arXiv subjects

Stephen S. -T. Yau

Publications and source records attributed to Stephen S. -T. Yau.

At least 19 recordsLinked to original sources

arXiv2026

Yau's number-theoretic conjecture and sharp spectral bounds for quantum harmonic oscillators

We prove Yau's number-theoretic conjecture on positive integral points in real right simplices in every dimension $n\ge3$, and apply the resulting sharp polynomial bound to quantum harmonic oscillators. The lattice-point bound holds for every choice of positive real intercepts for which the count is nonzero, with equality precisely when all intercepts equal the same integer. For oscillators in dimensions $n\ge3$, we obtain explicit upper bounds for eigenvalue counts and spectral remainders for arbitrary positive frequencies at every threshold at or above the lowest eigenvalue. For frequencies with rational ratios in dimensions $n\ge2$, an additional lattice coordinate and interpolation give bounds for energy sums, or first Riesz means, at every threshold. When all frequencies are equal (the isotropic case), the oscillator attains these bounds: the counting bound is exact at its energy levels, and interpolation on its natural energy grid recovers the energy sum at every threshold. In three dimensions, the exact formula displays the periodic contribution associated with the resonance discussed by Fefferman and Seco. The proof builds on the slicing method of Yau and Zhang. Arithmetic and geometric consequences include explicit bounds for smooth numbers and lattice-point discrepancies.

math.NT↗

arXiv2026

Nakai conjectures for isolated weighted homogeneous hypersurface singularities

The long-standing Nakai Conjecture asks whether differential operators detect singularities on algebraic varieties. On a smooth variety over a field of characteristic zero, the ring of differential operators is generated by first order derivations. Nakai asked whether the converse holds. In this paper, we prove that for the coordinate ring of an isolated weighted homogeneous hypersurface singularity, there always exists a second order derivation that is not generated by first order derivations. This verifies the Nakai Conjecture for isolated weighted homogeneous hypersurface singularities.

math.AG↗

arXiv2026

Pellarin's Identity, Carlitz Period, and Anderson Generating Functions over Arbitrary Curves

Let $A$ be the coefficient ring of a smooth projective curve over $\mathbb{F}_q$ with a closed point at infinity of arbitrary degree $\mathrm{N}=°(\infty)\ge 1$, and let $φ$ be a rank-one Drinfeld $A$-module. In this paper, we prove three main results concerning the module of special functions and its relation to Pellarin's $\mathcal L(1)$-series. First, the module $\mathrm{sf}(φ)$ of special functions contains a Tate-algebra unit exactly when it is free of rank one, equivalently when its period lattice is isomorphic to the module of regular differentials. This settles the Gazda--Maurischat conjecture. The proof evaluates Cauchy kernels; on the period lattice all such evaluations agree, and their common characteristic residue recovers the period. Second, for suitable isogeny data to a principal-period target $ψ$, a strictly normalized shtuka product, together with one characteristic residue, gives an explicit formula for the fundamental period $\tildeπ$ (the generator of the free lattice $Λ_ψ$). The same defect equation also yields the residue formula for the Drinfeld logarithm. Third, the first Frobenius twist of the normalized shtuka differential is identified with a pairing built from finite torsion traces, whose coefficients are finite étale traces over the field of definition. As an application, Pellarin's $\mathcal L(1)$-series is expressed exactly as the first Frobenius twist of the generating differential at a fundamental period. A notable feature of these results is that they hold for arbitrary $\mathrm{N} \geq 1$, and the source period lattice need not be free.

math.NT↗

arXiv2026

Well-posedness of Filtering Equations in Weighted Sobolev Spaces with Unbounded System Coefficients

Nonlinear filtering problem is one of the core subjects in modern control theory. In this paper, we will study the well-posedness of the three fundamental evolution equations arising in continuous-time nonlinear filtering--the robust Duncan-Mortensen-Zakai (DMZ) equation, the stochastic DMZ equation, and the Kushner-Stratonovich equation--within a unified buffered weighted formulation. An exponential-type weight function and the corresponding weighted Sobolev spaces are introduced to enable a variational treatment of the filtering equations in a more general setting, in which the coefficients of the filtering system may be unbounded with polynomial growth. Under mild and easily verifiable assumptions, we first establish the well-posedness of the weak solution to the robust DMZ equation in these weighted spaces. Using the gauge (exponential) transformation and its inverse, these results are then transferred to the stochastic DMZ equation and the Kushner-Stratonovich equation, whose solutions are shown to exist and be unique in buffered weighted Sobolev spaces, yielding a unified treatment of all three filtering equations. Sufficient conditions for the well-posedness are also summarized, which illustrate the wide applicability of the proposed framework to general nonlinear filtering systems.

