We analyze the existence of Brownian motion tilted by a potential of full support on hyperbolic spaces $\mathbb{H}^d$. On compact spaces, it is classical that these path limits, called Q-processes, exist and can be directly defined using the ground state of the corresponding Schrödinger operator. On non-compact spaces like $\mathbb{H}^d$, the existence fails in general. We show that for \emph{stationary random} potentials on $\mathbb{H}^d$ with suitable spectral and sup norm bounds, the Q-processes exist a.s. For potentials that are factors of a Poisson point process, the method works up to sup norm $(d-1)^2/8$. In this case, we also show that the path limit can be approximated by periodic potentials. As a tool, we use the foliated space defined by the point process. It turns out that the global ground state of this foliated space serves as a substitute for the non-existing $L^2$ ground states on the leaves of the foliation. Restricting the global ground state to a leaf gives a generalized eigenwave that can be plugged into the usual machinery to get the Q-process.

We investigate the convex hulls of the eight dihedral symmetry classes of $n \times n$ alternating sign matrices, i.e., ASMs invariant under a subgroup of the symmetry group of the square. Extending the prefix-sum description of the ASM polytope, we develop a uniform core--assembly framework: each symmetry class is encoded by a set of core positions and an affine assembly map that reconstructs the full matrix from its core. This reduction transfers polyhedral questions to lower-dimensional core polytopes, which are better suited to the tool set of polyhedral combinatorics, while retaining complete information about the original symmetry class. For the vertical, vertical--horizontal, half-turn, diagonal, diagonal--antidiagonal, and total symmetry classes, we give explicit polynomial-size linear inequality descriptions of the associated polytopes. In these cases, we also determine the dimension and provide facet descriptions. The quarter-turn symmetry class behaves differently: the natural relaxation admits fractional vertices, and we need to extend the system with a structured family of parity-type Chvátal--Gomory inequalities to obtain the quarter-turn symmetric ASM polytope. Our framework leads to efficient algorithms for computing minimum-cost ASMs in each symmetry class and provides a direct link between the combinatorics of symmetric ASMs and tools from polyhedral combinatorics and combinatorial optimization.

Designing robust trajectories under uncertainties is an emerging technology that may represent a key paradigm shift in space mission design. As we pursue more ambitious scientific goals (e.g., multi-moon tours, missions with extensive components of autonomy), it becomes more crucial that missions are designed with navigation (Nav) processes in mind. The effect of Nav processes is statistical by nature, as they consist of orbit determination (OD) and flight-path control (FPC). Thus, this mission design paradigm calls for techniques that appropriately quantify statistical effects of Nav, evaluate associated risks, and design missions that ensure sufficiently low risk while minimizing a statistical performance metric; a common metric is Delta-V99: worst-case (99%-quantile) Delta-V expenditure including statistical FPC efforts. In response to the need, this paper develops an algorithm for risk-constrained trajectory optimization under operational uncertainties due to initial state dispersion, navigation error, maneuver execution error, and imperfect dynamics modeling. We formulate it as a nonlinear stochastic optimal control problem and develop a computationally tractable algorithm that combines optimal covariance steering and sequential convex programming (SCP). Specifically, the proposed algorithm takes a block-Cholesky approach for convex formulation of optimal covariance steering, and leverages a recent SCP algorithm, SCvx*, for reliable numerical convergence. We apply the developed algorithm to risk-constrained, statistical trajectory optimization for exploration of dwarf planet Ceres with a Mars gravity assist, and demonstrate the robustness of the statistically-optimal trajectory and FPC policies via nonlinear Monte Carlo simulation.

This article explores Rota-Baxter operators on finite-dimensional $ω$-Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, $ω$-Lie algebras, and Hom-Lie algebras via compatible Rota-Baxter operators on a given $ω$-Lie algebra. We also study the geometric structures of compatible Rota-Baxter operators of weight $0$ and isometric Rota-Baxter operators of weight $1$ over the field of complex numbers. In particular, we prove that the affine variety of all isometric Rota-Baxter operators of weight $1$ on any finite-dimensional non-Lie complex simple $ω$-Lie algebra is $1$-dimensional. Furthermore, we show that for every $4$-dimensional non-Lie complex $ω$-Lie algebra, there always exists a nilpotent compatible Rota-Baxter operator of weight $0$ such that the induced Hom-Lie algebra is nonabelian but solvable.

