Papers

In this article, we prove the irrationality of at least one of the three numbers $\zeta(5), \zeta(7)$ and $\zeta(9)$ is irrational. We find linear forms in $1,\zeta(5),\zeta(7)$ and $\zeta(9)$ and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity $\Phi$ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $\zeta(5), \zeta(7)$ and $\zeta(9)$ is irrational.

Automated segmentation of CT images has become increasingly important to enhance the reliability of simulations through the generation of high fidelity numerical models. This study addresses the challenging task of semi-automatically tracking textile reinforcements in fan blade dry preforms using X-ray CT images captured at coarse resolutions (i.e., above 140 $\mu$m). Our approach offers a scalable, slice-based analysis conducted on planes orthogonal to the main yarn directions, applied to a large-scale real industrial component. This enables accurate identification and tracking of yarn paths while requiring minimal training. The method models three key yarn properties statistically: their typical cross-section shape, their continuity and movement in the 3D space, and their spatial relative arrangement with respect to neighboring yarns. These statistical properties are integrated into a tracking framework via a variational formulation that optimizes all yarn center positions in successive cross-section planes. The method tracks more than 3,000 warp yarns across 1,500 slices and achieves a tracking success rate above 90%. Overall, this work demonstrates a promising approach toward large-scale, automated textile reinforcement annotation, paving the way for more efficient material characterization in complex composite structures.

Integrated sensing and communications (ISAC) is a cornerstone of sixth-generation (6G) wireless networks: a single waveform, aperture, and hardware platform simultaneously conveys information and senses the physical environment. Its literature, however, is spread across radar signal processing, estimation and information theory, and wireless communications, which makes it difficult for a newcomer to assemble the full mathematical picture. This tutorial fills that gap. Starting from complex-baseband signal models, we build the required foundations step by step, with self-contained derivations. We develop the sensing toolkit (matched filter, ambiguity function, maximum-likelihood estimation, Fisher information, the Cramér-Rao bound (CRB), and Neyman-Pearson detection) and the communication toolkit (Shannon and MIMO capacity, water-filling, and multiuser precoding), and then unify them into the fundamental limits of ISAC: the CRB-rate region, the capacity-distortion function, and the deterministic-random tradeoff. On these foundations we treat waveform design (OFDM radar with two-dimensional FFT processing, OTFS, and jointly optimized waveforms), transmit beamforming as explicit optimization, including semidefinite relaxation, and receiver-side estimation and detection. We then survey the architectures that scale ISAC to the network level (millimeter-wave and terahertz operation, reconfigurable intelligent surfaces, and cell-free networks), together with applications, standardization, and security. A worked MIMO-OFDM design example with concrete numerology ties the mathematics together. Throughout, notation is unified and tabulated, key results are derived rather than merely cited, and seminal contributions are placed in context, so that the reader finishes with both the mathematical fluency and the literature map required to begin research in ISAC.

Let $\Omega$ be the curvilinear square bounded by the four circular arcs $|\zeta\mp1|=\sqrt{2}$, $|\zeta\mp i|=\sqrt{2}$, and let $W=\sqrt{1+14\zeta^{4}+\zeta^{8}}$. Put \[ \theta(\zeta)=\arcsin\frac{2(1+i)\zeta} {\sqrt{1+4i\zeta^{2}-\zeta^{4}}}, \] \[ f=\tfrac14\bigl(-iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr),\quad \quad g=\tfrac14\bigl(\ \ iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr), \] \[ \mathfrak h=(2-\sqrt3)\, F\bigl[\arcsin\bigl(i(2+\sqrt3)\zeta^{2}\bigr),(2-\sqrt3)^{4}\bigr]. \] These are, verbatim, the three functions of our companion paper $\mathrm{[D]}$ on the diamond surface $\mathrm{D}$: the two papers start from the same point, and differ only in which real parts are taken. With $X=(\kappa\,\operatorname{Re}f,\ \kappa\,\operatorname{Re}g,\ \tfrac12+\kappa\,\operatorname{Im}\mathfrak{h})$ and $\kappa=3/(2K[1/9])$, the map $X$ parametrizes a fundamental patch of Schwarz's primitive surface $\mathrm{P}$, the conjugate of $\mathrm{D}$. Part I builds the algebraic apparatus $\ldots$ the normalizing constant is identified exactly: $\varpi:=\kappa^{-1}=K[-3]=\tfrac12 K[3/4]=\tfrac23 K[1/9]$. Part II carries out the elimination and completes the proof of \[ (\star)\qquad \operatorname{sn}\bigl(\varpi(x{+}y),-3\bigr)\,\operatorname{sn}\bigl(\varpi(x{-}y),-3\bigr) =\tfrac13\operatorname{sn}\bigl(3\varpi(z-\tfrac12),\tfrac19\bigr). \] Part III shows that $(\star)$ is, after all, additively separable $\ldots$ for an overview of this paper and $\mathrm{[D]}$, please see arXiv:2609.14206.

