In this paper, it is shown, for the first time, that centralized performance is achievable in decentralized learning without sharing the local datasets. Specifically, when clients adopt an empirical risk minimization with relative-entropy regularization (ERM-RER) learning framework and a forward-backward communication between clients is established, it suffices to share the locally obtained Gibbs measures to achieve the same performance as that of a centralized ERM-RER with access to all the datasets. The core idea is that the Gibbs measure produced by client~$k$ is used, as reference measure, by client~$k+1$. This effectively establishes a principled way to encode prior information through a reference measure. In particular, achieving centralized performance in the decentralized setting requires a specific scaling of the regularization factors with the local sample sizes. Overall, this result opens the door to novel decentralized learning paradigms that shift the collaboration strategy from sharing data to sharing the local inductive bias via the reference measures over the set of models.

It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.

This paper develops a co-state based fusion frame work for spacecraft navigation, consistency monitoring, and hazard forecasting. A differential algebraic co-state is introduced as an instantaneous Lagrange multiplier that enforces measurement dynamics compatibility at the differential level and provides a physically interpretable signal of geometric inconsistency. On a longer time scale, co-state and innovation trajectories are used to learn a continuous time Markov generator governing transitions between coarse behavioural regimes, enabling intrinsic probabilistic risk forecasting through mode probabilities and mean first-passage time (MFPT). The resulting architecture unifies geometric projection, stochastic inference, and probabilistic risk assessment in a single online pipeline without requiring predefined fault models, labelled failure data, or heuristic thresholds. The framework is demonstrated on real lunar powered-descent telemetry, where it detects structural internal model inconsistency significantly earlier than physical divergence or statistical inconsistency in an Extended Kalman Filter (EKF). The results show that geometric inconsistency, stochastic drift, and probabilistic risk rise coherently prior to failure, yielding interpretable and operationally meaningful early-warning capability for autonomous landing systems.

We derive a topological decoupling of the equations of modified nodal analysis (MNA) to a semi-explicit index one differential-algebraic equation. The decoupling explicitly allows for controlled sources, which play a crucial role in engineering design workflows. Furthermore, the proof is constructive and provides a graph-based algorithmic framework for the computation of the decoupling, enabling its application to a variety of industry problems. These include the generation of consistent initial conditions, model order reduction, (scientific) machine learning, as well as speeding up conventional circuit simulation. In addition, the decoupling preserves the structure of MNA, i.e. the resulting systems remain sparse and key parts remain positive definite. We illustrate the decoupling using multiple examples, including some of the most common subcircuits containing controlled sources. Lastly, we also provide a first software implementation of the decoupling.

We address a short-wave asymptotic for one class of quasi-linear second order PDE systems involving the cross-diffusion described by the so-called Patlak--Keller--Segel law. It is common to employ these equations for modelling the predator--prey community with the prey-taxis that means the interactions of two species of particles or cells or anything else through which the species called "predators" is capable of moving directionally while searching for the other species called "prey." However, we suppose the predators to be sensitive not to the prey density but to a driving signal produced by the prey. Additionally, the production of the driving signal is assumed to be sensitive to the intensity of an external field, which is independent from the community state. This is what we call the external signal. It can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. We assume that the external signal takes a general short-wave form and construct a complete asymptotic expansion for the short-wave solutions with no restrictions on the spatial dimension or kinetics of inter/intraspecific reactions. Further, we apply the short wave asymptotic to studying the stability or instability induced by the external signal following Kapitza' theory for the upside-down pendulum. Applying the general results to some special classes external signals, we get examples of suppressing the taxical transport, examples of robustness of the species equilibrium to the signal or, oppositely, blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium. These results contribute to filling the gap in the literature, since the theory and techniques for the asymptotic integration of systems described above represent a weakly charted area.

We consider the problem on the existence of two dimensional superintegrable systems in the presence of a magnetic field in the two dimensional Euclidean space. We assume the existence of two integrals of motion, besides the Hamiltonian, that are quadratic polynomials in the momenta. This problem was already studied in the cases where one integral is of Cartesian or polar type [J. Bérubé, and P. Winternitz, J. Math. Phys., 45(5): 1959-1973, 2004]. We continue the investigation by assuming that one of the integrals is of parabolic type and the second integral is of elliptic or (''non-standard'') parabolic type, confirming so far that, on the Euclidean plane, the only two dimensional superintegrable system with quadratic integrals is the one with constant magnetic field and constant electrostatic potential.

