Novel Linear Transport Equation

• Novel Linear Transport Equation is a framework for generalized and pseudo-inverses that extends classical methods to indefinite, tropical, and high-dimensional algebraic systems.
• It employs hyperbolic pseudoinverses, pseudo-core inverses, and pseudo-reversing techniques to achieve coordinate invariance, stability, and sparsity in applications such as robotics and signal processing.
• Its unifying approach establishes reverse-order and absorption laws, guiding robust computational methods for multiscale analysis and high-dimensional statistical inference.

The novel linear transport equation refers to a family of non-standard generalized inverses, pseudo-inverse constructions, and regularized solutions in linear and multilinear algebraic systems. These methods arise across several mathematical frameworks: indefinite inner product spaces, rings with involution, tropical/supertropical algebra, Wiener algebras, and high-dimensional covariance estimation. Each application adapts or extends the role of the classical Moore–Penrose pseudoinverse to settings where classical conditions fail or alternative algebraic/geometric structures prevail.

1. Hyperbolic Pseudoinverses and Indefinite Metrics

A significant development is the theory of hyperbolic pseudoinverses for the Euclidean group $SE(3)$, crucial in robot kinematics where the Jacobian $J: \mathbb{R}^m \to \mathfrak{se}(3)$ maps joint velocities to twists (Donelan et al., 2017). When $J$ is rectangular or singular (i.e., $m \ne 6$ or at singularities), the Moore–Penrose pseudoinverse $J^+$ is often used but is not invariant under frame transformations due to its reliance on the standard Euclidean inner product.

Instead, the Lie algebra $\mathfrak{se}(3)$ carries a pencil of adjoint-invariant bilinear forms $Q_h$,

$$Q_h = \begin{pmatrix} -2 I_3 & I_3 \ I_3 & 0 \end{pmatrix} + h \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix},$$

parametrized by pitch $h$. Each $Q_h$ is indefinite (signature $(3,3)$) and coordinate-invariant, providing invariance under $SE(3)$ transformations. The associated hyperbolic pseudoinverse is: $J^{+h} = (J^T Q_h J)^{-1} J^T Q_h,$ whenever the generalized Gram matrix $G_h = J^T Q_h J$ is invertible. The existence criterion is $J$ full column rank and $\det(G_h) \ne 0$; the set of singular $h$ (principal pitches) depends on the screw system $S = \operatorname{Im} J$. For $m<6$, $J^{+h}$ is generically unique and invariant under frame changes, directly contrasting the coordinate-dependent $J^+$.

2. Rings with Involution: Pseudo-Core and Drazin-Type Inverses

Generalized inverses in rings with involution $\ast$, notably pseudo-core and pseudo n-strong Drazin inverses, extend partial inversion to elements of arbitrary index (Gao et al., 2016, Cui et al., 2023). For $a \in R$ a $*$-ring, $a$ is pseudo-core invertible if there exists $x$ satisfying: $$x a^{m+1} = a^m, \qquad a x^2 = x, \qquad (a x)^* = a x,$$ for some $m \ge 1$; $x$ is unique and denoted $a^{\circled{D}}$. The construction generalizes core and core-EP inverses from the matrix case to arbitrary rings and incorporates the Drazin inverse $a^D$: $a^{\circled{D}} = a^D a^k (a^k)^{(1,3)},$ for $k \ge m$ and $a^{(1,3)}$ any $\{1,3\}$-inverse. Pseudo n-strong Drazin inverses further relax the nilpotency requirement to radical-nilpotency: $a^n - a x \in \sqrt{J(R)}$, allowing existence in "pseudo $\pi$-polar" rings, which are characterized by all elements admitting some pseudo n-strong Drazin inverse.

Extended absorption and additive laws hold for these inverses. For instance, in rings with pseudo-core inverses $a^{\circled{D}}, f^{\circled{D}}$, the absorption law is equivalent to invertibility of $1 + a^{\circled{D}} b$ ($b = f - a$), with explicit additive formula,

$f^{\circled{D}} = (1 + a^{\circled{D}} b)^{-1} a^{\circled{D}}.$

These extend to minimal weak Drazin inverses, group inverses, and related outer inverses (Zhou et al., 6 Aug 2025).

