Elliptic Point Research Articles

On the automorphisms of the Drinfeld modular groups

Let [Formula: see text] be the ring of elements in an algebraic function field [Formula: see text] over [Formula: see text] which are integral outside a fixed place [Formula: see text]. In contrast to the classical modular group [Formula: see text] and the Bianchi groups, the Drinfeld modular group [Formula: see text] is not finitely generated and its automorphism group [Formula: see text] is uncountable. Except for the simplest case [Formula: see text] not much is known about the generators of [Formula: see text] or even its structure. We find a set of generators of [Formula: see text] for a new case. On the way, we show that every automorphism of [Formula: see text] acts on both the cusps and the elliptic points of [Formula: see text]. Generalizing a result of Reiner for [Formula: see text] we describe for each cusp an uncountable subgroup of [Formula: see text] whose action on [Formula: see text] is essentially defined on the stabilizer of that cusp. In the case where [Formula: see text] (the degree of [Formula: see text]) is [Formula: see text], the elliptic points are related to the isolated vertices of the quotient graph [Formula: see text] of the Bruhat–Tits tree. We construct an infinite group of automorphisms of [Formula: see text] which fully permutes the isolated vertices with cyclic stabilizer.

Unmanned Aerial Vehicle Obstacle Avoidance Based Custom Elliptic Domain

The velocity obstacles (VO) method is widely employed in real-time obstacle avoidance research for UAVs due to its succinct mathematical foundation and rapid, dynamic planning abilities. Traditionally, VO assumes a circle protection domain with a fixed radius, leading to issues such as excessive conservatism of obstacle avoidance areas, longer detour paths, and unnecessary avoidance angles. To overcome these challenges, this paper firstly reviews the fundamentals and pre-existing defects of the VO methodology. Next, we explore a scenario involving UAVs in head-on conflicts and introduce an elliptic velocity obstacle method tailored to the UAV’s current flight state. This method connects the protection domain size directly to the UAV’s flight state, transitioning from the conventional circle domain to a more efficient elliptic domain. Additionally, to manage the computational demands of Minkowski sums and velocity obstacle cones, an approximation algorithm for discretizing elliptic boundary points is introduced. A strategy to mitigate unilateral velocity oscillation had is developed. Comparative validation simulations in MATLAB R2022a confirm that, based on the experimental results for the first 10 s, the apex angle of the velocity obstacle cone for the elliptical domain is, on average, reduced by 0.1733 radians compared to the circular domain per unit simulation time interval, saving an airspace area of 13,292 square meters and reducing the detour distance by 14.92 m throughout the obstacle avoidance process, facilitating navigation in crowded situations and improving airspace utilization.

A High Speed and Area Efficient Processor for Elliptic Curve Scalar Point Multiplication for GF(2 m )

A High Speed and Area Efficient Processor for Elliptic Curve Scalar Point Multiplication for GF(2<i> ^m </i>)

Contact Cylindrical Surfaces and a Projection of a Surface Around a Parabolic Point

Contact Cylindrical Surfaces and a Projection of a Surface Around a Parabolic Point

Stability of Stuart vortices in rotating stratified fluids

The linear stability of the Stuart vortices, which is a model of arrays of vortices often observed in the atmosphere and the oceans, in rotating stratified fluids is investigated by local and modal stability analysis. As in the case of the two-dimensional (2-D) Taylor–Green vortices, five types of instability appear in general: the pure-hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate derived by Hattori &amp; Hirota (J. Fluid Mech., vol. 967, 2023, A32) are shown to also be useful for the Stuart vortices, which supports their applicability to general flows. The properties of each instability depend on stratification and rotation in a way similar to the case of the 2-D Taylor–Green vortices. For the Stuart vortices, however, the centrifugal instability and the elliptic instability become more dominant than the three hyperbolic instabilities in comparison to the 2-D Taylor–Green vortices; this is explained by the larger ratios of the maximum vorticity and the strain rate at the elliptic stagnation points to the strain rate at the hyperbolic stagnation points. Direct correspondence between the modal and local stability results is further established by comparing unstable modes to solutions to the local stability equations; this is useful for identifying the types of modes since the mechanism of instability is readily known in the local stability analysis. This helps us to discover the modes of the ring-type elliptic instability, which have been predicted only theoretically.

