In this paper we study the pricing of European options in a financial market modeled by the α -hypergeometric stochastic volatility model. We consider a setting that accommodates for parameter uncertainty, which is introduced by assuming that the model parameters are not known exactly but rather estimated from statistical data. The uncertainty is represented by a compact set of admissible parameters and we derive conservative bounds for the option price by treating the problem as an optimal control problem.
Many real-world processes and phenomena are affected by short-lived external disturbances during their evolution. Because these disturbances are brief compared to the overall duration of the processes, they are often considered instantaneous and modeled as impulsive effects. This work focuses on exploring a new class of 𝒫𝒞_1-κ mild solutions for Hilfer fractional feedback control systems with instantaneous impulses with history dependent operators. To do this, we utilize semigroup theory, impulsive effects, fractional calculus, and fixed-point techniques. First, we prove the existence of solutions under certain assumptions. Then, the Cesari property and Filippov’s theorem are used to establish conditions for the existence of feasible control-state pairs and optimal controls in feedback systems with history dependent operators. To demonstrate the applicability and effectiveness of the developed theoretical results, a representative example is presented at the end of the study.
The powerful connection between the classical sampling series and the exponential sampling series is well-established, linked by a logarithmic change of variable. This transference allows theoretical results from the Fourier domain to be mapped directly to the Mellin domain. In this paper, we first formalize this transference principle. We then posit that this principle can be generalized to other variable transformations. As a primary contribution, we apply this generalized principle using specific diffeomorphisms. We introduce the “trigonometric sampling series” for periodic-like intervals and the “algebraic sampling series” for finite intervals with boundary singularities. We provide explicit formulations and demonstrate that these operators inherit the convergence properties of the classical series. Numerical experiments confirm that the algebraic operators significantly outperform uniform sampling in reconstructing singular functions on bounded domains.
This paper is dedicated to studying an inverted finite element method (IFEM) for elliptic optimal control problems (OCPs) in cylindrical unbounded domains. Initially, weighted Sobolev spaces are introduced for the cylindrical unbounded domains, and the properties of these spaces are further investigated. Subsequently, the IFEM, specifically designed for the weighted Sobolev spaces, is proposed. Based on the node interpolation operator, we rigorously derive interpolation error estimates from the weighted Sobolev spaces to the inverted finite element space. In addition, these results are used for elliptic OCPs in cylindrical unbounded domains for the first time, and an error estimate of the IFEM is obtained successfully. Finally, the correctness of our findings is verified through two numerical examples, fully demonstrating the effectiveness and accuracy of the IFEM in tackling elliptic OCPs in cylindrical unbounded domains.
Recently, a new degree–based topological invariant, called the Euler Sombor index ( EUS ), has been introduced. For a graph G , the Euler Sombor index is defined by EUS(G)=∑_v_i v_j∈ E(G)√(d_i^2+d_j^2+d_i d_j), where d_i and d_j denote the degrees of the end vertices of the edge v_i v_j . Determining extremal values of this index under the constraint of a fixed diameter has become an active area of research. Very recently, Sekar et al. [Maximum Euler Sombor Index of Unicyclic Graphs with Given Diameter, MATCH Commun. Math. Comput. Chem. 95 (2026) 759–776] posed the open problem of determining the second, third, and fourth maximum values of the EUS index among all trees with a prescribed diameter. In this paper, we give a unified characterization of the trees attaining the first through ninth maximum EUS values, while the new contribution concerns the third through ninth values.
We study conditions under which partial stability holds for the solutions of a time-varying linear dynamic system of the form ξ^Δ(t) = A(t)ξ(t), ξ = (ξ_1, ξ_2) ∈ℝ^n_1×ℝ^n_2, where only the variables ξ_1 are required to be stable. We give sufficient conditions for various types of partial stability, including Lyapunov-type criteria involving functions that are positive definite only with respect to ξ_1 , and eigenvalue conditions that ensure exponential decay of the ξ_1 -component. As a concrete application, we show how these results can be used in the stability analysis of certain compartmental epidemic models in particular, for establishing the stability of the disease-free equilibrium with respect to the infected compartments only, without requiring stability in the direction of the susceptible or recovered populations. Finally, numerical simulations of an epidemic model illustrate and confirm the theoretical stability criteria.
