Conjugate Points Research Articles

Conjugate points of dynamic pairs and control systems

We study the geometry of dynamic pairs $(X,\VV)$ on a manifold $M$, where $X$ is a vector field and $\VV$ is a distribution on $M$, both satisfying a regularity condition. Special cases are pairs defined by systems of second order ODEs, geodesic sprays in Riemannian, Finslerian and Lagranian geometries, semi-Hamiltonian systems and control-affine systems. Analogs of conjugate points from the calculus of variations are defined for the pair $(X,\VV)$. The main results give estimates for the position of conjugate points in terms of a curvature operator, analogously to the Cartan--Hadamard and Bonet--Myers theorems. Contrary to classical cases, no metric is given a priori, the distribution $\VV$ may be nonintegrable and the curvature operator is defined in terms of $(X,\VV)$.

Conjugate points along spherical harmonics

Utilizing structure constants, we present a version of the Misiolek criterion for identifying conjugate points. We propose an approach that enables us to locate these points along solutions of the quasi-geostrophic equations on the sphere S2. We demonstrate that for any spherical harmonics Ylm with 1≤|m|≤l, except for Y1±1 and Y2±1, conjugate points can be determined along the solution generated by the velocity field elm=∇⊥Ylm. Subsequently, we investigate the impact of the Coriolis force on the occurrence of conjugate points. Moreover, for any zonal flow generated by the velocity field ∇⊥Yl10, we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where l1=2k+1.∈N Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.

Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points

This paper focuses on finding the upper bounds of the second Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for a new subclass of starlike functions with respect to symmetric conjugate points associated with the exponential function defined by subordination. Results on initial Taylor coefficients and logarithmic coefficients of inverse functions for a new subclass are also presented. This study may inspire others to focus further to the coefficient functional problems associated with the inverse functions of various classes of univalent functions.

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

Finding origins of CMB anomalies in the inflationary quantum fluctuations

In this paper, we present compelling evidence for the parity asymmetry (a discrete symmetry separate from isotropy) in the Cosmic Microwave Background (CMB) map, measured through two-point temperature correlations. Any asymmetry associated with discrete symmetries, such as parity, challenges our understanding of quantum physics associated with primordial physics rather than LCDM (Λ Cold-Dark-Matter) itself. We commence by conducting a comprehensive analysis of the Planck CMB, focusing on the distribution of power in low-multipoles and temperature anticorrelations at parity conjugate points in position space. We find tension with the near scale-invariant power-law power spectrum of Standard Inflation (SI), with p-values of the order 𝒪(10-4 - 10-3). Alternatively, we explore the framework of direct-sum inflation (DSI), where a quantum fluctuation arises as a direct sum of two components evolving forward and backward in time at parity conjugate points in physical space. This mechanism results in a parity-asymmetric scale-dependent power spectrum, particularly prominent at low-multipoles, without any additional free model parameters. Our findings indicate that DSI is consistent with data on parity asymmetry, the absence of power at θ > 60°, and power suppression at low-even-multipoles which are major data anomalies in the SI model. Furthermore, we discover that the parameters characterizing the hemispherical power asymmetry anomaly become statistically insignificant when the large SI quadrupole amplitude is reduced to align with the data. DSI explains this low quadrupole with a p-value of 3.5%, 39 times higher than SI. Combining statistics from parameters measuring parity and low-ℓ angular power spectrum, we find that DSI is 50-650 times more probable than SI. In summary, our investigation suggests that while CMB temperature fluctuations exhibit homogeneity and isotropy, they also display parity-asymmetric behavior consistent with predictions of DSI. This observation provides a tantalizing evidence for the quantum mechanical nature of gravity.

A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points

A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points

Triplet Epipolar Images for Spaceborne Three Linear Cameras

Abstract. Epipolar images can reduce conjugate points searching from a two-dimensional to a one-dimensional space, which significantly improves the robustness and efficiency of dense matching. However, two images matched the lack of redundancy and suffered from occlusions. The third image could introduce two more stereos and improve the accuracy and completeness of the three-dimension reconstruction, which were adopted by spaceborne three linear cameras (TLCs), the ALOS PRSIM and ZY-3. In this study, we propose a new triplet epipolar images generation method for spaceborne TLCs. Triplet epipolar images were defined as any two of the three epipolar image pairs. The triplet epipolar geometry requires the imaging rays of any conjugate point to be coplanar. For high-resolution satellite images, parallel projection can be used to approximate the imaging geometry of satellite images. Therefore, a coplanar principal optical axis is a fundamental requirement for spaceborne TLCs. We propose a general workflow to generate triplet epipolar images (TEIs), which includes free-net bundle adjustment of TLCs, building a triplet epipolar geometry, correcting the y-parallaxes, and generating a rational function model (RFM) for TEIs. The ZY3-02 satellite images were used to validate the proposed method. The root means square error (RMSEs) of the free-net bundle adjustment in the image space was 0.185 pixels, which proved the fine intrinsic accuracy. After compensation, the RMSEs of the y parallaxes of the three epipolar image pairs were 0.295, 0.310, and 0.370 pixels. Owing to the simple geometry of the TEIs, the RMSEs of the RFM replacements were within 0.001 pixels.

