That is, maps from A to B∗ are given by integral kernels in (A⊗B)∗. However, the validity of this adjunction depends on existence of a genuine tensor product. We recall in an appendix the demonstration that infinitedimensional Hilbert spaces do not have tensor products. Also, we must specify the topology on the duals B∗ and (A⊗B)∗. The strongest conclusion gives these the strong topology, as co...

We study F-theory duals of six dimensional heterotic vacua in extreme regions of moduli space where the heterotic string is very strongly coupled. We demonstrate how to use orientifold limits of these F-theory duals to regain a perturbative string description. As an example, we reproduce the spectrum of a T /ZZ4 orientifold as an F-theory vacuum with a singular K3 fibration. We relate this vacu...

Certain null singularities in ten dimensional supergravity have natural holographic duals in terms of Matrix Theory and generalizations of the AdS/CFT correspondence. In many situations the holographic duals appear to be well defined in regions where the supergravity develops singularities. We describe some recent progress in this area. Based on talks given at CTP Symposium, Cairo, Egypt (March...

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray category data. These can be viewed as a generalisation of ribbon diagrams. The Gray categories present two types of duals, which are extended to functor...

Abstract. A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable such as population or GDP. In recent years several algorithms for the automated construction of rectangular cartograms have been proposed, some of which are based on rectangular duals of the dual graph of the input map. ...

It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interior-point methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a s...

In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable r order Σ∆ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual correspo...

Using the integrable spin chain picture we study the one-loop anomalous dimension of certain single trace scalar operators of N = 4 SYM expected to correspond to semi-classical string states on AdS 5 × S 5 with three large angular momenta (J 1 , J 2 , J 3) on S 5. In particular, we investigate the analyticity structure encoded in the Bethe equations for various distributions of Bethe roots. In ...

Approximately dual frames are studied in the Hilbert space setting. Approximate duals are easier to construct than classical dual frames, and can be tailored to yield almost perfect reconstruction. Bounds on the deviation from perfect reconstruction are obtained for approximately dual frames constructed via perturbation theory. An alternative bound is derived for the rich class of Gabor frames,...

If the integer translates of a function φ with compact support generate a frame for a subspace W of L 2 (R), then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis belonging to W. We demonstrate that considerable freedom can be obtained by considering oblique duals, i.e., duals not necessarily belonging to W. For example, we present a condition for the existenc...