Papers

As mobile robots become increasingly integrated into everyday environments, privacy risks arising from onboard cameras have become a growing concern. Ultra-low-resolution (ULR) RGB can mitigate visual privacy exposure at the source, but ULR appearance alone substantially limits semantic and spatial understanding. We therefore introduce a privacy-preserving asymmetric sensing setting that combines high-resolution (HR) depth with ULR RGB, preserving dense geometry while restricting fine-grained visual information. To address the severe information imbalance between HR depth and ULR RGB, we propose a joint 2D framework using HR geometry to guide semantic-oriented RGB reconstruction and RGB-D segmentation. Despite reliable frame-level predictions, consistent scene-level understanding remains challenging under the asymmetric HR depth–ULR RGB setting. We therefore develop an end-to-end 2D-to-3D pipeline that consolidates 2D semantic features for 3D segmentation. Experiments on ScanNet show that our method achieves the best 2D and 3D segmentation performance among privacy-preserving approaches and delivers the strongest zero-shot transfer to SUN RGB-D and SceneNN. Privacy recoverability analysis shows that our proposed HR depth–ULR RGB input reduces the recoverability of sensitive data, and real-robot experiments demonstrate the utility of the resulting 3D semantics for object-goal navigation.

Hierarchical composite endpoints combining survival and longitudinal functional outcomes are increasingly used in clinical trials, especially when death precludes subsequent functional assessment. The Finkelstein–Schoenfeld strategy analyzes such endpoints through prioritized pairwise comparisons, with survival compared before function. In amyotrophic lateral sclerosis (ALS), this strategy is implemented in the Combined Assessment of Function and Survival (CAFS), which combines survival with the ALS Functional Rating Scale Revised. Although such endpoints provide a clinically meaningful summary of overall treatment benefit, investigators may also want to understand whether the treatment effect is driven by survival, functional outcome, or both. Motivated by the estimand framework, we use ALS-informed simulations from a joint longitudinal–survival model to study settings in which treatment effects on survival and function align or conflict. The simulations show that decomposing the composite win probability into survival and survivor-based functional contributions clarifies their relative roles and separates the composite treatment-benefit question from function-focused questions, including the while-alive comparison and conceptual alternatives based on hypothetical and always-survivor estimands. The functional win probability underlying the while-alive comparison can be subject to survivor-selection bias when treatment affects survival, and inverse-probability weighting can attenuate selection induced by measured predictors under appropriate assumptions. Brief supporting analyses based on principal stratification and multiply robust estimation illustrate the always-survivor estimand. This framework provides practical guidance for reporting and interpreting hierarchical composite endpoints, with CAFS in ALS serving as a concrete motivating example.

Microscaling quantization techniques are increasingly used to represent neural network parameters with 8 bits or fewer while preserving near-full precision accuracy. However, applying these methods efficiently in convolutional layers is not straightforward. A naive approach transfers full-precision weights and activations to processing units and quantizes each tensor twice, resulting in much more memory movement than expected. Additional overhead comes from the activation tensors, whose sizes grow substantially because of the im2col transformation applied before quantization. We propose MicroQonv, a way to combine microscaling with convolutional layers' forward and backward operations by quantizing each tensor only once and quantizing the activation tensor before applying a modified version of im2col: channel-batch-first im2col. MicroQonv reduces the quantization cost by a factor of $\times2$ for weights and gradients, and by up to $\times9$ for activations, at a negligible accuracy cost. It reduces memory movement and storage by up to $\times7.53$ compared to their full-precision counterparts. This way, MicroQonv reduces microscaling-quantized activation memory movement by $\times3.5$ for state-of-the-art object detection models YOLOV8nano and $\times2.2$ for YOLOV26nano. It also enables 4-bit microscaling in a quantized latent replay strategy for continual learning at the edge, improving accuracy by +5.7% to +11%.

