Papers

Data-consistent inversion (DCI) constructs probability measures whose push-forward distributions agree with observed data, while iterative data-consistent inversion (iDCI) extends this framework to generalized stochastic inverse problems by enforcing multiple push-forward constraints sequentially. Although iDCI avoids the direct approximation of high-dimensional joint densities, its relationship to the original joint DCI solution has remained unclear. In this work, we establish this relationship through copula theory. Using Sklar's theorem, we derive a factorization of the DCI update into separate marginal and dependence transformations and show that the discrepancy remaining after convergence of the iDCI algorithm is entirely characterized by the copulas associated with the observed and predicted joint distributions. This characterization motivates a copula-transformed iDCI solution, and we prove that an exact copula transformation recovers the original DCI solution. We further establish convergence results for approximate copula transformations under converging sequences of reference measures and progressively enriched feasible sets. Numerical examples demonstrate how the geometry induced by the quantity-of-interest map governs the importance of the copula transformation, illustrate an adaptive reference-measure refinement strategy for improving computational accuracy under a fixed sampling budget, and demonstrate the progressive refinement of generalized stochastic inverse problems through heterogeneous, asynchronously acquired experiments.

The divisible sandpile on $\mathbb{Z}^d$ starts from i.i.d. masses at each site, and, in each discrete time step, a site with mass above one keeps one unit and sends the excess equally to its neighbors. Levine, Murugan, Peres and Ugurcan (2016) showed that at mean one this process explodes, with every site emitting infinite mass. We show that the mass emitted from a site by time $t$ is of order $t^{(4-d)/4}$ for $d\leq3$, of order $\log t$ for $d=4$, and a tail-dependent, divergent rate for $d\geq5$. We further show that the mass emitted, after diffusive rescaling, converges to a Brownian optimal-stopping value for $d\leq3$ and to tail-dependent, weighted membrane fields for $d\geq5$, while at the critical dimension $d=4$, after superdiffusive rescaling, it converges to the membrane model. Using these estimates, we prove that, for every $d\geq2$, the set of sites that topple contains an infinite component at some mean below one, hence it has a non-trivial percolation phase transition. This answers a variant of a question of Fey, Meester and Redig (2009). The proof adapts ideas from the theory of level-set percolation of strongly correlated Gaussian fields.

Let $R$ be a discrete valuation ring. We prove that every proper flat finitely presented commutative $R$-group scheme $P$ fits into an exact sequence \[ 0\longrightarrow E\longrightarrow P\longrightarrow B\longrightarrow0, \] where $E$ is finite flat and $B$ is an abelian scheme. For a quasiabelian model—that is, a proper flat finitely presented $R$-group scheme with abelian generic fibre—we prove a finer structure theorem: its normalization is an abelian scheme, and the model is obtained from it by a pushout involving finite flat group schemes. An analogous abelian quotient exists for proper flat commutative group stacks with finite flat inertia. These structure theorems give explicit quotient presentations for duals. Over an arbitrary base, we show that the dual of a proper flat finitely presented commutative group algebraic space is algebraic, proper, flat, and finitely presented, and biduality holds. When $2$ is invertible, the corresponding result for group stacks proves Brochard's duality conjecture. As an application over a discrete valuation ring, Polishchuk's kernel-algebra Fourier–Mukai argument gives equivalences between the unbounded derived categories of quasi-coherent sheaves on every proper flat finitely presented commutative group scheme and its dual, and between their bounded derived categories of coherent sheaves, with no tameness or residue-characteristic hypothesis. For a quasiabelian model, the dual category is a finite-flat equivariant category on its abelian normalization.

We study the category of dual operator spaces equipped with an action of a locally compact quantum group $\mathbb{G}$. The Fubini crossed product functor $-\rtimes^\mathcal{F} \mathbb{G}$ and the weak$^*$-crossed product functor $-\bar{\rtimes}\mathbb{G}$ are shown to be equal if and only if $\mathbb{G}$ has the approximation property of Haagerup and Kraus. Using the natural isomorphism $-\rtimes^\mathcal{F}\mathbb{G}\cong {}_{L^1(\mathbb{G})}\mathcal{CB}(B(L^2(\mathbb{G}))_*, -)$, this leads to a characterization of the approximation property of $\mathbb{G}$ via an $L^1(\mathbb{G})$-module approximation property for $B(L^2(\mathbb{G}))_*$. Finally, exactness of the Fubini crossed product functor is investigated and related to amenability properties of $\mathbb{G}$.

