Papers

We study the offline gap between deterministic calibration distance C and its fractional relaxation L for binary unit-weight sequences under total absolute-change cost. We sharpen the offline comparison C <= L + O(sqrt(T)) (Qiao and Zheng, 2024, Theorem 2) to the sharp worst-case order Theta(T^(1/3)). If Delta_T is the supremum of C - L over length-T inputs, then T^(1/3)/1000 <= Delta_T <= 41T^(1/3) for T >= 216. The upper bound holds for every input, while each T >= 216 has a rational lower-bound input. For every input with m distinct forecasts, C <= L + m, and the unrestricted-sample worst-case sparse order is Theta(m). For rational forecasts and accuracy, with binary-encoded multiplicities of separately assignable unit identities, a grid-free polynomial-bit-time procedure returns B <= L <= U, U - B < eta, and an exactly calibrated compact repair of cost at most U + m <= L + m + eta.

Most humanoid loco-manipulation controllers require human motion data to learn whole-body coordination and posture, leaving policies reliant on external sources to provide this data. We present OCLO (Online-posture Compliant LOco-manipulation), a humanoid loco-manipulation system trained without human motion data and commanded only through two end-effector targets. Because these targets do not uniquely determine whole-body posture, OCLO generates pelvis height and torso orientation online using an analytic reachability prior, further refined through policy-in-the-loop sampling with a task-agnostic cost. OCLO also learns whole-body compliance by displacing end-effector references according to measured forces through a spring-damper model, encouraging the legs, waist, and pelvis to yield to external loads. In simulation, using the reachability prior leads to a 77.8% success rate in acquiring the commanded reference, a vast improvement over the 37.8% success rate accomplished without the prior. Further, refinement reduces end-effector orientation error across all evaluated tasks. The same posture module improves a pretrained SONIC controller on four of five tasks. Without compliance training, policies tend to lose balance under disturbances rather than sacrifice tracking. On a Unitree G1, OCLO maintains balance under end-effector disturbances that cause its ablations to fail and performs seven loco-manipulation tasks, including crouched walking and picking up a box from a low surface. Project website: https://oclo-humanoid.github.io/

Multi-robot imitation learning, particularly in settings where visuomotor policies are deployed in a communication-free, onboard decentralised style, represents an attractive paradigm. However, its realisation remains insufficiently understood, largely due to the difficulty of collecting collective demonstrations, since a single operator cannot control many robots simultaneously. Meanwhile, unlike coupled collaborative manipulation, many coordinated tasks achieve system-wide efficiency primarily through minimising inter-robot interference. This structure motivates us to study whether data collected by a teleoperated single-robot can be leveraged for large-scale coordinated box-pushing as a testbed. We systematically investigate dataset creation strategies and lightweight policy architectures. In particular, experiments with up to 40 robots highlight the difficulty of acquiring effective coordination solely through passive observation of other operating robots, revealing a concrete bottleneck for multi-robot research.

Groups of individuals can solve collective problems more accurately than any single member, by aggregating their opinions. Recent theoretical work has identified individual-level reward schemes that allow uninformed individuals to evolve collective intelligence from the bottom up, through social learning. Yet these results are restricted to linear prediction problems and simple averaging, while the decision tasks that real groups confront are often non-linear, and the institutions that aggregate opinions are seldom single-layer averages: districts elect representatives who in turn vote on policy, referees advise editors who decide on publication. Here we develop a framework for the evolution of collective intelligence in multi-layer voting populations, where individuals observe limited information and groups recursively aggregate their opinions by majority rule. We prove that single-layer voting cannot solve non-linear classification problems under any individual reward scheme. We then identify a "marginal feedback" payoff structure, which rewards individuals only when their opinion is pivotal in their group, and at every layer above them. This reward scheme induces a layered population to evolve accurate collective solutions to complex, non-linear decision tasks through individual-level peer imitation alone. The collective behavior that emerges is equivalent to a multi-layer perceptron in machine learning. Our results provide a naturalistic account of hierarchical institutions, in which the outsize importance of swing voters is the incentive that sustains collective accuracy; and they identify the credit-assignment rule in machine learning as not just an engineered solution but a natural evolutionary outcome.

