Two-dimensional unconventional magnetism has recently attracted growing interest due to its intriguing physical properties and promising applications in spintronics. However, existing studies on stacking-induced unconventional magnetism mainly focus on specific materials and stacking configurations. Here, we develop a general symmetry-based stacking theory for two-dimensional magnets. We first introduce spin layer groups as the fundamental symmetry framework, providing the essential magnetic symmetry information for the stacking theory. Based on this framework, we construct the complete set of 448 collinear spin layer groups for describing two-dimensional collinear magnets. Subsequently, we develop a general magnetic stacking theory applicable to arbitrary magnetic systems and derive its general solutions. Using CrF$_3$ as an illustrative example, we show how this theory enables designs of two-dimensional unconventional magnetism, as validated by first-principles calculations. We realize two-dimensional fully compensated ferrimagnetism through our stacking theory. Our work provides a general symmetry-guided platform for discovering and designing stacking-induced unconventional magnetism.

We consider a dilute quantum gas of interacting spin-1/2 fermions in the thermodynamic limit. For a trial state that resolves the ground state energy up to the precision of the Huang--Yang formula, we rigorously derive its momentum distribution. Our result agrees with the formal perturbative argument of Belyakov (Sov. Phys. JETP 13: 850--851 (1961)).

In the era of large language models, Text-to-SQL, as a natural language interface for databases, is playing an increasingly important role. The sota Text-to-SQL models have achieved impressive accuracy, but their performance critically relies on expert-written evidence, which typically clarifies schema and value linking that existing models struggle to identify. Such limitations stem from the ambiguity of user queries and, more importantly, the complexity of comprehending large-scale and dynamic database values. Consequently, in real-world scenarios where expert assistance is unavailable, existing methods suffer a severe performance collapse, with execution accuracy dropping by over 10%. This underscores their lack of robustness. To address this, we propose DIVER, a robust system that automates evidence reasoning with dynamic interactive value linking. It leverages a compatible toolbox containing diverse tools to probe the database. Then, restricted by a structured workspace (CoTF, Chain of Thoughts and Facts), it reflects based on probe results and selects a new tool for next round of probing. Through this automatically iterative process, DIVER identifies schema and value linking missed by existing methods. Based on these accurate linkings, DIVER is able to infer correct usage of SQL functions and formulas and generate high-quality evidence, achieving robust Text-to-SQL without expert assistance. Extensive experiments demonstrate that: 1) The DIVER system significantly enhances the robustness of various Text-to-SQL models, improving performance by up to 10.82% in Execution Accuracy (EX) and 16.09% in Valid Efficiency Score (VES). 2) Our dynamic interactive value linking significantly improves the robustness of existing systems and the accuracy of schema and value linking, especially when confronted with challenges posed by large-scale, dynamic database values.

This paper rigorously analyzes bond failure in the peridynamic theory of solid mechanics, which is a fundamental component of fracture modeling. We compare analytically and numerically two common bond-failure criteria:~{\em critical stretch} and~{\em critical energy density}. In the former, bonds fail when they stretch to a critical value, whereas in the latter, bonds fail when the bond energy density exceeds a threshold. By focusing the analysis on bond-based models, we prove mathematically that the critical stretch criterion and the critical energy density criterion are not equivalent in general and result in different bond-breaking and fracture phenomena. Numerical examples showcase the striking differences between the effect of the two criteria on crack dynamics, including the crack tip evolution, crack propagation, and crack branching.

The B335 protostar has undergone a major outburst detected in the scattered light of its outflow cavity that has not yet ended. B335 therefore offers the rare opportunity to study its effect on the jet of a protostellar object. Photometry of background stars behind B335 is used to map visual continuum extinction and H$_2$O ice absorption and demonstrates that the outflow has carved out a cavity. Precise proper motions of the shock fronts emerging from the B335 protostar were obtained. The kinematic age of the most prominent shock front (3E) corresponds to the early phases of the ongoing outburst of the B335 protostar. Shock 3E shows strong CO gas emission, as well as H$_2$ and [\ion{Fe}{2}] emission. Older shock fronts show diminished CO emission and are dominated by H$_2$ and [\ion{Fe}{2}]. The emission feature 0E, closest to the protostar, is distinct in proper motion and radial velocity from the other shock fronts in the jet. In the span of 4\arcsec\ closest to the protostar, the continuum extinction in front of the outflow cavity increases by A$_V$~$\approx$~200 mag. The CO-line-removed spectra close to the protostar show the unsaturated absorption features of $^{13}$CO$_2$, OCN$^-$, and OCS have strongly increasing column densities toward the protostar. The ice characteristics are overall similar to those found in lines of sight with less extinction. The central regions of the bipolar nebula show CO gas emission, but at distances of a few arcsec from the protostar, absorption by CO gas is also detected.

