Papers

[Bass and Burdzy 1999] introduced a stochastic flow of homeomorphisms of the real line $\mathcal R_t:x\mapsto \mathcal R_t(x)$, driven by a Brownian motion. Ray-Knight theorems for the process $\mathcal R_t^{-1}(0)$ were later obtained by [Hu and Warren 2000]. We extend their results by proving an integral representation driven by a common white noise for the local times of $\mathcal R_t^{-1}(0)$, in analogy with the case of the standard Ray-Knight theorems [Aïdékon, Hu and Shi 2024]. We generalize this representation to the setting of a Bass-Burdzy flow driven by a drifted Brownian motion perturbed at its maximum, and use this framework to answer a question of [Aïdékon, Hu and Shi 2026] concerning the description of the conditional law of a perturbed reflecting Brownian motion given its occupation measure in terms of Bass-Burdzy flows.

The parabolic-elliptic Keller-Segel system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - \mu + u, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega u, \end{array} \right. \qquad \qquad (\star) \] is considered under homogeneous Neumann boundary conditions in the ball $\Omega=B_R(0)\subset {\mathbb R}^n$, where $n\ge 2$ and $R>0$. The main results identify a mass level $m_\star(n,R)>0$ as critical in the sense that $\bullet$ whenever $0<m<m_\star(n,R)$, the constant equilibrium of ($\star$) given by $u\equiv v\equiv \frac{m}{|B_R|}$ is locally asymptotically stable with respect to the norm in $L^\infty(B_R)$, but that $\bullet$ if $m>m_\star(n,R)$, then any solution of ($\star$) emanating from nonnegative and nonconstant continuous radial initial data $u_0$ with $\int_{B_R} u_0=m$ and $\frac{1}{|B_r|} \int_{B_r} u_0 \ge \frac{m}{|B_R|}$ for all $r\in (0,R)$ blows up in finite time. The value of $m_\star(n,R)$ can be determined in an essentially explicit manner in dependence on principal zeroes of appropriate Bessel functions of the first kind; in particular, $m_\star(2,R)\approx 14.68197 \cdot \pi$ for all $R>0$. Accordingly, as a supplement to known mass-dependent blow-up dichotomies these findings reveal a genuinely secondary critical mass phenomenon for ($\star$) especially also in its two-dimensional version in which no precedent of this form seems known.

Chromatic villainy is the maximum, over rearrangements of optimal graph colorings, of the minimum Hamming distance to a proper coloring with the same color multiplicities. Weak villainy allows the repair to change these multiplicities. We give exact composition laws for disjoint unions and determine the eventual growth under repetition. Write $mG$ for the disjoint union of $m$ copies of $G$. For every graph $G$, with $k=\chi(G)$, weak villainy satisfies $B_{\mathrm w}(mG)=mF_k(G)$ whenever $k\mid m$, where $F_k(G)$ is the maximum distance to a proper $k$-coloring over all assignments of $k$ color labels to vertices. Its bounded defect from $mF_k(G)$ is eventually periodic; we give an explicit stabilization bound and identify the least period, which divides $k$. For ordinary villainy, we obtain a computable rational slope $\Lambda(G)\ge F_k(G)$ such that $B(mG)=m\Lambda(G)+O_G(1)$, where the error is bounded independently of $m$ and eventually periodic, with a computable period and threshold. Consequently the difference between the two values is bounded exactly when $\Lambda(G)=F_k(G)$, and otherwise grows linearly. The slope is determined by a finite optimization using at most $k$ source assignments. We also answer the unrestricted superadditivity questions negatively: for every $C>0$, a connected graph can lose more than $C$ in weak villainy after union with a triangle, and some connected $3$-chromatic graph $G$ satisfies $B(G\sqcup G)<2B(G)-C$. Finally, both exact values on an $n$-vertex graph can be computed in time $f(B_{\mathrm w}(G))(n+1)^{O(1)}$, without either value being supplied. Threshold recognition is fixed-parameter tractable as well.

