Papers

We study the well-posedness for the intermediate nonlinear Schrödinger equation (INLS) with periodic boundary conditions. Using a gauge transform, we obtain large data local well-posedness in $H^{s}(\mathbb{T})$ for any $s\geq \frac 12$. We extend this result to global well-posedness under a small $L^2$-norm constraint by exploiting the complete integrability of the continuum Calogero-Moser equation (CCM). We also establish additional results such as the unconditional well-posedness in the energy space and the convergence of solutions to INLS to those of CCM in the infinite-depth limit.

A distance-k dominating set of a graph is a set of vertices such that every vertex lies within distance k of some vertex in the set; its minimum size is the distance-k domination number gamma_k. We study gamma_k for triangular matchstick graphs T_d. Harris et al. (2020) claimed two upper bounds for the cases k=1 and k=2. These bounds are shown to be incorrect and are replaced by a corrected general upper bound for gamma_k(T_d) obtained by refining their tiling method. For k=1 and d=7q+beta, where 0 <= beta <= 6, we further sharpen this construction and prove gamma(T_{7q+beta}) <= 3.5q^2+(beta+4.5)q+(beta-1). Finally, we determine the radius of T_d as r_d = ceil(2d/3), which implies gamma_k(T_d)=1 whenever k >= r_d.

Recent work has analyzed the convergence of first-order methods under non-uniform smoothness assumptions that better model the loss landscape in machine learning tasks. We generalize this assumption to objectives whose curvature is an affine function of the objective value. This property is satisfied by a broad class of problems, including logistic regression, generalized linear models with a logistic link function, softmax policy gradient in reinforcement learning, and a class of neural networks. Under this assumption and gradient domination conditions, we establish a general convergence rate for the steepest descent method, and deterministic, diagonal variants of RMSProp and Adam. Our results imply that for logistic regression on separable data and the softmax policy gradient objective, sign GD converges linearly and is provably faster than GD. Furthermore, we show that for a class of two-layer neural networks on separable data, RMSProp and Adam can converge at a linear rate with a constant step-size and momentum parameter. Finally, we present a lower bound demonstrating that, under our assumption, RMSProp and Adam are provably faster than AdaGrad, AMSGrad, gradient descent, and heavy-ball momentum.

In this paper, we review the theory of the Fourier–Bessel transform. The Fourier–Bessel transform is an integral transform on the half-line that generalizes the Fourier cosine transform and is dependent upon a parameter $\nu > -1$. Many of the standard results for the Fourier–Bessel transform have become folklore results and are difficult to find single sources for or predate modern functional analysis and instead rely on hard analysis techniques that miss the elegance of more modern machinery. This review aims to centralize much of the basic theory while providing some new proofs and approaches that can be applied to other integral transforms beyond the Fourier and Fourier–Bessel transform. Particular care is taken to illustrate the distinctions in the theory of the Fourier–Bessel transform when $-1 < \nu < -\frac{1}{2}$ and $-\frac{1}{2} < \nu$.

We present a method to determine admissible configurations that lead to relative equilibria in the full gravitational N-body problem. The method exploits the SO(3) symmetry by working with its fundamental invariants and does not rely on restrictive assumptions about mass distributions or on truncating the gravitational potential. As a result, it yields necessary conditions that apply to arbitrary rigid-body configurations. This work is presented in two parts. In the first (this) part, we assume every body is axisymmetric to give a clear, pedagogical exposition of the approach; in a forthcoming paper, we remove that restriction and treat general triaxial bodies (the method remains valid, but the number of variables increases). As an illustration, we systematically recover all known relative-equilibrium families for the two-body problem consisting of one sphere and one axisymmetric body, confirm earlier results in the literature, and derive new necessary conditions for certain configurations, including additional constraints for the arrow and non-Lagrangian types. In the special case of a sphere and an axisymmetric body, we also obtain sufficient conditions for relative equilibria under a monotonicity assumption on the potential.

We study high-dimensional inference problems with tensor-structured data and establish conditions under which their free energy can be approximated by that of a Gaussian comparison model. Our framework applies to models with independent observations and mismatch between the data-generating distribution and the statistical model. The results extend prior work beyond matrix settings and accommodate scaling regimes where the model parameters depend on the dimension. A key technical contribution is the use of generic chaining to control remainder terms arising from likelihood expansions over tensor-structured parameter spaces. As an application, we establish free energy universality for binary hypergraph models under the minimal assumption of diverging average degree, showing that their asymptotic behavior coincides with that of a Gaussian tensor model, even under model mismatch.

