ MW $-, for brevity) topology. The Alexandroff compactification of the $ MW $-topological plane is called the infinite $ MW $-topological sphere up to homeomorphism. We first prove that under the $ MW $-topology on $ {\mathbb Z}^2 $ the connectedness of $ X(\subset {\mathbb Z}^2) $ with $ X^\sharp\geq 2 $ implies the semi-openness of $ X $. Besides, for the infinite $ MW $-topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the $ MW $-topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite $ MW $-topological sphere is a semi-regular space; thus, it is a semi-$ T_3 $-space because it is a semi-$ T_1 $-space. Hence we finally conclude that an Alexandroff compactification of the $ MW $-topological plane preserves the semi-$ T_3 $ separation axiom.</p></abstract>

Seismic Facies Visualization Analysis Method of SOM Corrected by Uniform Manifold Approximation and Projection

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">As a common seismic facies visualization analysis method, SOM projects the waveform or seismic attribute vectors into a two-dimensional topological plane in a nonlinear way, which can effectively and efficiently discover the topological structure of a dataset. SOM does not need to set the number of classes in prior, and has friendly visualization characteristics and excellent generalization, which are conducive to seismic facies interpretation using unlabeled data. However, due to the competitive learning used in SOM and the imbalance of data distribution in real world, the samples from majority classes are expanded on the topological plane and the minority classes are compressed. As a result, the plane cannot accurately describe the global structure of data distribution. To improve the visualization precision by modifying the topological relationship of the prototype vectors of SOM, we utilize UMAP (Uniform Manifold Approximation and Projection), a novel manifold learning technique for dimension reduction, to correct the prototype vectors generated by SOM. By combing the advantages of SOM and UMAP in the representation of data topological structure, the global relationship between seismic data samples can be properly established, and the internal relative spatial structure of majority class samples can be retained as much as possible, resulting in a more reliable classification. Meanwhile, the framework maintains the advantages of SOM in visualization. In the modeling tests and real data experiments, we have demonstrated the effectiveness and rationality of UMAP-SOM on the spatial structure representation of three-dimensional seismic data</i> .

Semi-Jordan curve theorem on the Marcus-Wyse topological plane

&lt;abstract&gt;&lt;p&gt;The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ into exactly two components. The semi-Jordan curve theorem on the $ MW $-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.&lt;/p&gt;&lt;/abstract&gt;

Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet

Despite recent efforts to advance spintronics devices and quantum information technology using materials with non-trivial topological properties, three key challenges are still unresolved1–9. First, the identification of topological band degeneracies that are generically rather than accidentally located at the Fermi level. Second, the ability to easily control such topological degeneracies. And third, the identification of generic topological degeneracies in large, multisheeted Fermi surfaces. By combining de Haas–van Alphen spectroscopy with density functional theory and band-topology calculations, here we show that the non-symmorphic symmetries10–17 in chiral, ferromagnetic manganese silicide (MnSi) generate nodal planes (NPs)11,12, which enforce topological protectorates (TPs) with substantial Berry curvatures at the intersection of the NPs with the Fermi surface (FS) regardless of the complexity of the FS. We predict that these TPs will be accompanied by sizeable Fermi arcs subject to the direction of the magnetization. Deriving the symmetry conditions underlying topological NPs, we show that the 1,651 magnetic space groups comprise 7 grey groups and 26 black-and-white groups with topological NPs, including the space group of ferromagnetic MnSi. Thus, the identification of symmetry-enforced TPs, which can be controlled with a magnetic field, on the FS of MnSi suggests the existence of similar properties—amenable for technological exploitation—in a large number of materials.

