Papers

Data-driven model predictive control based on Koopman operator theory is a promising approach for constrained control of nonlinear systems with unknown dynamics. This paper proposes a robust model predictive control framework for such systems using bilinear Koopman realizations with state- and input-dependent proportional approximation-error bounds that vanish at the target equilibrium. Since prediction in lifted coordinates with a finite-dimensional Koopman realization need not remain on the manifold of valid lifted states, multi-step prediction may leave the region where one-step error certificates apply. We avoid this difficulty by reprojecting each predicted lifted state onto the original state coordinates and lifting it again, yielding an error-aware discrete-time control-affine predictor in the original state space without assuming invariance of the Koopman dictionary. For this predictor, we develop a homothetic tube construction based on a discrete-time robust control contraction metric whose radius explicitly accounts for the proportional approximation error bounds. The tube tightening handles arbitrary continuously differentiable nonlinear constraints, and the resulting model predictive control problem includes terminal ingredients and a tube-radius penalty that exploits the vanishing uncertainty near the target. We prove robust satisfaction of the original nonlinear constraints, recursive feasibility, and exponential stability of the sampled true closed-loop system with high probability over the training data without requiring a globally optimal solution to the model predictive control problem. Numerical examples, including nonlinear obstacle-avoidance constraints, demonstrate robust stabilization and higher performance of the proposed approach compared to existing Koopman-based model predictive control methods in terms of smaller closed-loop cost and flexibility.

On the upper hemisphere, we use the Obata-Escobar argument to establish an Obata theorem for $\sigma_k$-flat and positive constant boundary $\mathcal B_k$-curvature equation, which was initiated by Case-Wang [Adv. Math. 337 (2018), 83-106].

Neural Stochastic Differential Equations (Neural SDEs) provide flexible continuous-time generative models, but generic neural drift and diffusion networks are costly to simulate on long horizons and can give unstable gradients when the training signal is a path functional rather than a pointwise observation. We introduce SLiSDE, a family of Neural SDE models built from structured linear stochastic layers. Parallel-in-time simulation is obtained at the layer level, while expressivity is recovered by gated in-flow stacking: previous-layer paths modulate the next layer's latent flow through learned gates. For functional calibration tasks in which rare paths dominate the loss, we add an optional Girsanov tilt that acts as a learned importance sampler with an exact likelihood-ratio correction. We prove well-posedness, a discretisation error bound, validity of the change of measure, and a universality result: the terminal laws of the gated stack are dense in the space of square-integrable laws. Experiments on functional calibration benchmarks show that the structured model outperforms fully neural SDE baselines while retaining parallel-time simulation and stable importance weights.

We study deformations of foliated complex analytic structures defined by locally free subsheaves of tangent sheaves and locally free subsheaves of cotangent sheaves on the basis of Kodaira-Spencer's deformation theory. We prove theorems of existence and completeness for deformations of foliated complex analytic structures defined by locally free subsheaves of tangent sheaves as analogues of theorems of existence and completeness for deformations of complex analytic structures by Kodaira-Spencer. We prove theorems of existence and completeness for deformations of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We also prove theorems of existence and completeness for simultaneous deformations when singular holomorphic foliations on compact complex manifolds are defined by locally free subsheaves of tangent and cotangent sheaves at the same time.

We extend Sullivan's rational homotopy theory by constructing geometrically and algebraically minimal models for diagrams of simply connected spaces over a class of finite indexing categories. Both classes classify rational diagram homotopy types, but their minimality conditions can differ. We also show that, in contrast to classical rational homotopy theory, H-diagrams need not split into products of Eilenberg–MacLane diagrams, and characterize the indexing categories for which every H-diagram admits such a splitting. Also we apply this to computation of automorphism groups of rational homotopy types of certain diagrams of spaces.