In this paper, we first extend the explicit formula \cite{gerard2023explicit} for the classical Benjamin-Ono equation to each flow of the Benjamin-Ono hierarchy on line. We then use this representation to derive two main applications. First, we obtain a complete classification of traveling wave solutions for all higher-order flows in the hierarchy. Second, we analyze the zero-dispersion limit for the corresponding small-dispersion flows. For every fixed time $t\in\mathbb R$, we prove that, at any time, the solution converges weakly in $L^2(\mathbb R)$ as the dispersion parameter tends to $0$, and we provide a geometric characterization of the limit in terms of an alternating sum, which yields the higher-order analogue of the formula obtained in \cite{miller2011zero}, \cite{Gerard2025small} for the Benjamin-Ono equation.

We study how strategic interaction can arise from controlled quantum dynamics rather than being imposed as an external mathematical structure. We introduce a class of interaction-defined quantum games in which players are represented by distinguishable quantum walkers, strategies correspond to local coin operations, and payoffs are defined as expectation values of physical observables. Using interacting discrete-time quantum walks as a concrete platform, we demonstrate numerically that competitive, cooperative, and asymmetric games admit stable stationary strategy profiles when the walkers are coupled, while no non-trivial equilibria exist in the absence of interaction. To clarify the game-theoretic structure, we derive an analytic perturbative decomposition of the payoff function in the weak-interaction regime, showing explicitly that strategic coupling originates from interaction-induced interference terms in the joint probability distribution. For a collision-based phase interaction, the payoff becomes non-separable at first order in the interaction strength and generically admits stationary points satisfying the Nash conditions. Our results provide a physically explicit realization of strategic interdependence in quantum transport processes and establish interacting quantum walks as a minimal platform for studying game-theoretic behavior emerging from unitary dynamics.

We discuss some mathematical aspects of the Mori-Zwanzig projection operator formalism. The core of the Mori-Zwanzig formalism is the generalised Langevin equation, which is typically derived from the Dyson-Duhamel identity. We derive the projection operator formalism for Mori's projection by means of semigroup theory, and we illustrate where rigorous methods fail for the case of Zwanzig's projection. For bounded perturbations of the time-evolution operator (e.g. for Mori's projection), the Dyson-Duhamel identity coincides with the variation of constants formula. For unbounded perturbations (e.g. for Zwanzigs's projection), the Dyson-Duhamel identity should be considered an equation for the orthogonal dynamics, for which the existence of unique solutions has yet to be established. Then we recall that all properties of Mori's generalised Langevin equation follow directly from the well-posedness of Volterra equations, irrespective of the projection operator formalism. Further, we discuss the use of Mori's generalised Langevin equation as a coarse-grained model. Finally, we illustrate that the memory term is a coupling term that is not necessarily related to memory. To this end, we introduce projections onto subspaces of 'fast' and 'slow' variables that are associated with the spectral decomposition of skew-adjoint operators. For these projections, the memory term vanishes.

We prove that if $X$ is a smooth Fano threefold and $L$ is an ample $\mathbb{Q}$-divisor such that $(X,L)$ is K-polystable, then the automorphism group $\operatorname{Aut}(X)$ is reductive. This verifies the reductivity statement predicted by the Yau--Tian--Donaldson conjecture in the setting of smooth Fano threefolds with arbitrary ample polarisation.

We study stochastic extinction for a class of Markov processes motivated by models in ecology and epidemiology. Extinction is often characterized by a boundedness condition and a condition on boundary Lyapunov exponents (invasion rates). While the latter is typically sharp, the former is often restrictive and can be improved. Building on the ideas initiated in \cite{benaim2018stochastic}, we develop a streamlined approach that relaxes this boundedness condition and yields concise and accessible criteria for extinction. In particular, we establish extinction criteria in two settings: with and without a linearly bounded quadratic variation condition. In the first case, our result is comparable to, and slightly improves upon, the main results in \cite{foldes2024stochastic}. In the second case, where the quadratic variation is not linearly bounded, we obtain new extinction results that fall outside the scope of existing frameworks. Several examples are provided to illustrate the applicability of our results and to highlight situations where previous conditions are not practically verifiable.

