While prior work established a verifier-based polynomial-time framework for NP, explicit deterministic machines for concrete NP-complete problems have remained elusive. In this paper, we construct fully specified deterministic Turing Machines (DTMs) for SAT and Subset-Sum within an improved NP verifier simulation framework. A key contribution of this work is the development of a functional implementation that bridges the gap between theoretical proofs and executable software. Our improved feasible-graph construction yields a theoretical reduction in the asymptotic polynomial degree, while enhanced edge extension mechanisms significantly improve practical execution speed. We show that these machines generate valid witnesses, extending the framework to deterministic FNP computation without increasing complexity. The complete Python implementation behaves in accordance with the predicted polynomial-time bounds, and the source code along with sample instances are available in a public online repository.

We present a detailed benchmark comparing two state-of-the-art phase-field implementations for simulating alloy solidification under experimentally relevant conditions. The study investigates the directional solidification of Al-3wt%Cu under high-velocity solidification conditions and SCN-0.46wt% camphor under microgravity conditions from National Aeronautics and Space Administration (NASA) DECLIC-DSI-R experiments. Both codes, one employing finite-difference discretization with uniform mesh and GPU-acceleration (GPU-PF) and the other one employing finite-element discretization with adaptive-mesh and CPU-parallelization (PRISMS-PF), solve the same quantitative phase-field formulation that incorporates an anti-trapping current for the solidification of dilute alloys. We evaluate the predictions of each code for dendritic morphology, primary spacing, and tip dynamics in both 2D and 3D, as well as their numerical convergence and computational performance. While existing benchmark problems have primarily focused on simplified or small-scale simulations, they do not reflect the computational and modeling challenges posed by employing experimentally relevant time and length scales. Our results provide a practical framework for assessing phase-field code performance as well as validating and facilitating their application in integrated computational materials engineering (ICME) workflows that require integration with realistic experimental data.

In an early theoretical work published in 1965, Belen'kii & Fradkin proposed a turbulent diffusivity model for Rayleigh--Taylor (RT) mixing. We review its derivation and present alternative arguments leading to the same final similarity equation. The original work then introduced an approximation that led to a simplified ordinary differential equation (ODE), which was used primarily to derive the important scaling result, $h \sim (\ln R)gt^2$. Here, we extend the analysis by examining the solutions to both the full similarity ODE and the simplified ODE in detail. It is shown that the full similarity equation captures many now well-known features of non-Boussinesq RT flows, including asymmetric spike and bubble growth and a systematic shift of velocity statistics toward the light-fluid side. Comparisons of the theoretical model with numerical and experimental studies show reasonable agreement in both spatial profiles and growth trends of mixing layer heights. We further show that a global mass correction applied to the simplified solution closely approximates the full solution, highlighting that, to leading order, RT mixing is governed by the competing dynamics between diffusion of $\ln \barρ$ and mass conservation.

We study the almost sure convergence of the occupation measure of evolution models where mutation rates decrease over time. We show that if the mutation parameter vanishes at a controlled rate, then the empirical occupation measure converges almost surely to a specific invariant distribution of a limiting Markov chain. Our results are obtained through the analysis of a larger class of time-inhomogeneous Markov chains with finite state space, where the control on the mutation parameter is explained by the energy barrier of the limit process. Additionally, we derive an explicit convergence rate, explained through the tree-optimality gap, that may be of independent interest.

We study the correlation functions of a conformally coupled $ϕ^4$-interacting theory in AdS$_3$ and its dual CFT$_2$. The one-loop diagram is not expressible in terms of known transcendental functions, but is shown to be expressible as an infinite sum of previously well-studied tree-level diagrams, and we compute this sum using several number-theoretic conjectures. This enables us to extract recursively, the analytic expressions of anomalous dimensions of all dual double-trace operators. In the $s$-channel various consistency checks were performed against established bootstrap method, while our results in the $t$- and $u$-channel are not available in any previous literature to our knowledge.

