Space Curves Research Articles

Noise-resistant Landau-Zener sweeps from geometrical curves

Landau-Zener physics is often exploited to generate quantum logic gates and to perform state initialization and readout. The quality of these operations can be degraded by noise fluctuations in the energy gap at the avoided crossing. We leverage a recently discovered correspondence between qubit evolution and space curves in three dimensions to design noise-robust Landau-Zener sweeps through an avoided crossing. In the case where the avoided crossing is purely noise-induced, we prove that operations based on monotonic sweeps cannot be robust to noise. Hence, we design families of phase gates based on non-monotonic drives that are error-robust up to second order. In the general case where there is an avoided crossing even in the absence of noise, we present a general technique for designing robust driving protocols that takes advantage of a relationship between the Landau-Zener problem and space curves of constant torsion.

In-Plane Anisotropic Plasmons in Van Der Waals Thin Films of WTe2

Anisotropic plasmonic surface supports elliptic, hyperbolic and even flattened polaritons, which is quite interesting for the diffractionless and highly collimated propagation of infrared light at the nanoscale. However, direct real-space near-field observation of anisotropic plasmons as well as frequency dependent topological transitions in natural materials have not been realized. In this paper, we theoretically investigate real-space anisotropic plasmons in WTe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> thin films by using a phenomenological cavity model, anisotropic near-field plasmonic images with specific interference patterns and isofrequency curves in momentum space have been demonstrated. Due to the frequency selective forbidden of plasmons along <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> axis, a topological transition from the elliptic to the hyperbolic regime is manifested. Moreover, the plasmons as well as topological transition present significant electrostatic-gating tunability. Our studies provide new insights into WTe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> based plasmonic components for the manipulation of plasmon propagation, which capable of tailoring anisotropic two-dimensional light confinement in the far-infrared regime and can be applied to investigate other anisotropic materials.

Visco-Da Rios Equation in 3-Dimensional Riemannian Manifold

In this paper, we study two classes of a space curve evolution in terms of Frenet frame for the visco-Da Rios equation in a 3-dimensional Riemannain manifold. Also, we obtain the connection between the visco-Da Rios equation and nonlinear Schrödinger equation for two classes in a 3-dimensional Riemannain manifold with constant sectional curvature. Finally, we give the Bäcklund transformations of space curve with the visco-Da Rios equation.

Maximal estimates for averages over space curves

Let M be the maximal operator associated to a smooth curve in $${\mathbb {R}}^3$$ which has nonvanishing curvature and torsion. We prove that M is bounded on $$L^p$$ if and only if $$p>3$$ .

A property of the spherical derivative of an entire curve in complex projective space

We establish a type of the Picard’s theorem for entire curves in $$P^n({\mathbb {C}})$$ whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union D of finite number of hypersurfaces in the complex projective space $$P^n({\mathbb {C}})$$ such that for every entire curve f in $$P^n({\mathbb {C}})$$ , if the spherical derivative $$f^{\#}$$ of f is bounded on $$ f^{-1}(D)$$ , then $$f^{\#}$$ is bounded on the entire complex plane, and hence, f is a Brody curve.

Hough Transform for Detecting Space Curves in Digital 3D Models

We present and analyse the Hough transform (HT) to recognise and approximate space curves in digital models, a problem that is not currently addressed by the standard HT. Our method works on meshes and point clouds and applies to models even incomplete or affected by noise, thus being suitable for the analysis of digital models deriving from 3D scans. In our approach we take advantage of a recent HT formulation for algebraic curves to define both parametric and implicit space curve representations. We also provide a comparative analysis of the HT-based method when dealing with both curve representations, discussing the computational performance and the approximation accuracy of both strategies.

Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-informed Deep Generative Models

We propose a new method for inferring the governing stochastic ordinary differential equations (SODEs) by observing particle ensembles at discrete and sparse time instants, i.e., multiple "snapshots". Particle coordinates at a single time instant, possibly noisy or truncated, are recorded in each snapshot but are unpaired across the snapshots. By training a physics-informed generative model that generates "fake" sample paths, we aim to fit the observed particle ensemble distributions with a curve in the probability measure space, which is induced from the inferred particle dynamics. We employ different metrics to quantify the differences between distributions, e.g., the sliced Wasserstein distances and the adversarial losses in generative adversarial networks (GANs). We refer to this method as generative "ensemble-regression" (GER), in analogy to the classic "point-regression", where we infer the dynamics by performing regression in the Euclidean space. We illustrate the GER by learning the drift and diffusion terms of particle ensembles governed by SODEs with Brownian motions and Levy processes up to 100 dimensions. We also discuss how to treat cases with noisy or truncated observations. Apart from systems consisting of independent particles, we also tackle nonlocal interacting particle systems with unknown interaction potential parameters by constructing a physics-informed loss function. Finally, we investigate scenarios of paired observations and discuss how to reduce the dimensionality in such cases by proving a convergence theorem that provides theoretical support.

