A generalized expression for the ratio of the critical temperature to the critical pressure with the van der Waals surface area

Using an extensive database of experimental critical properties for heavy compounds, which have been compiled mostly from recent literature sources, it is shown that the ratio T_c: P_c (critical temperature over critical pressure) can be expressed in terms of the van der Waals surface area (Q_w), which is readily available for any compound from the group contributions of Bondi (given also in UNIFAC tables). The proposed correlation is based on the hole theory of Kurata and Isida for n-paraffin liquids, which is mathematically equivalent to Flory's theory of polymer solutions. The method is suitable for medium to high molecular weight compounds with unknown critical constants. For example, if only one of the two critical constants is available, then the proposed generalized equation offers a useful rapid procedure for the estimation of the other critical property for use in corresponding states, and other relevant applications where knowledge of the critical properties is required. Furthermore, the T_c: P_c method can be used in many cases for identifying the most suitable among the existing group contribution methods for estimating the critical properties of heavy and complex compounds for which experimental values are, very often, not available.     

Cationic phosphorus-carbon-pnictogen cages isolobal to [C5R5]+

The cationic cages nido-[C2Bu(t)2P2E]+ (E = As, Sb), which are isolobal to the cyclopentadienyl cation, adopt square based pyramidal structures with the heavy pnictogen atom at the apex; NMR and computational methods have been used to probe the dynamic behaviour of the complexes.     

Spectral Exterior Calculus

A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space dimension. To achieve this, the exterior calculus is reformulated entirely in terms of the eigenvalues and eigenfunctions of the Laplacian operator on functions. The exterior derivatives of these eigenfunctions (and their wedge products) are shown to form a frame (a type of spanning set) for appropriate L² spaces of k -forms, as well as higher-order Sobolev spaces. Formulas are derived to express the Laplace-de Rham operators on forms in terms of the eigenfunctions and eigenvalues of the Laplacian on functions. By representing the Laplace-de Rham operators in this frame, spectral convergence results are obtained via Galerkin approximation techniques. Numerical examples demonstrate accurate recovery of eigenvalues and eigenforms of the Laplace-de Rham operator on 1-forms. The correct Betti numbers are obtained from the kernel of this operator approximated from data sampled on several orientable and non-orientable manifolds, and the eigenforms are visualized via their corresponding vector fields. These vector fields form a natural orthonormal basis for the space of square-integrable vector fields, and are ordered by a Dirichlet energy functional which measures oscillatory behavior. The spectral framework also shows promising results on a non-smooth example (the Lorenz 63 attractor), suggesting that a spectral formulation of exterior calculus may be feasible in spaces with no differentiable structure. © 2020 Wiley Periodicals, Inc.     

Exploring the design space of reliable multicast protocols for wireless mesh networks

Many important applications in wireless mesh networks require reliable multicast communication, i.e., with 100% packet delivery ratio (PDR). Previously, numerous multicast protocols based on automatic repeat request (ARQ) have been proposed to improve the packet delivery ratio. However, these ARQ-based protocols can lead to excessive control overhead and drastically reduced throughput. In this paper, we present a comprehensive exploration of the design space for developing high-throughput, reliable multicast protocols that achieve 100% PDR.Motivated by the fact that 802.11 MAC layer broadcast, which is used by most wireless multicast protocols, offers no reliability, we first examine if better hop-by-hop reliability provided by unicasting the packets at the MAC layer can help to achieve end-to-end multicast reliability. We then turn to end-to-end solutions at the transport layer. Previously, forward error correction (FEC) techniques have been proved effective for providing reliable multicast in the Internet, by avoiding the control packet implosion and scalability problems of ARQ-based protocols. In this paper, we examine if FEC techniques can be equally effective to support reliable multicast in wireless mesh networks. We integrate four representative reliable schemes (one ARQ, one FEC, and two hybrid) originally developed for the Internet with a representative multicast protocol ODMRP and evaluate their performance.Our experimental results via extensive simulations offer an in-depth understanding of the various choices in the design space. First, compared to broadcast-based unreliable ODMRP, using unicast for per-hop transmission only offers a very small improvement in reliability under low load, but fails to improve the reliability under high load due to the significantly increased capacity requirement which leads to congestion and packet drop. Second, at the transport layer, the use of pure FEC can significantly improve the reliability, increasing PDR up to 100% in many cases, but can be inefficient in terms of the number of redundant packets transmitted. In contrast, a carefully designed ARQ–FEC hybrid protocol, such as RMDP, can also offer 100% reliability while improving the efficiency by up to 38% compared to a pure FEC scheme. To our best knowledge, this is the first in-depth study of high-throughput, reliable multicast protocols that provide 100% PDR for wireless mesh networks.