Let $(M, [g])$ be a $3$-dimensional conformal manifold. The space of knots $\mathrm{Kn}(M)$ in $M$ is an infinite-dimensional manifold that is known to carry an almost complex structure. This structure is formally integrable by a result of Brylinski. We study finite dimensional holomorphic submanifolds in $\mathrm{Kn}(M)$. We give a definition of an holomorphic family of knots in $(M, [g])$ parametrised by a finite-dimensional complex manifold $(X, I_X)$, and construct several families of examples. We show that the base $(X, I_X)$ is Kähler, and if $X$ is compact, it is a projective variety of complex dimension at most $2$. Finally, we prove that if an holomorphic family of knots in $(M, [g])$ over a compact base $(X, I_X)$ defines a foliation on the spherisation of the tangent bundle of $M$, then $(X, I_X) \simeq \mathbb{C}\mathbf{P}^1 \times \mathbb{C}\mathbf{P}^1$, the manifold $(M, [g])$ is conformally equivalent to either $S^3$ or $\mathbb{R}\mathbf{P}^3$ with round metric, and all knots are geodesic in some round metric in the class.

Potential functions are a key tool in theoretical computer science with applications ranging from the runtime analysis of algorithms and data structures, through the analysis of the expected behavior of random processes and search heuristics, to proving the existence of equilibrium states in strategic games. Typically, proofs that employ potential functions are short, elegant, and easy to verify, yet very powerful. Moreover, potential functions are essential ingredients for constructive proofs, in particular in algorithmic game theory. There, a key question is the existence of equilibrium states, but the most powerful theorem in the field -- Nash's theorem -- is unfortunately non-constructive. For many strategic games, potential functions come to the rescue by enabling constructive proofs that sometimes even yield efficient algorithms for finding equilibria. We add to this by providing a novel class of entropy-inspired log-multinomial potential functions for natural game-theoretic settings where rational agents of different types strategically choose actions to maximize their utility. In particular, we consider utility functions that are based on the fraction of same- and other-type agents taking the same action. We demonstrate the versatility of the new potential function class by presenting simple equilibrium existence proofs for two recent game-theoretic models, for which only involved technical proofs were previously known. Even better, the new potential function class yields efficient algorithms for constructing equilibria for much more general models. Thereby, we positively resolve several open problems.

Most TinyML hardware accelerators focus on supporting Quantized Neural Networks (QNNs) to meet stringent constraints on power consumption and size. Despite this, the security aspects of quantization within TinyML hardware remain largely unexplored. Although previous studies indicate that QNNs demonstrate similar or enhanced robustness when compared to full-precision Deep Neural Networks (DNNs) against typical evasion attacks, no attack strategies tailored specifically for TinyML hardware have been proposed yet. This paper addresses this shortfall by demonstrating how a two-step attack pipeline can surpass the current state-of-the-art in the QNN context and shows the need for more hardware-aware security research.

A linear optimal control law for orbital station-keeping in the Earth-Moon Restricted Three Body Problem (R3BP) is developed via Linear Quadratic Regulator (LQR) theory. First, the cost function is established considering a periodic state-weight matrix, leveraging stability information of the target orbits retrieved through Floquet theory. Then, the resulting periodic Riccati differential equation is solved and local asymptotic stability guarantees are shown. Finally, the performance of the proposed LQR when tracking periodic orbits in the circular and elliptic R3BPs is analyzed numerically.

We introduce version 1.0 of the SmoQyElPhQMC package, an open-source Julia code for performing scalable quantum Monte Carlo simulations of electron-phonon coupled model Hamiltonians. SmoQyElPhQMC is built upon the SmoQyDQMC codebase and implements improved versions of the algorithms presented in [B. Cohen-Stead \textit{et al}., Phys. Rev. E {\bf 105}, 065302 (2022)] to enable linear-scaling simulations of a broad class of uncorrelated $e$-ph models both in system size and inverse temperature. By extending the functionality of the flexible scripting interface introduced in SmoQyDQMC, the SmoQyElPhQMC package continues to allow users to adapt it to different workflows and interface with other software packages in the Julia ecosystem. The code for this package can be downloaded from our GitHub repository at this https URL or installed using the Julia package manager. The online documentation, including examples, can be obtained from our documentation page at this https URL.

