Quantum simulation is a cornerstone application for quantum computing, yet standard methods face a trade-off between circuit depth and accuracy: Trotterization depth scales with the number of Hamiltonian terms $L$, while sampling-based qDRIFT is restricted to $O(t^2)$ error scaling. Here, We introduce qSHIFT, an adaptive sampling protocol that overcomes these limitations. By adaptively updating sampling distributions, qSHIFT maintains $L$-independent gate complexity while achieving an improved error scaling of $O(t^{1+r})$ for an adjustable parameter $r$. This performance is enabled by a classical subroutine solving $L^r$ linear equations per sampling round. Numerical demonstrations confirm the $O(t^{1+r})$ scaling, showcasing qSHIFT as a resource-efficient framework for high-precision quantum simulation. Furthermore, the protocol's reduced circuit depth enhances its compatibility with physical error mitigation, making it a promising candidate for implementation on near-term quantum devices. In addition to its role as a standalone algorithm, qSHIFT can provide a high-precision foundation for modular quantum frameworks such as qSWIFT or Krylov quantum diagonalization.

Baryonic physics is anticipated to be a major source of systematic uncertainty in current and future large-scale cosmological surveys. We investigate how baryonic effects on halo density profiles vary with secondary halo properties at fixed halo mass, using the large-volume MillenniumTNG hydrodynamical simulation and its dark matter-only counterpart. We focus on the impact of halo concentration and large-scale environment on the ratio of density profiles of matched halos in the hydrodynamical and dark matter-only simulations. At redshift $z = 0.0$, we find a strong dependence on halo concentration, especially at lower halo mass ($12.5 < \log(M_h/h^{-1}M_{\odot}) < 13.0$), where more concentrated halos exhibit weaker inner enhancement and stronger intermediate-radius suppression at fixed halo mass, with variations reaching $\sim 15\%$ at small scales and decreasing toward larger scales. This trend weakens and reverses at higher halo mass. In contrast, the secondary dependence on large-scale environment is weaker ($\sim 2\%$) and largely scale-independent, with halos in denser regions exhibiting slightly weaker intermediate suppression. By separating internal profile redistribution from total mass suppression, we show that concentration impacts both components, whereas the environmental dependence is primarily associated with an overall mass shift. These secondary dependencies persist at $ z = 0.5$ and correlate with variations in internal baryonic properties. We examine additional halo properties, including halo spin and velocity dispersion, and find significant secondary dependence. Overall, our results highlight the important role of secondary halo properties in modulating baryonic effects on halo density profiles, with potential implications for future modeling efforts.

These notes are written for a memorial Session dedicated to George Lazarides. They revisit a joint work on the cosmology of a gauged axion and place it in a broader line of ideas connecting anomalous gauge symmetries, orientifold effective actions, Stueckelberg fields and dark matter. In models with an anomalous extra $U(1)$ symmetry, the Stueckelberg pseudoscalar participates in the restoration of gauge invariance through Wess-Zumino counterterms and, after electroweak symmetry breaking, may leave a physical axion-like state. Its cosmological history differs from that of an ordinary Peccei-Quinn axion: the physical field appears only after Higgs-Stueckelberg mixing, is subject to sequential electroweak and QCD misalignment, and can give an appreciable dark-matter relic abundance only when the Stueckelberg scale is sufficiently large. This perspective connects naturally with George's earlier insight that the vacuum structure of axion models must be understood together with the gauge structure in which it is embedded. I dedicate these notes to his memory, with gratitude for the collaboration and for the clarity with which he connected particle physics to the early universe.

Almost every galaxy in the local Universe is observed to have a massive black hole in the centre. The properties of these black holes are observed to tightly correlate with those of their host galaxy which has been interpreted as coevolution regulated by black hole feedback. This coevolution spans most of cosmic history, as the first active black holes, so-called active galactic nuclei, are already observed as early as $z\sim10$. In this chapter, we lay out how we can find supermassive black holes, review what we know about the population of black holes and their host galaxies from observations, and summarise what we have learned about their coevolution across cosmic time from both observations and simulations.

