Papers

Data-driven reduced-order modeling is an essential component in the computer-aided design of control systems. In this work, we present a novel symmetric Hermite formulation of the quadrature-based balanced truncation algorithm that constructs linear reduced-order models from evaluations of the full-order system's transfer function and its derivative. Significantly, the Hermite formulation preserves desirable qualitative properties of the system used to generate the data, such as state-space Hermiticity and, consequently, asymptotic stability.

Neural operators are often reported to exhibit zero-shot super-resolution, a phenomenon in which a model trained on coarse grids produces accurate predictions on finer testing grids without additional retraining. Despite strong empirical evidence, the theoretical foundations of this phenomenon remain unclear. In this work, we provide a systematic theoretical study of zero-shot super-resolution in operator learning. We first show that zero-shot super-resolution can be information-theoretically impossible even in benign settings such as when the input functions are available over the entire continuum and the ground truth is a simple rank-one linear operator. We then identify Hölder smoothness of the output functions as a sufficient condition for zero-shot super-resolution and derive corresponding generalization bounds. Finally, we also validate the identified failure modes through experimental results.

In 2010, Steurer conjectured that any family of $n$ unit-norm vectors $v_1,\dots,v_n$ with polynomially small average correlation $\mathbb{E}_{i,j}|\langle v_i,v_j\rangle|\leq n^{-\epsilon}$ contains linear-sized constant-separated sets. We refute this conjecture in a strong sense using the machinery of sparse high-dimensional expanders: such vector families do not even have linear-sized $\frac{1}{\log^{1/4-o(1)}(n)}$-separated sets. Consequently, we show that there are families of vertex expanders on $n$ vertices for which the (average) $L_2$-mixing time to the uniform distribution of any reweighted simple random walk is at least $\log^{5/4-o(1)} n$.

The inducibility constant $\lambda_{F}$ of a graph $F$ is the asymptotically maximum induced density of $F$ in a growing sequence of graphs. This paper systematically investigates the case when $F$ has 6 vertices (and there are 78 cases to consider up to isomorphism and complementation). We show that flag algebras can compute the sharp upper bound on $\lambda_F$ in 36 cases of which, as far as the authors know, 30 are new results. In each of the solved cases, we also prove results about the structure of large (almost) extremal graphs. In particular, we establish perfect stability in all 32 cases when the extremal construction has no quasirandom parts. We also present conjectures about the value of $\lambda_{F}$ for 12 further cases (where the upper and lower bounds are very close to each other).

We present a constructive version of the Brøndsted-Rockafellar theorem in general geodesic metric spaces. Applications include a constructive form of the Caristi theorem and quantitative relations between metric slope error bounds and the global growth of functionals.

We derive dispersive-dissipative estimates for multipliers associated to a general class of wave-type equations, in particular for the viscous Boussinesq equation. The dispersion is related to the geometric hypotheses on the phase func tion and on the degeneracies that may happen at low and high frequencies. The dissipation interacts with the dispersion, influencing the decay rate of the solution.

In the design of computational methods for Bayesian inverse problems, costly forward model evaluations make it difficult to sample from or compute the posterior. This motivates the need for approximate forward models that are cheaper to evaluate. We consider reduced-order forward models which exploit the lower-dimensional structure in the Bayesian inverse problem by projecting to the "likelihood-informed subspace" of the parameter space where the prior-to-posterior update is significant. However, the theoretical properties of these reduced-order forward models and their impact on the solution of the Baysian inverse problem are not always well-understood. In this work we consider linear Gaussian inverse problems with a possibly singular prior covariance matrix. We analyse a recently proposed reduced-order model which uses a Petrov-Galerkin projection to likelihood-informed subspaces that arise in optimal low-rank approximations of the posterior covariance matrix. We bound the error in the resulting approximation of the root prior-preconditioned Hessian of the data misfit. Based on this we also bound the errors of the approximate posterior covariance and mean. Our analysis shows that this reduced-order model recovers the exact posterior when the rank of the reduced-order model is equal to the "intrinsic dimension" of the inverse problem, i.e. the rank of the prior-preconditioned Hessian. Two numerical experiments from structural engineering illustrate the performance of our bounds.

