# Closed surfaces

## Top PDF Closed surfaces: ### N body dynamics on closed surfaces : the axioms of mechanics

We have shown how to formulate the dynamics of point masses on closed surfaces. A key aspect of the analysis is to properly account for the mathematical requirement that the mass integrated over the surface must vanish. This leads to significant differences from previous formulations, developed for punctured surfaces (not truly closed). For example, on a sphere, the radius of the sphere scales out of the gravitational potential, but on a punctured sphere it does not. This has a profound influence on the resulting equations of motion. ### The motion of point vortices on closed surfaces

We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings, pp. 94–108; Dritschel 1985 J. Fluid Mech. 157, 95–134 (doi:10.1017/S0022112088003088)), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech. 255, 35–64 (doi:10.1017/S0022112093002381)). ### Multi-scale 3-D Surface Description: Open and Closed Surfaces

A novel technique for multi-scale smoothing of a free-form 3-D surface is presented. Complete tri- angulated models of 3-D objects are constructed automatically and using a local parametrization technique, are then smoothed using a 2-D Gaus- sian filter. Our method for local parametrization makes use of semigeodesic coordinates as a natu- ral and efficient way of sampling the local surface shape. The smoothing eliminates the surface noise together with high curvature regions such as sharp edges, therefore, sharp corners become rounded as the object is smoothed iteratively. Our technique for free-form 3-D multi-scale surface smoothing is independent of the underlying triangulation. It is also argued that the proposed technique is pre- ferrable to volumetric smoothing or level set meth- ods since it is applicable to incomplete surface data which occurs during occlusion. Our technique was applied to closed as well as open 3-D surfaces and the results are presented here. ### Second order splitting for a class of fourth order equations

Outline of paper. In Section 2 we define an abstract saddle point system consisting of two coupled vari- ational equations in a Banach space setting using three bilinear forms {c,b,m}. Well posedness is proved subject to Assumptions 2.1 and 2.2. An abstract finite element approximation is defined in Section 3. Nat- ural error bounds are proved under approximation assumptions. Section 4 details some notation for surface calculus and surface finite elements. Section 5 details results about a useful bilinear form b(·, ·) used in the examples of fourth order surface PDEs studied in later sections. Examples of two fourth order PDEs on closed surfaces satisfying the assumptions of Section 2 are given in Section 6 and the analysis of the application of the surface finite element method to the saddle point problem is studied in Section 7. Finally a couple of numerical examples are given in Section 8 which verify the proved convergence rates. ### Surface homeomorphisms : the interplay between topology, geometry and dynamics

The following problem naturally presents itself: given a class of surfaces, are the quasiconformality constants of these surfaces uniformly bounded away from 1? Natural classes to consider are genus zero surfaces, i.e. those surfaces that can be embedded in the Riemann sphere and closed surfaces of genus g ≥ 2. In chapter 4, we focus our attention to the former class, i.e. genus zero surfaces. Our main result states that there exists a universal lower bound K > 1 such that if M is any hyperbolic genus zero surface, then K ≥ K . The proof of this result makes essential use of the fact that the genus of the surface is zero and the idea of the proof is as follows. ### A stable transient BEM for diffuser scattering

A Transient Boundary Element Model has been described. These are efficient when a broadband result is required, but are iterative so can be unstable. The combined field boundary condition improves stability for closed surfaces compared to the pressure or velocity boundary conditions used alone. ### Minimal Nielsen Root Classes and Roots of Liftings

Gonc¸alves and Aniz 3 answered this question for maps from CW complexes into closed manifolds, both of same dimension greater or equal to 3. Here, we study this problem for maps from 2-dimensional CW complexes into closed surfaces. In this context, we present several examples of maps having liftings through some covering space and not having all Nielsen root classes with minimal cardinality. ### Lefschetz fixed point theorem for digital images

Boxer et al. [] expanded the knowledge of simplicial homology groups of digital images. They studied the simplicial homology groups of certain minimal simple closed surfaces, extended an earlier deﬁnition of the Euler characteristics of a digital image, and com- puted the Euler characteristic of several digital surfaces. Demir and Karaca [] computed ### Investigation of SH Wave Fundamental Modes in Piezoelectromagnetic Plate: Electrically Closed and Magnetically Closed Boundary Conditions

