### Table 1: High-order behaviour of perturbation expansion coe cients (see also explanations in the text)

"... In PAGE 5: ... Ogievetsky found the Borel sum in the form Z 1 0 e?m2ta(t; e)dt (6) with a(t; e) = ? 1 8 2t3[etH cot(etH) ? 1 ? (etH)2=3]; (7) which coincides with the compact expression obtained by Schwinger [5]. This important result shows that a divergent perturbative expansion does not signal an inconsistency in a theory; it also shows that there are special - but realistic - cases of Borel summability in QED, although general considerations indicate Borel non-summability (see [7, 9, 10], and Table1 and a discussion in section 2 of the present paper). Gradually, the Borel summation techniques became widely adopted in quantum theory.... In PAGE 8: ... It should be considered as very fortunate that, simultaneously, analyticity plays a crucial role also as a mathematical condition reducing the ambiguity of asymptotic series. In Section 2 of the present paper, we discuss in detail the interplay between large-order behaviour of a series (as listed in Table1 ) and the analyticity properties of the function expanded; it turns out that a balance between these two concepts is needed for a unique determination of f(z) from (3), in the sense that if more analyticity of f(z) is available, one can a ord a more violent behaviour of the an, and vice versa. In Section 3 we focus on some practical aspects of the operator-product expansion, in particular on the problem of how the remainder after subtraction of the rst n terms from the function expanded depends on the distance from euclidean region, provided that an estimate on the remainder in euclidean region is known.... In PAGE 9: ... There are types of diagrams for which the amplitude itself grows like n! [21]. A survey of the large-order behaviour of expansion coe cients in some typical theories and models is given in the Table1 . As subtle cancellations among higher-order graphs may occur, the expressions in the third column of Table 1 may sometimes give an upper bound rather than the actual high-order behaviour of the coe cients.... In PAGE 9: .... There are types of diagrams for which the amplitude itself grows like n! [21]. A survey of the large-order behaviour of expansion coe cients in some typical theories and models is given in the Table 1. As subtle cancellations among higher-order graphs may occur, the expressions in the third column of Table1 may sometimes give an upper bound rather than the actual high-order behaviour of the coe cients. Table 1 is intended for rst information and should not be used for systematic anal- yses because some important conditions or restrictions could not be mentioned.... In PAGE 9: ... As subtle cancellations among higher-order graphs may occur, the expressions in the third column of Table 1 may sometimes give an upper bound rather than the actual high-order behaviour of the coe cients. Table1 is intended for rst information and should not be used for systematic anal- yses because some important conditions or restrictions could not be mentioned. A brief explanation of its use is given below.... In PAGE 12: ... To organize the diagrams in classes, the expansion parameter 1=Nf is used, where Nf is the number of fermion species; as a consequence, diagrams suppressed in the 1=Nf expansion are not suppressed for large n and, consequently, no nite order in the 1=Nf expansion provides the correct behaviour in n in the full theory. Table1 shows the large-order behaviour of the vacuum polarization, rn being the coe cient of i n+1 in the perturbative expansion and 2 = 99=(8N2 f ) . The authors discuss extension of the formalism to non-abelian gauge theories and expect a similar result.... In PAGE 12: ... The series is not Borel summable, all its terms being positive. A look at the third column of Table1 shows that most of the theories listed are characterized by an n! large-order behaviour. This does not mean that all of them can be cured by the same resummation method: large-order behaviour is just one of aspects which determine the summation procedure.... In PAGE 12: ... This does not mean that all of them can be cured by the same resummation method: large-order behaviour is just one of aspects which determine the summation procedure. To each power series with coe cients listed in the 3rd column of Table1 , there is a whole class of functions f(z) having the same asymptotic expansion. To specify the asymptotic expansion, one has to establish the angle (ray(s)) along which z approaches the origin; further, to pick out one function f(z) of this class, one has to add some additional information, according to the theory in question.... In PAGE 17: ...i.e., = 1, (n) = n!)) plays no privileged role among the variety of possible summation methods. In many practical problems, the Borel method nevertheless seems to be preferable, because most of the large-order estimates suggest an n! behaviour of the perturbative coe cients (see Table1 ). But this method simultaneously requires analyticity and the bound (22) in the z plane in an opening angle that is equal to .... In PAGE 28: ...rom subsection 2.1 are satis ed. The condition 1) would be violated if the an were to grow faster than n!. As follows from Table1 , this is not the case in typical situations. We generally do not know the nature or distribution of singularities to assess the validity of the condition 2).... In PAGE 31: ...A further generalization of Borel transformation The functions B (t) and M(t) de ned in Table 2 are generalizations of the Borel transform, which can be used in the various situations listed in Table1 to reduce non-uniqueness, provided some additional information is available. More about the properties of B (t) and M(t) can be found in [38, 39, 40, 42] and in references therein.... ..."