Feller random measures generalize the Feller diffusion (the CIR process) by giving it memory. They arise as the scaling limits of nearly unstable Hawkes processes, and include the rough CIR process and its hyper-rough and discontinuous relatives. We show that every Feller random measure is the occupation measure of a Dawson–Watanabe superprocess whose spatial motion is a killed Lévy subordinator, integrated over the branching time. This is a continuum analogue of the Hawkes–Oakes cluster representation. It yields existence, stability in the parameters, and stochastic equations driven by an explicit noise. The structure of this noise depends on whether the kernel has an atom at the origin. Without an atom, the noise is a Brownian motion time-changed by the distribution function of the measure. This leads to a martingale characterization and to sharp results on densities and their Hölder regularity. With an atom, the noise is a compensated inverse-Gaussian process time-changed by the compensator of that distribution function, and the measure is purely atomic.

In GPS-denied or cluttered urban environments, vision-based landing is essential for reliable UAV missions. Real-world landing sites are often unstructured and highly variable, requiring strong generalization by the perception system. Deep Neural Networks (DNNs) trained with synthetic data augmentation offer a scalable solution for learning landing-site features across diverse vehicle and environmental states. However, computationally expensive DNN retraining, along with challenging performance validation via test flights, limits exhaustive model fine-tuning and necessitates an optimized retraining pipeline. In this work, we deploy a Bayesian data augmentation framework integrated with a photorealistic simulator featuring high-fidelity vehicle dynamics to iteratively retrain the helipad detector DNN, maximizing landing performance as the objective function. We validate our framework with experiments in a photorealistic simulator under different environmental conditions and vehicle states, demonstrating improved landing performance and tighter confidence intervals on predicted landing outcomes.

Accurate and low-complexity indoor localization is important for location-based services in fifth generation (5G) and sixth generation (6G) networks, where positioning devices operate under limited computational budgets and non conditions. Indoor localization has been studied widely using traditional signal-level localization approaches. However, these techniques often show degraded performance in on-line-of-sight (NLoS) scenarios. Recently, artificial intelligence (AI)-based techniques, including transformers, have been applied to address these challenges. While transformer-based architectures can capture the dependencies within the measurements collected from distributed access points (APs), their high computational complexity results in a large number of multiply-accumulate operations and long inference time. In contrast, lightweight recurrent and convolutional models trade this cost for degraded accuracy. In this paper, we propose LocAttMamba, a low-complexity localization framework in which the channel impulse response (CIR) and time-based features of each AP are processed by a separate Mamba encoder with near-linear complexity, and the resulting per-AP embeddings are fused through a multi-head attention layer that weights each AP according to its importance at every time step. The framework jointly predicts the user location and its per-axis uncertainty, which is refined through a post-hoc calibration step. We evaluate the proposed framework using two real-world 5G and ultra-wideband (UWB) measurement datasets. Our numerical results reveal that LocAttMamba obtains a mean two-dimensional (2-D) positioning error of 1.004 m and 0.599 m, respectively, on 5G and UWB datasets, outperforming the second-best benchmark by 9.79% and 5.82%, while requiring the fewest multiply-accumulate operations among all evaluated models and being 4.4-16 times faster than the transformer-based benchmarks.

