460 research outputs found
On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation
For an integer , let be the partition of the unit
interval into equal subintervals, and let be the class
of piecewise linear maps on with constant slope on each element of
. We investigate the effect on mixing properties when is composed with the interval exchange map given by a
permutation interchanging the subintervals of
. This extends the work in a previous paper [N.P. Byott, M.
Holland and Y. Zhang, DCDS, {\bf 33}, (2013) 3365--3390], where we considered
only the "stretch-and-fold" map .Comment: 27 pages 6 figure
On complex power nonnegative matrices
Power nonnegative matrices are defined as complex matrices having at least
one nonnegative integer power. We exploit the possibility of deriving a Perron
Frobenius-like theory for these matrices, obtaining three main results and
drawing several consequences. We study, in particular, the relationships with
the set of matrices having eventually nonnegative powers, the inverse of M-type
matrices and the set of matrices whose columns (rows) sum up to one
On Primitivity of Sets of Matrices
A nonnegative matrix is called primitive if is positive for some
integer . A generalization of this concept to finite sets of matrices is
as follows: a set of matrices is
primitive if is positive for some indices
. The concept of primitive sets of matrices comes up in a
number of problems within the study of discrete-time switched systems. In this
paper, we analyze the computational complexity of deciding if a given set of
matrices is primitive and we derive bounds on the length of the shortest
positive product.
We show that while primitivity is algorithmically decidable, unless it
is not possible to decide primitivity of a matrix set in polynomial time.
Moreover, we show that the length of the shortest positive sequence can be
superpolynomial in the dimension of the matrices. On the other hand, defining
to be the set of matrices with no zero rows or columns, we give
a simple combinatorial proof of a previously-known characterization of
primitivity for matrices in which can be tested in polynomial
time. This latter observation is related to the well-known 1964 conjecture of
Cerny on synchronizing automata; in fact, any bound on the minimal length of a
synchronizing word for synchronizing automata immediately translates into a
bound on the length of the shortest positive product of a primitive set of
matrices in . In particular, any primitive set of
matrices in has a positive product of length
Computable convergence rate bound for ratio consensus algorithms
The objective of the paper is to establish a computable upper bound on the
almost sure convergence rate for a class of ratio consensus algorithms. Our
result extends the works of Iutzeler et al. (2013) on similar bounds that have
been obtained in a more restrictive setup with limited conclusions. It also
complements the results of Gerencs\'er and Gerencs\'er (2021) that identified
the exact convergence rate which is however not computable in general
Recommended from our members
Copositivity and Complete Positivity
A real matrix is called copositive if holds for all . A matrix is called completely positive if it can be factorized as , where is an entrywise nonnegative matrix. The concept of copositivity can be traced back to Theodore Motzkin in 1952, and that of complete positivity to Marshal Hall Jr. in 1958. The two classes are related, and both have received considerable attention in the linear algebra community and in the last two decades also in the mathematical optimization community. These matrix classes have important applications in various fields, in which they arise naturally, including mathematical modeling, optimization, dynamical systems and statistics. More applications constantly arise.
The workshop brought together people working in various disciplines related to copositivity and complete positivity, in order to discuss these concepts from different viewpoints and to join forces to better understand these difficult but fascinating classes of matrices
Flow networks: A characterization of geophysical fluid transport
We represent transport between different regions of a fluid domain by flow
networks, constructed from the discrete representation of the Perron-Frobenius
or transfer operator associated to the fluid advection dynamics. The procedure
is useful to analyze fluid dynamics in geophysical contexts, as illustrated by
the construction of a flow network associated to the surface circulation in the
Mediterranean sea. We use network-theory tools to analyze the flow network and
gain insights into transport processes. In particular we quantitatively relate
dispersion and mixing characteristics, classically quantified by Lyapunov
exponents, to the degree of the network nodes. A family of network entropies is
defined from the network adjacency matrix, and related to the statistics of
stretching in the fluid, in particular to the Lyapunov exponent field. Finally
we use a network community detection algorithm, Infomap, to partition the
Mediterranean network into coherent regions, i.e. areas internally well mixed,
but with little fluid interchange between them.Comment: 16 pages, 15 figures. v2: published versio
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
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