We investigate the optical response of Kekulé-Y graphene under uniaxial strain and carrier doping. Using a low-energy effective Hamiltonian, we show that strain reshapes the low-energy electronic structure of the Kekulé-Y phase and induces Van Hove singularities at energies well below those of pristine graphene. Within the Kubo formalism, we calculate the optical conductivity and identify multiple anisotropic interband features, with a pronounced resonance arising from strain-induced Van Hove singularities. The pronounced resonance is strongly anisotropic and robust against moderate thermal broadening and disorder, providing a clear optical signature of Kekulé-Y ordering. We further derive analytical expressions for the low-energy optical conductivity and the Drude weight, providing a detailed characterization of the strain- and doping-dependent optical response. Our results establish strain engineering as an effective route for controlling valley-dependent optical properties in Kekulé-Y graphene, originating from the Kekulé-induced coupling of the Dirac valleys, and suggest feasible optical probes for the experimental identification of the Kekulé-Y phase.

A locally polyhedral manifold $M$ is a manifold which is locally diffeomorph to open subsets of convex polytopes. Manifolds with corners are a special case of locally polyhedral manifolds. We turn the group $\text{Diff}^{\infty}(M)$ of all smooth diffeomorphisms $f\colon M\to M$ for compact smooth $M$ into a regular Lie group.

Quantum information science has traditionally relied on nonclassical resources, such as entangled photon pairs and squeezed states, to achieve measurement performance beyond classical limits. Here, we revisit the multi-pass photonic scheme reported in Nature 450, 393 (2007) to clarify the physical origin of the observed superresolution and the associated claim of supersensitivity. Using a rigorous Jones-matrix formalism, we show that the round-trip evolution of the HQMQ linear optics unit is equivalent to the product of two reflections in polarization space, resulting in an effective rotation operator. This equivalence reveals that the accumulated phase arises from coherent polarization-state rotation on the Poincare'e sphere. The resulting phase accumulation is interpreted geometrically as a progressive realignment of the polarization state during successive forward and backward propagations. To validate the theoretical model, a classical-wave implementation is experimentally conducted, analyzed, and compared with the corresponding Jones-matrix solution. Finally, the scaling behavior of the Fisher information is analyzed to examine the origin of the claimed supersensitivity. The results are further compared with a recently developed coherence de Broglie wavelength framework, which achieves identical superresolution through repeated coherent interactions in a cascaded interferometeric architecture.

This paper investigates the impact of reciprocity calibration errors on the downlink spectral efficiency (SE) of multi-user large antenna systems. Specifically, we consider two calibration approaches: (a) global calibration, in which all antennas (can be distributed access-points (APs)) in the system cooperatively perform calibration, and (b) local calibration, wherein only a subset of antennas involved in downlink beamforming performs calibration. We derive the downlink SE considering the use-and-then-forget bound and side-information bound, and then demonstrate that, when downlink pilots are employed (in the case of side-information bound), the global calibration outperforms local calibration for arbitrary calibration topologies.

