Continuous variable

In the classical limit, the triplet of quantum numbers can be replaced by a continuous variable tiirough the transformation... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

The semiclassical version is obtained by the substitution mvb = ( + so that = ( + i) in tenns of the impact parameter b. Regarding as a continuous variable,... 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

Name Formula for continuous variables Formula for binary... 

Analog signal. A signal that can be expressed as a continuously variable mathematical function of time. 

For continuous variables without constraints, optimum values can be found mathematically by usiag advanced calculus. Numerically, such optimum values can be found by computer programs, a number of which are available with different degrees of sophistication (46,47). 

Quahtative description of physical behaviors require that each continuous variable space be quantized. Quantization is typically based on landmark values that are boundary points separating qualitatively distinct regions of continuous values. By using these qualitative quantity descriptions, dynamic relations between variables can be modeled as quahtative equations that represent the struc ture of the system. The... 

When X represents a continuous variable quantity, it is sometimes convenient to take the total or relative frequency of occurrences within a given range of x values. These frequencies can then be plotted against the midvalues of x to form a histogram. In this case, the ordinate should be the frequency per unit of width x. This makes the area under any bar proportional to the probability that the value of x will he in the given range. If the relative frequency is plotted as ordinate, the sum of the areas under the bars is unity. 

If X is a continuous variable and the interval ranges are made smaller and smaller, a smooth cui ve will eventually result. The area under such a cui ve between X and Xo represents the probabihty that a randomly selected item will have a value of x lying in the range X to Xo-This is the information that is desired. 

Most often the hypothesis H concerns the value of a continuous parameter, which is denoted 0. The data D are also usually observed values of some physical quantity (temperature, mass, dihedral angle, etc.) denoted y, usually a vector, y may be a continuous variable, but quite often it may be a discrete integer variable representing the counts of some event occurring, such as the number of heads in a sequence of coin flips. The expression for the posterior distribution for the parameter 0 given the data y is now given as... 

Table 2.1-1 compares the ordinary algebra of continuous variables with the Boolean algebra of 1 s and Os. This table uses the symbols and -h for the operations of intersection (AND) and union (OR) which mathematicians represent by n and u respectively. The symbols and -f which are the symbols of multiplication and addition, are used because of the similarity of their use to AND and OR in logic. 

Analog A continuously variable function in a control system ranging from off to full flow. 

Note that if Bn is zero, then T13 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D11 =D22- Because Tn, T12, and T22 are functions of both m and n, no simple conclusion can be drawn about the value of n at buckling as could be done for specially orthotropic laminated plates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of N with respect to m and n. 

Figure 19.8.5 displays the graph of tliis cdf. Anodier example of a cdf of a continuous random time variable is shown in Figure 19.8.6. A cdf of a continuous variable (a normal distribudon - to be reviewed in the next chapter) is provided in Figure 19.8.7.

The orbitals and orbital energies produced by an atomic HF-Xa calculation differ in several ways from those produced by standard HF calculations. First of all, the Koopmans theorem is not valid and so the orbital energies do not give a direct estimate of the ionization energy. A key difference between standard HF and HF-Xa theories is the way we eoneeive the occupation number u. In standard HF theory, we deal with doubly oecupied, singly occupied and virtual orbitals for which v = 2, 1 and 0 respectively. In solid-state theory, it is eonventional to think about the oecupation number as a continuous variable that can take any value between 0 and 2. 

Beachley, N. H., and Frank, A. A. (1980). Principles and Definitions for Continuously Variable Transmissions, with Emphasis on Automotive Applications. A.SME paper 80-C2/DET-95. 

This memory erasure problem is sometimes called the credit assignment problem [peret92l. Fortunately, there is an easy way out. We merely generalize the binary (on/off) McCulloch-Pitts neuronal values to continuous variables by smoothing out the step-function threshold. 

The amount of energy a molecule contains is not continuously variable but is quantized. That is, a molecule can stretch or bend only at specific frequencies corresponding to specific energy levels. Take bond-stretching, for example. Although we usually7 speak of bond lengths as if they were fixed, the numbers... 

Heating mantles. These consist of a flexible knitted fibre glass sheath which fits snugly around a flask and contains an electrical heating element which operates at black heat. The mantle may be supported in an aluminium case which stands on the bench, but for use with suspended vessels the mantle is supplied without a case. Electric power is supplied to the heating element through a control unit which may be either a continuously variable transformer or a thyristor controller, and so the operating temperature of the mantle can be smoothly adjusted... 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

We shall in this chapter be most concerned with the following example of Hilbert space. Each element /> is a complex-valued numerical function f(x) of one or more continuous variables represented collectively by the symbol x, such that the integral of its square modulus exists ... 

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