4 research outputs found
Generalized Perron--Frobenius Theorem for Nonsquare Matrices
Full text linkThe celebrated Perron--Frobenius (PF) theorem is stated for irreducible
nonnegative square matrices, and provides a simple characterization of their
eigenvectors and eigenvalues. The importance of this theorem stems from the
fact that eigenvalue problems on such matrices arise in many fields of science
and engineering, including dynamical systems theory, economics, statistics and
optimization. However, many real-life scenarios give rise to nonsquare
matrices. A natural question is whether the PF Theorem (along with its
applications) can be generalized to a nonsquare setting. Our paper provides a
generalization of the PF Theorem to nonsquare matrices. The extension can be
interpreted as representing client-server systems with additional degrees of
freedom, where each client may choose between multiple servers that can
cooperate in serving it (while potentially interfering with other clients).
This formulation is motivated by applications to power control in wireless
networks, economics and others, all of which extend known examples for the use
of the original PF Theorem.
We show that the option of cooperation between servers does not improve the
situation, in the sense that in the optimal solution no cooperation is needed,
and only one server needs to serve each client. Hence, the additional power of
having several potential servers per client translates into \emph{choosing} the
best single server and not into \emph{sharing} the load between the servers in
some way, as one might have expected.
The two main contributions of the paper are (i) a generalized PF Theorem that
characterizes the optimal solution for a non-convex nonsquare problem, and (ii)
an algorithm for finding the optimal solution in polynomial time
On problematic practice of using normalization in Self-modeling/Multivariate Curve Resolution (S/MCR)
Full text linkThe paper is briefly dealing with greater or lesser misused normalization in
self-modeling/multivariate curve resolution (S/MCR) practice. The importance of
the correct use of the ode solvers and apt kinetic illustrations are
elucidated. The new terms, external and internal normalizations are defined and
interpreted. The problem of reducibility of a matrix is touched. Improper
generalization/development of normalization-based methods are cited as
examples. The position of the extreme values of the signal contribution
function is clarified. An Executable Notebook with Matlab Live Editor was
created for algorithmic explanations and depictions
