Papers

We study exact simulation of diffusions via rejection sampling on path space using unbiased estimators of the density ratio obtained from Girsanov's theorem. When applied to the underdamped Langevin diffusion, it yields an algorithm for sampling from a strongly log-concave and log-smooth distribution with condition number $\kappa$, in dimension $d$, to accuracy $\varepsilon$ in Rényi divergence, in $\widetilde O(\kappa^{2/3} d^{1/3}\,\mathrm{polylog}(1/\varepsilon))$ queries. Under a third derivative bound, the dimension dependence improves to $d^{1/5}$. This improves substantially over the prior state-of-the-art complexity of $\widetilde O(\kappa d^{1/2}\,\mathrm{polylog}(1/\varepsilon))$ for the Metropolis-adjusted Langevin algorithm, and over the $d^{1/4}$ dimension dependence of Metropolized Hamiltonian Monte Carlo under the same third derivative bound. We also present applications to the mirror Langevin diffusion, and for obtaining Fisher information bounds in the non-log-concave case.

We obtain a motivic and a cohomological McKay correspondence for finite linearly reductive group schemes in arbitrary characteristic. In particular, we prove that if $V$ is a finite dimensional vector space and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then the Euler number of any crepant resolution of $V/G$ is equal to the number of irreducible algebraic representations of $G$. We obtain these McKay correspondences as a consequence of a motivic change of variables formula applied to $[V/G] \to V/G$. If $G$ is non-reduced, as can happen in positive characteristic, the stack quotient $[V/G]$ is not Deligne-Mumford. Therefore in order to prove this change of variables formula and the resulting McKay correspondences, we generalize the authors' theory of motivic integration for Artin stacks to arbitrary characteristic, which may be of independent interest.

Every SVAR result is conditional on two choices: the restrictions that identify the shock and the variables on which they operate. The literature disciplines the first; the second is chosen by hand. We develop a Bayesian methodology that constructs information sets, uses an out-of-sample criterion, and retains the largest system it admits. Under recursive identification, output rises with housing production rather than household credit alone. For monetary policy, an anchor-free joint Bayesian proxy SVAR with multiple instruments strengthens the credit spread channel. A core system augmented with the selected corporate spread identifies expected default risk as a potent transmission margin.

In this paper, we introduce the notion of subquandle filtrations for finite quandles as a foundation to construct algebraic and combinatorial link invariants. A nested sequence of subquandles naturally induces a filtration of the associated quandle coloring quiver by subquivers. By decategorifying these subquivers, we define two new polynomial-valued invariants of links: the disrespectful polynomial and the graded disrespectful polynomial, which measure how a chosen set of quandle endomorphisms respects and disrespects subquandle structure. Finally, we illustrate the distinguishing power of these new tools by comparing them to both classical quandle counting invariants and the quiver in-degree polynomial.

We study the law of the exit time and exit point of a Brownian motion in a two-dimensional domain with reflecting boundary conditions, except on small disjoint exit windows through which the stochastic process can escape the domain. In the limit of infinitely small exit windows, it is natural to assume that the process starts from the quasi-stationary distribution. In this setting, we obtain a precise description of the exit event.

Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\in K_2(A;\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the integral Gersten conjecture but it rules out a naive reduction to finite coefficients.

We study the least stable state determined by a causal response program and its quantitative calibration. For nondominated families of continuous Hilbert-valued semimartingale laws with uniform drift and trace-clock bounds, the calibrated quadratic Itô program selects a state $\mathcal A_2$ with canonical coordinates $(X,A,Q)$, where $A$ is Hilbert–Schmidt skew area and $Q$ is trace-class covariance. Brownian experiments identify the Hilbert–Schmidt/operator coefficient gauges and hence the dual $\mathcal S_2/\mathcal S_1$ state geometry. A lossless encoder–decoder pair shows that every joint realization factors through $\mathcal A_2$. A single sequence of total Borel causal approximants constructs the common state, while the laws verify its classical semantics. A dimension-free defect–energy estimate gives independence from the finite-variation regularization, and spatial trace tightness yields restart-stable compact capacity cores in infinite dimensions. Each fixed higher signature level is a continuous readout of $\mathcal A_2$. Rough flows inherit corewise continuity, while Euler schemes construct a jointly Borel raw-causal Itô field outside one parameter-independent polar set. On compact covariance-envelope subclasses, covariance responses select a backward first jet which, under the stated completion and continuation hypotheses, has modelwise Galtchouk–Kunita–Watanabe semantics and agrees with the solver's martingale coordinate.

