Programme
Sunday 16 October 2011
19:00  Informal gettogether in “Brady's Clock House Pub” (finger food will be served) 
Monday 17 October 2011
8:00  Welcome breakfast (at the Hamilton Institute) 
9:00  Opening address by R. Shorten and D. Heffernan 
9:15 
Shmuel Friedland: From nonnegative matrices to nonnegative tensors
Abstract
In this talk we will discuss a number of generalizations of results on nonnegative matrices to nonnegative tensors as: irreducibility and weak irreducibility, PerronFrobenius theorem, CollatzWielandt characterization, Kingman's inequality, KarlinOst and Friedland theorems, tropical spectral radius, diagonal scaling, FriedlandKarlin inequality, nonnegative multilinear forms.

10:00 
Raphael Loewy: Maximal exponents of polyhedral cones
Abstract
Let K be a proper (i.e., closed, pointed, full and convex) cone in R^{n}. We consider A∈R^{n×n} which is Kprimitive, that is, there exists a positive integer l such that A^{l}x ∈ int K for every 0≠x∈K. The smallest such l is called the exponent of A, denoted by γ(A).
For a polyhedral cone K, the maximum value of γ(A), taken over all Kprimitive matrices A, is denoted by γ(K). Our main result is that for any positive integers m,n, 3 ≤ n ≤ m, the maximum value of γ(K), as K runs through all ndimensional polyhedral cones with m extreme rays, equals ( n  1 )( m  1 ) + ½( 1 + (1)^{(n1)m} ). We will consider various uniqueness issues related to the main result as well as its connections to known results. This talk is based on a joint work with Micha Perles and BitShun Tam. 
10:45  Coffee 
11:15 
Thomas J. Laffey: Some relationships between formal power series and nonnegative matrices
Abstract
Let σ = (λ_{1},…,λ_{n}) be a list of complex numbers which we aim to realize constructively as
the spectrum of a nonnegative matrix. Most constructions available in the literature rely on building matrices related to companion
matrices from the polynomial f(x) = (x–λ_{1})…(x–λ_{n}).
Kim, Ormes and Roush (JAMS 2000) showed how certain formal power series related to f(x), which have all coefficients,
other than the leading one, negative, can be used in finding constructions over the semiring of polynomials with nonnegative coefficients,
while, in joint work, Šmigoc and this author (ELA 17 (2008) 333342, LAMA 58 (2010), 10531059) have used polynomials having all their
nonleading coefficients negative, to find realizations when σ has not more than two entries with positive real parts. Beginning with
the observation that if λ_{1},…,λ_{n} are all positive, then the Taylor expansion of the nth root
of F(t) = (1–λ_{1}t)…(1–λ_{n}t) about t=0 has all its nonleading
coefficients negative, we present a number of results on the negativity of the coefficients of power series and their applications to
nonnegative matrices.

12:00 
Patrizio Colaneri: Essentially Negative News About Positive Systems
Abstract
In this paper the discretisation of switched and nonswitched linear positive systems using
Padé approximations is considered. Padé approximations to the matrix exponential
are sometimes used by control engineers for discretising continuous time systems and
for control system design. We observe that this method of approximation is not suited
for the discretisation of positive dynamic systems, for two key reasons. First, certain
types of Lyapunov stability are not, in general, preserved. Secondly, and more seriously,
positivity need not be preserved, even when stability is. Finally we present an alternative
approximation to the matrix exponential which preserves positivity, and linear and
quadratic stability.
This talk is based on joint work with Steve Kirkland, Annalisa Zappavigna & Robert Shorten 
12:45  Lunch 
14:00 
KarlHeinz Förster: On the Block Numerical Range of Operators in Banach Spaces
Abstract
In this talk following topics will be discussed:

14:45 
Helena Šmigoc: The Symmetric Nonnegative Inverse Eigenvalue Problem
Abstract
The question of which lists of complex numbers are the spectra of nonnegative matrices, is known as the nonnegative inverse eigenvalue
problem, and the same question posed for symmetric nonnegative matrices is called the symmetric nonnegative inverse eigenvalue problem.
In the talk we will present an overview of some recent results on the symmetric nonnegative inverse eigenvalue problem.
Joint work with T. J. Laffey. 
15:30  Coffee 
16:00 
Steve Kirkland: Load balancing for Markov chains
Abstract
A square matrix T is called stochastic if its entries are nonnegative and its row sums are all equal to one. Stochastic matrices are the centrepiece of the theory of discretetime, time homogenous Markov chains on a finite state space. If some power of the stochastic matrix T has all positive entries, then there is a unique left eigenvector for T, known as the stationary distribution, to which the iterates of the Markov chain converge, regardless of what the initial distribution for the chain is. Thus, in this setting, the stationary distribution can be thought of as giving the probability that the chain is in a particular state over the long run.
In many applications, the stochastic matrix under consideration is equipped with an underlying combinatorial structure, which can be recorded in a directed graph. Given a stochastic matrix T, how are the entries in the stationary distribution influenced by the structure of the directed graph associated with T? In this talk we investigate a question of that type by finding the minimum value of the maximum entry in the stationary distribution for T, as T ranges over the set of stochastic matrices with a given directed graph. The solution involves techniques from matrix theory, graph theory, and nonlinear programming. 
16:45 
Abraham Berman: Diagonal Stability and Completely Positive Matrices
Abstract
In this paper a general notion of common diagonal Lyapunov matrix is formulated for a collection of n×n matrices A_{1},…,A_{s} and polyhedral cones k_{1},…,k_{s} in R^{n}. Necessary and sufficient conditions are derived for the existence of a common diagonal Lyapunov matrix in this setting.
This talk is based on joint work with Christopher King & Robert Shorten 
19:30 
Speakers' Banquet in “The Avenue” Any nonspeakers please contact if you would like to attend, the cost is 30 EUR p.p. 