Scintillators convert ionizing radiation into visible photons, enabling applications from cosmic ray detection to medical imaging. Two independent strategies for improving scintillator performance via nanoscale patterning have recently been demonstrated: engineering material properties to enhance absorption of ionizing radiation and integrating nanophotonic structures to enhance the spontaneous emission rate ("nanophotonic scintillators"). Here, we propose a nanophotonic scintillator that simultaneously enhances both the initial energy conversion and the spontaneous emission rate, by periodically stacking a fast-emitting scintillator and a visible-light-transparent material with strong X-ray attenuation ("stopping layer") to form a one-dimensional (1D) photonic crystal (PhC) scintillator. Photoelectric absorption in the stopping layer increases the number of photoelectrons that deposit energy in neighboring scintillator layers and contribute to scintillation. At the same time, the spontaneous emission rate is enhanced by the nanophotonic structuring itself. We design a 1D PhC comprising an organic scintillator and indium tin oxide (ITO) as the stopping layer and numerically simulate the enhancement in scintillation yield and decay rate. The total detected light output is enhanced by up to a factor of 700 compared to a bulk organic scintillator of equal thickness. We further investigate a 1D PhC structure integrating inorganic and organic scintillators for time-of-flight positron emission tomography (TOF-PET): replacing the non-scintillating stopping layer with an inorganic scintillator further increases the light yield, and the coincidence time resolution (CTR) is enhanced up to 3.5 times compared to a bulk inorganic scintillator of equal thickness. Our work presents a unified approach to improve key scintillation parameters within a single nanophotonic structure.

Angle power spectrum (APS) characterizes the directional distribution of received signal power and is directly relevant to beam management and MIMO processing. While environment-aware learning has been widely studied for radio maps and path loss, direct map-to-APS prediction still lacks a standardized large-scale benchmark. This paper presents Map2APS, a physically grounded benchmark constructed from intelligent ray-tracing (IRT) path-level propagation records. Map2APS covers 51 equal-height urban maps and approximately 2.55 million Tx--Rx samples, with a strict cross-map split for evaluating generalization to unseen urban layouts. We benchmark representative model families and introduce MS-AReg as a strong reference baseline. On the full held-out test set of 249{,}993 samples, MS-AReg achieves a cosine similarity of 0.948, a peak location error of 1.20$^\circ$, and an inference latency of 0.101 ms/sample. We further report dominant-direction metrics, including Top-1 dominant peak hit rate and dominant peak recall, to evaluate whether predicted spectra preserve decision-relevant arrival directions. The benchmark, code, and evaluation scripts are released at https://github.com/UNIC-Lab/aps-data.

Positronium-based imaging requires realistic modelling of positronium (Ps) decay in matter. We introduce a modular Ps decay model implemented in GATE 9.4 and GATE 10, enabling the definition of an arbitrary number of decay channels characterised by lifetime, branching fraction, annihilation multiplicity (2g/3g), and optional prompt photon emission. The model is validated through analytical and numerical benchmarks, including lifetime distributions, branching fraction consistency, photon kinematics, and prompt photon emission. Its practical applicability is demonstrated using simulations of mixed annihilation scenarios and the NEMA IEC phantom with a large field-of-view PET system. The proposed model accurately reproduces input lifetime distributions as weighted sums of exponential components and correctly samples decay channel fractions. Simulated two- and three-photon annihilation kinematics are consistent with theoretical expectations. Complex mixtures of decay channels, including varying 3g-to-2g ratios and multi-component ortho-positronium lifetimes, are correctly modelled, with observable signatures reflected in both temporal and energy distributions. Phantom simulations demonstrate the capability to generate realistic positronium-sensitive datasets. This work provides the first general-purpose, multi-channel positronium decay model integrated into GATE, enabling realistic simulations of positronium behaviour in complex media. The model supports the development and optimisation of positronium-based imaging techniques, including PLI and multi-photon PET, and applies to medical imaging, industrial tomography, and fundamental physics studies. Its public availability and compatibility with standard GATE workflows make it a valuable tool for the broader research community.

