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On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians
Bart Rosenzweig, Jonathan Stanfill
https://arxiv.org/abs/2606.04225 https://arxiv.org/pdf/2606.04225 https://arxiv.org/html/2606.04225
arXiv:2606.04225v1 Announce Type: new
Abstract: We answer in the affirmative a question posed by V. Maz'ya of whether one can continue as a meromorphic function of $t$ the series representation of the fundamental solution of a certain nonlocal parabolic equation associated to a logarithmic Laplacian on the circle, which arises in the study of boundary value problems associated to the ordinary Laplacian on domains with thin cavities. The $a\ln(n) O(1)$ growth of the eigenvalues of the integral operator, together with explicit formulas for the eigenfunctions and the subleading asymptotic behavior of the eigenvalues, allows us to show that the fundamental solution is reminiscent of a sum of shifted Riemann zeta functions or polylogarithms, depending on the spatial variable. We show an analogous result for an operator related to a different logarithmic Laplacian on the interval, whose structure is similar. Along the way we are led to prove and to conjecture a number of curious identities involving Bell polynomials and Bernoulli numbers related to the exponential of the digamma function which are of independent interest.
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Text2Villa: Hierarchical Generation of 3D Indoor Environments with Physics-Aware Analysis-by-Synthesis
Xiang Tang, Ruotong Li, Xiaopeng Fan
https://arxiv.org/abs/2607.17145 https://arxiv.org/pdf/2607.17145 https://arxiv.org/html/2607.17145
arXiv:2607.17145v1 Announce Type: new
Abstract: Generating 3D indoor scenes from natural language holds tremendous potential, yet existing methods predominantly fail to generate multi-room structures with vertical connectivity and arbitrary polygonal boundaries. Furthermore, they lack a deep grounding in continuous 3D physical laws, leading to severe geometric penetrations and floating artifacts. In this work, we propose Text2Villa, a novel hierarchical generative framework. At the macro level, we construct a multi-story dataset to fine-tune an autoregressive layout generator, ensuring the direct parsing of text into 3D building foundations featuring polygonal boundaries and multi-story connectivity. To enforce physical laws during micro-level asset arrangement, we introduce the Affordance-driven Physical-Semantic Scene Graph (A-PSSG) to explicitly abstract physical affordances (such as support surfaces and containment cavities) into node attributes, establishing strict geometric and semantic edge constraints. Guided by the A-PSSG, we formulate scene instantiation as a constrained closed-loop optimization problem following the analysis-by-synthesis paradigm. By integrating an underlying geometric collision detection engine with the high-level semantic reasoning of multimodal large language models (MLLMs), our heuristic solver dynamically executes physics-aware actions under the observation-evaluation-modification mechanism to effectively resolve mesh collisions, floating artifacts, and fine-grained cavity containment failures. Extensive experiments demonstrate that Text2Villa outperforms previous methods across various metrics, robustly generating high-fidelity and physically plausible villa-level 3D environments from text, thereby providing a reliable and interactive 3D content foundation for downstream applications.
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