We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians.

To reduce user costs and maximize cluster utilization, large model training increasingly leverages volatile but inexpensive GPU capacity, such as spot instances and reclaimable resources in shared clusters. Yet, capitalizing on these economic benefits requires jobs to adapt within the short warning windows that many such environments provide. Existing elastic training systems still treat reconfiguration as stop-and-restart: they externalize distributed state through checkpoints, rebuild the distributed runtime on a new topology, and restart training, turning each resize event into a storage-heavy recovery procedure that incurs substantial downtime from checkpoint I/O, process restart, CUDA initialization, and communicator setup. We present LiveR, a live reconfiguration runtime for elastic LLM training that replaces storage-backed restart with a live, bounded-memory handoff between mixed-parallel training worlds. While the current world continues training, LiveR asynchronously prepares the target world, bootstraps newly added workers in isolation to keep heavyweight initialization off the critical path, and streams model state directly over high-bandwidth interconnects while reshaping it online across tensor, pipeline, and data parallel dimensions. Once the target world is ready, LiveR performs a lightweight commit that switches training to the new configuration without stop-and-restart on the live path. We implement LiveR atop Megatron-LM and PyTorch and evaluate it end-to-end on a multi-node GPU cluster. Across diverse reconfiguration scenarios, LiveR reduces downtime from minutes to seconds, accelerates reconfiguration by 14$\times$-23$\times$ over checkpoint/restart baselines, incurs minimal steady-state overhead, and sustains up to 99% training goodput under volatile-resource conditions, making volatile low-cost GPU capacity far more practical for LLM training.

Stenting is among the most common transcatheter interventions for congenital heart disease (CHD). Patient-specific computational fluid dynamics (CFD) simulations can predict hemodynamic outcomes of intervention scenarios but require post-operative vascular geometries that reflect stent-induced shape changes, which existing tools either model inadequately or require extensive time or manual effort to generate. We present SDFStent, a signed distance function (SDF) based mesh deformation method for virtual stenting that operates in real time, maintains mesh integrity, and preserves junction geometry. The stent is modeled as a pipe surface composed of piecewise-capsule SDFs joined by a smooth-minimum operator. Mesh vertices near the expanding SDF surface are displaced along the SDF gradient with a compactly supported fall-off function and an alpha blending mask. SDFStent was benchmarked against three existing approaches and validated on three tetralogy of Fallot (ToF) patients and three coarctation of the aorta (CoA) patients using rigid-wall steady-state CFD simulations against clinical catheterization measurements. Against a prescribed diameter of 6.0 mm, the method produced a mean stented diameter of 5.92 $\pm$ 0.08 mm in 1.5 s, over 100$\times$ faster than the best stenting-specific comparator. All output meshes were watertight and self-intersection-free. CFD-simulated post-operative pressure drops agreed with clinical measurements within 4 mmHg (mean error 2 mmHg). SDFStent produces simulation-ready post-stent models that match prescribed stent dimensions at interactive speeds, from pre-operative anatomy and catheterization data alone. The implementation is open-source and available in 3D Slicer. Its scriptable architecture enables automated generation of large synthetic cohorts for data-driven surrogate modeling.

Intelligent network operation and maintenance systems in modern networks continuously generate large volumes of multi-modal operational data. However, Wi-Fi fault diagnosis under heterogeneous operational environments remains insufficiently understood. We build a real-world Wi-Fi testbed deployed in campus working environments with an automated fault injection system, and collect a multi-modal Wi-Fi fault dataset containing over 10,000 fault samples across diverse wireless scenarios. To the best of our knowledge, this is among the first publicly available datasets jointly capturing heterogeneous cross-layer operational observations for Wi-Fi fault diagnosis. Based on this dataset, we establish a unified benchmark spanning multiple diagnosis tasks, operational modalities, and representative diagnosis paradigms. Experimental results indicate that effectively leveraging heterogeneous operational data remains challenging for existing diagnosis approaches. We further evaluate emerging LLM-based approaches and develop a reasoningoriented evaluation framework to assess the consistency between generated diagnostic analyses and actual network conditions. Our findings suggest several important considerations for future multi-modal Wi-Fi diagnosis.

