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This manuscript describes three activities connecting the Tower of Hanoi puzzle to three familiar fractal forms. The first connects coin flipping, paper folding, and the Tower of Hanoi to the Dragon Curve. The second illustrates mathematician Ian Stewart's method for showing how the relationships between possible states of the Tower of Hanoi are related to stages in Sierpinski's Gasket. The final activity compares right-left moves of Tower of Hanoi disks to iterations of the Von Koch Curve. [ABSTRACT FROM AUTHOR]

Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this very property makes point set topology perfect for teaching and learning abstract mathematical concepts. It clears our minds of preconceived intuitions and expectations and forces us to think in new and creative ways. In this paper, we present guided investigations into topology through questions and thinking strategies that open up fascinating problems. They are intended for faculty who already teach or are thinking about teaching a class in topology or abstract mathematical reasoning for undergraduates. They can be used to build simple to challenging projects in topology, proofs, honors programs, and research experiences.

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. We show that in many cases, researchers using these tools have derived conclusions that are incorrect, trivial, or limited. [ABSTRACT FROM AUTHOR]

For a segmented dynamic wireless power transfer (DWPT) system, when electric vehicles (EVs) move over two adjacent transmitting coils, the received power fluctuates greatly and has some loss. Arming at this problem, this paper proposes a dual receiver segmented DWPT topology to reduce the received power fluctuation of the receiving coil passing through the rail junction during the moving process of the EV. Through magnetic field analysis in this paper, tuning the current phase difference between two adjacent transmitting coils to 180° can enhance the magnetic field. In this paper, a dual receiver series topology (DRST) is designed to pick up power, which consists of four fully controlled components and four diodes. According to EVs' real-time position, the segment DWPT system has three working modes and six working states under the control of DRST. Finally, experiments are carried out. Compared to the single-receiving-coil system, DRST is effective in improving the average output power from 2.007 to 5.657 W and significantly reducing the power fluctuation. [ABSTRACT FROM AUTHOR]

New grid equipment startup programs are widely used in various countries to regulate the commissioning of new equipment; these programs have unique differences in terms of strictness, information asymmetry, and complexity relative to other types of startup programs. With respect to rule-based generation methods, because the method of revising the rules weakens their migration ability, it is difficult to adapt these methods to the status quo of high-speed power grid construction; moreover, most of the current generation methods based on deep learning improve upon the rule-based methods but do not eliminate the rules of the constraints. Therefore, this paper presents a fusion topology for generating a new grid equipment startup scheme, which generates the scheme from end to end. The method utilizes the powerful processing capabilities of the GATv2 model and the ERNIE-GEN model for topology and text, respectively. The device type-based coding strategy and the scheme complexity-based self-attention layer selection strategy are used in the GATv2-based device identification model to address information asymmetry and complexity variability, and the device information modification strategy is applied to solve the strictness variability problem in the ERNIE-GEN-based scheme generation model. Finally, through the testing and verification of field data from four types of new equipment startup schemes in real power grids, it is verified that the method can effectively generate new equipment startup schemes for power grids, and the reasonableness and efficiency of the three strategies are verified through ablation experiments, which verify that the method can effectively generate new equipment startup schemes for power grids that meet the requirements of real power grids. [ABSTRACT FROM AUTHOR]

Optimizing the configuration of acquisition terminals can maximize the observability and state estimation accuracy of the distribution grid achieving comprehensive perception of the distribution grid. However, the existing optimization configuration methods typically target a single topology structure. For regional distribution grids with dynamic topology changes, it cannot meet the observability requirements of all their topologies. In this regard, this paper proposes an optimal configuration scheme for regional distribution grid acquisition terminals considering dynamic observability. Firstly, the regional distribution grid considering the change of contact switch is modeled. Based on the observation redundancy and state estimation accuracy, the dynamic observability index of regional distribution grid is proposed. Then, a multi-objective optimal configuration model of acquisition terminal is constructed with the objective function of maximizing dynamic observability and minimizing configuration cost. Finally, the effectiveness of the proposed approach is validated with the simulation model. [ABSTRACT FROM AUTHOR]