We prove that the set of time-frequency shifts $\{e^{2\pi i \beta l t} g(t-\alpha k) : k,l \in \mathbb{Z}\}$ with a continuous, integrable totally positive function $g$ and lattice parameters $\alpha,\beta>0$ generates a frame for $L^2(\mathbb{R})$ if and only if $\alpha\beta<1$. This fully settles the so-called frame set problem for the class of totally positive functions. As a closely related result we prove a sharp Kadets-type theorem for every shift-invariant space generated by a continuous totally positive function. The proofs are based on Fredholm theory and limit-operator theory. A formalization of our main result in Lean 4 is also provided.

We consider the computation of internal solutions for a time domain plasma wave equation with an unknown potential $q$ from boundary response data. The internal solutions are computed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian, or mass matrix. It was recently shown that in one dimension these data generated internal solutions converge in $L^2$ at order $\sqrt{\tau}$ for well chosen initial waves. Here we study the internal solution reconstruction in two dimensions, where a multiple input/multiple output (MIMO) setup and a block Gramian are needed, with the number of boundary sources increasing as the time sampling is refined. We show that the general error bound carries over to this setting: the distance between the data generated solutions and the best causal approximation from background snapshots is controlled by the best approximation mass matrix mismatch. Numerical refinement studies on a square domain measure the convergence of the data generated solutions alongside the best causal approximation from the background snapshots, with the relative errors appearing to go to zero at rate $\sqrt{\tau}$, and the absolute errors going to zero in $L^1$. We also show that the solution reconstructions remain accurate for high contrast composite media. Finally, since the mass matrix assembled from noisy response data can fail to be positive definite, regularization is needed; comparing a diagonal shift with an eigenvalue floor, we find the floor more robust, with reconstructions that remain more accurate than the unperturbed background field with noise that is up to ten percent of the root mean square data amplitude.

A graph $\Gamma$ is $(c,t)$-sparse for $c > 0$ and $t \ge 1$ if for every pair of vertex subsets $A, B \subseteq V(\Gamma)$ with $|A|, |B| \ge t$, the number of edges $e(A,B)$ between them satisfies $ e(A,B) \le (1 - c)|A||B|$. In this paper, we prove that for every integer $\ell\ge2$, there are $\varepsilon > 0, C, C' > 0$ such that if an $n$-vertex graph $\Gamma$ is $(1-\varepsilon,t)$-sparse for some $t$, and has at least $Ct^{1-1/\ell}n^{1+1/\ell}$ edges, then $\Gamma$ contains at least $C'n^2t^{2\ell-2}$ induced copies of $C_{2\ell}$. This partially resolves a problem of Ding, Gao, Liu, Luan, and Sun.

We establish Weyl-type subconvexity bounds in the level aspect for cuspidal automorphic representations of $\mathrm{GL}_2/F$ with simple supercuspidal ramification at a prime ideal $\mathfrak{q}$. More precisely, for the family $\mathcal{F}_t^\zeta(\mathfrak{q}^3;\omega)$ consisting of representations of conductor $\mathfrak{q}^3$, central character $\omega$, and prescribed simple supercuspidal local component, we prove the fourth moment estimate \begin{align*} \sum_{\substack{\pi \in \mathcal{F}_{t}^{\zeta}(\mathfrak{q}^3;\omega) \\ C_v(\pi) \leq \mathbf{C}_v,\ v \mid \infty}} |L(1/2,\pi)|^4 \ll_{F,\varepsilon} \mathbf C_\infty^{1+\varepsilon} N_F(\mathfrak{q})^{2+\varepsilon}. \end{align*} As a consequence, we deduce the Weyl-type bound \begin{align*} L(1/2,\pi) \ll_{F,\varepsilon} C_{\infty}(\pi)^{1/4+\varepsilon} C_{\mathrm{fin}}(\pi)^{1/6+\varepsilon}. \end{align*} In particular, this bound applies to a genuinely non-self-dual family of odd conductor exponent, beyond the reach of the cubic moment method.

In this paper, we disprove a conjecture of Liu, Lou, and Zhu concerning Carleson embeddings of $\mathcal Q_p$ spaces into tent spaces for $0<p<1$. More precisely, we construct a finite positive $p$-Carleson measure $\mu$ on the unit disc $\mathbb D$ such that the canonical embedding $$ \operatorname{id}:\mathcal Q_p \longrightarrow \mathcal T_{p,2}^2(\mu) $$ is not bounded. The main ingredient is a new family of $\mathcal Q_p$ test functions that encodes Cantor-type structures on the unit circle $\mathbb T$ into the analytic behavior of functions in $\mathcal Q_p$. This construction is inspired by ideas developed in a recent work of the first and third authors on composition operators on $\mathcal Q_p$ spaces.

