Deterministic functions

It is possible to limit our choice for stochastic modeling by stationary, linear, nonlinear, and ergodic models in combination with deterministic function. In this case the following well studied models can be proposed for the accepted concept [1] ... 

State transitions are therefore local in both space and time individual cells evolve iteratively according to a fixed, and usually deterministic, function of the current state of that cell and its neighboring cells. One iteration step of the dynamical evolution is achieved after the simultaneous application of the rule (p to each cell in the lattice C. 

For the present purposes the deterministic function generator yields a constant signal x - 0, which means the summation output is identical with that of the noise generator. 

This is already an ordinary differential equation of the first order (/(x) is a known deterministic function), but nobody knows how to find G(x, t). 

In general, hie conditional fluxes are deterministic functions of V, x, and t. On hie right-hand side of these equations, the conditional fluxes are evaluated at hie current location of the notional particle in velocity-composition-physical space V = U (0, tp = ), x = X ( ). 

Note that, with this choice, the initial mean composition field is a deterministic function (see Fig. 6.4) 137... 

Specifying 5 as a deterministic function implies that the inevitable randomness in S is negligible when compared to the mean emission rate and to the effect of the randomness of the turbulent field on c.)... 

Currin, C., Mitchell, T., Morris, M., and Ylvisaker, D. (1991). Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments. Journal of the American Statistical Association, 86, 953-963. 

When we evaluate a deterministic function in a set of given points we assume that the true function y(x) is approximated by a function f(x) with some error ... 

Alternatively, one can represent a probabilistic function as a deterministic function with an additional random input. Then one writes /(/, r), where r is chosen from a given probability space, and (i) is a random variable on the original space. Usually, r is uniformly distributed on a given set, such as the bit strings of a certain length, because it represents results of unbiased coin flips. ... 

Definition 7.3 (Functional notation). The names of the algorithms in Definitions 7.1 and 7.2 can also be used for the (probabilistic or deterministic) functions realized by these algorithms. However, this notation is only used with test and verify. For the three other components, the following probabilistic functions are defined ... 

Consistency of initialization is achieved automatically in standard fail-stop signature schemes because the result is a deterministic function of broadcast messages. 

For discussions of the many pitfalls lying between this intuitive idea and a formal definition, see [Beav91, MiRo91]. (Furthermore, [MiRo91] only considers deterministic functions.)... 

Under these conditions, the functional version of the definitions in Section 7.1 already has the necessary stmcture with a common probability space where all the random variables can be defined All values are deterministic functions of the results of key generation, and thus ultimately random variables in the probability space defined by the random strings used in key generation (because the protocol itself and the parameters have been fixed). Some of these random variables are now given names. 

Remark 11.3. As each correct signature is a deterministic function of the part of the secret key (in the functional version) that is actually used and the message sequence. Rule (5) implies fory = 1,. .., i,... 

Proof. Let i < W+1 be fixed. The success probability of A is at least as large as the probability that its intermediate result proofs is a valid proof of forgery, because no unsuccessful stop for a smaller value i is possible. The functional version of the corresponding part of is the following deterministic function A m/- On input (par, acc, idspi / gut, Pk, 10 with par = ( V, V, 1 ) ... 

It is now used that is a deterministic function of SK. By Rule (5), this... 

In general, die conditional fluxes are deterministic functions of V, ip, x, and r. On the right-hand side of... 

An important consequence of the linear energy relations (12.31) is that the rate coefficients, are deterministic functions of the adsorption heat s = - AH ... 

The expressions derived so far are only valid if the noise is assumed to be stationary. However, it is unfortunate that this is not always the case. Particularly in a technique like chromatography, a non-stationary baseline drift is often present due to, for instance, stripping of the column, contamination of detectors, etc. The non-stationary drift, not to be confused with stationary low frequency noise with properties defined in probalistic terms, can often be considered as a deterministic function (linear, exponential, etc.). 

We can view the above problem as the sample average approximation of the true (or expected value) problem (11). The function g//x) is random in the sense that it depends on the corresponding sample. However, note that once the sample is generated, gf/(x) becomes a deterministic function whose values and derivatives can be compirted for a given vMue of the argument x. Consequently, problem (32) becomes a deterministic optimization problem and one can solve it with an appropriate deterministic optimization algorithm. 

The stochastic process d% (z) is driftless and thereupon a local martingale, if the deterministic function A t,z) solves the following ODE... 

Property (1) ensures that the paths are continuous in t and T. The properties (2) and (3) imply that we have a martingale for each time t together with unit variance. Finally, property (4) ensures that we work with a deterministic function c T,V) = dW(t, T)dW(t,V) fulfilling the attributes of a correlation function. 

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