We introduce two reconstruction schemes that enable the recovery of a function in the entire Euclidean space $\mathbb{R}^n$ from local data $(u|_W, [(-Δ)^s u]|_W)$, where $W$ is an arbitrarily small nonempty open subset of $\mathbb R^n$ and $(-Δ)^s$ denotes the fractional Laplace operator of order $s\in (0,1)$. These procedures rely crucially on the weak Unique Continuation Property (UCP) for the fractional Laplacian. We apply these schemes to two distinct inverse problems. Following the seminal work from Ghosh et al., the first one concerns the recovery of a potential (Calderón-type problem) from the fractional Schrödinger equation under nonlocal Robin-type exterior conditions. The second one involves recovering the solution of the space-fractional heat equation in $\mathbb{R}^n$ from localized time-dependent measurements within a ball. To tackle these problems, we introduce new analytical tools such as a generalized weak Kelvin transform and a fractional Robin-to-Robin map. Finally, we provide numerical simulations for one of the reconstruction methods, illustrating the stability issues and the severe ill-posedness inherent to such inverse problems.

The information bottleneck (IB) method seeks a compressed representation of data that preserves information relevant to a target variable for prediction while discarding irrelevant information from the original data. In its classical formulation, the IB method employs mutual information to evaluate the compression between the original and compressed data and the utility of the representation for the target variable. In this study, we investigate a generalized IB problem, where the evaluation of utility is based on the $\mathcal{H}$-mutual information that satisfies the concave (\texttt{CV}) and averaging (\texttt{AVG}) conditions. This class of information measures admits a statistical decision-theoretic interpretation via its equivalence to the expected value of sample information. Based on this interpretation, we derive an alternating optimization algorithm to assess the tradeoff between compression and utility in the generalized IB problem.

Time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$ are discretised in time using collocation methods, which assume that the Caputo derivative of the computed solution is piecewise-polynomial. For such discretisations of any order $m\ge 0$, with any choice of collocation points, we give sufficient conditions for existence and uniqueness of collocation solutions. Furthermore, we investigate the applicability and performance of such schemes in the context of the a-posteriori error estimation and adaptive time stepping algorithms.

We study the codegree Turán density of $\mathcal{C}_\ell^r$, the $r$-uniform hypergraph tight cycle of length $\ell$. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is even, then the codegree Turán density of $\mathcal{C}_\ell^r$ is $1/2$. We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Turán density. That is, if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is odd, then the codegree Turán density of $\mathcal{C}_\ell^r$ can be at most $1/3$. Moreover, this bound is tight for infinitely many uniformities $r$ and all sufficiently large $\ell$ in the corresponding residue classes modulo $r$. Our proof makes use of a group-theoretic connection between Turán-type theorems for tight cycles and ``oriented colorings'' of the edge set of a hypergraph.

We propose PRISM-FCP (Partial shaRing and robust calIbration with Statistical Margins for Federated Conformal Prediction), a Byzantine-resilient federated conformal prediction framework that utilizes partial model sharing to improve robustness against Byzantine attacks during both model training and conformal calibration. Existing approaches address adversarial behavior only in the calibration stage, leaving the learned model susceptible to poisoned updates. In contrast, PRISM-FCP mitigates attacks end-to-end. During training, clients partially share updates by transmitting only $M$ of $D$ parameters per round. This attenuates the expected energy of an adversary's perturbation in the aggregated update by a factor of $M/D$, yielding lower mean-square error (MSE) and tighter prediction intervals. During calibration, clients convert nonconformity scores into characterization vectors, compute distance-based maliciousness scores, and downweight or filter suspected Byzantine contributions before estimating the conformal quantile. Extensive experiments on both synthetic data and the UCI Superconductivity dataset demonstrate that PRISM-FCP maintains nominal coverage guarantees under Byzantine attacks while avoiding the interval inflation observed in standard FCP with reduced communication, providing a robust and communication-efficient approach to federated uncertainty quantification.

