Search Results (112)

This research explores the existence of solutions for a class of random fractional differential equations characterized by bounded delay, specifically within the context of Fréchet spaces. Random fractional differential equations serve as powerful mathematical tools for modeling complex phenomena subjected to stochastic perturbations and hereditary effects. Despite their significance, establishing solution existence in infinite-dimensional spaces remains a challenging task. By integrating the properties of the noncompactness measures with a generalized Darbo fixed point approach, we establish new existence results for the associated Darboux-type problem under milder compactness conditions. To illustrate the practical utility of these analytical results and demonstrate the validity of our theoretical framework, a representative example is provided. Full article

Background/Objectives: While human epidermal growth factor receptor 2 (HER2) immunohistochemistry (IHC) is pivotal for breast cancer management, its reliance on additional tissue processing beyond routine H&E staining remains a clinical burden. Although virtual staining offers a potential solution, current methods often fail to explicitly account for HER2 score-specific expression patterns. To address this gap, we developed a score-aware framework designed for the precise generation of virtual HER2 IHC images. Methods: We introduce the non-contrastive multi-task (NCMT) framework, which integrates negative-free patch alignment, style–content constraints, and auxiliary HER2 score supervision for high-fidelity H&E-to-IHC translation. For rigorous evaluation, the model was validated on the BCI dataset, utilizing an official split of 3896 training and 977 independent test images derived from 51 whole-slide images. Results: NCMT demonstrated superior virtual staining performance, achieving a Fréchet Inception Distance (FID) of 38.8, a Kernel Inception Distance (KID) of 5.6, and an average Perceptual Hash Value (PHV) of 0.439. In downstream HER2 scoring tasks, while virtual IHC images alone yielded an accuracy of 83.01%, the fusion of H&E and virtual IHC further elevated performance to 97.85% accuracy and a 98.23% F1 score. These findings suggest that our framework effectively preserves diagnostic features while providing complementary information to H&E-based morphological analysis. Conclusions: NCMT enables HER2 score-aware virtual IHC generation from H&E and can serve as a complementary tool for HER2 assessment in digital pathology. Full article

Variational models describe deformation and stability through the first and second variations in an underlying functional, but the relationship between these responses is seldom expressed as an intrinsic equilibrium quantity of the model itself. A canonical curvature–strain representation for equilibrium ratios arising in variational field settings is developed. For a twice Fréchet differentiable functional and an admissible perturbation generator, strain is defined as normalized first-order response and curvature as normalized second-order response along the generator direction. Their quotient defines a curvature–strain ratio that measures proportional balance between deformation and curvature within the model. The main result shows that this curvature–strain ratio is a canonical representative of a response ratio already implicit in the variational data. Under canonical normalization, the curvature–strain ratio coincides with the quotient of second- and first-order response, and stationarity of the curvature–strain ratio is equivalent to proportional stationarity of that response quotient along the admissible flow. A further theorem establishes transfer of local isolation: when the second-variation operator satisfies standard hypotheses such as compact resolvent and non-degeneracy of the constrained extremum, isolated equilibrium ratios persist in the curvature–strain representation for the same operator-theoretic reasons. Quadratic scalar and Maxwell-type models illustrate the construction. The paper establishes a mathematically controlled curvature–strain representation of equilibrium ratios within ordinary variational theory, with emphasis on the analysis of variational response and equilibrium balance. 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

Demand-responsive transit (DRT) requires real-time vehicle assignment under dynamically arriving requests, where each decision may alter multi-stop routes and affect both onboard and newly arriving passengers. However, DRT simulations often face three key limitations: rapidly increasing computational complexity as fleet size and demand grow, insufficient integration of traffic congestion into routing decisions, and limited consideration of passenger-oriented service quality in final vehicle assignment. To address these issues, this study proposes an integrated DRT simulation incorporating three core algorithms: Fréchet Distance-based Candidate Vehicle Selection (FD-CVS), Congestion-Aware Path Planning (CA-PP), and Satisfaction-Aware Vehicle Assignment (SA-VA). FD-CVS reduces computational burden by filtering candidate vehicles based on route similarity. CA-PP extends conventional path planning by incorporating congestion-adjusted travel costs derived from public transportation data. SA-VA determines the final vehicle assignment by jointly evaluating passenger waiting time, in-vehicle travel time, and capacity constraints. The algorithms are implemented within a discrete-event simulation environment using real-world data. Experimental results demonstrate that FD-CVS significantly reduces execution time under high-demand conditions, while SA-VA improves passenger waiting time and acceptance rates. Overall, the proposed three-algorithm framework enables more realistic and computationally efficient DRT system evaluation. Full article

