Ronald Umble

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We construct an explicit diagonal on the permutahedra {Pn}. Related diagonals on the multiplihedra {Jn} and the associahedra {Kn} are induced by Tonks' projection θ : Pn → K n+1 [19] and its factorization through Jn. We use the diagonal on {Kn} to define the tensor product of A∞-(co)algebras. We introduce the notion of a permutahedral set Z, observe that(More)
An A∞-bialgebra is a DGM H equipped with structurally compatible operations ω j,i : H ⊗i → H ⊗j such that H, ω 1,i is an A∞-algebra and H, ω j,1 is an A∞-coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End(More)
Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combina-torial structure of Q(I) and obtain a homeomorphic cellular complex P (I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H * (P (I)).(More)
We introduce the notion of a matron M = {Mn,m} whose sub-modules M * ,1 and M 1, * are non-Σ operads. We construct a functor from PROP to matrons and its inverse, the universal enveloping functor. We define the free matron H∞, generated by a singleton in each bidegree (m, n) = (1, 1), and define an A∞-bialgebra as an algebra over H∞. We realize H∞ as the(More)
We compute the structure relations in special A∞-bialgebras whose operations are limited to those defining the underlying A∞-(co)algebra sub-structure. Such bialgebras appear as the homology of certain loop spaces. Whereas structure relations in general A∞-bialgebras depend upon the com-binatorics of permutahedra, only Stasheff's associahedra are required(More)
An A∞-bialgebra of type (m, n) is a Hopf algebra H equipped with a " compatible " operation ω n m : H ⊗m → H ⊗n of positive degree. We determine the structure relations for A∞-bialgebras of type (m, n) and construct a purely algebraic example for each m ≥ 2 and m + n ≥ 4. Let R be a (graded or ungraded) commutative ring with unity, and let m and n be(More)
We introduce the notion of a matrad M = {Mn,m} whose sub-modules M * ,1 and M 1, * are non-Σ operads. We define the free matrad H∞ generated by a singleton θ n m in each bidegree (m, n) and realize H∞ as the cellular chains on matrahedra {KKn,m = KKm,n} , of which KK n,1 = KK 1,n is the associahedron Kn. We construct the universal enveloping functor from(More)
1. INTRODUCTION. The trajectory of a billiard ball in motion on a frictionless billiards table is completely determined by its initial position, direction, and speed. When the ball strikes a bumper, we assume that the angle of incidence equals the angle of reflection. Once released, the ball continues indefinitely along its trajectory with constant speed(More)