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Juan Bory-Reyes

Publications and source records attributed to Juan Bory-Reyes.

At least 19 recordsLinked to original sources
arXiv2026

Covariance property for a quaternionic fractional Cauchy-Riemann operator under affine transformation

In this paper, we continue the development of the foundations of a quaternionic function theory that is associated to a fractional proportional Cauchy-Riemann type operator acting on quaternionic functions. We introduce a new class of fractional hyperholomorphic functions adapted to affine transformations of the domain, and study the corresponding quaternionic right module structure. In particular, we establish versions of the Stokes, Borel-Pompeiu and Cauchy formulas for this function theory. A covariance property of the fractional operator under affine transformations is proved, showing that the composition with an affine map, weighted by a suitable quaternionic factor, preserves the class of fractional hyperholomorphic functions. The result is shown to include the case considered previously by the authors, and extends their covariance property to the setting of affine transformations.

math.CV↗
arXiv2026

On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves

Hyperanalytic functions, in the sense established by the mathematician Avron Douglis, are Douglis algebra-valued functions defined via a hypercomplex structure rather than the standard Cauchy-Riemann equations characteristic of traditional complex analysis. The classes of polyhyperanalytic and meta-hyperanalytic functions represent advanced generalizations of Douglis's analysis. They are employed in the study of partial differential equations and elasticity, extending the concept of the classical holomorphic function through higher-order iterations and non-homogeneous terms. The aim of this work is to find solvability conditions for a fundamental Riemann-type boundary value problem for spaces of poly-hyperanalytic and meta-hyperanalytic functions defined on an open, bounded, simply connected subset of the complex plane, where the boundary need only be a closed d-summable curve. In fractal geometry, d-summability is a geometric property used to define the boundaries of fractal domains, enabling advanced mathematical integration and calculus on complex structures defined by Jenny Harrison and Alec Norton.

math.CV↗
arXiv2026

Exceptional supersphere integration and logarithmic Pizzetti kernels

We study orthosymplectically invariant supersphere integration at the exceptional superdimensions $M=-2u$, where the harmonic Fischer structure becomes nonsemisimple and the Pizzetti pairing degenerates. For the meromorphically continued homogeneous inverse kernels we obtain the generating function \[ \mathscr G_\mu(\rho;x,y) =\frac{\Gamma(\mu/2)}{2\pi^{\mu/2}} \bigl(1+\rho\{x,y\}+\rho^2x^2y^2\bigr)^{-\mu/2}. \] At $\mu=-2u$, its Laurent expansion has a polynomial residue and a logarithmic finite part. We prove that these coefficients recover the complete degreewise duality structure on a fixed superspace with nonzero bosonic dimension. In degrees $k\le u$, the residue inverts a canonical renormalized pairing on $\mathcal P_k$. In the collision range $u<k\le2u$, the ordinary pairing has radical $(x^2)^{k-u}\mathcal P_{2u-k}$; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For $k\ge2u+1$, the finite part is the ordinary inverse kernel. We also establish the nondegenerate head--socle pairing on the generalized harmonic modules. As an application, we derive covariant right--left radial $q$-monogenic zonal symbols and identify precise degree-one obstructions to transferring scalar Pizzetti reproduction through a one-sided $q$-Fischer projection.

math.CV↗
arXiv2026

A connection between quaternionic slice regular and Fueter hyperholormorphic functions

This paper presents some results for the slice regular functions and hyperholomorphic functions theories, as consequences of new relationships between the global operator $G$, Fueter operator $\mathcal D$ and Laplacian operator $\Delta_{\mathbb{R}^{4}}$. Although our results are presented in the quaternionic setting, a fine interpretation in terms of vector calculus of the slice regular function theory yields an interesting relationship between this theory and harmonic analysis. Moreover, we introduce and study a submodule of the hyperholomorphic functions associated to the unit ball $\mathbb B^4(0,1)$ given in terms of new series expansions of the functions.

math.CV↗
arXiv2026

Right q-vector calculus at integral superdimension: localized decompositions and resonance

