Topological Classification Research Articles

Metrization of soft metric spaces and its application to fixed point theory

<abstract><p>Soft set theory has attracted many researchers from several different branches. Sound theoretical improvements are accompanied with successful applications to practical solutions of daily life problems. However, some of the attempts of generalizing crisp concepts into soft settings end up with completely equivalent structures. This paper deals with such a case. The paper mainly presents the metrizability of the soft topology induced by a soft metric. The soft topology induced by a soft metric is known to be homeomorphic to a classical topology. In this work, it is shown that this classical topology is metrizable. Moreover, the explicit construction of an ordinary metric that induces the classical topology is given. On the other hand, it is also shown that soft metrics are actually cone metrics. Cone metrics are already proven to be an unsuccessful attempt of generalizing metrics. These results clarify that most, if not all, properties of soft metric spaces could be directly imported from the related classical theory. The paper concludes with an application of the findings, i.e., a new soft fixed point theorem is stated and proven with the help of the obtained homemorphism.</p></abstract>

Finite soft-open sets: characterizations, operators and continuity

<abstract><p>In this paper, we present a novel family of soft sets named "finite soft-open sets". The purpose of investigating this kind of soft sets is to offer a new tool to structure topological concepts that are stronger than their existing counterparts produced by soft-open sets and their well-known extensions, as well as to provide an environment that preserves some topological characteristics that have been lost in the structures generated by celebrated extensions of soft-open sets, such as the distributive property of a soft union and intersection for soft closure and interior operators, respectively. We delve into a study of the properties of this family and explore its connections with other known generalizations of soft-open sets. We demonstrate that this family strictly lies between the families of soft-clopen and soft-open sets and derive under which conditions they are equivalent. One of the unique features of this family that we introduce is that it constitutes an infra soft topology and fails to be a supra soft topology. Then, we make use of this family to exhibit some operators in soft settings, i.e., soft $ fo $-interior, $ fo $-closure, $ fo $-boundary, and $ fo $-derived. In addition, we formulate three types of soft continuity and look at their main properties and how they behave under decomposition theorems. Transition of these types between realms of soft topologies and classical topologies is examined with the help of counterexamples. On this point, we bring to light the role of extended soft topologies to validate the properties of soft topologies by exploring them for classical topologies and vice-versa.</p></abstract>

Efficiency Evaluation of Series Connected DC/DC Converters in On-Grid Solar PV System

The grid-connected photovoltaic (PV) system is an important technology for renewable energy applications. It usually consists of three main stages: a PV module, a DC/DC converter and a DC/AC inverter. In these PV systems, Power Conditioning System should have high efficiency and low cost. In this paper, a classification of possible power converter topologies for PV panel integration is introduced. Out of different topologies of DC/DC converter, series connection of boost converters is more suitable converter for PV applications. In order to simulate the steady state efficiencies of the series connection of boost converters and the inverter non-ideal averaged model of these power converters is developed. Then the different elements and control of the (PV) system are implemented in Matlab/Simulink. Using loss analysis, the benefits and drawbacks of series connection of boost converters are shown. This analysis allowed the conception of an optimized architecture constituted by serialized power converters.

Neural-ABC: Neural Parametric Models for Articulated Body With Clothes.

In this paper, we introduce Neural-ABC, a novel parametric model based on neural implicit functions that can represent clothed human bodies with disentangled latent spaces for identity, clothing, shape, and pose. Traditional mesh-based representations struggle to represent articulated bodies with clothes due to the diversity of human body shapes and clothing styles, as well as the complexity of poses. Our proposed model provides a unified framework for parametric modeling, which can represent the identity, clothing, shape and pose of the clothed human body. Our proposed approach utilizes the power of neural implicit functions as the underlying representation and integrates well-designed structures to meet the necessary requirements. Specifically, we represent the underlying body as a signed distance function and clothing as an unsigned distance function, and they can be uniformly represented as unsigned distance fields. Different types of clothing do not require predefined topological structures or classifications, and can follow changes in the underlying body to fit the body. Additionally, we construct poses using a controllable articulated structure. The model is trained on both open and newly constructed datasets, and our decoupling strategy is carefully designed to ensure optimal performance. Our model excels at disentangling clothing and identity in different shape and poses while preserving the style of the clothing. We demonstrate that Neural-ABC fits new observations of different types of clothing. Compared to other state-of-the-art parametric models, Neural-ABC demonstrates powerful advantages in the reconstruction of clothed human bodies, as evidenced by fitting raw scans, depth maps and images. We show that the attributes of the fitted results can be further edited by adjusting their identities, clothing, shape and pose codes. The dataset and trained parametric model will be available at https://ustc3dv.github.io/NeuralABC/.

