Extension algebras of standard modules

Let $A$ be a finite dimensional $k$-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E= \text{Ext}_A^{\ast} (\Delta, \Delta)$ of standard modules, characterize the stratification property of $E$ for $\leqslant$ and $\leqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).Comment: The final version accepted by Comm. Algebr

A generalized Koszul theory and its application

Let $A$ be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that $A_0$ is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described.Comment: The revised version accepted by Tran. AM

A NEW FUNCTION AND ITS MEAN VALUE

The main purpose of this paper is using the elementary method to study the mean value properties of a new function for n, and give a sharp asymptotic formula for it