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[ascl:1602.001] Automark: Automatic marking of marked Poisson process in astronomical high-dimensional datasets

Automark models photon counts collected form observation of variable-intensity astronomical sources. It aims to mark the abrupt changes in the corresponding wavelength distribution of the emission automatically. In the underlying methodology, change points are embedded into a marked Poisson process, where photon wavelengths are regarded as marks and both the Poisson intensity parameter and the distribution of the marks are allowed to change.

[ascl:1812.015] AUTOSPEC: Automated Spectral Extraction Software for integral field unit data cubes

AUTOSPEC provides fast, automated extraction of high quality 1D spectra from astronomical datacubes with minimal user effort. AutoSpec takes an integral field unit (IFU) datacube and a simple parameter file in order to extract a 1D spectra for each object in a supplied catalogue. A custom designed cross-correlation algorithm improves signal to noise as well as isolates sources from neighboring contaminants.

[ascl:1612.014] AUTOSTRUCTURE: General program for calculation of atomic and ionic properties

AUTOSTRUCTURE calculates atomic and ionic energy levels, radiative rates, autoionization rates, photoionization cross sections, plane-wave Born and distorted-wave excitation cross sections in LS- and intermediate-coupling using non- or (kappa-averaged) relativistic wavefunctions. These can then be further processed to form Auger yields, fluorescence yields, partial and total dielectronic and radiative recombination cross sections and rate coefficients, photoabsorption cross sections, and monochromatic opacities, among other properties.

[ascl:1109.016] aXe: Spectral Extraction and Visualization Software

aXe is a spectroscopic data extraction software package that was designed to handle large format spectroscopic slitless images such as those from the Wide Field Camera 3 (WFC3) and the Advanced Camera for Surveys (ACS) on HST. aXe is a PyRAF/IRAF package that consists of several tasks and is distributed as part of the Space Telescope Data Analysis System (STSDAS). The various aXe tasks perform specific parts of the extraction and calibration process and are successively used to produce extracted spectra.

[ascl:1605.004] BACCHUS: Brussels Automatic Code for Characterizing High accUracy Spectra

BACCHUS (Brussels Automatic Code for Characterizing High accUracy Spectra) derives stellar parameters (Teff, log g, metallicity, microturbulence velocity and rotational velocity), equivalent widths, and abundances. The code includes on the fly spectrum synthesis, local continuum normalization, estimation of local S/N, automatic line masking, four methods for abundance determinations, and a flagging system aiding line selection. BACCHUS relies on the grid of MARCS model atmospheres, Masseron's model atmosphere thermodynamic structure interpolator, and the radiative transfer code Turbospectrum (ascl:1205.004).

[ascl:1708.010] BAGEMASS: Bayesian age and mass estimates for transiting planet host stars

BAGEMASS calculates the posterior probability distribution for the mass and age of a star from its observed mean density and other observable quantities using a grid of stellar models that densely samples the relevant parameter space. It is written in Fortran and requires FITSIO (ascl:1010.001).

[ascl:1312.008] BAMBI: Blind Accelerated Multimodal Bayesian Inference

BAMBI (Blind Accelerated Multimodal Bayesian Inference) is a Bayesian inference engine that combines the benefits of SkyNet (ascl:1312.007) with MultiNest (ascl:1109.006). It operated by simultaneously performing Bayesian inference using MultiNest and learning the likelihood function using SkyNet. Once SkyNet has learnt the likelihood to sufficient accuracy, inference finishes almost instantaneously.

[ascl:1408.020] bamr: Bayesian analysis of mass and radius observations

bamr is an MPI implementation of a Bayesian analysis of neutron star mass and radius data that determines the mass versus radius curve and the equation of state of dense matter. Written in C++, bamr provides some EOS models. This code requires O2scl (ascl:1408.019) be installed before compilation.

[ascl:1905.014] Bandmerge: Merge data from different wavebands

Bandmerge takes in ASCII tables of positions and fluxes of detected astronomical sources in 2-7 different wavebands, and write out a single table of the merged data. The tool was designed to work with source lists generated by the Spitzer Science Center's MOPEX software, although it can be "fooled" into running on other data as well.

