Modeling a class of state-dependent routing in flexible manufacturing systems

We develop a closed queueing network model for flexible manufacturing systems (FMSs), where parts routing follows a probabilistic shortest-queue (PSQ) scheme, i.e. parts are routed to the shortest queue (or the most empty station) with the highest probability. We allow limited local buffer at each work station. We prove that with the PSQ routing, the Markovian queue-length process satisfies time reversibility and has a product-form equilibrium distribution. An algorithm is developed to compute the solutions to the model. The model can be used as a performance evaluation tool to study FMSs.     

On convergence of closed convex sets

In this paper we introduce a convergence concept for closed convex subsets of a finite-dimensional normed vector space. This convergence is called C-convergence. It is defined by appropriate notions of upper and lower limits. We compare this convergence with the well-known Painlevé-Kuratowski convergence and with scalar convergence. In fact, we show that a sequence (An)_n∈NC-converges to A if and only if the corresponding support functions converge pointwise, except at relative boundary points of the domain of the support function of A, to the support function of A.     

Markov processes with darning and their approximations

In this paper, we study darning of general symmetric Markov processes by shorting some parts of the state space into singletons. A natural way to construct such processes is via Dirichlet forms restricted to the function spaces whose members take constant values on these collapsing parts. They include as a special case Brownian motion with darning, which has been studied in details in Chen (2012), Chen and Fukushima (2012) and Chen et al. (2016). When the initial processes have discontinuous sample paths, the processes constructed in this paper are the genuine extensions of those studied in Chen and Fukushima (2012). We further show that, up to a time change, these Markov processes with darning can be approximated in the sense of finite-dimensional distributions by introducing additional jumps with large intensity among these compact sets to be collapsed into singletons. For diffusion processes, it is also possible to get, up to a time change, diffusions with darning by increasing the conductance on these compact sets to infinity. To accomplish these, we give a version of the semigroup characterization of Mosco convergence to closed symmetric forms whose domain of definition may not be dense in the $L^2$-space. The latter is of independent interest and potentially useful to study convergence of Markov processes having different state spaces. Indeed, we show in Section 5 of this paper that Brownian motion in a plane with a very thin flag pole can be approximated by Brownian motion in the plane with a vertical cylinder whose horizontal motion on the cylinder is a circular Brownian motion moving at fast speed.     

Minimizing geodesic nets and critical points of distance

We establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the n-sphere. This result is used to give conditions under which a minimizing geodesic flower degenerates into a simple closed geodesic.     

A formula for gap between two closed operators

Aim of this short note is to obtain a formula for the gap between two densely defined unbounded closed operators. It is interesting to note that the formula is very similar to the corresponding formula for the gap between two bounded operators.     

Optimal 1-planar graphs which triangulate other surfaces

We show that, for any given non-spherical orientable closed surface F², there exists an optimal 1-planar graph which can be embedded on F² as a triangulation. On the other hand, we prove that there does not exist any such graph for the nonorientable closed surfaces of genus at most 3.     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

半广义不定映射

在L-拓扑空间中引入半广义不定映射,强半广义连续映射,全半广义连续映射,同时讨论它们一些性质。     

On IP-graphs of association schemes and applications to group theory

Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger (1993) [5] studied the common divisor graph of (G,X). For a group G and its subgroup A, based on the results in Isaacs and Praeger (1993) [5], Kaplan (1997) [6] proved that if A is stable in G and the common divisor graph of (A,G) has two components, then G has a nice structure. Motivated by the notion of the common divisor graph of (G,X), Camina (2008) [3] introduced the concept of the IP-graph of a naturally valenced association scheme. The common divisor graph of (G,X) is the IP-graph of the association scheme arising from the action of G on X. Xu (2009) [8] studied the properties of the IP-graph of an arbitrary naturally valenced association scheme, and generalized the main results in Isaacs and Praeger (1993) [5] and Camina (2008) [3]. In this paper we first prove that if the IP-graph of a naturally valenced association scheme (X,S) is stable and has two components (not including the trivial component whose only vertex is 1), then S has a closed subset T such that the thin residue O?(T) and the quotient scheme (X/O?(T),S//O?(T)) have very nice properties. Then for an association scheme (X,S) and a closed subset T of S such that S//T is an association scheme on X/T, we study the relations between the closed subsets of S and those of S//T. Applying these results to schurian schemes and common divisor graphs of groups, we obtain the results of Kaplan [6] as direct consequences.