We study solutions to the Navier-Stokes equations in the class $L^\infty_t C^α_x$. Landau and Lifshitz [LL87] predicted that the Eulerian and Lagrangian temporal structure functions for turbulence exhibit $1/3$ and $1/2$ scaling laws, respectively. These laws were justified for the Euler equations in [Ise23,Ise25], assuming the spatial structure functions satisfies a $1/3$ scaling law. We demonstrate them in a viscous setting by proving that the $C^α_{t,x}$-norm of the solution and the $C^{1/(1-α)}$-norm of any fluid trajectory can be estimated by the $L^\infty_tC^α_x$-norm independently of the viscosity parameter $ν>0$, for times bounded away from zero by a positive power of $ν$.

Conformal methods provide prediction sets for outcomes with confidence guarantees. We study their use in a selective inference setting, where inference is performed only when the prediction set is informative. The analyst may consider as informative, for example, cases with prediction sets that are sufficiently small, exclude null values, or satisfy other appropriate monotone constraints. Because inference is typically restricted to informative cases in practical applications, accounting for the resulting selection bias is crucial to maintaining false coverage rate (FCR) control. A general framework for constructing such informative conformal prediction sets while controlling the FCR on the selected sample was suggested in Gazin et al. (2025). In this work we focus on oracle-guided procedures. We derive the optimal decision policy under a suitable power objective in the oracle setting where the probability of belonging to each prediction set can be computed. In practice, of course, only estimated probabilities are available. We therefore introduce a calibration procedure that adjusts the oracle policy to maintain finite sample FCR control. We show that this approach can achieve substantially higher power than available alternatives. We demonstrate the effectiveness of our new methods for classification outcomes on both real and simulated data.

For $l > 1$, the $l$-edge-connectivity $κ'_l(G)$ of a connected graph $G$ is defined as the minimum number of edges whose removal leaves a graph with at least $l$ components. A graph is minimally $(k,l)$-edge-connected if $κ'_l(G)\geq k$ but for any edge $e\in E(G)$ satisfies that $κ'_l(G-e)< k$. Motivated by two foundational extremal problems: Brualdi and Solheid's problem [SIAM J. Algebra Discrete Methods (1986)] for graphs of fixed order: determine sharp upper bounds for the spectral radius over graph families and characterize extremal graphs; and its fixed size analogue proposed by Brualdi and Hoffman [Linear Algebra Appl. (1985)], we resolve both problems for minimally $(k,k)$-edge-connected graphs. Building on the structural framework of Hennayake, Lai, Li, and Mao [J. Graph Theory (2003)], we combine edge-switching method and double eigenvectors skill to characterize the graphs maximizing the spectral radius among all minimally $(k,k)$-edge-connected graphs of prescribed order or size. Our results generalize the $k=2$ cases established by Lou, Min, and Huang [Electron. J. Comb. (2023)] and Chen and Guo [Discrete Math. (2019)].

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that the mixed Donaldson-Futaki invariant is non-increasing along the birational procedure. We also introduce a notion of Ding stability for adjoint Fano foliated structures which we show is equivalent to our notion of K-stability. We then introduce mixed alpha, beta and delta-invariants and use the reduction theorem to establish valuative criteria for the K-stability of adjoint Fano foliated structures. To conclude, as an application, we show that K-semistable adjoint Fano foliated structures with bounded volume form a bounded family.

In this paper, using extension theory and cohomological approach we introduce the notion of the obstruction class for an inner post-Lie algebra being induced by a Rota-Baxter operator, and show that an inner post-Lie algebra is induced by a Rota-Baxter operator if and only if the obstruction class is trivial. Similarly, we introduce the notion of the obstruction class for an inner post-group being induced by a Rota-Baxter operator, and prove a parallel result. Finally, we give some applications of inner post-Lie algebras and inner post-groups.

This chapter reviews the application of Artificial Intelligence (AI) techniques to the study of galaxy clusters, covering both theoretical developments and their use as tools to infer cluster properties from a variety of observational tracers. We discuss recent advances in mass estimation from SZ, X-ray, optical, and dynamical data, highlighting the ability of AI methods to capture non-linear features, projection effects, and complex cluster morphologies beyond more classical approaches. In addition, we present other emerging applications, including the emulation of baryonic physics from N-body simulations, the characterization of dynamical states and mergers, and the analysis of the diffuse components such as the intracluster light. Particular emphasis is placed on the role of simulations in training these models, the impact of baryonic modelling, and the need for a robust uncertainty quantification and interpretability. Finally, we outline current limitations and future prospects, stressing the importance of combining flexible simulation strategies with AI techniques to fully exploit next-generation surveys for precision cosmology.

