Real response function

In the limit of small pressure perturbations, any kinetic equation modeling the response of a catalyst surface can be reduced to first order. Following Yasuda s derivation C, the system can be described by a set of functions which describe the dependence of pressure, coverage amplitude, and phase on T, P, and frequency. After a mass balance, the equations can be separated Into real and Imaginary terms to yield a real response function, RRF, and an Imaginary response function, IRF ... 

In the first class, the created ROM aims at approximating the real response function of the system as function of the input parameters. Once it is built, it is used to search for the points that are in the proximity of the limit surface using contour reconstruction based algorithms. Response function can be built using Support Vector Machines (SVM) (Mandelli Smith 2012) or Kriging based interpolators (Mandelli, Smith, Rabiti, Alfonsi, et al. 2013). 

The response-factor approach is based on a method in which the response factors represent the transfer functions of the wall due to unit impulse excitations. The real excitation is approximated by a superposition of such impulses (mostly of triangular shape), and the real response is determined by the superposition of the impulse responses (see Figs. 11.33 and 11.34). ... 

The response functions are obtained as derivatives of the real part of the time-averaged quasienergy Lagrangian ... 

Si is the laminar flame velocity, the function Z(co) is the heat response function Equation 5.1.16, whose real part is plotted in Figure 5.1.10. The function f(r, giJ is a dimensionless acoustic structure factor that depends only on the resonant frequency, a , the relative position, r, of the flame, and the density ratio Pb/Po-... 

Equations (4.15)—(4.17) and subsequent theoretical expressions for r(t) are the true anisotropy, which is defined here as the fluorescence response to an instantaneous light pulse when measured by an instrument with infinitely rapid temporal response. In a real experiment this is convoluted with the instrument response function, as discussed in a later section. 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

The deconvolution method we propose here is also parametric and is based on direct integral parameter estimation (ref. 27). We consider a "hypothetical" linear system S with input u = h, where h is the known weighting function of the real system S, and the output of S is assumed to be = , the known response function. Then by (5.66) we have... 

To determine excitation energies, it is sufficient to consider the real KS orbitals within the theory of the preceding section. In the response function (16) we replace the complex orbitals with real ones and use the fact that the terms with the occupied orbitals (j)ia and real form of the response function (we use the indices i and j for the occupied orbitals, while the indices a and b are used for the virtual ones)... 

Here, / is the fraction of the delayed nonlinear response, and 7Z is the memory function of the stimulated Raman effect. Parameterization by 7Z(t) sin Qr)e rT is often sufficient for ultrashort pulses [7], This simple formula has the advantage of easy implementation that avoids explicit calculation of the convolution integral. Often, an even simpler, exponential memory function is used, 7Z(r) e rr in simulations (see e.g. [28]). If the real memory function is sufficiently complex, a numerical convolution approach must be used to calculate the convolution. This is e.g. the case in silica [29],... 

The change in the population distribution ATj(t) of the i th level due to the pump excitation is ANi(t). Let us assume for simplicity that a single AN(t) population is induced. There is a new absorption associated with this population, given by Aa(ui) = a(uj)AN(t). [Usually many excited states i are involved, and more than one transition starts from each state with rates proportional to aij(u>)ANi, so that the real equation is a sum over i and j] AN(t) depends on the material and is represented by the impulsive response function A(t) (i.e., the response to a delta-like pulse). Given the finite duration of the pulse, shorter but not negligible compared with A(t), the real change in a is described by a convolution ... 

The spectral response function x(ffl), a function of the angular frequency co, is the sum of an in-phase real part Xi(even function of co, and an out-of-phase (dissipative) imaginary part 2(ft)), which is an odd function... 

The time-gated heterodyne signal obtained in a phase-locked measurement gives the real and the imaginary parts of the response function ... 

Competing processes are another concern in real experiments. These processes result from interactions with different time orderings of the pulses and with perturbation-theory pathways proceeding through nonresonant states. They correspond to the constant nonresonant background seen in CARS and other frequency-domain spectroscopies. These nonresonant interactions are only possible when the excitation and probe pulses are overlapped in time, so they add an instantaneous component to the total material response function... 

When written with the help of the Tl matrix as in (19), from (20) the OR parameter and other linear response properties are seen to afford singularities where co = coj, just like in the SOS equation (2). Therefore, at and near resonances the solutions of the TDDFT response equations (and response equations derived for other quantum chemical methods) yield diverging results that cannot be compared directly to experimental data. In reality, the excited states are broadened, which may be incorporated in the formalism by introducing dephasing constants 1 such that o, —> ooj — iT j for the excitation frequencies. This would lead to a nonsingular behavior of (20) near the coj where the real and the imaginary part of the response function varies smoothly, as in the broadened scenario at the top of Fig. 1. 

Ideally a single-particle counter should have a response function monotonically dependent on particle size exemplified by that portion of Figure I for n = 1.57 — 0.56i and a < 17 but with zero response for all other a. The real situation however is quite different as indicated. Two types of deviations from desired behavior can be identified. 

Figure 1.2. (A) A schematic diagram depicting the processes in close proximity to metals (< 10 nm) involved in Metal-Enhanced Fluorescence enhanced absorption and coupling to surface piasmons. (B) Emission spectra of FITC deposited onto SIFs and glass. The inset shows the real-color photographs of FITC emission from these surfaces. (C) Intensity decays for FITC on both glass and SiFs. IRF Instrument Response Function.

Response functions. The elementary Cooper and Peierls logarithmic divergences (19) of the interacting electron gas are also present order by order in the perturbation theory of response functions in the 2kf density-wave and superconducting channels. A scaling procedure can thus be applied in order to obtain the asymptotic properties of the real part of the retarded response functions which we will note Xm.( )- is convenient to introduce auxiliary response functions noted x ( ) [107], which are defined... 

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