Geometric analysis of the generalized surface quasi-geostrophic equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach

ACCURACY ASSESSMENT OF THE EFFECT OF DIFFERENT FEATURE DESCRIPTORS ON THE AUTOMATIC CO-REGISTRATION OF OVERLAPPING IMAGES

This research seeks to assess the effect of different selected feature descriptors on the accuracy of an automatic image registration scheme. Three different feature descriptors were selected based on their peculiar characteristics, and implemented in the process of developing the image registration scheme. These feature descriptors (Modified Harris and Stephens corner detector (MHCD), the Scale Invariant Feature Transform (SIFT) and the Speeded Up Robust Feature (SURF)) were used to automatically extract the conjugate points common to the overlapping image pairs used for the registration. Random Sampling Consensus (RANSAC) algorithm was used to exclude outliers and to fit the matched correspondences, Sum of Absolute Differences (SAD) which is a correlation-based feature matching metric was used for the feature match, while projective transformation function was used for the computation of the transformation matrix (T). The obtained overall result proved that the SURF algorithm outperforms the other two feature descriptors with an accuracy measure of -0.0009 pixels, while SIFT with a cumulative signed distance of 0.0328 pixels also proved to be more accurate than MHCD with a cumulative signed distance of 0.0457 pixels. The findings affirmed the importance of choosing the right feature descriptor in the overall accuracy of an automatic image registration scheme.

Geodesic flows of compact higher genus surfaces without conjugate points have expansive factors

In this paper we show that a geodesic flow of a compact surface without conjugate points of genus greater than one is time-preserving semi-conjugate to a continuous expansive flow which is topologically mixing and has a local product structure. As an application we show that the geodesic flow of a compact surface without conjugate points of genus greater than one has a unique measure of maximal entropy. This gives an alternative proof of Climenhaga–Knieper–War Theorem.

Lorentzian Calderón problem near the Minkowski geometry

We study a Lorentzian version of the well-known Calderón problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map on its timelike boundary. In the earlier work of the authors it was shown that zeroth order coefficients can be uniquely determined under a two-sided spacetime curvature bound and the additional assumption that there are no conjugate points along null or spacelike geodesics. In this paper we show that uniqueness for the zeroth order coefficient holds for manifolds satisfying a weaker curvature bound and for spacetime perturbations of such manifolds. This relies on a new enhanced optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calderón problem near the Minkowski geometry.

Automatic calibration of airborne LiDAR installation angle based on point cloud conjugate matching

Airborne LiDAR plays an important role in island detection, shallow ocean topography mapping and intertidal zone detection. The installation angle error usually leads to deviation between overlapping strips, which affects the accuracy of LiDAR point cloud data. However, previous methods had complex requirements for routes and calibration field, leading to inefficient calibration. This paper proposes an automatic installation angle error calibration algorithm based on point cloud conjugate matching, which does not require strict route design and field selection. In the algorithm, the conjugated point cloud is extracted by selecting routes and processing point cloud, and the installation angle error is calculated by matching the conjugate point cloud. Airborne LiDAR data from Shanghai Institute of Optics and Fine Mechanics is utilized to verify the method which show that the overlap degree of conjugate point clouds can reach 90% in different routes. This method is expected to dramatically improve the efficiency of system calibration and the convenience of measurement.

Модель проходження повороту автомобілем

The car turning model is considered, in particular at an X-shaped intersection with an arbitrary intersection angle of the tracks. The conditions and restrictions that are imposed when the car enters a turn are analyzed. It is shown that, when the car moves from straight to curved sections, angular accelerations acting relative to its vertical axis have a significant influence on the redistribution of the forces of interaction between the wheels and the road surface and, accordingly, on the stability and controllability of the car. The analysis of motion trajectories, which consist of conjugate rectilinear and curvilinear sections and are described by the equations of a circle, a parabola, and a hyperbolic cosine, is provided. It is shown that choosing a trajectory according to the law of parabola and hyperbolic cosine slightly reduces the turning radius of the car, but significantly reduces the curvature gradient in the conjugation zone and, accordingly, reduces angular accelerations and increases the resistance of the car to rotation relative to the vertical axis. For a smooth transition from a straight path to a curved one, a special logistic dependency was used to connect (stitch) different sections of the route. This made it possible to describe the trajectory of the car by a smooth function, the first and second derivatives of which are also smooth functions. For the selection of the trajectory of passing turns with a slight curvature of the route, a dependence in the form of a fourth-degree polynomial, the curvature of which at the point of conjugation is equal to zero, is proposed, which ensures a smooth transition from a straight to a curved section and ensures passing a turn with minimal dynamic loads. 
 The developed model allows you to design the trajectory of turning for various types of intersections in real time and can be used to build dynamic models of car movement along given trajectories, as well as to build simulators for unmanned vehicles.

