A coalgebraic, equational approach to the specification of observational structures allowing for a choice in the result type of observations is presented. Coalgebraic operations whose result type is structured as a coproduct of basic types are considered, and notions of coterm, covariable and coequation, dual to the algebraic notions of term, variable and equation, are used to specify structure...

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usu...

A coalgebraic, equational approach to the speci(cation of observational structures allowing for a choice in the result type of observations is presented. Observers whose result type is structured as a coproduct of basic types are considered, and notions of covariable, coterm and coequation, dual to the algebraic notions of variable, term and equation are used to specify the associated structure...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

ion has been found not to occur in the D + H20 reaction.22 Recently, the back reaction OH+D2 -HOD +D has been studied,12 and the HOD product shows pronounced back scattering, which is consistent with the proposed transition-state geometry. In addition to these investigations, two groups have reported experimental studies of the effect of vibrational excitation on the abstraction reaction. Crim ...

Morava E-theory is a much-studied theory important in understanding K(n)-local stable homotopy theory, but it is not a homology theory in the usual sense. The object of this paper is to show that the usual computational methods, spectral sequences, used to compute with homology theories also apply to Morava E-theory. Conceptually, we show that Morava E-theory is n derived functors away from bei...

We calculate the S-matrix in the gauge-fixed sigma-model on AdS 5 × S 5 to the leading order in perturbation theory, and analyze how supersymmetry is realized on the scattering states. A mild nonlocality of the supercharges implies that their action on multi-particle states does not follow the Leibniz rule, which is replaced by a nontrivial coproduct. The plane wave symmetry algebra is thus nat...

In his work on P -partitions, Stembridge defined the algebra of peak functions Π, which is both a subalgebra and a retraction of the algebra of quasisymmetric functions. We show that Π is closed under coproduct, and therefore a Hopf algebra, and describe the kernel of the retraction. Billey and Haiman, in their work on Schubert polynomials, also defined a new class of quasi-symmetric functions ...