In this article we give several characterizations for various transitivity properties for linear operators. We define a general form of `Hypercyclicity Criterion' using a Furstenberg family $\mathcal{F}$ to characterize $\mathcal{F}$-transitive operators. In particular, we find an equivalent characterization for mixing operators. We study proximal and asymptotic relations for linear operators and prove that the difference between mixing operators and Kitai's Criterion can be presented through these relations. Finally, we find an equivalent characterization of strongly transitive abd strongly product transitive operators.

We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the two-dimensional case we can use a powerful tool of continued fractions. We develop an analog of Jarn\'ık's theory dealing with inequalities between the ordinary and uniform Diophantine exponents, which turned out to be related to mutual behaviour of irrationality measure functions for two real numbers.

The dual concepts of `universality' and `hypercyclicity' are better understood and studied as `topological transitivity'. In this article we consider transitivity properties of skew products, essentially with non-compact fibers. We study the `Universality Conditions' and `Hypercyclicity Criterion' associated with the dynamical properties of transitivity, weakly mixing and mixing for these skew products.

Many system management runtimes (SMRs), such as resource management and power management techniques, rely on quality-of-service (QoS) metrics, such as tail latency or throughput, as feedback. These QoS metrics are generally neither observable with hardware performance counters nor directly observable within the OS kernel. This introduces complexity and overhead in instrumenting the application and integrating QoS performance metric feedback with many management runtimes. To bridge this gap, we introduced eBeeMetrics, an eBPF-based library framework to accurately observe application-level metrics derived from only eBPF-observable events, such as system calls. eBeeMetrics can be used as a drop-in replacement to decouple system management runtimes from QoS metric feedback reporting, or can supplement existing QoS metrics to better identify server-side dynamics. eBeeMetrics achieves a strong correlation with real-world measured throughput and latency metrics across various latency-sensitive workloads. The eBeeMetrics tool is open-source; the source code is available at: https://github.com/Ibnathism/eBeeMetrics.

Capturing the dynamics of quantum many-body systems under time-dependent driving protocols is a central challenge for numerical simulations. Existing methods such as tensor networks and time-dependent neural quantum states, however, must be re-run for every protocol. In this work, we introduce the Neural Operator Quantum State (NOQS) as a foundation model for quantum dynamics. Rather than solving the Schrödinger equation for individual trajectories, our approach aims to \emph{learn the solution operator} that maps entire driving protocols to time-evolved quantum states. Once trained, the NOQS predicts time evolution under unseen protocols in a single forward pass, requiring no additional optimization. We validate NOQS on the two-dimensional Ising model with time-dependent longitudinal and transverse fields, demonstrating accurate prediction not only for unseen in-distribution protocols, but also for qualitatively different, out-of-distribution functional forms of driving. Further, a single NOQS model can be transferred between different temporal resolutions, and can be efficiently fine-tuned with sparse experimental measurements to improve predictions across all observables at negligible cost. Our work introduces a new paradigm for quantum dynamics simulation and provides a practical computational-experimental interface for driven quantum systems.

Tidal disruption events (TDEs) are powerful probes of supermassive black hole (SMBH) properties and accretion physics. The existing light curve fitting tools assume that the disrupted stars are on parabolic orbits, which may introduce systematic biases in derived parameters. In this work, we develop a non-parabolic TDE model that incorporates orbital energy of the disrupted star as a free parameter ($\tildeε_{\rm orb}$) to modify the debris mass distribution and mass fallback rate. We apply this model to 30 TDEs from the ZTF-I survey and compare the results with those from a standard parabolic model. We find that neglecting orbital energy leads to biased black hole mass estimates: for eccentric (hyperbolic) orbits, parabolic models systematically underestimate (overestimate) the black hole mass. Additionally, we measure orbital eccentricities ($e$) and penetration factors ($β$) of the disrupted stars in this sample, enabling an investigation of their origins via the $e$-$β$ parameter space. Most events (24/30) are consistent with production via two-body relaxation in spherical nuclear star clusters, but six outliers with high $β$ and $e<1$ suggest alternative mechanisms. Our results highlight the importance of accounting for orbital energy in TDE modeling to improve the accuracy of SMBH mass measurements and to better understand the dynamical origin of the disrupted stars.