The boundary conditions in the case when the treated material simultaneously possesses the piezoelectric, piezomagnetic, and magnetoelectric effects are perfectly described in . To obtain the dispersion relations for the case of the mechanically free, electrically closed, and magnetically closed surfaces of the piezoelectromag- netic plate, the following points must be passed through: ### Closed essential surfaces in hyperbolizable acylindrical 3 manifolds

finitely generated fundamental group, using a standard trick. Let Γ be a finitely generated torsion- free Kleinian group which has infinite co-volume and which contains no Z ⊕ Z subgroups. By the Core Theorem of Scott , there exists a compact submanifold M of H 3 /Γ whose inclusion is a homotopy equivalence. Since Γ has infinite co-volume, the boundary of M is non-empty; since Γ contains no Z ⊕ Z subgroups, M cannot contain an incompressible torus, and so by Thurston’s uniformization theorem (see for example Morgan ), there exists a convex co-compact Kleinian group Φ uniformizing M . If it happens that M is acylindrical, Theorem 4.2 implies that there exists a closed immersed essential surface S in M , and hence that there exists a closed essential surface S in M Γ . Note that this argument works in the presence of parabolics, though the fundamental ### Particle Orbit Analysis in the Finite Beta Plasma of the Large Helical Device using Real Coordinates

2 . 7%. In ( ψ − θ ) coordinates, the orbit of the passing par- ticle in β = 2.7% almost agrees with that in β = 0% (Fig. 5 (a)). In the rotating helical coordinate system, how- ever, there is a significant diﬀerence in the orbits of the passing particles between β = 0% and β = 2.7%. In contrast, the orbit of banana-orbit particles in β = 2.7% significantly diﬀers from that in β = 0% (Fig. 5 (b)). In β = 2.7%, the banana-orbit particle moves across the flux surfaces and reaches the neighborhood of the LCFS. Therefore, most of the banana-orbit particles become re- entering particles in the case of β = 2.7%. On the other hand, the banana-orbit particles in β = 0% follow the or- bit nearly concentrically with the flux surfaces in (ψ − θ) coordinates. Thus, there are no “re-entering banana-orbit particles” with the exception of the particles traced from the starting points near the LCFS. ### Characterizations of g*-open functions in topology

(i) Generalized closed set ( in brief, g-closed) set  if Cl(A)u whenever A  u and u is open in x . (ii) Generalized semiclosed (in brief, gs-closed) set  if sCl(A)u whenever A  u and u is open in x . (iii) Generalized semipreclosed (in brief, gsp-closed) set  if spCl(a)u whenever A  u and u is open in x . (iv) Generalized preclosed (in brief, gp-closed) set  if pCl(A)u whenever a  u and u is open in x . (v) Gpr--closed set  if pCl(A)  u whenever A  u and u is r-open in x . ### REGULAR HOMOTOPY OF CLOSED CURVES ON SURFACES

The use of rotation numbers in the classification of regular closed curves in the plane up to regular homotopy sparked the investigation of winding numbers to classify regular closed curves on other surfaces. Chillingworth  defined winding numbers for regular closed curves on particular surfaces and used them to classify orientation preserving regular closed curves that are based at a fixed point and direction. We define geometrically a group structure of the set of equivalence classes of regular closed curves based at a fixed point and direction. We prove this group structure coincides with the one introduced by Smale  via a weak homotopy equivalence. The set of equivalence classes of orientation preserving regular closed curves is a subgroup. This thesis investigates the relationship between this subgroup and the winding number of each element. Specifically, it is proven that this subgroup is isomorphic to the direct product of the integers with the group of orientation preserving closed curves up to homotopy where the isomorphism sends an equivalence class to its winding number and corresponding homotopy class. Using this result, we describe the subgroup for several surfaces by depicting representatives of generators. ### Helicoidal Surfaces and Their Relationship to Bonnet Surfaces

An important question that arises is which surfaces in three-space admit a mean curvature preserving isometry which is not an isometry of the whole space. This leads to a class of surface known as a Bonnet surface in which the number of noncongruent immersions is two or infinity. The intention here is to present a proof of a theorem using an approach which is based on differen- tial forms and moving frames and states that helicoidal surfaces necessarily fall into the class of Bonnet surfaces. Some other results are developed in the same manner. ### Determining the Closed Flux Surface in a Helical Plasma in TOKASTAR-2 with an Electrostatic Probe

