Camphor is a well-studied material capable of generating self-propelled motion at a water surface, and the resulting dynamics can exhibit a wide range of behaviors. Here, we analyze a one-dimensional model describing a mobile camphor disk perturbed by a second localized camphor source. The interaction between the rotor and the perturbing disk is represented by a distance-dependent potential. The study is motivated by experiments in which a camphor rotor interacts with a fixed camphor disk placed on the water surface. Numerical simulations of the model reproduce the essential features of the experimentally observed position-dependent rotor velocity for all considered forms of the potential. For weak perturbations, we derive analytical solutions valid for arbitrary potential profiles. Both the simulations and the analytical results demonstrate a pronounced asymmetry in the rotor velocity depending on whether the rotor approaches or recedes from the perturbation.

This paper considers confidence intervals (CIs) for the autoregressive (AR) parameter in an AR model with an AR parameter that may be close or equal to one. Existing CIs rely on the assumption of a stationary or fixed initial condition to obtain correct asymptotic coverage and good finite sample coverage. When this assumption fails, their coverage can be quite poor. In this paper, we introduce a new CI for the AR parameter whose coverage probability is completely robust to the initial condition, both asymptotically and in finite samples. This CI pays only a small price in terms of its length when the initial condition is stationary or fixed. The new CI also is robust to conditional heteroskedasticity of the errors.

Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary. In this note, we answer a question posed by Steinerberger: can one remove the $\sqrt{\log n}$ factor in the two-dimensional Green--Wasserstein inequality while keeping the unrenormalized off-diagonal Green term? We show that this is impossible on any compact connected surface: there is no inequality of the same form that holds uniformly over point sets with an $O(n^{-1/2})$ remainder for all $n$. We argue by contradiction and combine a second-moment estimate for the random Green energy of i.i.d. samples with the semi-discrete random matching asymptotics of Ambrosio--Glaudo.

In this short note we provide a characterisation of ball quotients among all minimal smooth projective varieties of general type purely in terms of their characteristic numbers. This generalises earlier work of Miyaoka, Yau and Greb--Kebekus--Peternell--Taji.

We introduce the Accretion-Modulated Internal Shock model (AMIS) as a possible framework for explaining the observational properties of long gamma-ray burst (GRB) prompt emission. In this scenario, the envelope of the prompt light curve follows the time-dependent mass-supply history to the central engine, associated with stellar collapse and, where applicable, fallback accretion, whose early-time onset can be approximated by $\dot{M}\propto t^{0-1/2}$ and which subsequently may decay as $\dot{M}\propto t^{-5/3}$, producing a photon count rate with a single fast-rise-exponential-decay (FRED)-like profile. In general, the prompt-emission envelope is regulated by a time-dependent mass supply to the central engine, while internal shocks produce the rapid variability. Since we only aim to introduce this framework here, we focus on the simplest single-FRED shape of the prompt emission profiles, while more complex cases involving multiple episodes and interacting shocks will be explored in forthcoming studies. The model indicates correlations between spectral evolution, FRED-pulse narrowing at high energies, and the mass-supply-controlled envelope. Stochastic Lorentz factor variations of ejected mass- or rate-driven shells, superimposed on the Accretion-Modulated envelope, explain the coexistence of smooth global trends and irregular short-timescale features, such as the widths of individual pulses in long GRB light curves, offering diagnostic tools for probing the inner engine activity.

We prove that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an $L^\infty$-close (bi-Lipschitz) smooth metric with two-sided Ricci curvature bounds and a uniform positive lower bound on injectivity radius. This answers Question 2 in the Morgan--Pansu list of open problems from the conference Modern Trends in Differential Geometry (São Paulo, 2018), proposed by L. Bandara. In the proof, we rely on controlled smoothing with Croke's universal local volume lower bound and the Cheeger--Gromov--Taylor injectivity radius estimate.

