### 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 2: Coe cients for the perturbative expansions, in powers of P(qm;n), of small Wilson loops. Scale qm;n is the average momentum carried by the gluon in the rst-order correction.

1997

"... In PAGE 11: ... The scale qm;n is the average gluon momentum in the rst-order contribution to Wm;n, computed directly from the Feynman diagrams as described in [17, 18]. In Table2 we list the perturbative coe cients through third order for nf =0, and through second order for nf =2 [19]. Unfortunately, the nf dependence of the third-order coe cients has not yet been computed.... ..."

### Table 7: Values of (5) MS(MZ) from several operators and various tunings of the QCD simulation. The two uncertainties listed are due to uncertainties in the inverse lattice spacing, and to truncation errors in perturbative expansions.

1997

"... In PAGE 17: ... To further facilitate comparisons with other analyses, we have numerically integrated the third-order perturbative evolution equation for MS and applied appropriate matching conditions at quark thresholds [27] to evolve it to the mass of the Z0. The results for our ten determinations are shown in Table7 . For matching we assumed MS masses of 1.... ..."

### Table 2: Total time (in seconds) of computing the surface (including the perturbation).

2003

"... In PAGE 5: ... Table 1 shows the time it took to find valid pole directions for several molecules with the original implementation as well as with the improved implementation. Table2 shows the total time it took to build the spherical arrangements (including the perturbation time) with the original implementation as well as with the improved implementation. The new implementation has other technical improvements in addition to the improvement described above (mainly due to inline expansion and passing variables by reference instead of by value, as well as caching the results of some calculations that occur more than once).... ..."

Cited by 1

### Table 4. Behaviour of Taylor apos;s series of the cost function with a bad triangulation Order of expansion Triangulation 2.3 Triangulation 2.4 Triangulation 2.5

"... In PAGE 19: ...249 We give in Table4 the results of the computing of j(I + V ) obtained when using Taylor apos;s expansion of j at the point I for these di erent perturbations, which have to be compared on the one hand with the direct computation of j(I + V ) on those perturbations with bad triangulation (triangulations 2.3, 2.... ..."

### Table 1: The expansion coe cients bg;(0) k

"... In PAGE 3: ... [13]. Here we list for brevity only the numerical values of the rst 15 coe cients bg;(0) k and bg;(1) k for the Lx and NLx series for Nf = 4, see Table1 . Note that the new terms bg;(1) 0;1 agree with the corresponding results from xed-order perturbation theory already taking into account the `irreducible apos; part of (1) gg only.... ..."

### Table 2: Per time step noise injection terms and error function expansion terms for cumulative and non- cumulative noise for both the additive and multiplicative cases. v = min(t + 1; u + 1). W is the noise-free weight set. of small weight perturbations, and dropped the rst order terms by assuming the network is at convergence. As in [24], we do not assume the networks are at convergence and consider both rst and second order terms.

1996

Cited by 4

### Table 1: Expansion terms, their orders of magnitude and the associated trees.

"... In PAGE 6: ... It follows that e m(t)y0 = y(t) + O?tm+2 and we obtain an order-(m + 1) numerical scheme (Iserles amp; N rsett 1997). As a matter of fact, terms corresponding to trees with 3m + 1 vertices might be of higher order of magnitude, the eighth term in Table1 being a case in point. It is possible to use graph theory in a more subtle manner, to identify exactly the order of magnitude of each term and this will be a subject of a forthcoming paper.... In PAGE 10: ... Letting = [ ; ], we rst observe that the leading term of a(t); R t 0 a( ) d is s s @?? + t + t ) ?1 2 t2: As a notational convention, we henceforth replace the subtree associated with a(t); R t 0 a( ) d by a fat vertex. It is apparent from Table1 that there are ve trees with eleven vertices (recall that we have added a root to each tree) and these are the... In PAGE 12: ... Note that one of the ve trees with eleven vertices results in an O?t7 perturbation, hence need not be considered in our error estimate. This is in line with the estimate in Table1 . As a matter of fact, the phenomenon whereby the order of magnitude of a term is larger (often, much larger) than the order of the associated tree is increasingly prevalent as the order increases.... ..."

### Table 1: Expansion terms, their orders of magnitude and the associated trees.

"... In PAGE 7: ... It follows that e m(t)y0 = y(t) + O?tm+2 and we obtain an order-(m + 1) numerical scheme (Iserles amp; N rsett 1997). As a matter of fact, terms corresponding to trees with 3m + 1 vertices might be of higher order of magnitude, the eighth term in Table1 being a case in point. It is possible to use graph theory in a more subtle manner, to identify exactly the order of magnitude of each term and this will be a subject of a forthcoming paper.... In PAGE 11: ... Letting = [ ; ], we rst observe that the leading term of a(t); R t 0 a( ) d is s s @?? + t + t ) ?1 2 t2: As a notational convention, we henceforth replace the subtree associated with a(t); R t 0 a( ) d by a fat vertex. It is apparent from Table1 that there are ve trees with eleven vertices (recall that we have added a root to each tree) and these are the... In PAGE 13: ... Note that one of the ve trees with eleven vertices results in an O?t7 perturbation, hence need not be considered in our error estimate. This is in line with the estimate in Table1 . As a matter of fact, the phenomenon whereby the order of magnitude of a term is larger (often, much larger) than the order of the associated tree is increasingly prevalent as the order increases.... ..."

### Table 1: A comparison of CPU time required to reach a particular value of the norm (19), for r = 0:1; HX = 0:1.

"... In PAGE 9: ...In order to outline how the multi-grid algorithm becomes important as the number of particles increases we give in Table1 the CPU time required by the two methods to solve the discretized problem to a value of the residual norm k kM 1 10?14 : (19) 5 Structure and Use of the Program As already mentioned in the introduction the program consists of two parts: the core, which resolves the TBA-equations (7) and the periphery, which on the one hand con- structs the kernel, and the initial solution , and on the other hand extracts from the solution the central charge, the dimension y and the coe cients fi of the perturbation expansion (10).... ..."