Papers

We relate the generalized Ricci curvature of left-invariant generalized metrics on a Lie group with the Ricci curvature of certain metric Lie algebras. As an application, we prove subconvergence of rescaled generalized Ricci flows to expanding generalized Ricci solitons on simply connected Lie groups with positive semi-definite Killing form. Under the same hypotheses, subconvergence of pluriclosed flows to expanding pluriclosed solitons is also shown. We further prove that left-invariant Bismut-Ricci flat metrics arise as critical points for the generalized scalar curvature functional on unimodular Lie groups. In view of this, we establish dynamical stability for the generalized Ricci flow of so-called standard bi-invariant, Bismut-flat metrics on compact, simply connected, semisimple Lie groups and construct examples of dynamically unstable Bismut-flat metrics which are non-standard.

We present a discrete Morse theoretic approach to proving high connectivity or contractibility of a Vietoris-Rips complex using distance to a fixed point as an initial measurement. In particular we focus on the case of a finitely generated group using word length. As a proof-of-concept application we prove that $\mathcal{VR}_2(A_\Gamma)$ is contractible for $A_\Gamma$ a right-angled Artin group on a triangle-free graph $\Gamma$. We also prove an interesting sufficient condition for a group to be finitely presented that only requires checking connectivity of certain finite complexes.

We classify all characters of quantum symmetric pair coideal subalgebras for generalised Satake diagrams of finite and affine type.

In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism $\alpha \dd M \to \CC(F,Z)$ to exhibit a functor $Z\dd \CA\to \CC$ as an absolute $M$-weighted colimit of a functor ${F\dd \CB\to \CC}$. We also review, with short proofs, the various criteria for the weight $M$ itself to be absolute, in the sense that any $M$-weighted colimit is absolute. Finally, we prove that any absolute $M$-weighted colimit can be viewed as a colimit weighted by an absolute weight $M'$.

We construct a categorical framework in which we can view the direct integral as a functor taking as input objects in a certain category of Abstract Banach bundles and outputting a Banach space. We do this by constructing a quasi-abelian category of abstract Banach bundles, and we prove certain foundational and theoretical results about this category and its objects. As a consequence, we are able to define a notion of direct integral of sheaves.

Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlevé equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlevé equations $\rm{VI},\rm{IV},\rm{II},\rm{I}$. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlevé equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlevé equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.

A positive integer $n$ is called a practical number if every positive integer less than or equal to $n$ can be expressed as a sum of distinct positive divisors of $n$. In this paper, we study an open conjecture proposed by Wang and Sun concerning the quadratic representations of practical numbers. Specifically, we provide a short proof of the second part of the conjecture, demonstrating that for any positive integers $b$ and $c$ with $2 \nmid b$ and $2 \mid c$, there exists an integer $n$ satisfying $1 < n \le \max\{b, c\}$ such that $n^2 + bn + c$ is a practical number. Combined with the work of Somu, Li, and Kukla, this completely settles the conjecture of Wang and Sun.

We study a two-particle lattice Schrödinger operator describing a fermion-boson pair on the one-dimensional lattice $\mathbb Z$ with zero-range on-site interaction of strength $\mu\in\mathbb R$ and mass ratio $\gamma>0$. Using relative coordinates and fiber decomposition with respect to the total quasi-momentum $k\in\mathbb T:=(-\pi,\pi]$, we obtain a real symmetric fiber Hamiltonian $H_{\mu,\gamma}(k)$ in $\ell^2(\mathbb Z)$ with essential spectrum $$ [\,2(1+\gamma)-2a_\gamma(k),\,2(1+\gamma)+2a_\gamma(k)\,], \qquad a_\gamma(k)=\sqrt{1+2\gamma\cos k+\gamma^2}. $$ For $a_\gamma(k)>0$ and $\mu\ne0$, the operator has a unique simple discrete eigenvalue $$ E_\gamma(k,\mu)=2(1+\gamma)+\operatorname{sgn}(\mu)\sqrt{\mu^2+4a_\gamma(k)^2}, $$ lying below the band for $\mu<0$ and above it for $\mu>0$, with an exponentially decaying eigenfunction. At the exceptional fiber $(\gamma,k)=(1,\pi)$, the band collapses to ${4}$ and the unique simple eigenvalue is $E_1(\pi,\mu)=4+\mu$. At the critical coupling $\mu=0$ and $a_\gamma(k)>0$, both spectral edges are threshold resonances with bounded non-square-integrable solutions. As $\mu\to0^\pm$, the discrete eigenvalue approaches the corresponding threshold with $$ |E_\gamma(k,\mu)-E_{\mathrm{thr}}^\pm(k)| =\frac{\mu^2}{4a_\gamma(k)}+O(\mu^4), $$ and the normalized eigenfunction converges to the resonant solution. For strong coupling, $E_\gamma(k,\mu)=\mu+2(1+\gamma)+O(|\mu|^{-1})$, while the eigenfunction localizes at the interaction site. We also classify the joint limit $\mu\to0^\pm$, $(\gamma,k)\to(1,\pi)$ according to the relative scale of $|\mu|$ and $a_\gamma(k)$, revealing the nonuniformity of the weak-coupling threshold asymptotics.

