Global Diffeomorphism Research Articles

Surjectivity of linear operators and semialgebraic global diffeomorphisms

Surjectivity of linear operators and semialgebraic global diffeomorphisms

Continuous sliding mode control for the translational oscillator with a rotating actuator system

The stabilization and disturbance rejection of the translational oscillator with a rotating actuator (TORA) are considered in this paper. To deal with the control issues, a novel continuous sliding mode control method is designed for the TORA system. Compared with existing sliding mode control methods for the TORA system, the proposed method here is continuous. Specifically, first, a global diffeomorphism is introduced for the model of the TORA system. Then, an elaborate sliding manifold is constructed, and a continuous sliding mode control scheme is developed to ensure the convergence of the sliding manifold. Furthermore, rigorous theoretical analysis is given. Finally, simulation tests are carried out, and the obtained simulation results demonstrate that the proposed method exhibits superior stabilization control performance and strong robustness.

A weight-homogenous condition to the real Jacobian conjecture in

AbstractIt is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

Confinement in 4D: An Attempt at Classical Understanding

In this review, we revisit our approach to constructing an effective theory for Abelian and Non-Abelian gauge theories in 4D. Our goal is to have an effective theory that provides a simple classical picture of the main qualitatively important features of these theories. We set out to ensure the presence of the massless photons—Goldstone bosons in Abelian theory and their disappearance in the Non-Abelian case—accompanied by the formation of confining strings between charged states. Our formulation avoids using vector fields and instead operates with the basic degrees of freedom that are the scalar fields of a nonlinear σ-model. The Mark 1 model we study turns out to have a large global symmetry group-the 2D diffeomorphism invariance in the Abelian limit, which is isomorphic to the group of all canonical transformations in the classical two dimensional phase space. This symmetry is not present in QED, and we eliminate it by “gauging” this infinite dimensional global group. Introducing additional modifications to the model (Mark 2), we are able to prove that the “Abelian” version is equivalent to the theory of a free photon. Achieving the desired property in the “Non-Abelian” regime turns out to be tricky. We are able to introduce a perturbation that leads to the formation of confining strings in our Mark 1 model. These strings have somewhat unusual properties, in that their profile does not decay exponentially away from the center of the string. In addition, the perturbation explicitly breaks the diffeomorphism invariance. Preserving this invariance in the gauged model as well as achieving confining strings in Mark 2 model remains an open question.

Globally diffeomorphic sigma-harmonic mappings

Given a two-dimensional mapping U whose components solve a divergence structure elliptic equation,we give necessary and sufficient conditions on the boundary so that U is a global diffeomorphism.

Perturbation Theory of Transformed Quantum Fields

We consider a scalar quantum field ϕ with arbitrary polynomial self-interaction in perturbation theory. If the field variable ϕ is repaced by a global diffeomorphism ϕ(x) = ρ(x) + a1ρ2(x) + …, this field ρ obtains infinitely many additional interaction vertices. We propose a systematic way to compute connected amplitudes for theories involving vertices which are able to cancel adjacent edges. Assuming tadpole graphs vanish, we show that the S-matrix of ρ coincides with the one of ϕ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. The diffeomorphism can be tuned to cancel all contributions of an underlying ϕt-type self interaction at one fixed external offshell momentum, rendering ρ a free theory at this momentum. Finally, we mention one way to extend the diffeomorphism to a non-diffeomorphism transformation involving derivatives without spoiling the combinatoric structure of the global diffeomorphism.

A Global Diffeomorphism Theorem for Fréchet Spaces

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fr\'{e}chet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fr\'{e}chet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fr\'{e}chet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.

On the existence of a global diffeomorphism between Fréchet spaces

We provide sufficient conditions for the existence of a global diffeomorphism between tame Fr\'echet spaces. We prove a version of mountain pass theorem which plays a key ingredient in the proof of the main theorem.

THE SCHWARZIAN DERIVATIVE AND CONFORMAL TRANSFORMATION ON FINSLER MANIFOLDS

Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ \mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.

Finite Time Cascade Control Scheme for Tracking Control of an Underwater Exploring Robot

In this paper, we discuss the curve trajectory tracking problem of an underactuated underwater robot in the horizontal plane. We use global diffeomorphism transformation method to transform the sixth-order dynamic tracking error model into a nonlinear cascaded system, by using the cascaded systems’ finite time stability theory, along with the back stepping technique to design the controller. Two discontinuous finite time controllers are designed for the two subsystems respectively. In the existing literature, finite time control is used for the first order nonholonomic system, we extended this method to two order nonholonomic system in this paper. The finite time controller can tune not only the gain, but also the fractional power, it is more convenience for engineering implementation. The results show the control laws can achieve the control objective with good performance for the horizontal trajectory tracking control.