Immediately following the commentary below, the previously published article by R. Thom is printed in its entirety: R. Thom, Ensembles et morphismes stratifyíes, Bulletin of the American Mathematical Society 75 (1969), no. 2, 240-284 (French). This is followed by the first publication of the 1970 lecture notes of J. Mather, Notes on topological stability. [ABSTRACT FROM AUTHOR]

Through tuning the conformations of N-donor ligands and the coordinative preferences of metal ions, three new metal–organic frameworks with different dimensionalities, namely, {[Co(L1)(bpp)(H2O)]·0.63H2O}n(1), [Ni(L1)(bpp)(H2O)(CH3OH)]n(2) and {Co2(L1)2(bpe)2(H2O)2}·(H2O)(CH3CN)}n(3) (H2L1 = 1,2-bis(4-carboxy-phenoxy)ethane, bpp = 1,3–bis(4-pyridyl)propane and bpe = 1,2–bis(4-pyridyl)ethene), have been synthesized and structurally characterized. Compound 1exhibits a 2D → 2D parallel interpenetrating motif containing two kinds of double helixes. Polymer 2displays 2D → 3D interpenetrating framework. Compound 3is composed of 3D unusual self-interpenetrating net with (4,6)-connected (54·68)(56·65·72·82) topology. This work evidently indicates that the effect of secondary ligands and geometric preferences of metal centers are critical in construction of polymeric arrangement. In addition, the stabilities of these compounds and the magnetic behaviour of 3are discussed. [ABSTRACT FROM AUTHOR]

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The deployment of multi-agent systems (MASs) is widely used in the fields of unmanned agricultural machineries, unmanned aerial vehicles, intelligent transportation, etc. To make up for the defect that the existing PDE-based results are overly idealistic in terms of system models and control strategies, we study the PDE-based deployment of clustered nonlinear first-order and second- order MASs over a finite-time interval (FTI). By designing special communication protocols, the collective dynamics of numerous agents are modeled by simple fist-order and second-order PDEs. Two practical factors, external disturbance and Markov switching topology, are considered in this paper to better match actual situations. Besides, T-S fuzzy technology is used to approximate the unknown nonlinearity of MASs. Then, by using boundary control scheme with collocated measurements, two theorems are obtained to ensure the finite-time H∞ deployment of first-order and second-order agents, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

In recent years, factorization combined with attribute information has played an important role in attribute community detection. However, previous studies focused more on connecting the original shallow topological and attribute data. They ignored the potential representation structure of the network. To solve the problem that shallow information cannot fully represent the network structure, this paper proposes an attribute node classification method based on Attribute Edges weights Fusion and graph Embedding Factorization, called AEFEF. First, AEFEF converts topological information and attribute information into corresponding matrices representing node associations. Then, AEFEF constructs a new adjacency matrix by increasing the weights of shared edges between the topology structure and attribute structure. This operation can strengthen the tightness between nodes. Second, to explore the potential community structure, feature embedding is obtained by factorizing the attribute similarity matrix. Meanwhile, the new adjacency matrix is designed as a weight matrix to make the feature embedding between related nodes more similar. Finally, semi Non-negative Matrix Factorization (NMF) is introduced to modify the feature embedding by converting the negative values into positive values. Then the embedding is factorized to generate the membership matrix. At the same time, the network with a rich structure is decomposed with topological data as the main component. Otherwise, attribute information is the main component of NMF used to increase the accuracy of node classification. AEFEF is compared with 10 state-of-the-art algorithms on 7 real network datasets. The results reveal that AEFEF can improve the precision of attribute community detection. [ABSTRACT FROM AUTHOR]

