Algebra

# Algebraic Number Theory - Papers Contributed for the Kyoto by S. Iyanaga (Editor)

By S. Iyanaga (Editor)

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The divided polynomial DG–algebra Γ (w, 2n), n ≥ 1, generated by γ1 (w) = w (γ0 (w) = 1) with product the one given by γk (w)γh (w) = (k+h)! k! h! γk+h (w); Given a connected DG–algebra A, one can construct the reduced bar cons¯ ¯ A typical element of truction of A, B(A), whose underlying module is T (sA). ⊗n ¯ ¯ B(A), is denoted by a ¯ = [a1 | · · · |an ] ∈ (sA) . The total diﬀerential dB¯ is given by dB¯ = dt + ds , being dt the natural one on the tensor module and ds the simplicial diﬀerential, that depends on the product on A.

Let A and A be two commutative connected DG–algebras. There is a con¯ ⊗ A ), B(A) ¯ ¯ ), fB⊗ , gB⊗ , φB⊗ } (see [7]), whose ⊗ B(A traction cB⊗ : {B(A formulas (for the connected case) are recalled here: Reducing Computational Costs in the BPL 45 • fB⊗ is null except for the case fB⊗ [a1 ⊗ 1| · · · |ai ⊗ 1|1 ⊗ ai+1 | · · · |1 ⊗ an ] = [a1 | · · · |ai ] ⊗ [ai+1 | · · · |an ] . • gB⊗ ([a1 | · · · |an ] ⊗ [a1 | · · · |am ]) = [a1 | · · · |an ] [a1 | · · · |am ] . • φB⊗ ¯ ([a1 ⊗ a1 | · · · |an−q ⊗ an−q |an−q+1 | · · · |an ]) n−q−1 ±[a1 ⊗ a1 | · · · |an¯ −1 ⊗ an¯ −1 | = p=0 π (5) (an¯ ∗A · · · ∗A an−q )|cπ(0) | · · · |cπ(p+q) ] , where π runs over the {(p + 1, q)-shuﬄes}, n ¯ = n − p − q and (c0 , .

Homology, Homotopy Appli. 2 (2000) 51–88 20. : Homotopy Associativity of H-spaces I, II. Trans. N. A. A. A. I. Vinitsky2 1 Belgorod State University, Studentcheskaja St. ru Abstract. A general scheme of a symbolic-numeric approach for solving the eigenvalue problem for the one-dimensional Shr¨ odinger equation is presented. The corresponding algorithm of the developed program EWA using a conventional pseudocode is described too. With the help of this program the energy spectra and the wave functions for some Schr¨ odinger operators such as quartic, sextic, octic anharmonic oscillators including the quartic oscillator with double well are calculated.