An orthogonal affine-precoded superimposed training sequence (AP-STS)-based framework is conceived for cyclic prefix (CP)-assisted multiple-input multiple-output (MIMO) direct-current-biased orthogonal time frequency space (DCOOTFS) visible light communication (VLC) links using arbitrary transmit-receive pulse shaping for transmission over doubly selective channels. For each light-emitting diode (LED), the pilot and data matrices are jointly affine-precoded and overlaid in the delay-Doppler (DD)-domain. Then, a unified end-to-end DD-domain input-output relationship is derived. At each photodiode (PD), orthogonal precoders are utilized to separate the pilot and data components, thereby suppressing mutual interference. Building on this model, an expectation-maximization (EM)-driven DD-domain pilot-aided Bayesian learning (DD-PBL) scheme is developed to estimate the channel state information (CSI). A DD-domain data-aided Bayesian learning (DD-DBL) procedure is then proposed for jointly refining the CSI and detecting data by exploiting the detected symbols as virtual pilots in the spirit of decision-directed channel estimation. The linear minimum mean square error (LMMSE) detector harnessed explicitly accounts for CSI uncertainty due to realistic estimation errors. In addition, Bayesian Cramer-Rao lower bounds (BCRLBs) are derived for the MIMO DCO-OTFS VLC setting considered. Numerical results confirm improved normalized mean-square-error (NMSE), reduced pilot overhead, and mitigated symbol error-rate (SER) relative to recent benchmarks.

The representation number of a graph is the smallest integer $k$ such that the graph can be represented by a word in which each vertex appears exactly $k$ times, and two distinct vertices $x$ and $y$ alternate in the word if and only if they are adjacent in the graph. We extend known results on the representation number for various graph classes to chessboard graphs - namely, king, queen, rook, bishop, and knight graphs. We provide a complete classification for queen graphs and partial classifications or observations for the other classes. As a consequence of our study, we obtain a characterization of all chessboard graphs that are circle graphs. Our work also leads to several interesting open problems.

Coupling-matrix synthesis of cross-coupled resonator filters is difficult when the desired topology is noncanonical. Automatic differentiation removes the need for problem-specific sensitivity expressions, but does not by itself overcome the nonconvex optimization landscape. This work develops a staged differentiable synthesis framework and identifies the importance of its initial objective through a controlled comparison that varies only the objective of the first optimization phase. For tenth- and sixteenth-order benchmarks, directly matching a frequency-sampled response under the prescribed topology reaches the target from $0$ of $24$ and $0$ of $12$ random starts, respectively. Replacing the first-stage objective with characteristic-polynomial matching increases the success rate to $18$ of $24$ and $12$ of $12$ (exact paired $p=7.6\times10^{-6}$ and $4.9\times10^{-4}$). The observed difference is not explained by local conditioning and is consistent with saturation of the bounded response residual far from the solution. A differentiable front end converts symmetric multiband specifications into target polynomials and, when applied end to end to a fabricated dual-band filter specification, finds a realization with one fewer cross-coupling than the published design. Finally, differentiating through Monte Carlo tolerance simulations enables yield-driven design centering, increasing the model-predicted yield from $50.9\%$ to $67.0\pm0.5\%$ under an additive coupling-error model, an improvement of $16.1$ percentage points.

