The Periodic Function

Emile de Chancourtois, co-discoverer of elemental periodicity, claimed [5] that 

His claim was vindicated with the discovery of atomic number, but the theme remained undeveloped until it was conjectured by Plichta [6] that the electron configuration of atoms is mapped by the distribution of prime numbers. Based on the observation that all prime numbers 3 are of the type 6n 1, he defined a prime-number cross that intersects a display of natural numbers on a set of concentric circles with a period of 24. In Fig. 4, the construct is shown, rearranged as a number spiral. Noting that the numbers on each cycle add up to 

Reinterpretation of the sums as electron pairs over all stable nuclides suggests  

This interpretation is supported [7] by analysis of the neutron imbalance of stable atomic species as a function of mass number, shown in Fig. 5. The region of nuclide stability is demarcated here by two zigzag lines with deflection points at common values of mass number A. Vertical hemlines through the deflection points divide the fleld into 11 segments of 24 nuclides each, in line with condition (c). This theme is developed in more detail in the paper on Atomic Structure in this volume. Defining neutron imbalance as either Z/N or (N — Z )jZ, the isotopes of each element, as shown in Fig. 6, map to either circular segments or straight lines that intersect where 

This result provides the exact value of the convergence limit of Z/N first identified by Harkins [8] as 0.62 according to the curves in Fig. 5. 

---

The rows and columns of Mendeleev s table are meant to reflect the periodic function asserted to exist by this periodic law. 

We shall now see how we can arrive at the same conclusion using the methods derived above, which, it should be emphasized, are applicable to a much larger class of functions than the periodic functions. 

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

Abstract. The relativistic periodically driven classical and quantum rotor problems are studied. Kinetical properties of the relativistic standard map is discussed. Quantum rotor is treated by solving the Dirac equation in the presence of the periodic -function potential. The relativistic quantum mapping which describes the evolution of the wave function is derived. The time-dependence of the energy are calculated. 

The periodic function F r) can always be expanded into a two-dimensional Fourier series,... 

The continuum limit of the Hamiltonian representation is obtained as follows. One notes that if the friction function y(t) appearing in the GLE is a periodic function with period T then Eq. 4 is just the cosine Fourier expansion of the friction function. The frequencies coj are integer multiples of the fundamental frequency and the coefficients Cj are the Fourier expansion coefficients. In practice, the friction function y(t) appearing in the GLE is a decaying function. It may be used to construct the periodic function y(t T) = Y(t TiT)0(t-... 

The continuous function consisting of only N terms can at best only approximate the most general (continuous) function, for, as we know, the Fourier series of the most general periodic function requires an infinite number of terms. However, consider the periodic function under examination made up of only N or fewer terms of its Fourier series, that is, a band-limited function. Then, because of the linear independence of the sinusoidal terms, the coefficients of the DFT are exactly the same as those of the Fourier series of the continuous function. That is, we find... 

The indices hkl of the reflection give the three frequencies necessary to describe the Fourier term as a simple wave in three dimensions. Recall from Chapter 2, Section VI.B, that any periodic function can be approximated by a Fourier series, and that the approximation improves as more terms are added to the series (see Fig. 2.14). The low-frequency terms in Eq. (5.18) determine gross features of the periodic function p(x,y,z), whereas the high-frequency terms improve the approximation by filling in fine details. You can also see in Eq. (5.18) that the low-frequency terms in the Fourier series that describes our desired function p(x,y,z) are given by reflections with low indices, that is, by reflections near the center of the diffraction pattern (Fig. 5.2). 

On a plot of Z/N vs Z for all stable nuclides the field of stability is outlined very well by a profile, defined by the special points of the periodic table derived from 4. Furthermore, hem lines that divide the 264 nuclides into 11 groups of 24 intersect the convergence line, Z/N = r, at most of the points that define the periodic function. If the hem lines are extended to intersect the line Z/N = 0.58, a different set of points are projected out and found to match the periodicity, derived from the wave-mechanical model. 

Fig. 15 The ion yield of Cr(CO)3+ obtained using a 270 nm pulse of 30 fs duration showing the periodic functions and a Fourier transform of these functions in the inset showing coherent oscillations at 96 and 350 cm-1. Adapted from [15]...

It is assumed that the curve in figure 15 represents the maximum possible number of stable elements, under any conditions. Because atomic numbers 43 and 61 occur in no sequences generated by a-particle synthesis, this number becomes decreased to 100. Outside of massive stars not all elements are stable and the symmetry of the periodic function (eq. 1) is hidden. Such a situation is shown in figure 3, in which it is necessary to distinguish between atomic numbers (1-83) and symmetry numbers (1-102), and also in figure 16. 

The fringe order ambiguity the periodical function leads to equally probable... 

When a y is detected in one detector a clock is started. If a y2 is detected in one of the remaining five detectors within 5-10 times the lifetime of the intermediate level (85ns for " Cd) the time and angle (90° or 180°) between y and yi is recorded. In a PAC measurement a large number of such yi - y2 coincidences are recorded, and the data are transformed into the so-called perturbation fimction, that is, the periodic function containing the transition frequencies described above. That is, the perturbation function holds the information about the NQI and therefore, about the local electronic and molecular structure that can be determined in... 

