We prove that if an nth degree rational Bézier curve has a singular point, then it belongs to the two (n− 1)th degree rational Bézier curves defined in the (n− 1)th step of the de Casteljau algorithm. Moreover, both curves are tangent at the singular point. A procedure to construct Bézier curves with singularities of any order is given.  2001 Elsevier Science B.V. All rights reserved.

For binary polynomial optimization problems (POPs) of degree d with n variables, we prove that the ⌈(n+ d− 1)/2⌉th semidefinite (SDP) relaxation in Lasserre’s hierarchy of the SDP relaxations provides the exact optimal value. If binary POPs involve only even-degree monomials, we show that it can be further reduced to ⌈(n + d − 2)/2⌉. This bound on the relaxation order coincides with the conject...

Given a sequence d→=(d1,…,dk) of natural numbers, we consider the Lie subalgebra h gl(d,F), where d=d1+⋯+dk and F is field characteristic 0, generated by two block upper triangular matrices D E partitioned according to d→, study problem computing nilpotency degree m nilradical n h. We obtain complete answer when belong certain family that arises naturally attempting classify indecomposable modu...

We prove a criteria for nilpotency of left braces in terms the $\star$-central series and also discuss Noetherian braces, obtaining some their elementary properties. show that if finitely generated brace $A$ is Smoktunowicz-nilpotent, then additive multiplicative groups are likewise generated.