Papers

We prove the hyperplane winning property of weighted badly approximable vectors inside nondegenerate analytic manifolds and affine subspaces satisfying almost optimal Diophantine conditions. This, in particular, answers a conjecture of Badziahin–Velani for higher-dimensional manifolds, including affine cases that were understood only in $\mathbb{R}^2$.

We prove that for each $d \geq 2$, there exist infinitely many symmetric abstract $d$-rigidity matroids that are not the $d$-rigidity matroid, the $d$-hyperconnectivity matroid, or the $d$-cofactor matroid. When $d=2$, our constructed matroids are pathological, in the sense that they are not contained within any abstract 2-rigidity matroid family. Our construction additionally provides a counter-example to a conjecture of Tyomkyn.

Archetypal Analysis (AA) represents observations as convex combinations of extremal data-driven profiles, yielding interpretable low-dimensional descriptions of complex datasets. Classical AA relies on a least-squares objective, which is poorly suited to discrete observations such as binary, count, and categorical data. We introduce an efficient likelihood-based framework for AA supporting Bernoulli, Poisson, and multinomial observation models. Our optimization scheme employs local quadratic approximations of the negative log-likelihood, enabling constrained updates through sequential minimal optimization (SMO) and an active-set method. Scalability is improved by bounding the active set while preserving simplex feasibility. We further introduce a cross-validated predictive likelihood criterion for selecting the number of archetypes, providing a principled alternative to reconstruction-error heuristics and stability-based diagnostics. Synthetic experiments demonstrate computational efficiency and accurate recovery of model complexity. Applications to single-cell RNA sequencing, microbiome composition, and somatic mutation data show that the learned archetypes capture interpretable domain-specific structures while achieving competitive likelihood fits and stable solutions. Overall, the proposed framework enables efficient likelihood-based archetypal analysis of discrete data, complemented by predictive likelihood-based model selection.

We study the capacity of the primitive diamond relay channel and derive an improved upper bound compared with existing upper bounds by employing the Csiszár–Körner–Marton sum identity and Gallager-type identification. For Gaussian primitive diamond relay channels, we show that our proposed bound coincides with the existing strengthened Gaussian upper bound of Wu, Özgür, Peleg, and Shamai. For discrete memoryless channels, our bound can be strictly tighter than both the cut-set bound and the existing upper bound, and we assess the improvement using different channel instances, together with a decode–forward/compress–forward time-sharing inner bound.

Robotic systems operating over extended tasks must maintain a world state assembled from observations arriving at different times, with varying confidence and potential revisions. Conventional representations emphasize latest estimates, hindering fact provenance, decision reproduction, or execution auditing. We present Traceable World State (TWS), a middleware-neutral semantic representation and reference runtime for provenance-aware robot world state. A TWS snapshot captures entities, relations, observations, confidence, and revision metadata. Validated update operations transform snapshots immutably, ordered updates support deterministic replay, and a canonical SHA-256 hash chain ensures tamper-evident logs. We evaluate TWS through schema conformance, complete state lifecycles, deterministic replay, and fault injection. Passing 38 tests across Python 3.10-3.14, the framework detects record corruptions, broken hash links, sequence discontinuities, and world mismatches. Across ten public BEHAVIOR-1K task definitions, TWS imported 153 entities and 146 relations with successful validation. On 103 NVIDIA Unitree G1 simulated trajectories containing 78,369 frames, TWS achieved exact terminal-state replay in all episodes and detected 412/412 injected corruptions with a 1.72% storage overhead over Plain JSONL.

We show that there exists $c > 0$ such that any subset of $\{1,\ldots,N\}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quantitative bounds over integers were obtained.

Controlling the magnitude and polarity of spin-orbit torque (SOT) is essential for manipulating magnetization in spintronic devices. Here, we realize bulk spin-orbit torque (SOT) in a single-layer Mn1.6Co1.4Ga (MCG) Heusler alloy and achieve lattice strain-controlled reversal of SOT polarity. SOT-driven magnetization switching is achieved over a thickness range of 3-20 nm, with the switching polarity reversed between the strongly strained ultrathin regime and thicker films. First-principles calculations reveal a sizable intrinsic spin Hall conductivity (SHC) in MCG originating from its topological band structure, which is enhanced by the tetragonal distortion without changing sign. Together with the thickness-dependent structural evolution, we demonstrate that the reversal of SOT polarity does not originate from a sign change of the SHC, but rather from lattice distortion induced symmetry breaking that modifies the conversion of spin current into a net bulk SOT, thereby controlling its polarity. These results establish strain-controlled structural symmetry as an additional degree of freedom for tuning bulk SOT in magnetic single layers.

