Search Results (300)

Inverse scattering problems seek anomalies in a medium given data measured after the interaction with emitted waves. Due to noise, predictions about the nature of these inclusions should be complemented with uncertainty estimates. To this end, we propose a progressive framework for inverse scattering from low- to high-dimensional Bayesian formulations depending on the prior information and the problem complexity. We aim to reduce computational costs by exploiting educated prior information. When we look for a few well-separated inclusions in a known medium with information about their number, we resort to low-dimensional parameterizations in terms of a few random variables representing their shape and material constants. We test this approach detecting anomalies in tissues and deposits in stratified subsoils. In more complex situations where the anomalies may overlap, we propose high-dimensional parameterizations obtained from Karhunen–Loève (KL) or Fourier expansions of the density and velocity fields. We employ these methods to characterize oil and gas reservoirs in a salt dome configuration, where the screening effect of the dome cap prevents the obtention of adequate prior information. We characterize the posterior probability by means of affine invariant ensemble and functional ensemble MCMC samplers depending on dimensionality. This provides information on configurations with the highest a posteriori probability and the uncertainty around them, identifying factors that could reduce the uncertainty. In high-dimensional setups, techniques based on KL developments are more effective and stable. A recurring issue is the choice of the a priori covariance (which strongly affects the results) and the choice of its hyperparameters. Here, we use educated choices. Formulations that include them as additional parameters could be a next step at a higher cost. Full article

In this paper, the uniform approximation of continuous functions on $[0,1]$ by rational functions with prescribed poles and bounded multiplicities is studied. A classical theorem of Fichera characterizes density in $C\left(\right[0,1\left]\right)$ through the divergence of a conformally invariant series involving the pole distribution. A modern reformulation of this result is developed and it is given an operator-theoretic interpretation in which the approximation property is equivalent to cyclicity and to the absence of nontrivial invariant subspaces in an associated Hardy-space model. In this framework, the classical Blaschke condition emerges as the fundamental obstruction to density, linking rational approximation to the structure of model spaces and non-selfadjoint operator algebras. The density criterion is interpreted in terms of symmetry: divergence corresponds to a balanced distribution of poles compatible with the conformal geometry of the slit domain, while convergence induces symmetry breaking and the emergence of invariant structures. Numerical models illustrate the sharpness of the criterion and provide a concrete manifestation of the Blaschke obstruction and cyclicity mechanism. This new approach places Fichera’s theorem within a broader operator-theoretic and spectral framework, connecting classical approximation theory with Hardy spaces, invariant subspace theory, and modern rational approximation methods. Full article

Electroencephalography (EEG) provides a non-invasive and high-temporal-resolution modality for decoding cognitive states, but high-density recordings remain challenging for Transformer-based models because self-attention scales quadratically with the number of channels. In addition, conventional Euclidean representations do not fully capture the intrinsic geometry of EEG covariance features, which may limit robustness in cross-subject settings. To address these issues, we propose EEG-RCformer, a Riemannian geometry-informed channel clustering Transformer for EEG decoding. The model first computes per-channel symmetric positive definite (SPD) covariance matrices from windowed EEG features and uses the affine-invariant Riemannian metric (AIRM) to identify trial-specific functional hubs. These hubs are then integrated with capacity-constrained spatial clustering to generate anatomically plausible and computationally efficient channel groups, which are encoded as tokens for a Transformer classifier. We evaluated EEG-RCformer on the MODMA and SEED datasets under both subject-dependent and -independent paradigms, achieving area under the curve (AUC) values of 0.9802 and 0.7154 on MODMA and 0.8541 and 0.8011 on SEED, respectively. Paired statistical tests further showed significant gains for MODMA in both the subject-dependent and -independent settings and for SEED in the subject-dependent setting, while SEED still showed a positive but non-significant mean improvement in the subject-independent setting. Full article

Chirality is a pervasive and functionally critical feature of biological macromolecules, yet its distributed and emergent forms remain poorly quantified in complex systems such as membrane proteins. We present Chirobiophore, a novel paradigm for capturing biochirality across scales—from atomic geometries to global structural asymmetries. Unlike traditional stereochemical metrics, Chirobiophore employs a multidimensional model-independent vector comprising Local Tetrahedral Asymmetry (LTA), Helical Path Curvature (HPC), Asymmetric Environment Score (AES), Directional Density Profile (DDP), Leaflet Asymmetry Index (LAI), and Orientation Twist Score (OTS). This framework enables coordinate-invariant comparisons of structurally diverse proteins in a continuous chirality space. We demonstrate its application to canonical, GPCR, and topologically complex membrane proteins, revealing distinct chirality signatures and functional clustering. Furthermore, we map Chirobiophore descriptors to tissue-level asymmetry indices, providing a bridge between molecular structure and morphogenetic patterning. Chirobiophore offers a unified, extensible platform for structural biology, synthetic design, and developmental modeling of chirality. Full article

