Some topological properties of halfgroupoids technical report no. 8
Some topological properties of halfgroupoids
MATHEMATICS↗
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Linear systems with structure group and their feedback invariants
A general method described by Hermann and Martin (1976) for the study of the feedback invariants of linear systems is considered. It is shown that this method, which makes use of ideas of topology and algebraic geometry, is very useful in the investigation of feedback problems for which the classical methods are not suitable. The transfer function as a curve in the Grassmanian is examined. The general concepts studied in the context of specific systems and applications are organized in terms of the theory of Lie groups and algebraic geometry. Attention is given to linear systems which have a structure group, linear mechanical systems, and feedback invariants. The investigation shows that Lie group techniques are powerful and useful tools for analysis of the feedback structure of linear systems.
Martin, C.↗
Toward Time Synchronization in Delay Tolerant Network based Solar System Internetworking
The expanding presence in space will place an increased dependency on networked communications– a scalable communications infrastructure; that is, the Solar System Internet (SSI). Upcoming developments towards a SSI include NASA’s upcoming LunaNet, or lunar Internet, which provides multi-hop multi-path communications using Delay Tolerant Networking (DTN). DTN has been an active area of research and development, particularly in routing, security, and optimization. DTNs are marked by mobility, disconnection, and a wide variance of latencies (propagation and processing delays). In this paper, we outline progress towards a theory of time synchronization across such a network. An underlying assumption of DTN is that the network is time synchronized already, rather than synchronization being provided as a service. While this is necessary for schedule-based routing, which is necessarily prevalent in DTNs, it is so deeply ingrained as to be built into the primary unit of data in DTNs– the bundle. Indeed, a bundle’s creation timestamp and its time to live (called the lifetime) are based on time, and there are special recommendations for systems that lack accurate clocks. The assumption of time synchronization makes sense when limiting considerations to smaller-scale and more traditional space communication. However, just as end-to-end connectivity cannot be guaranteed in DTNs, neither can access to a reference or authoritative clock. In this more general case, it might be necessary to synchronize over time-varying meshes, and perhaps even to consider relativistic effects. Moreover, by imposing synchronization restrictions in order to sustain a network, the effectiveness of the network to achieve scalability will be necessarily muted. To work towards a time synchronization theory for DTNs, we build upon past successes in modeling DTNs using time-varying graphs and sheaves. This includes error and limitation estimation, which allows one to define domains over which schedule-based routing is possible, up to some threshold sensitivity. Despite the theoretical nature of these results, the approaches taken are also algorithmic, and hence lend themselves to practical implementations. The paper concludes with comparisons of the various methods along with suggestions for future work.
Delay Tolerant Networking↗
Revealing topology in metals using experimental protocols inspired by K-theory
Topological metals are conducting materials with gapless band structures and nontrivial edge-localized resonances. Their discovery has proven elusive because traditional topological classification methods require band gaps to define topological robustness. Inspired by recent theoretical developments that leverage techniques from the field of C*-algebras to identify topological metals, here, we directly observe topological phenomena in gapless acoustic crystals and realize a general experimental technique to demonstrate their topology. Specifically, we not only observe robust boundary-localized states in a topological acoustic metal, but also re-interpret a composite operator—mathematically derived from the K-theory of the problem—as a new Hamiltonian whose physical implementation allows us to directly observe a topological spectral flow and measure the topological invariants. Our observations and experimental protocols may offer insights for discovering topological behaviour across a wide array of artificial and natural materials that lack bulk band gaps.
36 MATERIALS SCIENCE↗
Automatic algebraic coordinate generation
A computer software system has been developed to automatically generate two-dimensional coordinates from algebraic transformations. For topologically complex regions, a smooth assembly of the transformations can be used to automatically produce a composite mesh where a general gridded format is retained. The algebraic mesh generation system consists of a collection of operator subroutines which are applied to an established data structure and which automatically perform the necessary parts of mesh construction from a sequence of multisurface transformations. The system operators are discussed, taking into account the data base, the order of application, direct surface generators, geometric surface operators, surface generators from existing surfaces, transverse operators, mesh operators, assembly operators, and data visualization operators. Attention is given to applications related to airfoils.
