Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.

Quantum gases with short-range attractive interaction have a tendency to form pairs. For time-dependent interaction we find that the pairing amplitude at small separation satisfies a fractional differential equation (FDE). We derive analytic solutions of the pairing evolution for sudden interaction quenches and power-law drives toward resonant scattering. We observe universal short-time dynamics governed by a conformal fixed point at which the momentum distribution exhibits nonthermal, self-similar scaling in time, in quantitative agreement with experiment. At longer times, many-body effects induce relaxation toward an equilibrium state. In this limit, the FDE turns into a Müller-Israel-Stewart type equation that describes a hydrodynamic attractor approaching equilibrium.

In distributed-parameter inverse problems in computational mechanics, spatially varying fields are inferred from noisy, indirect, and heterogeneous observations. The relevant identifiability question concerns which spatial perturbation patterns of the field are distinguishable under a specified sensing and excitation programme. This paper develops a local information-operator framework for this purpose. Around a nominal parameter field, the parameter-to-observation map is linearized and the likelihood contribution to posterior precision is interpreted as an operator on parameter-field perturbations. For locally linearized Gaussian models with parameter-independent covariance, this operator is equivalently Fisher information, Gauss-Newton data-misfit curvature, and a noise-weighted sensitivity Gramian. The framework separates pointwise visibility from spatial identifiability. The diagonal gives a coordinate-dependent local information density, while the full kernel and metric- or prior-preconditioned spectra rank spatial patterns that are strongly visible, weakly visible, or locally invisible. Heterogeneous observation blocks are assembled in a common parameter space; information is additive only under conditional independence, whereas correlated errors require the full joint covariance. Model discrepancy, nuisance parameters, and prior information modify the same geometry through covariance inflation, Schur-complement information loss, and prior-preconditioned modes. Examples cover analytic beam kernels, two-span support coupling, static-dynamic fusion for flexural-rigidity identification, and two-dimensional damage-field reconstruction in a leading information subspace. The operator view supports interpretation of identifiability, sensor complementarity, and reduced reconstruction in distributed-parameter inverse problems.

We run a numerical linked-cluster expansion with a quantum algorithm (NLCE+QA), computing ground-state energies and one quasi-particle dispersions in the thermodynamic limit using a 20-qubit trapped-ion quantum processing unit (QPU). The NLCE+QA framework extracts thermodynamic-limit properties from small-cluster calculations, making it naturally suited for near-term quantum devices. Projector-based block-diagonalization schemes such as projective cluster-additive transformation (PCAT) are essential to NLCE+QA, and they involve matrix inversion and square root operations that amplify measurement noise. A central question is therefore whether current hardware can provide expectation values that are accurate enough to withstand non-linear classical post-processing. We explore this challenge for the transverse-field Ising model (TFIM) in one dimension, on a ladder geometry, as well as in a longitudinal field in one dimension. For the quantum algorithm, we consider adiabatic state preparation (ASP), as well as a variational quantum eigensolver (VQE) trained on a classical device. The final expectation values are obtained from the QPU, using a novel alternative to the Hadamard test that we name the CX-test. We explore the regimes currently attainable on quantum devices and comment on the improvements needed for quantum computers to achieve results beyond classical reach.

The twisted Alexander polynomials of a space, associated to a linear representation $σ$ of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they arise naturally from a new family of invariants -- the twisted homology jump loci -- which extend the rank-one characteristic varieties to higher-rank local systems. Using the tropical geometry of these twisted loci, we obtain sharper upper bounds for the Bieri--Neumann--Strebel--Renz (BNSR) $Σ$-invariants. For compact orientable $3$-manifolds with toroidal or empty boundary, we use a theorem of Friedl--Vidussi to show that the closure of the union of these twisted tropical bound is sharp: it recovers the fibered faces of the Thurston norm ball exactly, a result that fails without twisting. For compact Kähler manifolds, we prove that the $Σ^1$-invariant of $π_1(X)$ is controlled by the orbifold fibrations of $X$ for any representation $σ$, and that the twisted Alexander polynomial $Δ^σ(X)$ must equal $0$ or $1$. Both results provide obstructions to geometric realizability that are strictly stronger than their classical untwisted counterparts.

