d-Transformation: Methods & Applications

• d-Transformation is a multi-disciplinary concept that recodes problems—via coordinate changes, duality actions, or domain transforms—into more tractable forms.
• Its applications span fractional PDEs, string theory, tensor algebra, integrable systems, and numerical analysis, each preserving key structural invariants.
• The transformation methods simplify complex equations and models, enabling efficient analytical solutions and improved numerical convergence.

Searching arXiv for the papers on arXiv to ground the article with current metadata and citations. arxiv_search(query="(Ghosh et al., 2017) OR (Çatal-Özer et al., 2019) OR (Paliathanasis, 2021) OR (Qi et al., 2021) OR (Rolanía et al., 2019) OR (Schlederer et al., 2015) OR (Busser et al., 2013) OR (Shemyakova, 2012) OR (Kolomiytseva, 2018) OR (Chihara, 2024) OR (Chihara, 2021) OR (Xu et al., 2011) OR (Takahashi et al., 2021) OR (Sidi, 2017)", max_results=15, sort_by="relevance") d-Transformation is a field-dependent term used for several non-equivalent constructions in contemporary arXiv literature. In the sources considered here it denotes, among other things, a fractional coordinate transformation for fractional wave equations, a coordinate-dependent $O(d,d)$ duality transformation, a general linear/unitary transform along the third mode in tensor–tubal algebra, a Geronimus-type transformation for $d$-orthogonal polynomials, an exact canonical Lanczos transformation from a $d$-dimensional lattice to a one-dimensional chain, a $D$-dimensional cyclic transformation on orbital-angular-momentum modes, the directional technology distance function derived from a transformation function, the $d$-plane transform on Euclidean space, several Darboux transformations for operators of order $d$, and the $\tilde d^{(m)}$ convergence accelerator (Ghosh et al., 2017, Çatal-Özer et al., 2019, Qi et al., 2021, Rolanía et al., 2019, Busser et al., 2013, Schlederer et al., 2015, Kolomiytseva, 2018, Chihara, 2024, Shemyakova, 2012, Sidi, 2017).

1. Terminological scope

The term does not name a single standard object across mathematics and physics. In the literature represented here, it is attached to transformations whose common feature is not a shared formula but a shared role: they recast a problem into a more tractable domain, preserve a structural invariant, or generate a new object within a controlled class.

Domain	Meaning of “d-Transformation”	Representative source
Fractional PDEs	Complex fractional transformation $(x,t)\mapsto (X,T)$	(Ghosh et al., 2017)
String theory / DFT	Coordinate-dependent $O(d,d)$ matrix	(Çatal-Özer et al., 2019)
Teleparallel cosmology	Discrete duality transformation from $O(d,d)$ symmetry	(Paliathanasis, 2021)
Scalar–tensor gravity	Invertible generalized disformal transformation	(Takahashi et al., 2021)
Tensor algebra	General transform $d$0 along the third mode	(Qi et al., 2021)
Orthogonal polynomials	Geronimus-type transformation scheme for d-OPS	(Rolanía et al., 2019)
Integrable systems	Darboux transformation of order $d$1 or $d$2-fold DT	(Shemyakova, 2012, Xu et al., 2011)
Quantum impurity problems	Lanczos $d$3 dimensional transformation	(Busser et al., 2013)
Quantum optics	$d$4-dimensional cyclic unitary	(Schlederer et al., 2015)
Production theory	Directional technology distance function from a transformation function	(Kolomiytseva, 2018)
Integral geometry	$d$5-plane transform on $d$6	(Chihara, 2024, Chihara, 2021)
Numerical analysis	$d$7 transformation for convergence acceleration	(Sidi, 2017)

A plausible implication is that “d-Transformation” is best read locally: its meaning is fixed by the surrounding formalism, not by the phrase alone.

2. Fractional and asymptotic transformations

In fractional wave theory, the phrase refers to the complex fractional transformation used to convert Jumarie-type fractional derivatives into ordinary derivatives. For the fractional transport and wave equations,

$d$8

the paper introduces

$d$9

with the conversion rules

$d$0

Under this map the fractional wave equation becomes a standard wave equation in $d$1, so D’Alembert’s formula applies verbatim in the transformed variables, and the final solution is a travelling-wave solution in scaled coordinates

$d$2

with effective speed $d$3 (Ghosh et al., 2017).

