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.

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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)).

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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.

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.

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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.

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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.

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.

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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

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The boundary conditions in the case when the treated material simultaneously possesses the piezoelectric, piezomagnetic, and magnetoelectric effects are perfectly described in [60]. 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:

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 [21], 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 [20]), 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

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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.

(i) Generalized **closed** set ( in brief, g-**closed**) set [13] if Cl(A)u whenever A u and u is open in x . (ii) Generalized semiclosed (in brief, gs-**closed**) set [3] if sCl(A)u whenever A u and u is open in x . (iii) Generalized semipreclosed (in brief, gsp-**closed**) set [8] if spCl(a)u whenever A u and u is open in x . (iv) Generalized preclosed (in brief, gp-**closed**) set [21] if pCl(A)u whenever a u and u is open in x . (v) Gpr--**closed** set [10] if pCl(A) u whenever A u and u is r-open in x .

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 [1] 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 [9] 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.

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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.

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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.

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.

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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 [001] 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**,

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**.

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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.

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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.