The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere is covered by d + 1 closed sets, then at least one of the sets must contain a pair of antipodal points. In this paper, we prove a combinatorial version of this theorem for hypercubes. It is not hard to show that for any cover of the facets of a d-cube by d sets of facets, at least one such set contains a pair of antipo...

The aim of this paper is to investigate geometrically nonlinear static analysis of axially functionally graded cantilever beam subjected to transversal non follower load. The considered problem is solved by finite element method with total Lagrangian kinematic approach. The material properties of the beam vary along the longitudinal direction according to the power law function. The finite elem...

Coquand’s cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. We show that the underlying cube category is the opposite of the Lawvere theory of De Morgan algebras. The topos of cubical sets itself classifies the theory of ‘free De Morgan algebras’...

For a given point set S in a plane, we develop a distributed algorithm to compute the α−shape of S. α−shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for the shape of S. We assume that the distances between pairs of points which are closer than a certain distance r > 0 are provided, and we show constructively that this informatio...

Cell aggregates are a tool for in vitro studies of morphogenesis, cancer invasion, and tissue engineering. They respond to mechanical forces as a complex rather than simple liquid. To change an aggregate's shape, cells have to overcome energy barriers. If cell shape fluctuations are active enough, the aggregate spontaneously relaxes stresses ("fluctuation-induced flow"). If not, changing the ag...

Natural convection in cubical enclosure with hot surface geometry and partial partitions has been analyzed. The geometry is a cube with wavy hot surface (three undulations) and three partitions. The investigation has been performed for different partitions lengths and Rayleigh number while the Prandtl number kept constant. This problem is solved by using the partial differential equations which...

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be proc...

Direct numerical simulation of the flow in a lid-driven cubical cavity has been carried out at high Reynolds numbers (based on the maximum velocity on the lid), between 1.2 104 and 2.2 104. An efficient Chebyshev spectral method has been implemented for the solution of the incompressible Navier-Stokes equations in a cubical domain. The resolution used up to 5.0 million Chebyshev collocation nod...

We define a cubical version of categorical group 2-bundles with connection and consider their twodimensional parallel transport with the aim of defining Wilson surface functionals.