3,045 research outputs found
An algorithm for real and complex rational minimax approximation
Rational minimax approximation of real functions on real intervals is an
established topic, but when it comes to complex functions or domains, there
appear to be no algorithms currently in use. Such a method is introduced here,
the {\em AAA-Lawson algorithm,} available in Chebfun. The new algorithm solves
a wide range of problems on arbitrary domains in a fraction of a second of
laptop time by a procedure consisting of two steps. First, the standard AAA
algorithm is run to obtain a near-best approximation and a set of support
points for a barycentric representation of the rational approximant. Then a
"Lawson phase" of iteratively reweighted least-squares adjustment of the
barycentric coefficients is carried out to improve the approximation to
minimax
Magnetic Actuators and Suspension for Space Vibration Control
The research on microgravity vibration isolation performed at the University of Virginia is summarized. This research on microgravity vibration isolation was focused in three areas: (1) the development of new actuators for use in microgravity isolation; (2) the design of controllers for multiple-degree-of-freedom active isolation; and (3) the construction of a single-degree-of-freedom test rig with umbilicals. Described are the design and testing of a large stroke linear actuator; the conceptual design and analysis of a redundant coarse-fine six-degree-of-freedom actuator; an investigation of the control issues of active microgravity isolation; a methodology for the design of multiple-degree-of-freedom isolation control systems using modern control theory; and the design and testing of a single-degree-of-freedom test rig with umbilicals
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Passive stabilization for large space systems
The optimal tuning of multiple tuned-mass dampers for the transient vibration damping of large space structures is investigated. A multidisciplinary approach is used. Structural dynamic techniques are applied to gain physical insight into absorber/structure interaction and to optimize specific cases. Modern control theory and parameter optimization techniques are applied to the general optimization problem. A design procedure for multi-absorber multi-DOF vibration damping problems is presented. Classical dynamic models are extended to investigate the effects of absorber placement, existing structural damping, and absorber cross-coupling on the optimal design synthesis. The control design process for the general optimization problem is formulated as a linear output feedback control problem via the development of a feedback control canonical form. The techniques are applied to sample micro-g and pointing problems on the NASA dual keel space station
More on the O(n) model on random maps via nested loops: loops with bending energy
We continue our investigation of the nested loop approach to the O(n) model
on random maps, by extending it to the case where loops may visit faces of
arbitrary degree. This allows to express the partition function of the O(n)
loop model as a specialization of the multivariate generating function of maps
with controlled face degrees, where the face weights are determined by a fixed
point condition. We deduce a functional equation for the resolvent of the
model, involving some ring generating function describing the immediate
vicinity of the loops. When the ring generating function has a single pole, the
model is amenable to a full solution. Physically, such situation is realized
upon considering loops visiting triangles only and further weighting these
loops by some local bending energy. Our model interpolates between the two
previously solved cases of triangulations without bending energy and
quadrangulations with rigid loops. We analyze the phase diagram of our model in
details and derive in particular the location of its non-generic critical
points, which are in the universality classes of the dense and dilute O(n)
model coupled to 2D quantum gravity. Similar techniques are also used to solve
a twisting loop model on quadrangulations where loops are forced to make turns
within each visited square. Along the way, we revisit the problem of maps with
controlled, possibly unbounded, face degrees and give combinatorial derivations
of the one-cut lemma and of the functional equation for the resolvent.Comment: 40 pages, 9 figures, final accepted versio
Comments on the Links between su(3) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards
We examine the proposal made recently that the su(3) modular invariant
partition functions could be related to the geometry of the complex Fermat
curves. Although a number of coincidences and similarities emerge between them
and certain algebraic curves related to triangular billiards, their meaning
remains obscure. In an attempt to go beyond the su(3) case, we show that any
rational conformal field theory determines canonically a Riemann surface.Comment: 56 pages, 4 eps figures, LaTeX, uses eps
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