Operating-room allocation is a major challenge for public hospitals with limited surgical capacity and large elective waiting lists. This work proposes a two-stage operating-room allocation framework for specialty-block scheduling environments. The methodology combines a mixed-integer linear programming model for medical specialty block allocation with a priority-based patient allocation procedure. The framework was evaluated through one-week and multi-week simulation scenarios using parameters estimated from historical surgical and waiting-list data. The proposed methodology was compared against an integrated mixed-integer linear programming baseline adapted from the literature under realistic operational disruptions, including failed patient confirmations and surgery suspensions. Results show that the proposed framework achieves a more balanced distribution between offered and demanded surgical time across specialties while maintaining high operating-room utilization and substantially lower computational times than the integrated baseline. In the 20-week scenario, the proposed methodology achieved lower waiting times and operated a larger number of patients while preserving operational flexibility. These results suggest that decomposition-based operating-room allocation strategies provide a practical and computationally efficient alternative for reducing surgical waiting lists in public hospitals.

We prove that the standard degree-two and degree-three vortex solutions of the Ginzburg-Landau equation are nondegenerate. Their Morse indices are also computed. The proof relies on new explicit upper and lower bounds of the modulus of these solutions and a comparison argument. It is expected that our method can be generalized to study higher degree solutions.

Bergemann and Morris (2016) show that one information structure is more informative than another exactly when it induces a smaller set of Bayes correlated equilibrium outcomes in every game. We build the quantum analogue. An information structure becomes a family of density operators indexed by the payoff state, which the mediator observes. We show that obedience is equivalent to a Loewner domination between operators on one player's subsystem. The equilibrium set is then a nonempty compact spectrahedron computable by semidefinite programming, classical structures embed exactly, and under quantum individual sufficiency more information shrinks the equilibrium set in every game.

We associate height function models to a range of generalized two-dimensional XY models and prove that delocalisation of the height function model rules out exponential decay of the nematic order parameter in the primal spin model. The argument is based on a generalized loop representation which relates the variance of a height difference between two faces to the nematic order parameter. This generalizes the results by van Engelenburg and Lis (2023). The range of models also includes the much-studied generalized XY model as proposed by Korshunov and independently by Lee and Grinstein, where neighbouring spins interact ferromagnetically favouring parallel alignment and nematically favouring parallel or antiparallel alignment. Moreover, we recover the existence of a nematic region by a classical comparison argument of Pfister (1982).

We study an inverse problem of recovering a time-dependent convection term and density coefficient of the convection-diffusion equation from partial data on a certain type of compact Riemannian manifold of dimension at least three. We prove that the knowledge of a partial input-output operator determines both coefficients uniquely up to a natural gauge. Our geometric setting is conformally transversally anisotropic manifolds, that is, compact Riemannian manifolds with boundary that are conformally embedded into a product of the Euclidean line and a transversal manifold. Additionally, we assume that the attenuated geodesic ray transforms of one-forms and functions are both injective on the transversal manifold.

Let $G$ be a connected real semisimple linear algebraic group with finite center and no compact factors, let $X$ denote its associated Riemannian symmetric space, and let $h_{\mathrm{vol}}(X)$ be the volume growth entropy of $X$. We show that there exists an $\epsilon = \epsilon(G) > 0$ so that the following holds: if $\Gamma < G$ is a discrete subgroup with critical exponent greater than $h_{\mathrm{vol}}(X) - \epsilon$, then $\Gamma$ is Zariski dense in $G$. If $G$ is further assumed to be isomorphic to one of the isometry groups of the real, complex, or quaternionic hyperbolic spaces, we determine the smallest possible value of the critical exponent to guarantee Zariski density of the associated subgroup (in other words, the largest possible value of $\epsilon = \epsilon(G)$). In the setting of discrete subgroups of real semisimple Lie groups with no compact factors, this generalizes Borel's Density Theorem for lattices. The key ingredient is input from the theory of unitary representations, particularly the recent work of Benoist–Liang (which was inspired by earlier work of Benoist–Kobayashi) on general temperedness criteria for the quasi-regular representation of $L^2(G/H)$ for any closed subgroup $H$ of $G$.

We classify central extensions of the dg Lie algebra of derived global sections of the tangent sheaf on the punctured, formal d-disk for d > 2. (The d=2 case has appeared in previous work). We also prove a local, universal form of the Grothendieck-Riemann-Roch theorem for families of d-dimensional complex varieties.

