This research demonstrates high-repetition-rate laser-accelerated ion beams via dual, intersecting, counterpropagating laser-driven blast waves to precisely shape underdense gas into long-lived near-critical density targets. The collision of the shock fronts compresses the gas and forms steep density gradients with scale lengths of a few tens of microns. The compressed target persists for several nanoseconds, eliminating laser synchronization constraints. Measurements of multi-MeV ion energy spectra are reported. 3D hydrodynamic simulations are used to optimize the density profile and assess the influence of the Amplified Spontaneous Emission of the femtosecond accelerating laser pulse. A synthetic optical probing model is applied to directly compare simulations with experimental data. 3D Particle-In-Cell simulations reveal the formation of multi-kT, azimuthal magnetic fields, indicating Magnetic Vortex Acceleration as the main acceleration mechanism.

Cosmic strings, topological defects predicted by high-energy theories, may contribute to the late-time expansion of the Universe, effectively mimicking dynamical dark energy. We investigate four phenomenological extensions of the $Λ$CDM model involving a residual string network: (i) a non-relativistic component with positive energy density (Model~1), (ii) a velocity-dependent extension (Model~2), (iii) a non-relativistic string network with energy density allowed to take both positive and negative values (Model~3), and (iv) a general scenario with free energy and velocity parameters (Model~4). These models are constrained using \textit{Planck} CMB data, SDSS or DESI baryon acoustic oscillations, and Type Ia supernovae observations. Models~1 and~2 yield strong upper bounds on the string density, for example, $Ω_{\mathrm{s}} < 0.00901$ at 95\% CL from the CMB+DESI+DESY5 combination for Model~2, and mildly shift the inferred value of $H_0$ upward, though they are not favored by Bayesian evidence. For the same combination, the bulk velocity is bound as $v_{\mathrm{s}} < 0.569$. Models~3 and~4 exhibit a consistent preference for slightly negative values of $Ω_{\mathrm{s}}$, with CMB-only data yielding $Ω_{\mathrm{s}} = -0.038^{+0.029}_{-0.022}$ and $v_{\mathrm{s}}< 0.574$ in Model~4, and a best-fit improvement of $Δχ^2 = -6.07$. However, these improvements are not sufficient to overcome the Occam penalty, and the Bayesian evidence continues to favor $Λ$CDM. These findings demonstrate the power of current data to constrain exotic energy components and encourage further exploration of string-inspired extensions to $Λ$CDM, particularly those involving negative-tension networks.

The Generalized Independent Set (GIS) problem extends the classical maximum independent set problem by incorporating profits for vertices and penalties for edges. This generalized problem has been identified in diverse applications in fields such as forest harvest planning, competitive facility location, social network analysis, and even machine learning. However, solving the GIS problem in large-scale, real-world networks remains computationally challenging. In this paper, we explore data reduction techniques to address this challenge. We first propose 14 reduction rules that can reduce the input graph with rigorous optimality guarantees. We then present a reduction-driven local search (RLS) algorithm that integrates these reduction rules into the pre-processing, the initial solution generation, and the local search components in a computationally efficient way. The RLS is empirically evaluated on 278 graphs arising from different application scenarios. The results indicates that the RLS is highly competitive -- For most graphs, it achieves significantly superior solutions compared to other known solvers, and it effectively provides solutions for graphs exceeding 260 million edges, a task at which every other known method fails. Analysis also reveals that the data reduction plays a key role in achieving such a competitive performance.

We study the asymptotic distribution of norm ball averages along orbits of a lattice $Γ\subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in $\mathbb{R}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.

We consider the problem of estimating the partition function $Z(β)=\sum_x \exp(β(H(x))$ of a Gibbs distribution with the Hamiltonian $H:Ω\rightarrow\{0\}\cup[1,n]$. As shown in [Harris & Kolmogorov 2024], the log-ratio $q=\ln (Z(β_{\max})/Z(β_{\min}))$ can be estimated with accuracy $ε$ using $O(\frac{q \log n}{ε^2})$ calls to an oracle that produces a sample from the Gibbs distribution for parameter $β\in[β_{\min},β_{\max}]$. That algorithm is inherently sequential, or {\em adaptive}: the queried values of $β$ depend on previous samples. Recently, [Liu, Yin & Zhang 2024] developed a non-adaptive version that needs $O( q (\log^2 n) (\log q + \log \log n + ε^{-2}) )$ samples. We improve the number of samples to $O(\frac{q \log^2 n}{ε^2})$ for a non-adaptive algorithm, and to $O(\frac{q \log n}{ε^2})$ for an algorithm that uses just two rounds of adaptivity (matching the complexity of the sequential version). Furthermore, our algorithm simplifies previous techniques. In particular, we use just a single estimator, whereas methods in [Harris & Kolmogorov 2024, Liu, Yin & Zhang 2024] employ two different estimators for different regimes.

