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Register a new userThe Business of Height Pairings
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In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining a motivating example situation, we introduce new directions in this subject.
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We study the effect of the lattice structure on the spin-fluctuation mediated superconductivity in the iron pnictides adopting the five-band models of several virtual lattice structures of LaFeAsO as well as actual materials such as NdFeAsO and LaFePO obtained from the maximally-localized Wannier orbitals. Random phase approximation is applied to the models to solve the Eliashberg equation. This reveals that the gap function and the strength of the superconducting instability are determined by the cooperation or competition among multiple spin fluctuation modes arising from several nestings among disconnected pieces of the Fermi surface, which is affected by the lattice structure. Specifically, the appearance of the Fermi surface $gamma$ around $(pi,pi)$ in the unfolded Brillouin zone is sensitive to the pnictogen height $h_{rm Pn}$ measured from the Fe plane, where $h_{rm Pn}$ is shown to act as a switch between high-$T_c$ nodeless and low-$T_c$ nodal pairings. We also find that reduction of the lattice constants generally suppresses superconductivity. We can then combine these to obtain a generic superconducting phase diagram against the pnictogen height and lattice constant. This suggests that NdFeAsO is expected to exhibit a fully-gapped, sign-reversing s-wave superconductivity with a higher $T_c$ than in LaFeAsO, while a nodal pairing with a low $T_c$ is expected for LaFePO, which is consistent with experiments.
It is well-known that both the pathwidth and the outer-planarity of a graph can be used to obtain lower bounds on the height of a planar straight-line drawing of a graph. But both bounds fall short for some graphs. In this paper, we consider two other parameters, the (simple) homotopy height and the (simple) grid-major height. We discuss the relationship between them and to the other parameters, and argue that they give lower bounds on the straight-line drawing height that are never worse than the ones obtained from pathwidth and outer-planarity.
Recommender Systems are nowadays successfully used by all major web sites (from e-commerce to social media) to filter content and make suggestions in a personalized way. Academic research largely focuses on the value of recommenders for consumers, e.g., in terms of reduced information overload. To what extent and in which ways recommender systems create business value is, however, much less clear, and the literature on the topic is scattered. In this research commentary, we review existing publications on field tests of recommender systems and report which business-related performance measures were used in such real-world deployments. We summarize common challenges of measuring the business value in practice and critically discuss the value of algorithmic improvements and offline experiments as commonly done in academic environments. Overall, our review indicates that various open questions remain both regarding the realistic quantification of the business effects of recommenders and the performance assessment of recommendation algorithms in academia.
We give an elementary and self-contained introduction to pairings on elliptic curves over finite fields. For the first time in the literature, the three different definitions of the Weil pairing are stated correctly and proved to be equivalent using Weil reciprocity. Pairings with shorter loops, such as the ate, ate$_i$, R-ate and optimal pairings, together with their twisted variants, are presented with proofs of their bilinearity and non-degeneracy. Finally, we review different types of pairings in a cryptographic context. This article can be seen as an update chapter to A. Enge, Elliptic Curves and Their Applications to Cryptography - An Introduction, Kluwer Academic Publishers 1999.
In contemporary cryptographic systems, secret keys are usually exchanged by means of methods, which suffer from mathematical and technology inherent drawbacks. That could lead to unnoticed complete compromise of cryptographic systems, without a chance of control by its legitimate owners. Therefore a need for innovative solutions exists when truly and reliably secure transmission of secrets is required for dealing with critical data and applications. Quantum Cryptography (QC), in particular Quantum Key Distribution (QKD) can answer that need. The business white paper (BWP) summarizes how secret key establishment and distribution problems can be solved by quantum cryptography. It deals with several considerations related to how the quantum cryptography innovation could contribute to provide business effectiveness. It addresses advantages and also limitations of quantum cryptography, proposes a scenario case study, and invokes standardization related issues. In addition, it answers most frequently asked questions about quantum cryptography.
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