Irradiation-driven fragmentation and chemical transformations of organometallic molecules play a central role in nanofabrication techniques based on the use of focused charged-particle beams. In this paper, the electron irradiation-induced fragmentation dynamics of W(CO)$_6$, a commonly used precursor for focused electron beam-induced deposition (FEBID), is investigated using the irradiation-driven molecular dynamics (IDMD) method. Simulations are performed for gas-phase systems with different precursor densities and under different irradiation conditions. The results reveal progressive fragmentation of W(CO)$_6$ molecules into W(CO)$_n$ species, accompanied by the formation of W-rich molecular clusters. The evolution of fragment abundances shows a strong dependence on both precursor density and electron fluence. Higher densities and larger fluences result in more extensive fragmentation and promote the aggregation of tungsten atoms into small metal clusters. Under certain irradiation conditions, the studied molecular systems evolve towards a steady state characterised by slightly varying fragment abundances. The obtained scaling relations between irradiation parameters and fragment distributions provide guidance for selecting simulation parameters in IDMD simulations of the FEBID process, ensuring a quantitative description of precursor fragmentation dynamics.

Cosmological filaments, galaxy clusters, and galaxies are magnetized reservoirs of cosmic rays (CRs). The exchange of CRs across these structures is usually modeled assuming that they remain charged and magnetically confined. At high energies, hadronic interactions can convert CR protons to neutrons. This physics is routinely included in air-shower and ultra-high-energy (UHE) CR propagation Monte Carlo simulations used for composition studies but is rarely treated explicitly in propagation models of CR transport and exchange between magnetized reservoirs. CR neutrons are not affected by magnetic fields and can propagate ballistically over kpc-Mpc distances before decaying back into protons, with relativistic time dilation extending their effective decay length. We show how such charged-neutral switching modifies CR confinement and escape in four representative environments: a Milky Way-like galaxy, a starburst galaxy, a galaxy cluster, and a cosmological filament. By solving the transport of a confined CR proton population in each structure using a diffusion/streaming propagation approach with hadronic pp and p$γ$ interactions, and treating neutron production and decay as a stochastic Poisson ''jump'' process, we find that neutron-mediated steps can allow additional CR escape from large-scale cosmological structures at energies where charged-particle transport alone would predict strong CR confinement and attenuation in ambient radiation fields. These effects imply a qualitative shift in how ultra-high-energy CRs are transferred from embedded sources into filaments and voids once intermediate neutron propagation is considered, with consequences for the partitioning of CRs across the large-scale structure of the Universe.

We present an analysis of the light tensor ($J^{PC}=2^{++}$) hybrid meson mass and coupling from QCD Laplace Sum Rules where the next-to-leading order (NLO) perturbative (PT) corrections and the contributions of the non-perturbative (NP) condensates up to dimension-six ($D=6$) are included. NLO leading-logarithms corrections due to the condensates which contribute in the chiral limit are considered. We obtain the mass $M_{2^+}= (2038\pm 190)$ MeV and a relatively small coupling $f_{2^+}=(10.5\pm 2.9)$ MeV normalized as $f_π=93$ MeV. Our results suggest that the $f_2(1950)$ or/and the $f'_2(2010)$ may have a sizeable $\bar qqg$ hybrid component. We also compute the tensor hybrid topological charge (value of the two-point function at zero momentum) and find (for the first time) at NLO: $Π_{qg}(0)=(2.41\pm 0.43) \times 10^{-4}{\rm GeV}^6$ which could be checked from some lattice QCD or/and low energy theorems (LET).

This paper develops a direct data-driven framework for constructing reduced-order models (ROMs) of discrete-time linear dynamical systems with unknown dynamics and process disturbances. The proposed scheme enables controller synthesis on the ROM and its refinement to the original system by an interface function designed using noisy data. To achieve this, the notion of simulation functions (SFs) is employed to establish a formal relation between the original system and its ROM, yielding a quantitative bound on the mismatch between their output trajectories. To construct such relations and interface functions, we rely on data collected from the unknown system. In particular, using noise-corrupted input-state data gathered along a single trajectory of the system, and without identifying the original dynamics, we propose data-dependent conditions, cast as a semidefinite program, for the simultaneous construction of ROMs, SFs, and interface functions. Through a case study, we demonstrate that data-driven controller synthesis on the ROM, combined with controller refinement via the interface function, enables the enforcement of complex specifications beyond stability.