Static spherically symmetric perfect-fluid solutions of Einstein's equations play a central role in relativistic astrophysics and stellar structure theory. While many exact solutions satisfy Einstein's equations mathematically, only a limited subset satisfies physically acceptable conditions such as regularity, positivity of matter variables, and causal sound propagation. In this work, the classical concept of physical acceptability is extended to include thermodynamic considerations. Using relativistic equilibrium thermodynamics, entropy functionals, and the Tolman temperature relation, we formulate a set of thermodynamic acceptability conditions for relativistic stellar models. The Tolman IV solution is analyzed as an explicit example. We show that this solution admits a finite and positive equilibrium entropy functional consistent with the Tolman equilibrium condition. This analysis suggests that thermodynamic consistency provides a natural extension of the Delgaty-Lake acceptability program and may constitute an essential criterion in the classification of relativistic interior solutions.

We report stroboscopic time-resolved photoluminescence (PL) microscopy and spectroscopy revealing the transverse near field of an edge excitation in a $ν=1/3$ fractional quantum Hall (FQH) state. Time-resolved $y$-$t$ maps reveal an immediate PL response extending more than $30~μ\mathrm{m}$ into the bulk transverse to the edge when the edge magnetoplasmon (EMP) passes the mesa boundary. The nearly instantaneous nature of this long-range response identifies it as the non-radiative, quasi-electrostatic near field, revealing the EMP as a spatially extended collective excitation rather than a strictly one-dimensional charge-density oscillation. We also observe secondary bulk-side responses distinct from the immediate transverse near-field response. The coexistence of these immediate and secondary responses shows that electrically launched edge excitations produce bulk-side dynamics on widely separated time scales.

Conversational recommender systems aim to provide personalized recommendations via natural language interactions. However, existing approaches either decouple recommendation from dialog generation or rely on retrieval-based pipelines, limiting the integration between recommendation and response generation and leading to suboptimal modeling of user intent. In this paper, we propose a fully generative conversational recommender system that unifies recommendation and dialog generation within a single autoregressive framework. Our approach represents items as discrete semantic IDs and integrates them directly into the generation process, enabling joint prediction of items and responses via next-token modeling. We further introduce a structured generation paradigm that factorizes conversational recommendation into a sequence of interdependent decisions, where the model first predicts the response intent and the recommendation target, and then generates the response conditioned on them. This design enables end-to-end optimization, enforces a more coherent dependency structure, and supports faithful item generation via constrained decoding. Extensive experiments demonstrate that our method consistently improves recommendation performance, achieving gains of up to 29% on Recall@1 over strong baselines, while maintaining competitive dialog quality.

Collisionless shocks are common in astrophysical systems and stand as sites of particle acceleration. While particles at perpendicular shocks may not return to the upstream region, at oblique shocks a fraction of energetic electrons manage to escape the shock and travel upstream. An extended region known as the electron foreshock is formed, where these reflected particles drive various instabilities that may promote electron acceleration. Here we present the first 2D3V particle-in-cell (PIC) simulations of electron-ion non-relativistic oblique shocks that explore the interaction of the foreshock with pre-existing compressive turbulence with relative amplitude of 15% based on interstellar medium estimates. We find that pre-existing turbulence influences the emergence and behavior of the whistler-wave instability, as it enhances the amplitudes of the magnetic-field fluctuations and leads to larger nonlinear structures. This impacts the dynamics of the reflected electrons, resulting in a shorter and hotter electron foreshock. At the end of our simulations, with pre-existing upstream turbulence we observe non-thermal electrons that are more numerous, reach higher energies, and carry a larger portion of the total energy.

We study a two degrees of freedom Hamiltonian system describing the motion of a particle in a potential field of the form of $S^1$ symmetric double well, namely $V = - (x_1^2 + x_2^2) + (x_1^2 + x_2^2)^2$, known also as a champagne bottle potential. This system is completely integrable. The champagne bottle is the simplest member of a class of integrable systems that have no global action variables due to a non-trivial monodromy, Bates (1991). Beyond that, the geometric and dynamical properties of the system near the equilibrium are of primary interest. We calculate the Birkhoff normal form and the nontrivial action near the focus-focus singularity and obtain the semi-global symplectic invariant near focus-focus point, which is introduced by Vũ Ng\d{o}c (2003). Examples of such calculations are still few. We compare our result with the semi-global symplectic invariant of the spherical pendulum, calculated by Dullin (2013).