In this paper, we extend some Bohr-Rogosinski type inequalities of analytic functions in $\mathbb{C}$ to the cases of holomorphic mappings with values in higher-dimensional complex space. First, we obtain the general Bohr-Rogosinski radius for holomorphic mappings with values in the closure of the unit polydisc in $\mathbb{C} ^n$. Furthermore, we consider the corresponding problems for holomorphic mappings with value in the closure of the unit ball of a JB$^*$-triple. All the radius are optimal.

We introduce and study the class of almost subtractive submonoids of additive monoids. This property relaxes the strict condition of subtractivity while still allowing for a simulated subtraction operation sufficient for many algebraic applications. We explore the structural properties of these monoids and demonstrate their significance in the theory of modules over semirings, specifically regarding quadratic forms and their companions.

This paper presents a high-fidelity direct numerical simulation (DNS)-fluid-structure interaction (FSI) framework for rigid-liquid-flexible spacecraft dynamics under microgravity conditions. The liquid-gas flow is simulated with the incompressible two-phase solver implemented in DIVA, validated against FLUIDICS experiments conducted aboard the International Space Station (ISS). The flexible appendages are described by a rotating assumed-mode plate model that accounts for geometric stiffening. The fluid and structural operators are coupled through a Dirichlet-Neumann fixed-point algorithm with Aitken relaxation, and a closed-system mechanical energy balance is used as an a posteriori diagnostic to assess the energy imbalance of the partitioned discretisation. The coupling strategy is validated against an experimental free-decay sloshing benchmark, and its numerical consistency is assessed through spatial sensitivity studies of the energy-balance defect. Prescribed-motion, rigid open-loop, and flexible open-loop simulations of a spin-up manoeuvre are compared to isolate the effect of structural feedback on the sloshing response. Reduced liquid models identified from the different simulation architectures exhibit different predictive capabilities when embedded in the same rigid-flexible plant. A controller synthesized from the reduced model identified from the flexible simulation is replayed in the nonlinear CFD-FSI environment. The reduced model reproduces the principal attitude and actuator responses for the considered manoeuvre but does not recover the detailed nonlinear sloshing-load history. The framework provides a high-fidelity environment for analysing coupled spacecraft dynamics, identifying control-oriented models, and assessing reduced-model-based control strategies beyond their linear design representation.

In this paper, we investigate the values of zeta functions of root systems at non-positive integer points. In particular, we focus on the zeta functions of type $\mathrm{A}_r$. Since non-positive integer points are included in the set of singularities of zeta functions of root systems, we introduce the ordered limit values to define those values. We study four particular ordered limit values and give explicit formulas for them, or recurrence formulas that allow us to calculate them inductively. We then prove that three ordered limit values of zeta functions of type $\mathrm{A}_r$ vanish under some conditions that can be regarded as a multivariable analogue of the trivial zeros of the Riemann zeta function. Finally, we discuss the other zeros of zeta functions of type $\mathrm{A}_r$ at non-positive integer points and a possible connection of such zeros and non-trivial identities among classical Eisenstein series.

Continuous-aperture and holographic MIMO systems motivate signaling that operates directly on large electromagnetic apertures rather than on a few antenna ports. This paper is the empirical companion to the operator-theoretic curvature-domain framework: a reproducible benchmark of curvature-domain signaling under a common scalar aperture-channel model, stress-testing the theory's dual-budget generalized water-filling law against the modal bases used in near-field and holographic MIMO. All methods share the same apertures, quadrature, Fresnel or Green-function propagation, phase-only constraints, power, phase-energy and curvature-energy budgets, receiver noise, phase quantization and training assumptions. The compared coordinates are curvature-regularized eigenmodes, raw phase coefficients, Fourier phase modes, polynomial and Zernike-like wavefront modes, near-field matched-focus profiles, random and optimized RIS phase codebooks, and SVD water-filling upper bounds. The claim is deliberately limited: curvature-domain signaling is a gauge-invariant, physically realizable coordinate system that can approach the phase-space SVD water-filling bound with fewer stable modes where derivative noise, phase quantization, sampling density or ill conditioning limit conventional bases. It is not a claim of new electromagnetic physics, nor that curvature modes dominate every baseline. We prove the SVD upper-bound relation for the discretized phase-control tangent space, derive pairwise-error and perturbation bounds, and report capacity, retained modes, symbol error, robustness, quantization, sampling, regularization, conditioning and cost, with 95% bootstrap confidence intervals on Monte Carlo results. A verdict table locates the regimes where curvature-domain signaling is engineering-relevant: moderate-to-high phase noise, coarse phase quantization, and non-Fourier-diagonal channels.

