The maximum likelihood degree of a very affine variety

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to Varchenko's conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern-Schwartz-MacPherson class. The strengthened version recovers the geometric deletion-restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko's theorem on the Newton polytope for nondegenerate hypersurfaces.Comment: Improved readability. Final version, to appear in Compositio Mathematic

Interplay between network topology and synchrony-breaking bifurcation: homogeneous four-cell coupled networks

Complex networks are studied across many fields of science. Much progress has been made on static and statistical features of networks, such as small world and scale-free networks. However, general studies of network dynamics are comparatively rare. Synchrony is one commonly observed dynamical behaviour in complex networks. Synchrony breaking is where a fully synchronised network loses coherence, and breaks up into multiple clusters of self-synchronised sub-networks. Mathematically this can be described as a bifurcation from a fully synchronous state, and in this thesis we investigate the effect of network topology on synchrony-breaking bifurcations. Coupled cell networks represent a collection of individual dynamical systems (termed cells) that interact with each other. Each cell is described by an ordinary differential equation (ODE) or a system of ODEs. Schematically, the architecture of a coupled cell network can be represented by a directed graph with a node for each cell, and edges indicating cell couplings. Regular homogeneous networks are a special case where all the nodes/cells and edges are of the same type, and every node has the same number of input edges, which we call the valency of the network. Classes of homogeneous regular networks can be counted using an existing group theoretic enumeration formula, and this formula is extended here to enumerate networks with more generalised structures. However, this does not generate the networks themselves. We therefore develop a computer algorithm to display all connected regular homogeneous networks with less than six cells and analysed synchrony-breaking bifurcations for four-cell regular homogeneous networks. Robust patterns of synchrony (invariant synchronised subspaces under all admissible vector fields) describe how cells are divided into multiple synchronised clusters, and their existence is solely determined by the network topology. These robust patterns of synchrony have a hierarchical relationship, and can be treated as a partially ordered set, and expressed as a lattice. For each robust pattern of synchrony (or lattice point) we can reduce the original network to a smaller network, called a quotient network, by representing each cluster as a single combined node. Therefore, the lattice for a given regular homogeneous network provides robust patterns of synchrony and corresponding quotient networks. Some lattice structures allow a synchrony breaking bifurcation analysis based solely on the dynamics of the quotient networks, which are lifted to the original network using the robust patterns of synchrony. However, in other cases the lattice structure also tells us of the existence and location of additional synchrony-breaking bifurcating branches not seen in the quotient networks. In conclusion the work undertaken here shows that the invariant synchronised subspaces that arise from a network topology facilitate the classification of synchrony-breaking bifurcations of networks

The algorithm for generation of structured grids in deformed volumes of revolution

For the volume of revolution deformed by another volume of revolution, a grid generation algorithm is suggested. The algorithm is designed for multimaterial hydrodynamic simulation and for solving other physical and engineering problems. The algorithm represents the non-stationary procedure generating three-dimensional structured grids in domains with moving boundaries. The algorithm is developed within the variational approach for constructing optimal curvilinear grids. The volume of revolution is obtained by the rotation about the axis through 180° of a plane generatrix curve consisting of straight line segments and arcs of circles. The non-stationary algorithm is the iterative process at each stage of which the deformation of a grid and then its optimization are carried out. The deformation of a grid is implemented within the geometrical approach, and the optimization of a grid within the variational approach. Iterations are continued until the necessary deformation is reached. The algorithm is realized in the computer code written in C++. © Published under licence by IOP Publishing Ltd