Anderson localization is usually understood as a transition between extended and localized phases, with criticality confined to a single mobility edge. Recent advances predict that quasiperiodic systems can instead host a finite critical window bounded by mobility edges, in which localized, critical and extended states coexist. Yet both the experimental realization of this regime and whether interactions can provide controlled access to it remain unknown. Here, we realize such a tripartite phase in a nonlinear quasiperiodic photonic lattice and show that Kerr nonlinearity, acting as an effective interaction, enables state-selective access to the critical window. By tracking wavepacket dynamics, we distinguish localized, critical and extended transport regimes and uncover a state-selective response: rather than simply reinforcing localization through self-trapping, weak nonlinearity drives low-energy localized states into the critical window, whereas stronger nonlinearity restores localization. By contrast, critical, extended and high-energy localized states evolve monotonically towards self-trapped behaviour. Our results reveal a state-selective mechanism by which interactions provide controlled access to a pre-existing critical window in quasiperiodic systems.

In this work, we fully explore three refined convergence structures of the lowest-order rectangular Raviart-Thomas element in solving the Laplace eigenvalue problem. Firstly, the scheme possesses a property of supercloseness between the discrete eigenfunctions and the interpolated ones, so that post-processing can be easily constructed to improve the accuracy at most by one order. The essentially skillful method is the integral expansion for interpolation terms. Secondly, based on the supercloseness property, we derive the error expansions for not only simple eigenvalues but also multiple eigenvalues, and provide a rigorous proof for them, based on which Richardson extrapolation can be performed. As a byproduct, we prove that all eigenvalues converge from above. Moreover, by utilizing the supercloseness property and Rayleigh quotient analysis, we give a rigorous proof for the convergence behavior for multiple eigenvalues on uniform meshes for the problem on the square domain. Thirdly, the equivalence between the lowest-order rectangular Raviart-Thomas element and the enriched rotated bilinear element is also indicated. At the last of this work, several numerical experiments are designed to demonstrate our theory.

In this paper, we introduce and investigate a novel subclass $Σ(θ, λ, γ)$ of meromorphic functions defined in the punctured unit disk ${D}^*$. This class is constructed utilizing a specialized generalized operator $W_{α, β}$ associated with Wright function. We derive the exact integral representation and establish necessary and sufficient convolution (Hadamard product) conditions. Furthermore, sufficient conditions involving strict inequalities are provided for functions to be members of this class $Σ(θ, λ, γ)$. Additionaly, by employing the properties of Carathéodory functions and the principle of mathematical induction, we establish coefficient estimates for functions belonging to this new class. Finally, as an applications, we the established coefficient bounds, we determine the precise radii of meromorphic starlikeness and meromorphic convexity of order $ρ$. The results presented in this study generalize several existing outcomes in geometric Function Theory.

Geometric modelling has been a crucial component of the design process ever since the introduction of the first Computer-Aided Design (CAD) systems. Additive Manufacturing (AM) pushes design freedom to previously unachievable limits. AM allows the manufacturing of lattice structures which are otherwise close to impossible to be manufactured conventionally. Yet, the geometric modelling of heterogeneous lattice structures is still greatly limited. Thus, the AM industry is now in a situation where the manufacturing capabilities exceed the geometric modelling capabilities. While there have been advancements in the modelling of heterogeneous lattice structures, the review of relevant literature revealed critical limitations of the existing approaches. These limitations include their inability to model non-linear variation of geometric parameters, as well as the limited amount of controllable geometric parameters. This work presents a novel geometric modelling methodology based on function representation as an attempt to bridge this gap. The proposed approach avoids the manual definition of geometric parameters and provides a method to control them with mathematical functions instead. A software prototype implementing the proposed approach is presented, and several use-cases are analysed.

