We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators $f^1,\dots,f^k$ to the drift $f$ with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate $f^k$ and many evaluations of the less costly $f^1,\dots,f^{k-1}$. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires $ε^{-γ}$ compute to be $ε$-approximated for some $γ>2$, then ML-EM $ε$-approximates the solution of the SDE with $ε^{-γ}$ compute, improving over the traditional EM rate of $ε^{-γ-1}$. In other terms it allows us to solve the SDE at the same cost as a single evaluation of the drift. In the context of diffusion models, the different levels $f^{1},\dots,f^{k}$ are obtained by training UNets of increasing sizes, and ML-EM allows us to perform sampling with the equivalent of a single evaluation of the largest UNet. Our numerical experiments confirm our theory: we obtain up to fourfold speedups for image generation on the CelebA dataset downscaled to 64x64, where we measure a $γ\approx2.5$. Given that this is a polynomial speedup, we expect even stronger speedups in practical applications which involve orders of magnitude larger networks.

The freight industry is undergoing a digital revolution, with an ever-growing volume of transactions being facilitated by digital marketplaces. A core capability of these marketplaces is the fulfillment of demand for truckload movements (loads) by procuring the services of carriers who execute them. Notably, these services are procured both through long-term contracts, where carriers commit capacity to execute loads (e.g., contracted fleet of drivers or lane-level commitments), and through short-term spot marketplaces, where carriers can agree to move individual loads for the offered price. This naturally couples two canonical problems of the transportation industry: contract assignment and spot pricing. In this work, we model and analyze the problem of coordinating long-term contract supply and short-term spot supply to minimize total procurement costs. We develop a Dual Frank Wolfe algorithm to compute shadow prices which allow the spot pricing policy to account for the committed contract capacity. We show that our algorithm achieves small relative regret against the optimal -- but intractable -- dynamic programming benchmark when the size of the market is large. Importantly, our Dual Frank Wolfe algorithm is computationally efficient, modular, and only requires oracle access to spot-pricing protocols, making it ideal for large-scale markets. Finally, we evaluate our algorithm on semi-synthetic data from a major Digital Freight Marketplace, and find that it yields significant savings ($\approx 10\%$) compared to a popular status-quo method.

We show that the homology of strata of abelian differentials stabilizes in a range where the number of simple zeros is large relative to the homological degree. In this range, we show that the rational cohomology agrees with the restriction of the tautological classes to the stratum, and that the rational Picard group is trivial for unprojectivized strata. Our proof method is to develop an $h$-principle for these strata, valid in a range of homological degrees that increases with the number of simple zeros. The same approach also applies to higher order differentials.

Given any compact connected matrix Lie group $G$ and any lattice dimension $d\ge 2$, we construct a massive Gaussian scaling limit for the $G$-valued lattice Yang-Mills-Higgs theory in the "complete breakdown of symmetry" regime. This limit arises as the lattice spacing tends to zero and the (inverse) gauge coupling constant tends to infinity sufficiently fast, causing the theory to "abelianize" and yield a Gaussian limit. This complements a recent work by Chatterjee (arXiv:2401.10507), which obtained a similar scaling limit in the special case $G= SU(2)$.

The unconditional, i.e. without assuming validity of RH, sharp limit relationship (as p tends to infinity) is found between the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals and maximal values of Gronwall numbers G(N) among all integers whose greatest prime factor is p and which are divided by any prime q<p. The proofs are based on the properties of G(N) studied in previous author's preprints.

In this paper, we analyze $m$-dimensional ($m$D) convolutional codes with finite support, viewed as a natural generalization of one-dimensional (1D) convolutional codes to higher dimensions. An $m$D convolutional code with finite support consists of codewords with compact support indexed in $\mathbb{N}^m$ and taking values in $\mathbb{F}_{q}[z_1,\ldots,z_m]^n$, where $\mathbb{F}_{q}$ is a finite field with $q$ elements. We recall a natural upper bound on the free distance of an $m$D convolutional code with rate $k/n$ and degree~$δ$, called $m$D generalized Singleton bound. Codes that attain this bound are called maximum distance separable (MDS) $m$D convolutional codes. As our main result, we develop new constructions of MDS $m$D convolutional codes based on superregularity of certain matrices. Our results include the construction of new families of MDS $mD$ convolutional codes of rate $1/n$, relying on generator matrices with specific row degree conditions. These constructions significantly expand the set of known constructions of MDS $m$D convolutional codes.

