In this paper, we investigate the dynamics and gravitational-wave signatures of periodic orbits of spinning test particles moving in the equatorial plane around static, spherically symmetric black holes within the framework of Deser-Woodard nonlocal gravity. Based on the Mathisson-Papapetrou-Dixon equations, combined with the Tulczyjew spin supplementary condition, we derive the orbital dynamic equations for spinning particles moving in the equatorial plane and impose a timelike constraint to exclude unphysical superluminal trajectories. By comparing with the classical Schwarzschild black hole, we systematically analyze the effects of the nonlocal gravitational parameters $ζ$ and $b$ on the effective potential governing the radial motion of particles and the innermost stable circular orbit. In addition, gravitational waveforms exhibit significant phase differences: an increase in $ζ$ induces a phase delay, whereas an increase in $b$ results in a phase advance. A one-year simulation of the orbital evolution of an extreme mass ratio inspiral demonstrates that when $b=2$ and $ζ\approx10^{-6}$, the mismatch between the gravitational waveforms predicted for the nonlocal gravity black hole and those for the Schwarzschild black hole reaches the distinguishable threshold ($\mathcal{M}=0.0125$), providing a basis for observational discrimination between general relativity and nonlocal gravity.

The magnetic groundstate of RuO$_2$ remains controversial, with experimental evidence for a nonmagnetic groundstate of ideal bulk material and indications of a magnetic state in strained thin films. Here, I investigate the Néel temperature of the (hypothetical) altermagnetic state of bulk RuO$_2$, stabilized via the DFT$+U$ technique, by mapping on a Heisenberg Hamiltonian. The Néel temperature scales monotonously with the magnetic moment up to the point where a large $+U$ term opens a band gap and turns RuO$_2$ semiconducting. The maximum Néel temperature obtained by this procedure is 408\,K at $U=3$\,eV, and much smaller values for smaller $U$. A reciprocal-space eigenvalue analysis reveals a helimagnetic groundstate of the spin model due to intra-sublattice antiferromagnetic coupling. This situation resembles the isostructural $β$-MnO$_2$, which is a prototype helimagnet. Further comparison with calculations on CrO$_2$ and altermagnetic MnF$_2$ taking $U$ as an adjustable parameter supports the validity of the spin model analysis.

We present a fast and exact method for computing CMB mode-coupling matrices based on an optimised evaluation of Wigner-3j symbols. The method exploits analytic structure in the relevant Wigner-3j symbol configurations appearing in temperature and polarization coupling matrices, expressing all required quantities in terms of a small set of recurrence-generated values which are precomputed and stored in lookup tables. This approach reduces the computational cost of constructing the full coupling matrices whilst maintaining numerical accuracy. We demonstrate the performance of the threej_cosmo implementation using realistic survey masks from current CMB experiments. Relative to standard recursion-based approaches used in existing pseudo-C_l pipelines, the method achieves speedups of 6-25x in practical coupling-matrix constructions, with the largest gains occurring at high multipoles. The algorithm admits efficient parallelisation on both CPUs and GPUs, the latter providing additional acceleration, up to a further order of 50 on modern hardware, without altering the underlying formalism. Beyond full matrix construction, the approach is naturally suited to applications in which only a restricted set of l3 modes is required for each (l1,l2) pair, such as in the computation of band-limited coupling matrices and analytic covariance terms. These features make threej_cosmo a practical backend for pseudo-C_l estimation and related calculations in next-generation CMB analysis pipelines.

The tracheal flow field shapes particle transport into the lower airways and thus influences both the spread of inhaled pathogens and the effectiveness of aerosol-based therapies. Identifying how different inhalation routes modify the flow field is therefore crucial for understanding lower-airway disease transmission and for guiding targeted drug delivery. To gain a detailed understanding of the influence of the inhalation route on the flow structures in the human trachea, the flow field in the trachea is investigated in vitro in a non-compliant, refractive-index matched silicone model of the human respiratory tract. The investigations comprise steady inhalation, and oscillatory flow to simulate calm breathing. A realistic breathing pattern is approximated by a sinusoidal waveform for two Reynolds numbers of $Re_{Tr} = [400, 1200]$, based on the bulk velocity at maximum volume flux and the hydraulic diameter of the trachea and two Womersley numbers of $Wo = [3, 4.5]$, representing the oscillation time scales. To capture the inherently three-dimensional and asymmetric nature of the flow field, 3D particle-tracking velocimetry measurements are performed using the Shake-The-Box algorithm. Using a refractive-index matched fluid consisting of water and glycerin, the complex flow structures inside the trachea are fully resolved. The PTV measurements confirm that the nasal and/or oral cavity must be considered when analyzing the flow field in the lower respiratory tract. In particular, we find that the presence of both cavities significantly alters the flow field compared to idealised, fully developed inflow conditions. However, velocity profiles in the sagittal and coronal plane in the trachea as well as contour plots of the of the normalized velocity magnitude evidence nearly identical flow structures for oral and nasal inhalation, indicating minimal influence of the inhalation route.

