Vectors are denoted by boldfaced lower-case letters, and matrices by boldfaced upper-case letters. Elements of vectors and matrices are not bold-faced. The transposition of a matrix A is denoted by A and the inverse of A is denoted by A−1. The n × n identity matrix is denoted by In. We use Eij to denote a matrix in which (i, j) element is one and all the others are zero. The n-dimensional vecto...

Theorem (Generalized Dirichlet Density). Let f : Y → X be a generically finite and Galois morphism of integral separated schemes of finite type over Z. Let K and L denote the function fields of X, respectively Y , and let G := Gal(L|K) be the group of (rational) automorphisms of Y over X. Then the following hold: 1) There exists an open sub-scheme U ⊂ X such that f is étale above U , and if V =...

Reusable water bottles are growing in popularity; thus, possible chemical release from the internal surface into should be carefully considered to control related health risks. We experimentally evaluated deionized of 40 elements, six phthalates, and bisphenol-A for 20 different reusable by simulating use real world scenario. The bottles, identified as those most purchased Italy, were made vari...

In this paper we study the iterated absolute values of differences between consecutive elements of a periodic sequence of real algebraic numbers.

Let i be a natural number and let x be a real number. Then i 7→ x is a finite sequence of elements of R. Let x be a real number and let n be a natural number. The functor xn is defined by: (Def. 1) xn = ∏(n 7→ x). Let x be a real number and let n be a natural number. Observe that xn is real. Let x be a real number and let n be a natural number. Then xn is a real number. We now state several pro...

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.