arXiv subjects

Daniel Alayón-Solarz

Publications and source records attributed to Daniel Alayón-Solarz.

10 recordsLinked to original sources

arXiv2026

Moduli of the Sourceless Framed Beltrami-Vekua Normal Form

We study the moduli of sourceless framed Beltrami-Vekua equations under recombinations of the unknown, scalings, and orientation-preserving changes of variables. On every bounded simply connected domain, such an equation reduces to $w_{\bar z}=B\bar w$ on the unit disk, with residual symmetries given exactly by zero-free holomorphic gauges and Möbius transformations. When $B$ is zero-free, the equation is completely classified by two data modulo Möbius: the hyperbolic mass density $\vartheta=\tfrac14(1-|z|^2)^2|B|^2$ and the phase-curvature current $K=Δ\arg B$, whose hyperbolic density at $C^2$ regularity is $κ=Δ_{\mathrm{hyp}}\arg B$. We determine the exact range of these invariants: every positive Hölder density and every phase current arising as the Laplacian of a Hölder phase occur. For fields with zeros, the classification extends to the phase-integrable sector through the triple $(\vartheta,dη,d{\star}η)$, where $η=\operatorname{Im}(dB/B)$. On the tame sector, $dη$ is the atomic charge measure, while $d{\star}η$ carries the remaining phase curvature. Thus the pseudo-analytic mass and charge are numerical projections of a larger infinite-dimensional moduli space. Explicit equal-mass, equal-charge, inequivalent equations are exhibited.

math.CV↗

arXiv2026

The Pseudo-Analytic Charge

The framed Beltrami--Vekua equation $Φ(w_{\bar z} - μw_z) + Ψ(\overline{w_z} - μ\overline{w_{\bar z}}) + \mathfrak{a}w + \mathfrak{b}\bar w = \mathfrak{f}$, with $|μ|<1$ and $|Φ|>|Ψ|$, carries a numerator field $N = Φ\mathfrak{b} - Ψ\mathfrak{a} - W_L(Φ,Ψ)$ whose weighted modulus integrates to the pseudo-analytic mass. This paper extracts the integer carried by the same field. When the zero set of $N$ is compactly contained in a bounded simply connected domain, the winding number of $N$ along any enclosing curve -- the pseudo-analytic charge $n \in \mathbb{Z}$ -- is invariant under every recombination $w = φw' + ψ\bar w'$ of the unknown, every scaling of the equation, and every orientation-preserving $C^1$ change of variables: recombinations multiply $N$ by the positive factor $|φ|^2 - |ψ|^2$, so their invariance is exact, while on multiply connected domains the other two actions fix the component charges only in $\mathbb{Z}/2\mathbb{Z}$ and the total charge exactly. The charge is a Brouwer degree: it localizes at the zeros of $N$, vortices which no action of the class creates or destroys; an isolated vortex persists under perturbation of the data precisely when its local charge is non-zero. It involves the Beltrami coefficient only through the $L$-Wronskian of the frame, and is $μ$-independent wherever $W_\partial(Φ,Ψ) \equiv 0$ -- in particular at the trivial frame, where $N = \mathcal{B}$ and the charge is the gauge-invariant winding of the coefficient of the Beltrami--Vekua equation. Mass and charge are independent: every pair in $(0,\infty)\times\mathbb{Z}$ is realized.

math.CV↗

arXiv2026

The Framed Beltrami-Vekua Normal Form and its Pseudo-Analytic Mass

We normalize a first-order real planar elliptic system, by pointwise algebra, to a framed Beltrami-Vekua equation $Φ(w_{\bar z} - μw_z) + Ψ(\overline{w_z} - μ\,\overline{w_{\bar z}}) + \mathfrak{a} w + \mathfrak{b} \bar w = \mathfrak{f}$, with $|μ| < 1$ and $|Φ| > |Ψ|$, and compute the closed transformation laws of its data under the recombination of unknowns $w \mapsto φw + ψ\bar w$ and under orientation-preserving $C^1$ changes of variables. The 2-form $Θ= \frac{\bigl|\,Φ\,\mathfrak{b} - Ψ\,\mathfrak{a} - (Φ\, LΨ- Ψ\, LΦ)\,\bigr|^2}{\bigl(|Φ|^2 - |Ψ|^2\bigr)^2\,\bigl(1 - |μ|^2\bigr)}\; dx\, dy$, with $L = \bar\partial - μ\,\partial$, is invariant under the recombination and covariant under the changes of variables. The total mass $\mathcal{M} = \int_ΩΘ$ is therefore an invariant of the equivalence class. One recombination and one scaling carry any framed equation, in closed form, onto the trivial-frame slice - a Beltrami-Vekua equation over the same $μ$ - there identifying $Θ$ with the pseudo-analytic mass density of the unframed equation. We then show all of this persists at measurable regularity: it suffices that $μ$ be measurable and locally elliptic and that the frame lie in $W^{1,2}_{\mathrm{loc}} \cap L^\infty_{\mathrm{loc}}$, the changes of variables then being quasiconformal homeomorphisms. In that class every equation with $\|μ\|_\infty < 1$ is quasiconformally equivalent, of equal mass, to one over $μ= 0$.

