Poisson transform

The choice of sampling rate is sometimes governed by a tradeoff between frequency aliasing (low rate) and noise sensitivity (high rate). Quick and Bolgiano (1976) have employed the Poisson transformation to avert this complication. Consult Piovoso and Bolgiano (1970) for additional background on the Poisson transform. 

The notation Poi(k,rji Id) is used to denote a Poisson distribution in k with mean rji/p. p(k) is a function of the intensity at the detector and the probability distribution of the intensitypC/r,). The constant rjj is proportional to the detection efficiency (the proportionality constant between the amount of light falling on the detector and the average number of photon counts < k> detected) and incorporates the sampling time. This distribution is thus the Poisson transform of the light intensity distribution at the detector. Thus, for a perfectly steady light source (i.e. one in which pil ) is a delta function), the distribution p(k) will be Poissonian [8,9] and can therefore be described in terms of just the mean count number = t/iZd ... 

H. M. Quiney. Poisson-transformed density fitting in relativistic four-component Dirac-Kohn-Sham theory. /. Chem. Phys.,... 

The erf term of Eq. (5) is now subjected to a Poisson transformation, adding a contribution from v = 0 if it was missing and then subtracting it again external to the transformation. Before the Poisson transformation, this term, which we call t/ps(R), can be written as an integral ... 

In the right-hand side of Eq. (7) appear the component of R in the direction of periodicity, denoted R, and the two-dimensional remainder of R perpendicular to that direction, denoted Ro. Reaching Eq. (7) requires a number of steps, but they are similar to those used for Vat and other lattice sums in Ref. [10]. The function Kq is an incomplete Bessel function that always occurs when a lattice of GTO s is subjected to a Poisson transformation, with integral representation... 

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

If a gaussian function is chosen for the charge spread function, and the Poisson equation is solved by Fourier transformation (valid for periodic... 

The generation of photons obeys Poisson statistics where the variance is N and the deviation or noise is. The noise spectral density, N/, is obtained by a Fourier transform of the deviation yielding the following at sampling frequency,... 

To describe the simple phenomena mentioned above, we would hke to have only transparent approximations as in the Poisson-Boltzmann theory for ionic systems or in the van der Waals theory for non-coulombic systems [14]. Certainly there are many ways to reach this goal. Here we show that a field-theoretic approach is well suited for that. Its advantage is to focus on some aspects of charged interfaces traditionally paid little attention for instance, the role of symmetry in the effective interaction between ions and the analysis of the profiles in terms of a transformation group, as is done in quantum field theory. 

At the instant of contact between a sphere and a flat specimen there is no strain in the specimen, but the sphere then becomes flattened by the surface tractions which creates forces of reaction which produce strain in the specimen as well as the sphere. The strain consists of both hydrostatic compression and shear. The maximum shear strain is at a point along the axis of contact, lying a distance equal to about half of the radius of the area of contact (both solids having the same elastic properties with Poisson s ratio = 1/3). When this maximum shear strain reaches a critical value, plastic flow begins, or twinning occurs, or a phase transformation begins. Note that the critical value may be very small (e.g., in pure simple metals it is zero) or it may be quite large (e.g., in diamond). 

From Poisson s equation (265), we get for the Fourier transform of the potential ... 

Colorless or light yellow metal at ordinary temperatures it occurs in hexagonal close-packed crystalline form, known as alpha-gadolinium alpha form transforms to a body-centered cubic allotropic form, beta-gadolinium upon heating at 1,262°C density 7.90 g/cm melting point 1,313°C vaporizes at 3,266°C vapor pressure 9.0 torr at 1,800°C (calculated) electrical resistivity 134.0 microhm-cm at 25°C Poisson ratio 0.259 modulus of elasticity 8.15x106 psi thermal neutron absorption cross section 46,000 barns insoluble in water dissolves in acid (reacts). 

Silvery white metal soft and malleable hexagonal closed pack crystal system transforms to face-centered cubic crystals at 310°C which further transforms to a body-centered cubic allotropic modification at 868°C density 6.166 g/cm3 Brinnel hardness (as cast) 37 melts at 918°C vaporizes at 3,464°C vapor pressure 1 torr at 2,192°C electrical resistivity 56.8 x 10 ohm-cm at 25°C Young s modulus 3.84 x lO- dynes/cm Poisson s ratio 0.288 thermal neutron cross section 8.9 bams. 

The next level of complexity in the treatment is to orient the apphed stress at an angle, 6, to the lamina fiber axis, as illustrated in Figure 5.119. A transformation matrix, [T], must be introduced to relate the principal stresses, o, 02, and tu, to the stresses in the new x-y coordinate system, a, ay, and t j, and the inverse transformation matrix, [T]- is used to convert the corresponding strains. The entire development will not be presented here. The results of this analysis are that the tensile moduli of the composite along the x and y axes, E, and Ey, which are parallel and transverse to the applied load, respectively, as well as the shear modulus, Gxy, can be related to the lamina tensile modulus along the fiber axis, 1, the transverse tensile modulus, E2, the lamina shear modulus, Gu, Poisson s ratio, vn, and the angle of lamina orientation relative to the applied load, 6, as follows ... 

Fit data to a recognized mathematical distribution (e.g., normal, Poisson, binomial). When appropriate, transform the data (e.g., log 10 transformation). Calculate confidence limits. 

Poisson s equation (6.9) can be solved directly by writing the potential, K(r), in terms of its Fourier transform, K(q), that is... 

Solve the Poisson-Boltzmann equation for a spherically symmetric double layer surrounding a particle of radius Rs to obtain Equation (38) for the potential distribution in the double layer. Note that the required boundary conditions in this case are at r = Rs, and p - 0 as r -> oo. (Hint Transform p(r) to a new function y(r) = r J/(r) before solving the LPB equation.)... 

The Poisson equation can also be solved using the discrete sine transform (Zenger and Bader 2004). The discrete sine transform of a two-dimensional function f(x, y) defined on... 

Thus, the Poisson equation may be solved by applying the discrete sine transform, multiplying the result by... 

Transformation based on square root from data X = /X is applied when the test values and variances are proportional as in Poisson s distribution. If the data come from counting up and the number of units is below 10 transformation form X --fX + 1 and text X =s/X + /X I 1 is used. If the test averages and their standard deviations are approximately proportional, we use the logarithm transformation X =log X. If there are data with low values or they have a zero value, we use X =log (X+l). When the squares of arithmetical averages and standard deviations are proportional we use the reciprocal transformation X =l/X or X =1/(X+1) if the values are small or are equal to zero. The transformation arc sin [X is used when values are given as proportions and when the distribution is Binomial. If the test value of the experiment is zero then instead of it we take the value l/(4n), and when it is 1, l-l/(4n) is taken as the value and n is the number of values. Transforming values where the proportion varies between 0.30 and 0.70 is practically senseless. This transformation is done by means of special tables suited for the purpose. 

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