This paper develops a high-order compact block-centered finite difference method for two-dimensional nonlinear advection diffusion reaction equations under both periodic and Dirichlet boundary conditions. The proposed scheme is mass-conservative and simultaneously achieves fourth-order spatial accuracy and second-order temporal accuracy for both the solution and fluxes. A rigorous convergence analysis is established for periodic boundary conditions. Numerical experiments verify the theoretical analysis and confirm the predicted convergence rates for both periodic and Dirichlet boundary conditions.
In this study, we investigate the impact of relapses on the behavior of a stochastic Susceptible-Infected-Recovered model by incorporating a phase-type semi-Markov process. We propose a framework that extends the standard Markov regime-switching model to a more comprehensive approach for modeling stochastic transitions. Our method utilizes a Coxian phase-type semi-Markov process to capture the dynamics of evolving environments. We determine the threshold ( R_0^* ) of our system by integrating relapse factors in a phase-type semi-Markov switching environment. Furthermore, we identify the conditions under which the disease will either be eliminated or persist. Moreover, we propose an index to quantify the effect of relapses, and our results show that relapses substantially increase the disease’s potential for spread. We perform numerical simulations to validate our theoretical findings. Both the analytical and practical results underscore the importance of incorporating phase-type semi-Markov switching into the stochastic modeling of disease dynamics.
In this paper, we propose and analyze a five-compartment smoking epidemic model incorporating heterogeneous smoking intensity and dual-stage cessation. The population is divided into potential smokers, occasional smokers, chain smokers, temporary quitters, and permanent quitters. The model includes relapse from temporary cessation and is formulated as a nonlinear system of ordinary differential equations. We establish positivity and boundedness of solutions and investigate the smoking-free and smoking-present equilibria. The basic reproduction number ℛ_0 governs the threshold dynamics of the system. Local stability, global stability of the smoking-free equilibrium, and uniform persistence are analyzed. A center manifold analysis proves the occurrence of a forward bifurcation at ℛ_0=1 . Sensitivity analysis reveals that transmission-related parameters promote smoking persistence, whereas quitting-related parameters reduce smoking prevalence. An optimal control problem involving prevention and treatment strategies is formulated and characterized via Pontryagin’s maximum principle. Numerical simulations using a non-standard finite difference (NSFD) scheme illustrate the theoretical results and demonstrate that combined intervention strategies effectively suppress smoking dynamics. A comparison with the classical fourth-order Runge–Kutta method further confirms the stability and computational efficiency of the NSFD approach.
With the rapid development of data-intensive applications, including large models for spine analysis, efficient stochastic optimization methods have become increasingly important. A new stochastic quasi-Newton method (SQN) is proposed for solving stochastic optimization problems. Unlike classical stochastic BFGS-type methods, an adaptive scaling damped BFGS strategy is introduced, thereby improving the stability and reliability of curvature information in stochastic settings. In addition, a mini-batch technique is incorporated into the algorithm to reduce the computational cost while preserving useful stochastic gradient information. The global convergence of the proposed method is established under diminishing step sizes and Armijo line search. A notable feature of the theoretical analysis is that the boundedness of H_k, which is commonly imposed as an additional assumption in classical methods, can be obtained automatically during the proof process. Moreover, the boundedness analysis of the approximate Hessian matrix B_k is significantly simplified through a concise and transparent proof framework. Numerical experiments are done, and the results show the superiority of proposed algorithm.
