73,603 research outputs found
Minimal realizations of linear systems: The "shortest basis" approach
Given a controllable discrete-time linear system C, a shortest basis for C is
a set of linearly independent generators for C with the least possible lengths.
A basis B is a shortest basis if and only if it has the predictable span
property (i.e., has the predictable delay and degree properties, and is
non-catastrophic), or alternatively if and only if it has the subsystem basis
property (for any interval J, the generators in B whose span is in J is a basis
for the subsystem C_J). The dimensions of the minimal state spaces and minimal
transition spaces of C are simply the numbers of generators in a shortest basis
B that are active at any given state or symbol time, respectively. A minimal
linear realization for C in controller canonical form follows directly from a
shortest basis for C, and a minimal linear realization for C in observer
canonical form follows directly from a shortest basis for the orthogonal system
C^\perp. This approach seems conceptually simpler than that of classical
minimal realization theory.Comment: 20 pages. Final version, to appear in special issue of IEEE
Transactions on Information Theory on "Facets of coding theory: From
algorithms to networks," dedicated to Ralf Koette
A general realization theorem for matrix-valued Herglotz-Nevanlinna functions
New special types of stationary conservative impedance and scattering
systems, the so-called non-canonical systems, involving triplets of Hilbert
spaces and projection operators, are considered. It is established that every
matrix-valued Herglotz-Nevanlinna function of the form
V(z)=Q+Lz+\int_{\dR}(\frac{1}{t-z}-\frac{t}{1+t^2})d\Sigma(t) can be realized
as a transfer function of such a new type of conservative impedance system. In
this case it is shown that the realization can be chosen such that the main and
the projection operators of the realizing system satisfy a certain
commutativity condition if and only if L=0. It is also shown that with
an additional condition (namely, is invertible or L=0), can be realized as
a linear fractional transformation of the transfer function of a non-canonical
scattering -system. In particular, this means that every scalar
Herglotz-Nevanlinna function can be realized in the above sense.
Moreover, the classical Livsic systems (Brodskii-Livsic operator
colligations) can be derived from -systems as a special case when
and the spectral measure is compactly supported. The realization
theorems proved in this paper are strongly connected with, and complement the
recent results by Ball and Staffans.Comment: 28 page
Differential Geometry of Quantum States, Observables and Evolution
The geometrical description of Quantum Mechanics is reviewed and proposed as
an alternative picture to the standard ones. The basic notions of observables,
states, evolution and composition of systems are analised from this
perspective, the relevant geometrical structures and their associated algebraic
properties are highlighted, and the Qubit example is thoroughly discussed.Comment: 20 pages, comments are welcome
Localization and Pattern Formation in Quantum Physics. I. Phenomena of Localization
In these two related parts we present a set of methods, analytical and
numerical, which can illuminate the behaviour of quantum system, especially in
the complex systems. The key points demonstrating advantages of this approach
are: (i) effects of localization of possible quantum states, more proper than
"gaussian-like states"; (ii) effects of non-perturbative multiscales which
cannot be calculated by means of perturbation approaches; (iii) effects of
formation of complex quantum patterns from localized modes or classification
and possible control of the full zoo of quantum states, including (meta) stable
localized patterns (waveletons). We'll consider calculations of Wigner
functions as the solution of Wigner-Moyal-von Neumann equation(s) corresponding
to polynomial Hamiltonians. Modeling demonstrates the appearance of (meta)
stable patterns generated by high-localized (coherent) structures or
entangled/chaotic behaviour. We can control the type of behaviour on the level
of reduced algebraical variational system. At the end we presented the
qualitative definition of the Quantum Objects in comparison with their
Classical Counterparts, which natural domain of definition is the category of
multiscale/multiresolution decompositions according to the action of
internal/hidden symmetry of the proper realization of scales of functional
spaces. It gives rational natural explanation of such pure quantum effects as
``self-interaction''(self-interference) and instantaneous quantum interaction.Comment: LaTeX2e, spie.cls, 13 pages, 15 figures, submitted to Proc. of SPIE
Meeting, The Nature of Light: What is a Photon? Optics & Photonics, SP200,
San Diego, CA, July-August, 200
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