We introduce a triangular array of integers defined by a(n, k) = k(k + 1)n−k for n ≥ 1 and 1 ≤ k ≤ n, which we call the researcher’s triangle. The construction is multiplicative and differs from classical additive triangles such as Pascal’s. We give combinatorial interpretations using a leaders-followers model and a rooted hierarchical structure model. We identify several diagonal sequences as classical figurate numbers, including powers of two (A000079), natural numbers (A000027), and oblong numbers (A002378), and establish connections to triangular and tetrahedral numbers. We derive a bivariate generating function and a recurrence relation, and we discuss the row sums.

We introduce the prime density triangle, a multiplicative array constructed by a modified rule of indices. Two display formats are given: a right-angle triangle with converted and unconverted entries, and an equilateral triangle obtained by mirroring the right-angle triangle about the central term xn. We derive the row-sum generat- ing function and provide worked examples using both direct computation and the generating function. Using a four-rule division scheme applied to the unconverted rows we obtain the sequence whose integer parts sum to the magic numbers for two spin orientations. Replacing the repeated end integers by 1,1 yields integer parts that give the standard harmonic oscillator (HO) magic numbers and, in conjunction with the two-spin-orientation nucleon magic numbers, gives precisely the standard nuclear magic numbers 2, 8, 20, 28, 50, 82, 126, 184. Furthermore, digital-root reduc- tion of the converted triangle reveals a distinct “4,9,9” pattern whose diagonals concatenate into large primes, establishing a direct arithmetic link between prime distribution and the symmetry conditions of nuclear shell closures. Subshell filling up to x9, deductions, and evidence from nuclear physics are included. A recurrence M = m ± [(n1n2) + 2] is presented for generating further magic numbers.The sequences appear as OEIS A005897, OEIS A018226 and OEIS A007290.

This note presents two original and creative divisibility tests for the integers 2 and 5. Both rules use the factorial of the last digit of a number, combined with a small correction when the factorial value would disrupt the required modular congruence. The resulting tests are mathematically equivalent to the classical last-digit rules but reach the same conclusion through an unconventional factorial-based approach. Each theorem includes a short proof sketch, an explanation of the mechanism, and four worked examples. The methods are valid precisely because 2 and 5 divide the base 10, allowing the last digit to fully determine the remainder modulo of these divisors. The work is intended as an accessible piece of recreational number theory for educational purposes.

We propose the backward phase flow method to implement the FourierBrosIagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schrdinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in [12]. In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying the FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to

We propose a fast local level set method for the inverse problem of gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x₃-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructively, the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes. To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set. The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domain . To overcome this difficulty we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain