In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

In this paper the Poincaré-Hopf inequalities are shown to be necessary and sufficient conditions for an abstract Lyapunov graph L to be continued to an abstract Lyapunov graph of Morse type. The Lyapunov graphs considered may represent smooth flows on closed orientable n-manifolds, n ≥ 2. The continuation which is presented by means of a constructive algorithm, is shown to be unique in dimensions two and three. In all other dimensions, the exact number of possible continuations of L are presented. Downloaded: 05 Apr 2015 IP address: 150.131.192.151 2 M. A. Bertolim et al Lyapunov graphs were first introduced by Franks [6]and have proven to be an excellent bookkeeping device of dynamical and topological information of the flow and the phase space. A Lyapunov graph is a finite, connected, oriented graph with no oriented cycles and with labeled vertices and edges.A Lyapunov function f : M → R associated to a flow, determines a Lyapunov graph by the following equivalence relation on M: x ∼ f y if and only if x and y belong to the same connected component of a level set of f . Therefore, M/ ∼ f is a Lyapunov graph. It is possible to choose f so that each critical level contains a component R of R. A point on M/ ∼ f is a vertex point if under the equivalence relation it corresponds to a level set containing a component R of R. All other points are edge points. Each edge represents a codimension one submanifold Q of M times an open bounded interval I , Q × I .In order to retain some topological information of Q × I , the edges will be labeled with the Betti numbers of Q. All homology groups will be computed with Z 2 coefficients.An abstract Lyapunov graph is an oriented graph with no oriented cycles such that each vertex v is labeled with a list of non-negative integersWhenever h j (v) = 0, it will be omitted from the list. Also, the labels on each edge {β 0 = 1, β 1 , . . . , β n−2 , β n−1 = 1} must be a collection of non-negative integers satisfying Poincaré duality and if n − 1 is even then β (n−1)/2 is even.An abstract Lyapunov graph of Morse type is defined later, but roughly speaking, it is an abstract Lyapunov graph with all vertices labeled with non-degenerate singularities, i.e. {h j (v) = 1} for some j .Next, the notion of vertex explosion is established in order to define continuation of abstract Lyapunov graphs. Let v be a vertex on an abstract Lyapunov graph labeled with {h 0 (v), h 1 (v), . . . , h n (v)}. A vertex v can be exploded if v can be removed and replaced by an abstract Lyapunov graph I of Morse type with k vertices. The graph I must respect the orientations and labels of the incoming and outgoing edges of v. In other words, the new graph obtained must be oriented and with no oriented cycles. The incoming (outgoing) edges of v, must be incoming (outgoing) edges on vertices of I and all labels on the edges must respect the restrictions of the Morse type vertices. Moreover,An abstract Lyapunov graph admits a continuation to an abstract Lyapunov graph of Morse type if each vertex can be exploded.The...

In this article we study algorithms that arise in both topological and dynamical settings, namely, the Spectral Sequence Sweeping Algorithm (SSSA) and the Row Cancellation Algorithm (RCA) for a filtered Morse chain complex on a manifold M n . Both algorithms have as input a connection matrix and the results obtained in this article make it possible to establish a correspondence between the algebraic cancellations in SSSA and the dynamical cancellations in RCA.where n(x, y) denotes the intersection number of x and y.A spectral sequence E = (E r , d r ) r≥0 is a sequence, such that E r is a bigraded R-module over a principal ideal domain R, i.e., an indexed collection of R-modules E r p,q for all pair of integers p and q; and d r is a differential of bidegree (−r, r − 1), i.e., it is a collection of homomorphisms d r : E r p,q → E r p−r,q+r−1 for all p and q such thatThe correctness of the sweeping algorithm over Z rests on Proposition 4, an extension of Proposition 8 of [5] that encompasses both general and grouped connection matrices. Before proceeding, we introduce a technical lemma that facilitates the extension of the results previously developed for grouped connection matrices.Lemma 3 Let N be an upper triangular m × m matrix with nonzero diagonal entries and column/row partitionThen N is invertible, its inverse is upper triangular and also has zero entries outside the setAdditionally, if j ∈ J k , for some k ∈ {0, . . . , b}, I = J k ∩ {j, . . . , m} and J = J k ∩ {1, . . . , j}, I and J denote their respective complements with respect to {1, . . . , m}, then (N −1 ) II = 0 and (N −1 ) JJ = 0.6 Temporary marks are erased at the end of the iterative step.for all r ≥ 1. Since the update of ∆(k) involves only the post-multiplication step, only elementary column operations are performed.Lemma 12 implies that ∆(1) m J0J1 = ∆ m J0J1 and the change-of-basis and primary pivots marked during the application of the algorithm to ∆(1) coincide with the ones marked in columns in J 1 when the algorithm is applied to ∆. Assume by induction that ∆1 and change-of-basis and primary pivots of ∆(k − 1) agree with the ones in columns in J k−1 marked when ∆ is swept. Also by induction, J k−1 = N k−1 . Proposition 10 implies rows of ∆ m in J k−1 are zero. By construction, rows of ∆(k) in J k−1 are also zero, and, since ∆(k) suffers only elementary column operations during the application of the Incremental Sweeping over F, these rows are not changed. So ∆ m J k−1 J k = ∆(k) m J k−1 J k . By Lemma 11, entries in the rows of ∆ in J k−1 are not subjected to elementary row operations during the application of the Incremental Sweeping Algorithm over F thereto. Furthermore, all change-of-basis and primary pivots in columns in J k occur in positions in J k−1 × J k . Hence changes to entries in these rows are only due to elementary column operations, as also happens when the Incremental Sweeping is applied to ∆(k). Since ∆ J k−1 J k = ∆(k) J k−1 J k and the elementary column operations on columns in J k are solely dependent on the entries...

We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on Griewank and Walther's [Evaluating derivatives, in Principles and Techniques of Algorithmic Differentiation, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008] state transformations synthesizes the calculation of the Hessian as a formula. These dual points of view, graphical and algebraic, lead to a new framework for Hessian computation. This is illustrated by developing edge_pushing, a new truly reverse Hessian computation algorithm that fully exploits the Hessian's symmetry. Computational experiments compare the performance of edge_pushing on 16 functions from the CUTE collection [I. Bongartz et al. Cute: constrained and unconstrained testing environment, A.H. Gebremedhin et al. Efficient computation of sparse Hessians using coloring and automatic differentiation, INFORMS J. Comput. 21(2) (2009), pp. 209-223], and the results are very promising.

Abstract. In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.