Using the results of Nikulin and Vinberg on the groups of isometries generated by reflections, we construct a subvariety called the Nikulin-Vinberg locus in the moduli space of polarized hyperkahler manifolds. It is obtained as a finite union of components of higher Noether-Lefschetz loci which parameterize manifolds with certain special Neron-Severi lattices. The Nikulin-Vinberg locus is the closure of the set of hyperkahler manifolds with Picard number $\geq 3$ which have finite groups of birational automorphisms. Using this construction and a refinement of an argument by Oguiso, we show that any non-trivial family of projective deformations of a hyperkahler manifold with $b_2(M)\geq 6$ has a dense set of fibers which have an infinite group of birational automorphisms.

We compute the two-loop massless QCD corrections to the amplitude of Higgs boson decay to bottom quark pair and a gluon ($H \rightarrow b + \bar{b} + g$) in the higher powers of the dimensional regularization parameter $ε$. The calculation is performed by projecting the amplitude onto the appropriate Lorentz structures related to the process. We also show the numerical behaviour of the form factors for a few sample phase-space points. These amplitudes are necessary ingredients for computing the three-loop virtual corrections to bottom-quark annihilation to Higgs plus jet production at the hadron collisions.

Distinct Clifford orbits of magic states can exhibit different stabilizer ranks at small tensor powers. We establish this for qutrits, where the single-qutrit Clifford group has four inequivalent orbits of magic states: Strange, Norrell, Hadamard-eigenstate, and the qutrit T-state, but a nontrivial upper bound on the asymptotic exponent had been pinned down for only the qutrit T-state. For the other three orbits we give explicit stabilizer decompositions, yielding upper bounds on the per-copy asymptotic stabilizer-rank exponent: $γ_S \le \log_3(2)/2 \approx 0.316$ for the Strange state, and $γ_{H_3}, γ_N \le \log_3(4)/3 \approx 0.421$ for the Hadamard-eigenstate and Norrell orbits, all strictly below the prior $γ_{T_3} \le 1/2$ baseline. We also prove the first nontrivial $Ω(m / \log m)$ asymptotic lower bounds for the Hadamard-eigenstate and Norrell orbits, and exhibit two-qutrit Clifford circuits that convert two copies of these states into an injectable phase state with constant success probability, enabling constant-overhead injection of one non-Clifford diagonal gate per orbit. In the case of qubits, we give a closed-form decomposition of the qubit T-type orbit at four copies matching the existing $γ_T \le \log_2(3)/4 \approx 0.396$ exponent via a direct algebraic identity rather than an entangled cat-state construction. An open-source library stabrank accompanies the paper, with Lean 4 proof formalizations of all the decompositions.

We prove two conjectures on the spaces generated by multiple $q$-zeta values. More precisely, we show that the spaces $Z_q^{\mathrm{o}}$ and $Z_{q,1}^{\mathrm{o}}$ already generate the larger spaces $Z_q$ and $Z_{q,1}$, respectively. Our result is stronger than the equality of $\mathbb{Q}$-vector spaces: for every generator in the larger spaces, we construct an explicit expression with integer coefficients in terms of the smaller generating families. We first establish these formulas at the finite level, where suitable finite $q$-analogues admit recursive descriptions through generating series, and then pass to the infinite limit.

In recent work, we have explored the novel possibility that the geodesic of a spacetime density flow in GR could be upgraded to an amplitude $ψ$ with density $|ψ|^2$. We show how this arises from generally-covariant quantum mechanics and the Klein-Gordon operator in a semiclassical analysis, and how it connects to the Raychaudhuri equations. In this setting, we demonstrate an Aharonov-Bohm type effect for the phase of $ψ$ in motion approaching a black hole.