In this paper, by using the generalized Borel theorems in $\mathbb{CP}^2$, we show the hyperbolicity of Fermat type curves and their complement in $\mathbb{CP}^2$. This improves Noguchi-Shirosaki's and Demailly-El Goul's degree bounds.

Certifying and accelerating execution times of nonlinear model predictive control (NMPC) implementations are two core requirements. Execution-time certificate guarantees that the NMPC controller returns a solution before the next sampling time, and achieving faster worst-case and average execution times further enables its use in a wider set of applications. However, NMPC produces a nonlinear program (NLP) for which it is challenging to derive its execution time certificates. Our previous works, \citep{wu2025direct,wu2025time} provide data-independent execution time certificates (certified number of iterations) for box-constrained quadratic programs (BoxQP). To apply the time-certified BoxQP algorithm \citep{wu2025time} for state-input constrained NMPC, this paper i) learns a linear model via Koopman operator; ii) proposes a dynamic-relaxation construction approach yields a structured BoxQP rather than a general QP; iii) exploits the structure of BoxQP, where the dimension of the linear system solved in each iteration is reduced from $5N(n_u+n_x)$ to $Nn_u$ (where $n_u, n_x, N$ denote the number of inputs, states, and length of prediction horizon), yielding substantial speedups (when $n_x \gg n_u$, as in PDE control).

A mixed integer maximization problem involving several additional constraints defined with both a lower and an upper bound is considered. It is assumed that one of such constraints is more restrictive than the others. As it can be seen as a resource window constraint, it defines the so-called window-constrained problem. From a bi-objective perspective, a 2-phase algorithm, called BORWin, is devised. It stands for Bi-Objective Relaxation for Window-constrained problems. The first phase is generic for any window-constrained problem and provides a family of upper bounds based on a bi-objective relaxation of the additional constraints. It is shown that the latter bounds strongly relate to the Lagrangian dual bounds. The second phase is derived for a variant involving a graph structure, namely the window-constrained longest-path problem on an acyclic graph. The aim is to take advantage of the upper bounds to devise an efficient label extension algorithm. It is shown that complementary upper bounds could be derived to further improve performance in some special cases. A typical example is when the additional constraints have special knapsack structures. This is the case for the Hydro-Unit Commitment problem with a single plant (1-HUC). From numerical experiments for the 1-HUC, BOR-Win appears to be very efficient compared to state-of-the-art approaches.

Testing functionality in Software-Defined Vehicles is challenging because requirements are written in natural language, specifications combine text, tables, and diagrams, while test assets are scattered across heterogeneous toolchains. Large Language Models and Vision-Language Models are used to extract signals and behavioral logic to automatically generate Gherkin scenarios, which are then converted into runnable test scripts. The Vehicle Signal Specification (VSS) integration standardizes signal references, supporting portability across subsystems and test benches. The pipeline uses retrieval-augmented generation to preselect candidate VSS signals before mapping. We evaluate the approach on the safety-relevant Child Presence Detection System, executing the generated tests in a virtual environment and on an actual vehicle. Our evaluation covers Gherkin validity, VSS mapping quality, and end-to-end executability. Results show that 32 of 36 requirements (89\%) can be transformed into executable scenarios in our setting, while human review and targeted substitutions remain necessary. This paper is a feasibility and architectural demonstration of an end-to-end requirements-to-test pipeline for SDV subsystems, evaluated on a CPDS case in simulation and Vehicle-in-the-Loop settings.

We consider the radiation of two distinguishable soft quark-antiquark pairs ($\mathrm{q}\bar{\mathrm{q}}\mathrm{Q}\bar{\mathrm{Q}}$) in a generic process for multiparton hard scattering in QCD. We evaluate the corresponding soft current at tree level in terms of an independent-emission contribution and an irreducible correlation component, which includes strictly non-abelian terms and also terms with an abelian character. The squared current for soft $\mathrm{q}\bar{\mathrm{q}}\mathrm{Q}\bar{\mathrm{Q}}$ emission produces colour dipole and colour tripole interactions between the hard-scattering partons, with structures similar to soft gluon-quark-antiquark emission. The colour tripole interactions are odd under charge conjugation and lead to charge asymmetry effects. We extend our analysis by including QED interactions, the emission of two distinguishable soft lepton-antilepton pairs, and mixed quark-antiquark-lepton-antilepton soft emission.

