Article
Multidisciplinary Sciences
Amartya Mandal, Pragya Tiwari, Paul K. Upputuri, Venkata R. Dantham
Summary: This article reports the theoretical investigation on the photonic nanojets (PNJs) of single dielectric microspheres illuminated by focused broadband radiation from various sources. The study explores the effects of incident beam waist, refractive index of the surrounding medium, and radius of the microsphere on the characteristic parameters of PNJs. Interestingly, the characteristic parameters of PNJs obtained from solid microspheres under broadband and monochromatic radiation are found to be close. Additionally, the study examines the characteristic parameters of PNJs from core-shell microspheres of different thicknesses illuminated by polychromatic radiation from commonly used sources. For each thickness, an appropriate wavelength of monochromatic radiation is found to generate PNJs with similar characteristic parameters to those obtained from polychromatic radiation.
SCIENTIFIC REPORTS
(2022)
Article
Statistics & Probability
Jan Rataj
Summary: The translative intersection formula of integral geometry provides an expression for the mean Euler characteristic of a stationary random closed set intersected with a fixed observation window. This result is formulated in the setting of sets with positive reach and using flag measures, which yield curvature measures as marginals. As an application, the study focuses on excursion sets of stationary random fields with C-1,C-1 realizations, such as stationary Gaussian fields, and extends known results from the literature.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2021)
Article
Physics, Mathematical
Meina Sun
Summary: The Riemann solutions for a simplified two-phase flow model with the logarithmic equation of state are analyzed to identify and understand the phenomena of cavitation and concentration through the vanishing pressure limit. The solutions converge to specific configurations as the perturbed parameter tends to zero, showing different patterns in the behavior of shock waves and rarefaction waves. Additionally, Dirac delta measures develop simultaneously in the densities of liquid and gas in the limiting situation.
JOURNAL OF MATHEMATICAL PHYSICS
(2021)
Article
Statistics & Probability
Radomyra Shevchenko, Anna Paola Todino
Summary: This paper considers a sequence of needlet random fields, which are defined as weighted averaged forms of spherical Gaussian eigenfunctions. The main result is a Central Limit Theorem for the boundary lengths of their excursion sets in the high energy setting. The result is obtained using Wiener chaos expansion and Stein-Malliavin techniques, with a careful analysis of the variances of each chaotic component of the boundary length.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2023)
Article
Statistics & Probability
Ali Reza Taheriyoun, Khalil Shafie
Summary: Fractal dimension serves as a useful characteristic for measuring irregularity in random fields. In this study, the Euler characteristic is used to estimate the fractal dimension of random fields, with a weighted smooth version employed to address issues with non-differentiable Gaussian fields. Consistency of the estimator is confirmed through simulation and real data application under certain assumptions.
CANADIAN JOURNAL OF STATISTICS-REVUE CANADIENNE DE STATISTIQUE
(2022)
Article
Statistics & Probability
Charles Bordenave, Djalil Chafai, David Garcia-Zelada
Summary: As the dimension of the matrix approaches infinity, the spectral radius is probabilistically equivalent to the square root of the dimension. The proof establishes the convergence of the reciprocal characteristic polynomial and a random analytic function related to a hyperbolic Gaussian function outside the unit circle. This proof differs from traditional approaches and relies on tightness and joint central limit phenomena.
PROBABILITY THEORY AND RELATED FIELDS
(2022)
Article
Statistics & Probability
Viet-Hung Pham
Summary: The study focuses on the conjunction probability of Gaussian fields with smooth sample paths and provides an asymptotic formula for this probability as the threshold level approaches infinity. The findings partially confirm the effectiveness of the Euler characteristic method by analyzing the shape of the excursion set of the Gaussian field.
ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS
(2023)
Article
Statistics & Probability
Andrew M. Thomas, Takashi Owada
Summary: This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes, focusing on points drawn from a nonhomogeneous Poisson process with highly connected and nontrivial topological characteristics. The study establishes two 'functional-level' limit theorems for the appropriately normalized Euler characteristic process, including a strong law of large numbers and a central limit theorem.
ADVANCES IN APPLIED PROBABILITY
(2021)
Article
Statistics & Probability
Anna Paola Todino
Summary: This paper investigates geometric functionals for band-limited Gaussian and isotropic spherical random fields in dimension 2, focusing on the area of excursion sets and its behavior in the high energy limit. The results are based on Wiener chaos expansion for non linear transform of Gaussian fields and on derivation of the high-frequency limit of the covariance function of the field. As a corollary, the Central Limit Theorem for the excursion area is also established.
ELECTRONIC COMMUNICATIONS IN PROBABILITY
(2022)
Article
Statistics & Probability
Johannes Krebs, Benjamin Roycraft, Wolfgang Polonik
Summary: The study focuses on the approximation theorems for the Euler characteristic of the Vietoris-Rips and Cech filtration, obtained from Poisson or binomial sampling schemes in the critical regime. The results are applied to the smooth bootstrap of the Euler characteristic to determine its rate of convergence in the Kantorovich-Wasserstein distance and the Kolmogorov distance.
ELECTRONIC JOURNAL OF STATISTICS
(2021)
