Positive Geometries and Differential Forms with Non-Logarithmic Singularities I

Positive geometries encode the physics of scattering amplitudes in flat space-time and the wavefunction of the universe in cosmology for a large class of models. Their unique canonical forms, providing such quantum mechanical observables, are characterised by having only logarithmic singularities along all the boundaries of the positive geometry. However, physical observables have logarithmic singularities just for a subset of theories. Thus, it becomes crucial to understand whether a similar paradigm can underlie their structure in more general cases. In this paper we start a systematic investigation of a geometric-combinatorial characterisation of differential forms with non-logarithmic singularities, focusing on projective polytopes and related meromorphic forms with multiple poles. We introduce the notions of covariant forms and covariant pairings. Covariant forms have poles only along the boundaries of the given polytope; moreover, their leading Laurent coefficients along any of the boundaries are still covariant forms on the specific boundary. Whereas meromorphic forms in covariant pairing with a polytope are associated to a specific (signed) triangulation, in which poles on spurious boundaries do not cancel completely, but their order is lowered. These meromorphic forms can be fully characterised if the polytope they are associated to is viewed as the restriction of a higher dimensional one onto a hyperplane. The canonical form of the latter can be mapped into a covariant form or a form in covariant pairing via a covariant restriction. We show how the geometry of the higher dimensional polytope determines the structure of these differential forms. Finally, we discuss how these notions are related to Jeffrey-Kirwan residues and cosmological polytopes.Comment: 47 pages, figures in Tik

Type Annotation for Adaptive Systems

We introduce type annotations as a flexible typing mechanism for graph systems and discuss their advantages with respect to classical typing based on graph morphisms. In this approach the type system is incorporated with the graph and elements can adapt to changes in context by changing their type annotations. We discuss some case studies in which this mechanism is relevant.Comment: In Proceedings GaM 2016, arXiv:1612.0105

Conditions, constraints and contracts: on the use of annotations for policy modeling.

Organisational policies express constraints on generation and processing of resources. However, application domains rely on transformation processes, which are in principle orthogonal to policy specifications and domain rules and policies may evolve in a non-synchronised way. In previous papers, we have proposed annotations as a flexible way to model aspects of some policy, and showed how they could be used to impose constraints on domain configurations, how to derive application conditions on transformations, and how to annotate complex patterns. We extend the approach by: allowing domain model elements to be annotated with collections of elements, which can be collectively applied to individual resources or collections thereof; proposing an original construction to solve the problem of annotations remaining orphan , when annotated resources are consumed; introducing a notion of contract, by which a policy imposes additional pre-conditions and post-conditions on rules for deriving new resources. We discuss a concrete case study of linguistic resources, annotated with information on the licenses under which they can be used. The annotation framework allows forms of reasoning such as identifying conflicts among licenses, enforcing the presence of licenses, or ruling out some modifications of a licence configuration

Ictal epileptic headache. an old story with courses and appeals

The term "ictal epileptic headache" has been recently proposed to classify the clinical picture in which headache is the isolated ictal symptom of a seizure. There is emerging evidence from both basic and clinical neurosciences that cortical spreading depression and an epileptic focus may facilitate each other, although with a different degree of efficiency. This review address the long history which lead to the 'migralepsy' concept to the new emerging pathophysiological aspects, and clinical and electroencephalography evidences of ictal epileptic headache. Here, we review and discuss the common physiopathology mechanisms and the historical aspects underlying the link between headache and epilepsy. Either experimental or clinical measures are required to better understand this latter relationship: the development of animal models, molecular studies defining more precise genotype/phenotype correlations as well as multicenter clinical studies with revision of clinical criteria for headache-/epilepsy-related disorders represent the start of future research. Therefore, the definition of ictal epileptic headache should be used to classify the rare events in which headache is the only manifestation of a seizure. Finally, using our recently published criteria, we will be able to clarify if ictal epileptic headache represents an underestimated phenomenon or not

Cross correlations of the American baby names

The quantitative description of cultural evolution is a challenging task. The most difficult part of the problem is probably to find the appropriate measurable quantities that can make more quantitative such evasive concepts as, for example, dynamics of cultural movements, behavior patterns and traditions of the people. A strategy to tackle this issue is to observe particular features of human activities, i.e. cultural traits, such as names given to newborns. We study the names of babies born in the United States of America from 1910 to 2012. Our analysis shows that groups of different correlated states naturally emerge in different epochs, and we are able to follow and decrypt their evolution. While these groups of states are stable across many decades, a sudden reorganization occurs in the last part of the twentieth century. We think that this kind of quantitative analysis can be possibly extended to other cultural traits: although databases covering more than one century (as the one we used) are rare, the cultural evolution on shorter time scales can be studied thanks to the fact that many human activities are usually recorded in the present digital era.Comment: submitted for consideration to PNA

Profit Shifting by Debt Financing in Europe

This article aims at analyzing the link between subsidiaries’ capital structure and taxation in Europe. First we introduce a trade-off model, which studies a MNCs’ financial strategy and shows how debt policy allows multinational groups to shift profits from low-tax to high-tax jurisdictions. By letting the MNC choose both leverage and the percentage of profit shifting, we depart from the relevant literature which has mainly focused on the latter. Using the AMADEUS dataset we show that: i) subsidiaries’ leverage increases with the statutory tax rate, levied in the country where it operates; ii) this positive effect is lower, the higher the parent company tax rate is. Furthermore, an increase in the parent company’s tax rate is estimated to raise its subsidiaries’ leverage.capital structure, default, multinationals, profit shifting, taxation

Scaling and asymptotic scaling in two-dimensional CPN−1CP^{N-1} models

Two-dimensional $CP^{N-1}$ models are investigated by Monte Carlo methods on the lattice, for values of $N$ ranging from 2 to 21. Scaling and rotation invariance are studied by comparing different definitions of correlation length $\xi$. Several lattice formulations are compared and shown to enjoy scaling for $\xi$ as small as $2.5$. Asymptotic scaling is investigated using as bare coupling constant both the usual $\beta$ and $\beta_E$ (related to the internal energy); the latter is shown to improve asymptotic scaling properties. Studies of finite size effects show their $N$-dependence to be highly non-trivial, due to the increasing radius of the $\bar z z$ bound states at large $N$.Comment: 5 pages + 12 figures (PostScript), report no. IFUP-TH 46/9

Mosaic multi-state scenario vs. one-state description of supercooled liquids

According to the mosaic scenario, relaxation in supercooled liquids is ruled by two competing mechanisms: surface tension, opposing the creation of local excitations, and entropy, providing the drive to the configurational rearrangement of a given region. We test this scenario through numerical simulations well below the Mode Coupling temperature. For an equilibrated configuration, we freeze all the particles outside a sphere and study the thermodynamics of this sphere. The frozen environment acts as a pinning field. Measuring the overlap between the unpinned and pinned equilibrium configurations of the sphere, we can see whether it has switched to a different state. We do not find any clear evidence of the mosaic scenario. Rather, our results seem compatible with the existence of a single (liquid) state. However, we find evidence of a growing static correlation length, apparently unrelated to the mosaic one.Comment: 4 pages, 3 figures, final version accepted in PR

A Termination Criterion for Graph Transformations with Negative Application Conditions

Termination of graph transformations is in general undecidable, but it is possible to prove it for specific systems by checking for sufficient conditions. In the presence of rules with negative application conditions, the difficulties increase. In this paper we propose a different approach to the identification of a (sufficient) criterion for termination, based on the construction of a labelled transition system whose states represent overlaps between the negative application condition and the right hand side that can give rise to cycles