Papers

3D scene inpainting aims to recover missing or occluded regions in edited 3D scenes, while ensuring geometric and textural consistency. Existing approaches, however, typically require accurately calibrated camera poses, which restricts their applicability in casual, in-the-wild scenarios and introduces additional preprocessing overhead. To overcome this limitation, we present FreeInpaint, a novel feed-forward framework that generates complete and 3D-consistent scenes directly from unposed multi-view images with masked regions. At its core, FreeInpaint extends a 3D foundation model to propagate masked regions from a reference view to other unposed views, bridging 3D reconstruction and scene inpainting while preserving the model's native ability to recover camera poses and scene geometry. Our method addresses two key challenges in adapting feed-forward 3D foundation models to masked inputs. First, masked regions can corrupt cross-view correspondence reasoning, degrading pose estimation and geometry recovery. To address this, we introduce a Learnable Mask Attention mechanism that preserves the spatial anchoring of reliable observations while allowing masked regions to progressively absorb useful context in deeper layers. Second, under severe occlusions, a single forward pass often lacks sufficient appearance evidence for high-fidelity completion. Therefore, we propose a Support Token Refinement strategy, which injects diffusion-generated support evidence as confidence-weighted auxiliary tokens to refine under-observed regions while preserving the original spatial anchor. Extensive experiments across diverse datasets demonstrate that FreeInpaint achieves superior inpainting quality, eliminating the reliance on pre-computed camera poses while keeping a fast inference speed. The project page is https://rorisis.github.io/FreeInpaint/.

We report, to the best of our knowledge, the first controlled, like-for-like experimental mapping of how four practical design parameters–seed-cavity topology, seed power, amplifier pumping geometry, and seed-cavity free spectral range (FSR)–jointly shape the radio-frequency (RF)-resolved relative intensity noise (RIN) of high-power continuous-wave (CW) Yb-doped fiber lasers. We compare a standalone bidirectional fiber Bragg grating (FBG) oscillator, an FBG-seeded master-oscillator power amplifier (MOPA), and unidirectional ring-seeded MOPAs with FSRs of approximately 11.5 and 5 MHz, all near 1064 nm at a common 72 W output, with RIN measured from 10 kHz to 1 GHz. Across all settings, the low-frequency RIN is technical-noise limited and nearly architecture-independent, whereas RIN above 160 kHz is governed by the seed-cavity topology and its transfer through the amplifier. Higher seed power suppresses the mid- and high-frequency RIN, but the seed-alone and amplified-output comparison reveals opposite transfer for the two topologies: for FBG seeding the 2 MHz-1 GHz RMS RIN increases through amplification (3.66 percent to 4.14 percent at 15 W seed), whereas for ring seeding it decreases (1.47 percent to 1.35 percent). Forward pumping gives the best power-scaling/noise trade-off, while reducing the ring FSR from approximately 11.5 to 5 MHz preserves the high-frequency RIN advantage, confirming it is not a mode-spacing effect. At matched conditions, ring seeding lowers the integrated RMS RIN by 2-3 times (160 kHz-2 MHz) and approximately 3 times (2 MHz-1 GHz, approximately 9-10 dB to near -127.4 dBc/Hz). These results yield band-resolved design rules linking each band to its dominant lever. The unidirectional ring-seeded MOPA thus offers a simple, scalable route to broadband pump-RIN suppression for nonlinear and frequency-conversion photonics.

Extra spatial dimensions arise naturally in theories beyond four-dimensional general relativity (GR), but direct observational evidence remains elusive. Gravitational waves offer a distinctive probe: geometric projection onto the observable three-dimensional spatial subspace can produce apparent tensor, vector, and scalar polarizations. We consider one additional spatial dimension and a two-mode tensor-wave ansatz for which the projected polarization components are fixed linear combinations of the same two underlying GR tensor waveforms. These components enter the measured strain through the standard detector antenna patterns. A frequency-domain Bayesian analysis of five compact-binary events constrains two rotation angles for two noncommuting orderings of rotations that preserve propagation within the observed subspace. Two events are further analyzed for four orderings that allow propagation to tilt into the extra dimension. For GW250114, the maximum GR quantiles among the tested rotation orderings are $Q_{\mathrm{GR}}\simeq0.6\%$ for the two-angle fits and $Q^{\mathrm{marg}}_{\mathrm{GR}}\simeq4.6\%$ for the three-angle fits after marginalizing over the third angle. Across the full event sample, the GR reference point lies inside the estimated 90% highest-posterior-density region in every case.

