We consider an overdetermined elliptic problem known as the hollow vortex problem. We prove that the solutions to this problem are in 1:1 correspondence with minimal graphs bounded by horizontal symmetry lines. We use this correspondence to give various examples of domains with hollow vortices.
A family of periodic solutions to the hollow vortex problem was constructed by Baker, Saffman and Sheffield in @cite_6 . It corresponds to the family of horizontal Scherk surfaces (see Figure , top left) -- the authors were of course not aware of that relationship. The same solution was derived again by Crowdy and Green in @cite_3 , together with another family of solutions called "staggered vortex streets". The corresponding minimal graphs are periodic and take on two different values on the boundary as in Case II. However, they are asymptotic to half-planes of non-zero slope at infinity, so extending these surfaces by reflection yields complete minimal surfaces which are not embedded.
@cite_3 @cite_6
1966911335 2049968098
New analytical solutions are presented for steadily translating von Karman vortex streets made up of two infinite rows of hollow vortices. First, the solution for a single row of hollow vortices due to ["Structure of a linear array of hollow vortices of finite cross-section," J. Fluid Mech. 74, 469 (1976)] is rederived, in a modified form, and using a new mathematical approach. This approach is then generalized to find relative equilibria for both unstaggered and staggered double hollow vortex streets. The method employs a combination of free streamline theory and conformal mapping ideas. The staggered hollow vortex streets are compared with analogous numerical solutions for double streets of vortex patches due to Saffman and Schatzman [“Properties of a vortex street of finite vortices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 285 (1981)] and several common features are found. In particular, within the two different inviscid vortex models, the same street aspect ratio of approxi... Free-streamline theory is employed to construct an exact steady solution for a linear array of hollow, or stagnant cored, vortices in an inviscid incompressible fluid. If each vortex has area A and the separation is L, there are two possible shapes if A[1 2] L is less than a critical value 0.38 and none if it is larger. The stability of the shapes to two-dimensional, periodic and symmetric disturbances is considered for hollow vortices. The more deformed of the two possible shapes is found to be unstable while the less deformed shape is stable. We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal surfaces. In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc. Preface to the Classics Edition Preface Symbols 1. The Riesz-Fredholm theory for compact operators 2. Regularity properties of surface potentials 3. Boundary-value problems for the scalar Helmholtz equation 4. Boundary-value problems for the time-harmonic Maxwell equations and the vector Helmholtz equation 5. Low frequency behavior of solutions to boundary-value problems in scattering theory 6. The inverse scattering problem: exact data 7. Improperly posed problems and compact families 8. The determination of the shape of an obstacle from inexact far-field data 9. Optimal control problems in radiation and scattering theory References Index. We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases. c Decorative reliefs are widely used for e.g. packaging and porcelain design. In periodic reliefs, the relief repeats a pattern, for example all the way around an underlying surface of revolution. Reverseengineering of existing reliefs allows them to be re-applied to different base surfaces; we show here how to segment a single repeat unit of a periodic relief starting from a scanned triangle mesh. We first briefly review how we segment the relief from the background surface using our previous work. The rest of the paper then concentrates on how we extract a single repeat unit from the relief. To do so, the user provides two points on one relief boundary which are in approximate correspondence on consecutive repeats of the relief. We first refine the relative locations of these points, and then determine a third corresponding point using relief boundary information. These are used to determine three initial cutting planes across the relief. Then surface registration strategies are utilised to refine the correspondence between adjacent repeat units. Finally, we refine the exact locations of the cutting planes by considering only surface information close to the cutting planes. This allows a repeat unit of the periodic relief to be extracted. We demonstrate that our algorithm is successful and practical, using various real scanned models: user input can be quite imprecise, and we can cope with hand-made reliefs in which the pattern units are only approximately copies of each other. We present a novel shape completion technique for creating temporally coherent watertight surfaces from real-time captured dynamic performances. Because of occlusions and low surface albedo, scanned mesh sequences typically exhibit large holes that persist over extended periods of time. Most