### Table 2: Invariants for simple germs F : (C3; 0) ! (C3; 0)

"... In PAGE 12: ... From these two equations the equality of the theorem follows. Table2 shows the generalised discriminant Milnor numbers and the invariants not included in Table 1 of [10]. Remark 3.... ..."

### Table 1. The embedded graph invariant

"... In PAGE 2: ... In the process, simple formulae, such as the de nition (1), become apparent, and the properties and examples can be developed rapidly. Table1 gives some examples of the evaluation of the invariant for 4-valent graphs and links.... ..."

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### Table 23: NC21, NC12 family. Cross sections in pb. In this subsection only those processes are given that were treated within the semi-analytic approach with a Simple Cuts on the invariant mass of any fermion-antifermion pair, which could be coupled to the photon. The latter cut value is chosen to be equal 5 GeV. Every table contains two sets of numbers which are computed: 1. in the Born approximation and without gluon exchange diagrams for non-leptonic processes; 2. with the ISR radiation (SF) and with gluon exchange diagrams for non-leptonic processes.

### Table VI: Maximal regular sub(super)algebras of the basic Lie superalgebras. Some of the singular subsuperalgebras of the basic Lie superalgebras can be found by the folding technique. Let G be a basic Lie superalgebra, with non-trivial outer au- tomorphism (Out(G) does not reduce to the identity). Then, there exists at least one Dynkin diagram of G which has the symmetry given by Out(G). One can notice that each symmetry described on that Dynkin diagram induces a direct construction of the subsuperalgebra G0 invariant under the G outer automorphisms associated to . Indeed, if the simple root is transformed into ( ), then 1 2( + ( )) is -invariant since 2 = 1, and appears as a simple root of G0 associated to the generators E + E ( ), the generator E (resp. E ( ) corresponding to the root (resp. ( )). A Dynkin diagram of G0 will therefore be obtained by folding the Z2-symmetric Dynkin diagram of G, that is by trans- forming each couple ( ; ( )) into the root 12( + ( )) of G0. One obtains the following invariant subsuperalgebras (which are singular): superalgebraG

### Table 1. The classification of complexities of patterns from a psychological point of view and complexities to be detected in each module of a rotation-invariant neocognitron or a standard neocognitron. The lower layer detects comparatively simple patterns (mental- rotation-free patterns) and the higher layer detects complicated patterns (mental-rotation- required patterns).

1999

"... In PAGE 4: ... Lower modules, a12 a0 and a12 a1 , detect comparatively simple patterns and higher modules, a12 a2 and a12a16a13 , complicated patterns. Examples of patterns to be detected in each module is depicted in Table1 . We can regard patterns detected in a12 a0 and a12 a1 as mental-rotation-free patterns because those patterns are oriented segments or composed of two or more oriented segments and patterns detected in a12 a2 and a12a16a13 as mental-rotation-required patterns because those require a mental rotation for correct recognition.... In PAGE 4: ... We can regard patterns detected in a12 a0 and a12 a1 as mental-rotation-free patterns because those patterns are oriented segments or composed of two or more oriented segments and patterns detected in a12 a2 and a12a16a13 as mental-rotation-required patterns because those require a mental rotation for correct recognition. We can see from Table1 that the psychological classification of complexities of pat- terns corresponds to the classification by the serial number of modules in the rotation- invariant neocognitron or the standard neocognitron. Therefore, the model of a bottom-up type recognition system have two modules: (i) the lower modules are composed of ones of a rotation-invariant neocognitron, in which all the simple rotated patterns in any angles (mental-rotation-free patterns) are detected, and (ii) the higher modules are composed of a standard neocognitron, in which only standard complicated patterns are detected.... ..."

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### Table 1. The classification of complexities of patterns from a psychological point of view and complexities to be detected in each module of a rotation-invariant neocognitron or a standard neocognitron. The lower layer detects comparatively simple patterns (mental- rotation-free patterns) and the higher layer detects complicated patterns (mental-rotation- required patterns).

"... In PAGE 4: ... Lower modules, U1 and U2, detect comparatively simple patterns and higher modules, U3 and U4, complicated patterns. Examples of patterns to be detected in each module is depicted in Table1 . We can regard patterns detected in U1 and U2 as mental-rotation-free patterns because those patterns are oriented segments or composed of two or more oriented segments and patterns detected in U3 and U4 as mental-rotation-required patterns because those require a mental rotation for correct recognition.... In PAGE 4: ... We can regard patterns detected in U1 and U2 as mental-rotation-free patterns because those patterns are oriented segments or composed of two or more oriented segments and patterns detected in U3 and U4 as mental-rotation-required patterns because those require a mental rotation for correct recognition. We can see from Table1 that the psychological classification of complexities of pat- terns corresponds to the classification by the serial number of modules in the rotation- invariant neocognitron or the standard neocognitron. Therefore, the model of a bottom-up type recognition system have two modules: (i) the lower modules are composed of ones of a rotation-invariant neocognitron, in which all the simple rotated patterns in any angles (mental-rotation-free patterns) are detected, and (ii) the higher modules are composed of a standard neocognitron, in which only standard complicated patterns are detected.... ..."

### Table 2: The first six differential invariants for nonlinear systems in the plane V = K2 with quadratic coefficients. All sums are from 1 to 2.

in The Symbolic Computation of Differential Invariants of Polynomial Vector Field Systems Using Trees ∗

1995

"... In PAGE 2: ...easily computed in terms of a few basic operations on the space of trees. Our main result is expressed in Theorem 5 and illustrated in Figure 2 and Table2 . It provides a simple and direct combinatorial means of computing dif- ferential invariants.... In PAGE 4: ... Differential invariants are naturally expressed and easily computed in terms of a few basic operations on the space of trees. Our main result is expressed in Theorem 5 and illustrated in Figure 2 and Table2 . It provides a simple and direct combinatorial means of computing dif- ferential invariants.... In PAGE 8: ... This can be done either by hand or using DIFF-INV. See [11], for example and Table2 . Given the differential polynomials, one can then compute a basis using standard symbolic packages.... In PAGE 10: ...for planar systems with quadratic coefficients listed in Table2 . Except for I2, this is the same invariant basis as in Sibursky [11].... ..."

### Table 2. Results(%) in DynTex dataset.( 2 riu is rotation invariant uniform) Features 2 riu

2006

"... In PAGE 10: ...Table2 presents the overall classification rates, while Table 3 provides more de- tailed results for each test dataset. When using the simple 2 1,2,1 riu VLBP , we get good re- sults of over 85%.... ..."

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### Table 3. Time (in sec) for Invariant Checking on the Industrial Circuits using the non-deterministic and the deterministic program

"... In PAGE 10: ...e. Table3 (b). However, the performance of the deterministic program is better than the non-deterministic version in the hard circuits in Table 3 (a).... In PAGE 10: ...he deterministic program in the simple circuits, i.e. Table 3 (b). However, the performance of the deterministic program is better than the non-deterministic version in the hard circuits in Table3 (a). Therefore, we strongly prefer the deterministic version to the non-deterministic version.... ..."