Analytic Boundary Research Articles

Channel‐Spanning Logjams and Reach‐Scale Hydraulic Resistance in Mountain Streams

AbstractLogjams create an upstream backwater of deepened, slower water, locally reducing bed shear stress. We compared hydraulic impact of logjam series across 37 geomorphically diverse reaches of mountain streams observed over 11 years in the US Southern Rockies. To enable reach‐scale comparison of logjam structure and spacing, we identified the modeled best‐fit effective resistance coefficient minimizing difference between outflow exiting a 1D channel with logjams present, and the same model channel with elevated channel resistance. Effective resistance increased with ratio of jam upstream depth to depth without a logjam, ratio of backwater length to average spacing, and decreased for randomly distributed jams due to close spacing, which reduced backwater impact. An analytic approximation and boundaries for region of relative spacing with steepest increase in effective resistance are provided. Our results can assist in targeting interventions to areas where hydraulic impact is greatest, providing value for money in nature‐based solution design.

A computation of two-loop six-point Feynman integrals in dimensional regularization

We compute three families of two-loop six-point massless Feynman integrals in dimensional regularization, namely the double-box, the pentagon-triangle, and the hegaxon-bubble family. This constitutes the first analytic computation of two-loop master integrals with eight scales. We use the method of canonical differential equations. We describe the corresponding integral basis with uniform transcendentality, the relevant function alphabet, and analytic boundary values at a particular point in the Euclidean region up to the fourth order in the regularization parameter ϵ. The results are expressed as one-fold integrals over classical polylogarithms. We provide a set of supplementary files containing our results in machine-readable form, including a proof-of-concept implementation for numerical evaluations of the one-fold integrals valid within a subset of the Euclidean region.

Polynomial and rational convergence rates for Laplace problems on planar domains

Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is non-convex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates and the theory of crowding of conformal maps.

Families of non-tiling domains satisfying Pólya’s conjecture

We show the existence of classes of non-tiling domains satisfying Pólya’s conjecture in any dimension, in both the Euclidean and non-Euclidean cases. This is a consequence of a more general observation asserting that if a domain satisfies Pólya’s conjecture eventually, that is, for a sufficiently large order of the eigenvalues, and may be partioned into p non-overlapping isometric sub-domains, with p arbitrarily large, then there exists an order p0 such that for p larger than p0 all such sub-domains satisfy Pólya’s conjecture. In particular, this allows us to show that families of sectors of domains of revolution with analytic boundary, and thin cylinders satisfy Pólya’s conjecture, for instance. We also improve upon the Li–Yau constant for general cylinders in the Dirichlet case.

Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions

AbstractThe time-harmonic Maxwell equations at high wavenumberkin domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly inkand an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of orderpon a mesh with mesh sizehis shown under thek-explicit scale resolution condition that (a)kh/pis sufficient small and (b)$$p/\ln k$$p/lnkis bounded from below.

Singularities almost always scatter: Regularity results for non‐scattering inhomogeneities

AbstractIn this paper we examine necessary conditions for an inhomogeneity to be non‐scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out by Williams in order to establish that a domain, which does not possess the Pompeiu Property, has a real analytic boundary. That analysis, as well as ours, relies crucially on classical free boundary regularity results due to Kinderlehrer and Nirenberg, and Caffarelli.

Well-Posedness of a Hirota–Satsuma System Posed on a Half Line

This paper studies the initial boundary value problem of the Hirota–Satsuma system posed on the half line. u t − α u x x x + 6 u u x = 2 β v v x , x > 0 , t > 0 , v t + v x x x + 3 u v x = 0 , x > 0 , t > 0 , u x , 0 = u 0 x , v x , 0 = v 0 x , x > 0 , u 0 , t = f t , v 0 , t = g t , t > 0. . For − 1 / 8 < s < 3 / 2 , s ≠ 1 / 2 , we demonstrate that the abovementioned system is locally well-posed in H s ℝ × H 1 + s ℝ by utilizing several analytic boundary forcing operators.

Dirichlet energy-minimizers with analytic boundary

Dirichlet energy-minimizers with analytic boundary

Simultaneous recovery of attenuation and source density in SPECT

We show that under a certain non-cancellation condition the attenuated Radon transform uniquely determines piecewise constant attenuation $ a $ and piecewise $ C^2 $ source density $ f $ with jumps over real analytic boundaries possibly having corners. We also look at numerical examples in which the non-cancellation condition fails and show that unique reconstruction of multi-bang $ a $ and $ f $ still appears to be possible although not yet explained by theoretical results.