Dynamics of 2 × 2 matrix non-Hermitian quantum systems on Bloch sphere

By casting evolution to the Bloch sphere, the dynamics of 2 × 2 matrix non-Hermitian systems are investigated in detail. This investigation reveals that there are four kinds of dynamical modes for such systems. The different modes are classified by different kinds of fixed points, namely, the elliptic point, spiral point, critical node, and degenerate point. The Hermitian systems and the unbroken PT non-Hermitian cases belong to the category with elliptic points. The degenerate point just corresponds to the systems with exceptional point (EP). The topological properties of the fixed point are also discussed. It is interesting that the topological charge for the degenerate point is two, while the others are one.

Flux Landscape with enhanced symmetry not on SL(2, ℤ) elliptic points

We study structures of solutions for SUSY Minkowski F-term equations on two toroidal orientifolds with h2,1 = 1. Following our previous study [1], with fixed upper bounds of a flux D3-brane charge Nflux, we obtain a whole Landscape and a distribution of degeneracies of physically-distinct solutions for each case. In contrast to our previous study, we consider a non-factorizable toroidal orientifold and its Landscape on which SL(2, ℤ) is violated into a certain congruence subgroup, as it had been known in past studies. We find that it is not the entire duality group that a complex-structure modulus U enjoys but its outer semi-direct product with a “scaling” outer automorphism group. The fundamental region is enlarged to include the |U| < 1 region. In addition, we find that high degeneracy is observed at an elliptic point, not of SL(2, Z) but of the outer automorphism group. Furthermore, ℤ2-enhanced symmetry is realized on the elliptic point. The outer automorphism group is exceptional in the sense that it is consistent with a symplectic basis transformation of background three-cycles, as opposed to the outer automorphism group of SL(2, ℤ). We also compare this result with Landscape of another factorizable toroidal orientifold.

Quasi-inner automorphisms of Drinfeld modular groups

Let A be the set of elements in an algebraic function field K over \mathbb{F}_{q} which are integral outside a fixed place \infty . Let G=\mathrm{GL}_{2}(A) be a Drinfeld modular group . The normalizer of G in \mathrm{GL}_{2}(K) , where K is the quotient field of A , gives rise to automorphisms of G , which we refer to as quasi-inner . Modulo the inner automorphisms of G , they form a group \mathrm{Quinn}(G) which is isomorphic to \mathrm{Cl}(A)_{2} , the 2 -torsion in the ideal class group \mathrm{Cl}(A) . The group \mathrm{Quinn}(G) acts on all kinds of objects associated with G . For example, it acts freely on the cusps and elliptic points of G . If \mathcal{T} is the associated Bruhat–Tits tree, the elements of \mathrm{Quinn}(G) induce non-trivial automorphisms of the quotient graph G \backslash \mathcal{T} , generalizing an earlier result of Serre. It is known that the ends of G \backslash \mathcal{T} are in one-to-one correspondence with the cusps of G . Consequently, \mathrm{Quinn}(G) acts freely on the ends. In addition, \mathrm{Quinn}(G) acts transitively on those ends which are in one-to-one correspondence with the vertices of G \backslash \mathcal{T} whose stabilizers are isomorphic to \mathrm{GL}_{2}(\mathbb{F}_{q}) .

Dynamically Characterizing the Structures of Dirac Points via Wave Packets

Topological non-trivial band structures are the core problem in the field of topological materials. We investigate the topological band structure in a system with controllable Dirac points from the perspective of wave packet dynamics. By adding a third-nearest-neighboring coupling to the graphene model, additional pairs of Dirac points emerge. The emergence and annihilation of Dirac points result in hybrid and parabolic points, and we show that these band structures can be revealed by the dynamical behaviors of wave packets. In particular, for the gapped hybrid point, the motion of the wave packet shows a one-dimensional Zitterbewegung motion. Furthermore, we also show that the winding number associated with the Dirac point and parabolic point can be determined via the center of mass and spin texture of wave packets, respectively. The results of this work could motivate new experimental methods to characterize a system’s topological signatures through wave packet dynamics, which may also find applications in systems of other exotic topological materials.

A discontinuous Galerkin discretization of elliptic problems with improved convergence properties using summation by parts operators

Nishikawa (2007) proposed to reformulate the classical Poisson equation as a steady state problem for a linear hyperbolic system. This results in optimal error estimates for both the solution of the elliptic equation and its gradient. However, it prevents the application of well-known solvers for elliptic problems. We show connections to a discontinuous Galerkin (DG) method analyzed by Cockburn, Guzmán, and Wang (2009) that is very difficult to implement in general. Next, we demonstrate how this method can be implemented efficiently using summation by parts (SBP) operators, in particular in the context of SBP DG methods such as the DG spectral element method (DGSEM). The resulting scheme combines nice properties of both the hyperbolic and the elliptic point of view, in particular a high order of convergence of the gradients, which is one order higher than what one would usually expect from DG methods for elliptic problems.