The goal of this work is to study the combinatorics of the lattices of higher torsion classes of the higher Auslander and Nakayama algebras of type A. Combinatorial descriptions of these higher torsion classes were recently obtained by August et al. (2025), and it was observed that the lattices that they form are not semidistributive. We study in some depth these lattices, proving in particular that the lattices of higher torsion classes of the higher Auslander algebras of type A are join-semidistributive, join-extremal and left modular. We also prove that the lattices of higher torsion classes of the higher Nakayama algebras of type A are lattice quotients of them. In order to prove these results, we define a general construction that produces a lattice, which we call $(P,\phi)$-Tamari, for any choice of poset $P$ and chain $\phi$ in $P$. We prove the lattice results for this general construction. When $P=\phi$ is a chain, we recover the Tamari lattice, whereas the lattices of higher torsion classes that we study are obtained for another very particular choice.

We show that Grothendieck's real constant $K_G$ can be upper bounded by projecting vectors onto a random plane through the origin and thresholding a degree five Hermite polynomial. This resolves a conjecture of Braverman-Makarychev-Makarychev-Naor from 2011, who required an extra randomization step in their rounding scheme and proved $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-500}$. As a corollary of our result, we prove the bound $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-217}$ by thresholding degree three Hermite polynomials in the plane. We finally give a rigorous computer-assisted proof that $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-5}$ using interval arithmetic and degree three Hermite polynomial thresholding.

The Fourier Entropy-Influence (FEI) conjecture states that the Fourier entropy of Boolean functions is uniformly bounded by their total influence. It has been verified for canonical examples such as disjoint tribes and for some classes of Boolean functions such as symmetric functions and read-$k$ decision trees (with a constant that depends linearly on $k$). In this note we present new classes of Boolean functions that verify the FEI conjecture. The key element is an inequality controlling the difference between the entropy of a function $f$ and the average of the entropies of $f^{\pm}$, the sub-functions obtained by setting $x_m=\pm1$ for some $m$, by the $m$-influence of $f$. If this key inequality were to hold for Boolean functions, then the full FEI conjecture would follow by induction. We introduce the notion of a stopping binary tree and observe that functions that satisfy the key inequality at the branching nodes of the tree and the FEI conjecture at the stopping nodes will satisfy the FEI conjecture. We identify some classes of functions that fit this framework: the $\delta$-tribes functions, the monotone Boolean functions with the tribe separation property, and the Boolean functions with the semi-separation property, and, and, along the way, demonstrate some results that we hope the experts in this fascinating field might find useful.

We develop a new algorithm for computing the second discrete homology group of a graph which is much faster when compared to existing algorithms. To do so, we identify five basic shapes, which are quotient graphs of the 3-cube with the property that the injective maps from them detect all possible 2-boundaries in the singular chain complex computing discrete homology.

Density estimation is often presented as a choice among parametric summaries, finite mixtures, and nonparametric smoothers. This review argues for a complementary view: a data set can be studied through a path of densities indexed by smoothing scale, diffusion time, model complexity, density level, or noise level. We call this perspective density evolution. Under this lens, Gaussian kernel density estimation is heat flow from the empirical measure; scale-space methods, critical bandwidths, mode trees, and derivative-significance displays describe the evolution of modal and derivative structure; finite mixtures and mixture reduction provide compressed representations of kernel-like estimates; and cluster trees and persistent homology summarize evolving level-set topology. We review these connections and discuss inference for feature lifetimes, high-dimensional complications, and links with score-based generative diffusion. We also include three elementary structural results: nondegenerate modes move along smooth branches, a natural moment-preserving Gaussianization semigroup is forced to be Ornstein–Uhlenbeck, and shared-covariance Gaussian mixtures become log-concave once component means are sufficiently concentrated. Together, these ideas shift attention from choosing one density estimate to studying the multiscale probability landscape.

