Papers

Linnik's approximation to Goldbach's conjecture concerns the representation of sufficiently large even integers as the sum of two primes and a bounded number of powers of two. We prove that $6$ powers of $2$ suffice unconditionally and that $5$ suffice under the GRH, improving previous works.

The Hanbury Brown-Twiss (HBT) effect marked a significant milestone in quantum optics by illustrating that intensity correlations can reveal inherently quantum features of light. Motivated by the structure of the original 8-mode HBT interferometer, we develop a simplified 4-mode toy-model that captures its essential physical ingredients while significantly reducing analytical complexity. Our toy-model provides an intuitive and well-structured framework for examining the desired correlation functions. We investigate whether our toy-model can exhibit genuinely non-classical behaviour by formulating and applying a Bell-type inequality tailored to its specific measurement configuration. Our analysis shows that the correlations generated by the toy-model violate the inequality, indicating the presence of non-local features that are incompatible with any classical hidden-variable description. The results highlight the power of simplified models in uncovering fundamental aspects of non-classicality in optical systems.

Vision-language navigation requires agents to follow natural-language instructions to reach target locations, making high-quality language supervision essential. Since human annotation is costly, prior work has explored scalable data synthesis methods. However, existing methods typically either use separate models for navigation and instruction generation or couple the two tasks through task-specific components; most also annotate the full trajectory corpus with a fixed model before updating, which can cause generated supervision to lag behind the evolving policy. Therefore, we introduce RSI-Nav, a data-centric recursive self-improvement framework for navigation via policy-synchronized language-action dual bootstrapping, which unifies vision-language-to-action instruction following and vision-action-to-language instruction generation within a shared model. Specifically, after supervised fine-tuning establishes fundamental capabilities, RSI-Nav progressively processes unlabeled trajectories in small shards. The current model generates instructions for these trajectories, executes them to evaluate goal completion and path efficiency, retains qualified samples, and immediately updates itself before processing the next shard. This shard-wise update keeps newly generated supervision aligned with the evolving policy. We also incorporate spatial understanding data, since stronger spatial reasoning benefits navigation. Extensive experiments show that RSI-Nav reaches 68.6% SR on R2R-CE and 72.4% SR on RxR-CE, while providing stronger instruction-generation utility and modestly improved spatial understanding after recursive evolution.

Training-free pruning removes weights from pretrained large language models to reduce their deployment cost. However, existing approaches typically consider only the total number of retained weights. Consequently, their pruning decisions do not account for how target hardware stores and processes those weights. These methods select a sparsity structure in advance and apply it uniformly to all projections, rather than accounting for how each structure trades pruning error against latency and memory across sublayers. We propose Heterogeneous Structured Pruning (HSP), a hardware-aware pruning framework that combines two components: (i) a heterogeneous search space of dense, unstructured, 2:4, block-sparse, and head / channel (H/C) pruning options and (ii) structure-regularising constraints based on latency and memory models. Our contribution generalises existing importance-based methods: a global solver selects one option per sublayer to minimise proxy pruning loss under a resource budget, thereby assigning each projection a structure and density. We evaluate HSP on OPT and LLaMA models for an NVIDIA Jetson Orin Nano Super edge device, where it improves the quality-resource frontiers over training-free baselines. A project page and updated links for the related code is maintained at: cameronbarker.me/publications/hsp.

The Flow Matching objective supervises the velocity field at intermediate states, whereas waveform metrics are evaluated after decoding and may be non-differentiable. To align Flow Matching models with waveform-level and perceptual quality objectives during post-training, FlowGRPO optimizes their trajectories using endpoint rewards. In this work, we focus on music separation and use multi-resolution short-time Fourier transform (MRSTFT), scale-invariant signal-to-distortion ratio (SI-SDR), and Virtual Speech Quality Objective Listener (ViSQOL) as reward components. Before policy optimization, we analyze the relationship between data quality factors and the within-group gap between the maximum and mean rollout rewards for each component. This analysis motivates a revised ViSQOL reward that matches channel-wise root-mean-square (RMS) levels and evaluates the interval from the first to the last active frame, retaining inactive frames within that interval. We also apply a sample-quality filter based on predefined waveform checks. Together, the revised reward and quality filter enable FlowGRPO to improve the pretrained vocal separator in both separation performance and perceptual quality.