conventional dynamic shape reconstruction techniques rely on template models or assume slow deformations in the input data. Our framework sidesteps these requirements and directly initializes shape completion with topology derived from the visual hull. To seal the holes with patches that are consistent with the subject's motion, we first minimize surface bending energies in each frame to ensure smooth transitions across hole boundaries. Temporally coherent dynamics of surface patches are obtained by unwarping all frames within a time window using accurate interframe correspondences. Aggregated surface samples are then filtered with a temporal visibility kernel that maximizes the use of nonoccluded surfaces. A key benefit of our shape completion strategy is that it does not rely on long-range correspondences or a template model. Consequently, our method does not suffer error accumulation typically introduced by noise, large deformations, and drastic topological changes. We illustrate the effectiveness of our method on several high-resolution scans of human performances captured with a state-of-the-art multiview 3D acquisition system. The problem is to calculate an approximate solution of an initial value problem for a scalar autonomous differential equation. A generalized notion of a nonlinear Runge-Kutta (NRK) method is defined. We show that the order of anys-stage NRK method cannot exceed 2s−1; hence, the family of NRK methods due to Brent has the maximal order possible. Using this result, we derive complexity bounds on the problem of finding an approximate solution with error not exceeding e. We also compute the order which minimizes these bounds, and show that this optimal order increases as e decreases, tending to infinity as e tends to zero. A median of a family of vertices in a graph is any vertex whose distance-sum to that family is minimum. In the framework of metric spaces the problem of minimizing a distance-sum is often referred to as the Fermat problem. On the other hand, medians have been studied from a purely order-theoretic or combinatorial point of view (for instance, in statistics, or in Jordan’s work [12] on trees). The aim of this paper is to investigate the mutual relationship of the metric and the ordinal combinatorial approaches to the median problem in the class of median graphs. A connected graph is a median graph if any three vertices admit a unique median (see Avann [l]). Note that trees and the covering graphs of distributive lattices are median graphs. Very little is known about medians in arbitrary graphs (cf. Slater [20]); so far, only trees (Zelinka [22], and many others) and the covering graphs of distributive lattices (Barbut [4]) have been considered. In both cases we get that (i) the medians of any family form an interval (a path in a tree, an order-theoretic interval in a distributive lattice), and (ii) medians of odd numbered families are unique (see Slater [19] for trees, and Barbut [4] for distributive lattices). These results point to the fact that (i) and (ii) must be true for any median graph. After recalling some basic definitions and facts concerning median graphs and median semilattices (for further information, see Bandelt and Hedlikova [3]), we establish (i) and (ii) for arbitrary median graphs. Our results are based on theorems of Avann, Sholander, and Barbut. In trees medians have nice local properties (cf. [7]). Indeed, median sets are related to mass centers (Zelinka [22]) and security centers (Slater [18]). In Section 3 this is extended to median graphs. The study of medians applies to social choice theory (see Barbut 151, and Barthelemy and Monjardet [8]). The median procedure is strongly related to the simple majority rule: the median of a family (A,, . . . , Azk+ ,) of subsets of a set X may be written as U n Ai (Barbut’s formula). The popular h-index used to measure scientific output can be described in terms of a pool of evaluated objects (the papers), a quality function on the evaluated objects (the number of citations received by each paper) and a sentencing line crossing the origin, whose intersection with the graph of the quality function yields the index value (in the h-index this is a line with slope 1). Based on this abstraction, we present a new index, the c-index, in which the evaluated objects are the citations received by an author, a group of authors, a journal, etc., the quality function of a citation is the collaboration distance between the authors of the cited and the citing papers, and the sentencing line can take slopes between 0 and ∞. As a result, the new index counts only those citations which are significant enough, where significance is proportional to collaboration distance. Several advantages of the new c-index with respect to previous proposals are discussed. We present a robot-pose-registration algorithm, which is entirely based on large planar-surface patches extracted from point clouds sampled from a three-dimensional (3-D) sensor. This approach offers an alternative to the traditional point-to-point iterative-closest-point (ICP) algorithm, its point-to-plane variant, as well as newer grid-based algorithms, such as the 3-D normal distribution