Topological Problems Research Articles

A subsystem-based analysis approach for fixed-time consensus of multi-agent systems with local pinning strategy

This paper investigates the global consensus of multi-agent systems with unknown dynamics in a fixed time. The disconnected network topology is considered, where the connectivity or initial connectivity of the network is not assumed. To achieve the global consensus of all the agents, a switching control protocol is proposed by adding the controls for the type II uninformed agents from their closest informed agent. The subsystem-based analysis approach is given based on the change of the topology by constructing the multiple Lyapunov functions, where the Lyapunov function can be divided into two parts with and without leader information. It is proven that the total energy is non-monotonic but bounded all the time, and the system can converge in a fixed time eventually. The upper bound of the convergence time can be estimated with the convergence of each part of the Lyapunov function, which only depends on the control protocol and the topology of network. Finally, numerical examples are used to verify the theoretical results, and show that the proposed method provides an applicable method for the control of the disconnected topology problems.

A Novel LED Driving Power Supply with High Power Factor and Low Total Harmonic Distortion

proposing a new type of LED driving power supply which is based on buck-boost cascade quad and analysing it in this paper. The main circuit topology structure is based on a quadratic buck and a buck-boost converter. They share the same switch through cascading which not only simplify the topology but also control strategy, reducing the cost of the control circuit. Eliminate the input current dead-time problem of the original quadratic buck topology by adopting two-stage cascade structure. The main circuit topology improves the power factor and ameliorates the input current THD as well. Meanwhile, making the duty cycle of the switch operates in a more plausible area. Finally, relevant experiments demonstrate that the theoretical analysis is correct.

Asymptotic identities for additive convolutions of sums of divisors

AbstractIn a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $\sigma_a(n)$ and $\sigma_b(n)$ , and proved an asymptotic formula for it when a and b are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real a and b. Ramanujan’s conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term.In this paper, we give a new proof of Ramanujan’s conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author’s thesis and which served as our initial motivation in studying this sum.

On a topological classification of multidimensional polar flows

The work solves the classification problem for structurally stable flows, which goes back to the classical works of Andronov, Pontryagin, Leontovich and Mayer. One of important examples of such flows is so-called Morse-Smale flow, whose non-wandering set consists of a finite number of fixed points and periodic trajectories. To date, there are exhaustive classification results for Morse-Smale flows given on manifolds whose dimension does not exceed three, and a very small number of results for higher dimensions. This is explained by increasing complexity of the topological problems that arise while describing the structure of the partition of a multidimensional phase space into trajectories. In this paper authors investigate the class G(Mn) of Morse-Smale flows on a closed connected orientable manifold Mn whose non-wandering set consists of exactly four points: a source, a sink, and two saddles. For the case when the dimension n of the supporting manifold is greater or equal than four, it is additionally assumed that one of the invariant manifolds for each saddle equilibrium state is one-dimensional. For flows from this class, authors describe the topology of the supporting manifold, estimate minimum number of heteroclinic curves, and obtain necessary and sufficient conditions of topological equivalence. Authors also describe an algorithm that constructs standard representative in each class of topological equivalence. One of the surprising results of this paper is that while for n=3 there is a countable set of manifolds that admit flows from class G(M3), there is only one supporting manifold (up to homeomorphism) for dimension n>3.

LPTD: a novel linear programming-based topology determination method for cryo-EM maps.

Topology determination is one of the most important intermediate steps toward building the atomic structure of proteins from their medium-resolution cryo-electron microscopy (cryo-EM) map. The main goal in the topology determination is to identify correct matches (i.e. assignment and direction) between secondary structure elements (SSEs) (α-helices and β-sheets) detected in a protein sequence and cryo-EM density map. Despite many recent advances in molecular biology technologies, the problem remains a challenging issue. To overcome the problem, this article proposes a linear programming-based topology determination (LPTD) method to solve the secondary structure topology problem in three-dimensional geometrical space. Through modeling of the protein's sequence with the aid of extracting highly reliable features and a distance-based scoring function, the secondary structure matching problem is transformed into a complete weighted bipartite graph matching problem. Subsequently, an algorithm based on linear programming is developed as a decision-making strategy to extract the true topology (native topology) between all possible topologies. The proposed automatic framework is verified using 12 experimental and 15 simulated α-β proteins. Results demonstrate that LPTD is highly efficient and extremely fast in such a way that for 77% of cases in the dataset, the native topology has been detected in the first rank topology in <2 s. Besides, this method is able to successfully handle large complex proteins with as many as 65 SSEs. Such a large number of SSEs have never been solved with current tools/methods. The LPTD package (source code and data) is publicly available at https://github.com/B-Behkamal/LPTD. Moreover, two test samples as well as the instruction of utilizing the graphical user interface have been provided in the shared readme file. Supplementary data are available at Bioinformatics online.

Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence ofPin⁡(2)\operatorname {Pin}(2)-equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing thePin⁡(2)\operatorname {Pin}(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem.We prove our theorem by analyzing maps between certain finite spectra arising fromBPin⁡(2)B\operatorname {Pin}(2)and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and thejj-based Atiyah–Hirzebruch spectral sequence.

Ecodesign with topology optimization

In order to mitigate the impact of the transportation sector on climate change, light and ecological parts must be designed. A lifecycle oriented design methodology with C02 footprint minimization of parts used in various transports is presented in this work. Only material production and use phase are considered in this work, to have a better understanding of the different contributions. Simultaneous topological design and material choice are investigated for 2D examples. The results show that considering out-of-plane thickness as a variable, both problems can now be decoupled for simple load cases. It is shown that a very simple material index depending only on the type of transport can be used. An optimal volume fraction is obtained, specific only to each topology problem, but unrelated to the material chosen or the loads applied. The method is promising for fast ecodesign and its simple implementation enables easy future improvements.

Some open problems in equivariant infinite-dimensional topology

In this paper we present some open problems in the field mentioned in the title. All problems here discussed arose in the investigation of the author. Special attention is paid to such topics as Extension of equivariant maps, Characterization of G-ANRs, Topology of the Banach-Mazur compacta, Group actions on hyperspaces, and Invariant metric problem.

Acoustic Tunneling Study for Hexachiral Phononic Crystals Based on Dirac-Cone Dispersion Properties

Acoustic tunneling is an essential property for phononic crystals in a Dirac-cone state. By analyzing the linear dispersion relations for the accidental degeneracy of Bloch eigenstates, the influence of geometric parameters on opening the Dirac-cone state and the directional band gaps’ widths are investigated. For two-dimensional hexachiral phononic crystals, for example, the four-fold accidental degenerate Dirac point emerges at the center of the irreducible Brillouin zone (IBZ). The Dirac cone properties and the band structure inversion problem are discussed. Finally, to verify acoustic transmission properties near the double-Dirac-cone frequency region, the numerical calculation of the finite-width phononic crystal structure is carried out, and the acoustic transmission tunneling effect is proved. The results enrich and expand the manipulating method in the topological insulator problem for hexachiral phononic crystals.

Quantum variational optimization: The role of entanglement and problem hardness

Quantum variational optimization has been posed as an alternative to solve optimization problems faster and at a larger scale than what classical methods allow. In this paper we study systematically the role of entanglement, the structure of the variational quantum circuit, and the structure of the optimization problem, in the success and efficiency of these algorithms. For this purpose, our study focuses on the variational quantum eigensolver (VQE) algorithm, as applied to quadratic unconstrained binary optimization (QUBO) problems on random graphs with tunable density. Our numerical results indicate an advantage in adapting the distribution of entangling gates to the problem's topology, specially for problems defined on low-dimensional graphs. Furthermore, we find evidence that applying conditional value at risk type cost functions improves the optimization, increasing the probability of overlap with the optimal solutions. However, these techniques also improve the performance of Ans\"atze based on product states (no entanglement), suggesting that a new classical optimization method based on these could outperform existing NISQ architectures in certain regimes. Finally, our study also reveals a correlation between the hardness of a problem and the Hamming distance between the ground- and first-excited state, an idea that can be used to engineer benchmarks and understand the performance bottlenecks of optimization methods.

The Role of Topoisomerase II in DNA Repair and Recombination in Arabidopsis thaliana.