3. Pseudoinverses in Banach Algebras and Multiscale Analysis

The concept of pseudo-reversing in the Wiener algebra $\mathcal{A}(\mathbb{T})$ provides non-standard pseudoinverse constructions for convolution/refinement operators in multiscale analysis (Mattar et al., 2023). Given a mask symbol $\alpha(z)$, its pseudo-reverse is constructed by regularizing zeros on the unit circle: $$\alpha^{-\dagger}_\xi(z) = \prod_{r\in\Lambda\setminus\mathbb{T}}(z-r) \prod_{r\in\Lambda\cap\mathbb{T}} (z - (1+\xi) r)$$ and

$$\alpha^\dagger_\xi(z) = [\alpha^{-\dagger}_\xi(z)]^{-1},$$

so pseudo-reversing always exists (Wiener's lemma) and improves condition number $\kappa(\alpha^\dagger_\xi) \leq (1+2/\xi)^n$. There is no Hilbert-space or least-squares interpretation, but these pseudo-inverses provide stable, localized transforms, especially for manifold-valued data, and ensure tunable synthesis error bounds.

4. Generalized Pseudoinverses for Sparsity and High-Dimensional Statistics

High-dimensional linear algebra and statistics confront singular or poorly conditioned matrices. Several contributions define pseudo-inverses beyond the classical Moore–Penrose, either to induce sparsity or to regularize estimation (Fuentes et al., 2016, Ponte et al., 2023, Bodnar et al., 23 Mar 2024).

• Sparse generalized inverses: Norm-minimizing pseudoinverses (1-norm, or mixed $2,1$-norm) subject to Penrose or relaxed conditions yield row/entry-wise sparse solutions suitable for large least-squares problems. Second-order cone programming (SOCP) formulations efficiently find generalized inverses $H$ minimizing $\|H\|_{2,1}$ under $AHA = A$; they automatically satisfy reflexivity and $ah$-symmetry, producing sparse and computationally tractable pseudoinverses for block-structured and network applications.
• Asymptotic analysis: Ridge-type inverses and Moore–Penrose inverses are analyzed under minimal assumptions (no identity covariance, no normality), with deterministic limit formulas for trace moments via resolvent methods and partial exponential Bell polynomials. These explicit asymptotics enable provably optimal shrinkage estimators for precision matrices, achieving almost sure minimization of quadratic loss in the large-$p$, large-$n$ regime (Bodnar et al., 23 Mar 2024).

5. Pseudo-Inverses in Tropical and Supertropical Algebra

In tropical and supertropical algebra, invertibility is dramatically restricted. The "supertropical pseudo-inverse" $A^\nabla$ is defined via the adjugate and tropical permanent, $A^\nabla = \mathrm{adj}(A)/\mathrm{det}(A)$ (Niv, 2013). Whereas only permutation/diagonal matrices are invertible tropically, $A^\nabla$ extends partial inversion, satisfying

$$A A^\nabla \models_{gs} I, \qquad A^\nabla A \models_{gs} I,$$

with diagonal entries tangible and off-diagonals ghost-corrected. For definite matrices, $A^\nabla$ coincides with the tropical closure $A^*$ and stabilizes powers. Under tropical similarity ($B' = A^\nabla B A$), the characteristic polynomials and eigenvalues exhibit a reciprocal phenomena: $$\det(A) f_{A^\nabla}(x) \models_{gs} x^n f_A(x^{-1}),$$ with eigenvalues and characteristic roots reflected, supporting a robust representation theory in max-plus settings.

6. Functional Analysis and Weak Pseudo-Inverses of Monotone Maps

For monotone, non-strictly monotone maps $t: [a, b] \to [m, n]$, the weak pseudo-inverse $t^{[-1]}$ is defined as: $$t^{[-1]}(y) = \sup\{ x \in [a, b] \mid (t(x) - y)(t(b) - t(a)) \leq 0\}.$$ This construction recovers the classical inverse when $t$ is bijective and strictly monotone, but properly extends inversion to account for flat zones and plateaux, which classical pseudo-inverses do not invert correctly (Chen et al., 27 Oct 2024). Such extensions are necessary to recover associativity in functional compositions.

7. Unifying Themes and Algebraic Properties

These non-standard linear transport equations and their associated pseudo-inverses preserve, extend, or adapt core identities:

• Reverse-order law: generalized inverses in a ring often satisfy $(ab)^{\circled{D}} = b^{\circled{D}} a^{\circled{D}}$ under suitable commutation.
• Additive/absorption laws: Explicit conditions link the sum of inverses and absorbability to invertibility of certain commutators and ideal-generated elements (Zhou et al., 6 Aug 2025).
• Stability and spectral theory: The periodicity, stabilization, and closure under composition for both algebraic and analytic pseudo-inverses support the development of robust solution theories in complex and infinite-dimensional systems.

These constructions form the foundation for contemporary advances in algebraic analysis, computational mathematics, multiscale modeling, and high-dimensional inference, with applications spanning robotics, signal processing, numerical linear algebra, control, and stochastic estimation.