We introduce GINKAKU, a new cosmological $N$-body code developed for the Dark Quest II (DQ2) simulation campaign and designed for controlled ensemble production across the cosmological model space required by next-generation galaxy surveys, including massive neutrinos and clustering dark energy. Built on the FDPS framework, GINKAKU couples a TreePM gravity solver with a linear-response treatment of external source terms for components not evolved as $N$-body particles, formulated in the $N$-body gauge. This design incorporates massive-neutrino perturbations, general-relativistic corrections, early-time radiation perturbations, and dark-energy clustering with non-unit effective sound speed at the linear level, while preserving Newtonian particle dynamics on subhorizon scales. The code is validated through internal convergence studies and cross-comparisons with GADGET, PKDGRAV3, and RAMSES on shared initial conditions: code-to-code differences in the nonlinear power spectrum can be reduced below $\sim1\%$ level by tuning internal accuracy parameters, and we identify a production-grade fiducial setting achieving this control at modest cost. We apply GINKAKU to an initial set of DQ2 production runs -- eight cosmological models with $3,000^3$ particles in boxes up to $4\,h^{-1}\mathrm{Gpc}$ -- processed by a renewed post-processing pipeline that reduces the inter-resolution spread of the halo mass function to $\sim 1\%$ and includes halo-shape measurements for intrinsic-alignment statistics. The scale-dependent-growth cosmologies reproduce the expected nonlinear signatures of massive neutrinos and clustering dark energy, demonstrating suitability for emulator-scale production. A total matter power spectrum emulator from these runs is presented in an accompanying paper. (abridged)

We investigate perturbative dynamics, tidal effects, and relativistic frequency shifts in a dyonic Kalb-Ramond black hole generated by a Lorentz-violating antisymmetric tensor background. The geometry is controlled by the mass $M$, the electric charge $Q$, the magnetic charge $p$, and the Lorentz-violating parameter $\ell$, with the dyonic sector entering through the effective combination $P_{\ell}^{2}=Q^{2}/(1-\ell)^{2}+p^{2}/(1-2\ell)$. First, we analyze the gravitational Doppler effect for radial signal exchange between freely falling and static observers, showing how the dyonic charges weaken the redshift by shifting the frequency ratio toward unity. We then compute the radial and angular tidal forces in a freely falling frame and determine the characteristic radii at which the usual stretching and compression patterns are reversed. The gravitational time delay is also evaluated for null trajectories, showing that the electric and magnetic sectors reduce the delay relative to the reference configuration. In the perturbative sector, we derive the scalar, vector, tensor, and spinor effective potentials and compute the corresponding quasinormal frequencies through the sixth-order WKB method. The numerical spectra indicate that the Lorentz-violating parameter gives the dominant correction, increasing the oscillation frequencies and modifying the damping rates, while the dyonic charges produce milder shifts. Finally, the time-domain profiles confirm the presence of damped quasinormal ringing followed by late-time power-law tails.

Effects of a third rank permeability term in chiral solids are studied. Fluid flow through such materials acquires vorticity upon emergence from the material. Materials of interest include chiral surface lattices such as the gyroid, chiral rib lattices, and granular materials comprised of sugar crystals, quartz sand, wheat or beans. A characteristic length scale is associated with the chirality. The length scale can be obtained by several methods. Contacts with nonlocal permeability, elasticity and piezoelectricity are explored.

High-temperature superconducting (HTS) coated conductors (CCs) can be wound into no-insulation (NI) coils, in which electrical current can partially bypass local normal zones via turn-to-turn contact layers (T2TCLs). Accurate magneto-thermal simulation of such coils, therefore, requires an efficient representation of the electrical and thermal behavior of the T2TCLs. This paper introduces a magneto-thermal surface contact approximation (SCA) for finite element analysis of NI HTS coils. The formulation is derived as a special case of the more general thin shell approximation (TSA) by introducing suitable approximations such as negligible tangential surface currents and eddy-current effects inside the T2TCL. The resulting SCA formulation replaces the thin volumetric contact layer with a dedicated surface weak formulation based on the electric contact resistance and thermal contact conductance. In contrast, the TSA formulation requires the definition of electric resistivities and thermal conductivities as well as the thickness of the T2TCL. The SCA is implemented in the Pancake3D module of the free and open-source Finite Element Quench Simulator. It is verified through transient magneto-thermal simulations of a model NI pancake coil. Numerical results are compared against the established TSA formulation. The results show that the SCA accurately reproduces the relevant electromagnetic and thermal behavior. For the TSA, there is a trade-off between choosing large (potentially unphysical) thicknesses with low resistivities leading to inaccurate results, or small thicknesses with large resistivities making the linear system harder to solve, increasing the computational effort. In contrast, the SCA, thanks to using contact resistances and conductances directly without the necessity to define a thickness, is easy to use and robust.