Recommender systems are critical for delivering personalized content across digital platforms, and recent advances in Large Language Models (LLMs) offer new opportunities to enhance them with richer world knowledge and explicit reasoning capabilities. With the help of reasoning knowledge, recommendations can better infer users' underlying intents, adapt to evolving preferences, and leverage semantic relationships for improved accuracy and interpretability. However, existing reasoning-based recommendation methods often fail to fully align the LLM's reasoning process with recommendation-specific objectives due to structural disruption during integration and difficulties in translating free-form generation into accurate item predictions. In this paper, we introduce RPORec, a reinforced preference optimization framework that unifies an LLM backbone's reasoning ability with a dedicated recommendation head (Rechead) for precise item retrieval. RPORec comprises two stages: (1) Reasoning-Augmented Recommendation Modeling, where high-quality Chain-of-Thought (CoT) reasoning is generated and used as auxiliary knowledge to guide the Rechead in learning recommendation-specific representations; and (2) Advanced Reasoning Refinement and Alignment, in which the trained Rechead produces verifiable rewards to fine-tune the LLM backbone via reinforcement learning, enhancing reasoning quality, structural consistency, and task relevance. Extensive experiments on public benchmarks and large-scale online deployments show that RPORec consistently outperforms state-of-the-art LLM-based recommendation methods, demonstrating the effectiveness of reasoning-augmented recommendation modeling in real-world systems.

Minimising EPR consumption is the dominant objective when routing a quantum circuit on a distributed quantum computer (DQC). We present dSABRE, a SABRE-style router for multi-core processors that, on each iteration of a lookahead-driven loop, first resolves any intra-core front-layer gates by SWAP scoring and only falls back to scoring inter-core teleportation candidates when the intra-core front is empty. Three mechanisms drive the improvement over the state of the art: a five-term gate-centric teleportation score that generalises the local SWAP heuristic to the inter-core setting, whose explicit capacity-penalty term keeps the scorer from teleporting into saturated cores; a proactive congestion-relief pass that redistributes idle qubits out of high-demand cores before deadlock; and a BFS-layer construction of the inter-core extended set that respects DAG dependencies layer by layer rather than mixing wires in topological order. Across 18 MQT-Bench circuits at 25, 36, and 64 logical qubits, dSABRE reduces geometric-mean EPR consumption by 41-44% over TeleSABRE and by 16-68% over the gate-teleportation-based pytket-dqc, using standard Qiskit SabreLayout for the initial layout. A large-circuit QFT sweep at 100-360 qubits confirms scalability. Code and online appendices are available at https://github.com/ebony72/dsabre.

We demonstrate how the relative phase between the multiple time periodic driving can decide the emergence and the favorable localization direction of non-Hermitian skin modes. For the static non-Hermitian quantum chain with parity time symmetry, non-Hermitian skin effects (NHSEs) can be prohibited. As the dynamical driving is turned on, NHSEs get artificially reactivated, where the relative phase can serve as the controlling switch by breaking the temporal symmetry constraints. Meanwhile, a change of relative phase can also alter the spatial structures of the long-time averaged effective Hamiltonian, which will consequently lead to the variation of skin localization direction for systems of higher dimensions. Our formalisms can be generally realized in diverse optical and mechanical platforms, and will pave the way for realizing tunable skin density profiles.

Artificial Intelligence (AI) agents can now orchestrate cyberattacks. This development is already increasing the speed and scale of cyber attacks, decreasing attack costs, and improving the operational autonomy of cyber capabilities. To defend against these emerging threats, actors must first develop the capability to detect them. This report frames the offensive cyber agent detection challenge by outlining the coming detection gap between offensive cyber agents and traditional cyber capabilities; introducing detection-in-depth, a strategic framework to guide policymakers and defenders responding to this detection gap; and presents five actionable detection mechanisms to support policymakers, industry, and defenders when putting this strategic framework into practice. These include (1) Agent Identifiers for Critical Infrastructure,(2) Agent Honeypots; (3) AI-Automated Alert Analysis and Triage: systems that use AI to filter, prioritize, and interpret the growing volume of detection signals expected from autonomous cyber operations; (4) An Agentic Security Alert Standard: A reporting standard model that providers can use to communicate agentic threats, improving the speed, consistency, and actionability of reports; (5) An Agentic Cybersecurity Exchange (ACE): an institution modeled on the Global Signal Exchange that brings together model and cloud providers to detect offensive cyber agent threats at their origin point and coordinate ecosystem-wide agentic threat disruption.

