13,100 research outputs found
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
An Algebraic Criterion for the Ultraviolet Finiteness of Quantum Field Theories
An algebraic criterion for the vanishing of the beta function for
renormalizable quantum field theories is presented. Use is made of the descent
equations following from the Wess-Zumino consistency condition. In some cases,
these equations relate the fully quantized action to a local gauge invariant
polynomial. The vanishing of the anomalous dimension of this polynomial enables
us to establish a nonrenormalization theorem for the beta function ,
stating that if the one-loop order contribution vanishes, then will
vanish to all orders of perturbation theory. As a by-product, the special case
in which is only of one-loop order, without further corrections, is
also covered. The examples of the N=2,4 supersymmetric Yang-Mills theories are
worked out in detail.Comment: 1+32 pages, LaTeX2e, typos correcte
Efficient algorithms for deciding the type of growth of products of integer matrices
For a given finite set of matrices with nonnegative integer entries
we study the growth of We show how to determine in polynomial time whether the growth with
is bounded, polynomial, or exponential, and we characterize precisely all
possible behaviors.Comment: 20 pages, 4 figures, submitted to LA
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
On the complexity of computing the capacity of codes that avoid forbidden difference patterns
We consider questions related to the computation of the capacity of codes
that avoid forbidden difference patterns. The maximal number of -bit
sequences whose pairwise differences do not contain some given forbidden
difference patterns increases exponentially with . The exponent is the
capacity of the forbidden patterns, which is given by the logarithm of the
joint spectral radius of a set of matrices constructed from the forbidden
difference patterns. We provide a new family of bounds that allows for the
approximation, in exponential time, of the capacity with arbitrary high degree
of accuracy. We also provide a polynomial time algorithm for the problem of
determining if the capacity of a set is positive, but we prove that the same
problem becomes NP-hard when the sets of forbidden patterns are defined over an
extended set of symbols. Finally, we prove the existence of extremal norms for
the sets of matrices arising in the capacity computation. This result makes it
possible to apply a specific (even though non polynomial) approximation
algorithm. We illustrate this fact by computing exactly the capacity of codes
that were only known approximately.Comment: 7 pages. Submitted to IEEE Trans. on Information Theor
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