This paper develops a power control framework driven by delayed signal-to-interference-plus-noise ratio (SINR) feedback for orthogonal time frequency space (OTFS) unmanned aerial vehicle (UAV) communications operating under high mobility, with reliability and fairness as the primary design targets.A base station with a uniform linear array serves several UAVs on a common OTFS frame, while the path delays, Doppler shifts, and inter-UAV interference are determined by the three-dimensional propagation geometry and the base-station array response rather than by a postulated coupling model. In place of instantaneous channel state information, the proposed controller refreshes the transmit-power vector from delayed SINR measurements alone, which matches the practical limitations of fast-fading aerial links. A prediction-smoothing-projection rule mixes a reliability share, a fairness share and a spectral-efficiency share, each normalized separately, so that the utility weights control the closed loop directly. Simulations show that the effective SINR of OTFS changes 37.8% less per frame than that of orthogonal frequency division multiplexing (OFDM) at 70 m/s under the same numerology, and that the resulting controller raises the average minimum SINR by 1.21 dB over equal power allocation at 40 m/s while lifting Jain's fairness index from 0.833 to 0.970, at a sum-rate cost of 24.6% that the utility weights keep under the designer's control. The margin of OTFS over OFDM within the same controller widens from 0.18 dB at 10 m/s to 0.81 dB at 90 m/s, showing that the waveform contribution to the usefulness of stale SINR feedback increases with mobility.

Coding-agent benchmarks usually evaluate implementation after the target behavior has been specified in text, code, or demonstrations. Existing research has extensively evaluated the ability of coding agents to generate programs from textual specifications. However, under black-box conditions where neither source code nor documentation is available, it remains underexplored whether an agent can induce the rules solely through visual observation and active interaction and reproduce the target system as a verifiable executable system. To this end, we present GameReplica, a closed-loop evaluation framework for end-to-end black-box game replication that covers the full perception, exploration, induction, reproduction, and verification pipeline. GameReplica comprises 125 tasks spanning 25 games across 5 core mechanism families, with each game instantiated at five difficulty levels. The tasks require an agent to access the target game only through screenshots and an action interface, induce the key visual elements and gameplay rules from pixel feedback and interaction outcomes, and generate a self-contained, runnable game replica that can be automatically verified by an external program. Experiments show that current coding agents still face substantial challenges in end-to-end black-box replication: the best-performing model (Claude Opus 4.8) achieves an overall score of 71.6%, while the remaining models score only 4.0%–42.9%. Further analysis reveals a consistent pattern across all models: visual-fidelity scores are substantially higher than implementation- and rule-consistency scores, indicating that agents replicate visual appearance more readily than game mechanics. The difficulty levels further amplify the performance gap: from L1 to L5, the overall score of weaker agents drops sharply, whereas that of the best-performing agent declines only slightly.

This paper investigates a reconfigurable intelligent surface (RIS)-enabled secure integrated sensing and communication (ISAC) system, where the direct links between the base station (BS) and users are blocked, and the mobile eavesdropper is treated as the sensing target. The mobility of the eavesdropper induces time-varying wiretap channels, potentially compromising secure transmission when outdated eavesdropper information is used. To address this challenge, the BS dynamically tracks the eavesdropper over consecutive time slots and exploits the predicted state information to adapt its transmission strategy. A sum secrecy rate maximization problem is formulated by jointly optimizing the BS beamforming, artificial noise, and RIS reflection coefficients, subject to a constraint on the tracking error covariance to ensure the required sensing accuracy. To tackle the resulting nonconvex optimization problem with temporal coupling, an iterative algorithm integrating an extended Kalman filter (EKF) with an alternating optimization framework is developed. Specifically, the EKF recursively predicts and updates the eavesdropper's state, while the joint optimization problem is decomposed into four tractable subproblems that are solved iteratively. Simulation results demonstrate that the proposed algorithm achieves improved secrecy performance while effectively tracking the trajectory of the mobile eavesdropper, compared with benchmark schemes.

In this paper, the unified generalized $q$-Stirling matrices are defined in terms of the $q$-analogues of the unified generalized Stirling numbers. Several matrix relations involving these matrices are derived, and matrix decompositions of the unified generalized $q$-Stirling matrices are established.

Let $5\leq p_1<\cdots<p_r$ be primes, put $P=p_1\cdots p_r$, and let $g_P(2n)$ count the unordered decompositions $2n=h+k$ for which both summands are coprime to $6P$. We derive an explicit finite inclusion–exclusion formula for $g_P(2n)$ by parametrizing the admissible residue classes and combining the local restrictions through the Chinese remainder theorem. The associated complete-modulus correlation is a specialization of Nagell's totient function and determines both the finite local density and the affine increment $g_P(2(n+6P))-g_P(2n)=\kappa_P(n)$. A parameter-density decomposition gives a residue-sensitive positivity criterion and the uniform sufficient condition $n\geq 3(3^r-1)\prod_{i=1}^{r}\frac{p_i}{p_i-2}+2\Longrightarrow g_P(2n)>0$. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime $p$ gives an explicit threshold $\mathcal N(p)$ satisfying $\log\mathcal N(p)\sim(\log3)p/\log p$. We also describe the complete prime cutoff at $\sqrt{2n}$ and its exact additive-prime interpretation. We also evaluate $g_P(2tP)$ explicitly using the real character of period $3$. The positivity thresholds are elementary and fully explicit; lower-bound sieves provide smaller asymptotic scales, and the present thresholds do not establish positivity at the complete cutoff for all $n\geq13$.