The electron temperature and density were measured with an electrostatic probe in a helical plasma in the TOKASTAR-2 device in order to determine the location and the shape of the last closed flux surface (LCFS). The electron density inside the calculated LCFS was found to be higher in a helical plasma than in a plasma without a helical field when the electron-cyclotron-resonance layer was located inside the LCFS. Although errors in the manufacturing and installation of coils have been a concern, this result indicates that the LCFS formed in this device does not diﬀer greatly from the calculated LCFS. ### On Locally b#-Closed Sets and Weakly b#-Closed Sets

Theorem 5.8. Let (X, ) and (Y, ) be the topological spaces. A mapping f: (X, ) (Y, ) is weakly b # -closed if and only if for each sub set B of Y and for each open set G containing f -1 (B) there exists a weakly b # -open set F of Y such that B F and f -1 (F) G. ### Effect of Surface Treatment and Crystal Orientation on Microstructural Changes in Aluminized Ni Based Single Crystal Superalloy

The so-called 4th generation Ni-based single crystal superalloy, TMS-138, whose composition is Ni-6.7Co- 3.6Cr-1.9Mo-2.1W-13.8Al-2.1Ta-0.04Hf-1.7Re-1.2Ru in at%, was used as a substrate material. After the precise determination of crystal orientation by the back Laue reﬂection method, the substrate superalloy was cut into an octahedral rod by the electron discharge machining (EDM) method, of which width is 1.9 mm and length is 18 mm. The  direction was set to the longer direction. According to this method, each of the side surfaces faces to either {100} or {110} plain group. All the side surfaces were then mechan- ically polished followed by the electro-polishing in order to get rid of the surface residual stress. As shown in Fig. 1, among the 8 side surfaces of the octagonal rod, 5 surfaces, ### Bounding surfaces in a barchan dune: annual cycles of deposition? Seasonality or erosion by superimposed bedforms?

On the 200 MHz GPR profile along the length of the dune X-X’ (Figure 6), the steeply dipping, planar-inclined reflections within the dune are interpreted as sets of cross-strata. The lower-angle reflections are interpreted as bounding surfaces within the dune (Figure 6). Erosional bounding surfaces are picked where they truncate underlying reflections and also where they are downlapped by overlying reflections (Figure 6). The truncation of reflections marks a period of erosion, either due to reshaping of the dune by wind erosion or erosion in the lee of a superimposed bedform. The 200 MHz data have a higher resolution and record more detail of the dune strata and images a greater number of bounding surfaces than the 100 MHz data. Using the satellite imagery to reconstruct the position of the toe of the dune slipface in 2009 appears to confirm the presence of a winter reactivation surface. However, additional bounding surfaces are revealed, and it is obvious that these cannot all be attributed to an annual cycle given the location and migration of the dune recorded from satellite images. It is not possible from the profile along the dune alone to determine the cause of these erosion surfaces. ### Experimental study of contact transition control incorporating joint acceleration feedback

The experiment is carried out on the third joint of a three-link direct-drive robot (Fig. 3). A one-dimensional force sensor and a linear accelerometer are implemented at the end of the last link, and other sensors equipped include the current sensors, tachometers, and encoders. The accelerometer used has a sen- sitive band (0–200 Hz) and sensitivity of 10 g (the gravity acceleration), while the force sensor used has inherent resonant frequency of 20 kHz and sensitivity of 1 mv/v. The force closed loop is built on a Pentium 100 personal computer at a sampling rate of 1 kHz. With respect to elastic (sponge), less elastic (card- board), and hard (steel plate) contact surfaces, the closed-loop transition control without velocity and acceleration feedbacks, with only velocity feedback, and with both velocity and acceler- ation feedbacks are investigated, respectively. The postcontact force tracking performance of the closed-loop system incorpo- rating the velocity and acceleration feedbacks are also investi- gated experimentally. ### Minimal closed sets and maximal closed sets

The symbol Λ \ Γ means di ﬀ erence of index sets, namely, Λ \ Γ = Λ − Γ , and the cardi- nality of a set Λ is denoted by | Λ | in the following arguments. A subset M of a topological space X is called a pre-open set if M ⊂ Int(Cl(M)) and a subset M is called a pre-closed set if X − M is a pre-open set.