### Table 1. DIFFERENTIAL TOPOLOGY QUANTUM THEORY (n ? 1)-dimensional manifold Hilbert space

2001

"... In PAGE 12: ... Table1 . Analogy between di erential topology and quantum theory I shall explain this analogy between di erential topology and quantum theory fur- ther in Section 5.... In PAGE 17: ... Recall from Section 3 that a TQFT maps each manifold S representing space to a Hilbert space Z(S) and each cobordism M: S ! S0 representing spacetime to an operator Z(M): Z(S) ! Z(S0), in such a way that composition and identities are preserved. We may summarize all this by saying that a TQFT is a functor Z: nCob ! Hilb: In short, category theory makes the analogy in Table1 completely precise. In terms of this analogy, many somewhat mysterious aspects of quantum theory correspond to easily understood facts about spacetime! For example, the noncommutativity of operators in quantum theory corresponds to the noncommutativity of composing cobordisms.... ..."

Cited by 3

### Table 3: Ordering and orientation tables used to map between Hilbert and Morton space- fllling curve in two dimensions. Row i determines the ordering and orientation of ofispring when a parent with orientation i is reflned.

2003

"... In PAGE 13: ... For example, consider the initial template and flrst reflnement of the Hilbert curve (Fig- ure 13). The root quadrant has orientation 0, so the ofispring at the flrst level are ordered according to the Morton index sequence given in row 0 of the ordering Table3 : f0 1 3 2g. These ofispring are assigned orientations according to row 0 of the orientation Table 3: f1... In PAGE 13: ... The root quadrant has orientation 0, so the ofispring at the flrst level are ordered according to the Morton index sequence given in row 0 of the ordering Table 3: f0 1 3 2g. These ofispring are assigned orientations according to row 0 of the orientation Table3 : f1... In PAGE 14: ... The next reflnement uses this orientation information to determine the order and orientation of their ofispring. For example, the orientation in quadrant 0 is 1, so we use row 1 of the tables in Table3 to determine the ofispring order and orientation as f0 2 3 1g and... In PAGE 15: ...21 23 22 32 30 12 10 11 13 31 33 1 2 3 0 00 01 03 02 1 2 2 2 3 0 Orient: 1 Orient: 0 Orient: 0 Orient: 2 0 3 1 1 1 0 0 0 0 2 Figure 13: The use of the ordering and orientation tables to generate the flrst two levels of the two-dimensional Hilbert ordering. The left diagram shows the reflnement of the root quadrant, with child ordering f0 1 3 2g and orientations f1 0 0 2g given by row 0 of the tables in Table3 . The right shows the next level of reflnement, guided in each quadrant by the row of Tables 3 corresponding to the orientation of the parent in that quadrant.... ..."

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### Table 4.2 Molien factorizations of the Hilbert series of crystallographic point groups in the R6 space of the Cauchy tensor.

### Table 3: Comparison Between Di erent Schemes that uses Hilbert order 16

1993

"... In PAGE 16: ... The di erences between the alternative methods are small. However, from Table3 we see that (2D-c) does better, especially for large queries. The next best method is the (4D-cd), which uses a 4-d hilbert curve on the parameter space (center-x, center-y, diameter-x, diameter-y).... ..."

Cited by 203

### Table 3: Comparison Between Di erent Schemes that uses Hilbert order 16

1993

"... In PAGE 16: ... The di erences between the alternative methods are small. However, from Table3 we see that (2D-c) does better, especially for large queries. The next best method is the (4D-cd), which uses a 4-d hilbert curve on the parameter space (center-x, center-y, diameter-x, diameter-y).... ..."

Cited by 203

### Table 2: The Hilbert series of the WDVV ring W n for 3 n 8.

2005

"... In PAGE 11: ... 6 Concluding remarks The author is currently studying the Hilbert series of the WDVV ring. The rst few values are given in Table2 . The symmetry follows immediately from Poincar e duality and the fact the moduli space Mg;n is smooth and compact.... ..."

Cited by 1

### Table 1: De nition of Symbols The following lemma shows that the edges of d di erent orientations approach the uniform distribution as the order of the Hilbert curve approximation grows into in nity. Notation 3.2 Let quot;i;k denote the number of i-oriented edges in a (d-oriented) Hd k. Lemma 2 In a d-dimensional space, for any i and j (1 i; j d), quot;i;k= quot;j;k approaches unity as k grows to in nity.

1996

"... In PAGE 7: ... 3 Asymptotic Analysis In this section, we give an asymptotic formula for the clustering property of the Hilbert space- lling curve for general polyhedra in a d-dimensional space. The symbols used in this section are summarized in Table1 . The polyhedra we consider here are not necessarily convex, but are rectilinear in the sense that any (d-1)-dimensional polygonal surface is perpendicular to one of the d coordinate axes.... ..."

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### Table 1: Examples of RKHS Kernels and the decision surfaces they define.

"... In PAGE 8: ... Another important question is: which Hilbert space and kernels correspond to standard approximation schemes used conventionally. Table1 gives examples of kernel functions corresponding to some RKHS and the type of decision surface they describe, recovering some well known approximation schemes like Gaussian RBF, MLP... ..."

### Table 5: H 2 basis set: x-y separable basis set and interpolation functions for n0ct to Hilbert transform of

1991

"... In PAGE 40: ...0 haveodd symmetry about tap 0; the others haveeven symmetry. These n0clters were taken from Table5 , with a sample spacing of 0.... ..."

Cited by 560

### Table 9: H 4 basis set: x-y separable basis set and interpolation functions for n0ct to Hilbert transform of

1991

"... In PAGE 42: ...0 haveodd symmetry about tap 0; the others haveeven symmetry. These n0clters were taken from Table9 , with a sample spacing of 0.... ..."

Cited by 560