In this short note we discuss how to solve the quantum harmonic oscillator. There is a well known traditional method of solving the quantum harmonic oscillator. Here we present a matrix-eigenvalue method. We also present another equivalent matrix-eigenvalue method, which has pedagogical value.

The Riemann-Roch theorem is a classical result which forms a beautiful algebraic connection between complex analysis on a compact Riemann surface and a global topological property of that surface (the genus). We present a survey of the theorem and its many variants and generalizations. We also provide an alternative elementary proof of the Riemann-Roch formula for algebraic curves and outline some applications of the formula.

The notions of an $\inf$-hesitant fuzzy right (left, lateral) ideal, an $\inf$-hesitant fuzzy ideal, a $(\sup, \inf)$-hesitant fuzzy right (left, lateral) ideal, and a $(\sup, \inf)$-hesitant fuzzy ideal, which are generalizations of an interval-valued fuzzy ideal of a ternary semigroup, are introduced and their properties are investigated. Conditions for a hesitant fuzzy set to be an $\inf$-hesitant fuzzy right (left, lateral) ideal, an $\inf$-hesitant fuzzy ideal, a $(\sup, \inf)$-hesitant fuzzy right (left, lateral) ideal, and a $(\sup, \inf)$-hesitant fuzzy ideal of a ternary semigroup are provided in terms of sets, fuzzy sets, Pythagorean fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets. Furthermore, characterizations of an ideal of a ternary semigroup are studied via a generalization of the characteristic hesitant and the characteristic interval-valued fuzzy set.

In this paper we study the analysis of conformable fractional differentiable vector valued functions of several real variables.

We build a unification of several variants of the kernel of a set in a generalized topological space endowed with a hereditary class, which is a fundamental concept to introduce new modifications of important concepts as open sets and closed sets. This new theoretical framework leads to the study in unified form of separation properties in a context much more general and versatile than the case of a topological space provided with an ideal.

The notion of $R$-regular module is introduced. We will show that if $M$ is a torsion-free module over a commutative ring, then $M$ is $R$-regular module if and only if $M$ is a strongly regular module. We will also show that if $M$ is a cyclic $R$-regular $IFP$ module, then the submodule $P$ is completely prime if and only if $P$ is maximal.

We consider a new version of the Composition-Diamond Lemma for dialgebras in order to obtain an explicit Gröbner-Shirshov basis and a normal form for the HNN-extensions of dialgebras.

The purpose of this paper is to introduce the concepts of $e^{*}$-supplement submodules, weakly $e^{*}$-supplement submodules, ample $e^{*}$-supplement submodules, co-$e^{*}$essential submodules, $e^{*}$-supple-mented modules, weakly $e^{*}$-supplemented modules, and amply $e^{*}$-supplemented modules as generalizations of the concepts of supplement submodules, weakly supplement submodules, ample supplement submodules, co-essential submodules, supplemented modules, weakly supplemented modules, and amply supplemented modules, respectively. We will prove some properties of these concepts.

In mathematics, the prime counting function $\pi(x)$ is defined as the function yielding the number of primes less than or equal to a given number $x$. In this paper, we prove that the asymptotic limit of a summation operation performed on a unique subsequence of the prime numbers yields the prime number counting function $\pi(x)$ as $x$ approaches $\infty$. We also show that the prime number count $\pi(n)$ can be estimated with a notable degree of accuracy by performing the summation operation on the subsquence up to a limit $n$.

An exact $r$-coloring of a set $S$ is a surjective function $c:S\to \{1,2,\cdots, r\}$. Given an equation $eq$, a solution in $S$ is a rainbow solution if each element is colored distinctly by the coloring $c$. The rainbow number of a set $S$ for equation $eq$ is the smallest integer $r$ such that every exact $r$-coloring of $S$ contains a rainbow solution to $eq$. The rainbow numbers of $\mathbb{Z}_p$, for prime $p$, for the equation $x_1 + x_2 = 4x_3$ are known to be either $3$ or $4$. This paper investigates which primes yield rainbow number $3$ or $4$. Additionally, the rainbow numbers of $\mathbb{Z}_n$ for this equation are discussed.

This work presents the concept of the neutrosophic set on an incline algebra and provides some basic results concerning the cartesian product and projections on neutrosophic filters on an incline.