[ascl:1801.001] BANYAN_Sigma: Bayesian classifier for members of young stellar associations

BANYAN_Sigma calculates the membership probability that a given astrophysical object belongs to one of the currently known 27 young associations within 150 pc of the Sun, using Bayesian inference. This tool uses the sky position and proper motion measurements of an object, with optional radial velocity (RV) and distance (D) measurements, to derive a Bayesian membership probability. By default, the priors are adjusted such that a probability threshold of 90% will recover 50%, 68%, 82% or 90% of true association members depending on what observables are input (only sky position and proper motion, with RV, with D, with both RV and D, respectively). The algorithm is implemented in a Python package, in IDL, and is also implemented as an interactive web page.

[ascl:1402.025] BAOlab: Baryon Acoustic Oscillations software

Using the 2-point correlation function, BAOlab aids the study of Baryon Acoustic Oscillations (BAO). The code generates a model-dependent covariance matrix which can change the results both for BAO detection and for parameter constraints.

[ascl:1403.013] BAOlab: Image processing program

BAOlab is an image processing package written in C that should run on nearly any UNIX system with just the standard C libraries. It reads and writes images in standard FITS format; 16- and 32-bit integer as well as 32-bit floating-point formats are supported. Multi-extension FITS files are currently not supported. Among its tools are ishape for size measurements of compact sources, mksynth for generating synthetic images consisting of a background signal including Poisson noise and a number of pointlike sources, imconvol for convolving two images (a “source” and a “kernel”) with each other using fast fourier transforms (FFTs) and storing the output as a new image, and kfit2d for fitting a two-dimensional King model to an image.

[ascl:1810.002] Barcode: Bayesian reconstruction of cosmic density fields

Barcode (BAyesian Reconstruction of COsmic DEnsity fields) samples the primordial density fields compatible with a set of dark matter density tracers after cosmic evolution observed in redshift space. It uses a redshift space model based on the analytic solution of coherent flows within a Hamiltonian Monte Carlo posterior sampling of the primordial density field; this method is applicable to analytically derivable structure formation models, such as the Zel'dovich approximation, but also higher order schemes such as augmented Lagrangian perturbation theory or even particle mesh models. The algorithm is well-suited for analysis of the dark matter cosmic web implied by the observed spatial distribution of galaxy clusters, such as obtained from X-ray, SZ or weak lensing surveys, as well as that of the intergalactic medium sampled by the Lyman alpha forest. In these cases, virialized motions are negligible and the tracers cannot be modeled as point-like objects. Barcode can be used in all of these contexts as a baryon acoustic oscillation reconstruction algorithm.

[ascl:1608.004] BART: Bayesian Atmospheric Radiative Transfer fitting code

BART implements a Bayesian, Monte Carlo-driven, radiative-transfer scheme for extracting parameters from spectra of planetary atmospheres. BART combines a thermochemical-equilibrium code, a one-dimensional line-by-line radiative-transfer code, and the Multi-core Markov-chain Monte Carlo statistical module to constrain the atmospheric temperature and chemical-abundance profiles of exoplanets.

[ascl:1807.018] BARYCORR: Python interface for barycentric RV correction

BARYCORR is a Python interface for ZBARYCORR (ascl:1807.017); it requires the measured redshift and returns the corrected barycentric velocity and time correction.

[ascl:1808.001] Barycorrpy: Barycentric velocity calculation and leap second management

barycorrpy (BCPy) is a Python implementation of Wright and Eastman's 2014 code (ascl:1807.017) that calculates precise barycentric corrections well below the 1 cm/s level. This level of precision is required in the search for 1 Earth mass planets in the Habitable Zones of Sun-like stars by the Radial Velocity (RV) method, where the maximum semi-amplitude is about 9 cm/s. BCPy was developed for the pipeline for the next generation Doppler Spectrometers - Habitable-zone Planet Finder (HPF) and NEID. An automated leap second management routine improves upon the one available in Astropy. It checks for and downloads a new leap second file before converting from the UT time scale to TDB. The code also includes a converter for JDUTC to BJDTDB.

[ascl:1601.017] BASCS: Bayesian Separation of Close Sources

BASCS models spatial and spectral information from overlapping sources and the background, and jointly estimates all individual source parameters. The use of spectral information improves the detection of both faint and closely overlapping sources and increases the accuracy with which source parameters are inferred.