We specialize the intrinsic right $q$-vector derivative on radial algebras to integral superdimension. The formal dimension is encoded by an independent coefficient $Q$, and formal radial superspace of superdimension $M=m-2n$ is obtained by the coefficient specialization $Q\mapsto q^M$. This gives a rigorous universal calculus and, on finite blocks containing at most $m$ abstract vectors, a faithful coordinate realization on $\mathbb{R}^{m|2n}$. The localized exterior result is formulated as a Green decomposition by complementary projector images. Whenever the specialized finite determinant is nonzero, full left multiplication yields a determinant-localized right-monogenic Fischer decomposition. Beyond this base-change theory, we determine the exceptional one-vector calculus completely: for $M=-2\ell$ there is one additional singular monomial and one missing image monomial, whereas all other integral superdimensions give a surjective derivative with constants as its kernel. We then prove that the degree-zero Fischer operator is exactly diagonal on exterior blades, obtain its determinant explicitly, classify all support-resonance values in $0<q<1$, and give an exact kernel-rank formula as a sum of support multiplicities. On a block with $N$ auxiliary vectors, an even support rank $p$ has pure multiplicity $\binom Np$ on the support truncation. At an odd-support root, every lower odd factor is nonzero and any simultaneous lower resonance is unique, even, and characterized by one strictly monotone scalar equation. These results distinguish persistent nonpositive-even-superdimension defects from isolated support-dependent $q$-resonances. An appendix records that constant scalar projection of two independent orthogonal right $q$-vector derivatives does not descend to the Hermitian quotient.

math.CV↗
arXiv2026

A Non-integer Dimensional Space Approach to the Moisil-Teodorescu Operator

The vector calculus in non-integer dimensional space (NIDS), including the NIDS version of the standard vector differential operators (gradient, divergence, and curl) is well-known. A deformation of the quaternionic Moisil-Teodorescu operator, written in terms of NIDS derivatives is the main purpose of this article. Along similar lines, we consider the NIDS reformulation of the quaternionic Bitsadze operator and the Lam\'e-Navier operator of the three-dimensional elasticity theory. Also, a quaternionic reformulation of a NIDS time-harmonic Maxwell system is introduced, whose solutions are directly related with those of the perturbed NIDS Moisil-Teodorescu operator. Finally, a generalized approach to the study is addressed.

math.CV↗
arXiv2026

On the Negation of a Hyperbolic-Valued Probability Distribution

In the context of hyperbolic numbers we define the concept of negation of finite hyperbolicvalued probability distributions that is based on the partial order induced by the idempotent structure of hyperbolic numbers. Then, a hyperbolic majorization and general hyperbolic negators are introduced. For a broad class of generated negators, we prove that the original distribution majorizes its negation. This comparison yields that entropy increase for the strong hyperbolic Shannon entropy and the hyperbolic Gini-Simpson entropy, and it implies component-wise uniformization of the iterated negation. Finally, we analyze involutive property of hyperbolic negators and prove that are structurally distinct from the generated negators responsible for the entropy increase. These results show that hyperbolic probabilistic negation is not merely a component-wise copy of the real case, but a theory governed by the interaction between idempotent decomposition, partial order, and entropy measure.

math.CV↗
arXiv2026

Intrinsic \(q\)-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras

We construct intrinsic right q-radial vector derivatives on radial algebras. Here intrinsic means that the construction uses only the abstract radial variables and their central scalar invariants, without choosing coordinates, a Clifford representation, a quadratic form, or an ambient dimension, and that it is compatible with enlargement of the finite parameter set. Throughout we use the right-handed convention, whose classical limit is the standard right action \([F]\partial_x\). A coordinatewise replacement of ordinary partial derivatives by Jackson derivatives does not preserve even the one-vector radial subalgebra, so the deformation must act on the mixed scalar invariants associated with a distinguished vector variable. After extending scalars to \(\B=\R[q,q^{-1},Q,(1-q)^{-1}]\), where \(Q\) records the formal dimension through the specialization \(Q=q^M\), we define for every finite set \(Y\subset S\setminus\{x\}\) a relative right \(q\)-vector derivative \(\partial^{Y,\mathrm R}_{x,q}\) on \(R_{\B}(\{x\}\cup Y)\). These operators are equivariant under relabelling and compatible with inclusions of finite sets; hence they induce a direct-limit operator on the scalar extension of every radial algebra.