This paper addresses the high-frequency signal transmission problem of high-speed differential lines on four-layer printed circuit boards (PCBs). It establishes a mathematical model of high-speed differential lines in conjunction with modified Kron's methodology (MKM), a nontraditional circuit modeling method. The article builds the model through diakoptics of differential lines, then generates the corresponding topology maps, and finally creates the model through tensorial analysis of the network (TAN). The differential line model is simulated and optimized by HFSS. This paper mainly analyzes the influence of differential line spacing and grounding vias on the signal transmission of differential lines. Secondly, it analyzes the problem of multi-group differential line arrangement based on the above work. Finally, the experimental results obtained are consistent with the simulation ones. [ABSTRACT FROM AUTHOR]

The topological Dehn twists have several applications in mathematical sciences as well as in physical sciences. The interplay between homotopy theory and Dehn twists exposes a rich set of properties. This paper generalizes the Dehn twists by proposing the notion of pre-twisted space, orientations of twists and the formation of pointed based space under a homeomorphic continuous function. It is shown that the Dehn twisted homotopy under non-retraction admits a left lifting property (LLP) through the local homeomorphism. The LLP extends the principles of Hurewicz fibration by avoiding pullback. Moreover, this paper illustrates that the Dehn twisted homotopy up to a base point in a based space can be formed by considering retraction. As a result, two disjoint continuous functions become point-wise continuous at the base point under retracted homotopy twists. Interestingly, the oriented Dehn twists of a pre-twisted space under homotopy retraction mutually commute in a contractible space. [ABSTRACT FROM AUTHOR]

This paper presents a 13-level switched capacitor inverter with a novel modulation method designed to minimize the number of switches and significantly reduce switching losses. The inverter stands out for its simplicity, requiring only ten semiconductor switches to generate 13 levels. A key feature is the inherent self-voltage balancing of the capacitors, which eliminates the need for additional control mechanisms. The inverter's unique architecture, comprising high-voltage and low-voltage modules, enables modulation using a hybrid PWM approach that combines step waveform modulation with level-shifted PWM (LS-PWM). This innovative technique dramatically lowers the switching frequency of the high-voltage module's switches, independent of the carrier frequency, thereby limiting the number of switches in high-frequency operation and achieving substantial reductions in switching losses. This paper provides a detailed analysis of the inverter's operating modes, voltage-balancing mechanisms, and parameter calculations. The advantages of the topology presented in this paper are demonstrated by comparison. Finally, the simulation and experimental results confirm the practicality and effectiveness of the proposed inverter and its modulation strategy. [ABSTRACT FROM AUTHOR]

Multi-level inverters have characteristics suitable for high-voltage and high-power applications through various topology configurations. These reduce harmonic distortion and improve the quality of the output waveform by generating a multi-level output voltage waveform. In particular, an active neutral-point-clamped topology is one of the multi-level inverters advantageous for high-power and medium-voltage applications. It has the advantage of controlling the output waveform more precisely by actively clamping the neutral point using an active switch and diode. However, it has a problem, which is that an unwanted zero-crossing current may occur if an inaccurate switching signal is applied at the time when the polarity of the output voltage changes. In this paper, a control strategy to suppress the zero-crossing current of a single-phase half-bridge three-level active neutral-point-clamped inverter is proposed. The operating principle of a single-phase half-bridge three-level active neutral-point-clamped inverter is identified through an operation mode analysis. In addition, how the switching signal is reflected in an actual digital signal processor is analyzed to determine the situation in which the zero-crossing current occurs. Through this, a control strategy capable of suppressing zero-crossing current is designed. The proposed method prevents a zero-crossing current by appropriately modifying the update timing of reference voltages at the point where the polarity of the output changes. The validity of the proposed method is verified through simulation and experiments. Based on the proposed method, the total harmonic distortion of the output current is significantly reduced from 12.15% to 4.59% in a full-load situation. [ABSTRACT FROM AUTHOR]