We study the standard four-stroke regenerative quantum Stirling heat engine cycle, which assumes local thermal equilibrium at each stage, within the standard weak-coupling, Markovian open quantum system framework. We point out that the regeneration process is not thermodynamically free in a reduced open-system description, and we treat the required work input as an explicit regeneration cost by modifying the cycle efficiency accordingly. We consider two working substances--a single spin-$1/2$ and a pair of interacting spin-$1/2$ particles--and investigate the cycle performance by taking the regeneration cost at its minimum value set by the Carnot heat-pump limit. For comparison, we also analyze the conventional Stirling cycle without regeneration under the same conditions. The super-Carnot efficiencies reported under the cost-free regeneration assumption disappear once the regeneration cost is included: the modified efficiency stays below the Carnot bound, while still remaining higher than the efficiency of the conventional Stirling cycle. For the conventional Stirling cycle, we provide a rigorous Carnot bound using quantum relative entropy, whereas for the regenerative cycle we derive a sufficient lower bound on the regeneration cost that guarantees thermodynamic consistency. Finally, we suggest three candidate quantum regenerator models for future work.

We are interested in proving input-output properties of functions that handle infinite data such as streams or non-wellfounded trees. We provide a finitary refinement type system which is (sound and) complete for Scott-open properties defined in a fixpoint-like logic. Working on top of Abramsky's Domain Theory in Logical Form, we build from the well-known fact that the Scott domains interpreting recursive types are spectral spaces. The usual symmetry between Scott-open and compact-saturated sets is reflected in logical polarities: positive formulae allow for least fixpoints and define Scott-open sets, while negative formulae allow for greatest fixpoints and define compact-saturated sets. A realizability implication with the expected (contra)variance on polarities allows for non-trivial input-output properties to be formulated as positive formulae on function types.

Traditionally, Quantum Information, and Quantum Communication specifically, have been focused on qubit-based architectures. Recent results, however, highlighted that higher dimensional architectures (qudit-based) may present advantages both in terms of communication and computation; a family of channels called Multi-level Amplitude Damping (MAD) channels, which are a possible qudit generalization of the well known Amplitude Damping Channels, is able to model energy decay processes that may happen during signal transmission. In this work, the Quantum Capacity of 4-dimensional MAD's is studied, relying on a technique for computing it even outside of degradable and antidegradable conditions. We also characterized the complete region of antidegradability and degradability in the parameter space for a generic d-dimensional MAD using both analytical and semi-numerical methods.

Radiative strength functions (RSFs) model the bulk electromagnetic response of highly-excited nuclei and are critical inputs for statistical reaction codes. In this paper, we present a definition of the RSF that is consistent with Hauser-Feshbach reaction codes and that can be efficiently computed with the shell model using the Lanczos strength-function (LSF) method. We introduce a variant of the shell model LSF method that exploits the energy-localized Brink-Axel hypothesis, which makes it possible to compute both electric and magnetic RSFs across all energies relevant to capture reactions. We verify agreement with the conventional definition of RSFs with benchmark calculations of $^{24}$Mg, then present novel results for $^{56}$Fe. For $^{56}$Fe we find that: (i) the M1 RSF shape evolves smoothly with excitation energy, consistent with the energy-localized Brinkl-Axel hypothesis, (ii) both M1 and E1 transitions contribute significantly to the radiative strength below the photo-absorption threshold, and (iii) within the sdpf model space, the strength below 3 MeV observed in Oslo-type experiments cannot be fully reproduced. These results pave the way for a coherent microscopic description of the RSFs and further motivate the use of energy-dependent RSFs in modern reaction codes.

The ability to directly measure radiative-association rate coefficients for reactions between ions and neutral molecules has long challenged chemical physics laboratories, yet radiative association is one of the most important processes occurring in cold, diffuse regions of space. A reaction kinetics instrument has been developed for the investigation of ion--molecule radiative-association reactions, aimed at measuring slow, effective reaction rate coefficients for species relevant to astrophysical objects. The instrument consists of a glow-discharge ion source for production of bright and stable ion currents, a quadrupole mass filter for mass selection and detection, and a quadrupole ion trap capable of trapping reactants and products for the long times needed to measure slow kinetics. The performance and adaptability of the glow-discharge ion source has been evaluated using several configurations. To assess the feasibility of measuring reaction rate coefficients, the reaction of Ag$^{+}$ and O$_{2}$ was studied under pseudo-first-order conditions in the ion trap at room temperature. We present the first pressure-dependent study of this reaction and extract a lower limit of $1 \times 10^{-15}$ cm$^3$ molecule$^{-1}$ s$^{-1}$ for the Ag$^{+}$ + O$_{2}$ effective radiative-association rate coefficient. Measurements of effective radiative-association rate coefficients are possible for diverse atomic and molecular ions that react with neutral molecules over a range of rates in this versatile new instrument.

