2,379,419 research outputs found
Chebyshev constants for the unit circle
It is proven that for any system of n points z_1, ..., z_n on the (complex)
unit circle, there exists another point z of norm 1, such that
Equality holds iff the point system is a
rotated copy of the nth unit roots.
Two proofs are presented: one uses a characterisation of equioscillating
rational functions, while the other is based on Bernstein's inequality.Comment: 11 page
Nonlinear behaviors of first and second order complex digital filters with two’s complement arithmetic
For first order complex digital filters with two’s complement arithmetic, it is proved in this paper that overflow does not occur at the steady state if the eigenvalues of the system matrix are inside or on the unit circle. However, if the eigenvalues of the system matrix are outside the unit circle, chaotic behaviors would occur. For both cases, a limit cycle behavior does not occur. For second order complex digital filters with two’s complement arithmetic, if all eigenvalues are on the unit circle, then there are two ellipses centered at the origin of the phase portraits when overflow does not occur. When limit cycle occurs, the number of ellipses exhibited on the phase portraits is no more than two times the periodicity of the symbolic sequences. If the symbolic sequences are aperiodic, some state variables may exhibit fractal behaviors, at the same time, irregular chaotic behaviors may occur in other phase variables
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Transferring Unit Cell Based Tissue Scaffold Design to Solid Freeform Fabrication
Designing for the freeform fabrication of heterogeneous tissue scaffold is always a challenge in
Computer Aided Tissue Engineering. The difficulties stem from two major sources: 1)
limitations in current CAD systems to assembly unit cells as building blocks to form complex
tissue scaffolds, and 2) the inability to generate tool paths for freeform fabrication of unit cell
assemblies. To overcome these difficulties, we have developed an abstract model based on
skeletal representation and associated computational methods to assemble unit cells into an
optimized structure. Additionally we have developed a process planning technique based on
internal architecture pattern of unit cells to generate tool paths for freeform fabrication of tissue
scaffold. By modifying our optimization process, we are able to transfer an optimized design to
our fabrication system via our process planning technique.Mechanical Engineerin
Anomalous Light Scattering by Topological -symmetric Particle Arrays
Robust topological edge modes may evolve into complex-frequency modes when a
physical system becomes non-Hermitian. We show that, while having negligible
forward optical extinction cross section, a conjugate pair of such complex
topological edge modes in a non-Hermitian -symmetric system can
give rise to an anomalous sideway scattering when they are simultaneously
excited by a plane wave. We propose a realization of such scattering state in a
linear array of subwavelength resonators coated with gain media. The prediction
is based on an analytical two-band model and verified by rigorous numerical
simulation using multiple-multipole scattering theory. The result suggests an
extreme situation where leakage of classical information is unnoticeable to the
transmitter and the receiver when such a -symmetric unit is
inserted into the communication channel.Comment: 16 pages, 8 figure
Defect states emerging from a non-Hermitian flat band of photonic zero modes
We show the existence of a flat band consisting of photonic zero modes in a
gain and loss modulated lattice system, as a result of the underlying
non-Hermitian particle-hole symmetry. This general finding explains the
previous observation in parity-time symmetric systems where non-Hermitian
particle-hole symmetry is hidden. We further discuss the defect states in these
systems, whose emergence can be viewed as an unconventional alignment of a
pseudo-spin under the influence of a complex-valued pseudo-magnetic field.
These defect states also behave as a chain with two types of links, one rigid
in a unit cell and one soft between unit cells, as the defect states become
increasingly localized with the gain and loss strength. A realistic photonic
design is presented based on coupled InP/InGaAsP waveguides, and we also extend
the discussion to two- and three-dimensional lattices.Comment: 10 pages, 12 figure
Fast recursive filters for simulating nonlinear dynamic systems
A fast and accurate computational scheme for simulating nonlinear dynamic
systems is presented. The scheme assumes that the system can be represented by
a combination of components of only two different types: first-order low-pass
filters and static nonlinearities. The parameters of these filters and
nonlinearities may depend on system variables, and the topology of the system
may be complex, including feedback. Several examples taken from neuroscience
are given: phototransduction, photopigment bleaching, and spike generation
according to the Hodgkin-Huxley equations. The scheme uses two slightly
different forms of autoregressive filters, with an implicit delay of zero for
feedforward control and an implicit delay of half a sample distance for
feedback control. On a fairly complex model of the macaque retinal horizontal
cell it computes, for a given level of accuracy, 1-2 orders of magnitude faster
than 4th-order Runge-Kutta. The computational scheme has minimal memory
requirements, and is also suited for computation on a stream processor, such as
a GPU (Graphical Processing Unit).Comment: 20 pages, 8 figures, 1 table. A comparison with 4th-order Runge-Kutta
integration shows that the new algorithm is 1-2 orders of magnitude faster.
The paper is in press now at Neural Computatio
The quantum group, Harper equation and the structure of Bloch eigenstates on a honeycomb lattice
The tight-binding model of quantum particles on a honeycomb lattice is
investigated in the presence of homogeneous magnetic field. Provided the
magnetic flux per unit hexagon is rational of the elementary flux, the
one-particle Hamiltonian is expressed in terms of the generators of the quantum
group . Employing the functional representation of the quantum group
the Harper equation is rewritten as a systems of two coupled
functional equations in the complex plane. For the special values of
quasi-momentum the entangled system admits solutions in terms of polynomials.
The system is shown to exhibit certain symmetry allowing to resolve the
entanglement, and basic single equation determining the eigenvalues and
eigenstates (polynomials) is obtained. Equations specifying locations of the
roots of polynomials in the complex plane are found. Employing numerical
analysis the roots of polynomials corresponding to different eigenstates are
solved out and the diagrams exhibiting the ordered structure of one-particle
eigenstates are depicted.Comment: 11 pages, 4 figure
Model based methodology development for energy recovery in ash heat exchange systems
Flash tank evaporation combined with a condensing heat exchanger can be used when heat exchange is required between two streams and where at least one of these streams is difficult to handle (in terms of solid particles content, viscosity, pH, consistency etc.). To increase the efficiency of heat exchange, a cascade of these units in series can be used. Heat transfer relationships in such a cascade are very complex due to their interconnectivity, thus the impact of any changes proposed is difficult to predict. In this report, a mathematical model of a single unit ash tank evaporator combined with a condensing heat exchanger unit is proposed. This model is then developed for a chain of the units. The purpose of this model is to allow an accurate evaluation of the effect and result of an alteration to the system. The resulting model
is applied to the RUSAL Aughinish Alumina digester area
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