In this paper, a complete characterization of the boundedness, compactness, norm and essential norm of the differential Hilbert operator $\mathcal{H}_2:A^2_\alpha\to A^2_\beta$ is obtained. More precisely, $\mathcal H_2:A^2_\alpha \to A^2_\beta$ is bounded if and only if $-1<\alpha<0$ and $\beta\geq\alpha+2$ and it is compact if and only if $-1<\alpha<0$ and $\beta>\alpha+2$. The norm and essential norm of $ \|\mathcal H_2\|_{A^2_\alpha\to A^2_{\beta}}$ are also investigated. In particular, when $\beta=\alpha+2$, \[ \|\mathcal H_2\|_{A^2_\alpha\to A^2_{\alpha+2}} =\|\mathcal H_2\|_{\mathrm e,A^2_\alpha\to A^2_{\alpha+2}} =\frac{\pi\sqrt{(\alpha+2)(\alpha+3)}} {\sin(\pi(\alpha+2)/2)},\qquad -1<\alpha<0. \] When $\beta>\alpha+2$, $\|\mathcal H_2\|_{\mathrm e,A^2_\alpha\to A^2_{\beta}}=0$. Furthermore, for every $1\leq p<\infty$, the operator $\mathcal H_2$ belongs to the Schatten class $\mathcal S_p$ if and only if it is compact.

Automatic prompt optimization (APO) has been widely employed to adapt large language models without updating their weights, yielding promising results. However, existing methods such as GEPA and OPRO assume hundreds to thousands of subject-model calls, far more than is practical behind paid, rate-limited APIs. Under tight budgets they fail in two ways: multi-stage pipelines can exhaust the budget and return the seed prompt unchanged, while single-stage methods compare candidates on fixed-size minibatches, regardless of each task's noise. As a remedy, we introduce BudgetAPO, a single-stage optimizer for the tight-budget regime. BudgetAPO incorporates (1) a noise-adaptive rule that sizes the evaluation slice to each task's noise, measured by a short probe; (2) a fixed slice that turns every accept/reject decision into a paired comparison; and (3) a reflective operator that rewrites reasoning strategy and output format jointly. Extensive results across seven benchmarks and five subject models demonstrate that BudgetAPO ranks first on every subject and beats every baseline under Holm-corrected paired tests, while returning the seed in 13% of runs at 250 calls against 86% for GEPA. On GPT-OSS-20B, GEPA needs 5 times as many calls to match \method's 100-call score.

Generative recommendation (GR) represents items with semantic IDs (i.e., discrete token sequences) and generates target item tokens as recommendations. Despite its promising results, existing methods predominantly optimize for accuracy while neglecting the credibility of the recommendations they generate. This oversight inevitably exposes users to uncredible content (e.g., fake news) with serious societal consequences, including user distrust, reputation harm to platforms, and broader social instability. To address this critical yet underexplored challenge, we propose CreGR, the first credible GR model that jointly tackles content credibility across the two core stages of GR: tokenization and generation. In the tokenization stage, we design a new credibility-aware tokenizer that explicitly encourages the model to learn discriminative tokens respectively for credible and uncredible items, thereby disentangling credibility signals at the token level. Building on this, in the generation stage, we propose a novel accuracy-preserving and credibility-oriented generator grounded in discrete diffusion. Specifically, we introduce an asymmetric masking probability reduction strategy that selectively diminishes the contribution of tokens associated with uncredible content to the generation process, while leaving tokens encoding user preference signals unaffected so as to preserve recommendation accuracy. Experiments on three real-world datasets demonstrate the effectiveness of CreGR.

Communication using quantum channels generally requires encoding and decoding on many identical uses of a quantum channel. In practical settings, decoherence may severely limit our ability to do so, potentially rendering each channel use nonidentical. Given $n$ nonidentical channels, we present an entanglement-assisted encoding and decoding strategy that yields a channel with higher worst-case fidelity than all of the $n$ given channels. The strategy is universal, in the sense that the improvement holds regardless of what channels are given, as long as they satisfy mild assumptions on their initial entanglement fidelities. Our protocol is optimal when the given channels are depolarizing channels, and is essentially the unique one that is universal. This idea can also be extended to classical channels, where shared randomness between the sender and the receiver allows universal enhancement of the probability of correct transmission. We then consider general quantum resource theories, providing a necessary condition and a sufficient condition for the possibility of such universal improvement of quantum resources.