For the second-order problem, the transformed solution is

$d$4

and for $d$5 this reduces, after inversion, to the classical D’Alembert formula written in the scaled variables. The paper explicitly states that when $d$6, the formula reduces to the standard D’Alembert solution. It also states that the plots for $d$7 show that “the solution depends on the order of fractional derivative; with the increase of order […] the solution pattern changes.” The word “complex” is purely nominal in this setting: the method is described there as essentially a real-valued fractional coordinate transform (Ghosh et al., 2017).

A distinct asymptotic use appears in numerical analysis as the $d$8-transformation, a member of a broader family of $d$9-transformations for accelerating convergence or summing series whose terms satisfy

$D$0

The construction is based on the remainder model

$D$1

for $D$2. Truncating this expansion at sample indices $D$3 yields a linear system whose solution defines the transformed approximants $D$4. The paper emphasizes implementation by the recursive W-algorithm, and states that the method applies whether the series converge or diverge, provided the relevant asymptotic structure is present. It also states that the same framework applies efficiently to infinite products $D$5 when $D$6 with $D$7 (Sidi, 2017).

3. Duality, disformal, and cosmological uses

In string theory and Double Field Theory, the phrase is used in the sense of an $D$8 duality transformation acting on the $D$9 isometry directions of a Green–Schwarz background. For homogeneous Yang–Baxter deformation, the deformed NS–NS fields satisfy

$d$0

and the paper identifies the corresponding $d$1 element as the pure $d$2-shift

$d$3

This realizes Yang–Baxter deformation as a coordinate-dependent $d$4 transformation. In the Gauged Double Field Theory interpretation, the same transformation is a duality twist $d$5, and the resulting fluxes satisfy

$d$6

when the $d$7-matrix satisfies the classical Yang–Baxter equation. The paper further states that unimodularity controls the trace of the $d$8-flux and the need for a generalized dilaton linear in winding coordinates, thereby distinguishing ordinary supergravity from the generalized supergravity frame (Çatal-Özer et al., 2019).

In teleparallel dark energy, the same $d$9 language appears as a discrete duality transformation in minisuperspace. For the teleparallel dilaton model,

$d$0

the paper constructs the symmetry

$d$1

with $d$2 given explicitly in terms of the parameter $d$3. In the variables $d$4, the same transformation becomes the exchange $d$5, and the transformed Lagrangian is

$d$6

which the paper interprets as revealing the $d$7 origin of the duality. It also states that in the limit of large $d$8, the transformation becomes

$d$9

namely the Gasperini–Veneziano scale-factor duality in $\tilde d^{(m)}$0 (Paliathanasis, 2021).

A further generalization occurs in scalar–tensor theory as an invertible generalized disformal transformation. The higher-derivative version considered there is

$\tilde d^{(m)}$1

with $\tilde d^{(m)}$2, $\tilde d^{(m)}$3, and $\tilde d^{(m)}$4. The paper formulates sufficient conditions for invertibility and group closure: $\tilde d^{(m)}$5 where

$\tilde d^{(m)}$6

Under these conditions the inverse map is again of generalized disformal form, and the paper uses this to generate new ghost-free scalar–tensor theories containing third- or higher-order derivatives of the scalar field and higher-derivative couplings to curvature (Takahashi et al., 2021).

4. Algebraic, polynomial, and Darboux constructions

In tensor algebra, the term denotes the general linear/unitary transform along the third mode used to define a generalized t-product and T-SVD. For a tube $\tilde d^{(m)}$7, the transform is

$\tilde d^{(m)}$8

and the induced tubal product is

$\tilde d^{(m)}$9

where the right-hand $(x,t)\mapsto (X,T)$0 is the Hadamard product. On tubal matrices, this yields the transformed-domain identity

$(x,t)\mapsto (X,T)$1

The paper then formulates T-SVD with respect to $(x,t)\mapsto (X,T)$2,

$(x,t)\mapsto (X,T)$3

and proves two Eckart–Young-like theorems whenever $(x,t)\mapsto (X,T)$4 is a doubly real-preserving unitary transformation. The normalized DFT, the DCT, any orthogonal matrix, and the product of the normalized DFT with an orthogonal matrix are explicitly listed as examples (Qi et al., 2021).

In the theory of $(x,t)\mapsto (X,T)$5-orthogonal polynomials, the relevant object is a Geronimus-type transformation scheme adapted to a d-orthogonal setting. Starting from a vector of functionals $(x,t)\mapsto (X,T)$6, the transformed vectors are defined recursively by

$(x,t)\mapsto (X,T)$7

This produces $(x,t)\mapsto (X,T)$8 new vectors $(x,t)\mapsto (X,T)$9 and, when regularity holds, $O(d,d)$0 new $O(d,d)$1-orthogonal polynomial sequences $O(d,d)$2. At the operator level the associated Hessenberg matrices satisfy Darboux-type factorizations such as

$O(d,d)$3

and, for the full $O(d,d)$4-step chain,

$O(d,d)$5

The paper treats this chain itself as the $O(d,d)$6-transformation (Rolanía et al., 2019).

