Let A and B be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials p which satisfy the following property: a map T from A onto B such that the equality σ(p(Tf, T g)) = σ(p(f, g)), f, g ∈ A holds is an algebra isomorphism.

Let A be a C *-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (A t) t∈T be a continuous field of operators in A such that the function t → A t is norm continuous on T and the function t → A t is integrable. Then the following equality including Bouchner integrals holds T A t − T A s dP 2 dP = T |A t | 2 dP − T A t dP 2. (0.1) This equality is relat...

In this paper we give a direct proof of the equality of certain generating function associated with tensor product multiplicities of Kirillov-Reshetikhin modules for each simple Lie algebra g. Together with the theorems of Nakajima and Hernandez, this gives the proof of the combinatorial version of the Kirillov-Reshetikhin conjecture, which gives tensor product multiplicities in terms of restri...

We present a generic method for establishing interpolation properties by ‘borrowing’ across logical systems. The framework used is that of the so-caled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles ...

Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, Vasconcelos shows that both can be compared in case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained i...

Metaset is a new concept of set with partial membership relation. It is designed to represent and process vague, imprecise data – similarly to fuzzy sets. Metasets are based on the classical set theory primitive notions. At the same time they are directed towards efficient computer implementations and applications. The degrees of membership for metasets are expressed as finite binary sequences,...

This paper analyses the role of technology in the development of crucial knowledge and competencies in algebra, such as those involved in the comprehension and use of the notions of equivalence and equality. The hypothesis of our research is that technology can be exploited to make available new operative and representative possibilities to structure a new phenomenological space in order to inv...

In this paper, we make a substantial step towards an encoding of Cubical Type Theory (CTT) in the Dedukti logical framework. Type-checking CTT expressions features decision procedure de Morgan algebra that so far could not be expressed by rewrite rules Dedukti. As alternative, 2 Layer Theories are variants Martin-L\"of where all or part definitional equality can represented terms so-called exte...

The lambda-calculus is a fundamental syntax in computer science. It resists an algebraic treatment because of captureavoidance side-conditions. Nominal algebra is a logic of equality designed with formalisation of specifications involving binding in mind. In this paper we axiomatise the lambda-calculus using nominal algebra, demonstrate how proofs with these axioms reflect the informal argument...

Abstract. In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, this paper introduces the notion of C-sets by considering the tests from an abstract C-algebra. When the C-algebra is an ada, the axiomatization is shown to be complete by obtaining a subdirect representation of C-sets. Further, this paper considers the equality test with t...