Constructing reducible Brill–Noether curves

A fundamental problem in the theory of algebraic curves in projective space is to understand which reducible curves arise as limits of smooth curves of general moduli. Special cases of this question and variants have been critical in the resolution of many problems in the theory of algebraic curves over the past half century; examples include Sernesi's proof of the existence of components of the Hilbert scheme with the expected number of moduli when the Brill–Noether number is negative [ E. Sernesi , Invent. Math. 75, 25–57 (1984; Zbl 0541.14024)], and Ballico's proof the Maximal Rank Conjecture for quadrics [ E. Ballico , Lith. Math. J. 52, No. 2, 134–137 (2012; Zbl 1282.14054)]. In this paper, we give close-to-optimal bounds on this problem when the nodes are general points and the components are general in moduli. The results given here significantly extend those cases established by [Sernesi, loc. cit.], [Ballico, loc. cit.], and others. As explained in [ Eric Larson , "Degenerations of curves in projective space and the maximal rank conjecture", Preprint (2018); ], they also play a key role in the author's proof of the Maximal Rank Conjecture [ Eric Larson , "The maximal rank conjecture", Preprint (2017); ].

Curvature Singularity of Space Curves and Its Relationship to Computational Mechanics

Abstract Curve geometry plays a fundamental role in many aspects of analytical and computational mechanics, particularly in developing new data-driven science (DDS) approaches. Furthermore, curvature and torsion of space curves serve as deformation measures that need to be properly interpreted, shedding light on the significance of relationship between differential-geometry curve framing methods and computational-mechanics motion description. Alternate space-curve framing methods were proposed to address the existence of Frenet frame at isolated zero-curvature points. In this paper, both mechanics and differential-geometry approaches are used to establish Frenet-frame continuity and the existence of Serret-Frenet equations at curvature-vanishing points for curves with arbitrary parameterization. Frenet–Euler angles, referred to for brevity as Frenet angles, are used to define curve geometry, with particular attention given to the definition of Frenet bank angle used to prove the existence of curve normal and binormal vectors at curvature-vanishing points. Solving curvature-singularity problem and using mechanics description based on Frenet angles contributes to successful development and computer implementation of new DDS approaches based on analysis of recorded motion trajectories (RMT). Centrifugal-inertia force is always in direction of curve normal vector, and in most applications, this force is continuous and approaches zero value as curve curvature approaches zero. Discontinuity in definition of Frenet frame can negatively impact the quality of numerical results that define RMT curves. The study also demonstrates that Frenet-frame curvature singularity can be solved without need for integrating curve torsion, which is not, in general, an exact differential.

Smooth Three-Dimensional Route Planning for Fixed-Wing Unmanned Aerial Vehicles With Double Continuous Curvature

This paper presents a smooth flight path planner for maneuvering in a 3D Euclidean space, which is based on two new space curves. The first one is called “Elementary Clothoid-based 3D Curve (ECb3D)”, which is built by concatenating two symmetric Clothoid-based 3D Curves (Cb3D). The combination of these curves allows to reach an arbitrary orientation in 3D Euclidean space. This new curve allows to generate continuous curvature and torsion profiles that start and finish with a null value, which means that they can be concatenated with other curves, such as straight segments, without generating discontinuities on those variables. The second curve is called “Double Continuous Curvature 3D Curve (DCC3D)” which is built as a concatenation of three straight line segments and two ECb3D curves, allowing to reach an arbitrary configuration in position and orientation in the 3D Euclidean space without discontinuities in curvature and torsion. This trajectory is applied for autonomous path planning and navigation of unmanned aerial vehicles (UAVs) such as fixed-wing aircrafts. Finally, the results are validated on the FlightGear 2018 flight simulator with the UAV kadett 2400 platform.

Geometrical dynamics of edge-driven accretive surface growth

Accretion of mineralized thin wall-like structures via localized growth along their edges is observed in physical and biological systems ranging from molluscan and brachiopod shells to carbonate–silica composite precipitates. To understand the shape of these mineralized structures, we develop a mathematical framework that treats the thin-walled shells as a smooth surface left in the wake of the growth front that can be described as an evolving space curve. Our theory then takes an explicit geometric form for the prescription of the velocity of the growth front curve, along with compatibility relations and a closure equation related to the nature of surface curling. Solutions of these equations capture a range of geometric precipitate patterns seen in abiotic and biotic forms across scales. In addition to providing a framework for the growth and form of these thin-walled morphologies, our theory suggests a new class of dynamical systems involving moving space curves that are compatible with non-Euclidean embeddings of surfaces.