We analyze a thermodynamically consistent diffuse interface model for incompressible two-phase viscous flows with unmatched densities in a smooth bounded domain $\Omega\subset\mathbb{R}^d$ $(d=2,3)$. This model consists of the Navier-Stokes-Voigt equations for the velocity field and a convective Cahn-Hilliard equation with a singular potential for the phase-field variable. The resulting hydrodynamic system is subject to a generalized Navier slip boundary condition for the velocity field and Cahn-Hilliard-type dynamic boundary conditions for the phase-field variable and the chemical potential. With the aid of the Voigt regularization, we establish the existence of global quasi-strong solutions that satisfy an energy equality. This is achieved through a combination of delicate approximation schemes and a semi-Galerkin method.

We prove that the Bernstein-Sato polynomials of ideals satisfy the multiplicative Thom-Sebastiani property. This affirmatively settles a question of Shi and Zuo that extends a question independently formulated by Budur and Popa.

A method is developed for structural analysis in which parts of an analysis domain are modeled by reduced-order models (ROMs) and the remaining part is modeled by a high-dimensional model (HDM). The multiple ROMs and HDM are combined based on an iterative domain decomposition method (DDM) with an arbitrary number of non-overlapping subdomains. A diagonal preconditioner is derived for the iterative DDM. The method is implemented based on a framework of the existing DDM-based analysis code. The ROM is implemented on GPU. Snapshot generation methods for the ROM are also proposed. Static structural analyses of nuclear power plant models are conducted using the combined multiple ROMs and HDM with multiple-subdomains model. The analyses are performed in parallel by the hybrid MPI-CUDA implementation. The convergence property of the iterative DDM is improved by the preconditioner.

For a set of $n$ alternatives, the best–worst choice polytope is the convex hull of the deterministic best–worst choice vectors induced by all linear rankings of the alternatives. The question of determining the dimension of this polytope as a function of $n$ has been raised by Doignon \cite{Doignon2023}. In this paper we answer this question by finding an explicit formula for this dimension. Our proof is motivated by a representation-theoretic interpretation of the relevant rank problem for the symmetric group, but is presented entirely in elementary linear-algebraic terms. The connection with representation theory will be explored elsewhere.

Variable-metric proximal methods accelerate composite convex optimization, but the scaled proximal map induced by a quasi-Newton metric rarely has a closed form. We develop Decomposed Bilevel Search (DBS), based on a diagonal-plus-rank-one factor \(X=D+uv^\top\) whose diagonal scaling satisfies a weak secant equation. The induced inverse metric \(B^{-1}=XX^\top\) recovers zero-memory DFP/BFGS-type Broyden members, while the factor form reduces each scaled proximal step to a two-dimensional monotone residual system. Each residual evaluation requires one diagonal-metric proximal map, and a certified bilevel solve reaches target accuracy \(\epsilon\) in \(\mathcal O((d+T_p)\log^2(1/\epsilon))\) work, where \(T_p\) is the cost of that proximal map. Under strong convexity, the outer method converges linearly with exact and inexact inner solves. In the scalar specialization \(D=\alpha I\), the oracle uses only ordinary proximal evaluations of the regularizer and no generalized Jacobian or active-set information. Experiments on ordered-weighted \(\ell_1\) (SLOPE/OWL) and group-lasso logistic regression evaluate both the inner oracle and the full outer method. The oracle solves high-dimensional SLOPE scaled proximal subproblems of condition number up to \(4\times10^6\) using a few hundred ordinary proximal evaluations. In SLOPE outer benchmarks, the warm-started oracle keeps the scaled-proximal overhead controlled and DBS reaches stringent targets with far fewer gradient evaluations than Lipschitz-normalized FISTA; on real-sim this becomes a clear target-time advantage. On group-lasso logistic regression, DBS is reliable on synthetic correlated instances and fastest on a real-data grouped-copy instance.