An exact solution for geophysical internal waves with underlying current in modified equatorial β-plane approximation

In this paper, a modification of the standard geophysical equatorial β-plane model equations, incorporating a gravitational-correction term in the tangent plane approximation, is derived. We present an exact solution to meet the modified governing equations, whose form is explicit in the Lagrangian framework and which represents internal oceanic waves in the presence of a constant underlying current. It is rigorously established, that the solution is dynamically possible, by way of analytical and degree-theoretical considerations. In the sense that the mapping it prescribes from Lagrangian to Eulerian coordinates is a global diffeomorphism. In addition, the paper an analysis of the mean flow velocities and related mass transport are presented in this paper, they are induced by certain geophysical internal waves. In particular, we examine an exact solution to the geophysical governing equations in the modified β-plane approximation at the equator which incorporates a constant underlying current.

On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative

In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems.

Solvability of Abstract Semilinear Equations by a Global Diffeomorphism Theorem

In this work we develop a method towards unique solvability of abstract semilinear equations. We use a global diffeomorphism theorem for which we provide a simplified proof. Applications to second order partial differential equations are given. Some additional technical tools about the properties of the Niemytskij operator are also given.

A global diffeomorphism theorem and a unique weak solution of Dirichlet problem

ABSTRACTUsing an improved version of the infinite dimensional theorem on diffeomorphisms, we obtain conditions for the existence of the unique weak solution of Dirichlet problem, where is a bounded domain with a -boundary and is a -Carathéodory function. It is also obtained that the solution operator, which assigns to each the unique solution of the equation, is a diffeomorphism.

Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions

We present a rigorous analysis of the nonlinear surface waves in the presence of a zonal current under the effects of Earth’s rotation derived by Constantin and Monismith (J Fluid Mech 820:511–528, 2017). It is shown that the three-dimensional Lagrangian flow-map describing this exact solution is a global diffeomorphism, resulting in a flow description that is dynamically possible.

Counterexamples in calculus of variations in L∞ through the vectorial Eikonal equation

We show that, for any regular bounded domain Ω⊆Rn, n=2,3, there exist infinitely many global diffeomorphisms equal to the identity on ∂Ω that solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the ∞-Laplace system arising in vectorial calculus of variations in L∞ does not suffice to characterise either limits of p-Harmonic maps as p→∞ or absolute minimisers in the sense of Aronsson.

Black hole evaporation, quantum hair and supertranslations

In a black hole, hair and quantum information retrieval are interrelated phenomena. The existence of any new form of hair necessarily implies the existence of features in the quantum-mechanically evaporated radiation. Therefore, classical supertranslation hair can be only distinguished from global diffeomorphisms if we have access to the interior of the black hole. Indirect information on the interior can only be obtained from the features of the quantum evaporation. We demonstrate that supertranslations (T^-,T^+) in BMS_{-}otimes BMS_{+} can be used as bookkeepers of the probability distributions of the emitted quanta where the first element describes the classical injection of energy and the second one is associated to quantum-mechanical emission. However, the connection between T^- and T^+ is determined by the interior quantum dynamics of the black hole. We argue that restricting to the diagonal subgroup is only possible for decoupled modes, which do not bring any non-trivial information about the black hole interior and therefore do not constitute physical hair. It is shown that this is also true for gravitational systems without horizon, for which both injection and emission can be described classically. Moreover, we discuss and clarify the role of infrared physics in purification.

Global diffeomorphism theorem applied to the solvability of discrete and continuous boundary value problems

Using the global diffeomorphism theorem we consider the existence of solutions to the following Dirichlet problem , , where is a jointly continuous function subject to some further growth conditions. Together with a given problem we consider the family of its discretizations and as a result we prove the existence of a non-spurious solution.

Initial Value Problem Method for Diffeomorphism and Its Applications

A set of sufficient conditions related to a class of initial value problems are given for the global diffeomorphism of a nonlinear mapping F. In addition, the results obtained are used to discuss the existence and uniqueness of solution for the boundary value problems.

On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers

In this article we analyze the global diffeomorphism property of polynomial maps $$F:\mathbb {R}^n\rightarrow \mathbb {R}^n$$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials $$\Vert F\Vert _2^2$$ . This allows us to identify a class of polynomial maps F for which their global diffeomorphism property on $$\mathbb {R}^n$$ is equivalent to their Jacobian determinant $$\det JF$$ vanishing nowhere on $$\mathbb {R}^n$$ . In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.