Homotopy Research Articles

A novel process for fabricating Mg-AZ31B/Al 6063 composite clad sheets that significantly improved mechanical behavior and microstructure via combining continuous shear deformation and direct extrusion

A novel process for fabricating Mg-AZ31B/Al 6063 composite clad sheets that significantly improved mechanical behavior and microstructure via combining continuous shear deformation and direct extrusion

A global Morse index theorem and applications to Jacobi fields on CMC surfaces

In this paper, we establish a “global” Morse index theorem. Given a hypersurface [Formula: see text] of constant mean curvature, immersed in [Formula: see text]. Consider a continuous deformation of “generalized” Lipschitz domain [Formula: see text] enlarging in [Formula: see text]. The topological type of [Formula: see text] is permitted to change along [Formula: see text], so that [Formula: see text] has an arbitrary shape which can “reach afar” in [Formula: see text], i.e. cover any preassigned area. The proof of the global Morse index theorem is reduced to the continuity in [Formula: see text] of the Sobolev space [Formula: see text] of variation functions on [Formula: see text], as well as the continuity of eigenvalues of the stability operator. We devise a “detour” strategy by introducing a notion of “set-continuity” of [Formula: see text] in [Formula: see text] to yield the required continuities of [Formula: see text] and of eigenvalues. The global Morse index theorem thus follows and provides a structural theorem of the existence of Jacobi fields on domains in [Formula: see text].

Comparative lithology and alteration mineral chemistry of host rocks at the Seabee Gold Operation, northern Saskatchewan, Canada: Implications for a connected orogenic gold-mineralizing system

The Seabee Gold Operation, located in northern Saskatchewan, Canada, hosts the Santoy and Seabee gold deposits, as well as the Fisher property; the latter is an important exploration target. This gold mineralization is hosted in rocks of the Pine Lake greenstone belt that is within the Glennie domain of the larger Paleoproterozoic Trans-Hudson orogen. Mineralization is focused within shear zones, including the Santoy shear system that extends south onto the Fisher property. The role of ore-adjacent intrusions, systematics of gold mineralization, and relationship between the mineralization of the properties are largely undefined. In this study, hydrothermally-altered samples from locations along the Santoy shear system, namely the Santoy 7, Santoy Gap, and the Fisher property, were analyzed to identify and compare the mineralogical and geochemical characteristics related to sulfide-gold mineralization and associated alteration.The altered host rocks primarily comprise upper greenschist to lower amphibolite-grade sheared metabasalt to chlorite-hornblende-biotite ± garnet schist, and intrusive rocks dominated by granodiorite and tonalite. These lithologies are cut by similar quartz vein generations and are associated with similar alteration assemblages and mineralization, including (i) syn-late-D3 “distal” (∼0.4 to 3 m wide) sericite-biotite-chlorite-clinozoisite ± albite, titanite to vein-adjacent coarse-grained diopside-K-feldspar-albite-actinolite ± apatite, titanite calc-silicate alteration, associated with main-stage Py2a, 2b-Cpy2-Po1-Au (syn-late-D3) mineralization; (ii) remobilized, fracture-filling Py3-Po2-Cpy3-Au-BiTel (late-D3) mineralization within the calc-silicate assemblage; and (iii) early colloform Py1a (pre-syn-D3) and intrusion-hosted, inclusion-rich Py1b (pre-D3). Microprobe analyses reveal similar mineralogical trends between Fisher and Santoy alteration minerals typical of orogenic gold systems. Empirical geothermobarometric calculations of gold-associated vein-adjacent and vein-hosted chlorite and biotite exhibit orogenic-style formation conditions (∼320–330 °C, 3 kbar, ∼9 km depth). As such, these analyses suggest that the Fisher property and Santoy Mine Complex underwent inter-connected, multi-stage hydrothermal processes. These stages comprise (i) a potential early introduction of gold into the system closer to the timing of plutonism (Py1b magmatic-hydrothermal; pre-D1) or during the onset of gold mineralization (Py1a; orogenic; late-D2 to early-syn-D3) during the evolution of the Reindeer zone; (ii) orogenic main-stage mineralization (syn-late-D3) concurrent with late- to post- collisional deformation throughout the Reindeer zone; and (iii) remobilization during continuous deformation (late-D3) where elevated cyclical fluid fluxes and higher fracture density at Santoy produced higher-grade mineralization.

