Papers

We prove that every discrete faithful representation of the surfcae group into SL(2,C) is the monodromy of a holomorphic connection on the trivial rank-2 vector bundle over a Riemann surface. As an application, we answer the question posed by Ghys and Huckleberry-Winkelmann (known as the Margulis' problem) by proving that every compact quotient of SL(2,C) contains a holomorphic curve of genus at least two. The main tools we use are the Non-Abelian Hodge correspondence, the WKB analysis, and the Morgan-Shalen compactification.

The Szemerédi Regularity Lemma, in combination with the Blow-up Lemma, form the Regularity Method, a fundamental tool in graph embeddings, albeit restricted to very large and dense graphs. We propose an alternative vertex-partitioning framework that remains effective even as the density tends to zero and without requiring astronomically large vertex sets. This approach, while narrower in scope, extends regularity-type techniques to relatively small graphs previously inaccessible to the Regularity Method. As an application, we use this novel vertex-partitioning method for bipartite packing problems.

We study the global convergence of policy gradient for infinite-horizon entropy-regularized Markov decision processes (MDPs) with continuous state and action spaces. We consider log-linear softmax policies with linear function approximation, which extend the tabular softmax parameterization while retaining a tractable policy class. Under $Q^\pi_\tau$-realizability for the regularized state-action value function, we first establish a non-uniform Polyak–Łojasiewicz (PŁ) inequality. The non-uniformity arises through degeneracy of constants associated with the policy geometry, namely the Fisher information matrix or an uncentered feature covariance matrix. We then identify two feature regimes under which this non-uniform constant can be bounded along the gradient flow. For full-affine-span features, we prove radial unboundedness of the KL regularizer and show that the smallest eigenvalue of the Fisher information matrix remains bounded below by an initialization-dependent positive constant. For simplex-valued features, we prove an analogous radial unboundedness result in the subspace orthogonal to the all-ones vector and obtain a uniform lower bound for the smallest eigenvalue of the uncentered covariance matrix. These results imply global linear convergence of the regularized objective along the gradient flow, i.e. suboptimality decaying as $\mathcal{O}(e^{-Ct})$ for some $C>0$. Our analysis extends the global convergence theory of entropy-regularized softmax policy gradient beyond the tabular setting of Agarwal et al. (2020); Bhandari and Russo (2024); Mei et al. (2020).

In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a logarithmic Sobolev inequality. In this regime, the classical Euler–Maruyama scheme underlying the unadjusted Langevin algorithm (ULA) is known to be unstable. We analyze the KL-accelerated tamed unadjusted Langevin algorithm (kTULA) and introduce a new tamed randomized midpoint scheme, termed tRLMC. Building on the shifted-composition approach of \cite{chewi2024local}, we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE – in KL divergence for kTULA, and in total variation for tRLMC – valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimal $\widetilde{O}(\varepsilon^{-1/2})$ iteration complexity for kTULA in KL divergence, with corresponding guarantees in total variation and Wasserstein distance. We further establish, for the first time, a non-asymptotic guarantee in total variation for a tamed randomized Langevin scheme under super-linear drift growth, together with the corresponding Wasserstein-distance bound, both with $\widetilde{O}(\varepsilon^{-1})$ complexity for tRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.

We first characterise graphs with binomial edge ideals of König type as those for which the path covering number is equal to a minor variant of the scattering number. This enables us to apply known graph-theoretic results to immediately deduce that several classes of graphs have binomial edge ideals of König type. In particular, we show this for cocomparability graphs, or weakly closed graphs in the language of Matsuda. Along with work of LaClair and McCullough, this allows us to prove that an unmixed binomial edge ideal is of König type if and only if G is weakly closed. We then conjecture that AT-free graphs have binomial edge ideals of König type.