In this paper, we provide an algebraic proof of the condition for the existence of modular form solutions to the Kaneko-Zagier differential equation of weight $k$. Unlike previous representation-theoretic approaches relying on $SL_2(\mathbb{Z})$, our method employs the connection matrices of the principal congruence subgroup $\Gamma(N)$. By deriving a condition for simultaneous triangularizability from the commutators of the representation matrices and applying Galois theory, we prove that the associated two-dimensional representation of $\Gamma(N)$ is irreducible when the denominator $m$ of the fraction $(k+1)/6 = n/m$ satisfies $m=1$ or $m \ge 7$. Consequently, we identify the weights that admit no modular form solutions and obtain a complete classification of the dimensions of the spaces of such solutions.

A series of 11.6-d-duration photon-counting experiments, employing a broad variety of light sources with different statistical properties and optical spectra, were carried out over a 1.5-yr period. All of the photon-count spectra at baseband followed a common $1/f^2$ form over the frequency range $1 \times 10^{-6} \leqslant f \leqslant 5 \times 10^{-4}$ Hz, corresponding to a timescale range $33$ min $ \leqslant T_f \leqslant 11.6$ d, where $T_f \equiv 1/f$. The lower and upper timescale limits were established by the photodetector noise floor and the duration of the individual experiments, respectively. Unlike ordinary Brownian motion, all of the measured photon-count sample paths exhibited irregular long-timescale fluctuations, with durations ranging from hours to days. It has been established that the photon-count spectra cannot plausibly be ascribed to fluctuations of the current, voltage, or temperature of the source or the detector, nor to technical sources of noise associated with the optical system or local environment. Although the physical origin of the photon-count fluctuations remains unresolved, several heterodox hypotheses are set forth. From a statistical point-of-view, the photon counts appear to follow a doubly stochastic Poisson process with a slowly varying random intensity. Analogous experiments that rely on ionizing radiation and direct-conversion solid-state radiation detectors are proposed.

Distributed optimization over networks requires agents to repeatedly exchange decision variables with their neighbors. When the decision dimension $d$ is large, these exchanges dominate the communication cost, which is critical for bandwidth-constrained agents. Existing remedies quantize or sparsify the exchanged vectors, yet each message still scales with $d$ and the compression error must be compensated by additional states. To address this limitation, we propose a scalar-communication mechanism in which every neighbor message carries a single real number regardless of $d$. Agents regenerate a common random direction from a shared seed, transmit only the inner product of their state with that direction, and act on the resulting rank-one surrogate of their neighbors' states while retaining full local gradients. We develop and analyze the mechanism for an existing continuous-time distributed optimization algorithm. For strongly convex local costs with Lipschitz gradients, we show that the optimizer remains the unique consensus equilibrium, that a fixed direction admits spurious equilibria, and that refreshing the direction at a sufficiently high rate yields exponential mean-square and almost-sure convergence with constant gains and no residual error. The framework admits any isotropic fixed-norm direction distribution, including Rademacher, scaled-coordinate, and sphere-normalized Gaussian directions; all three attain lower fresh-encoding variance than unnormalized Gaussian directions. The effects of the direction distribution and the refresh interval are illustrated in~simulations.

Partly motivated by enumeration of parking functions and their variants, there is a long-standing interest in lattice paths refined by several statistics, including the notable $\mathsf{run}$ and $\mathsf{return}$. We consider bounded lattice paths, which are bounded by a given lattice path and are related to vector parking functions, for which we generalize $\mathsf{run}$ to composition runs, parameterized by a composition. By constructing involutions on these paths, we establish symmetries relating composition runs to some generalized return statistics. As an application, we settle an open problem of Dai, Fu, and Qiu on rational Dyck paths. The symmetries between generalized $\mathsf{run}$ and $\mathsf{return}$ are then transferred to vector parking functions, which suggest a new notion of prime decomposition. In the special case of $(a, b)$-parking functions, we compute the generating function refined by $\mathsf{run}$ and $\mathsf{pri}$, which is new even for classical parking functions.