A programmable linear resistor with a compact footprint would have profound implications for microelectronics, enabling efficient in-sensor analog signal processing and in-memory computing. Non-volatile memory offers a potential solution but suffers from limitations due to the programming mechanisms that confine switching to nanoscale constrictions or field-sensitive semiconductor junctions, leading to non-linear current-voltage relationships and errors. Here, we introduce a tunable resistor that is programmed into non-volatile, high-precision resistance states spanning nine orders of magnitude, with linear current-voltage characteristics across the entire range -- significantly improving the performance and widening the application space of resistive memory. A key advance is an electrothermal gate that simultaneously spreads heat and electrochemical reactions during programming to enable large, bulk composition modulation. The volumetric modulation can host thousands of linear resistance states with 100x lower conductance errors than other memory. This enables direct processing of analog signals with high fidelity, and we demonstrate variable-gain amplification, division, and multiplication. Integration with CMOS is used to show resilience to electrical and thermal disturb in arrays and to demonstrate retention of analog levels at <1% average loss for more than 2 months across 100 devices. Simulations indicate matrix multiplication efficiency could approach >1,000 TOPS/W.

The Josephson junction is the fundamental nonlinear building block of superconducting quantum technologies. Its macroscopic quantum tunneling physics underpins superconducting quantum computing, sensing, and communication, but scaling these platforms to utility-scale architectures places increasingly stringent demands on junction materials, interfaces, and fabrication. In quantum computing, these demands include high reproducibility, low dissipation, tunability, compact device footprint, and resilience to noise and defects. This review surveys how advances in materials science, device characterization, and nanofabrication are addressing these challenges and redefining the figures of merit for next-generation Josephson junctions. We also examine the evolution of fabrication strategies, from conventional multi-angle evaporation to foundry-compatible superconducting processes and the integration of emerging junction materials. Progress along these directions will determine how rapidly Josephson junctions move from laboratory-scale components to the foundation of industrial-scale quantum processors.

Intent-based network (IBN) is a promising solution to automate network operation and management. IBN aims to offer human-tailored network interaction, allowing the network to communicate in a way that aligns with the network users' language, rather than requiring the network users to understand the technical language of the network/devices. Nowadays, different applications interact with the network, each with its own specialized needs and domain language. Creating semantic languages (i.e., ontology-based languages) and associating them with each application to facilitate intent translation lacks technical expertise and is neither practical nor scalable. To tackle the aforementioned problem, we propose a context-aware AI framework that utilizes machine reasoning (MR), retrieval augmented generation (RAG), and generative AI technologies to interpret intents from different applications and generate structured network intents. The proposed framework allows for generalized/domain-specific intent expression and overcomes the drawbacks of large language models (LLMs) and vanilla-RAG framework. The experimental results show that our proposed intent-RAG framework outperforms the LLM and vanilla-RAG framework in intent translation.

Several theoretical models predict that spatial patterning increases ecosystem resilience. However, these predictions rely on simplifying assumptions, such as assuming isotropic and infinitely large ecosystems, and empirical evidence directly linking spatial patterning to enhanced resilience remains scarce. We introduce a unifying framework, encompassing existing models for vegetation pattern formation in water-stressed ecosystems, that relaxes these assumptions. This framework incorporates finite vegetated areas surrounded by desert and anisotropic environmental conditions that lead to non-reciprocal plant interactions. Under these more realistic conditions, we identify a novel desertification mechanism, known as nonlinear convective instability in physics but largely overlooked in ecology. These instabilities form when non-reciprocal interactions destabilize the vegetation-desert interface and can trigger desertification fronts even under stress levels where isotropic models predict stability. Importantly, ecosystems exhibiting periodic vegetation patterns are more susceptible to nonlinear convective instabilities than those with homogeneous vegetation, suggesting that spatial patterning may reduce, rather than enhance, resilience. These findings challenge the prevailing view that self-organized patterning enhances ecosystem resilience and provide a new framework for investigating how spatial dynamics shape the stability and resilience of ecological systems under changing environmental conditions.