In the parking model, each site of the $d$-dimensional lattice independently starts with one car with probability $p$ or one parking spot with probability $1-p$. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime $p=1/2$, the expected number of visits to a site in $n$ rounds is of order $n^{(4-d)/4}$ for $d\leq3$ and $\log n$ for $d\geq4$. We also prove that in the subcritical regime $p\in(0,1/2)$, the parking-time tail is bounded above and below by stretched exponentials with exponent $d/(d+2)$. As $p\uparrow1/2$, we also determine the divergence of the expected total number of visits to a site: its order is $(1-2p)^{-3}$, $(1-2p)^{-1}$ and $(1-2p)^{-1/3}$ in dimensions one, two and three, respectively, and $\log(1/(1-2p))$ in dimensions four and higher. Our proof uses a representation of the parking process as the divisible sandpile of Levine and Peres plus a martingale-type term. These results answer questions posed by Damron, Gravner, Junge, Lyu and Sivakoff (2019).

Let $k$ be a number field and let $G$ be a connected linear algebraic group over $k$. We compare three approaches to strong approximation for the classifying stack $BG$ with respect to its full Brauer group: the homogeneous-space method of \cite{DhillonClassifying}, Kottwitz's local–global sequence, and, for reductive groups, Borovoi's localization theorem. Under the natural identification \[ Br(BG)/Br(k)\simeq Pic(G), \] we identify the Kottwitz and Borovoi obstruction maps with Brauer evaluation. The three approaches therefore give the same exact description of the localization image as the projected full Brauer–Manin set. We then study strong approximation with respect to the ordinary unramified Brauer group. We prove \[ Br^{un}(BG)/ Br(k)\simeq Sha^1_{\mathrm{cyc}}(k,\widehat G), \qquad \widehat G=X^*(G_{\bar{k}}), \] and give an exact local criterion for unramified Brauer approximation off a finite set of places. Examples show that this approximation can fail even when a finite place is omitted, and exhibit a torus for which the unramified Brauer group cuts out the global image as a proper subset of the adelic space.

We give an algorithm, based on second-order necessary conditions, to test whether a point is isolated on an algebraic set. Using this subroutine, we develop an algorithm that bounds the local dimension of an algebraic set at a point. The principal computation in our isolation test is governed by the nullity of the Jacobian, avoiding the combinatorial growth of classical methods. For exact input, the bound returned by the algorithm is certified. We compare our approach with classical methods. As an application, we show that the realization space of the 24-cell has the expected local dimension at its symmetric regular realization, completing the local dimension program for regular 4-polytopes initiated by Rastanawi, Sinn, and Ziegler.

The late-time ringdown dynamics of near-extremal black holes (BHs) are expected to be dominated by zero-damped modes (ZDMs), whose decay rates are parametrically suppressed relative to those of ordinary quasinormal modes. In this paper, we demonstrate that exact WKB methods provide an exceptionally powerful framework for analyzing the ZDM spectrum of near-extremal Reissner–Nordström (RN) BHs. Focusing on massless, neutral scalar modes propagating on an RN background, we present the full Stokes geometry derived from the radial eigenvalue problem and establish the corresponding exact quantization condition (EQC). Our analytic computation of the Voros symbols entering the EQC achieves higher-order accuracy for the ZDM spectrum compared to previous studies and is systematically improvable. Ergo, this work serves as a proof of concept for investigations of ZDM spectra of other systems using exact WKB methods.

A standard approach for studying magic angle twisted bilayer graphene (MATBG)'s correlated electronic phase diagram is to project the Coulomb interaction down to effective models only involving electrons in single-particle flat bands and some nearby remote bands. We provide a novel systematic derivation of a single-particle continuum model of MATBG's single-particle properties which incorporates structural relaxation while remaining in the Lagrangian frame. We project the Coulomb interactions down to electrons occupying the flat bands of this model and compute the Hartree-Fock many-body ground states at fillings $\nu = \pm 2$. We find that incorporating relaxation effects drives the model into a semi-metallic phase at $- 2$ because of particle-hole asymmetry in the relaxed model's single-particle dispersion and because the flat band wavefunctions become more concentrated leading to an enhanced Hartree potential. Our results corroborate recent ab initio density functional theory studies which also found semi-metallic phases at $-2$. We discuss potential explanations for why such phases have not been seen in experiments.

