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On the exponential convergence of Kobayashi geodesics in strongly convex domains
Kingshook Biswas, Sanjoy Chatterjee
https://arxiv.org/abs/2607.17259 https://arxiv.org/pdf/2607.17259 https://arxiv.org/html/2607.17259
arXiv:2607.17259v1 Announce Type: new
Abstract: In this paper, we have proved a qualitative version of the approaching geodesic property for certain convex domains. More precisely, we have proved that that if $\Omega \subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $\gamma_{1}, \gamma_{2}:[0, \infty) \to \Omega$ are two geodesics such that $\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega$. Then there exists $A\big(\gamma_{1}(0), \gamma_{2}(0) \big)>0$ and $T \in \mathbb{R}$ such that
$$K_{\Omega}(\gamma_{1}(t), \gamma_{2}(t T))\leq Ae^{-\frac{t}{2}} ~~~\,\hspace{1em} \forall t \geq 0.$$ Furthermore, using this property we provided a characterization of strongly pseudoconvex domain via a biholomorphic invariant function namely generalized squeezing function. We have proved that: For every $\alpha>2$ there exists $\epsilon(d,\alpha)>0$ such that the following holds: if $\Omega \subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,\alpha}$-boundary and \[ T_{\Omega}^{D}(z)\geq 1-\epsilon \] outside a compact subset of $\Omega$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_{\Omega}^{D}$ is the squeezing function of $\Omega$ with respect to the domain $D$ then $\Omega$ is strongly pseudoconvex.
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Decode-Time Grammars: Constrained LLM Generation over a Refinement Order of Grammar Fragments
Shuoming Zhang, Ruiyuan Xu, Haofeng Li, Qiuchu Yu, Yangyu Zhang, Chunwei Xia, Xiaobing Feng, Chenxi Wang, Huimin Cui, Jiacheng Zhao
https://arxiv.org/abs/2607.18357 https://arxiv.org/pdf/2607.18357 https://arxiv.org/html/2607.18357
arXiv:2607.18357v1 Announce Type: new
Abstract: Large language models now write a growing share of the world's code, increasingly inside agents and serving systems that compile, execute, or dispatch generated code without line-by-line review. This works well for mainstream languages but remains brittle for low-resource programming surfaces such as domain-specific languages, custom library APIs, and command-line tools. Even under grammar-constrained decoding, a model can still produce references invalid in the current environment: a buffer never declared, a column absent from the schema, a function the library does not provide, or an unsupported CLI option.
This paper introduces decode-time grammars: grammar fragments instantiated during generation from a runtime environment Gamma. A region-specific policy selects a fragment for each hole, and a tightening operator replaces open reference positions with Gamma-typed slots whose candidates are exactly the names, fields, APIs, or options available at that point. Newly generated declarations enter Gamma before later regions are decoded, so the constraining grammar can depend on the prefix already generated. This ensures not only grammatical correctness but also semantic correctness, by preventing references to undefined symbols.
We formalize grammar fragments as environment-indexed grammars ordered by refinement, prove No-Ghost soundness for Gamma-slotted fragments, show that refinement preserves this support-set guarantee, and characterize the boundary of mask-enforceable properties. We implement the approach in gproj with offline grammar induction and online policy resolution. Across TileLang, SQL, and P4, with models from 0.6B to 236B parameters, gproj eliminates ghost references by construction at moderate overhead over standard constrained decoding.
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Functional inequalities for starlike and convex functions associated with $g$-derivative operator
An Huang, Pinhong Long, Halit Orhan, Huo Tang
https://arxiv.org/abs/2607.18976 https://arxiv.org/pdf/2607.18976 https://arxiv.org/html/2607.18976
arXiv:2607.18976v1 Announce Type: new
Abstract: In this paper we first introduce a new class of derivative operators, termed $g$-derivative operators, which unifies the $q$-derivative, $(p,q)$-derivative and $(\alpha,\beta,\gamma)$-derivative operators, even classical derivative in the literature. Based on this generalized framework, we define two novel function classes, specifically $g$-starlike functions and $g$-convex functions. Further, we employ the subordination principle of analytic functions to conduct the coefficient estimations for such function classes, and subsequently derive the bounds for the corresponding Fekete-Szeg\"{o} inequality, Toeplitz determinants and Hankel determinants.
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