Order Topology Research Articles

Design Considerations for Power-Efficient Fully Integrated 3:1 Switched Capacitor DC-DC Converter for PV Modules

This article presents a power-efficient DC-DC converter based on a switched-capacitor (SC) cell in power management systems supplied for fully integrated photovoltaic (PV) modules. These modules shall provide high-performance point-of-load voltage regulation. The primary objective of this study is to better utilize capacitance and switches by selecting a proper SC topology in order to improve the power efficiency of SC converters. A general steady-state performance model is investigated to optimize and compare a variety of SC DC-DC topologies. The investigation method relies on a charge-multiplier approach and considers the impact of area constraint on capacitors. To identify the most suitable topology for a given conversion ratio, the performance-limit metrics of SC converters are calculated. The analysis provides framework to determine optimum switch size and switching frequency for a two-phase 3:1 series–parallel converter for a target load current of 10 mA implemented on a 22 nm process technology. The results shows that a minimum of 250 MHz switching frequency is desirable for achieving a target efficiency greater than 85% while maintaining the minimum output voltage of 0.34 V. The analysis results are verified through MATLAB and PSpice-based simulations.

Additive Manufacturing, Numerical and Experimental Analyses for Pentamode Metamaterials

Pentamodes are lattice structures composed of beams. Their main property is the low ratio of the shear to bulk modulus, making them suitable for aerospace, antiseismic, and bioengineering applications. At first, in our study, pentamode structures were fabricated using three-dimensional printing and were tested in a laboratory. Then, computational analyses of bulk strength have been performed. In addition, several preliminary computational analyses have been considered, comparing different pentamodes’ dimensions and topologies in order to understand their behaviour under different loading conditions. Experimental results have been compared with the numerical results in order to validate the forces applied to the lattice structures. Our new contribution is that for the first time, the experimental and numerical results are investigated up to the failure of the specimens, the effective Young’s modulus has been calculated for different pentamode lattice structures, and our results are also compared with analytical equations.

Tunable quantum transport in topological semimetal candidates LaxSr1-xMnSb2

The magnetic semimetal SrMnSb2 contains a nonsymmorphic crystal symmetry that could protect Dirac crossings, offering the coexistence of a quasi-two-dimensional characteristic, magnetic order and nontrivial topology in bulk single crystals. Here, we report the evolution of Fermi surfaces by probing quantum oscillations in the La x Sr1-x MnSb2 single crystals. Hall measurements reveal that the conduction process changes from a hole-dominated multiband nature to a single hole-band nature after La doping. Compared with parent-SrMnSb2, the Fermi surface corresponding to F α is slightly enlarged and the small Fermi surface corresponding to F γ appears with a tunable feature in La-doped crystals. The topological semimetal candidates La x Sr1-x MnSb2 provide a fine-tuning platform for potential applications in the future.

Structural Features and Phylogenetic Implications of Crinoid Echinoderms Based on Thirteen Novel Mitochondrial Genomes

Crinoids, as integral echinoderms, play a crucial ecological role in benthic communities, serving as significant indicators reflecting the health of marine ecosystems. However, the phylogenetic relationships within crinoids are unclear. More molecular data can help to facilitate biodiversity assessment and elucidate evolutionary relationships by the phylogenetic tree. In this study, 13 complete mitochondrial genomes of the Crinoidea class were sequenced, annotated, and compared with other same class species available on NCBI. The results reveal five different gene order patterns among these mitochondrial genomes, indicating that crinoids have undergone gene rearrangements during evolution. The complete mitochondrial genome length of crinoids ranges from 15,772 bp to 16,850 bp. High A + T content, ranging from 64.5% to 74.2%, was observed. Additionally, our analysis of protein-coding genes highlights a preference for A + T nucleotides, along with specific start and stop codon usage, offering insights into codon bias and its implications for protein synthesis and function. The phylogenetic topology shows that the stalkless crinoid and stalked crinoid are distinct, and the phylogenetic trees generated based on maximum likelihood and Bayesian inference are almost identical at the family and order topology levels. The phylogenetic relationships of each family were fully clarified in four orders. A total of eleven positive selection sites were detected within six genes: cytb, nad2, nad3, nad4, nad4L and nad5. This study reveals the phylogenetic relationships of crinoid species, the mitochondrial gene differences, and the selective pressure on the evolution of stalked crinoids. This study significantly enhanced the crinoid mitochondrial genome database and contributed to a better understanding of the phylogenetic relationships among crinoid echinoderms.