In this paper we study dicritical foliations having a family of cusps as invariant curves. We give conditions to assure that all the curves of the family have the same semimodule of differential values. We also give an expression to compute the conductor of a cuspidal semimodule.

Many biophysical search processes employ searchers which enter or "immigrate" into the domain progressively over time. Existing search time estimates can become unphysical for fast immigration since they imply that searchers move at infinite speed. In this paper, we investigate search times of immigrating searchers that move at bounded speed. In the fast immigration limit, we determine the full probability distribution and all the moments of the $k$th search time in terms of the early time probability distribution of a single searcher. We apply these rigorous mathematical results to several canonical models of stochastic search. We further analyze different models of "diffusion" and use these results to investigate when and how the minutiae of searcher dynamics affect search times. We compare our theory to numerical simulations.

Robust and efficient state estimation is crucial for perception, navigation, and control in robotics. State estimation problems are conveniently modeled using the factor-graph framework as enabled by modern software packages such as GTSAM or g2o. However, the standard solvers included in such frameworks are local and may converge to poor local minima, posing significant safety concerns. Conversely, techniques based on convex relaxations have been shown to provide a means of globally solving or certifying many state estimation problems. However, these relaxations 1) often require substantial effort to formulate, and 2) may incur significantly higher cost compared to efficient local solvers, as they require solving a large semidefinite program (SDP). In this work, we address both shortcomings by 1) creating a new procedure within the GTSAM framework for automatically constructing convex SDP relaxations for any factor graphs with common factor and variable types, and by 2) exploiting the Bayes tree constructions native to GTSAM to decompose the SDP problem, leading to significant speedup in solver time for chordally sparse problems. We demonstrate the favorable scaling of this structure-exploiting global estimator compared to standard local solvers for two case studies: A 3D pose-graph SLAM problem with a ring factor graph and a 2D localization problem with a chain factor graph. The software framework is available at https://github.com/borglab/gtsam.

The purpose of this paper is to define normal Gaussian variables in the setting of hyperbolic probabilities, and introduce an associated Brownian motion, when both the index and the values of the process lie in the real algebra $\mathbb{H}$ of hyperbolic numbers. In Hida's white noise space, we construct two probability measures (say $P_1$ and $P_2$), and associate to them two families of $N(0,1)$ variables $(Z_n)_{n\in\mathbb N_0}$ (independent with respect to $P_1$) and $(W_n)_{n\in\mathbb N_0}$ (independent with respect to $P_2$). An important feature is that the $Z_n$ and $W_m$ need not be mutually independent either with respect to $P_1$ or $P_2$. An hyperbolic normal Gaussian variable is constructed (in non-degenerate cases) from two classical Gaussian variables and the hyperbolic Brownian motion is, in general, composed from two copies of the classical Brownian motion. Using the associated Gelfand triples we also compute the derivative of the hyperbolic Brownian motion as a stochastic distribution. The argument extends to the $\mathbb{H}$-valued fractional Brownian motion, and more generally to a wide family of $\mathbb{H}$-valued stationary-increment second order processes.

This paper develops a regression framework for analyzing integrated conditional distribution and quantile functions. The proposed method, termed rectified linear unit (ReLU) regression, projects the ReLU-transformed outcome onto covariates and admits a closed-form estimator. Its population regression function is the best linear approximation to the integrated conditional distribution function of the outcome, and the corresponding convex conjugate, obtained via the Legendre-Fenchel transform, approximates the integrated conditional quantile function. Both the regression and its conjugate require only mild distributional assumptions and accommodate non-continuous outcomes. We establish the asymptotic distribution of the estimator and develop inference for the conjugate functional via the delta method for Hadamard directionally differentiable maps. Building on these results, we establish identification and inference for average quantile treatment effects over arbitrary subintervals of probability levels. This broadens the set of distributional parameters available to empirical work.

In 2007, Adamczewski and Bugeaud introduced the notion of the Diophantine exponent of an infinite word as a quantitative measure of repetition, leading to new transcendence results for real numbers whose expansions in an integer base are sufficiently simple. In the present article, we introduce the refined Diophantine exponent, which detects weaker forms of repetition while preserving the full strength of the classical approach. This new exponent applies in situations where repetition is partially obscured by some form of noise. Related ideas already appear in the work of Corvaja and Zannier in 2002 and, more recently, in the works of Kebis, Luca, Ouaknine, Scoones, and Worrell. Our approach provides a unified framework that recovers and extends these results, as well as those of Adamczewski and Bugeaud. We also develop quantitative refinements of this method, leading to results about transcendence measures. The recent breakthrough of Bell, Diller, and Jonsson in the context of algebraic dynamics is partly based on a similar idea, which also served as a motivation for the present work.