We prove a sharp comparison, with Obata-type rigidity, for all Neumann eigenvalues of one-dimensional $\mathrm{CD}(1,2)$ spaces against the Legendre model, under a convexity condition on the density. It follows that minimizers of the $k$-th Laplace eigenvalue among closed surfaces of Gaussian curvature at least $1$ cannot collapse in the measured Gromov-Hausdorff completion, for every $k\ge2$. We also give a variational proof that smooth minimizers are round.

We review the paper `Models for the Eremenko–Lyubich class' by Chris Bishop, which appeared in the Journal of the London Mathematical Society in 2015, the first of two LMS papers for which Bishop received the LMS Senior Berwick Prize in 2024. This is one of a series of papers where Bishop introduced his new technique of quasiconformal folding, which transformed the range of approaches to constructing examples of transcendental entire functions leading to the resolution of many open problems, in particular in transcendental dynamics.

We investigate the ratio asymptotics and the asymptotic zero distribution of a sequence of polynomials that satisfy a recurrence relation of order $r+1$ with all recurrence coefficients, except the last one, equal to zero. Such a sequence is part of a system of multiple orthogonal polynomials and it satisfies the symmetry property $P_n(\omega_{r+1} z) = \omega_{r+1}^n P_n(z)$, where $\omega_{r+1}$ is the primitive $(r+1)$th root of unity. We consider the unbounded regime in which the recurrence coefficients exhibit algebraic growth and, after division by $n^{\gamma}$ become asymptotically periodic and bounded. After the appropriate scaling, we establish ratio asymptotics and characterize the limiting ratio as the distinguished solution of an algebraic equation. We then determine the limiting zero distribution through its Stieltjes transform and investigate the associated \(\St\)-transform, which in several cases yields connections with hypergeometric polynomial sequences and distributions arising in free probability. The recurrence is represented by a two-diagonal non-self-adjoint Hessenberg operator, so that the limiting zero measure also admits a natural interpretation as a limiting empirical spectral distribution of its rescaled finite sections. Our analysis is based solely on the positivity and asymptotic behavior of the recurrence coefficients and requires no explicit knowledge of the underlying orthogonality measures.

We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.

In a smoothly bounded domain $\Om\subset\R^n$ with $n\geq 1$ and $a>0$, we consider an initial-boundary value problem for the general viscous wave equation \bas h(u,u_t) u_{tt} = \Del u_t + a\Del u + f(u,u_t,\na u,\na u_t) \eas which appears in models of nonlinear acoustics wave propagation; well-established equations of Kuznetsov and Westervelt type form particular examples.\abs % While the existing literature offers extensive results on global solutions for sufficiently small initial data $(u_{0}, u_{0t})=(u, u_{t})|_{t=0}$ in second- and higher-order Sobolev spaces it appears to remain open how far global solvability can be established under smallness conditions involving only first-order Sobolev spaces. The present manuscript addresses this question by proving the existence of global classical solutions together with exponential decay of the pair $(u,u_{t})$ in $ W^{1,r}\times W^{1,p}$-Sobolev spaces whenever the nonlinearities $h$ and $f$ are sufficiently smooth and are such that $h(0,0)>0$ as well as $f(0,0,0,0)=0$ and $\na f(0,0,0,0)=0$.

We prove that the standard cohomogeneity-one action of \(SU(3)\) on \(S^7\), with principal orbit the Wallach flag manifold \(SU(3)/T^2\), admits infinitely many invariant Einstein metrics. The proof uses a detection approach to the Einstein boundary-value problem. Starting from one singular orbit, we follow the Einstein solutions only to a canonical hypersurface and measure there how far they are from closing smoothly. When the singular-orbit scale becomes small, this closing problem is governed by a Ricci-flat limiting solution. The linearisattion about its limiting cone has an oscillatory mode. As the scale shrinks, this oscillation repeatedly changes the sign of the closing error, producing infinitely many parameter values for which the metric closes. The closing scales satisfy an asymptotic logarithmic quantization law, with successive ratio tending to \(e^{-2\pi/\sqrt{15}}\). We also determine the geometry of the resulting sequence. Away from the two singular orbits the metrics converge to the singular sine cone over the non-normal Einstein metric on \(SU(3)/T^2\), while after rescaling by the square of the singular-orbit scale at either end they converge to the same complete Ricci-flat threshold metric. Their curvature is of order \(b_n^{-2}\), so the logarithmic phase law induces a corresponding quantization of the focal curvature scale.