In the Darboux-transform setting, one paper studies invertible Darboux transformations for bivariate LPDOs of arbitrary order $O(d,d)$7. With

$O(d,d)$8

and first-order auxiliary operator $O(d,d)$9 or $O(d,d)$0, the induced map $O(d,d)$1 is invertible exactly when

$O(d,d)$2

For $O(d,d)$3, the paper states the criterion

$O(d,d)$4

and presents this as the higher-order analogue of the classical invertible Laplace transformation (Shemyakova, 2012).

A related integrable-systems use appears in the derivative nonlinear Schrödinger equation, where the $O(d,d)$5-fold Darboux transformation is a $O(d,d)$6 polynomial matrix $O(d,d)$7 whose entries are written as ratios of $O(d,d)$8 and $O(d,d)$9 determinants built from eigenfunctions of the Kaup–Newell Lax pair. The transformed fields $d$00 and $d$01 are then generated in determinant form, and under the reduction $d$02 the construction yields explicit DNLS solutions including bright soliton, dark soliton, breather solution, periodic solution, rational traveling solution, and rogue wave (Xu et al., 2011).

5. Dimensional reduction and integral geometry

In quantum impurity theory, the phrase designates an exact canonical Lanczos transformation that maps a quantum impurity problem in a $d$03-dimensional lattice to an equivalent one-dimensional system. Starting from

$d$04

with seed state

$d$05

the Lanczos recursion

$d$06

tridiagonalizes the noninteracting host Hamiltonian. In the new basis,

$d$07

so the impurity couples only to the first site of an effective chain. The paper states that this dimensional reduction decreases the scaling of the entanglement entropy by a factor $d$08, thereby making DMRG practical for large $d$09D and $d$10D hosts (Busser et al., 2013).

In integral geometry, the term is the $d$11-plane transform on $d$12, also described as the $d$13-dimensional Radon transform. For $d$14,

$d$15

with $d$16 and $d$17. The adjoint is a backprojection over all $d$18-planes through a point, and the normal operator satisfies

$d$19

at the symbol level, with symbol

$d$20

This yields the filtered backprojection formula

$d$21

A subsequent microlocal analysis treats $d$22 as an elliptic Fourier integral operator, writes down its canonical relation explicitly, and uses this to analyze metal streaking artifacts generated by products such as $d$23 under filtered backprojection (Chihara, 2024, Chihara, 2021).

6. Cyclic, directional, and production-theoretic meanings

In high-dimensional quantum optics, a $d$24-dimensional cyclic transformation is a unitary operator $d$25 on a $d$26-dimensional subspace such that

$d$27

The paper experimentally implements a four-dimensional cycle on orbital-angular-momentum modes

$d$28

with permutation matrix

$d$29

The optical realization uses a spiral phase hologram, two OAM beamsplitters, and a reflection, and the global OAM map is

$d$30

The same paper explicitly notes that “an $d$31-fold cyclic transformations is an $d$32-root-of-unity transformation” (Schlederer et al., 2015).

In production theory, the relevant construction is the directional technology distance function derived from a transformation function. For technology set $d$33, direction vector $d$34, and production point $d$35, the directional technology distance function is

$d$36

The paper emphasizes two defining properties: $d$37 and

$d$38

It then proves that the standard quadratic specification used in the empirical literature does not satisfy homogeneity of degree $d$39 in the direction vector and therefore “is not the directional technology distance function.” To construct valid functional forms, the paper derives the DTDF from a symmetric transformation function $d$40 satisfying F1–F4, solves the associated optimization problem, and obtains a piecewise explicit expression $d$41 that satisfies D1–D6 (Kolomiytseva, 2018).

Taken together, these usages show that d-Transformation is not a single doctrine but a family of field-specific constructions. In one direction it means changing coordinates so that a fractional PDE becomes classical; in another it means acting by a duality group; elsewhere it means choosing a transform domain, building a Darboux or Geronimus chain, reducing a $d$42-dimensional problem to one dimension, integrating over $d$43-planes, cycling a $d$44-level basis, or measuring directional distance to a production frontier. The shared theme is structural recoding: a d-Transformation replaces the original representation by one in which the governing object—equation, metric, tensor, recurrence, Hamiltonian, image, or technology set—has a more analyzable form.