This is an exposition of some of the constructions which have arisen in higher-dimensional category theory. We start with a review of the general theory of operads and multicategories (cf. Bur], Her], Lei1]). Using this we give a new deenition of n-category (a variation on Batanin's); we also give an informal dee-nition in pictures. Next we discuss Gray-categories and their place in coherence p...

In recent years, especially in the subject of harmonic analysis, there has been interest in geometric phenomena of R as N → +∞. In the present paper we examine several specific geometric phenomena in Euclidean space and calculate the asymptotics as the dimension gets large. 0 Introduction Typically when we do geometry we concentrate on a specific venue in a particular space. Often the context i...

We introduce higher dimensional hypergraphs, which is a generalization of Baez-Dolans’s opetopic sets and Hermida-Makkai-Power’s mutigraphs. This is based on a simple combinatorial structure called shells and the formal composites of pasting diagrams based on the closure of open shells. We give two types of graphical representation of higher dimensional cells which show effectively the relation...

Higher dimensional automata (HDA) are a model of concurrency that can express most of the traditional partial order models like Mazurkiewicz traces, pomsets, event structures, or Petri nets. Modal logics, interpreted over Kripke structures, are the logics for reasoning about sequential behavior and interleaved concurrency. Modal logic is a well behaved subset of first-order logic; many variants...

We study the cosmology induced on a brane probing a warped throat region in a CalabiYau compactification of type IIB string theory. For the case of a BPS D3-brane probing the Klebanov-Strassler warped deformed conifold, the cosmology described by a suitable brane observer is a bouncing, spatially flat Friedmann-Robertson-Walker universe with timevarying Newton’s constant, which passes smoothly ...

A bstract bouncing scenario of a flat homogeneous and isotropic universe is explored by using the reconstruction technique for power-law parametrization Hubble parameter in modified gravity theory with higher-order curvature trace energy-momentum tensor terms. It demonstrated that criteria are satisfied so cosmological initial singularity can be avoided. In addition, it shown equation state cro...