This paper is concerned with qualitative properties of solutions to an initial value problem for nonlinear implicit fractional differential equations involving the Caputo-Hadamard derivative. Existence of solutions is obtained under sufficient conditions by means of Schauder’s fixed point theorem. A Gronwall-type inequality is then applied to derive estimates of solutions and to establish continuous dependence on the initial data, from which uniqueness follows. The Ulam-type stability is also investigated by a fixed point approach in a generalized metric space. Examples are given to illustrate the obtained results.
The Polak–Ribière–Polyak (PRP) conjugate gradient method, while renowned for its practical efficiency among classical conjugate gradient algorithms, suffers from certain theoretical limitations that have motivated numerous modifications. One notable variant, the Rivaie–Mustafa–Ismail–Leong (RMIL) method, modifies the denominator of the PRP conjugate parameter and retains the anti-jamming property, but guarantees the sufficient descent condition—and consequently global convergence—only under exact line searches. To fully preserve the computational strengths of the RMIL scheme while overcoming its theoretical shortcomings, we introduce a novel spectral modification integrated within the non-linear conjugate gradient framework. Motivated by quasi–Newton principles and the classical secant condition, we derive a dynamic spectral parameter that enables the systematic projection of the search direction onto a subspace orthogonal to the objective gradient. This projection mechanism guarantees a strict sufficient descent condition at every iteration, completely independent of the line search strategy or the convexity of the objective function. A rigorous global convergence proof is provided for general non-linear objective functions, and the theoretical worst-case computational complexity of the proposed algorithm is established under the Armijo line search framework. We comprehensively evaluate its performance against state-of-the-art conjugate gradient baselines on a standard benchmark suite of CUTEr test functions, utilizing Dolan–Moré performance profiles for rigorous structural comparison. To further demonstrate its practical utility, the framework is applied to a representative real-world task in compressed sensing sparse signal recovery. The empirical findings conclusively highlight the superior convergence acceleration, numerical robustness, and practical efficiency of the proposed algorithm.
The primary objective of this article is to introduce a sequence of positive linear operators constructed via the Riemann–Liouville fractional integral and Hermite polynomials. Several estimates involving test functions and central moments are established to derive the rate of convergence and the order of approximation. Furthermore, uniform convergence is established using a Korovkin-type theorem, while quantitative estimates are given in terms of the first modulus of smoothness. Approximation results are also investigated through Peetre’s K -functional, the second modulus of continuity, and Lipschitz-type function spaces. The weighted approximation properties of these operators are further established in terms of the modulus of continuity. The analysis is subsequently extended to a two-dimensional setting, where the corresponding approximation properties are rigorously examined. To support the theoretical developments, numerical examples are presented that illustrate the effectiveness of the proposed operators and confirm the theoretical error estimates. Finally, concluding remarks and possible directions for future research are provided.
We study supersingular ℓ -isogeny graphs as finite regular multigraphs through the lens of non-backtracking spectral graph theory. Using the Hashimoto operator, we show that traces of its powers count cyclically non-backtracking closed isogeny chains, and we derive an exact Möbius inversion formula for the associated primitive cycle numbers. This leads to a finite primitive cycle certificate for supersingular isogeny graphs, invariant under graph isomorphism and computable from the oriented-edge data or, equivalently, from the Ihara–Bass determinant formula. The resulting framework provides a rigorous discrete and algorithmic invariant for measuring primitive cycle structure in arithmetic regular graphs.
In this study, a fractional-order formulation of the Stepanova tumor-immune model is developed to explore complex tumor dynamics. To address the limitations of continuous time systems in capturing chaotic behavior in low dimensional settings, the model is transformed into a discrete time structure via piecewise constant arguments. The resulting system admits biologically meaningful equilibrium states, and their local stability is examined. Variations in the tumor carrying capacity parameter lead to a Neimark-Sacker bifurcation, generating quasi-periodic oscillations and, for larger values, chaos, as confirmed by Lyapunov exponent analysis. To mitigate these irregular behaviors, a feedback control mechanism incorporating chemotherapy and immunotherapy effects is proposed. Numerical simulations show that the control strategy effectively suppresses chaos and drives the system toward a tumor dormant state. The proposed framework offers a useful tool for analyzing and regulating complex tumor dynamics.
