3 Polynomial Optimization, Sums of Squares, and Applications 47 Pablo A. Parrilo 3.1 Nonnegative Polynomials and Sums of Squares . . . . . . . . . . 48 3.2 Applications of Sum of Squares Programs . . . . . . . . . . . . . 76 3.3 Special Cases and Structure Exploitation . . . . . . . . . . . . . 86 3.4 Infeasibility Certificates . . . . . . . . . . . . . . . . . . . . . . . 106 3.5 Duality and S...

For non-negative integers a,b, and n, let N(a,b;n) be the number of representations n as a sum squares with coefficients 1 or 3 (a ones b threes). Let N⁎(a,b;n) odd We have that N⁎(a,b;8n+a+3b) is triangular numbers It known for satisfying 1≤a+3b≤7, we haveN⁎(a,b;8n+a+3b)=22+(a4)+abN(a,b;8n+a+3b) a+3b=8, haveN⁎(a,b;8n+a+3b)=22+(a4)+ab(N(a,b;8n+a+3b)−N(a,b;(8n+a+3b)/4)). Such identities are not ...

1.8. Investigate algorithms for expressing an integer as a sum of squares. We have determined exactly which integers are sums of two squares. Later on in the course we will prove which integers are sums of four squares (Lagrange’s theorem) and state without proof which integers are sums of three squares (Legendre-Gauss theorem). But these methods do not give efficient algorithms for actually fi...

1. Overview of the Madelung Problem 2. Analytic Consequences of the Andrews Identity 3. Connection with Multiple Zeta Sums 4. An -Series for M(s) 5. Integral Representations for the Madelung Constant 6. Number-Theoretical Implications: Sums of Three Squares 7. Open Problems Acknowledgments References From a modern theta-function identity of G. E. Andrews we derive new representations for the ce...

Among the invariant factors g of a positive semidefinite analytic function f on R, those g whose zero set Y is a curve are called special. We show that if each special g is a sum of squares of global meromorphic functions on a neighbourhood of Y , then f is a sum of squares of global meromorphic functions. Here sums can be (convergent) infinite, but we also find some sufficient conditions to ge...