Narrow Results

For a closure space (P, φ) with φ(ø) = ø, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P, φ), extending the poset Clop(P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces,

• Reg(P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity.

• If Reg(P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice.

• Clop(P, φ) is a lattice if and only if every regular closed set is clopen.

The extended permutohedron R(G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R(G) and Reg S are bounded homomorphic images of free lattices.

This paper is concerned with the generalized hematopoiesis model with discontinuous harvesting terms. Under the framework of Filippov solution, by means of the differential inclusions and the topological degree theory in set-valued analysis, we have established the existence of the bounded positive periodic solutions for the addressed models. After that, based on the nonsmooth analysis theory with Lyapunov-like approach, we employ a novel argument and derive some new criteria on the uniqueness, global exponential stability of the addressed models and convergence of the corresponding autonomous case of the addressed models. Our results extend previous works on hematopoiesis model to the discontinuous harvesting terms and some corresponding results in the literature can be enriched and extended. In addition, typical examples with numerical simulations are given to illustrate the feasibility and validity of obtained results.

This paper studies the features of the USD/HKD exchange rate process by assessing the conformity of its dynamics to that of a random walk. This is not a trivial task since we consider the period within which the rate is confined to a specified corridor. This is achieved via analysis of its fractal dimension by means of the Hurst exponent as estimated using the rescaled range method. The conformity can be quantified by the difference between the estimated Hurst exponent and the random walk Hurst exponent of ½. At least two distinct Hurst exponents are identified, one corresponding to a random walk while the other, to an anti-persistent process. Partitioning the rate in state space associates the anti-persistence with proximity to the lower boundary of the corridor so the rate can be modeled using a random walk when sufficiently distant from the boundary.

We review several recent results showing that small piecewise smooth perturbations of integrable systems may exhibit unstable behavior on the set of initial condition of large measure. We also present open questions related to this subject.