For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if-and-only-if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characte...

Variations to S-box and MixColumn transformations of Rijndael are discussed in this paper. These variations are designed over the Galois field GF(2) generated by the primitive polynomial x + x + x + x + 1. The proposed S-box is constructed by computing the powers of a primitive root in F257 and taking the multiplicative inverse of the resulting entries as in Rijndael’s algorithm. The new MixCol...

We consider some inverse min-max (or max-min) network problems. Such an inverse problem is to modify the weights with bound constraints so that a given feasible solution becomes an optimal solution of a min-max (or max-min) network problem, and the deviation of the weights, measured by the weighted l1 norm or weighted l∞ norm, is minimum. In this paper, we present strongly polynomial time algor...

We obtain inequalities involving many topological indices in classical graph products by using the f-polynomial. In particular, we work with lexicographic product, Cartesian sum and first Zagreb, forgotten, inverse degree lordeg indices.

We study the asymptotic behaviour of a finitely presented Rees quotient S = Inv〈A | ci = 0 (i = 1, . . . , k) 〉 of a free inverse semigroup over a finite alphabet A. It is shown that if the semigroup S has polynomial growth then S is monogenic (with zero) or k ≥ 3. The three relator case is fully characterised, yielding a sequence of two-generated three-relator semigroups whose Gelfand-Kirillov...

In this paper an inverse problem of the weighted shortest arborescence problem is discussed. That is. given a directed graph G and a set of nonnegative costs on its arcs. we need to modify those costs as little as possible to ensure that T, a given (.I-arborescence of G, is the shortest one. It is found that only the cost of T needs modifying. An O(rz”) combinatorial algorithm is then proposed....

We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simp...