Radio waves propagating through the interstellar medium are influenced by variations in plasma density. For spatially localised plasma structures along the line of sight, time-delay Doppler analyses of pulsars often reveal scintillation arcs in the secondary spectrum, frequently exhibiting a parabolic morphology. In the thin-screen approximation, the arc curvature is commonly used to infer the distance to the plasma concentration, which is modelled - via Kirchhoff-Fresnel diffraction theory - as an effective phase screen imposed by the column density of a localised disturbance. Here, we identify several limitations of the thin-screen model that necessitate a fully three-dimensional treatment, without reducing the problem to a projected screen density. We show that the arc curvature can vary depending on the three-dimensional structure of the plasma, rendering it a less reliable indicator of distance. Moreover, when volume propagation is considered, asymmetries and a richer variety of features emerge in the secondary spectrum compared to those predicted by the thin-screen approximation. We conjecture that these phenomena are linked to the onset of branched flow produced by a sequence of weak but correlated scattering events.

By implementing the Bogoliubov-de Gennes (BdG) formalism of population-imbalanced atomic Fermi gases with pairing interactions in a thin spherical shell, we characterize the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state in such a compact geometry. We first construct a phase diagram showing where uniform solutions of spin-polarized Fermi superfluid from the BdG equation cease to exist due to the vanishing order parameter. Near the boundary, various LOFF states with spatially modulating order parameters and density profiles can survive as convergent solutions to the BdG equation. When both uniform and LOFF solutions are present, we compare their grand potentials to determine the energetically favorable state and find that the LOFF states with multiple nodes in the order parameter become more stable at higher spin polarization. However, the LOFF state only survives close to the phase boundary where the uniform solutions vanish, indicating fragility of the LOFF state on a spherical surface. We also briefly discuss possible implications.

The main purpose of the present paper is to investigate the symmetry properties of a Kähler manifold involving the Ricci tensor. In this context, the most symmetric manifolds are Kähler-Einstein spaces, and their natural generalizations are Ricci-parallel Kähler manifolds, Ricci-semisymmetric Kähler manifolds and holomorphically Ricci-pseudosymmetric Kähler manifolds. Unlike their Riemannian counterparts, we prove that all these conditions also admit a characterization solely in terms of holomorphic planes, analogously to the symmetries related to the Riemannian curvature tensor in Kähler manifolds. A key finding is that the concept of holomorphic Ricci pseudosymmetry is distinct from the classical Ricci-pseudosymmetric condition introduced by Deszcz. By carefully analyzing the interplay between these definitions, we clarify the precise geometric role of the so-called Ricci curvature of Deszcz. Additionally, we also present a geometric interpretation of the complex Tachibana-Ricci tensor and we establish a new criterion for a Kähler manifold to be Einstein based on holomorphic planes.

We investigate a scalar-vector-tensor theory in which matter is minimally coupled to a Jordan-frame metric, while a massive vector sector interacts with the baryonic current. We show that the conformal scalar coupling modifies the physical expansion rate measured by matter observers, leading to a late-time enhancement of the effective Hubble constant. By constructing a phenomenological scalar evolution that becomes relevant only at low redshifts, the model provides a purely late-time mechanism for alleviating the Hubble tension without significantly affecting early-universe cosmology. The scalar potential naturally acts as a dynamical dark-energy sector, while the vector contribution behaves effectively as a pressureless component at cosmological scales through a density-dependent vector mass. Hence, the framework connects late-time scalar dynamics, effective dark-energy evolution, and Hubble-tension alleviation within a unified setup. Finally, local gravitational constraints can be suppressed through a chameleon-type screening mechanism, allowing the theory to remain compatible with Solar-System tests while retaining nontrivial cosmological effects.