The Recursive Structures of Manin Symbols over Q, Cusps and Elliptic Points on X0 (N)

Firstly, we present a more explicit formulation of the complete system D(N) of representatives of Manin’s symbols over Q, which was initially given by Shimura. Then, we establish a bijection between D(M)×D(N) and D(MN) for (M,N)=1, which reveals a recursive structure between Manin’s symbols of different levels. Based on Manin’s complete system Π(N) of representatives of cusps on X0(N) and Cremona’s characterization of the equivalence between cusps, we establish a bijection between a subset C(N) of D(N) and Π(N), and then establish a bijection between C(M)×C(N) and C(MN) for (M,N)=1. We also provide a recursive structure for elliptical points on X0(N). Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over Q, cusps, and elliptical points on X0(N). This may give rise to more efficient algorithms for modular elliptic curves. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and prove constructively formulas of the numbers of D(N), cusps, and elliptic points on X0(N).

Refined horoball counting and conformal measure for Kleinian group actions

Parabolic fixed points form a countable dense subset of the limit set of a non-elementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the 'inverse cost' of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small \(r&gt;0\) there are \(r^{-\delta}\) many horoballs of size approximately \(r\), where \(\delta\) is the Poincaré exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately \(r\) inside a given ball \(B(z,R)\). Roughly speaking, if \(r \lesssim R^2\), then we obtain an analogue of the Stratmann-Velani result (normalised by the Patterson-Sullivan measure of \(B(z,R)\)). However, for larger values of \(r\), the count depends in a subtle way on \(z\).&#x0D; &#x0D; Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several 'fractal dimensions' of certain \(s\)-conformal measures for \(s&gt;\delta\) and use this to examine continuity properties of \(s\)-conformal measures at \(s=\delta\).&#x0D;

Proof of the transverse intersection of the stable and unstable manifolds in the two-dimensional area preserving map

Abstract In this paper, we study the area preserving map T included in the Hénon family of maps. The map T defined on the plane is characterized by a parameter a (≥ 0). There exist the saddle fixed point P and the elliptic point Q at a &amp;gt; 0. Let u be the primary homoclinic point at which the unstable manifold Wu starting from the saddle fixed point P intersects the symmetry axis denoted by the dominant axis of Q. The stable manifold Ws also intersects Wu at u. At u, let ξu(u) be the inclination of Wu and ξs(u) be that of Ws. Using a new geometric method, we prove the transversality of the intersection for any value of a (&amp;gt; 0). Here, the transversality means that ξu(u) &amp;lt; ξs(u) holds.

Topological data analysis of noise: Uniform unimodal distributions

Random point clouds determine random simplicial complexes which can be characterized via homology. In this work we consider Vietoris–Rips complexes built on top of random point clouds in the D-dimensional Euclidian space in two setups: (i) a homogeneous Poisson point process (HPPP) and (ii) N i.i.d. points sampled from a uniform elliptical distribution. We derive expected topological summaries for such complexes in terms of the average Betti numbers (ABN) and of the newly introduced average integral persistence landscape (AIPL). The latter summary is derived from the average persistence landscape (APL) and preserves its 1-norm. In the HPPP model, we derive general analytical expressions for the density of ABN and AIPL and their 1-norms. For finite samples, we establish the functional form of decline of the 1-norm of the APL with the decrease of the sample size and/or increase of the cloud dimensionality. Furthermore, we derive analytical estimates of boundary effects in a disk, an ellipse, a D-dimensional ball, and 3-dimensional spheroids. We develop asymptotic models accounting for degeneration of a highly eccentric ellipse or a prolate spheroid to a line-segment and of an oblate spheroid to a disk, which describe the decline or growth of the 1-norm of the APL upon degeneration of an elliptic or spheroidal point cloud to a one- or two-dimensional cloud, respectively. Our results can serve as a “null model” for selection of the hyperparameters of the method D and N, as well as for the interpretation of observations in a broad range of real-world problems.

Invariant regions in piecewise linear area-preserving map

This paper studies regular islands (invariant regions) associated with elliptic periodic points of a piecewise linear area-preserving map. We reveal why regular island chains are interspersed in a chaotic sea, which is typical of the behavior exhibited by the area-preserving map. We find that regular islands may be ellipses or polygons and this depends on whether the rotation number of the rotation matrix that defines the linear map is irrational or rational. We also present a method for determining the boundary of these regular islands and chaotic seas (the largest invariant set). We show that critical lines play an important role in determining the boundaries of regular islands and chaotic seas.

