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We explain how to deduce from the multi-height analysis of rational points on a toric stack (respectively on a toric variety) the asymptotic study of the number of rational points of bounded orbifold anticanonical height (respectively bounded anticanonical height), using a general version of the hyperbola method developed by Marta Pieropan and Damaris Schindler.

In this paper, we investigate the Krein space numerical range of \\(2\\)-by-\\(2\\) block matrices, with diagonal blocks as scalar multiples of the identity. For these matrices, we specifically investigate the cases when the respective boundary generating curves consist of hyperbolas. This provides a unified approach to derive established and new results concerning the numerical range hyperbolic shape.

We give an asymptotic formula for the number of rational points of bounded height on algebraic varieties defined by systems of multihomogeneous diagonal equations. The proof uses the Hardy-Littlewood circle method and the hyperbola method developed by Blomer and Brüdern.

The mining-induced strata movement and surface subsidence are closely related to the dip angle of coal seam. However, most surface subsidence prediction methods are empirical, and only suitable for nearly flat coal seam mining. In this paper, a new theoretical method is proposed to predict the strata movement boundary and surface subsidence caused by inclined coal seam mining, which considers the influence of key strata, rock quality and coal seam dip angle. The strata movement caused by inclined coal seam mining is generalized and described by three models: analogous hyperbola model (AHM), analogous hyperbola-funnel model (AHFM), and analogous funnel model (AFM). Considering the rock quality of roof and floor strata, the rock mass rating system is adopted to calculate the surface maximum subsidence and its location. Additionally, the distinct element method was used to assess the performance of the theoretical models. The numerical simulation results match well with theoretical predictions of strata movement boundary and surface subsidence. It is discovered that the appearance of surface subsidence troughs is obviously asymmetric. As the dip angle increases, the surface maximum subsidence decreases and its location is laterally displaced. When the dip angle is greater than 50°, the double subsidence troughs can be visualized clearly. Furthermore, the theoretical predictions of surface subsidence are verified by field measurements of two cases. As a result, the theoretical predictions of surface subsidence are greatly improved by comparing with the empirical method.

For a third-order complex polynomial with its roots bounded by a circle, Marden’s theorem provides a useful way to evaluate the locus of its critical points. We construct a fourth-order complex polynomial by restricting its roots to ellipse and hyperbola. The locus of its critical points can be expressed by a formula. In addition, if the roots of complex polynomials are not restricted, their critical points will not occupy some regions.

Spontaneous electric polarization of solid ferroelectrics follows aligning directions of crystallographic axes. Domains of differently oriented polarization are separated by domain walls (DWs), which are predominantly flat and run along directions dictated by the bulk translational order and the sample surfaces. Here we explore DWs in a ferroelectric nematic (N F ) liquid crystal, which is a fluid with polar long-range orientational order but no crystallographic axes nor facets. We demonstrate that DWs in the absence of bulk and surface aligning axes are shaped as conic sections. The conics bisect the angle between two neighboring polarization fields to avoid electric charges. The remarkable bisecting properties of conic sections, known for millennia, play a central role as intrinsic features of liquid ferroelectrics. The findings could be helpful in designing patterns of electric polarization and space charge. Defect lines shaped as conic sections are common in smectic liquid crystals, where they manifest equidistance of molecular layers curled in space. Here authors present hyperbolas and parabolas as domain walls in ferroelectric nematics, which are shaped so to avoid being electrically charged.

We combine the split torsor method and the hyperbola method for toric varieties to count rational points and Campana points of bounded height on certain subvarieties of toric varieties.

Structural anisotropy in crystals is crucial for controlling light propagation, particularly in the infrared spectral regime where optical frequencies overlap with crystalline lattice resonances, enabling light-matter coupled quasiparticles called phonon polaritons (PhPs). Exploring PhPs in anisotropic materials like hBN and MoO 3 has led to advancements in light confinement and manipulation. In a recent study, PhPs in the monoclinic crystal β-Ga 2 O 3 (bGO) were shown to exhibit strongly asymmetric propagation with a frequency dispersive optical axis. Here, using scanning near-field optical microscopy (s-SNOM), we directly image the symmetry-broken propagation of hyperbolic shear polaritons in bGO. Further, we demonstrate the control and enhancement of shear-induced propagation asymmetry by varying the incident laser orientation and polariton momentum using different sizes of nano-antennas. Finally, we observe significant rotation of the hyperbola axis by changing the frequency of incident light. Our findings lay the groundwork for the widespread utilization and implementation of polaritons in low-symmetry crystals. Hyperbolic phonon polaritons occurring in anisotropic materials exhibit strong light confinement and propagation directionality. Matson et al. report real-space imaging and control of recently discovered hyperbolic shear phonon-polaritons in beta-Ga2O3, arising from symmetry breaking in the dielectric response.

We consider the following elliptic system with Neumann boundary: equation cases - u + u=v^p, &in \\\- v + v=u^q, &in \\\ u n = v n = 0, &on \\>0,v>0, &in cases equation where \\( R^N\\) is a smooth bounded domain, \\(\\) is a positive constant and \\((p,q)\\) lies in the critical hyperbola:$$ \\dfrac{1}{p+1} + \\dfrac{1}{q+1} =\\dfrac{N-2}{N}. $$By using the Lyapunov-Schmidt reduction technique, we establish the existence of infinitely many solutions to above system. These solutions have multiple peaks that are located on the boundary \\( \\). Our results show that the geometry of the boundary \\( \\) especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.

Due to the way line drive, the continuous endoscope robot will have a certain lag in the process of movement, which will affect the accuracy and flexibility of the operation. In addition, the hysteresis and return difference caused by wire rope transmission will also increase the hysteresis effect of continuous robots. In this paper, the motion characteristics of the continuous endoscopic robot are analyzed, the hysteresis of the robot is modeled theoretically based on the Preisach model, and the effectiveness of the model is verified. Preisach hysteretic nonlinear hyperbola model predicts the bending changes under different positive and negative drives. The experimental results show that the maximum error between the model and the theoretical prediction is 3.18 degrees. The hysteresis model can predict the hysteresis characteristics of the continuous endoscope robot well.