Fast preparation of quantum error-correcting codes is essential for scalable quantum memories, but geometric locality and $U(1)$ charge conservation impose an unavoidable transport constraint. We combine exact complementary-channel geometry, charge-sector Haar analysis, and a gate-resolved connected-moment expansion to study one-dimensional covariant encoders under flagged erasure. Charge-Haar codes attain the universal adjacent-charge lower bound up to exponentially small corrections, yielding an exact $n^{-1/2}$ extensive-erasure law and a sharp half-erasure transition. For local number-conserving brickwork circuits, diffusion of the logical charge enforces an $\Omega(n^2)$ encoding-time lower bound; we also prove an $O(n^3)$ mixing bound for the classical component and reduce the remaining full-channel upper bound to a source-restricted low-support operator-spreading problem. These results identify diffusion as an operational limit on symmetry-constrained quantum coding and establish a route to its exact formation time.

We develop a new analytical framework for the local deformation theory of compact cohomogeneity-one Einstein metrics. The framework combines an intrinsic reformulation of the Einstein boundary-value problem with an auxiliary theory of equivariant harmonic maps and leads to what we call the Einstein Detection Principle. The auxiliary harmonic maps generate a rich variational, elliptic and spectral framework while introducing no additional Einstein degrees of freedom.

Chirality plays an important role in many optical phenomena, and it is crucial to have efficient and accurate numerical algorithms to simulate solutions in this setting. In this paper, we discuss a High-Order Spectral method coupled to geometric regular perturbation theory, which results in just such a method. We view the chirality of the laterally periodic, constant permittivity/permeability structure as deviating from a background value, and demonstrate analyticity of the field scattered by a plane electromagnetic wave with respect to this deformation. This analyticity is not only with respect to chirality deformations of arbitrarily large real size, but also joint in the spatial variables. We also show how the High–Order Perturbation of Envelopes (HOPE) recursions, which we used to establish these results, can be implemented as a rapid, robust, and reliable numerical algorithm.

Unified Multimodal Relation Extraction (UMRE) aims to identify intra-modal and cross-modal relations between textual entities and visual objects. However, existing UMRE studies still encounter two critical issues: ignoring inherent aleatoric uncertainty causes noise propagation, and deep-seated heterogeneity between distinct modal distributions hinders alignment. To address these issues, we propose the Uncertainty-Guided UMRE Network (UG-UMRE). Specifically, we design an Uncertainty-Driven Unimodal Augmentation (UDUA) module, which models features as Gaussian distributions based on the Variational Information Bottleneck. By incorporating an uncertainty-aware self-supervised contrastive learning mechanism, UDUA effectively filters out noise while maintaining semantic consistency. Furthermore, we introduce the Joint Aleatoric Uncertainty Alignment (JAUA) module as a global semantic pre-calibration mechanism. JAUA leverages probabilistic distribution consistency to construct a shared latent space, eliminating the distributional gap by synchronizing cross-modal statistical properties, thereby laying a robust foundation for fine-grained interaction. Experiments on three benchmark datasets (UMRE, MORE, and MNRE) demonstrate that UG-UMRE achieves state-of-the-art performance. Further analysis validates the pluggable and effective performance of the proposed UDUA and JAUA modules.

In this article, we research the mean value of integral of exponential sum $S(\alpha)=\sum_{u\in I}e(\alpha u^k)$, where $I$ is a short interval whose length is $2N^{\theta},\theta<1$, and the broken arc $\mathfrak{m^*}$ is a subset of a following minor arcs \[ \mathfrak{m}=\bigcap_{j\leq k-1}\left\{\alpha:\forall q<(\log N)^A,h<q,(h,q)=1,\Big|\alpha-\frac{h}{q}\Big|>\frac{1}{qN^{(k-j-1/2)\theta}}\right\} \] which has measure at least $c>0$. By setting $m$ is a sufficiently large number, $N$ is be sufficiently large in terms of $m$. When $k>\log m$ and $\frac{\log k}{\log m}<1/2$ we set the following estimate: \[ \int_{\mathfrak{m^*}}\bigg|\sum\limits_{{N_1}<u<{N_2}}{e(zu^k)}\bigg|^{m}\mathrm{d}z\ll_cN^{\theta m(\frac{2k+1}{2k+2}+o(1))} \] We can find this estimate moving beyond the even-odd restriction of powers. To get this bound, we first set a strong estimate for almost $\alpha$ by Diophantine approximation and Vinogradov's main value theorem. Then, combining this result, we construct a refined Weyl differencing argument by partitioning the differences step into large and small range, which significantly outperforms the classical one. By the version of probability, we can calculate the multiplicity of each sum. Put them together and we can complete the proof.

We determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched Rényi conditional entropies for every order $\alpha\in[\frac12,1)$. If two bipartite states are within trace distance $\delta$, then both conditional entropies differ by at most $\frac{1}{1-\alpha} \log[(1-\varepsilon)^{\alpha} +(D-1)^{1-\alpha}\varepsilon^{\alpha}]$, where $\varepsilon := \min\{\delta,1-1/D\}$ and $D$ is the effective dimension, given by the dimension of the first subsystem times the largest possible Schmidt rank. For every distance constraint $\delta\in[0,1]$, the bound is attained by an isotropic pair with a maximally entangled anchor. Taking $\alpha\uparrow1$ recovers the recent sharp continuity bound of quantum conditional entropy by Berta et al. [arXiv:2607.24687]. The proof linearizes the relevant concave Rényi functional at a comparison point dictated by the isotropic equality family. Schmidt-rank domination extends the equality geometry to an arbitrary anchor state, after which trace-distance duality and a noncommutative calibration estimate control the perturbation and anchor term without weakening the sharp constant. The latter estimate requires matrix analysis and is assisted by ChatGPT 5.6 Sol.

We propose a deep learning framework for image restoration from images degraded by both multiplicative Gamma noise and blur. Unlike conventional deep equilibrium (DEQ) models that rely on implicit neural regularization, the proposed method learns an explicit and interpretable regularizer parameterized by geometric priors associated with surface area and mean curvature. To minimize the resulting variational model, we develop a mirror descent algorithm tailored to the commonly used Gamma-noise fidelity terms. Leveraging the Kurdyka-Lojasiewicz property for functions defined in $o$-minimal structures, we establish the global convergence of the generated iterates to a critical point. Experimental results on both grayscale and color image restoration demonstrate that the proposed method consistently outperforms representative model-based approaches while achieving performance comparable to state-of-the-art DEQ models based on implicit regularization, despite requiring substantially fewer trainable parameters.

We start by rephrasing some well-known characterizations of fit and subfit locales in terms of closure and interior operators on the system of sublocales. Using these characterizations, we are able to extend the notions of fitness and subfitness to a more general setting. This approach not only allows us to obtain more general results, but also to present new properties of fitness and subfitness (such as a formula for the computation of supplements of sublocales). Using this approach we were also able to simplify the proofs of some well-known results.

We show that every symmetric fusion category over $\mathbb{R}$ is equivalent to the category of finite-dimensional semi-linear representations of a $\mathbb{Z}_2$-graded finite super group. The proof uses Galois descent for tensor categories over $\mathbb{C}/\mathbb{R}$, reducing the classification to semi-linear $\mathbb{Z}_2$-actions on symmetric fusion categories over $\mathbb{C}$. As a further structural result, we establish a Tannaka-Krein type correspondence between symmetric fusion categories over $\mathbb{R}$ and finite groupoids with a $\mathbb{Z}_2 \times \mathrm{B} \mathbb{Z}_2$-action. This gives a complete real analogue of Deligne's classification result.

We determine the complete local Bloch spectrum bifurcating from the origin for small-amplitude periodic hydroelastic Stokes waves in infinite depth, under the combined effects of gravity, surface tension, and elastic bending. Away from the Wilton-type resonance set, we construct a real-analytic Stokes-wave branch and analyze the four eigenvalues emerging from the defective zero eigenvalue of the linearized hydroelastic Euler system. Using analytic spectral perturbation theory and Hamiltonian-reversible reductions, we decouple them into a Benjamin–Feir pair and a long-wave pair. The long-wave pair remains purely imaginary and has the singular scale $\cO(\sqrt{|\mu|})$, whereas the Benjamin–Feir pair is governed by an explicit discriminant whose leading sign yields a sharp criterion for modulational stability and instability. We derive the exact non-resonant phase diagram in the surface-tension-bending parameter plane and identify a bounded stability island generated by elastic bending. In the unstable region, and away from a drift degeneracy, the Benjamin–Feir branches form a local figure-eight curve. In the zero-bending limit, the reduced coefficients recover the known deep-water gravity and gravity-capillary results, while the change from the finite-depth $\cO(|\mu|)$ long-wave scale to $\cO(\sqrt{|\mu|})$ shows that the infinite-depth problem is singular.

We study mean-field doubly reflected forward-backward stochastic differential equations with two optional barriers satisfying a strong Mokobodzki condition. For $L^p$-data, $p\in(1,2]$, we prove existence and uniqueness on sufficiently short time horizons when the coefficients may depend on the joint law of $(X,Y,Z)$. Under an additional monotonicity condition and using an exponentially weighted norm, we also obtain a global-in-time result for $p=2$. The setting is motivated by recursive mean-field Dynkin games and game-option valuation with irregular payoff barriers.