Let $X$ be a complete toric variety and let $e$ be the identity of its open torus. The Cox ring of ${\rm Bl}_eX$ is described by saturated powers of the toric-point lattice ideal $I_X$. We give a finite criterion for generation in Rees multiplicity one in terms of analytic spreads of projected lattice ideals and zero divisors in the associated graded ring. We apply this criterion first to fake weighted projective spaces, proving that multiplicity-one generation is equivalent to $I_X$ being a complete intersection. For projective toric surfaces, we prove that multiplicity-one generation holds if and only if it holds for every Picard-number-one toric surface dominated by $X$.

Stochastic gradient descent (SGD) is typically analyzed at a deterministic horizon chosen before the algorithm is run, even though practical stopping decisions are made adaptively by inspecting the evolving trajectory. This mismatch creates a fundamental certification problem: fixed-time guarantees do not generally remain valid at data-dependent stopping times, while deterministic horizons derived from worst-case bounds can be highly conservative. We address this problem for strongly convex stochastic optimization by constructing fully observable, trajectory-adaptive upper confidence sequences for the squared distance of the last iterate to the optimizer and the suboptimality of a weighted average. These bounds hold simultaneously over time, attain the optimal $1/t$ decay rate up to iterated-logarithmic factors in the worst case, and adapt to the realized stochastic gradients, allowing SGD to stop as soon as a prescribed accuracy is certified without sacrificing statistical validity. Our approach treats the evolving SGD trajectory as a sequential experiment whose observations provide evidence about the unknown optimization error. To formalize this perspective, we develop new recursive confidence-sequence techniques and a general time-uniform empirical Bernstein inequality for adapted processes with time-varying conditional means and predictable ranges that may grow without bound. We further extend these confidence-sequence constructions to minibatch SGD, with the empirical Bernstein bounds exploiting the realized second-moment structure within each minibatch. Numerical experiments show that the resulting stopping rules can require several orders of magnitude fewer iterations than natural deterministic horizons.

Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.

We study pair creation for quantum fields of spin $0$, $1/2$, and $1$ in the presence of weak gravitational and gauge backgrounds. Using resummed heat-kernel techniques, we obtain closed expressions for the pair creation probability up to quadratic order in the “generalized curvatures”. In addition, for gravitational backgrounds, we recast the pair production probability in terms of the matter distribution required for consistency with the Einstein equation. Finally, to illustrate the powerfulness of our methods, we analyze the effect of pair creation in a binary star system.

Computational models of tumor growth form a contemporary area of study. Diffuse interface models have been proposed as an important tool for cancer modeling. In this work, we consider preconditioners for numerical simulations of two tumor growth models based on diffuse interface models. For numerical efficiency, we propose block preconditioners that rely on using effective Schur-complement approximations. We show that these methods combined with state-of-the-art adaptive finite element discretizations lead to robust simulations in two and three dimensions. In extensive numerical experiments, we show that the proposed methods show robust convergence behavior.

Given a probability distribution $p = (p_1, \dots, p_n)$ and an integer $1\leq m \leq n$, a contiguous aggregation of $p$ is a probability distribution $q = (q_1, \dots, q_m)$ such that each $q_i$ is a sum of consecutive elements of $p$. Given $p$ and a positive number $R$, we consider the problem of computing a maximum entropy contiguous aggregation $q$ of $p$, under the constraint that its Shannon entropy $H(q)$ is at most $R$. We devise a dynamic programming algorithm that solves the problem exactly, and two time-efficient greedy algorithms that provide close-to-optimal solutions. We discuss a few scenarios where our problem arises.

A sharpened version of Carlson's integral inequality is established, which features a remainder term involving a relative distance to the set of extremal functions. A similar approach leads to an alternative derivation of Carlson's and Landau's discrete inequalities and provides another characterization of the optimal constant in Carlson's inequality for finite sums.

