. In this paper, we investigate a singular fractional Kirchhoff-type problem involving a general nonlo cal integrodifferential operator with variable exponents. The considered model combines a Kirchhoff term, a singular nonlinearity, and a fractional operator of elliptic type defined through a measurable kernel satisfying suitable structural assumptions. Working within the framework of generalized fractional Sobolev spaces with variable exponents, we establish the existence of at least one nontrivial weak solution. The analysis relies on variational techniques and minimization arguments. First, the associated energy functional is shown to be coercive, bounded from below, and weakly lower semicontinuous. Then, by proving the existence of a negative energy level and applying direct methods of the calculus of variations, the existence of a global minimizer is obtained, which yields a nontrivial weak solution to the problem. An illustrative example is also provided to demonstrate the applicability of the main theorem.
We study the approximation of functions from the generalized Ho & uml;lder classes B(H ohm k([0, 11d)) in the Lp([0, 11d) metric for all 1 < p < infinity. We establish the exact orders of complexity for this problem in the deterministic, randomized and quantum settings. The optimal convergence rates in all three settings are ohm(n-1/d), where n is the number of function evaluations involved in the computational process. The results show that quantum and randomized algorithms yield no improvement in terms of the optimal convergence rates. Furthermore, we apply our general result to the special case ohm(t) = ta (log2 (2+ 1))b with 0 < a < k and b is an element of R, and derive the corret sponding complexity orders.
. In this article, we have proposed a numerical scheme for solvwhich arises in the case of jump-diffusion model. The fractional derivative in time is discretized by classical L1-scheme. The cubic spline method is used to discretize the spatial derivative and composite trapezoidal rule is employed for discretization of integral part. Further, the stability and convergence results are proved. The numerical experiments for both Merton and Kou model are performed to validate the theoretical estimates and the effect of fractional derivative is shown on the option price. Also, we have compared our existing method with the real-world market data for two stocks showing that jumpdiffusion model performs better than the classical Black-Scholes model.
. Variational inference (VI) has emerged as a cornerstone methodology for approximating posterior distributions in complex Bayesian models. However, conventional mean-field VI is inherently constrained by its factorization assumption, which significantly limits its expressiveness, particularly in high-dimensional settings. Recent approaches, including Implicit Variational Inference (IVI) and Semi-Implicit Variational Inference (SIVI), attempt to alleviate these restrictions by incorporating implicit distributions; nevertheless, they remain hindered by challenges such as adversarial training instability and dependence on surrogate evidence lower bounds (ELBOs). In this work, we propose Diffusion Implicit Variational Inference (DIVI), a novel framework that seamlessly integrates diffusion models with implicit variational inference to enhance both the flexibility and fidelity of posterior approximation. Specifically, DIVI exploits the iterative generative mechanism of diffusion processes to progressively refine posterior samples, thereby effectively capturing intricate structures in high-dimensional and multimo dal distributions. We further derive a tractable and explicit ELBO for DIVI, enabling efficient optimization via stochastic gradient ascent without resorting to adversarial schemes or surrogate objectives. To extend the applicability of DIVI to high-dimensional image modeling, we introduce a conditional reverse diffusion formulation, wherein deep neural encoders are employed to parameterize data-dependent transition kernels, ensuring scalability to large-scale and complex datasets. Extensive experiments demonstrate that DIVI consistently outperforms existing methods in terms of accuracy, stability, and scalability across diverse Bayesian inference tasks, particularly in regimes where approaches such as SIVI and adversarial VI exhibit notable limitations. Overall, this work presents a more robust and efficient paradigm for variational inference, delivering substantial improvements over prior art.
This paper focuses on the Stancu modification of operators defined via multiple Appell polynomials and examines their approximation properties. We present results concerning the convergence of these operators and determine their approximation rates using classical techniques, the Lipschitz class, and Voronovskaja-type theorem. Finally, we compare the convergence of the operators for specific functions through graphs and table.
Multi-modal image registration (MIR) remains a challenging problem due to severe appearance discrepancies across imaging modalities and the limited adaptability of handcrafted similarity measures. While deep learning methods have shown strong empirical performance, many existing architectures do not explicitly incorporate physical deformation priors. In this work, we propose DUM-Net, a model-driven deep unrolling framework for multi-modal image registration. Starting from a variational MIR model with a similarity term and a regularization term, we unroll a time-discrete gradient descent process into a layer-wise trainable architecture. DUM-Net consists of three components: an initialization module (IM), an adaptive similarity module (ASM), and a linear elasticity module (LEM). The IM provides deformation initialization to accelerate convergence. The ASM employs a deep neural network to learn an adaptive descent direction for cross-modal matching, thereby overcoming the limitations of handcrafted similarity functionals. The LEM introduces a parameterized linear-elastic prior during forward propagation to encourage smooth and physically plausible deformation fields. In this way, DUM-Net combines the flexibility of learned similarity modeling with the structural guidance of variational unrolling and explicit physical regularization. Experiments on T1/T2 MR, CT/MR, and FL/IR registration tasks demonstrate that DUM-Net consistently outperforms existing methods, achieving Dice score improvements of 3.74% for small and 6.16% for large deformations on T1/T2 MR image pairs, together with strong performance on CT/MR and FL/IR datasets. The source code is publicly available at https://github.com/QDUJackson/DUM-Net.
