Exponential Map Research Articles

Controllability and diffeomorphism groups on manifolds with boundary

Abstract In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.

Hermite interpolation with retractions on manifolds

AbstractInterpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. A novel procedure relying on the general concept of retractions is proposed to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. The well-posedness of the method is analyzed by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. A classical result on the asymptotic interpolation error of Hermite interpolation is extended to the manifold setting. Finally numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns illustrate these results and the effectiveness of the method.

An enhanced key expansion module based on 2D hyper chaotic map and Galois field

Key expansion is an essential component in block cryptography, which serves the round function. By analyzing key expansion module of AES, it is found that the round-keys are highly correlated, and current round-key can be derived from its previous or the next round-key. To address these weaknesses, first, a 2D exponential chaotic map (2D-ECM) that exhibits ergodicity and superior randomness was constructed. Then, the Lyapunov exponents (LEs) were calculated based on the singular value decomposition (SVD) method. In addition, TestU01 test results showed that the sequences generated by 2D-ECM have better randomness. Further, an enhanced key expansion module was designed utilizing 2D-ECM and primitive polynomial over GF(2n), which has irreversibility and parallelism, and the round-keys are independent of each other. Simulation results and performance analysis demonstrated the effectiveness of the proposed enhanced key expansion module.

SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres

We mimic the conventional explicit Total Variation Diminishing Runge–Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 4$$\\end{document}. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method.

On constraint-conforming numerical discretizations in constitutive material modeling

AbstractFor the modelling of complex materials, internal variables are usually introduced which characterize the microstructural state. Then, evolution equations describe the change of the internal variables due to varying external loading conditions. These equations can be derived, for instance, on the basis of variational principles. The consideration of characteristic observations, such as the preservation of the volume during a change in the microstructural state, can significantly improve the accuracy of the evolution equations. We present a Hamilton principle that provides a unique way to derive evolution equations that obey holonomic constraints and opens up new possibilities for their algorithmic treatment. This is demonstrated for isochoric finite plasticity and phase transformation based on Backward-Euler time discretization. The models presented are efficient and are characterized by simple implementation compared to the exponential map, for example, without suffering a loss of accuracy due to unfulfilled constraints.

Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization

Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of Rn, and prove convergence results.We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.

Algebraic structure of the renormalization group in the renormalizable QFT theories

In this paper, we consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant, its generators [Formula: see text] with [Formula: see text] satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the [Formula: see text]-function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map.

On the v-Picard Group of Stein Spaces

Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $\mathbf{Z}_{p}$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $1$.

An Exponential Map-based Whale Optimization Algorithm (Exp-WOA) for Optimization

This research work offers three variations of the Whale Optimization Algorithm (WOA) based on exponential chaotic maps, namely Logistic-Exponential-Logistic WOA (LEL-WOA), Logistic-Exponential-Sinusoidal WOA (LES-WOA), and Logistic-Exponential-Tent WOA (LET-WOA). The WOA with an exponential chaos-based mechanism is developed in this study to overcome the poor rate of convergence of the WOA and to prevent getting caught in local optimal solutions while dealing with the challenges. An exponential chaotic mechanism was employed in this research to initialize the agents and control the parameters of the exploration and exploitation phases of WOA. The proposed methodologies (Exp-WOA) are evaluated using twenty-three widely recognized test functions. The results demonstrate that the given solutions can enhance the performance of WOA by achieving optimal (minimum) values. The findings also indicate that LEL-WOA and LES-WOA exhibit faster convergence than WOA.

Finite deformation peridynamics shell theory: Application to mechanical metasurfaces

A finite deformation Simo-Reissner peridynamics (PD) shell theory is developed to investigate wave propagation and crack growth in shell structures. Simo-Reissner deformation approximation is utilized to derive the governing equations of the proposed theory from the three-dimensional PD equation. Deformation states for the PD shell theory are identified from the reduced stress power equation. Classical shell constitutive equations are seamlessly incorporated in PD framework by constitutive correspondence approach where a particular form of the classical reduced stress power expression suitable for constitutive correspondence is used. Rotation tensor is updated using an exponential map. A new bond breaking criterion is proposed for PD shell based on critical stretch and critical relative rotation. The solutions of the PD shell governing equations for quasi-static and dynamic cases are obtained through the Newton-Raphson method and the Newmark-beta method, respectively. Numerical simulations include finite deformation of cylindrical shell under both quasi-static and dynamic loading conditions. The PD shell model is validated against finite element solutions obtained using ANSYS®. Crack propagation through the cylindrical shell is simulated under quasi-static and dynamic loading. Wave propagation through cylindrical shell embedded with periodically arranged holes and cracks is investigated. Significant wave attenuation and large bandgap demonstrates the efficacy of the present proposal.

