Coincidence function

One more function needs to be defined, the coincidence function, symbolized by 0- As the name implies, this function gives a 1 output whenever both inputs are the same, either all O s or all I s. Its symbol is also illustrated in Figure 23.1 IB. The reader should construct its logic diagram using AND, OR, NAND, NOR, and INVERT gates in a manner similar to that just presented for the XOR function. 

Using this inner product, we then define the following metric or coincidence function ... 

FIGURE 4.3 Neighbor-joining tree of all explicitly named NIAID Category A, B, or C bacterial pathogens as resolved by base-specific fragmentation of the Lane-AB amplicon and spectral distances derived from the presented coincidence function Separation of some of the unresolved clusters may be improved by further mass spectrometric analysis of the Lane-BC sequence region. 

Fig. 8.8 Sensitivity functions of enzyme Cln2 normalised to unit maximum. The thick solid line indicates 10 coinciding functions having shape A, whilst the dotted line shows 38 coinciding functions having shape B. The shapes of other 9 sensitivity functions (thin solid line) are not similar to either shapes A or B (Lovrics et al. 2008)...

Finally, the geotextile must have the required filtration/hydraulic characteristics to provide the coincident functions of separation, filtration, and drainage. Properties required for separation and filtration functions are related to opening characteristics... 

Apolar stationary phases having no dipolar moments, that is their center of gravities of their positive and negative electric charges coincide. With this type of compound, the components elute as a function of their increasing boiiing points. The time difference between the moment of injection and the moment the component leaves the column is called the retention time. 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes (a, b, c). In order to determine the parity of the molecule through inversions in SF, we first rotate all the displacement vectors... 

The initial conditions of system (20) coincide with those for the original equations X/,(0) = X" and V/i(0) = V . Appropriate treatments, as discussed in [72], are essential for the random force at large timesteps to maintain thermal equilibrium since the discretization S(t — t ) => 6nml t is poor for large At. This problem is alleviated by the numerical approach below because the relevant discretization of the Dirac function is the inner timestep At rather than a large At. 

In the overview to this chapter we noted that the experimentally determined end point should coincide with the titration s equivalence point. For an acid-base titration, the equivalence point is characterized by a pH level that is a function of the acid-base strengths and concentrations of the analyte and titrant. The pH at the end point, however, may or may not correspond to the pH at the equivalence point. To understand the relationship between end points and equivalence points we must know how the pH changes during a titration. In this section we will learn how to construct titration curves for several important types of acid-base titrations. Our... 

Let a solid body occupy the domain fl C with the smooth boundary T. The solid particle coincides with the point x = xi,X2,xs) G fl. An elastic solid is described by the following functions ... 

Now we intend to derive nonpenetration conditions for plates and shells with cracks. Let a domain Q, d B with the smooth boundary T coincide with a mid-surface of a shallow shell. Let L, be an unclosed curve in fl perhaps intersecting L (see Fig.1.2). We assume that F, is described by a smooth function X2 = i ixi). Denoting = fl T we obtain the description of the shell (or the plate) with the crack. This means that the crack surface is a cylindrical surface in R, i.e. it can be described as X2 = i ixi), —h < z < h, where xi,X2,z) is the orthogonal coordinate system, and 2h is the thickness of the shell. Let us choose the unit normal vector V = 1, 2) at F,, ... 

The inequality like (1.59) is called a variational inequality. It was obtained from a minimization problem of the functional J over the set K. In the sequel we will look more attentively at a connection between a minimization problem and a variational inequality. Now we want to underline one essential point. We see that the problem (1.58) is more general in comparison with the minimization problem on the whole space V. It is well known that the necessary condition in the last problem coincides with the Euler equation. The variational inequality (1.59) generalizes the Euler equation. Moreover, ior K = V the Euler equation follows from (1.59). To obtain it we take U = Uq +u and substitute in (1.59) with an arbitrary element u gV. It gives... 

The set K in Theorem 1.11 may coincide with the space V. For a differentiable functional J it guarantees the solvability of the Euler equation... 

Observe that variational inequality (3.106) is valid for every function X G 82- It means that a solution % to problem (3.106) with 9 G Si coincides with the unique solution to problem (3.100) with the same 9] i.e. problems (3.100) and (3.106) are equivalent. For small 5, we write down an extra variational inequality for which a solution exists, and demonstrate that the solution coincides with the solution of variational inequality (3.98). 

Let us consider a thin homogeneous isotropic beam of thickness 2s. We assume that the beam mid-line coincides with the segment (0,1) of the axis X. At the point y = 1/2, the beam has an inclined cut as a segment having the angle a with the vertical line, 0 < tana < 2e). We look for the function % = (W, w) of horizontal displacements W(x) and vertical displacements w(x) provided that the external forces g(x),f(x) are given. The condition of clamped edges... 

The Ubbelohde viscometer is shown in Figure 24c. It is particularly useful for measurements at several different concentrations, as flow times are not a function of volume, and therefore dilutions can be made in the viscometer. Modifications include the Caimon-Ubbelohde, semimicro, and dilution viscometers. The Ubbelohde viscometer is also called a suspended-level viscometer because the Hquid emerging from the lower end of the capillary flows down only the walls of the reservoir directly below it. Therefore, the lower Hquid level always coincides with the lower end of the capillary, and the volume initially added to the instmment need not be precisely measured. This also eliminates the temperature correction for glass expansion necessary for Cannon-Fen ske viscometers. 

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