This paper investigates the so-called reward-balancing methods, a novel class of algorithms for solving discounted-return reinforcement learning (RL) problems. These methods consist of iteratively adjusting the reward function to transform the RL problem into an equivalent one in which the optimal policies are greedy. For this procedure, referred to as normalization process, we provide a theoretical analysis of the involved transformations, emphasizing their algebraic structure. Then, we introduce a control-theoretic reformulation, recasting the reward-balancing procedure into an optimal control framework. The approach is further extended to address model uncertainty through stochastic model sampling, yielding normalization guarantees and probabilistic bounds on stochastic fluctuations. Using the proposed optimal control framework within a scenario model predictive control (MPC) setting, we demonstrate, through simulation studies, performance improvements over the current state-of-the-art.

Reconfigurable distributed antennas and reflecting surface (RDARS) has emerged as a transformative solution to address the stringent requirements of future wireless networks. By combining distributed active antennas with reconfigurable passive reflecting surfaces, RDARS integrates the advantages of both active transmission and passive wave control in a cost-effective and energy-efficient manner. This hybrid architecture enables enhanced coverage, improved spectral efficiency, and seamless support for integrated communication and sensing. In this article, we first introduce the fundamental architecture and working principles of RDARS, followed by practical benefits and comparisons with recently proposed intelligent surface variants. We then highlight the signal-to-noise ratio (SNR) gains in representative applications of RDARS-aided communication and sensing scenarios, where RDARS demonstrates clear advantages over conventional reconfigurable intelligent surfaces. Finally, we outline key challenges related to practical implementation and resource allocation, and discuss potential research directions. With its unique hybrid mode synergy, RDARS is envisioned to play a pivotal role in shaping the evolution of next-generation intelligent communication systems.

In this paper, we consider a parabolic counterpart of Serrin's overdetermined problem, in which the overdetermined condition (constant flux condition) is imposed only on a discrete infinite set of time values. We show that, under suitable regularity assumptions on the domain, such a discrete-time overdetermined problem admits a solution if and only if the domain is a ball. Remarkably, depending on the temporal scale, the same overdetermined condition captures either geometric or spectral information, yet both mechanisms lead to the same rigidity conclusion. We study both the case in which the constant flux condition is imposed on the boundary and the case in which the constant flux condition is imposed on an interior surface. We remark that the methods employed in our analysis do not depend on the location of the overdetermined surface but only on whether the sequence of time instants accumulates away from zero. Finally, we will show how this problem generalizes to complete Riemannian manifolds.

In this paper, we study finite subgroups $G\subset\mathrm{Aut}(\mathbb{P}^n)$ such that $\mathbb{P}^n$ is $G$-birationally rigid. For each $n\geqslant 3$, we prove that $\mathrm{Aut}(\mathbb{P}^n)$ contains at most finitely many such subgroups up to conjugation. For $n=4$, we prove that $\mathbb{P}^4$ is $G$-birationally superrigid if $G\simeq\mathrm{PSp}_{4}(\mathbf{F}_3)$.

We classify pairs $(X,G)$ consisting of a (possibly singular) cubic threefold $X\subset\mathbb{P}^4$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally rigid, i.e., $X$ is a $G$-Mori fiber space (over a point), and $X$ is not $G$-birational to any $G$-Mori fibre space that is not $G$-biregular to $X$.

We complete the classification of regular generically free actions of finite groups on del Pezzo surfaces, up to birational equivalence. As a byproduct, we settle several open problems in equivariant birational geometry, e.g., we classify birationally rigid actions on del Pezzo surfaces.

We develop a likelihood-based inference for finite-state birth-death processes with composite birth rates, in which multiple distinct mechanisms contribute additively to the total birth intensity. Our main motivating example is an SIS epidemic model with pairwise and higher-order transmission. The process is observed through a single aggregate trajectory, and in the main setting of interest, birth events are unmarked. This creates a deconvolution problem in event space: the state is one-dimensional, but the mechanism underlying each birth is latent. We formulate the inference under a Doob $h$-transformed $Q$-process, which is time-homogeneous and ergodic and which provides a time-homogeneous asymptotic surrogate for the law of the original process conditioned on long survival. We derive the corresponding conditional likelihood and study both the conditional maximum likelihood estimator and a quasi-maximum likelihood estimator which is based on a simplified working score. Under the Doob-transform law, we prove consistency and asymptotic normality for both estimators, with asymptotic covariance determined by the inverse Fisher and inverse Godambe information matrices, respectively. We also showcase a practical one-dimensional test for the presence of a specific higher-order birth mechanism.

