For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disc, and they generalise the conjugacy class example of Alekseev, Malkin and Meinrenken (which appears in the simple pole case). Using the `fusion product' in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy/Stokes data of meromorphic connections over arbitrary genus Riemann surfaces, together with a new proof of the symplectic nature of isomonodromic deformations of such connections.

Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate $C^*$-algebras.

Let $(\mathcal{O}_n, \mathfrak{m})$ denote the ring of germs of holomorphic functions $\mathbb{C}^n\to \mathbb{C}$, and let $I\subseteq \mathcal{O}_n$ be an $\mathfrak{m}$-primary ideal. Demailly and Pham showed that $\mathrm{lct}(I) \geq \frac{1}{e_1(I)} + \dots + \frac{e_{n-1}(I)}{e_n(I)}$, where $e_j(I)$ is the mixed multiplicity $e(I,\dots, I, \mathfrak{m},\dots, \mathfrak{m})$, with $I$ repeated $j$ times and $\mathfrak{m}$ repeated $n-j$ times. We generalize the lower bound to the case of an arbitrary ideal of an excellent regular local (or standard-graded) ring of equal characteristic, with $\mathrm{lct}(I)$ replaced by the $F$-threshold $c^{\mathfrak{m}}(I)$ in positive characteristic. Our main result is a classification of homogeneous ideals in polynomial rings for which the lower bound is attained, resolving a conjecture of Bivià-Ausina in the graded case.

In this paper, we derive general bright-dark soliton solutions to the coupled Sasa-Satsuma (CSS) equation using the Kadomtsev-Petviashvili (KP) reduction method. Since the CSS equation is a special case of the four-component Hirota equation, our approach begins with the construction of two-bright-two-dark soliton solutions for the four-component Hirota equation. By imposing specific parameter constraints, these solutions are subsequently reduced to the bright-dark soliton solutions of the CSS equation. Finally, the dynamical behaviors of the one- and two-bright-dark soliton solutions are thoroughly analyzed and illustrated.

Let $f:2^{E} \rightarrow \mathbb{Z}_+$ be a submodular function on a ground set $E = [n]$, and let $P(f)$ denote its extended polymatroid. Given a direction $d \in \mathbb{Z}^n$ with at least one positive entry, the line search problem is to find the largest scalar $λ$ such that $λd \in P(f)$. The best known strongly polynomial-time algorithm for this problem is based on the discrete Newton's method and requires $\tilde{O}(n^2 \log n)\cdot$ SFM time, where SFM is the time for exact submodular function minimization under the value oracle model. In this work, we study the first weakly polynomial-time algorithms for this problem. We reduce the number of calls to the exact submodular minimization oracle by exploiting a dual formulation of the parametric line search problem and recent advances in cutting plane methods. We obtain a running time of \[ O\bigl(n^2 \log(nM\|d\|_1)\cdot \text{EO} + n^3 \log(nM\|d\|_1)\bigr) + O(1)\cdot \text{SFM}, \] where $M = \|f\|_\infty$ and EO is the cost of evaluating $f$ at a set. Note that when $\log \|d\|_1 = O(\log (nM))$, this matches the current best weakly polynomial running time for submodular function minimization [Lee, Sidford, Wong '15], and therefore, one cannot hope to improve this running time. Our approach proceeds by deriving a dual formulation that minimizes the Lovász extension $F$ over a hyperplane intersecting the unit hypercube, and then solving this dual problem approximately via cutting-plane methods, after which we round to the exact intersection using the integrality of $f$ and $d$.

We characterise Dedekind rings among not necessarily Noetherian domains by a property of their module homomorphisms. Our proof relies on a homological algebra argument.

We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct Monge--Ampère measures, first associated with piecewise affine functions, and then we extend it to polyhedrally plurisubharmonic functions. We investigate polyhedral Monge--Ampère equations on balanced polyhedral spaces via a variational approach, providing sufficient conditions for the existence of solutions as well as explicit counterexamples. Finally, we relate our framework to non-archimedean pluripotential theory and explore its connection with the non-archimedean Monge--Ampère equation.

