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...

Task: A polynomial can be factorized to the product of linear polynomials, but only for orders up to 4. Numerical solution is possible, but it is imprecise and division leaves a remainder. Also this is a subject of numerical anaysis. Task: Linear algebra: “A system of linear equations Ax = b has a unique solution x = A−1b if and only if det A 6= 0.” Programming: “Real numbers cannot be tested f...

In the debate over the power of connectionist models to handle linguistic phenomena, considerable attention has been focused on the learning of simple morphophonemic rules. Rumelhart and McClelland’s celebrated model of the acquisition of the English past tense (1986), which used a simple pattern associator to learn mappings from stems to past tense forms, was advanced as evidence that networks...