The (naive) de Rham cohomology of a zero-dimensional scheme is the homology of the Kähler differential algebra of its coordinate ring, viewed as a complex. Since it is well-known to vanish in higher degrees if the ring is quasi-homogeneous, we concentrate on the affine case. For a general affine $K$-algebra $R=P/I$, where ${\rm char}(K)=0$ and $P=K[x_1,\dots,x_n]$, we prove a vanishing theorem for $H_{\rm dR}^m(R)$ based on the shape of a Macaulay basis of $dI\wedge \Omega^m_{P/K}$. For an Artinian local algebra $A=P/\langle f_1,\dots,f_n\rangle$, where $\{f_1,\dots,f_n\}$ is a super regular sequence, we show that $H_{\rm dR}^{\bullet}(A)$ is non-trivial in general, but trivial when the natural system of generators of the relation module of $\Omega^m_{A/K}$ is a standard basis. Moreover, we provide a detailed study of the dimension of $H_{\rm dR}^0(A)=\ker(d_A)$ for Artinian local rings $A=P/I$. The case of arbitrary affine zero-dimensional schemes is reduced to this case using a Galois splitting, and as a result we obtain the de Rham cohomology of a fat point scheme. Many explicitly computed examples and counterexamples support the results and indicate how subtle the de Rham cohomology of a zero-dimensional scheme is in general.

We consider the following singular Liouville equation $-\Delta v=\lambda V(x)|x|^2e^v \quad \text{in } B_1, \quad v=0 \quad \text{on } \partial B_1, $ where $B_1\subset\mathbb R^2$ is the unit disk, $\lambda>0$ is a small parameter, and $V$ is a positive smooth function. We first prove a non-degeneracy result for the bubbling solutions constructed in \cite{D-W-Z2026} by the local Pohozaev identities. Then we use this non-degeneracy result to establish the local uniqueness of bubbling solutions whose concentration parameters are sufficiently close.

Given a monoid that can be embedded in a group and that acts partially on a set, we show that under certain conditions the category associated with the partial action has a groupoid of fractions. By imposing some topological conditions on the action, we show that the groupoid is actually étale. This étale groupoid arises naturally in $C^{\ast}$-algebra theory.

We study finite-time concentration and convergence rates for projected two-time-scale stochastic approximation driven by a controlled Markov chain. The averaged fast map is contractive, while the slow iterate is projected onto a compact convex polyhedron. The associated projected ordinary differential equation may have a discontinuous vector field at the boundary, preventing a direct application of standard analyses based on Lipschitz vector fields. Using the Skorokhod map, we establish explicit high-probability bounds for tracking the moving fast equilibrium and the projected slow dynamics. These bounds separate martingale fluctuations, Markov-noise residuals, and the bias due to time-scale separation. A Lipschitz Lyapunov function satisfying a uniform decrease condition over fixed time intervals yields almost-sure convergence, with explicit last-iterate rates when the decrease admits a power lower bound. Under uniform Lyapunov contraction, polynomial step sizes yield joint fast-tracking and slow Lyapunov-error exponents arbitrarily close to $1/3$. Under the additional assumption that the reduced slow update map is a Euclidean contraction, logarithmically separated step sizes improve the joint rate to $O(n^{-1/2}\log n)$ almost surely, including for boundary equilibria. The same rate holds under a distinct geometric condition involving a strictly attracting face of a box and a fast equilibrium that is constant on that face. An actor-critic application achieves an almost-sure value-gap rate of $O(n^{-1}\log n)$ relative to the optimum within the constrained policy class. Further applications include projected TD(0) and projected stochastic gradient descent. We also extend the analysis to projection of the fast recursion under Euclidean contractivity.

We classify smoothly immersed Lagrangian $n$-balls, $n\geq 2$, in the unit ball of $\mathbb{C}^n$ with conformal Maslov form and Legendrian capillary boundary. The image of every such immersion is either an equatorial Lagrangian disk or is contained in a Whitney sphere centered at the origin. In dimension two, this confirms a conjecture of Li, Wang and Weng [Sci. China Math. 2021].