The paper proposes a new multilevel inverter (MLI) topology that is designed to overcome the challenges posed by conventional MLIs, which can have high costs and complexity due to the large number of power components. The proposed MLI is designed to work with both symmetric and asymmetric DC sources, and uses fewer power semiconductor switches compared to recently propose reduced components MLIs. The paper presents the ratio of voltage levels generated per power semiconductor switch, and the gate pulses for the inverter are generated using a DSPACE1103 controller. A prototype of the 7-level inverter is developed, and experimental outcomes are provided to evaluate the simulated results. The paper highlights the importance of reducing the number of power components in MLIs to reduce costs and improve reliability. The proposed MLI offers a promising solution to this challenge, while also maintaining the advantages of MLIs such as reduced harmonic distortion and electromagnetic interference. Overall, the research offers valuable insights into the design and performance of multilevel inverters, and the proposed topology has the potential to be a useful tool for researchers and engineers working in the field of power electronics. [ABSTRACT FROM AUTHOR]

W* is the category of the title. For G ∈ W*, we have the canonical compact space YG, and Yosida representation G ≤ C(YG), thus, for g ∈ G, one has the cozero-set coz(g) in YG. The ideals at issue in G include the principal ideals and polars, G(g) and g⊥⊥, respectively, and the W*-kernels of W*-morphisms from G. The "coincidences of types" include these properties of G: (M) Each G(g) = g⊥⊥; (Y) Each G(g) is a W*-kernel; (CR) Each g⊥⊥ is a W*-kernel (iff each coz(g) is regular open). For each of these, we give numerous "rephrasings" and examples, and note that (M) = (Y) ∩ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on W* brings topology to bear, and sharpens the view. [ABSTRACT FROM AUTHOR]

The idea of nano topology was proposed by M.L. Thivagar. Since then, various researchers have worked on generalizing this theory. Bi-ideal nano topology was generated by approximation using two ideals. Motivated by the fact that this technique of approximation is better than the existing ones due to higher accuracy and more versatility, there is a need to investigate some new forms of bi-ideal nano topology. The aim of this paper is to generate novel forms of bi-ideal nano open sets, namely bi-ideal nano semi-open, bi-ideal nano pre-open, bi-ideal nano regular open, and bi-ideal nano ff open sets. The interrelation of all these weak forms is studied. A practical application of nano topology is also discussed towards the end of this paper to find the main factors responsible for covid-19 disease. [ABSTRACT FROM AUTHOR]

Featured Application: This paper proposes a method to determine the dynamic response of a planar mechanism with a practical application to a flap and wing control mechanism of a light sport aircraft. For this, a topological model and its related theory are used. This paper presents a more general method to study the dynamic behavior of a planar mechanism with joints. To do this, Graph Theory is used. A topological description is presented based on which a corresponding graph is attached to any mechanism. This method can be used for both kinematic and dynamic study in the same model, and many of the descriptors used in kinematic analysis can also be used for dynamic analysis. As a final result, the motion equations of the studied mechanical system can be obtained. An application is made to a flap and wing control system of a light sport aircraft. The graph associated with the mechanism first used to determine the field of velocities and accelerations of the mechanism elements is then used to obtain the motion equation for the control system. In this way, Graph Theory proves useful for the parallel study of both the kinematic and dynamic study of such problems. [ABSTRACT FROM AUTHOR]

We consider standard T-interval dynamic networks, under the synchronous timing model and the broadcast CONGEST model. In a T-interval dynamic network, the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the topology may change arbitrarily from round to round, as determined by some adversary and subject to the following constraint: For every T consecutive rounds, the topologies in those rounds must contain a common connected spanning subgraph. Let H r to be the maximum (in terms of number of edges) such subgraph for round r through r + T - 1 . We define the backbone diameterd of a T-interval dynamic network to be the maximum diameter of all such H r 's, for r ≥ 1 . We use n to denote the number of nodes in the network. Within such a context, we consider a range of fundamental distributed computing problems including Count/Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood. Existing algorithms for these problems all have time complexity of Ω (n) rounds, even for T = ∞ and even when d is as small as O(1). This paper presents a novel approach/framework, based on the idea of massively parallel aggregation. Following this approach, we develop a novel deterministic Count algorithm with O (d 3 log 2 n) complexity, for T-interval dynamic networks with T ≥ c · d 2 log 2 n . Here c is a (sufficiently large) constant independent of d, n, and T. To our knowledge, our algorithm is the very first such algorithm whose complexity does not contain a Θ (n) term. This paper further develops novel algorithms for solving Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood, while incurring O (d 3 polylog (n)) complexity. Again, for all these problems, our algorithms are the first ones whose time complexity does not contain a Θ (n) term. [ABSTRACT FROM AUTHOR]

