The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. Finally, we use this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon. The paper ends wi...

We prove that any hyperplane H in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem (or this direction of his theorem) is a corollary of Gromov’s link condition. We also give new arguments establishing some combinatorial properties of hyperplanes. We sho...

Any diagram group over a +nite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satis+es the Baum–Connes conjecture. Any diagram group over a +nite presentation ...

Cubical type theory is an extension of Martin-Löf type theory recently proposed by Cohen, Coquand, Mörtberg and the author which allows for direct manipulation of n-dimensional cubes and where Voevodsky’s Univalence Axiom is provable. In this paper we prove canonicity for cubical type theory: any natural number in a context build from only name variables is judgmentally equal to a numeral. To a...

the heterogeneous photo-fenton degradation of crystal violet under visible light has been investigated usingcopper modified iron oxide. the photocatalyst has been prepared by coprecipitation method. the rate ofphotocatalyic degradation of dye was monitored spectrophotometrically. it has been observed thatphotocatalytic degradation follows pseudo first order kinetics. the effect of various param...

The zinc(II) complex [Zn2(dmdtc)2(μ-dmdtc)2] has been synthesized directly from thiram ligand, containing a disulfide bond {dmdtc = N,N-dimethyldithiocarbamate; thiram = N,N-tetramethylthiuram disulfide}, and characterized by elemental analysis and spectroscopic methods. Surprisingly thiram, undergoes a reductive disulfide bond scission upon reaction with Zn2+ in methanolic media to give the [Z...

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes weak equivalences, special cases appropriate for the respective Lipschitz settings. Cubical additional compatible structures categories (co)fibrant objects. A categorical equivalence between lifts to full faithful embedding from an associated catego...