### 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 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 1. Comparison of different appearance-based feature sets

in On Appearance-Based Feature Extraction Methods for Writer-Independent Handwritten Text Recognition

### Table 1. Comparison of feature-selection techniques in the appearance-based expression recognition system of Littlewort et al (2006). Three feature selection options are compared using LDA and SVMs as the classifier

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

2004

### Table 6.4. Code lengths in bytes and average bit rates in bits per pixel (in the lossless mode) for the significant bits for the initialized heuristic JPEG2000 models and the high-order optimized models of different size for the set of test images and the training set.

2004

### Table IV lists the recognition rate, averaging those of 20 simulations, using the top 1 match. The PKPCA/IPS algorithm attains the best performance since it combines the discriminative power of the IPS model and the merit of PKPCA. However, compared to PPCA/IPS, the improvement is not significant, indicating that second-order statistics might be enough after IPS modeling for the face recognition problem. However, PKPCA may be more effective since it also takes into account high-order statistics. Another observation is that variations in illumination are easier to model than facial expression using subspace methods.

### Table 1: DC-DC converter example. Error of the DC out- put voltage for hp-type high-order, p-type high-order, and low- order method.

2000

"... In PAGE 6: ... 3. The result is oscillation free and very accurate, as noted from the comparisons of the error in the output voltage presented in Table1 . The second order Gear method requires two orders of mag- nitude more timepoints to reach comparable accuracy, while using global high-order basis functions, such as used in harmonic bal- ance, leads to large errors.... ..."

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