4,586 research outputs found
Complexity of Model Testing for Dynamical Systems with Toric Steady States
In this paper we investigate the complexity of model selection and model
testing for dynamical systems with toric steady states. Such systems frequently
arise in the study of chemical reaction networks. We do this by formulating
these tasks as a constrained optimization problem in Euclidean space. This
optimization problem is known as a Euclidean distance problem; the complexity
of solving this problem is measured by an invariant called the Euclidean
distance (ED) degree. We determine closed-form expressions for the ED degree of
the steady states of several families of chemical reaction networks with toric
steady states and arbitrarily many reactions. To illustrate the utility of this
work we show how the ED degree can be used as a tool for estimating the
computational cost of solving the model testing and model selection problems
Qudits of composite dimension, mutually unbiased bases and projective ring geometry
The Pauli operators attached to a composite qudit in dimension may
be mapped to the vectors of the symplectic module
( the modular ring). As a result, perpendicular vectors
correspond to commuting operators, a free cyclic submodule to a maximal
commuting set, and disjoint such sets to mutually unbiased bases. For
dimensions , and 18, the fine structure and the incidence
between maximal commuting sets is found to reproduce the projective line over
the rings , , ,
and ,
respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A
Mathematical and Theoretical (2008) accepte
On the average condition number of tensor rank decompositions
We compute the expected value of powers of the geometric condition number of
random tensor rank decompositions. It is shown in particular that the expected
value of the condition number of tensors with a random
rank- decomposition, given by factor matrices with independent and
identically distributed standard normal entries, is infinite. This entails that
it is expected and probable that such a rank- decomposition is sensitive to
perturbations of the tensor. Moreover, it provides concrete further evidence
that tensor decomposition can be a challenging problem, also from the numerical
point of view. On the other hand, we provide strong theoretical and empirical
evidence that tensors of size with all have a finite average condition number. This suggests there exists a gap
in the expected sensitivity of tensors between those of format and other order-3 tensors. For establishing these results, we show
that a natural weighted distance from a tensor rank decomposition to the locus
of ill-posed decompositions with an infinite geometric condition number is
bounded from below by the inverse of this condition number. That is, we prove
one inequality towards a so-called condition number theorem for the tensor rank
decomposition
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