Gravitational waves from merging compact binaries are sensitive to line-of-sight acceleration (LOSA) induced by a massive companion in their vicinity. Interestingly, the leading-order contributions of LOSA and residual orbital eccentricity to the Fourier phase of the inspiral waveform exhibit similar frequency dependence, raising the possibility that a small eccentricity could mimic LOSA effects in transient GW events such as GW190814. We perform Bayesian inference using the IMRPhenomXPHM waveform family as the baseline LIGO-Virgo-KAGRA waveform model, augmented with leading-order LOSA and residual eccentricity corrections while using 32 seconds of data associated with GW190814. For a LOSA-only analysis, we find no evidence for a non-zero LOSA effect in GW190814, with a Bayes factor relative to the baseline model of approximately 0.22, consistent with the findings of Hendriks et al. and in tension with the claim by Yang et al., who employed only 4 seconds of GW190814 data. In a joint analysis that includes both leading-order LOSA and eccentricity effects, we obtain informative posteriors for both parameters, with representative values a/c approximately -2.8 x 10^{-3} s^{-1} and e_0 approximately 0.11. However, the corresponding Bayes factor relative to the baseline model is approximately 0.64, suggesting that the 32-second data do not provide significant evidence for either LOSA or residual eccentricity in GW190814. Further, our Bayesian runs reveal a strong correlation between the LOSA and eccentricity parameters, indicating a significant degeneracy in their imprint on the inspiral phase. This finding is consistent with our theoretical arguments and most likely explains the non-zero parameter estimates obtained in the joint analysis.

As large language models (LLMs) continue to advance, retrieval-augmented generation (RAG) has become the key mechanism for expanding model knowledge and reducing hallucinations. Central to RAG is approximate nearest neighbor search (ANNS), which retrieves database vectors most similar to a given query. However, distance calculation over high-dimensional vectors is inherently memory-bound, causing retrieval performance to be constrained by I/O bandwidth on mainstream platforms such as CPUs and GPUs. Although many prior early exiting (EE) techniques attempt to reduce memory accesses by only computing partial dimensions, the partial distance converges too slowly to the EE threshold, which ultimately limits their performance gains. To address these challenges, we propose NASZIP, a hardware-software co-designed framework that integrates near data processing (NDP) with a novel feature-level early exiting guided by statistics-based principal component analysis (PCA). Instead of relying solely on partial distances, NASZIP incorporates estimation and correction parameters to approximate full dimensional distances accurately, enabling earlier exiting without compromising accuracy. We further introduce a bit-level NDP-aware dynamic-float scheme that significantly reduces memory access for vector data. On the hardware side, we develop a data aware neighbor list mapping strategy that reduces neighbor retrieval latency and inter-channel communication overhead, complemented by a dedicated cache that exploits data locality and enhances prefetch efficiency. With these co-optimized techniques, NASZIP delivers speedups of up to $8.4\times$ / $1.4\times$ over CPU baseline and state-of-the-art GPU implementation at equal accuracy. Relative to the state-of-the-art NDP ANNS accelerator ANSMET, NASZIP achieves $1.69\times$ performance improvement.

We determine, up to lower-order terms in the exponent, the best possible deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix. If $A\succeq 0$ has no zero diagonal entry, $d=\operatorname{rank}(A)$, $A=VV^\dagger$ with $V\in\mathbb{C}^{n\times d}$ full column rank, and $v_1,\ldots,v_n$ are the rows of $V$, define \[ Φ(V)=\max_{X\succ 0} \left\{\sum_{i=1}^n \log(v_i^\dagger Xv_i)+\log\det X-\operatorname{tr} X+d\right\}, \qquad \widehat P(A)=e^{Φ(V)}. \] We prove the exact sandwich \[ e^{-γn}\widehat P(A)\le \operatorname{per}(A)\le \widehat P(A). \] Here $γ$ is the Euler--Mascheroni constant. Since the maximization is concave, this gives a deterministic polynomial-time $e^{(γ+\varepsilon)n}$-approximation for every $\varepsilon>0$. Combined with the previous $e^{(γ-\varepsilon)n}$-hardness of approximation for positive semidefinite permanents, this resolves the optimal exponential approximation ratio for deterministic polynomial-time algorithms as $e^{(γ+o(1))n}$, assuming $\mathrm{P}\ne\mathrm{NP}$. The proof is an entropy argument applied to the standard Wick integral formula for $\operatorname{per}(A)$; the loss is exactly $γ$ per factor because $\mathbb{E}[\log T]=-γ$ for $T\sim\operatorname{Exp}(1)$. The result was obtained through interactions with GPT 5.5 Pro Extended: the first author's interaction was one-shot, while the second author's was a separate multi-turn interaction with high-level guidance. Both authors verified the theorem and proof. Codex was used to assemble and typeset the manuscript.

