Broken circuit complexes and hyperplane arrangements

We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.Comment: 21 page

Positivity of hexagon perturbation theory

The hexagon-form-factor program was proposed as a way to compute three- and higher-point correlation functions in $\mathcal{N}=4$ super-symmetric Yang-Mills theory and in the dual AdS$_5\times$S$^5$ superstring theory, by exploiting the integrability of the theory in the 't Hooft limit. This approach is reminiscent of the asymptotic Bethe ansatz in that it applies to a large-volume expansion. Finite-volume corrections can be incorporated through L\"uscher-like formulae, though the systematics of this expansion is largely unexplored so far. Strikingly, finite-volume corrections may feature negative powers of the 't Hooft coupling $g$ in the small-$g$ expansion, potentially leading to a breakdown of the formalism. In this work we show that the finite-volume perturbation theory for the hexagon is positive and thereby compatible with the weak-coupling expansion for arbitrary $n$-point functions.Comment: v2: misprints corrected, further details on physical magnons adde

Theorems of Carath\'{e}odory, Minkowski-Weyl, and Gordan up to symmetry

In this paper we extend three classical and fundamental results in polyhedral geometry, namely, Carath\'{e}odory's theorem, the Minkowski-Weyl theorem, and Gordan's lemma to infinite dimensional spaces, in which considered cones and monoids are invariant under actions of symmetric groups.Comment: 19 page

Equivariant lattice bases

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in $\mathbb{Z}^{(\mathbb{N}\times[c])}$, where $c\in\mathbb{N}$, has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature.Comment: 31 page

A profile-driven dynamic risk assessment framework for connected and autonomous vehicles

The Internet of Things has already demonstrated clear benefits when applied in many areas. In connected and autonomous vehicles (CAV), IoT data can help the autonomous systems make better decisions for safer and more secure transportation. For example, different IoT data sources can extend CAV's risk awareness, while the incoming data can update these risks in real-time for faster reactions that may mitigate possible damages. However, the current state of the art CAV research has not addressed this matter well enough. This paper proposes a profile-driven approach to manage IoT data in the context of CAV systems through a dynamic risk management framework. Unlike the current inflexible risk assessment strategies, the framework encourages more flexible investigation of risks through different risk profiles, each representing risk knowledge through a set of risk input considerations, assessment methods and optimal reaction strategies. As the risks change frequently with time and location, there will be no single profile that can cover all the risks that CAVs face on the road. The uses of different risk profiles, therefore can help interested parties to better understand the risks and adapt to various situations appropriately. Our framework includes the effective management of IoT data sources to enable the run-time risk assessment. We also describe a case study of using the proposed framework to manage the risks for the POD being developed in the Innovate UK-funded CAPRI project