Papers

We study the critical points $U_\varepsilon$ of the two-dimensional Ginzburg–Landau energy on domains with corners $\Omega$ and imposed Dirichlet boundary condition which vanishes precisely at the vertices $V$. We isolate the forced scalar boundary layer $\rho_\varepsilon$ by the weighted Lassoued–Mironescu splitting. Assuming the reduced logarithmic energy bound and vertex logarithmic tightness, we prove that after passing to a subsequence, the normalized maps $v_\varepsilon = U_\varepsilon/\rho_\varepsilon$ converges in $C^k_{\mathrm{loc}}(\Omega\setminus A)$ $\forall k\ge0$ away from a finite set of interior vortices $A$. The limiting map is a canonical $S^1$-valued harmonic map $u_*$ containing integer interior vortex factors and fixed fractional corner factors. The renormalized energy is defined by removing discs around the interior vortices and sectors around the vertices and subtracting the corresponding logarithmic divergences. The interior vortex locations of the limiting map are critical points of this renormalized energy.

Graph clustering aims to partition the vertices of a graph into groups with dense intra-cluster connections and sparse inter-cluster connections. In this paper, we study a quadratic $K$-medoids formulation for graph clustering from the perspective of CM optimization. By identifying the binary feasible set as a discrete CM set, we establish a connection between the quadratic $K$-medoids model and CM optimization. Rather than solving the original binary problem directly, we consider the convex hull of its feasible set and introduce a negative squared-norm penalty to promote extreme-point solutions, resulting in an extreme point pursuit model over a simple convex set. The efficient projection onto this convex set enables the use of projected-gradient-based optimization for large-scale medoid selection. Based on this formulation, we propose a two-stage clustering method. In Stage~1, a penalty-continuation projected gradient solver is used to approximately solve the penalized medoid-selection problem. In Stage~2, the selected medoids are converted into a graph partition through a unified two-level assignment rule, where Jaccard distance is used as the primary criterion and shortest-path distance is used as a secondary criterion when needed. Numerical experiments on synthetic PPM and SBM graphs, together with real-world graph data, demonstrate the effectiveness and computational efficiency of our method. The experiments also show that a smaller objective value in the medoid-selection problem does not necessarily lead to a more accurate final graph partition, highlighting the importance of evaluating both optimization performance and clustering quality after assignment.

We define a symplectic structure on singular 1-form densities and a presymplectic structure on augmented triples, a compressible analog of the Marsden–Weinstein symplectic structure on vortex membranes in the incompressible case. Equivalence classes of augmented triples can be regarded as special vorticities in compressible fluids, i.e., singular elements of the dual to the Lie algebra of the group of all diffeomorphisms of a manifold, while Marsden–Weinstein symplectic structure is related to singular vorticities for the group of volume-preserving diffeomorphisms. This structure on singular 1-form densities occupies an intermediate position between two other settings, which we revisit, the Weinstein symplectic structure on weighted isotropic submanifolds of symplectic manifolds and the Marsden–Weinstein symplectic structure on vortex membranes.

In this paper, we establish a volume estimate and an associated rigidity result for volume-minimizing free boundary hypersurfaces in an $(n+1)$-dimensional Riemannian manifold $M$. First, we derive a volume estimate for stable minimal free boundary hypersurfaces in terms of the scalar curvature of $M$ and the second fundamental form of the boundary $\partial M$. We then prove rigidity results of $M$ in the corresponding equality cases.

We study conditional fluctuations of smooth mean-field systems on a torus with endogenous symmetric stable common jumps of index $0<\alpha<1$. A positive-strip estimate makes empirical Poisson acceptance errors negligible at the sampling scale. The conditional laws given the full Poisson master converge in probability to a centered Gaussian kernel in a negative Sobolev path space. In dimension one, quantile coupling preserves this limit for microscopic marks with relative tail remainder $O(x^{-\rho})$, $\rho>\alpha/2$. A quantified second-order tail expansion gives the endogenous rank response: it shifts the conditional Gaussian mean at criticality and, when nonzero, dominates sampling below criticality. We also identify a separate stable-to-Brownian transition for the rank error. Finally, in a translation model, rank and clock couplings give the same microscopic laws. The rank coupling has a functional Gaussian limit. The clock coupling has no diverging additive path normalization at the limiting center.