We present the first quantum-hardware implementation of a Hamiltonian simulation algorithm that produces signed vector-field solutions to the time-domain Maxwells equations using a Schrodingerisation-based approach. The electromagnetic fields are discretized using finite-difference operators, and the resulting non-unitary matrices are mapped to Bell-basis Trotter blocks, enabling efficient circuit construction. We introduce a measurement procedure that retrieves not only field amplitudes, but also physical directions of the electric and magnetic field values at select spatial points. Implementing this logic on quantum hardware relies on relative-phase-based sign reconstruction. Numerical results obtained using IonQ QPU, show good agreement with analytical solutions of benchmark problems in two dimensions and on simulators; in three dimensions. We further extend our approach to compute fields scattered from simple bodies, by enforcing appropriate boundary conditions. Our work lays the foundational steps towards realizing quantum-hardware solutions for computational electromagnetics.

We analyze the nonabelian surface holonomy on a bipartite hypercubic lattice following a proposal in arXiv:1002.4636 [hep-th]. The bipartite structure of the lattice enables us to introduce spike string configurations. These spikes play a crucial role for the time evolution of the string when the total number of color indices changes.

Demonstrating quantum advantage for combinatorial optimization requires more than standalone algorithmic results; it calls for end-to-end case studies that integrate problem modelling, quantum execution, and classical refinement into practical workflows. This paper presents a sandbox platform for experimenting with hybrid quantum-classical workflows in graph optimization, enabling the systematic study of end-to-end optimization pipelines. Using our platform, we investigate three classically intractable and mutually reducible graph problems -- Minimum Vertex Cover, Maximum Independent Set, and Maximum Clique -- by transforming them into an unconstrained problem and solving the resulting instances with QAOA on IBM platforms. Our workflow combines classical pre-processing to reduce instance size, quantum optimization on the reduced problem, and classical postprocessing to map quantum outputs to high-quality feasible solutions, thereby avoiding direct constraint encoding in the quantum circuit. We evaluate the approach on synthetic graphs, benchmark instances, and real-world networks, and report hardware experiments on IBM Quantum System One at PINQ2 in Bromont, Quebec, powered by IBM's 156-qubit Heron r2 processor on graphs up to 128 vertices, with circuits involving up to 128 qubits and 13,555 two-qubit gates. The results illustrate how sandbox-style end-to-end experimentation can expose bottlenecks, clarify the role of classical-quantum workload partitioning, and provide domain experts and practitioners with a practical guide for interpreting quantum optimization outputs and assessing quantum utility on the road to quantum advantage in combinatorial optimization.

Bayesian inference in hidden Markov models (HMMs) can be challenging due to the presence of multimodality in the likelihood function, and consequently in the joint posterior distribution, even after correcting for label switching. The parallel tempering (PT) algorithm, a state-space augmentation method, is a widely used approach for dealing with multimodal distributions. Nevertheless, standard implementation of the PT algorithm may not always be sufficient to effectively explore the high-dimensional, complex multimodal posterior distributions that arise in HMMs. In this work, we demonstrate common pitfalls when implementing the PT algorithm for HMMs, approaches to remedy them, and introduce new non-informative prior distributions that facilitate effective posterior distribution exploration. We analyse time series of blue whale dive data with two 3-state HMMs in a Bayesian framework, one of which includes a categorical covariate in the transition probability matrix to account for the effect of sound stimuli on the whale's behavior. We demonstrate how effective implementation of the modified PT algorithm for Bayesian inference leads to effective exploration of the resultant multimodal posterior distribution and how that affects inference for the underlying movement patterns of the blue whales.

The compound SrNi$_2$P$_2$ can exhibit multiple crystal structures with no P-P pairs bonded (uncollapsed tetragonal, or ucT, state), with one-third of the P-P pairs bonded (one-third collapsed orthorhombic, or tcO, state), or with all P-P pairs bonded (collapsed tetragonal, or cT, state) across the Sr layers. The system can be tuned into its different states by changing temperature, mechanical stress, or chemical composition. Changes in bonding may manifest in changes of macroscopic properties of the material, such as its shape, electrical conductivity, or magnetism. In this work, we show that SrNi$_2$P$_2$ can be tuned among the three states by changing Cu substitution and temperature. We present temperature-dependent resistance and single-crystal x-ray diffraction results in Sr(Ni$_{1-x}$Cu$_x$)$_2$P$_2$ single-crystals that show that Cu substitution favors the P-P bonding, stabilizing the cT state at ambient pressure. We construct a $T-x$ phase diagram that shows how all of these transition temperatures increase with increasing Cu fraction, $x$. The transition between the tcO state and the cT state exhibits a very large thermal hysteresis, which can be tuned to temperatures close to room temperature. In particular, the properties of Sr(Ni$_{0.963}$Cu$_{0.037}$)$_2$P$_2$ may make it suitable for applications as a shape memory material at room temperature.