Large Language Models (LLMs) have renewed interest in autoformalization. Yet, when First-Order Logic (FOL) is considered as the target formalism, the field still lacks a unified task formulation and a systematic survey. This paper addresses this gap: we first provide a principled definition for the FOL-autoformalization task by distinguishing Ontology Extraction from Logical Translation, showing how their conflation obscures (cross-study) evaluation; we review existing datasets, evaluation metrics, and LLM-based methods, including fine-tuning, prompting, and verification-based refinement; we identify open challenges in benchmarking, semantic evaluation, ontology-aware methods, and end-to-end applications.

Embedding subsets of Wasserstein spaces into linear spaces is a problem of both theoretical and computational interest. In this work, we prove that three classical and natural embeddings of Wasserstein spaces into Banach spaces of non-trivial type are bi-Hölder on subsets of Wasserstein spaces given by moment bounds. These embeddings, all considered in multiple applications, are the linearized optimal transport embedding, the sliced-Wasserstein embedding, and a particular kernel mean embedding. Our results contrast with known obstructions to embeddin the full Wasserstein space into Banach spaces of non-trivial type. We also show the sharpness of our results regarding the moments required for our conclusions to hold.

Consider a finite population of agents with decoupled Markov transition dynamics and empirical-density feedback, subject to the following constraints: with probability at least $1-\delta_r$, at least a fraction $\alpha_r$ of agents must reach a target region at some time $t^*$, while, at each time up to $t^*$, the unsafe population fraction must remain below $\beta_u$ with probability at least $1-\delta_u$. However, standard mean-field methods enforce these constraints only in expectation, which fails to account for stochastic fluctuations at finite fleet size $N$. To address this control problem, we propagate the second-order moment (variance) of the empirical density alongside the mean-field trajectory via a discrete-time Lyapunov recursion, and apply the Cantelli inequality to convert chance constraints into tractable deterministic conditions on the moments of the empirical density. We then incorporate these moment-based surrogate constraints into a gradient-based sequential convex approximation procedure for density-feedback policy synthesis. We further introduce additional moment-error bounds to construct a rigorous finite-$N$ certificate. The method is evaluated on a gridworld environment and a power-system EV-charging aggregation problem and compared with a standard deterministic population-level LP baseline.

The oscillator groups are four-dimensional solvable Lie groups, extensions of the Heisenberg Lie group. We present them as circle bundles of the standard contact sub-Riemannian structure on $\mathbb{R}^{3}$. We define sub-Riemannian and sub-Lorentzian structures on them, describe their geodesics explicitly and determine which of them are periodic. We also study a generalization: Let $G$ be a semisimple Lie group and $K$ a compact subgroup such that $\left( G,K\right) $ is a symmetric pair of the compact or the noncompact type. Let $\mathfrak{g}$ and $% \mathfrak{k}$ be the Lie algebras of $G$ and $K$, respectively. We consider on $\mathfrak{g}$ the usual nilpotent Lie group structure with center $\mathfrak{k}$ and call it $N\left( \mathfrak{g},\mathfrak{k}\right)$. A suitable semidirect product with $\operatorname{Ad}\left( K\right) $ yields a solvable Lie group $\operatorname{Osc}\left( G,K\right) $, which generalizes the oscillator groups and is also quadratic (that is, it possesses bi-invariant metrics; in particular, a canonical one). We define nonholonomic pseudo-Riemannian structures on it and find their geodesics explicitly. In doing so, we obtain a result that may not be merely auxiliary: formulas for the monoparametric subgroups of $\operatorname{Osc}\left( G,K\right) $ and for the sub-Riemannian geodesics of $N\left( \mathfrak{g},\mathfrak{k}\right) $ with the standard left-invariant distribution $\mathcal{D}$. Moreover, when $G/K$ is compact with rank one, we realize $\operatorname{Osc}\left( G,K\right) $ as a Stiefel bundle of $\left( N\left( \mathfrak{g},\mathfrak{k}\right) ,\mathcal{D}\right) $, up to finite coverings.