We introduce a novel approximation to the same marginal Schrödinger bridge using the Langevin diffusion. As $\varepsilon \downarrow 0$, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schrödinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is $\varepsilon$ times the gradient of the marginal log density (i.e., the score function), in $\mathbf{L}^2$. More generally, we show that the family of Markov operators, indexed by $\varepsilon > 0$, derived from integrating test functions against the conditional density of the static Schrödinger bridge at temperature $\varepsilon$, admits a derivative at $\varepsilon=0$ given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures.

Symmetry-protected topological (SPT) phases are fundamental features of cluster states, serving as key resources for measurement-based quantum computation (MBQC). Generating large-scale cluster states and verifying their SPT phases are essential steps toward practical MBQC, which however still presents significant experimental challenges. In this work, we address these challenges by utilizing advanced superconducting hardware with optimized gate operations, enhanced readout fidelity, and error mitigation techniques. We successfully generate and verify 95-qubit one-dimensional and 72-qubit two-dimensional genuine entangled cluster states, achieving fidelities of $0.5603 \pm 0.0084$ and $0.5519 \pm 0.0054$, respectively. Leveraging these high-fidelity cluster states, we investigate SPT phases through quantum teleportation across all 95 qubits and demonstrate input-state-dependent robustness against symmetry-breaking perturbations, highlighting the practicality and intrinsic robustness of MBQC enabled by the SPT order. Our results represent a significant advancement in large-scale entanglement generation and topological phase simulation, laying the foundation for scalable and practical MBQC using superconducting quantum systems.

The non-Hermiticity of the system gives rise to a distinct knot topology in the complex eigenvalue spectrum, which has no counterpart in Hermitian systems. In contrast, the singular values of a non-Hermitian (NH) Hamiltonian are always real by definition, meaning that they can also be interpreted as the eigenvalues of some underlying Hermitian Hamiltonian. In this work, we demonstrate that if the singular values of an NH Hamiltonian are treated as eigenvalues of prototype translational invariant Hermitian models that undergo a topological phase transition between two distinct topological phases, the complex eigenvalues of the NH Hamiltonian will also undergo a {\it{first order knot transition}} between different knot structures. Unlike the usual knot transition, this transition is not accompanied by an Exceptional point (EP); in contrast, the real and complex parts of the eigenvalues of the NH Hamiltonian show a discrete jump at the transition point. We emphasize that the choice of an NH Hamiltonian whose singular values match the eigenvalues of a Hermitian model is not unique. However, our study suggests that this connection between the NH and Hermitian models remains robust as long as the periodicity in lattice momentum is the same for both. Furthermore, we provide an example showing that a change in the topology of the Hermitian model implies a transition in the underlying NH knot topology, but a change in knot topology does not necessarily signal a topological transition in the Hermitian system.

The frequency $K_i$s ($i\in[4,n]$) are studied for symmetric traveling salesman problem ($TSP$) to characterize the structure properties of the edges in optimal Hamiltonian cycle ($OHC$). For a given $K_i$ in the complete graph $K_n$, the frequency $K_i$ is computed with the set of ${{i}\choose{2}}$ optimal $i$-vertex paths with fixed endpoints (optimal $i$-vertex paths) in the $K_i$. Given an $OHC$ edge related to $K_i$, it has certain frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ in the frequency $K_i$, and that of an ordinary edge not in $OHC$ is smaller than $2(n-3)$. Moreover, given a frequency $K_i$ containing an $OHC$ edge related to $K_n$, the frequency of the $OHC$ edge is bigger than $\frac{1}{2}{{i}\choose{2}}$ in the average case. It also found that the probability that an $OHC$ edge is contained in the optimal $i$-vertex paths increases according to $i\in [4, n]$ or keeps stable if it decreases from $i$ to $i+1\leq n$. As the frequency $K_i$s are used to compute the frequency of an edge, each $OHC$ edge reaches its own peak frequency at $i=P_0$ where $P_0=\frac{n}{2} + 2$ for even $n$ or $\frac{n+1}{2} + 1$ for odd $n$. For each ordinary edge out of $OHC$, the probability that they are contained in the optimal $i$-vertex paths decreases according to $i$, respectively, in the average case. Moreover, the probability of an ordinary edge definitely decreases if $i \geq i_d$ where $i_d = O(n^{\frac{4}{7}})$ is the smallest number meeting the inequality $\frac{(n-2)(n-3) - (i_d-2)(i_d-3)}{(n-2)(n-3) - (i_d-1)(i_d-2)} \geq \sqrt{1 + \frac{2}{i_d(i_d+1)}}$. Based on these findings, an algorithm is presented to find $OHC$ in $O(n^2i_d^42^{i_d})$ time using dynamic programming. The experiments are executed to verify these findings with various $TSP$ instances.

