1,197 research outputs found
Generalizations of Bounds on the Index of Convergence to Weighted Digraphs
We study sequences of optimal walks of a growing length, in weighted
digraphs, or equivalently, sequences of entries of max-algebraic matrix powers
with growing exponents. It is known that these sequences are eventually
periodic when the digraphs are strongly connected. The transient of such
periodicity depends, in general, both on the size of digraph and on the
magnitude of the weights. In this paper, we show that some bounds on the
indices of periodicity of (unweighted) digraphs, such as the bounds of
Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply
to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure
Generalized exponents of non-primitive graphs
AbstractThe exponent of a primitive digraph is the smallest integer k such that for each ordered pair of (not necessarily distinct) vertices x and y there is a walk of length k from x to y. As a generalization of exponent, Brualdi and Liu (Linear Algebra Appl. 14 (1990) 483–499) introduced three types of generalized exponents for primitive digraphs in 1990. In this paper we extend their definitions of generalized exponents from primitive digraphs to general digraphs which are not necessarily primitive. We give necessary and sufficient conditions for the finiteness of these generalized exponents for graphs (undirected, corresponding to symmetric digraphs) and completely determine the largest finite values and the exponent sets of generalized exponents for the class of non-primitive graphs of order n, the class of connected bipartite graphs of order n and the class of trees of order n
Bounds on the exponent of primitivity which depend on the spectrum and the minimal polynomial
AbstractSuppose A is an n × n nonnegative primitive matrix whose minimal polynomial has degree m. We conjecture that the well-known bound on the exponent of primitivity (n − 1)2 + 1, due to Wielandt, can be replaced by (m − 1)2 + 1. The only case for which we cannot prove the conjecture is when m ⩾ 5, the number of distinct eigenvalues of A is m − 1 or m, and the directed graph of A has no circuits of length shorter than m − 1, but at least one of its vertices lies on a circuit of length not shorter than m. We show that m(m − 1) is always a bound on the exponent, this being an improvement on Wielandt's bound when m < n. For the case in which A is also symmetric, the bound which we obtain is 2(m − 1). To obtain our results we prove a lemma which shows that for a (general) nonnegative matrix, the number of its distinct eigenvalues is an upper bound on the length of the shortest circuit in its directed graph
Uniform Sampling for Matrix Approximation
Random sampling has become a critical tool in solving massive matrix
problems. For linear regression, a small, manageable set of data rows can be
randomly selected to approximate a tall, skinny data matrix, improving
processing time significantly. For theoretical performance guarantees, each row
must be sampled with probability proportional to its statistical leverage
score. Unfortunately, leverage scores are difficult to compute.
A simple alternative is to sample rows uniformly at random. While this often
works, uniform sampling will eliminate critical row information for many
natural instances. We take a fresh look at uniform sampling by examining what
information it does preserve. Specifically, we show that uniform sampling
yields a matrix that, in some sense, well approximates a large fraction of the
original. While this weak form of approximation is not enough for solving
linear regression directly, it is enough to compute a better approximation.
This observation leads to simple iterative row sampling algorithms for matrix
approximation that run in input-sparsity time and preserve row structure and
sparsity at all intermediate steps. In addition to an improved understanding of
uniform sampling, our main proof introduces a structural result of independent
interest: we show that every matrix can be made to have low coherence by
reweighting a small subset of its rows
Recovering modular forms and representations from tensor and symmetric powers
We consider the problem of determining the relationship between two
representations knowing that some tensor or symmetric power of the original
represetations coincide. Combined with refinements of strong multiplicity one,
we show that if the characters of some tensor or symmetric powers of two
absolutely irreducible -adic representation with the algebraic envelope of
the image being connected, agree at the Frobenius elements corresponding to a
set of places of positive upper density, then the representations are twists of
each other by a finite order character.Comment: 18 pages; this is a revised version of a paper submitted to the old
Number Theory archive as ANT-035
The local exponent sets of primitive digraphs
AbstractLet D=(V,E) be a primitive digraph. The local exponent of D at a vertex u∈V, denoted by expD(u), is defined to be the least integer k such that there is a directed walk of length k from u to v for each v∈V. Let V={1,2,…, n}. The vertices of V can be ordered so that expD(1)⩽expD(2)⩽⋯⩽expD(n)=γ(D). We define the kth local exponent set En(k):={expD(k)∣D∈PDn}, where PDn is the set of all primitive digraphs of order n. It is known that En(n)={γ(D)∣D∈PDn} has been completely settled by K. Zhang [Linear Algebra Appl. 96 (1987) 102–108]. In 1998, En(1) was characterized by J. Shen and S. Neufeld [Linear Algebra Appl. 268 (1998) 117–129]. In this paper, we describe En(k) for all n,k with 2⩽k⩽n−1. So the problem of local exponent sets of primitive digraphs is completely solved
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