Elliptic Case Research Articles

Convergence of the kinetic annealing for general potentials

The convergence of the kinetic langevin simulated annealing is proven under mild assumptions on the potential U for slow logarithmic cooling schedules, which widely extends the scope of the previous results of [14]. Moreover, non-convergence for fast logarithmic and non-logarithmic cooling schedules is established. The results are based on an adaptation to non-elliptic non-reversible kinetic settings of a localization/local convergence strategy developed by Fournier and Tardif in [6] in the overdamped elliptic case, and on precise quantitative high order Sobolev hypocoercive estimates.

Explicit height bounds for K-rational points on transverse curves in powers of elliptic curves

Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper algebraic subgroups of $E^N$. The method gives a totally explicit version of the Manin-Dam'janenko Theorem in the elliptic case and it is a generalisation of previous results only proved when $E$ does not have Complex Multiplication.

Integrability, normal forms, and magnetic axis coordinates

Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation entirely consisting of invariant submanifolds. A compact regular leaf (a flux surface) of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves, including elliptic and hyperbolic magnetic axes. In this paper, the existence of exact smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydrodynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields: they are presymplectic systems.

Normal forms and near-axis expansions for Beltrami magnetic fields

A formal series transformation to Birkhoff–Gustavson normal form is obtained for toroidal magnetic field configurations in the neighborhood of a magnetic axis. Bishop's rotation minimizing coordinates are used to obtain a local orthogonal frame near the axis in which the metric is diagonal, even if the curvature has zeros. We treat the cases of vacuum and force-free (Beltrami) fields in a unified way, noting that the vector potential is essentially the Poincaré–Liouville one-form of Hamiltonian dynamics, and the resulting magnetic field corresponds to the canonical two-form of a non-autonomous one-degree-of-freedom system. Canonical coordinates are obtained and Floquet theory is used to transform to a frame in which the lowest order Hamiltonian is autonomous. The resulting magnetic axis can be elliptic or hyperbolic, and resonant elliptic cases are treated. The resulting expansion for the field is shown to be well-defined to all orders, and is explicitly computed to degree four. An example is given for an axis with constant torsion near a 1:3 resonance.

Neumann problem for Monge-Ampere type equations revisited.

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.

The continuation method and the real analyticity of the accessory parameters: the general elliptic case

We apply the Le Roy–Poincaré continuation method to prove the real analytic dependence of the accessory parameters on the position of the sources in the Liouville theory in presence of any number of elliptic sources. The treatment is easily extended to the case of the torus with any number of elliptic singularities. A discussion is given of the extension of the method to parabolic singularities and higher genus surfaces.

Essential self-adjointness of real principal type operators

We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.

Lipschitz Bounds and Nonautonomous Integrals

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

Superconvergent HDG methods for Maxwell’s equations via the [formula omitted]-decomposition

The concept of the M-decomposition was introduced by Cockburn et al. (2017) to provide criteria to guarantee optimal convergence rates for the Hybridizable Discontinuous Galerkin (HDG) method for coercive elliptic problems. In that paper they systematically constructed superconvergent hybridizable discontinuous Galerkin (HDG) methods to approximate the solutions of elliptic PDEs on unstructured meshes. In this paper, we use the M-decomposition to construct HDG methods for the Maxwell’s equations on unstructured meshes in two dimension. In particular, we show the any choice of spaces having an M-decomposition, together with sufficiently rich auxiliary spaces, has an optimal error estimate and superconvergence even though the problem is not in general coercive. Motivated by the elliptic case, we obtain a superconvergent rate for the curl and flux of the solution, and this is confirmed by our numerical experiments.

Riemann–Hilbert problem on a hyperelliptic surface and uniformly stressed inclusions embedded into a half-plane subjected to antiplane strain

An inverse problem of the elasticity of n elastic inclusions embedded into an elastic half-plane is analysed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the conditions of ideal contact hold in the interfaces between the inclusions and the half-plane. The shapes of the inclusions are not prescribed and have to be determined by enforcing uniform stresses inside the inclusions. The method of conformal mappings from a slit domain onto the ( n + 1 ) -connected physical domain is worked out. It is shown that to recover the map and the shapes of the inclusions, one needs to solve a vector Riemann–Hilbert problem on a genus- n hyperelliptic surface. In a particular case of loading, the vector problem reduces to two scalar Riemann–Hilbert problems on n + 1 slits on a hyperelliptic surface. In the elliptic case, in addition to three parameters of the model, the conformal map possesses a free geometric parameter. The results of numerical tests in the elliptic case show the impact of these parameters on the inclusion shape.

Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems

We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale varepsilon >0, we establish homogenization error estimates of the order varepsilon in case dgeqq 3, and of the order varepsilon |log varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence varepsilon ^delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,alpha } regularity theory is available.

