A supplement to the condition of J. Douglas

In this article, we revisit some results published by several mathematicians regarding existence and non-existence of doubly connected minimal surfaces between two circles in parallel planes. In particular, we propose new analytical interpretations and results for catenoidal solutions formed between two coaxial circles in parallel planes of arbitrary radii. We conclude by discussing the nature of the pair of the catenoidal solutions that arise from our proposed algorithm.

We present a method for constructing harmonic immersions in [Formula: see text], known as the Enneper-type representation. We also prove that any harmonic immersion in [Formula: see text] can be obtained using this approach. Furthermore, we determine the number of non-planar rotational harmonic immersions in [Formula: see text] that connect two coaxial circles in parallel planes, where both circles have the same radius [Formula: see text] and are separated by a distance [Formula: see text].

Application of an Inertial Measurement Unit for quantitative assessment of Single Leg Squat in recreational runners

Let shape P be any simply-connected set in the plane, bounded by a Jordan curve, that is not a circular disk. We say that a set of points I on the boundary of P immobilize the shape if any rigid motion of P in the plane causes at least one point of I to penetrate the interior of P. We prove that four points always suffice to immobilize any shape. For a large class of shapes, which includes polygons without parallel edges, three points are sufficient to immobilize. An O(n log n) algorithm is given that finds a set 3 points that immobilize a given polygon without parallel edges. The algorithm becomes linear for convex polygons. Some results are generalized for d-dimensional polytopes, where 2d points are always sufficient and sometimes necessary to immobilize.

In this work, for given H, we investigate the existence of radial cmc H annulus spanning two given non necessarily convex Jordan curves in parallel planes of \({\mathbb{R}^{3}}\). We established some existence results under hypotheses relating the geometry of the curves and the distance between the planes.

The problem of the asymptotic behaviour of unbounded connected components of plane sections of a 3-periodic surface in R and the structure of the associated foliations on a surface in the three-torus T = R/Z was posed by S. P. Novikov in [1]. This problem comes from the theory of normal metals and is formulated as follows. In the space R with coordinates (x, x, x) one fixes an embedded surface M that is invariant under shifts by vectors in Z ⊂ R and has null homologous projection on T = R/Z. For all covectors H = (H1, H2, H3) one considers the sections of M by planes ⟨H, x⟩ = const (we call them H-sections) and asks about the asymptotic behaviour of their unbounded regular connected components (if any). This question has been studied in a number of papers [2]–[4]. Physical consequences of the results are discussed in [5]. There are the following three possibilities for the behaviour of the H-sections, and moreover, the type of a section is the same for parallel planes. Trivial case. All connected components are bounded. Integrable case. Every regular non-closed component of an H-section is a finitely deformed straight line: γ(t) = t·v+O(1), v ∈ R\{0}. Moreover, there is a covector LM,H , defined up to sign, with relatively prime integral coordinates (L1, L2, L3) that annihilates the vector v. (If we assume our three-space is Euclidean, we can write v = LM,H ×H.) The projection of such a component on T is contained in an embedded two-torus whose homology class is Poincare dual to the covector LM,H . We denote the projective class (L1 : L2 : L3) ∈ RP of LM,H by lM,H . Chaotic case. The closure of the projection on T of any unbounded component is a surface of genus greater than two. The behaviour of such components has not been studied. In some explicitly described examples such a curve ‘wanders all around the plane’, that is, the d-neighbourhood of the curve is the whole plane for some finite d. Apparently, such behaviour is typical for the chaotic case, but this has not been proved. If H1, H2, H3 are linearly dependent over Q, then the definitions above have to be refined. Moreover, in this case the covector LM,H may be not uniquely defined. We skip these details in the present paper. For a fixed surface M and a rational point l ∈ QP ⊂ RP let DM,l denote the set DM,l = {(H1 : H2 : H3) ∈ RP; lM,H = l}. It is known that for a generic 3-periodic surface M the sets DM,l are disjoint closed domains with piecewise smooth boundary, and the set of directions H with chaotic H-sections has measure zero. (Of course, for some l ∈ QP the set DM,l may be empty.) The set of directions H with trivial H-sections is open. We call the non-empty domains DM,l stability zones. For studying the stability zones it is useful to consider a 1-parameter family Mc = {x ∈ R; f(x) = c} of level surfaces of a fixed smooth function and to introduce generalized stability zones Df,l = ⋃ cDMc,l. They are also domains with piecewise smooth boundary. If l = l′, then the zones Df,l and Df,l′ can intersect only at the boundary, and moreover, the set of their common points is at most countable (in all examples known to us, two zones have at most one common point). If there are at least two zones, then there must

On edges crossing few other edges in simple topological complete graphs

Schottky Groups and Circles

THEOREM 1. Let the region R of the extended (x, y)-plane be bounded by a finite number of mutually disjoint Jordan curves Co, G., C2, • • • , Cn. Let the function u(x, y) be harmonic in R, continuous in the corresponding closed region, equal to zero on Co and to unity on G, C2, • • • , Cn. Denote by Ro the region bounded by Co containing the curves Ci, C2, • • • , Cn in its interior; define noneuclidean straight lines in Ro as the images of arcs of circles orthogonal to the unit circle, when Ro is mapped conformally onto the interior of the unit circle. If II is any non-euclidean convex region in R0 which contains all the curves Ci, C2> * • * » Cn» then II also contains all critical points of u(x, y) in R.

