The Standard Metric

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

The Effect of Water Hammer on Pressure Oscillation of Hydrogen Natural Gas Transient Flow

The unexpected closing of a control valve or pumping, which can be planned or accidental could lead to pressure oscillation commonly known as water hammer. Pressure waves oscillation in the fluid flow interaction leads to unsteady state phenomenon. Numerical computation on transient flow model of hydrogen natural gas mixture (HCNG) in a pipeline where different types of valves are considered for effective delivery and safety. Reduced order modeling (ROM) technique is employed in the transient analysis of high pressured HCNG in a short pipe. The flow is governed by a system of Euler equations that are discretized using the implicit Steger-Warming flux vector splitting method (FSM). Eigenmode analysis and the subsequent construction of ROM use only a few dominant flow eigenmodes. The boundary conditions are imposed using the closure law at the upstream and downstream of the pipeline. The effect of closure function is analyzed and the results are presented on different mass ratios and HCNG flow parameters. The transient pressure oscillations of HCNG at different closure functions are compared so as to test the best type of valve for HCNG transportation. The highest and lowest value of transient pressure wave occurs when instantaneous and convex closure law is considered, respectively.

Locally convex topologies and the convex compactness property

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.