Search Results (24)

In this paper, the fixed points of involutions on the moduli space of principal $E_6$-bundles over a compact Riemann surface X are investigated. In particular, it is proved that the combined action of a representative $\sigma$ of the outer involution of $E_6$ with the pull-back action of a surface involution $\tau$ admits fixed points if and only if a specific topological obstruction in $H^2\left(X/\tau ,{\pi }_0\left(E_6^{\sigma }\right)\right)$ vanishes. For an involution $\tau$ with $2k$ fixed points, it is proved that the fixed point set is isomorphic to the moduli space of principal H-bundles over the quotient curve $X/\tau$, where H is either $F_4$ or $PSp(8,\mathbb{C})$ and it consists of $2^{g-k+1}$ components. The complex dimensions of these components are computed, and their singular loci are determined as corresponding to H-bundles admitting non-trivial automorphisms. Furthermore, it is checked that the stability of fixed $E_6$-bundles implies the stability of their corresponding H-bundles over $X/\tau$, and the behavior of characteristic classes is discussed under this correspondence. Finally, as an application of the above results, it is proved that the fixed points correspond to octonionic structures on $X/\tau$, and an explicit construction of these octonionic structures is provided. Full article

Let X be a compact Riemann surface of genus $g\ge 2$, G be a complex semisimple Lie group, and ${\mathcal{M}}_G\left(X\right)$ be the moduli space of stable principal G-bundles. This paper studies the fixed point set of involutions on ${\mathcal{M}}_G\left(X\right)$ induced by an anti-holomorphic involution $\tau$ on X and a Cartan involution $\theta$ of G, producing an involution $\sigma ={\theta }_{\ast }\circ {\tau }_{\ast }$. These fixed points are shown to correspond to stable $G_{\mathbb{R}}$-bundles over the real curve $(X_{\tau },\tau )$, where $G_{\mathbb{R}}$ is the real form associated with $\theta$. The fixed point set ${\mathcal{M}}_G{\left(X\right)}^{\sigma }$ consists of exactly $2^r$ connected components, each a smooth complex manifold of dimension ${\displaystyle \frac{(g-1)\dim G}{2}}$, where r is the rank of the fundamental group of the compact form of G. A cohomological obstruction in $H^2(X_{\tau },{\pi }_1\left(G_{\mathbb{R}}\right))$ characterizes which bundles are fixed. A key result establishes a derived equivalence between coherent sheaves on ${\mathcal{M}}_G{\left(X\right)}^{\sigma }$ and on the fixed point set of the dual involution on the moduli space of $G^{\vee }$-local systems, where $G^{\vee }$ denotes the Langlands dual of G. This provides an extension of the Geometric Langlands Correspondence to settings with involutions. An application to the Chern–Simons theory on real curves interprets ${\mathcal{M}}_G{\left(X\right)}^{\sigma }$ as a $(B,B,B)$-brane, mirror to an $(A,A,A)$-brane in the Hitchin system, revealing new links between real structures, quantization, and mirror symmetry. Full article

Recently, with policies aimed at strengthening domestic manufacturing and technological innovation, the robotics industry has been growing rapidly, and its applications are expanding across various industrial fields. Accordingly, the importance of high-performance speed reducers with flexibility and precision is gradually increasing. The tooth profiles used in conventional harmonic reducers have structural limitations, such as meshing discontinuity, restrictions on the radius of curvature of the tooth base, and distortion of the contact trajectory, especially when the number of teeth is small. These problems limit the design freedom of the reducer and make it difficult to secure contact stability and durability under precision driving conditions. To solve these problems, this paper proposes a new tooth profile design equation, the IH (Involute Harmonic) tooth profiles and the HTPs (Hybrid Tooth Profiles), using the cycloid curve to overcome the structural limitations of the conventional harmonic tooth profile, which is difficult to design under small-tooth-number conditions, and to enable tooth design without restrictions on the number of teeth. HTP tooth profile is a new gear tooth profile design method that utilizes IH tooth profile and cycloid curve to optimize the meshing characteristics of gears. A tooth profile design tool based on the HTP equation was developed using Python 3.13.3. The tool’s effectiveness was validated through simulations assessing tooth meshing and interference. Using the tool, an R21_z3 reducer with a single-stage high reduction ratio was designed to evaluate practical applicability. A prototype was fabricated using 3D printing with PLA material, and experimental testing confirmed the absence of meshing or interference issues, consistent with simulation results. Through this study, we verified the usefulness of the HTP tooth profile design formula and design tool using the IH tooth profile and cycloid curve, and it is expected that the proposed HTP tooth profile can be utilized as a tooth profile applicable to various reducer designs. Full article

We introduced the concept of involute partner-ruled surfaces, which are formed by the involutes of spacelike curves and additional conditions ensuring the presence of definite surface normals in Minkowski three-space. First, we provided the criteria for each couple of involute partner-ruled surfaces to be simultaneously developable and minimal. Then, we established the requirements for the coordinate curves lying on these surfaces to be geodesic, asymptotic, and lines of curvature. We also expanded this paper with an example by providing graphical illustrations of the involute partner-ruled surfaces. Full article

