E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate texts in Mathematics. Springer-Verlag, New York, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....I rst heard of it from L.C. Evans [31] in the context of his attempts to prove a version of his partial regularity theorem [32] for quasiconvex integrals that would be valid for elastic energies satisfying (3. 4) He remarked to me that the existing literature on simplicial approximation (see e.g. [46]) did not cover the case of mappings in Sobolev spaces, since the techniques used relied on composition of mappings, and mappings in Sobolev spaces are not closed under composition. Consider the simplest case n = 2. For a continuous y 2 W 1;p with det Dy(x) 0 a.e. a natural algorithm is to ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate texts in Mathematics. Springer-Verlag, New York, 1977.

....Two interior points that belong to the same connected component can be connected by a semi algebraic curve lying entirely in the S ffi . It is well known that a uniformly continuous curve (such as a semi algebraic one) can be arbitrarily closely approximated by a piecewise linear curve [9]. B. Termination. To establish termination we use a refined version of Collins s Cylindrical Algebraic Decomposition (CAD) 4, 1] Collins s CAD, when applied to a semi algebraic set S, shows the existence of a decomposition of S in a finite number of cells, where each cell is either a point, a ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

....homeomorphic semi algebraic sets are also H equivalent, but the converse does not hold. For example, we will see later that if A consists of a single closed disk, and B consists of two separate closed disks, then A and B are H equivalent. Isotopy invariance and equivalence. It is known (e.g. [6, 13, 16]) that any orientation preserving homeomorphism of R 2 is isotopic to the identity mapping of R 2 . We will therefore, for reasons of convenience, refer to an orientation preserving homeomorphism of R 2 as an isotopy of R 2 . The prototypical example of a homeomorphism that is not an ....

....the same way. Condition (i) of Property 1 implies h 2 (S 1 (p; 1 ) S 1 (p; 1 ) So, h 2 induces an homeomorphism of S 1 (p; 1 ) To complete the proof, it suffices to show that h 2 is orientation preserving. This follows directly from a classical result by J.W. Alexander (see, e.g. [13], page 81) A homeomorphism of B 2 (p; 2 ) that is the identity on S 1 (p; 2 ) is isotopic to the identity mapping, and therefore orientation preserving. Let C be the set of all possible cones. We define: Definition 3 Let A be a closed semi algebraic set in R 2 . The point structure of ....

E.E. Moise. Geometric Topology in Dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer-Verlag, 1977.

.... p and q are part of a region in the interior of S, they can be connected by a semi algebraic curve lying entirely in the interior of S [3] Since uniformly continuous curves (such as semi algebraic ones) can be arbitrarily closely approximated by a piece wise linear curve with the same endpoints [15], p and q can be connected by a piece wise linear curve lying entirely in the interior of S, we are done. 1.b. The curves in the decomposition are either part of the boundary of S or vertical lines belonging to S. In the latter case, the vertical line itself connects the two points. For the ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

.... p and q are part of a region in the interior of S, they can be connected by a semi algebraic curve fl lying entirely in the interior of S [3] Since uniformly continuous curves (such as semi algebraic ones) can be arbitrarily closely approximated by a piecewise linear curve with the same endpoints [14], p and q can be connected by a piecewise linear curve lying entirely in the interior of S, we are done. 1.b. The curves in the decomposition are either part of the boundary of S or vertical lines belonging to S. In the latter case, the vertical line itself connects the two points. For the former ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

.... p and q are part of a region in the interior of S, they can be connected by a semi algebraic curve c lying entirely in the interior of S [3] Since uniformly continuous curves (such as semi algebraic ones) can be arbitrarily closely approximated by a piecewise linear curve with the same endpoints [12], p and q can be connected by a piecewise linear curve lying entirely in the interior of S, we are done. 1.b. The curves in the decomposition are either part of the boundary of S or vertical lines in the interior of S. In the latter case, the vertical line itself connects the two points. For ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

....curves in Collins s CAD. be connected by a curve, also defined in terms of polynomial inequalities, lying entirely in the the interior of S [3, 5] It is also known that such a uniformly continuous curve can be arbitrarily closely approximated by a piecewise linear curve with the same endpoints [11]. 1.b. Let p and q be two points on a curve c in the cell decomposition of S. We have remarked that the curves in the decomposition are part of the solution set of at least one of the polynomials that appear in the description of S. Therefore c is part of an algebraic curve of degree at most ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