5G promises enhanced performance-not only in bandwidth and capacity, but also latency and security. Its ultra-reliable low-latency configuration targets round-trip times below 1 ms, while optional security controls extend protection across all interfaces, making 5G attractive for mission-critical applications. A key enabler of low latency is the disaggregation of network components, including the RAN, allowing user-plane functions to be deployed nearer to end users. However, this split introduces additional interfaces, whose protection increases latency overhead. In this paper, guided by discussions with a network operator and a 5G manufacturer, we evaluate the latency overhead of enabling optional 5G security controls across internal RAN interfaces and the 5G user plane. To this end, we deploy the first testbed implementing a disaggregated RAN with standardized optional security mechanisms. Our results show that disaggregated RAN deployments retain a latency advantage over monolithic designs, even with security enabled. However, achieving sub-1 ms round-trip times remains challenging, as cryptographic overhead alone can already exceed this target.

Cyclic proof theory breaks tradition by allowing certain infinite proofs: those that can be represented by a finite graph, while satisfying a soundness condition. We reconcile cyclic proofs with traditional finite proofs: we extend abstract cyclic proof systems with a well-founded induction principle, and transform any cyclic proof into a finite proof in the extended system. Moreover, this transformation preserves the structure of the cyclic proof. Our results leverage an annotated representation of cyclic proofs, which allows us to extract induction hypotheses and to determine their introduction order. The representation is essentially a reset proof with one key modification: names must be covered in a uniform way before a reset. This innovation allows us to handle cyclic proofs where the underlying inductive sort is non-linear. Our framework is general enough to cover recursive functions satisfying the size-change termination principle, which are viewed as cyclic proofs under the Curry-Howard correspondence.

This paper presents a GPU-accelerated simulation package, TRED, for next-generation neutrino detectors with pixelated charge readout, leveraging community-driven software ecosystems to ensure sustainability and extensibility. We introduce two generic contributions: (i) an effective-charge calculation based on Gaussian quadrature rules for numerical integration, and (ii) a sparse, block-binned tensor representation that enables efficient FFT-based computation of induced signals on readout electrodes for sparsely activated detector volumes. The former captures sub-grid structure without requiring dense sampling, while the latter achieves low memory usage and scalable runtime, as demonstrated in benchmark studies. The underlying data representation is applicable to large-scale detectors and to other computational problems involving sparse activity.

We present a comprehensive interstellar scintillation (ISS) study of the radio-loud magnetar XTE~J1810$-$197, based on six years of multi-frequency monitoring (2018$-$2024) with the Shanghai Tian Ma Radio Telescope (TMRT) at 7.0, 8.6, and 14.0~GHz. The scintillation parameters--decorrelation bandwidth $Δν_{\rm d}$, decorrelation time $Δτ_{\rm d}$, and drift rate $dt/dν$--are fully characterized. Our measured $Δτ_{\rm d}$ implies $Δτ_{\rm d} < 4$~s at 575-725~MHz under a Kolmogorov spectrum, which is shorter than the magnetar's 5.54~s spin period. This result naturally explains the previously reported absence of pulse-to-pulse coherence at these frequencies. Kinematic modeling locates the dominant scattering screen at $1.6\pm0.1$~kpc away from the Earth, within the Sagittarius Arm. The screen coincides with the HII region JCMTSE~J180921.2$-$201932 and is unrelated to the magnetar's 2018 outburst suggested by earlier studies. A scintillation arc detected at 14.0~GHz represents the highest-frequency arc observed to date. The asymmetry of arcs is linearly correlated with a dispersion-measure gradient across the screen ($r = 0.959$, $p < 10^{-8}$). We also measure its refractive scintillation timescale, which is only $1.21\pm0.19$~d. Clear DISS at 14~GHz effectively resolves the debate over a possible strong-to-weak scattering transition at this frequency. These results extend the ISS characterization of magnetars to previously unexplored frequencies and provide a precise probe of the ionized interstellar medium in the Sagittarius Arm.