math.OC↗

arXiv2026

Derivations of Generalized Moduli Algebras of Isolated Hypersurface Singularities

Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in\mathbb{C}\{x_1,\ldots,x_n\}$. Let $A(V)$ be the moduli(Tjurina) algebra, let $A^*(V)$ be the generalized moduli algebra, and let $$ L(V)=\mathrm{Der}_{\mathbb{C}}(A(V),A(V)),\qquad L^*(V)=\mathrm{Der}_{\mathbb{C}}(A^*(V),A^*(V)) $$ be the Yau algebra and the new Yau algebra of $V$, respectively. Write $λ(V)=\dim_{\mathbb{C}}L(V)$ and $λ^*(V)=\dim_{\mathbb{C}}L^*(V)$. We prove that the difference between these two dimensions is determined by the Hessian corank. In the Morse case, the generalized moduli algebra is zero, and hence both derivation Lie algebras are zero. We have $λ^*(V)=λ(V)-1$ if $\mathrm{corank}(\mathrm{Hess}(f)(0))=1$, and $λ^*(V)=λ(V)$ otherwise. In particular, $λ^*(V)=λ(V)$ when $n\ge2$ and $\mathrm{mult}(f)\ge3$. This proves Conjecture~1.1 proposed in \cite{ChenHussainYauZuo2020}. We construct an exact sequence whose two end terms are copies of the socle of the Milnor algebra. This homological approach reveals the detailed algebraic structure underlying the dimension count. We also give an alternative proof that pushes the original arguments in the proof of \cite[Theorem~C]{ChenHussainYauZuo2020} further. The general case is obtained by combining Saito's criterion for the non-quasi-homogeneous case.

math.AG↗

arXiv2026

A Lie-algebraic approach to non-Markovian quantum dynamics

In this paper, we study the non-Markovian quantum dynamics in quantum computations from the perspective of a Lie algebraic approach based on numerical analysis. By vectorizing the density matrix of quantum states, the non-Markovian evolutions can be represented with high-dimensional linear time-varying equations, where the time-varying parameters arise from the non-Markovian interactions between the quantum system and environment. We study the Magnus expansion of such linear time-varying quantum dynamics and clarify how the truncation errors for the first- and second-order Magnus expansions are influenced by the non-Markovian properties. Besides, when the quantum states are measured for filtering, the dynamics can be modeled as time-varying stochastic differential equations due to the existence of measurement noise. The Magnus expansions based on quantum stochastic filtering are different when the quantum measurement noises are modeled in an {Itô} or Stratonovich approach, rendering different truncation errors. Based on this, numerical simulations further demonstrate the efficiency of Magnus expansions in simulating non-Markovian quantum dynamics without or with stochasticity, and how the truncation errors are influenced by the Lie algebras in the Liouville space.

quant-ph↗

arXiv2026

Tensor train methods for high-dimensional nonlinear filtering problems with correlated noise

Nonlinear filtering with correlated noise leads to a Duncan-Mortensen-Zakai (DMZ) equation in the form of a stochastic partial differential equation (SPDE). Unlike the independent noise case, the presence of correlation prevents the classical invertible transformation that reduces the DMZ equation to a deterministic partial differential equation, requiring a direct numerical treatment of the SPDE. This paper develops a tensor train (TT) based framework for solving medium- to high-dimensional DMZ equations with correlated noise. Spatial discretization transforms the SPDE into a high-dimensional stochastic differential system, which is efficiently compressed using TT approximation. A semi-implicit Milstein scheme is employed for temporal integration to ensure stability and accuracy. Under suitable regularity assumptions, we establish a convergence analysis of the proposed method. In particular, the spatial error is controlled by both the mesh size and the prescribed TT approximation accuracy. In the temporal direction, the convergence is proved by estimating stochastic integrals involving drifted observations, without invoking a change-of-measure argument. Numerical experiments demonstrate that the proposed method achieves stable and accurate performance for cubic sensor problems. In challenging multi-modal settings, where particle filter and extended Kalman filter deteriorate, the proposed method maintains accuracy and effectively captures the posterior distribution.