A pulser R&D for the HEPS booster bumper magnet

The High Energy Photon Source (HEPS) is a fourth generation photon source, including a storage ring, a booster ring and a Linac. Due to the small dynamic aperture of the storage ring, a novel on-axis swap-out injection scheme was chosen. Here, the 6GeV booster acts as an accumulating ring during that injection process. To extract 6 GeV beam from the booster before injection into the storage ring, four slow bumper magnets are applied to assist the extraction kicker to accomplish. The bumper pulse magnetic field waveform is a half-sine wave with 1ms pulse bottom width. Depending on the simulation and test, a classic LC resonance circuit topology with IGBT switching in series with fast recovery diodes is adopted. In addition, an energy recycle circuit and capacitor charging circuit are designed, to decrease power loss and reduce the influence on the output pulse current waveform during the capacitor re-charge process. A pulsed power supply prototype has been completed, and the testing results show that the bumper pulser can fully meet the all requirements of HEPS booster high energy extraction system.

Platoon Control of Connected Vehicles from a Networked Control Perspective: Literature Review, Component Modeling, and Controller Synthesis

The platooning of connected and automated vehicles has the potential to significantly benefit the road traffic, including enhancing highway safety, improving traffic capacity, and reducing fuel consumption. This paper presents a four-component analysis framework for platoon systems from a networked control perspective, including a literature review by network awareness, unified models of key components, and two application cases for controller synthesis. The networked control perspective naturally decomposes a platoon into four interrelated components, namely, 1) node dynamics (ND), 2) information flow topology (IFT), 3) formation geometry (FG), and 4) distributed controller (DC). The existing literature is categorized under this framework and analyzed according to the component features. The unified mathematical models are derived for platoons with linear dynamics and distributed controllers. As a case study, a distributed controller synthesis method is introduced for homogeneous platoons, which guarantees the internal stability in the presence of a broad class of topologies with/without uniform time-delays. The effectiveness is demonstrated by numerical simulations.

Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surface

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the "source-sink" form, that is, by singling out a dual attractor-repeller pair, consisting of the attracting and repelling sets for all other trajectories of the system. A choice of an attractor-repeller dual pair so that the space of orbits in their complement (the characteristic space of orbits) is connected paves the way for finding complete topological invariants of the dynamical system. In this way, in particular, several classification results for Morse-Smale systems were obtained. Thus, a complete topological classification of Morse-Smale 3-diffeomorphisms is essentially based on the existence of a connected characteristic space of orbits associated with the choice of a one-dimensional dual attractor-repeller pair. For Morse-Smale diffeomorphisms with heteroclinic points on surfaces, there are examples in which the characteristic spaces of orbits are disconnected in all cases. In this paper, we prove a criterion for the existence of a connected characteristic space of orbits for gradient-like (without heteroclinic points) diffeomorphisms on surfaces. This result implies, in particular, that any orientation-preserving diffeomorphism admits a connected characteristic space. For an orientable surface of any kind, we also construct an orientation-changing gradient-like diffeomorphism that does not have a connected characteristic space. On any non-orientable surface of any kind, we also construct a gradient-like diffeomorphism which does not admit a connected characteristic space.

Global Analysis of Riccati Quadratic Differential Systems

In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait.

Explicit Topology Optimization of Voronoi Foams.

