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We construct an explicit diagonal \Delta_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization… (More)

Let C_*(K) denote the cellular chains on the Stasheff associahedra. We construct an explicit combinatorial diagonal \Delta : C_*(K) --> C_*(K) \otimes C_*(K); consequently, we obtain an explicit… (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)… (More)

- Andrew M. Baxter, Ronald Umble
- The American Mathematical Monthly
- 2005

We give a complete solution of the following problem: Find, classify and count the (classes of) periodic orbits on an equilateral triangle. We prove that Fagnano's period 3 orbit is the only periodic… (More)

Abstract Let p be an odd prime. We show that when n ≥ 3 , each tensor factor E ⊗ Γ of H ∗ ( Z , n ; Z p ) is an A ∞ -bialgebra with non-trivial structure. We give explicit formulas for the structure… (More)

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… (More)

- Ronald Umble
- 2004

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… (More)

- Ronald Umble
- 2011

At the 1996 conference honoring Jim Stasheff in the year of his 60th birthday, I initiated the search for A ∞ -bialgebras in a talk entitled “In Search of Higher Homotopy Hopf Algebras.” The idea in… (More)