A pentagonal geometry PENT(k,r) is a k-uniform, r-regular partial linear space with the property that the set of points not collinear with a given point x is itself a line in the geometry. The existence spectrum of these geometries is known; we begin the study of automorphisms of these structures in the case k=3. We determine the spectrum of pentagonal geometries PENT(3,r) with either a cyclic or reverse automorphism to be r = 1 or 10 (mod 12). In addition, we show that there are 584 pairwise non-isomorphic PENT(3,7)s and classify them by the orders of their automorphism groups.

In this paper, a unmanned aerial vehicle (UAV) corridor-assisted network is proposed to study the joint sensing and communication (JCAS) coverage probability (CP) of a terrestrial (BS) equipped with a realistic three-dimensional (3D) 3GPP antenna pattern. By employing the 3D binomial point process (BPP) to model the spatial locations of the UAV-UEs, a fixed number of UAV-UEs are deployed in an aerial corridor modeled as a finite cylinder. The BS exploits radar sensing capabilities to track a known UAV-UE under a 3D beam misalignment error, which follows a multivariate normal distribution. Subsequently, under clutter and uplink communication interference, exact- form analytical expressions for JCAS CP are derived and used to analyze the performance at the terrestrial BS. The results highlight corridor-length saturation at high UAV heights and the beamwidth trade-off between interference suppression and misalignment sensitivity.

We consider polynomial processes, i.e., continuous-time Markov processes that preserve polynomials without increasing degree, and augment them with resetting. Resetting means that at the jump times of an indepen- dent renewal process, the polynomial process is instantaneously reset to a new state, which can be chosen deterministically, randomly or in state-dependent fashion. We show that under certain types of resets ('affine resetting') the moments of the resulting process remain tractable and can be calculated by matrix exponentials (for Poissonian reset times) or by a matrix renewal equa- tion (for more general reset times). Our framework encompasses many models that have been considered in the existing literature, such as growth-collapse processes with deterministic growth or with Lévy inflow, and Brownian motion with resets to the origin, to a fixed distribution or by multiplying by a fixed proportion.

In unmanned aerial vehicle (UAVs) networks, joint communication and sensing (JCAS) is emerging as a key enabler to support extended and continuous communication and sensing capabilities for sixth-generation (6G) services among UAVs operating in swarms. Within this framework, aerial corridors provide a structured environment for supporting coordinated and reliable operations. In this paper, a comprehensive frame- work is presented to investigate the JCAS coverage probability (CP) of a UAV-base station (BS) in an aerial corridor populated by UAV-user equipments (UEs). The corridor is modeled as a finite cylinder, within which a fixed number of UAV-UEs are spatially distributed according to a three-dimensional (3D) binomial point process (BPP). The UAV-BS is assumed to be equipped with a realistic 3D third generation partnership project (3GPP) antenna pattern and exploits radar sensing capabilities to track a known UAV-UE. Subsequently, the tracked UAV-UE is assumed to perform uplink communication with the UAV-BS. Accordingly, the JCAS CP at the UAV-BS is analyzed under the presence of clutter and uplink communication interference, and exact-form analytical expressions are derived. To the best of our knowledge, this is the first work to develop a stochastic geometry framework for the rigorous analysis of JCAS performance in aerial corridors, with UAV-UE locations modeled as a 3D BPP. Among several insights, results show that increasing the directivity of the UAV-BS antenna beams leads to notable JCAS performance gains, particularly for shorter UAV corridors.

