Results in Nonlinear Analysis

Self-similar sets with single-point intersections violating OSC

M. B. Kadirova — 2026-04-08

In the one-dimensional Euclidean space, it was shown that there exists a self-similar set with a single-point intersection that does not satisfy the open set condition (OSC). In the present paper, we prove the existence of a self-similar set with analogous properties in which all mappings are of the form $S_i(x)=q_ix+a_i$ with $q_i>0$.

Artificial Intelligence and Neural Network-Driven Quantum Calculus Framework for Nonlinear Optimization of Fuzzy Partial Differential Equations in Fluid Dynamics

S. Manjula — 2026-04-04

This paper introduces a mathematically sound and computationally unified approach to the solution of nonlinear fuzzy partial differential equations in fluid mechanics with quantum calculus and neural network approximation combined together. The intended model redefines the fuzzy nonlinear fluid equation to make use of q-time derivatives and a representation of an equivalent integral operator in a Banach space framework. The existence and uniqueness of the solutions are proved through Banach fixed-point theorem with Lipschitz continuity assumptions whereas the exponential stability is proved by Lyapunov functional analysis. To improve the accuracy of the solutions, a nonlinear optimization functional is presented and a scheme of neural network approximation is integrated into the analytical structure to enhance a faster convergence without breaking any theoretical assurances. It is demonstrated that the neural approximation error can decrease with network size in a polylogarithmic manner, as can be expected in approximation theory. The sensitivity analysis shows that the quantum parameter q has a direct effect on stability decay rates and minimization of residual, which can be used as a controllable balance of discrete-continuous dynamics. The presence of limited uncertainty propagation, consistent optimization paths, and enhanced convergence behavior with respect to different levels of fuzziness and q-parameters are proved through numerical experiments on fuzzy representations on the α-level. These findings confirm that quantum operator theory, nonlinear optimization, and neural approximation have a stable, convergent and uncertainty consistent computational framework of nonlinear fuzzy fluid systems.

Quantum calculus-based nonlinear analysis and neural approximation of fuzzy partial differential models in fluid dynamics

Bharani Sethupandian S — 2026-04-08

This paper constructs a firm quantum calculus based nonlinear analytical framework of fuzzy partial differentiable models that are encountered in fluid dynamics. The fuzzy nonlinear equation governing is expressed in terms of a q-time derivative operator and reduced to a corresponding operator equation in a Banach space. The locality of solutions and existence are proved through contraction mapping principles on the appropriate Lipschitz and coerciveness conditions. The exponential stability is obtained using a Lyapunov functional method and provides explicit decay rates which are proportional to viscosity and nonlinear growth parameters. In an attempt to improve computationally the tractability of computations and maintain analytical properties, a residual-based scheme of
neural approximations is proposed. A nonlinear residual functional is then formulated so that it would be consistent with the governing fuzzy quantum PDE and convergence estimates are derived in the correct Sobolev norms. The contribution of the quantum parameter is calculated in a systematic manner and its impact on the stability rates, minimization of the residual and damping behaviour
have been identified.

The theoretical findings are confirmed by the numerical experiments that show that neural approximation error decays polynomially, fuzzy uncertainty propagation is bounded by 8 levels of the 8-level, and the optimization dynamics is stable on a variety of quantum parameter space. The suggested framework combines the quantum operator theory, nonlinear functional analysis, and neural approximation into a consistent approach to uncertain modelling of nonlinear fuzzy fluid flow systems.

AI-Assisted quantum computing and neural network approaches for graph-theoretic nonlinear optimization of fuzzy partial differential equation models

S. Pandikumar — 2026-04-08

This work comes up with a synthetic computational framework, which incorporates quantum-assisted optimization, neural network approximation, and graph-theoretic modeling of the nonlinear fuzzy partial differential equation (FPDE) systems. The strategy transforms nonlinear FPDEs into graph-based operator systems, where the spatial time interactions are represented discretely under the uncertainty of fuzziness. The methods of quantum computing such as variational quantum eigen solvers, quantum approximate optimization algorithms are applied to find solutions to high-dimensional nonlinear optimization subproblems, which are the results of discretized fuzzy operators. Neural networks are instantiated to provide approximations on nonlinear residual mappings and fuzzy membership evolution and constitute a hybrid quantum-classical architecture. The suggested framework is tested on reference nonlinear fuzzy diffusion reaction systems and convective transport equations. The convergence stability, computational complexity and the uncertainty propagation robustness are improved as shown by performance metrics. Spectral encoding based on the graph Laplacian makes it possible to arrange the quantum circuit parameters in a structured way and to enforce consistency of fuzzy boundary constraints with the help of neural residual correction. It is experimentally verified that the hybrid model provides a better optimization accuracy when compared to classical deterministic solvers.

M-systems of semirings applied to the bi-semirings via prime k-ideals

Omaima Alshanqiti — 2026-04-04

This paper aims to investigate certain notable classes of k-ideals of bi-semirings such as 0-prime, 1-prime, and 2-prime. Twenty sufficient and necessary conditions are established for unique type k-ideals to be type prime k-ideals in bi-semiring. A T 1 − KId (T 2 − KId) in a bi-semiring was shown to correspond to a prime T 1 − KId (prime T2 − KId) in bi-semiring whenever it is either 0-prime 1-prime
or 2-prime. There are numerous necessary and sufficient conditions for a T 1 − KId (T 2 − KId) to be a T 1 − PKId (T 2 − PKId). We show that when its complement is a τ2 2-system (τ1 2), T 1 − KId (T 2 − KId) is T 1 − PKId (T 2 − PKId). There exists a T 1 − PKId (T 2 − PKId) P such that P ∩  = ϕ and A ⊆ P for every T 1−KId (T 2 − KId) A in, where A ∩  = ϕ where  is τ2 2-system (τ3 2-system).

Fekete–SzegÖ inequalities and initial coefficient bounds for Bi-univalent functions involving fractional q-differential operator subordination to q-hermite polynomials

Sarab Kazim Hassan — 2026-04-12

In this paper, we introduce new subclasses of bi-univalent functions in the open unit disk defined via the q-fractional difference operator related to q-Hermite polynomials. Using subordination, we define the starlike class  and the convex class   . For functions in these classes, we derive upper bounds for the first two Taylor coefficients |a2| and |a3|, and we establish upper Fekete–Szegö inequalities for 2 aa 32 - b and 2 aa 32 - m , extending classical results to the q-fractional setting. We focus particularly on comparing the initial coefficients of these functions, providing insight into their geometric behavior and the influence of the q-fractional operator on the coefficient bounds.