This paper presents a data-driven approach to lossless and lossy source compression using polar codes. Classical polar source coding achieves fundamental limits with low complexity for binary memoryless sources. However, for unknown q-ary sources with memory, successive-cancellation (SC) decoding requires an explicit source model and can be computationally prohibitive. We present analytical SC recursions and show that exact implementations scale quadratically with alphabet size and cubically with state-space size. To overcome these limitations, we develop neural polar decoders trained directly from data to approximate conditional distributions of synthetic polar coordinates without requiring an explicit source model. These learned distributions enable estimation of entropy rates, mutual information, and rate-distortion functions, and support explicit polar compression schemes. For lossless compression, we identify high-entropy polarized coordinates and reconstruct the remaining coordinates using the learned SC recursion. For lossy compression, we combine neural polar decoders with a neural test channel to estimate and optimize the mutual information-distortion tradeoff and construct an explicit lossy polar code. We also extend the framework to Wyner-Ziv compression with decoder side information by learning conditional polarized distributions and a reconstruction mapping. Experiments on memoryless sources, sources with memory, and real genomic data show that the learned decoders closely match analytical SC decoders when available and remain applicable when conventional methods are computationally infeasible or the source model is unavailable.

We continue the systematic study initiated in arXiv:2602.05185 of the relationship between the spectral theory and descriptive combinatorics of bounded-degree Borel pmp graphs, particularly in the non-regular setting. We first adapt a classical result of Nikiforov to give a new spectral lower bound on the approximate measurable chromatic number. For certain non-regular graphs, this bound improves on the approximate measurable version of Hoffman's classical bound established in arXiv:2602.05185. Next, we strengthen a characterization in arXiv:2602.05185 of approximate measurable bipartiteness in terms of the spectrum of the adjacency operator by generalizing to the case of non-regular graphs. Finally, we introduce the measurable domination number and adapt several classical results of Lu, Liu, and Tian to give a bound on the spectral gap of the Laplacian operator in terms of the measurable domination number.

These are notes to accompany my public lecture at the International Congress of Mathematicians, in Philadelphia in July, 2026. The goal is to discuss some developments at the interface of AI and mathematics in the context of the long conversation of mathematics.

We study a stationary ergodic mean-field game of scalar type where the representative player controls a diffusion through two-sided singular controls exerted only at the jump times of an independent Poisson process. For each mean-field parameter, we solve the novel constrained ergodic singular control problem, proving that the optimal policy is of threshold type: the diffusion is reset at Poisson arrival times to the nearest boundary point of an interval. Our explicit probabilistic derivation of the stationary distributions induced by Poissonian threshold singular controls is a key contribution of independent interest. This distribution is used to construct the mean-field fixed-point map. We prove existence and uniqueness of the stationary mean-field equilibrium by combining the stationary density with the monotonicity and continuity of the optimal thresholds in the mean-field parameter. The model provides a tractable framework for a case study of partially reversible investment with singular adjustments under exogenous intervention constraints.

Spherical harmonics have been studied extensively and are used in many fields of application. We consider the probably earliest application of spherical harmonics, namely for a representation of the solution of the Laplace equation on the $d$-dimensional unit sphere where the boundary data are represented in a spherical harmonics basis. We consider general $d \geq 2$. For the solution $u$ of the Laplace equation the fundamental maximum principle holds. If one approximates the boundary data in the finite dimensional space spanned by all spherical harmonics of degree at most $N$, the resulting discrete solution, denoted by $u_N$, does in general not satisfy the maximum principle. In this paper we derive sharp estimates for $r_N^{\rm max}$ such that the maximum principle holds for $u_N(x)$ for all $x \in \Bbb{R}^d$ with $\|x\| \leq r_N^{\rm max}$. We show that for $d=3$ one has $r_N^{\rm max} \gtrsim 1- 3 N^{-1} \ln\big( 2^\frac13 N \big) $. Similar estimates are derived for $d \neq 3$. We also study the sharpness of these estimates.

