Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Homotopies exploiting Newton polytopes for solving sparse polynomial systems

Abstract: This paper is concerned with the problem of finding all isolated solutions of a polynomial system The BKK bound, defined as the mixed volume of the Newton polytopes of the polynomials in the system, is a sharp upper bound for the number of isolated solutions in $\mathbb{C}_0^n ,\mathbb{C}_0 = \mathbb{C} \backslash \{ 0\} $, of a polynomial system with a sparse monomial structure First an algorithm is described for computing the BKK bound Following the lines of Bernshtei˘n’s proof, the algorithmic construction of the cheater’s homotopy or the coefficient homotopy is obtained The mixed homotopy methods can be combined with the random product start systems based on a generalized Bezout number Applications illustrate the effectiveness of the new approach

Methods and systems for object identification and interaction

Abstract: A method of facilitating an interaction between a user and a product item, the product item having an identity and the method including a number of steps. The first step involves providing the user with an interface surface associated with the product item and containing information relating to the product item. The interface surface includes coded data indicative of the identity of the product item and of a number of reference points of the interface surface. The next step involves receiving, in a computer system, indicating data from a sensing device regarding the identity of the product item and a position of the sensing device relative to the interface surface. The sensing device, when placed in an operative position relative to the interface surface, sensing the coded data in the vicinity of the sensing device and generating the indicating data using the sensed coded data. The next step involves facilitating, in the computer system and with reference to the indicating data, the interaction between the user and the product item.

Uniqueness of enhancement for triangulated categories

Abstract: The paper contains general results on the uniqueness of a DG enhancement for trian- gulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded de- rived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de- rived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.