Search Results (1,319)

We identify a fundamental tension between general relativity and spectral geometry arising from the global, nonlocal character of spectral data versus the local causal dynamics of spacetime. To resolve this, we postulate spectral invariance, $\delta {\Lambda }_n=0$, requiring the eigenvalues of the Laplace–Beltrami operator to remain fixed under physical evolution. This condition yields a compensatory relation between metric deformations and eigenfunction amplitudes, suggesting a kinematic coupling linking energy distribution to spacetime curvature. From the second variation of the associated energy functional, we derive a rank-4 tensor proportional to the inverse DeWitt supermetric on superspace and a contracted rank-2 tensor proportional to the spacetime metric, and we recover a invariance law of the energy functional in configuration space. Spectral invariance may suggest a framework in which geometry and energy are co-defined through fixed spectral data. Full article

Plant viruses from the Potyviridae family have a significant impact on crop productivity worldwide. We conducted a bioinformatic analysis of the VPg sequences from several members of the Potyviridae family. All analyzed primary structures of VPg contain an invariant arginine, which, according to the model we proposed earlier, is located in the functionally important α1–α2 hairpin of the viral protein and forms a recognition contact during the formation of its complex with the eIF4E host cell. Among the amino acid mutations observed in the sequences of VPg PVY, we separately considered those associated with adaptation to the host plant. Several strain-specific mutations were identified, the functional roles of which are currently unclear. For each of the Potyviridae species considered, a consensus VPg sequence was determined. 3D-models of the corresponding proteins were constructed by de novo molecular modelling using the consensus amino acid sequences. Cross-comparative analysis of the theoretical models and the experimental VPg PVY structure obtained by NMR showed that all these proteins share a high degree of structural homology and contain the conserved arginine within the α1–α2 hairpin. However, the spatial position of this arginine may vary across models, which apparently reflects species-specific differences in the VPg recognition module. Full article

Precise segmentation of cardiac substructures in magnetic resonance imaging is pivotal for diagnosis and treatment planning but remains impeded by anatomical scale heterogeneity and the scarcity of high-quality pixel-level annotations. Existing deep learning paradigms often struggle to simultaneously resolve the global geometry of ventricular cavities and the fine-grained boundaries of the myocardium, particularly in low-data regimes. To address these challenges, we propose SAS-SemiUNet++, a holistic semi-supervised segmentation framework. This architecture incorporates two novel mechanisms: (1) The Scale-Aware Semantic Recalibration (SASR) unit, which functions as a dynamic semantic gate to adaptively adjust receptive fields, mimicking a radiologist’s variable-focus mechanism to capture multi-scale anatomical details, and (2) Stochastic Consistency Regularization (SCR), a dual-path perturbation strategy that enforces geometric invariance on unlabeled data, thereby mitigating overfitting to noisy pseudo-labels. Comprehensive evaluations on the ACDC benchmark demonstrate that SAS-SemiUNet++ significantly outperforms state-of-the-art methods, achieving superior segmentation accuracy and boundary fidelity, particularly in reducing the 95% Hausdorff distance. This study presents a data-efficient and robust solution for cardiac image analysis, offering potential for scalable clinical deployment. Full article

With the growing demand for trustworthy multi-party data sharing, federated learning has demonstrated broad potential in cross-entity collaborative modeling. However, it still faces challenges such as insufficient participant engagement, inaccurate contribution assessment, and the lack of dynamic profit-sharing mechanisms. Traditional incentive schemes, which typically rely on game-theoretic models or static rules, struggle to accommodate dynamic client participation and heterogeneous data distributions, thereby degrading the convergence efficiency and generalization performance of the global model. To address these issues, we propose a budget-aware closed-loop incentive allocation for federated learning with deep deterministic policy gradient (DDPG). The proposed approach constructs a DDPG-driven closed-loop framework in which the server manages system states, incentive decisions, and model aggregation, while clients autonomously adjust their data contribution levels. By formulating incentive allocation as a sequential decision-making problem, the mechanism jointly optimizes policy and value functions. A permutation method is introduced to ensure invariance to client ordering, and an Ornstein–Uhlenbeck process is employed to enhance exploration, thereby improving the adaptiveness and overall effectiveness of incentive allocation. Experimental results show that the proposed method significantly increases cumulative rewards and improves client data-sharing rates in high-dimensional dynamic environments. Compared with traditional fixed incentive schemes, the mechanism demonstrates clear advantages in adaptiveness, incentive effectiveness, and model performance. Full article