In this paper, we study Hölder regularity of local weak super-solutions to the parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ where $\text{p.v.}$ denotes the principal value, $p\in(1,\infty)$ and the kernel $J(t;x,y)$ is comparable (uniformly with respect to $(t;x,y)\in\R\times\R^d\times\R^d$) to the fractional $p$-Laplacian kernel $|x-y|^{-d-sp}$ with $s\in(0,1)$. Our contribution extends the fundamental work of Kassmann and Weidner to the nonlinear case $p\ne 2$. In particular, we provide an explicit nonlocal parabolic tail condition in terms of an $L^{1+\varepsilon}$-norm in the time variable (for a suitable $\varepsilon>0$) for all $p\in(1,\infty)$, under which Hölder regularity holds. Our proof is based on the method of expansion of positivity, which is the key ingredient in the reduction-of-oscillation argument.

In one dimension, Anderson localization by a disordered potential sets in at any strength, whereas localization by a quasiperiodic potential in the Aubry-André model occurs through a transition at a critical strength. Disordered photonic time crystals exhibit a temporal counterpart of this Anderson localization, in which light, instead of decaying in space, stops propagating and grows in time. Whether a counterpart of the Aubry-André transition exists in photonic time quasicrystals remains an open question. Here we show that it does, and that the transition occurs at mobility edges in momentum rather than in energy. These edges arise even under impulsive modulation, where the field obeys the self-dual almost-Mathieu equation exactly, and for general analytic modulations Avila's global theory reveals the condition under which they form. The Aubry-André transition thus extends from spatial lattices to the time domain, where it sorts light by wave number into components that propagate and components that stop and grow.

A Hsiang cubic is a cubic form on $\mathbb{R}^{n}$ whose zero set is a minimal cone; equivalently, it is the cubic form of a commutative nonassociative metrized algebra satisfying a single identity of degree four. We show that the intrinsic geometry of such an algebra, encoded in its idempotents and their Peirce decompositions, is the geometry of a Chebyshev-type extremal problem on the sphere. A Hsiang cubic equioscillates on $S^{n-1}$: its maximum set is the set of normalized idempotents, and on suitable great circles through an idempotent it restricts to the Chebyshev polynomial $T_3$. For the spherical analogue of the problem of least deviation, the minimization of $\|p\|_\infty^2/\|p\|_2^2$ over harmonic cubics, a sharp lower bound based on Kellogg's inequality shows that Cartan's isoparametric cubics are the global minimizers in dimensions $5$, $8$, $14$ and $26$. In general, we compute the ratio of every Hsiang cubic in terms of the Killing form and characterize stationarity by a Kolmogorov criterion: it amounts to a positive measure on the idempotent variety. Such measures are given explicitly in several cases which together cover every Hsiang cubic known to exist; in particular, all of them are stationary. These measures give quadrature identities on the idempotent variety whose moments are invariants of the algebra; they are spherical designs exactly in the Einstein (Killing metrized) case, i.e.\ when the Killing form is proportional to the inner product. Distinguished low-dimensional Hsiang cubics are strict local minimizers, whereas a polar Hsiang cubic is a stationary non-minimizer.

We study the zeros of the spectrogram of complex white noise analyzed with Hermite windows of order n, extending the classical Gaussian-window correspondence with Gaussian Analytic Functions to the polyanalytic setting. Using the polyanalytic Bargmann transform, these zeros split into two families according to the sign of the local Jacobian, giving rise to a natural notion of charge, and we show that these zeros are almost surely simple, isolated, and form a translation and rotation-invariant point process. We derive closed-form expressions for the one-point intensity, decomposed by charge, and also for the unsigned, signed, and hybrid two-point intensities. Besides, we prove that unlike the total repulsion observed with a Gaussian window, the Hermite window of order n generates a pair correlation function that converges at the origin to a strictly positive constant depending on n. Finally, we propose a numerical method for detecting spectrogram zeros via the winding number around candidate local minima.