We experimentally investigate the dynamics of a sphere rolling down a granular slope by varying the initial velocity, slope angle, and sphere density. The results show that the sphere rolls down with constant deceleration while sinking into the granular bed. $δ/R$ (the sinking depth $δ$ normalized to the sphere radius $R$) is scaled by the sphere density normalized by the bulk density of the granular layer. To evaluate the translational energy dissipation, we introduce an effective friction coefficient $μ_\mathrm{d}$. We demonstrate that $μ_\mathrm{d}$ decreases with increasing the slope angle and the slip ratio. Furthermore, systematic measurements over a wide range of sphere densities reveal that $μ_\mathrm{d}$ increases linearly with $δ/R$ : $μ_\mathrm{d}=β(δ/R)+μ_0$. The value of $μ_0$ is linearly decreasing with slip ratio and its coefficient $β(\simeq0.41)$ does not vary significantly. The results suggest that the normalized depth and slip ratio determine the effective friction of a rolling sphere.

As online reviews increasingly drive consumer decisions, the impact of review interface design on rating authenticity remains under-explored. This research investigates how evaluation sequence ("Rating-First" vs. "Review-First") influences consumer ratings through three experiments and a large-scale secondary data analysis. The results reveal a significant polarization effect: in high-quality service contexts, the "Rating-First" sequence (vs. "Review-First") increases overall ratings, whereas in low-quality contexts, it leads to significantly lower ratings. This mechanism is driven by a serial mediation path of affective heuristics and cognitive effort. Furthermore, product attributes moderate this effect, with hedonic products amplifying the rating extremity compared to utilitarian ones. Secondary data from Yelp and Letterboxd confirm these findings, showing that the "Rating-First" platform (Yelp) exhibits a polarized bimodal distribution, while the "Review-First" platform (Letterboxd) shows more concentrated ratings. In conclusion, this research reveals how evaluation sequence shapes consumer ratings through affective and cognitive paths from an information-processing perspective. These findings extend the theoretical understanding of the online review formation process and offer practical insights for platforms to optimize interface design and enhance rating authenticity and credibility.

This paper investigates leader-follower linear-quadratic stochastic graphon games, which consist of a single leader and a continuum of followers. The state equations of the followers interact through graphon coupling terms, with their diffusion coefficients depending on the state, the graphon aggregation term, and the control variables. The diffusion term of the leader's state equation depends on its state and control variables. Within this framework, a hierarchical decision-making structure is established: for any strategy adopted by the leader, the followers compete to attain a Nash equilibrium, while the leader optimizes its own cost functional by anticipating the followers' equilibrium response. This work develops a rigorous mathematical model for the game, proves the existence and uniqueness of solutions to the system's state equations under admissible control sets, and constructs a Stackelberg-Nash equilibrium for the continuum follower game. By employing the continuity method, we establish the existence, uniqueness, and stability of solutions to the associated forward-backward stochastic differential equation with a graphon aggregation term.

Characterizing protostellar outflows is fundamental to understanding star formation feedback, yet traditional methods are often hindered by projection effects and complex morphologies. We present a multi-modal deep learning framework that jointly leverages spatial and spectral information from CO observations to infer protostellar mass, inclination, and position angle ($PA$). Our model, trained on synthetic ALMA observations generated from 3D magnetohydrodynamic simulations, utilizes a cross-attention fusion mechanism to integrate morphological and kinematic features with probabilistic uncertainty estimation. Our results demonstrate that Vision Transformer architectures significantly outperform convolutional networks, showing remarkable robustness to reduced spatial resolution. Interpretability analysis reveals a physically consistent hierarchy: spatial features dominate across all parameters, whereas spectral profiles provide secondary constraints for mass and inclination. Applied to observational ALMA data, the framework delivers stable mass and $PA$ estimates with exceptionally tightly constrained inclination angles. This study establishes multi-modal deep learning as a powerful, interpretable tool for overcoming projection biases in high-mass star formation studies.