Tree search is a central abstraction behind many language-agent reasoning and decision-making tasks: agents must explore actions, remember failures, and backtrack toward promising alternatives. Yet, we lack a theoretical understanding of how transformer-based policies acquire such search capabilities from the training dynamics of reinforcement learning (RL). We study this question in a stochastic $k$-ary tree environment, where an agentic transformer observes only its trajectory history through interaction and receives a terminal reward for reaching a hidden leaf goal node. We first construct a two-head transformer that implements randomized depth-first search (DFS): one head tracks previous actions, while the other detects failure outcomes and triggers backtracking. We then analyze the training dynamics of policy gradient under a depth-wise curriculum, showing that this same DFS mechanism emerges in stages from sparse reinforcement feedback without expert demonstrations. The resulting policy exhibits depth generalization: after training only on depth-$1$ and depth-$2$ trees, it succeeds on deeper full trees. We further show that, under imbalanced goal distributions, discounting the return leads to a ranked DFS policy that prioritizes higher-probability branches. Overall, our results identify a mechanistic normal form for transformer-based search, in which attention heads specialize and cooperate to extract decision-relevant traces from context and convert them into agentic action selection via RL training.

In this paper, we investigate contraction analysis for nonlinear time-delay systems described by functional differential equations. We first extend the concept of Lyapunov-Krasovskii functionals within the differential framework. We then show that its existence is equivalent to that of an incremental Lyapunov-Krasovskii functional and guarantees uniform incremental exponential stability. Next, we extend the concept of Lyapunov-Razumikhin functions within the differential framework, whose existence also ensures uniform incremental exponential stability. As an application of our results, we formulate stabilizing feedback control design for nonlinear time-delay systems with single delays in terms of linear matrix inequalities.

Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, $P({\cal N})$, of the stochastic number of e-folds, ${\cal N}$, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determine $P({\cal N})$. First we apply this method to the simple case of quantum diffusion along a flat potential without any classical drift. We recover the expression for the PDF that has previously been found using characteristic functions, with an exponential tail, and a power-law behaviour, $P({\cal N}) \propto {\cal N}^{-3/2}$, in the intermediate regime between the peak and the tail of the PDF. Finally we apply the method to constant drift inflation, in the narrow- and broad-well limits. In the narrow-well limit, there is an analytic solution and the PDF is similar to the drift-free case, with a mildly suppressed tail. In the broad-well limit, determining the full set of eigenvalues and eigenfunctions requires a piecewise construction of the spectrum, and the broad-well PDF is qualitatively different, with an enhanced peak and a strongly suppressed tail.

In this work we study the general relativistic massless Boltzmann equation on Friedmann-Lemaître-Robertson-Walker spacetimes with spatial topology $\mathbb{T}^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^{\mathfrak{q}}$ with $\mathfrak{q}\in [0,1]$. The massless Boltzmann equation on these backgrounds admits non-stationary Maxwell-Jüttner equilibria of the form $\exp(- |t^{2\mathfrak{q}}p|)$. For $0 \leq \mathfrak{q} \leq 1$, we prove future global-in-time existence and uniqueness of small perturbations of these equilibria in the case of hard ball interaction without symmetry assumptions. For $0\leq \mathfrak{q} < 1/3$, we prove that the perturbation – measured in a suitable $L^2_p$ based energy norm – decays at the superpolynomial time-decay rate of $t^{-3\mathfrak{q}}\exp(-t^{1-3\mathfrak{q}})$, whereas for $1/3< \mathfrak{q} \leq 1$ we obtain the polynomial time-decay rate of $t^{-3\mathfrak{q}}$. In the borderline case $\mathfrak{q}=1/3$, we show the time-decay of $t^{-3\mathfrak{q} -c}$ with a uniform constant $c>0$. Finally, for $\frac{1}{3}< \mathfrak{q}\leq 1$, we prove future global-in-time existence and uniqueness of small perturbations of the vacuum solution on $\mathbb{T}^3$.