Eiseman, P. R.↗
On families of systems and deformations
Until recently, there has not been any systematic effort towards a theory of systems with parameter variations. It is presently suggested that the concept of families of systems is basic to such an effort, and entails the techniques of Lie theory, differential geometry, and algebraic geometry. Attention is given to the geometric characterization of certain families of systems that appear in control and identification problems. The ways in which families of systems degenerate as parameter variations become large are isolated, using the topological rather than algebraic geometric case.
Krishnaprasad, P. S.↗
Densification and coarsening during solid state sintering of ceramics: A review of the models. II - Grain growth
Two processes occur simultaneously during the sintering of a ceramic powder compact: densification and coarsening (or grain growth). Both processes have as their driving force the reduction of the excess free surface energy of the powder particles. Several different mechanisms of atom transport, operating concurrently or consecutively, may be responsible for the two processes. Algebraic, geometric and topological models have been proposed and refined in attempts to determine the mechanism, or mechanisms, responsible for densification under defined processing conditions. These efforts have met with varying degrees of success. Recently, it has become apparent that more attention must be paid to the coarsening processes during sintering. The models for both densification and coarsening during solid state sintering are reviewed with particular emphasis on their applicability to engineering ceramics.
Shaw, Nancy J.↗
Topological Rigidity and Non-Abelian Defect Junctions in Chiral Nematic Systems with Effective Biaxial Symmetry
We study topologically stable defect structures in systems where the defect line classification in three dimensions and associated algebra of interactions (the fundamental group) are governed by the non-Abelian eight-element group, the quaternions 𝑄 8 . The non-Abelian character of the defect algebra leads to a topological rigidity of bound defect pairs, and trivalent junctions which are the building blocks of multijunction trivalent networks. We realize such structures in laboratory chiral nematics and analyze their behavior analytically, along with numerical modeling.
Liquid crystals↗
Hypernetwork Science: From Multidimensional Networks to Computational Topology
As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra, lattice theory, and topology. They are able to represent complex systems interactions more faithfully than graphs and networks, while also being some of the simplest classes of systems representing topological structures as collections of multidimensional objects connected in a particular pattern. In this paper we discuss the role of (undirected) hypergraphs in the science of complex networks, and provide a mathematical overview of the core concepts needed for hypernetwork modeling, including duality and the relationship to bicolored graphs, quantitative adjacency and incidence, the nature of walks in hypergraphs, and available topological relationships and properties. We close with a brief discussion of two example applications: biomedical databases for disease analysis, and domain-name system (DNS) analysis of cyber data.
Joslyn, Cliff A.↗
Newton-Euler Dynamic Equations of Motion for a Multi-body Spacecraft
The Magnetospheric MultiScale (MMS) mission employs a formation of spinning spacecraft with several flexible appendages and thruster-based control. To understand the complex dynamic interaction of thruster actuation, appendage motion, and spin dynamics, each spacecraft is modeled as a tree of rigid bodies connected by spherical or gimballed joints. The method presented facilitates assembling by inspection the exact, nonlinear dynamic equations of motion for a multibody spacecraft suitable for solution by numerical integration. The building block equations are derived by applying Newton's and Euler's equations of motion to an "element" consisting of two bodies and one joint (spherical and gimballed joints are considered separately). Patterns in the "mass" and L'force" matrices guide assembly by inspection of a general N-body tree-topology system. Straightforward linear algebra operations are employed to eliminate extraneous constraint equations, resulting in a minimum-dimension system of equations to solve. This method thus combines a straightforward, easily-extendable, easily-mechanized formulation with an efficient computer implementation.
Stoneking, Eric↗
Strategies Toward Automation of Overset Structured Surface Grid Generation
An outline of a strategy for automation of overset structured surface grid generation on complex geometries is described. The starting point of the process consists of an unstructured surface triangulation representation of the geometry derived from a native CAD, STEP, or IGES definition, and a set of discretized surface curves that captures all geometric features of interest. The procedure for surface grid generation is decomposed into an algebraic meshing step, a hyperbolic meshing step, and a gap-filling step. This paper will focus primarily on the high-level plan with details on the algebraic step. The algorithmic procedure for the algebraic step involves analyzing the topology of the network of surface curves, distributing grid points appropriately on these curves, identifying domains bounded by four curves that can be meshed algebraically, concatenating the resulting grids into fewer patches, and extending appropriate boundaries of the concatenated grids to provide proper overlap. Results are presented for grids created on various aerospace vehicle components.
