Homotopy Research Articles

From ray tracing to waves of topological origin in continuous media

Continuous media commonly support a discrete number of wave modes that are trapped along interfaces defined by spatially varying parameters. In the case of multicomponent wave problems, those trapped modes fill a frequency gap between different wave bands. When they are robust against continuous deformations of parameters, such waves are said to be of topological origin. It has been realized over the last decades that waves of topological origin can be predicted by computing a single topological invariant, the first Chern number, in a dual bulk wave problem that is much simpler to solve than the original wave equation involving spatially varying coefficients. The correspondence between the simple bulk problem and the more complicated interface problem is usually justified by invoking an abstract index theorem. Here, by applying ray tracing machinery to the paradigmatic example of equatorial shallow water waves, we propose a physical interpretation of this correspondence. We first compute ray trajectories in the phase space given by position and wavenumber of the wave packet, using Wigner-Weyl transforms. We then apply a quantization condition to describe the spectral properties of the original wave operator. This bridges the gap between previous work by Littlejohn and Flynn showing manifestation of Berry curvature in ray tracing equations, and more recent studies that computed the Chern number of flow models by integrating the Berry curvature over a closed surface in parameter space. We find that an integral of Berry curvature over this closed surface emerges naturally from the quantization condition, which allows us to recover the bulk-interface correspondence.

Sharing Traffic Priorities via Cyber–Physical–Social Intelligence: A Lane-Free Autonomous Intersection Management Method in Metaverse

Replacing traffic signals with roadside vehicle-to-infrastructure systems in the era of connected and autonomous vehicles (CAVs) is promising. Managing CAVs in a signal-free intersection, known as autonomous intersection management (AIM), controls the driving behavior of each intersection-traverse CAV to maximize the throughput. Although AIM improves the gross throughput, the fairness of each individual vehicle in its right of way is not seriously considered. This study sets up an AIM system in the cyber–physical–social space to trade traverse priorities quantitatively and fairly. To that end, one needs an AIM method that is optimal and stable, otherwise no convincing trades of traverse priorities could be made. This study proposes a near-optimal lane-free AIM method based on numerical optimal control, wherein log-exp functions are deployed to convexify nondifferentiable collision-avoidance constraints. Besides that, a parameterized social force model (SFM) is proposed to provide a tunable initial guess for numerical optimal control. By tuning the urgency weights in SFM, one may get cooperative trajectories in different homotopy classes, which are further utilized to decide the amount of virtual currency to reward those CAVs who tend to share their traverse priorities. The overall method improves the traverse throughput with individual fairness respected. In experiencing this system, passengers learn how to behave with politeness when they drive manually. Experiments show the efficiency and robustness of the AIM method and also show the efficacy of the overall priority-sharing system.

Dislocation behavior in Cu single crystal joints under the ultrasonically excited high-strain-rate deformation

• The interfacial evolution for ultrasonically welded joints was used to clarify the coupling effects of ultrasonic excitation and high-strain-rate deformation. • Transmission Kikuchi diffraction and transmission electron microscopy were used to analyze grain orientation and deformed microstructures simultaneously. • Ultrasonic excitation dramatically weakened the constraint of high-strain-rate and characteristic orientations on dislocation motion. • The first-principle calculation and applied strain-rate analysis were conducted to discuss the preferred deformation mechanism to accommodate the imposed plastic strain. The coupling effects of ultrasonic excitation and high-strain-rate deformation are the core factors for weld formation during ultrasonic welding. However, interfacial deformation behavior still shrouds in uncertainty because of the contradictory features between mutual dislocation retardation caused by severely frictional deformation and ultrasonic-accelerated dislocation motion. [101] and [111]-oriented Cu single crystals which tended to form geometrically necessary boundaries (GNBs) were selected as the welding substrates to trace the uniquely acoustoplastic effects in the interfacial region under the ultrasonically excited high-strain-rate deformation. It was indicated that for a low energy input, micro-welds localized at the specific interface region, and equiaxed dislocation cells substituting for GNBs dominated in the initial single crystal rotation region. As the welding energy increased, continuous shear deformation drove the dynamic recrystallization region covered by equiaxed grains to spread progressively. Limited discrete dislocations inside the recrystallized grains and nascent dislocation cells at the grain boundaries were observed in [101] and [111] joints simultaneously, suggesting that the ultrasonic excitation promoted motion of intragranular dislocation and pile-up along the sub-grain boundaries. The interfacial morphology before and after expansion of recrystallization region all exhibited the weakening of orientation constraint on dislocation motion, which was also confirmed by the similar micro-hardness in joint interface. The first-principle calculation and applied strain-rate analysis further revealed that ultrasonic excitation enhanced dislocation slipping, and enabled dislocation motion to accommodate severe plastic deformation at a high-strain-rate.

