Stability of quaternionic linear systems

The main goal of this paper is to characterize stability and bounded-input-bounded-output (BIBO)-stability of quaternionic dynamical systems. After defining the quaternion skew-field, algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated. Having established these results, the Smith and the Smith-McMillan forms of quaternionic matrices are introduced and studied. Finally, all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework

Dynamical properties of quaternionic behavioral systems

In this paper we study behavioral systems whose trajectories are given as solutions of quaternionic difference equations. As happens in the commutative case, it turns out that quaternionic polynomial matrices play an important role in this context. Therefore we focus our attention on such matrices and derive new results concerning their Smith form. Based on these results, we obtain characterizations of system theoretic properties of quaternionic behaviors

Linear fractional discrete-time systems

Mathematicians have been discussing about the existence (and the meaning) of derivatives and integrals of fractional order since the beginnings of differential calculus. Various concepts of fractional calculus have been developed and some of them were already applied to dynamical systems. In particular, the author already proposed a way to consider systems defined by linear differential equations of fractional order within the so-called behavioral approach. In this paper, it is shown how to generalize, analogously, discrete-time linear systems by defining a certain type of difference equations of fractional order. Some of the ideas and techniques which will be used belong to the theory of dynamical systems on time scales

On the stability of linear fractional difference systems

A fractional linear system is defined by differential or difference equations of non-integer order. A well-known result about the stability of fractional differential systems will be extended to discrete-time systems defined by fractional difference equations. This will be accomplished using time scales, which permit to unify continuous and discrete-time systems

Antisymmetry in Portuguese Ceramic Tile Facades

In this work we explore the presence of antisymmetry in Portuguese ceramic tile facades, showing examples from different styles and different antisymmetry groups. We further present some experi-ments with school children who explored symmetry and antisymmetry in a creative way. Children used a particular tile design, the Truchet tile, to create specific types of symmetry and antisymmetry. Truchet tiles can be found in several Portuguese facades from north to south.publishe

Crossings. Flying over Antisymmetric Truchet Friezes

Symmetry and antisymmetry are resources used by humans since pre-historic times in their artistic creations. In this work we start by introducing antisymmetry and the 17 possible associated groups. We then present some counts of friezes, built with a particular module, the Truchet tile, which is itself antisymmetric. These counts take into consideration the frieze antisymmetry group and the number of tiles composing the unitary cell which originates the frieze by translation. These Truchet friezes led to the creation of a set of 5 paintings, named Crossings, which illustrate all the possible groups, in an artistic context.publishe

Algebraic tools for the study of quaternionic behavioral systems

In this paper we study behavioral systems whose trajectories are given as solutions of quaternionic difference equations. As happens in the commutative case, it turns out that quaternionic polynomial matrices play an important role in this context. Therefore we pay special attention to such matrices and derive new results concerning their Smith form. Based on these results, we obtain a characterization of system theoretic properties such as controllability and stability of a quaternionic behavior

Rank metric convolutional codes

In this contribution, we propose a first general definition of rank-metric convolutional codes for multi-shot network coding. To this aim, we introduce a suitable concept of distance and we establish a generalized Singleton bound for this class of codes

MRD Rank Metric Convolutional Codes

So far, in the area of Random Linear Network Coding, attention has been given to the so-called one-shot network coding, meaning that the network is used just once to propagate the information. In contrast, one can use the network more than once to spread redundancy over different shots. In this paper, we propose rank metric convolutional codes for this purpose. The framework we present is slightly more general than the one which can be found in the literature. We introduce a rank distance, which is suitable for convolutional codes, and derive a new Singleton-like upper bound. Codes achieving this bound are called Maximum Rank Distance (MRD) convolutional codes. Finally, we prove that this bound is optimal by showing a concrete construction of a family of MRD convolutional codes