‎In this paper‎, ‎we study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎The equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎Such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

Using principles of the theory of smoothness spaces we give systematic constructions of scales of inverse-closed subalgebras of a given Banach algebra with the action of a d-parameter automorphism group. In particular we obtain the inverse-closedness of Besov algebras, Bessel potential algebras and approximation algebras of polynomial order in their defining algebra. By a proper choice of the g...

We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit...

It is shown that a polynomial map F = (P, Q) of C with finite fibres is a polynomial automorphism if for all (a : b) ∈ P the polynomials aP + bQ are irreducible and the curves aP + bQ = 0 are rational curves. In consequence, a non-zero constant Jacobian polynomial map F of C is a polynomial automorphism if the inverse images F(l) of complex lines l ⊂ C passing through (0, 0) are irreducible rat...

We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate poly-nomials). In particular, we show that the inverse sumset theorems of Fre˘ ıman type are equivalent to the known inverse results for the Gowers U 3 norms, and mo...

Bit-parallel finite field multiplication in F2m using polynomial basis can be realized in two steps: polynomial multiplication and reduction modulo the irreducible polynomial. In this article, we prove that the modular polynomial reduction can be done with (r − 1)(m − 1) bit additions, where r is the Hamming weight of the irreducible polynomial. We also show that a bit-parallel squaring operati...

Definition 2.1. A commutative ring R is Noetherian if every chain of ideals in R I0 ⊂ I1 ⊂ I2 ⊂ · · · terminates after a finite number of steps (i.e., there is an interger k such that Is = Ik if s ≥ k). Remark 2.2. Polynomial rings R [X1, . . . , Xn] are Noetherian if R is. In particular, S (p) = C [p∗] is Noetherian. Theorem 2.3. If R is a Noetherian ring, and M is a finitely generated R-modul...