Assessment of radial transport induced by Alfvénic resonances in tokamaks and stellarators

In a symmetric plasma confinement device, a perturbation such as an Alfvén mode produces a resonance island which flattens the local energy gradient due to induced rotation about the elliptic points of the resonance. The location of the resonance is mode frequency and particle energy dependent, and particles away from the resonance are not affected. A projection onto the Poincaré subspace allows the evaluation of resonance widths and internal rotation rates. In a stellarator without symmetry because of the toroidal dependence of the equilibrium magnetic field, a mode instead produces chaos in all orbits passing where the amplitude is large and is, thus, much more effective in producing radial transport and loss. The mode growth rate and saturation are also affected by the local chaos.

Pseudo-prolongations in the qualitative theory of dynamical systems

This paper considers the qualitative behaviour of the flow in a neighbourhood of closed invariant sets of dynamical systems. The properties of compactness, invariance, and connectivity of pseudo-prolongations are investigated. A rather deep analysis of the flow in the vicinity of a compact invariant set of asymptotically compact phase spaces is presented. The connection of pseudo-prolongation with the first positive prolongation of T. Ura and the set of weakly elliptic points is refined.

Microcontroller Implementation, Controls, and Synchronization of Three-Dimensional Autonomous System with a Parabolic Equilibrium Point

The microcontroller implementation, controls, and synchronization of a three-dimensional (3D) autonomous system with a parabolic equilibrium point are investigated in this paper. The system in question displays a reverse period doubling route hidden chaotic attractors with two different shapes. Then, the partial and total amplitude controls of the system are achieved by inserting two parameters. A microcontroller implementation is performed in order to confirm the results obtained from the numerical simulations. It is found that the results from the numerical simulations and microcontroller implementation qualitatively agree with each other. The sliding mode controllers are designed to control chaos in the system under study. With the sliding mode control method, the numerical simulations confirm that chaos can be controlled in the 3D autonomous system with a parabolic equilibrium point. In addition, two chaotic 3D autonomous systems with a parabolic equilibrium point and the same parameters are synchronized by the use of a unidirectional linear error feedback coupling scheme. Finally, an active control technique is applied to bring about chaos synchronization between two chaotic 3D autonomous systems with a parabolic equilibrium and different parameters.

Identity Access Management via ECC Stateless Derived Key Based Hierarchical Blockchain for the Industrial Internet of Things

Recently, the adoption of the industrial Internet of things (IIoT) has optimized many industrial sectors and promoted industry “smartization.” Smart factories and smart industries connect the real and virtual worlds through cyber-physical systems (CPS). However, these linkages will increase the cyber security danger surface to new levels, putting millions of dollars' worth of assets at risk if communications in big network systems like IIoT settings are left unsecured. To solve these problems, the fundamental method is security, such as authentication and confidentiality, and it should require the encryption key. However, it is challenging the security performance with the limited performance of the sensor. Blockchain-based identity management is emerging for lightweight, integrity and persistence. However, the key generation and management issues of blockchain face the same security performance issues. First, through blockchain smart contracts and hierarchical deterministic (HD) wallets, hierarchical key derivation efficiently distributes and manages keys by line and group in the IIoT environment. Second, the pairing verification value based on an elliptic curve single point called Root Signature performs efficient public key certificate registration and verification and improves the key storage space. Third, the identity log recorded through the blockchain is the global transparency of the key lifecycle, providing system reliability from various security attacks. Keyless Signature Infrastructure (KSI) is adopted to perform efficiently via hash-based scheme (hash calendar, hash tree etc.). We analyze our framework compared to hash-based state commitment methods. Accordingly, our method achieves a calculation efficiency of O(nlog N) and a storage space saving of 60% compared to the existing schemes.

Global solvability and global hypoellipticity of complex vector fields on surfaces

We consider smooth, nonvanishing complex (not essentially real) vector fields L=X+iY that satisfy the Nirenberg-Treves condition (P) and are allowed to possess some closed one-dimensional orbits in the sense of Sussmann that we call clean orbits. Concerning global solvability, we deal with two different cases —depending on whether the surface is compact or not— and prove two positive results which extend and unify several known results on the subject.In our positive results, it is assumed that X∧Y vanishes of order greater than 1 on all closed one-orbits, on the other hand, we prove that global solvability cannot hold if there exists at least one closed one-orbit on which X∧Y vanishes of order 1.We also give a characterization of globally hypoelliptic complex vector fields L in terms of the topological type of the equivalence classes determined by a standard equivalence relation defined on the set of non elliptic points of L. Some of the consequences are that globally hypoelliptic vector fields are globally solvable and that if a surface M carries a globally hypoelliptic vector field, it must be parallelizable.