We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetildeΘ(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

Nonlinear Schwarz methods are a type of nonlinear domain decomposition method used as an alternative to Newton's method for solving discretized nonlinear partial differential equations. In this article, the first parallel implementation of a two-level nonlinear Schwarz method leveraging the GDSW-type coarse spaces from the Fast and Robust Overlapping Schwarz (FROSch) framework in Trilinos is presented. This framework supports both additive and hybrid two-level nonlinear Schwarz methods and makes use of modifications to the coarse spaces constructed by FROSch to further enhance the robustness and convergence speed of the methods. Efficiency and excellent parallel performance of the software framework are demonstrated by applying it to two challenging nonlinear problems: the two-dimensional lid-driven cavity problem at high Reynolds numbers, and a Neo-Hookean beam deformation problem. The results show that two-level nonlinear Schwarz methods scale exceptionally well up to 9\,000 subdomains and are more robust than standard Newton-Krylov-Schwarz solvers for the considered Navier-Stokes problems with high Reynolds numbers or, respectively, for the nonlinear elasticity problems and large deformations. The new parallel implementation provides a foundation for future research in scalable nonlinear domain decomposition methods and demonstrates the practical viability of nonlinear Schwarz techniques for large-scale simulations.

Barren plateaus present a major challenge in the training of variational quantum algorithms (VQAs), particularly for large-scale discretizations of nonlinear partial differential equations. In this work, we introduce a domain decomposition framework to mitigate barren plateaus by localizing the cost functional. Our strategy is based on partitioning the spatial domain into overlapping subdomains, each associated with a localized parameterized quantum circuit and measurement operator. Numerical results for the time-independent Gross-Pitaevskii equation show that the domain-decomposed formulation, allowing subdomain iterations to be interleaved with optimization iterations, exhibits improved solution accuracy and stable optimization compared to the global VQA formulation.

Given a $\vartheta$-Anosov representation into a real reductive group $G$, we construct a natural resonance spectrum associated with the representation. This spectrum is a complex analytic variety of codimension $1$ in $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$, the complexified dual of the split component of the associated Levi group $L_\vartheta < G$. We reinterpret several objects from the theory of Anosov representations within this spectral framework and investigate, in higher rank, questions that are classically related to Ruelle-Pollicott theory in the rank-one setting. In particular, the ``leading resonance'' -- which is now a hypersurface -- is identified with the critical hypersurface of the representation. As a corollary of our work, we prove that the zeta functions and Poincaré series associated with Anosov representations admit a meromorphic extension to $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$. We also establish sharp mixing estimates for the refraction flow under a Diophantine condition on the representation. Most of our results concerning Anosov representations are obtained as a byproduct of a general theory of free Abelian cocycles over hyperbolic flows. This article is intended as a foundational work toward more advanced results such as higher-rank quantum/classical correspondence, the detection of topological invariants of representations via the value at zero of Poincaré series or the order of vanishing of zeta functions, sharp counting results for the Lyapunov spectrum, etc.

Convolutional codes are a class of error-correcting codes that performs very well over erasure channels with low delay requirements. In particular, Maximum Distance Profile (MDP) convolutional codes, which are defined to have optimal column distances, are able to correct a maximal number of erasures in decoding windows of fixed sizes. However, the required field size in the known constructions for MDP convolutional codes increases rapidly with the code parameters. On the other hand, if the code parameters are small, larger bursts of erasures cannot be corrected. In this paper, we present a new class of convolutional codes, which we call Pseudo-MDP convolutional codes. By definition these codes can correct large bursts of erasures within a prescribed time-delay and still keep part of the advantageous properties of MDP convolutional codes, in the sense that we require some but not all column distances to be optimal. This release in the condition on the column distances allows us to construct Pseudo-MDP convolutional codes over fields of smaller size than those required for MDP convolutional codes with the same code parameters.

A conjecture of Verstraëte states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell = 4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

We study the linear-quadratic optimal control problem for infinite-dimensional dissipative systems with possibly indefinite cost functional. Under the assumption that a storage function exists, we show that this indefinite optimal control problem is equivalent to a linear-quadratic optimal control problem with a nonnegative cost functional. We establish the relationship between the corresponding value functions and present the associated operator Lur'e equation. Finally, we illustrate our results with several examples.