Objective. Long-term potentiation (LTP) is a fundamental mechanism underlying learning and memory, yet its investigation at the network level in vitro remains challenging, particularly when optogenetic stimulation is used. The objective of this work is to develop a robust experimental and analytical framework for inducing and quantifying optogenetically driven LTP in neuronal cultures recorded with microelectrode arrays (MEAs). Approach. We first systematically investigate the effect of widefield optogenetic stimulation frequency on evoked neuronal activity, to identify a test-stimulus that reliably probes network responses without inducing activity modulation. By analyzing spike-rate dynamics during repeated stimulation, we characterize frequency-dependent response adaptation consistent with Channelrhodopsin-2 photocycle kinetics. Based on these results, an optimized low-frequency test-stimulus is selected and combined with a spatially confined tetanic optogenetic stimulation to induce LTP. Network responses are quantified using post-stimulus time histograms and a normalized efficacy metric, enabling electrode-wise and network-level analysis of plasticity. Main results. Low-frequency optical stimulation (<= 0.2 Hz) preserves stable evoked responses, whereas higher frequencies induce a pronounced sigmoid-like decay in firing rate. Following tetanic stimulation, a subset of electrodes exhibits robust and long-lasting potentiation, persisting for several hours. Significance. This work provides a systematic methodology for studying activity-dependent plasticity in optogenetically driven neuronal networks.

We study the non-Hermitian topology of multi-terminal transport in a quantum spin-Hall device described by the Bernevig-Hughes-Zhang model. We show that breaking time-reversal symmetry alone does not imply non-reciprocal transport or a non-Hermitian conductance matrix. Instead, non-Hermitian topology arises only when transport becomes directionally imbalanced. We identify two distinct mechanisms that generate such a response: spin-selective coupling at the contacts and an out-of-plane Zeeman field that unbalances the counter-propagating helical edge modes. We show, for unpolarized leads, that the spin polarization-dependent response to Zeeman fields, provides a transport-based probe of the intrinsic spin polarization of the helical edge states. Moreover, we demonstrate that non-Hermitian skin effect is more sensitive than conductance elements to detect the spin polarization of the edge states. Our results clarify the conditions required for non-Hermitian topology in quantum spin-Hall transport and establish non-Hermitian skin effect as a diagnostic tool for spin-selective coupling and edge-state polarization.

We investigate the local boundedness of solutions $u:Ω_T\to\mathbb{R}$ to parabolic equations of the form \begin{equation*} \partial_tu-\mathrm{div}\,\mathcal{A}(x,t,Du)=0 \qquad\mbox{in }Ω_T=Ω\times(0,T) \end{equation*} that satisfy $p,q$-growth conditions and have degenerate coefficients. More precisely, we assume structure conditions of the type \begin{align*} |\mathcal{A}(x,t,ξ)|&\le b(x,t)(μ^2+|ξ|^2)^{\frac{q-1}{2}},\\ \langle \mathcal{A}(x,t,ξ),ξ\rangle&\ge a(x,t)(μ^2+|ξ|^2)^{\frac {p-2}{2}}|ξ|^2, \end{align*} for $2\le p\le q$ and $μ\in[0,1]$, where the functions $a^{-1}, b:Ω_T\to\mathbb{R}$ are possibly unbounded and only satisfy some integrability condition. Under a certain assumption on the gap between $p$ and $q$, we prove two main results. First, we show that subsolutions that are contained in the natural energy space are locally bounded from above. Second, for parabolic equations with a variational structure, we use these bounds to show the existence of locally bounded variational solutions.