This paper studies Markov-switching (MS) models with time-varying transition probabilities (TVTP) under various specifications of the transition probability matrix. Especially, we extend the two-regime common-variance setting of the Generalized Autoregressive Score (GAS) model from (Bazzi et al., 2017) to the general $K$-regime case with regime-specific means and variances. Our study contains comprehensive Monte Carlo simulations and we developed an open-source R package, \texttt{multiregimeTVTP}, for data simulation and parameter estimation. We find that the regime means, variances, and transition probabilities are reliably recovered, whereas the TVTP driving coefficients are harder to identify. Another finding from our paper is that the GAS score coefficient appears to be statistically non-identifiable, due to a ridge in the joint likelihood surface $(σ^2,A)$. In addition, we find that one-step point forecasts are remarkably robust to TVTP misspecification, but filtered regime probabilities are not, so correct specification matters most for characterizing regime dynamics rather than short-horizon forecasting. An empirical application to U.S. Treasury zero-coupon yield changes at four maturities (1961-2024) shows that an exogenous specification driven by the lagged yield level dominates the constant and lagged-change models in fit, while the GAS specification fails to converge, with $\hat{A}$ collapsing to zero, reflecting the same identifiability issue observed in simulation.

A set $\mathcal{G}$ of integers is called a $g$-Golomb ruler of length $n$ if the difference between any two distinct elements of $\mathcal{G}$ is repeated at most $g$ times. If $g=1$, these are also called $B_2$-sets, Sidon sets, and Babcock sets. We define $G(g,n)$ to represent the minimum diameter of a $g$-Golomb Ruler. In this paper, we prove that for all $b\ge 1$, if $g \ge \frac{7}{4}\left(b^{3/2} -b\right)+1,$ then $G(g,g+b)=g+2b-2$. Sharper bounds are given for $b\le 18$. The main technique is through an arithmetic property of the integers that are \emph{not} in a $g$-Golomb ruler, leading us to introduce LM rulers, a new class of rulers where every distance $d$ occurs as a difference at most $d-1$ times. We show that the minimum diameter of an $n$-element LM ruler $L(n)$ is $\sqrt{8/9} \cdot (n-1)^{3/2} \le L(n) \le \frac{7}{4}\left((n+1)^{3/2}-(n+1)\right).$

We present a new, calibrated $G$-band relationship between pulsation period $P$, Fourier parameter $ϕ_{31}$, and metallicity [Fe/H] for galactic RR Lyrae stars from the Gaia survey. A set of 72 fundamental mode RR Lyrae stars were identified for deriving the relation in the $G$-band, after visual examination of their light curves. Unlike recent large-scale calibrations, our relation prioritizes calibration purity by anchoring exclusively to a homogeneously analyzed sample of high-resolution spectroscopic metallicities from the literature. Our best fit relation is $\text{[Fe/H]} = (-6.93 \pm 0.58) - (6.04 \pm 0.37)P + (1.65 \pm 0.11)ϕ_{31}$. We compare the [Fe/H] predicted by our relation for the stars in our calibration sample with that obtained from previously established relations in the $G$-band using different approaches. Our calibrated $G$-band $P$-$ϕ_{31}$-[Fe/H] relationship demonstrates high reliability when validated against spectroscopic data, achieving a negligible bias of $0.00$ dex and an empirical RMS scatter of 0.26 dex. Furthermore, by applying an Orthogonal Distance Regression (ODR) routine that fully propagates parameter covariance, we establish a mathematically strict empirical baseline whose theoretical uncertainties perfectly align with this observed dispersion. We find that the inclusion of the $R_{21}$ Fourier parameter offers no significant improvement in metallicity estimation. Comparisons with literature confirm that our linear relation aligns closely with other Gaia DR3-based studies, while offering improved precision over older DR2-based relations.

Ramanujam's theorem states that any connected finite-dimensional subgroup of the automorphism group $\mathrm{Aut}(X)$ of an irreducible variety $X$ is an algebraic group, in a natural way. In this note, we discuss the notion of dimension and extend Ramanujam's theorem to arbitrary (not necessarily irreducible) varieties.