A central open question in gravitational-wave detector characterization is whether the O4a observing run has introduced glitch morphologies not present in earlier runs. We present gravi-signal-ml, an open-source pipeline for unsupervised morphological characterization of instrumental noise transients (glitches) in LIGO gravitational-wave data, applied to 1,277 hours of public O4a strain data from the Hanford and Livingston detectors. The pipeline extracts 384-dimensional visual embeddings from Q-transform spectrograms using a frozen DINOv2 Vision Transformer with register tokens (ViTS/14), requiring no labeled training data. Embeddings are projected via PCA and UMAP with cosine metric, then clustered using a Dirichlet Process Mixture Model (DPMM). Cluster robustness is systematically assessed through ablation studies, stability analysis across hyperparameter perturbations, and morphological cross-check against an in-domain Gravity Spy O3b reference index. A time-slide background test excludes statistically significant H1--L1 coincidences ($p \geq 0.1$) in all sessions. Across 188,000+ spectrograms, no morphologically novel glitch candidates were identified -- all anomalous clusters map to known Gravity Spy classes with cosine similarity $> 0.98$. L1 embeddings show consistently high robustness (ablation ARI $> 0.90$ in all four sessions), while H1 exhibits lower and more variable grayscale ablation ARI ($\sim 0.68$--$0.90$), suggesting a structural difference in the H1 noise manifold under DINOv2 feature extraction. This null result, obtained with a fully validated pipeline, establishes a reproducible baseline for zero-shot glitch morphology characterization in O4a data. The pipeline and all results are publicly available at https://github.com/lucacirfeta/dante-gravi-signal-ml DOI: https://doi.org/10.5281/zenodo.20121860.

Despite warnings that LLMs can make mistakes, users often develop inappropriate trust and accept incorrect answers without critical evaluation. Uncertainty quantification (UQ), displaying LLMs' confidence, has emerged as a promising approach to calibrate user trust. However, prior empirical studies on uncertainty communication have treated uncertainty as a single numerical score or simple natural language expression. This simplification fails to capture a key property of LLM outputs: a single response often comprises multiple claims and reasoning steps, each with distinct levels of uncertainty. To address this gap, this study investigates uncertainty granularity (i.e., the extent to which uncertainty is expressed at different levels within an LLM response) and examines its impact on LLM-assisted decision-making. We conducted a large-scale, between-subjects study (N=192) in which participants answered medical questions using LLMs that displayed uncertainty at three different granularities: output-level (entire response), relation-level (individual reasoning steps), and token-level (specific words). Our findings reveal distinct behavioral effects as a function of uncertainty granularity. Token-level uncertainty increased users' agreement with the AI, whereas output- and relation-level uncertainty did not increase agreement but instead reduced users' confidence in their own answers. Notably, relation-level uncertainty also reduced external verification (i.e., internet searches, checking provided URLs), steering users away from independent fact-checking and toward reliance on the LLM and its accompanying uncertainty cues. Our findings demonstrate that uncertainty granularity significantly shapes how users interact with and verify LLM outputs, providing concrete design guidance for building responsible LLM applications that encourage appropriate skepticism and verification behaviors.

We prove that every concatenation of $10$ or more binary squares contains an overlap. The bound $10$ is best possible. In contrast, over a ternary alphabet, there are infinitely long overlap-free words that consist of a concatenation of squares.