The solution of sparse linear systems constitutes the dominant computational bottleneck in interior point methods (IPMs), frequently consuming over 70% of the total solution time. As optimization problems scale to millions of variables, direct solvers encounter prohibitive fill-in, excessive memory consumption, and limited parallel scalability. We present SDSL-Solver, a scalable distributed sparse linear solver framework designed for IPMs. SDSL-Solver employs Krylov subspace methods, combined with numerics-based sparse filtering and diagonal correction techniques that produce high-quality preconditioners. To accommodate diverse problem characteristics, SDSL-Solver offers two complementary distributed parallel methods: Block Jacobi for diagonally dominant matrices, and Bordered Block Diagonal (BBD) for general or ill-conditioned matrices requiring globally coupled preconditioning via Schur complement techniques. A preconditioner reuse strategy further amortizes construction costs across consecutive IPMs iterations. We evaluate SDSL-Solver on benchmark problems with matrix dimensions ranging from tens of thousands to over five million on multi-node clusters equipped with X86 processors. The experimental results show that under the Block Jacobi and BBD distributed methods, SDSL-Solver on a four-node configuration achieves average speedups of 6.23 times and 7.77 times, respectively, compared to PETSc running on the same number of nodes. Relative to the single-node PARDISO, the average speedups reach 97.54 times and 5.85 times, respectively.

We present the first data release (DR1) of the Spectroscopic Classification of Astronomical Transients (SCAT) survey, covering the first $\approx 5$ years of observations (March 2018 - January 2023). DR1 includes 1810 spectra of 1330 transients, which we sort into broad spectroscopic classes including supernovae (SNe), transients originating in galactic nuclei, and stellar variability. We collect multi-filter light curves from imaging surveys and fit them with phenomenological models to estimate peak brightnesses and the time of explosion/first-light. Extragalactic transients are matched to candidate host galaxies, and we compare host-galaxy luminosities and projected offsets by SN type. SNe appear to be a reliable way to augment the redshift coverage of nearby ($z\lesssim 0.1$) galaxies in tandem with dedicated redshift surveys. We present new redshifts for roughly half of the SN host galaxies, most of which are low-luminosity dwarfs similar to the Magellanic Clouds ($M_r \gtrsim -18$ mag). This set of transient spectra, light curves, luminosities, redshifts, and host galaxies offers an excellent testbed for real-time photometric/light curve classification pipelines in the modern era of deep and large-area surveys. We conclude with a brief discussion of the provided data products and status of the SCAT survey.

We study a family of scale-invariant $p$-densities of knot types in $R^3$, defined as the ratio of length to an $L^p$-type spread of pairwise distances along a curve. The first point of the paper is that the unconstrained theory has a strong degeneration. Local knotting shows that, for every $p\in(-1,\infty]$ and every knot type $K$, the unconstrained $p$-density of $K$ is no larger than that of the unknot. Using the sharp mean-chord inequality of Exner--Harrell--Loss, we show that this degeneration is complete throughout the range $-1<p\le2$: for $p\ne0$ one has \[ ρ_p(K)= π\left( \fracπ{\int_0^π\sin^pθ\,dθ} \right)^{1/p}, \] while $ρ_0(K)=2π$. At the endpoint $p=\infty$, one also has $ρ_\infty(K)=2$ for every knot type $K$. The remaining finite range $p>2$ is analytically different: the round circle is not the relevant extremal curve in general, and knot-type independence in this range is left as a separate extremal problem. These degenerations motivate a constrained refinement. We introduce ropelength-windowed $p$-densities by imposing the thickness normalization $Thi(γ)\ge 1$ and the length bound $len(γ)\le λRop(K)$. These constraints prevent the collapse caused by arbitrarily small local knotting. We prove basic monotonicity properties and an existence theorem for minimizers of the ropelength-windowed problem. We also retain the polygonal approximation theorem for the unconstrained densities, showing that the continuous and polygonal theories agree asymptotically as the number of edges tends to infinity. The paper concludes with a list of open questions concerning the finite high-exponent range, constrained density spectra, thickness-controlled polygonal approximation, and regularized inverse-power extensions.