This study highlights the distribution of Standard Arabic (SA; H(igh) variety) and Colloquial Arabic (CA; L(ow) variety) across X and Facebook. 16754 public posts were collected via Python, with 10000 retained as the net dataset. Posts were classified into 7 discourse categories: *politics, technology, science, business, culture, fun,* and *sports*. Bivariate analyses, including Chi-square tests and Cramer's V (CV), examined associations among platform, discourse category, and diglossic choice, while binary logistic regression with Platform x Discourse Category interactions tested whether these associations varied across platforms. Findings reveal that there are significant associations between discourse category and diglossic choice on X, chi-square(6, *N* = 5000) = 600.35, p < .001, CV = .347, and Facebook, chi-square(6, N = 5000) = 1249.52, p < .001, CV = .500. Across platforms, platform was also associated with diglossic choice, chi-square(1, N = 10000) = 262.16, p < .001, CV = .162. Binary logistic regression further shows higher odds of SA use on X than Facebook in the political reference category (*OR* = 1.31, p = .0028), with significant platform-by-domain interactions for Culture (OR = 2.65), Fun (*OR* = 6.34), Sports (*OR* = 26.71), Science (OR = 0.41), and Technology (OR = 0.71). The study concludes that the diglossic use of SA and CA contributes to the growing body of research on digital discourse, unveiling that the digital age reshapes but does not erode diglossic boundaries, giving rise instead to a reconfigured digital diglossia.

In this paper, we investigate the family of GHZ diagonal states, which are mixed multipartite quantum states that are diagonal in a GHZ type basis. We show that the class of GHZ diagonal states is highly symmetric and exhibits rich entanglement properties, making it relevant for applications in multipartite quantum key distribution.

It is well known that an Eisenstein series $E_k$ cannot be written as a product of two lower-weight Eisenstein series, except for $E_{14}=E_4 E_{10}= E_4^2 E_6= E_6 E_8$. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer $k\ge 4$ and a negative integer $m$, the Poincaré series $G_{k}(z,m)$ of weight $k$ and index $m$ is a weakly holomorphic modular form having a pole of order $-m$ at $\iota\infty$. It is immediate that the Poincaré series $G_k(z,m)$ for $m<0$ can not be written as a product of two lower-weight Poincaré series of the same index $m$. In this article, we investigate the possible equalities among the monomials of the Poincaré series $G_k(z,m)$ for arbitrary $m<0$. In particular, we show that for any $m<0$ such that $0.06\le\{-2m\pi\}\le0.99$, two monomials composed of $G_k(z,m)$ of the weights $k\ge 50$ are never equal, where $\{x\}$ denotes the fractional part of a real number $x$. In view of Weyl's equidistribution criterion, at least $93\%$ of $m<0$ satisfies the above inequality.

