### Table 4: Series Expansion Coe cients for the 2 2 Lattice

### Table 5: Series Expansion Coe cients for the 3 3 Lattice

### TABLE I Comparison between the polynomial phase coefficients and the Taylor series expansion coefficients

2001

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### Table 2: The fractional errors on the various f(1) computed using the series expansions up to order 10, for values of 1:05 z 1:4.

"... In PAGE 14: ... Adding (43) and (45) gives the series for the j = 0 total cross section and adding (44), (46) gives the series for the j = 2 total cross section. In Table2 we present the fractional error on the series for f(1)(+; +), f(1)(+; ?), f(1) unp relative to the result obtained using numerical integration, for various values of z in the region 1:05 z 1:4. For z .... ..."

### Table 1: Seidel aberrations are described by power series expansion of the aberration function (n; m). The position of best focus is shifted by (Rf cos o; Rf sin o; Zf).

"... In PAGE 6: ... As can be seen from (7) five terms with a + b + c = 2 exist. In Table1 a summary of the Seidel aberrations including the resulting image shift is described. Although the expansion of (7) is a general representation of the aberration function in that any aberra- tion function (n; m) can be represented, other polynomials are more suited to describe higher-order aberration terms.... ..."

### Table 1: Series expansions for the zero{ eld renormalized magnetization M0 = Pm M0 m m and two{point correlation function E0 = Pm E0 m m. Only the non{zero contributions are displayed.

1994

"... In PAGE 9: ...s odd (even). And if it is vertical, Ly = 2L ? 2 and Lx = (L + 1)=2 (Lx = (L + 2)=2). In this way we have been able to obtain the series (5a, 5b) up to order O( 30). The result is displayed in Table1 . In this algorithm we need to deal with very large numbers, much larger than the precision of the computer (32 bits in our case).... ..."

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### Table 2 shows the results of applying the stability tests from Theorems 2, 3, 4 (using a nite series expansion of the multiplier), and the o -axis circle criterion, for various values of H1 = L in the observer-based anti-windup scheme. The corresponding multipliers X ? W (s)

1995

"... In PAGE 23: ... Table2 : Application of various AWBT stability conditions establishing stability for the four cases above, using Theorem 4 and the nite dimensional approximation of the multiplier, as discussed in x3.2.... ..."

Cited by 4

### TABLES TABLE I. The coe cients in the expansions of nk, vk, Q, S, VQ and V for fully penetrable disks. Recall that n1 = e?4 from (29) and that v1 = V1 trivially. The series expansions for vk, VQ and VS are expressed in units of V1, the area of a single disk. The expansions for Q, S, VQ and VS are derived from (10), (12), (14) and (15), respectively. 0 1 2 3 4

1996

Cited by 1

### Table 1 Threshold values for random K-SAT. Bold numbers are the results of the population dynamics algorithm. (0) d is the value predicted by the rst moment expansion of the cavity equations (sec. 6.3), (r) c is the result of a series expansion in quot; = 2 K of the cavity equations up to order r (secs. 6.2 and A). Note that all reported values c(K) fall between the best rigorously known upper and lower bounds.

2006

"... In PAGE 22: ...he cavity equations up to order r (secs. 6.2 and A). Note that all reported values c(K) fall between the best rigorously known upper and lower bounds. Table1 shows the results. Since c for K = 3 is the most \prominent quot; thresh- old we spent a bit more CPU power to increase its accuracy.... In PAGE 27: ... Fig. 8 and Table1 show that the seventh order expansion gets actually very close to the numerical values. For K = 3 the deviation of the seventh order asymptotic expansion from the numerical value is less than 1%, and for K 4 this deviation is even smaller.... In PAGE 28: ... (39). The results are the values for (0) d in Table1 . The values for (0) d agree perfectly with the exact values d (within the error bars of the latter), even for K = 3, although the non-trivial distributions A(x) and B(y) that appear right above d are not -like.... In PAGE 35: ... The quality of the expansion up to seventh order can be seen in Fig. 8 and Table1 . Note that there exist also nonanalytic terms in quot;, because we dropped some corrections of order which in turn behaves as quot;1=(2 quot;).... ..."

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### Table 1: Frequency contents of the function f1(t) The representation of a periodic function (or of a function that is de ned only on a nite interval) as the linear combination of sines and cosines, is known as the Fourier series expansion of the function. The Fourier transform is a tool for obtaining such frequency and amplitude information for sequences and functions, which are not necessarily periodic. (Note that sequences are just a special case of functions.) 2

1995

"... In PAGE 3: ... The function f1(t) consists of sines and cosines of 3 frequencies. 1 2 3 4 5 c -6 -4 -2 0 2 4 6 1 2 3 4 5 d -6 -4 -2 0 2 4 6 1 2 3 4 5 a -6 -4 -2 0 2 4 6 1 2 3 4 5 b -6 -4 -2 0 2 4 6 Figure 1: A plot of f1(t), (d), and its components (a; b; c), for t = 0::5 Thus, the frequency analysis of f1(t), can be summarized in a table such as Table1 , which provides for each frequency of f1 the amplitude of the sine wave and of the cosine wave... In PAGE 4: ... Both are easily derived from the Taylor series expansion of cos, sin, and e . Through addition and subtraction they can be rewritten as: cos( ) = ei + e?i 2 sin( ) = ei ? e?i 2i (3) Hence, we can substitute the sin and cos expressions of equation 1 by the respective expressions of equation 3 and get: f(t) = n X k=1[Ak 2 (e2 i!kt + e?2 i!kt) + Bk 2i (e2 i!kt ? e?2 i!kt)] (4) If we denote: Ck = Ak?iBk 2 k gt; 0 Ck = Ak+iBk 2 k lt; 0 C0 = 0 !k = ?!?k k lt; 0 (5) we can again rewrite f(t): f(t) = n X k=?n[Cke2 i!kt] (6) Under this new notation we can rewrite the frequency analysis of Table1 as shown in Table 2. k Frequency (!k) Ck ?3 ?1 2 ?2 ?2 2i ?1 ?1=2 i=4 0 0 0 1 1=2 ?i=4 2 2 ?2i 3 1 ?2 Table 2: Another form of frequency contents of the function f1(t) Further manipulation of equation 6 is based on using the polar notation for complex numbers, that is: x + iy = r(cos( ) + isin( )) = rei where r = jx + iyj = qx2 + y2 and tan( ) = y x... ..."

Cited by 9