Axioms

In this work, a high-order meshless framework is developed for the numerical resolution of the temporal–fractional Black–Scholes equation arising in option pricing with long-memory effects. The spatial discretization is carried out with a local radial basis function produced finite difference (RBF–FD) method on seven-node stencils. Analytical differentiation weights are constructed by employing closed-form second integrations of a variant of the inverse multiquadric kernel, which yields sparse differentiation matrices. Explicit formulas are derived for both first- and second-order operators, and a detailed truncation error analysis confirms sixth-order convergence in space. Numerical experiments for European options discuss better accuracy per spatial node than standard finite difference schemes.

We formalize an inverse, data-conditioned variant of the Variational Quantum Eigensolver (VQE) for clinical biomarker discovery. Given patient-encoded quantum states, we construct a task-specific Hamiltonian whose coefficients are inferred from clinical associations and interpret its expectation value as a calibrated energy score for prognosis and treatment monitoring. The method integrates coefficient estimation, ansatz specification with basis rotations, commuting-group measurements, and a practical shot budget analysis. Evaluated on public infectious disease datasets under severe class imbalance, the approach yields consistent gains in balanced accuracy and precision–recall over strong classical baselines, with stability across random seeds and feature ablations. This variational energy scoring framework bridges Hamiltonian learning and clinical risk modeling, offering a compact, interpretable, and reproducible route to biomarker prioritization and decision support.

This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using ${\ell }_p$-spherical distributions on ${\mathbb{R}}^d$ with . The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any $p\ge 1$. Additionally, the mean squared errors (MSEs) of the gradient estimators are bounded by $K_0{N}^{-1}d$ for any , and by $K_1{N}^{-1}{d}^{2/p}$ when with N the sample size and $K_0,K_1$ some constants. Taking allows for achieving dimension-free upper-bounds of MSEs. In the case where , the upper-bound is reached with $K_2$ a constant. Such results lead to dimension-free MSEs of the proposed estimators, which boil down to estimators of the traditional gradient when the variables are independent. Numerical comparisons show the efficiency of the proposed approach.

In this paper, we investigate optimization problems for a Hermitian quadratic matrix-valued function involving two variable matrices. We derive algebraic formulas for the maximal and minimal ranks and partial inertias of this function based on a linearization method and specific block matrix rank and inertia transformations. Further, we establish necessary and sufficient conditions for the existence of solutions satisfying the associated matrix equation and various Hermitian inequalities. As applications, we obtain extremal ranks and inertias for the Hermitian generalized Schur complement with respect to Hermitian reflexive generalized inverses, and provide conditions for Hermitian reflexive generalized inverses to satisfy specific Hermitian properties. In addition, we apply these results to analyze optimization, stability, and congestion in a traffic flow network modeled by this quadratic function.