Reinforcement learning (RL) methods such as GRPO substantially improve large language model reasoning but often suffer from policy entropy collapse: the loss of sampling diversity weakens exploration and limits further improvement. Existing methods address this issue either through algorithm-level interventions, such as reward modification and entropy/KL regularization, or through token-level reweighting. We investigate a complementary perspective: entropy collapse can also be mitigated by changing which generated rollouts contribute to policy updates. Under the same sampling budget, not all rollouts contribute positively to an update, and selectively excluding some can improve learning. To address this, we propose GRPODropout: before the standard update, we use a simple strategy that selectively removes a small number of high-probability positive-advantage rollouts and recenters the retained advantages. To motivate this design, we develop a rollout-level theoretical analysis that guides method design and threshold selection. The method changes only rollout usage, and adds negligible computational overhead. Experiments show higher accuracy than original GRPO and higher actor entropy while using fewer rollout samples for updates, illustrating "less is more." This work provides insight into RL rollout usage: removing some rollouts can improve performance. Code is available at https://github.com/hexuandeng/GRPODropout/.

Modern neural networks are heavily over-parameterized. This redundancy incurs substantial compute and memory overhead during training and inference. Existing pruning methods rely on post-hoc magnitude thresholds or static initialization heuristics. Consequently, they often require manual per-layer sparsity targets or expensive retraining cycles. We propose Dynamic Activity-Dependent Pruning (DADP), a biologically inspired structural plasticity mechanism. During training, DADP measures connection importance via the accumulated product of pre-synaptic activations and post-synaptic error gradients. Using a single global threshold instead of fixed layer budgets, DADP dynamically allocates sparsity across network depth while naturally inducing neuron- and channel-level pruning. Across MLP, VGG-16, ResNet-18, BiLSTM-CRF, and MiniBERT architectures, DADP matches or outperforms Magnitude, SNIP and RigL, retaining 73.67% accuracy (dense baseline: 76.06%) at 99% sparsity on ResNet-18. Finally, matrix-based Shannon entropy and effective rank measurements confirm that DADP preserves latent feature diversity at extreme sparsities without representation collapse.

The Hydrogen Plasma Smelting Reduction (HPSR) is a promising technology for low-CO$_2$ steel production. However, unstable arc conditions limit energy efficiency, hydrogen utilization, and electrode wear. In this paper, we discuss the influence of the electrical supply characteristics, which can be a source of additional instability. For active stabilization, a fast current controller using a buck converter with an adaptive state controller is used. For analyzing arc stability, a stereoscopic measurement system for reconstructing the arc is presented. The results show a clear correlation between the arc lengths and the measured arc voltage at constant current. Therefore, arc stability can be determined by the measured voltage. These findings build the basis for dynamic arc control by fast electrical interventions.

We study social welfare rules with strict individual and social preferences. A preference profile is dictatorial for a rule if the social preference coincides with some agent's preference and no agent with a different preference can change the social preference unilaterally. We establish that (i) a rule is dictatorial if and only if all of its profiles are dictatorial; and (ii) if a rule satisfies unanimity and independence (of irrelevant alternatives), then all its profiles must be dictatorial. These results constitute a proof of Arrow's Impossibility Theorem.

Finding the right video often requires distinguishing similar scenes in which different events occur. Joint matching improves retrieval, but processing rich video representations for each query is costly. VEDJE compresses features within sampled frames while keeping their representations separate in a reusable cache. Feature-change prediction supplies an auxiliary training signal that improves retrieval from the compressed cache without adding work at query time. On MSR-VTT, MSVD, DiDeMo, and ActivityNet, VEDJE improves R@1 over matched first-stage retrievers in both retrieval directions. On MSR-VTT, it reaches 59.8 text-to-video R@1 with a fine-tuned VideoCLIP-XL first stage. In the VideoPrism configuration, shrinking the per-video cache fourfold to 12 KiB preserves text-to-video recall within 0.2 points. These results show that accurate video search can operate on compact evidence, encoded once and reused as new queries arrive.