A permutation is $n$-universal if it contains every permutation of length $n$ as a pattern. We prove that a uniformly random permutation of length $32n^2$ is $n$-universal with probability at least $1-\exp(-c n/\log n)$, for an absolute constant $c>0$ and all sufficiently large $n$. This removes a $\log\log n$ factor from an earlier result of He and Kwan~\cite{He-Kwan} and resolves an open problem of Alon~\cite{Alon2016}. The proof was found by GPT-6 Pro.

For every smooth projective generalised Markoff surface $X_m$, every hyperplane section $H$, and every integer $n\ge2$, we establish Campana strong approximation with the Brauer–Manin obstruction for $\bigl(X_m,(1-1/n)H\bigr)$.

In machine learning, a self-attention dynamics describes the attention mechanism of transformers, and is obtained treating the layer index of a transformer as time index. Even though naturally discrete-time, this dynamical system is most often studied in continuous-time. Scope of this paper is to provide a thorough investigation of the discrete-time version of the self-attention dynamics: a complete classification of its equilibria is given, and their stability properties are studied, in both the single-head and the multihead case. We show that in both cases the main attractor of the dynamics is a consensus equilibrium aligned with the principal eigenvector of the value matrix (in the single-head case, a corresponding matrix of weights in the multihead case), even though other locally asymptotically stable bipartite equilibria can be show to exist at least in the single-head situation. The behavior is confirmed by looking at the learned weight matrices of a real transformer.

In this paper, we investigate the weak invariance principle in some Besov spaces for strictly stationary sequences. In particular, we provide sufficient conditions on the moments and the dependence of a strictly stationary sequence to guaranty the weak invariance principle in these Besov spaces. Several dependence structures are considered, including projective conditions, $\alpha$-dependent, $\rho$-mixing, and $\tau$ -dependent conditions. The main tools in the proofs are martingale approximation and deviation inequalities for weakly dependent sequences. Our results cover a large range of strictly stationary sequences that were not treated previously in the literature. Moreover, the presented conditions are sharp.

We study the asymptotic behavior of eigenvalues of nonlocal convolution-type operators $$L_\varepsilon\rho=-\frac{1}{\varepsilon^{d}}\int_\Omega J\left({\frac{|x-y|^2}{\varepsilon^2}}\right)\rho(y)dy+a(x)\rho(x),$$ (with a continuous sufficiently fast decaying at infinity function $J\geq 0$) when the scaling parameter $\varepsilon>0$ tends to zero. Assuming that the minimum of the function $a(x)$ is attained at a point $x_0\in \overline{\Omega}\subset \mathbb{R}^d$ and is non-degenerate, we establish several terms in the asymptotic expansion of the first eigenvalues $\lambda_\varepsilon^{(k)}$ of $L_\varepsilon$. If $x_0$ is an interior point of the domain $\Omega$, we show that $\lambda_\varepsilon^{(k)}=a(x_0)-1+\varepsilon\mu^{(k)}+\overline{o}(\varepsilon)$, where $\mu^{(k)}$ are eigenvalues of a harmonic oscillator (obtained by the blow up analysis), the corresponding eigenfunctions of $\mathcal{L}_\varepsilon$ are concentrated near $x_0$ on the scale $\varepsilon^{1/2}$. If $x_0\in\partial \Omega$ then the asymptotic expansion of the eigenvalues of $L_\varepsilon$ has the form $\lambda_\varepsilon^{(k)}=a(x_0)-1+\varepsilon^{2/3}\Lambda_0+\varepsilon(\mu^{(k)}+\alpha p_0)+\overline{o}(\varepsilon)$. The corresponding eigenfunctions are represented as the product of a function depending on the distance to $\partial\Omega$ and concentrated on the scale $\varepsilon^{2/3}$, and functions localized on the scale $\varepsilon^{1/2}$ and asymptotically independent of the distance to $\partial\Omega$. The latter functions converge (after rescaling) to eigenfunctions of a harmonic oscillator (in tangential variables), $\mu^{(k)}$ being their eigenvalues. The former function is given, to the leading term, by the scaled Airy function, and $\Lambda_0$, $\alpha$ and $p_0$ depend on $J(|x|^2)$ and $a(x)$.