Data aggregation and routing in Mobile Ad hoc network: Introduction to Self-Adaptive Tasmanian Devil Optimization

Mobile Ad-Hoc Network (MANETs) is referred to as the mobile wireless nodes that make up ad hoc networks. The network topology may fluctuate on a regular basis due to node mobility. Each node serves as a router, passing traffic throughout the network, and they construct the network’s infrastructure on their own. MANET routing protocols need to be able to store routing information and adjust to changes in the network topology in order to forward packets to their destinations. While mobile networks are the main application for MANET routing techniques, networks with stationary nodes and no network infrastructure can also benefit from using them. In this paper, we proposed a Self Adaptive Tasmanian Devil Optimization (SATDO) based Routing and Data Aggregation in MANET. The first step in the process is clustering, where the best cluster heads are chosen according to a number of limitations, such as energy, distance, delay, and enhanced risk factor assessment on security conditions. In this study, the SATDO algorithm is proposed for this optimal selection. Subsequent to the clustering process, routing will optimally take place via the same SATDO algorithm introduced in this work. Finally, an improved kernel least mean square-based data aggregation method is carried out to avoid data redundancy. The efficiency of the suggested routing model is contrasted with the conventional algorithms via different performance measures.

On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics

We prove that the smallest minimizer σ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma (f)$$\\end{document} of a real convex function f is less than or equal to a real point x if and only if the right derivative of f at x is non-negative. Similarly, the largest minimizer τ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au (f)$$\\end{document} is greater than or equal to x if and only if the left derivative of f at x is non-positive. From this simple result we deduce measurability and semi-continuity of the functionals σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} and τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. Furthermore, if f has a unique minimizing point, so that σ(f)=τ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma (f)=\ au (f)$$\\end{document}, then the functional is continuous at f. With these analytical preparations we can apply Continuous Mapping Theorems to obtain several Argmin theorems for convex stochastic processes. The novelty here are statements about classical distributional convergence and almost sure convergence, if the limit process does not have a unique minimum point. This is possible by replacing the natural topology on R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} with the order topologies. Another new feature is that not only sequences but more generally nets of convex stochastic processes are allowed.

A review of energy losses in power electronics converters and developments in related application fields

In recent years, the world has been paying more and more attention to the power electronics converter in the power system. The power electronics converter has made great development in its own structure and control strategy, but there are still concerns of significant energy consumption and insufficient energy saving in many fields of application, the purpose of this paper is to let researchers grasp the current situation of the power electronics converter research and advance the development of the industry. Firstly, we review and evaluate some relevant methods that can evaluate the energy consumption of power electronic converters and introduce several cases of improving the topology in order to achieve energy saving. Afterwards, we introduce and summarize the development and application of power electronic converters in specific scenarios. The review and discussion presented in this paper concludes with some limitations and concerns in the current development of power electronics converters.

Second order topology in a band engineered Chern insulator

Haldane model is a celebrated tight binding toy model of a Chern insulator in a 2D honeycomb lattice that exhibits quantized Hall conductance in the absence of an external magnetic field. In our work, we deform the bands of the Haldane model smoothly by varying one of its three nearest neighbour hopping amplitudes (t1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_1$$\\end{document}), while keeping the other two (t) fixed. This breaks the C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} symmetry of the Hamiltonian, while the Mx∗T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_x*T$$\\end{document} symmetry is preserved. The symmetry breaking causes the Dirac cones to shift from the K and the K′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$'$$\\end{document} points in the Brillouin zone (BZ) to an intermediate M point. This is evident from the Berry curvature plots which show a similar shift in the corresponding values as a function of the deformation parameter, namely t1t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{t_1}{t}$$\\end{document}. We observe two different topological phases of which, one is a topological insulator (TI) and the other is a second order topological insulator (SOTI). The Chern number (C) remains perfectly quantized at a value of C=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=1$$\\end{document} for the TI phase and it goes to zero in the SOTI phase. Furthermore, the evolution of the Wannier charge center (WCC) as the band is smoothly deformed shows a jump in the TI phase indicating the presence of conducting edge modes. We also study the SOTI phase and diagonalize the real space Hamiltonian on a rhombic supercell which shows the presence of in-gap zero energy corner modes. The polarization of the system, namely px\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_x$$\\end{document} and py\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_y$$\\end{document}, are evaluated, along the x and the y directions, respectively. We see that both px\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_x$$\\end{document} and py\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_y$$\\end{document} are quantized in the SOTI phase owing to the presence of the inversion symmetry of the system. Finally we establish the SOTI phase as an example of a topological phase with zero Berry curvature and provide an analogy with the two dimensional Su–Schrieffer–Heeger model.

