New mathematical structures in renormalizable quantum field theories

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.Comment: 26p,4figs., Invited Contribution to Annals of Physics, minor typos correcte

Towards building information modelling for existing structures

The transformation of cities from the industrial age (unsustainable) to the knowledge age (sustainable) is essentially a ‘whole life cycle’ process consisting of; planning, development, operation, reuse and renewal. During this transformation, a multi-disciplinary knowledge base, created from studies and research about the built environment aspects is fundamental: historical, architectural, archeologically, environmental, social, economic, etc is critical. Although there are a growing number of applications of 3D VR modelling applications, some built environment applications such as disaster management, environmental simulations, computer aided architectural design and planning require more sophisticated models beyond 3D graphical visualization such as multifunctional, interoperable, intelligent, and multi-representational. Advanced digital mapping technologies such as 3D laser scanner technologies can be are enablers for effective e-planning, consultation and communication of users’ views during the planning, design, construction and lifecycle process of the built environment. For example, the 3D laser scanner enables digital documentation of buildings, sites and physical objects for reconstruction and restoration. It also facilitates the creation of educational resources within the built environment, as well as the reconstruction of the built environment. These technologies can be used to drive the productivity gains by promoting a free-flow of information between departments, divisions, offices, and sites; and between themselves, their contractors and partners when the data captured via those technologies are processed and modelled into BIM (Building Information Modelling). The use of these technologies is key enablers to the creation of new approaches to the ‘Whole Life Cycle’ process within the built and human environment for the 21st century. The paper describes the research towards Building Information Modelling for existing structures via the point cloud data captured by the 3D laser scanner technology. A case study building is elaborated to demonstrate how to produce 3D CAD models and BIM models of existing structures based on designated technique

Strong forms of self-duality for Hopf monoids in species

A vector species is a functor from the category of finite sets with bijections to vector spaces (over a fixed field); informally, one can view this as a sequence of $S_n$-modules. A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. A vector species has a basis if and only if it is given by a sequence of $S_n$-modules which are permutation representations. We say that a Hopf monoid is freely self-dual if it is connected and finite-dimensional, and if it has a basis in which the structure constants of its product and coproduct coincide. Such Hopf monoids are self-dual in the usual sense, and we show that they are furthermore both commutative and cocommutative. We prove more specific classification theorems for freely self-dual Hopf monoids whose products (respectively, coproducts) are linearized in the sense that they preserve the basis; we call such Hopf monoids strongly self-dual (respectively, linearly self-dual). In particular, we show that every strongly self-dual Hopf monoid has a basis isomorphic to some species of block-labeled set partitions, on which the product acts as the disjoint union. In turn, every linearly self-dual Hopf monoid has a basis isomorphic to the species of maps to a fixed set, on which the coproduct acts as restriction. It follows that every linearly self-dual Hopf monoid is strongly self-dual. Our final results concern connected Hopf monoids which are finite-dimensional, commutative, and cocommutative. We prove that such a Hopf monoid has a basis in which its product and coproduct are both linearized if and only if it is strongly self-dual with respect to a basis equipped with a certain partial order, generalizing the refinement partial order on set partitions.Comment: 42 pages; v2: a few typographical errors corrected and references updated; v3: discussion in Sections 3.1 and 3.2 slightly revised, Theorem A corrected to include hypothesis about ambient field, final versio