Pyrrolidine-2,5-dione and maleimide are important scaffolds of many organic
substances, and their derivatives are now attracting more and more interests
from researchers in organic synthesis, medicinal chemistry, and drug development.
Tosyloxy (-OTs) group is an important functional group widely used
in organic synthesis, because it can be readily prepared from alcohols and is
an excellent leaving group. However, surprisingly, substances bearing tosyloxy
groups on pyrrolidine-2,5-dione or maleimide scaffolds are very rare. In this
study, we discovered that, when treated with TsCl/Et_{3}N,？trans-3,4-dihydroxypyrrolidine-2,5- dione will eliminate a TsOH molecule to form monotosyloxymaleimide.
Thermodynamic and kinetic factors affecting this reaction were
investigated by theoretical computation using density functional theory
(DFT), and the possible reaction mechanism was proposed based on the
computation results. Our results showed that tosylates of trans -3,4-dihydroxypyrrolidine-
2,5-dione, either monotosylate or ditosylate, are thermodynamically
instable and may spontaneously convert to maleimides. This knowledge
could be useful in understanding the properties of pyrrolidine-2,5-diones and
maleimides, as well as the related organic synthesis.

Abstract:
The mass stream function of mean meridional circulation is calculated from the ECMWF and NCEP/NCAR reanalysis data sets using a superposition computation scheme. The comparison of results shows that the common ascending leg of the Hadley cell calculated from the ECMWF data is strong and narrow, and averagely lies more north of the equator in comparison with its counterpart from the NCEP/NCAR data, and furthermore the Hadley cell from the ECMWF data shows an obvious double-layer structure. Therefore, there are obvious differences between Hadley cells displayed by the two objective analysis data sets.

Abstract:
Let $R$ be a left noetherian ring, $S$ a right noetherian ring and $_RU$ a generalized tilting module with $S={\rm End}(_RU)$. The injective dimensions of $_RU$ and $U_S$ are identical provided both of them are finite. Under the assumption that the injective dimensions of $_RU$ and $U_S$ are finite, we describe when the subcategory $\{{\rm Ext}_S^n(N, U)|N$ is a finitely generated right $S$-module$\}$ is closed under submodules. As a consequence, we obtain a negative answer to a question posed by Auslander in 1969. Finally, some partial answers to Wakamatsu Tilting Conjecture are given.

Abstract:
We prove that for a left and right Noetherian ring $R$, $_RR$ satisfies the Auslander condition if and only if so does every flat left $R$-module, if and only if the injective dimension of the $i$th term in a minimal flat resolution of any injective left $R$-module is at most $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the $i$th term in a minimal injective coresolution (resp. flat resolution) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$ plus $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the injective envelope (resp. flat cover) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$, and if and only if any of the opposite versions of the above conditions hold true. Furthermore, we prove that for an Artinian algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if the subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. As applications, we get some equivalent characterizations of Auslander-Gorenstein rings and Auslander-regular rings.

Abstract:
Let $R$ be an arbitrary ring and $(-)^+=\Hom_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given.

Abstract:
We introduce relative preresolving subcategories and precoresolving subcategories of an abelian category and define homological dimensions and codimensions relative to these subcategories respectively. We study the properties of these homological dimensions and codimensions and unify some important properties possessed by some known homological dimensions. Then we apply the obtained properties to special subcategories and in particular to module categories. Finally we propose some open questions and conjectures, which are closely related to the generalized Nakayama conjecture and the strong Nakayama conjecture.

Abstract:
Let $\mathscr{A}$ be an abelian category and $\mathscr{C}$ an additive full subcategory of $\mathscr{A}$. We provide a method to construct a proper $\mathscr{C}$-resolution (resp. coproper $\mathscr{C}$-coresolution) of one term in a short exact sequence in $\mathscr{A}$ from that of the other two terms. By using these constructions, we answer affirmatively an open question on the stability of the Gorenstein category $\mathcal{G}(\mathscr{C})$ posed by Sather-Wagstaff, Sharif and White; and also prove that $\mathcal{G}(\mathscr{C})$ is closed under direct summands. In addition, we obtain some criteria for computing the $\mathscr{C}$-dimension and the $\mathcal{G}(\mathscr{C)}$-dimension of an object in $\mathscr{A}$.

Abstract:
Let $\Lambda$ be a quasi $k$-Gorenstein ring. For each $d$th syzygy module $M$ in mod $\Lambda$ (where $0 \leq d \leq k-1$), we obtain an exact sequence $0 \to B \to M \bigoplus P \to C \to 0$ in mod $\Lambda$ with the properties that it is dual exact, $P$ is projective, $C$ is a $(d+1)$st syzygy module, $B$ is a $d$th syzygy of Ext$_{\Lambda}^{d+1}(D(M), \Lambda)$ and the right projective dimension of $B^*$ is less than or equal to $d-1$. We then give some applications of such an exact sequence as follows. (1) We obtain a chain of epimorphisms concerning $M$, and by dualizing it we then get the spherical filtration of Auslander and Bridger for $M^*$. (2) We get Auslander and Bridger's Approximation Theorem for each reflexive module in mod $\Lambda ^{op}$. (3) We show that for any $0 \leq d \leq k-1$ each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation. As an immediate consequence of (3), we have that, if $\Lambda$ is a commutative noetherian ring with finite self-injective dimension, then for any non-negative integer $d$, each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation, which generalizes an Evans and Griffith's result to much more general setting.

Abstract:
Let $\Lambda$ be a left and right noetherian ring and $\mod \Lambda$ the category of finitely generated left $\Lambda$-modules. In this paper we show the following results: (1) For a positive integer $k$, the condition that the subcategory of $\mod \Lambda$ consisting of $i$-torsionfree modules coincides with the subcategory of $\mod \Lambda$ consisting of $i$-syzygy modules for any $1\leq i \leq k$ is left-right symmetric. (2) If $\Lambda$ is an Auslander ring and $N$ is in $\mod \Lambda ^{op}$ with $\grade N=k<\infty$, then $N$ is pure of grade $k$ if and only if $N$ can be embedded into a finite direct sum of copies of the $(k+1)$st term in a minimal injective resolution of $\Lambda$ as a right $\Lambda$-module. (3) Assume that both the left and right self-injective dimensions of $\Lambda$ are $k$. If $\grade {\rm Ext}_{\Lambda}^k(M, \Lambda)\geq k$ for any $M\in\mod \Lambda$ and $\grade {\rm Ext}_{\Lambda}^i(N, \Lambda)\geq i$ for any $N\in\mod \Lambda ^{op}$ and $1\leq i \leq k-1$, then the socle of the last term in a minimal injective resolution of $\Lambda$ as a right $\Lambda$-module is non-zero.

Abstract:
In this paper we give a sufficient condition of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations. We also introduce property (W$^{k}$). As an application of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations we give a connection between the finitistic dimension conjecture, the Auslander-Reiten conjecture and property (W$^{k}$).