Underlying Vector Field Research Articles

On the lack of Gaussian tail for rough line integrals along fractional Brownian paths

We show that the tail probability of the rough line integral ∫01ϕ(Xt)dYt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{0}^{1}\\phi (X_{t})dY_{t}$$\\end{document}, where (X, Y) is a 2D fractional Brownian motion with Hurst parameter H∈(1/4,1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\in (1/4,1/2)$$\\end{document} and ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} is a Cb∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{b}^{\\infty }$$\\end{document}-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}-Weibull tail with any exponent γ>2H+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma >2H+1$$\\end{document}. In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class Cb∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{b}^{\\infty }$$\\end{document}. This also demonstrates that the well-known upper tail estimate proved by Cass–Litterer–Lyons in 2013 is essentially sharp.

Mathematical Model of Two-Dimensional Disease Progression and Categorization of Underlying Vector Fields with Machine Learning

A mathematical model was created for simulating disease progression in two-dimension, by specifying a vector field. It was equivalent to observing how a lesion pattern on the skin, could change its shape over time. This model had a parameter that might control the resolution and signal of the pattern. By specifying two different vector fields, we established two sets of subtly different image sets. And, three different supervised machine learning algorithms such as nearest neighbor, decision tree, and support vector machine were used for binary classification. This study was done with Python's numerical (numpy), plotting (matplotlib), and machine learning (sklearn) libraries on the Google Colab platform. All machine learning algorithms were able to distinguish subtle differences produced by different vector fields. Furthermore, the performance of the algorithms were improved by concatenating the beginning and end stages of the pattern and helping the algorithms to pick up the temporal changes. These results demonstrated how AI and machine learning could be adopted in medicine for accurately diagnosing underlying diseases from images.

Existence and structure of P-area minimizing surfaces in the Heisenberg group

We study existence and structure of P-area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field N that characterized existence and structure of P-area minimizing surfaces. This vector field exists even if there is no P-area minimizing surface satisfying the prescribed boundary conditions. We prove that if ∂Ω satisfies a so called Barrier condition, it is sufficient to guarantee existence of such surfaces. Our approach is completely different from previous methods in the literature and makes major progress in understanding existence of P-area minimizing surfaces.

Neural Flow Map Reconstruction

AbstractIn this paper we present a reconstruction technique for the reduction of unsteady flow data based on neural representations of time‐varying vector fields. Our approach is motivated by the large amount of data typically generated in numerical simulations, and in turn the types of data that domain scientists can generate in situ that are compact, yet useful, for post hoc analysis. One type of data commonly acquired during simulation are samples of the flow map, where a single sample is the result of integrating the underlying vector field for a specified time duration. In our work, we treat a collection of flow map samples for a single dataset as a meaningful, compact, and yet incomplete, representation of unsteady flow, and our central objective is to find a representation that enables us to best recover arbitrary flow map samples. To this end, we introduce a technique for learning implicit neural representations of time‐varying vector fields that are specifically optimized to reproduce flow map samples sparsely covering the spatiotemporal domain of the data. We show that, despite aggressive data reduction, our optimization problem — learning a function‐space neural network to reproduce flow map samples under a fixed integration scheme — leads to representations that demonstrate strong generalization, both in the field itself, and using the field to approximate the flow map. Through quantitative and qualitative analysis across different datasets we show that our approach is an improvement across a variety of data reduction methods, and across a variety of measures ranging from improved vector fields, flow maps, and features derived from the flow map.

Trajectory Planning in Unknown 2D Workspaces: A Smooth, Reactive, Harmonics-Based Approach

A novel reactive method for robot trajectory planning within unknown 2D workspaces is presented in this letter. The trajectories provided by this method stem from an underlying potential field and are provably safe, with asymptotic convergence to the desired position. Given an initially unknown workspace, and a sensing process for obtaining boundary information, the scope of this work is to provide an underlying vector field that can be used as a reference signal for higher order, non-linear dynamical systems (robots). This can be achieved by tracking, either the field itself, or the trajectories that result from the latter. In this spirit, our method is advantageous due to: a) its reactivity, as open-loop approaches may exhibit large errors during the transient phase within unknown workspaces, whereas our method provides a velocity profile for each position of the robot within the workspace, b) its smoothness, and c) its safety and convergence guarantees. We prove the asserted claims and provide rigorous comparative simulations in various benchmark workspaces, as well as in a high-fidelity ROS-Gazebo implementation.