Sampling from discrete distributions is a ubiquitous task in machine learning, recently revisited by the emergence of discrete diffusion models. While Langevin algorithms constitute the state of the art for continuous spaces, discrete versions lack similar theoretical guarantees when the step-size becomes small. In this paper, we address this limitation by interpreting discrete sampling algorithms as discretizations of continuous-time dynamics on the hypercube. In particular, we describe several score functions for discrete algorithms which result in approximations of Glauber dynamics for the correct target distribution. We also compute upper bounds for the contraction of these algorithms, with or without Metropolis adjustment.

We study the problem of detecting a planted star in the Erd{ő}s--R{é}nyi random graph $G(n,m)$, formulated as a hypothesis test. We determine the scaling window for critical detection in $m$ in terms of the star size, and characterize the asymptotic total variation distance between the null and alternative hypotheses in this window. In the course of the proofs we show a condensation phase transition in the likelihood ratio that closely resembles that of the random energy model from spin glass theory.

In systems with fast periodic driving, there are special subsets of (resonant) states, which behavior can be described with effective, time-independent Hamiltonian in a rotating reference frame. Here, we show that experimentally feasible system of ultracold bosonic atoms on a ring with rapidly oscillating scattering length can be used to simulate time-independent two-component atomic mixture with exotic, long-range interactions.

The Frank Lie algebras are simple Lie algebras that only occur over fields of characteristic 3. These come equipped with distinguished inner derivations that make them algebras in the category $\textbf{Rep}(α_3)$. We apply the semisimplification functor to these Frank Lie algebras and obtain modular contact Lie superalgebras. We also obtain a class of simple $J$-ternary algebras whose associated Jordan algebras are not simple.

Nozomi was Japan's first space mission to Mars, launched on 3 July 1998 UT. It was equipped with the Mars Dust Counter (MDC) which was an impact ionisation dust detector. MDC detected 96 dust particle impacts when the spacecraft was in Earth orbit and later in interplanetary space, before its operation ended in April 2002 due to a technical failure on board. We compare the Nozomi dust measurements with the dust measurements obtained with the dust detector on board the Ulysses spacecraft. Impact speeds and masses of dust particles measured by Nozomi MDC are overall consistent with the measurements obtained by Ulysses in the same region of interplanetary space. Based on the impact speeds measured while Nozomi was in Earth orbit, MDC detected neither dust particles of natural origin that were bound to the Earth nor space debris. The dust impact rate measured in interplanetary space varied by approximately a factor of 2, consistent with theoretical predictions by the Interplanetary Meteoroid Engineering Model. The particle impact direction was concentrated towards the ecliptic plane, in agreement with an interplanetary origin of the majority of the measured dust particles. No impacts of cometary trail particles could positively be identified during known cometary trail crossings of Nozomi. The Nozomi dust data may become a valuable reference for the dust measurements to be obtained in the same region of interplanetary space with future space missions like, for example, MMX and DESTINY$^+$.

We develop a microscopic theory of nonlinear magneto-optical injection currents in multi-Weyl semimetals subjected to a uniform magnetic field. Using the Landau-level spectrum of a tilted multi-Weyl Hamiltonian with arbitrary monopole charge $ν$ as a starting point, we formulate a Kubo-type nonlinear response theory in the Landau-level basis and derive the second-order conductivity tensor. We identify distinct contributions originating from chiral-chiral, chiral-bulk, and bulk-bulk optical transitions, revealing characteristic monopole-charge scaling and sharp resonant structures governed by Landau-level selection rules and tilt-induced asymmetry. In the untilted limit, closed-form analytical expressions emerge that expose universal frequency thresholds and provide clear experimental signatures of higher-order Weyl topology. Our results establish nonlinear magneto-optical injection currents as a direct transport probe of chiral Landau levels and multi-Weyl topological charge.

