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 se...

Having addressed the issue ofgenerating octrees from raster scan efficiently using parallel computation, it is worthwhile to ensure that the encoding is accomplished without much loss of information. Knowing that each cube usually stores an average ofthe intensity value in the cubical subspace, the encoding could lead to a "blocky" (as against ''patchy'') solid during regeneration. Retaining th...

It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists, is uniquely determined by the corresp...

We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, me...

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a refreshing view on this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in R in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. ...

lif, mg and ti cubical tld chips (known as tld-100) are widely used for dosimetry purposes. the repeatability of tl dosimetry is investigated by exposing them to doses of (81, 162 and 40.5 mgy) with 662kev photons of cs-137. a group of 40 cubical tld chips was randomly selected from a batch and the values of element correction coefficient (ecc) were obtained 4 times by irradiating them to doses...

Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky’s univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some...

A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature” etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial “Theorema Egregium”...

Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical convex d-polytope C d whose r-skeleton is combinatorially equivalent to that of the n-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂C d of a neighborly cubical polytope C n d maximizes the f -vector among all cubical (d− 1)-spheres with 2 vertices. While we sh...