Graph burning is a discrete-time process that models the spread of social contagion. Initially, all vertices are unburned. In each round, one unburned vertex is selected and burned, while any unburned vertex that has a burned neighbour from the previous round also becomes burned. The burning number of a graph is the minimum number of rounds needed to burn the entire graph. In this paper, we study the burning number of graph powers. First, we show that for a connected graph $G$, its graph power $G^k$ contains a $(k+1)^+$-branching tree as a spanning tree. A $(k+1)^+$-branching tree is one whose internal vertices have degree at least $k+1$. We then show that $(k+1)^+$-branching trees on $n$ vertices have burning number at most $\left\lceil{\sqrt{\frac{4(k-1)n}{k^2}}}~\right\rceil$. As the burning number of a graph is at most the burning number of any of its spanning trees, this gives an upper bound on the burning number of graph powers. We also derive an explicit bound building on the results of Bastide et al., and identify the ranges of $k$ and $n$ for which our bound outperforms theirs. Finally, we show that $b(G^k) \le (1+o(1))\sqrt{n/k}$ based on the asymptotic burning number bound of Norin and Turcotte.

We study how the color code under decoherence gives rise to an intrinsic mixed-state topological order (imTO), which has no counterpart in pure ground states of local gapped Hamiltonians. For decoherence induced by XX-type operators on red links of the honeycomb lattice, we show that the resulting mixed state inherits half of the topological properties of the color code, including anyon content, logical operators, and topological entanglement structure. Using a gauging procedure for mixed stabilizer states, we identify the emergent phase as closely related to a single toric code. We characterize this phase by topological entanglement negativity (TEN) and perform efficient stabilizer-formalism simulations. While the pure color code has ${\rm TEN} = 2 \ln 2$, the maximally decohered state has ${\rm TEN} = \ln 2$, indicating emergence of a single toric code. By tuning the decoherence strength, we find a smooth crossover in TEN accompanied by a pronounced, nearly system-size-independent peak in its variance. We further show that the negativity exhibits characteristic scaling only for subsystem partitions commensurate with the triangular lattice of the emergent toric code. Our results demonstrate that negativity-based quantities provide powerful probes of mixed-state topological order generated by decoherence.

We introduce the notion of uniform exactness, or uniform amenability at infinity, for discrete groups and prove it for a wide class of groups containing free groups and their limit groups. This shows a novel strong convergence phenomenon that any convergent sequence of such groups in the space of marked groups converges strongly in the operator algebraic sense. In particular, convergence of the spectral radius formula is uniform over probability measures on such groups whose supports have a fixed cardinality.

Galaxy evolution is commonly described through the time evolution of observational statistics such as luminosity functions and stellar mass functions. However, these quantities are projections of an underlying multivariate galaxy state space rather than fundamental dynamical variables. We develop a unified framework in which galaxy evolution is formulated as the time evolution of a probability measure on the galaxy manifold. Representing galaxy states by latent variables $θ\in\mathcal{M}$ and the population by a density $ρ(θ,t)$, the evolution is governed by a general equation containing continuous transport and nonlocal jump processes. By reinterpreting manifold learning as the pushforward of measures, we distinguish observational, representation, and physical measures, and emphasize that manifold coordinates themselves need not carry direct physical meaning. In this picture, luminosity functions and stellar mass functions arise as projected observables of a single underlying dynamics, and generally do not form closed equations in observational space. The framework contains existing models as limiting cases: reduction to a single mass variable yields continuity-equation models, while additive post-merger states recover the Smoluchowski coagulation equation. We further show that luminosity-function evolution is naturally described within the Schechter family, whose apparent stability is interpreted as an effective consequence of projection. Since observables are projections of measures, inference of galaxy evolution becomes a statistical inverse problem of recovering manifold dynamics from data. This framework shifts the focus from fitting observed statistics directly to inferring the underlying state-space dynamics, thereby bridging manifold learning and physical theory.

