Summation formula

The reciprocals of the terms of the arithmetic-progression series are called harmonic progression. No general summation formulas are available for this series. 

In Eqs. (4.43)—(4.47), use is made of the well-known rclation(23)P = K/kBT, where k is the bending rigidity. Equation (4.45) differs from the corresponding BZ result by the inclusion of the D L / term. Normally, this is negligible on the fluorescence time scale, but for sufficiently short filaments it could make a significant contribution. It appears with the coefficient 1.0 instead of 2.0 because the correction term from the Euler-McLaurin summation formula actually cancels out one of the two D t terms in Eq. (4.43). Equation (4.43) or (4.45) is then inserted in Eq. (4.26) to obtain the BZ tumbling correlation function for the filament. 

The general case of 3n7-coefficients is considered in detail in [9, 11], There are many summation formulas for various products of 3ny-... 

Note that Eq. (4.15) is simply the linearized equation of motion for the classical upside-down barrier (55/5 = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15) describes how the trajectory escapes from the instanton solution, when deviated from it. The parameter A, referred to as the stability angle [Gutzwiller, 1967 Rajaraman, 1975], generalizes the harmonic oscillator phase a>,/3, which would stand in (4.16), if w, were constant. The fact that A is real is a reminder of the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det(-dj + w2) is a function of A only, and using the Poisson summation formula, we are able to write... 

It was with support of the Mosbauer spectroscopy327 that a long-lasting argument could be decided 328,329 Turnbull s Blue, Fe3[Fe(CN)6]2, is actually the same as Berlin Blue, Fe4[Fe(CN)6]3, even if the summation formulas suggest they are different. As a matter of fact, the summation formula of Berlin Blue is closest to the reality In the ideal Iron Blue crystal, 16 molecules of coordination water are included ... 

Another common name for the threshold value 0 is bias. The idea is that each neuron may have its own built-in bias term, independent of the input. One way of handling this pictorially and computationally is to add an extra unit to the input layer that always has a value of -1. Then the weight of the connections between this unit and the neurons in the next layer is the threshold or bias values for those neurons and the summation operation includes the bias term automatically. Then the summation formula becomes... 

Two standard methods are in common use in the MD community the reaction field method [79,80] and the Ewald summation technique [72,81-83]. There are also various hierarchical algorithms which are quite attractive in principle, but have proved to be difficult to implement efficiently in practice [67,84-87]. An alternative and potentially development interesting complement, is the summation formula developed by Lekner [88,89] which has been given an alternative and more general derivation by Sperb [90]. 

R. Sperb, Extension and Simple Proof of Lekner s Summation Formula for Coulomb Forces, Mol. Simul., 13 (1994), 189-193. 

The question raised by the quasicrystal debate is much deeper than whether they exist or not. To see this, we recall that the interpretation of diffraction experiments on all known translationally invariant crystals, however complicated, depends ultimately on the existence of the Poisson summation formula. This relation asserts that the Fourier trtinsform of the periodic delta function is itself a periodic delta function, whence the term reciprocal space. Explicitly, the Poisson summation formula is... 

Now the Poisson summation formula is at the core of all mathematical analysis [33]. It is equivalent in fact to the calculus, the Jacobi theta function transformations, and to a statement of the Riemann relation connecting the... 

There has been no basic formula analogous to the Poisson summation formula, characteristic of translational invariance, on which to base an analysis of quasicrystal diffraction patterns. Here successive values of reciprocal space have geometric ratios instead of the arithmetic spacing of the peaked functions observed with ordinary crystalline diffraction. Fig. 2.15 illustrates a two-dimensional section in reciprocal space of a diffraction pattern. The five-fold symmetry is exact, and typically six indices instead of three are required to index each point, with the choice of origin arbitrary, and for assigiunent of indices, ambiguous. The features of interest are ... 

J. Kolafa and J. W. Perram (1992) Cutoff errors in the ewald summation formulae for point charge systems. Molecular Simulation 9(5), pp. 351-68... 

R. Sperb (1994) Extension and simple proof of lekner s summation formula for coulomb forces. Molecular Simulation 13, pp. 189-193... 

All this formula instructs us to do is to take the contribution to the total stress of each dislocation using the results of eqn (8.38) and to add them up, dislocation by dislocation. We follow Landau and Lifshitz (1959) in exploiting the Poisson summation formula to evaluate the sums. Note that while it would be simple enough to merely quote the result, it is fun to see how such sums work out explicitly. If we use dimensionless variables a = x/D and yS = y/D, then the sum may be rewritten as... 

With these clever algebraic machinations behind us, the original problem has been reduced to that of evaluating the sum J a, ). Recall that the Poisson summation formula tells us... 

For the problem at hand, the Poisson summation formula allows us to rewrite our sum as... 

The semiclassical approximation is reached in three different steps (Berry and Mount, 1972). The first stage is to transform the summation over the discrete values of / into an integration over the continuous variable A = (/ + ). The proper way to do this is to use the Poisson summation formula which gives... 

An immediate application of Abels partial summation formula yields... 

The effectiveness factors are found with high accuracy via the summation formula which requires knowledge of the solution only at the interior collocation points and at the boundary ... 

---