The critical brain hypothesis proposes that neural systems operate near a phase transition to optimize information processing. A key method for investigating this hypothesis is the phenomenological renormalization group (pRG), which looks for scale-invariant features across levels of coarse-graining. One such feature is the power-law scaling of eigenvalues of covariance matrices of coarse-grained variables. However, the estimation of this scaling exponent, $\mu$, often relies on linear regression over arbitrarily selected ranges of the plot of eigenvalues versus rank. This heuristic “eyeballing” introduces uncontrolled bias and complicates the interpretation of observed scaling relationships. In order to obtain a more robust estimation of $\mu$, we do not fit the standard eigenvalue-vs-rank relationship, but rather the density of eigenvalues, for which standard protocols exist for fitting power laws to empirical data distributions. We demonstrate this approach using a synthetic model that replicates the scaling signatures of neural data while providing control over the system’s exponents as well as neural data obtained from publicly available Neuropixels recordings. We also establish standards for the minimal data required to quantify power-law behavior in a pRG eigenvalue analysis. Our approach contributes a tool for understanding the fundamental limitations imposed by spatial and temporal constraints of experimental datasets, which is required to rigorously assess the neural criticality hypothesis. Full article

The advent of fault-tolerant quantum computers poses an existential threat to the current blockchain technology, which relies on cryptographic primitives like elliptic-curve cryptography and SHA-256 hashing. This manuscript surveys the emerging field of quantum-secure blockchains, categorizing the main research directions into two paradigms. The first, post-quantum blockchain, seeks to replace classical cryptographic elements with quantum-resistant algorithms. The second, more radical approach aims to construct an intrinsically quantum blockchain, where the ledger’s security and state are encoded directly in quantum mechanical principles. We delve into three promising intrinsic schemes: those based on Greenberger–Horne–Zeilinger (GHZ) states and entanglement in time, those leveraging multi-time states and pseudo-density matrices, and hypergraph-based approaches. As the principal original contribution of this work, we present a comprehensive theoretical framework for a topological quantum blockchain based on non-Abelian anyons, providing the first detailed encoding scheme mapping classical blockchain data to braiding sequences. We further develop the connection to Chern–Simons theory, establishing a field-theoretic foundation where the blockchain’s history is encoded in Wilson loops, and its immutability follows from topological and gauge invariance. Extending this framework, we introduce a holographic AdS/CFT interpretation, revealing that the topological blockchain can be understood as a dual description of a black hole analog in anti-de Sitter space, where the blockchain’s history is encoded in the microstates of a black hole and linking braids between blocks correspond to wormholes. We provide a detailed physical and mathematical analysis of each scheme, comparing their security assumptions, resource requirements, and feasibility in the near and long terms. The topological approach, in particular, offers a compelling new path toward a blockchain with inherent fault tolerance, where the chain’s history is encoded in the topology of anyon worldlines, making it naturally resistant to decoherence and local tampering. Full article

The coupling of non-stationary marine noise and complex ship-radiated signals makes high-fidelity signal recovery exceptionally difficult. Existing deep learning methods often prioritize objective metrics, such as the Scale-Invariant Signal-to-Noise Ratio (SI-SNR), but fail to maintain the integrity of narrow-band line spectral data. We propose FocuS-MN, an end-to-end framework that combines learnable resampling with Feedforward Sequential Memory Network (FSMN)-based temporal modeling for precise waveform reconstruction. The model is optimized using a two-stage training strategy to ensure stable magnitude estimation and waveform consistency. On the ShipsEar dataset, FocuS-MN shows strong generalization to unseen vessel types. At a −5 dB Signal-to-Noise Ratio (SNR), it achieves a Signal-to-Distortion Ratio (SDR) of 3.77 dB and a Segmental Signal-to-Noise Ratio (SSNR) of 3.83 dB. Power Spectral Density (PSD) analysis further confirms that FocuS-MN recovers fine-grained line spectral structures, proving its effectiveness in both noise suppression and signal fidelity. Full article

The string-inspired running vacuum model (StRVM) of inflation is based on a Chern–Simons (CS) gravity effective action in which the only four-spacetime-derivative-order term is a gravitational anomalous CS–Pontryagin density coupled to an axion. In this work, we revisit curvature-squared string-inspired effective actions from the point of view of appropriate local field redefinitions, leaving the perturbative string scattering matrices invariant. We require simultaneously unitarity and torsion interpretation of the field strength of the Kalb–Ramond antisymmetric tensor, features characterizing the (3+1)-dimensional StRVM cosmology. Unlike the higher-dimensional case, the above features are possible in the context of (3+1)-dimensional spacetimes, obtained after string compactification. We demonstrate that the unitarity and torsion interpretation requirements lead to a single type of extra four-derivative terms in the effective gravitational action, not discussed in the previous literature on StRVM, which is, however, shown to be subleading by many orders of magnitude compared to the terms of the StRVM framework. Hence, its presence has no practical implications for the relevant inflationary (and, hence, postinflationary) physics of the StRVM. This demonstrates the phenomenological completeness of the StRVM cosmological scenario, which is thus fully embeddable in the UV-complete (quantum gravity-compatible) string theory framework. Full article

