11 research outputs found
Counting Semilinear Endomorphisms Over Finite Fields
For a finite field k and a triple of integers g \ge r \ge s \ge 0, we count
the number of semilinear endomorphisms of a g-dimensional k-vector space which
have rank r and stable rank s. Such endomorphisms show up naturally in the
classification of finite flat group schemes of p-power order over k which are
killed by p and have p-rank s, via Dieudonne theory
The Wiener index and the Wiener Complexity of the zero-divisor graph of a ring
We calculate the Wiener index of the zero-divisor graph of a finite
semisimple ring. We also calculate the Wiener complexity of the zero-divisor
graph of a finite simple ring and find an upper bound for the Wiener complexity
in the semisimple case
The probability that an operator is nilpotent
Choose a random linear operator on a vector space of finite cardinality N:
then the probability that it is nilpotent is 1/N. This is a linear analogue of
the fact that for a random self-map of a set of cardinality N, the probability
that some iterate is constant is 1/N. The first result is due to Fine, Herstein
and Hall, and the second is essentially Cayley's tree formula. We give a new
proof of the result on nilpotents, analogous to Joyal's beautiful proof of
Cayley's formula. It uses only general linear algebra and avoids calculation
entirely.Comment: 5 pages, title change
Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers
Using the structure of Singer cycles in general linear groups, we prove that
a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special
case, and outline a plausible approach to prove it in the general case. This
conjecture is about the number of primitive -LFSRs of a given order
over a finite field, and it generalizes a known formula for the number of
primitive LFSRs, which, in turn, is the number of primitive polynomials of a
given degree over a finite field. Moreover, this conjecture is intimately
related to an open question of Niederreiter (1995) on the enumeration of
splitting subspaces of a given dimension.Comment: Version 2 with some minor changes; to appear in Designs, Codes and
Cryptography
Counting symmetric nilpotent matrices
We determine the number of nilpotent matrices of order n over Fq that are self-adjoint for a given nondegenerate symmetric bilinear form, and in particular find the number of symmetric nilpotent matrices. Keywords: Nilpotent; symmetric; matrix; endomorphism; enumeratio
A q-analogue of spanning trees : nilpotent transformations over finite fields
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 67).The main result of this work is a q-analogue relationship between nilpotent transformations and spanning trees. For example, nilpotent endomorphisms on an n-dimensional vector space over Fq is a q-analogue of rooted spanning trees of the complete graph Kn. This relationship is based on two similar bijective proofs to calculate the number of spanning trees and nilpotent transformations, respectively. We also discuss more details about this bijection in the cases of complete graphs, complete bipartite graphs, and cycles. It gives some refinements of the q-analogue relationship. As a corollary, we find the total number of nilpotent transformations with some restrictions on Jordan block sizes.by Jingbin Yin.Ph.D