Classical electrodynamics is one of the most well-tested and understood theories in physics. After more than a century of history, it may be surprising that such an established theory still makes new predictions that have not yet been experimentally verified. A noteworthy example is the memory effect—a prediction that an electromagnetic wave can leave a lasting influence long after it has passed. This influence is manifested in a velocity “kick” on a test charge. This simple remark lies at the heart of modern investigations of the low-energy behavior of gravity, electrodynamics, and gauge theories and awaits confirmation (or refutation) through experiments. From a pedagogical perspective, this current research topic beautifully epitomizes how standard concepts from undergraduate electrodynamics can still lead to new physics. In this work, we use the Liénard–Wiechert solutions for the electromagnetic fields of moving charges to understand what the memory effect is, where it comes from, and how it could be experimentally probed in the near future. We also discuss the connections between the memory effect and other important topics in fundamental physics, as well as the search for memory in modern gravitational wave observatories.

The clique partition number of a graph is the minimum number of complete subgraphs whose edge sets partition its edge set. We study graphs in which, in every induced subgraph and outside every prescribed clique, some vertex has a neighbourhood that becomes a clique after deleting at most $s$ vertices. The case $s=0$ is exactly the class of chordal graphs. For each fixed $s$, we prove that the maximum clique partition number at all sufficiently large orders $n$ is $\lfloor(n+s)(n+s+1)/6\rfloor-\binom{s+1}{2}$, and determine all equality graphs. The same expression is an upper bound up to an additive constant depending only on $s$ at every order. In particular, every chordal graph has clique partition number at most $n^2/6+n/6+O(1)$, answering a question of Erdős, Ordman and Zalcstein. We also prove structural stability for sublinear defect and a sharp finite-order theorem for integer signed clique functionals. The proof combines signed fractional localization with an edge-disjoint triangle construction; only a qualitative fractional-packing approximation is required.

We introduce the Internet Archive Music Dataset (IAMD), a large-scale collection of captioned music segments derived from the Internet Archive. To the best of our knowledge, IAMD constitutes the largest publicly available music-caption dataset to date with over 34,000 hours of audio, providing a valuable benchmark for training and evaluating music understanding and generative models. The dataset is built from content declared to be distributed under Creative Commons licenses, and cross-referencing with MusicBrainz is done to improve license information reliability. To annotate IAMD, we present an automatic captioning pipeline that augments base captions produced by an audio-language model (ALM) with textual metadata sourced from the Internet Archive and imputed metadata obtained using audio classification models. Caption quality is assessed objectively and subjectively, and results indicate that the annotation pipeline is reliable and does not degrade caption quality with scaling.

We present TAPe+ML v3, a compact computer vision system based on TAPe (Theory of Active Perception), a structured representation that encodes relations among perceptual elements before recognition. Instead of operating directly on pixel tensors, the system uses a shared TAPe representation and a modular recognition architecture for image classification, object detection, and instance segmentation. TAPe+ML v3 combines background and contour processing, local object localization, prototype-based classification, and a coordinator for specialized submodels. Across the reported experiments, it uses fewer than 100,000 parameters. On COCO object detection, it obtains 84.7 mAP50 and 65.3 mAP50-95. On COCO instance segmentation, it obtains 80.7 mask mAP50 and 58.4 mask mAP50-95. In classification experiments, it reaches 92 percent validation accuracy on Imagenette under an identical-training comparison with a raw-pixel baseline, and 89.9 percent Top-1 accuracy on ImageNet-Real. We also evaluate compactness in video scene detection and adaptation under distribution shift in an industrial pilot. The results suggest that shifting part of the modeling burden from network parameters to a structured input representation can support compact multi-task vision systems with reduced data, memory, and compute requirements.

