We conjecture that every oriented graph G on n vertices with δ(G), δ−(G) ≥ 5n/12 contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing of transitive triangles in an oriented graph. A link between Ramsey numbers and perfect packings of transitive tournaments is also considered. 1.1. Powers of Hamilton cycles. One of th...

Recent work on path integral stochastic optimal control theory Theodorou et al. (2010a); Theodorou (2011) has shown promising results in planning and control of nonlinear systems in high dimensional state spaces. The path integral control framework relies on the transformation of the nonlinear Hamilton Jacobi Bellman (HJB) partial differential equation (PDE) into a linear PDE and the approximat...

In this article, we discuss some aspects of the search for chessknight closed tours in a chessboard, as in [1], [g], [10], [14] and [16] and give an elementary application to the interaction between graph theory and group theory. These closed tours will be herewith denoted as chessknight Hamilton cycles. In particular, we prove the following: If n and r are integers>0withn4r> 2,thenthedifferenc...

We deal with a class of semilinear parabolic PDEs on the space continuous functions that arise, for example, as Kolmogorov equations associated to infinite-dimensional lifting path-dependent SDEs. investigate existence smooth solutions through their representation via forward–backward stochastic systems, which we provide necessary regularity theory. Because lack smoothing properties operators a...

ôbjective: many studies evaluated the efficacy of exercise on some depression indices, but the effect of physical exercise in exhilarating milieu on urine 3-methoxy-4-hydroxyphenylglycol (mhpg) sulfate-the main metabolite of norepinephrine is not clear. the purpose of this research was to study the effect of a six-week intermittent walking in the water on 24-hour urine mhpg sulfate in depressed...

The path-integral control, which stems from the stochastic Hamilton–Jacobi–Bellman equation, is one of methods to control nonlinear systems. This paper gives a new insight into optimal problems perspective Koopman operators. When finite-dimensional dynamical system nonlinear, corresponding operator linear. Although infinite-dimensional, adequate approximation makes it tractable and useful in so...

All graphs considered here are simple and finite unless otherwise stated. LetCk (resp.Pk) denote the cycle (resp. path) on k vertices. For a graphG, if its edge set E(G) can be partitioned into E1,E2, . . . ,Ek such that 〈Ei〉 ∼= H, for all i, 1 ≤ i ≤ k, then we say thatH decomposesG.A k-factor ofG is a k-regular spanning subgraph of it. A k-factorization of a graphG is a partition of the edge s...

Multi-UAV systems are safety-critical, and guarantees must be made to ensure no unsafe configurations occur. Hamilton-Jacobi (HJ) reachability is ideal for analyzing such safety-critical systems; however, its direct application is limited to small-scale systems of no more than two vehicles due to an exponentially-scaling computational complexity. Previously, the sequential path planning (SPP) m...

Stochastic shortest path (SSP) problems arise in a variety of discrete stochastic control contexts. An optimal solutions to such a problem is typically computed using the value function, which can be found by solving the corresponding dynamic programming equations. In the deterministic case, these equations can be often solved by the highly efficient label-setting methods (such as Dijkstra’s an...

Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald [L.H. Clark, N.C. Wormald, Hamiltonian-like indices of graphs, ARS Combinatoria 15 (1983) 131–148] defined hc(G) to be the least integerm such that the iterated line graph Lm(G) is Hamilton-connected. Let diam(G) be the diameter of G and k be the length of a longest path whose internal vertices, if any, have degr...