The Levenberg-Marquardt (LM) algorithm is the most widely used method for solving nonlinear least-squares problems, as it combines the robustness of steepest descent with the fast local convergence of the Gauss-Newton method. However, its computational cost can become prohibitive for large-scale problems because each iteration requires solving a large damped linear system, and conventional step acceptance strategies may require repeated solves as the damping parameter is adjusted. Despite this computational challenge, many large-scale least-squares problems exhibit effective low-dimensional structure, with only a small number of parameter-space directions strongly informed by the data. We propose an adaptive hybrid subspace Levenberg-Marquardt (HSLM) algorithm that constructs a low-dimensional subspace from complementary sources of gradient, memory, Krylov-subspace, and randomized curvature information and computes a spectrally damped LM step within this subspace. A distinguishing feature of the method is a deterministic adequacy monitor that quantifies how much descent information is captured by the reduced space and adaptively enriches the subspace when necessary. Step acceptance is decoupled from damping adjustment: Armijo backtracking determines the accepted step length, while the ratio of actual to predicted reduction is used solely to update the damping parameter, thereby avoiding repeated damped-system solves during step acceptance. For the HSLM algorithm, we establish global convergence to stationarity and prove local linear and superlinear convergence. Numerical experiments on neural-network training problems show that HSLM achieves convergence behavior comparable to classical and Krylov subspace LM (KSLM) while substantially reducing per-iteration computational cost, with increasing advantages observed as the parameter dimension grows.

This paper investigates inertial primal-dual dynamics with implicit Hessian-driven damping for strongly convex optimization problems with linear equality constraints. We first establish fast convergence rates for the objective function value error, the feasibility measure, and both the trajectory and its corresponding velocity vector. By appropriately adjusting parameters, we show that the proposed system achieves exponential convergence rates. Through implicit time discretization of the dynamical system, we derive an inertial accelerated primal-dual algorithm for solving the strongly convex optimization problems. Using Lyapunov-based method, we show that the proposed algorithm achieves convergence rates consistent with those of its continuous-time counterpart. We also extend the obtained results to non-smooth convex optimization case. Furthermore, we conduct numerical experiments to illustrate theoretical results.

In our previous work \cite{SZ2026}, we established a class of abstract Orlicz-Morrey spaces endowed with a ball-basis and introduced the notion of \(\Psi\)-bounded oscillation operators. The present paper further develops this theory by investigating sparse domination and weighted estimates for such operators and their commutators within the framework of abstract Orlicz-Morrey spaces endowed with a ball-basis. We obtain pointwise sparse domination results for the operators and establish their boundedness on weighted Orlicz-Morrey spaces. The hypotheses of these conclusions differ from those in the classical Orlicz-Morrey setting and do not rely on dyadic decompositions of Euclidean structure. Moreover, we study the duality of these spaces and prove that the dual of the associated block space is exactly the Orlicz-Morrey space under consideration.

This paper investigates joint beamforming and port selection in multi-cell fluid antenna-assisted (FAS) networks. In such networks, active beamforming and discrete FA port selection are coupled through intra-cell and inter-cell interference and are jointly optimized to maximize the weighted sum-rate (WSR). We develop a federated representation learning (FedRep) framework with a position-aware dual-branch deep neural network (PA-DNN). The PA-DNN uses channel state information and port positional encoding as inputs, and jointly outputs beamforming vectors and port selections through two task-specific branches. To support decentralized training across heterogeneous cells, the FedRep framework shares global beamforming-related parameters among base stations while keeping port-selection parameters local for cell-specific adaptation. Simulation results show that the proposed scheme achieves a higher weighted sum-rate than conventional FL and port-selection benchmark schemes.

We prove the leading low-temperature free-energy asymptotics of the nearest-neighbour quantum Heisenberg ferromagnet on $\mathbb Z^2$. For every fixed $S\in\{\frac12,1,\frac32,\ldots\}$, the free energy per site satisfies \[ \lim_{\beta\to\infty}\beta^2S f_2(\beta,S)=-\frac{\pi}{24}. \] This formula was predicted by Takahashi's modified spin-wave theory [15]. Napiórkowski and Seiringer proved the corresponding upper bound but left the matching lower bound in two dimensions open [13], a gap later highlighted again by Seiringer [14]. We prove this missing bound through a new comparison between the magnon exclusion dynamics and free bosons, based on extending wave functions to collision configurations with controlled kinetic cost.