The rotating vector model (RVM) is widely used to interpret pulsar radio polarization under the assumption of a dipolar magnetic field. Motivated by tentative evidence for non-dipolar magnetic structure in millisecond pulsars provided by recent modeling of surface thermal X-ray pulsations, we examine how a global quadrupole component modifies position-angle (PA) behavior within the classical RVM framework. We numerically integrate magnetic field lines in static vacuum magnetospheres composed of a centered dipole component that is inclined with respect to the spin axis, and a centered quadrupole component aligned with it. We then compute the open field line regions (OFLRs), emission skymaps, and PA curves for a range of dipole inclinations, relative quadrupole strengths, and light cylinder radii corresponding to two trial spin periods of 1\,ms and 5\,ms. We find that quadrupole-modified magnetospheres produce a wide variety of PA behaviors and substantially alter the morphology and total area of OFLRs, with the OFLR area varying non-monotonically with quadrupole strength and, in some cases, exceeding that of the corresponding pure dipole. Depending sensitively on the field configuration and viewing geometry, three broad PA regimes can be distinguished: a) nearly dipolar PA swings that are largely insensitive to emission altitude; b) moderately distorted, altitude-dependent PA tracks that can bias dipolar RVM fits; and c) strongly dispersed PA behavior associated with emission from different OFLRs, which can remain incompatible with a single dipolar PA track even within narrow emission altitude ranges. These results demonstrate that dipole-like PA curves do not uniquely imply a purely dipolar magnetic field, while global quadrupole components alone are insufficient to reproduce the full complexity of observed millisecond pulsar PA profiles.

Under GRPO against a frozen, reference-free LLM judge, RL exploits the judge's false positives. Does the policy-judge capability gap or the judge itself set how fast and how far? We train 0.5B and 1.5B Qwen2.5 policies on GSM8K against four judges at least as large, and the 1.5B policy also against one smaller judge. In a post-hoc analysis, judge acceptance rose, but no cell of 12 gained more than 0.04 in seed-mean held-out greedy accuracy. A gold-reward control with the same recipe raised the 1.5B policy's accuracy by +0.054 (for the 0.5B policy, gold reward gave no detectable gain either). The depth of judge hacking tracks the judge: the plateau correlates with judge AUC (Spearman -0.69 over 12 cells), and within a family the larger judge is fooled less deeply. For the pace, two pre-specified pairs contradict the gap account: the cell with the larger gap was fooled 3.5x sooner at the pre-specified threshold. With 12 cells, however, we cannot formally separate judge AUC from the gap for either endpoint. A blind test of step-0 predictors on three held-out judges was inconclusive. At 7-8B, judges from three families plateaued between 0.37 and 0.85.

Minimum product-set problems ask how small a product of two subsets of prescribed cardinalities can be in a finite group. Although the problem is completely understood for finite abelian groups, exact results in the nonabelian setting are available only for particular classes. We determine the minimum product-set function $\mu_G$ for every finite metacyclic group, without assuming that the relevant cyclic extension splits. Our proof converts the extremal problem into a finite recursive arithmetic description of the feasible cardinality vectors of subset trios. The recursion combines the subgroup structure of finite metacyclic groups with the structural classification of maximal critical subset trios. The resulting characterization recovers the previously known formulas for abelian and nilpotent groups, dihedral groups, and nonabelian groups of order $pq$. It also yields metacyclic examples in which the minimum product-set size lies strictly between the classical lower and upper arithmetic bounds for finite solvable groups. For faithful split metacyclic groups, we further show that requiring one factor to be symmetric, contain the identity, and generate the whole group can force a strictly larger product set than the unrestricted minimum. The latter constraint admits a vertex-isoperimetric interpretation on connected Cayley graphs.

We identify the free-facility gaps of greedy online assignment on a cycle with the discrete additive coalescent and use this connection to determine the exact expected assignment cost. The cycle has circumference $n$, with one unit-capacity facility at each integer point and $n$ independent uniform arrivals. Each request is assigned immediately to a nearest free facility. The conditional one-step cost is the sum of squared gap lengths divided by $4n$. A uniform cyclic-order argument identifies the unordered gap process with additive coalescence. The classical Cayley-tree representation then reduces the accumulated cost to a two-vertex distance functional, yielding \[ G_n=\frac n4\sum_{d=0}^{n-1}\prod_{j=1}^{d}\left(1-\frac jn\right) \sim\frac14\sqrt{\frac\pi2}\,n^{3/2}. \] The formula requires a linear number of scalar recurrence steps to evaluate. We also obtain an asymptotic upper bound of $\pi\sqrt3/2$ on the ratio of expected greedy cost to expected offline optimum under this arrival model. An exact rational Bellman calculation shows that greedy is optimal for three facilities but exceeds the optimal online expected cost by $1/512$ for four facilities. Independent exact recurrences and finite tree enumeration provide reproducible checks of the analysis.

Horizontally explicit, vertically implicit (HEVI) time integration removes the acoustic time-step restriction imposed by fine vertical grids in nonhydrostatic atmospheric models, but requires fast solvers for the resulting vertical linear systems, especially on GPUs. We present GPU-accelerated nested Schur-complement solvers for these systems for a split-form discontinuous Galerkin spectral element method on Legendre-Gauss-Lobatto (LGL) nodes applied to the compressible Euler equations with gravity. In time, we use a Rosenbrock-W method, a linearly implicit Runge-Kutta scheme that retains its order and stability with approximate Jacobians. This allows a sparse vertical Jacobian that captures vertical acoustic waves and gravity on curvilinear meshes with orography and decouples into independent column systems. The solvers are nested: a global Schur complement (static condensation) exploits the LGL sparsity to condense each column onto a few element-edge unknowns, and a local Schur complement eliminates all variables but the vertical momentum within each element, exploiting that the pressure depends only on the potential-temperature density. We propose and analyze two condensation strategies, which lead to banded and block-tridiagonal reduced systems. The resulting HEVI schemes conserve total mass and potential-temperature density to round-off. The solvers are implemented for CPUs and GPUs in TrixiAtmo.jl, the atmospheric extension of Trixi.jl. We verify high-order convergence with linear inertia-gravity waves in a 2D channel and demonstrate robustness on the sphere with the baroclinic instability, a mountain-induced baroclinic instability, and the DCMIP-2025 gap-flow and vortex-shedding cases. For the baroclinic instability without orography, the scheme achieves 18.8 and 2.9 simulated years per day at 100 and 50 km resolution on one NVIDIA GH200 GPU in double precision.

In this paper, we study two bosonic modes that exchange pairs of quanta, which realize two-axis twisting exactly: within a sector of fixed particle number the exchange operator is proportional to the two-axis twisting generator, with a fixed factor of two. Because any real nonlinear medium supplies a local Kerr nonlinearity alongside this operator, the two merge into one twisting Hamiltonian governed by a single dimensionless ratio $r=\chi_s/\chi_e$ of the Kerr strength to the exchange strength. Two conditions then decide how much of the ideal behavior survives this constraint, and they must be met separately. The first concerns speed: the exponential squeezing rate follows in closed form, vanishes exactly at $r=\pm1$, and recovers as the square root of the distance from that point. The second concerns depth: the smallest attainable squeezing parameter depends only on the anisotropy of the mean-field saddle, attaining the full two-axis-twisting value at $r=0$ and $r=\pm3$ regardless of the rate, so that settings sharing the same anisotropy but differing twofold in rate reach the same depth. The two conditions address different targets, and a system may squeeze quickly yet terminate at a shallow optimum; only $r=0$ is optimal in both respects. The closed-form Gaussian solution reveals a further systematic bias: because the early-time decay rate of the variance exceeds twice the exponential rate, a fit to short-time data overestimates the rate by more than a factor of two. Since an isotropic cubic nonlinearity imposes $r=1$, the value at which the rate vanishes, the two strengths must be mismatched, and a moderate mismatch already restores most of the optimal depth. Finally, the constraint is invisible at low excitation: at two quanta the Kerr term shifts both coupled states equally, so the two Hamiltonians are indistinguishable until four quanta.