Obtaining reliable inferences with traditional difference-in-differences (DiD) methods can be difficult. Problems can arise when both outcomes and errors are serially correlated, when there are few clusters or few treated clusters, when cluster sizes vary greatly, and in various other cases. In recent years, recognition of the ``staggered adoption'' problem has shifted the focus away from inference towards consistent estimation of treatment effects. One of the most popular new estimators is the CSDID procedure of Callaway and Sant'Anna (2021). We find that the issues of over-rejection with few clusters and/or few treated clusters are at least as severe for CSDID as for traditional DiD methods. We also propose using a cluster jackknife for inference with CSDID, which simulations suggest greatly improves inference. We provide software packages in Stata csdidjack and R didjack to calculate cluster-jackknife standard errors easily.

Preparing arbitrary quantum states requires exponential resources. Matrix Product States (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50% and CNOT counts by 33%. We evaluate the full pipeline on datasets of varying complexity across systems of 19-50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.

Modeling high-order feature interactions efficiently is a central challenge in click-through rate and conversion rate prediction. Modern industrial recommender systems are predominantly built upon deep learning recommendation models, where the interaction backbone plays a critical role in determining both predictive performance and system efficiency. However, existing interaction modules often struggle to simultaneously achieve strong interaction capacity, high computational efficiency, and good scalability, resulting in limited ROI when models are scaled under strict production constraints. In this work, we propose MLCC, a structured feature interaction architecture that organizes feature crosses through hierarchical compression and dynamic composition, which can efficiently capture high-order feature dependencies while maintaining favorable computational complexity. We further introduce MC-MLCC, a Multi-Channel extension that decomposes feature interactions into parallel subspaces, enabling efficient horizontal scaling with improved representation capacity and significantly reduced parameter growth. Extensive experiments on three public benchmarks and a large-scale industrial dataset show that our proposed models consistently outperform strong DLRM-style baselines by up to 0.52 AUC, while reducing model parameters and FLOPs by up to 26$\times$ under comparable performance. Comprehensive scaling analyses demonstrate stable and predictable scaling behavior across embedding dimension, head number, and channel count, with channel-based scaling achieving substantially better efficiency than conventional embedding inflation. Finally, online A/B testing on a real-world advertising platform validates the practical effectiveness of our approach, which has been widely adopted in Bilibili advertising system under strict latency and resource constraints.

Stacked intelligent metasurfaces (SIMs) have recently emerged as a promising metasurface-based physical-layer paradigm for wireless communications, enabling wave-domain signal processing through multiple cascaded metasurface layers. However, conventional SIM designs rely on rigid planar layers with fixed interlayer spacing, which constrain the propagation geometry and can lead to performance saturation as the number of layers increases. This paper investigates the potential of introducing structural flexibility into SIM-enabled communication systems. Specifically, we consider two flexible SIM architectures: distance-adaptive SIM (DSIM), where interlayer distances are optimized, and stacked flexible intelligent metasurface (SFIM), where each metasurface layer is fully morphable. We jointly design the meta-atom positions and responses together with the transmit beamformer to maximize the system sum rate under per-user rate, quantization, morphing, and interlayer distance constraints. An alternating optimization framework combining gradient projection, penalty-based method, and successive convex approximation is developed to address the resulting non-convex problems. Perturbation analysis reveals that the flexibility gains of both DSIM and SFIM scale approximately linearly with the morphing range, with SFIM exhibiting a faster growth rate. Simulation results demonstrate that flexible SIM designs mitigate performance saturation with increasing layers and achieve significant transmit power savings compared to rigid SIMs.

We analyze strategic communication when advice is generated by a reinforcement-learning algorithm rather than by a fully rational sender. Building on the cheap-talk framework of Crawford and Sobel (1982), an advisor adapts its messages based on payoff feedback, while a decision maker best-responds. We provide a theoretical analysis of the long-run communication outcomes induced by such reward-driven adaptation. With aligned preferences, we establish that learning robustly leads to informative communication even from uninformative initial policies. With misaligned preferences, no stable outcome exists; instead, learning generates cycles that sustain highly informative communication and payoffs exceeding those of any static equilibrium.

We revisit the Ligon--Schaaf regularization of the Kepler problem and identify the geometric origin of the rotation appearing in their transformation. We show that this rotation is determined by the eccentric anomaly of the Kepler motion, providing a transparent dynamical interpretation of the angle that renders the Kepler flow uniform on $T^{*}S^{3}$. Building on this insight, we extend the construction to positive and zero energies via the corresponding hyperbolic and parabolic anomalies, obtaining a unified geometric description of the Kepler flow across all energy levels.