Vision-and-Language Navigation (VLN) policies increasingly benefit from strong semantic priors provided by large vision-language models (VLMs). However, standard action supervision does not explicitly encourage intermediate representations to preserve scene geometry, relative orientation, or global episode progress. Incorporating depth estimators, explicit maps, point clouds, or geometry foundation models at inference can provide such structure but introduces additional computation, memory overhead, and architectural dependence during deployment. We introduce StageVLN, a training framework that shapes navigation representations through privileged spatial and trajectory guidance while preserving the original inference pathway. A frozen geometry foundation model provides multi-level spatial guidance to hierarchical navigator states, while relative-heading and expert-route progress objectives provide complementary trajectory-state supervision. All auxiliary components are used only during training and removed at deployment. On R2R-CE validation-unseen, StageVLN achieves 56.3% SR and 51.4% SPL with a 4B-parameter backbone, without an additional geometry encoder at inference. On RxR-CE, it achieves 54.3% SR without additional navigation training data or a geometry encoder at inference.

We study the hitting time index, a graph invariant defined in terms of expected hitting times of simple random walks on finite connected graphs, through its relation with the Kirchhoff index. Using the commute time identity and Tetali's formula, we decompose the hitting time index into a Kirchhoff-index term and a nonnegative asymmetry term expressed by degree-weighted effective resistance sums. We derive lower and upper bounds for the asymmetry term, together with their equality conditions, and obtain a lower bound for the hitting time index in terms of the range of the degree-weighted resistance sums. For trees, we obtain lower and upper bounds for the asymmetry term, with the upper bound expressed in terms of the Wiener index, and characterize the equality case in the lower bound. In particular, we show that among all trees of fixed order, the hitting time index is minimized uniquely by the star graph. Finally, we apply the decomposition to complete bipartite graphs, paths, and graphs obtained by conjoining complete graphs, and recover several known hitting-time formulas.

We develop a Bayesian generalization of the previous method to constrain a wide variety of mass-concentration (M-C) relations motivated by both theoretical and observational studies. The framework incorporates the intrinsic scatter in concentration, nonlinear weak-lensing (WL) calibration relations, the WL measurement-error covariance, and the underlying parent mass distribution. We employ Hamiltonian Monte Carlo for computational efficiency. Our sample consists of 213 X-ray-selected clusters with WL measurements for the eRASS1, XXL, LoCuSS, and MCXC catalogs at 0<z< 0.9. For each subsample, we account for its specific WL calibration relation and underlying population distribution. We consider a conventional single power-law model, an empirical double power-law model, and four theoretical models calibrated against numerical simulations. The four theoretical M-C models comprise two in which concentration decreases with mass, distinguished by the sign of d^2ln c/d(ln M)^2, and two in which the concentration decreases with increasing mass before turning upward at high masses. Because the mass and redshift ranges of our cluster sample do not extend into the upturn regime predicted by numerical simulations, we distinguish among the models primarily through the curvature of their M-C relations. Regardless of which model is adopted, the inferred M-C relation is in good agreement with the results of recent numerical simulations. We find no statistically significant differences among the models above the $4\sigma$ level. However, the result shows a tendency toward a gradual flattening of the mass dependence toward higher masses. Specifically, a monotonically decreasing model with positive curvature is preferred over the single power-law relation and a monotonically decreasing model with negative curvature at the $2\sigma$ level and over three double power-law models at the $0.5-1\sigma$ level.

Let $s(n):=\sum_{d\mid n,\,d < n} d$ denote the sum of the proper divisors of $n$. Distinct positive integers $n,m$ form an amicable pair if $s(n)=m$ and $s(m)=n$. Let $A(x)$ count the positive integers not exceeding $x$ that belong to an amicable pair. We report on a proof, found by ChatGPT Astra, that $$ A(x)\le x\exp\{-(1/2+o(1))\log x\,\log_3 x/\log_2 x\}$$ as $x\to\infty$. Here $\log_k$ denotes the $k$th iterate of the natural logarithm.

