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For balloon-borne telescopes, the primary mirror is the most important optical element, but designing a primary mirror with an excellent overall performance is a challenge. To comprehensively consider the contradictory objectives of the root mean square (RMS) surface error under gravity in the X and Z directions, the mass and fundamental frequency of the primary mirror, a parametric primary mirror design using the compromise programming method based on topology optimization is proposed. The parametric design of the compromise programming method based on topology optimization is used to find the optimal solution for X-direction RMS (RMSx), Z-direction RMS (RMSz), mass, and fundamental frequency. Compared with the initial primary mirror structure designed according to traditional experience, the overall performance is improved. Results show that the respective mass of the primary mirror, the RMSx and the RMSz decreased by 8.5%, 14.3% and 10.5% compared to those before optimization. Comprehensive consideration can prove the effectiveness of parametric design based on the topology optimization of the primary mirror. This method provides a reference for the design of other primary mirrors for balloon-borne telescope and space cameras. [ABSTRACT FROM AUTHOR]

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In the paper we generalize the singular cubical homology theory of digraphs and introduce a collection of cell singular homologies that are parametrized by a natural parameter of the category of digraphs. The natural parameter corresponds to the "length" of a directed line digraph that corresponds to the unit segment in algebraic topology. The introduced earlier singular cubical homology is a particular case of this collection that corresponds to the line digraph with one arrow. We describe relations between these theories for different parameters given by a natural number and study their basic properties. In particular, we prove the functoriality and homotopy invariance of these theories and compare the theory with the path homology theory that was introduced in our previous papers. Then we describe the transfer of the cell singular homology theories to the categories of graphs, quivers, and multigraphs. [ABSTRACT FROM AUTHOR]

Models of ceramic-graphene nanocomposites are used to study how the manufacturing process-dependent arrangement of reduced graphene oxide (rGO) inclusions governs nano-crack network development. The work builds upon recent studies of such composites where a novel combinatorial approach was used to investigate the effect of rGO arrangements on electrical conductivity and porosity. This approach considers explicitly the discrete structure of the composite and represents it as a collection of entities of different dimensions - grains, grain boundaries, triple junctions, and quadruple points. Here, the combinatorial approach is developed further by considering the effects of rGO agglomerations, stress concentrators and adhesion energies on intergranular cracking. The results show that the fracture networks can be effectively controlled by the local ordering of rGO inclusions to allow for a concurrent increase in the strength and conductivity of the ceramic composites. It is shown that the ratio of local stress concentrators related to rGO inclusions and cracks is the most significant factor affecting the nano-crack network topology. The local spatial arrangement of rGO inclusions becomes an effective tool for controlling nano-crack network topology only when this ratio approaches one. It is anticipated that these results will inform future design of toughness-enhanced composites. • Effect of rGO spatial distribution on composite cracking revealed by discrete methodology. • Fracture network controlled by the ratio of local stress concentrators due to inclusions and cracks. • Diversity of network topologies achieved by increasing the number of rGO agglomerations. • The effect of the initial rGO fraction on the fracture network topology is not monotonous. [ABSTRACT FROM AUTHOR]

Abstract Graph distance and graph embedding are two fundamental tasks in graph mining. For graph distance, determining the structural dissimilarity between networks is an ill-defined problem, as there is no canonical way to compare two networks. Indeed, many of the existing approaches for network comparison differ in their heuristics, efficiency, interpretability, and theoretical soundness. Thus, having a notion of distance that is built on theoretically robust first principles and that is interpretable with respect to features ubiquitous in complex networks would allow for a meaningful comparison between different networks. For graph embedding, many of the popular methods are stochastic and depend on black-box models such as deep networks. Regardless of their high performance, this makes their results difficult to analyze which hinders their usefulness in the development of a coherent theory of complex networks. Here we rely on the theory of the length spectrum function from algebraic topology, and its relationship to the non-backtracking cycles of a graph, in order to introduce two new techniques: Non-Backtracking Spectral Distance (NBD) for measuring the distance between undirected, unweighted graphs, and Non-Backtracking Embedding Dimensions (NBED) for finding a graph embedding in low-dimensional space. Both techniques are interpretable in terms of features of complex networks such as presence of hubs, triangles, and communities. We showcase the ability of NBD to discriminate between networks in both real and synthetic data sets, as well as the potential of NBED to perform anomaly detection. By taking a topological interpretation of non-backtracking cycles, this work presents a novel application of topological data analysis to the study of complex networks.