Comment on ‘An efficient code to solve the Kepler equation: elliptic case’

ABSTRACTIn a recent MNRAS article, Raposo-Pulido and Pelaez (RPP) designed a scheme for obtaining very close seeds for solving the elliptic Kepler equation with the classical and modified Newton–Raphson methods. This implied an important reduction in the number of iterations needed to reach a given accuracy. However, RPP also made strong claims about the errors of their method that are incorrect. In particular, they claim that their accuracy can always reach the level of ∼5ε, where ε is the machine epsilon (e.g. ε = 2.2 × 10−16 in double precision), and that this result is attained for all values of the eccentricity e < 1 and the mean anomaly M ∈ [0, π], including for e and M that are arbitrarily close to 1 and 0, respectively. However, we demonstrate both numerically and analytically that any implementation of the classical or modified Newton–Raphson methods for Kepler’s equation, including those described by RPP, has a limiting accuracy of the order of ${\sim}\varepsilon /\sqrt{2(1-e)}$. Therefore the errors of these implementations diverge in the limit e → 1, and differ dramatically from the incorrect results given by RPP. Despite these shortcomings, the RPP method can provide a very efficient option for reaching such limiting accuracy. We also provide a limit that is valid for the accuracy of any algorithm for solving Kepler equation, including schemes like bisection that do not use derivatives. Moreover, similar results are also demonstrated for the hyperbolic Kepler equation. The methods described in this work can provide guidelines for designing more accurate solutions of the elliptic and hyperbolic Kepler equations.

Fractional linear maps in general relativity and quantum mechanics

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.

Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms – Part II

We study the Selmer group associated to a p-ordinary newform f∈S2r(Γ0(N)) over the anticyclotomic Zp-extension of an imaginary quadratic field K/Q. Under certain assumptions, we prove that this Selmer group has no proper Λ-submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants of congruent modular forms by the present authors. Along the way, we prove a general result on the vanishing of several anticyclotomic μ-invariants attached to f.

Random Walks with Invariant Loop Probabilities: Stereographic Random Walks.

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

Trajectory Design and Maintenance of the Martian Moons eXploration Mission Around Phobos

The Martian Moons eXploration mission, currently under development by the Japan Aerospace Exploration Agency (JAXA), will be launched in 2024 with the goal of retrieving pristine samples from the surface of Phobos. Soon after arrival, the spacecraft will inject into retrograde relative trajectories known as quasi-satellite orbits and study the geophysical environment of the Martian moon for more than three years. This paper presents the orbit design and maintenance strategy of the Martian Moons eXploration mission in the framework of the elliptic Hill problem with ellipsoidal secondary. This paper first introduces a numerical continuation procedure on the eccentricity of Phobos to replace purely periodic solutions with families of quasi-periodic invariant tori. Two-dimensional torus maps can be then constructed and used to represent physical quantities of interest, as well as to generate reference trajectories at arbitrary epochs. Sensitivity and stability analyses are carried out to investigate the dynamic properties of retrograde relative trajectories in the elliptic case. Finally, a linear quadratic regulator is implemented in order to assess the robustness of the computed trajectories under injection, navigation, and execution errors. Monte Carlo simulations demonstrate that the baseline quasi-satellite orbits of the Martian Moons eXploration mission can be maintained with as low as per month.

Elliptic q,t matrix models

The Gaussian matrix model is known to deform to the q,t-matrix model. We consider further deformation to the elliptic q,t matrix model by properly deforming the Gaussian density as well as the Vandermonde factor. Properties of an associated basis of symmetric functions that provide the matrix model property <char>∼char in the deformed elliptic case are discussed.

Regularity for the fully nonlinear parabolic thin obstacle problem

We prove [Formula: see text] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.

Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations

In this paper, we study the boundedness and Holder continuity of local weak solutions to the following nonhomogeneous equation $$\begin{aligned} \partial _tu(x,t)+\mathrm{P.V.}\int _{{\mathbb {R}}^N}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}\big (u(x,t)-u(y,t)\big )dy= f(x,t,u) \end{aligned}$$ in $$Q_T=\Omega \times (0,T)$$ , where the symmetric kernel K(x, y, t) has a generalized form of the fractional p-Laplace operator of s-order. We impose some structural conditions on the function f and use the De Giorgi-Nash-Moser iteration to establish the boundedness of local weak solutions in the a priori way. Based on the boundedness result, we also obtain Holder continuity of bounded solutions in the superquadratic case. These results can be regarded as a counterpart to the elliptic case due to Di Castro et al. (Ann Inst H Poincare Anal Non Lineaire, 2016).

Modular transformations of elliptic Feynman integrals

We investigate the behaviour of elliptic Feynman integrals under modular transformations. This has a practical motivation: Through a suitable modular transformation we can achieve that the nome squared is a small quantity, leading to fast numerical evaluations. Contrary to the case of multiple polylogarithms, where it is sufficient to consider just variable transformations for the numerical evaluations of multiple polylogarithms, it is more natural in the elliptic case to consider a combination of a variable transformation (i.e. a modular transformation) together with a redefinition of the master integrals. Thus we combine a coordinate transformation on the base manifold with a basis transformation in the fibre. Only in the combination of the two transformations we stay within the same class of functions.