Masked diffusion models generate sequences in rounds, predicting several masked positions at once. Fewer rounds can accelerate decoding, but sampling positions independently can lose their dependence and bias the output. How few rounds suffice for a prescribed error budget? Beyond error bounds for individual decoding rules, we study sharp lower bounds on the number of rounds required for a given target distribution. We introduce serial depth: the minimum number of rounds, averaged over target sequences, needed to meet a forward Kullback-Leibler error budget. With exact conditional probabilities, we prove these bounds for all deterministic rules that may adapt to generated values and construct matching schedules for several distribution families, with depths ranging from $\log\log n$ to linear in sequence length $n$. To narrow the gap to trained models, we show that these lower bounds remain valid for learned predictors and give sufficient accuracy conditions on visited contexts for schedules to retain their error guarantees. On two pretrained models across five text domains, a schedule suggested by the theory achieves higher average log-likelihood on held-out text in five rounds than random selection in eight. A preregistered confirmation study certifies a relative improvement on independent dialogue data.

Accurate online traffic prediction is essential for intelligent transportation systems, where forecasting must be performed continuously under imperfect sensing conditions. Missing observations and anomalous disturbances make this task challenging, particularly when prediction relies on a single traffic view. This paper proposes a Multi-View Coupled Tensor Decomposition (MVCTD) model for online traffic prediction from imperfect multi-view observations, such as speed, flow, and occupancy. The proposed model uses coupled tensor decomposition to build a structured latent forecasting space, in which shared spatial structures across traffic views and view-specific temporal dynamics are jointly modeled. A group sparse regularization is further introduced to capture correlated abnormal responses induced by real traffic anomalies and thus reduce their influence on forecasts. For streaming deployment, MVCTD performs iterative refinement only on the current latent tensor, while the remaining model variables are updated by lightweight closed-form steps based on summarized historical information, thereby avoiding repeated optimization over the full historical sequence. Experiments on real-world traffic datasets demonstrate that MVCTD achieves accurate forecasts with favorable runtime under severe missingness, confirming its suitability for online traffic prediction.

It is proved that, for every $\delta \in (0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q \geq (1+\delta)\Delta$ and the underlying graph has girth at least $5$ and maximum degree $\Delta = \Omega_\delta(1)$. This result also extends to general multi-spin systems satisfying a $\textit{local spectral contraction}$ condition, including the anti-ferromagnetic Potts model with $q\geq (1+\delta)(1-\beta)\Delta$. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.

We study the fine geometry of collision measures associated with independent Markov processes on a common metric measure space. These measures are natural space-time random measures encoding collision times and collision points. Under small-scale Ahlfors regularity and standard short-time heat-kernel estimates, we determine the local dimensions of their temporal and spatial marginals, the Hausdorff dimensions of their supports and of the collision sets themselves, and logarithmic limsup laws for their local masses at typical collisions and at exceptional thick collisions. Through this analysis, we identify exact Hausdorff measure functions for the collision-time sets and, in a natural regime, for the collision-point sets. Moreover, we prove that the corresponding Hausdorff measures are comparable, uniformly over all Borel subsets, to the temporal and spatial marginals of the collision measures. This provides a broad solution to the Hausdorff-measure part of an open question posed by Xiao more than two decades ago. Our framework covers symmetric stable processes, canonical diffusions on affine nested fractals such as the Sierpiński gasket, and stable-like jump processes on $d$-sets.

Functional data analysis is an important statistical field that treats data as random functions. In practice, the random functions are often not fully observed but instead measured at discrete times. While simpler problems, such as mean and covariance estimation, have been widely studied for discretely observed data, optimal estimation of linear regression for this data type has remained unsolved for over two decades. To tackle this fundamental challenge, we propose a novel approach, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method by incorporating the unbiased estimation of operators based on discretely observed measurements from all subjects. This unified estimation framework enables us to achieve minimax optimality in prediction risk in arbitrary sampling schemes ranging from sparse to dense designs, for both scalar-on-function and function-on-function regression models. Such methodological and theoretical advances are obtained for the first time and accurately reveal the influence of discrete sampling. For scalar-on-function regression, the phase transition occurs once, separating the convergence behavior into two distinct regimes. Remarkably, for function-on-function regression, up to three phase transitions may occur, determined by the sampling frequencies of the predictor/response functions. Finally, simulation experiments and two real data examples provide empirical support for the proposed methods.