High\text{--}resolution scientific data, such as geomagnetic index streams, often exhibit complex temporal dependencies that can be modeled through functional data analysis. Conventional functional time series (FTS) methods typically partition continuous processes into non-overlapping segments, which artificially fragments temporal continuity and can limit estimation efficiency and stability. This is particularly evident in geomagnetic time series prediction due to their noisy, sudden, and large\text{--}scale changes. This study presents a robust multivariate FTS forecasting framework for multi\text{--}dimensional time series with inter\text{--}series correlations and the existence of exogenous predictors. We introduce an overlapping rolling\text{--}window scheme that preserves temporal coherence and mitigates boundary information loss, thereby enriching the effective sample size for a more efficient and stable estimation. We integrate functional principal component analysis for dimension reduction with a vector autoregressive model with exogenous inputs to capture latent dynamics across correlated series. We also construct computationally efficient conformal prediction intervals for uncertainty quantification. The framework is motivated by and applied to the simultaneous forecasting of five critical geomagnetic indices, Kp, Dst, SYM\text{--}H, SME, and SMR, using solar wind parameters as predictors. Empirical results show that this approach outperforms state\text{--}of\text{--}the\text{--}art machine learning baselines, extends forecast horizons to 6\text{--}24 hours, and provides calibrated uncertainty bounds.

Vizing's conjecture from 1963, considered by many the most important open problem in the field of graph domination, states that all graphs $G$ and $H$ satisfy $$\gamma(G\square H)\ge \gamma(G)\gamma(H),$$ where $\gamma$ denotes the domination number and $\square$ the Cartesian product. In a seminal result, Clark and Suen (2000) proved an approximate form of the conjecture, namely that $\gamma(G\square H)\ge \frac{1}{2}\gamma(G)\gamma(H)$ for all graphs $G$ and $H$. Despite several lower-order improvements of this bound and improvements for special classes of graphs $G$ and $H$, no absolute constant $c>\frac{1}{2}$ such that $\gamma(G\square H)\ge c\gamma(G)\gamma(H)$ for all graphs $G$ and $H$, has been known thus far. In this paper, we obtain the first constant-factor improvement of the Clark-Suen bound by proving that for all graphs $G$ and $H$, we have $$\gamma(G\square H)\ge c\gamma(G)\gamma(H),$$ where $$c=\frac{5+\sqrt{73}}{24}\approx 0.5643.$$ Along the way, we prove another lower bound on $\gamma(G\square H)$ which outperforms the above bound for many graphs.

We show that \textsc{number of bistellar moves and sparse degree-two edge collapses for 3-sphere} is NP-hard. It follows that a similar problem for an arbitrary 3-manifold is NP-hard as well. This is the first NP-hardness result concerning moves between two triangulations of a 3-manifold.

Reciprocity-based downlink beamforming is imperative for a scalable time-division duplex massive multiple-input multiple-output~(MIMO) deployment. Specifically, for a dual-antenna repeater-assisted massive MIMO system, a mismatch between forward and reverse path gains at the repeater can exacerbate the overall calibration error between the user equipments (UEs) and the base station (BS), which potentially also contains calibration errors of their individual radio-frequency chains. This paper models the effects of such calibration errors, underpins the relations between the uplink and downlink channels for repeater-assisted systems with calibration errors clubbed with the over-the-air channel estimation errors, and derives analytical expressions of the downlink spectral efficiency. The presented results can then be simplified to several special cases, underscoring situations wherein such errors can become pronounced.

We design and evaluate Fabula, an interactive app for fiction writers. Fabula uses detailed narrative plans informed by general narratological theory. Stories are structured hierarchically into scenes and beats that can be (re)generated and revised at script and story plan level. Using participatory AI, we critically evaluate and improve Fabula with casual and published writers, via design interviews and writing sessions with 42 experts, and large-scale internal and external testing. We interrogate our design choices: (1) whether a language model-based auto-evaluator, optimized on human experts' preferences, can improve story quality, (2) whether users want UI that exposes the detailed narrative plan alongside the story script, (3) to what extent our narratology assumptions fit localised storytelling traditions and serve screenwriters or playwrights, and (4) whether convergent iteration over the story plan supports writers' creativity. Building on critical feedback and concerns, we use Fabula as a cultural probe in adversarial design, and identify potentials for writing feedback and for interactive storytelling.