We investigate elliptic fractional equations in the whole space, involving zero order perturbations of the fractional Laplacian $(-Δ)^s$, $0<s<1$. Our main objective is to determine appropriate radiation conditions at infinity that ensure existence and uniqueness of solutions to the fractional type Helmholtz equation. Extending classical scattering theory for the Helmholtz equation, we introduce and analyze suitable Sommerfeld type radiation conditions for fractional orders. A central contribution is the explicit computation of the outgoing free space Greens function for the operator $(-Δ)^s-k^{2s}$, for all $0<s<1$, any dimension and $k>0$, obtained via contour integration and a limiting absorption principle. We show that its asymptotic behavior at infinity coincides with a rescaled version of the classical Helmholtz fundamental solution, thereby justifying the standard Sommerfeld radiation condition for compactly supported sources. In addition, using resolvent estimates and a limiting absorption framework, we establish existence and uniqueness of outgoing solutions for compactly supported data, and for weighted sources. We further derive a convolution representation of the solution in terms of the outgoing fundamental solution. For inhomogeneous media with compactly supported perturbations, we reformulate the problem as a Lippmann Schwinger integral equation of Fredholm type and prove unique solvability away from a discrete set of frequencies. Our analysis provides a rigorous foundation for scattering theory of fractional Helmholtz operators and offers a framework suitable for numerical implementation of these nonlocal wave propagation models.

In this paper, we study the contractivity of nonlinear stochastic differential equations (SDEs) driven by deterministic inputs and Brownian motions. Given a weighted $\ell_2$-norm for the state space, we show that an SDE is incrementally noise- and input-to-state stable if its vector field is uniformly contracting in the state and uniformly Lipschitz in the input. This result is applied to error estimation for time-varying equilibrium tracking in the presence of noise affecting both the system dynamics and the input signals. We consider both Ornstein-Uhlenbeck processes modeling unbounded noise and Jacobi diffusion processes modeling bounded noise. Finally, we turn our attention to the associated Fokker-Planck equation of an SDE. For this context, we prove incremental input-to-state stability with respect to an arbitrary $p$-Wasserstein metric when the drift vector field is uniformly contracting in the state and uniformly Lipschitz in the input with respect to an arbitrary norm.

In this work we introduce a new family of symmetric tensors generalizing Killing tensors, that we call magnetic Killing symmetric tensors. We make use of them to construct first integrals for the magnetic flow associated to a given magnetic field. We apply the results to prove integrability of some invariant magnetic flows (either exact or non-exact) on some 2-step nilmanifolds: the Kodaira-Thurston manifold and Heisenberg nilmanifolds of higher dimensions.

In this article, we study the Schrödinger equation posed in the Euclidean space. We prove observability inequalities for measurable sets that are thick with respect to decaying densities. The proof relies on quantitative uncertainty principles adapted to decaying densities, notably those established by Shubin, Vakilian, Wolff, and Kovrijkine.

Given a uniformly expanding transitive Markov interval map, we show that within the set of ergodic measures the set of nonadapted ergodic measures is residual in with respect to the topology induced by the $\overline{d}$-metric. This set of measures is also shown to be path connected in many cases.

It has been proved by Boros and Makino that there is no output-polynomial-time algorithm enumerating the minimal redundant sets or the maximal irredundant sets of a hypergraph, unless P=NP. The same question was left open for graphs, with only a few tractable cases known to date. In this paper, we focus on graph classes that capture incidence relations such as bipartite, co-bipartite, and split graphs. Concerning maximal irredundant sets, we show that the problem on co-bipartite graphs is as hard as in general graphs and tractable in split and strongly orderable graphs, the latter being a generalization of chordal bipartite graphs. As for minimal redundant sets enumeration, we first show that the problem is intractable in split and co-bipartite graphs, answering the aforementioned open question, and that it is tractable on $(C_3,C_5,C_6,C_8)$-free graphs, a class of graphs incomparable to strongly orderable graphs, and which also generalizes chordal bipartite graphs.