Mechanical properties and microstructure of Al–Mg (5052) alloy processed by equal-channel angular pressing (ECAP) with variation of ECAP routes and heat treatment

Equal-Channel Angular Pressing (ECAP) has become an effective technique of severe plastic deformation designed to produce ultrafine grain metals with improved mechanical properties, such as a good combination of strength and ductility. A report on the effect of ECAP routes on the mechanical and microstructure of commercial 5052 aluminum alloy needs also to be included. This work has been undertaken, in order to obtain the results. In this work, several deformation routes were used to process the Al – Mg (5052) alloy, namely A, Ba, Bc and C. Deformation route A involved repeatedly pushing the sample into the ECAP die without rotation, route Ba was performed by rotating the sample through 90° in alternate directions between each pass, route Bc by rotating the sample 90° in the same sense between each pass and route C by rotating the sample 180° between passes. The addition of the pass number decreases the grain size of ECAP-processed samples when compared to the as-annealed sample. It also confirmed that the microstructure of the 8-pass samples shows a finer grain size than the as-annealed sample. Furthermore, the Bc route (samples rotated in the same sense by 90° between each pass) has been proven to be the most effective deformation route, in order to obtain equiaxed ultrafine grain structure when compared to other deformation routes. This phenomenon takes place due to the continuous deformation in all cubic planes. The restoration after the 4-pass number will lead to the rapid evolution of sub-grains to high-angle grain boundaries, forming equiaxed grains. The characterization of the hardness number also shows that the addition of the ECAP pass number increases the hardness number of 5052 aluminum alloy, where samples processed with the Bc route indicate the highest hardness number at 168.4 HB. Moreover, a similar phenomenon also suggests that the tensile strength of all ECAP deformation routes has comparable values. The effect of heat treatment for samples with the Bc route also shows that 200 °C annealed samples have the highest hardness number and tensile strength when compared to other samples.

“Subvolume Splitting” DVC applied for accurate discontinuous deformation measurement and fracture parameter extraction

As a powerful experimental tool for quantifying internal full-field mechanical response, digital volume correlation (DVC) has been increasingly used for extracting crucial fracture parameters. Nevertheless, due to the limitations of its regularly-used shape functions in describing discontinuous deformation, routine subvolume-based DVC encounters challenges in accurately capturing displacement maps around discontinuities, thus impeding reliable and accurate extraction of fracture parameters. To address this issue, a “subvolume splitting” DVC (SS-DVC) method is proposed. SS-DVC first splits subvolumes into two subblocks along a pre-calculated fitted crack plane and then only tracks the subblock with continuous deformation, thereby circumventing the limitations associated with discontinuities. Additionally, a straightforward and effective computational strategy is devised to ensure correct displacement measurements around discontinuities. For validation, both numerical simulation and real three-point bending tests were conducted to examine the performance of SS-DVC in terms of displacement measurement accuracy and extraction of fracture parameters. The experimental results demonstrate that, in comparison to routine DVC, SS-DVC exhibits a significant reduction in measurement errors of displacement around discontinuities, thus enabling more accurate determination of material fracture mechanical properties.

Finkelstein–Rubinstein constraints from ADHM data and rational maps

We establish simple formulae for computing Finkelstein–Rubinstein signs for Skyrme fields generated in two ways: from instanton ADHM data, and from rational maps. This may be used to compute homotopy classes of general loops in the configuration spaces of skyrmions, and as a result provide a useful tool for a quantum treatment beyond rigid-body quantisation of skyrmions.