We revisit the classical problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss. Framing the task as a stochastic convex optimization problem over the space of $ M $-component mixture distributions, we propose a family of estimators derived from the stochastic mirror descent (SMD) algorithm. This optimization-based approach provides a principled and flexible framework that generalizes traditional estimators and proposes a variety of novel estimators through the choice of Bregman divergences. A key advantage of our method is that it scales efficiently with the number of candidate components $ f_i $; that is, one can employ a large set of basis distributions in the mixture model without incurring significant computational overhead. This enables richer approximations and improved estimation accuracy. Moreover, in the case of categorical distribution (discrete outcomes) our estimators do not require a strict lower bound, in other words our framework does not require the precise knowledge of the support of the distribution. We demonstrate that, under mild conditions, the proposed $ \varphi $-SMD estimators achieve near-optimal convergence rates in both Kullback-Leibler (KL) divergence and $ \ell_2 $-norm and offer practical benefits when computation is expensive. Our numerical analysis highlights improved performance guaranties over classical estimators, particularly in terms of sample efficiency and scalability.

We investigate the structural and dynamical properties of a class of quantum channels on von Neumann algebras induced by averaging over operator groups via the Pettis integral. Utilizing the classical Yosida–Hewitt decomposition, we focus on the interplay between the set-theoretic properties of the underlying space and the topological nature of the resulting quantum states. We establish a suffucient condition under which a channel preserves $\sigma$-additivity and exhibits a singularising property, completely suppressing the normal component of any incoming state. In conjunction with the theory of Ulam real-valued measurable cardinals, this framework reveals a novel phenomenon: the existence of quantum channels that transform normal states into singular yet strictly $\sigma$-additive states. Furthermore, we analyze the structural constraints on the preservation of state purity imposed by the cardinality of the continuum, and extend our constructions to invariant measures on groups and their unitary representations, establishing the convergence of their Cesàro averages in the strong operator topology.

In this paper, we characterize all eigenfunctions corresponding to nonpositive eigenvalues of the Jacobi operator of the link $M$ of the Lawson-Osserman cone $\mathbf{C}$ in $\mathbb{R}^7$. In particular, we prove that $\mathbf{C}$ is integrable, i.e., all Jacobi fields on $\mathbf{C}$ of homogeneous degree 1 and 0, are generated by rotations and translations in $\mathbb{R}^7$. As applications, we prove that $M$ is rigid as minimal submanifolds in $\mathbb{S}^6$, and derive the optimal decay order for minimal submanifolds in $\mathbb{R}^7$ asymptotic to $\mathbf{C}$ at infinity.

Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\pm 1$ and discriminant $\Delta=P^2+4Q>0$. We study Diophantine equations \[ A y^k=\prod_{i=1}^r U_{n_i}(P,Q), \qquad k\geq 2, \] where the indices $n_1,\ldots,n_r$ are pairwise coprime. The strong divisibility property implies that the factors $U_{n_i}$ are pairwise coprime, and hence a global $k$-th power condition separates into local valuation conditions on the individual factors. For $k=2$, this gives a termwise square-class restriction: each $U_{n_i}$ has signed squarefree part supported on the primes dividing $A$. In particular, the equation $\Delta y^2=U_mU_n$, with $\gcd(m,n)=1$, reduces to a finite square-class compatibility condition together with an integrality condition. Assuming the number-field $abc$ conjecture over $\mathbb Q(\sqrt{\Delta})$, we prove that only finitely many Lucas terms have squarefree part supported on a fixed finite set of rational primes. Consequently, the coprime product equations above admit an $abc$-conditional finite reduction. We also give the corresponding $k$-th power analogue and a primitive-divisor obstruction.