Transfer and multi-nucleon transfer reactions are essential tools for probing nuclear structure and reaction dynamics, requiring precise determination of the identity, energy, and emission angles of reaction products. The increasing granularity of modern silicon telescope arrays enhances experimental capabilities but challenges detector calibration, as conventional channel-by-channel approaches become inefficient and difficult to scale. In this work, we present a fully automated, physics-informed calibration framework based on neural networks, specifically designed for highly segmented silicon detector arrays. The method formulates calibration as a global optimization problem, in which detector gains and geometrical corrections are determined simultaneously by minimizing the width of the reconstructed excitation energy under two-body kinematics constraints. The approach relies exclusively on experimental data and well-established physical principles, without requiring explicit modeling of detector response. A distinctive feature is the use of multiple neural network sub-models sharing a common loss function with embedded physics constraints, enabling coherent and self-consistent calibration across all detector channels. This strategy ensures scalability, robustness, and reproducibility, making it particularly suitable for next-generation detector systems with increasing complexity. The performance of the method is demonstrated using experimental data from the Particle-Identification Silicon-Telescope Array (PISTA) in high-resolution fission studies in inverse kinematics. The results show excellent agreement with theoretical kinematics, high-quality particle identification, and a significant improvement in calibration efficiency. The proposed framework provides a general and adaptable solution for the calibration of complex detector systems in modern nuclear physics experiments.

Physics has traditionally advanced through two interrelated activities: inferring the laws of nature from observations and testing theoretical predictions. This Perspective proposes a new research paradigm – adversarial physics – that subjects physical limits regarded as insurmountable to active stress testing. We introduce the heuristic concept of a "cosmic bug" to denote an exploitable physical anomaly: a reproducibly realizable physical configuration that grants an observer new capabilities previously considered outside the set of achievable physical operations. The proposed paradigm currently encompasses at least three levels: (1) actively constructing complex systems under established physical laws to seek and exploit unforeseen loopholes in specific configurations and combinations of operations; (2) actively seeking, testing, and exploiting anomalous mechanisms beyond the explanatory scope of currently known theories, prompting revisions to our understanding of physical laws or the discovery of new physics; and (3) exploring how loopholes might be exploited to control or modify local physical rules. We further argue that traditional searches centered on the energy, intensity, and precision frontiers may have comparatively neglected another dimension – the configuration frontier: the vast space of physical configurations comprising complex temporal sequences, phase correlations, coherent and entangled states, feedback protocols, topological structures, multiscale organization, and engineered spacetime or vacuum conditions. Artificial intelligence provides an important tool for exploring this high-dimensional configuration space. The ultimate aim is not merely to map nature's capability boundaries, but to discover and exploit possible loopholes in the laws governing it – including physically realizable routes to controlling or modifying local physical rules.

For a ferromagnetic Ising model on a graph of maximum degree $\Delta\ge3$, we prove a bound of order $\sqrt n$ on every row of the influence matrix at the tree uniqueness threshold. The estimate is uniform in the degree, the external fields, and all pinnings. More generally, if the couplings are bounded by $\beta$ and $\varepsilon=((\Delta-1)\tanh\beta-1)_+$, the bound is $C(\sqrt n+n\varepsilon)$. The proof combines a pointwise cavity bound with a positive-series magnetization tilt and the field comparison theorem of Ding, Song and Sun. The critical estimate removes the logarithm in recent general graphical bounds for the ferromagnetic case. As a consequence, zero-field single-site Glauber dynamics mixes in polynomial time throughout the supercritical window $\varepsilon=O(\sqrt{\log n/n})$, with the polynomial degree depending on the window size.

We present a covariant modified gravity theory calibrated to match all the galactic phenomenology usually attributed to dark matter in the low acceleration regime, including the flat rotation curves of massive tracers, and the deflection angles of photons as inferred through gravitational lensing. We first derive a fully covariant description of galactic space-times under a spherically symmetric and static approximation. This description accounts for flat rotation curves, observed gravitational lensing phenomenology and the Tully-Fisher mass scalings of the above. The resulting covariant description is then used to constrain the parameters of a minimally coupled gravitational action using power laws for both the Ricci scalar and the matter Lagrangian. The action presented hence leads to field equations whose trace, has as solutions metric coefficients which reproduce flat rotation curves and gravitational lensing observations, without including any dark matter. Using the same action for a FLRW metric results in a cosmology where a dust component behaves as effective dark energy with an equation of state $\omega=-5/6$, without the explicit inclusion of any dark energy component.

