Zero-range interaction in arbitrary dimensions and in the presence of external forces

We analyze zero-range interactions in arbitrary dimensions and in the presence of external forces in terms of non-local separable potentials. We show that in one-, two- and three-dimensional cases this approach is equivalent to the standard method with a renormalization operator. Our analysis indicates that the method of a projection operator is also useful for studying processes occurring under the influence of additional forces. As an example we present exact solutions for a particle interacting with a static electric field.     

Antidirected Hamiltonian paths and directed cycles in tournaments

We prove that in a tournament of odd order $n\ge 5$, the number of antidirected Hamiltonian paths starting with a forward arc and the number of Hamiltonian circuits have the same parity.     

Differential equations driven by rough paths with jumps

We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.     

Reconfiguration graphs of shortest paths

For a graph $G$ and$a,b\in V\left(G\right)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and$b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and$b$ in $G$. Two vertices in $V\left(S(G,a,b)\right)$ are adjacent, if their corresponding paths in $G$ differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We include an infinite family of well structured examples, showing that the shortest path graph of a grid graph is an induced subgraph of a lattice.     

Dyck paths with a first return decomposition constrained by height

We study the enumeration of Dyck paths having a first return decomposition with special properties based on a height constraint. We exhibit new restricted sets of Dyck paths counted by the Motzkin numbers, and we give a constructive bijection between these objects and Motzkin paths. As a byproduct, we provide a generating function for the number of Motzkin paths of height $k$ with a flat (resp. with no flats) at the maximal height.     

Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any $2$ longest paths have a vertex in common. For $k\ge 7$, Skupień in 1966 obtained a connected graph in which some $k$ longest paths have no common vertex, but every $k?1$ longest paths have a common vertex. It is not known whether every $3$ longest paths in a connected graph have a common vertex and similarly for 4, 5, and $6$ longest path. Fujita et al. in 2015 give an upper bound on distance among $3$ longest paths in a connected graph. In this paper we give a similar upper bound on distance between $4$ longest paths and also for $k$ longest paths, in general.     

On the spanning connectivity of the generalized Petersen graphs P(n,3)

In this paper, we employed lattice model to describe the three internally vertex-disjoint paths that span the vertex set of the generalized Petersen graph $P(n,3)$. We showed that the $P(n,3)$ is 3-spanning connected for odd $n$. Based on the lattice model, five amalgamated and one extension mechanisms are introduced to recursively establish the 3-spanning connectivity of the $P(n,3)$. In each amalgamated mechanism, a particular lattice trail was amalgamated with the lattice trails that was dismembered, transferred, or extended from parts of the lattice trails for $P(n?6,3)$, where a lattice tail is a trail in the lattice model that represents a path in $P(n,3)$.     

Sensitivity of rough differential equations: An approach through the Omega lemma

The Itô map gives the solution of a Rough Differential Equation, a generalization of an Ordinary Differential Equation driven by an irregular path, when existence and uniqueness hold. By studying how a path is transformed through the vector field which is integrated, we prove that the Itô map is Hölder or Lipschitz continuous with respect to all its parameters. This result unifies and weakens the hypotheses of the regularity results already established in the literature.     

On stochastic calculus with respect to q-Brownian motion

Following the approach and the terminology introduced in Deya and Schott (2013) [6], we construct a product Lévy area above the q-Brownian motion (for $q\in [0,1)$) and use this object to study differential equations driven by the process.We also provide a detailed comparison between the resulting “rough” integral and the stochastic “Itô” integral exhibited by Donati-Martin (2003) [7].