Quantum computing is moving from research laboratories towards early commercialization and broader socio-technical adoption, supported by sustained hardware progress and a rapidly expanding open-source software ecosystem. This momentum is especially visible on GitHub, where many quantum and hybrid software projects coexist around frameworks such as Qiskit, Cirq, PennyLane and Amazon Braket. However, this ecosystem remains fragmented, making it difficult to understand who shapes quantum software, where expertise is concentrated, how collaboration flows across organizations and disciplines, and which actors connect otherwise separated communities. This paper presents Entangle, a data-driven analysis of the open-source quantum computing ecosystem on GitHub. Starting from 71 domain keywords, Entangle identifies more than 1,500 quantum repositories, 27,000 contributors and 400 organizations, revealing an ecosystem strongly organized around four leading industrial vendors, but also supported by 2,387 contributors who connect projects, organizations and domains. These findings provide practical evidence for responsible quantum innovation by making visible patterns of influence, dependency, collaboration and knowledge transfer. They also offer actionable indicators for strategic decisions on investment, hiring, partnerships, ecosystem stewardship and capacity building. More broadly, Entangle shows how open-source intelligence can support a more transparent, measurable and governable quantum software ecosystem, helping align technical development with responsible innovation, public–private coordination and long-term sustainability.

Supercooled cosmological first-order phase transitions (FOPTs) generate a large temperature inhomogeneity due to their stochastic reheating process. If a baryon asymmetry production predates the FOPT and is conserved during it, the temperature contrast is transferred onto the baryon-to-photon ratio $\eta$ or the baryon-to-entropy ratio $Y_{B}$, producing an inhomogeneity probed by big bang nucleosynthesis (BBN). We extend a recent formalism for this effect by including the reheating dependence of the induced $Y_{B}$ fluctuations, combining the transition-time power spectrum from finite-bubble statistics with proton and neutron diffusion at the BBN epoch. When the nucleation temperature is in the range $10^{-2}\,{\rm GeV} \lesssim T_{n} \lesssim 10^{2}\,{\rm GeV}$, the precise measurement of deuterium abundances disfavors parameter regions with a sizable latent heat and a long transition duration at the $1 \sigma$ level. The bound weakens towards higher $T_{n}$ as diffusion becomes more efficient. We further show that this bound disfavors FOPT parameter regions compatible with the NANOGrav gravitational wave signal. This conclusion assumes that the baryon asymmetry predates the transition and that the fluctuation spectrum remains valid down to sub-bubble scales. Therefore, in such FOPT explanations of the NANOGrav signal, the baryon asymmetry cannot simply predate the transition. It can instead be generated after reheating or during the transition if the local baryon production tracks the local entropy injection. As a result, the timing of baryogenesis may be constrained by combining gravitational wave observations with light element abundance measurements.

The RMn$_\text{6}$Sn$_\text{6}$ (R = rare earth) family of kagome magnets exhibits a rich variety of magnetic states and unconventional electronic and magnetic properties. While the static magnetic properties of these materials have been extensively investigated, their magnetization dynamics remain less explored. Here, we investigate the magnetic anisotropy and dynamic magnetization of ferromagnetic YbMn$_\text{6}$Sn$_\text{6}$ using angle-dependent ferromagnetic resonance (FMR) spectroscopy and DC magnetization measurements. The angular dependence of FMR reveals the expected uniaxial anisotropy along the crystallographic c-axis, which is the global magnetically hard axis. In addition, we identify a pronounced twofold anisotropy within the ab-plane, which is independently confirmed by angle-dependent DC magnetization measurements. The observed uniaxial anisotropy in the ab-plane indicates a breaking of the expected sixfold rotational symmetry in the magnetic response that is attributed to growth induced anisotropy.

Understanding the dynamic structural evolution of metallic nanoparticles is crucial for tailoring their functional properties. In this study, we utilized three-dimensional coherent X-ray diffraction to investigate the morphological transformation of Au nanocrystalline domains in a multi-domain nanoparticle during in-situ annealing. By combining Bragg coherent diffraction imaging with autocorrelation function analysis, we observed a transformation from an anisotropic multi-domain configuration toward a single crystalline equilibrium morphology. Above 600 degrees Celsius, a small subsidiary domain merged into the primary domain, which was continuously coarsened at 625 degrees Celsius, resulting in a more isotropic morphology. Finally, at 635 degrees Celsius, the low-energy {111} and {100} facets emerged, and the crystal shape converged toward an equilibrium truncated octahedral geometry. This work demonstrates a framework for interpreting complex domain dynamics, offering fundamental insights into domain growth and equilibrium shape formation.