Simulating multi-material specimen manufacturing from VZh159 and CuCr1Zr alloys via SLM method: Computational modeling and experimental findings

Manufacturing of multi-material products through layer-by-layer synthesis poses various challenges encompassing process parameter optimization, equipment calibration, and the mitigation of warping and internal stresses within the manufactured parts. The article investigates the feasibility of simulating the selective laser melting (SLM) process for manufacturing multi-material components, exemplified through specimens composed of the VZh159 nickel alloy and CuCr1Zr copper alloy. The study entails numerical simulations of the printing process, which were then validated against real specimens produced through SLM. Each test specimen was vertically divided into three parts: the top and bottom sections consisted of the VZh159 alloy, while the central part was composed of the CuCr1Zr alloy. Simulations involved using identical process parameters as employed in the printing process. Thermal and mechanical analyses for each part of the multi-material specimen were sequentially addressed, transferring the outcomes of the preceding analysis as initial conditions for subsequent calculations. The study concludes that while the obtained simulation results are indicative, they do not precisely capture the deformation observed in the specimens manufactured via the SLM method. The numerical values of deformations derived from simulation results slightly underestimate the actual deformations, attributed to limitations in the chosen calculation algorithms. For future utilization of numerical computer simulation in the SLM manufacturing of multi-material specimens, the study suggests the necessity of implementing a seamless, continuous simulation process without transitions between different parts of the specimen. This entails considering the entire manufacturing process without segregating sections, ensuring a comprehensive account of continuous deformation and stress accumulation throughout fabrication.

Comparative study of Ni-TaC composites via high-energy ball milling activation and spark plasma sintering: Reinforcement, densification, oxidation resistance, and mechanical property

In this study, metal matrix composites (MMCs) were produced using nickel (Ni) and tantalum carbide (TaC) powders by high-energy ball milling, and their structural characteristics and microstructural evolution against reinforcement interactions were investigated. The nominal compositions of the Ni-TaC composites were Ni (5, 10, 15, and 20) vol% TaC. The formation of the ductile-brittle (Ni-TaC) components was affected by the continued severe deformation caused by the high-energy ball milling. As a result, a Ni0.92Ta0.08 solid solution was formed, which influenced the lattice distortion and peak shift. The Ni-TaC composites were densely consolidated by spark plasma sintering at 900°C and 60 MPa. The effect of the TaC on the sinterability and mechanical properties of Ni-TaC composites was analyzed based on the shrinkage rate, densification strain, densification strain, uniform distribution of carbide phase in the Ni matrix, grain size, and morphology of crack propagation. Oxidation behavior was analyzed by scanning electron microscopy with energy dispersive X-ray diffraction. The results indicate that the formation of the NiO and TaO2 passive layer for Ni- 15 vol% TaC composites resulted in a 12 µm (thick) oxide layer. This was the thinnest oxide layer on surface at oxidation 1000°C.

Morse theory for group presentations

We introduce a novel combinatorial method to study Q ∗ ∗ Q^{**} -transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplified Morse CW-complex, with an explicit description of the attaching maps and bounds on the dimension of the complexes involved in the deformation. We apply this technique to show that some known potential counterexamples to the Andrews–Curtis conjecture do satisfy the conjecture.

3D Printing of Capacitive Pressure Sensors with Tuned Wide Detection Range and High Sensitivity Inspired by Bio-Inspired Kapok Structures.