The current paper introduces new positive linear operators via the modified Kantorovich-Szasz-type Erdelyi-Kober fractional operator involving a Boas-Buck-type polynomial. We investigate some results enclosed with the rate of approximation for new operator with respect to Ditizian-Totik modulus of smoothness of higher order, Peetre's K-functional, and the Lipschitz maximal function. Finally, some approximation estimates concerning the Korovkin theorem, a quantitative Voronovskya-type theorem, and a Gruss-Voronovskaya-type type theorem are derived.
. This study aims to explore the innovative aspects of Bernstein-type operators based on the Polya-Eggenberger distribution for function approximation over the interval [alpha o, beta o]. As a preliminary step, the moments of the new operators are derived to facilitate further analysis. The study presents several significant results, namely point-wise convergence, an asymptotic formula, and estimate of the rate of convergence in terms of the second-order modulus of smoothness. Finally, we employ Mathematica to demonstrate numerical examples that visually confirm the findings.
. This study aims to determine the source term within a subdiffusion model using an artificial neural network approach, leveraging additional data. The core strategy involves replacing the unknown source term with a neural network, thereby converting the inverse source problem into an optimal control problem. By employing this approach, we establish the existence of an optimal solution for the control problem and compute the associated optimality conditions using the alternating direction multiplier method. Numerical solutions are derived via this proposed methodology, showcasing its effectiveness through a series of numerical tests on both regular and singular examples. Our results demonstrate the efficacy of the artificial neural network method, particularly evident in its comparison with established techniques like gradient descent and alternating direction multiplier method. This comparative analysis reinforces the strength and reliability of the artificial neural network-based approach in solving the source term determination problem within the sub diffusion model.
The capped l(q)-norm, as a representative example of capped folded concave functions, has been widely adopted in sparse optimization problems due to its effectiveness in mitigating the bias issue associated with the l(1)-norm. For the specific case of q = 1/2, the proximal operator of the capped l(q)-norm was characterized by Lili Pan and Xiaojun Chen [Group sparse optimization for images recovery using capped folded concave functions, SIAM J. Imaging Sci., 14(1):1-25, 2021]. However, applying their characterization generally requires comparing function values among several candidate points, which can be computationally cumbersome. To address this limitation, we derive an explicit characterization for the proximal operator of the capped l(q)-norm for each 0 < q < 1. Numerical experiments in compressed sensing reveal that capped l(q)-norm regularization enhances the performance of the corresponding l(q)-norm regularization and outperforms other widely used sparsity-promoting functions.
We present a novel Composite PDHG method for addressing a class of deterministic and stochastic saddle-point problems (SPP), in particular designed for efficiently fixing large-scale, non-smooth convex inverse problems. The basic idea of this algorithm lies in its integration of gradient updates with proximal operators, permitting it to handle the smooth component of the objective function effectively, a capability that distinguishes it from the Chambolle-Pock algorithm, which is tailored exclusively for non-smooth functions. By leveraging a strategy that updates a randomly selected subset of dual variables in each iteration, the Stochastic Composite PDHG (SCPDHG) considerably reduces computational overhead while maintaining convergence stability. We establish its almost-sure convergence through foundational lemmas inspired by the framework of the Three-Operator Splitting with Stochastic Primal-Dual Hybrid Gradient (TOS-SPDHG). Additionally, we exhibit the practical utility of SCPDHG inside the context of image deblurring, illustrating its robustness and adaptability across complex data environments. This work highlights the capacity of SCPDHG to enhance performance and scalability in challenging real-world applications.
Condorcet's jury theorem establishes a foundational principle: a group's majority decision becomes infallibly accurate as its size grows, provided each member is independently more likely than not to be correct. This theorem, however, presents a critical practical dilemma: under finite resources, should one invest in enlarging the group or in enhancing its members' individual competence? We address this question by first revisiting the theorem's binary-choice core. We provide a rigorous, non-asymptotic analysis using the Berry-Esseen inequality, precisely quantifying the probability of a correct group decision as a function of group size and individual accuracy. This yields a closed-form expression for the minimal group size needed to achieve any desired reliability threshold. Building upon this theoretical cornerstone, we extend the investigation to multi-winner approval voting, a prevalent mechanism for selecting representative committees. Through large-scale Monte Carlo simulations, we systematically analyze the interplay between electorate size and ballot accuracy, measuring outcome effectiveness against a known true ranking. Our results delineate the trade-off between electorate size and ballot accuracy, identify conditions where enhancing one factor yields diminishing returns, and demonstrate that flexible, upper-bounded ballot designs significantly outperform fixed-length designs in information aggregation efficiency. The findings provide a mathematical framework for optimizing the design of collective decision-making systems under constraints, offering principled guidance on when to prioritize the "quantity" of participants versus the "quality" of their judgments.
In this article, the analytical bounds of the solution as well as its derivatives of the singularly perturbed parabolic delay differential equation are analyzed. The solution exhibits both a boundary layer and an interior layer due to the existence of perturbation and delay effect on the equation. Consequently, the solution is decomposed into regular and singular components, and bounds are established for both components. For the regular component, bounds for mixed derivatives are obtained directly, while for the singular component, bounds for spatial, temporal, and mixed derivatives are presented separately. The results simultaneously illustrate the effect of the small perturbation parameter and the space delay parameter itself on the layers while providing a theoretical basis for accurate numerical approximation.
This paper is devoted to the existence of multiple pairs of positive solutions for a degenerate nonlo cal Robin problem governed by the p-Laplacian operator with an indefinite potential. The analysis is based on sub-super solution methods, topological methods and variational tools. By allowing the nonlo cal term to change sign, we establish the existence of multiple pairs of positive solutions, where the number of solutions doubles the number of zeros of the degenerate term. The solutions are ordered with respect to their Lq-norms.