We show that generic automorphisms of stable groups are supertight in a strong sense. In particular, we obtain the existence of supertight automorphisms. We also answer a question concerning the relationship between supertight automorphisms of $\mathrm{PGL}_2(K)$ and generic automorphisms of the underlying field $K$. Moreover, we provide partial evidence-already suggested by Hrushovski-toward the principle that ``fixed points are pseudofinite'' in the setting of generic automorphisms of simple groups of finite Morley rank.

Gaussian processes are widely used for accurate emulation of unknown surfaces in sequential design of expensive simulation experiments. Integrated mean squared error (IMSE) is an effective acquisition function for sequential designs based on Gaussian processes. However, existing approaches struggle with its implementation because the required integrals often lack closed-form expressions for most kernel functions. We propose a novel and computationally efficient Hilbert space Gaussian process approximation for the IMSE acquisition function, where a truncated eigenbasis representation of the integral enables closed-form evaluation. We establish sharp global non-asymptotic bounds for both the approximation error of isotropic kernels and the resulting error in the acquisition function. In a series of numerical experiments with $γ$-stabilizing, the proposed method achieves substantially lower prediction error and reduced computation time compared to existing benchmarks. These results demonstrate that the proposed Hilbert space Gaussian process framework provides an accurate and computationally efficient approach for Gaussian process based sequential design.

We construct a free Poisson algebra endowed with a Rota-Baxter operator. The same construction works for a free Poisson algebra endowed with a Nijenhuis operator.

We show that the $ L^2({\mathbb R}) $-unconditional well-posedness, that is well-known for the KdV equation, is shared by KdV type equations with weaker dispersion. This is despite the difference in the nature of these equations, which are quasilinear while KdV is semilinear. More precisely we prove that the low dispersion fractional KdV equation $$ \partial_t u -D_x^α\partial_x u +\partial_x(u^2)=0 $$ is unconditionally globally well-posed in $L^2({\mathbb R}) $ for $α\in ]\frac{55}{38},2] $. Our method of proof combined refined bilinear estimates with the energy method enhanced with Bourgain's type estimates developed in Molinet-Vento (2015).

Let $τ(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} τ\left( \left[ \frac{x}{n_1 n_2} \right] \right) = \sum_{n \leqslant x} τ\left( \left[ \frac{x}{n} \right] \right) τ(n), $$ where $[t]$ denotes the integral part of the real number $t$. By establishing new estimates for a class of three-dimensional exponential sums with constant perturbation, we obtain an improved asymptotic formula for $T(x)$. In particular, we show that for any $\varepsilon > 0$, the error term in the asymptotic expansion of $T(x)$ is bounded by $O(x^{17/30+\varepsilon})$. This result breaks the $4/7$-barrier which corresponds to the application of the classical divisor problem conjecture $1/4+\varepsilon$.

Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of matrices which are obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for multivariable Vandermonde determinants as a sum of completely factorising terms, each of which is a Vandermonde determinant in fewer variables. As an application, we deduce an elementary proof of the multiplicativity of the transfinite diameter for products of compact sets.

We study the $l_p$ Hausdorff distance from convex hull of a compact set $A\subset\mathbb{R}^n$, which is the distance \begin{equation*} d^{(l_p)}(A):=\sup_{x\in conv(A)}\inf_{a\in A}\|x-a\|_p, \end{equation*} where $\|\cdot\|_p$ is the $l_p$-norm on $\mathbb{R}^n$. We prove that when $n=2$ and $1\leq p<\infty$, the function $(d^{(l_p)})^p$ is subadditive with respect to Minkowski summation, up to multiplication by the factor $\max\{1,2^{p-2}\}$, and we observe that this bound is sharp.