The production of a $D^+ D^-$ pair by photon - photon interactions at the Large Hadron Collider (LHC) is investigated considering ultraperipheral proton - proton ($pp$), proton - lead ($pPb$) and lead - lead ($PbPb$) collisions. Assuming that the scattering amplitude for the $γγ\rightarrow D^+ D^-$ process can be described by the Brodsky - Lepage formalism in the heavy - quark approximation, we derive the associated differential distributions and total cross - sections. In particular, in addition to the rapidity and transverse momentum distributions, usually presented in the literature, we also present predictions for the total transverse momentum and momentum imbalance distributions. Our results indicate that a future experimental analysis of this final state is, in principle, feasible during the high luminosity run of LHC.

Controlling the spatial incorporation of carbon into hexagonal boron nitride (hBN) is essential for engineering optically active defects, yet existing approaches lack nanoscale precision and control over the carbon supply. Here, we demonstrate a method for carbon doping of hBN using electron-beam irradiation in a low-pressure methane atmosphere, where the beam simultaneously generates vacancies and decomposes methane into individual carbon and hydrogen atoms. Using annular dark-field scanning transmission electron microscopy, we show that increasing the methane partial pressure suppresses pore growth and drives the formation of triangular boron-terminated pores through preferential hydrogen etching of nitrogen. Time-resolved electron energy-loss spectroscopy (EELS) mapping reveals progressive carbon incorporation into the lattice, accompanied by boron and nitrogen depletion. Carbon clustering occurs predominantly within the irradiated area: 84+-7% of carbon-rich regions are confined to the area exposed to the electron beam, while some carbon atoms are also found to diffuse up to an average distance of 4.7+-0.5 nm beyond it. The incorporated carbon atoms arrange in a hexagonal pattern within the lattice, forming patches that do not exceed ~1 nm in size. Analysis of the EELS fine structure indicates modifications to the local electronic environment within these regions, with implications for the optical properties of the resulting carbon-related defects.

In this short note, I show that in a McCall (1970) model with risk-neutral agents, a system of wage insurance combined with unemployment insurance financed by a lump-sum tax on the employed can be replicated -- in the ex ante sense -- by a system of unemployment insurance that depends on the agent's wage when last employed and a tax that depends on the agent's wage when employed. This holds with or without endogenous search when unemployed. I also show that wage insurance is not binding until the last earned wage exceeds a threshold, which may be of independent interest.

Optical wireless communication (OWC) leveraging single-photon avalanche diode (SPAD) arrays offers exceptional sensitivity for photon-starving links. However, the inherent dead time of SPADs critically limits achievable data rates by introducing non-linear photon-counting distortions: blocking loss within a symbol duration and inter-symbol interference (ISI) across durations. This paper proposes a unified analytical framework capturing both distortions across all operational speed regimes for pulse-amplitude modulation (PAM), by establishing comprehensive statistical models for SPAD array receivers. For low and medium-speed systems (symbol duration longer than dead time), we derive exact closed-form expressions for the photon counts probability distribution using renewal theory, explicitly incorporating blocking loss and ISI. For high-speed systems (symbol duration shorter than dead time), we develop a Markov chain model characterizing the steady-state operational states and integrate it with trigger probability to obtain the exact binomial photon counts distribution. Furthermore, we propose low-complexity, near-optimal threshold detection schemes based on these models. This work provides essential theoretical tools for designing and optimizing high-performance SPAD-based OWC systems employing PAM.

Test equating using covariates may be applied to provide comparable scores from multiple test forms when no anchor items are available. However, its performance may be compromised if some of the covariates themselves are measured using different test forms. In this work, we propose sequential generalized kernel equating to account for possible differences in the distribution of covariates used in the NEC design. We evaluate the proposed approach through a simulation study within the kernel equating framework. Results indicate that equating the covariate reduces bias in equated test scores, particularly when the covariate distributions differ and the correlation between the covariate and the test score is strong. A real data example from a national high school leaving examination further demonstrates the practical application.