We give an algorithm for pure state tomography with near-optimal copy and time complexity using only single-qubit measurements. Specifically, given $\widetilde{O}(2^n/ε)$ copies of an unknown $n$-qubit pure state $|ψ\rangle$, the algorithm performs only nonadaptive Pauli measurements, runs in time $\widetilde{O}(2^n/ε)$, and outputs $|\widehatψ \rangle$ with fidelity at least $1-ε$ with $|ψ\rangle$ with high probability. This is the first algorithm for pure state tomography that achieves near-optimal running time.

The microscopic and fundamental origin of slide electrification, where droplets of water move across insulating surfaces accumulating and depositing electrical charges, is still debated. Charge transfer is often attributed to ion transfer at the receding contact line. However, it is still unclear whether ion transfer alone can fully account for the observed charge separation. We examined slide electrification of two polar, self-ionizing liquids (water, formamide) and two non-polar liquids (diiodomethane, bromonaphthalene). By cooling below the melting temperature, we were able to compare this process to tribocharging of the respective frozen components. Despite reduced ion mobility at sub-freezing temperatures, the ice of the polar liquids continues to accumulate significant charge. Non-polar liquids exhibit lower charging (<25% of polar liquids) and nearly identical charging behaviour in both their liquid and frozen phases on five different substrates. Since non-polar liquids contain few free ions, these observations indicate an alternative charging mechanism, which could be electron transfer. Our findings suggest that slide electrification operates through at least two mechanisms, with the dominant charge transfer pathway shifting between ions and electron transfer depending on the electronegativity, phase, and temperature.

Let $\mathsf{E}$ be the event space of an experiment that can be indefinitely repeated. A natural question arises: given a countable cardinal $κ$, which is the event space of the $κ$-times repeated experiment? In the case of classical experiments, where $\mathsf{E}$ is a (complete) Boolean algebra on some set $S$, i.e. a classical or distributive logic, the answer is more or less known: the (complete) Boolean algebra on $S^κ$ generated by $\mathsf{E}^κ$. But, what if $\mathsf{E}$ is not a Boolean algebra? In this paper we give a constructive answer to this question for any $κ$ and in the context of general orthocomplemented complete lattices, i.e. general logics. Concretely, given a general logic $\mathsf{E}$ defining the event space of a given experiment, we construct a logic $\mathsf{U}_κ\left(\mathsf{E}\right)$ representing the event space of the $κ$-times repeated experiment, in such a way that $\mathsf{U}_κ\left(\mathsf{E}\right)$ and $\mathsf{E}$ are isomorphic if $κ=1$, and such that $\mathsf{U}_κ\left(\mathsf{E}\right)$ is distributive if and only if so is $\mathsf{E}$. We also extend our construction to the case in which the event space changes from one repetition to another and the cardinal $κ$ is arbitrary. This gives rise to tensor products $\bigotimes_{α\inκ}\mathsf{E}_α$ of families $\left\{ \mathsf{E}_α\right\} _{α\inκ}$ of orthocomplemented complete lattices, in terms of which $\mathsf{U}_κ\left(\mathsf{E}\right)=\bigotimes_{α\inκ}\mathsf{E}$.

In this manuscript we study the subdivisions of the permutahedron $Π_n$ into two subpolytopes corresponding to flags of positroids, which are in particular flags of lattice path matroids (LPFMs). A subpolytope $P_{[u,v]}$ of $Π_n$ is a Bruhat Interval Polytope (BIP) if $P_{[u,v]}$ is the convex hull of all the permutations (viewed as points in $\RR^n$) in the interval $[u,v]$ in the Bruhat order of $§_n$. We show that the coarsest subdivisions we obtain into LPFMs are the only subdivisions of $Π_n$ via hyperplane splits, into subpolytopes corresponding to BIPs. More specifically, we describe the hyperplanes whose intersection with $Π_n$ give rise to BIPs. Hence, these subdivisions are polytopes coming from points in the complete nonnegative flag variety.