In the present paper, we discuss for the first time the theoretical Kendall correlation coefficient for non-identical bivariate data. In the non-identical case, we first introduce a theoretical Kendall correlation coefficient $τ_n$ and show that the expected value of the rank Kendall correlation coefficient $\tildeτ_n$ is equal to $τ_n$. We then prove that $\tildeτ_n$ converges in probability to $τ=\lim_{n\rightarrow\infty} τ_n$. These facts enable us to state that $τ_n$ is a correctly defined theoretical Kendall correlation coefficient for the non-identical case. We also support our theoretical results by simulation experiments.

Photonic bound states in the continuum (BICs) provide a revolutionary paradigm for boosting light-matter interactions in integrated nanocavity systems. Nevertheless, precise manipulation of open cavity-emitter architectures still faces critical challenges, especially in realizing deterministic directional radiation and suppressing the perturbation of intrinsic cavity modes induced by emitters as local impurities. Conventional investigations on cavity-emitter coupling are predominantly based on ensemble measurements, which inevitably mask the intrinsic physics underlying individual light-matter interactions. Here, we propose a robust strategy to control the upconversion and emission of a single-particle emitter using a topological plasmonic cavity with broken σh mirror symmetry. This structured design enables the transition from symmetry-protected BICs to a multi-BIC regime with finite but ultrahigh confinement, where nontrivial phase evolution and hybridization of transverse electric and magnetic modes open a well-defined far-field radiation channel for directional emission. Leveraging this scheme, we experimentally demonstrate dramatically enhanced radiation intensity from a single point-like emitter, together with uniform and deterministic directional emission, while achieving excellent structural robustness against local perturbations. This work establishes a general framework for engineering coherent directional light emission at the nanoscale, which lays a solid foundation for high-performance chip-scale integrated nanophotonic applications.

In this work, we investigate the critical behavior of photon rings in the Kerr-Bertotti-Robinson spacetime, describing a rotating black hole immersed in a background magnetic field. We analyze the radial and angular motions of photons under the small magnetic field approximation. Focusing on unstable spherical orbits, we determine three key parameters, $γ$, $δ$, and $τ$, which characterize radial compression, azimuthal advancement, and time delay. We then examine how these parameters depend on the black hole spin, magnetic field strength, and observer inclination for both on-axis and off-axis observers, and we further analyze the properties of higher-order images through near-critical lens equations. The results show that the magnetic field modifies the geodesic structure, and leads to observable changes in the fine structure of photon rings, providing a useful framework for probing magnetized black hole environments.

With the rapid evolution of the Industrial Internet of Things (IIoT), the boundaries and scale of the Internet are continuously expanding. Consequently, the limitations of traditional certificate-based Public Key Infrastructure (PKI) have become increasingly evident, particularly in scenarios requiring large-scale certificate storage, verification, and frequent transmission. These challenges are expected to be further amplified by the widespread adoption of post-quantum cryptography. In this paper, we propose a novel identity-based public key management framework for PKI based on post-quantum cryptography, termed \textit{IPK-pq}. This approach implements an identity key generation protocol leveraging NIST ML-DSA and random matrix theory. Building on the concept of the Composite Public Key (CPK), \textit{IPK-pq} addresses the linear collusion problem inherent in CPK through an enhanced identity mapping mechanism. Furthermore, it simplifies the verification of the declared public key's authenticity, effectively reducing the complexity associated with certificate-based key management. We also provide a formal security proof for \textit{IPK-pq}, covering both individual private key components and the composite private key. To validate our approach, formally, we directly implement and evaluate \textit{IPK-pq} within a typical PKI application scenario: Resource PKI (RPKI). Comparative experimental results demonstrate that an RPKI system based on \textit{IPK-pq} yields significant improvements in efficiency and scalability. These results validate the feasibility and rationality of \textit{IPK-pq}, positioning it as a strong candidate for next-generation RPKI systems capable of securely managing large-scale routing information.

