Hamiltonian spectral invariants, symplectic spinors and Frobenius structures II

In this article, we continue our study of 'Frobenius structures' and symplectic spectral invariants in the context of symplectic spinors. By studying the case of $C^1$-small Hamiltonian mappings on symplectic manifolds $M$ admitting a metaplectic structure and a parallel $\hat O(n)$-reduction of its metaplectic frame bundle we derive how the construction of 'singularly rigid' resp. 'self-dual' pairs of irreducible Frobenius structures associated to this Hamiltonian mapping $\Phi$ leads to a Hopf-algebra-type structure on the set of irreducible Frobenius structures. We then generalize this construction and define abstractly conditions under which 'dual pairs' associated to a given $C^1$-small Hamiltonian mapping emerge, these dual pairs are esssentially pairs $(s_1, J_1), (s_2, J_2)$ of closed sections of the cotangent bundle $T^*M$ and (in general singular) comptaible almost complex structures on $M$ satisfying certain integrability conditions involving a Koszul bracket. In the second part of this paper, we translate these characterizing conditions for general 'dual pairs' of Frobenius structures associated to a $C^1$-small Hamiltonian system into the notion of matrix factorization. We propose an algebraic setting involving modules over certain fractional ideals of function rings on $M$ so that the set of 'dual pairs' in the above sense and the set of matrix factorizations associated to these modules stand in bijective relation. We prove, in the real-analytic case, a Riemann Roch-type theorem relating a certain Euler characteristic arising from a given matrix factorization in the above sense to (integral) cohomological data on $M$ using Cheeger-Simons-type differential characters, derived from a given pair $(s_1, J_1), (s_2, J_2)$. We propose extensions of these techniques to the case of 'geodesic convexity-smallness' of $\Phi$ and to the case of general Hamiltonian systems on $M$.Comment: 24 pages, minor corrections in statement and proof of Theorem 1.11, this paper builds in content, notation and referencing on arXiv:1411.423

Applications of finite geometry in coding theory and cryptography

We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory

Reasoning About Pragmatics with Neural Listeners and Speakers

We present a model for pragmatically describing scenes, in which contrastive behavior results from a combination of inference-driven pragmatics and learned semantics. Like previous learned approaches to language generation, our model uses a simple feature-driven architecture (here a pair of neural "listener" and "speaker" models) to ground language in the world. Like inference-driven approaches to pragmatics, our model actively reasons about listener behavior when selecting utterances. For training, our approach requires only ordinary captions, annotated _without_ demonstration of the pragmatic behavior the model ultimately exhibits. In human evaluations on a referring expression game, our approach succeeds 81% of the time, compared to a 69% success rate using existing techniques

On the non-minimality of the largest weight codewords in the binary Reed-Muller codes

The study of minimal codewords in linear codes was motivated by Massey who described how minimal codewords of a linear code define access structures for secret sharing schemes. As a consequence of his article, Borissov, Manev, and Nikova initiated the study of minimal codewords in the binary Reed-Muller codes. They counted the number of non-minimal codewords of weight 2d in the binary Reed-Muller codes RM(r, in), and also gave results on the non-minimality of codewords of large weight in the binary Reed-Muller codes RM(r, in). The results of Borissov, Manev, and Nikova regarding the counting of the number of non-minimal codewords of small weight in RM(r,m) were improved by Schillewaert, Storme, and Thas who counted the number of non-minimal codewords of weight smaller than 3d in RM(r,m). This article now presents new results on the non-minimality of large weight codewords in RM(r, m)