Microstructural characteristics and mechanical response of diffusion bonding Inconel 617 superalloy

This study focuses on the microstructural characteristics and mechanical response of the diffusion-bonded joints of Inconel 617 superalloy. The joints exhibited two types of interfacial microstructure involving the straight interface (i.e., bond line) and the interface with partial migration of grain boundaries (GB). The straight interface was composed of the GB of γ-Ni, M23C6 carbide and α-Al2O3 oxide. The M23C6 carbide was precipitated with a semi-coherence of 33̅3M23C6//111̅γ−Ni, and an amorphous interface was detected between the α-Al2O3 and γ-Ni phase. The migration of GB across the interface was mainly caused by dynamic recrystallization and strain-induced GB migration. In addition, the α-Al2O3 oxide was remained inside the migrated grain. The formation of the M23C6 and α-Al2O3 was governed by diffusion mechanism, and the γ′-Ni3Al phase was formed surrounding the α-Al2O3 due to the abundant diffusion of Al. The straight interface, particularly embedded with the hard particles (i.e., the M23C6 carbide and α-Al2O3 oxide), restricted GND slipping, resulting in concentration of stresses and initiation of cracks. The migrated interface, in contrast, effectively prevented crack propagation and induced crack deflection, supporting more continuous and uniform plastic deformation. The elongation of 51.2 % was obtained for the joint with partial migrated interface, reaching 71 % of the base metal. Nevertheless, the optimum impact toughness of the joints (20.38 J/cm2) was merely 24 % of the base metal, mainly affected by the remained straight interface.

Effects of temperature on the deformation of 6H–SiC during nanoscratching

With the aid of large-scale molecular dynamics simulations, this paper comprehensively investigated the effects of processing temperature on the deformation of single crystal 6H–SiC during nanoscratching. The effect was investigated across a wide range of temperature variations, from 1 K to 900 K. It was found that with increasing the processing temperature, the material removal mechanism of 6H–SiC experience a transition from intermittent cleavage to continuous plastic deformation. The deformation mechanism of 6H–SiC is achieved by the combination of surface amorphization and subsurface dislocations that reside mainly in the cubic diamond structural layer. The processing temperature significantly affects the dislocation distribution, groove morphology, scratching forces and coefficient of friction. The anisotropic effect on chip formation decreases with increasing the processing temperature. By evaluating surface roughness and the maximum subsurface damage depth, the investigation concludes that the scratched surface/subsurface quality of 6H–SiC deteriorates with increasing processing temperature.

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Given a closed smooth manifold M of even dimension 2n≥6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n\\ge 6$$\\end{document} with finite fundamental group, we show that the classifying space BDiff(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\ extrm{Diff}(M)$$\\end{document} of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold N⊂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\subset M$$\\end{document} of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.