This paper studies optimal consumption and saving decisions under uncertainty about the transition dynamics of the economic environment. We consider a general optimal savings problem in which the exogenous state governing discounting, capital returns, and nonfinancial income follows a Markov process with unknown transition probability, and agents update their beliefs over time through Bayesian learning. Despite the added endogenous state from belief updating, we establish the existence, uniqueness, and key structural properties of the optimal policy, including monotonicity and concavity. We also develop an efficient computational method and use it to study how transition uncertainty and learning interact with precautionary motives and wealth accumulation, highlighting a dynamic mechanism through which uncertainty about regime persistence shapes consumption dynamics and long-run household wealth.

This paper considers structure-preserving model order reduction (MOR) techniques for port-Hamiltonian (pH) systems, which are typically derived from energy-based modelling. To keep favorable properties of pH systems such as stability and passivity in a reduced order model (ROM), we use structure-preserving methods in the reduction process. There exists an extensive literature on structure-preserving MOR methods of pH systems, however, to the best of our knowledge, there does not exist an intrusive structure-preserving MOR method for nonlinear pH systems on the base of general nonlinear approximation maps. To close this gap, we propose a MOR method for pH systems based on the idea of the generalized manifold Galerkin (GMG) reduction. The resulting MOR method can be applied to both linear and nonlinear pH systems resulting in ROMs, which are again of pH form. For the numerical examples, we employ a linear and a nonlinear mass-spring-damper system and the results show that the proposed MOR methods have lower relative reduction error compared to existing methods.

Inquisitive team logic is a variant of inquisitive logic interpreted in team semantics, which has been argued to provide a natural setting for the regimentation of dependence claims. With respect to sentences, this logic is known to be expressively equivalent with first-order logic. In this article we show that, on the contrary, the expressive power of open formulas in this logic properly exceeds that of first-order logic. On the way to this result, we show that if inquisitive team logic is extended with the range-generating universal quantifier adopted in dependence logic, the resulting logic can express finiteness, and as a consequence, it is neither compact nor recursively axiomatizable. We further extend our results to standard inquisitive first-order logic, showing that some sentences of this logic express non first-order properties of models.

Distributed machine learning (ML) over wireless networks hinges on accurate channel state information (CSI) and efficient exchange of high-dimensional model updates. These demands are governed by channel coherence time and bandwidth, which vary across devices (links) due to heterogeneous mobility and scattering, causing degraded downlink delivery and distorted uplink over-the-air (OTA) aggregation. We propose a coherence-aware federated learning (FL) framework that jointly addresses impairments on downlink and uplink with communication-efficient strategies. In the downlink, we employ product superposition to multiplex global model symbols for long-coherence (static) devices onto the pilot tones required by short-coherence (dynamic) devices for channel estimation, turning pilot overhead into payload while preserving estimation fidelity. In the proposed scheme, an orthogonal frequency-division multiplexing (OFDM) super-block is partitioned into sub-blocks aligned with the smallest coherence time and bandwidth, enabling consistent channel estimation and stabilizing OTA aggregation across heterogeneous devices. Partial model reception at dynamic devices is mitigated via previous local model filling (PLMF), which reuses prior updates. We establish convergence guarantees under heterogeneous link impairments, imperfect CSI, and aggregation noise. The proposed framework enables efficient scheduling under coherence heterogeneity; analysis and experiments demonstrate notable gains in communication efficiency, latency, and learning accuracy over conventional FL baselines.

We give the first and lowest order examples of 3-regular 3-edge-colored graphs that demonstrate the non-factorization of tensor model invariants in the large N limit of Gaussian random tensors, as proven on general grounds in [Gurau R., Joos F. and Sudakov B., Lett. Math. Phys., 115 (2025), arXiv:2506.15362 [math-ph]]. This non-factorization is in stark contrast to the well-known large N factorization for random matrices.

In this article, we will compute the expectation value of observables (which appear as Wilson loops) in $\mathrm{U}(1)^n$ Chern-Simons theory for closed oriented $3$-manifolds. We will show how the various topological sectors of the observable affect the expectation value and confirm that it is a topological invariant. We will also exhibit in this case as well a form of the CS duality introduced in previous works. Finally, to complete the treatment of this theory, we will compute its zero modes and the equations of motion.

