3,531 research outputs found
On minimal realisations of dynamical structure functions
Motivated by the fact that transfer functions do not contain structural
information about networks, dynamical structure functions were introduced to
capture causal relationships between measured nodes in networks. From the
dynamical structure functions, a) we show that the actual number of hidden
states can be larger than the number of hidden states estimated from the
corresponding transfer function; b) we can obtain partial information about the
true state-space equation, which cannot in general be obtained from the
transfer function. Based on these properties, this paper proposes algorithms to
find minimal realisations for a given dynamical structure function. This helps
to estimate the minimal number of hidden states, to better understand the
complexity of the network, and to identify potential targets for new
measurements
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
The equilibrium landscape of the Heisenberg spin chain
We characterise the equilibrium landscape, the entire manifold of local
equilibrium states, of an interacting integrable quantum model. Focusing on the
isotropic Heisenberg spin chain, we describe in full generality two
complementary frameworks for addressing equilibrium ensembles: the functional
integral Thermodynamic Bethe Ansatz approach, and the lattice regularisation
transfer matrix approach. We demonstrate the equivalence between the two, and
in doing so clarify several subtle features of generic equilibrium states. In
particular we explain the breakdown of the canonical Y-system, which reflects a
hidden structure in the parametrisation of equilibrium ensembles.Comment: 31 pages, revised versio
Mathematical models for dispersive electromagnetic waves: an overview
In this work, we investigate mathematical models for electromagnetic wave
propagation in dispersive isotropic media. We emphasize the link between
physical requirements and mathematical properties of the models. A particular
attention is devoted to the notion of non-dissipativity and passivity. We
consider successively the case of so-called local media and general passive
media. The models are studied through energy techniques, spectral theory and
dispersion analysis of plane waves. For making the article self-contained, we
provide in appendix some useful mathematical background.Comment: 46 pages, 16 figure
Selection of sampling rate for digital control of aircrafts
The considerations in selecting the sample rates for digital control of aircrafts are identified and evaluated using the optimal discrete method. A high performance aircraft model which includes a bending mode and wind gusts was studied. The following factors which influence the selection of the sampling rates were identified: (1) the time and roughness response to control inputs; (2) the response to external disturbances; and (3) the sensitivity to variations of parameters. It was found that the time response to a control input and the response to external disturbances limit the selection of the sampling rate. The optimal discrete regulator, the steady state Kalman filter, and the mean response to external disturbances are calculated
Influence of branch points in the complex plane on the transmission through double quantum dots
We consider single-channel transmission through a double quantum dot system
consisting of two single dots that are connected by a wire and coupled each to
one lead. The system is described in the framework of the S-matrix theory by
using the effective Hamiltonian of the open quantum system. It consists of the
Hamiltonian of the closed system (without attached leads) and a term that
accounts for the coupling of the states via the continuum of propagating modes
in the leads. This model allows to study the physical meaning of branch points
in the complex plane. They are points of coalesced eigenvalues and separate the
two scenarios with avoided level crossings and without any crossings in the
complex plane. They influence strongly the features of transmission through
double quantum dots.Comment: 30 pages, 14 figure
Black hole determinants and quasinormal modes
We derive an expression for functional determinants in thermal spacetimes as
a product over the corresponding quasinormal modes. As simple applications we
give efficient computations of scalar determinants in thermal AdS, BTZ black
hole and de Sitter spacetimes. We emphasize the conceptual utility of our
formula for discussing `1/N' corrections to strongly coupled field theories via
the holographic correspondence.Comment: 28 pages. v2: slightly improved exposition, references adde
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