An FDM-Based Dynamic Zoning Method for Disturbed Rock Masses above a Longwall Mining Panel

Underground longwall mining can seriously disturb the surrounding rock masses above the panel. A surface cracking zone, continuous deformation zone, fractured zone, and caved zone can be formed in the overlying strata (termed the “four zones”), which may further result in the spontaneous combustion of coal seams and water inrush. It is essential to study and predict the development characteristics of the four zones induced by longwall mining to guarantee mining safety. These four zones are developed during the mining process, and the mechanical properties in different regions correspondingly differ. Thus, the dynamic zoning characteristics of the disturbed rock masses should be considered in any simulations. In this paper, an FDM-based dynamic zoning method for disturbed rock masses above a longwall mining panel is proposed. This method is mainly composed of four stages: (1) establishing a simplified complete stress-strain curve; (2) determining the zoning criteria; (3) adaptively adjusting the mechanical parameters of the disturbed rock mass; and (4) numerically modeling the longwall mining based on the FDM. The proposed method was applied to a study site in the Taixi coal mine. The dynamic development process of the four zones induced by longwall mining was clearly observed in the modeling procedure. The numerical modeling results achieved in this work, including the periodical coal-seam roof caving and the dynamic development characteristics of the four zones, are consistent with the observed distribution and other studies. The heights of the caved and fractured zones are basically consistent with the empirical formula. Thus, the dynamic zoning method can analyze and predict the dynamic development characteristics of four regions.

G∞–ring spectra and Moore spectra for β–rings

In this paper, we introduce the notion of $G_\infty$-ring spectra. These are globally equivariant homotopy types with a structured multiplication, giving rise to power operations on their equivariant homotopy and cohomology groups. We illustrate this structure by analysing when a Moore spectrum can be endowed with a $G_\infty$-ring structure. Such $G_\infty$-structures correspond to power operations on the underlying ring, indexed by the Burnside ring. We exhibit a close relation between these globally equivariant power operations and the structure of a $\beta$-ring, thus providing a new perspective on the theory of $\beta$-rings.

Self-supporting electrodes with in situ built aniline on carbon fibers and reduced graphene oxide covalently for stable flexible supercapacitors

Practical applications of flexible electronics have challenged the stability and cycle performance of flexible supercapacitor under continuous mechanical deformation, thus flexible electrode with stable structure has to be considered to maintain high-quality performance. A covalent modification strategy is present here to improve the stability of electrodes. Graphitized carbon cloth (GCC) and reduced graphene oxide (rGO) that modified with aniline by Suzuki reaction, are covalently bonded together with the polymerization of aniline (PANI). The electrode (GCC-PANI-rGO) modified by covalent bonds is stable enough to remain 97.73 % initial specific capacitance after charge-discharge 15,000 times at 40 mA/cm2. The capacitance retention of asymmetric flexible solid supercapacitor (FSSC) prepared by GCC-PANI-rGO and GCC is 108.78 % after 10,000 times mechanical folding. The electrode formed a π-π conjugation system through Suzuki reaction possesses low charge transfer resistance (0.428 Ω), and exhibits high specific capacitance and energy density that up to 11.825 F/cm2 and 1.642 mWh/cm2 at 1 mA/cm2. The energy density of FSSC is 0.266 mWh/cm2 at 1 mA/cm2 and surpasses most FSSC based on PANI and graphene. This work gives an essential covalently synthetic strategy for flexible supercapacitor with stable electrochemical performance in application.

Constraints on families of smooth 4–manifolds from Pin−(2)–monopole

Using the Seiberg-Witten monopole equations, Baraglia recently proved that for most of simply-connected closed smooth $4$-manifolds $X$, the inclusions $\mathrm{Diff}(X) \hookrightarrow \mathrm{Homeo}(X)$ are not weak homotopy equivalences. In this paper, we generalize Baraglia's result using the $\mathrm{Pin}^{-}(2)$-monopole equations instead. We also give new examples of $4$-manifolds $X$ for which $\pi_{0}(\mathrm{Diff}(X)) \to \pi_{0}(\mathrm{Homeo}(X))$ are not surjections.