Recent results have shown that domain-uniform stabilizability and detectability imply spatial exponential decay of perturbations in optimally controlled hyperbolic PDEs on one-dimensional domains. This domain-uniform stabilizability and detectability can be achieved only if the control domain is distributed over the whole spatial domain such that the distance between neighboring control intervals is bounded from above. Motivated by these insights we investigate whether analogous effects can be observed for optimal boundary control in higher dimensions. For this purpose we conduct a simulation study on a two-dimensional wave equation on expanding square domains which is driven by a localized perturbation of the initial displacement. We compare two control geometries: a control acting only on the outer boundary and a control acting on a regular grid of interior line interfaces combined with the boundary. The problem is discretized by conforming P1 finite elements in space and the implicit midpoint rule in time. Instead of assembling the full space-time KKT system, we use a condensed formulation and solve the reduced optimality system by preconditioned conjugate gradients. We show that domain-uniform spatial decay of the perturbation can only be observed in the scenario with a regular grid of line controls. This is because finite propagation velocity requires a uniform bound on the distance that a wave can travel without reaching the control domain.

Hong-Ou-Mandel (HOM) interference has become a central resource for quantum sensing and metrology owing to its sensitivity to temporal delay and photon indistinguishability. However, existing HOM-based sensing schemes generally rely on inserting a sample into one arm of the interferometer, making the measurement vulnerable to optical loss, alignment instability, and bandwidth-dependent distortion of the interference profile. Here, we demonstrate a symmetry-controlled quantum sensing scheme based on continuous transitions between symmetric (bosonic-like) and antisymmetric (fermionic-like) Bell states in two-photon interference. By imprinting a geometric phase onto the classical pump beam and transferring it to polarization-entangled photons generated via spontaneous parametric down-conversion, we coherently tune the exchange symmetry of the entangled state without altering the temporal or spectral indistinguishability of the photons. The HOM response evolves continuously from bunching to antibunching with a sine square phase dependence, producing a coincidence modulation of approximately 10 * 10^4 counts s^-1 counts/s. In contrast to conventional HOM sensing, the phase-modulation linewidth remains fixed at pi/2, independent of photon bandwidth. Using a birefringent crystal placed directly in the pump beam, we measure thermo-dispersive birefringence with a resolution of the order of 10^{-6} over a broad temperature range. Our results establish exchange symmetry as a controllable resource for robust quantum sensing and symmetry-engineered photonic quantum information processing.

We prove that, for a classically free vertex algebra, the $C_2$-algebra is naturally isomorphic to the associated graded algebra of Zhu's algebra. As an application, we show that non-trivial holomorphic VOAs of CFT type are not classically free. This includes the affine vertex operator algebra of $E_8$ at level one, as well as the Moonshine VOA.

Clinical decision-making is augmented by decision-support systems (DSSs). To counter overreliance on DSSs, several methods have been proposed that create friction in order to promote cognitive engagement and reflection. In this paper, we investigate how two such forms of friction, namely data-driven questions and `what-if' analysis, are perceived by medical experts. For a real-world decision task, we replicated a DSS used in clinical practice and gathered clinicians' feedback on a prototype through in-situ interviews (n=7). Our findings suggest that while the questions were perceived as unhelpful for reflective thinking, they could serve as reminders to consider relevant information. Furthermore, inspecting `what-if' hypotheticals was found useful for potentially improving patient care. Clinicians saw our prototype as a promising training tool for novice clinicians. From the clinicians' feedback, we make recommendations for designing friction in work practices. Our work contributes to human-AI interaction research, which aims to encourage reflection to mitigate AI overreliance.