We consider the substitution subshift generated by the Thue-Morse substitution $0\to01$, $1\to10$. We prove that the topological full group of the subshift contains a subgroup of intermediate growth. Namely, one group from the family known as the Grigorchuk groups embeds into this group. To obtain our main result, we prove an embedding theorem for topological full groups, and also develop a technique to prove isomorphism of groups using the Schreier graphs.

The intersection cut framework is a versatile tool for generating valid inequalities in optimization. Its main ingredients are so-called $S$-free sets: convex sets whose interiors do not intersect a given set $S$. Among these, inclusion-wise maximal $S$-free sets are particularly important, as they yield the strongest intersection cuts. In the integer programming setting, maximal lattice-free sets are well studied and admit explicit characterizations. In the quadratic optimization context, Muñoz, Paat, and Serrano (2025) characterized maximal $S$-free sets when $S$ is defined by a homogeneous quadratic inequality. In this work, we characterize maximal $S$-free sets when $S$ is defined by an inhomogeneous quadratic inequality. As in the homogeneous case, our characterization is built using non-expansive functions. Together with the results in the homogeneous case, our results complete a characterization of $every$ maximal quadratic-free set via non-expansive functions.

Randomized sketching is a central tool for compressing large-scale optimization problems while preserving accuracy. In particular, sketches that are based on structured matrices, such as the Hadamard matrix, can be applied efficiently and often yield solutions that approximate those of the original problem at much lower computational cost. In differential privacy (DP), Gaussian sketching has been used to solve DP linear regression, beginning with \citet{sheffet2017differentially, sheffet2019old} and later refined by \citet{lev2025gaussianmix, lev2026near}. However, although these methods achieve strong utility guarantees, they usually do not improve runtime over classical DP approaches. In this work, we introduce a new DP sketching mechanism based on fast transforms, which, in certain cases, matches the runtime of classical fast sketching methods. We prove state-of-the-art privacy guarantees for this mechanism and show that, in favorable regimes, they match those of the Gaussian sketch up to a constant factor. As an application, we combine this mechanism with recent sketch-based methods for DP linear regression to obtain a new algorithm with strong utility and improved runtime. We establish privacy and accuracy guarantees for this algorithm, yielding, to the best of our knowledge, the first fast method for DP ordinary least squares.

We establish a sharp three-particle fractional Hardy inequality for the Laplacian of order $s\in(0,1)$ in dimension $d\geq 4-2s$ (Theorem 1.1), involving an explicit intrinsically three-body interaction potential $V_{s,3}$. The inequality holds with the optimal two-particle fractional Hardy constant $C_{fH}(d,s)$, which is shown to be sharp relative to the fixed potential $V_{s,3}$. This potential $V_{s,3}$ strictly dominates the standard pairwise Coulomb-type interaction and captures genuine three-body effects. As a consequence, we derive a nontrivial many-particle fractional Hardy inequality for $N\geq 3$, and, in the regime $N>d+1$, obtain an improved Coulomb-type inequality with a strictly larger constant, agreeing in spirit with the results of Hoffmann-Ostenhof et al. [M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom, "Many-particle Hardy inequalities", J. Lond. Math. Soc. 77 (2008), no. 2, 99-115] and Lundholm [D. Lundholm, "Geometric extensions of many-particle Hardy inequalities", J. Phys. A: Math. Theor. 48 (2015), no. 17, 175203]. The proof relies on a fractional ground-state representation method adapted to three-particle interactions, combined with an explicit evaluation of the resulting nonlocal interaction term. This evaluation is achieved through a new singular integral identity of Selberg-type (Theorem 1.2), extending the three-fold formula of Grafakos-Morpurgo [L. Grafakos and C. Morpurgo, "A Selberg integral formula and applications", Pacific J. Math. 191 (1999), no. 1, 85-94] beyond the radial setting. This identity provides the analytic mechanism underlying the emergence of the three-body potential and may be of independent interest in harmonic analysis.

