Topological Tools Research Articles

Vitreous Carbon, Geometry and Topology: A Hollistic Approach.

Glass-like carbon (GLC) is a complex structure with astonishing properties: isotropic structure, low density and chemical robustness. Despite the expanded efforts to understand the structure, it remains little known. We review the different models and a physical route (pulsed laser deposition) based on a well controlled annealing of the native 2D/3D amorphous films. The many models all have compromises: neither all bad nor entirely satisfactory. Properties are understood in a single framework given by topological and geometrical properties. To do this, we present the basic tools of topology and geometry at a ground level for 2D surface, graphene being the best candidate to do this. With this in mind, special attention is paid to the hyperbolic geometry giving birth to triply periodic minimal surfaces. Such surfaces are the basic tools to understand the GLC network architecture. Using two theorems (the classification and the uniformisation), most of the GLC properties can be tackled at least at a heuristic level. All the properties presented can be extended to 2D materials. It is hoped that some researchers may find it useful for their experiments.

Topological field theory approach to intermediate statistics

Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are characterized by the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szeg"{o}’s limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of U(N)U(N) Chern-Simons theory on S^3S3, and the SFF of this ensemble is proportional to the HOMFLY invariant of (2n,2)(2n,2)-torus links with one component in the fundamental and one in the antifundamental representation. This is one example of a large class of ensembles with intermediate statistics arising from topological field and string theories. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.

Automatic Overcurrent Protection Coordination after Distribution Network Reconfiguration Based on Peer-To-Peer Communication

Electrical power systems represent a fundamental part of society, and their efficient operations are of vital importance for social and economic development. Power systems have been designed to withstand interruptions under already provided safety and quality principles; however, there are some extreme and not so frequent events that could represent inconveniences for the correct operation of the entire system. For this reason, in recent years the term resilience, which serves to describe the capacity of a system to recover from an unwanted event, has been analyzed on planning, operation and remedial actions. This work is focused on the implementation of a topological reconfiguration tool, which is oriented to change the structure of primary feeders based on changing the status of switchgears. Once the distribution network has been reconfigured, an algorithm of protection coordination is executed based on communication peer-to-peer between Matlab and PowerFactory, which develops an adaptive calculation to determine the current setting and the time multiplier setting. The reconfiguration and coordination protection algorithms could be implemented and evaluated on different distribution networks, areas and locations.

A computational framework for connection matrix theory

The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals: 1) We cast the connection matrix theory into appropriate categorical, homotopy-theoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions. 2) We describe an algorithm for this computation based on algebraic-discrete Morse theory.

Shape-based Feature Engineering for Solar Flare Prediction

Solar flares are caused by magnetic eruptions in active regions (ARs) on the surface of the sun. These events can have significant impacts on human activity, many of which can be mitigated with enough advance warning from good forecasts. To date, machine learning-based flare-prediction methods have employed physics-based attributes of the AR images as features; more recently, there has been some work that uses features deduced automatically by deep learning methods (such as convolutional neural networks). We describe a suite of novel shape-based features extracted from magnetogram images of the Sun using the tools of computational topology and computational geometry. We evaluate these features in the context of a multi-layer perceptron (MLP) neural network and compare their performance against the traditional physics-based attributes. We show that these abstract shape-based features outperform the features chosen by the human experts, and that a combination of the two feature sets improves the forecasting capability even further.

Cooperativity and Anticooperativity in Ion-Water Interactions: Implications for the Aqueous Solvation of Ions.