Form grinding is a precision machining method for the modified ZI worms, and the grinding accuracy mainly depends on the dressing accuracy of the grinding wheel’s profile. A mathematical model of the modified involute helicoid of ZI worms is established based on the curve superposition method. Subsequently, the normal vector of the tooth surface is derived. After that, space meshing theory and matrix transformation methods are applied. Thus, the meshing equation between the grinding wheel and the tooth surface during the form grinding is constructed. Based on the equal error principle, an interpolation algorithm for the modified involute is proposed. The nonlinear meshing equations are solved using MATLAB R2019b software to obtain the discrete point coordinates of the worm end section profile and the grinding wheel axial section profile. The derivative of the discrete points is calculated by using the difference method, and the motion trajectory of the diamond wheel during the grinding wheel dressing process is solved based on the equidistant curve theory. The proposed methods holds certain reference value for calculating the profile of grinding wheels used in the form grinding of modified ZI worms. Full article

A widespread distribution of gas-regulating stations creates challenges in energy recovery during pressure reduction. This study employs a scroll expander as a central mechanism to enhance the efficiency of residual gas pressure recovery, demonstrating its adaptability. We conducted experimental tests on its power generation capabilities and numerically studied the expansion characteristics. Our results indicate significant improvements as the inlet pressure was increased from 85 kPa to 375 kPa: the generator speed increased from 1238 rpm to 4615 rpm, the power output increased from 9.64 W to 165.48 W, and the temperature difference between the inlet and outlet flows changed from 5.41 K to 27.3 K. Turbulent dissipative and wall friction were identified as primary contributors to the energy loss, overcoming the temperature and viscosity loss, and increasing together with the radial and axial clearances. A comparative analysis of scroll designs reveals that the scrolls modified with higher order and arc curves display a reduced torque compared with the traditional circular involutes, and more scrolls are beneficial for handling high-pressure gas. These findings offer insights into scroll expander design, enhancing the energy efficiency of microscale gas residual pressure recovery systems. Full article

This paper investigates the design methodologies of pure rolling helical gear pumps with various tooth profiles, based on the active design of meshing lines. The transverse active tooth profile of a pure rolling helical gear end face is composed of various function curves at key control points. The entire transverse tooth profile consists of the active tooth profile and the Hermite curve as the tooth root transition, seamlessly connecting at the designated control points. The tooth surface is created by sweeping the entire transverse tooth profile along the pure rolling contact curves. The fundamental design parameters, tooth profile equations, tooth surface equations, and a two-dimensional fluid model for pure rolling helical gears were established. The pressure pulsation characteristics of pure rolling helical gear pumps and CBB-40 involute spur gear pumps, each with different tooth profiles, were compared under specific working pressures. This comparison encompassed the maximum effective positive and negative pressures within the meshing region, pressure fluctuations at the midpoints of both inlet and outlet pressures, and pressure fluctuations at the rear sections of the inlet and outlet pressures. The results indicated that the proposed pure rolling helical gear pump with a parabolic tooth profile exhibited 42.81% lower effective positive pressure in the meshing region compared to the involute spur gear pump, while the maximum effective negative pressure was approximately 27 times smaller than that of the involute gear pump. Specifically, the pressure pulsations in the middle and rear regions of the inlet and outlet pressure zones were reduced by 33.1%, 6.33%, 57.27%, and 69.61%, respectively, compared to the involute spur gear pump. Full article

Due to the dense crop residue in the Huang-Huai-Hai region, challenges such as large resistance, increased power consumption, and straw backfilling arise in the process of no-till seeding under the high-speed operations. This paper presents the design of a straw treatment device to address these issues. The cutting edge of a straw-cutting disc is optimized using an involute curve, and the key structural parameters of the device are designed by analyzing the process of stubble cutting and clearing. In this study, the Discrete Element Method (DEM) was employed to construct models of compacted soil and hollow, flexible wheat straw, forming the foundation for a comprehensive interaction model between the tool, soil, and straw. Key experimental variables, including working speed, rotation speed, and installation centre distance, were selected. The power consumption of the straw-cutting disc (PCD) and the straw-clearing rate (SCR) were used as evaluation metrics. Response surface methodology was applied to develop regression models linking the experimental factors with the evaluation indexes using Design-Expert 12 software. Statistical significance was assessed through ANOVA (p < 0.05), and factor interactions were analyzed via response surface analysis. The optimal operational parameters were found to be a working speed of 14 km/h, a rotation speed of 339.2 rpm, and an installation centre distance of 100 cm. Simulation results closely matched the predicted values, with errors of 1.59% for SCR and 9.68% for PCD. Field validation showed an SCR of 86.12%, improved machine passability, and favourable seedling emergence. This research provides valuable insights for further parameter optimization and component development. Full article

