Discrete bifurcation diagrams and persistence

Download

index.pdf

Date

2018

Author

Örnek, Türkmen

Metadata

Show full item record

Item Usage Stats

249
views

91
downloads

Let fti : M → R be a discrete Morse function on a cell complex M for each t0 < t1 < ... < tn = 1. Let us denote slice as Mi = M ×{ti} ⊂ M × I and let Vi be the discrete vector ﬁeld on each slice. After extending the discrete vector ﬁeld on each slice to a discrete vector ﬁeld on all of M ×I, a discrete bifurcation diagram is obtained by connecting critical cells of the slices. In”Birth and Death in Discrete MorseTheory”(King,Knudson,Mramor), a solution about ﬁnding the discrete bifurcation diagram has been presented. In this article, two cases have been researched. In the ﬁrst case, triangulation of each slice is the same andM×I is regularly celldecomposed. For the ﬁrst case,there is a known algorithm about extending discrete vector ﬁeld on each slice to a gradient vector ﬁeld on all of M×I. In the second case, triangulation on each slice of cell complex M is different. It has seen that the way to extend the vector ﬁeld on each slice to the vector ﬁeld on all of M ×I is not obvious. In this case, we have to choose a cell structure of M ×I compatible with slices. The algorithm how one could deﬁne a discrete vector ﬁeld on the resulting cell complex which does not have any closed paths is not known. In this thesis, we intend to explain how to deﬁne the extended vector ﬁeld when the triangulation on M is different and obtain the discrete bifurcation diagram. Furthermore, we want to get discrete bifurcation diagrams by using the persistence diagrams and compare these two discrete bifurcation diagrams.

Subject Keywords

Morse theory., Homology theory., Bifurcation theory.

URI

http://etd.lib.metu.edu.tr/upload/12622428/index.pdf
https://hdl.handle.net/11511/27745

Collections

Graduate School of Natural and Applied Sciences, Thesis

Suggestions

OpenMETU
Core

Decomposing perfect discrete Morse functions on connected sum of 3-manifolds Kosta, Neza Mramor; Pamuk, Mehmetcik; Varli, Hanife (2019-06-15) In this paper, we show that if a closed, connected, oriented 3-manifold M = M-1 # M-2 admits a perfect discrete Morse function, then one can decompose this function as perfect discrete Morse functions on M-1 and M-2. We also give an explicit construction of a separating sphere on M corresponding to such a decomposition.
Quasi-constricted linear operators on Banach spaces Wolff, MPH; Emel'yanov, Eduard Yu. (2001-01-01) Let X be a Banach space over C. The bounded linear operator T on X is called quasi-constricted if the subspace X-0 := {x epsilon X : lim(n --> infinity) parallel toT(n)x parallel to = 0} is closed and has finite codimension. We show that a power bounded linear operator T epsilon L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness chi parallel to (.)parallel to (1) (A) )over bar>T is mean ergodic for all lambda in the peripheral spectrum sigma (pi)(T) of T is constric...
Invariant subspaces for Banach space operators with an annular spectral set Yavuz, Onur (2008-01-01) Consider an annulus Omega = {z epsilon C : r(0) 0 such that parallel to p(T)parallel to <= K sup{vertical bar p(lambda)vertical bar : vertical bar lambda vertical bar <= 1} and parallel to p(r(0)T(-1))parallel to <= K sup{vertical bar p(lambda)vertical bar : vertical bar lambda vertical bar <= 1} for all polynomials p. Then there exists a nontrivial common invariant subspace for T* and T*(-1).
Invariant subspaces for banach space operators with a multiply connected spectrum Yavuz, Onur (Springer Science and Business Media LLC, 2007-07-01) We consider a multiply connected domain Omega = D \U (n)(j= 1) (B) over bar(lambda(j), r(j)) where D denotes the unit disk and (B) over bar(lambda(j), r(j)) subset of D denotes the closed disk centered at lambda(j) epsilon D with radius r(j) for j = 1,..., n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains delta Omega and does not contain the points lambda(1),lambda(2),...,lambda(n), and the operators T and r(j)( T -lambda I-j)(-1) are polynomially bounded, then th...
An obstruction to finding algebraic models for smooth manifolds with prescribed algebraic submanifolds CELIKTEN, A; Ozan, Yıldıray (2001-03-01) Let N ⊆ M be a pair of closed smooth manifolds and L an algebraic model for the submanifold N. In this paper, we will give an obstruction to finding an algebraic model X of M so that the submanifold N corresponds in X to an algebraic subvariety isomorphic to L.

Citation Formats

T. Örnek, “Discrete bifurcation diagrams and persistence,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.