DNA entanglements and supercoiling arise frequently during normal DNA metabolism. DNA topoisomerases are highly conserved enzymes that resolve the topological problems that these structures create. Topoisomerase II (TOPII) releases topological stress in DNA by removing DNA supercoils through breaking the two DNA strands, passing a DNA duplex through the break and religating the broken strands. TOPII performs key DNA metabolic roles essential for DNA replication, chromosome condensation, heterochromatin metabolism, telomere disentanglement, centromere decatenation, transmission of crossover (CO) interference, interlock resolution and chromosome segregation in several model organisms. In this study, we reveal the endogenous role of Arabidopsis thaliana TOPII in normal root growth and cell cycle, and mitotic DNA repair via homologous recombination. Additionally, we show that the protein is required for meiotic DSB repair progression, but not for CO formation. We propose that TOPII might promote mitotic HR DNA repair by relieving stress needed for HR strand invasion and D-loop formation.

Communication Topology Assignment and Control Co-design for Vehicle Platoons in LTE-V2V Network

In this study, we investigate the problem of inter-vehicle communication topology (IVCT) assignment and control for vehicular platoons in LTE-V2V networks based on the cooperative awareness message dissemination mechanism. A sampled-data feedback controller is proposed for platoons to eliminate the effect of stochastic packet dropouts and external disturbance, where the controller gain depends on the IVCT. To guarantee the stability requirement of a platoon with the minimized cost function, a unified control framework is established to jointly determine the optimal IVCT from all the available ones and the associated feedback controller gain. This co-design procedure is based on the optimal control and dynamic programming technique, where both fixed and periodic switching IVCTs are available. A useful algorithm is proposed to implement the established co-design framework. Furthermore, we present numerical examples to demonstrate the effectiveness of the proposed method.

Application loader in the RW. Ring platform

When developing decision support systems in agriculture, the task often arises of creating applications that include a large number of different components. These components can have dependencies on each other, so you need to load them in the correct order. This boils down to solving the classic topological sorting problem. However, in addition to the purely algorithmic part, the loader must correctly interact with the environment, which poses a large number of other technology-specific tasks for its developer. These are the tasks of obtaining and storing information about dependencies, ensuring that components are loaded in the user interface thread where necessary, as well as ensuring the most responsive program behavior so that loading an application does not annoy the user, as well as ensuring the extensibility of the decision support system without recompiling. This work is devoted to the description of the solution of these problems in the RW.Ring platform based on the .NET technological stack and intended for the development of such software systems.

Nucleolar translocation of human DNA topoisomerase II by ATP depletion and its disruption by the RNA polymerase I inhibitor BMH-21

DNA topoisomerase II (TOP2) is a nuclear protein that resolves DNA topological problems and plays critical roles in multiple nuclear processes. Human cells have two TOP2 proteins, TOP2A and TOP2B, that are localized in both the nucleoplasm and nucleolus. Previously, ATP depletion was shown to augment the nucleolar localization of TOP2B, but the molecular details of subnuclear distributions, particularly of TOP2A, remained to be fully elucidated in relation to the status of cellular ATP. Here, we analyzed the nuclear dynamics of human TOP2A and TOP2B in ATP-depleted cells. Both proteins rapidly translocated from the nucleoplasm to the nucleolus in response to ATP depletion. FRAP analysis demonstrated that they were highly mobile in the nucleoplasm and nucleolus. The nucleolar retention of both proteins was sensitive to the RNA polymerase I inhibitor BMH-21, and the TOP2 proteins in the nucleolus were immediately dispersed into the nucleoplasm by BMH-21. Under ATP-depleted conditions, the TOP2 poison etoposide was less effective, indicating the therapeutic relevance of TOP2 subnuclear distributions. These results give novel insights into the subnuclear dynamics of TOP2 in relation to cellular ATP levels and also provide discussions about its possible mechanisms and biological significance.

Topologies Generated by Two Ideals and the Corresponding j -Approximations Spaces with Applications

Ideal is a fundamental concept in topological spaces and plays an important role in the study of topological problems. This motivated us to use two ideals to generate different topologies to take the advantage of the two ideals at the same time. Two ideals represent two opinions instead of one opinion which is very useful for using the insights of two groups of experts to study the problem and elicit decisions based on their common vision. Topology is a rich source for constructs that is helpful to enrich the original model of approximations spaces. Rough set theory has inbuilt topological concepts. Hence, the main purpose of this paper is to point out that the concept of rough sets has a purely topological aspects nature. To do so, new approximations spaces are introduced and defined based on the topologies generated by two ideals. The results in this paper show that the topological concepts can be a powerful method to study rough set models. The basic properties of these approximations are studied and compared to the previous ones and shown to be more general. The importance of the current paper is not only introducing a new kind of rough set based on bi-ideals, increasing the accuracy measure, and reducing the boundary region of the sets which is the main aim of rough set but also introducing a chemical application to explain the concepts.