In Chinese computer science and engineering, safety and security have long been translated by the same word, "anquan". This convention is concise in ordinary communication, but it creates persistent conceptual compression in standards interpretation, interdisciplinary collaboration, risk analysis and academic writing. When researchers need to discuss both whether a system is free from intolerable non-adversarial harm and whether it can resist adversarial threats, the single word "anquan" often cannot carry the distinction. This article argues that, while established legal and standards titles should be retained, scholarly and engineering writing should translate security as "anbao", and reserve "anquan" mainly for safety. This is not a cosmetic translation preference, but a proposal for terminological governance in scientific cognition, engineering risk communication and assurance argumentation. The article first surveys the conceptual boundary between safety and security in international and Chinese standards, and analyzes how the current translation overload affects functional safety, SOTIF, information security, cybersecurity, automotive cybersecurity and AI governance. It then uses recent work on AI assurance, safety-security co-assurance and security-informed safety to show why precise terminology is fundamental to scientific arguments that can be examined, challenged and communicated. Finally, it proposes a staged, dual-track writing practice for Chinese technical discourse.

Objective. Hypoxic tumor subvolumes resist radiation through elevated oxygen enhancement ratios (OER), yet no computational OER model is simultaneously particle-specific, mechanistically grounded, and fast enough for voxel-scale treatment planning. We present the VOxel-Aware Oxygen Model (VOxA) to address all three requirements. Approach. An Oxygen Model (OM) encodes particle-specific LET-OER dependence through dual sigmoidal transitions constrained to increase monotonically with atomic number Z, combined with Michaelis-Menten oxygen kinetics. A Voxel-Aware (VA) extension resolves per-DSB local energy heterogeneity via a calibrated particle-specific sensitivity parameter. Calibrated on 233 OER observations from 29 sources across 10 particle types (LET = 0.2-654 keV/um); DSB coordinates from TOPAS-nBio simulations. Main results. The OM achieves $R^2 = 0.719$ and MAE = 0.300 retention OER units; theoretical OER maximum 3.32 (2.4% from measurement), bootstrap median 3.37 [3.18, 4.09]. The composite $K_{\rm fix} + K_{\rm repair} = 2.82$ mmHg is tightly constrained despite high collinearity (r = 0.935). On the Furusawa heavy-ion subset, VOxA achieves 28.4% lower survival OER MAE than the clinical standard (63.1% on helium, 24.0% on carbon) and reproduces He < C < Ne Z-ordering that universal models cannot capture. The VA extension passes 18 tests confirming sample-size-invariant within-nucleus coefficient of variation of the per-DSB retention probability. VOxA evaluates in under $10^{-3}$ ms per voxel, more than $10^6$ times faster than Monte Carlo chemistry. Significance. VOxA is the first particle-specific OER model to reproduce Z-ordering analytically at clinical planning speed, validated on the largest OER calibration dataset for this model class. Committed-break coordinates at whole-nuclear scale provide the input for inter-break topological analysis and hypoxic LET painting.

Two-body reduced density matrices (2RDMs) encode the essential two-electron physics of electronic states, but their quartic storage cost poses a major limitation in practical workflows. We investigate a simple protocol to compress both transition and non-transition 2RDMs into a lower-rank representation that preserves their wedge-product structure and physical symmetries under truncation. The resulting decomposition couples Coulomb and exchange channels through a common set of low-rank factors, yielding a more compact rank-sparse representation than single-channel factorizations. For correlated states, the effective rank scales linearly with system size, achieving a $\sim99$\% compression for the coupled-cluster 2RDM of octane while retaining chemical accuracy. We apply this to the recently introduced {\em ab initio} eigenvector continuation workflows, where many-body wave functions are interpolated across nuclear geometries with mean-field cost. Here, 2RDMs between training states act as projectors into a subspace but their memory scaling limits applications to larger systems. The compression scheme reduces the memory cost from quartic to quadratic for a fixed error per electron. Metrics to systematically control the decomposition are investigated, enabling statistically resolved structural, dynamical and spectroscopic observables from nonadiabatic molecular dynamics simulations of photoexcited H$_{28}$ chains, interpolating from compressed near-exact DMRG training data. This establishes these structure-preserving compressed intermediates for practical correlated electronic structure workflows.

