We enumerate two-dimensional walks inside a band parallel to an axis of symmetry, using an algebraic approach. The restrictions along the boundary lines can be more complicated than just initial values of zeroes; we consider recursive initial values prescribing which predecessors a position at the boundary can have. More general, the initial conditions are expressed by certain linear operators ...

This paper presents a generalization of the minimax state estimation approach for singular linear Differential-Algebraic Equations (DAE) with uncertain but bounded input and observation’s noise. We apply generalized Kalman Duality principle to DAE in order to represent the minimax estimate as a solution of a dual control problem for adjoint DAE. The latter is then solved converting the adjoint ...

A logic extending classical reasoning and supporting quantum reasoning is presented. The quantum logic is obtained by applying the exogenous semantics approach to propositional logic. The design is guided by the postulates of quantum mechanics and inspired by applications in quantum computation and information. The models of the quantum logic are superpositions of classical valuations. In order...

The algebraic form of Zolotarev polynomials refraining from their parametric representation is introduced. A recursive algorithm providing the coefficients for a Zolotarev polynomial of an arbitrary order is obtained from a linear differential equation developed for this purpose. The corresponding narrow-band, notch and complementary pair FIR filters are optimal in Chebyshev sense. A recursion ...

Abstract Representations are key to mathematical activities and meaning-making processes as they part of modeling, connecting, communicating, understanding ideas concepts. The current study sought examine a group novice algebra learners’ interactions with different representations from an onto-semiotic approach. A case method was employed understand how algebraic practices (abstracting, general...

Recursive analysis is the most classical approach to model and discuss computations over the real numbers. Recently, it has been shown that computability classes of functions in the sense of recursive analysis can be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (...

We present recent results on the enumeration of q-coloured planar maps, where each monochromatic edge carries a weight ν. This is equivalent to weighting each map by its Tutte polynomial, or to solving the q-state Potts model on random planar maps. The associated generating function, obtained by Olivier Bernardi and the author, is differentially algebraic. That is, it satisfies a (nonlinear) di...

Solutions of recursive program schemes over a given signature Σ were characterized by Bruno Courcelle as precisely the context-free (or algebraic) Σ-trees. These are the finite and infinite Σ-trees yielding, via labelling of paths, context-free languages. Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad C “generated” by solutions of recursive ...

Abstract. We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for...