It is well known that the choice of basis functions in hp-FEM heavily influences the stability and the computational cost in order to obtain an approximate solution. For simplicial elements in two and three space dimensions, tensor-product-like basis functions built from Jacobi polynomials with different weights yield optimal properties with respect to condition number and sparsity. In this paper we construct such a high order basis for the Nédélec spaces and for the Raviart-Thomas and Brezzi-Douglas-Marini spaces, modifying existing definitions of Zaglmayr so that the functions belong to the Nédélec spaces of first kind and the Raviart-Thomas spaces. The bases are designed so that the Nédélec space of second kind and the BDM space are extensions of the respective other spaces. In the second part of the paper we introduce biorthogonal basis functions for $H(\text{div}, {\Omega})$ continuing previous research for $H^1({\Omega})$ and $H(\text{curl}, {\Omega})$. These functions can be expressed in closed form as sums of tensor products of Jacobi polynomials, which allows for fast computation of the $L^2$-projection.

Audio-language models (ALMs) integrate acoustic perception with the knowledge encoded in language models, enabling contextual understanding of auditory events. Making these capabilities practical on devices with limited memory and computation motivates our focus on small ALMs with fewer than 200M parameters. We introduce a recipe that brings together architecture, data, and three-stage training to build Mizar, a 159.3M-parameter ALM. Its architecture connects a compact CED-Small audio encoder to SmolLM2-135M through a frequency-merging mapper. With supervision drawn from ReasonAQA, AudioMCQ, and AVQA, the model undergoes three training stages: audio-language alignment (Stage 1), audio-dependent fine-tuning (Stage 2), and post-training (Stage 3) aimed at strengthening weak skills while retaining learned capabilities. Across five random seeds, Mizar achieves mean accuracies of 52.92% on MMAU, 42.42% on MMAR, and 36.02% on ADQA-clean, surpassing the previous best-performing ALM below 200M parameters on all three benchmarks. It also supports local inference on a single CPU: on questions from the MMAU benchmark, the mean latency from opening the audio file to generating a complete answer is 1.09 seconds. Code and checkpoints are available at https://github.com/KaiyangLi1992/Mizar_159M.

The growing use of spatial audio in applications such as augmented and virtual reality has driven the development of binaural reproduction methods for wearable arrays with a limited number of microphones. Binaural signal matching (BSM) is one such method, producing high-quality binaural signals under a diffuse-field assumption, but degrading at high direct-to-reverberant ratios (DRR) where the direct sound dominates. Previous extensions incorporate Field-of-View (FoV) weighting, either with fixed apertures or based on explicit source localization, but these approaches are limited by coarse spatial coverage or reliance on localization estimation accuracy. This paper introduces FoV-BSM-Net, a signal-dependent FoV-BSM formulation that avoids explicit source estimation by learning the FoV parameters end-to-end from the microphone signals using a Convolutional Recurrent Neural Network. The method is evaluated in simulated rooms across varying reverberation conditions, and compared against BSM and a fixed FoV-BSM baseline. Results show that FoV-BSM-Net consistently improves over BSM, with gains that grow with DRR in both binaural NMSE and interaural cue errors, and are further supported by perceptual evaluation showing a substantial advantage over both baselines across low and high DRR conditions.