Racism remains a persistent societal issue, increasingly amplified by the structure and dynamics of online social networks. In this work, we propose a three-state compartmental model to study the spreading and suppression of racist content, drawing from epidemic-like dynamics and interaction-driven transitions. We analyze the model on fully-connected (homogeneous mixing) networks using a set of coupled differential equations, and on Barabási-Albert (BA) scale-free and Watts-Strogatz (WS) small-world networks through agent-based simulations. The system exhibits three distinct stationary regimes: two racism-free absorbing states and one active phase with persistent racist content. We identify and characterize the phase transitions between these regimes, discuss the role of network topology, and highlight the emergence of absorbing states. Our findings illustrate how statistical physics tools can help uncover the macroscopic consequences of microscopic social interactions in digital environments.

The performance of tensor network methods has seen constant improvements over the last few years. We add to this effort by introducing a new algorithm that efficiently applies tree tensor network operators to tree tensor network states inspired by the density matrix method and the Cholesky decomposition. This application procedure is a common subroutine in tensor network methods. We explicitly include the special case of tensor train structures and demonstrate how to extend methods commonly used in this context to general tree structures. We compare our newly developed method with the existing ones in a benchmark scenario with random tensor network states and operators. We find our Cholesky-based compression (CBC) performs equivalently to the current state-of-the-art method, while outperforming most established methods by at least an order of magnitude in runtime. We then apply our knowledge to perform circuit simulation of tree-like circuits, in order to test our method in a more realistic scenario. Here, we find that more complex tree structures can outperform simple linear structures and achieve lower errors than those possible with the simple structures. Additionally, our CBC still performs among the most successful methods, showing less dependence on the different bond dimensions of the operator.

Transfer learning (TL) has emerged as a powerful tool for improving estimation and prediction performance by leveraging information from related datasets, with the offset TL (O-TL) being a prevailing implementation. In this paper, we adapt the control-variates (CVS) method for TL and develop CVS-based estimators for scalar-on-function regression, one of the most fundamental models in functional data analysis. These estimators rely exclusively on dataset-specific summary statistics, thereby avoiding the pooling of subject-level data and remaining applicable in privacy-restricted or decentralized settings. We establish, for the first time, a theoretical connection between O-TL and CVS-based TL, showing that these two seemingly distinct TL strategies adjust local estimators in fundamentally similar ways. We further derive convergence rates that explicitly account for the unavoidable but typically overlooked smoothing error arising from discretely observed functional predictors, and clarify how similarity among covariance functions across datasets governs the performance of TL. Numerical studies support the theoretical findings and demonstrate that the proposed methods achieve competitive estimation and prediction performance compared with existing alternatives.

We explore the feasibility of using Lyman-$α$ (Ly$α$) forests to calibrate the ensemble redshift distribution of the high-redshift tail ($2<z<3$) of photometric galaxies. We use \texttt{CoLoRe} simulations to create mock DESI 5-year Ly$α$ forests and Rubin Observatory LSST 10-year photometric galaxies up to $z=3$, and measure the galaxy redshift distribution via their angular cross-correlations. Due to large redshift-space distortions in the Ly$α$ forest, the conventional $n(z)$ estimator for clustering redshifts does not apply, and we develope a theoretical framework to model the angular cross-correlation directly. Using the simulations, we explore effects of instrumental noise, continuum fitting, and contamination in the Ly$α$ forest, cross-correlation angular scales ($θ$), and redshift bin size ($Δz$) on the signal-to-noise (SNR) of the measurements. We find that continuum fitting methods strongly impact the SNR of the measurements. With our baseline continuum fitting method, \texttt{LyCAN}, at angular scales $θ\sim10$ arcmin and $Δz=0.1$, we measure the cross-correlation signal at $24σ$. If the shape of the redshift distribution and galaxy bias evolution are known well for $z<2$, the cross-correlation can constrain the mean redshift of the galaxy sample to $σ_z/(1+\bar{z}) = 0.006$ at a mean redshift of $\bar{z}=2$. This demonstrates that Ly$α$ cross-correlation is a reliable and promising method to calibrate the high-redshift tails of photometric Stage IV galaxy surveys.