We present a convergence analysis of a finite volume (FV) scheme for the multicomponent compressible Euler system in the framework of dissipative weak (DW) solutions. DW solutions were introduced as a generalized solution framework in computational fluid dynamics and have recently gained considerable attention. They extend the well-known Lax Equivalence Theorem to nonlinear settings, meaning that if a numerical scheme is both consistent and stable, it will also converge. The FV scheme under consideration preserves key physical properties of the fluid mixture, in particular, positivity of partial densities, pressure, and temperature. Using uniform stability bounds and consistency estimates, we prove that the numerical solutions converge in the framework of DW solutions of the multicomponent Euler system. Applying the relative entropy method and the weak-strong uniqueness principle, we further show that the approximate solutions converge strongly to the classical solution as long as it exists. Numerical experiments confirm the theoretical results, not only for low-order FV methods but also through extended numerical investigations of a higher-order, structure-preserving discontinuous Galerkin scheme.

Itenberg-Katzarkov-Mikhalkin-Zharkov gave an isomorphism of tropical cohomology and cohomology of some maximally degenerate algebraic varieties. Their proof was based on tropical analogs of Steenbrink's geometric monodromy-weight spectral sequences. These were generalized to the non-realizable case by Amini-Piquerez. In this paper, we give a new construction of these tropical spectral sequences in the same way as Steenbrink's ones. For this purpose, we introduce reductions of tropical varieties. We also show that eigenwave actions are given by tropical Gauss-Manin connections.

We are concerned with the fast simulation of random fields on closed surfaces in $\mathbb{R}^3$ which are generated by the (Whittle-) Matérn class of covariance functions. To this end, we solve the underlying fractional stochastic partial differential equation with additive white noise by using an isogeometric finite element method on the surface in combination with the Balakrishnan integral representation of the solution. The solution of the underlying linear system of equations is performed by means of a geometric multigrid method that naturally underlies the isogeometric approach. Numerical results are presented to demonstrate the approach.

Large language models (LLMs) are trained on vast datasets that may contain sensitive information. Differential privacy (DP), the de facto standard for formal privacy guarantees, provides a principled framework for training LLMs with provable privacy protection. However, state-of-the-art DP training implementations rely on fast gradient clipping techniques with memory overhead $O(B \min\{T^2, d^2\})$, where $B$ is the batch size, $T$ is the sequence length, and $d$ is the model width. This becomes prohibitive as both model size and context length grow. We propose DP-SGD-RC, a novel variant of DP-SGD with randomized clipping that reduces memory and compute complexity. DP-SGD-RC leverages stochastic trace estimation methods, specifically Hutchinson's estimator[Hutchinson, 1989] and its improved variant, Hutch++[Meyer et al., 2021], to reduce the memory footprint of per-sample gradient norm estimation. We provide a tight privacy analysis showing that DP-SGD-RC achieves noise multipliers competitive with deterministic clipping. Experiments fine-tuning Llama~3.2-1B on long-context benchmarks spanning classification, question answering, and summarization tasks demonstrate that DP-SGD-RC matches baseline utility while significantly reducing memory and compute requirements.

We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net (IV-Net), realizes a mapping from the input coefficients and righthand side to the corresponding solution field. The architecture of IV-Net is informed by, and closely resembles, a V-cycle multigrid solver. The IV-Net model is parameterized via convolutional layers defined in the physical domain. For coercive problems with highly heterogeneous coefficients, the proposed network exhibits superior performance relative to a proper orthogonal decomposition (POD) approach and several existing neural operator architectures. For low-frequency oscillatory Helmholtz problems with smooth coefficients, its performance is similar to that of a Fourier neural operator. We analyze the approximation error and convergence behavior of IV-Net, its data efficiency, and its dependence on the underlying discretization mesh. Furthermore, we demonstrate the practical effectiveness of the architecture through a series of numerical experiments, including applications to uncertainty quantification, inverse problems, and prediction of quantities of interest.

