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In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: MSC: Primary: 18D50 (operads); 55P20 (Eilenberg–Mac Lane spaces); 55P43 (spectra with… (More)

- SAMSON SANEBLIDZE, RONALD UMBLE
- 2002

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 → Kn+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 the… (More)

- RONALD UMBLE
- 2005

We introduce the notion of a matron M = {Mn,m} whose submodules M∗,1 and M1,∗ 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) 6= (1, 1), and define an A∞-bialgebra as an algebra over H∞. We realize H∞ as the… (More)

- RONALD N. UMBLE
- 1996

Let H be a differential graded Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the Hochschild-Cartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H(q)-structure. The direct limit of all such truncations is the appropriate setting for… (More)

- Rocío González-Díaz, Javier Lamar, Ronald Umble
- IWCIA
- 2011

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 combinatorial 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)). The… (More)

- RONALD UMBLE
- 2004

An A∞-bialgebra is a DGM H equipped with structurally compatible operations { ω : H → H } such that ( H, ω ) is an A∞-algebra and ( H, ω ) 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 (TH).… (More)

Let p be an odd prime. When n ≥ 3, we show that each tensor factor of form E ⊗ Γ in H∗ (Z, n;Zp) is an A∞-infinity bialgebra with nontrivial structure. We give explicit formulas for the structure maps and the quadratic relations among them. Thus E⊗Γ is a naturally occurring example of an A∞-bialgebra whose internal structure is well-understood.

- RONALD UMBLE
- 2006

We introduce the notion of a matrad M = {Mn,m} whose submodules M∗,1 and M1,∗ are non-Σ operads. We construct a functor from PROP to matrads and its inverse, the universal enveloping functor. We define the free matrad H∞, generated by a singleton in each bidegree (m, n) 6= (1, 1), and define an A∞-bialgebra as an algebra over H∞. We realize H∞ as the… (More)

- RONALD UMBLE
- 2005

A general A∞-infinity bialgebra is a DG module (H, d) equipped with a family of structurally compatible operations ωj,i : H ⊗i → H,where i, j ≥ 1 and i+ j ≥ 3 (see [6]). In special A∞-bialgebras, ωj,i = 0 whenever i, j ≥ 2, and the remaining operationsmi = ω1,i and ∆j = ωj,1 define the underlying A∞-(co)algebra substructure. Thus special A∞-bialgebras have… (More)

- Andrew M. Baxter, Ronald Umble
- The American Mathematical Monthly
- 2008

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)