arXiv subjects

Xiaoduo Wang

Publications and source records attributed to Xiaoduo Wang.

6 recordsLinked to original sources

arXiv2026

Definability of Hausdorff Limits for Lipschitz Cells in O-minimal Structures

We study Hausdorff limits of definable families over arbitrary models of o-minimal expansions of real closed fields. Over the real field, van den Dries proved that Hausdorff limits of definable families are definable, giving a geometric interpretation of the Marker--Steinhorn theorem. We prove a non-Archimedean analogue for definable families which are Lipschitz cells with a fixed cell presentation and a uniform Lipschitz bound. The proof replaces compactness of closed and bounded subsets of $\mathbb R^n$ by dense completions and long Cauchy sequences, and treats the Hausdorff distance as a metric valued in an ordered completion. We show that Hausdorff limits of such families are standard parts of external fibers over tame extensions, and use stable embeddedness of tame pairs to prove that every such limit is definable in the dense completion of the base model. We also prove a uniform version: the collection of these Hausdorff limits forms a definable family in the dense completion.

math.LO↗

arXiv2026

Delta-Cell Decomposition and Curve Selection

We develop a cell decomposition framework for o-minimal structures equipped with a generic derivation. To a $\delta$-cell we associate source cells and finite configurations in ordinary o-minimal sorts, allowing differential-topological questions to be studied through finite jet spaces. We then introduce a metric space of definable curve germs and identify its local half-space pieces with Cartesian powers of the maximal ideal of the Hardy field of definable germs. Using this germ-space description, we prove an abstract curve selection theorem for the $\delta$-topology. In the case of closed ordered differential fields, we further describe concrete asymptotic representatives for the abstract curve germs.

math.LO↗

arXiv2026

Axiomatizations of Presburger Arithmetic With Predicates For Powers

We give a complete first-order axiomatization of the structure $(\mathbb{Z},+,(\ell^{\mathbb{N}})_{\ell\in L})$, where $L \subseteq \mathbb{Z}_{\ge 2}$ is a set of pairwise multiplicatively independent integers and $\ell^{\mathbb{N}} = \{\ell^n : n\in \mathbb{N}\}$. Using recent work of Karimov et al., we obtain that this axiomatization is computable for $|L|=2$, which proves that $(\mathbb{Z},+,k^{\mathbb{N}}, \ell^{\mathbb{N}})$ is decidable for $k, \ell\in \mathbb{Z}_{\ge 2}$. Furthermore, we give an axiomatization of the universal theory of $(\mathbb{Z},+,<,(\ell^{\mathbb{N}})_{\ell\in L})$.

math.LO↗

arXiv2025

T-Convexity, Tame Extensions and Definability of Hausdorff Limits in O-minimal Structures with Generic Derivations

We study the combination of two o-minimal extensions of the theory of real closed fields: one by a T-convex subring and the other by a T-derivation. Let T be a complete, model complete o-minimal extension of RCF. We show that the combined theory T_convex^delta has a model completion T_g,convex^delta. By adding a definable unary function st, we obtain a relative quantifier elimination result for tame pairs (M, delta^M, st^M, N, delta^N, st^N), where st is the standard part map and N is Dedekind complete in M. As an application, we prove the stable embedding property for tame pairs of T_g^delta. We also associate a sequence of definable metric topologies with models of T_g^delta and prove the Marker-Steinhorn Theorem for T_g^delta. As a consequence, Hausdorff limits of definable families are definable. A special case of our framework recovers Borotta's results on CODF with convex valuation subrings and tame pairs.

math.LO↗

arXiv2023

Deep Learning Enables Large Depth-of-Field Images for Sub-Diffraction-Limit Scanning Superlens Microscopy

Scanning electron microscopy (SEM) is indispensable in diverse applications ranging from microelectronics to food processing because it provides large depth-of-field images with a resolution beyond the optical diffraction limit. However, the technology requires coating conductive films on insulator samples and a vacuum environment. We use deep learning to obtain the mapping relationship between optical super-resolution (OSR) images and SEM domain images, which enables the transformation of OSR images into SEM-like large depth-of-field images. Our custom-built scanning superlens microscopy (SSUM) system, which requires neither coating samples by conductive films nor a vacuum environment, is used to acquire the OSR images with features down to ~80 nm. The peak signal-to-noise ratio (PSNR) and structural similarity index measure values indicate that the deep learning method performs excellently in image-to-image translation, with a PSNR improvement of about 0.74 dB over the optical super-resolution images. The proposed method provides a high level of detail in the reconstructed results, indicating that it has broad applicability to chip-level defect detection, biological sample analysis, forensics, and various other fields.

physics.optics↗

arXiv2021

Sobolev Orthogonal Polynomials on the Sierpinski Gasket

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $L^2$, $L^\infty$ and Sobolev norms, and study their asymptotic behaviour. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.

math.CA↗