Let Q be a finite quiver with vertex set I and arrow set Q1, k a field, and k Q its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr(k Q) := Gr(k Q)/Fdim(k Q) of graded k Q-modules modulo those that are the sum of their finite dimensional submodules, namely QGr(k Q) ≡ ModS(Q) ≡ GrL(Q) ≡ ModL(Q◦)0 ≡ QGr(k Q (n)). Here S(Q) =...

Consider a field k of characteristic zero. Let Q = (qij) be a n× n-matrix with the entries qij ∈ k∗ or qij is an indeterminate, satisfying qijqji = qii = 1. The algebra k[Q] is generated by k and q ij , i, j = 1, . . . , n. Denote k(Q) = Fract(k[Q]) and Γ is the subgroup in k(Q) generated by k and the entries of Q. Throughout the paper R is a k-algebra with a unit. Definition 1.1. We say that R...

let $g$ be a finite group and $pi(g)$ be the set of all the prime‎ ‎divisors of $|g|$‎. ‎the prime graph of $g$ is a simple graph‎ ‎$gamma(g)$ whose vertex set is $pi(g)$ and two distinct vertices‎ ‎$p$ and $q$ are joined by an edge if and only if $g$ has an‎ ‎element of order $pq$‎, ‎and in this case we will write $psim q$‎. ‎the degree of $p$ is the number of vertices adjacent to $p$ and is‎ ...

This is a set of lecture notes for the first author’s lectures on difference equations in 2019 at Institute Advanced Study Mathematics Zhejiang University. We focus explicit computations and examples. The convergence local solutions discussed.

Let Mq(k) be the maximum length of MDS codes with parameters q, k. In this paper, the properties of Mq(k) are studied, and some new upper bounds ofMq(k) are obtained. Especially we obtain thatMq(q− 1) ≤ q+2(q ≡ 4(mod 6)), Mq(q− 2) ≤ q+1(q ≡ 4(mod 6)), Mq(k) ≤ q + k − 3 (q = 36(5s+ 1), s ∈ N and k = 6, 7).

Let n, k, α be integers, n, α > 0, p be a prime and q = pα. Consider the complete q-uniform family F(k, q) = {K ⊆ [n] : |K| ≡ k (mod q)}. We study certain inclusion matrices attached to F(k, q) over the field Fp. We show that if ` ≤ q − 1 and 2` ≤ n then rankFp I(F(k, q), (