We give a Hasse-diagram characterization of when a clustering system $\mathcal C$ on a finite taxa set $X$ is the hardwired clustering system $C_N$ of a rooted level-$k$ network. For each non-trivial block $B$ of $H=\mathcal H[\mathcal C]$, we define a parameter $μ(B)$ using minimum families of clusters that generate all overlap-intersections inside $B$. The main theorem proves that there exists a rooted level-$k$ network $N$ with $C_N=\mathcal C$ if and only if $μ(B)\le k$ for every non-trivial block $B$ of $H$. The necessity proof shows that overlap-intersection pieces must be represented by non-root hybrid vertices in any realizing block. The sufficiency proof is constructive: starting from the Hasse diagram, it iteratively splits selected hybrid vertices, preserves the hardwired clustering system, and terminates with a realization whose level is bounded by the block-wise values of $μ$.

Free-electron-based nanoscopy enables the study of optical excitations in materials with deep-subwavelength spatial resolution, with cathodoluminescence (CL) being one of the resulting radiation signals. When combined with an optical collection system, CL measurements can access multidimensional information of light; yet the phase of the emitted optical fields has remained largely elusive. Here, we demonstrate a reference-free phase retrieval approach for far-field CL wavefronts using the Gerchberg-Saxton algorithm implemented with real-space and angular-space CL intensity data. Applying this approach to representative nanostructures, including a planar surface, nanosphere, plasmonic crystal, and nanowire, we reconstruct distinct phase distributions that reveal their underlying radiation mechanisms. This reference-free framework offers a robust and flexible route for retrieving the phase of electron-beam-excited optical fields without relying on a reference wave, making it readily extendable to a wide range of nanostructures.

Background: Automated code summarisation supports program comprehension and documentation, yet the relative strengths and limitations of deterministic (heuristic-based) and probabilistic (LLM-based) pipelines remain unclear. Aims: This paper presents a controlled empirical comparison of these paradigms for intent-oriented design-pattern code summarisation. Method: Using design-pattern-centric Java code as a structured testbed (150 files from three open-source repositories covering nine patterns), we compare a rule-based natural language generation (NLG) pipeline, a Software Word Usage Model (SWUM)-based approach, and a probabilistic pipeline based on the Mixtral LLM. Summaries are evaluated against human references using BERTScore and cosine similarity, complemented by rubric-based judgements produced by Llama 3 across five dimensions: accuracy, conciseness, adequacy, code-context awareness, and design-pattern fidelity. Statistical analysis includes Wilcoxon signed-rank tests (with effect sizes), Friedman tests with post-hoc corrections, and Spearman correlation for sensitivity analysis of rubric consistency. Results: Probabilistic summaries show stronger semantic alignment and richer contextual coverage, while deterministic approaches produce more concise and fully reproducible outputs. Prompt-sensitivity and multi-run analyses indicate variability in LLM outputs, though relative trends remain stable. Conclusions: A clear trade-off emerges: probabilistic methods favour semantic depth and contextual accuracy, whereas deterministic pipelines are preferable for brevity and reproducibility. These findings provide practical guidance for selecting code summarisation techniques.

Cryogenic calorimetric detectors are a powerful tool in the search for rare events such as neutrinoless double beta decay ($0νββ$), due to their excellent energy resolution and low intrinsic background. The performance of these detectors depends critically on a precise understanding of their energy scale and energy resolution. Recent studies suggest that both energy scale and energy resolution may vary depending on the spatial location and topology of energy deposition within the detector, indicating the presence of previously uncharacterized systematic effects. The Cryogenic Experiment to Reconstruct Energy Systematics (CERES) is a dedicated experiment designed to directly measure the position dependence of calorimetric response in Tellurium Dioxide (TeO$_{2}$) crystals. This paper details the experimental design, current status, and future upgrade plans for CERES.