[ascl:1608.007] BASE-9: Bayesian Analysis for Stellar Evolution with nine variables

The BASE-9 (Bayesian Analysis for Stellar Evolution with nine variables) software suite recovers star cluster and stellar parameters from photometry and is useful for analyzing single-age, single-metallicity star clusters, binaries, or single stars, and for simulating such systems. BASE-9 uses a Markov chain Monte Carlo (MCMC) technique along with brute force numerical integration to estimate the posterior probability distribution for the age, metallicity, helium abundance, distance modulus, line-of-sight absorption, and parameters of the initial-final mass relation (IFMR) for a cluster, and for the primary mass, secondary mass (if a binary), and cluster probability for every potential cluster member. The MCMC technique is used for the cluster quantities (the first six items listed above) and numerical integration is used for the stellar quantities (the last three items in the above list).

[ascl:1208.010] BASE: Bayesian Astrometric and Spectroscopic Exoplanet Detection and Characterization Tool

BASE is a novel program for the combined or separate Bayesian analysis of astrometric and radial-velocity measurements of potential exoplanet hosts and binary stars. The tool fulfills two major tasks of exoplanet science, namely the detection of exoplanets and the characterization of their orbits. BASE was developed to provide the possibility of an integrated Bayesian analysis of stellar astrometric and Doppler-spectroscopic measurements with respect to their binary or planetary companions’ signals, correctly treating the astrometric measurement uncertainties and allowing to explore the whole parameter space without the need for informative prior constraints. The tool automatically diagnoses convergence of its Markov chain Monte Carlo (MCMC[2]) sampler to the posterior and regularly outputs status information. For orbit characterization, BASE delivers important results such as the probability densities and correlations of model parameters and derived quantities. BASE is a highly configurable command-line tool developed in Fortran 2008 and compiled with GFortran. Options can be used to control the program’s behaviour and supply information such as the stellar mass or prior information. Any option can be supplied in a configuration file and/or on the command line.

[ascl:1308.006] BASIN: Beowulf Analysis Symbolic INterface

BASIN (Beowulf Analysis Symbolic INterface) is a flexible, integrated suite of tools for multiuser parallel data analysis and visualization that allows researchers to harness the power of Beowulf PC clusters and multi-processor machines without necessarily being experts in parallel programming. It also includes general tools for data distribution and parallel operations on distributed data for developing libraries for specific tasks.

[ascl:1510.002] batman: BAsic Transit Model cAlculatioN in Python

batman provides fast calculation of exoplanet transit light curves and supports calculation of light curves for any radially symmetric stellar limb darkening law. It uses an integration algorithm for models that cannot be quickly calculated analytically, and in typical use, the batman Python package can calculate a million model light curves in well under ten minutes for any limb darkening profile.

[ascl:1612.021] BaTMAn: Bayesian Technique for Multi-image Analysis

Bayesian Technique for Multi-image Analysis (BaTMAn) characterizes any astronomical dataset containing spatial information and performs a tessellation based on the measurements and errors provided as input. The algorithm iteratively merges spatial elements as long as they are statistically consistent with carrying the same information (i.e. identical signal within the errors). The output segmentations successfully adapt to the underlying spatial structure, regardless of its morphology and/or the statistical properties of the noise. BaTMAn identifies (and keeps) all the statistically-significant information contained in the input multi-image (e.g. an IFS datacube). The main aim of the algorithm is to characterize spatially-resolved data prior to their analysis.

[ascl:1505.027] BAYES-X: Bayesian inference tool for the analysis of X-ray observations of galaxy clusters

The great majority of X-ray measurements of cluster masses in the literature assume parametrized functional forms for the radial distribution of two independent cluster thermodynamic properties, such as electron density and temperature, to model the X-ray surface brightness. These radial profiles (e.g. β-model) have an amplitude normalization parameter and two or more shape parameters. BAYES-X uses a cluster model to parametrize the radial X-ray surface brightness profile and explore the constraints on both model parameters and physical parameters. Bayes-X is programmed in Fortran and uses MultiNest (ascl:1109.006) as the Bayesian inference engine.

[ascl:1407.015] BayesFlare: Bayesian method for detecting stellar flares

BayesFlare identifies flaring events in light curves released by the Kepler mission; it identifies even weak events by making use of the flare signal shape. The package contains functions to perform Bayesian hypothesis testing comparing the probability of light curves containing flares to that of them containing noise (or non-flare-like) artifacts. BayesFlare includes functions in its amplitude-marginalizer suite to account for underlying sinusoidal variations in light curve data; it includes such variations in the signal model, and then analytically marginalizes over them.