math.CV↗

arXiv2026

The complex form of Vekua's characteristic factor: a derivation, and two sign corrections in §7 of Generalized Analytic Functions

In \S7 of \emph{Generalized Analytic Functions} \cite{vekua}, the reduction of a first-order elliptic system to canonical form proceeds through a factor of the characteristic equation, which Vekua selects in real form~(7.13) and then restates, without derivation, in complex (Beltrami) form~(7.14). We supply that conversion. With the standard Wirtinger convention used below, the complex form of~(7.13) is the negative of the coefficient printed in~(7.14) (p.~126, 1962 Pergamon edition), and we confirm the correct sign against Vekua's own factorization~(7.12) and his canonical coefficient~(7.17). A related sign defect appears in the second-order Beltrami coefficient~(7.23) (p.~127): a coordinate solving (7.23) as printed reduces the equation to the canonical form~(7.26) only in the special symmetric case $a=c$. In both instances the error is confined to the displayed coefficient and leaves the surrounding reduction, carried out independently of it, intact; we record the corrected coefficient in each case.

math.CV↗

arXiv2026

The Absorption Theorem for the Beltrami-Vekua Normal Form

The Beltrami-Vekua normal form assigns to every smooth first-order real planar elliptic system a complex equation $w_{\bar z}-μw_z+\mathcal{A}w+\mathcal{B}\bar w=\mathcal{F}$ by an explicit pipeline. A companion paper showed that the density $Θ=|\mathcal{B}|^2/(1-|μ|^2)\,dx\,dy$ and its total mass are invariants under multiplicative gauges $w\mapstoϕw$ and orientation-preserving diffeomorphisms. The real system carries a larger symmetry: its unknowns may be recombined by any pointwise invertible real-linear substitution $w=φv'+ψ\bar v'$, the complex gauges being the case $ψ\equiv0$. We prove the absorption theorem: re-normalizing through the pipeline after any such substitution returns to the gauge orbit of the original equation, with a universal explicit gauge $\tildeφ=-iλ/(φ-ψ)$, where $λ$ is the spectral root of the structure polynomial.

math.CV↗

arXiv2026

The Pseudo-Analytic Mass of a Beltrami-Vekua Equation

Every smooth first-order real planar elliptic system admits a universal complex form $w_{\bar z} - μw_z + \mathcal{A} w + \mathcal{B} \bar w = \mathcal{F}$, which we call the Beltrami-Vekua equation: the data $(μ, \mathcal{A}, \mathcal{B}, \mathcal{F})$ are produced from the original system by algebraic operations and differentiations, with no auxiliary PDE. On this space we study the joint action of multiplicative gauges $w \mapsto ϕw$ and orientation-preserving diffeomorphisms. Our main result is that the 2-form $Θ= |\mathcal{B}|^2 / (1 - |μ|^2) \, dx \, dy$ is gauge-invariant and pulls back covariantly under diffeomorphisms; its form is forced, with $|\mathcal{B}|^2$ the unique $\mathcal{B}$-quadratic combination invariant under $\mathcal{B} \mapsto \mathcal{B}ϕ/\barϕ$ and $1 - |μ|^2$ the conformal distortion factor from the diffeomorphism law for $μ$. The total mass $\mathcal{M}(D) = \int_ΩΘ$, the \emph{pseudo-analytic mass}, vanishes precisely on the analytic class $\mathcal{B} \equiv 0$ and separates a continuous family of pairwise inequivalent pseudo-analytic equations on the disk. As a by-product, Vekua's two-stage reduction - uniformization then gauge elimination - requires only one variable-coefficient PDE solve: the Beltrami diffeomorphism supplies the integrating factor for a flat $\bar\partial$-equation.