Music-understanding benchmarks should identify both the evidence available to a model and the operations a correct answer requires. We introduce HMU-Bench, a bilingual benchmark with notation and audio components. ABC Q&A contains 1,029 Chinese-English question pairs across six musical domains, labeled as extraction, single-step inference, or multi-step inference. Audio Q&A contains 460 question pairs on 92 recordings, spanning five scope-difficulty configurations. All music is Suno-generated, and questions and references are human-reviewed. Reference-based evaluation of seven omni models yields high notation scores for the strongest systems, but no model averages above 1.4 on the 0-4 audio scale. Notation and audio rankings show little correlation. A transposition case study also reveals octave errors missed by LLM grading. These findings motivate separate modality reporting and event-level checks for symbolic outputs. Data, annotations, evaluator prompts, and graded responses will be released at https://github.com/stepaudiollm/hmu-bench.

Recent focus in agentic reinforcement learning (RL) seeks to improve credit assignment in long-horizon tasks. One promising direction combines reinforcement learning with verifiable rewards (RLVR) and on-policy self-distillation (OPSD), integrating trajectory-level advantages with dense token-level supervision. This line of work often conditions a teacher model on privileged context, such as agentic skills, to score tokens generated by the student policy. Despite their progress, they largely overlook the on-policy evolution of agentic skills and the reliability of teacher supervision. Towards this end, we introduce Recursive Self-Improving Policy Optimization (RSIPO), a self-evolving agentic RL framework that enables agentic skills and the policy model to co-evolve throughout the training process. Specifically, RSIPO enables the policy itself to extract skills from completed trajectories and explicitly optimizes its skill-generation capability through trajectory replay. It further employs empirical Bayes shrinkage to dynamically calibrate the student's trust in teacher scores, accounting for the fact that skill-enhanced teacher guidance can sometimes be less reliable. Experiments with Qwen-family models on embodied interaction and search-based agentic tasks demonstrate that RSIPO outperforms all the compared baseline approaches. With Qwen2.5-7B as the base model, RSIPO achieves an average success rate of 98.6% on ALFWorld, yielding absolute improvements of 2.5% and 17.4% over the second-best baseline and standard GRPO, respectively.

We consider singular Schrödinger operators with an attractive $\delta'$ interaction of strength $\beta$ supported by a star graph and prove that the ground-state eigenvalue is uniquely maximized by stars of the maximum symmetry. Two different proofs are presented, one for an even number of edges using a comparison to the $\delta$ interaction star, the other general based on Birman-Schwinger technique.

Let $(T_n)_{n\ge1}$ be the Tribonacci sequence, defined by $T_1=T_2=1$, $T_3=2$, and $T_n=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\ge4$. In this paper, we study the Tribonacci numbers whose decimal representation is a palindromic concatenation of two distinct repdigits, that is, numbers of the shape $\overline{d_1\cdots d_1\,d_2\cdots d_2\,d_1\cdots d_1}$ with $1\le d_1\le9$, $0\le d_2\le9$, $d_1\ne d_2$. We prove that no Tribonacci number is a palindromic concatenation of two distinct repdigits. The proof combines lower bounds for linear forms in logarithms, the Baker-Davenport reduction method, and a finite computational verification.