Removing an object from a real image requires more than synthesizing plausible content within a mask: the method must suppress residual object features, preserve the unedited scene, and generate replacement content that is consistent with the surrounding background. This paper approaches object removal from a stage-based perspective and proposes a zero-shot framework for constrained latent inpainting with a frozen pretrained Stable Diffusion model, requiring no task-specific training or model fine-tuning. The method integrates SAM-based mask construction, BLIP image-caption conditioning, DDIM inversion, background-weighted masked null-text optimization, decoder self-attention masking, hard outside-mask latent anchoring, and localized renoise–denoise refinement into a unified pipeline. The method is evaluated through qualitative examples, quantitative local-consistency metrics, and ablation studies. The results demonstrate effective object removal and context-consistent replacement content. The ablations indicate that background-weighted masked NTI is particularly beneficial for structurally complex backgrounds, whereas the no-NTI variant is sufficient in other evaluated examples. Repeated refinement further reduces object remnants and boundary artifacts remaining after the primary editing pass.

We prove that for every window $w$ of the form $w(x) = e^{-\pi |x|^2} h(x)$, where $h$ is a polynomial, there exists a uniformly discrete set of points $\mathcal{S} \subset \mathbb R^{2d}$ such that the magnitude of the short-time Fourier transform $V_w f$ on $\mathcal{S}$ determines every $f \in L^2(\mathbb R^d)$ up to a constant phase factor. The separation distance can be chosen independent of the degree of $h$ and proportional to the square root of the dimension. The proof combines a discrete norming inequality, based on a multidimensional Remez inequality and VC-dimension bounds, with tail estimates for the reproducing kernel. A formalization of our main result in Lean 4 is also provided.

We examine the interchangeability between two duplex systems with opinion dynamics: one characterized by different dynamics parameters $a_1$ and $a_2$ on each layer and full node overlap, and the second with the same parameter $a_1$ on both layers but only partial node overlap $r$. We introduce a general framework for identifying the critical line between the ordered and disordered phases, thereby translating one system into another. The description is supported by several examples of opinion dynamics models and a rigorous measure of the distance between the stationary solutions of the rate equations in both settings. Our results show that even large values of $r$ are sufficient to obtain very good correspondence between the two settings.

Pretrained generative Diffusion Transformers (DiTs) capture rich pixel-level visual and language-conditioned structure through large-scale image and video generation training. A growing line of robot policies builds on this generative prior, but how it should be transferred to control remains unclear, and existing approaches commonly instantiate this transfer through future visual prediction. We ask a more basic question: what a pretrained generative DiT actually contributes to action learning, and how this prior should be adapted for control. We introduce NowWAM, a future-target-free co-training formulation that denoises the current observation and predicts robot actions from the same visual stream, directly coupling the native generative objective to the action-facing representation across the denoising trajectory. Under matched controlled settings, past and future visual targets perform comparably, while restricting training to the clean endpoint substantially reduces robustness, suggesting that a separate future target is not essential for generative adaptation, while the denoising trajectory remains an effective interface for control. On LIBERO-Plus, NowWAM reaches 87.7% with FLUX2-Klein, improving over the future-target co-training baseline by 6.1 points while halving training visual tokens (784 to 392) and reducing step time from 2.85 s to 1.63 s, a 1.8x speedup. With the pure text-to-image Z-Image backbone, NowWAM further reaches 87.8%, showing that strong control adaptation is not tied to video generation or image-editing backbones.

Local geometric mixing localizes geometric mixing by requiring geometric convergence to equilibrium in total variation only over finitely many transitions. It accommodates local convergence rates and captures rapid local equilibration, even when global mixing is much slower. We establish and discuss local geometric mixing bounds through Dobrushin contraction. We then apply this approach to Diffusion Path Monte Carlo, a recently proposed Markov chain Monte Carlo method, aimed at leveraging advances in score-based modeling, whose ideal transitions coincide with those of the Proximal Sampler. Our analysis covers both the ideal method and its implementable Metropolis-adjusted counterpart, providing mixing guarantees under minimal assumptions. For the ideal method, these guarantees complement recent spectral gap estimates, which we develop into mixing time bounds.

In this work we study the long term behavior of the curve shortening flow of asymptotically conical curves with convex projections in $\mathbb{R}^3$.

