Simply connected region

T. W. Shield and D. M. Bogy, Some Axisymmetric Problems for Layered Elastic Media Part I — Multiple Region Contact Solutions for Simply-Connected Indenters, Thansac-tions of the ASME, vol. 56, pp. 798-806, Dec 1989. 

Simply Connected Region. A region of space is described as simply connected when all circuits joining any two points are reconcilable or any loop drawn within the region is reducible. For instance, regions located inside or outside a finite surface are individually, simply connected spaces. 

In the case of an infinitely long solenoid, space is halved and dissociated into two distinct regions (1) a simply connected region inside the solenoid and (2) a doubly connected region outside the solenoid. 

Traditionally, physics emphasizes the local properties. Indeed, many of its branches are based on partial differential equations, as happens, for instance, with continuum mechanics, field theory, or electromagnetism. In these cases, the corresponding basic equations are constructed by viewing the world locally, since these equations consist in relations between space (and time) derivatives of the coordinates. In consonance, most experiments make measurements in small, simply connected space regions and refer therefore also to local properties. (There are some exceptions the Aharonov-Bohm effect is an interesting example.)... 

It is the present writer s opinion that the existence of the obstruction changes the situation entirely. Without the existence of the solenoid in the interferometer, the loop of the two paths can be reduced to a single point and the region occupied by the interferometer is then simply connected. But with the existence of the solenoid, the loop of the two paths cannot be reduced to a single point and the region occupied by this special interferometer is multiply connected. The Aharonov-Bohm effect only exists in the multiply connected scenario. But we should note that the Aharonov-Bohm effect is a physical effect and simple and multiple connectedness are mathematical descriptions of physical situations. 

In conventional, that is normal U(l) or simply connected situations, the fact that a vector field, viewed axially, is pointing in one direction, if penetrated from one direction on one side, and is pointing in the opposite direction, if penetrated from the same direction, but on the other side, is of no consequence at all— because that field is of U(l) symmetry and can be reduced to a single point. Therefore in most cases which are of U(l) symmetry, we do not need to distinguish between the direction of the vectors of a field from one region to another of that field. However, the Aharonov-Bohm situation is not conve-... 

Skinny molecular range, [af, a< ) af is defined above, whereas is the maximum threshold at and below which all locally nonconvex domains on the surface of density domains are simply connected. In simpler terms, in the skinny molecular range all nuclei are found within a single density domain, but there are formal "neck regions on the surface of density domains. In the terminology of shape group analysis [2], rings of D) type can be found on the surface of density domains. 

Fig. 30. Packing density versus electrode potential at a given orientation in various supporting electrolytes. , n2-orientation , transition region (HQ concentrations were 0.36,0.36, 0.25, and 0.37 mM in HCIO, H2SO., H 1PO, and CsCIO, respectively) , nK-orientation. The solid lines simply connect experimental points and do not assume any theoretical curve. Bottom figures thin-layer cyclic current potential curves for clean Pt electrode in the respective supporting electrolytes. Thin-layer volume, V 3.29/il electrode surface area, A = 1.16cm2 rate of potential sweep, r = 2.00 mV s 1 temperature, T = 23 t 1°C. The average standard deviation in T was 1 3% below 1 mM and l 6% above 1 mM. Reprinted from ref. 61.

Theorem (Dulac criterion). Suppose that (3.1) is two-dimensional. Let T be a simply connected region in and let (x) be a continuously differentiable scalar function defined on P. If V(f(x)0(x)) is of one sign (excluding zero) in the region V then there are no periodic orbits in P. 

Since the region x,y >0 is simply connected and g and f satisfy the required smoothness conditions, Dulac s criterion implies there are no closed orbits in the positive quadrant. ... 

Here G is the n-fold direct sum of an arbitrary, simply connected region g in the ordinary 3-space 1, z> g, the physical region of the system where at this stage there is no boundedness requirement for g. Note, however, that in the strict quantum mechanical sense g should be taken as the entire 3-space R, a fact that will require special attention in the proof of the holographic electron density theorem. 

Figure 3.12 Property 4. The AR is unique, simply connected, and a single region, (a) Three separate ARs ARj, ARj, and ARj, (b) mixing between the three regions, and (c) the final (single) region obtained contains aU regions and is simply connected.

Simply connected It appears as though both regions consist of single subspaces in... 

The region R must be simply connected that is, a single closed curve which does not cross itself or a region without holes. 

The interior of the bounded surface is defined to be n t from any point on the boundary, where n is the surface normal and t the boundary curve tangent vector at this point. The region so defined must be simply connected. 

The first discretization step of the finite integration method consists in the restriction of the electromagnetic field problem, which represent an open boundary problem, to a simply connected and bounded space domain fl containing the region of interest. The next step consists in the decomposition of the computational domain into a finite number of volume elements (cells). This decomposition yields the volume element complex G, which serves as computational grid. Assuming that i is brick-shaped, we have the volume element complex... 

The core of the network is the region of the rail network in which the branch line of two transformer substations or of any other branch line is less than 2 km distant in a direct line. All branch lines outside the core of the network are termed outlying lines. In a branched rail network, not only the stretches of line within a circle with a radius of 2 km from the most negative return current point of a transformer substation belong to the core of the network, but also connecting branch lines that are less than 2 km from each other [1]. The area of a network core can be simply determined on a track plan with the help of a circular template as in Fig. 15-1. 

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