For \(n\ge3\) and a graph \(H\) on at most \(n\) vertices, let \(q(H)\) be the expectation threshold for its containment in \(G(n,p)\). We prove that there are absolute constants \(a,L>0\) such that, for every \(C>0\), every graph of degeneracy at most \(C\log n/\log\log n\) has a deterministic edge decomposition into at most \(\lceil a(C+1)\rceil\) pieces, each with ordinary containment threshold at most \(Lq(H)\). This removes the maximum-degree hypothesis from a theorem of Ascoli, He, Park, and Talagrand in their approach to Talagrand's discrete convexity problem. With constants depending on the fixed parameters, a structural extension allows the addition of \(O(\log n)\) vertices with arbitrary incident edges. The uniform main bound also gives an \(O(1+\log\log n)\)-piece decomposition for every target graph, with an absolute threshold multiplier. The main ingredient converts Li's two-set coupling for spread measures into a partition of a prescribed neighborhood list, fixed before the random host is sampled. It allows arbitrary overlaps and repetitions and controls all Hall matching conditions simultaneously by bounding target demand and host supply through common two-sided approximations under the biased host measure.

The answer is "Yes". As it has been shown in the Ref.[1] (22 Sep.2017, see also the comments), useing a new definite mathematical axiomatic-algebraic matrix approach, all the fundamental laws of nature could be derived uniquely (where the axiom of "no zero divisors" of the ring of integers has been generilzed and written in a new definite formulation, then basically assuming that all the physical quantities could only and only take the rational values). Based on this new mathematical approach along with the C, P and T symmetries of the derived field equations, it is concluded that the universe could be realized solely with the (2+1) and (3+1)-dimensional space-times. Moreover it is shown that the (3+1) dimensional cases of the directly determinated general covariant field equations (including two definite classes: a two indexes and a four indexes tensor fields), respectively, represent two new massive forms of the bispinor fields of spin-1and spin-2 particles; and the (2+1)-dimensional cases of the drived equations (including: a two indexes and a four indexes tensor fields), represent (asymptotically) two new massive forms of the bispinor fields of spin-3/2 and spin-1/2 particles, respectively. As a particular result, based on the formulation of the derived Electromagnetic Maxwell equations (representing by the bispinor fields of spin-1 particles, including new field equations - corresponding to the YangMills equations - compatible with two specified forms of the gauge symmetry groups), it has been concluded that magnetic monopoles could not exist in the nature to any extend. Furthermore, as the only elementary particles that could be existed in nature, along with the all discovered particles, eight new particles, including four charge-less right-handed spin-1/2 fermions (two leptons and two quarks), and a spin-3/2 fermion, and also three spin-1 massive bosons are also solely predicted.

For a finite simple graph G, let alpha(G) denote its independence number and let l_avg(G) = (1 / |V(G)|) sum_{v in V(G)} alpha(G[N_G(v)]) be the average independence number of its open neighbourhoods. Written on the Wall II Conjecture 194 asserts that every simple connected graph on n > 1 vertices satisfying alpha(G) <= 1 + l_avg(G) has a Hamiltonian path. We give a four-parameter family of counterexamples. Its principal two-parameter subfamily satisfies the proposed inequality with equality: for every pair of integers s >= 1 and t >= 3 it has (s + 1)t^2 vertices, independence number t + 1, l_avg(G) = t, and minimum degree s, but has no Hamiltonian path. This entire infinite subfamily is machine-checked in Lean 4: one universally quantified theorem certifies its order, connectivity, independence number, average neighbourhood independence, minimum degree, conjecture hypothesis, and failure of traceability. Thus no fixed lower bound on the minimum degree repairs the conjecture. The case (s,t) = (1,3) has 18 vertices, but the formal certificate is parametric rather than a verification of that one graph alone.

AthenaZero is a bimanual manipulator designed to minimize inertia without compromising control authority. By utilizing quasi-direct drive actuation and transmission remotization techniques, the system achieves an effective endpoint mass comparable to that of a human—about an order of magnitude less than conventional robot manipulators. This characteristic, combined with its inherent torque transparency, makes AthenaZero exceptionally well-suited for dynamic manipulation. We describe the methodology} that led to this design and demonstrate the robot's capabilities on three baseball-inspired tasks: throwing, catching, and batting, which showcase complex interactions on human-comparable timescales where milliseconds matter. AthenaZero was capable of throwing at speeds in excess of 30 m/s, with catching and batting at speeds in excess of 14 m/s over a short 7.3 m distance. Batting practice and a game of catch were subsequently performed in robot-to-robot and human-to-robot variations, showcasing the efficacy and adaptability of our system in tasks that require high acceleration.