We prove Conjecture~2 of Bondarenko, Ortega-Cerdà, Radchenko, and Seip for the three-term recurrence attached to the Hörmander--Bernhardsson extremal function $φ$. More precisely, define \[ \widetilde u_{-1}=0,\qquad \widetilde u_0=1, \] and \[ \widetilde u_{n+1} = \frac{4n+2}{n+1}\bigl(n(n+1)-λ\bigr)\widetilde u_n + \frac{4n}{n+1}x\,\widetilde u_{n-1}. \] Then \[ \widetilde u_n(x,λ)\in\mathbb Z[x,λ] \qquad(n\ge0). \] The proof is a determinant comparison in the scaled Legendre basis. After sign reversal and central-binomial normalization, the recurrence becomes exactly the continuant recurrence of a finite tridiagonal compression. In particular, if $T_n(a,λ)$ denotes the $n$th BOCRS tridiagonal truncation, then \[ \widetilde u_{n+1}(a^2,λ)=\binom{2n+2}{n+1}\det T_n(a,λ). \] As consequences, we derive that \[ \left(\fracπ{4C}\right)^2 \quad\text{and}\quad -\frac{L_τ(1)}{2C} \] are not simultaneously rational, where \(C\) is the sharp point-evaluation constant for $PW^1$, $\pmτ_n$ are the nonzero zeros of $φ$, and $ L_τ(1)=\sum_{n\ge1}\frac{(-1)^n}{τ_n}.$ Finally, if we write $φ(z)=\sum_{n\ge0}c_n z^{2n},$ then \[ c_n\in C^n\,\mathbb Z[π^2,C,L_τ(1)] \qquad(n\ge0). \]

There is a well understood way of generating random coverings of a fixed manifold by sampling homomorphisms from the fundamental group of this manifold into the symmetric group. We prove a central limit theorem for the number of connected components of these random coverings when the fundamental group is nilpotent. This provides a nonabelian generalization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least two. Our result relies on the work of du Sautoy and Grunewald on the subgroup growth zeta functions of nilpotent groups, and on Delange's generalization of the Wiener-Ikehara Tauberian theorem.

We study an inverse problem for the viscoacoustic wave equation, an integro-differential model describing wave propagation in viscoacoustic media with memory in the leading order term. The medium is characterized by a spatially varying sound speed and a space-time dependent memory kernel. Assuming that waves are generated by sources supported outside the region of interest, we consider exterior measurements encoded by the source-to-solution map. To study this inverse problem, we construct solutions concentrating near fixed geodesics and establish a corresponding propagation of singularities result for the semiclassical wave front set. These results are valid without any restriction on the underlying sound speed. Then, under certain geometric conditions, we prove that the exterior data uniquely determine not just the sound speed inside the domain but also all time derivatives at zero of the memory kernel. This involves a reduction to the lens rigidity and geodesic ray transform inverse problems. As an application, we establish uniqueness for the recovery of variable parameters in the extended Maxwell model.

While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $\mathbb{R}^D$. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness $α$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{α+1-δ}{2α+d}}$ for arbitrarily small $δ> 0$, demonstrating that the generative algorithm fully adapts to the intrinsic dimension $d$. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.

Uniform laws of large numbers form a cornerstone of Vapnik--Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying distribution that are compatible with the product structure. Specifically, we assume that the distribution is absolutely continuous with respect to the product of its marginals, a condition that captures many natural settings, including product distributions, sparse mixtures of product distributions, distributions with low mutual information, and more. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the linear VC dimension of the family is finite. The linear VC dimension is defined as the maximum size of a shattered set that lies on an axis-parallel line, namely, a set of vectors that agree on all but at most one coordinate. This dimension is always at most the classical VC dimension, yet it can be arbitrarily smaller. For instance, the family of convex sets in $\mathbb{R}^d$ has linear VC dimension $2$, while its VC dimension is infinite already for $d\ge 2$. Our proofs rely on estimator that departs substantially from the standard empirical mean estimator and exhibits more intricate structure. We show that such deviations from the standard empirical mean estimator are unavoidable in this setting. Throughout the paper, we propose several open questions, with a particular focus on quantitative sample complexity bounds.

We study cyclic adjoint modules arising from the relative locally finite part of the adjoint action of a quantum Levi subalgebra on a quantized enveloping algebra. We analyze the realization of irreducible modules inside the quantized enveloping algebra via cyclic generators and describe embeddings of a fixed type. This leads to a natural map to isomorphism classes, whose fibers reflect the non-uniqueness of such realizations. We further introduce a partial order on cyclic adjoint modules and relate its minimal elements to irreducible submodules. In addition, we show that every cyclic adjoint module is generated by finitely many irreducible submodules.

We study a variational problem modeling equilibrium configurations of charged liquid droplets resting on a surface under a convexity constraint. In the two-dimensional case with Coulomb interactions, we establish the validity of Young's law for the contact angle for small enough charges.

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.