The applicability of a highly efficient sixth-order convergent method, originally proposed by Kansal et al., is extended in this study to a Banach space setting. The initial development of this method relied upon Taylor series expansions in ${\mathbb{R}}^n$ and the assumption that the nonlinear operator is sufficiently differentiable. This vague condition implies the existence of high-order derivatives that are not actually utilized by the algorithm. This study transcends these limitations by establishing convergence based solely on generalized continuity conditions of the first Fréchet derivative. By dispensing with these strong smoothness requirements, the domain of applicability is significantly widened. We derive computable radii for the ball of convergence and establish error bounds under local analysis. Furthermore, a rigorous semi-local convergence analysis is presented, a feature previously absent in the literature for this specific scheme, utilizing a majorizing sequence technique to guarantee the existence and uniqueness of the solution. The theoretical results are validated through numerical experiments, which demonstrate that the method converges even when the standard sufficiently differentiable conditions are violated. Full article

This paper investigates the optimal portfolio control problem under a stochastic volatility model, whose dynamics are governed by a highly nonlinear Hamilton–Jacobi–Bellman equation. We employ a separable value function and introduce a novel exponential approximation technique to simplify the nonlinear terms of the auxiliary function. The simplified HJB equation is solved numerically using the advanced Fréchet–Newton method, which is known for its rapid convergence properties. We rigorously analyze the numerical outcomes, demonstrating that the iterative sequence converges quickly to the trivial fixed point ($g^{*}=1$) under zero risk and zero excess return conditions. This convergence is mathematically justified through rigorous functional analysis, including the principles of contraction mapping and the Kantorovich theorem, which validate the stability and efficiency of the proposed numerical scheme. The results offer theoretical insight into the behavior of the HJB equation in simplified solution spaces. Full article

Traditional Turkish marbling (Ebru) art is an intangible cultural heritage characterized by highly asymmetric, fluid, and non-reproducible patterns, making its long-term preservation and large-scale dissemination challenging. It is highly sensitive to environmental conditions, making it enormously difficult to mass produce while maintaining its original aesthetic qualities. A data-driven generative model is therefore required to create unlimited, high-fidelity digital surrogates that safeguard this UNESCO heritage against physical loss and enable large-scale cultural applications. This study introduces a deep generative modeling framework for the digital reconstruction of traditional Turkish marbling (Ebru) art using a Deep Convolutional Generative Adversarial Network (DCGAN). A dataset of 20,400 image patches, systematically derived from 17 original marbling works, was used to train the proposed model. The framework aims to mathematically capture the asymmetric, fluid, and stochastic nature of Ebru patterns, enabling the reproduction of their aesthetic structure in a digital medium. The generated images were evaluated using multiple quantitative and perceptual metrics, including Fréchet Inception Distance (FID), Kernel Inception Distance (KID), Learned Perceptual Image Patch Similarity (LPIPS), and PRDC-based indicators (Precision, Recall, Density, Coverage). For experimental validation, the proposed DCGAN framework is additionally compared against a Vanilla GAN baseline trained under identical conditions, highlighting the advantages of convolutional architectures for modeling marbling textures. The results show that the DCGAN model achieved a high level of realism and diversity without mode collapse or overfitting, producing images that were perceptually close to authentic marbling works. In addition to the quantitative evaluation, expert qualitative assessment by a traditional Ebru artist confirmed that the model reproduced the organic textures, color dynamics, and compositional asymmetrical characteristic of real marbling art. The proposed approach demonstrates the potential of deep generative models for the digital preservation, dissemination, and reinterpretation of intangible cultural heritage recognized by UNESCO. Full article

Understanding surface textures through visual cues is crucial for applications in haptic rendering and virtual reality. However, accurately translating visual information into tactile feedback remains a challenging problem. To address this challenge, this paper presents a class-conditional Generative Adversarial Network (cGAN) for cross-modal translation from texture images to vibrotactile spectrograms, using samples from the LMT-108 dataset. The generator is adapted from pix2pix and enhanced with Conditional Batch Normalization (CBN) at the bottleneck to incorporate texture class semantics. A dedicated label predictor, based on a DenseNet-201 and trained separately prior to cGAN training, provides the conditioning label. The discriminator is derived from pix2pixHD and uses a multi-scale architecture with three discriminators, each comprising three downsampling layers. A grid search over multi-scale discriminator configurations shows that this setup yields optimal perceptual similarity measured by Learned Perceptual Image Patch Similarity (LPIPS). The generator is trained using a hybrid loss that combines adversarial, $L_1$, and feature matching losses derived from intermediate discriminator features, while the discriminators are trained using standard adversarial loss. Quantitative evaluation with LPIPS and Fréchet Inception Distance (FID) confirms superior similarity to real spectrograms. GradCAM visualizations highlight the benefit of class conditioning. The proposed model outperforms pix2pix, pix2pixHD, Residue-Fusion GAN, and several ablated versions. The generated spectrograms can be converted into vibrotactile signals using the Griffin–Lim algorithm, enabling applications in haptic feedback and virtual material simulation. Full article