Humanoid robots are a promising platform for general-purpose manipulation. Recent Vision-Language-Action (VLA) policies learn actions directly from multimodal observations, while World Action Models (WAMs) further incorporate future visual prediction to improve action generation. However, in hierarchical humanoid systems, VLA and WAM policies output reference actions that are subsequently realized through whole-body control, robot dynamics, balance, and contact. This hierarchy creates an action–execution gap: the reference produced by the policy can differ from the motion realized by the robot. Without explicitly modeling the realized body state, future visual prediction must jointly explain scene evolution and discrepancies between reference actions and executed motion, making it difficult to associate an action with its physical outcome. We propose HWAM, a Humanoid World Action Model with joint state–action generation, which makes the robot's post-execution proprioceptive state an explicit prediction target. By jointly generating reference actions and their realized body states, HWAM directly incorporates supervision of executed motion into action learning. HWAM is trained through three complementary conditional paths. The Policy path jointly denoises state–action trajectories conditioned only on current observations, matching deployment conditions. Forward Dynamics Modeling (FDM) predicts future visual observations conditioned on actions and post-execution states, while Inverse Dynamics Modeling (IDM) reconstructs the joint trajectory from visual transitions. Together, these paths connect policy references, realized body motion, and visual outcomes. HWAM achieves the highest success rate among evaluated baselines on three real-robot tasks on the LimX OLI humanoid. On Candy Picking, HWAM achieves a 70.6% success rate, compared with 43.3% for Fast-WAM.

The paper develops a theory for selecting the rolling window when generating out-of-sample forecasts with factor-augmented regression (FAR) models in the presence of structural instabilities. It shows how to select a rolling window by minimizing the conditional mean squared forecast error (MSFE) while accounting for uncertainty in factor estimation. Because the conditional MSFE is unobserved and the factors are latent, this paper proposes a feasible version of the criterion and derives conditions under which the new method is asymptotically loss-efficient. A simulation experiment documents the procedure's performance.

Securing bandwidth-constrained Internet of Things (IoT) and edge networks during post-quantum migration creates an authentication trade-off: ML-KEM provides post-quantum key establishment, whereas post-quantum signatures increase certificate-chain size, verification cost, and handshake latency. We present HPQ-AKE, a sign-less hybrid authenticated key exchange for RSA-enabled IoT gateways and edge/cloud services that replaces transcript signatures with dual KEMs. HPQ-AKE combines ML-KEM-768 for post-quantum session secrecy and forward secrecy with RSA-OAEP for implicit mutual authentication, preserving existing RSA-enabled infrastructure during migration. We analyze HPQ-AKE in an extended Bellare-Rogaway model, separating classical authentication in the Random Oracle Model from session-key secrecy in the Quantum Random Oracle Model. Under the Module-LWE and RSA assumptions, it establishes AKE security, forward secrecy, and conditional KCI resistance under the long-term-key-only exposure assumptions. Evaluation combines x86 micro-benchmarking, analytical latency modeling, and 10,000-iteration Monte Carlo simulation. HPQ-AKE reduces modeled handshake transmission from 13,009 to 5,668 Bytes, a 56.4% reduction relative to a Hybrid TLS 1.3 Full handshake, while requiring 7.11 ms of total local computation on the measured x86 testbed. Under a simulated 50 kbps satellite-like link with 600 ms RTT and stochastic jitter, median latency is 1,914 ms, 31.4% below the 2,791 ms baseline. At 20 ms RTT, modeled break-even thresholds are 0.31 Mbps against the pre-cached baseline and 4.08 Mbps against the full baseline; both decrease as RTT increases. These x86-based results motivate evaluation on gateway-class IoT and edge platforms; performance on ARM gateways and unaccelerated microcontrollers remains unvalidated.