Human demonstrations can capture task-relevant manipulation without using the target robot, but the recorded hand motions cannot directly serve as robot action supervision. Hand reconstruction and motion retargeting provide a direct route to such supervision, yet recovering hand motion alone does not ensure physically consistent hand-object interactions. To address this issue, we introduce Hand2Robot, an object-centric 4D hand reconstruction framework for robot policy post-training. Given videos of humans performing the target task, Hand2Robot extracts interaction clips, reconstructs object geometry, and generates robot action supervision through hand-object interaction optimization and motion retargeting. For object tracking, we combine observation confidence with temporal motion constraints to select a globally consistent object pose trajectory from per-frame pose candidates. For hand refinement, we use object geometry and motion as a reference and combine staged 2D alignment with object-local contact constraints to refine hand trajectories and finger joint configurations. Experiments on DexYCB and HO3D-v3 show improved object pose accuracy and hand-object interaction quality. Experiments on three real-robot dexterous manipulation tasks further show that reconstructed and retargeted human demonstrations can effectively supplement limited robot data for target-task policy post-training.

A long-standing challenge in climate science since the Charney Report in 1979 is to determine how much the Earth's climate warms in response to a unit external energy input at the top of the atmosphere (TOA). In this study, we show from first principles that temperature feedback within an atmosphere-surface column operates as a set of feedback circuits, each governed by an intrinsic gain less than unity, leading to geometric amplification of energy through successive feedback cycles. In particular, the surface-centered temperature feedback circuit determines the amplification of surface energy per unit TOA forcing. The surface energy amplification by non-temperature feedback is estimated from the amplification of solar energy absorption by downward longwave radiation at the surface in the climate mean state. Climate sensitivity then emerges as the product of these two amplifications scaled by the surface Planck feedback parameter. The climate sensitivity derived from the 1980-2000 climate mean state is 1.15 K (W m$^{-2}$)$^{-1}$. It yields a warming of 0.44 K from 1980-2000 to 2000-2020, consistent with the observed warming of 0.41 K, and equilibrium warmings of 3.02 and 5.83 K for CO$_2$ doubling and quadrupling, respectively, the former consistent with the Charney estimate near 3 degrees C. The framework provides a basis for predicting global surface temperature responses to other external forcing scenarios from the climate mean state.

Levenshtein's classical theorem shows that any code that can uniquely decode from \(t\) deletions can also uniquely decode from \(t\) insertions. This symmetry, however, does not generally hold in the list-decoding regime. Several prior results on the capacity and radii of list-decoding from insertions and deletions suggest that insertions are easier to handle than deletions. In an effort to better understand this asymmetry between insertions and deletions under list-decoding, we study an analogous version of McEliece's average list size problem. More precisely, given a binary code \(\mathcal{C}\) of length \(n\) that uniquely corrects \(t'\) insertions (or deletions), we are interested in the average number of codewords returned by a \(t\)-insertion (or deletion) list decoder, where \(t' < t\). Focusing on the case of \(t'=1\) and \(t=2\), we find that under two insertions, this quantity is at most \((n+12)/4\) for any VT code of length \(n\), whereas under two deletions it is at least \(n/2 - 10\) when averaged over all length-\(n\) VT codes. The nearly factor-of-two gap between these upper and lower bounds serves to confirm the relative ease of list-decoding of insertions compared to deletions from an average-case perspective.