We study abeloid varieties over the adic space $\operatorname{Spa}(K, K^+)$, where $K$ is a complete non-Archimedean field and $K^+$ is a valuation ring of arbitrary rank. We extend the well-known Raynaud uniformization to this setting. We also show the existence of suitable Néron models in algebraic and formal situations over higher rank valuation rings: We prove the existence of algebraic (lft-)Néron models for semi-abelian varieties over valuation rings with algebraically closed fraction fields, as well as, up to finite separable base change, over complete rank one valuation rings with algebraically closed residue field. Building on this result, we show the existence of formal marked Néron models for abeloid varieties over $\operatorname{Spa}(K, K^+)$ after finite separable base change. We also investigate the connection between the higher rank Raynaud extension and the formal Néron model.

We give an $\infty$-categorical enhancement of Neisendorfer's equivalences between the homotopy categories of rational simplicial sets, commutative dg algebras (cdgas), cocommutative dg coalgebras (cdgcs) and dg Lie algebras (dgls), where these objects have respective hypotheses of connectedness, nilpotency and finite type. We do so by proving that linear duality is a partial Quillen equivalence between weak finite type connected cdgcs and cdgas, and that Quillen's Chevalley–Eilenberg adjunction is a partial crossed derivable equivalence between cdgcs and dgls, a concept we introduce in this paper to axiomatize precisely the behavior of this adjunction.

Following Miyamoto, we consider Q-charts, a class of subductions $\phi\colon X\to Y$ of diffeological spaces along which Blohmann's elasticity descends. We show that the Riemannian apparatus of elastic diffeology—vertical, horizontal and full connections, Riemannian metrics, Levi-Civita connections and geodesics—also descends along Q-charts. More precisely, structures on $Y$ correspond bijectively to structures on $X$ that are invariant under the pseudogroup $\Psi(\phi)$ of fibre-preserving transitions, and torsion-freeness, effectivity, flatness, fullness, the Levi-Civita property and the geodesic equation are preserved in both directions. The key input is that the iterated tangent bundles $T^nX$ and their fibre products $T_kX$ are the pullbacks along $\phi$ of the corresponding bundles over $Y$. For quotients by principal actions of diffeologically discrete groups, we lift geodesics globally, prove that the class of elastic spaces on which $\R$ is a curve object is closed under such quotients, and show that complete geodesic flows descend. As an application, we determine the Riemannian geometry of the irrational tori $\Tor_\alpha=\R/(\Z+\alpha\Z)$ completely. Let $\vartheta$ be the one-form on $\Tor_\alpha$ induced by $dt$ on $\R$. Every Riemannian metric is a positive constant multiple of $\vartheta^2$, and the vertical connections form a one-parameter family of flat effective connections, a single member of which is the unique Levi-Civita connection of every metric. This connection has a complete geodesic flow, yet the induced pseudo-distance vanishes identically.

Reinforcement learning offers the prospect of a reusable sequential decision-making mechanism for spacecraft trajectory design, motivating policy interfaces that connect learned decisions to the underlying maneuver geometry. This paper develops Reachability Analysis-Informed Reinforcement Learning (RARL) for deterministic multi-impulse interplanetary transfers, placing intermediate waypoint selection at the center of the learned decision process. Local first-order reachability maps bounded velocity perturbations into an ellipsoidal set of next-node positions, within which the policy selects its waypoint. Lambert reconstruction then determines the corresponding maneuver to reach this selected waypoint along a dynamically consistent ballistic arc, coupling learned transfer-geometry selection with classical astrodynamics. A terminal two-impulse reconstruction completes the rendezvous, supported by a linear maneuver-demand assessment used for reward shaping. Numerical studies characterize this interface on a two-body Earth-Mars benchmark. Across three independent training runs, RARL achieves a mean maneuver cost of 10.23 km/s, 1.72% above a validated local sequential convex programming reference. Training over dispersed initial states extends policy reuse across a departure family with fixed target state and transfer duration. Each of the three independently trained multi-state policies completes all 10,000 held-out Monte Carlo departures without impulse-cap violations, compared with a mean feasibility rate of 6.49% for single-state policies. This broader sampled feasibility is accompanied by a 0.61% increase in mean nominal maneuver cost, without further training across departures. These results demonstrate that a reachability-informed decision interface supports benchmark-quality trajectory construction and policy reuse across dispersed departure conditions.