Nonabelian Ramified Coverings and $$L^p$$-boundedness of Bergman Projections in $${\mathbb {C}}^2$$

In this work we explore the theme of $L^p$-boundedness of Bergman projections of domains that can be covered, in the sense of ramified coverings, by "nice" domains (e.g. strictly pseudoconvex domains with real analytic boundary). In particular, we focus on two-dimensional normal ramified coverings whose covering group is a finite unitary reflection group. In an infinite family of examples we are able to prove $L^p$-boundedness of the Bergman projection for every $p\in(1,\infty)$.

Social Clinics and Analytic Boundaries

In this reflection the author makes connections between the themes of the special issue and an early series of seminars held at the Freud Museum London, in 1992, ‘Psychotherapy Black and White’, organized in cooperation with the intercultural therapy centre Nafsiyat. These seminars were invested in the social mission of psychoanalysis, and were intended not only for psychotherapists but for social workers, counsellors, psychiatrists, probation officers, community health workers, teachers and others. The author reflects on psychotherapy as an ecosystem and on the issue of the ‘boundaries’ of psychoanalysis: when they are excessively policed, this limits the domain of psychoanalytic knowledge, including by undermining the legitimacy of social clinics.

On the Automorphism Group of Certain Short ℂ2’s

Abstract For a Hénon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not = 0$, it is known that the sub-level sets of the Green’s function $G^+_H$ associated with $H$ are Short $\mathbb {C}^2$’s. For a given $c> 0$, we study the holomorphic automorphism group of such a Short $\mathbb {C}^2$, namely $\Omega _c = \{ G^+_H < c \}$. The unbounded domain $\Omega _c \subset \mathbb {C}^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that $\Omega _c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, $\textrm {Aut}(\Omega _c)$, cannot be too large. On the other hand, examples are provided to show that these automorphism groups are non-trivial in general. We also obtain necessary and sufficient conditions for such a pair of Short $\mathbb {C}^2$’s to be biholomorphic.

NEWTON POLYHEDRA IN ESTIMATES FOR THE FOURIER TRANSFORM OF CHARACTERISTIC FUNCTIONS AND CONVOLUTION OPERATORS

In this paper, we prove analog of Randol Theorem, on estimates for the Fourier transform of indicator functions of convex domains with analytic boundaries, for some classes of nonconvex domains. Also, we obtain estimates for convolution operators related to solutions to the strictly hyperbolic equations.

Inhomogeneous global minimizers to the one-phase free boundary problem

Given a global 1-homogeneous minimizer U 0 to the Alt-Caffarelli energy functional, with we provide a foliation of the half-space with dilations of graphs of global minimizers with analytic free boundaries at distance 1 from the origin.

Mach limits in analytic spaces on exterior domains

<p style='text-indent:20px;'>We address the Mach limit problem for the Euler equations in an exterior domain with an analytic boundary. We first prove the existence of tangential analytic vector fields for the exterior domain with constant analyticity radii and introduce an analytic norm in which we distinguish derivatives taken from different directions. Then we prove the uniform boundedness of the solutions in the analytic space on a time interval independent of the Mach number, and Mach limit holds in the analytic norm. The results extend more generally to Gevrey initial data with convergence in a Gevrey norm.</p>

Empirical stability boundary for hierarchical triples

Abstract The three-body problem is famously chaotic, with no closed-form analytical solutions. However, hierarchical systems of three or more bodies can be stable over indefinite timescales. A system is considered hierarchical if the bodies can be divided into separate two-body orbits with distinct time and length scales, such that one orbit is only mildly affected by the gravitation of the other bodies. Previous work has mapped the stability of such systems at varying resolutions over a limited range of parameters, and attempts have been made to derive analytic and semi-analytic stability boundary fits to explain the observed phenomena. Certain regimes are understood relatively well. However, there are large regions of the parameter space which remain unmapped, and for which the stability boundary is poorly understood. We present a comprehensive numerical study of the stability boundary of hierarchical triples over a range of initial parameters. Specifically, we consider the mass ratio of the inner binary to the outer third body ( $q_\mathrm{out}$ ), mutual inclination (i), initial mean anomaly and eccentricity of both the inner and outer binaries ( $e_\mathrm{in}$ and $e_\mathrm{out}$ respectively). We fit the dependence of the stability boundary on $q_\mathrm{ out}$ as a threshold on the ratio of the inner binary’s semi-major axis to the outer binary’s pericentre separation $a_\mathrm{in}/R_\mathrm{p, out} \leq 10^{-0.6 + 0.04q_\mathrm{out}}q_\mathrm{out}^{0.32+0.1q_\mathrm{out}}$ for coplanar prograde systems. We develop an additional factor to account for mutual inclination. The resulting fit predicts the stability of $10^4$ orbits randomly initialised close to the stability boundary with $87.7\%$ accuracy.