A compact time projection chamber (TPC) prototype was studied for charged-pion tracking in the annihilation detector proposed for the HIBEAM experiment at the European Spallation Source. HIBEAM aims to search for neutron-antineutron oscillations, where a produced antineutron would annihilate on a carbon target and produce a pionic final state. A tracking algorithm was developed and validated using cosmic-particle measurements expected to be dominated by atmospheric cosmic muons, proton elastic-scattering data from the Cyclotron Centre Bronowice, and simulated TPC event data. Cosmic measurements provided a minimum-ionising reference for pion-like tracks, while proton-scattering data tested reconstruction in a more highly ionising environment. Simulated crossing-track events were used as a stress test of complex topologies. The algorithm performed reliably for well-separated tracks, while crossing or overlapping tracks and dense proton-scattering events reduced performance. In experimental data, reconstructed tracks showed sub-millimetre residual widths, robust charge sharing supporting the zigzag-shaped readout, and uniform measured dE/dx across the drift volume. The truncated-mean track-level dE/dx distribution in cosmic data showed an approximately Gaussian peak with a relative width of about 16%, demonstrating how energy-loss information can be used for particle identification. Tilted proton tracks showed broader residuals in yz, because the pad row signal was extended in time. A sub-centroid refit technique was proposed, using the internal ADC time profile within a pad row instead of a single charge-weighted time position. With four equal-y sub-centroids per pad row, the yz residual width was reduced from 1.30 mm to 0.55 mm. These results validate the initial tracking capabilities of the algorithm and prototype, and provide feedback for the next-generation prototype.

Internal waves with frequency larger than twice the local minimum allowable wave frequency can be susceptible to parametric subharmonic instability (PSI). This instability draws energy from the wave and provides a mechanism for generating small-scale turbulence and mixing. In the ocean, strongly baroclinic flows at the submesoscale adjust the minimum frequency of internal waves such that it is possible for PSI to occur for locally near-inertial waves. One setting where this may occur is in baroclinic bottom boundary layers along sloping topography, where near-bottom interior flows in the sense of Kelvin-wave propagation lead to a reduction of bottom boundary layer Ertel potential vorticity, and consequently lower the minimum frequency sufficiently to allow PSI. Linear stability analysis, and nonlinear simulations, show that PSI grows at a rate determined by the vertical stratification of the bottom boundary layer, and the slope Burger number. Wave shear production is the primary energy source for the instability, with additional contributions from buoyancy production that depend on the slope parameters. A partially compensating loss of energy to geostrophic shear production becomes increasingly important as the flow approaches the marginally stable state. These results suggest PSI as a potential mechanism for generating near-bottom mixing in the ocean.

Positive systems describing networks with inherently non-negative states and inputs arise naturally in routing, logistics, and compartmental modelling. We consider problems modelled as positive linear systems in incidence form with linear cost. The addition of capacity constraints on states (storage) and inputs (flows between nodes) significantly increases the problem complexity. Leveraging the analytic structure of the unconstrained problem, an explicit suboptimal admissible controller is constructed. This yields graph-computable performance bounds and a minimum stabilising horizon length for a model predictive controller without terminal conditions. A convex program enables efficient computation of the optimal bound and horizon. These results highlight how system structure enables explicit MPC guarantees that are typically not available.

This paper presents a unified Cramér-Rao lower bound (CRLB) framework for signal-level parameters in integrated sensing and communications (ISAC)-enabled radar systems. Starting from the generic signal model, we analyze the coupling between delay and Doppler in the Fisher information matrix (FIM), which is unsolved and often overlooked in relevant studies. Addressing this issue, we derive the conditions under which the coupling terms can be eliminated and demonstrate that these conditions are typically satisfied for ISAC-enabled waveforms. Afterward, the CRLBs of representative ISAC waveforms are derived within the unified framework, enabling consistent and comparable analysis across the waveforms and avoiding model-dependent discrepancies. Further, the framework is extended to virtual array (VA) sensing systems, where the impact of different multiplexing schemes is analyzed. Simulation results demonstrate the consistency between the CRLBs derived from the proposed framework and those obtained from waveform-specific analyses. The proposed framework shows strong generality, waveform-compatibility, and flexibility, offering a versatile tool for the CRLB analysis of various waveforms, including those lacking existing analytical results.