Non-Markovian (renewal) epidemic simulation on multi-million-node contact networks is essential for realistic forecasting under general age-dependent holding-time distributions (log-normal, Weibull, Erlang, and similar), but the age-dependent hazard forces dense per-step updates that render the sparse event-queue strategies of standard CPU methods ineffective. We present FlashSpread, a GPU framework that consolidates the per-step renewal pipeline (CSR traversal, numerically stable erfcx-based hazard evaluation, Bernoulli tau-leaping, state transition, and next-step infectivity write-back) into a single fused Triton kernel whose intermediates never leave streaming-multiprocessor registers, with block-scalar skips that preserve CUDA Graph capture and a degree-aware CSR dispatch (thread / warp / edge-merge) that keeps the peak throughput on scale-free graphs. On an NVIDIA A100 the fused CUDA-Graph engine reaches 8.09 Giga-NUPS at N = 10^6 on a uniform-degree graph, a 217x strict hardware speedup over optimised CPU tau-leaping at the same N; on a Barabasi-Albert graph of the same size the merge-based dispatch recovers 4.5x (0.45 to 2.0 Giga-NUPS) over the default kernel, and the framework scales to N = 10^8 on a single A100 (40 GB), with a mixed-precision storage path that extends the L2-reachable scale by roughly 3x and delivers a 1.15x throughput lift at the far bandwidth-bound end. Validation against an exact non-Markovian Gillespie reference shows a structural-bias floor of approximately 6% on peak infection and approximately 7% on final attack rate that does not detectably decrease as epsilon nears 0 across two decades of tolerance, comfortably within typical epidemiological parameter uncertainty. Code: https://github.com/Shakeri-Lab/FlashSpread.

Who gets to decide how generative AI tools enter students' classrooms? We report on a five-week participatory design program in which three 11th-grade Latinx students and three high school teachers in California negotiated how generative AI tools would be used and taught about in learning environments. Drawing on video recordings and designed artifacts, we ask: what critical AI literacy practices emerged as students and teachers jointly designed how generative AI tools would be used and taught about? Our analysis reveals three practices: collectively unsettling assumptions about AI, mutual learning through complementary expertise, and grounding AI critique in cultural knowledge and creative practice. Students and teachers developed these practices through the design work itself. This case contributes strategies for designing with youth around an emergent technology like generative AI toward critical AI literacy. It extends work on youth as protagonists by showing how this approach enables students to shape both the adoption and the interrogation of these tools in their learning environments.

Systematic control errors remain a primary obstacle to realizing high-fidelity single-qubit gates. We introduce composite pulse sequences that implement X and Hadamard gates while simultaneously compensating amplitude (Rabi-frequency), detuning (frequency), and duration errors. Our construction uses two complementary strategies: (i) derivative-based cancellation of error terms in the full unitary (not just the transition probability), formulated via the Cayley-Klein parametrization, and (ii) direct minimization of the average gate infidelity over prescribed error ranges. We derive symmetric five-pulse solutions with closed-form phases that cancel all first-order terms (including the mixed derivative), and numerically optimize longer sequences -- up to 15 pulses -- to achieve higher-order suppression. We also show that standard ``universal'' five-pulse sequences (U5a/U5b) emerge as simple phase-shifted instances of our symmetric solutions, yielding broad robustness to both detuning and amplitude errors. Finally, we construct variable-area sequences for $R_x(π/2)$, which, up to virtual Z rotations, benchmark the Hadamard gate. Across all families we observe the expected trade-off between sequence length and robustness window, with substantial boosts in fidelity over large error domains.

We develop a mathematically controlled framework for Yang--Baxter integrability in pseudo-Hermitian quantum impurity systems arising from periodic driving of a Dirac-like bath. The effective impurity Hamiltonian possesses a dynamically generated $\PT$ symmetry and exhibits exceptional points (EPs) where it becomes non-diagonalizable. We construct the Yang--Baxter generator as a rank-one operator on the two-particle contact space, built from biorthogonal impurity eigenvectors, and prove that it satisfies the Temperley-Lieb relations. Its standard Baxterization gives an $R$-matrix, an RLL relation, an RTT structure,and a commuting family of transfer matrices. At the exceptional point(EP), the semisimple biorthogonal eigenvector construction is replaced by a Jordan-chain contact vector, while the Hamiltonian itself develops a nilpotent Jordan block. Within this framework we derive biorthogonal Bethe equations and show that the Gaudin matrix becomes defective at the EP, establishing that the smallest singular value $σ_N(G)\to0$ at the EP while remaining $\OO(1)$ at the Kondo critical point,providing a sharp algebraic diagnostic. We further prove that Bethe rapidities exhibit square-root coalescence and $\mathbb{Z}_2$ monodromy at the EP, reflecting the underlying Jordan structure, and that the effective pseudo-Hermitian Hamiltonian emerges from the periodically driven microscopic system by adiabatic coarse graining of off-shell angular-momentum modes, with corrections controlled by the auxiliary-mode gap.