We study the automorphism ind-group of a Stanley-Reisner variety $X_\Delta$ through a combinatorial toolkit on the underlying complex: a facet closure operator, its Demazure roots, and the resulting dichotomy between exposed and hidden facets. Our main theorem is that a fully exposed complex, that is, one in which every facet has a private vertex, is recovered from the ind-group: $\mathrm{Aut}(X_\Delta)\cong\mathrm{Aut}(X_{\Delta'})$ forces $\Delta\cong\Delta'$. The hypothesis cannot be dropped, but it holds after one stabilization, so for arbitrary $\Delta,\Delta'$ an isomorphism $\mathrm{Aut}(X_\Delta\times\mathbb{A}^1)\cong\mathrm{Aut}(X_{\Delta'}\times\mathbb{A}^1)$ already forces $\Delta\cong\Delta'$. At the opposite extreme, a fully hidden $\Delta$ gives $\mathrm{Aut}(X_\Delta)=T_0\rtimes S(\Delta)$, never isomorphic to the ind-group of a non-rigid Stanley-Reisner variety. The criterion decides graphs and skeleta, and every complex is homotopy equivalent to a fully exposed one.

We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace–Neumann eigenvalue. This is the quadrilateral case of the Neumann analogue of the long-standing Pólya–Szegő conjecture, which asserts that the regular $n$-gon minimizes the first Laplace–Dirichlet eigenvalue among $n$-gons of equal area. The proof uses a Rayleigh–Ritz lower bound, obtained from the span of the first five nonconstant Neumann modes of the square, on a smooth auxiliary functional whose minimization implies the original maximization. Local maximality at the square follows from a symmetry-reduced Hessian, computed in closed form, together with a certified second-order difference-quotient test that establishes the local inequality on an explicit ball around the square. Global maximality is then established by a certified box covering of the parameter space, certifying on each box a direct upper bound on the first Rayleigh–Ritz eigenvalue. Both certifications operate entirely on integrals over a fixed reference square and avoid a posteriori finite-element bounds on the perturbed quadrilateral.

Let $H\in C^2(T^*M)$ be a Tonelli Hamiltonian on a closed connected manifold and let $u_\lambda$ solve \[ \lambda u_\lambda+H(x,Du_\lambda)=c(H)\qquad\text{in }M. \] We study the convergence rate of $u_\lambda$ to the selected critical solution $u_0$. Assume that the lifted Aubry set is a finite union $\widetilde{A}=\Gamma_1\sqcup\cdots\sqcup\Gamma_N$, where each $\Gamma_i$ is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove \[ -C\lambda\le u_\lambda-u_0\le C\lambda|\log\lambda|. \] Let $\mu_i$ be the projected Mather measure associated with $\Gamma_i$. If \[ \int_M u_0\,\mathrm d\mu_i=0\qquad \text{for every }i, \] then \[ \|u_\lambda-u_0\|_\infty\le C\lambda. \] In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is $O(\lambda)$. We give examples showing that both convergence rates $O(\lambda)$ and $O(\lambda|\log\lambda|)$ are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order $\lambda^{1/(2r-1)}$ with $r\ge2$. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.

We prove that the proportion of $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes that are normal tends to zero at least at a double exponential rate as $d\to\infty$. As a consequence, the same holds for any of the following classes given by the type of triangulation possible: (a) quadratic, (b) flag unimodular, (c) regular unimodular, or (d) unimodular, among others. We classify the $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes for $d\leq5$ according to whether they admit a unimodular, flag unimodular, or quadratic triangulation. In dimension five, exactly $175$ out of $1{,}226{,}525$ classes have a flag unimodular triangulation, but no quadratic triangulation. Among them, there are polytopes whose toric rings are not Koszul; thus, we find the first polytopes that have a flag unimodular triangulation, but whose toric ring is not Koszul. In contrast with the matroid case, we exhibit a delta-matroid polytope that is not normal.