Decarbonizing aviation remains challenging because energy-dense jet fuels dominate beyond short-range operations, while batteries impose severe range and payload penalties. Here we evaluate a new infrastructure pathway in which utility-scale solar farms equipped with solar phased arrays wirelessly beam microwave power to hybrid-electric aircraft during cruise. Integrating 143,152 U.S. flight trajectories, 5,712 solar farms and wireless power transfer models, we quantify the spatial, temporal, and operational potential of this concept at continental scale. We find that benefits are highly concentrated in solar-rich, traffic-dense states and are dominated by short- and medium-range flights, accounting for nearly all delivered energy and cost savings. Schedule optimization and higher cruise altitudes further increase value by improving alignment between aircraft demand and beaming availability. Market penetration analysis reveals non-linear scaling between solar farm and flight adoption. These results show that wireless power beaming is best understood as a corridor-specific strategy complementing other aviation decarbonization pathways.

The character codegree of an irreducible character of a finite group $G$ is given by the index of its kernel in $G$ upon the character degree. We compute the codegrees of irreducible characters of VZ and Camina $p$-groups, and also obtain the character codegrees set of $p$-groups of order $\leq p^5$, where $p$ is an odd prime.

We classify the Einstein metrics on closed 4-manifolds whose isometry group has dimension at least 3. Consequently, we show that the cohomogeneity-one Einstein metrics on a closed $4$-manifold are locally symmetric or homothetic to the Page metric on $\mathbf{CP}^2 \sharp \overline{\mathbf{CP}}^2$.

The Wiener index of a connected graph is defined as the sum of distances between all its unordered pairs of vertices. Characterising graphs on $n$ vertices with a fixed diameter that maximise the Wiener index is a long-standing open problem. This problem has been resolved fully for trees on $n$ vertices with diameter $d \in \{1,2,3,4,n-3,n-2,n-1\}$ while partial results are available for $d=5$ and $6$. In this context, a conjecture proposed by DeLaViña and Waller has remained open for the last 18 years. In this paper, we establish a necessary condition for a tree to attain the maximum Wiener index among all trees on $n$ vertices with a given diameter. Using this condition, we characterise the maximal trees for diameter $n-4$ and $n-5$. Furthermore, we prove the DeLaViña Waller conjecture for the classes of graphs having $0,1,2,3$ or $n-4$ cut vertices.

This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion technique. In the core section of the latter half, we present a determinant representation of Tribonacci numbers in a slightly modified Toeplitz–Hessenberg form derived from Bell polynomials.

Evaluating whether large language models (LLMs) capture the structure of natural language beyond local fluency remains an open challenge. Existing evaluation methods, largely based on task performance or short-context behavior, provide limited insight into the long-range statistical organization of generated text. We propose a complementary evaluation framework based on repeated subsequences. By analyzing their distribution across scales and relating it to higher-order Rényi entropies, we probe how texts reuse previously established structure under finite-length conditions. Experiments on human-written texts and length-matched GPT-generated texts show that, while power-law models can describe restricted ranges of block length, the observed entropy growth is often equally or better characterized by logarithmic–power forms. Across datasets, natural language exhibits stable entropy-growth patterns over accessible ranges, with consistent average behavior despite variability across individual texts. In contrast, GPT-generated texts show systematic and statistically significant shifts in estimated exponents with model size. These results demonstrate that repeated-subsequence entropy provides a quantitative structural diagnostic that reveals systematic differences in long-range organization, distinguishing natural language from state-of-the-art LLM outputs beyond surface-level fluency.

We introduce the Markov Distributional Conformal Prediction (MDCP) method that extends the distributional conformal prediction (previously developed for regression) to the setting of a strictly stationary Markov process. Instead of relying on a specific model structure to do prediction, the idea of distributional conformal prediction interval aligns with the Model-Free (MF) Prediction Principle. In analogy to MF prediction of Markov processes, our method exploits the probability integral transform based on estimated transition distribution functions to transform the Markov data to an i.i.d.~dataset. We show a non-asymptotic error bound of MDCPs unconditional coverage rate under a $\beta$-mixing condition and other standard assumptions on the kernel estimators. The asymptotic validity of the conditional prediction interval is also verified. In addition, we show that our conditional prediction interval is still asymptotically valid with Markov processes being $L^p$-$m$-approximable instead of satisfying the mixing property. Numerical simulations and real data experiments are deployed to empirically illustrate the finite-sample performance of MDCP, and compare it with the MF bootstrap prediction method.