A proper-time method for constructing models of dynamic gravitational-wave fields is presented. Using the proper-time method, analytical (not numerical) models of secondary gravitational waves are constructed as perturbative solutions of linearized field equations against the background of the exact wave solution of Einstein's equations for the vacuum in the Bianchi VI universe in a privileged wave coordinate system. Relations for the proper time of test particles against the background of a strong gravitational wave are used. The analytical form of the metric components for secondary gravitational waves is found from compatibility conditions for the field equations. From the field equations, an explicit form of ordinary differential equations and their solutions is obtained for functions included in small corrections to the metric for secondary gravitational waves. It is shown that there exists a continuum of gravitational wave parameters for which the perturbative solutions are stable.

We derive an explicit formula for the ADM mass of asymptotically locally Euclidean (ALE) almost Kähler manifolds. The formula expresses the mass in terms of the total Hermitian scalar curvature and topological data associated with the underlying almost complex structure, extending a result of Hein and LeBrun in the Kähler ALE case. Our approach is based on a spin$^\mathbb {C}$ adaptation of Witten's proof of the positive mass conjecture in the spin case and is therefore distinct from previous complex-geometric methods. In dimension $4$, we prove a positive mass theorem and a Penrose-type inequality for asymptotically Euclidean (AE) almost Kähler manifolds. We also study rigidity phenomena of almost Kähler ALE manifolds. We prove that an almost Kähler-Einstein ALE manifold with nonnegative scalar curvature and certain decay assumptions is necessarily Kähler-Einstein. In particular, any four dimensional Ricci-flat almost Kähler manifold with maximal volume growth and curvature in $L^2$ is Kähler, yielding new evidence towards the Bando--Kasue--Nakajima conjecture. We also discuss analogous rigidity results for asymptotically locally flat (ALF) manifolds.

Muller and Schupp introduced the concept of context-free graphs (originating from Cayley graphs of context-free groups). These graphs are always tree-like (i.e. quasi-isometric to a tree) and in this paper we investigate the subclass of bona fide context-free trees. We show that they have a finite-state description using multi-edge NFAs and that this specializes to certain partial DFAs in the case of deterministic graphs. We investigate this form of encoding algorithmically and show that the isomorphism problem for deterministic context-free trees is NL-complete in the rooted and the non-rooted case.

Let ${\mathcal{D}}'({\mathbb{R}}^d)$ denote the space of distributions on ${\mathbb{R}}^d$. For a linear partial different equation $p(\frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_d}, \frac{\partial}{\partial t}) u=0$ (briefly $D_pu=0$) corresponding to a polynomial $p\in \mathbb{C}[ξ_1,\cdots, ξ_d,τ]$, let $S_p:=\{u\in C(\mathbb{R}, {\mathcal{D}}'({\mathbb{R}}^d)):D_pu=0\}$. The set $S_p$ has the `concatenability property' if whenever $u_1,u_2\in S_p\cap C^1(\mathbb{R}, {\mathcal{D}}'({\mathbb{R}}^d))$ are such that $u_1(0)=u_2(0)$, their concatenation $u_1\& u_2$ (defined to be $u_1(t)$ for $t\le 0$, and $u_2(t)$ for $t\ge 0$) belongs to $S_p$. It is shown that for $p=a_0+a_1τ+\cdots+a_{d}τ^{d}\in \mathbb{C}[ξ_1,\cdots, ξ_d][τ]$, where $a_0,\cdots, a_{d}\in \mathbb{C}[ξ_1,\cdots, ξ_d]$ and $d\in \mathbb{N}$, $S_p$ has the concatenation property if and only if $d=1$.

Federated learning (FL) over wireless networks is fundamentally constrained by unreliable communication links, particularly when uplink channels suffer from blockage, fading, or weak line-of-sight (LoS) conditions. Pinching-antenna systems (PASSs) offer a new physical-layer capability to dynamically reposition radiating points along a dielectric waveguide, enabling controllable LoS connectivity and significantly improved channel quality. This paper develops FedPASS, a novel framework for low-latency wireless FL assisted by PASS. We formulate a multi-objective optimization problem that jointly minimizes the end-to-end round latency and an upper bound on the FL optimality gap. The resulting formulation is a mixed-integer nonlinear program subject to practical constraints on scheduling, transmit power, local CPU frequency, and PA placement. To address the resulting computational challenges, we develop a two-tier iterative algorithm: an outer loop that updates scheduling, communication time allocation, and power control via block coordinate descent, and an inner loop that optimizes PA locations using a Gauss-Seidel-based coordinate update with grid search under spacing constraints. Numerical results on MNIST and CIFAR-10 demonstrate that FedPASS achieves accuracy comparable to idealized FL baselines while drastically reducing the total training latency compared to conventional wireless FL.