We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovskiĭ\ operators to construct such approximations, thereby extending the approximation theory developed for smooth bounded domains and the simultaneous approximation framework introduced by \emph{Fefferman, Hajduk, and Robinson, {Proc. Lond. Math. Soc.} (3) \ {125} (2022), no.~4, 759-777}. As an application, we establish energy equality for Leray-Hopf weak solutions of the incompressible convective Brinkman-Forchheimer (CBF) equations on $\mathbb{R}^d$, $d\in\{2,3\}$ covering both the critical and supercritical regimes. For general unbounded domains, we employ the resolvent operator associated with the Stokes operator, developed by \emph{Farwig, Kozono and Sohr, {Acta Math.}, {195} (2005), 21-53}, to obtain simultaneous approximation results. We also establish a generalized version of the classical Lions-Magenes lemma, which is of independent interest. Finally, by combining this result with the simultaneous approximation framework for unbounded domains, we establish energy equality for weak solutions of the CBF equations on general unbounded domains.

We extend the setup of Lai-Nakano-Xiang's quantum wreath product algebras to symmetric monoidal categories and deformed braid relations. Under some mild assumptions, we drastically simplify the criterion for such algebras to have PBW basis; curiously, the new criterion is phrased in terms of associative and quantum Yang-Baxter equations. As an application, we clean up some basis results in the existing literature, and propose a certain generalization of quiver Hecke algebras.

For a tournament $T$ and a vertex ordering $\prec$, let $T^{\prec}$ be the graph of backward arcs in $\prec$. The diclique number of a tournament is $\domega(T)=\min_{\prec}\omega(T^{\prec})$, and the dichromatic number is $\dchi(T)=\min_{\prec}\chi(T^{\prec})$. We prove a mixed parameter transfer theorem: for all $k$ and $r$, if $\tww(T)\le k$ and $\omega(T^{\prec})\le r$, then the ordered twin-width of $(T^{\prec},\prec)$ is bounded by a function of $k$ and $r$. The proof combines the regular-semigrid theorem for ordered graphs with permutation-encoding obstructions to bounded twin-width in tournaments. Together with polynomial $\chi$-boundedness of graphs of bounded twin-width, this implies that tournaments of bounded twin-width are $\dchi$-bounded by $\domega$, resolving a conjecture of Aboulker, Aubian, Charbit, and Lopes.

One of the most common descriptions of solutions of singular SPDEs is their characterisation as the limit of solutions of renormalised smooth random PDEs. We quantify the speed of this convergence in the case of the KPZ equation. In particular, we show that the naïve guess that the rate is given by the distance of the noise regularity from the endpoint regularity for well-posedness is not correct. Instead, we obtain convergence at the larger rate $1/2$ and show that this rate is sharp. This is achieved by considering the equation satisfied by the rescaled error, which is critical for variance blowup, and showing that this equation has a limit given by an affine linear singular SPDE driven by a new, independent noise. This result can be alternatively interpreted as identifying the asymptotic size and law of fluctuations of the solutions of the KPZ equation driven by mollified noise around their singular limit.

We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert–Schmidt regime, we prove $N$-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman–Fredholm determinant of the centered interaction operator. We show that the Hilbert–Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two $U$-statistics. As consequences, we establish entropic commutator estimates with the sharp $O(N^{-1})$ additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert–Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.

A basic question in adaptive control is how rapidly a discrete-time nonlinear plant with unknown parameters may grow while remaining stabilizable. When the nonlinear growth exponent is known and only a scalar coefficient is unknown, existing theory identifies 4 as the exact critical exponent for stabilizability. We determine how this critical exponent changes when the growth exponent is also unknown. For any positive disturbance bound, one feedback law stabilizes all plants in some neighborhood of a nominal parameter pair if and only if the nominal exponent is below $3\sqrt{3}/2$. At equality, every compact parameter set whose exponents do not exceed this value remains stabilizable. If the exponent belongs to a known finite set, the critical exponent remains 4 for every nondegenerate compact coefficient interval. Hence finite exponent sets and arbitrarily short exponent intervals can have different critical exponents. Uncertainty in the growth rate therefore changes the range of growth that feedback can stabilize, not merely the size of the uncertainty set. The critical exponent remains $3\sqrt{3}/2$ under a known positive state-dependent multiplier bounded above and away from zero. For a system whose coefficient and exponent are known functions of an unknown parameter, a compact parameter family is stabilizable when its exponents do not exceed this value. At an interior parameter point where the Jacobian of these functions has rank two and the exponent is at least this value, no compact neighborhood is stabilizable.