### Table 6.2: Sizes of the high-order raw models.

2004

### Table 12. Models Improved by Branch and Bound Using Full Polynomial Set Reduced by High-Order Correlations

"... In PAGE 71: ... (Note that many terms reappear as the complex correlations subsume simpler ones.) Removing the constant (to match the original investigators) leaves the following 18 features of the inputs, sufficiently different from one another to allow an optimal algorithm to be used: { a, b, c, a2, b2, c2, d2, ab, ac, ad, bc, bd, cd, a2c, b2c, bcd, bd2, c2d } (37) Now the best model of each size, 1 - 18, of this orthogonal subset of terms (see Table12 ) is almost surely the best representative of that size from the full, but unusable, set of terms.21 21Judging each by the MDL criterion (e.... ..."

### Table 7.1: The generalization and size of constructed high order perceptrons.

### Table 6 Summary of advection schemes used in the various ocean modelsa

"... In PAGE 36: ... The recent paper by Hecht et al. (2000) provides a useful overview with ocean model tests of the various schemes, and Table6 summarizes the schemes commonly used or available in the various ocean models. All models employ flux form advection schemes, hence allowing for trivial conservation of total tracer amount.... In PAGE 37: ... For q-models, it has long been recognized that it is essential to move away from the conser- vative centered schemes, since layer thickness must maintain positive-definiteness, and centered schemes cannot guarantee such. Consequently, q-model developers have traditionally paid close attention to developments in numerical advection schemes and have found the schemes listed in Table6 to be of use. Many studies of advection consider the eC128ects of various schemes on the transport of an ide- alized tracer patch or structure (e.... ..."

### Table 2: Grid convergence of high-order approximations of ut + ux = 0 using projection implementation of boundary conditions. A fourth-order Runge-Kutta method was used with CFL= 0:25 for fourth-order schemes, and CFL = 0:10 for the sixth-order scheme.

1996

"... In PAGE 11: ... We now use the projection method (16) and the SAT method (17) to solve the scalar model equation. Table2 and 3 shows a grid re nement study, performed as in section 1.2, for the model problem (5)-(7).... ..."

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### Table 1: Grid convergence of high-order approximations of ut + ux = 0, using conventional implementation of boundary conditions. A fourth-order Runge-Kutta method was used with CFL= 0:25 for fourth-order schemes, and CFL= 0:10 for sixth-order schemes.

1996

"... In PAGE 5: ... Figure 1 shows the stability region for the standard fourth-order Runge-Kutta method. Table1 shows a grid re nement study for the model problem (5) using three di erent di erence operators. The fourth-order diagonal norm di erence operator (D4) and the fourth-order restricted full norm di erence operator (RF4) can be found in [11].... ..."

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### Table 1. Evaluation of the efficiency on the three synthetic surfaces. First order Lax-Friedrichs scheme High order scheme

2007

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### Table 5: Position of the free boundary. nx - Number of grid lines D = 0:1 10?3;Q = 1 10?3 Finally we present the numerical solution to some problems, where positive in ux conditions compete with the reaction induced dissolution. In this general case, the solutions ci are given by c1 = 12(pc2 4 + 4K + c4) , c2 = 12(pc2 4 + 4K ? c4) in zone II. Extending these functions to zone I, we obtain the data for the (FBP) for u = Ci ? ci in zone I. We calculate the solution c4(x; t) of problem (12) - (15) using a standard MOL scheme and construct a bicubic spline interpolation for the functions c4(x; t) and c4x(x; t) for the evaluation of the concentrations in zone II, the source term q(x; t), the function h(t) and the function (x; t). The integral in (24) is approximated by a high-order Gauss-Formula. Constant in ux conditions satisfying cF lt; 2pK result in a continuous but retarded dissolution.

### Table 5: Grid convergence of high-order approximations of ut = ux using projec- tion implementation of boundary conditions. A fourth-order Runge-Kutta method was used with CFL= 0:25 for fourth-order schemes, and CFL= 0:10 for the sixth- order scheme.

1996

"... In PAGE 20: ... However, as numerical experiments will show, we lose accuracy compared with the projection and SAT method because of the incorrect boundary treatment. Table5 and 6 show the grid re nement study for the projection method (25) and the SAT method (27). The error is chosen such that the error of the time discretization is smaller than the error of the space discretization and such that the condition (18) holds.... ..."

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