Discrete Morse theory is an e cient method which allows one to study the topology of discrete objects. The instrumental tool in the algebraic setting of this theory is the notion of admissible discrete vector eld. In this paper, we present a formally veri ed implementation of an algorithm in charge of building an admissible discrete vector eld from a digital image. Such a program will play a ke...

Coupled cell systems are systems of ODEs, defined by ‘admissible’ vector fields, associated with a network whose nodes represent variables and whose edges specify couplings between nodes. It is known that non-isomorphic networks can correspond to the same space of admissible vector fields. Such networks are said to be ‘ODE-equivalent’. We prove that two networks are ODE-equivalent if and only i...

Two conjectures on admissible control operators by George Weiss are disproved in this paper. One conjecture says that an operator B defined on an infinite-dimensional Hilbert space U is an admissible control operator if for every element u ∈ U the vector Bu defines an admissible control operator. The other conjecture says that B is an admissible control operator if a certain resolvent estimate ...

with fixed initial data x(t = x where x is the n-dimensional o) o' state vector, u is the r-dimensional vector control function, A(t) , B(t) are the nxn, nxr matrix functions, respectively. A(t), B(t), are assumed to be piecewise continuous for t ~ to on any finite in­ terval. A control function u(t) is said to be admissible if it is measur­ able over any finite interval and takes its values fr...

The concept of elementary flux vector is valuable in a number of applications of metabolic engineering. For instance, in metabolic flux analysis, each admissible flux vector can be expressed as a non-negative linear combination of a small number of elementary flux vectors. However a critical issue concerns the total number of elementary flux vectors which may be huge because it combinatorially ...

A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems are represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. Symmetry of coupled cell systems can lead to synchronized cells. We show that symmetry is not the only mechanism that can create such states in a coupled cell system. The first m...

Given a strongly connected directed graph D, let SD denote the set of all stochastic matrices whose directed graph is a spanning subgraph of D. We consider the problem of completely describing the set of stationary vectors of irreducible members of SD . Results from the area of convex polytopes and an association of each matrix with an undirected bipartite graph are used to derive conditions wh...

We prove asymptotic completeness for operators of the form H = −∆+L on L(R), d ≥ 2, where L is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials V ∈ L(R), q ∈ [d/2, (d + 1)/2] (if d = 2 then we require q ∈ (1, 3/2]), as well as real-valued potentials V satisfying a global Kato condition. The class of admissible ...