Download Meromorphic continuation of higher-rank Eisenstein series by Garrett P. PDF

Show description

A non-empty subset P of S is said to be a Smarandache pseudo right (left) ideal (S- pseudo right (left) ideal) of the semiring S if the following conditions are true. i. ii. e. P⊂ A, A is a semifield in S. For every p ∈ P and every a∈ A; ap ∈ P (pa ∈ P). If P is simultaneously both a S-pseudo right and left ideal we say P is a Smarandache pseudo ideal (S-pseudo ideal). We define Smarandache pseudo dual ideal of S as follows: A non-empty subset P of S is said to be a Smarandache pseudo dual ideal (S-pseudo dual ideal) if the following conditions hold good.

7: Qo is a semifield and Ro is a semivector space over Qo. 8: Let Mn × n = {(aij)  aij ∈ Zo}, the set of all n × n matrices with entries from Zo. Clearly Mn × n is a semivector space over Zo. e. 0 < a1 < a2 < … < an-2 < 1). Several interesting properties about semivector spaces can be had from [81 and 122]. The main property which we wish to state about semivector spaces is the concept of basis. 9: A set of vectors (v1, v2, …, vn) in a semivector space V over a semifield S is said to be linearly dependent if there exists a non-trivial relation among them; otherwise the set is said to be linearly independent.

9: A set of vectors (v1, v2, …, vn) in a semivector space V over a semifield S is said to be linearly dependent if there exists a non-trivial relation among them; otherwise the set is said to be linearly independent. (The main difference between vector spaces and semivector spaces is that we do not have negative terms in a semifield over which semivector spaces are built). 10: Let V be a semivector space over the semifield S. For any subset A of V the set of all linear combinations of vectors in A is called the set spanned by A and we shall denote it by 〈A〉.