Willmore defined embedded surfaces on $f:S\to E^3$, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of $E^3$, inducing a Riemannian structure on $f\left(S\right)$. The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature $H^2$ on $f\left(S\right).$ Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the $\stackrel{\sim }{D}$-scroll of the curve–surface pair and involute $\stackrel{\sim }{D}$-scroll of the curve–surface pair with some differential geometric elements and found ${\stackrel{\sim }{D}}_{(\alpha ,M)}\left(s\right)$ and ${\stackrel{\sim }{D^{\ast }}}_{(\alpha ,M)}\left(s\right)$-scrolls of the curve–surface pair $(\alpha ,M)$. Full article

We propose a new hierarchy of the vector derivative nonlinear Schrödinger equations and consider the simplest multiphase solutions of this hierarchy. The study of the simplest solutions of these equations led to the following results. First, the three-leaf spectral curves $\Gamma =\left\{\right(\mu ,\lambda \left)\right\}$ of the simplest multiphase solutions have a quite simple symmetry. They are invariant with respect to holomorphic involution $\tau$. The type of this involution depends on the genus of the spectral curve. Or the involution has the form $\tau :(\mu ,\lambda )\to (\mu ,-\lambda )$, or $\tau :(\mu ,\lambda )\to (-\mu ,-\lambda )$. The presence of symmetry leads to the fact that the dynamics of the solution is determined not by the entire spectral curve $\Gamma$, but by its factor $\Gamma /\tau$, which has a smaller genus. Secondly, it turned out that the dynamics of the two-component vector $\mathbf{p}={(p_1,p_2)}^t$ is determined, first of all, by the dynamics of its length $\left|\mathbf{p}\right|$. Independent equations determine the dependence of the direction of the vector $\mathbf{p}$ from its length. In cases where the direction of the vector $\mathbf{p}$ is fixed, the corresponding spectral curve splits into separate components. In conclusion, we note that, as in the case of the Manakov system, the equation of the spectral curve is invariant with respect to the orthogonal transformation of the vector solutions. I.e., the solution can be found from the spectral curve up to the orthogonal transformation. This fact indicates that the spectral curve does not depend on the individual components of the solution, but on their symmetric functions. Thus, the spectral data of multiphase solutions have two symmetries. These symmetries make it difficult to reconstruct signals from their spectral data. The work contains examples illustrating these statements. Full article

To meticulously examine the repercussions of nonlinear vibrations on fretting damage within aero-engine involute spline pairs, a dynamic model was constructed rooted in well-established theories and methodologies. MATLAB was engaged to resolve the model, where the vibration displacement function was treated under Fourier transformation. The emergent sub-model was then integrated into finite element analysis software to scrutinize the distribution curves of fretting damage over the external spline tooth surface. The analysis included a comprehensive comparison of the axial and radial distributions, in addition to scenarios with and without vibration interferences. Further, an empirical platform was devised to authenticate the outcomes harvested through finite element simulation. The results indicate that the principal mode of fretting damage failure in aero-engine involute spline pairs fundamentally comprises fretting wear. This wear occurs throughout the rotational period of the fretting cycle and reciprocally interacts with fretting fatigue phenomena. Significantly, it was ascertained that acute nonlinear vibrations escalate the magnitude of fretting damage and the quantity of worn teeth within aero-engine spline pairs. Beyond that, angular misalignment was recognized as an aggravating factor that compounds fretting damage in the secondary bond teeth of involute spline pairs. These newfound insights are of paramount significance for the strategic design of involute splines to combat wear. Full article

In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that there is no quasi-evolute curve in Galilean three-space. Also, we introduce quasi-Smarandache curves in Galilean three-space. Finally, we demonstrate an illustrated example to present our findings. Full article

A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in fields such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this support plane is about an axis in Euclidean 3-space. In addition, some examples of the conchoidal twisted surface are given and the graphs of the surfaces are presented. The Gaussian and mean curvatures of this conchoidal twisted surface are calculated. Afterward, the conchoidal twisted surface formed by an involute curve and the conchoidal twisted surface formed by a Bertrand curve pair are given. Thanks to the results obtained in our study, we have added a new type of surface to the literature. Full article

In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. This paper follows Jacobi’s prophetic observations by introducing certain “meta” variational problems on semi-simple reductive groups G having a compact subgroup K. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi’s geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal. Full article

In this paper, novel non-involute cylindrical gears are designed based on a curved path of contact. Firstly, a parabolic curve is predesigned as the contact path of novel gears. Then, the tooth profiles of the novel gears are calculated using differential geometry and spatial meshing theory. Secondly, the three-dimensional tooth models of the novel non-involute gears are established according to a machining simulation. Thirdly, a simulation model (including misalignment error and longitudinal modification) is established to analyze the performance of the novel non-involute gears. Finally, an example is given, and the results show that the presented novel non-involute gears have greater load-carrying capacity compared with the involute gears. Moreover, whether modified or not, with or without misalignment error, the stresses of the presented novel gears are lower than those of involute gears. Full article