Flexible pressure sensors have drawn considerable attention for their potential applications as electronic skins with both sensitivity and pressure response range. Although the introduction of surface microstructures effectively enhances sensitivity, the confined volume of their compressible structures results in a limited pressure response range. To address this issue, a biomimetic kapok structure is proposed and implemented for constructing the dielectric layer of flexible capacitive pressure sensors employing 3D printing technology. The structure is designed with easily deformable concave and rotational structures, enabling continuous deformation under pressure. This design results in a significant expansion of the pressure response range and improvement in sensitivity. Further, the study purposively analyses crucial parameters of the devised structure that affect its compressibility and stability. These include the concave angle θ, height ratio d1/d2, rotation angle α, and width k. As a result, the ultimate pressure sensors demonstrate remarkable features such as high sensitivity (≈2.38kPa-1 in the range of 0-10kPa), broad detection range (734kPa), fast response time (23ms), and outstanding pressure resolution (0.4% at 500kPa). This study confirms the viability of bionic structures for flexible sensors, and their potential to expand the scope of wearable electronic devices.

Electromechanical Coupling Model for Ionic Liquid Gel Soft Actuators

A soft robot is composed of soft materials, which exhibit continuous deformation and driving structure integration and can arbitrarily change shapes and sizes over wide ranges. It shows strong adaptability to unstructured environments and has broad application prospects in military reconnaissance, medical rescues, agricultural production, etc. Soft robots based on ionic electroactive polymers (EAPs) have low-driving voltages, large-actuation displacements, fast responses, light weights, and low powers and have become a hot research field of bionic robots. Ionic liquid gels (ILGs) are new ionic EAPs. In this study, a new soft actuator was designed based on an ILG, and the electromechanical coupling model of an ILG soft actuator was studied in detail. Based on the system transfer function method, a mechatronic coupling model for the soft actuator was developed. According to the material characteristics and current response law of the ILG-containing EAP, an equivalent circuit model was used to describe transfer of the output current and input voltage. Based on the equivalent transformer model for ionic polymer–metal composite (IPMC) actuators proposed by Claudia Bonomo, the electromechanical coupling equation and a driving equation of the ILG soft actuator were established. The least-squares method was used with the coupling model of an ILG soft actuator to identify the system parameters for the model, and the effects of the structural parameters on the end displacement and driving force of the soft actuator were analyzed.

Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

Study on damage mechanism and damage distribution of the rear plate under impact of debris cloud

The debris cloud generated by the hypervelocity impact (HVI) of orbiting space debris directly threatens the spacecraft. A full understanding of the damage mechanism of rear plate is useful for the optimal design of protective structures. In this study, the hypervelocity yaw impact of a cylindrical aluminum projectile on a double-layer aluminum plate is simulated by the FE-SPH adaptive method, and the damage process of the rear plate under the impact of the debris cloud is analyzed based on the debris cloud structure. The damage process can be divided into the main impact stage of the debris cloud and the structural response of the rear plate. The main impact stage lasts a short time and is the basis of the rear plate damage. In the stage of structure response, the continuous deformation and inertial motion of the rear plate dominate the perforation of the rear plate. We further analyze the damage mechanism and damage distribution characteristics of the rear plate in detail. Moreover, the connection between velocity space and position space of the debris cloud is established, which promotes the general analysis of the damage law of debris cloud. Based on the relationship, the features of typical damage areas are identified by the localized fine analysis. Both the cumulative effect and structural response cause the perforation of rear plate; in the non-perforated area, cratering by the impact of hazardous fragments is the main damage mode of the rear plate.

Spheres over fields, their entire rational maps and applications

The paper summarizes some results on algebraic geometry presence in the homotopy theory. For the homotopy group πm(Sn), denote by πalgm(Sn) its subset of homotopy classes represented by ℝ-entire rational maps Sm→Sn of spheres. The main result of this paper concerns to the study of πn+kalg(Sn) for k=0,...,7.

Homotopy equivalent boundaries of cube complexes

A finite-dimensional CAT(0) cube complex X is equipped with several well-studied boundaries. These include the Tits boundary∂TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _TX$$\\end{document} (which depends on the CAT(0) metric), the Roller boundary∂RX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _R}X$$\\end{document} (which depends only on the combinatorial structure), and the simplicial boundary∂▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _\ riangle X$$\\end{document} (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of ∂RX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _R}X$$\\end{document} to define a simplicial Roller boundary R▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}_\ riangle X$$\\end{document}. Then, we show that ∂TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _TX$$\\end{document}, ∂▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _\ riangle X$$\\end{document}, and R▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}_\ riangle X$$\\end{document} are all homotopy equivalent, Aut(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Aut}(X)$$\\end{document}-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.