For a sequence of independent identically distributed random variables having a distribution function with an unknown parameter from a set $Θ\subset \mathbf{R}^d$, we prove an analogue of the lower bound of Bahadur asymptotic efficiency for the zone of moderate deviation probabilities. The assumptions coincide with assumptions conditions under which the locally asymptotically minimax lower bound of Hajek-Le Cam was proved. The lower bound for local Bahadur asymptotic efficiency is a special case of this lower bound.

Contrastive Learning (CL) enhances the training of sequential recommendation (SR) models through informative self-supervision signals. Existing methods often rely on data augmentation strategies to create positive samples and promote representation invariance. Some strategies such as item reordering and item substitution may inadvertently alter user intent. Supervised Contrastive Learning (SCL) based methods find an alternative to augmentation-based CL methods by selecting same-target sequences (interaction sequences with the same target item) to form positive samples. However, SCL-based methods suffer from the scarcity of same-target sequences and consequently lack enough signals for contrastive learning. In this work, we propose to use similar sequences (with different target items) as additional positive samples and introduce a Relative Contrastive Learning (RCL) framework for sequential recommendation. RCL comprises a dual-tiered positive sample selection module and a relative contrastive learning module. The former module selects same-target sequences as strong positive samples and selects similar sequences as weak positive samples. The latter module employs a weighted relative contrastive loss, ensuring that each sequence is represented closer to its strong positive samples than its weak positive samples. We apply RCL on two mainstream deep learning-based SR models, and our empirical results reveal that RCL can achieve 4.88% improvement averagely than the state-of-the-art SR methods on five public datasets and one private dataset.

Large language models have gained significant traction and popularity in recent times, extending their usage to code-generation tasks. While this field has garnered considerable attention, the exploration of testing and evaluating the robustness of code generation models remains an ongoing endeavor. Previous studies have primarily focused on code generation models specifically for the Python language, overlooking other widely used programming languages. In this work, we conduct a comprehensive comparative analysis to assess the robustness performance of several prominent code generation models and investigate whether robustness can be improved by repairing perturbed docstrings using an LLM. Furthermore, we investigate how their performance varies across different programming languages. To accomplish this, we introduce perturbations in four key areas of the prompt: DocString, function name, syntax, and format. We have compiled and released a dedicated dataset for this purpose. Our results show that all models consistently degrade under perturbations across all three languages, but vary in magnitude depending on the language and perturbation type. Larger model size does not reliably improve robustness, and semantic perturbations prove at least as disruptive as syntactic ones. Our LLM-based docString repair yields only marginal gains for simple perturbations and can degrade performance for semantic ones, highlighting the limits of prompt-level mitigation.

This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in 2D and edge values in 3D), in addition to the typical degrees of freedom in the mesh and on the hyperfaces in the HHO literature. This approach enables the characteristic commuting property of the hybrid high-order methodology in any space dimension. The main results are guaranteed lower eigenvalue bounds of higher order. Furthermore, we derive quasi-best approximation estimates as well as reliable and efficient a~posteriori error estimators under minimal regularity assumptions on the exact solution. The latter motivates an adaptive mesh-refining algorithm that empirically recovers optimal convergence rates for singular solutions.

