Trees with Maximum p-Reinforcement Number

Let $G=(V,E)$ be a graph and $p$ a positive integer. The $p$-domination number \g_p(G) is the minimum cardinality of a set $D\subseteq V$ with $|N_G(x)\cap D|\geq p$ for all $x\in V\setminus D$. The $p$-reinforcement number $r_p(G)$ is the smallest number of edges whose addition to $G$ results in a graph $G'$ with \g_p(G')<\g_p(G). Recently, it was proved by Lu et al. that $r_p(T)\leq p+1$ for a tree $T$ and $p\geq 2$. In this paper, we characterize all trees attaining this upper bound for $p\geq 3$

Heterogeneous flow structure and gas-solid transport of riser

This study aims to understand physical mechanisms of gas-solid transport and riser flow, investigate heterogeneous flow structures of gas-solid transport and their formation mechanism of the in riser flows, both in axial and radial directions. It provides sound interpretation for the experimental observation and valuable suggestion to riser reactor design. Chemical reaction is also coupled with flow hydrodynamics to board the industrial applications. This study mainly focuses on mathematical modeling approach based upon physical mechanism, and endeavor to validate model prediction against available experimental data. First of all, most important physical mechanisms including inter-particle collision force, gas/solid interfacial force and wall boundary effects, which are believed to be most important aspects of the flow hydrodynamics, have been investigated in this part. An energy-based mechanistic model was developed to analyze the partitions of the axial gradient of pressure by solids acceleration, collision-induced energy dissipation and solids holdup in gas-solid riser flows. Thought this part of study, important understanding of the inter-particle collision force (Fc), gas/solid interfacial force (FD) inside the momentum equations and energy dissipation (F), especially in dense and acceleration region, has been reached, Based on these understandings, a mechanistic riser hydrodynamic model was developed on the basis of gas-solid continuity and momentum equations, along with the better formulated drag force correlation and new formulation for moment dissipation of solids due to solids collisions. The proposed model is capable of yielding the coupled hydrodynamic parameters of solid volume fraction, gas and solid velocity, and pressure distribution along the whole riser. At the same time, special considerations are given to solids back-mixing and resultant cross- section area variation for the upward flow, which is especially prominent for low solids mass flow condition. With the further understanding of solid collision, gas/solid interfacial and wall boundary effects, in order to soundly interpret the well-known core-annulus 2-zone flow structure, newly discovered core-annulus-wall 3-zone structure and provide reasonable explanation for the choking phenomena, a comprehensive modeling of continuous gas-solids flow structure both in radial and axial directions has been presented. This model, assuming one-dimensional two-phase flow in each zone along the riser, consists of a set of coupled ordinary-differential equations developed from the conservation laws of mass, momentum, and energy of both gas and solids phases. This part of study not only provides reasonable explanation for the 2-zone and 3-zone structure , but also finds out the potential reasons for the choking phenomenon. In order to investigate the different riser inlet configuration\u27s effects on gas-solid mixing in dense region and improve the uniform inlet condition assumption in above models, a systemically study regarding with different inlet conditions have been done based on commercial package, Those simulation results are directly combined with model approach which reached the conclusion that riser flow structure an flow stability are weakly dependent on the type of solids feeding configuration. This part of study is specifically focused on chemical reaction coupled gas-solid transport flow hydrodynamics. The aim of this work is to develop a generic modeling approach which can fully incorporate multiphase flow hydrodynamics with chemical reaction process. This modeling approach opens up a new dimension for making generic models suitable for the analysis and control studies of chemical reaction units. The chemical reaction model was represented by a relatively simple four-lump based FCC reaction kinetic model, which will not bring us too complicated mathematical derivation without losing its popular acceptance. As a first endeavor to consider the significant mutual coupling between the flow hydrodynamics and cracking reaction, a localized catalyst to oil ratio is introduced. The new developed chemical reaction coupled hydrodynamic model was capable of quickly evaluating the flow parameters including gas and solid phase velocity and concentration, temperature and reaction yield profiles as the function of riser height

Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2

In this paper various representations of the exceptional Lie algebra G_2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined on the space of the universal enveloping algebra of G_2, are explicitly determined. From this master representation, different indecomposable representations defined on invariant subspaces or quotient spaces with respect to these invariant subspaces are discussed. Especially, the elementary representations of G_2 are investigated in detail, and the corresponding six-boson realization is given. After obtaining explicit forms of all twelve extremal vectors of the elementary representation with the highest weight {\Lambda}, all representations with their respective highest weights related to {\Lambda} are systematically discussed. For one of these representations the corresponding five-boson realization is constructed. Moreover, a new three-fermion realization from the fundamental representation (0,1) of G_2 is constructed also.Comment: 29 pages, 4 figure

Electron-nuclear entanglement in the cold lithium gas

We study the ground-state entanglement and thermal entanglement in the hyperfine interaction of the lithium atom. We give the relationship between the entanglement and both temperature and external magnetic fields.Comment: 7 pages, 3 figure