Increasingly, AI interviewers are being developed to elicit open-ended responses in applications like market research, public polling, preference elicitation, and social science research. However, evaluating AI interviewers is challenging because they function in extended, multi-turn interactions where they must adapt to participant behaviors. To address this need, we develop InterviewPlayground, a simulation environment for evaluating AI interviewers using simulated study participants whose behaviors are grounded in social theory. Simulated studies in InterviewPlayground produce an InterviewReportCard, which assesses the performance of AI interviewers using a suite of validated measures. To test whether our simulation-based evaluations predict performance with human participants, we conduct 15 real qualitative studies with five AI interviewers, three interview topics, and 450 human participants and compare them to simulated studies in InterviewPlayground. We find that AI interviewer performance in InterviewPlayground predicts performance in human studies with an average Pearson correlation of 0.86 across 12 measures, and the simulated interactions from InterviewPlayground reproduce key findings from behavioral analysis of AI interviewers in the human studies. Together, these findings support the validity of InterviewPlayground in assessing AI interviewer performance and examining potential failure modes. Our work contributes a simulation environment for AI interviewers supported with empirical validation, and more broadly, a roadmap for future work to develop validated, simulation-based evaluations of conversational AI systems.

Large language models (LLMs) are increasingly deployed in sociotechnical systems where social attribution, the reasoning process attributing external events to the causes and reasons of agents' social behaviors, plays a critical role. These processes involve judgments of social cause, responsibility, and blame/credit to agents. Although attributional models are well-studied in social psychology and cognition through Attribution Theory, social attribution remains underexplored in AI, particularly LLM social reasoning. This paper provides the first systematic exploration of LLM social attribution. Our work focuses on responsibility and blame attributions, examining current LLMs' judgments and their underlying internal mechanisms. Guided by attribution theory, we construct a social attribution benchmark consisting of a Vignette subset based on classic scenarios from attribution theory research and a Reality subset based on real-world social narratives, yielding 7,639 responsibility/blame judgment questions. On this basis, we evaluate 32 representative LLMs and 5 basic non-LLM baselines. To further explore the internal mechanisms underlying the LLM judgment process, we develop a probing-based methodology to investigate the latent-space representations of 5 key attribution dimensions and the consistency of their influences on LLM judgments compared to those in human social attribution. Our research findings reveal that current LLMs exhibit measurable but incomplete agreement with human responsibility and blame judgments, and meanwhile, this agreement is positively correlated with model size. Some attribution dimensions are systematically decodable from specific positions in LLM hidden states, and their influences on the final judgment are consistent with those indicated by human Attribution Theory. The dataset and associated code are available at https://github.com/Yuzhaoxin946/SAB-Bench.

We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and $\infty$, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors–Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to $\mathbb{C}$ have the form $\nu(z)=a(\bar z/z)^{n-1}+\nu_0(z)$, where $a\in(-1,1)$, $n\ge1$ is an integer, and $\nu_0\in L^r(\mathbb{C})$ for $0<r<2$. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.

Ionisation control remains a key challenge in plasma-assisted processes and more particularly in magnetron sputtering. In this context, the former directly governs species transport and thin-film properties. Although HiPIMS substantially enhances the ionisation rate of sputtered species, its performance remains constrained by the coupling between sputtering and ionisation, which limits discharge tunability. Here, we introduce Fast High Voltage for Ionisation Improvement (FHiVI$^{2}$), a new transient electrical regime designed to improve the ionisation rate without requiring additional elements. By applying an ad hoc very short, high-voltage pulse during plasma sputtering processes, FHiVI$^{2}$ regime is expected to induce a strongly anisotropic electric-field burst intended to enhance sputtered metallic species ionisation while limiting back-attraction losses and sputtering. To assess the ionising potential of this regime, optical emission spectroscopy together with mass spectrometry were carried out. More particularly, mass spectrometry measurements show an improvement of the ionisation rate by a factor of 2.7 compared with the HiPIMS regime, for an applied voltage of 2 kV which correspond to an additional mean power of only 1.2 W, using a 2-inch tungsten target. These findings suggest that FHiVI$^{2}$ provides a flexible route towards improved ionisation control in magnetron sputtering and opens new perspectives for plasma-assisted processes.