Modern GPUs increasingly rely on specialized hardware units and asynchronous coordination mechanisms, so performance depends on orchestrating data movement, tensor-core computation, and synchronization rather than exposing more thread-level parallelism. This creates a programming-model tension: if too much execution structure is hidden, the compiler must catch up to new hardware mechanisms; if too much is exposed, the burden of orchestration falls back onto the programmer. We present TLX (Triton Low-level Language Extensions), built around MIMW (Multi-Instruction, Multi-Warp), which expresses orchestration at warp-group granularity while preserving Triton's productive blocked programming model for regular computation. TLX realizes this idea as an embedded extension to Triton, exposing explicit interfaces for multi-warp execution, local-memory orchestration, asynchronous operations, and cluster-aware control. Our evaluation shows that TLX supports substantial customization with limited development effort while remaining competitive with state-of-the-art implementations. TLX-authored kernels have been deployed in large-scale training and inference production systems. Our code is open sourced at https://github.com/facebookexperimental/triton.

Similarity metrics are central in the theory of large networks and graph limits. For bounded-degree graphs, the Benjamini--Schramm metric records the distribution of rooted neighbourhoods, while the stronger colored-neighbourhood metric gives rise to local-global convergence. In this paper we show that this intricate topology is already determined by much smaller observations. For technical convenience and greater generality, we work with graphings, which are measurable generalizations of finite graphs and include all finite graphs as special cases. We prove that, for graphings of uniformly bounded degree, convergence of all colored degree distributions, or equivalently of all colored star statistics, is equivalent to local-global convergence. We also introduce an even more economical sampling procedure, the colored cherry metric, in which one observes only the root and two randomly chosen neighbours, and prove that it induces the same topology. Thus the full local-global structure can be reconstructed, at the level of topology, from families of very small colored observations. Our star-observation theorem was previously announced in the work of Backhausz and the author as an important ingredient in the proof that the so-called action convergence unifies dense graph limit theory with local-global convergence, thereby providing a general graph limit theory for sparse, dense, and intermediate-density graphs.

Traditional federated learning (FL) relies on a central aggregator server, which can create performance bottlenecks and privacy risks. Decentralized mix-and-forward designs remove the server, but repeated local mixing can attenuate global information under heterogeneity and expose peer-to-peer neighborhoods as a privacy attack surface. To preserve FedAvg-style aggregation semantics over updates reconstructable by the round deadline while scaling dissemination, we present FLTorrent, a BitTorrent-based dissemination layer for serverless FL with a short warm-up. Warm-up hardens within-round source unlinkability, a dissemination-layer goal orthogonal to content protections such as DP or secure aggregation, via pre-round obfuscation, randomized lags, and coordination-only non-owner-first scheduling with the tracker off the data path, before switching to vanilla BitTorrent swarming. We upper-bound the per-transfer attribution posterior by the fraction of owner chunks in a sender's eligible cover set, and derive a tighter high-probability bound that improves with early non-owner mass. A simple heuristic, GreedyFastestFirst, attains about 92% of a bandwidth-optimal max-flow upper bound, while warm-up remains a stable about 12% share of a round across 100-500 peers. Under an observation-only local adversary, FLTorrent drives attribution success close to neighborhood-level random guessing for typical nodes, improves with network size, and remains robust under collusion. In LLM-scale dissemination stress tests over 7-10 Gbps access links, FLTorrent adds only about 6-10% round-time overhead relative to BitTorrent-only. Overall, FLTorrent shows that within-round unlinkability and BitTorrent-level efficiency can co-exist with predictable, low overheads at scale.

Many papers have studied inequalities for Andrews and Paule's broken $k$-diamond partition function $Δ_{k}(n)$ when $k=1$ or $2$. In this paper, we derive an exact formula for $Δ_{k}(n)$ when $k\geq 1$. Building on this result, we also derive an asymptotic formula for $Δ_{k}(n)$ with an explicit error bound. Using this formula, we prove that for $k\geq 1$ and sufficiently large $n$, $Δ_{k}(n)$ satisfies the Turán and Laguerre inequalities of any order and exhibits asymptotic complete monotonicity. Define $n_k:=\max\left\{\left\lceil8k^{3}+\frac{k+1}{12}\right\rceil,526\right\}$. Furthermore, we show that $Δ_{k}(n)$ is log-concave for $k\ge3$ and $n\ge n_k$. Consequently, it follows that $Δ_{k}(a)Δ_{k}(b)\geΔ_{k}(a+b)$ for $k\ge3$ and $a,b \ge n_k$.

