Alpha-Beta-Gamma (ABG): Scope & Applications

• Alpha-Beta-Gamma (ABG) is a domain-dependent acronym with definitions fixed by local context, ranging from ordered triplets to parameterized frameworks and eponymic attributions.
• It is used to denote physical phases, measurement channels, and model parameters in fields such as materials science, detector physics, and wireless propagation.
• Practical applications span condensed matter studies, spectroscopic analysis, variational inference, and gravitational physics, each employing ABG with distinct technical nuances.

Alpha-Beta-Gamma (ABG) is a domain-dependent acronym rather than a single technical object. In current arXiv usage, it appears in three principal forms: as an ordered triplet of labels such as $\alpha,\beta,\gamma$ phases or channels; as a parameterized framework built from alpha, beta, and gamma coefficients or divergences; and as an eponymic abbreviation formed from author surnames, most prominently Arkhipov–Bezrukavnikov–Ginzburg and Ayón-Beato–García. The term therefore has no field-independent meaning, and its interpretation is fixed by local context (Lima et al., 7 Aug 2025, Cichocki et al., 2014, Hodge et al., 2016, Cai et al., 2021).

1. Nomenclature and principal senses

The breadth of ABG usage is best understood as a problem of scientific nomenclature.

Sense of ABG	Expansion	Representative domain
Ordered triplet	$\alpha$, $\beta$, $\gamma$ labels	Carbon phases, radiation classes, Balmer lines
Parameter triplet	Alpha-Beta-Gamma coefficients or linked divergence families	Path-loss models, matrix divergences, variational objectives
Eponymic acronym	Author surnames	ABG induction theorem, ABG black holes

In materials science, detector physics, and spectroscopy, ABG usually denotes three related members of a family or three measurement channels. In information geometry and wireless propagation, it denotes a family parameterized by coefficients named $\alpha$, $\beta$, and $\gamma$. In representation theory and gravitation, by contrast, ABG is not alphabetical at all: it abbreviates surnames. A recurrent source of confusion is the assumption that ABG always expands to “alpha-beta-gamma”; that assumption is false in the eponymic cases (Hodge et al., 2016, Cai et al., 2021).

2. Three-phase and three-channel physical uses

A recent condensed-matter use of ABG is the family of $\alpha$-, $\beta$-, and $\gamma$-TODD-Graphene, three planar porous 2D carbon allotropes derived from TODD-G. The reported structures use mixed $\alpha$0 bonding and non-hexagonal rings: $\alpha$1- and $\alpha$2-TODD-G are described as 3-8-12-16 networks, while $\alpha$3-TODD-G is built from 3-4-8-12 rings. Their optimized lattice constants are $\alpha$4 Å, $\alpha$5 Å for $\alpha$6; $\alpha$7 Å, $\alpha$8 Å for $\alpha$9; and $\beta$0 Å, $\beta$1 Å for $\beta$2. Stability is supported by cohesive energies of $\beta$3, $\beta$4, and $\beta$5 eV/atom, phonon spectra with no imaginary frequencies, and ab initio molecular dynamics at 1000 K for 5 ps. All three phases are metallic, with states near $\beta$6 dominated by $\beta$7-orbital character; $\beta$8-TODD-G shows pronounced tilted Dirac cones, whereas $\beta$9-TODD-G shows more symmetric Dirac-like features. Mechanically, the anisotropy trend is $\gamma$0, while optically $\gamma$1-TODD-G is distinguished by strong infrared absorption near 0.8 eV and $\gamma$2- and $\gamma$3-TODD-G mainly absorb in the visible and ultraviolet ranges (Lima et al., 7 Aug 2025).

