Stieltjes integral

All difficulties of this nature could be avoided in subsequent discussions by replacing integrals of the form J 4(()Px(()< ( hy their corresponding Lebesque-Stieltjes integrals... 

In most cases of interest, this n + m order derivative can be written as an ordinary n + mth order derivative and some Dirac delta functions. Situations do exist in which this is not true, but they do not seem to have any physical significanpe and we shall ignore them. In any event, all difficulties of this nature could be avoided by replacing integrals involving probability density functions by their corresponding Lebesque-Stieltjes integrals. 

The Stieltjes integral appearing in Eq. (3-239) can be rewritten (at least formally) in the form... 

Moments of this conditional distribution can be written as standard Riemann Integrals of the pdf fx(z (N)) or as Stieltjes integrals of the cdf Fx(zf(N)) For example, the conditional expectation is written ... 

Comparing equation (46.15) with equation (46.7) we see the relation between II [x) and 6 x). It may be seen from these equations that is not a function l ut a Stieltjcs measure, and thnt the use of the Dirac delta function could be avoided entirely by a systematic use of Stieltjes integration. 

But there is some complication arising from the double Stieltjes integral. 

For enumeration of many of the Stieltjes integral properties, cf. [605] (p.105). In the following, we present some useful convolution relationships ... 

The space structure via the scalar product as the generalized Stieltjes integral... 

These zeros uk of QK(u) coincide with the eigenvalues of both the evolution matrix U and the corresponding Hessenberg matrix H from Eqs. (131) and (130), respectively. The zeros of Qk(u) are called eigenzeros. The structure of CM is determined by its scalar product for analytic functions of complex variable z or u. For any two regular functions/(m) and g(u) from CM, the scalar product in CM is defined by the generalized Stieltjes integral ... 

The solution (299) is the continued fraction of the order n. The system (293) is encountered in statistical mechanics [69] when applying the method of power moments to obtain a sequence of approximations to the partition function Q(/3) defined as the Stieltjes integral /0°° exp (-pE)dcp(E), where fi is a parameter proportional to the reciprocal temperature of the investigated system, d[Pg.218]

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r —> 0 (Stratonovich sense) the coefficients read [50]... 

Although in many homogeneous systems fluorophores have distinct and discrete decay constants for their fluorescence, in heterogeneous systems the luminescent molecules have different environments and consequently different energy levels and also pathways for the energy dissipation. Moreover, in the RET processes the distance between the donor and acceptor is not constant but may vary slightly. Then it can be expected that the lifetimes are not sharply defined but they are actually continuously distributed. Mathematically this means that instead of the sum in (40), we have to use a Riemann-Stieltjes integral ... 

Equations 9 and 10 can be solved for arbitrary strain (stress) histories to obtain the material stress (strain) response. Solution for the step-deformations requires use of the unit Heaviside function and is discussed in detail by Tschoegl (10) and by Findley, Onaran and co-workers. (21) Wineman and Rajagopal (22) deal with the step-strains, using Riemann-Stieltjes integrals. Other histories are more directly solvable. Also, in the linear theory the limits on the integral can be written from 0 to rather than from -00 to t (9,10,21,22). 

In view of (3.2.10) the right-hand side of this equation may be conveniently represented in terms of a Stieltjes integral converting the equation into the following evolution equation for breakage processes ... 

The definition of the Stieltjes integral can be found in any treatment of integral calculus. See, for example, Taylor (1955), p. 532. 

This theorem has been rigorously proved by Riemann and Stiel es. The integral is called the Riemann-Stieltjes integral. The term h itself is also a function of x. If /(x) and h x) are both bounded on a closed internal [a, b], the Riemann-Stiel es integral is then in the form jj /(x) Ah x). 

The stationary random processes can be expressed in terms of Fourier-Stieltjes integrals [e.g., 3, 21]. Hence, the buffeting loads, the bridge responses, and the active control variables v(a,t) and a(a,t) are expressed as follows ... 

Substituting Eqs. (16)-(17), (21)-(22) and (26)-(27) into Eqs. (1) and (2), and expressing the bridge responses, control force variables and the buffeting loads in terms of Fourier-Stieltjes integrals, one obtains the following results... 

The integral in (1.2.5) is really a special case (where e t) is differentiable) of a Stieltjes integral. Gurtin and Sternberg (1962) base their rigorous formulation of Linear Viscoelasticity on constitutive equations which have this Stieltjes form. We adopt a convenient notation of theirs, and write (1.2.5) as... 

A nonstationaiy stochastic process NSSP) X(t) can be expressed in the general form of a Fourier-Stieltjes integral as (Priestley 1987)... 

After the divisimi of the interval [0, t onto disjoint, contiguous subintervals, X(t) can be written down as the Riemann-Stieltjes sum. The limit, in the mean-square sense, of the sequence of such sums, is the mean-square Riemann-Stieltjes integral with respect to the counting process N t) or the stochastic integral ... 

In the Priestley spectral representation of nonstationary processes, a sample of the nonstationary stochastic process is defined by the Fourier-Stieltjes integral as follows ... 

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