Tanuj Gupta | Searcharxiv

arXiv2025

Diederich-Forn\ae ss index and global regularity of the complex Green operator: domains with comparable Levi eigenvalues

Let $\Omega\subset \mathbb{C}^{n}$, with $n \geq 3$, be a smooth bounded pseudoconvex domain satisfying the symmetric eigenvalue comparability condition $D(q_0)$ for some $1\le q_0\le n-2$. We show that if the Diederich-Fornaess-index of $\Omega$ is one, then the complex Green operator $G_q$, associated with $\Omega$, is globally regular for $q$ in the range $\min\{q_0,\, n - 1 - q_0\} \leq q \leq \max\{q_0,\, n - 1 - q_0\}$.

math.CV↗

arXiv2024

AI-driven Inverse Design of Band-Tunable Mechanical Metastructures for Tailored Vibration Mitigation

On-demand vibration mitigation in a mechanical system needs the suitable design of multiscale metastructures, involving complex unit cells. In this study, immersing in the world of patterns and examining the structural details of some interesting motifs are extracted from the mechanical metastructure perspective. Nine interlaced metastructures are fabricated using additive manufacturing, and corresponding vibration characteristics are studied experimentally and numerically. Further, the band-gap modulation with metallic inserts in the honeycomb interlaced metastructures is also studied. AI-driven inverse design of such complex metastructures with a desired vibration mitigation profile can pave the way for addressing engineering challenges in high-precision manufacturing. The current inverse design methodologies are limited to designing simple periodic structures based on limited variants of unit cells. Therefore, a novel forward analysis model with multi-head FEM-inspired spatial attention (FSA) is proposed to learn the complex geometry of the metastructures and predict corresponding transmissibility. Subsequently, a multiscale Gaussian self-attention (MGSA) based inverse design model with Gaussian function for 1D spectrum position encoding is developed to produce a suitable metastructure for the desired vibration transmittance. The proposed AI framework demonstrated outstanding performance corresponding to the expected locally resonant bandgaps in a targeted frequency range.

cs.LG↗

arXiv2023

Modifications of the Levi core

We construct a family of subdistributions of the Levi core $\mathfrak{C}(\mathcal{N})$ called modified Levi cores $\{\mathcal{M}\mathfrak{C}_{\mathcal{A}}\}_{\mathcal{A}}$ indexed over closed distributions $\mathcal{A}$ that contain the Levi null distribution $\mathcal{N}$ and are contained in the complex tangent bundle $T^{1, 0}b\Omega$ of a smooth bounded pseudoconvex domain $\Omega$. We show that Catlin's Property ($P$) holds on $b\Omega$ if and only if Property ($P$) holds on the support of one, and hence all, of the modified Levi cores. In $\mathbb{C}^2$, all of the modified Levi cores coincide. For a smooth bounded pseudoconvex complete Hartogs domain in $\mathbb{C}^2$ that satisfies Property ($P$), we show that its modified Levi core is trivial. This contrasts with $\mathfrak{C}(\mathcal{N})$, which can be nontrivial for such domains.

math.CV↗