We establish the local Lipschitz regularity of the local minimizers of non autonomous integral funtionals of the form \[ \int_ΩF(x, Dz)\,dx, \] where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \ge 2$. The energy density $F(x,ξ)$ satisfies $(p,q)-$growth conditions with respect to the gradient variable and belongs to the Sobolev class $W^{1,ϕ}$, with $ϕ(t)=t^r\log^α(e+t),$ $r\ge n$, $α\ge 0$, as a function of the $x$ variable, under the condition $$ 1\le\frac{q}{p} \le 1 + \frac{1}{n} - \frac{1}{r}. $$ We present a unified approach that covers the limit case $$ \frac{q}{p} = 1 + \frac{1}{n} - \frac{1}{r} $$ and retrieves the results in \cite{EMM16} and in \cite{CGHPdN20}.

The stability of mixed halide perovskites against phase separation is crucial for their optoelectronic applications, yet difficult to rationalize due to the interplay of enthalpic, configurational, and dynamical effects. Here we present a simple thermodynamic framework for multicomponent halide perovskites of composition FA$_{1-x}$MA$_{x-y}$Cs$_y$Pb(I$_{1-z}$Br$_z$)$_3$, based on \textit{ab initio} molecular dynamics. By decomposing the free energy of mixing into enthalpic, configurational, and rotational entropic contributions, we show that although the enthalpy of mixing is generally positive, the solid solutions are thermodynamically stable against phase separation due to the large configurational entropy associated with random substitution on cation and halide sublattices. Mixing reduces the rotational entropy of the organic cations, partially offsetting the configurational stabilization. However, within our model, this rotational penalty is not sufficient to overcome the configurational driving force, and a curvature analysis within a regular-solution model does not predict a miscibility gap for any of the mixing channels considered. Analysis of hydrogen-bond dynamics shows that MA--Y (Y = I, Br) interactions are more persistent than FA--Y interactions, while the dominant FA-donated N$-$H$\cdots$I hydrogen bonds remain nearly composition-invariant. Cs-containing mixtures, in which Cs$^{+}$ forms no hydrogen bonds, can nevertheless be thermodynamically stable. These results demonstrate that hydrogen bonding does not control thermodynamic stability in mixed halide perovskites. Instead, phase stability is governed by the balance between strong configurational entropy and a smaller, systematically destabilizing rotational-entropy correction.

In finance, sequential decision problems are often faced, for which reinforcement learning (RL) emerges as a promising tool for optimisation without the need of analytical tractability. However, the objective of classical RL is the expected cumulated reward, while financial applications typically require a trade-off between return and risk. In this work, we focus on settings where one cares about the time split of the total return, ruling out most risk-aware generalisations of RL which optimise a risk measure defined on the latter. We notice that a preference for homogeneous splits, which we found satisfactory for hedging, can be unfit for other problems, and therefore propose a new risk metric which still penalises uncertainty of the single rewards, but allows for an arbitrary planning of their target levels. We study the properties of the resulting objective and the generalisation of learning algorithms to optimise it. Finally, we show numerical results on toy examples.

A merge tree is a fundamental topological structure used to capture the sub-level set (and similarly, super-level set) topology in scalar data analysis. The interleaving distance is a theoretically sound, stable metric for comparing merge trees. However, computing this distance exactly is NP-hard. First fixed-parameter tractable (FPT) algorithm for it's exact computation introduces the concept of an $\varepsilon$-good map between two merge trees, where $\varepsilon$ is a candidate value for the interleaving distance. The complexity of their algorithm is $O(2^{2τ}(2τ)^{2τ+2}\cdot n^2\log^3n)$ where $τ$ is the degree-bound parameter and $n$ is the total number of nodes in both the merge trees. Their algorithm exhibits exponential complexity in $τ$, which increases with the increasing value of $\varepsilon$. In the current paper, we propose an improved FPT algorithm for computing the $\varepsilon$-good map between two merge trees. Our algorithm introduces two new parameters, $η_f$ and $η_g$, corresponding to the numbers of leaf nodes in the merge trees $M_f$ and $M_g$, respectively. This parametrization is motivated by the observation that a merge tree can be decomposed into a collection of unique leaf-to-root paths. The proposed algorithm achieves a complexity of $O\!\left(n^2\log n+η_g^{η_f}(η_f+η_g)\, n \log n \right)$. To obtain this reduced complexity, we assume that number of possible $\varepsilon$-good maps from $M_f$ to $M_g$ does not exceed that from $M_g$ to $M_f$. Notably, the parameters $η_f$ and $η_g$ are independent of the choice of $\varepsilon$. Compared to their algorithm, our approach substantially reduces the search space for computing an optimal $\varepsilon$-good map. We also provide a formal proof of correctness for the proposed algorithm.