overset↗
Characterizing Topological Order with Matrix Product Operators
One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of $\mathsf G$- and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Classifying photonic topology using the spectral localizer and numerical K -theory
Recently, the spectral localizer framework has emerged as an efficient approach for classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts such as topological solitons and topological dynamics, where a system’s occupation induces a local change in its topology due to nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of C*-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches for classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework’s application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell’s equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical K-theory, two mathematical concepts that form the foundation for the spectral localizer framework. Finally, we provide some mathematically oriented comments on the C*-algebraic origins of this framework, including a discussion of real C*-algebras and graded C*-algebras that are necessary for incorporating physical symmetries. Looking forward, we hope that this tutorial will serve as an approachable starting point for learning the foundations of the spectral localizer framework.
97 MATHEMATICS AND COMPUTING↗
Generalized symmetries of the graviton
We find the set of generalized symmetries associated with the free graviton theory in four dimensions. These are generated by gauge invariant topological operators that violate Haag duality in ring-like regions. As expected from general QFT grounds, we find a set of “electric” and a dual set of “magnetic” topological operators and compute their algebra. To do so, we describe the theory using phase space gauge-invariant electric and magnetic dual variables constructed out of the curvature tensor. Electric and magnetic fields satisfy a set of constraints equivalent to the ones of a stress tensor of a 3d CFT. The constraints give place to a group $\mathbb{R}^{20}$ of topological operators that are charged under space-time symmetries. Finally, we discuss similarities and differences between linearized gravity and tensor gauge theories that have been introduced recently in the context of fractonic systems in condensed matter physics.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Noncompact atomic insulators
Here we study the conditions for Bloch bands to be spanned by symmetric and strictly compact Wannier states that have zero overlap with all lattice sites beyond a certain range. Similar to the characterization of topological insulators in terms of an algebraic (rather than exponential) localization of Wannier states, we find that there may be impediments to the compact localization even of topologically “trivial” obstructed atomic insulators. These insulators admit exponentially localized Wannier states centered at unoccupied orbitals of the crystalline lattice. First, we establish a sufficient condition for an insulator to have a compact representative. Second, for $\mathscr{C}_2$ rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for $\mathscr{C}_4$ symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a $\mathscr{C}_3$-symmetric atomic insulator, we explicitly derive that there are no compact Wannier states overlapping with less than 18 lattice sites. We conjecture that this obstruction generalizes to all finite Wannier sizes. Our results can be regarded as the stepping stone to a generalized theory of Wannier states beyond dipole or quadrupole polarization.
36 MATERIALS SCIENCE↗
Semilinear (topological) spaces and applications
Semivector spaces are defined and some of their algebraic aspects are developed including some structure theory. These spaces are then topologized to obtain semilinear topological spaces for which a hierarchy of local convexity axioms is identified. A number of fixed point and minmax theorems for spaces with various local convexity properties are established. The spaces of concern arise naturally as various hyperspaces of linear and semilinear (topological) spaces. It is indicated briefly how all this can be applied in socio-economic analysis and optimization.
Prakash, P.↗
Nonreciprocal superfluidlike topological spin transport
We study superfluidlike spin transport facilitated by thermal diffusion of magnetic domain walls, where the positive and negative chiralities of domain walls act as opposite topological charges. The topological charge conservation leads to algebraic decay of the spin current carried by domain walls, allowing for the transport of spin over extended distances. We demonstrate that the presence of the Dzyaloshinskii-Moriya interaction can lead to nonreciprocity in spin flow, thus effectively realizing a spin ratchet. In one scenario, the nonreciprocity arises due to diode-like behavior where the nucleation of domain walls is governed by thermal activation for one direction of spin current and by viscous injection for the other direction of spin current. Here, we confirm our predictions by micromagnetic simulations of domain walls in TmIG nanowire.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry
We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D. The gauge group is constructed from the volume-preserving diffeomorphisms of space. We show that the intriguing physics of the lowest Landau level (LLL) limit can be interpreted as the consequences of the symmetry. We derive the renowned Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee term within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL effective action of vortex crystal in rotating Bose gas as well as Wigner crystal of electron in an applied magnetic field. We show that the nonlinear sigma models of ferromagnets in 2+1D and 3+1D exhibit the higher-rank gauge symmetries that we introduce in this paper. We interpret the fractonic behavior of the excitations on the lowest Landau level and of skyrmions in ferromagnets as the consequence of the higher-rank gauge symmetry.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