To investigate the impact of time delay and booster vaccinations on pertussis transmission, this paper establishes a model of pertussis infection incorporating the incubation delay of pertussis and the effect of control measures based on the mechanisms of pertussis spread. Through theoretical analysis, we demonstrate that when the control reproduction number is less than 1, the disease-free equilibrium of the system is globally asymptotically stable. When the control reproduction number exceeds 1, the system possesses a unique endemic equilibrium that is globally asymptotically stable. Furthermore, sensitivity analysis indicates that reducing the effective contact rate, increasing first-dose vaccination coverage and enhancing early detection and treatment of confirmed cases are crucial for reducing transmission. Numerical simulation results reveal: (a) the peak size of pertussis infections decreases as the time delay increases, and the peak onset time is delayed accordingly; (b) when the treatment recovery rate increases from 0.5 to 0.9, the peak size of the total infected population decreases by 92.4 % ; (c) when the coverage rate of the first vaccine dose is raised from 0.6 to 0.9, the peak size of total infections decreases by 8.58 % ; in contrast, an increase in the booster dose coverage from 0.2 to 0.5 resulted in a 1.57 % reduction in the peak size of total infected individuals. The findings suggest that increasing first-dose and booster vaccine coverage, as well as improving timely treatment rates, are all effective strategies for significantly reducing pertussis transmission.
The weighted essentially non-oscillatory (WENO) scheme is a high-order numerical technique widely used for accurately approximating solutions that contain discontinuities. A crucial component in the construction of a WENO scheme is the formulation of nonlinear weights corresponding to each substencil. The primary challenge lies in substantially diminishing the contribution of substencils that contain discontinuities, while accurately retaining the linear weights in smooth regions. In this context, the present study proposes a novel strategy for constructing nonlinear weights in fifth-order WENO schemes. The key idea of our approach is to modify the unnormalized nonlinear weights in a way that amplifies the scale differences among them when a discontinuity exists within the global stencil. Accordingly, compared to existing fifth-order methods, the proposed technique exhibits an enhanced capability to distinguish non-smooth substencils from the smooth ones. In addition, we verify that the resulting WENO scheme maintains fifth-order accuracy, even in the presence of critical points. Several numerical results are provided to demonstrate the improved performance of the suggested WENO method.
In a two-dimensional space, we investigate a spatiotemporally discrete predator-prey model based on coupled map lattice, with diffusion and advection being its core characteristics. By considering diffusion and advection, we explore and derive the conditions for the occurrence of pure Turing instability, Neimark-Sacker-Turing instability, and flip-Turing instability. These instabilities triggered the evolution of the system from bifurcation to chaos. Additionally, we analyze the dynamical behavior of the model under spatially homogeneous steady-state conditions, including stability analysis, codimension-1 bifurcations, and codimension-2 bifurcations. Through numerical simulations, we verify the obtained results and use the maximum Lyapunov exponent to characterize the transition from bifurcation to chaos. The complex spatiotemporal dynamic behaviors exhibited by the system are of great significance for our deeper understanding of ecosystems and for their effective management.
This study introduces a new three term hybrid conjugate gradient method for large-scale unconstrained optimization and applies it in image restoration problems. The new conjugate gradient variant is defined by a convex combination of the AlBayati–AlAssady (BA), Fletcher–Reeves (FR), and Polak–Ribière–Polyak (PRP) formulas. The method satisfies the sufficient descent property and its global convergence is established under the strong Wolfe line search. The algorithm is implemented in Python, and its numerical performance is evaluated using Dolan–Moré performance profiles against some classical methods on a collection of well known test functions. As an application, the proposed method is further assessed on image restoration problems in terms of both reconstruction quality and computational efficiency, and is compared with the BA, FR, and PRP algorithms.
