For a circle action on a compact almost complex manifold with a fixed point, the lower bound on the number of fixed points is known in dimension up to 12 except 10. In this paper, we show that if the circle group acts on a 10-dimensional compact almost complex manifold with a fixed point, then there are at least 6 fixed points. This minimum is attained by ℂℙ⁵ and S⁶ × ℂℙ² . We establish this lower bound by showing that there does not exist a circle action on a 10-dimensional compact almost complex manifold with 4 fixed points.

We study memory effects generated by a general time-dependent input through a convolution operator with an integrable kernel. Focusing on the role of the input rather than the specific kernel structure, we analyze how qualitative properties of g(t) determine the long-term behavior of the resulting memory response. We establish general conditions under which the memory output converges to a constant equilibrium when the input approaches a steady value, showing that the equilibrium depends only on the total mass of the kernel. For time-dependent inputs, including periodic functions, we prove the existence of steady responses that attract all transient dynamics. Numerical simulations illustrate representative phenomena such as transient overshoot, phase lag, and memory-induced smoothing. This work provides a kernel-agnostic framework for understanding input-driven memory dynamics with broad relevance to biological and medical applications.

The purpose of this study is to analyse the value of Vulnerable Digital Exchange Options (VDEOs), which are defined as financial instruments that incorporate both issuer default risk and stochastic interest rates. The contract under consideration is of a digital exchange nature, and it reflects the relative performance of two underlying assets. Additionally, it incorporates an option which accounts for the possibility of default arising from the issuer's credit risk. Also, the short-term interest rates are modelled using the Hull-White model, which captures term structure volatility and correlations between asset and credit factors while remaining interpretable. A high-dimensional partial differential equation (PDE) for the VDEO price was derived utilising the Feynman-Kac theorem. Also, we applied a variable transformation to isolate the bond component and reduce the dimension of the PDE. Furthermore, a computable semi-analytic price expression was derived via a double Mellin transformation. In addition the accuracy of the pricing formula is verified through comparison with Monte Carlo simulations.

We study the problem of approximating the principal square root A^1/2 of a nonsingular matrix A ∈ ℂ^n×n with no eigenvalues on ℝ≤0. Classical iterative methods-Newton, Denman-Beavers, Halley, productform, Padé, and the Schur method of Björck-Hammarling-each require O(n³) work per step and may converge slowly near the branch cut. We cast the map F : A 7→ A^1/2 as an empirical risk minimization problem over deep ReLU networks, establish a Rademacher-complexity generalization bound, and show that the principal square-root map is qualitatively approximable by ReLU networks on the compact spectral band A_n,α,β. Two hybrid algorithms are proposed in which a trained network warm-starts Newton or Denman-Beavers refinement; their iteration savings follow directly from the quadratic convergence of Newton's method without new convergence proofs.

This paper concerns the multiplicity of positive solutions to the second order differential systems with strongly coupled integral boundary value conditions. The Avery-Peterson fixed point theorem is used for the main results.

This paper presents the 𝑎-number of superelliptic curves of genus 3 over a finite field using the action of the Cartier operator on H⁰ (C, Ω¹ ). The 𝑎-number is an invariant of the p-torsion part of their Jacobians.