We consider the construction of confidence intervals for population means when observations are subject to missingness. To accommodate more general missingness mechanisms than missing completely at random (MCAR), we adopt a reparametrised version of the realisable contamination model of Ma et al. (2026), which is a mixture of an MCAR version and a missing not at random version of the same base distribution $P$. We characterise the minimax length of confidence intervals that can adapt to potentially unknown parameters of the model, including the contamination fraction, for Gaussian base distributions and for nonparametric classes satisfying certain tail or symmetry assumptions. In all of these settings, we provide explicit constructions of simple, practical and finite-sample valid adaptive confidence intervals that attain the corresponding minimax rates. Finally, we provide implications of our results for causal inference.

Claims about AI safety reach audiences well beyond the AI community, yet many rely on opaque evidence or static assessments, when supporting evidence is accessible at all. We present the Systemic Risk Index, an open evaluation pipeline and dashboard built to make empirical evidence more transparent and traceable to the public. Our work organizes 19 public benchmarks into four systemic-risk categories defined by the EU GPAI Code of Practice—CBRN, cyber offense, harmful manipulation, and loss of control—and evaluates models using harm-preserving perturbations and simulated deployment contexts. The interactive dashboard lets users alternate between average and worst-case aggregation, vary how model capability affects the aggregate score, and trace each risk rating to its benchmark evidence. Across 18 models, scores fall by 14 to 37 points under worst-case aggregation, highlighting information that can be hidden by an average assessment of model risk. LLM judges show agreement with human graders comparable to human–human agreement ($\kappa = 0.78\text{--}0.82$), and a blind audit finds that $83\%$ of sampled transformations preserve the original harm. In a survey ($N = 21$), most participants report that scores are easy to understand and that the dashboard encouraged them to view model evaluations under different settings

The scotogenic model generates sub-eV neutrino masses radiatively through small Yukawa couplings $\sim\mathcal{O}(10^{-3})$ that also set the decay length of the heavy neutral fermion $\Ntwo$; in the compressed regime a separate, much smaller Yukawa entry $\sim\mathcal{O}(10^{-6})$ additionally renders the charged scalar $\etapm$ long-lived. We study the compressed-spectrum regime of its two-inert-doublet variant ($\DMcomp = \mHpm - \mNtwo \simeq 200\MeV$), in which small Yukawa entries render both the charged scalar $\etapm$ and the heavy neutral fermion $\Ntwo$ long-lived and produce a nested cascade at the High-Luminosity LHC~(HL-LHC): $\etapm$ leaves a disappearing track~(DT), decays to an invisible $\Ntwo$, which travels macroscopically before forming a displaced dilepton vertex~(DV). The two slow flights combine into a 4D time-of-flight delay $\dt\sim200$–$800\ps$ resolvable by the CMS MIP Timing Detector, while the DT-to-DV displacement vector inherits the parent track direction, yielding a spatial back-pointing correlation with no SM analogue. These two handles are expected to suppress the background to $B\sim10^{-5}$–$10^{-3}$ events, and the discovery remains $5\sigma$-significant for any background up to $B\sim10$ events. At the benchmark ($\mNtwo=150\GeV$ and proper decay lengths $\ctauH=300\mm$, $\ctauN=189\mm$) the strategy yields $N_s\approx1212$ signal events at $3000\ifb$ and establishes sensitivity across $15\mm\lesssim\ctauH\lesssim300\mm$, $10\mm\lesssim\ctauN\lesssim1000\mm$, a correlated region left uncovered by any standalone DT or DV search. Because the $\Ntwo$ decay length is controlled by the same Yukawa couplings that generate sub-eV neutrino masses, mapping this nested cascade turns the HL-LHC into a direct collider probe of the interaction responsible for neutrino mass.