Integrating Automatic Speech Recognition (ASR) into Speech Emotion Recognition (SER) enhances modeling by providing linguistic context. However, conventional feature fusion faces performance bottlenecks, and multi-task learning often suffers from optimization conflicts. While task vectors and model merging have addressed such conflicts in NLP and CV, their potential in speech tasks remains largely unexplored. In this work, we propose an Adaptive Layer-wise Task Vector Merging (AdaLTM) framework based on WavLM-Large. Instead of joint optimization, we extract task vectors from in-domain ASR and SER models fine-tuned on emotion datasets. These vectors are integrated into a frozen base model using layer-wise learnable coefficients. This strategy enables depth-aware balancing of linguistic and paralinguistic knowledge across transformer layers without gradient interference. Experiments on the MSP-Podcast demonstrate that the proposed approach effectively mitigates conflicts between ASR and SER.

AstroSat/LAXPC and SXT observed the persistent neutron star low-mass X-ray binary 4U 1820-30 between 2016 and 2022. During these observations, the hardness-intensity diagram (HID) and color-color diagram (CCD) indicated that the source was in the banana state. We divided the CCD into 11 segments for spectral and timing analyses. For each segment in the CCD, we modeled the spectral data using two distinct approaches over the 0.7-20.0 keV band. A combination of a multi-color-disk component with an inner disk temperature of around 0.6 keV and Comptonized emission from the boundary layer (BL)/ hot corona provided the best description of the X-ray spectral data of this source. The truncation radius was found to be in the range of $\sim$ 19-40 km. The Comptonized component has an optical depth in the range of $\sim 7 - 13$ with electron temperature in the range of $\sim 2.5 - 3.8$ keV. The optical depth of the corona varies significantly along the position on the CCD, while $\sim$ 80\% of the X-ray flux comes from the Comptonized component. We discuss possible physical scenarios to explain the relationship between the spectral evolution and motion of the source along the CCD. The timing analysis revealed kHz QPOs peaks at $\sim 710$ Hz and $\sim 740$ Hz in the lower left banana branch. An energy-dependent study indicates that these QPOs are stronger in the high-energy band.

We present results from a mini-survey of dust spectral features arising in galaxies at redshifts $0.5 < z < 1.2$ in our James Webb Space Telescope (JWST) mid-infrared spectra of physically-unrelated background quasars. We analyze the JWST Mid-infrared Instrument (MIRI) Medium-Resolution Spectrometer (MRS) spectra of five quasars presented in Klimenko, Kulkarni, \& Aller 2025a (Paper I) to determine the best-fit silicate mineralogies. Template profile fits to the 10 $μ$m feature suggest the possible presence of crystalline silicates in three of the galaxies. This contrasts with the predominately amorphous silicate grains in the Milky Way diffuse interstellar medium (ISM). We also measure the extinction curves using existing data from UV to mid-IR. Combining our results with past Spitzer IRS studies, we find that (i) the 10~$μ$m silicate peak optical depth ($τ_{10}$) is about three times stronger than expected for the local diffuse ISM over the range $A_V =0.1-2.0$, with $τ_{10}/A_V$=$0.17\pm0.09$. (ii) The relative strength of the UV bump is similar to that in the local ISM. However, the ratio $τ_{10}/A_{2175}$ is larger ($\sim0.1-1$), and appears to decrease with $A_V$, approaching the Galactic ISM value ($\sim 0.1$) at $A_V\sim1.5-2$. (iii) No significant correlation of $τ_{10}/A_V$ with $R_V$. (iv) $τ_{10}$ is strongly correlated with the gas-phase Mg~II absorption strength for the quasar sightlines. Possible interpretations include that some quasar sightlines probe dust in the circumgalactic medium (CGM), and that dust grains may have been significantly reprocessed in the ISM and CGM under conditions that may differ from those in the local ISM.

Positional encoding has become a standard component in graph learning, especially for graph Transformers and other models that must distinguish structurally similar nodes, yet its fundamental identifiability remains poorly understood. In this work, we study node localization under a hybrid positional encoding that combines anchor-distance profiles with quantized low-frequency spectral features. We cast localization as an observation-map problem whose difficulty is controlled by the number of distinct codes induced by the encoding and establish an information-theoretic converse identifying an impossibility regime jointly governed by the anchor number, spectral dimension, and quantization level. Experiments further support this picture: on random $3$-regular graphs, the empirical crossover is well organized by the predicted scaling, while on two real-world DDI graphs identifiability is strongly graph-dependent, with DrugBank remaining highly redundant under the tested encodings and the Decagon-derived graph becoming nearly injective under sufficiently rich spectral information. Overall, these results suggest that positional encoding should be understood not merely as a heuristic architectural component, but as a graph-dependent structural resolution mechanism.