The time-periodic problem on the Boltzmann equation with a given time-periodic external force in the three-dimensional whole space has remained open since it was first studied in [15] for only spatial dimensions not less than five. The goal of this paper is to give an affirmative answer to this problem provided that the external force is sufficiently small in the function space $\mathcal{C}(\mathbb{R};\dot{B}^{-3/2}_{2,\infty}\cap\dot{H}^N)$ with $N\geq 4$. The proof is based on Serrin's method through studying the global-in-time stability of the Cauchy problem with time-periodic external forces. As a direct consequence, the result also yields the existence and stability of stationary solutions to the physically realistic three-dimensional Boltzmann equation when the external force is time-independent.

We present a simple hierarchical communication scheme for distributed Fast Multipole Methods (FMMs) based on MPI neighborhood collectives and uniform trees. The method targets the common case of extending an existing high-performance shared-memory uniform-tree FMM implementation to distributed memory with minimal redesign while preserving any shared memory optimizations. Benchmarks on the ARCHER2 supercomputer demonstrate that our method can scale to very large problem sizes, we demonstrate weak-scaling up to 3.2e10 uniformly distributed points on 512 nodes of the machine in our largest runs. Our simplifications based on uniform trees result in worse asymptotic scaling for non-uniform points, however we still obtain practically useful runtimes due to the ability to retain our shared memory optimizations.

We characterize those finite groups for which the bounded derived category of finite dimensional representations over an algebraically closed field of characteristic $p$ has distributive lattice of thick subcategories: they are precisely the $p$-nilpotent groups. Along the way we give necessary and sufficient criteria for the bounded derived category and perfect complexes of a finite dimensional $k$-algebra to have distributive lattices of thick subcategories.

We propose CHRONOS, a hardware-assisted framework that decouples the cryptographic setup required for private gradient aggregation from the active training phase. CHRONOS executes a once-per-epoch server-relayed Diffie-Hellman key exchange during a device's idle window. It generates ephemeral keypairs and derives PRG keys entirely within an ARM TrustZone enclave, ensuring private keys never exist in Normal World memory. Pairwise secrets are sealed in the enclave, and Shamir secret shares of the ephemeral private key are distributed to peers. During training, clients mask gradients with a single stream-cipher evaluation and transmit them in one communication round. A hardware-backed round counter enforces single-use freshness. If clients drop out mid-round, the server reconstructs their masks from peer-held Shamir shares, preserving correct aggregation without repeating the round. Evaluation on Rock Pi 4 devices using OP-TEE demonstrates that CHRONOS achieves OS-level compromise resistance and thwarts state-of-the-art gradient inversion attacks. It reduces active-phase aggregation latency by up to 74% compared to synchronous secure aggregation for 20 clients. The system maintains a persistent Secure World storage footprint of fewer than 700 bytes per device, scaling independently of model dimension.

The non-Hermitian skin effect can lead to directional amplification of response, with the associated end-to-end Green's function generally exhibiting size dependence. Any deviation in length or local disorder can drastically alter the amplification factor, rendering the response fragile in practical implementations. In this work, we identify a new type of scale-free, topological, and directionally amplified response in a Hatano-Nelson model under perturbed open boundary conditions. The scale-free response can be attributed to the first order boundary effect and characterized by a winding number defined on a continuous generalization of the finite-size Brillouin zone-a concept introduced in this work. Such scale-free behavior endows the end-to-end Green's function with significant robustness and making it promising for practical applications.