In detector physics, ABG commonly denotes alpha, beta, and gamma interactions. A TeO$\gamma$4 bolometer equipped with a coincident light detector exploited this distinction by measuring Cherenkov-like light from $\gamma$5 events but not from $\gamma$6 events: the mean light signal was $\gamma$7 eV for the 2615 keV $\gamma$8Tl $\gamma$9 line and $\alpha$0 eV for the 2310 keV $\alpha$1Sm $\alpha$2 line, yielding a $\alpha$3 $\alpha$4 separation and an optimum light threshold of 148 eV in the CUORE sensitivity model (Beeman et al., 2011). The GeSparK alpha-beta/gamma coincidence detector, which combines HPGe spectroscopy with liquid-scintillator pulse-shape discrimination, added a six-scintillator muon veto and reported $\alpha$5 background reduction in the 20–3000 keV range and $\alpha$6 reduction above 3000 keV; the optimized geometry was simulated to reach $\alpha$7 veto efficiency (Barresi et al., 17 Feb 2025). In CRESST-II Phase 2, the TUM40 CaWO$\alpha$8 module reported an average beta/gamma rate of $\alpha$9 in 1–40 keV and a total intrinsic alpha activity of $\beta$0 mBq/kg (Strauss et al., 2014). For 0.1% diphenylbutadiene-doped para-terphenyl, the measured light yield was $\beta$1 times that of EJ-200, the attenuation length was $\beta$2 mm, the alpha quenching factor was $\beta$3, and the rejection power for 660 keV photons relative to same-energy electrons ranged from 3% to 11% depending on threshold (Angelone et al., 2013).

These physical uses share a tripartite $\beta$4 labeling scheme, but their formal content is distinct. In one case ABG indexes structural phases of a carbon monolayer; in another it indexes particle-interaction classes or detection channels.

3. Spectroscopic and astronomical uses

In stellar spectroscopy, the alpha-beta-gamma sequence can denote the Balmer lines H$\beta$5, H$\beta$6, and H$\beta$7. For $\beta$8-type RR Lyrae stars, template radial-velocity curves were constructed separately for these three lines from high-precision du Pont 2.5 m echelle measurements of six field stars. The template formalism is

$\beta$9

with the templates normalized so that the systemic-velocity reference occurs at phase $\gamma$0. The line-specific amplitude calibrations against Johnson $\gamma$1-band amplitude are

$\gamma$2

$\gamma$3

$\gamma$4

with corresponding $\gamma$5 values of $\gamma$6, $\gamma$7, and $\gamma$8. The templates are line-specific because the Balmer curves differ in both shape and amplitude; the paper states that replacing the generic H$\gamma$9 curve of X Ari with matched-amplitude Balmer templates can reduce systemic-velocity uncertainties by up to $\alpha$0 (Sesar, 2012).

This use is structurally closer to detector-physics ABG than to eponymic ABG. The letters identify three observational channels within a common spectroscopic series, not a theorem, model family, or author set.

4. Divergence families, variational objectives, and fuzzy algebra

A major mathematical use of ABG occurs in the unified log-determinant divergence framework for symmetric positive definite matrices. The Alpha-Beta log-det divergence is defined for $\alpha$1 by

$\alpha$2

for $\alpha$3, $\alpha$4, and $\alpha$5. In generalized-eigenvalue form it depends only on the spectrum of $\alpha$6, is invariant under congruence transformations, and recovers several standard quantities: at $\alpha$7 it yields the squared Affine Invariant Riemannian Metric, at $\alpha$8 the S-divergence or JBLD divergence, and in the $\alpha$9 or $\beta$0 limits Stein’s loss. The same paper introduces a Gamma divergence for Gaussian densities that decomposes into an AB covariance term plus a Mahalanobis-type mean term, thereby making the “ABG” linkage explicit at the distributional level (Cichocki et al., 2014).

In variational inference, the scale-invariant Alpha-Beta divergence provides a two-parameter objective

$\beta$1

Its special cases include KL at $\beta$2, reverse KL at $\beta$3, Rényi on the line $\beta$4, and Gamma divergence on the slice $\beta$5. The paper reparameterizes with $\beta$6, interpreting $\beta$7 as controlling robustness and $\beta$8 as controlling the mass-covering versus mode-seeking tradeoff, and it emphasizes direct Monte Carlo estimation and divergence minimization rather than ELBO maximization (Regli et al., 2018).

A different algebraic usage appears in $\beta$9-semigroup theory, where “gamma” refers not to a third scalar parameter but to the ambient algebraic structure. A $\gamma$0-semigroup is a set $\gamma$1 with a ternary product $\gamma$2, and the paper defines $\gamma$3-fuzzy subsemigroups and bi-ideals using the quasi-coincidence relation $\gamma$4 iff $\gamma$5. One of the main equivalences is that $\gamma$6 is an $\gamma$7-fuzzy subsemigroup iff

$\gamma$8

and the characteristic function of an ordinary subsemigroup or bi-ideal is an $\gamma$9-fuzzy subsemigroup or bi-ideal, respectively (Sardar et al., 2011).