The error of regression using noisy function values naturally decomposes into an approximation error from the choice of model space and stochastic error from the noise on the observations. Statistical theory is mostly concerned with the stochastic error while approximation theory focuses on the approximation error, often subject to the assumption of noiseless observations. We combine the tools from approximation theory and statistics into a framework for bounding both errors in generalized least squares regression. The results cover weighted sup-norms and $L^2(\mu)$-norms, real- and complex-valued functions, and correlated noise under a range of tail and moment assumptions. Weighted Lebesgue functions and constants control the amplification of the best approximation error, while concentration inequalities provide bounds on the stochastic error. We organize these results into reusable bounds with explicit constants and establish sharpness where possible. For the common strategy of randomized regression we prove concentration of the Lebesgue function and constant around those of the corresponding $L^2(\mu)$-projection. For Fourier and Chebyshev regression on standard sampling grids, we obtain explicit, asymptotically tight $\mathcal O(\log m)$ bounds on the Lebesgue constants where $m$ is the model dimension. This improves upon the previous $\mathcal O(\sqrt{m})$ bound obtained for the regression setting and matches the bounds known for interpolation. A catalogue of approximation bounds and worked examples illustrates how to turn these results into explicit regression guarantees.

The distance between two points is provided by the length of their test connecting path. However, there is no unique way to attain this path in the space $R^2$, which is split by a straight line $L$ into two regions, $\Omega_p$ and $\Omega_q$, with $\ell_{p}$- and $\ell_q$-norms, respectively, where $1<p\leq q $ and $L\subset\Omega_q$. In this paper, given a point $P\in \Omega_p$, the locus of points $Q\in\Omega_p$, called $p$-curve, is found, such that there are two ways to attain the distance between $P$ and $Q$. For each $P\in \Omega_p$, there are two $p$-curves which are two branches of two different $p$-parabola. These two branches split $\Omega_p$ into three subregions using a three-link shortest path in two of these regions. The implicit equation of each $p$-parabola is given. Furthermore, by using the $p$-curve, an improved algorithm, called P-MFP, based on the MFP algorithm, is developed in order to solve the single-facility location problem in two regions with $\ell_{p}$ and $\ell_{q}$-norms. The results obtained with the new algorithm are better than those obtained with the MFP algorithm.ike subheadings, citations, or equations are permitted.

Every scalar ordinary differential equation of order at least three with rational right-hand side that is locally linearizable by a point transformation admits a linear form with rational coefficients over the same coefficient field. We prove this by restricting the derived symmetry algebra to a coordinate line and recovering a scalar differential operator from rational symmetry-jet data. The construction uses differential elimination and linear algebra; it does not require the symmetry generators or a linearizing transformation to be solved for. A single integer parameter suffices to choose the line.

We construct solutions to the Allen-Cahn equation $\Delta u+u-u^3=0$ in $\mathbb{R}^{N+1}$, with $N\ge 3$, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying cone is required. Moreover, we calculate the morse index of such solutions. In particular, our main result generalizes a result by Pacard and Wei (2013) dealing with stable solutions to the Allen-Cahn equation in high dimensions. More precisely, we give a positive answer to a remark raised in the paper by Pacard and Wei about the possibility to relax the assumption for the underlying cone is area-minimizing or at least stable. As a corollary, we also obtain solutions with infinite morse index and connected zero level set.