Quantum relations in the sense of Weaver are $M'$-bimodules, for a von Neumann algebra $M$, these generalising actual relations on a set $X$ when $M=\ell^\infty(X)$. Similarly, relations between two sets can be generalised as bimodules over the commutants of two algebras. We make an explicit study of this idea, developing some tools to check that constructions are well-defined. Motivation comes from Kornell's concept of a Quantum Set (for algebras which are sums of matrix algebras), and we find that $*$-homomorphisms correspond to certain quantum relations, extending unpublished work of Kornell. We find a functor from completely positive maps to quantum relations, related to the idea of taking a noisy communication channel and reducing it to its underlying ``relation''. As with Quantum Graphs, at least in finite-dimensions, quantum relations correspond to ``adjacency operators'', certain CP maps depending on a choices of faithful functional on the algebras. We develop some tools to deal with the non-Schur-idempotent case, and show links with our functor from CP maps, and work of Verdon. We explicitly compute the adjacency operator of a $*$-homomorphism.

In this article, we characterize the range of $α$ for which the helical maximal function is bounded from $L^p(|x|^α)$ to itself for $3<p<\infty$. Our result is optimal for $4\leq p<\infty,$ except possibly at end-points.

In this paper, we establish a sharp lower bound for the spectrum of the Hodge Laplacian on Kähler hyperbolic manifolds. This bound is expressed explicitly in terms of the supremum norm of the 1-form associated with the Kähler hyperbolic structure. As an application, we obtain explicit spectral lower bounds for bounded symmetric domains.

For a left artinian ring A, we study the lattice torsA of torsion pairs in the category of finitely generated A-modules by considering an isomorphic lattice formed by certain closed sets in a topological space associated to A, the Ziegler spectrum of the unbounded derived category of ModA. Torsion pairs in torsA turn out to be adjacent if and only if the associated closed sets are related by an operation which is induced by mutation of cosilting complexes. We describe this operation from several perspectives and present a number of applications in the case when A is a finite dimensional algebra.

We prove new Sobolev type inequalities on compact Kähler manifolds with positive Ricci curvature. A proof of an already existing Sobolev inequality in the classical Bidaut-Véron and Véron approach is also discussed.

Building practical filtrations on objects to detect topological and geometric features is an important task in the field of Topological Data Analysis (TDA). In this paper, leveraging the ability of the Laplacian of Gaussian operator to enhance the boundaries of medical images, we define the G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration to generate the features more suitable for multi-parameter persistence module. By modeling volumetric images as bounded functions, then we prove the interleaving distance on the persistence modules obtained from our bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions. Finally, we conduct experiments on the MedMNIST dataset, comparing our bi-filtration against single-parameter filtration and the established deep learning baselines, including Google AutoML Vision, ResNet, AutoKeras and auto-sklearn. Experiments results demonstrate that our bi-filtration significantly outperforms single-parameter filtration. Notably, a simple Multi-Layer Perceptron (MLP) trained on the topological features generated by our bi-filtration achieves performance comparable to complex deep learning models trained on the original dataset.

Consider the budget-constrained random graph process introduced by Frieze, Krivelevich and Michaeli, where each time an edge is offered through the (standard) random graph process we must irrevocably decide whether to "purchase" this edge or not, with our goal being to construct a graph which satisfies some property within a given time $t$ and while purchasing at most $b$ edges. We consider the problem of constructing graphs containing certain fixed small subgraphs. We provide an optimal strategy for building a graph which contains a copy of $K_4$, showing that budget $b=ω(\max\{n^8/t^5,n^2/t\})$ suffices and that if $b=o(\max\{n^8/t^5,n^2/t\})$ then no strategy can a.a.s. produce a graph containing a copy of $K_4$. This resolves a problem raised by Iľkovič, León and Shu. More generally, we obtain analogously tight results for containing a wheel of any fixed size, or a graph consisting of a tree plus one additional universal vertex. We also tackle the problem of constructing graphs containing a copy of $K_5$, obtaining both lower and upper bounds on the optimal budget, though a gap remains in this case.

