Correction. Brownian models of open processing networks: canonical representation of workload

Due to a printing error the above mentioned article [Annals of Applied Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous equations appearing incorrectly in the print version of this paper. The entire article follows as it should have appeared. IMS apologizes to the author and the readers for this error. A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an ``equivalent workload formulation'' of a Brownian network model. Denoting by $Z(t)$ the state vector of the original Brownian network, one has a lower dimensional state descriptor $W(t)=MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of ``heavy traffic'' for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix $M$. To be specific, rows of the canonical $M$ are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix $M$ is shown to be nonnegative, and another natural condition is identified which ensures that $M$ admits a factorization related to the notion of resource pooling.Comment: Published at http://dx.doi.org/10.1214/105051606000000583 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Arkansas Open Carry: Understanding Law Enforcement’s Legal Capability Under a Difficult Statute

“There seems to us no doubt, on the basis of both text and history, that the Second Amendment conferred an individual right to keep and bear arms.”1 Although the United States Supreme Court in District of Columbia v. Heller established a fundamental understanding that individuals have a right to own a gun for personal use, the Court recognized that, as with all fundamental rights, the individual right to keep and bear arms is “not unlimited.”2 A few limits the Court mentioned included “prohibitions on the possession of firearms by felons and the mentally ill, or laws forbidding the carrying of firearms in sensitive places such as schools and government buildings, or laws imposing conditions and qualifications on the commercial sale of arms.”3 Naturally, the Heller decision left us with this question: What are the constitutionally sound restrictions, and how far can the government go?

Formalization of Complex Vectors in Higher-Order Logic

Complex vector analysis is widely used to analyze continuous systems in many disciplines, including physics and engineering. In this paper, we present a higher-order-logic formalization of the complex vector space to facilitate conducting this analysis within the sound core of a theorem prover: HOL Light. Our definition of complex vector builds upon the definitions of complex numbers and real vectors. This extension allows us to extensively benefit from the already verified theorems based on complex analysis and real vector analysis. To show the practical usefulness of our library we adopt it to formalize electromagnetic fields and to prove the law of reflection for the planar waves.Comment: 15 pages, 1 figur