Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.     

A Family of Graded Modules Associated to a Module

In this article we introduce a certain family of graded modules associated to a given module. These modules provide a natural extension of the notion of the associated graded ring of an ideal. We will investigate their properties. In particular, we will try to extend Ree' theorem on the associated graded ring of an ideal generated by a regular sequence to this context.     

蛭石对高钠高钙准东煤结渣特性影响研究

采用一维沉降炉,辅以灰熔点仪的实验方法,将蛭石与高钠高钙准东煤掺混燃烧,研究其对高钠高钙准东煤结渣特性的影响。结果表明,随着蛭石掺烧量的提高,灰熔点温度呈现先降低后升高的趋势,当掺烧量为6%时灰熔点温度最低;蛭石掺烧量越高,高钠高钙准东煤结渣情况改善越明显,当掺烧量达到4%时,渣样变得疏松多孔,质地变脆,渣样与沉积探针之间的黏附性较弱,极易通过吹灰除去,建议蛭石掺烧量为4%;煤灰中原始矿物质以石英、钙铝黄长石或钙黄长石以及辉石类的低熔点矿物质为主,掺烧蛭石后,含钠的绿辉石矿物质被转化为韭闪石,含铁的斜辉石、赤铁矿等矿物质被转化为铁橄榄石,渣样中的矿物质均以高熔点的镁橄榄石为主;蛭石具有固钠的作用,取样温度越低,蛭石掺混量越高,其固钠效果越明显。     

Comparative Adsorption of Acetone on Water and Ice Surfaces

Small organic molecules on ice and water surfaces are ubiquitous in nature and play a crucial role in many environmentally relevant processes. Herein, we combine surface‐specific vibrational spectroscopy and a controllable flow cell apparatus to investigate the molecular adsorption of acetone onto the basal plane of single‐crystalline hexagonal ice with a large surface area. By comparing the adsorption of acetone on the ice/air and the water/air interface, we observed two different types of acetone adsorption, as apparent from the different responses of both the free O?H and the hydrogen‐bonded network vibrations for ice and liquid water. Adsorption on ice occurs preferentially through interactions with the free OH group, while the interaction of acetone with the surface of liquid water appears less specific.     

Protection‐Group‐Free Synthesis of Sequence‐Defined Macromolecules via Precision λ‐Orthogonal Photochemistry

An advanced light‐induced avenue to monodisperse sequence‐defined linear macromolecules via a unique photochemical protocol is presented that does not require any protection‐group chemistry. Starting from a symmetrical core unit, precision macromolecules with molecular weights up to 6257.10 g mol^?1 are obtained via a two‐monomer system: a monomer unit carrying a pyrene functionalized visible light responsive tetrazole and a photo‐caged UV responsive diene, enabling an iterative approach for chain growth; and a monomer unit equipped with a carboxylic acid and a fumarate. Both light‐induced chain growth reactions are carried out in a λ‐orthogonal fashion, exciting the respective photosensitive group selectively and thus avoiding protecting chemistry. Characterization of each sequence‐defined chain (size‐exclusion chromatography (SEC), high‐resolution electrospray ionization mass spectrometry (ESI‐MS), and NMR spectroscopy), confirms the precision nature of the macromolecules.     

Topological Structure of Entropy of 4-Dimensional Axisymmetric Black Holes

Using the relationship between the entropy and the Euler characteristic, an entropy density is introduced to describe the inner topological structure of the entropy of 4-dimensional axisymmetric black holes. It is pointed out that the density of entropy is determined by the singularities of the timelike Killing vector field of spacetime, and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, which are topological invariants. At last, Kerr–Newman black hole as an example of axisymmetric black holes is given. What’s more, the entropy and the latent heat of the topological phase transition of the black hole mentioned above are calculated and the latent heat just lies in the range of the energy of gamma ray bursts. This work is supported in part by the NSFs of China under Grant No. 10575068 and of Shanghai Municipal Committee of Science and Technology under Grant No. 04ZR14059 and Shanghai Leading Academic Discipline Project under Project Number: T0104.     

Characteristic Numbers of Matrix Lie Algebras

A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.     

Deriving Energy-Gap of Some Nonlinear Hamiltonians by Invariant Eigen-operator Method

By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for some Hamiltonians describing nonlinear processes in particle physics. In this way the energy-gap of the Hamiltonians can be naturally obtained. The characteristic polynomial theory has been fully employed in our derivation.