We investigate different treatments of dark matter (DM) distributions surrounding extreme mass ratio inspirals (EMRIs) to assess their impact on orbital evolution and gravitational wave emission. Density profiles derived from the mass current and from the energy-momentum tensor using a distribution function yield consistent results, but both differ substantially from profiles obtained using an anisotropic fluid model based on Einstein cluster ansatz. We find that the inclusion of radial pressure significantly modifies both the orbital dynamics and the resulting gravitational wave waveforms. By analyzing waveform dephasing and mismatches, we show that a fully relativistic treatment of DM distributions can substantially alter the detectability thresholds of DM halos. Our results demonstrate that radial pressure and relativistic modeling of DM are essential for accurately describing the dynamics and observational signatures of EMRIs embedded in DM halos.

Understanding how biological and synthetic systems achieve robust function in noisy environments remains a fundamental challenge across the physical and life sciences. To connect robust behavior with non-trivial topological features present already in the dynamics of individual units, here we introduce the topological chiral random walker (TCRW) model. While exploring the system, a TCRW locates edges and boundaries in the system and develops topologically protected edge currents even in the presence of defects and disorder. Drawing on the bulk-boundary correspondence found in hard condensed matter systems allows us to rationalize the emergence of robust edge currents through topological features of the dynamic spectrum. We show that chiral motion and rotational noise with opposite chirality are two crucial components in our inherently non-Hermitian model. As proofs of principle, we first show that a topological walker outperforms diffusive motion to efficiently solve complex mazes due to its property of remaining on the edge with some rare detachments. Second, we use this model to design building blocks that can perform efficient self-assembly overcoming the timescale bottlenecks of diffusion-limited growth and reducing self-assembly times by approximately 80%.

This paper presents adaptive behavioral predictive control (ABPC), an indirect adaptive predictive control framework operating on streaming data. An LPV--ARX predictor is identified online via kernel--recursive least squares and used to compute closed-form predictive control sequences over a finite horizon, avoiding batch Hankel constructions and iterative optimization. Nonlinear kernel dictionaries extend model expressiveness within a behavioral formulation. Numerical studies on Hammerstein and NARX systems demonstrate effective performance when the dictionary aligns with the plant class and highlight conditioning and feature-selection effects. The paper emphasizes numerical simulation, computational feasibility, and reproducibility.

Space-borne gravitational wave detectors will expand the scope of gravitational wave astronomy to the milli-Hertz band in the near future. The development of data analysis software infrastructure at the current stage is crucial. In this paper, we introduce \texttt{pespace} which can be used for the full Bayesian parameter estimation of massive black hole binaries with detectors including LISA, Taiji, and Tianqin. The core computations are implemented using the high-performance parallel programming framework \texttt{taichi-lang} which enables automatic differentiation and hardware acceleration across different architectures. We also reimplement the waveform models \texttt{PhenomXAS} and \texttt{PhenomXHM} in the separate package \texttt{tiwave} to integrate waveform generation within the \texttt{taichi-lang} scope, making the entire computation accelerated and differentiable. To demonstrate the functionality of the tool, we use a typical signal from a massive black hole binary to perform the full Bayesian parameter estimation with the complete likelihood function for three scenarios: including a single detector using the waveform with only the dominant mode; a single detector using the waveform including higher modes; and a detector network with higher modes included. The results demonstrate that higher modes are essential in breaking degeneracies, and coincident observations by the detector network can significantly improve the measurement of source properties. Additionally, automatic differentiation provides an accurate way to obtain the Fisher matrix without manual fine-tuning of the finite difference step size. Using a subset of extrinsic parameters, we show that the approximated posteriors obtained by the Fisher matrix agree well with those derived from Bayesian parameter estimation.