Liquid droplets deform from spherical shape due to aerodynamic variation of pressure along the surface as the droplet moves through a gas. The deformation is predicted for axisymmetric droplets translating through a gas with low Weber numbers, We < 1, and Reynolds number Re = O(10). That deformation analysis is based on the relations between local pressure jump and the two radii of curvature. A thin boundary layer on both sides of the gas-liquid interface is considered with a surface-velocity jump due to pressure-gradient-driven flow with a large density jump and a pressure jump due to surface tension. A near-ellipsoidal shape is predicted using $We$ as a perturbation parameter. Then, the quasi-steady internal liquid-phase stream function and velocity field are predicted, describing internal circulation and a vortex ring structure with vorticity distributed through an inviscid liquid. The gas-phase flow over the oblate droplet is described using a ring doublet as an image within the droplet. The ring-doublet radius is related to We. Gas potential flow results are presented and compared using both the exact analytical solution and a perturbation analysis based on the square root of We. The perturbation analysis provides a lower computational cost. Three analyses for local curvature, liquid circulation, and gas potential flow are matched to yield the velocity and pressure fields. The appropriate radius for the image ring doublet is matched to the square root of We. Liquid-phase stream function, two velocity components in each fluid, and gas potential field are predicted. S Some comments on droplet drag are presented.

The exploration of non-hexagonal two-dimensional topologies has opened new possibilities for tailoring the properties of group III-V monolayers beyond those accessible through conventional honeycomb phases. In this context, we have investigated the structural, mechanical, electronic, and optical properties of T-GaN, a two-dimensional tetragonal gallium nitride monolayer composed of alternating four- and eight-membered rings featuring coexisting homoelemental (Ga-Ga, N-N) and heteropolar (Ga-N) bonds, using density functional theory (DFT) within the generalized gradient approximation (GGA/PBE) and the hybrid HSE06 functional. The dynamical stability of T-GaN was confirmed by phonon dispersion calculations, which revealed the absence of imaginary frequencies throughout the Brillouin zone, and was further supported by ab initio molecular dynamics (AIMD) simulations. The mechanical characterization reveals a pronounced in-plane anisotropy, with critical strains of approximately 16.5% and 7.0% along the $x$- and $y$-directions, respectively. The electronic band structure analysis indicates that T-GaN is a nonmagnetic semiconductor with an indirect band gap of 0.35~eV (PBE) and 1.15~eV (HSE06), with the valence band maximum dominated by nitrogen 2p orbitals and the conduction band minimum governed by gallium 4s and 4p states. The optical response was evaluated along three crystallographic directions, exhibiting considerable anisotropy in the absorption coefficient, refractive index, and reflectivity. These findings provide new insights into the physical properties of tetragonal group III-V monolayers and suggest that T-GaN may serve as a promising candidate for anisotropic nanoelectronic and optoelectronic applications.

Atomic vacancies and vacancy aggregates control the thermodynamic stability and the functional response of graphene, yet the configurational space spanned by many vacancies at variable concentration and separation is too large to be mapped exhaustively by first-principles methods. Here, we map and rationalize this stability landscape by combining semiempirical atomistic thermodynamics, interpretable machine learning, and symbolic regression. Several hundred defective supercells, built from a 72-atom cell by varying the vacancy concentration and the inter-vacancy distance up to the fourth neighbor, were relaxed with the PM7 Hamiltonian in MOPAC, and the heat of formation was adopted as the stability metric. Each structure was encoded with the Dynamic Collision Fingerprint, a translationally and rotationally invariant descriptor that maps the local topology onto transport-like statistics of virtual probe particles. A gradient boosted decision tree model, optimized by Bayesian hyperparameter search, reproduces the heat of formation of an independent test set with a root mean squared error of approximately 23.5~kcal/mol and no evidence of overfitting, and a SHAP analysis identifies the defect concentration and the inter-vacancy distance as the two variables that dominate the stability. Symbolic regression then condenses the learned mapping into a compact closed-form expression that reproduces the heat of formation with a coefficient of determination of $R^{2} = 0.9966$, a mean absolute error of 15.51~kcal/mol, and a root mean squared error of 24.18~kcal/mol. The workflow turns a high-dimensional structure–stability problem into an interpretable analytical law, providing a transferable route to rational defect engineering in two-dimensional materials.