Photonic quantum technologies utilize various degrees of freedom (DOFs) of light, such as polarization, frequency, and spatial modes, to encode quantum information. In the effort of further improving channel capacity of quantum communication, and for increasing the complexity of available quantum operations, high-dimensional and hyperentangled states are now gaining interest. However, efficiently measuring these high dimensional states is challenging due to the large number of measurements required for reconstructing the full density matrix via quantum state tomography (QST), and the fact that each measurement requires some modification in the experimental setup. Here, we propose a framework for reconstructing the density matrix of a single-photon hyperentangled across multiple DOFs using a single intensity-measurement obtainable from traditional cameras, and discuss extensions for multiphoton hyperentangled states. Our method hinges on the spatial DOF of the photon and uses it to encode the quantum information from the other DOFs. We numerically demonstrate this method for single-photon OAM-spin and OAM-frequency entangled states using an ideal coupler and a multimode fiber, to perform the information mixing and transfer the encoding to spatial information, where it is detected using a simple camera. This technique simplifies the experimental setup and reduces acquisition time compared to traditional QST-based methods. Moreover, it allows the recovery of DOFs that conventional cameras cannot detect, such as polarization, thus eliminating the need for projection measurements.

We formulate a time-dependent Fluctuating Local Field (TD-FLF) method for correlated fermion dynamics, extending the stationary FLF approach. The wavefunction is approximated as an ensemble of non-interacting states subject to a classical fluctuating field, with dynamics encoded in the field's time-dependent distribution. This reduces the time-dependent Schrödinger equation to a generalized eigenvalue problem in a significantly reduced basis. Applied to half-filled 2D Hubbard lattices, TD-FLF yields highly accurate results, outperforming mean-field theory and capturing oscillation frequencies and amplitudes in good agreement with exact diagonalization. Its low computational cost and flexibility make TD-FLF a promising tool for simulating driven correlated systems.

We explore applications of the celebrated construction of the Milnor connecting homomorphism from the odd to the even K-groups in the context of Hopf--Galois theory. For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, we provide an explicit formula computing the clutching $K_1$-class in terms of the representation matrix defining the module. Thus, the module is determined by an explicit Milnor idempotent. We apply this new tool to the K-theory of quantum complex projective planes to determine their $K_0$-generators in terms of modules associated to noncommutative Hopf fibrations. On the other hand, using explicit homotopy between unitaries, we express the $K_0$-class of the Milnor idempotents in terms of elementary projections in the Toeplitz C*-algebra. This allows us to infer that all our generators are in the positive cone of the $K_0$-group, which is a purely quantum phenomenon absent in the classical case.

We introduce a method for effectively identifying bound states in the continuum (BICs) - notably without computing the imaginary part of the eigenvalues - thereby simplifying the modeling and potentially reducing computation time. In real, open, physical systems, wave decay must be taken into account. This phenomenon is captured by complex-valued solutions of the harmonic wave equation, the so-called quasi-normal modes (QNMs). BICs, however, constitute a limiting class of solutions that do not radiate energy to infinity and are therefore, by their very nature, insensitive to the region surrounding the physical structure. Building on this observation, we identify BICs by varying the external boundary conditions that close the computational domain; the resulting behavior is displayed in the form of spectral histograms. We demonstrate the effectiveness of this procedure by comparing it with conventional QNM analysis employing perfectly matched layers. Two representative examples are considered: a periodic system of permeable inclusions supporting guided Rayleigh-Bloch waves, and a whispering-gallery resonator constructed from this configuration. Finally, we provide a mathematical explanation for the method's validity by deriving integral reciprocity statements.