To the extent that science seeks to explain the mechanism of physical phenomena with mathematically expressible laws, it reduces the data of concrete observation in particular events to the status of pure abstractions. The abstractions existed antecedently to the physical phenomena they were found to describe. The complex of ideas surrounding the periodic functions had to be worked out, as pure mathematical theory, before their relations to such physical phenomena as the motion of a pendulum, the movements of the planets, and the physical properties of a vibrating string could be discerned. The point is that as mathematics became more abstract, it acquired an ever-increasing practical application to diverse concrete phenomena. Thus, abstraction, characterized by numerical operations, became the dominant conceptual mode used to describe concrete facts. 

Any periodic function (such as the electron density in a crystal which repeats from unit cell to unit cell) can be represented as the sum of cosine (and sine) functions of appropriate amplitudes, phases, and periodicities (frequencies). This theorem was introduced in 1807 by Baron Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist who pioneered, as a result of his interest in a mathematical theory of heat conduction, the representation of periodic functions by trigonometric series. Fourier showed that a continuous periodic function can be described in terms of the simpler component cosine (or sine) functions (a Fourier series). A Fourier analysis is the mathematical process of dissecting a periodic function into its simpler component cosine waves, thus showing how the periodic function might have been been put together. A simple... 

Here Dx and /), are the window scales in the x- and y- directions. In (A.l) we assumed that the slits move in the x-direction with velocity V, practically it can be achieved using a rotating disk with radial slits, if the radius of the disk is large compared with the size of an analysed image. The periodic function R(x) can be represented by a Fourier series as... 

Solution (6.10) corresponds to a relatively small detuning. In this case, Aip converges to a constant for t —> oo, and the system settles on a stable CW solution. When the detuning 5 becomes larger 2rj, then Atp is the periodic function (considered modulo 27t) (6.11) with the period Tsyn = 2-kj. The growth rate of Api is determined by the term 2Tt[t /5 - Aif /2Tv i, i.e. the averaged frequency of self-pulsations is Afi = /5 — Arf. Note that their period tends to infinity as 5 —> 2 . 

Draw a graph of the periodic function represented by the series. 

In practice, the friction function y(t) appearing in the STGLE is not periodic but a decaying function. However, one may use it to construct the periodic function y(t r) = x "y(/ — m)6(t — nT)0[(n + 1)t — r] where 0(x) is the unit step function. The continuum limit is obtained when the period t goes to o°. In any numerical discretization of the STGLE care must always be taken not to extend the dynamics beyond the chosen value of the period t, as beyond this time one is following the dynamics of a system which is considerably different from the continuum STGLE. 

From the mathematical definition of the Fourier transform, it is advisable that the process determines the frequency content of the signal by obtaining the contribution of each sine and cosine function at each frequency. However, if a transient phenomenon occurred in the signal or it is non-stationary, then the Fourier transform fails either in localizing the anomaly in time domain or representing the signal by the summation of the periodic functions. 

We will show in the following that each principal maximum in the periodic function converges with increasing N towards a <5 distribution. Indeed, the width of the maximum is proportional to 1/iV, whereas the integral of the maximum tends to 1 ... 

Sampled at any ratio of Z/N < 1 the predicted periodicity in terms of N or A remains invariant. In contrast, the periodic function, defined by atomic munber Z, assmnes distinctive canonical forms at Z/N = 0, 0.58, r and 1, obviously related to the variability of extranuclear electronic configurations with environmental pressme. 

The unexplained features of only 81, rather than 100, stable elements observed in Nature, and the relationship between the periodic functions that occur at Z/N = 1.04,1.0,0.62 and 0.58 are explained later on. 

Figure 5.7 The Periodic Function mapped to a Mobius hand...

The graphical representation of the way in which chemical periodicity varies continuously as a function of the limiting ratio (Figure 5.3), 1 < Z/N < 0, appears strangely unsymmetrical, despite perfect symmetry at the extreme values. By adding an element of mirror symmetry a fully symmetrical closed function, that now represents matter and antimatter, is obtained. To avoid self overlap the graphical representation of the periodic function is transferred to the double cover of a Mobius band, which in closed form defines a projective plane. 

There are theoretical reasons for this consideration, which have been largely confirmed in practice also. The extended trapezoid method is often superior to the Gauss integration only for the periodical functions that have a period equal to the integration interval. 

For liquids in contact with a face of a periodic crystal, the correlation functions can be represented as Steele s expansion into sums of Fourier components periodic on the surface lattice [11, 12]. The Ornstein-Zernike (OZ) integral equation then reduces to a linear matrix equation for the expansion coefficients dependent on the distance to the surface [13, 14, 15]. This approach, however, is not very convenient since the surface symmetry entirely determines a particular form of the periodic functional basis. 

---