We give non-asymptotic operator-norm tail bounds for random multi-matrix models with quadratic plus globally Lipschitz interactions. More precisely, let $\mu_f$ be the probability measure on $(\mathbb{M}_n)_{\operatorname{sa}}^m$ with density proportional to $\exp(-n^2(\lVert x \rVert_2^2/2+f(x)))$. If $f$ is $L$-Lipschitz with respect to $\lVert \cdot \rVert_1$, we construct a coupling of $X\sim\mu_f$ and an $m$-tuple $Z$ of independent GUE matrices such that $\lVert X-Z \rVert_\infty\leq L$ almost surely, where $\lVert \cdot \rVert_\infty$ is the maximum of the operator norms of the matrix tuple. This is an operator-norm analog of the $L^\infty$-Wasserstein bound of Khudiakova, Maas, and Pedrotti (2025). We give a new proof based on the variational formula of Boué and Dupuis (1998), the associated Hamilton–Jacobi–Bellman equation, and the duality between entropy and pressure.

We study linear bandits with memory, where past actions induce endogenous nonstationarity through an arbitrary known, bounded matrix-valued memory map. To trade off exploration and exploitation while accounting for the memory dynamics, we develop RSM-LinUCB, a Bellman-centric algorithm that learns as in linear bandits and plans as in reinforcement learning. This design admits a novel regret decomposition which separates the memory-induced error from the cumulative reward estimation error along the learner's trajectory. We prove a high-probability regret bound of $\widetilde O\big(dRS(M+1)+\sigma d\sqrt T\big)$, where $T$ is the learning horizon, $d$ is the parameter dimension, $M$ is the memory length, $R$ and $S$ bound the memory-map operator norm and reward-parameter norm, respectively, and $\sigma$ is the sub-Gaussian noise scale. Our results reveal that the multiplicative memory-horizon coupling in prior bounds is not intrinsic: memory only contributes an additive cost, up to logarithmic factors. We also prove a matching minimax lower bound, establishing near-optimality. We further extend the algorithm to generalized linear rewards, preserving this separation with near-optimal memory and leading statistical dependence. Our algorithms outperform the baselines in numerical experiments on synthetic instances and semi-synthetic KV- and semantic-cache tasks.

Learning-based wireless research increasingly combines simulation, external model training, and benchmarking. Rebuilding these stages for every experiment is time-consuming, while differences introduced between them can silently change the conditions of a comparison. We present NOEMA, an open-source toolkit that prepares model development and baseline evaluation from a shared, machine-checkable wireless scenario. From the same scenario, NOEMA can prepare benchmark execution, capture aligned training data, and export differentiable components for model development. Researchers can therefore focus on training and selecting their models, then return the selected component to the same controlled experiment for evaluation. NOEMA checks the returned model and verifies that declared comparison conditions, including available information, random conditions, and failure accounting, remain consistent through benchmarking and reporting. An included agentic-control example applies these contracts to a language-model supervisor that selects allocation policies and power budgets from historical link feedback. All 43 regression tests of the workflow checks produced their expected outcomes, including rejection of all 31 cases containing known inconsistencies. Separate fidelity tests showed that captured data and returned-model behavior were preserved. In a 20-replicate quadrature phase-shift keying phase-tracking study, the mean bit-error-rate difference between the learned receiver and five- pilot smoothing was -0.01526 (95% confidence interval: -0.01547 to -0.01506). NOEMA therefore provides both a reusable path for developing learned wireless components and a way to keep their final comparisons inspectable and checkable.

Global magnetohydrodynamic (MHD) simulations provide the complete plasma state of the heliosphere, but their computational cost limits applications requiring repeated model evaluations. We introduce SNOW (Solar Neural Operator for Wind), a fast and differentiable surrogate for global MHD solar wind. SNOW combines a rotational ballistic approximation, a spherical harmonic neural operator, and causal radial context to autoregressively propagate all eight MHD variables from 20 to 240 Rs through consecutive shell to shell transitions. Trained and evaluated using AWSoM simulations spanning a solar cycle, SNOW remains stable across the complete rollout. The model successfully reproduces the large scale plasma and magnetic field structures, including fast and slow wind streams, magnetic polarity sectors, vector orientations, fieldline connectivity, and bulk-flow structures. Additionally, the magnetic divergence and induction diagnostics remain comparable to the reference MHD solutions. The complete three-dimensional solution is generated in less than one second on a single GPU. These results demonstrate the feasibility of fast neural emulation of global MHD solar wind solutions while retaining physically meaningful plasma and magnetic field structure.