We investigate the formation of multiple spiral modes in Milky Way-like disk-halo systems without explicitly exciting perturbations. We explore how numerical resolution, the level of local disk stability, and the presence of a live halo influence both the initial appearance and the subsequent evolution of these modes. To characterize spiral structure, we compute Fourier amplitudes for modes $m=1$-$6$. In marginally unstable, lower-resolution disks ($N_\star=5\times10^6$, $N_{\rm DM}=1.14\times10^7$), faint features appear within the first $0.5$ Gyr due to numerical noise, in contrast to high-resolution models where perturbations emerge later. Across all sufficiently resolved, live-halo models with $m_{\rm DM}/m_\star \le 10$, the spirals exhibit a cascading sequence in both mode number and radius: higher-$m$ modes form and decay first, followed by the delayed emergence of lower-$m$ modes, with an inward drift of the activity's epicenter. This behavior reflects a combination of local swing amplification, which explains the initial growth of short-wavelength modes, and interference between coexisting long-lived spiral modes, which accounts for the recurrent short-timescale amplitude modulations. In contrast, models with a fixed halo potential or coarse halo resolution ($N_{\rm DM}=1.14\times10^6$ and $m_{\rm DM}/m_\star=100$) show strong early spirals but lack this coherent cascading behavior, owing to excessive shot noise and insufficient halo responsiveness. The $m=3$ mode plays a transitional role, marking the onset of angular-momentum transport in the inner disk that precedes bar formation, a process absent in fixed-potential models. Our results show that a live halo with appropriate mass resolution provides the gravitational response needed to sustain and regenerate multi-mode spiral structure, even though the total angular-momentum exchange remains small.

Optically generated microwave signals exhibit some of the lowest phase noise and timing jitter of any microwave-generating technology to date. The success of octave-spanning optical frequency combs in down-converting ultrastable optical frequency references has motivated the development of compact, robust and highly manufacturable optical systems that maintain the ultralow microwave phase noise of their tabletop counterparts. Two-point optical frequency division using chip-scale components and ~1 THz-spanning microcombs has been quite successful, but with stringent requirements on the comb source's free-running noise and feedback control dynamics. Here we introduce a major simplification of this architecture that replaces feedback control of the frequency comb in favor of electronic feedforward noise cancelation that significantly relaxes the comb requirements. Demonstrated with both a high repetition rate solid-state mode-locked laser and a microcomb, feedforward on a 10 GHz carrier results in more robust operation with phase noise as low as -153 dBc/Hz at offsets >10 kHz, femtosecond timing jitter, and elimination of the large "servo bump" noise increase at high offset frequency. The system's compatibility with a variety of highly manufacturable mode-locked laser designs and its resilience and straightforward implementation represents an important step forward towards a fully chip-scale implementation of optically generated microwaves, with applications in radar, sensing, and position, navigation and timing.

In a significant conceptual revision to the tenfold classification scheme for topological insulators and superconductors, it was recently demonstrated that most three-dimensional (3D) classes are simultaneously "localizable" in two distinct, but intricately connected ways: (1) There is no obstruction to Wannier localization of all bulk eigenstates, and (2) Almost all surface states can be Anderson localized by arbitrarily weak symmetry-preserving quenched disorder. Here we consider the localizable class CI in 3D, and numerically investigate the stability of surface states. We demonstrate that surface states of a bulk class-CI topological lattice model are fragile in that they can be Anderson localized by the combination of weak quenched randomness and hybridization with an additional trivial 2D band (a trivializing proximity effect, TPE). With the TPE, stronger disorder is more destructive to the surface states of the bulk lattice model. By contrast, without additional bands the surface states remain extended, exhibiting robust spectrum-wide quantum criticality. We also investigate the fragility of surface states in effective 2D class-CI continuum Dirac theories, including the chiral limit of the Bistritzer-MacDonald model for twisted bilayer graphene. Although the continuum models exhibit signs of Anderson localization near gap edges for weak disorder, stronger disorder instead appears to heal the surface, restoring criticality whilst filling in spectral energy gaps. Our results provide further evidence that effective continuum field theories fail to capture key aspects of surface-state physics in localizable topological phases.

We place bounds on the order of enhanced discrete gauge symmetries that act on massless fields and thus arise at subloci of the moduli space in supergravity theories. We focus on supersymmetric theories with 8 or more supercharges which in some cases lead to sharp upper bounds realized by specific string constructions.