A Grassmann manifold handbook: basic geometry and computational aspects

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.

Numerical Solution of a Left-Invariant Sub-Riemannian Problem on the Group $\mathrm{SO}(3)$

We consider a left-invariant sub-Riemannian problem on the Lie group of rotations of a three-dimensional space. We find the cut locus numerically, in fact we construct the optimal synthesis numerically, i. e., the shortest arcs. The software package nutopy designed for the numerical solution of optimal control problems is used. With the help of this package we investigate sub-Riemannian geodesics, conjugate points, Maxwell points and diffeomorphic domains of the exponential map. We describe some operating features of this software package.

ГЛАДКОЕ СОПРЯЖЕНИЕ ДВУХ ПЛОСКИХ КРИВЫХ БЕЗЬЕ

In this paper we consider a mathematical method which solves the problem of conjugating a Bézier curve with two given curves at boundary points. At the conjugation points, a smoothness (continuity) order must be preserved corresponding to the degree of the given curves. To solve the problem, a system of linear equations is composed in which the equations are conditions of equality of derivatives of the conjugated curves up to the order equal to the degree of the given Bézier curves. The solution of this system is the coordinates of the vertices of the defining polygon of the conjugated curve. Numerical calculations are performed in the Python programming language.

Generic properties of conjugate points in optimal control problems

Generic properties of conjugate points in optimal control problems

Quasi subordinations for bi-univalent functions with symmetric conjugate points

Many researchers have recently acquainted and researched several interesting subfamilies of bi-univalent function family δ and they have found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a_2 | and |a_3 |. In this current work, the subfamily F_(δ,q)^sc (α,Ξ) of bi-univalent functions in the sense of symmetric conjugate points with quasi subordination is defined. The Maclaurin coefficients |a_2 |, |a_3 | and besides related with these coefficients |a_3-〖a_2〗^2 | for functions belonging to this subfamily are derived. Further some corollaries are also presented.

The Hemispheric Difference in Electric Potential and Electron Precipitation Observed by DMSP in the Auroral Zone

AbstractIn this case study, we focus on how hemispheric differences in the electric potential distribution in the auroral zones are related to hemispheric differences in the energetic particle precipitation that indicate the presence of field‐aligned potential drops. Utilizing data during the spring and fall equinox in 2013 and 2014 from the Defense Meteorological Satellite Program (DMSP) F16 and F17 satellites, we systematically examine simultaneous measurements of plasma drift velocity and particle precipitation in magnetically conjugate regions across the auroral zone in both hemispheres. This allows us to establish the degree to which the interhemispheric differences in ionospheric electric potential may be accounted for by field‐aligned potential drops. Thirty‐seven eligible cases examined show accelerated electron energy spectra at conjugate locations in the northern and/or southern hemispheres. Of these 37 cases, 23 show conjugate differences in electric potential that agree qualitatively with the observed electron acceleration and average energy of precipitating electrons in each hemisphere. This result suggests that the hemispheric differences in ionospheric electric potential in the auroral zone may be most frequently accounted for by field‐aligned potential drops along the magnetic flux tubes connecting conjugate points to the equatorial plane. The spatial and temporal variability of the potential distribution, electron precipitation, and magnetic field configuration could account for deviations from the normally observed situation in the remaining cases. More magnetically conjugate observations at various altitudes in both hemispheres are needed to investigate this hemispheric coupling in more detail.

Algorithm for Creating 3d Scenes of Recognized Objects from Depth Maps

Purpose of research. Development of an algorithm for constructing 3d scenes of recognized objects from synthesized depth maps in order to improve the speed of real-time image processing.Methods. The 3d scene construction algorithm is based on the method of stereo image construction using a threelevel fuzzy depth map construction model. At the first level of this model the boundaries of objects are determined using a modified Canny algorithm, at the second level the values of disparity are calculated on the basis of the sum of absolute differences algorithm modified by fuzzy logic methods, and at the final level the gradients of distances from the boundaries of images to the edges of recognized objects are calculated first and then according to the obtained values of disparity at the second and third levels of the fuzzy hierarchical model, the refined values of disparity are calculated, which are used to carry out the analysis of the depth map.Results. An algorithm for constructing 3d scenes of recognized objects using synthesized depth maps has been developed. It was determined that the proposed algorithm has better performance compared to existing depth map algorithms such as conjugate point algorithm and pyramidal algorithm.Conclusion. The experimental results showed that the proposed algorithm has a lower complexity compared to the analyzed algorithms (conjugate points and pyramidal). The minimum average execution time of the 3d scene construction operation was about 1-2 minutes, which is almost 120 times better compared to the conjugate point algorithms.