We construct an exact NUT-charged generalization of the Ellis-Bronnikov wormhole in the Einstein-phantom-scalar system. The solution is stationary, asymptotically locally flat at both ends and free of curvature singularities. It inherits characteristic features of the Taub-NUT geometry, including the Dirac-Misner string and closed timelike curves, while the absence of Killing horizons eliminates the horizon-induced infinite winding of geodesics and renders the spacetime geodesically complete. By allowing the phantom field to have a nontrivial scalar potential with an infinite sequence of AdS vacua, we also obtain an exact asymptotically locally AdS wormhole with a NUT charge, which interpolates between different vacua at its two asymptotic ends. Using the covariant phase space formalism, we show that the Hamiltonian charges associated with the two asymptotic regions are finite under appropriate scalar boundary conditions. By performing a double Wick rotation to Euclidean signature, we demonstrate that the conformal class of the resulting metric admits an ambi-Kähler structure.

In this paper we consider the 4th order (or biharmonic) nonlinear Schrödinger (NLS) equation in dimensions 1, 2, and 3, where the potential term is expressed as a power non-linearity (for any positive power) and the dispersion operator has the fourth order combined with the lower second order. The fourth order NLS equation has recently attracted attention of researchers since quartic solitons have been experimentally obtained in optics, and thus, mathematical theory of solutions to the 4th order NLS equation in physical dimensions $N=1,2,3$ is timely to develop. In this work we show the local well-posedness of the 4th order or bi-harmonic NLS equation on a weighted subset of a Sobolev space as well as in $H^s$ spaces.

For a finite group $G$, we define a constant spectral $G$-Mackey functor as a genuine $G$-spectrum in which the restrictions between categorical fixed points are equivalences. We further identify the full subcategory of constant spectral $G$-Mackey functors with the category of modules over a certain $\mathbb{E}_1$-ring spectrum $R_G$. When $G=C_{p^n}$ is a cyclic $p$-group, we provide an explicit construction of the $\mathbb{E}_1$-ring spectrum $R_{C_{p^n}}$. The proof uses a reconstruction theorem for genuine $C_{p^n}$-spectra, which might be of independent interest. As an application, we compute the $\mathbb{Z}$-graded $C_{p^n}$-equivariant mod $p$ Steenrod algebra.

Graphs provide a natural representation of many complex systems, ranging from social platforms to ecosystems. However, the development of graph-based machine learning methods is often constrained by the limited availability of large and diverse graph datasets. In this paper, we introduce $\texttt{DCBA}$, a model-based approach to graph data augmentation that infers the configuration of a synthetic graph generator from an observed network. We instantiate the proposed framework using the $\texttt{ABCD}$ generator, which produces scale-free networks with community structure. Our model learns a joint representation of graphs and generator parametrisations using a multi-positive contrastive objective with soft negative weighting. The learned representation enables the prediction of an $\texttt{ABCD}$ configuration whose stochastic realisations preserve the macrostructural properties encoded by the generator. Experiments show that $\texttt{DCBA}$ recovers generator parameters more accurately and robustly than an algorithmic inverse-modelling baseline. Its downstream utility is further demonstrated in community detection, where inferred configurations used to fine-tune $\texttt{PRoCD}$ improve AMI on average by $161\%$ on synthetic and $273\%$ on real-world networks.