A key challenge of gravitational-wave physics is distinguishing the nature of compact objects involved in binary coalescences, particularly whether they are black holes or neutron stars. Neutron stars are distinguished from black holes by a stronger tidal response, with both static and dynamical aspects directly linked to their rich internal physics. Measurements of the tidal response through gravitational observations constrains the neutron-star equation of state and provides insight into the physics of high-density matter. However, defining the tidal response of neutron stars in general relativity is challenging due to coordinate ambiguities and the complexity of connecting the star's response to binary dynamics and the associated gravitational waveforms. In this paper, we show how the dynamical tidal response of a neutron star can be systematically defined within the worldline effective field theory (EFT) framework, connecting the problem to gravitational-wave scattering off an isolated neutron star. These scattering amplitudes are computed both within the EFT, using standard quantum field-theory techniques, and within stellar perturbation theory (the corresponding ultraviolet theory), where the coupled metric and matter perturbation equations are solved in the stellar interior within general relativity and matched to the analytical Mano-Suzuki-Takasugi (MST) solutions in the exterior. We match the scattering amplitude between effective theory and the ultraviolet theory to obtain the dynamical tidal response. We show the result to be consistent with known expectations, such as the static limit and the behaviour near the neutron star's resonant modes, while also recovering the imaginary part of the dominant oscillation mode induced by gravitational-wave dissipation. We conclude with a discussion of potential future improvements within both the EFT and the perturbation theory.

Recent advances in quantum computing pose a looming threat to most current Byzantine fault-tolerant (BFT) consensus protocols, which rely on quantum-vulnerable public-key signature schemes such as Ed25519 and BLS12-381. Instead of switching to much more expensive post-quantum secure signature schemes, an alternative is to use signature-free protocols, which rely only on cheap, post-quantum secure authenticated channels. In this paper, we ask whether signature-free BFT consensus protocols can match the performance of current state-of-the-art, quantum-vulnerable BFT consensus protocols. While previous work on the Sailfish++ protocol showed that state-of-the-art throughput is attainable signature-free, the question of latency is still open. Several recent signature-free protocols have low latency in theory, but they are all very intricate, and no practical implementation has so far been presented. In this work, we propose Simple-IT, a new leader-based, signature-free BFT consensus protocol that achieves a theoretical latency of 4 message delays (one more than the optimum), and only 3 on its optimistic path. Crucially, Simple-IT is simple enough to be amenable to implementation and to practical optimizations such as speculative pipelining, and, as we show experimentally in a geo-distributed testbed, it achieves both throughput and latency competitive with state-of-the-art quantum-vulnerable protocols.

Effective modeling of endangered marine mammal species, such as the North Atlantic Right Whale, is critical for balancing marine conservation with the growing blue economy. Passive acoustic monitoring data collected by autonomous underwater vehicles provide new opportunities for localized marine species detection and oceanographic sensing, but introduce complex statistical challenges such as zero inflation, imperfect detection, and intricate dependence structures. In response, we propose the Deep Zero-Inflated Bernoulli (DeepZIB) model--a deep statistical method which jointly models latent species presence and conditional detection probabilities while learning complex habitat relationships from heterogeneous covariate information. We establish theoretical results on the model's structural properties and conduct simulation experiments to demonstrate its ability to recover underlying parameters and latent presence fields. Application to real-world passive acoustic monitoring data on the North Atlantic Right Whale along the U.S. East Coast demonstrates improved model adequacy and predictive performance in capturing the species' dynamic and spatially varying habitat. A key advantage of DeepZIB is its ability to generate high-resolution, spatially and temporally varying presence maps, providing valuable insights for targeted and risk-aware management of blue economy industries, ranging from offshore and marine energy, to fisheries management and maritime transport.

