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| Main Author: | |
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| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
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| Subjects: | |
| Online Access: | https://doi.org/10.5281/zenodo.19599672 |
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Table of Contents:
- This paper develops a multidimensional sequel to the author's hyper-residue program for improper integrals. The central shift is geometric: instead of a single Mellin variable with poles at isolated points, we build a several-variable spectral-Mellin calculus whose singular set consists of coordinate planes and genuine affine hyperplanes. The first part of the paper proves an AbuGhuwaleh tensor spectral solver for Laplace-orbit kernels on the positive orthant and derives a rectangular meromorphic continuation theorem for gapped product spectra. The second part proves an AbuGhuwaleh simplex-radial solver: when the kernel depends on a weighted simplex radius $\rho_\lambda(x)=\lambda_1x_1+\cdots+\lambda_dx_d$, the $d$-dimensional improper integral collapses exactly to a one-dimensional spectral symbol evaluated at the combined weight $\Sigma(z)+\tau$, thereby producing affine hyperplane poles $\Sigma(z)+\tau=-n$. The third part pushes the construction further by monomial changes of variables. This yields oblique linear-pole geometries of the form $(A^{-T}z)_j=-n$ and $\langle A^{-1}\one,z\rangle+\tau=-n$, together with explicit hyperplane residues and regular values. We also prove oscillatory pole-collapse, moment-cancellation, frequency-transport, scaling, and logarithmic differentiation theorems. Benchmark examples include algebraic radial kernels, one-atom hyperplane models, cancellation of the first affine pole, tensor-product Beta families, and a genuinely oblique two-variable monomial family. The resulting framework is substantially stronger than a one-dimensional gapped theory because it creates an explicit multidimensional residue geometry for improper integrals while remaining completely spectral and constructive.