Let $(M,\omega)$ be a closed monotone integral symplectic manifold with monotonicity constant $\tau>0$, and let $D\subset M$ be a symplectic hyperplane section of degree $0<k<\tau$. By Biran's decomposition theorem, $M$ is the union of a symplectic disk bundle over $D$ and the skeleton of the complement $M\setminus D$ equipped with its Liouville structure. In this paper, we compute the relative symplectic cohomology of circle subbundles of this disk bundle over $\mathbb{Z}$ in terms of the quantum cohomology of $D$. This relative symplectic cohomology depends on the radius of the circle bundle relative to a certain critical radius. For a broad class of pairs $(M,D)$, we prove that the relative symplectic cohomology over $\mathbb{Z}$ does not vanish for supercritical radii. Therefore, circle bundles of supercritical radii are heavy subsets, which yields infinite dimensional quasi-flats in the Hamiltonian diffeomorphism group with respect to both the Hofer metric and the spectral metric over $\mathbb{Z}$. We also establish that if the degree $k$ of $D$ is greater than one, the skeleton of $M\setminus D$ is SH-full over $\mathbb{Z}/k\mathbb{Z}$, and hence non-displaceable in $M$ by any symplectomorphism.

We investigate finite equational bases for full power semirings of finite semigroups, including the empty set, in the constant-free signature with addition and multiplication. For a nontrivial finite nilpotent semigroup, we associate a finite relational structure recording the ordered products that are nonzero. We prove that the full power semiring is finitely based if and only if this structure has finite duality, and relate this condition to first-order definability and dismantling of the square of its core. Quantitative bounds connect obstruction size with the number of variables required in an identity basis. In the commutative case of nilpotency index $d$, the criterion reduces to the existence of an element with nonzero $(d-1)$st power. We also establish a nonfinite-basis obstruction for semigroups with a two-element group ideal and prove a finite lifting theorem. These results yield a direct-product criterion and a five-element counterexample to sufficiency of the identity-fibre condition. The arguments use equational logic, finite relational duality, and explicit algebraic constructions.

For integers $m\ge 2$ and rows $N>m$ divisible by $m$, consider the restricted binomial greatest common divisor $G(N;m)=\gcd\{\binom Nk:0<k<N,\ m\mid k\}$. Fix a prime $p$ with $p\nmid m$, and let $r_p(m)$ be the least positive integer $r$ such that $m<p^r$. We prove that the largest possible value of $v_p(G(N;m))$, as $N$ ranges over all admissible rows, is exactly $r_p(m)$, and we give a constructive equality row. Attainment is established before the least extremal row $T_p(m)$ is defined. For the special family $m=p^a+1$ with $a\ge2$, we determine that least row exactly: $T_p(p^a+1)=p^{3a}+1$. The proof combines Kummer's carry theorem with an explicit leading-digit witness for the universal upper bound, a multiplicative-order construction for equality, and a separate strict-minimality argument below $p^{3a}+1$ with a fallback witness for the unique worst leading-digit pattern. The selected-GCD family itself is known in the literature; the relation to earlier results and the limits of the documented literature search are stated explicitly.

This paper presents a control allocation framework with adaptive augmentation for a trailing edge morphing aircraft, providing stability guarantees under uncertainty while exploiting the available morphing degrees of freedom to optimize aerodynamic efficiency. A baseline controller is designed using the nominal aircraft model to establish the desired closed-loop reference dynamics. For the aerodynamics, the wing shape is optimized dependent on the flight state to achieve a target elliptical lift distribution corresponding to minimum induced drag. The adaptive augmentation is tailored to the control allocation problem to account for the uncertain system dynamics and stabilize the aircraft around the reference dynamics. The resulting stabilization condition is formulated as hard constraint in the allocation problem while minimizing deviations from the corresponding state-dependent aerodynamically optimal wing shape. Numerical results demonstrate that the adaptive augmentation compensates for matched uncertainties while the control allocation optimizes the wing shape with respect to the reference elliptical lift distribution.

