Numbers absolute value

As we have seen, if is a root of a polynomial equation, then z is also a root. Recall that for real numbers, absolute value refers to the magnitude of a number, independent of its sign. Thus, 3.14 = — 3.14 = 3.14. We can also write - 3.14 = - 3.14. The absolute value of a complex number z, also called its magnitude or modulus, is likewise written as z - It is defined by... 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

The evolution of the nuclear wavepacket is also traced by a number of snapshots of the absolute values of the wavepacket, again integrating over the... 

While studying these results we have to pay attention to two features (1) In each case, these numbers must, in absolute value, be as close as possible to 1 and (2) two of these numbers have to be negative. Then, we also have to be able to justify the fact that it is the first two diagonal elements that have to be negative and it is the third one that must be positive. Note that these D terms are reasonably close to fulfilling the expected features just mentioned ... 

Set this threshold Lo a small positive constant (the default value is 10 Hartrcc), Tli is tli resh old is used by HyperCh cm to igu ore all two-cicetron repulsion in tegrals with an absolute value less th an th is value. Tli is option controls the performance of the SCF itera-lious and the accuracy of the wave function and energies since it can decrease the number of ealeulatcd Iwo-elcclrou integrals. 

Hamiltonian contains (fe2/2me r ) 32/3y2 whereas the potential energy part is independent of Y, the energies of the moleeular orbitals depend on the square of the m quantum number. Thus, pairs of orbitals with m= 1 are energetieally degenerate pairs with m= 2 are degenerate, and so on. The absolute value of m, whieh is what the energy depends on, is ealled the X quantum number. Moleeular orbitals with = 0 are ealled a orbitals those with = 1 are 7i orbitals and those with = 2 are 5 orbitals. 

The phenomena we discuss, phase separation and osmotic pressure, are developed with particular attention to their applications in polymer characterization. Phase separation can be used to fractionate poly disperse polymer specimens into samples in which the molecular weight distribution is more narrow. Osmostic pressure experiments can be used to provide absolute values for the number average molecular weight of a polymer. Alternative methods for both fractionation and molecular weight determination exist, but the methods discussed in this chapter occupy a place of prominence among the alternatives, both historically and in contemporary practice. 

Favorable Vapoi Liquid Equilibria. The suitabiHty of distiUation as a separation method is strongly dependent on favorable vapor—Hquid equiHbria. The absolute value of the key relative volatiHties direcdy determines the ease and economics of a distillation. The energy requirements and the number of plates required for any given separation increase rapidly as the relative volatiHty becomes lower and approaches unity. For example given an ideal binary mixture having a 50 mol % feed and a distillate and bottoms requirement of 99.8% purity each, the minimum reflux and minimum number of theoretical plates for assumed relative volatiHties of 1.1,1.5, and 4, are... 

Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms. 

In the real-number system a greater than h a > b) and b less than c(b < c) define an order relation. These relations have no meaning for complex numbers. The absolute value is used for ordering. Some important relations follow bl > x bl > y z Z9 z- + bgl bi - Zol Ibil — zo z- > (bl -I- lyl)/V2. Parts of the complex plane, commonly called regions or domains, are described by using inequalities. 

The assumption of small deformations is now made. Let the absolute value of the largest component of A be equal to or less than a small number c... 

For this to be constant, if L/dp doubles then must become one half, or u to be (1/2)° = 0.71 over the smaller pellets. Since the reaction rate is the same and the catalyst volumes are the same, the number of moles converted are the same. This constant number of moles will result in the 71% flow over the smaller pellets in 1/0.71 = 1.41 times larger reactant decrease than over the larger pills. The flow volumes are relative, not absolute values. 

It will be apparent from the discussions in the previous sections that an absolute value of reliability is at best an educated guess. However, the risk of failure determined is a quantitative measure in terms of safety and reliability by which various parts can be defined and compared (Freudenthal et al., 1966). In developing a reliable product, a number of design schemes should be generated to explore each for their... 

The frequency of occurrence of winds from each of the 36 direchons for the two sites is given in the second and third columns of Table 21-6. Knoxville has its maximum from 240°, with frequent winds also from adjacent points. There is a secondary maximum across the compass (in the vicinity of 60°). Nashville, on the other hand, seems to have only one principal maximum (from 180°). To explore the magnitude of the differences, the fourth column is the difference for each direchon. Summing the absolute values of the differences and expressing them as a percentage of the total number of hours of observations shows that the differences constitute 62% of the total. 

Note that the absolute value of Pp - p has been assumed. The negative value of this difference indicates that the droplet displacement is centripetal. The value of the Reynolds number corresponding to Ar = 3650 from Figure 10 is 45. Hence we can determine the radial settling velocity from the definition of the Reynolds number ... 

Random error varies in an unpredictable manner in absolute value and in sign when a large number of measurements of the same value of a quantity are made under essentially identical conditions. 

The pA a of dicyanamide is less than 1, so it exists mainly as the anion under the reaction conditions. aHA is a negative number, so the absolute value is taken.) This equation constitutes the R step. The quantity 3 uc deseribes the fraetional extent of the R 1 step, which is... 

To obtain the predicted shift for the carbon atom in methane, we subtract its absolute value from that of the reference molecule, resulting in a predicted shift of -3.9 ppm, which is in reasonable agreement with the experimental value of -7.0. Note the sign convention for shifts a negative number indicates that there is more shielding in the specified molecule than in the reference molecule, and a positive number indicates that there is less shielding than in the reference molecule. 

A variety of procedures can be used to determine Z, as a function of composition.2 Care must be taken if reliable values are to be obtained, since the determination of a derivative or a slope is often difficult to do with high accuracy. A number of different techniques are employed, depending upon the accuracy of the data that is used to calculate Z, and the nature of the system. We will now consider several examples involving the determination of V,- and Cpj, since these are the properties for which absolute values for the partial molar quantity can be obtained. Only relative values of //, and can be obtained, since absolute values of H and G are not available. For H, and we determine H, — H° or — n°, where H° and are values for H, and in a reference or standard state. We will delay a discussion of these quantities until we have described standard states. 

If AH% is negative, with an absolute value smaller than that of A//f, then the quantity of A// -I- Ais a negative number. In such a case the rate constant for the small reaction will have an apparent negative activation enthalpy (energy). That is, the rate will decrease with increasing temperature. 

In this expression, e is the fundamental charge (the absolute value of the charge of an electron), Z and z2 are the charge numbers of the two ions, r12 is the distance between the centers of the ions, and E0 is the vacuum permittivity (see inside the back cover for the value of this fundamental constant). 

Thus at small Pol the growth rate of the oscillations is negative and the capillary flow is stable. The absolute value of sharply increases with a decrease of the capillary tube diameter. It also depends on the thermal diffusivity of the liquid and the vapor, as well as on the value of the Prandtl number. 

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