On the Riemannian Schwarzschild–Tangherlini metric of dimension $n\geq 4$, we show that the destabilizing TT tensor of Ozuch lies in the image of the “TT-tensor machine”. By continuity, it can then be approximated, in suitable weighted Sobolev spaces, by smooth compactly supported TT tensors, on which the stability quadratic form is still negative. This proves instability in the usual sense for noncompact manifolds. To our knowledge, no exact smooth compactly supported destabilizing TT tensor was known before, even in 4D.

Semi-Markov networks provide a natural framework for describing stochastic transitions between coarse-grained states with non-exponential dwell times, with applications ranging from biophysics and neuroscience to epidemiology and finance. Here, it is shown that finite semi-Markov networks admit explicit combinatorial representations of their Laplace-transformed occupation probabilities and hitting-time distributions as sums over weighted spanning trees and forests of an augmented graph. These representations yield asymptotic results, including a generalization of the Markov Chain Tree Theorem to semi-Markov networks, in which the rate constants replaced by pseudo-rates given by the inverse mean sojourn time weighted by the exit probability through this channel, as well as the long-time scaling of hitting-time survival probabilities in the presence of infinite-mean sojourn times. They also extend recently derived combinatorial formulas for hitting-time moments to the semi-Markov setting. In particular, the mean hitting-time formula coincides with that of a Markov network with the corresponding pseudo-rates, which also yields a short proof of the constancy of Kemeny's constant for finite-mean semi-Markov networks. Finally, we use the same framework to show that weak ergodicity breaking occurs generically for time-additive state observables that are nonconstant on the support of the limiting distribution in heavy-tailed semi-Markov networks.

We describe a family of two-dimensional metric deformations that preserve integrability. The seeds are (classically) conformal integrable field theories and the deformations are directly related to different classes of auxiliary field sigma models. We introduce a master action connecting deformations by Lie-algebra-valued auxiliary fields, the so-called $\nu$-frame, to deformations by two Lorentz-charged scalars. These auxiliary scalar fields can be identified as Beltrami differentials of an auxiliary two-dimensional metric, providing a direct connection with the metric approach to $T\bar{T}$-like deformations. For Weyl-invariant seed matter theories, the auxiliary metric sector is described by a reduced gravitational potential. We show that the resulting framework preserves classical Lax integrability whenever the seed matter theory admits a Lax representation of its equations of motion on a generic curved two-dimensional spacetime. Finally, within the Lorentz-invariant family and the zweibein deformations of conformal matter theories considered here, we identify $T\bar{T}$ as the only deformation that can be interpreted as a field-dependent coordinate transformation.

In order to control epidemics of mosquito-borne diseases such as dengue, for which there is still no vaccine available, several strategies aim to directly target the mosquito population. One technique involves introducing a symbiotic bacterium called Wolbachia into the population, which blocks the transmission of pathogens. A mathematical model of the introduction of this bacterium and the replacement of the mosquito population with a population carrying this bacterium has been developed, proving that in a homogeneous environment, there is spatial propagation of the replacement. The objective of this paper is to study the blocking of this propagation in cases where the environment is not homogeneous. In particular, we are considering scenarios where the environmental carrying capacity varies significantly across different regions of the spatial domain. To achieve this objective, we first reduce the original 2x2 reaction-diffusion system to a single closed equation, providing a simplified framework for studying the dynamics of Wolbachia propagation. We then investigate the blocking of the propagation in the population replacement process, focusing on the effects of spatial variations in the environmental carrying capacity. Numerical simulations have been performed to validate the theoretical results.