In a recent paper, arXiv:2509.19899, we presented a new method to test the consistency between uncalibrated BAO and SNIa data through a common parameter, the Alcock-Paczynski variable. Using Gaussian Processes, we can determine if various datasets are consistent, independently of dark energy or modified gravity models, and of the sound horizon and SNIa peak magnitude. We found that the DES-Y5 SNIa data showed non-negligible tension with other datasets. However, the recent update DES-Dovekie removes this tension. We find that all uncalibrated data from DESI DR2 BAO and three SNIa datasets, Union3, Pantheon+, and DES-Dovekie, are consistent with each other within $\sim 1σ$.

We propose a bootstrap approximation method for the Hermitian one-matrix model that does not rely on positivity constraints. The theoretical foundation of this method is that the one-matrix model admits an eigenvalue distribution $ρ(λ)$, and that the moments $w_n$ generated from it satisfy the loop equations. Our framework is designed to numerically determine a self-consistent pair of $ρ(λ)$ and $w_n$ that simultaneously satisfies these two requirements. In the concrete implementation the least-squares method is employed, and since the sign problem is absent in this formulation, the method can be formally applied to the Minkowski one-matrix model as well, provided that the one-cut structure of the resolvent is assumed. Actual numerical calculations show that this bootstrap approximation reproduces, with very high accuracy, the exact solutions for Euclidean-type models and the perturbative results for Minkowski-type models.

Given finite groups $H \leq G$, a representation $σ$ of $G$ is called center-preserving on $H$ if the only elements of $H$ that become central under $σ$ are those that were already central in $G$. We prove that if $H$ has a faithful irreducible representation $ρ$, then at least one of the irreducible components of the induction $\operatorname{Ind}_H^G(ρ)$ is center-preserving on $H$. In consequence, $H$ has a faithful irreducible representation if and only if every finite group $G$ containing $H$ as a subgroup has an irreducible representation whose restriction to $H$ is faithful, and which is center-preserving on $H$. In addition, we give examples illustrating the sharpness of the statement, and discuss the connection with projective representations.

The increasing number of control lines connecting quantum processors to external electronics constitutes a major bottleneck in the realization of large-scale quantum computers. Frequency-division multiplexing is expected to enable control of multiple qubits through a single microwave cable; however, interference from off-resonant microwave tones hinders precise qubit control. Here, we propose an active interference suppression method for frequency-division-multiplexed simultaneous gate operations. We demonstrate that the deliberate incorporation of off-resonant microwave tones improves single-qubit gate fidelity. In particular, the gate infidelity scales inversely with the square of the number of microwave tones when off-resonant orthogonal or quasi-orthogonal tones are incorporated. Furthermore, we show that fast oscillations, neglected under the rotating wave approximation, degrade the gate fidelity, and that this degradation can be mitigated through optimized frequency allocation. The proposed approach is simple and effective for improving the performance of frequency-division-multiplexed quantum gates.

We investigate non-perturbative multi-pion contributions to nucleon flavor asymmetry within the framework of Light-Front Hamiltonian Effective Field Theory (LFHEFT). Utilizing a Fock sector expansion, we systematically incorporate pionic degrees of freedom, with the nucleon-pion interactions governed by a scalar variant of chiral effective field theory. Our results demonstrate that the non-perturbatively calculated longitudinal momentum distributions exhibit significant deviations from leading-order perturbative predictions, emphasizing the importance of higher-order Fock components in describing the proton's sea quark structure. Furthermore, we demonstrate the feasibility of extending this framework to investigate nuclear effects in light nuclei, such as the deuteron. This unified approach provides a consistent basis for analyzing the interplay between intrinsic nucleon structure and nuclear modifications, potentially offering new insights into the flavor asymmetry observed in fixed-target and collider experiments.

Here, we report a scanning tunneling microscopy study on an s-wave superconductor NaAlSi, revealing the coexistence of stripe order and superconductivity. This stripe order manifests as a unidirectional spatial charge modulation with a commensurate period of four times the lattice constant. This modulation undergoes a phase shift in the differential conductance maps under opposite bias voltages, while its period remains approximately constant over an energy range of $\pm$50 meV. These features suggest that this stripe is likely a static charge order. Furthermore, we find that the stripe order imposes a periodic modulation on the intensity of the superconducting coherence peaks. This work provides new perspectives on the intricate interplay between stripe order and s-wave superconductivity.