We study online fair division, where indivisible goods arrive one at a time and must be allocated immediately and irrevocably among $n\ge2$ agents. Against an oblivious adversary that fixes the input in advance, we ask what fairness guarantees randomized online algorithms can achieve. We present the Deficit Allocation (DA) algorithm, which improves the previously known high-probability approximation to PROP1 from inverse-logarithmic to inverse-doubly-logarithmic in $n/\delta$, where $\delta$ is the failure probability. DA does not require $\delta$ as input. With probability at least $1-\delta$, a refined variant achieves an arbitrarily close approximation to proportionality when no single item is too valuable relative to an agent's proportional share; without any restriction on item values, it obtains an arbitrarily close approximation to PROP$k$ for $k$ doubly logarithmic in $n/\delta$. For $n2^{-2n}\le\delta\le1/2$, our bounds on the approximation factor and $k$ are asymptotically tight for the stronger fractional benchmark that divides each item equally among the agents who value it positively. We also prove that algorithms that use only which agents value each item positively, but not by how much, cannot improve upon the logarithmic dependence of uniformly random allocation. However, other fairness notions admit weaker guarantees. For EF1, every randomized online algorithm has an input on which any positive approximation succeeds with probability at most $e^{-\Omega(n)}$. For EFX, this probability is at most $1/n!$ with only $n+1$ items, tight within a factor of $(n+1)/2$; for MMS, it can be at most $5/6$. Finally, allocating each item uniformly among the agents who value it positively gives, with probability at least $1-\delta$, an arbitrarily close approximation to EF$k$ for $k$ logarithmic in $n/\delta$, and this dependence is necessary for this rule.

A classical theorem of Beckner, Carbery, Semmes, and Soria states that the Fourier restriction operator for the sphere does not satisfy the restricted weak-type estimate at the conjectured endpoint. Their proof uses the Besicovitch compression phenomenon and proceeds by contradiction, as in Fefferman's disproof of the ball multiplier conjecture. As a result, it reveals little about how badly the estimate fails. We develop a general framework that converts Besicovitch compression into a quantitative failure at an endpoint. Given a family of sets that compresses, our framework produces an explicit set that violates the bound, with a rate governed by the amount of compression. The rate is sharp within the admissible family of sets. We apply this to the sphere and the paraboloid and construct a deterministic sequence of sets along which the restricted weak-type inequality degrades at an explicitly computed rate. We also show that our construction does not provide blowup for the moment curve in dimensions three and higher, where the (much stronger) restricted strong-type bound is known to hold at the endpoint.

Entropy has emerged as a key concept at the intersection of gravity, quantum theory, and cosmology, revealing deep connections between geometry, information, and thermodynamics. The realization that black holes are thermodynamic systems, characterized by an entropy proportional to their horizon area, has provided evidence that spacetime itself may possess a microscopic structure. Since then, entropy has acquired a central role across quantum field theory, holography, quantum gravity, and cosmology. In this review, we present a unified account of entropy in gravitational and cosmological contexts. We first survey entropy in gravitational systems, focusing on black-hole thermodynamics, entropy bounds, the holographic principle, entanglement entropy, and the area law, together with the main corrections and generalizations arising from quantum field theory, generalized uncertainty principles, modified statistics, and extended theories of gravity. We then discuss three conceptually distinct ways in which entropy is employed in cosmology. The first is through the spacetime-thermodynamics conjecture, in which gravitational dynamics emerge from thermodynamic relations applied to cosmological horizons, and generalized entropy forms consequently lead to modified cosmological evolution. The second is the holographic approach, in which entropy bounds constrain the vacuum energy of the Universe, giving rise to the holographic dark energy scenario and its various extensions. The third is entropic gravity, where gravity is interpreted as an emergent phenomenon of entropic origin, leading to interesting cosmological phenomenology. By reviewing the theoretical foundations and cosmological implications of these entropic frameworks, we aim to clarify the role of entropy, discuss open issues and outline directions toward a deeper thermodynamic and informational understanding of gravity and cosmology.