Model collapse arises when generative models are trained on synthetic data produced by earlier models. The phenomenon has attracted considerable attention because of its societal and technical implications. However, previous studies have reached seemingly contradictory conclusions: replacing real data with synthetic data causes collapse (Shumailov et al.), yet accumulating real data alongside synthetic data can prevent it. For diffusion models, we study an intermediate regime typical of finite-budget pipelines: all past datasets and the real data are kept, but each new model is trained on a fixed-size sample from this growing pool, so the real fraction vanishes without any data being removed. Experiments on a 2D spiral dataset as well as the image benchmarks (MNIST, Fashion-MNIST, and CIFAR-10) show that replacement protocol degrades dataset rapidly as in the literature, whereas the fixed budget degrades only partially, sparing some features. A linear-response model of the multi-generational parameter dynamics, analyzed by stochastic recursion, confirms that the two protocols differ: some features will be fragile and lost within a few generations for both protocols, while some will be robust and preserved over practically unbounded horizons under the fixed budget protocol.

Disentangling shared (joint) structures from source-specific (individual) variations is a fundamental task in multi-source data integration. Existing joint-individual models often rely on computationally intensive optimization or loose spectral bounds, leading to suboptimal separation accuracy and poor scalability. In this paper, we propose the Multi-Source Sequential Alignment Test (MSSAT). MSSAT leverages the geometric observation that true joint components manifest as closely aligned score subspaces across different data sources. By deriving the asymptotic null distribution of our alignment statistic, we develop a rigorous, resampling-free sequential testing procedure to accurately estimate the joint rank. Extensive simulations and real data applications, including a TCGA multi-omics dataset, demonstrate that MSSAT achieves superior separation accuracy and substantially faster computation compared to competing methods.

We prove that every finite abelian group arises as the Schur multiplier of a finite solvable group, from which it can be deduced that every finite abelian group is the knot group of some extension of number fields.

Existing trajectory designs for covert communications with uncrewed aerial vehicles (UAVs) typically rely on point-mass kinematics, which overlook the inherent coupling between flight maneuvers and antenna orientation. Motivated by recent studies linking UAV acceleration to attitude, we investigate dynamics-consistent trajectory and attitude co-design for rotary-wing UAV covert communications. We consider a UAV equipped with a directional antenna transmitting confidential information to a legitimate ground receiver in the presence of a warden. The acceleration-induced reduced attitude of the UAV is explicitly modeled together with its impact on the attitude-dependent directional channel gain. To ensure covertness, we derive a per-slot constraint based on the Kullback–Leibler (KL) divergence at the warden. We then formulate a joint trajectory and attitude optimization problem to maximize the average achievable covert rate subject to covertness, motion, thrust-magnitude, attitude-smoothness, and acceleration-domain safety constraints induced by roll and pitch limits. The resulting problem is non-convex because the channel gains depend jointly on position, acceleration, and attitude. To solve it, we develop a minorization–maximization (MM)-based algorithm that constructs a concave lower bound on the transmission rate and a convex upper bound on the covert constraint at each iteration. We further extend the framework to imperfect attitude control and develop a robust design against attitude tracking errors caused by actuator limitations, sensor noise, and control delays. Numerical results demonstrate the effectiveness of the proposed dynamics-consistent design and highlight the benefit of jointly optimizing UAV trajectory and attitude for covert communications.

We prove stability of concentration in the Paley-Wiener space for each fixed time-bandwidth product. The concentration deficit controls the squared $L^2(\mathbb R)$-distance from a translated first prolate spheroidal wave function and the squared normalized symmetric difference from an interval at the same center. The obtained exponent for the function distance is optimal. The main idea is the introduction of the set discrepancy weighted by the best optimizers in place of the symmetric difference. The proof then combines elementary Hilbert space arguments with classical properties of the prolate spheroidal wave functions.