New classes of C 1-robustly transitive maps with persistent critical points

Recently, the authors proved in [C. Lizana and W. Ranter, Topological obstructions for robustly transitive endomorphisms on surfaces, Adv. Math. 390 (2021), pp. 107901] that every C 1 -robustly transitive toral endomorphism displaying critical points must be homotopic to a linear endomorphism having at least one eigenvalue with modulus greater than one. Here, we exhibit some examples of C 1 -robustly transitive surface endomorphisms displaying critical points in certain homotopy classes.

Quandle colorings vs. biquandle colorings

Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for virtual links. However, we have not found an essentially refined invariant for classical/surface links so far. In this paper, we give an explicit one-to-one correspondence between biquandle colorings and quandle colorings for classical/surface links. We also show that biquandle homotopy invariants and quandle homotopy invariants are equivalent. As a byproduct, we can interpret biquandle cocycle invariants in terms of shadow quandle cocycle invariants.

Measuring Dam Deformation of Long-Distance Water Transfer Using Multi-Temporal Synthetic Aperture Radar Interferometry: A Case Study in South-to-North Water Diversion Project, China

Long-distance water transfer is a critical engineering measure to rectify disparities in water resource distribution across regions. The effective operation and safety of such projects are paramount to their success, as localized issues can have cascading consequences, potentially disrupting the entire network. Conventional ground-based monitoring methods have limitations in measuring the deformation of large-scale structures. In this paper, InSAR is employed to monitor the deformation of the Shuangwangcheng (SWC) Reservoir, which features a long embankment dam as part of the South-to-North Water Diversion Project in China. We utilize data from both Sentinel-1 and TerraSAR-X satellites to derive 7-year deformation. Results reveal that the entire dam experiences continuous subsidence, with the maximum deformation in the line-of-sight direction measuring ~160 mm. While minor differential settlements are noted in different sections of the dam, the gradient is not significant due to the dam’s substantial length. The InSAR deformation results from multiple geometries demonstrate good consistency, with the highest correlation observed between the Sentinel-1 ascending and descending datasets, exceeding 0.9. Validation against the GNSS observations of the three sites on the SWC Dam shows the accuracy of InSAR displacements is ~8 mm. Water level changes do impact deformation, but consolidation settlement appears to be the primary controlling factor during the monitoring period. This study underscores the potential of InSAR in long-distance water transfer projects and highlights that spatially continuous deformation is the most significant advantage.

Morse theory for complexes of groups

We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence.

An efficient Kriging-based calibration framework for FDEM

The combined finite-discrete element method (FDEM) is a powerful numerical tool to simulate the transition of materials from continuous deformation to discontinuous fracturing and fragmentation. Calibrating parameters using laboratory data is crucial for applying FDEM to engineering-scale analysis. Considering the high computational costs in existing calibration schemes, this study proposes an effective FDEM calibration framework employing the Kriging surrogate model rooted in Uniaxial Compressive Strength (UCS) and Brazilian Tensile Strength (BTS) tests. The sensitivity analysis is first conducted on mesh size, loading rate, and numerical parameters to find a suitable set that balances computational efficiency and accuracy. Subsequently, an efficient mapping relationship between input fracture parameters and simulated strengths is established based on the Kriging surrogate model, replacing the expensive FDEM program for forward calculations. With the fracture parameters treated as unknown variables and experimental strengths available, the objective function is formulated and then solved to estimate fracture parameters. The effectiveness of the proposed method is validated through a numerical example. Compared to the conventional methods, the proposed approach requires fewer simulations of UCS and BTS, which highlights a significant computational efficiency advantage.