Emergence of Universal Computations Through Neural Manifold Dynamics.

There is growing evidence that many forms of neural computation may be implemented by low-dimensional dynamics unfolding at the population scale. However, neither the connectivity structure nor the general capabilities of these embedded dynamical processes are currently understood. In this work, the two most common formalisms of firing-rate models are evaluated using tools from analysis, topology, and nonlinear dynamics in order to provide plausible explanations for these problems. It is shown that low-rank structured connectivities predict the formation of invariant and globally attracting manifolds in all these models. Regarding the dynamics arising in these manifolds, it is proved they are topologically equivalent across the considered formalisms. This letter also shows that under the low-rank hypothesis, the flows emerging in neural manifolds, including input-driven systems, are universal, which broadens previous findings. It explores how low-dimensional orbits can bear the production of continuous sets of muscular trajectories, the implementation of central pattern generators, and the storage of memory states. These dynamics can robustly simulate any Turing machine over arbitrary bounded memory strings, virtually endowing rate models with the power of universal computation. In addition, the letter shows how the low-rank hypothesis predicts the parsimonious correlation structure observed in cortical activity. Finally, it discusses how this theory could provide a useful tool from which to study neuropsychological phenomena using mathematical methods.

A Hybrid Vehicle Configuration with Zero Emission.

The search for alternatives to petroleum economy, in automotive field, must satisfy the energy requirements and to offer, at least, the same performance than conventional vehicles. This work proposes a vehicle configuration that satisfies its real energy requirements and also obtains a zero level of noxious emissions. This topology is constituted by a fuel cell, a battery and a bank of ultracapacitors. Once the energetic supply is guaranteed, the second problem consists on the power delivery control strategy. The importance of this control is to optimize the hybrid vehicle components operation mode. By means of simulations, this work shows the deficiencies of current control systems with respect to the supply and the energetic efficiency. Moreover, it suggests using a fuzzy control in the proposed topology in order to optimize the efficiency. This control policy is currently being developed by the author.

DEVELOPMENT OF WIRELESS SENSOR NETWORKS EFFICIENT CLUSTERIZATION METHOD

Wireless sensor networks are a critical element of the Internet of Things and are widely used in various fields. Effective clustering is key to energy efficiency and network stability in wireless sensor networks. Optimize the density of clusters, their number and size to increase the stability and service life of the network – It is one of the tasks of clustering wireless sensor networks. Another problem – effectively organize the network topology in order to balance the load and extend the network lifetime. Clustering has proven to be an effective approach for organizing a network into a cohesive hierarchy. However, there are several key issues that affect the practical application of clustering methods in sensor network applications. These issues include the rationale for developing different clustering approaches, classification of proposed approaches based on their goals and design principles, and problems associated with clustering. K-means algorithm is one of the unsupervised machine learning methods for clustering tasks and is widely used in wireless sensor networks. This article proposes a modified K-means clustering algorithm that eliminates the shortcomings of the basic K-means algorithm. The modified algorithm increases network lifetime, reduces power consumption and improves network stability.

Asymmetric edge supercurrents in MoTe2 Josephson junctions.

To investigate the higher order topology in MoTe2, the supercurrent interference phenomena in Nb/MoTe2/Nb planar Josephson junctions have been systematically studied. By analyzing the obtained interference pattern of the critical supercurrents and performing a comparative study of the edge-touched and untouched junctions, it's found that the supercurrent is dominated by the edges, rather than the bulk or surfaces of MoTe2. An asymmetric Josephson effect with a field-tunable sign is also observed, indicating the nontrivial origin of the edge states. These results not only provide initial evidence for the hinge states in the higher order topological insulator MoTe2, but also demonstrate the potential applications of MoTe2-based Josephson junctions in rectifying the supercurrent.