We study a class of degenerate parabolic equations with boundary point degeneracy in dimensions N>=2 and investigate the associated boundary observability problem by means of shape design. While one-dimensional degenerate models have been treated in the literature, the genuinely higher-dimensional case remains much more delicate because the degeneracy occurs at a boundary point and the boundary normal trace cannot be extracted directly near the singularity. We approximate the degenerate equation by a family of uniformly parabolic problems on truncated domains obtained by removing a small neighborhood of the degenerate point. Under a geometric condition on the boundary, we establish uniform estimates for the approximate problems, prove convergence to the solution of the original degenerate equation, and identify the convergence of the boundary normal derivatives under additional regularity. We then combine this approximation scheme with a parabolic Carleman estimate for the approximate backward equations and derive a boundary observability inequality for the limiting degenerate equation. In this way, we obtain a higher-dimensional parabolic counterpart of the shape-design program previously developed for degenerate hyperbolic equations.

Sequential recommendation models, particularly those based on attention, achieve strong accuracy but incur quadratic complexity, making long user histories prohibitively expensive. Sub-quadratic operators such as Hyena provide efficient alternatives in language modeling, but their potential in recommendation remains underexplored. We argue that Hyena faces challenges in recommendation due to limited representation capacity on sparse, long user sequences. To address these challenges, we propose HyenaRec, a novel sequential recommender that integrates polynomial-based kernel parameterization with gated convolutions. Specifically, we design convolutional kernels using Legendre orthogonal polynomials, which provides a smooth and compact basis for modeling long-term temporal dependencies. A complementary gating mechanism captures fine-grained short-term behavioral bursts, yielding a hybrid architecture that balances global temporal evolution with localized user interests under sparse feedback. This construction enhances expressiveness while scaling linearly with sequence length. Extensive experiments on multiple real-world datasets demonstrate that HyenaRec consistently outperforms Attention-, Recurrent-, and other baselines in ranking accuracy. Moreover, it trains significantly faster (up to 6x speedup), with particularly pronounced advantages on long-sequence scenarios where efficiency is maintained without sacrificing accuracy. These results highlight polynomial-based kernel parameterization as a principled and scalable alternative to attention for sequential recommendation.

Inspired by the recent observation of the orbital Seebeck effect in alpha-quartz, we identify an intrinsic amplification mechanism for thermally generated phonon angular momentum and phonon magnetic moment in chiral insulators. We propose a Heisenberg-type long-range coupling between phonon angular momenta, referred to here as L-L coupling, which opens a self-consistent feedback channel and strongly enhances the bare thermal response within linear response. Our calculations reveal a pronounced temperature- and size-dependent amplification, dominated by the off-diagonal channel, with the total phonon angular momentum enhanced by up to nearly two orders of magnitude as the system approaches the threshold from below. These findings suggest that L-L coupling may provide a microscopic origin of giant phonon magnetic moment the recently observed orbital Seebeck effect in alpha-quartz.

We present the first polylogarithmic-round algorithm for sampling a random spanning tree in the (Broadcast) Congested Clique model. For any constant $c > 0$, our algorithm outputs a sample from a distribution whose total variation distance from the uniform spanning tree distribution is at most $O(n^{-c})$ in at most $c \cdot \log^{O(1)}(n)$ rounds. The exponent hidden in $\log^{O(1)}(n)$ is an absolute constant independent of $c$ and $n$. This is an exponential improvement over the previous best algorithm of Pemmaraju, Roy, and Sobel (PODC 2025) for the Congested Clique model.

Causal representation learning seeks to uncover causal relationships among high-level latent variables from low-level, entangled, and noisy observations. Existing approaches often either rely on deep neural networks, which lack interpretability and formal guarantees, or impose restrictive assumptions like linearity, continuous-only observations, and strong structural priors. These limitations particularly challenge applications with a large number of discrete latent variables and mixed-type observations. To address these challenges, we propose discrete causal representation learning (DCRL), a generative framework that models a directed acyclic graph among discrete latent variables, along with a sparse bipartite graph linking latent and observed layers. This design accommodates continuous, count, and binary responses through flexible measurement models while maintaining interpretability. Under mild conditions, we prove that both the bipartite measurement graph and the latent causal graph are identifiable from the observed data distribution alone. We further propose a three-stage estimate-resample-discovery pipeline: penalized estimation of the generative model parameters, resampling of latent configurations from the fitted model, and score-based causal discovery on the resampled latents. We establish the consistency of this procedure, ensuring reliable recovery of the latent causal structure. Empirical studies on educational assessment and synthetic image data demonstrate that DCRL recovers sparse and interpretable latent causal structures.