Like a crêpe resting on a plate, a thin elastic sheet can fold smoothly under its own weight, forming reversible shapes without creases or imposed hinges. Such soft folds arise from a balance between elastic bending and gravity, yet their stability, packing limits, and dynamics remain poorly understood. Here we show that these behaviors are governed by a single physical length scale, the elasto-gravity length $\ell_{eg}$. Using experiments and heavy-elastica theory, we demonstrate that $\ell_{eg}$ sets the characteristic fold geometry, determines when a fold becomes unstable and unfolds, and limits how many reversible folds can be stacked in rectangular and circular sheets. In particular, when lengths are rescaled by $\ell_{eg}$, fold shapes and stability thresholds collapse across materials and thicknesses. We further show that unfolding follows a universal speed scaling $v \sim \sqrt{g\,\ell_{eg}}$, revealing a gravity-controlled time scale for the release of stored bending energy. Together, these results establish a unified physical framework for reversible folding, compact storage, and gravity-assisted deployment of thin elastic sheets.

The recent demonstrations of cascaded PDC (CPDC) and the hopeful prospects of realizing third-order PDC (TOPDC) for the generation of three-photon entangled states are paving the way for experimental studies on genuine three-photon interference. In this article, we formulate three-photon interference in terms of ``each three-photon interfering only with itself.'' We show that although a generalized two-alternative three-photon interference setup based on CPDC or TOPDC involves eight different length parameters, the interference can be fully characterized in terms of only three independent parameters. The first parameter is the three-photon path-length difference, which has a direct analog in the one-photon and two-photon cases, and the other two parameters quantify the path-asymmetry length. Unlike two-photon interference, which requires only one parameter to quantify path-asymmetry, two independent parameters are needed in three-photon interference. This results in a broader class of nonclassical three-photon effects, including three-photon HOM-type effects. Our work provides the theoretical basis for existing and future three-photon interference experiments exploring the rich and complex quantum correlations associated with three-particle entanglement and potentially enabling the development of novel protocols for harnessing those correlations.

In this work, we investigate in detail the thermodynamic properties of a spherically symmetric ModMax-AdS black hole sourced by a global monopole within the Kalb-Ramond gravity. We derive the key thermodynamic quantities, including the Hawking temperature, Gibbs free energy, and specific heat capacity, and analyze how the geometric parameters influence these physical quantities. The first law of thermodynamics and the corresponding Smarr formula are explicitly verified. Furthermore, we study the thermodynamic criticality of the system by deriving the critical points and examining the effects of the space-time geometric parameters. We also obtain the inversion temperature and demonstrate that the minimum inversion temperature is modified by the space-time parameters. In addition, the sparsity of Hawking radiation and thermal fluctuations of the system are investigated, highlighting the effects of the parameters on the entropy corrections. Finally, we analyze the optical properties of the black hole, in particular the photon sphere and shadow radius, showing how these parameters influence these features.

We analyze the behavior of Urysohn width of manifolds under a connected sum operation, specifically, bounding widths of summands in terms of widths of the sum and vice versa. Our methods also apply to the universal covers of these spaces, and to more general type of surgeries. Lastly, we provide examples that show the optimality of constants in our estimates.

Understanding how supermassive black holes (BHs) couple to their host galaxies across a vast spatial and temporal dynamic range remains a central challenge in galaxy evolution. Using the multizone framework -- designed to capture bidirectional inflow--outflow from the event horizon to the Bondi scale -- we present a suite of long-duration GRMHD simulations spanning BH spins $|a_\ast|=0$--0.9 and Bondi radii $R_B/r_g=4\times10^2$--$2\times10^6$. From these simulations we derive spin-dependent subgrid prescriptions from first principles, applicable to hot accretion flows with low-Eddington ratios ($f_{\rm Edd}\lesssim10^{-3}$), for adoption in cosmological simulations and semi-analytic models. We provide compact analytic fits for the time-averaged accretion rate $\dot M(R_B,a_\ast)$ and feedback power $\dot E_{\rm fb}(R_B,a_\ast)$ with respect to the Bondi rate $\dot{M}_B$, which are largely insensitive to the initial gas configuration and magnetic field strength. To capture intrinsic time-variability, we also quantify the full distributions of $\dot M$ and feedback efficiency $η$, both well described by lognormal statistics, with widths that increase toward larger $R_B$. We further measure self-consistent spin evolution in the hot accretion mode, finding that the spin-up parameter varies as $s(a_\ast)\simeq -3.7\,a_\ast$, which implies a very long spindown timescale $t_s\simeq 12(10^{-3}/f_{\rm Edd})\,{\rm Gyr}$. Thus, BH spins are effectively frozen during phases of quiescent accretion. Compared to conventional small-domain GRMHD calculations, our simulations, which reach dynamical equilibrium across horizon-to-galaxy scales, yield systematically different long-term accretion, feedback, and spin properties, cautioning against direct extrapolation from small-scale GRMHD simulations when constructing galactic-scale subgrid models.