Finite Control Set Model Predictive Control (FCS-MPC) for Enhancing the Performance of a Single-Phase Inverter in a Renewable Energy System (RES)

A single-phase five-level T-type topology has been investigated in this article. This topology has emerged as a viable option for renewable energy systems (RES) due to its inherent benefits. The finite control set model predictive control (FCS-MPC) strategy has been implemented to this topology in order to improve the performance and overall reliability of the system. This control strategy empowers the inverter to predict future behavior based on a discrete set of control signals, enabling precise modulation and high-speed response to system dynamics. In the realm of RES, integration of FCS-MPC with multilevel inverters (MLIs) holds great potential to enhance energy conversion efficiency, grid integration, and overall system reliability. The article is structured to present an overview of the evolving landscape of power electronic systems, and the advantages of FCS-MPC. This paper provides a comprehensive analysis of the FCS-MPC control strategy applied to the single-phase five-level T-type topology. The study covers various aspects including the theoretical framework, hardware development, and experimental evaluation. It is obvious from the analysis that this inverter topology is reliable. Several redundant states make it fault-tolerant which helps in maintaining the output voltage at the same level even in the fault conditions. Additionally, the results show that the output load voltage is maintained at the same level irrespective of load change. Also, output load voltage has maintained the high-quality sinusoidal characteristics as the total harmonic distortion (THD) is very low. With all these features, this system is suitable within the framework of RES.

The complementary contribution of each order topology into the synchronization of multi-order networks.

Higher-order interactions improve our capability to model real-world complex systems ranging from physics and neuroscience to economics and social sciences. There is great interest nowadays in understanding the contribution of higher-order terms to the collective behavior of the network. In this work, we investigate the stability of complete synchronization of complex networks with higher-order structures. We demonstrate that the synchronization level of a network composed of nodes interacting simultaneously via multiple orders is maintained regardless of the intensity of coupling strength across different orders. We articulate that lower-order and higher-order topologies work together complementarily to provide the optimal stable configuration, challenging previous conclusions that higher-order interactions promote the stability of synchronization. Furthermore, we find that simply adding higher-order interactions based on existing connections, as in simple complexes, does not have a significant impact on synchronization. The universal applicability of our work lies in the comprehensive analysis of different network topologies, including hypergraphs and simplicial complexes, and the utilization of appropriate rescaling to assess the impact of higher-order interactions on synchronization stability.

Interplay of magnetism and band topology in Eu[formula omitted]Ca[formula omitted]Mg[formula omitted]Bi[formula omitted] (x=0, 0.5, 0.67) from first principles study

Recent discovery of the time reversal symmetry breaking magnetic Weyl semimetals (WSMs) has created a huge surge of activities in the field of quantum topological materials. In this work, we have studied systematically various magnetic orders, electronic structure and the interplay between the magnetic order and band topology in one such material, EuMg2Bi2 (EMB) and its Ca doped variant using first principle methods within the framework of density functional theory (DFT). Our thorough investigation reveals the existence of various magnetic order driven topological phases (for example, topological insulator in the A-type antiferromagnetic (A-AFM) phase with Eu moments aligned along the crystallographic a or b direction, Dirac semimetal in the A-AFM phase when Eu moments are parallel to c direction and Weyl semimetal in the ferromagnetic (FM) phase with Eu moments pointing along the c direction) in this material. These phases are found to be energetically very close and hence are expected to be tunable from one to the other by an external handle such as magnetic field or chemical substitution. Most importantly, we observe the existence of a single pair of Weyl points (WPs) connecting valence and conduction band very close to the Fermi level (FL) along Γ-A direction in the FM state of EuMg2Bi2 with Eu moments aligned along the c direction making it an ideal Weyl semimetal in its FM state exactly similar to EuCd2As2. On doping 50% and 67% Ca at Eu sites, we observe the single pair of WPs to move even closer to the FL which is highly desirable for application purposes. Further, we observe that the separation between the WPs in the pair decreases in doped compounds compared to that in the parent compound which has direct consequence on anomalous Hall conductivity (AHC). Our first principles calculation of AHC shows high peak values exactly at these WPs and the peak height decreases when we dope the system with Ca. Therefore, Ca doping can be a good external handle to tune AHC in this system.

Computational Studies on Diverse Characterizations of Molecular Descriptors for Graphyne Nanoribbon Structures.