A finite bordered Riemann surface 9-with s boundary components and interior genus g has first Berti number r = 2g + s -1.Let a be any interior point of 9, and e" denote evaluation at a on the usual hypo-Dirichlet algebra associated with 9. We establish some connections between the topological and, more strongly, the conformai type of 9, and the geometry of 91t" the set of representing measures for e".For example, we show that if 91t" has an isolated extreme point, then ft must be a planar surface.Several questions posed by Sarason are answered through exhausting the possibilities for the case r = 2.1. Let fi be a finite open Riemann surface with boundary oR consisting of a finite nonzero number s of disjoint Jordan curves (see Ahlfors and Sario [4, p. 117]).The first Betti number of R, which we denote by r, equals 2g + s -1, where g is the interior genus of R. We take 9fi to be oriented with the induced orientation from R. Lei A be the hypo-Dirichlet algebra (see [2], [15]) of functions continuous on oR and possessing continuous extensions to T = R U oR which are holomorphic on R. Fix a point a E R and let ea denote evaluation on A at a; i.e., ea(f) = f(a) for/ G A. Let 9ILa be the compact convex set of representing measures for ea.A measure m E 91ta is a positive Borel measure of total mass one supported on 3fi such that/(a) = Smfdm for all/ G A.In this paper we shall show some connections between the topological and, more strongly, the conformai type of R and the geometry of 91t,,.We will let S2 denote the Riemann sphere and Re be the real axis.The set <Dlta will be called strictly convex if 991ta consists entirely of extreme points.The principal results are stated below.Theorem 1.1.If 911,, has an isolated extreme point, then R is conformally equivalent to S2 minus a finite number of slits contained in Re.The next theorem exhausts the possibilities for 91ta when r = 2, and thus answers Questions 1 and 3 on p. 376 of [12].

Let D be a plane region of arbitrary connectivity (> 1) for which the Dirichlet problem is solvable.There exists a conformai map of D onto a region bounded by two level loci of H, a nontrivial harmonic measure.H is essentially the difference of two logarithmic potentials.The two measures involved are mutually singular probability measures.Further properties of these measures, and of H, are derived.The special case in which D is of connectivity 2 is the classical theorem which states that an annular region is conformally equivalent to a region bounded by two circles.The case in which D is of finite connectivity was treated by J. L. Walsh in 1956.A similar generalization of the Riemann mapping theorem is also established.Finally, converses of the above results are also valid.In 1956, J. L. Walsh showed that a plane region bounded by finitely many, mutually disjoint Jordan curves could be mapped conformally onto a region bounded by two level loci of a function harmonic in the plane with the exception of a finite number of poles, one in each component of the complement of the region [11].This is presented below as Theorem 1, where t log T(Z) is harmonic, and the region A is defined by 0<t log T(Z)< 1.A special case of Theorem 1 is the classical theorem which states that a doubly connected region can be mapped conformally onto an annulus.Furthermore, Theorem 3 below, due also to J. L. Walsh, is a similar generalization of the Riemann mapping theorem.At least three other proofs of these theorems have since been given (H.Grunsky, [4]; J. A. Jenkins, [7]; H. J. Landau, [8]).It is the object of this paper to show in Theorems 2 and 4 below that the theorems of Walsh extend further to Dirichlet regions of arbitrary connectivity.Finally, Theorem 5 below can be regarded as a converse of Theorem 2. The existence of the conformai map in Theorem 2 characterizes the Dirichlet region, and thus may prove useful in the study of such regions.Here, a Dirichlet region is one for which the Dirichlet problem is solvable.See [3, p. 53] for a totally unrelated definition of Presented in part to the Society, January 27, 1963, under the title On conformai maps of regions of infinite connectivity; received by the editors November 20, 1969.

1. The basic theory of approximate calculation by the use of subareas of the capacitance of a plane area and of the distribution of charge density over it is outlined. 2. The method of subareas is employed to obtain an accurate value for the capacitance of an annulus of ratio of outer to inner radius of ro/ri=1.5. 3. The fourth approximation to the capacitance of a specified circular disk, as calculated by the method of subareas, is found to be in good agreement with the known exact value. As a circular disk is an annulus of ratio of radii ro/ri=∞, it is to be concluded that the fourth approximation for the much narrower annulus of ratio 1.5 is very nearly the exact value. This conjecture is substantiated by the curve of Figure 3. 4. Comparison in Figure 6 of the charge distribution on a circular disk as determined both from the known equation and by the method of subareas indicates that calculation of charge distribution by use of subareas affords a good approximation to the actual distribution. Accordingly, the charge distribution of Figure 7 for the much narrower annulus is to be considered as a close approximation to the actual distribution. 5. The universal curve of Figure 8 yields the capacitance of an annulus of any stated ratio of external to internal radii.

Thermal stress in rotating functionally graded hollow circular disks