A cascade model is a rule induction methodology using levelwise expansion of an itemset lattice, where the explanatory power of a rule set and its constituent rules are quantitatively expressed. The sum of squares for a categorical variable has been decomposed to within-group and between-group sum of squares, where the latter provides a good representation of the power concept in a cascade mode...

In this paper, we investigate property testing whether or not a degree d multivariate polynomial is a sum of squares or is far from a sum of squares. We show that if we require that the property tester always accepts YES instances and uses random samples, nΩ(d) samples are required, which is not much fewer than it would take to completely determine the polynomial. To prove this lower bound, we ...

We present a faster interior-point method for optimizing sum-of-squares (SOS) polynomials, which are central tool in polynomial optimization and capture convex programming the Lasserre hierarchy. Let $$p = \sum _i q^2_i$$ be an n-variate SOS of degree 2d. Denoting by $$L:= \left( {\begin{array}{c}n+d\\ d\end{array}}\right) $$ $$U:= {\begin{array}{c}n+2d\\ 2d\end{array}}\right) dimensions vector...

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...

This paper proposes predictive inference for the multiple regression model with independent normal errors. The distributions of the sample regression vector (SRV) and the residual sum of squares (RSS) for the model are derived by using invariant differentials. Also the predictive distributions of the future regression vector (FRV) and the future residual sum of squares (FRSS) for the future reg...

Hilbert showed that for most (n, m) there exist positive semidefinite forms p(x1, . . . , xn) of degree m which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form h so that hp is a sum of squares of forms; that is, p is a sum of squares of rational functions with denominator h. We show that, for every such (n, m) there does not exis...

It is shown that the set of prime integers in Q( √ 2) is partitioned into two sets with respect to their representation as a sum of squares: 1) a set S0 of primes that cannot be represented as a sum of squares; 2) a set S2 of primes that can be represented as a sum of two squares. Moreover, we give an effective, polynomial-time Euclidean Algorithm for the ring of integers of the cyclotomic fiel...

Let φ denote Euler’s totient function. The frequency with which φ(n) is a perfect square has been investigated by Banks, Friedlander, Pomerance, and Shparlinski, while the frequency with which φ(n) is a sum of two squares has been studied by Banks, Luca, Saidak, and Shparlinski. Here we look at the corresponding threesquares question. We show that φ(n) is a sum of three squares precisely sevene...

Jacobi’s four squares theorem asserts that the number of representations of a positive integer n as a sum of four squares is 8 times the sum of the positive divisors of n, which are not multiples of 4. A formula expressing an infinite product as an infinite sum is called a product-to-sum identity. The product-to-sum identities in a single complex variable q from which Jacobi’s four squares form...