Materials made of graphyne, graphyne oxide, and graphyne quantum dots have drawn a lot of interest due to their potential uses in medicinal nanotechnology. Their remarkable physical, chemical, and mechanical qualities, which make them very desirable for a variety of prospective purposes in this area, are mostly to blame for this. In the subject of mathematical chemistry, molecular topology deals with the algebraic characterization of molecules. Molecular descriptors can examine a compound's properties and describe its molecular topology. By evaluating these indices, researchers can predict a molecule's behavior including its reactivity, solubility, and toxicity. Amidst the captivating realm of carbon allotropes, γ-graphyne has emerged as a mesmerizing tool, with exquisite attention due to its extraordinary electronic, optical, and mechanical attributes. Research into its possible applications across numerous scientific and technological fields has increased due to this motivated attention. The exploration of molecular descriptors for characterizing γ-graphyne is very attractive. As a result, it is crucial to investigate and predict γ-graphyne's molecular topology in order to comprehend its physicochemical characteristics fully. In this regard, various characterizations of γ-graphyne and zigzag γ-graphyne nanoribbons, by computing and comparing distance-degree-based topological indices, leap Zagreb indices, hyper leap Zagreb indices, leap gourava indices, and hyper leap gourava indices, are investigated.

Energy-Efficient Hierarchical Resource Allocation in Uplink–Downlink Decoupled NOMA HetNets

The dense deployment of small cell networks is a key feature of next-generation mobile networks aimed at providing the necessary capacity increase. It is noteworthy that small cell networks employ high-capacity backhaul links on millimeter-wave bands to develop multi-hop topologies in order to mitigate data transmission costs. The current static backhaul infrastructures cannot control severe fluctuating network traffic. To resolve this problem, this paper proposed a novel adaptive backhaul topology with the ability to adapt to different traffic patterns. Based on the graph theory, the adaptive system dynamically allows changes to the hybrid millimeter-wave backhaul architecture, and it also provides the possibility of effective channel allocation to each backhaul link to meet capacity and QoS demands. Also, regarding the importance of green networking in integrated-access-and-backhaul networks we proposed a dynamic optimization model which minimizes the overall energy consumption of UL/DL Decoupled NOMA heterogeneous networks in addition to providing the essential coverage and capacity. The proposed model optimizes user association/power utilization and presents an effective modular and scalable framework for analytical technology-oriented modeling of integrated multi-hop backhauls. The numerical results proved that the joint power optimization and hybrid backhaul architecture can increase the total network throughput by 18 percent compared to the current optimized static architectures. It can also reduce the energy consumption level by 30 percent, and enhance users’ quality satisfaction by 24.5 percent with respect to user distribution patterns.

Consensus of fractional-order multi-agent systems with sampled position data

In this paper, the consensus of fractional-order multi-agent systems with the order α satisfying α∈(0,1] is investigated. A novel distributed control protocol is constructed, in which only the past sampled position data of neighbors are utilized. Then, based on Laplace transform, Mittag-Leffler function, and matrix theory, necessary and sufficient criteria depending on the order, coupling gains, sampling period, and communication topology, are established to attain consensus for the systems. Meanwhile, the intervals of coupling gains and sampling period associated with the maximum eigenvalue of Laplacian matrix over an undirected graph are presented to attain consensus. Finally, the availability of theoretical results is verified by numerical simulations.

Topological Ordered Rings and Measures

Given a ring endowed with a ring order, we provide sufficient conditions for the order topology induced by the ring order to become a ring topology (analogous results for module orders are consequently derived). Finally, the notions of Radon and regular measures are transported to the scope of module-valued measures through module orders. Classical characterizations of these measures are obtained as well as the hereditariness of regularity for conditional ring-valued measures.

Higher order topology in a Creutz ladder

A Creutz ladder, is a quasi one dimensional system hosting robust topological phases with localized edge modes protected by different symmetries such as inversion, chiral and particle-hole symmetries. Non-trivial topology is observed in a large region of the parameter space defined by the horizontal, diagonal and vertical hopping amplitudes and a transverse magnetic flux that threads through the ladder. In this work, we investigate higher order topology in a two dimensional extrapolated version of the Creutz ladder. To explore the topological phases, we consider two different configurations, namely a torus (periodic boundary) and a ribbon (open boundary) to look for hints of gap closing phase transitions. We also associate suitable topological invariants to characterize the non-trivial sectors. Further, we find that the resultant phase diagram hosts two different topological phases, one where the higher order topological excitations are realized in the form of robust corner modes, along with (usual) first order excitations demonstrated via the presence of edge modes in a finite lattice, for the other.