Adaptive randomized experiments update treatment probabilities as data accrue, but still require an end-of-study interval for the average treatment effect (ATE) at a prespecified horizon. Under adaptive assignment, propensities can keep changing, so the predictable quadratic variation of AIPW/DML score increments may remain random. When no deterministic variance limit exists, Wald statistics normalized by a single long-run variance target can be conditionally miscalibrated given the realized variance regime. We assume no interference, sequential randomization, i.i.d. arrivals, and executed overlap on a prespecified scored set, and we require two auditable pipeline conditions: the platform logs the executed randomization probability for each unit, and the nuisance regressions used to score unit $t$ are constructed predictably from past data only. These conditions make the centered AIPW/DML scores an exact martingale difference sequence. Using self-normalized martingale limit theory, we show that the Studentized statistic, with variance estimated by realized quadratic variation, is asymptotically N(0,1) at the prespecified horizon, even without variance stabilization. Simulations validate the theory and highlight when standard fixed-variance Wald reporting fails.

Radio wave propagation can be perturbed by compact ionized gas clumps through plasma lensing, which induces frequency dependent magnification and may distort the observed number counts of background sources. The quasar (QSO) number densities are a powerful probe for understanding the effects of intervening material. Absorption lines in QSO spectra reveal the presence of interstellar and intergalactic gas, which can change observed fluxes through dust extinction and plasma lensing. By combining observations from radio (VLASS), infrared (WISE), and optical bands (DESI), we assembled a sample of QSOs: ~4000 sources with MgII absorbers, and ~12, 000 non-absorbers. In the radio band, the MgII sample shows a moderate excess at the bright end of the flux distribution, which is broadly consistent with plasma lensing predications. In the optical, the MgII sample turns over at higher g-band fluxes and exhibits a steeper decline at the faint end than the non-MgII sample. Control samples were constructed by matching in redshift, infrared (W1), and optical (g) luminosities. In these comparisons, the radio excess becomes less prominent, suggesting that the apparent magnification may not be robust evidence for plasma lensing. Nevertheless, a weak contribution cannot be ruled out, especially given residual excess observed at the bright end relative to the non-MgII sample. Dust extinction along the line-of-sight remains a plausible alternative. Regardless of the dominant mechanism, the multi-wavelength differences offer a valuable probe of the physical state of the intervening medium.

Reliable detection and tracking in active sonar require accurate and efficient learning of the acoustic multipath background environment. Conventionally, background learning is performed after transforming measurements into the range-Doppler domain, a step that is computationally expensive and can obscure phase-coherent structure useful for monitoring and tracking. This paper proposes a framework for learning and tracking the multipath background directly in the raw measurement domain. Starting from a wideband Doppler linearization of the impulse response of a time-varying multipath channel, a state-space model with a heteroscedastic measurement equation is derived. This model enables channel tracking using an extended Kalman filter (EKF), and unknown model parameters are learned from the marginalized likelihood. The statistical adequacy of the proposed models is assessed via a p-value significance test. Finally, this paper integrates the learned channel model into a sequential likelihood-ratio test for target detection. BELLHOP-based simulations show that the proposed model better captures channel dynamics induced by sea-surface fluctuations and transmitter and receiver drift, yielding more reliable detection in time-varying shallow-water environments

Urban infrastructure degrades over time, necessitating periodic renovation to maintain functionality and safety. When renovation is delayed beyond the infrastructure's remaining lifespan, costly emergency interventions become necessary to prevent failure. Decision makers must therefore balance expected emergency intervention costs against traffic congestion impacts. We formalize this trade-off as a road network maintenance scheduling problem with uncertain deadlines, which presents optimization challenges including computationally expensive evaluation and an exponentially growing solution space. To address these challenges, this paper contributes a hybrid optimization approach combining machine learning with genetic algorithms for large-scale infrastructure renovation scheduling under uncertainty. We formulate the problem as a bi-level multi-objective optimization problem that explicitly accounts for uncertain infrastructure lifespans through probabilistic failure models. We develop a progressive lower bound evaluation method that integrates machine learning surrogate models with a multi-objective genetic algorithm to improve solution quality by enabling more iterations within fixed computational budgets. We demonstrate the method's effectiveness on substantially larger problem instances (76 projects) than previously addressed in the literature, achieving statistically significant improvements across multiple performance metrics by increasing computational efficiency up to 40 times compared to standard approaches.

