Threshold singularity

If a singularity in the medium modified few-body T matrix is obtained, it may be taken to indicate the formation of a quantum condensates. Different kinds of quantum condensates are also considered [7, 8], They become obvious if the binding energy of nuclei is investigated [9], Correlated condensates are found to give a reasonable description of near-threshold states of na nuclei [10], The contribution of condensation energy to the nuclear matter EOS would be of importance and has to be taken into account not only in mean-field approximation but also considering correlated condensates. 

For the case of particles of light, after crossing the necessary number of beamsplitters to reach the threshold, the energy of the photon singularity begins to decrease. 

B60 IF EI=1 THEN LPRINT " REQUIRED THRESHOLD NOT ATTAINED" LPRINT iLPRINT 4B62 IF ES=1 THEN LPRINT " SINGULAR CROSS PRODUCT MATRIX" iLPRINT iLPRINT 4864 FOR 1=1 TO NT... 

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

Butcher K, Parsons M, Baird T et al (2003) Perfusion thresholds in acute stroke thrombolysis. Stroke 34 2159-2164 Calamante F, Gadian DG, Connelly A (2000) Delay and dispersion effects in dynamic susceptibility contrast MRI simulations using singular value decomposition. Magn Reson Med 44 466-473... 

As the region near an X-ray absorption edge is scanned in energy, the ejected photoelectron sequentially probes the empty electronic levels of the material. Theoretically, interest in core-state excitation has developed considerably since the work of Mahan (179) and Nozieres and De Dominicis (219) on the singular response of the conduction electrons (in metals) to the sudden potential created by removal of a core electron. The resulting electron-hole pair excitations lead to a threshold edge asymmetry. 

The X-ray singularity problem was originally solved in the asymptotic limit and the complicated many-body problem was turned into an effective one-particle problem (219). For the X-ray photon frequency threshold frequency (o0, the absorption spectrum g(m) for the process in which a deep, structureless core electron is excited to the conduction band by the absorption of an X-ray of frequency w is expressed by the power law... 

A similar model was analyzed by Pikios and Luss (283). They analyzed the same set of reaction steps with the coverage-dependent activation energy interpreted in terms of surface heterogeneity. They derived criteria for the occurrence of oscillations as did Belyaev et al. (154,162). They also found a singular steady state, which became a limit cycle for values of the surface heterogeneity lying above a certain threshold value, and they performed numerical analyses of these oscillatory states. 

Wadsworth and Wysong made a detailed assessment of the threshold line model (and other dissociation models) for hydrogen dissociation by making comparison with quasiclassical trajectory computations. They found that the original forms of the threshold line models proposed by Macheret and Rich have to be extended to more complete forms in order to avoid singularities at specific values of the collision energies. Some of their findings are shown in the results section. 

By far, singular value decomposition (SVD) is the most popular algorithm to estimate the rank of the data matrix D. As a drawback of SVD, the threshold that separates significant contributions from noise is difficult to settle. Other eigenvalue-based and error functions can be utilized in a similar way, but the arbitrariness in the selection of the significant factors still persists. For this reason, additional assays may be required, especially in the case of complex data sets. 

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