Theory for distribution matchers (DMs) is extended to distributions with memory. The divergence-minimizing DM is shown to select the sequences with the highest target probability, as in the memoryless case. A scaling law for divergence is extended to distributions driven by innovation processes. The theory is applied to parallel additive white Gaussian noise channels. A modified enumerative sphere-shaping (ESS) method with a weighted energy constraint is used in implementations. An illustrative example with three channels shows that joint ESS across channels reduces the rate loss by a large factor compared to product DMs at short blocklengths. The gains are confirmed by simulations with probabilistic amplitude shaping and a 5G-NR low-density parity-check code.

A graph $G$ is point determining, if no two vertices have the same neighborhoods. In this paper, we show that this property is recognizable from the deck of cards of a graph.

We study Bregman proximal gradient (BPG) algorithms under relative smoothness for convex, relatively strongly convex, and nonconvex objectives. Existing accelerated BPG algorithms for convex objectives typically require additional assumptions on Bregman divergences, most notably triangle-scaling conditions, which can lead to slower convergence rates. We propose a family of geometry-accelerated BPG algorithms that exploit Bregman geometry in both proximal-gradient and mirror-space updates, without imposing additional geometric conditions such as triangle-scaling conditions. Our methods adapt the stepsizes and mirror-space updates through local backtracking and computable acceptance criteria, without a global relative-smoothness constant as input. We derive convergence bounds in terms of the parameters accepted during the iterations. These bounds yield an $\O(k^{-2})$ rate for convex objectives and a linear rate for relatively strongly convex objectives when the geometry-acceleration parameters remain uniformly bounded. For nonconvex objectives, we establish an $\O(k^{-1})$ rate for a stationarity measure without requiring a lower Bregman bound or a full-domain Bregman divergence. Numerical experiments on inverse problems, entropy-regularized least squares, D-optimal design, and nonnegative matrix factorization demonstrate faster practical convergence than established Bregman baselines.

A compact formulation of magnetostatics is presented. Without using constitutive relations, including the aether relations, an expression is proposed for the potential energy of a region carrying a distribution of magnetic dipoles under a vector potential field, where the dipole distribution is specified by a current density potential field. Assuming that during a virtual motion of the dipole distribution the virtual work expended by the field is equal to minus the time derivative of the potential energy, an expression for the mechanical force functional is derived. In addition to the Lorentz and Kelvin force densities, the force functional contains an asymmetric active stress field, the skew-symmetric part of which is the mechanical couple density, and a pressure term.

This paper focuses on robust beamforming for orthogonal time frequency space (OTFS)-enabled massive multiple-input multiple-output (MIMO) systems under channel state information (CSI) uncertainty. In high-mobility satellite communications, uncertain CSI severely degrades beamforming accuracy and poses a major challenge to meeting users' quality of service (QoS) requirements. To address this challenge, we first formulate a chance-constrained optimization problem aiming to minimize the total transmit power while guaranteeing a predefined outage probability. Building on a 3D delay-Doppler-angle (DDA) channel representation, we propose a data-driven approach using support vector clustering (SVC) to model the uncertain CSI as an asymmetric uncertainty set. We then derive a robust counterpart that reformulates the intractable chance constraints into a deterministic semidefinite program. Finally, we design a graphics processing unit (GPU)-accelerated parallelizable alternating direction method of multipliers (ADMM) algorithm to address the computational complexity of large-scale antenna arrays. Simulation results show that the proposed SVC-based design reduces transmit power compared with conventional beamforming schemes, and that the GPU-accelerated ADMM achieves a speedup. These results confirm that the proposed framework achieves improved robustness, energy efficiency, and efficient computation in dynamic massive MIMO networks.