Generative adversarial networks (GANs) are increasingly applied to mitigate extreme class imbalance in intrusion detection systems, yet reported improvements often obscure role augmentation intensity and adversarial stability. This paper presents a controlled ablation study that isolates the impact of adversarial objective choice, augmentation ratio, and training duration on GAN-based minority data augmentation for highly imbalanced tabular cybersecurity data. Using the UWF-ZeekData22 dataset, nine MITRE ATT&CK tactic-versus-benign classification tasks are evaluated under augmentation ratios of 0.25 and 0.50 and training durations of 400 and 800 epochs. Four GAN variants—Vanilla GAN, Conditional GAN (cGAN), WGAN, and WGAN-GP—are assessed using stratified cross-validation and five classical classifiers representing diverse inductive biases. The results reveal consistent structural patterns. Moderate augmentation (r = 0.25) with controlled training (400 epochs) yields the most stable and reliable improvement in minority recall. Wasserstein-based objectives demonstrate superior stability under aggressive augmentation and prolonged training, while conditional GANs frequently exhibit recall collapse in ultra-sparse regimes. Increasing augmentation volume does not uniformly improve performance and may introduce distributional overlaps that degrade linear and margin-based classifiers. Tree-based classifiers remain largely invariant once sufficient minority density is achieved. These findings demonstrate that adversarial calibration is more important than architectural complexity for improving the detection of rare attacks. The study provides practical guidance for designing robust GAN-based augmentation pipelines under extreme cybersecurity class imbalance. Full article

Shannon’s differential entropy for continuous variables suffers from a fundamental limitation: it is not invariant under scale transformations. This makes entropy values dependent on the choice of measurement units rather than reflecting intrinsic properties of distributions. While Jaynes proposed the limiting density of discrete points (LDDP) as a theoretical solution, a concrete method for computing the required invariant measure has been lacking. This paper establishes a rigorous connection between Kullback–Leibler divergence and the invariant measure, providing theoretical proofs of invariance under affine transformations and a practical computational method. We prove that entropy normalized by the median of k-nearest neighbor distances is invariant under affine transformations (Theorems 1 and 2). The non-negativity of the resulting entropy has been validated empirically across all tested distribution families, though a complete theoretical proof remains an open question. This approach extends naturally to multivariate settings, enabling scale-invariant mutual information estimation. We provide open-source implementations in Julia (EntropyInvariant.jl) and Python (entropy_invariant) and demonstrate their advantages over traditional approaches, particularly for variables with disparate scales. Full article

A two-fluid model can be described by an anisotropic fluid matter, and we introduced the notion of an anisotropic fluid spacetime. The algebraic and differential properties of an anisotropic fluid spacetime equipped with several forms of the stress–energy tensor is the focus of this research. We show that an anisotropic fluid spacetime with a radial pressure $p^{\Vert }$, transverse pressure $p^{\perp }$, and the energy density $\rho$ is a generalized quasi-Einstein spacetime. We prove that a dark matter era or an anisotropic fluid spacetime with vanishing vorticity is represented by an anisotropic fluid spacetime endowed with a covariant constant stress–energy tensor; on the contrary, a dark matter era or the expansion scalar vanishes is represented by an anisotropic fluid spacetime endowed with a Codazzi type of stress–energy tensor, as long as $\mathcal{A}$ stays invariant under the velocity vector field $\zeta$. Furthermore, we use the Killing velocity vector field, parallel vector fields to characterize Ricci Semi-Symmetric, $\mathcal{T}$-recurrent, Pseudo-Ricci symmetric, and $\widehat{\mathbb{R}}$-harmonic anisotropic fluid spacetime. We find that the anisotropic fluid spacetime reflect a stiff matter and a radiation era with these geometric symmetries. Finally, we provide findings for an anisotropic fluid spacetime with a divergence-free matter tensor and the vanishing space-matter tensor and explore the dynamical aspects of cosmological epoch of an anisotropic fluid spacetime coupled with $f\left(\mathcal{R}\right)$-gravity. Full article