The geometric signatures of macroscopic interfaces in the two-dimensional critical Ising model strictly adhere to Schramm–Loewner Evolution (SLE) theory. In this study, we propose a physics-driven generative approach using Super-Resolution Generative Adversarial Networks (SRGANs) to approximate the inverse coarse-graining operation to generate larger configurations. From the perspective of Geometric Deep Learning (GDL), we leverage the geometric priors of Convolutional Neural Networks (CNNs)—specifically their translational and rotational symmetries—to effectively encode the universal physical laws of the Ising Hamiltonian. This inductive bias allows the model to be trained on small scales yet be generalized to large-scale systems ($2048 \times 2048$) while preserving physical conservation. To accommodate spin discreteness, we employ an L1-based loss function to maintain domain wall sharpness. SLE analysis and long-range correlation functions confirm that the model reproduces critical dynamics and conformal invariance, successfully serving as a physics-preserving inverse coarse-graining transformation framework. Full article

This article proposes a single-deformation scheme for probability measures that simultaneously encompasses two classical deformations from free probability: the $V_a$ deformation ($a\in \mathbb{R}$) and the $T_c$ deformation ($c\in \mathbb{R}$). The associated operator, written as ${\mathcal{W}}_{(a,c)}$, is introduced via a functional relation involving the Cauchy–Stieltjes transform and is constructed so as to recover the initial deformations as special cases, namely ${\mathcal{W}}_{(0,c)}=T_c$ and ${\mathcal{W}}_{(a,0)}=V_a$. Working within the concept of Cauchy–Stieltjes kernel families, we analyze the action of ${\mathcal{W}}_{(a,c)}$ on variance functions and establish an explicit expression for the variance function induced by this deformation. This approach leads to a structural invariance property demonstrating that the free Meixner class is preserved under the action of ${\mathcal{W}}_{(a,c)}$. In addition, the operator provides a new perspective on the semicircle distribution, yielding a characterization that reflects the symmetric nature of the deformation and its compatibility with fundamental distributions in free probability. Full article

We analyze the spherically symmetric complex diffusion and special type of the complex reaction–diffusion equations. These equations are form invariant to the free Schrödinger equations and to the Schrödinger equations with power-law space-dependent potentials. Our new type of solutions are important because we found a new realm of solutions which lie between the solutions of the classical regular diffusion equation and the usual quantum mechanical solutions of the Schrödinger equation. As the solution method, we applied the the self-similar Ansatz, which reduces the original partial differential equation (PDE) to an ordinary differential equation (ODE) which can be solved analytically. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation which has to be used in ordinary quantum mechanics for time-independent Hamiltonian. For the complex diffusion equation—without any additional source term—the solutions are the Kummer’s M and Kummer’s U functions. For some parameter values we found $L^2$ integrability, as in the Cartesian case. We interpret that this property can be a “quantum mechanical heritage” and can be a far relation to ordinary quantum mechanics. Therefore, in this sense, our solutions might have quantum mechanical interest in the future. For the complex reaction–diffusion-type equation we derived the Whittaker M and Whittaker W functions as solutions. These solutions have no $L^2$ integrability at all. All derived solutions have complex quadratic arguments. These kind of analytic solutions are new and cannot be found in the existing scientific literature. Finally, the role of the complex angular momentum was investigated as well. Full article

Proportional–integral–derivative (PID) controllers are always a preferred choice of control strategy in industrial and biomedical systems due to their simplicity, reliability, and easy implementation. However, the systematic tuning of PID parameters for nonlinear, constrained, and safety-critical systems remains challenging, particularly in the presence of disturbances and actuator limitations. This paper presents a unified surrogate-based optimization framework for tuning PID controllers for linear and nonlinear dynamical systems. The tuning problem is formulated as a constrained optimization task, where performance objectives and safety requirements are explicitly incorporated into the cost function. A surrogate-based optimization via clustering (SBOC) approachis employed to efficiently explore the PID parameter space while reducing the number of expensive closedloop simulations. The proposed framework is first applied to the first- and second-order linear time-invariant systems to check its feasibility and then to the nonlinear systems to demonstrate its robustness under nonlinearity and saturation. The approach is further applied to safety-critical systems considering the case of glucose regulation in type 1 diabetes under realistic meal disturbances and insulin delivery constraints. The simulation results show that the surrogate-optimized PID controller achieves stable regulation with improved tracking performance while strictly satisfying safety requirements, including control effort penalties to limit actuator wear and the avoidance of hypoglycemia and hyperglycemia in glucose regulation problems. Full article