5. Algorithmic and propagation-model usages

In wireless propagation, ABG is a standard large-scale path-loss model. Its canonical form is

$\alpha$00

where $\alpha$01 is transmitter-receiver distance, $\alpha$02 is carrier frequency, $\alpha$03 is the distance-slope parameter, $\alpha$04 is the intercept, $\alpha$05 is the frequency-slope parameter, and $\alpha$06 is a zero-mean fluctuation term. The weighted generalization WABG uses $\alpha$07 to combine heterogeneous datasets, while the Extended Weighted ABG (EWABG) model adds second-order terms in $\alpha$08, $\alpha$09, and their cross-term, adopts Theil-Sen for robust outlier handling, and explicitly removes and restores atmospheric-gas attenuation using ITU-R P.676-12. In UMiSC, UMiOS, and UMa NLOS evaluations, EWABG was reported to obtain the best accuracy and to keep increment error rates below 1% relative to the non-outlier condition, outperforming ABG and WABG in noisy settings with outliers (Casillas-Pérez et al., 17 Jan 2026).

Game-tree search supplies a nearby but distinct terminology. “SSS* = Alpha-Beta + TT” does not use ABG as a formal acronym; instead it reformulates Stockman’s SSS* as repeated null-window Alpha-Beta searches with a transposition table, yielding AB-SSS*. The core driver repeatedly calls $\alpha$10 to compute a sequence of upper bounds, and the paper’s main conclusion is that SSS* is best understood as an Alpha-Beta enhancement rather than a separate search paradigm. It also reports that iterative-deepening Alpha-Beta can sometimes expand fewer leaf nodes than iterative-deepening SSS* because of dynamic move re-ordering (Plaat et al., 2014).

This juxtaposition is instructive: in path-loss modeling ABG is a genuine three-parameter model class, whereas in minimax search the relevant acronym remains AB, not ABG.

6. Eponymic ABG in representation theory and gravitation

In representation theory, ABG refers to Arkhipov–Bezrukavnikov–Ginzburg. The ABG Induction Theorem concerns Lusztig quantum groups at a root of unity and identifies the principal block of the full algebra with a derived category on the Borel side. In the notation used by Hodge–Karuppuchamy–Scott, the quantum statement is

$\alpha$11

and the modular analogue for a semisimple simply connected algebraic group $\alpha$12 in characteristic $\alpha$13 is

$\alpha$14

The paper presents a proof strategy based on generators, Ext-isomorphisms, twisted injectives, parity properties of $\alpha$15-cohomology, and translation functors, while also repairing gaps in the original ABG treatment (Hodge et al., 2016).

In gravitational physics, ABG refers to Ayón-Beato–García. A 2021 paper constructs a five-parameter ABG-related regular black-hole family in Einstein gravity coupled to nonlinear electrodynamics, with regularity conditions

$\alpha$16

The saturated case $\alpha$17, $\alpha$18 yields a three-parameter family $\alpha$19 whose quasinormal modes were computed with sixth-order WKB and the eikonal null-geodesic method; the shadow radius is $\alpha$20, and the $\alpha$21 shadow data were used to constrain $\alpha$22, leading to an allowed fundamental $\alpha$23 frequency range of approximately $\alpha$24 (Cai et al., 2021). A related STVG extension defines a generalized ABG STVG black hole with parameters $\alpha$25, regular when $\alpha$26 and $\alpha$27, and verifies the improved eikonal-shadow correspondence

$\alpha$28

for odd-parity gravitational perturbations (Cai et al., 2020).

The eponymic cases are conceptually independent of alphabetical triplets. “ABG black hole” and “ABG induction theorem” denote historically attributed constructions, not alpha-beta-gamma phase families or parameter scans.

Across these domains, ABG functions less as a unified concept than as a compact naming device for triads, parameterizations, or attributed results. The decisive interpretive question is therefore not what ABG means in general, but which disciplinary grammar—alphabetical, parametric, or eponymic—governs the specific usage under consideration.