math.NA↗

arXiv2026

AEM: Adaptive Entropy Modulation for Multi-Turn Agentic Reinforcement Learning

Reinforcement learning (RL) has substantially improved the ability of large language model (LLM) agents to interact with environments and solve multi-turn tasks. However, effective agentic RL remains challenging: sparse outcome-only rewards provide limited guidance for assigning credit to individual steps within long interaction trajectories. Existing approaches often introduce dense intermediate supervision, such as process reward models or auxiliary self-supervised signals, which increases supervision and tuning complexity and may limit generalization across tasks and domains. We present AEM, a supervision-free credit assignment method that adaptively modulates entropy dynamics during RL training to improve the exploration-exploitation trade-off. Since in agentic RL the environment is typically affected by a complete response, rather than an individual token, our analysis lifts entropy dynamics from the token level to the response level, aligning uncertainty estimation with the effective action granularity of LLM agents and reducing sensitivity to token-level sampling noise. We further show that entropy drift under natural-gradient updates is governed by the interaction between the sampled-response advantage and its relative surprisal. Motivated by this result, AEM derives a practical response-level uncertainty proxy and uses it to rescale advantages, leveraging the evolving balance between positive and negative samples to naturally transition from exploration to exploitation. Extensive experiments on ALFWorld, WebShop, and SWE-bench-Verified with models ranging from 1.5B to 32B demonstrate that AEM consistently improves strong RL baselines, including a +1.4\% gain when integrated into a state-of-the-art software-engineering RL training framework.

cs.AI↗

arXiv2026

Anderson generating function of rank-one Drinfeld Module over rational function fields

We establish a fundamental breakthrough in rank-one Drinfeld module arithmetic by deriving explicit formulas over the integral domain $\A = H^{0}(\mathbb{P}^1-P_ρ, \mathcal{O}_{\mathbb{P}^1})$, which generalizes the classical polynomial ring ($N=1$) to the projective line associated with an infinite place of degree $N \geqslant 2$. This fills a longstanding gap by developing a comprehensive parallel to Carlitz module theory foundational in positive characteristic arithmetic for the understudied case of infinite places of degree $>1$. We construct Anderson generating functions for these modules and link them to the Carlitz period via Pellarin's series, exponential torsion modules, and logarithmic deformations. These constructions provide powerful tools for studying such Drinfeld modules and their associated $L$-series, central to modern number theory. A key result reveals a critical distinction from Carlitz theory: the standard Anderson generating function residue formula fails due to Galois group action. We resolve this obstruction by introducing an exponential action, enabling simultaneous study of all twisted exponential functions a major methodological advance. We further show that Anderson generating function computation involves the dual of Drinfeld modules, leading to an appropriate residue formula modification. Notably, our natural approach generalizes to arbitrary Dedekind domains, extending our results beyond $\A$ and opening new avenues in Drinfeld module theory.

math.NT↗

arXiv2026

Classification of unimodal isolated complete intersection singularities in positive characteristic

In this paper we classify the unimodal isolated complete intersection singularities in arbitrary characteristic under contact equivalence. The classification over $\mathbb{C}$ has already done by A. Dimca and C.G. Gibson. We continue and generalize their work. To complete the classification, we generalized the complete transversal method into positive characteristic field, which is also useful in many other classification problem.

math.AG↗

arXiv2026

Nonlinear Bayesian Filtering with Natural Gradient Gaussian Approximation

Practical Bayes filters often assume the state distribution of each time step to be Gaussian for computational tractability, resulting in the so-called Gaussian filters. When facing nonlinear systems, Gaussian filters such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) typically rely on certain linearization techniques, which can introduce large estimation errors. To address this issue, this paper reconstructs the prediction and update steps of Gaussian filtering as solutions to two distinct optimization problems, whose optimal conditions are found to have analytical forms from Stein's lemma. It is observed that the stationary point for the prediction step requires calculating the first two moments of the prior distribution, which is equivalent to that step in existing moment-matching filters. In the update step, instead of linearizing the model to approximate the stationary points, we propose an iterative approach to directly minimize the update step's objective to avoid linearization errors. For the purpose of performing the steepest descent on the Gaussian manifold, we derive its natural gradient that leverages Fisher information matrix to adjust the gradient direction, accounting for the curvature of the parameter space. Combining this update step with moment matching in the prediction step, we introduce a new iterative filter for nonlinear systems called \textit{N}atural Gr\textit{a}dient Gaussia\textit{n} Appr\textit{o}ximation filter, or NANO filter for short. We prove that NANO filter locally converges to the optimal Gaussian approximation at each time step. Furthermore, the estimation error is proven exponentially bounded for nearly linear measurement equation and low noise levels through constructing a supermartingale-like property across consecutive time steps.