We investigate the existence of the three-body molecular state composed of $DD^*\bar{K}^*$ within the one-boson-exchange (OBE) model. A major challenge is that while the pseudoscalar-meson couplings are well-determined, the couplings for scalar- and vector-meson exchanges render significant model dependence. To ensure the reliability of our predictions and reduce model dependence, we recalibrate the coupling constants of the OBE model. We treat the pole position of $Z_c(3900)$, or equivalently the scalar $σ$-exchange coupling constant, as the only unknown parameter. The coupling constants for the vector $ρ$- and $ω$-exchanges are determined by the pole positions of the well established states $X(3872)$ and $T_{cc}(3875)$. We demonstrate that these parameter sets also successfully describe the $T_{cs0}(2870)$ without further tuning. For the three-body system, our results indicate that an $I\left(J^P\right)=1 / 2\left(0^{-}\right)$ three-body molecular bound state exists when $Z_c(3900)$ is a virtual state located within approximately $-10~\text{MeV}$ of the $D\bar{D}^*$ threshold. Furthermore, we extend our analysis to the complex energy plane using the complex scaling method to search for molecular resonances, though no evidence of resonances is found in considered channels. We also apply this formalism to the bottom analog $BB^*K^*$ system. In this sector, the conditions for the existence of a three-body bound state are more relaxed, as a $Z_c(3900)$ virtual state located within $-25~\text{MeV}$ below the threshold suffices, although three-body molecular resonances remain absent. We suggest that future experiments precisely measure the pole position of $Z_c(3900)$ or search for the three-body bound state in $DD\bar{K}ππ$ and $DD\bar{K}$ channels, as these efforts would mutually illuminate the nature of the associated states.

This paper is concerned with the numerical approximation of the $L^2$ Dirichlet eigenpairs of the operator $-Δ+ V$ on a simply connected $C^2$ bounded domain $Ω\subset \mathbb{R}^2$ containing the origin, where $V$ is a radial potential. We propose a mesh-free method inspired by the Method of Particular Solutions for the Laplacian (i.e. $V=0$). Extending this approach to general $C^1$ radial potentials is challenging due to the lack of explicit basis functions analogous to Bessel functions. To overcome this difficulty, we consider the equation $-Δu + V u = λu$ on a ball containing $Ω$, without imposing boundary conditions, for a collection of values $λ$ forming a fine discretisation of the interval in which eigenvalues are sought. By rewriting the problem in polar coordinates and applying a Fourier expansion with respect to the angular variable, we obtain a decoupled system of ordinary differential equations. These equations are solved numerically using a one-dimensional Finite Element Method, yielding a family of basis functions that are solutions of the equation $-Δu + V u = λu$ on the ball and are independent of the domain $Ω$. Dirichlet eigenvalues of $-Δ+ V$ are then approximated by minimising the boundary values on $\partial Ω$ among linear combinations of the basis functions and identifying those values of $λ$ for which the computed minimum is sufficiently small. The proposed method is highly memory-efficient compared to the standard Finite Element approach.

We give characterizations of the bounded subanalytic $\mathscr{C}^\infty$ submanifolds $M$ of $\mathbb{R}^n$ for which the space of Neumann regular functions is dense in Sobolev spaces. By ``Neumann regular function'', we mean a function which is smooth at almost every boundary point and whose gradient is tangent to the boundary. In the case $p\in [1,2]$, we prove that the Neumann regular elements of $\mathscr{C}^\infty(\overline{M})$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at almost every boundary point. In the case $p$ large, we show that the Neumann regular Lipschitz elements of $\mathscr{C}^\infty(M)$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at every boundary point. The proof involves the construction of Lipschitz Neumann regular partitions of unity, which is of independent interest.

We present a novel approach to the proof of Peng's maximum principle for McKean-Vlasov stochastic differential equations (SDE). The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE, to accommodate the dualization of quadratic terms containing two independent copies of the first-order variational process. This is an intrinsic extension of the maximum principle from Peng for standard SDE and gives a conceptually consistent proof. Our approach will be useful in further extensions to the common noise setting and the infinite dimensional setting.