Data analytics offer a great opportunity for organizations within the cold-chain industry to become more sustainable and reduce their environmental footprint. Yet, despite recent progress in data sciences, they still face challenges to ensure data quality and streamline data processes through adapted full-scale ecosystems, limiting their ability to perform sustainable analytics. Most data techniques and tools stem from the data-science community and are not always suitably adapted to domain specificities. For instance, nearly all data-quality frameworks are designed to individually consider entities, i.e. representations of real-world elements. However, achieving environmental impact reduction in industrial settings, particularly within the cold-chain sector, often requires modelling and studying entire systems comprised of multiple interdependent entities. This publication challenges the application of conventional data science and engineering concepts in data ecosystems within the cold-chain sector and proposes an alternative approach to sustainably address prevalent challenges and thus optimize analytical processes.

In this short communication we answer negatively to a question posed by the authors in their work \cite[Section 1.4]{Brue2022} about the complex interpolation nature of the spaces $BV^{s}(\R^n)$, introduced by G.E. Comi and G. Stefani in \cite{ComiStefani2019}.

We study the maximum likelihood estimation of the coefficient {\beta} in well-specified Poisson regression. Using tools from empirical process theory and random conic geometry, we show that the probability of existence of the maximum likelihood estimator (MLE) exhibits a sharp phase transition at the threshold n > d. We then determine a minimum threshold exponential in the norm of {\beta} on the sample size n to guarantee with high probability an excess risk of the asymptotic order d/n. We reveal the existence of an intermediate regime in Poisson regression, when n is larger than d but smaller than this exponential threshold, where the MLE exists but does not achieve the optimal rate d/n. We close the gap between the two regimes up to a d^{1+\epsilon} term with {\epsilon} \in (0, 1) by providing an upper bound on the distance between the MLE and {\beta} whenever n > d^{1+\epsilon}. Along the way, we provide two generalizations of well-known PAC-Bayes inequalities regarding sub-Gamma random vectors and sub-Gamma random matrices that are of independent interest and that we use extensively to prove the main results of the present paper.

We solve the asymptotic Plateau problem in hyperbolic space for the curvature given by the normalized geometric mean of the $p$-fold sums of the principal curvatures. For every $n\ge3$, $2\le p\le n-1$, and $\sigma\in(0,1)$, each bounded domain $\Omega\subset\mathbb{R}^n$ with smooth mean-convex boundary admits a unique complete admissible vertical graph with prescribed curvature $\sigma$ and asymptotic boundary $\partial\Omega\times\{0\}$. The result extends the known cases $(n,p)=(3,2)$ and $(4,3)$ to the stated range. The key curvature estimate is uniform with respect to the positive boundary height in the approximating Dirichlet problems. The proof establishes two lower bounds for the full third-derivative quadratic form under the linearized constraints: a sharp universal bound with coefficient $2/p$ and a stronger direction-dependent bound when the relevant principal curvature lies in a specified range.

In this paper we completely classify the minimizers of a measure constraint global vectorial free boundary problem with a linear datum. More precisely, we show that there exists a unique minimizer whose coincidence set is an ellipsoid. We are motivated by the connection of the problem with the blow-ups of the vectorial Bernoulli free boundary problem and the obstacle problem.

In this paper, we study a numeration system with a non–integer base $a>1$ over the binary alphabet $A=\{0,1\}$: $$\left[0;\frac{r}{a-1}\right]\ni x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n} \equiv \Delta^{r_a}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}, \quad \alpha_n\in A.$$ We investigate the geometry of $r_a$–representation of numbers, including the geometric interpretation of digits, the structure of cylinder overlaps, and the associated metric properties. The main object of our study is a nowhere monotone function of unbounded variation defined by $$f\left(x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{2^n}\right) =\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n},$$ where the classical binary representation of the argument is assumed not to end with the infinite period $(1)$. We derive a system of two functional equations satisfied by $f$. We prove that the function has fractal level sets and a self–affine graph. We also evaluate the integral of $f$ over the interval $[0,1]$.