Local coordinate-wise linear (LCL) properties provide a unified framework for studying list decoding, list recovery and other local properties of linear codes. We prove that random one-point algebraic geometry (AG) codes have the same LCL threshold rates as random linear codes under explicit alphabet and sampling conditions. More precisely, for a family of local profiles with random linear code threshold $R_{\mathcal P}$, a random one-point AG code avoids all such profiles below $R_{\mathcal P}-\varepsilon$ and contains one above $R_{\mathcal P}+\varepsilon$, with explicit failure bounds on both sides. Our proof also removes an auxiliary lower bound on the number of available rational places that arises in a direct adaptation of the Reed–Solomon argument and requires no additional genus-dependent rate gap. We also observe that the previously claimed above threshold proof for random Reed–Solomon codes does not establish the required probability lower bound. The containment relation used in that argument gives the probability comparison in the opposite direction from what is needed. We give a different above threshold argument, which in particular also yields an alternative proof in the genus zero case. Combining our threshold theorem with recent results for random linear codes gives a collection of new list decoding, average weight list decoding and list recovery guarantees for random one-point AG codes. We further obtain new correlated agreement and proximity gap results for polynomial curves. These results extend the LCL threshold framework from random linear and Reed–Solomon codes to random one-point AG codes.

Spintronic logic-in-memory provides a route to reducing data-transfer energy consumption and accelerating computation by integrating nonvolatile storage and logic functions within a single physical unit. Here, we investigate field-free spin-orbit-torque (SOT) switching and reconfigurable spin logic in a vertically composition-graded MnxCoAl/Pt heterostructure. In contrast to the uniform-composition Mn1.8CoAl/Pt and Mn2.5CoAl/Pt control devices, the Mn1.8-2.5CoAl gradient device exhibits deterministic SOT switching under zero external magnetic field. Dzyaloshinskii-Moriya interaction (DMI)-related measurements reveal a finite characteristic effective-field scale, indicating the involvement of chiral magnetization-reversal processes. Meanwhile, the vertical composition gradient introduces magnetic inhomogeneity that can modify domain nucleation and propagation, and the combined effects are consistent with the observed field-free switching behavior. The composition-gradient device further exhibits accumulative multistate switching with stable intermediate Hall-resistance states under zero magnetic field, whereas the uniform-composition controls do not show comparable stable multistate behavior. By tailoring the amplitude, polarity, and sequence of current pulses, the device can be tuned between binary and multilevel switching modes, enabling the experimental realization of AND, OR, NAND, and NOR Boolean operations within the same Hall-bar device without any external magnetic field. These results extend Mn-Co-Al Heusler heterostructures from field-assisted SOT switching toward field-free magnetization control, multistate operation, and reconfigurable spin logic through vertical composition engineering.

We consider reduced basis approximations of non-equilibrium steady states of parameter-dependent Lindbladians, whose ambient matrix space grows exponentially with system size. We give a unified structural view of the problem through its characterizations as an eigenvalue problem or as a trace-constrained source problem. Building on these characterizations, we propose a residual-driven reduced basis framework for both formulations. Under the assumption of a unique steady state, we derive residual-based state-error estimates, and establish (sub-)exponential convergence of the greedy algorithms under real-analytic parameter dependence of the problem data. Numerical experiments on boundary-driven Fermi–Hubbard and dissipative XYZ chains support the analysis and suggest that the trace-constrained greedy method offers the best overall balance of accuracy and computational cost.

World models for control must capture which aspects of the environment respond to the agent's actions and which are relevant to reward. Generative world models such as Dreamer 4 consist of a video tokenizer, which encodes each frame into a latent, and a dynamics model, which is pretrained to predict future latents from past latents and actions. Yet the tokenizer is trained with a reconstruction objective, without action or reward supervision, so its latent provides no explicit mechanism to separate controllable, uncontrollable, reward-relevant, and reward-irrelevant information. We propose CausalDreamer, which keeps the tokenizer frozen and re-encodes its latent into a factored representation of four groups along two axes: controllability, where only the two controllable groups receive the action, and reward relevance, learned by predicting the reward from the two reward-relevant groups. The pretrained dynamics model is then fine-tuned to predict the factored representation. We evaluate CausalDreamer and the pretrained world model it starts from with model-predictive planning on 20 MMBench2 tasks: 10 clean tasks seen during training and 10 unseen tasks, of which 6 are manipulated variants of clean tasks with a changed background, object, or maze layout, and 4 are new environments. We normalize returns so that a policy taking uniformly random actions scores 0 and an expert scores 1. CausalDreamer achieves a 14% higher normalized score than the pretrained world model on the clean tasks (0.199 vs.\ 0.175) and a 25% higher score on the manipulated variants (0.307 vs.\ 0.246), while neither model scores meaningfully above the random policy in the new environments. Additionally, our analysis shows that the factored representation separates reward-irrelevant changes, such as a changed background, from its reward-relevant groups.

