In this paper, we show that if a closed, connected, oriented 3-manifold M=M1#M2 admits a perfect discrete Morse function, then one can decompose this function as perfect discrete Morse functions on ...M1 and M2. We also give an explicit construction of a separating sphere on M corresponding to such a decomposition.
Birth and death in discrete Morse theory King, Henry; Knudson, Kevin; Mramor Kosta, Neža
Journal of symbolic computation,
January-February 2017, 2017-01-00, Letnik:
78
Journal Article
Recenzirano
Odprti dostop
Suppose M is a finite cell decomposition of a space X and that for 0=t0<t1<⋯<tr=1 we have a discrete Morse function Fti:M→R. In this paper, we study the births and deaths of critical cells for the ...functions Fti and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the cell decomposition of X is the same for each ti, and then generalize to the case where they may differ. This has potential applications in topological data analysis, where one has function values at a sample of points in some region in space at several different times or at different levels in an object.
We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse–Smale decomposition of a smooth manifold with respect to a ...smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological disks. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.
In Lie sphere geometry, a cycle in \RR^n is either a point or an oriented
sphere or plane of codimension 1, and it is represented by
a point on a projective surface \Omega\subset \PP^{n+2}. The Lie ...product, a bilinear form on the space of homogeneous coordinates \RR^{n+3}, provides an algebraic description of geometric properties of cycles and their mutual position in \RR^n. In this paper, we discuss geometric objects which correspond to the intersection of
\Omega with projective subspaces of \PP^{n+2}. Examples of such
objects are spheres and planes of codimension~2 or more, cones and tori.
The algebraic framework which Lie geometry provides gives rise to simple and efficient
computation of invariants of these objects, their properties and their mutual position in \RR^n.
GEOMETRIC CONSTRUCTIONS ON CYCLES IN ℝ ZLOBEC, BORUT JURČIČ; KOSTA, NEŽA MRAMOR
The Rocky Mountain journal of mathematics,
01/2015, Letnik:
45, Številka:
5
Journal Article
Recenzirano
In Lie sphere geometry, a cycle in R𝑛 is either a point or an oriented sphere or plane of codimension 1, and it is represented by a point on a projective surface Ω ⊂ 𝕡𝑛+2. The Lie product, a ...bilinear form on the space of homogeneous coordinates ℝ𝑛+3, provides an algebraic description of geometric properties of cycles and their mutual position in ℝ𝑛. In this paper, we discuss geometric objects which correspond to the intersection of Ω with projective subspaces of ℙ𝑛+2. Examples of such objects are spheres and planes of codimension 2 or more, cones and tori. The algebraic framework which Lie geometry provides gives rise to simple and efficient computation of invariants of these objects, their properties and their mutual position in ℝ𝑛.
The degree of maps of free G-manifolds Jaworowski, Jan; Kosta, Neža Mramor
Journal of fixed point theory and applications,
12/2007, Letnik:
2, Številka:
2
Journal Article
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