Let$H$^{∞}be the algebra of bounded analytic functions in the unit disk$D$. Let$I=I(f$_{1},...,$f$_{N}) be the ideal generated by$f$_{1},...,$f$_{N}∈$H$^{∞}and$J=J(f$_{1},...,$f$_{N}) the ideal of the functions$f∈H$^{∞}for which there exists a constant$C=C(f)$such that |$f(z)|≤C(|f$_{1}$(z)|+$...;$+|f$_{N}$(z)$|),$z$∈$D$. It is clear that $$I \subseteq J$$ , but an example due to J. Bourgain shows that$J$is not, in general, in the norm closure of$I$. Our first result asserts that$J$is included in the norm closure of$I$if$I$contains a Carleson-Newman Blaschke product, or equivalently, if there exists$s$>0 such that $$\mathop {\inf }\limits_{z \in D} \sum\limits_{k = 0}^s {(1 - |z|)^k } \sum\limits_{j = 1}^N {|f_j^{(k)} (z)| > 0.} $$

String theory contains solutions with SL(2,R)_R x U(1)_L-invariant warped AdS3 (WAdS3) factors arising as continuous deformations of ordinary AdS3 factors. We propose that some of these are holographically dual to the IR limits of nonlocal dipole-deformed 2D D-brane gauge theories, referred to as "dipole CFTs". Neither the bulk nor boundary theories are currently well-understood, and consequences of the proposed duality for both sides is investigated. The bulk entropy-area law suggests that dipole CFTs have (at large N) a high-energy density of states which does not depend on the deformation parameter. Putting the boundary theory on a spatial circle leads to closed timelike curves in the bulk, suggesting a relation of the latter to dipole-type nonlocality.

Narain lattices are unimodular lattices {\it in} $\R^{r, s} $, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2, 2} $ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2, 2} $, a new interpretation of the Gauss product of binary forms brings new light to a number of related objects--rank 4 rational Narain lattices, over-lattices, rank 2 primitive sublattices of an abstract rank 4 even unimodular lattice U^ 2, and isomorphisms of discriminant groups of rank 2 lattices.

In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper-and the lower-level problems are polynomials. We present methods for finding its global minimizers and global minimum using a sequence of semidefinite programming (SDP) relaxations and provide convergence results for the methods. Our scheme for problems with a convex lower-level problem involves solving a transformed equivalent single-level problem by a sequence of SDP relaxations, whereas our approach for general problems involving a nonconvex polynomial lower-level problem solves a sequence of approximation problems via another sequence of SDP relaxations.

We consider a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space,through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.