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We first present a detailed geometric and analytical foundation for intrinsic quantization on the unit sphere, including definitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation, the Fréchet mean, spherical Voronoi regions, centroid conditions, and quantization dimensions. Building upon this framework, we develop explicit continuous and discrete quantization models on spherical curves, namely great circles, small circles, and great circular arcs—supported by rigorous derivations and pedagogical exposition. For uniform continuous distributions, we compute optimal sets of n-means and the associated quantization errors on these curves; for discrete distributions, we analyze antipodal, equatorial, tetrahedral, and finite uniform configurations, illustrating convergence to the continuous model. The central conclusion is that for a uniform probability distribution supported on a one-dimensional geodesic subset of total length L, the optimal n-means form a uniform partition and the quantization error satisfies $V_n=L^2/\left(12n^2\right)$.The exposition emphasizes geometric intuition, detailed derivations, and clear step-by-step reasoning, making it accessible to beginning graduate students and researchers entering the study of quantization on manifolds. This article is intended as an expository and tutorial contribution, with the main emphasis on geometric reformulation and pedagogical clarity of intrinsic quantization on spherical curves, rather than on the development of new asymptotic quantization theory. Full article

A geometric-analytic approach to studying invariants of solvable nonlinear ordinary differential equations is developed. In particular, there is described in detail a general scheme of constructing solvable nonlinear ordinary differential equations, based on a linear differential spectral problem and its related invariants. Examples of nonlinear differential equations applications are discussed, generalizing those previously studied in the literature. The analytical properties of the invariants and determining the Noether-Lax evolution equation, including its asymptotic properties, are analyzed in detail. Some interesting from a practical point examples of the second ordinary differential equations are analyzed in detail, including the classical Van der Pol and Painlevé equations. The backgrounds of the isolvability problem are also presented and applied to ordinary super-differential equations on the superaxis, which are of interest for research in the field of modern quantum physics. Full article

Estimating extreme quantiles and the number of future exceedances is an important task in financial risk management. More important than estimating the quantile itself is to insure zero coverage error, which implies the quantile estimate should, on average, reflect the desired probability of exceedance. In this research, we show that for unconditional distributions isomorphic to the exponential, a Bayesian quantile estimate results in zero coverage error. This compares to the traditional maximum likelihood method, where the coverage error can be significant under small sample sizes even though the quantile estimate is unbiased. More generally, we prove a sufficient condition for an unbiased quantile estimator to result in coverage error and we show our result holds by virtue of using a Jeffreys prior for the unknown parameters and is independent of the true prior. We derive a new, predictive distribution, and the moments, for the number of quantile exceedances, and highlight its superior performance. We extend our results to the conditional tail of distributions with asymptotic Paretian tails and, in particular, those in the Fréchet maximum domain of attraction which are typically encountered in finance. We illustrate our results using simulations for a variety of light and heavy-tailed distributions. Full article

This paper investigates the boundedness of multiplication operators $M_{\psi }$ between natural $\mu$-Bloch-type spaces $B_{\mu ,\mathrm{nat}}(B_X)$ (or their little $\mu$-Bloch counterparts) and natural $\omega$-Bloch-type spaces $B_{\omega ,\mathrm{nat}}(B_X)$ on the unit ball $B_X$ of a complex Banach space X. We establish complete characterizations for the boundedness of $M_{\psi }$ under varying conditions on the weight functions $\mu$ and $\omega$, including specific cases such as logarithmic and power-weighted Bloch spaces. The results extend classical operator theory to infinite-dimensional settings, unifying prior work on finite-dimensional domains. Full article

We study the regularity properties of the unique solution of a generalized mean-field G-SDE. More precisely, we consider a generalized mean-field G-SDE with a square-integrable random initial condition, establish its first- and second-order Fréchet differentiability in the stochastic initial condition, and specify the G-SDEs of the respective Fréchet derivatives. The first- and second-order Fréchet derivatives are obtained for locally Lipschitz coefficients admitting locally Lipschitz first- and second-order Fréchet derivatives respectively. Our approach heavily relies on the Grönwall inequality, which leverages the Lipschitz continuity of the coefficients. Full article

The integration of a large number of distributed generation (DG) units has altered the grid structure and fault characteristics of distribution networks, posing significant challenges to conventional protection methods. To address this, this paper proposes an adaptive current differential protection method based on comparing the similarity of current waveforms at both ends of a line. After a fault occurs, the current waveforms at both ends of each line are first extracted and normalized. The Fréchet distance algorithm is then introduced to quantify the waveform similarity. Based on the calculated Fréchet distance, the restraint coefficient of the current differential protection is constructed and the protection criterion is improved. Finally, a logic function is derived from the improved criterion. A short-circuit fault within the section is identified when the sum of the logic functions across all sampling points exceeds 50% of the total number of sampling points. A simulation model built in PSCAD/EMTDC is used for validation. Simulation results demonstrate that the proposed method is unaffected by fault type, transition resistance, fault location, or DG grid-connected capacity, has low data synchronization requirements, and exhibits excellent reliability and selectivity. Full article