We develop an information-theoretic framework for generalization in next-token prediction under temporally dependent data. We consider independent trajectories generated by finite-memory Markov processes and distinguish algorithmic dependence, quantified by mutual information, from temporal dependence, characterized by mixing. For cross-entropy loss, we derive an expected generalization bound using the Donsker–Varadhan variational representation and a McDiarmid-type concentration inequality for Markov chains. A refinement captures the joint effect of context length and temporal mixing through the mixing properties of the history-state process. We then extend the bound through a rate–distortion formulation, replacing mutual information with the minimum information rate required to represent the learned model within a prescribed distortion in the generalization gap, yielding informative guarantees for deterministic algorithms over continuous hypothesis spaces. For margin-based prediction, we derive explicit bounds for linear and self-attention next-token predictors via noisy low-dimensional compression, revealing the roles of context length, model complexity, sample size, margin, and temporal mixing. Experiments on TinyStories and ETTh2 show that longer contexts can reduce both training and test losses, but typically reduce training loss more, enlarging the generalization gap. A complementary ETTh2 analysis identifies an effective predictive-memory scale near 24 hours, with no statistically supported improvement beyond this scale, offering a plausible explanation for test-performance saturation at larger contexts.

This paper focuses on the joint robust beamforming and resource allocation for space-air-ground integrated networks (SAGIN) under uncertain channel state information (CSI). In SAGIN, uncertain CSI undermines the precise adjustment of beamforming and resource allocation, posing a major challenge to meeting heterogeneous users' strict quality of service (QoS) requirements. To address this challenge, we first formulate a chance-constrained optimization problem to minimize the total transmit power while satisfying QoS requirements under a predefined outage probability. By leveraging semidefinite relaxation (SDR), the objective function is transformed into a linear function of the traces of the beamforming matrices. Then, we propose a deep neural network (DNN)-driven channel uncertainty learning to dynamically learn and model the uncertain CSI as an asymmetric uncertainty set. Under the constructed CSI uncertainty set, a robust counterpart method based on pre-trained network parameters is developed. It characterizes the uncertainty set as a finite union of convex sets, thereby providing a tractable approximation for the original chance constraints. Finally, we design an adaptive iterative algorithm to jointly optimize the resource allocation and beamforming vectors. Simulation results show that our proposed DNN-driven method outperforms traditional robust and non-robust methods, achieving a superior energy efficiency and robust reliability in SAGIN.

We formulate scattering experiments in terms of parameterized finite measures on outcome spaces and develop the associated information geometry defined by the Fisher metric. The decomposition of a scattering measure into total mass and normalized shape separates Fisher information into contributions from changes in overall scattering strength and redistribution among outcomes. Measurement processes are represented by probability or sub-probability kernels acting on the scattering measure. Such kernels contract Fisher information, placing a broad class of experimental operations within a common information-theoretic framework. This formulation connects scattering experiments to the theory of optimal experimental design through the Fisher geometry of scattering measures. The same finite-measure structure arises naturally in quantum measurements with incomplete detection, clarifying its place in the hierarchy from quantum measurement to classical outcome statistics.

Let $\mathcal{X}$ be a projective, geometrically irreducible, nonsingular algebraic curve of genus $g\ge2$ over an algebraically closed field of characteristic two. We classify cyclic subgroups of $\Aut(\mathcal{X})$ of order $N\ge2g+1$. Besides Kummer extensions of odd degree ramified over three points, precisely two cases occur: $y^2+y=x^m$, with $N=2m=4g+2$, and $y^2+y=c/(x^m+1)$, with $m$ odd, $c\ne0$, and $N=2m=2g+2$. In particular, $4\nmid N$. We then classify groups $H$ with a cyclic subgroup of index two and $|H|>4g+4$. They are precisely the groups $C_k\times D_{2M}$ on the curves $y^k=x^M+x^{-M}$, where $k,M\ge3$ are odd and coprime, and $2g=M(k-1)$. Their possible orders are $4g+2M$, where $M$ runs over certain odd divisors of $g$, and $|H|\le6g$. No curve in this family is ordinary. Hermitian curves occur in the family, and we give two sufficient conditions for maximality over finite fields. For dihedral groups the sharp bound is $4g+4$ in even genus and $4g$ in odd genus. The equality cases are given by explicit Artin–Schreier equations, and in each case the dihedral group is the full automorphism group.