Non-additive effects in hydrogen bonds (HB) take place as a consequence of electronic charge transfers. Therefore, it is natural to expect cooperativity and anticooperativity in ion-water interactions. Nevertheless, investigations on this matter are scarce. This paper addresses the interactions of (i) the cations Li+ , Na+ , K+ , Be2+ , Mg2+ , and Ca2+ together with (ii) the anions F- , Cl- , Br- , NO3- and SO42- with water clusters (H2 O)n , n=1-8, and the effects of these ions on the HBs within the complete molecular adducts. We used quantum chemical topology tools, specifically the quantum theory of atoms in molecules and the interacting quantum atoms energy partition to investigate non-additive effects among the interactions studied herein. Our results show a decrease on the interaction energy between ions and the first neighbouring water molecules with an increment of the coordination number. We also found strong cooperative effects in the interplay between HBs and ion-dipole interactions within the studied systems. Such cooperativity affects considerably the interactions among ions with their first and second solvation shells in aqueous environments. Overall, we believe this article provides valuable information about how ion-dipole contacts interact with each other and how they relate to other interactions, such as HBs, in the framework of non-additive effects in aqueous media.

On a class of controlled differential variational inequalities

By using various analytical and topological tools, weak sharp solutions are investigated for a class of controlled differential variational inequality involving convex functionals defined by path-independent curvilinear integrals. In order to illustrate the main results, an application is also provided.

Risk assessment of recent high-interest conjunctions

There have been several recent high-profile space events, notably the conjunctions of the Aeolus and Starlink spacecraft (both maneuverable) and the IRAS and GGSE-4 spacecraft (both dead). Here, we examine the timeline, activities, and risks associated with each of these conjunction events. To accomplish this, we fused observations from the Space Surveillance Network (SSN) to produce a sequence of high-accuracy orbit solutions and used that sequence to generate three-dimensional, dynamic progressions of each conjunction's miss distance, collision probability and its relation to the dilution threshold. Probability of Collision (Pc) is an essential tool for assessing conjunction threats. The Pc Topology tool characterizes the evolution of a conjunction's miss distance and collision probability, both of which are critical inputs to the maneuver avoidance decision-making process and exposing SSA deficiencies. We show the effect of combined hardbody radius (CHBR) on Pc as well as modeling with different shapes. Additionally, the Pc Topology tool reveals the calculation's placement in/out of the dilution region.The European Space Agency commanded the Aeolus spacecraft to perform an emergency maneuver to avoid this high-probability collision hazard. Our results indicate that this maneuver was warranted based upon ESA's decision threshold criteria and the Pc Topology trending analysis. While both collision events fortunately did not occur, we estimated the potential debris fields that these events may have generated had they occurred. Realizing that large debris production might be a factor in maneuver decision making, we then assess the impact of these fragmentation fields upon estimated operator workload and subsequent collision risk.

Ricci Curvature, Reeb Flows and Contact 3-Manifolds

Given a contact 3-manifold we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.

Persistent homology in cosmic shear: Constraining parameters with topological data analysis

In recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods such as peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same data set. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING-450 data: We extract the Betti functions from a suite of N-body simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainties. When comparing our results, we recover the input cosmology and achieve a constraining power on S8 ≡ σ8Ωm/0.3 that is 3% tighter than that on peak count statistics. Performing the same analysis on 100 deg2 of Euclid-like simulations, we are able to improve the constraints on S8 and Ωm by 19% and 12%, respectively, while breaking some of the degeneracy between S8 and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools for constraining cosmological parameters with lensing data.

Anisotropic Singular Neumann Equations with Unbalanced Growth

We consider a nonlinear parametric Neumann problem driven by the anisotropic (p, q)-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter λ varies. We also show the existence of minimal positive solutions u_{lambda }^{*} and determine the monotonicity and continuity properties of the map lambda mapsto u_{lambda }^{*}.

Design of additive manufactured passive heat sinks for electronics

Abstract During the last decades, in industrial applications the integration of electronic components shifted its focus on reliability. In fact, the high thermal power generated by electronic systems can cause system failure and even irreversible damage, due to their ongoing miniaturization and complexity. This work aims to explore solutions for the optimal design of passive heat sinks to be fabricated by additive manufacturing. The methodology involves the use of CAE and thermal topological optimization tools, to analyse the thermal loads and the heat sink performance, and consequently to optimize the heat sink design. A simple case study allows to compare three different geometrical heat sink configurations, corresponding to the optimal solutions for three different sets of constraints, namely only volume constraints, laser-powder bed process design and internal heat generation. The predominant role of heat conduction is demonstrated and significant weight reduction and high efficiency are combined in topology optimized passive heat sinks to be fabricated by AM.