Unlike classical optimal transport, weak transport costs depend nonlinearly on the conditional law of couplings. This feature is essential in problems involving barycenter, conditional moments, and martingale-type constraints. Meanwhile, such conditional dependence makes ordinary Wasserstein geometry insufficient and calls instead for an adapted Wasserstein viewpoint. In this paper, we investigate the entropy-regularized weak optimal transport via gradient flows in adapted Wasserstein space. We derive, from the formal tangent structure of adapted Wasserstein space and the projection onto the set of couplings with prescribed marginals, a coupled McKean–Vlasov SDE. A novel and subtle term is a projection that, at each $Y$-location, averages a weak-transport force that already depends on the conditional law of $Y$ given $X$, thereby preserving marginals while retaining the nonlinear weak-transport structure. Under mild integrability and regularity assumptions, we prove weak existence and uniqueness in law for this projected McKean–Vlasov equation. We then prove that the flow converges, in the adapted Wasserstein topology, to the unique minimizer of the entropic weak optimal transport problem. We also describe a particle approximation and illustrate the dynamics on optimal transport and martingale optimal transport examples.

This work presents a conforming finite-element scheme for non-isothermal phase-field systems coupled to the incompressible Navier-Stokes equations. The proposed numerical scheme preserves entropy production and total energy conservation exactly by variable transformations using entropy as main variable instead of temperature. Convergence tests in space are conducted, and representative examples are provided to demonstrate the scheme's effectiveness.

Given a BV theory on a space of fields split into two subspaces ("infrared" and "ultraviolet"), one has the BV pushforward map $P_*$, sending observables to observables of the effective theory on the infrared space. This note proves that $P_*$ is a quasi-isomorphism of BV complexes, by realizing it as a part of a strong deformation retraction constructed using the homological perturbation lemma. Two proofs are given: (i) comparing Feynman diagrams for $P_*$ with "cable diagrams" arising from homological perturbation theory and (ii) using topological quantum mechanics. This construction gives a formula for the quasi-inverse $i_\mathrm{int}$ of $P_*$ - the map lifting observables of the effective theory to the full theory. The topological quantum mechanics perspective - and its realization as an AKSZ theory - allows one to write $i_\mathrm{int}$ as a path integral (realizing cable diagrams for $i_\mathrm{int}$ as Feynman diagrams) and to study its classical limit.

I present diffusion models as part of a family of machine learning techniques that withhold information from a model's input and train it to guess the withheld information. I argue that diffusion's destroying approach to withholding is more flexible than typical hand-crafted information withholding techniques, providing a rich training playground that could be advantageous in some settings, notably data-scarce ones. I then address subtle issues that may arise when porting reinforcement learning techniques to the diffusion context, and wonder how such exploration problems could be addressed in more diffusion-native ways. I do not have definitive answers, but I do point my fingers in directions I deem interesting. A tutorial follows this thesis, expanding on the destroy-then-generate perspective. A novel kind of probabilistic graphical models is introduced to facilitate the tutorial's exposition.

In this paper, we establish the expected asymptotic formula for the number of rational lines on a diagonal cubic hypersurface in 18 or more variables, improving on recent work of the second author. This is achieved via a refined mean value estimate for minor arcs that non-trivially exploits a shifting variables argument in both underlying dimensions.

Motivated by the open problem of whether all one-relator groups have decidable Diophantine problem, in this paper we prove a collection of undecidability results about the Diophantine problem for several families of groups that are close to one-relator groups in various ways. We prove that there is a generalised Baumslag–Solitar group with an undecidable Diophantine problem. Using our example we show there is a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group. Also, we prove that there is a one-relator product of cyclic groups with an undecidable Diophantine problem. In addition, we show that there there is a one-relator group $G$, with a single fixed finite rank free subgroup $H$, such that the Diophantine problem for $G$ with $H$-constraints is undecidable. The related open question of whether there is a free-by-cyclic group with undecidable Diophantine problem is also discussed, and we prove that there is a free-by-free group of the form $F_3 \rtimes F_2$ with an undecidable Diophantine problem.

Bilevel optimization involves an upper-level and a lower-level decision maker. The lower-level optimization problem is nested within the constraints of the upper-level one. A point is said to be admissible for the bilevel problem if it satisfies all constraints and is optimal for the lower-level decision-maker. Bilevel derivative-free optimization (BL-DFO) algorithms address bilevel optimization problems in which the upper-level is solved using a derivative-free optimization method, and an inexact method it used to solve the lower- level. In this context, existing BL-DFO benchmarking techniques often do not rigorously validate the admissibility of proposed solutions, and do not adequately account for the computational effort deployed by the upper- and lower-level solvers. This work proposes a benchmarking methodology for BL-DFO algorithms. A post-optimization procedure, named the refereeing procedure, is introduced to discard non-admissible points and ensures a fair comparison between the algorithms. The computational effort deployed by upper- and lower-level solvers is also taken into account into the overall computational cost. Numerical experiments illustrate the benchmarking methodology.