The conformal dimension of a metric space $(X, d)$ is equal to the infimum of the Hausdorff dimensions among all metric spaces quasisymmetric to $(X, d)$. It is an important quasisymmetric invariant which lies non-strictly between the topological and Hausdorff dimensions of $(X, d)$. We consider the conformal dimension of the Brownian sphere (a.k.a. the Brownian map), whose law can be thought of as the uniform measure on metric measure spaces homeomorphic to the standard sphere $\mathbf S^2$ with unit area. Since the Hausdorff dimension of the Brownian sphere is $4$, its conformal dimension lies in $[2, 4]$. Our main result is that its conformal dimension is equal to $2$, its topological dimension.

We prove that the intersection cohomology of the Baily-Borel compactification of a complex Shimura variety is identified with the top weight quotient of the mixed Hodge structure on the reductive Borel-Serre compactification. This yields canonical cup products and functorial pullbacks on the intersection cohomology. As an application, we introduce canonical cycle classes associated to special cycles, relating analytic geometric volumes of non-compact Shimura varieties to topological terms.

We study Schrödinger operators on the real line whose potentials are generated by the Fibonacci substitution sequence and a rule that replaces symbols by compactly supported potential pieces. We consider the case in which one of those pieces is identically zero, and study the dimension of the spectrum in the large-coupling regime. Our results include a generalization of theorems regarding explicit examples that were studied previously and a counterexample that shows that the naïve generalization of previously established statements is false. In particular, in the aperiodic case, the local Hausdorff dimension of the spectrum does not necessarily converge to zero uniformly on compact subsets as the coupling constant is sent to infinity.

Sterile Insect Technique (SIT) is widely regarded as a promising, environmentally friendly and chemical-free strategy for the prevention and control of dengue and other vector-borne diseases. In this paper, we develop and analyze a spatio-temporal reaction-diffusion model describing the dynamics of three mosquito subpopulations involved in SIT-based biological control of Aedes aegypti mosquitoes. Our sex-structured model explicitly incorporates fertile females together with fertile and sterile males that compete for mating. Its key features include spatial mosquito dispersal and the incorporation of spatially heterogeneous external releases of sterile individuals. We establish the existence and uniqueness of global, non negative, and bounded solutions, guaranteeing the mathematical well-posedness and biological consistency of the system. A fully discrete numerical scheme based on the finite element method and an implicit-explicit time-stepping scheme is proposed and analyzed. Numerical simulations confirm the presence of a critical release-size threshold governing eradication versus persistence at a stable equilibrium with reduced total population size, in agreement with the underlying ODE dynamics. Moreover, the spatial structure of the model allows us to analyze the impact of spatial distributions, heterogeneous releases, and periodic impulsive control strategies, providing insight into the optimal spatial and temporal deployment of SIT-based interventions.

The Abelian sandpile model serves as a canonical example of self-organized criticality. This critical behavior manifests itself through large cascading events triggered by small perturbations. Such large-scale events, known as avalanches, are often regarded as stylized representations of catastrophic phenomena, such as earthquakes or forest fires. Motivated by this perspective, we study strategies to reduce avalanche sizes. We provide a first rigorous analysis of the impact of interventions in the Abelian sandpile model, considering a setting in which an external controller can perturb a configuration by removing sand grains at selected locations. We first develop and formalize an extended method to compute the expected size of an avalanche originating from a connected component of critical vertices, i.e., vertices at maximum height. Using this method, we characterize the structure of avalanches starting from square components and explicitly analyze the effect of interventions in such components. Our results show that the optimal intervention locations strike an interesting balance between reduction of largest avalanche sizes and increasing the number of mitigated avalanches.

In this paper, we obtain optimal asymptotic behavior of parabolically convex $C^{2,1}$ solution to the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f$ converges to $1$ at infinity with a slow rate. This result extends the elliptic estimate in \cite{lb5} to the parabolic setting.

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $ψ(t,x)=e^{iλt}φ\left(x-vt\right)$ to the nonlinear pseudo-relativistic Schrödinger equation \[ i\partial_t ψ=(\sqrt{-Δ+m^2})ψ- |ψ|^{\frac{2}{N}}ψ-μ|ψ|^{q}ψ~~\text{ on }\mathbb{R}^N, \] for $m\ge 0$ and $|v|<1$. To this end, we introduce and analyse an associated constrained variational problem, whose minimizers are termed boosted ground states and the parameter $λ$ is obtained as a Lagrangian multiplier. We first provide a complete classification for the existence and nonexistence of such boosted ground states. Based on this classification, we then study several limiting profiles, for which the exact blow-up rate is also established.

In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}^{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}_{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $\operatorname{Bir}_{\mathbb{C}}(\mathbb{P}^{2})$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$.