Let $\mathbb{F}_q$ denote a finite field of order $q$. A rational function $r(x)\in \mathbb{Q}(x)$ is said to be arithmetically exceptional if it induces a permutation on $\mathbb{P}^1(\mathbb{F}_p)$ for infinitely many primes $p$. Based on some computational results, Odabaş conjectured that for each $k\in \mathbb{N}$, the $k$-th Lattès map attached to an elliptic curve $E/\mathbb{Q}$ is arithmetically exceptional if and only if $E$ has no $k$-torsion point whose $x$-coordinate is rational. In this paper, we prove that this conjecture is true for any elliptic curve $E/\mathbb{Q}$ having complex multiplication by an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-11}).$ On the other hand, we show that the conjecture becomes invalid if $E$ has CM by $\mathbb{Q}(\sqrt{-11})$ and $6\mid k$. Partial results for non-CM elliptic curves are also given.

Let $R$ be a domain and $B=R[x_1^{\pm1},\ldots,x_n^{\pm1}]$ the Laurent polynomial ring over $R$. In this paper we study pre-factorially closed (pfc) and quasi-factorially closed (qfc) $R$-subalgebras of $B$, which generalize the notion of factorially closed subalgebras. We first establish a localization criterion for the qfc property. Using this criterion, we investigate monoid algebras $A=R[M]$ associated with submonoids $M\subset \mathbb{Z}^n$. We prove that $R[M]$ is qfc in $B$ if and only if the group generated by $M$ is a direct summand of $\mathbb{Z}^n$. This provides a complete characterization of the qfc property in terms of the lattice structure of the associated group. As a consequence, when $n=1$ and $M\subset\mathbb{N}$, the algebra $R[M]$ is qfc in $B$ precisely when $M$ is a numerical semigroup. For a general $R$-subalgebra $A\subset B$, we introduce an invariant $\mathrm{Gap}(A)$. We show that if $\mathrm{Gap}(A)$ is finite, then $A$ is qfc in $B$. Moreover, we clarify how the pfc and qfc conditions are related to other notions that naturally appear for subalgebras, such as retracts, being algebraically closed in $B$, and normality.

For $λ\ge0$, the so-called $λ$-analytic functions are defined in terms of the (complex) Dunkl operators $D_{z}$ and $D_{\bar{z}}$. In the paper we introduce a Bloch type space on the disk ${\mathbb D}$ associated with $λ$-analytic functions, called the $λ$-Bloch space and denoted by ${\mathfrak{B}}_λ({\mathbb D})$. Various properties of the $λ$-Bloch space ${\mathfrak{B}}_λ({\mathbb D})$ are proved. We give a characterization of functions in ${\mathfrak{B}}_λ({\mathbb D})$ by means of the higher-order operators $(D_z\circ z)^n$ for $n\ge2$. A general integral operator is proved to be bounded from $L^{\infty}({\mathbb D})$ onto ${\mathfrak{B}}_λ({\mathbb D})$, and as an application, the dual relation of ${\mathfrak{B}}_λ({\mathbb D})$ and the $λ$-Bergman space ($p=1$) is verified.

State-of-the-art Learned Sparse Retrieval (LSR) models, such as Splade, typically employ a Language Modeling (LM) head to project latent hidden states into a lexically-anchored logit matrix. This intermediate matrix is subsequently transformed into a sparse lexical representation through element-wise operations (ReLU, Log1P) and max-pooling over the sequence dimension. Despite its effectiveness, the LM head creates a massive memory bottleneck due to the sheer size of the vocabulary (V), which can range from 30,000 to over 250,000 tokens in recent models. Materializing this matrix creates a significant memory bottleneck, limiting model scaling. The resulting I/O overhead between operators further throttles throughput and runtime performance. In this paper, we propose Sparton, a fast memory-efficient Triton kernel tailored for the LM head in LSR models. Sparton utilizes a fused approach that integrates the tiled matrix multiplication, ReLU, Log1P, and max-reduction into a single GPU kernel. By performing an early online reduction directly on raw logit tiles, Sparton avoids materializing the full logit matrix in memory. Our experiments demonstrate that the Sparton kernel, in isolation, achieves up to a 4.8x speedup and an order-of-magnitude reduction in peak memory usage compared to PyTorch baselines. Integrated into Splade (|V| ~ 30k), Sparton enables a 33% larger batch size and 14% faster training with no effectiveness loss. On a multilingual backbone (|V| ~ 250k), these gains jump to a 26x larger batch size and 2.5x faster training.