Perturbations of exponential maps: Non-recurrent dynamics

AbstractWe study perturbations of non-recurrent parameters in the exponential family. It is shown that the set of such parameters has Lebesgue measure zero. This particularly implies that the set of escaping parameters has Lebesgue measure zero, which complements a result of Qiu from 1994. Moreover, we show that non-recurrent parameters can be approximated by hyperbolic ones.

Wasserstein principal component analysis for circular measures

We consider the 2-Wasserstein space of probability measures supported on the unit-circle, and propose a framework for Principal Component Analysis (PCA) for data living in such a space. We build on a detailed investigation of the optimal transportation problem for measures on the unit-circle which might be of independent interest. In particular, building on previously obtained results, we derive an expression for optimal transport maps in (almost) closed form and propose an alternative definition of the tangent space at an absolutely continuous probability measure, together with fundamental characterizations of the associated exponential and logarithmic maps. PCA is performed by mapping data on the tangent space at the Wasserstein barycentre, which we approximate via an iterative scheme, and for which we establish a sufficient a posteriori condition to assess its convergence. Our methodology is illustrated on several simulated scenarios and a real data analysis of measurements of optical nerve thickness.

Unscented Schmidt-Kalman filter on Lie groups with application to spacecraft attitude estimation

This paper addresses the estimation problem of nonlinear systems evolving on Lie groups with unknown parameters. More precisely, some parameters in the equations of motion or sensor measurements are unknown, such as gravitational anomalies and measurement biases, and are infeasible to estimate with available observations. The unscented Schmidt-Kalman filter (USKF) approach in Euclidean space is incorporated with exponential maps from Lie algebra to Lie groups, to develop USKF algorithms on Lie groups. Two types of USKFs are derived, respectively, from left-invariant and right-invariant state estimation errors. The two USKFs, not only account for the effect of unknown parameters but also provide estimates preserving the geometry of state manifold. They are advantageous over the extended Schmidt-Kalman filter for nonlinear systems in the sense of avoiding the computation of Jacobian and achieving higher or comparable estimation accuracy depending on the magnitude of parameters uncertainties. The proposed method is then applied to a spacecraft attitude estimation problem based on quaternion representation, where the magnitude of the gyroscope bias noise is unknown. Simulations are conducted to illustrate the effectiveness of the proposed algorithms in comparison with other methods.

Research on Variable Parameter Color Image Encryption Based on Five-Dimensional Tri-Valued Memristor Chaotic System.

To construct a chaotic system with complex characteristics and to improve the security of image data, a five-dimensional tri-valued memristor chaotic system with high complexity is innovatively constructed. Firstly, a pressure-controlled tri-valued memristor on Liu's pseudo-four-wing chaotic system is introduced. Through analytical methods, such as Lyapunov exponential map, bifurcation map and attractor phase diagram, it is demonstrated that the new system has rich dynamical behaviors with periodic limit rings varying with the coupling parameter of the system, variable airfoil phenomenon as well as transient chaotic phenomenon of chaos-periodic depending on the system parameter and chaos-quasi-periodic depending on the memristor parameter. The system is simulated with dynamic circuits based on Simulink. Secondly, the differently structured synchronous controls of chaotic systems are realized using a nonlinear feedback control method. Finally, based on the newly constructed five-dimensional chaotic system, a variable parameter color image encryption scheme is proposed to iteratively generate varying chaotic pseudo-random sequences by varying the system parameters, which will be used for repetition-free disambiguation, additive modulo left-shift diffusion and DNA encryption for the three components of RGB of the color image after chunking. The simulation results are analyzed by histogram, information entropy, adjacent pixel correlation, etc., and the images are tested using differential attack, noise attack and geometric attack, as well as analyzing the PSNR and SSIM of the decrypted image quality. The results show that the encryption method has a certain degree of security and can be applied to medical, military and financial fields with more complex environmental requirements.

