Change of representation paradigm in the light of semiotic consideration

2018 ◽  
Vol 55 ◽  
pp. 281-308
Author(s):  
ChangYeon Jo
Keyword(s):  
Change Of Representation

scholarly journals A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials

2019 ◽  
Author(s):  
Wolter Groenevelt
Keyword(s):  
Quantum Algebra ◽  
Matrix Elements ◽  
Primitive Elements ◽  
Difference Problem ◽  
Change Of Representation ◽  
Algebra Approach ◽  
Orthogonality Relations ◽  
The Matrix ◽  
Wilson Polynomials

Abstract We study matrix elements of a change of basis between two different bases of representations of the quantum algebra ${\mathcal{U}}_q(\mathfrak{s}\mathfrak{u}(1,1))$. The two bases, which are multivariate versions of Al-Salam–Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman’s multivariate Askey–Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey–Wilson polynomials are solutions of a multivariate bispectral $q$-difference problem.

scholarly journals A representation theorem for second-order functionals

2015 ◽  
Vol 25 ◽  
Author(s):  
MAURO JASKELIOFF ◽  
RUSSELL O'CONNOR
Keyword(s):  
Wide Class ◽  
Category Theory ◽  
Representation Theorem ◽  
Second Order ◽  
The Other ◽  
Functional Language ◽  
Representation Theorems ◽  
Complex Objects ◽  
Change Of Representation ◽  
Generic Representation

AbstractRepresentation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a datatype-generic representation theorem. More precisely, we prove a representation theorem for a wide class of second-order functionals which are polymorphic over a class of functors. Types polymorphic over a class of functors are easily representable in languages such as Haskell, but are difficult to analyse and reason about. The concrete representation provided by the theorem is easier to analyse, but it might not be as convenient to implement. Therefore, depending on the task at hand, the change of representation may prove valuable in one direction or the other. We showcase the usefulness of the representation theorem with a range of examples. Concretely, we show how the representation theorem can be used to prove that traversable functors are finitary containers, how coalgebras of a parameterised store comonad relate to very well-behaved lenses, and how algebraic effects might be implemented in a functional language.

scholarly journals Automating Change of Representation for Proofs in Discrete Mathematics (Extended Version)

2016 ◽  
Vol 10 (4) ◽  
pp. 429-457 ◽  
Author(s):  
Daniel Raggi ◽  
Alan Bundy ◽  
Gudmund Grov ◽  
Alison Pease
Keyword(s):  
Discrete Mathematics ◽  
Extended Version ◽  
Change Of Representation
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