Frequency mapping

Piazza, A. Menozzi, P. Cavall-Sforza, L. "The Making and Testing of Geographic Gene Frequency Maps" Dept, of Genetics, School of Medicine, Stanford University Stanford, California, 1979. 

Frequency mapping ui —> Q uj) that is used to construct the artificial water susceptibility is shown in long-dashed line in Fig. 13.8 (the dotted... 

Fig. 4.1. Frequency maps of two types of ischemic damage complete infarction (left) and scattered neuronal injury (right). Histology was obtained 24 h after left MCA occlusion in rats. Total number of animals was 9. Pseudocolor representation denotes the number of animals that showed the respective type of injury at this pixel. Note the widespread distribution of incomplete infarction over the affected left hemisphere. [Adapted from Alexis et al. (1996)]...

Arnold diffusion is a motion along a resonance. One can observe motion across the resonance by using a frequency map [20], Also, motion at the crossing points of resonances are interesting [21-23],... 

We have noted the importance of incorporating calculations of IR intensities in the analysis of spectra. This approach is certain to prove fruitful in a number of areas determination of the dependence of amide mode intensities on conformation influence of size and perfection of structure on intensities correlation of intensities with hydrogen-bond geometry (Cheam and Krimm, 1986). Just as it is possible to develop a conformational (, i/ )-frequency map (Hsu et al., 1976), it should be possible to compute a conformational (, i/ )-intensity map, which could be useful in analyzing the spectra of unordered polypeptide chain structures. Of course, nothing has yet been done on the calculation of Raman intensities of polypeptides, and this area is ripe for future development. 

Issa NP, Trepel C, Stryker MP. 2000. Spatial frequency maps... 

All studies mentioned in this section have been single-wavelength/single-frequency maps, and thus restricted to the imaging of changes in band intensities, such as the Si phonon band of strained silicon. Important information that can... 

In our definition, real 2D NMR data are either homo- or hetero-nuclear data, where Fourier transformation is performed in both directions and a chemical shift map is created, as seen in COSY, TOCSY, NOESY, J-RES, HSQC, etc. In the case of HSQC, a short delay is being incremented during which the signals in the ID spectrum are being coded with the frequency of the corresponding hetero-nucleus and 2D Fourier transformation is employed to create the frequency map. An example of a real 2D NMR spectrum can be seen in Fig. 3A. With homo-nuclear data a similar picture is obtained, but the plot is symmetrical on the diagonal axis. 

Laskar, J. (1995). Introduction to frequency map analysis. In Simo, C., editor. Hamiltonian systems with three or m,ore degrees of freedom, NATO ASI series C, 533. Kluwer Academic Publishers, Dordrecht-Boston-London. 

Different methods have been developed either for a rapid computation of the LCIs (Cincotta and Simo 2000) or for detecting the structure of the phase space (chaotic zones, weak chaos, regular resonant motion, invariant tori). Especially for this last purpose we quote the frequency map analysis (Laskar 1990, Laskar et al. 1992, Laskar 1993, Lega and Froeschle 1996), the sup-map method (Laskar 1994, Froeschle and Lega 1996), and more recently the fast Lyapunov indicator (hereafter FLI, Froeschle et al. 1997, Froeschle et al. 2000) and the Relative Lyapunov Indicator (Sandor et al. 2000). The definitions and comparisons between different methods including a preliminary version of the FLI have been discussed in Froeschle and Lega (1998, 1999). 

Using the frequency map analysis we visualized (Lega and Froeschle 1996) the predicted result on the topology of the neighborhood of noble tori for values of the perturbing parameter well above the ones allowed by the mathematical demonstration. Moreover, we have measured the size of the complementary set of tori, showing that the size of islands and chaotic zones decreases exponentially when the distance to the noble torus goes to zero. 

Figure 8. Variation of the quantity log(a/6) versus the order for the set of Fibonacci orbits of the standard map with e = 0.9715 analyzed in Figure 6. The points labeled with a triangle stand for the log(a/6) obtained by the FLI computation. The points labeled by a square correspond to the log(a/6), with a = it/q and b measured with the frequency-map analysis by Lega and Froeschle 1996. The points labeled by a circle correspond to the LMA (explained in appendix A and in Locatelli et al. 2000).

Laskar, J. (1994). Frequency map analysis of an Hamiltonian system. Workshop on Non-Linear Dynamics in Particle Accelerators, Arcidosso, September 1994. 

Lega, E. and Froeschle, C. (1996). Numerical investigations of the structure around an invariant KAM torus using the frequency map analysis. Physica D, 95 97-106. 

We are beginning to understand chaotic structure in a way not seen before. Numerical methods of measuring chaotic and regular behaviour such as Fast Liapunov Indicators, sup-maps, twist-angles, Frequency Map Analysis, fourier spectal analysis are providing lucid maps of the global dynamical behaviour of multidimensional systems. Fourier spectral analysis of orbits looks to be a powerful tool for the study of Nekhoroshev type stability. Identification of the main resonances and measures of the diffusion of trajectories can be found easily and quickly. Applied to the full N-body problem without simplification, use of these tools is beginning to explain the observed behaviour of real physical systems. 

