The 3-j and 6-j Symbols

Shortley, George; Rotenberg, Mordechai; Bivins, Robert L.; Metropolis, N.; Wooten, J. K.

doi:10.2307/2003907

Cited by 20 publications

(41 citation statements)

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“…3) where the temperature T and the E and B modes represent any spin-0 and spin-2 fields respectively. The coupling terms can be simplified using the symmetry and orthogonality properties of the Wigner-3j symbols [120]. Our implementation of the non-zero mode-coupling terms is based on equations (A12) -(A17) in [118], and is given by the following set of expressions:…”

Section: Pseudo-c Estimationmentioning

confidence: 99%

Fast lightcones for combined cosmological probes

Sgier¹,

Fluri²,

Herbel³

et al. 2021

J. Cosmol. Astropart. Phys.

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The combination of different cosmological probes offers stringent tests of the ΛCDM model and enhanced control of systematics. For this purpose, we present an extension of the lightcone generator UFALCON first introduced in Sgier et al. [1], enabling the simulation of a self-consistent set of maps for different cosmological probes. Each realization is generated from the same underlying simulated density field, and contains full-sky maps of different probes, namely weak lensing shear, galaxy overdensity including RSD, CMB lensing, and CMB temperature anisotropies from the ISW effect. The lightcone generation performed by UFALCON is parallelized and based on the replication of a large periodic volume simulated with the GPU-accelerated N-Body code PKDGRAV3. The post-processing to construct the lightcones requires only a runtime of about 1 walltime-hour corresponding to about 100 CPU-hours. We use a randomization procedure to increase the number of quasi-independent full-sky UFALCON map-realizations, which enables us to compute an accurate multi-probe covariance matrix. Using this framework, we forecast cosmological parameter constraints by performing a multi-probe likelihood analysis for a combination of simulated future stage-IV-like surveys. We find that the inclusion of the cross-correlations between the probes significantly increases the information gain in the parameter constraints. We also find that the use of a non-Gaussian covariance matrix is increasingly important, as more probes and cross-correlation power spectra are included. A version of the UFALCON package currently including weak gravitational lensing is publicly available.

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Section: Pseudo-c Estimationmentioning

confidence: 99%

Fast lightcones for combined cosmological probes

Sgier¹,

Fluri²,

Herbel³

et al. 2021

J. Cosmol. Astropart. Phys.

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“…(2) The integer values of the parameters of the 3j symbol which result in a nonvanishing coefficient are strictly limited by the laws governing the addition of angular momenta, and values have been extensively tabulated, e.g. [45,46].…”

Section: The Sw Description Of Lightmentioning

confidence: 99%

The plane- and spherical-wave descriptions of electromagnetic radiation: a comparison and discussion of their relative merits

Jones

Grinter

2018

Eur. J. Phys.

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During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 month embargo period.

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“…where s d = 1/2, and l d = 2. The numerical values of 6j symbols in the above expressions (and 3j , 6j symbols in the following sections) can be calculated from the relations given in [12]. The values of the reduced matrix elements of the unit tensor operator U (k) and double tensor operator V (11) have been tabulated by Nielson and Koster [2], and those for V (1r) with r > 1 can be calculated by the formula (11.68) of [11] in terms of the coefficients of fractional parentage (cfps) in [2].…”

Section: Matrix Elements Of the Coulomb Interaction H C With The Abo...mentioning

confidence: 99%

Theoretical analysis and intensity calculation for the f d absorption spectrum of U³⁺ in the LiY F₄ crystal

Ning

Jiang

Xia

et al. 2003

J. Phys.: Condens. Matter

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The absorption spectrum of U3+ in LiY F4 has been well calculated using the model proposed by Reid for calculations of spectra. The relevant formulae for the matrix element calculations which were omitted in this model are now described in detail, and the values of the direct and exchange coefficients associated with the f–d Coulomb interactions within the f2d configuration are derived and listed. The amount of reduction for the f–d Coulomb interaction parameters from the free-ion values is found to be , which is much larger than the value of 26% for the isoelectronic Nd3+ lanthanide ion in the same host.

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The 3-j and 6-j Symbols

Cited by 20 publications

References 0 publications

Fast lightcones for combined cosmological probes

Fast lightcones for combined cosmological probes

The plane- and spherical-wave descriptions of electromagnetic radiation: a comparison and discussion of their relative merits

Theoretical analysis and intensity calculation for the f d absorption spectrum of U³⁺ in the LiY F₄ crystal

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The 3-j and 6-j Symbols

Cited by 20 publications

References 0 publications

Fast lightcones for combined cosmological probes

Fast lightcones for combined cosmological probes

The plane- and spherical-wave descriptions of electromagnetic radiation: a comparison and discussion of their relative merits

Theoretical analysis and intensity calculation for the f d absorption spectrum of U3+ in the LiY F4 crystal

Contact Info

Product

Resources

About

Theoretical analysis and intensity calculation for the f d absorption spectrum of U³⁺ in the LiY F₄ crystal