transform (NDT). The simpler case of known plane correspondences is tackled first by deriving expressions for least-squares pose estimation considering plane-parameter uncertainty computed during plane extraction. Closed-form expressions for covariances are also derived. To round-off the solution, we present a new algorithm, which is called minimally uncertain maximal consensus (MUMC), to determine the unknown plane correspondences by maximizing geometric consistency by minimizing the uncertainty volume in configuration space. Experimental results from three 3-D sensors, viz., Swiss-Ranger, University of South Florida Odetics Laser Detection and Ranging, and an actuated SICK S300, are given. The first two have low fields of view (FOV) and moderate ranges, while the third has a much bigger FOV and range. Experimental results show that this approach is not only more robust than point- or grid-based approaches in plane-rich environments, but it is also faster, requires significantly less memory, and offers a less-cluttered planar-patches-based visualization. Mesh partitioning and skeletonisation are fundamental for many computer graphics and animation techniques. Because of the close link between an object’s skeleton and its boundary, these two problems are in many cases complementary. Any partitioning of the object can assist in the creation of a skeleton and any segmentation of the skeleton can infer a partitioning of the object. In this paper, we consider these two problems on a wide variety of meshes, and strive to construct partitioning and skeletons which remain consistent across a family of objects, not a single one. Such families can consist of either a single object in multiple poses and resolutions, or multiple objects which have a general common shape. To achieve consistency, we base our algorithms on a volume-based shape-function called the shape-diameter-function (SDF), which remains largely oblivious to pose changes of the same object and maintains similar values in analogue parts of different objects. The SDF is a scalar function defined on the mesh surface; however, it expresses a measure of the diameter of the object’s volume in the neighborhood of each point on the surface. Using the SDF we are able to process and manipulate families of objects which contain similarities using a simple and consistent algorithm: consistently partitioning and creating skeletons among multiple meshes. In this article, we study small perturbations of the family of Friedmann–Lemaitre–Robertson–Walker cosmological background solutions to the 1 + 3 dimensional Euler–Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state ( p = c^2_s ) , ( 0 < c^2_s < 1 3 ) , the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure ( [0, ) T ^3 ) , are future-causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably defined energy currents. Abstract Let S be a set of N grid points in the plane, no two of which lie on the same horizontal or vertical line, and let G be a graph with n vertices ( n ⩽ N ). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is a chain of horizontal and vertical segments with bends on grid points whose length is equal to the Manhattan distance of its end vertices. We study the following problem. Given a family F of trees, what is the minimum value f ( n ) such that every n -vertex tree in F admits an orthogeodesic point-set embedding on every set of grid points of size f ( n ) such that no two points lie on the same horizontal or vertical line? We provide polynomial upper bounds on f ( n ) for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L -shaped. Abstract We make explicit the geometric content of Mel’nikov’s method for detecting heteroclinic points between transversally hyperbolic periodic orbits. After developing the general theory of intersections for pairs of families of Lagrangian submanifolds N e ± , with N 0 + = N 0 − and constrained to live in an auxiliary family of submanifolds, we explain how the heteroclinic orbits of a given Hamiltonian system are detected by the zeros of the Mel’nikov 1-form. This 1-form admits an integral expression which is non-convergent in general. We discuss different solutions to this convergence problem. We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains open. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open. A new notion of solutions is introduced to study degenerate nonlinear parabolic equations in one space dimension whose diffusion effect is so strong at particular slopes of the unknowns that the equation is no longer a partial differential equation. By extending the theory of viscosity solutions, a comparison principle is established. For periodic continuous initial data a unique global continuous solution (periodic in space) is constructed. The theory applies to motion of interfacial curves by crystalline energy or more generally by anisotropic interfacial energy with corners when the curves are the graphs of functions. Even if the driving force term (homogeneous in space) exists, the initial-value problem is solvable for general nonadmissible continuous (periodic) initial data. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability a...