The two Blanuša snarks $B_1$ and $B_2$ are 3-regular graphs on 18 vertices. Dantas, Lordelo, Niedermaier and Nogueira (Discrete Appl. Math. 361, 2025, pp. 336-346) established the first systematic bounds $23 \le π(B_i) \le 34$ for $i=1,2$. Bridi, Marquezino and Figueiredo (arXiv:2505.16050, 2025) then sharpened the upper side to $π(B_1) \le 31$ and $π(B_2) \le 30$ via a Weight Function Lemma heuristic. We push the upper bounds further to $π(B_1) \le 28$ and $π(B_2) \le 29$. The route is again Hurlbert's Weight Function Lemma, but applied one automorphism orbit at a time, with optimal weight functions coming from a linear program over a corpus of roughly $30{,}000$ rooted-subtree strategies per target. For the lower bound $π(B_i) \ge 23$ we re-derive the witnesses of Dantas et al. and re-verify them with two independent oracles: an exhaustive forward state-space search, and a sound-and-complete MILP encoding whose acyclicity constraint is motivated by the Milans-Clark No-Cycle Lemma. The interval for $B_1$ shrinks from $[23, 31]$ to $[23, 28]$, and for $B_2$ from $[23, 30]$ to $[23, 29]$.

A graph $G$ on $m$ edges is graceful if there is an injection $f : V(G) \to \{0, 1, \ldots, m\}$ whose induced edge labels $\{|f(u) - f(v)| : uv \in E(G)\}$ are exactly $\{1, 2, \ldots, m\}$. Ringel and Kotzig conjectured in 1964 that every tree is graceful. A computer check has confirmed this for all trees on at most 35 vertices (Fang 2010), but no general proof is known. Here we exhibit an infinite family of trees that escapes the named spider results of Bahls--Lake--Wertheim, Panpa--Poomsa-ard, and Panpa--Imnang--Wasuanankul: the family $T(k, m)$ of spiders with $k$ legs of length $3$ together with $m$ pendant leaves at the centre. We prove every such tree is graceful for all $k \ge 1$ and $m \ge 0$. The argument splits into two short lemmas. The first is a pendant-extension lemma that applies whenever the underlying graceful labeling sends the centre to $0$; the second is the base case, namely that $S_{k, 3}$ admits exactly such an apex-zero labeling, a fact already implicit in Bahls--Lake--Wertheim (2010). What is new is the explicit identification of $T(k, m)$ and the observation that the family is closed under both pendant addition and the apex-zero condition, so it includes infinitely many trees not handled by the named theorems above.

Phylogenetic trees are the primary framework for conveying evolutionary relationships. While many tools exist for visualizing phylogenetic trees, most are limited to static graphics, require coding expertise, or are developed for a specific website and not easily reusable or extensible. To address these limitations, we developed heat-tree, a collection of software packages in JavaScript, R, and Python for interactive visualization, manipulation, and editing of phylogenetic trees and their associated metadata. Heat-tree allows for the creation of customizable, web-compatible tree visualizations that can be easily embedded in R Markdown, Jupyter Notebooks, and Quarto documents, as well as directly in HTML/JavaScript applications and websites. The package supports radial and rectangular tree layouts, automated translation of metadata values into visual encodings on the tree, interactive tree editing, and export capabilities for publication-quality figures. All visualization parameters are definable programmatically or interactively using the comprehensive graphical user interface included with each visualization. Heat-tree was designed to be a user-friendly software package for interactive tree viewing, manipulation, editing, and self-contained, embeddable visualization across software environments.

We explore the possibility of explaining dark matter through six-dimensional (6D) primordial black holes (PBHs) in a theory with two extra dimensions. Interestingly, in this scenario the fundamental energy scale is of the order of $\sim 10$ TeV, accessible by future experiments. We analyse the viability of charged and rotating 6D black holes under standard Hawking evaporation as well as the memory burden scenario. In the case of pure Hawking evaporation, only PBHs with masses $M > 10^8$ g survive to present, while the lifetime of near-extremal configurations is extended by a factor $1/β^{1/2}$, where the parameter $β$ characterizes small deviations from extremality. In the memory burden scenario evaporation is enormously suppressed, and sub-gram mass PBHs can survive to the present epoch. At future colliders such as the Future Circular Collider, these micro black holes produce characteristic high multiplicity events, $\langle N \rangle \sim 21$, with thermal spectra, enabling direct probes of the fundamental scale and the number of extra dimensions. We find that the memory burden mechanism opens a broad new mass window for light PBH dark matter, while the Kaluza-Klein mass splitting $Δm$ aligns with the atmospheric neutrino scale, suggesting a unified framework between Swampland constraints, cosmology, collider physics, and low energy phenomenology.