We prove that hypergraphs defined by low-degree polynomial inequalities contain large homogeneous subsets. Formally, let $H$ be an $r$-uniform hypergraph on $N$ vertices that is semialgebraic of constant description complexity, and each defining polynomial has degree at most $D$. Then $H$ contains a clique or an independent set of size $n$, where $N\leq \mbox{tw}_{3D^3}(n)$.

The work investigates the problem of whether a context-free language is a subset of a group language. A.~V. Anisimov has shown that the problem of determining the unambiguity of finite automata is a special case of this problem. Then the question of finding polynomial algorithm verifying the inclusion of context-free languages in group languages naturally arises. The article focuses on this open problem. For the purpose, the paper describes an unconventional method of description of context-free languages, namely a representation with the help of a finite digraph whose arcs are labelled with a specially defined monoid $\mathcal{U}$. Also, we define a semiring $\mathcal{S}_\mathcal{U}$ whose elements are the set $2^\mathcal{U}$ of all subsets of $\mathcal{U}$ and with operations - product and union of the elements of $2^\mathcal{U}$. The described algorithm executes no more than $O(n^3)$ operations in $\mathcal{S}_\mathcal{U}$.

In \cite{NZ25}, the authors resolved the rational function analogue of the finiteness results for greatest common divisors of iterates of polynomials established in \cite{HT17}. These results may be viewed as dynamical generalizations of a classical problem concerning upper bounds for the greatest common divisors (GCDs) of two integer sequences studied by Bugeaud, Corvaja, and Zannier. The most delicate case arises when the maps involved are automorphisms, where the methods of \cite{NZ25} and \cite{HT17} rely heavily on Diophantine approximation and asymptotic analysis. In the present paper, we develop an alternative approach to the automorphism case. This method is more powerful, allowing us to give complete answers to the further questions posed in \cite{HT17}. In particular, we strengthen the main theorem of \cite{HT17} and provide an alternative proof of the main theorem of \cite{NZ25} in the automorphism setting. Moreover, we relate this dynamical GCD problem to a special case of a higher-dimensional generalization of the Dynamical Mordell--Lang Conjecture proposed by Junyi Xie. We establish this generalized conjecture when the dynamics arise from algebraic group actions. In addition, we resolve the corresponding special case associated with dynamical GCD questions when the maps involved are polynomials.

Chain-of-thought (CoT) monitors are LLM-based systems that analyze reasoning traces to detect when outputs may exhibit attributes of interest, such as test-hacking behavior during code generation. In this paper, we use information-theoretic analysis to show that non-zero mutual information between CoT and output is a necessary but not sufficient condition for CoT monitorability. We identify two sources of approximation error that may undermine the performance of CoT monitors in practice: information gap, which measures the extent to which the monitor can extract the information available in CoT, and elicitation error, which measures the extent to which the monitor approximates the optimal monitoring function. We further demonstrate that CoT monitorability can be systematically improved through targeted training objectives. To this end, we propose two complementary approaches: (a) an oracle-based method that directly rewards the monitored model for producing CoTs that maximize monitor accuracy, and (b) a more practical, label-free approach that maximizes conditional mutual information between outputs and CoTs. Across multiple different environments, we show both methods significantly improve monitor accuracy while preventing CoT degeneration even when training against a monitor, thereby mitigating reward hacking when the task reward is imperfectly specified.

