Staircase function

We begin by showing that (3-33) holds for the special case where is a staircase function i.e. 

The concluding step in the argument is to note that many can be approximated, as closely as desired, by means of a staircase function. This means that, given such a function , it is possible to find a sequence of staircase functions fa with the property... 

The justification of Eq. (3-119) is similar in every respect to the one presented in the one-dimensional case. One first verifies, by direct calculation, that Eq. (3-119) is valid for n + m dimensional staircase functions—functions that are constant over n + m-dimensional intervals of the form an < xn < bn, cm < ym <, dm—and then argues that, since any function can be approximated as closely as desired by staircase functions, Eq. (3-119) must also hold for all . 

This means that the integral in Eq. (3-239) converges because all the increments of N(t) are, by definition, finite. Moreover, many h(t) of interest can be approximated as closely as desired by a staircase function so, at least for practical purposes, there is little loss of generality in restricting our attention to h(t ) s of the form (3-244). 

The nearest neighbor level spacings are defined as Si = -E)+i — Ei, where Ei are the energies of the unfolded levels, which are obtained by the following way The spectrum Ei is separated into smoothed average part and fluctuating parts. Then the number of levels below E is counted and the following staircase function is defined ... 

A hold device is always needed in a sampled-data process control system. The hold converts the sequence of impulses of an impulse-sampled function/(, into a continuous (usually staircase) function/g,). There are several types of mathematical holds, but the only one that is of any practical interest is called a zero-order hold. This type of hold generates the stair-step function described above. 

Figure 20. Total integral level density N E) as obtained for Model IVa. The mapping calculation (y = 1, upper line) is seen to match the quantum staircase function almost perfectly, while the mean-field trajectory results (y = 0, lower line) underestimate the correct level density considerably.

Since the staircase function is a sum of Heaviside functions and the derivative of a Heaviside function is a 5 function, the density p E) is given by... 

The staircase function N E) is interpolated by a mean staircase M E), a smooth fit to the actual staircase function N E). The mean staircase is shown as the full fine interpolating the steps in Fig. 4.1. The mean staircase allows us to define the mean level density p E) as... 

It can be shown that, under the assumption that Riemann s conjecture holds, the staircase function of Riemann zeros is given to leading order by (Whittaker and Watson (1927))... 

Dc polarograms, as obtained in historic practice, are records of the current flow at a DME as the potential is scanned linearly with time, but sufficiently slowly (1-3 mV/s) that the potential remains essentially constant during the lifetime of each drop. This constancy of potential is the basis for the descriptor dc in the name of the method. In more modem practice, the potential is applied as a staircase function, such that there is a small shift in potential (normally 1-10 mV) at the birth of each drop, but the potential otherwise... 

In fact, many instruments now generate linear scan waveforms as staircase functions with very small (< 0.2 mV) A , because it is simpler to do so with digital control systems. When ISE is reduced below the level of the noise on the waveform, the distinction between staircase and linear functions is lost. 

Figure 25 Product translational energy distributions for the fragments NO and O in NO2 photodissociation. experimental ref. 188), continuous line classical mkrocanonical equilibrium distribution, staircase function from the statistical adiabatic charnel model ref 101)...

Selection of scan parameters. As mentioned earher, the scan rate significantly influences the shape and quantitative features of voltammograms. Usually, a variation of v gives important information. There are, however, other scan-related parameters that may affect the current/potential curves. In computer controlled instruments, the ideal linear variation of E is often simulated using a staircase function (Fig. 7). The response from such an excitation function... 

Among the types of functions (linear function, power functions, exponential functions, step (staircase) function, sigmoid functions) that have been tested the last two variants are of particular interest. Step function has only two parameters the upper limit and the point of jump discontinuity (a certain critical value of... 

O Figure 2.12a shows the theoretical single-neutron energies (e, staircase function) versus neutron number (N)- The smooth, average curve e (N )removes local fluctuations. The shell... 

The trajectory of the neutron in terms of its kinetic energy is determined from its speed over each straight-line portion of its path. Thus, the trajectory in energy-space is a staircase function, as illustrated in Fig. 2.2. Note that each scattering collision causes a jump change in the neutron energy. As we have already mentioned, the history of any one neutron is of little importance to the over-all neutron problem. We are concerned only with the average behavior of the neutrons, and this can be... 

Here, we discuss a simple and informative plot for analyzing recurrent repair data. The plotting method provides a non-parametric graphical estimate of the number/cost of repairs per unit versus age. This estimate can be used to evaluate the repair rate of the population with respect to age and predict future number/cost of repairs. The MCF plot reveals most of the information that we are searching for in recurrent data (Nelson, 2003). A key advantage of the MCF plot is that it provides a single summary plot for a group of units, which is easily compared with other MCF plots of different populations. The MCF plot is the population of staircase functions for all the studied units. This staircase is a consequence of... 

Chose an equally-spaced increasing sequence of stress Oj = /Ao (i = 1, 2, 3,...) and denote the associated static and creep time to failure fs(f) = fs(Oi) and t (i) = 4(0i), respectively. During static tests, the strain rates are kept constant and the force deflection curves are considered linear until failure is reached. Therefore, the stress increase is assumed to be linear during the static test. The linear stress gradient is approximated by a staircase function with the steps o O3,05,..., shown in Figure 6.15. This means that the overall degradation in a CSR test can be considered as series of creep loads. From Eq. 6.9 it follows ... 

Information about the ionic states can be gained more easily by scanning with the second (ionizing) laser and monitoring the total ion intensity. This method is called REMPI-photoionization efficiency (REMPI-PIE) spectroscopy. The REMPI-PIE spectrum is a staircase function, and the conventional photoelectron spectrum can be obtained from its differentiation. 

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