In the present work, an algorithm for ngerprints verication based on an application of the algebraic topology is presented. Specically, we propose a representation of impressions as simplicial complexes and the denition of local structures based on local ltrations ordering from the complexes. This ltrations are determined by neighboring minutiae. It is also proposed the extraction of a set of features based on the analysis of the homology variation in these ltrations. The features combine information about the quantity and connectivity of papillary ridges in the local structures. In addition, a matching method based on the extracted topological information is presented. This paper shows that this information is discriminative and can be used in combination with classic geometric features to improve the description of local structures of the impressions and the accuracy in the comparison.

The story of the friendship and collaboration between Vladimir Abramovich Rokhlin and the Nizhny Novgorod mathematician Dmitry Andreevich Gudkov during the last period of Rokhlin's mathematical biography, when he worked in the topology of real algebraic varieties, in which he obtained remarkable results. The paper is based on the correspondence of 1971–1982 preserved in Gudkov's archive containing 15 letters by V. A.Rokhlin and 8 letters by D. A. Gudkov. [ABSTRACT FROM AUTHOR]

Most robotic applications demand lightweight and high-speed manipulators for considerably reducing the consumed power and achieving high production rates. The two ways for seeking such high-speed arms are; applying advanced control algorithms and/or performing an extensive optimization of the arm structure itself. Therefore, the topology optimization technique is proposed here for obtaining an optimal robot arm design from both structure and control viewpoints. Results of some researches, that have been previously accomplished by size and shape optimization, were encouraging enough to extend and propose this optimization approach. The method of moving asymptotes (MMA) as an optimization algorithm, the finite element analysis (FEA) by ANSYS, and the time-optimal control method are integrated to gain an optimum design capable of attaining the minimum traveling time. The proposed methodology focuses on performing different comparisons between the proposed optimum topological designs and their initial designs for different robot arms' sizes and materials. It also distinguishes between the proposed optimum design and the previously achieved one by size optimization under the same operational conditions. Therefore, the significance of the proposed technique is emphasized. It shows that the traveling time is reduced by 44.8%, while the previous work only achieved 23.5%. In addition, the mass is reduced to nearly half of its initial value, taking into account the air damping as the real case in all terrestrial applications. [ABSTRACT FROM AUTHOR]

Both spectral sequences and persistent homology are tools in algebraic topology defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. A recent work has shown the details of the relation between both concepts. Moreover, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are also related, generalizing results valid in the case of filtrations over Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a birth-death descriptor and a new invariant providing information on multifiltrations. This new invariant, in some cases, is able to provide more information than the rank invariant. We show some applications of our algorithms to spaces of infinite type via the effective homology technique, where the performance has also been improved by means of discrete vector fields. [ABSTRACT FROM AUTHOR]

Flat metalenses have attracted attention due to an increasing demand for compact electromagnetic devices. For such applications, broadband metalenses are highly desirable; however, conventional metalenses show relatively narrow band operation. Here, we propose a design method of free-form metalenses using topology optimization to operate with enhanced bandwidths. In contrast with preceding reports of topology optimization methods for metalenses, we developed a topology optimization method based on the time domain formulation to deal with broadband frequencies simultaneously. For this purpose, a group delay of optical pulses in the time domain, which is equivalent to the broadband phase matching condition in the frequency domain, is employed in the objective function. A level set based topology optimization method is applied to obtain a clear optimal configuration. To demonstrate the effectiveness of the proposed method, we provide design examples of metalens unit cells at millimeter frequency. We confirm that optimized unit cells of metalenses show superior performance compared to the conventional unit cells for both transmittance efficiency and phase error in broadband wavelength. [ABSTRACT FROM AUTHOR]

In this paper, we study some algebraic topology aspects of Stringc structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of Spinc(n). We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1-40] to correspond to Stringc structures of various levels and give vanishing results for them. [ABSTRACT FROM AUTHOR]

Cloaking structure designs have been widely performed in a form of metamaterial development. However, physics-based design approach makes it difficult to define complex material profiles. This study proposes an alternative design method for multi-directional cloak design via topology optimization method using a dielectric material. The phase field design method, which uses the reaction-diffusion equations combined with the double well potentials as the update scheme for design variables, is employed for systematic derivation of a topological configuration of the cloaking structure. In addition, the concept of symmetric design sensitivity mapping is introduced for multi-directional cloaking effect. Design results of four-, six- and eight-directional cloak are presented, and experimental verification was performed for six-directional cloak using the fabricated structure through 3D printing technology. [ABSTRACT FROM AUTHOR]