This paper develops an operator-reduction framework for a class of coupled implicit Caputo-Hadamard fractional differential systems subject to nonlocal Hadamard integral constraints. The system, involving fractional derivatives in both state and auxiliary variables, is resolved through a pointwise contraction argument that eliminates the auxiliary components and reduces the problem to a two-dimensional fixed-point operator acting on a Banach space of continuous functions. This reduction overcomes the compactness obstruction that arises in direct multi-component formulations. Under explicit growth and smallness conditions, the existence of at least one solution is established via Mönch’s fixed-point theorem. By imposing strengthened Lipschitz hypotheses, the reduced operator becomes a strict contraction on an invariant ball, yielding uniqueness and Ulam-Hyers stability with explicit constant $C_{UH}=1/(1-\Lambda )$. A fully computed example demonstrates the verifiability of the theoretical assumptions and illustrates how the smallness condition $\Lambda <1$ governs both existence and stability. The results establish a systematic operator-based approach for implicit Caputo-Hadamard systems with nonlocal integral constraints. Full article

Lorentz invariance violation (LIV) can alter the group velocity of photons by modifying their dispersion relation, manifesting as differences in the arrival times of photons with different energies. This effect can accumulate over long propagation distances, making gamma-ray bursts (GRBs) a key tool for probing Lorentz invariance violation. By analyzing spectral lag data from 360 measurements across 90 GRBs using Markov Chain Monte Carlo (MCMC) sampling, and under the assumption that all GRBs share a common intrinsic time delay function, we report a maximum a posteriori value of the energy scale of quantum gravity at linear order $E_{QG}=8.96\times {10}^{14}$ GeV, though the data are also compatible with Lorentz invariance ($E_{\mathrm{QG}}=\infty$) to within $2.8\sigma$. Furthermore, we are $95\%$ confident that $E_{QG}\ge 6.67\times {10}^{14}$ GeV. Full article

Background/Objectives: As life expectancy increases, chronic diseases have become more prevalent, often leading to poorer health in later years. Maintaining cognitive functioning is therefore essential for preserving independence in older adulthood. Within the framework of cognitive enrichment, research highlights the protective role of healthy lifestyles and engagement in social and intellectual activities on cognitive functioning. This study aimed to provide evidence of the moderator effect of diagnosis group (including healthy condition, dementia, Parkinson’s, and stroke) on a predictive model of cognitive function. Methods: Data employed in this study came from the 9th wave of the Survey of Health, Ageing and Retirement in Europe (SHARE) project, including 17,105 individuals aged 50 years and older from 27 European countries. Cognitive functioning was assessed through numeracy, temporal orientation, verbal fluency, and memory. Physical inactivity, social participation, intellectual activities, age, gender, and education were included as predictors. A measurement invariance routine across diagnostic groups was tested. Results: The model demonstrated excellent fit in the general sample and partial invariance across groups. Physical inactivity was negatively associated with numeracy in all groups, with stronger effects in clinical populations, particularly stroke and dementia. Intellectual activities were positively associated with numeracy across groups, with the largest effects observed in dementia. Temporal orientation, physical inactivity and intellectual activities showed significant associations mainly in clinical groups, whereas age demonstrated a consistent negative effect across all groups. Conclusions: Lifestyle factors show differential associations with cognitive domains depending on diagnostic condition. These findings support the heterogeneity of cognitive aging and highlight the importance of tailored, person-centered intervention strategies. Full article