Following \cite{BR} we study hyponormal operators in the strictly larger class $\beta(H)$ of operators whose restriction to every non-zero reducing subspace is norm attaining. The Weyl-type characterization of positive operators via the essential spectrum used in the absolutely norm attaining case is replaced by a well-ordering condition on the eigenvalues (Theorem~\ref{thm:positive}). We prove a transfinite, block-diagonal representation $T=\bigoplus_i\lambda_iS_i$ (with isometries $S_i$) for the $\beta(H)$-operators whose $|T|$-eigenspaces are reducing, equivalently the quasinormal operators with $|T|\in\beta(H)$, and we show by examples that hyponormality alone does not give such a decomposition. For normal operators we obtain a complete representation, and for hyponormal $\beta(H)$-operators we show that normality is equivalent to the spectrum having planar Lebesgue measure zero. An analogous area-zero criterion for normality is proved for hyponormal operators attaining the minimum modulus on every non-zero reducing subspace.

This paper considers the problem of estimating the ratio of variance components of a balanced one-way random effects model. Existing estimators of ratio of variance, such as the ANOVA and maximum likelihood estimators, can take negative values and are inadmissible under quadratic loss. We therefore study the problem under a general class of scale-invariant loss functions. We characterize the class of scale-equivariant estimators of ratio of variance and obtain the best equivariant estimator. We then derive a class of estimators that dominate the BEE and extend this class to obtain further improved estimators. We propose a smooth improved estimator of ratio of variance. For interval estimation of ratio of variance, we derive an asymptotic confidence interval, bootstrap confidence intervals, a generalized confidence interval and a highest posterior density credible interval. The performance of the proposed improved point estimators and confidence interval procedures is assessed through an extensive simulation study. Finally, we provide a real data analysis for implementation purposes.

A pair of topological spaces $X$ and $Y$ is a Fourier–Mukai (FM) pair if there is an equivalence of $\infty$-categories of derived sheaves of abelian groups \[\operatorname{Shv}(X;\mathrm D(\mathbb Z))\simeq \operatorname{Shv}(Y;\mathrm D(\mathbb Z)).\] An FM pair $(X,Y)$ is non-trivial if $X$ and $Y$ are not homeomorphic. We consider the question of whether non-trivial FM pairs $(X,Y)$ exist for which $X$ and $Y$ are both "nice" spaces, which in this article means admitting a triangulation. We show that the answer is no if $X$ is second-countable and either (i) one-dimensional, (ii) a surface, or (iii) the 3-sphere. In order to prove this, we define two invariants of dualizable categories. The first invariant is an extension of Tate–Hochschild cohomology (aka singular Hochschild cohomology) to a class of dualizable categories satisfying a countability assumption. For a homology manifold $X$, we relate the Tate–Hochschild cohomology of $\operatorname{Shv}(X;\mathrm D(\mathbb Z))$ to the homology at infinity of $X$ (in the sense of proper homotopy theory). The second invariant takes the form of a subcategory $\mathcal D_{\mathrm{Gor}}\subseteq\mathcal D$ called the Gorenstein locus, defined for every dualizable stable category $\mathcal D$. For a triangularizable space $X$ and a field $k$, we show that the Gorenstein locus of $\operatorname{Shv}(X;\mathrm D(k))$ is equal to the full subcategory of sheaves supported on the $k$-homology manifold locus of $X$.

In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\times n$ Latin square, the probability that all rows are even is of the form $2^{-n+o(n)}$. This fills a gap left by recent work of Kwan, Petrova and Sawhney: they proved that the row parities behave like a sequence of independent coin flips in a variety of different senses, but they were not able to lower-bound the all-even probability, as their approach had a small number of "exceptional rows" whose parities they could not control. In fact, we prove a more general theorem on the probability of any possible prescription of row parities, column parities and symbol parities. Our approach extends the ideas of Kwan, Petrova and Sawhney, introducing a new way to "bootstrap" intercalate switchings to control more complex switching operations.