This work is devoted to a systematic study of slope stability for $\mathbf Z_p$-towers of the projective line over finite fields of characteristic $p$, ramified only at infinity. Our aim is to understand to what extent geometric stability of the genus sequence, or of the ramification break sequence, implies arithmetic stability of the Newton slopes of the zeta functions of the curves in the tower. We prove positive results and construct counterexamples that clarify the boundary between slope stability and its failure, with sharp bounds in several cases. For towers with eventual minimal ramification break ratios, a condition stronger than genus stability, Kosters and Zhu proved slope stability under an additional degree-gap condition. We construct towers with minimal break ratios that are not slope stable, answering negatively both their question of whether the gap can be removed and their broader question of whether genus stability implies slope stability. Under the minimal-break hypothesis alone, we prove exact partial slope stability on growing intervals approaching the first block, with uniform comparisons over explicit boundary annuli and finite flat spectral pieces. Further counterexamples distinguish partial from complete first-block propagation, first-block from two-block propagation, and full classical stability from a uniform annular slope law below one. We obtain improved bounds for the classical degree-gap constant, the partial-stability loss, and the sufficient starting levels, all independent of the constant-field degree. The proof uses a separate precision theorem for the semilinear Frobenius on finite contact blocks.

Mission-critical edge applications in 6G-and-beyond networks, such as autonomous systems, disaster response, and infrastructure monitoring, require that the edge decision-maker's estimate of a monitored process remain correct, not merely up to date. Satellite-ground integrated networks (SAGIN) often provide the only connectivity in infrastructure-limited or disaster-affected regions, yet satellite handover and shadowing interrupt links, during which the process may change state several times, leaving the edge node's estimate substantially wrong. Age of information (AoI) tracks only elapsed time and cannot distinguish a harmless delay from a dangerous error. We instead adopt the age of incorrect information (AoII), which penalizes both the duration and magnitude of estimation error, and formulate, to our knowledge, the first network-level, multi-node AoII minimization problem over SAGIN under stochastic handover and shadowing. We propose SENTINEL, a distributed hybrid quantum-classical framework that jointly schedules update rates, satellite-to-base-station associations, and bandwidth allocation to minimize the worst-case time-average AoII. Because AoII is history dependent, it resists per-slot optimization; a renewal-interval decomposition that separates source dynamics from channel disruption yields a closed-form AoII cost per inter-delivery interval. The resulting robust scheduling problem, VANGUARD, is formulated as a QUBO, mapped to an Ising Hamiltonian, and solved via distributed QAOA with ADMM-based coordination across satellite and ground domains. Every returned schedule carries a certified worst-case AoII over all disruption scenarios. Simulations show that SENTINEL outperforms learning-based and random baselines, matches a state-aware threshold policy in small networks while additionally providing a worst-case guarantee, and remains close to an exact minimax reference.

When auction bidders misperceive their rivals, a seller can change not only the amount of information rent but also the states in which it is delivered. We ask when this payment-design opportunity survives a finite bidding interface, nonnegative-payment restrictions, and comparison with conventional auctions under the same beliefs. We formulate payment design as a linear program in nonnegative bidder rents. A single state can safely carry a report's rent when it is both cheapest for its intended type and least attractive to potential mimics. We characterize this alignment, bound the cost of a restricted payment architecture when alignment is imperfect, and use shadow prices when the rankings conflict. Own-type projection yields a unique optimal expected-payment rule for a fixed allocation. Contiguous bid bands implement truthful reporting at every value, but create pooling losses and within-band surplus. Under a separate smooth-belief model, feasible truthful rules and revenue upper bounds support comparisons with a first-price equilibrium under the same beliefs. A calibration using 1,000 eBay proof-set auction records illustrates a partition-dependent reversal: first price dominates all exactly truthful, no-subsidy rules on ten equal-probability bands in one scenario, while a richer contingent rule on ten equal-dollar bands outperforms it. An operator should choose interface resolution, payment dependence, and integrity tolerances jointly. Validate the probabilities that support incentives, compare formats under the same beliefs, and distinguish redesign from deployment. Any receipt advantage must cover implementation costs before it can imply a profit advantage.