We show that every triple of integers $(a,b,c)$ satisfying $1\leq a\leq b\leq c$ is realized as the smooth slice genus, smooth concordance genus, and Seifert genus of a knot, respectively. This proves a conjecture of Kearney.

The rapid proliferation of Large Language Models (LLMs) necessitates reliable watermarking techniques to identify AI-generated text and ensure appropriate attribution. While token-based watermarks are vulnerable to paraphrasing, a central challenge for semantic watermarking is to turn sentence meanings into a stable, well-calibrated document-level signal. To this end, we introduce SemField, a simple, training-free semantic watermark that embeds a continuous, linear signal directly into the sentence embedding space. Using a shared secret key to define a specific Gaussian direction, SemField iteratively evaluates candidate sentences and selects those that maximize the alignment of the cumulative document aggregate with this targeted direction. We also propose SemField-PL, a polarity-locked variant that first determines the optimal orientation from the initial sentence and continuously reinforces it throughout the generation process. The document-level aggregated formulation guarantees exact invariance to sentence reordering and provides theoretical bounds against structural tampering, such as sentence insertion and deletion. Lastly, with extensive evaluations across three models, we demonstrate that both variants outperform 12 recent baselines. Across clean detection and four paraphrasing attacks, both variants achieve a mean True Positive Rate (TPR) of 88.2% to 90.4% at a 1% False Positive Rate (FPR), all while maintaining comparable perplexity and naturalness as human-generated content. We open-source our implementation at https://github.com/declare-lab/SemField

We introduce an asymmetric variant of a classic problem by Roth about colourings of integers and midpoints. The new problem cannot be solved with the standard approach by Erdös-Sárközy-Sós which uses symmetry. We describe a Fourier Analytic approach which gives asymptotically tight bounds. In the $\mathbb{F}_3^n$ setting, the density increment we show is efficient and together with the Freiman-Ruzsa Theorem provides an affine subspace of near optimal codimension within the distinguished set. This approach adopted to the setting of the integers from $1$ to $N$ via Bohr sets and Bogolyubov-Ruzsa Lemma gives a progression of length being a power of $N$ which only depends on the number of colours. We then use Chang's Lemma and Balog-Szemerédi-Gowers Theorem to further improve the dependence on the number of colours.

Open-Vocabulary Audio-Visual Event Localization (OV-AVEL) labels each video segment with an event class, including classes that were never seen during training. The dominant pipeline uses a frozen multimodal foundation model (e.g. ImageBind) to embed the visual frame, the audio mel-spectrogram, and each candidate class name into a shared space, then computes two cosine similarities for each segment against each class: visual-text and audio-text. Existing methods then collapse this pair into a single scalar score with a fixed rule (geometric mean, weighted average) before taking the argmax. Instead, we compute complex-valued similarities and learn their fusion using a complex-valued neural network (CVNN). Each modality's standard representation becomes the real part of our pipeline, and a paired companion stream supplies the imaginary part. We use imaginary part of iHSV for visual modality and CycleGAN-translated phase spectrogram for audio modality as these companion streams. This results in two complex similarities, which are then fused. While the vision and audio encoders remain frozen, only the temporal-attention blocks and the fusion CVNN are trained. The four-stream complex architecture sets a new state of the art on both OV-AVEL benchmarks. On the open (unseen-class) split of OV-AVEBench we reach 66.5/59.1/54.1% Acc/Seg-F1/Event-F1 (+1.6/+4.1/+6.6 over the previously reported fine-tuned baseline), with consistent gains for seen classes as well. We also modify AVE dataset for this task and observe that our architecture reaches 60.7/51.9/50.4% Acc/Seg-F1/Event-F1, achieving state-of-the-art OV-AVEL results on it as well. We also propose a two-stream alternative, which also sees great improvements over the baseline.