....topological data model consists of a finite set of labeled points, a finite set of labeled lines and a finite set of labeled areas. Each point name is assigned to a distinct point in the plane R 2 . Each line name is assigned to a distinct injective and continuous curve (a simple Jordan curve) [31] in the plane that starts and ends in a labeled point and does not contain any other labeled points except these. Distinct curves only meet in labeled points. Each area label is assigned to a distinct area formed by the labeled lines. Figure 3 gives an example of such a database. We apply the ....

E.E. Moise. Geometric Topology in Dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1997.

....require that B is the image of A under a homeomorphism rather than an isotopy. The only difference between the two notions is that the latter considers mirror images to be the same, while the former does not. Indeed, every homeomorphism either is an isotopy itself, or is isotopic to a reflection [13]. All the results we will present under isotopies have close analogues under homeomorphisms. 8 D(p; is the closed disk with center p and radius ; C(p; is its bordering circle. 9 If we project R 2 stereographically onto a sphere, the point at infinity corresponds to the missing point ....

E.E. Moise. Geometric Topology in Dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

....not used. In all these cases, similar definitions and notations are used. Figure 1: Two homeomorphic, but not isotopic semi algebraic sets. of a single closed disk, and B consists of two separate closed disks, then A and B are H equivalent. Isotopy invariance and equivalence. It is known (e.g. [10, 16, 21]) that any orientation preserving homeomorphism of R 2 is isotopic to the identity mapping of R 2 . We will therefore, for reasons of convenience, refer to an orientation preserving homeomorphism of R 2 as an isotopy of R 2 . The prototypical example of a homeomorphism that is not an ....

....same way. Condition (i) of Property 1 implies h 2 (S 1 (p; 1 ) S 1 (p; 1 ) So, h 2 induces an homeomorphism of S 1 (p; 1 ) To complete the proof, it suffices to show that h 2 is orientation preserving. This follows directly from a classical result by J.W. Alexander (see, e.g. [16], page 81) A homeomorphism of B 2 (p; 2 ) that is the identity on S 1 (p; 2 ) is isotopic to the identity mapping, and therefore orientation preserving. Let C be the set of all possible cones. We define: Definition 3 Let A be a closed semi algebraic set in R 2 . The point structure of ....

E.E. Moise. Geometric Topology in Dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer-Verlag, 1977.

....is always such a curve lying entirely in S ffi . Therefore there exists an ffl 0 such that the ffl environment of is contained in S ffi . It is well known that a uniformly continuous curve (such as a semi algebraic one) can be arbitrarily closely approximated by a piecewise linear curve [9]. Hence, the pair of points can be connected by a piecewise linear curve lying entirely in S ffi . 2. Let p be a borderpoint of S ffi . Locally around p, the border of S ffi consists of a finite number of lines leaving p. Because p is a borderpoint of S ffi , at least one of the sectors ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

....Two databases A and B are called isotopic if there is an isotopy h such that h(A) B. Intuitively, this means that A can be continuously deformed into B without leaving the plane. The prototypical example of a homeomorphism that is not an isotopy is a reflection. As a matter of fact, it is known [15] that every homeomorphism either is an isotopy, or is isotopic to a reflection. Hence, when two databases A and B are homeomorphic, either A is actually isotopic to B, or A is isotopic to the mirror image of B. For example, Figure 1 shows two databases that are mirror images of each other but that ....

E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.

....key lemma in this context is the following. Lemma 1 It is decidable whether two movie frames are isotopic 1 , and also whether they are homeomorphic. Proof. Sketch) Let A and B be two movie frames. A and B are homeomorphic if and only if A is isotopic to B or to a reflection of B (see, e.g. [18, 12, 6]) It therefore suffices to prove that it is decidable whether A is isotopic to B. The algorithm to decide isotopy first computes for X = A, respectively B the labeled planar graph embedding GX as follows. The nodes of GX are the singular points of X, i.e. the points that do not belong to ....

E.E. Moise. Geometric Topology in Dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer-Verlag, 1977.

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