We investigate the spreading of entanglement, and entanglement memory effects, in two-dimensional conformal field theory (CFT) propagating on evaporating black hole backgrounds. Memory effects leading to late-time spikes in mutual information for widely separated intervals are well known in CFTs admitting a quasiparticle description. In this work we examine the effect of black hole scrambling on late time mutual information spikes for disjoint intervals in free fermion CFT prepared in a thermofield double state. Late-time entanglement revival is driven by island-induced purification of modes in the union of the intervals. We show across two distinct 2d gravity models, Jackiw-Teitelboim (JT) gravity and the Russo-Susskind-Thorlacius (RST) model, that parametrically dialing up black hole scrambling time smooths out and suppresses entanglement spikes until they disappear at a critical scale, interpolating between free quasiparticle and maximal scrambling pictures. At the critical point, the interval lengths are exponential in black hole scrambling time. We further find a very closely related effect manifest as an entanglement dip for a single interval in a single-sided evaporating RST black hole.

A finite set $S \subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero residue arises exactly once as a difference of two elements of $B$. Erdős's \$1000 conjecture -- that every finite Sidon set extends to a finite PDS -- was disproved by Alexeev and Mixon (arXiv:2510.19804, October 2025), via the size-5 counterexamples $\{1,2,4,8,13\}$ and Hall's earlier $\{1,3,9,10,13\}$; they then asked: what is the smallest size $s$ of a non-extending Sidon set? The trivial bounds give $3 \le s \le 5$. Our evidence points to $s = 4$. We exhibit two integer Sidon sets, \[ A = \{0, 1, 3, 11\}, \qquad B = \{0, 1, 4, 11\}, \] together with the apparent infinite family of dilations $kA$, $kB$ and their reflections, all of which fail to extend for every prime power $q \le 317$ via the Singer affine-orbit check (rigorous under Hall's 1947 uniqueness for Desarguesian cyclic planes through $q \le 40$ and under the prime-power conjecture beyond that), and unconditionally for every modulus $v \le 133$ via brute-force depth-first search. We also report the exact density $N_{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$ of non-extending size-4 Sidon sets in $[0, N]$ for $N \le 50$ -- the match is exact, which suggests the $kA, kB$ family is complete in this range. A complete proof, perhaps in the spirit of Alexeev--Mixon's polarity argument or via a multiplier descent, remains open.

Using the higher covariant derivative on a manifold $ M $ equipped with a torsion-free connection, we define a natural surjective bundle map $ Φ$ from $ (\otimes(TM))\otimes (\wedge(TM)) $ to the vector bundle $ \mathcal{U}(M) $ of de Rham currents on $ M $ supported in a single (variable) point. The resulting quotient bundle can be thought of as a bundle of generalized Weyl algebras, with the symplectic form replaced with the Riemannian curvature tensor. The fibers of the bundle $ \mathcal{U}(M) $ are differential co-algebras, and the boundary, co-product and co-unit stitch together to form bundle maps which lift via $ Φ$ to commuting bundle maps on $ (\otimes(TM))\otimes (\wedge(TM)) $. Interior product, higher-order covariant differentiation, and their $ L^2 $ adjoints also form bundle maps on $ \mathcal{U}(M) $ which lift via $ Φ$. The higher-order covariant derivative in particular is an $ \mathbb{R} $-algebra representation of the space $ C^\infty(\otimes(TM)) $ equipped with a non-standard, \emph{covariant product}. Its composition with interior product yields a quantization of $ \mathcal{U}(M) $ corresponding to a Hopf-algebraic smash product. Finitely supported and locally finitely supported sections functors can be applied to $ \mathcal{U}(M) $, yielding the spaces of finitely supported and locally finitely supported currents, respectively. In particular, the finitely supported currents on a smooth manifold are a filtered differential graded co-algebra in duality with differential forms.