Equidistribution of Expanding Translates of Curves in Homogeneous Spaces with the Action of (SO(n, 1)k

Equidistribution of Expanding Translates of Curves in Homogeneous Spaces with the Action of (SO(n, 1)k

Coincidence of set functions in quasiconformal analysis

It is known that mappings occurring in quasiconformal analysis can be defined in several equivalent ways: 1) as homeomorphisms inducing bounded composition operators between Sobolev spaces; 2) as Sobolev-class homeomorphisms with bounded distortion whose operator distortion function is integrable; 3) as homeomorphism changing the capacity of the image of a condenser in a controllable way in terms of the weighted capacity of the condenser in the source space; 4) as homeomorphism changing the modulus of the image of a family of curves in a controllable way in terms of the weighted modulus of the family of curves in the source space. A certain set function, defined on open subsets, can be associated with each of these definitions. The main result consists in the fact that all these set functions coincide. Bibliography: 48 titles.

Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

&lt;p style='text-indent:20px;'&gt;This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [&lt;xref ref-type="bibr" rid="b26"&gt;26&lt;/xref&gt;] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.&lt;/p&gt;

Based on Artificial Intelligence Neural Network CNN Method Analysis and Processing of Dynamic Optical Breast Lesion Images

Breast cancer is the most common cancer in women. At present, the methods of examining lesions are generally mammography, or B-scanning and other methods with certain radioactive sources, which may lead to aggravation of breast lesions in young women. The dynamic optical breast scanning method uses infrared light to avoid the harm of X-rays to the human body. According to current research, doctors can use Dynamic Optical Breast Image (DOBI) to determine whether a patient has breast cancer. Studies have shown that convolutional neural networks (CNN) have higher detection accuracy in determining whether a patient has breast cancer. In this paper, we use an artificial intelligence neural network approach to analyze and process dynamic optical breast lesion images: we model the clinical lesion breast images in 3D, and use the VoxelMorph algorithm to segment the 3D images into 2D images; The time, space, location, and pathological trend curves in the image are analyzed and processed. We compared classification, sensitivity, and specific characteristics with the original dynamic breast lesion image scoring analysis system. The experimental results show that the accuracy is improved. At the same time, the problem of the original system's signature in the ROI area and leaf curve is solved. The use of CNN improves the analysis and processing speed, shortens the processing time, and increases the accuracy of the diagnostic reference from 83% to 90%.

On geometry of focal surfaces due to B-Darboux and type-2 Bishop frames in Euclidean 3-space

&lt;abstract&gt;&lt;p&gt;In Euclidean 3-space $ {\mathrm{E}}^3 $, a canonical subject is the focal surface of such a cliched space curve, which would be a two-dimensional corrosive with Lagrangian discontinuities. The tubular surfaces with respect to the B-Darboux frame and type-2 Bishop frame in $ {\mathrm{E}}^3 $ are given. These tubular surfaces' focal surfaces are then defined. For such types of surfaces, we acquire some results becoming Weingarten, flat, linear Weingarten conditions and we demonstrate that in $ {\mathrm{E}}^3 $, a tubular surface has no minimal focal surface. We also provide some examples of these types of surfaces.&lt;/p&gt;&lt;/abstract&gt;

Minimal surface generating flow for space curves of non-vanishing torsion

&lt;p style='text-indent:20px;'&gt;This article introduces new geometric flow for space curves with positive curvature and torsion. Curves evolving according to this motion law trace out a zero mean curvature surface. First, the geometry of surfaces defined as trajectories of moving curves is analyzed and the minimal surface generating flow is derived. Then, an upper bound for the area of the generated minimal surface and for the terminating time of the flow is provided.&lt;/p&gt;

Hypersurface families with Smarandache curves in Galilean 4-space

In this paper, we study the hypersurface families with Smarandache curves in 4-dimensional Galilean space G 4 and give the conditions for different Smarandache curves to be parameter and the curve which generates the Smarandache curves is geodesic on a hypersurface in G 4 . Also, we investigate three types of marching-scale functions for one of these hypersurfaces and construct an example for it.

Characterizations of Framed Curves in Four-Dimensional Euclidean Space

Framed curves in Euclidean space are used to investigate singular curves and are important for singularity theory. In this study, framed curves in four-dimensional Euclidean space are introduced and new results are obtained. The relation of framed curves with Frenet curves in four-dimensional Euclidean space is given and Bishop-type frame of framed curves is introduced with the help of Euler angles. In addition, by using Bishop-type framed curves in four-dimensional Euclidean space, framed rectifying curves, framed osculating curves and framed normal curves are introduced. Also, some characterizations depending on framed curvatures are obtained.

A Different View on Dynamics of Space Curves Geometry

In this study, we define the X-torque curves, X-equilibrium curves, X-moment conservative curves, X-gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where Xin left{ Tleft( sright) , Nleft( sright) , Bleft( sright) right} and we examine these curves and we give their properties.