The systematic analysis of non-adiabatic effect on convective mode has been conducted using wave energy relation. In the adiabatic analysis, the "propagation diagram" for convective mode is proposed as a useful tool to see its behavior. In the non-adiabatic analysis, it is found that for strongly non-adiabatic case, a monotonically growing convective mode becomes oscillatory. In this phase, the radial displacement and the distribution of wave energy show only one bump, in which the distribution of entropy energy eS almost overlaps with the distribution of gravity energy eg. Entropy energy eS seems to act as potential energy of oscillatory convection. In addition to this, this change occurs not gradually, but abruptly with change of non-adiabatic indicator.

We investigate quantum correlations in a hybrid qubit-qutrit system subject to both axial and planar single-ion anisotropies, dipolar spin-spin interactions, and Dzyaloshinskii-Moriya (DM) coupling. Using Negativity, Measurement-Induced Non-locality (MIN), Uncertainty-Induced Nonlocality (UIN), and Bell nonlocality (as quantified by the CHSH inequality) as measures, we analyze the interplay between anisotropy parameters, magnetic fields, and temperature on the survival of quantum correlations. Our results demonstrate that Bell nonlocality and entanglement (Negativity) are highly sensitive to temperature and anisotropy, exhibiting sudden death under thermal noise, whereas MIN and UIN are significantly more robust. In particular, these discord-like and information-theoretic measures provide the largest baseline and persist even in parameter regions where entanglement vanishes, highlighting their suitability as a quantumness witness in realistic conditions. Notably, our Bell nonlocality study is tailored to the asymmetric qubit-qutrit setting by exploiting a recently developed qubit-qudit CHSH maximization framework. However, Bell nonlocality is confirmed to be the most fragile, surviving only in narrow parameter windows at low temperature. A key finding of this work is that we observe the fragility hierarchy: Bell nonlocality $\subseteq$ Negativity $\subseteq$ UIN(MIN) in the qubit-qutrit setting. These results provide deeper insight into the relative robustness of distinct quantum resources in anisotropic qubit-qutrit models, suggesting that quantum discord-like measures, such as MIN and UIN, may serve as more practical resources than entanglement for quantum information tasks in thermally active spin systems.

Planet population synthesis is an integral tool for linking exoplanets to their formation environments. Most planet population synthesis studies have focused on the carbon-to-oxygen ratio (C/O) in gas or solids, yet more insight into planet formation may be afforded by considering a wider suite of elements. Sulfur is one such key element. It has been assumed to be entirely refractory in population synthesis models, restricting it to being a tracer of accreted rocky solids. However, sulfur also has a volatile reservoir dominant at the onset of star and planet formation, which is then converted into refractories. We investigate sulfur's wider potential as a formation history tracer by implementing a gas-grain chemical conversion, the first multi-phase treatment of S in a planet population synthesis model. We also present the planet formation module of SPONCHPOP and its first predicted planet growth tracks and populations. We apply these to explore the diversity of the planetary sulfur budget. We show that planets can inherit a wide range of core and envelope sulfur content, depending on their formation environment and accretion history including late-stage infall, demonstrating sulfur's new potential as a diagnostic tool for planet formation. Our models predict that some rocky planets are born sulfur-poor, which may have significant implications for their geochemistry and habitability. Enhanced sulfur abundances in gas-giant atmospheres, such as in our solar system, may result not only from accretion of rocky planetesimals, but also from formation beyond the H2S iceline.