We identify the $r^{-3}$ curvature-decay rate as a universal geometric threshold separating compact from non-compact perturbations of Laplace-type operators on asymptotically flat manifolds. For the coupled Einstein--Maxwell system, we prove that the linearized operator $\mathcal{L}$ is essentially self-adjoint and that curvature and field strengths decaying faster than $r^{-3}$ act as relatively compact perturbations, while decay exactly at $r^{-3}$ places $0\inσ_{\mathrm{ess}}(\mathcal{L})$ through delocalized zero modes. This threshold mechanism unifies the infrared behavior of spin-1, spin-2, and mixed spin-$(1\oplus2)$ fields, linking the onset of spectral delocalization with the appearance of gravitational and electromagnetic memory. Finite-difference simulations corroborate the analytic scaling and reproduce the characteristic quadrupolar and dipolar sky maps predicted for the coupled memory fields. These results demonstrate that curvature decay at $r^{-3}$ constitutes a fundamental geometric boundary underlying infrared universality in gauge and gravitational theories, providing a spectral counterpart to the asymptotic-symmetry and soft-theorem formulations of memory.

In offline data-driven multi-objective optimization (MOO), optimization is performed using surrogate models trained only on an offline dataset. These surrogate models contain inherent errors and uncertainty. This epistemic uncertainty can lead to incorrect dominance judgments, thereby misleading the search process. Existing methods mitigate this issue by incorporating uncertainty estimates from Gaussian Process Regression (GPR) to correct dominance judgments; however, they are restricted to GPR, and their optimization strategies cannot be scaled to other uncertainty quantification methods. In addition, GPR-based surrogates suffer from high computational cost. We propose a simple yet effective dual-ranking strategy that flexibly leverages both predictive results and uncertainty estimates from different surrogate models. By performing non-dominated sorting on candidate solutions using both surrogate-based fitness values and uncertainty-aware fitness values, the proposed method prioritizes candidate solutions that are simultaneously high-quality and reliable. Through extensive experimental evaluations, including ablation, sensitivity, and comparative experiments, we demonstrate the effectiveness and robustness of the proposed dual-ranking strategy working with different surrogates. Our dual-ranking framework offers more robust solutions for data-limited, real-world applications.

We consider the problem of choosing prices of a set of products so as to maximize profit, taking into account self-elasticity and cross-elasticity, subject to constraints on the prices. We show that this problem can be formulated as maximizing the sum of a convex and concave function. We compare three methods for finding a locally optimal approximate solution. The first is based on the convex-concave procedure, and involves solving a short sequence of convex problems. Another one uses a custom minorization-maximization method, and involves solving a sequence of quadratic programs. The final method is to use a general purpose nonlinear programming method. In numerical examples all three converge reliably to the same local maximum, independent of the starting prices, leading us to believe that the prices found are likely globally optimal.

Freeze-thaw cycles can be regularly observed in nature in water and are essential in industry and science. Objects present in the medium will interact with either an advancing solidification front during freezing or a retracting solidification front, i.e., an advancing melting front, during thawing. It is well known that objects show complex behaviours when interacting with the advancing solidification front, but the extent to which they are displaced during the retraction of the solid-liquid interface is less well understood. To study potential hysteresis effects during freeze-thaw cycles, we exploit experimental model systems of oil-in-water emulsions and polystyrene (PS) particle suspensions, in which a water-ice solidification front advances and retracts over an individual immiscible (and deformable) oil droplet or over a solid PS particle. We record several interesting hysteresis effects, resulting in non-zero relative displacements of the objects between freezing and thawing. PS particles tend to migrate further and further away from their initial position, whereas oil droplets tend to return to their starting positions during thawing. We rationalize our experimental findings by comparing them to our prior theoretical model of Meijer, Bertin & Lohse, Phys. Rev. Fluids (2025), yielding a qualitatively good agreement. Additionally, we look into the reversibility of how the droplet deforms and re-shapes throughout one freeze-thaw cycle, which will turn out to be remarkably robust.

