Domain D0 Research Articles

Creation mechanism of Devil’s Staircase surface and unstable and stable periodic orbits in the anisotropic Kepler problem

In the two-dimensional Anisotropic Kepler Problem (AKP), a longstanding question concerns the uniqueness of an unstable periodic orbit (PO) for a given binary code (modulo symmetry equivalence). In this paper, a finite-level (N) surface defined by the binary coding of the orbit is considered over the initial-value domain D0. A tiling of D0 by base ribbons of the surface steps is shown to be proper, i.e., the surface height increases monotonously when the ribbons are traversed from left to right. The mechanism of creating a level-(N+1) tiling from the level-N tiling is clarified in the course of the proof. There are two possible cases depending on the code and the anisotropy. (A) Every ribbon shrinks to a line as . Here, the uniqueness holds. (B) When future (F) and past (P) ribbons become tangential to each other, they escape from the shrinking. Then, the initial values of a stable PO (S) and an unstable PO (U) sharing the same code co-exist inside the overlap of the F and P non-shrinking ribbons. This corresponds to Broucke’s PO. When the anisotropy is high, only case A is observed; however, as the anisotropy decreases, a bifurcation of the form occurs along with the emergence of a non-shrinking ribbon. (Here, R and NR denote self-retracing and non-self-retracing POs, respectively). We conjecture that, from a classification based on topology and symmetry, case B occurs only for odd-rank, Y-symmetric POs. We report two applications. First, the classification is applied successfully to the successive bifurcation of a high-rank PO (n = 15), where the above bifurcation is followed by . Second, enhancing the sensitivity to the co-existence of S and U POs through ribbon tiling, we examine the high-anisotropy region. A new symmetry-type POs (O-type) are found and, at γ = 0.2, all POs are shown to be unstable and unique. An investigation of 13648 POs at rank 10 verifies that Gutzwiller’s action formula works with amazing accuracy.

Persistence of steady flows of a two-dimensional perfect fluid in deformed domains

The robustness of steady solutions of the Euler equations for two-dimensional, incompressible and inviscid fluids is examined by studying their persistence for small deformations of the fluid-domain boundary. Starting with a given steady flow in a domain D0, we consider the class of flows in a deformed domain D that can be obtained by rearrangement of the vorticity by an area-preserving diffeomorphism.We provide conditions for the existence and (local) uniqueness of a steady flow in this class when D is sufficiently close to D0 in Ck,α, k ⩾ 3 and 0 < α < 1. We consider first the case where D0 is a periodic channel and the flow in D0 is parallel and show that the existence and uniqueness are ensured for flows with non-vanishing velocity. We then consider the case of smooth steady flows in a more general domain D0. The persistence of the stability of steady flows established using the energy–Casimir or, in the parallel case, the energy–Casimir–momentum method, is also examined. A numerical example of a steady flow obtained by deforming a parallel flow is presented.

On a Construction of Diffusion Processes on Moving Domains

We construct time inhomogeneous diffusion processes on a space-time domain D. For each t, the t-section D t of D is assumed to be an image of a fixed domain D 0 by a homeomorphic map. After constructing a diffusion process on a fixed domain D 0 by means of the general theory of time dependent Dirichlet forms, transformations by a multiplicative functional and a homeomorphic map are used to get the desired processes. In the case of reflecting difusion process on D, we also give Skorohod type representations of the process by means of the local time on the boundary of D.

The Axial α-Helices and Radial Spokes in the Core of the Cryo-negatively Stained Complex Flagellar Filament of Pseudomonas rhodos: Recovering High-resolution Details from a Flexible Helical Assembly

Of the two known “complex” flagellar filaments, those of Pseudomonas are far more flexible than those of Rhizobium. Their diameter is larger and their outer three-start ridges and grooves are more prominent. Although the symmetry of both complex filaments is similar, the polymer's linear mass density and the flagellin molecular mass of the latter are lower. A recent comparison of a three-dimensional reconstruction of the filament of Pseudomonas rhodos to that of Rhizobium lupini indicates that the outer flagellin domain (D3) is missing in R.lupini. Here, we concentrate on the structure of the inner core of the filament of P.rhodos using field emission cryo-negative staining electron microscopy and a hybrid helical/single particle reconstruction technique. Averaging 158 filaments caused the density band corresponding to the radial spokes to nearly average out due to their variability and inferred flexibility. Treating the Z=0 cross-sections through the aligned individual three-dimensional density maps as images, classifying them by correspondence analysis (using a mask containing the radial spokes domain) and re-averaging the subclasses (using helical reconstruction techniques) allowed a recovery of the radial spokes and resolved the α-helices in domain D0 and the triple α-helical bundles in domain D1 at a resolution of 1/7Å−1. Although the perturbed components of the helical lattice are present along the entire filament's radius, the interior of the complex filament is similar to that of the plain one, whereas it's exterior is altered. Reconstructions of vitrified and cryo-negatively stained plain, right-handed filaments of Salmonella typhimurium SJW1655 prepared and imaged under conditions identical with those used for P.rhodos confirm the similarity of their inner cores and that the secondary structures in the interior of the flagellar filament can, under critical conditions of image recording and correction, be resolved in negative stain.