Electrochemical doping offers dynamic control of the electronic properties of organic semiconductors, and it is the enabling feature of a range of technologies, including electrochemical transistors, energy-storage devices, light-emitting electrochemical cells (LECs), and bioelectronics. Electrochemical doping is commonly controlled by the selection of the constituents in the active material of the device or the applied voltage bias, but herein we report that the incorporation of Au nanoparticles (Au-NPs) at an electrode interface can constitute an alternative control parameter. The LEC features balanced p- and n-type electrochemical doping that forms a p-n junction doping structure in its active material, and we find that it is possible to reshape this doping profile by incorporating Au-NPs at an electrode interface. Specifically, we establish that the inclusion of neat non-capped Au-NPs at the anodic interface shifts the p-n junction (i.e., the emission zone) away from the anode. In contrast, the inclusion of Au-NPs capped with sodium citrate is found to reverse this behavior, so that the emission zone is instead moved towards the anode. We utilize this control parameter to shift the emission zone towards a position of constructive (destructive) interference, as manifested in a strong increase (decrease) of the LEC emission efficiency. Our findings establish an interfacial strategy for modulating the spatial profile of electrochemical doping and tuning device performance without altering the chemistry of the active material, relying instead on the surface modification of one electrode. This approach is important because it provides a versatile and minimally invasive route to optimize electrochemical devices while preserving the intrinsic properties and formulation of the active material.

This article studies the stability of feedback interconnections of linear time-invariant systems based on frequency-dependent passivity indices. Using these frequency-dependent passivity indices, we show that the feedback interconnection of two systems can be certified to be stable even if both systems have a lack of passivity in terms of their scalar passivity indices. The main contribution of this paper is a new stability theorem based on frequency-dependent passivity indices. Moreover, we discuss the connection of the proposed feedback stability theorem to prior results based on scalar passivity indices. A numerical case study showcases the advantages of frequency-dependent passivity indices over scalar indices for feedback interconnections of linear systems.

This paper investigates the fundamental limits and optimal signal distribution design for Integrated Sensing and Communication (ISAC) systems operating under strictly noncoherent conditions. Unlike conventional coherent frameworks that rely on perfect channel state information, we consider a block-fading MIMO channel where the channel realizations are unknown to both the transmitter and the receiver. We adopt a realization-wise perspective to characterize the noncoherent performance tradeoff across different signal-to-noise ratio (SNR) regimes. In the high-SNR regime, we derive a lower bound for the noncoherent mutual information and define a metric, termed sensing-induced rate loss, to quantify the communication penalty incurred by sensing-oriented beamforming. We then employ a projected gradient algorithm to optimize the spatial power allocation, balancing the conflict between the unitary space-time modulation-based structure for communication and the task-oriented spatial power allocation for sensing. Conversely, in the low-SNR regime, we perform a first-order asymptotic analysis of the ergodic minimum mean squared error (EMMSE). Our theoretical derivation reveals a fundamental synergy: the sensing-optimal strategy collapses to a rank-one transmission along the dominant eigenvector of the target response, which incurs no first-order communication loss in the low-SNR regime. This result demonstrates that the conflicting tradeoff observed at high SNR vanishes asymptotically at low SNR, enabling perfect alignment between sensing and communication objectives.

This paper continues our previous work on duality for Anderson $t$-modules. We study two dimensional triangular t-modules in a specific reduce form. Computer - assisted symbolic computations with matrices over a skew field revealed a reduction pattern for t-modules satisfying the ALD condition. Using this pattern, we prove that such t-modules are isomorphic to their double duals and extend the validity of the Cartier-Nishi theorem and the Weil-Barsotti formula to a substantially broader class of t-modules.

We present a morphokinematic analysis based on high-resolution long-slit echelle spectroscopy of the \nii$\lambda6583$ line and narrowband imaging. Position-velocity diagrams reveal asymmetric expansion and localized kinematic features. We derive a systemic velocity of $V_{\rm sys}^{\rm LSR} = -25\pm1$\kms\ ($V_{\rm sys}^{\rm HEL} = -34 \pm 1$\kms) and a main shell expansion velocity of $V_{\rm exp} = 22 \pm 1$\kms. Three-dimensional modeling indicates an ellipsoidal main body surrounded by a thin shell, two ear-like protrusions, and additional small-scale structures. The corresponding kinematic ages are $3\,600 \pm 700$ yr for the ellipsoid and ring, and $7\,500 \pm 1\,000$ yr and $8\,800 \pm 1\,500$ yr for the two opposite ear-like protrusions, respectively, indicating that these outer structures predate the main nebular envelope. The kinematic asymmetry and enhanced emission regions suggest evolution within a non-uniform ambient medium. At the same time, the presence of collimated ear-like structures is consistent with shaping influenced by binary interaction, where earlier outflows preceded the ejection of the dense shell. NGC\,6563 therefore appears to be a dynamically evolved system shaped by the combined effects of episodic mass ejection and environmental interaction.