Spin in clinical trials includes reporting practices that distort the presentation of results. This is particularly critical in medicine, where spin is present in more than 50% of randomized controlled trials that fail to reach statistical significance. The comparison of primary and reported outcomes is crucial for detecting several types of spin, including outcome switching. We used 300 pairs of outcomes labeled with semantic similarity to develop a system for automatic detection of outcome switching. We evaluated baseline text similarity models and open-source LLMs using generated similarity scores and the Youden index to determine the classification threshold. The proposed approach involves prompt engineering, classification based on token probabilities, and majority voting for the final decision. The results on the test set of 2,496 examples with an F1 score of 0.78 and an accuracy of 0.90 outperform baseline text similarity models but trail behind fine-tuned versions of BERT. We used LLMs to generate natural language explanations for the classified instances and manually assessed their quality.

Using notions from graded Poisson geometry, we develop a framework that allows for concise formulations of field theories such as Einstein-Cartan and Palatini-Cartan gravity. We show that Einstein-Cartan gravity is equivalent to a variant of Palatini-Cartan gravity in which the vielbein fields are constrained to satisfy appropriate symmetry relations, which we refer to as constrained vielbein gravity. To that end, we relate these theories via a diffeomorphism of graded manifolds. We then show how to obtain the full Palatini-Cartan theory by removing the symmetry constraint. Finally, we show how our graded geometric framework allows for a direct incorporation of the minimal BV extension of the graded constrained vielbein and Palatini-Cartan theories.

A compromised LLM inference server can leak model weights by encoding payload bits in otherwise plausible token choices. A replay of the same prompt in a trusted server can expose such deviations, but benign numerical nondeterminism also causes token mismatches. Patient attackers can therefore hide within normal variation unless evidence is combined across responses. We introduce a prompt-level e-process that calibrates whole-response mismatch events on trusted benign traffic and accumulates evidence sequentially while, under a calibration-transfer assumption, controlling the probability of any false alarm over an unbounded monitoring horizon. We evaluate it on four models against a seed-blind attack and a stronger seed-aware attack that hides payload bits only in near-ties to remain stealthy, analyzing the channel capacity vs detectability trade-off. Compared with a hard per-token alarm, the e-process combines weak evidence across responses while providing explicit anytime false-alarm control.

Ad-hoc collaboration often requires agents to identify and adhere to some shared convention within a cooperative task. Existing work on reinforcement learning (RL) for ad-hoc collaboration focuses on training agents that adapt to the conventions established by their partners. These methods fail to consider the possibility that while some partners might follow only a single fixed convention, others may themselves be capable of adapting to multiple conventions. Here we present ConventionPlay, an RL-based approach that teaches agents to discover their partner's optimal convention by training against a learned population of partners that exhibit different degrees of adaptability across conventions. Some of these partners follow a single, fixed convention, while others are able to adapt to a subset of the possible conventions for the task in question. The existence of partners that support a limited subset of conventions forces agents trained against this population to actively probe their partner's capabilities, and steer their partner towards the most effective joint strategy that they are capable of following. Our experimental results demonstrate that agents trained via ConventionPlay achieve superior performance to existing ad-hoc collaboration methods against test populations of partners that are compatible with multiple conventions.

We present the postshock R package, which implements and extends a donor-based framework for forecasting when a structural shock is known and the target response of interest is not yet observed. The package estimates shock effects from historical donor episodes, balances donors using specified matching features, and transfers the resulting adjustment to a target-series forecast. It provides integrated workflows for conditional mean forecasting through ARIMA and ARIMAX models and conditional variance forecasting through GARCH-X models. Additional functionality includes structured donor pools, control-shock regressors, automated GARCH-X order selection, processed data objects, and reproducible empirical workflows.