In this article we find locally an eigenfunctions for a particular nonlinear hyperbolic differential operator $Δ_H u^{n}$, where $Δ_H$ is the hyperbolic Laplacian in the half-plane of Poincairé. We have proved that these eigenfunctions are an analytic and non-exact whose coefficients satisfy a specific algebraic recursive rule. The existence of these eigenfunctions allows us to find non-exact solutions respecting the spatial coordinate of nonlinear diffusive PDEs on the Poincairé half-plane, which could describe a possible one-dimensional physical model.

We investigate the role of policy heterogeneity in enhancing the olfactory search capabilities of cooperative agent swarms operating in complex, real-world turbulent environments. Using odor fields from direct numerical simulations of the Navier-Stokes equations, we demonstrate that heterogeneous groups, with exploratory and exploitative agents, consistently outperform homogeneous swarms where the exploration-exploitation tradeoff is managed at the individual level. Our results reveal that policy diversity enables the group to reach the odor source more efficiently by mitigating the detrimental effects of spatial correlations in the signal. These findings provide new insights into collective search behavior in biological systems and offer promising strategies for the design of robust, bioinspired search algorithms in engineered systems.

Disordered superconducting materials with high kinetic inductance are an important resource to generate nonlinearity in quantum circuits and create high-impedance environments. In thin films fabricated from these materials, the combination of disorder and the low effective dimensionality leads to increased order parameter fluctuations and enhanced kinetic inductance values. Among the challenges of harnessing these compounds in coherent devices are their proximity to the superconductor-insulator phase transition, the presence of broken Cooper pairs, and the two-level systems located in the disordered structure. In this work, we fabricate tungsten silicide wires from quasi-two-dimensional films with one spatial dimension smaller than the superconducting coherence length and embed them into microwave resonators and fluxonium qubits, where the kinetic inductance provides the inductive part of the circuits. We study the dependence of loss on the frequency, disorder, and geometry of the device, and find that the loss increases with the level of disorder and is dominated by the localized quasiparticles trapped in the spatial variations of the superconducting gap.

Integrated Sensing and Communication (ISAC) will be one key feature of future 6G networks, enabling simultaneous communication and radar sensing. The radar sensing geometry of ISAC will be multistatic since that corresponds to the common distributed structure of a mobile communication network. Within this framework, micro-Doppler analysis plays a vital role in classifying targets based on their micromotions, such as rotating propellers, vibration, or moving limbs. However, research on bistatic micro-Doppler effects, particularly in ISAC systems utilizing OFDM waveforms, remains limited. Existing methods, including electromagnetic simulations, often lack scalability for generating the large datasets required to train machine learning algorithms. To address this gap, this work introduces an OFDM-based bistatic micro-Doppler model for multi-propeller drones. The proposed model adapts the classic thin-wire model to include bistatic sensing configuration with an OFDM-like signal. Then, it extends further by incorporating multiple propellers and integrating the reflectivity of the drone's static parts. Measurements were performed to collect ground truth data for verification of the proposed model. Validation results show that the model generates micro-Doppler signatures closely resembling those obtained from measurements, demonstrating its potential as a tool for data generation. In addition, it offers a comprehensive approach to analyzing bistatic micro-Doppler effects.

In this work, we develop a system {\it plus} environment approach for interacting chiral molecules under a quantum-classical description of the spin-spin model using a Caldeira-Leggett-like coupling. After presenting the interacting model, we show that a long-ranged parity-nonconserving interaction, encoded within a non-linear Schrödinger equation, produces an energy difference between the two enantiomers of the central chiral molecule when it interacts with a chiral environment. Three examples of such interactions are considered, with particular focus on $Z^0$-photon vacuum polarization. Finally, we reveal a {\it chirality transmission effect} phenomenon, where the time-averaged population difference of the central molecule is amplified when the chiral asymmetry of the environment is considered.

Constraints on Wilson coefficients (WCs) corresponding to dimension-six operators of the standard model effective field theory (SMEFT) are determined from a simultaneous fit to seven sets of CMS measurements probing Higgs boson, electroweak vector boson, top quark, and multijet production. Masurements of electroweak precision observables are also included and provide complementary constraints to those from CMS experiment. The CMS measurements, using LHC proton-proton collision data at $\sqrt{s}$ = 13 TeV, corresponding to integrated luminosities of 36.3 or 138 fb$^{-1}$, are chosen to provide sensitivity to a broad set of operators, for which consistent SMEFT predictions can be derived. These are primarily measurements of differential cross sections which are parameterized as functions of the WCs. In measurements targeting $\text{t}(\bar{\text{t}})\text{X}$ production, SMEFT effects are modelled at the detector level. Individual constraints on 64 WCs, and constraints on 43 linear combinations of WCs, are obtained.