For the fractional heat equation $\partial_t u + (-\Delta)^{\beta/2}u=0$ on $\mathbb{R}^d$ with $\beta\in(0,2)$, Weber and Zacher introduced the optimal constant $C_{\rm LY}(\beta,d)$ in the Li–Yau type estimate $$(-\Delta)^{\beta/2}\log u(t,\cdot)\leq \frac {C_{\rm LY}(\beta,d)}{t},\quad t>0.$$ They asked whether $C_{\rm LY}(\beta,d)$ converges to the classical Li–Yau constant $d/2$ as $\beta\uparrow2$. In this note we prove that the answer is affirmative. More precisely, for every fixed $d\geq1$ there exists a constant $K_d>0$ such that $$\frac{d}{\beta}\leq C_{\rm LY}(\beta,d)\leq\frac{d}{\beta}+K_d(2-\beta)\log\frac{e}{2-\beta},\quad 1\leq\beta<2.$$ Consequently, $\lim_{\beta\uparrow2}C_{\rm LY}(\beta,d)=d/2$. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber–Zacher's formula for $C_{\rm LY}(\beta,d)$ is attained at the origin. Second, the factor $2-\beta$ is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy–Itô decomposition plays an important role.

Let $\mathcal E$ be a separable Hilbert space, $B\in H^\infty_1(L(\mathcal E))$, and $\alpha>-1$. Associated with the Toeplitz operator $T_B$ on the vector-valued weighted Bergman space $A^2_{\alpha,\mathcal E}$ are the sub-Bergman spaces $\mathcal A_\alpha(B)$ and $\mathcal A_\alpha(B^*)$. In this paper, we prove that \[ \mathcal A_\alpha(B)\supset A^2_{\alpha-1,\mathcal H(B(0))} \quad\text{and}\quad \mathcal A_\alpha(B^*)\supset A^2_{\alpha-1,\mathcal H(B(0)^*)}. \] This confirms a conjecture of \cite{GLM26}. We also construct an infinite-dimensional two-sided inner function $B$ such that $\|B(0)\|<1$ and \[ \mathcal A_\alpha(B)\approx \mathcal A_\alpha(B^*)\approx A^2_{\alpha-1,\mathcal E}, \] but $B$ is not a finite Blaschke-Potapov product. This example shows that the finite-dimensional rigidity theorem of \cite{GLM26} does not extend to infinite-dimensional coefficient spaces.

We investigate two natural formulations of equal-length Littlewood–Paley–Rubio de Francia (LPR) inequalities on arbitrary Vilenkin systems \(\Gm\). The canonical formulation uses canonical intervals in \(\mathbb N\) of equal cardinality, while Vilenkin's formulation uses translates of a common initial interval under the group operation on the dual of \(\Gm\). In this paper, our main results show that the definitions of these two types of intervals exhibit dramatically different properties. More precisely, for the canonical formulation, we construct a counterexample showing that, for every \(2<p<\infty\), the Schatten class \(S^p\) fails the equal-length LPR property, in sharp contrast to the corresponding result on the torus; for Vilenkin's formulation, we prove that, for every \(2\le p<\infty\), a Banach space \(X\) has the equal-length LPR property if and only if it is UMD and has type~\(2\). The sufficiency proof combines a transference argument from the torus to finite cyclic groups with an embedding of Vilenkin intervals into digital rectangular envelopes, yielding bounds independent of the generating sequence.

We give a new proof that if the natural numbers are colored in two colors, then there exists a monochromatic Pythagorean triple. The proof was found by ChatGPT Astra.

We study wave packet density estimates in the local smoothing problem for the wave equation. Building on the framework of Gan, He, Li, and Wu, we refine the broad-incidence argument to obtain an improved local smoothing range in even dimensions. We then investigate the limitations of wave packet density methods, showing that genuine geometric obstructions already arise in the Euclidean setting.