Combinatorial counting problems pervade artificial intelligence, statistics, and discrete mathematics. Whether the task is enumerating subsets, multisets, permutations, partitions, or compositions under structural and arithmetic constraints, solving it remains a stubbornly manual exercise. Closed-form derivations are powerful but brittle, while naive encodings to propositional model counting or constraint satisfaction destroy the exchangeability that makes counting tractable in the first place. We present Cofola (COmbinatorial counting LAnguage with First-Order logic), a typed declarative language whose primitives are the combinatorial objects that recur in everyday counting questions, including sets, bags, tuples, sequences, circles, partitions, and compositions, together with natural relational and arithmetic constraints over them. A denotational semantics maps every Cofola program to a well-defined combinatorial counting problem, and a three-phase compilation pipeline (preprocessing, decomposition, and symmetry-preserving encoding) reduces this problem to a weighted first-order model counting (WFOMC) instance augmented with coefficient-extraction constraints. To stay inside known domain-liftable fragments whenever possible, the encoding groups indistinguishable entities, breaks the symmetry of unordered groupings lexicographically, and encodes sequences and circles via order axioms. On a suite of representative combinatorial counting problems, ranging from textbook math problems to multi-object scenarios that the closest prior framework cannot express, Cofola produces concise specifications and a uniform solving pipeline that is practical end-to-end.

Spectral projectors and the reflectors derived from them are basic objects in numerical linear algebra. This paper studies the prescribed-index reflector I-2P_k, where P_k is the spectral projector associated with the first k eigenvectors of a symmetric matrix. If a shift s lies in the target spectral gap, then this reflector is exactly the shifted matrix sign sign(H-sI). The exact identity is elementary, but its algorithmic consequences are not: all admissible shifts give the same exact reflector, while finite-step sign filters can have very different errors. We analyze odd sign-preserving spectral filters, prove local inheritance and discrete stability for the induced reflector iterations, derive a gap-dependent Newton-Schulz operator bound, and give deterministic admissibility certificates for inexact and reused shifts. The analysis identifies the shifted spectral margin as the quantity controlling finite-step reflector accuracy and explains why the midpoint shift is the natural default. Numerical experiments separate the matrix-function issues from the outer saddle-search dynamics: controlled spectra verify the margin predictions, low-dimensional tests distinguish shifted signs from raw signs, target-index scans probe non-small k, and Allen-Cahn and dense timing tests identify the regimes in which full-matrix sign filters are useful and the stiff regimes in which stronger sign engines are needed.

This paper establishes a sharp, expanded wave-breaking criterion for a class of nonlinear nonlocal Whitham-type equations, significantly generalizing the classical threshold introduced by Seliger. While the system of inequalities governing the spatial extrema of the fluid deviation has been extensively studied, establishing a sharp condition has remained a challenge. This difficulty is primarily due to the non-cooperative nature of the system, which precludes the application of standard comparison principles. We rigorously overcome this analytical obstacle by identifying the exact nonlinear threshold that confines the solutions of the inequality system and analyzing its time evolution.

We study invariants and structures of Poisson fields of rational functions in two variables. For four particular families, we classify the members, establish criteria for isomorphisms and, with the exception of the Weyl Poisson field, describe the automorphism groups. Embeddings are also investigated, along with an analog of the Dixmier Conjecture: For which Poisson fields is every Poisson endomorphism an automorphism? The answer is negative for the first family, but positive answers are obtained for several subclasses of the other families. Finally, we exhibit a Poisson field which is not isomorphic to any Poisson field $\Bbbk(x,y)$ for which $\{x,y\}$ is a polynomial in $\Bbbk[x,y]$.