Topology optimization can maximally leverage the high DOFs and mechanical potentiality of porous foams but faces challenges in adapting to free-form outer shapes, maintaining full connectivity between adjacent foam cells, and achieving high simulation accuracy. Utilizing the concept of Voronoi tessellation may help overcome the challenges owing to its distinguished properties on highly flexible topology, natural edge connectivity, and easy shape conforming. However, a variational optimization of the so-called Voronoi foams has not yet been fully explored. In addressing the issue, a concept of explicit topology optimization of open-cell Voronoi foams is proposed that can efficiently and reliably guide the foam's topology and geometry variations under critical physical and geometric requirements. Taking the site (or seed) positions and beam radii as the DOFs, we explore the differentiability of the open-cell Voronoi foams w.r.t. its seed locations, and propose a highly efficient local finite difference method to estimate the derivatives. During the gradient-based optimization, the foam topology can change freely, and some seeds may even be pushed out of shape, which greatly alleviates the challenges of prescribing a fixed underlying grid. The foam's mechanical property is also computed with a much-improved efficiency by an order of magnitude, in comparison with benchmark FEM, via a new material-aware numerical coarsening method on its highly heterogeneous density field counterpart. We show the improved performance of our Voronoi foam in comparison with classical topology optimization approaches and demonstrate its advantages in various settings.

Virtual Topologies for Populating Overhead Low-Voltage Broadband over Powerlines Topology Classes by Exploiting Neural Network Topology Generator Methodology (NNTGM) - Part 1: Theory

Based on a set of indicative overhead low voltage broadband over power lines (OV LV BPL) topologies of respective OV LV BPL topology classes, Neural Network Topology Generator Methodology (NNTGM) is theoretically proposed in this paper, so that its generated OV LV BPL topologies (NNTGM OV LV BPL topologies) can populate the OV LV BPL topology classes. Given the indicative topology of the OV LV BPL topology class, the NNTGM OV LV BPL topologies can be statistically familiar with the corresponding indicative OV LV BPL topology in terms of the theoretical channel attenuation behavior. Actually, NNTGM is based on the reverse procedure of the Neural Network Identification Methodology for the distribution line and branch Line Length Approximation (NNIM-LLA); say, NNTGM generates theoretical channel attenuation behaviors given the number of branches, the distribution line lengths and the branch line lengths, when appropriate NNTGM default operation settings are assumed. The statistical familiarity between the examined indicative OV LV BPL topology and an NNTGM OV LV BPL topology is examined after the application of appropriate channel attenuation metrics. On the basis of the channel attenuation metrics, class maps are theoretically defined so that the relative positions of indicative OV LV BPL topologies and their respective NNTGM OV LV BPL topologies (NNTGM virtual topologies) can be further studied.

Big Data and Neural Networks in Smart Grid - Part 2: The Impact of Piecewise Monotonic Data Approximation Methods on the Performance of Neural Network Identification Methodology for the Distribution Line and Branch Line Length Approximation of Overhead Low-Voltage Broadband over Powerlines Networks

Τhe impact of measurement differences that follow continuous uniform distributions (CUDs) of different intensities on the performance of the Neural Network Identification Methodology for the distribution line and branch Line Length Approximation (NNIM-LLA) of the overhead low-voltage broadband over powerlines (OV LV BPL) topologies has been assessed in [1]. When the αCUD values of the applied CUD measurement differences remain low and below 5dB, NNIM-LLA may internally and satisfactorily cope with the CUD measurement differences. However, when the αCUD values of CUD measurement differences exceed approximately 5dB, external countermeasure techniques against the measurement differences are required to be applied to the contaminated data prior to their handling by NNIM-LLA. In this companion paper, the impact of piecewise monotonic data approximation methods, such as L1PMA and L2WPMA of the literature, on the performance of NNIM-LLA of OV LV BPL topologies is assessed when CUD measurement differences of various αCUD values are applied. The key findings that are going to be discussed in this companion paper are: (i) The crucial role of the applied numbers of monotonic sections of the L1PMA and L2WPMA for the overall performance improvement of NNIM-LLA approximations as well as the dependence of the applied numbers of monotonic sections on the complexity of the examined OV LV BPL topology classes; and (ii) the performance comparison of the piecewise monotonic data approximation methods of this paper against the one of more elaborated versions of the default operation settings in order to reveal the most suitable countermeasure technique against the CUD measurement differences in OV LV BPL topologies.

Feedback Linearization Control for a Class of Transformerless Common-Ground VSI

As is well known in practice, conventional transformerless inverters may suffer the risk of high-frequency leakage currents between the photovoltaic (PV) array or battery energy storage systems (BESS) and the ground due to the stray capacitances. This operational condition can generate several electrical problems. Recently, we proposed in [1] a new method for designing transformerless topologies of single-phase voltage source inverters (VSI) with common ground derived from four classical DC-DC converters. As a sequence of that previous work, a novel control technique based on a feedback linearization approach is proposed to control that class of transformerless VSI with common ground in this paper. Unlike the classical local linearization techniques commonly used in the technical literature, this approach allows to obtain large-signal linear equations, by means of the feedback linearization of the non-linear terms. Real-time simulation results using Hardware-in-the-Loop allow validation of the proposed discrete control technique applied to this class of VSI topologies.