We consider establishing the interpretability theory of deep learning through constructing a corresponding relationship between the renormalization group (RG) method in statistical physics and the training process of deep neural networks (DNNs). We have proved the constructed relationship using the one-dimensional Ising model as the input data. In this paper we generalize our results to the case of continuous input data, which is a necessary preparation for applying the corresponding framework to real-world data. To be representative, we consider a class of data distribution in the exponential family. We prove that when the parameters of fully connected (FC) DNNs achieve their optimal value after training, the characteristic parameters of the feature layer output of DNNs are equal to the fixed points of the characteristic parameters of input data under RG method for continuous fields. This conclusion shows that the training process of DNNs is equivalent to RG calculation on this kind of data and therefore the network can extract main features from the input data just like RG. Also, the equivalence further validates the correspondence framework we have established, providing an explanation for the outstanding performance of DNNs on real-world data.

Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in $\mathcal{Z}(\text{Vec}_G^\omega)$, i.e. in group-theoretical topological orders for finite groups $G$ with anomaly $\omega$. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.

We introduce the concept of twin phases for a symmetry $\mathcal{S}$, defined as inequivalent phases, whose order parameters are part of the same generalized charge under $\mathcal{S}$. Stable, direct transitions between such twin phases are never spontaneous-symmetry-breaking transitions, even after (partially) gauging the initial symmetry $\mathcal{S}$: they are phase transitions without hidden symmetry breaking. We illustrate this with an (anomalous) finite group symmetry in 1+1d, which exhibits such intrinsically beyond Landau transition, i.e. an intrinsically Deconfined Quantum Critical Point (DQCP).

Empirical Bayes analyses routinely model noisy measurements of latent parameters as normal, justifying this by an informal appeal to the central limit theorem (CLT). This paper puts this heuristic appeal on firmer analytical grounds. We show that the denoising regret of the nonparametric maximum likelihood estimator (NPMLE) and related sieve methods is controlled by the rate attained under exact normality, plus a term reflecting the quality of the CLT approximation. The CLT need only hold marginally for each coordinate, and moreover only on average, without needing high-dimensional normal approximations. We identify two asymptotic regimes in which the normal approximation is adequate and the empirical Bayesian prior remains informative, and we show that our guarantees are robust to dependence and to variance estimation.

Communication costs are a major bottleneck in distributed learning and first-order optimization. A common approach to alleviate this issue is to compress the gradient information exchanged between agents. However, such compression typically degrades the convergence guarantees of gradient-based methods. Error feedback mechanisms provide a simple and computationally cheap remedy for this issue, but numerous variants have been proposed, and their relative performance remains poorly understood. This paper provides tight convergence analyses for two of the main error-feedback algorithms from the literature, the classic Error Feedback method (EF) and Error Feedback 21 (EF21), by identifying optimal step-size choices and constructing optimal Lyapunov functions tailored to each method. The results hold independently of the number of agents and recover the known best guarantees possible in the single-agent regime.

The crucial feature of resurgence theory is the ambiguity of non-perturbative behavior, reflected either in the different choices of integration contours or in the existence of several solutions to Ward identities. This is well illustrated by considering exactly solvable models, of which the prominent example is Chern-Simons theory. Its important chapter, which should have a direct generalization to arbitrary Yang-Mills, is the consideration of Wilson averages in the double-scaling limit of large representation and small coupling. For historical reasons, we call it a Kashaev limit. It possesses a natural interpretation in terms of quasiclassical/WKB approximation, which is, however, somewhat peculiar and thus sheds new light on the old story. The crucial point is the appearance of Alexander polynomials $\Delta$ in two seemingly opposite roles: the classical $A$-polynomials have common roots with $\Delta$, while Jones polynomials tend to $\Delta^{-1}$ in the perturbative expansion. The consistency is provided by the peculiar form of the quantum $A$-polynomial, and the resolution of the puzzle is the co-existence of two different branches (phases) in the quasiclassical limit – with non-trivial and with vanishing classical actions. The first leads to classical $A$-polynomials and hyperbolic volumes, the second – to inverse Alexanders.