An updated 10-year frequency map for ASP in the United States is available at the Harmful Algae Page, which is supported by a National Oceanic and Atmospheric Administration Center for Sponsored Coastal Ocean Research Coastal Ocean Program grant to the National Office for Harmful Algal Blooms [133]. 

Due to the complex structure of odor space, it is an extremely interesting and challenging question whether the olfactory system has a correlate of retinotopic maps in vision or frequency maps in audition. And if it does, what would be the organizing principle of such an odor map Furthermore, can we learn from this organization of biological olfactory systems to build artificial chemosensor systems that perform at levels comparable to the performance of the former in general olfactory sensing tasks ... 

Human hearing arises from airborne waves alternating 50 to 20,000 times a second about the mean atmospheric pressure. These pressure variations induce vibrations of the tympanic membrane, movement of the middle-ear ossicles connected to it, and subsequent displacements of the fluids and tissues of the cochlea in the inner ear. Biomechanical processes in the cochlea analyze sounds to frequency-mapped vibrations along the basilar membrane, and approximately 3,500 inner hair cells modulate transmitter release and spike generation in 30,000 spiral ganghon cells whose proximal processes make up the auditory nerve. This neural activity enters the central auditory system and reflects sound patterns as temporal and spatial spike patterns. The nerve branches and synapses extensively in the cochlear nuclei, the first of the central auditory nuclei. Subsequent brainstem nuclei pass auditory information to the medial geniculate and auditory cortex (AC) of the thalamocortical system. 

As suggested previously, it is inaccurate to view central auditory pathways as a frequency-mapped sensory conduit to higher centers. The auditory system, indeed the entire brain operates to ensure survival by promoting effective behaviors and storing experience in memory for future decisions. In essence, audition identifies critical sound events and associates them with appropriate responses. This is evident in many observations at the cerebral cortex, but examples for bats and human speech suggest the range and complexity of processing in auditory pathways. 

TABLE4.12 Frequency Mapping of Network Elements Type Series Branch(es) Shunt Branch(es)... 

Once a LP prototype filter has been designed, a prototype HP, BP, or BS L-C filter is realized by replacing the LP network elements according to the scheme shown in Table 4.12. This technique is called frequency mapping. Recall that prototype means that the cutoff frequency of a frequency-mapped HP filter or the center frequency of a frequency-mapped BP or BS filter is 1 rad/s (Fig. 4.37) when operating with a l-f2 load. The bandwidth parameter B in Table 4.12 expresses bandwidth of a BS or BP filter as the ratio of bandwidth to center frequency. If, for example, the bandwidth and center frequency requirements for a BP or BS filter are 100 Hz and 1 kHz, respectively, then the prototype bandwidth parameter B is 0.1. The general network configurations for HP, BP, and BS L-C filters are shown in Fig. 4.38. 

The mathematicalbasis for frequency mapping is omitted here however, the technique involves replacing the LP transfer function variable (s or ) with the mapped variable, as presented in Table 4.13. The mapped amplitude response for a Butterworth filter or a Chebyshev filter is obtained by performing the appropriate operation from column two of Table 4.13 on Eq. (4.60) or (4.61), respectively. The mapped phase response... 

TABLE 4.13 Frequency Mapping of Transfer Functions Variables... 

Frequency mapping Procedure by which a lowpass prototype filter is converted to a highpass, bandpass, or bandstop prototype. 

Both IHCs and OHCs have precise patterns of stereocilia at the end held within the rigid plate formed by supporting cells. The tectorial membrane overlies the stereocilia, but while those of OHCs contact the tectorial membrane, the stereodlia of the IHCs do not. At the end opposite the stereociha, IHCs synapse with the distal processes of SCCs. The proximal processes of SCCs are generally myelinated and form most of the auditory nerve. The nerve is laid down as these fibers collect from the apex and base in an orderly spiral manner. The nerve retains the frequency map of the BM, that is, it is tonotopically organized, a characteristic that carries through much of the auditory system. 

Figure 8 Final result of a Kohonen ANN shown as a map of labels and as a frequency map. The 62 objects belonging to three different classes A, B, and C are clearly separated into three clusters. The frequency map shows group C to be quite homogenous, within group A six very similar samples can be found, while in group B one can identify two different sub-groups, each located in one comer...

In the actual 2D procedure, the first interferogram is formed from the first data point from each of the original spectra, and the second interferogram is formed from the second data point from each of the original spectra. Mathematically, this process is a simple matrix transposition. In fact, if these spectra are thought of as the rows of the data matrix S(tc, Fa) then the columns represent the variation of the intensity of a given frequency component Fa as a function of r. The result of this complete process is a 2D frequency map, on which the data are plotted in the form of a series of spectra (stacked or white-washed plot) or in the form of a contour plot in which each successive contour represents a higher intensity. 

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