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This paper explores physical layer group key generation in wireless relay networks with a star topology. In this setup, the relay node plays the role of either a trusted or untrusted central node, while one legitimate node (Alice) acts as the reference node. The channel between the relay and Alice serves as the reference channel. To enhance security during the channel measurement stage, a cooperative jamming-based scheme is proposed in this paper. This scheme allows the relay to obtain superimposed channel observations from both the reference channel and other relay channels. Then, a public discussion is utilized to enable all nodes to obtain estimates of the reference channel. Subsequently, the legitimate nodes can agree on a secret key (SK) that remains secret from the eavesdropper (Eve), or a private key (PK) that needs to be secret from both the relay and Eve. This paper also derives the lower and upper bounds of the SK/PK capacity. Notably, it demonstrates that there exists only a small constant difference between the SK/PK upper and lower bounds in the high signal-to-noise ratio (SNR) regime. Simulation results confirm the effectiveness of the proposed scheme for ensuring security and efficiency of group key generation. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing a new similarity measure for the neutrosophic binary set. Similarity measure are used in multi attribute desicion making problems to find the difference between the alternatives. In this paper a new measure based on Euclidean distance is introduced to find the measure between two binary single valued neutrosophic set. Further its is applied in a multi attribute desicion problem to see the attainability of the proposed measure. [ABSTRACT FROM AUTHOR]

Abstract  In this paper exact, analytical solutions are derived for another highly popular benchmark problem, namely, L-shaped domains having a horizontal line support and one or several point loads. The optimal topologies are obtained in the context of Michell structures, i.e., least-weight, stress, or compliance-controlled trusses with a single load condition. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we show that any graph map without periodic points has only one minimal set. We describe a class of graph maps without periodic points. Our main result is to give a structure theorem of graph maps without periodic points, which states that any graph map without periodic points must be topologically conjugate to one of the described class. In addition, we give some applications of the structure theorem. [Copyright &y& Elsevier]

In this paper, we characterize the capacity of a new class of discrete memoryless multicast networks having a tree topology. For achievability, a novel coding scheme is constructed where some relays employ a combination of decode-and-forward and compress-and-forward and the other relays perform a random binning such that codebook constructions and relay operations are independent for each node and do not depend on the network topology. For converse, a new technique of iteratively manipulating inequalities exploiting the tree topology is used. This class of multicast tree networks includes the class of diamond networks studied by Kang and Ulukus as a special case. [ABSTRACT FROM PUBLISHER]

The present paper studies digital topological rough set structures associated with Khalimsky (K -, for brevity) topology. We further explore their properties related to topological (closure and interior) operators. In addition, we find that the K -topological rough set structures have their own features different from certain theoretical properties of the H -rough set structure in [9] w.r.t. the topological operators. More precisely, whereas the H -rough set structure is a topological operator, K -topological concept approximations are proved not to be topological operators. Thus, this paper proposes an alternative approach to the K -topological concept approximations so that we finally have new types of topological operators. Furthermore, it appears that this new ones are smoothly matched with the earlier H -rough set structure from the viewpoint of topological operators. Therefore, they can facilitate the study of continuous-oriented or digital-oriented rough sets, which can be used in applied sciences without any limitations of the cardinalities of the universe and a target set. Besides, they can support certain decision rules with no restriction to anything continuous or discrete (or digital). In addition, we introduce the notion of K -definability of a target subset of a universal set and write an algorithm for estimating the roughness of the above new concept approximations. In this paper each set need not be finite and a covering is locally finite. [ABSTRACT FROM AUTHOR]