We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,\alpha}\cap L^2$, where $\alpha\in\left[\frac{1}{3},1\right)$. In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the $C^{1,\frac{1}{3}}$ threshold in this setting. Motivated by this, we define Lagrangian classes of coherent conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when $\alpha\geq\frac{1}{3}$. When $\alpha>\frac{1}{3}$, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint $\alpha=\frac{1}{3}$, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under transverse cusp-tail, longitudinal, off-clock, Dini, and suitable geometric coherence hypotheses, this singular value cannot collapse in finite time. In particular, we also show that the class of such coherent solutions includes the smooth ones locally in time. In the on-axis case, the criterion reduces to the scalar clock inequality $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3\alpha}$, which rules out Shkoller-type clock collapse for $\alpha\geq\frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case $\alpha>\frac{1}{3}$.

We study the global evolution problem for the Einstein equations under T2 symmetry on T3, allowing vacuum, scalar-field, and compressible-fluid matter models, governed by a general equation of state including isothermal and polytropic fluids. Under this symmetry, we obtain the first non-perturbative, global existence and stability theory with connection coefficients being merely square-integrable, which allows both impulsive gravitational waves and shock waves. In areal gauge, we introduce new fluid and geometric variables and reformulate the Einstein-Euler system as a first-order system of nonlinear balance laws with constraints and an entropy structure. The resulting formulation exhibits hyperbolicity, null forms, entropy currents, div-curl structure, maximum principles, and spacetime estimates. This leads to a notion of tame Einstein-Euler flow for which the essential geometric and fluid variables are square-integrable (finite energy), and the secondary variables are absolutely continuous (or, more generally, of bounded variation). In this non-perturbative and weak regularity setting, the equations remain meaningful even when the Weyl curvature concentrates into Dirac masses along timelike hypersurfaces, and the Ricci curvature remains only integrable. Our main results are a global existence theorem for areal foliations, a nonlinear stability theorem for well-prepared initial data, and a nonlinear instability theorem for geometrically oscillatory data, the latter producing measure corrections to the stress energy tensor. In the future-contracting regime, the areal foliation reaches a geometric singularity where the volume of T3 spatial slices degenerates to zero. The areal function reaches zero generically in the non-vacuum Gowdy-symmetric and vacuum torus-symmetric cases. In the future-expanding regime, the areal foliation is complete.

In this paper, we study the minimax risk of detecting a sparse, elevated-mean Gaussian submatrix planted in a larger noisy matrix. When the planted submatrix has size $n\times n$ and the ambient matrix has size $N\times N$ with $N=n^{1+\alpha}$, the seminal work of Butucea and Ingster (2013) identifies the sharp detection boundary around which the risk converges to $0$ or $1$. We extend that zero-one theory by determining sharp risk asymptotics throughout a two-parameter phase diagram. Above the boundary, the risk decays stretched- or super-exponentially fast, and we determine its precise exponent. Below the boundary, where the risk tends to $1$, we obtain matching absolute-constant bounds for its polynomial rate of convergence. In both regimes, the asymptotics change across the line $\alpha+\delta=1/2$, where $\delta$ parametrizes the signed distance from the detection boundary. Finally, on the boundary, we show that the minimax risk converges to the nondegenerate constant $\frac12$ in a very sparse regime where $n$ remains fixed and $N\to\infty$. Each of these rates corresponds to the risk of a suitably calibrated scan or sum test, whence follow the upper bounds. The main problem is to derive the matching regime-specific lower bounds. The mixture likelihood ratio is an average rather than a product and thus its logarithm lacks the additive structure required by standard large deviation formulations. On the other hand, classical second-moment estimates fail to be sharp due to overlap-induced correlations. To address these issues, we develop two general lower-bound principles based on truncated mixture likelihoods, tailored to the target risks on both sides of the detection boundary. The same framework yields sharp risk asymptotics for sparse subtensors with equal side lengths and also applies to more general testing problems including the detection of rectangular subtensors.

