in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

in this paper we introduce the concept of dirac structures on (hermitian) modules and vectorbundles and deduce some of their properties. among other things we prove that there is a one to onecorrespondence between the set of all dirac structures on a (hermitian) module and the group of allautomorphisms of the module. this correspondence enables us to represent dirac structures on (hermitian)mod...

Let  be a commutative ring and  be a unitary  module. We define a semi prime sub module of a module and consider various properties of it. Also we define semi-radical of a sub module of a module and give a number of its properties. We define modules which satisfy the semi-radical formula and present the existence of such a module.

We study the semi-infinite or BRST cohomology of affine Lie algebras in detail. This cohomology is relevant in the BRST approach to gauged WZNW models. Our main result is to prove necessary and sufficient conditions on ghost numbers and weights for non-trivial elements in the cohomology. In particular we prove the existence of an infinite sequence of elements in the cohomology for non-zero ghos...

2 Highest weight modules 4 2.1 Kac-Moody Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Integral Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Category of highest weight modules . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Enright functor for non-integral weights . . . . . . . . . . . . . . . . . . . . 11 2.5 Embeddings of Verma m...

Most learning systems applied to problem solving have the goal of learning knowledge for solving problems more eeciently. However, very few systems have focused on the acquisition of planning operator descriptions. In this paper, we present a system that learns operator deenitions from the interaction with a completely unknown environment. In order to achieve better learning convergence, severa...

Abstract We construct a $(\mathfrak {gl}_2, B(\mathbb {Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms the local cohomology at $0$ of sheaf on $\mathbb {P}^1$ , landing in compactly supported completed {C}_p$ -cohomology curve. The group is highest-weight Verma module, non-trivial vector for any form infinitesimal weight not equal to $1$ . For classical $k...

Most learning systems applied to problem solving have the goal of learning knowledge for solving problems more eeciently. However, very few systems have focused on the acquisition of planning operator descriptions. In this paper, we present a system that learns operator deenitions from the interaction with a completely unknown environment. In order to achieve better learning convergence, severa...