The existence of multiple topologically distinct solutions to $\sigma_{2,p}$-energy

Let ${\An} \subset \R^n$ be a bounded Lipschitz domain and consider the $\sigmap$-energy functional \begin{equation*} {{\mathbb F}_{\sigmap}}[u; {\An}] := \int_{\An} \big|{\wedge}^2 \nabla u\big|^p dx, \end{equation*} with $p\in\mathopen]1, \infty]$ over the space of measure preserving maps \begin{equation*} {\mathcal A}_p(\An) =\big\{u \in W^{1,2p}\big(\An, \R^n\big) : u|_{\partial \An} = {x},\ \det \nabla u =1 \mbox{ for ${\mathcal L}^n$-a.e.\ in $\An$} \big\}. \end{equation*} In this article we address the question of multiplicity {\it versus} uniqueness for {\it extremals} and {\it strong} local minimizers of the $\sigmap$-energy funcional $\mathbb F_{\sigmap}[\cdot; {\An}]$ in ${\mathcal A}_p({\An})$. We use a topological class of maps referred to as {\it generalised} twists and examine them in connection with the Euler-Lagrange equations associated with $\sigmap$-energy functional over ${\mathcal A}_p({\An})$. Most notably, we prove the existence of a countably infinite of topologically distinct twisting solutions to the later system in all {\it even} dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group ${\rm SO}(n)$. In sharp contrast in {\it odd} dimensions the only solution is the map $u\equiv x$. The result relies on a careful analysis of the {\it full} versus the {\it restricted} Euler-Lagrange equations. Indeed, an analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.

Topological equivalence classification and enumeration of n-input linearly separable Boolean functions

A perceptron, as an “artificial neuron”, plays a crucial role in the research of neural networks. As a mathematical model of linearly separable Boolean functions, perceptron neurons can realize any nonlinearly separable Boolean function. Linearly separable Boolean functions can be divided into a smaller number of the same functional classes. These functional classes are equivalent to the core basic units (AND, NOT, OR, and other components) that make up the circuit. Therefore, the research on the classification problem of linearly separable Boolean functions is very important. In this study, a novel algorithm, called a topological equivalence classification algorithm, is proposed for classifying balanced linearly separable Boolean functions. By the proposed algorithm, the total number of the topological equivalence classes and the number of balanced linearly separable Boolean functions in each of the topological equivalence classes are obtained. According to the one-to-one correspondence between n-input balanced linearly separable Boolean functions and n-1-input linearly separable Boolean functions, we get the topological classification of linearly separable Boolean functions. As for n=9, the accurate results of classification and counting the number of linearly separable Boolean functions are obtained: There are 53,063,448 topological equivalence classes, and the total number of linearly separable Boolean functions is 144,380,202,286,068,288.

Thermodynamic topology and photon spheres in the hyperscaling violating black holes

It was shown that a standard ring of light can be imagined outside the event horizon for stationary rotating four-dimensional black holes with axial symmetry using the topological method (Cunha, 2020; Wei and Shao-Wen, 2020). Based on this concept, in this paper, we investigate the topological charge and the conditions of existence of the photon sphere (PS) for a hyperscaling violation (HSV) black hole with various values of the parameters of this model. Then, after carrying out a detailed analysis, we show the conventional topological classes viz Q=−1 for the mentioned black hole and Q=0 for the naked singularities. Also, we propose a new topological class for naked singularities (Q=+1) with respect toz≥1. We also determined thatz≥2, it either shows a naked singularity form with total topological charge+1or has no solution. Therefore, we have the black hole solution only in1≤z<2. Then, we will use two different methods, namely the temperature (Duan’s topological current Φ-mapping theory) and the generalized Helmholtz free energy method, to study the topological classes of our black hole. By considering the black hole mentioned, we discuss the critical and zero points (topological charges and topological numbers) for different parameters of hyperscaling violating black holes, such as (z,θ¯) and other free parameters, and study their thermodynamic topology. We observe that for a given value of the parameters 1≤z<2, θ¯, and other free parameters, there exist two total topological charges (Qtotal=−1,0) with the same phase structure for the T method and total topological numbers (W=+1) for the generalized Helmholtz free energy method. Additionally, we summarize the results for each study as photon sphere, temperature, and generalized Helmholtz free energy in some figures and tables. Finally, we compare our findings with other related studies in the literature.