Quantized topological response in trapped quantum gases

In this letter, we propose a quantized topological response in trapped 1D quantum gases. The experimental protocol for the response requires the application of an instant optical pulse to a half-infinite region in an asymptotically harmonic trap and measuring the density distribution. We show that the corresponding linear response is described by a universal quantized formula in the thermal dynamical limit, which is invariant under local continuous deformations of the trapping potential $V$, atom distribution $f_\Lambda$, the spatial envelope of the optical pulse $\Theta_p$, and the measurement region $\Theta_m$. We test the statement by various numerical analysis, the result of which is consistent with the analytical prediction to high accuracy. We further show that a short but finite optical pulse duration only results in a violation of the quantization near the transition time, which suggests that quantized response could be observed in realistic experiments. We also generalize our results to non-linear quantized topological responses for atoms in higher dimensional harmonic traps.

Equivariant homotopy equivalence of homotopy colimits of G-functors

Given a group G and a G-category {textbf{C}}, we give a condition on a diagram of simplicial sets indexed by {textbf{C}} that allows us to define a natural action of G on its homotopy colimit, and some other simplicial sets defined in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to equivariant versions.

Effect of Deformation Degree on Microstructure and Texture of AZ31 Magnesium Alloy in Hot Drawing

Microstructure and texture evolution of the AZ31 magnesium alloy wire under different deformation degrees was investigated by continuous hot drawing deformation. With the increase of cumulative deformation degree, the grain refinement effect is remarkable. However, there is a critical deformation degree, and excessive deformation will increase the grain size. During the deformation process of hot drawing, the dynamic recrystallization mechanism of AZ31 magnesium alloy wire changed from twin dynamic recrystallization to rotary dynamic recrystallization. With the increase of cumulative deformation, the initial {0001} <11-20> silk texture gradually changed into {0001} <10-10> silk texture.

An innovative destressing technology and key parameters determination in both sides of a deep roadway

The general or single supporting theory and technology of the shallow surrounding rock of the roadway are not suitable for solving the problem of continuous large deformation of the both sides under the continuous migration of coal mass in the deep domain of the roadway side. Furthermore, the general destressing technology of dense drilling in the roadway destroys the shallow anchorage domain while releasing the stress. Therefore, this study proposes the “shallow supporting and deep destressing” synergism technology. This technology provides puissant supporting in the shallow domain of roadway sides, and at the same time, large destressing holes are excavated at the coal mass migration channel in deep stress peak domain far from the anchorage domain, conducting destressing regulation of roadway sides. This technology can shift stress peak domain of the roadway side to solid coal side of destressing hole without destroying the shallow anchorage domain, and at the same time, provide a buffer space for that coal mass in the deep domain of the roadway side continuously migrates to the anchorage surrounding rock, creating a beneficial stress circumstances for the roadway stability. The “shallow supporting and deep destressing” synergism technology can solve the contradiction between the shallow surrounding rock supporting and the continuous migration of coal mass in deep domain. The field application results show that the innovative destressing technology can effectively solve the problem of surrounding rock control in deep roadway.

Realisations of elliptic operators on compact manifolds with boundary

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.

A complete derived invariant for gentle algebras via winding numbers and Arf invariants

Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections of the associated surface. We associate to each gentle algebra a line field on the corresponding surface and prove that the derived equivalence class of the algebra is completely determined by the homotopy class of the line field up to homeomorphism of the surface. Then, based on winding numbers and the Arf invariant of a certain quadratic form over {mathbb {Z}}_2, we translate this to a numerical complete derived invariant for gentle algebras.

Influence of Discontinuous Linear Deformation on the Values of Continuous Deformations of a Mining Area and a Building Induced by an Exploitation of Hard Coal Seam

In this article, the impact of a ground step on a residential building and a mining terrain has been presented. Influence of a discontinuous linear deformation by the changes of the inclination and curvatures’ values (continuous deformations) has been observed. Discontinuous and continuous deformations have been caused by an exploitation of the 405/1 hard coal seam, located at a depth of 550 m. Extraction was carried out with the use of the longwall system to the south of the building. A discontinuous linear deformation occurred parallel to a north wall of the building and had a height of 0.2 m. The inclination and curvatures’ values have been calculated on the basis of the results of the geodesic surveys. Vertical displacements and horizontal distances between the measuring points in the ground and on the building’s walls have been measured. Points in two perpendicular directions (parallel and perpendicular to a ground step) have been stabilized. The observational network consisted of 12 points (3 points in each direction, in the ground and on the walls). Research has shown that inclinations and curvatures have different values on a terrain’s surface and in buildings, which means that a deformation process takes place in a different way in the ground and a building. The obtained results indicate that an occurrence of a discontinuous linear deformation near the building reduces the values of inclinations and curvatures of a terrain surface and a building in a parallel direction to its longitudinal axis, and it increases the values of the continuous deformations in the direction perpendicular to it.