eess.SY↗

arXiv2025

On the $k$-th Tjurina number of weighted homogeneous singularities

Let $ (X,0) $ denote an isolated singularity defined by a weighted homogeneous polynomial $ f $. Let $ \mathcal{O}$ be the local algebra of holomorphic function germs at the origin, with the maximal ideal $m $. We study the $k$-th Tjurina algebra, defined by $ A_k(f): = \mathcal{O} / \left( f , m^k J(f) \right) $, where $J(f)$ denotes the Jacobian ideal of $ f $. The zeroth Tjurina algebra is well known to represent the tangent space of the base space of the semi-universal deformation of $(X, 0)$. Motivated by this observation, we explore the deformation of $(X,0)$ with respect to a fixed $k$-residue point. We show that the tangent space of the corresponding deformation functor is a subspace of the $k$-th Tjurina algebra. Explicit calculation of the $k$-th Tjurina numbers, which correspond to the dimensions of the $k$-th Tjurina algebras, plays a crucial role in understanding these deformations. According to the results of Milnor and Orlik, the zeroth Tjurina number can be expressed explicitly in terms of the weights of the variables in $f$. However, we observe that for values of $k$ exceeding the multiplicity of $X$, the $k$-th Tjurina number becomes more intricate and is not solely determined by the weights of the variables. In this paper, we introduce a novel complex derived from the classical Koszul complex and obtain a computable formula for the $k$-th Tjurina numbers for all $ k \geqslant 0 $. As an application, we calculate the $k$-th Tjurina numbers for all weighted homogeneous singularities in three variables.

math.AG↗

arXiv2025

Computing singular simplicial homologies of digraphs and quivers

The dynamics of large complex systems are predominately modeled through pairwise interactions, the principle underlying structure being a network of the form of a digraph or quiver. Significant success has been obtained in applying the homology of the directed flag complex to study digraphs arising as networks within numerous scientific disciplines. This homology of directed cliques enjoys relative ease of computation when compared to other digraph homologies, making it preferable for use in applications concerning large networks. By extending the ideas of singular simplicial homology to quivers in categories of different morphism types, several new singular simplicial homology theories have recently been constructed. Computationally efficient homologies for quivers have in general not previously been seriously considered. In this paper we develop further the homotopy theory of quivers necessary to derive functors that realise isomorphisms between the singular simplicial quiver homologies and the homologies of certain spaces. The simplicial chains of these spaces arise in a conveniently compact form that is at least as convenient as the directed flag complex for computations. Moreover, our constructions are natural with respect to the isomorphisms on homology making them suitable for applications in conjunction with persistent homology for practical use. In particular, for each of the singular simplicial homologies considered, we provide efficient algorithms for the computation of their persistent homology.

math.AT↗

arXiv2024

Normal surface singularities of small degrees

The notion of the Yau sequence was introduced by Tomaru, as an attempt to extend Yau's elliptic sequence for (weakly) elliptic singularities to normal surface singularities of higher fundamental genera. In this paper, we obtain the canonical cycle using the Yau cycle for certain surface singularities of degree two. Furthermore, we obtain a formula of arithmetic genera and an upper bound of geometric genera for these singularities. We also give some properties about the classification of weighted dual graphs of certain surface singularities of degree two.

math.AG↗

arXiv2024

Groupoids derived from the simple elliptic singularities

K. Saito's classification of simple elliptic singularities includes three families of weighted homogeneous singularities: $ \tilde{E}_{6}, \tilde{E}_7$, and $ \tilde{E}_8 $. For each family, the isomorphism classes can be distinguished by K. Saito's $j$-functions. By applying the Mather-Yau theorem, which states that the isomorphism class of an isolated hypersurface singularity is completely determined by its $k$-th moduli algebra, M. Eastwood demonstrated explicitly that one can directly recover K. Saito's $j$-functions from the zeroth moduli algebras. This research aims to generalize M. Eastwood's result through meticulous computation of the groupoids associated with simple elliptic singularities. We not only directly retrieve K. Saito's $j$-functions from the $k$-th moduli algebras but also elucidate the automorphism structure within the $k$-th moduli algebras. We derive the automorphisms using the methodology of the $k$-th Yau algebra and establish a Torelli-type theorem for the $\tilde{E}_7 $-family when $k=1$. In contrast, we find that the Torelli-type theorem is inapplicable for the first Yau algebra in the $ \tilde{E}_6 $-family. By considering the first Yau algebra as a module rather than solely as a Lie algebra, we can impose constraints on the coefficients of the transformation matrices, which facilitates a straightforward identification of all isomorphisms. Our new approach also provides a simple verification of the result by Chen, Seeley, and Yau concerning the zeroth moduli algebras.

