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  • Bias reduction  (1)
  • Key words. Capillary-gravity waves, third harmonic resonance, multiple scales, evolution equation.  (1)
  • Key words. Capillary-gravity, mode interaction, O(2)-invariance, bifurcation, Lyapunov-Schmidt, universal unfolding.  (1)
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Keywords
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Years
  • 1
    Electronic Resource
    Electronic Resource
    Variable location and scale kernel density estimation (1994)
    Springer
    Annals of the Institute of Statistical Mathematics 46 (1994), S. 521-535 
    Details
    ISSN: 1572-9052
    Keywords: Bias reduction ; smoothing ; variable bandwidth
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Variable (bandwidth) kernel density estimation (Abramson (1982,Ann. Statist.,10, 1217–1223)) and a kernel estimator with varying locations (Samiuddin and El-Sayyad (1990,Biometrika,77, 865–874)) are complementary ideas which essentially both afford bias of orderh 4 as the overall smoothing parameterh → 0, sufficient differentiability of the density permitting. These ideas are put in a more general framework in this paper. This enables us to describe a variety of ways in which scale and location variation may be extended and/or combined to good theoretical effect. This particularly includes extending the basic ideas to provide new kernel estimators with bias of orderh 6. Technical difficulties associated with potentially overly large variations are fully accounted for in our theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00773515
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  • 2
    Electronic Resource
    Electronic Resource
    A codimension seven unfolding of a third harmonic resonant interaction between two contiguous continua (2000)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 51 (2000), S. 651-673 
    Details
    ISSN: 0044-2275
    Keywords: Key words. Capillary-gravity, mode interaction, O(2)-invariance, bifurcation, Lyapunov-Schmidt, universal unfolding.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. In this paper we study the capillary-gravity profiles which may occur at the interface of two fluids when the fundamental mode and its third harmonic interact with each other. The problem is treated as one in bifurcation theory by transforming it into a nonlinear operator equation between suitable function spaces. Using classical analytic methods and by exploiting the group invariances of the operator, we are able to describe the solution set of the equation for a selection of values of the parameters present and to allow for possible imperfections in the resonance.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00001513
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  • 3
    Electronic Resource
    Electronic Resource
    A nonlinear multiple scales approach to modelling almost third harmonic interfaces (1999)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 50 (1999), S. 47-63 
    Details
    ISSN: 0044-2275
    Keywords: Key words. Capillary-gravity waves, third harmonic resonance, multiple scales, evolution equation.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. The evolution of third harmonic resonant narrow bandwidth capillary–gravity waves on an interface of two fluids is considered. The method of multiple scales is utilised in order to derive a set of first order coupled nonlinear partial differential equations which model the evolution of the wavepacket. Some solution families are exhibited.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000330050138
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