ON THE SUFFICIENT CONDITION FOR A CONVEX DOMAIN TO CONTAIN ANOTHER IN R2n

In this paper, the author extends Ren's results in plane to the case of R2n, i.e., estimates the kinematic measure of a convex domain D1 with boundary Σ1 moving to another domain D0 with boundary Σ0 under the isometry group G in R2n. A sufficient condition for D0 contain D1 by using kinematic foundamatal formula and Poincare's formula in Rn is obtained.

On the Continuous Dependence of the Exterior Dirichlet Problem on the Boundary

We consider a sequence of exterior domains Dj,j∈ℕ0, and assume that the boundaries ∂Dj converge to ∂D0 with respect to the Hausdorff distance. We investigate solutions to the exterior Dirichlet problem for the Laplace equation and for the Helmholtz equation in these domains. Assuming convergence of the boundary data and Dj⊂D0, j∈ℕ, then, by essentially using the method of Perron, we show that the solutions in the domains Dj converge to the solution in the domain D0 with respect to the maximum norm. We prove the same result in case that the requirement Dj⊂D0,j∈ℕ, is replaced by an equicontinuity property of all barrier functions to all boundary points. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 707–716 (1997)

When can one domain enclose another in 3?

AbstractIn this paper, we give a sufficient condition (Theorem) in order that one domain D1 bounded by a C2-smooth boundary can be enclosed in, or enclose, another domain D0 bounded by the same kind of boundary. A same kind of sufficient condition for convex bodies (Corollary) is also obtained.

Computation of nash equilibrium pairs of a stochastic differential game

AbstractConsider the random motion of two points Me and Mp in an open and bounded domain D0 in the plane. Each of the velocities, u = (u1 u2)T of Me and v = (v1, v2)T of Mp, are perturbed by a corresponding R2‐valued Gaussian white noise. Let A and Dc be two disjoint closed subsets of D0. Suppose that at t = 0, Me is in A and Mp is anywhere in D0. Denote by ℰ1 and ℰ2 the following events: ℰ1 = {Mp intercepts Me in A before Me reaches the set Dc and before either Me or Mp has left D0}, and ℰ2 = {Me reaches the set Dc before being intercepted by Mp, while Mp is in A, and before either Mp or Me has left D0}.The problem dealt with here is to find a pair of velocity strategies (u*, v*) such that, in the sense of a Nash equilibrium point, the probabilities Prob(ℰ1) and Prob(ℰ2) will both be maximized on a given class of velocity strategies (u, v). Sufficient conditions on (u*, v*) are derived which require the existence of a smooth solution (V,Q) to a pair of coupled non‐linear partial differential equations. A finite‐difference scheme for solving these equations is suggested, and two examples are treated in detail.

Inverse problem for the reduced wave equation with fixed incident field

The inverse problem for the reduced wave equation Δu+k2n2(x) u=0 is examined when n (x) is assumed to be real continuous and equal to unity outside some prescribed compact domain D0 in R3. The case where a finite set of measurements of the scattered or total field (generated by a single incident wave at a fixed frequency) at points outside D0 is considered. It is shown that the inverse problem can be expressed in terms of a system of linear functional equations, plus a quadratic nonlinear integral equation. By imposing an additional criterion for uniqueness of the functional equations and placing certain restrictions on the size of the measured data, a solution of the system can be generated by successive approximations. It is shown that the iterative scheme yields correction terms to the solution that are obtained by the common linearized approximation where n (x) is assumed to be a very small perturbation of a known quantity and only linear terms are retained.

Poincaré analytic vectors in axiomatic quantum field theory

It is demonstrated that in a Wightman quantum field theory with a denumerable set of field operators over the test function space SR, transforming covariantly among themselves under a unitary representation U(Λ, a) of the covering group of the Poincaré group, the domain D0 of vectors generated by the polynomial ring over the field operators applied on the vacuum state contains a dense invariant set of analytic vectors for U(Λ, a).

Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel

A blob of Newtonian fluid is sandwiched in the narrow gap between two plane parallel surfaces so that, a t some initial instant, its plan-view occupies a simply connected domain D0. Further fluid, with the same material properties, is injected into the gap at some fixed point within D0, so that the blob begins to grow in size. The domain D occupied by the fluid at some subsequent time is to be determined.It is shown that the growth is controlled by the existence of an infinite number of invariants of the motion, which are of a purely geometric character. For sufficiently simple initial domains D0 these allow the problem to be reduced to the solution of a finite system of algebraic equations. For more complex initial domains an approximation scheme leads to a similar system of equations to be solved.