ДИНАМИЧЕСКИЙ АНАЛИЗ КОМПОЗИТНОГО ОБТЕКАТЕЛЯ РАКЕТЫ ПРИ ОТДЕЛЕНИИ С УЧЕТОМ РАССЛОЕНИЯ СТРУКТУРЫ

The dynamic processes in the rocket fairing when the pyrotechnic separation system is triggered are considered. The fairing construction is mixed and includes composite and metal elements. The main composite construction element is a fiberglass shell with regular and irregular winding zones. The speed acceleration required to separate the fairing occurs under the action of impulse pressure from the powder gases in the pyrotechnic system. The displacement of the fairing is made up of displacements of the movement as a rigid whole along its axis and vibrations caused by deformations. The calculation of the fairing movement is carried out according to a three-dimensional FEM model using software that uses a topologically regular discretization system. The problem solution in time is performed according to the implicit Wilson finite-difference scheme. When studying the fairing dynamics, it is allowed to break the structure of the shell in the form of lamination, which in the FEM scheme is modeled by a special method. A cut with double nodes is created on the surface of the proposed lamination along topological planes by transforming the finite element mesh. Modification of the stiffness matrix and mass matrix for the transformed mesh is performed based on the created information base of degenerate finite elements and formalized matrix operations. In numerical studies, two types of lamination from irregular zones of fiberglass winding are considered – the internal location from the flange and edge location with access to the fairing free edge. The results of calculating vibrations along the sides of lamination and data on the redistribution of dynamic stresses due to lamination are presented. Radial and axial displacements when passing through the lamination surface discontinue, the magnitude of which for internal lamination is much less, which is explained by the compression of deformation for this case, in contrast to the lamination that goes to the boundary. When estimating the relative axial displacements, the component of the displacement of a rigid whole, determined by a separate calculation, was excluded. The maximum radial displacements during lamination from the edge reach 3 mm, which is one and a half times higher than for an undamaged shell. Axial stresses are maximal from the action of inertial forces during acceleration. Its redistribution over the layers is significantly greater for the edge lamination, for which the maximum values increase almost two times concerning the undamaged shell, which determines this type of lamination as more dangerous.

The fixed point property of the infinite K-sphere in the set Con*((Z2)*)

In this paper the Alexandroff one point compactification of the 2-dimensional Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line) is called the infinite K-sphere (resp. the infinite K-circle). The present paper initially proves that the infinite K-circle has the fixed point property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of all continuous self-maps f of the infinite K-circle. Next, we address the following query which remains open: Under what condition does the infinite K-sphere have the FPP? Regarding this issue, we prove that the infinite K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally, we compare the FPP of the infinite K-sphere and that of the infinite M-sphere, where the infinite M-sphere means the one point compactification of the Marcus-Wyse topological plane.

Topologies of the quotient spaces induced by the M-topological plane and the infinite M-topological sphere

The present paper studies two topologies of the quotient spaces related to the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological plane. The topologies are exactly proved to be the cofinite particular point and the excluded two points topology. We also concern separation axioms and nowhere dense subset properties of the quotient spaces of the M-topological plane. Furthermore, we investigate both a particular point and an excluded point topological structure associated with the quotient spaces of the M-topological plane. Finally, we investigate the fixed point property of the compactification of the M-topological plane. Unlike the non-fixed point property of the Hausdorff compactification of the Euclidean topological plane, we prove that every continuous self-bijection of the compactification of the M-topological plane has a fixed point.

Rectifiability of planes and Alberti representations

We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let $X$ be a pointwise doubling metric measure space. Let $U$ be a Borel subset on which the blowups of $X$ are topological planes. We show that $U$ can admit at most $2$ independent Alberti representations. Furthermore, if $U$ admits $2$ Alberti representations, then the restriction of the measure to $U$ is $2$-rectifiable. This is a partial answer to the case $n=2$ of a question of the second author and Schioppa.