Recent advances in large language models (LLMs) have driven extensive evaluations in software engineering. however, most prior work concentrates on code-level tasks, leaving software design capabilities underexplored. To fill this gap, we conduct a comprehensive empirical study evaluating 29 LLMs on object-oriented design (OOD) tasks. Owing to the lack of standardized benchmarks and metrics, we introduce OODEval, a manually constructed benchmark comprising 50 OOD tasks of varying difficulty, and OODEval-Human, the first human-rated OOD benchmark, which includes 940 undergraduate-submitted class diagrams evaluated by instructors. We further propose CLUE (Class Likeness Unified Evaluation), a unified metric set that assesses both global correctness and fine-grained design quality in class diagram generation. Using these benchmarks and metrics, we investigate five research questions: overall correctness, comparison with humans, model dimension analysis, task feature analysis, and bad case analysis. The results indicate that while LLMs achieve high syntactic accuracy, they exhibit substantial semantic deficiencies, particularly in method and relationship generation. Among the evaluated models, Qwen3-Coder-30B achieves the best overall performance, rivaling DeepSeek-R1 and GPT-4o, while Gemma3-4B-IT outperforms GPT-4o-Mini despite its smaller parameter scale. Although top-performing LLMs nearly match the average performance of undergraduates, they remain significantly below the level of the best human designers. Further analysis shows that parameter scale, code specialization, and instruction tuning strongly influence performance, whereas increased design complexity and lower requirement readability degrade it. Bad case analysis reveals common failure modes, including keyword misuse, missing classes or relationships, and omitted methods.

The broad-line region (BLR) of active galactic nuclei (AGN) is an essential component, yet its small size keeps its origin, structure, and kinematics uncertain. Infrared interferometry with VLTI/GRAVITY is now resolving BLR-scale emission, with data for NGC 3783 consistent with a rotating, geometrically thick configuration. However, the processes shaping the spectra remain poorly constrained, and the cloud models are tuned phenomenologically rather than derived from first-principles predictions. We address this by coupling three-dimensional radiation-hydrodynamic (RHD) simulations of gas around a supermassive black hole with radiative-transfer calculations using Cloudy, comparing the results to the SINFONI Br$γ$ profile of NGC 3783. We find that Br$γ$ arises from ionized gas in the surface of the rotating thin disk, with electron temperatures of approximately $T_e \approx 10^4$ K and number densities of $n_e \approx 10^8-10^{11}$ cm$^{-3}$. However, the intrinsic line profile produced by the RHD kinematics is narrower than observed and displays substructure. An approximate treatment of the electron scattering suggests that scattering in surrounding diffuse ionized gas significantly broadens and smooths the intrinsic Br$γ$ profile, making it consistent with the observed profile. This scattering medium has an electron temperature of $10^4 - 10^5$ K, and a number density of $n \lesssim 10^8$ cm$^{-3}$. Although a best-fit viewing angle of $\approx 15$ deg is suggested, the scattered line is notably less sensitive to inclination than the intrinsic line. The observed BLR profiles may be understood as the intrinsic emission viewed through an electron-scattering haze, such that some spectral detail is plausibly redistributed rather than seen directly.

We study structural properties of self-gravitating fluid spheres made of a dilute, homogeneous and ultracold Bose gas assuming repulsive, short-range interactions. For the first time we include the Lee-Huang-Yang correction to the usual polytropic equation-of-state of index $n=1$, which goes beyond the Hartree mean-field approximation taking into account quantum fluctuations. We find that the correction has a considerable impact on the M-R relationships and other properties of condensate dark stars, such as factor of compactness and tidal Love numbers. The impact is more significant for equation-of-states that support larger highest stellar masses.

The James Webb Space Telescope (JWST) has spectroscopically confirmed galaxies up to $z\sim14$, 300 Myr after the Big Bang, and several candidates have been discovered at $z\sim15-25$, with one candidate as high as $z\sim30$, only 100 Myr after the Big Bang. Such objects are unexpected, since theoretical studies have not predicted the existence of detectable galaxies at $z\sim30$. While any $z\sim30$ candidates may be contaminants at lower redshifts, we explore whether such extreme redshift sources could be consistent with hyper-energetic transient events linked to the formation of the first, metal-free, stars. Specifically, we consider pair-instability supernovae (PISNe), a predicted class of extreme thermonuclear explosions that leave no remnant behind. Using cosmological simulations, we investigate an overdense cosmic region, where star formation and subsequent PISNe occur at $z\sim30-40$, even within standard cosmology. Assessing the likelihood of such a region, the corresponding number of PISNe at $z\gtrsim20$, and their observed flux, we find that JWST has a non-negligible chance to detect a PISN event at extremely high redshifts. If a transient event were confirmed at $z\sim30$, this would provide a direct glimpse into the epoch of first star formation, dramatically extending the empirical reach of astronomy.

