62,691 research outputs found
State space formulas for stable rational matrix solutions of a Leech problem
Given stable rational matrix functions and , a procedure is presented
to compute a stable rational matrix solution to the Leech problem
associated with and , that is, and . The solution is given in the form of a state space
realization, where the matrices involved in this realization are computed from
state space realizations of the data functions and .Comment: 25 page
An efficient algorithm for positive realizations
We observe that successive applications of known results from the theory of
positive systems lead to an {\it efficient general algorithm} for positive
realizations of transfer functions. We give two examples to illustrate the
algorithm, one of which complements an earlier result of \cite{large}. Finally,
we improve a lower-bound of \cite{mn2} to indicate that the algorithm is indeed
efficient in general
Order bound for the realization of a combination of positive filters
In a problem on the realization of digital ¯lters, initiated by Gersho and Gopinath [8], we extend and
complete a remarkable result of Benvenuti, Farina and Anderson [4] on decomposing the transfer function
t(z) of an arbitrary linear, asymptotically stable, discrete, time-invariant SISO system as a di®erence
t(z) = t1(z) ¡ t2(z) of two positive, asymptotically stable linear systems. We give an easy-to-compute
algorithm to handle the general problem, in particular, also the case of transfer functions t(z) with multiple
poles, which was left open in [4]. One of the appearing positive, asymptotically stable systems is always
1-dimensional, while the other has dimension depending on the order and, in the case of nonreal poles,
also on the location of the poles of t(z). The appearing dimension is seen to be minimal in some cases
and it can always be calculated before carrying out the realization
A new graph parameter related to bounded rank positive semidefinite matrix completions
The Gram dimension \gd(G) of a graph is the smallest integer
such that any partial real symmetric matrix, whose entries are specified on the
diagonal and at the off-diagonal positions corresponding to edges of , can
be completed to a positive semidefinite matrix of rank at most (assuming a
positive semidefinite completion exists). For any fixed the class of graphs
satisfying \gd(G) \le k is minor closed, hence it can characterized by a
finite list of forbidden minors. We show that the only minimal forbidden minor
is for and that there are two minimal forbidden minors:
and for . We also show some close connections to
Euclidean realizations of graphs and to the graph parameter of
\cite{H03}. In particular, our characterization of the graphs with \gd(G)\le
4 implies the forbidden minor characterization of the 3-realizable graphs of
Belk and Connelly \cite{Belk,BC} and of the graphs with of van
der Holst \cite{H03}.Comment: 31 pages, 6 Figures. arXiv admin note: substantial text overlap with
arXiv:1112.596
Teichmüller spaces and HR structures for hyperbolic surface dynamics
We construct a Teichmüller space for the C^{1+}-conjugacy classes of hyperbolic dynamical systems on surfaces. After introducing the notion of an HR structure which associates an affine structure with each of the stable and unstable laminations, we show that there is a one-to-one correspondence between these HR structures and the C^{1+}-conjugacy classes. As part of the proof we construct a canonical representative dynamical system for each HR structure. This has the smoothest holonomies of any representative of the corresponding C^{1+}-conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmüller space
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