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Inversions of statistical parameters of an acoustic signal in range-dependent environments with applications in ocean acoustic tomography
Get PDFOn the Analyticity of the Spectral Density for Semiclassical NLS
Get PDFIn a previous work, we have analyzed the semiclassical behavior of solutions to the focusing,completely integrable nonlinear Schroedinger equation, under the assumption of real analytic initial data (among others). We have provided global semiclassical asymptotics under the so-called ”finite gap” assumption. In a subsequent paper, we have justified the ”finite gap” assumption, again under several assumptions, the main assumption being that the limiting spectral density of the eigenvalues of the associated
Dirac operator has an analytic extension in the upper half-plane. In the present article, we show that this constraint is unnecessary. In fact, analyticity of the
neccessary quantities in the analysis can be recovered via the solution of a scalar Riemann-Hilbert problem
Implicitization of curves and (hyper)surfaces using predicted support
Get PDFWe reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
On cavitation in Elastodynamics
Get PDFMotivated by the works of Ball (1982) and Pericak-Spector and Spector (1988), we investigate singular solutions of the compressible nonlinear elastodynamics equations.
These singular solutions contain discontinuities in the displacement field and
can be seen as describing fracture or cavitation.
We explore a definition of singular solution via approximating sequences of smooth functions.
We use these approximating sequences to investigate the energy of such solutions, taking into account the energy needed to open a crack or hole.
In particular, we find that the existence of singular solutions and the finiteness of their energy
is strongly related to the behavior of the stress response function for infinite stretching, i.e.
the material has to display a sufficient amount of softening.
In this note we detail our findings in one space dimension
On cavitation in nonlinear elastodynamics
Get PDFMotivated by the works of Ball (1982) and Pericak-Spector and Spector (1988), we investigate singular solutions of the compressible nonlinear elastodynamics equations.
These singular solutions contain discontinuities in the displacement field and
can be seen as describing fracture or cavitation.
We explore a definition of singular solution via approximating sequences of smooth functions.
We use these approximating sequences to investigate the energy of such solutions, taking into account the energy needed to open a crack or hole.
In particular, we find that the existence of singular solutions and the finiteness of their energy
is strongly related to the behavior of the stress response function for infinite stretching, i.e.
the material has to display a sufficient amount of softening.
In this note we detail our findings in one space dimension
Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics
Get PDFWe propose a simple yet efficient method for generating abruptly autofocusing optical beams with arbitrary caustics. In addition we introduce a family of abruptly autodefocusing beams whose maximum intensity suddenly decreases by orders of magnitude right after the target. The method relies on appropriately modulating the phase of a circularly symmetric optical wavefront, such as that of a Gaussian, and subsequently on Fourier-transforming it by means of a lens. If two such beams are superimposed in a Bessel-like standing wave pattern, then a complete mirror-symmetric, with respect to the focal plane, caustic surface of revolution is formed that can be used as an optical bottle. We also show how the same method can be used to produce accelerating 1D or 2D optical beams with arbitrary convex caustics