Recent multimodal large language models (MLLMs) and vision–language–action models have provided robots with new capabilities for long-horizon task understanding and planning. In surgical robotics, however, end-to-end prediction of low-level actions remains limited by insufficient precision, opaque decisions, and scarce data. This paper presents SurgiCLAW, a hierarchical skill-based framework for precision surgical manipulation. SurgiCLAW confines an MLLM to intraoperative visual understanding, structured explanation, and atomic surgical-skill selection, while delegating physical execution to verifiable deterministic control programs. For brain–machine-interface implantation, a representative high-precision surgical task, we construct a state-to-skill invocation dataset and expand it through visual and state perturbations. The model first learns structured skill invocation through supervised fine-tuning and is then optimized using three rewards for action correctness, operation prerequisites, and format validity, improving action selection, operational safety, and output parseability. We evaluate subtask success, offline model comparison, real-robot performance, and ablations. On a fixed test set, SurgiCLAW achieves 94.6% skill-selection accuracy and 93.0% macro-averaged F1 (Macro-F1). Across 50 closed-loop real-robot implantation trials, it achieves an 88.0% task success rate.

Large language model agents are emerging as an important new class of users of desktop applications. However, existing desktop interfaces were designed for human end users or application developers, leaving agents to interpret visual controls or discover and compose low-level software operations. This mismatch can hinder even capable agents, yet it remains unclear what interface properties make desktop software easier for agents to use. To study this question, we treat software evolution as a natural source of interface variation and introduce DeskIF-Bench, a benchmark of 255 artifact-verified tasks across six desktop applications. By holding the task, agent, and verifier fixed while varying the software release, we measure how interface changes affect agent performance. Across two models and six applications, the newest release consistently outperforms the oldest. Trajectory analysis identifies three relevant interface properties: discoverability, granularity, and observability. Guided by these properties, we develop ANIS, a framework that synthesizes agent-native interfaces from the capabilities of installed applications. On an expanded evaluation of 368 tasks across eight applications, atomic interfaces improve success over native SDK use by 3.8 and 3.5 percentage points with DeepSeek V4 Flash and Claude Haiku 4.5, respectively, while reducing online token usage by approximately 73%. Goal-level commands further raise success to 74.5% and 67.4%, respectively.

Audio-Visual Segmentation (AVS) requires predicting pixel-wise masks of sounding objects by grounding acoustic signals in visual space. In multi-source scenes, concurrent sound sources often remain entangled within shared visual representations, leading to merged masks or unstable specialization when using standard alignment-focused transformer architectures. To address this, we present SlotAVS, an object-centric framework driven by competitive bi-modal slot attention. While robust foundation models provide rich semantic priors, SlotAVS shifts the paradigm from global cross-modal fusion to explicit competitive binding. By initializing discrete object slots from temporal audio sequences and enforcing strict slot-wise competition over spatial visual tokens, our method structurally mitigates the blending of distinct acoustic entities. Further, to stabilize instance-level learning, we introduce a Hungarian matching strategy that optimally aligns slots with ground-truth objects using spatial and boundary overlap. Coupled with a numerically stable, confidence-aware mask fusion mechanism, SlotAVS robustly aggregates multi-source predictions. Extensive experiments on the AVSBench dataset demonstrate the efficacy of our approach, achieving 88.6% mIoU on the Single-Source (S4) split and 71.7% mIoU on the Multi-Source (MS3) split. These results show that explicit competitive binding is critical for complex audio-visual scene understanding and effectively leveraging foundation-model priors without relying solely on their capacity.

Modern industrial automation demands autonomous collaborative systems capable of coordinated manipulation in workspaces with heterogeneous object arrangements. Existing systems rely on predetermined trajectories and rigid allocation strategies, causing suboptimal workspace utilization and collision risks. This paper presents a distance-based competitive task allocation framework for dual-arm systems, integrating real-time vision perception with collision-free motion planning. A temporal window mechanism enables each arm to evaluate object detections over five consecutive vision frames before committing to optimal distance-based selection, featuring dynamic reallocation and spatial conflict resolution at 0.01m threshold accuracy. An improved Artificial Potential Field (iAPF) framework provides coordinated dual-arm movements through exponential goal attraction, inter-arm repulsion, home-seeking, and velocity damping forces. Simulation results validate superiority over market-based, round-robin, and fixed allocation strategies in distance efficiency, collision risk, and workspace coverage. Hardware experiments using two 6-DOF Kinova Gen3 Lite manipulators demonstrate approximately 47% average velocity improvement over fixed allocation in homogeneous scenarios with zero collisions, while mixed object scenarios confirm universal handling with approximately 11% time overhead offset by superior safety margins. The framework integrates YOLOv8 OBB detection at 30 FPS with 10 Hz control loops, demonstrating practical deployment capability for industrial assembly and sorting operations.