Eco-green machining of superalloy A286: Assessment of tool wear morphology and surface topology

Unstable crude oil derivatives are among the main substances added to the mineral-based lubricants (MBL). However, the high purification and disposal costs of such lubricants and consequent environmental difficulties are the major drawbacks of cutting fluid consumption. The lower use of MBL or replacing them with less harmful ones, in principle, vegetable-based lubricants (VBL) are the prime importance of achieving less operating costs and pollution. Furthermore, the use of VBLs is much more recommended if machinability aspects, including tool life, surface, and edge quality, would also be improved. Therefore, in the present study, cutting parameters and lubricant types were used as the experimental variables in the MQL-turning of A286 superalloy. A286 is classified as a difficult-to-cut material with a wide range of applications in the energy and turbine industries. No experimental study was found on the machining of A286 under separate cooling and lubrication with two MQL systems simultaneously. The effects of cutting parameters on both average surface roughness Ra and tool wear morphology were evaluated. Experimental results denoted that the use of VBLs led to better surface topology. On the other hand, the tendency of the built-up layer (BUL) and built-up edge (BUE) phenomenon were intensified when MBL was used. On the word, better tool life is also expected when using VBL. This observation revealed that the use of VBL not only tends to improve productivity and environmental safety aspects but also machinability aspects; in principle, better surface topology and longer tool life are also expected.

Fluorination Effects in XPhos Gold(I) Fluorothiolates

Gold phosphine derivatives such as thiolates have been recently proposed as catalysts or catalyst precursors. The relevance of the supramolecular environment on the fine-tuning of the catalytical activity on these compounds incentivizes the use of tools that are convenient to characterize in detail the non-covalent landscape of the systems. Herein, we show the molecular and supramolecular diversity caused by the changes in the fluorination pattern in a family of new XPhos goldfluorothiolate derivatives. Furthermore, we studied the supramolecular interactions around the Au centers using quantum chemical topology tools, in particular the quantum theory of atoms in molecules (QTAIM) and the non-covalent interaction index. Our results give detailed insights into the fluorination effects on the strength of intramolecular and intermolecular interactions in these systems. We have also used QTAIM delocalization indexes to define a novel hapticity indicator. Finally, we assessed the trans influence of the fluorothiolates on the phosphine in terms of the change in the δ 31P-NMR. These results show the feasibility of the use of fluorination in the modulation of the electronic properties of Buchwald phosphine gold(I) compounds, and thereby its potential catalytic activity.

C1 density of stable ergodicity

We prove a C1 version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving Cr diffeomorphisms, r>1, the stably ergodic ones are C1-dense.To establish these results, we develop new perturbation tools for the C1 topology: linearization of horseshoes while preserving entropy, and creation of “superblenders” from hyperbolic sets with large entropy.

A practical redesign method for functional additive manufacturing

Additive Manufacturing (AM) liberates a lot for the hands of designers to define complex geometry and avoid many traditional manufacturing constraints. A set of advanced topological optimization and lattice structure generation tools are emerging to support the design for AM, especially in lightweight design. However, these tools are still in development and not so stable with quite limited operational maturity, which forms a barrier for practitioners to explore the design freedom brought by AM since intensive AM knowledge and expertized computation and numerical simulation skills are required. To solve this problem with an aim to enable an ordinary mechanical designer with basic AM knowledge and numerical modeling skills to design for AM at an operational level, this paper introduces a practical redesign method to decrease the height of the threshold of design for AM and realize functional design solutions. This method combines both subtractive and generative methods to define material distribution in the design space with consideration of AM manufacturability analysis and functional requirements. It can be implemented via a set of existing ordinary CAD tools, which implies its potential for a wide application. A design case on a redesign of a hydraulic manifold component is presented for method demonstration. The result shows that this method is feasible and practical for functional AM design with less AM knowledge on complex geometric or computational operations.

