Conversion between Generating Sets

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"Time-Delay Interferometry"

Massimo Tinto and Sanjeev V. Dhurandhar

	Abstract
1	Introduction
2	Physical and Historical Motivations of TDI
3	Time-Delay Interferometry
4	Algebraic Approach for Canceling Laser and Optical Bench Noises
	4.1	Cancellation of laser phase noise
	4.2	Cancellation of laser phase noise in the unequal-arm interferometer
	4.3	The module of syzygies
	4.4	Gröbner basis
	4.5	Generating set for the module of syzygies
	4.6	Canceling optical bench motion noise
	4.7	Physical interpretation of the TDI combinations
5	Time-Delay Interferometry with Moving Spacecraft
	5.1	The unequal-arm Michelson
	5.2	The Sagnac combinations
	5.3	Algebraic approach to second-generation TDI
	5.4	Solutions with one arm nonfunctional
6	Optimal LISA Sensitivity
	6.1	General application
	6.2	Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
7	Experimental Aspects of TDI
	7.1	Time-delays accuracies
	7.2	Clocks synchronization
	7.3	Clocks timing jitter
	7.4	Sampling reconstruction algorithm
	7.5	Data digitization and bit-accuracy requirement
8	Concluding Remarks
	Acknowledgement
A	Generators of the Module of Syzygies
B	Conversion between Generating Sets
	References
	Footnotes
	Updates
	Figures

B Conversion between Generating Sets

We list the three sets of generators and relations among them. We first list below $α$ , $β$ , $γ$ , $ζ$ :

α = (− 1,− z,− xz,1,xy, y), β = (− xy,− 1,− x,z,1,yz ), (121 ) γ = (− y,− yz,− 1,xz,x, 1), ζ = (− x,− y,− z,x,y,z ).

We now express the $ai$ and $∗ bj$ in terms of $α$ , $β$ , $γ$ , $ζ$ :

a1 = γ − zζ, a2 = α − zβ, a3 = − zα + β − xγ + xz ζ, a4 = zζ, (122 ) ∗ b1 = − yα + yzβ + γ − zζ, b∗2 = (1 − z2)β − xγ + xzζ, ∗ b3 = β − yζ.

Further, we also list below $α$ , $β$ , $γ$ , $ζ$ in terms of $X (A)$ :

α = X (3), (4) β = X , γ = − X (1) + zX (2), (123 ) (2) ζ = X .

This proves that since the $a i$ , $b∗ j$ generate the required module, the $α$ , $β$ , $γ$ , $ζ$ and $X (A )$ , $A = 1,2,3,4$ , also generate the same module.

The Gröbner basis is given in terms of the above generators as follows: $G (1) = ζ$ , $G(2) = X (1)$ , $G (3) = β$ , $G (4) = α$ , and $G(5) = a3$ .