We study controlled McKean–Vlasov contagion with state-dependent killing, common noise, loss feedback, and interacting populations. The main result is a comparison principle for the two-population killed-particle HJB on a decomposed state space of alive sub-probability measures and cemetery masses. The proof combines a Wasserstein smooth-gauge comparison argument with a killing-jump absorption estimate for mass transfer into the cemetery state. We also establish a multi-population mean-field limit, an explicit first-order particle convergence rate, conditional propagation of chaos, controlled well-posedness, and a steep-killing bridge to absorbing-boundary default. Finite-particle convergence tests and a two-population HJB feedback experiment illustrate the theory.

Full-potential electronic structure calculations for periodic systems retain the Coulomb singularity at the nuclei, which induces cusp behavior of the orbitals near the nuclei while leaving the interstitial region smooth. This multiscale regularity motivates a hybrid discretization framework that combines localized real-space discretizations in atomic patches with plane waves in the interstitial region. In this work, we employ tensor-product B-splines in the atomic patches and couple the two approximation spaces through a symmetric interior penalty discontinuous Galerkin formulation, yielding the isogeometric-plane wave (IGA-PW) method. To efficiently evaluate the restricted plane wave integrals, we develop a combined fast Fourier transform and Chebyshev correction strategy. We also construct a trace-block DG preconditioner to alleviate the conditioning deterioration caused by the SIPG penalty terms and accelerate the iterative eigensolver. In addition, we establish a priori error estimates for the corresponding linear eigenvalue problem, showing algebraic convergence near the nuclei and superalgebraic convergence in the interstitial region. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.

In this paper we present a new boundary integral equation formulation for the solution of the elastostatic traction boundary value problem in two and three dimensions. The approach relies on the introduction of new layer potentials, called string kernels, which are based on modifications of the Boussinesq-Cerruti family of half-space solutions. We prove that the resulting integral equations are second-kind integral equations, and show that they are well-behaved in the incompressible limit. We illustrate the performance of the method with several numerical examples.

This paper investigates a multilayered Helmholtz model in $\mathbb{R}^d$ ($d \ge 2$) characterized by concentric layers of materials with alternating positive and negative refractive indices. To overcome the loss of coercivity induced by the sign-changing material parameters, we construct a bespoke $\mathbb T$-coercivity operator to restore the coercive structure of the problem. Furthermore, to address the inherent lack of compactness on unbounded domains, we integrate a complex-wavenumber Dirichlet-to-Neumann (DtN) operator into this framework. By combining this variational synthesis with sharp a priori estimates, we rigorously establish the limiting absorption principle and prove the well-posedness of the corresponding transmission problem in appropriate function spaces. Crucially, we quantify the dependence of uniqueness on the domain geometry by explicitly analyzing the optimal trace constant, thereby providing a rigorous mathematical criterion for the design of multi-layer metamaterials.

In dynamic acoustic environments with time-varying interferers, effective beamforming requires identifying stationary regions over time. The Capon beamformer, a whitened matched filter constrained to maintain unity gain in the desired direction, theoretically relies on the instantaneous ensemble covariance matrix. Practical implementations rely on the batch Capon (or Sample Matrix Inversion), which estimates the sample covariance matrix (SCM) by averaging over a block of snapshots. This practical approach implicitly assumes that the data within the batch window is stationary and can be coherently combined. In non-stationary settings, a batch approach that averages over fixed or excessively long windows fails, as moving interferers smear the SCM and degrade the beamformer's nulling capabilities. To address this, this paper introduces a temporally segmented distortionless response beamformer. Inspired by the segmented least squares method, which fits piecewise polynomials to data while penalizing excessive segmentation to prevent overfitting, the framework extends practical Capon beamforming by incorporating data-driven temporal segmentation. This formulation minimizes output power while dynamically adapting the SCM estimation windows to local stationarity, offering a principled approach to tracking time-varying interferers.