Continuum Models and Morse Theory in the Simulation, Design, and Evaluation of Magnetic Skyrmion Devices

Nanomagnetic devices such as computer gates and memory devices based on magnetic skyrmions are close to becoming a reality. In this letter, we will explore the highly nonconvex nanomagnetic energy landscape in order to draw conclusions about the complexity of magnetic phenomena. Morse-theoretic arguments show that, in a bounded energy interval, the number of critical points of the energy functional grows exponentially. To show this, we exploit a hierarchy of models for the design of nanomagnetic devices to provide a solid foundation for the introduction of topological tools. To reason in terms of lattice models, one must make a distinction between two types of lattices: the quantum mechanical model of the actual physical lattice and the lattice model that can be associated with a discretization of a continuum model of the physics. By focusing on 1) the implications of Morse theory applied to lattice systems arising from the discretization of the continuum models and 2) the notion of “topological frustration,” we provide a framework for understanding complexity in the context of nanomagnetic systems. We conclude with some suggestions for making the analysis more qualitative.

Topological Dynamic Matter.

The principles of topology in condensed matter physics have expanded to areas such as photonics, acoustics, electronics, and mechanics. Their extension to dynamic (soft) matter could enable the control and design of topological thermodynamic (micro)states and nonreciprocal dynamics, potentially leading to paradigmatic applications in molecular and thermal waveguiding, logics, and energy management. This Perspective explores distinct topological concepts for dynamic matter and prospective function. Topological tools are exemplified and discussed for the study of nonlocal order parameters or invariants in dynamic molecular matter, toward the engineering of assemblies, reactions, and system chemistry with unconventional global properties-a scope which has the potential to push the frontiers of physical chemistry and transform chemical topology from form to function.

Topological optimization of the “Earring” element

In recent decades, computer technologies and software such as computer-aided design (CAD) have been actively developing. Thanks to this, modern machine-building enterprises increase the reliability and quality of their products while reducing their weight and complexity of manufacturing. In order to meet numerous requirements, leading companies are increasingly using topological optimization tools at various design stages. The use of this method for effective product design is growing rapidly, due to the continuously increasing computing power of computers and software capabilities. Modern software for topological optimization allows to design the shape of the part from scratch, setting only the conditions for fixing and touching surfaces, and also allows to improve existing structures by reducing their weight based on the set restrictions (equal strength with the original part, equal deformation, preservation of natural frequency, etc.). However, the result of topological optimization is often a complex spatial structure. Using the example of the Earring element, a topological optimization of the structure was performed to ensure that the strength requirements were met and that the minimum mass was obtained. As a result of optimization, the design of a reduced weight compared to the prototype is obtained. The results of the verification calculation showed the sufficiency of the values of the strength reserves of the final design of the Earring.

Understanding flow features in drying droplets via Euler characteristic surfaces—A topological tool

In this paper, we propose a mathematical picture of flow in a drying multiphase droplet. The system studied consists of a suspension of microscopic polystyrene beads in water. The time development of the drying process is described by defining the “Euler characteristic surface,” which provides a multiscale topological map of this dynamical system. A novel method is adopted to analyze the images extracted from experimental video sequences. Experimental image data are converted to binary data through appropriate Gaussian filters and optimal thresholding and analyzed using the Euler characteristic determined on a hexagonal lattice. In order to do a multiscale analysis of the extracted image, we introduce the concept of Euler characteristic at a specific scale r > 0. This multiscale time evolution of the connectivity information on aggregates of polysterene beads in water is summarized in a Euler characteristic surface and, subsequently, in a Euler characteristic level curve plot. We introduce a metric between Euler characteristic surfaces as a possible similarity measure between two flow situations. The constructions proposed by us are used to interpret flow patterns (and their stability) generated on the upper surface of the drying droplet interface. The philosophy behind the topological tools developed in this work is to produce low-dimensional signatures of dynamical systems, which may be used to efficiently summarize and distinguish topological information in various types of flow situations.