We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality à la Ben-Zvi--Sakellaridis--Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral of a Hecke eigensheaf. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin--Selberg case $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, we obtain a formula for higher derivatives of the Rankin--Selberg $L$-function. This provides a conceptual generalization of Yun--Zhang's higher Gross--Zagier formula to higher-dimensional spherical varieties.

Neural networks have proven practical for a synergistic combination of advanced control techniques. This work analyzes the implementation of rectified linear unit neural networks to achieve constrained control in differentially flat systems. Specifically, the class of flat systems enjoys the benefit of feedback linearizability, i.e., the systems can be linearized by means of a proper variable transformation. However, the price for linearizing the dynamics is that the constraint descriptions are distorted geometrically. Our results show that, by using neural networks, these constraints can be represented as a union of polytopes, enabling the use of mixed-integer programming tools to guarantee constraint satisfaction. We further analyze the integration of the characterization into efficient settings such as control Lyapunov function-based and model predictive control (MPC). Interestingly, this description also allows us to explicitly compute the solution of the MPC problem for the nonlinear system. Several examples are provided to illustrate the effectiveness of our framework.

Kazhdan and Lusztig introduced the $W$-graphs, which represent the multiplication action of the standard basis on the canonical bais in the Iwahori-Hecke algebra. In the Hecke algebra module, Marberg defined two generalied $W$-graphs, called the Gelfand $W$-graphs. The classification of the molecules of the type $A$ Gelfand $S_n$-graphs are determined by two RSK-like insertion algorithms. We finish the classification of cells by proving that every molecule in the $S_n$-graphs is indeed a cell.

We present an extension of local sensitivity analysis, also referred to as the perturbation approach for uncertainty quantification, to Bayesian inverse problems. More precisely, we show how moments of random variables with respect to the posterior distribution can be approximated efficiently by asymptotic expansions. This is under the assumption that the measurement operators and prediction functions are sufficiently smooth and their corresponding stochastic moments with respect to the prior distribution exist. Numerical experiments are presented to the illustrate the theoretical results.

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erdős and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible.

This study developed a new explainable artificial intelligence algorithm called PassAI, which classifies successful or failed passes in a soccer game and explains its rationale using both tracking and passer's seasonal stats information. This study aimed to address two primary challenges faced by artificial intelligence and machine learning algorithms in the sports domain: how to use different modality data for the analysis and how to explain the rationale of the outcome from multimodal perspectives. To address these challenges, PassAI has two processing streams for multimodal information: tracking image data and passer's stats and classifying pass success and failure. After completing the classification, it provides a rationale by either calculating the relative contribution between the different modality data or providing more detailed contribution factors within the modality. The results of the experiment with 6,349 passes of data obtained from professional soccer games revealed that PassAI showed higher classification performance than state-of-the-art algorithms by >5% and could visualize the rationale of the pass success/failure for both tracking and stats data. These results highlight the importance of using multimodality data in the sports domain to increase the performance of the artificial intelligence algorithm and explainability of the outcomes.

Given two populations from which independent binary observations are taken with parameters $p_1$ and $p_2$ respectively, estimators are proposed for the relative risk $p_1/p_2$, the odds ratio $p_1(1-p_2)/(p_2(1-p_1))$ and their logarithms. The sampling strategy used by the estimators is based on two-stage sequential sampling applied to each population, where the sample sizes of the second stage depend on the results observed in the first stage. The estimators guarantee that the relative mean-square error, or the mean-square error for the logarithmic versions, is less than a target value for any $p_1, p_2 \in (0,1)$, and the ratio of average sample sizes from the two populations is close to a prescribed value. The estimators can also be used with group sampling, whereby samples are taken in batches of fixed size from the two populations simultaneously, each batch containing samples from the two populations. The efficiency of the estimators with respect to the Cramér-Rao bound is good, and in particular it is close to $1$ for small values of the target error.