Persistent homology (PH) is a cornerstone of topological data analysis; however, standard filtrations often overlook the anisotropic structure in empirical data. We introduce Ellipse Cloud, a preprocessing pipeline that accentuates anisotropy within PH. The method builds a Vietoris-Rips filtration from ellipse tangency times rather than pairwise Euclidean distances, thereby accommodating anisotropy while preserving compatibility with existing computational tools. The central computational challenge-identifying the first contact times of expanding ellipses-is resolved with an efficient and provably stable numerical algorithm.
We study modified ruin probabilities in a Cramér-Lundberg model driven by a compound mixed Poisson process. In the heavy-tailed regime, if the integrated claim-size distribution is subexponential and the upper endpoint of the mixing distribution stays below the net-profit boundary, the modified and classical ruin probabilities are asymptotically equivalent. In the light-tailed regime, we prove a fixed-intensity ratio theorem and obtain both an endpoint-atom result and a sharp endpoint-density asymptotic with an explicit constant.
This paper first reviews the hypercircle method for a posteriori error estimation of finite element (FE) solutions to Poisson's equation, which originates from the classical Prager- Synge theorem and employs basic P-1 and H(div) elements. With new numerical results, we examine a simplified post-processing method for P-1 solutions that avoids mixed FE systems. While this approach offers practical advantages, it lacks rigorous error analysis. Although the numerical results are promising, the theoretical foundation remains open, and further investigation is needed.
This study aims to derive an integrable discretization of the Bernoulli equation that achieves arbitrary higher-order accuracy while preserving the original solution structure. Using the Fade approximation, we develop discretizations for nonhomogeneous first-order linear differential equations with both constant and variable coefficients. By transforming the independent variable, we obtain the discretization of the Bernoulli equation and its general solution.
The Ziggurat method is an efficient rejection sampling technique for generating one-dimensional normally distributed random numbers. This study proposes the pattern block method, a generalization of the Ziggurat method. The pattern block method enables the generation of random numbers from multimodal density functions and multidimensional distributions. The effectiveness of the pattern block method is demonstrated through several examples.
We perform some simulations of the semilinear Klein-Gordon equation with a power-law nonlinear term and propose each of the quantitative evaluation methods for the stability and convergence of numerical solutions. We also investigate each of the thresholds in the methods by varying the amplitude of the initial value and the mass, and propose appropriate values.
In this letter, we establish a recurrence relation for the characteristic polynomial of alternatingoriented tetragonal comb graphs. Using this relation, we express the polynomial in terms of Chebyshev polynomials of the second kind. This formulation allows us to explicitly determine the eigenvalues of the graphs. Furthermore, we analyze the geometric structure of the spectra in the complex plane and propose conjectures on generalized comb graphs involving higherorder polygons and various gluing configurations.
Parallel-in-time methods for time-dependent problems have risen to prominence over the past decade as massively parallel computers push core counts into the millions. Among them, the block e-circulant (BEC) preconditioned solver achieves outstanding convergence, yet the preconditioner itself dominates the solver runtime. We propose a mixed-precision strategy that uses single precision for BEC preconditioning and double precision elsewhere, thereby reducing the solver runtime especially on CPU-GPU systems. On an NVIDIA GH200 cluster, the mixed-precision BEC-GMRES solver achieves a 1.42x speedup for two-dimensional advectiondiffusion problems without significant loss of accuracy.
This paper presents a theorem of existence for an optimum point and methods to obtain component-wise verified solutions in constrained convex programming. The proposed method is based on the continuous Newton method and Kantorovich's theorem and slightly modifies Oishi and Tanabe's theorem for linear programming. Moreover, a theorem of existence for an optimum point for convex quadratic programming problems (CQPs) was also formulated. The CQP can be rewritten as linear complementarity problems (LCPs). A method to use component-wise verified solution method for LCPs is also presented. Numerical examples show that the proposed method is effective.