math.CV↗

arXiv2026

Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation

For the rigid subclass of variable elliptic structures -- characterized equivalently by the inviscid Burgers law $λ_x+λλ_y=0$ or the self-dilatation $μ_{\bar z}=μμ_z$ -- we show that the auxiliary Beltrami equation in the classical Vekua pipeline is unnecessary. The canonical coordinate $ξ=y-λx$, computed by arithmetic from the spectral parameter $λ$, reduces every rigid variable-algebra Vekua equation to a standard Vekua equation in $ξ$ on any open set where the characteristic Jacobian $Φ=\barξ_x+λ\barξ_y$ does not vanish, with global reduction on domains where $ξ$ is injective. No PDE is solved at any stage.

math.CV↗

arXiv2026

The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures

We introduce the Burgers transform $\mathcal{B}$, a nonlinear bijection between holomorphic functions $f\colon U\to\mathbb{C}^+$ and rigid variable elliptic structures on the plane, defined implicitly by $λ= f(y-λx)$. The output automatically satisfies the conservative complex Burgers equation $λ_x+λλ_y=0$. Our main result is that holomorphicity of the seed $f$ is necessary, not merely sufficient, for rigidity: any $C^1$ function whose implicit solution satisfies $λ_x+λλ_y=0$ must be holomorphic. This closes a gap in the existing literature and identifies $\operatorname{Hol}(U,\mathbb{C}^+)$ as the maximal seed space compatible with rigidity. The obstruction formula $H|_{x=0} = 2i\,(\operatorname{Im} f)\,f_{\bar{w}}$ quantifies the cost of non-holomorphicity at the level of the initial data. We characterise the domain of $\mathcal{B}$ through shock formation, its interaction with affine automorphisms of $\mathbb{C}^+$, and the infinitesimal structure: the propagator $\mathcal{P}_f = D\mathcal{B}_f$ satisfies a Jacobian-twisted multiplicativity that deforms the seed algebra by the density of characteristics. Four worked examples -- affine, exponential, inverse, and trigonometric seeds -- show that the complexity class of a seed and that of the resulting structure are generically unrelated.

math.CV↗

arXiv2026

Variable Elliptic Structures on the Plane: Transport Dynamics, Rigidity, and Function Theory

We develop a theory of variable elliptic structures on planar domains, in which the imaginary unit $i(x,y)$ is a moving generator of a rank-two real algebra bundle defined by a smoothly varying quadratic relation. Differentiating this relation produces an intrinsic obstruction $G = i_x + i\, i_y$ that governs all deviations from the constant-coefficient theory, such as the inhomogeneity of the generalized Cauchy-Riemann system and the forcing of a universal complex inviscid Burgers equation satisfied by the spectral parameter. The vanishing of $G$ -- rigidity -- selects the conservative regime of this transport law and simultaneously restores a coherent function theory: Cauchy-Pompeiu representation, covariant holomorphicity with gauge structure, a similarity principle, and a factorization of the variable Laplacian. A rigidity-flatness theorem shows that the only structure that is both rigid and Riemannian-flat is the constant one. Translated into Beltrami coordinates, the rigidity condition becomes $μ_{\bar{z}} = μ\, μ_z$: the structure map satisfies its own Beltrami equation, a self-dilatation property in the Poincaré disk. The central result is the Fundamental Independence Theorem: the Beltrami modulus $\|μ\|_{C^0}$ (zeroth order) and the transport obstruction $\|R(μ)\|_{C^{0,α}}$ (first order) are independently prescribable.

math.CV↗

arXiv2026

A Degenerate Elliptic System Solvable by Transport: A Cautionary Example

We exhibit a one-parameter family of first-order real elliptic systems on the plane whose ellipticity constant degenerates to zero as $δ\to 0$, with condition number $κ= O(δ^{-2})$. For any fixed elliptic solver operating at finite precision, the parameter $δ$ can be chosen small enough to defeat the solver; no uniform numerical scheme based on the ellipticity constant alone can handle the entire family. Despite this, every member of the family is explicitly solvable -- and its initial value problem well posed -- by elementary means once a transport-theoretic invariant is identified. The cost of the transport solution is independent of $δ$. The example serves as a cautionary tale: the ellipticity constant alone does not determine the practical difficulty of a first-order PDE. Before invoking an elliptic solver, one should compute the transport obstruction $G$; its vanishing -- or smallness -- signals structure that standard elliptic methods miss entirely.

math.AP↗