Let $\mathcal O$ be the ring of integers of a non-Archimedean local field with uniformizer $\varpi$, and let $\mathcal O_\ell=\mathcal O/\varpi^\ell\mathcal O$ for $\ell\geq1$. In this article, we study representations of $\mathrm{GL}_n(\mathcal O_\ell)$ parabolically induced from characters of the Borel subgroup $\mathrm{B}_n(\mathcal O_\ell)$. For $n=3$, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary $n$. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple $(\mathrm{GL}_n(\mathcal O_\ell),\mathrm{B}_n(\mathcal O_\ell),\chi)$ is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for $\mathrm{GL}_3(\mathcal O_\ell)$ when two relative conductors attain their maximal values. Using the decomposition theory for $\mathrm{GL}_2(\mathcal O_\ell)$, we describe the irreducible constituents and determine their dimensions.

For languages with limited resources, Sentiment Analysis remains a challenging problem due to limited annotated data and strong domain variability across textual sources. Marathi, despite being widely spoken, lacks extensive multi-domain sentiment analysis studies. Recent efforts by L3Cube have introduced multi-domain datasets in Marathi, enabling a better study of domain-aware sentiment classification. This study uses the pretrained MahaBERT model to improve sentiment classification for Marathi across different domains. Models are trained on individual domain datasets and domain-adaptive pretraining is applied using masked language modeling to help the model better understand domain-specific language. In addition, back-translation is used as a method of data augmentation to increase the diversity of training data. Results show that these methods lead to consistent improvements in accuracy across all domains. The improvements range from a minimum of about 0.12% to a maximum of about 1.74%, depending on the dataset. These results demonstrate that domain-adaptive pretraining and data augmentation are effective strategies for improving sentiment analysis models. This helps achieve more robust performance across different domains and makes the approach more suitable for low-resource languages like Marathi.

Multiscale structure is central to time series forecasting: temporal signals contain slowly varying trends, intermediate seasonal patterns, and high-frequency residuals. Traditional forecasters model this multiscale structure in Euclidean space, without representing the ordering it induces among components. We identify that cascaded decomposition orders components by their characteristic frequencies, forming a natural frequency hierarchy. This hierarchy aligns with the exponential volume growth of hyperbolic space, which provides increasing representational capacity with radius and naturally accommodates nested hierarchies. We propose HyperForecast, the first long-term time series forecasting (LTSF) framework operating in hyperbolic space. HyperForecast embeds decomposed temporal segments on the Poincaré ball where radial distance captures this nested structure, from low-frequency components near the origin to high-frequency components near the boundary. We model geodesic velocity through two manifold-dynamics variants: (a) autoregressive forecasting via Poincaré residual dynamics, and (b) non-autoregressive forecasting that predicts all future segments in parallel. HyperForecast consistently outperforms existing and state-of-the-art baselines on 9 of 11 long-term forecasting benchmarks, achieving up to 58% lower mean squared error at long horizons. Further analysis reveals that the learned radial embeddings recover the frequency hierarchy and that hyperbolic geometry contributes substantially to forecasting performance.

For every $0<s<1$ we construct a compact $s$-Ahlfors regular set $K\subset\mathbb C$ for which harmonic measure of $\mathbb C\setminus K$ with pole at infinity is comparable to $\mathcal H^s|_K$. In particular, there is no dimension drop for any dimension $s<1$. The construction is $N$-adic and based on configurations close to the $N$-th roots of unity. We adjust the positions of the children by small polynomial deformations so that the logarithmic potential of the natural Bernoulli measure becomes constant in the limit. A first-mode correction gives the range $s<2/3$, while corrections of arbitrary finite order yield the full range $0<s<1$.

We establish two quantitative results concerning additive decompositions of the set $\mathbb{P}$ of primes. First, we prove that, if $A,B\subseteq\mathbb N$ have at least two elements and $A+B$ differs from the primes by only finitely many elements, then $$ \frac{\sqrt{x}}{\log x}\ll A(x),B(x) \ll \sqrt{x}. $$ Second, we show that, if $A,B,C\subseteq\mathbb N$ have at least two elements each, then $$ ((A+B+C)\mathbin{\bigtriangleup}\mathbb P)(x) \gg \frac{\sqrt{x}}{(\log x)^2}. $$ An analogous estimate is obtained for sumsets of $k\ge 3$ sets. The proofs use Montgomery's large sieve, Gallagher's larger sieve, a mean-value argument for a certain multiplicative function, and a submultiplicative sumset inequality.

Sentiment analysis is a key task in Natural Language Processing that determines the polarity of opinions in text. For low-resource languages like Marathi–where limited data prevents the direct use of standard statistical methods–models are often trained on single-domain datasets. While this yields strong in-domain performance, it leads to poor generalization to unseen domains. Recent studies highlight the need for domain-robust systems that can handle diverse real-world content through multi-domain modeling. This study investigates how multi-domain training improves sentiment classification for unseen Marathi domains. Models are trained using single-domain, two-domain, and three-domain dataset combinations. Results show that two-domain training improves accuracy on unseen domains by about 8% compared to single-domain training. Adding a third domain provides further improvement, though the gain is smaller, averaging around 2% for some domains. These results suggest that training with data from multiple domains helps build more robust sentiment analysis models for low-resource languages, though the benefits reduce as more domains are added.

Yoshikawa's Manipulability Index is generally used as a classical measure to define an optimal morphology of a manipulator. However, any such optimization is almost always studied as a purely kinematic quantity in simulation and is rarely validated on hardware. We present a reinforcement-learning framework with RSL-RL PPO that optimizes the morphology of a 2R SCARA-type manipulator arm with Yoshikawa's kinematic manipulability index as a core reward, and the analytically derived closed-form optimum for the circular task is used to verify that our method recovers it. Then the same method is applied to custom elliptical and rectangular paths having no analytical optimum. The RL-optimized morphologies are then transferred to a 2R hardware manipulator to validate its effect in reduction of joint load in the hardware. Compared to the worst morphology, the optimum morphology delivered by our method reduces second joint load by 38.4 +/- 0.6% for the ellipse path and 41.3 +/- 0.3% for the rectangle, and first joint load by 15%.

Hybrid attention-SSM serving combines growing attention KV with fixed-size recurrent and convolution state whose allocation units, reuse boundaries, and lifetimes differ. We implement four memory organizations in a modified vLLM runtime: a coarse shared-page baseline (S0), checkpoint-internal recurrent paging (S1), paging with convolution-slot packing (S2), and native dual pools (D0). Nine evaluation questions cover storage costs and serving behavior. The cumulative S0-S1-S2 refinement improves completed-output throughput and tail completion time at the main operating point. D0-S2 performance varies with memory budget, offered load, and admission headroom. Shared-prefix workloads yield larger stepwise gains; gains also persist in a no-sharing control with equal recorded input-work totals. State traces identify recurrent-checkpoint availability as the limit on joint reuse in the tested S2 runs. Extending producer-side checkpoint lifetime increases reusable depth at a fixed page budget. A fresh-seed SHARE/ISOLATE comparison estimates a positive retention interaction, with a primary confidence interval that includes zero. Retained-state continuation correctness remains unresolved. These findings motivate evaluating physical state representation together with request admission and checkpoint lifetime when designing memory management for concurrent hybrid-model serving.