Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight 2. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

On-axis joint transform correlation based on a four-level power spectrum

We propose a method to obtain a single centered correlation with use of a joint transform correlator. We analyze the required setup to carry out the whole process optically, and we also present experimental results.

Effects of thresholding level variation in fringe binarization of multiobject joint transform correlation

It is possible to improve the fringe binarization method of joint transform correlation by choosing a suitable threshold level.

Controlled-intensity detection peaks in a binary joint transform correlator

In multiobject pattern recognition the height of the correlation peaks should be controlled when the power spectrum of ajoint transform correlator is binarized. In this paper a method to predetermine the value of detection peaks is demonstrated. The technique is based on a frequency-variant threshold in order to remove the intraclass terms and on a suitable factor to normalize the binary joint power spectrum. Digital simulations and experimental hybrid implementation of this method were carried out.

Nonlinear filtering in object and Fourier space in a joint transform optical correlator: comparison and experimental realization

The use of different kinds of nonlinear filtering in a joint transform correlator are studied and compared. The study is divided into two parts, one corresponding to object space and the second to the Fourier domain of the joint power spectrum. In the first part, phase and inverse filters are computed; their inverse Fourier transforms are also computed, thereby becoming the reference in the object space. In the Fourier space, the binarization of the power spectrum is realized and compared with a new procedure for removing the spatial envelope. All cases are simulated and experimentally implemented by a compact joint transform correlator.

Reduction of the effect of aberrations in a joint transform correlator

We report the study of the influence of optical aberrations in a joint-transform correlator: The wave aberration of the optical system is computed from data obtained by ray tracing. Three situations are explored: We consider the aberration only in the first diffraction stage (generation of power spectrum), then only in the second (transformation of the power spectrum into correlation), and finally in both stages simultaneously. The results show that the quality of the correlation is determined mostly by the aberrations of the first diffraction stage and that we can optimize the setup by moving the cameras along the optical axis to a suitable position. The good agreement between the predicted data and the experimental results shows that the method explains well the behavior of optical diffraction systems when aberrations are taken into account.

Mecánica clásica en el espacio de Hilbert

We continue our study of classical mechanics using the methods of quantum mechanics. A Hilbert space is introduced, new conservation laws deduced, and the possibility of representing by new methods the many body classical problem discussed.

Applications of Fourier transform infrared spectroscopy

This article summarizes the basic principles of Fourier Transform Infrared Spectroscopy, with examples of methodologies and applications to different field sciences.

Fourier transform convolution integrals applied to generalized Born molecular volume.

Generalized Born methods are currently among the solvation models most commonly used for biological applications. We reformulate the generalized Born molecular volume method initially described by (Lee et al, 2003, J Phys Chem, 116, 10606; Lee et al, 2003, J Comp Chem, 24, 1348) using fast Fourier transform convolution integrals. Changes in the initial method are discussed and analyzed. Finally, the method is extensively checked with snapshots from common molecular modeling applications: binding free energy computations and docking. Biologically relevant test systems are chosen, including 855-36091 atoms. It is clearly demonstrated that, precision-wise, the proposed method performs as good as the original, and could better benefit from hardware accelerated boards.

Cross-frequency coupling in EEG revealed by adaptive oscillation tracking.