MORSE AND CODIMENSION 1 FUNCTIONS ON THE PROJECTIVE PLANE

The topological properties of functions on two-dimensional spaces are described using an invariant called the Reeb graph. In the case of Morse functions on closed oriented two-dimensional manifolds, this serves as a complete topological invariant for homotopy equivalence. For non-oriented two-dimensional manifolds and manifolds with boundaries, additional information is required to construct a complete topological invariant. If a linear order is imposed on the set of vertices, it becomes a complete topological invariant with respect to topological equivalence. The research explores the topological classification of Morse functions and functions of corank 1 on the projective plane. In these cases, the Reeb graph is an oriented rooted tree. An analytical expression is derived for the number of oriented Reeb graphs with a root. For functions of corank 1 and a fixed number of critical points, all possible Reeb graphs are constructed.

Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives

In this article, we investigate the thermodynamic topology of some black holes, namely AdS Reissner Nordstrom (R–N), AdS Einstein–Gauss–Bonnet (EGB), and AdS Einstein-power-Yang–Mills (EPYM), from different frameworks: bulk-boundary (BB) and restricted phase space (RPS). Using the generalized off-shell Helmholtz free energy method, we calculate the thermodynamic topology of the selected black holes in each space separately and determine their topological classifications. We show that the addition of GB terms, dimensions, and other factors do not affect the topological classes of black holes in both spaces. The calculations and plots indicate that the AdS R–N and AdS EGB black holes show similar behavior and their topological numbers sets in both spaces, i.e., BB and RPS, are similar and equal to (W=+1). However, AdS EPYM black holes show an interesting behavior. In addition to BBT and RPS, we also consider the extended phase space thermodynamics (EPST) and evaluate the thermodynamic topology for AdS EPYM black hole. The changing (r−τ) in both spaces shows similar behavior. Also, the topological number and the total topological numbers for this black hole in the BB, RPS and EPS thermodynamics are completely same, i,e., (ωBBT=ωRPS=ωEPST=+1,−1) or WBBT=WRPS=WEPST=0. An important point is that the Einstein–Yang–Mills black hole has thermodynamic topology equivalence in three spaces. The present result may be due to the non-linear YM charge parameter and the difference between the gauge and gravity corrections in the above black holes.

Localic separation and the duality between closedness and fittedness

There are a number of localic separation axioms which are roughly analogous to the T1-axiom from classical topology. For instance, besides the well-known subfitness and fitness, there are also Rosický–Šmarda's T1-locales, totally unordered locales and, more categorically, the recently introduced F-separated locales (i.e., those with a fitted diagonal) — a property strictly weaker than fitness.It has recently been shown that the strong Hausdorff property and F-separatedness are in a certain sense dual to each other. In this paper, we provide further instances of this duality — e.g., we introduce a new first-order separation property which is to F-separatedness as the Johnstone–Sun-shu-Hao–Paseka–Šmarda conservative Hausdorff axiom is to the strong Hausdorff property, and which can be of independent interest. Using this, we tie up the loose ends of the theory by establishing all the possible implications between these properties and other T1-type axioms occurring in the literature. In particular, we show that the strong Hausdorff property does not imply F-separatedness, a question which remained open and shows a remarkable difference with its counterpart in the category of topological spaces.

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403–453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.

Hierarchical Cyclic Pursuit: Algebraic Curves Containing the Laplacian Spectra

The paper addresses the problem of multiagent communication in networks with regular directed ring structure. These can be viewed as hierarchical extensions of the classical cyclic pursuit topology. We show that the spectra of the corresponding Laplacian matrices allow exact localization on the complex plane. Furthermore, we derive a general form of the characteristic polynomial of such matrices, analyze the algebraic curves its roots belong to, and propose a way to obtain their closed-form equations. In combination with frequency domain consensus criteria for high-order SISO linear agents, these curves enable one to analyze the feasibility of consensus in networks with varying number of agents.