Simulation and design of isostatic thick origami structures

Thick origami structures are considered here as assemblies of polygonal panels hinged to each other along their edges according to a corresponding origami crease pattern. The determination of the internal actions in equilibrium with the external loads in such structures is not an easy task, owing to their high degree of static indeterminacy, and the likelihood of unwanted self-balanced internal actions induced by manufacturing imperfections. Here, we present a method for reducing the degree of static indeterminacy which can be applied to several thick origami structures to make them isostatic. The method utilizes sliding hinges, which allow relative translation along the hinge axis, to replace conventional hinges. After giving the analytical description of both types of hinges and describing a rigid folding simulation procedure based on the integration of the exponential map, we present the static analysis of a series of noteworthy examples based on the Miura-ori pattern, the Yoshimura pattern, and the Kresling pattern. Our method, based on kinematic-static duality, provides a novel design paradigm that can be applied for the design and realization of thick origami structures with adequate strength to resist external actions.

Thumbnail-preserving encryption by sum-preserving within blocks based on exponential chaotic map

Thumbnail-preserving encryption by sum-preserving within blocks based on exponential chaotic map

On rigidity of Pham-Brieskorn surfaces

It is well known that, over an algebraically closed field k of characteristic zero, for any three integers a,b,c≥2, any Pham-Brieskorn surface B(a,b,c):=k[X,Y,Z]/(Xa+Yb+Zc) is rigid when at most one of a,b,c is 2 and stably rigid when 1a+1b+1c≤1. In this paper we consider Pham-Brieskorn domains over an arbitrary field k of characteristic p≥0 and give sufficient conditions on (a,b,c) for which any Pham-Brieskorn domain B(a,b,c) is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on k[X,Y,Z,T]/(XmY+Tprq+Zpe)=k[x,y,z,t], for m,q>1, p∤mq and e>r≥1, fixing y, a crucial result used in [10] to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine 3-space in positive characteristic.We also provide a sufficient condition for B(a,b,c) to be stably rigid. Along the way we prove that for integers a,b,c≥2 with gcd(a,b,c)=1 and for F(Y)∈k[Y], the ring k[X,Y,Z]/(XaYb+Zc+F(Y)) is a rigid domain.

The Wasserstein Distance for Ricci Shrinkers

Abstract Let $(M^{n},g,f)$ be a Ricci shrinker such that $\text{Ric}_{f}=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4\pi )^{-\frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_{p}M$, namely the Gaussian measure $\gamma $ and the measure $\overline{\nu }$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_{0}$ between the measures $\overline{\nu }$ and $\gamma $, and which also elucidates the rigidity implications resulting from this estimate.

Analysis of the Dynamics of a Cubic Nonlinear Five-Dimensional Memristive Chaotic System and the Study of Reduced-Dimensional Synchronous Masking

To improve the complexity of the chaotic system and achieve the effective transmission of image information, in this paper, a five-dimensional memristive chaotic system with cubic nonlinear terms is constructed, which has four pairs of symmetric coordinates. First, the cubic nonlinear memristive chaotic system is analyzed using the Lyapunov exponential map, bifurcation map, and attractor phase diagram. The experimental results show that under four pairs of symmetric coordinates, the system exists not only parameter-dependent symmetric rotational coexisting attractor and transient chaotic phenomena but also exists super-multistationary with alternating chaotic cycles dependent on the initial value of the memristor. Then, it is proposed to add a constant term to the linear state variable to explore the effect of the offset increment of the linear state variable on the system in four pairs of symmetric coordinates, while circuit simulation of the five-dimensional chaotic system is carried out using Simulink to verify its existence and realisability. Finally, the synchronization of the dimensionality reduction system and the confidential transmission of the image are achieved, using the control voltage of the system to replace the internal state variables of the memristor to achieve the one-dimensional reduction process, and an adaptive synchronization controller is designed to synchronize the system before and after the dimensionality reduction. Based on the above, an image to be transmitted is modulated into a one-dimensional array and then subjected to the fractional and cyclic operations and combined with the linear encryption and decryption functions and the chaotic masking technique, the simple encryption and decryption of the image processes are realized.

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