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In this paper, we investigate the convergence of the Faedo-Galerkin approximations, in a strong sense, to a strong T-periodic solution of the torso-coupled bidomain model where $T$ is the period of activation of the inner wall of heart. First, we define the torso-coupled bi-domain operator and prove some of its more important properties for our work. After, we define the abstract evolution system of equations associated with torso-coupled bidomain model and give the definition of strong solution. We prove that the Faedo-Galerkin's approximations have the regularity of a strong solution, and we find that some restrictions can be imposed over the initial conditions, so that this sequence of Faedo-Galerkin fully converge to a global strong solution of the Cauchy problem. Finally, this results are used for showing the existence a strong $T$-periodic solution., Comment: 22 pages

Frequency is a central concept in Mathematics, Physics, and Signal Processing. It is the main tool for describing the oscillatory behavior of signals, which is usually argued to be the manifestation of some of their key features, depending on their nature. For instance, this is the case of Electroencephalographic signals. Hence, frequency is substantially present in the most common methodologies for analyzing signals, as the Fourier Analysis or the Time-Frequency Analysis. However, in spite of its importance as a keystone in Signal Processing, and its seemingly simple meaning, its mathematical foundation is not as straightforward as it may seem at first glance. A naive interpretation of the different mathematical concepts modelling frequency can be misleading, as their actual meanings essentially differ from the intuitive notion which are supposed to represent. In our opinion, this circumstance should be taken into account in order to develop appropriate signal analyzing and processing tools in some applications. In the current text we discuss this topic, with the main goal to draw the attention of the mathematical and engineering community to this point,often overlooked., Comment: 8 pages

In the present work, we introduce and study the concepts of state and quantum channel on spaces equipped with an indefinite metric. Exclusively, we will limit our analysis to the matricial framework. As it will be confirmed below, from our research it is noticed that, when passing to the spaces with indefinite metric, the use of the adjoint of a matrix with respect to the indefinite metric is required in the construction of states and quantum channels; which prevents us to consider the space of matrices of certain order $M_{n}(\mathbb{C})$ as a $C^{\ast}$-algebra. In our case, this adjoint is defined through a $J$-metric, where the matrix $J$ is a fundamental symmetry of $M_{n}(\mathbb{C})$. In our paper, for quantum operators, we include the general setting in the which, these operators map $J_{1}$-states into $J_{2}$-states, where $J_{2}\neq \pm J_{1}$ are two arbitrary fundamental symmetries. In the middle of this program, we carry out a study of the completely positive maps between two different positive matrices spaces by considering two different indefinite metrics on $\mathbb{C}^{n}$.

In this paper, we find relations between the ionic parameters and the diffusion parameters which are sufficient to ensure the existence of a periodic solution for a well-known monodomain model in a weak sense. We make use of the method of approximation of Faedo-Galerkin to prove the existence of weak periodic solutions of the monodomain model for the electrical activity of the heart assuming that it is periodically activated in its boundaries. Actually, this periodic solution has the same period of activation. Finally, we reflect on how these ionic-diffusive relations are useful to explain the pathophysiology of some rhythm disorders., Comment: 28 pages, 1 figure, 1 tables

In this paper, we study transfer functions corresponding to parametric linear systems whose coefficients are block matrices. Thus, these transfer functions constitute Laurent polynomials whose coefficients are square matrices. We assume that block matrices defining the parametric linear systems are solutions of an integrable hierarchy called for us, the block matrices version of the finite discrete KP hierarchy, which is introduced and studied with certain detail in this paper. We see that the linear system defined by means of the simplest solution of the integrable system is controllable and observable. Then, as a consequence of this fact, it is possible to verify that any solution of the integrable hierarchy, obtained by the dressing method of the simplest solution, defines a parametric linear system which is also controllable and observable.