We propose a Bayesian propensity score-augmented latent factor model for causal inference with time-series cross-sectional data. The framework explicitly models the treatment assignment mechanism by incorporating latent factor loadings, while the outcome model flexibly incorporates the propensity score, for example through stratification. Relative to existing approaches, the proposed method provides greater flexibility and captures additional heterogeneity across propensity-score strata, enabling more credible comparisons between treated and control units within each stratum. For estimation and inference, we adopt an approximate Bayesian procedure to address the model feedback problem common in Bayesian propensity score analysis. We demonstrate the performance of the proposed method through Monte Carlo simulations and an empirical application examining the effect of political connections on firm value.

Let $V$ be an $n$-dimension real vector space with a direct sum decomposition $V = V_1 \oplus \cdots \oplus V_r$. Let $\mathcal{P} = \{(A_i, B_i) : i \in [m]\}$ be a skew Bollobás system of subspaces of $V$ such that each $i\in [m]$, $ A_i = \bigoplus_{k=1}^r (A_i \cap V_k)$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V_k)$. We prove that $$\sum_{i=1}^{m} \prod_{k=1}^{r} \left[ \binom{a_{i,k} + b_{i,k}}{a_{i,k}} (1 + a_{i,k} + b_{i,k})^{-1} \right] \leq 1,$$ where $a_{i,k} = \dim(A_i \cap V_k)$ and $b_{i,k} = \dim(B_i \cap V_k)$. This extends a recent result of Yue from set systems to finite dimensional subspaces. We then consider Tuza's theorem on weak Bollobás system for $d$-tuples. We give an alternative proof of the original set version of Tuza, and also establish its vector space analogue. Precisely, let $\mathcal{P} = \{(A_i^{(1)}, \ldots, A_i^{(d)}) : i \in [m]\}$ be a skew Bollobás system of $d$-tuples of subspaces of finite dimensional space $V$ with $a^{(\ell)}_i=\dim (A_i^{(\ell)})$. Then, for any positive real numbers $p_1, \ldots, p_d$ satisfying $p_1 + \cdots + p_d = 1$, we prove that $ \sum_{i=1}^{m} \prod_{\ell=1}^{d} p_{\ell}^{a_i^{(\ell)}} \leq 1. $

Programming is a core skill in computer science and software engineering (SE), yet identifying and resolving code errors remains challenging for both novice and experienced developers. While Large Language Models (LLMs) have shown remarkable capabilities in natural language understanding and generation tasks, their potential in domain-specific, complex scenarios, such as multi-label classification (MLC) of programming errors, remains underexplored. Recognizing this less-explored area, this study proposes a multi-label error classification (MLEC) framework for source code that leverages fine-tuned LLMs, including CodeT5-base, GraphCodeBERT, CodeT5+, UniXcoder, RoBERTa, PLBART, and CoTexT. These LLMs are integrated with deep learning (DL) architectures such as GRU, LSTM, BiLSTM, and BiLSTM with an additive attention mechanism (BiLSTM-A) to capture both syntactic and semantic features from a real-world student-written Python code error dataset. Extensive experiments across 32 model variants, optimized using Optuna-based hyperparameter tuning, have been evaluated using comprehensive multi-label metrics, including average accuracy, macro and weighted precision, recall, F1-score, exact match accuracy, One-error, Hamming loss, Jaccard similarity, and ROC-AUC (micro, macro, and weighted). Results show that the CodeT5+\_GRU model achieved the strongest performance, with a weighted F1-score of 0.8243, average accuracy of 91.84\%, exact match accuracy of 53.78\%, Hamming loss of 0.0816, and One error of 0.0708. These findings confirm the effectiveness of combining pretrained semantic encoders with efficient recurrent decoders. This work lays the foundation for developing intelligent, scalable tools for automated code feedback, with potential applications in programming education (PE) and broader SE domains.