In this article, we show that the Sidon subsets of $[N]^d$ are Fourier uniform, and we prove a dense model lemma for Sidon sets. Using these results and a higher-dimensional version of Rado's theorem, we prove that given any partition regular linear equation in $s \geq 5$ variables with nonzero coefficients, and any finite partition of a sufficiently dense Sidon subset $S$ of $[N]^d$, the number of monochromatic solutions to this equation is $\gg |S|^s N^{-d}$ for all large $N$. As a corollary of the Fourier uniformity result, we show that dense Sidon subsets of $[N]^d$ are equidistributed in certain arithmetic and Bohr structures. Our proofs are motivated by the arguments of Ortega and Prendiville.

Searches for Charged Lepton Flavor Violation (CLFV) are powerful tools to probe physics beyond the Standard Model (BSM). This chapter provides a comprehensive survey of CLFV theory, focusing primarily on low-energy muon and tau transitions while outlining complementary searches from high-energy colliders. Grounded in the Effective Field Theory (EFT) framework, we systematically review key observables including radiative and purely leptonic decays of muon and tau as well as $\mu\to e$ conversion in nuclei and $\tau$ hadronic decay channels. CLFV involving light new physics particles is also discussed. This review captures the modern synergy between EFTs, novel light-mediator scenarios, and hadronic and nuclear physics in the ongoing hunt for BSM signals.

Conclusive identification of an altermagnetic state requires going beyond mere symmetry arguments. We illustrate this in a combined computational and experimental study of the rutile-like material CuF$_2$, which is on the list of predicted altermagnets. Using ab initio and linear spin-wave calculations supplied by magnetization measurements, we show that CuF$_2$ in its experimental monoclinic structure can be described by a spin-$\frac12$ model of weakly coupled square-lattice layers with the in-plane coupling $J_1\simeq 115$ K and two synergistic antiferromagnetic interplane couplings amounting to 4% and 8% of $J_1$, respectively. Driven by long-range superexchange, these interlayer couplings are oblique to the square planes, resulting in the unit-cell doubling in the magnetically ordered state, thus effectively suppressing any altermagnetic band splitting. Concurrently, we identify unusually strong Dzyaloshinskii-Moriya interactions, $|\mathbf D|/J_1\simeq 0.3$, that produce spin canting and, together with order-by-disorder effect, pin the Néel vector to the crystallographic $b$-axis. Additionally, DM anisotropy promotes magnon band splitting, but these bands remain non-chiral. Our results highlight the importance of relativistic effects even in $3d$ magnets with altermagnetic symmetries.

The strength of a polymer network is orders of magnitude below that of a polymer chain, because the network concentrates high tension in a small fraction of polymer strands. Here we show that the strength of a network can be greatly amplified by recruiting a larger fraction of strands to bear high tension. We develop approaches to fabricate hydrogels of exceptional strength while maintaining low hysteresis. The strength of the hydrogel increases from ~0.05 MPa for a regular network, to ~1 MPa for a highly entangled network, and further to ~10 MPa for a prestretched interpenetrating network. Similar amplifications of strength are achieved for elastomers. Furthermore, experimental data suggest a scaling relation between strength and strand length. This work provides design principles for creating elastic and strong polymer networks.

We present a boundary integral equation (BIE) framework for two-dimensional time-harmonic elastic transmission problems. Using the Helmholtz decomposition, the elastodynamic displacement fields are expressed in terms of scalar pressure and shear potentials, so that only the four classical Helmholtz boundary integral operators are needed. The key step is a rewriting of the Navier traction operator in the local normal–tangential frame of the interface, in which second-order normal derivatives are eliminated in favour of tangential derivatives and curvature terms. This leads to a one-parameter family of indirect combined field integral equations, depending on a coupling constant $\alpha\ne 0$, which we prove to be uniquely solvable. The principal part of the resulting operator is, however, defective, and the discretized systems are severely ill-conditioned. Using the pseudodifferential calculus of the Helmholtz BIOs, we construct explicit left and right analytical preconditioners and prove that the preconditioned operator is a compact perturbation of the identity. Numerical experiments based on high-order Nyström discretizations confirm that the preconditioner clusters the spectrum: the number of GMRES iterations becomes independent of the discretization and is reduced by one order of magnitude or more, also in the high-frequency regime and for non-convex scatterers.

