Pair exponentiation

The simplest way to consider the formation of ion pairs was discussed in Ref. [29]. It was shown that the concentration of ion pairs exponentially increases with the decrease of the dielectric constant of the solvent . This effect should be taken into account in the theory of collapse of polyelectrolyte networks, because effective values of in the collapsed state are usually much less than in the swollen state (e depends mainly on the water content which is much larger in swollen networks). This effect has not been taken into account in the theories developed so far. 

The basic functions with a bundling property in the discrete-logarithm case are simply products of several exponentiations. This is called tuple exponentiation or, if the number of components is known to be jti,/x-tuple exponentiation. Pair exponentiation seems to have been first used like this in [B0CT88], larger tuples in [ChHP92]. 

In this section, only 2-tuple or pair exponentiation is treated other statements follow where they are needed. 

After all the mathematical and computational preparations, it is quite clear that pair exponentiation in a family of groups of prime order where the discrete logarithm is hard yields a collision-intractable family of bundling homomorphisms. (It was first used in this way in [HePe93].) It only remains to be decided how the security parameters are related, because the family of groups has only one, k, whereas the family of bundling homomorphisms has two, say k and T for distinction. The following facts are known ... 

The bundling degree of pair exponentiation is the group order q (Corollary 8.4). 

Construction 8.46. Let a family of groups of prime order be given. The corresponding family of pair exponentiations as hiding homomorphisms... 

Now tuple exponentiation is turned into collision-intractable families of fixed-size hash functions. This was first done in [ChHP92] the construction was extended for the use in incremental signature schemes in [BeGG94]. In particular, one can use pair exponentiation, but larger tuples tiun out to be more efficient. 

Of course, in the present context, the scheme is constructed by using the family of pair exponentiations as bundling homomorphisms from Section 8.5.3 in the general construction framework from Section 9.2. 

BundFam, the collision-intractable family of bundling homomorphisms, is given by Construction 8.44, i.e., pair exponentiation in the given family of groups. Its... 

Main key generation Generation of 4 random numbers and two pair exponentiations, which require less than 4k modular multiplications of numbers of length 1. 

Verifying proofs of forgery Either one pair exponentiation to compute gX-x g y-y (jgss tjjgjj 2k modular multiplications of numbers of length /) or, with Remark 9.13, one exponentiation. ... 

Equation (B2.4.13) is a pair of first-order differential equations, so its fonnal solution is given by equation (B2.4.14)), in which exp() means the exponential of a matrix. 

The above potential is referred to as a Lennard-Jones or 6-12 potential and is summed over all nonbonded pairs of atoms ij. The first positive term is the short range repulsion and the second negative term is the long range attraction. The parameters of the interaction are Aj and B... The convenient analytical form of the 6-12 potential means that it is often used, although an exponential repulsion term is usually considered to be a more accurate representation of the repulsive forces (as used in MM-t). 

Alder and Wainwright gave MD treatments of particles whose pair potential was very simple, typically the square well potential and the hard sphere potential. Rahman (1964) simulated liquid argon in 1964, and the subject has shown exponential growth since then. The 1970s saw a transition from atomic systems... 

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

Markovian perturbation theory as well as impact theory describe solely the exponential asymptotic behaviour of rotational relaxation. However, it makes no difference to this theory whether the interaction with a medium is a sequence of pair collisions or a weak collective perturbation. Being binary, the impact theory holds when collisions are well separated (tc < to) while the perturbation theory is broader. If it is valid, a new collision may start before the preceding one has been completed when To < Tc TJ = t0/(1 - y). 

Here AR is R(n) — J (l), in A., and n is the number of shared electron pairs involved in the bond. This logarithmic relation is, of course, to be expected in consequence of the exponential character of interatomic forces. 

Figure 3. (+)-Anatoxin-a (AnTx) and ACh induced single ion channel currents in isolated frog muscle fibers. Open channels with 32 pS conductance are downward deflections (inward current at hyperpolarized potentials). The currents shown on the left are all at one potential. The duration of channel open events had a similar voltage-dependence for both ACh and (+)-anatoxin-a. With ACh, the events were most often singular, while with (+)-anatoxin-a the events were shorter and were more frequently paired so that the mean duration of the exponentially distributed open times and selected membrane holding potentials was approximately one-half, independent of the concentration of the agonist applied.

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