Robotic Coverage Path Planning for Ultrasonic Inspection

Automatic robotic inspection of arbitrary free-form shapes is relevant for many quality control applications in different industries. We propose a method for planning the motion of an industrial robot to perform ultrasonic inspection of varying 3D shapes. Our method starts with the calculation of a set of sub-paths. These sub-paths are derived from streamlines. The underlying vector field is deduced from local curvature of the inspected geometry. Intermediate robot motions are planned to connect individual sub-paths to obtain a single complete inspection path. Coverage is calculated via ray tracing to simulate the propagation of ultrasound signals. This simulation enables the algorithm to proceed adaptively and to find a good trade-off between path length and coverage. We report experiments for four different geometries. The results indicate that shorter paths are achieved by using ray tracing for adaptive adjustment of streamline density. Our algorithm is tailored to ultrasonic inspection. However, the main concept of exploiting local surface curvature and streamlines for coverage path planning generalizes to other robotic inspection problems.

Topological Route to New and Unusual Coulomb Spin Liquids.

Coulomb spin liquids are topological magnetic states obeying an emergent Gauss law. Little distinction has been made between different kinds of Coulomb liquids. Here we show how a series of distinct Coulomb liquids can be generated straightforwardly by varying the constraints on a classical spin system. This leads to pair creation, and coalescence, of topological defects of an underlying vector field. The latter makes higher-rank spin liquids, of recent interest in the context of fracton theories, with attendant multifold pinch points in the structure factor, appear naturally. New Coulomb liquids with an abundance of pinch points also arise. We thus establish a new and general route to uncovering exotic Coulomb liquids, via the manipulation of topological defects in momentum space.

On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds

In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the SVI via the subgradient extragradient approach, where each algorithm consists of two parts which are of symmetric structure mutually. It is proven that, if the underlying vector fields are of monotonicity, then the sequences constructed by these algorithms converge to a solution of the SVI. We also discuss applications of these algorithms for approximating solutions to the VIP. Our theorems complement some recent and important ones in the literature.

Averaging principle for stochastic differential equations in the random periodic regime

We present the validity of stochastic averaging principle for non-autonomous slow–fast stochastic differential equations (SDEs) whose fast motions admit random periodic solutions. Our investigation is motivated by some problems arising from multi-scale stochastic dynamical systems, where configurations are time dependent due to nonlinearity of the underlying vector fields and the onset of time dependent random invariant sets. Averaging principle with respect to uniform ergodicity of the fast motion is no longer available in this scenario. The ergodicity of time periodic measures of the fast motion on certain minimal Poincaré section is used to identify the averaging limit.

Lipschitz continuity for isotropic matrix functions

We prove a sharp Lipschitz bound for isotropic matrix functions in terms of the Lipschitz constant for the underlying vector field.

An introduction to mixed hemivariational inequality problems on Hadamard manifolds

In this article, we introduce a category of mixed hemivariational inequality problems on Hadamard manifolds. Using Fan-KKM lemma we initiate the existence of solutions of mixed hemivariational inequality problems (MHVIP) when the underlying vector field is monotone. We also establish the proximal point algorithm for solving mixed hemivariational inequality problems on these nonlinear spaces. Our outcomes and schemes may stimulate further exploration in this fascinating and interesting area of analysis.

Parallel Tseng’s Extragradient Methods for Solving Systems of Variational Inequalities on Hadamard Manifolds

The aim of this article is to study two efficient parallel algorithms for obtaining a solution to a system of monotone variational inequalities (SVI) on Hadamard manifolds. The parallel algorithms are inspired by Tseng’s extragradient techniques with new step sizes, which are established without the knowledge of the Lipschitz constants of the operators and line-search. Under the monotonicity assumptions regarding the underlying vector fields, one proves that the sequences generated by the methods converge to a solution of the monotone SVI whenever it exists.

Approximation of Lyapunov functions from noisy data

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.

Modified Tseng's extragradient methods for variational inequality on Hadamard manifolds

ABSTRACT This paper is devoted to two efficient algorithms for solving variational inequality on Hadamard manifolds. The algorithms are inspired by Tseng's extragradient methods with new step sizes, established without the knowledge of the Lipschitz constant of the mapping. Under a pseudomonotone assumption on the underlying vector field, we prove that the sequence generated by the methods converges to a solution of variational inequality, whenever it exists.

Predictability of escape for a stochastic saddle-node bifurcation: When rare events are typical.

Transitions between multiple stable states of nonlinear systems are ubiquitous in physics, chemistry, and beyond. Two types of behaviors are usually seen as mutually exclusive: unpredictable noise-induced transitions and predictable bifurcations of the underlying vector field. Here, we report a different situation, corresponding to a fluctuating system approaching a bifurcation, where both effects collaborate. We show that the problem can be reduced to a single control parameter governing the competition between deterministic and stochastic effects. Two asymptotic regimes are identified: When the control parameter is small (e.g., small noise), deviations from the deterministic case are well described by the Freidlin-Wentzell theory. In particular, escapes over the potential barrier are very rare events. When the parameter is large (e.g., large noise), such events become typical. Unlike pure noise-induced transitions, the distribution of the escape time is peaked around a value which is asymptotically predicted by an adiabatic approximation. We show that the two regimes are characterized by qualitatively different reacting trajectories with algebraic and exponential divergences, respectively.

