Annals of Mathematics and Computer Science

Controllability of Variable-Order Conformable Systems

2026-07-05T13:20:56+00:00

We study controllability of semilinear evolution systems with infinite memory governed by a generalized conformable derivative of variable order. Since the time-change reduction to semigroups fails, we construct a two-parameter evolution family and the corresponding integral representation in a Banach space. Under Lipschitz and Hale-Kato hypotheses we prove existence and uniqueness of mild solutions, introduce a variable-order controllability Gramian, and characterize exact and approximate controllability. We also obtain a quantitative observability inequality with explicit constants. Approximate controllability in the semilinear case follows from a regularized feedback and a fixed point scheme. An application and a numerical illustration are included.

Radial Distance Functions for Partitioning Soft Spaces

2026-07-20T18:28:14+00:00

In this paper, we introduce a scalar measure for soft sets and define reference-based radial distances from a fixed reference soft set to every soft set in a soft space. We establish their fundamental properties, including non-negativity, identity of indiscernibles, and conditional monotonicity. The normalized radial distances induce a partition of the soft space into clusters determined by prescribed distance intervals. We formulate the partitioning task as a computational problem, develop an algorithm for constructing the induced partition, and establish its worst-case time and space complexities. The proposed framework is illustrated through a course outcomes assessment example

Duality and Reproducing Kernels for Dirichlet-Type Spaces

2026-08-01T14:37:09+00:00

Using an isometric differentiation isomorphism between the weighted Dirichlet-type spaces and the weighted Bergman space, we determine the duals of the weighted Dirichlet-type spaces of the unit disk and the upper half-plane. Further, we compute the reproducing kernels for the Hilbert version of these spaces.

Devaney Chaos of a Proportional Caputo Derivative

2026-08-06T23:38:21+00:00

In this article, we analyze the dynamical behaviour of a proportional Caputo-type complex fractional derivative. We prove, by using the Bayart–Grivaux eigenvector-field theorem, that this operator is Devaney chaotic in a suitable weighted Mittag–Leffler–Caputo space.

Weak Solutions of Fractional p-Kirchhoff Problem with Nonlocal Neumann Condition

2026-08-28T15:22:15+00:00

The existence of a weak solution to a Schrödinger-Kirchhoff- type problem involving the fractional p-Laplacian with a homogeneous Neumann type boundary condition is the focus of this essay. This p-Neumann boundary condition arises from a simple probabilistic consideration. The Berkovits degree theory is used to establish the existence of weak solutions under certain suitable conditions.

Leakage-Aware Explainable Framework for Breast Cancer Prognosis

2026-07-10T13:56:37+00:00

Breast cancer remains a leading cause of cancer-related mortality among women, necessitating reliable and interpretable prognostic models for personalized treatment planning. This study developed a leakage-aware, explainable machine learning framework for breast cancer survival prediction using clinicopathological data from 4,024 patients in the SEER database. The framework integrated feature engineering, leakage detection, hyperparameter optimization, calibration assessment, and explainable artificial intelligence. Among five evaluated algorithms, CatBoost achieved the best performance, with a ROC-AUC of 0.725 and a five-fold cross-validated ROC-AUC of 0.746 (95% CI: 0.708–0.785). SHAP analysis identified Node Ratio, Age, Hormone Index, Tumor Burden, and Grade as the most influential predictors. The proposed framework provides transparent, reliable, and clinically relevant prognostic predictions for breast cancer risk stratification.

Operator Theoretic Analysis of Coupled Decomposition Methods

2026-07-25T08:34:42+00:00

Coupled Kharrat-Toma-Mohand Transform and Adomian Decomposition Techniques for Nonlinear Differential and Integro-Differential Equations have been studied using a combination of an integral transform with Adomian Decomposition Method. The convergence property has been proved for specific problems to date. In this paper, we give a general proof for this issue. Assuming boundedness of the operators in the Banach space of continuous functions, we will prove that the proposed combination technique is nothing but Picard Iteration of an operator T. By virtue of the contraction property of the operator T, existence, uniqueness, error estimate, and stability of the method are proved. Some logistic and diffusion problem examples illustrate the divergence of the constant at blow-up point.

Fractional Burgers Equations with Mixed Derivatives

2026-08-17T11:13:25+00:00

This paper studies a nonlinear fractional Burgers equation involving an Atangana–Baleanu time derivative and a Caputo space derivative. We establish existence and uniqueness results under suitable assumptions and formulate a Laplace variational iteration scheme for constructing approximate solutions. The convergence of the iterative procedure is analyzed under explicit conditions. An illustrative example is presented to examine the influence of the fractional order on the solution behavior and to recover the corresponding classical Burgers model as the fractional order approaches one.

Horizontal Gradient Flows on the Heisenberg Group

2026-08-16T10:21:18+00:00

This paper studies horizontal gradient flows on the Heisenberg group, an important example of a sub-Riemannian manifold. The analysis focuses on horizontal differential operators and their role in describing evolution processes governed by the intrinsic geometry of the group. The work highlights the structure of the horizontal gradient and related operators, providing insight into gradient-driven dynamics and partial differential equations in non-commutative geometric settings.

Surrogate-accelerated Landweber iteration for stiffness identification

2026-09-27T19:37:49+00:00

Identifying stiffness degradation from sparse sensor data is a central inverse problem in structural health monitoring, and neural surrogates are increasingly used to replace the finite element solves it requires. Such replacements are rarely accompanied by guarantees that the resulting identification procedure converges or that its error is controlled. We study a Landweber iteration in which both the parameter-to-observation map and its derivative are replaced by a neural surrogate trained on finite element data. For an elliptic structural model with a finite-dimensional damage parameterization, we establish explicit Lipschitz bounds on the forward map and its derivative in terms of material bounds, load and sensor functionals. We then show that a surrogate whose empirical value and derivative errors are small on a random design attains uniform accuracy on the whole admissible set, with an explicit dependence on the fill distance of the design and on the mesh size. Under a tangential cone condition, which we prove holds locally whenever the sensor layout renders the linearized problem identifiable, the surrogate-based iteration is monotone, terminates after finitely many steps under a discrepancy principle that accounts for the surrogate error, and returns an estimate whose error is bounded by a constant multiple of the measurement noise, the empirical training error, the fill distance and the squared mesh size, divided by the smallest singular value of the linearized sensor map. The analysis yields quantitative criteria for sensor placement and for derivative-informed training. A numerical study on a bridge-deck model problem confirms the predicted trends and quantifies the conservatism of the certificate.