Thin fillers in the cubical nerves of omega-categories

It is shown that the cubical nerve of a strict omega-category is a sequence of sets with cubical face operations and distinguished subclasses of thin elements satisfying certain thin filler conditions. It is also shown that a sequence of this type is the cubical nerve of a strict omega-category unique up to isomorphism; the cubical nerve functor is therefore an equivalence of categories. The sequences of sets involved are in effect the analogues of cubical T-complexes appropriate for strict omega-categories. Degeneracies are not required in the definition of these sequences, but can in fact be constructed as thin fillers. The proof of the thin filler conditions uses chain complexes and chain homotopies.Comment: Revised version to appear in Theory and Applications of Categories; changed terminology; additional figures, examples and references; 27 page

The algebraic structure of the universal complicial sets

The nerve of a strict omega-category is a simplicial set with additional structure, making it into a so-called complicial set, and strict omega-categories are in fact equivalent to complicial sets. The nerve functor is represented by a sequence of strict omega-categories, called orientals, which are associated to simplexes. In this paper we give a detailed algebraic description of the morphisms between orientals. The aim is to describe complicial sets algebraically, by operators and equational axioms.Comment: 25 pages. As to appear in Journal of Pure and Applied Algebra. Minor corrections, addtional explanations, some propositions reclassified as lemma

Omega-categories and chain complexes

There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented directed complexes to omega-categories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases. The omega-categories equivalent to augmented directed complexes with good bases include the omega-categories associated to globes, simplexes and cubes; thus the morphisms between these omega-categories are determined by morphisms between chain complexes. It follows that the entire theory of omega-categories can be expressed in terms of chain complexes; in particular we describe the biclosed monoidal structure on omega-categories and calculate some internal homomorphism objects.Comment: 18 pages; as published, with minor changes from version

Opetopes and chain complexes

We give a simple algebraic description of opetopes in terms of chain complexes, and we show how this description is related to combinatorial descriptions in terms of treelike structures. More generally, we show that the chain complexes associated to higher categories generate graphlike structures. The algebraic description gives a relationship between the opetopic approach and other approaches to higher category theory. It also gives an easy way to calculate the sources and targets of opetopes.Comment: 20 pages. Revised version has an extra figur

Simple omega-categories and chain complexes

The category of strict omega-categories has an important full subcategory whose objects are the simple omega-categories freely generated by planar trees or by globular cardinals. We give a simple description of this subcategory in terms of chain complexes, and we give a similar description of the opposite category, the category of finite discs, in terms of cochain complexes. Berger has shown that the category of simple omega-categories has a filtration by iterated wreath products of the simplex category. We generalise his result by considering wreath products of categories of chain complexes over the simplex category.Comment: 14 pages; v2 has minor corrections and a little additional materia

Some Measured Characteristics of Severe Storm Turbulence

Measurements of atmospheric turbulence obtained from airplane flights through severe storms in connection with the National Severe Storms Project will be discussed. Various characteristics of turbulence, such as differences in intensity between storms and the turbulence intensity with altitude and time will be indicated. These measurements for severe storm conditions will also be compared with other measurements for clear-air and non-storm weather conditions as a means of illustrating the relative severity of turbulence for various flight conditions. For these purposes, both derived gust velocities and power spectra of atmospheric turbulence will be used. The detailed nature of the vertical and horizontal flow patterns and the variations in atmospheric pressure as measured during several airplane traverses through storm centers will also be discussed

Graphene-Based Nanomaterials in the Design of Nerve Conduits for Regenerative Medicine Applications

Peripheral neuropathies are a debilitating problem in human and animal patients resulting in a diminished quality of life. The current gold standard methods for repair of critical size peripheral neuropathies have limitations that overall diminish the quality of life for patients. The use of nerve scaffolds composed of synthetic polymer-based materials to heal damaged nerves has become an attractive approach in regenerative medicine research. Studies have shown the biomaterial characteristics of graphene oxide to have potential in applications for regenerating damaged peripheral nerves. Studies have also shown that incorporating Mesenchymal Stem Cell (MSC) therapies into neural scaffold designs can significantly improve the quality of tissue healing as well. The hypothesis of this study is that a novel synthetic thin film composed of electro spun polycaprolactone (PCL) and modified with surface coating of Graphene Oxide (GO) and cultures of Human Mesenchymal Stem Cells (hMSC) will have the potential to regenerate a critical size peripheral nerve defect. The first objective studied the potential cytotoxic effect of graphene surfaces with different oxidative group saturation levels to adipose derived Rat MSC cultures. This objective also manufactured PCL materials of fibrous and smooth surface topographies using both electrospinning and polymer-drop techniques. The second objective assessed the in-vitro capabilities of High Oxygen Graphene (hGO) and GO surface modifications of both fibrous and smooth surface PCL material templates seeded with adipose derived hMSCs for materials effectiveness in supporting and guiding trans-differentiation of hMSC into a Schwann like cell lineage. The final objective involved the development of an approved critical nerve defect model in Rats to assess the in-vivo performance of electro-spun PCL films with GO surface modification and hMSC platform to stimulate nerve regeneration at a critical nerve defect. The degree of nerve regeneration was determined by exogenous detection of gait patterns in the rats during nerve repair and tissue identification/ measurements thru Histology sections. This study to date has shown that neural wraps composed of electro-spun PCL surface coated with GO can support the hMSC in both static and trans-differentiated forms and can stimulate nerve regeneration in a critical nerve defect rodent model