In frequency division duplex massive multiple-input multiple-output systems, downlink channel state information must be fed back within a limited uplink budget. While transform coding with Karhunen-Loeve transform and reverse water-filling is rate-distortion optimal for Gaussian channels, its performance is limited by basis mismatch between the user and base station. We analyze this mismatch and propose a practical architecture separating long-term basis feedback from short-term coefficient quantization. Using a random vector quantization, we derive a closed-form end-to-end mean square error expression. This allows us to characterize the optimal rate split and identify a phase transition threshold for basis updates. Simulations on correlated Gaussian and COST2100 channels demonstrate near-optimal performance, robustness to update overhead, and significant complexity reduction compared to deep-learning-based autoencoders.

In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$ on the boundary, the solution $g(t)$ to the normalized Ricci flow $(1.2)$ which is continuous up to the boundary, exists for all $t>0$ and converges locally uniformly in $B_{r_0}(0)$ to a complete hyperbolic metric as $t\to\infty$(see Theorem 1.2 for details). Moreover, the sectional curvature of $g(t)$ maintains less than $-1$ for $t>0$. For dimension $2$, to achieve such a convergence result, we need the additional assumption that the mean curvature on the boundary increases in a certain speed to infinity as $t\to\infty$.

We study the rate-cost tradeoff in rate-limited control of general stochastic control systems, including nonlinear systems, over a finite horizon. At each time step, an encoder observes the state and transmits a description to a controller, which then selects the control action. For an average control-cost threshold $D$, we characterize the minimum achievable communication rate $R_n(D)$ via a nonasymptotic bound: $R_n(D)$ lies within an additive logarithmic gap of the optimal value of a directed-information minimization $F_n(D)$, namely, we show that $F_n(D) \le R_n(D) \le F_n(D)+\log \bigl(F_n(D)+3.4\bigr)+2+\frac{1}{n}$, in bits. This establishes directed information as the operationally relevant quantity governing rate-limited control, thereby broadening its utility beyond its previously established roles in causal source coding and linear quadratic Gaussian (LQG) control to general nonlinear control systems. We prove the upper bound constructively by building an encoding-and-control policy using the strong functional representation lemma at each time step. As special cases of our setting, our framework yields nonasymptotic bounds for sequential (causal) rate-distortion and LQG control.

Through adaptive antenna repositioning, the movable antenna (MA) technology enables on-demand reconfiguration of wireless channels, thereby creating an additional spatial degree of freedom in improving communication performance. This paper investigates a multiuser uplink communication system aided by MAs, where a base station (BS) equipped with multiple MAs serves multiple single-antenna users. Specifically, given that an optimized array geometry cannot guarantee rate fairness, we focus on designing antenna trajectory at the BS to maximize the minimum achievable rate among all users over a finite time period. The resulting optimization problem is fundamentally challenging to solve due to the continuous-time nature. To address it, we first examine an ideal case with infinitely fast MA movement and demonstrate that the relaxed problem can be optimally solved via the Lagrangian dual method. The obtained trajectory solution reveals that the BS should employ a finite set of MA deployment patterns, each allocated an optimal time duration. Building on this, we then study the general case with limited MA movement speed and propose a heuristic trajectory design inspired by the optimal patterns identified in the ideal scenario. Several insights are also gained by examining the simplified special case. Finally, numerical results are provided to validate the effectiveness of the proposed designs compared to competitive benchmarks.

Solving inverse problems requires appropriate regularization techniques to ensure well-posedness and stability. In recent years, denoiser-driven methods have emerged as effective regularization strategies, achieving state-of-the-art performance in various imaging applications. However, their stability and convergence within iterative regularization frameworks remain largely unexplored. In this work, we extend the framework of Regularization by Denoising (RED) by introducing a novel denoiser-driven iterative regularization scheme, referred to as \texttt{DDIR}, that incorporates a new regularization functional based on averaged denoisers. The proposed approach employs an adaptive step-size strategy together with an \emph{a posteriori} stopping rule to ensure stability while alleviating oscillatory behavior and semi-convergence effects induced by noise. As our main theoretical contribution, we prove that the resulting reconstruction method constitutes a stable and convergent regularization scheme in the classical sense. To the best of our knowledge, this provides the first rigorous justification of \texttt{DDIR} within the framework of regularization theory. Finally, we demonstrate the performance of the proposed method through numerical experiments on image deblurring and phase retrieval Computed Tomography (CT) using three denoisers, namely median, TNRD, and TV proximal. The results highlight the effectiveness of the method in terms of reconstruction accuracy and computational efficiency.