In 1960, G. B. Robison discovered the general equations relating roads and wheels, where either can have an unusual shape (e.g., the square wheel rolls smoothly on a catenary). But he used some inobvious assumptions regarding the meaning of rolling. Here we derive the equations for the road appropriate for a given wheel using only the single assumption that rolling occurs with no slipping. We do not require that the wheel be differentiable, so this allows the construction of a wheel-road pair when the wheel is a continuous nowhere differentiable function.

Background: Short sequence substrings of a fixed length k, called k-mers, are a ubiquitous computational primitive in bioinformatics, used across sequence indexing, read mapping, genome assembly, metagenomic classification, and comparative genomics. Spaced k-mers generalize this concept by selecting only a subset of positions within a k-mer, improving robustness to mismatches and sequencing errors. While k-mers are computationally highly efficient, spaced k-mers require additional work to be extracted from a sequence, which has slowed down existing methods. Results: We present a collection of efficient algorithms for extracting spaced k-mers from nucleotide sequences, optimized for different hardware architectures. They are based on bit manipulation instructions at CPU level, making them both simpler to implement and up to an order of magnitude faster than existing methods. We further evaluate common pitfalls in k-mer processing, which can cause substantial inefficiencies. Conclusions: Our approaches allow the utilization of spaced k-mers in high-performance bioinformatics applications without major performance degradation compared to regular k-mers, achieving a throughput of up to 750MB of sequence data per second per core. Availability: The implementation in C++20 is published under the MIT license, and freely available at https://github.com/lczech/fisk

Although continuous symmetry theory has attracted increasing attention in modern chemistry, local symmetry remains under-investigated. As a consequence, the relationship between symmetry and chemical behavior is often obscured, limiting the practical use of fuzzy symmetry measures. In this study, we introduce a novel framework for evaluating local symmetry based on electron density localization, and present continuous symmetry representations for several representative molecules. Our approach not only quantitatively captures global symmetry, but also reveals distinctive features of symmetry in a local chemical environment. The related concept, local chirality or chirotopicity, is also discussed. Overall, the proposed local symmetry and chirality measures provide valuable insights into molecular structure and structure-property relationships.

Card-based cryptography uses physical playing cards to construct protocols for secure multi-party computation. Existing card-based protocols employ various types of shuffles, some of which are easy to implement in practice while others are considerably more complex. In this paper, we classify shuffle operations into several levels according to their implementation complexity. We motivate this hierarchy from both practical and theoretical perspectives, and prove separation results between several levels by showing that certain shuffles cannot be realized using only operations from lower levels. Finally, we propose a new complexity measure for evaluating card-based protocols based on this hierarchy.

Vectorization is a compiler optimization that replaces multiple operations on scalar values with a single operation on vector values. Although common in traditional compilers such as rustc, clang, and gcc, vectorization is not common in the Verilog ecosystem. This happens because, even though Verilog supports vector notation, the language provides no semantic guarantee that a vectorized signal behaves as a word-level entity: synthesis tools still resolve multiple individual assignments and a single vector assignment into the same set of parallel wire connections. However, vectorization brings important benefits in other domains. In particular, it reduces symbolic complexity even when the underlying hardware remains unchanged. Formal verification tools such as Cadence Jasper operates at the symbolic level: they reason about Boolean functions, state transitions, and equivalence classes, rather than about individual wires or gates. When these tools can treat a bus as a single symbolic entity, they scale more efficiently. This paper supports this observation by introducing a Verilog vectorizer. The vectorizer, built on top of the CIRCT compilation infrastructure, recognizes several vectorization patterns, including inverted assignments, assignments involving complex expressions, and inter-module assignments. It has been experimented with some Electronic design automation (EDA) tools, and for Jasper tool, it improves elaboration time by 28.12% and reduces memory consumption by 51.30% on 1,157 designs from the ChiBench collection.