Conceptualising knowledge for access in the sciences: academic development from a social realist perspective

Whilst arguing from a social realist perspective that knowledge matters in academic development (AD) curricula, this paper addresses the question of what knowledge types and practices are necessary for enabling epistemological access. It presents a single, in-depth, qualitative case study in which the curriculum of a science AD course is characterised using Legitimation Code Theory (LCT). Analysis of the course curriculum reveals legitimation of four main categories of knowledge types along a continuum of stronger to weaker epistemic relations: disciplinary knowledge, scientific literacies knowledge, general academic practices knowledge and everyday knowledge. These categories are ‘mapped’ onto an LCT(Semantics) (how meaning relates to both context and empirical referents) topological plane to reveal a curriculum that operates in three distinct but interrelated spaces by facing towards both the field of science and the practice of academia. It is argued that this empirically derived differentiated curriculum framework offers a conceptual means for considering the notion of access to ‘powerful’ knowledge in a range of AD and mainstream contexts.

Migration strategies for FTTx solutions based on active optical networks

AON, one of the most deployed fiber access solutions in Europe, needs to be upgraded in order to satisfy the ever growing bandwidth demand driven by new applications and services. Meanwhile, network providers want to reduce both capital expenditures and operational expenditures to ensure that there is profit coming from their investments. This article proposes several migration strategies for AON from the data plane, topology, and control plane perspectives, and investigates their impact on the total cost of ownership

Three complexes of Cu(I) cluster with flexible and rigid ligands: Synthesis, characterization and photoluminescent properties

Three new Cu(I) cluster complexes, viz. [(Cu4I4)(Cu2I2)(dimb)3]n (1; dimb=1,4-diimidazol-1-ylbutane), [(Cu3I2)(dimb)(dmtz)]n (2; dmtz=3,5-dimethyl-1,2,4-triazole), and [Cu6(mbt)6] (3; mbt=2-mercaptobenzothiazole), have been solvothermally synthesized and structurally characterized. In 1, a Cu4I4 cubane core as a 4-connecting node, connects the neighboring nodes either through single dimb or μ2-[(Cu2I2)(dimb)2] linkers, affording an undulated 2D (4,4) net. Parallel interpenetration occurs between the adjacent nets and thus the overall 2D→3D network is formed. Complex 2 is constructed by 2D (4,4) topological plane grid layers of AB stacking. The core, a distorted triangular bipyramidal Cu3I2 cluster, is acted as a 4-connecting node and connected with dimb and μ3-dmtz to form the layer. Complex 3 contains a (Cu6S6) core in discrete paddle-wheel molecule, which serves as a 4-connecting node to link equivalent ones via π···π interaction, forming 2D (4,4) layers. Solid-state luminescence properties and thermogravimetric analyses of 1, 2 and 3 were investigated.

A Topology Visualization Early Warning Distribution Algorithm for Large‐Scale Network Security Incidents

It is of great significance to research the early warning system for large-scale network security incidents. It can improve the network system's emergency response capabilities, alleviate the cyber attacks' damage, and strengthen the system's counterattack ability. A comprehensive early warning system is presented in this paper, which combines active measurement and anomaly detection. The key visualization algorithm and technology of the system are mainly discussed. The large-scale network system's plane visualization is realized based on the divide and conquer thought. First, the topology of the large-scale network is divided into some small-scale networks by the MLkP/CR algorithm. Second, the sub graph plane visualization algorithm is applied to each small-scale network. Finally, the small-scale networks' topologies are combined into a topology based on the automatic distribution algorithm of force analysis. As the algorithm transforms the large-scale network topology plane visualization problem into a series of small-scale network topology plane visualization and distribution problems, it has higher parallelism and is able to handle the display of ultra-large-scale network topology.

Two-parameter characterization of elastic–plastic crack front fields: Surface cracked plates under uniaxial and biaxial bending

In this paper the J–Q two-parameter characterization of elastic–plastic crack front fields is examined for surface cracked plates under static uniaxial and biaxial bending. Extensive three-dimensional elastic–plastic finite element analyses are performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial bending conditions, covering from small-scale to large-scale yielding. Surface cracks with aspect ratios a/c=0.2, 1.0 and relative depths a/t=0.2, 0.6, corresponding to four specimen configurations, are investigated. The full stress field solutions are obtained in topological planes perpendicular to the crack fronts. The question of J–Q characterization is addressed by comparing these elastic–plastic crack-front stress fields with J–Q family of stress fields. It is found that the J–Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from the finite element analyses. Complete distributions of the J-integral and Q-factors for the wide range of geometry and loading conditions are obtained. Size requirements in terms of crack depth (for shallow cracks) or ligament size (for deep cracks) that are necessary to ensure J–Q characterization of crack front fields in surface cracked plates are also discussed.