In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation \[ f\big(f(-x)+x\big)=f\big(-f(x)\big)+f(x),\qquad x\in\mathbb{R}. \] Matkowski conjectured that all continuous solutions must necessarily be linear on both the negative and the positive half-line. We show, however, that the family of continuous solutions to the equation in question is far richer than anticipated: there exist continuous solutions that admit an arbitrary part. In addition, we provide a sufficient condition which, in the continuous setting, enforces the conclusion predicted by Matkowski's Conjecture.

We introduce a systematic framework for three-qubit entanglement classification using a cascaded architecture of Support Vector Machine (SVM) classifiers. Leveraging the well defined three-qubit structure with the four nested entanglement classes (S, B, W, and GHZ), we construct three distinct witness models ($\mathcal{M}_{B}$, $\mathcal{M}_{W}$, and $\mathcal{M}_{GHZ}$) that sequentially discriminate between these classes. The proposed Cascaded model achieves an overall classification accuracy of $95\%$ on a comprehensive dataset of mixed states. The framework's robustness and generalization capabilities are confirmed through rigorous testing against out-of-distribution (OOD) entangled states and various quantum noise channels, where the model maintains high performance. A key contribution of this research is an optimization protocol based on systematic feature importance analysis. This approach yields a tunable framework that significantly reduces the number of required features, while maintaining reliable model accuracy.

We study joint transmit-waveform and receive-filter design for a multi-user downlink integrated sensing and communication (ISAC) system under practical constant-modulus and similarity constraints. We cast the design as a unified multi-objective program that balances communication sum rate and sensing signal-to-interference-plus-noise ratio (SINR). To address this, we introduce an efficient algorithm that use consensus alternating direction method of multipliers (ADMM) framework to alternately update the transmit waveform and radar filter. The proposed method effectively handles the non-convex fractional sensing's SINR formulation and ensures fast convergence. Simulation results demonstrate that the proposed approach achieves better trade-offs between communication sum rate and sensing's SINR compared to existing benchmark schemes.

In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we assume differentiability on each involved function, eliminate a former condition on $ g'_1 $ and $ g'_2$, moreover we determine a brand new family of solutions. We also present a particular member of this class as an example. In order to achieve this, we strengthen known results about certain auxiliary functional equations as well.

Interpolating scattered CFD datasets onto a uniform Cartesian grid can distort the true geometry, producing a convex-hull type envelope and activating nonphysical regions. This work presents a reconstruction framework that recovers physically consistent masks before exporting CNN-ready fields. It introduces two novel strategies, distance-based masking and an adaptive alpha-shape formulation that normalizes alpha using local data resolution, and evaluates them against classical alpha-shape boundary recovery. A quantitative, topology-aware metric suite is introduced to assess retention, suppression of unsupported regions, overlap consistency, and connectivity. The novel distance-based method is robust across the geometries considered under the same threshold rule, with tau set to the minimum CFD grid spacing, and achieves 500-800 times speedups over classical alpha-shapes. The adaptive alpha-shape remains stable when its control parameter is set to 1 and is 1.7-2.6 times faster than the classical variant, which requires geometry-specific alpha tuning. A lightweight boundary inflation post-process using a minimal dilation further improves retention by up to 2.96% with negligible unsupported activation (less than 0.08%). Overall, the distance-based method is recommended as the default due to its accuracy, stability, minimal tuning, and low cost, while the adaptive alpha-shape is a strong alternative when grid-spacing information for threshold selection is unavailable. A companion web application operationalizes the workflow end to end, enabling 2D ASCII dataset upload, parameter tuning, mask and boundary generation, and export of CNN-ready outputs.

The shape of the undepleted volume of a p-type high-purity Broad Energy Germanium detector, dependent on the bias voltage, has been imaged by measuring spatially-resolved Compton-scattering efficiency. The bias voltage was raised stepwise from $-50\,\text{V}$ to the full-depletion voltage. The geometric acceptance was determined at full depletion. Below full depletion, the relative acceptance observed for $2\times2\times2\,\text{mm}^3$ voxels was used to create the image of the undepleted volume for each bias voltage. The images were used to extract the impurity density profile of the detector by fitting predictions of the open-source software package SolidStateDetectors$.$jl to the images. The result is shown and compared to the impurity density profile deduced from capacitance measurements. This is the first time that three-dimensional images of the undepleted volumes of a germanium detector have become available and have been used to deduce an impurity density profile.