We study the $L^q$-regularity of the density of barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). According to a previous result by the first author and collaborators, if one marginal is absolutely continuous, so is the $W_p$-barycenter. The next natural question is whether the $L^q$- regularity on the marginals is also preserved for any $q > 1$, as in the classical case ($p=2$) of Agueh--Carlier, or for $W_p$-geodesics ($N=2$). Here we prove that this is the case if one marginal belongs to $L^q$ and the supports of all the marginals satisfy suitable geometric assumptions. However, we show that, as soon as $N>2$, it is possible to find examples of $W_p$-barycenters which are not $q$-integrable, even if one marginal is compactly supported and bounded, thus highlighting the role played by the geometry of the supports. Furthermore, we provide a general estimate of the $L^q$-norm, including a detailed study of the sources of singularities, and a characterization of the $W_p$-barycenters à la Agueh--Carlier in terms of the associated Kantorovich potentials. Finally, we explicitly compute the $W_p$-barycenters of measures obtained as push-forward of special affine transformations. In this case, regularity holds without any additional requirement on the supports.

We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-Δ_{g_\mathcal{M}} u = f(r)$ in a model manifold $\mathcal{M} = [0,S) \times_h \mathbb S^{N-1}$ with warping function $h$. The variable $r$ ranges in the interval $[0,S)$, whose endpoint $S$ is positive and possibly infinite. The first part of the paper deals with the problem \[ \begin{array}{ll} -Δ_{g_\mathcal{M}} {u}=f(r) &\mbox{in $Ω$}, u=φ(r) &\mbox{on $\partial Ω$}, \frac{\partial u}{\partial ν} = κ(r) &\mbox{on $\partial Ω$}, \end{array} \] where $Ω\subset \mathcal{M}$ is a bounded domain containing the point $O \in \mathcal{M}$ corresponding to $r = 0$, $ν$ is the exterior unit normal vector on $\partial Ω$, and $f$, $φ$, $κ$ are three prescribed functions. In the second part of the paper, we consider a similar overdetermined problem for the exterior Bernoulli problem in a domain $Ω\setminus \overline B_{R_0}(O)$, where $B_{R_0}(O)$ denotes the geodesic ball centered at $O$ with radius $R_0$, within the class of functions that vanish on $\partial B_{R_0}(O)$. In both cases, we give conditions on $f$, $φ$ and $κ$ implying that the solution $u$ is radial and $Ω$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms $\mathbb{R}^N$, $\mathbb{H}^N$ and $\mathbb{S}^N$.

We show that the images of the Bethe subalgebras of the Yangians $Y(\mathfrak{gl}_{n})$ and $Y(\mathfrak{gl}_{m})$ under the homomorphisms to $U(\mathfrak{gl}_{n+m})$ given by the Olshanski centralizer construction coincide. We use this result to obtain the $(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality of the trigonometric Gaudin model and the XXX-spin chain. The duality is obtained in an explicit way relating the generating differential operator on one side and the generating difference operator on the other, thus agreeing with the result of Mukhin, Tarasov and Varchenko arXiv:math/0605172.

Deploying multiple beyond diagonal reconfigurable intelligent surfaces (BD-RISs) can potentially improve the communication performance thanks to inter-element connections of each BD-RIS and inter-surface cooperative beamforming gain among BD-RISs. However, a major issue for multi-BD-RISassisted communication lies in the channel estimation overhead - the channel coefficients associated with the off-diagonal elements in each BD-RIS's scattering matrix as well as those associated with the reflection links among BD-RISs have to be estimated. In this paper, we propose an efficient channel estimation framework for double-BD-RIS-assisted multi-user multipleinput multiple-output (MIMO) systems. Specifically, we reveal that high-dimensional cascaded channels are characterized by five low-dimensional matrices by exploiting channel correlation properties. Based on this novel observation, in the ideal noiseless case, we develop a channel estimation scheme to recover these matrices sequentially and characterize the closed-form overhead required for perfect estimation as a function of the numbers of users and each BD-RIS's elements and channel ranks, which is with the same order as that in double-diagonal-RIS-aided communication systems. This exciting result implies the superiority of cooperative BD-RIS-aided communication over the diagonal- RIS counterpart even when channel estimation overhead is considered. We further extend the proposed scheme to practical noisy scenarios and provide extensive numerical simulations to validate its effectiveness.