In contemporary IoT edge devices with real-time requirements, security is primarily enforced through design-time parameters associated with security tasks, leading to mechanisms that operate in an \emph{opportunistic} manner. As a result, security checks are often performed as secondary operations. This approach can result in systems where no security tasks are executed due to high utilization by other tasks. An alternative approach taken in prior work is to add security mechanisms to every task in the system, resulting in substantially lower performance than that of a system with no security. These approaches have resulted in an \emph{all-or-nothing} scenario for edge device security, motivating numerous studies on the safety-security trade-off in real-time cyber-physical systems (RT-CPS). This study introduces an analytical framework -- REPOSE -- for evaluating the security feasibility of real-time control systems at runtime. REPOSE is developed for \textit{weakly-hard} real-time control systems that facilitate a ``bounded trade-off'' between safety and security. In contrast to imposing additional (pessimistic) design-time overhead as considered in some real-time security literature, REPOSE performs security operations in both \textit{proactive} and \textit{reactive} manners based on the task's current behavior. Our evaluations show that REPOSE can effectively add security operations to RT-CPS with a feasibility overhead of $0.06\%$ at $80\%$ utilization, compared to a $ 29\%$ overhead observed in systems with hard constraints. Through a case study of a classic control system, we also demonstrate that REPOSE provides a robust framework to \textit{analyze and calculate} the safety-security tradeoff.

Public blockchains continue to struggle with scalability because improving throughput is not as simple as increasing block size or reducing block interval. Larger blocks increase validation and transmission cost, while shorter intervals raise the likelihood of propagation delays, forks, and stale blocks. These limits motivate sharding, where transaction processing is divided across multiple parallel shard groups. In this work, we present a configurable SimPy-based discrete-event simulator for evaluating sharded blockchain architectures under controlled workload and network assumptions. The simulator models mining, verification, inter-shard coordination, block dissemination, measured throughput, average block time, and communication overhead. Our simulator achieves 1.6M TPS at 256 shards under a local datacenter-like setup and 0.6M TPS in a global WAN setup, showing strong throughput gains from parallel execution. However, the gains are not unbounded: beyond a certain number of shards, coordination traffic, synchronization, and network overhead begin to dominate, leading to diminishing returns.

We present bijections connecting tree records, the girth of a connected endofunction, and the genesis sequence (the first sequence in OEIS). Using these, we derive generating functions for tree and forest record numbers in terms of Cayley's tree function and give a new proof of Cayley's forest formula.

In the late 20th century, the Hubble Space Telescope (HST) revolutionized astronomy, showing the Universe with a detail never seen before in the ultraviolet (UV), optical and infrared (IR) bands. In the early 2020s, the James Webb Space Telescope started a similar revolution in the IR. The launch of Roman in late 2026 challenges the reign of HST in the optical band, but even after Roman's launch, HST will remain as the only telescope capable of high-quality imaging in the UV band. In the optical bands, HST provides superior resolution than Roman for point sources. Although equipped with more sensitive MOS, Roman's sensors have a pixel size about 3 times larger than HST's CCDs, hence undersampling the point-spread-function, and resulting in a worse spatial resolution. The UV-capable and higher-resolution of HST in the UV and optical band, makes HST the best instrument for specific science cases. This paper responds to the "Building a Roadmap for Hubble science into the 2030s" call and focuses on science with lensed hot stars at $z>0.5$ in the UV and optical bands, exploiting the features that makes HST the best instrument in the UV/optical until the launch of the Habitable World Observatory in the 2040s.