Singular functions

While for L-> 00, < >x at = 0 has a discontinuous jump and x, hence exhibits a delta-function singularity, for finite L this singularity is rounded in a finite peak ... 

This implies that the exponents and y defined above are 0 = y = 2( = d) for a first-order transition. Since the symmetry around if = 0 is preserved for finite L, there is no shift of the transition. This feature is different, however, if we consider temperature-driven first-order transitions , since there is no symmetry between the disordered high-temperature phase and the ordered low-temperature phase. In order to understand the rounding of the delta-function singularity of the specific heat, which measures the latent heat for L- oo, it now is useful to consider the energy distribution, for which again a double Gaussian approximation applies ... 

If a first-order transition in a small system is driven by the temp>erature, for example, a transition taking place along an isobar, then it is typically spread out over a finite range of temperatures. The size of the transition region can be defined as the width AT of the finite p>eak in the curve of specific heat versus temperature. This peak replaces the -function singularity at the tran-... 

Fig. 2. Isobaric relationship between enthalpy and temperature in the liquid, glassy, and crystalline states. is the melting temperature, and Fg the glass transition temperature. The lower diagram shows the behavior of the isobaric heat capacity. The arrow indicates the -function singularity due to latent heat at a first-order phase transition. (From Debenedetti, 1996.)...

Note that where the spectral density has a S function singularity, the correlation function has a constant (in time) background. 

Outside the bands, Im G exhibits a -function singularity at the energies of the bound states. The weight factor of this singularity gives the contribution to the charge-bond-order matrix and to Mulliken s gross population for the various basis orbitals from the localized states at E,. 

Applying eqn (2.91) to our GCMC results for the 0-Pt(321) system, we obtain the results in Figure 2.17, where we plot v5. J at a simulation chemical potential of -0.7 eV/O. This chemical potential is shown because it most clearly exhibits a sharp maximum arormd 800 K, whereas most other results not shown exhibit broad maxima rather than sharp peaks. The presence of a gradual increase and decrease of the heat capacity around the maximum rather than a sharp delta function singularity is a consequence of the finite system size being unable to capture the true behavior at phase transitions. ... 

In this figure the next definitions are used A - projection operator, B - pseudo-inverse operator for the image parameters a,( ), C - empirical posterior restoration of the FDD function w(a, ), E - optimal estimator. The projection operator A is non-observable due to the Kalman criteria [10] which is the main singularity for this problem. This leads to use the two step estimation procedure. First, the pseudo-inverse operator B has to be found among the regularization techniques in the class of linear filters. In the second step the optimal estimation d (n) for the pseudo-inverse image parameters d,(n) has to be done in the presence of transformed noise j(n). 

The question of determination of the phase of a field (classical or quantal, as of a wave function) from the modulus (absolute value) of the field along a real parameter (for which alone experimental determination is possible) is known as the phase problem [28]. (True also in crystallography.) The reciprocal relations derived in Section III represent a formal scheme for the determination of phase given the modulus, and vice versa. The physical basis of these singular integral relations was described in [147] and in several companion articles in that volume a more recent account can be found in [148]. Thus, the reciprocal relations in the time domain provide, under certain conditions of analyticity, solutions to the phase problem. For electromagnetic fields, these were derived in [120,149,150] and reviewed in [28,148]. Matter or Schrodinger waves were... 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

This condition is fulfilled as long as the components of t are analytic functions at the point under consideration (in case part of them become singular at this point, curl X is not defined). 

The biasing function is applied to spread the range of configurations sampled such that the trajectory contains configurations appropriate to both the initial and final states. For the creation or deletion of atoms a softcore interaction function may be used. The standard Lennard-Jones (LJ) function used to model van der Waals interactions between atoms is strongly repulsive at short distances and contains a singularity at r = 0. This precludes two atoms from occupying the same position. A so-called softcore potential in contrast approaches a finite value at short distances. This removes the sin-... 

This discussion will be limited to functions of one variable that can be plotted in 2-space over the interval considered and that constitute the upper boundar y of a well-defined area. The functions selected for illustration are simple and well-behaved, they are smooth, single valued, and have no discontinuities. When discontinuities or singularities do occur (for example the cusp point of the Is hydrogen orbital at the nucleus), we shall integrate up to the singularity but not include it. 

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