Journal of Applied & Computational Mathematics

Objectives:The paper aims to present a survey of both time-honored and contemporary studies on triangular number, factorial, relationship between the 2, and a few other numbers related to them.

Methods: The research is expository in nature. It focuses on expositions regarding the triangular number, its multiplicative analog – the factorial and other numbers associated with them.

Findings: Much had been studied about triangular numbers, factorials and other numbers involving sums of triangular numbers or sums of factorials. However, it seems that no-one had explored the properties of the sums of corresponding factorials and triangular numbers. Hence, explorations on these integers, called factoriangular numbers, were conducted. Series of experimental mathematics resulted to the characterization of factoriangular numbers on its parity, compositeness, number and sum of positive divisors and other minor characteristics. The sequence of factoriangular numbers may be a recurring sequence and it's a rational closed-form of exponential generating function. These numbers were also characterized on when a factoriangular number are often expressed as a sum of two triangular numbers and/or as a sum of two squares.

Application/ Improvement: The introduction of factoriangular number and expositions on this sort of number may be a novel contribution to the idea of numbers. Surveys, expositions and explorations on existing studies may still be a serious undertaking in number theory.

Hybrid quantum-classical systems combine both classical and quantum degrees of freedom. Typically, in chemistry, molecular physics, or saterials science, the classical degrees of freedom describe atomic nuclei (or cations with frozen core electrons), whereas the quantum particles are the electrons. Although many possible hybrid dynamical models exist, the essential one is that the so-called Ehrenfest dynamics that results from the simple partial classical limit applied to the complete quantum Schrödinger equation. Few numerical methods are developed specifically for the mixing of this sort of systems. Here we present a preliminary study of the performance of a family of recently developed propagators: the (quasi) commutator-free Magnus expansions. These methods, however, were initially designed for nonautonomous linear equations. We employ them for the nonlinear Ehrenfest system, by approximating the state value at whenever step within the propagation, using an extrapolation from previous time steps.

We have created a formula to calculate the number of primes less than or equal to any given positive integer ‘n'. It is denoted by π (n). This is a fundamental concept in number theory and it is difficult to calculate. A prime number can be divided by 1 and itself . Therefore the set of primes (2,3,5,7,11,13,17.). The Prime Counting Function was conjectured the end of the 18th century by Gauss and by Legendre to be approximately x/Ln(x) But in this paper we are presenting the real formula, by applying the modern approach that is applying the basic concept of set theory.

Sensitivity analysis is that the study of how the uncertainty within the output of a mathematical model or system (numerical or otherwise) are often divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty analysis, which features a greater specialise in uncertainty quantification and propagation of uncertainty; ideally, uncertainty and sensitivity analysis should be run in tandem.

Detecting outliers is an integral part of data analysis that sheds light on points that do not conform to the rest of the data. Whereas in univariate data, outliers appear at the extremes of the ordered sample, in the multivariate case they may be defined in many ways and are not generally based on an assumed statistical model. We present here methods for detecting multivariate outliers based on various definitions and illustrate their features by applying them to two sets of data. No single approach can be recommended over others, since each one aims at detecting outliers of a particular kind.