[ascl:1209.001] Bayesian Blocks: Detecting and characterizing local variability in time series

Bayesian Blocks is a time-domain algorithm for detecting localized structures (bursts), revealing pulse shapes within bursts, and generally characterizing intensity variations. The input is raw time series data, in almost any form. Three data modes are elaborated: (1) time-tagged events, (2) binned counts, and (3) measurements at arbitrary times with normal errors. The output is the most probable segmentation of the observation interval into sub-intervals during which the signal is perceptibly constant, i.e. has no statistically significant variations. The idea is not that the source is deemed to actually have this discontinuous, piecewise constant form, rather that such an approximate and generic model is often useful. Treatment of data gaps, variable exposure, extension to piecewise linear and piecewise exponential representations, multi-variate time series data, analysis of variance, data on the circle, other data modes, and dispersed data are included.

This implementation is exact and replaces the greedy, approximate, and outdated algorithm implemented in BLOCK.

[ascl:1711.004] BayesVP: Full Bayesian Voigt profile fitting

BayesVP offers a Bayesian approach for modeling Voigt profiles in absorption spectroscopy. The code fits the absorption line profiles within specified wavelength ranges and generates posterior distributions for the column density, Doppler parameter, and redshifts of the corresponding absorbers. The code uses publicly available efficient parallel sampling packages to sample posterior and thus can be run on parallel platforms. BayesVP supports simultaneous fitting for multiple absorption components in high-dimensional parameter space. The package includes additional utilities such as explicit specification of priors of model parameters, continuum model, Bayesian model comparison criteria, and posterior sampling convergence check.

[ascl:1805.022] BCcodes: Bolometric Corrections and Synthetic Stellar Photometry

BCcodes computes bolometric corrections and synthetic colors in up to 5 filters for input values of the stellar parameters Teff, log(g), [Fe/H], E(B-V) and [alpha/Fe].

[ascl:1907.011] beamconv: Cosmic microwave background detector data simulator

beamconv simulates the scanning of the CMB sky while incorporating realistic beams and scan strategies. It uses (spin-)spherical harmonic representations of the (polarized) beam response and sky to generate simulated CMB detector signal timelines. Beams can be arbitrarily shaped. Pointing timelines can be read in or calculated on the fly; optionally, the results can be binned on the sphere.

[ascl:1905.006] beamModelTester: Model evaluation for fixed antenna phased array radio telescopes

beamModelTester enables evaluation of models of the variation in sensitivity and apparent polarization of fixed antenna phased array radio telescopes. The sensitivity of such instruments varies with respect to the orientation of the source to the antenna, resulting in variation in sensitivity over altitude and azimuth that is not consistent with respect to frequency due to other geometric effects. In addition, the different relative orientation of orthogonal pairs of linear antennae produces a difference in sensitivity between the antennae, leading to an artificial apparent polarization. Comparing the model with observations made using the given telescope makes it possible evaluate the model's performance; the results of this evaluation can provide a figure of merit for the model and guide improvements to it. This system also enables plotting of results from a single station observation on a variety of parameters.

[ascl:1104.013] BEARCLAW: Boundary Embedded Adaptive Refinement Conservation LAW package

The BEARCLAW package is a multidimensional, Eulerian AMR-capable computational code written in Fortran to solve hyperbolic systems for astrophysical applications. It is part of AstroBEAR, a hydrodynamic & magnetohydrodynamic code environment designed for a variety of astrophysical applications which allows simulations in 2, 2.5 (i.e., cylindrical), and 3 dimensions, in either cartesian or curvilinear coordinates.

[ascl:1908.013] BEAST: Bayesian Extinction And Stellar Tool

BEAST (Bayesian Extinction and Stellar Tool) fits the ultraviolet to near-infrared photometric SEDs of stars to extract stellar and dust extinction parameters. The stellar parameters are age (t), mass (M), metallicity (M), and distance (d). The dust extinction parameters are dust column (Av), average grain size (Rv), and mixing between type A and B extinction curves (fA).

[ascl:1306.006] BEHR: Bayesian Estimation of Hardness Ratios

BEHR is a standalone command-line C program designed to quickly estimate the hardness ratios and their uncertainties for astrophysical sources. It is especially useful in the Poisson regime of low counts, and computes the proper uncertainty regardless of whether the source is detected in both passbands or not.

[submitted] BELLAMY: A cross-matching package for the cynical astronomer

BELLAMY is a cross-matching algorithm designed primarily for radio images, that aims to match all sources in the supplied target catalogue to sources in a reference catalogue by calculating the probability of a match. BELLAMY utilises not only the position of a source on the sky, but also the flux data to calculate this probability, determining the most probable match in the reference catalog to the target source. Additionally, BELLAMY attempts to undo any spatial distortion that may be affecting the target catalogue, by creating a model of the offsets of matched sources which is then applied to unmatched sources. This combines to produce an iterative cross-matching algorithm that provides the user with an obvious measure of how confident they should be with the results of a cross-match.

[ascl:1306.013] Bessel: Fast Bessel Function Jn(z) Routine for Large n,z

Bessel, written in the C programming language, uses an accurate scheme for evaluating Bessel functions of high order. It has been extensively tested against a number of other routines, demonstrating its accuracy and efficiency.

[ascl:1901.009] bettermoments: Line-of-sight velocity calculation

bettermoments measures precise line-of-sight velocities from Doppler shifted lines to determine small scale deviations indicative of, for example, embedded planets.

[ascl:1402.015] BF_dist: Busy Function fitting

The "busy function" accurately describes the characteristic double-horn HI profile of many galaxies. Implemented in a C/C++ library and Python module called BF_dist, it is a continuous, differentiable function that consists of only two basic functions, the error function, erf(x), and a polynomial, |x|^n, of degree n >= 2. BF_dist offers great flexibility in fitting a wide range of HI profiles from the Gaussian profiles of dwarf galaxies to the broad, asymmetric double-horn profiles of spiral galaxies, and can be used to parametrize observed HI spectra of galaxies and the construction of spectral templates for simulations and matched filtering algorithms accurately and efficiently.

[ascl:1504.020] BGLS: A Bayesian formalism for the generalised Lomb-Scargle periodogram

BGLS calculates the Bayesian Generalized Lomb-Scargle periodogram. It takes as input arrays with a time series, a dataset and errors on those data, and returns arrays with sampled periods and the periodogram values at those periods.

[ascl:1806.002] BHDD: Primordial black hole binaries code

BHDD (BlackHolesDarkDress) simulates primordial black hole (PBH) binaries that are clothed in dark matter (DM) halos. The software uses N-body simulations and analytical estimates to follow the evolution of PBH binaries formed in the early Universe.

[ascl:1206.005] bhint: High-precision integrator for stellar systems

bhint is a post-Newtonian, high-precision integrator for stellar systems surrounding a super-massive black hole. The algorithm makes use of the fact that the Keplerian orbits in such a potential can be calculated directly and are only weakly perturbed. For a given average number of steps per orbit, bhint is almost a factor of 100 more accurate than the standard Hermite method.

[ascl:1802.013] BHMcalc: Binary Habitability Mechanism Calculator

BHMcalc provides renditions of the instantaneous circumbinary habital zone (CHZ) and also calculates BHM properties of the system including those related to the rotational evolution of the stellar components and the combined XUV and SW fluxes as measured at different distances from the binary. Moreover, it provides numerical results that can be further manipulated and used to calculate other properties.

[ascl:9910.006] BHSKY: Visual distortions near a black hole

BHSKY (copyright 1999 by Robert J. Nemiroff) computes the visual distortion effects visible to an observer traveling around and descending near a non-rotating black hole. The codes are general relativistically accurate and incorporate concepts such as large-angle deflections, image magnifications, multiple imaging, blue-shifting, and the location of the photon sphere. Once star.dat is edited to define the position and orientation of the observer relative to the black hole, bhsky_table should be run to create a table of photon deflection angles. Next bhsky_image reads this table and recomputes the perceived positions of stars in star.num, the Yale Bright Star Catalog. Lastly, bhsky_camera plots these results. The code currently tracks only the two brightest images of each star, and hence becomes noticeably incomplete within 1.1 times the Schwarzschild radius.

[ascl:1501.009] BIANCHI: Bianchi VIIh Simulations

BIANCHI provides functionality to support the simulation of Bianchi Type VIIh induced temperature fluctuations in CMB maps of a universe with shear and rotation. The implementation is based on the solutions to the Bianchi models derived by Barrow et al. (1985), which do not incorporate any dark energy component. Functionality is provided to compute the induced fluctuations on the sphere directly in either real or harmonic space.

[ascl:1908.021] bias_emulator: Halo bias emulator

bias_emulator models the clustering of halos on large scales. It incorporates the cosmological dependence of the bias beyond the mapping of halo mass to peak height. Precise measurements of the halo bias in the simulations are interpolated across cosmological parameter space to obtain the halo bias at any point in parameter space within the simulation cloud. A tool to produce realizations of correlated noise for propagating the modeling uncertainty into error budgets that use the emulator is also provided.

[ascl:1312.004] BIE: Bayesian Inference Engine

The Bayesian Inference Engine (BIE) is an object-oriented library of tools written in C++ designed explicitly to enable Bayesian update and model comparison for astronomical problems. To facilitate "what if" exploration, BIE provides a command line interface (written with Bison and Flex) to run input scripts. The output of the code is a simulation of the Bayesian posterior distribution from which summary statistics e.g. by taking moments, or determine confidence intervals and so forth, can be determined. All of these quantities are fundamentally integrals and the Markov Chain approach produces variates $ heta$ distributed according to $P( heta|D)$ so moments are trivially obtained by summing of the ensemble of variates.

[ascl:1711.021] Bifrost: Stream processing framework for high-throughput applications

Bifrost is a stream processing framework that eases the development of high-throughput processing CPU/GPU pipelines. It is designed for digital signal processing (DSP) applications within radio astronomy. Bifrost uses a flexible ring buffer implementation that allows different signal processing blocks to be connected to form a pipeline. Each block may be assigned to a CPU core, and the ring buffers are used to transport data to and from blocks. Processing blocks may be run on either the CPU or GPU, and the ring buffer will take care of memory copies between the CPU and GPU spaces.

[ascl:1208.007] Big MACS: Accurate photometric calibration

Big MACS is a Python program that estimates an accurate photometric calibration from only an input catalog of stellar magnitudes and filter transmission functions. The user does not have to measure color terms which can be difficult to characterize. Supplied with filter transmission functions, Big MACS synthesizes an expected stellar locus for your data and then simultaneously solves for all unknown zeropoints when fitting to the instrumental locus. The code uses a spectroscopic model for the SDSS stellar locus in color-color space and filter functions to compute expected locus. The stellar locus model is corrected for Milky Way reddening. If SDSS or 2MASS photometry is available for stars in field, Big MACS can yield a highly accurate absolute calibration.

[ascl:1901.011] Bilby: Bayesian inference library

Bilby provides a user-friendly interface to perform parameter estimation. It is primarily designed and built for inference of compact binary coalescence events in interferometric data, such as analysis of compact binary mergers and other types of signal model including supernovae and the remnants of binary neutron star mergers, but it can also be used for more general problems. The software is flexible, allowing the user to change the signal model, implement new likelihood functions, and add new detectors. Bilby can also be used to do population studies using hierarchical Bayesian modelling.

[ascl:1710.008] Binary: Accretion disk evolution

Binary computes the evolution of an accretion disc interacting with a binary system. It has been developed and used to study the coupled evolution of supermassive BH binaries and gaseous accretion discs.

[ascl:1811.003] binaryBHexp: On-the-fly visualizations of precessing binary black holes

binaryBHexp (binary black hole explorer) uses surrogate models of numerical simulations to generate on-the-fly interactive visualizations of precessing binary black holes. These visualizations can be generated in a few seconds and at any point in the 7-dimensional parameter space of the underlying surrogate models. These visualizations provide a valuable means to understand and gain insights about binary black hole systems and gravitational physics such as those detected by the LIGO gravitational wave detector.

[ascl:1312.012] BINGO: BI-spectra and Non-Gaussianity Operator

The BI-spectra and Non-Gaussianity Operator (BINGO) code, written in Fortran, computes the scalar bi-spectrum and the non-Gaussianity parameter fNL in single field inflationary models involving the canonical scalar field. BINGO can calculate all the different contributions to the bi-spectrum and the parameter fNL for an arbitrary triangular configuration of the wavevectors.

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