Classical central limit theorems (CLTs) for log-partition functions of Ising models on Erdős-Rényi random graphs lead to degenerate limits in the high-temperature regime when self-loops are absent. In this paper, we fill this surprising gap in the more general context of Ising models on inhomogeneous random graphs. In particular, we show that if $G(N,W)$ is an inhomogeneous random graph without self-loops generated by a graphon $W$, then, in the high-temperature regime, the log-partition function of the corresponding Ising model satisfies a CLT on the $N$-scale, with a limiting variance that incorporates contributions from cycles of all orders. This behavior contrasts sharply with that of the corresponding model in which self-loops are allowed, where the analogous Gaussian fluctuations occur on the $\sqrt{N}$-scale, with their asymptotic variance determined entirely by the diagonal profile of $W$. Our analysis yields, as a byproduct of independent interest, a CLT for the log-determinant of the associated resolvent matrix, established by combining the Lindeberg CLT for triangular arrays with suitable bounds on higher-order trace terms.

Data-driven model order reduction methods such as operator inference enable the efficient construction of reduced-order models directly from high-dimensional time-domain data. Standard operator inference typically seeks a Galerkin-type reduced model in a prescribed low-dimensional subspace by identifying its reduced operators from projected snapshot data. The resulting inference problem is formulated as a least-squares problem admitting an efficient closed-form solution. However, it is well known from intrusive model order reduction for linear time-invariant systems that Petrov-Galerkin projections can additionally preserve important system properties such as stability and passivity. To overcome the limitations of standard operator inference, we extend the framework for linear time-invariant systems to incorporate Petrov-Galerkin projections and provide explicit error expressions and bounds between the intrusive and nonintrusive reduced operators, thus generalizing results from the literature. We demonstrate the proposed approach in the context of dissipative and port-Hamiltonian systems. Furthermore, we introduce a novel convex optimization formulation that explicitly enforces the port-Hamiltonian structure on the inferred operators. The effectiveness of the proposed methods is demonstrated on several well-established benchmark problems, including the CD player, an atmospheric model, a mass-spring-damper system, and a poroelasticity system.

We revisit the problem of defining natural torsion and curvature tensors for generalised connections and of using them to reconstruct the effective action for the massless modes of the closed string from generalised geometry. The starting point is the description of a Courant algebroid as a differential graded (dg) symplectic manifold. A systematic procedure for split dg manifolds singles out a generalised Lie bracket as part of a larger Lie 2-algebroid structure, with the right properties to unambiguously define torsion and Riemann curvature tensors together with their corresponding Bianchi identities. This framework eliminates the structural circularity and ambiguity problems of previous approaches in the literature. We show that a unique generalised metric compatible generalised connection with fixed (con)torsion exists for any choice of ordinary connection that splits the dg manifold, while at the same time all Lie 2-algebroids obtained in this way are equivalent up to $L_{\infty}$ quasi-isomorphisms. For a given Lie 2-algebroid we demonstrate that the unique compatible generalised connection reproduces the effective closed string action from a geometric action based on the new generalised Ricci tensor upon making a canonical choice of torsion.

We study an inverse boundary value problem for the Helmholtz equation on a smooth compact non-trapping Riemannian manifold with strictly convex boundary. We prove that, given two such metrics, for sufficiently large but fixed frequency $\lambda$, equality of their Dirichlet-to-Neumann maps implies equality of their lens data, up to a smooth boundary-fixing diffeomorphism. This establishes a high-frequency bridge between the Calderón problem and lens rigidity. We also prove a quantitative version of this result uniformly valid in a suitable bounded set of Riemannian metrics. Crucially, we show that Gaussian beam type solutions concentrating on maximal geodesics, split near the boundary into outgoing/incoming parts, such that the phase of the outgoing part contains information on exit point, direction, and travel time. The recovery of lens data then proceeds by a careful stationary phase analysis and a boundary integral identity.