Software is under constant evolution: developers continuously add features, fix bugs, and refactor code, and any of these changes may break existing functionality. Regression testing guards against such effects by capturing expected behavior in test cases. LLM-based test generation aims to automate this process by generating regression tests directly from the code under test. This is beneficial when the implementation is correct, but problematic when the code contains faults: the generated tests may then encode and preserve incorrect behavior. To investigate this risk, we apply LLM-based regression test generation to pull requests merged into the main branch of software projects and study the impact of the generated tests on subsequent project evolution. We distinguish between fault-revealing tests, which assert correctly implemented behavior, and fault-enforcing tests, which assert faulty behavior. Across 145 pull requests from SciPy, Qiskit, and pandas, 8%-17% of the generated tests are fault-enforcing, while only 2.4%-4.8% reveal faults. Fault-enforcing tests persist over time: after several subsequent commits, 83%-91% of them are still relevant and pass. They also accumulate: when the faults of all pull requests are combined in one codebase, 83%-92% remain enforced at the end of the commit history, and the developer-written test suite detects only 14%-30% of them. Our results reveal a fundamental risk of LLM-generated regression tests: without manual validation, they may encode faulty behavior as expected behavior, allowing bugs to persist across software revisions and largely evade developer-maintained test suites. LLM-based regression testing can thus give rise to a new form of technical debt.

We study the sample complexity of noisy one-bit compressed sensing for signals drawn from a prior distribution. By characterizing the effective distributional complexity of the prior via its approximate covering number, we prove that posterior sampling achieves accurate recovery with high probability when the number of measurements scales with the logarithm of the approximate covering number, up to a one-bit separation gap factor. This upper bound is robust to learned prior mismatch. Specifically, we show that posterior sampling with an approximate prior remains reliable, provided that the learned prior distribution is sufficiently close to the true signal distribution in Wasserstein distance. In addition, we establish a sample complexity lower bound for any reliable method of noisy one-bit compressed sensing, showing that our upper bound is nearly matched in its main prior dependent term. To approximate the ideal posterior sampling process for real world scenarios, we instantiate posterior sampling through a plug-and-play algorithm with diffusion priors. Experiments on the FFHQ and ImageNet datasets demonstrate the effectiveness of our proposed approach.

Neural networks trained on modular addition tasks often develop Fourier-structured representations that support exact generalization. While prior work has identified these Fourier circuits, the mechanism by which gradient-based training selects them from the data distribution remains unclear. We address this question using probability signatures, which express leading gradient interactions through conditional statistics of the training distribution. For modular addition, these signatures are cyclic shift operators and are diagonalized by the discrete Fourier transform, yielding approximately decoupled Fourier-mode dynamics. This explains the emergence of Fourier sparsity, frequency matching, and phase alignment. The same framework resolves a puzzle under label noise: corrupted examples can show faster early loss decrease than clean examples, despite lacking a coherent generalization rule. We show that noise increases conditional label collisions, strengthening early shared-coordinate reinforcement. Finally, this method can be applied to other operators. Taking XOR as an example, we observed the predicted frequency in experiments.

Supernova remnants (SNRs) are widely regarded as the primary sources of Galactic cosmic ray acceleration and the particles are energized at the shock front of SNRs through the diffusive shock acceleration (DSA) mechanism, gaining energy by repeatedly crossing the shock. Magnetic turbulence plays a crucial role in scattering these particles back and forth, making the investigation of turbulence essential to understanding the acceleration process. In this study, we apply the two-point correlation method to derive the magnetic energy spectrum from the non-thermal X-ray flux image of Tycho's SNR. This image was obtained using Poissonian-based Generalized Morphological Component Analysis (pGMCA). The turbulence length scale observed in the resulting spectrum is shorter than the turbulence scale derived from previous studies of radio observations of Tycho's SNR. Additionally, we use the X-ray rim thickness measured across Tycho which is an indicator of magnetic field strength to estimate the magnetic energy spectrum. The turbulence scale inferred from this method is consistent with that obtained from the non-thermal X-ray flux analysis.