For an abelian variety $X$ and $\alpha \in Br(X)$, we propose a new invariance $Ind_{SH}(\alpha)$ that refines the known period index relations. It is closely related to the geometry of $\mathcal{X}$, the $\mathbb{G}_m$-gerbe over $X$ that corresponds to $\alpha$: we study the minimal trivializing isogenies for $\mathcal{X}$ via its $\mu_n$-lifts and the $1-$twisted semi-homogeneous vector bundles on $\mathcal{X}$. As an application, we show that the period index conjecture holds true for products of elliptic curves of any dimension.

Rational functions make sense over any commutative ring, as long as the denominators are invertible. When there is no subtraction involved, they even apply over semirings (rings without subtraction). It is particularly worthwhile to evaluate them over the *tropical semiring*, in which the roles of addition and multiplication are played by maxima and addition. Over this semiring, algebraic results often acquire combinatorial meaning. We give a few examples.

We present a conservation-based feedback-circuit decomposition specifically for general linear forced systems. In a role parallel to that of eigenvalues and eigenvectors for initial-value problems, the complete set of independent intrinsic circuit gains and their associated forcing-transformation vectors provide a complete analytical representation of both transient and equilibrium forced solutions. The sign of intrinsic circuit gains determines whether successive feedback cycles exhibit monotonic or oscillatory convergence to transformed forcing, while the forcing-transformation vectors determine the structure of transformed forcing. The exact transient and equilibrium solutions are represented analytically through the convergence of the finite-cycle forcing-transformation kernel to the equilibrium forcing-transformation kernel, which is guaranteed regardless of whether the magnitudes of circuit gains exceed one or unstable modes exist in the system. The feedback-circuit decomposition provides a new generic foundational mathematical tool for understanding, predicting, and controlling forced responses in a broad range of coupled linear systems across science and engineering.

It is well-known that first-order methods can offer accelerated convergence rates in the presence of growth structures. Restarting schemes provide a general tool for such speed-ups. These schemes typically either require unrealistic problem knowledge, incur logarithmic overhead factors in oracle complexity, and/or have a nontrivial initial burn-in phase. We present a parameter-free approach for restarting any first-order method, avoiding these three drawbacks. Our approach dynamically deploys parallel instances of a given first-order method communicating progress in the style of Renegar and Grimmer. Our optimized scheme avoids expensive burn-ins and only requires $O(\log\log(1/\epsilon))$ parallel processes when the accelerated rate is sublinear.

For the Sparre Andersen non-life insurance model with investments in an arbitrary Lévy process, we establish the exact power-law asymptotics of the ruin probability $\Psi(u)\sim C^* u^{-\beta}$ as $u\to\infty$ with a positive finite constant $C^*$; the exponent $\beta$ is the Cramér root of the Laplace exponent of the Lévy process describing the logarithm of the risky asset price. This strengthens previously known two-sided estimates of the order – the existence of an exact limit had remained an open question. The proof combines a reduction to discrete time, the one-dimensional Kesten-Goldie theorem for the stationary measure of the associated affine recursion, and Goldie's result on the asymptotics of the supremum of a perpetuity.

We provide analysis for the posterior distribution and expectation of $(p_1, p_2)$ where $Y|p_1,p_2 \sim \hbox{Binomial}(n, p_1 p_2)$ and $ (p_1, p_2)$ is uniformly distributed on the unit square $[0,1]^2$. We exhibit interesting expressions in terms of a truncated Beta distribution, a finite mixture of Beta distributions and harmonic numbers, and derive a simple large sample size $n$ approximation for the posterior expectations $\mathbb{E}(p_i|y)$ as well as for the normalization constant in the posterior joint density.

Let $G$ be an arbitrary group and let $\mathbb{F}$ be a finite field. In this paper, we determine bases for the $T_G$-ideals of graded polynomial identities of the algebra $M_2(\mathbb{F})$ for all possible $G$-gradings. The bases obtained consist of finitely many non-trivial graded identities, and are finite whenever $G$ is finite.