The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform

This note reviews complex and real techniques in harmonic analysis. We describe a common source of both approaches rooted in the covariant transform generated by the affine group. Keywords: wavelet, coherent state, covariant transform, reconstruction formula, the affine group, ax+b-group, square integrable representations, admissible vectors, Hardy space, fiducial operator, approximation of the identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal function, vertical maximal functions, non-tangential maximal functions, intertwining operator, Cauchy-Riemann operator, Laplace operator, singular integral operator, SIO, boundary behaviour, Carleson measure.Comment: 31 pages, AMS-LaTeX, no figures; v2: a major revision, sections on representations of the ax+b group and transported norms are added; v3: major revision: an outline section on complex and real variables techniques are added, numerous smaller improvements; v4: minor correction

Anonymous and Adaptively Secure Revocable IBE with Constant Size Public Parameters

In Identity-Based Encryption (IBE) systems, key revocation is non-trivial. This is because a user's identity is itself a public key. Moreover, the private key corresponding to the identity needs to be obtained from a trusted key authority through an authenticated and secrecy protected channel. So far, there exist only a very small number of revocable IBE (RIBE) schemes that support non-interactive key revocation, in the sense that the user is not required to interact with the key authority or some kind of trusted hardware to renew her private key without changing her public key (or identity). These schemes are either proven to be only selectively secure or have public parameters which grow linearly in a given security parameter. In this paper, we present two constructions of non-interactive RIBE that satisfy all the following three attractive properties: (i) proven to be adaptively secure under the Symmetric External Diffie-Hellman (SXDH) and the Decisional Linear (DLIN) assumptions; (ii) have constant-size public parameters; and (iii) preserve the anonymity of ciphertexts---a property that has not yet been achieved in all the current schemes

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding "admissible faces". Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in Journal of Combinatorial Theory A

Heegaard Floer homology as morphism spaces

In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukaya-categorical constructions. The result also leads to various dualities in bordered Floer homology.Comment: 57 pages, 14 figures; v2: many updates, including changing orientation conventions, which changed the signs in many theorem