We develop a family of control protocols for finite-time transitions between active nonequilibrium steady states using the noise-color (correlation rate) as the sole control parameter. By reverse-engineering the second moment dynamics of an active Ornstein-Uhlenbeck process, we determine the time-dependent correlation rate required to realize a prescribed evolution of the system's state. We experimentally implement these protocols with a micrometer-sized optically trapped particle coupled to an engineered active bath, demonstrating transitions substantially faster than the natural relaxation while keeping the confining potential, noise amplitude, and temperature fixed. Our approach extends engineered swift-equilibration methods to far-from-equilibrium steady states by directly controlling the temporal correlations of an active environment. We show how physical constraints impose a speed limit on finite-time transitions, while the freedom in choosing the prescribed system evolution can be exploited to eliminate control discontinuities, minimize the admissible transition time, or optimize a thermodynamic cost.

Prussian Blue (Fe-PBA) and its Co and Ni analogues (Co-PBA and Ni-PBA) were synthesized by an additive-free aqueous co-precipitation route using K$_4$[Fe(CN)$_6$] and FeCl$_3$, CoCl$_2$, or NiCl$_2$, respectively. The three powders were systematically compared by X-ray diffraction, ATR-FTIR and Raman spectroscopy, FEG-SEM/EDS, and TG/SDTA. All compositions exhibit the characteristic cubic cyanide-bridged PBA framework, with apparent lattice parameters of 10.10 +/- 0.02, 10.01 +/- 0.02, and 10.10 +/- 0.03 Angstrom for Fe-PBA, Co-PBA, and Ni-PBA, respectively. The dominant C$\equiv$N stretching band in ATR-FTIR shifts from 2062 to 2071 and 2087 cm$^{-1}$ across the Fe-Co-Ni series, indicating composition-dependent changes in the local cyanide environment. Fe-PBA contains less potassium than the Co- and Ni-containing powders and exhibits a substantially larger low-temperature mass loss. Sharp reflections assigned to crystalline KCl are observed in all three diffraction patterns, while additional unassigned reflections in Ni-PBA indicate the presence of at least one further crystalline phase. These results show that differences among Fe-, Co-, and Ni-based PBAs cannot be attributed solely to transition-metal identity, as precursor oxidation state and washing efficiency also influence the composition and thermal response of the resulting powders.

Let \(b(M)\), \(d(M)\), and \(r(M)\) denote the number of bases, the number of circuits, and the rank of a matroid \(M\), respectively. We prove that every nonempty simple binary matroid with no coloops satisfies \[ 2b(M)\ge(r(M)+1)d(M), \] with equality if and only if \(M\) is isomorphic to the Fano matroid. This confirms a conjecture recorded by Oxley in 1983. For the same class, we prove that deleting any element leaves at least as many bases as there are circuits in the original matroid: \(b(M\backslash e)\ge d(M)\) for every \(e\in E(M)\). We determine all equality cases and deduce a sharp linear lower bound for basis growth under successive series extensions. We also give a circuit-size exclusion condition guaranteeing distinct bases containing \(C-e\) for circuit-element pairs \((C,e)\) with prescribed circuit sizes.

Conventional statistical modeling typically assumes a one-to-one mapping between input and output variables, where one-to-one is used in the predictive sense that a specified input results in a unique output. Many scientific and engineering systems, however, exhibit non-one-to-one(noto) input-output relations. In a noto relation, the same input may correspond to multiple admissible outputs, so the conventional statistical models may not be appropriate. This paper develops manifold regression, a parametric modeling framework for such noto input-output relations, which represents the underlying input-output relation through a latent manifold. The manifold is specified upto an unknown finite dimensional parameter and we estimate the unknown parameters from ordinary input-output observations by regularized profile optimization with a robust solver. Once the manifold has been learned, prediction is obtained by slicing the estimated manifold along a specified coordinate value. Because a slice may contain multiple latent roots, the resulting manifold prediction is naturally set-valued. This formulation extends predictive learning beyond conventional one-to-one statistical models while containing classical regression and inverse prediction as special cases. Theoretical results are established to justify the manifold regression model and its sliced prediction sets; and case studies are used to demonstrate stable performance across representative noto cases.