Generalized Chessboard Structures Whose Effective Conductivities Are Integer Valued

Generalized Chessboard Structures Whose Effective Conductivities Are Integer Valued

Generalized Rings of Integer-valued Polynomials

The classical ring of integer-valued polynomials Int(Z) consists of the polynomials in Q[X] that map Z into Z. We consider a generalization of integervalued polynomials where elements of Q[X] act on sets such as rings of algebraic integers or the ring of n× n matrices with entries in Z. The collection of polynomials thus produced is a subring of Int(Z), and the principal question we consider is...

About polynomials whose divided differences are integer valued on prime numbers

We show here how to construct bases of the Z-module Int(P,Z) of polynomials that are integer-valued on the prime numbers together with their finite divided difference, that is, Int(P,Z) = { f ∈ Q[x] | ∀p, q ∈ P f(p) ∈ Z and f(p)− f(q) p− q ∈ Z } .

What are Rings of Integer-Valued Polynomials?

Every integer is either even or odd, so we know that the polynomial f(x) = x(x− 1) 2 is integervalued on the integers, even though its coefficients are not in Z. Similarly, since every binomial coefficient ( k n ) is an integer, the polynomial ( x n ) = x(x− 1)...(x− n+ 1) n! must also be integervalued. These polynomials were used for polynomial interpolation as far back as the 17 century. Inte...

Heronian Triangles Whose Areas Are Integer Multiples of Their Perimeters

We present an improved algorithm for finding all solutions to Goehl’s problem A = mP for triangles, i.e., the problem of finding all Heronian triangles whose area (A) is an integer multiple (m) of the perimeter (P ). The new algorithm does not involve elimination of extraneous rational triangles, and is a true extension of Goehl’s original method.