math.CV↗
arXiv2026

Magnetic Dirichlet Laplacian on a perturbed twisted tube

It is well known that the spectrum of the Dirichlet Laplacian for a compact perturbation of a three-dimensional, periodically twisted tube is unstable with respect to domain deformations. This means that if the periodically twisted tube is unperturbed, then the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, the perturbation of this domain produces eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a magnetic field. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small and local deformations of the domain.

math.SP↗
arXiv2025

On slice regular fractal-fractional Dirichlet type spaces

In this paper, we study some families of right modules of quaternionic slice regular functions induced by a generalized fractal-fractional derivative with respect to a truncated quaternionic exponential function on slices. Important Banach spaces of slice regular functions, namely the Bergman and the Dirichlet modules, are two important elements of one of our families.

math.CV↗
arXiv2025

Three-dimensional magnetic Schr\"odinger operator with the potential supported in a tube

In this paper, we study the following magnetic Schr\"odinger operator in $\mathbb{R}^3$: \[ H=(i \nabla +A)^2- \tilde{V}, \] where $\tilde{V}$ is non-negative potential supported over the tube built along a curve which is a local deformation of a straight one, and $B:=\mathrm{rot}(A)$ is a non-zero and local (i.e., a compact supported) magnetic field. Based on some new strategies, we first prove that the magnetic field does not change the essential spectrum of this system. Finally, in the last section of this paper, we establish the sufficient condition such that the discrete spectrum is empty.

math.SP↗
arXiv2025

On quaternionic analysis and a certain generalized fractal-fractional $\psi$-Fueter operator

This paper introduce a fractal {\psi}-Fueter operator in the quaternionic context inspired in the concepts of proportional derivative and Hausdorff derivative (fractal derivative) of a function with respect to a fractal measure. Moreover, we establish the corresponding Stokes and Borel- Pompeiu formulas associated to this generalized fractal {\psi}-Fueter operator with respect to a vector valued function induced by truncated exponential functions.

math.CV↗
arXiv2024

A Borel-Pompeiu formula in a $(q,q')$-model of quaternionic analysis

The study of $\psi-$hyperholomorphic functions defined on domains in $\mathbb R^4$ with values in $\mathbb H$, namely null-solutions of the $\psi-$Fueter operator, is a topic which captured great interest in quaternionic analysis. This class of functions is more general than that of Fueter regular functions. In the setting of $(q,q')-$calculus, also known as post quantum calculus, we introduce a deformation of the $\psi-$Fueter operator written in terms of suitable difference operators, which reduces to a deformed $q$ calculus when $q'=1$. We also prove the Stokes and Borel-Pompeiu formulas in this context. This work is the first investigation of results in quaternionic analysis in the setting of the $(q,q')-$calculus theory.

math.CV↗
arXiv2023

A bicomplex proportional fractional $(\vartheta,\varphi)-$weighted Cauchy-Riemann operator using Riemann-Liouville derivatives with respect to an hyperbolic-valued function

Based on the Riemann-Liouville derivatives with respect to functions taking values in the set of hyperbolic numbers, we consider a novel bicomplex proportional fractional $(\vartheta,\varphi)-$weighted Cauchy-Riemann operator, involving weights hyperbolic orthogonal bicomplex functions. This operator is defined for the first time here, and its associated fractional Borel-Pompeiu formula is proved as the main result.

math.CV↗
arXiv2023

Fractional slice regular functions of a quaternionic variable

The theory of slice regular functions of a quaternionic variable on the unit ball of the quaternions was introduced by Gentili and Struppa in 2006 and nowadays it is a well established function theory, especially in view of its applications to operator theory. In this paper, we introduce the notion of fractional slice regular functions of a quaternionic variable defined as null-solutions of a fractional Cauchy-Riemann operators. We present a fractional Cauchy-Riemann operator in the sense of Riemann-Liouville and then in the sense of Caputo, with orders associated to an element of $(0,1)\times \mathbb R \times (0,1)\times \mathbb R$ for some axially symmetric slice domains which are new in the literature. We prove a version of the representation theorem, of the splitting lemma and we discuss a series expansion.

math.FA↗