Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups

This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler–Arnold formalism in the holomorphic setting. We study the real Lie group SL(2, R) and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2, ℂ).The first author was financed by FCT - Fundação para a Ciência e Tecnologia, I.P. (Portugal) - through the PhD scholarship PD/BD/143019/2018. The second author was partially supported by FCT, Portugal through the sabbatical grant SFRH/BSAB/135549/2018 and through CMAT, Portugal under the project UID/MAT/00013/2013. The third author was partially supported by CMUP, Portugal, member of LASI, which is financed by national funds through FCT under the project UIDB/00144/2020 and also by CIMI, France through the project “Complex dynamics of group actions, Halphen and Painlevé systems”. Finally, all three authors benefited from CNRS (France) support through the PICS project “Dynamics of Complex ODEs and Geometry”

Detection of helicopters using neural nets

Artificial neural networks (ANNs), in combination with parametric spectral representation techniques, are applied for the detection of helicopter sound. Training of the ANN detectors was based on simulated helicopter sound from four helicopters and a variety of nonhelicopter sounds. Coding techniques based on linear prediction coefficients (LPCs) have been applied to obtain spectral estimates of the acoustic signals. Other forms of the LPC parameters such as reflection coefficients, cepstrum coefficients, and line spectral pairs (LSPs) have also been used as feature vectors for the training and testing of the ANN detectors. We have also investigated the use of wavelet transform for signal de-noising prior to feature extraction. The performance of various feature extraction techniques is evaluated in terms of their detection accurac

Fuzzy controllers design using space-filling curves

We present a clustering technique for fuzzy rules based on Hilbert space-filling curves (SFC). SFC scans an n-dimensional space and reduces it to a curve, i.e. a one-dimensional line. We first introduce the Hilbert space-filling curves, and outline the algorithms for clustering and adaptive clustering which demonstrate SFC efficient self-organizing features. We then propose a SFC fuzzy inference model based on clustering the object space. The SFC fuzzy model is then used to design a fuzzy controller. The proposed method achieves a dramatic reduction of the complexity of fuzzy controller by reducing the multivariable fuzzification problem to a one dimensional spac

Fast Methods Fbr Split Codebooks

This paper presents a fast method for building and searching split codebooks for vector quantization. The proposed method is evaluated in near transparent quality vector quantization of Line Spectral Frequencies (LSF) at 24-bit per frame. The method is based on a family of fractals called Space-Filling Curves (SFC). The SF curves achieve a significant saving in the complexity of vector quantization by reducing the problem to quantization in one-dimensional space. The paper presents algorithms for the generation of the SFC mapping utilizing the self-replication feature of the curves, and a number of simulation experiments to demonstrate the effectiveness of the method. It is shown that the SFC can reduce the search complexity of split codebooks by a factor of 8-32 times with a slight degradation in the vector quantization performance

Fast Methods Fbr Split Codebooks

This paper presents a fast method for building and searching split codebooks for vector quantization. The proposed method is evaluated in near transparent quality vector quantization of Line Spectral Frequencies (LSF) at 24-bit per frame. The method is based on a family of fractals called Space-Filling Curves (SFC). The SF curves achieve a significant saving in the complexity of vector quantization by reducing the problem to quantization in one-dimensional space. The paper presents algorithms for the generation of the SFC mapping utilizing the self-replication feature of the curves, and a number of simulation experiments to demonstrate the effectiveness of the method. It is shown that the SFC can reduce the search complexity of split codebooks by a factor of 8-32 times with a slight degradation in the vector quantization performance

Fuzzy Controllers Design Using Space-Filling Curves

In this paper we present a clustering technique for fuzzy rules based on Hilbert Space-filling Curves (SFC). SFC scans an n-dimensional space and reduces it to a curve, i.e. a one-dimensional line. The paper introduces first the Hilber Space-filling curves, and outlines algorithms for clustering and adaptive clustering which demonstrate the SFC efficient self-organizing features. We then propose a SFC fuzzy inference model based on clustering the object space. The SFC fuzzy model is then used to design a fuzzy controller. The proposed method achieves a dramatic reduction of the complexity of fuzzy controller by reducing the multivariable fuzzification problem to a one dimentional space

Lie groups with all left-invariant semi-Riemannian metrics complete

For each left-invariant semi-Riemannian metric $g$ on a Lie group $G$, we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of $g$. When the adjoint representation of $G$ satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any $g$. We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products $K \ltimes_\rho \mathbb{R}^n$ , where $K$ is the direct product of a compact and an abelian Lie group and $\rho(K)$ is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the the absence of linear growth and suggest new questions