Velocity distribution plays an important role in the performance of microchannel devices. However, traditional microchannel plate design processes are based on experience with the existing models, which limits the ability to achieve optimal performance. Here, topology optimization is introduced into microchannel plate design, achieving a new microchannel plate with lower power dissipation and more uniform velocity distribution. A detailed design procedure was performed and an ANSYS Fluent analysis was carried out to further confirm the accuracy. [ABSTRACT FROM AUTHOR]

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones C 1 , C 2 , their minimal tensor product is the cone generated by products of the form x 1 ⊗ x 2 , where x 1 ∈ C 1 and x 2 ∈ C 2 , while their maximal tensor product is the set of tensors that are positive under all product functionals φ 1 ⊗ φ 2 , where φ 1 | C 1 ⩾ 0 and φ 2 | C 2 ⩾ 0 . Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled. [ABSTRACT FROM AUTHOR]

Amphidrome refers to an oceanic point where the amplitude of one harmonic constituent of the tidal system is zero, and the phase of the harmonic constituent is undetermined. The concept of amphidrome can also be used in climatological studies because of the existence of annual amphidromes, points of zero amplitude, and ill‐defined phase of annual cycle. This study investigated the global atmospheric geopotential height from the ERA‐interim to identify annual amphidromes at different isobaric altitudes. Many well‐defined annual amphidromes were identified. These amphidromic points appear always as twins with different rotating directions with respect to phases. This phenomenon can be explained mathematically with the basic theory of spherical algebraic topology, suggesting amphidrome twins as a common feature for periodic variables in the atmosphere. Due to the spatial continuity of atmospheric parameters, amphidromes at different isobaric altitudes can be concatenated into amphidromic lines. Amphidromic lines have a three‐dimensional structure with both clockwise and anticlockwise vertical branches connected by one or two horizontal components. Particularly, some amphidromic lines in the atmosphere can be closed loops when they are connected by two horizontal components. The rotary phases around the amphidromic loops are similar to the magnetic induction lines around a closed electromagnetic coil. Plain Language Summary: Amphidrome refers to an oceanic point that is encircled by one harmonic constituent of rotary tides. The amplitude of the amphidromic point is zero and its phase is undetermined. Recent studies showed that an amphidrome is not a tide‐only phenomenon but widely exists in oceanic and atmospheric systems. This study investigated annual amphidromes in the atmospheric pressure field. Many well‐defined amphidromic points can be found at different heights, and four interesting features of these amphidromes were identified: (1) Amphidromes are very common in many periodic atmosphere features and always appear in pairs of two amphidromes that rotate in opposite directions, termed amphidromic twins. (2) The location of amphidromes in different layers changes slowly with height forming so‐called amphidromic lines. (3) Amphidromic lines have a three‐dimensional (3D) structure with both clockwise and anticlockwise vertical branches, connected by one or two horizontal components. The amphidromic twins are, in fact, only two vertical branches. (4) The 3D structure of specific amphidromic lines, sometimes, can be a closed loop named amphidromic loop. The phase rotating around this loop is similar to magnetic induction lines around a closed electromagnetic coil. Key Points: Amphidromes for periodic variables in the atmosphere are a common feature and always appear as twins with opposite rotating directionsAmphidromes at different heights form amphidromic lines with cyclone and anticyclone branches, connected by one or two horizontal componentsThe three‐dimensional structure of amphidromic lines in the atmosphere can sometimes be a closed loop [ABSTRACT FROM AUTHOR]

This article presents an extended algorithm for topology optimization of compliant mechanisms and structures with design-dependent pressure loadings using the moving iso-surface threshold (MIST) method. In this algorithm, the fluid-structure interface is modeled using the finite element method via considering equivalent virtual strain energy and work and is tracked by an element-based searching scheme. Design-dependent pressure loads are directly applied on interface boundary and are calculated as virtual work equivalent nodal forces in the interface elements based on the finite element formulation. Several numerical examples are presented for topology optimization of mean compliance and compliant mechanisms. The present algorithm is validated through benchmarking with the results in literature and/or full finite element analysis (FEA) results of the optimum compliant mechanism and structure designs. [ABSTRACT FROM AUTHOR]