Non-parametric information geometry has long faced an “intractability barrier”: in the infinite-dimensional setting, the Fisher–Rao metric is a weak Riemannian metric functional that lacks a bounded inverse, rendering classical optimization and estimation techniques computationally inaccessible. This paper resolves this barrier by building the statistical manifold on the Orlicz space $L_0^{\mathsf{\Phi }}\left(P_f\right)$ (the Pistone–Sempi manifold), which provides the necessary exponential integrability for score functions and a rigorous Fréchet differentiability for the Kullback–Leibler divergence. We introduce a novel Structural Decomposition of the Tangent Space ($T_{f}M=S\oplus S^{\perp }$), where the infinite-dimensional space is split into a finite-dimensional covariate subspace (S)—representing the observable system—and its orthogonal complement ($S^{\perp }$). Through this decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as $G_f$, which acts as the computable “Hilbertian slice” of the otherwise intractable metric functional. Key theoretical contributions include proving the Trace Theorem ($H_G\left(f\right)=\mathrm{Tr}\left(G_f\right)$) to identify G-entropy as a fundamental geometric invariant; demonstrating the Geometric Invariance of the Covariate Fisher Information Matrix (cFIM) as a covariant $(0,2)$-tensor under reparameterization; establishing the cFIM as the local Hessian of the KL-divergence; and characterizing the Efficiency Standard through a generalized Cramer–Rao Lower Bound for semi-parametric inference within the Orlicz manifold. Furthermore, we demonstrate that this framework provides a formal mathematical justification for the Manifold Hypothesis, as the structural decomposition naturally identifies the low-dimensional subspace where information is concentrated. By shifting the focus from the intractable global manifold to the tractable covariate geometry, this framework proves that statistical information is not a property of data alone, but an active geometric interaction between the environment (data), the system (covariate subspace), and the mechanism (Fisher–Rao connection). Full article

In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces $\mathrm{U}/\mathrm{H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on $\mathrm{U}/\mathrm{H}$ provided by generalized thermodynamics à la Souriau. Our main result is the proof that $\mathrm{U}/\mathrm{H}$.s supporting such Gibbs distributions are only the Kähler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of $\mathrm{U}$ of a positivity domain in the Cartan subalgebra $C_c\subset \mathbb{H}$ of the maximal compact subalgebra $\mathbb{H}\subset \mathbb{U}$. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi–Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao’s, Chentsov’s, and Amari’s Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincaré plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against $\mathrm{U}$ transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group $\mathrm{H}\subset \mathrm{U}$. Full article

The generative mechanisms underlying multifractal scaling in complex systems remain a fundamental unsolved problem, limiting our ability to distinguish healthy from pathological dynamics, predict system failures, or understand how scale-invariant organization emerges across vastly different physical domains. We resolve this challenge by introducing threshold sensitivity analysis—an extension of Chhabra–Jensen’s direct method—as a framework that classifies cascade types by examining how scaling reliability varies across moment orders q. Different q values systematically probe weak fluctuations (negative q) versus strong fluctuations (positive q), and the coefficient of determination ($r^2$) of partition function regressions quantifies scaling reliability at each q. Analyzing $r^2\left(q\right)$ patterns in 280 cardiac recordings (healthy controls through fatal heart failure), 200 financial time series (global equity markets and currencies, 2000–2025), and 80 climate stations (tropical to continental zones, 2000–2025), we discover a universal diagnostic signature: symmetric expansion of valid scaling behavior under relaxed $r^2$ thresholds, spanning both weak and strong fluctuations. This threshold sensitivity fingerprint—predicted by synthetic cascade simulations but never before validated empirically—uniquely identifies additomultiplicative cascades, hybrid processes that randomly alternate between additive stabilization and multiplicative amplification. Critically, this symmetric signature persists universally across domains: cardiac dynamics maintain consistent patterns across health and disease states, financial markets show varying robustness across asset classes (currencies more variable than US equities) while preserving a hybrid structure, and climate systems exhibit geographical variations (subtropical/continental stronger than tropical) without altering fundamental cascade type. These findings suggest that additomultiplicative organization is a unifying feature of complex adaptive systems, offering a resolution to decades of debate between additive and multiplicative models. The $r^2\left(q\right)$ profiling provides a mechanistic diagnostic capable of detecting early dysfunction, assessing system resilience, and revealing how environmental constraints shape—but do not determine—the fundamental principles governing multifractal complexity. Full article

This paper develops a comprehensive theory of multi-slice decompositions via convex functions, extending the classical framework of slices determined by linear functionals to arbitrary convex functions with disjoint zero sets. We establish a fundamental structure theorem that completely characterizes the convex component decomposition of multi-slices, showing that under natural conditions of pairwise disjoint zero sets and convex separation, the multi-slice decomposes canonically into convex components that correspond precisely to the individual functions in the family. Our results reveal several key properties: the component-wise exposing nature of the supremum function, the closedness of components in appropriate topologies, the maximality of the resulting decomposition, and the affine invariance of convex component structures under injective affine maps. These contributions significantly extend the existing theory of multi-slices and convex components, providing new tools for understanding the geometric structure of convex sets under nonlinear constraints, with potential applications in optimization theory, high-dimensional data analysis, and modern convex geometry. Full article