Quotients of projective spaces and spheres

Abstract In this paper we prove results on topological properties of quotients of manifolds, in particular of projective spaces and spheres. Specifically, under suitable conditions we exhibit results on homology and homotopy groups of these quotients. As an application we generalize the characterization of topological translation planes obtained by Löwen, Steinke, and Van Maldeghem in [Adv. Geom.: S59–S74, 2003] to all possible (finite) dimensions. Our approach also yields a more unified proof for this characterization result. Most of the results in this paper are obtained by means of standard results from algebraic topology.

Two-parameter characterization of elastic–plastic crack front fields: Surface cracked plates under tensile loading

In this paper the J–Q two-parameter characterization of elastic–plastic crack front fields is examined for surface cracked plates under uniaxial and biaxial tensile loadings. Extensive three-dimensional elastic–plastic finite element analyses were performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. Surface cracks with aspect ratios a/c=0.2, 1.0 and relative depths a/t=0.2, 0.6 were investigated. The loading levels cover from small-scale to large-scale yielding. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. In order to facilitate the determination of Q-factors, modified boundary layer analyses were also conducted. The J–Q two-parameter approach was then used in characterizing the elastic–plastic crack front stress fields along these 3D crack fronts. Complete distributions of the J-integral and Q-factors for a wide range of loading conditions were obtained. It is found that the J–Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis. These results are suitable for elastic–plastic fracture mechanics analysis of surface cracked plates.

On the quantification of the constraint effect along a three-dimensional crack front

Abstract In this paper we examined the characterization of constraint effects for surface cracked plates under uniaxial and biaxial tension loadings. First, three-dimensional (3D) modified boundary layer analyses were conducted using the finite element method to study the constraint effect at a typical 3D crack front. The analyses were carried out using small geometry change formulation and deformation plasticity material model. Elastic-plastic crack front stress fields at a constant J and various T-stress levels were obtained. Three-dimensional elastic-plastic analyses were performed for semi-circular surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. Then, J-Q and J-T two-parameter approaches are used in characterizing the elastic-plastic crack-tip stress fields along the 3D crack front. It is found that the J-Q characterization provides good estimate for the constraint effect for crack-tip stress fields. Reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis.

Singular solitary waves associated with homoclinic orbits

Different from what we have expected before, when a homoclinic orbit intersects with a quadratic singular curve on the topological phase plane derived from a generalized KdV equation, corresponding to the homoclinic orbit, there exist a few types of solitary waves, including peakons and antipeakons as well as periodic waves and furthermore, new types of solitary waves with peaks. The investigation shows that, when a trajectory along the homoclinic orbit moves at the intersection points between the homoclinic orbit and the singular curve, it may jump between these intersection points, which forms peaks on the waves, implying the nonsmooth solitary waves occur.

Peaked singular wave solutions associated with singular curves

Abstract We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit.

Topological affine planes with affine connections

Abstract We consider the question whether the system of lines of a two-dimensional topological plane can be described as the system of geodesics of a Riemannian metric or an affine connection. In [4] we have shown that non-classical compact projective planes do not admit Riemannian metrics. Here we use different methods that are suited to two-dimensional planes in general. We apply them to three families of topological affine planes with large collineation groups. It turns out that for the standard models of the skew-parabola planes and the planes over cartesian fields no affine connection, and hence no Riemannian metric exist. However, for the Moulton planes affine connections do exist and we determine all of them. Among them is at least one Riemannian connection. We also give an example where no Riemannian metric exists. Moreover, we derive a characterization of the classical affine plane in the case that ℝ2 acts by vector space translations as a subgroup of the collineation group.