Consider \begin{equation*}\ \left\{ \begin{aligned} -\mathcal_{\lambda,\Lambda}^{+}(D^{2}u)&=\mu u+a(x)u^{p}\quad\text{in}~\Omega,\\ u&=0\quad\text{on}~~\partial\Omega, \end{aligned} \right. \end{equation*} where $ \ Omega$ is a smooth bounded domain, $a$ is a continuous function which is allowed to change sign. Moreover, the set of zeros of $a$ may have a non-empty interior. The main objective of this article is to establish a uniform bound on the set of positive solutions of the above problem. This approach requires a suitable Liouville-type theorem. We first prove the Liouville-type theorem for $-\mathcal_{\lambda,\Lambda}^{+}(D^{2}v)=h(x_{1})f(v),$ where \(h\) is a strictly increasing one-dimensional weight and \(f\) is positive and nondecreasing.

We prove the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the large-distance behaviour of the two-point correlation function of the $\mathsf{Sine}_\beta$ point process for all $\beta>0$, confirming the first term of an asymptotic expansion predicted by Forrester.

In this paper we propose a novel regularisation strategy for structured linear inverse problems of the form $Ax=F \circ G x = z$. Instead of using the operator $A$ or a complex scheme involving both $F$ and $G$ simultaneously, combining already existing and simpler regularisations for $F$ and $G$ allows the use of established and optimised programs for high-speed computations. As classical parameter choice rules fail in the scenario when no source conditions are known because intermediate error bounds may be unavailable, we provide both a-priori and a-posteriori rules with a convergence speed analysis given source conditions. We test the proposed method on the pre-clinical medical imaging technique of magnetic particle imaging with different choices of regularisation and on a convolution problem for an analysis of the convergence speed.

Nonlinear model predictive control (NMPC) can be applied to large nonlinear systems with complex objectives while enforcing constraints, but the cost of solving the underlying optimization problems limits real-time application for systems with fast dynamics or many states. Neural networks can reduce this online cost by approximating the control law, but may violate constraints and typically require many precomputed optimizer solutions for training. The learning-based iterative solver for constrained optimization (LISCO) instead learns to predict primal-dual steps toward satisfaction of the Karush-Kuhn-Tucker (KKT) conditions through self-supervised training without precomputed solutions, but its dense, monolithic architecture grows with the prediction horizon. We remove this horizon dependence by exploiting the temporal structure of the problem in the solver architecture, which makes the network size independent of the horizon while leaving the training loss and the solver iteration unchanged. A closed-loop resampling strategy further aligns the training distribution with closed-loop operation at no additional sampling cost. On an economic NMPC benchmark of a CSTR cascade, the proposed solver scales to larger problems than monolithic LISCO and achieves median speedups of up to a factor of four over IPOPT. In closed loop, it achieves comparable control performance to IPOPT with no observed constraint violations.

Let $\Omega$ ___ C n be a bounded hyperconvex domain and $\Phi$\,: $\Omega$ $\rightarrow$ R a maximal plurisubharmonic function in $\Omega$ (possibly unbounded). We introduce a new class E 1 $\Phi$ ($\Omega$) of plurisubharmonic functions with boundary values equal to $\Phi$ in the sense of Cegrell and finite $\Phi$-energy on $\Omega$. Then given a Borel measure $\mu$ on $\Omega$, we apply a variational approach to prove that $\mu$ is the Monge-Ampère measure of a function u $\in$ E 1 $\Phi$ ($\Omega$) if and only if it satisfies the integrability condition We also establish a quantitative version of this condition extending previous results of Cegrell and others. Using a logarithmic transformation, we obtain analogous results for the real Monge-Ampère equations with (unbounded) boundary data.