Restricted Partitions

We prove a known partitions theorem by Bell in an elementary and constructive way. Our proof yields a simple recursive method to compute the corresponding Sylvester polynomials for the partition. The previous known methods to obtain these polynomials are in general not elementary. 1. Proof of Bell's theorem. The main purpose of this section is to prove the following theorem, originally proved by Bell in [1], by elementary methods. Theorem 1.1. For a fixed positive integer n, let A1,...,An be positive integers and let M be their least common multiple. For a fixed integer r , the number of nonnegative solutions Xn,...,X1 of An·Xn+···+A1·X1 = M K +r , which we indicate by D n (M K + r), is given by a polynomial in K, which is either the zero polynomial or a polynomial with rational coefficients of degree n − 1. First, we need the following known result. Lemma 1.2. For N ≥ 0 and m ≥ 1, H m (N) = 0 m + 1 m +···+N m is a polynomial in N of degree m + 1 with rational coefficients. Besides, H m (−1) = 0. For example, we have H 1 (N) = 1 2 N 2 + 1 2 N, H 2 (N) = 1 3 N 3 + 1 2 N 2 + 1 6 N. There exist several elementary methods to obtain the polynomials H m (N). We will see that D n (M K +r) is a polynomial as a direct consequence of Lemma 1.2. The proof of Theorem 1.1. We are going to prove Bell's theorem by mathematical induction. The theorem is clearly true for n = 1 since in this case, the number of solutions to the equation A1 · X1 = A1 · K + r ' is given by the polynomials D 1 (A1 · K + r) = 1 if r is multiple of A1, D 1 (A1 · K + r) = 0 if r is not a multiple of A1. (1.2) Let n ≥ 1 be a given, and assume Theorem 1.1 holds for n − 1; we will prove it is also true for n.

Theorem 1.1.For a fixed positive integer n, let A1,...,An be positive integers and let M be their least common multiple.For a fixed integer r , the number of nonnegative solutions Xn,...,X1 of An•Xn+•••+A1•X1 = M K +r , which we indicate by D n (M K + r ), is given by a polynomial in K, which is either the zero polynomial or a polynomial with rational coefficients of degree n − 1.
For example, we have There exist several elementary methods to obtain the polynomials H m (N).
We will see that D n (M K +r ) is a polynomial as a direct consequence of Lemma 1.2.
The proof of Theorem 1.1.We are going to prove Bell's theorem by mathematical induction.The theorem is clearly true for n = 1 since in this case, the number of solutions to the equation A1 • X1 = A1 • K + r ' is given by the polynomials (1.2) Let n ≥ 1 be a given, and assume Theorem 1.1 holds for n − 1; we will prove it is also true for n.
The equation corresponding to n is From the inductive hypothesis, we know the polynomials D n−1 (MK +r ) describing the number of solutions to where M is the least common multiple of A(n − 1), A(n − 2),...,A1.
We can write where 0 ≤ b < An and 0 ≤ a < α.
Letting the variable Xn run through all possible values of n ∈ {0, 1, 2,...,α(K +c)+ a}, we obtain (1.5) In order to directly use the induction hypothesis, we need to express each of the terms Ann + b or the form MK + r , for suitable K and r .
For that purpose, consider the set partition 0, 1,...,α(K + c) + a Letting β = M /M, we have An • α = Mβ and, by (1.5), we obtain ) is, by induction hypothesis, a polynomial in S of degree n − 2 or the zero polynomial.The proof of Theorem 1.1 now follows directly from Lemma 1.2.
Note that (1.7) yields a recursive method to obtain D n from previous D n−1 ,D n−2 ,...,D 1 , which we will demonstrate in the next section.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.