We investigate certain Liouville integrable systems constructed earlier via reduction of the quasi-Hamiltonian double of $\mathrm{SU}(n)$. These systems live on compact connected symplectic manifolds of dimension $2(n-1)$ and can be interpreted as compactified trigonometric Ruijsenaars--Schneider systems. Depending on the value of a parameter $0<y< π$, they arise in two drastically different forms: in type (i) these are toric systems, while in the type (ii) cases they possess globally continuous action variables that generate a Hamiltonian torus action (only) on a dense open subset of the phase space. The principal goal of the paper is to study those fibers of the action map (alias the $\mathbb{T}^{n-1}$ momentum map) which are contained in the complement of the domain of the densely defined torus action occurring in the type (ii) cases. We demonstrate that all such `singular fibers' are smooth connected isotropic submanifolds. We also work out a model of the fibers as quotient spaces of certain subgroups of $\mathrm{SU}(n)$ with respect to an action of another subgroup. The general results are exemplified by determining the vertices of the polytope filled by the action variables in the simplest type (ii) cases that appear for any $n\geq 4$ with $π/(n-1) <y < π/(n-2)$, and proving that the fibers over the `singular vertices' are diffeomorphic to $S^3 \simeq \mathrm{SU}(2)$ in these cases. In this way, our findings enrich the set of examples of Liouville integrable systems with spherical singularities.

In this article, we introduce the notion of Quaternion Boostlet Transform (QBT), a hypercomplex framework designed to unify the analysis of multi-component wavefields by merging the algebraic richness of quaternions with the relativistic, hyperbolic geometry of the boostlet system. By treating coupled physical phenomena such as acoustic pressure with particle velocity or orthogonally polarized elastic displacements as single quaternion-valued entities, the QBT preserves intrinsic geometric correlations that are typically lost in component-wise processing. We also establish a rigorous mathematical foundation for the transform, including the admissibility condition, a convolution-based representation, a Plancherel theorem for energy conservation, and an explicit inversion formula ensuring perfect signal reconstruction. Furthermore, the work derives a comprehensive set of uncertainty principles namely Heisenberg, logarithmic, and Pitt's inequalities that define the precise localization constraints of QBT coefficients in the augmented phase space. The theoretical development is substantiated with illustrative examples, wherein the QBT is applied to a quaternion-valued plane wave featuring coupled pressure-velocity components and to a Gaussian-modulated circularly polarized elastic wave packet. These examples demonstrate how the transform naturally encodes wavefront orientation and polarization state through quaternion phase, offering a physically coherent and sparse dictionary for vector-valued wavefield analysis in acoustics and seismology.

We explore what may be deduced about the neutrino mass ordering problem from the observation of core-collapse supernova burst neutrinos in modern terrestrial detectors. We employ ternary plots in a novel way to visualize the time evolution of the flavor composition of various supernova neutrino flux models from the SNEWPY software package. Through our analysis of several models using a simplified unfolding process, we have explored potential robust discriminants between the normal and inverted mass orderings. We find that the normal and inverted mass orderings tend to occupy different regions in ternary space across different models.

Programming is about automation in a wide variety of domains. Developing itself is one of those. As a side-effect, progress in automated coding may make people less willing to learn computer programming. This could become an issue, if the skill of computational problem solving is not only for the immediate economic benefit, but an important part of our knowledge about the world. We suggest that weakened incentives can be countered by lowering the entry barrier. We plan to shorten learning time by reducing the accidental complexity of the programming language and its runtime system. We describe a session plan that introduces programming and computing fundamentals for novices, assuming only basic mathematical background. This requires a non-mainstream, functional and concatenative language. This language, CON-CAT, is a by-product of research in category theory. It provides direct access to fundamental ideas like recursion and advanced ones like Gödel-encoding in an entertaining puzzle-like manner.

Commit signing is a principal mechanism for verifying the origin of code in software supply chains. Security frameworks treat it as a core trust signal, assuming developers sign their commits consistently with keys they control and keep those keys in good standing over time. Whether this assumption holds in practice has not been evaluated at ecosystem scale. This study addresses this gap. We present the first developer-centric, ecosystem-scale measurement of commit signing on GitHub, covering the platform's full history, spanning 71,694 active developers, 16.1 million commits, and 874,198 repositories. To summarize our findings: (1) overall signing adoption rates are misleading, as most signed commits come from automatic platform signing rather than deliberate developer action; (2) developers who do sign locally rarely keep it up consistently across repositories or over time; (3) signing lapse rates rise alongside account age rather than falling, making sustained coverage structurally unlikely; and (4) developers manage their signing keys poorly, leaving expired keys unrevoked and letting credential debt grow over time. Our findings show that the assumptions underlying supply-chain security frameworks do not hold in practice, establishing that progress requires either signing credentials redesigned to travel with developer identity while remaining outside platform control, or frameworks built on defenses that do not depend on every developer managing their own keys.