This paper takes an interesting approach to conceptualize some power sum inequalities and uses them to develop limits on possible solutions to some Diophantine equations. In this work, we introduce how to apply center of mass of a $k$-mass-system to discuss a class of Diophantine equations (with fixed positive coefficients) and a class of equations related to Fermat's Last Theorem. By a constructive method, we find a lower bound for all positive integers that are not the solution for these type of equations. Also, we find an upper bound for any possible integral solution for these type of equations. We write an alternative expression of Fermat's Last Theorem for positive integers in terms of the product of the centers of masses of the systems of two fixed points (positive integers) with different masses. Finally, by assuming the validity of Beal's conjecture, we find an upper bound for any common divisor of $x$, $y$, and $z$ in the expression $ax^m+by^n = z^r$ in terms of $a, b, m({\rm or} ~n), r$, and the center of mass of the $k$-mass-system of $x$ and $y$.

In this paper, we apply the concept of cubic sets to implicative ideals of $BCK$-algebras, and then characterize their basic properties. We discuss relations among cubic implicative ideals, cubic subalgebras and cubic ideals of $BCK$-algebras. We provide a condition for a cubic ideal to be a cubic implicative ideal. We define inverse images of cubic implicative ideals and establish how the inverse images of a cubic implicative ideal become a cubic implicative ideal. Finally we introduce products of cubic $BCK$-algebras.

In this paper, the concept of quasi-coincidence of a bipolar fuzzy point within a bipolar fuzzy set is introduced. The notion of $(\in,\in\vee q)$-bipolar fuzzy subalgebras and ideals of $BCK/BCI$-algebras are introduced and their related properties are investigated by some examples. We study bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-ideals by their level subalgebras and level ideals. We also provide the relationship between $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-subalgebras, and $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-ideals and bipolar fuzzy $BCK/BCI$-ideals by counter examples.

Given a complex-valued polynomial of the form $p(z) = (z-1)^k(z-r_1)^m(z-r_2)^n$ with $|r_1|=|r_2|=1$; $k,m,n \in \mathbb{N}$ and $m \neq n$, where are the critical points? The Gauss-Lucas Theorem guarantees that the critical points of such a polynomial will lie within the unit disk. This paper further explores the location and structure of these critical points. Surprisingly, the unit disk contains two ‘desert’ regions in which critical points cannot occur, and each $c$ inside the unit disk and outside of the desert regions is the critical point of exactly two such polynomials.

We prove the infinitude of prime numbers by the principle of contradiction, that is different from Euclid's proof in a way that it uses an explicit property of prime numbers. A sieve method that applies the inclusion-exclusion principle is used to give the property of the prime numbers in terms of the prime counting function.

We define and completely explore the so-called WJU rings. This class properly encompasses the class of JU rings, introduced and studied by the present author in detail in Toyama Math. J. (2016).

Molodtsov initiated soft set theory which has provided a general mathematical framework for handling uncertainties that occur in various real life problems. The aim of this paper is to provide fuzzy soft algebraic tools in considering many problems that contain uncertainties. In this article, the notion of $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy soft $BCI$-subalgebra of $BCI$-algebra is introduced. Some operational properties on $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy soft $BCI$-subalgebras are discussed as well as lattice structures of this kind of fuzzy soft set on $BCI$-subalgebras are derived.

An exercise in Harary [1, p. 100] states that the product of the vertex independence number and the vertex covering number is an upper bound on the number of edges in a bipartite graph. In this note, we extend the bound to triangle-free graphs, and show that equality holds if and only if the graph is complete bipartite.

We show, with simple combinatorics, that if the dimples on a golf ball are all 5-sided and 6-sided polygons, with three dimples at each “vertex”, then no matter how many dimples there are and no matter the sizes and distribution of the dimples, there will always be exactly twelve 5-sided dimples. Of course, the same is true of a soccer ball and its faces.