Quantum fidelity kernels and quantum extreme learning machines share the same basic architecture: a fixed quantum device generates features that are combined through a classically trained linear model. We refer to this broad non-variational setting as quantum linear regression. A defining aspect of these models is that the quantum features are not directly available, but must be estimated from a finite number of shots. We show that finite statistics makes quantum linear regression unavoidably ill-conditioned. Ideal feature vectors are confined to a subspace determined by the dimension of the Hilbert space, whereas finite-shot estimates generically acquire components outside this subspace. Training then attempts to fit directions that disappear in the infinite-statistics limit, making the learned model increasingly sensitive to sampling noise. We analyze this mechanism for quantum extreme learning machines trained on $n_{\rm tr}$ states with $N$ shots per state. Despite the ill-conditioning, predictions remain controlled when the training targets are exact. Finite sampling produces a systematic prediction error that decreases quadratically with $N$, while fluctuations induced by the random training measurements decrease inversely with the total training budget. Thus, unlike in standard linear regression with exactly known features, increasing the dataset size alone cannot remove the error caused by sampling noise. If the training targets are also noisy, due for instance to state-preparation errors, the otherwise hidden directions become active. The test error can then increase with $N$, a behavior that we ascribe to a form of quantum overfitting. Our results expose a fundamental tradeoff between dataset size and per-state statistics in non-variational quantum learning and provide a baseline for resource-aware training and regularization strategies.

We prove that Brandt semirings over nontrivial groups with at least two indices admit no identity basis with a bounded number of variables. A uniform family of finite witnesses has size linear in the variable bound and applies to groups of arbitrary cardinality. A commutative construction yields the corresponding result for flat extensions of groups of unbounded exponent and gives an exact finite basis criterion for arbitrary flat group extensions. Structural representations lead to a classification of finite natural inverse semirings and a single identity characterizing locally Clifford natural inverse structure among regular semirings. We also describe products, orthogonal sums, minimum generators, colored graph identities and linear variable complexity. Further results concern infinite generating algebras and inherent nonfinite basability.

We prove an equidistribution result for Galois characters in the graded Lie components $L_n$ associated to the $\mathbf{Q}_p$-fundamental group of a projective hyperbolic curve $X$ over a number field $k$ with geometrically CM Jacobian. That is, we consider the $p$-adic completion of a fixed character lattice, where we define measures $\mu_n$ counting normalized character multiplicities in $L_n$, and show that $(\mu_n)_n$ weakly converges to the Haar measure. The key idea to prove convergence is to write the Fourier coefficients of $\mu_n$ in terms of traces of operators on $L_n$ induced by symplectic operators on the Tate module, and control them with a known symplectic character formula, followed by Weyl's criterion for compact abelian groups. Our main arithmetic application is to combine equidistribution with the Iwasawa-theoretic method of Coates–Kim and a compactness argument for moving analytic loci, giving a dimension inequality for unipotent Selmer schemes: $\dim_{\mathbf{Q}_p}H^1_f(G_{k,T},U_n)< \dim_{\mathbf{Q}_p}\prod_{v\mid p}H^1_f(G_v,U_n)$ for all sufficiently large $n,$ with the quotient having limit superior at most $1/2.$ This answers a particular case of a conjecture by Kim, from which the finiteness of $X(\mathbf{Q})$ can be recovered when $k=\mathbf{Q}.$

We prove interior $C^{1,\alpha}$ estimates for weak solutions of the fractional $p$-Poisson equation, with $p\geq2$. For right-hand sides in $L^q$, we obtain the estimate when $p<(1-d/q)/(1-s)$. For $\gamma$-Hölder continuous right-hand sides, the range is $p<(1+\gamma)/(1-s)$. Both ranges of parameters are optimal.

We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on $\mathbb R^2$. More precisely, for every $1<s<2$, $T>0$ and $\varepsilon>0$, we construct initial data $\theta_0\in H^s(\mathbb R^2)$ with $\|\theta_0\|_{H^s}\leq\varepsilon$ that generate a solution $\theta$ on $[0,T]$ such that \[ \theta\in L^\infty([0,T];H^{1+\gamma}(\mathbb R^2)), \quad \theta(t)\notin H^\sigma(\mathbb R^2) \quad\text{for every }t\in(0,T],\ \sigma>\sigma_*(s), \] where $\gamma>0$ and $\sigma_*(s)$ satisfy \[ 1+\gamma<\sigma_*(s):= \frac{s(3-s)}{1+2s-s^2}<s. \] The solution is unique in a determined family of classical solutions with initial datum $\theta_0$.