This paper develops a functional theory for multi-target detection, where a compactly supported signal is recovered from a single noisy observation containing many unknown translations of the signal. Our formulation allows continuous, off-grid translations and correlated stationary Gaussian process noise, extending beyond the discrete, grid-aligned, white-noise models common in prior work. We analyze two uninitialized recovery algorithms based on autocorrelation analysis; in particular, both algorithms first estimate the signal's bispectrum via a debiased third-order empirical autocorrelation. The signal is then recovered from the estimated bispectrum using either a functional frequency marching scheme or a Kotlarski-type deconvolution formula. For both algorithms, we prove non-asymptotic recovery guarantees for compactly supported signals without bandlimiting assumptions. The resulting error bounds depend on the smoothness of the signal and the accuracy of bispectrum estimation, with the latter governed by the noise characteristics and the number of signal occurrences. Numerical experiments validate our theory and demonstrate accurate recovery in low-SNR regimes.

In this article, a one-off option for temporary service rate control for the M/M/1 queue is considered. After taking this option, during a single exponentially distributed period, two service rates are available for use. Once service rate control is lost, the system continues with a fixed service rate $\mu$. The objective is to minimise the sum of holding costs and service costs. We approximate the expected total saved cost by taking the one-off option, depending on the starting state or starting distribution. Using the Value Iteration algorithm with $M$-uniform geometric recurrence, we present methods to approximate the expected total saved future cost, as well as the expected total saved discounted future cost. Furthermore, we obtain theoretical results on the structure of optimal policies and strong Blackwell optimality. The paper is concluded by numerically applying the methods to various instances of the model.

Our purpose is to revisit the screw theory in light of the affine tensor formalism, introducing the co-momentum and momentum tensors. Our target-applications of the Euler-Poincaré equation are problems of mechanics such as the motion of the rigid body or the statics and the dynamics of Cosserat arches, in relation to the concept of momentum tensor. Using the framework of Ehresmann connections on the principal bundle of affine frames, we show that the Euler-Poincaré equation means that the momentum tensor is parallel-transported.

We consider the Boolean model of continuum percolation, where points are placed in $\mathbb{R}^d$ by a Poisson point process and pairs of points with distance at most 1 are connected by an edge. The time constant is the limiting ratio of the chemical distance (i.e. graph distance) to the Euclidean distance for pairs of distant connected points. Yao, Chen, and Guo established the existence of a time constant in the supercritical regime. We show that above the critical intensity, the time constant is a Lipschitz continuous function of the intensity. The proof adapts a recent argument of Can, Nakajima, and Nguyen to the continuous setting.

We introduce a stochastic interacting particle system in separable Hilbert spaces together with its associated mean-field formulation. The model is shown to retain the characteristic consensus-driven structure of classical Consensus-Based Optimization, while accounting for the analytical challenges of infinite-dimensional dynamics. We establish well-posedness of the proposed dynamics and analyze the associated consensus mechanism. Furthermore, we derive convergence guarantees under suitable assumptions on the objective functional, showing concentration of the dynamics toward the minimizer in the long-time regime. This extends the applicability of the method to a broad class of infinite-dimensional optimization problems. In addition, we study the corresponding finite-particle system relevant for numerical implementation and propose a practical algorithm.