We consider on a Riemannian manifold $M$ the Leibenson equation \begin{equation*}\label{eqabs}\partial _{t}u=\Delta _{p}u^{q},\end{equation*} where $p>1$ and $q>0$. When $q(p-1)\geq 1$, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound $V(x_0, r)\leq \exp\left(C r^{\frac{p}{p-1}}\right)$ for some $x_{0}\in M$ and all large enough $r>0$. When $q(p-1)< 1$, we prove this property assuming $V(x_0, r)\leq Cr^{N}$ and $p>N[1-q(p-1)]$, which matches the threshold in $\mathbb{R}^{n}$ with $N=n$. We also show that solutions on the hyperbolic space $\mathbb{H}^{n}$ have a finite extinction time in the case $q(p-1)< 1$, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case $q(p-1)=1$, we also prove a $L^{p-1}$- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.

We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize stochastic stability through the critical learning rate, defined as the largest learning rate for stable training, and obtain $\eta_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $\eta_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-\beta)\}$, and $\eta_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^q(1-\beta)\}$, where $B$ is the batch size, $\beta\in[0,1)$ is the momentum factor, and \(q>1\) is the capacity exponent governing spectral decay. Within the stable region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient through power-law decay to a noise floor. We then optimize the final-step risk under a fixed data budget, obtaining a three-regime phase diagram that reveals how the benefits of momentum change with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the optimal small-batch data-scaling exponent, thereby allowing fewer sequential updates without sacrificing data efficiency. In contrast, Nesterov improves the optimal data-scaling exponent in the large-batch regime through look-ahead-induced suppression of noise amplification. Experiments on PLK regression quantitatively support the theoretical scaling predictions, while neural-network experiments provide qualitative evidence for the predicted batch-size dependence and stability behavior.

We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well. By using stability estimates between weighted Hölder spaces, the one-step approximation error can be propagated through the iterations which yields universal approximation of the corresponding semigroup. Second, we introduce the more specialized class of envelope-neural operators for envelope semigroups which allows us to derive quantitative approximation rates. Finally, we illustrate the effectiveness of these neural operators in several numerical examples arising from non-linear partial differential equations, stochastic optimal control and stochastic processes under model uncertainty.

We consider stochastic multi-objective optimization over a nonempty closed convex set, where every objective is an expectation and only sample-gradient information is available. We develop a line-search-free and function-value-free adaptive projected-gradient algorithm for the sample-average approximation (SAA) problem. Each iteration computes a feasible regularized multi-gradient step and updates the regularization parameter from the projected step length. A normal-cone-based certificate yields descent estimates and an explicit complexity bound for the Pareto-stationarity residual of the SAA problem. The consistency of SAA gradients then transfers vanishing SAA residuals to Pareto stationarity for the population problem, while an additional concentration argument gives a finite-sample residual bound on compact sets. Experiments on synthetic problems, classification, portfolio selection, multi-task learning, and robot control illustrate the practical performance of our algorithm.

We consider the numerical solution of the linear Vlasov transport equation on bounded spatial domains with inflow boundary conditions based on low-rank approximation. We combine a finite volume discretization with a rank-adaptive step-and-truncate scheme for the resulting matrix ODE. The spatial and velocity meshes may be unstructured, while suitable numerical fluxes retain a separated space-velocity representation. For homogeneous inflow, we show that the low-rank scheme inherits the $L_2$ stability and CFL restriction of the underlying full finite volume forward Euler method. In addition, the low-rank approximation error is bounded explicitly in terms of the truncation tolerances, avoiding the modeling error associated with tangent-space projections in dynamical low-rank approximation. Numerical experiments in 1d1v and 2d2v confirm the predicted error behavior. In 2d2v, the method handles an unstructured spatial mesh with nonzero inflow and a full tensor-product discretization of approximately $5.8\cdot10^{10}$ phase-space cells while the numerical rank is at most twelve.