Predictive representation learning from photoplethysmography (PPG) can violate causal information access even with causal attention, as normalization, nonlocal transforms, or companion views may depend on withheld samples. We introduce PulseBound, a PPG representation learner combining physiologically structured future-beat prediction with an explicit stored-window information boundary. A content-independent cutoff separates the visible prefix from the prediction target. Prefix-only normalization, suffix replacement before derived-view construction, and aligned masking ensure that encoder inputs depend only on the visible prefix and cutoff. This yields stored-suffix invariance: with fixed model state, randomness, prefix, and cutoff, changing the stored suffix cannot change the forecast context. A shared horizon-conditioned head predicts nine rhythm and morphology descriptors for up to four extractor-valid future beats, using elementwise validity masks; optional ECG-derived pulse-arrival-time supervision is restricted to training. On MIMIC and VitalDB groups held out from PulseBound backbone pretraining, PulseBound reduces nine-state transformed-space MAE relative to last-visible-beat persistence by 28.06% and 22.22%, respectively, with gains in MAE, MAE-Skill, and Spearman correlation across all 40 source-cutoff-horizon cells. In a separate comparison of seven models on 13 downstream tasks, PulseBound achieves the best mean on nine frozen linear-probe and seven full-fine-tuning tasks. Stored-suffix interventions cause zero recorded changes in forecast contexts or predictions, with zero suffix-input gradients at audited precision under the stored-window interface. These findings separate three testable aspects of predictive physiological representation learning: information access, supervised future structure, and transfer.

Physics-informed machine learning has gained significant attention in recent years. In regimes of limited data, parametrized Gaussian processes have become popular. Existing approaches, however, often face limitations, such as requiring parametrizable (also called controllable) systems or a large number of output tasks. In this work, we introduce a systematic procedure we call "ghost tasking", using auxiliary tasks to circumvent these limitations. We prove that such ghost tasks can render any non-parametrizable system effectively parametrizable, enabling algorithmic construction of parametrized Gaussian Processes while keeping the number of required tasks (i.e. output dimensions) and latent functions low. We find that ghost tasking performs especially well in an inverse problem setting, even with very few available data. We show the usage and power of ghost tasking in three experiments, providing systematic comparisons to the only other currently available method applicable to all experiments. We provide necessary syntax and explications for two computer algebra programs that compute parametrizations for systems with polynomial or rational coefficients. Our theoretical results extend to systems with meromorphic functions.

Abstract argumentation frameworks (AFs) introduced by Dung provide a formal foundation for non-monotonic reasoning in artificial intelligence. While decision problems for general infinite AFs typically reside at high levels of the analytical hierarchy ($\Sigma_1^1$ or $\Pi_1^1$), restricting the framework to be computably finitary reduces some of the complexity to the arithmetical hierarchy. In this paper, we present a complexity mapping of grounded and preferred semantics in computably finitary AFs across standard decision problems: credulous acceptance ($\Cred$), skeptical acceptance ($\Skep$), extension existence ($\Ex$), uniqueness ($\Uni$), and non-empty existence ($\NE$). For grounded semantics, credulous and skeptical acceptance are already known to be $\Sigma_1^0$-complete. We show that non-empty existence is also $\Sigma_1^0$-complete, whereas existence and uniqueness are trivial. These classifications are understood within the domain of valid computably finitary representations. For preferred semantics, using a computably finitely branching computation tree, $\Cred_{\pref}$ is shown to be in $\Pi_1^0$-c and $\NE_{\pref}$ is $\Sigma_2^0$-c. However, it is insufficient to reduce universal quantification and global uniqueness, leaving $\Skep_{\pref}$ in $\Pi_1^1$ and $\UniPref$ in $\Sigma_2^1$-c. Our results show the precise boundary where finitarity succeeds to bring reasoning down to the arithmetical hierarchy and where second-order quantification forces problems back into the analytical hierarchy.