The combinatorial Hopf algebra of motivic dissection polylogarithms

We introduce a family of periods of mixed Tate motives called dissection polylogarithms, that are indexed by combinatorial objects called dissection diagrams. The motivic coproduct on the former is encoded by a combinatorial Hopf algebra structure on the latter. This generalizes Goncharov's formula for the motivic coproduct on (generic) iterated integrals. Our main tool is the study of the relative cohomology group corresponding to a bi-arrangement of hyperplanes.     

The hierarchical basis multigrid method

Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

Commutative free loop space models at large primes

Let p be a prime number and a natural number. If E is a r-connected finite CW-complex of dimension at most pr, then E is an example of a p -Anick space. For p > 2 we construct a commutative cochain algebra over that is an -model of the free loop space on a p-Anick space, i.e., its cohomology algebra is isomorphic to the mod p cohomology of the free loop space. For p-Anick spaces that are p-formal, such as spheres and projective spaces, we define an even simpler commutative free loop space model that applies for all primes p. We then use the simplified model to compute the cohomology algebras of a number of free loop spaces explicitly. Received: 23 June 1999; in final form: 8 September 2000 // Published online: 7 April 2003     

Improving the accuracy of carbon-to-hydrogen ratio determination for P,N, S,O, Cl,and Br-containing organic compounds using atomic emission detection

The objective of this work was to investigate the dependence of atomic emission detector C and H response on microwave-induced plasma conditions and to improve the accuracy of carbon-to-hydrogen ratio determination for trialkylphosphates, herbicides, chlorophenols, and sulfur-containing organic compounds. Compounds which differed structurally from the analytes were used as reference compounds. It was found that when the oxygen concentration in the helium was the maximum for the instrument (9%) relative errors in carbon-to-hydrogen ratio determination were 3–8%, irrespective of analyte and reference compound structure, whereas when working in the mode of operation recommended by the manufacturer of the instrument (1.5% oxygen in helium) the respective errors were 10–20% or higher. This improvement in the accuracy of carbon-to-hydrogen ratio determination was accompanied by a factor of ten decrease in sensitivity.     

The Crystal Structure of Kx(MgxIn1–x)F3 (x = 0.38): a New Magnéli‐Bronze Type Fluoride

K_x(Mg_xIn_1–x)F₃ (x = 0.38) is monoclinic, pseudo tetragonal: a = 12.781(2) Å, b = 12.787(2) Å, c = 7.930(1) Å, β = 90,00(1)°, Z = 20. The crystal structure was solved in the space group P2₁/a (No. 14), subgroup of the tetragonal space group P4/mbm (No. 127), from X‐ray single crystal data using 4302 unique reflections (1770 with F_o/σ(F_o) > 4). The final observed R factor is 0.053. K_x(Mg_xIn_1–x)F₃ has the Magnéli‐bronze structural type, which consists in a tridimensional framework of mixed [(Mg_xIn_1–x)F₆] octahedra linked together by corners. The potassium ions are mainly located in large almost fully occupied 15‐coordinated sites and in practically empty 12‐coordinated cavities.