Granular materials subjected to complex stress histories exhibit pronounced path dependence, multi-scale heterogeneity and scale-invariant characteristics, especially when particle breakage leads to gradation evolution with fractal features. Discrete element simulations are performed on granular assemblies with prescribed idealized fractal gradations under constant stress ratio loading–turning paths, while maintaining identical solid volume and initial relative density. The results show that, for a given gradation, both the peak strength envelope and the critical state line exhibit high consistency and are effectively independent of the examined stress paths, which are supported by high R² values from regression. At the critical state, microstructural parameters together with energy measures consistently follow stable power–law relationships with mean effective stress. For different gradations, the critical stress ratio remains nearly unchanged, whereas peak strength increases with increasing fractal dimensions; although critical state points remain nearly collinear in the deviatoric stress (q) –mean effective stress (p) plane, the critical state line in void ratio (e)–p plane shifts downward as the particle size distribution becomes broader. The evolution of microstructural and energy-related power–law relationships with fractal dimension exhibits a clear saturation trend. This study demonstrates that, within the simulated framework, fractal gradation primarily governs the position of the critical state in e–p space without altering its fundamental path-independent nature, providing fundamental insights into the multi-scale mechanics of graded granular materials under complex loading. Full article

This paper presents a set of survey-style notes linking core themes of pure algebra with central topics in algebraic and analytic number theory. We begin with finite extensions of $\mathbb{Q}$ and describe algebraic number fields through their realization as finite-dimensional $\mathbb{Q}$-algebras (via multiplication operators and matrix representations), leading naturally to the arithmetic invariants—trace, norm, and discriminant—and to the ring of integers, ideals, Dedekind domains, and the ideal class group. We then develop the classical theory of cyclotomic fields, emphasizing their Galois structure and their role in abelian extensions of $\mathbb{Q}$. Next, we discuss ramification in general extensions, including decomposition and inertia groups, the Frobenius element, and the Chebotarev density theorem. The exposition continues with a concise algebraic introduction to elliptic curves and their L-functions, and it places key conjectural links (including Birch and Swinnerton-Dyer) in context. Finally, a collection of examples highlights a common operational language between fractional calculus and number theory: Laplace and Mellin transforms turn convolution-type operators into multiplication, clarifying the appearance of $\Gamma$-factors, Dirichlet series, and zeta- and L-function structures in both settings. Full article

Conventional wisdom posits that smog suppresses outdoor activity while shifting peoples’ activities indoors. Using anonymized Mobile Phone Data Provider Records fused with Point-of-Interest (POI) data sourced from the Gaode (Amap) open database for Beijing (2–22 February 2015), we test this substitution hypothesis at an hourly resolution across 12 POI-defined activity categories. We estimate the adjusted population density (APD) from mobile phone data via usage-bias calibration, interpolate city-wide AQI (Air Quality Index) and PM2.5 fields, and identify associations with a two-way fixed-effects design (Voronoi polygon (VP), day × hour model. We also handle time-invariant POI activities, while factoring in weather and day types. We find a dual suppression of both outdoor and indoor physical activities: worsening air quality is associated with lower participation in most outdoor and indoor activities. Effects are heterogeneous across categories and hours; shopping shows all-day negative marginal effects, whereas a few categories (e.g., sightseeing) display positive correlations in select afternoon hours consistent with congestion-avoidance rather than health-driven indoor substitution. Quantitatively, a 100-point AQI increase is associated with an order of 1–5 persons/km2 decline at peak hours for most activities. A Comprehensive Impact Index (CII) summarizes the spatial heterogeneity across the city. POI venue operators should anticipate city-wide activity reduction both indoors and outdoors under heavy pollution, rather than plan solely for outdoor-to-indoor activity shifts. Full article

Global Navigation Satellite Systems (GNSSs) observables, such as those of the Global Positioning System (GPS), are frequently affected by multipath effects that cause unpredictable signal interference at the receiver, posing significant challenges for accurate state estimation in complex environments with non-Gaussian noise or outliers. The traditional extended Kalman filter (EKF), based on the minimum mean square error (MMSE) criterion, assumes Gaussian noise distributions and exhibits degraded performance under non-Gaussian conditions. To overcome this limitation, the minimum error entropy (MEE) criterion was proposed to reduce random uncertainty in estimation error distributions; however, due to its translation invariance property, MEE may inadvertently increase bias when errors contain systematic offsets, leading to poor convergence. In contrast, the maximum correntropy criterion (MCC) concentrates the error probability density function (PDF) around zero, enabling effective entropy adjustment even in the presence of bias and achieving superior error convergence. This paper presents the centered error entropy (CEE) extended Kalman filter (CEE-EKF) that integrates the complementary merits of both MEE and MCC approaches to overcome their individual limitations. Experimental validation in complex nonlinear GPS environments with non-Gaussian noise demonstrates that the CEE-EKF significantly outperforms individual algorithms in noise suppression, particularly exhibiting enhanced robustness and accuracy when handling outliers. These results offer an effective approach to enhancing the reliability of GPS navigation in challenging real-world environments, and the algorithm can be readily extended to other GNSS applications. Full article