Overlying main roof breaking characteristic and its effect on the stability of gob-side entry

In this study, to figure out the deformation mechanism, a particle flow model was used to simulate the deformation of surrounding rock in a deeply buried entry. The fracture evolution, stress, and porosity of surrounding rock in the process of the main roof breaking, meanwhile, the displacement after entry being excavated were recorded and analyzed in the simulation. The results show that the main roof breaking is different from the ideally elastic–plastic simply supported beam for it has two more plastic zones on both sides of the beam and its support points tend to move. In addition, the stress reduction zone in the coal seam is about 10 m where the coal body and surrounding rock are almost completely broken and porosity is surprisingly increasing from 0.01 to 0.39 in coal and 0.23 in the main roof. The displacement concentrated on the ribs of entry, especially on the coal pillar. The continuous and large deformation that appears in the stress reduction zone results from the nearly completely broken state of the surrounding rock. The tradeoff between the stress reduction and the broken state of the surrounding rock should be considered when choosing the location of the gob-side entry.

The Hercynian tectonics in the Tassili-n-Ajjers area, Algeria: A possible continuous stress-strain regime?

This work shed some light on the multiple tectonic phases occurred during the late Hercynian Orogeny that is related to the global plate tectonic geodynamics and their effects in the Tassili-n-Ajjers plateau area, which is located in the south-eastern edge of Saharan platform in Algeria, and known to be the southern margin of the Paleozoic Illizi Basin, wherewell preserved Paleozoic sedimentary rocks outcrop. Three Hercynian tectonic phases have been evidenced in the area are: Early Carboniferous (E1), Late Carboniferous-Permian (E2), and a Late Permian-Early Triassic (E3). The Hercynian tectonic regime in the Tassili-n-Ajjers area that was dominated by strike-slip movements during E1 and E2 tectonic phases changed into an extensional regime during E3 tectonic phase. Paleo-strain determination, folds axis distribution and the kinematic analyses of the strain field of the Hercynian active tectonic show a continuous deformation regime rather than discrete tectonic phases. Moreover, the progressive deformation regime is strongly supported by the absence of tectonic unconformities between Late Devonian/Early Carboniferous - Early Permian. At a large-scale, this continuous deformation appears as a result of a progressive closure of Rheic and western Paleotethys Oceans, and the clockwise rotation of African plate which occurred during the Hercynian/Variscan orogeny. Therefore, we could consider that the most part of Algerian Saharan platform has registered the far-field effects of this orogeny and provides a suitable illustration of an intracontinental deformation.

Completely Elastic Deformation of Hydrogenated Ta Thin Films

Nanostructured metal hydrides could play a key role in a hydrogen economy. The nanostructuring or confinement of these materials as, e.g., thin films significantly affects the structural and functional properties. For tantalum hydride, a versatile hydrogen sensing material, we show that the confinement of tantalum as a thin film extends the solubility limit by suppressing the phase transition observed in bulk upon hydrogenation. Different from bulk, the body centered cubic unit cell continuously deforms with unequal lattice constants and angles between lattice vectors. This deformation ensures that the volumetric expansion is realized in the out-of-plane direction, and surprisingly, completely elastic in nature. The first-order phase transition suppression combined with the continuous elastic deformation of the tantalum unit cell over an extraordinary wide solubility range ensures the superb performance of tantalum and its alloys as a hysteresis-free optical hydrogen sensing range over a hydrogen pressure/concentration range of over 7 orders of magnitude.