Topology optimization based on the high-order numerical manifold method by implementing a four-node quadrilateral element

In this study, the numerical manifold method (NMM), coupled with material interpolation, is applied to the topology optimization of continuum structures. Quadrilateral element (Q4) meshes are used in the NMM for mathematical cover. The order of the local approximation function is increased to improve the accuracy of the analysis. This method usually leads to linear dependency of the global degrees of freedom, as reflected in the rank deficiency of the global approximation space. However, using Q4 elements, the rank deficiency value is not fixed with mesh refinement. The presented solution handles this problem, and the efficiency of the proposed method is investigated with four examples. The results indicate that the high-order numerical manifold method (HONMM) has great accuracy in the analysis and works well for topology optimization problems. One advantage of using the combined HONMM with material interpolation in topology problems is chequerboard-free patterns, owing to mathematical cover and high-order analysis.

New Results on the Aggregation of Norms

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

On the hardness of finding normal surfaces

For many fundamental problems in computational topology, such as unknot recognition and 3-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms for these problems is normal surface theory. However, we currently have a poor understanding of the computational complexity of problems in normal surface theory: many such problems are still not known to have polynomial-time algorithms, yet proofs of \(\mathrm {NP}\)-hardness also remain scarce. We give three results that provide some insight on this front. A number of modern normal surface theoretic algorithms depend critically on the operation of finding a non-trivial normal sphere or disc in a 3-dimensional triangulation. We formulate an abstract problem that captures the algebraic and combinatorial aspects of this operation, and show that this abstract problem is \(\mathrm {NP}\)-complete. Assuming \(\mathrm {P}\ne \mathrm {NP}\), this result suggests that any polynomial-time procedure for finding a non-trivial normal sphere or disc will need to exploit some geometric or topological intuition. Another key operation, which applies to a much wider range of topological problems, involves finding a vertex normal surface of a certain type. We study two closely-related problems that can be solved using this operation. For one of these problems, we give a simple alternative solution that runs in polynomial time; for the other, we prove \(\mathrm {NP}\)-completeness.

Controlling the topology of mammalian mitochondrial DNA.

The genome of mitochondria, called mtDNA, is a small circular DNA molecule present at thousands of copies per human cell. MtDNA is packaged into nucleoprotein complexes called nucleoids, and the density of mtDNA packaging affects mitochondrial gene expression. Genetic processes such as transcription, DNA replication and DNA packaging alter DNA topology, and these topological problems are solved by a family of enzymes called topoisomerases. Within mitochondria, topoisomerases are involved firstly in the regulation of mtDNA supercoiling and secondly in disentangling interlinked mtDNA molecules following mtDNA replication. The loss of mitochondrial topoisomerase activity leads to defects in mitochondrial function, and variants in the dual-localized type IA topoisomerase TOP3A have also been reported to cause human mitochondrial disease. We review the current knowledge on processes that alter mtDNA topology, how mtDNA topology is modulated by the action of topoisomerases, and the consequences of altered mtDNA topology for mitochondrial function and human health.

Eliminating topological errors in neural network rotation estimation using self-selecting ensembles

Many problems in computer graphics and computer vision applications involves inferring a rotation from a variety of different forms of inputs. With the increasing use of deep learning, neural networks have been employed to solve such problems. However, the traditional representations for 3D rotations, the quaternions and Euler angles, are found to be problematic for neural networks in practice, producing seemingly unavoidable large estimation errors. Previous researches has identified the discontinuity of the mapping from SO(3) to the quaternions or Euler angles as the source of such errors, and to solve it, embeddings of SO(3) have been proposed as the output representation of rotation estimation networks instead. In this paper, we argue that the argument against quaternions and Euler angles from local discontinuities of the mappings from SO(3) is flawed, and instead provide a different argument from the global topological properties of SO(3) that also establishes the lower bound of maximum error when using quaternions and Euler angles for rotation estimation networks. Extending from this view, we discover that rotation symmetries in the input object causes additional topological problems that even using embeddings of SO(3) as the output representation would not correctly handle. We propose the self-selecting ensemble, a topologically motivated approach, where the network makes multiple predictions and assigns weights to them. We show theoretically and with experiments that our methods can be combined with a wide range of different rotation representations and can handle all kinds of finite symmetries in 3D rotation estimation problems.