An Extension of Lyapunov’s First Method to Nonlinear Systems With Non-Continuously Differentiable Vector Fields

This letter investigates the extension of Lyapunov’s first method to nonlinear systems in the case where the $\mathscr {C}^{{1}}$ -regularity assumption, i.e., the underlying vector field is continuously differentiable, is not satisfied. It is shown that if this regularity assumption is not fulfilled, the Hurwitz nature of the Jacobian matrix, if it exists, does not guarantee the stability of the original nonlinear system. Under weaker assumptions than the $\mathscr {C}^{{1}}$ -regularity, namely the existence of the directional derivatives of the vector field, conditions for guaranteeing the local exponential stability of the nonlinear system are derived.

Poincaré gauge gravity: An emergent scenario

The Poincar\'e gauge gravity (PGG) with the underlying vector fields of tetrads and spin-connections is perhaps the best theory candidate for gravitation to be unified with the other three elementary forces of nature. There is a clear analogy between local frame in PGG and local internal symmetry space in the Standard Model. As a result, the spin-connection fields, gauging the local frame Lorentz symmetry group SO(1,3)_{LF}, appear in PGG much as photons and gluons appear in SM. We propose that such an analogy may follow from their common emergent nature allowing to derive PGG in the same way as conventional gauge theories. In essence, we start with an arbitrary theory of some vector and fermion fields which possesses only global spacetime symmetries, such as Lorentz and translational invariance, in flat Minkowski space. The two vector field multiplets involved are proposed to belong, respectively, to the adjoint (A_{{\mu}}^{ij}) and vector (e_{{\mu}}^{i}) representations of the starting global Lorentz symmetry. We show that if these prototype vector fields are covariantly constrained, A_{{\mu}}^{ij}A_{ij}^{{\mu}}=M_{A} and e_{{\mu}}^{i}e_{i}^{{\mu}}=M_{e}, thus causing a spontaneous violation of the accompanying global symmetries (M_{A,e} are their proposed violation scales), then the only possible theory compatible with these length-preserving constraints is turned out to be the gauge invariant PGG, while the corresponding massless (pseudo)Goldstone modes are naturally collected in the emergent gauge fields of tetrads and spin-connections. In a minimal theory case being linear in a curvature we unavoidably come to the Einstein-Cartan theory. The extending theories with propagating spin-connection and tetrad modes are also considered and their possible unification with the Standard Model is briefly discussed.

Phase portrait analysis for automatic initialization of multiple snakes for segmentation of the ultrasound images of breast cancer

Segmentation of ultrasound (US) images of breast cancer is one of the most challenging problems of modern medical image processing. A number of popular codes for US segmentation are based on the active contours (snakes) and on a variety of modifications of gradient vector flow. The snakes have been used to locate objects in various applications of medical images. However, the main difficulty in applying the method is initialization. Therefore, we suggest a new method for automatic initialization of active contours based on phase portrait analysis (PPA) of the underlying vector field and a sequential initialization of trial multiple snakes. The PPA makes it possible to exclude the noise and artifacts and properly initialize the multiple snakes. In turn, the trial snakes allow us to differentiate between the seeds initialized inside and outside the desired object. While preceding methods require the manual selection of at least one seed point inside the object or rely on the particular distribution of the gray levels, the proposed method is fully automatic and robust to the noise, as can be seen from the tests with synthetic and real images.

A Projection-Type Method for Set Valued Variational Inequality Problems on Hadamard Manifolds

In this paper, we develop a projection-type algorithm for set-valued variational inequalities on Hadamard manifolds. The proposed method is well defined whether the solution set of the problem is non-empty or not. Under pseudomonotonicity assumptions on the underlying vector field, our method is convergent to a solution of the given set-valued variational inequality. The results presented in this paper generalize and improve some known results introduced by Tang et al. (Optimization 64(5):1081–1096, 2015).

Shock waves: The Maxwell-Cattaneo case.

Several continuum theories for shock waves give rise to a set of differential equations in which the analysis of the underlying vector field can be done using the tools of the theory of dynamical systems. We illustrate the importance of the divergences associated with the vector field by considering the ideas by Maxwell and Cattaneo and apply them to study shock waves in dilute gases. By comparing the predictions of the Maxwell-Cattaneo equations with shock wave experiments we are lead to the following conclusions: (a) For low compressions (low Mach numbers: M) the results from the Maxwell-Cattaneo equations provide profiles that are in fair agreement with the experiments, (b) as the Mach number is increased we find a range of Mach numbers (1.27 ≈ M(1) < M < M(2) ≈ 1.90) such that numerical shock wave solutions to the Maxwell-Cattaneo equations cannot be found, and