math.AG↗

arXiv2024

On the Convergence Analysis of Yau-Yau Nonlinear Filtering Algorithm: from a Probabilistic Perspective

At the beginning of this century, a real time solution of the nonlinear filtering problem without memory was proposed in [1, 2] by the third author and his collaborator, and it is later on referred to as Yau-Yau algorithm. During the last two decades, a great many nonlinear filtering algorithms have been put forward and studied based on this framework. In this paper, we will generalize the results in the original works and conduct a novel convergence analysis of Yau-Yau algorithm from a probabilistic perspective. Instead of considering a particular trajectory, we estimate the expectation of the approximation error, and show that commonly-used statistics of the conditional distribution (such as conditional mean and covariance matrix) can be accurately approximated with arbitrary precision by Yau-Yau algorithm, for general nonlinear filtering systems with very liberal assumptions. This novel probabilistic version of convergence analysis is more compatible with the development of modern stochastic control theory, and will provide a more valuable theoretical guidance for practical implementations of Yau-Yau algorithm.

math.OC↗

arXiv2024

Convolutional Bayesian Filtering

Bayesian filtering serves as the mainstream framework of state estimation in dynamic systems. Its standard version utilizes total probability rule and Bayes' law alternatively, where how to define and compute conditional probability is critical to state distribution inference. Previously, the conditional probability is assumed to be exactly known, which represents a measure of the occurrence probability of one event, given the second event. In this paper, we find that by adding an additional event that stipulates an inequality condition, we can transform the conditional probability into a special integration that is analogous to convolution. Based on this transformation, we show that both transition probability and output probability can be generalized to convolutional forms, resulting in a more general filtering framework that we call convolutional Bayesian filtering. This new framework encompasses standard Bayesian filtering as a special case when the distance metric of the inequality condition is selected as Dirac delta function. It also allows for a more nuanced consideration of model mismatch by choosing different types of inequality conditions. For instance, when the distance metric is defined in a distributional sense, the transition probability and output probability can be approximated by simply rescaling them into fractional powers. Under this framework, a robust version of Kalman filter can be constructed by only altering the noise covariance matrix, while maintaining the conjugate nature of Gaussian distributions. Finally, we exemplify the effectiveness of our approach by reshaping classic filtering algorithms into convolutional versions, including Kalman filter, extended Kalman filter, unscented Kalman filter and particle filter.

stat.ML↗

arXiv2020

Inverted repeats in coronavirus SARS-CoV-2 genome and implications in evolution

The coronavirus disease (COVID-19) pandemic, caused by the coronavirus SARS-CoV-2, has caused 60 millions of infections and 1.38 millions of fatalities. Genomic analysis of SARS-CoV-2 can provide insights on drug design and vaccine development for controlling the pandemic. Inverted repeats in a genome greatly impact the stability of the genome structure and regulate gene expression. Inverted repeats involve cellular evolution and genetic diversity, genome arrangements, and diseases. Here, we investigate the inverted repeats in the coronavirus SARS-CoV-2 genome. We found that SARS-CoV-2 genome has an abundance of inverted repeats. The inverted repeats are mainly located in the gene of the Spike protein. This result suggests the Spike protein gene undergoes recombination events, therefore, is essential for fast evolution. Comparison of the inverted repeat signatures in human and bat coronaviruses suggest that SARS-CoV-2 is mostly related SARS-related coronavirus, SARSr-CoV/RaTG13. The study also reveals that the recent SARS-related coronavirus, SARSr-CoV/RmYN02, has a high amount of inverted repeats in the spike protein gene. Besides, this study demonstrates that the inverted repeat distribution in a genome can be considered as the genomic signature. This study highlights the significance of inverted repeats in the evolution of SARS-CoV-2 and presents the inverted repeats as the genomic signature in genome analysis.

q-bio.OT↗