### Table 3. Ansatzes and reduced equations

"... In PAGE 16: ... The last step of the reduction algorithm is the construction of ansatzes for the function u(t; x) and reduction of both invariant equation and initial conditions to systems of ordinary di erential equations and algebraic equations, correspondingly. We present the list of ob- tained results in Table3 , where Fi = Fi(t; apos;1(t); apos;2(t)); i = 1; 2, the functions apos;1(t); apos;2(t) are new dependent variables and the symbol g stands for the converse of the function f(ux), i.e.... ..."

### Table 4.2 and 4.3 show the results for two matching tasks, the localization of faces and hand postures. For both tasks matching within the approach described in this chapter is compared to the matching with bunch graphs as described in (Wiskott et al., 1997; Kr uger et al., 1997; Triesch and von der Malsburg, 1996). The extremely di cult face test set contains 120 frontal faces with uncontrolled illu-

### Table 1: This table lists the aI and the K coe cients for di erent telepar- allel Lagrangians. GRk, spelled out in the rst column, represents a viable gravitational model, the same is true for the von der Heyde case. Obviously, the Kaniel-Itin Lagrangian, in the framework of the conventional variational procedure, is not viable. We have KI = YM + YMy and vdH = YM ? YMy.

"... In PAGE 3: ... 4 we give a short overview of teleparallelism theories and the rele- vant quadratic Lagrangians. We will discuss the viable set of Lagrangians and display the results in Table1 . We will show that the KI-Lagrangian, for arbitrary variations, is not viable.... In PAGE 15: ... Therefore, on a phenomenological level, all viable teleparallel theories are indistinguishable. In Table1 we have listed some quadratic torsion Lagrangians. It follows from the end of the last subsection, see also (63), that only the teleparallel equivalent of Einstein apos;s general relativity is both viable and locally Lorentz invariant.... In PAGE 18: ...18 Its Euler-Lagrange equation can be read o from (56) by substituting the coe cients (59): ?2`2 = 2 # ^ d ? ?# ^ d# + 4 d ?d# + 2e c ?d# ^ ?d# ? 4 ?e cd# ^ ?d# ? 2d# ^ ? ?# ^ d# ? e c # ^ d# ^ ? (d# ^ # ) + 2 ?e cd# ^ # ^ ? (d# ^ # ) : (65) In the rst line, we displayed the second derivatives of the gravitational potential. As we already saw in Table1 , this eld equation is not viable. If we use the the constrained variations in (64), then we commute and the star ? and nd by simple algebra: `2 (VKI + Lmat) = # ^ ?? + `2 = 0 : (66) Since the variations are constrained, we have to turn to our proposition and to use (35): # = ! # , with !( ) = 0.... ..."

### Table 6: Solution of the Riemann problems for nonpolytropic van der Waals gas provided by integrating the Euler equations with the numerical method [15].

"... In PAGE 26: ...245020 Table 5: Solution of the Riemann problems for the nonpolytropic van der Waals gas provided by the proposed method. These values are compared with those calculated by the approximate Riemann solver for nonideal gases of Guardone and Vigevano [15] using a uniform grid of 1000 points, shown in Table6 . Since in the numerical solution there is a slight jump in velocity and in pressure across the contact discontinuity, the corresponding arithmetic average is reported in the table.... ..."

### Table 2: zur Verf ugung-verb collocations as prenominal modi ers Tokens Occurrence Frequency

### Table 2: Results for the NEO planning domain.

2001

"... In PAGE 4: ...1 34.8 16% Table2 summarizes the results. The helicopter transport preference yields plans that have shorter execution durations (38.... ..."

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