Monitoring the in-service aerodynamic performance of airliners is critical for operational efficiency and safety, but using operational Quick Access Recorder (QAR) data for this purpose presents significant challenges. This paper first establishes that the absence of key parameters, particularly aircraft moments of inertia, makes conventional state-propagation filters fundamentally unsuitable for this application. This limitation necessitates a decoupled, Equation-Error Method (EEM). However, we then demonstrate through a comparative analysis that standard recursive estimators with time-varying gains, such as Recursive Least Squares (RLS), also fail within an EEM framework, exhibiting premature convergence or instability when applied to low-excitation cruise data. To overcome these dual challenges, we propose and validate the Constant-Gain Equation-Error Method (CG-EEM). This framework employs a custom estimator with a constant, Kalman-like gain, which is perfectly suited to the stationary, low-signal-to-noise characteristics of cruise flight. The CG-EEM is extensively validated on a large, multi-fleet dataset of over 200 flights, where it produces highly consistent, physically plausible aerodynamic parameters and correctly identifies known performance differences between aircraft types. The result is a robust, scalable, and computationally efficient tool for fleet-wide performance monitoring and the early detection of performance degradation.

The standard fuzzy regression discontinuity (FRD) estimator is a ratio of differences of local polynomial estimators. I show that this estimator does not possess any finite integer moments, regardless of local polynomial degree, kernel function, or bandwidth. The estimator is heavy-tailed in small samples or when the treatment probability discontinuity at the cutoff is small. I present a generalized class of FRD estimators which preserves all finite moments from the data, indexed by a single tuning parameter, and nesting both standard FRD and sharp (SRD) estimators. Simple deterministic values of the tuning parameter lead to substantial improvements in median bias, median absolute deviation, and root mean squared error. Confidence intervals typically give reliable small-sample coverage in simulations. Estimator stability and performance are demonstrated using data on class size effects on educational attainment.

In many proofs of Fermat's Two Squares Theorem, the smallest least residue solution $x_0$ of the quadratic congruence $x^2 \equiv -1 \bmod p$ plays an essential role; here $p$ is prime and $p \equiv 1 \bmod 4$. Such an $x_0$ is called a Størmer number, named after the Norwegian mathematician and astronomer Carl Størmer (1874-1957). In this paper, we establish necessary and sufficient conditions for $x_0 \in \mathbb{N}$ to be a Størmer number of some prime $p \equiv 1 \bmod 4$. Størmer's main interest in his investigations of Størmer numbers stemmed from his study of identities expressing $π$ as finite linear combinations of certain values of the Gregory-MacLaurin series for $\arctan(1/x)$. Since less than 600 digits of $π$ were known by 1900, approximating $π$ was an important topic. One such identity, discovered by Størmer in 1896, was used by Yasumasa Kanada and his team in 2002 to obtain 1.24 trillion digits of $π$. We also discuss Størmer's work on connecting these numbers to Gregory numbers and approximations of $π$.

Transfer RNAs (tRNAs) are essential components of the translational machinery. Their abundance and diversity shape decoding capacity and protein synthesis efficiency and accuracy. Because tRNA abundance is encoded in the genome through tDNA copy number, chromosomal organization, and cis-regulatory sequences controlling transcription, these features are expected to influence translational. However, the principles governing nuclear tDNA organization remain poorly understood. Here, we analyzed nuclear tDNA repertoires across 53 photosynthetic eukaryotes spanning major Archaeplastida lineages and secondary endosymbionts, along with seven non-plant eukaryotic outgroups, using comparative genomic approaches at sequence, chromosomal, and genome-wide scales. To support these analyses and enable interactive exploration of tDNA organization, we developed ShinytRNA (https://nebula.ibmp.unistra.fr/shinytRNA/), a web application for genome-wide exploration of tDNA organization. Nuclear tDNA copy numbers vary by more than two orders of magnitude across species, yet the relative representation of tRNA families corresponding to each amino acid remains strikingly conserved across lineages, revealing strong evolutionary constraints on tDNA dosage. In angiosperms, tDNAs show reinforced cis-regulatory features linked to RNA polymerase III transcription, including expanded AT-rich regions, enriched CAA motifs, and extended poly(T) tracts. At the chromosomal scale, tDNAs are predominantly dispersed along chromosome arms, with homogeneous spacing that scales with genome size, while also showing non-random chromosomal distribution, exclusion from centromeric regions, and limited clustering. Together, these patterns reveal conserved yet lineage-specific principles governing nuclear tDNA organization in plants, and highlight how multiple genomic constraints shape the evolution of nuclear tDNA repertoires.