Bayesian optimization (BO) is a popular method for solving optimization problems involving expensive objective functions. Although BO has been applied across various fields, its use in structural optimization area is in its early stages. Origami folding structures provide a complex design space where the use of an efficient optimizer is critical. In this work for the first time we demonstrate the ability of BO to solve origami-inspired design problems. We use a Gaussian process (GP) as the surrogate model that is trained to mimic the response of the expensive finite element (FE) objective function. The ability of this BO-FE framework to find optimal designs is verified by applying it to well-known origami design problems. We compare the performance of the proposed approach to traditional gradient-based optimization techniques and genetic algorithm methods in terms of ability to discover designs and computational efficiency. BO has many user-defined components/parameters and intuitions for these for structural optimization are currently limited. In this work, we study the role of hyperparameter tuning and the sensitivity of Bayesian optimization to the quality and size of the initial training set. Taking a holistic view of the computational expense, we propose various heuristic approaches to reduce the overall cost of optimization. Our results show that Bayesian optimization is an efficient alternative to traditional methods. It allows for the discovery of optimal designs using fewer finite element solutions, which makes it an attractive choice for the non-convex design space of origami fold mechanics. [ABSTRACT FROM AUTHOR]

This paper presents a feature modeling approach to address the 3D structural topology design optimization with feature constraints. In the proposed algorithm, various features are formed into searchable shape features by the feature modeling technology, and the models of feature elements are established. The feature elements that meet the design requirements are found by employing a feature matching technology, and the constraint factors combined with the pseudo density of elements are initialized according to the optimized feature elements. Then, through controlling the constraint factors and utilizing the optimization criterion method along with the filtering technology of independentmesh, the structural design optimization is implemented. The present feature modeling approach is applied to the feature-based structural topology optimization using empirical data. Meanwhile, the improved mathematical model based on the density method with the constraint factors and the corresponding solution processes are also presented. Compared with the traditional methodwhich requires complicated constraint processing, the present approach is flexibly applied to the 3D structural design optimization with added holes by changing the constraint factors, thus it can design a structure with predetermined features more directly and easily. Numerical examples show effectiveness of the proposed feature modeling approach, which is suitable for the practical engineering design. [ABSTRACT FROM AUTHOR]

By including acoustic-elastic interactions in a topology optimization based on the covariance matrix adaptation evolution strategy, we developed acoustic cloaks of optimal design that render an object unobservable through airborne and water-borne sounds. This strategy helps in exploring optimal topologies that minimize the scattering of airborne and water-borne sounds around acoustic cloaks made from acrylonitrile butadiene styrene copolymers frequently used as ink in 3D printers. By applying level set methods, our designed cloaks are expressed as iso-surfaces representing the sharp structural boundaries between the acoustic and elastic media. On these boundaries, conditions that couple the elastic and acoustic aspects of the scattering are imposed. Furthermore, our cloak designs ensure that scatterers are unobservable to sound waves of multiple frequencies incident from various angles by minimizing the fitness incorporating multiple objective functions under several structural symmetries. [ABSTRACT FROM AUTHOR]

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p2 weighted persistent homology provided with some information on the mod p weighted persistent homology. [ABSTRACT FROM AUTHOR]

Persistent homology is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general pipeline to apply persistent homology to study time series, particularly the instantaneous heart rate time series for the heart rate variability (HRV) analysis. The first step is capturing the shapes of time series from two different aspects—the persistent homologies and hence persistence diagrams of its sub-level set and Taken's lag map. Second, we propose a systematic and computationally efficient approach to summarize persistence diagrams, which we coined persistence statistics. To demonstrate our proposed method, we apply these tools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movement and non rapid eyeball movement) and sleep-REM-NREM classification problems. The proposed algorithm is evaluated on three different datasets via the cross-database validation scheme. The performance of our approach is better than the state-of-the-art algorithms, and the result is consistent throughout different datasets. [ABSTRACT FROM AUTHOR]

Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X→ Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d−2 and Y is (d−1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S¹ it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or computing the ℤ2-index—everything in the stable range. Outside the stable range, the extension problem is undecidable. [ABSTRACT FROM AUTHOR]

As Crux said the word, the translator lit up with a red light to indicate there was no translation. Well, it started out as applications of algebra to topology, but now it's really just about the algebra." "I know algebra and I know topology, but what's algebraic topology?" You going?" Crux asked. [Extracted from the article]