Inverse, ill-posed problems, essentially different solutions and explicit formulas for solutions

A request for an inverse problem, as well as for an incorrect problem produces tens of millions of answers in the Internet. In the past few decades, hundreds of international conferences on these topics have been held annually. Problems of this kind are quite involved, and their numerical analysis requires the development of special methods and numerical algorithms. Explicit formulas provide the main tool for testing these methods and numerical algorithms. The Cauchy problem for an elliptic equation is a classical ill-posed problem, which serves as a model for many inverse and incorrect problems. In the present paper we give a numerically realizable explicit formula for solving the Cauchy problem in a two-dimensional domain for a general second-order linear elliptic equation with analytic coefficients and the Cauchy analytic data on the analytic boundary.

On the non-existence of stepped-pressure equilibria far from symmetry

The Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502) has been successful in the construction of equilibria in 3D configurations that contain a mixture of flux surfaces, islands and chaotic magnetic field lines. In this model, the plasma is sliced into sub-volumes separated by ideal interfaces, and in each volume the magnetic field is a Beltrami field. In the cases where the system is far from possessing a continuous symmetry, such as in stellarators, the existence of solutions to a stepped-pressure equilibrium with given constraints, such as a multi-region relaxed MHD minimum energy state, is not guaranteed but is often taken for granted. Using SPEC, we have studied two different scenarios in which a solution fails to exist in a slab with analytic boundary perturbations. We found that with a large boundary perturbation, a certain interface becomes fractal, corresponding to the break up of a Kolmogorov–Arnold–Moser (KAM) surface. Moreover, an interface can only support a maximum pressure jump while a solution of the magnetic field consistent with the force balance condition can be found. An interface closer to break-up can support a smaller pressure jump. We discovered that the pressure jump can push the interface closer to being non-smooth through force balance, thus significantly decreasing the maximum pressure it can support. Our work shows that a convergence study must be performed on a SPEC equilibrium with interfaces close to break-up. These results may also provide insights into the choice of interfaces and have applications in finding out the maximum pressure a machine can support.

The Stability Boundary of the Distant Scattered Disk

Abstract The distant scattered disk is a vast population of trans-Neptunian minor bodies that orbit the Sun on highly elongated, long-period orbits. The orbital stability of scattered-disk objects (SDOs) is primarily controlled by a single parameter—their perihelion distance. While the existence of a perihelion boundary that separates chaotic and regular motion of long-period orbits is well established through numerical experiments, its theoretical basis as well as its semimajor axis dependence remain poorly understood. In this work, we outline an analytical model for the dynamics of distant trans-Neptunian objects and show that the orbital architecture of the scattered disk is shaped by an infinite chain of exterior 2:j resonances with Neptune. The widths of these resonances increase as the perihelion distance approaches Neptune’s semimajor axis, and their overlap drives chaotic motion. Within the context of this theoretical picture, we derive an analytic criterion for instability of long-period orbits, and demonstrate that rapid dynamical chaos ensues when the perihelion drops below a critical value, given by q crit = a N ln ( ( 24 2 / 5 ) ( m N / M ⊙ ) a / a N 5 / 2 ) 1 / 2 . This expression constitutes an analytic boundary between the “detached” and actively “scattering” subpopulations of distant trans-Neptunian minor bodies. Additionally, we find that within the stochastic layer, the Lyapunov time of SDOs approaches the orbital period, and show that the semimajor axis diffusion coefficient is approximated by  a ∼ ( 8 / ( 5 π ) ) ( m N / M ⊙ )  M ⊙ a N exp ​ − q / a N 2 / 2 . We confirm our results with direct N-body simulations and highlight the connections between scattered-disk dynamics and the Chirikov Standard Map. Implications of our results for the long-term evolution of minor bodies in the distant solar system are discussed.

Algorithms for subelliptic multipliers in ℂ²

We give examples of pseudoconvex domains of finite type in C 2 \mathbb {C}^2 where the Kohn algorithm for subelliptic estimates fails to yield an effective lower bound for the order of subellipticity in terms of the type. We show how to modify the algorithm to obtain an effective procedure to prove subellipticity on domains of finite type in C 2 \mathbb {C}^2 with real analytic boundary satisfying a condition slightly stronger than pseudoconvexity. We close with a generalization to higher dimensions.