We demonstrate existence of optimal controls with respect to a quadratic cost criterion for the equation u t ? u = F(u; ') | rst for F aanely dependent on the control ' and then for a more fully nonlinear setting in which weak convergence for ' no longer suuces to give consistency in the limit.

multi objective quadratic fractional programming (moqfp) problem involves optimization of several objective functions in the form of a ratio of numerator and denominator functions which involve both contains linear and quadratic forms with the assumption that the set of feasible solutions is a convex polyhedral with a nite number of extreme points and the denominator part of each of the object...

Quadratic programming can be seen both as a general approach to linear programming and a special class of non-linear programming. Moreover, quadratic programming problems are of utmost importance in an increasing variety of practical fields, such as, regression, efficient production and portfolio selection. As ambiguity and vagueness are natural and ever-present in real-life situations requirin...

We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by 2s for orientable equation and by 12s for non-orientable equation, where s is the sum of the lengths of the coefficients.

We reformulate the dual gauge model of the long-distance Yang-Mills theory in terms of the two-point Wightman functions formalism. In the flux-tube scheme of Abelian dominance and monopole condensation, the analytic expressions of both monopole-and dual gauge boson-fields propagators are obtained. Formally, new features of higher order quadratic equations for a monopole field are given. Finally...

Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds.

The type II1 unprojection is, by definition, the generic complete intersection type II unprojection, in the sense of [P] Section 3.1, for the parameter value k = 1, and depends on a parameter n ≥ 2. Our main results are the explicit calculation of the linear relations of the type II1 unprojection for any value n ≥ 2 (Theorem 3.16) and the explicit calculation of the quadratic equation for the c...

We investigate the class of regular-ordered word equations. The sides of such an equation contain both exactly the same variables, occurring in the same order (but separated by potentially distinct constant factors). Surprisingly, we obtain that solving even such simple equations is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general qua...