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The two-fold singularity has played a significant role in our understanding of uniqueness and stability in piecewise smooth dynamical systems. When a vector field is discontinuous at some hypersurface, it can become tangent to that surface…

We show that Horrock's criterion for the splitting of vector bundles on $\PP^n$ can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool…

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially…

We consider $C^2$ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic…

A coupled cell system is an ODE system associated with a coupled cell network, where the dimension is determined by the number of cells. A heteroclinic connection is a set of solution trajectories between two equilibria of an ODE system. A…

In this paper, we survey constructions of and nonexistence results on combinatorial/geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on…

Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies these are nondegenerate maxima, minima, and…

We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive…

In this paper, we investigate saddle-node to saddle separatrix--loops that we term SNICeroclinic bifurcations. They are generic codimension-two bifurcations involving a heteroclinic loop between one non-hyperbolic and one hyperbolic saddle.…

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincar\'e map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits…

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links,…

We find an upper bound to the maximal number of limit cycles, which bifurcate from a hamiltonian two-saddle loop of an analytic vector field, under an analytic deformation.

We study the geometry of complex Poisson bivectors over smooth manifolds. We show that under mild regularity conditions any complex Poisson bivector has associated a complex presymplectic foliation. After that, we use techniques of Dirac…

Network geometry is currently a topic of growing scientific interest as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However the field is…

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…

The {\it two-fold singularity} has played a significant role in our understanding of uniqueness and stability in piecewise smooth dynamical systems. When a vector field is discontinuous at some hypersurface, it can become tangent to that…

We study behaviour of trajectories near a type Z heteroclinic network which is a union of two cycles. Analytical and numerical studies indicate that attractiveness of this network can be associated with various kinds of dynamics in its…

For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…

We describe relations between hyperbolic geometry and codimension two knots or, more exactly, between varieties of conjugacy classes of discrete faithful representations of the fundamental groups of hyperbolic n-manifolds M into…

The linear transports along paths in vector bundles introduced in Ref. [1] are applied to the special case of tensor bundles over a given differentiable manifold. Links with the transports along paths generated by derivations of tensor…