Density of primes

Prekey generation is of the same order of complexity as key generation in ordinary digital signature schemes It is dominated by the primality tests needed for the generation of two primes, q and p. This means approximately one exponentiation per number tested for primality with the Rabin-Miller test. Hence the number of exponentiations is determined by the density of primes of the chosen size (see Section 8.1.5) however, many numbers can be excluded by trial division as usual. 

Clearly, some paths are more strategic than others. A prime factor of N cannot exceed It makes little sense, however, to commence a factoring job by searching for and testing primes near This is because the density of primes scales as 1/ loge(A). For arbitrary N, it is more likely that low-valued integers have prime status and prove to be viable factors. After all, 50% of integers have 2 as a prime factor. 

M. J. Jacobson Jr. and H. C. Williams, New qnadratic polynomials with high densities of prime valnes. Math. Comput. 72 (2003) 499-519. 

The prime function of the saturated acid is to space out the double bonds and thus reduce the density of cross-linking. Phthalic anhydride is most commonly used for this purpose because it provides an inflexible link and maintains the rigidity in the cured resin. It has been used in increasing proportions during the past decade since its low price enables cheaper resins to be made. The most detrimental effect of this is to reduce the heat resistance of the laminates but this is frequently unimportant. It is usually produced by catalytic oxidation of o-xylene but sometimes naphthalene and is a crystalline solid melting at 131°C. 

Selecting a rigorous and convenient quantitahve parameter characterizing the catalyhc achvity, A, is of prime importance when studying electrocatalytic phenomena and processes. The parameter usually selected is the current density, i (in AJan ), at a specified value of electrode poteuhal, E. The current density is referred to the electrode s true working surface area [which can be measured by the Brunauer-Emmett-TeUer (BET) or other methods]. Closely related to this true current density is another parameter, known as the turnover number y (in s ), and indicating the number of elementary reachon acts performed or number of electrons transferred in unit time per surface atom (or catalytic surface site) of the catalyst. 

Table 13.1). In the solid P(CH4) > P(CD4) but the curves cross below the melting point and the vapor pressure IE for the liquids is inverse (Pd > Ph). For water and methane Tc > Tc, but for water Pc > Pc and for methane Pc < Pc- As always, the primes designate the lighter isotopomer. At LV coexistence pliq(D20) < Pliq(H20) at all temperatures (remember the p s are molar, not mass, densities). For methane pliq(CD4) < pLiq(CH4) only at high temperature. At lower temperatures Pliq(CH4) < pliq(CD4). The critical density of H20 is greater than D20, but for methane pc(CH4) < pc(CD4). Isotope effects are large in the hydrogen and helium systems and pLIQ/ < pLiQ and P > P across the liquid range. Pc < Pc and pc < pc for both pairs. Vapor pressure and molar volume IE s are discussed in the context of the statistical theory of isotope effects in condensed phases in Chapters 5 and 12, respectively. The CS treatment in this chapter offers an alternative description.

Substituted nickel oxides, such as LiNii j /3ojAl/l2, are prime candidates for the cathode of advanced lithium batteries for use in large-scale systems as required for hybrid electric vehicles. On charging these mixed oxides the nickel is oxidized first to Ni + then the cobalt to Co +. SAFT has constructed cells with these substituted nickel oxides that have been cycled 1000 times at 80% depth of discharge with an energy density of 120—130 Wh/kg. ... 

In applications where high power density or thermal management is of prime importance, hard- switched converters are not feasible using conventional Si components. In these cases, resonant or quasi-resonant (also termed soft-switching ) topologies can be used. The electrical resonance is obtained through parasitic... 

In the second example we already treated a problem where the gravitational constant is of prime importance, due to extreme difference in densities in the gas/liquid system, provided that the Froude nrunber is low Fr < 0.65. 

The proper choice of a solvent for a particular application depends on several factors, among which its physical properties are of prime importance. The solvent should first of all be liquid under the temperature and pressure conditions at which it is employed. Its thermodynamic properties, such as the density and vapour pressure, and their temperature and pressure coefficients, as well as the heat capacity and surface tension, and transport properties, such as viscosity, diffusion coefficient, and thermal conductivity also need to be considered. Electrical, optical and magnetic properties, such as the dipole moment, dielectric constant, refractive index, magnetic susceptibility, and electrical conductance are relevant too. Furthermore, molecular characteristics, such as the size, surface area and volume, as well as orientational relaxation times have appreciable bearing on the applicability of a solvent or on the interpretation of solvent effects. These properties are discussed and presented in this Chapter. 

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