We study nonexistence results and gradient estimates for solutions of \[ Δ_p v + a v^{q}=0 \] defined on complete Riemannian manifolds satisfying a \emph{$χ$-type Sobolev inequality}. We establish a Liouville theorem under the assumptions that the underlying manifold $(M,g)$ supports a \emph{$χ$-type Sobolev inequality} and that the $L^{\fracχ{χ-1}}$-norm of $\ric_-(x)$ is bounded above by a constant depending only on $\dim(M)$, the Sobolev constant $\mathbb{S}_χ(M)$, and the volume growth rate of geodesic balls $B_r\subset M$. This extends and improves several recent results of Ciraolo, Farina, and Polvara \cite{CFP}; our approach, however, differs from their ``$P$-function'' method. In addition, for manifolds satisfying a \emph{$χ$-type Sobolev inequality}, we obtain a lower bound on the volume growth of geodesic balls. We also derive a local logarithmic gradient estimate for positive solutions, assuming that $\ric_-(x)\in L^γ$ for some $γ> \fracχ{χ-1}$. Some geometric and topological applications of our main result are also presented in this article (see \thmref{end}, \thmref{main4}, and \corref{main5}). In particular, we prove the following. Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension $n\ge 3$ on which the Sobolev inequality \eqref{chi-n} holds, and assume that $\ric(x)\ge 0$ outside some geodesic ball $B(o,R_0)$. Then there exists a positive constant $C(n)$, depending only on $n$, such that if \[ \|\ric_-\|_{L^{\frac{n}{2}}}\leq C(n)\,\mathbb{S}_{\frac{n}{n-2}}(M), \] then $(M,g)$ has exactly one end.

This paper presents an embedded EEG instrumentation platform for real-time steady-state visually evoked potential (SSVEP) decoding based on an ESP32-S3 microcontroller and an ADS1299 analog front end. The system performs $8$-channel EEG acquisition, zero-phase bandpass filtering, and canonical correlation analysis entirely on-device, while supporting wireless communication and closed-loop operation without external computation. A central contribution is the quantitative characterization of the platform's measurement integrity. Reported results demonstrate a stable shorted-input noise floor ($\approx 0.08~μ\text{V}_{\text{RMS}}$), tightly bounded sampling jitter ($0.56~μ\text{s}$ standard deviation), and negligible long-term drift ($< 1~\text{ppm}$). Numerical fidelity analysis shows $100\%$ decision agreement between the mixed-precision embedded pipeline and a $64$-bit double-precision reference. Effective common-mode attenuation exceeded $112~\text{dB}$ under balanced conditions, with a localized $26.9~\text{dB}$ degradation observed under source-impedance mismatch. Closed-loop validation achieved $99.17\%$ online accuracy and an information transfer rate of $27.66~\text{bits/min}$. These results position the proposed system as a quantitatively characterized embedded EEG measurement and processing platform for real-time SSVEP decoding.

Nematic aerogel (nAG) supports so-called polar phase in liquid 3He. The experiment [Dmitriev et al, JETP Lett. 112, 780 (2020)] showed that the onset of polar phase inside the nAG is accompanied by emergence of a sound wave with frequency quickly growing with cooling down from transition temperature and reaching a plateau. To describe this behavior, we start by calculating the elastic properties of the dry nematic AG that appear to depend only on Young's modulus of the parent material (e.g. mullite), the volume fraction of the solid phase and the aspect ratio of the representative volume of nAG. The elastic constants are then used to solve elasto-hydrodynamic equations for various sound vibrations of nAG filled with 3He. The (isotropic) first sound and anisotropic second sound in the polar phase are strongly hybridized with fourth sound and standard elastic modes in nAG. The hybrid second and the transverse fourth sound start with zero velocity at the transition, similar to pure 3He, and quickly grow with lowering temperature until they hit the sample finite size cutoff.