A digraph consisting of a set of vertices $V$ and a set of arcs $E$ is called an interval digraph if there exists a family of closed intervals $\{I_u,J_u\}_{u \in V}$ such that $uv$ is an arc if and only if the intersection of $I_u$ and $J_v$ is non-empty. Interval digraphs naturally generalize interval graphs, by extending the classical interval intersection model to directed graphs. Several subclasses of interval digraphs have been studied in the literature-such as balanced, chronological and catch interval digraphs-each characterized by admitting interval representations that satisfy specific restrictions. Among these, interval nest digraphs are the ones that admit an interval representation in which $J_u$ is contained in $I_u$ for all vertices $u$ of $V$. In this work, we provide a complete characterization of interval nest digraphs in terms of vertex linear orderings with forbidden patterns, which we call nest orderings. This result completes the picture of vertex-ordering characterizations among the main subclasses of interval digraphs.

Path transformations are fundamental to the study of Brownian motion and related stochastic processes, offering elegant constructions of the Brownian bridge, meander, and excursion. Central to this theory is the well-established link between Brownian motion and the $3$-dimensional Bessel process ${\rm BES}(3)$. This paper is specifically motivated by Pitman and Yor (2003), who showed that a ${\rm BES}(3)$ process can be constructed by excising the excursions of a Brownian path below its past maximum that reach zero and concatenating the remaining excursions. Our main result shows that a similar excision procedure, when applied to a Brownian bridge, can be related to a $3$-dimensional Bessel bridge.

We give a new proof of a recent generalization to Shimura curve of genus 0 of the work of Gross and Zagier in `On singular moduli'. This generalization was conjectured by Giampietro and Darmon and proved by Daas by using $p$-adic $Θ$-functions as an analogue of the $j$-invariant. Instead of working $p$-adically, we prove this result by evaluating Green's function at CM points on the Shimura curve. Our strategy is inspired by the analytic proof of Gross--Zagier. We put a special emphasis on both the similarities and the differences with the $p$-adic proof.

Recent work has noted that a space-sharing argument proves the tightness of the entropic quantum Singleton bounds, which was left open in the literature for various settings involving only-quantum messages, only-classical messages, or both classical and quantum messages. Focusing on the setting of entanglement-assisted classical coding (EACC), in this letter we first elaborate upon the space-sharing argument and the tight Singleton bound for this setting, and then establish a new tight entropic Singleton bound for EACC codes with entanglement assistance distributed across a subset of encoders when only local quantum operations are allowed at each encoder.

This paper studies two-variable compressions of shifts associated to rational inner functions on the bidisk; these generalize the classical compressions of the shift associated to finite Blasckhe products and are unitarily equivalent to one-variable, matrix-valued Toeplitz operators. This paper proves that a rational inner function is almost completely determined by these Toeplitz operator symbols but provides examples showing that (unlike in the one-variable case) rational inner functions are not determined by the numerical ranges of their compressed shifts. This paper also investigates related questions including methods of constructing these compressed-shift Toeplitz operators and when the associated numerical ranges are open and closed.

Driven by the rapid growth of the low-altitude economy, integrated sensing and communication (ISAC) technologies are essential to meet the stringent demands for reliable connectivity and situational awareness. Within this context, multipath-based simultaneous localization and mapping has emerged as a promising approach by leveraging radio frequency (RF) multipath to reconstruct environment maps alongside agent localization. Nevertheless, existing studies largely confine themselves to bistatic non-line-of-sight links and assume purely specular reflections from smooth surfaces, overlooking the monostatic sensing capability inherent in ISAC systems and the diffuse scattering effects induced by non-ideal outdoor building facades. To address these limitations, this paper presents the first Bayesian multipath-based environment mapping framework for ISAC that integrates monostatic and bistatic measurements under non-ideal surface propagation. We establish geometric relationships linking both sensing modes to a common reflective surface, enabling their association with the same physical feature for data-level fusion. Building on this formulation, we design two complementary Bayesian frameworks with corresponding factor-graph representations, allowing flexible adaptation to different scene requirements. The effectiveness of the proposed approach is validated through synthetic RF data, demonstrating that the fusion of monostatic and bistatic links consistently yields environment maps with higher accuracy, greater robustness and faster convergence than single-link baselines.