Background and Objectives: Spatial mapping retains the spatial distribution of radiomic features, but computational cost and fragmented software limit its use. We developed Flash-Radiomics with scalar extraction and spatial mapping, a central processing unit (CPU) backend, a hybrid Compute Unified Device Architecture (CUDA) backend, consistent feature names, and Hierarchical Data Format version 5 (HDF5) storage. Methods: We evaluated Image Biomarker Standardisation Initiative (IBSI) compliance, CPU-CUDA concordance, and end-to-end processing time. Compliance testing included 825 chapter 1 (IBSI-1) tests covering 165 high-consensus features and 323 chapter 2 (IBSI-2) tests with numerical references. Concordance testing included 1,148 scalar pairs and 93 spatial-map pairs. End-to-end processing time was measured five times per input volume of interest (VOI) size. Comparisons included the Medical Image Radiomics Processor (MIRP) and PyRadiomics for 102 shared scalar features and PyRadiomics for 93 shared spatial maps. Results: Both backends passed all 1,148 IBSI tests, and all paired results were concordant. At the largest scalar input, CPU required 76.343 s and hybrid CUDA 81.915 s; CPU was 4.7 times faster than MIRP and 190.7 times faster than PyRadiomics. At the largest spatial input completed by both backends, hybrid CUDA reduced processing time by 68.6% relative to CPU (79.280 versus 252.791 s). At PyRadiomics' largest completed spatial input, hybrid CUDA was 84.9 times faster. Conclusions: Flash-Radiomics unified standardized scalar extraction, spatial mapping, concordant CPU-CUDA results, and HDF5 storage. CPU processing time was similar or shorter for scalar extraction, whereas hybrid CUDA was faster for spatial mapping under the tested conditions.

Design verification remains one of the most resource-intensive stages of hardware development, often consuming up to 70% of the total design effort. While recent work has explored using Large Language Models (LLMs) to automate testbench generation, most existing approaches focus narrowly on functional correctness, overlooking the critical aspect of coverage quality. To bridge this gap, we present CovR, an agentic framework for automated testbench generation that combines self-reflection loops with simulation-based feedback to maximize coverage. Using this pipeline, we construct a large-scale dataset of 16,514 natural specification RTL reasoning testbench tuples with a strong teacher model, enabling coverage-aware supervision. Building on this, we propose a reinforcement learning (RL) framework tailored for coverage-driven testbench generation, leveraging tool-derived rewards from simulation and coverage feedback to optimize a student model. Experimental results show that the CovR finetuned model achieves 93.81% cov@10 on VerilogEval and RTLLM V2.0, and 87.76% cov@10 on CVDP, outperforming state-of-the-art approaches by 7.97% and 3.59%, respectively. Furthermore, deploying the finetuned model back into the agentic refinement pipeline further improves cov@10 to 94.27% on VerilogEval and RTLLM V2.0 and 91.39% on CVDP. Moreover, when integrated as a plug-in stimulus engine for full verification workflows, CovR improves coverage by 18.95% and mutation detection score by 1.19%, while revealing 4.46% undetected failures, highlighting the importance of optimizing for coverage in LLM-based hardware verification.

We study umbilic slopes and the global classification of cubic Weingarten surfaces. For a smooth surface in Euclidean three-space, a nonconstant principal curvature germ with a unique secant tangent has slope zero, minus one, infinity, an odd integer at least three, or its reciprocal. At a nonconstant umbilic germ, a smooth regular relation between mean and Gaussian curvature supplies such a tangent after a continuous labeling of the principal curvatures; the ordered curvature image has at most two limiting directions, both in the same set. We also classify compact immersed surfaces without boundary of class $C^4$ satisfying $\kappa_2=c\kappa_1^3$ in some order at every point, for a fixed real $c$: each connected component is embedded as an ellipsoid of revolution. The arguments share a classification of homogeneous Hessian identities at an umbilic. Scalar unique continuation treats infinite-order contact in the smooth slope problem, whereas an exact fourth-order growth estimate and angular energy estimates yield the cubic classification.