We give conditions on the rational numbers a, b, c which imply that there are infinitely many triples (x, y, z) of rational numbers such that x+ y + z = a+ b+ c and xyz = abc. We do the same for the equations x + y + z = a + b + c and x + y + z = a + b + c. These results rely on exhibiting families of positive-rank elliptic curves.

Stembridge, J.R., Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Mathematics 99 (1992) 307-320. Let Z denote the Coxeter complex of S,, and let X(X) denote the associated toric variety. Since the Betti numbers of the cohomology of X(Z) are Eulerian numbers, the additional presence of an &-module structure permits the definition of an isotypic refinement of these ...

The most basic concept is that of an infinite sequence (of real or complex numbers in these notes). For p ∈ Z, let Np = {k ∈ Z : k ≥ p}. An infinite sequence of (complex) numbers is a function a : Np → C. Usually, for n ∈ Np we write a(n) = an, and denote the sequence by a = {an}n=p. The sequence {an}n=p is said to converge to the limit A ∈ C provided that for each > 0 there is an N ∈ Z such th...

j j j k z T z T z = = ∑ where T is a 2D similarity transformation. Proof: Similarity transformations can be represented using a linear polynomial over the complex plane in the following way. If the similarity transformation T consists of rotation by positive angle θ, uniform scale s and a translation t = tx + ity, then according to the rules of complex numbers: ( ) ( ) i T x iy T z se z t z θ +...

This paper studies the integers that belong the multiplicative semigroup W generated by { 2n+1 : n ≥ 0} and 1 2 . They form a multiplicative semigroup W(Z) of integers which we call the wild integer semigroup, and the wild numbers are the generators of W(Z). It presents convincing evidence that the wild numbers consist of all prime numbers excluding 3.

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun Z.-H. Sun. For instance, by using a transformation formula due to Chan Zudilin, we show that for any prime p ≥ 5 , ∑ k = 0 − 1 3 + ( 32 ) ≡ 2 E mod 4 is regarded as -adic analogue the interesting / π Rogers: ∞ . Here n are famous numbers Euler numbers.

This paper presents a method to construct more general fuzzy complex numbers and sets from ordinary fuzzy complex numbers by introduced the ordered sequences of membership functions. It is concluded that multifuzzy complex set is an extension of Buckley fuzzy complex set. Also, two types of multi-fuzzy complex numbers based on the forms z x iy = + and z re = θ are investigated. Multi-Fuzzy Comp...