This paper extends the study of resonance phenomenon initiated by the authors in~\cite{LS} to the case of doubly degenerate embedded eigenvalues (i.e. eigenvalue of multiplicity two). A fundamentally new concept is introduced to resolve the difficulties that arise in this study, beyond the methods of \cite{LS}. We apply a differential topological technique, namely the Morse Lemma, to study the present case. This allows us to understand rank-two self-adjoint perturbations of the Laplacian on $L^{2}(\mathbb{R}^{3})$, and along with methods of \cite{LS}, we obtain asymptotic results for the spectral density near a doubly degenerate embedded eigenvalue. Importantly, we are able to easily handle the threshold eigenvalue case. \par We also analyze important properties which explain such resonance phenomenon, viz., asymptotic behaviour of the sojourn time, scattering cross-section and time delay.

We propose Generative Adversarial Regression (GAR), a framework for learning conditional risk scenarios through generators aligned with downstream risk objectives. GAR builds on a regression characterization of conditional risk for elicitable functionals, including quantiles, expectiles, and jointly elicitable pairs. We extend this principle from point prediction to generative modeling by training generators whose policy-induced risk matches that of real data under the same context. To ensure robustness across all policies, GAR adopts a minimax formulation in which an adversarial policy identifies worst-case discrepancies in risk evaluation while the generator adapts to eliminate them. This structure preserves alignment with the risk functional across a broad class of policies rather than a fixed, pre-specified set. We illustrate GAR through a tail-risk instantiation based on jointly elicitable $(\mathrm{VaR}, \mathrm{ES})$ objectives. Experiments on S\&P 500 data show that GAR produces scenarios that better preserve downstream risk than unconditional, econometric, and direct predictive baselines while remaining stable under adversarially selected policies.

This paper builds stochastic geometry frameworks for analyzing downlink electromagnetic field (EMF) exposure and efficiency in 5G multi-connectivity networks, using 5G E-UTRAN New Radio - Dual Connectivity (EN-DC) configuration as a representative use case. The Poisson point process (PPP) and the beta-Ginibre point process (beta-GPP) are used to model the spatial distribution of base stations (BSs), where beta-GPP effectively captures the repulsion observed in real deployments. We derive tractable expressions for the distribution of EMF exposure and validate the framework through both Monte Carlo simulations and real BS data from Paris. In addition to conventional metrics, we introduce the Radiated Energy per Bit Transmitted in the Downlink (REBT-DL), which accounts for throughput and received power. Results show that network configuration significantly affect exposure and REBTDL, highlighting the relevance of energy-aware deployment strategies and confirming the proposed approach as a comprehensive tool for sustainable network evaluation. The results also confirm that \b{eta}-GPP provides a more accurate fit to practical deployments than PPP.

Let $E/\mathbb{Q}$ be an elliptic curve and let $ρ_E \colon \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}(2, \widehat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. We describe and implement an algorithm to compute the image of $ρ_E$ in $\operatorname{GL}(2, \widehat{\mathbb{Z}})$ (up to conjugation) for an elliptic curve $E/\mathbb{Q}$ with complex multiplication (CM) and $j$-invariant not $0$ or $1728$. In the process, we prove certain entanglement results between division fields of elliptic curves over $\mathbb{Q}$ with CM.

We study Bayesian inference of an unknown matching $π^*$ between two correlated random point sets $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$ in $[0,1]^d$, under a critical scaling $\|X_i-Y_{π^*(i)}\|_2 \asymp n^{-1/d}$, in both an exact matching model where all points are observed and a partial matching model where a fraction of points may be missing. Restricting to the simplest setting of $d=1$, in this work, we address the questions of (1) whether the posterior distribution over matchings is approximable by a local algorithm, and (2) whether marginal statistics of this posterior have a well-defined limit as $n \to \infty$. We answer both questions affirmatively for partial matching, where a decay-of-correlations arises for large $n$. For exact matching, we show that the posterior is approximable locally only after a global sorting of the points, and that defining a large-$n$ limit of marginal statistics requires a careful indexing of points in the Poisson point process limit of the data, based on a notion of flow. We leave as an open question the extensions of such results to dimensions $d \geq 2$.

We study a generalization of tropical oriented matroids by Ardila and Develin, and show that they are in bijection with subdivisions of root polytopes, which are sub-polytopes of a product of two simplices.