### Table 1: Overview of results on bifurcations from periodic solutions with spatiotem- poral symmetry and spatial symmetry in ?-equivariant dynamical systems

1999

"... In PAGE 6: ... When = = Zm, the periodic solution P is called a discrete rotating wave. A brief overview of some key papers on bifurcation from rotating waves and discrete rotating waves is sketched in Table1 . In this paper, we con ne ourselves to discussing bifurcation from isolated periodic solutions with compact spatiotemporal symmetry.... ..."

Cited by 9

### Table 1: Overview of results on bifurcations from periodic solutions with spatiotem- poral symmetry and spatial symmetry in ?-equivariant dynamical systems

1999

"... In PAGE 6: ... When = = Zm, the periodic solution P is called a discrete rotating wave. A brief overview of some key papers on bifurcation from rotating waves and discrete rotating waves is sketched in Table1 . In this paper, we con ne ourselves to discussing bifurcation from isolated periodic solutions with compact spatiotemporal symmetry.... ..."

Cited by 9

### Table 14. Therefore the normal form of the D4-equivariant map f in (1), with IRq = V is

"... In PAGE 31: ...= [x1; x2; 0; 0]; b2 = [0; 0; x1; x2]; b3 = [x3; x4; 0; 0]; b4 = [0; 0; x3; x4]; b5 = [x13; x23; 0; 0]; b6 = [0; 0; x13; x23]; b7 = [x12x3; x22x4; 0; 0]; b8 = [0; 0; x12x3; x22x4]; b9 = [x1x32; x2x42; 0; 0]; b10 = [0; 0; x1x32; x2x42]; b11 = [x33; x43; 0; 0]; b12 = [0; 0; x33; x43]; b13 = [x1x2x4; x1x2x3; 0; 0]; b14 = [0; 0; x1x2x4; x1x2x3]; b15 = [x2x4x3; x1x3x4; 0; 0]; b16 = [0; 0; x2x4x3; x1x3x4]; b17 = [x23x1x4; x13x2x3; 0; 0]; b18 = [0; 0; x23x1x4; x13x2x3]; b19 = [x13x32; x23x42; 0; 0]; b20 = [0; 0; x13x32; x23x42]; b21 = [x23x4x3; x13x3x4; 0; 0]; b22 = [0; 0; x23x4x3; x13x3x4]; b23 = [x12x33; x22x43; 0; 0]; b24 = [0; 0; x12x33; x22x43]; b25 = [x2x1x43; x1x2x33; 0; 0]; b26 = [0; 0; x2x1x43; x1x2x33]; b27 = [x2x43x3; x1x33x4; 0; 0]; b28 = [0; 0; x2x43x3; x1x33x4]; b29 = [x23x1x43; x13x2x33; 0; 0]; b30 = [0; 0; x23x1x43; x13x2x33]; b31 = [x23x43x3; x13x33x4; 0; 0]; b32 = [0; 0; x23x43x3; x13x33x4] Table14 : Generating set for the equivariants when D4 acts as #5 + #5. The parameters will be introduced later in order to unfold the linear part of (28); in which case at most B1; :::; B4 will depend on them.... ..."

### Table 6: Timings for computation of equivariants for various variants.

### Table 1: Several examples of continuous transformation groups and their equivariant function

"... In PAGE 5: ... Putting it yet another way, an equivariant function space is a function space that is closed under the transformation group. Table1 lists the equivariant function spaces under di erent one-parameter transformation groups. Multi-parameter groups can be constructed by combining several of these groups.... In PAGE 13: ... Thus, using the conventional SVD method, the singular v alue decomposition of a 128 128 matrix was computed. For the cascade basis reduction method, the sinusoids (and co-sinusoids) with integer frequencies over the domain [;1;; 1] were used as the equivariant functions (see Table1 ). A total of 21 were required to approximate the Gaussian over this interval (one DC component, and 10 pairs of sinusoids and co-sinusoids of increasing integral frequencies).... ..."

### Table 3: Timings for completeness of invariants and equivariants for O(2) S1.

### Table 11: Generating set for the equivariants when D4 acts as #3 + #5.

"... In PAGE 26: ... ii) A Hilbert basis for the ring of smooth IR-valued D4-invariant functions on V is given by i, i = 1; :::; 4, in Table 10. iii) The generators for the module of D4-equivariant smooth mappings from V to V , over the above ring of invariants, are given by bi in Table11 . Therefore, the normal form for the D4-equivariant map f in (1) is f(x) = 6 X i=1 Ai( 1(x); :::; 4(x))bi(x) (27) where, Ai are smooth functions.... ..."

### Table 1: Developments in equivariant bifurcation theory from (relative) equilibria and (relative) periodic solutions.

1998

"... In PAGE 2: ...Table 1: Developments in equivariant bifurcation theory from (relative) equilibria and (relative) periodic solutions. The references in Table1 are not intended to be complete. Extensive references to previous partial results on bifurcation from isolated periodic solutions can be found in [25].... In PAGE 2: ...l. [34]. These results are necessarily partial since at that time the results in [25] were not available. In this paper, we show that the ideas in [25, 34], together with certain group theoretic results, yield a systematic approach to bifurcation from relative periodic solutions. We now describe in more detail the results outlined in Table1 . The rst systematic results in equivariant bifurcation theory were obtained in [18] for bifurcations from fully symmetric equilibria under the assumption that the group of symmetries ? is a compact Lie group.... In PAGE 20: ... Recall that V0; denotes the one dimensional irreducible representations of D 2`, and Vj, 1 j lt; `=2 denotes the two dimensional irreducible representations. Our results are almost identical to those given in [25, Table1 ] and are tabulated in Table 5. We note that the rst ve columns and most of the eighth column can be read o directly from [25, Table 1].... In PAGE 20: ... Our results are almost identical to those given in [25, Table 1] and are tabulated in Table 5. We note that the rst ve columns and most of the eighth column can be read o directly from [25, Table1 ]. The remaining entries are straightforward.... ..."

Cited by 6

### Table 1: Basic equivariant adaptive learning algorithms for ICA. Some of these algorithms require pre-whitening.

2006

Cited by 1