Abstract The graphical description of morphisms in rigid monoidal categories, in particular in ribbon categories, is summarized. It is illustrated with various examples of algebraic structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras, and modules and bimodules. Nakayama automorphisms of Frobenius algebras are introduced; they are inner iff the algebra is symmetric.

We study a new type of higher categorical structure, called weakly globular n-fold category, previously introduced by the author. We show that this structure is a model of weak ncategories by proving that it is suitably equivalent to the TamsamaniSimpson model. We also introduce groupoidal weakly globular nfold categories and show that they are algebraic models of n-types.

Bidirectional model transformation plays an important role in maintaining consistency between two models, and has many potential applications in software development, including model synchronization, round-trip engineering, software evolution, multiple-view software development, and reverse engineering. However, unclear bidirectional semantics, weak bidirectionalization method, and lack of syst...

Homotopy 3-types can be modelled algebraically by Tamsamani’s weak 3-groupoids as well as, in the path connected case, by cat-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two different semistrict algebraic models of connected 3-types using Tamsamani’s model. Both are then related to Gray groupoids.

We observe that the function introduced by Z. Tu and Y. Deng in the ePrint Archive paper 2009/272 is weak against fast algebraic attacks. We propose an alternative function sharing all the properties of the Tu-Deng function but having not this weakness.

The objective of this paper is to introduce the fundamental algebro-geometric constructions over the extended tropical semi-ring. The study of tropical varieties, co-varieties and ideals over this extension eventually yields the theorem of the weak tropical Nullstellensatz and gives an algebraic interpretation of the tropical Nullstellensatz.

The aim of this note is to show that weak relative hyperbolicity of a group relative to a subgroup (or relative hyperbolicity in the sense of Farb) does not imply any natural analogues of some well-known algebraic properties of ordinary hyperbolic groups. Our main tools are combination theorems for weakly relatively hyperbolic groups.

We study Brauer–Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields. A technique for constructing Azumaya algebra representatives of Brauer group elements is given, and this is applied to the computation of obstructions.

A stabilized weak formulation of the radiative transfer equation is presented, which is stable independent of the scattering coeecient. This enables the use of standard nite element discretizations without further algebraic constraints. Furthermore, a weighted residual-based a posteriori error estimate is derived for the discrete solution. An example demonstrates the eeciency of the new method.

Let C(X) be the ring of real continuous functions on a completely regular Hausdorff space. In this paper an almost discrete space is determined by the algebraic structure of C(X). The intersection of essential weak ideal in C(X) is also studied.