This paper reports a deliberately small executable proof for a DAG-TOML contract: six "Hello, world!" implementations in Rust, Go, C, Java, TypeScript, and AWK are linked to one observable-output contract, one implementation DAG, one traceability file, one readiness gate, and one evidence matrix. The load-bearing contract requires the exact UTF-8 byte sequence `Hello, world!\n`, zero stderr bytes, and exit code 0. On the runner used for this paper, the witness harness reported five PASS outcomes, one SKIP for Java because `javac/java` was not on `PATH`, and zero FAIL outcomes. Two sidecar witnesses exercise narrower source-analysis claims: a convoluted Go rewrite hides the contiguous greeting literal but remains visible to sqry at the declared AST symbol and simple-edge level, while an indirect AWK rewrite uses a declared source profile because AWK is not in the repository's sqry-backed validator language set. The contribution is not a benchmark, a claim of general semantic equivalence, or a production assurance system. It is a compact, falsifiable artifact that shows how a contract, implementation graph, traceability chain, and review gate can be checked against executable witnesses.

PhyloFrame is a Python library for phylogenetic computation targeting the gap between specialist, compiler-optimized operations and flexible, script-based workflows -- with emphasis on fast, memory-efficient operations for very large tree sizes (e.g., $\geq$ 300,000 taxa). PhyloFrame is built around a DataFrame-based tree representation, where each row corresponds to a node and columns record ancestor relationships, branch lengths, taxon labels, and any user-defined attributes. Crucial for scalability, such array-backed storage allows both library and end-user code alike to seamlessly harness Just-in-Time (JIT) compilation (e.g., Numba) and vectorized execution (e.g., NumPy, Polars). At large tree sizes, performance generally matches or exceeds Python libraries backed by native code -- notably, achieving strong performance in topological-order traversals and Newick I/O. DataFrame-based representation affords several additional conveniences, including: - succinct bulk operations (e.g., NumPy); - powerful queries and transformations (e.g., Polars expressions, Pandas indexing, SQL-style joins and merges); - compatibility with modern tabular data formats that are compression-friendly, type-aware, nullable, and highly portable (e.g., Parquet); and - broad interoperation with table-oriented data science tools (e.g., Seaborn, Plotly, Vega-Altair, tidyverse, Excel). Current library features include tree input/output, synthetic tree generation, taxon-based queries, tree traversals, tree metrics, tree manipulation, tree downsampling, and tree comparison. Most functionality supports both Pandas and Polars DataFrames, and is available through programmatic and CLI-based interfaces.

A new two-parameter asymptotically flat (AF) toric gravitational instanton is identified as a special case of the Euclidean double Kerr-NUT solution, by imposing certain symmetry and regularity conditions on its rod structure. These conditions are solved explicitly, except for one which takes the form of a fifth-order polynomial. This gravitational instanton has Euler number $χ=4$ and Hirzebruch signature $τ=0$, and its global topology is $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$ with a circle $S^1$ removed appropriately. It is the third of an infinite sequence of AF toric gravitational instantons that was proved to exist by Li and Sun, the first two being the Kerr and Chen--Teo instantons. It is also the first known example of a Ricci-flat gravitational instanton that is not Hermitian.

Understanding the electron beam distribution in the longitudinal phase space (LPS) is crucial for free electron laser (FEL) facilities. Conventionally, LPS diagnostics utilize radio frequency (RF) deflecting structures to streak the electron beam transversely, mapping the longitudinal bunch distribution onto a transverse plane for observation. However, RF structures are complex and costly, especially for high-energy machines like the European XFEL. Wakefield structures have emerged as a promising alternative, offering simplicity in construction and minimal maintenance costs. However, they suffer from nonlinear streaking, requiring image reconstruction for LPS distribution. Several iterative algorithms have been developed for LPS reconstruction using passive wakefield streakers in recent years. This paper proposes a simple, computationally efficient method tailored for cases with known beam current profiles.