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Exact Methods for PDEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Numerical implementations Even when an analytic conformal map cannot be found, there are fast numerical techniques for finding an approximate conformal map. Riemann's mapping theorem states that all bounded simply connected plane regions can be conformally mapped onto the unit disk, and all bounded doubly connected plane regions can be conformally mapped onto an annulus. Using Poisson's formula (see page 346) exact solutions can be written down for these two geometries. See Fornberg [447] or Trefethen [1192] for details.Numerical implementation of the Schwartz–Christoffel transformation can fail on some seemingly simple polygons. Mapping a rectangle with an aspect ratio of 20 to 1, or another region with a similar degree of elongation, onto a half-plane may cause problems because the points in the transformed plane will be very close together. (This is known as the “crowding phenomenon.”)MATLAB has a package for conformal mapping, see Trefethen [1199].Algorithms for conformal mappings are described in Gopal and Trefethen [510], their numerical implementation is in Trefethen [1196].
Preliminaries
Published in Patrick Knupp, Stanly Steinberg, Fundamentals of Grid Generation, 2020
Patrick Knupp, Stanly Steinberg
It is assumed that the physical region is connected, that is, made up of a single piece. Technically, the physical region is assumed to be a domain, that is, the physical region is the closure of an open connected set. In one dimension, this means the region is an interval. In two dimensions, connected regions can be more complicated; for example, an annulus is connected (an annulus is the region between two concentric circles). In this book, it is also assumed that all regions are simply connected. Intuitively, a region is not simply connected if it has a hole in it. The definition of “simply connected” is given in most texts on topology (see Armstrong [13]). It is easy to see that the logical region is simply connected. In one dimension, only intervals are simply connected. In the plane, a connected region is simply connected if its complement is connected. Thus, the annulus is not simply connected. The situation in three dimensions is even more complicated. For example the region between two concentric spheres or that within a torus (doughnut) are not simply connected.
The Exact Form and General Integrating Factors
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
That region ℛ is said to be simply connected if each and every simple closed curve (i.e., loop) in ℛ encloses only points in ℛ. If any simple closed curve in ℛ encloses any point not in ℛ, then we will say that ℛ is not simply connected. If you think about it, you will realize that saying a region is simply connected is just a precise way of saying that the region has no “holes”. And if you think a little more about the situation, you will realize that, if our open region ℛ has “holes” (i.e., is not simply connected), then it is probably because these are points where M(x, y) or N(x, y) or their partial derivatives fail to exist.
Some reflections on defects in liquid crystals: from Amerio to Zannoni and beyond
Published in Liquid Crystals, 2018
The property of connectedness in a region of space is much considered by pure mathematicians and geometers. Let us briefly review the key ideas (I apologise to those already familiar!). A region is simply connected if any closed curve or surface inside it can be smoothly shrunk to a point. An example would be the inside of a sphere. A multiply connected region does not share this property. The simplest example is the inside of a donut (known as a torus to mathematicians). In this case, a closed curve stretching around the axis of the donut is unshrinkable. The study and classification of such objects is known as topology. Topologists can divide all closed regions into classes, depending, roughly speaking, on how many holes or handles the region encompasses.
Les vertus des défauts: The scientific works of the late Mr Maurice Kleman analysed, discussed and placed in historical context, with particular stress on dislocation, disclination and other manner of local material disbehaviour
Published in Liquid Crystals Reviews, 2022
But all this was far from Volterra's mind when, around 1904, he began to consider the properties of elastic materials in multiply-connected bodies. The property of connectedness in a region of space is much considered by pure mathematicians and geometers. Let us briefly recapitulate the key ideas. A region is simply connected if any closed curve or surface inside it can be smoothly shrunk to a point. An example would be the inside of a sphere. A multiply connected region does not share this property. The simplest example is the inside of a donut (known as a torus to mathematicians). In this case a closed curve stretching around the axis of the donut is unshrinkable. Examples are shown in Figure 6 below.