Introduction: Neuronal oscillations have been the focus of increasing interest in the neuroscientific community, in part because they have been considered as a possible integrating mechanism through which internal states can influence stimulus processing in a top-down way (Engel et al., 2001). Moreover, increasing evidence indicates that oscillations in different frequency bands interact with one other through coupling mechanisms (Jensen and Colgin, 2007). The existence and the importance of these cross-frequency couplings during various tasks have been verified by recent studies (Canolty et al., 2006; Lakatos et al., 2007). In this study, we measure the strength and directionality of two types of couplings - phase-amplitude couplings and phase-phase couplings - between various bands in EEG data recorded during an illusory contour experiment that were identified using a recently-proposed adaptive frequency tracking algorithm (Van Zaen et al., 2010). Methods: The data used in this study have been taken from a previously published study examining the spatiotemporal mechanisms of illusory contour processing (Murray et al., 2002). The EEG in the present study were from a subset of nine subjects. Each stimulus was composed of 'pac-man' inducers presented in two orientations: IC, when an illusory contour was present, and NC, when no contour could be detected. The signals recorded by the electrodes P2, P4, P6, PO4 and PO6 were averaged, and filtered into the following bands: 4-8Hz, 8-12Hz, 15-25Hz, 35-45Hz, 45-55Hz, 55-65Hz and 65-75Hz. An adaptive frequency tracking algorithm (Van Zaen et al., 2010) was then applied in each band in order to extract the main oscillation and estimate its frequency. This additional step ensures that clean phase information is obtained when taking the Hilbert transform. The frequency estimated by the tracker was averaged over sliding windows and then used to compare the two conditions. Two types of cross-frequency couplings were considered: phase-amplitude couplings and phase-phase couplings. Both types were measured with the phase locking value (PLV, Lachaux et al., 1999) over sliding windows. The phase-amplitude couplings were computed with the phase of the low frequency oscillation and the phase of the amplitude of the high frequency one. Different coupling coefficients were used when measuring phase-phase couplings in order to estimate different m:n synchronizations (4:3, 3:2, 2:1, 3:1, 4:1, 5:1, 6:1, 7:1, 8:1 and 9:1) and to take into account the frequency differences across bands. Moreover, the direction of coupling was estimated with a directionality index (Bahraminasab et al., 2008). Finally, the two conditions IC and NC were compared with ANOVAs with 'subject' as a random effect and 'condition' as a fixed effect. Before computing the statistical tests, the PLV values were transformed into approximately normal variables (Penny et al., 2008). Results: When comparing the mean estimated frequency across conditions, a significant difference was found only in the 4-8Hz band, such that the frequency within this band was significantly higher for IC than NC stimuli starting at ~250ms post-stimulus onset (Fig. 1; solid line shows IC and dashed line NC). Significant differences in phase-amplitude couplings were obtained only when the 4-8 Hz band was taken as the low frequency band. Moreover, in all significant situations, the coupling strength is higher for the NC than IC condition. An example of significant difference between conditions is shown in Fig. 2 for the phase-amplitude coupling between the 4-8Hz and 55-65Hz bands (p-value in top panel and mean PLV values in the bottom panel). A decrease in coupling strength was observed shortly after stimulus onset for both conditions and was greater for the condition IC. This phenomenon was observed with all other frequency bands. The results obtained for the phase-phase couplings were more complex. As for the phase-amplitude couplings, all significant differences were obtained when the 4-8Hz band was considered as the low frequency band. The stimulus condition exhibiting the higher coupling strength depended on the ratio of the coupling coefficients. When this ratio was small, the IC condition exhibited the higher phase-phase coupling strength. When this ratio was large, the NC condition exhibited the higher coupling strength. Fig. 3 shows the phase-phase couplings between the 4-8Hz and 35-45Hz bands for the coupling coefficient 6:1, and the coupling strength was significantly higher for the IC than NC condition. By contrast, for the coupling coefficient 9:1 the NC condition gave the higher coupling strength (Fig. 4). Control analyses verified that it is not a consequence of the frequency difference between the two conditions in the 4-8Hz band. The directionality measures indicated a transfer of information from the low frequency components towards the high frequency ones. Conclusions: Adaptive tracking is a feasible method for EEG analyses, revealing information both about stimulus-related differences and coupling patterns across frequencies. Theta oscillations play a central role in illusory shape processing and more generally in visual processing. The presence vs. absence of illusory shapes was paralleled by faster theta oscillations. Phase-amplitude couplings were decreased more for IC than NC and might be due to a resetting mechanism. The complex patterns in phase-phase coupling between theta and beta/gamma suggest that the contribution of these oscillations to visual binding and stimulus processing are not as straightforward as conventionally held. Causality analyses further suggest that theta oscillations drive beta/gamma oscillations (see also Schroeder and Lakatos, 2009). The present findings highlight the need for applying more sophisticated signal analyses in order to establish a fuller understanding of the functional role of neural oscillations.

SAR Interferometric phase noise reduction using wavelet transform

The problem of synthetic aperture radar interferometric phase noise reduction is addressed. A new technique based on discrete wavelet transforms is presented. This technique guarantees high resolution phase estimation without using phase image segmentation. Areas containing only noise are hardly processed. Tests with synthetic and real interferograms are reported.

On the inverse windowed Fourier transform

The inversion problem concerning the windowed Fourier transform is considered. It is shown that, out of the infinite solutions that the problem admits, the windowed Fourier transform is the "optimal" solution according to a maximum-entropy selection criterion.